Diffractometry through multiple scattering media

Optics Communications 87 (1992) 5-8 North-Holland

OPTICS COM MUNICATIONS

Diffractometry through multiple scattering media Isaac F r e u n d

Depamnent of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel Received 6 August 1991

It is shown that the diffraction pattern of an object may be observed when both the source and the viewingscreen are separated from the object by a multiple scattering turbid medium.

The diffraction pattern of an object often carries sufficient information for a complete determination of the shape of the object. But what if the object is hidden behind a multiple scatterer with both the source and the detector blocked by the intervening turbid medium? Under such circumstances it would appear impossible to obtain the object's diffraction pattern. This is not the case, however, and I show here that under the assumption of ensemble averaging, recording the diffraction pattern in the presence of the turbid medium is scarcely more complicated than recording the pattern in its absence. The turbid medium does exact a price, however: fine detail in the pattern in smeared out by convolution with a function whose width approximately equals the thickness of the medium. Since the analytical form of this smearing function is known, and the (often unknown) properties of the scatterers comprising the turbid medium are found not to be relevant, deconvolution techniques may be applied to recover some or all of the detail apparently missing from the observed diffraction pattern. When this is possible, a high resolution image of an object hidden behind the multiple scattering medium may be obtained. For simplicity, the planar geometry of fig. 1 will be assumed. Since thin optical diffusers are often called screens, it appears appropriate to call the thick multiple scatterer of interest here a wall. A simple scalar wave treatment will first be used to extract the main results, and then the (often non-essential) complications due to the depolarizing effects associated with multiple scattering will be considered. Similarly, an Elsevier Science Publishers B.V.

Wall SourCeroi.:~' i: . r'

Ob ect l

,R

~

Diffraction~il ~

Pattern ~ , , r

b"

L =I~

d

,I

Fig. 1. Diffractiongeometry. ensemble average over scatterer configurations will be assumed initially, after which a single realization will be discussed. Consider first (fig. 1 ) the case in which the wall is absent. Then, the measured intensity is the usual diffraction pattern for illumination with a point source, 2

I ( r ) = l E ( r ) 1 2 = ] f deRG(ro, R,r) O(R)

,

(l)

where O(R) is the complex reflectance of the object. In the Huygens-Fresnel-Khirchhoff approximation

G(r', R, r" ) = ( 1/i2d) 2 exp (i2kd) ×exp{(ik/2d)[(R-r')2+(R-r")2]},

(2)

where k = 2 n / 2 , 2 is the wavelength, and an inclination factor of unity has been assumed. 5

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In the presence of the wall, the field E(r) may be written

E(r)= f d2r' f d2R I d2r" T(ro,r') G(r',R,r" ) XO(R) T(r", r ) ,

(3)

where T ( r, r' ) = T ( r' , r) is the complex (real space) transfer function which describes optical propagation through the wall. Far from localization, the optical field at different points in a multiple scattering medium may be considered as having completely random, independent amplitudes and phases [1]. Accordingly, we may write for the fourth moment

(T(ro, r') T*(ro, s') T(r",r) T*(s',r)> = ( ] T(ro,

r')[2)< IT(r, r" ) 12> 6(s'--r')

6(s"--r") . (4)

( I T ( r , r ' ) 1 2 ) represents the probability that light injected at r will exit at r'. On average, this is independent of the point of injection and depends only on the separation of the entrance and exit points a = r ' - r . Thus,

(IT(r, r ' ) l 2 ) = ( I T I 2 ) D ( o ") ,

(5)

where ( I T] 2 ) is the mean intensity transmittance of the wall, and the diffusive spreading function of the light D(a) has been normalized to unit. Since on average there is no preferred direction in the random medium, D(a)=D(a), where a= la]. I note that D(a) is the radial distribution function of the light intensity in the blur circle on the object side of the wall produced by the point source at ro on the image side of the wall (fig. 1). Since light that diffuses through the wall thickness L can also diffuse sideways by the same amount, the radius of the blur circle is of order L. Taking the origin of coordinates at the source location to, the ensemble average intensity in the observed diffraction pattern may be written

( l(r) ) = ( I TI252

f d2v~(r-v) I(v)

,

(6a)

where ~ ( f l ) = ~ d2aO(fl-•) d

D(*).

(6b)

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Eq. (6), our central result, says that the measured diffraction pattern ( l ( r ) ) is the free-space pattern l(r) smeared by the diffusive spreading of the light as it propagates through the wall. This result is so transparent that it can be written down almost by inspection. Consider, for example, the case in which the point source is located on the object side of the wall, but the diffraction pattern is viewed from the image side of the wall. Since the object is illuminated by a fully coherent point source, the normal free-space diffraction pattern l(r) is produced. However, on the image side of the wall a smeared pattern ( I ( r ) ) is observed because of the diffusive spreading of the light in the turbid medium as described by the impulse response function D(a). So in this case the convolution in eq. (6a) would be with

~o(r-v) = D ( r - v ) . Suppose we now view the diffraction pattern on the object side of the wall, so that the smearing described above is absent, but we keep the point source on the image side of the wall. Because of the diffusive spreading of the light as it propagates through the wall, the source seen by the object is no longer a coherent point, but rather a spatially incoherent blur circle described by D(a). The van Cittert-Zernike theorem [2] implies that such a source has a finite transverse coherence length at the object which is determined by the intensity distribution D(a). Closely spaced object points which lie within a single coherence area interfere fully, so that the widely spaced diffraction fringes they produce are unaffected by the finite coherence length. But interference between widely spaced object points which produce closely spaceddiffraction fringes is damped out. The effect of a finite transverse coherence length is easily calculated, and leads to the convolution in eq. (6a) with ~o(r-v) once again equal to D(r-v). In the general case where both the source and the diffraction pattern are on the image side of the wall, the above two smearing mechanisms operate simultaneously. This leads to a convolution of the individual smearing functions which produces the form for ~o(r-v) given in eq. (6b). The width of this function is also of order the wall thickness L. It is now clear that eq. (6) is a robust result whose range of validity extends well beyond the boundaries of the simplifying approximations used here. The

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smearing of the diffraction pattern as viewed through the wall is, of course, independent of any assumptions about how the pattern is created. The effects of finite source coherence are dependent upon the assumptions made, but as long as the van Cittert-Zernike theorem is applicable, eq. (6) can be expected to provide a good description of the observed diffraction pattern. In particular, the assumption of a planar object (fig. 1 ) may be relaxed, as long as the depth of the object is much less than the wall-to-object distance. I note for completeness that with no approximations regarding the free-space Green's function G(r', R, r" ), and thus with no restrictions on the object or its location,

( l ( r ) ) = ( l T l 2 ) 2 f d2r'f d2r " ×D(ro

- r ' ) D ( r - r " ) I(r', r") ,

(7)

where l(r', r")=I(r", r') is the free-space diffraction pattern at r" produced by a point source at r'. In solving the convolution in eq. (6) for I(r) it is usually most convenient to have the Fourier transform ~ of the smearing function ~o. This may be easily obtained from previous work [3,4] as

O(qL) = f d2fl exp(iq.B) ¢P(B) = [(qL)/sinh(qL) ]4.

(8)

I note that ~ is extremely smooth and should lend itself well to numerical analysis. An estimate of I ( r ) obtained from eqs. (6) and (8) should also prove useful in solving the double convolution in eq. (7) in circumstances where this more accurate form is required. When the vector nature of the light is included the analysis becomes more complicated. Assuming for simplicity that both the source and the detector are x-polarized, eq. (3) becomes

E,.(r)=~d2r'fd2Rfd2r

"

×T,-i(ro, r') G(r',R,r") Ov(R) Tjx(r",r) , (9) where the sums are over i, j=x, y the incident, scattered polarization directions. When the object's diffraction pattern is not polarization sensitive, and the wall is thick enough to depolarize the light (a few

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transport mean free paths /* generally suffice), the sums in eq. (3) are innocuous. In the general case, the observed pattern is a superposition of the object's various polarization-dependent diffraction patterns weighted by polarization correlation parameters which depend upon the properties of the wall. Should the numerous polarization combinations which exist in the general case be required, these may be enumerated with the aid of recent results for multiple scattering polarization correlations [ 5 ]. For a single realization of the wall, a highly speckled diffraction patterns results. There are two sources for this speckling. When used as a viewing screen, the multiple scattering medium introduces sample specific fluctuations into the pattern. The length scale of these fluctuations is very much less than the wall thickness L, however, so that the diffraction fringes (whose apparent width is of order L) are not seriously perturbed and should still be visible. On the other hand, due to the scattering of light from the point source during propagation through the wall, the object is illuminated by a stationary speckle pattern. As a result, the diffraction pattern becomes highly speckled and tends to be obscured. Sweeping the source frequencyfover a range 8f/f> l*/kL 2<< 1, can, however, serve as the equivalent of ensemble averaging [6 ]. This technique may be particularly useful in the microwave region. In addition to the diffusely transmitted light, there is always a directly transmitted beam consisting of photons which have not been scattered. If sufficiently intense, this direct beam can be used for direct imaging of the object. In comparing the direct and diffuse transmitted intensities, two different mean free paths need to be considered. The first, the scattering mean free path or extinction depth z, is the distance the light travels before the direct intensity is attenuated by l/e. The second mean free path is the transport mean free path /*, the characteristic distance over which the initial photon direction is fully randomized [ 7 ]. These two mean free paths are related b y / * = z / ( 1 - c o s 0 ) , where 0 is the angular deflection due to a single scattering event. Typically, for a dense concentration of scatterers whose size is of order or larger than the wavelength of light, l*~ (5-10) z ~ 10-100 ~tm [8]. For a wall of thickness L, the twice transmitted direct beam (through the wall to the object and back again through the wall

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to the detector) has an intensity Id~rec,/lo=exp( - 2L/ r), where Io is the initial intensity. The twice transmitted diffuse intensity, on the other hand, is approximately Id~m,se/Io=(l*/L) z. If L = I 0 / * and l*=5r, then /di .... / l o = e x p ( - 1 0 0 ) = 4 × 1 0 -44. In contrast, Id~ffuse/Io=lO -2. Accordingly, diffractomerry through multiple scattering media would appear to have significant potential for practical application. I am pleased to acknowledge the support of the U.S.-Isreal Binational Science Foundation (BSF), Jerusalem.

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References [ 1 ] M.J. Stephen, Phys. Rev. Lett. 56 (1986) 1809. [2] M. Born and E. Wolf, Principles of optics (Pergamon, New York, 1959) sect. 10. [3] S. Feng, C. Kane, P.A. Lee and A.D. Stone, Phys. Rev. Lett. 61 (1988) 834. [4] I. Freund, M. Rosenbluh and R. Berkovits, Phys. Rev. B 39 (1989) 12403. [ 5 ] I. Freund and M. Rosenbluh, Optics Comm. 82 ( 1991 ) 362. [6] I. Edrei and M. Kaveh, Phys. Rev. B 38 (1988) 950. [ 71 P.M. Morse and H. Feschbach, Methods of theoretical physics ( McGraw-Hill, New York, 1953 ). [8] G.H. Watson, P.A. Fleury and S.L. McCall, Phys. Rev. Lett. 58 (1987) 945.