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 aresimpler 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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  1. K. Halbach, "Matrix representation of Gaussian optics," Am. J. Phys. 3, 90–108 (1964).
  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.
  4. H. H. Arsenault, "Generalization of the principal plane concept in matrix optics," Am. J. Phys. 66, 397–399 (1980).
  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).
  7. M. Nazarathy and J. Shamir, "Holography described by operator algebra," J. Opt. Soc. Am. 71, 529–541 (1981).
  8. M. Nazarathy and J. Shamir, "First-order optics—a canonical operator representation: lossless systems," J. Opt. Soc. Am. 72, 356–364 (1982).
  9. J. Shamir, "Cylindrical lens systems described by operator algebra," Appl. Opt. 18, 4195–4202 (1979).
  10. L. W. Casperson, "Synthesis of Gaussian beam optical systems," Appl. Opt. 20, 2243–2249 (1981).
  11. J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976).
  12. 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).
  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).

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).

H. H. Arsenault, "Generalization of the principal plane concept in matrix optics," Am. J. Phys. 66, 397–399 (1980).

1979 (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.

J. Shamir, "Cylindrical lens systems described by operator algebra," Appl. Opt. 18, 4195–4202 (1979).

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).

1978 (1)

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.

1965 (1)

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

1964 (1)

K. Halbach, "Matrix representation of Gaussian optics," Am. J. Phys. 3, 90–108 (1964).

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).

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).

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.

12. 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.

12. 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).

Frazer, R. A.

12. 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).

Kogelnik, H.

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.

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

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).

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).

Am. J. Phys. (2)

K. Halbach, "Matrix representation of Gaussian optics," Am. J. Phys. 3, 90–108 (1964).

H. H. Arsenault, "Generalization of the principal plane concept in matrix optics," Am. J. Phys. 66, 397–399 (1980).

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).

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).

J. Opt. Soc. Am. (3)

Other (5)

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.

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.

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.

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