Abstract

We study the properties of Quasi-Homogeneous Isotropic Electromagnetic (QuHIEM) Sources, a model for partially-coherent secondary light sources beyond the scalar and paraxial approximations. Our results include polarization properties in the far zone and the realizability condition. We demonstrate these results for sources with a degree of coherence described by Gaussians.

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  1. A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am.58, 1256–1259 (1968).
    [CrossRef]
  2. E. Wolf, “Coherence and radiometry,” J. Opt. Soc. Am.68, 6–17 (1978).
    [CrossRef]
  3. A. T. Friberg, ed., Selected Papers on Coherence and Radiometry, SPIE Milestone Series MS 69 (SPIE, 1993).
  4. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, 2007).
  5. J. H. Lambert, Photometria sive de mensure et gradibus luminis colorum et umbra (Bassel, 1760).
  6. W. H. Carter and E. Wolf, “Coherence properties of Lambertian and non-Lambertian sources,” J. Opt. Soc. Amer.65, 1067–1071 (1975).
    [CrossRef]
  7. E. Wolf, “Invariance of the spectrum of light on propagation,” Phys. Rev. Lett.56, 1370–1372 (1986).
    [CrossRef] [PubMed]
  8. E. Wolf, “Non-cosmological redshifts of spectral lines,” Nature326, 363–365 (1987).
    [CrossRef]
  9. E. Wolf and D. F. V. James, “Correlation-induced spectral changes,” Rep. Prog. Phys.59, 771–818 (1996).
    [CrossRef]
  10. D. F. V. James, “Change of polarization of light-beams on propagation in free-space,” J. Opt. Soc. Am. A111641–1643 (1994).
    [CrossRef]
  11. F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, “Beam coherence-polarization matrix,” Pure and Applied Optics7941–951 (1998).
    [CrossRef]
  12. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A312, 263–267 (2003).
    [CrossRef]
  13. J. Tervo, T. Setälä, and A. T. Friberg, “Theory of partially coherent electromagnetic fields in the space-frequency domain,” J. Opt. Soc. Am. A212205–2215 (2004).
    [CrossRef]
  14. O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun.246, 35–43 (2005).
    [CrossRef]
  15. D. F. V. James, “Polarization of light radiated by black-body sources,” Opt. Commun.109, 209 (1994).
    [CrossRef]
  16. G. K. Batchelor, The Theory of Homogeneous Turbulence (Cambridge Science Classics, 1990).
  17. A. C. Schell, “A technique for determination of the radiation pattern of a partially coherent aperture,” IEEE Trans. Antennas Propag.AP15187–188 (1967).
    [CrossRef]
  18. G. Toraldo di Francia, Electromagnetic Waves (Interscience, 1955), pp. 218–221.
  19. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1975), p. 437.
  20. M. A. Alonso, O Korotkova, and E. Wolf, “Propagation of the electric correlation matrix and the van CittertZernike theorem for random electromagnetic fields,” J. Mod. Opt.53, 969–978 (2006).
    [CrossRef]
  21. F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell-model sources,” J. Opt. Soc. Am A25, 1016 (2008).
    [CrossRef]

2008

F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell-model sources,” J. Opt. Soc. Am A25, 1016 (2008).
[CrossRef]

2006

M. A. Alonso, O Korotkova, and E. Wolf, “Propagation of the electric correlation matrix and the van CittertZernike theorem for random electromagnetic fields,” J. Mod. Opt.53, 969–978 (2006).
[CrossRef]

2005

O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun.246, 35–43 (2005).
[CrossRef]

2004

2003

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A312, 263–267 (2003).
[CrossRef]

1998

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, “Beam coherence-polarization matrix,” Pure and Applied Optics7941–951 (1998).
[CrossRef]

1996

E. Wolf and D. F. V. James, “Correlation-induced spectral changes,” Rep. Prog. Phys.59, 771–818 (1996).
[CrossRef]

1994

D. F. V. James, “Change of polarization of light-beams on propagation in free-space,” J. Opt. Soc. Am. A111641–1643 (1994).
[CrossRef]

D. F. V. James, “Polarization of light radiated by black-body sources,” Opt. Commun.109, 209 (1994).
[CrossRef]

1987

E. Wolf, “Non-cosmological redshifts of spectral lines,” Nature326, 363–365 (1987).
[CrossRef]

1986

E. Wolf, “Invariance of the spectrum of light on propagation,” Phys. Rev. Lett.56, 1370–1372 (1986).
[CrossRef] [PubMed]

1978

1975

W. H. Carter and E. Wolf, “Coherence properties of Lambertian and non-Lambertian sources,” J. Opt. Soc. Amer.65, 1067–1071 (1975).
[CrossRef]

1968

1967

A. C. Schell, “A technique for determination of the radiation pattern of a partially coherent aperture,” IEEE Trans. Antennas Propag.AP15187–188 (1967).
[CrossRef]

Alonso, M. A.

M. A. Alonso, O Korotkova, and E. Wolf, “Propagation of the electric correlation matrix and the van CittertZernike theorem for random electromagnetic fields,” J. Mod. Opt.53, 969–978 (2006).
[CrossRef]

Batchelor, G. K.

G. K. Batchelor, The Theory of Homogeneous Turbulence (Cambridge Science Classics, 1990).

Borghi, R.

F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell-model sources,” J. Opt. Soc. Am A25, 1016 (2008).
[CrossRef]

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, “Beam coherence-polarization matrix,” Pure and Applied Optics7941–951 (1998).
[CrossRef]

Carter, W. H.

W. H. Carter and E. Wolf, “Coherence properties of Lambertian and non-Lambertian sources,” J. Opt. Soc. Amer.65, 1067–1071 (1975).
[CrossRef]

Friberg, A. T.

Gori, F.

F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell-model sources,” J. Opt. Soc. Am A25, 1016 (2008).
[CrossRef]

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, “Beam coherence-polarization matrix,” Pure and Applied Optics7941–951 (1998).
[CrossRef]

Guattari, G.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, “Beam coherence-polarization matrix,” Pure and Applied Optics7941–951 (1998).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1975), p. 437.

James, D. F. V.

E. Wolf and D. F. V. James, “Correlation-induced spectral changes,” Rep. Prog. Phys.59, 771–818 (1996).
[CrossRef]

D. F. V. James, “Change of polarization of light-beams on propagation in free-space,” J. Opt. Soc. Am. A111641–1643 (1994).
[CrossRef]

D. F. V. James, “Polarization of light radiated by black-body sources,” Opt. Commun.109, 209 (1994).
[CrossRef]

Korotkova, O

M. A. Alonso, O Korotkova, and E. Wolf, “Propagation of the electric correlation matrix and the van CittertZernike theorem for random electromagnetic fields,” J. Mod. Opt.53, 969–978 (2006).
[CrossRef]

Korotkova, O.

O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun.246, 35–43 (2005).
[CrossRef]

Lambert, J. H.

J. H. Lambert, Photometria sive de mensure et gradibus luminis colorum et umbra (Bassel, 1760).

Ramírez-Sánchez, V.

F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell-model sources,” J. Opt. Soc. Am A25, 1016 (2008).
[CrossRef]

Santarsiero, M.

F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell-model sources,” J. Opt. Soc. Am A25, 1016 (2008).
[CrossRef]

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, “Beam coherence-polarization matrix,” Pure and Applied Optics7941–951 (1998).
[CrossRef]

Schell, A. C.

A. C. Schell, “A technique for determination of the radiation pattern of a partially coherent aperture,” IEEE Trans. Antennas Propag.AP15187–188 (1967).
[CrossRef]

Setälä, T.

Tervo, J.

Toraldo di Francia, G.

G. Toraldo di Francia, Electromagnetic Waves (Interscience, 1955), pp. 218–221.

Vicalvi, S.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, “Beam coherence-polarization matrix,” Pure and Applied Optics7941–951 (1998).
[CrossRef]

Walther, A.

Wolf, E.

M. A. Alonso, O Korotkova, and E. Wolf, “Propagation of the electric correlation matrix and the van CittertZernike theorem for random electromagnetic fields,” J. Mod. Opt.53, 969–978 (2006).
[CrossRef]

O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun.246, 35–43 (2005).
[CrossRef]

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A312, 263–267 (2003).
[CrossRef]

E. Wolf and D. F. V. James, “Correlation-induced spectral changes,” Rep. Prog. Phys.59, 771–818 (1996).
[CrossRef]

E. Wolf, “Non-cosmological redshifts of spectral lines,” Nature326, 363–365 (1987).
[CrossRef]

E. Wolf, “Invariance of the spectrum of light on propagation,” Phys. Rev. Lett.56, 1370–1372 (1986).
[CrossRef] [PubMed]

E. Wolf, “Coherence and radiometry,” J. Opt. Soc. Am.68, 6–17 (1978).
[CrossRef]

W. H. Carter and E. Wolf, “Coherence properties of Lambertian and non-Lambertian sources,” J. Opt. Soc. Amer.65, 1067–1071 (1975).
[CrossRef]

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, 2007).

IEEE Trans. Antennas Propag.

A. C. Schell, “A technique for determination of the radiation pattern of a partially coherent aperture,” IEEE Trans. Antennas Propag.AP15187–188 (1967).
[CrossRef]

J. Mod. Opt.

M. A. Alonso, O Korotkova, and E. Wolf, “Propagation of the electric correlation matrix and the van CittertZernike theorem for random electromagnetic fields,” J. Mod. Opt.53, 969–978 (2006).
[CrossRef]

J. Opt. Soc. Am A

F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell-model sources,” J. Opt. Soc. Am A25, 1016 (2008).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Soc. Amer.

W. H. Carter and E. Wolf, “Coherence properties of Lambertian and non-Lambertian sources,” J. Opt. Soc. Amer.65, 1067–1071 (1975).
[CrossRef]

Nature

E. Wolf, “Non-cosmological redshifts of spectral lines,” Nature326, 363–365 (1987).
[CrossRef]

Opt. Commun.

O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun.246, 35–43 (2005).
[CrossRef]

D. F. V. James, “Polarization of light radiated by black-body sources,” Opt. Commun.109, 209 (1994).
[CrossRef]

Phys. Lett. A

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A312, 263–267 (2003).
[CrossRef]

Phys. Rev. Lett.

E. Wolf, “Invariance of the spectrum of light on propagation,” Phys. Rev. Lett.56, 1370–1372 (1986).
[CrossRef] [PubMed]

Pure and Applied Optics

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, “Beam coherence-polarization matrix,” Pure and Applied Optics7941–951 (1998).
[CrossRef]

Rep. Prog. Phys.

E. Wolf and D. F. V. James, “Correlation-induced spectral changes,” Rep. Prog. Phys.59, 771–818 (1996).
[CrossRef]

Other

A. T. Friberg, ed., Selected Papers on Coherence and Radiometry, SPIE Milestone Series MS 69 (SPIE, 1993).

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, 2007).

J. H. Lambert, Photometria sive de mensure et gradibus luminis colorum et umbra (Bassel, 1760).

G. Toraldo di Francia, Electromagnetic Waves (Interscience, 1955), pp. 218–221.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1975), p. 437.

G. K. Batchelor, The Theory of Homogeneous Turbulence (Cambridge Science Classics, 1990).

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Figures (2)

Fig. 1
Fig. 1

Geometry for diffraction of Electromagnetic sources. The source lies in the xy plane, containing the origin O; a typical position vector of a point in the source is ρ = {x, y} (not shown). The field is observed at a far-zone point P, with position vector rs and spherical polar coordinates {r, θ, ϕ}. The field at P is conveniently described using the unit vectors {êr, êθ, êϕ}, given by eqs.(8)(10).

Fig. 2
Fig. 2

Normalized Radiation Patterns and Degrees of Polarization versus θ. This plot is based on the Gaussian model QuHIEMS defined by eq.(31). The blue solid line shows the normalized radiation pattern ��(∞)(θ), defined by eq.(34); the red dashed line is the degree of polarization P(∞)(θ) defined by eq.(35). The parameters used were as follows: (i) incoherent limit (a → 0) with μ = (a2b2)/a2 = 1; (ii) incoherent limit with μ = 2; (iii) incoherent limit with μ → ∞; (iv) a = 1, b = 0.0; (v) a = 1, b = 1.35; (vi) a = 1, b = 1.9318 (the largest value consistent with the realizability condition); (vii) a = 5, b = 0.0; (viii) a = 5, b = 1.3; (ix) a = 5, b = 5.1486.

Equations (38)

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W ( 0 ) ( ρ 1 , ρ 2 , ω ) S ( 0 ) ( ρ 1 + ρ 2 2 , ω ) μ ( 0 ) ( ρ 2 ρ 1 , ω ) .
J ( r s , ω ) = ( 2 π k ) 2 S ˜ ( 0 ) ( 0 , ω ) μ ˜ ( 0 ) ( k s , ω ) cos 2 θ ,
μ ˜ ( 0 ) ( f , ω ) = 1 ( 2 π ) 2 μ ( 0 ) ( ρ , ω ) exp ( i f ρ ) d 2 ρ
W α , β ( 0 ) ( ρ , ρ 2 , ω ) = S ( ρ 1 + ρ 2 2 , ω ) μ α , β ( ρ 2 ρ 1 , ω ) , ( α , β = x , y )
μ ( α , β ) ( ρ , ω ) = δ α , β A ( k ρ ) + ρ α ρ β ρ 2 B ( k ρ ) .
( r s , ω ) = i k exp ( i k r ) 2 π r E ( 0 ) ( ρ , ω ) exp ( i k s ρ . ) d 2 ρ .
E ( ) ( r s , ω ) = { s z x ( r s , ω ) , s z y ( r s , ω ) , s x x ( r s , ω ) s y y x ( r s , ω ) } .
s e ^ r = { sin θ cos ϕ , sin θ sin ϕ , cos θ } ,
e ^ θ = { cos θ cos ϕ , cos θ sin ϕ , sin θ } ,
e ^ ϕ = { sin ϕ , cos ϕ , 0 } .
[ E θ ( ) ( r s , ω ) E ϕ ( ) ( r s , ω ) ] = P ( θ ) R ( ϕ ) [ x ( r s , ω ) y ( r s , ω ) ] ,
P ( θ ) = [ 1 0 0 cos θ ]
R ( ϕ ) = [ cos ϕ sin ϕ sin ϕ cos ϕ ]
𝒲 α , β ( r 1 s 1 , r 2 s 2 , ω ) α * ( r 1 s 1 , ω ) β ( r 2 s 2 , ω ) = k 2 exp [ i k ( r 2 r 1 ) ] ( 2 π ) 2 r 1 r 2 W α , β ( 0 ) ( ρ 1 , ρ 2 , ω ) exp [ i k ( s 1 ρ 1 s 2 ρ 2 ) ] d 2 ρ 1 d 2 ρ 2 .
J = [ J θ , θ J θ , ϕ J ϕ , θ J ϕ , ϕ ] [ E θ ( ) * E θ ( ) E θ ( ) * E ϕ ( ) E ϕ ( ) * E θ ( ) E ϕ ( ) * E ϕ ( ) ] = 𝒫 ( θ ) ( ϕ ) [ 𝒲 x , x 𝒲 x , y 𝒲 y , x 𝒲 y , y ] ( ϕ ) T 𝒫 ( θ ) T ,
𝒲 α , β ( r s , r s , ω ) = ( 2 π k ) 2 r 2 S ˜ ( 0 ) ( 0 , ω ) μ ˜ α , β ( 0 ) ( k s , ω ) .
μ ˜ α , β ( 0 ) ( k s , ω ) = 1 k 2 { δ α , β [ I 1 ( θ ) + 1 2 I 2 ( θ ) ] s α s β sin 2 θ I 2 ( θ ) } , ( α , β = x , y )
I 1 ( θ ) = 1 2 π 0 [ A ( ξ ) + 1 2 B ( ξ ) ] J 0 ( ξ sin θ ) ξ d ξ I 2 ( θ ) = 1 2 π 0 B ( ξ ) J 2 ( ξ sin θ ) ξ d ξ ,
= 1 2 π 0 [ 2 ξ sin 2 θ B ( ξ ) B ( ξ ) ] J 0 ( ξ sin θ ) ξ d ξ
J ( r s , ω ) = ( 2 π ) 2 r 2 S ˜ ( 0 ) ( 0 , ω ) [ I 1 ( θ ) 1 2 I 2 ( θ ) 0 0 { I 1 ( θ ) + 1 2 I 2 ( θ ) } cos 2 θ ] .
S ( ) ( r s , ω ) = s ε 0 c 2 Tr { J ( r s , ω ) } = s ( 2 π 2 ε 0 c r 2 ) S ˜ ( 0 ) ( 0 , ω ) { ( 1 + cos 2 θ ) I 1 ( θ ) 1 2 sin 2 θ I 2 ( θ ) } .
P ( ) ( θ ) = 1 4 Det { J ( r s , ω ) } Tr { J ( r s , ω ) } 2 = | sin 2 θ I 1 ( θ ) 1 2 ( 1 + cos 2 θ ) I 2 ( θ ) | ( 1 + cos 2 θ ) I 1 ( θ ) 1 2 sin 2 θ I 2 ( θ ) .
| μ ˜ x y ( f , ω ) | μ ˜ x x ( f , ω ) μ ˜ y y ( f , ω ) ,
( I 1 ( θ ) ± I 2 ( θ ) 2 ) 0 .
( I 1 ( π / 2 ) I 2 ( π / 2 ) 2 ) = 0 .
J ( r e ^ z , ω ) = ( 2 π ) 2 r 2 S ˜ ( 0 ) ( 0 , θ ) I 1 ( 0 ) [ 1 0 0 1 ] .
A ( k ρ ) = j 0 ( k ρ ) j 1 ( k ρ ) / k ρ
B ( k ρ ) = j 2 ( k ρ ) .
𝒮 ( ) ( θ ) = cos θ .
A ( k ρ ) = 1 2 exp ( k 2 ρ 2 / 2 a 2 ) ,
B ( k ρ ) = B 0 k 2 ρ 2 exp ( k 2 ρ 2 / 2 b 2 ) ,
I 1 ( θ ) = 1 4 π k 2 [ a 2 exp ( 1 2 a 2 sin 2 θ ) + B 0 b 4 ( 2 b 2 sin 2 θ ) exp ( 1 2 b 2 sin 2 θ ) ]
I 2 ( θ ) = 1 2 π k 2 B 0 b 6 sin 2 θ exp ( 1 2 b 2 sin 2 θ ) .
𝒮 ( ) ( θ ) = ( 1 b 2 ) ( 1 + cos 2 θ ) exp ( a 2 cos 2 θ / 2 ) ( 1 + cos 2 θ b 2 sin 2 θ ) exp ( b 2 cos 2 θ / 2 ) 2 [ ( 1 b 2 ) exp ( a 2 / 2 ) exp ( b 2 / 2 ) ]
P ( ) ( θ ) = | ( 1 b 2 ) sin 2 θ [ exp ( a 2 cos 2 θ / 2 ) exp ( b 2 cos 2 θ / 2 ) ] | ( 1 b 2 ) ( 1 + cos 2 θ ) exp ( a 2 cos 2 θ / 2 ) ( 1 + cos 2 θ b 2 sin 2 θ ) exp ( b 2 cos 2 θ / 2 )
𝒮 ( ) ( θ ) = cos 2 θ μ ( 5 + cos 2 θ ) 4 6 μ 4
P ( ) ( θ ) = μ sin 2 θ μ ( 5 + cos 2 θ ) 4 ,
μ = a 2 b 2 a 2 ,

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