International Journal of Theoretical Physics

, Volume 44, Issue 1, pp 95–125 | Cite as

Wavelength-Dependent Modifications in Helmholtz Optics



The Helmholtz wave equation is linearized using the Feshbach–Villars procedure used for linearizing the Klein–Gordon equation, based on the close algebraic analogy between the Helmholtz equation and the Klein–Gordon equation for a spin-0 particle. The Foldy–Wouthuysen iterative diagonalization technique is then applied to the linearized Helmholtz equation to obtain a Hamiltonian description for a system with varying refractive index. The Hamiltonian has a wavelength-dependent part absent in the traditional descriptions. Besides reproducing all the traditional quasi-paraxial terms, our method leads to additional contributions dependent on the wavelength. Applied to the axially symmetric graded-index fiber, this method results in wavelength-dependent modifications of the paraxial behavior and the aberration coefficients to all orders. Explicit expression for the modified aberration coefficients to the third order are presented. Sixth- and eighth-order Hamiltonians are also presented.


Scalar wave optics Helmholtz equation wavelength-dependent effects beam propagation Hamiltonian description aberrations graded-index fiber mathematical methods of optics Feshbach–Villars linearization Foldy–Wouthuysen transformation 


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  1. Acharya, R. and Sudarshan, E. C. G. (1960). Front description in relative quantum mechanics, Journal of Mathematical Physics 1, 532–536.Google Scholar
  2. Ambrosini, D., Ponticiello, A., Schirripa Spagnolo, G., Borghi, R., and Gori, F. (1997). Bouncing light beams and the Hamiltonian analogy. European Journal of Physics 18, 284–289.Google Scholar
  3. Bellman, R. and Vasudevan, R. (1986). Wave Propagation: An Invariant Imbedding Approach, Reidel, Dordrecht.Google Scholar
  4. Bjorken, J. D. and Drell, S. D. (1964). Relativistic Quantum Mechanics, McGraw-Hill, New York, San Francisco.Google Scholar
  5. Born, M. and Wolf, E. (1999). Principles of Optics, 7th edn., Cambridge University Press, United Kingdom.Google Scholar
  6. Chen, P. (1999) (ed.). Proceedings of the 15th Advanced ICFA Beam Dynamics Workshop on Quantum Aspects of Beam Physics, 4–9 January 1998, Monterrey, California, USA, World Scientific, Singapore,; Chen, P. (2002) (ed.), Proceedings of the 18th Advanced ICFA Beam Dynamics Workshop on Quantum Aspects of Beam Physics, 15–20 October 2000, Capri, Italy, World Scientific, Singapore,; Chen, P. (2003) (ed.). Proceedings of the Joint 28th ICFA Advanced Beam Dynamics and Advanced & Novel Accelerators Workshop on QUANTUM ASPECTS OF BEAM PHYSICS and Other Critical Issues of Beams in Physics and Astrophysics, 7–11 January 2003, Hiroshima University, Japan, World Scientific, Singapore,; Workshop Reports: ICFA Beam Dynamics Newsletter 16, (1988) 22–25; ibid 23, (2000) 13–14; ibid 30, (2003) 72–75; Bulletin of the Association of Asia Pacific Physical Societies 13(1), (2003) 34–37.Google Scholar
  7. Conte, M., Jagannathan, R., Khan, S. A., and Pusterla, M. (1996). Beam optics of the Dirac particle with anomalous magnetic moment. Particle Accel. 56, 99–126.Google Scholar
  8. Dattoli, G., Renieri, A., and Torre, A. (1993). Lectures on the Free Electron Laser Theory and Related Topics, World Scientific, Singapore.Google Scholar
  9. Dragt, A. J., Forest, E., and Wolf, K. B. (1986). Lie Methods in Optics, Lecture Notes in Physics, Vol. 250, Springer Verlag, pp. 105–157.Google Scholar
  10. Dragt, A. J. and Forest, E. (1986). Adv. Imag. Electron Phys. 67, 65–120; Dragt, A. J., Neri, F., Rangarajan, G., Douglas, D. R., Healy, L. M., and Ryne, R. D. (1988). Lie algebriac treatment of linear and nonlinear beam dynamics. Ann. Rev. Nucl. Part. Sci. 38, 455–496; Forest, E. and Hirata, K. (1992). A Contemporary Guide to Beam Dynamics, KEK Report 92-12, National Laboratory for High Energy Physics, Tsukuba, Japan; Forest, E., Berz, M., and Irwin, J. (1989). Particle Accel. 24, 91–97; Rangarajan, G., Dragt, A. J., and Neri, F. (1990). Solvable map representation of a nonlinear sympletic map. Particle Accel. 28, 119–124; Ryne, R. D. and Dragt, A. J. (1991). Magnetic optics calculations for cylinderically symmetric beams. Particle Accel. 35, 129–165.Google Scholar
  11. Dragt, A. J. (1998). Lie algebriac theory of geometrical optics and optical aberrations. J. Opt. Soc. Am 72, (1982) 372; Lie Algebraic Method for Ray and Wave Optics, University of Maryland Physics Department Report.Google Scholar
  12. Fedele, R. and Man’ko, V. I. (1999). The role of semiclassical description in the quantum-like theory of light rays. Physical Review E 60, 6042–6050.Google Scholar
  13. Feshbach, H. and Villars, F. M. H. (1958). Elementary relativistic wave mechanics of spin 0 and spin 1/2 particles. Reviews of Modern Physics 30, 24–45.Google Scholar
  14. Fishman, L. and McCoy, J. J. (1984). Derivation and application of extended parabolic wave theories. Part I. The factored Helmholtz equation. Journal of Mathematical Physics 25, 285–296.Google Scholar
  15. Foldy, L. L. and Wouthuysen, S. A. (1950). On the Dirac theory of spin 1/2 particles and its non-relativistic limit. Physical Review 78, 29–36.Google Scholar
  16. Goodman, J. W. (1996). Introduction to Fourier Optics, 2nd edn., McGraw-Hill, New York.Google Scholar
  17. Hawkes, P. W. and Kasper, E. (1989). Principles of Electron Optics, Vols. I and II, Academic Press, London; Hawkes, P. W. and Kasper, E. (1994) Principles of Electron Optics. Vol. 3: Wave Optics, Academic Press, London and San Diego.Google Scholar
  18. Jagannathan, R., Simon, R., Sudarshan, E. C. G., and Mukunda, N. (1989). Quantum theory of magnetic electron lenses based on the Dirac equation. Physics Letters A 134, 457–464.Google Scholar
  19. Jagannathan, R. (1990). Quantum theory of electron lenses based on the Dirac equation. Physics Letters A 42, 6674–6689.Google Scholar
  20. Jagannathan, R. (1993). Dirac equation and electron optics. In Dutt, R. and Ray, A. K. (eds.), Dirac and Feynman: Pioneers in Quantum Mechanics, Wiley Eastern, New Delhi, India, pp. 75–82.Google Scholar
  21. Jagannathan, R. and Khan, S. A. (1996). Quantum theory of the optics of charged particles. In Hawkes Peter, W. (ed.), Advances in Imaging and Electron Physics, Vol. 97, Academic Press, San Diego, pp. 257–358.Google Scholar
  22. Jagannathan, R. and Khan, S. A. (1997). Quantum mechanics of accelerator optics. ICFA Beam Dynamics Newsletter 13, 21–27 (ICFA: International Committee for Future Accelerators).Google Scholar
  23. Jagannathan, R. and Khan, S. A. (1998). Several articles in Proceedings/E-Prints on the Quantum theory of charged-particle beam optics, arXiv: physics/9803042; arXiv: physics/0101060; arXiv: physics/9809032; arXiv: physics/9904063; arXiv: physics/0112085; arXiv: physics/0112086 and arXiv: physics/0304099.Google Scholar
  24. Khan, S. A. and Jagannathan, R. (1995). On the quantum mechanics of charged particle beam transport through magnetic lenses. Physical Review E 51, 2510–2515.Google Scholar
  25. Khan, S. A. (1997). Quantum Theory of Charged-Particle Beam Optics, Ph.D. Thesis, University of Madras, Chennai, India.Google Scholar
  26. Khan, S. A., Jagannathan, R., and Simon, R. (2002). Foldy-Wouthuysen transformation and a quasiparaxial approximation scheme for the scalar wave theory of light beams, arXiv: physics/0209082 (communicated).Google Scholar
  27. Khan, S. A. (2002). Analogies between light optics and charged-particle optics. ICFA Beam Dynamics Newsletter 27, 42–48; arXiv: physics/0210028 (ICFA: International Committee for Future Accelerators).Google Scholar
  28. Magnus, W. (1954). On the exponential solution of differential equations for a linear operator. Communications on Pure and Applied Mathematics 7, 649–673.Google Scholar
  29. Pryce, M. H. L. (1948). The mass-centre in the restricted theory of relativity and its connexion with the quantum theory of elementary particles. Proceedings of the Royal Society of London, Series A: Mathematical and Physical Sciences 195, 62–81.Google Scholar
  30. Tani, S. (1951). Connection between particle models and field theories. I. The case spin 1/2. Progress of Theoretical Physics 6, 267–285.Google Scholar
  31. Todesco, E. (1999). Overview of single-particle nonlinear dynamics, CERN-LHC-99-1-MMS, 16pp; Talk given at 16th ICFA Beam Dynamics Workshop on Nonlinear and Collective Phenomena in Beam Physics, Arcidosso, Italy, 1–5 September 1998; AIP Conference Proceedings 468, 157–172.Google Scholar
  32. Turchetti, G., Bazzani, A., Giovannozzi, M., Servizi, G., and Todesco, E. (1989). Normal forms for symplectic maps and stability of beams in particle accelarators. In Proceedings of the Dynamical symmetries and Chaotic Behaviour in Physical Systems, Bologna, Italy, pp. 203–231.Google Scholar
  33. Wilcox, R. M. (1967). Exponential operators and parameter differentiation in quantum physics. Journal of Mathematical Physics 8(4), 962–982.Google Scholar

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© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Middle East College of Information Technology (MECIT), Technowledge CorridorKnowledge Oasis MuscatMuscatSultanate of Oman

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