Casimir force in non-planar geometric configurations

Casimir Force in Non-Planar Geometric Configurations Sung Nae Cho Dissertation submitted to the Faculty of the Virginia ...

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Casimir Force in Non-Planar Geometric Configurations Sung Nae Cho Dissertation submitted to the Faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics

Tetsuro Mizutani, Chair John R. Ficenec Harry W. Gibson A. L. Ritter Uwe C. Tauber April 26, 2004 Blacksburg, Virginia Keywords: Casimir Effect, Casimir Force, Dynamical Casimir Force, Quantum Electrodynamics (QED), Vacuum Energy c 2004, Sung Nae Cho Copyright °

Casimir Force in Non-Planar Geometric Configurations Sung Nae Cho (ABSTRACT) The Casimir force for charge-neutral, perfect conductors of non-planar geometric configurations have been investigated. The configurations were: (1) the plate-hemisphere, (2) the hemisphere-hemisphere and (3) the spherical shell. The resulting Casimir forces for these physical arrangements have been found to be attractive. The repulsive Casimir force found by Boyer for a spherical shell is a special case requiring stringent material property of the sphere, as well as the specific boundary conditions for the wave modes inside and outside of the sphere. The necessary criteria in detecting Boyer’s repulsive Casimir force for a sphere are discussed at the end of this thesis.

Acknowledgments I would like to thank Professor M. Di Ventra for suggesting this thesis topic. The continuing support and encouragement from Professor J. Ficenec and Mrs. C. Thomas are gracefully acknowledged. Thanks are due to Professor T. Mizutani for fruitful discussions which have affected certain aspects of this investigation. Finally, I express my gratitude for the financial support of the Department of Physics of Virginia Polytechnic Institute and State University.

Contents

Abstract

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Acknowledgments

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List of Figures

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1. Introduction 1.1. Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3. Developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 3

2. Casimir Effect 2.1. Quantization of Free Maxwell Field . . . . . . . . . . . . . . . . . . . . . 2.2. Casimir-Polder Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Casimir Force Calculation Between Two Neutral Conducting Parallel Plates 2.3.1. Euler-Maclaurin Summation Approach . . . . . . . . . . . . . . . 2.3.2. Vacuum Pressure Approach . . . . . . . . . . . . . . . . . . . . . 2.3.3. The Source Theory Approach . . . . . . . . . . . . . . . . . . . .

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3. Reflection Dynamics 3.1. Reflection Points on the Surface of a Resonator . . . . . . 3.2. Selected Configurations . . . . . . . . . . . . . . . . . . . 3.2.1. Hollow Spherical Shell . . . . . . . . . . . . . . . 3.2.2. Hemisphere-Hemisphere . . . . . . . . . . . . . . 3.2.3. Plate-Hemisphere . . . . . . . . . . . . . . . . . . 3.3. Dynamical Casimir Force . . . . . . . . . . . . . . . . . . 3.3.1. Formalism of Zero-Point Energy and its Force . . 3.3.2. Equations of Motion for the Driven Parallel Plates

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4. Results and Outlook 4.1. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1. Hollow Spherical Shell . . . . . . . . . . . . . . . . . . . . . 4.1.2. Hemisphere-Hemisphere and Plate-Hemisphere . . . . . . . . 4.2. Interpretation of the Result . . . . . . . . . . . . . . . . . . . . . . . 4.3. Suggestions on the Detection of Repulsive Casimir Force for a Sphere 4.4. Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1. Sonoluminescense . . . . . . . . . . . . . . . . . . . . . . . 4.4.2. Casimir Oscillator . . . . . . . . . . . . . . . . . . . . . . .

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Appendices on Derivation Details

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A. Reflection Points on the Surface of a Resonator

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B. Mapping Between Sets (r, θ, φ) and (r0 , θ0 , φ0 )

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iv

Contents C. Selected Configurations C.1. Hollow Spherical Shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.2. Hemisphere-Hemisphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.3. Plate-Hemisphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74 74 76 81

D. Dynamical Casimir Force D.1. Formalism of Zero-Point Energy and its Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.2. Equations of Motion for the Driven Parallel Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91 91 95

E. Extended List of References

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Bibliography

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v

List of Figures

2.1. Two interacting molecules through induced dipole interactions. . . . . . . . . . . . . . . . . . . . . 2.2. A cross-sectional view of two infinite parallel conducting plates separated by a gap distance of z = d. The lowest first two wave modes are shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. A cross-sectional view of two infinite parallel conducting plates. The plates are separated by a gap distance of z = d. Also, the three regions have different dielectric constants εi (ω) . . . . . . . . . . .

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3.1. The plane of incidence view of plate-hemisphere configuration. The waves that through ° are supported ° ° ~0 ° 0 ~ 1° . . . . . . . internal reflections in the hemisphere cavity must satisfy the relation λ ≤ 2 °R 2 − R

19

3.2. The thick line shown here represents the intersection between hemisphere surface and the plane of inci° °−1 P ° ° 3 0 0 dence. The unit vector normal to the incident plane is given by nˆ0 p,1 = − °n~0 p,1 ° i=1 ²ijk k1,j r0,k eˆi . 21 3.3. The surface of the hemisphere-hemisphere configuration can be described relative to the system origin ~ or relative to the hemisphere centers through R ~ 0. . . . . . . . . . . . . . . . . . . . . . . through R, 0 ~ ~ 0 i induces a series of reflections that 3.4. Inside the cavity, an incident wave k i on first impact point R ~0 propagate throughout the entire inner cavity. Similarly, a wave k~0 i incident on the°impact ° point R i + ° ° aRˆ0 i , where a is the thickness of the sphere, induces reflected wave of magnitude °k~0 i ° . The resultant ~ 0 i and the resultant wave direction in the resonator is wave direction in the external region is along R 0 ~ along −R i due to the fact there is exactly another wave vector traveling in opposite direction in both regions. In both cases, the reflected and incident waves have equal magnitude due to the fact that the sphere is assumed to be a perfect conductor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5. The dashed line vectors represent the situation where only single internal reflection occurs. The dark line vectors represent the situation where multiple internal reflections occur. . . . . . . . . . . . . . . 3.6. The orientation of a disk is given through the surface unit normal nˆ0 p . The disk is spanned by the two unit vectors θˆ0 p and φˆ0 p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7. The plate-hemisphere configuration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8. The intersection between oscillating plate, hemisphere and the plane of incidence whose normal is ° °−1 P ° ° 3 nˆ0 p,1 = − °n~0 p,1 ° ²ijk k 0 r0 eˆi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i=1

1,j 0,k

3.9. Because there are more vacuum-field modes in the external regions, the two charge-neutral conducting plates are accelerated inward till the two finally stick. . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10. A one dimensional driven parallel plates configuration. . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Boyer’s configuration is such that a sphere is the only matter in the entire universe. His universe extends to the infinity, hence there are no boundaries. The sense of vacuum-field energy flow is along the radial vector rˆ, which is defined with respect to the sphere center. . . . . . . . . . . . . . . . . . 4.2. Manufactured sphere, in which two hemispheres are brought together, results in small non-spherically symmetric vacuum-field radiation inside the cavity due to the configuration change. For the hemispheres made of Boyer’s material, these fields in the resonator will eventually get absorbed by the conductor resulting in heating of the hemispheres. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. The process in which a configuration change from hemisphere-hemisphere to sphere inducing virtual photon in the direction other than rˆ is shown. The virtual photon here is referred to as the quanta of energy associated with the zero-point radiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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List of Figures 4.4. A realistic laboratory has boundaries, e.g., walls. These boundaries have effect similar to the field modes between two parallel plates. In 3D, the effects are similar to that of a cubical laboratory, etc. . 4.5. The schematic of sphere manufacturing process in a realistic laboratory. . . . . . . . . . . . . . . . . ~0 ~0 4.6. The pnet k = ° vacuum-field ° wave vectors k i,b and k i,f impart a net momentum of the magnitude k~ ° ~0 ° 0 ~ ~ °k i,b − k i,f ° /2 on differential patch of an area dA on a conducting spherical surface. . . . . . . 4.7. To deflect away as much possible the vacuum-field radiation emanating from the laboratory boundaries, the walls, floor and ceiling are constructed with some optimal curvature to be determined. The apparatus is then placed within the “Apparatus Region.” . . . . . . . . . . . . . . . . . . . . . . . . 4.8. The original bubble shape shown in dotted lines and the deformed bubble in solid line under strong acoustic field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9. The vacuum-field radiation energy flows are shown for closed and unclosed hemispheres. For the hemispheres made of Boyer’s material, the non-radial wave would be absorbed by the hemispheres. . A.1. A simple reflection of incoming wave k~0 i from the surface defined by a local normal n~0 . . . . . . . . ° °−1 P ° ° 3 0 0 . . . . . . . . . A.2. Parallel planes characterized by a normal nˆ0 p,1 = − °n~0 p,1 ° i=1 ²ijk k1,j r0,k eˆi . 0 0 ~ 1 and R ~ 2 are connected through the angle ψ1,2 . A.3. The two immediate neighboring reflection points R 0 ~ ~ 0 i+2 are connected through the angle Similarly, the two distant neighbor reflection points R i and R Ωψi,i+1 ,ψi+1,i+2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1. Introduction The introduction is divided into three parts: (1) physics, (2) applications, and (3) developments. A brief outline of the physics behind the Casimir effect is discussed in item (1). In the item (2), major impact of Casimir effect on technology and science is outlined. Finally, the introduction of this thesis is concluded with a brief review of the past developments, followed by a brief outline of the organization of this thesis and its contributions to the physics.

1.1. Physics When two electrically neutral, conducting plates are placed parallel to each other, our understanding from classical electrodynamics tell us that nothing should happen for these plates. The plates are assumed to be that made of perfect conductors for simplicity. In 1948, H. B. G. Casimir and D. Polder faced a similar problem in studying forces between polarizable neutral molecules in colloidal solutions. Colloidal solutions are viscous materials, such as paint, that contain micron-sized particles in a liquid matrix. It had been thought that forces between such polarizable, neutral molecules were governed by the van der Waals interaction. The van der Waals interaction is also referred to as the Lennard-Jones interaction. It is a long range electrostatic interaction that acts to attract two nearby polarizable molecules. Casimir and Polder found to their surprise that there existed an attractive force which could not be ascribed to the van der Waals theory. Their experimental result could not be correctly explained unless the retardation effect was included in the van der Waals’ theory. This retarded van der Waals interaction or Lienard-Wiechert dipole-dipole interaction [1] is now known as the Casimir-Polder interaction [2]. Casimir, following this first work, elaborated on the Casimir-Polder interaction in predicting the existence of an attractive force between two electrically neutral, parallel plates of perfect conductors separated by a small gap [3]. This alternative derivation of the Casimir force is in terms of the difference between the zero-point energy in vacuum and the zero-point energy in the presence of boundaries. This force has been confirmed by experiments and the phenomenon is what is now known as the “Casimir Effect.” The force responsible for the attraction of two uncharged conducting plates is accordingly termed the “Casimir Force.” It was shown later that the Casimir force could be both attractive or repulsive depending on the geometry and the material property of the conductors [4, 5, 6]. The Casimir effect is regarded as macroscopic manifestation of the retarded van der Waals interaction between uncharged polarizable atoms. Microscopically, the Casimir effect is due to interactions between induced multipole moments, where the dipole term is the most dominant contributor if it is non-vanishing. Therefore, the dipole interaction is exclusively referred to, unless otherwise explicitly stated, throughout the thesis. The induced dipole moments can be qualitatively explained by quantum fluctuations in matter which leads to the energy imbalance 4E due to charge-separation between virtual positive and negative charge contents that lasts for a time interval 4t consistent with the Heisenberg uncertainty principle 4E4t ≥ h/4π, where h is the Planck constant. The fluctuations in the induced dipoles then result in fluctuating zero-point electromagnetic fields in the space around conductors. It is the presence of these fluctuating vacuum fields that lead to the phenomenon of the Casimir effect. However, the dipole strength is left as a free parameter in the calculations because it cannot be readily calculated. Its value must be determined from experiments. Once this idea is accepted, one can then move forward to calculate the effective, temperature averaged, energy due to the dipole-dipole interactions with the time retardation effect folded in. The energy between the dielectric (or conducting) media is obtained from the allowed modes of electromagnetic waves determined by the Maxwell equations together with the boundary conditions. The Casimir force is then obtained by taking the negative gradient of the energy in space. This approach, as opposed to full atomistic treatment of the dielectrics (or conductors), is justified as long as the most significant field wavelengths determining the interaction are large when compared with the spacing of the lattice points in the media. The effect of all the multiple dipole scattering by atoms in the dielectric (or conducting) media simply enforces the macroscopic reflection laws of electromagnetic waves. For instance, in the case of the two parallel plates, the most significant wavelengths are those of the order of the plate gap distance. When this wavelength is large compared with the interatomic distances, the macroscopic electromagnetic theory can be used

1

1. Introduction with impunity. But, to handle the effective dipole-dipole interaction Hamiltonian, the classical electromagnetic fields have to be quantized. Then the geometric configuration can introduce significant complications, which is the subject matter this study is going to address. Finally, it is to be noticed that the Casimir force on two uncharged, perfectly conducting parallel plates originally calculated by H. B. G. Casimir was done under the assumption of absolute zero temperature. In such condition, the occupational number ns for photon is zero; and hence, there are no photons involved in Casimir’s calculation for his parallel plates. However, the occupation number convention for photons refers to those photons with electromagnetic energy in quantum of Ephoton = ~ω, where ~ is the Planck constant divided by 2π and ω, the angular frequency. The zero-point quantum of energy, Evac = ~ω/2, involved in Casimir effect at absolute zero temperature is also of electromagnetic origin in nature; however, we do not classify such quantum of energy as a photon. Therefore, this quantum of electromagnetic energy, Evac = ~ω/2, will be simply denoted “zero-point energy” throughout this thesis. By convention, the lowest energy state, the vacuum, is also referred to as a zero-point.

1.2. Applications In order to appreciate the importance of the Casimir effect from industry’s point of view, we first examine the theoretical value for the attractive force between two uncharged conducting parallel plates separated by a gap of distance d : FC = −240−1 π 2 d−4 ~c, where c is the speed of light in vacuum and d is the plate gap distance. To get a sense of the magnitude of this force, two mirrors of an area of ∼ 1 cm2 separated by a distance of ∼ 1 µm would experience an attractive Casimir force of roughly ∼ 10−7 N, which is about the weight of a water droplet of half a millimeter in diameter. Naturally, the scale of size plays a crucial role in the Casimir effect. At a gap separation in the ranges of ∼ 10 nm, which is roughly about a hundred times the typical size of an atom, the equivalent Casimir force would be in the range of 1 atmospheric pressure. The Casimir force have been verified by Steven Lamoreaux [7] in 1996 to within an experimental uncertainty of 5%. An independent verification of this force have been done recently by U. Mohideen and Anushree Roy [8] in 1998 to within an experimental uncertainty of 1%. The importance of Casimir effect is most significant for the miniaturization of modern electronics. The technology already in use that is affected by the Casimir effect is that of the microelectromechanical systems (MEMS). These are devices fabricated on the scale of microns and sub-micron sizes. The order of the magnitude of Casimir force at such a small length scale can be enormous. It can cause mechanical malfunctions if the Casimir force is not properly taken into account in the design, e.g., mechanical parts of a structure could stick together, etc [9]. The Casimir force may someday be put to good use in other fields where nonlinearity is important. Such potential applications requiring nonlinear phenomena have been demonstrated [10]. The technology of MEMS hold many promising applications in science and engineering. With the MEMS soon to be replaced by the next generation of its kind, the nanoelectromechanical systems or NEMS, understanding the phenomenon of the Casimir effect become even more crucial. Aside from the technology and engineering applications, the Casimir effect plays a crucial role in accurate force measurements at nanometer and micrometer scales [11]. As an example, if one wants to measure the gravitational force at a distance of atomic scale, not only the subtraction of the dominant Coulomb force has to be done, but also the Casimir force, assuming that there is no effect due to strong and weak interactions. Most recently, a new Casimir-like quantum phenomenon have been predicted by Feigel [12]. The contribution of vacuum fluctuations to the motion of dielectric liquids in crossed electric and magnetic fields could generate velocities of ∼ 50 nm/s. Unlike the ordinary Casimir effect where its contribution is solely due to low frequency vacuum modes, the new Casimir-like phenomenon predicted recently by Feigel is due to the contribution of high frequency vacuum modes. If this phenomenon is verified, it could be used in the future as an investigating tool for vacuum fluctuations. Other possible applications of this new effect lie in fields of microfluidics or precise positioning of micro-objects such as cold atoms or molecules. Everything that was said above dealt with only one aspect of the Casimir effect, the attractive Casimir force. In spite of many technical challenges in precision Casimir force measurements [7, 8], the attractive Casimir force is fairly well established. This aspect of the theory is not however what drives most of the researches in the field. The Casimir effect also predicts a repulsive force and many researchers in the field today are focusing on this phenomenon yet to be confirmed experimentally. Theoretical calculations suggest that for certain geometric configurations, two neutral conductors would exhibit repulsive behavior rather than being attractive. The classic result that started it all is that of Boyer’s work on the Casimir force calculation for an uncharged spherical conducting shell [4]. For a spherical conductor, the net electromagnetic radiation pressure, which constitute the Casimir force, has a positive sign, thus

2

1. Introduction being repulsive. This conclusion seems to violate fundamental principle of physics for the fields outside of the sphere take on continuum in allowed modes, where as the fields inside the sphere can only assume discrete wave modes. However, no one has been able to experimentally confirm this repulsive Casimir force. The phenomenon of Casimir effect is too broad, both in theory and in engineering applications, to be completely summarized here. I hope this informal brief survey of the phenomenon could motivate people interested in this remarkable area of quantum physics.

1.3. Developments Casimir’s result of attractive force between two uncharged, parallel conducting plates is thought to be a remarkable application of QED. This attractive force have been confirmed experimentally to a great precision as mentioned earlier [7, 8]. However, it must be emphasized that even these experiments are not done exactly in the same context as Casimir’s original configuration due to technical difficulties associated with Casimir’s idealized perfectly flat surfaces. Casimir’s attractive force result between two parallel plates has been unanimously thought to be obvious. Its origin can also be attributed to the differences in vacuum-field energies between those inside and outside of the resonator. However, in 1968, T. H. Boyer, then at Harvard working on his thesis on Casimir effect for an uncharged spherical shell, had come to a conclusion that the Casimir force was repulsive for his configuration, which was contrary to popular belief. His result is the well known repulsive Casimir force prediction for an uncharged spherical shell of a perfect conductor [4]. The surprising result of Boyer’s work has motivated many physicists, both in theory and experiment, to search for its evidence. On the theoretical side, people have tried different configurations, such as cylinders, cube, etc., and found many more configurations that can give a repulsive Casimir force [5, 13, 14]. Completely different methodologies were developed in striving to correctly explain the Casimir effect. For example, the “Source Theory” was employed by Schwinger for the explanation of the Casimir effect [14, 15, 16, 17]. In spite of the success in finding many boundary geometries that gave rise to the repulsive Casimir force, the experimental evidence of a repulsive Casimir effect is yet to be found. The lack of experimental evidence of a repulsive Casimir force has triggered further examination of Boyer’s work. The physics and the techniques employed in the Casimir force calculations are well established. The Casimir force calculations involve summing up of the allowed modes of waves in the given resonator. This turned out to be one of the difficulties in Casimir force calculations. For the Casimir’s original parallel plate configuration, the calculation was particularly simple due to the fact that zeroes of the sinusoidal modes are provided by a simple functional relationship, kd = nπ, where k is the wave number, d is the plate gap distance and n is a positive integer. This technique can be easily extended to other boundary geometries such as sphere, cylinder, cone or a cube, etc. For a sphere, the functional relation that determines the allowed wave modes in the resonator is kro = αs,l , where ro is the radius of the sphere; and αs,l , the zeroes of the spherical Bessel functions js . In the notation αs,l denotes lth zero of the spherical Bessel function js . The same convention is applied to all other Bessel function solutions. The allowed wave modes of a cylindrical resonator is determined by a simple functional relation kao = βs,l , where ao is the cylinder radius and βs,l are now the zeroes of cylindrical Bessel functions Js . One of the major difficulties in the Casimir force calculation for nontrivial boundaries such as those considered in this thesis is in defining the functional relation that determines the allowed modes in the given resonator. For example, for the hemisphere-hemisphere boundary configuration, the radiation originating from one hemisphere would enter the other and run through a complex series of reflections before escaping the°hemispherical vacuum° cavity. The allowed ° ° ° ~0 ° ~0 0 ° ~ ~ 01° field modes in the resonator is then governed by a functional relation k °R − R = nπ, where R − ° 2 R ° is 2 1° ~ 0 1 and R ~ 0 2 of the resonator, as is illustrated in Figure 3.1. As the distance between two successive reflection points R ° ° ° ~0 ~0 ° will be shown in the subsequent sections, the actual functional form for °R 2 − R 1 ° is not simple even though the ° ° ° ~0 0 ° ~ physics behind °R − R 2 1 ° is particularly simple: the application of the law of reflections. The task of obtaining ° ° ° ~0 ~0 ° the functional relation k °R 2 − R 1 ° = nπ for the hemisphere-hemisphere, the plate-hemisphere, and the sphere configuration formed by bringing in two hemispheres together is to the best of my knowledge my original development. It constitutes the major part of this thesis. This thesis is not about questioning the theoretical origin of the Casimir effect. Instead, its emphasis is on applying the Casimir effect as already known to determine the sign of Casimir force for the realistic experiments. In spite of a

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1. Introduction number of successes in the theoretical study of repulsive Casimir force, most of the configurations are unrealistic. In order to experimentally verify Boyer’s repulsive force for a charge-neutral spherical shell made of perfect conductor, one should consider the case where the sphere is formed by bringing in two hemispheres together. When the two hemispheres are closed, it mimics that of Boyer’s sphere. It is, however, shown later in this thesis that a configuration change from hemisphere-hemisphere to a sphere induces non-spherically symmetric energy flow that is not present in Boyer’s sphere. Because Boyer’s sphere gives a repulsive Casimir force, once those two closed hemispheres are released, they must repulse if Boyer’s prediction were correct. Although the two hemisphere configuration have been studied for decades, no one has yet carried out its analytical calculation successfully. The analytical solutions on two hemispheres, existing so far, was done by considering the two hemispheres that were separated by an infinitesimal distance. In this thesis, the consideration of two hemispheres is not limited to such infinitesimal separations. The three physical arrangements being studied in this thesis are: (1) the plate-hemisphere, (2) the hemispherehemisphere and (3) the sphere formed by brining in two hemispheres together. Although there are many other boundary configurations that give repulsive Casimir force, the configurations under consideration were chosen mainly because of the following reasons: (1) to be able to confirm experimentally the Boyer’s repulsive Casimir force result for a spherical shell, (2) the experimental work involving configurations similar to that of the plate-hemisphere configuration is underway [10]; and (3) to the best of my knowledge, no detailed analytical study on these three configurations exists to date. My motivation to mathematically model the plate-hemisphere system came from the experiment done by a group at the Bell Laboratory [10] in which they bring in an atomic-force-probe to a flopping plate to observe the Casimir force which can affect the motion of the plate. In my derivations for equations of motion, the configuration is that of the “plate displaced on upper side of a bowl (hemisphere).” The Bell Laboratory apparatus can be easily mimicked by simply displacing the plate to the under side of the bowl, which I have not done. The motivation behind the hemisphere-hemisphere system actually arose from an article by Kenneth and Nussinov [18]. In their paper, they speculate on how the edges of the hemispheres may produce effects such that two arbitrarily close hemispheres cannot mimic Boyer’s sphere. This led to their heuristic conclusion which stated that Boyer’s sphere can never be the same as the two arbitrarily close hemispheres. To the best of my knowledge, two of the geometrical configurations investigated in this thesis work have not yet been investigated by others. They are the plate-hemisphere and the hemisphere-hemisphere configurations. This does not mean that these boundary configurations were not known to the researchers in the field, e.g., [18]. For the case of the hemisphere-hemisphere configuration, people realized that it could be the best way to test for the existence of a repulsive Casimir force for a sphere as predicted by Boyer. The sphere configuration investigated in this thesis, which is formed by bringing two hemispheres together, contains non-spherically symmetric energy flows that are not present in Boyer’s sphere. In that regards, the treatment of the sphere geometry here is different from that of Boyer. The basic layout of the thesis is as follows: (1) Introduction, (2) Theory, (3) Calculations, and (4) Results. The formal introduction of the theory is addressed in chapters (1) and (2). The original developments resulting from this thesis are contained in chapters (3) and (4). The brief outline of each chapter is the following: In chapter (1), a brief introduction to the physics is addressed; and the application importance and major developments in this field are discussed. In chapter (2), the formal aspect of the theory is addressed, which includes the detailed outline of the Casimir-Polder interaction and brief descriptions of various techniques that are currently used in Casimir force calculations. In chapter (3), the actual Casimir force calculations ° pertaining ° to the boundary geometries considered in ° ~0 0 ° ~ this thesis are derived. The important functional relation for °R − R 2 1 ° is developed here. The dynamical aspect of the Casimir effect is also introduced here. Due to the technical nature of the derivations, many of the results presented are referred to the detailed derivations contained in the appendices. In chapter (4), the results are summarized. Lastly, the appendices have been added in order to accommodate the tedious and lengthy derivations to keep the text from losing focus due to mathematical details. To the best of my knowledge, everything in the appendices represent original developments, with a few indicated exceptions. The goal in this thesis is not to embark so much on the theory side of the Casimir effect. Instead, its emphasis is on bringing forth the suggestions that might be useful in detecting the repulsive Casimir effect originally initiated by Boyer on an uncharged spherical shell. In concluding this brief outline of the motivation behind this thesis work, I must add that if by any chance someone already did these work that I have claimed to represent my original developments, I was not aware of their work at the time of this thesis was being prepared. And, should that turn out to be the case, I would like to express my apology for not referencing their work in this thesis.

4

2. Casimir Effect The Casimir effect is divided into two major categories: (1) the electromagnetic Casimir effect and (2) the fermionic Casimir effect. As the titles suggest, the electromagnetic Casimir effect is due to the fluctuations in a massless Maxwell bosonic fields, whereas the fermionic Casimir effect is due to the fluctuations in a massless Dirac fermionic fields. The primary distinction between the two types of Casimir effect is in the boundary conditions. The boundary conditions appropriate to the Dirac equations are the so called “bag-model” boundary conditions, whereas the electromagnetic Casimir effect follows the boundary conditions of the Maxwell equations. The details of the fermionic force can be found in references [14, 17]. In this thesis, only the electromagnetic Casimir effect is considered. As it is inherently an electromagnetic phenomenon, we begin with a brief introduction to the Maxwell equations, followed by the quantization of electromagnetic fields.

2.1. Quantization of Free Maxwell Field There are four Maxwell equations: ³

´

³

´

~ •E ~ R, ~ t = 4πρ R, ~ t , ∇

³

´

~ •B ~ R, ~ t = 0, ∇

³ ´ ~ R, ~ t ∂B 1 ~ ×E ~ R, ~ t =− ∇ , c ∂t ³

´

³ ´ ~ R, ~ t ³ ´ 4π ³ ´ 1 ∂E ~ ×B ~ R, ~ t = ~ t + ∇ J~ R, , c c ∂t

(2.1)

(2.2)

where the Gaussian system of units have been adopted. The electric and the magnetic field are defined respectively by ~ = −∇Φ ~ − c−1 ∂t A ~ and B ~ =∇ ~ × A, ~ where Φ is the scalar potential and A ~ is the vector potential. Equations (2.1) E and (2.2) are combined to give ( · ¸ 3 3 X X 4π 1 4π 2 4π∂l ρ + 2 ∂t Jl − ∂m ∂l Φ + ∂t Al + ²ljk ∂j Ak + ²lmn ∂m Jn c c c m=1 l=1 · ¸ ¾ 1 1 1 + 2 ∂t2 ∂l Φ + ∂t Al + 2 ²ljk ∂j ∂t2 Ak eˆl = 0, c c c where the Einstein summation convention is assumed for repeated indices. Because the components along basis direction eˆl are independent of each other, the above vector algebraic relation becomes three equations: · ¸ 3 X 1 4π 4π 2 ∂m ∂l Φ + ∂t Al + ²ljk ∂j Ak + ²lmn ∂m Jn 4π∂l ρ + 2 ∂t Jl − c c c m=1 · ¸ 1 2 1 1 + 2 ∂t ∂l Φ + ∂t Al + 2 ²ljk ∂j ∂t2 Ak = 0, c c c

(2.3)

where l = 1, 2, 3. To understand the full implications of electrodynamics, one has to solve the above set of coupled differential equations. Unfortunately, they are in general too complicated to solve exactly. The need to choose an appropriate gauge to approximately solve the above equations is not only an option, it is a must. Also, for what is concerned with the vacuum-fields, that is, the radiation from matter when it is in its lowest energy state, information about the charge density ρ and the current density J~ must be first prescribed. Unfortunately, to describe properly the charge and current

5

2. Casimir Effect densities of matter is a major difficulty in its own. Therefore, the charge density ρ and the current density J~ are set to ~ •A ~ = 0, is adopted. Under these conditions, equation be zero for the sake of simplicity and the Coulomb gauge, ∇ 2 −2 2 (2.3) is simplified to ∂l Al − c ∂t Al = 0, where l = 1, 2, 3. The steady state monochromatic solution is of the form ³ ´ ³ ´ ³ ´ ~ R, ~ t = α (t) A ~0 R ~ + α∗ (t) A ~ ∗0 R ~ A ³ ´ ³ ´ ~ + α∗ (0) exp (iωt) A ~ ∗0 R ~ , ~0 R = α (0) exp (−iωt) A ³ ´ ³ ´ ³ ´ ~0 R ~ is the solution to the Helmholtz equation ∇2 A ~0 R ~ + c−2 ω 2 A ~0 R ~ = 0 and α (t) is the solution where A ´ ³ ~ t , the electric and the ~ R, of the temporal differential relation satisfying α ¨ (t) + ω 2 α (t) = 0. With the solution A magnetic fields are found to be ³ ´ h ³ ´ ³ ´i ~ R, ~ t = −c−1 α˙ (t) A ~0 R ~ + α˙ ∗ (t) A ~ ∗0 R ~ E and ³ ´ ³ ´ ³ ´ ~ R, ~ t = α (t) ∇ ~ ×A ~0 R ~ + α∗ (t) ∇ ~ ×A ~∗ R ~ . B 0 The electromagnetic field Hamiltonian becomes: Z h i 2 1 ~∗ • E ~ +B ~∗ • B ~ d3 R = k kα (t)k2 , HF = E (2.4) 8π R 2π ³ ´ R ~ have been normalized such that ~0 R A (R) d3 R = 1 with A0,l (R) representwhere k is a wave number and A R 0,l ³ ´ ~0 R ~ . ing the lth component of A We can transform HF into the “normal coordinate representation” through the introduction of “creation” and “annihilation” operators, a† and a. The resulting field Hamiltonian HF of equation (2.4) is identical in form to that of the canonically transformed simple harmonic oscillator, HSH ∝ p2 + q 2 → KSH ∝ a† a. For the free electromagnetic 2 field Hamiltonian, the canonical transformation is to follow the sequence KSH ∝ kα (t)k → HSH ∝ E 2 + B 2 under a properly chosen generating function. The result is that with the following physical quantities, i q (t) = √ [α (t) − α∗ (t)] , c 4π

k p (t) = √ [α (t) + α∗ (t)] , 4π

the free field Hamiltonian of equation (2.4) becomes HF =

¤ 1£ 2 p (t) + ω 2 q 2 (t) , 2

(2.5)

which is identical to the Hamiltonian of the simple harmonic oscillator. Then, through a direct comparison and observation with the usual simple harmonic oscillator Hamiltonian in quantum mechanics, the following replacements are made r r 2π~c2 2π~c2 † ∗ α (t) → a (t) , α (t) → a (t) , ω ω ³ ´ ³ ´ ³ ´ ~ R, ~ t ,E ~ R, ~ t and B ~ R, ~ t are found, and, the quantized relations for A ³ ´ ~ R, ~ t = A

r

³ ´ ³ ´i 2π~c2 h ~0 R ~ + a† (t) A ~ ∗0 R ~ , a (t) A ω

³ ´ h ³ ´ ³ ´i √ ~ R, ~ t = i 2π~ω a (t) A ~0 R ~ − a† (t) A ~ ∗0 R ~ , E

6

2. Casimir Effect r

´ ³ ~ R, ~ t = B

³ ´ ³ ´i 2π~c2 h ~ ×A ~0 R ~ + a† (t) ∇ ~ ×A ~ ∗0 R ~ , a (t) ∇ ω ³ ´ ³ ´ ³ ´ ~ R, ~ R, ~ R, ~ t ,E ~ t and B ~ t are now quantum mechanical operators. where it is understood that A The associated field Hamiltonian operator for the photon becomes · ¸ 1 † ˆ HF = ~ω a (t) a (t) + , 2

(2.6)

where the hat (∧) over HF now denotes an operator. The quantum mechanical expression for the free electromagnetic field energy per mode of angular frequency ω 0 , summed over all occupation numbers becomes HF ≡

¸ ∞ D ∞ · ¯ ¯ E X X 1 ¯ˆ ¯ ~ω 0 , n s ¯H n = n + ¯ F s s 2 n =0 n =0 s

s

where ω 0 ≡ ω 0 (n) and ns is the occupation number corresponding to the quantum state |ns i . Summation over all angular frequency modes n and polarizations Θω0 gives (· ) ¸ ∞ ∞ ∞ X X X 1 0 HF = ns + ~Θω0 ω (n) ≡ Hn0 s , 2 n =0 n=0 n =0 s

s

where Hn0 s is defined by (

·

Hn0 s

¸ ∞ X 1 = ns + ~Θω0 ω 0 (n) , 2 n=0

ns = 0,°1, 2, 3, °· · · , ° ° ω 0 (n) ≡ °ω~0 (n)° > 0.

(2.7)

° ° P3 ° ° Here ω 0 (n) ≡ °ω~0 (n)° is the magnitude of nth angular frequency of the electromagnetic field, ω~0 (n) = i=1 ωi0 (ni ) eˆi , and Θω0 is the number of independent polarizations of the field. The energy equation (2.7) is valid for the case where the angular frequency vector ω~0 n happens to be parallel to one of the coordinate axes. For the general case where ω~0 n is not necessarily parallel to any one of coordinate axes, the angular frequency is given by ω 0 (n) = nP o1/2 2 3 0 . The stationary energy is therefore i=1 [ωi (ni )] )1/2 ( 3 ¸ ∞ X ∞ X ∞ X X 1 2 0 [ki (ni , Li )] , ~cΘk0 = ns + 2 n =0 n =0 n =0 i=1 ·

Hn0 s ,b

≡

Hn0 s

1

2

(2.8)

3

where the substitution ωi0 (ni ) = cki0 (ni , Li ) have been made. Here Li is the quantization length, Θω0 has been been changed to Θk0 , and the subscript b of Hn0 s ,b denotes bounded space. When the dimensions of boundaries are such that the difference, 4ki0 (ni , Li ) = ki0 (ni + 1, Li ) − ki0 (ni , Li ) , is infinitesimally small, we can replace the summation in equation (2.8) by integration, Z

∞ X ∞ X ∞ X

∞

Z

∞

Z

−1

∞

Z

∞

Z

∞

dn1 dn2 dn3 → [f1 (L1 ) f2 (L2 ) f3 (L3 )]

→

n1 =0 n2 =0 n3 =0

Z

∞

n1 =0

n2 =0

0

n3 =0

0

0

dk10 dk20 dk30 ,

where in the last step the functional definition for ki0 ≡ ki0 (ni , Li ) = ni fi (Li ) have been used to replace dni by dki0 /fi (Li ) . In free space, the electromagnetic field energy for quantum state |ns i is given by Hn0 s ,u

≡

Hn0 s

)1/2 £ ¤ Z ∞ Z ∞ Z ∞ (X 3 ns + 12 ~cΘk0 2 0 = [ki (ni , Li )] dk10 dk20 dk30 , f1 (L1 ) f2 (L2 ) f3 (L3 ) 0 0 0 i=1

(2.9)

where the subscript u of Hn0 s ,u denotes free or unbounded space, and the functional fi (Li ) in the denominator is equal −1 to ζzero n−1 i Li for a given Li . Here ζzero is the zeroes of the function representing the transversal component of the

7

2. Casimir Effect Induced dipoles

p

d,1

S

p

d,2

R1

R2 Reference origin

Figure 2.1.: Two interacting molecules through induced dipole interactions. electric field.

2.2. Casimir-Polder Interaction The phenomenon referred to as Casimir effect has its root in van der Waals interaction between neutral particles that are polarizable. The Casimir force may be regarded as a macroscopic manifestations of the retarded van der Waals ~ is Hd = −~ ~ When the pd • E. force. The energy associated with an electric dipole moment p~d in a given electric field E involved dipole moment p~d is that of the induced rather than that of the permanent one, the induced dipole interaction ~ energy is reduced by a factor of two, Hd = −~ pd • E/2. The factor of one half is due to the fact that Hd now represents the energy of a polarizable particle in an external field, rather than a permanent dipole. The role of an external field here is played by the vacuum-field. Since the polarizability is linearly proportional to the external field, the average value leads to a factor of one half in the induced dipole interaction energy. Here the medium of the dielectric is assumed to be linear. Throughout this thesis, the dipole moments induced by vacuum polarization are considered as a free parameters. The interaction energy between two induced dipoles shown in Figure 2.1 are given by °−5 ½ ° °2 ´i¾ ´i h ³ h ³ 1° °~ ° °~ ° ~ ~ ~ ~ ~ ~ Hint = °R − R [~ p • p ~ ] R − R , R − R R − R p ~ • − 3 p ~ • ° ° ° 2 1 d,1 d,2 2 1 2 1 2 1 d,2 d,1 2 D E ~ i is the position of ith dipole. For an isolated system, the first order perturbation energy H(1) vanishes due where R int to the fact that dipoles are randomly oriented, i.e., h~ p i = 0. The first non-vanishing perturbation energy is that of the d,i D E P (2) −1 second order, Uef f,static = Hint = m6=0 h0 |Hint | mi hm |Hint | 0i [E0 − Em ] , which falls off with respect ° °−6 °~ ~ ° . This is the classical result obtained by F. London for short to the separation distance like Uef f,static ∝ °R 2 − R1 ° distance electrostatic fields. F. London employed quantum mechanical perturbation approach to reach his result on a static van der Waals interaction without retardation effect in 1930. The electromagnetic interaction can only propagate as fast as the speed of light in a given medium. This retardation effect due to propagation time was included by Casimir and Polder in their consideration. It led to their surprising ° °−7 °~ ~ ° . It became the now well known Casimirdiscovery that the interaction between molecules falls off like °R 1 − R2 ° Polder potential [2], Uef f,retarded = − (i)

°−7 n h i h io ~c ° °~ (1) (2) (1) (2) (1) (2) (1) (2) ~ 1° 23 αE αE + αM αM − 7 αE αM + αM αE , °R2 − R ° 4π

(i)

where αE and αM represents the electric and magnetic polarizability of ith particle (or molecule). To understand the Casimir effect, the physics behind the Casimir-Polder (or retarded van der Waals) interaction is ~ ~ ω for the Fourier essential. In the expression of the induced dipole energy Hd = −~ pd • E/2, we rewrite p~d = α (ω) E ~ ω of the field. Here α (ω) is the polarizability. component of the dipole moment induced by the Fourier component E † ~ ~ The induced dipole field energy becomes Hd = −α (ω) Eω · Eω /2, where the (·) denotes the matrix multiplication

8

2. Casimir Effect instead of the vector dot product (•) . Summing over all possible modes and polarizations, the field energy due to the induced dipole becomes Hd,1 = −

³ ´ ³ ´ 1X ~† ~ 1, t · E ~ ~ ~ 1, t , α1 (ωk ) E R R ~ 1, k,λ 1,k,λ 2 ~ k,λ

³ ´ where the subscripts (1) and 1, ~k, λ denote that this is the energy associated with the induced dipole moment p~d,1 ³ ´ ³ ´ ~ 1 as shown in Figure 2.1. The total electric field E ~ ~ ~ 1 , t in mode ~k, λ acting on p~d,1 is given at location R R 1,k,λ by ´ ´ ³ ´ ³ ³ ~ 1, t , ~ 1, t + E ~ ~ ~ 1, t = E ~ ~ ~ ~ R R R E 2,k,λ o,k,λ 1,k,λ ³ ´ ³ ´ ~ ~ ~ 1 , t is the vacuum-field at location R ~ 1 and E ~ ~ ~ 1 , t is the induced dipole field at R ~ 1 due to where E R R o,k,λ 2,k,λ ~ 2 . The effective Hamiltonian becomes the neighboring induced dipole p~d,2 located at R h ³ ´ ³ ´ ³ ´ ³ ´ 1 Xn ~† ~ 1, t · E ~ ~ ~ 1, t + E ~† ~ 1, t · E ~ ~ ~ 1, t α1 (ωk ) E R R R R o,k,λ 2,k,λ o,~ k,λ 2,~ k,λ 2 ~ k,λ ³ ´ ³ ´ ³ ´ ³ ´io ~† ~ 1, t · E ~ ~ ~ 1, t + E ~† ~ 1, t · E ~ ~ ~ 1, t +E R R R R ~ ~ 2,k,λ o,k,λ

Hd,1 = −

o,k,λ

2,k,λ

= Ho + Hp~d,2 + Hp~d,1 ,~pd,2 , where Ho = −

³ ´ ³ ´ 1X ~† ~ 1, t · E ~ ~ ~ 1, t , α1 (ωk ) E R R o,k,λ o,~ k,λ 2 ~ k,λ

Hp~d,2 = −

´ ´ ³ ³ 1X ~ 1, t , ~ 1, t · E ~ ~ ~† R R α1 (ωk ) E 2,k,λ 2,~ k,λ 2 ~ k,λ

Hp~d,1 ,~pd,2 = −

h ³ ´ ³ ´ ³ ´ ³ ´i 1X ~† ~ 1, t · E ~ ~ ~ 1, t + E ~† ~ 1, t · E ~ ~ ~ 1, t . α1 (ωk ) E R R R R ~ ~ 2, k,λ o, k,λ o,k,λ 2,k,λ 2 ~ k,λ

Because only the interaction between the two induced dipoles is relevant to the Casimir effect, the Hp~d,1 ,~pd,2 term is ´ ³ ~ ~ ~ 1 , t is expressed as a sum: considered solely here. In the language of field operators, the vacuum-field E R o,k,λ

³

´

~ ~ ~ ~ (+) E o,k,λ R1 , t = E ~

o,k,λ

³

´

~ 1, t + E ~ (−) R ~

o,k,λ

³

´ ~ 1, t , R

where ³

~ (+) E ~

o,k,λ

~ (−) E o,~ k,λ

³

r ´ ³ ´ 2π~ωk ~ ~ 1 eˆ~ , a~k,λ (0) exp (−iωk t) exp i~k • R R1 , t ≡ i k,λ V ´

r

~ 1 , t ≡ −i R

³ ´ 2π~ωk † ~ 1 eˆ~ . a~ (0) exp (iωk t) exp −i~k • R k,λ k,λ V

In the above expressions, a~†

and a~k,λ are the creation and annihilation operators respectively; and V, the quantization ³ ´ ~ (+) R ~ 1 , t is called the positive frequency (annihilation) operator volume; eˆ~k,λ , the polarization. By convention, E ~ k,λ

o,k,λ

9

2. Casimir Effect ³ ´ ~ 1 , t is called the negative frequency (creation) operator. R ´ ³ ~ ~ ~ The field operator E 2,k,λ R1 , t has the same form as the classical field of an induced electric dipole, (−) o,~ k,λ

~ and E

°¸ ³ ´ n h i o· 1 1 ° °˙ ° ~ ~ ~ 1 , t = 3 pˆd,2 • Sˆ Sˆ − pˆd,2 E R k~ p (t − r/c)k + p ~ (t − r/c) ° ° d,2 d,2 2,k,λ r3 cr2 ° h i o° 1 n ° ° − 2 pˆd,2 − pˆd,2 • Sˆ Sˆ °p~¨d,2 (t − r/c)° , c r ° ° ° ° ° h i ° °~ ° °~° ˆ °~ ° ~ ~ ~ ~ where r = °R − R ≡ S , S = R − R / R − R ˆd,2 = p~d,2 / k~ pd,2 k as shown in Figure 2.1, and c ° ° ° ° 2 1 2 1 2 1° , p ~ is the speed of light in vacuum. Because the ³ dipole ´ moment is expressed as p~d = α (ω) Eω , the appropriate dipole ~ ~ ~ moment in the above expression for E 2,k,λ R1 , t is to be replaced by p~d,2 =

X

h ³ ´ ³ ´i ~ (+) R ~ 2, t + E ~ (−) R ~ 2, t , α2 (ωk ) E ~ ~ o,k,λ

~ k,λ

o,k,λ

where α2 (ωk ) is now the polarizability³of the´molecule or atom associated with the induced dipole moment p~d,2 at the ~ 2 . With this in place, E ~ ~ ~ 1 , t is now a quantum mechanical operator. location R R 2,k,λ

ˆ p~ ,~p can be written as The interaction Hamiltonian operator H d,1 d,2 ˆ p~ ,~p = − 1 H d,1 d,2 2

X ~ k,λ

hD α1 (ωk )

~ (+) E ~

o,k,λ

³

´ ³ ´E D ³ ´ ³ ´Ei ~ 1, t · E ~ ~ ~ ~ ~ ~ ~ (−) R ~ 1, t R + E , 2,k,λ R1 , t 2,k,λ R1 , t · E ~ o,k,λ

³ ´ ³ ´ ~ (+) R ~ 2 , t |vaci = hvac| E ~ (−) R ~ 2 , t = 0. It was shown in [17] where we have taken into account the fact that E o,~ k,λ o,~ k,λ in great detail that the interaction energy is given by ³ ´ X ® 2π~ U (r) ≡ Hp~d,1 ,~pd,2 = − RE k 3 ωk α1 (ωk ) α2 (ωk ) exp (−ikr) exp i~k • ~r V ~ k,λ ¾½ ½ h ¾¸ ·½ i2 i2 ¾ 1 h 1 i ˆ ˆ + 3 eˆ~k,λ • S − 1 + 2 2 . × 1 − eˆ~k,λ • S kr k3 r3 k r In the limit of r ¿ c/ |ωmn | , where ωmn is the transition frequency between the ground state and the first excited energy state, or the resonance frequency, the above result becomes · ¸ 3 −1 2 2 ∼ U (r) = − ~ωo α r−6 , α = 2 [3~ωo ] khm |~ pd | 0ik . 4 This was also the non-retarded van der Waals potential obtained by F. London. Here ωo is the transition frequency, and α is the static (ω = 0) polarizability of an atom in the ground state. Once the retardation effect due to light propagation is taken into account, the Casimir-Polder potential becomes, ¸ · 23 ~cα (ω) α (ω) r−7 . U (r) ∼ − = 1 2 4π What we try to emphasize in this brief derivation is that both retarded and non-retarded van der Waals interaction may be regarded as a consequence of the fluctuating vacuum-fields. It arises due to a non-vanishing corre° ° °~ ° ~ lation of the vacuum-fields over distance of r = °R2 − R1 ° . The non-vanishing correlation here is defined by ¯ ³ ´¯ ³ ´ E D ¯ ~ (+) ~ ~ 2 , t ¯¯ vac 6= 0. In more physical terms, the vacuum-fields induce fluctuating dipole ~ (−) R R , t · E vac ¯E 1 o,~ k,λ o,~ k,λ moments in polarizable media. The correlated dipole-dipole interaction is the van der Waals interaction. If the retardation effect is taken into account, it is called the “Casimir-Polder” interaction. In the Casimir-Polder picture, the Casimir force between two neutral parallel plates of infinite conductivity was

10

2. Casimir Effect

z=d

z=0

Figure 2.2.: A cross-sectional view of two infinite parallel conducting plates separated by a gap distance of z = d. The lowest first two wave modes are shown. found by a simple summation of the pairwise intermolecular forces. It can be shown that such a procedure yields for the force between two parallel plates of infinite conductivity [17] 207~c 2 F~ (d; L, c)Casimir−P older = − L . 640π 2 d4

(2.10)

When this is compared with the force of equation (2.11) computed with Casimir’s vacuum-field approach, which will be discussed in the next section, the agreement is within ∼ 20% [17]. In other words, one can obtain a fairly reasonable estimate of the Casimir effect by simply adding up the pairwise intermolecular forces. The recent experimental verification of the Casimir-Polder force can be found in reference [19]. The discrepancy of ∼ 20% between the two force results of equations (2.10) and (2.11) can be attributed to the fact that the force expression of equation (2.10) had been derived under the assumption that the intermolecular forces were additive in the sense that the force between two molecules is independent of the presence of a third molecule [17, 20]. The van der Waals forces are not however simply additive (see section 8.2 of reference [17]). And, the motivation behind the result of equation (2.10) is to illustrate the intrinsic connection between Casimir-Polder interaction and the Casimir effect, but without any rigor put into the derivation. It is this discrepancy between the microscopic theories assuming additive intermolecular forces, and the experimental results reported in the early 1950s, that motivated Lifshitz in 1956 to develop a macroscopic theory of the forces between dielectrics [21, 22]. Lifshitz theory assumed that the dielectrics are characterized by randomly fluctuating sources. From the assumed delta-function correlation of these sources, the correlation functions for the field were calculated, and from these in turn the Maxwell stress tensor was determined. The force per unit area acting on the two dielectrics was then calculated as the zz component of the stress tensor. In the limiting case of perfect conductors, the Lifshitz theory correctly reduces to the Casimir force of equation (2.11).

2.3. Casimir Force Calculation Between Two Neutral Conducting Parallel Plates Although the Casimir force may be regarded as a macroscopic manifestation of the retarded van der Waals force between two polarizable charge-neutral molecules (or atoms), it is most often alternatively derived by the consideration of the vacuum-field energy ~ω/2 per mode of frequency ω rather than from the summation of the pairwise intermolecular forces. Three different methods widely used in Casimir force calculations are presented here. They are: (1) the Euler-Maclaurin sum approach, (2) the vacuum pressure approach by Milonni, Cook and Goggin, and lastly, (3) the source theory by Schwinger. The main purpose here is to exhibit their different calculational techniques.

2.3.1. Euler-Maclaurin Summation Approach For pedagogical reasons and as a brief introduction to the technique, the Casimir’s original configuration (two chargeneutral infinite parallel conducting plates) shown in Figure 2.2 is worked out in detail.

11

2. Casimir Effect Since the electromagnetic fields are sinusoidal functions, and the tangential component of the electric fields vanish at the conducting surfaces, the functions fi (Li ) have the form fi (Li ) = πL−1 i . The wave numbers are given by ki0 (ni , Li ) = ni fi (Li ) = ni πL−1 . For n = 0 in equation (2.8), the ground state radiation energy is given by s i Hn0 s ,b

( 3 )1/2 ∞ X ∞ X ∞ X X 1 n2i π 2 L−2 . = ~cΘk0 i 2 n =0 n =0 n =0 i=1 1

2

3

For the arrangement shown in Figure 2.2, the dimensions are such that L1 À L3 and L2 À L3 , where (L1 , L2 , L3 ) corresponds to (Lx , Ly , Lz ) . The area of the plates are given by L1 × L2 . The summation over n1 and n2 can be replaced by an integration, Hn0 s ,b =

1 ~cL1 L2 π −2 Θk0 2

Z

∞

Z

0

∞

0

∞ n o1/2 X £ ¤2 2 [kx0 ] + ky0 + n23 π 2 L−2 dkx0 dky0 . i n3 =0

For simplicity and without any loss of generality, the designation of L1 = L2 = L and L3 = d yields the result Hn0 s ,b

Θ k 0 L2 (d) = ~c 2 2 π

Z 0

∞

Z

∞ 0

∞ ½ X n3 =0

2 [kx0 ]

£ ¤2 n2 π 2 + ky0 + 3 2 d

¾1/2 dkx0 dky0 .

Here Hn0 s ,b (d) denotes the vacuum electromagnetic field energy for the cavity when plate gap distance is d. In the limit the gap distance becomes arbitrarily large, the sum over n3 is also replaced by an integral representation to yield µ Z ∞Z ∞Z ∞n ¶ o1/2 £ 0 ¤2 Θk0 L2 d 0 0 2 0 2 0 0 0 Hns ,b (∞) = ~c 2 lim [kx ] + ky + [kz ] dkx dky dkz . 2 π d→∞ π 0 0 0 This is the electromagnetic field energy inside an infinitely large cavity, i.e., free space. The work required to bring in the plates from an infinite separation to a final separation of d is then the potential energy, U (d) = Hn0 s ,b (d) − Hn0 s ,b (∞) "Z Z ¾ ∞ ½ ∞ ∞ X £ ¤2 n2 π 2 1/2 0 0 Θ k 0 L2 2 ~c 2 dkx dky = [kx0 ] + ky0 + 3 2 2 π d 0 0 n3 =0 µ Z ∞Z ∞Z ∞n ¶¸ o £ ¤2 d 2 2 1/2 − lim [kx0 ] + ky0 + [kz0 ] dkx0 dky0 dkz0 . d→∞ π 0 0 0 The result is a grossly divergent function. Nonetheless, with a proper choice of the cutoff function (or regularization function), a finite value for U (d) can be obtained. In the polar coordinates representation (r, θ) , we define r2 = £ ¤2 2 [kx0 ] + ky0 and dkx0 dky0 = rdrdθ, then "Z r ∞ π/2 Z ∞ X Θk0 ~cL2 n2 π 2 U (d) = r2 + 3 2 rdrdθ 2 2π d θ=0 r=0 n =0 3 !# Ã Z Z π/2 Z ∞ q d ∞ 2 0 2 0 r + [kz ] rdrdθdkz , − lim d→∞ π kz0 =0 θ=0 r=0 where the integration over θ is done in the range 0 ≤ θ ≤ π/2 to ensure kx0 ≥ 0 and ky0 ≥ 0. For convenience, the integration over θ is carried out first, "Z Ã Z !# r ∞ ∞ X ∞ Z ∞ q 2 π2 Θk0 ~cL2 d n 2 0 3 U (d) = r2 + 2 rdr − lim r2 + [kz0 ] rdrdkz . d→∞ 4π d π kz0 =0 r=0 r=0 n =0 3

As mentioned earlier, U (d) in current form is grossly divergent. The need to regularize this divergent function through

12

2. Casimir Effect some physically intuitive cutoff function is not a mere mathematical convenience, it is a must; otherwise, such a grossly in physics. A cutoff (or regularizing) function in the form of f (k 0 ) = µq divergent¶function is meaningless ³p ´ 2 0 0 r2 + [kz0 ] (or f (k 0 ) = f r2 + n23 π 2 d−2 ) such that f (k 0 ) = 1 for k 0 ¿ kcutof f f and f (k ) = 0 for 0 0 k 0 À kcutof f is chosen. Mathematically speaking, this cutoff function f (k ) is able to regularize the above divergent function. Physically, introduction of the cutoff takes care of the failure at small distance of the assumption that plates 0 are perfectly conducting for short wavelengths. It is a good approximation to assume kcutof f ∼ 1/ao , where ao is the Bohr radius. In this sense, one is inherently assuming that Casimir effect is primarily a low-frequency or long wavelength effect. Hence, with the cutoff function substituted in U (d) above, the potential energy becomes " ∞ Z Ãr ! r 2 π2 2 π2 n n Θk0 ~cL2 X ∞ r2 + 3 2 f r2 + 3 2 U (d) = rdr 4π d d r=0 n3 =0 Ã Z !# µq ¶ Z ∞q d ∞ 2 2 0 r2 + [kz0 ] f r2 + [kz0 ] rdrdkz − lim . d→∞ π kz0 =0 r=0

R∞ P∞ The summation n3 =0 and the integral r=0 in the first term on the right hand side can be interchanged. The interchange of sums and integrals is justified due to the absolute convergence in the presence of the cutoff function. In terms of the new definition for the integration variables x = r2 d2 π −2 and κ = kz0 dπ −1 , the above expression for U (d) is rewritten as " µ q ¶ ∞ Z q π 1 1 X ∞ 2 2 2f 2 dx x + n x + n U (d) = Θk0 ~cL π 3 3 8 d3 n =0 x=0 d 3 µ Z ∞ Z ∞ p ¶¸ ³π p ´ 1 2f 2 dxdκ − lim x + κ x + κ d→∞ d3 κ=0 x=0 d " # Z ∞ ∞ X 1 2 2 1 ≡ Θk0 ~cL π F (0) + F (n3 ) − F (κ) dκ , 8 2 κ=0 n =1 3

where F (n3 ) ≡

1 d3

Z

q

∞

x=0

x + n23 f

µ q ¶ π x + n23 dx, d

and µ F (κ) ≡ lim

d→∞

1 d3

Z

∞

p

x + κ2 f

x=0

´ ¶ x + κ2 dx .

³πp d

Then, according to the Euler-Maclaurin summation formula [23, 24], ∞ X n3 =1

Z F (n3 ) −

∞

1 1 dF (0) 1 d3 F (0) F (κ) dκ = − F (0) − + + ··· 2 12 dκ 720 dκ3 κ=0

¡ ¢ R∞√ ¡ √ ¢ for F (∞) → 0. Noting that from F (κ) = κ2 rf πd r dr and dF (κ) /dκ = −2κ2 f πd κ , one can find dF (0) /dκ = 0, d3 F (0) /dκ3 = −4, and all higher order derivatives vanish if one assumes that all derivatives of the cutoff function vanish at κ = 0. Finally, the result for the vacuum electromagnetic potential energy U (d) becomes U (d; L, c) = −Θk0

~cπ 2 2 L . 1440d3

This result is finite, and it is independent of the cutoff function as it should be. The corresponding Casimir force for

13

2. Casimir Effect the two infinite parallel conducting plates is given by ∂U (d; L, c) 3~cπ 2 2 F~ (d; L, c) = − = −Θk0 L . ∂d 1440d4 The electromagnetic wave has two possible polarizations, Θk0 = 2, therefore, ~cπ 2 2 F~ (d; L, c) = − L . 240d4

(2.11)

This is the Casimir force between two uncharged parallel conducting plates [3]. It is to be noted that the Euler-Maclaurin summation approach discussed here is just one of the many techniques that can be used in calculating the Casimir force. One can also employ dimensional regularization to compute the Casimir force. This technique can be found in section 2.2 of the reference [14].

2.3.2. Vacuum Pressure Approach The Casimir force between two perfectly conducting plates can also be calculated from the radiation pressure exerted by a plane wave incident normally on one of the plates. Here the radiation pressure is due to the vacuum electromagnetic fields. The technique discussed here is due to Milonni, Cook and Goggin [25]. The Casimir force is regarded as a consequence of the radiation pressure associated with the zero-point energy of ~ω/2 per mode of the field. The main idea behind this techniques is that since the zero-point fields have the momentum p0i = ~ki0 /2, the pressure exerted by an incident wave normal to the plates is twice the energy H per unit volume of the incident field. The pressure imparted to the plate is twice that of the incident wave for perfect conductors. If the wave has an angle of incidence θinc , the radiation pressure is P = F A−1 = 2H cos2 θinc . Two factors of cos θinc appear here because (1) the normal component of the linear momentum imparted to the plate is proportional to cos θinc , and (2) the element of area A is increased by 1/ cos θinc compared with the case of normal incidence. It can be shown then ° 1 1 ~ω 0 2 ° ° °−2 P = 2H cos2 θinc = 2 × × ~ω × V −1 × cos2 θinc = [kz ] °k~0 ° , 2 2 2V where the factor of half have been inserted because the zero-point field energy of a mode of energy ~ω/2 is divided equally between waves propagating toward ° ° and away from each of the plates. The cos θinc factor have been rewritten ° ° using the fact that kz0 = k~0 • eˆz = °k~0 ° cos θinc , where eˆz is the unit vector normal to the plate on the inside, ° ° ° ~0 ° °k ° = ω/c and V is the quantization volume. The successive reflections of the radiation off the plates act to push the plates apart through a pressure P. For large plates where kx0 , ky0 take on a continuum of values and the component along the plate gap is kz0 = nπ/d, where n is a positive integer, the total outward pressure on each plate over all possible modes can be written as Pout =

Z ∞ ∞ Z Θk0 ~c X ∞ q 2π 2 d n=1 ky0 =0 kx0 =0

2

[nπ/d] dkx0 dky0 , £ ¤2 2 2 [kx0 ] + ky0 + [nπ/d]

where Θk0 is the number of independent polarizations. to the plates, the allowed field modes take on a continuum of values. Therefore, by the replacement of R∞ PExternal ∞ −1 d k0 =0 in the above expression, the total inward pressure on each plate over all possible modes is given n=1 → π z by Pin =

Θk0 ~c 2π 3

Z

∞

Z

∞

Z

2

∞

[kz0 ] dkx0 dky0 dkz0 . £ ¤2 2 2 0 0 0 [kx ] + ky + [kz ]

q kz0 =0

ky0 =0

0 =0 kx

14

2. Casimir Effect Both Pout and Pin are infinite, but their difference has physical meaning. After some algebraic simplifications, the difference can be written as "∞ # Z ∞ Z ∞ Z Θk0 π 2 ~c X 2 ∞ dx u2 √ √ Pout − Pin = − dxdu . n 8d4 x + n2 x + u2 x=0 u=0 x=0 n=1 An application of the Euler-Maclaurin summation formula [23, 24] leads to the Casimir’s result Pout − Pin = −

π 2 ~c , 240d4

where Θk0 = 2 for two possible polarizations for zero-point electromagnetic fields.

2.3.3. The Source Theory Approach

The Casimir effect can also be explained by the source theory of Schwinger [14, 15, 17]. An induced dipole p~d in a ~ has an energy Hd = −~ ~ field E pd • E/2. The factor of one half comes from the fact that this is an induced dipole energy. ~ When there are N dipoles per unit volume,Dthe associated E polarization is P = N p~d and the expectation value of the R 3 ~ ~ Here the polarizability in p~d is left as a free parameter energy in quantum theory is hHd i = − p~d • E/2 d R. which needs to be determined from the experiment. The expectation value of the energy is then Z E 1 D ~ ~ (+) + E ~ (−) • p~d d3 R, p~d • E hHd i = − 2 ³ ´ ³ ´ ³ ´ ~ s(±) is the field due to other ~ t +E ~ s(±) R, ~ t . Here E ~ v(±) is the vacuum-field and E ~ (±) R, ~ t =E ~ v(±) R, where E ³ ´ ³ ´ ~ t |vaci = hvac| E ~ v(−) R, ~ t = 0, the above expectation value of the energy can be written ~ v(+) R, sources. Since E as Z E 1 D ~ (+) d3 R ~ + c.c., hHd i = − p~d • E (2.12) 2 where c.c. denotes complex ³ ´ conjugation. From the fact that electric field operator can be written as an expansion in ~α R ~ , the mode functions A ~ (+) = i E

h ³ ´ ³ ´i Xp ~α R ~ − a†α (t) A ~ ∗α R ~ , 2π~ωα aα (t) A α

the Heisenberg equation of motion for a˙ α (t) and aα,s (t) are obtained as r Z ³ ´ ³ ´ 2πωα ~∗ R ~ • p~d R, ~ t d3 R, ~ A a˙ α (t) = −iωα aα (t) + α ~ r aα,s (t) =

2πωα ~

Z

t

Z exp (iωα [t0 − t]) dt0

0

15

´ ³ ´ ³ ~ ~ ∗α R ~ • p~d R, ~ t0 d3 R, A

2. Casimir Effect where³aα,s´(t) is the source contribution part of aα (t) . The “positive frequency” or the photon annihilation part of ~ s(+) R, ~ t can then be written as E Z ³ ´ ³ ´Z t ³ ´ ³ ´ X ~ s(+) R, ~ t = 2πi ~α R ~ ~ ∗α R ~ 0 • p~d R ~ 0 , t0 d3 R ~0 E ωα A exp (iωα [t0 − t]) dt0 A α

= 2πi

Z Z tX 0

0

³ ´ ³ ´ ³ ´ ~ A ~ ∗α R ~ 0 exp (iωα [t0 − t]) • p~d R ~ 0 , t0 dt0 d3 R ~0 ~α R ωα A

α

Z Z t ←−→ ³ ´ ³ ´ ~ R ~ 0 ; t, t0 • p~d R ~ 0 , t0 dt0 d3 R ~ 0, ≡ 8π G(+) R, 0

´ ←−→ ³ ~ R ~ 0 ; t, t0 is a dyadic Green function where G(+) R, ´ ³ ´ ³ ´ X ←−→ ³ ~ R ~ 0 ; t, t0 = i ~α R ~ A ~∗ R ~ 0 exp (iωα [t0 − t]) . G(+) R, ωα A α 4 α

(2.13)

Equations (2.12) and (2.13) lead to the result Z Z Z t ←−→ ³ ´D ³ ´ ³ ´E (+) ~ ~ 0 ~ t • p~d,i R ~ 0 , t0 dt0 d3 R ~ 0 d3 R, ~ Gij R, R ; t, t0 p~d,j R, hHd i = −8πRE ~ R

~0 R

0

where the summation over repeated indices is understood, and RE denotes the real part. The above result is the energy of the induced dipoles in a medium due to the source fields produced by the dipoles. It can be further shown that for the infinitesimal variations in energy, Z Z Z tZ ∞ ³ ´D ³ ´ ³ ´E ~ R ~ 0 , ω p~d,j R, ~ t • p~d,i R ~ 0 , t0 exp (iω [t0 − t]) dωdt0 d3 R ~ 0 d3 R, ~ hδHd i = −4RE Γij R, ~ R

³

~0 R

0

0

´

´ ←−→ ³ ~ R ~ 0 , ω is related to G(+) R, ~ R ~ 0 ; t, t0 through the relation where Γij R, ij Z ∞ ´ ³ ´ ←−→ ³ 1 (+) ~ ~ 0 0 ~ R ~ 0 , ω exp (iω [t0 − t]) dω. G R, R ; t, t = Γij R, 2π 0 The force per unit area can then be shown to be Z ∞Z ³ ´ i~ ~ F (d) = [ε − ε ] Γ d, d, k , ω d2~k⊥ dω, 2 3 jj ⊥ 8π 3 0 ~ k⊥ ³ ´ where the factor [ε2 − ε3 ] Γjj d, d, ~k⊥ , ω is given by ³ [ε2 − ε3 ] Γjj

(2.14)

(µ·

¸· ¸ ¶−1 K2 + K3 K1 + K3 exp (2K3 d) − 1 K1 − K3 K2 − K3 µ· ¸· ¸ ¶−1 ) ε3 K1 + ε1 K3 ε3 K2 + ε2 K3 + exp (2K3 d) − 1 . ε3 K1 − ε1 K3 ε3 K2 − ε2 K3

´ d, d, ~k⊥ , ω = 2 [K3 − K2 ] + 2K3

2 Here K 2 ≡ k⊥ − c−2 ω 2 ε (ω) and εi is the dielectric constant corresponding to the region i. The plate configuration corresponding to the source theory description discussed above is illustrated in Figure 2.3.

16

2. Casimir Effect

ε1

ε2

ε3

z=d

z=0

Figure 2.3.: A cross-sectional view of two infinite parallel conducting plates. The plates are separated by a gap distance of z = d. Also, the three regions have different dielectric constants εi (ω) . The expression of force, equation (2.14), is derived from the source theory of Schwinger, Milton and DeRaad [14, 15]. It reproduces the result of Lifshitz [21, 22], which is a generalization of the Casimir force involving perfectly conducting parallel plates to that involving dielectric media. The details of this brief outline of the source theory description can be found in references [14, 17].

17

3. Reflection Dynamics Once the idea of physics of vacuum polarization is taken for granted, one can move forward to calculate the effective, temperature-averaged energy due to the dipole-dipole interactions with the time retardation effect folded into the van der Waals interaction. The energy between the dielectric or conducting media is then obtained from the allowed modes of electromagnetic waves determined by the Maxwell equations together with the electromagnetic boundary conditions, granted that the most significant zero-point electromagnetic field wavelengths determining the interaction are large when compared with the spacing of the lattice points in the media. Under such an assumption, the effect of all the multiple dipole scattering by atoms in the dielectric or conducting media is to simply enforce the macroscopic reflection laws of electromagnetic waves; and this allows the macroscopic electromagnetic theory to be used with impunity in calculation of the Casimir force, granted the classical electromagnetic fields have been quantized. The Casimir force is then simply obtained by taking the negative gradient of the energy in space. In principle, the atomistic approach utilizing the Casimir-Polder interaction explains the Casimir effect observed between any system. Unfortunately, the pairwise summation of the intermolecular forces for systems containing large number of atoms can become very complicated. H. B. G. Casimir, realizing the linear relationship between the field and the polarization, devised an easier approach to the calculation of the Casimir effect for large systems such as two perfectly conducting parallel plates. This latter development is the description of the Euler-Maclaurin summation approach shown previously, in which the Casimir force have been found by utilizing the field boundary conditions only. The vacuum pressure approach originally introduced by Milonni, Cook and Goggin [25] is a simple elaboration of Casimir’s latter invention utilizing the boundary conditions. The source theory description of Schwinger is an alternate explanation of the Casimir effect which can be inherently traced to the retarded van der Waals interaction. Because all four approaches which were previously mentioned, (1) the Casimir-Polder interaction, (2) the EulerMaclaurin summation, (3) the vacuum pressure and (4) the source theory, stem from the same physics of vacuum polarization, they are equivalent. The preference of one over another mainly depends on the geometry of the boundaries being investigated. For the type of physical arrangements of boundary configurations that are being considered in this thesis, the vacuum pressure approach provides the most natural route to the Casimir force calculation. The three physical arrangements for the boundary configurations considered in this thesis are: (1) the plate-hemisphere, (2) the hemisphere-hemisphere and (3) a sphere formed by brining two hemispheres together. Because the geometric configurations of items (2) and (3) are special versions of the more general, plate-hemisphere configuration, the basic reflection dynamics needed for the plate-hemisphere case is worked out first. The results can then be applied to the hemisphere-hemisphere and the sphere configurations later. The vacuum-fields are subject to the appropriate boundary conditions. For boundaries made of perfect conductors, the transverse components of the electric field are zero at the surface. For this simplification, the skin depth of penetration is considered as zero. The plate-hemisphere under consideration is shown in Figure 3.1. The solutions to the³vacuum-fields are °´ ´ ³ that ´ of the Cartesian version of the free Maxwell field vector potential differential ³ ° equation °~° 2~ ~ −2 2 ~ ~ ~ ~ ∇ A R − c ∂t A R = 0, where the Coulomb gauge ∇ • A = 0 and the absence of the source Φ ρ, °R ° =0 ~ ~ have been imposed. The electric and the magnetic field component of the vacuum-field are given by E = −c−1 ∂t A ~ ~ ~ ~ and B = ∇ × A, where A is the free field vector potential. The zero value requirement for the transversal ° °´ ³ component ~ is in the form of E ~ ∝ sin 2πλ−1 ° ~° of the electric field at the perfect conductor surface implies the solution for E °L ° , ° ° °~ ° where λ is the wavelength and °L ° is the path length between the boundaries. The wavelength is restricted by the ° ° ° ~0 ~0 ° ~0 ~0 condition λ ≤ 2 °R 2 − R 1 ° ≡ 2ξ2 , where R 2 and R 1 are two immediate reflection points in the hemisphere cavity of Figure 3.1. In order to compute the modes allowed inside the hemisphere resonator, a detailed knowledge of the reflections occurring in the hemisphere cavity is needed. This is described in the following section.

18

3. Reflection Dynamics

θ1 a

Infinity

^ θ kξ 1 k 1 θi θ r

R0R i

R

R1

θ

b R0

Ri θ2 θr

^z

k

^y

ξ2 ^k 2

R2 θr

Infinity

Figure 3.1.: The plane of incidence view of plate-hemisphere configuration. The waves through ° that are supported ° ° ~0 0 ° ~ internal reflections in the hemisphere cavity must satisfy the relation λ ≤ 2 °R − R . ° 2 1

3.1. Reflection Points on the Surface of a Resonator

The wave vector directed along an arbitrary direction in Cartesian coordinates is written as  0 3 , eˆ1 = x ˆ,  i = 1 → k1,x X ¢ ¡ 0 0 0 0 0 0 0 ~ , eˆ2 = yˆ i = 2 → k1,y k1,i = k 1 k1,x , k1,y , k1,z = k1,i eˆi ,  0 i=1 i = 3 → k1,z , eˆ3 = zˆ.

(3.1)

° °−1 P ° ° 3 0 ~0 ~0 Hence, the unit wave vector, kˆ0 1 = °k~0 1 ° i=1 k1,i eˆi . Define the initial position R 0 for the incident wave k 1 , 3 ¡ ¢ X 0 ~ 0 0 r0 , r0 , r0 R = r0,i eˆi , 0,x 0,y 0,z i=1

0 r0,i

 0 ,  i = 1 → r0,x 0 i = 2 → r0,y , =  0 i = 3 → r0,z .

(3.2)

~ 0 0 really has only components r0 and r0 . But nevertheless, one can always set Here it should be noted that R 0,x 0,z 0 r0,y = 0 whenever needed. Since no particular wave vectors with specified wave lengths are prescribed initially, it is desirable to employ a parameterization scheme to represent these wave vectors. The line segment traced out by this

19

3. Reflection Dynamics wave vector kˆ0 1 is formulated in the parametric form ~ 00 = ~ 0 1 = ξ1 kˆ0 1 + R R

¸ 3 · ° °−1 X ° ~0 ° 0 0 r0,i + ξ1 °k 1 ° k1,i eˆi ,

(3.3)

i=1

where the variable ξ1 is a positive definite parameter. The restriction ξ1 ≥ 0 is necessary because the direction of the ~ 0 1 is the first reflection point on the hemisphere. In terms of spherical coordinate wave propagation is set by kˆ0 1 . Here R ~ 0 1 takes the form variables, R  0 3  Λ1,1 = sin θ10 cos φ01 , X 0 0 0 0 0 0 ~ Λ0 = sin θ10 sin φ01 , R 1 (ri , θ1 , φ1 ) = ri Λ1,i eˆi , (3.4)  01,2 i=1 Λ1,3 = cos θ10 , ~ 0 1 at the first where ri0 is the hemisphere radius, θ10 and φ01 are the polar and the azimuthal angle respectively of R 0 reflection point. Notice that subscript i of ri denotes “inner radius” not a summation index. By combining equations (3.3) and (3.4), we can solve for the parameter ξ1 . It can be shown that rh ° °2 i2 0 0 ˆ ~ ~ 0 0 + [r0 ]2 − ° ~ 00° kˆ0 1 • R ξ1 ≡ ξ1,p = −k 1 • R 0 + °R ° , i

(3.5)

where the positive root for ξ1 have been chosen due to the restriction ξ1 ≥ 0. The detailed proof of equation (3.5) is given in Appendix A, where the same equation is designated as equation (A.11). Substituting ξ1 in equation (3.3), the first reflection point off the inner hemisphere surface is expressed as ¸ 3 · ° °−1 ³ ´ X ° ~0 ° 0 0 0 0 0 ˆ ~ ~ R 1 ξ1,p ; R 0 , k 1 = r0,i + ξ1,p °k 1 ° k1,i eˆi ,

(3.6)

i=1

where ξ1,p is from equation (3.5). The incoming wave vector k~0 i can always be decomposed into parallel and perpendicular with respect to the local ~0 reflection surface components, k~0 i,k and h k i,⊥ . iIt is shown in equation (A.14) of Appendix A that the reflected wave vector k~0 r has the form k~0 r = αr,⊥ nˆ0 × k~0 i × nˆ0 − αr,k nˆ0 • k~0 i nˆ0 , where the quantities αr,k and αr,⊥ are the reflection coefficients and nˆ0 isna unithsurface normal. For ithe perfect reflecting surfaces, αr,k = αr,⊥ = 1. In o P3 0 0 0 0 0 0 0 0 0 0 ~ ˆ0 component form, k r = l=1 αr,⊥ nn ki,l nn − nl ki,n nn − αr,k nn ki,n nl eˆl , where it is understood that n is ~0 already normalized and Einstein summation convention is applied to the index n. The°second ° reflection point R 2 is ° ° ~ 0 1 and by using the expression k~0 r ≡ k~0 r / °k~0 r ° , found then by repeating the steps done for R

~ 02 R

h i αr,⊥ nˆ0 × k~0 i × nˆ0 − αr,k nˆ0 • k~0 i nˆ0 ~ 0 1 + ξ2,p kˆ0 r = R ~ 0 1 + ξ2,p ° °, h i =R ° ° °αr,⊥ nˆ0 × k~0 i × nˆ0 − αr,k nˆ0 • k~0 i nˆ0 °

where ξ2,p is the new positive definite parameter for the second reflection point. The incidence plane of reflection is determined solely by the incident wave k~0 i and the local normal n~0 i of the reflecting surface. It is important to recognize the fact that the subsequent successive reflections of this incoming wave will be confined to this particular incident plane. This incident plane can be characterized by a unit normal vector. For ~ 0 0 . The unit vector which represents the incident the system shown in Figure 3.1, k~0 i = k~0 1 and n~0 n0i ,1 = −ξ1,p kˆ0 1 − R ° °−1 P ° ° 3 0 0 plane is given by nˆ0 p,1 = − °n~0 p,1 ° i=1 ²ijk k1,j r0,k eˆi , where the summations over indices j and k are implicit. If the plane of incidence is represented by a scalar function f (x0 , y 0 , z 0 ) , then its unit normal vector nˆ0 p,1 will satisfy

20

3. Reflection Dynamics

^z

Intercept

ξ i ^k i θi θr R0

Ri ψi,i+1

^

ξ i+1 k i+1 θ i+1

R i+1

^x

n^ p,i

θ r+1

^y

ψi+1,i+2 R i+2 θ i+2 ξ i+3 ^k i+3 θ r+2

ξ i+2 ^k i+2

The x coordinate is coming out of the page!

Figure 3.2.: The thick line shown here represents the intersection between hemisphere surface and the plane of inci° °−1 P ° ° 3 dence. The unit vector normal to the incident plane is given by nˆ0 p,1 = − °n~0 p,1 ° ²ijk k 0 r0 eˆi . i=1

~ 0 fp,1 (x0 , y 0 , z 0 ) . It is shown from equation (A.43) of Appendix A that the relationship nˆ0 p,1 ∝ ∇  3 ° °−1 X  1 → ν10 = x0 , ° ~0 ° 0 0 0 0 0 0 2 → ν20 = y 0 , fp,1 (ν1 , ν2 , ν3 ) = − °n p,1 ° ²ijk k1,j r0,k νi , i=  i=1 3 → ν30 = z 0 ,

1,j 0,k

(3.7)

where −∞ ≤ {ν10 = x0 , ν20 = y 0 , ν30 = z 0 } ≤ ∞. P3 2 2 The surface of a sphere or hemisphere is defined through the relation fhemi (x0 , y 0 , z 0 ) = [ri0 ] − i=1 [νi0 ] , 0 where ri is the radius of sphere and the subscript i denotes the inner surface. The intercept of interest is shown in Figure 3.2. The intersection between the hemisphere surface and the incidence plane fp,1 (ν10 , ν20 , ν30 ) is given by fhemi (x0 , y 0 , z 0 ) − fp,1 (x0 , y 0 , z 0 ) = 0. After substitution of fp,1 (x0 , y 0 , z 0 ) and fhemi (x0 , y 0 , z 0 ) , we have  ¾ 3 ½ ° °−1  1 → ν10 = x0 , X 2 2 ° ~0 ° 0 0 0 0 0 2 → ν20 = y 0 , [νi ] − °n p,1 ° ²ijk k1,j r0,k νi − [ri ] = 0, i=  i=1 3 → ν30 = z 0 . P3 £ 0 ¤2 2 2 0 0 0 0 0 0 The term [ri0 ] can be rewritten in the form [ri0 ] = i=1 ri,i , where ri,1 = ri,x 0 , ri,2 = ri,y 0 and ri,3 = ri,z 0 . 0 Solving for νi , it can be shown from equation (A.51) of Appendix A that νi0

°−1 1° ° ° 0 0 = °n~0 p,1 ° ²ijk k1,j r0,k ± 2

(· )1/2 ¸2 ° £ 0 ¤2 1° ° ~0 °−1 0 0 , °n p,1 ° ²ijk k1,j r0,k + ri,i 2

i = 1, 2, 3,

(3.8)

where ²ijk is the Levi-Civita coefficient. The result for νi0 shown above provide a set of discrete reflection points found by the intercept between the hemisphere and the plane of incidence. 0 0 0 ~ 0 1 can be Using spherical coordinate representations for the variables ri,1 , ri,2 and ri,3 , the initial reflection point R

21

3. Reflection Dynamics

θ ^z

d

θ ri

R

ro

R

x^

φ

^y

RT

a

2a

φ

Figure 3.3.: The surface of the hemisphere-hemisphere configuration can be described relative to the system origin ~ or relative to the hemisphere centers through R ~ 0. through R,

expressed in terms of the spherical coordinate variables (ri0 , θ10 , φ01 ) (equation (A.109) of Appendix A),  0 3 = ri0 sin θ10 cos φ01 ,  1 → ν1,1 X 0 0 0 0 0 0 0 0 0 ~ 2 → ν1,2 = ri0 sin θ10 sin φ01 , R 1 (ri , θ1 , φ1 ) = ν1,i (ri , θ1 , φ1 ) eˆi , i=  0 i=1 = ri0 cos θ10 , 3 → ν1,3

(3.9)

where ri0 is the hemisphere radius, φ01 and θ10 , the polar and azimuthal angle, respectively. They are defined in equations (A.102), (A.103), (A.107) and (A.108) of Appendix A. Similarly, the second reflection point on the inner hemisphere surface is given by equation (A.151) of Appendix A:  0 3 = ri0 sin θ20 cos φ02 ,  1 → ν2,1 X 0 0 0 0 0 0 0 0 0 ~ 2 → ν2,2 = ri0 sin θ20 sin φ02 , (3.10) i= ν2,i (ri , θ2 , φ2 ) eˆi , R 2 (ri , θ2 , φ2 ) =  0 i=1 = ri0 cos θ20 , 3 → ν2,3 where the spherical angles φ02 and θ20 are defined in equations (A.143), (A.144), (A.148) and (A.149) of Appendix A. In general, leaving the details to Appendix A, the N th reflection point inside the hemisphere is, from equation (A.162) of Appendix A,  0 0 = ri0 sin θN cos φ0N , 3  1 → νN,1 X 0 0 0 0 0 0 0 0 0 0 ~ 0 N (ri , θN , φN ) = 2 → νN,2 = ri sin θN sin φ0N , R νN,i (ri , θN , φN ) eˆi , i= (3.11)  0 0 i=1 3 → νN,3 = ri0 cos θN , 0 where the spherical angles θN and φ0N are defined in equations (A.158), (A.159), (A.160) and (A.161) of Appendix A. The details of all the work shown up to this point can be found in Appendix A.

~ 01, R ~ 0 2 and R ~ 0 N ) were described relative to the hemisphere center. In The previously shown reflection points (R many cases, the preferred choice for the system origin, from which the variables are defined, depend on the physical arrangements of the system being considered. For a sphere, the natural choice for the origin is its center from which the spherical variables (ri0 , θ0 , φ0 ) are prescribed. For more complicated configuration shown in Figure 3.3, the preferred choice for origin really depends on the problem at hand. For this reason, a set of transformation rules between (ri0 , θ0 , φ0 ) and (ri , θ, φ) is sought. Here the primed set is defined relative to the sphere center and the unprimed set is ~ and R ~0 defined relative to the origin of the global configuration. In terms of the Cartesian variables, the two vectors R

22

3. Reflection Dynamics describing an identical point on the hemisphere surface are expressed by ~ (ν1 , ν2 , ν3 ) = R

3 X

~ 0 (ν 0 , ν 0 , ν 0 ) = R 1 2 3

νi eˆi ,

i=1

3 X

νi0 eˆi ,

(3.12)

i=1

~ and R ~ 0 are where (ν1 , ν2 , ν3 ) → (x, y, z) , (ν10 , ν20 , ν30 ) → (x0 , y 0 , z 0 ) and (eˆ1 , eˆ2 , eˆ3 ) → (ˆ x, yˆ, zˆ) . The vectors R P P 3 3 0 ~ ~ (ν1 , ν2 , ν3 ) = connected through the relation R i=1 [νT,i + νi ] eˆi with RT ≡ Pi=1 νT,i eˆi which represents the 3 position of hemisphere center relative to the system origin. As a result, we have i=1 [νi − νT,i − νi0 ] eˆi = 0. In terms of the spherical coordinate representation for (ν1 , ν2 , ν3 ) and (ν10 , ν20 , ν30 ) , we can solve for θ and φ. As shown from equations (B.10) and (B.12) of Appendix B, the result is µ ¶ νT,2 + ri0 sin θ0 sin φ0 0 0 0 ` φ ≡ φ (ri , θ , φ , νT,1 , νT,2 ) = arctan , (3.13) νT,1 + ri0 sin θ0 cos φ0 ³ ´ ~T θ ≡ θ` ri0 , θ0 , φ0 , R   0 0 0 0 0 0 −1 {ν + ν + r sin θ [cos φ + sin φ ]} [ν + r cos θ ] T,3 i ´´ ³ ³ ´´  , ³ T,1 ³ T,2 0 i 0 = arctan  νT ,2 +ri0 sin θ 0 sin φ0 ν ,2 +ri sin θ sin φ0 + sin arctan cos arctan νTT,1 +r 0 sin θ 0 cos φ0 νT ,1 +r 0 sin θ 0 cos φ0 i

(3.14)

i

where the notation φ` and θ` indicates that φ and θ are explicitly expressed in terms of the primed variables, respectively. It is to be noticed that for the configuration shown in Figure 3.3, the hemisphere center is only shifted along yˆ by an amount of νT,2 = a, which leads to νT,i6=2 = 0. Nevertheless, the derivation have been done for the case where νT,i 6= 0, i = 1, 2, 3 for the generalization purpose. ° ° nP o1/2 °~° 3 0 0 2 With the magnitude °R , where Λ01 (θ0 , φ0 ) = sin θ0 cos φ0 , Λ02 (θ0 , φ0 ) = sin θ0 sin φ0 °= i=1 [νT,i + ri Λi ] ´ ³ ~` Λ ~ 0, R ~ T is given by equation (B.13) of Appendix B as ~ r0 , Λ, and Λ03 (θ0 ) = cos θ0 , the vector R i ´ ³ ~` Λ ~ 0, R ~T = ~ ri0 , Λ, R

(

3 X

)1/2 [νT,i + ri0 Λ0i ]

2

i=1

3 X i=1

` i eˆi , Λ

 ³ ´ ` φ` = sin θ` cos φ, ` ` 1 θ,  Λ    ³ ´ ` φ` = sin θ` sin φ, ` ` 2 θ, Λ  ³ ´   ` ` 3 θ` = cos θ.  Λ

(3.15)

The details of this section can be found in Appendices A and B.

3.2. Selected Configurations Having found all of the wave reflection points in the hemisphere resonator, the net momentum imparted on both the inner and outer surfaces by the incident wave is computed for three configurations: (1) the sphere, (2) the hemispherehemisphere and (3) the plate-hemisphere. The surface element that is being impinged upon by´an incident wave would ³ ~ 0 s,1 , R ~ 0 s,0 on the inner side, and experience the net momentum change in an amount proportional to 4k~0 inner ; R ³ ´ ~ 0 s,1 + aRˆ0 s,1 on the outer side of the surface. The quantities 4k~0 inner and 4k~0 outer are due to the 4k~0 outer ; R ³ ´ ~ 0 s,1 , R ~ 0 s,0 of 4k~0 inner contribution from a single mode of wave traveling in particular direction. The notation ; R ~ 0 s,1 on the surface and the initial crossing point R ~ 0 s,0 of denotes that it is defined in terms of the initial reflection point R ³ ´ ~ 0 s,1 + aRˆ0 s,1 of 4k~0 outer implies the outer the hemisphere opening (or the sphere cross-section). The notation ; R surface reflection point. The total resultant imparted momentum on the hemisphere or sphere is found by summing over all modes of wave, over all directions.

23

3. Reflection Dynamics

The initial wave−number vector

^z

ki

kr

ki

ki

ψ

i,i+1

L kr

R0 Ri

R i+1

a

^ y

R i+2

~ 0 i induces a series of reflections that propagate Figure 3.4.: Inside the cavity, an incident wave k~0 i on first impact point R 0 ~ ~ 0 i + aRˆ0 i , where throughout the entire inner cavity. Similarly, a wave k i incident on the point R ° impact ° ° ~0 ° a is the thickness of the sphere, induces reflected wave of magnitude °k i ° . The resultant wave direction ~ 0 i and the resultant wave direction in the resonator is along −R ~ 0 i due to in the external region is along R the fact there is exactly another wave vector traveling in opposite direction in both regions. In both cases, the reflected and incident waves have equal magnitude due to the fact that the sphere is assumed to be a perfect conductor.

3.2.1. Hollow Spherical Shell A sphere formed by bringing in two hemispheres together is shown in Figure 3.4. The resultant change in wave vector direction upon reflection at the inner surface of the sphere is from the equation (C.4) of Appendix C1, ½ ³ ´ 4nπ cos θinc 0 ≤ θinc < π/2, 0 0 0 0 ~ ~ ~ ˆ ° ´ ³ ´° ³ 4k inner ; R s,1 , R s,0 = − ° R s,1 , (3.16) ° n = 1, 2, · · · , 0 0 0 0 ~ s,2 r , Λ ~ ~ ~ °R s,2 − Rs,1 ri , Λs,1 ° i ´ ´ ³ ³ 0 ~0 ~ s,1 r0 , Λ ~0 ~ where θinc is from equation (A.115); R s,1 and Rs,2 ri , Λs,2 follow the generic form shown in the equai tion (C.1) of Appendix C1, ¡ 0 ¢  0 0 0 0 3  Λs,N,1 ¡θs,N , φs,N ¢ = sin θs,N cos φs,N , ³ ´ X 0 ~0 0 0 0 0 0 0 0 ~ Λ θ ,φ = sin θs,N sin φs,N , Rs,N ri , Λs,N = ri Λs,N,i eˆi , (3.17)  s,N,2 0 s,N ¡ 0s,N ¢ 0 i=1 Λs,N,3 θs,N = cos θs,N . 0 Here the label s have been attached to denote a sphere and the obvious index changes in the spherical variables θs,N 0 and φs,N are understood from the set of equations (A.158), (A.159), (A.160) and (A.161). Similarly, the resultant change in wave vector direction upon reflection at the outer surface of the sphere is from equation (C.5) of Appendix C1, ½ ° ° ³ ´ 0 ≤ θinc < π/2, ° 0 0 ~ ~ 0 s,1 + aRˆ0 s,1 = 4 ° ˆ 4k~0 outer ; R k cos θ R , (3.18) ° i,f ° inc s,1 n = 1, 2, · · · .

24

3. Reflection Dynamics The details of this section can be found in Appendix C1.

3.2.2. Hemisphere-Hemisphere For the hemisphere, the changes in wave vector directions after the reflection at a point Rˆ0 h,1 inside the resonator, or ~ 0 h,1 + aRˆ0 h,1 outside the hemisphere, can be found from equations (3.16) and (3.18) after the reflection at location R with obvious subscript changes, ½ ³ ´ 4nπ cos θinc 0 ≤ θinc < π/2, 0 0 0 0 ~ ~ ~ ˆ ° ´ ´° ³ ³ 4k inner ; R h,1 , R h,0 = − ° R h,1 , (3.19) ° n = 1, 2, · · · ; 0 0 0 0 ~ ~ ~ ~ h,2 r , Λ °R i h,2 − Rh,1 ri , Λh,1 ° ° ° ³ ´ ° ~ 0 h,1 + aRˆ0 h,1 = 4 ° 4k~0 outer ; R °k~0 i,f ° cos θinc Rˆ0 h,1 ,

½

0 ≤ θinc < π/2, n = 1, 2, · · · ,

(3.20)

³ ´ ~ h,N r0 , Λ ~` h,N , Λ ~0 ,R ~ T,h follows the generic form as shown in equation (C.6) of where the reflection location R i h,N Appendix C2, ³ ´ ~ h,N ri0 , Λ ~` h,N , Λ ~ 0h,N , R ~ T,h = R

( 3 )1/2 3 X£ X ¤2 0 0 ` h,N,i eˆi . νT,h,i + ri Λh,N,i Λ i=1

(3.21)

i=1

In the above equation, the subscript h denotes the hemisphere; and  ³ ´ ` h,N,1 θ`h,N , φ`h,N = sin θ`h,N cos φ`h,N ,  Λ    ³ ´ ` h,N,2 θ`h,N , φ`h,N = sin θ`h,N sin φ`h,N , Λ  ³ ´   ` h,N,3 θ`h,N = cos θ`h,N .  Λ The expressions for Λ0h,N,i , i = 1, 2, 3, are defined identically in form. The angular variables in spherical coordinates, θ`h,N and φ`h,N , can be obtained from equations (3.13) and (3.14), where the obvious notational changes are 0 understood. The implicit angular variables, θh,N and φ0h,N , are the sets defined in Appendix A, equations (A.158) and 0 (A.159) for θs,N , and the sets from equations (A.160) and (A.161) for φ0s,N . Unlike the sphere situation, the initial wave vector could eventually escape the hemisphere resonator after some maximum number of reflections. It is shown in the Appendix C2 that this maximum number for internal reflection is given by equation (C.8), Nh,max = [Zh,max ]G , where the greatest integer function [Zh,max ]G is defined in equation (C.7) of Appendix C2, · µ ½ ° ° ¾¶¸ ° ° h ° °i−1 −1 1 1 0 ° ~0 ° 0 −1 ° ~ 0 ° 0 ° ~0 ° 2 π − arccos r °R 0 ° + [ri ] °R 0 ° − ri °R 0 ° ξ1,p . Zh,max = π − 2θinc 2 i

(3.22)

(3.23)

Here ξ1,p is given in equation (3.5) and θinc is from equation (A.115). ³ ´ ³ ´ ~ 0 h,1 , R ~ 0 h,0 and 4k~0 outer ; R ~ 0 h,1 + aRˆ0 h,1 have been derived based on the The above results of 4k~0 inner ; R fact that there are multiple internal reflections. For a sphere, the multiple internal reflections are inherent. However, for a hemisphere, it is not necessarily true that all incoming waves would result in multiple internal reflections. Naturally, the criteria for multiple internal reflections are in order. If the initial direction of the incoming wave vector, kˆ0 1 , is given, the internal reflections can be either single or multiple depending upon the location of the entry point in the ~ 0 0 . As shown in Figure 3.5, these are two reflection dynamics where the dashed vectors represent the single cavity, R reflection case and the non-dashed vectors represent multiple reflections case. Because the whole process occurs in the ~ 0 f = −λ0 R ~ 0 0 where λ0 > 0. The multiple or single internal reflection criteria same plane of incidence, the vector R

25

3. Reflection Dynamics

^z

Intercept

ξp ^k 1

R0

R1

n^ p,1 ^y

k2 Rf

Figure 3.5.: The dashed line vectors represent the situation where only single internal reflection occurs. The dark line vectors represent the situation where multiple internal reflections occur.

can be summarized by the relation found in equation (C.21) of Appendix C2: −1 # 3 3 " # " 3 3 n 3 °  X ° 1° ° X X X ° °2 X £ 0 ¤2 2 ° ~0 ° ° ~0 ° ° ~0 ° 0 0 0 0 0 0 [ri0 ] − r0,l r0,j k1,l k1,n k1,m r0,l °R f ° = °R °k 1 ° − k1,l 0°   2 n=1 m=1 j=1 l=1 l=1  #−1 3 " # " 3 3  ° °2 ° °2 X X X ° ~0 ° ° ~0 ° 0 0 0 0 0 0 0 ~ 0 0 • k~0 1 r0,l . (3.24) r0,i − °R k1,l k1,m +2R °k 1 ° − k1,i 0 ° k1,l − 2r0,l  m=1

i=1

l=1

Finally, because the hemisphere opening has a radius ri0 , the following criteria are concluded:  ° ° ° ~0 °  R Single − Internal − Ref lection,  ° ° < ri0 , f  (3.25)

° °   ~ 0f °  ° °R ° ≥ ri0 ,

M ultiple − Internal − Ref lections,

° ° ° ~0 ° where °R f ° is defined in equation (3.24). The details of this section can be found in Appendix C2.

3.2.3. Plate-Hemisphere ˆ0 A surface is represented by a unit ³ vector n p , ´which is normal to the surface locally. For the circular plate shown in Figure 3.6, its orthonormal triad nˆ0 p , θˆ0 p , φˆ0 has the form p

nˆ0 p =

P3 i=1

Λ0p,i eˆi ,

P3 θˆ0 p = i=1

∂Λ0p,i ˆi , ∂θp0 e

26

P3 φˆ0 p = i=1

0 1 ∂Λp,i ˆi , sin θp0 ∂φ0p e

3. Reflection Dynamics

θp

^z ^ n ^θ φ^ ^x p

p

^ y

p

φp Figure 3.6.: The orientation of a disk is given through the surface unit normal nˆ0 p . The disk is spanned by the two unit vectors θˆ0 p and φˆ0 p . ¡ ¢ ¡ ¢ ¡ ¢ where Λ0p,1 θp0 , φ0p = sin θp0 cos φ0p , Λ0p,2 θp0 , φ0p = sin θp0 sin φ0p and Λ0p,3 θp0 = cos θp0 . ~ p on the plane and For the plate-hemisphere configuration shown in Figure 3.7, it can be shown that the element R ~ its velocity dRp /dt are given by (see equation (3.27) and (C.30) in Appendix C3: ~p R

³

 " #2 1/2 3 0 3  X 0 0 ´ X νp,φ 0 ∂Λ ∂Λ ` p,i p,i p 0 ~ T,p = ~ p, Λ ~ 0p , R ` p,i eˆi , νT,p,i + νp,θ Λ + Λ 0 0 ∂φ0 p ∂θ 0   sin θ p p p i=1 i=1

 #2 −1/2 3 3 µ· " 0 3 X  0 0 νp,φ XX ~p 0 ∂Λ ∂Λ0p,k ∂Λ dR ˙~ p,i p,i p 0 0 Rp ≡ + = νT,p,i + νp,θ ν + ν 0 0 T,p,k p,θ p ∂θ 0 p ∂θ 0   dt sin θp0 ∂φ0p p p i=1 j=1 k=1 # " Ã !) ( 0 0 0 νp,φ νp,φ 0 ∂Λ 0 ∂Λ0p,k ∂ 2 Λ0p,k ∂ 2 Λ0p,k p,k p p 0 0 + − cot θ ν ˙ + ν + θ˙0 p 0 £ ¤ T,p,k p p,θp 2 sin θp0 ∂φ0p sin θp0 ∂θp0 ∂φ0p ∂φ0p ∂ θp0 ) # ( 0 2 0 0 2 0 ˙0 p,φ0 ∂Λ0 νp,φ 0 ∂ Λ ν ∂Λ ∂ Λ p,k p,k p,k p,k p p 0 ` p,j + φ˙0 p + ν˙0 p,θp0 + Λ + νp,θ 0 £ ¤ p ∂φ0 ∂θ 0 sin θp0 ∂ φ0 2 ∂θp0 sin θp0 ∂φ0p p p p " #2 " # 0 3 0 0 ν X ` ` ` ` 0 ∂Λ ∂Λ p,φ ∂ Λ ∂ θ ∂ Λ ∂ φ p,i p,i p,j p ˙0 p,j p ˙0  p 0 eˆj , + νT,p,i + νp,θ + θ + φ 0 0 ∂φ0 p ∂θ 0 `p ∂φ0p p `p ∂φ0p p sin θ ∂ θ ∂ φ p p p i=1

(3.26)

(3.27)

³ ´ ` p,1 , Λ ` p,2 , Λ ` p,3 is defined in equation (C.31) and the angles φ`p and θ`p are defined in equations (C.27) and where Λ (C.28) of Appendix C3. The subscript p of φ`p and θ`p indicates that these are spherical variables for the points on the ` p,3 are independent of φ0p and plate of Figure 3.7, not that of the hemisphere. It is also understood that Λ0p,3 and Λ φ`p , respectively. Therefore, their differentiation with respect to φ0p and φ`p respectively vanishes. The quantities θ˙0 p and φ˙0 p are the angular frequencies, and ν˙ T,p,i is the translation speed of the plate relative to the system origin. The quantities ν˙0 p,θp0 and ν˙0 p,φ0p are the lattice vibrations along the directions θˆ0 p and φˆ0 p respectively. For the static plate without lattice vibrations, ν˙0 p,θ0 and ν˙0 p,φ0 vanishes. p

p

In the cross-sectional view of the plate-hemisphere system shown in Figure 3.8, the initial wave vector k~0 i traveling toward the hemisphere would go through a complex series of reflections according to the law of reflection and finally exit the cavity. It would then continue toward the plate, and depending on the orientation of plate at the time of impact, the wave-vector, now reflecting off the plate, would either escape to infinity or re-enter the hemisphere. The process repeats successively. In order to determine whether the wave that just escaped from the hemisphere cavity can reflect

27

3. Reflection Dynamics

R R R θ R ^x ^n

Plate

θ

Point on plate

^z

p

p

Plate Origin

R

θ

T,p

p

n

R ^ y

RT φ

p

φ

φp

Figure 3.7.: The plate-hemisphere configuration.

back from the plate and re-enter the hemisphere or escape to infinity, the exact location of reflection on the plate must be known. This reflection point on the plate is found to be, from equation (C.54) of Appendix C3,  · ¸ 2 1/2  ° ° P3 ∂Λ0p,i   ° ~0 °−1   0 0 0   Λp,i + °n p,1 ° ²ijk k1,j r0,k 3 X i=1 ∂φ0p  ∂Λ0p,s ∂Λ0p,s   ~   ¸ Rp = ° °−1  ∂φ0 − P3 ∂Λ0 ·  ∂θp0   ° p p,l   0 + °n 0 0 0 ~ s=1   Λ r ² k ° ° p,1 lmn 1,m 0,n   l=1 ∂θp0 p,l 3 h iX −1 −1 −1 −1 −1 ` p,i eˆi , × Cβ Cγ Aγ Aβ + γo Cβ Cγ Bγ Aβ + Cβ Bζ Bβ Λ

(3.28)

i=1

where the translation parameter νT,p,j = 0 and the terms (Aζ , Bζ , Cζ ) , (Aγ , Bγ , Cγ ) , (Aβ , Bβ , Cβ ) and γo are defined in equations (C.46), (C.49), (C.50) and (C.52) of Appendix C3. It is to be noticed that for a situation where ` becomes identical to Λ0 in form. Results for Λ ` can be obtained from Λ0 by a the translation parameter νT,p,j = 0, Λ simple replacement of primed variables with the unprimed ones. ~p Leaving the details to the relevant Appendix, the criterion whether the wave reflecting off the plate at location R can re-enter the hemisphere cavity or simply escape to infinity is found from the result shown in equation (C.58) of Appendix C3,   · ¸ 2 1/2  ° °−1 P3 ∂Λ0p,i   ° °   0 0 0 0 ~  X  Λp,i + °n p,1 ° ²ijk k1,j r0,k 3  0   0 i=1 ∂φ0p   ∂Λ ∂Λ p,s p,s 0   · ¸ ξκ,i =  ν + r − − ° °−1 0,i  T,h,i  ∂φ0 P3 ∂Λ0p,l  ∂θp0   ° ° p    0 0   Λ0p,l + °n~0 p,1 ° ²lmn k1,m r0,n s=1  l=1 ∂θ 0 p

Ã 3 h i ´ X © £ ` p,i × Cβ−1 Cγ−1 Aγ Aβ + γo Cβ−1 Cγ−1 Bγ Aβ + Cβ−1 Bζ Bβ Λ αr,⊥ kNh,max +1,i n0p,k n0p,k −n0p,i kNh,max +1,k n0p,k

¤

−

αr,k n0p,k kNh,max +1,k n0p,i

28

ª¢−1

k=1

,

(3.29)

3. Reflection Dynamics

Plane of Incidence

Hemisphere Cross Section

^z Initial Wave Vector

Plate Cross Section

ξi ^ki ^n p

R0

Ri ψi,i+1

ξ i+1^k i+1 θ i+1

ψi+1,i+2 θ i+2

New Initial Wave Vector

R h,i+3

Ri+3 θr+2

^n p

n^ p,1

θ Ri+1 r+1

R h,i+2

Rp

k^ i+3

θi θr

Hemisphere Origin System Origin

Plate Origin

^x

^y

ξ i+2 ^k i+2

^ ξi+3 k i+3 R i+2

Figure 3.8.: The intersection between oscillating plate, hemisphere and the plane of incidence whose normal is nˆ0 p,1 = ° °−1 P ° ° 3 − °n~0 p,1 ° ²ijk k 0 r0 eˆi . i=1

1,j 0,k

P3 where i = 1, 2, 3 and ξκ,i is the component of the scale vector ξ~κ = i=1 ξκ,i eˆi explained in the Appendix C3. ~ 0 ≤ r0 . This implies r0 ≤ r0 , where r0 is the radius In the above re-entry criteria, it should be noticed that R i 0,i i i of hemisphere. It is then concluded that all waves re-entering the hemisphere cavity would satisfy the condition ξκ,1 = ξκ,2 = ξκ,3 . On the other hand, those waves that escapes to infinity cannot have all three ξκ,i equal to a single constant. The re-entry condition ξκ,1 = ξκ,2 = ξκ,3 is just another way of stating the existence of a parametric line along the vector ~kr,Nh,max +1 that happens to pierce through the hemisphere opening. In case such a line does not exist, the initial wave direction has to be rotated into a new direction such that there is a parametric line that pierces through the hemisphere opening. That is why all three ξκ,i values cannot be equal to a single constant. The re-entry criteria are summarized here for bookkeeping purpose: ½ ξκ,1 = ξκ,2 = ξκ,3 → W ave − ReEnters − Hemisphere, (3.30) ELSE → W ave − Escapes − to − Inf inity, where ELSE is the case where ξκ,1 = ξκ,2 = ξκ,3 cannot be satisfied. The details of this section can be found in Appendix C3.

3.3. Dynamical Casimir Force The phenomenon of Casimir effect is inherently a dynamical effect due to the fact that it involves radiation, rather than static fields. One of my original objectives in studying the Casimir effect was to investigate the physical implications of vacuum-fields on movable boundaries. Consider the two parallel plates configuration of charge-neutral, perfect conductors shown in Figure 3.9. Because there are more wave modes in the outer region of the parallel plate res-

29

3. Reflection Dynamics

z=d

z=0

Figure 3.9.: Because there are more vacuum-field modes in the external regions, the two charge-neutral conducting plates are accelerated inward till the two finally stick. onator, two loosely restrained (or unfixed in position) plates will accelerate inward until they finally meet. The energy conservation would require that the energy initially confined in the resonator when the two plates were separated be transformed into the heat energy that acts to raise the temperatures of the two plates. Davies in 1975 [26], followed by Unruh in 1976 [27], have asked the similar question and came to a conclusion that when an observer is moving with a constant acceleration in vacuum, the observer perceives himself to be immersed ¨ [2πck 0 ] , where R ¨ is the acceleration of the observer and k 0 , the wave in a thermal bath at the temperature T = ~R/ number. The details of the Unruh-Davies effect can also be found in the reference [17]. The other work that dealt with the concept of dynamical Casimir effect is due to Schwinger in his proposals [14, 16] to explain the phenomenon of sonoluminescense. Sonoluminescense is a phenomenon in which when a small air bubble filled with noble gas is under a strong acoustic-field pressure, the bubble will emit an intense flash of light in the optical range. Although the name “dynamical Casimir effect” have been introduced by Schwinger, the motivation and derivation behind the dynamical Casimir force in this thesis did not stem from that of Schwinger’s work. Therefore, the dynamical Casimir force here should not have any resemblance to Schwinger’s work to the best of my knowledge. I have only found out of Schwinger’s proposals on sonoluminescense after my work on dynamical Casimir force have already begun. The terminology “dynamical Casimir force” seemed to be appealing enough, I have personally used it at the beginning of my work. After discovering Schwinger’s work on sonoluminescense, I have learned that Schwinger had already introduced the terminology “dynamical Casimir effect” in his papers. My original development to the dynamical Casimir force formalism is briefly presented in the following sections. The details of the derivations pertaining to the dynamical Casimir force can be found in Appendix D.

3.3.1. Formalism of Zero-Point Energy and its Force For massless fields, the energy-momentum relation is Hn0 s ≡ ET otal = pc, where p is the momentum, c the speed of light, and Hn0 s is the quantized field energy for the harmonic fields of equation (2.8) for the bounded space, or equation (2.9) for the free space. For the bounded space, the quantized field energy Hn0 s ≡ Hn0 s ,b of equation (2.8) is a function of the wave number ki0 (ni ) , which in turn is a function of hthe wave mode value i ni and the boundary ~i = R ~ 0 2 • eˆi − R ~ 0 1 • eˆi eˆi . Here R ~ 0 1 and R ~ 02 functional fi (Li ) , where Li is the gap distance in the direction of L are the position vectors for the involved boundaries. As an illustration with the two plate configuration shown in ~ 0 1 may represent the plate positioned at z = 0 and R ~ 0 2 may correspond to the plate at the position z = d. Figure 3.9, R When the position of these boundaries are changing in time, the quantized field energy Hn0 s ≡ Hn0 s ,b will be modified accordingly because the wave number functional ki0 (ni ) is varying in time, dki0 ∂ki0 dni ∂fi dLi ∂k 0 ∂fi ˙ = fi (Li ) + ni = fi (Li ) i n˙ i + ni Li . dt ∂ni dt ∂Li dt ∂ni ∂Li Here the term proportional to n˙ i refers to the case where the boundaries remain fixed throughout all times but the number of wave modes in the resonator are being driven by some active external influence. The term proportional to L˙ i represents the changes in the number of wave modes due to the moving boundaries. For an isolated system, there are no external influences, hence n˙ i = 0. Then, the dynamical force arising from the

30

3. Reflection Dynamics

Lx,1

Rrp,1

Lx,2

1

R dpl

Right Plate

Left Plate

Left Driven Plate

2 R rp,2

3 Rlp,3

Rlp,2

^y ^x

Right Driven Plate

R dpr

^Z axis is out of the page!

Figure 3.10.: A one dimensional driven parallel plates configuration.

fact that the time variation of the boundaries is given by equation (D.17) of Appendix D1, ( " µ · ¸ ¶· ¸# 3 X ∂ 2 Hn0 s 1 1 0 ∂fi 0 ~ Fα= Cα,5 + (1 − δiα ) Cα,6 − Cα,7 ns + ki ns + L˙ i ni 0 ]2 ∂L 2 2 i ∂ [k i i=1  3  2 0 X ∂ H ∂fj ns ˙ + L eˆα , (1 − δij ) Cα,5 nj j ∂Lj ∂kj0 ∂ki0 

(3.31)

j=1

where Cα,1 , Cα,2 , Cα,3 , Cα,4 , Cα,5 , Cα,6 and Cα,7 are defined in equations (D.6), (D.9), (D.14), (D.15) and (D.16) of Appendix D1. The force shown in the above expression vanishes for the one dimensional case. This is an expected result. To understand why the force vanishes, we have to refer to the starting point equation £ (D.4) ¤in the Appendix D1. The summation there obviously runs only once to arrive at the expression, ∂Hn0 s /∂ki0 = ns + 12 ~c. This is a classic situation where P3 2 the problem has been over specified. For the 3D case, equation (D.4) is a combination of two constraints, i=1 [p0i ] 0 0 and Hns . For the one dimensional case, there is only one constraint, Hns . Therefore, equation (D.4) becomes an over specification. In order to avoid the problem caused by over specifications in this formulation, the one dimensional force expression can be obtained directly by differentiating equation (D.1) instead of using the above formulation for the three dimensional case. The 1D dynamical force expression for an isolated, non-driven systems then becomes (see equation (D.18) of Appendix D1) n ∂f ∂Hn0 s ˙ F~ 0 = Lˆ e, c ∂L ∂k 0

(3.32)

where F~ 0 is an one dimensional force. Here the subscript α of F~ 0 α have been dropped for simplicity, since it is a one dimensional force. The details of this section can be found in Appendix D1.

3.3.2. Equations of Motion for the Driven Parallel Plates The Unruh-Davies effect states that heating up of an accelerating conductor plate is proportional to its acceleration ¨ [2πck 0 ] , where R ¨ is the plate acceleration. A one dimensional system of two paralthrough the relation T = ~R/ lel plates, shown in Figure 3.10, can be used as a simple model to demonstrate the complicated sonoluminescense phenomenon for a bubble subject to a strong acoustic field. The dynamical force for the 1D, linear coupled system can be expressed with equation (3.32), ¨ 1 − η1 R˙ 1 − η2 R˙ 2 = ξrp , R

¨ 2 − η3 R˙ 2 − η4 R˙ 1 = ξlp , R

(3.33)

where the quantities η1 , η2 , η3 , η4 , ξrp , ξlp , R1 , R2 are defined in equation (D.31) of Appendix D2. Here R1 represents the center of mass position for the “Right Plate” and R2 represents the center of mass position for the “Left Plate” as illustrated in Figure 3.10. With a slight modification, equation (3.33) for this linear coupled system can be written in

31

3. Reflection Dynamics the matrix form, (see equations (D.33), (D.34) and (D.35) of Appendix D2): Z t Z t R1 = R3 dt0 , R2 = R4 dt0 , t0

t0

and ¸ · ¸ · ¸ ¸ · R3 R˙ 3 η1 η2 ξrp · , = + η4 η3 R4 ξlp R˙ 4 {z } | {z } | {z } | {z } | ·

~˙ η R

where

~η R

fη M

(3.34)

ξ~

 

R˙ 1 = R3 , R˙ 2 = R4 , ¨ 1 = ξrp + η1 R˙ 1 + η2 R˙ 2 = ξrp + η1 R3 + η2 R4 , R˙ 3 = R  ˙ ¨ 2 = ξlp + η3 R˙ 2 + η4 R˙ 1 = ξlp + η3 R4 + η4 R3 . R4 = R

The matrix equation has the solutions given by equations (D.51) and (D.52) of Appendix D2: ¸−1 · ψ11 (t, t0 ) R˙ rp,cm,α (t0 ) + ψ12 (t, t0 ) R˙ lp,cm,α (t0 ) λ4 (; t0 ) − η1 (; t0 ) −1 R˙ rp,cm,α (t) = λ3 (; t0 ) − η1 (; t0 ) exp ([λ3 (; t0 ) + λ4 (; t0 )] t0 ) Z t 0 0 ψ22 (t , t0 ) ξrp (t ) − ψ12 (t0 , t0 ) ξlp (t0 ) + ψ11 (t, t0 ) dt0 + ψ12 (t, t0 ) 0 0 0 0 t0 ψ11 (t , t0 ) ψ22 (t , t0 ) − ψ12 (t , t0 ) ψ21 (t , t0 ) Z t ψ11 (t0 , t0 ) ξlp (t0 ) − ψ21 (t0 , t0 ) ξrp (t0 ) × dt0 , 0 0 0 0 t0 ψ11 (t , t0 ) ψ22 (t , t0 ) − ψ12 (t , t0 ) ψ21 (t , t0 ) · ¸−1 λ4 (; t0 ) − η1 (; t0 ) ψ21 (t, t0 ) R˙ rp,cm,α + ψ22 (t, t0 ) R˙ lp,cm,α (t0 ) ˙ Rlp,cm,α (t) = −1 λ3 (; t0 ) − η1 (; t0 ) exp ([λ3 (; t0 ) + λ4 (; t0 )] t0 ) Z t ψ22 (t0 , t0 ) ξrp (t0 ) − ψ12 (t0 , t0 ) ξlp (t0 ) + ψ21 (t, t0 ) dt0 + ψ22 (t, t0 ) 0 0 0 0 t0 ψ11 (t , t0 ) ψ22 (t , t0 ) − ψ12 (t , t0 ) ψ21 (t , t0 ) Z t ψ11 (t0 , t0 ) ξlp (t0 ) − ψ21 (t0 , t0 ) ξrp (t0 ) × dt0 , 0 , t ) ψ (t0 , t ) − ψ (t0 , t ) ψ (t0 , t ) ψ (t 11 0 22 0 12 0 21 0 t0

(3.35)

(3.36)

where the terms λ3 and λ4 are defined in equation (D.37); and ψ11 (t, t0 ) , ψ12 (t, t0 ) , ψ21 (t, t0 ) and ψ22 (t, t0 ) are defined in equations (D.43) through (D.46) in Appendix D2. The quantities R˙ rp,cm,α and R˙ lp,cm,α are the speed of the center of mass of “Right Plate” and the speed of the center of mass of the “Left Plate,” respectively, and α defines the particular basis direction.

The corresponding positions Rrp,cm,α (t) and Rlp,cm,α (t) are found by integrating equations (3.35) and (3.36) with respect to time, ¸−1 Z t " · ψ11 (τ, t0 ) R˙ rp,cm,α (t0 ) + ψ12 (τ, t0 ) R˙ lp,cm,α (t0 ) λ4 (; t0 ) − η1 (; t0 ) −1 Rrp,cm,α (t) = λ3 (; t0 ) − η1 (; t0 ) exp ([λ3 (; t0 ) + λ4 (; t0 )] t0 ) t0 Z τ 0 0 ψ22 (t , t0 ) ξrp (t ) − ψ12 (t0 , t0 ) ξlp (t0 ) + ψ11 (τ, t0 ) dt0 + ψ12 (τ, t0 ) 0 0 0 0 t0 ψ11 (t , t0 ) ψ22 (t , t0 ) − ψ12 (t , t0 ) ψ21 (t , t0 ) ¸ Z τ ψ11 (t0 , t0 ) ξlp (t0 ) − ψ21 (t0 , t0 ) ξrp (t0 ) 0 × dt dτ + Rrp,cm,α (t0 ) , (3.37) 0 0 0 0 t0 ψ11 (t , t0 ) ψ22 (t , t0 ) − ψ12 (t , t0 ) ψ21 (t , t0 )

32

3. Reflection Dynamics ¸−1 Z t " λ4 (; t0 ) − η1 (; t0 ) ψ21 (τ, t0 ) R˙ rp,cm,α (t0 ) + ψ22 (τ, t0 ) R˙ lp,cm,α (t0 ) Rlp,cm,α (t) = −1 λ3 (; t0 ) − η1 (; t0 ) exp ([λ3 (; t0 ) + λ4 (; t0 )] t0 ) t0 Z τ 0 0 ψ22 (t , t0 ) ξrp (t ) − ψ12 (t0 , t0 ) ξlp (t0 ) + ψ21 (τ, t0 ) dt0 + ψ22 (τ, t0 ) 0 , t ) ψ (t0 , t ) − ψ (t0 , t ) ψ (t0 , t ) ψ (t 11 0 22 0 12 0 21 0 t0 ¸ Z τ ψ11 (t0 , t0 ) ξlp (t0 ) − ψ21 (t0 , t0 ) ξrp (t0 ) × dt0 dτ + Rlp,cm,α (t0 ) . (3.38) 0 0 0 0 t0 ψ11 (t , t0 ) ψ22 (t , t0 ) − ψ12 (t , t0 ) ψ21 (t , t0 ) ·

The remaining integrations are straightforward and the explicit forms will not be shown here. One may argue that for the static case, R˙ rp,cm,α (t0 ) and R˙ lp,cm,α (t0 ) must be zero because the conductors seem to be fixed in position. This argument is flawed, for any wall totally fixed in position upon impact would require an infinite amount of energy. One has to consider the conservation of momentum simultaneously. The wall has to have moved by the amount 4Rwall = R˙ wall 4t, where 4t is the total duration of impact, and R˙ wall is calculated from the momentum conservation and it is non-zero. The same argument can be applied to the apparatus shown in Figure 3.10. For that system  ° ° ~˙ lp,3 (t0 ) + R ~˙ rp,2 (t0 )°  R˙ rp,cm,α (t0 ) = ° °R °, 1 0 ° ° k~ pvirtual−photon k = Hns ,< (t0 ) , ° ° ˙ ˙  R˙ lp,cm,α (t0 ) = °R c ~ rp,1 (t0 ) + R ~ lp,2 (t0 )° . For simplicity, assuming that the impact is always only in the normal direction, R˙ rp,cm,α (t0 ) =

° 2 ° °Hn0 ,3 (t0 ) − Hn0 ,2 (t0 )° , s s mrp c

R˙ lp,cm,α (t0 ) =

° 2 ° °Hn0 ,1 (t0 ) − Hn0 ,2 (t0 )° , s s mlp c

where the differences under the magnitude symbol imply field energies from different regions counteract the other. The details of this section can be found in Appendix D2.

33

4. Results and Outlook The results for the sign of Casimir force on non-planar geometric configurations considered in this thesis will eventually be compared with the classic repulsive result obtained by Boyer decades earlier. For this reason, it is worth reviewing Boyer’s original configuration as shown in Figure 4.1. Edge of Universe

Sphere

Infinity Sphere vacuum−field Poynting vector field lines

Figure 4.1.: Boyer’s configuration is such that a sphere is the only matter in the entire universe. His universe extends to the infinity, hence there are no boundaries. The sense of vacuum-field energy flow is along the radial vector rˆ, which is defined with respect to the sphere center. T. H. Boyer in 1968 obtained a repulsive Casimir force result for his charge-neutral, hollow spherical shell of a perfect conductor [4]. For simplicity, his sphere is the only object in the entire universe and, therefore, no external boundaries such as laboratory walls, etc., were defined in his problem. Furthermore, the zero-point energy flow is always perpendicular to his sphere. Such restriction can be a very stringent condition for the material property that a sphere has to meet. For example, if one were to look at Boyer’s sphere, he would not see the whole sphere; but instead, he would see a small spot on the surface of a sphere that happens to be in his line of sight. This happens because the sphere in Boyer’s configuration can only radiate in a direction normal to the surface. One could equivalently argue that Boyer’s sphere only responds to the approaching radiation at normal angles of incidence with respect to the surface of the sphere. When the Casimir force is computed for such restricted radiation energy flow, the result is repulsive. This can be attributed to the fact that closer to the sphere origin, the spherically symmetric radiation energy flow becomes more dense and this density decreases as it gets further away from the sphere center. As an illustration, Boyer’s sphere is shown in Figure 4.1. For the rest of the thesis, “Boyer’s sphere” would be strictly referred to as the sphere made of such material property that it only radiates or responds to vacuum-field radiations at normal angle of incidence with respect to its surface. The formation of a sphere by bringing together two nearby hemispheres satisfying the material property of Boyer’s sphere is illustrated in Figure 4.2. Since Boyer’s material property only allow radiation in the normal direction to its surface, the radiation associated with each hemisphere would necessarily go through the corresponding hemisphere centers. For clarity, let us define the unit radial basis vector associated with the left and right hemispheres by rˆL and rˆR , respectively. If the hemispheres are made of normal conductors the radiation from one hemisphere entering the other hemisphere cavity would go through a complex series of reflections before escaping the cavity. Here, a conductor with Boyer’s stringent material property is not considered normal. Conductors that are normal also radiate in directions non-normal to their surface, whereas Boyer’s conductor can only radiate normal to its surface. Due to the fact that Boyer’s conducting materials can only respond to radiation impinging at a normal angle of incidence with respect to its surface, all of the incoming radiation at oblique angles of incidence with respect to the local surface normal is absorbed by the host hemisphere. This suggests that for the hemisphere-hemisphere arrangement made of

34

4. Results and Outlook Edge of Universe Edge of Universe

Non−radial Poynting vector fields due to configurational change

Infinity

Sphere Poynting vector field line from right hemisphere

Infinity

Sphere vacuum−field Poynting vector field lines

Poynting vector field line from left hemisphere

Figure 4.2.: Manufactured sphere, in which two hemispheres are brought together, results in small non-spherically symmetric vacuum-field radiation inside the cavity due to the configuration change. For the hemispheres made of Boyer’s material, these fields in the resonator will eventually get absorbed by the conductor resulting in heating of the hemispheres.

η

δ

ξ

A virtual photon along one of the Poynting vector field lines from left hemisphere

Figure 4.3.: The process in which a configuration change from hemisphere-hemisphere to sphere inducing virtual photon in the direction other than rˆ is shown. The virtual photon here is referred to as the quanta of energy associated with the zero-point radiation.

Boyer’s material shown in Figure 4.2, the temperature of the two hemispheres would rise indefinitely over time. This does not happen with ordinary conductors. This suggests that Boyer’s conducting material, of which his sphere is made, is completely hypothetical. Precisely because of this material assumption, Boyer’s Casimir force is repulsive. For the moment, let us relax the stringent Boyer’s material property for the hemispheres to that of ordinary conductors. For the hemispheres made of ordinary conducting materials, there would result a series of reflections in one hemisphere cavity due to those radiations entering the cavity from nearby hemisphere. For simplicity, the ordinary conducting material referred to here is that of perfect conductors without Boyer’s hypothetical material property requirement. Furthermore, only the radiation emanating normally with respect to its surface is considered. The idea is to illustrate that the “normally emanated radiation” from one hemisphere results in elaboration of the effects of “obliquely emanated radiation” on another hemisphere cavity. Here the obliquely emanated radiation means those radiation emanating from a surface not along the local normal of the surface. When two such hemispheres are brought together to form a sphere, there would exist some radiation trapped in the sphere of which the radiation energy flow lines are not spherically symmetric with respect to the sphere center. To see how a mere change in configuration invokes such non-spherically symmetric energy flow, consider the illustration shown in Figure 4.3. For clarity, only one “normally emanated radiation” energy flow line from the left hemisphere is shown. When one brings together the two hemispheres just in time before that quantum of energy escapes the hemisphere cavity to the right, the trapped energy quantum would continuously go through series of complex reflections in the cavity obeying the reflection law. But how fast or how slow one brings in two hemispheres is irrelevant in

35

4. Results and Outlook Poynting vector field lines originating from lab boundaries

Lab boundary

Figure 4.4.: A realistic laboratory has boundaries, e.g., walls. These boundaries have effect similar to the field modes between two parallel plates. In 3D, the effects are similar to that of a cubical laboratory, etc.

Poynting vector field lines from lab boundary

Trapped Poynting vector field lines originally from laboratory boundary and due to the configurational changes going from hemisphere−hemisphere to a sphere.

Figure 4.5.: The schematic of sphere manufacturing process in a realistic laboratory. invoking such non-spherically symmetric energy flow because the gap δ can be chosen arbitrarily. Therefore, there would always be a stream of energy quanta crossing the hemisphere opening with ξ 6= 0 as shown in Figure 4.3. In other words, there is always a time interval 4t within which the hemispheres are separated by an amount δ before closure. The quanta of vacuum-field radiation energy created within that time interval 4t would always be satisfying the condition ξ 6= 0, and this results in reflections at oblique angle of incidence with respect to the local normal of the walls of inner sphere cavity. Only when the two hemispheres are finally closed, would then ξ = 0 and the radiation energy produced in the sphere after that point would be spherically symmetric and the reflections would be normal to the surface. However, those trapped quantum of energy that were produced prior to the closure of the two hemispheres would always be reflecting from the inner sphere surface at oblique angles of incidence. Unlike Boyer’s ideal laboratory, realistic laboratories have boundaries made of ordinary material as illustrated in Figure 4.4. One must then take into account, when calculating the Casimir force, the vacuum-field radiation pressure contributions from the involved conductors, as well as those contributions from the boundaries such as laboratory walls, etc. We will examine the physics of placing two hemispheres inside the laboratory. For simplicity, the boundaries of the laboratory as shown in Figure 4.5 are assumed to be simple cubical. Normally, the dimension of conductors considered in Casimir force experiment is in the ranges of microns. When this is compared with the size of the laboratory boundaries such as the walls, the walls of the laboratory can be treated as a set of infinite parallel plates and the vacuum-fields inside the the laboratory can be treated as simple plane waves with impunity. The presence of laboratory boundaries induce reflection of energy flow similar to that between the two parallel plate arrangement. When the two hemisphere arrangement shown in Figure 4.2 is placed in such a laboratory, the result is to elaborate the radiation pressure contributions from obliquely incident radiations on external surfaces of the two hemispheres. If the two hemispheres are made of conducting material satisfying Boyer’s material property, the vacuum-field radiation impinging on hemisphere surfaces at oblique angles of incidence would cause heating of the hemispheres. It means that Boyer’s hemispheres placed in a realistic laboratory would continue to rise in temperature

36

4. Results and Outlook

k outer k i,b

k i,f k inner

^ R s,1

dA

L

~0 ~0 Figure 4.6.: The pnet k = ° vacuum-field ° wave vectors k i,b and k i,f impart a net momentum of the magnitude k~ ° ~0 ° 0 ~ ~ °k i,b − k i,f ° /2 on differential patch of an area dA on a conducting spherical surface.

as a function of time. However, this does not happen with ordinary conductors. If the two hemispheres are made of ordinary perfect conducting materials, the reflections of radiation at oblique angles of incidence from the laboratory boundaries would elaborate on the radiation pressure acting on the external surfaces of two hemispheres at oblique angles of incidence. Because Boyer’s sphere only radiates in the normal direction to its surface, or only responds to impinging radiation at normal incidence with respect to the sphere surface, the extra vacuum-field radiation pressures considered here, i.e., the ones involving oblique angles of incidence, are missing in his Casimir force calculation for the sphere.

4.1. Results T. H. Boyer in 1968 have shown that for a charge-neutral, perfect conductor of hollow spherical shell, the sign of the Casimir force is positive, which means the force is repulsive. He reached this conclusion by assuming that all vacuum-field radiation energy flows for his sphere are spherically symmetric with respect to its center. In other words, only the wave vectors that are perpendicular to his sphere surface were included in the Casimir force calculation. In the following sections, the non-perpendicular wave vector contributions to the Casimir force that were not accounted for in Boyer’s work are considered.

4.1.1. Hollow Spherical Shell As shown in Figure 4.6, the vacuum-field radiation imparts upon a differential patch of an area dA on the inner wall of the conducting spherical cavity a net momentum of the amount ½ ³ ´ 1 2nπ~ cos θinc 0 ≤ θinc < π/2, ~ 0 s,1 , R ~ 0 s,0 = ° ³ ´ ³ ´° Rˆ0 s,1 , 4~ pinner = − ~4k~0 inner ; R °~ ° n = 1, 2, 3, · · · , 2 0 0 0 0 ~ ~ ~ °Rs,2 ri , Λs,2 − Rs,1 ri , Λs,1 ° ³ ´ ~ 0 s,1 , R ~ 0 s,0 is from equation (3.16). The angle of incidence θinc is from equation (A.115); where 4k~0 inner ; R ³ ´ ³ ´ 0 ~0 ~ s,1 r0 , Λ ~0 ~ R s,1 and Rs,2 ri , Λs,2 follow the generic form shown in equation (3.17). i Similarly, the vacuum-field radiation imparts upon a differential patch of an area dA on the outer surface of the conducting spherical shell a net momentum of the amount ½ ° ° ³ ´ 1 0 ≤ θinc < π/2, ° 0 0 ~ ~ 0 s,1 + aRˆ0 s,1 = −2~ ° ˆ k cos θ R , 4~ pouter = − ~4k~0 outer ; R ° i,f ° inc s,1 n = 1, 2, 3, · · · , 2

37

4. Results and Outlook ³ ´ ~ 0 s,1 + aRˆ0 s,1 is from equation (3.18). where 4k~0 outer ; R The net average force per unit time, per initial wave vector direction, acting on differential element patch of an area dA is given by ¶ µ 4~ pouter 4~ pinner ~ Fs,avg = lim + 4t→1 4t 4t or 

~s,avg F

 ° ° ~0 ° ˆ0 ´ ³ ´° − ° ³ = 2~ cos θinc  ° ° °k i,f ° R s,1 , °~ 0 0 0 0 ~ ~ ~ °Rs,2 ri , Λs,2 − Rs,1 ri , Λs,1 ° nπ

½

0 ≤ θinc < π/2, n = 1, 2, 3, · · · .

~s,avg is called a force per initial wave vector direction because it is computed for k~0 i,b and k~0 i,f along Notice that F ~ s,0 as shown specific initial directions. Here k~0 i,b denotes a particular initial wave vector k~0 i entering the resonator at R 0 0 ~ ~ in Figure 3.4. The subscript b for k i,b denotes the bounded space inside the resonator. The k i,f denotes a particular ~ 0 s,1 + aRˆ0 s,1 as shown initial wave vector k~0 i impinging upon the surface of the unbounded region of sphere at point R 0 ~ in Figure 3.4. The subscript f for k i,f denotes the free space external to the resonator. ° ° ° ° Because the wave vector k~0 i,f resides in free or unbounded space, its magnitude °k~0 i,f ° can take on a continuum ° ° ° ° ° ´° ³ ´ ³ ° ° ° °~ ° °~ 0 ~0 0 ~0 ~ of allowed modes, whereas °k~0 i,b ° have been restricted by °L ° = °R s,2 ri , Λs,2 − Rs,1 ri , Λs,1 ° of equation (C.2). ° °The free space limit is the case where the radius of the sphere becomes very large. Therefore, by designating ° ~0 ° °k i,f ° as ° ° nπ ° ~0 ° ³ ´ ³ ´° , °k i,f ° = 0lim ° ° ° ri →∞ ~ 0 ~0 ~0 ~ °Rs,2 ri0 , Λ s,2 − Rs,1 ri , Λs,1 ° and summing over all allowed modes, the total average force per unit time, per initial wave vector direction, per unit area is given by   ∞ ∞ X X nπ2~ cos θ nπ2~ cos θ inc inc ~s,avg =  ° ° ³ ´ ³ ´° − lim ³ ´ ³ ´°  Rˆ0 s,1 . F °~ °~ ° ri0 →∞ ° 0 0 0 0 0 0 0 0 ~ ~ ~ ~ ~ ~ − Rs,1 r , Λ − Rs,1 r , Λ ° ° n=1 °Rs,2 r , Λ n=1 °Rs,2 r , Λ i

s,2

i

s,1

i

s,2

i

s,1

R∞ P∞ In the limit ri0 → ∞, the second summation to the right can be replaced by an integration, n=1 → 0 dn. Hence, we have   Z ∞ ∞ X 2~nπ cos θ 2~nπ cos θ inc inc ~s,avg =  ° ° ³ ´ ³ ´° − lim ³ ´ ³ ´° dn Rˆ0 s,1 , F °~ ° ri0 →∞ 0 ° ~ ° 0 0 0 0 0 0 0 0 ~ ~ ~ ~ ~ ~ °Rs,2 ri , Λs,2 − Rs,1 ri , Λs,1 ° n=1 °Rs,2 ri , Λs,2 − Rs,1 ri , Λs,1 ° or with the following substitutions, nπ 0 ³ ´ ³ ´° , ki,f ≡° °~ ° 0 ~0 ~0 ~ °Rs,2 ri0 , Λ s,2 − Rs,1 ri , Λs,1 °

dn =

³ ´ ³ ´° 1° °~ 0 ~ 0s,2 − R ~ s,1 ri0 , Λ ~ 0s,1 ° , °Rs,2 ri0 , Λ ° dki,f π

the total average force per unit time, per initial wave vector direction, per unit area is written as  ∞ X nπ ~s,avg = 2~ cos θinc  ° ³ ´ ³ ´° F °~ ° 0 ~0 0 ~0 ~ n=1 °Rs,2 ri , Λs,2 − Rs,1 ri , Λs,1 ° ¸ ° ³ ´ ³ ´° Z ∞ 1 °~ ° 0 ~0 0 ~0 0 0 ~ ki,f dki,f Rˆ0 s,1 , − 0lim °Rs,2 ri , Λs,2 − Rs,1 ri , Λs,1 ° π ri →∞ 0

38

(4.1)

4. Results and Outlook where 0 ≤ θinc < π/2 and n = 1, 2, 3, · · · . The total average vacuum-field radiation force per unit time acting on the uncharged conducting spherical shell is therefore Z X ~s,avg • dS ~sphere ~ Fs,total = F S {k~0 i,b ,k~0 i,f ,R~0 s,0 } or X

Z



∞ X

2nπ~ cos θinc 2~ ° ³ ´ ³ ´° − cos θinc ° ° π ~ s,2 r0 , Λ ~0 ~ s,1 r0 , Λ ~0 ° S n=1 °R − R 0 0 0 ~ ~ ~ s,2 s,1 i i k , k , R { i,b i,f s,0 } ¸ ° ³ ´ ³ ´° Z ∞ °~ ° 0 ~0 0 ~0 0 0 ~ ~sphere , × 0lim °R ki,f dki,f Rˆ0 s,1 • dS s,2 ri , Λs,2 − Rs,1 ri , Λs,1 °

F~s,total =



ri →∞

(4.2)

0

R ~sphere is a differential surface element of a sphere and the integration where dS is over the spherical surface. The S ~ 0 s,0 is the initial crossing point inside the sphere as defined in equation (3.2). The notation P ~0 ~0 ~0 term R {´k i,b ,k i,f ,R s,0 } ³ ´ ³ 0 0 ~ ~ imply the summation over all initial wave vector directions for both inside k i,b and outside k i,f of the sphere, ~ 0 s,0 . over all crossing points given by R ~ It is easy to see that Fs,avg of equation (4.1) is an “unregularized” 1D Casimir force expression for the parallel plates (see the vacuum pressure approach by Milonni, Cook and Goggin [25]). It becomes more apparent with the substitution 4t = d/c. An application of the Euler-Maclaurin summation formula [23, 24] leads to the regularized, ~s,avg is attractive because finite force expression. The force F cos θinc > 0 and ° ³ ´ ³ ´° nπ 1 °~ ° 0 ~0 0 ~0 ~ ° ´ ³ ´° < ³ lim R r , Λ − R r , Λ ° s,2 s,1 i s,2 i s,1 ° °~ ° π ri0 →∞ 0 0 0 0 ~ ~ ~ n=1 °Rs,2 ri , Λs,2 − Rs,1 ri , Λs,1 ° ∞ X

Z 0

∞

0 0 ki,f dki,f ,

° ´° ´ ³ ³ ° °~ 0 ~0 0 ~0 ~0 ~ where °R s,2 ri , Λs,2 − Rs,1 ri , Λs,1 ° is a constant for a given initial wave k i,b and the initial crossing point ~ 0 s,0 in the cross-section of a sphere (or hemisphere). The total average force F~s,total , which is really the sum of R ~s,avg over all R ~ 0 s,0 and all initial wave directions, is therefore also attractive. For the sphere configuration of Figure F 3.4, where the energy flow direction is not restricted to the direction of local surface normal, the Casimir force problem becomes an extension of infinite set of parallel plates of a unit area.

4.1.2. Hemisphere-Hemisphere and Plate-Hemisphere Similarly, for the hemisphere-hemisphere and plate-hemisphere configurations, the expression for the total average force per unit time, per initial wave vector direction, per unit area is identical to that of the hollow spherical shell with modifications,  ∞ X nπ ~h,avg = 2~ cos θinc  ° ³ ´ ³ ´° F °~ ° 0 0 0 ~0 ~ ~ n=1 °Rh,2 ri , Λh,2 − Rh,1 ri , Λh,1 ° ¸ ° ³ ´ ³ ´° Z ∞ 1 °~ ° 0 ~0 0 ~0 0 0 ~ ˆ0 r , Λ − R r , Λ k dk (4.3) − 0lim °R ° h,2 h,1 i h,2 i h,1 i,f i,f R h,1 , π ri →∞ 0 ´ ³ ~ h,1 r0 , Λ ~0 where θinc ≤ π/2 and n = 1, 2, 3, · · · . The incidence angle θinc is from equation (A.115); R i h,1 and ³ ´ ~ h,2 r0 , Λ ~0 R i h,2 follow the generic form shown in equation (3.21). This force is attractive for the same reasons as

39

4. Results and Outlook discussed previously for the hollow spherical shell case. The total radiation force averaged over unit time, over all possible initial wave vector directions, acting on the uncharged conducting hemisphere-hemisphere (plate-hemisphere) surface is given by  Z X ∞ X 2~ 2nπ~ cos θinc  ° ´ ³ ´° − ³ cos θinc F~h,total = ° ° π 0 0 0 0 ~ ~ ~ ~ S n=1 °Rh,2 r , Λ − R r , Λ ° h,1 0 0 0 ~ ~ ~ i i h,2 h,1 {k i,b ,k i,f ,R h,0 } ¸ ° ³ ´ ³ ´° Z ∞ °~ ° 0 ~0 0 ~0 0 0 ~ ~hemisphere , × 0lim °R ki,f dki,f Rˆ0 h,1 • dS (4.4) h,2 ri , Λh,2 − Rh,1 ri , Λh,1 ° ri →∞

0

R ~hemisphere is now a differential surface element of a hemisphere and the integration where dS is over the surface S 0 ~ of the hemisphere. The term R h,0 is the initial crossing point of the hemisphere opening as defined in equation ³ (3.2). ´ P The notation {k~0 i,b ,k~0 i,f ,R~0 h,0 } imply the summation over all initial wave vector directions for both inside k~0 i,b ³ ´ and outside k~0 i,f of the hemisphere-hemisphere (or the plate-hemisphere) resonator, over all crossing points given ~ 0 h,0 . by R It should be remarked that for the plate-hemisphere configuration, the total average radiation force remains identical to that of the hemisphere-hemisphere configuration only for the case where the gap distance between plate and the center of is more than the hemisphere radius ri0 . When the plate is placed closer, the boundary quantization ° hemisphere ° °~ ° length °L° must be chosen carefully to be either ° ° ° ´° ´ ³ ³ °~ ° ° ~ ~ 0h,1 ° ~ 0h,2 − R ~ h,1 ri0 , Λ °L° = °Rh,2 ri0 , Λ ° or

° ° ° ³ ´ ³ ´° °~ ° ° ~ ° ~ 0p − R ~ h,N ~ 0h,N ri0 , Λ °L° = °Rp ri0 , Λ °. h,max h,max

³ ´ ~ p r0 , Λ ~ 0p is from equation They are illustrated in Figure 3.8. The proper one to use is the smaller of the two. Here R i (C.54) of Appendix C3 and Nh,max is defined in equation (C.8) of Appendix C2.

4.2. Interpretation of the Result Because only the specification of boundary is needed in Casimir’s vacuum-field approach as opposed to the use of a polarizability parameter in Casimir-Polder interaction picture, the Casimir force is sometimes regarded as a configurational force. On the other hand, the Casimir effect can be thought of as a macroscopic manifestation of the retarded van der Waals interaction. And the Casimir force can be equivalently approximated by a summation of the constituent molecular forces employing Casimir-Polder interaction. This practice inherently relies on the material properties of the involved conductors through the use of polarizability parameters. In this respect, the Casimir force can be regarded as a material dependent force. Boyer’s material property is such that the atoms in his conducting sphere are arranged in such manner to respond only to the impinging radiation at local normal angle of incidence to the sphere surface, and they also radiate only along the direction of local normal to its surface. When the Casimir force is calculated for a sphere made of Boyer’s fictitious material, the force is repulsive. Also, in Boyer’s original work, the laboratory boundary did not exist. When Boyer’s sphere is placed in a realistic laboratory, the net Casimir force acting on his sphere becomes attractive because the majority of the radiation from the laboratory boundaries acts to apply inward pressure on the external surface of sphere when the angle of incidence is oblique with respect to the local normal. If the sphere is made of ordinary perfect conductors, the impinging radiation at oblique angles of incidence would be reflected. In such cases the total radiation pressure applied to the external local-sphere-surface is twice the pressure exerted by the incident wave, which is the force found in equation (4.2) of the previous section. However, Boyer’s sphere cannot radiate along the direction that is not normal to the local-sphere-surface. Therefore, the total pressure applied to Boyer’s sphere is half of the force given in equation (4.2) of the previous section. This peculiar incapability of emission of a Boyer’s sphere would lead to

40

4. Results and Outlook

Apparatus Region Figure 4.7.: To deflect away as much possible the vacuum-field radiation emanating from the laboratory boundaries, the walls, floor and ceiling are constructed with some optimal curvature to be determined. The apparatus is then placed within the “Apparatus Region.”

the absorption of the energy and would cause a rise in the temperature for the sphere. Nonetheless, the extra pressure due to the waves of oblique angle of incidence is large enough to change the Casimir force for Boyer’s sphere from being repulsive to attractive. The presence of the laboratory boundaries only act to enhance the attractive aspect of the Casimir force on a sphere. The fact that Boyer’s sphere cannot irradiate along the direction that is not normal to the local-sphere-surface, whereas ordinary perfect conductors irradiate in all directions, implies that his sphere is made of extraordinarily hypothetical material, and this may be the reason why the repulsive Casimir force have not been experimentally observed to date. In conclusion, (1) the Casimir force is both boundary and material property dependent. The particular shape of the conductor, e.g. sphere, only introduces the preferred direction for radiation. For example, radiations in direction normal to the local surface has bigger magnitude than those radiating in other directions. This preference for the direction of radiation is intrinsically connected to the preferred directions for the lattice vibrations. And, the characteristic of lattice vibrations is intrinsically connected to the property of material. (2) Boyer’s sphere is made of extraordinary conducting material, which is why his Casimir force is repulsive. (3) When the radiation pressures of all angles of incidence are included in the Casimir force calculation, the force is attractive for a charge-neutral sphere made of ordinary perfect conductor.

4.3. Suggestions on the Detection of Repulsive Casimir Force for a Sphere The first step in detecting the repulsive Casimir force for a spherical configuration is to find a conducting material that most closely resembles the Boyer’s material to construct two hemispheres. It has been discussed previously that even Boyer’s sphere can produce attractive Casimir force when the radiation pressures due to oblique incidence waves are included in the calculation. Therefore, the geometry of the laboratory boundaries have to be chosen to deflect away as much as possible the oblique incident wave as illustrated in Figure 4.7. Once these conditions are met, the experiment can be conducted in the region labeled “Apparatus Region” to observe Boyer’s repulsive force.

4.4. Outlook The Casimir effect has influence in broad range of physics. Here, we list one such phenomenon known as “sonoluminescense,” and, finally conclude with the Casimir oscillator.

41

4. Results and Outlook Original Bubble

L 21

R1 R2

Acoustic field

Bubble under strong acoustic field

Figure 4.8.: The original bubble shape shown in dotted lines and the deformed bubble in solid line under strong acoustic field.

4.4.1. Sonoluminescense The phenomenon of sonoluminescense remains a poorly understood subject to date [28, 29]. When a small air bubble of radius ∼ 10−3 cm is injected into water and subjected to a strong acoustic field of ∼ 20 kHz under pressure roughly ∼ 1 atm, the bubble emits an intense flash of light in the optical range, with total energy of roughly ∼ 107 eV. This emission of light occurs at minimum bubble radius of roughly ∼ 10−4 cm. The flash duration has been determined to be on the order of 100 ps [30, 31, 32]. It is to be emphasized that small amounts of noble gases are necessary in the bubble for sonoluminescense. The bubble in sonoluminescense experiment can be thought of as a deformed sphere under strong acoustic pressure. ~ The ° dynamical ° Casimir effect arises due to the deformation of the shape; therefore, introducing a modification to L21 = °~ ° ~ 1 ° from that of the original bubble shape. Here L ~ 21 is the path length for the reflecting wave in the original °R2 − R ° ° ~ 21 ≡ L ~ 21 (t) = ° ~ 2 (ri (t) , θ (t) , φ (t)) − R ~ 1 (ri (t) , θ (t) , φ (t))° bubble shape. In general L °R ° . From the relations ~ 1 (ri (t) , θ (t) , φ (t)) and R ~ N (ri (t) , θ (t) , φ (t)) , together with found in this thesis work for the reflection points R the dynamical Casimir force expression of equation (3.31), the amount of initial radiation energy converted into heat energy during the deformation process can be found. The bubble deformation process shown in Figure 4.8 is a three dimensional heat generation problem. Current investigation seeks to determine if the temperature can be raised sufficiently to cause deuterium-tritium (D-T) fusion, which could provide an alternative approach to achieve energy generation by this D-T reaction (threshold ∼ 17 KeV ) [33]. Its theoretical treatment is similar to that discussed on the 1D problem shown in Figure 3.9.

4.4.2. Casimir Oscillator If one can create a laboratory as shown in Figure 4.7, and place in the laboratory hemispheres made of Boyer’s material, then the hemisphere-hemisphere system will execute an oscillatory motion. When two such hemispheres are separated, the allowed wave modes in the hemisphere-hemisphere confinement would no longer follow Boyer’s ° ° spherical Bessel °~ ° ~ ~ 1 and R ~2 function restriction. Instead it will be strictly constrained by the functional relation of °R2 − R1 ° , where R are two neighboring reflection points. Only when the two hemispheres are closed, would the allowed wave modes obey Boyer’s spherical Bessel function restriction. Assuming that hemispheres are made of Boyer’s material and the laboratory environment is that shown in Figure 4.7, the two closed hemispheres would be repulsing because Boyer’s Casimir force is repulsive. Once the two hemispheres are separated, the allowed wave modes are governed by the internal reflections at oblique angle of incidence. Since the hemispheres made of Boyer’s material are “infinitely unresponsive” to oblique incidence waves, all these temporary non-spherical symmetric waves would be absorbed by the Boyer’s hemispheres and the hemispheres would heat up. The two hemispheres would then attract each other and the oscillation cycle repeats. Such a mechanical system may have application.

42

4. Results and Outlook

R 2 R1

R2 R 1

Closed Unclosed Figure 4.9.: The vacuum-field radiation energy flows are shown for closed and unclosed hemispheres. For the hemispheres made of Boyer’s material, the non-radial wave would be absorbed by the hemispheres.

43

Appendices These are my original derivations and developments that were too tedious and lengthy to be included in the main body of the thesis. There are five appendices: (1) Appendix A, (2) Appendix B, (3) Appendix C, (4) Appendix D and Appendix E. The appendices C and D are further divided into subparts C.1, C.2, C.3, D.1 and D.2. The title and the layout of the appendices closely follow the main body of the thesis. Finally, the appendix E have been added to provide further list of references pertaining to the Casimir effect, but which were not explicitly used by this thesis.

44

A. Reflection Points on the Surface of a Resonator In this appendix, the original derivations and developments of this thesis pertaining to the reflection dynamics are included. It is referenced by the text of this thesis to supply all the details. For the configuration shown in Figure 3.1, the wave vector directed along an arbitrary direction in Cartesian coordinates is written as  0 3 , eˆ1 = x ˆ,  i = 1 → k1,x X ¡ 0 ¢ 0 0 0 0 0 0 ~ i = 2 → k1,y , eˆ2 = yˆ k 1 k1,x , k1,y , k1,z = k1,i eˆi , k1,i = (A.1)  0 i=1 i = 3 → k1,z , eˆ3 = zˆ. The unit wave vector is given by 3 ° °−1 X ° ° 0 kˆ0 1 = °k~0 1 ° eˆi . k1,i

(A.2)

i=1

~ 0 0 of hemisphere opening for the incident wave k~0 1 is defined as The initial crossing position R  0 3 ,  i = 1 → r0,x X ¢ ¡ 0 0 0 0 0 0 0 ~ , i = 2 → r0,y r0,i = r0,i eˆi , R 0 r0,x , r0,y , r0,z =  0 i=1 . i = 3 → r0,z

(A.3)

~ 0 0 has only two components, r0 and r0 . But nevertheless, one can always set It should be noticed here that R 0,z 0,x 0 r0,y = 0 whenever needed. Since no particular wave with certain wavelength is prescribed initially, it is desirable to employ a parameterization scheme to represent these wave vectors. The line segment traced out by the wave vector kˆ0 1 is formulated in the parametric form ~ 0 1 = ξ1 kˆ0 1 + R ~ 00 = R

3 · X

0 r0,i

¸ ° °−1 ° ~0 ° 0 + ξ1 °k 1 ° k1,i eˆi ,

(A.4)

i=1

where the real variable ξ1 is a positive definite parameter. The restriction ξ1 ≥ 0 is a necessary condition since the ~ 0 1 denotes the first reflection point on the hemisphere. In terms direction of the wave propagation is set by kˆ0 1 . Here R ~ 0 1 takes the form of spherical coordinates, R  0 3  Λ1,1 = sin θ10 cos φ01 , X 0 0 0 0 0 0 ~ Λ0 = sin θ10 sin φ01 , R 1 (ri , θ1 , φ1 ) = ri Λ1,i eˆi , (A.5)  01,2 i=1 Λ1,3 = cos θ10 , where ri0 is the hemisphere radius, θ10 and φ01 are the polar and azimuthal angles respectively of the first reflection ~ 0 1 . The subscript i of r0 denotes “inner radius” and it is not a summation index. Equations (A.4) and (A.5) are point R i combined as ¸ 3 · ° °−1 X ° ~0 ° 0 0 0 r0,i + ξ1 °k 1 ° k1,i − ri Λ1,i eˆi = 0.

(A.6)

i=1

Because the basis vectors eˆi are independent of each other, the above relations are only satisfied when each coefficients

45

A. Reflection Points on the Surface of a Resonator of eˆi vanish independently, ° °−1 ° ° 0 0 r0,i + ξ1 °k~0 1 ° k1,i − ri0 Λ1,i = 0,

i = 1, 2, 3.

(A.7)

The three terms Λ1,i=1 , Λ1,i=2 and Λ1,i=3 satisfy an identity 3 X

Λ21,i = 1.

(A.8)

i=1

From equation (A.7), Λ21,i is computed for each i : ½ Λ21,i

=

−2 [ri0 ]

¾ ° °−2 £ ¤ ° °−1 £ 0 ¤2 2 ° ~0 ° 2 ° ~0 ° 0 0 0 r0,i + ξ1 °k 1 ° k1,i + 2r0,i ξ1 °k 1 ° k1,i ,

i = 1, 2, 3.

Substituting the above result of Λ21,i into equation (A.8) and after rearrangement, one obtains ξ12

3 ° 3 3 ° °−1 X ° X X £ 0 ¤2 2 ° ~0 °−2 £ 0 ¤2 ° ° 0 0 k1,i + 2ξ1 °k~0 1 ° r0,i k1,i + r0,i − [ri0 ] = 0. °k 1 ° i=1

i=1

(A.9)

i=1

Further simplifying, it becomes ° °2 2 ~ 0 0 ξ1 + ° ~ 00° ξ12 + 2kˆ0 1 • R °R ° − [ri0 ] = 0.

(A.10)

There are two roots, ξ1,a

~ 00 − = −kˆ0 1 • R

rh

i2

° °2 ° ~0 ° 2 + [ri0 ] − °R 0°

rh

i2

° °2 ° ~0 ° 2 + [ri0 ] − °R 0° .

~ 00 kˆ0 1 • R

and ξ1,b =

−kˆ0

1•

~0 R

0+

~ 00 kˆ0 1 • R

~ 0 0 ≤ r0 , The root to be used should have a positive value. For the wave reflected within the hemisphere, R i rh ° °2 ¯ ¯ i2 ¯ ~ 0 0 + [r0 ]2 − ° ~ 00° ~ 0 0 ¯¯ ≥ −kˆ0 1 • R ~ 00 kˆ0 1 • R °R ° ≥ ¯kˆ0 1 • R i

¯ ¯ ¯ ~ 0 0 ¯¯ = −kˆ0 1 • R ~ 0 0 happens when kˆ0 1 • R ~ 0 0 ≤ 0. Therefore, ξ1,a ≤ 0 and ξ1,b ≥ 0; the where the equality ¯kˆ0 1 • R positive root ξ1,b should be selected. For bookkeeping purposes, ξ1,b is designated as ξ1,p : rh ° °2 i2 0 0 ˆ ~ 0 0 + [r0 ]2 − ° ~ 00° ~ kˆ0 1 • R (A.11) ξ1,p = −k 1 • R 0 + °R ° . i Using this positive root, the first reflection point of the inner hemisphere is found to be ¸ 3 · ° °−1 ³ ´ X ° ~0 ° 0 0 0 0 0 ˆ ~ ~ R 1 ξ1,p ; R 0 , k 1 = r0,i + ξ1,p °k 1 ° k1,i eˆi .

(A.12)

i=1

The incident wave k~0 i , shown in Figure A.1 and where i here stands for incident wave, can always be decomposed

46

A. Reflection Points on the Surface of a Resonator

^n ki

θi k r

n

^n θi

Surface

Figure A.1.: A simple reflection of incoming wave k~0 i from the surface defined by a local normal n~0 .

into components parallel and perpendicular to the vector n~0 normal to the reflecting surface, h i n~0 × k~0 i × n~0 ~0 • k~0 i n k~0 i = k~0 i,k + k~0 i,⊥ = n~0 + . n~0 • n~0 n~0 • n~0 If the local normal nˆ0 is already normalized to unity, the above expression reduces to h i k~0 i = nˆ0 • k~0 i nˆ0 + nˆ0 × k~0 i × nˆ0 .

(A.13)

Here the angle between k~0 i and nˆ0 is π − θi . The action of reflection only modifies k~0 i,k in the reflected wave. The reflected wave part of k~0 i in equation (A.13) is i h (A.14) k~0 r = αr,⊥ k~0 i,⊥ − αr,k k~0 i,k = αr,⊥ nˆ0 × k~0 i × nˆ0 − αr,k nˆ0 • k~0 i nˆ0 , where k~0 i,k have been rotated by 180o on the plane of incidence. The new quantities αr,k and αr,⊥ are the reflection coefficients. For a perfect reflecting surfaces, αr,k = αr,⊥ = 1. Because of the frequent usage of thehcomponent i for 0 0 0 ~ ~ ˆ k r , equation (A.14) is also written in component form. The component of the double cross product n × k i × nˆ0 is computed first, nh i o h i 0 0 nˆ0 × k~0 i × nˆ0 = ²lmn nˆ0 × k~0 i n0n = ²lmn ²mqr n0q ki,r n0n = ²nlm ²qrm n0q ki,r n0n l

m

0 0 0 = [δnq δlr − δnr δlq ] n0q ki,r n0n = δnq δlr n0q ki,r n0n − δnr δlq n0q ki,r n0n 0 0 = n0n ki,l n0n − n0l ki,n n0n

or 3 h i X £ 0 0 0 ¤ 0 nn ki,l nn − n0l ki,n n0n eˆl , nˆ0 × k~0 i × nˆ0 =

(A.15)

l=1

where the summation over the index n is implicit. In component form, k~0 r is hence expressed as k~0 r =

3 X ©

£ ¤ ª 0 0 0 αr,⊥ n0n ki,l n0n − n0l ki,n n0n − αr,k n0n ki,n n0l eˆl ,

l=1

where it is understood nˆ0 is already normalized.

47

(A.16)

A. Reflection Points on the Surface of a Resonator ~ 0 2 is found by repeating the steps done for R ~ 01, The second reflection point R h i αr,⊥ nˆ0 × k~0 i × nˆ0 − αr,k nˆ0 • k~0 i nˆ0 ~ 02 = R ~ 0 1 + ξ2,p kˆ0 r = R ~ 0 1 + ξ2,p ° °, h i R ° ° °αr,⊥ nˆ0 × k~0 i × nˆ0 − αr,k nˆ0 • k~0 i nˆ0 ° where ξ2,p is the new positive parameter corresponding to the second reflection point. The procedure can be repeated for any reflection point. Although this technique is sound, it can be noticed immediately that the technique suffers from the lack of elegance. For this reason, the scalar field technique will be exclusively used in studying the reflection dynamics. For a simple plane, the scalar field function can be inferred rather intuitively. However, for more complex surfaces, one has to work it out to get the corresponding scalar field. For the purpose of generalization of the technique to any arbitrary surfaces, we derive the scalar field functional for the plane in great detail.

In simple reflection dynamics, there exists a plane of incidence in which all reflections occur. The plane of incidence is determined by the incident wave k~0 i and the local surface normal n~0 i . For the system shown in Figure 3.1, k~0 i and n~0 i are given by ³ ´ ~ 0 1 ξ1,p ; R ~ 0 0 , kˆ0 1 = −ξ1,p kˆ0 1 − R ~ 00. k~0 i = k~0 1 , n~0 n0i ,1 ≡ −R The normal to the incidence plane is characterized by the cross product, 3 h i X 0 0 ~ 0 0 = −k~0 1 × R ~ 00 = − n~0 p,1 = k~0 1 × n~0 n0i ,1 = k~0 1 × −ξ1,p kˆ0 1 − R ²ijk k1,j r0,k eˆi , i=1

where the summation over the indices j and k are implicit. The normal to the incidence plane is normalized as 3 ° °−1 X ° ° 0 0 nˆ0 p,1 = − °n~0 p,1 ° ²ijk k1,j r0,k eˆi .

(A.17)

i=1

In order to take advantage of the information given above, the concept of scalar fields in mathematical sense is in order. A functional f (x0 , y 0 , z 0 ) is a scalar field if to each point (x0 , y 0 , z 0 ) of a region in space, there corresponds a number λ. The study of a scalar field is a study of scalar valued functions of three variables. Scalar fields are connected to its normals, e.g., equation (A.17), through the relation  3  1 → ν10 = x0 , X ∂ 0 0 0 0 0 0 0 0 ~ ˆ 2 → ν20 = y 0 , (A.18) i= n p,1 ∝ ∇ fp,1 (x , y , z ) = eˆi 0 fp,1 (x , y , z ) ,  ∂ν i i=1 3 → ν30 = z 0 . Introducing a constant proportionality factor βp,1 , equation (A.18) becomes nˆ0 p,1 = βp,1

3 X i=1

eˆi

∂ fp,1 (x0 , y 0 , z 0 ) . ∂νi0

(A.19)

~ 0 fp,1 . Because the vector nˆ0 p,1 is a The proportionality factor βp,1 is intrinsically connected to the normalization of ∇ unit vector, its magnitude squared is 2 βp,1

¸2 3 · X ∂ 0 0 0 f (x , y , z ) =1 p,1 ∂νi0 i=1

(

→

βp,1

¸2 )−1/2 3 · X ∂ 0 0 0 =± fp,1 (x , y , z ) . ∂νi0 i=1

~ 0 fp,1 are intrinsically built in. Therefore, the proportionality In equation (A.19), the directions for vectors nˆ0 p,1 and ∇

48

A. Reflection Points on the Surface of a Resonator factor βp,1 has to be a positive quantity, ( βp,1 =

¸2 3 · X ∂ 0 0 0 fp,1 (x , y , z ) ∂νi0 i=1

)−1/2 .

(A.20)

Unfortunately, the exact form of the proportionality coefficient βp,1 requires the knowledge of fp,1 , which is yet to be determined. However, we can use it formally for now until the solution for fp,1 is found. ~ 0 fp,1 into equation (A.19), and using equations (A.17) and (A.19), one arrives Substituting the gradient function ∇ at ¸ 3 · ° °−1 X ∂ ° −1 ° ~0 0 0 0 0 0 fp,1 (x , y , z ) + βp,1 °n p,1 ° ²ijk k1,j r0,k eˆi = 0, ∂νi0 i=1

  1 → ν10 = x0 , 2 → ν20 = y 0 , i=  3 → ν30 = z 0 .

Because the basis vectors eˆi are linearly independent, the equation for each component is obtained as   1 → ν10 = x0 = α, ° ° −1 ∂ ° −1 ° ~0 0 0 2 → ν20 = y 0 = β, fp,1 (α, β, γ) + βp,1 °n p,1 ° ²ijk k1,j r0,k = 0, i=  ∂νi0 3 → ν30 = z 0 = γ.

(A.21)

(A.22)

Integrating both sides of equation (A.22) over the variable νi0 = α, Z α Z α ° °−1 ∂ ° −1 ° ~0 0 0 0 0 f (α , β, γ) dα = − β n r0,k dα0 , ° ° ²α0 jk k1,j p,1 p,1 p,1 0 ∂α α0 α0 ° ° ° ° 0 0 where the dummy variable α0 is introduced for integration purpose. The terms ²α0 jk k1,j r0,k , βp,1 and °n~0 p,1 ° are independent of the dummy variable α0 , and they can be moved out of the integrand, Z α Z α ° °−1 ∂ ° −1 ° ~0 0 0 0 0 f (α , β, γ) dα = −β n r dα0 . (A.23) ² k αjk 1,j 0,k p,1 ° p,1 ° 0 p,1 ∂α α0 α0 Because the total differential of fp,1 is given by dfp,1 =

∂fp,1 ∂fp,1 0 ∂fp,1 dα + dβ + dγ, ∂α0 ∂β ∂γ

α0 6= β 6= γ,

(A.24)

the term [∂fp,1 /∂α0 ] dα0 can be written as ∂fp,1 ∂fp,1 ∂fp,1 0 dα = dfp,1 − dβ − dγ. 0 ∂α ∂β ∂γ 0 The integration over the variable νi0 = α in equation (A.23), with variables νi0 6= α fixed, can be carried out with dβ = dγ = 0,

∂ fp,1 (α0 , β, γ) dα0 = dfp,1 (α0 , β, γ) ∂α0

(A.25)

as Z

α α0

Z ° °−1 ° −1 ° ~0 0 0 dfp,1 (α0 , β, γ) = −βp,1 r0,k °n p,1 ° ²αjk k1,j

α

dα0

α0

to give ° °−1 ° −1 ° ~0 0 0 fp,1 (α, β, γ) = βp,1 r0,k [α0 − α] + fp,1 (α0 , β, γ) . °n p,1 ° ²αjk k1,j

(A.26)

° °oi h n ° ° 0 0 The two terms ²αjk k1,j r0,k / βp,1 °n~0 p,1 ° α0 and fp,1 (α0 , β, γ) are independent of α. These terms can only

49

A. Reflection Points on the Surface of a Resonator assume values of νi0 = β or νi0 = γ. By re-designating α independent terms, ° °−1 ° −1 ° ~0 0 0 hp,1 (β, γ) = βp,1 r0,k α0 + fp,1 (α0 , β, γ) , °n p,1 ° ²αjk k1,j

(A.27)

equation (A.26) can be rewritten for bookkeeping purposes as ° °−1 ° −1 ° ~0 0 0 fp,1 (α, β, γ) = hp,1 (β, γ) − βp,1 r0,k α. °n p,1 ° ²αjk k1,j

(A.28)

Substituting the result into equation (A.22) and performing a differentiation with respect to the variable νi0 = β gives ¸ · ° °−1 ° °−1 ∂ ° ° −1 ° ~0 −1 ° ~0 0 0 0 0 r0,k =0 hp,1 (β, γ) − βp,1 °n p,1 ° ²αjk k1,j r0,k α + βp,1 °n p,1 ° ²βjk k1,j ∂β or ° °−1 ∂ ° −1 ° ~0 0 0 r0,k . hp,1 (β, γ) = −βp,1 °n p,1 ° ²βjk k1,j ∂β The integration of both sides with respect to the variable νi0 = β yields the result Z

β

β0

∂ hp,1 (β 0 , γ) dβ 0 = − ∂β 0

Z

β β0

Z ° °−1 ° °−1 ° ° −1 ° ~0 −1 ° ~0 0 0 0 0 r0,k dβ 0 = −βp,1 r0,k βp,1 °n p,1 ° ²β 0 jk k1,j °n p,1 ° ²βjk k1,j

β

dβ 0 ,

β0

° ° ° ° 0 0 where the dummy variable β 0 is introduced for integration purpose and the terms ²βjk k1,j r0,k , βp,1 and °n~0 p,1 ° have been taken out of the integrand because they are independent of β 0 . Following the same procedure used in equations (A.24) through (A.25), the integrand [∂hp,1 /∂β 0 ] dβ 0 on the left hand side of the integral is ∂ hp,1 (β 0 , γ) dβ 0 = dhp,1 (β 0 , γ) . ∂β 0 Consequently, hp,1 (β, γ) is given by ° °−1 ° −1 ° ~0 0 0 hp,1 (β, γ) = βp,1 r0,k [β0 − β] + hp,1 (β0 , γ) . °n p,1 ° ²βjk k1,j

(A.29)

° °oi h n ° ° 0 0 The two terms ²βjk k1,j r0,k / βp,1 °n~0 p,1 ° β0 and hp,1 (β0 , γ) are independent of β. The β independent terms can be re-designated as ° °−1 ° −1 ° ~0 0 0 gp,1 (γ) = βp,1 r0,k β0 + hp,1 (β0 , γ) . °n p,1 ° ²βjk k1,j

(A.30)

For bookkeeping purposes, equation (A.29) is rewritten as ° °−1 ° −1 ° ~0 0 0 hp,1 (β, γ) = gp,1 (γ) − βp,1 r0,k β. °n p,1 ° ²βjk k1,j

(A.31)

Substitution of hp,1 (β, γ) into equation (A.28) gives ° °−1 £ ¤ ° −1 ° ~0 0 0 0 0 fp,1 (α, β, γ) = gp,1 (γ) − βp,1 ²βjk k1,j r0,k β + ²αjk k1,j r0,k α . °n p,1 °

(A.32)

Once more substituting fp,1 (α, β, γ) into equation (A.22), and performing the differentiation with respect to the variable νi0 = γ, where γ 6= α 6= β, we obtain · ¸ ° °−1 © ° °−1 ª d ° ° −1 ° ~0 −1 ° ~0 0 0 0 0 0 0 gp,1 (γ) − βp,1 ²βjk k1,j r0,k β + ²αjk k1,j r0,k α + βp,1 r0,k =0 °n p,1 ° °n p,1 ° ²γjk k1,j dγ

50

A. Reflection Points on the Surface of a Resonator or ° °−1 d ° −1 ° ~0 0 0 gp,1 (γ) = −βp,1 r0,k , °n p,1 ° ²γjk k1,j dγ where the differentiation have been changed from ∂ to d because gp,1 is a function of single variable. The integration of both sides with respect to the variable νi0 = γ then gives Z γ Z γ Z γ ° °−1 ° °−1 d ° ° −1 ° ~0 −1 ° ~0 0 0 0 0 0 0 0 βp,1 °n p,1 ° ²γ 0 jk k1,j r0,k dγ = −βp,1 °n p,1 ° ²γjk k1,j r0,k g (γ ) dγ = − dγ 0 , 0 p,1 γ0 dγ γ0 γ0 0 0 where the dummy variable γ 0 have been introduced for integration purpose and the terms ²γjk k1,j r0,k , βp,1 and ° ° ° ~0 ° 0 0 °n p,1 ° have been taken out of the integrand because they are independent of γ . Knowing [dgp,1 /dγ ] dγ 0 = dgp,1 , the integration is carried out to yield

° °−1 ° −1 ° ~0 0 0 gp,1 (γ) = βp,1 r0,k [γ0 − γ] + gp,1 (γ0 ) . °n p,1 ° ²γjk k1,j

(A.33)

° °oi n h ° ° 0 0 The two terms ²γjk k1,j r0,k / βp,1 °n~0 p,1 ° γ0 and gp,1 (γ0 ) are independent of γ. The γ independent terms are re-designated as ° °−1 ° −1 ° ~0 0 0 b0 = βp,1 r0,k γ0 + gp,1 (γ0 ) . °n p,1 ° ²γjk k1,j

(A.34)

For bookkeeping purposes, equation (A.33) is rewritten as ° °−1 ° −1 ° ~0 0 0 gp,1 (γ) = b0 − βp,1 r0,k γ. °n p,1 ° ²γjk k1,j Substituting gp,1 (γ) in equation (A.32), the result for fp,1 (α, β, γ) is found to be  3 ° °−1 X  1 → ν10 = α = x0 , ° −1 ° ~0 0 0 0 2 → ν20 = β = y 0 , fp,1 (α, β, γ) = b0 − βp,1 °n p,1 ° ²ijk k1,j r0,k νi , i=  i=1 3 → ν30 = γ = z 0 .

(A.35)

(A.36)

The cross product expressed in terms of the Levi-Civita symbol is expanded to give 0 0 0 0 0 0 ²x0 jk k1,j r0,k = k1,j=y 0 r0,k=z 0 − k1,k=z 0 r0,j=y 0 ,

(A.37)

0 0 0 0 0 0 ²y0 jk k1,j r0,k = k1,j=z 0 r0,k=x0 − k1,k=x0 r0,k=z 0 ,

(A.38)

0 0 0 0 0 0 ²z0 jk k1,j r0,k = k1,j=x 0 r0,k=y 0 − k1,k=y 0 r0,j=x0 .

(A.39)

It is important to understand that the functional fp,1 in equation (A.36) is a scalar field description of an infinite family of parallel planes characterized by the normal given in equation (A.17), 3 ° °−1 X ° ° 0 0 nˆ0 p,1 = − °n~0 p,1 ° ²ijk k1,j r0,k eˆi . i=1

~ 0 0 , the surface represented by the scalar field Because the normal nˆ0 p,1 is a cross product of the two vectors k~0 1 and R fp,1 is a plane spanned by all the scattered wave vectors. The graphical plot of the functional fp,1 is illustrated in Figure A.2. The three coefficients, 0 0 , ²αjk k1,j r0,k

0 0 , ²βjk k1,j r0,k

51

0 0 , ²γjk k1,j r0,k

A. Reflection Points on the Surface of a Resonator

^n p,1 ^z

^n p,1

^y ^x

n^ p,1 ^n p,1

b 0=2

b0=1

b0=0

b 0=−1

° °−1 P ° ° 3 0 0 Figure A.2.: Parallel planes characterized by a normal nˆ0 p,1 = − °n~0 p,1 ° i=1 ²ijk k1,j r0,k eˆi .

of the independent variables α, β and γ define the slopes along the respective bases α ˆ , βˆ and γˆ . The integration 0 ˆ constant b0 provides infinite set of parallel planes whose common normal is n p,1 . For what is concerned with here, only one of them containing the coordinate origin is required. It is convenient to choose the plane with b0 = 0. With the plane of b0 = 0 chosen, the scalar field of equation (A.36) is rewritten for the sake of bookkeeping purposes:  3 ° °−1 X  1 → ν10 = α = x0 , ° −1 ° ~0 0 0 0 2 → ν20 = β = y 0 , (A.40) fp,1 (α, β, γ) = −βp,1 n ² k r ν , i = ° p,1 ° ijk 1,j 0,k i  i=1 3 → ν30 = γ = z 0 . where −∞ ≤ {α, β, γ} ≤ ∞. ~ 0 0 are the two initially known vectors As mentioned before, fp,1 of equation (A.40) is a scalar field where k~0 1 and R 0 0 ~ ~ which span locally the reflection surface. Other than k 1 and R 0 , any member vectors of the spanning set for the incidence plane can also be used to determine the orientation of a surface. Then it is always true that ³ ´ ³ ´ ³ ´ ~ 0 0 = fp,2 α, β, γ; k~0 2 , R ~ 0 1 = · · · = fp,N α, β, γ; k~0 N , R ~ 0N , fp,1 α, β, γ; k~0 1 , R (A.41) ~ 0 N is where the integer index N of k~0 N is used to enumerate the sequence of reflections; and the integer index N of R used to enumerate the sequence of reflection points. What is still yet undetermined in equation (A.40) is the proportionality factor βp,1 . From equation (A.20), the βp,1 has an algebraic definition  ( 3 · ¸2 )−1/2  1 → ν10 = α, X ∂ 2 → ν20 = β, βp,1 = , i= 0 fp,1 (α, β, γ)  ∂ν i i=1 3 → ν30 = γ.

52

A. Reflection Points on the Surface of a Resonator The partial derivatives ∂fp,1 /∂νi0 are calculated with the solution of fp,1 in equation (A.40), 3 °−1 ∂ X °−1 ° ° ∂ ° ° −1 ° ~0 −1 ° ~0 0 0 0 0 0 f (α, β, γ) = −β n ² k r ν = −β n r0,k ; ° ° ° ° ²ijk k1,j p,1 p,1 ijk p,1 1,j 0,k i p,1 p,1 ∂νi0 ∂νi0 i=1

which reduces to 

( 3 )−1/2  ° ° X £ ¤ 2 ° ° 0 0  = 0. βp,1 1 − °n~0 p,1 ° ²ijk k1,j r0,k

(A.42)

i=1

The two possible solutions are either βp,1 = 0, or the term enclosed in the outermost square bracket°must °van° ° ish. Because βp,1 = 0 is a useless trivial solution, the second solution has to be adopted. Since °n~0 p,1 ° = ½ i2 ¾1/2 P3 h 0 0 , the equation (A.42) is already satisfied for any value of βp,1 . Therefore, βp,1 is an arbii=1 ²ijk k1,j r0,k ~ 0 fp,1 automatically normalized. trary quantity, and it is simply chosen to be unity which makes the gradient function ∇ The scalar field solution fp,1 of equation (A.40) is now restated with βp,1 = 1,  3 ° °−1 X  1 → ν10 = α = x0 , ° ~0 ° 0 0 0 2 → ν20 = β = y 0 , fp,1 (α, β, γ) = − °n p,1 ° ²ijk k1,j r0,k νi , i= (A.43)  i=1 3 → ν30 = γ = z 0 , where −∞ ≤ {ν10 = α = x0 , ν20 = β = y 0 , ν30 = γ = z 0 } ≤ ∞. The intercept between the plane of incidence, defined by equation (A.43), and the hemisphere is found through the algebraic relation 2

x02 + y 02 + z 02 = [ri0 ] , which can be written as 2

[ri0 ] −

3 X

  1 → ν10 = x0 , 2 → ν20 = y 0 , i=  3 → ν30 = z 0 ,

2

[νi0 ] = 0,

i=1

(A.44)

where the ri0 is the radius of sphere and the index i denotes the radius of the inner surface. The intercept of interest is shown in Figure 3.2. One may be tempted to incorporate the surface vibration into equation (A.44) through a slight modification 2

[ri0 (t)] −

3 X

2

[νi0 ] = 0,

i=1

where the vibration have been introduced through the time variations in radius. Some have employed such a model in describing the “Casimir radiation,” as well as the phenomenon of sonoluminescense mainly due to its simplicity from the mathematical point of view [14, 29]. In general, if one wishes to incorporate the vibration of a surface, the description of such system could be represented in the form 2

[ri0 (θ0 , φ0 , t)] −

3 X

2

[νi0 (θ0 , φ0 )] = 0.

i=1

Since the radius function varies with θ0 , φ0 and t, its treatment has to be postponed until the surface function can be found in later sections. In the present discussion, the hemisphere is regarded as having no vibration. Returning from the above short digression, the surface function of the sphere is expressed as the null function from

53

A. Reflection Points on the Surface of a Resonator equation (A.44), 2

fhemi (x0 , y 0 , z 0 ) = [ri0 ] −

3 X

  1 → ν10 = x0 , 2 → ν20 = y 0 , i=  3 → ν30 = z 0 .

2

[νi0 ] = 0,

i=1

(A.45)

The intersection between the two surfaces, the plane of incidence defined in equation (A.43) and the hemisphere defined in equation (A.45), is found through the relation fp,1 (x0 , y 0 , z 0 ) − fhemi (x0 , y 0 , z 0 ) = 0,

(A.46)

which is equivalent to setting the scalar function fp,1 (x0 , y 0 , z 0 ) = 0. Substituting expressions for fp,1 (x0 , y 0 , z 0 ) and fhemi (x0 , y 0 , z 0 ) given in equations (A.43) and (A.45), we arrive at  ¾ 3 ½  1 → ν10 = α = x0 , ° °−1 X ° ~0 ° 0 0 0 0 2 0 2 2 → ν20 = β = y 0 , i= (A.47) [νi ] − °n p,1 ° ²ijk k1,j r0,k νi − [ri ] = 0,  i=1 3 → ν30 = γ = z 0 . 2

It is convenient to rewrite [ri0 ] in the form 2 [ri0 ]

3 £ 0 ¤2 £ 0 ¤2 £ 0 ¤2 £ 0 ¤2 X ri,i , + ri,y0 + ri,z0 = = ri,x 0 i=1

 0 0 = ri,x 0,  1 → ri,1 0 0 2 → ri,2 = ri,y i= 0,  0 0 = ri,z 3 → ri,3 0.

(A.48)

Equation (A.47) can then be written as 3 ½ X

2 [νi0 ]

¾ ° ° £ 0 ¤2 ° ~0 °−1 0 0 0 − °n p,1 ° ²ijk k1,j r0,k νi − ri,i = 0,

i=1

 0 0 = ri,x 0,  1 → ν10 = α = x0 ; ri,1 0 0 0 0 2 → ν2 = β = y ; ri,2 = ri,y0 , i=  0 0 = ri,z 3 → ν30 = γ = z 0 ; ri,3 0.

(A.49)

Since the first two terms are already known, we can set each braced term equal to zero, ° °−1 £ 0 ¤2 2 ° ° 0 0 [νi0 ] − °n~0 p,1 ° ²ijk k1,j r0,k νi0 − ri,i = 0,

i = 1, 2, 3.

(A.50)

The above relation, equation (A.50), is valid in determining the set of discrete reflection points. The solutions of this quadratic equation are νi0

°−1 1° ° ° 0 0 = °n~0 p,1 ° ²ijk k1,j r0,k ± 2

(· )1/2 ¸2 ° £ 0 ¤2 1° ° ~0 °−1 0 0 , °n p,1 ° ²ijk k1,j r0,k + ri,i 2

i = 1, 2, 3,

(A.51)

where the summation over the indices j and k is implicit. The restriction of νi0 being real imposes the condition · ° ¸2 £ 0 ¤2 1 ° ~0 ° °−1 0 0 ≥ 0, °n p,1 ° ²ijk k1,j r0,k + ri,i 2

i = 1, 2, 3.

(A.52)

0 0 0 In spherical coordinates, the three radial vector components ri,1 , ri,2 and ri,3 are 0 ri,1 = ri0 sin θ0 cos φ0 ,

0 ri,2 = ri0 sin θ0 sin φ0 ,

0 ri,3 = ri0 cos θ0 ,

(A.53)

0 0 0 0 0 0 0 0 0 where ri,1 = ri,x 0 , ri,2 = ri,y 0 and ri,3 = ri,z 0 . Here the terms ri , θ and φ are the usual radial length, the polar and P 3 0 ~0 = the azimuthal angle. This guarantees that R i=1 ri,i eˆi is on the sphere, and justifies the step taken in equation 0 (A.50) since we are only interested in the conditions of the discriminants expressed by equation (A.52). With ri,i

54

A. Reflection Points on the Surface of a Resonator redefined in terms of spherical coordinates, the reality condition of νi0 in equation (A.52) becomes · ° ¸2 1 ° ~0 ° 2 °−1 0 0 n ² k r + [ri0 ] sin2 θ0 cos2 φ0 ≥ 0, ° p,1 ° 1jk 1,j 0,k 2

(A.54)

· ° ¸2 1 ° ~0 ° 2 °−1 0 0 r0,k + [ri0 ] sin2 θ0 sin2 φ0 ≥ 0, °n p,1 ° ²2jk k1,j 2

(A.55)

· ° ¸2 1 ° ~0 ° 2 °−1 0 0 °n p,1 ° ²3jk k1,j r0,k + [ri0 ] cos2 θ0 ≥ 0. 2

(A.56)

Equations (A.54) through (A.56) provide allowed range of νi0 to ensure its value being real. The solution of equation (A.56) is · cos2 θ0 ≥ −

¸2 ° 1 ° ° ~0 °−1 0 0 r , n ² k ° ° p,1 3jk 1,j 0,k 2ri0

(A.57)

which leads to two inequalities, cos θ0 ≥ i

° 1 ° ° ~0 °−1 0 0 , r0,k n ° ° ²3jk k1,j p,1 2ri0

cos θ0 ≤ −i

° 1 ° ° ~0 °−1 0 0 . r0,k n ° ° ²3jk k1,j p,1 2ri0

(A.58)

~ 0 0 and k~0 1 . We have to look for sin θ0 These two inequalities cannot be satisfied simultaneously by the two vectors R by combining equations (A.54) and (A.55) to give ° n ¤2 £ ¤2 o 1° 2 ° ~0 °−2 £ 0 0 0 0 ²1jk k1,j r0,k + ²2mn k1,m r0,n + [ri0 ] sin2 θ0 ≥ 0, °n p,1 ° 4 which yields ° °−2 n£ ¤2 £ ¤2 o 1 −2 ° ° 0 0 0 0 ²1jk k1,j r0,k + ²2mn k1,m r0,n . sin2 θ0 ≥ − [ri0 ] °n~0 p,1 ° 4

(A.59)

The solutions are again two inequalities, sin θ0 ≥

i 2ri0

sin θ0 ≤ −

° ° n ¤2 £ ¤2 o ° ~0 °−1 £ 0 0 0 0 ²1jk k1,j r0,k + ²2mn k1,m r0,n , °n p,1 °

i 2ri0

° ° n ¤2 £ ¤2 o ° ~0 °−1 £ 0 0 0 0 r0,n , r0,k + ²2mn k1,m ²1jk k1,j °n p,1 °

(A.60)

(A.61)

~ 0 0 and k~0 1 . We have to combine equations (A.57) and (A.59) which cannot be simultaneously satisfied by the vectors R to give £ ¤−2 n£ ¤2 £ ¤2 o 0 0 0 0 0 0 tan2 θ0 ≥ ²3qr k1,q r0,r ²1jk k1,j r0,k + ²2mn k1,m r0,n , (A.62) which leads to another two inequalities, £ ¤−1 n£ ¤2 £ ¤2 o1/2 0 0 0 0 0 0 tan θ0 ≥ ²3qr k1,q r0,r ²1jk k1,j r0,k + ²2mn k1,m r0,n ,

(A.63)

£ ¤−1 n£ ¤2 £ ¤2 o1/2 0 0 0 0 0 0 tan θ0 ≤ − ²3qr k1,q r0,r ²1jk k1,j r0,k + ²2mn k1,m r0,n .

(A.64)

55

A. Reflection Points on the Surface of a Resonator In the specified range for θ0 , 0 ≤ θ0 ≤ π, the tangent function has the limits µ ¶ 1 0 lim 0 ≤ θ ≤ π − |ε| ⇒ 0 ≤ tan θ0 ≤ ∞, ε→0 2 µ lim

ε→0

1 π + |ε| ≤ θ0 ≤ π 2

(A.65)

¶ ⇒ −∞ ≤ tan θ0 ≤ 0,

(A.66)

0 0 where ε is infinitesimal quantity introduced for limiting purposes. Since there is no guarantee that ²3qr k1,q r0,r > 0 in 0 0 0 0 equations (A.63) and (A.64), one has to consider both cases where ²3qr k1,q r0,r > 0 and ²3qr k1,q r0,r < 0. Therefore, 0 0 for the positive denominator case where ²3qr k1,q r0,r > 0, we have

  

¢ ¡ 0 0 ²3qr k1,q r0,r ≥ 0, limε→0 0 ≤ θ0 ≤ 21 π − |ε| , ¾ ! ½ i2 £ ¤2 1/2 £ ¤−1 h 0 0 0 0 0 0 0 ≤ ∞; tan θ ≥ ²3qr k1,q r0,r ²1jk k1,j r0,k + ²2mn k1,m r0,n

Ã

  0≤   

¡ ¢ 0 0 ²3qr k1,q r0,r ≥ 0, limε→0 21 π + |ε| ≤ θ0 ≤ π , ¾1/2 ! ½h i2 £ £ ¤ ¤ 2 −1 0 0 0 0 0 0 r0,k + ²2mn k1,m r0,n ≤ 0. tan θ0 ≤ − ²3qr k1,q r0,r ²1jk k1,j

(A.67)

Ã

  −∞ ≤

(A.68)

0 0 < 0, we rewrite equations (A.63) and (A.64) in the form r0,r For the negative denominator case where ²3qr k1,q ¯ ¯ 0 0 0 0 ¯ ²3qr k1,q r0,r ≤ 0 ⇒ − ¯²3qr k1,q r0,r ≤ 0,

¯ n£ ¯ ¤2 o1/2 ¤2 £ 0 0 0 0 ¯−1 0 0 r0,n , r0,k + ²2mn k1,m tan θ0 ≥ − ¯²3qr k1,q r0,r ²1jk k1,j

(A.69)

¯ n£ ¯ ¤2 o1/2 ¤2 £ 0 0 0 0 ¯−1 0 0 r0,n . r0,k + ²2mn k1,m tan θ0 ≤ ¯²3qr k1,q r0,r ²1jk k1,j

(A.70)

The tangent function in the domain 0 ≤ θ0 ≤ π has a discontinuity at θ0 = π/2, the inequality (A.69) has the limit 0 ≥ tan θ0 ≥ −∞, and the inequality (A.70) has the limit ∞ ≥ tan θ0 ≥ 0. Therefore, the limits for a negative 0 0 < 0, r0,r denominator case where ²3qr k1,q      0≤

¡ ¢ 0 0 ≤ 0, limε→0 0 ≤ θ0 ≤ 12 π − |ε| , r0,r ²3qr k1,q ½h ¾1/2 ! i2 £ ¯ ¯ ¤ −1 2 0 0 ¯ 0 0 0 0 ≤ ∞; tan θ0 ≤ ¯²3qr k1,q r0,r ²1jk k1,j r0,k + ²2mn k1,m r0,n

Ã

(A.71)

  

¡1 ¢ 0 0 0 ≤ 0, lim r ² k π + |ε| ≤ θ ≤ π , 3qr ε→0 0,r 1,q 2 Ã ¾ ! ½h i 2 ¯ ¯ ¤2 1/2 £ 0 0 0 0 ¯−1 0 0 0 ¯ ≤ 0. ²1jk k1,j r0,k + ²2mn k1,m r0,n   −∞ ≤ tan θ ≥ − ²3qr k1,q r0,r

(A.72)

Comparing equations (A.67), (A.68), (A.71) and (A.72), we see that ¯two of them¯are identical when rewritten in 0 0 0 0 ¯ terms of the later convention where ²3qr k1,q r0,r ≤ 0 is expressed as − ¯²3qr k1,q r0,r ≤ 0. The two tangent function inequality limits are summarized below for bookkeeping purposes:  ¢ ¡ limε→0 0 ≤ θ0 ≤ 21 π − |ε| ,   Ã ¾ ! ½h i2 £ ¯ ¯ ¤2 1/2 (A.73) 0 0 0 0 0 ¯−1 0 0 ¯ ≤ ∞; ²1jk k1,j r0,k + ²2mn k1,m r0,n   0 ≤ tan θ ≤ ²3qr k1,q r0,r

56

A. Reflection Points on the Surface of a Resonator   

¡ ¢ limε→0 12 π + |ε| ≤ θ0 ≤ π , ½h ¾ ! i2 £ ¯ ¯ ¤2 1/2 0 0 0 ¯−1 0 0 0 0 ¯ tan θ ≥ − ²3qr k1,q r0,r ²1jk k1,j r0,k + ²2mn k1,m r0,n ≤ 0.

Ã

  −∞ ≤

The corresponding arguments for inequalities in (A.73) and (A.74) are  ¡ ¢ limε→0 0 ≤ θ0 ≤ 21 π − |ε| ,   Ã ½h ¾ ! i2 £ ¯ ¯ ¤2 1/2 0 0 0 ¯−1 0 0 0 0 ¯ ²1jk k1,j r0,k + ²2mn k1,m r0,n ;   θ = arctan ²3qr k1,q r0,r   

¡ ¢ limε→0 21 π + |ε| ≤ θ0 ≤ π , ½ ¾ ! i2 £ ¯ ¯−1 h ¤2 1/2 0 0 0 0 0 0 0 ¯ ¯ ²1jk k1,j r0,k + ²2mn k1,m r0,n .   θ = arctan − ²3qr k1,q r0,r

(A.74)

(A.75)

Ã

(A.76)

The spherical coordinate representation is incomplete without the azimuthal angle φ0 . We have to solve for the allowed range for the azimuthal angle φ0 by combining equations (A.54) and (A.55). From equation (A.54), we have cos2 φ0 ≥ −

· ° ¸2 £ 0 2 0 ¤−2 1 ° ~0 ° °−1 0 0 n ² k r ri sin θ ° p,1 ° 1jk 1,j 0,k 2

and from equation (A.55), · ° ¸2 £ 0 2 0 ¤−2 1 ° ~0 ° °−1 0 0 sin φ ≥ − ri sin θ . °n p,1 ° ²2jk k1,j r0,k 2 2

0

They are combined to give tan2 φ0 =

£ ¤−2 £ ¤2 sin2 φ0 0 0 0 0 ≥ ²1jk k1,j r0,k ²2mn k1,m r0,n . cos2 φ0

(A.77)

The two inequalities are derived from the last equation, £ ¤−1 0 0 0 0 tan φ0 ≥ ²1jk k1,j r0,k ²2mn k1,m r0,n ,

£ ¤−1 0 0 0 0 tan φ0 ≤ − ²1jk k1,j r0,k ²2mn k1,m r0,n .

In the range of φ0 , 0 ≤ φ0 ≤ 2π, the tangent function has the limits ¶ µ 1 lim 0 ≤ φ0 ≤ π − |ε| ⇒ [0 ≤ tan φ0 ≤ ∞] , ε→0 2 µ

(A.78)

(A.79)

¶ 1 0 π + |ε| ≤ φ ≤ π − ε ⇒ [−∞ ≤ tan φ0 ≤ 0] , 2

(A.80)

¶ µ 3 0 lim π + |ε| ≤ φ ≤ π − |ε| ⇒ [0 ≤ tan φ0 ≤ ∞] , ε→0 2

(A.81)

lim

ε→0

µ lim

ε→0

3 π + |ε| ≤ φ0 < 2π − |ε| 2

¶ ⇒ [−∞ ≤ tan φ0 < 0] ,

(A.82)

where ε is an infinitesimal number used in the limiting process. Because discontinuities occur at φ0 = π/2 and φ0 = 3π/2, the inequalities (A.78) has the limits ³ ´ £ ¤−1 0 0 0 0 0 ≤ tan φ0 ≥ ²1jk k1,j r0,k ²2mn k1,m r0,n ≤ ∞,

57

A. Reflection Points on the Surface of a Resonator ³ ´ £ ¤−1 0 0 0 0 −∞ ≤ tan φ0 ≤ − ²1jk k1,j r0,k ²2mn k1,m r0,n ≤ 0.

and

The ranges for inequalities in (A.79) through (A.82) can now be expressed explicitly as  ¢ ¡ ¢ ¡ limε→0 π + |ε| ≤ φ¶0 ≤ 32 π − |ε| ,  limε→0 0 ≤µφ0 ≤ 12 π − |ε| , h i−1 0 0 0 0 r0,k ²2mn k1,m r0,n ≤ ∞; 0 ≤ tan φ0 ≥ ²1jk k1,j 

(A.83)

 ¡ ¡ ¢ ¢ 0 limε→0 32 π + |ε| ≤ φ¶0 < 2π − |ε| ,  limε→0 21 π + |ε| µ ≤ φ ≤ π −h ε , i−1 0 0 0 0 −∞ ≤ tan φ0 ≤ − ²1jk k1,j r0,k ²2mn k1,m r0,n < 0. 

(A.84)

The solutions for φ0 are  ¢ ¡ ¢ ¡ limε→0 π + |ε| ≤ φ0 ¶ ≤ 32 π − |ε| ,  limε→0 0 ≤ φ0 ≤ 21 π −µ|ε| , h i−1 0 0 0 0 φ0 = arctan ²1jk k1,j ; r0,k ²2mn k1,m r0,n 

(A.85)

 ¡ ¢ ¡ ¢ limε→0 32 π + |ε| ≤ φ¶0 < 2π − |ε| ,  limε→0 21 π + |ε| ≤ φ0 ≤µπ − ε , h i−1 0 0 0 0 φ0 = arctan − ²1jk k1,j r0,k ²2mn k1,m r0,n . 

(A.86)

In order to have νi0 values being real, the allowed range of θ0 is determined by equations (A.75) and (A.76) and the allowed range of φ0 is determined by equations (A.85) and (A.86). Having found valid ranges of θ0 and φ0 in which νi0 is real, the task is now shifted in locating reflection points on the inner hemisphere surface in spherical coordinates. 0 To distinguish one reflection point from the other, the notation νi0 is modified to νi0 → ν1,i in equation (A.51). The first 0 0 and the N th index 1 of ν1,i denotes the first reflection point. In this notation, the second reflection point would be ν2,i 0 0 reflection point, νN,i . Then equation (A.53) is used to rewrite ri,i in terms of spherical coordinates. The Cartesian coordinate variables x0 , y 0 and z 0 in equation (A.51) are expressed as 0 ν1,1

0 ν1,2

≡

x01

°−1 1° ° ° 0 0 = °n~0 p,1 ° ²1jk k1,j r0,k ± 2

≡

y10

°−1 1° ° ° 0 0 = °n~0 p,1 ° ²2jk k1,j r0,k ± 2

0 ν1,3

≡

z10

(·

(·

°−1 1° ° ° 0 0 = °n~0 p,1 ° ²3jk k1,j r0,k ± 2

¸2 ° 1° 2 ° ~0 °−1 0 0 °n p,1 ° ²1jk k1,j r0,k + [ri0 ] sin2 θ10 cos2 φ01 2 ¸2 ° 1° 2 ° ~0 °−1 0 0 °n p,1 ° ²2jk k1,j r0,k + [ri0 ] sin2 θ10 sin2 φ01 2 (·

° 1° ° ~0 °−1 0 0 r0,k °n p,1 ° ²3jk k1,j 2

)1/2 ,

(A.87)

,

(A.88)

)1/2

)1/2

¸2 +

2 [ri0 ]

cos

2

θ10

.

(A.89)

~ 0 1 in equation (A.12), it is not convenient to Although the first reflection point on hemisphere is fully described by R 0 0 ~ ~ use R 1 in its current form. The most effective representation of R 1 is in spherical coordinates. We set θ0 = θ10 and ~ 0 1 on the hemisphere. The subscript on the angular variables θ0 φ0 = φ01 that describe the same reflection point R 1 0 and φ1 denotes first reflection point on the hemisphere surface. In terms of Cartesian variables x0 , y 0 and z 0 , the first reflection point on hemisphere is given by  0 3 = x01 ,  1 → ν1,1 X 0 0 0 0 0 ~ 0 1 (x1 , y1 , z1 ) = 2 → ν1,2 = y10 , R ν1,i eˆi , i= (A.90)  0 i=1 3 → ν1,3 = z10 . The same point on the hemisphere, defined by equation (A.90), can be expressed in terms of a parametric representation

58

A. Reflection Points on the Surface of a Resonator of equation (A.12), ¸ 3 · 3 ° °−1 ´ X ³ X ° ~0 ° 0 0 0 0 0 ˆ ~ ~ r0,i + ξ1,p °k 1 ° k1,i eˆi = Υ1,i eˆi , R 1 ξ1,p ; R 0 , k 1 = i=1

(A.91)

i=1

where ° °−1 ° ° 0 0 Υ1,i = r0,i + ξ1,p °k~0 1 ° k1,i ,

i = 1, 2, 3.

(A.92)

³ ´ ~ 0 1 (x0 , y 0 , z 0 ) and R ~ 0 1 ξ1,p ; R ~ 0 0 , kˆ0 1 , describe the same point on the hemisphere. Therefore, Both representations, R 1 1 1 we have ³ ´ ~ 0 1 ξ1,p ; R ~ 0 0 , kˆ0 1 ~ 0 1 (x01 , y10 , z10 ) = R R

→

3 X

[ν1,i − Υ1,i ] eˆi = 0.

i=1

The components of the last equation are 0 ν1,i − Υ1,i = 0,

i = 1, 2, 3.

(A.93)

0 Substituting expression of ν1,i from equations (A.87), (A.88) and (A.89) into the above equation, we obtain

° 1° ° ~0 °−1 0 0 r0,k ± °n p,1 ° ²1jk k1,j 2 ° 1° ° ~0 °−1 0 0 r0,k ± °n p,1 ° ²2jk k1,j 2

(· )1/2 ¸2 ° 1° ° ~0 °−1 2 0 0 0 0 2 2 0 − Υ1,1 = 0, °n p,1 ° ²1jk k1,j r0,k + [ri ] sin θ1 cos φ1 2 (·

° 1° ° ~0 °−1 0 0 r0,k ± °n p,1 ° ²3jk k1,j 2

° 1° ° ~0 °−1 0 0 r0,k °n p,1 ° ²2jk k1,j 2

(A.94)

)1/2

¸2 +

2 [ri0 ]

sin

2

θ10

sin

2

φ01

− Υ1,2 = 0,

)1/2 (· ¸2 ° 1° 2 ° ~0 °−1 0 0 r0,k + [ri0 ] cos2 θ10 − Υ1,3 = 0, °n p,1 ° ²3jk k1,j 2

(A.95)

(A.96)

where Υ1,i is defined in equation (A.92). To solve for θ10 , equation (A.96) is first rearranged, (· ±

¸2 ° 1° 2 ° ~0 °−1 0 0 °n p,1 ° ²3jk k1,j r0,k + [ri0 ] cos2 θ10 2

)1/2 = Υ1,3 −

° 1° ° ~0 °−1 0 0 r0,k . °n p,1 ° ²3jk k1,j 2

Square both sides and solve for cos2 θ10 , the result is · ¸ ° ° ° ~0 °−1 2 0 0 −2 2 0 0 cos θ1 = [ri ] Υ1,3 − Υ1,3 °n p,1 ° ²3jk k1,j r0,k .

(A.97)

For reasons discussed earlier, θ10 information from the sine function is also needed. Following the earlier procedures, equations (A.94) and (A.95) are combined to yield the relation, (· )1/2 ¸2 ° ° 1° 1° ° ~0 °−1 ° ~0 °−1 2 0 0 0 0 0 0 2 2 0 − Υ1,1 °n p,1 ° ²1jk k1,j r0,k ± °n p,1 ° ²1jk k1,j r0,k + [ri ] sin θ1 cos φ1 2 2 (· )1/2 ¸2 ° ° 1° 1° ° ~0 °−1 ° ~0 °−1 2 0 2 0 0 0 0 0 0 2 + °n p,1 ° ²2mn k1,m r0,n ± − Υ1,2 = 0, °n p,1 ° ²2mn k1,m r0,n + [ri ] sin θ1 sin φ1 2 2

59

A. Reflection Points on the Surface of a Resonator The equation is not easy to solve for sin2 φ01 . Fortunately, there is another way to extract the sine function which requires the knowledge of φ01 . The solution of θ10 is postponed until a solution of φ01 is found. To solve for φ01 , it is desirable to solve for cos2 φ01 and sin2 φ01 from equations (A.94) and (A.95) first. Rearranging equation (A.94), (· )1/2 ¸2 ° °−1 1° 1° ° ~0 °−1 ° ° 2 0 0 0 0 2 2 0 0 0 ± = Υ1,1 − °n~0 p,1 ° ²1jk k1,j r0,k , °n p,1 ° ²1jk k1,j r0,k + [ri ] sin θ1 cos φ1 2 2 and followed by squaring both sides, then cos2 φ01 can be found to be · ¸ ° °−1 −2 ° ° 0 0 cos2 φ01 = [ri0 sin θ10 ] Υ21,1 − Υ1,1 °n~0 p,1 ° ²1jk k1,j . r0,k

(A.98)

Similarly, rearranging equation (A.95), )1/2 (· ¸2 ° °−1 1° 1° ° ~0 °−1 ° ° 2 0 2 0 0 0 0 2 0 0 = Υ1,2 − °n~0 p,1 ° ²2jk k1,j r0,k ± °n p,1 ° ²2jk k1,j r0,k + [ri ] sin θ1 sin φ1 2 2 and squaring both sides, then sin2 φ01 can be found to be · ¸ ° °−1 −2 ° ° 0 0 Υ21,2 − Υ1,2 °n~0 p,1 ° ²2jk k1,j r0,k . sin2 φ01 = [ri0 sin θ10 ]

(A.99)

Finally, tan2 φ01 can be obtained by combining equations (A.99) and (A.98), · tan

2

φ01

=

Υ21,1

¸−1 · ¸ ° ° ° ° ° ~0 °−1 ° ~0 °−1 0 2 0 0 0 − Υ1,1 °n p,1 ° ²1jk k1,j r0,k Υ1,2 − Υ1,2 °n p,1 ° ²2mn k1,m r0,n .

(A.100)

Finally, the azimuthal angle φ01 is found to be  

1/2  ° ° ° ~0 °−1 0 0 − Υ1,2 °n p,1 ° ²2mn k1,m r0,n    φ01 = arctan  ±    ° °  . ° ~0 °−1 2 0 0 Υ1,1 − Υ1,1 °n p,1 ° ²1jk k1,j r0,k 2  Υ1,2

(A.101)

The restriction of φ01 being real imposes the condition ° °−1 ° ° 0 0 Υ21,2 − Υ1,2 °n~0 p,1 ° ²2mn k1,m r0,n ≥ς ° °−1 ° ° 0 r0 Υ21,1 − Υ1,1 °n~0 p,1 ° ²1jk k1,j 0,k or ° °−1 ° °−1 ° ° ° ° 0 0 0 0 Υ21,2 − Υ1,2 °n~0 p,1 ° ²2mn k1,m r0,n ≥ ςΥ21,1 − ςΥ1,1 °n~0 p,1 ° ²1jk k1,j r0,k , where ς ≥ 0. Following equations (A.77) through (A.86), the following results are obtained:  ¢ ¡ ¢ ¡ 3 1 0 |ε| , limε→0 π + |ε| ≤ φ01 ≤ !  2 π − |ε| ,  limε→0 0 ≤ φ1 ≤ 2 πÃ− ¸1/2 · 2 ~0 p,1 k−1 ²2mn k0 r 0 Υ1,2 −Υ1,2 kn 1,m 0,n 0 ; φ1 = arctan   ~0 p,1 k−1 ²1jk k0 r 0 Υ21,1 −Υ1,1 kn 1,j 0,k     

limε→0

¢ ¢ ¡ + |ε| ≤ φ01Ã≤ π − ε , limε→0 32 π + |ε| ≤ φ01 < ! 2π − |ε| , · 2 ¸ 1/2 ~0 p,1 k−1 ²2mn k0 r 0 Υ −Υ1,2 kn 1,m 0,n φ01 = arctan − 1,2 . −1 ~0 p,1 k ²1jk k0 r 0 Υ21,1 −Υ1,1 kn 1,j 0,k

(A.102)

¡1

2π

60

(A.103)

A. Reflection Points on the Surface of a Resonator Having found the solution for φ01 , we can proceed to finalize the task of solving for the polar angle θ10 . Combining the results for cos2 φ01 and sin2 φ01 found in equations (A.98) and (A.99), sin2 θ10 can be found to be · ¸ ° ° ª ° ~0 °−1 © 2 0 0 −2 2 2 0 0 0 0 sin θ1 = [ri ] Υ1,1 + Υ1,2 − °n p,1 ° (A.104) Υ1,1 ²1jk k1,j r0,k + Υ1,2 ²2mn k1,m r0,n . Finally, tan2 θ10 is constructed with equations (A.97) and (A.104), ° °−1 n o ° ° 0 0 0 0 Υ21,1 + Υ21,2 − °n~0 p,1 ° r0,k + Υ1,2 ²2mn k1,m r0,n Υ1,1 ²1jk k1,j tan2 θ10 = . ° ° ° ~0 °−1 2 0 0 Υ1,3 − Υ1,3 °n p,1 ° ²3qr k1,q r0,r

(A.105)

Then θ10 is given by   ° ° n o 1/2  ° ~0 °−1 2 2 0 0 0 0 Υ1,1 ²1jk k1,j r0,k + Υ1,2 ²2mn k1,m r0,n     Υ1,1 + Υ1,2 − °n p,1 ° θ10 = arctan  ±    ° ° .  ° ~0 °−1 2 0 0 Υ1,3 − Υ1,3 °n p,1 ° ²3qr k1,q r0,r Following equations (A.62) through (A.76), we arrive at  ¢ ¡ limε→0 0 ≤ θ10 ≤ 21 π − |ε| ,   Ã· ¸1/2 ! ~0 p,1 k−1 {Υ1,1 ²1jk k0 r 0 +Υ1,2 ²2mn k0 r 0 } Υ21,1 +Υ21,2 −kn 1,j 0,k 1,m 0,n 0 ;   θ1 = arctan ~0 p,1 k−1 ²3qr k0 r 0 Υ21,3 −Υ1,3 kn 1,q 0,r

(A.106)

(A.107)

  

¢ ¡ limε→0 12 π + |ε| ≤ θ10 ≤ π , Ã · ¸1/2 ! 2 2 ~0 p,1 k−1 {Υ1,1 ²1jk k0 r 0 +Υ1,2 ²2mn k0 r 0 } Υ +Υ − n k 1,1 1,2 1,j 0,k 1,m 0,n .  θ10 = arctan −  ~0 p,1 k−1 ²3qr k0 r 0 Υ21,3 −Υ1,3 kn 1,q 0,r

(A.108)

The allowed angular values are all defined now: φ01 by equations (A.102) and (A.103); and θ10 by equations (A.107) and (A.108). The initial reflection point on the inner hemisphere surface can be calculated by the equation:  0 3 = ri0 sin θ10 cos φ01 ,  1 → ν1,1 X 0 0 0 0 0 0 0 0 0 ~ 2 → ν1,2 = ri0 sin θ10 sin φ01 , (A.109) R 1 (ri , θ1 , φ1 ) = ν1,i (ri , θ1 , φ1 ) eˆi , i=  0 i=1 = ri0 cos θ10 . 3 → ν1,3

We still have to determine the maximum wavelength that can fit the hemispherical cavity. It is determined from the distance between two immediate reflection points once they are found. We have to find expression describing the ~ 0 2 . In Figure 3.2, the angle ψ1,2 satisfies the relation second reflection point R ψ1,2 + θ2 + θr = π.

(A.110)

Angles θ2 and θr are equal due to the law of reflection, consequently θ2 = θr = θ1 ,

ψ1,2 = π − 2θi .

(A.111)

It is important not to confuse the angle θi above with that of spherical polar angle θi which was previously denoted ~ 0 i . The θi in equation (A.111) is an angle of incidence, not a with an index i to indicate particular reflection point R polar angle. In order to avoid any further confusion in notation, equation (A.111) is restated with modifications applied to the indexing convention for angle of incidence, θi+1 = θr = θinc ,

ψi,i+1 = π − 2θinc .

61

(A.112)

A. Reflection Points on the Surface of a Resonator ~ 0 1 is The relation that connects angle of incidence to known quantities k~0 1 and R ~ 01 = k~0 1 • R

3 X

° ° ° ° 0 0 k1,i ν1,i = ri0 °k~0 1 ° cos θinc ,

i=1

° ° ° ~0 ° 0 where °R 1 ° = ri , and the index i is not summed over. The incident angle θinc is given by θinc

Ã ! 3 h ° °i−1 X 0 ° ~0 ° 0 0 = arccos ri °k 1 ° k1,i ν1,i .

(A.113)

i=1 0 Substituting the explicit expression of ν1,i : 0 ν1,1 = x01 = ri0 sin θ10 cos φ01 ,

0 ν1,2 = y10 = ri0 sin θ10 sin φ01 ,

into equation (A.113), the incident angle is evaluated as  h 0  sin θ1

θinc = arccos  

kx0 0 1

cos φ01

ky0 0 1

sin φ01

0 ν1,3 = z10 = ri0 cos θ10 ,



i

kz0 0 1

+ + rh i h i h i2 2 2 kx0 0 + ky0 0 + kz0 0 1

1

(A.114)

cos θ10 

, 

(A.115)

1

0 0 0 ~ 0 2 has the form = kz0 0 . The second reflection point R = ky0 0 and k1,3 = kx0 0 , k1,2 where k1,1 1

1

1

3 ¡ 0 ¢ X ¡ 0 ¢ 0 0 0 0 0 ~ 0 2 ν2,1 R , ν2,2 , ν2,3 = ν2,i ν1,1 , ν1,2 , ν1,3 eˆi , i=1

 0 = x02 ,  1 → ν2,1 0 2 → ν2,2 = y20 , i=  0 = z20 , 3 → ν2,3

0 = x01 , ν1,1 0 ν1,2 = y10 , 0 = z10 . ν1,3

(A.116)

~ 0 1 and R ~ 0 2 is The relation that connects two vectors R ~ 01 • R ~ 0 2 = [ri0 ]2 cos ψ1,2 , R

(A.117)

° ° ° ° ° ~0 ° ° ~0 ° 0 where °R 1 ° = °R 2 ° = ri for a rigid hemisphere. Equivalently, this expression can be evaluated using ψ1,2 , given in equation (A.112), as 2

[ri0 ] cos (π − 2θinc ) −

3 X

0 0 ν1,i ν2,i = 0.

(A.118)

i=1

Equation (A.118) serves as one of the two needed relations. The other relation can be found from the cross product of ~ 0 1 and R ~ 02, R ~ 01 × R ~ 02 = R

3 X

0 0 ²ijk ν1,j ν2,k eˆi .

i=1

~ 0 1 and R ~ 0 2 span the plane of incidence whose unit normal is given by equation (A.17), Since R 3 ° °−1 X ° ° 0 0 nˆ0 p,1 = − °n~0 p,1 ° ²ijk k1,j r0,k eˆi , i=1

62

(A.119)

A. Reflection Points on the Surface of a Resonator ~ 0 1 and R ~ 0 2 can be equivalently expressed as the cross product of R 3 °−1 X ° ° 0 0 ~ 0 2 = −Γ1,2 ° ~ 01 × R ²ijk k1,j r0,k eˆi , R °n~0 p,1 °

(A.120)

i=1

where Γ1,2 is a proportionality factor. The factor Γ1,2 can be found simply by noticing ° ° ° ° ° ° ~ 01 × R ~ 0 2 = Γ1,2 nˆ0 p,1 → ° ~ 01 × R ~ 02° R °R ° = Γ1,2 °nˆ0 p,1 ° = Γ1,2 , which leads to

° ° ° ~0 0 2 0 ° ~ Γ1,2 = °R × R 1 2 ° = [ri ] sin (π − 2θinc ) .

(A.121)

Equations (A.119) and (A.120) are combined as 3 · X

¸ ° °−1 ° ° 0 0 0 0 ²ijk ν1,j ν2,k + Γ1,2 °n~0 p,1 ° ²ijk k1,j r0,k eˆi = 0.

(A.122)

i=1

The individual component equation is written ° °−1 ° ° 0 0 0 0 ²ijk ν1,j ν2,k + Γ1,2 °n~0 p,1 ° ²ijk k1,j r0,k = 0,

i = 1, 2, 3.

(A.123)

~ 0 2 in Equations (A.123) and (A.118) together provide the needed relations to specify the second reflection point R 0 0 0 ~ ~ ~ terms of the known quantities, R 0 , k 1 and R 1 . It is convenient to expand equations (A.118) and (A.123) as −[dΓ1,2 /dθinc ]/2

}| { z 2 0 0 0 0 0 0 [ri0 ] cos (π − 2θinc ) −ν1,1 ν2,1 − ν1,2 ν2,2 − ν1,3 ν2,3 = 0,

(A.124)

° °−1 ° ° 0 0 0 0 0 0 ν1,2 ν2,3 − ν1,3 ν2,2 + Γ1,2 °n~0 p,1 ° ²1jk k1,j r0,k = 0,

(A.125)

° °−1 ° ° 0 0 0 0 0 0 + Γ1,2 °n~0 p,1 ° ²2jk k1,j = 0, ν2,3 r0,k − ν1,1 ν2,1 ν1,3

(A.126)

° °−1 ° ° 0 0 0 0 0 0 ν1,1 ν2,2 − ν1,2 ν2,1 + Γ1,2 °n~0 p,1 ° ²3jk k1,j r0,k = 0.

(A.127)

Equations (A.124) and (A.125) are added to yield ° °−1 £ 0 ¤ 0 £ 0 ¤ 1 dΓ1,2 ° ° 0 0 0 0 0 0 ν1,1 ν2,1 + ν1,2 + ν1,3 ν2,2 + ν1,3 − ν1,2 ν2,3 = Γ1,2 °n~0 p,1 ° ²1jk k1,j . r0,k − 2 dθinc

(A.128)

Equations (A.124) and (A.126) are added to give ° °−1 £ 0 ¤ 0 £ 0 ¤ 0 1 dΓ1,2 ° ° 0 0 0 0 0 0 . ν1,1 − ν1,3 ν2,1 + ν1,2 ν2,2 + ν1,3 + ν1,1 ν2,3 = Γ1,2 °n~0 p,1 ° ²2jk k1,j r0,k − 2 dθinc

(A.129)

Similarly, equations (A.124) and (A.127) are combined to give ° °−1 £ 0 ¤ 0 £ 0 ¤ 0 1 dΓ1,2 ° ° 0 0 0 0 0 0 ν1,1 + ν1,2 . ν2,1 + ν1,2 − ν1,1 ν2,2 + ν1,3 ν2,3 = Γ1,2 °n~0 p,1 ° ²3jk k1,j r0,k − 2 dθinc

63

(A.130)

A. Reflection Points on the Surface of a Resonator Define the quantities   0 0  α1 = ν1,2 + ν1,3 ,    0 0 α3 = ν1,1 − ν1,3 ,      α5 = ν 0 + ν 0 , 1,1 1,2

0 0 α2 = ν1,3 − ν1,2 , 0 0 α4 = ν1,3 + ν1,1 , 0 0 α6 = ν1,2 − ν1,1 ,

°−1 ° ° ° 0 0 r0,k − ζ1 = Γ1,2 °n~0 p,1 ° ²1jk k1,j ° °−1 ° ~0 ° 0 0 r0,k − ζ2 = Γ1,2 °n p,1 ° ²2jk k1,j ° °−1 ° ~0 ° 0 0 ζ3 = Γ1,2 °n p,1 ° ²3jk k1,j r0,k −

1 dΓ1,2 2 dθinc , 1 dΓ1,2 2 dθinc ,

(A.131)

1 dΓ1,2 2 dθinc ,

where Γ1,2 is defined in equation (A.121). Equations (A.128), (A.129) and (A.130) form a reduced set 0 0 0 0 ν1,1 ν2,1 + α1 ν2,2 + α2 ν2,3 = ζ1 ,

0 0 0 0 α3 ν2,1 + ν1,2 ν2,2 + α4 ν2,3 = ζ2 ,

0 0 0 0 α5 ν2,1 + α6 ν2,2 + ν1,3 ν2,3 = ζ3 .

In matrix form it reads 

0 ν1,1  α3 α5 |

α1 0 ν1,2 α6 {z

  0    α2 ν2,1 ζ1 0  =  ζ2  , α4  ·  ν2,2 0 0 ζ3 ν1,3 ν2,3 }

(A.132)

f0 M

and its determinant is expressed as ³ ´ £ ¤ ¤ n£ 0 ¤2 £ 0 ¤2 £ 0 ¤2 o 2£ 0 0 0 0 0 0 f0 = ν1,1 + ν1,2 + ν1,3 . det M + ν1,2 + ν1,3 ν1,1 + ν1,2 + ν1,3 = [ri0 ] ν1,1 Three new matrices are then defined here as   0  ν1,1 ζ1 α1 α2 0 f f    α ζ ν α3 , M2 = M1 = 4 2 1,2 0 α5 ζ3 α6 ν1,3

ζ1 ζ2 ζ3

 α2 α4  , 0 ν1,3



0 ν1,1 f  α3 M3 = α5

α1 0 ν1,2 α6

(A.133)

 ζ1 ζ2  . ζ3

0 0 0 are solved with the Cramer’s Rule as and ν2,3 , ν2,2 The variables ν2,1

³ ´ ³ ´ 0 f2 / det M f0 , = det M ν2,2

³ ´ ³ ´ 0 f1 / det M f0 , = det M ν2,1

³ ´ ³ ´ 0 f3 / det M f0 . = det M ν2,3

Explicitly, they are given by ³ n £ 0 ¤ £ 0 ¤ £ 0 ¤2 £ 0 ¤2 o 0 0 0 0 0 0 0 ν`2,1 ≡ ν2,1 = ν1,1 ν1,1 − ν1,2 + ν1,3 ζ1 + ν1,1 ν1,2 − ν1,3 − ν1,2 − ν1,3 ζ2 n o ´£ £ ¤ £ ¤ £ ¤ ¤ 2 2 −1 0 −2 0 0 0 0 0 0 0 0 + ν1,1 ν1,2 + ν1,3 + ν1,2 + ν1,3 ζ3 ν1,1 + ν1,2 + ν1,3 [ri ] , n ¤ £ 0 ¤2 o ¤ £ 0 ¤2 £ 0 ¤2 o £ 0 £ 0 0 0 0 0 ν1,2 ν1,1 + ν1,3 + ν1,1 + ν1,3 ζ1 + ν1,2 ν1,1 − ν1,3 + ν1,2 ζ2 n o ´ ¤−1 0 −2 £ 0 ¤ £ 0 ¤2 £ 0 ¤2 £ 0 0 0 0 0 + ν1,2 + ν1,3 [ri ] , − ν1,1 − ν1,3 ζ3 ν1,1 + ν1,2 ν1,3 − ν1,1

0 0 = ≡ ν2,2 ν`2,2

(A.134)

³n

(A.135)

³n£ n£ ¤ 0 £ 0 ¤2 £ 0 ¤2 o ¤ 0 £ 0 ¤2 £ 0 ¤2 o 0 0 0 0 ν1,1 − ν1,2 ν1,3 − ν1,1 − ν1,2 ζ1 + ν1,1 + ν1,2 ν1,3 + ν1,1 + ν1,2 ζ2 n£ o ´ ¤ 0 £ 0 ¤2 £ 0 ¤−1 0 −2 0 0 0 0 + ν1,2 − ν1,1 ν1,3 + ν1,3 ζ3 ν1,1 + ν1,2 + ν1,3 [ri ] , (A.136)

0 0 ν`2,3 ≡ ν2,3 =

where  0 0 0 ν1,1 + ν1,2 + ν1,3 6= 0,

 0 ν1,1 = x01 (ri0 , θ1 , φ1 ) 0  ν1,2 = y10 (ri0 , θ1 , φ1 )  , 0 ν1,3 = z10 (ri0 , θ1 , φ1 )

64

¡ 0 ¢  0 0 0 ν2,1 ν1,1 , ν1,2 , ν1,3 = x02 ¡ ¢ 0 0 0 0 0   ν2,2 , ν1,2 , ν1,3 . ¡ν1,1 ¢ = y20 0 0 0 0 ν2,3 ν1,1 , ν1,2 , ν1,3 = z2 

(A.137)

A. Reflection Points on the Surface of a Resonator 0 0 In the above set of equations, ν`2,i has been used to indicate that ν2,i is now expressed explicitly in terms of the ¡ 0 ¢ 0 0 Cartesian coordinates ν1,1 , ν1,2 , ν1,3 instead of spherical coordinates corresponding to the second reflection point, (ri0 , θ2 , φ2 ) . The second reflection point inside the hemisphere is then from equation (A.116), 3 ¡ ¢ X 0 ~ 0 2 ν`0 , ν`0 , ν`0 R = ν`2,i eˆi , 2,1 2,2 2,3 i=1 0 where ν`2,i , i = 1, 2, 3 are given in equations (A.134) through (A.136) with restriction given in equation (A.137). In ~ 0 N can be expressed in generic form general, all subsequent reflection points R 3 ¡ 0 ¢ X 0 0 0 ~ 0 N ν`N,1 R , ν`N,2 , ν`N,3 = eˆi ν`N,i i=1 0 through iterative applications of the result ν`2,i , i = 1, 2, 3. This however proves to be very inefficient technique. A 0 ~ ~ 0 2 belongs to a spanning set for the plane better way is to express R N in terms of spherical coordinates. Because R 0 0 ˆ of incidence whose unit normal is n p,1 defined in equation (A.17), the component relations ν`2,i of equations (A.134), (A.135) and (A.136) satisfy the intercept relation given in equation (A.51),

0 ν`2,i

°−1 1° ° ° 0 0 = °n~0 p,1 ° ²ijk k1,j r0,k ± 2

(· )1/2 ¸2 ° £ 0 ¤2 1° ° ~0 °−1 0 0 , °n p,1 ° ²ijk k1,j r0,k + ri,i 2

i = 1, 2, 3,

0 0 0 = ri0 cos θ20 . Here the subscript 2 of angular variables θ20 = ri0 sin θ20 sin φ02 and ri,3 = ri0 sin θ20 cos φ02 , ri,2 where ri,1 0 0 , the and φ2 denote the second reflection point. In terms of the angular variables, using the above expression for ν`2,i 0 0 0 ν`2,1 , ν`2,2 and ν`2,3 are expressed as

(· )1/2 ¸2 ° °−1 1° 1° 2 ° ~0 °−1 ° ° 0 0 0 0 0 ∓ r0,k + [ri0 ] sin2 θ20 cos2 φ02 = °n~0 p,1 ° ²1jk k1,j r0,k − ν`2,1 , °n p,1 ° ²1jk k1,j 2 2 (· ∓

¸2 ° 1° 2 ° ~0 °−1 0 0 r0,k + [ri0 ] sin2 θ20 sin2 φ02 °n p,1 ° ²2jk k1,j 2

)1/2 =

° 1° ° ~0 °−1 0 0 0 r0,k − ν`2,2 , °n p,1 ° ²2jk k1,j 2

(· )1/2 ¸2 ° °−1 1° 1° ° ~0 °−1 ° ° 0 0 0 2 2 0 0 0 0 ∓ = °n~0 p,1 ° ²3jk k1,j r0,k − ν`2,3 . °n p,1 ° ²3jk k1,j r0,k + [ri ] cos θ2 2 2 Square both sides of equation (A.138), cos2 φ02 can be solved as (· ¸2 · ° ¸2 ) ° 1° 1 ° ~0 ° ° ~0 °−1 °−1 2 0 0 0 −2 0 0 0 0 0 cos φ2 = [ri sin θ2 ] . °n p,1 ° ²1jk k1,j r0,k − ν`2,1 − °n p,1 ° ²1jk k1,j r0,k 2 2 Similarly, square both sides of equation (A.139), sin2 φ02 can be solved as (· ¸2 · ° ¸2 ) ° °−1 °−1 1 1 −2 ° ° ° ° 0 0 0 0 0 r0,k − ν`2,2 − r0,k . sin2 φ02 = [ri0 sin θ20 ] °n~0 p,1 ° ²2jk k1,j °n~0 p,1 ° ²2jk k1,j 2 2

65

(A.138)

(A.139)

(A.140)

(A.141)

(A.142)

A. Reflection Points on the Surface of a Resonator The last two equations are divided to give ¸2 · ° ¸2 · ° °−1 °−1 ° ° 1 ° ~0 1 ° ~0 0 0 0 0 0 n ² k r − ν ` − n ² k r 2mn 1,m 0,n 2mn 1,m 0,n 2,2 2 ° p,1 ° 2 ° p,1 ° tan2 φ02 = · ° ¸ · ¸2 . 2 °−1 ° °−1 ° ° 1 ° ~0 1 ° ~0 0 0 0 0 0 ²1jk k1,j r0,k − ν`2,1 − 2 °n p,1 ° ²1jk k1,j r0,k 2 °n p,1 ° The azimuthal angle φ02 is obtained as  · ¸2 · ° ¸2 1/2  ° °−1 °−1   ° ° ° °   1 °n~0 ° ² 0 0 0 0 0    `2,2 − 12 °n~0 p,1 ° ²2mn k1,m r0,n p,1 2mn k1,m r0,n − ν    2   0 .  φ2 = arctan ± ¸2 · ° ¸2 · °  ° ° −1 −1        ° ° 1 ° ~0 1 ° ~0 0 r0 0 0 r0  n ² k − ν ` − n ² k   ° ° ° ° p,1 1jk p,1 1jk 2,1 1,j 0,k 1,j 0,k 2 2 Following the procedures used in equations (A.77) through (A.86), the following results are arrived at  ¡ ¢ ¡ ¢ limε→00 ≤ φ02 ≤ 21 π − |ε| , limε→0 π + |ε| ≤ φ02 ≤ 32 π − |ε| ,     (h i2 h i2 )1/2 −1 0 0 0 1 ~0 ~0 p,1 k−1 ²2mn k0 r 0 n p,1 k ²2mn k1,m r0,n −` ν2,2 − 12 kn k 1,m 0,n 2 ; i2 h i2 h φ0 = arctan   −1  1 ~0 0 0 r0 ~0 p,1 k−1 ²1jk k0 r 0  2 ²1jk k1,j ν2,1 − 12 kn 1,j 0,k 2 kn p,1 k 0,k −`

(A.143)

   

¢ ¡ ¢ ¡1 0 limε→0 32 π + |ε| ≤ φ02 < 2π − |ε| ,  limε→0  2 π + |ε| ≤ φ2 ≤ π − ε , (h i2 h i2 )1/2 −1 0 0 0 1 ~0 ~0 p,1 k−1 ²2mn k0 r 0 n p,1 k ²2mn k1,m r0,n −` ν2,2 − 12 kn k 1,m 0,n 2 . h i2 h i2  φ02 = arctan − −1  1 ~0 0 r0 0 ~0 p,1 k−1 ²1jk k0 r 0  ²1jk k1,j ν2,1 − 12 kn 1,j 0,k 2 kn p,1 k 0,k −`

(A.144)

The solution for φ02 forms a generic structure for any subsequent reflection points on the inner hemisphere surface. The N th azimuthal angle φ0N is found following a prescribed sequential steps φ01 → φ02 → φ03 → · · · → φ0N −1 → φ0N .

(A.145)

By reversing the direction of sequence, φ0N can be expressed in terms of the initial azimuthal angle φ01 , φ0N = φ0N (φ01 ) . The polar angle θ20 of the second reflection point can be found by squaring equation (A.140), which yields (· ¸2 · ° ¸2 ) ° °−1 °−1 1 1 −2 ° ° ° ° 0 0 0 0 0 cos2 θ20 = [ri0 ] r0,k − ν`2,3 − r0,k . (A.146) °n~0 p,1 ° ²3jk k1,j °n~0 p,1 ° ²3jk k1,j 2 2 Add together equations (A.141) and (A.142), sin θ20 can be solved as (· ¸2 · ° ¸2 ° 1 ° ~0 ° 1° ° ~0 °−1 °−1 2 0 0 −2 0 0 0 0 0 r0,k − ν`2,1 − r0,k sin θ2 = [ri ] °n p,1 ° ²1jk k1,j °n p,1 ° ²1jk k1,j 2 2 · ° · ¸ ¸2 ) 2 ° 1 ° ~0 ° 1° °−1 ° ~0 °−1 0 0 0 0 0 + . °n p,1 ° ²2mn k1,m r0,n − ν`2,2 − °n p,1 ° ²2mn k1,m r0,n 2 2

66

(A.147)

A. Reflection Points on the Surface of a Resonator Finally, by dividing equations (A.146) and (A.147), we get (· ¸2 · ° ¸2 ° 1° 1 ° ~0 ° ° ~0 °−1 °−1 2 0 0 0 0 0 0 tan θ2 = °n p,1 ° ²1jk k1,j r0,k − ν`2,1 − °n p,1 ° ²1jk k1,j r0,k 2 2 · ° ¸2 · ° ¸2 ) 1 ° ~0 ° 1 ° ~0 ° °−1 °−1 0 0 0 0 0 + °n p,1 ° ²2mn k1,m r0,n − ν`2,2 − °n p,1 ° ²2mn k1,m r0,n 2 2 (· ¸2 · ° ¸2 )−1 ° 1° 1 ° ~0 ° ° ~0 °−1 °−1 0 0 0 0 0 × . °n p,1 ° ²3qr k1,q r0,r − ν`2,3 − °n p,1 ° ²3qr k1,q r0,r 2 2 The polar angle θ20 is given by Ã (· ¸2 · ° ¸2 ° 1 ° ~0 ° 1° ° ~0 °−1 °−1 0 0 0 0 0 0 θ2 = arctan ± °n p,1 ° ²1jk k1,j r0,k − ν`2,1 − °n p,1 ° ²1jk k1,j r0,k 2 2 · ° ¸2 · ° ¸2 )1/2 °−1 1 1 ° ~0 ° °−1 ° ° 0 0 0 0 0 + r0,n − ν`2,2 − r0,n °n p,1 ° ²2mn k1,m °n~0 p,1 ° ²2mn k1,m 2 2  (· ¸2 · ° ¸2 )−1/2 °−1 °−1 1° 1 ° ~0 ° ° ~0 ° 0 0 0 0 0 . × r0,r − ν`2,3 − r0,r °n p,1 ° ²3qr k1,q °n p,1 ° ²3qr k1,q 2 2 Following the procedures given in equations (A.62) through (A.76), the following results are obtained ¢ ¡  limε→0 0 ≤ θ20 ≤ 21 π − |ε| ,  Ã(  · ¸ · ° ¸2  2 ° °−1 °−1   ° ° ° °  1 1 0 0 0 0 0 0 0 0 ~ ~  θ2 = arctan ²1jk k1,j r0,k − ν`2,1 − 2 °n p,1 ° ²1jk k1,j r0,k  2 °n p,1 °      ¸2 · ° · ° ¸2 )1/2 °−1 °−1 ° ° 1 ° ~0 1 ° ~0 0 0 0 0 0 + 2 °n p,1 ° ²2mn k1,m r0,n − ν`2,2 − 2 °n p,1 ° ²2mn k1,m r0,n       (·  ¸2 · ° ¸2 )−1/2  ° °−1 °−1   ° ° ° 1 ° ~0  0 0 0 0 0 ;  − ν`2,3 − 21 °n~0 p,1 ° ²3qr k1,q r0,r r0,r ²3qr k1,q  × 2 °n p,1 °  ¢ ¡  limε→0 21 π + |ε| ≤ θ20 ≤ π ,  Ã (  · ° ¸2 · ° ¸2  °−1 °−1   ° °  1 ° ~0 1 ° ~0 0 0 0 0 0 0  θ = arctan − n r − ν ` − n r ² k ² k ° ° ° ° p,1 1jk 1,j 0,k p,1 1jk 1,j 0,k  2 2,1 2 2      ¸2 · ° · ° ¸2 )1/2 °−1 °−1 ° ° 1 ° ~0 1 ° ~0 0 0 0 0 0 + 2 °n p,1 ° ²2mn k1,m r0,n − ν`2,2 − 2 °n p,1 ° ²2mn k1,m r0,n       (·  ¸2 · ° ¸2 )−1/2  ° °−1 °−1   ° ° ° ° 1 ~0  0 0 0 0 0 .  × − ν`2,3 − 21 °n~0 p,1 ° ²3qr k1,q r0,r r0,r ²3qr k1,q  2 °n p,1 ° 

(A.148)

(A.149)

The above result of θ20 forms a generic structure for any subsequent reflection points on the inner hemisphere surface. 0 The N th polar angle θN can be obtained by following the sequential steps 0 0 θ10 → θ20 → θ30 → · · · → θN −1 → θN .

(A.150)

0 Equivalently, reversing the direction of sequence, θN can be expressed as a function of the initial polar angle θ10 , 0 0 0 0 θN = θN (θ1 ) . With angular variable φ2 defined in equations (A.143) and (A.144), and θ20 defined in equations

67

A. Reflection Points on the Surface of a Resonator

^z

Intercept

ξ i ^k i θi R 0 ψ 0,i R i

θr

ψi,i+1

ξ i+1^k i+1

Ω ψi,i+1, ψ i+1,i+2

θ i+1

θ r+1 R i+1 ψ i+1,i+2 θ i+2 ξ i+2 ^k i+2

n^ p,1 ^y

R i+2 θ ^ ξ i+3 k i+3 r+2

~ 0 1 and R ~ 0 2 are connected through the angle ψ1,2 . Figure A.3.: The two immediate neighboring reflection points R 0 ~ i and R ~ 0 i+2 are connected through the angle Similarly, the two distant neighbor reflection points R Ωψi,i+1 ,ψi+1,i+2 .

(A.148) and (A.149), the second reflection point on the inner hemisphere surface is given by  0 3 = ri0 sin θ20 cos φ02 ,  1 → ν2,1 X 0 0 0 0 0 0 0 0 ~ 0 2 (ri , θ2 , φ2 ) = 2 → ν2,2 = ri0 sin θ20 sin φ02 , R ν2,i (ri , θ2 , φ2 ) eˆi , i=  0 i=1 = ri0 cos θ20 . 3 → ν2,3

(A.151)

~ 0 1 and R ~ 0 2 are related through ψ1,2 , which is the angle measured As shown in Figure A.3, two reflection points R ~ 0 j , where the index j = 1, 2, · · · , Nmax and Nmax is the last count of reflection before a between the two. Since R repeat in cycle, belongs to a spanning set for a plane of incidence whose unit normal is nˆ0 p,1 given in equation (A.17), all reflections occur on the same plane of incidence. The task of determining the N th subsequent reflection point ~ 0 N is therefore particularly simple. The needed connection formulae between the initial reflection point R ~ 0 1 and the R 0 ~ N th subsequent reflection point R N is found through both scalar and vector cross product relations similar to those ~ 0 2 to R ~ 0 N , recall the set from given in equations (A.117) and (A.119). In order to generalize the previous result for R equations (A.117) and (A.119),  P3 2 0 0 ¾ [ri0 ] cos ψ1,2 − i=1 ν1,i ν2,i = 0, ¸  0 0 ~ ~ · R 1•R2 ° ° −1 P3 (A.152) ⇔ ° ° 0 0 0 0 ~0 ~ 01 × R ~ 02  eˆi = 0, ²ijk k1,j r0,k R i=1 ²ijk ν1,j ν2,k + Γ1,2 °n p,1 ° n o 2 ~ 0 N ∈ S : nˆ0 p,1 , it is true that where Γ1,2 = [ri0 ] sin ψ1,2 and ψ1,2 = π − 2θinc . Because R ~ 01 × R ~ 0N ∝ R ~ 01 × R ~ 02. R

68

A. Reflection Points on the Surface of a Resonator Therefore, we can write ~ 01 × R ~ 0 N = Γ01,N R ~ 01 × R ~ 0 2 = Γ01,N Γ1,2 nˆ0 p,1 = Γ1,N nˆ0 p,1 , R where Γ1,N = Γ01,N Γ1,2 is a proportionality factor. Comparing the results, n o ~ 01 × R ~ 0 N = Γ1,N nˆ0 p,1 R

n o ~ 01 × R ~ 0 2 = Γ1,2 nˆ0 p,1 , R

~ 0N , one obtains a set of relations similar to equation (A.152) for R  P3 2 0 0 ¾ [ri0 ] cos ψ1,N − i=1 ν1,i νN,i = 0, ¸  ~ 01 • R ~ 0N · R ° ° −1 P ⇔ ° ° 3 0 0 0 0 ~0 ~ 01 × R ~ 0N  ²ijk k1,j r0,k eˆi = 0. R i=1 ²ijk ν1,j νN,k + Γ1,N °n p,1 °

(A.153)

0 0 ~ 0 N and ψ1,N is the In the above expression the index N on νN,i and νN,k denotes components corresponding to R ~ 0 1 and R ~ 0 N . The proportionality factor Γ1,N is found to be angle measured between R ° ° ° ° ° ° ~ 01 × R ~ 0 N = Γ1,N nˆ0 p,1 → ° ~ 01 × R ~ 0N ° R °R ° = Γ1,N °nˆ0 p,1 ° = Γ1,N ,

which yields ° ° ° ~0 0 2 ~0 ° Γ1,N = °R 1 × R N ° = [ri ] sin ψ1,N . The angle ψ1,N is contained in Ωψ1,2 ,ψN −1,N as shown in Figure A.3, ΩR~0 1 ,R~0 N ≡ Ωψ1,2 ,ψN −1,N = ψ1,2 + ψ2,3 + · · · + ψN −2,N −1 + ψN −1,N .

(A.154)

For each ψi,i+1 , the sum of inner angles of a triangle gives  ψ1,2 + θ2 + θr = π,     ψ  2,3 + θ3 + θr+1 = π,  .. .     ψN −2,N −1 + θN −1 + θr+N −2 = π,   ψN −1,N + θN + θr+N −1 = π. The law of reflection gives θ2 = θ3 = · · · = θN −1 = θN = θr = θr+1 = · · · = θr+N −2 = θr+N −1 = θinc . Hence, the angles ψi,i+1 are found to be ψ1,2 = ψ2,3 = · · · = ψN −2,N −1 = ψN −1,N = π − 2θinc .

(A.155)

The angle Ωψ1,2 ,ψN −1,N is expressed as Ωψ1,2 ,ψN −1,N = ψ1,2 + ψ2,3 + · · · + ψN −1,N = [π − 2θinc ] + [π − 2θinc ] + · · · + [π − 2θinc ] or ψ1,N ≡ Ωψ1,2 ,ψN −1,N = [N − 1] [π − 2θinc ] .

(A.156)

Hence for Γ1,N , we have the result: 2

Γ1,N = [ri0 ] sin ([N − 1] [π − 2θinc ]) .

69

(A.157)

A. Reflection Points on the Surface of a Resonator 0 ~ 0 N are given as The angular variables θN and φ0N corresponding to N th reflection point R ¡ ¢  0 limε→0 0 ≤ θN ≤ 12 π − |ε| , N ≥ 2,  Ã(  ¸2 · ¸2 · °  °−1 °−1 °   ° ° ° °  1 1 0 0 0 0 0 0 0 0 ~ ~  ²1jk k1,j r0,k − ν`N,1 − 2 °n p,1 ° ²1jk k1,j r0,k θN ≥2 = arctan  2 °n p,1 °      ¸2 · ° · ° ¸2 )1/2 °−1 °−1 ° ° 1 ° ~0 1 ° ~0 0 0 0 0 0 + 2 °n p,1 ° ²2mn k1,m r0,n − ν`N,2 − 2 °n p,1 ° ²2mn k1,m r0,n       (·  ¸2 · ° ¸2 )−1/2  ° °−1 °−1   ° ° °  1 ° ~0 0 0 0 0 0 ;  − 21 °n~0 p,1 ° ²3qr k1,q × ²3qr k1,q r0,r − ν`N,3 r0,r  2 °n p,1 ° 

¢ ¡  0 ≤π , N ≥ 2, limε→0 21 π + |ε| ≤ θN  Ã (  ¸2 · ° ¸2 · °  °−1 °−1   ° °  1 ° ~0 1 ° ~0 0 0 0 0 0 0  − r − ν ` r θ = arctan − n ² k n ² k ° ° ° ° p,1 1jk 1,j 0,k p,1 1jk 1,j 0,k  N,1 N ≥2 2 2      · ° ¸2 )1/2 ¸2 · ° °−1 °−1 ° ° 1 ° ~0 1 ° ~0 0 0 0 0 0 + 2 °n p,1 ° ²2mn k1,m r0,n − ν`N,2 − 2 °n p,1 ° ²2mn k1,m r0,n       (·  ¸2 )−1/2 ¸2 · °  ° °−1 °−1   ° ° ° °  1 ~0 0 0 0 0 0 .  × ²3qr k1,q r0,r − ν`N,3 − 21 °n~0 p,1 ° ²3qr k1,q r0,r  2 °n p,1 °   ¡ ¢ ¡ ¢ limε→0 0 ≤ φ0N ≤ 21 π − |ε| , limε→0 π + |ε| ≤ φ0N ≤ 32 π − |ε| , N ≥ 2,     (h i2 h i2 )1/2 −1 −1 0 0 0 1 ~0 ~0 p,1 k ²2mn k0 r 0 ²2mn k1,m r0,n −` νN,2 − 12 kn 1,m 0,n 2 kn p,1 k ; i2 h h i2 φ0N ≥2 = arctan   −1  1 1 0 0 0 ~0 p,1 k ²1jk k r −` ~0 p,1 k−1 ²1jk k0 r 0  ν − n n k k 1,j 0,k 1,j 0,k N,1 2 2    

limε→0

  

φ0N ≥2

¢ ¡ ¢ + |ε|≤ φ0N ≤ π − ε , limε→0 32 π + |ε| ≤ φ0N < 2π − |ε| , N ≥2, (h i2 h i2 )1/2 −1 −1 0 0 0 0 0 1 ~0 1 ~0 ²2mn k1,m r0,n −` νN,2 − 2 kn p,1 k ²2mn k1,m r0,n 2 kn p,1 k , h i2 h i2 = arctan − −1 1 ~0 0 0 0 ~0 p,1 k−1 ²1jk k0 r 0 νN,1 ²1jk k1,j r0,k −` − 12 kn 1,j 0,k 2 kn p,1 k

(A.158)

(A.159)

(A.160)

¡1

2π

(A.161)

© 0 ª 0 where ν`N,i : i = 1, 2, 3 are given in equations (A.134), (A.135) and (A.136) together with the modification ν`2,i → 0 ν`N,i , ζ1 (Γ1,2 ) → ζ1 (Γ1,N ) , ζ2 (Γ1,2 ) → ζ2 (Γ1,N ) and ζ3 (Γ1,2 ) → ζ3 (Γ1,N ) , where Γ1,N is given in equation 0 0 (A.157). With angular variable θN ≥2 defined in equations (A.158) and (A.159) and φN ≥2 defined in equations (A.160) and (A.161), the N th reflection point on the inner hemisphere surface is given by  0 0 0 0 3  1 → νN,1 = ri sin θN cos φN , X 0 0 0 0 0 0 0 0 0 0 0 ~ 0 N (ri , θN , φN ) = 2 → νN,2 = ri sin θN sin φN , (A.162) i= νN,i (ri , θN , φN ) eˆi , R  0 0 i=1 , 3 → νN,3 = ri0 cos θN ~ 0 1 is given in equation (A.109). where the initial reflection point R

For a sphere, the maximum number of reflections are given by the equation Ns,max ψN −2,N −1 = 2π, ~ 0 N −1 and R ~ 0 N −2 ; the subscript Ns,max where ψN −2,N −1 is the angle between two neighboring reflection points R denotes the maximum number of reflection points for a sphere. The above result is a statement that the sum of all angles is equal to 2π. Application of the rule shown in equation (A.155) for ψN −2,N −1 gives Nmax [π − 2θinc ] = 2π

→ Ns,max =

70

2π , π − 2θinc

(A.163)

A. Reflection Points on the Surface of a Resonator where θinc is given in equation (A.115). In explicit form Ns,max is given by 

 0  sin θ1

  Ns,max = 2π  π − 2 arccos 

−1 h i 0 0 0 0 0 0 kx0 cos φ1 + ky0 sin φ1 + kz0 cos θ1  1 1  . r1h i  h i2 h i2 2 0 0 0 kx0 + ky0 + kz0 1

71

1

1

(A.164)

B. Mapping Between Sets (r, θ, φ) and (r0, θ0, φ0) In this appendix, the original derivations and developments pertaining to the mapping between the sets (r, θ, φ) and (r0 , θ0 , φ0 ) used in this thesis are described in detail. For a sphere, the natural choice for origin is the sphere center from which the spherical coordinates (ri0 , θ0 , φ0 ) are prescribed. For more complicated configurations, as shown in Figure 3.3, the preferred choice for the origin depends upon the problem in hand. For this reason, this section is devoted in deriving a set of transformations between (ri0 , θ0 , φ0 ) and (ri , θ, φ) , where the primed set is defined relative to the sphere center, and the unprimed set is defined ~ and R ~ 0 describing an identical relative to the global configuration origin. In Cartesian coordinates, the two vectors R point on the hemisphere surface are expressed as ~ (ν1 , ν2 , ν3 ) = R

3 X

~ 0 (ν10 , ν20 , ν30 ) = R

νi eˆi ,

i=1

 ν1 = x  ν2 = y  , ν3 = z

νi0 eˆi ,

(B.1)

i=1

 ν10 = x0  ν20 = y 0  , ν30 = z 0



where

3 X





 eˆ1 = x ˆ  eˆ2 = yˆ  . eˆ3 = zˆ

~ and R ~ 0 are the position vectors of the same location relative to the system origin and the hemisphere center, Here R respectively. The two vectors are related through a translation, ~ (ν1 , ν2 , ν3 ) = R ~ T (νT,1 , νT,2 , νT,3 ) + R ~ 0 (ν10 , ν20 , ν30 ) = R

3 X

[νT,i + νi0 ] eˆi ,

(B.2)

i=1

~T = where R written as

P3 i=1

νT,i eˆi is the position of hemisphere center relative to the system origin. Equation (B.2) can be 3 X

[νi − νT,i − νi0 ] eˆi = 0.

(B.3)

i=1

and the component equations are νi − νT,i − νi0 = 0,

i = 1, 2, 3.

(B.4)

It is to be emphasized that in the configuration shown in Figure 3.3, the hemisphere center is only shifted along yˆ by the distance νT,2 = a, therefore νT,i6=2 = 0. Nevertheless, the derivation is done for the case where νT,i 6= 0, i = 1, 2, 3 for general purpose. In explicit forms, they are written as ν1 − νT,1 − ν10 = 0,

ν2 − νT,2 − ν20 = 0,

In spherical coordinates,   ν1 = ri sin θ cos φ = ri Λ1 (θ, φ)  ν2 = ri sin θ sin φ = ri Λ2 (θ, φ)  , ν3 = ri cos θ = ri Λ3 (θ)

ν3 − νT,3 − ν30 = 0.

 ν10 = ri0 sin θ0 cos φ0 = ri0 Λ01 (θ0 , φ0 )  ν20 = ri0 sin θ0 sin φ0 = ri0 Λ02 (θ0 , φ0 )  , ν30 = ri0 cos θ0 = ri0 Λ03 (θ0 )

(B.5)



(B.6)

equation (B.5) is written as ri sin θ cos φ − νT,1 − ri0 sin θ0 cos φ0 = 0,

72

(B.7)

B. Mapping Between Sets (r, θ, φ) and (r0 , θ0 , φ0 ) ri sin θ sin φ − νT,2 − ri0 sin θ0 sin φ0 = 0,

(B.8)

ri cos θ − νT,3 − ri0 cos θ0 = 0,

(B.9)

where the Cartesian variables {νi , νi0 : i = 1, 2, 3} were expressed in terms of the spherical coordinates. The cos φ and sin φ functions are obtained from equations (B.7) and (B.8), cos φ =

νT,1 + ri0 sin θ0 cos φ0 , ri sin θ

sin φ =

νT,2 + ri0 sin θ0 sin φ0 . ri sin θ

The azimuthal angle φ is given by µ φ ≡ φ` (ri0 , θ0 , φ0 , νT,1 , νT,2 ) = arctan

νT,2 + ri0 sin θ0 sin φ0 νT,1 + ri0 sin θ0 cos φ0

¶ ,

(B.10)

where the notation φ` indicates that φ is explicitly expressed in terms of primed variables. Combining equations (B.7) and (B.8), we have ri sin θ [cos φ + sin φ] − νT,1 − νT,2 − ri0 sin θ0 [cos φ0 + sin φ0 ] = 0 which leads to sin θ =

νT,1 + νT,2 + ri0 sin θ0 [cos φ0 + sin φ0 ] . ri [cos φ + sin φ]

From equation (B.9), we have cos θ = ri−1 [νT,3 + ri0 cos θ0 ] . Combining the above results for sin θ and cos θ; and, solving for the argument θ, ¶ µ νT,1 + νT,2 + ri0 sin θ0 [cos φ0 + sin φ0 ] , θ = arctan [cos φ + sin φ] [νT,3 + ri0 cos θ0 ]

(B.11)

where φ is to be substituted in from equation (B.10). For convenience, the above result for θ is rewritten explicitly in terms of primed variables,   ³ ´ 0 0 0 0 0 0 −1 {ν + ν + r sin θ [cos φ + sin φ ]} [νT,3 + ri cos θ ] ~ T = arctan  ³ T,1 ³ T,2 0 i 0 ´´ ³ ³ ´´  . (B.12) θ` ri0 , θ0 , φ0 , R ν ,2 +ri sin θ sin φ0 νT ,2 +ri0 sin θ 0 sin φ0 cos arctan νTT,1 + sin arctan 0 0 0 0 0 0 +r sin θ cos φ νT ,1 +r sin θ cos φ i

i

Here the notation θ` indicates that θ is explicitly expressed in terms of primed variables. The magnitude of a vector describing hemisphere relative to system origin is found from equation (B.2),  0 0 0 ( 3 )1/2  Λ1 (θ , φ ) = sin θ0 cos φ0 , ³ ´ ° ° X °~° 0 ~0 ~ 0 0 2 Λ0 (θ0 , φ0 ) = sin θ0 sin φ0 , ri ri , Λ , RT ≡ °R° = [νT,i + ri Λi ] ,  2 0 0 i=1 Λ3 (θ ) = cos θ0 . In terms of spherical coordinates, the position vector is expressed as ³ ´ ~ ri0 , Λ, ~` Λ ~ 0, R ~T = R

(

3 X

)1/2 [νT,i + ri0 Λ0i ]

2

i=1

3 X i=1

73

` i eˆi , Λ

 ³ ´ ` φ` = sin θ` cos φ, ` ` 1 θ,  Λ    ³ ´ ` φ` = sin θ` sin φ, ` ` 2 θ, Λ  ³ ´   ` ` 3 θ` = cos θ.  Λ

(B.13)

C. Selected Configurations In this appendix, the original derivations and developments in the thesis pertaining to the selected configurations: (1) the hollow spherical shell, (2) the hemisphere-hemisphere and (3) the plate-hemisphere are described in detail.

C.1. Hollow Spherical Shell For the reflection problem in a sphere as shown in Figure 3.4, the natural choice for a system origin is that of the ~ 0 = 0. The N th reflection point inside the sphere is given by equation (A.162) as sphere center, R  0 0 0 3  νs,N,1 = ri0 sin θs,N cos φs,N , X ¡ 0 0 ¢ ¡ ¢ 0 0 0 0 0 0 0 0 0 0 ~ ν = ri sin θs,N sin φs,N , R s,N ri , θs,N , φs,N = νs,N,i ri , θs,N , φs,N eˆi ,  s,N,2 0 0 i=1 = ri0 cos θs,N , νs,N,3 where the label s have been attached to denote the sphere. Keeping in mind the obvious index changes, the angular 0 variable θs,N is defined in equations (A.158) and (A.159), and φ0s,N , in equations (A.160) and (A.161). Staying with ~ 0 s,N is rewritten as the notation of equation (B.13), R ¡ 0 ¢  0 0 0 0 3  Λs,N,1 ¡θs,N , φs,N ¢ = sin θs,N cos φs,N , ³ ´ X 0 ~ s,N ri0 , Λ ~ 0s,N = ri0 Λ0 θ 0 , φ0 = sin θs,N sin φ0s,N , Λ0s,N,i eˆi , (C.1) R  s,N,2 0 s,N ¡ 0s,N ¢ 0 i=1 Λs,N,3 θs,N = cos θs,N , ¤2 P3 £ where the relations νT,s,i = 0, and i=1 Λ0s,N,i = 1 are used. The maximum number of internal reflections for a spherical cavity before a repeat in cycle is given by equation (A.163), Ns,max =

2π , π − 2θinc

where θinc is given ° °in equation (A.115). °~ ° The distance °L ° between two immediate neighboring reflection points on a sphere is ° ° ° ³ ´ ³ ´° °~ ° ° ~ ~ 0s,2 − R ~ s,1 ri0 , Λ ~ 0s,1 ° °L° = °Rs,2 ri0 , Λ °. It should be noted that

° ° ° ° °~ °~ ° ~ s,1 ° ~ − R °Rs,2 − R ° = °R s,j s,j−1 ° ,

j = 3, · · · , Ns,max .

(C.2)

(C.3)

~ s,1 and R ~ s,2 are used is for the purpose of convenience. The only reason that R ~ 0 s,1 , the incident wave is first decomposed into To compute the resultant wave vector, k~0 inner , acting at the point R components parallel and perpendicular to the inner surface normal vector, −Rˆ0 s,1 , h i h i k~0 i,+ ≡ k~0 i = k~0 ik + k~0 i⊥ = k~0 i • Rˆ0 s,1 Rˆ0 s,1 + Rˆ0 s,1 × k~0 i × Rˆ0 s,1 , ~ 0 s,1 where the subscript (+) of k~0 i,+ denotes the particular contribution where the incident wave k~0 i is approaching R ~ 0 s,0 . From equation (A.14) of Appendix A, the corresponding reflected wave vector can be expressed in terms from R

74

C. Selected Configurations of the incident wave as h nh i h i oi k~0 r,+ ≡ k~0 r = Rˆ0 s,1 × k~0 i • Rˆ0 s,1 Rˆ0 s,1 + Rˆ0 s,1 × k~0 i × Rˆ0 s,1 × Rˆ0 s,1 nh i h i o − Rˆ0 s,1 • k~0 i • Rˆ0 s,1 Rˆ0 s,1 + Rˆ0 s,1 × k~0 i × Rˆ0 s,1 Rˆ0 s,1 h h i nh i oi = Rˆ0 s,1 × k~0 i • Rˆ0 s,1 Rˆ0 s,1 + Rˆ0 s,1 × Rˆ0 s,1 × k~0 i × Rˆ0 s,1 × Rˆ0 s,1 h h i nh i oi − Rˆ0 s,1 • k~0 i • Rˆ0 s,1 Rˆ0 s,1 + Rˆ0 s,1 • Rˆ0 s,1 × k~0 i × Rˆ0 s,1 Rˆ0 s,1 , i nh i o h where αr,⊥ = αr,k = 1 and nˆ0 → −Rˆ0 s,1 . Because Rˆ0 s,1 ⊥ Rˆ0 s,1 × k~0 i × Rˆ0 s,1 and Rˆ0 s,1 k k~0 i • Rˆ0 s,1 Rˆ0 s,1 , the above expression is simplified to h nh i oi h i k~0 r,+ = Rˆ0 s,1 × Rˆ0 s,1 × k~0 i × Rˆ0 s,1 × Rˆ0 s,1 − k~0 i • Rˆ0 s,1 Rˆ0 s,1 . ~ 0 s,1 due to k~0 i,+ at location R ~ 0 s,0 is given by The changes in resultant wave vector k~0 inner at the point R ³ ´ ~ 0 s,1 , R ~ 0 s,0 = k~0 r,+ − k~0 i,+ 4k~0 inner,+ ; R h nh i oi = Rˆ0 s,1 × Rˆ0 s,1 × k~0 i × Rˆ0 s,1 × Rˆ0 s,1 h i h i − Rˆ0 s,1 × k~0 i × Rˆ0 s,1 − 2 k~0 i • Rˆ0 s,1 Rˆ0 s,1 h i = −2 k~0 i • Rˆ0 s,1 Rˆ0 s,1 , h nh i oi h i where Rˆ0 s,1 × Rˆ0 s,1 × k~0 i × Rˆ0 s,1 × Rˆ0 s,1 = Rˆ0 s,1 × k~0 i × Rˆ0 s,1 .

~ 0 s,1 from R ~ 0 s,2 , one has For the incident wave traveling in the opposite direction, i.e., approaching R k~0 i,− ≡ −k~0 r = −k~0 r,+ ,

k~0 r,− ≡ −k~0 i = −k~0 i,+ ,

~ 0 s,1 where the subscript (−) on k~0 i,− denotes the particular contribution where the incident wave k~0 i is approaching R ~ 0 s,2 . In this case, the changes in the resultant wave vector k~0 inner at the point R ~ 0 s,1 due to k~0 i,− at the location from R 0 ~ R s,0 is given by ³ ´ ³ ´ ~ 0 s,1 , R ~ 0 s,0 = k~0 r,− − k~0 i,− = −k~0 i,+ + k~0 r,+ = 4k~0 inner,+ ; R ~ 0 s,1 , R ~ 0 s,0 . 4k~0 inner,− ; R ~ 0 s,1 due to incident wave approaching R ~ 0 s,1 from R ~ 0 s,0 and the The resultant wave vector k~0 inner acting at the point R ~ 0 s,1 from R ~ 0 s,2 is therefore other incident wave approaching R ³ ´ ³ ´ ³ ´ ~ 0 s,1 , R ~ 0 s,0 ≡ 4k~0 inner,+ ; R ~ 0 s,1 , R ~ 0 s,0 + 4k~0 inner,− ; R ~ 0 s,1 , R ~ 0 s,0 4k~0 inner ; R h i = −4 k~0 i,b • Rˆ0 s,1 Rˆ0 s,1 , where the subscript b of k~0 i,b denotes the wave vector for ambient fields inside cavity. ° ° ° ° The wave number °k~0 i,b ° that can be fit in the bounded space of a resonator is restricted by the boundary condition ° ° ° °−1 ° ³ ´ ³ ´°−1 ° ~0 ° °~ ° °~ ° 0 ~0 0 ~0 ~ . °k i,b ° = nπ °L ° = nπ °R s,2 ri , Λs,2 − Rs,1 ri , Λs,1 °

75

C. Selected Configurations The scalar product of k~0 i,b and Rˆ0 s,1 is ° ° ° ° k~0 i,b • Rˆ0 s,1 = °k~0 i,b ° cos θinc , ~ s,1 is equal to the angle of incidence θinc , as shown in equation where the angle between the two vectors k~0 i,b and R (A.115). The momentum transfer is proportional to ½ ³ ´ 4nπ cos θinc 0 ≤ θinc < π/2, 0 0 0 0 ~ ~ ~ ˆ ³ ´ ³ ´° R s,1 , 4k inner ; R s,1 , R s,0 = − ° (C.4) °~ ° n = 1, 2, · · · . 0 ~0 ~0 ~ °Rs,2 ri0 , Λ s,2 − Rs,1 ri , Λs,1 ° ~ 0 s,1 + aRˆ0 s,1 on the outer spherical surface, where a is Similarly, the resultant wave vector k~0 outer acting at point R the sphere thickness parameter, is given by i ³ ´ h ~ 0 s,1 + aRˆ0 s,1 = −4 k~0 i,f • Rˆ0 s,1 Rˆ0 s,1 , 4k~0 outer ; R where the subscript f of k~0 i,f denotes the wave vector of the ambient fields in free space, and the factor ° 4 is ° due to ° ~0 ° the fact there are two incidence wave vectors from opposite directions. The free space wave number °k i,f ° has no quantization restriction due to the boundary. And, the scalar product of k~0 i,f and Rˆ0 s,1 is ° ° ° ° ° ° ° ° k~0 i,f • Rˆ0 s,1 = °k~0 i,f ° cos (π − θinc ) = − °k~0 i,f ° cos θinc Rˆ0 s,1 . The momentum transfer is proportional to ° ° ³ ´ ° ~ 0 s,1 + aRˆ0 s,1 = 4 ° 4k~0 outer ; R °k~0 i,f ° cos θinc Rˆ0 s,1 ,

½

0 ≤ θinc < π/2, n = 1, 2, · · · .

(C.5)

C.2. Hemisphere-Hemisphere ~ = 0. The N th For the hemisphere-hemisphere configuration, the preferred choice for a system origin is that of R internal reflection point is given by equation (B.13), ~ h,N R

³

~` h,N , Λ ~ 0h,N , R ~ T,h ri0 , Λ

´

)1/2 3 ( 3 X X£ ¤2 0 0 ` h,N,i eˆi , = νT,h,i + ri Λh,N,i Λ i=1

(C.6)

i=1

where the label h denotes hemisphere; and  ³ ´ ` h,N,1 θ`h,N , φ`h,N = sin θ`h,N cos φ`h,N ,  Λ    ³ ´ ` h,N,2 θ`h,N , φ`h,N = sin θ`h,N sin φ`h,N , Λ  ³ ´   ` h,N,3 θ`h,N = cos θ`h,N .  Λ ³ ´ The definitions for Λ0h,N,i , i = 1, 2, 3 are identical in form. The angular variables θ`h,N , φ`h,N are given in equations 0 (B.10) and (B.12), where the obvious notational changes are understood. The implicit angular variable θs,N is defined 0 in equations (A.158) and (A.159); and φs,N , defined in equations (A.160) and (A.161).

We have to determine the maximum number of internal reflections of the wave in the hemisphere cavity before its ~ 0 0 , ξi kˆ0 i and R ~ 0 h,i ≡ R ~ 0 i shown in Figure A.3 satisfy the relation escape. Three vectors R ~ 0 h,i=1 − R ~ 0 0 = ξi=1 kˆ0 i=1 , R

76

C. Selected Configurations ~ 0 h,i=1 denotes the hemisphere. The path length squared is given by where the notation h of R ° °2 ° ° °2 ° ° ° ° ° ° ~0 ° ° ~0 ° ~ 0 °2 ° ~ 0 °2 0 2 0 ° ~0 ° ~0 ~ 00° ~0 °R h,i=1 − R ° = °R h,1 ° + °R 0 ° − 2R h,1 • R 0 = [ri ] + °R 0 ° − 2ri °R 0 ° cos ψ0,1 . ° °2 ° °2 ° ~0 ° 2 ˆ0 ° ~0 ° Since °R h,i=1 − R 0 ° = °ξi=1 k i=1 ° = ξ1 , the angle ψ0,1 is found from the last equation to be ψ0,1

µ ½ ° ° ° ° h ° °i−1 ¾¶ −1 1 ° ~0 ° 0 ° ~0 ° 0 −1 ° ~ 0 ° = arccos r °R 0 ° + [ri ] °R 0 ° − ri0 °R ξ12 , 0° 2 i

~ 0 h,1 and R ~ 0 h,2 is where ξ1 = ξ1,p is given in equation (A.11). The angle ψ1,2 measured between the two vectors R Ã 3 ! ´ ³ X 0 −2 ~ 0 0 0 0 ~ Λ , ψ1,2 = arccos [r ] R h,1 • R h,2 = arccos Λ i

h,1,i

h,2,i

i=1

~ 0 h,1 and R ~ 0 h,2 have been explicitly written for N = 1, 2 in equation (C.1), or equivalently, where R ψ1,2 = π − 2θinc from equation (A.112). Because a hemisphere is just a sphere halve, it is convenient to define a quantity Zh,max =

1 [π − ψ0,1 ] , ψ1,2

or in explicit expression, Zh,max

· µ ½ ° ° ¾¶¸ ° ° h ° °i−1 −1 1 1 0 ° ~0 ° 0 −1 ° ~ 0 ° 0 ° ~0 ° 2 = π − arccos r °R 0 ° + [ri ] °R 0 ° − ri °R 0 ° ξ1,p , π − 2θinc 2 i

(C.7)

where ξ1,p is given in equation (A.11) and θinc is given in equation (A.115). The maximum number of internal reflections is then simply Nh,max = [Zh,max ]G ,

(C.8)

where the notation [Zh,max ]G is the greatest integer of Zh,max and it is defined in equation (C.7). ° ° °~ ° The distance °L ° between the two immediate neighboring reflection points of the hemisphere is ° ° ° ´° ´ ³ ³ °~ ° ° ~ ~` h,1 , Λ ~ 0h,1 , R ~ T,h ° ~ T,h − R ~ h,1 ri0 , Λ ~` h,2 , Λ ~ 0h,2 , R °L° = °Rh,2 ri0 , Λ °. It should be noted that

° ° ° ° °~ °~ ° ~ h,1 ° ~ − R °Rh,2 − R ° = °R h,j h,j−1 ° ,

j = 3, · · · , Nh,max

(C.9)

(C.10)

~ h,1 and R ~ h,2 are used is for the purpose of convenience. and the only reason that R The change in wave vector direction upon reflection at the point Rˆ0 h,1 inside the resonator, or at the location 0 ~ R h,1 + aRˆ0 h,1 outside of the hemisphere, is given by results found for the sphere case, equations (C.4) and (C.5), with obvious subscript changes, ½ ³ ´ 4nπ cos θinc 0 ≤ θinc < π/2, 0 0 0 0 ~ ~ ~ ˆ ° ³ ´ ³ ´° 4k inner ; R h,1 , R h,0 = − ° (C.11) ° R h,1 , n = 1, 2, · · · 0 0 0 0 ~ ~ ~ ~ °Rh,2 ri , Λh,2 − Rh,1 ri , Λh,1 °

77

C. Selected Configurations and ½

° ° ´ ³ ° ~ 0 h,1 + aRˆ0 h,1 = 4 ° 4k~0 outer ; R °k~0 i,f ° cos θinc Rˆ0 h,1 ,

0 ≤ θinc < π/2, n = 1, 2, · · · .

(C.12)

³ ´ ³ ´ ~ 0 h,1 , R ~ 0 h,0 and 4k~0 outer ; R ~ 0 h,1 + aRˆ0 h,1 have been derived based upon the The above results on 4k~0 inner ; R fact that there are multiple internal reflections. For a sphere, the multiple internal reflections are inherent. However, for a hemisphere, it is not necessarily true that all incoming waves would result in multiple internal reflections. The criteria for multiple internal reflections are to be established. For a given initial incoming wave vector kˆ0 1 , there can ~ 0 0 . Shown be multiple or single internal reflections depending upon the location of point of entry into the cavity, R in Figure 3.5 are the two such reflections dynamics where the dashed vectors represent the single reflection case and the non-dashed vectors represent the multiple reflection case. Because all the processes occur in the same plane of ~ 0 f = −λ0 R ~ 0 0 with λ0 ≥ 0 has to be true. Therefore, we will have incidence, the relationship R ~ 01 = R ~ 0 0 + ξp kˆ0 1 , R

~ 0 f ≡ −λ0 R ~ 00 = R ~ 0 1 + k~0 2 . R

(C.13)

~ 0 1 from the last two equations, we find After eliminating R h i ~ 0 0 = − [1 + λ0 ]−1 ξp kˆ0 1 + k~0 2 . R

(C.14)

The direction of the reflected wave vector k~0 2 cannot be arbitrary because it has to obey the reflection law. The relationship between the directions of an incident and the associated reflection wave is shown in equation (A.14). ~ 0 1 /r0 , k~0 r → k~0 2 and k~0 i → k~0 1 , the reflected wave vector k~0 2 can be written in the form Designating nˆ0 = −R i i h −2 ~ 0 ~ 0 1 − αr,k [ri0 ]−2 R ~ 0 1 • k~0 1 R ~ 01 k~0 2 ∝ αr,⊥ [ri0 ] R 1 × k~0 1 × R or, introducing a proportionality factor λ2 , it becomes h i −2 ~ 0 ~ 0 1 • k~0 1 R ~ 01. ~ 0 1 − λ2 αr,k [r0 ]−2 R k~0 2 = λ2 αr,⊥ [r0 ] R 1 × k~0 1 × R i

i

~ 0 f , or λ0 , in terms of R ~ 0 0 . Substituting the expression for R ~ 0 1 from equation (C.13), we arrive at The goal is to relate R ° ° n h i h io −2 −2 ° ~0 ~ 0 0 × k~0 1 × kˆ0 1 − αr,k R ~ 0 0 • k~0 1 kˆ0 1 + ° k~0 2 = −ξp2 λ2 αr,k [ri0 ] k~0 1 + ξp λ2 [ri0 ] αr,⊥ R °k~0 1 ° R 0 n h i o −2 ~ 0 0 × k~0 1 × R ~ 0 0 − αr,k R ~ 0 0 • k~0 1 R ~ 00 . + λ2 [ri0 ] αr,⊥ R (C.15) Finally, equations (C.14) and (C.15) are combined to yield ° ° h i o n h i ° ~0 −1 0 2 ˆ0 ~ 0 0 • k~0 1 kˆ0 1 + ° ~ 0 0 × k~0 1 × kˆ0 1 − αr,k R + λ [r ] k ξp2 αr,k k~0 1 − ξp αr,⊥ R °k~0 1 ° R 0 1 i 2 h i 2 ~ 0 0 • k~0 1 R ~ 0 0 − αr,⊥ R ~ 0 0 × k~0 1 × R ~ 0 0 − λ−1 [ri0 ] [1 + λ0 ] R ~ 0 0 = 0. +αr,k R 2

(C.16)

h i nh i h i o ~×B ~ ×C ~ = P3 ~•C ~ Bl − B ~ •C ~ Al eˆl , the cross product are evaluated as Utilizing the formula A A l=1 ¾ ½° ° h  h ° °−1 h ° ° i i i −1 P3 ° ° ° ° ° °  0 0 0 0 0 0 0 0 0 0 0 0 0 ~ ˆ ~ ~ ~ ~ ~ ~ ~ ~ ~  R 0 × k 1 × k 1 ≡ °k 1 ° R 0 × k 1 × k 1 = l=1 °k 1 ° R 0 • k 1 k1,l − °k 1 ° r0,l eˆl ,    ½° ° ¾ h i h i P3 ° ~ 0 °2 0 0 0 0 0 0 0 ~ ~ ~ ~ ~ R 0 × k 1 × R 0 = l=1 °R 0 ° k1,l − k 1 • R 0 r0,l eˆl ,     P3  0 ~ 0 0 = P3 r0 eˆl . eˆl , R k~0 1 = l=1 k1,l l=1 0,l

78

C. Selected Configurations Rewriting equation (C.16) then 3 ½ X

0 ξp2 αr,k k1,l

¸ · µ · ° °−1 ° ° ° °−1 ° ~0 ° ° ~0 ° 0 ° 0 0 0 0 ~ ~ ~ 0 0 • k~0 1 ° + ξp αr,k R 0 • k 1 °k 1 ° k1,l + °k 1 ° r0,l − αr,⊥ R °k~0 1 ° k1,l

l=1

¶ ° ° ° °−1 i ° ~0 ° 0 −1 0 2 ° ~0 ° 0 0 ~ 0 0 • k~0 1 r0,l − °k 1 ° r0,l − λ2 [ri ] °k 1 ° k1,l + αr,k R − αr,⊥ ·° ° ¸ ¾ ° ~ 0 °2 0 −1 0 2 0 0 0 0 ~ ~ × °R 0 ° k1,l − k 1 • R 0 r0,l − λ2 [ri ] [1 + λ0 ] r0,l eˆl = 0,

which leads to the component equations µ · ¸ · ° °−1 ° ° ° °−1 ° ° ° 0 ° 0 0 0 ~ 0 0 • k~0 1 ° ~ 0 0 • k~0 1 ° ξp2 αr,k k1,l + ξp αr,k R + °k~0 1 ° r0,l − αr,⊥ R °k~0 1 ° k1,l °k~0 1 ° k1,l ¶ ° °−1 ° ° i 2° ° ° ° 0 0 0 ~ 0 0 • k~0 1 r0,l + αr,k R − αr,⊥ − °k~0 1 ° r0,l − λ−1 [ri0 ] °k~0 1 ° k1,l 2

·° ° ¸ ° ~ 0 °2 0 −1 0 2 0 0 •R 0 r0 ~ ~ − k × °R k ° 0 1 0 0,l − λ2 [ri ] [1 + λ0 ] r0,l = 0, 1,l where l = 1, 2, 3. For an isotropic system, αr,⊥ = αr,k = αr , the last equation reduces to · ¸ ° ° ° °−1 ° ° 0 0 0 2 ° ~0 ° 0 + ξp 2αr °k~0 1 ° r0,l ] − λ−1 [r k ξp2 αr k1,l k ° ° 1 i 1,l 2 ° °2 ° ~0 ° 0 −1 0 2 0 0 ~ 0 0 • k~0 1 r0,l − αr °R +2αr R 0 ° k1,l − λ2 [ri ] [1 + λ0 ] r0,l = 0, where l = 1, 2, 3. Because there are three such relations of the above, all three component equations are added to yield " ξp2 " +

3 X

+ ξp

#−1 0 k1,l

3 X l=1

#−1 0 k1,l

¸ 3 · ° ° °−1 ° X ° ° 0 0 2 ° ~0 ° 0 2 °k~0 1 ° r0,l ] − λ−1 [r k k i ° 1° 1,l 2 l=1

3 ½ X

¾ ° °2 ° ~0 ° 0 −1 0 2 0 0 0 0 ~ ~ 2R 0 • k 1 r0,l − °R 0 ° k1,l − λ2 [ri ] [1 + λ0 ] r0,l = 0,

(C.17)

l=1

l=1

i−1 hP 3 0 k and αr = 1 have been chosen for where the both sides of the above equation have been multiplied by l=1 1,l simplicity. Since ξp is just a positive root of ξ1 of equation (A.9), it satisfies the equation 1

z }| { 3 ° 3 3 °−2 £ ¤ ° °−1 X X X £ 0 ¤2 2 2 ° ~0 ° ° ° 0 0 0 ξ12 k1,l +2ξ1 °k~0 1 ° r0,l k1,l + r0,l − [ri0 ] = 0, °k 1 ° l=1

l=1

l=1

where the index i have been changed to l in equation (A.9). The reflection coefficient αr have been set to a unity in equation (C.17) for the very reason that αr = 1 had been already imposed in the above relation, the equation (A.9). Because ξ1 ≡ ξp , and the fact that coefficient of ξp2 = ξ12 = 1 in equations (A.9) and (C.17), the two polynomials must be identical. Therefore, subtracting equation (A.9) from equation (C.17), we obtain "  #−1 3 · ¸ 3 3  X ° ° °−1 ° °−1 X  X ° 2 ° ° 0 ° ~0 ° 0 0 2 ° ~0 ° 0 0 0 ξp k1,l 2 °k~0 1 ° r0,l − λ−1 [r ] r k + [ri0 ] k k − 2 k ° ° ° ° 1 1 i 0,l 1,l 1,l 2   l=1

l=1

" +

3 X l=1

#−1 0 k1,l

l=1

3 ½ X

¾ ° °2 £ 0 ¤2 ° ~0 ° 0 −1 0 2 0 0 ~ 0 0 • k~0 1 r0,l 2R − °R = 0. 0 ° k1,l − λ2 [ri ] [1 + λ0 ] r0,l − r0,l

l=1

79

C. Selected Configurations Because ξp is a particular value for the root of ξ1 , for the case where ξp 6= 0, the above equation is satisfied only when the coefficients of the different powers of ξp vanish independently. This is another way of stating that each coefficients of the different powers of ξp in equations (A.9) and (C.17) must be proportional to each other. For the situation here, they must be identical due to the fact that coefficients of ξp2 = ξ12 = 1. Hence, we have the conditions:  ¸ ° °−1 ° °−1 P hP i−1 P · ° °  ° ~0 ° 0 ° ~0 ° 3 3 3 −1 0 2 ° ~0 ° 0 0 0 0  2 k r k − λ [r ] k k − 2  ° ° ° ° °k 1 ° 1 1 i 2 0,l 1,l l=1 1,l l=1 l=1 r0,l k1,l = 0, (C.18) ° °2 i−1 P ½ hP h i2 ¾  2 3 3 −1 0 2 0 0 0 0 ~ 0 0 • k~0 1 r0 − ° ~ 00°  k 2 R R k − λ [r ] [1 + λ ] r − r = 0.  [ri0 ] + ° ° 0 0,l i 2 0,l 1,l 0,l l=1 l=1 1,l From the first expression of equation (C.18), we find " λ−1 2

=

−2 2 [ri0 ]

3 X

#−1 0 k1,l

l=1

# " 3 3 ° °2 X X ° ~0 ° 0 0 0 . k1,m r0,l °k 1 ° − k1,l

(C.19)

m=1

l=1

Solving for λ0 from the second expression of equation (C.18), we have ¾ ½ ° °2 h i2 P3 ° ~0 ° 0 −1 0 0 0 −2 0 0 0 −2 0 −2 0 0 ~ ~ − °R 0 ° k1,l [ri ] − λ2 r0,l − r0,l [ri ] l=1 k1,l + 2R 0 • k 1 r0,l [ri ] λ0 = P3 0 λ−1 2 l=1 r0,l or, substituting the expression of λ−1 2 given in equation (C.19), we have the result: −1 # 3 3 " # 3 3 3 n  ° °  X X X X X 2 £ 0 ¤2 1 2 ° ~0 ° 0 0 0 0 0 0 λ0 = k1,n k k r r − k k1,l [ri0 ] − r0,l ° 1° 1,m 0,l 0,j 1,l   2 n=1 m=1 j=1 l=1 l=1  #−1 3 " # " 3 3 ° °2 ° °2  X X X ° ~0 ° 0 ° ~0 ° 0 0 0 0 0 0 ~ 0 0 • k~0 1 r0,l +2R − °R r . k k k k − 2r − k ° ° ° 0 1 0,i 1,l 1,m 1,l 0,l 1,i  "

i=1

l=1

(C.20)

m=1

~ 0 f through the relation R ~ 0 f ≡ −λ0 R ~ 0 0 . Therefore, the Referring back to Figure 3.5, the term λ0 is connected to R criterion for waves to have multiple or single internal reflection is contained in the controlled quantity λ0 . The vector ~ 0 0 is a quantity that must be specified initially. Because λ0 is a positive definite scalar, we can rewrite it as R ( )−1/2 ° °° ° ° ° 3 ° ~ 0 ° ° ~ 0 °−1 ° ~ 0 ° X £ 0 ¤2 λ0 = °R f ° °R 0 ° = °R f ° r0,i . i=1

° ° ° ~0 ° Substituting the above definition of λ0 into equation (C.20), the quantity °R f ° can be solved as −1 # 3 3 " # " 3 3 n 3  X X X ° °2 ° ° 1° ° X X £ 0 ¤2 2 ° ~0 ° ° ~0 ° ° ~0 ° 0 0 0 0 0 0 k1,n k1,m r0,l r0,j k1,l [ri0 ] − r0,l °k 1 ° − k1,l °R f ° = °R 0 °   2 n=1 m=1 j=1 l=1 l=1  " 3 #−1 3 " # 3 ° °2 ° °2  X X X ° ~0 ° 0 ° ~0 ° 0 0 0 0 0 0 ~ 0 0 • k~0 1 r0,l +2R − °R k1,l k1,m r0,i . (C.21) °k 1 ° − k1,i 0 ° k1,l − 2r0,l  i=1

l=1

m=1

Because the hemisphere opening has a radius ri0 , the following criteria are concluded:  ° ° ° ~0 ° 0  R Single − Internal − Ref lection,  ° f ° < ri ,  ° °   ~ 0f °  ° M ultiple − Internal − Ref lections, °R ° ≥ ri0 ,

80

(C.22)

C. Selected Configurations ° ° ° ~0 ° 0 where °R f ° is given in equation (C.21) and ri is the radius of a hemisphere.

C.3. Plate-Hemisphere The description of a surface is a study of the orientation of its local normal nˆ0 p , which is shown in Figure 3.6. In spherical coordinates, the unit vectors nˆ0 p , θˆ0 p and φˆ0 p are expressed as nˆ0 p =

P3 i=1

P3 θˆ0 p = i=1

Λ0p,i eˆi ,

∂Λ0p,i ˆi , ∂θp0 e

P3 φˆ0 p = i=1

0 1 ∂Λp,i ˆi , sin θp0 ∂φ0p e

(C.23)

where ¡ ¢ Λ0p,1 θp0 , φ0p = sin θp0 cos φ0p ,

¡ ¢ Λ0p,2 θp0 , φ0p = sin θp0 sin φ0p , ³

¡ ¢ Λ0p,3 θp0 = cos θp0 .

(C.24)

´

It is easy to show that the set of unit vectors nˆ0 p , θˆ0 p , φˆ0 p forms an orthonormal coordinates. Therefore, the points on plane can be described by a 2D coordinate system made of θˆ0 p and φˆ0 , p

~ 0p = R

0 ˆ0 νp,θ 0 θ p p

+

0 ˆ0 νp,φ 0 φ p p

=

3 X

" 0 νp,θ 0 p

i=1

# 0 0 νp,φ 0 ∂Λ ∂Λ0p,i p,i p + eˆi . ∂θp0 sin θp0 ∂φ0p

(C.25)

If the plane’s orientation constantly changes in time about its origin, the points on the plane experience the velocity ~ 0 p /dt, dR ( Ã !) " 0 3 νp,φ X ~ 0p 0 ∂ 2 Λ0p,i ∂Λ0p,i ∂ 2 Λ0p,i ∂Λ0p,i dR ˙0 p 0 0 ~ 0 ˙ Rp ≡ + νp,θp0 £ ¤2 + − cot θp = θ˙0 p ν p,θp0 0 0 0 ∂φ0 0 0 dt ∂θ sin θ ∂θ ∂φ p p p p p ∂ θ i=1 p ( ) # 0 2 0 0 2 0 0 ˙ νp,φ 0 ν p,φ0p ∂Λp,i ∂ Λ ∂ Λ p,i p,i p 0 + φ˙0 p eˆi , (C.26) + νp,θ + 0 £ ¤ p ∂φ0 ∂θ 0 sin θp0 ∂φ0p sin θp0 ∂ φ0 2 p p p

where θ˙0 p , φ˙0 p are the angular frequencies and ν˙0 p,θp0 , ν˙0 p,φ0p are the lattice vibrations along the directions θˆ0 p and φˆ0 p , respectively. Here, it is understood that Λ0p,3 is independent of φ0p . Therefore, the differentiation of Λ0p,3 with respect to the φ0p vanishes. For the static plate in which there are no lattice vibrations, ν˙0 p,θp0 and ν˙0 p,φ0p vanishes. For the case of plate-hemisphere configuration shown in Figure 3.7, the points on the plate are represented by the ~ p relative to the system origin. Making the correspondence in equation (B.2), R ~ → R ~ n, R ~T → R ~ T,p and vector R ¡ 0 0¢ ~ 0 → nˆ0 p , the two angular variable sets θp , φp and (θ, φ) are connected through the relations given in equations R ° ° ° ° (B.10) and (B.12) with r0 → 1. Here r0 → 1 because °nˆ0 p ° = 1. Therefore, we obtain i

i

¡ ¢ φ`p θp0 , φ0p , νT,p,1 , νT,p,2 = arctan

³

~ T,p θ`p θp0 , φ0p , R

´

µ

νT,p,2 + sin θp0 sin φ0p νT,p,1 + sin θp0 cos φ0p

¶ ,

(C.27)

 £ ¤ª £ ¤−1 νT,p,1 + νT,p,2 + sin θp0 cos φ0p + sin φ0p νT,p,3 + cos θp0 ´´ ³ ³ ´´  , (C.28) ³ ³ = arctan  νT ,p,2 +sin θ 0 sin φ0 νT ,p,2 +sin θ 0 sin φ0 + sin arctan νT ,p,1 +sin θ0p cos φp0 cos arctan νT ,p,1 +sin θ0p cos φp0 

©

p

p

p

p

where the subscript p of φ`p and θ`p indicates that these are the spherical variables for the points on the plate shown in ~ p becomes Figure 3.7, and they are not that of the hemisphere. The vector R " # 0 3 0 0 νp,φ X 0 ∂Λ ∂Λ p,i p,i p 0 ~p = R ~ T,p (νT,p,1 , νT,p,2 , νT,p,3 ) + R ~ 0p = R νT,p,i + νp,θp0 eˆi . (C.29) + ∂θp0 sin θp0 ∂φ0p i=1

81

C. Selected Configurations ° ° °~ ° The magnitude °R p ° is given by ³

~ 0p , R ~ T,p rp Λ

´

 " #2 1/2 0 3 ° ° X  0 0 ν 0 ∂Λp,i p,φp ∂Λp,i °~ ° 0 ≡ °R νT,p,i + νp,θ . + 0 p° = p ∂θ 0   sin θp0 ∂φ0p p i=1

~ p is expressed by In terms of the spherical coordinates, R ~p R

³

 " #2 1/2 3 0 3  X 0 0 ´ X νp,φ 0 ∂Λ ∂Λ ` p,i p,i p 0 ~ p, Λ ~ 0p , R ~ T,p = ` p,i eˆi , Λ νT,p,i + νp,θ + Λ 0 0 ∂φ0 p ∂θ 0   sin θ p p p i=1 i=1

(C.30)

where ³ ´ ` p,1 θ`p , φ`p = sin θ`p cos φ`p , Λ

³ ´ ` p,2 θ`p , φ`p = sin θ`p sin φ`p , Λ

³ ´ ` p,3 θ`p = cos θ`p . Λ

(C.31)

~ p indicates that the vector R ~ p describe the points on the plate. If the plane’s orientation Here, the subscript p in R constantly changes in time about its origin, then the same orientation change observed relative to the system origin is ~ p /dt, given by the velocity dR  #2 −1/2 3 3 µ· " 0 3 0 0   ν XX X ~ 0 ∂Λ0p,k ∂Λp,i p,φp ∂Λp,i 0 0 ~˙ p ≡ dRp = + ν + ν R νT,p,i + νp,θ 0 0 T,p,k p,θ p ∂θ 0 p ∂θ 0   dt sin θp0 ∂φ0p p p j=1 k=1 i=1 #" ( Ã !) 0 0 νp,φ0p ∂Λ0p,k νp,φ0p ∂Λ0p,k ∂ 2 Λ0p,k ∂ 2 Λ0p,k 0 0 − cot θ + ν ˙ + ν θ˙0 p + 0 £ ¤ T,p,k p p,θ 2 p sin θp0 ∂φ0p sin θp0 ∂θp0 ∂φ0p ∂φ0p ∂ θp0 ( ) # 0 2 0 νp,φ 0 ∂ Λ ν˙0 p,φ0p ∂Λ0p,k ∂ 2 Λ0p,k ∂Λ0p,k p,k p 0 ` p,j + νp,θp0 + φ˙0 p + ν˙0 p,θp0 + Λ £ ¤ ∂φ0p ∂θp0 sin θp0 ∂ φ0 2 ∂θp0 sin θp0 ∂φ0p p # " #2 " 0 3 0 0 ν X ` ` ` ` 0 ∂Λ ∂Λ p,φ ∂ Λ ∂ Λ ∂ θ ∂ φ p,i p,j p,i p,j p ˙0 p ˙0  p 0 eˆj , + + θ + φ νT,p,i + νp,θ 0 0 ∂φ0 p ∂θ 0 `p ∂φ0p p `p ∂φ0p p sin θ ∂ θ ∂ φ p p p i=1

(C.32)

` p,3 are independent of φ0p and φ`p , respectively; and as a consequence, their difwhere it is understood that Λ0p,3 and Λ ferentiation with respect to φ0p and φ`p vanishes. Here θ˙0 p , φ˙0 p are angular frequencies and ν˙ T,p,i is the translation speed of plate relative to system origin. Also, ν˙0 p,θp0 , ν˙0 p,φ0p are lattice vibrations along directions θˆ0 p and φˆ0 p , respectively. For a static plate in which there are no lattice vibrations, ν˙0 p,θp0 and ν˙0 p,φ0p vanishes. A cross-sectional view of the plate-hemisphere system is shown in Figure 3.8. The initial wave vector k~0 i traveling toward the hemisphere would go through reflections according to the law of reflection and finally exit. It then continues toward the plate and reflect from it. Depending on the orientation of plate at the time of impact, the wave would either escape to infinity or re-enter the hemisphere to repeat the process all over again. The equation (C.25) defines points on a plate, the Figure 3.7, relative to the plate origin. If Sp is a set of points on ~ 0 p of equation (C.25), the wave reflection dynamics off the plate involve only a plate whose members are defined by R those points of Sp in the intersection between the plate and the plane of incidence whose unit normal is nˆ0 p,1 given in equation (A.17). In order to determine the intersection between the plate and the incidence plane, the plate is first represented by a scalar field. From equation (C.23), the unit plate normal is nˆ0 p =

3 X i=1

82

Λ0p,i eˆi .

C. Selected Configurations The scalar field corresponding to the unit normal nˆ0 p satisfies the relation ~ 0 fp (ν10 , ν20 , ν30 ) ≡ ∇

3 X

3

X ∂fp eˆi 0 = Λ0p,i eˆi ∂ν i i=1 i=1

→

3 · X ∂fp i=1

∂νi0

¸ −

Λ0p,i

eˆi = 0.

The individual component of the equation is given by ∂fp − Λ0p,i = 0, ∂νi0

i = 1, 2, 3.

Notice that Λ0p,i is independent of νi0 . An integration with respect to νi0 yields the result fp (ν10 , ν20 , ν30 ) =

3 X

Λ0p,i νi0 ,

(C.33)

i=1

where the integration constant is set to zero because the plate contains its local origin. The intersection between the plane of incidence and the plate shown in Figure 3.6 satisfies the relation fp (ν10 , ν20 , ν30 )

−

fp,1 (ν10 , ν20 , ν30 )

=0

¸ 3 · ° ° X ° ~0 °−1 0 0 0 Λp,i + °n p,1 ° ²ijk k1,j r0,k νi0 = 0,

→

(C.34)

i=1

where fp,1 (ν10 , ν20 , ν30 ) is given in equation (A.43), and νi0 is a scalar corresponding to the basis eˆi , of course. We have, from equation (C.25), 0 νi0 = νp,θ 0 p

0 0 νp,φ 0 ∂Λ ∂Λ0p,i p,i p + , ∂θp0 sin θp0 ∂φ0p

i = 1, 2, 3.

(C.35)

0 Substituting νi0 into equation (C.34), νp,θ 0 is solved as p

0 νp,θ 0 = − p

0 νp,φ 0 p

P3

sin θp0 P3

∂Λ0p,l l=1 ∂θp0

·

¸ ° °−1 ° ° 0 0 Λ0p,i + °n~0 p,1 ° ²ijk k1,j r0,k · ¸, ° ° ° ~0 °−1 0 0 0 Λp,l + °n p,1 ° ²lmn k1,m r0,n

∂Λ0p,i i=1 ∂φ0p

(C.36)

0 0 where the summation over indices j, k, m and n is implicit, also the quotient νp,φ 0 / sin θp has been moved out of the p ~ 0 p given in equation (C.25) is then rewritten as summation. The R

~ 0p = R

0 νp,φ 0 p

 

P3

sin θp0 i=1

· ¸  ° °−1 ° °  0 0  Λ0p,i + °n~0 p,1 ° ²ijk k1,j r0,k ∂Λ0p,i  · ¸ eˆi . ° ° ∂θp0  ° ~0 °−1  0 0 0  Λp,l + °n p,1 ° ²lmn k1,m r0,n

∂Λ0p,i i=1 ∂φ0p

3   X

∂Λ0p,i − P3  ∂φ0p   l=1

∂Λ0p,l ∂θp0

(C.37)

~ p given in equation (C.30) is rewritten as Similarly, R      3 X ν0 0  ~p = νT,p,i + p,φp R   sin θp0    i=1 ×

3 X



P3

¸ 2 1/2 ° °  ° ~0 °−1  0 0  + °n p,1 ° ²ijk k1,j r0,k ∂Λ0p,i     · ¸ ° °−1 ∂θp0   ° °  0 0  Λ0p,l + °n~0 p,1 ° ²lmn k1,m r0,n 

∂Λ0p,i i=1 ∂φ0p

 ∂Λ0p,i   ∂φ0 − P3 p

∂Λ0p,l l=1 ∂θp0

` p,i eˆi , Λ

·

Λ0p,i

(C.38)

i=1

83

C. Selected Configurations ` p,i is given in equation (C.31). If Nh,max is the maximum count of reflections within the hemisphere before where Λ ~ = 0, is the wave escapes, the direction of the escaping wave, measured with respect to the system origin R ~kN ~ h,N ~ h,N =R −R , h,max +1 h,max +1 h,max

(C.39)

~ h,N ~ h,i+3 and R ~ h,N where ~kNh,max +1 = ξNh,max +1 k~0 Nh,max +1 . Similarly, by the correspondence R →R → h,max +1 h,max ~ h,i+2 in Figure 3.8, the direction of the escaping wave vector ~k is equivalently described by the relation R ~p − R ~ h,N ζ~kNh,max +1 = R h,max

(C.40)

~ p is solved as where ζ is an appropriate positive scale factor. Combining equations (C.39) and (C.40), R ~ p = ζR ~ h,N ~ h,N R + [1 − ζ] R . h,max +1 h,max

(C.41)

~ p and ~kN Because both R belong to a spanning set for the plane of incidence whose unit normal is nˆ0 p,1 given h,max +1 in equation (A.17), we observe that the following relationship n o ~ p × ~kN ~ h,N ~ h,N R = ζR + [1 − ζ] R × ~kNh,max +1 = γ nˆ0 p,1 (C.42) h,max +1 h,max +1 h,max hold, where γ is a proportional constant. Substituting the explicit form for nˆ0 p,1 from equation (A.17) into equation (C.42), it simplifies into the following equation 3 · X

¸ ° ° ° ~0 °−1 0 0 ζ²ijk Rh,Nh,max +1,j kNh,max +1,k + [1 − ζ] ²ijk Rh,Nh,max ,j kNh,max +1,k + γ °n p,1 ° ²ijk k1,j r0,k eˆi = 0,

i=1

and its component equations are given by ° °−1 ° ° 0 0 ζ²ijk Rh,Nh,max +1,j kNh,max +1,k + [1 − ζ] ²ijk Rh,Nh,max ,j kNh,max +1,k + γ °n~0 p,1 ° ²ijk k1,j r0,k = 0, where kNh,max +1,k = Rh,Nh,max +1,k − Rh,Nh,max ,k as described in equation (C.39). Finally, the scale factor ζ is solved as · ¸ ° ° ° ~0 °−1 0 0 ζ ≡ ζi = ²ijk Rh,Nh,max ,j Rh,Nh,max +1,k − ²ijk Rh,Nh,max ,j Rh,Nh,max ,k + γ °n p,1 ° ²ijk k1,j r0,k £ × ²ijk Rh,Nh,max ,j Rh,Nh,max +1,k − ²ijk Rh,Nh,max ,j Rh,Nh,max ,k − ²ijk Rh,Nh,max +1,j Rh,Nh,max +1,k ¤−1 +²ijk Rh,Nh,max +1,j Rh,Nh,max ,k , (C.43) where i = 1, 2, 3; j = 1, 2, 3 and k = 1, 2, 3. Here, the notation ζi have been adopted in place of ζ. It should be understood that for irrotational 3D vectors, ζ1 = ζ2 = ζ3 = ζ. For vectors in 2D and 1D space, it is understood then ζ3 , ζ2 are absent, respectively. In current form, equation (C.43) is incomplete because γ is still arbitrary. This happens 0 0 ~ because νp,θ 0 and νp,φ0 of Rp , equation (C.30), still needs to be related to the scale parameter ζi . Substituting ζi for ζ p p in equation (C.41), it is rewritten as ~ p = ζi R ~ h,N ~ h,N R + [1 − ζi ] R h,max +1 h,max ~ h,N ~ h,N or using equation (C.6) to explicitly substitute for R and R for N = Nh,max + 1, N = Nh,max , h,max +1 h,max

84

C. Selected Configurations respectively; and, regrouping the terms   1/2 3 3 h i2  X  X ~p = ` h,N R νT,h,j + ri0 Λ0h,Nh,max +1,j Λ + [1 − ζi ] ζi h,max +1,i   i=1

×

 3 h X 

j=1

νT,h,j + ri0 Λ0h,Nh,max ,j

j=1

1/2 i2  

  ` h,N Λ eˆ , h,max ,i  i

(C.44)

~ p with that where ζ1 = ζ2 = ζ3 = ζ. The subscript i of ri0 is not a summation index. Equating the above result for R of equation (C.38), we arrive at  ¸ ·  2 1/2  ° °−1 P3 ∂Λ0p,x   ° °   0 0 r0,z Λ0p,x + °n~0 p,1 ° ²xyz k1,y 0 3  3  0 0   x=1 ∂φ0p νp,φ X 0  ∂Λ X  ∂Λ p,j p,j p    νT,p,j + · ¸ − ° °   P3 ∂Λ0p,l sin θp0  ∂φ0p ∂θp0   ° °−1  0 0 i=1    Λ0p,l + °n~0 p,1 ° ²lmn k1,m r0,n  j=1 l=1 ∂θp0  1/2 3 h X i2  ` p,i − ζi ` h,N ` h,N ×Λ νT,h,j + ri0 Λ0h,Nh,max +1,j Λ + [ζi − 1] Λ h,max +1,i h,max ,i   j=1 1/2   3 h X i2   νT,h,j + ri0 Λ0h,Nh,max ,j ×  eˆi = 0,   j=1

and its component equations are      3 X  ν0 0 νT,p,j + p,φp   sin θp0   j=1



P3

¸ 2 1/2 ° °  ° ~0 °−1  0 0 + °n p,1 ° ²xyz k1,y r0,z  0   ∂Λp,j  · ¸ ° °−1 ∂θp0   ° °  0 0  Λ0p,l + °n~0 p,1 ° ²lmn k1,m r0,n 

∂Λ0p,x x=1 ∂φ0p

 ∂Λ0p,j   ∂φ0 − P3 p

∂Λ0p,l l=1 ∂θp0

·

Λ0p,x

1/2  3 h X i2  ` h,N ` h,N ` p,i − ζi Λ + [ζi − 1] Λ νT,h,j + ri0 Λ0h,Nh,max +1,j ×Λ h,max +1,i h,max ,i   j=1

 1/2 3 h X i2  = 0, × νT,h,j + ri0 Λ0h,Nh,max ,j  

(C.45)

j=1

where i = 1, 2, 3. Introducing the following definitions for convenience,  ½ ½ i2 ¾1/2 i2 ¾1/2 P3 h P3 h  0 0 0 0  , , Bζ = Aζ =  j=1 νT,h,j + ri Λh,Nh,max ,j j=1 νT,h,j + ri Λh,Nh,max +1,j    µ · ¸¶  ° ° P3 ∂Λ0p,x ° °−1 0 0 Cζ = − Λ0p,x + °n~0 p,1 ° ²xyz k1,y r0,z x=1 ∂φ0p   ¸¶−1 µ ·  ° °   P3 ∂Λ0p,l ° ~0 °−1  0 0 0  + n ² k r , × Λ ° p,1 ° lmn 1,m 0,n p,l l=1 ∂θ 0 p

(C.46)

85

C. Selected Configurations the relation shown in equation (C.45) is rewritten as  " 3 X 

νT,p,j

j=1

#2 1/2 µ 0 0 0 ¶  νp,φ 0 ∂Λ ∂Λ p,j p,j p ` p,i − ζi Aζ Λ ` h,N + − Cζ Λ h,max +1,i  sin θp0 ∂φ0p ∂θp0 ` h,N + [ζi − 1] Bζ Λ = 0, h,max ,i

where i = 1, 2, 3. There are three such relations, one for each value of i. It is convenient to combine additively all three relations to form  " #2 1/2 3 µ 0 0 3 3 X 0 ¶  νp,φ X X 0 ∂Λ ∂Λ p,j p,j p ` ` h,N Λ − ζ A Λ νT,p,j + − C p,i i ζ ζ h,max +1,i   sin θp0 ∂φ0p ∂θp0 i=1

j=1

i=1

+ [ζi − 1] Bζ

3 X

` h,N Λ = 0. h,max ,i

i=1

After regrouping the terms and squaring both sides, it becomes 3 X

" νT,p,j

j=1

|

#2 P3 ` P3 ` µ 0 ¶#2 " 0 νp,φ 0 ∂Λp,j ∂Λ0p,j ζi Aζ i=1 Λ h,Nh,max +1,i − [ζi − 1] Bζ p i=1 Λh,Nh,max ,i + − Cζ . = P3 ` sin θp0 ∂φ0p ∂θp0 l=1 Λp,l {z } LP

The summation labeled LP is rewritten as LP

3 h i2 X 0 = νp,φ 0 p j=1

· 0 ¸2 µ ¶ ¸ 3 · X ∂Λp,j ∂Λ0p,j ∂Λ0p,j 1 2νT,p,j ∂Λ0p,j 0 2 − Cζ − Cζ + νp,φ0p + νT,p,j . ∂θp0 sin θp0 ∂φ0p ∂θp0 sin2 θp0 ∂φ0p j=1

0 The above equation is simplified into a quadratic equation of νp,φ 0 , p 3 h i2 X 0 νp,φ0p j=1

· 0 ¸2 µ ¶ ¸ 3 · X ∂Λp,j ∂Λ0p,j ∂Λ0p,j 1 2νT,p,j ∂Λ0p,j 0 2 − Cζ − Cζ + νp,φ0p + νT,p,j ∂θp0 sin θp0 ∂φ0p ∂θp0 sin2 θp0 ∂φ0p j=1 #2 " P3 ` P3 ` ζi Aζ i=1 Λ h,Nh,max +1,i − [ζi − 1] Bζ i=1 Λh,Nh,max ,i = 0. − P3 ` l=1 Λp,l

0 The two roots νp,φ 0 are given by p

   µ 0 ¸  µ 0 ¶ ¸ 2 3 · 3 · 1 X 0 ¶ X ∂Λp,j ∂Λp,j ∂Λ0p,j νT,p,j ∂Λp,j 1 2 2ν  T,p,j 2   = − − Cζ − Cζ + νT,p,j ± + νT,p,j 0 0 0 0 0 0  sin θ ∂φ ∂θ 2 4 sin θ ∂φ ∂θ p p p p p p  j=1 j=1 

0 νp,φ 0 p

+

3 X j=1

 ×

1 sin2 θp0

3 X j=1

·

1 sin2 θp0

∂Λ0p,j ∂φ0p ·

−

∂Λ0p,j Cζ ∂θp0

¸2 "

∂Λ0p,j ∂Λ0p,j − Cζ 0 ∂φp ∂θp0

¸2

ζi Aζ

P3 i=1

` h,N Λ − [ζi − 1] Bζ h,max +1,i P3 ` l=1 Λp,l

P3 i=1

` h,N Λ h,max ,i

#2 1/2  

  

−1 

,

(C.47)

where Aζ , Bζ and Cζ are defined in equation (C.46). It is understood that one does not mix summation indices of Aζ , 0 Bζ and Cζ with those already present above. The result for νp,φ 0 is still incomplete because the factor γ in ζi needs p

86

C. Selected Configurations to be fixed by normalization. Unfortunately, the translation property of the plate, νT,p,j , makes it difficult to extract ζi 0 out of the radical. Besides the stated difficulty regarding ζi , νp,φ 0 is still ambiguous in deciding which of the two roots p correspond to the actual reflection point on the plate. Fortunately, for the plate-hemisphere system of Figure 3.7, the choice of system origin is arbitrary. One can always choose the plate origin to be the system origin and the translation of the plate can be equivalently simulated by a translation of the hemisphere origin in the opposite direction. Then, in 0 the rest frame of the plate, the translational motion of the plate is zero, i.e., νT,p,j = 0. In this frame, νp,φ 0 takes on p much simplified form 0 νp,φ 0 p

=

ζi Aζ ± sin θp0

P3 ` ` h,N Λ − [ζi − 1] Bζ i=1 Λ h,Nh,max ,i h,max +1,i . ¾1/2 ½ i 2 0 P3 h ∂Λ0p,j P ∂Λp,j 3 ` j=1 l=1 Λp,l ∂φ0 − Cζ ∂θ 0

P3

i=1

p

p

0 For the sign ambiguity in νp,φ 0 , it can be quickly fixed by noting that for νT,p,j = 0, equation (C.45) yields p

0 νp,φ 0 = p

ζi

P3

h i P3 ` ` h,N ` h,N Aζ Λ − Bζ Λ + Bζ i=1 Λ h,Nh,max ,i h,max +1,i h,max ,i , ½ ¾1/2 h i £ ¤−1 P3 P3 ` ∂Λ0p,j ∂Λ0p,j 2 0 sin θp j=1 l=1 Λp,l ∂φ0 − Cζ ∂θ 0

i=1

p

νT,p,j = 0,

(C.48)

p

where Aζ , Bζ and Cζ are defined in equation (C.46) with νT,p,j = 0. It is to be noticed that for a situation where ` becomes identical to Λ0 in form. One can obtain Λ ` simply by replacing the primed variables with the νT,p,j = 0, Λ 0 unprimed ones in Λ . For convenience, ζi of equation (C.43) is rewritten as ζi = Cγ−1 Aγ + γCγ−1 Bγ , where (

° °−1 £ ¤ ° ° 0 0 Aγ = ²ijk Rh,Nh,max ,j Rh,Nh,max +1,k − Rh,Nh,max ,k , Bγ = °n~0 p,1 ° ²ijk k1,j r0,k , £ ¤£ ¤ Cγ = ²ijk Rh,Nh,max ,j − Rh,Nh,max +1,j Rh,Nh,max +1,k − Rh,Nh,max ,k .

Furthermore introducing the definitions, i  P3 ` P3 h ` `  Bβ = i=1 Λ h,Nh,max ,i , h,Nh,max +1,i − Bζ Λh,Nh,max ,i ,  Aβ = i=1 Aζ Λ ¾ ½ 1/2 i h P3 ` P3 ∂Λ0p,j 2 ∂Λ0p,j   Cβ = l=1 Λp,l , j=1 ∂φ0 − Cζ ∂θ 0 p

(C.49)

(C.50)

p

0 the νp,φ 0 of equation (C.48) is rewritten as p

h i −1 −1 −1 −1 −1 0 0 νp,φ C A A + γC C B A + C B B 0 = C γ β γ β ζ β sin θp , γ γ β β β p

νT,p,j = 0.

0 Substituting the last expression for νp,φ 0 into equation (C.38), we arrive at p

( 3 · ¸2 )1/2 X 3 h i X ∂Λ0p,i ∂Λ0p,i −1 −1 −1 −1 −1 ~ ` p,i eˆi , Rp = Cβ Cγ Aγ Aβ + γCβ Cγ Bγ Aβ + Cβ Bζ Bβ Λ − C ζ 0 0 ∂φ ∂θ p p i=1 i=1 ~ p × ~kN where νT,p,i = 0. The vector cross product R is given by h,max +1 ~ p × ~kN R = h,max +1

3 X

²ijk Rp,j kNh,max +1,k eˆi

i=1

87

(C.51)

C. Selected Configurations or ~ p × ~kN R h,max +1

1/2  3 · 0 0 ¸2  h i X ∂Λp,j ∂Λp,j = Cβ−1 Cγ−1 Aγ Aβ + γCβ−1 Cγ−1 Bγ Aβ + Cβ−1 Bζ Bβ − Cζ   ∂φ0p ∂θp0 j=1

×

3 X

` p,j kN eˆ . ²ijk Λ h,max +1,k i

i=1

~ p × ~kN Finally, substituting above vector cross product R into equation (C.42) and regrouping the terms, it h,max +1 becomes   1/2 3 3 · 0 ¸2  0 i X X h ∂Λ ∂Λ p,j p,j −1 −1 −1 −1 −1 − Cζ  Cβ Cγ Aγ Aβ + γCβ Cγ Bγ Aβ + Cβ Bζ Bβ   ∂φ0p ∂θp0 i=1

j=1

¶ ° ° ° ~0 °−1 0 0 ` p,j kN ×²ijk Λ − γ n ² k r ° ° +1,k p,1 ijk 1,j 0,k eˆi = 0, h,max ° °−1 P ° ° 3 0 0 where nˆ0 p,1 = °n~0 p,1 ° i=1 ²ijk k1,j r0,k eˆi have been used. And for the component equations  1/2 3 · 0 0 ¸2  h i X ∂Λ ∂Λ p,j p,j Cβ−1 Cγ−1 Aγ Aβ + γCβ−1 Cγ−1 Bγ Aβ + Cβ−1 Bζ Bβ − Cζ   ∂φ0p ∂θp0 j=1

° ° ° ~0 °−1 0 0 ` p,j kN ×²ijk Λ − γ n r0,k = 0, ° ° ²ijk k1,j +1,k p,1 h,max where i = 1, 2, 3. There are three such relations and they are additively combined to yield 1/2  3 · 0 ¸2  0 h i X ∂Λ ∂Λ p,j p,j − Cζ Cβ−1 Cγ−1 Aγ Aβ + γCβ−1 Cγ−1 Bγ Aβ + Cβ−1 Bζ Bβ   ∂φ0p ∂θp0 j=1

×

3 X

3 ° ° ° ~0 °−1 X 0 0 ` p,j kN ²ijk Λ − γ n ²ijk k1,j r0,k = 0. ° p,1 ° h,max +1,k

i=1

i=1

Finally, γ is solved to give the result  1/2  3 3 · 0 ¸2  ° °−1 X X 0 ∂Λ ∂Λ ° ° p,j p,j 0 0 − Cζ γ ≡ γo = °n~0 p,1 ° ²ijk k1,j r0,k − Cβ−1 Cγ−1 Bγ Aβ   ∂φ0p ∂θp0 i=1

×

3 X

j=1

!−1 ` p,j kN ²ijk Λ h,max +1,k

i=1

×

 3 · X ∂Λ0p,j 

j=1

∂φ0p

−

∂Λ0p,j Cζ ∂θp0

³h i Cβ−1 Cγ−1 Aγ Aβ + Cβ−1 Bζ Bβ

 ¸2 1/2 X 3 

  ` p,j kN ²ijk Λ . h,max +1,k 

(C.52)

i=1

0 The parameter νp,φ 0 is now completely defined, p

h i −1 −1 −1 −1 −1 0 νp,φ C A A + γ C C B A + C B B sin θp0 , 0 = C γ β o γ β ζ β γ γ β β β p

νT,p,j = 0,

(C.53)

where (Aζ , Bζ , Cζ ) , (Aγ , Bγ , Cγ ) , (Aβ , Bβ , Cβ ) and γo are given by equations (C.46), (C.49), (C.50) and (C.52),

88

C. Selected Configurations 0 respectively. With νp,φ 0 defined in equation (C.53), the reflection point on the plate is obtained from equation (C.38), p

 ¸ ·  2 1/2 °−1 ° P3 ∂Λ0p,i   ° °   0 0 0 0 ~   Λp,i + °n p,1 ° ²ijk k1,j r0,k 3 X i=1 ∂φ0p  ∂Λ0p,s ∂Λ0p,s   ~p =   · ¸ R − ° °−1  ∂φ0 P3 ∂Λ0p,l  ∂θp0   ° ° p   0 0   Λ0p,l + °n~0 p,1 ° ²lmn k1,m r0,n s=1  l=1 ∂θ 0 p

3 h iX ` p,i eˆi , × Cβ−1 Cγ−1 Aγ Aβ + γo Cβ−1 Cγ−1 Bγ Aβ + Cβ−1 Bζ Bβ Λ

(C.54)

i=1

` becomes identical to Λ0 in form, and Λ ` where νT,p,j = 0. It should be noticed that for a situation where νT,p,j = 0, Λ can be obtained simply by replacing the primed variables with the unprimed ones.

~ p re-enters the hemisphere cavity or escape to infinity, we To see if the wave reflected from the plate at location R consider the reflected wave ~kr,Nh,max +1 , h i ~kr,N = αr,⊥ nˆ0 p × ~kNh,max +1 × nˆ0 p − αr,k nˆ0 p • ~kNh,max +1 nˆ0 p , (C.55) h,max +1 where the relation found in equation (A.14) have been used. As always, it is convenient to express vectors in component forms. Making the changes in variables (l, m, n) → (i, j, k) , (m, q, r) → (j, l, m) , nˆ0 → nˆ0 p and k~0 i → ~kNh,max +1 , the component result of equation (A.16) is used to get ~kr,N = h,max +1

3 X 3 X ©

£ ¤ αr,⊥ kNh,max +1,i n0p,k n0p,k − n0p,i kNh,max +1,k n0p,k

i=1 k=1 ª −αr,k n0p,k kNh,max +1,k n0p,i eˆi ,

(C.56)

where n0p,i and n0p,k are coefficients of the normalized nˆ0 p . All wave vectors entering the hemisphere cavity satisfy the relation ~p + R

3 h X

ξ~κ • eˆi

ih i ~kr,N ~ 0 = 0, • e ˆ eˆi − R +1 i h,max

i=1

~0 = R

3 X £ ¤ 0 νT,h,i + r0,i eˆi ,

(C.57)

i=1

~ 0 is the points on the opening face of hemisphere. The scale vector where ξ~κ is a real-valued positive scale vector and R ξ~κ has the form ξ~κ =

3 X

ξκ,i eˆi .

i=1

~ p and ~kr,N With the scale vector ξ~κ defined above; and, R defined in equations (C.54) and (C.56), respectively, h,max +1 equation (C.57) is rewritten in component form  · ¸  2 1/2 ° ° P3 ∂Λ0p,i   ° ~0 °−1   0 0 0    Λp,i + °n p,1 ° ²ijk k1,j r0,k 3 3 i=1 ∂φ0p X X  ∂Λ0p,s ∂Λ0p,s      ¸ ° °−1   ∂φ0 − P3 ∂Λ0 · ∂θp0   ° p,l p  0 + °n 0 0 0 ~ s=1 i=1    Λ ² k r ° ° p,1 lmn 1,m 0,n   l=1 ∂θp0 p,l 3 h i X © £ ` p,i + ξκ,i × Cβ−1 Cγ−1 Aγ Aβ + γo Cβ−1 Cγ−1 Bγ Aβ + Cβ−1 Bζ Bβ Λ αr,⊥ kNh,max +1,i

×n0p,k n0p,k

−

n0p,i kNh,max +1,k n0p,k

¤

−

k=1 ª 0 αr,k np,k kNh,max +1,k n0p,i

89

¢ 0 − νT,h,i − r0,i eˆi = 0,

C. Selected Configurations which yields the component equations,  · ¸  2 1/2 ° °−1 P3 ∂Λ0p,i   ° °   0 0 0 0 ~   Λp,i + °n p,1 ° ²ijk k1,j r0,k 3 X i=1 ∂φ0p  ∂Λ0p,s ∂Λ0p,s   h −1 −1   · ¸ Cβ Cγ Aγ Aβ − ° °−1  ∂φ0 P3 ∂Λ0p,l  ∂θp0   ° ° p   0 0 0 0 ~ s=1   Λp,l + °n p,1 ° ²lmn k1,m r0,n   l=1 ∂θ 0 p

3 i X © £ ` p,i + ξκ,i +γo Cβ−1 Cγ−1 Bγ Aβ + Cβ−1 Bζ Bβ Λ αr,⊥ kNh,max +1,i n0p,k n0p,k k=1

¤ ª 0 −n0p,i kNh,max +1,k n0p,k − αr,k n0p,k kNh,max +1,k n0p,i − νT,h,i − r0,i = 0, where i = 1, 2, 3. Finally, ξκ,i is solved as   ¸ ·  2 1/2 ° °−1 P3 ∂Λ0p,i   ° °   0 0 0 0 ~   Λp,i + °n p,1 ° ²ijk k1,j r0,k 3 0 X 0 0   i=1 ∂φ   p ∂Λp,s ∂Λp,s  0   ¸ ξκ,i =  ° °−1 νT,h,i + r0,i −   ∂φ0 − P3 ∂Λ0 · ∂θp0   ° ° p p,l    0 0 0 0 ~ s=1   Λp,l + °n p,1 ° ²lmn k1,m r0,n   l=1 ∂θ 0 p

Ã 3 h ´ X i © £ −1 −1 −1 −1 −1 ` × Cβ Cγ Aγ Aβ + γo Cβ Cγ Bγ Aβ + Cβ Bζ Bβ Λp,i αr,⊥ kNh,max +1,i n0p,k n0p,k −n0p,i kNh,max +1,k n0p,k

¤

−

αr,k n0p,k kNh,max +1,k n0p,i

ª¢−1

k=1

,

(C.58)

where i = 1, 2, 3. ~ 0 ≤ r0 , which implies r0 ≤ r0 , where The above result can be applied in setting the re-entry criteria. Notice that R i 0,i i 0 ri is the radius of the hemisphere. It can be concluded then that all waves re-entering hemisphere cavity would satisfy the condition ξκ,1 = ξκ,2 = ξκ,3 . On the other hand, those waves that escapes to infinity cannot have all three ξκ,i equaling to a same constant. The re-entry condition ξκ,1 = ξκ,2 = ξκ,3 is just another way of stating the existence of parametric line along the vector ~kr,Nh,max +1 that happens to pierce through a hemisphere opening. When such a line does not exist, the initial wave vector direction has to be rotated accordingly to a new direction, such that in its rotated direction there is a parametric line that pierces through the hemisphere opening; it leads to the condition that all three ξκ,i cannot equal to a same constant. The re-entry criteria are now summarized for bookkeeping purpose, ½ ξκ,1 = ξκ,2 = ξκ,3 → W ave − ReEnters − Hemisphere, (C.59) ELSE → W ave − Escapes − to − Inf inity, where ELSE is the case where ξκ,1 = ξκ,2 = ξκ,3 cannot be satisfied.

90

D. Dynamical Casimir Force The original derivations and developments of this thesis pertaining to the dynamical Casimir force are included in this appendix. It is referenced by the text of the thesis to supply all the fine details.

D.1. Formalism of Zero-Point Energy and its Force For massless fields, the energy-momentum relation is given by Hn0 s ≡ ET otal = pc,

(D.1)

wherePp is the momentum and c the speed of light. The field propagating in an arbitrary direction has a momentum 3 p~0 = i=1 p0i eˆi . The associated field energy-momentum relation is hence Hn0 s

−c

( 3 X

)1/2 2 [p0i ]

= 0.

i=1

The differentiation of the above equation gives  ( 3 )1/2  ( 3 )1/2 X X 2 0 0 0 2 0   [pi ] ≡ dHns − cd [pi ] d Hns − c = 0. i=1

(D.2)

i=1

The total differential energy dHn0 s is dHn0 s =

¸ ¶−1 3 µ· 3 X ∂Hn0 s 0 ∂Hn0 s ∂ki0 0 X 1 dp = n + dpi , ~ s i 0 0 ∂ki ∂pi 2 ∂ki0 i=1 i=1

(D.3)

£ ¤ where the relation p0i = ns + 12 ~ki0 has been used. The total differential momentum is d

( 3 X i=1

)1/2 2 [p0i ]

=

( 3 X i=1

)−1/2 2 [p0i ]

3 X

p0i dp0i .

i=1

The combined result is  ( 3 )−1/2  ¸ ¶−1 µ· 3 X X ∂Hn0 s 1 2  ns + ~ [p0i ] cp0i  dp0i = 0. 0 − 2 ∂k i i=1 i=1 Because all the momentum differentials are linearly independent, their coefficients are zero, µ·

( 3 )−1/2 ¸ ¶−1 X ∂Hn0 s 1 2 ~ ns + − [p0i ] cp0i = 0, 2 ∂ki0 i=1

91

i = 1, 2, 3.

(D.4)

D. Dynamical Casimir Force There are three such equations. Then, additively combining the three relations, and rearranging the terms, we have ( 3 X

)1/2 2 [p0i ]

i=1

3 µ· X

ns +

i=1

¸ ¶−1 3 X ∂Hn0 s 1 ~ = cp0i . 2 ∂ki0 i=1

Squaring both sides to get rid of the radical leads to "

3 µ· X i=1

The summations

P3

2

i=1

3 X

[p0i ] and

2

hP 3 i=1

2

[p0i ] = [p0α ] +

i=1

"

3 X

#2 3 " 3 #2 ¸ ¶−1 X ∂Hn0 s X 0 2 1 2 0 ~ ns + [pi ] = c pi . 2 ∂ki0 i=1 i=1

3 X

p0i

i2 are rewritten as 2

2

(1 − δiα ) [p0i ] = [p0α ] +

i=1

#2 p0i

" =

p0α

+

i=1

3 X

#2 δiα ) p0i

=

2 [p0α ]

+2

i=1

=

2 [p0α ]

3 X

µ· (1 − δiα )

ns +

i=1

(1 −

(D.5)

3 X i=1

" (1 −

δiα ) p0i p0α

+

¸ ¶2 1 2 ~ [ki0 ] , 2

3 X

#2 (1 −

δiα ) p0i

i=1

#2 ¸ µ· ¸ ¶2 "X 3 1 1 0 0 0 ~ki pα + ns + ~ (1 − δiα ) ki , +2 (1 − δiα ) ns + 2 2 i=1 i=1 ·

3 X

£ ¤ where p0i has been replaced by ns + 12 ~ki0 . Substituting the result into equation (D.5) and rearranging the terms in powers of p0α , we have "  #2 µ· · ¸ ¶2 ¸ X µ· ¸ ¶2 3 3 X ∂Hn0 s 1 1 1 0 2   [pα ] − 2 ns + − ns + ~c ~ (1 − δiα ) ns + ~c ki0 p0α 0 ∂k 2 2 2 i i=1 i=1 "

#2 3 µ· µ· ¸ ¶2 #2 "X ¸ ¶2 3 ∂Hn0 s X 1 1 2 0 − (1 − δiα ) ns + (1 − δ ) n + ~c ki + ~ [ki0 ] = 0. iα s 0 2 ∂k 2 i i=1 i=1 i=1 3 X

Defining the following quantities, ( P3 Cα,1 = i=1

¡£ ¤ ¢2 P3 , Cα,2 = i=1 (1 − δiα ) ns + 12 ~c ki0 , ¡£ ¤ ¢2 P3 2 = i=1 (1 − δiα ) ns + 12 ~ [ki0 ] ,

∂H0ns ∂ki0

Cα,3

(D.6)

the above quadratic equation is rewritten as " ¸ ¶2 # · ¸ µ· 1 1 2 2 2 2 [p0α ] − 2 ns + Cα,1 − ns + ~c ~Cα,2 p0α − Cα,2 + Cα,1 Cα,3 = 0. 2 2 Finally, the root p0α is found to be  1/2 £ ¤ £ ¤   1 2 2 2  2 2 1 ns + 2 ~ Cα,2 Cα,2 − Cα,1 Cα,3  ns + 2 ~Cα,2 0 pα = , ¤ ¢2 + h ¤ ¢2 ¡£ ¡£ ¡£ ¤ ¢2 i2 + 2 2 −  2 Cα,1 Cα,1 − ns + 21 ~c  ns + 21 ~c   Cα,1 − ns + 12 ~c

(D.7)

where the positive root have been chosen since p0α is an α component magnitude of the total field momentum p~0 , therefore it is a positive scalar, p0α ≥ 0.

92

D. Dynamical Casimir Force By definition, force is equal to the change in momentum per unit time, ¸ X 3 3 3 · 3 X X d X 0 dp0α dp0α d ~0 0 deˆα 0 ~ F = p = pα eˆα = eˆα + pα = eˆα = F~ 0 α . dt dt α=1 dt dt dt α=1 α=1 α=1 The explicit expression for F~ 0 α is found to be    C C £C 2 C − C 2 ¤ 2 £n + 1 ¤2 ~2 C C 2 C Cα,1 Cα,3 Cα,4 s α,1 α,2 α,4  α,1 α,4 α,1 α,3 α,2 2 0 F~ α =  h ¡£ ¤ ¢2 ¡£ ¤ ¢2 i2 − h 2 ¡£ ¤ ¢2 i3 − 2  1 1 2 Cα,1 − ns + 21 ~c  Cα,1 − ns + 2 ~c Cα,1 − ns + 2 ~c   £ ¤ £ ¤2 1 2 ns + 2 ~Cα,1 Cα,2  dCα,1  ns + 12 ~2 Cα,2 Cα,4 Cα,2 Cα,4 −h + h i2  i2 + 2 ¡£ ¤ ¢2 ¡£ ¤ ¢ ¡£ ¤ ¢ 2 2 dt 2 − 2 − Cα,1 − ns + 21 ~c Cα,1 ns + 12 ~c Cα,1 ns + 21 ~c ! ) £ ¤ 1 2 ns + 21 ~ dCα,2 dCα,3 2 Cα,1 Cα,4 − eˆα , (D.8) + ¤ ¢2 ¤ ¢2 ¡£ ¡£ dt dt C 2 − ns + 1 ~c C 2 − ns + 1 ~c α,1

2

α,1

2

where −1/2 £ ¤ 1 2 2 2 2 2 ns + 2 ~ Cα,2 Cα,2 − Cα,1 Cα,3   = h . ¡£ i2 + 2 ¤ ¢2  ¡£ ¤ ¢ 2 2 − Cα,1 − ns + 21 ~c ns + 12 ~c Cα,1 

Cα,4

(D.9)

Before computing the three time derivatives dCα,1 /dt, dCα,2 /dt and dCα,3 /dt, we should notice that ki0 (ni ) = ni fi (Li ) . Hence, the derivative dki0 /dt can be written as ∂ki0 dni ∂fi dLi ∂k 0 ∂fi ˙ dki0 = fi (Li ) + ni = fi (Li ) i n˙ i + ni Li . dt ∂ni dt ∂Li dt ∂ni ∂Li

(D.10)

The three derivatives dCα,1 /dt, dCα,2 /dt and dCα,3 /dt are given by ¸ · 3 X 3 3 X 3 X X ∂kj0 ∂ 2 Hn0 s dkj0 ∂ 2 Hn0 s dCα,1 ∂fj ˙ Lj = = fj (Lj ) n˙ j + nj dt ∂kj0 ∂ki0 dt ∂kj0 ∂ki0 ∂nj ∂Lj i=1 j=1 i=1 j=1 ( · ¸ X 3 3 X 3 X ∂ 2 Hn0 s ∂ 2 Hn0 s ∂ki0 ∂fi ˙ = f (L ) n ˙ + n L + (1 − δ ) i i i i i ij 0 2 ∂ni ∂Li ∂kj0 ∂ki0 i=1 ∂ [ki ] i=1 j=1 · ¸¾ ∂kj0 ∂fj ˙ × fj (Lj ) n˙ j + nj Lj , ∂nj ∂Lj

(D.11)

· ¸· ¸ 3 X 1 ∂k 0 ∂fi ˙ dCα,2 = (1 − δiα ) ns + fi (Li ) i n˙ i + ni Li , dt 2 ∂ni ∂Li i=1

(D.12)

· ¸2 · ¸ 3 X dCα,3 1 ∂k 0 ∂fi ˙ =2 (1 − δiα ) ns + ki0 fi (Li ) i n˙ i + ni Li , dt 2 ∂ni ∂Li i=1

(D.13)

where Cα,1 , Cα,2 and Cα,3 are defined in equation (D.6). It is noted that the derivative dki0 /dt, and also each of dCα,1 /dt, dCα,2 /dt and dCα,3 /dt, consists of two contributing parts, one is proportional to n˙ i and the other involves L˙ i . The force expression in equation (D.8) has then two contributing parts. The force contribution involving n˙ i has a physical meaning that the boundaries are being driven to generate the extra wave modes that would otherwise be

93

D. Dynamical Casimir Force missing when such drivers were not present. The force contribution involving L˙ i is the effect of feedbacks from the moving boundaries. This feedback effect due to the moving boundaries tends to either cool or heat the conducting boundaries. For an isolated, non-driven conducting boundaries, the force contribution proportional to n˙ i vanishes. For what is concerned with in this thesis, only isolated systems are studied; and, therefore, n˙ i = 0. The expression of force is then rewritten as    C C £C 2 C − C 2 ¤ 2 £n + 1 ¤2 ~2 C C 2 C Cα,1 Cα,3 Cα,4 s α,1 α,2 α,4  α,1 α,4 α,1 α,3 α,2 2 0 ~ Fα=  h ¡£ ¤ ¢2 i2 − h i3 − 2 ¡£ ¡£ ¤ ¤ ¢ ¢ 2 2  2 − 2 − Cα,1 − ns + 12 ~c  ns + 21 ~c ns + 21 ~c Cα,1 Cα,1   £ ¤ 3 3 X 3 1 X 2 ns + 2 ~Cα,1 Cα,2  X ∂ 2 Hn0 s ∂fi ˙ ∂ 2 Hn0 s ∂f j ˙   −h n L + (1 − δij ) 0 0 nj Lj ¡£ ¤ ¢2 i2  0 ]2 i ∂Li i ∂k ∂k ∂L j ∂ [k 1 2 j i i i=1 i=1 j=1 Cα,1 − ns + 2 ~c   £ £ ¤ ¤ 3 1 1 2 2 ns + 2 ~ Cα,2 Cα,4 X  ns + 2 ~ Cα,2 Cα,4 + h + + (1 − δiα )  i ¡£ ¤ ¢ ¡£ ¤ ¢ 2 2 ¡£ ¤ ¢2 2 2 − 2 − 2 − Cα,1 ns + 12 ~c Cα,1 ns + 12 ~c i=1 Cα,1 ns + 21 ~c ) ¸ ¸2 · · 3 1 2 X 1 ∂fi ˙ 1 ∂fi ˙ 2 Cα,1 Cα,4 0 ni k i ni × ns + Li − (1 − δiα ) ns + Li eˆα . ¡£ ¤ ¢2 2 2 ∂Li 2 ∂Li C 2 − ns + 1 ~c i=1 α,1

2

It can be simplified with the following definitions, ¤ £ ¤2 £ 2 2 2 Cα,4 Cα,3 − Cα,2 2 ns + 12 ~2 Cα,1 Cα,2 Cα,1 Cα,4 Cα,1 − Cα,5 = h i i h 2 ¡£ ¤ ¢ ¡£ ¤ ¢ 2 2 3 2 − 2 − ns + 21 ~c ns + 12 ~c Cα,1 Cα,1 £ ¤ 2 ns + 12 ~Cα,1 Cα,2 Cα,1 Cα,3 Cα,4 − ¤ ¢2 − h ¡£ ¡£ ¤ ¢2 i2 , 2 − 2 − Cα,1 ns + 12 ~c ns + 12 ~c Cα,1

Cα,6

£ ¤2 £ ¤ ns + 21 ~2 Cα,2 Cα,4 ns + 21 ~ Cα,2 Cα,4 =h ¡£ ¡£ ¤ ¢2 + 2 ¤ ¢2 , ¡£ ¤ ¢2 i2 + 2 2 − Cα,1 − ns + 21 ~c Cα,1 − ns + 12 ~c ns + 12 ~c Cα,1

Cα,7 =

2 Cα,4 Cα,1 ¡£ ¤ ¢2 . 2 Cα,1 − ns + 21 ~c

The dynamical force can then be rewritten as ( " µ · ¸ ¶· ¸# 3 X ∂ 2 Hn0 s ∂fi 1 0 1 0 ~ Fα= ni Cα,5 L˙ i ki ns + + (1 − δiα ) Cα,6 − Cα,7 ns + 0 ]2 ∂L 2 2 i ∂ [k i i=1  3 2 0 X ∂fj ∂ Hns ˙  + (1 − δij ) Cα,5 nj Lj eˆα , ∂Lj ∂kj0 ∂ki0 

(D.14)

(D.15)

(D.16)

(D.17)

j=1

where Cα,5 , Cα,6 and Cα,7 are defined in equations (D.14), (D.15) and (D.16). The force equation (D.17) vanishes for the 1D case, which is an expected result. The reason is explained as follow: Recall that equation (D.4) reads  ( 3 )−1/2  ¸ ¶−1 µ· 3 X X ∂Hn0 s 1 2  ns + ~ [p0i ] cp0i  dp0i = 0. 0 − 2 ∂k i i=1 i=1

94

D. Dynamical Casimir Force For the 1D case, the summation runs only once and the above expression simplifies to "µ· # ¸ ¶−1 µ· ¸ ¶−1 ∂Hn0 s ∂Hn0 s 1 1 0 ns + ~ − c dpi = 0 → ns + ~ − c = 0. 0 2 ∂ki 2 ∂ki0 This is a classic situation where P the problem has been over specified. For the 3D case, equation (D.4) is really a 2 3 combination of two constraints, i=1 [p0i ] and Hn0 s . For the 1D case, there is only one constraint, Hn0 s . Hence, equation (D.4) becomes an over specification. In order to avoid the problem caused by over specifications in this formulation, the one dimensional force expression can be obtained directly by differentiating equation (D.1) instead of using the above formulation for the three dimensional case. We have then for the force expression in 1D case: · ¸ dp0 1 ∂Hn0 s dk 0 1 ∂Hn0 s ∂k 0 ∂f ˙ 1 = = f (L) n ˙ + n p0 = Hn0 s → L . c dt c ∂k 0 dt c ∂k 0 ∂n ∂L For an isolated, non-driven systems, n ∂f ∂Hn0 s ˙ Lˆ e, F~ 0 = c ∂L ∂k 0

(D.18)

where F~ 0 is the force expression in 1D space. Here the subscript α of F~ 0 α have been dropped for simplicity, since it is a one dimensional force.

D.2. Equations of Motion for the Driven Parallel Plates Consider the one dimensional system of two i parallel plates shown in Figure 3.10. Defining the boundary length Li,< h ~ as the magnitude of a vector eˆi L< • eˆi , where < denotes the region, the following relation is found from Figure 3.10, ~< = R ~ rp,< − R ~ lp,< = L

3 h i X ~ rp,< • eˆi − R ~ lp,< • eˆi eˆi . R

(D.19)

i=1

~ < /dt is Hence, the velocity dL " # 3 X ~< ~ rp,< ~ lp,< ~ lp,< ~ rp,< dL dR dR dR dR = − = • eˆi − • eˆi eˆi dt dt dt dt dt i=1

(D.20)

and the corresponding component magnitude is given by ~< ~ rp,< ~ lp,< dL dR dR L˙ i,< ≡ • eˆi = • eˆi − • eˆi . dt dt dt

(D.21)

Substituting the result L˙ i,< of equation (D.21) for L˙ α in the one dimensional dynamical force expression of equation (D.18), " # ~ rp,< ~ lp,< ∂Hn0 s ,< dR ∂f d R n α,< α,< • eˆα − • eˆα eˆα , (D.22) F~ 0 α,< = 0 c ∂Lα,< ∂kα,< dt dt where L˙ i,< ≡ L˙ α and i ≡ α. The subscript < denotes the corresponding quantities associated with the region < = 1, 2, 3, e.g., Hn0 s ,< denotes the field energy in region <. For simplicity, the following notational convention is adopted ~a dR R˙ a,b = • eˆb , dt

2~ ¨ a,b = d Ra • eˆb , R dt2

95

D. Dynamical Casimir Force gα,<

nα,< = c

µ

∂fα,< ∂Lα,<

¶Ã

∂Hn0 s ,< 0 ∂kα,<

! .

(D.23)

The force expression of equation (D.22) is then rewritten as h i F~ 0 α,< = gα,< R˙ rp,<,α − R˙ lp,<,α eˆα ,

(D.24)

Before writing down equations of motion for each plates illustrated in Figure 3.10, the associated center of mass ~ rp,<,α for each plates needs to be determined. The center of mass point point relative the the surface point vectors R ~ ~ rp,< through a relation Rrp,cm for plate labeled “right plate” in Figure 3.10 is related to the surface point vector R ~ rp,cm (t) = R ~ rp,<=2 (t) + R ~ rp,cm−< (t) , R ~ rp,cm−< (t) ≡ R ~ rp,cm−2 (t) is a displacement between surface and the center of mass point. The α component where R ~ rp,cm is then of the center of mass point R ~ rp,cm (t) = eˆα • R ~ rp,2 (t) + eˆα • R ~ rp,cm−2 (t) = Rrp,2,α (t) + Rrp,cm−2,α (t) . Rrp,cm,α (t) ≡ eˆα • R

(D.25)

The component of the center of mass point speed is given by R˙ rp,cm,α (t) = R˙ rp,2,α (t) + R˙ rp,cm−2,α (t) .

(D.26)

~ lp,<=2 (t) Similarly, for the plate labeled “left plate,” the center of mass point is related to the surface vector point R by ~ lp,cm (t) = R ~ lp,<=2 (t) − R ~ lp,cm−< (t) , R and the component along the direction eˆα is R˙ lp,cm,α (t) = R˙ lp,2,α (t) − R˙ lp,cm−2,α (t) .

Rlp,cm,α (t) = Rlp,2,α (t) − Rlp,cm−2,α (t) ,

(D.27)

Using the above center of mass relations, equations (D.25), (D.26) and (D.27), along with the force equation (D.24), the net force acting on a plate labeled “right plate” along the direction of eˆα in the configuration shown in Figure 3.10 is i h ¨ rp,cm,α = F~ 0 α,<=2 + F~ 0 α,<=3 • eˆα mrp R or h i ¨ rp,cm,α = gα,2 R˙ rp,cm,α − R˙ lp,cm,α − R˙ rp,cm−2,α − R˙ lp,cm−2,α mrp R h i + gα,3 R˙ dpr,cm,α − R˙ rp,cm,α − R˙ dpr,cm−2,α − R˙ rp,cm−2,α

(D.28)

where mrp is the mass of the “right plate.” If the plate surface is not vibrating longitudinally along the direction of eˆα , the displacements Rrp,cm−2,α and Rdpr,cm−2,α are constants; hence, R˙ rp,cm−2,α = R˙ dpr,cm−2,α = 0. For static surfaces, the above net force relation simplifies to h i h i ¨ rp,cm,α = srp,2 gα,2 R˙ rp,cm,α − R˙ lp,cm,α + srp,3 gα,3 R˙ dpr,cm,α − R˙ rp,cm,α , mrp R (D.29) where srp,2 and srp,3 have been inserted for convenience due to the force sign convention to be set later. Similarly, for the plate labeled “left plate” in Figure 3.10, the net force relation along the direction of eˆα is h i ¨ lp,cm,α (t) = F~ 0 α,β,<=1 + F~ 0 α,β,<=2 • eˆα mlp R

96

D. Dynamical Casimir Force or, for the case where plate surfaces do not have longitudinal vibrations, h i h i ¨ lp,cm,α = slp,1 gα,1 R˙ lp,cm,α − R˙ dpl,cm,α + slp,2 gα,2 R˙ rp,cm,α − R˙ lp,cm,α , mlp R

(D.30)

where mlp is a mass of “left plate” and the terms slp,1 and slp,2 have been inserted for convenience due to the force sign convention to be set later. We have now the two coupled differential equations, ¨ rp,cm,α + srp,3 gα,3 R˙ rp,cm,α − srp,2 gα,2 R˙ rp,cm,α + srp,2 gα,2 R˙ lp,cm,α = srp,3 gα,3 R˙ dpr,cm,α , mrp R ¨ lp,cm,α + slp,2 gα,2 R˙ lp,cm,α − slp,1 gα,1 R˙ lp,cm,α − slp,2 gα,2 R˙ rp,cm,α = −slp,1 gα,1 R˙ dpl,cm,α . mlp R Introducing the following definitions,  η1 = m−1  rp [srp,2 gα,2 − srp,3 gα,3 ] ,   η = m−1 [s g − s g ] , 3

  

lp,1 α,1

lp

η2 = −srp,2 gα,2 m−1 rp η4 = slp,2 gα,2 m−1 lp , ξlp = −slp,1 gα,1 m−1 lp , R2 = Rlp,cm,α ,

lp,2 α,2

˙ ξrp = srp,3 gα,3 m−1 rp Rdpr,cm,α , R1 = Rrp,cm,α ,

(D.31)

the coupled differential equations are rewritten as ¨ 1 − η1 R˙ 1 − η2 R˙ 2 = ξrp , R

¨ 2 − η3 R˙ 2 − η4 R˙ 1 = ξlp . R

(D.32)

The equations of motion shown in equation (D.32) are a system of two linear second-order inhomogeneous differential equations. In order to rewrite the coupled linear inhomogeneous differential equation (D.32) into a set of first-order linear inhomogeneous equation, a set of new variables are defined first,  R˙ 1 = R3 , R˙ 2 = R4 ,  ˙ ¨ ˙ ˙ (D.33) R = R1 = ξrp + η1 R1 + η2 R2 = ξrp + η1 R3 + η2 R4 ,  ˙3 ¨ 2 = ξlp + η3 R˙ 2 + η4 R˙ 1 = ξlp + η3 R4 + η4 R3 . R4 = R Using these new variables defined in equation (D.33), equation (D.32) can be cast into first-order inhomogeneous equation in matrix form,

~˙ = M f·R ~ + ξ~ → R

 ˙ R1  R˙ 2   R˙ 3 R˙ 4





0   0 =   0 0

0 0 0 0

1 0 η1 η4

  0 R1  R2 1  · η2   R3 η3 R4

The above first-order inhomogeneous equation is equivalent to Z t Z t 0 R1 = R3 dt , R2 = R4 dt0 , t0



 0   0  +    ξrp  . ξlp 

(D.34)

t0

and ·

¸ · ¸ ¸ · ¸ · R˙ 3 ξrp η1 η2 R3 + = · . ξlp η4 η3 R4 R˙ 4 {z } | {z } | {z } | {z } | ~˙ η R

~η R

fη M

97

ξ~

(D.35)

D. Dynamical Casimir Force For the homogeneous system ¸ ¸ · ¸ · R˙ 3 η1 η2 R3 · , = η4 η3 R4 R˙ 4 | {z } | {z } | {z } ·

~η R

fη M

~˙ η R

(D.36)

the eigenvalues are found from the root of the characteristic equation ³ ´ fη ≡ λ2 − [η1 + η3 ] λ + η1 η3 − η2 η4 = 0. det λIe − M The two eigenvalues are η1 + η3 λ3 = + 2

½

1 2 [η1 − η3 ] + η2 η4 4

¾1/2

η1 + η3 λ4 = − 2

,

½

1 2 [η1 − η3 ] + η2 η4 4

¾1/2 .

(D.37)

And, the two corresponding eigenvectors are found to be · ~λ = R `4 R 3

η2 λ3 −η1

(·

¸ ,

1

`4 = R

η2 λ3 − η 1

)−1/2

¸2 +1

,

(D.38)

,

(D.39)

and · ~λ = R `3 R 4

1 λ4 −η1 η2

(

¸ ,

`3 = R

·

λ4 − η1 1+ η2

¸2 )−1/2

` 3 and R ` 4 are the normalization constants. The solutions for the matrix equation (D.36) are then where R · η2 ¸ · ¸ exp (λ4 t) exp (λ3 t) λ −η ~ ~ ~ ` ~ ` 3 1 φλ3 = Rλ3 exp (λ3 t) = R4 , φλ4 = Rλ4 exp (λ4 t) = R3 λ4 −η1 . exp (λ4 t) exp (λ3 t) η2 h i ~ λ (t) , φ ~ λ (t) is given by e (t) = φ The fundamental matrix solution Φ 3 4 " e (t) = Φ

η2 ` λ3 −η1 R4

exp (λ3 t) ` 4 exp (λ3 t) R

` 3 exp (λ4 t) R λ4 −η1 ` η2 R3 exp (λ4 t)

e (t) has an inverse The fundamental matrix solution Φ " λ4 −η1 ` 1 η2 R3 exp (λ4 t) e −1 (t) = ³ ´ Φ ` 4 exp (λ3 t) −R e (t) det Φ

# .

` 3 exp (λ4 t) −R η2 ` λ3 −η1 R4 exp (λ3 t)

(D.40)

# ,

where ¸ ´ ·λ − η 4 1 e `3R ` 4 exp ([λ3 + λ4 ] t) . det Φ (t) = −1 R λ3 − η1 ³

e (t, t0 ) = Φ e (t) · Φ e −1 (t0 ) of equation (D.35) becomes then The principal matrix solution Ψ · ¸ 1 ψ11 (t, t0 ) ψ12 (t, t0 ) e (t, t0 ) = ³ ´ Ψ , ψ21 (t, t0 ) ψ22 (t, t0 ) e (t0 ) det Φ

98

(D.41)

(D.42)

D. Dynamical Casimir Force where ·

¸ λ4 − η1 exp (λ3 t + λ4 t0 ) − exp (λ4 t + λ3 t0 ) , λ3 − η 1

(D.43)

¸ η2 η2 exp (λ4 t + λ3 t0 ) − exp (λ3 t + λ4 t0 ) , λ3 − η 1 λ3 − η1

(D.44)

· ¸ λ4 − η 1 λ4 − η1 ` ` ψ21 (t, t0 ) = R3 R4 exp (λ3 t + λ4 t0 ) − exp (λ4 t + λ3 t0 ) , η2 η2

(D.45)

¸ · λ4 − η 1 ` ` ψ22 (t, t0 ) = R3 R4 exp (λ4 t + λ3 t0 ) − exp (λ3 t + λ4 t0 ) . λ3 − η 1

(D.46)

`3R `4 ψ11 (t, t0 ) = R · `3R `4 ψ12 (t, t0 ) = R

e (t, t0 ) is The inverse of principal matrix solution Ψ e −1 (t, t0 ) = Ψ

1 ³ ´ ³ ´ e e (t0 ) det Ψ (t, t0 ) det Φ

·

ψ22 (t, t0 ) −ψ21 (t, t0 )

−ψ12 (t, t0 ) ψ11 (t, t0 )

¸ ,

(D.47)

where ³ ´ h ³ ´i−2 e (t, t0 ) = det Φ e (t0 ) det Ψ [ψ11 (t, t0 ) ψ22 (t, t0 ) − ψ12 (t, t0 ) ψ21 (t, t0 )] .

(D.48)

Using a variation-of-parameters technique, the solution to the inhomogeneous first-order differential equation (D.35) is Z t ~ η (t) = Ψ e (t, t0 ) · R ~ η (t0 ) + Ψ e (t, t0 ) · e −1 (t0 , t0 ) · ξ~ (t0 ) dt0 , R Ψ t0

where it is understood the multiplications are that of the matrix operations. Substituting into this integral equation the ~ η (t) , ξ~ (t0 ) , Ψ e (t) and Ψ e −1 (t0 ) given by equations (D.35), (D.42) and (D.47), results for R · ¸ µ· ¸ · ¸ · ¸ 1 R3 (t) ψ11 (t, t0 ) ψ12 (t, t0 ) R3 (t0 ) ψ11 (t, t0 ) ψ12 (t, t0 ) ³ ´ = · + R4 (t) ψ21 (t, t0 ) ψ22 (t, t0 ) R4 (t0 ) ψ21 (t, t0 ) ψ22 (t, t0 ) e (t0 ) det Φ    ¸ · ¸ · Z t 0 0 0 1 ψ22 (t , t0 ) −ψ12 (t , t0 ) ξrp (t ) ³ ´ dt0  · · 0 0 0 −ψ (t , t ) ψ (t , t ) ξ (t )   lp 21 0 11 0 0 e t0 det Ψ (t , t ) 0

or  

1 ψ11 (t, t0 ) ´ ψ11 (t, t0 ) R3 (t0 ) + ψ12 (t, t0 ) R4 (t0 ) + ³ ´  e e (t0 ) det Φ (t0 ) det Φ   Z t Z t 0 0 0 0 ψ22 (t , t0 ) ξrp (t ) 0 ψ12 (t , t0 ) ξlp (t ) 0  ψ12 (t, t0 ) ³ ´ dt − ³ ´ dt + ³ ´ × 0 0 e e e (t0 ) t0 det Ψ (t , t0 ) t0 det Ψ (t , t0 ) det Φ   Z t Z t  0 0 0 0 ψ (t , t ) ξ (t ) ψ (t , t ) ξ (t ) 11 21 ³ 0 lp ´ dt0 − ³ 0 rp ´ dt0  , ×  e (t0 , t0 ) e (t0 , t0 ) t0 det Ψ t0 det Ψ

R3 (t) =

³

99

(D.49)

D. Dynamical Casimir Force   1 ψ21 (t, t0 ) ³ ´ ψ21 (t, t0 ) R3 (t0 ) + ψ22 (t, t0 ) R4 (t0 ) + ³ ´ R4 (t) =  e (t0 ) e (t0 ) det Φ det Φ   Z t Z t 0 0 0 0 ψ (t , t ) ξ (t ) ψ (t , t ) ξ (t ) ψ22 (t, t0 ) 22 12 ³ 0 rp ´ dt0 − ³ 0 lp ´ dt0  + ³ ´ × e (t0 , t0 ) e (t0 , t0 ) e (t0 ) t0 det Ψ t0 det Ψ det Φ   Z t Z t 0 0 0 0 ψ11 (t , t0 ) ξlp (t ) 0 ψ21 (t , t0 ) ξrp (t ) 0  ³ ³ ´ dt − ´ dt × .  e (t0 , t ) e (t0 , t ) t0 det Ψ t0 det Ψ 0

(D.50)

0

It is noted from equation (D.34), R3 (t0 ) and R4 (t0 ) are initial speeds, R˙ rp,cm,α (t0 ) ≡ R˙ 1 (t0 ) = R3 (t0 ) ,

R˙ lp,cm,α (t0 ) ≡ R˙ 2 (t0 ) = R4 (t0 ) .

Hence, ¸−1 · ψ11 (t, t0 ) R˙ rp,cm,α (t0 ) + ψ12 (t, t0 ) R˙ lp,cm,α (t0 ) λ4 (; t0 ) − η1 (; t0 ) −1 R˙ rp,cm,α (t) = λ3 (; t0 ) − η1 (; t0 ) exp ([λ3 (; t0 ) + λ4 (; t0 )] t0 ) Z t 0 0 ψ22 (t , t0 ) ξrp (t ) − ψ12 (t0 , t0 ) ξlp (t0 ) + ψ11 (t, t0 ) dt0 + ψ12 (t, t0 ) 0 , t ) ψ (t0 , t ) − ψ (t0 , t ) ψ (t0 , t ) ψ (t 11 0 22 0 12 0 21 0 t0 Z t ψ11 (t0 , t0 ) ξlp (t0 ) − ψ21 (t0 , t0 ) ξrp (t0 ) dt0 , × 0 0 0 0 t0 ψ11 (t , t0 ) ψ22 (t , t0 ) − ψ12 (t , t0 ) ψ21 (t , t0 )

(D.51)

· ¸−1 λ4 (; t0 ) − η1 (; t0 ) ψ21 (t, t0 ) R˙ rp,cm,α + ψ22 (t, t0 ) R˙ lp,cm,α (t0 ) ˙ Rlp,cm,α (t) = −1 λ3 (; t0 ) − η1 (; t0 ) exp ([λ3 (; t0 ) + λ4 (; t0 )] t0 ) Z t 0 0 ψ22 (t , t0 ) ξrp (t ) − ψ12 (t0 , t0 ) ξlp (t0 ) + ψ21 (t, t0 ) dt0 + ψ22 (t, t0 ) 0 0 0 0 t0 ψ11 (t , t0 ) ψ22 (t , t0 ) − ψ12 (t , t0 ) ψ21 (t , t0 ) Z t ψ11 (t0 , t0 ) ξlp (t0 ) − ψ21 (t0 , t0 ) ξrp (t0 ) × dt0 , (D.52) 0 , t ) ψ (t0 , t ) − ψ (t0 , t ) ψ (t0 , t ) ψ (t 11 0 22 0 12 0 21 0 t0 ³ ´ ³ ´ e (t0 ) and det Ψ e (t0 , t0 ) from equations (D.41) and where substitutions have been made for the determinants det Φ (D.48). It is to be understood that the notation (; t0 ) on η1 , λ3 and λ4 implies implicit time dependence for these terms. Finally, integration of both sides of equations (D.51) and (D.52) with respect to time gives the results ¸−1 Z t " · λ4 (; t0 ) − η1 (; t0 ) ψ11 (τ, t0 ) R˙ rp,cm,α (t0 ) + ψ12 (τ, t0 ) R˙ lp,cm,α (t0 ) Rrp,cm,α (t) = −1 λ3 (; t0 ) − η1 (; t0 ) exp ([λ3 (; t0 ) + λ4 (; t0 )] t0 ) t0 Z τ ψ22 (t0 , t0 ) ξrp (t0 ) − ψ12 (t0 , t0 ) ξlp (t0 ) + ψ11 (τ, t0 ) dt0 + ψ12 (τ, t0 ) 0 , t ) ψ (t0 , t ) − ψ (t0 , t ) ψ (t0 , t ) ψ (t 11 0 22 0 12 0 21 0 t0 ¸ Z τ ψ11 (t0 , t0 ) ξlp (t0 ) − ψ21 (t0 , t0 ) ξrp (t0 ) 0 × dt dτ + Rrp,cm,α (t0 ) , (D.53) 0 0 0 0 t0 ψ11 (t , t0 ) ψ22 (t , t0 ) − ψ12 (t , t0 ) ψ21 (t , t0 ) ¸−1 Z t " ψ21 (τ, t0 ) R˙ rp,cm,α (t0 ) + ψ22 (τ, t0 ) R˙ lp,cm,α (t0 ) λ4 (; t0 ) − η1 (; t0 ) Rlp,cm,α (t) = −1 λ3 (; t0 ) − η1 (; t0 ) exp ([λ3 (; t0 ) + λ4 (; t0 )] t0 ) t0 Z τ ψ22 (t0 , t0 ) ξrp (t0 ) − ψ12 (t0 , t0 ) ξlp (t0 ) dt0 + ψ22 (τ, t0 ) + ψ21 (τ, t0 ) 0 , t ) ψ (t0 , t ) − ψ (t0 , t ) ψ (t0 , t ) ψ (t 11 0 22 0 12 0 21 0 t0 ¸ Z τ ψ11 (t0 , t0 ) ξlp (t0 ) − ψ21 (t0 , t0 ) ξrp (t0 ) 0 × dt dτ + Rlp,cm,α (t0 ) , (D.54) 0 0 0 0 t0 ψ11 (t , t0 ) ψ22 (t , t0 ) − ψ12 (t , t0 ) ψ21 (t , t0 ) ·

100

D. Dynamical Casimir Force where the terms ψ11 , ψ12 , ψ21 and ψ22 are defined in equations (D.43), (D.44), (D.45) and (D.46). The remaining integrations are straightforward; hence, their explicit forms are not shown. As a closing remark of this section, one may argue that for the static case, R˙ rp,cm,α (t0 ) and R˙ lp,cm,α (t0 ) must be zero because the conductors seem to be fixed in position. This argument is flawed for any wall totally fixed in position upon impact would require an infinite amount of energy. One has to consider the conservation of momentum simultaneously. The wall has to have moved by the amount 4Rwall = R˙ wall 4t, where 4t is the total duration of impact, and R˙ wall is calculated from the momentum conservation and it is non-zero. The same argument can be applied to the apparatus shown in Figure 3.10. For that system °  ° ˙ rp,cm,α (t0 ) = ° ~˙ lp,3 (t0 ) + R ~˙ rp,2 (t0 )°  R R °,  °  1 0 k~ pvirtual−photon k = Hns ,< (t0 ) , ° °  c   R˙ lp,cm,α (t0 ) = ° ~˙ rp,1 (t0 ) + R ~˙ lp,2 (t0 )° °R ° or, for simplicity, assuming impact without any angle, R˙ rp,cm,α (t0 ) =

° 2 ° °Hn0 ,3 (t0 ) − Hn0 ,2 (t0 )° , s s mrp c

R˙ lp,cm,α (t0 ) =

° 2 ° °Hn0 ,1 (t0 ) − Hn0 ,2 (t0 )° , s s mlp c

where the difference under the magnitude symbol implies that the energies from different regions act to counteract each other.

101

E. Extended List of References These are the additional references pertaining to the Casimir effect which have not been explicitly used in this thesis. Nonetheless, these references were listed here to provide further information to those with possible research interest in the field of physics concerning the Casimir phenomenon.

Larry Spruch, “Long-Range Casimir Interactions,” Science 272, 1452-1455 (1996). Peter W. Milonni and Mei-Li Shih, “Casimir Forces,” Contemporary Physics 33, No. 5, 313-322 (1992). I. J. R. Aitchison, “Nothing’s plenty: The Vacuum in Modern Quantum Field Theory,” Contemporary Physics 26, No. 4, 333-391 (1985). M. H. Wallis, “Essay Review: Field Theorists Strike Back—Stochastic Electrodynamics,” Contemporary Physics 39, 483-486 (1998). R. Weigand and J. M. Guerra, “The Vacuum Field Energy in a Constant Volume Cavity,” European J. Phys. 18, 40-42 (1997). E. Elizalde and A. Romeo, “Essentials of the Casimir Effect and its Computation,” Am. J. Phys. 59, No. 8, 711-719 (1991). V. M. Mostepanenko and N. N. Trunov, “The Casimir Effect and its Applications,” Oxford (1997). G. Barton, “New Aspects of the Casimir Effect: Fluctuations and Radiative Reaction,” Advances in Atomic and Molecular Physics, Suppl. 2, P. R. Berman, ed., Academic Press, 425-458, New York (1994). Marc-Thierry Jaekel and Serge Reynaud, “Movement and Fluctuations of the Vacuum,” Rep. Prog. Physics 60, 863887 (1997). I. E. Dzyaloshinskii, E. M. Lifshitz, and L. P. Pitaevskii, “The General Theory of van der Waals Forces,” Advances in Physics 10, 165-209 (1961). G. Bressi, G. Garugno, R. Onofrio, and G. Ruoso, “Measurement of the Casimir Force Between Parallel Metallic Surfaces,” Phys. Rev. Lett. 88, 041804 (2002). T. Ederth, “Template-stripped Gold Surfaces with 0.4-nm RMS Roughness Suitable for Force Measurements: Application to the Casimir Force in the 20-100-nm Range,” Phys. Rev. A 62, 062104 (2000). M. Bordag, B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko, “Stronger Constraints for Nanometer Scale Yukawa-type Hypothetical Interactions the new Measurement of the Casimir Force,” Phys. Rev. D 60, 055004 (1999). V. B. Svetovoy and M. V. Lokhanin, “Precise Calculation of the Casimir Force Between Gold Surfaces,” Mod. Phys. Lett. A 15, 1437-1444 (2000). F. Chen, U. Mohideen, G. L. Klimchitskaya and V. M. Mostepanenko, “Demonstration of the Lateral Casimir Force,” Phys. Rev. Lett. 88, 101801 (2002).

102

E. Extended List of References

M. Bordag, B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko, “Constraints for Hypothetical Interactions from a Recent Demonstration of the Casimir Force and Some Possible Improvements,” Phys. Rev. D 58 (1998). G. L. Klimchitskaya, Anushree Roy, U. Mohideen, and V. M. Mostepanenko, “Complete Roughness and Conductivity Corrections for Casimir Force Measurements,” Phys. Rev. A 60, 3487-3495 (1999). V. B. Bezerra, G. L. Klimchitskaya, and V. M. Mostepanenko, “Higher-order Conductivity Corrections to the Casimir Force,” Phys. Rev. A 62, 014012 (2000). T. Emig, A. Hanke, R. Golestanian, and M. Kardar, “Probing the Strong Boundary Shape Dependence of the Casimir Force,” Phys. Rev. Lett. 87, 260402 (2001). R. Balian and B. Duplantier, “Electromagnetic Waves Near Perfect Conductors. I. Multiple Scattering Expansions. Distribution of Modes,” Annals of Physics 104, 300-335, New York (1977). R. Balian and B. Duplantier, “Electromagnetic Waves Near Perfect Conductors. II. Casimir Effect,” Annals of Physics 112, 165-208, New York (1978). M. Bordag and J. Lindig, “Radiative Corrections to the Casimir Force on a Sphere,” Phys. Rev. D 58, 1-16 (1998). V. Hushwater, “Repulsive Casimir Force as a Result of Vacuum Radiation Pressure,” American Journal of Physics 65, 381-384 (1997). Martin Schaden, Larry Spruch, and Fei Zhou, “Unified Treatment of Some Casimir Energies and Lamb Shifts: A Dielectric Between Two Ideal Conductors,” Phys. Rev. A 57, 1108-1120 (1998). Martin Schaden and Larry Spruch, “Infinity-free Semiclassical Evaluation of Casimir Effects,” Phys. Rev. A 58, 935953 (1998). A. Bulgac and A. Wirzba, “Casimir Interaction Among Objects Immersed in a Fermionic Environment,” Phys. Rev. Lett. 87 (2001). V. B. Bezerra, G. L. Klimchitskaya, and C. Romero, “Casimir Force Between a Flat Plate and a Spherical Lens: Application to the Result of a New Experiment,” Mod. Phys. Lett. A 12, 2613-2622 (1997). V. B. Bezerra, G. L. Klimchitskaya, and C. Romero, “Surface Roughness Contribution to the Casimir Interaction Between an Isolated Atom and a Cavity Wall,” Phys. Rev. A 61 (2000). C. Genet, A. Lambrecht, and S. Reynaud, “Correlation Between Plasma and Temperature Corrections to the Casimir Force,” Int. J. Mod. Phys. A 17, 761-766 (2002). J. Feinberg, A. Mann, M. Revzen, “Casimir Effect: The Classical Limit,” Annals of Physics 288, 103, New York (2001). L. Spruch, “Classical Casimir Interactions of Some Simple Systems at Very High Temperature,” Phys. Rev. A 66, 022103 (2002). M. Schaden and L. Spruch, “Classical Casimir Effect: The Interaction of Ideal Parallel Walls at a Finite Temperature,” Phys. Rev. A 65, 034101 (2002). M. Schaden and L. Spruch, “Semiclassical Casimir Energies at Finite Temperature,” Phys. Rev. A 65, 022108 (2002).

103

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