Abstract

In the course of developing an operator formalism in a previous paper we established commutation relations between pairs of (2×2) matrix operators representing various physical devices. These commutation relations were established under the stringent requirement that the outgoing Jones vector remains unchanged, independent of the order in which the operators operate on an arbitrary (quasimonochromatic) incoming field. In this paper we seek a wider class of operators that behave as though they commute, by imposing weaker requirements: (1) that the outgoing Jones vector remains unchanged only up to an arbitrary constant phase factor. In this case the outgoing coherency matrix (or the Stokes parameters), however, remains unchanged. Independent of this, we can impose a still weaker requirement: (2) that the outgoing coherency matrix remains unchanged only as far as the degree of polarization and the state of polarization is concerned but may be allowed to change in intensity. This paper deals with these weaker requirements. Pertinent examples of physical devices and their series combinations are also given.

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  1. A. S. Marathay, J. Opt. Soc. Am. 55, 969 (1965).
  2. M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959).
  3. E. Wolf, Nuovo Cimento 13, 1165 (1959).
  4. G. Parrent and P. Roman, Nuovo Cimento 15, 370 (1960).
  5. N. Wiener, Acta Math. 55, 118 (1930).
  6. E. Wolf, Nuovo Cimento 12, 884 (1954).
  7. R. Barakat, J. Opt. Soc. Am. 53, 317 (1963).
  8. E. L. O'Neill, Introduction to Statistical Optics (Addison-Wesley Publishing Co., Reading, Mass., 1963).
  9. R. C. Jones, J. Opt. Soc. Am. 31, 488 (1941).
  10. H. Hurwitz and R. C. Jones, J. Opt. Soc. Am. 31, 493 (1941).
  11. The 4×4 Mueller-matrix M⌃A representation of the 2×2 operator  is given by the relation M⌃A= T(Â×Â*)T-1, and similarly M⌃B = T (B⌃×B⌃*) T-1, where, T, apart from a constant multiplicative factor, is a 4×4 unitary matrix which transforms the elements of the coherency matrix to the Stokes parameters of the optical field (see Refs. 1 and 7). Now when the operators  and B⌃ have the property given in (3.1) we can easily show that M⌃ AM⌃ B = |γ|2M⌃BM⌃A. It therefore follows that our condition (1) is automatically taken care of in the Mueller method.
  12. U. Fano, Phys. Rev. 93, 121 (1956).
  13. P. Roman, Elementary Particles (North-Holland Publishing Company, Amsterdam, 1961).
  14. The refraction R and translation T matrices used in geometrical optics form an example of two linearly independent operators Â. They satisfy the conditions in (4.3). We have shown here that there is no operator B⌃ that satisfies (3.10) for γ ≠ 1. For the R and T matrices we refer to Ref. 8 and to W. Brouwer, Matrix Methods in Optical Instrument Design (W. A. Benjamin, Inc., New York, 1964).
  15. Although matrix operators of higher dimensions do not concern us in the present paper, we make the following remark for the sake of completeness. The procedure outlined in Section 3 is cumbersome and unmanageable for determining paracommutation relations for n×n matrix operators. But the relations established in this section clearly indicate that, at least as far as the unitary and Hermitian operators of dimension higher than two are concerned, the paracommuting operators can be determined. They are, in fact, the solution Jij given in (2.2). The definition given in (2.2) is applicable in any dimension.

Barakat, R.

R. Barakat, J. Opt. Soc. Am. 53, 317 (1963).

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959).

Fano, U.

U. Fano, Phys. Rev. 93, 121 (1956).

Hurwitz, H.

H. Hurwitz and R. C. Jones, J. Opt. Soc. Am. 31, 493 (1941).

Jones, R. C.

H. Hurwitz and R. C. Jones, J. Opt. Soc. Am. 31, 493 (1941).

R. C. Jones, J. Opt. Soc. Am. 31, 488 (1941).

Marathay, A. S.

A. S. Marathay, J. Opt. Soc. Am. 55, 969 (1965).

O’Neill, E. L.

E. L. O'Neill, Introduction to Statistical Optics (Addison-Wesley Publishing Co., Reading, Mass., 1963).

Parrent, G.

G. Parrent and P. Roman, Nuovo Cimento 15, 370 (1960).

Roman, P.

G. Parrent and P. Roman, Nuovo Cimento 15, 370 (1960).

P. Roman, Elementary Particles (North-Holland Publishing Company, Amsterdam, 1961).

Wiener, N.

N. Wiener, Acta Math. 55, 118 (1930).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959).

E. Wolf, Nuovo Cimento 12, 884 (1954).

E. Wolf, Nuovo Cimento 13, 1165 (1959).

Other (15)

A. S. Marathay, J. Opt. Soc. Am. 55, 969 (1965).

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959).

E. Wolf, Nuovo Cimento 13, 1165 (1959).

G. Parrent and P. Roman, Nuovo Cimento 15, 370 (1960).

N. Wiener, Acta Math. 55, 118 (1930).

E. Wolf, Nuovo Cimento 12, 884 (1954).

R. Barakat, J. Opt. Soc. Am. 53, 317 (1963).

E. L. O'Neill, Introduction to Statistical Optics (Addison-Wesley Publishing Co., Reading, Mass., 1963).

R. C. Jones, J. Opt. Soc. Am. 31, 488 (1941).

H. Hurwitz and R. C. Jones, J. Opt. Soc. Am. 31, 493 (1941).

The 4×4 Mueller-matrix M⌃A representation of the 2×2 operator  is given by the relation M⌃A= T(Â×Â*)T-1, and similarly M⌃B = T (B⌃×B⌃*) T-1, where, T, apart from a constant multiplicative factor, is a 4×4 unitary matrix which transforms the elements of the coherency matrix to the Stokes parameters of the optical field (see Refs. 1 and 7). Now when the operators  and B⌃ have the property given in (3.1) we can easily show that M⌃ AM⌃ B = |γ|2M⌃BM⌃A. It therefore follows that our condition (1) is automatically taken care of in the Mueller method.

U. Fano, Phys. Rev. 93, 121 (1956).

P. Roman, Elementary Particles (North-Holland Publishing Company, Amsterdam, 1961).

The refraction R and translation T matrices used in geometrical optics form an example of two linearly independent operators Â. They satisfy the conditions in (4.3). We have shown here that there is no operator B⌃ that satisfies (3.10) for γ ≠ 1. For the R and T matrices we refer to Ref. 8 and to W. Brouwer, Matrix Methods in Optical Instrument Design (W. A. Benjamin, Inc., New York, 1964).

Although matrix operators of higher dimensions do not concern us in the present paper, we make the following remark for the sake of completeness. The procedure outlined in Section 3 is cumbersome and unmanageable for determining paracommutation relations for n×n matrix operators. But the relations established in this section clearly indicate that, at least as far as the unitary and Hermitian operators of dimension higher than two are concerned, the paracommuting operators can be determined. They are, in fact, the solution Jij given in (2.2). The definition given in (2.2) is applicable in any dimension.

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