Abstract

Off-axis light-beam propagation in optical systems representable by complex ABCD matrices is considered in detail. The results are presented in terms of modal parameter transformations similar to Kogelnik’s ABCD law. The modes emphasized here can be expressed in terms of polynomial-Gaussian functions of complex argument. These modes may be both astigmatic and asymmetric and can describe the propagation of off-axis beams of light in lenslike media with spatial gain or loss variation.

© 1991 Optical Society of America

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References

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  1. G. D. Boyd, J. P. Gordon, “Confocal multimode resonator for millimeter through optical wavelength masers,” Bell Syst. Tech. J. 40, 489–508 (1961).
  2. L. W. Casperson, “Beam modes in complex lenslike media and resonators,”J. Opt. Soc. Am. 66, 1373–1379 (1976).
    [CrossRef]
  3. H. Kogelnik, “On the propagation of Gaussian beams of light through lenslike media, including those with a loss or gain variation,” Appl. Opt. 4, 1562–1599 (1965).
    [CrossRef]
  4. L. W. Casperson, A. Yariv, “The Gaussian mode in optical resonators with a radial gain profile,” Appl. Phys. Lett. 12, 355–357 (1968).
    [CrossRef]
  5. H. Zucker, “Optical resonators with variable reflectivity mirrors,” Bell Syst. Tech. J. 49, 2349–2376 (1970), and references therein.
  6. J. A. Arnaud, “Nonorthogonal optical waveguides and resonators,” Bell Syst. Tech. J. 49, 2311–2348 (1970).
  7. J. A. Arnaud, “Mode coupling in first order optics,”J. Opt. Soc. Am. 61, 751–758 (1971).
    [CrossRef]
  8. J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), especially Subsecs. 2.16, 2.17, and 4.18–4.20.
  9. A. E. Siegman, “Hermite–Gaussian functions of complex argument as optical-beam eigenfunctions,”J. Opt. Soc. Am. 63, 1093–1094 (1973).
    [CrossRef]
  10. R. Pratesi, L. Ronchi, “Generalized Gaussian beams in free space,”J. Opt. Soc. Am. 67, 1274–1276 (1977).
    [CrossRef]
  11. M. Nazarathy, A. Hardy, J. Shamir, “Generalized mode propagation in first-order optical systems with loss or gain,”J. Opt. Soc. Am. 72, 1409–1420 (1982).
    [CrossRef]
  12. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), especially Subsec. 20.5.
  13. L. W. Casperson, “Beam propagation in periodic quadratic-index waveguides,” Appl. Opt. 24, 4395–4403 (1985).
    [CrossRef] [PubMed]
  14. L. W. Casperson, “Mode stability of lasers and periodic optical systems,” IEEE J. Quantum Electron. QE-10, 629–634 (1974).
    [CrossRef]
  15. L. W. Casperson, “Synthesis of Gaussian beam optical systems,” Appl. Opt. 20, 2243–2249 (1981).
    [CrossRef] [PubMed]
  16. C. C. Su, “On the scalar approximation in fiber optics,” IEEE Trans. Microwave Theory Tech. 36, 1100–1103 (1988).
    [CrossRef]
  17. C. Yeh, L. W. Casperson, “Scalar-wave approach for single-mode inhomogeneous fiber problems,” Appl. Phys. Lett. 34, 460–462 (1979).
    [CrossRef]
  18. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), pp. 131–137.
  19. J. T. Verdeyen, Laser Electronics (Prentice-Hall, Englewood Cliffs, N.J., 1981).
  20. L. W. Casperson, “Gaussian light beams in inhomogeneous media,” Appl. Opt. 12, 2434–2441 (1973).
    [CrossRef] [PubMed]

1988

C. C. Su, “On the scalar approximation in fiber optics,” IEEE Trans. Microwave Theory Tech. 36, 1100–1103 (1988).
[CrossRef]

1985

1982

1981

1979

C. Yeh, L. W. Casperson, “Scalar-wave approach for single-mode inhomogeneous fiber problems,” Appl. Phys. Lett. 34, 460–462 (1979).
[CrossRef]

1977

1976

1974

L. W. Casperson, “Mode stability of lasers and periodic optical systems,” IEEE J. Quantum Electron. QE-10, 629–634 (1974).
[CrossRef]

1973

1971

1970

H. Zucker, “Optical resonators with variable reflectivity mirrors,” Bell Syst. Tech. J. 49, 2349–2376 (1970), and references therein.

J. A. Arnaud, “Nonorthogonal optical waveguides and resonators,” Bell Syst. Tech. J. 49, 2311–2348 (1970).

1968

L. W. Casperson, A. Yariv, “The Gaussian mode in optical resonators with a radial gain profile,” Appl. Phys. Lett. 12, 355–357 (1968).
[CrossRef]

1965

1961

G. D. Boyd, J. P. Gordon, “Confocal multimode resonator for millimeter through optical wavelength masers,” Bell Syst. Tech. J. 40, 489–508 (1961).

Arnaud, J. A.

J. A. Arnaud, “Mode coupling in first order optics,”J. Opt. Soc. Am. 61, 751–758 (1971).
[CrossRef]

J. A. Arnaud, “Nonorthogonal optical waveguides and resonators,” Bell Syst. Tech. J. 49, 2311–2348 (1970).

J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), especially Subsecs. 2.16, 2.17, and 4.18–4.20.

Boyd, G. D.

G. D. Boyd, J. P. Gordon, “Confocal multimode resonator for millimeter through optical wavelength masers,” Bell Syst. Tech. J. 40, 489–508 (1961).

Casperson, L. W.

L. W. Casperson, “Beam propagation in periodic quadratic-index waveguides,” Appl. Opt. 24, 4395–4403 (1985).
[CrossRef] [PubMed]

L. W. Casperson, “Synthesis of Gaussian beam optical systems,” Appl. Opt. 20, 2243–2249 (1981).
[CrossRef] [PubMed]

C. Yeh, L. W. Casperson, “Scalar-wave approach for single-mode inhomogeneous fiber problems,” Appl. Phys. Lett. 34, 460–462 (1979).
[CrossRef]

L. W. Casperson, “Beam modes in complex lenslike media and resonators,”J. Opt. Soc. Am. 66, 1373–1379 (1976).
[CrossRef]

L. W. Casperson, “Mode stability of lasers and periodic optical systems,” IEEE J. Quantum Electron. QE-10, 629–634 (1974).
[CrossRef]

L. W. Casperson, “Gaussian light beams in inhomogeneous media,” Appl. Opt. 12, 2434–2441 (1973).
[CrossRef] [PubMed]

L. W. Casperson, A. Yariv, “The Gaussian mode in optical resonators with a radial gain profile,” Appl. Phys. Lett. 12, 355–357 (1968).
[CrossRef]

Gordon, J. P.

G. D. Boyd, J. P. Gordon, “Confocal multimode resonator for millimeter through optical wavelength masers,” Bell Syst. Tech. J. 40, 489–508 (1961).

Hardy, A.

Kogelnik, H.

Nazarathy, M.

Pratesi, R.

Ronchi, L.

Shamir, J.

Siegman, A. E.

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), pp. 131–137.

Su, C. C.

C. C. Su, “On the scalar approximation in fiber optics,” IEEE Trans. Microwave Theory Tech. 36, 1100–1103 (1988).
[CrossRef]

Verdeyen, J. T.

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

Yariv, A.

L. W. Casperson, A. Yariv, “The Gaussian mode in optical resonators with a radial gain profile,” Appl. Phys. Lett. 12, 355–357 (1968).
[CrossRef]

Yeh, C.

C. Yeh, L. W. Casperson, “Scalar-wave approach for single-mode inhomogeneous fiber problems,” Appl. Phys. Lett. 34, 460–462 (1979).
[CrossRef]

Zucker, H.

H. Zucker, “Optical resonators with variable reflectivity mirrors,” Bell Syst. Tech. J. 49, 2349–2376 (1970), and references therein.

Appl. Opt.

Appl. Phys. Lett.

L. W. Casperson, A. Yariv, “The Gaussian mode in optical resonators with a radial gain profile,” Appl. Phys. Lett. 12, 355–357 (1968).
[CrossRef]

C. Yeh, L. W. Casperson, “Scalar-wave approach for single-mode inhomogeneous fiber problems,” Appl. Phys. Lett. 34, 460–462 (1979).
[CrossRef]

Bell Syst. Tech. J.

G. D. Boyd, J. P. Gordon, “Confocal multimode resonator for millimeter through optical wavelength masers,” Bell Syst. Tech. J. 40, 489–508 (1961).

H. Zucker, “Optical resonators with variable reflectivity mirrors,” Bell Syst. Tech. J. 49, 2349–2376 (1970), and references therein.

J. A. Arnaud, “Nonorthogonal optical waveguides and resonators,” Bell Syst. Tech. J. 49, 2311–2348 (1970).

IEEE J. Quantum Electron.

L. W. Casperson, “Mode stability of lasers and periodic optical systems,” IEEE J. Quantum Electron. QE-10, 629–634 (1974).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

C. C. Su, “On the scalar approximation in fiber optics,” IEEE Trans. Microwave Theory Tech. 36, 1100–1103 (1988).
[CrossRef]

J. Opt. Soc. Am.

Other

J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), especially Subsecs. 2.16, 2.17, and 4.18–4.20.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), especially Subsec. 20.5.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), pp. 131–137.

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

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Equations (89)

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× E = - i ω μ H ,             × H = i ω n 2 μ c 2 E ,
E Re { E ( x , y , z ) e i ω t }
2 E + k 2 n 2 E = 0 ,
n n o ( z ) - ½ n ( T , z ) ,
2 E + ( k 2 n o 2 - k 2 n o n ) E = 0 ,
E A ( T , z ) exp ( - i k n o d z )
T 2 A - 2 i k n o A z - ( i k d n o d z + k 2 n o n ) A = 0 ,
n = n 1 x ( z ) x + n 1 y ( z ) y + n x y ( z ) x y + n 2 x ( z ) x 2 + n 2 y ( z ) y 2 .
T 2 A - 2 i k n o A z - [ i k d n o d z + k 2 n o ( n 1 x x + n 1 y y + n x y x y + n 2 x x 2 + n 2 y y 2 ) ] A = 0.
A ( x , y , z ) = H m [ a x ( z ) x + b x ( z ) ] H n [ a y ( z ) y + b y ( z ) ] × exp { - i [ Q x ( z ) 2 x 2 + Q y ( z ) 2 y 2 + Q x y ( z ) x y + S x ( z ) x + S y ( z ) y + P ( z ) ] } ,
Q x 2 + k n o d Q x d z + k 2 n o n 2 x = 0 ,
Q y 2 + k n o d Q y d z + k 2 n o n 2 y = 0 ,
Q x S x + k n o d S x d z + 1 2 k 2 n o n 1 x = 0 ,
Q y S y + k n o d S y d z + 1 2 k 2 n o n 1 y = 0 ,
2 k n o d P d z + S x 2 + S y 2 + i ( Q x + Q y ) + i k d n o d z = - 2 ( m a x 2 + n a y 2 ) ,
( a x Q x + k n o d a x d z ) ( 1 a x ) = - i a x 2 ,
( a x Q x + k n o d a x d z ) ( - b x a x ) + ( a x S x + k n o d b x d z ) = 0 ,
( a y Q y + k n o d a y d z ) ( 1 a y ) = - i a y 2 ,
( a y Q y + k n o d a y d z ) ( - b y a y ) + ( a y S y + k n o d b y d z ) = 0.
E m , n = E o , m n exp ( k n o i d z - 1 2 Q x i d x a 2 - 1 2 Q y i d y a 2 + P i ) × { ( H m r 2 + H m i 2 ) 1 / 2 exp [ 1 2 Q x i ( x - d x a ) 2 ] } × { ( H n r 2 + H n i 2 ) 1 / 2 exp [ 1 2 Q y i ( y - d y a ) 2 ] } × cos { ( P r + k n o r d z - 1 2 Q x r d x p 2 - 1 2 Q y r d y p 2 + ϕ o , m n - ω t ) + [ 1 2 Q x r ( x - d x p ) 2 - tan - 1 H m i / H m r ] + [ 1 2 Q y r ( y - d y p ) 2 - tan - 1 H n i / H n r ] } ( p 1 i x + p 2 i y ) ,
Q α k n o 1 q α = C α + D α / q o α A α + B α / q o α ,
S α = S o α A α + B α / q o α ,
a α = a o α A α + B α / q o α ( 1 + 2 i a o α 2 k n o B α A α + B α / q o α ) - 1 / 2 ,
b α = ( b o α - a o α S o α k n o B α A α + B α / q o α ) × ( 1 + 2 i a o α 2 k n o B α A α + B α / q o α ) - 1 / 2 ,
P - P o = i 2 [ m ln ( 1 + 2 i a o x 2 k n o B x A x + B x / q o x ) + n ln ( 1 + 2 i a o y 2 k n o B y A y + B y / q o y ) ] - 1 2 k n o ( S o x 2 B x A x + B x / q o x + S o y 2 B y A y + B y / q o y ) - i 2 ln ( A x + B x / q o x ) - i 2 ln ( A y + B y / q o y ) ,
1 A + B / q o d z = B A + B / q o ,
r α + n 2 α ( z ) n o r α = 0
E l p = E o , l p exp ( - i k n o z ) { [ a ( z ) x + b x ( z ) ] 2 + [ a ( z ) y + b y ( z ) ] 2 } l / 2 × ( cos { l tan - 1 [ a ( z ) y + b y ( z ) a ( z ) x + b x ( z ) ] } sin { l tan - 1 [ a ( z ) y + b y ( z ) a ( z ) x + b x ( z ) ] } ) × exp { - i [ Q ( z ) 2 ( x 2 + y 2 ) + S x ( z ) x + S y ( z ) y + P ( z ) ] } × L p l { [ a ( z ) x + b x ( z ) ] 2 + [ a ( z ) y + b y ( z ) ] 2 } ,
P - P o = i 2 [ ( 2 p + l ) ln ( 1 + 2 i a o 2 k n o B A + B / q o ) ] - 1 2 k n o [ ( S o x 2 + S o y 2 ) B A + B / q o ] - i ln ( A + B / q o ) .
E l p = E o , l p exp ( - i k n o z ) a l ( z ) r l { cos l ϕ sin l ϕ } × exp { - i [ Q ( z ) r 2 / 2 + P ( z ) ] } L p l ( a 2 r 2 ) .
E 10 = E o , 10 2 ( a r 2 + a i 2 ) 1 / 2 [ ( x + a r b r + a i b i a r 2 + a i 2 ) 2 + ( a r b i - a i b r a r 2 + a i 2 ) 2 ] 1 / 2 exp [ 1 2 Q x i ( x - d x a ) 2 ] × exp [ 1 2 Q y i ( y - d y a ) 2 ] cos { ( ϕ o , m n - ω t ) + 1 2 Q x r ( x - d x p ) 2 - tan - 1 [ a i a r ( x + b i / a i x + b r / a r ) ] + 1 2 Q y r ( y - d y p ) 2 } ( p 1 i x + p 2 i y )             ( z = z o ) .
E g o = 2 E o , 10 ,
Q k n o = 1 q = 1 R - i 2 k n o w g 2 ,
d a = - S i / Q i ,
d p = - S r / Q r ,
E h o = E g o ( b i a r - b r a i ) ( a r 2 + a i 2 ) 1 / 2 ,
w h = ( a r 2 + a i 2 ) - 1 / 2 ,
d a h = - a r b r + a i b i a r 2 + a i 2 ,
d p o = - b i / a i ,
a i = 1 w h ( 1 + Δ 2 ) 1 / 2 ,
a r = Δ w h ( 1 + Δ 2 ) 1 / 2 ,
b i = - d p o w h ( 1 + Δ 2 ) 1 / 2 ,
b r = ( d p o 1 + Δ 2 - d a h ) ( 1 + Δ 2 ) 1 / 2 Δ w h ,
Δ E g o E h o d a h - d p o w h .
H m + 1 ( x ) = 2 x H m ( x ) - 2 m H m - 1 ( x ) .
H m + 1 ( a x + b ) = 2 [ ( a r x + b r ) H m r ( a x + b ) - ( a i x + b i ) H m i ( a x + b ) - m H m - 1 , r ( a x + b ) ] + 2 i [ ( a r x + b r ) H m i ( a x + b ) - ( a i x + b i ) H m r ( a x + b ) - m H m - 1 , i ( a x + b ) ] ,
a 1 = a 0 1 - i z / z o ( 1 + 2 i a 0 2 k n o z 1 - i z / z o ) - 1 / 2 ,
b 1 = a 1 [ b 0 a 0 ( 1 - i z / z o ) - S 0 z k n o ] ,
a 1 = i a 0 z o z ( 1 - 2 a 0 2 z o k n o ) - 1 / 2             ( z z o ) ,
b 1 = ( b o - i a 0 z o S 0 k n o ) ( 1 - 2 a 0 2 z o k n o ) - 1 / 2             ( z z o ) .
d a h = ( a o i b o r - a o r b o i ) w h 0 2 ( z / z o )             ( z z o ) ,
w h = w h 0 z z o [ 1 - 4 z o k n o ( a 0 r 2 - a 0 i 2 ) + 4 z o 2 w h 0 4 k 2 n o 2 ] 1 / 4             ( z z o ) ,
θ h 2 d d z w h = w h 0 z o [ 1 - 4 z o k n o ( a o r 2 - a o i 2 ) + 4 z o 2 w h o 4 k 2 n o 2 ] 1 / 4             ( z z o ) .
γ ( n 2 / n o ) 1 / 2 ,
a 1 = a 0 [ A 2 + 2 ( 1 q 0 + i a 0 2 k n o ) A B + 1 q 0 ( 1 q 0 + 2 i a 0 2 k n o ) B 2 ] 1 / 2 .
a 1 2 = a 0 2 1 2 [ 1 + 1 q 0 γ 2 ( 1 q 0 + 2 i a 0 2 k n o ) ] + 1 γ [ 1 q 0 + i a 0 2 k n o ] sin 2 γ z + 1 2 [ 1 - 1 γ 2 q 0 ( 1 q 0 + 2 i a 0 2 k n o ) ] cos 2 γ z .
cos 2 ( γ r + i γ i ) z = cos 2 γ r z cosh 2 γ i z - i sin 2 γ r z sinh 2 γ i z ,
sin 2 ( γ r + i γ i ) z = sin 2 γ r z cosh 2 γ i z + i cos 2 γ r z sinh 2 γ i z
p α = π / γ r ,             l α = 1 / ( 2 γ i ) ,
1 q s s = - i a s s 2 k n o = i γ ,
1 q o = 1 R o - i λ m π w o 2 .
1 q = C o + D o / q o A o + B o / q o .
1 q 2 = C 1 + D 1 / q 1 A 1 + B 1 / q 1 .
1 q 2 = ( C 1 D o + D 1 C o ) + ( C 1 B o + D 1 D o ) / q o ( A 1 A o + B 1 C o ) + ( A 1 B o + B 1 D o ) / q o ,
1 q 2 = C + D / q o A + B / q o .
[ A B C D ] = [ A 1 B 1 C 1 D 1 ] [ A 0 B 0 C 0 D 0 ] = [ A 1 A 0 + B 1 C 0 C 1 A 0 + D 1 C 0 A 1 B 0 + B 1 D 0 C 1 B 0 + D 1 D 0 ] .
a 1 = a 0 A 0 + B 0 / q 0 ( 1 + 2 i a 0 2 k n o B 0 A 0 + B 0 / q 0 ) - 1 / 2 ,
a 2 = a 1 A 1 + B 1 / q 1 ( 1 + 2 i a 1 2 k n o B 1 A 1 + B 1 / q 1 ) - 1 / 2 .
a 2 = a o ( A 1 A 0 + B 1 C 0 ) + ( A 1 B 0 + B 1 D 0 ) / q 0 ( 1 + 2 i a 0 2 k n o B o A 0 + B 0 / q 0 ) - 1 / 2 × [ 1 + 2 i a 0 2 k n o B 1 A 0 + B 0 / q 0 1 ( A 1 A 0 + B 1 C 0 ) + ( A 1 B 0 + B 1 D 0 ) / q 0 1 ( 1 + 2 i a 0 2 k n o B 0 A 0 + B 0 / q 0 ) ] - 1 / 2 .
a 2 = a o ( A 1 A 0 + B 1 C 0 ) + ( A 1 B 0 + B 1 D 0 ) / q 0 × [ 1 + 2 i a 0 2 k n o A 1 B 0 + B 1 D 0 ( A 1 A 0 + B 1 C 0 ) + ( A 1 B 0 + B 1 D 0 ) / q 0 ] - 1 / 2 .
a 2 = a 0 A 0 + B 0 / q 0 ( 1 + 2 i a 0 2 k n o B 0 A 0 + B 0 / q 0 ) - 1 / 2 ,
b 1 a 1 = b 0 a 0 ( A 0 + B 0 / q 0 ) - S 0 B 0 k n o ,
b 2 a 2 = b 1 a 1 ( A 1 + B 1 / q 1 ) - S 1 B 1 k n o .
S 1 = S 0 A 0 + B 0 / q 0 .
b 2 a 2 = b 0 a 0 [ ( A 1 A 0 + B 1 C 0 ) + ( A 1 B 0 + B 1 D 0 ) / q 0 ] - S 0 k n o { B 0 [ ( A 1 A 0 + B 1 C 0 ) + ( A 1 B 0 + B 1 D 0 ) / q 0 ] + B 1 A 0 + B 0 / q 0 } .
b 2 a 2 = b 0 a 0 [ ( A 1 A 0 + B 1 C 0 ) - ( A 1 B 0 + B 1 D 0 ) / q 0 ] - S 0 k n o ( A 1 B 0 + B 1 D 0 ) .
b 2 a 2 = b 0 a 0 ( A + B / q 0 ) - S 0 B k n o .
P 1 - P 0 = i 2 [ m ln ( 1 + 2 i a 0 x 2 k n o B 0 x A 0 x + B 0 x / q 0 x ) + n ln ( 1 + 2 i a 0 y 2 k n o + B 0 y A 0 y + B 0 y / q 0 y ) ] - 1 2 k n o ( S 0 x 2 B 0 x A 0 x + B 0 x / q 0 x + S 0 y 2 B 0 y A 0 y + B 0 y / q 0 y ) - i 2 ln ( A 0 x + B 0 x / q 0 x ) - i 2 ln ( A 0 y + B 0 y / q 0 y ) ,
P 2 - P 1 = i 2 [ m ln ( 1 + 2 i a 1 x 2 k n o B 1 x A 1 x + B 1 x / q 1 x ) + n ln ( 1 + 2 i a 1 y 2 k n o + B 1 y A 1 y + B 1 y / q 1 y ) ] - 1 2 k n o ( S 1 x 2 B 1 x A 1 x + B 1 x / q 1 x + S 1 y 2 B 1 y A 1 y + B 1 y / q 1 y ) - i 2 ln ( A 1 x + B 1 x / q 1 x ) - i 2 ln ( A 1 y + B 1 y / q 1 y ) ,
( P 2 - P 0 ) 1 st term = i ( m / 2 ) ln F ,
F ( 1 + 2 i a 0 2 k n o B 0 A 0 + B 0 / q 0 ) ( 1 + 2 i a 1 2 k n o B 1 A 1 + B 1 / q 1 )
F = 1 + 2 i a 0 2 k n o B 0 A 0 + B 0 / q 0 + 2 i a 0 2 k n o × 1 A 0 + B 0 / q 0 B 1 ( A 1 A 0 + B 1 C 0 ) + ( A 1 B 0 + B 1 D 0 ) / q 0 ,
F = 1 + 2 i a 0 2 k n o B A + B / q 0 .
( P 2 - P 0 ) 3 rd term = - 1 2 k n o ( S 0 2 B 0 A 0 + B 0 / q 0 + S 1 2 B 1 A 1 + B 1 / q 1 ) ,
( P 2 - P 0 ) 3 rd term = - 1 2 k n o S 0 2 A 0 + B 0 / q 0 × [ B 0 + B 1 ( A 1 A 0 + B 1 C 0 ) + ( A 1 B 0 + B 1 D 0 ) / q 0 ] ,
( P 2 - P 0 ) 3 rd term = - 1 2 k n o S 0 2 B A + B / q 0 ,
( P 2 - P 0 ) 5 th term = - ( i / 2 ) ln [ ( A 0 + B 0 / q 0 ) ( A 1 + B 1 / q 1 ) ] ,
( P 2 - P 0 ) 5 th term = - ( i / 2 ) ln [ ( A 1 A 0 + B 1 C 0 ) + ( A 1 B 0 + B 1 D 0 ) / q 0 ] ,
( P 2 - P 0 ) 5 th term = - ( i / 2 ) ln ( A + B / q 0 ) ,

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