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1 - Introduction

Published online by Cambridge University Press:  05 April 2014

E. R. Tracy ,
A. J. Brizard ,
A. S. Richardson  and
A. N. Kaufman
E. R. Tracy
Affiliation:
College of William and Mary, Virginia
A. J. Brizard
Affiliation:
Saint Michael's College, Vermont
A. S. Richardson
Affiliation:
US Naval Research Laboratory (NRL)
A. N. Kaufman
Affiliation:
University of California, Berkeley
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Summary

The science of optics, like every other physical science, has two different directions of progress, which have been called the ascending and the descending scale, the inductive and the deductive method, the way of analysis and of synthesis. In every physical science, we must ascend from facts to laws, by the way of induction and analysis; and must descend from laws to consequences, by the deductive and synthetic way. We must gather and groupe appearances, until the scientific imagination discerns their hidden law, and unity arises from variety: and then from unity must re-deduce variety, and force the discovered law to utter its revelations of the future.

William Rowan Hamilton (1805–1865)

It is a fact of immediate importance to our everyday experience that light nearly always travels in straight lines from the source to our eyes, perhaps scattering off some object along the way. Without the ability to assume this as a fact about the world around us, our extraordinary talent for instinctively comprehending spatial relationships in everyday life would be severely compromised. Consider how much computer power must be expended to disentangle the multiple images of distant galaxies to map the dark matter distribution in the visible universe [MRE+07]. Imagine what life would be like if we had to do similar mental computations just to navigate around the furniture in our living room.

How do we build upon this insight that light nearly always travels in straight lines in order to develop a theory with predictive power?

Type
Chapter
Information
Ray Tracing and Beyond
Phase Space Methods in Plasma Wave Theory
 , pp. 1 - 61
Publisher: Cambridge University Press
Print publication year: 2014

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References

[AL97] S, Antoci and D-E, Liebscher. The third way to quantum mechanics is the forgotten first. arXiv:physics/9704028 [physics.hist-ph], April 1997.
[Ari12] Aristophanes. Aristophanes: Clouds. Cambridge Translations from Greek Drama. Cambridge University Press, Cambridge, 2012.
[Ber77a] MV, Berry. Regular and irregular semiclassical wavefunctions. Journal of Physics A: Mathematical and General, 10(12):2083, 1977.Google Scholar
[Ber77b] MV, Berry. Semi-classical mechanics in phase space: a study of Wigner's function. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 287(1343):237–271, 1977.Google Scholar
[Ber07] L, Berggren. Ibn sahl: Abū sa'dal-‘alā’ ibn Sahl. In T, Hockey, V, Trimble, TR, Williams, K, Bracher, RA, Jarrell, Jordan D, Marché II, FJ, Ragep, J, Palmeri, and M, Bolt, editors, The Biographical Encyclopedia of Astronomers, pages 567–567. Springer, New York, 2007.
[Bog87] BM, Boghosian. Covariant Lagrangian methods of relativistic plasma theory. Ph.D. thesis, University of California, Davis, 1987. Uploaded to arXiv in 2003.
[Bri26] L, Brillouin. La mécanique ondulatoire de Schrödinger; une méthode generale de resolution par approximations successives. Comptes rendus, 138:24–26, 1926. Proceedings of the French Academy of Sciences.Google Scholar
[Bri09] AJ, Brizard. Variational principles for reduced plasma physics. Journal of Physics: Conference Series, 169(1):012003, 2009.Google Scholar
[BS91] FA, Berezin and MA, Shubin. The Schrödinger Equation, volume 66 of Mathematics and Its Applications (Sowaiet Series). Kluwer Academic Publishers, Dordrecht, 1991.
[BT76] MV, Berry and M, Tabor. Closed orbits and the regular bound spectrum. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 349(1656):101–123, 1976.Google Scholar
[BT09] AJ, Brizard and ER, Tracy. Kaufman Fest 2007: Plasma theory, wave kinetics, and nonlinear dynamics. Journal ofPhysics: Conference Series, 169(1):012001, 2009.Google Scholar
[BU80] MV, Berry and C, Upstill. Catastrophe optics: morphologies of caustics and their diffraction patterns. Progress in Optics, 18:257–346, 1980.Google Scholar
[Bur45] HE, Burton. The optics of Euclid. Journal of the Optical Society of America, 35(5):357–372, 1945.Google Scholar
[Bye98] N, Byers. E. Noether's discovery of the deep connection between symmetries and conservation laws. arXiv preprint physics/9807044, 1998.
[CF73] S, Choudhary and L, Felsen. Asymptotic theory for inhomogeneous waves. Antennas and Propagation, IEEE Transactions on, 21(6):827–842, 1973.Google Scholar
[CK77] JR, Cary and AN, Kaufman. Ponderomotive force and linear susceptibility in Vlasov plasma. Physical Review Letters, 39:402–404, August 1977.Google Scholar
[CK81] JR, Cary and AN, Kaufman. Ponderomotive effects in collisionless plasma: a Lie transform approach. Physics of Fluids, 24(7):1238–1250, 1981.Google Scholar
[CKTF93] DR, Cook, AN, Kaufman, ER, Tracy, and T, Fla. Collective-wave spin-off and the gyroballistic continuum in gyroresonant absorption. Physics Letters A, 175(5):326–333, 1993.Google Scholar
[Dar12] O, Darrigol. A History of Optics from Greek Antiquity to the Nineteenth Century. Oxford University Press, 2012.
[DB84] RR, Dominguez and HL, Berk. Variational structure of the Vlasov equation in multidimensional systems. Physics of Fluids, 27(5):1142–1147, 1984.Google Scholar
[Dew70] RL, Dewar. Interaction between hydromagnetic waves and a time-dependent, inhomogeneous medium. Physics of Fluids, 13(11):2710–2720, 1970.Google Scholar
[Dew72a] RL, Dewar. A Lagrangian theory for nonlinear wave packets in a collisionless plasma. Journal of Plasma Physics, 7:267–284, March 1972.Google Scholar
[Dew72b] RL, Dewar. A Lagrangian derivation of the action-conservation theorem for density waves. The Astrophysical Journal, 174:301, 1972.Google Scholar
[Dew73] RL, Dewar. Oscillation center quasilinear theory. Physics of Fluids, 16(7):1102–1107, 1973.Google Scholar
[Dew76] RL, Dewar. Renormalised canonical perturbation theory for stochastic propagators. Journal of Physics A: Mathematical and General, 9(12):2043, 1976.Google Scholar
[Dew77] RL, Dewar. Energy and momentum tensors for dispersive electromagnetic waves. Australian Journal of Physics, 30(6):533–576, 01/1977.Google Scholar
[DF76] AJ, Dragt and JM, Finn. Lie series and invariant functions for analytic symplectic maps. Journal of Mathematical Physics, 17(12):2215–2227, 1976.Google Scholar
[DF79] AJ, Dragt and JM, Finn. Normal form for mirror machine Hamiltonians. Journal of Mathematical Physics, 20(12):2649–2660, 1979.Google Scholar
[DKOL83] DHE, Dubin, JA, Krommes, CA, Oberman, and WW, Lee. Nonlinear gyrokinetic equations. Physics of Fluids, 26(12):3524–3535, 1983.Google Scholar
[DMMN77] C, DeWitt-Morette, A, Maheshwari, and B, Nelson. Path integration in phase space. General Relativity and Gravitation, 8(8):581–593, 1977.Google Scholar
[Dou70] JP, Dougherty. Lagrangian methods in plasma dynamics: I. General theory of the method of the averaged Lagrangian. Journal of Plasma Physics, 4:761–785, December 1970.Google Scholar
[Dou74] JP, Dougherty. Lagrangian methods in plasma dynamics: 2. Construction of Lagrangians for plasmas. Journal of Plasma Physics, 11:331–346, April 1974.Google Scholar
[FB42] RP, Feynman and LM, Brown. Feynman's Thesis: A New Approach to Quantum Theory. World Scientific Publishing Company, 1942.
[FG86] L, Friedland and G, Goldner. Reduction of order in the geometric optics of plasmas. Physics of Fluids, 29(12):4073–4084, 1986.Google Scholar
[FGK87] L, Friedland, G, Goldner, and AN, Kaufman. Four-dimensional eikonal theory of linear mode conversion. Physical Review Letters, 58(14):1392–1394, April 1987.Google Scholar
[FGNO09] PGO, Freund, CJ, Goebel, Y, Nambu, and R, Oehme. Gregor Wentzel 1898–1978. Biographical Memoir. National Academy of Sciences. April 2009.
[FK87] L, Friedland and AN, Kaufman. Congruent reduction in geometric optics and mode conversion. Physics of Fluids, 30(10):3050–3058, 1987.Google Scholar
[Fre00] A, Fresnel. Fresnel's prize memoir on the diffraction on light. In H, Crew, editor, The Wave Theory of Light: Memoirs ofHuygens, Young, and Fresnel, volume 15 of Scientific Memoirs, pages 81–107. American Book Company, New York, 1900.
[GB05] DA, Gurnett and A, Bhattacharjee. Introduction to Plasma Physics: With Space and Laboratory Applications. Cambridge University Press, New York, 2005.
[GC77] JJ, Galloway and FW, Crawford. Microscopic Lagrangian description of warm plasmas: 3. Nonlinear wave-particle interaction. Radio Science, 12(6):965–975, 1977.Google Scholar
[GK83] C, Grebogi and AN, Kaufman. Multidimensional canonical/symplectic maps for gyroresonance crossing. In CW, Horton Jr, LE, Reichl, and VG, Szebehely, editors, Long-Time Prediction in Dynamics, volume 2 of Nonequilibrium Problems in the Physical Sciences and Biology. Wiley, New York, 1983.
[GKL79] C, Grebogi, AN, Kaufman, and RG, Littlejohn. Hamiltonian theory of pondero-motive effects of an electromagnetic wave in a nonuniform magnetic field. Physical Review Letters, 43:1668–1671, November 1979.Google Scholar
[Ham28] WR, Hamilton. Theory of systems of rays. Transactions of the Royal Irish Academy, 15:69–174, 1828.Google Scholar
[Ham33] WR, Hamilton. On a general method of expressing the paths of light, and of the planets, by the coefficients of a characteristic function. Dublin University Review and Quarterly Magazine, 1:795–826, 1833.Google Scholar
[Jac98] JD, Jackson. Classical Electrodynamics. Wiley, New York, 1998.
[JK78] S, Johnston and AN, Kaufman. Lie-operator approach to mode coupling in nonuniform plasma. Physical Review Letters, 40:1266–1269, May 1978.Google Scholar
[JKJ78] S, Johnston, AN, Kaufman, and GL, Johnston. Beat Hamiltonians and generalized ponderomotive forces in hot magnetized plasma. Journal of Plasma Physics, 20:365–390, December 1978.Google Scholar
[Joh76] S, Johnston. Oscillation-center formulation of the classical theory of induced scattering in plasma. Physics of Fluids, 19(1):93–107, 1976.Google Scholar
[Kau72] AN, Kaufman. Reformulation of quasi-linear theory. Journal of Plasma Physics, 8:1–5, August 1972.Google Scholar
[Kau82] AN, Kaufman. Natural Poisson structures of nonlinear plasma dynamics. Physica Scripta, 1982(T2B):517, 1982.Google Scholar
[Kau87] AN, Kaufman. Phase-space-Lagrangian action principle and the generalized K – χ theorem. Physical Review A, 36:982–984, July 1987.Google Scholar
[Kau91] AN, Kaufman. Phase-space plasma-action principles, linear mode conversion, and the generalized Fourier transform. In W, Rozmus and JA, Tuszynski, editors, Nonlinear and Chaotic Phenomena in Plasmas, Solids, and Fluids, pages 160–192. CAP-NSERC Summer Institute in Theoretical Physics, Edmonton, Alberta, Canada, 16–27 July 1990. World Scientific, New Jersey, 1991.
[Kau09] AN, Kaufman. A half-century in plasma physics. Journal of Physics: Conference Series, 169(1):012002, 2009.Google Scholar
[KB84] AN, Kaufman and BM, Boghosian. Lie-transform derivation of the gyrokinetic Hamiltonian system. In JE, Marsden, editor, Fluids and Plasmas: Geometry and Dynamics, volume 28 of Contemporary Mathematics, pages 169–176. American Mathematical Society, 1984.
[KC77a] H, Kim and FW, Crawford. Microscopic Lagrangian description of warm plasmas: 1. Linear wave propagation. Radio Science, 12(6):941–951, 1977.Google Scholar
[KC77b] H, Kim and FW, Crawford. Microscopic Lagrangian description of warm plasmas: 2. Nonlinear wave interactions. Radio Science, 12(6):953–963, 1977.Google Scholar
[Kel58] JB, Keller. Corrected Bohr-Sommerfeld quantum conditions for nonseparable systems. Annals of Physics, 4(2):180–188, 1958.Google Scholar
[Kel85] JB, Keller. Semiclassical mechanics. SIAM Review, 27(4):485–504, 1985.Google Scholar
[KF87] AN, Kaufman and L, Friedland. Phase-space solution of the linear mode-conversion problem. Physics Letters A, 123(8):387–389, 1987.Google Scholar
[KH84] AN, Kaufman and DD, Holm. The Lie-transformed Vlasov action principle: relativistically covariant wave propagation and self-consistent ponderomotive effects. Physics Letters A, 105(6):277–279, 1984.Google Scholar
[KR60] JB, Keller and SI, Rubinow. Asymptotic solution of eigenvalue problems. Annals of Physics, 9(1):24–75, 1960.Google Scholar
[Kra26] HA, Kramers. Wellenmechanik und halbzahlige quantisierung. Zeitschrift fur Physik, 39(10–11):828–840, 1926.Google Scholar
[Kru65] M, Kruskal. Asymptotology. In Lectures presented at the Trieste Seminar on Plasma Physics, volume 1, page 373. Prentice-Hall, Englewood Cliffs, NJ, 1965.
[Lit79] RG, Littlejohn. A guiding center Hamiltonian: a new approach. Journal of Mathematical Physics, 20(12):2445–2458, 1979.Google Scholar
[Lit81] RG, Littlejohn. Hamiltonian formulation of guiding center motion. Physics of Fluids, 24(9):1730–1749, 1981.Google Scholar
[Lit82] RG, Littlejohn. Hamiltonian perturbation theory in noncanonical coordinates. Journal of Mathematical Physics, 23(5):742–747, 1982.Google Scholar
[Lit83] RG, Littlejohn. Variational principles of guiding centre motion. Journal of Plasma Physics, 29:111–125, February 1983.Google Scholar
[Lit86] RG, Littlejohn. The semiclassical evolution of wave packets. Physics Reports, 138(4–5):193–291, 1986.Google Scholar
[McD88] SW, McDonald. Phase-space representations of wave equations with applications to the eikonal approximation for short-wavelength waves. Physics Reports, 158(6):337–416, 1988.Google Scholar
[MF02] VP, Maslov and MV, Fedoriuk. Semi-Classical Approximation in Quantum Mechanics. Mathematical Physics and Applied Mathematics Series. Springer, 2002.
[MGK85] SW, McDonald, C, Grebogi, and AN, Kaufman. Locally coupled evolution of wave and particle distribution in general magnetoplasma geometry. Physics Letters A, 111(1–2):19–21, 1985.Google Scholar
[MK79] SW, McDonald and AN, Kaufman. Spectrum and eigenfunctions for a Hamiltonian with stochastic trajectories. Physical Review Letters, 42:1189–1191, April 1979.Google Scholar
[MK82] SW, McDonald and AN, Kaufman. Hamiltonian kinetic theory of plasma ponderomotive processes. AIP Conference Proceedings, 88(1):117–120, 1982.Google Scholar
[MK85] SW, McDonald and AN, Kaufman. Weyl representation for electromagnetic waves: the wave kinetic equation. Physical Review A, 32(3):1708–1713, September 1985.Google Scholar
[MK88] SW, McDonald and AN, Kaufman. Wave chaos in the stadium: statistical properties of short-wave solutions of the Helmholtz equation. Physical Review A, 37:3067–3086, April 1988.Google Scholar
[MMPF13] O, Maj, A, Mariani, E, Poli, and D, Farina. The wave energy flux of high frequency diffracting beams in complex geometrical optics. Physics of Plasmas, 20(4):042122, 2013.Google Scholar
[Moo94] W, Moore. A Life of Erwin Schrodinger. Cambridge University Press, Cambridge, 1994.
[Mor05] PJ, Morrison. Hamiltonian and action principle formulations of plasma physics. Physics of Plasmas, 12(5):058102, 2005.Google Scholar
[MRE+07] R, Massey, et al.Dark matter maps reveal cosmic scaffolding. Nature, 445(7125):286–290, 2007.Google Scholar
[New10] I, Newton. Opticks: Or a Treatise of the Reflections, Refractions, Inflections and Colours of Light. EBook No. 33504. Project Gutenberg, August 2010.
[Omo86] SM, Omohundro. Geometric Perturbation Theory in Physics. World Scientific, 1986.
[OSZ+13] J, Oettler, VS, Schmid, N, Zankl, O, Rey, A, Dress, J, Heinze. Fermat's principle of least time predicts refraction of ant trails at substrate borders. PLoS ONE, 8(3):e59739, 2013.Google Scholar
[Per77] IC, Percival. Semiclassical theory of bound states. Advances in Chemical Physics, 36(1), 1977.Google Scholar
[Pla97] D, Plantzos. Crystals and lenses in the Graeco-Roman world. American Journal of Achaeology, 101(3):451–464, 1997.Google Scholar
[PPP99] E, Poli, GV, Pereverzev, and AG, Peeters. Paraxial Gaussian wave beam propagation in an anisotropic inhomogeneous plasma. Physics of Plasmas, 6(1):5–11, 1999.Google Scholar
[PPP01] E, Poli, AG, Peeters, and GV, Pereverzev. TORBEAM, a beam tracing code for electron-cyclotron waves in tokamak plasmas. Computer Physics Communications, 136(1–2):90–104, 2001.Google Scholar
[Ric08] AS, Richardson. Topics in mode conversion theory and the group theoretical foundations of path integrals. Ph.D. thesis, William & Mary, 2008.
[Sch07] RA, Schorn. Ole, Romer. In T, Hockey, V, Trimble, TR, Williams, K, Bracher, RA, Jarrell, Jordan D, Marché II, FJ, Ragep, J, Palmeri, and M, Bolt, editors, The Biographical Encyclopedia of Astronomers, page 983. Springer, New York, 2007.
[Sim85] PL, Similon. Conservation laws for relativistic guiding-center plasma. Physics Letters A, 112(1–2):33–37, 1985.Google Scholar
[SK84] PL, Similon and AN, Kaufman. Theory of ponderomotive stabilization of a magnetically confined plasma. Physical Review Letters, 53:1061–1064, September 1984.Google Scholar
[SKH86] PL, Similon, AN, Kaufman, and DD, Holm. Oscillation center theory and pon-deromotive stabilization of low-frequency plasma modes. Physics of Fluids, 29(6):1908–1922, 1986.Google Scholar
[SSS01] RA, Shelby, DR, Smith, and S, Schultz. Experimental verification of a negative index of refraction. Science, 292(5514):77–79, 2001.Google Scholar
[TB09] ER, Tracy and AJ, Brizard. Allan Kaufman's contributions to plasma wave theory. Journal of Physics: Conference Series, 169(1):012008, 2009.Google Scholar
[TBK96] ER, Tracy, AJ, Brizard, and AN, Kaufman. Generalized Case-van Kampen modes in a multidimensional nonuniform plasma with application to gyrores-onance heating. Journal of Plasma Physics, 55(03):449–486, 1996.Google Scholar
[TK90] ER, Tracy and AN, Kaufman. Wave-kinetic formulation of incoherent linear mode conversion. Physical Review Letters, 64(14):1621–1624, April 1990.Google Scholar
[Wen24] G, Wentzel. Zur Quantenoptik. Zeitschrift für Physik, 22(1):193–199, December 1924.Google Scholar
[Wen26a] G, Wentzel. Eine Verallgemeinerung der Quantenbedingugen fur die Zwecke der Wellenmechanik. Zeitschrift fur Physik, 38, 1926.
[Wen26b] G, Wentzel. Zur Theorie des photoelektrischen Effekts. Zeitschrift fur Physik, 40, 1926.
[Wen26c] G, Wentzel. Zwei Bemerkungen uber die Zerstreuung korpuskularer Strahlen als Beugungserscheinung. Zeitschrift fur Physik, 40, 1926.
[Wey27] H, Weyl. Quantenmechanik und Gruppentheorie. Zeitschrift fur Physik, 46:1–46, 1927.Google Scholar
[Wey31] H, Weyl. The Theory of Groups and Quantum Mechanics. EP Dutton & Company, New York, second edition, 1931.
[Wey70] H, Weyl. Emmy Noether. Beihefte zur Zeitschrift (Elemente der Mathematik), 13:53–72, 1970.Google Scholar
[WFO80] J-M, Wersinger, JM, Finn, and E, Ott. Bifurcation and “strange” behavior in instability saturation by nonlinear three-wave mode coupling. Physics of Fluids, 23(6):1142–1154, 1980.Google Scholar
[Whi74] GB, Whitham. Linear and Nonlinear Waves. Pure and Applied Mathematics. Wiley,New York, 1974.
[Wil08] R, Willach. The Long Route to the Invention of the Telescope. American Philosophical Society, 2008.
[Wol07] E, Wolf. Introduction to the Theory ofCoherence and Polarization ofLight. Cambridge University Press, Cambridge, 2007.
[Ye90] H, Ye. Wave dynamics in phase-space and ion gryoresonant absorption. Ph.D. thesis, UC Berkeley, 1990.
[YK88a] H, Ye and AN, Kaufman. Analytic solution for second-harmonic gyroresonant absorption and mode conversion. Physical Review Letters, 61:2762–2765, December 1988.Google Scholar
[YK88b] H, Ye and AN, Kaufman. Wave propagation in £-space and the linear ion-cyclotron echo. Physical Review Letters, 60(16):1642–1644, April 1988.Google Scholar
[YK92] H, Ye and AN, Kaufman. Self-consistent theory for ion gyroresonance. Physics of Fluids B: Plasma Physics, 4(7):1735–1753, 1992.Google Scholar

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