Abstract

A detailed examination of the propagation of Gaussian–Schell model sources in one-dimensional, possibly nonlossless, first-order systems is constructed. The laws of focusing are derived. The conditions for periodicity of the Gaussian–Schell model source are derived. This result generalizes the well-known result −2 ≤ A + D ≤ 2 for confinement of a perfectly coherent Gaussian beam to the partially coherent nonlossless case. When loss or gain is present several conditions must be satisfied simultaneously for periodicity. The self-consistent solutions are derived and the perturbation stability of the solutions is studied. A physical realization of an arbitrary nonlossless one-dimensional ABCD system is derived, which yields a convenient formula for deciding whether the ABCD system has loss or gain. Special attention is devoted to real and ripple systems.

© 1993 Optical Society of America

Full Article  |  PDF Article

Corrections

Mark Kauderer, "Gaussian Schell model sources in one-dimensional first-order systems with loss or gain: errata," Appl. Opt. 32, 3923-3924 (1993)
https://www.osapublishing.org/ao/abstract.cfm?uri=ao-32-21-3923

References

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  1. R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized rays in first-order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
    [CrossRef]
  2. A. T. Friberg, J. Turunen, “Imaging of Gaussian Schell-model sources,” J. Opt. Soc. Am. A 5, 713–720 (1988).
    [CrossRef]
  3. M. Kauderer, “First order sources in first order systems: second order correlations,” Appl. Opt. 30, 1025–1035 (1991); Errata, Appl. Opt. 30, 3788 (1991).
    [CrossRef] [PubMed]
  4. M. Nazarathy, J. Shamir, “First order optics—a canonical operator representation: lossless systems,” J. Opt. Soc. Am. 72, 356–364 (1982); “First order optics: operator representation for systems with loss or gain,” J. Opt. Soc. Am. 72, 1398–1408 (1982).
    [CrossRef]
  5. P. Kramer, M. Moshinsky, T. H. Seligman, “Complex extensions of canonical transformations in quantum mechanics,” in Group Theory and Its Applications, E. M. Loebl, ed. (Academic, New York, 1975), Vol. III.
  6. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
  7. R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD law,” Opt. Commun. 65, 322–328 (1988).
    [CrossRef]
  8. M. Nazarathy, J. Shamir, A. Hardy, “Nonideal phase conjugate resonators—a canonical operator analysis,” J. Opt. Soc. Am. 73, 587–593 (1983).
    [CrossRef]
  9. L. W. Casperson, “Mode stability of lasers and periodic optical systems,” IEEE J. Quantum Electron. QE-10, 629–634 (1974).
    [CrossRef]
  10. V. Magni, “Multielement stable resonators containing a variable lens,” J. Opt. Soc. Am. A 4, 1962–1969 (1987).
    [CrossRef]
  11. M. Kauderer, Symplectic Matrices, First Order Systems, and Special Relativity (World Scientific, Singapore, 1992).

1991 (1)

1988 (2)

A. T. Friberg, J. Turunen, “Imaging of Gaussian Schell-model sources,” J. Opt. Soc. Am. A 5, 713–720 (1988).
[CrossRef]

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD law,” Opt. Commun. 65, 322–328 (1988).
[CrossRef]

1987 (1)

1984 (1)

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized rays in first-order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[CrossRef]

1983 (1)

1982 (1)

1974 (1)

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

Casperson, L. W.

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

Friberg, A. T.

Hardy, A.

Kauderer, M.

Kramer, P.

P. Kramer, M. Moshinsky, T. H. Seligman, “Complex extensions of canonical transformations in quantum mechanics,” in Group Theory and Its Applications, E. M. Loebl, ed. (Academic, New York, 1975), Vol. III.

Magni, V.

Moshinsky, M.

P. Kramer, M. Moshinsky, T. H. Seligman, “Complex extensions of canonical transformations in quantum mechanics,” in Group Theory and Its Applications, E. M. Loebl, ed. (Academic, New York, 1975), Vol. III.

Mukunda, N.

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD law,” Opt. Commun. 65, 322–328 (1988).
[CrossRef]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized rays in first-order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[CrossRef]

Nazarathy, M.

Seligman, T. H.

P. Kramer, M. Moshinsky, T. H. Seligman, “Complex extensions of canonical transformations in quantum mechanics,” in Group Theory and Its Applications, E. M. Loebl, ed. (Academic, New York, 1975), Vol. III.

Shamir, J.

Siegman, A. E.

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

Simon, R.

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD law,” Opt. Commun. 65, 322–328 (1988).
[CrossRef]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized rays in first-order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[CrossRef]

Sudarshan, E. C. G.

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD law,” Opt. Commun. 65, 322–328 (1988).
[CrossRef]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized rays in first-order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[CrossRef]

Turunen, J.

Appl. Opt. (1)

IEEE J. Quantum Electron. (1)

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

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD law,” Opt. Commun. 65, 322–328 (1988).
[CrossRef]

Phys. Rev. A (1)

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized rays in first-order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[CrossRef]

Other (3)

P. Kramer, M. Moshinsky, T. H. Seligman, “Complex extensions of canonical transformations in quantum mechanics,” in Group Theory and Its Applications, E. M. Loebl, ed. (Academic, New York, 1975), Vol. III.

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

M. Kauderer, Symplectic Matrices, First Order Systems, and Special Relativity (World Scientific, Singapore, 1992).

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Figures (3)

Fig. 1
Fig. 1

Focusing problem. For a given ABCD system find the distance z from the output plane of the ABCD system to the beam waist.

Fig. 2
Fig. 2

Example of periodic system. A Gaussian aperture (jcGi, cGi > 0) followed by free space of a distance z1 followed by a Gaussian amplifier (jcLi, cLi < 0) followed by free space of a distance z2.

Fig. 3
Fig. 3

Resonator with curved mirrors, a Gaussian aperture (jcGi, cGi > 0), and a Gaussian amplifier (jcLi, cLi < 0). Distance between the mirrors is z.

Equations (246)

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T { [ A B C D ] } = Q [ D B - 1 ] F [ B - 1 ] Q [ B - 1 A ]             ( B 0 )
= Q [ C A - 1 ] K [ A - 1 ] R [ A - 1 B ]             ( A 0 )
= R [ A C - 1 ] F [ - C ] R [ C - 1 D ]             ( C 0 )
= R [ B D - 1 ] K [ D ] Q [ D - 1 C ] ,             ( D 0 )
Q [ c ] = exp ( j k x 2 c / 2 ) = T { [ 1 0 c 1 ] } ,
R [ d ] u ( x ) 1 ( j λ d ) - 1 / 2 - d x exp [ j k ( x - x ) 2 2 d ] u ( x ) 1 = T { [ 1 d 0 1 ] } u ( x ) 1 ,
F [ b ] u ( x ) 1 [ b / ( j λ ) ] - 1 / 2 - d x exp ( - j k x b x ) u ( x ) 1 = T { [ 0 b - 1 - b 0 ] } u ( x ) 1             ( b 0 ) ,
K [ a ] u ( x ) 1 u ( a x ) 1 a 1 / 2 = T { [ a - 1 0 0 a ] } u ( x ) 1             ( a 0 ) .
A D - B C = 1 ,
A r D r - B r C r - A i D i + B i C i = 1 ,
A r D i + A i D r = B r C i + B i C r .
C r = 1 B 2 [ B i ( A r D i + A i D r ) - B r ( 1 - A r D r + A i D i ) ]             ( B 0 ) ,
C i = 1 B 2 [ B r ( A r D i + A i D r ) + B i ( 1 - A r D r + A i D i ) ]             ( B 0 ) .
M [ A B C D ] = [ A 2 B 2 C 2 D 2 ] [ 1 0 c 1 ] [ A 1 B 1 C 1 D 1 ] = [ A 2 B 2 C 2 D 2 ] [ A 1 B 1 C 1 D 1 ] + c [ B 2 A 1 B 2 B 1 D 2 A 1 D 2 B 1 ] ,
M r = [ A r B r C r D r ] = [ A 2 B 2 C 2 D 2 ] [ A 1 B 1 C 1 D 1 ] ,
M i = [ A i B i C i D i ] = c i [ B 2 A 1 B 2 B 1 D 2 A 1 D 2 B 1 ] = c i ( B 2 D 2 ) ( A 1 , B 1 ) .
M i det ( M i ) = A i D i - B i C i = 0.
M = [ A B C D ] = 1 2 [ 1 i i 1 ]
M = [ A B C D ] = [ 1 0 c 1 ] [ A 2 B 2 C 2 D 2 ] [ 1 0 c 1 ] [ A 1 B 1 C 1 D 1 ] ,
M r = [ A 2 B 2 C 2 D 2 ] [ A 1 B 1 C 1 D 1 ] - B 2 c i c i [ 0 0 A 1 B 1 ] ,
M i = c i [ 0 0 A 2 A 1 + B 2 C 1 A 2 B 1 + B 2 D 1 ] + c i [ B 2 A 1 B 2 B 1 D 2 A 1 D 2 B 1 ] .
A r = A 2 A 1 + B 2 B 1 , B r = A 2 B 1 + B 2 D 1 , D r = C 2 B 1 + D 2 D 1 - c i c i B 2 B 1 ,
A i = c i B 2 A 1 , B i = c i B 2 B 1 , D i = c i ( A 2 B 1 + B 2 D 1 ) + c i D 2 B 1 .
[ A B C D ] = [ 1 0 j - D i A i + B i C i A × B 1 ] [ 0 - A × B 1 A × B - D i A r + B r C i ] × [ 1 0 - j A × B 1 ] [ A i B i - A r A × B - B r A × B ]             ( A × B 0 ) ,
A × B A r B i - A i B r = ( A r , A i ) [ 0 1 - 1 0 ] ( B r B i ) .
T { [ A B C D ] } = Q [ j - D i A i + B i C i A × B ] Q [ D i A r - B r C i A × B ] × F [ - 1 A × B ] Q [ - j A × B ] × T { [ A i B i - A r A × B - B r A × B ] }             ( A × B 0 ) ,
T { [ A i B i - A r A × B - B r A × B ] } ,
T { [ A i B i - A r A × B - B r A × B ] } = R [ - A i A × B A r ] F [ A r A × B ] R [ B r A r ]             ( A × B 0 , A r 0 )
= Q [ - B r B i ( A × B ) ] F [ B i - 1 ] Q [ A i B i ]             ( A × B 0 , B i 0 )
= R [ - B i A × B B r ] K [ - B r A × B ] Q [ A r B r ]             ( A × B 0 , B r 0 ) .
[ A B C D ] = [ 0 1 - 1 0 ] [ - C - D A B ] ,
T { [ A B C D ] } = F T { [ - C - D A B ] } = F Q [ j - D i A i + B i C i C × D ] Q [ - D i A r + B r C i C × D ] × F [ - 1 C × D ] Q [ - j C × D ] T { [ - C i - D i C r C × D D r C × D ] }             ( C × D 0 ) .
T { [ A B C D ] } = T { [ B r B × D B i D r B × D D i ] } Q [ j B × D ] F [ - 1 B × D ] × Q [ - D r A i + B r C i B × D ] Q [ j D i A i - B i C i B × D ]             ( B × D 0 ) .
T { [ A B C D ] } = T { [ - A r A × C - A i - C r A × C - C i ] } Q [ j A × C ] F [ 1 A × C ] × Q [ - D i A r + B i C r A × C ] Q [ j D i A i - B i C i A × C ] F             ( A × C 0 ) .
A × B < 0 , C × D < 0 , B × D > 0 , or A × C > 0 , and D i A i B i C i ,             loss ;
A × B > 0 , C × D > 0 , B × D < 0 , or A × C < 0 , and D i A i B i C i ,             gain ;
D i A i < B i C i ,             mixed
A , B , C , D real ,             lossless .
K [ a ] = F Q [ a - 1 ] F - 1 Q [ a ] F Q [ a - 1 ] .
F [ b ] = Q [ b ] F Q [ b - 1 ] F Q [ b ] ,
[ j a i j b i j c i j d i ] = [ j 0 0 - j ] [ a i b i - c i - d i ]
T [ j a i j b i j c i j d i ] = K [ - j ] T { [ a i b i - c i - d i ] } .
M = [ A B C D ] = [ A B r A r A C r A r A D r A r A ] ,
T { [ A r j B i j C i D r ] } = Q [ j C i A r ] F [ - 1 A r B i ] Q [ - j A r B i ] F [ 1 B i ]             ( A r B i 0 , A B C D ripple )
= K [ - C i ] Q [ j A r C i ] F [ 1 A r C i ] Q [ j - B i A r ]             ( A r C i 0 , A B C D ripple ) ;
A r B i < 0 , C i D r > 0 , B i D r < 0 or A r C i > 0 , and 0 B i C i ,             loss ;
A r B i > 0 , C i D r < 0 , B i D r > 0 , or A r C i < 0 , and 0 B i C i ,             gain ;
0 < B i C i ,             mixed .
R [ d r + j d i ] = F [ - 1 d i ] Q [ - j d i ] T { [ 0 d i - 1 d i - d r d i ] } ,             d i > 0 , gain ; d i < 0 , loss ;
T { 1 2 [ 1 j j 1 ] } = Q [ j ] F [ - 2 ] Q [ - 2 j ] F [ 2 ] ,             mixed .
T { [ j A i B r C r j D i ] } = Q [ j D i B r ] F [ 1 A i B r ] Q [ j A i B r ] K [ 1 A i ]             ( A i B r 0 ) ,
A i B r > 0 and D i A i 0 loss ; A i B r < 0 and D i A i 0 gain ; D i A i < 0 mixed .
[ A B C D ] = [ cos [ ( a r + j a i ) d ] sin [ ( a r + j a i ) d ] n 0 ( a r + j a i ) - n 0 ( a r + j a i ) sin [ ( a r + j a i ) d ] cos [ ( a r + j a i ) d ] ] .
cos ( a r + j a i ) = cos ( a r ) cos ( j a i ) - sin ( a r ) sin ( j a i ) = cos ( a r ) cosh ( a i ) - j sin ( a r ) sinh ( a i ) , sin ( a r + j a i ) = sin ( a r ) cos ( j a i ) + cos ( a r ) sin ( j a i ) = sin ( a r ) cosh ( a i ) + j cos ( a r ) sinh ( a i ) .
E F = E F * F F * = E · F - j E F F · F
A r = D r = cos ( d a r ) cosh ( d a i ) , A i = D i = - sin ( d a r ) sinh ( d a i ) , B r = sin ( d a r ) cosh ( d a i ) a r + cos ( d a r ) sinh ( d a i ) a i n 0 ( a r 2 + a i 2 ) , B i = - sin ( d a r ) cosh ( d a i ) a i + cos ( d a r ) sinh ( d a i ) a r n 0 ( a r 2 + a i 2 ) , C r = - n 0 [ a r sin ( d a r ) cosh ( d a i ) - a i cos ( d a r ) sinh ( d a i ) ] , C i = - n 0 [ a r cos ( d a r ) sinh ( d a i ) + a i sin ( d a r ) cosh ( d a i ) ] ,
A × B = - sin ( 2 d a r ) a i + sinh ( 2 d a i ) a r 2 n 0 ( a r 2 + a i 2 ) , - A i D i + B i C i = - a r 2 sinh 2 ( d a i ) - a i 2 sin 2 ( d a r ) a r 2 + a i 2 , A r D i - B r C i = a r a i [ cos 2 ( d a r ) sinh 2 ( d a i ) + sin 2 ( d a r ) cosh 2 ( d a i ) ] a r 2 + a i 2 .
T { [ cos [ ( a r + j a i ) d ] sin [ ( a r + j a i ) d ] n 0 ( a r + j a i ) - n 0 ( a r + j a i ) sin [ ( a r + j a i ) d ] cos [ ( a r + j a i ) d ] ] } = Q [ j - 2 n 0 [ a r 2 sinh 2 ( d a i ) + a i 2 sin 2 ( d a r ) ] - sin ( 2 d a r ) a i + sinh ( 2 d a i ) a r ] Q [ 2 n 0 a r a i [ cos 2 ( d a r ) sinh 2 ( d a i ) + sin 2 ( d a r ) cosh 2 ( d a i ) ] - sin ( 2 d a r ) a i + sinh ( 2 d a i ) a r ] F [ - 2 n 0 ( a r 2 + a i 2 ) - sin ( 2 d a r ) a i + sinh ( 2 d a i ) a r ] Q [ j - 2 n 0 ( a r 2 + a i 2 ) - sin ( 2 d a r ) a i + sinh ( 2 d a i ) a r ] R [ tan ( d a r ) tanh ( d a i ) - sin ( 2 d a r ) a i + sinh ( 2 d a i ) a r 2 n 0 ( a r 2 + a i 2 ) ] F [ cos ( d a r ) cosh ( d a i ) 2 n 0 ( a r 2 + a i 2 ) - sin ( 2 d a r ) a i + sinh ( 2 d a i ) a r ] R [ sin ( d a r ) cosh ( d a i ) a r + cos ( d a r ) sinh ( d a i ) a i n 0 ( a r 2 + a i 2 ) cos ( d a r ) cosh ( d a i ) ] ( for A × B 0 , A r 0 ) .
- sin ( 2 d a r ) a i + sinh ( 2 d a i ) a r < 0 , loss ; - sin ( 2 d a r ) a i + sinh ( 2 d a i ) a r > 0 , gain .
c ( z ) = ( C + D c ) ( A + B c ) * - D B * r i 2 A + B c 2 - B r i 2 = C · A + c r ( A · D + C · B ) - c i ( A D + C B ) + ( D · B ) ( c · c - r i 2 ) A + B c 2 - B r i 2 - j C × A + c r ( C × B - A × D ) + c i ( C · B - A · D ) + ( D × B ) ( c · c - r i 2 ) A + B c 2 - B r i 2 ,
r ( z ) = r A + B c 2 - B r i 2 ,
γ ( z ) A + B c 2 - B r i 2 ,
c f r = 0             ( condition for beam waist or focusing ) .
[ A B C D ] = [ 1 z 0 1 ] [ A B C D ] = [ A + C z B + z D C D ] ,
c f = ( C + D c ) ( A + B c ) * - D B * r i 2 + z ( C + D c 2 - D 2 r i 2 ) A + B c 2 - B r i 2 + 2 z [ ( A + B c ) · ( C + D c ) - ( B · D ) r i 2 ] + z 2 ( C + D c 2 - D 2 r i 2 ) .
z = - ( C + D c ) · ( A + B c ) - ( D · B ) r i 2 C + D c 2 - D 2 r i 2 = - C · A + c r ( A · D + C · B ) - c i ( A × D + C × B ) + ( D · B ) ( c · c - r i 2 ) C + D c 2 - D 2 r i 2 .
c f i = - ( C + D c ) × ( A + B c ) + ( D × B ) r i 2 A + B c 2 - B r i 2 + 2 z [ ( A + B c ) · ( C + D c ) - ( B · D ) r i 2 ] + z 2 ( C + D c 2 - D 2 r i 2 ) = - [ C × A + c r ( C × B - A × D ) + c i ( C · B - A · D ) + ( D × B ) ( c · c - r i 2 ) ] γ f ,
r f i = r i γ f ,
γ f = ( C + D c 2 - D 2 r i 2 ) - 1 × { ( A + B c 2 - B r i 2 ) ( C + D c 2 - D 2 r i 2 ) - [ ( C + D c ) · ( A + B c ) - ( D · B ) r i 2 ] 2 } = ( C + D c 2 - D 2 r i 2 ) - 1 × [ [ ( A + B c ) × ( C + D c ) ] 2 + r i 4 ( D × B ) 2 + r i 2 ( - 1 - 2 { [ D r A r - B r C r + c i ( B × D ) ] 2 + [ B r C i - D r A i + c r ( B × D ) ] 2 - [ D r A r - B r C r + c i ( B × D ) ] } ) ] .
z = - ( C + D c r ) ( A + B c r ) + D B ( c i 2 - r i 2 ) C + D c 2 - D 2 r i 2 = - C A + c r ( A D + C B ) + D B ( c · c - r i 2 ) C 2 + 2 C D c r + D 2 ( c · c - r i 2 )             ( A B C D real ) ,
γ f = c i 2 - r i 2 C + D c 2 - D 2 r i 2             ( A B C D real ) ,
c f i = c i γ f = C 2 + 2 C D c r + D 2 { c r 2 + c i 2 [ 1 - ( r i / c i ) 2 ] } c i [ 1 - ( r i / c i ) 2 ]             ( A B C D real ) ,
r f i = r i γ f = C 2 + 2 C D c r + D 2 { c r 2 + c i 2 [ 1 - ( r i / c i ) 2 ] } r i [ ( c i / r i ) 2 - 1 ]             ( A B C D real ) .
c f i 2 - r f i 2 = c i 2 - r i 2 ( γ f ) 2 = [ C 2 + 2 C D c r + D 2 ( c r 2 + c i 2 - r i 2 ) ] 2             ( A B C D real ) ,
b f = b b 2 C 2 + 2 C D B 2 c r + D 2 ( b 2 c r 2 + 1 )             ( A B C D real ) ,
b f = b b 2 C 2 + D 2 .
z = - C A b 2 + D B C 2 b 2 + D 2
z = - c r C i 2 - 2 C i D c r + D 2 { c i 2 + c r 2 [ 1 - ( r i / c r ) 2 ] }             ( A B C D ripple ) ,
c f i = c i γ f = - D B i { c i 2 + c r 2 [ 1 - ( r i / c r ) 2 ] } - A C i + c i ( B i C i - A D ) γ f             ( A B C D ripple ) ,
r f i = r i γ f             ( A B C D ripple ) ,
γ f = ( C i 2 - 2 C i D c r + D 2 { c i 2 + c r 2 [ 1 - ( r i / c r ) 2 ] } ) - 1 × ( [ A C i + c i ( A D - B i C i ) - D B i ( c r 2 + c i 2 ) ] 2 + r i 4 D 2 B i 2 + r i 2 { - 1 - 2 [ ( D A - c i D B i ) 2 + ( c r B i D ) 2 - ( D A - c i B i D ) ] } ) ( A B C D ripple ) .
W ( x 1 , x 2 ; z ) = w ( x 1 ; z ) w * ( x 2 ; z ) = exp [ j k ( ½ x 1 2 c - ½ x 2 2 c * + r x 1 x 2 ) ] ,
c = c r + i c i , r = i r i ,
I ( x ; z ) W ( x , x ; z ) = exp [ j k ½ x 2 ( c - c * + 2 i r i ) ] = exp [ - k x 2 ( c i + r i ) ] = Q x [ c I ] ,
c I 2 i ( c i + r i ) .
c i + r i > 0             ( finite energy condition ) .
u ( x 1 , x 2 ; z ) W ( x 1 , x 2 ; z ) [ I ( x 1 ; z ) I ( x 2 ; z ) ] 1 / 2 = exp [ j k ( ½ x 1 2 c c - ½ x 2 2 c c * + r x 1 x 2 ) ] ,
c c c - c I / 2 = c r - i r i , r = i r i .
Re [ u ( x 1 , x 2 ; z ) ] = exp [ k ( ½ x 1 2 r i - ½ x 2 2 r i - r i x 1 x 2 ) ] = exp [ k ( x 1 - x 2 ) 2 r i ] .
r i 0             ( finite coherence condition ) ,
c = c I / 2 + c c , - Im ( c c ) = r i ,
c ( z ) = c ( 0 ) = c ; r ( z ) = r ( 0 ) = r .
A 2 + B 2 ( c 2 - r 2 ) + 2 Re ( A B * c * ) = 1 ,
( C + D c ) ( A + B c ) * - D B * r i 2 = c .
( A + B c ) ( A + B c ) * - 1 = B B * r i 2 .
( C + D c ) ( A + B c ) * - D B - 1 [ ( A + B c ) ( A + B c ) * - 1 ] = c ,             for B 0.
2 Re ( B c ) D - A * = D r - A r .
D i = - A i .
c i > 0 , c i - r i c i > 0 , c i 2 - r i 2 > 0.
0 B 2 ( c i 2 - r i 2 ) = 1 - A 2 - B 2 c r 2 - 2 A r Re ( B c ) - 2 A i ( B r c i + B i c r ) .
- B 2 c r 2 - C r B r + C i B i - 2 A i ( B r c i + B i c r ) 0.
c i = B r c r B i + A r - D r 2 B i             ( B i 0 ) ,
- c r 2 - 2 c r A i B i + C i B i 0             ( B i 0 ) .
A i 2 + B i C i 0             ( B i 0 ) .
c r ( - ) c r c r ( + )             ( B i 0 ) ,
c r ( ± ) - A i B i ± [ ( A i B i ) 2 + C i B i ] 1 / 2 .
- B r 2 c r 2 - C r B r - 2 A i B r c i 0             ( B i = 0 ) ,
( A r + D r 2 ) 2 1 - 2 A i B r c i - A i 2 ( B i = 0 , B r 0 ) ,
c i 1 - A i 2 - ( A r + D r 2 ) 2 2 A i B r ,             if A i B r > 0 , B i = 0 ;
c i 1 - A i 2 - ( A r + D r 2 ) 2 2 A i B r ,             if A i B r < 0 , B i = 0.
( A r + D r 2 ) 2 1 ,             if A i = B i = 0 , B r 0.
c i = 1 B i ( B r c r + A r - D r 2 )             ( B r 0 , B i 0 ) .
c i ( - ) c i c i ( + ) ,
c i ( ± ) A r - D r 2 B i - A i B r B i 2 ± [ B r 2 B i 4 ( A i 2 + C i B i ) ] 1 / 2 .
c r = D r - A r 2 B r             ( A B C D real ) ,
( A r + D r 2 ) 2 1             ( A B C D real ) ,
1 = ( A r + B r c r ) 2 + B r 2 ( c i 2 - r i 2 )             ( A B C D real )
c i 2 - r i 2 = 1 - ( A r + B r c r ) 2 B r 2 = 1 - ( A r + D r ) 2 / 4 B r 2             ( A B C D real ) .
c = D r - A r 2 B r + j 1 B r [ 1 - ( A r + D r 2 ) 2 ] 1 / 2             ( A B C D real ) .
c i = A r - D r 2 B i             ( A B C D ripple ) ,
B i C i 0 or 1 A r D r             ( A B C D ripple ) ,
- ( C i B i ) 1 / 2 c r ( C i B i ) 1 / 2             ( A B C D ripple ) .
1 - ( A r - B i c i ) 2 + B i 2 ( c r 2 - r i 2 )             ( A B C D ripple ) ,
c r 2 - r i 2 = 1 - ( A r - B i c i ) 2 B i 2 = 1 - ( A r + D r ) 2 / 4 B i 2             ( A B C D ripple ) .
c = [ 1 - ( A r + D r ) 2 / 4 B i 2 ] 1 / 2 + j A r - D r 2 B i .             ( A B C D ripple )
A 2 - B 2 r i 2 - 1 + B 2 ( c r 2 + c i 2 ) + A B * ( c r - i c i ) + A * B ( c r + i c i ) = 0.
c i 2 B 4 + c i [ B 2 B i ( D r - A r ) + 2 B r B i ( A · B ) - 2 B r 2 ( A × B ) ] + [ B r 2 ( A 2 - B 2 r i 2 - 1 ) + B 2 ( D r - A r 2 ) 2 + B r ( D r - A r ) ( A · B ) ] = 0.
c i 2 B 4 + c i B 2 [ B i ( D r - A r ) + 2 B r A i ] + [ B r 2 ( A 2 - B 2 r i 2 - 1 ) + B 2 ( D r - A r 2 ) 2 + B r ( D r - A r ) ( A · B ) ] = 0.
c r 2 B 4 + c r B 2 [ - B r ( D r - A r ) + 2 B i A i ] + [ B i 2 ( A 2 - B 2 r i 2 - 1 ) + B 2 ( D r - A r 2 ) 2 + B i ( D r - A r ) ( A × B ) ] = 0.
c i - c r , B r - B i , B i B r .
Re ( A B * ) Im ( A B * ) ,             Im ( A B * ) - Re ( A B * ) ,             A · B - A × B , A × B A · B .
c i - c r , c r c i , B r - B i , B i B r , r i r i , A A .
[ A B C D ] [ 1 0 0 - i ] [ A B C D ] [ 1 0 0 i ] = [ A i B - i C D ] , ( c r c i ) [ 0 1 - 1 0 ] ( c r c i ) = ( c i - c r ) .
4 B 4 B r 2 [ 1 + B 2 r i 2 - ( D r + A r 2 ) 2 ] 0 ,
4 B 4 B i 2 [ 1 + B 2 r i 2 - ( D r + A r 2 ) 2 ] 0.
r i 2 1 B 2 [ ( D r + A r 2 ) 2 - 1 ]             for B 0 ,
B = 0.
1 B 2 [ ( D r + A r 2 ) 2 - 1 ] r i 2             ( B 0 )
c i = 1 B 2 { - B i D r - A r 2 - B r A i ± B r [ 1 + B 2 r i 2 - ( D r + A r 2 ) 2 ] 1 / 2 } ,             for B 0 ,
c r = 1 B 2 { B r D r - A r 2 - B i A i ± B i [ 1 + B 2 r i 2 - ( D r + A r 2 ) 2 ] 1 / 2 } ,             for B 0.
M r Re [ A B C D ] = [ A r B r C r D r ]
c = 1 B ( T r ( M r ) 2 - A ± i { 1 + B 2 r i 2 - [ T r ( M r ) 2 ] 2 } 1 / 2 ) ,             for B 0.
c = 1 B ( T r ( M ) 2 - A ± i { 1 - [ T r ( M ) 2 ] 2 } 1 / 2 ) ,             for B 0.
c i = ± [ 1 + B r 2 r i 2 - ( D r + A r 2 ) 2 ] 1 / 2 / B r c i Re             ( A B C D real ) ,
c r = D r - A r 2 B r c r Re             ( A B C D real ) .
c i = A r - D r 2 B i c i Ri             ( A B C D ripple ) ,
c r = ± [ 1 + B i 2 r i 2 - ( D r + A r 2 ) 2 ] 1 / 2 / B i c r Ri             ( A B C D ripple ) .
C A * + D A * c = c ,             for B = 0.
0 = A i ( D r - A r ) , 1 = A r D r + A i 2 ,             for B = 0.
- c r ,             0 c i ,             for A i = B = 0             ( A B C D = ± I ) .
C r A r + C i A i = 2 A i ( c r A i - c i A r ) , C i A r - C r A i = 2 A i ( c r A r + c i A i ) ,             for A i 0 , B = 0.
c i = - C r 2 A i , c r = C i 2 A i ,             for A i 0 , B = 0.
B B = C C = D - A D - A
c + δ = [ C + D ( c + δ ) ] [ A + B ( c + δ ) ] * - D B * ( r i + ) 2 A + B ( c + δ ) 2 - B 2 ( r i + ) 2 ,
r i + = r i + A + B ( c + δ ) 2 - B 2 ( r i + ) 2 ,
δ = δ ( A + B c ) * 2 + δ * B * 2 r i 2 - 2 r i B * ( A + B c ) * ( A + B c 2 - B 2 r i 2 ) 2 ,
= ( A + B c ) 2 + B 2 r i 2 ) - 2 r i Re [ δ B ( A + B c ) * ] ( A + B c 2 - B 2 r i 2 ) 2 .
A + B c = T r ( M r ) 2 ± i { 1 + B 2 r i 2 - [ T r ( M r ) 2 ] 2 } 1 / 2             ( B 0 ) ,
A + B c 2 = 1 + B 2 r i 2             ( B 0 ) ,
δ = δ ( A + B c ) * 2 + δ * B * 2 r i 2 - 2 r i B * ( A + B c ) * ,
= ( 1 + 2 B 2 r i 2 ) - 2 r i Re [ δ B ( A + B c ) * ] .
δ = δ ( A + B c ) 2 , = A + B c 2             ( r i = 0 ) ,
δ = δ ( A + B c ) * 2 + δ * B * 2 r i 2 , = - 2 r i Re [ δ B ( A + B c ) * ]             ( = 0 ) .
δ = - 2 r i B * ( A + B c ) * , = ( 1 + 2 B 2 r i 2 )             ( δ = 0 ) .
δ 2 = δ 2 ( A + B c 4 + B 4 r i 4 ) + 4 r i 2 B 2 A + B c 2 - 4 r i Re [ δ B ( A + B c ) * ] × ( B 2 r i 2 + A + B c 2 ) + 2 r i 2 Re [ δ 2 B 2 ( A + B c ) * 2 ] ,
2 = 2 ( A + B c 4 + B 4 r i 4 + 2 A + B c 2 B 2 r i 2 ) + 4 r i 2 { Re [ δ B ( A + B c ) * ] } 2 - 4 r i Re [ δ B ( A + B c ) * ] × ( B 2 r i 2 + A + B c 2 ) .
δ 2 - 2 = ( δ 2 - 2 ) ( - 2 A + B c 2 B 2 r i 2 ) ,
δ 2 - 2 δ 2 - 2 = - 2 ( 1 + B 2 r i 2 ) B 2 r i 2 .
c i r i = c i + δ i r i + = c i + δ i r i + = const .             ( A B C D real ) ,
= δ i r i c i = δ i δ i             ( A B C D real ) ,
δ 2 - 2 = δ r 2 + δ i 2 [ 1 - ( r i / c i ) 2 ] = { δ r 2 + δ i 2 [ 1 - ( r i / c i ) 2 ] } × [ - 2 ( 1 + B 2 r i 2 ) B 2 r i 2 ]             ( A B C D real ) .
r i 2 3 - 1 2 B 2             ( condition for perturbation stability for real systems ) .
c r r i = c r + δ r r i + = c r + δ r r i + = const .             ( A B C D ripple ) ,
= δ r r i c r = δ r δ r             ( A B C D ripple ) ,
δ 2 - 2 = δ r 2 1 - ( r i / c r ) 2 + δ i 2 = [ δ r 2 1 - ( r i / c r ) 2 + δ i 2 ] × [ - 2 ( 1 + B 2 r i 2 ) B 2 r i 2 ]             ( A B C D ripple ) .
r i 2 3 - 1 2 B i 2             ( condition for perturbation stability for ripple systems ) .
r i 2 = 3 - 1 2 B 2 .
1 B i 2 [ ( D r + A r 2 ) 2 - 1 ] r i 2 3 - 1 2 B i 2             ( B 0 ,             A B C D ripple ) .
[ Tr ( M ) 2 ] 2 3 + 1 2             ( B 0 ,             A B C D ripple ) .
T { [ A B C D ] } = Q [ j A 1 2 + B i C i A × B ] Q [ - A i A r - B r C i A × B ] × F [ - 1 A × B ] Q [ - j A × B ] × T { [ A i B i - A r A × B - B r A × B ] }             ( A × B 0 ) .
T = R [ z 2 ] Q [ c L ] R [ z 1 ] Q [ c G ] [ 1 - z 1 z 2 c Li c Gi + j [ z 1 c Gi + z 2 ( c Gi + c Li ) ] z 1 + z 2 + j z 1 z 2 c Li - z 1 c Li c Gi + j ( c Gi + c Li ) 1 + j z 1 c Li ] .
- c Li = c Gi
M rt [ A B C D ] [ D B C A ] [ A B C D ] = [ D A + B C 2 D B 2 A C D A + B C ] .
( D A + B C ) 2 = ( D A - B C ) 2 + 4 A B C D = 1 + 4 A B C D ,
[ Re ( D A + B C ) ] 2 - [ Im ( D A + B C ) ] 2 = 1 + 4 [ ( A D ) r ( B C ) r - ( A D ) i ( B C ) i ] ,
Re ( D A + B C ) Im ( D A + B C ) = 2 [ ( A D ) r ( B C ) i + ( A D ) i ( B C ) r ] .
Tr [ Im ( M rt ) ] = A i + D i = 0             iff ( A D ) i = 0 iff ( B C ) i = 0 ,
( D A + B C ) 2 = 1 + 4 ( A D ) r ( B C ) r + 4 ( A D ) i 2 , 0 = ( A D ) i [ ( B C ) r - ( A D ) r ] .
4 ( D A ) r 2 = 1 + 4 ( A D ) r 2 + 4 ( A D ) i 2 ,
[ ( D A ) r + ( B C ) r ] 2 - ( B C ) i 2 = 1 + 4 ( A D ) r ( B C ) r , [ ( D A ) r + ( B C ) r ] ( B C ) i = 2 ( A D ) r ( B C ) i ,
( B C ) i [ ( B C ) r - ( A D ) r ] = 0.
[ T r ( M rt ) / 2 ] 2 1
1 = A 2 - B C = A 2 - ( B C ) r ; 0 = ( B C ) i = B r C i + B i C r .
( B C ) r = B r C r - B i C i = C r B r B 2 .
( A D ) i = 0 or ( B C ) i = 0 ,
( B D ) i ( A C ) i 0
T { [ A B C D ] } = Q [ j C i A r ] Q [ - B r C i A r B i ] F [ - 1 A r B i ] × Q [ - j A r B i ] Q [ - B r A r ( B i ) 2 ] F [ 1 B i ] ( A r B i 0 , linear resonator ) .
c = 1 B [ ± i ( 1 + B 2 r i 2 - A 2 ) 1 / 2 ] = 1 2 B D { ± i [ 1 + 4 B D 2 r i 2 - ( A D + B C ) 2 ] 1 / 2 } .             ( B = 2 B D 0 , linear resonator ) .
T = Q [ - 1 R + j c Li ] R [ z ] Q [ - 1 R + j c Gi ] [ 1 - z R + j z c Gi z - 1 R - 1 R + z R R - z c Li c Gi + j [ c Li + c Gi - z ( c Li R + c Gi R ) ] 1 - z R + j z c Li ] .
c Gi + c Li - z ( c Li R + c Gi R ) = 0 ,
z = c Li + c Gi c Li R + c Gi R = R R ( c Li + c Gi ) c Li R + c Gi R > 0.
B i = 2 ( B D ) i = 2 B r D i = 2 z 2 c Li < 0 ,
( A C ) i = A i C r = z c Gi ( - 1 R - 1 R + z R R - z c Li c Gi ) .
- 1 R - 1 R + z R R - z c Li c Gi 0 ,
z ( 1 R R - c Li c Gi ) 1 R + 1 R = R + R R R ,
c Li + c Gi c Li R + c Gi R ( 1 - R R c Li c Gi ) R + R R R .
0 < - c Gi c Li c Li R 2 ( c Gi + c Li )             ( R = ) ,
c Li > c Gi             ( R = ) .
z = R ( c Li + c Gi ) c Li > 0             ( R = ) ,
R > 0 ,             ( R = ) ,
0 < - c Gi c Li c Gi R 2 ( c Gi + c Li )             ( R = ) ,
c Li < c Gi             ( R = ) .
z = R ( c Li + c Gi ) c Gi > 0             ( R = ) ,
R > 0             ( R = ) .
R = 1 , z = 1 / 30 , c Li = - 3 , c Gi = 2.9 ,
R = 1 / 100 , z = 1 / 200 , c Li = - 4 , c Gi = 2.
U V * = ( U r + j U i ) ( V r - j V i ) = U r V r + U i V i - j ( U r V i - U i V r ) U · V - j U × V ,
U · V U r V r + U i V i = ( U r , U i ) ( V r V i ) = ( U r , U i ) [ 1 0 0 1 ] [ V r V i ] ,
U × V U r V i - U i V r = ( U r , U i ) [ 0 1 - 1 0 ] ( V r V i ) .
U · U = U 2 .
U × V = - V × U ,
U × U = 0.
U + V 2 = U 2 + V 2 + 2 U · V .
a ( U · V ) = ( a U ) · V = U · ( a V )             ( a real ) ,
a ( U × V ) = ( a U ) × V = U × ( a V )             ( a real )
( j U ) · V = - U · ( j V ) = U × V ,
( j U ) × V = - U × ( j V ) = - U . V .
( U + V ) · W = U · W + V · W ,
( U + V ) × W = U × W + V × W .
U · ( V W ) = ( U · V ) W r - ( U × V ) W i ,
( V W ) · U = ( V · U ) W r + ( V × U ) W i ,
U × ( V W ) = ( U × V ) W r + ( U · V ) W i ,
( V W ) × U = ( V × U ) W r - ( V · U ) W i .
( U Z ) · ( V W ) = [ ( U Z ) · V ] W r - [ ( U Z ) × V ] W i = [ ( U · V ) Z r + ( U × V ) Z i ] W r - [ ( U × V ) Z r - ( U · V ) Z i ] W i = ( U · V ) ( Z · W ) - ( U × V ) ( Z × W ) .
( U Z ) × ( V W ) = [ j ( U Z ) ] · ( V W ) = [ ( j U ) Z ] · ( V W ) = [ ( j U ) · V ] ( Z · W ) - [ ( j U ) × V ] ( Z × W ) = ( U × V ) ( Z · W ) + ( U · V ) ( Z × W ) .
( U W ) · ( V W ) = ( U · V ) ( W · W ) ,
( U W ) × ( V W ) = ( U × V ) ( W · W ) .
V i U × V + V r U · V = U r V 2 ,
V i U · V - V r U × V = U i V 2 .
U V 2 = U 2 V 2 = ( U · V ) 2 + ( U × V ) 2 .
D A 2 + B C 2 - 2 ( D · B ) ( C · A ) = 2 [ ( D r A r - B r C r ) 2 + ( B r C i - D r A i ) 2 - ( D r A r - B r C r ) ] + 1 ,
- ( D · C ) ( A × B ) + ( A · B ) ( D × C ) - ( D · A + B · C ) ( B r C i - D i A r ) = ( - D r A i + B r C i ) { 2 ( D r A r - B r C r ) - 1 } ,
B 2 D × C - D 2 A × B - 2 ( D · B ) ( B r C i - D i A r ) = ( B × D ) { 2 ( D r A r - B r C r ) - 1 } ,
- C 2 A × B + A 2 D × C - 2 ( C · A ) ( B r C i - D i A r ) = - ( C × A ) { 2 ( D r A r - B r C r ) - 1 } ,
C 2 B · A + A 2 D · C - ( C · A ) ( D · A + B · C ) = 2 ( A × C ) ( - A i D r + C i B r ) ,
B 2 D · C + D 2 B · A - ( D · B ) ( D · A + B · C ) = 2 ( B × D ) ( - A i D r + C i B r ) .

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