Abstract

Solutions to Maxwell’s wave equation have been derived for the propagation of the fundamental (Gaussian) mode of a laser beam in a fluid electrolyte which is in contact with an active electrode. An electrochemical or photoelectrochemical reaction at the electrolyte–electrode interface is assumed to generate a concentration gradient of the product in the electrolyte, which results in an inhomogeneous refractive-index profile. The analytic solutions for the propagation of the beam explicitly demonstrate the dependence of the displacement of the intensity centroid and of the spot shape on the electrochemical parameters of the system.

© 1984 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. J. Bard, L. R. Faulkner, Electrochemical Methods (Wiley, New York, 1980).
  2. J. O’M. Bockris, S. U. M. Kahn, Quantum Electrochemistry (Plenum, New York, 1979).
    [CrossRef]
  3. S. R. Morrison, Electrochemistry at Semiconductor and Oxidized Metal Electrodes (Plenum, New York, 1980).
    [CrossRef]
  4. B. S. H. Royce, S. Sanchez-Sinencio, R. Goldstein, R. Muratore, R. Williams, W. M. Yim, “Studies of Photocorrosion at the ZnSe-Electrolyte Interface by Photothermal Deflection Spectroscopy,” J. Electrochem. Soc. 129, 2393 (1982).
    [CrossRef]
  5. J. G. Mendoza-Alvarez, B. S. H. Royce, F. Sanchez-Sinencio, O. Zelaya-Angel, C. Menzes, R. Triboulet, “Optical. Properties of CdTe Thin Films Studied by Photothermal Deflection Spectroscopy,” Thin Solid Films 102, 259 (1983).
    [CrossRef]
  6. J. P. Roger, D. Fournier, A. C. Boccara, at Third International Conference on Photoacoustic and Photothermal Spectroscopy, Paris, Apr. 1983.
  7. A. C. Boccara, D. Fournier, J. Badoz, “Thermo-Optical Spectroscopy: Detection by the Mirage Effect,” Appl. Phys. Lett. 36, 130 (1980).
    [CrossRef]
  8. J. C. Murphy, L. C. Aamodt, “Photothermal Spectroscopy Using Optical Beam Probing: Mirage Effect,” J. Appl. Phys. 51, 4580 (1980); J. Appl. Phys. 52, 4903 (1981).
    [CrossRef]
  9. A. Mandelis, “Absolute Optical Absorption Coefficient Measurements Using Transverse Photothermal Deflection Spectroscopy,” J. Appl. Phys. 54, 3404 (1983).
    [CrossRef]
  10. J. CrankThe Mathematics of Diffusion (Oxford U.P., London, 1975), Chap. 2.
  11. W. A. Roth, K. Scheel, Eds., Landolt-Boernstein, Physikatisch-Chemische Tabellen, Vol. 2 (Springer, Berlin, 1923), Tables 174, 179, 185, 186.
  12. R. H. Muller, Advances in Electrochemistry and Electrochemical Engineering, Vol. 9 (Wiley, New York, 1973), pp. 281–368.
  13. E. A. J. Marcatili, “Modes in a Sequence of Thick Astigmatic Lens-Like Focusers,” Bell Syst. Tech. J. 44, 2887 (1964).
  14. H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids (Oxford U.P., London, 1959), Appendix II.
  15. A. Yariv, Quantum Electronics (Wiley, New York, 1975), Chap. 6.5.
  16. L. W. Casperson, “Gaussian Light Beams in Inhomogeneous Media,” Appl. Opt. 12, 2434 (1973).
    [CrossRef] [PubMed]
  17. D. L. Kreider, R. G. Kuller, D. R. Ostberg, Elementary Differential Equations (Addison-Wesley, Reading, Mass., 1968), p. 64.
  18. L. W. Casperson, A. Yariv, “The Gaussian Mode in Optical Resonators with a Radial Gain Profile,” Appl. Phys. Lett. 12, 355 (1968).
    [CrossRef]
  19. J. T. Verdeyen, Laser Electronics (Prentice-Hall, Englewood Cliffs, N.J., 1981), Chap. 1.
  20. P. Rossi, C. W. McCurdy, R. L. McCreery, “Diffractive Spectroelectrochemistry. Use of Diffracted Light for Monitoring Electrogenerated Chromophores,” J. Am. Chem. Soc. 103, 2524 (1981).
    [CrossRef]
  21. R. Pruiksma, R. L. McCreery, “Observation of Electrochemical Concentration Profiles by Absorption Spectroelectro-chemistry,” Anal. Chem. 51, 2253 (1979).
    [CrossRef]
  22. J. T. Knudtson, K. L. Ratzlaff, “Laser Beam Spatial Profile Analysis Using a Two-Dimensional Photodiode Array,” Rev. Sci. Instrum. 54, 856 (1983).
    [CrossRef]

1983 (3)

J. G. Mendoza-Alvarez, B. S. H. Royce, F. Sanchez-Sinencio, O. Zelaya-Angel, C. Menzes, R. Triboulet, “Optical. Properties of CdTe Thin Films Studied by Photothermal Deflection Spectroscopy,” Thin Solid Films 102, 259 (1983).
[CrossRef]

A. Mandelis, “Absolute Optical Absorption Coefficient Measurements Using Transverse Photothermal Deflection Spectroscopy,” J. Appl. Phys. 54, 3404 (1983).
[CrossRef]

J. T. Knudtson, K. L. Ratzlaff, “Laser Beam Spatial Profile Analysis Using a Two-Dimensional Photodiode Array,” Rev. Sci. Instrum. 54, 856 (1983).
[CrossRef]

1982 (1)

B. S. H. Royce, S. Sanchez-Sinencio, R. Goldstein, R. Muratore, R. Williams, W. M. Yim, “Studies of Photocorrosion at the ZnSe-Electrolyte Interface by Photothermal Deflection Spectroscopy,” J. Electrochem. Soc. 129, 2393 (1982).
[CrossRef]

1981 (1)

P. Rossi, C. W. McCurdy, R. L. McCreery, “Diffractive Spectroelectrochemistry. Use of Diffracted Light for Monitoring Electrogenerated Chromophores,” J. Am. Chem. Soc. 103, 2524 (1981).
[CrossRef]

1980 (2)

A. C. Boccara, D. Fournier, J. Badoz, “Thermo-Optical Spectroscopy: Detection by the Mirage Effect,” Appl. Phys. Lett. 36, 130 (1980).
[CrossRef]

J. C. Murphy, L. C. Aamodt, “Photothermal Spectroscopy Using Optical Beam Probing: Mirage Effect,” J. Appl. Phys. 51, 4580 (1980); J. Appl. Phys. 52, 4903 (1981).
[CrossRef]

1979 (1)

R. Pruiksma, R. L. McCreery, “Observation of Electrochemical Concentration Profiles by Absorption Spectroelectro-chemistry,” Anal. Chem. 51, 2253 (1979).
[CrossRef]

1973 (1)

1968 (1)

L. W. Casperson, A. Yariv, “The Gaussian Mode in Optical Resonators with a Radial Gain Profile,” Appl. Phys. Lett. 12, 355 (1968).
[CrossRef]

1964 (1)

E. A. J. Marcatili, “Modes in a Sequence of Thick Astigmatic Lens-Like Focusers,” Bell Syst. Tech. J. 44, 2887 (1964).

Aamodt, L. C.

J. C. Murphy, L. C. Aamodt, “Photothermal Spectroscopy Using Optical Beam Probing: Mirage Effect,” J. Appl. Phys. 51, 4580 (1980); J. Appl. Phys. 52, 4903 (1981).
[CrossRef]

Badoz, J.

A. C. Boccara, D. Fournier, J. Badoz, “Thermo-Optical Spectroscopy: Detection by the Mirage Effect,” Appl. Phys. Lett. 36, 130 (1980).
[CrossRef]

Bard, A. J.

A. J. Bard, L. R. Faulkner, Electrochemical Methods (Wiley, New York, 1980).

Boccara, A. C.

A. C. Boccara, D. Fournier, J. Badoz, “Thermo-Optical Spectroscopy: Detection by the Mirage Effect,” Appl. Phys. Lett. 36, 130 (1980).
[CrossRef]

J. P. Roger, D. Fournier, A. C. Boccara, at Third International Conference on Photoacoustic and Photothermal Spectroscopy, Paris, Apr. 1983.

Bockris, J. O’M.

J. O’M. Bockris, S. U. M. Kahn, Quantum Electrochemistry (Plenum, New York, 1979).
[CrossRef]

Carslaw, H. S.

H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids (Oxford U.P., London, 1959), Appendix II.

Casperson, L. W.

L. W. Casperson, “Gaussian Light Beams in Inhomogeneous Media,” Appl. Opt. 12, 2434 (1973).
[CrossRef] [PubMed]

L. W. Casperson, A. Yariv, “The Gaussian Mode in Optical Resonators with a Radial Gain Profile,” Appl. Phys. Lett. 12, 355 (1968).
[CrossRef]

Crank, J.

J. CrankThe Mathematics of Diffusion (Oxford U.P., London, 1975), Chap. 2.

Faulkner, L. R.

A. J. Bard, L. R. Faulkner, Electrochemical Methods (Wiley, New York, 1980).

Fournier, D.

A. C. Boccara, D. Fournier, J. Badoz, “Thermo-Optical Spectroscopy: Detection by the Mirage Effect,” Appl. Phys. Lett. 36, 130 (1980).
[CrossRef]

J. P. Roger, D. Fournier, A. C. Boccara, at Third International Conference on Photoacoustic and Photothermal Spectroscopy, Paris, Apr. 1983.

Goldstein, R.

B. S. H. Royce, S. Sanchez-Sinencio, R. Goldstein, R. Muratore, R. Williams, W. M. Yim, “Studies of Photocorrosion at the ZnSe-Electrolyte Interface by Photothermal Deflection Spectroscopy,” J. Electrochem. Soc. 129, 2393 (1982).
[CrossRef]

Jaeger, J. C.

H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids (Oxford U.P., London, 1959), Appendix II.

Kahn, S. U. M.

J. O’M. Bockris, S. U. M. Kahn, Quantum Electrochemistry (Plenum, New York, 1979).
[CrossRef]

Knudtson, J. T.

J. T. Knudtson, K. L. Ratzlaff, “Laser Beam Spatial Profile Analysis Using a Two-Dimensional Photodiode Array,” Rev. Sci. Instrum. 54, 856 (1983).
[CrossRef]

Kreider, D. L.

D. L. Kreider, R. G. Kuller, D. R. Ostberg, Elementary Differential Equations (Addison-Wesley, Reading, Mass., 1968), p. 64.

Kuller, R. G.

D. L. Kreider, R. G. Kuller, D. R. Ostberg, Elementary Differential Equations (Addison-Wesley, Reading, Mass., 1968), p. 64.

Mandelis, A.

A. Mandelis, “Absolute Optical Absorption Coefficient Measurements Using Transverse Photothermal Deflection Spectroscopy,” J. Appl. Phys. 54, 3404 (1983).
[CrossRef]

Marcatili, E. A. J.

E. A. J. Marcatili, “Modes in a Sequence of Thick Astigmatic Lens-Like Focusers,” Bell Syst. Tech. J. 44, 2887 (1964).

McCreery, R. L.

P. Rossi, C. W. McCurdy, R. L. McCreery, “Diffractive Spectroelectrochemistry. Use of Diffracted Light for Monitoring Electrogenerated Chromophores,” J. Am. Chem. Soc. 103, 2524 (1981).
[CrossRef]

R. Pruiksma, R. L. McCreery, “Observation of Electrochemical Concentration Profiles by Absorption Spectroelectro-chemistry,” Anal. Chem. 51, 2253 (1979).
[CrossRef]

McCurdy, C. W.

P. Rossi, C. W. McCurdy, R. L. McCreery, “Diffractive Spectroelectrochemistry. Use of Diffracted Light for Monitoring Electrogenerated Chromophores,” J. Am. Chem. Soc. 103, 2524 (1981).
[CrossRef]

Mendoza-Alvarez, J. G.

J. G. Mendoza-Alvarez, B. S. H. Royce, F. Sanchez-Sinencio, O. Zelaya-Angel, C. Menzes, R. Triboulet, “Optical. Properties of CdTe Thin Films Studied by Photothermal Deflection Spectroscopy,” Thin Solid Films 102, 259 (1983).
[CrossRef]

Menzes, C.

J. G. Mendoza-Alvarez, B. S. H. Royce, F. Sanchez-Sinencio, O. Zelaya-Angel, C. Menzes, R. Triboulet, “Optical. Properties of CdTe Thin Films Studied by Photothermal Deflection Spectroscopy,” Thin Solid Films 102, 259 (1983).
[CrossRef]

Morrison, S. R.

S. R. Morrison, Electrochemistry at Semiconductor and Oxidized Metal Electrodes (Plenum, New York, 1980).
[CrossRef]

Muller, R. H.

R. H. Muller, Advances in Electrochemistry and Electrochemical Engineering, Vol. 9 (Wiley, New York, 1973), pp. 281–368.

Muratore, R.

B. S. H. Royce, S. Sanchez-Sinencio, R. Goldstein, R. Muratore, R. Williams, W. M. Yim, “Studies of Photocorrosion at the ZnSe-Electrolyte Interface by Photothermal Deflection Spectroscopy,” J. Electrochem. Soc. 129, 2393 (1982).
[CrossRef]

Murphy, J. C.

J. C. Murphy, L. C. Aamodt, “Photothermal Spectroscopy Using Optical Beam Probing: Mirage Effect,” J. Appl. Phys. 51, 4580 (1980); J. Appl. Phys. 52, 4903 (1981).
[CrossRef]

Ostberg, D. R.

D. L. Kreider, R. G. Kuller, D. R. Ostberg, Elementary Differential Equations (Addison-Wesley, Reading, Mass., 1968), p. 64.

Pruiksma, R.

R. Pruiksma, R. L. McCreery, “Observation of Electrochemical Concentration Profiles by Absorption Spectroelectro-chemistry,” Anal. Chem. 51, 2253 (1979).
[CrossRef]

Ratzlaff, K. L.

J. T. Knudtson, K. L. Ratzlaff, “Laser Beam Spatial Profile Analysis Using a Two-Dimensional Photodiode Array,” Rev. Sci. Instrum. 54, 856 (1983).
[CrossRef]

Roger, J. P.

J. P. Roger, D. Fournier, A. C. Boccara, at Third International Conference on Photoacoustic and Photothermal Spectroscopy, Paris, Apr. 1983.

Rossi, P.

P. Rossi, C. W. McCurdy, R. L. McCreery, “Diffractive Spectroelectrochemistry. Use of Diffracted Light for Monitoring Electrogenerated Chromophores,” J. Am. Chem. Soc. 103, 2524 (1981).
[CrossRef]

Royce, B. S. H.

J. G. Mendoza-Alvarez, B. S. H. Royce, F. Sanchez-Sinencio, O. Zelaya-Angel, C. Menzes, R. Triboulet, “Optical. Properties of CdTe Thin Films Studied by Photothermal Deflection Spectroscopy,” Thin Solid Films 102, 259 (1983).
[CrossRef]

B. S. H. Royce, S. Sanchez-Sinencio, R. Goldstein, R. Muratore, R. Williams, W. M. Yim, “Studies of Photocorrosion at the ZnSe-Electrolyte Interface by Photothermal Deflection Spectroscopy,” J. Electrochem. Soc. 129, 2393 (1982).
[CrossRef]

Sanchez-Sinencio, F.

J. G. Mendoza-Alvarez, B. S. H. Royce, F. Sanchez-Sinencio, O. Zelaya-Angel, C. Menzes, R. Triboulet, “Optical. Properties of CdTe Thin Films Studied by Photothermal Deflection Spectroscopy,” Thin Solid Films 102, 259 (1983).
[CrossRef]

Sanchez-Sinencio, S.

B. S. H. Royce, S. Sanchez-Sinencio, R. Goldstein, R. Muratore, R. Williams, W. M. Yim, “Studies of Photocorrosion at the ZnSe-Electrolyte Interface by Photothermal Deflection Spectroscopy,” J. Electrochem. Soc. 129, 2393 (1982).
[CrossRef]

Triboulet, R.

J. G. Mendoza-Alvarez, B. S. H. Royce, F. Sanchez-Sinencio, O. Zelaya-Angel, C. Menzes, R. Triboulet, “Optical. Properties of CdTe Thin Films Studied by Photothermal Deflection Spectroscopy,” Thin Solid Films 102, 259 (1983).
[CrossRef]

Verdeyen, J. T.

J. T. Verdeyen, Laser Electronics (Prentice-Hall, Englewood Cliffs, N.J., 1981), Chap. 1.

Williams, R.

B. S. H. Royce, S. Sanchez-Sinencio, R. Goldstein, R. Muratore, R. Williams, W. M. Yim, “Studies of Photocorrosion at the ZnSe-Electrolyte Interface by Photothermal Deflection Spectroscopy,” J. Electrochem. Soc. 129, 2393 (1982).
[CrossRef]

Yariv, A.

L. W. Casperson, A. Yariv, “The Gaussian Mode in Optical Resonators with a Radial Gain Profile,” Appl. Phys. Lett. 12, 355 (1968).
[CrossRef]

A. Yariv, Quantum Electronics (Wiley, New York, 1975), Chap. 6.5.

Yim, W. M.

B. S. H. Royce, S. Sanchez-Sinencio, R. Goldstein, R. Muratore, R. Williams, W. M. Yim, “Studies of Photocorrosion at the ZnSe-Electrolyte Interface by Photothermal Deflection Spectroscopy,” J. Electrochem. Soc. 129, 2393 (1982).
[CrossRef]

Zelaya-Angel, O.

J. G. Mendoza-Alvarez, B. S. H. Royce, F. Sanchez-Sinencio, O. Zelaya-Angel, C. Menzes, R. Triboulet, “Optical. Properties of CdTe Thin Films Studied by Photothermal Deflection Spectroscopy,” Thin Solid Films 102, 259 (1983).
[CrossRef]

Anal. Chem. (1)

R. Pruiksma, R. L. McCreery, “Observation of Electrochemical Concentration Profiles by Absorption Spectroelectro-chemistry,” Anal. Chem. 51, 2253 (1979).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (2)

L. W. Casperson, A. Yariv, “The Gaussian Mode in Optical Resonators with a Radial Gain Profile,” Appl. Phys. Lett. 12, 355 (1968).
[CrossRef]

A. C. Boccara, D. Fournier, J. Badoz, “Thermo-Optical Spectroscopy: Detection by the Mirage Effect,” Appl. Phys. Lett. 36, 130 (1980).
[CrossRef]

Bell Syst. Tech. J. (1)

E. A. J. Marcatili, “Modes in a Sequence of Thick Astigmatic Lens-Like Focusers,” Bell Syst. Tech. J. 44, 2887 (1964).

J. Am. Chem. Soc. (1)

P. Rossi, C. W. McCurdy, R. L. McCreery, “Diffractive Spectroelectrochemistry. Use of Diffracted Light for Monitoring Electrogenerated Chromophores,” J. Am. Chem. Soc. 103, 2524 (1981).
[CrossRef]

J. Appl. Phys. (2)

J. C. Murphy, L. C. Aamodt, “Photothermal Spectroscopy Using Optical Beam Probing: Mirage Effect,” J. Appl. Phys. 51, 4580 (1980); J. Appl. Phys. 52, 4903 (1981).
[CrossRef]

A. Mandelis, “Absolute Optical Absorption Coefficient Measurements Using Transverse Photothermal Deflection Spectroscopy,” J. Appl. Phys. 54, 3404 (1983).
[CrossRef]

J. Electrochem. Soc. (1)

B. S. H. Royce, S. Sanchez-Sinencio, R. Goldstein, R. Muratore, R. Williams, W. M. Yim, “Studies of Photocorrosion at the ZnSe-Electrolyte Interface by Photothermal Deflection Spectroscopy,” J. Electrochem. Soc. 129, 2393 (1982).
[CrossRef]

Rev. Sci. Instrum. (1)

J. T. Knudtson, K. L. Ratzlaff, “Laser Beam Spatial Profile Analysis Using a Two-Dimensional Photodiode Array,” Rev. Sci. Instrum. 54, 856 (1983).
[CrossRef]

Thin Solid Films (1)

J. G. Mendoza-Alvarez, B. S. H. Royce, F. Sanchez-Sinencio, O. Zelaya-Angel, C. Menzes, R. Triboulet, “Optical. Properties of CdTe Thin Films Studied by Photothermal Deflection Spectroscopy,” Thin Solid Films 102, 259 (1983).
[CrossRef]

Other (11)

J. P. Roger, D. Fournier, A. C. Boccara, at Third International Conference on Photoacoustic and Photothermal Spectroscopy, Paris, Apr. 1983.

A. J. Bard, L. R. Faulkner, Electrochemical Methods (Wiley, New York, 1980).

J. O’M. Bockris, S. U. M. Kahn, Quantum Electrochemistry (Plenum, New York, 1979).
[CrossRef]

S. R. Morrison, Electrochemistry at Semiconductor and Oxidized Metal Electrodes (Plenum, New York, 1980).
[CrossRef]

J. CrankThe Mathematics of Diffusion (Oxford U.P., London, 1975), Chap. 2.

W. A. Roth, K. Scheel, Eds., Landolt-Boernstein, Physikatisch-Chemische Tabellen, Vol. 2 (Springer, Berlin, 1923), Tables 174, 179, 185, 186.

R. H. Muller, Advances in Electrochemistry and Electrochemical Engineering, Vol. 9 (Wiley, New York, 1973), pp. 281–368.

J. T. Verdeyen, Laser Electronics (Prentice-Hall, Englewood Cliffs, N.J., 1981), Chap. 1.

H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids (Oxford U.P., London, 1959), Appendix II.

A. Yariv, Quantum Electronics (Wiley, New York, 1975), Chap. 6.5.

D. L. Kreider, R. G. Kuller, D. R. Ostberg, Elementary Differential Equations (Addison-Wesley, Reading, Mass., 1968), p. 64.

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 (7)

Fig. 1
Fig. 1

Schematic diagram of laser beam propagation in electrochemical medium near the electrode–electrolyte interface in the presence of a refractive-index gradient ∇cn due to an electrochemical species concentration gradient: C0, species concentration at electrode surface; D, species diffusion coefficient in the electrolyte; n0, refractive index of the electrolyte in the absence of a concentration gradient.

Fig. 2
Fig. 2

Beam intensity centroid displacement according to perturbation theoretical expression, Eq. (77). A = 5.9 × 10−4; D = 1.49 × 10−5 cm2/sec; z = 3.0 cm. t0 corresponds to the intensity profile of Fig. 6.

Fig. 3
Fig. 3

Beam deflection according to perturbation theoretical expression, Eq. (78). A = 5.9 × 10−4; D = 1.49 × 10−5 cm2/sec; z = 3.0 cm.

Fig. 4
Fig. 4

Fundamental mode He–Ne laser beam spot shapes, Eq. (79), and beam centroid displacement for two positions of the intensity profile detector. x0 = y0 = 0.0176 cm; D = 1.49 × 10−5 cm2/sec; A = 5.9 × 10−3; t = 5 sec. The dashed line circle in the lower spot shape has been drawn to emphasize its elliptic character.

Fig. 5
Fig. 5

Fundamental mode He–Ne laser beam spot shapes, Eq. (79), for three positions of the intensity profile detector. All spot shapes were drawn concentrically to emphasize the spatial development of the ellipse. x0 = y0 = 0.0176 cm; D = 1.49 × 10−5 cm2/sec; A = 5.9 × 10−3; t = 5 sec.

Fig. 6
Fig. 6

Fundamental mode He–Ne laser beam spot shapes, Eq. (79), and beam centroid displacements for three positions of the intensity profile detector. x0 = y0 0.0176 cm; D = 1.49 × 10−5 cm2/sec; A = 5.9 × 10−4; t = 30 sec (see Fig. 2).

Fig. 7
Fig. 7

Qualitative illustration of the intensity spatial profile and displacement of a fundamental Gaussian mode propagation in an optically inhomogeneous electrolyte fluid.

Equations (123)

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

C ( x , t ) = C 0 erfc ( x 2 D t )
n ( x , t ) n 0 + C 0 ( n C ) C = C 0 erfc ( x 2 D t ) ,
C 0 ( n C ) C = C 0 1
2 e x ( x , y , z ; t ) - μ 0 [ ɛ ( x , t ) 2 t 2 e x ( x , y , z ; t ) + t ɛ ( x , t ) t e x ( x , y , z ; t ) ] = 0 ,
n 2 ( x , t ) = μ 0 ɛ ( x , t ) .
e x ( x , y , z ; t ) = E x ( x , y , z ; t ) exp ( i ω t ) ,
2 E x ( x , y , z ; t ) + ω 2 μ 0 [ ɛ ( x , t ) - i ɛ ( x , t ) ω t ] E x ( x , y , z ; t ) = 0.
( ɛ / t ) t ω - 1 ɛ ( t ) 1.
2 E x ( x , y , z ; t ) + k 0 2 n 2 ( x , t ) E x ( x , y , z ; t ) = 0 ,
n ( x , t ) t > τ d n 0 [ 1 - A π ( x D t ) + 0 ( x D t ) 3 ] ,
τ d x 2 / ( 4 D ) ,
A C 0 ( n / C ) C = C 0 n 0 + C 0 ( n / C ) C = C 0 ( C 0 n 0 ) ( n C ) C = C 0 1.
2 E x + k 0 2 n 0 2 [ 1 - ( 2 A π D t ) x + ( A 2 π D t ) x 2 ] E x = 0 ,
E x ( x , y , z ; t ) = G ( x , y , z ; t ) exp ( - i k 0 n 0 z )
2 z 2 G ( x , y , z ; t ) 0.
G ( x , y , z ; t ) = exp { - i [ ½ Q x ( z , t ) x 2 + ½ Q y ( z , t ) y 2 + S x ( z , t ) x + S y ( z , t ) y + P ( z , t ) ] } ,
Q y 2 ( z , t ) + k 0 n 0 z Q y ( z , t ) = 0 ;
Q x 2 ( z , t ) + k 0 n 0 z Q x ( z , t ) + k 0 n 0 k 2 x ( t ) = 0 ;
Q y ( z , t ) S y ( z , t ) + k 0 n 0 z S y ( z , t ) = 0 ;
Q x ( z , t ) S x ( z , t ) + k 0 n 0 z S x ( z , t ) + ½ k 0 n 0 k 1 x ( t ) = 0 ;
z P ( z , t ) + i 2 k 0 n 0 [ Q x ( z , t ) + Q y ( z , t ) ] + 1 2 k 0 n 0 [ S x 2 ( z , t ) + S y 2 ( z , t ) ] = 0.
k 1 x ( t ) 2 k 0 A π D t ,
k 2 x ( t ) - k 0 A 2 π D t .
k k 0 n 0 ;
m ( t ) A 2 π D t 1.
Q y ( z ) = k z + q 0 y ,
Q x ( z , t ) = k m ( sinh m z + ζ cosh m z cosh m z + ζ sinh m z ) ,
q 0 y k Q y ( z = 0 ) ,
ζ ( t ) Q x ( z = 0 ) k m ( t ) = 1 q 0 x m ( t ) .
S y ( z ) = S y ( 0 ) ( q 0 y z + q 0 y ) .
S x ( z , t ) = S x ( z , t ) - 2 m Q x ( z , t )
S x ( z , t ) = S x ( 0 ) + k [ ζ ( 1 - cosh m z ) - sinh m z ] cosh m z + ζ sinh m z .
P ( z , t ) = P ( z , t ) - 1 m S x ( z , t ) - 1 2 m Q x ( z , t ) - k z 2
z P ( z , t ) + i 2 k [ Q x ( z , t ) + Q y ( z ) ] + 1 2 k [ ( S x ) 2 + S y 2 ] ( z , t ) = 0.
P ( z , t ) = P ( 0 ) + 1 m S x ( 0 ) + k 2 m q 0 x - i 2 [ ln ( cosh m z + ζ sinh m z ) + ln ( 1 + z q 0 y ) ] - 1 2 k 0 S y 2 ( 0 ) ( z q 0 y z + q 0 y ) - [ S x ( 0 ) + k ζ ] 2 2 k m ( sinh m z cosh m z + ζ sinh m z ) - 1 m { S x ( 0 ) + k [ ζ ( 1 - cosh m z ) - sinh m z ] cosh m z + ζ sinh m z } - ½ k z - k 2 m ( sinh m z + ζ cosh m z cosh m z + ζ sinh m z ) .
Q x ( z ) = Q x r ( z , t ) + i Q x i ( z , t ) ;
Q y ( z ) = Q y r ( z ) + i Q y i ( z ) ;
S x ( z , t ) = S x r ( z , t ) + i S x i ( z , t ) ;
S y ( z ) = S y r ( z ) + i S y i ( z ) ;
P ( z , t ) = P r ( z , t ) + i P i ( z , t ) .
1 q x ( z , t ) Q x ( z , t ) k 1 R x ( z , t ) - i λ π W x 2 ( z , t ) ,
1 q y ( z ) Q y ( z ) k 1 R y ( z ) - i λ π W y 2 ( z ) ,
R x ( z , t ) = m z 0 x 2 1 + m z 0 x 2 [ coth m z + ( 1 m z 0 x 2 ) tanh m z ] ,
R y ( z ) = z [ 1 + ( z 0 y z ) 2 ] .
W x 2 ( z , t ) = W 0 x 2 [ cosh 2 m z + ( 1 m z 0 x 2 ) sinh 2 m z ] ,
W y 2 ( z ) = W 0 y 2 [ 1 + ( z z 0 y ) 2 ] .
q 0 x i z 0 x = i π W 0 x 2 λ ,
q 0 y i z 0 y = i π W 0 y 2 λ .
G ( x , y , z ; t ) = exp ( - i { ½ Q x r ( z , t ) [ x - x p ( z , t ) ] 2 + ½ Q y r ( z ) ( y - y p ) 2 - ½ Q x r ( z , t ) x p 2 ( z , t ) - ½ Q y r ( z ) y p 2 + P r ( z , t ) } ) exp { ½ Q x i ( z , t ) [ x - x a ( z , t ) ] 2 + ½ Q y i ( z ) ( y - y a ) 2 - ½ Q x i ( z , t ) x a 2 ( z , t ) - ½ Q y i ( z ) x y 2 + P i ( z , t ) } ,
x p ( z , t ) - S x r ( z , t ) Q x r ( z , t ) ,
x a ( z , t ) - S x i ( z , t ) Q x i ( z , t ) ,
Q x r ( z , t ) = k R x ( z , t ) ;
Q x i ( z , t ) = - 2 W x 2 ( z , t ) ;
Q y r ( z ) = k R y ( z ) ;
Q y i ( z ) = - 2 W y 2 ( z ) ;
S x r ( z , t ) = k { [ s 0 x + g ( 1 - cosh m z ) ] g - cosh m z m ( 1 + g 2 ) R x ( z , t ) cosh m z } ,
S x ( 0 ) - i k s 0 x ,
g i ζ = 1 / m z 0 x .
S x i ( z , t ) = - k [ W 0 x W x ( z , t ) ] 2 [ ( s 0 x + g ) cosh m z - g ] ,
S y r ( z ) = k s 0 y z 0 y R y ( z ) ,
S y ( 0 ) - i k s 0 y .
S y i ( z ) = - s 0 y z 0 y W y 2 ( z ) .
P r ( z , t ) = P r ( 0 ) - ½ [ η x ( z , t ) + η y ( z ) ] + k ( s 0 y z 0 y ) 2 2 R y ( z ) - k z 2 + k [ s 0 x 2 + 2 g 2 + 1 + 2 g ( s 0 x + g ) ( cosh m z - 1 ) 2 m ( 1 + g 2 ) R x ( z , t ) cosh m z ] ,
P i ( z , t ) = P i ( 0 ) - ¼ { ln ( cosh 2 m z + g 2 sinh 2 m z ) + ln [ 1 + ( z z 0 y ) 2 ] } + ( s 0 y z ) 2 W y 2 ( z ) + ( z 0 x / m ) W x 2 ( z , t ) [ ( s 0 x 2 + 1 ) g sinh 2 m z + 2 ( s 0 x + g ) × ( 1 - cosh m z ) cosh m z ] ,
η x ( z , t ) tan - 1 ( g tanh m z ) ,
η y ( z ) tan - 1 ( z / z 0 y ) .
cosh x 1 + x 2 2 ;             x 1 ,
sinh x x ;             x 1 ,
R x ( z , t ) z [ ( 1 + m z 0 x 2 ) + ( z 0 x / z ) 2 1 + m ( z 0 x 2 + ½ z 2 ) ] ;
W x 2 ( z , t ) W 0 x 2 [ ( 1 + m z 2 ) + ( z / z 0 x ) 2 ] ;
S x r ( z , t ) k R x ( z , t ) [ s 0 x z 0 x - m ( z 0 x 2 + z 2 2 ) 1 + m ( z 0 x 2 + z 2 2 ) ] ;
S x i ( z , t ) - k [ W 0 x W x ( z , t ) ] 2 [ s 0 x ( 1 + m z 2 2 ) + m 2 z 2 z 0 x ] ;
P r ( z , t ) P r ( 0 ) - ½ { tan - 1 [ z / z 0 x 1 + m ( z 2 / 2 ) ] + tan - 1 ( z / z 0 y ) } + k ( s 0 y z 0 y ) 2 2 R y ( z ) - k z 2 + k 2 R x ( z , t ) { [ z 0 x 2 ( 1 + s 0 x 2 ) + z 2 + m s 0 x z 0 x z 2 + ( m z 0 x 2 z / 2 ) ( 1 + s 0 x 2 ) ] / [ 1 + m ( z 0 x 2 + z 2 / 2 ) ] } ;
P i ( z , t ) = P i ( 0 ) - ¼ ln { [ ( 1 + m z 2 ) + ( z / z 0 x ) 2 ] [ 1 + ( z / z 0 y ) 2 ] } + ( s 0 y z ) 2 W y 2 ( z , t ) + z 2 ( s 0 x 2 - m s 0 x z 0 x - m z 2 / 2 ) W x 2 ( z , t ) .
x a ( z , t ) = m - 1 / 2 - ( s 0 x z 0 x + m - 1 / 2 ) cosh m z ,
x p ( z , t ) = m - 1 / 2 [ ( 1 + m z 0 x 2 ) cosh m z - 1 - m s 0 x z 0 x ( 1 + m z 0 x 2 ) cosh m z ] .
x a ( z , t ) - s 0 x z 0 x ( 1 + m z 2 / 2 ) - m z 2 / 2 ,
x p ( z , t ) - s 0 x z 0 z + m ( z 0 x 2 + z 2 2 ) 1 + m ( z 0 x 2 + z 2 2 ) .
E x ( x , y , z ; t ) = E 0 [ W 0 x W 0 y W x ( z , t ) W y ( z ) ] 1 / 2 exp [ y a 2 + s 0 y 2 z 2 W y 2 ( z ) + x a 2 + z 2 ( s 0 x 2 - m s 0 x z 0 x - m z 2 / 2 ) W x 2 ( z , t ) + P i ( 0 ) ] × exp [ - ( x - x a ) 2 W x 2 ( z , t ) - ( y - y a ) W y 2 ( z , t ) ] .
x a ( 0 , t ) = x 0 ,
y a ( 0 , t ) = y 0 ,
s 0 x = - x 0 / z 0 x             s 0 y = - y 0 / z 0 y ,
x a ( z , t ) = x 0 ( 1 + m z 2 2 ) - m 2 z 2 ,
y a ( z ) = y 0 ,
x p ( z , t ) = x 0 + m ( z 0 x 2 + z 2 / 2 ) 1 + m ( z 0 x 2 + z 2 / 2 ) ,
y p ( z ) = y 0 .
- ( x 0 2 W 0 x 2 + y 0 2 W 0 y 2 ) ,
E x ( x , y , z ; t ) = E 0 [ W 0 x W 0 y W x ( z , t ) W y ( z ) ] 1 / 2 × exp { - [ x - x a ( z , t ) ] 2 W x 2 ( z , t ) - ( y - y 0 ) 2 W y 2 ( z ) } ,
ϕ ( x , y , z ; t ) = ½ Q x r ( z , t ) ( x - x p ) 2 + ½ Q y r ( z ) ( y - y p ) 2 - ½ Q x r ( z , t ) x p 2 - ½ Q y r ( z ) y p 2 + P r ( z , t ) + k z .
ϕ ( x , y , z ; t ) = k 2 ( [ x - x p ( z , t ) ] 2 R x ( z , t ) + ( y - y 0 ) 2 R y ( z ) + z 2 R x ( z , t ) { ( 1 - m x 0 ) 2 [ 1 + ( z 0 x / z ) 2 ] 1 + 2 m ( z 0 x 2 + z 2 / 2 ) } ) + ½ ( k z - tan - 1 [ ( z / z 0 x ) 1 + m ( z 2 / 2 ) ] - tan - 1 ( z / z 0 y ) ) .
I ( x , y , z ; t ) = 1 2 η E x · E x * ( W / m 2 ) ,
I ( x , y , z ; t ) = I 0 [ W 0 x W 0 y W x ( z , t ) W y ( z ) ] × exp ( - 2 { x - x a ( z , t ) ] 2 W x 2 ( z , t ) + ( y - y 0 ) 2 W y 2 ( z ) } ) .
2 z 2 x a ( z , t ) + [ k 2 x ( t ) k ] x a ( z , t ) + ½ [ k 1 x ( t ) k ] = 0.
n ( x , t ) n 0 { 1 - ( n 1 x 2 n 0 ) x - ( n 2 x 2 n 0 ) x 2 } ,
n 1 x ( t ) = k 1 x ( t ) / k 0 ,
n 2 x ( t ) = k 2 x ( t ) / k 0 .
n 0 2 z 2 x a ( z , t ) = - ½ n 1 x ( t ) - n 2 x ( t ) x a ( z , t )
n 0 2 z 2 x a ( z , t ) = - x n ( x , t ) x = x a ( z , t ) ,
x a ( z , t ) = m - 1 / 2 + [ x a ( 0 , t ) - m - 1 / 2 ] cosh m z + m - 1 / 2 ( x a z | z = 0 ) sinh m z .
x a ( 0 , t ) = x 0 , for the initial displacement , x a z | z = 0 x 0 = 0 , for the initial beam deflection ,
x a ( z , t ) = x 0 ( 1 + m z 2 2 ) - m 2 z 2 ,
θ ( x , t ) = z x a ( z , t ) = - ( 1 - m x 0 ) sinh m z + x 0 cosh m z .
θ ( z , t ) - ( A π D t - A 2 x 0 π D t ) z .
d 2 x a ( z , t ) d z 2 + A π D t exp [ - x a 2 ( z , t ) 4 D t ] = 0.
x a ( z , t ) = k = 0 δ k X k ( z , t )
X 0 ( 0 , t ) = x 0 ,
X j ( 0 , t ) = 0 ;             j 0 ,
d d z X k ( z , t ) z = 0 = 0 ;             k = 0 , 1 , 2 , .
d 2 d z 2 X 0 ( z , t ) = 0 ;
d 2 d z 2 X 1 ( z , t ) + exp [ - X 0 2 ( z , t ) 4 D t ] = 0 ;
d 2 d z 2 X 2 ( z , t ) - X 0 ( z , t ) X 1 ( z , t ) 2 D t exp [ - X 0 2 ( z , t ) 4 D t ] = 0.
x a ( z , t ) = x 0 = - m z 2 2 ( 1 + m x 0 z 2 24 D t ) exp ( - x 0 2 4 D t )
θ ( z , t ) = - m z [ 1 + m x 0 z 2 12 D t ] exp ( - x 0 2 4 D t ) .
n ( x , ) = n 0 + C 0 ( n C ) C = C 0 n 0
τ d 0 = x 0 2 / 4 D ,
x n ( x , t ) = - ( A / π D t ) exp ( - x 2 4 D t ) ,
exp ( - x 0 2 4 D t ) 1
n x - ( A / π D t ) ,
[ x - x a ( z * , t ) ] 2 W x 2 ( z * , t ) + ( y - y 0 ) 2 W y 2 ( z * ) = 1
d W x 2 ( z , t 0 ) d ( z 2 ) = W 0 x 2 ( A 2 π D t 0 + 1 z 0 x 2 ) .
d W x 2 ( z * , t ) d ( 1 / t ) = ( z * ) 2 W 0 x 2 A 2 π D .
d x a ( z , t 0 ) d ( z 2 ) = - A 2 π D t ,
d x a ( z * , t ) d ( 1 / t ) = - A / 2 π D .

Metrics