Abstract

A rigorous spectral analysis is given for the nonspecular reflection of a three-dimensional Gaussian beam at a dielectric isotropic planar structure. For the first time all independent nonspecular effects are derived in a self-consistent manner for the three-dimensional case. It is shown that the longitudinal nonspecular effects in the incidence plane, that is, the lateral and focal shifts of the beam waist position, the angular rotation of the reflected-beam axis, and the modifications of the beam waist width and complex amplitude, have their direct analogies in the plane transverse to the incidence and interface planes that gives transverse nonspecular effects. Moreover, the existence of the other, not yet reported, effect of nonspecular modification of the beam polarization is also proved. A role for TM and TE polarizations in reflected-beam formation is indicated. The results show that, up to the symmetric second-order terms in approximation of Fresnel coefficients, each of the longitudinal and transverse beam factors independently preserves its shape under reflection at the expense of changes of the beam reference frame, width, amplitude, and polarization parameters.

© 1996 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. W. G. Wagner, H. A. Haus, J. H. Marburger, “Large-scale self-trapping of optical beams in the paraxial ray approximation,” Phys. Rev. 175, 256–266 (1968).
    [CrossRef]
  2. S. A. Akhmanov, R. V. Khokhlov, A. P. Sukhorukov, “Self-focusing, self-defocusing and self-modulation of laser beams,” in Laser Handbook, F. T. Arecchi, E. O. Schultz-Dubois, eds. (North-Holland, Amsterdam, 1972), pp. 1152–1228.
  3. J. H. Marburger, “Self-focusing: theory,” Prog. Quant. Electron. 4, 35–110 (1975).
    [CrossRef]
  4. M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
    [CrossRef]
  5. J. Picht, “Beitrag zur Theorie der total reflection,” Ann. Phys. Leipzig 5, 433–496 (1929).
    [CrossRef]
  6. F. Goos, H. Hänchen, “Ein neuer and fundamentaler Versuch zur total Reflection,” Ann. Phys. Leipzig 1, 333–345 (1947).
    [CrossRef]
  7. K. V. Artmann, “Berechung der Seitenversetzung des total-reflektierten Strahles,” Ann. Phys. Leipzig 2, 87–102 (1948).
    [CrossRef]
  8. C. Fragstein, “Zur Seitenversetzung des totalreflektierten Lichtstrahles,” Ann. Phys. Leipzig 4, 271–278 (1949).
    [CrossRef]
  9. H. K. V. Lotsch, “Beam displacement at total reflection: the Goos–Hänchen effect,” Part I, Optik (Stuttgart) 32, 116–137 (1970); Part II, Optik (Stuttgart) 32, 189–204 (1970); Part III, Optik (Stuttgart) 32, 299–319 (1970); Part IV, Optik (Stuttgart) 32, 553–569 (1971).
  10. J. W. Ra, H. L. Bertoni, L. B. Felsen, “Reflection and transmission of beams at a dielectric interface,” SIAM J. Appl. Math. 24, 396–413 (1973).
    [CrossRef]
  11. M. McGuirk, C. K. Carniglia, “An angular spectrum representation approach to the Goos–Hänchen shift,” J. Opt. Soc. Am. 67, 103–107 (1977).
    [CrossRef]
  12. T. Tamir, “Nonspecular phenomena in beam fields reflected by multilayered media,” J. Opt. Soc. Am. A 3, 558–565 (1986).
    [CrossRef]
  13. R. H. Renard, “Total reflection: a new evaluation of the Goos – Hänchen shift,” J. Opt. Soc. Am. 54, 1190–1197 (1964).
    [CrossRef]
  14. B. R. Horovitz, T. Tamir, “Lateral displacement of a light beam at a dielectric interface,” J. Opt. Soc. Am. 61, 586–594 (1971).
    [CrossRef]
  15. T. Tamir, H. L. Bertoni, “Laternal displacement of optical beams at multilayered and periodic structures,” J. Opt. Soc. Am. 61, 1397–1413 (1971).
    [CrossRef]
  16. H. Shih, N. Bloembergen, “Phase-matched critical total reflection and the Goos–Hänchen shift in second-harmonic generation,” Phys. Rev. A 3, 412–420 (1971).
    [CrossRef]
  17. Y. M. Antar, W. M. Boerner, “Gaussian beam interaction with a planar dielectric interface,” Can. J. Phys. 52, 962–972 (1974).
  18. I. A. White, A. W. Snyder, C. Pask, “Directional change of beams undergoing partial reflection,” J. Opt. Soc. Am. 67, 703–705 (1977).
    [CrossRef]
  19. W. Nasalski, “Modified reflectance and geometrical deformations of Gaussian beams reflected at a dielectric interface,” J. Opt. Soc. Am. A 6, 1447–1454 (1989).
    [CrossRef]
  20. F. Falco, T. Tamir, “Improved analysis of nonspecular phenomena in beams reflected from stratified media,” J. Opt. Soc. Am. A 7, 185–190 (1990).
    [CrossRef]
  21. F. I. Fedorov, “K teorii polnovo otrazenija,” Dok. Akad. Nauk SSSR 105, 465–467 (1955).
  22. C. Imbert, “Calculation and experimental proof of the transverse shift induced by total internal reflection of a circularly polarized light beam,” Phys. Rev. D 5, 787–796 (1972).
    [CrossRef]
  23. O. Costa de Beauregard, “Translational internal spin effect with photons,” Phys. Rev. 139, B1443–B1446 (1965).
    [CrossRef]
  24. H. Schilling, “Die Strahlversetzung bei der Reflection linear oder elliptisch polarisierter ebener Wellen an der Trennebene zwischen absorbierenden Medien,” Ann. Phys. Leipzig 16, 122–134 (1965).
    [CrossRef]
  25. J. Richard, “Courbes de flux d’énergie de l’onde évanescente et nouvelle explication du dêplacement d’un faisceau lumineux dants la réflexion totale,” Nouv. Rev. Opt. 1, 275–286 (1970).
  26. O. Costa de Beauregard, C. Imbert, “Quantized longitudinal and transverse shifts associated with total internal reflection,” Phys. Rev. D 7, 3555–3563 (1972).
    [CrossRef]
  27. Y. Levy, C. Imbert, “Amplification de déplacements à la réflection totale,” Opt. Commun. 13, 43–47 (1975).
    [CrossRef]
  28. J. P. Hugonin, R. Petit, “Étude généralie des déplacements à la réflection totale,” J. Opt. (Paris) 8, 73–87 (1977).
    [CrossRef]
  29. R. G. Turner, “Shifts of coherent light beams on reflection at plane interfaces between isotropic media,” Aust. J. Phys. 33, 319–335 (1980).
  30. J. J. Greffet, C. Baylard, “Nonspecular astigmatic reflection of a 3D Gaussian beam on an interface,” Opt. Commun. 93, 271–276 (1992).
    [CrossRef]
  31. E. F. Y. Kou, T. Tamir, “Excitation of surface plasmons by finite width beams,” Appl. Opt. 28, 1169–1177 (1989).
    [CrossRef] [PubMed]
  32. S. Zhu, A. W. Yu, D. Hawley, R. Roy, “Frustrated total internal reflection: a demonstration and review,” Am. J. Phys. 54, 601–606 (1986).
    [CrossRef]
  33. T. Tamir, “Inhomogeneous wave types at planar structures: I. The lateral wave,” Optik (Stuttgart) 36, 209–232 (1972); “II. Surface waves,” Optik (Stuttgart) 37, 204–228 (1973); “III. Leaky waves,” Optik (Stuttgart) 38, 269–297 (1973).
  34. A. E. Kaplan, P. W. Smith, W. J. Tomlison, “Nonlinear waves and switching effects at nonlinear interfaces,” in Nonlinear Surface Electromagnetic Phenomena, H.-E. Ponath, G. I. Stegeman, eds. (North-Holland, Amsterdam, 1991), Chap. 4, pp. 323–351.
    [CrossRef]
  35. W. Nasalski, “Nonspecular bistability versus diffraction at nonlinear hybrid interfaces,” Opt. Commun. 77, 443–451 (1990); “Bistable switching effects at nonlocal nonlinear interfaces,” in Proceedings of the Progress in Electromagnetics Research Symposium, JPL Publication 93–17, J. J. van Zyl, ed. (Jet Propulsion Laboratory, Pasadena, Calif., 1993), p. 339.
    [CrossRef]
  36. A. Puri, D. N. Pattanayak, J. L. Birman, “Resonance effects on total internal reflection and lateral (Goos–Hänchen) beam displacement at the interface between nonlocal and local dielectric,” Phys. Rev. B 28, 5877–5886 (1983).
    [CrossRef]
  37. E. Pfleghaar, A. Marseille, A. Weis, “Quantitative investigation of the effect of resonant absorbers on the Goos–Hänchen shift,” Phys. Rev. Lett. 70, 2281–2284 (1993).
    [CrossRef] [PubMed]
  38. S. Schiller, I. I. Yu, M. M. Fejer, R. L. Byer, “Fused-silica monolitic total-internal-reflection resonator,” Opt. Lett. 17, 378–380 (1992); S. Schiller, R. L. Byer, “Quadruply resonant optical parametric oscillation in a monolithic total-internal-reflection resonator,” J. Opt. Soc. Am. B 10, 1696–1707 (1993); K. Fiedler, S. Schiller, R. Paschotta, P. Kürz, J. Mlynek, “Highly efficient frequency doubling with a doubly resonant monolithic total-internal-reflection ring resonator,” Opt. Lett. 18, 1786–1788 (1993).
    [CrossRef] [PubMed]
  39. H. A. Haus, J. G. Fujimoto, E. P. Ippen, “Analytic theory of additive pulse and Kerr lens mode locking,” IEEE J. Quantum Electron. 10, 2086–2096 (1992).
    [CrossRef]
  40. F. Bretenaker, A. Le Floch, L. Dutriaux, “Direct measurement of the optical Goos–Hänchen effect in lasers,” Phys. Rev. Lett. 68, 931–933 (1992); L. Dutriaux, A. Le Floch, F. Bretenaker, “Goos–Hänchen effect in the dynamics of laser eigenstates,” J. Opt. Soc. Am. B 9, 2283–2289 (1992); L. Dutriaux, A. Le Floch, F. Bretenaker, “Measurement of the transverse displacement at total reflection by helicoidal laser eigenstates,” Europhys. Lett. 24, 345–349 (1993).
    [CrossRef] [PubMed]
  41. W. Nasalski, “Ray analysis of Gaussian beam nonspecular scattering,” Opt. Commun. 92, 307–314 (1992).
    [CrossRef]
  42. R. C. Jones, “New calculus for the treatment of optical systems,” J. Opt. Soc. Am. 31, 488–493 (1941).
    [CrossRef]
  43. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chaps. 15–23, pp. 581–922.
  44. W. Nasalski, T. Tamir, L. Lin, “Displacement of the intensity peak in narrow beams reflected at a dielectric interface,” J. Opt. Soc. Am. A 5, 132–140 (1988).
    [CrossRef]
  45. R. Simon, T. Tamir, “Nonspecular phenomena in partly coherent beams reflected by multilayered structures,” J. Opt. Soc. Am. A 6, 18–22 (1989).
    [CrossRef]
  46. S. Zhang, T. Tamir, “Spatial modifications of Gaussian beams diffracted by reflection gratings,” J. Opt. Soc. Am. A 6, 1368–1381 (1989).
    [CrossRef]
  47. G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
    [CrossRef]
  48. L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N.J., 1973), Chap. 5, pp. 506–538.
  49. S. Zeroug, L. B. Felsen, “Nonspecular reflection of two-and three-dimensional acoustic beams from fluid-immersed plane-layered elastic structures,” J. Acoust. Soc. Am. 95, 3075–3089 (1994).
    [CrossRef]
  50. L. B. Felsen, “Geometrical theory of diffraction, evanescent waves, complex rays and Gaussian beams,” Geophys. J. R. Astron. Soc. 79, 77–88 (1984).
    [CrossRef]
  51. P. D. Einzinger, S. Raz, “Wave solutions under complex space–time shifts,” J. Opt. Soc. Am. A 4, 2–10 (1987).
  52. W. Nasalski, “Linear formulation of nonlinear propagation of optical beams and pulses,” Opt. Appl. 24, 205–208 (1994); “Complex ray tracing of nonlinear propagation,” Opt. Commun. 119, 218–226 (1995).
  53. D. Huang, M. Ulman, H. Acioli, H. A. Haus, J. G. Fujimoto, “Self-focusing-induced saturable loss for laser mode locking,” Opt. Lett. 17, 511–513 (1992).
    [CrossRef] [PubMed]

1994 (2)

S. Zeroug, L. B. Felsen, “Nonspecular reflection of two-and three-dimensional acoustic beams from fluid-immersed plane-layered elastic structures,” J. Acoust. Soc. Am. 95, 3075–3089 (1994).
[CrossRef]

W. Nasalski, “Linear formulation of nonlinear propagation of optical beams and pulses,” Opt. Appl. 24, 205–208 (1994); “Complex ray tracing of nonlinear propagation,” Opt. Commun. 119, 218–226 (1995).

1993 (1)

E. Pfleghaar, A. Marseille, A. Weis, “Quantitative investigation of the effect of resonant absorbers on the Goos–Hänchen shift,” Phys. Rev. Lett. 70, 2281–2284 (1993).
[CrossRef] [PubMed]

1992 (6)

S. Schiller, I. I. Yu, M. M. Fejer, R. L. Byer, “Fused-silica monolitic total-internal-reflection resonator,” Opt. Lett. 17, 378–380 (1992); S. Schiller, R. L. Byer, “Quadruply resonant optical parametric oscillation in a monolithic total-internal-reflection resonator,” J. Opt. Soc. Am. B 10, 1696–1707 (1993); K. Fiedler, S. Schiller, R. Paschotta, P. Kürz, J. Mlynek, “Highly efficient frequency doubling with a doubly resonant monolithic total-internal-reflection ring resonator,” Opt. Lett. 18, 1786–1788 (1993).
[CrossRef] [PubMed]

H. A. Haus, J. G. Fujimoto, E. P. Ippen, “Analytic theory of additive pulse and Kerr lens mode locking,” IEEE J. Quantum Electron. 10, 2086–2096 (1992).
[CrossRef]

F. Bretenaker, A. Le Floch, L. Dutriaux, “Direct measurement of the optical Goos–Hänchen effect in lasers,” Phys. Rev. Lett. 68, 931–933 (1992); L. Dutriaux, A. Le Floch, F. Bretenaker, “Goos–Hänchen effect in the dynamics of laser eigenstates,” J. Opt. Soc. Am. B 9, 2283–2289 (1992); L. Dutriaux, A. Le Floch, F. Bretenaker, “Measurement of the transverse displacement at total reflection by helicoidal laser eigenstates,” Europhys. Lett. 24, 345–349 (1993).
[CrossRef] [PubMed]

W. Nasalski, “Ray analysis of Gaussian beam nonspecular scattering,” Opt. Commun. 92, 307–314 (1992).
[CrossRef]

J. J. Greffet, C. Baylard, “Nonspecular astigmatic reflection of a 3D Gaussian beam on an interface,” Opt. Commun. 93, 271–276 (1992).
[CrossRef]

D. Huang, M. Ulman, H. Acioli, H. A. Haus, J. G. Fujimoto, “Self-focusing-induced saturable loss for laser mode locking,” Opt. Lett. 17, 511–513 (1992).
[CrossRef] [PubMed]

1990 (2)

W. Nasalski, “Nonspecular bistability versus diffraction at nonlinear hybrid interfaces,” Opt. Commun. 77, 443–451 (1990); “Bistable switching effects at nonlocal nonlinear interfaces,” in Proceedings of the Progress in Electromagnetics Research Symposium, JPL Publication 93–17, J. J. van Zyl, ed. (Jet Propulsion Laboratory, Pasadena, Calif., 1993), p. 339.
[CrossRef]

F. Falco, T. Tamir, “Improved analysis of nonspecular phenomena in beams reflected from stratified media,” J. Opt. Soc. Am. A 7, 185–190 (1990).
[CrossRef]

1989 (4)

1988 (1)

1987 (1)

P. D. Einzinger, S. Raz, “Wave solutions under complex space–time shifts,” J. Opt. Soc. Am. A 4, 2–10 (1987).

1986 (2)

T. Tamir, “Nonspecular phenomena in beam fields reflected by multilayered media,” J. Opt. Soc. Am. A 3, 558–565 (1986).
[CrossRef]

S. Zhu, A. W. Yu, D. Hawley, R. Roy, “Frustrated total internal reflection: a demonstration and review,” Am. J. Phys. 54, 601–606 (1986).
[CrossRef]

1984 (1)

L. B. Felsen, “Geometrical theory of diffraction, evanescent waves, complex rays and Gaussian beams,” Geophys. J. R. Astron. Soc. 79, 77–88 (1984).
[CrossRef]

1983 (1)

A. Puri, D. N. Pattanayak, J. L. Birman, “Resonance effects on total internal reflection and lateral (Goos–Hänchen) beam displacement at the interface between nonlocal and local dielectric,” Phys. Rev. B 28, 5877–5886 (1983).
[CrossRef]

1980 (1)

R. G. Turner, “Shifts of coherent light beams on reflection at plane interfaces between isotropic media,” Aust. J. Phys. 33, 319–335 (1980).

1977 (3)

1975 (3)

J. H. Marburger, “Self-focusing: theory,” Prog. Quant. Electron. 4, 35–110 (1975).
[CrossRef]

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Y. Levy, C. Imbert, “Amplification de déplacements à la réflection totale,” Opt. Commun. 13, 43–47 (1975).
[CrossRef]

1974 (1)

Y. M. Antar, W. M. Boerner, “Gaussian beam interaction with a planar dielectric interface,” Can. J. Phys. 52, 962–972 (1974).

1973 (1)

J. W. Ra, H. L. Bertoni, L. B. Felsen, “Reflection and transmission of beams at a dielectric interface,” SIAM J. Appl. Math. 24, 396–413 (1973).
[CrossRef]

1972 (3)

O. Costa de Beauregard, C. Imbert, “Quantized longitudinal and transverse shifts associated with total internal reflection,” Phys. Rev. D 7, 3555–3563 (1972).
[CrossRef]

C. Imbert, “Calculation and experimental proof of the transverse shift induced by total internal reflection of a circularly polarized light beam,” Phys. Rev. D 5, 787–796 (1972).
[CrossRef]

T. Tamir, “Inhomogeneous wave types at planar structures: I. The lateral wave,” Optik (Stuttgart) 36, 209–232 (1972); “II. Surface waves,” Optik (Stuttgart) 37, 204–228 (1973); “III. Leaky waves,” Optik (Stuttgart) 38, 269–297 (1973).

1971 (4)

B. R. Horovitz, T. Tamir, “Lateral displacement of a light beam at a dielectric interface,” J. Opt. Soc. Am. 61, 586–594 (1971).
[CrossRef]

T. Tamir, H. L. Bertoni, “Laternal displacement of optical beams at multilayered and periodic structures,” J. Opt. Soc. Am. 61, 1397–1413 (1971).
[CrossRef]

H. Shih, N. Bloembergen, “Phase-matched critical total reflection and the Goos–Hänchen shift in second-harmonic generation,” Phys. Rev. A 3, 412–420 (1971).
[CrossRef]

G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
[CrossRef]

1970 (2)

H. K. V. Lotsch, “Beam displacement at total reflection: the Goos–Hänchen effect,” Part I, Optik (Stuttgart) 32, 116–137 (1970); Part II, Optik (Stuttgart) 32, 189–204 (1970); Part III, Optik (Stuttgart) 32, 299–319 (1970); Part IV, Optik (Stuttgart) 32, 553–569 (1971).

J. Richard, “Courbes de flux d’énergie de l’onde évanescente et nouvelle explication du dêplacement d’un faisceau lumineux dants la réflexion totale,” Nouv. Rev. Opt. 1, 275–286 (1970).

1968 (1)

W. G. Wagner, H. A. Haus, J. H. Marburger, “Large-scale self-trapping of optical beams in the paraxial ray approximation,” Phys. Rev. 175, 256–266 (1968).
[CrossRef]

1965 (2)

O. Costa de Beauregard, “Translational internal spin effect with photons,” Phys. Rev. 139, B1443–B1446 (1965).
[CrossRef]

H. Schilling, “Die Strahlversetzung bei der Reflection linear oder elliptisch polarisierter ebener Wellen an der Trennebene zwischen absorbierenden Medien,” Ann. Phys. Leipzig 16, 122–134 (1965).
[CrossRef]

1964 (1)

1955 (1)

F. I. Fedorov, “K teorii polnovo otrazenija,” Dok. Akad. Nauk SSSR 105, 465–467 (1955).

1949 (1)

C. Fragstein, “Zur Seitenversetzung des totalreflektierten Lichtstrahles,” Ann. Phys. Leipzig 4, 271–278 (1949).
[CrossRef]

1948 (1)

K. V. Artmann, “Berechung der Seitenversetzung des total-reflektierten Strahles,” Ann. Phys. Leipzig 2, 87–102 (1948).
[CrossRef]

1947 (1)

F. Goos, H. Hänchen, “Ein neuer and fundamentaler Versuch zur total Reflection,” Ann. Phys. Leipzig 1, 333–345 (1947).
[CrossRef]

1941 (1)

1929 (1)

J. Picht, “Beitrag zur Theorie der total reflection,” Ann. Phys. Leipzig 5, 433–496 (1929).
[CrossRef]

Acioli, H.

Akhmanov, S. A.

S. A. Akhmanov, R. V. Khokhlov, A. P. Sukhorukov, “Self-focusing, self-defocusing and self-modulation of laser beams,” in Laser Handbook, F. T. Arecchi, E. O. Schultz-Dubois, eds. (North-Holland, Amsterdam, 1972), pp. 1152–1228.

Antar, Y. M.

Y. M. Antar, W. M. Boerner, “Gaussian beam interaction with a planar dielectric interface,” Can. J. Phys. 52, 962–972 (1974).

Artmann, K. V.

K. V. Artmann, “Berechung der Seitenversetzung des total-reflektierten Strahles,” Ann. Phys. Leipzig 2, 87–102 (1948).
[CrossRef]

Baylard, C.

J. J. Greffet, C. Baylard, “Nonspecular astigmatic reflection of a 3D Gaussian beam on an interface,” Opt. Commun. 93, 271–276 (1992).
[CrossRef]

Bertoni, H. L.

J. W. Ra, H. L. Bertoni, L. B. Felsen, “Reflection and transmission of beams at a dielectric interface,” SIAM J. Appl. Math. 24, 396–413 (1973).
[CrossRef]

T. Tamir, H. L. Bertoni, “Laternal displacement of optical beams at multilayered and periodic structures,” J. Opt. Soc. Am. 61, 1397–1413 (1971).
[CrossRef]

Birman, J. L.

A. Puri, D. N. Pattanayak, J. L. Birman, “Resonance effects on total internal reflection and lateral (Goos–Hänchen) beam displacement at the interface between nonlocal and local dielectric,” Phys. Rev. B 28, 5877–5886 (1983).
[CrossRef]

Bloembergen, N.

H. Shih, N. Bloembergen, “Phase-matched critical total reflection and the Goos–Hänchen shift in second-harmonic generation,” Phys. Rev. A 3, 412–420 (1971).
[CrossRef]

Boerner, W. M.

Y. M. Antar, W. M. Boerner, “Gaussian beam interaction with a planar dielectric interface,” Can. J. Phys. 52, 962–972 (1974).

Bretenaker, F.

F. Bretenaker, A. Le Floch, L. Dutriaux, “Direct measurement of the optical Goos–Hänchen effect in lasers,” Phys. Rev. Lett. 68, 931–933 (1992); L. Dutriaux, A. Le Floch, F. Bretenaker, “Goos–Hänchen effect in the dynamics of laser eigenstates,” J. Opt. Soc. Am. B 9, 2283–2289 (1992); L. Dutriaux, A. Le Floch, F. Bretenaker, “Measurement of the transverse displacement at total reflection by helicoidal laser eigenstates,” Europhys. Lett. 24, 345–349 (1993).
[CrossRef] [PubMed]

Byer, R. L.

Carniglia, C. K.

Costa de Beauregard, O.

O. Costa de Beauregard, C. Imbert, “Quantized longitudinal and transverse shifts associated with total internal reflection,” Phys. Rev. D 7, 3555–3563 (1972).
[CrossRef]

O. Costa de Beauregard, “Translational internal spin effect with photons,” Phys. Rev. 139, B1443–B1446 (1965).
[CrossRef]

Deschamps, G. A.

G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
[CrossRef]

Dutriaux, L.

F. Bretenaker, A. Le Floch, L. Dutriaux, “Direct measurement of the optical Goos–Hänchen effect in lasers,” Phys. Rev. Lett. 68, 931–933 (1992); L. Dutriaux, A. Le Floch, F. Bretenaker, “Goos–Hänchen effect in the dynamics of laser eigenstates,” J. Opt. Soc. Am. B 9, 2283–2289 (1992); L. Dutriaux, A. Le Floch, F. Bretenaker, “Measurement of the transverse displacement at total reflection by helicoidal laser eigenstates,” Europhys. Lett. 24, 345–349 (1993).
[CrossRef] [PubMed]

Einzinger, P. D.

P. D. Einzinger, S. Raz, “Wave solutions under complex space–time shifts,” J. Opt. Soc. Am. A 4, 2–10 (1987).

Falco, F.

Fedorov, F. I.

F. I. Fedorov, “K teorii polnovo otrazenija,” Dok. Akad. Nauk SSSR 105, 465–467 (1955).

Fejer, M. M.

Felsen, L. B.

S. Zeroug, L. B. Felsen, “Nonspecular reflection of two-and three-dimensional acoustic beams from fluid-immersed plane-layered elastic structures,” J. Acoust. Soc. Am. 95, 3075–3089 (1994).
[CrossRef]

L. B. Felsen, “Geometrical theory of diffraction, evanescent waves, complex rays and Gaussian beams,” Geophys. J. R. Astron. Soc. 79, 77–88 (1984).
[CrossRef]

J. W. Ra, H. L. Bertoni, L. B. Felsen, “Reflection and transmission of beams at a dielectric interface,” SIAM J. Appl. Math. 24, 396–413 (1973).
[CrossRef]

L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N.J., 1973), Chap. 5, pp. 506–538.

Fragstein, C.

C. Fragstein, “Zur Seitenversetzung des totalreflektierten Lichtstrahles,” Ann. Phys. Leipzig 4, 271–278 (1949).
[CrossRef]

Fujimoto, J. G.

H. A. Haus, J. G. Fujimoto, E. P. Ippen, “Analytic theory of additive pulse and Kerr lens mode locking,” IEEE J. Quantum Electron. 10, 2086–2096 (1992).
[CrossRef]

D. Huang, M. Ulman, H. Acioli, H. A. Haus, J. G. Fujimoto, “Self-focusing-induced saturable loss for laser mode locking,” Opt. Lett. 17, 511–513 (1992).
[CrossRef] [PubMed]

Goos, F.

F. Goos, H. Hänchen, “Ein neuer and fundamentaler Versuch zur total Reflection,” Ann. Phys. Leipzig 1, 333–345 (1947).
[CrossRef]

Greffet, J. J.

J. J. Greffet, C. Baylard, “Nonspecular astigmatic reflection of a 3D Gaussian beam on an interface,” Opt. Commun. 93, 271–276 (1992).
[CrossRef]

Hänchen, H.

F. Goos, H. Hänchen, “Ein neuer and fundamentaler Versuch zur total Reflection,” Ann. Phys. Leipzig 1, 333–345 (1947).
[CrossRef]

Haus, H. A.

H. A. Haus, J. G. Fujimoto, E. P. Ippen, “Analytic theory of additive pulse and Kerr lens mode locking,” IEEE J. Quantum Electron. 10, 2086–2096 (1992).
[CrossRef]

D. Huang, M. Ulman, H. Acioli, H. A. Haus, J. G. Fujimoto, “Self-focusing-induced saturable loss for laser mode locking,” Opt. Lett. 17, 511–513 (1992).
[CrossRef] [PubMed]

W. G. Wagner, H. A. Haus, J. H. Marburger, “Large-scale self-trapping of optical beams in the paraxial ray approximation,” Phys. Rev. 175, 256–266 (1968).
[CrossRef]

Hawley, D.

S. Zhu, A. W. Yu, D. Hawley, R. Roy, “Frustrated total internal reflection: a demonstration and review,” Am. J. Phys. 54, 601–606 (1986).
[CrossRef]

Horovitz, B. R.

Huang, D.

Hugonin, J. P.

J. P. Hugonin, R. Petit, “Étude généralie des déplacements à la réflection totale,” J. Opt. (Paris) 8, 73–87 (1977).
[CrossRef]

Imbert, C.

Y. Levy, C. Imbert, “Amplification de déplacements à la réflection totale,” Opt. Commun. 13, 43–47 (1975).
[CrossRef]

C. Imbert, “Calculation and experimental proof of the transverse shift induced by total internal reflection of a circularly polarized light beam,” Phys. Rev. D 5, 787–796 (1972).
[CrossRef]

O. Costa de Beauregard, C. Imbert, “Quantized longitudinal and transverse shifts associated with total internal reflection,” Phys. Rev. D 7, 3555–3563 (1972).
[CrossRef]

Ippen, E. P.

H. A. Haus, J. G. Fujimoto, E. P. Ippen, “Analytic theory of additive pulse and Kerr lens mode locking,” IEEE J. Quantum Electron. 10, 2086–2096 (1992).
[CrossRef]

Jones, R. C.

Kaplan, A. E.

A. E. Kaplan, P. W. Smith, W. J. Tomlison, “Nonlinear waves and switching effects at nonlinear interfaces,” in Nonlinear Surface Electromagnetic Phenomena, H.-E. Ponath, G. I. Stegeman, eds. (North-Holland, Amsterdam, 1991), Chap. 4, pp. 323–351.
[CrossRef]

Khokhlov, R. V.

S. A. Akhmanov, R. V. Khokhlov, A. P. Sukhorukov, “Self-focusing, self-defocusing and self-modulation of laser beams,” in Laser Handbook, F. T. Arecchi, E. O. Schultz-Dubois, eds. (North-Holland, Amsterdam, 1972), pp. 1152–1228.

Kou, E. F. Y.

Lax, M.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Le Floch, A.

F. Bretenaker, A. Le Floch, L. Dutriaux, “Direct measurement of the optical Goos–Hänchen effect in lasers,” Phys. Rev. Lett. 68, 931–933 (1992); L. Dutriaux, A. Le Floch, F. Bretenaker, “Goos–Hänchen effect in the dynamics of laser eigenstates,” J. Opt. Soc. Am. B 9, 2283–2289 (1992); L. Dutriaux, A. Le Floch, F. Bretenaker, “Measurement of the transverse displacement at total reflection by helicoidal laser eigenstates,” Europhys. Lett. 24, 345–349 (1993).
[CrossRef] [PubMed]

Levy, Y.

Y. Levy, C. Imbert, “Amplification de déplacements à la réflection totale,” Opt. Commun. 13, 43–47 (1975).
[CrossRef]

Lin, L.

Lotsch, H. K. V.

H. K. V. Lotsch, “Beam displacement at total reflection: the Goos–Hänchen effect,” Part I, Optik (Stuttgart) 32, 116–137 (1970); Part II, Optik (Stuttgart) 32, 189–204 (1970); Part III, Optik (Stuttgart) 32, 299–319 (1970); Part IV, Optik (Stuttgart) 32, 553–569 (1971).

Louisell, W. H.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Marburger, J. H.

J. H. Marburger, “Self-focusing: theory,” Prog. Quant. Electron. 4, 35–110 (1975).
[CrossRef]

W. G. Wagner, H. A. Haus, J. H. Marburger, “Large-scale self-trapping of optical beams in the paraxial ray approximation,” Phys. Rev. 175, 256–266 (1968).
[CrossRef]

Marcuvitz, N.

L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N.J., 1973), Chap. 5, pp. 506–538.

Marseille, A.

E. Pfleghaar, A. Marseille, A. Weis, “Quantitative investigation of the effect of resonant absorbers on the Goos–Hänchen shift,” Phys. Rev. Lett. 70, 2281–2284 (1993).
[CrossRef] [PubMed]

McGuirk, M.

McKnight, W. B.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Nasalski, W.

W. Nasalski, “Linear formulation of nonlinear propagation of optical beams and pulses,” Opt. Appl. 24, 205–208 (1994); “Complex ray tracing of nonlinear propagation,” Opt. Commun. 119, 218–226 (1995).

W. Nasalski, “Ray analysis of Gaussian beam nonspecular scattering,” Opt. Commun. 92, 307–314 (1992).
[CrossRef]

W. Nasalski, “Nonspecular bistability versus diffraction at nonlinear hybrid interfaces,” Opt. Commun. 77, 443–451 (1990); “Bistable switching effects at nonlocal nonlinear interfaces,” in Proceedings of the Progress in Electromagnetics Research Symposium, JPL Publication 93–17, J. J. van Zyl, ed. (Jet Propulsion Laboratory, Pasadena, Calif., 1993), p. 339.
[CrossRef]

W. Nasalski, “Modified reflectance and geometrical deformations of Gaussian beams reflected at a dielectric interface,” J. Opt. Soc. Am. A 6, 1447–1454 (1989).
[CrossRef]

W. Nasalski, T. Tamir, L. Lin, “Displacement of the intensity peak in narrow beams reflected at a dielectric interface,” J. Opt. Soc. Am. A 5, 132–140 (1988).
[CrossRef]

Pask, C.

Pattanayak, D. N.

A. Puri, D. N. Pattanayak, J. L. Birman, “Resonance effects on total internal reflection and lateral (Goos–Hänchen) beam displacement at the interface between nonlocal and local dielectric,” Phys. Rev. B 28, 5877–5886 (1983).
[CrossRef]

Petit, R.

J. P. Hugonin, R. Petit, “Étude généralie des déplacements à la réflection totale,” J. Opt. (Paris) 8, 73–87 (1977).
[CrossRef]

Pfleghaar, E.

E. Pfleghaar, A. Marseille, A. Weis, “Quantitative investigation of the effect of resonant absorbers on the Goos–Hänchen shift,” Phys. Rev. Lett. 70, 2281–2284 (1993).
[CrossRef] [PubMed]

Picht, J.

J. Picht, “Beitrag zur Theorie der total reflection,” Ann. Phys. Leipzig 5, 433–496 (1929).
[CrossRef]

Puri, A.

A. Puri, D. N. Pattanayak, J. L. Birman, “Resonance effects on total internal reflection and lateral (Goos–Hänchen) beam displacement at the interface between nonlocal and local dielectric,” Phys. Rev. B 28, 5877–5886 (1983).
[CrossRef]

Ra, J. W.

J. W. Ra, H. L. Bertoni, L. B. Felsen, “Reflection and transmission of beams at a dielectric interface,” SIAM J. Appl. Math. 24, 396–413 (1973).
[CrossRef]

Raz, S.

P. D. Einzinger, S. Raz, “Wave solutions under complex space–time shifts,” J. Opt. Soc. Am. A 4, 2–10 (1987).

Renard, R. H.

Richard, J.

J. Richard, “Courbes de flux d’énergie de l’onde évanescente et nouvelle explication du dêplacement d’un faisceau lumineux dants la réflexion totale,” Nouv. Rev. Opt. 1, 275–286 (1970).

Roy, R.

S. Zhu, A. W. Yu, D. Hawley, R. Roy, “Frustrated total internal reflection: a demonstration and review,” Am. J. Phys. 54, 601–606 (1986).
[CrossRef]

Schiller, S.

Schilling, H.

H. Schilling, “Die Strahlversetzung bei der Reflection linear oder elliptisch polarisierter ebener Wellen an der Trennebene zwischen absorbierenden Medien,” Ann. Phys. Leipzig 16, 122–134 (1965).
[CrossRef]

Shih, H.

H. Shih, N. Bloembergen, “Phase-matched critical total reflection and the Goos–Hänchen shift in second-harmonic generation,” Phys. Rev. A 3, 412–420 (1971).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chaps. 15–23, pp. 581–922.

Simon, R.

Smith, P. W.

A. E. Kaplan, P. W. Smith, W. J. Tomlison, “Nonlinear waves and switching effects at nonlinear interfaces,” in Nonlinear Surface Electromagnetic Phenomena, H.-E. Ponath, G. I. Stegeman, eds. (North-Holland, Amsterdam, 1991), Chap. 4, pp. 323–351.
[CrossRef]

Snyder, A. W.

Sukhorukov, A. P.

S. A. Akhmanov, R. V. Khokhlov, A. P. Sukhorukov, “Self-focusing, self-defocusing and self-modulation of laser beams,” in Laser Handbook, F. T. Arecchi, E. O. Schultz-Dubois, eds. (North-Holland, Amsterdam, 1972), pp. 1152–1228.

Tamir, T.

Tomlison, W. J.

A. E. Kaplan, P. W. Smith, W. J. Tomlison, “Nonlinear waves and switching effects at nonlinear interfaces,” in Nonlinear Surface Electromagnetic Phenomena, H.-E. Ponath, G. I. Stegeman, eds. (North-Holland, Amsterdam, 1991), Chap. 4, pp. 323–351.
[CrossRef]

Turner, R. G.

R. G. Turner, “Shifts of coherent light beams on reflection at plane interfaces between isotropic media,” Aust. J. Phys. 33, 319–335 (1980).

Ulman, M.

Wagner, W. G.

W. G. Wagner, H. A. Haus, J. H. Marburger, “Large-scale self-trapping of optical beams in the paraxial ray approximation,” Phys. Rev. 175, 256–266 (1968).
[CrossRef]

Weis, A.

E. Pfleghaar, A. Marseille, A. Weis, “Quantitative investigation of the effect of resonant absorbers on the Goos–Hänchen shift,” Phys. Rev. Lett. 70, 2281–2284 (1993).
[CrossRef] [PubMed]

White, I. A.

Yu, A. W.

S. Zhu, A. W. Yu, D. Hawley, R. Roy, “Frustrated total internal reflection: a demonstration and review,” Am. J. Phys. 54, 601–606 (1986).
[CrossRef]

Yu, I. I.

Zeroug, S.

S. Zeroug, L. B. Felsen, “Nonspecular reflection of two-and three-dimensional acoustic beams from fluid-immersed plane-layered elastic structures,” J. Acoust. Soc. Am. 95, 3075–3089 (1994).
[CrossRef]

Zhang, S.

Zhu, S.

S. Zhu, A. W. Yu, D. Hawley, R. Roy, “Frustrated total internal reflection: a demonstration and review,” Am. J. Phys. 54, 601–606 (1986).
[CrossRef]

Am. J. Phys. (1)

S. Zhu, A. W. Yu, D. Hawley, R. Roy, “Frustrated total internal reflection: a demonstration and review,” Am. J. Phys. 54, 601–606 (1986).
[CrossRef]

Ann. Phys. Leipzig (5)

H. Schilling, “Die Strahlversetzung bei der Reflection linear oder elliptisch polarisierter ebener Wellen an der Trennebene zwischen absorbierenden Medien,” Ann. Phys. Leipzig 16, 122–134 (1965).
[CrossRef]

J. Picht, “Beitrag zur Theorie der total reflection,” Ann. Phys. Leipzig 5, 433–496 (1929).
[CrossRef]

F. Goos, H. Hänchen, “Ein neuer and fundamentaler Versuch zur total Reflection,” Ann. Phys. Leipzig 1, 333–345 (1947).
[CrossRef]

K. V. Artmann, “Berechung der Seitenversetzung des total-reflektierten Strahles,” Ann. Phys. Leipzig 2, 87–102 (1948).
[CrossRef]

C. Fragstein, “Zur Seitenversetzung des totalreflektierten Lichtstrahles,” Ann. Phys. Leipzig 4, 271–278 (1949).
[CrossRef]

Appl. Opt. (1)

Aust. J. Phys. (1)

R. G. Turner, “Shifts of coherent light beams on reflection at plane interfaces between isotropic media,” Aust. J. Phys. 33, 319–335 (1980).

Can. J. Phys. (1)

Y. M. Antar, W. M. Boerner, “Gaussian beam interaction with a planar dielectric interface,” Can. J. Phys. 52, 962–972 (1974).

Dok. Akad. Nauk SSSR (1)

F. I. Fedorov, “K teorii polnovo otrazenija,” Dok. Akad. Nauk SSSR 105, 465–467 (1955).

Electron. Lett. (1)

G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
[CrossRef]

Geophys. J. R. Astron. Soc. (1)

L. B. Felsen, “Geometrical theory of diffraction, evanescent waves, complex rays and Gaussian beams,” Geophys. J. R. Astron. Soc. 79, 77–88 (1984).
[CrossRef]

IEEE J. Quantum Electron. (1)

H. A. Haus, J. G. Fujimoto, E. P. Ippen, “Analytic theory of additive pulse and Kerr lens mode locking,” IEEE J. Quantum Electron. 10, 2086–2096 (1992).
[CrossRef]

J. Acoust. Soc. Am. (1)

S. Zeroug, L. B. Felsen, “Nonspecular reflection of two-and three-dimensional acoustic beams from fluid-immersed plane-layered elastic structures,” J. Acoust. Soc. Am. 95, 3075–3089 (1994).
[CrossRef]

J. Opt. (Paris) (1)

J. P. Hugonin, R. Petit, “Étude généralie des déplacements à la réflection totale,” J. Opt. (Paris) 8, 73–87 (1977).
[CrossRef]

J. Opt. Soc. Am. (6)

J. Opt. Soc. Am. A (7)

Nouv. Rev. Opt. (1)

J. Richard, “Courbes de flux d’énergie de l’onde évanescente et nouvelle explication du dêplacement d’un faisceau lumineux dants la réflexion totale,” Nouv. Rev. Opt. 1, 275–286 (1970).

Opt. Appl. (1)

W. Nasalski, “Linear formulation of nonlinear propagation of optical beams and pulses,” Opt. Appl. 24, 205–208 (1994); “Complex ray tracing of nonlinear propagation,” Opt. Commun. 119, 218–226 (1995).

Opt. Commun. (4)

W. Nasalski, “Ray analysis of Gaussian beam nonspecular scattering,” Opt. Commun. 92, 307–314 (1992).
[CrossRef]

Y. Levy, C. Imbert, “Amplification de déplacements à la réflection totale,” Opt. Commun. 13, 43–47 (1975).
[CrossRef]

J. J. Greffet, C. Baylard, “Nonspecular astigmatic reflection of a 3D Gaussian beam on an interface,” Opt. Commun. 93, 271–276 (1992).
[CrossRef]

W. Nasalski, “Nonspecular bistability versus diffraction at nonlinear hybrid interfaces,” Opt. Commun. 77, 443–451 (1990); “Bistable switching effects at nonlocal nonlinear interfaces,” in Proceedings of the Progress in Electromagnetics Research Symposium, JPL Publication 93–17, J. J. van Zyl, ed. (Jet Propulsion Laboratory, Pasadena, Calif., 1993), p. 339.
[CrossRef]

Opt. Lett. (2)

Optik (Stuttgart) (1)

T. Tamir, “Inhomogeneous wave types at planar structures: I. The lateral wave,” Optik (Stuttgart) 36, 209–232 (1972); “II. Surface waves,” Optik (Stuttgart) 37, 204–228 (1973); “III. Leaky waves,” Optik (Stuttgart) 38, 269–297 (1973).

Part I, Optik (Stuttgart) (1)

H. K. V. Lotsch, “Beam displacement at total reflection: the Goos–Hänchen effect,” Part I, Optik (Stuttgart) 32, 116–137 (1970); Part II, Optik (Stuttgart) 32, 189–204 (1970); Part III, Optik (Stuttgart) 32, 299–319 (1970); Part IV, Optik (Stuttgart) 32, 553–569 (1971).

Phys. Rev. (2)

W. G. Wagner, H. A. Haus, J. H. Marburger, “Large-scale self-trapping of optical beams in the paraxial ray approximation,” Phys. Rev. 175, 256–266 (1968).
[CrossRef]

O. Costa de Beauregard, “Translational internal spin effect with photons,” Phys. Rev. 139, B1443–B1446 (1965).
[CrossRef]

Phys. Rev. A (2)

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

H. Shih, N. Bloembergen, “Phase-matched critical total reflection and the Goos–Hänchen shift in second-harmonic generation,” Phys. Rev. A 3, 412–420 (1971).
[CrossRef]

Phys. Rev. B (1)

A. Puri, D. N. Pattanayak, J. L. Birman, “Resonance effects on total internal reflection and lateral (Goos–Hänchen) beam displacement at the interface between nonlocal and local dielectric,” Phys. Rev. B 28, 5877–5886 (1983).
[CrossRef]

Phys. Rev. D (2)

O. Costa de Beauregard, C. Imbert, “Quantized longitudinal and transverse shifts associated with total internal reflection,” Phys. Rev. D 7, 3555–3563 (1972).
[CrossRef]

C. Imbert, “Calculation and experimental proof of the transverse shift induced by total internal reflection of a circularly polarized light beam,” Phys. Rev. D 5, 787–796 (1972).
[CrossRef]

Phys. Rev. Lett. (2)

E. Pfleghaar, A. Marseille, A. Weis, “Quantitative investigation of the effect of resonant absorbers on the Goos–Hänchen shift,” Phys. Rev. Lett. 70, 2281–2284 (1993).
[CrossRef] [PubMed]

F. Bretenaker, A. Le Floch, L. Dutriaux, “Direct measurement of the optical Goos–Hänchen effect in lasers,” Phys. Rev. Lett. 68, 931–933 (1992); L. Dutriaux, A. Le Floch, F. Bretenaker, “Goos–Hänchen effect in the dynamics of laser eigenstates,” J. Opt. Soc. Am. B 9, 2283–2289 (1992); L. Dutriaux, A. Le Floch, F. Bretenaker, “Measurement of the transverse displacement at total reflection by helicoidal laser eigenstates,” Europhys. Lett. 24, 345–349 (1993).
[CrossRef] [PubMed]

Prog. Quant. Electron. (1)

J. H. Marburger, “Self-focusing: theory,” Prog. Quant. Electron. 4, 35–110 (1975).
[CrossRef]

SIAM J. Appl. Math. (1)

J. W. Ra, H. L. Bertoni, L. B. Felsen, “Reflection and transmission of beams at a dielectric interface,” SIAM J. Appl. Math. 24, 396–413 (1973).
[CrossRef]

Other (4)

S. A. Akhmanov, R. V. Khokhlov, A. P. Sukhorukov, “Self-focusing, self-defocusing and self-modulation of laser beams,” in Laser Handbook, F. T. Arecchi, E. O. Schultz-Dubois, eds. (North-Holland, Amsterdam, 1972), pp. 1152–1228.

A. E. Kaplan, P. W. Smith, W. J. Tomlison, “Nonlinear waves and switching effects at nonlinear interfaces,” in Nonlinear Surface Electromagnetic Phenomena, H.-E. Ponath, G. I. Stegeman, eds. (North-Holland, Amsterdam, 1991), Chap. 4, pp. 323–351.
[CrossRef]

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chaps. 15–23, pp. 581–922.

L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N.J., 1973), Chap. 5, pp. 506–538.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1
Fig. 1

Geometry of the problem. a: Principal rays or x3 axes in the principal incidence plane (A); b: local rays or x ¯ 3 axes in the local incidence plane (B); a′: local rays or x ¯ 3 axes in the local incidence plane coincident with plane A. Incident and reflected beams are on the left-hand and right-hand sides, respectively.

Fig. 2
Fig. 2

Longitudinal nonspecular effects in the incidence plane (j = 1, y = 0). δx1 is the lateral shift and δx31 is the focal shift of the beam waist location (marked by ●), and δϑ1 is the angular shift of the actual reflected-beam axis xr31 of the longitudinal beam factor Ψ1. All the nonspecular changes are with respect to the waist location and direction of the x3 axis of the g-o reflected beam.

Fig. 3
Fig. 3

Transverse nonspecular effects in the transverse plane (j = 2, z = 0). δx2 is the lateral shift and δx32 is the focal shift of the beam waist location (marked by ●), and δϑ2 is the angular shift of the actual reflected-beam axis xr32 of the transverse beam factor Ψ2. All the nonspecular changes are with respect to the waist location and direction of the x3 axis of the g-o reflected beam.

Equations (84)

Equations on this page are rendered with MathJax. Learn more.

α = cos ϑ 1 ,             γ = sin ϑ 1 sin ϑ 2 ,             β = sin ϑ 1 cos ϑ 2 ,
x ¯ 1 = + x sin ϑ 1 ( y sin ϑ 2 + z cos ϑ 2 ) cos ϑ 1 ,
x ¯ 2 = + y cos ϑ 2 - z sin ϑ 2 ,
x ¯ 3 = ± x cos ϑ 1 + ( y sin ϑ 2 + z cos ϑ 2 ) sin ϑ 1 ,
α ¯ 1 = α sin ϑ 1 - ( γ sin ϑ 2 + β cos ϑ 2 ) cos ϑ 1 ,
α ¯ 2 = γ cos ϑ 2 - β sin ϑ 2 ,
α ¯ 3 = α cos ϑ 1 + ( γ sin ϑ 2 + β cos ϑ 2 ) sin ϑ 1 .
E ( x 1 , x 2 , x 3 ) = - + E ( α 1 , α 2 , x 3 ) × exp [ i k ( ± α 1 x 1 + α 2 x 2 + α 3 x 3 ) ] × k 2 d α 1 d α 2 .
E ( α 1 , α 2 , x 3 ) = [ + α ¯ 3 E p ( α 1 , α 2 , x 3 ) , E s ( α 1 , α 2 , x 3 ) , α ¯ 1 E p ( α 1 , α 2 , x 3 ) ]
E x ( α 1 , α 2 , x 3 ) = + E p ( α 1 , α 2 , x 3 ) sin ϑ 1 ,
E y ( α 1 , α 2 , x 3 ) = + E s ( α 1 , α 2 , x 3 ) cos ϑ 2 E p ( α 1 , α 2 , x 3 ) × sin ϑ 2 cos ϑ 1 ,
E z ( α 1 , α 2 , x 3 ) = E p ( α 1 , α 2 , x 3 ) cos ϑ 2 cos ϑ 1 - E s ( α 1 , α 2 , x 3 ) sin ϑ 2 ,
E ( x 1 , x 2 , x 3 ) = s exp ( i k x 3 ) Ψ 1 ( x 1 , x 3 ; w 01 ) Ψ 2 ( x 2 , x 3 ; w 02 )
Ψ j ( x j , x 3 ; w 0 j ) = ( w 0 j / v j ) exp [ - ( x j / v j ) 2 ] ,             j = 1 , 2.
v j v j ( x 3 ) = w 0 j ( 1 + i x 3 / z D j ) 1 / 2
s = ( s s ) = ( χ + 1 ) - 1 / 2 ( χ 1 ) ,
χ = E / E = η exp ( i σ ) ,
E ( x 1 , 0 , x 3 ) = s exp ( i k x 3 ) v 2 - 1 ( x 3 ) Ψ 1 ( x 1 , x 3 ; w 01 ) ,
E ( 0 , x 2 , x 3 ) = s exp ( i k x 3 ) v 1 - 1 ( x 3 ) Ψ 2 ( x 2 , x 3 ; w 02 ) .
α 3 1 - ( 1 / 2 ) α 1 2 - ( 1 / 2 ) α 2 2
E ( x 1 , x 2 , x 3 ) ( 4 π ) - 1 w 01 w 02 × s - + exp [ - ( k / 2 ) ( z D 1 α 1 2 + z D 2 α 2 2 ) ] × exp [ i k ( α 1 x 1 + α 2 x 2 + α 3 x 3 ) ] k 2 d α 1 d α 2 .
E r ( x 1 , x 2 , x 3 ) = - + E r ( α 1 , α 2 , x 3 ) × exp [ i k ( - α 1 x 1 + α 2 x 2 + α 3 x 3 ) ] k d α 1 d α 2 ,
E r ( α 1 , α 2 , x 3 ) = R ( α 1 , α 2 , x 3 ) E ( α 1 , α 2 , x 3 ) ,
E r ( α 1 , α 2 , x 3 ) = R ( α 1 , α 2 , x 3 ) E ( α 1 , α 2 , x 3 ) ,
R ( α 1 , α 2 , x 3 ) = + R p ( α 1 , α 2 ) [ cos ϑ 2 - χ ¯ - 1 ( α 1 , α 2 , x 3 ) × sin ϑ 2 ] cos ϑ 2 - R s ( α 1 , α 2 ) × [ χ ¯ - 1 ( α 1 , α 2 , x 3 ) cos ϑ 2 + sin ϑ 2 ] sin ϑ 2 ,
R ( α 1 , α 2 , x 3 ) = + R s ( α 1 , α 2 ) [ cos ϑ 2 + χ ¯ ( α 1 , α 2 , x 3 ) × sin ϑ 2 ] cos ϑ 2 + R p ( α 1 , α 2 ) × [ χ ¯ ( α 1 , α 2 , x 3 ) cos ϑ 2 - sin ϑ 2 ] sin ϑ 2 .
χ ¯ ( α 1 , α 2 , x 3 ) = E z ( α 1 , α 2 , x 3 ) / E ( α 1 , α 2 , x 3 ) = [ sin ϑ 2 + κ ( α 1 , α 2 , x 3 ) cos ϑ 2 cos ϑ 1 ] / [ - cos ϑ 2 + κ ( α 1 , α 2 , x 3 ) sin ϑ 2 cos ϑ 1 ] ,
κ ( α 1 , α 2 , x 3 ) = E p ( α 1 , α 2 , x 3 ) / E s ( α 1 , α 2 , x 3 )
χ ¯ ( α 1 , α 2 , x 3 ) = χ ( α 1 , α 2 , x 3 ) cos ϑ 01 ,
χ ( α 1 , α 2 , x 3 ) = χ 0 + χ 1 ( α 1 - α 10 ) + χ 2 α 2 + ,
χ 0 χ ( 0 , 0 , x 3 ) = E p ( 0 , 0 , x 3 ) / E s ( 0 , 0 , x 3 ) = E ( 0 , 0 , x 3 ) / E ( 0 , 0 , x 3 ) ,
χ χ 0 .
R ( α 1 , α 2 ) = [ R ( α 1 , α 2 ) 0 0 R ( α 1 , α 2 ) ] ,
E r ( x 1 , x 2 , x 3 ) ( 4 π ) - 1 w 01 w 02 - + [ R ( α 1 , α 2 ) s ] × exp [ - ( k / 2 ) ( α 1 2 z D 1 + α 2 2 z D 2 ) ] × exp [ i k ( α 1 x 1 + α 2 x 2 + α 3 x 3 ) ] k 2 d α 1 d α 2 .
E 0 ( x 1 , x 2 , x 3 ) = ( R 0 s ) exp ( i k x 3 ) Ψ 1 ( x 1 , x 3 ; w 01 ) × Ψ 2 ( x 1 , x 3 ; w 02 ) ,
R ( α 1 , α 2 ) = exp { ln [ R ( α 1 , α 2 ) ] } = R 0 exp [ α 1 R 1 + α 2 R 2 + ( 1 / 2 ) α 1 2 R 11 + ( 1 / 2 ) α 2 2 R 22 + α 1 α 2 R 12 + ] ,
R 0 = R ( 0 , 0 ) ,             R j = R j / R ,             R i j = R i j / R - R i R j / R 2 ,
R ( α 1 , α 2 ) R 0 exp { - i k [ α 1 L 1 + α 2 L 2 - ( 1 / 2 ) α 1 2 F 1 - ( 1 / 2 ) α 2 2 F 2 ] } ,
L j = ( i / k ) R j ,
F j = - ( i / k ) R j j ,
E r ( x 1 , x 2 , x 3 ) = ( R 0 s ) exp ( i k x 3 ) Ψ 1 ( x 1 - L 1 , x 3 - F 1 ; w 01 ) × Ψ 2 ( x 2 - L 2 , x 3 - F 2 ; w 02 ) .
x r j = ( x j - δ x j ) cos δ ϑ j - ( x 3 - δ x 3 j ) sin δ ϑ j ,
x r 3 j = ( x j - δ x j ) sin δ ϑ j + ( x 3 - δ x 3 j ) cos δ ϑ j .
α r j = α j cos δ ϑ j - α 3 sin δ ϑ j ,
α r 3 = α j sin δ ϑ j + α 3 cos δ ϑ j .
E r ( x r 1 , x r 2 , x 3 ) = ( a R 0 s ) exp ( i k x 3 ) Ψ 1 ( x r 1 , x r 31 ; ν 1 w 01 ) × Ψ 2 ( x r 2 , x r 32 ; ν 2 w 02 ) ,
α j ( x j - L j ) + α 3 ( x 3 - F j - i z D j ) = α r j x r j + α r 3 ( x r 3 - i ν j 2 z D j ) .
δ x j = Re ( L j ) ,
δ x 3 j = Re ( F j ) ,
ν j 2 sin δ ϑ j = Im ( L j ) / z D j ,
ν j 2 cos δ ϑ j = 1 + Im ( F j ) / z D j .
δ ϑ j Im ( L j ) / z D j ,
δ ν j 2 Im ( F j ) / z D j .
r = R p / R s = ρ exp ( i μ ) ,             ρ = ρ p / ρ s ,             μ = μ p - μ s ,
R p = ρ p exp ( i μ p ) ,             R s = ρ s exp ( i μ s ) .
L 1 = k - 1 ( - σ s 1 + i ρ s 1 / ρ s ) ,
L 1 = k - 1 ( - σ p 1 + i ρ p 1 / ρ p ) ,
F 1 = k - 1 { - σ s 11 + i [ ρ s 11 / ρ s - ( ρ s 11 / ρ s ) 2 ] } ,
F 1 = k - 1 { - σ p 11 + i [ ρ p 11 / ρ p - ( ρ p 11 / ρ p ) 2 ] } ;
L 2 = - ( i / k ) ( 1 + r ) χ ¯ ( sin ϑ 01 ) - 1 ,
L 2 = ( i / k ) ( 1 + r - 1 ) χ ¯ - 1 ( sin ϑ 01 ) - 1 ,
F 2 = ( i / k ) [ 2 r + ( 1 + r ) 2 χ ¯ 2 ] ( sin ϑ 01 ) - 2 ,
F 2 = ( i / k ) [ 2 r - 1 + ( 1 + r - 1 ) 2 χ ¯ - 2 ] ( sin ϑ 01 ) - 2 .
δ ¯ x j ~ z D j - 1 ,             δ ¯ ϑ j ~ z D j - 2 ,
δ ¯ x 3 j ~ z D j - 2 ,             δ ¯ ν j 2 ~ z D j - 3 ,
a = [ a 0 0 a ] = a [ a 0 0 1 ] ,
a = a 1 / a 2 = τ exp ( i υ ) ,             a j = τ j exp ( i υ j ) ,
τ = τ 1 / τ 2 ,             τ j = ν j - 1 / 2 cos δ ϑ j ,
υ = υ 1 / υ 2 ,             υ j = - i k z D j ν j 2 ( 1 - cos δ ϑ j ) .
E 0 ( x 1 , x 2 , x 3 ) = A 0 s 0 exp ( i k x 3 ) Ψ 1 ( x 1 , x 3 ; w 01 ) × Ψ 2 ( x 2 , x 3 ; w 02 ) ,
E r ( x r 1 , x r 2 , x 3 ) = A r s r exp ( i k x 3 ) Ψ 1 ( x r 1 , x r 31 ; ν 1 w 01 ) × Ψ 2 ( x r 2 , x r 32 ; ν 2 w 02 ) ,
A r = A A 0 ,
s r = S s 0 ,
a = A S .
A 0 = R ( ρ 2 η 2 + 1 ) 1 / 2 ( η 2 + 1 ) - 1 / 2 ,
A r = a ( ρ 2 η 2 τ 2 + 1 ) 1 / 2 ( η 2 + 1 ) - 1 / 2 ,
s 0 = ( ρ 2 η 2 + 1 ) - 1 / 2 ( r χ 1 ) ,
s r = ( ρ 2 η 2 τ 2 + 1 ) - 1 / 2 ( r a χ 1 ) ,
A = a ( ρ 2 η 2 τ 2 + 1 ) 1 / 2 ( ρ 2 η 2 + 1 ) - 1 / 2 ,
S = ( ρ 2 η 2 + 1 ) 1 / 2 ( ρ 2 η 2 τ 2 + 1 ) - 1 / 2 [ a 0 0 1 ] .
δ 3 = k ( x 3 - x r 3 ) k δ x 3 ,
δ 3 = k ( x 3 - x r 3 ) k δ x 3 ,
A ˜ = A exp ( i δ 3 ) ,
S ˜ = S [ exp ( i δ 3 - i δ 3 ) 0 0 1 ] .

Metrics