Abstract

Stochastic fields do not generally possess a Fourier transform. This makes the second-order statistics calculation very difficult, as it requires solving a fourth-order stochastic wave equation. This problem was alleviated by Wolf who introduced the coherent mode decomposition and, as a result, space-frequency statistics propagation of wide-sense stationary fields. In this paper we show that if, in addition to wide-sense stationarity, the fields are also wide-sense statistically homogeneous, then monochromatic plane waves can be used as an eigenfunction basis for the cross spectral density. Furthermore, the eigenvalue associated with a plane wave, exp[i(krωt)], is given by the spatiotemporal power spectrum evaluated at the frequency (k, ω). We show that the second-order statistics of these fields is fully described by the spatiotemporal power spectrum, a real, positive function. Thus, the second-order statistics can be efficiently propagated in the wavevector-frequency representation using a new framework of deterministic signals associated with random fields. Analogous to the complex analytic signal representation of a field, the deterministic signal is a mathematical construct meant to simplify calculations. Specifically, the deterministic signal associated with a random field is defined such that it has the identical autocorrelation as the actual random field. Calculations for propagating spatial and temporal correlations are simplified greatly because one only needs to solve a deterministic wave equation of second order. We illustrate the power of the wavevector-frequency representation with calculations of spatial coherence in the far zone of an incoherent source, as well as coherence effects induced by biological tissues.

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References

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  1. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995), pp. xxvi, 1166 p.
  2. J. W. Goodman, Statistical Optics, Wiley classics library ed., Wiley classics library (Wiley, 2000), pp. xvii, 550 p.
  3. G. Popescu, Quantitative Phase Imaging of Cells and Tissues, McGraw-Hill biophotonics (McGraw-Hill, 2011), p. 385.
  4. Z. Wang and G. Popescu, “Quantitative phase imaging with broadband fields,” Appl. Phys. Lett.96(5), 051117 (2010).
    [CrossRef]
  5. R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev.130(6), 2529–2539 (1963).
    [CrossRef]
  6. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th expanded ed. (Cambridge University Press, 1999).
  7. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, 2007), pp. xiv, 222 p.
  8. E. Wolf, “New spectral representation of random sources and of the partially coherent fields that they generate,” Opt. Commun.38(1), 3–6 (1981).
    [CrossRef]
  9. E. Wolf, “New theory of partial coherence in the space-frequency domain. 1. Spectra and cross spectra of steady-state sources,” J. Opt. Soc. Am.72(3), 343–351 (1982).
    [CrossRef]
  10. E. Wolf, “New theory of partial coherence in the space-frequency domain. 2. Steady-state fields and higher-order correlations,” J. Opt. Soc. Am. A3, 76–85 (1986).
    [CrossRef]
  11. D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science254(5035), 1178–1181 (1991).
    [CrossRef] [PubMed]
  12. T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy,” Nat. Phys.3(2), 129–134 (2007).
    [CrossRef]
  13. C. A. Puliafito, M. R. Hee, J. S. Schuman, and J. G. Fujimoto, Optical Coherence Tomography of Ocular Diseases (Slack, Inc., 1995).
  14. D. Lim, K. K. Chu, and J. Mertz, “Wide-field fluorescence sectioning with hybrid speckle and uniform-illumination microscopy,” Opt. Lett.33(16), 1819–1821 (2008).
    [CrossRef] [PubMed]
  15. Z. Wang, L. J. Millet, M. Mir, H. Ding, S. Unarunotai, J. A. Rogers, M. U. Gillette, and G. Popescu, “Spatial light interference microscopy (SLIM),” Opt. Express19(2), 1016–1026 (2011).
    [CrossRef] [PubMed]
  16. R. Zhu, S. Sridharan, K. Tangella, A. Balla, and G. Popescu, “Correlation-induced spectral changes in tissues,” Opt. Lett.36(21), 4209–4211 (2011).
    [CrossRef] [PubMed]
  17. J. Shamir, “Optical Systems and Processes, Vol,” PM65 of the SPIE Press Monographs (SPIE, 1999).
  18. P. Langevin, “On the theory of Brownian motion,” C. R. Acad. Sci. (Paris)146, 530 (1908).

2011

2010

Z. Wang and G. Popescu, “Quantitative phase imaging with broadband fields,” Appl. Phys. Lett.96(5), 051117 (2010).
[CrossRef]

2008

2007

T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy,” Nat. Phys.3(2), 129–134 (2007).
[CrossRef]

1991

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science254(5035), 1178–1181 (1991).
[CrossRef] [PubMed]

1986

1982

1981

E. Wolf, “New spectral representation of random sources and of the partially coherent fields that they generate,” Opt. Commun.38(1), 3–6 (1981).
[CrossRef]

1963

R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev.130(6), 2529–2539 (1963).
[CrossRef]

1908

P. Langevin, “On the theory of Brownian motion,” C. R. Acad. Sci. (Paris)146, 530 (1908).

Balla, A.

Boppart, S. A.

T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy,” Nat. Phys.3(2), 129–134 (2007).
[CrossRef]

Carney, P. S.

T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy,” Nat. Phys.3(2), 129–134 (2007).
[CrossRef]

Chang, W.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science254(5035), 1178–1181 (1991).
[CrossRef] [PubMed]

Chu, K. K.

Ding, H.

Flotte, T.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science254(5035), 1178–1181 (1991).
[CrossRef] [PubMed]

Fujimoto, J. G.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science254(5035), 1178–1181 (1991).
[CrossRef] [PubMed]

Gillette, M. U.

Glauber, R. J.

R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev.130(6), 2529–2539 (1963).
[CrossRef]

Gregory, K.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science254(5035), 1178–1181 (1991).
[CrossRef] [PubMed]

Hee, M. R.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science254(5035), 1178–1181 (1991).
[CrossRef] [PubMed]

Huang, D.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science254(5035), 1178–1181 (1991).
[CrossRef] [PubMed]

Langevin, P.

P. Langevin, “On the theory of Brownian motion,” C. R. Acad. Sci. (Paris)146, 530 (1908).

Lim, D.

Lin, C. P.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science254(5035), 1178–1181 (1991).
[CrossRef] [PubMed]

Marks, D. L.

T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy,” Nat. Phys.3(2), 129–134 (2007).
[CrossRef]

Mertz, J.

Millet, L. J.

Mir, M.

Popescu, G.

Puliafito, C. A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science254(5035), 1178–1181 (1991).
[CrossRef] [PubMed]

Ralston, T. S.

T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy,” Nat. Phys.3(2), 129–134 (2007).
[CrossRef]

Rogers, J. A.

Schuman, J. S.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science254(5035), 1178–1181 (1991).
[CrossRef] [PubMed]

Sridharan, S.

Stinson, W. G.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science254(5035), 1178–1181 (1991).
[CrossRef] [PubMed]

Swanson, E. A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science254(5035), 1178–1181 (1991).
[CrossRef] [PubMed]

Tangella, K.

Unarunotai, S.

Wang, Z.

Wolf, E.

Zhu, R.

Appl. Phys. Lett.

Z. Wang and G. Popescu, “Quantitative phase imaging with broadband fields,” Appl. Phys. Lett.96(5), 051117 (2010).
[CrossRef]

C. R. Acad. Sci. (Paris)

P. Langevin, “On the theory of Brownian motion,” C. R. Acad. Sci. (Paris)146, 530 (1908).

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Nat. Phys.

T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy,” Nat. Phys.3(2), 129–134 (2007).
[CrossRef]

Opt. Commun.

E. Wolf, “New spectral representation of random sources and of the partially coherent fields that they generate,” Opt. Commun.38(1), 3–6 (1981).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev.

R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev.130(6), 2529–2539 (1963).
[CrossRef]

Science

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science254(5035), 1178–1181 (1991).
[CrossRef] [PubMed]

Other

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995), pp. xxvi, 1166 p.

J. W. Goodman, Statistical Optics, Wiley classics library ed., Wiley classics library (Wiley, 2000), pp. xvii, 550 p.

G. Popescu, Quantitative Phase Imaging of Cells and Tissues, McGraw-Hill biophotonics (McGraw-Hill, 2011), p. 385.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th expanded ed. (Cambridge University Press, 1999).

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, 2007), pp. xiv, 222 p.

C. A. Puliafito, M. R. Hee, J. S. Schuman, and J. G. Fujimoto, Optical Coherence Tomography of Ocular Diseases (Slack, Inc., 1995).

J. Shamir, “Optical Systems and Processes, Vol,” PM65 of the SPIE Press Monographs (SPIE, 1999).

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Figures (2)

Fig. 1
Fig. 1

Field propagation from an extended source. At the observation plane, field U, contains the maximum spatial frequency, kM, which is set by the angle θ subtended by the source.

Fig. 2
Fig. 2

(a) The optical path-length map of a tissue biopsy sample, imaged with SLIM using a 40X objective with 0.75 NA. A close-up of the sample is shown on the right as a demonstration of the imaging ability (quantitative and high-resolution). (b) The scattering geometry for this experiment. (c) The spectrum measured from sample shown in (a) through 2D Fourier transform and radial averaging. (d) Normalized optical spectrum for light propagated after the tissue. (e) The variation of effective spectral width of the angular spectrum.

Equations (43)

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W( r 1 , r 2 ,ω )= n λ n 2 ( ω ) ψ n ( r 1 ,ω ) ψ n ( r 2 ,ω ) ,
D W( r 1 , r 2 ,ω ) ψ n ( r 1 ,ω ) d 3 r 1 = λ n 2 ( ω ) ψ n ( r 2 ,ω ),
W( r 1 , r 2 ,ω )= W ( r 2 r 1 ,ω ).
W( r 1 , r 2 ,ω)=W'( r 2 r 1 ,ω) = n c n (ω) ψ n ( r 2 r 1 ,ω) = n c n (ω) e i( k n ( r 2 r 1 )ωt) = n c n (ω) e iωt e i( k n r 1 ωt) e i( k n r 2 ωt) = n c n (ω) e iωt ψ n ( r 1 ,ω) ψ n ( r 2 ,ω) .
D W( r 1 , r 2 ,ω ) ψ m ( r 1 ,ω ) d 3 r 1 = D ( n c n (ω) e iωt ψ n ( r 1 ,ω) ψ n ( r 2 ,ω) ) ψ m ( r 1 ,ω ) d 3 r 1 = c m (ω) e iωt ψ m ( r 2 ,ω).
λ n 2 (ω)= c n (ω) e iωt =( D W'(r,ω) ψ n * (r,ω)dr ) e iωt = D W'(r,ω) e i k n r dr = D W( r 1 , r 2 ,ω) e i k n ( r 2 r 1 ) d r 2 S( k n ,ω).
V ˜ ( k,ω )= S( k,ω ) ,
V( r,t )= V k V ˜ ( k,ω ) e i( ωtkr ) dω d 3 k ,
V ˜ ( k,ω )= V r V( r,t ) e i( ωtkr ) dt d 3 r .
V ˜ U ( k,ω )= V ˜ s ( k,ω ) β 0 2 k 2 ,
2 V U ( r,t ) 1 c 2 2 V U ( r,t ) t 2 = V s ( r,t ).
UU= V U V U .
J( r 1 , r 2 ,t )= U( r 1 ,t ) U ( r 2 ,t ) .
J( ρ )= U( r,t ) U ( r+ρ,t ) .
J( ρ )=Γ( ρ,τ=0 ).
J( ρ )=Γ( ρ,τ=0 ) = W( ρ,ω )dω .
2 V U ( r,t ) 1 c 2 2 V U ( r,t ) t 2 = V s ( x,y,t )δ( z ),
V ˜ U ( k,ω )= V ˜ S ( k ,ω ) β 0 2 k 2 ,
1 β 0 2 k 2 = 1 q 2 k z 2 = 1 2q [ 1 q k z + 1 q+ k z ],
V ˜ U ( k,ω )= V ˜ S ( k ,ω ) 2q( q k z ) .
V ˜ U ( k ,z,ω )=i V ˜ S ( k ,ω ) e iqz 2q .
( β 0 2 k 2 ) S U ( k ,ω )= 1 4 S s ( k ,ω ).
k 2 ( ω ) = A k k 2 S U ( k ,ω ) d 2 k A k S U ( k ,ω ) d 2 k ,
k 2 ( ω ) = β 0 2 A k S s ( k ,ω ) d 2 k A k S s ( k ,ω ) β 0 2 k 2 d 2 k .
k 2 = β 0 2 ( 1 1 1+ k M 2 /2 β 0 2 ) k M 2 /2,
A c =1/ k 2 = 2 π λ 2 Ω ,
2 V U (r,ω)+ β 2 V U (r,ω)= β 0 2 ( n 2 (r) n(r) r 2 ) V s (r,ω) = β 0 2 χ(r) V s (r,ω),
V ˜ U ( k , k z ,ω)= β 0 2 ( β 2 k 2 ) k z 2 [ χ ˜ (k) k V ˜ s (k,ω) ] = β 0 2 2q [ 1 q+ k z + 1 q k z ][ χ ˜ (k) k V ˜ s (k,ω) ],
V ˜ U ( k ,z,ω)= β 0 2 2 e iqz q z [ χ ˜ ( k ,z) V ˜ s ( k ,z,ω) ],
V ˜ U ( k ,z,ω)= β 0 2 2 e iqz q [ χ ˜ (k) k V ˜ s (k,ω) ] k z =q .
S U (θ,ω)= | V ˜ (θ,ω) | 2 = β 0 2 4 cos 2 θ | [ χ ˜ (k) k V ˜ s (k,ω) ] | k z = β 0 cosθ 2 .
Δω( θ ) 2 = (ω ω 0 ) 2 S U (θ,ω)dω S U (θ,ω)dω .
2 U( r,t ) 1 c 2 2 U( r,t ) t 2 =s( r,t ).
[ 1 2 U( r,t ) 1 c 2 2 U( r,t ) t 2 ][ 2 2 U ( r+ρ,t+τ ) 1 c 2 2 U ( r+ρ,t+τ ) ( t+τ ) 2 ] = s( r,t ) s ( r+ρ,t+τ ) = Γ s ( ρ,τ ),
1 2 = 2 2 = ρ x 2 + ρ y 2 + ρ z 2 t 2 = ( t+τ ) 2 = τ 2 .
( 2 1 c 2 τ 2 )( 2 1 c 2 τ 2 ) Γ U ( ρ,τ )= Γ s ( ρ,τ )
( β 0 2 k 2 )( β 0 2 k 2 ) S U ( k,ω )= S s ( k,ω ),
S U ( k,ω )= S s ( k,ω ) ( β 0 2 k 2 ) 2 .
τ c = 1 Δω , A c = 1 Δ k 2 = 1 Δ k x 2 +Δ k y 2 ,
Δ ω 2 ( k )= ( ω ω ) 2 S( k ,ω )dω S( k ,ω )dω = ω 2 ( k ) ω ω( k ) ω 2 ,
Δ k x 2 ( ω )= A k | k x k x | 2 S( k ,ω ) d 2 k S( k ,ω ) d 2 k = k x 2 (ω) k k x (ω) k 2 .
Δ ω 2 k = A k ( ω ω ) 2 S( k ,ω )dω d 2 k A k S( k ,ω )dω d 2 k ,
Δ k x 2 ω = A k ( k x k x ) 2 S( k ,ω ) d 2 k dω S( k ,ω ) d 2 k dω .

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