Advertisement

Numerical Methods for Computing Casimir Interactions

  • Steven G. JohnsonEmail author
Chapter
Part of the Lecture Notes in Physics book series (LNP, volume 834)

Abstract

We review several different approaches for computing Casimir forces and related fluctuation-induced interactions between bodies of arbitrary shapes and materials. The relationships between this problem and well known computational techniques from classical electromagnetism are emphasized. We also review the basic principles of standard computational methods, categorizing them according to three criteria—choice of problem, basis, and solution technique—that can be used to classify proposals for the Casimir problem as well. In this way, mature classical methods can be exploited to model Casimir physics, with a few important modifications.

Keywords

Imaginary Frequency Perfectly Match Layer Casimir Force Casimir Energy Perfect Electric Conductor 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

This work was supported in part by the Army Research Office through the ISN under contract W911NF-07-D-0004, by the MIT Ferry Fund, and by the Defense Advanced Research Projects Agency (DARPA) under contract N66001-09-1-2070-DOD. We are especially grateful to our students, A. W. Rodriguez, A. P. McCauley, and H. Reid for their creativity and energy in pursuing Casimir simulations. We are also grateful to our colleagues F. Capasso, D. Dalvit, T. Emig, R. Jaffe, J. D. Joannopoulos, M. Kardar, M. Levin, M. Lončar, J. Munday, S. J. Rahi, and J. White, for their many suggestions over the years.

References

  1. 1.
    Chew, W.C., Jian-Ming, J., Michielssen, E., Jiming, S.: Fast and Efficient Algorithms in Computational Electromagnetics. Artech, Norwood, MA (2001)Google Scholar
  2. 2.
    Taflove, A., Hagness, S.C.: Computational Electrodynamics: The Finite-Difference Time-Domain Method. Artech, Norwood, MA (2000)zbMATHGoogle Scholar
  3. 3.
    Volakis, J.L., Chatterjee, A., Kempel, L.C.: Finite Element Method Electromagnetics: Antennas, Microwave Circuits, and Scattering Applications. IEEE Press, New York (2001)Google Scholar
  4. 4.
    Zhu, Y., Cangellaris, A.C.: Multigrid Finite Element Methods for Electromagnetic Field Modelling. John Wiley and Sons, Hooboke, NJ (2006)CrossRefGoogle Scholar
  5. 5.
    Yasumoto, K. (ed.): Electromagnetic Theory and Applications for Photonic Crystals. CRC Press, Boca Raton, FL (2005)Google Scholar
  6. 6.
    Joannopoulos, J.D., Johnson, S.G., Winn, J.N., Meade, R.D.: Photonic Crystals: Molding the Flow of Light, 2nd edn. Princeton University Press, Princeton, NJ (2008)Google Scholar
  7. 7.
    Jin, J.: The Finite Element Method in Electromagnetics, 2nd edn. Wiley, New York (2002)Google Scholar
  8. 8.
    Rao, S.M., Balakrishnan, N.: Computational electromagnetics. Curr. Sci. 77(10), 1343–1347 (1999)Google Scholar
  9. 9.
    Reid, M.T.H., Rodriguez, A.W., White, J., Johnson, S.G.: Efficient computation of three-dimensional Casimir forces. Phys. Rev. Lett. 103(4), 040–401 (2009)CrossRefGoogle Scholar
  10. 10.
    Reynaud, S., Maia Neto, P.A., Lambrecht, A.: Casimir energy and geometry: Beyond the proximity force approximation. J. Phys. A: Math. Theor. 41, 164–004 (2008)CrossRefGoogle Scholar
  11. 11.
    Rodriguez, A., Ibanescu, M., Iannuzzi, D., Capasso, F., Joannopoulos, J.D., Johnson, S.G.: Computation and visualization of Casimir forces in arbitrary geometries: Non-monotonic lateral-wall forces and failure of proximity force approximations. Phys. Rev. Lett. 99(8), 080–401 (2007)CrossRefGoogle Scholar
  12. 12.
    Rodriguez, A., Ibanescu, M., Iannuzzi, D., Joannopoulos, J.D., Johnson, S.G.: Virtual photons in imaginary time: Computing Casimir forces in arbitrary geometries via standard numerical electromagnetism. Phys. Rev. 76(3), 032–106 (2007)Google Scholar
  13. 13.
    Rodriguez, A.W., McCauley, A.P., Joannopoulos, J.D., Johnson, S.G.: Casimir forces in the time domain: Theory. Phys. Rev. A 80(1), 012–115 (2009)CrossRefGoogle Scholar
  14. 14.
    Emig, T., Graham, N., Jaffe, R.L., Kardar, M.: Casimir forces between arbitrary compact objects. Phys. Rev. Lett. 99, 170–403 (2007)CrossRefGoogle Scholar
  15. 15.
    Emig, T., Hanke, A., Golestanian, R., Kardar, M.: Probing the strong boundary shape dependence of the Casimir force. Phys. Rev. Lett. 87, 260–402 (2001)CrossRefGoogle Scholar
  16. 16.
    Emig, T., Jaffe, R.L., Kardar, M., Scardicchio, A.: Casimir interaction between a plate and a cylinder. Phys. Rev. Lett. 96, 080–403 (2006)CrossRefGoogle Scholar
  17. 17.
    Lambrecht, A., Maia Neto, P.A., Reynaud, S.: The Casimir effect within scattering theory. New J. Phys. 8, 243 (2008)CrossRefGoogle Scholar
  18. 18.
    Lambrecht, A., Marachevsky, V.I.: Casimir interactions of dielectric gratings. Phys. Rev. Lett. 101, 160–403 (2008)CrossRefGoogle Scholar
  19. 19.
    Xiong, J.L., Chew, W.C.: Efficient evaluation of Casimir force in z-invariant geometries by integral equation methods. Appl. Phys. Lett. 95, 154–102 (2009)Google Scholar
  20. 20.
    Xiong, J.L., Tong, M.S., Atkins, P., Chew, W.C.: Efficient evaluation of Casimir force in arbitrary three-dimensional geometries by integral equation methods. Phys. Lett. A 374(25), 2517–2520 (2010)MathSciNetADSzbMATHCrossRefGoogle Scholar
  21. 21.
    Rahi, S.J., Emig, T., Jaffe, R.L., Kardar, M.: Casimir forces between cylinders and plates. Phys. Rev. A 78, 012–104 (2008)CrossRefGoogle Scholar
  22. 22.
    Rahi, S.J., Emig, T., Jaffe, R.L., Kardar, M.: Scattering theory approach to electrodynamic casimir forces. Phys. Rev. D 80, 085–021 (2009)CrossRefGoogle Scholar
  23. 23.
    Pasquali, S., Maggs, A.C.: Numerical studies of Lifshitz interactions between dielectrics. Phys. Rev. A. 79, 020–102 (2009)CrossRefGoogle Scholar
  24. 24.
    Maia Neto, P.A., Lambrecht, A., Reynaud, S.: Roughness correction to the Casimir force: Beyond the proximity force approximation. Europhys. Lett. 69, 924–930 (2005)ADSCrossRefGoogle Scholar
  25. 25.
    Maia Neto, P.A., Lambrecht, A., Reynaud, S.: Casimir energy between a plane and a sphere in electromagnetic vacuum. Phys. Rev. A 78, 012–115 (2008)CrossRefGoogle Scholar
  26. 26.
    Kenneth, O., Klich, I.: Casimir forces in a T-operator approach. Phys. Rev. B 78, 014–103 (2008)CrossRefGoogle Scholar
  27. 27.
    McCauley, A.P., Rodriguez, A.W., Joannopoulos, J.D., Johnson, S.G.: Casimir forces in the time domain: Applications. Phys. Rev. A 81, 012–119 (2010)CrossRefGoogle Scholar
  28. 28.
    Casimir, H.B.G.: On the attraction between two perfectly conducting plates. Proc. K. Ned. Akad. Wet. 51, 793–795 (1948)zbMATHGoogle Scholar
  29. 29.
    Lamoreaux, S.K.: Demonstration of the Casimir force in the 0.6 to 6\(\mu\)m range. Phys. Rev. Lett. 78, 5–8 (1997)ADSCrossRefGoogle Scholar
  30. 30.
    Derjaguin, B.V., Abrikosova, I.I., Lifshitz, E.M.: Direct measurement of molecular attraction between solids separated by a narrow gap. Q. Rev. Chem. Soc. 10, 295–329 (1956)CrossRefGoogle Scholar
  31. 31.
    Bordag, M.: Casimir effect for a sphere and a cylinder in front of a plane and corrections to the proximity force theorem. Phys. Rev. D 73, 125–018 (2006)CrossRefGoogle Scholar
  32. 32.
    Bordag, M., Mohideen, U., Mostepanenko, V.M.: New developments in the Casimir effect. Phys. Rep. 353, 1–205 (2001)MathSciNetADSzbMATHCrossRefGoogle Scholar
  33. 33.
    Casimir, H.B.G., Polder, D.: The influence of retardation on the London-van der Waals forces. Phys. Rev. 13(4), 360–372 (1948)ADSCrossRefGoogle Scholar
  34. 34.
    Sedmik, R., Vasiljevich, I., Tajmar, M.: Detailed parametric study of Casimir forces in the Casimir polder approximation for nontrivial 3d geometries. J. Comput. Aided Mater. Des. 14(1), 119–132 (2007)ADSCrossRefGoogle Scholar
  35. 35.
    Tajmar, M.: Finite element simulation of Casimir forces in arbitrary geometries. Intl. J. Mod. Phys. C 15(10), 1387–1395 (2004)ADSzbMATHCrossRefGoogle Scholar
  36. 36.
    Jaffe, R.L., Scardicchio, A.: Casimir effect and geometric optics. Phys. Rev. Lett. 92, 070–402 (2004)CrossRefGoogle Scholar
  37. 37.
    Hertzberg, M.P., Jaffe, R.L., Kardar, M., Scardicchio, A.: Casimir forces in a piston geometry at zero and finite temperatures. Phys. Rev. D 76, 045–016 (2007)Google Scholar
  38. 38.
    Zaheer, S., Rodriguez, A.W., Johnson, S.G., Jaffe, R.L.: Optical-approximation analysis of sidewall-spacing effects on the force between two squares with parallel sidewalls. Phys. Rev. A 76(6), 063–816 (2007)CrossRefGoogle Scholar
  39. 39.
    Scardicchio, A., Jaffe, R.L.: Casimir effects: An optical approach I. Foundations and examples. Nucl. Phys. B 704(3), 552–582 (2005)MathSciNetADSzbMATHCrossRefGoogle Scholar
  40. 40.
    Van Enk, S.J.: The Casimir effect in dielectrics: A numerical approach. J. Mod. Opt. 42(2), 321–338 (1995)MathSciNetADSzbMATHCrossRefGoogle Scholar
  41. 41.
    Jackson, J.D.: Classical Electrodynamics, 3rd edn. Wiley, New York (1998)Google Scholar
  42. 42.
    Schaubert, D.H., Wilton, D.R., Glisson, A.W.: A tetrahedral modeling method for electromagnetic scattering by arbitrarily shaped inhomogeneous dielectric bodies. IEEE Trans. Antennas Propagat. 32, 77–85 (1984)ADSCrossRefGoogle Scholar
  43. 43.
    Bienstman, P., Baets, R.: Optical modelling of photonic crystals and VCSELs using eigenmode expansion and perfectly matched layers. Opt. Quantum Electron. 33(4–5), 327–341 (2001)CrossRefGoogle Scholar
  44. 44.
    Bienstman, P., Baets, R.: Advanced boundary conditions for eigenmode expansion models. Opt. Quantum Electron. 34, 523–540 (2002)Google Scholar
  45. 45.
    Willems, J., Haes, J., Baets, R.: The bidirectional mode expansion method for two dimensional waveguides. Opt. Quantum Electron. 27(10), 995–1007 (1995)CrossRefGoogle Scholar
  46. 46.
    Moharam, M.G., Grann, E.B., Pommet, D.A., Gaylord, T.K.: Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings. J. Opt. Soc. Am. A 12, 1068–1076 (1995)ADSCrossRefGoogle Scholar
  47. 47.
    Moharam, M.G., Pommet, D.A., Grann, E.B., Gaylord, T.K.: Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: Enhanced transmittance matrix approach. J. Opt. Soc. Am. A 12, 1077–1086 (1995)ADSCrossRefGoogle Scholar
  48. 48.
    Katsenelenbaum, B.Z., Mercaderdel Río, L., Pereyaslavets, M., Sorolla Ayza, M., Thumm, M.: Theory of Nonuniform Waveguides: The Cross-Section Method (1998)Google Scholar
  49. 49.
    Christ, A., Hartnagel, H.L.: Three-dimensional finite-difference method for the analysis of microwave-device embedding. IEEE Trans. Microwave Theory Tech. 35(8), 688–696 (1987)ADSCrossRefGoogle Scholar
  50. 50.
    Strikwerda, J.: Finite Difference Schemes and Partial Differential Equations. Wadsworth and Brooks/Cole, Pacific Grove, CA (1989)zbMATHGoogle Scholar
  51. 51.
    Bonnet, M.: Boundary Integral Equation Methods for Solids and Fluids. Wiley, Chichester (1999)Google Scholar
  52. 52.
    Hackbush, W., Verlag, B.: Integral Equations: Theory and Numerical Treatment. Birkhauser Verlag, Basel, Switzerland (1995)Google Scholar
  53. 53.
    Boyd, J.P.: Chebychev and Fourier Spectral Methods, 2nd edn. Dover, New York (2001)Google Scholar
  54. 54.
    Ditkowski, A., Dridi, K., Hesthaven, J.S.: Convergent Cartesian grid methods for Maxwell’s equations in complex geometries. J. Comp. Phys. 170, 39–80 (2001)MathSciNetADSzbMATHCrossRefGoogle Scholar
  55. 55.
    Oskooi, A.F., Kottke, C., Johnson, S.G.: Accurate finite-difference time-domain simulation of anisotropic media by subpixel smoothing. Opt. Lett. 34, 2778–2780 (2009)ADSCrossRefGoogle Scholar
  56. 56.
    Stratton, J.A.: Electromagnetic Theory. McGraw-Hill, New York (1941)zbMATHGoogle Scholar
  57. 57.
    Johnson, S.G., Joannopoulos, J.D.: Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis. Opt. Express 8(3), 173–190 (2001)ADSCrossRefGoogle Scholar
  58. 58.
    Kuo, S.H., Tidor, B., White, J.: A meshless, spectrally accurate, integral equation solver for molecular surface electrostatics. ACM J. Emerg. Technol. Comput. Syst. 4(2), 1–30 (2008)CrossRefGoogle Scholar
  59. 59.
    Sladek, V., Sladek, J.: Singular Integrals in Boundary Element Methods. WIT Press, Southampton, UK (1998)zbMATHGoogle Scholar
  60. 60.
    Taylor, D.J.: Accurate and efficient numerical integration of weakly singular integrals in Galerkin EFIE solutions. IEEE Trans. Antennas Propagat. 51, 1630–1637 (2003)ADSCrossRefGoogle Scholar
  61. 61.
    Tong, M.S., Chew, W.C.: Super-hyper singularity treatment for solving 3d electric field integral equations. Microwave Opt. Tech. Lett. 49, 1383–1388 (2006)CrossRefGoogle Scholar
  62. 62.
    Nédélec, J.C.: Mixed finite elements in \({\mathbb{R}}^3\). Numerische Mathematik. 35, 315–341 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  63. 63.
    Rao, S.M., Wilton, D.R., Glisson, A.W.: Electromagnetic scattering by surfaces of arbitrary shape. IEEE Trans. Antennas Propagat. 30, 409–418 (1982)ADSCrossRefGoogle Scholar
  64. 64.
    Cai, W., Yu, Y., Yuan, XC.: Singularity treatment and high-order RWG basis functions for integral equations of electromagnetic scattering. Intl. J. Numer. Meth. Eng. 53, 31–47 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  65. 65.
    Trefethen, L.N., Bau, D.: Numerical Linear Algebra. SIAM, Philadelphia (1997)CrossRefGoogle Scholar
  66. 66.
    Davis, T.A.: Direct Methods for Sparse Linear Systems. SIAM, Philadelphia (2006)CrossRefGoogle Scholar
  67. 67.
    Barrett, R., Berry, M., Chan, T.F., Demmel, J., Donato, J., Dongarra, j., Eijkhout, V., Pozo, R., Romine, C., Vander Vorst, H.: Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd edn. SIAM, Philadelphia (1994)CrossRefGoogle Scholar
  68. 68.
    Bai, Z., Demmel, J., Dongarra, J., Ruhe, A., VanDer Vorst, H.: Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide. SIAM, Philadelphia (2000)zbMATHCrossRefGoogle Scholar
  69. 69.
    Anderson, E., Bai, Z., Bischof, C., Blackford, S., Demmel, J., Dongarra, J., Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A., McKenney, A.: LAPACK Users’ Guide, 3rd edn. SIAM, Philadelphia (1999)CrossRefGoogle Scholar
  70. 70.
    Duff, I.S., Erisman, A.M., Reid, J.K.: On George’s nested dissection method. SIAM J. Numer. Anal. 13, 686–695 (1976)MathSciNetADSzbMATHCrossRefGoogle Scholar
  71. 71.
    Phillips, J.R., White, J.K.: A precorrected-FFT method for electrostatic analysis of complicated 3-D structures. IEEE Trans. Comput. Aided Des. 16, 1059–1072 (1997)CrossRefGoogle Scholar
  72. 72.
    Greengard, L., Rokhlin, V.: A fast algorithm for particle simulations. J. Comp. Phys. 73, 325–348 (1987)MathSciNetADSzbMATHCrossRefGoogle Scholar
  73. 73.
    Greengard, L., Gueyffier, D., Martinsson, P.G., Rokhlin, V.: Fast direct solvers for integral equations in complex three-dimensional domains. Acta Numer 18, 243–275 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  74. 74.
    Milonni, P.W.: The Quantum Vacuum: An Introduction to Quantum Electrodynamics. Academic Press, San Diego (1993)Google Scholar
  75. 75.
    Cohen-Tannoudji, C., Din, B., Laloë, F.: Quantum Mechanics. Hermann, Paris (1977)Google Scholar
  76. 76.
    Strang, G.: Computational Science and Engineering. Wellesley-Cambridge Press, Wellesley, MA (2007)Google Scholar
  77. 77.
    Lamoreaux, S.K.: The Casimir force: background, experiments, and applications. Rep. Prog. Phys. 68, 201–236 (2005)ADSCrossRefGoogle Scholar
  78. 78.
    Ford, L.H.: Spectrum of the Casimir effect and the Lifshitz theory. Phys. Rev. A 48, 2962–2967 (1993)ADSCrossRefGoogle Scholar
  79. 79.
    Nesterenko, V.V., Pirozhenko, I.G.: Simple method for calculating the Casimir energy for a sphere. Phys. Rev. D 57, 1284–1290 (1998)ADSCrossRefGoogle Scholar
  80. 80.
    Cognola, G., Elizalde, E., Kirsten, K.: Casimir energies for spherically symmetric cavities. J. Phys. A 34, 7311–7327 (2001)MathSciNetADSzbMATHCrossRefGoogle Scholar
  81. 81.
    Lifshitz, E.M., Pitaevskii, L.P.: Statistical Physics: Part 2. Pergamon, Oxford (1980)Google Scholar
  82. 82.
    Boyd, J.P.: Exponentially convergent Fourier–Chebyshev quadrature schemes on bounded and infinite intervals. J. Sci. Comput. 2, 99–109 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  83. 83.
    Stroud, A.H., Secrest, D.: Gaussian Quadrature Formulas. Prentice-Hall, Englewood Cliffs, NJ (1966)zbMATHGoogle Scholar
  84. 84.
    Piessens, R., de Doncker-Kapenga, E., Uberhuber, C., Hahaner, D.: QUADPACK: A Subroutine Package for Automatic Integration. Springer-Verlag, Berlin (1983)zbMATHGoogle Scholar
  85. 85.
    Buhmann, S.Y., Scheel, S.: Macroscopic quantum electrodynamics and duality. Phys. Rev. Lett. 102, 140–404 (2009)CrossRefGoogle Scholar
  86. 86.
    Gel’fand, I.M., Shilov, G.E.: Generalized Functions. Academic Press, New York (1964)zbMATHGoogle Scholar
  87. 87.
    Milton, K.A., Wagner, J., Parashar, P., Brevek, I.: Casimir energy, dispersion, and the Lifshitz formula. Phys. Rev. D 81, 065–007 (2010)CrossRefGoogle Scholar
  88. 88.
    Economou, E.N.: Green’s Functions in Quantum Physics, 3rd edn. Springer, Heidelberg (2006)Google Scholar
  89. 89.
    Zhao, J.S., Chew, W.C.: Integral equation solution of Maxwell’s equations from zero frequency to microwave frequencies. IEEE Trans. Antennas Propagat. 48, 1635–1645 (2000)MathSciNetADSCrossRefGoogle Scholar
  90. 90.
    Epstein, C.L., Greengard, L.: Debye sources and the numerical solution of the time harmonic Maxwell equations. Commun. Pure Appl. Math. 63, 413–463 (2009)MathSciNetGoogle Scholar
  91. 91.
    Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes in C: The Art of Scientific Computing, 2nd edn. Cambridge Univ. Press,   (1992)Google Scholar
  92. 92.
    Hoye, J.S., Brevik, I., Aarseth, J.B., Milton, K.A.: What is the temperature dependence of the casimir effect? J. Phys. A: Math. Gen. 39(20), 6031–6038 (2006)MathSciNetADSCrossRefGoogle Scholar
  93. 93.
    Rodriguez, A.W., Munday, J., Dalvit, D.A.R., Capasso, F., Joannopoulos, J.D., Johnson, S.G.: Stable suspension and dispersion-induced transition from repulsive Casimir forces between fluid-separated eccentric cylinders. Phys. Rev. Lett. 101(19), 190–404 (2008)CrossRefGoogle Scholar
  94. 94.
    Pitaevskii, L.P.: Comment on Casimir force acting on magnetodielectric bodies in embedded in media. Phys. Rev. A 73, 047–801 (2006)CrossRefGoogle Scholar
  95. 95.
    Munday, J., Capasso, F., Parsegian, V.A.: Measured long-range repulsive Casimir-lifshitz forces. Nature 457, 170–173 (2009)ADSCrossRefGoogle Scholar
  96. 96.
    Munday, J.N., Capasso, F.: Precision measurement of the Casimir-Lifshitz force in a fluid. Phys. Rev. A 75, 060–102 (2007)CrossRefGoogle Scholar
  97. 97.
    Chew, W.C., Weedon, W.H.: A 3d perfectly matched medium from modified Maxwell’s equations with stretched coordinates. Microwave Opt. Tech. Lett. 7(13), 599–604 (1994)ADSCrossRefGoogle Scholar
  98. 98.
    Zhao, L., Cangellaris, A.C.: A general approach for the development of unsplit-field time-domain implementations of perfectly matched layers for FDTD grid truncation. IEEE Microwave and Guided Wave Lett. 6(5), 209–211 (1996)CrossRefGoogle Scholar
  99. 99.
    Teixeira, F.L., Chew, W.C.: General closed-form PML constitutive tensors to match arbitrary bianisotropic and dispersive linear media. IEEE Microwave and Guided Wave Lett. 8(6), 223–225 (1998)CrossRefGoogle Scholar
  100. 100.
    Ward, A.J., Pendry, J.B.: Refraction and geometry in Maxwell’s equations. J. Mod. Opt. 43(4), 773–793 (1996)MathSciNetADSzbMATHCrossRefGoogle Scholar
  101. 101.
    Farjadpour, A., Roundy, D., Rodriguez, A., Ibanescu, M., Bermel, P., Joannopoulos, J.D., Johnson, S.G., Burr, G.: Improving accuracy by subpixel smoothing in FDTD. Opt. Letters. 31, 2972–2974 (2006)ADSCrossRefGoogle Scholar
  102. 102.
    Kottke, C., Farjadpour, A., Johnson, S.G.: Perturbation theory for anisotropic dielectric interfaces, and application to sub-pixel smoothing of discretized numerical methods. Phys. Rev. E 77, 036–611 (2008)CrossRefGoogle Scholar
  103. 103.
    Zhao, S.: High order matched interface and boundary methods for the Helmholtz equation in media with arbitrarily curved interfaces. J. Comp. Phys. 229, 3155–3170 (2010)ADSzbMATHCrossRefGoogle Scholar
  104. 104.
    Golub, G.H., Loan, C.F.V.: Matrix Computations, 3rd edn. Johns Hopkins University Press, Baltimore, MD (1996)Google Scholar
  105. 105.
    Trottenberg, U., Ooseterlee, C., Schüller, A.: Multigrid. Academic Press, London (2001)zbMATHGoogle Scholar
  106. 106.
    Rengarajan, S.R., Rahmat-Samii, Y.: The field equivalence principle: Illustration of the establishment of the non-intuitive null fields. IEEE Antennas Propag. Mag. 42, 122–128 (2000)ADSCrossRefGoogle Scholar
  107. 107.
    Schelkunoff, S.A.: Some equivalence theorems of electromagnetic waves. Bell Syst. Tech. J. 15, 92–112 (1936)zbMATHGoogle Scholar
  108. 108.
    Stratton, J.A., Chu, L.J.: Diffraction theory of electromagnetic waves. Phys. Rev. 56, 99–107 (1939)ADSCrossRefGoogle Scholar
  109. 109.
    Love, A.E.H.: The integration of equations of propagation of electric waves. Phil. Trans. Roy. Soc. London A 197, 1–45 (1901)ADSzbMATHCrossRefGoogle Scholar
  110. 110.
    Harrington, R.F.: Boundary integral formulations for homogeneous material bodies. J. Electromagn. Waves Appl. 3, 1–15 (1989)ADSCrossRefGoogle Scholar
  111. 111.
    Umashankar, K., Taflove, A., Rao, S.: Electromagnetic scattering by arbitrary shaped three-dimensional homogeneous lossy dielectric objects. IEEE Trans. Antennas Propagat. 34, 758–766 (1986)ADSCrossRefGoogle Scholar
  112. 112.
    Medgyesi-Mitschang, L.N., Putnam, J.M., Gedera, M.B.: Generalized method of moments for three-dimensional penetrable scatterers. J. Opt. Soc. Am. A 11, 1383–1398 (1994)ADSCrossRefGoogle Scholar
  113. 113.
    Trefethen, L.N.: Is Gauss quadrature better than clenshaw–curtis. SIAM Review 50, 67–87 (2008)MathSciNetADSzbMATHCrossRefGoogle Scholar
  114. 114.
    Cools, R.: Advances in multidimensional integration. J. Comput. Appl. Math 149, 1–12 (2002)MathSciNetADSzbMATHCrossRefGoogle Scholar
  115. 115.
    Cools, R.: An encyclopaedia of cubature formulas. J. Complexity 19, 445–453 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  116. 116.
    Berntsen, J., Espelid, T.O., Genz, A.: An adaptive algorithm for the approximate calculation of multiple integrals. ACM Trans. Math. Soft. 17, 437–451 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  117. 117.
    Atkinson, K., Sommariva, A.: Quadrature over the sphere. Elec. Trans. Num. Anal. 20, 104–118 (2005)MathSciNetzbMATHGoogle Scholar
  118. 118.
    Le Gia, Q.T., Mhaskar, H.N.: Localized linear polynomial operators and quadrature formulas on the sphere. SIAM J. Num. Anal. 47, 440–466 (2008)MathSciNetCrossRefGoogle Scholar
  119. 119.
    Oskooi, A.F., Roundy, D., Ibanescu, M., Bermel, P., Joannopoulos, J.D., Johnson, S.G.: Meep: A flexible free-software package for electromagnetic simulations by the FDTD method. Comp. Phys. Comm. 181, 687–702 (2010)ADSzbMATHCrossRefGoogle Scholar
  120. 120.
    Rodriguez, A.W., McCauley, A.P., Joannopoulos, J.D., Johnson, S.G.: Theoretical ingredients of a Casimir analog computer. Proc. Nat. Acad. Sci. 107, 9531–9536 (2010)ADSCrossRefGoogle Scholar
  121. 121.
    McCauley, A.P., Rodriguez, A.W., Johnson, S.G.: Casimir Meep wiki. http://ab-initio.mit.eduGoogle Scholar
  122. 122.
    Werner, G.R., Cary, J.R.: A stable FDTD algorithm for non-diagonal anisotropic dielectrics. J. Comp. Phys. 226, 1085–1101 (2007)ADSzbMATHCrossRefGoogle Scholar
  123. 123.
    Li, H., Kardar, M.: Fluctuation-induced forces between rough surfaces. Phys. Rev. Lett. 67, 3275–3278 (1991)ADSCrossRefGoogle Scholar
  124. 124.
    Reid, M.T.H.: Fluctuating surface currents: A new algorithm for efficient prediction of Casimir interactions among arbitrary materials in arbitrary geometries. Ph.D. Thesis, Department of Physics, Massachusetts Institute of Technology (2010)Google Scholar
  125. 125.
    Emig, T.: Casimir forces: An exact approach for periodically deformed objects. Europhys. Lett. 62, 466 (2003)ADSCrossRefGoogle Scholar
  126. 126.
    Gies, H., Klingmuller, K.: Worldline algorithms for Casimir configurations. Phys. Rev. D 74, 045–002 (2006)CrossRefGoogle Scholar
  127. 127.
    Gies, H., Langfeld, K., Moyaerts, L.: Casimir effect on the worldline. J. High Energy Phys. 6, 018 (2003)MathSciNetADSCrossRefGoogle Scholar
  128. 128.
    Emig, T., Graham, N., Jaffe, R.L., Kardar, M.: Orientation dependence of Casimir forces. Phys. Rev. A 79, 054–901 (2009)CrossRefGoogle Scholar
  129. 129.
    Waterman, P.C.: The T-matrix revisited. J. Opt. Soc. Am. A 24(8), 2257–2267 (2007)MathSciNetADSCrossRefGoogle Scholar
  130. 130.
    Emig, T.: Fluctuation-induced quantum interactions between compact objects and a plane mirror. J. Stat. Mech. p. P04007 (2008)Google Scholar
  131. 131.
    Emig, T., Jaffe, R.L.: Casimir forces between arbitrary compact objects. J. Phys. A 41, 164–001 (2008)MathSciNetCrossRefGoogle Scholar
  132. 132.
    Bimonte, G.: Scattering approach to Casimir forces and radiative heat transfer for nanostructured surfaces out of thermal equilibrium. Phys. Rev. A 80, 042–102 (2009)CrossRefGoogle Scholar
  133. 133.
    McCauley, A.P., Zhao, R., Reid, M.T.H., Rodriguez, A.W., Zhao, J., Rosa, F.S.S., Joannopoulos, J.D., Dalvit, D.A.R., Soukoulis, C.M., Johnson, S.G.: Microstructure effects for Casimir forces in chiral metamaterials. Phys. Rev. B 82, 165108 (2010)ADSCrossRefGoogle Scholar
  134. 134.
    Marcuse, D.: Theory of Dielectric Optical Waveguides, 2nd edn. Academic Press, San Diego (1991)Google Scholar
  135. 135.
    Johnson, S.G., Bienstman, P., Skorobogatiy, M., Ibanescu, M., Lidorikis, E., Joannopoulos, J.D.: Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals. Phys. Rev. E. 66, 066–608 (2002)CrossRefGoogle Scholar
  136. 136.
    Skorobogatiy, M., Yang, J.: Fundamentals of Photonic Crystal Guiding. Cambridge University Press, Cambridge (2009)Google Scholar
  137. 137.
    Inui, T., Tanabe, Y., Onodera, Y.: Group Theory and Its Applications in Physics. Springer, Heidelberg (1996)zbMATHGoogle Scholar
  138. 138.
    Tinkham, M.: Group Theory and Quantum Mechanics. Dover, New York (2003)Google Scholar
  139. 139.
    Rodriguez, A.W., Woolf, D., McCauley, A.P., Capasso, F., Joannopoulos, J.D., Johnson, S.G.: Achieving a strongly temperature-dependent Casimir effect. Phys. Rev. Lett. 105, 060–401 (2010)Google Scholar
  140. 140.
    Milton, K.A.: The Casimir effect: recent controveries and progress. J. Phys. A: Math. Gen. 37, R209–R277 (2004)MathSciNetADSzbMATHCrossRefGoogle Scholar
  141. 141.
    Najafi, A., Golestanian, R.: Forces induced by nonequilibrium fluctuations: The soret–Casimir effect. Europhys. Lett. 68, 776–782 (2004)ADSCrossRefGoogle Scholar
  142. 142.
    Obrecht, J.M., Wild, R.J., Antezza, M., Pitaevskii, L.P., Stringari, S., Cornell, E.A.: Measurement of the temperature dependence of the Casimir-Polder force. Phys. Rev. Lett. 98(6), 063–201 (2007)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg  2011

Authors and Affiliations

  1. 1.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA

Personalised recommendations