Abstract

We establish conditions under which a legitimate degree of coherence of a statistically stationary beam-like field raised to a power results in a novel legitimate degree of coherence. The general results and examples relate to scalar beams having uniform and non-uniform correlations.

© 2015 Optical Society of America

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References

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  1. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  2. F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007).
    [Crossref] [PubMed]
  3. H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011).
    [Crossref] [PubMed]
  4. Z. Tong and O. Korotkova, “Nonuniformly correlated light beams in uniformly correlated media,” Opt. Lett. 37(15), 3240–3242 (2012).
    [PubMed]
  5. H. Lajunen and T. Saastamoinen, “Non-uniformly correlated partially coherent pulses,” Opt. Express 21(1), 190–195 (2013).
    [Crossref] [PubMed]
  6. Z. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013).
    [Crossref] [PubMed]
  7. S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012).
    [Crossref] [PubMed]
  8. O. Korotkova, “Random sources for rectangular far fields,” Opt. Lett. 39(1), 64–67 (2014).
    [Crossref] [PubMed]
  9. F. Wang, C. Liang, Y. Yuan, and Y. Cai, “Generalized multi-Gaussian correlated Schell-model beam: from theory to experiment,” Opt. Express 22(19), 23456–23464 (2014).
    [Crossref] [PubMed]
  10. O. Korotkova, S. Sahin, and E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A 29(10), 2159–2164 (2012).
    [Crossref] [PubMed]
  11. Z. Mei and O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. 38(14), 2578–2580 (2013).
    [Crossref] [PubMed]
  12. L. Pan, C. Ding, and H. Wang, “Diffraction of cosine-Gaussian-correlated Schell-model beams,” Opt. Express 22(10), 11670–11679 (2014).
    [Crossref] [PubMed]
  13. Z. Mei, “Light sources generating self-splitting beams and their propagation in non-Kolmogorov turbulence,” Opt. Express 22(11), 13029–13040 (2014).
    [Crossref] [PubMed]
  14. O. Korotkova and E. Shchepakina, “Random sources for optical frames,” Opt. Express 22(9), 10622–10633 (2014).
    [Crossref] [PubMed]
  15. Z. Mei, “Two types of sinc Schell-model beams and their propagation characteristics,” Opt. Lett. 39(14), 4188–4191 (2014).
    [Crossref] [PubMed]
  16. M. Santarsiero, G. Piquero, J. C. G. de Sande, and F. Gori, “Difference of cross-spectral densities,” Opt. Lett. 39(7), 1713–1716 (2014).
    [Crossref] [PubMed]
  17. F. Gori and M. Santarsiero, “Difference of two Gaussian Schell-model cross-spectral densities,” Opt. Lett. 39(9), 2731–2734 (2014).
    [Crossref] [PubMed]
  18. Z. Mei, O. Korotkova, and Y. Mao, “Products of Schell-model cross-spectral densities,” Opt. Lett. 39(24), 6879–6882 (2014).
    [Crossref] [PubMed]
  19. J. Zhang, C. Liang, and Y. Chen, “A new family of windows-convolution windows and their applications,” Sci. China Ser. E 48(4), 468–480 (2005).
    [Crossref]
  20. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

2014 (9)

2013 (3)

2012 (3)

2011 (1)

2007 (1)

2005 (1)

J. Zhang, C. Liang, and Y. Chen, “A new family of windows-convolution windows and their applications,” Sci. China Ser. E 48(4), 468–480 (2005).
[Crossref]

Cai, Y.

Chen, Y.

J. Zhang, C. Liang, and Y. Chen, “A new family of windows-convolution windows and their applications,” Sci. China Ser. E 48(4), 468–480 (2005).
[Crossref]

de Sande, J. C. G.

Ding, C.

Gori, F.

Korotkova, O.

Lajunen, H.

Liang, C.

F. Wang, C. Liang, Y. Yuan, and Y. Cai, “Generalized multi-Gaussian correlated Schell-model beam: from theory to experiment,” Opt. Express 22(19), 23456–23464 (2014).
[Crossref] [PubMed]

J. Zhang, C. Liang, and Y. Chen, “A new family of windows-convolution windows and their applications,” Sci. China Ser. E 48(4), 468–480 (2005).
[Crossref]

Mao, Y.

Mei, Z.

Pan, L.

Piquero, G.

Saastamoinen, T.

Sahin, S.

Santarsiero, M.

Shchepakina, E.

Tong, Z.

Wang, F.

Wang, H.

Yuan, Y.

Zhang, J.

J. Zhang, C. Liang, and Y. Chen, “A new family of windows-convolution windows and their applications,” Sci. China Ser. E 48(4), 468–480 (2005).
[Crossref]

J. Opt. Soc. Am. A (1)

Opt. Express (5)

Opt. Lett. (11)

Z. Mei, “Two types of sinc Schell-model beams and their propagation characteristics,” Opt. Lett. 39(14), 4188–4191 (2014).
[Crossref] [PubMed]

M. Santarsiero, G. Piquero, J. C. G. de Sande, and F. Gori, “Difference of cross-spectral densities,” Opt. Lett. 39(7), 1713–1716 (2014).
[Crossref] [PubMed]

F. Gori and M. Santarsiero, “Difference of two Gaussian Schell-model cross-spectral densities,” Opt. Lett. 39(9), 2731–2734 (2014).
[Crossref] [PubMed]

Z. Mei, O. Korotkova, and Y. Mao, “Products of Schell-model cross-spectral densities,” Opt. Lett. 39(24), 6879–6882 (2014).
[Crossref] [PubMed]

F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007).
[Crossref] [PubMed]

H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011).
[Crossref] [PubMed]

Z. Tong and O. Korotkova, “Nonuniformly correlated light beams in uniformly correlated media,” Opt. Lett. 37(15), 3240–3242 (2012).
[PubMed]

Z. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013).
[Crossref] [PubMed]

S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012).
[Crossref] [PubMed]

O. Korotkova, “Random sources for rectangular far fields,” Opt. Lett. 39(1), 64–67 (2014).
[Crossref] [PubMed]

Z. Mei and O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. 38(14), 2578–2580 (2013).
[Crossref] [PubMed]

Sci. China Ser. E (1)

J. Zhang, C. Liang, and Y. Chen, “A new family of windows-convolution windows and their applications,” Sci. China Ser. E 48(4), 468–480 (2005).
[Crossref]

Other (2)

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

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

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Figures (11)

Fig. 1
Fig. 1 Illustration of the degree of coherence, the top row corresponding to the CSD of the Nth order modified GSM source and the bottom row corresponding to the CSD of the Nth order modified NUGSM source for several values of parameter N, N = 1 on the left hand side, N = 2 in the middle column and N = 6 on the right hand side.
Fig. 2
Fig. 2 Illustration of the degree of coherence, the top row corresponding to the CSD of the Nth order modified SSM sources and the bottom row corresponding to the CSD of the Nth modified NUSSM sources for several values of parameter N, N = 1 on the left hand side, N = 2 in the middle column and N = 6 on the right hand side.
Fig. 3
Fig. 3 Illustration of the degree of coherence, the top row corresponding to the CSD of the Nth order modified CGSM sources and the bottom row corresponding to the CSD of the Nth modified NUCG sources for several values of parameter N, N = 1 on the left hand side, N = 2 in the middle column and N = 6 on the right hand side.
Fig. 4
Fig. 4 Evolution of the spectral intensity S of the Nth order modified GSM beam with several values of parameter N on propagation in free space, the top row corresponding to N = 2 and the bottom row corresponding to N = 6.
Fig. 5
Fig. 5 Changes on-axis spectral intensity S of the Nth order modified GSM beam (a) and the Nth order modified NUGSM beam (b) with x0 = 0 for several values of parameter N.
Fig. 6
Fig. 6 Evolution of the spectral intensity S of the Nth order modified NUGSM beam with several values of parameter N on propagation in free space, the top row corresponding to N = 2 and the bottom row corresponding to N = 6.
Fig. 7
Fig. 7 Evolution of the spectral intensity S of the Nth order modified SSM beam with several values of parameter N on propagation in free space, the top row corresponding to N = 2 and the bottom row corresponding to N = 6.
Fig. 8
Fig. 8 Changes on-axis spectral intensity S of the Nth order modified SSM beam (a) and the Nth order modified NUSSM beam (b) with x0 = 0 for several values of parameter N.
Fig. 9
Fig. 9 Evolution of the spectral intensity S of the Nth order modified NUSSM beam with several values of parameter N on propagation in free space, the top row corresponding to N = 2 and the bottom row corresponding to N = 6.
Fig. 10
Fig. 10 Evolution of the spectral intensity S of the Nth order modified CGSM beam with several values of parameter N on propagation in free space, the top row corresponding to N = 1 and the bottom row corresponding to N = 2.
Fig. 11
Fig. 11 Evolution of the spectral intensity S of the Nth order modified NUCG beam with several values of parameter N on propagation in free space, the top row corresponding to N = 1 and the bottom row corresponding to N = 2.

Equations (40)

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W (0) ( x 1 , x 2 )= p(v) H 0 ( x 1 ,v) H 0 ( x 2 ,v)dv,
H 0 ( x ,v)=F( x )exp[ 2πivg( x ) ],
W (0) ( x 1 , x 2 )= F ( x 1 )F( x 2 ) p ˜ [ g( x 1 )g( x 2 ) ] = F ( x 1 )F( x 2 )μ[ g( x 1 )g( x 2 ) ],
H 0 ( x ,v)=F( x )exp( 2πiv x ),
H 0 ( x ,v)=F( x )exp[ 2πiv ( x x 0 ) 2 ],
p N (v)= p 1 (v) p 2 (v) p N (v).
W (0) ( x 1 , x 2 )= F ( x 1 )F( x 2 ) i=1 N p ˜ i [ g( x 1 )g( x 2 ) ] = F ( x 1 )F( x 2 ) i=1 N μ i [ g( x 1 )g( x 2 ) ] .
F( x )=exp[ x 2 /(2 σ 2 ) ].
p g (v)= 2π δ 2 exp(2 π 2 δ 2 v 2 ).
W (0) ( x 1 , x 2 )=exp( x 1 2 + x 2 2 2 σ 2 )exp[ ( x 1 x 2 ) 2 2 δ 2 ],
W (0) ( x 1 , x 2 )=exp( x 1 2 + x 2 2 2 σ 2 )exp{ [ ( x 2 x 0 ) 2 ( x 1 x 0 ) 2 ] 2 δ 4 },
p gN (v)= p g (v) p g (v) p g (v) N .
p gN (v)= 2π δ 2 /N exp(2 π 2 δ 2 v 2 /N).
W (0) ( x 1 , x 2 )=exp( x 1 2 + x 2 2 2 σ 2 )exp[ N ( x 1 x 2 ) 2 2 δ 2 ].
W (0) ( x 1 , x 2 )=exp( x 1 2 + x 2 2 2 σ 2 )exp{ N [ ( x 2 x 0 ) 2 ( x 1 x 0 ) 2 ] 2 δ 4 }.
p s (v)= 1 c rect( v c )={ 1/c, |v|c/2, 0, |v|>c/2,
W (0) ( x 1 , x 2 )=exp( x 1 2 + x 2 2 2 σ 2 )sinc( c x 1 c x 2 ).
W (0) ( x 1 , x 2 )=exp( x 1 2 + x 2 2 2 σ 2 )sinc[ c ( x 1 x 0 ) 2 c ( x 2 x 0 ) 2 ],
p 1 (v)= p 2 (v)== p N (v) p s (v).
P sN (v)= p s (v) p s (v) p s (v) N .
P s2 (v)= p s (v) p s (v) = 1 c Tri(v/c)= 1 c { 0, |v|c, 1| v |/c, |v|<c,
P s3 (v)= p s (v) p s (v) p s (v) = 1 2!c { 0, | v |3c/2, (3/2| v |/c) 2 , c/2| v |<3c/2, (3/2| v |/c) 2 3 (1/2| v |/c) 2 , 0| v |<c/2.
P s6 (v)= p s (v) p s (v) p s (v) p s (v) p s (v) p s (v) = 1 5!c { 0, | v |3c, (3| v |/c) 5 , 2c| v |<3c, (3| v |/c) 5 6 (2| v |/c) 5 , c| v |<2c, 9 (1| v |/c) 5 2 (2| v |/c) 5 4 (| v |/c) 5 + (1+| v |/c) 5 +5!(1| v |/c), | v |<c.
W (0) ( x 1 , x 2 )=exp( x 1 2 + x 2 2 2 σ 2 )sin c N ( c x 1 c x 2 ),
W (0) ( x 1 , x 2 )=exp( x 1 2 + x 2 2 2 σ 2 )sin c N [ c ( x 1 x 0 ) 2 c ( x 2 x 0 ) 2 ],
p c (v)= 2π δ 2 cosh[a (2π) 3/2 δv]exp(2 π 2 δ 2 v 2 a 2 π),
W (0) ( x 1 , x 2 )=exp( x 1 2 + x 2 2 2 σ 2 )cos[ a 2π ( x 1 x 2 ) δ ]exp[ ( x 1 x 2 ) 2 2 δ 2 ].
W (0) ( x 1 , x 2 )=exp( x 1 2 + x 2 2 2 σ 2 )cos{ 2a π [ ( x 2 x 0 ) 2 ( x 1 x 0 ) 2 ] δ 2 } ×exp{ [ ( x 2 x 0 ) 2 ( x 1 x 0 ) 2 ] 2 δ 4 }.
P cN (v)= p c (v) p c (v) p c (v) N .
W (0) ( x 1 , x 2 )=exp( x 1 2 + x 2 2 2 σ 2 ) cos N [ a 2π ( x 1 x 2 ) δ ]exp[ N ( x 1 x 2 ) 2 2 δ 2 ],
W (0) ( x 1 , x 2 )=exp( x 1 2 + x 2 2 2 σ 2 ) cos N { 2a π [ ( x 2 x 0 ) 2 ( x 1 x 0 ) 2 ] δ 2 } ×exp{ N [ ( x 2 x 0 ) 2 ( x 1 x 0 ) 2 ] 2 δ 4 }.
W( x 1 , x 2 ,z)= W (0) ( x 1 , x 2 ) K z * ( x 1 , x 1 ) K z ( x 2 , x 2 )d x 1 d x 2 ,
K z = k 2πz exp[ ik 2z ( x x ) 2 ],
W( x 1 , x 2 ,z)= p(v) H z * ( x 1 ,v) H z ( x 2 ,v)dv,
H z (x,v)= H 0 ( x ,v) K z (x, x )d x .
| H z (x,v) | 2 = σ w(z) exp[ (x+2πzv/k) 2 / w 2 (z) ],
w 2 (z)= σ 2 + z 2 /( k 2 σ 2 ).
| H z (x,v) | 2 = σ w(z,v) exp[ (x4πvz x 0 /k) 2 w 2 (z,v) ],
w 2 (z,v)= σ 2 (14πzv/k) 2 + z 2 /( k 2 σ 2 ).
S(x,z)= p(v) | H z (x,v) | 2 dv .

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