January 1, 2014 / Vol. 39, No. 1 / OPTICS LETTERS
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Prediction of electric field frequency correlations for randomly scattering slabs in the nondiffusive regime with the scalar Bethe–Salpeter equation Vaibhav Gaind, Aung K. San, Dergan Lin, and Kevin J. Webb* School of Electrical and Computer Engineering, Purdue University, 465 Northwestern Ave., West Lafayette, Indiana 47907, USA *Corresponding author: [email protected] Received July 17, 2013; revised September 20, 2013; accepted October 10, 2013; posted October 11, 2013 (Doc. ID 194157); published December 16, 2013 We show that a scalar Bethe–Salpeter equation model captures the measured copolarized electric field frequency correlation magnitude for randomly scattering slabs in the weakly scattering, nondiffusive regime. Consequently, the model could be used to form images of tissue on the millimeter and submillimeter length scale, and for environmental sensing with comparable scatter, as dictated by the optical scattering length in relation to the scattering domain size. © 2013 Optical Society of America OCIS codes: (030.6140) Speckle; (260.2110) Electromagnetic optics; (290.4210) Multiple scattering; (110.0113) Imaging through turbid media. http://dx.doi.org/10.1364/OL.39.000001
The specific nature of depolarization with increasing scatter in randomly scattering medium characteristics can depend on the particular type of polarization (linear, circular) [1–3]. In the weakly scattering, nondiffusive regime, some information about the initial polarization of light is preserved. This means that multiple polarization state measurements can provide additional information about the medium or the incident field. Polarization information coupled with spatial and spectral data will allow enhanced imaging of scattering media, yield information about the scatterers, and allow increased communication capacity [4]. With sufficiently heavy scatter in the regime where the light is depolarized [2], and with weakly interacting scatterers, the diffusion model predicts the mean intensity as a function of position and time or temporal frequency [5,6]. However, the important regime where there is too little scatter for the convenient diffusion model to hold requires another description of the scattering medium. The vector Bethe–Salpeter equation (BSE) describes coherent optical scatter and hence captures the wave nature of light [7,8]. Because the BSE is field based, it can predict all measurements, regardless of the degree of scatter, subject only to the specific approximations used in the solution. The solution can be represented using scattering (Feynman) diagrams, which depict different scattering processes [9]. However, solving the vector BSE can be tedious. Here, we show the surprising result that the scalar BSE, which has been used to calculate field frequency correlations in the ultrasonic regime to explore the ballistic to diffusive medium transition [10], predicts the copolarized field correlation over frequency, and hence the copolarized temporal impulse response, with weakly scattered light. This result indicates that the scalar BSE is a suitable forward model in optimization-based imaging of the weakly scattering regime using polarized light [11]. We solve the scalar BSE under the independent scatterer approximation, also known as the ladder approximation [9]. This approximation is only valid when the density of scatterers is low (which is true for the scattering 0146-9592/14/010001-04$15.00/0
medium we consider). The radiative transfer equation (RTE) [12,13], which describes the conservation of energy at each scattering event, could also be used to model the frequency response of a weakly scattering medium. However, for sufficiently weak scatter, or dense scattering media where the scatterers can no longer be considered independent, the RTE will not hold. We show the suitability of the scalar BSE model in predicting fieldfrequency correlations in a weakly scattering medium, and hence the merits of this forward model for imaging in this regime. Extensions of the model allow for modeling other scatter regimes. The scattered partial waves in a random medium have a distribution for the times of flight that we denote by the probability density function pt that is polarization dependent and is related to the Fourier transform of the field frequency correlation as [2,14] F fptg hEνE ν Δνi∕hIi;
(1)
where E is the electric field, is complex conjugate, Δν and t are the conjugate variables, ν is the optical frequency, Δν is a deviation frequency, F is the Fourier transform, h·i represents configurational averaging over scatterer position, and the mean intensity hIi is assumed constant over Δν (or the fields are appropriately normalized). Measures of pt provide a basis for imaging and characterization of scattering media. We assume uniformly distributed scatterers of dielectric constant ϵr in a homogeneous background ϵb , with wavenumber kb . The mean-field Green’s function, hGrd ; rs ; νi, satisfies the Dyson equation [9], Z hGrd ; rs ; νi G0 rd ; rs ; ν
G0 rd ; r0 ; ν
· Mr0 ; r00 ; νhGr00 ; rs ; νidr0 dr00 ;
(2)
where rd and rs represent the detector and source positions, respectively, and G0 is the free space Green’s function. Mr0 ; r00 ; ν is a spatially dependent effective scattering function (with position vectors r0 and r00 ) © 2014 Optical Society of America
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OPTICS LETTERS / Vol. 39, No. 1 / January 1, 2014
representing the effect of multiple scatter. Equation (2) suggests that the scattering medium can be represented by a spatially dependent effective medium, with which P the total mean field interacts. The function M· i M i is the configurational average of the sum of all possible use only M 1 , with scattering processes P R M i0 [9]. We 00 Sr − R ; r − R M 1 r0 ; r00 ; ν N j j ; νpRj dRj , where j1 V S· is the scattering function for the jth scatterer with centroid at Rj . Here, we use a lossless point scatterer representation, with Sr0 − Rj ;r00 − Rj ;ν f δr0 − Rj δr00 − Rj , where f represents the strength of scatter, with scattering cross-section σ s jf j2 . Substituting M 1 · into Eq. (2), hGrd ; rs ; νi G0 rd ; rs ; ν
Z N X f G0 rd ; r0 ; ν
00
· δr − Rj δr − Rj hGr00 ; rs ; νi · pRj dr0 dr00 dRj :
(3)
For an infinite medium, Eq. (3) has an analytical solution given by [15,16] hGrd ; rs ; νi A
eikjrd −rs j ; 4πjrd − rs j
(4)
where A describes strength of scatter and k q 2 kb ikb ∕ls is the effective medium wave number with ls the average distance between scattering events, where ls nσ s −1 , with n being the density of the scatterers. We approximate the mean-field Green’s function for a scattering slab, as used in the experiment, with the mean-field Green’s function of an infinite scattering medium. Note that the mean field described by Eq. (4) decays with increasing distance due to multiple scatter, approaching zero for jrd − rs j ≫ ls , suggesting zero-mean circular Gaussian statistics hold in that limit [17]. The BSE is the Dyson equation [Eq. (2)] for field correlation [9], and substituting electric field correlation for G in Eq. (2) gives
N Z X
Sr0 −Rj ; r¯ 0 −Rj ;ν
j1
·S r00 −Rj ; r¯ 00 −Rj ;νΔνpRj dRj Z μs δr0 −Rδ¯r0 −Rδr00 −R ·δ¯r00 −RdR;
(6)
hEr; νE r; ν Δνi hEr; νihE r; ν Δνi Z μs hGr; R; νihG r; R; ν Δνi ⋅hER; νE R; ν ΔνidR:
(7)
Treating the scattering medium as a homogenized effective medium with a complex wavenumber k, the mean field for a Gaussian beam is hEr⊥ ; z; νi exp−jr⊥ j2 ∕σ 2 · expikb z · exp−z∕2ls ; (8) where r⊥ defines the transverse coordinates and σ describes the beam width. Note that the mean field approaches zero as z → ∞. Setting hGr; R; νi hGr⊥ − R⊥ ; z; Z; νi [see Eq. (4)], ϕr; ν; ν Δν hEr; νE r; ν Δνi, ψr; ν; ν Δν hEr; νihE r; ν Δνi, and Kr; R; ν; ν Δν hGr; R; νihG r; R; ν Δνi, and taking the spatial Fourier transform of Eq. (7) in the transverse (x–y) direction, ϕq⊥ ; z; ν; ν Δν ψq⊥ ; z; ν; ν Δν Zd Kq⊥ ; z; Z; ν; ν Δν μs 0
hErd1 ; νE rd2 ; ν Δνi
· ϕq⊥ ; Z; ν; ν ΔνdZ;
hErd1 ; νihE rd2 ; ν Δνi Z hGrd1 ; r0 ; νihG rd2 ; r00 ; ν Δνi ⋅Ur0 ; r¯ 0 ; r00 ; r¯ 00 ; ν; ν Δν ⋅hE¯r0 ; νE ¯r00 ; ν Δνidr0 d¯r0 dr00 d¯r00 ;
U 1 r0 ; r¯ 0 ;r00 ; r¯ 00 ;ν;νΔν
where we have used pR V −1 and nσ s μs l−1 s . Substituting Eq. (6) into Eq. (5) and using the sifting property of the Dirac delta function, we find
j1 0
the product of means on the right of Eq. (5). We also assume that there is a single detector, so that rd1 rd2 r. Therefore, U in Eq. (5) can be written as
(5)
where rd1 and rd2 are detector position vectors and U P i U i is the counterpart of the scattering function M in the Dyson equation. U can be interpreted as a four-port network, with two outputs at positions r0 and r¯ 0 and two inputs at r00 and r¯ 00 . We solve Eq. (5) under the ladder approximation, where the scatterer positions are independent of each other and there are no scattering loops, and the ascribed paths taken by the two fields at frequencies ν and ν Δν have at least one common scatterer. The case of no common scatterers is encompassed by
(9)
where d is the slab thickness. We solve Eq. (9), a Fredholm integral equation of the second kind in Z, by discretizing the region Z ∈ 0; d into L uniform subdomains and linearly interpolating ϕq⊥ ; z; ν; ν Δν between ϕq⊥ ; zj ; ν; ν Δν and ϕq⊥ ; zj1 ; ν; ν Δν, where j is an integer, zj jδz, and δz d∕L. Following the solution of the linear equations in Eq. (9) for ϕq⊥ ; z; ν; ν Δν at z d, a Hamming-windowed (with window bandwidth equal to 8∕σ) inverse Fourier transform with respect to q⊥ is taken to obtain hEr⊥ ; d; νE r⊥ ; d; ν Δνi. The large bandwidth of the Hamming window (relative to that of the incident Gaussian beam) ensured that the effect of the window on the correlation results was minimal. In the BSE model of Eq. (9), with a truncated spectral representation in the transverse spatial frequencies, we assume the slab to be
January 1, 2014 / Vol. 39, No. 1 / OPTICS LETTERS
p infinite in the transverse dimensions. We assume ϵb 1.5 and σ 0.5 mm for the incident Gaussian field defined in Eq. (8), consistent with the experiment, and L 30 in the solution to Eq. (9). The spectral domain interferometer experiment shown in Fig. 1 allows the scattered electric field to be determined [2]. The laser source is an external-cavity laser diode (New Focus Vortex II 6917) with a center wavelength of 850 nm that can be scanned over approximately 60 GHz. In the arrangement of Fig. 1, the aperture just to the right of the scattering sample controls the size of the speckle (the intensity correlation area), and the detector aperture (defined by the pinhole) is small relative to the speckle spot size. The linear copolarized and crosspolarized transmitted electric field was measured for commercial white acrylic samples (Cyro Industries, Acrylite FF) composed of small TiO2 particles of average diameter (approximately 50 nm) in a clear acrylic background. Data was collected as a function of position by scanning only the scattering sample (to 50 positions), to form averages. The two scattering materials used had reduced scattering coefficients [2] of μ0s 4 cm−1 and μ0s 14 cm−1 , and μ0s μs because the scatterers are small and the scatter is isotropic. The samples had transverse dimensions 12 cm × 12 cm, much greater than the thicknesses used. Experimental data showed that all samples had zero-mean circular Gaussian field statistics, implicit in Eq. (8), which shows the mean approaches zero with increasing scatter. Figure 2 shows the measured copolarized and crosspolarized electric field correlation magnitude as a function of scan frequency, along with the BSE results, for various thicknesses of a μ0s 4 cm−1 sample. Notice from Fig. 2 that the scalar BSE captures the copolarized nondiffusive light transport properties effectively, and that the diffusion model result (see [18,19] for the method) shown does a somewhat reasonable job of describing
Fig. 1. Spectral interferometer for measuring the randomly scattered electric field. HWP is a half-wave plate, LP is a linear polarizer, PBS is a polarizing beam splitter, and the attenuator is composed of a HWP–LP pair. The reference interferometer gives the scan frequency. The copolarized or cross-polarized light is selected by the LP in front of the photodetector.
Cross−Pol
Co−Pol
Diffusion
BSE
Cross−Pol
1
1
0.5
0.5
0 0
20 40 Frequency (GHz)
Cross−Pol
Co−Pol
Diffusion
60
0 0
1
1
0.5
0.5
0 0
20 40 Frequency (GHz)
60
0 0
Diffusion
20 40 Frequency (GHz)
Cross−Pol
BSE
Co−Pol
Co−Pol
Diffusion
5 10 Frequency (GHz)
3 BSE
60
BSE
15
Fig. 2. Comparison of the measured electric field frequency correlation magnitude for copolarized and cross-polarized light with the BSE and diffusion models for a slab with μ0s 4 cm−1 and a thickness of (a) 6 mm, (b) 9 mm, (c) 12 mm, and (d) 18 mm. Standard error bars for estimation of the mean from the experimental data are given.
the cross-polarized correlation. The copolarized response decorrelates more slowly with frequency than the crosspolarized field correlation because that light is less scattered. With increasing scatter, the responses for the two polarizations approach one another and the diffusion regime is entered. It is the more weakly scattering domains, and in the prediction of the copolarized response, that the scalar BSE solution appears to offer a viable forward model. We also made measurements on another sample having μ0s 14 cm−1 , and Fig. 3 shows a comparison with the BSE results. The BSE again predicts the copolarized field correlations. Also, note again the interesting differences between the copolarized and cross-polarized responses for weaker scatter that diminish with increasing sample thickness. The data in Figs. 2 and 3 show the transition between nondiffusive and diffusive light propagation, becoming diffusive for the 18 mm μ0s 4 cm−1 sample and 6 mm for the μ0s 14 cm−1 case. In our case, where the scatterers have a size much less than the wavelength, the nondiffusive to diffusive transition occurs when the slab thickness is about 7–8 times the transport length, approximately in agreement with prior estimates [3,10]. We find it interesting that the scalar BSE solution in the ladder approximation captures measured copolarized electric field correlations over frequency for the weakly scattering slabs studied having small scatterers. The influence of polarization of the incident light and scatterer size on the back-scattered intensity from a BSE model [20] indicates that vector and perhaps scalar solutions may yield a method to determine scatterer information (size or anisotropy) when used with free variables to predict experimental data. With use of a moment theorem that relies on zero mean circular Gaussian field statistics [21], measured intensity correlations can be used in conjunction with a forward BSE model for imaging applications. It also appears that the scalar BSE model with measured
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OPTICS LETTERS / Vol. 39, No. 1 / January 1, 2014 Co−Pol
Diffusion
1
0.5
0 0
Cross−Pol
BSE
||
||
Cross−Pol
20 40 Frequency (GHz)
60
Co−Pol
Diffusion
BSE
1
0.5
0 0
10 20 Frequency (GHz)
30
Fig. 3. Comparison of the measured electric field correlation magnitude for copolarized and cross-polarized light with the BSE and diffusion models for a slab with μ0s 14 cm−1 and a thickness of (a) 3 mm and (b) 6 mm. Error bars are shown.
copolarized data, combined with Raman or fluorescence spectroscopy, could lead to new molecular imaging methods at the subsurface tissue level. While we used electric field frequency correlations, our measurement method can be extended to allow spatial correlations in addition to frequency correlations. This combination of frequency and spatial correlation data could be used to form images of the spatially varying scatter, as has been done in optical diffusion tomography [11]. We acknowledge funding from the National Science Foundation (NSF) under awards 0854249, 0915966, 1028610, and 1218909. References 1. D. Bicout, C. Brosseau, A. S. Martinez, and J. M. Schmitt, Phys. Rev. E 49, 1767 (1994). 2. T. D. Gerke, M. A. Webster, A. M. Weiner, and K. J. Webb, J. Opt. Soc. Am. A 22, 2691 (2005).
3. Z. Wang, M. A. Webster, A. M. Weiner, and K. J. Webb, J. Opt. Soc. Am. A 23, 3045 (2006). 4. A. L. Moustakas, H. U. Baranger, L. Balents, A. M. Sengupta, and S. H. Simon, Science 287, 287 (2000). 5. S. R. Arridge, Appl. Opt. 34, 7395 (1995). 6. M. A. Webster, K. J. Webb, and A. M. Weiner, Phys. Rev. Lett. 88, 033901 (2002). 7. M. J. Stephen and G. Cwilich, Phys. Rev. B 34, 7564 (1986). 8. K. J. Peters, Phys. Rev. B 46, 801 (1992). 9. U. Frisch, “Wave propagation in random media,” in Probability Methods in Applied Mathematics, A. Bharucha-Reid, ed. (Academic, 1968), Vol. 1, pp. 176–198. 10. Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Weitz, and P. Sheng, Phys. Rev. E 60, 4843 (1999). 11. A. B. Milstein, S. Oh, J. S. Reynolds, K. J. Webb, C. A. Bouman, and R. P. Millane, Opt. Lett. 27, 95 (2002). 12. M. I. Mishchenko, Appl. Opt. 41, 7114 (2002). 13. L. A. Apresyan and Y. A. Kravtsov, Radiation Transfer: Statistical and Wave Aspects (Gordon and Breach, 1996). 14. A. Z. Genack and J. M. Drake, Europhys. Lett. 11, 331 (1990). 15. M. B. van der Mark, M. P. van Albada, and A. Lagendijk, Phys. Rev. B 37, 3575 (1988). 16. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1977), Vol. 2. 17. J. Goodman, Statistical Optics (Wiley, 1985). 18. M. S. Patterson, B. Chance, and B. C. Wilson, Appl. Opt. 28, 2331 (1989). 19. C. A. Thompson, K. J. Webb, and A. M. Weiner, Appl. Opt. 36, 3726 (1997). 20. V. L. Kuz’min and I. V. Meglinskiı˘, J. Exp. Theor. Phys. 110, 742 (2010). 21. I. S. Reed, IRE Tran. Inf. Theory 8, 194 (1962).
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Prediction of electric field frequency correlations for randomly scattering slabs in the nondiffusive regime with the scalar Bethe–Salpeter equation Vaibhav Gaind, Aung K. San, Dergan Lin, and Kevin J. Webb* School of Electrical and Computer Engineering, Purdue University, 465 Northwestern Ave., West Lafayette, Indiana 47907, USA *Corresponding author: [email protected] Received July 17, 2013; revised September 20, 2013; accepted October 10, 2013; posted October 11, 2013 (Doc. ID 194157); published December 16, 2013 We show that a scalar Bethe–Salpeter equation model captures the measured copolarized electric field frequency correlation magnitude for randomly scattering slabs in the weakly scattering, nondiffusive regime. Consequently, the model could be used to form images of tissue on the millimeter and submillimeter length scale, and for environmental sensing with comparable scatter, as dictated by the optical scattering length in relation to the scattering domain size. © 2013 Optical Society of America OCIS codes: (030.6140) Speckle; (260.2110) Electromagnetic optics; (290.4210) Multiple scattering; (110.0113) Imaging through turbid media. http://dx.doi.org/10.1364/OL.39.000001
The specific nature of depolarization with increasing scatter in randomly scattering medium characteristics can depend on the particular type of polarization (linear, circular) [1–3]. In the weakly scattering, nondiffusive regime, some information about the initial polarization of light is preserved. This means that multiple polarization state measurements can provide additional information about the medium or the incident field. Polarization information coupled with spatial and spectral data will allow enhanced imaging of scattering media, yield information about the scatterers, and allow increased communication capacity [4]. With sufficiently heavy scatter in the regime where the light is depolarized [2], and with weakly interacting scatterers, the diffusion model predicts the mean intensity as a function of position and time or temporal frequency [5,6]. However, the important regime where there is too little scatter for the convenient diffusion model to hold requires another description of the scattering medium. The vector Bethe–Salpeter equation (BSE) describes coherent optical scatter and hence captures the wave nature of light [7,8]. Because the BSE is field based, it can predict all measurements, regardless of the degree of scatter, subject only to the specific approximations used in the solution. The solution can be represented using scattering (Feynman) diagrams, which depict different scattering processes [9]. However, solving the vector BSE can be tedious. Here, we show the surprising result that the scalar BSE, which has been used to calculate field frequency correlations in the ultrasonic regime to explore the ballistic to diffusive medium transition [10], predicts the copolarized field correlation over frequency, and hence the copolarized temporal impulse response, with weakly scattered light. This result indicates that the scalar BSE is a suitable forward model in optimization-based imaging of the weakly scattering regime using polarized light [11]. We solve the scalar BSE under the independent scatterer approximation, also known as the ladder approximation [9]. This approximation is only valid when the density of scatterers is low (which is true for the scattering 0146-9592/14/010001-04$15.00/0
medium we consider). The radiative transfer equation (RTE) [12,13], which describes the conservation of energy at each scattering event, could also be used to model the frequency response of a weakly scattering medium. However, for sufficiently weak scatter, or dense scattering media where the scatterers can no longer be considered independent, the RTE will not hold. We show the suitability of the scalar BSE model in predicting fieldfrequency correlations in a weakly scattering medium, and hence the merits of this forward model for imaging in this regime. Extensions of the model allow for modeling other scatter regimes. The scattered partial waves in a random medium have a distribution for the times of flight that we denote by the probability density function pt that is polarization dependent and is related to the Fourier transform of the field frequency correlation as [2,14] F fptg hEνE ν Δνi∕hIi;
(1)
where E is the electric field, is complex conjugate, Δν and t are the conjugate variables, ν is the optical frequency, Δν is a deviation frequency, F is the Fourier transform, h·i represents configurational averaging over scatterer position, and the mean intensity hIi is assumed constant over Δν (or the fields are appropriately normalized). Measures of pt provide a basis for imaging and characterization of scattering media. We assume uniformly distributed scatterers of dielectric constant ϵr in a homogeneous background ϵb , with wavenumber kb . The mean-field Green’s function, hGrd ; rs ; νi, satisfies the Dyson equation [9], Z hGrd ; rs ; νi G0 rd ; rs ; ν
G0 rd ; r0 ; ν
· Mr0 ; r00 ; νhGr00 ; rs ; νidr0 dr00 ;
(2)
where rd and rs represent the detector and source positions, respectively, and G0 is the free space Green’s function. Mr0 ; r00 ; ν is a spatially dependent effective scattering function (with position vectors r0 and r00 ) © 2014 Optical Society of America
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OPTICS LETTERS / Vol. 39, No. 1 / January 1, 2014
representing the effect of multiple scatter. Equation (2) suggests that the scattering medium can be represented by a spatially dependent effective medium, with which P the total mean field interacts. The function M· i M i is the configurational average of the sum of all possible use only M 1 , with scattering processes P R M i0 [9]. We 00 Sr − R ; r − R M 1 r0 ; r00 ; ν N j j ; νpRj dRj , where j1 V S· is the scattering function for the jth scatterer with centroid at Rj . Here, we use a lossless point scatterer representation, with Sr0 − Rj ;r00 − Rj ;ν f δr0 − Rj δr00 − Rj , where f represents the strength of scatter, with scattering cross-section σ s jf j2 . Substituting M 1 · into Eq. (2), hGrd ; rs ; νi G0 rd ; rs ; ν
Z N X f G0 rd ; r0 ; ν
00
· δr − Rj δr − Rj hGr00 ; rs ; νi · pRj dr0 dr00 dRj :
(3)
For an infinite medium, Eq. (3) has an analytical solution given by [15,16] hGrd ; rs ; νi A
eikjrd −rs j ; 4πjrd − rs j
(4)
where A describes strength of scatter and k q 2 kb ikb ∕ls is the effective medium wave number with ls the average distance between scattering events, where ls nσ s −1 , with n being the density of the scatterers. We approximate the mean-field Green’s function for a scattering slab, as used in the experiment, with the mean-field Green’s function of an infinite scattering medium. Note that the mean field described by Eq. (4) decays with increasing distance due to multiple scatter, approaching zero for jrd − rs j ≫ ls , suggesting zero-mean circular Gaussian statistics hold in that limit [17]. The BSE is the Dyson equation [Eq. (2)] for field correlation [9], and substituting electric field correlation for G in Eq. (2) gives
N Z X
Sr0 −Rj ; r¯ 0 −Rj ;ν
j1
·S r00 −Rj ; r¯ 00 −Rj ;νΔνpRj dRj Z μs δr0 −Rδ¯r0 −Rδr00 −R ·δ¯r00 −RdR;
(6)
hEr; νE r; ν Δνi hEr; νihE r; ν Δνi Z μs hGr; R; νihG r; R; ν Δνi ⋅hER; νE R; ν ΔνidR:
(7)
Treating the scattering medium as a homogenized effective medium with a complex wavenumber k, the mean field for a Gaussian beam is hEr⊥ ; z; νi exp−jr⊥ j2 ∕σ 2 · expikb z · exp−z∕2ls ; (8) where r⊥ defines the transverse coordinates and σ describes the beam width. Note that the mean field approaches zero as z → ∞. Setting hGr; R; νi hGr⊥ − R⊥ ; z; Z; νi [see Eq. (4)], ϕr; ν; ν Δν hEr; νE r; ν Δνi, ψr; ν; ν Δν hEr; νihE r; ν Δνi, and Kr; R; ν; ν Δν hGr; R; νihG r; R; ν Δνi, and taking the spatial Fourier transform of Eq. (7) in the transverse (x–y) direction, ϕq⊥ ; z; ν; ν Δν ψq⊥ ; z; ν; ν Δν Zd Kq⊥ ; z; Z; ν; ν Δν μs 0
hErd1 ; νE rd2 ; ν Δνi
· ϕq⊥ ; Z; ν; ν ΔνdZ;
hErd1 ; νihE rd2 ; ν Δνi Z hGrd1 ; r0 ; νihG rd2 ; r00 ; ν Δνi ⋅Ur0 ; r¯ 0 ; r00 ; r¯ 00 ; ν; ν Δν ⋅hE¯r0 ; νE ¯r00 ; ν Δνidr0 d¯r0 dr00 d¯r00 ;
U 1 r0 ; r¯ 0 ;r00 ; r¯ 00 ;ν;νΔν
where we have used pR V −1 and nσ s μs l−1 s . Substituting Eq. (6) into Eq. (5) and using the sifting property of the Dirac delta function, we find
j1 0
the product of means on the right of Eq. (5). We also assume that there is a single detector, so that rd1 rd2 r. Therefore, U in Eq. (5) can be written as
(5)
where rd1 and rd2 are detector position vectors and U P i U i is the counterpart of the scattering function M in the Dyson equation. U can be interpreted as a four-port network, with two outputs at positions r0 and r¯ 0 and two inputs at r00 and r¯ 00 . We solve Eq. (5) under the ladder approximation, where the scatterer positions are independent of each other and there are no scattering loops, and the ascribed paths taken by the two fields at frequencies ν and ν Δν have at least one common scatterer. The case of no common scatterers is encompassed by
(9)
where d is the slab thickness. We solve Eq. (9), a Fredholm integral equation of the second kind in Z, by discretizing the region Z ∈ 0; d into L uniform subdomains and linearly interpolating ϕq⊥ ; z; ν; ν Δν between ϕq⊥ ; zj ; ν; ν Δν and ϕq⊥ ; zj1 ; ν; ν Δν, where j is an integer, zj jδz, and δz d∕L. Following the solution of the linear equations in Eq. (9) for ϕq⊥ ; z; ν; ν Δν at z d, a Hamming-windowed (with window bandwidth equal to 8∕σ) inverse Fourier transform with respect to q⊥ is taken to obtain hEr⊥ ; d; νE r⊥ ; d; ν Δνi. The large bandwidth of the Hamming window (relative to that of the incident Gaussian beam) ensured that the effect of the window on the correlation results was minimal. In the BSE model of Eq. (9), with a truncated spectral representation in the transverse spatial frequencies, we assume the slab to be
January 1, 2014 / Vol. 39, No. 1 / OPTICS LETTERS
p infinite in the transverse dimensions. We assume ϵb 1.5 and σ 0.5 mm for the incident Gaussian field defined in Eq. (8), consistent with the experiment, and L 30 in the solution to Eq. (9). The spectral domain interferometer experiment shown in Fig. 1 allows the scattered electric field to be determined [2]. The laser source is an external-cavity laser diode (New Focus Vortex II 6917) with a center wavelength of 850 nm that can be scanned over approximately 60 GHz. In the arrangement of Fig. 1, the aperture just to the right of the scattering sample controls the size of the speckle (the intensity correlation area), and the detector aperture (defined by the pinhole) is small relative to the speckle spot size. The linear copolarized and crosspolarized transmitted electric field was measured for commercial white acrylic samples (Cyro Industries, Acrylite FF) composed of small TiO2 particles of average diameter (approximately 50 nm) in a clear acrylic background. Data was collected as a function of position by scanning only the scattering sample (to 50 positions), to form averages. The two scattering materials used had reduced scattering coefficients [2] of μ0s 4 cm−1 and μ0s 14 cm−1 , and μ0s μs because the scatterers are small and the scatter is isotropic. The samples had transverse dimensions 12 cm × 12 cm, much greater than the thicknesses used. Experimental data showed that all samples had zero-mean circular Gaussian field statistics, implicit in Eq. (8), which shows the mean approaches zero with increasing scatter. Figure 2 shows the measured copolarized and crosspolarized electric field correlation magnitude as a function of scan frequency, along with the BSE results, for various thicknesses of a μ0s 4 cm−1 sample. Notice from Fig. 2 that the scalar BSE captures the copolarized nondiffusive light transport properties effectively, and that the diffusion model result (see [18,19] for the method) shown does a somewhat reasonable job of describing
Fig. 1. Spectral interferometer for measuring the randomly scattered electric field. HWP is a half-wave plate, LP is a linear polarizer, PBS is a polarizing beam splitter, and the attenuator is composed of a HWP–LP pair. The reference interferometer gives the scan frequency. The copolarized or cross-polarized light is selected by the LP in front of the photodetector.
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Fig. 2. Comparison of the measured electric field frequency correlation magnitude for copolarized and cross-polarized light with the BSE and diffusion models for a slab with μ0s 4 cm−1 and a thickness of (a) 6 mm, (b) 9 mm, (c) 12 mm, and (d) 18 mm. Standard error bars for estimation of the mean from the experimental data are given.
the cross-polarized correlation. The copolarized response decorrelates more slowly with frequency than the crosspolarized field correlation because that light is less scattered. With increasing scatter, the responses for the two polarizations approach one another and the diffusion regime is entered. It is the more weakly scattering domains, and in the prediction of the copolarized response, that the scalar BSE solution appears to offer a viable forward model. We also made measurements on another sample having μ0s 14 cm−1 , and Fig. 3 shows a comparison with the BSE results. The BSE again predicts the copolarized field correlations. Also, note again the interesting differences between the copolarized and cross-polarized responses for weaker scatter that diminish with increasing sample thickness. The data in Figs. 2 and 3 show the transition between nondiffusive and diffusive light propagation, becoming diffusive for the 18 mm μ0s 4 cm−1 sample and 6 mm for the μ0s 14 cm−1 case. In our case, where the scatterers have a size much less than the wavelength, the nondiffusive to diffusive transition occurs when the slab thickness is about 7–8 times the transport length, approximately in agreement with prior estimates [3,10]. We find it interesting that the scalar BSE solution in the ladder approximation captures measured copolarized electric field correlations over frequency for the weakly scattering slabs studied having small scatterers. The influence of polarization of the incident light and scatterer size on the back-scattered intensity from a BSE model [20] indicates that vector and perhaps scalar solutions may yield a method to determine scatterer information (size or anisotropy) when used with free variables to predict experimental data. With use of a moment theorem that relies on zero mean circular Gaussian field statistics [21], measured intensity correlations can be used in conjunction with a forward BSE model for imaging applications. It also appears that the scalar BSE model with measured
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OPTICS LETTERS / Vol. 39, No. 1 / January 1, 2014 Co−Pol
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Fig. 3. Comparison of the measured electric field correlation magnitude for copolarized and cross-polarized light with the BSE and diffusion models for a slab with μ0s 14 cm−1 and a thickness of (a) 3 mm and (b) 6 mm. Error bars are shown.
copolarized data, combined with Raman or fluorescence spectroscopy, could lead to new molecular imaging methods at the subsurface tissue level. While we used electric field frequency correlations, our measurement method can be extended to allow spatial correlations in addition to frequency correlations. This combination of frequency and spatial correlation data could be used to form images of the spatially varying scatter, as has been done in optical diffusion tomography [11]. We acknowledge funding from the National Science Foundation (NSF) under awards 0854249, 0915966, 1028610, and 1218909. References 1. D. Bicout, C. Brosseau, A. S. Martinez, and J. M. Schmitt, Phys. Rev. E 49, 1767 (1994). 2. T. D. Gerke, M. A. Webster, A. M. Weiner, and K. J. Webb, J. Opt. Soc. Am. A 22, 2691 (2005).
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