Abstract

The reverse theorem that governs backward propagation through optical systems represented by transfer matrices is examined for various matrix theories. We extend several reverse theorems to allow for optical systems represented by matrices that may or may not be unimodular and that may be 2 × 2 or take on an augmented 3 × 3 form. As an example, we use the 3 × 3 form of the reverse theorem to study a laser with intracavity misaligned optics. It is shown that, by tilting one of the laser’s mirrors, we can align the laser output arbitrarily, and the mirror tilt angle is calculated.

© 1994 Optical Society of America

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References

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  1. A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, New York, 1975).
  2. H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).
  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, 1562–1569 (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. L. W. Casperson, “Gaussian light beams in inhomogeneous media,” Appl. Opt. 12, 2434–2441 (1973).
    [CrossRef] [PubMed]
  6. L. W. Casperson, S. Lunnam, “Gaussian modes in high loss laser resonators,” Appl. Opt. 14, 1193–1199 (1975).
    [CrossRef] [PubMed]
  7. R. C. Jones, “A new calculus for the treatment of optical systems. I. Description and discussion of the calculus,” J. Opt. Soc. Am. 31, 488–493 (1941).
    [CrossRef]
  8. R. C. Jones, “A new calculus for the treatment of optical systems. II. Proof of three general equivalence theorems,” J. Opt. Soc. Am. 31, 493–499 (1941).
    [CrossRef]
  9. R. C. Jones, “A new calculus for the treatment of optical systems. III. The Sohncke theory of optical activity,” J. Opt. Soc. Am. 31, 500–503 (1941).
    [CrossRef]
  10. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), pp. 66–70.
  11. R. A. Craig, “Method for analysis of the characteristic matrix in optical systems,” J. Opt. Soc. Am. A 4, 1092–1096 (1987).
    [CrossRef]
  12. J. Hong, W. Huang, T. Makino, “On the transfer matrix method for distributed-feedback waveguide devices,” J. Lightwave Technol. 10, 1860–1868 (1992).
    [CrossRef]
  13. S. P. Dijaili, A. Dienes, J. S. Smith, “ABCDmatrices for dispersive pulse propagation,” IEEE J. Quantum Electron. 26, 1158–1164 (1990).
    [CrossRef]
  14. J. L. A. Chilla, O. E. Martinez, “Spatial-temporal analysis of the self-mode-locked Ti:sapphire laser,” J. Opt. Soc. Am. B 10, 638–643 (1993).
    [CrossRef]
  15. J. L. Rosner, “The Smith chart and quantum mechanics,” Am. J. Phys. 61, 310–316 (1993).
    [CrossRef]
  16. R. A. Plastock, G. Kalley, Schaum’s Outline Series, Theory and Problems of Computer Graphics (McGraw-Hill, New York, 1986).
  17. See, for example, B.-G. Kim, E. Garmire, “Comparison between the matrix method and the coupled-wave method in the analysis of Bragg reflector structures,” J. Opt. Soc. Am. A 9, 132–136 (1992).
    [CrossRef]
  18. J. Krasinski, D. F. Heller, Y. B. Band, “Multipass amplifiers using optical circulators,” IEEE J. Quantum Electron. 26, 950–958 (1990).
    [CrossRef]
  19. N. Vansteenkiste, P. Vignolo, A. Aspect, “Optical reversibility theorems for polarization: application to remote control of polarization,” J. Opt. Soc. Am. A 10, 2240–2245 (1993).
    [CrossRef]
  20. P. Yeh, “Extended Jones matrix method,” J. Opt. Soc. Am. 72, 507–513 (1982).
    [CrossRef]
  21. J. Capmany, M. A. Muriel, “A new transfer matrix formalism for the analysis of fiber ring resonators: compound coupled structures for FDMA demultiplexing,” J. Lightwave Technol. 8, 1904–1919 (1990).
    [CrossRef]
  22. L. W. Casperson, “Synthesis of Gaussian beam optical systems,” Appl. Opt. 20, 2243–2249 (1981).
    [CrossRef] [PubMed]
  23. O. E. Martinez, “Matrix formalism for pulse compressors,” IEEE J. Quantum Electron. 24, 2530–2536 (1988).
    [CrossRef]
  24. O. E. Martinez, P. Thiagarajan, M. C. Marconi, J. J. Rocca, “Correction to magnified expansion and compression of subpicosecond pulses from a frequency-doubled Nd:YAG laser,” IEEE J. Quantum Electron. 26, 1676–1679 (1990).
  25. O. E. Martinez, “Matrix formalism for dispersive laser cavities,” IEEE J. Quantum Electron. 25, 296–300 (1989).
    [CrossRef]
  26. L. W. Casperson, “Beam propagation in tapered quadratic-index waveguides: analytical solutions,” J. Lightwave Technol. LT-3, 264–272 (1985).
    [CrossRef]
  27. L. W. Casperson, “Beam propagation in periodic quadratic-index waveguides,” Appl. Opt. 24, 4395–4403 (1985).
    [CrossRef] [PubMed]
  28. P. I. Richards, Manual of Mathematical Physics (Pergamon, New York, 1959).
  29. A. E. Siegman, “Orthogonality properties of optical resonator eigenmodes,” Opt. Commun. 31, 369–373 (1979).
    [CrossRef]
  30. L. W. Casperson, P. M. Scheinert, “Multipass resonators for annular gain lasers,” Opt. Quantum Electron. 13, 193–199 (1981).
    [CrossRef]
  31. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, New York, 1987), pp. 84–91.
  32. L. W. Casperson, “Mode stability of lasers and periodic optical systems,” IEEE J. Quantum Electron. QE-10, 629–634 (1974).
    [CrossRef]
  33. C. Fog, “Synthesis of optical systems,” Appl. Opt. 21, 1530–1531 (1982).
    [CrossRef] [PubMed]
  34. V. Magni, “Multielement stable resonators containing a variable lens,” J. Opt. Soc. Am. A 4, 1962–1969 (1987).
    [CrossRef]

1993 (3)

1992 (2)

J. Hong, W. Huang, T. Makino, “On the transfer matrix method for distributed-feedback waveguide devices,” J. Lightwave Technol. 10, 1860–1868 (1992).
[CrossRef]

See, for example, B.-G. Kim, E. Garmire, “Comparison between the matrix method and the coupled-wave method in the analysis of Bragg reflector structures,” J. Opt. Soc. Am. A 9, 132–136 (1992).
[CrossRef]

1990 (4)

J. Krasinski, D. F. Heller, Y. B. Band, “Multipass amplifiers using optical circulators,” IEEE J. Quantum Electron. 26, 950–958 (1990).
[CrossRef]

S. P. Dijaili, A. Dienes, J. S. Smith, “ABCDmatrices for dispersive pulse propagation,” IEEE J. Quantum Electron. 26, 1158–1164 (1990).
[CrossRef]

J. Capmany, M. A. Muriel, “A new transfer matrix formalism for the analysis of fiber ring resonators: compound coupled structures for FDMA demultiplexing,” J. Lightwave Technol. 8, 1904–1919 (1990).
[CrossRef]

O. E. Martinez, P. Thiagarajan, M. C. Marconi, J. J. Rocca, “Correction to magnified expansion and compression of subpicosecond pulses from a frequency-doubled Nd:YAG laser,” IEEE J. Quantum Electron. 26, 1676–1679 (1990).

1989 (1)

O. E. Martinez, “Matrix formalism for dispersive laser cavities,” IEEE J. Quantum Electron. 25, 296–300 (1989).
[CrossRef]

1988 (1)

O. E. Martinez, “Matrix formalism for pulse compressors,” IEEE J. Quantum Electron. 24, 2530–2536 (1988).
[CrossRef]

1987 (2)

1985 (2)

L. W. Casperson, “Beam propagation in tapered quadratic-index waveguides: analytical solutions,” J. Lightwave Technol. LT-3, 264–272 (1985).
[CrossRef]

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

1982 (2)

1981 (2)

L. W. Casperson, P. M. Scheinert, “Multipass resonators for annular gain lasers,” Opt. Quantum Electron. 13, 193–199 (1981).
[CrossRef]

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

1979 (1)

A. E. Siegman, “Orthogonality properties of optical resonator eigenmodes,” Opt. Commun. 31, 369–373 (1979).
[CrossRef]

1975 (1)

1974 (1)

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

1973 (1)

1968 (1)

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 (2)

H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).

H. Kogelnik, “On the propagation of Gaussian beams of light through lenslike media including those with a loss or gain variation,” Appl. Opt. 4, 1562–1569 (1965).
[CrossRef]

1941 (3)

Aspect, A.

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, New York, 1987), pp. 84–91.

Band, Y. B.

J. Krasinski, D. F. Heller, Y. B. Band, “Multipass amplifiers using optical circulators,” IEEE J. Quantum Electron. 26, 950–958 (1990).
[CrossRef]

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, New York, 1987), pp. 84–91.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), pp. 66–70.

Burch, J. M.

A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, New York, 1975).

Capmany, J.

J. Capmany, M. A. Muriel, “A new transfer matrix formalism for the analysis of fiber ring resonators: compound coupled structures for FDMA demultiplexing,” J. Lightwave Technol. 8, 1904–1919 (1990).
[CrossRef]

Casperson, L. W.

L. W. Casperson, “Beam propagation in tapered quadratic-index waveguides: analytical solutions,” J. Lightwave Technol. LT-3, 264–272 (1985).
[CrossRef]

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]

L. W. Casperson, P. M. Scheinert, “Multipass resonators for annular gain lasers,” Opt. Quantum Electron. 13, 193–199 (1981).
[CrossRef]

L. W. Casperson, S. Lunnam, “Gaussian modes in high loss laser resonators,” Appl. Opt. 14, 1193–1199 (1975).
[CrossRef] [PubMed]

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]

Chilla, J. L. A.

Craig, R. A.

Dienes, A.

S. P. Dijaili, A. Dienes, J. S. Smith, “ABCDmatrices for dispersive pulse propagation,” IEEE J. Quantum Electron. 26, 1158–1164 (1990).
[CrossRef]

Dijaili, S. P.

S. P. Dijaili, A. Dienes, J. S. Smith, “ABCDmatrices for dispersive pulse propagation,” IEEE J. Quantum Electron. 26, 1158–1164 (1990).
[CrossRef]

Fog, C.

Garmire, E.

Gerrard, A.

A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, New York, 1975).

Heller, D. F.

J. Krasinski, D. F. Heller, Y. B. Band, “Multipass amplifiers using optical circulators,” IEEE J. Quantum Electron. 26, 950–958 (1990).
[CrossRef]

Hong, J.

J. Hong, W. Huang, T. Makino, “On the transfer matrix method for distributed-feedback waveguide devices,” J. Lightwave Technol. 10, 1860–1868 (1992).
[CrossRef]

Huang, W.

J. Hong, W. Huang, T. Makino, “On the transfer matrix method for distributed-feedback waveguide devices,” J. Lightwave Technol. 10, 1860–1868 (1992).
[CrossRef]

Jones, R. C.

Kalley, G.

R. A. Plastock, G. Kalley, Schaum’s Outline Series, Theory and Problems of Computer Graphics (McGraw-Hill, New York, 1986).

Kim, B.-G.

Kogelnik, H.

H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).

H. Kogelnik, “On the propagation of Gaussian beams of light through lenslike media including those with a loss or gain variation,” Appl. Opt. 4, 1562–1569 (1965).
[CrossRef]

Krasinski, J.

J. Krasinski, D. F. Heller, Y. B. Band, “Multipass amplifiers using optical circulators,” IEEE J. Quantum Electron. 26, 950–958 (1990).
[CrossRef]

Lunnam, S.

Magni, V.

Makino, T.

J. Hong, W. Huang, T. Makino, “On the transfer matrix method for distributed-feedback waveguide devices,” J. Lightwave Technol. 10, 1860–1868 (1992).
[CrossRef]

Marconi, M. C.

O. E. Martinez, P. Thiagarajan, M. C. Marconi, J. J. Rocca, “Correction to magnified expansion and compression of subpicosecond pulses from a frequency-doubled Nd:YAG laser,” IEEE J. Quantum Electron. 26, 1676–1679 (1990).

Martinez, O. E.

J. L. A. Chilla, O. E. Martinez, “Spatial-temporal analysis of the self-mode-locked Ti:sapphire laser,” J. Opt. Soc. Am. B 10, 638–643 (1993).
[CrossRef]

O. E. Martinez, P. Thiagarajan, M. C. Marconi, J. J. Rocca, “Correction to magnified expansion and compression of subpicosecond pulses from a frequency-doubled Nd:YAG laser,” IEEE J. Quantum Electron. 26, 1676–1679 (1990).

O. E. Martinez, “Matrix formalism for dispersive laser cavities,” IEEE J. Quantum Electron. 25, 296–300 (1989).
[CrossRef]

O. E. Martinez, “Matrix formalism for pulse compressors,” IEEE J. Quantum Electron. 24, 2530–2536 (1988).
[CrossRef]

Muriel, M. A.

J. Capmany, M. A. Muriel, “A new transfer matrix formalism for the analysis of fiber ring resonators: compound coupled structures for FDMA demultiplexing,” J. Lightwave Technol. 8, 1904–1919 (1990).
[CrossRef]

Plastock, R. A.

R. A. Plastock, G. Kalley, Schaum’s Outline Series, Theory and Problems of Computer Graphics (McGraw-Hill, New York, 1986).

Richards, P. I.

P. I. Richards, Manual of Mathematical Physics (Pergamon, New York, 1959).

Rocca, J. J.

O. E. Martinez, P. Thiagarajan, M. C. Marconi, J. J. Rocca, “Correction to magnified expansion and compression of subpicosecond pulses from a frequency-doubled Nd:YAG laser,” IEEE J. Quantum Electron. 26, 1676–1679 (1990).

Rosner, J. L.

J. L. Rosner, “The Smith chart and quantum mechanics,” Am. J. Phys. 61, 310–316 (1993).
[CrossRef]

Scheinert, P. M.

L. W. Casperson, P. M. Scheinert, “Multipass resonators for annular gain lasers,” Opt. Quantum Electron. 13, 193–199 (1981).
[CrossRef]

Siegman, A. E.

A. E. Siegman, “Orthogonality properties of optical resonator eigenmodes,” Opt. Commun. 31, 369–373 (1979).
[CrossRef]

Smith, J. S.

S. P. Dijaili, A. Dienes, J. S. Smith, “ABCDmatrices for dispersive pulse propagation,” IEEE J. Quantum Electron. 26, 1158–1164 (1990).
[CrossRef]

Thiagarajan, P.

O. E. Martinez, P. Thiagarajan, M. C. Marconi, J. J. Rocca, “Correction to magnified expansion and compression of subpicosecond pulses from a frequency-doubled Nd:YAG laser,” IEEE J. Quantum Electron. 26, 1676–1679 (1990).

Vansteenkiste, N.

Vignolo, P.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), pp. 66–70.

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, P.

Am. J. Phys. (1)

J. L. Rosner, “The Smith chart and quantum mechanics,” Am. J. Phys. 61, 310–316 (1993).
[CrossRef]

Appl. Opt. (6)

Appl. Phys. Lett. (1)

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

Bell Syst. Tech. J. (1)

H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).

IEEE J. Quantum Electron. (6)

J. Krasinski, D. F. Heller, Y. B. Band, “Multipass amplifiers using optical circulators,” IEEE J. Quantum Electron. 26, 950–958 (1990).
[CrossRef]

S. P. Dijaili, A. Dienes, J. S. Smith, “ABCDmatrices for dispersive pulse propagation,” IEEE J. Quantum Electron. 26, 1158–1164 (1990).
[CrossRef]

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

O. E. Martinez, “Matrix formalism for pulse compressors,” IEEE J. Quantum Electron. 24, 2530–2536 (1988).
[CrossRef]

O. E. Martinez, P. Thiagarajan, M. C. Marconi, J. J. Rocca, “Correction to magnified expansion and compression of subpicosecond pulses from a frequency-doubled Nd:YAG laser,” IEEE J. Quantum Electron. 26, 1676–1679 (1990).

O. E. Martinez, “Matrix formalism for dispersive laser cavities,” IEEE J. Quantum Electron. 25, 296–300 (1989).
[CrossRef]

J. Lightwave Technol. (3)

L. W. Casperson, “Beam propagation in tapered quadratic-index waveguides: analytical solutions,” J. Lightwave Technol. LT-3, 264–272 (1985).
[CrossRef]

J. Capmany, M. A. Muriel, “A new transfer matrix formalism for the analysis of fiber ring resonators: compound coupled structures for FDMA demultiplexing,” J. Lightwave Technol. 8, 1904–1919 (1990).
[CrossRef]

J. Hong, W. Huang, T. Makino, “On the transfer matrix method for distributed-feedback waveguide devices,” J. Lightwave Technol. 10, 1860–1868 (1992).
[CrossRef]

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (4)

J. Opt. Soc. Am. B (1)

Opt. Commun. (1)

A. E. Siegman, “Orthogonality properties of optical resonator eigenmodes,” Opt. Commun. 31, 369–373 (1979).
[CrossRef]

Opt. Quantum Electron. (1)

L. W. Casperson, P. M. Scheinert, “Multipass resonators for annular gain lasers,” Opt. Quantum Electron. 13, 193–199 (1981).
[CrossRef]

Other (5)

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, New York, 1987), pp. 84–91.

P. I. Richards, Manual of Mathematical Physics (Pergamon, New York, 1959).

R. A. Plastock, G. Kalley, Schaum’s Outline Series, Theory and Problems of Computer Graphics (McGraw-Hill, New York, 1986).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), pp. 66–70.

A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, New York, 1975).

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Figures (1)

Fig. 1
Fig. 1

Schematic demonstration that the reverse of a product of matrices is the product of reverse matrices in reverse order independent of matrix theory, i.e., (T4T3T2T1)R = T1RT2RT3RT4R.

Tables (2)

Tables Icon

Table 1 Physical Interpretation of Some Simple Matrix Operationsa

Tables Icon

Table 2 Reverse Matrices for Several Matrix Theories

Equations (62)

Equations on this page are rendered with MathJax. Learn more.

( X 2 Y 2 ) = [ A 1 B 1 C 1 D 1 ] ( X 1 Y 1 ) ,
( X 3 Y 3 ) = [ A 2 B 2 C 2 D 2 ] ( X 2 Y 2 ) .
( X 3 Y 3 ) = [ A 2 B 2 C 2 D 2 ] [ A 1 B 1 C 1 D 1 ] ( X 1 Y 1 ) .
T system = T n T n - 1 T 3 T 2 T 1 .
[ 1 0 0 1 ] ,
[ 1 χ 0 1 ]
[ 1 0 χ 1 ] ,
[ χ X 0 0 χ Y ] ,
[ cos θ χ sin θ - χ - 1 sin θ cos θ ] ,
[ cos θ - sin θ sin θ cos θ ] .
[ 1 0 0 - 1 ] ,
[ - 1 0 0 1 ] .
[ - 1 0 0 - 1 ] ,
( X 2 Y 2 1 ) = [ A B E C D F 0 0 1 ] ( X 1 Y 1 1 ) .
[ 1 0 χ X 0 1 χ Y 0 0 1 ] ,
[ A B C D ] s = 1 sin θ [ A sin ( s θ ) - sin [ ( s - 1 ) θ ] B sin ( s θ ) C sin ( s θ ) D sin ( s θ ) - sin [ ( s - 1 ) θ ] ] ,
cos θ A + D 2 .
[ cos θ χ sin θ - χ - 1 sin θ cos θ ] s = [ cos ( s θ ) χ sin ( s θ ) - χ - 1 sin ( s θ ) cos ( s θ ) ] .
T R = [ D B C A ] .
T R = [ A C B D ] .
T s = [ 1 0 0 1 ] ,
A + D 2 = cos ( 2 k π / s ) ,
0 k s / 2
[ A B C D ] = [ 1 ( A - 1 ) / C 0 1 ] [ 1 0 C 1 ] [ 1 ( D - 1 ) / C 0 1 ] ,
[ A B C D ] = [ 1 0 ( D - 1 ) / B 1 ] [ 1 B 0 1 ] [ 1 0 ( A - 1 ) / B 1 ]
[ 1 0 0 1 ] = ( [ 1 0 - 3 / l 1 ] [ 1 l 0 1 ] ) 3 = ( ± [ 1 0 0 1 ] ) 2 = ( ± [ 1 0 0 - 1 ] ) 2 = [ 0 γ γ - 1 0 ] 2 = - [ 0 γ - γ - 1 0 ] 2 .
Z X Y ,
Z 2 = A Z 1 + B C Z 1 + D .
A + B Z = A + D 2 ± i [ 1 - ( A + D 2 ) 2 ] 1 / 2
= exp ( ± i θ ) ,
cos θ A + D 2 .
A + B Z > 1 ,
( T s T 2 T 1 ) R = T 1 R T 2 R T s R .
( T s ) R = ( T R ) s .
T R T = ( T T R ) R .
T R Input going backward given Output going backward .
[ X X τ ] 2 = T [ X X τ ] 1 ,
[ X X τ ] 1 = T - 1 [ X X τ ] 2 .
[ 1 0 0 - 1 ] [ X X ( - τ ) ] 1 = T - 1 [ 1 0 0 - 1 ] [ X X ( - τ ) ] 2 .
[ 1 0 0 - 1 ] [ 1 0 0 - 1 ] = [ 1 0 0 1 ]
[ X X ( - τ ) ] 1 = [ 1 0 0 - 1 ] T - 1 [ 1 0 0 - 1 ] [ X X ( - τ ) ] 2 .
T R = [ 1 0 0 - 1 ] T - 1 [ 1 0 0 - 1 ] .
T R = 1 A D - B C [ 1 0 0 - 1 ] [ D - B - C A ] [ 1 0 0 - 1 ] = 1 A D - B C [ D B C A ] .
T R = [ 1 0 0 0 - 1 0 0 0 1 ] T - 1 [ 1 0 0 0 - 1 0 0 0 1 ] = 1 A D - B C [ 1 0 0 0 - 1 0 0 0 1 ] × [ D - B B F - D E - C A C E - A F 0 0 A D - B C ] [ 1 0 0 0 - 1 0 0 0 1 ] = 1 A D - B C [ D B B F - D E C A A F - C E 0 0 A D - B C ] .
T = [ A B E C D F 0 0 1 ] .
[ A x exp ( i ϕ x ) A y exp ( i ϕ y ) 1 ] 2 = T [ A x exp ( i ϕ x ) A y exp ( i ϕ y ) 1 ] 1 ,
[ A x exp ( i ϕ x ) A y exp ( i ϕ y ) 1 ] 1 = T - 1 [ A x exp ( i ϕ x ) A y exp ( i ϕ y ) 1 ] 2 .
[ A x exp [ i ( - ϕ x ) ] A y exp [ i ( - ϕ y ) ] 1 ] 1 * = T - 1 [ A x exp [ i ( - ϕ x ) ] A y exp [ i ( - ϕ y ) ] 1 ] 2 * ,
[ A x exp [ i ( - ϕ x ) ] A y exp [ i ( - ϕ y ) ] 1 ] 1 = ( T - 1 ) * [ A x exp [ i ( - ϕ x ) ] A y exp [ i ( - ϕ y ) ] 1 ] 1 .
T R = ( T - 1 ) * .
T T = ( A D - B C ) [ 0 1 - 1 0 ] - 1 [ A B C D ] - 1 [ 0 1 - 1 0 ] .
[ A + A - 1 ] 2 = T [ A + A - 1 ] 1 .
[ A + A - 1 ] 1 = T - 1 [ A + A - 1 ] 2 .
[ 0 1 0 1 0 0 0 0 1 ] [ A - A + 1 ] 1 = T - 1 [ 0 1 0 1 0 0 0 0 1 ] [ A - A + 1 ] 2 .
[ A - A + 1 ] 1 = [ 0 1 0 1 0 0 0 0 1 ] T - 1 [ 0 1 0 1 0 0 0 0 1 ] [ A - A + 1 ] 2 .
T R = [ 0 1 0 1 0 0 0 0 1 ] T - 1 [ 0 1 0 1 0 0 0 0 1 ] .
T R = 1 A D - B C [ A - C - B D ] .
T mirror = [ 1 0 0 0 1 tan ( 2 θ ) 0 0 1 ] .
T round trip = T system T mirror ( T system ) R
= [ A B E C D F 0 0 1 ] [ 1 0 0 0 1 tan ( 2 θ ) 0 0 1 ] × [ D B B F - D E C A - ( C E - A F ) 0 0 1 ]
= [ A D + B C 2 A B B [ 2 ( A F - C E ) + tan ( 2 θ ) ] 2 C D A D + B C D [ 2 ( A F - C E ) + tan ( 2 θ ) ] 0 0 1 ] ,
tan ( 2 θ ) = 2 ( C E - A F ) .

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