Abstract

Based on the 2×2 (electric field) cross-spectral density matrix, a model for an electromagnetic J0-correlated Schell-model beam is given that is a generalization of the scalar J0-correlated Schell-model beam. The conditions that the matrix for the source to generate an electromagnetic J0-correlated Schell-model beam are obtained. The condition for the source to generate a scalar J0-correlated Schell-model beam can be considered as a special case.

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References

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  1. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), Section 5.6.
  2. E. Collett and E. Wolf, Opt. Lett. 2, 27 (1978).
    [CrossRef] [PubMed]
  3. O. Korotkova, M. Salem, and E. Wolf, Opt. Lett. 29, 1173 (2004).
    [CrossRef] [PubMed]
  4. F. Gori and G. Guattari, Opt. Commun. 64, 311 (1987).
    [CrossRef]
  5. C. Palma, R. Borghi, and G. Cincotti, Opt. Commun. 125, 113 (1996).
    [CrossRef]
  6. J. Turunen, A. Vasara, and A. T. Friberg, J. Opt. Soc. Am. A 8, 282 (1991).
    [CrossRef]
  7. C. Palma and G. Cincotti, IEEE J. Quantum Electron. 33, 1032 (1997).
    [CrossRef]
  8. E. Wolf, Phys. Lett. A 312, 263 (2003).
    [CrossRef]
  9. E. Wolf, Opt. Lett. 28, 1078 (2003).
    [CrossRef] [PubMed]
  10. M. Salem and E. Wolf, Opt. Lett. 33, 1180 (2008).
    [CrossRef] [PubMed]
  11. G. Wu, Q. Lou, and J. Zhou, Opt. Express 16, 6417 (2008).
    [CrossRef] [PubMed]
  12. O. Korotkova, M. Salem, and E. Wolf, Opt. Commun. 233, 225 (2004).
    [CrossRef]
  13. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed. (Academic, 1994), p. 690, Eq. (6.615).
  14. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1972), p. 375, Eq. (9.6.3).
  15. G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, 1994), p. 16.

2008

2004

O. Korotkova, M. Salem, and E. Wolf, Opt. Commun. 233, 225 (2004).
[CrossRef]

O. Korotkova, M. Salem, and E. Wolf, Opt. Lett. 29, 1173 (2004).
[CrossRef] [PubMed]

2003

1997

C. Palma and G. Cincotti, IEEE J. Quantum Electron. 33, 1032 (1997).
[CrossRef]

1996

C. Palma, R. Borghi, and G. Cincotti, Opt. Commun. 125, 113 (1996).
[CrossRef]

1991

1987

F. Gori and G. Guattari, Opt. Commun. 64, 311 (1987).
[CrossRef]

1978

Abramowitz, M.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1972), p. 375, Eq. (9.6.3).

Borghi, R.

C. Palma, R. Borghi, and G. Cincotti, Opt. Commun. 125, 113 (1996).
[CrossRef]

Cincotti, G.

C. Palma and G. Cincotti, IEEE J. Quantum Electron. 33, 1032 (1997).
[CrossRef]

C. Palma, R. Borghi, and G. Cincotti, Opt. Commun. 125, 113 (1996).
[CrossRef]

Collett, E.

Friberg, A. T.

Gori, F.

F. Gori and G. Guattari, Opt. Commun. 64, 311 (1987).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed. (Academic, 1994), p. 690, Eq. (6.615).

Guattari, G.

F. Gori and G. Guattari, Opt. Commun. 64, 311 (1987).
[CrossRef]

Korotkova, O.

O. Korotkova, M. Salem, and E. Wolf, Opt. Lett. 29, 1173 (2004).
[CrossRef] [PubMed]

O. Korotkova, M. Salem, and E. Wolf, Opt. Commun. 233, 225 (2004).
[CrossRef]

Lou, Q.

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), Section 5.6.

Palma, C.

C. Palma and G. Cincotti, IEEE J. Quantum Electron. 33, 1032 (1997).
[CrossRef]

C. Palma, R. Borghi, and G. Cincotti, Opt. Commun. 125, 113 (1996).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed. (Academic, 1994), p. 690, Eq. (6.615).

Salem, M.

Stegun, I.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1972), p. 375, Eq. (9.6.3).

Turunen, J.

Vasara, A.

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, 1994), p. 16.

Wolf, E.

M. Salem and E. Wolf, Opt. Lett. 33, 1180 (2008).
[CrossRef] [PubMed]

O. Korotkova, M. Salem, and E. Wolf, Opt. Commun. 233, 225 (2004).
[CrossRef]

O. Korotkova, M. Salem, and E. Wolf, Opt. Lett. 29, 1173 (2004).
[CrossRef] [PubMed]

E. Wolf, Phys. Lett. A 312, 263 (2003).
[CrossRef]

E. Wolf, Opt. Lett. 28, 1078 (2003).
[CrossRef] [PubMed]

E. Collett and E. Wolf, Opt. Lett. 2, 27 (1978).
[CrossRef] [PubMed]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), Section 5.6.

Wu, G.

Zhou, J.

IEEE J. Quantum Electron.

C. Palma and G. Cincotti, IEEE J. Quantum Electron. 33, 1032 (1997).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

F. Gori and G. Guattari, Opt. Commun. 64, 311 (1987).
[CrossRef]

C. Palma, R. Borghi, and G. Cincotti, Opt. Commun. 125, 113 (1996).
[CrossRef]

O. Korotkova, M. Salem, and E. Wolf, Opt. Commun. 233, 225 (2004).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Lett. A

E. Wolf, Phys. Lett. A 312, 263 (2003).
[CrossRef]

Other

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), Section 5.6.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed. (Academic, 1994), p. 690, Eq. (6.615).

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1972), p. 375, Eq. (9.6.3).

G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, 1994), p. 16.

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Equations (30)

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W ͇ ( 0 ) ( ρ 1 , ρ 2 , ω ) = [ W x x ( 0 ) ( ρ 1 , ρ 2 , ω ) W x y ( 0 ) ( ρ 1 , ρ 2 , ω ) W y x ( 0 ) ( ρ 1 , ρ 2 , ω ) W y y ( 0 ) ( ρ 1 , ρ 2 , ω ) ] ,
W i j ( 0 ) ( ρ 1 , ρ 2 , ω ) = E i * ( ρ 1 , ω ) E j ( ρ 2 , ω ) ,
( i , j = x , y ) ,
W i j ( ρ 1 , z 1 ; ρ 2 , z 2 ; ω ) = k 4 W ̃ i j ( 0 ) ( f 1 , f 2 , ω ) × exp [ i ( f 2 ρ 2 + f 1 ρ 1 + k m 2 z 2 k m 1 * z 1 ) ] d 2 f 1 d 2 f 2 ,
W ̃ i j ( 0 ) ( f 1 , f 2 , ω ) = 1 ( 2 π ) 4 W i j ( 0 ) ( ρ 1 , ρ 2 , ω ) × exp [ i ( f 1 ρ 1 + f 2 ρ 2 ) ] d 2 ρ 1 d 2 ρ 2 ,
( i , j = x , y ) .
E ( ) ( r s , ω ) = 2 π i k cos θ a ( s x , s y , ω ) e i k r r .
W i j ( ) ( r 1 s 1 , r 2 s 2 , ω ) = ( 2 π k ) 2 cos θ 1 cos θ 2 { W ̃ i j ( 0 ) ( k s 1 , k s 2 , ω ) exp [ i k ( r 2 r 1 ) ] r 1 r 2 } ,
( i , j = x , y ) .
W i j ( 0 ) ( ρ 1 , ρ 2 , ω ) = S i ( 0 ) ( ρ 1 , ω ) S j ( 0 ) ( ρ 2 , ω ) μ i j ( 0 ) ( ρ 2 ρ 1 , ω ) ,
( i , j = x , y ) ,
S j ( 0 ) ( ρ , ω ) = A j 2 exp [ ρ 2 2 σ 2 ] , ( j = x , y ) ,
μ i j ( 0 ) ( ρ 2 ρ 1 , ω ) = B i j J 0 ( β i j ρ 2 ρ 1 ) .
B i j 1 when i = j ,
B i j 1 when i j ,
B i j = B j i * .
ρ = ρ 2 ρ 1 , ρ = ρ 2 + ρ 1 ,
f = f 2 f 1 , f = f 2 + f 1 ,
W ̃ i j ( 0 ) ( f , f , ω ) = A i A j B i j 4 ( 2 π ) 4 exp [ ρ 2 + ρ 2 8 σ 2 ] × J 0 ( β i j ρ ) exp [ i 2 ( ρ f + ρ f ) ] d 2 ρ d 2 ρ .
W ̃ i j ( 0 ) ( f , f , ω ) = A i A j B i j σ 4 ( 4 π ) 2 exp [ 2 σ 2 β i j 2 ] exp [ σ 2 f 2 2 ] exp [ σ 2 f 2 2 ] J 0 [ 2 i σ 2 β i j f ] , ( i , j = x , y ) .
0 exp [ α x ] J ν [ 2 β x ] J ν [ 2 γ x ] d x = 1 α I ν [ 2 β γ α ] exp [ β 2 + γ 2 α ] , [ Re ν > 1 ] ,
I n ( x ) = ( i ) n J n ( i x ) ,
W i j ( ) ( r 1 s 1 , r 2 s 2 , ω ) = k 2 cos θ 1 cos θ 2 exp [ i k ( r 2 r 1 ) ] r 2 r 1 A i A j B i j σ 4 4 exp [ 2 σ 2 β i j 2 ] exp [ k 2 σ 2 ] J 0 [ 2 i σ 2 k β i j ] ,
( i , j = x , y ) .
S ( ) ( r , ω ) = E x * ( r , ω ) E x ( r , ω ) + E y * ( r , ω ) E y ( r , ω ) = Tr [ W ͇ ( ) ( r , r , ω ) ] ,
S ( ) ( r , ω ) = k 2 cos 2 θ σ 4 4 r 2 × { A x 2 exp [ 2 σ 2 β x x 2 ] exp [ k 2 σ 2 ] J 0 [ 2 i σ 2 k β x x ] + A y 2 exp [ 2 σ 2 β y y 2 ] exp [ k 2 σ 2 ] J 0 [ 2 i σ 2 k β y y ] } .
J n ( x ) 1 n ! x 2 n exp [ x 2 ] ,
S ( ) ( r , ω ) k 2 cos 2 θ σ 4 4 r 2 { A x 2 exp [ 2 σ 2 β x x 2 ] exp [ k 2 ( σ 2 + σ 4 β x x 2 ) ] + A y 2 exp [ 2 σ 2 β y y 2 ] exp [ k 2 ( σ 2 + σ 4 β y y 2 ) ] } .
1 σ 2 + σ 4 β x x 2 k 2 , 1 σ 2 + σ 4 β y y 2 k 2 .
1 σ 2 + σ 4 β scalar 2 k 2 .

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