Abstract

It is shown that the two-by-two Jones-matrix formalism for polarization optics is a six-parameter two-by-two representation of the Lorentz group. The attenuation and phase-shift filters are represented, respectively, by the three-parameter rotation subgroup and the three-parameter Lorentz group for two spatial dimensions and one time dimension. The Lorentz group has another three-parameter subgroup, which is like the two-dimensional Euclidean group. Optical filters that may have this Euclidean symmetry are discussed in detail. It is shown that the Jones-matrix formalism can be extended to some of the nonorthogonal polarization coordinate systems within the framework of the Lorentz-group representation.

© 1997 Optical Society of America

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  1. E. Wigner, “On unitary representations of the inhomogeneous Lorentz group,” Ann. Math. 40, 149–204 (1939).
    [CrossRef]
  2. V. Bargmann, “Irreducible unitary representations of the Lorentz group,” Ann. Math. 48, 568–640 (1947).
    [CrossRef]
  3. P. A. M. Dirac, “Forms of relativistic dynamics,” Rev. Mod. Phys. 21, 392–399 (1949).
    [CrossRef]
  4. P. A. M. Dirac, “A remarkable representation of the 3+2 de Sitter group,” J. Math. Phys. 4, 901–909 (1963).
    [CrossRef]
  5. H. P. Yuen, “Two-photon coherent states of the radiation fields,” Phys. Rev. A 13, 2226–2243 (1976).
    [CrossRef]
  6. C. M. Caves, B. L. Schumaker, “New formalism for two-photon quantum optics. I. Quadrature phases and squeezed light,” Phys. Rev. A 31, 3068–3092 (1985);B. L. Schumaker, C. M. Caves, “New formalism for two-photon quantum optics. II. Mathematical foundation and compact notation,” Phys. Rev. A 31, 3093–3111 (1985).
    [CrossRef] [PubMed]
  7. Y. S. Kim, M. E. Noz, Phase Space Picture of Quantum Mechanics (World Scientific, Singapore, 1991), pp. 77–122.
  8. H. Bacry, M. Cadilhac, “Metaplectic group and Fourier optics,” Phys. Rev. A 23, 2533–2536 (1981).
    [CrossRef]
  9. E. C. G. Sudarshan, R. Simon, N. Makunda, “Paraxial-wave optics and relativistic front description. I. The scalar theory,” Phys. Rev. A 28, 2921–2932 (1983);N. Makunda, R. Simon, E. C. G. Sudarshan, “Paraxial-wave optics and relativistic front description. II. The vector theory,” Phys. Rev. A 28, 2933–2942 (1983).
    [CrossRef]
  10. D. Han, Y. S. Kim, M. E. Noz, “Wavelets, windows, and photons,” Phys. Lett. A 206, 299–304 (1995).
    [CrossRef]
  11. P. Pellat-Finet, M. Buasset, “What is common to both polarization optics and relativistic kinematics?” Optik (Stuttgart) 90, 101–106 (1992).
  12. M. Gutierrez, J. C. Minano, C. Vega, P. Benitez, “Application of Lorentz geometry to non-imaging optics: new three-dimensional ideal contractors,” J. Opt. Soc. Am. A 13, 532–542 (1996).
    [CrossRef]
  13. S. R. Cloude, “Group theory and polarization algebra,” Optik 75, 26–36 (1986).
  14. R. Y. Chiao, T. F. Jordan, “Lorentz-group Berry phases in squeezed light,” Phys. Lett. A 132, 77–81 (1988).
    [CrossRef]
  15. M. Kitano, T. Yabuzaki, “Observation of Lorentz-group Berry phases in polarization optics,” Phys. Lett. A 142, 321–325 (1989).
    [CrossRef]
  16. T. Opatrny, J. Perina, “Non-image-forming polarization optical devices and Lorentz transformations—analogy,” Phys. Lett. A 181, 199–202 (1993).
    [CrossRef]
  17. D. Han, Y. S. Kim, M. E. Noz, “Polarization optics and bilinear representation of the Lorentz group,” Phys. Lett. A 219, 26–32 (1996).
    [CrossRef]
  18. R. C. Jones, “New calculus for the treatment of optical systems,” J. Opt. Soc. Am. 31, 488–493 (1941).
    [CrossRef]
  19. W. Swindell, Polarized Light (Dowden, Hutchinson, and Ross, Stroudsburg, Pa., 1975).
  20. F. L. Pedrotti, L. S. Pedrotti, Introduction to Optics, 2nd ed. (Prentice-Hall, Englewood Cliffs, N. J., 1993), pp. 280–297.
  21. D. Han, Y. S. Kim, D. Son, “E(2)-like little group for massless particles and polarization of neutrinos,” Phys. Rev. D 26, 3717–3725 (1982).
    [CrossRef]
  22. Y. S. Kim, L. Yeh, “E(2)-symmetric sheared states,” J. Math. Phys. 33, 1237–1246 (1992).
    [CrossRef]
  23. A. S. Chirkin, A. A. Orlov, D. Yu. Parashchuk, “Quantum theory of two-mode interactions in optically anisotropic media with cubic nonlinearities: generation of quadrature- and polarization-squeezed light,” Quantum Electron. 23, 870–874 (1993).
    [CrossRef]
  24. R. Abraham, J. E. Marsden, Foundations of Mechanics, 2nd ed. (Benjamin/Cummings, Reading, Mass., 1978).
  25. V. Guillemin, S. Sternberg, Symplectic Techniques in Physics (Cambridge U. Press, Cambridge, 1984).
  26. Y. S. Kim, M. E. Noz, Theory and Applications of the Poincaré Group (Reidel, Dordrecht, The Netherlands, 1986).
  27. G. P. Parent, P. Roman, “On the matrix formulation of the theory of partial polarization in terms of observables,” Nuovo Cimento 15, 370–388 (1960);R. Barakat, “Theory of the coherency matrix for light of arbitrary bandwidth,” J. Opt. Soc. Am. 53, 317–323 (1963);H. Takenaka, “A unified formalism for polarization optics by using group theory,” Nouv. Res. Opt. 4, 37–41 (1973).
    [CrossRef]
  28. C. E. Baum, H. N. Kritikos, Electromagnetic Symmetry (Taylor & Francis, Washington, D.C., 1995).
  29. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), pp. 545–553.
  30. J. Perina, Coherence of Light (Van Nostrand-Reinhold, London, 1971), pp. 77–88.
  31. D. Han, Y. S. Kim, D. Son, “Photons, neutrinos, and gauge transformations,” Am. J. Phys. 54, 818–821 (1986).
    [CrossRef]
  32. S. Baskal, Y. S. Kim, “Four-potentials and Maxwell field tensors from SL(2, C) spinors as infinite-momentum/zero-mass limits of their massive counterparts,” Los Alamos E-print Archive hep-th/9512088 (Los Alamos National Laboratory, Los Alamos, N. Mex., 1995).
  33. S. Weinberg, “Feynman rules for any spin II massless particles.” Phys. Rev. 134, B882–B896 (1964).
    [CrossRef]

1996 (2)

D. Han, Y. S. Kim, M. E. Noz, “Polarization optics and bilinear representation of the Lorentz group,” Phys. Lett. A 219, 26–32 (1996).
[CrossRef]

M. Gutierrez, J. C. Minano, C. Vega, P. Benitez, “Application of Lorentz geometry to non-imaging optics: new three-dimensional ideal contractors,” J. Opt. Soc. Am. A 13, 532–542 (1996).
[CrossRef]

1995 (1)

D. Han, Y. S. Kim, M. E. Noz, “Wavelets, windows, and photons,” Phys. Lett. A 206, 299–304 (1995).
[CrossRef]

1993 (2)

T. Opatrny, J. Perina, “Non-image-forming polarization optical devices and Lorentz transformations—analogy,” Phys. Lett. A 181, 199–202 (1993).
[CrossRef]

A. S. Chirkin, A. A. Orlov, D. Yu. Parashchuk, “Quantum theory of two-mode interactions in optically anisotropic media with cubic nonlinearities: generation of quadrature- and polarization-squeezed light,” Quantum Electron. 23, 870–874 (1993).
[CrossRef]

1992 (2)

Y. S. Kim, L. Yeh, “E(2)-symmetric sheared states,” J. Math. Phys. 33, 1237–1246 (1992).
[CrossRef]

P. Pellat-Finet, M. Buasset, “What is common to both polarization optics and relativistic kinematics?” Optik (Stuttgart) 90, 101–106 (1992).

1989 (1)

M. Kitano, T. Yabuzaki, “Observation of Lorentz-group Berry phases in polarization optics,” Phys. Lett. A 142, 321–325 (1989).
[CrossRef]

1988 (1)

R. Y. Chiao, T. F. Jordan, “Lorentz-group Berry phases in squeezed light,” Phys. Lett. A 132, 77–81 (1988).
[CrossRef]

1986 (2)

S. R. Cloude, “Group theory and polarization algebra,” Optik 75, 26–36 (1986).

D. Han, Y. S. Kim, D. Son, “Photons, neutrinos, and gauge transformations,” Am. J. Phys. 54, 818–821 (1986).
[CrossRef]

1985 (1)

C. M. Caves, B. L. Schumaker, “New formalism for two-photon quantum optics. I. Quadrature phases and squeezed light,” Phys. Rev. A 31, 3068–3092 (1985);B. L. Schumaker, C. M. Caves, “New formalism for two-photon quantum optics. II. Mathematical foundation and compact notation,” Phys. Rev. A 31, 3093–3111 (1985).
[CrossRef] [PubMed]

1983 (1)

E. C. G. Sudarshan, R. Simon, N. Makunda, “Paraxial-wave optics and relativistic front description. I. The scalar theory,” Phys. Rev. A 28, 2921–2932 (1983);N. Makunda, R. Simon, E. C. G. Sudarshan, “Paraxial-wave optics and relativistic front description. II. The vector theory,” Phys. Rev. A 28, 2933–2942 (1983).
[CrossRef]

1982 (1)

D. Han, Y. S. Kim, D. Son, “E(2)-like little group for massless particles and polarization of neutrinos,” Phys. Rev. D 26, 3717–3725 (1982).
[CrossRef]

1981 (1)

H. Bacry, M. Cadilhac, “Metaplectic group and Fourier optics,” Phys. Rev. A 23, 2533–2536 (1981).
[CrossRef]

1976 (1)

H. P. Yuen, “Two-photon coherent states of the radiation fields,” Phys. Rev. A 13, 2226–2243 (1976).
[CrossRef]

1964 (1)

S. Weinberg, “Feynman rules for any spin II massless particles.” Phys. Rev. 134, B882–B896 (1964).
[CrossRef]

1963 (1)

P. A. M. Dirac, “A remarkable representation of the 3+2 de Sitter group,” J. Math. Phys. 4, 901–909 (1963).
[CrossRef]

1960 (1)

G. P. Parent, P. Roman, “On the matrix formulation of the theory of partial polarization in terms of observables,” Nuovo Cimento 15, 370–388 (1960);R. Barakat, “Theory of the coherency matrix for light of arbitrary bandwidth,” J. Opt. Soc. Am. 53, 317–323 (1963);H. Takenaka, “A unified formalism for polarization optics by using group theory,” Nouv. Res. Opt. 4, 37–41 (1973).
[CrossRef]

1949 (1)

P. A. M. Dirac, “Forms of relativistic dynamics,” Rev. Mod. Phys. 21, 392–399 (1949).
[CrossRef]

1947 (1)

V. Bargmann, “Irreducible unitary representations of the Lorentz group,” Ann. Math. 48, 568–640 (1947).
[CrossRef]

1941 (1)

1939 (1)

E. Wigner, “On unitary representations of the inhomogeneous Lorentz group,” Ann. Math. 40, 149–204 (1939).
[CrossRef]

Abraham, R.

R. Abraham, J. E. Marsden, Foundations of Mechanics, 2nd ed. (Benjamin/Cummings, Reading, Mass., 1978).

Bacry, H.

H. Bacry, M. Cadilhac, “Metaplectic group and Fourier optics,” Phys. Rev. A 23, 2533–2536 (1981).
[CrossRef]

Bargmann, V.

V. Bargmann, “Irreducible unitary representations of the Lorentz group,” Ann. Math. 48, 568–640 (1947).
[CrossRef]

Baskal, S.

S. Baskal, Y. S. Kim, “Four-potentials and Maxwell field tensors from SL(2, C) spinors as infinite-momentum/zero-mass limits of their massive counterparts,” Los Alamos E-print Archive hep-th/9512088 (Los Alamos National Laboratory, Los Alamos, N. Mex., 1995).

Baum, C. E.

C. E. Baum, H. N. Kritikos, Electromagnetic Symmetry (Taylor & Francis, Washington, D.C., 1995).

Benitez, P.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), pp. 545–553.

Buasset, M.

P. Pellat-Finet, M. Buasset, “What is common to both polarization optics and relativistic kinematics?” Optik (Stuttgart) 90, 101–106 (1992).

Cadilhac, M.

H. Bacry, M. Cadilhac, “Metaplectic group and Fourier optics,” Phys. Rev. A 23, 2533–2536 (1981).
[CrossRef]

Caves, C. M.

C. M. Caves, B. L. Schumaker, “New formalism for two-photon quantum optics. I. Quadrature phases and squeezed light,” Phys. Rev. A 31, 3068–3092 (1985);B. L. Schumaker, C. M. Caves, “New formalism for two-photon quantum optics. II. Mathematical foundation and compact notation,” Phys. Rev. A 31, 3093–3111 (1985).
[CrossRef] [PubMed]

Chiao, R. Y.

R. Y. Chiao, T. F. Jordan, “Lorentz-group Berry phases in squeezed light,” Phys. Lett. A 132, 77–81 (1988).
[CrossRef]

Chirkin, A. S.

A. S. Chirkin, A. A. Orlov, D. Yu. Parashchuk, “Quantum theory of two-mode interactions in optically anisotropic media with cubic nonlinearities: generation of quadrature- and polarization-squeezed light,” Quantum Electron. 23, 870–874 (1993).
[CrossRef]

Cloude, S. R.

S. R. Cloude, “Group theory and polarization algebra,” Optik 75, 26–36 (1986).

Dirac, P. A. M.

P. A. M. Dirac, “A remarkable representation of the 3+2 de Sitter group,” J. Math. Phys. 4, 901–909 (1963).
[CrossRef]

P. A. M. Dirac, “Forms of relativistic dynamics,” Rev. Mod. Phys. 21, 392–399 (1949).
[CrossRef]

Guillemin, V.

V. Guillemin, S. Sternberg, Symplectic Techniques in Physics (Cambridge U. Press, Cambridge, 1984).

Gutierrez, M.

Han, D.

D. Han, Y. S. Kim, M. E. Noz, “Polarization optics and bilinear representation of the Lorentz group,” Phys. Lett. A 219, 26–32 (1996).
[CrossRef]

D. Han, Y. S. Kim, M. E. Noz, “Wavelets, windows, and photons,” Phys. Lett. A 206, 299–304 (1995).
[CrossRef]

D. Han, Y. S. Kim, D. Son, “Photons, neutrinos, and gauge transformations,” Am. J. Phys. 54, 818–821 (1986).
[CrossRef]

D. Han, Y. S. Kim, D. Son, “E(2)-like little group for massless particles and polarization of neutrinos,” Phys. Rev. D 26, 3717–3725 (1982).
[CrossRef]

Jones, R. C.

Jordan, T. F.

R. Y. Chiao, T. F. Jordan, “Lorentz-group Berry phases in squeezed light,” Phys. Lett. A 132, 77–81 (1988).
[CrossRef]

Kim, Y. S.

D. Han, Y. S. Kim, M. E. Noz, “Polarization optics and bilinear representation of the Lorentz group,” Phys. Lett. A 219, 26–32 (1996).
[CrossRef]

D. Han, Y. S. Kim, M. E. Noz, “Wavelets, windows, and photons,” Phys. Lett. A 206, 299–304 (1995).
[CrossRef]

Y. S. Kim, L. Yeh, “E(2)-symmetric sheared states,” J. Math. Phys. 33, 1237–1246 (1992).
[CrossRef]

D. Han, Y. S. Kim, D. Son, “Photons, neutrinos, and gauge transformations,” Am. J. Phys. 54, 818–821 (1986).
[CrossRef]

D. Han, Y. S. Kim, D. Son, “E(2)-like little group for massless particles and polarization of neutrinos,” Phys. Rev. D 26, 3717–3725 (1982).
[CrossRef]

Y. S. Kim, M. E. Noz, Theory and Applications of the Poincaré Group (Reidel, Dordrecht, The Netherlands, 1986).

S. Baskal, Y. S. Kim, “Four-potentials and Maxwell field tensors from SL(2, C) spinors as infinite-momentum/zero-mass limits of their massive counterparts,” Los Alamos E-print Archive hep-th/9512088 (Los Alamos National Laboratory, Los Alamos, N. Mex., 1995).

Y. S. Kim, M. E. Noz, Phase Space Picture of Quantum Mechanics (World Scientific, Singapore, 1991), pp. 77–122.

Kitano, M.

M. Kitano, T. Yabuzaki, “Observation of Lorentz-group Berry phases in polarization optics,” Phys. Lett. A 142, 321–325 (1989).
[CrossRef]

Kritikos, H. N.

C. E. Baum, H. N. Kritikos, Electromagnetic Symmetry (Taylor & Francis, Washington, D.C., 1995).

Makunda, N.

E. C. G. Sudarshan, R. Simon, N. Makunda, “Paraxial-wave optics and relativistic front description. I. The scalar theory,” Phys. Rev. A 28, 2921–2932 (1983);N. Makunda, R. Simon, E. C. G. Sudarshan, “Paraxial-wave optics and relativistic front description. II. The vector theory,” Phys. Rev. A 28, 2933–2942 (1983).
[CrossRef]

Marsden, J. E.

R. Abraham, J. E. Marsden, Foundations of Mechanics, 2nd ed. (Benjamin/Cummings, Reading, Mass., 1978).

Minano, J. C.

Noz, M. E.

D. Han, Y. S. Kim, M. E. Noz, “Polarization optics and bilinear representation of the Lorentz group,” Phys. Lett. A 219, 26–32 (1996).
[CrossRef]

D. Han, Y. S. Kim, M. E. Noz, “Wavelets, windows, and photons,” Phys. Lett. A 206, 299–304 (1995).
[CrossRef]

Y. S. Kim, M. E. Noz, Phase Space Picture of Quantum Mechanics (World Scientific, Singapore, 1991), pp. 77–122.

Y. S. Kim, M. E. Noz, Theory and Applications of the Poincaré Group (Reidel, Dordrecht, The Netherlands, 1986).

Opatrny, T.

T. Opatrny, J. Perina, “Non-image-forming polarization optical devices and Lorentz transformations—analogy,” Phys. Lett. A 181, 199–202 (1993).
[CrossRef]

Orlov, A. A.

A. S. Chirkin, A. A. Orlov, D. Yu. Parashchuk, “Quantum theory of two-mode interactions in optically anisotropic media with cubic nonlinearities: generation of quadrature- and polarization-squeezed light,” Quantum Electron. 23, 870–874 (1993).
[CrossRef]

Parashchuk, D. Yu.

A. S. Chirkin, A. A. Orlov, D. Yu. Parashchuk, “Quantum theory of two-mode interactions in optically anisotropic media with cubic nonlinearities: generation of quadrature- and polarization-squeezed light,” Quantum Electron. 23, 870–874 (1993).
[CrossRef]

Parent, G. P.

G. P. Parent, P. Roman, “On the matrix formulation of the theory of partial polarization in terms of observables,” Nuovo Cimento 15, 370–388 (1960);R. Barakat, “Theory of the coherency matrix for light of arbitrary bandwidth,” J. Opt. Soc. Am. 53, 317–323 (1963);H. Takenaka, “A unified formalism for polarization optics by using group theory,” Nouv. Res. Opt. 4, 37–41 (1973).
[CrossRef]

Pedrotti, F. L.

F. L. Pedrotti, L. S. Pedrotti, Introduction to Optics, 2nd ed. (Prentice-Hall, Englewood Cliffs, N. J., 1993), pp. 280–297.

Pedrotti, L. S.

F. L. Pedrotti, L. S. Pedrotti, Introduction to Optics, 2nd ed. (Prentice-Hall, Englewood Cliffs, N. J., 1993), pp. 280–297.

Pellat-Finet, P.

P. Pellat-Finet, M. Buasset, “What is common to both polarization optics and relativistic kinematics?” Optik (Stuttgart) 90, 101–106 (1992).

Perina, J.

T. Opatrny, J. Perina, “Non-image-forming polarization optical devices and Lorentz transformations—analogy,” Phys. Lett. A 181, 199–202 (1993).
[CrossRef]

J. Perina, Coherence of Light (Van Nostrand-Reinhold, London, 1971), pp. 77–88.

Roman, P.

G. P. Parent, P. Roman, “On the matrix formulation of the theory of partial polarization in terms of observables,” Nuovo Cimento 15, 370–388 (1960);R. Barakat, “Theory of the coherency matrix for light of arbitrary bandwidth,” J. Opt. Soc. Am. 53, 317–323 (1963);H. Takenaka, “A unified formalism for polarization optics by using group theory,” Nouv. Res. Opt. 4, 37–41 (1973).
[CrossRef]

Schumaker, B. L.

C. M. Caves, B. L. Schumaker, “New formalism for two-photon quantum optics. I. Quadrature phases and squeezed light,” Phys. Rev. A 31, 3068–3092 (1985);B. L. Schumaker, C. M. Caves, “New formalism for two-photon quantum optics. II. Mathematical foundation and compact notation,” Phys. Rev. A 31, 3093–3111 (1985).
[CrossRef] [PubMed]

Simon, R.

E. C. G. Sudarshan, R. Simon, N. Makunda, “Paraxial-wave optics and relativistic front description. I. The scalar theory,” Phys. Rev. A 28, 2921–2932 (1983);N. Makunda, R. Simon, E. C. G. Sudarshan, “Paraxial-wave optics and relativistic front description. II. The vector theory,” Phys. Rev. A 28, 2933–2942 (1983).
[CrossRef]

Son, D.

D. Han, Y. S. Kim, D. Son, “Photons, neutrinos, and gauge transformations,” Am. J. Phys. 54, 818–821 (1986).
[CrossRef]

D. Han, Y. S. Kim, D. Son, “E(2)-like little group for massless particles and polarization of neutrinos,” Phys. Rev. D 26, 3717–3725 (1982).
[CrossRef]

Sternberg, S.

V. Guillemin, S. Sternberg, Symplectic Techniques in Physics (Cambridge U. Press, Cambridge, 1984).

Sudarshan, E. C. G.

E. C. G. Sudarshan, R. Simon, N. Makunda, “Paraxial-wave optics and relativistic front description. I. The scalar theory,” Phys. Rev. A 28, 2921–2932 (1983);N. Makunda, R. Simon, E. C. G. Sudarshan, “Paraxial-wave optics and relativistic front description. II. The vector theory,” Phys. Rev. A 28, 2933–2942 (1983).
[CrossRef]

Swindell, W.

W. Swindell, Polarized Light (Dowden, Hutchinson, and Ross, Stroudsburg, Pa., 1975).

Vega, C.

Weinberg, S.

S. Weinberg, “Feynman rules for any spin II massless particles.” Phys. Rev. 134, B882–B896 (1964).
[CrossRef]

Wigner, E.

E. Wigner, “On unitary representations of the inhomogeneous Lorentz group,” Ann. Math. 40, 149–204 (1939).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), pp. 545–553.

Yabuzaki, T.

M. Kitano, T. Yabuzaki, “Observation of Lorentz-group Berry phases in polarization optics,” Phys. Lett. A 142, 321–325 (1989).
[CrossRef]

Yeh, L.

Y. S. Kim, L. Yeh, “E(2)-symmetric sheared states,” J. Math. Phys. 33, 1237–1246 (1992).
[CrossRef]

Yuen, H. P.

H. P. Yuen, “Two-photon coherent states of the radiation fields,” Phys. Rev. A 13, 2226–2243 (1976).
[CrossRef]

Am. J. Phys. (1)

D. Han, Y. S. Kim, D. Son, “Photons, neutrinos, and gauge transformations,” Am. J. Phys. 54, 818–821 (1986).
[CrossRef]

Ann. Math. (2)

E. Wigner, “On unitary representations of the inhomogeneous Lorentz group,” Ann. Math. 40, 149–204 (1939).
[CrossRef]

V. Bargmann, “Irreducible unitary representations of the Lorentz group,” Ann. Math. 48, 568–640 (1947).
[CrossRef]

J. Math. Phys. (2)

P. A. M. Dirac, “A remarkable representation of the 3+2 de Sitter group,” J. Math. Phys. 4, 901–909 (1963).
[CrossRef]

Y. S. Kim, L. Yeh, “E(2)-symmetric sheared states,” J. Math. Phys. 33, 1237–1246 (1992).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Nuovo Cimento (1)

G. P. Parent, P. Roman, “On the matrix formulation of the theory of partial polarization in terms of observables,” Nuovo Cimento 15, 370–388 (1960);R. Barakat, “Theory of the coherency matrix for light of arbitrary bandwidth,” J. Opt. Soc. Am. 53, 317–323 (1963);H. Takenaka, “A unified formalism for polarization optics by using group theory,” Nouv. Res. Opt. 4, 37–41 (1973).
[CrossRef]

Optik (1)

S. R. Cloude, “Group theory and polarization algebra,” Optik 75, 26–36 (1986).

Optik (Stuttgart) (1)

P. Pellat-Finet, M. Buasset, “What is common to both polarization optics and relativistic kinematics?” Optik (Stuttgart) 90, 101–106 (1992).

Phys. Lett. A (5)

D. Han, Y. S. Kim, M. E. Noz, “Wavelets, windows, and photons,” Phys. Lett. A 206, 299–304 (1995).
[CrossRef]

R. Y. Chiao, T. F. Jordan, “Lorentz-group Berry phases in squeezed light,” Phys. Lett. A 132, 77–81 (1988).
[CrossRef]

M. Kitano, T. Yabuzaki, “Observation of Lorentz-group Berry phases in polarization optics,” Phys. Lett. A 142, 321–325 (1989).
[CrossRef]

T. Opatrny, J. Perina, “Non-image-forming polarization optical devices and Lorentz transformations—analogy,” Phys. Lett. A 181, 199–202 (1993).
[CrossRef]

D. Han, Y. S. Kim, M. E. Noz, “Polarization optics and bilinear representation of the Lorentz group,” Phys. Lett. A 219, 26–32 (1996).
[CrossRef]

Phys. Rev. (1)

S. Weinberg, “Feynman rules for any spin II massless particles.” Phys. Rev. 134, B882–B896 (1964).
[CrossRef]

Phys. Rev. A (4)

H. P. Yuen, “Two-photon coherent states of the radiation fields,” Phys. Rev. A 13, 2226–2243 (1976).
[CrossRef]

C. M. Caves, B. L. Schumaker, “New formalism for two-photon quantum optics. I. Quadrature phases and squeezed light,” Phys. Rev. A 31, 3068–3092 (1985);B. L. Schumaker, C. M. Caves, “New formalism for two-photon quantum optics. II. Mathematical foundation and compact notation,” Phys. Rev. A 31, 3093–3111 (1985).
[CrossRef] [PubMed]

H. Bacry, M. Cadilhac, “Metaplectic group and Fourier optics,” Phys. Rev. A 23, 2533–2536 (1981).
[CrossRef]

E. C. G. Sudarshan, R. Simon, N. Makunda, “Paraxial-wave optics and relativistic front description. I. The scalar theory,” Phys. Rev. A 28, 2921–2932 (1983);N. Makunda, R. Simon, E. C. G. Sudarshan, “Paraxial-wave optics and relativistic front description. II. The vector theory,” Phys. Rev. A 28, 2933–2942 (1983).
[CrossRef]

Phys. Rev. D (1)

D. Han, Y. S. Kim, D. Son, “E(2)-like little group for massless particles and polarization of neutrinos,” Phys. Rev. D 26, 3717–3725 (1982).
[CrossRef]

Quantum Electron. (1)

A. S. Chirkin, A. A. Orlov, D. Yu. Parashchuk, “Quantum theory of two-mode interactions in optically anisotropic media with cubic nonlinearities: generation of quadrature- and polarization-squeezed light,” Quantum Electron. 23, 870–874 (1993).
[CrossRef]

Rev. Mod. Phys. (1)

P. A. M. Dirac, “Forms of relativistic dynamics,” Rev. Mod. Phys. 21, 392–399 (1949).
[CrossRef]

Other (10)

Y. S. Kim, M. E. Noz, Phase Space Picture of Quantum Mechanics (World Scientific, Singapore, 1991), pp. 77–122.

W. Swindell, Polarized Light (Dowden, Hutchinson, and Ross, Stroudsburg, Pa., 1975).

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Equations (86)

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Ex=A cos(kz-ωt+ϕ1),
Ey=B cos(kz-ωt+ϕ2),
ExEy=A exp[i(kz-ωt+ϕ1)]B exp[i(kz-ωt+ϕ2)].
EyEx=BAexp[i(ϕ2-ϕ1)],
w=r exp(iϕ),
r=B/A,ϕ=ϕ2-ϕ1.
1000,
P(0, δ)=exp(-iδ/2)00exp(iδ/2),
R(θ)=cos(θ/2)-sin(θ/2)sin(θ/2)cos(θ/2).
exp(-η1)00exp(-η2)=exp[-(η1+η2)/2]×exp(η/2)00exp(-η/2),
S(0, η)=exp(η/2)00exp(-η/2).
P(0, δ)=exp(-iδJ1),
J1=12 100-1.
R(θ)=exp(-iθJ3),
J3=12 0-ii0.
S(0, η)=exp(-iηK1),
K1=i2 100-1.
R(θ)=cos(θ/2)-sin(θ/2)sin(θ/2)cos(θ/2),
S(θ, η)=R(θ)S(0, η)R(-θ)=exp(η/2)cos2(θ/2)+exp(-η/2)sin2(θ/2)[exp(η/2)-exp(-η/2)]cos(θ/2)sin(θ/2)[exp(η/2)-exp(-η/2)]cos(θ/2)sin(θ/2)exp(-η/2)cos2(θ/2)+exp(η/2)sin2(θ/2).
S(θ2, η2)S(θ1, η1)=S(θ3, η3)R(ψ),
[J3, K1]=iK2.
exp(iδ1)00exp(iδ2)=exp[-i(δ1+δ2)/2]×exp(-iδ/2)00exp(iδ/2),
P(0, δ)=exp(-iδ/2)00exp(iδ/2).
P(θ, δ)=R(θ)P(0, δ)R(-θ)=exp(-iδ/2)cos2(θ/2)+exp(iδ/2)sin2(θ/2)[exp(-iδ/2)-exp(iδ/2)]cos(θ/2)sin(θ/2)[exp(-iδ/2)-exp(iδ/2)]cos(θ/2)sin(θ/2)exp(iδ/2)cos2(θ/2)+exp(-iδ/2)sin2(θ/2).
L=αβγρ,
α2β2γ2ρ2 α1β1γ1ρ1=α2α1+β2γ1α2β1+β2ρ1γ2α1+ρ2γ1γ2β1+ρ2ρ1.
w=ρw+γβw+α.
w1=ρ1w+γ1β1w+α1,w2=ρ2w1+γ2β2w1+α2.
w2=(γ2β1+ρ2ρ1)w+(γ2α1+ρ2γ1)(α2β1+β2ρ1)w+(α2α1+β2γ1).
αβγρ ExEy=αEx+βEyγEx+ρEy,
EyEx=γEx+ρEyαEx+βEy.
w=γ+ρwα+βw.
αβ-β*α*.
w=[exp(-iδ/2)]wexp(iδ/2)+βw.
exp(-iδ/2)β0exp(iδ/2).
N2=J2+K3,N3=J3-K2,
N2=0100,N3=0-i00.
[J1, N2]=iN3,[J1, N3]=-iN2,
[N2, N3]=0.
Q=P(0, π/2)=exp(-iπ/4)00exp(iπ/4).
J2=QJ3Q-1,K3=-QK2Q-1.
N2=QN3Q-1.
T2(τ)=exp(-iτN2)=1iτ01,
T3(τ)=exp(-iτN3)=1-τ01.
1-τ01 ExEy=Ex-τEyEy.
P(0, δ)=exp(-iδ/2)00exp(iδ/2).
P(0, δ)T3(τ)=1-exp(-iδ/2)τ01,
T3(τ)P(0, δ)=1-exp(iδ/2)τ01,
P(0, -δ)T3(τ)P(0, δ)=T3[exp(iδ)τ]=1-exp(iδ)τ01.
1β1011β201=1β1+β201.
10β1.
1β01
xy=cosh(η/2)sinh(η/2)sinh(η/2)cosh(η/2) xy.
xy=1b01 xy.
αβ0γρ0001,
1000cos ξ-sin ξ0sin ξcos ξ.
cos θ-sin θ00sin θcos θ0000100001.
J3=0-i00i00000000000.
[Ji, Jj]=iijkJk.
cosh η00sinh η01000010sinh η00cosh η,
K3=000i00000000i000.
K1=0i00i00000000000,K2=00i00000i0000000.
[Ji, Kj]=iijkKk,[Ki, Kj]=-iijkJk.
σ1=100-1,σ2=0110,σ3=0-ii0.
Ji=12 σi,
[Ji, Jj]=iijkJk.
Ki=i2 σi,
[Ki, Kj]=-iijkJk.
[Ji, Kj]=iijkKk.
J˙i=12 σi,K˙i=-i2 σi.
L=exp-i2 i=13(θiσi+iηiσi),
L˙=exp-i2 i=13(θiσi-iηiσi).
uu˙=-(x-iy),vv˙=(x+iy),
uv˙=(t+z),vu˙=-(t-z),
C=uv˙-uu˙vv˙-vu˙=uv(v˙-u˙),
v˙-u˙
g=-iσ3=0-110.
gσig-1=-(σi)T,
{g-1L˙g}T=g-1L˙Tg.
C=Ex*ExEy*ExEx*EyEy*Ey,
C=1r exp(iδ)r exp(-iδ)r2
C=t+zx-iyx+iyt-z,
C=LCL,
Lz=-ix y-y x,
Px=-i x,Py=-i y.
[Lz, Px]=iPy,[Lz, Py]=-iPx,[Px, Py]=0.

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