Abstract

Coupled contradirectional mode equations are derived to describe Hermite-Gaussian beam interactions in corrugated thin-film waveguides when the grating teeth are curved and/or tilted. Coupling coefficients are evaluated for identical and hybrid modes. The problem of focusing a guided Gaussian beam by reflection from a curved grating is considered. It is shown that the grating curvature should serve to match the phase fronts of the contradirectional Gaussians and the grating should extend axially to include the region where the two beam widths are approximately equal.

© 1979 Optical Society of America

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References

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  1. H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
    [Crossref]
  2. W. Streifer, D. R. Scifres, and R. D. Burnham, “Coupling coefficients for distributed feedback single- and double-heterostructure lasers,” IEEE J. Quantum. Electron. QE-11, 867–873 (1975).
    [Crossref]
  3. M. A. Di Forto, M. Papuchon, C. Puech, P. Lallemand, and D. B. Ostrowsky, “Tunable optically pumped GaAs-GaAlAs distributed-feedback lasers,” IEEE. J. Quantum Electron.,  QE-14, 560–562, (1978).
    [Crossref]
  4. R. J. Capik and P. K. Tien, “Use of curved line gratings for diffraction of light in an optical waveguide,” J. Opt. Soc. Am. 67, 1392A (1977).
  5. P. K. Tien, “Method of forming novel curved-line gratings and their use as reflectors and resonators in integrated optics”, Opt. Lett. 1, 64–66 (1977).
    [Crossref] [PubMed]
  6. W. Streifer and A. Hardy, “Analysis of two dimensional waveguides with misaligned or curved gratings,” IEEE J. Quantum. Electron. QE-14, 936–943 (1978).
  7. E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J. 48, 2071–2102 (1969).
    [Crossref]
  8. H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1567 (1966).
    [Crossref] [PubMed]
  9. A. E. Siegman, An Introduction to Lasers and Masers, (McGraw-Hill, New York, 1971).
  10. A. Yariv, “Coupled-Mode Theory for Guided Wave Optics,” IEEE J. Quantum Electron. QE-9, 919–933 (1973).
    [Crossref]
  11. H. Kogelnik, “Filter response of nonuniform almost-periodic structures”B.S.T.J. 55, 109–126 (1976).
  12. W. Streifer, D. R. Scifres, and R. D. Burnham, “Perturbation analysis of nonuniform almost-periodic Bragg reflectors,” J. Opt. Soc. Am. 66, 1359–1363 (1976).
    [Crossref]
  13. L. B. W. Jolley, Summation of Series (Dover, New York, 1961). See number 1069, p. 198.
  14. G. A. Campbell and R. M. Foster, Fourier Integrals for Practical Applications (Van Nostrand, New York, 1948) p. 87.

1978 (2)

M. A. Di Forto, M. Papuchon, C. Puech, P. Lallemand, and D. B. Ostrowsky, “Tunable optically pumped GaAs-GaAlAs distributed-feedback lasers,” IEEE. J. Quantum Electron.,  QE-14, 560–562, (1978).
[Crossref]

W. Streifer and A. Hardy, “Analysis of two dimensional waveguides with misaligned or curved gratings,” IEEE J. Quantum. Electron. QE-14, 936–943 (1978).

1977 (2)

R. J. Capik and P. K. Tien, “Use of curved line gratings for diffraction of light in an optical waveguide,” J. Opt. Soc. Am. 67, 1392A (1977).

P. K. Tien, “Method of forming novel curved-line gratings and their use as reflectors and resonators in integrated optics”, Opt. Lett. 1, 64–66 (1977).
[Crossref] [PubMed]

1976 (2)

1975 (1)

W. Streifer, D. R. Scifres, and R. D. Burnham, “Coupling coefficients for distributed feedback single- and double-heterostructure lasers,” IEEE J. Quantum. Electron. QE-11, 867–873 (1975).
[Crossref]

1973 (1)

A. Yariv, “Coupled-Mode Theory for Guided Wave Optics,” IEEE J. Quantum Electron. QE-9, 919–933 (1973).
[Crossref]

1972 (1)

H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[Crossref]

1969 (1)

E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J. 48, 2071–2102 (1969).
[Crossref]

1966 (1)

Burnham, R. D.

W. Streifer, D. R. Scifres, and R. D. Burnham, “Perturbation analysis of nonuniform almost-periodic Bragg reflectors,” J. Opt. Soc. Am. 66, 1359–1363 (1976).
[Crossref]

W. Streifer, D. R. Scifres, and R. D. Burnham, “Coupling coefficients for distributed feedback single- and double-heterostructure lasers,” IEEE J. Quantum. Electron. QE-11, 867–873 (1975).
[Crossref]

Campbell, G. A.

G. A. Campbell and R. M. Foster, Fourier Integrals for Practical Applications (Van Nostrand, New York, 1948) p. 87.

Capik, R. J.

R. J. Capik and P. K. Tien, “Use of curved line gratings for diffraction of light in an optical waveguide,” J. Opt. Soc. Am. 67, 1392A (1977).

Di Forto, M. A.

M. A. Di Forto, M. Papuchon, C. Puech, P. Lallemand, and D. B. Ostrowsky, “Tunable optically pumped GaAs-GaAlAs distributed-feedback lasers,” IEEE. J. Quantum Electron.,  QE-14, 560–562, (1978).
[Crossref]

Foster, R. M.

G. A. Campbell and R. M. Foster, Fourier Integrals for Practical Applications (Van Nostrand, New York, 1948) p. 87.

Hardy, A.

W. Streifer and A. Hardy, “Analysis of two dimensional waveguides with misaligned or curved gratings,” IEEE J. Quantum. Electron. QE-14, 936–943 (1978).

Jolley, L. B. W.

L. B. W. Jolley, Summation of Series (Dover, New York, 1961). See number 1069, p. 198.

Kogelnik, H.

H. Kogelnik, “Filter response of nonuniform almost-periodic structures”B.S.T.J. 55, 109–126 (1976).

H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[Crossref]

H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1567 (1966).
[Crossref] [PubMed]

Lallemand, P.

M. A. Di Forto, M. Papuchon, C. Puech, P. Lallemand, and D. B. Ostrowsky, “Tunable optically pumped GaAs-GaAlAs distributed-feedback lasers,” IEEE. J. Quantum Electron.,  QE-14, 560–562, (1978).
[Crossref]

Li, T.

Marcatili, E. A. J.

E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J. 48, 2071–2102 (1969).
[Crossref]

Ostrowsky, D. B.

M. A. Di Forto, M. Papuchon, C. Puech, P. Lallemand, and D. B. Ostrowsky, “Tunable optically pumped GaAs-GaAlAs distributed-feedback lasers,” IEEE. J. Quantum Electron.,  QE-14, 560–562, (1978).
[Crossref]

Papuchon, M.

M. A. Di Forto, M. Papuchon, C. Puech, P. Lallemand, and D. B. Ostrowsky, “Tunable optically pumped GaAs-GaAlAs distributed-feedback lasers,” IEEE. J. Quantum Electron.,  QE-14, 560–562, (1978).
[Crossref]

Puech, C.

M. A. Di Forto, M. Papuchon, C. Puech, P. Lallemand, and D. B. Ostrowsky, “Tunable optically pumped GaAs-GaAlAs distributed-feedback lasers,” IEEE. J. Quantum Electron.,  QE-14, 560–562, (1978).
[Crossref]

Scifres, D. R.

W. Streifer, D. R. Scifres, and R. D. Burnham, “Perturbation analysis of nonuniform almost-periodic Bragg reflectors,” J. Opt. Soc. Am. 66, 1359–1363 (1976).
[Crossref]

W. Streifer, D. R. Scifres, and R. D. Burnham, “Coupling coefficients for distributed feedback single- and double-heterostructure lasers,” IEEE J. Quantum. Electron. QE-11, 867–873 (1975).
[Crossref]

Shank, C. V.

H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[Crossref]

Siegman, A. E.

A. E. Siegman, An Introduction to Lasers and Masers, (McGraw-Hill, New York, 1971).

Streifer, W.

W. Streifer and A. Hardy, “Analysis of two dimensional waveguides with misaligned or curved gratings,” IEEE J. Quantum. Electron. QE-14, 936–943 (1978).

W. Streifer, D. R. Scifres, and R. D. Burnham, “Perturbation analysis of nonuniform almost-periodic Bragg reflectors,” J. Opt. Soc. Am. 66, 1359–1363 (1976).
[Crossref]

W. Streifer, D. R. Scifres, and R. D. Burnham, “Coupling coefficients for distributed feedback single- and double-heterostructure lasers,” IEEE J. Quantum. Electron. QE-11, 867–873 (1975).
[Crossref]

Tien, P. K.

R. J. Capik and P. K. Tien, “Use of curved line gratings for diffraction of light in an optical waveguide,” J. Opt. Soc. Am. 67, 1392A (1977).

P. K. Tien, “Method of forming novel curved-line gratings and their use as reflectors and resonators in integrated optics”, Opt. Lett. 1, 64–66 (1977).
[Crossref] [PubMed]

Yariv, A.

A. Yariv, “Coupled-Mode Theory for Guided Wave Optics,” IEEE J. Quantum Electron. QE-9, 919–933 (1973).
[Crossref]

Appl. Opt. (1)

B.S.T.J. (1)

H. Kogelnik, “Filter response of nonuniform almost-periodic structures”B.S.T.J. 55, 109–126 (1976).

Bell Syst. Tech. J. (1)

E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J. 48, 2071–2102 (1969).
[Crossref]

IEEE J. Quantum Electron. (1)

A. Yariv, “Coupled-Mode Theory for Guided Wave Optics,” IEEE J. Quantum Electron. QE-9, 919–933 (1973).
[Crossref]

IEEE J. Quantum. Electron. (2)

W. Streifer and A. Hardy, “Analysis of two dimensional waveguides with misaligned or curved gratings,” IEEE J. Quantum. Electron. QE-14, 936–943 (1978).

W. Streifer, D. R. Scifres, and R. D. Burnham, “Coupling coefficients for distributed feedback single- and double-heterostructure lasers,” IEEE J. Quantum. Electron. QE-11, 867–873 (1975).
[Crossref]

IEEE. J. Quantum Electron. (1)

M. A. Di Forto, M. Papuchon, C. Puech, P. Lallemand, and D. B. Ostrowsky, “Tunable optically pumped GaAs-GaAlAs distributed-feedback lasers,” IEEE. J. Quantum Electron.,  QE-14, 560–562, (1978).
[Crossref]

J. Appl. Phys. (1)

H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[Crossref]

J. Opt. Soc. Am. (2)

R. J. Capik and P. K. Tien, “Use of curved line gratings for diffraction of light in an optical waveguide,” J. Opt. Soc. Am. 67, 1392A (1977).

W. Streifer, D. R. Scifres, and R. D. Burnham, “Perturbation analysis of nonuniform almost-periodic Bragg reflectors,” J. Opt. Soc. Am. 66, 1359–1363 (1976).
[Crossref]

Opt. Lett. (1)

Other (3)

A. E. Siegman, An Introduction to Lasers and Masers, (McGraw-Hill, New York, 1971).

L. B. W. Jolley, Summation of Series (Dover, New York, 1961). See number 1069, p. 198.

G. A. Campbell and R. M. Foster, Fourier Integrals for Practical Applications (Van Nostrand, New York, 1948) p. 87.

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Figures (3)

FIG. 1
FIG. 1

Cross section of a corrugated slab waveguide.

FIG. 2
FIG. 2

Top view of arbitrarily curved and tilted grating teeth together with an ideal straight, parallel, periodic reference grating.

FIG. 3
FIG. 3

Illustrating the geometry for reflection grating focusing of a Gaussian beam.

Equations (84)

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n 2 ( x , z ) = q = A q ( x ) exp ( i 2 π q z / Λ ) ,
A q ( x ) = { n 2 2 ( n 2 2 n 1 2 ) [ w 2 ( x ) w 1 ( x ) ] / Λ , q = 0 n 2 2 n 1 2 i 2 π q [ exp i 2 π q w 2 ( x ) Λ exp i 2 π q w 1 ( x ) Λ ] , q 0
A q ( x ) = A q * ( x ) .
n d 2 ( x , y , z ) = n 2 [ x , z Δ ( y , z ) ] ,
n d 2 ( x , y , z ) = q = A q ( x ) exp { ( i 2 π q / Λ ) [ z Δ ( y , z ) ] } , = q = A q ( x ) N q ( y , z ) e i 2 π q z / Λ , 0 < x < g
N q ( y , z ) = exp [ ( i 2 π q / Λ ) Δ ( y , z ) ]
N q * = N q .
( 2 x 2 + 2 y 2 + 2 z 2 + k 0 2 n d 2 ( x , y , z ) ) E y ( x , y , z ) = 0
n d 2 ( x , y , z ) = n 0 2 ( x ) + q = q 0 A q ( x ) N q ( y , z ) exp ( i 2 π q z / Λ ) ,
n 0 2 ( x ) = { n 1 2 , x < 0 A 0 ( x ) , 0 < x < g n 2 2 , g < x < t 2 n 3 2 , t 2 < x .
E y ( x , y , z ) = φ m ( x ) ψ p ( y , z ) e ± i β m z .
φ m ( x ) φ m ¯ ( x ) d x = { 1 , m = m ¯ 0 , m m ¯ .
2 ψ p y 2 ± 2 i β ( m ) ψ p z = 0
ψ p ( y , z ) = ( 2 π ) 1 / 4 1 [ 2 p p ! W ( z ) ] 1 / 2 H p ( 2 1 / 2 y / W ) × exp { ± i [ ( β y 2 / 2 q ( z ) ) Φ p ( z ) ] } , p = 0 , 1 , 2 , .
1 q ( z ) = 1 ρ b ( z ) ± i 2 β ( m ) W 2 ( z ) ,
Φ p = ( p + 1 / 2 ) tan 1 ( 2 z / β ( m ) W 0 2 ) ,
ρ b ( z ) = z [ 1 + ( β ( m ) W 0 2 2 z ) 2 ]
W ( z ) = W 0 [ 1 + ( 2 z / β ( m ) W 0 2 ) 2 ] 1 / 2
ψ p ( y , z ) ψ p ¯ * ( y , z ) d y = { 1 , p = p ¯ , 0 , p p ¯ .
β ̂ r + β ̂ s = 2 π M / Λ .
δ r = β r β ̂ r β ̂ r
δ s = β s β ̂ s β ̂ s
E y = φ r ( x ) e i β ̂ r z p = 0 R p ( z ) ψ p ( r ) ( y , z ) + φ s ( x ) e i β ̂ s z × p = 0 S p ( z ) ψ p ( s ) ( y , z ) ,
φ r ( x ) p = 0 ( d R p d z i δ r R p ) ψ p ( r ) ( y , z ) i k 0 2 A M ( x ) N M ( y , z ) 2 β ̂ r φ s ( x ) p = 0 ψ p ( s ) ( y , z ) S p = 0
φ s ( x ) p = 0 ( d S p d z i δ s S p ) ψ p ( s ) ( y , z ) i k 0 2 A M ( x ) N M ( y , z ) 2 β ̂ s φ r ( x ) p = 0 ψ p ( r ) ( y , z ) R p = 0 .
d R u d z i δ r R u i υ = 0 κ r s ( u , υ ) S υ = 0 , u = 0 , 1 , 2 , ,
d S υ d z i δ r S υ i u = 0 κ s r ( υ , u ) R u = 0 , υ = 0 , 1 , 2 , ,
κ r s ( u , υ ) = k 0 2 2 β ̂ r 0 g φ r A M φ s d x ψ υ ( s ) N M ψ u ( r ) * d y
κ s r ( υ , u ) = k 0 2 2 β ̂ s 0 g φ r A M φ s d x ψ u ( r ) N M ψ υ ( s ) * d y .
β r κ r s = β s κ s r * .
κ r s ( u , υ ) = κ x κ y ( z ) ,
κ x = k 0 2 2 β ̂ r 0 g φ r ( x ) A M ( x ) φ s ( x ) d x
κ y ( z ) = ψ υ ( s ) ( y , z ) N M ( y , z ) ψ u ( r ) * ( y , z ) d y .
κ y ( z ) = | κ y ( z ) | e i γ ( z ) ,
N M = exp ( i 2 π M Λ Δ ( y , z ) ) ,
| κ y ( z ) | 1 .
ψ u ( r ) = ( 2 π ) 1 / 4 1 ( 2 υ υ ! W s ) 1 / 2 × e y 2 / W r 2 H u ( 2 1 / 2 y W r ) e i [ ( β r y 2 / 2 ρ r ) Φ u ]
ψ υ ( s ) = ( 2 π ) 1 / 4 1 ( 2 υ υ ! W s ) 1 / 2 × e y 2 / W s 2 H υ ( 2 1 / 2 y W s ) e i [ ( β s y 2 / 2 ρ s ] Φ υ )
β r ( W 0 ( r ) ) 2 = β s ( W 0 ( s ) ) 2 β W 0 2
κ y ( z ) = ( 2 / π ) 1 / 2 e i [ Φ u ( z ) + Φ υ ( z ) ] [ 2 u + υ u ! υ ! W r ( z ) W s ( z ) ] 1 / 2 × e y 2 ( 1 / W r 2 + 1 / W s 2 ) H u ( 2 1 / 2 y W r ) H υ ( 2 1 / 2 y W s ) × e i [ ( y 2 / 2 ρ b ) ( β r + β s ) + ( 2 π M / Λ ) Δ ] d y .
Δ ( y , z ) = y 2 2 ρ b ( z ) ( β r + β s ) ( β ̂ r + β ̂ s ) + ( U + 1 ) ( β ̂ r + β ̂ s ) tan 1 ( 2 z β W 0 2 )
Δ ( y , z ) y 2 2 ρ b ( z ) + ( U + 1 ) ( β ̂ r + β ̂ s ) tan 1 ( 2 z β W 0 2 ) ,
σ ( z ) = ( U + 1 ) ( β ̂ r + β ̂ s ) tan 1 ( 2 z β W 0 2 ) ,
κ y ( z ) = ( 2 / π ) 1 / 2 e i [ Φ u ( z ) + Φ υ ( z ) ] ( 2 u + υ u ! υ ! ) 1 / 2 W ( z ) × e 2 y 2 / W 2 H u ( 2 1 / 2 y W ) H υ ( 2 1 / 2 y W ) e i [ ( y 2 β / ρ b ) + ( 2 π M Δ ) / Λ ] d y .
Δ ( y , z ) = y 2 2 ρ b ( z ) β β ̂ + ( U + 1 ) 2 β ̂ tan 1 ( 2 z β W 0 2 )
Δ ( y , z ) y 2 2 ρ b ( z ) + ( U + 1 ) 2 β ̂ tan 1 ( 2 z β W 0 2 ) ,
K y ( z ) = { exp [ i ( 2 u U ) tan 1 ( 2 z / β W 0 2 ) ] , u = υ , 0 , u υ ,
ψ 0 ( r ) = ( 2 π ) 1 / 4 1 W r 1 / 2 e y 2 / w r 2 e i [ ( β y 2 / 2 ρ r ) Φ 0 r ] ,
W r ( z ) = W 0 r [ 1 + ( 2 ( z z r ) β ( W 0 r ) 2 ) 2 ] 1 / 2 ,
ρ r ( z ) = ( z z r ) [ 1 + ( β ( W 0 r ) 2 2 ( z z r ) ) ] ,
Φ 0 r ( z ) = 1 2 tan 1 ( 2 ( z z r ) β ( W 0 r ) 2 ) .
W s ( z ) = W 0 s [ 1 + ( 2 ( z z s ) β ( W 0 s ) 2 ) 2 ] 1 / 2 ,
ρ s ( z ) = ( z z s ) [ 1 + ( β ( W 0 s ) 2 2 ( z z s ) ) 2 ] ,
Φ 0 ( s ) = 1 2 tan 1 ( 2 ( z z s ) β ( W 0 s ) 2 ) .
ψ 0 ( s ) = ( 2 π ) 1 / 4 1 W s 1 / 2 e y 2 / w s 2 e i [ ( β y 2 / 2 ρ s ) Φ 0 s ] .
κ y ( 0 , 0 , z ) = ( 2 π ) 1 / 2 e i ( Φ 0 s + Φ 0 r ) ( W r W s ) 1 / 2 e y 2 ( 1 / W s 2 + 1 / W r 2 ) × e i [ ( β y 2 / 2 ρ s ) + ( β y 2 / 2 ρ r ) + ( 2 π M Δ / Λ ) ] d y .
Δ ( y , z ) = Λ 4 π M β y 2 ( 1 ρ r + 1 ρ s ) + Λ 2 π M ( Φ 0 s + Φ 0 r )
κ y ( 0 , 0 , z ) = ( 2 W r W s W r 2 + W s 2 ) 1 / 2 .
κ y ( 0 , υ , z ) = ( 2 π ) 1 / 2 e i ( Φ r s Φ 0 s ) ( 2 υ υ ! W r W s ) 1 / 2 × H υ ( 2 y W s ) e y 2 ( 1 / W s 2 + 1 / W r 2 ) d y
κ y ( 0 , υ , z ) = κ y ( 0 , 0 , z ) e i ( Φ v s Φ 0 s ) ( ( υ ! ) 1 / 2 2 v / 2 ( v / 2 ) ! ) ( W r 2 W s 2 W r 2 + W s 2 ) υ / 2 , υ = 2 , 4 , 6 , .
( W r 2 W s 2 W r 2 + W s 2 ) υ / 2
κ y ( 0 , 0 , z ) ( 2 W s ( z ) W r ( z ) ) 1 / 2 .
Φ υ s ( z ) Φ 0 s ( z ) = υ tan 1 ( 2 ( z z s ) β ( W 0 s ) 2 )
Δ ( y , z ) = α ( z ) y 2 + b ( z ) y + c ( z )
κ y ( u , υ , z ) = e i [ Φ u + Φ υ ( 2 π M / Λ ) c ( z ) ] ( π 2 u + υ w r w s ) 1 / 2 I ( z ) ,
I ( z ) = H υ ( η / w s ) H u ( η / w r ) e ( 1 + i A ) η 2 e i 2 B ¯ η d η .
η = 2 y / W ,
1 / W 2 ( z ) = ½ ( 1 / W r 2 + 1 / W s 2 ) ,
w r = W r / W ,
w s = W s / W
A ( z ) = W 2 ( z ) 2 ( β s 2 ρ s ( z ) + β r 2 ρ r ( z ) + 2 π M Λ a ( z ) ) ,
B ( z ) = [ π M W ( z ) / Λ ] b ( z ) .
exp ( 2 ζ t t 2 ) = p = 0 ( 1 / p ! ) H p ( ζ ) t p
exp [ γ ( i 2 π f + λ ) ] exp ( i 2 π f g ) d f = [ 1 / ( 2 π 1 / 2 γ 1 / 2 ) ] exp ( λ g g 2 / 4 γ ) ,
κ y ( u , υ , z ) = e i [ Φ u + Φ υ ( 2 π M / Λ ) c ( z ) ] ( u ! υ ! ) 1 / 2 [ 2 u υ w r 2 u + 1 w s 2 υ + 1 ( 1 + i A ) ] 1 / 2 × e B 2 / [ 2 ( 1 + i A ) ] F ( u , υ , z ) ,
F ( u , υ , z ) = m = 0 [ υ / 2 ] p = 2 m υ n = 0 [ ( u υ + p ) / 2 ] × ( 1 ) n + m ( i 2 B ) u υ + 2 ( p m n ) ( 1 + i A ) m + n p u 2 p m ! ( p 2 m ) ! ( υ p ) ! n ! ( u υ + p 2 n ) ! × [ w r 2 ( 1 + i A ) 1 ] n [ w s 2 ( 1 + i A ) 1 ] m
u υ + 2 ( p m n ) = 0
F ( u , υ , z ) = { m = 0 [ υ / 2 ] ( 1 ) ( u υ ) / 2 ( 1 + i A ) ( u + υ ) / 2 [ w r 2 ( 1 + i A ) 1 ] m + ( u υ ) / 2 [ w s 2 ( 1 + i A ) 1 ] m 2 2 m m ! ( υ 2 m ) ! [ m + ( u υ ) / 2 ] ! , u , υ same parity 0 , u , υ differing parity
F ( u , υ , z ) = p = 0 υ ( i 2 B ) u υ + 2 p 2 p p ! ( υ p ) ! ( u υ + p ) !
F ( u , υ , z ) = { 1 / u ! , u = υ 0 , u υ
F ( u , υ , z ) = { m = 0 [ υ / 2 ] ( 1 ) ( u υ ) / 2 ( w r 2 1 ) m + ( u υ ) / 2 ( w s 2 1 ) m 2 2 m m ! ( υ 2 m ) ! [ m + ( u υ ) / 2 ] , u , υ same parity 0 , u , υ differing parity
F ( u , 0 , z ) = n = 0 [ u / 2 ] ( 1 ) n ( i 2 B ) u 2 n ( 1 + i A ) n u n ! ( u 2 n ) ! × [ w r 2 ( 1 + i A ) 1 ] n .
F ( u , 0 , z ) = { ( 1 ) u / 2 ( u / 2 ) ! ( w r 2 1 ( 1 + i A ) ) u / 2 , u even 0 , u odd
F ( u , 0 , z ) = { ( 1 ) u / 2 ( w r 2 1 ) u / 2 , ( u / 2 ) ! u even 0 , u odd .