Abstract

Extending the work of earlier papers on the relativistic-front description of paraxial optics and the formulation of Fourier optics for vector waves consistent with the Maxwell equations, we generalize the Jones calculus of axial plane waves to describe the action of the most general linear optical system on paraxial Maxwell fields. Several examples are worked out, and in each case it is shown that the formalism leads to physically correct results. The importance of retaining the small components of the field vectors along the axis of the system for a consistent description is emphasized.

© 1985 Optical Society of America

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References

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  1. R. C. Jones, “A new calculus for the treatment of optical systems I. Description and discussion of the calculus,” J. Opt. Soc. Am. 31, 488–493 (1941);H. Hurwitz, R. C. Jones, “A new calculus for the treatment of optical systems. II. Proof of three general equivalence theorems,” J. Opt. Soc. Am. 31, 493–499 (1941).All the papers of this series and several other important contributions to polarization optics have been reprinted in W. Swindell, ed., Polarized Light (Dowden, Hutchinson and Ross, Stroudsburg, Pa., 1975).
    [CrossRef]
  2. See, for example, S. Pancharatnam, “Generalized theory of interference and its applications,” Proc. Ind. Acad. Sci. A 44, 247–262, 398–417 (1956);Proc. Ind. Acad. Sci. A 45, 402–411 (1957);Proc. Ind. Acad. Sci. A 46, 1–18 (1957).
  3. E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento 13, 1165–1181 (1959);G. B. Parrent, P. Roman, “On the matrix formulation of the theory of partial polarization in terms of observables,” Nuovo Cimento 15, 370–388 (1960);N. Wiener, “Generalized harmonic analysis,” Acta Math. 55, 182–195 (1930).
    [CrossRef]
  4. G. G. Stokes, “On the composition and resolution of streams of polarized light from different sources,” Trans. Cambridge Phil. Soc. 9, 399–416 (1852);H. Mueller, “The foundations of optics,” J. Opt. Soc. Am. 38, 661 (1948);S. Chandrasekhar, Radiative Transfer (Dover, New York, 1950), Chap. 1.
  5. R. Simon, “The connection between Mueller and Jones matrices of polarization optics,” Opt. Commun. 42, 293–297 (1982);R. Barakat, “Bilinear constraints between elements of the 4 × 4 Mueller–Jones transfer matrix of polarization theory,” Opt. Commun. 38, 159–161 (1981).
    [CrossRef]
  6. E. C. G. Sudarshan, R. Simon, N. Mukunda, “Paraxial wave optics and relativistic front description I: the scalar theory,” Phys. Rev. A 28, 2921–2932 (1983).
    [CrossRef]
  7. N. Mukunda, R. Simon, E. C. G. Sudarshan, “Paraxial wave optics and relativistic front description II: the vector theory,” Phys. Rev. A28, 2933–2942 (1983).See also H. Bacry, “Group theory and paraxial optics,” invited talk at the XIIIth International Colloquium on Group Theoretical Methods in Physics, University of Maryland, College Park, Md., May 1984.
    [CrossRef]
  8. P. A. M. Dirac, “Forms of relativistic dynamics,” Rev. Mod. Phys. 21, 392–399 (1949).
    [CrossRef]
  9. N. Mukunda, R. Simon, E. C. G. Sudarshan, “Fourier optics for the Maxwell field: formalism and applications,” J. Opt. Soc. Am. A 2, 416–426 (1985).
    [CrossRef]
  10. See in this connection M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
    [CrossRef]
  11. P. Roman, “Generalized Stokes parameters for waves of arbitrary form,” Nuovo Cimento 13, 974–982 (1959);G. Ramachandran, M. V. N. Murthy, K. S. Mallesh, “SU(3) representation of the polarization of light,” Pramāna 15, 357–369 (1980).
    [CrossRef]
  12. E. C. G. Sudarshan, “Quantum electrodynamics and light rays,” Physica 96A, 315–320 (1979);“Quantum theory of radiative transfer,” Phys. Rev. A 23, 2802–2809 (1981).

1985

1983

E. C. G. Sudarshan, R. Simon, N. Mukunda, “Paraxial wave optics and relativistic front description I: the scalar theory,” Phys. Rev. A 28, 2921–2932 (1983).
[CrossRef]

1982

R. Simon, “The connection between Mueller and Jones matrices of polarization optics,” Opt. Commun. 42, 293–297 (1982);R. Barakat, “Bilinear constraints between elements of the 4 × 4 Mueller–Jones transfer matrix of polarization theory,” Opt. Commun. 38, 159–161 (1981).
[CrossRef]

1979

E. C. G. Sudarshan, “Quantum electrodynamics and light rays,” Physica 96A, 315–320 (1979);“Quantum theory of radiative transfer,” Phys. Rev. A 23, 2802–2809 (1981).

1975

See in this connection M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

1959

P. Roman, “Generalized Stokes parameters for waves of arbitrary form,” Nuovo Cimento 13, 974–982 (1959);G. Ramachandran, M. V. N. Murthy, K. S. Mallesh, “SU(3) representation of the polarization of light,” Pramāna 15, 357–369 (1980).
[CrossRef]

E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento 13, 1165–1181 (1959);G. B. Parrent, P. Roman, “On the matrix formulation of the theory of partial polarization in terms of observables,” Nuovo Cimento 15, 370–388 (1960);N. Wiener, “Generalized harmonic analysis,” Acta Math. 55, 182–195 (1930).
[CrossRef]

1956

See, for example, S. Pancharatnam, “Generalized theory of interference and its applications,” Proc. Ind. Acad. Sci. A 44, 247–262, 398–417 (1956);Proc. Ind. Acad. Sci. A 45, 402–411 (1957);Proc. Ind. Acad. Sci. A 46, 1–18 (1957).

1949

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

1941

1852

G. G. Stokes, “On the composition and resolution of streams of polarized light from different sources,” Trans. Cambridge Phil. Soc. 9, 399–416 (1852);H. Mueller, “The foundations of optics,” J. Opt. Soc. Am. 38, 661 (1948);S. Chandrasekhar, Radiative Transfer (Dover, New York, 1950), Chap. 1.

Dirac, P. A. M.

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

Jones, R. C.

Lax, M.

See in this connection M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Louisell, W. H.

See in this connection M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

McKnight, W. B.

See in this connection M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Mukunda, N.

N. Mukunda, R. Simon, E. C. G. Sudarshan, “Fourier optics for the Maxwell field: formalism and applications,” J. Opt. Soc. Am. A 2, 416–426 (1985).
[CrossRef]

E. C. G. Sudarshan, R. Simon, N. Mukunda, “Paraxial wave optics and relativistic front description I: the scalar theory,” Phys. Rev. A 28, 2921–2932 (1983).
[CrossRef]

N. Mukunda, R. Simon, E. C. G. Sudarshan, “Paraxial wave optics and relativistic front description II: the vector theory,” Phys. Rev. A28, 2933–2942 (1983).See also H. Bacry, “Group theory and paraxial optics,” invited talk at the XIIIth International Colloquium on Group Theoretical Methods in Physics, University of Maryland, College Park, Md., May 1984.
[CrossRef]

Pancharatnam, S.

See, for example, S. Pancharatnam, “Generalized theory of interference and its applications,” Proc. Ind. Acad. Sci. A 44, 247–262, 398–417 (1956);Proc. Ind. Acad. Sci. A 45, 402–411 (1957);Proc. Ind. Acad. Sci. A 46, 1–18 (1957).

Roman, P.

P. Roman, “Generalized Stokes parameters for waves of arbitrary form,” Nuovo Cimento 13, 974–982 (1959);G. Ramachandran, M. V. N. Murthy, K. S. Mallesh, “SU(3) representation of the polarization of light,” Pramāna 15, 357–369 (1980).
[CrossRef]

Simon, R.

N. Mukunda, R. Simon, E. C. G. Sudarshan, “Fourier optics for the Maxwell field: formalism and applications,” J. Opt. Soc. Am. A 2, 416–426 (1985).
[CrossRef]

E. C. G. Sudarshan, R. Simon, N. Mukunda, “Paraxial wave optics and relativistic front description I: the scalar theory,” Phys. Rev. A 28, 2921–2932 (1983).
[CrossRef]

R. Simon, “The connection between Mueller and Jones matrices of polarization optics,” Opt. Commun. 42, 293–297 (1982);R. Barakat, “Bilinear constraints between elements of the 4 × 4 Mueller–Jones transfer matrix of polarization theory,” Opt. Commun. 38, 159–161 (1981).
[CrossRef]

N. Mukunda, R. Simon, E. C. G. Sudarshan, “Paraxial wave optics and relativistic front description II: the vector theory,” Phys. Rev. A28, 2933–2942 (1983).See also H. Bacry, “Group theory and paraxial optics,” invited talk at the XIIIth International Colloquium on Group Theoretical Methods in Physics, University of Maryland, College Park, Md., May 1984.
[CrossRef]

Stokes, G. G.

G. G. Stokes, “On the composition and resolution of streams of polarized light from different sources,” Trans. Cambridge Phil. Soc. 9, 399–416 (1852);H. Mueller, “The foundations of optics,” J. Opt. Soc. Am. 38, 661 (1948);S. Chandrasekhar, Radiative Transfer (Dover, New York, 1950), Chap. 1.

Sudarshan, E. C. G.

N. Mukunda, R. Simon, E. C. G. Sudarshan, “Fourier optics for the Maxwell field: formalism and applications,” J. Opt. Soc. Am. A 2, 416–426 (1985).
[CrossRef]

E. C. G. Sudarshan, R. Simon, N. Mukunda, “Paraxial wave optics and relativistic front description I: the scalar theory,” Phys. Rev. A 28, 2921–2932 (1983).
[CrossRef]

E. C. G. Sudarshan, “Quantum electrodynamics and light rays,” Physica 96A, 315–320 (1979);“Quantum theory of radiative transfer,” Phys. Rev. A 23, 2802–2809 (1981).

N. Mukunda, R. Simon, E. C. G. Sudarshan, “Paraxial wave optics and relativistic front description II: the vector theory,” Phys. Rev. A28, 2933–2942 (1983).See also H. Bacry, “Group theory and paraxial optics,” invited talk at the XIIIth International Colloquium on Group Theoretical Methods in Physics, University of Maryland, College Park, Md., May 1984.
[CrossRef]

Wolf, E.

E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento 13, 1165–1181 (1959);G. B. Parrent, P. Roman, “On the matrix formulation of the theory of partial polarization in terms of observables,” Nuovo Cimento 15, 370–388 (1960);N. Wiener, “Generalized harmonic analysis,” Acta Math. 55, 182–195 (1930).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Nuovo Cimento

E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento 13, 1165–1181 (1959);G. B. Parrent, P. Roman, “On the matrix formulation of the theory of partial polarization in terms of observables,” Nuovo Cimento 15, 370–388 (1960);N. Wiener, “Generalized harmonic analysis,” Acta Math. 55, 182–195 (1930).
[CrossRef]

P. Roman, “Generalized Stokes parameters for waves of arbitrary form,” Nuovo Cimento 13, 974–982 (1959);G. Ramachandran, M. V. N. Murthy, K. S. Mallesh, “SU(3) representation of the polarization of light,” Pramāna 15, 357–369 (1980).
[CrossRef]

Opt. Commun.

R. Simon, “The connection between Mueller and Jones matrices of polarization optics,” Opt. Commun. 42, 293–297 (1982);R. Barakat, “Bilinear constraints between elements of the 4 × 4 Mueller–Jones transfer matrix of polarization theory,” Opt. Commun. 38, 159–161 (1981).
[CrossRef]

Phys. Rev. A

E. C. G. Sudarshan, R. Simon, N. Mukunda, “Paraxial wave optics and relativistic front description I: the scalar theory,” Phys. Rev. A 28, 2921–2932 (1983).
[CrossRef]

See in this connection M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Physica

E. C. G. Sudarshan, “Quantum electrodynamics and light rays,” Physica 96A, 315–320 (1979);“Quantum theory of radiative transfer,” Phys. Rev. A 23, 2802–2809 (1981).

Proc. Ind. Acad. Sci. A

See, for example, S. Pancharatnam, “Generalized theory of interference and its applications,” Proc. Ind. Acad. Sci. A 44, 247–262, 398–417 (1956);Proc. Ind. Acad. Sci. A 45, 402–411 (1957);Proc. Ind. Acad. Sci. A 46, 1–18 (1957).

Rev. Mod. Phys.

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

Trans. Cambridge Phil. Soc.

G. G. Stokes, “On the composition and resolution of streams of polarized light from different sources,” Trans. Cambridge Phil. Soc. 9, 399–416 (1852);H. Mueller, “The foundations of optics,” J. Opt. Soc. Am. 38, 661 (1948);S. Chandrasekhar, Radiative Transfer (Dover, New York, 1950), Chap. 1.

Other

N. Mukunda, R. Simon, E. C. G. Sudarshan, “Paraxial wave optics and relativistic front description II: the vector theory,” Phys. Rev. A28, 2933–2942 (1983).See also H. Bacry, “Group theory and paraxial optics,” invited talk at the XIIIth International Colloquium on Group Theoretical Methods in Physics, University of Maryland, College Park, Md., May 1984.
[CrossRef]

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Equations (50)

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E a a b B b ,
E 3 i k a E a ,
B 3 i k a B a .
A a i k E a , A 0 i 2 k E 3 ,
E ( x ) = [ E 1 ( x ) E 2 ( x ) E 3 ( x ) ] [ E 1 ( x ) E 2 ( x ) i k a E a ( x ) ]
B ( x ) = [ B 1 ( x ) B 2 ( x ) B 3 ( x ) ] [ E 2 ( x ) E 1 ( x ) i k a b a E b ( x ) ] .
E = B = 0
1 c t B + E = 0 .
E T ( x ) = [ E 1 ( x ) E 2 ( x ) 0 ] .
P a = i k x a
G 1 = ( 0 0 0 0 0 0 i 0 0 ) , G 2 = ( 0 0 0 0 0 0 0 i 0 ) .
Q a = x a + i k G a .
G a G b = 0 .
[ Q a , Q b ] = [ P a , P b ] = 0 , [ Q a , P b ] = i k δ a b .
E = exp ( i G a P a ) E T .
E T E T = [ E 1 ( x ) E 2 ( x ) 0 ] = Ω T [ E 1 ( x ) E 2 ( x ) 0 ] = Ω T E T ,
( Ω T ) 31 = ( Ω T ) 32 = 0 .
E = Ω E Ω = exp ( i G a P a ) Ω T exp ( i G b P b ) .
J = exp ( i 2 f x 2 ) × 1 ,
J = exp ( ikd 2 P 2 ) × 1 .
J = j 0 ( x , P ) × 1 2 × 2 + r = 1 3 j r ( x , P ) σ r .
j 0 = 1 2 Tr J , j r = 1 2 Tr J σ r .
( Ω T ) 33 = j 0 ( x , P ) .
Ω T = j 0 ( x , P ) × 1 3 × 3 + r = 1 3 j r ( x , P ) σ ¯ r .
S r = exp ( i G a P a ) σ ¯ r exp ( i G b P b ) ;
Ω = j 0 ( Q , P ) + r = 1 3 j r ( Q , P ) S r .
σ ¯ r exp ( i G a P a ) = σ ¯ r ,
B = exp ( i G a P a ) B T , B T = ( B 1 B 2 0 ) = ( E 2 E 1 0 ) .
E = Ω E , B = Ω B B , Ω = j 0 ( Q , P ) + r = 1 3 j r ( Q , P ) S r , Ω B = j 0 ( Q , P ) + r = 1 3 r j r ( Q , P ) S r , 1 = 3 = 1 , 2 = 1 .
J = ( cos θ sin θ sin θ cos θ ) , J 0 = cos θ , j 1 = j 3 = 0 , j 2 = i sin θ .
E = ( 0 1 0 ) u , B = ( 1 0 α ) u , u ( x , t ) = u 0 exp [ i k ( x 3 + α x 1 ) i ω t ] .
P 1 u = α u , P 2 u = 0 .
E = Ω E = ( sin θ cos θ α sin θ ) u , B = Ω B = ( cos θ sin θ α cos θ ) u .
θ = π / 2 : E = B , B = E .
J = ( j 0 + j 3 0 0 j 0 j 3 ) .
j 0 = cos δ / 2 , j 3 = i sin δ / 2 ,
0 j 0 ± j 3 1 .
E = 1 2 ( 1 1 α β ) υ , υ ( x , t ) = υ 0 exp [ i k ( α x 1 + β x 2 + x 3 ) i ω t ] .
P 1 υ = α υ , P 2 υ = β υ .
E = Ω E = 1 2 ( exp ( i δ / 2 ) exp ( i δ / 2 ) α exp ( i δ / 2 ) β exp ( i δ / 2 ) ) υ .
E = 1 2 ( 1 1 α ) u .
E = Ω E = 1 2 ( 0 1 0 ) u ,
j 0 = exp ( i k 2 f x 2 ) , j r = 0 .
Ω = Ω B = j 0 ( Q ) = exp [ i k 2 f ( x + i k G ) 2 ] = exp ( i k 2 f x 2 ) ( 1 0 0 0 1 0 x 1 f x 2 f 1 ) .
| P a ( x ) | P 3 ( x ) .
[ E a ( x ) E 3 ( x ) B a ( x ) B 3 ( x ) ] exp ( i k 2 f x 2 ) [ E a ( x ) E 3 ( x ) + x a f E a ( x ) B a ( x ) B 3 ( x ) + x a f B a ( x ) ] .
P 3 ( x ) P 3 ( x ) , P a ( x ) P a ( x ) x a f P 3 ( x ) .
P ( x ) = [ 1 u ( x a ) , 1 ] .
P ( x ) = [ 1 u ( x a ) x f , 1 ] = [ 1 υ ( b x ) , 1 ]
1 u + 1 υ = 1 f , b = υ u a .

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