Abstract

Mode conversion due to fluctuations caused by the dielectric-constant inhomogeneities of gaussian autocorrelation in a lens-like medium is discussed. Mean-square value of the mode-conversion coefficient, i.e., the power occupation of the higher mode in the total converted power, is obtained by means of the perturbation technique with the assumption of the fluctuations being smaller than square-law variation of the dielectric constant. The results show that the mode conversion increases in an undulating manner along the length of the gaseous medium, and is proportional to the longitudinal correlation length, whereas it decreases as the transverse correlation length becomes larger.

© 1969 Optical Society of America

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References

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  1. S. E. Miller, Bell System Tech. J.,  44, 2017 (1965).
    [Crossref]
  2. D. Marcuse, Bell System Tech. J.,  45, 1345 (1966).
    [Crossref]
  3. E. A. J. Marcatili, Bell System Tech. J.,  46, 149 (1967).
    [Crossref]
  4. M. Imai, Ph.D. dissertation, Hokkaido University, 1969.
  5. Y. Suematsu and K. Iga, J. Inst. Electron. Comm. Engrs. Japan 49, 1645 (1966).
  6. Y. Suematsu, K. Iga, and S. Ito, IEEE Trans. MTT-14, 657 (1966).
  7. S. E. Miller and L. C. Tillotson, Proc. IEEE 54, 1300 (1966).
    [Crossref]
  8. M. Imai, K. Itoh, and T. Matsumoto, J. Inst. Electron. Comm. Engrs. Japan B-51, 185 (1968).
  9. E. A. J. Marcatili, Bell System Tech. J. 43, 2887 (1964).
    [Crossref]
  10. In a actual gas lens, the fluctuations of the gaseous medium can be kept uniform in a cross section of the gas lens by suitably adjusting the roughness of the heating wall. Hence the gaussian autocorrelation may be more valid than in the open air.4
  11. S. Kikuchi, M. Imai, and T. Matsumoto, Bull. Faculty Engineering, Hokkaido University53, 181 (1969).
  12. Y. Kinoshita, T. Asakura, and M. Suzuki, J. Opt. Soc. Am. 58, 798 (1968).
    [Crossref]
  13. A. Erdélyi, Ed., Tables of Integral Transforms (McGraw–Hill Book Co., New York, 1954), pp. 288–292.
  14. A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions (McGraw–Hill Book Co., New York, 1953), p. 189.

1968 (2)

M. Imai, K. Itoh, and T. Matsumoto, J. Inst. Electron. Comm. Engrs. Japan B-51, 185 (1968).

Y. Kinoshita, T. Asakura, and M. Suzuki, J. Opt. Soc. Am. 58, 798 (1968).
[Crossref]

1967 (1)

E. A. J. Marcatili, Bell System Tech. J.,  46, 149 (1967).
[Crossref]

1966 (4)

Y. Suematsu and K. Iga, J. Inst. Electron. Comm. Engrs. Japan 49, 1645 (1966).

Y. Suematsu, K. Iga, and S. Ito, IEEE Trans. MTT-14, 657 (1966).

S. E. Miller and L. C. Tillotson, Proc. IEEE 54, 1300 (1966).
[Crossref]

D. Marcuse, Bell System Tech. J.,  45, 1345 (1966).
[Crossref]

1965 (1)

S. E. Miller, Bell System Tech. J.,  44, 2017 (1965).
[Crossref]

1964 (1)

E. A. J. Marcatili, Bell System Tech. J. 43, 2887 (1964).
[Crossref]

Asakura, T.

Erdélyi, A.

A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions (McGraw–Hill Book Co., New York, 1953), p. 189.

Iga, K.

Y. Suematsu, K. Iga, and S. Ito, IEEE Trans. MTT-14, 657 (1966).

Y. Suematsu and K. Iga, J. Inst. Electron. Comm. Engrs. Japan 49, 1645 (1966).

Imai, M.

M. Imai, K. Itoh, and T. Matsumoto, J. Inst. Electron. Comm. Engrs. Japan B-51, 185 (1968).

S. Kikuchi, M. Imai, and T. Matsumoto, Bull. Faculty Engineering, Hokkaido University53, 181 (1969).

M. Imai, Ph.D. dissertation, Hokkaido University, 1969.

Ito, S.

Y. Suematsu, K. Iga, and S. Ito, IEEE Trans. MTT-14, 657 (1966).

Itoh, K.

M. Imai, K. Itoh, and T. Matsumoto, J. Inst. Electron. Comm. Engrs. Japan B-51, 185 (1968).

Kikuchi, S.

S. Kikuchi, M. Imai, and T. Matsumoto, Bull. Faculty Engineering, Hokkaido University53, 181 (1969).

Kinoshita, Y.

Magnus, W.

A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions (McGraw–Hill Book Co., New York, 1953), p. 189.

Marcatili, E. A. J.

E. A. J. Marcatili, Bell System Tech. J.,  46, 149 (1967).
[Crossref]

E. A. J. Marcatili, Bell System Tech. J. 43, 2887 (1964).
[Crossref]

Marcuse, D.

D. Marcuse, Bell System Tech. J.,  45, 1345 (1966).
[Crossref]

Matsumoto, T.

M. Imai, K. Itoh, and T. Matsumoto, J. Inst. Electron. Comm. Engrs. Japan B-51, 185 (1968).

S. Kikuchi, M. Imai, and T. Matsumoto, Bull. Faculty Engineering, Hokkaido University53, 181 (1969).

Miller, S. E.

S. E. Miller and L. C. Tillotson, Proc. IEEE 54, 1300 (1966).
[Crossref]

S. E. Miller, Bell System Tech. J.,  44, 2017 (1965).
[Crossref]

Oberhettinger, F.

A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions (McGraw–Hill Book Co., New York, 1953), p. 189.

Suematsu, Y.

Y. Suematsu and K. Iga, J. Inst. Electron. Comm. Engrs. Japan 49, 1645 (1966).

Y. Suematsu, K. Iga, and S. Ito, IEEE Trans. MTT-14, 657 (1966).

Suzuki, M.

Tillotson, L. C.

S. E. Miller and L. C. Tillotson, Proc. IEEE 54, 1300 (1966).
[Crossref]

Tricomi, F. G.

A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions (McGraw–Hill Book Co., New York, 1953), p. 189.

Bell System Tech. J. (4)

S. E. Miller, Bell System Tech. J.,  44, 2017 (1965).
[Crossref]

D. Marcuse, Bell System Tech. J.,  45, 1345 (1966).
[Crossref]

E. A. J. Marcatili, Bell System Tech. J.,  46, 149 (1967).
[Crossref]

E. A. J. Marcatili, Bell System Tech. J. 43, 2887 (1964).
[Crossref]

IEEE Trans. (1)

Y. Suematsu, K. Iga, and S. Ito, IEEE Trans. MTT-14, 657 (1966).

J. Inst. Electron. Comm. Engrs. Japan (2)

Y. Suematsu and K. Iga, J. Inst. Electron. Comm. Engrs. Japan 49, 1645 (1966).

M. Imai, K. Itoh, and T. Matsumoto, J. Inst. Electron. Comm. Engrs. Japan B-51, 185 (1968).

J. Opt. Soc. Am. (1)

Proc. IEEE (1)

S. E. Miller and L. C. Tillotson, Proc. IEEE 54, 1300 (1966).
[Crossref]

Other (5)

M. Imai, Ph.D. dissertation, Hokkaido University, 1969.

A. Erdélyi, Ed., Tables of Integral Transforms (McGraw–Hill Book Co., New York, 1954), pp. 288–292.

A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions (McGraw–Hill Book Co., New York, 1953), p. 189.

In a actual gas lens, the fluctuations of the gaseous medium can be kept uniform in a cross section of the gas lens by suitably adjusting the roughness of the heating wall. Hence the gaussian autocorrelation may be more valid than in the open air.4

S. Kikuchi, M. Imai, and T. Matsumoto, Bull. Faculty Engineering, Hokkaido University53, 181 (1969).

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Figures (9)

Fig. 1
Fig. 1

Total mode-conversion coefficients 〈|Ct|2〉 divided by π1/2δ2k2a3/4g vs the normalized propagation distance gz for gaussian input of s/w0 = 1.0.

Fig. 2
Fig. 2

Mode-conversion coefficients 〈|Cr|2〉 (shown by solid lines), total mode-conversion coefficient 〈|Ct|2〉 (solid line) and 〈|C0|2〉 (shown by dotted line), which are all divided by π 1 2 δ 2 × a 3 z ( k / 2 ) 2 vs the normalized correlation length a/w0, for gaussian input, s/w0 = 1.0.

Fig. 3
Fig. 3

The ratio R defined by Eq. (23) which is obtained by dividing Eq. (22) by Eq. (18) with a3 = a, corresponding to the case of the isotropic fluctuations, vs the anisotropic parameter a2/a, for gaussian input, s/w0 = 1.0.

Fig. 4
Fig. 4

Total mode-conversion coefficients 〈|Ct|2〉 divided by π 1 2 δ 2 ( k / 2 g ) 2, vs the normalized longitudinal correlation length ga3, for gaussian input, s/w0 = 1.0. The normalized correlation length a/w0 = 10.0 is chosen as an example.

Fig. 5
Fig. 5

Total mode-conversion coefficients 〈|Ct|2〉 divided by π 1 2 δ 2 k 2 a 3 / 4 g vs gz for a beam of relatively small spot size, i.e., s/w0 = 0.5.

Fig. 6
Fig. 6

Total mode-conversion coefficients 〈|Ct|2〉 divided by π 1 2 δ 2 k 2 a 3 / 4 g vs gz for a beam of relatively large spot size, i.e., s/w0 = 4.0.

Fig. 7
Fig. 7

Mode-conversion coefficients 〈|Cr|2〉 (solid lines), total mode-conversion coefficient 〈|Ct|2〉 (solid line) and 〈|C0|2〉 (dotted line), which are all divided by π 1 2 δ 2 a 3 z ( k / 2 ) 2 vs a/s. The normalized propagation distance gz = 0.1.

Fig. 8
Fig. 8

Mode-conversion coefficients 〈|Cr|2 (solid lines), 〈|Ct|2〉 (solid line) and 〈|C0|2〉 (dotted line), which are all alike divided by π 1 2 δ 2 a 3 z ( k / 2 ) 2 vs a/s, for the input spot size s/w0 = 0.5 at gz = π/2.

Fig. 9
Fig. 9

Mode-conversion coefficients 〈|Cr|2〉 (solid lines), 〈|Ct|2〉 (solid line), and 〈|C0|2〉 (dotted line), which are all alike divided by π 1 2 δ 2 a 3 z ( k / 2 ) 2 vs a/s, for the input spot size s/w0 = 4.0 at gz = π/2.

Equations (53)

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( r ) = 0 [ 1 - ( g 1 x ) 2 - ( g 2 y ) 2 - δ ( r ) ] ,
2 Φ ( r ) - ( r ) 0 c 2 2 t 2 Φ ( r ) = 0 ,
Φ ( r ) = ϕ ( r ) exp [ j ( ω t - k z ) ] ,
L [ ϕ ( r ) ] = - k 2 δ ( r ) ϕ ( r ) ,
L = 2 x 2 + 2 y 2 - 2 j k z - k 2 [ ( g 1 x ) 2 + ( g 2 y ) 2 ] .
ϕ ( r ) = ϕ 0 ( r ) + δ ϕ ( r ) ,
L [ ϕ 0 ( r ) ] = 0
L [ δ ϕ ( r ) ] = - f ( r ) ,             f ( r ) = k 2 δ ( r ) ϕ 0 ( r ) .
δ ϕ ( r ) = d r 1 f ( r 1 ) G ( r , r 1 ) ,
G ( r , r 1 ) = { ( g 1 g 2 ) 1 2 4 π ( sin g 1 ( z - z 1 ) sin g 2 ( z - z 1 ) ) 1 2 exp [ j 2 x x 1 - ( x 2 + x 1 2 ) cos g 1 ( z - z 1 ) 2 w 01 2 sin g 1 ( z - z 1 ) + j 2 y y 1 - ( y 2 + y 1 2 ) cos g 2 ( z - z 1 ) 2 w 02 2 sin g 2 ( z - z 1 ) ] when z z 1 0 when z z 1 ,
δ ϕ ( r ) = p = 0 q = 0 C p q ϕ p q ( r ) ,
ϕ 0 ( r ) = 1 ( π w 01 w 02 ) 1 2 exp { - 1 2 ( ( x w 01 ) 2 + ( y w 02 ) 2 ) + j 2 ( g 1 + g 2 ) z } ;
ϕ p q ( r ) = A p A q H p ( x w 01 ) H q ( y w 02 ) × exp { - 1 2 [ ( x w 01 ) 2 + ( y w 02 ) 2 ] + j [ ( p + 1 2 ) g 1 + ( q + 1 2 ) g 2 ] z } ,
C p q = - j k 2 π w 01 w 02 1 ( 2 p + q p ! q ! ) 1 2 × 0 z d z 1 exp { - j ( p q 1 + q g 2 ) z 1 } - d x 1 d y 1 δ ( r 1 ) × H p ( x 1 w 01 ) H q ( y 1 w 02 ) exp { - [ ( x 1 w 01 ) 2 + ( y 1 w 02 ) 2 ] } .
C p q 2 = ( k 2 π w 01 w 02 ) 2 1 2 p + q p ! q ! 0 z d z 1 d z 2 × exp { - j ( p g 1 + q g 2 ) ( z 1 - z 2 ) } - d x 1 d x 2 d y 1 d y 2 × δ ( r 1 ) δ ( r 2 ) H p ( x 1 w 01 ) H p ( x 2 w 01 ) H q ( y 1 w 02 ) H q ( y 2 w 02 ) × exp ( - x 1 2 + x 2 2 w 01 2 - y 1 2 + y 2 2 w 02 2 ) .
δ ( r 1 ) δ ( r 2 ) = δ 2 × exp { - ( x 1 - x 2 a 1 ) 2 - ( y 1 - y 2 a 2 ) 2 - ( z 1 - z 2 a 3 ) 2 } ,
C p q 2 = k 2 δ 2 a 1 a 2 w 01 w 02 ( 2 p - 1 ) ! ! ( 2 q - 1 ) ! ! p ! q ! × 1 [ ( a 1 / w 01 ) 2 + 2 ] p + 1 2 [ ( a 2 / w 02 ) 2 + 2 ] q + 1 2 × 0 z / 2 d z -     exp { - ( 2 z - a 3 ) 2 } cos 2 ( p g 1 + q g 2 ) z - ,
( 2 n - 1 ) ! ! = ( 2 n ) ! / ( 2 n n ! ) = ( 2 n - 1 ) ( 2 n - 3 ) ( 2 n - 5 ) 3.1.
C p q 2 = ( k 2 ) 2 δ 2 π 1 2 a 3 z a 1 a 2 w 01 w 02 ( 2 p - 1 ) ! ! ( 2 q - 1 ) ! ! p ! q ! × 1 [ ( a 1 / w 01 ) 2 + 2 ] p + 1 2 [ ( a 2 / w 02 ) 2 + 2 ] q + 1 2 × exp { - 1 4 [ a 3 2 ( p q 1 + q g 2 ) 2 ] } .
C p q 2 = π 1 2 δ 2 ( k 2 ) 2 ( 2 p - 1 ) ! ! ( 2 q - 1 ) ! ! p ! q ! × exp { - a 3 2 g 2 ( p + q ) 2 4 } a 3 z ( a / w 0 ) 2 [ ( a / w 0 ) 2 + 2 ] p + q + 1 ,
C r 2 = π 1 2 δ 2 ( k 2 ) 2 a 3 z ( a / w 0 ) 2 2 r [ ( a / w 0 ) 2 + 2 ] r + 1 ,
C t 2 = r = 1 C r 2 = π 1 2 δ 2 ( k 2 ) 2 a 3 z 2 ( a / w 0 ) 2 + 2 .
C t 2 = π 1 2 δ 2 ( k 2 ) 2 a z × { 1 - a a 2 / w 0 2 [ ( a / w 0 ) 2 + 2 ] 1 2 [ ( a 2 / w 0 ) 2 + 2 ] 1 2 } ,
R = 1 2 { [ ( a w 0 ) 2 + 2 ] - a a 2 w 0 2 [ ( a / w 0 ) 2 + 2 ( a 2 / w 0 ) 2 + 2 ] 1 2 } .
C t 2 = 1 2 k 2 δ 2 z ( a w 0 ) 2 ( ( a w 0 ) 2 + 2 ) 2 0 z / 2 d z -     exp { - ( 2 z - a 3 ) 2 } cos 2 g z - - 2 / [ ( a / w 0 ) 2 + 2 ] 1 - 4 cos 2 g z - / [ ( a / w 0 ) 2 + 2 ] + { 2 / [ a / w ) 0 2 + 2 ] } 2 ,
ϕ 0 ( r ) = 1 π 1 2 s     exp [ - x 2 + y 2 2 w 0 2 ( w 0 2 / s 2 ) cos g z - j sin g z cos g z - j ( w 0 2 / s 2 ) sin g z ] / [ cos g z - j ( w 0 2 / s 2 ) sin g z ]
ϕ p q ( r ) = 1 ( 2 p + q p ! q ! π s 2 ) 1 2 ( cos g z + j ( w 0 2 / s 2 ) sin g z cos g z - j ( w 0 2 / s 2 ) sin g z ) ( p + q ) / 2 exp [ - x 2 + y 2 2 w 0 2 ( w 0 2 / s 2 ) cos g z - j sin g z cos g z - j ( w 0 2 / s 2 ) sin g z ] / [ cos g z - j ( w 0 2 / s 2 ) sin g z ] H p { x s [ cos 2 g z + ( w 0 4 / s 4 ) sin 2 g z ] 1 2 } H q { y s [ cos 2 g z + ( w 0 4 / s 4 ) sin 2 g z ] 1 2 } .
G ( r , r 1 ) = { - j 1 2 k p = 0 q = 0 ϕ p q ( r ) ϕ p q * ( r 1 ) when             z z 1 0 when             z z 1 ,
C p q = - j k 2 d r 1 δ ( r 1 ) ϕ 0 ( r 1 ) ϕ p q * ( r 1 ) .
C p q 2 = k 2 δ 2 ( a s ) 2 ( 2 p - 1 ) ! ! ( 2 q - 1 ) ! ! p ! q ! 0 z d z + - z / 2 z / 2 d z - exp { - ( 2 z - a 3 ) 2 } F ( z + , z - ) ,
F ( z + , z - ) = [ cos g ( z + + z - ) cos g ( z + - z - ) + ( w 0 4 / s 4 ) sin g ( z + + z - ) sin g ( z + - z - ) - j ( w 0 2 / s 2 ) sin 2 g z - ] p + q { a 2 / s 2 + cos 2 g ( z + + z - ) + cos 2 g ( z + - z - ) + w 0 4 / s 4 [ sin 2 g ( z + + z - ) + sin 2 g ( z + - z - ) ] } p + q + 1 .
F ( z + , z - ) = F ( z + , 0 ) + z - F ( z + , z - ) z - | z - = 0 + .
C p q 2 = π 1 2 δ 2 ( k 2 ) 2 a 3 ( 2 p - 1 ) ! ! ( 2 q - 1 ) ! ! p ! q ! 0 z d z + × a 2 / s 2 [ cos 2 g z + + ( w 0 4 / s 4 ) sin 2 g z + ] p + q { a 2 / s 2 + 2 [ cos 2 g z + + ( w 0 4 / s 4 ) sin 2 g z + ] } p + q + 1 .
C r 2 = π 1 2 δ 2 ( k 2 ) 2 ( a s ) 2 a 3 0 z d z + × 2 r [ cos 2 g z + + ( w 0 4 / s 4 ) sin 2 g z + ] r { a 2 / s 2 + 2 [ cos 2 g z + + ( w 0 4 / s 4 ) sin 2 g z + ] } r + 1 .
C t 2 = π 1 2 δ 2 ( k 2 ) 2 a 3 × 0 z d z + 2 [ cos 2 g z + + ( w 0 4 / s 4 ) sin 2 g z + ] a 2 / s 2 + 2 [ cos 2 g z + + ( w 0 4 / s 4 ) sin 2 g z + ] .
L [ G ( r , r 1 ) ] = - δ ( x - x 1 ) δ ( y - y 1 ) δ ( z - z 1 ) .
δ ( z - z 1 ) = 1 2 π - d κ exp { j κ ( z - z 1 ) }
δ ( x - x 1 ) = p = 0 C p H p ( x w 01 ) exp ( - x 2 2 w 01 2 )
δ ( y - y 1 ) = q = 0 C q H q ( y w 02 ) exp ( - y 2 2 w 02 2 ) .
C p = 1 2 p p ! π 1 2 w 01 H p ( x 1 w 01 ) exp ( - x 1 2 2 w 01 2 )
C q = 1 2 q q ! π 1 2 w 02 H q ( y 1 w 02 ) exp ( - y 1 2 2 w 02 2 ) .
δ ( x - x ) 1 = 1 π 1 2 w 01 p = 0 1 2 p p ! H p ( x w 01 ) × H p ( x 1 w 01 ) exp ( - x 2 + x 1 2 2 w 01 2 )
δ ( y - y 1 ) = 1 π 1 2 w 02 q = 0 1 2 q q ! H q ( y w 02 ) × H q ( y 1 w 02 ) exp ( - y 2 + y 1 2 2 w 02 2 ) .
G ( r , r 1 ) = - k ( g 1 g 2 ) 1 2 2 π 2 - d κ × exp [ j κ ( z - z 1 ) ] p = 0 q = 0 C p q κ H p ( x w 01 ) H q ( y w 02 ) × exp { - 1 2 [ ( x w 01 ) 2 + ( y w 02 ) 2 ] } .
C p q κ = H p ( x 1 / w 01 ) H q ( y 1 / w 02 ) exp { - 1 2 [ ( x 1 / w 01 ) 2 + ( y 1 / w 02 ) 2 ] } k ( κ - ( p + 1 2 ) g 1 - ( q + 1 2 ) g 2 ) 2 p + q + 1 p ! q ! .
- d κ exp { j κ ( z - z 1 ) } κ - ( p + 1 2 ) g 1 - ( q + 1 2 ) g 2 = { 2 π j exp { j [ ( p + 1 2 ) g 1 + ( q + 1 2 ) g 2 ] ( z - z 1 ) } , when             z z 1 0 , when             z z 1 .
G ( r , r 1 ) = { - j ( g 1 g 2 ) 1 2 π p = 0 q = 0 H p ( x w 01 ) H p ( x 1 w 01 ) H q ( y w 01 ) H q ( y 1 w 02 ) exp { - 1 2 ( x 2 + x 1 2 w 01 2 + y 2 + y 1 2 w 02 2 ) } / [ 2 p + q + 1 p ! q ! ] exp { j ( ( p + 1 2 ) g 1 ( q + 1 2 ) g 2 ) ( z - z 1 ) } when             z z 1 0 when             z z 1 .
C p q 2 = ( k 2 π w 01 w 02 ) 2 δ 2 2 p + q p ! q ! 0 z d z 1 d z 2 exp { - ( z 1 - z 2 a 3 ) 2 - j ( p g 1 + q g 2 ) ( z 1 - z 2 ) } × - d x 1 d x 2 H p ( x 1 w 01 ) H p ( x 2 w 01 ) exp { - ( x 1 - x 2 a 1 ) 2 - x 1 2 + x 2 2 w 01 2 } × - d y 1 d y 2 H q ( y 1 w 02 ) H q ( y 2 w 02 ) exp { - ( y 1 - y 2 a 2 ) 2 - y 1 2 + y 2 2 w 02 2 } .
- d x H n ( a x ) exp { - ( x - y ) 2 } = π 1 2 ( 1 - a 2 ) n / 2 H n ( a y ( 1 - a 2 ) 1 2 )
- d x H n ( a x ) H n ( b x ) exp ( - x 2 ) = π 1 2 2 n ( 2 n - 1 ) ! ! ( a b ) n ,             for             a 2 + b 2 = 1.
C p q 2 = ( k 2 ) 2 δ 2 a 1 a 2 w 01 w 02 ( 2 p - 1 ) ! ! ( 2 q - 1 ) ! ! p ! q ! { [ ( a 1 w 01 ) 2 + 2 ] p + 1 2 [ ( a 2 w 02 ) 2 + 2 ] q + 1 2 } - 1 × 0 z d z 1 d z 2 exp { - ( z 1 - z 2 a 3 ) 2 - j ( p g 1 + q g 2 ) ( z 1 - z 2 ) } .
0 z d z 1 0 z d z 2 2 0 z d z + - z / 2 z / 2 d z - ,
0 z d z 1 d z 2 exp { - ( z 1 - z 2 a 3 ) 2 - j ( p g 1 + q g 2 ) ( z 1 - z 2 ) } = 4 z 0 z / 2 d z - exp { - ( 2 z - a 3 ) 2 } cos 2 ( p g 1 + q g 2 ) z - .