Abstract

Extending the work of part I of this series [ J. Opt. Soc. Am. A 10, 2008– 2016 ( 1993)], we analyze the structure of the eigenvalue spectrum as well as the propagation characteristics of the twisted Gaussian Schell-model beams. The manner in which the twist phase affects the spectrum, and hence the positivity property of the cross-spectral density, is brought out. Propagation characteristics of these beams are simply deduced from the elementary properties of their modes. It is shown that the twist phase lifts the degeneracy in the eigenvalue spectrum on the one hand and acts as incoherence in disguise on the other. An abstract Hilbert-space operator corresponding to the cross-spectral density of the twisted Gaussian Schell-modelbeam is explicitly constructed, bringing out the useful similarity between these cross-spectral densities and quantum-mechanical thermal-state-density operators of isotropic two-dimensional oscillators, with a term proportional to the angular momentum added to the Hamiltonian.

© 1993 Optical Society of America

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References

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  1. R. Simon, K. Sundar, N. Mukunda, “Twisted Gaussian Schell-model beams: I. Symmetry and normal-mode spectrum,” J. Opt. Soc. Am. A 10, 2008–2016 (1993).
    [CrossRef]
  2. R. Simon, N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10, 95–109 (1993).
    [CrossRef]
  3. F. Gori, “Mode propagation of the field generated by Collett–Wolf Schell-model sources,” Opt. Commun. 46, 149–154 (1983); A. Starikov, E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and of their radiation fields,”J. Opt. Soc. Am. 72, 923–928 (1982).
    [CrossRef]
  4. J. Schwinger, “On angular momentum,” in Quantum Theory of Angular Momentum, L. C. Biedenharn, H. Van Dam, eds. (Academic, New York, 1965), pp. 229–279.
  5. See, for example, A. E. Siegman, Lasers (Oxford U. Press, Oxford, 1986), Chap. 16.
  6. F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
    [CrossRef]
  7. R. Simon, E. C. G. Sudarshan, N. Mukunda, “Partially coherent beams and a generalised abcd-law,” Opt. Commun. 65, 322–328 (1988), and references therein.
    [CrossRef]
  8. R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized rays in first-order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
    [CrossRef]
  9. R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985), App. A in particular.
    [CrossRef] [PubMed]
  10. R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian Wigner distributions in quantum mechanics and optics,” Phys. Rev. A 36, 3868–3880 (1987), and references therein.
    [CrossRef] [PubMed]

1993 (2)

1988 (1)

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Partially coherent beams and a generalised abcd-law,” Opt. Commun. 65, 322–328 (1988), and references therein.
[CrossRef]

1987 (1)

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian Wigner distributions in quantum mechanics and optics,” Phys. Rev. A 36, 3868–3880 (1987), and references therein.
[CrossRef] [PubMed]

1985 (1)

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985), App. A in particular.
[CrossRef] [PubMed]

1984 (1)

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized rays in first-order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[CrossRef]

1983 (1)

F. Gori, “Mode propagation of the field generated by Collett–Wolf Schell-model sources,” Opt. Commun. 46, 149–154 (1983); A. Starikov, E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and of their radiation fields,”J. Opt. Soc. Am. 72, 923–928 (1982).
[CrossRef]

1980 (1)

F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

Gori, F.

F. Gori, “Mode propagation of the field generated by Collett–Wolf Schell-model sources,” Opt. Commun. 46, 149–154 (1983); A. Starikov, E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and of their radiation fields,”J. Opt. Soc. Am. 72, 923–928 (1982).
[CrossRef]

F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

Mukunda, N.

R. Simon, K. Sundar, N. Mukunda, “Twisted Gaussian Schell-model beams: I. Symmetry and normal-mode spectrum,” J. Opt. Soc. Am. A 10, 2008–2016 (1993).
[CrossRef]

R. Simon, N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10, 95–109 (1993).
[CrossRef]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Partially coherent beams and a generalised abcd-law,” Opt. Commun. 65, 322–328 (1988), and references therein.
[CrossRef]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian Wigner distributions in quantum mechanics and optics,” Phys. Rev. A 36, 3868–3880 (1987), and references therein.
[CrossRef] [PubMed]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985), App. A in particular.
[CrossRef] [PubMed]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized rays in first-order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[CrossRef]

Schwinger, J.

J. Schwinger, “On angular momentum,” in Quantum Theory of Angular Momentum, L. C. Biedenharn, H. Van Dam, eds. (Academic, New York, 1965), pp. 229–279.

Siegman, A. E.

See, for example, A. E. Siegman, Lasers (Oxford U. Press, Oxford, 1986), Chap. 16.

Simon, R.

R. Simon, N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10, 95–109 (1993).
[CrossRef]

R. Simon, K. Sundar, N. Mukunda, “Twisted Gaussian Schell-model beams: I. Symmetry and normal-mode spectrum,” J. Opt. Soc. Am. A 10, 2008–2016 (1993).
[CrossRef]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Partially coherent beams and a generalised abcd-law,” Opt. Commun. 65, 322–328 (1988), and references therein.
[CrossRef]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian Wigner distributions in quantum mechanics and optics,” Phys. Rev. A 36, 3868–3880 (1987), and references therein.
[CrossRef] [PubMed]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985), App. A in particular.
[CrossRef] [PubMed]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized rays in first-order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[CrossRef]

Sudarshan, E. C. G.

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Partially coherent beams and a generalised abcd-law,” Opt. Commun. 65, 322–328 (1988), and references therein.
[CrossRef]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian Wigner distributions in quantum mechanics and optics,” Phys. Rev. A 36, 3868–3880 (1987), and references therein.
[CrossRef] [PubMed]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985), App. A in particular.
[CrossRef] [PubMed]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized rays in first-order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[CrossRef]

Sundar, K.

J. Opt. Soc. Am. A (2)

Opt. Commun. (3)

F. Gori, “Mode propagation of the field generated by Collett–Wolf Schell-model sources,” Opt. Commun. 46, 149–154 (1983); A. Starikov, E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and of their radiation fields,”J. Opt. Soc. Am. 72, 923–928 (1982).
[CrossRef]

F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Partially coherent beams and a generalised abcd-law,” Opt. Commun. 65, 322–328 (1988), and references therein.
[CrossRef]

Phys. Rev. A (3)

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized rays in first-order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[CrossRef]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985), App. A in particular.
[CrossRef] [PubMed]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian Wigner distributions in quantum mechanics and optics,” Phys. Rev. A 36, 3868–3880 (1987), and references therein.
[CrossRef] [PubMed]

Other (2)

J. Schwinger, “On angular momentum,” in Quantum Theory of Angular Momentum, L. C. Biedenharn, H. Van Dam, eds. (Academic, New York, 1965), pp. 229–279.

See, for example, A. E. Siegman, Lasers (Oxford U. Press, Oxford, 1986), Chap. 16.

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Equations (34)

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W ( ρ , ρ ; ν ) = 1 2 π σ s 2 exp [ - 1 4 σ s 2 ( ρ 2 + ρ 2 ) - 1 2 σ g 2 ( ρ - ρ ) 2 - i u ( ν ) ρ ρ ] .
λ j , m = 2 a I a + b + ( c / 2 ) ( a + b - ( c / 2 ) a + b + ( c / 2 ) ) j ( 2 b + u 2 b - u ) m ,
a = ( 4 σ s 2 ) - 1 ,             b = ( 2 σ g 2 ) - 1 , c = 2 ( a 2 + 2 a b + u 2 / 4 ) 1 / 2 .
a + b c / 2 , b ≥∣ u / 2.
( c / 2 ) 2 = ( a + b ) 2 - ( b 2 - u 2 / 4 ) ,
- σ g - 2 u / 2 σ g - 2
λ j , m = 2 a I a + b + c / 2 [ b 2 - u 2 / 4 ( a + b + c / 2 ) 2 ] j ( b + u / 2 b - u / 2 ) m = 2 a I a + b + c / 2 ( b + u / 2 a + b + c / 2 ) j + m ( b - u / 2 a + b + c / 2 ) j - m .
[ ( b - u / 2 ) / ( a + b + c / 2 ) ] j - m δ j , m , λ j , m a I a + b ( b a + b ) 2 j δ j , m .
[ ( b + u / 2 ) / ( a + b + c / 2 ) ] j + m δ j , - m , λ j , m a I a + b ( b a + b ) 2 j δ j , m .
W ( ρ , ρ ; ν ) = j , m λ j , m Φ j , m ( ρ ) Φ j , m * ( ρ ) ,
t r W ^ = W ( ρ , ρ ; ν ) d 2 ρ = I = j , m λ j , m .
λ j , m = 2 a I a + b + c / 2 ( b a + b + c / 2 ) j + m ( b a + b + c / 2 ) j - m ,
Ω 0 = 1 ,             Ω 1 / 2 = [ 1 2 i 2 i 2 1 2 ] , Ω 1 = [ 1 2 i 2 - 1 2 i 2 0 i 2 - 1 2 i 2 1 2 ] , Ω 3 / 2 = [ 1 2 2 i 3 2 2 - 3 2 2 - i 2 2 i 3 2 2 - 1 2 2 i 2 2 - 3 2 2 - 3 2 2 i 2 2 - 1 2 2 i 3 2 2 - i 2 2 - 3 2 2 i 3 2 2 1 2 2 ] .
Ω m . m j = Ω - m , - m j ,             Ω m , m j = Ω m , m j .
Φ 0 , 0 ( ρ , θ ) = c / π exp [ - ( 1 / 2 ) c ρ 2 ] , Φ 1 / 2 , 1 / 2 ( ρ , θ ) = ( c ρ / π ) exp [ - ( 1 / 2 ) c ρ 2 ] exp ( i θ ) , Φ 1 / 2 , - 1 / 2 = i ( c ρ / π ) exp [ - ( 1 / 2 ) c ρ 2 ] exp ( - i θ ) , Φ 1 , 1 ( ρ , θ ) = ( c 3 / 2 ρ 2 / 2 π ) exp [ - ( 1 / 2 ) c ρ 2 ] exp ( 2 i θ ) , Φ 1 , 0 ( ρ , θ ) = - i c / π ( 1 - c ρ 2 ) exp [ - ( 1 / 2 ) c ρ 2 ] , Φ 1 , - 1 ( ρ , θ ) = - ( c 3 / 2 ρ 2 / 2 π ) exp [ - ( 1 / 2 ) c ρ 2 ] exp ( - 2 i θ ) .
c - 1 / 2 c - 1 / 2 ( z ) = M ( z ) c - 1 / 2 , M ( z ) = [ 1 + ( λ z / 2 π c - 1 ) 2 ] 1 / 2 .
R ( z ) = z [ 1 + ( 2 π c - 1 / λ z ) 2 ] ,
Φ j , m ( ρ ) Φ j , m ( ρ ; z ) = M ( z ) - 1 exp [ i η j ( z ) ] Φ j , m [ M ( z ) - 1 ρ ] × exp [ i π ρ 2 / λ R ( z ) ] .
W ( ρ , ρ ; ν ) = j = 0 , 1 / 2 , 1 , m = - j j λ j , m Φ j , m ( ρ ) Φ j m * ( ρ ) .
( a , b , c , u ) M ( z ) - 2 ( a , b , c , u ) .
σ s σ s ( z ) = M ( z ) σ s , σ g σ g ( z ) = M ( z ) σ g , u u ( z ) = M ( z ) - 2 u .
σ g ( z ) / σ s ( z ) = const . , u ( z ) σ s ( z ) 2 = const .
W ( ρ , ρ ; ν ; σ s , σ g , u ) W [ ρ , ρ , ν ; σ s ( z ) , σ g ( z ) , u ( z ) ] × exp [ i π λ R ( z ) ( ρ 2 - ρ 2 ) ] .
W [ ρ , ρ ; ν ; σ s ( z ) , σ g ( z ) , u ( z ) ] exp [ i π λ R ( z ) ( ρ 2 - ρ 2 ) ] = j m λ j , m Φ j , m ( ρ ; z ) Φ j , m * ( ρ ; z ) ,
Φ j , m ( ρ ; z ) = M ( z ) - 1 Φ j , m [ M ( z ) - 1 ρ ] × exp [ i π ρ 2 / λ R ( z ) ] exp [ i η j ( z ) ] .
c - 1 = ( 1 4 σ s 4 + 1 σ s 2 σ g 2 + u 2 4 ) 1 / 2 .
z R = 2 π c - 1 / λ , Δ θ = Lt z σ s ( z ) / z = λ σ s / 2 π c - 1 .
W ^ = W ^ ( J ^ 0 , J ^ 3 ) .
W ^ = λ ( J ^ 0 , J ^ 3 ) = 2 a I a + b + c / 2 ( a + b - c / 2 a + b + c / 2 ) J ^ 0 ( 2 b + u 2 b - u ) J ^ 3 = 2 a I a + b + c / 2 ( b + u / 2 a + b + c / 2 ) J ^ 0 + J ^ 3 ( b - u / 2 a + b + c / 2 ) J ^ 0 - J ^ 3 .
2 ξ = - ln ( a + b - c / 2 a + b + c / 2 ) , 2 ζ ξ = ln ( 2 b + u 2 b - u ) .
W ^ = 2 a I a + b + c / 2 exp [ - ξ ( 2 J ^ 0 - 2 ζ J ^ 3 ) ] .
2 J ^ 0 = α ^ x a ^ x + a ^ y a ^ y = H ^ 0 , 2 J ^ 3 = x ^ p ^ y - y ^ p ^ x = L ^ z ,
W ^ = 2 a I a + b + c / 2 exp ( - ξ H ^ ) ,
H ^ = H ^ 0 - ζ L ^ z .

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