Abstract

A novel 3 × 3 transfer-matrix method is developed to propagate off-axis Gaussian beams in astigmatic optical systems that may include tilted, displaced, or curved optical elements. Unlike in a previous generalized ray matrix formalism, optical elements that possess gain or loss such as Gaussian apertures, complex lenslike media, and amplifiers are included; and a new beam transformation is found. In addition, a novel exponential variable-reflectivity mirror, which displaces a Gaussian beam without changing its spot size, and a complex prismlike medium are introduced.

© 1995 Optical Society of America

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  1. A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, New York, 1975), pp. 24–26.
  2. A. Cayley, “A memoir on the theory of matrices,” Philos. Trans. R. Soc. London CXLVIII, 17–37 (1858); also published in The Collected Mathematical Papers of Arthur Cayley (Cambridge U. Press, Cambridge, 1889), Vol. 2, pp. 475–496.
  3. K. Halbach, “Matrix representation of Gaussian optics,” Am. J. Phys. 32, 90–108 (1964).
    [CrossRef]
  4. J. A. Arnaud, “Degenerate optical cavities. II. Effects of misalignments,” Appl. Opt. 8, 1909–1917 (1969).
    [CrossRef] [PubMed]
  5. A. Hardy, “Beam propagation through parabolic-index waveguides with distorted optical axis,” Appl. Phys. 18, 223–226 (1979).
    [CrossRef]
  6. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), pp. 607–616.
  7. H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).
  8. P. K. Tien, J. P. Gordon, J. R. Whinnery, “Focusing of a light beam of Gaussian field distribution in continuous and periodic lens-like media,” IEEE Proc. 53, 129–136 (1965).
    [CrossRef]
  9. L. W. Casperson, “Gaussian light beams in inhomogeneous media,” Appl. Opt. 12, 2434–2441 (1973).
    [CrossRef] [PubMed]
  10. L. W. Casperson, “Mode stability of lasers and periodic optical systems,” IEEE J. Quantum Electron. QE-10, 629–634 (1974).
    [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. A. Tovar, L. W. Casperson, “Off-axis complex-argument polynomial-Gaussian beams in optical systems,” J. Opt. Soc. Am. A 8, 60–68 (1991).
    [CrossRef]
  13. N. McCarthy, P. Lavigne, “Optical resonators with Gaussian reflectivity mirrors: misalignment sensitivity,” Appl. Opt. 22, 2704–2708 (1983).
    [CrossRef] [PubMed]
  14. N. McCarthy, M. Morin, “High-order transverse modes of misaligned laser resonators with Gaussian reflectivity mirrors,” Appl. Opt. 28, 2189–2191 (1989).
    [CrossRef] [PubMed]
  15. D. Marcuse, “The effect of the ∇n2term on the modes of an optical square-law medium,” IEEE J. Quantum Electron. QE-9, 958–960 (1973).
    [CrossRef]
  16. C. Yeh, L. W. Casperson, W. P. Brown, “Scalar-wave approach for single-mode inhomogeneous fiber problems,” Appl. Phys. Lett. 34, 460–462 (1979).
    [CrossRef]
  17. J. D. Love, C. D. Hussey, A. W. Snyder, R. A. Sammut, “Polarization corrections to mode propagation on weakly guiding fibers,” J. Opt. Soc. Am. 72, 1583–1591 (1982).
    [CrossRef]
  18. C. C. Su, “On the scalar approximation in fiber optics,” IEEE Trans. Microwave Theory Tech. 36, 1100–1103 (1988).
    [CrossRef]
  19. R. C. Jones, “A new calculus for the treatment of optical systems. V. A more general formulation and description of another calculus,” J. Opt. Soc. Am. 37, 107–110 (1947).
    [CrossRef]
  20. A. A. Tovar, L. W. Casperson, “Gaussian beam optical systems with high gain or high loss media,” IEEE Trans. Microwave Theory Tech. (to be published).
  21. W. R. Bennett, Appl. Opt. Suppl. 2, 3–33 (1965).
  22. B. N. Perry, P. Rabinowitz, M. Newstein, “Wave propagation in media with focused gain,” Phys. Rev. A 27, 1989–2002 (1983).
    [CrossRef]
  23. L. W. Casperson, “Beam modes in complex lenslike media and resonators,” J. Opt. Soc. Am. 66, 1373–1379 (1976).
    [CrossRef]
  24. L. W. Casperson, S. J. Sheldrake, “Beam deflection and isolation in laser amplifiers,” Opt. Commun. 12, 349–353 (1974).
    [CrossRef]
  25. L. W. Casperson, S. Lunnam, “Gaussian modes in high loss laser resonators,” Appl. Opt. 14, 1193–1199 (1975).
    [CrossRef] [PubMed]
  26. L. W. Casperson, “Beam propagation in tapered quadratic-index waveguides: analytical solutions,” J. Lightwave Technol. LT-3, 264–272 (1985).
    [CrossRef]
  27. A. A. Tovar, L. W. Casperson, “Beam propagation in parabolically tapered graded-index waveguides,” Appl. Opt. 33, 7733–7739 (1994).
    [CrossRef] [PubMed]
  28. L. W. Casperson, “Beam propagation in periodic quadratic-index waveguides,” Appl. Opt. 24, 4395–4403 (1985).
    [CrossRef] [PubMed]
  29. L. W. Casperson, J. L. Kirkwood, “Beam propagation in tapered quadratic-index waveguides: numerical solutions,” J. Lightwave Technol. LT-3, 256–263 (1985).
    [CrossRef]

1994

1991

1989

1988

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

1985

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

L. W. Casperson, J. L. Kirkwood, “Beam propagation in tapered quadratic-index waveguides: numerical solutions,” J. Lightwave Technol. LT-3, 256–263 (1985).
[CrossRef]

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

1983

B. N. Perry, P. Rabinowitz, M. Newstein, “Wave propagation in media with focused gain,” Phys. Rev. A 27, 1989–2002 (1983).
[CrossRef]

N. McCarthy, P. Lavigne, “Optical resonators with Gaussian reflectivity mirrors: misalignment sensitivity,” Appl. Opt. 22, 2704–2708 (1983).
[CrossRef] [PubMed]

1982

1979

A. Hardy, “Beam propagation through parabolic-index waveguides with distorted optical axis,” Appl. Phys. 18, 223–226 (1979).
[CrossRef]

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

1976

1975

1974

L. W. Casperson, S. J. Sheldrake, “Beam deflection and isolation in laser amplifiers,” Opt. Commun. 12, 349–353 (1974).
[CrossRef]

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

1973

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

D. Marcuse, “The effect of the ∇n2term on the modes of an optical square-law medium,” IEEE J. Quantum Electron. QE-9, 958–960 (1973).
[CrossRef]

1969

1965

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

P. K. Tien, J. P. Gordon, J. R. Whinnery, “Focusing of a light beam of Gaussian field distribution in continuous and periodic lens-like media,” IEEE Proc. 53, 129–136 (1965).
[CrossRef]

W. R. Bennett, Appl. Opt. Suppl. 2, 3–33 (1965).

1964

K. Halbach, “Matrix representation of Gaussian optics,” Am. J. Phys. 32, 90–108 (1964).
[CrossRef]

1947

1858

A. Cayley, “A memoir on the theory of matrices,” Philos. Trans. R. Soc. London CXLVIII, 17–37 (1858); also published in The Collected Mathematical Papers of Arthur Cayley (Cambridge U. Press, Cambridge, 1889), Vol. 2, pp. 475–496.

Arnaud, J. A.

Bennett, W. R.

W. R. Bennett, Appl. Opt. Suppl. 2, 3–33 (1965).

Brown, W. P.

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

Burch, J. M.

A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, New York, 1975), pp. 24–26.

Casperson, L. W.

A. A. Tovar, L. W. Casperson, “Beam propagation in parabolically tapered graded-index waveguides,” Appl. Opt. 33, 7733–7739 (1994).
[CrossRef] [PubMed]

A. A. Tovar, L. W. Casperson, “Off-axis complex-argument polynomial-Gaussian beams in optical systems,” J. Opt. Soc. Am. A 8, 60–68 (1991).
[CrossRef]

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

L. W. Casperson, J. L. Kirkwood, “Beam propagation in tapered quadratic-index waveguides: numerical solutions,” J. Lightwave Technol. LT-3, 256–263 (1985).
[CrossRef]

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

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

L. W. Casperson, S. J. Sheldrake, “Beam deflection and isolation in laser amplifiers,” Opt. Commun. 12, 349–353 (1974).
[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]

A. A. Tovar, L. W. Casperson, “Gaussian beam optical systems with high gain or high loss media,” IEEE Trans. Microwave Theory Tech. (to be published).

Cayley, A.

A. Cayley, “A memoir on the theory of matrices,” Philos. Trans. R. Soc. London CXLVIII, 17–37 (1858); also published in The Collected Mathematical Papers of Arthur Cayley (Cambridge U. Press, Cambridge, 1889), Vol. 2, pp. 475–496.

Gerrard, A.

A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, New York, 1975), pp. 24–26.

Gordon, J. P.

P. K. Tien, J. P. Gordon, J. R. Whinnery, “Focusing of a light beam of Gaussian field distribution in continuous and periodic lens-like media,” IEEE Proc. 53, 129–136 (1965).
[CrossRef]

Halbach, K.

K. Halbach, “Matrix representation of Gaussian optics,” Am. J. Phys. 32, 90–108 (1964).
[CrossRef]

Hardy, A.

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]

A. Hardy, “Beam propagation through parabolic-index waveguides with distorted optical axis,” Appl. Phys. 18, 223–226 (1979).
[CrossRef]

Hussey, C. D.

Jones, R. C.

Kirkwood, J. L.

L. W. Casperson, J. L. Kirkwood, “Beam propagation in tapered quadratic-index waveguides: numerical solutions,” J. Lightwave Technol. LT-3, 256–263 (1985).
[CrossRef]

Kogelnik, H.

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

Lavigne, P.

Love, J. D.

Lunnam, S.

Marcuse, D.

D. Marcuse, “The effect of the ∇n2term on the modes of an optical square-law medium,” IEEE J. Quantum Electron. QE-9, 958–960 (1973).
[CrossRef]

McCarthy, N.

Morin, M.

Nazarathy, M.

Newstein, M.

B. N. Perry, P. Rabinowitz, M. Newstein, “Wave propagation in media with focused gain,” Phys. Rev. A 27, 1989–2002 (1983).
[CrossRef]

Perry, B. N.

B. N. Perry, P. Rabinowitz, M. Newstein, “Wave propagation in media with focused gain,” Phys. Rev. A 27, 1989–2002 (1983).
[CrossRef]

Rabinowitz, P.

B. N. Perry, P. Rabinowitz, M. Newstein, “Wave propagation in media with focused gain,” Phys. Rev. A 27, 1989–2002 (1983).
[CrossRef]

Sammut, R. A.

Shamir, J.

Sheldrake, S. J.

L. W. Casperson, S. J. Sheldrake, “Beam deflection and isolation in laser amplifiers,” Opt. Commun. 12, 349–353 (1974).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), pp. 607–616.

Snyder, A. W.

Su, C. C.

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

Tien, P. K.

P. K. Tien, J. P. Gordon, J. R. Whinnery, “Focusing of a light beam of Gaussian field distribution in continuous and periodic lens-like media,” IEEE Proc. 53, 129–136 (1965).
[CrossRef]

Tovar, A. A.

Whinnery, J. R.

P. K. Tien, J. P. Gordon, J. R. Whinnery, “Focusing of a light beam of Gaussian field distribution in continuous and periodic lens-like media,” IEEE Proc. 53, 129–136 (1965).
[CrossRef]

Yeh, C.

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

Am. J. Phys.

K. Halbach, “Matrix representation of Gaussian optics,” Am. J. Phys. 32, 90–108 (1964).
[CrossRef]

Appl. Opt.

Appl. Opt. Suppl.

W. R. Bennett, Appl. Opt. Suppl. 2, 3–33 (1965).

Appl. Phys.

A. Hardy, “Beam propagation through parabolic-index waveguides with distorted optical axis,” Appl. Phys. 18, 223–226 (1979).
[CrossRef]

Appl. Phys. Lett.

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

Bell Syst. Tech. J.

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

IEEE J. Quantum Electron.

D. Marcuse, “The effect of the ∇n2term on the modes of an optical square-law medium,” IEEE J. Quantum Electron. QE-9, 958–960 (1973).
[CrossRef]

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

IEEE Proc.

P. K. Tien, J. P. Gordon, J. R. Whinnery, “Focusing of a light beam of Gaussian field distribution in continuous and periodic lens-like media,” IEEE Proc. 53, 129–136 (1965).
[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. Lightwave Technol.

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

L. W. Casperson, J. L. Kirkwood, “Beam propagation in tapered quadratic-index waveguides: numerical solutions,” J. Lightwave Technol. LT-3, 256–263 (1985).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

L. W. Casperson, S. J. Sheldrake, “Beam deflection and isolation in laser amplifiers,” Opt. Commun. 12, 349–353 (1974).
[CrossRef]

Philos. Trans. R. Soc. London

A. Cayley, “A memoir on the theory of matrices,” Philos. Trans. R. Soc. London CXLVIII, 17–37 (1858); also published in The Collected Mathematical Papers of Arthur Cayley (Cambridge U. Press, Cambridge, 1889), Vol. 2, pp. 475–496.

Phys. Rev. A

B. N. Perry, P. Rabinowitz, M. Newstein, “Wave propagation in media with focused gain,” Phys. Rev. A 27, 1989–2002 (1983).
[CrossRef]

Other

A. A. Tovar, L. W. Casperson, “Gaussian beam optical systems with high gain or high loss media,” IEEE Trans. Microwave Theory Tech. (to be published).

A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, New York, 1975), pp. 24–26.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), pp. 607–616.

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Tables (2)

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Table 1 Generalized Beam Matrices for Nonprofiled Elements

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Table 2 Generalized Beam Matrices for Profiled Elements Misaligned in the x Direction

Equations (113)

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2 E ¯ + k 2 E ¯ = - 2 ( k k · E ¯ ) .
E ¯ ( x , y , z ) = ψ ( x , y , z ) exp [ - i 0 z k 0 ( z ) d z ] × [ ( cos χ ) i x ¯ + ( sin χ ) i y ¯ ] ,
2 ψ x 2 + 2 ψ y 2 - 2 i k 0 ψ z + [ k 2 ( x , y , z ) - k 0 2 - i d k 0 d z ] ψ = 0 ,
| z ( ψ z ) | | - 2 i k 0 ψ z |
k 2 ( x , y , z ) = [ k 0 ( z ) - ½ k 1 x ( z ) x - ½ k 2 x ( z ) x 2 - ½ k 1 y ( z ) y - ½ k 2 y ( z ) y 2 ] 2
k 0 ( z ) [ k 0 ( z ) - k 2 x ( z ) x 2 - k 1 x ( z ) x - k 2 y ( z ) y 2 - k 1 y ( z ) y ] .
k ( x , y , z ) = [ β 0 + β 1 x 2 8 β 2 x + β 1 y 2 8 β 2 y - β 2 x 2 ( x + β 1 x 2 β 2 x ) 2 - β 2 y 2 ( y + β 1 y 2 β 2 y ) 2 ] + i [ α 0 + α 1 x 2 8 α 2 x + α 1 y 2 8 α 2 y - α 2 x 2 ( x + α 1 x 2 α 2 x ) 2 - α 2 y 2 ( y + α 1 y 2 α 2 y ) 2 ] ,
d x β ( z ) = - β 1 x ( z ) 2 β 2 x ( z ) ,
d y β ( z ) = - β 1 y ( z ) 2 β 2 y ( z ) ,
d x α ( z ) = - α 1 x ( z ) 2 α 2 x ( z ) ,
d y α ( z ) = - α 1 x ( z ) 2 α 2 y ( z ) .
ψ ( x , y , z ) = ψ ( x , y , z ) exp { - i [ ½ Q x ( z ) x 2 + ½ Q y ( z ) y 2 + S x ( z ) x + S y ( z ) y ] }
2 ψ x 2 - 2 i ( S x + Q x x ) ψ x + 2 ψ y 2 - 2 i ( S y + Q y y ) ψ y - ( S x 2 + S y 2 ) ψ - i ( Q x + Q y ) ψ - 2 i k 0 ψ z - i d k 0 d z ψ = 0 ,
Q x 2 + k 0 ( z ) d Q x d z + k 0 ( z ) k 2 x ( z ) = 0 ,
Q y 2 + k 0 ( z ) d Q y d z + k 0 ( z ) k 2 y ( z ) = 0 ,
Q x S x + k 0 ( z ) d S x d z + k 0 ( z ) k 1 x ( z ) 2 = 0 ,
Q y S y + k 0 ( z ) d S y d z + k 0 ( z ) k 1 y ( z ) 2 = 0.
2 k 0 d P d z + i ( Q x + Q y ) + ( S x 2 + S y 2 ) + i d k 0 d z = 0.
Q x ( z ) = β 0 R x ( z ) - i 2 w x 2 ( z ) .
d x a ( z ) = - S x i ( z ) / Q x i ( z ) ,
d x p ( z ) = - S x r ( z ) / Q x r ( z ) .
S x ( z ) = - Q x r ( z ) d x p ( z ) - i Q x i ( z ) d x a ( z ) .
d d x a d z = - α 2 x Q x i ( d x α - d x a ) + ( 1 - α λ Q x r / Q x i 1 + α λ 2 ) × ( d x a - d x p ) Q x r β 0 ,
= α 2 x w x 2 2 ( d x α - d x a ) + ( 1 + α λ π n w x 2 / λ R x 1 + α λ 2 ) × ( d x a - d x p R x ) ,
S x = - Q x r d x a + β 0 1 + α λ 2 1 - α λ Q x r / Q x i × [ d x a + α 2 x Q x i ( d x α - d x a ) ] - i Q x i d x a ,
S x = - Q x d x a + β 0 d x a .
Q x ( z ) k 0 ( z ) = 1 u x ( z ) d u x d z ,
d d z [ k 0 ( z ) d u x d z ] + k 2 x ( z ) u x ( z ) = 0 ,
d d z [ u x ( z ) S x ( z ) ] = - ½ k 1 x ( z ) u x ( z ) .
u x ( z ) = A x ( z ) u x ( 0 ) + B x ( z ) u x ( 0 ) .
u x ( z ) = C x ( z ) u x ( 0 ) + D x ( z ) u x ( 0 ) .
d d z [ k 0 ( z ) C x ( z ) ] = - k 2 x ( z ) A x ( z ) ,
d d z [ k 0 ( z ) D x ( z ) ] = - k 2 x ( z ) B x ( z ) .
A x ( z ) D x ( z ) - B x ( z ) C x ( z ) = k 0 ( 0 ) k 0 ( z ) ,
u x ( z ) S x ( z ) = u x ( 0 ) S x ( 0 ) - [ 1 2 0 z k 1 x ( z ) A x ( z ) d z ] u x ( 0 ) - [ 1 2 0 z k 1 x ( z ) B x ( z ) d z ] u x ( 0 ) .
G ( z ) - 1 2 0 z k 1 x ( z ) A ( z ) d z ,
H ( z ) - 1 2 0 z k 1 x ( z ) B ( z ) d z ,
( u x ( z ) [ Q x ( z ) / k 0 ( z ) ] u x ( z ) S x ( z ) u x ( z ) ) = [ A x ( z ) B x ( z ) 0 C x ( z ) D x ( z ) 0 G x ( z ) H x ( z ) 1 ] × ( u x ( 0 ) [ Q x ( 0 ) / k 0 ( 0 ) ] u x ( 0 ) S x ( 0 ) u x ( 0 ) ) ,
Q x ( z ) k 0 ( z ) = C x ( z ) + D x ( z ) Q x ( 0 ) / k 0 ( 0 ) A x ( z ) + B x ( z ) Q x ( 0 ) / k 0 ( 0 ) .
S x ( z ) = S x ( 0 ) A x ( z ) + B x ( z ) Q x ( 0 ) / k 0 ( 0 ) + G x ( z ) + H x ( z ) Q x ( 0 ) / k 0 ( 0 ) A x ( z ) + B x ( z ) Q x ( 0 ) / k 0 ( 0 ) .
T complex lenslike medium = [ cos ( γ x z ) γ x - 1 sin ( γ x z ) 0 - γ x sin ( γ x z ) cos ( γ x z ) 0 - 1 2 0 z k 1 x ( z ) cos ( γ x z ) d z - 1 2 γ x - 1 0 z k 1 x ( z ) sin ( γ x z ) d z 1 ] ,
γ x ( k 2 x k 0 ) 1 / 2 .
( u x 2 ( Q x 2 / k 02 ) u x 2 S x 2 u x 2 ) = [ A x B x 0 C x D x 0 G x H x 1 ] ( u x 1 ( Q x 1 / k 01 ) u x 1 S x 1 u x 1 ) ,
Q x 2 k 02 = C x + D x Q x 1 / k 01 A x + B x Q x 1 / k 01 .
S x 2 = S x 1 A x + B x Q x 1 / k 01 + G x + H x Q x 1 / k 01 A x + B x Q x 1 / k 01 .
T complex prismlike medium = [ 1 0 d k 0 ( 0 ) k 0 - 1 ( z ) d z 0 0 k 0 ( 0 ) k 0 - 1 ( z ) 0 - 1 2 0 d k 1 x ( z ) d z - 1 2 0 d k 1 x ( z ) 0 z k 0 ( 0 ) k 0 - 1 ( z ) d z d z 1 ] .
S x 2 = S x 1 + G x + H x Q x 1 k 01 ,
Q x 2 k 02 = C x + Q x 1 k 01 .
d x a 2 = Q x 1 i d x a 1 - G x i - H x i Q x 1 r / β 0 - H x r Q x 1 i / β 0 Q x 1 i + β 0 C x i ,
d x p 2 = Q x 1 r d x p 1 - G x r - H x r Q x 1 r / β 0 + H x i Q x 1 i / β 0 Q x 1 r + β 0 C x r ,
d x a = ( d x a - d x p ) Q x r β 0 .
d x a 2 = ( C x r Q x 1 i - C x i Q x 1 r Q x 1 i + β 0 C x i ) ( d x a 1 - H x r / β 0 ) + d x a 1 + G x r - H x i Q x 1 i / β 0 β 0 - ( Q x 1 r + β 0 C x r Q x 1 i + β 0 C x i ) × G x i + H x i Q x 1 r / β 0 β 0 .
d x a 2 = d x a 1 1 + ( w x 1 / w g a , x ) 2 ,
d x a 2 = - R x 1 - 1 ( w x 1 / w g a , x ) 2 1 + ( w x 1 / w g a , x ) 2 d x a 1 + d x a 1 ,
d x a 2 = d x a 1 - G x i Q x 1 i - H x r β 0 - Q x 1 r Q x 1 i H x i β 0 ,
d x a 2 = d x a 1 + G x r β 0 - Q x 1 r Q x 1 i G x i β 0 - ( 1 + Q x 1 r 2 Q x 1 i 2 ) Q x 1 i H x i β 0 2 .
d x a 2 = d x a 1 + x 0 ,
d x a 2 = d x a 1 + tan θ x ,
T axis transformation , x = [ 1 0 0 0 1 0 β 0 tan θ x - β 0 x 0 1 ] .
T tilted A B C D , x = [ 1 0 0 0 1 0 β 0 tan θ x - β 0 ( x 0 + L sin θ x ) 1 ] [ A x B x 0 C x D x 0 0 0 1 ] [ 1 0 0 0 1 0 - β 0 tan θ x β 0 x 0 1 ]
= [ A x B x 0 C x D x 0 β 0 [ ( A x - 1 ) tan θ x - C x ( x 0 + L sin θ x ) ] β 0 [ B x tan θ x - D x L sin θ x + ( 1 - D x ) x 0 ] 1 ] .
T spherical mirror , x = [ 1 0 0 - 2 R - 1 1 0 2 β 0 x 0 R - 1 0 1 ] .
{ E 0 exp [ - i ( Q x x 2 / 2 + Q y y 2 / 2 + S x x + S y y + P ) ] } out = { E 0 exp [ - i ( Q x x 2 / 2 + Q y y 2 / 2 + S x x + S y y + P ) ] } in × exp [ ( x - x 0 ) / ( w e a , x cos θ x ) ] .
Q x 2 = Q x 1 ,
S x 2 = S x 1 + i w g a , x cos θ x ,
P 2 = P 1 - i x 0 w g a , x cos θ x .
T exponential aperture , x = [ 1 0 0 0 1 0 i / ( w e a , x cos θ x ) 0 1 ] .
E out = E in exp ( - i k 01 z ) exp [ - i k 02 ( z - z ) ]
= E in exp ( - i k 02 z ) exp [ - i ( k 01 - k 02 ) z ] .
z = ( x - x 0 ) tan θ x ,
{ E 0 exp [ - i ( Q x x 2 / 2 + Q y y 2 / 2 + S x x + S y y + P ) ] } out = { E 0 exp [ - i ( Q x x 2 / 2 + Q y y 2 / 2 + S x x + S y y + P ) ] } in × exp ( - i k 02 z ) exp [ - i ( k 01 - k 02 ) ( x - x 0 ) tan θ x ] .
P 2 = P 1 - ( k 01 - k 02 ) x 0 tan θ x ,
T linear boundary , x = [ 1 0 0 0 k 01 / k 02 0 ( k 01 - k 02 ) tan θ x 0 1 ] .
T thin prism , x = [ 1 0 0 0 1 0 ( k 0 - β air ) ( tan θ x 2 - tan θ x 1 ) 0 1 ] .
{ E 0 exp [ - i ( Q x x 2 / 2 + Q y y 2 / 2 + S x x + S y y + P ) ] } out = { E 0 exp [ - i ( Q x x 2 / 2 + Q y y 2 / 2 + S x x + S y y + P ) ] } in × exp [ - ( x - x 0 ) 2 / ( w g a , x 2 cos 2 θ x ) ] exp ( - y 2 / w g a , y 2 ) .
Q x 2 2 = Q x 1 2 - i w g a , x 2 cos 2 θ x ,
S x 2 = S x 1 + i 2 x 0 w g a , x 2 cos 2 θ x ,
P 2 = P 1 - i x 0 2 w g a , x 2 cos 2 θ x .
T Gaussian aperture , x = [ 1 0 0 - 2 i / ( β air w g a , x 2 cos 2 θ x ) 1 0 2 i x 0 / ( w g a , x 2 cos 2 θ x ) 0 1 ] .
x 2 + y 2 + ( z + R ) 2 = R 2 .
z = - R [ 1 - ( 1 - x 2 + y 2 R 2 ) 1 / 2 ] .
z - x 2 2 R x - y 2 2 R y ,
( z x y 1 ) = [ 1 0 0 0 0 1 0 x 0 0 0 1 0 0 0 0 1 ] [ cos θ x sin θ x 0 0 - sin θ x cos θ x 0 0 0 0 1 0 0 0 0 1 ] ( z x y 1 ) ,
( z x y 1 ) = [ cos θ x - sin θ x 0 x 0 sin θ x + sin θ x cos θ x 0 - x 0 cos θ x 0 0 1 0 0 0 0 1 ] ( z x y 1 ) .
z = - x 2 2 R x - y 2 2 R y .
z = - x 2 2 R x cos θ x + ( tan θ x + x 0 cos θ x R x ) x - y 2 2 R y cos θ x - ( x 0 2 cos θ x 2 R x + x 0 tan θ x ) ,
| ( x - x 0 ) sin θ x R x | 1
T spherical boundary , x = [ 1 0 0 ( 1 - k 01 / k 02 ) R x - 1 cos θ x k 01 / k 02 0 ( k 01 - k 02 ) ( tan θ x + x 0 R x - 1 cos θ x ) 0 1 ] ,
T spherical boundary , y = [ 1 0 0 ( 1 - k 01 / k 02 ) ( R y cos θ x ) - 1 k 01 / k 02 0 0 0 1 ] ,
P 2 = P 1 - x 0 2 cos θ x 2 R x - x 0 tan θ x .
T thin lens , x = [ 1 0 0 - f active , x - 1 cos θ x 1 0 - β 0 x 0 f active , x - 1 cos θ x 0 1 ] ,
T thin lens , y = [ 1 0 0 - ( f active , x cos θ x ) - 1 1 0 0 0 1 ] ,
f active , x ( 1 - β air - 1 k 0 , lens ) ( R x 1 - 1 - R x 2 - 1 ) .
Q x S x Q x = β 0 R x - i 2 w x 2 S x = - Q x d x a + β 0 d x a Q x 2 k 02 = C x + D x Q x 1 / k 01 A x + B x Q x 1 / k 01 S x 2 = S x 1 A x + B x Q x 1 / k 01 + G x + H x Q x 1 / k 01 A x + B x Q x 1 / k 01 Thin lens Thin prism Gaussian aperture Exponential aperture Spherical mirror Tilted flat mirror Complex lenslike medium Complex prismlike medium
d x a 2 = d x a 1 + w x 1 2 2 w e a , x ,
d x a 2 = d x a 1 + w x 1 2 2 R x 1 w e a , x ,
Q x 2 k 02 C x + D x Q x 1 / k 01 A x + B x Q x 1 / k 01 ,
S x 2 = S x 1 A x + B x Q x 1 / k 01 + G x + H x Q x 1 / k 01 A x + B x Q x 1 / k 01 ,
Q x 2 β 0 = ( C x + D x Q x 1 r / β 0 ) + i ( D x Q x 1 i / β 0 ) ( A x + B x Q x 1 r / β 0 ) + i ( B x Q x 1 i / β 0 ) ,
S 2 x = ( S x 1 r + G x + H x Q x 1 r / β 0 ) + i ( S x 1 i + H x Q x 1 i / β 0 ) ( A x + B x Q x 1 r / β 0 ) + i ( B x Q x 1 i / β 0 ) ,
d x a 2 = - S x 2 i Q x 2 i ,
d x p 2 = - S x 2 r Q x 2 r ,
d x a 2 = Q x r 2 β 0 ( d x a 2 - d x p 2 ) ,
d x a 2 = A x d x a 1 + B x d x a 1 + B x G x - A x H x β 0 .
d x a 2 = C x d x a 1 + D x d x a 1 + D x G x - C x H x β 0 .
( d x a 2 d x a 2 1 ) = [ A x B x E x C x D x F x 0 0 1 ] ( d x a 1 d x a 1 1 ) ,
E x = ( B x G x - A x H x ) / β 0 ,
F x = ( D x G x - C x H x ) / β 0 .
G x = β 0 ( A x F x - C x E x ) ,
H x = β 0 ( B x F x - D x E x ) ,
E x = γ x 0 z d β x ( z ) sin [ γ x ( z - z ) ] d z ,
F x = γ x 0 z d β x ( z ) cos [ γ x ( z - z ) ] d z ,
γ x ( n 2 x n 0 ) 1 / 2 .

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