Abstract

A technique for experimental determining the coherent-mode structure of electromagnetic field is proposed. This technique is based on the coherence measurements of the field in some reference basis and represents a nontrivial vector generalization of the dual-mode field correlation method recently reported by F. Ferreira and M. Belsley [Opt. Lett. 38(21), 4350 (2013)]. The justifiability and efficiency of the proposed technique is illustrated by an example of determining the coherent-mode structure of some specially generated and experimentally characterized secondary electromagnetic source.

© 2014 Optical Society of America

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References

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  1. H. Gamo, “Matrix treatment of partial coherence,” in Progress in Optics III, E. Wolf, ed. (North-Holland, 1964).
  2. E. Wolf, “New theory of partial coherence in the space-frequency domain. Part 1: Spectra and cross spectra of steady-state source,” J. Opt. Soc. Am. A 72(3), 343–351 (1982).
    [Crossref]
  3. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  4. E. Wolf, Introduction to the Coherence and Polarization of Light (Cambridge University, 2007).
  5. A. S. Ostrovsky, Coherent-Mode Representations in Optics (SPIE, 2006).
  6. F. Gori, M. Santarsiero, R. Simon, G. Piquero, R. Borghi, and G. Guattari, “Coherent-mode decomposition of partially polarized, partially coherent sources,” J. Opt. Soc. Am. A 20(1), 78–84 (2003).
    [Crossref] [PubMed]
  7. J. Tervo, T. Setälä, and A. T. Friberg, “Theory of partially coherent electromagnetic fields in the space-frequency domain,” J. Opt. Soc. Am. A 21(11), 2205–2215 (2004).
    [Crossref] [PubMed]
  8. K. Kim and E. Wolf, “A scalar-mode representation of stochastic, planar, electromagnetic sources,” Opt. Commun. 261(1), 19–22 (2005).
    [Crossref]
  9. F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23(4), 241–243 (1998).
    [Crossref] [PubMed]
  10. J. Tervo, T. Setälä, and A. T. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11(10), 1137–1143 (2003).
    [Crossref] [PubMed]
  11. F. Ferreira and M. Belsley, “Holographic spatial coherence analysis of a laser,” Opt. Lett. 38(21), 4350–4353 (2013).
    [Crossref] [PubMed]
  12. A. Starikov, “Effective number of degrees of freedom of partially coherent source,” J. Opt. Soc. Am. A 72(11), 1538–1544 (1982).
    [Crossref]
  13. A. S. Ostrovsky, G. Rodríguez-Zurita, C. Meneses-Fabián, M. Á. Olvera-Santamaría, and C. Rickenstorff-Parrao, “Experimental generating the partially coherent and partially polarized electromagnetic source,” Opt. Express 18(12), 12864–12871 (2010).
    [Crossref] [PubMed]
  14. C. Rickenstorff and A. S. Ostrovsky, “Measurement of the amplitude and phase modulation of a liquid crystal spatial light modulator,” Superficies Vacio 23(S), 36–39 (2009).

2013 (1)

2010 (1)

2009 (1)

C. Rickenstorff and A. S. Ostrovsky, “Measurement of the amplitude and phase modulation of a liquid crystal spatial light modulator,” Superficies Vacio 23(S), 36–39 (2009).

2005 (1)

K. Kim and E. Wolf, “A scalar-mode representation of stochastic, planar, electromagnetic sources,” Opt. Commun. 261(1), 19–22 (2005).
[Crossref]

2004 (1)

2003 (2)

1998 (1)

1982 (2)

E. Wolf, “New theory of partial coherence in the space-frequency domain. Part 1: Spectra and cross spectra of steady-state source,” J. Opt. Soc. Am. A 72(3), 343–351 (1982).
[Crossref]

A. Starikov, “Effective number of degrees of freedom of partially coherent source,” J. Opt. Soc. Am. A 72(11), 1538–1544 (1982).
[Crossref]

Belsley, M.

Borghi, R.

Ferreira, F.

Friberg, A. T.

Gori, F.

Guattari, G.

Kim, K.

K. Kim and E. Wolf, “A scalar-mode representation of stochastic, planar, electromagnetic sources,” Opt. Commun. 261(1), 19–22 (2005).
[Crossref]

Meneses-Fabián, C.

Olvera-Santamaría, M. Á.

Ostrovsky, A. S.

Piquero, G.

Rickenstorff, C.

C. Rickenstorff and A. S. Ostrovsky, “Measurement of the amplitude and phase modulation of a liquid crystal spatial light modulator,” Superficies Vacio 23(S), 36–39 (2009).

Rickenstorff-Parrao, C.

Rodríguez-Zurita, G.

Santarsiero, M.

Setälä, T.

Simon, R.

Starikov, A.

A. Starikov, “Effective number of degrees of freedom of partially coherent source,” J. Opt. Soc. Am. A 72(11), 1538–1544 (1982).
[Crossref]

Tervo, J.

Wolf, E.

K. Kim and E. Wolf, “A scalar-mode representation of stochastic, planar, electromagnetic sources,” Opt. Commun. 261(1), 19–22 (2005).
[Crossref]

E. Wolf, “New theory of partial coherence in the space-frequency domain. Part 1: Spectra and cross spectra of steady-state source,” J. Opt. Soc. Am. A 72(3), 343–351 (1982).
[Crossref]

J. Opt. Soc. Am. A (4)

E. Wolf, “New theory of partial coherence in the space-frequency domain. Part 1: Spectra and cross spectra of steady-state source,” J. Opt. Soc. Am. A 72(3), 343–351 (1982).
[Crossref]

F. Gori, M. Santarsiero, R. Simon, G. Piquero, R. Borghi, and G. Guattari, “Coherent-mode decomposition of partially polarized, partially coherent sources,” J. Opt. Soc. Am. A 20(1), 78–84 (2003).
[Crossref] [PubMed]

J. Tervo, T. Setälä, and A. T. Friberg, “Theory of partially coherent electromagnetic fields in the space-frequency domain,” J. Opt. Soc. Am. A 21(11), 2205–2215 (2004).
[Crossref] [PubMed]

A. Starikov, “Effective number of degrees of freedom of partially coherent source,” J. Opt. Soc. Am. A 72(11), 1538–1544 (1982).
[Crossref]

Opt. Commun. (1)

K. Kim and E. Wolf, “A scalar-mode representation of stochastic, planar, electromagnetic sources,” Opt. Commun. 261(1), 19–22 (2005).
[Crossref]

Opt. Express (2)

Opt. Lett. (2)

Superficies Vacio (1)

C. Rickenstorff and A. S. Ostrovsky, “Measurement of the amplitude and phase modulation of a liquid crystal spatial light modulator,” Superficies Vacio 23(S), 36–39 (2009).

Other (4)

H. Gamo, “Matrix treatment of partial coherence,” in Progress in Optics III, E. Wolf, ed. (North-Holland, 1964).

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

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

A. S. Ostrovsky, Coherent-Mode Representations in Optics (SPIE, 2006).

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Figures (4)

Fig. 1
Fig. 1

Optical system for measuring coefficients c ij;kl : BS, beam splitter; M, mirror; P, polarizer; R, polarization rotator; SLM, spatial light modulator; L, lens.

Fig. 2
Fig. 2

Schematic illustration of the technique for generating the partially coherent and partially polarized electromagnetic source: BE, beam expander; BS, beam splitter; PBS, polarizing beam splitter; M, mirror; GGP, rotating ground glass plate.

Fig. 3
Fig. 3

Schematic illustration of modified Young experiment for measuring the cross-spectral density matrix of generated secondary source: BS, beam splitter; M, mirror; TP, translating pinhole; P, polarizer; R, polarization rotator. (The purpose of P and R is just the same as in technique sketched in Fig. 1.)

Fig. 4
Fig. 4

Normalized cross-spectral densities of generated secondary source measured in experiment (dotted curves) and determined in accordance with the proposed technique (solid curves) for ground glass plates with diffusion angles of 10° (a) and 30° (b).

Equations (31)

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W( x 1 , x 2 )=[ W xx ( x 1 , x 2 ) W xy ( x 1 , x 2 ) W yx ( x 1 , x 2 ) W yy ( x 1 , x 2 ) ],
W ij ( x 1 , x 2 )= E i ( x 1 ) E j ( x 2 ) ,(i,j=x,y),
η( x 1 , x 2 )= ( Tr[ W ( x 1 , x 2 )W( x 1 , x 2 )] TrW( x 1 , x 1 )TrW( x 2 , x 2 ) ) 1/2 ,
P(x)= ( 1 4DetW(x,x) [ TrW(x,x) ] 2 ) 1/2 ,
W( x 1 , x 2 )= n λ n W n ( x 1 , x 2 ),(n=0,1,2,...),
W ij;n ( x 1 , x 2 )= φ i;n ( x 1 ) φ j;n ( x 2 ).
j D W ij;n ( x 1 , x 2 ) φ j;n ( x 1 )d x 1 = λ n φ i;n ( x 2 ).
D φ i;n (x) φ i;m (x) dx= δ nm ,
E i (x)= k a i;k ψ k (x) ,
D ψ k (x) ψ l (x) dx= δ kl ,
a i;k = D E i (x) ψ k (x) dx.
W ij ( x 1 , x 2 )= k l c ij;kl ψ k ( x 1 ) ψ l ( x 2 ) ,
c ij;kl = a i;k a j;l .
j k l c ij;kl b j;n;k ψ l (x) = λ n φ i;n (x),
b j;n;k = D φ j;n (x) ψ k (x)dx .
j k c ij;kl b j;n;k = λ n b i;n;l .
[ C xx C xy C xy C yy ] [ B x;n B y;n ]= λ n [ B x;n B y;n ],
Det( [ C xx C xy C xy C yy ] λ n I )=0.
φ i;n (x)= k b i;n;k ψ k (x) .
t k (x)= t 0 +2| ψ k (x) |cos[ Arg( ψ k (x) )+2πx p 0 + β k ],
U ij;kl ( x )= D [ E i (x) t k (x)+ E j (x) t l (x) ]exp( i 2π λf x x )dx ,
U ij;kl = a i;k exp(i β k )+ a j;l exp(i β l ).
I ij;kl ( β kl )= | U ij;kl | 2 = c ii;kk + c jj;ll + c ij;kl exp(i β kl )+ c ij;kl exp(i β kl ),
Re( c ij;kl )= 1 2 I ij;kl (0) 1 8 I ii;kk (0) 1 8 I jj;ll (0),
Im( c ij;kl )= 1 2 I ij;kl (π/2) 1 8 I ii;kk (0) 1 8 I jj;ll (0).
t x(y) (x)=exp[ i ϕ x(y) (x) ],
ϕ x(y) ( x 1 ) ϕ x(y) ( x 2 ) = σ 2 exp( ( x 1 x 2 ) 2 2 γ x(y) 2 ),
ϕ x ( x 1 ) ϕ y ( x 2 ) =0,
σ= | ϕ x(y) (x) | 2 .
W SS ( x 1 , x 2 ) S 0 2 exp( x 1 2 + x 2 2 4 α 2 ) [ exp( ( x 1 x 2 ) 2 2 ( γ x /σ) 2 ) 0 0 exp( ( x 1 x 2 ) 2 2 ( γ y /σ) 2 ) ],
ψ k (x)= ( 1 π α 2 k k! ) 1/2 exp( x 2 2 α 2 ) H k ( x α ),

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