Abstract

In the paraxial approximation a symmetrical optical system may be represented by a 2 × 2 matrix. It has been the custom to describe each optical element by a transfer matrix representing propagation between the principal planes or through an interface for thin elements. If the focal-plane representation is used instead, any focusing element or combination of elements is represented by the same antidiagonal matrix whose nonzero elements are the focal lengths: The matrix represents propagation between the focal planes. For propagation between any two arbitrary planes, the system transfer matrix can be decomposed into the product of two upper triangular matrices and an antidiagonal matrix. This decomposition yields the above-mentioned focal-plane matrix, and the two upper triangular matrices represent propagation between the input and the output planes and the focal planes. Because the matrix decomposition directly yields the parameters of interest, the analysis and the synthesis of optical systems are simpler to carry out. Examples are given for lenses, diopters, mirrors, periodic sequences, resonators, lenslike media, and phase-conjugate mirror systems.

© 1983 Optical Society of America

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References

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  1. K. Halbach, “Matrix representation of Gaussian optics,” Am. J. Phys. 3, 90–108 (1964).
    [Crossref]
  2. A. Gerrard and J. M. Burch, Introduction to Matrix Method in Optics (Wiley, New York, 1975).
  3. H. Kogelnik, “Propagation of laser beams” in Applied Optics and Optical Engineering, R. R. Shannon and J. C. Wyant, eds. (Academic, New York, 1979), Vol. VII, pp. 155–190.
    [Crossref]
  4. H. H. Arsenault, “Generalization of the principal plane concept in matrix optics,” Am. J. Phys. 66, 397–399 (1980).
    [Crossref]
  5. D. Stoler, “Operator algebraic methods for laser cavity modes,” in Coherence and Quantum Optics IV, L. Mandel and E. Wolf, eds. (Plenum, New York, 1978), pp. 683–693.
  6. M. Nazarathy and J. Shamir, “Fourier optics described by operator algebra,” J. Opt. Soc. Am. 70, 150–159 (1980).
    [Crossref]
  7. M. Nazarathy and J. Shamir, “Holography described by operator algebra,” J. Opt. Soc. Am. 71, 529–541 (1981).
    [Crossref]
  8. M. Nazarathy and J. Shamir, “First-order optics—a canonical operator representation: lossless systems,” J. Opt. Soc. Am. 72, 356–364 (1982).
    [Crossref]
  9. J. Shamir, “Cylindrical lens systems described by operator algebra,” Appl. Opt. 18, 4195–4202 (1979).
    [Crossref] [PubMed]
  10. L. W. Casperson, “Synthesis of Gaussian beam optical systems,” Appl. Opt. 20, 2243–2249 (1981).
    [Crossref] [PubMed]
  11. J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976).
  12. See, e.g., R. A. Frazer, W. J. Duncan, and A. R. Collar, Elementary Matrices (Cambridge U. Press, Cambridge, 1938), pp. 78–79.
  13. H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell. Syst. Tech. J. 44, 455–494 (1965).
    [Crossref]
  14. J. AuYeung, D. Fekete, M. Pepper, and A. Yariv, “A theoretical and experimental investigation of the modes of optical resonators with phase-conjugate mirrors,” IEEE J. Quantum Electron. QE-15, 1180–1188 (1979).
    [Crossref]

1982 (1)

1981 (2)

1980 (2)

M. Nazarathy and J. Shamir, “Fourier optics described by operator algebra,” J. Opt. Soc. Am. 70, 150–159 (1980).
[Crossref]

H. H. Arsenault, “Generalization of the principal plane concept in matrix optics,” Am. J. Phys. 66, 397–399 (1980).
[Crossref]

1979 (2)

J. AuYeung, D. Fekete, M. Pepper, and A. Yariv, “A theoretical and experimental investigation of the modes of optical resonators with phase-conjugate mirrors,” IEEE J. Quantum Electron. QE-15, 1180–1188 (1979).
[Crossref]

J. Shamir, “Cylindrical lens systems described by operator algebra,” Appl. Opt. 18, 4195–4202 (1979).
[Crossref] [PubMed]

1965 (1)

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

1964 (1)

K. Halbach, “Matrix representation of Gaussian optics,” Am. J. Phys. 3, 90–108 (1964).
[Crossref]

Arnaud, J. A.

J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976).

Arsenault, H. H.

H. H. Arsenault, “Generalization of the principal plane concept in matrix optics,” Am. J. Phys. 66, 397–399 (1980).
[Crossref]

AuYeung, J.

J. AuYeung, D. Fekete, M. Pepper, and A. Yariv, “A theoretical and experimental investigation of the modes of optical resonators with phase-conjugate mirrors,” IEEE J. Quantum Electron. QE-15, 1180–1188 (1979).
[Crossref]

Burch, J. M.

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

Casperson, L. W.

Collar, A. R.

See, e.g., R. A. Frazer, W. J. Duncan, and A. R. Collar, Elementary Matrices (Cambridge U. Press, Cambridge, 1938), pp. 78–79.

Duncan, W. J.

See, e.g., R. A. Frazer, W. J. Duncan, and A. R. Collar, Elementary Matrices (Cambridge U. Press, Cambridge, 1938), pp. 78–79.

Fekete, D.

J. AuYeung, D. Fekete, M. Pepper, and A. Yariv, “A theoretical and experimental investigation of the modes of optical resonators with phase-conjugate mirrors,” IEEE J. Quantum Electron. QE-15, 1180–1188 (1979).
[Crossref]

Frazer, R. A.

See, e.g., R. A. Frazer, W. J. Duncan, and A. R. Collar, Elementary Matrices (Cambridge U. Press, Cambridge, 1938), pp. 78–79.

Gerrard, A.

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

Halbach, K.

K. Halbach, “Matrix representation of Gaussian optics,” Am. J. Phys. 3, 90–108 (1964).
[Crossref]

Kogelnik, H.

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

H. Kogelnik, “Propagation of laser beams” in Applied Optics and Optical Engineering, R. R. Shannon and J. C. Wyant, eds. (Academic, New York, 1979), Vol. VII, pp. 155–190.
[Crossref]

Nazarathy, M.

Pepper, M.

J. AuYeung, D. Fekete, M. Pepper, and A. Yariv, “A theoretical and experimental investigation of the modes of optical resonators with phase-conjugate mirrors,” IEEE J. Quantum Electron. QE-15, 1180–1188 (1979).
[Crossref]

Shamir, J.

Stoler, D.

D. Stoler, “Operator algebraic methods for laser cavity modes,” in Coherence and Quantum Optics IV, L. Mandel and E. Wolf, eds. (Plenum, New York, 1978), pp. 683–693.

Yariv, A.

J. AuYeung, D. Fekete, M. Pepper, and A. Yariv, “A theoretical and experimental investigation of the modes of optical resonators with phase-conjugate mirrors,” IEEE J. Quantum Electron. QE-15, 1180–1188 (1979).
[Crossref]

Am. J. Phys. (2)

H. H. Arsenault, “Generalization of the principal plane concept in matrix optics,” Am. J. Phys. 66, 397–399 (1980).
[Crossref]

K. Halbach, “Matrix representation of Gaussian optics,” Am. J. Phys. 3, 90–108 (1964).
[Crossref]

Appl. Opt. (2)

Bell. Syst. Tech. J. (1)

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

IEEE J. Quantum Electron. (1)

J. AuYeung, D. Fekete, M. Pepper, and A. Yariv, “A theoretical and experimental investigation of the modes of optical resonators with phase-conjugate mirrors,” IEEE J. Quantum Electron. QE-15, 1180–1188 (1979).
[Crossref]

J. Opt. Soc. Am. (3)

Other (5)

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

H. Kogelnik, “Propagation of laser beams” in Applied Optics and Optical Engineering, R. R. Shannon and J. C. Wyant, eds. (Academic, New York, 1979), Vol. VII, pp. 155–190.
[Crossref]

D. Stoler, “Operator algebraic methods for laser cavity modes,” in Coherence and Quantum Optics IV, L. Mandel and E. Wolf, eds. (Plenum, New York, 1978), pp. 683–693.

J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976).

See, e.g., R. A. Frazer, W. J. Duncan, and A. R. Collar, Elementary Matrices (Cambridge U. Press, Cambridge, 1938), pp. 78–79.

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Figures (12)

Fig. 1
Fig. 1

Compound optical system composed of two thin lenses.

Fig. 2
Fig. 2

Compound optical system composed of three thin lenses.

Fig. 3
Fig. 3

Simple focal-length doubler. The compound lens on the right is required to have twice the focal length of the simple lens on the left.

Fig. 4
Fig. 4

Spherical interface between two media with different indices of refraction.

Fig. 5
Fig. 5

For a thick lens, the analysis is simplified if the input plane is taken at the focus of one diopter and the output plane is taken at the focus of the other.

Fig. 6
Fig. 6

For a spherical mirror, the antidiagonal system matrix represents propagation from the focus to the mirror and back.

Fig. 7
Fig. 7

Periodic sequence of identical lenses.

Fig. 8
Fig. 8

A single segment of a periodic sequence may consist of two different lenses.

Fig. 9
Fig. 9

Optical resonator with spherical mirrors and an equivalent resonator with plane mirrors and lenses.

Fig. 10
Fig. 10

Resonator with spherical mirrors and an internal lens and an equivalent resonator with plane mirrors.

Fig. 11
Fig. 11

Afocal sequence with four lenses. Propagation through this system is equivalent to two round trips through the afocal resonator of Fig. 12.

Fig. 12
Fig. 12

Afocal resonator with two mirrors having equal curvatures.

Equations (83)

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x = A x + B ξ , ξ = C x + D ξ ,
[ x ξ ] = [ A B C D ] [ x ξ ] = M [ x ξ ] .
P M = L D U ,
P = [ 0 1 1 0 ] ,
P - 1 ( P M ) = P - 1 [ L ( P P - 1 ) D U ] ,
M = ( P - 1 L P ) ( P - 1 D ) U = U 1 Ā U 2 ,
M = [ A B C D ] ,
P M = L D U = [ 1 0 A / C 1 ] [ C 0 0 B - A D / C ] [ 1 D / C 0 1 ] ,
M = U 1 Ā U 2 = [ 1 a 0 1 ] [ 0 f * - 1 / f * 0 ] [ 1 b 0 1 ] ,
a = A / C ,             b = D / C ,             f * = - 1 / C .
L = [ 1 0 - 1 / f 1 ] ,
L e = [ 0 f - 1 / f 0 ]
f * = - f 1 f 2 d ,
k = - f 2 2 / d ,
k = - f 1 2 / d ,
M = 1 f 1 f 2 f 3 [ d 4 f 2 2 + d 2 f 3 2 - d 2 d 3 f 4 - d 1 d 2 d 3 d 4 + d 1 d 4 f 2 2 + d 1 d 2 f 3 2 + d 3 d 4 f 1 2 - f 1 2 f 3 2 f 2 2 - d 2 d 3 - d 1 d 2 d 3 + d 1 f 2 2 + d 3 f 1 2 ] .
M = [ 1 d 4 + d 2 f 3 2 f 2 2 - d 2 d 3 0 1 ] [ 0 - f 1 f 2 f 3 f 2 2 - d 2 d 3 f 2 2 - d 2 d 3 f 1 f 2 f 3 0 ] [ 1 d 1 + d 3 f 1 2 f 2 2 - d 2 d 3 0 1 ]
f = - f 1 f 2 f 3 f 2 2 - d 2 d 3 .
b = d 1 + d 3 f 1 2 f 2 2 - d 2 d 3
a = d 4 + d 2 f 3 2 f 2 2 - d 2 d 3
M = [ 1 0 - 1 / f 2 1 ] [ 1 d 0 1 ] [ 1 0 - 1 / f 1 1 ] = [ 1 - d / f 1 d d / f 1 f 2 - 1 / f 1 - 1 / f 2 1 - d / f 2 ] ;
[ 1 f 2 ( f 1 - d ) d - f 1 - f 2 0 1 ] [ 0 f 1 f 2 f 1 + f 2 - d - f 1 + f 2 + d f 1 f 2 0 ] [ 1 f 1 ( f 2 - d ) d - f 1 - f 2 0 1 ]
f = 2 f 1 ,
x = - f 1 .
f 1 f 2 f 1 + f 2 - d = 2 f 1 , f 2 ( f 1 - d ) d - f 1 - f 2 = - f 1 .
f 2 = - f 1 d = f 1 / 2.
M = [ 1 0 - n / f 1 ] .
[ 1 0 n 2 R 1 ] [ M ] [ 1 0 - n 1 R 1 ] = [ 1 0 0 1 ] ,
M = [ 1 0 n 1 - n 2 R 1 ] .
[ 1 - R n 2 - n 1 0 1 ] [ 0 R n 2 - n 1 n 1 - n 2 R 0 ] [ 1 - R n 2 - n 1 0 1 ] .
f 1 = n 1 R n 1 - n 2
f 2 = - n 2 R n 1 - n 2
M = [ 0 R 2 n 3 - n 2 n 2 - n 3 R 2 0 ] [ 1 1 n 2 ( d - n 2 R 1 n 2 - n 1 - n 2 R 2 n 3 - n 2 ) 0 1 ] [ 0 R 1 n 2 - n 1 n 1 - n 2 R 1 0 ] = [ R 2 ( n 2 - n 1 ) R 1 ( n 2 - n 3 ) 0 ( n 3 - n 2 ) ( n 2 - n 1 ) R 1 R 2 ( d n 2 - R 1 n 2 - n 1 - R 2 n 3 - n 2 ) R 1 ( n 3 - n 2 ) R 2 ( n 1 - n 2 ) ] .
[ 1 f R 2 ( n 1 - n 2 ) R 1 ( n 2 - n 3 ) 0 1 ] [ 0 f - 1 f 0 ] [ 1 f R 1 ( n 2 - n 3 ) R 2 ( n 1 - n 2 ) 0 1 ]
1 f = d ( n 3 - n 2 ) ( n 2 - n 1 ) n 2 R 1 R 2 - n 3 - n 2 R 2 - n 2 - n 1 R 1 .
D 1 = f n 1 R 1 ( n 3 - n 2 ) R 2 ( n 2 - n 1 )
D 2 = f n 2 R 2 ( n 2 - n 1 ) R 1 ( n 3 - n 2 )
n = n 0 - 1 2 n 2 r 2 ,             n 0 = constant .
M = [ cos [ d ( n 2 n 0 ) 1 / 2 ] 1 ( n 0 n 2 ) 1 / 2 sin [ d ( n 2 n 0 ) 1 / 2 ] - ( n 0 n 2 ) 1 / 2 sin [ d ( n 2 n 0 ) 1 / 2 ] cos [ d ( n 2 n 0 ) 1 / 2 ] ]
[ 1 - 1 ( n 0 n 2 ) cot [ d ( n 2 n 0 ) 1 / 2 ] 0 1 ] [ 0 1 ( n 0 - n 2 ) 1 / 2 sin [ d ( n 2 n 0 ) 1 / 2 ] - ( n 0 n 2 ) 1 / 2 sin [ d ( n 2 n 0 ) 1 / 2 ] 0 ] [ 1 - 1 ( n 0 n 2 ) 1 / 2 cot [ d ( n 2 n 0 ) 1 / 2 ] 0 1 ] .
f = 1 ( n 0 n 2 ) 1 / 2 sin [ d ( n 2 n 0 ) 1 / 2 ] ,
a = b = - 1 ( n 0 n 2 ) 1 / 2 cot [ d ( n 2 n 0 ) 1 / 2 ] .
T = [ A B C D ] ,
T * = [ D B C A ] .
M = [ 1 0 - 2 R 1 ] .
M = [ 1 - d R d - 1 R 1 ] [ 1 d - 1 R 1 - 3 R ] = [ 1 - 2 d R 2 d ( 1 - d R ) - 2 R 1 - 2 d R ] .
M = [ 0 R / 2 - 2 / R 0 ] .
M = [ 1 - 2 d 2 R d 1 + d 2 - 2 d 1 d 2 R - 2 R 1 - 2 d 1 R ] .
[ 1 d 2 - R 2 0 1 ] [ 0 R 2 - 2 R 0 ] [ 1 d 1 - R 2 0 1 ] .
M = [ 1 0 - 1 / f 1 ] [ 1 d 0 1 ] = [ 1 d - 1 / f 1 - d / f ] .
M n = M n = [ 1 d - 1 / f 1 - d / f ] n .
M = [ A B C D ] ,
M n = M n = cos n α · E - sin n α sin α · X ,
cos α = A + D 2 , X = [ D - A 2 - B - C A - D 2 ] .
M n = 1 sin α × [ A sin n α - sin ( n - 1 ) α B sin n α C sin n α D sin n α - sin ( n - 1 ) α ] = 1 sin α × [ sin n α - sin ( n - 1 ) α d sin n α - sin n α f ( 1 - d f ) sin n α - sin ( n - 1 ) α ] ,
cos α = 1 - d 2 f .
[ 1 f [ sin ( n - 1 ) α - sin n α sin n α 0 1 ] [ 0 f sin α sin n α - sin n α f sin α 0 ] [ 1 d + f [ sin ( n - 1 ) α - sin n α ] sin n α 0 1 ]
f * = f sin α sin n α .
b = d + f sin n α [ sin ( n - 1 ) α - sin n α ] , a = f sin n α [ sin ( n - 1 ) α - sin n α ] .
M = [ 1 0 - 1 / f 2 1 ] [ 1 d 2 0 1 ] [ 1 0 - 1 / f 1 1 ] [ 1 d 1 0 1 ] = [ 1 - d 2 f 1 d 1 + d 2 - d 1 d 2 f 1 - 1 f 1 - 1 f 2 + d 2 f 1 f 2 1 - d 1 f 2 - d 1 f 1 - d 2 f 2 + d 1 d 2 f 1 f 2 ] .
cos α = 1 - d 1 + d 2 2 ( 1 f 1 + 1 f 2 ) + d 1 d 2 2 f 1 f 2
f * = - f 1 f 2 d 2 - f 1 - f 2 · sin α sin n α
b = d 1 + f 1 ( f 2 - d 2 ) d 2 - f 1 - f 2 + f 1 f 2 f 1 + f 2 - d 2 · sin ( n - 1 ) α sin n α , a = f 2 ( f 1 - d 2 ) / ( d 2 - f 1 - f 2 ) + f 1 f 2 f 1 + f 2 - d 2 · sin ( n - 1 ) α sin n α .
- 2 < A + D < 2 ,
M = [ 1 - d R 1 d - 1 R 1 - 1 R 2 + d R 1 R 2 1 - d R 2 ] ,
M * = [ 1 - d R 2 d - 1 R 1 - 1 R 2 + d R 1 R 2 1 - d R 1 ] .
M * M = [ 1 - 2 d ( 1 R 1 + 1 R 2 - d R 1 R 2 ) 2 d ( 1 - d R 2 ) 2 ( 1 - d R 1 ) ( d R 1 R 2 - 1 R 1 - 1 R 2 ) 1 - 2 d ( 1 R 1 + 1 R 2 - d R 1 R 2 ) ] .
M = [ 1 - d 2 f - 1 R 1 ( d 1 + d 2 - d 1 d 2 f ) d 1 + d 2 - d 1 d 2 f - 1 R 2 - 1 f + d 2 f R 2 - 1 R 1 [ - d 1 R 2 + ( 1 - d 2 R 2 ) ( 1 - d 1 f ) ] - d 1 R 2 + ( 1 - d 2 R 2 ) ( 1 - d 1 f ) ] .
M P = [ B A D C ] = [ 1 0 D / B 1 ] [ B 0 0 - 1 / B ] [ 1 A / B 0 1 ] .
M P P - 1 = M = L D ( P - 1 P ) U P - 1 = L ( D P - 1 ) ( P U P - 1 ) = [ 1 0 D / B 1 ] ( 0 B - 1 / B 0 ] [ 1 0 A / B 1 ] .
[ 0 B - 1 / B 0 ] = [ 1 0 - 1 / B 0 ] [ 1 B 0 1 ] [ 1 0 - 1 / B 1 ] ,
M = [ 1 0 D - 1 B 1 ] [ 1 B 0 1 ] [ 1 0 A - 1 B 1 ] .
1 R l = 1 R 1 + d 2 f d 1 + f d 2 - d 1 d 2 / e , 1 R r = 1 R 2 + d 1 f d 1 + f d 2 - d 1 d 2
d 0 = d 1 + d 2 - d 1 d 2 f .
M m = [ 1 0 0 - 1 ] .
M = [ D B C A ] [ 1 0 0 - 1 ] [ A B C D ] = [ 1 0 0 - 1 ] ,
M = [ 1 0 - 2 / R 1 ] [ 1 0 0 - 1 ] = [ 1 0 - 2 / R - 1 ] .
[ 1 - R / 2 0 1 ] [ 0 - R / 2 - 2 / R 0 ] [ 1 R / 2 0 1 ]
[ 1 0 0 - 1 ] [ 1 0 0 - 1 ] [ 1 - R / 2 0 1 ] [ 1 0 0 - 1 ] [ 1 0 0 - 1 ] [ 1 - R / 2 - 2 / R 0 ] [ 1 R / 2 0 1 ] = [ 1 0 0 - 1 ] [ 1 R / 2 0 1 ] [ 0 - R / 2 2 / R 0 ] [ 1 R / 2 0 1 ] = [ 1 0 0 - 1 ] [ 1 0 2 / R 1 ] .
M = [ 1 0 0 - 1 ] [ A B C D ] ,
M 2 = [ 1 0 0 - 1 ] [ A B C D ] [ 1 0 0 - 1 ] [ A B C D ] = [ A - B - C D ] [ A B C D ] ,
M 2 = [ 1 0 - 2 / R 1 ] [ 1 0 2 / R 0 ] = [ 1 0 0 1 ] .
M 4 = [ 0 R - 1 / R 0 ] 4 = [ - 1 0 0 - 1 ] 2 = [ 1 0 0 1 ] .