Abstract

Vector wave solutions are obtained for the propagation of beams of light in media having slow spatial variations of the gain, loss, or index of refraction. The formalism developed here is applicable to a wide range of problems, and an example considered in detail is the propagation of off-axis beams in lenslike laser materials and optical waveguides. A procedure is also described for the diagnosis of localized dielectric inhomogeneities such as plasmas by means of Gaussian laser beams.

© 1973 Optical Society of America

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References

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  1. G. D. Boyd, J. P. Gordon, Bell Sys. Tech. J. 40, 489 (1961).
  2. G. Goubau, F. Schwering, IRE Trans. Antennas Propag. AP-9, 248 (1961).
    [CrossRef]
  3. B. W. McCaul, Appl. Opt. 9, 653 (1970).
    [CrossRef] [PubMed]
  4. H. Kogelnik, Appl. Opt. 4, 1562 (1965).
    [CrossRef]
  5. L. W. Casperson, A. Yariv, Appl. Phys. Lett. 12, 355 (1968).
    [CrossRef]
  6. L. W. Casperson, A. Yariv, Appl. Opt. 11, 462 (1972).
    [CrossRef] [PubMed]
  7. L. A. Schlie, J. T. Verdeyen, IEEE J. Quantum Electron. QE-5, 21 (1969).
    [CrossRef]
  8. L. M. Osterink, J. D. Foster, Appl. Phys. Lett. 12, 128 (1968).
    [CrossRef]
  9. D. W. Berreman, J. Opt. Soc. Am. 55, 239 (1965).
    [CrossRef]
  10. D. Marcuse, S. E. Miller, Bell Syst. Tech. J. 43, 1759 (1964).
  11. E. A. J. Marcatili, Bell Syst. Tech. J. 43, 2887 (1964).
  12. P. K. Tien, J. P. Gordon, J. R. Whinnery, Proc. IEEE 53, 129 (1965).
    [CrossRef]
  13. T. Uchida, M. Furukawa, J. Kitano, K. Koizumi, H. Matsumura, paper presented at the IEEE/Optical Society of America Joint Conference on Laser Engineering Applications, May 1969, Washington, D.C.
  14. A. Sommerfeld, J. Runge, Ann. Phys. 35, 290 (1911).
  15. A. N. Rosen, Appl. Opt. 11, 946 (1972).
    [CrossRef] [PubMed]
  16. C. N. Kurtz, W. Streifer, IEEE Trans. Microwave Theory Tech. MTT-17, 11 (1969).
    [CrossRef]
  17. H. Kogelnik, T. Li, Appl. Opt. 5, 1550 (1966).
    [CrossRef] [PubMed]
  18. G. Lampis, S. C. Brown, Phys. Fluids 11, 1137 (1968).
    [CrossRef]
  19. M. Nakatsuka, M. Yokoyama, Y. Izawa, K. Toyoda, C. Yamanaka, Phys. Lett. 37A, 169 (1971).
  20. A related transform is given by K. Bockasten, J. Opt. Soc. Am. 51, 943 (1961).
    [CrossRef]
  21. J. Shmoys, J. Appl. Phys. 32, 689 (1961). Equation (21) simplifies to v(w) = expexp[-w1/2π∫0wθ(x)dxx(w-x)1/2].
    [CrossRef]
  22. D. L. Jassby, M. E. Marhic, J. Appl. Phys. 43, 4586 (1972).
    [CrossRef]

1972 (3)

1971 (1)

M. Nakatsuka, M. Yokoyama, Y. Izawa, K. Toyoda, C. Yamanaka, Phys. Lett. 37A, 169 (1971).

1970 (1)

1969 (2)

L. A. Schlie, J. T. Verdeyen, IEEE J. Quantum Electron. QE-5, 21 (1969).
[CrossRef]

C. N. Kurtz, W. Streifer, IEEE Trans. Microwave Theory Tech. MTT-17, 11 (1969).
[CrossRef]

1968 (3)

L. W. Casperson, A. Yariv, Appl. Phys. Lett. 12, 355 (1968).
[CrossRef]

G. Lampis, S. C. Brown, Phys. Fluids 11, 1137 (1968).
[CrossRef]

L. M. Osterink, J. D. Foster, Appl. Phys. Lett. 12, 128 (1968).
[CrossRef]

1966 (1)

1965 (3)

1964 (2)

D. Marcuse, S. E. Miller, Bell Syst. Tech. J. 43, 1759 (1964).

E. A. J. Marcatili, Bell Syst. Tech. J. 43, 2887 (1964).

1961 (4)

G. D. Boyd, J. P. Gordon, Bell Sys. Tech. J. 40, 489 (1961).

G. Goubau, F. Schwering, IRE Trans. Antennas Propag. AP-9, 248 (1961).
[CrossRef]

A related transform is given by K. Bockasten, J. Opt. Soc. Am. 51, 943 (1961).
[CrossRef]

J. Shmoys, J. Appl. Phys. 32, 689 (1961). Equation (21) simplifies to v(w) = expexp[-w1/2π∫0wθ(x)dxx(w-x)1/2].
[CrossRef]

1911 (1)

A. Sommerfeld, J. Runge, Ann. Phys. 35, 290 (1911).

Berreman, D. W.

Bockasten, K.

Boyd, G. D.

G. D. Boyd, J. P. Gordon, Bell Sys. Tech. J. 40, 489 (1961).

Brown, S. C.

G. Lampis, S. C. Brown, Phys. Fluids 11, 1137 (1968).
[CrossRef]

Casperson, L. W.

L. W. Casperson, A. Yariv, Appl. Opt. 11, 462 (1972).
[CrossRef] [PubMed]

L. W. Casperson, A. Yariv, Appl. Phys. Lett. 12, 355 (1968).
[CrossRef]

Foster, J. D.

L. M. Osterink, J. D. Foster, Appl. Phys. Lett. 12, 128 (1968).
[CrossRef]

Furukawa, M.

T. Uchida, M. Furukawa, J. Kitano, K. Koizumi, H. Matsumura, paper presented at the IEEE/Optical Society of America Joint Conference on Laser Engineering Applications, May 1969, Washington, D.C.

Gordon, J. P.

P. K. Tien, J. P. Gordon, J. R. Whinnery, Proc. IEEE 53, 129 (1965).
[CrossRef]

G. D. Boyd, J. P. Gordon, Bell Sys. Tech. J. 40, 489 (1961).

Goubau, G.

G. Goubau, F. Schwering, IRE Trans. Antennas Propag. AP-9, 248 (1961).
[CrossRef]

Izawa, Y.

M. Nakatsuka, M. Yokoyama, Y. Izawa, K. Toyoda, C. Yamanaka, Phys. Lett. 37A, 169 (1971).

Jassby, D. L.

D. L. Jassby, M. E. Marhic, J. Appl. Phys. 43, 4586 (1972).
[CrossRef]

Kitano, J.

T. Uchida, M. Furukawa, J. Kitano, K. Koizumi, H. Matsumura, paper presented at the IEEE/Optical Society of America Joint Conference on Laser Engineering Applications, May 1969, Washington, D.C.

Kogelnik, H.

Koizumi, K.

T. Uchida, M. Furukawa, J. Kitano, K. Koizumi, H. Matsumura, paper presented at the IEEE/Optical Society of America Joint Conference on Laser Engineering Applications, May 1969, Washington, D.C.

Kurtz, C. N.

C. N. Kurtz, W. Streifer, IEEE Trans. Microwave Theory Tech. MTT-17, 11 (1969).
[CrossRef]

Lampis, G.

G. Lampis, S. C. Brown, Phys. Fluids 11, 1137 (1968).
[CrossRef]

Li, T.

Marcatili, E. A. J.

E. A. J. Marcatili, Bell Syst. Tech. J. 43, 2887 (1964).

Marcuse, D.

D. Marcuse, S. E. Miller, Bell Syst. Tech. J. 43, 1759 (1964).

Marhic, M. E.

D. L. Jassby, M. E. Marhic, J. Appl. Phys. 43, 4586 (1972).
[CrossRef]

Matsumura, H.

T. Uchida, M. Furukawa, J. Kitano, K. Koizumi, H. Matsumura, paper presented at the IEEE/Optical Society of America Joint Conference on Laser Engineering Applications, May 1969, Washington, D.C.

McCaul, B. W.

Miller, S. E.

D. Marcuse, S. E. Miller, Bell Syst. Tech. J. 43, 1759 (1964).

Nakatsuka, M.

M. Nakatsuka, M. Yokoyama, Y. Izawa, K. Toyoda, C. Yamanaka, Phys. Lett. 37A, 169 (1971).

Osterink, L. M.

L. M. Osterink, J. D. Foster, Appl. Phys. Lett. 12, 128 (1968).
[CrossRef]

Rosen, A. N.

Runge, J.

A. Sommerfeld, J. Runge, Ann. Phys. 35, 290 (1911).

Schlie, L. A.

L. A. Schlie, J. T. Verdeyen, IEEE J. Quantum Electron. QE-5, 21 (1969).
[CrossRef]

Schwering, F.

G. Goubau, F. Schwering, IRE Trans. Antennas Propag. AP-9, 248 (1961).
[CrossRef]

Shmoys, J.

J. Shmoys, J. Appl. Phys. 32, 689 (1961). Equation (21) simplifies to v(w) = expexp[-w1/2π∫0wθ(x)dxx(w-x)1/2].
[CrossRef]

Sommerfeld, A.

A. Sommerfeld, J. Runge, Ann. Phys. 35, 290 (1911).

Streifer, W.

C. N. Kurtz, W. Streifer, IEEE Trans. Microwave Theory Tech. MTT-17, 11 (1969).
[CrossRef]

Tien, P. K.

P. K. Tien, J. P. Gordon, J. R. Whinnery, Proc. IEEE 53, 129 (1965).
[CrossRef]

Toyoda, K.

M. Nakatsuka, M. Yokoyama, Y. Izawa, K. Toyoda, C. Yamanaka, Phys. Lett. 37A, 169 (1971).

Uchida, T.

T. Uchida, M. Furukawa, J. Kitano, K. Koizumi, H. Matsumura, paper presented at the IEEE/Optical Society of America Joint Conference on Laser Engineering Applications, May 1969, Washington, D.C.

Verdeyen, J. T.

L. A. Schlie, J. T. Verdeyen, IEEE J. Quantum Electron. QE-5, 21 (1969).
[CrossRef]

Whinnery, J. R.

P. K. Tien, J. P. Gordon, J. R. Whinnery, Proc. IEEE 53, 129 (1965).
[CrossRef]

Yamanaka, C.

M. Nakatsuka, M. Yokoyama, Y. Izawa, K. Toyoda, C. Yamanaka, Phys. Lett. 37A, 169 (1971).

Yariv, A.

L. W. Casperson, A. Yariv, Appl. Opt. 11, 462 (1972).
[CrossRef] [PubMed]

L. W. Casperson, A. Yariv, Appl. Phys. Lett. 12, 355 (1968).
[CrossRef]

Yokoyama, M.

M. Nakatsuka, M. Yokoyama, Y. Izawa, K. Toyoda, C. Yamanaka, Phys. Lett. 37A, 169 (1971).

Ann. Phys. (1)

A. Sommerfeld, J. Runge, Ann. Phys. 35, 290 (1911).

Appl. Opt. (5)

Appl. Phys. Lett. (2)

L. M. Osterink, J. D. Foster, Appl. Phys. Lett. 12, 128 (1968).
[CrossRef]

L. W. Casperson, A. Yariv, Appl. Phys. Lett. 12, 355 (1968).
[CrossRef]

Bell Sys. Tech. J. (1)

G. D. Boyd, J. P. Gordon, Bell Sys. Tech. J. 40, 489 (1961).

Bell Syst. Tech. J. (2)

D. Marcuse, S. E. Miller, Bell Syst. Tech. J. 43, 1759 (1964).

E. A. J. Marcatili, Bell Syst. Tech. J. 43, 2887 (1964).

IEEE J. Quantum Electron. (1)

L. A. Schlie, J. T. Verdeyen, IEEE J. Quantum Electron. QE-5, 21 (1969).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

C. N. Kurtz, W. Streifer, IEEE Trans. Microwave Theory Tech. MTT-17, 11 (1969).
[CrossRef]

IRE Trans. Antennas Propag. (1)

G. Goubau, F. Schwering, IRE Trans. Antennas Propag. AP-9, 248 (1961).
[CrossRef]

J. Appl. Phys. (2)

J. Shmoys, J. Appl. Phys. 32, 689 (1961). Equation (21) simplifies to v(w) = expexp[-w1/2π∫0wθ(x)dxx(w-x)1/2].
[CrossRef]

D. L. Jassby, M. E. Marhic, J. Appl. Phys. 43, 4586 (1972).
[CrossRef]

J. Opt. Soc. Am. (2)

Phys. Fluids (1)

G. Lampis, S. C. Brown, Phys. Fluids 11, 1137 (1968).
[CrossRef]

Phys. Lett. (1)

M. Nakatsuka, M. Yokoyama, Y. Izawa, K. Toyoda, C. Yamanaka, Phys. Lett. 37A, 169 (1971).

Proc. IEEE (1)

P. K. Tien, J. P. Gordon, J. R. Whinnery, Proc. IEEE 53, 129 (1965).
[CrossRef]

Other (1)

T. Uchida, M. Furukawa, J. Kitano, K. Koizumi, H. Matsumura, paper presented at the IEEE/Optical Society of America Joint Conference on Laser Engineering Applications, May 1969, Washington, D.C.

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Figures (4)

Fig. 1
Fig. 1

Axial Gaussian beam in a complex lenslike medium showing the normalized spot size w′ = 2−1/2(β0β2x)1/4w vs the normalized distance z′ = (β2x/β0)1/2z for (a) α2x/β2x = 0.1, (b) α2x/β2x = 0, and (c) α2x/β2x = −0.1. The dashed line in (c) marks the distance at which the spot size becomes infinite.

Fig. 2
Fig. 2

Off-axis beam in a lenslike medium showing the normalized beam displacement in the x direction vs z′ for (a) α2x/β2x = −0.1, (b) α2x/β2x = 0, and (c) α2x/β2x = −0.1.

Fig. 3
Fig. 3

Normalized beam displacement dxa = 2−3/4(α2xβ0)1/4dxa and spot size vs z″ = (a2x/2β0)1/2z for a gain-focused beam. Dashed lines are asymptotic limits. For α2x < 0 reverse direction of propagation.

Fig. 4
Fig. 4

Schematic drawing of a typical experiment involving the interaction of a Gaussian laser beam and a cylindrical plasma.

Equations (61)

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¯ × E ¯ = - i ω μ H ¯ ,             ¯ × H ¯ = i ω E ¯ ,
¯ × ¯ × E ¯ - ω 2 μ E ¯ = ( ¯ μ / μ ) × ¯ × E ¯ .
2 E ¯ + ω 2 μ E ¯ = - ¯ [ ( ¯ / ) · E ¯ ] - ( ¯ μ / μ ) × ¯ × E ¯ ,
2 E ¯ + k 2 E ¯ = 0 ,
k 2 ( x , y , z ) = k 0 ( z ) [ k 0 ( z ) - k 1 x ( z ) x - k 1 y ( z ) y - k 2 x ( z ) x 2 - 2 k x y ( z ) x y - k 2 y ( z ) y 2 ] .
E x = G ( x , y , z ) exp [ - i k 0 ( z ) d z ] .
( 2 G / x 2 ) + ( 2 G / y 2 ) - 2 i k 0 ( G / z ) - i ( d k 0 / d z ) G - k 0 ( k 1 x x + k 1 y y + k 2 x x 2 + 2 k x y x y + k 2 y y 2 ) G = 0 ,
G ( x , y , z ) = exp - i { [ Q x ( z ) x 2 / 2 ] + Q x y ( z ) x y + [ Q y ( z ) y 2 / 2 ] + S x ( z ) x + S y ( z ) y + P ( z ) } .
Q x 2 + Q x y 2 + k 0 ( d Q x / d z ) + k 0 k 2 x = 0 ,
Q y 2 + Q x y 2 + k 0 ( d Q y / d z ) + k 0 k 2 y = 0 ,
( Q x + Q y ) Q x y + k 0 ( d Q x y / d z ) + k 0 k x y = 0 ,
Q x S x + Q x y S y + k 0 ( d S x / d z ) + ( k 0 k 1 x ) / 2 = 0 ,
Q y S y + Q x y S x + k 0 ( d S y / d z ) + ( k 0 k 1 y ) / 2 = 0 ,
( d P / d z ) = - i [ ( Q x + Q y ) / 2 k 0 ] - [ ( S x 2 + S y 2 ) / 2 k 0 ] - ( i / 2 k 0 ) ( d k 0 / d z ) .
Q x 2 + k 0 ( d Q x / d z ) + k 0 k 2 x = 0 ,
Q y 2 + k 0 ( d Q y / d z ) + k 0 k 2 y = 0 ,
Q x S x + k 0 ( d S x / d z ) + ( k 0 k 1 x ) / 2 = 0 ,
Q y S y + k 0 ( d S y / d z ) + ( k 0 k 1 y ) / 2 = 0 ,
( d P / d z ) = - i [ ( Q x + Q y ) / ( 2 k 0 ) ] - [ ( S x 2 + S y 2 ) / ( 2 k 0 ) ] - ( i / 2 k 0 ) ( d k 0 / d z ) ,
G ( x , y , z ) = exp - i ( Q x x 2 / 2 + Q y y 2 / 2 + S x x + S y y + P ) .
1 / q = Q / k 0 = ( 1 / R ) - i [ λ m / ( π w 2 ) ] ,
G = exp { - i [ Q x r 2 ( x - d x p ) 2 + Q y r 2 ( y - d y p ) 2 - Q x r d x p 2 2 - Q y r d y p 2 2 + P r ] + [ Q x i 2 ( x - d x a ) 2 + Q y i 2 ( y - d y a ) 2 - Q x i d x a 2 2 - Q y i d y a 2 2 + P i ] }
Q x ( z ) k 0 = - ( k 2 x / k 0 ) 1 / 2 sin [ ( k 2 x / k 0 ) 1 / 2 z ] + [ Q x ( 0 ) / k 0 ] cos [ ( k 2 x / k 0 ) 1 / 2 z ] cos [ ( k 2 x / k 0 ) 1 / 2 z ] + [ Q x ( 0 ) / k 0 ] ( k 0 / k 2 x ) 1 / 2 sin [ ( k 2 x / k 0 ) 1 / 2 z ]
( d S x / d z ) + ( Q x S x / k 0 ) = 0 ,
( d S y / d z ) + ( Q y S y / k 0 ) = 0 ,
( d P / d z ) = - i [ ( Q x + Q y ) / ( 2 k 0 ) ] - [ ( S x 2 + S y 2 ) / ( 2 k 0 ) ] .
S x ( z ) = S x ( 0 ) / { cos [ ( k 2 x / k 0 ) 1 / 2 z ] + [ Q x ( 0 ) / k 0 ] ( k 0 / k 2 x ) 1 / 2 sin [ ( k 2 x / k 0 ) 1 / 2 z ] }
P ( z ) - P ( 0 ) = - ( i / 2 ) ln { cos [ ( k 2 x / k 0 ) 1 / 2 z ] + [ Q x ( 0 ) / k 0 ] ( k 0 / k 2 x ) 1 / 2 sin [ ( k 2 x / k 0 ) 1 / 2 z ] } - ( i / 2 ) ln { cos [ ( k 2 y / k 0 ) 1 / 2 z ] + [ Q y ( 0 ) / k 0 ] ( k 0 / k 2 y ) 1 / 2 sin [ ( k 2 y / k 0 ) 1 / 2 z ] } - [ S x ( 0 ) ] 2 2 k 0 × ( k 0 / k 2 x ) 1 / 2 sin [ ( k 2 x / k 0 ) 1 / 2 z ] cos [ ( k 2 x / k 0 ) 1 / 2 z ] + [ Q x ( 0 ) / k 0 ] ( k 0 / k 2 x ) 1 / 2 sin [ ( k 2 x / k 0 ) 1 / 2 z ] - [ S y ( 0 ) ] 2 2 k 0 × ( k 0 / k 2 y ) 1 / 2 sin [ ( k 2 y / k 0 ) 1 / 2 z ] cos [ ( k 2 y / k 0 ) 1 / 2 z ] + [ Q y ( 0 ) / k 0 ] ( k 0 / k 2 y ) 1 / 2 sin [ ( k 2 y / k 0 ) 1 / 2 z ] .
p x = π [ Re ( k 2 x / k 0 ) 1 / 2 ] - 1 , l x = [ 2 Im ( k 2 x / k 0 ) 1 / 2 ] - 1
1 / q x ( ) = i ( k 2 x / k 0 ) 1 / 2 = 1 / R x ( ) - i [ λ m / ( π w x 2 ( ) ) ] ,
Im ( k 2 x / k 0 ) > 0.
S x ( ) = - i [ ( k 1 x / 2 ) ( k 0 / k 2 x ) 1 / 2 ] .
d x a ( ) = - Im [ - i k 1 x 2 ( k 0 k 2 x ) 1 / 2 ] / Im [ - i ( k 0 k 2 x ) 1 / 2 ] - 1 2 Re ( k 1 x k 2 x - 1 / 2 ) Re k 2 x 1 / 2 ,
d x p ( ) = - Re [ - i k 1 x 2 ( k 0 k 2 x ) 1 / 2 ] / Re [ - i ( k 0 k 2 x ) 1 / 2 ] - 1 2 Im ( k 1 x k 2 x - 1 / 2 ) Im k 2 x 1 / 2 ,
( d / d z ) k 0 ( d d x a / d z ) = - ( k 1 x / 2 ) - ( k 2 z d x a ) .
( d / d z ) n 0 ( d d x a / d z ) = ( - n 1 x / 2 ) - n 2 x d x a = ( n / d x a ) .
( d / d z ) n 0 ( a d ¯ a / d z ) = ¯ n ,
( d 2 / d z 2 ) d x a + ( n 2 x d x a / n 0 ) = - n 1 x / 2 n 0 .
d x a ( z ) = [ d x a ( 0 ) + ( n 1 x / 2 n 2 x ) ] cos [ ( n 2 x / n 0 ) 1 / 2 z ] + d x a ( 0 ) ( n 0 / n 2 x ) 1 / 2 sin [ ( n 2 x / n 0 ) 1 / 2 z ] - ( n 1 x / 2 n 2 x ) ,
d x a ( z ) = d x a ( 0 ) cos [ ( n 2 x / n 0 ) 1 / 2 z ] + d x a ( 0 ) ( n 0 / n 2 x ) 1 / 2 sin [ ( n 2 x / n 0 ) 1 / 2 z ] .
d x a = d x a ( 0 ) + d x a ( 0 ) z - [ n 1 x / ( 4 n 0 ) ] z 2 .
P ( z ) = ( k 1 x 2 z / 8 k 2 x ) + ( k 1 y 2 z / 8 k 2 y ) + 1 2 ( k 2 x / k 0 ) 1 / 2 z + 1 2 ( k 2 y / k 0 ) 1 / 2 z .
c 2 E ¯ + k 2 E ¯ = 1 / r 2 { i ¯ r [ E r + 2 ( E ϕ / ϕ ) ] + i ¯ ϕ [ E ϕ - 2 ( E r / ϕ ) ] } - ¯ [ ( ¯ / ) · E ¯ ] - ( ¯ μ / μ ) × ¯ × E ¯ ,
c 2 E r + k 2 E r = 1 / r 2 [ E r + 2 ( E ϕ / ϕ ) ] ,
c 2 E ϕ + k 2 E ϕ = 1 / r 2 [ E ϕ - 2 ( E r / ϕ ) ] .
E r = i T ( r , z ) exp ( - i n ϕ ) ,             E ϕ = - T ( r , z ) exp ( - i n ϕ ) ,
( 2 T / r 2 ) + ( 1 / r ) ( T / r ) - [ ( n ± 1 ) 2 / r 2 ] T + ( 2 T / z 2 ) + k 2 T = 0.
T = U ( r , z ) exp [ - i k 0 ( z ) d z ]
( 2 U / r 2 ) + ( 1 / r ) ( U / r ) - [ ( n 1 ) 2 / r 2 ] U - i ( d k 0 / d z ) U - 2 i k 0 ( U / z ) - k 0 k 2 r 2 U = 0 ,
U = ( 2 r 2 / w 2 ) ( n 1 ) / 2 L m n 1 ( 2 r 2 / w 2 ) exp ( - r 2 / w 2 ) × exp [ i ( 2 m + 1 + n 1 ) ( k 2 / k 0 ) 1 / 2 z ] ,
H r = ( / μ ) 1 / 2 T exp ( - i n ϕ ) ,             H ϕ = i ( / μ ) 1 / 2 T exp ( - i n ϕ ) .
E z = 1 / k 0 { ( T / r ) + [ ( 1 n ) / r ] T } exp ( - i n ϕ ) , H z = - i / ω μ { ( T / r ) + [ ( 1 n ) / r ] T } exp ( - i n ϕ ) .
( T / r ) + [ ( 1 n ) / r ] T = 1 / r { [ 2 m + ( n + 1 ) × ( 1 1 ) - ( 2 r 2 / w 2 ) ] L m n 1 ( 2 r 2 / w 2 ) - 2 ( m + n 1 ) × L m - 1 n 1 ( 2 r 2 / w 2 ) } · ( 2 r 2 / w 2 ) ( n 1 ) / 2 exp ( - r 2 / w 2 ) × exp ( - i k 0 z ) exp [ i ( 2 m + 1 + n 1 ) ( k 2 / k 0 ) 1 / 2 z ] .
( d / d z ) ( 1 / q x ) = - ( 1 / q x ) 2 - ( k 2 x / k 0 ) .
Δ ( 1 / q ) = - ( k 2 x / k 0 ) d z .
Δ ( 1 q ) = 1 k 0 - 2 k ( x , z ) x 2 | x 0 d z ,
Δ [ 1 q ( x 0 ) ] = 1 k 0 - [ x 2 2 k r 2 r 2 + ( 1 r - x 2 r 3 ) k r ] x 0 d z = 2 k 0 x 0 { [ x 0 2 r d 2 k d r 2 + ( 1 - x 0 2 r 2 ) d k d r ] / ( r 2 - x 0 2 ) 1 / 2 } d r .
d θ ( x 0 , z ) / d z = - ( n 1 x / 2 n 0 ) ,
θ ( x 0 ) = - 1 2 n 0 - n 1 x d z = 1 n 0 - n x | x 0 d z .
θ ( x 0 ) = x 0 n 0 - 1 r d n d r d z = 2 x 0 n 0 x 0 ( d n / d r ) d r ( r 2 - x 0 2 ) 1 / 2 .
n ( r ) = - n 0 π r θ ( x 0 ) d x 0 ( x 0 2 - r 2 ) 1 / 2 .

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