Optical determination of field angular correlation for transmission through three-dimensional turbid media

Transcription

1 1040 J. Opt. Soc. Am. A/Vol. 16, No. 5/May 1999 Brian G. Hoover Optical determination o ield angular correlation or transmission through three-dimensional turbid media Brian G. Hoover Department o Electrical Engineering and Computer Science, University o Michigan, Ann Arbor, Michigan Received April 9, 1998; revised manuscript received October 28, 1998; accepted December 7, 1998 A method is presented or measuring the correlation between the optical ields scattered in dierent directions by an arbitrary three-dimensional turbid medium. The ield angular correlation unction is obtained by processing ensemble average intensity data, which are recorded experimentally by a single-channel Shack Hartmann wave-ront sensor. Some general properties o scattered light are expressed in terms o the ield angular correlation unction, and the correlation unction is measured or transmission through a suspension o microspheres under nonballistic transport conditions Optical Society o America [S (99) ] OCIS codes: , , , INTRODUCTION The majority o the research in photon migration has been based, at least in concept, on the application o the classical radiative transer equation. For instance, the seemingly ubiquitous classiication o light as either ballistic (unscattered), diuse (randomized), or subject only to small-angle scattering, then being called quasi-diuse or snake, is ormally based on the orm o the scattering unction that is chosen or the solution o the radiative transer equation. 1 Although the classical ormulation o photon transport has enabled considerable progress in many research areas, there are certain phenomena that arise in the interaction o a spatially coherent wave with an inhomogeneous medium that the classical theory cannot predict. Speciically, because the classical transport approach ignores the phase o the wave, it cannot account or certain coherence-preserving transport phenomena. Enhanced backscattering, which was irst reported in the 1930 s, 2 is one such phenomenon. Watson 3 was apparently the irst to suggest that enhanced backscattering is a coherent transport eect, wherein the spatial coherence o a wave is partially preserved, even ater transport through a multiple-scattering medium. Several other coherent transport eects have been reported in the literature as well. Léger and Perrin 4 used a single-scattering approximation to derive an expression or the ield angular correlation unction or scattering rom rough metal suraces and used the result or surace roughness characterization. Michel and O Donnell 5 used computer simulations to study the ield angular correlation unction or relection rom multiple-scattering metallic suraces with various one-dimensional roughness distributions. Feng and co-workers 6 predicted the memory eect or multiple-scattering static media, which describes the intensity correlation between the transmitted speckle patterns generated by angularly separated incident laser beams. Recently, Leith et al. 7 presented a technique or amplitude phase imaging through turbid media that relies on residual spatial coherence in the scattered wave. Also o note are recent realizations o strong (Anderson) localization o photons. 8 The present paper documents a phenomenological study o ield angular correlation or transmission through multiple-scattering, threedimensional turbid media. Such phenomena not only are o great theoretical interest but also have important potential applications such as characterization o and imaging through biological tissues. It should be noted that the coherent transport eects discussed in this paper are not the result o a ballistic, or specular, component o the scattered wave. Contrary to the classical viewpoint, coherence can be partially preserved even in the absence o a ballistic component. Consequently, although ballistic light is obviously coherent, coherent light is oten not ballistic but scattered, and the distinction should be made among ballistic, nonballistic coherent, and incoherent transport. This paper deals only with nonballistic coherent transport. Subsection 2.A outlines the general ormulation, and some properties o the scattered light are expressed in terms o the characteristic ield angular correlation unction o the scattering medium. To deine ield angular correlation, consider two coherent plane waves, labeled 1 and 2, incident on a scattering medium at angles i1 and i2, as shown in Fig. 1. The light rom each plane wave is scattered over a characteristic range o directions. The ield angular correlation is the cross correlation, averaging over the ensemble o scattered ields, between the ield rom plane wave 1 that is scattered in the direction s1 and the ield rom plane wave 2 that is scattered in the direction s2. It can be measured, in principle, by placing a Young s apparatus in the ar ield o Fig. 1. It is shown in this paper that a single-channel wave-ront sensor may be used to measure the ield angular correlation over an entire range o scattering angles simultaneously /99/ $ Optical Society o America

2 is the spatial requency. denotes the particular realization o the random process that describes the inhomogeneous medium. The medium may also vary with time t, in which case the random process is assumed to be ergodic; hence the variables and t may be reely interchanged in the calculation o average quantities. Here the medium is assumed to be time dependent, and t is adopted as the independent variable. Let h (, t; ) represent the complex Fourier distribution o the light o polarization on the exit surace at time t when a plane wave o spatial requency and polarization is incident on the slab. h (, t; ) may be thought o as the Fourier space point-spread unction (PSF) o the scatter- Brian G. Hoover Vol. 16, No. 5/May 1999/J. Opt. Soc. Am. A 1041 ing medium. Thus, at time t, the scattered wave may be expressed in Fourier space as a ield superposition: U s s, t U i i h s, t; i U i i h s, t; i d i. (2) Fig. 1. Ray-optical schematic depicting two plane waves simultaneously scattered by an inhomogeneous medium. In Subsection 2.B the ield angular correlation unction is derived rom the expression or the ensemble average intensity data recorded by the wave-ront sensor. Subsection 3.A describes the experimental procedure or measuring ield angular correlation, and Subsection 3.B presents experimental results, or transmission through a suspension o microspheres. 2. THEORY A. General Formulation The general problem is illustrated in Fig. 2. A narrowband, spatially coherent optical wave (e.g., laser light) is incident on an inhomogeneous medium rom the let, and a scattered wave emerges rom the exit surace o the medium at the right. The scattering medium is in the orm o a slab o side dimension s, and it is always assumed that s l, where l is the mean scattering length within the medium, so that a section o the slab may be treated as an ininite plane. Initially at issue is the character o the light directly on the exit surace o the slab; Subsection 2.B treats the propagation o the scattered wave through a simple optical system. To keep the equations brie, without loss o generality, the incident wave is restricted to vary in two dimensions only, that is, it may be written as u i (x, z); then only the scattered light with wave vector lying in the x z plane is considered. For a particular polarization state ( ), the incident and scattered waves may be completely speciied in Fourier (reciprocal) space by their complex Fourier transorms, U i ( ) and U s (, ), respectively, where In Eq. (2) and represent orthogonal polarization states. The second term in the integrand contributes i the scattering medium causes some amount o depolarization o the incident wave. Unless otherwise speciied, all o the integrations in Fourier space are over the interval [ 1/, 1/ ], which corresponds to the angular interval [ /2, /2]. A quantitative description o the propagation o the scattered light generally requires knowledge o the ield cross correlation, i.e., the coherence unction. Because the light is narrow band, the coherence unction will be considered under quasi-monochromatic conditions. 9 For simplicity, assume that the incident wave is polarized. Accordingly, rom Eq. (2), the mutual intensity in Fourier space is given by J s1, s2 U s s1, t U* s s2, t U i i1 h s1, t; i1 d i1 U* i i2 h* s2, t; i2 d i2 U i i1 U* i i2 h s1, t; i1 h* s2, t; i2 d i1 d i2. (3) In Eq. (3) denotes a time average. Because Eq. (1) associates s1 and s2 with angular directions o the scattered light, the unction h ( s1, t; i1 )h* ( s2, t; i2 ) is called the ield angular correlation unction, as discussed in Section 1. Analysis o this unction reveals that nonzero correlation may occur only i s1 i1 s2 i2. (4) The condition o Eq. (4) is necessary or correlation under sin (1) Fig. 2. Schematic o the general scattering problem, with an arbitrary two-dimensional (x, z) coherent input wave and a threedimensional scattering medium.

3 1042 J. Opt. Soc. Am. A/Vol. 16, No. 5/May 1999 Brian G. Hoover any transport conditions; it states that a photon that suers a net scattering delection o may be correlated only with another photon that has suered the same net delection. A proo o this condition is given in Appendix A. Applying Eq. (4) to Eq. (3) reveals that the ield angular correlation unction is, in act, the mutual intensity unction in Fourier space when the incident wave consists exclusively o two plane waves o spatial requencies i1 and i2. This result suggests adopting the notation J 1, 2 ;, h 1, t; h * 2, t; (5) or the ield angular correlation unction. Beore an examination o how the ield angular correlation unction may be measured, several important properties o the scattered light can be expressed by using Eqs. (3) (5). The ensemble average intensity o the scattered wave in Fourier space is generally given by I s I s I s, (6) where, rom Eqs. (3) and (4), or arbitrary polarization o the incident wave, I s J, U i i 2 h, t; i 2 U i i 2 h, t; i 2 2Re U i i U i * i J, ; i, i d i, (7) and likewise or I s ( ). The character o the scattered light in real space, described by the mutual intensity unction J(x 1, x 2 ) on the exit surace o the scattering medium, can be calculated rom Eq. (3) through the relation J x 1, x 2 J 1, 2 exp 2 j 1 x 1 2 x 2 d 1 d 2. (8) The derivation o Eq. (8) is given in Appendix B. The results o this section are closely related to those given by Marchand and Wol 10 in their treatment o partially coherent sources. B. Acquiring the Field Angular Correlation rom Wave- Front Sensor Data In this subsection it is shown analytically how the ield angular correlation is obtained rom intensity data recorded by a single-channel Shack Hartmann wave-ront sensor. The material in this subsection also orms the analytical basis or ensemble average wave-ront sensor imaging, a technique that was introduced in Re. 7. The arrangement is illustrated in Fig. 3. For brevity, as in Subsection 2.A, attention is restricted to two-dimensional input waves. For paraxial scattering angles, the lens o the wave-ront sensor orms, in the plane o, an instantaneous Fourier transorm o the sampled ield, similar in expression to Eq. (2), except that now a narrow slit is introduced directly on the exit surace o the scattering medium. h (, t; ) was deined in Subsection 2.A as the Fourier space PSF in the presence o an inhomogeneous Fig. 3. Schematic arrangement or measuring the ield angular correlation. The scattered waveront and the Fourier intensity distribution are shown or an instantaneous observation. The elements to the right o the scattering medium constitute a single-channel wave-ront sensor; those to the let are replaced in the laboratory by a Mach Zehnder intererometer. medium o ininite side dimension. The PSF o the system illustrated in Fig. 3 is straightorwardly related to h (, t; ). Assuming that the sampling aperture o the wave-ront sensor has width w and is centered at point x on the exit surace o the scattering medium, the amplitude PSF o the system o Fig. 3 is given by h, t; x rect 2 c h, t; *w sinc w exp 2 jx, (9) where * denotes a convolution. In Eq. (9), c is the cuto requency o the wave-ront sensor, which is typically determined by the area o the detector array. The nature o the light in the detection plane o may now be ound by substituting h (, t; x) or h (, t; ) in Eq. (3), with U i ( ) equal to the complex amplitude distribution, o polarization, ound in the plane i. Without loss o generality, suppose that the polarizer P in Fig. 3 is oriented to transmit -polarized scattered light, and suppose that the coherent distribution in i is likewise polarized. In this case the ensemble average intensity distribution in o may be expressed as I s x J, x w rect 2 2 i i1 U* i i2 c U h s1, t; i1 sinc w s1 exp 2 j s1 x d s1 h* s2, t; i2 sinc w s2 exp 2 j s2 x d s2 d i1 d i2 w rect 2 2 i i1 U* i i2 c U J s1, s2 ; i1, i2 sinc w s1 sinc w s2 exp 2 j s1 s2 x d s1 d s2 d i1 d i2. (10)

4 Brian G. Hoover Vol. 16, No. 5/May 1999/J. Opt. Soc. Am. A 1043 To break down this expression, irst consider integration over the line s1 s2. On this line the correlation unction o Eq. (10) becomes J ( s, s ; i1, i2 ), which, according to the condition expressed by Eq. (4), vanishes unless i1 i2. When i1 i2, with reerence to Eq. (5), J ( s, s ; i, i ) h ( s, t; i ) 2, which is just the intensity PSF o the system. It ollows that integration over the line s1 s2 will extract the incoherent portion o the distribution o Eq. (10): I inc s x w rect 2 2 c sinc 2 w * U i i 2 h, t; i 2 d i. (11) What remains o the integral o Eq. (10) must be due to correlation between the various scattered plane waves. Thereore the intensity distribution recorded by the waveront sensor can be decomposed into a sum o coherent and incoherent components: I s x I inc s x s x. (12) The coherent component, s ( x), may be simpliied by considering integration over conjugate pairs o points in the our-dimensional s1 s2 i1 i2 space, with the result that s x 2w rect 2 2 c Re i1 i2 i1 s1 s2 s1 U i i1 U* i i2 J s1, s2 ; i1, i2 sinc w s1 sinc w s2 exp 2 j s1 s2 x d s1 d s2 d i1 d i2. (13) Without the sampling aperture [w in Eq. (13)], the coherent term vanishes. Comparison o Eqs. (11) (13) with Eq. (7) o Subsection 2.A shows that the presence o the sampling aperture enables the wave-ront sensor to measure the ield angular correlation. According to the condition expressed by Eq. (4), the vast majority o the points in s1 s2 i1 i2 space do not contribute to the integral in Eq. (13). A change o variables enables reduction o the range o integration to only those points that satisy Eq. (4). The appropriate new variables are i i2 i1 2 i i2 i1 2, s s2 s1, (14) 2, s s2 s1. (15) 2 These variables are proportional to the hal-angle separation o two wave vectors and to the mean angle o two wave vectors. When these variables are used, the condition expressed by Eq. (4) can be imposed by writing the ield angular correlation unction as J s1, s2 ; i1, i2 J i, i, s s i, (16) where is the Dirac delta unction. Once Eq. (13) is converted to the new variables, the integration over s is eliminated, leaving s x 8w rect 2 U i i i 2 c Re i 0 i s U* i i i J i, i, s sinc w s i sinc w s i exp 4 j i x d sd id i (17) as the expression or the coherent component o the recorded intensity distribution or an arbitrary coherent input wave. For the purpose o measuring the ield angular correlation, the input to the system o Fig. 3 is a pair o correlated point sources, located at the points i1 and i2 ( i2 i1 ) in the plane i ; the polarization states o these sources can be changed to examine various correlation unctions. With the input o Eq. (17) expressed by U i i i U* i i i i i i1 i i i2 (18) [other terms do not contribute because i 0 or the integration o Eq. (17)], the integrations over i and i are eliminated, leaving s x 8w rect 2 2 c Re exp 4 j ix sj i, i, s sinc w s i sinc w s i d s, (19) where i and i are now ixed parameters. the deinition Finally, with L sinc w i sinc w i, (20) Eq. (19) may be written as a convolution: s x 8w rect 2 2 c Re exp 4 j i x L * J i, i,. (21) Writing J as the sum o a real and an imaginary part, i.e., J J jj, Eq. (21) may be recast as s x 8w rect 2 2 c L * J i, i, cos 4 i x J i, i, sin 4 i x. (22) Subsection 3.B explains how the ield angular correlation unction is obtained rom the coherent component o the wave-ront sensor data by solving a system o data equa-

5 1044 J. Opt. Soc. Am. A/Vol. 16, No. 5/May 1999 Brian G. Hoover tions o the orm o Eq. (22). The irst step o the solution is to subtract the incoherent component rom the recorded intensity distribution. When the Fourier transorm o the input is substituted into Eq. (11), the expression or the incoherent component reduces to I inc s x w rect 2 2 c sinc 2 w * h, t; i1 2 h, t; i2 2. (23) Thereore the incoherent component can be removed rom the recorded data by acquiring the intensity PSF s or the input points i1 and i2 individually and then digitally subtracting their sum rom the recorded data. In gen- Fig. 4. (a) Typical intensity distributions recorded by a waveront sensor system at two sampling coordinates. In each plot the dashed curve indicates the incoherent component, which is determined by the routine described in the text. (b) The coherent component o each intensity distribution in (a). eral, the incoherent component contributes a bias intensity, and the coherent component, through its dependence on the sampling coordinate x, provides inormation about the ield angular correlation and the phase o the input wave. This is illustrated in Fig. 4, where the recorded intensity distribution is shown or several values o the sampling coordinate x, and the coherent and incoherent components are indicated. 3. RESULTS A. Experimental Procedure The optical system used to measure ield angular correlation is illustrated in Fig. 5. Collimated light rom a HeNe laser ( nm) is sent through a Mach Zehnder intererometer to produce two correlated plane waves o known angular separation and polarization. Lenses L 2 and L 3 serve to expand the beams to a diameter o approximately 250 mm and are ollowed by a 45- mm-diameter iris, resulting in highly planar wave ronts incident on the scattering medium. A precision electronic shutter system is required to ensure that the incident energy per camera exposure is kept constant, which is critical since the typically dominant incoherent component must be subtracted rom the recorded intensity distributions. The turbid medium used or these experiments was an approximately 3-mm-wide glass tank containing an aqueous suspension o 5- m-diameter polystyrene divinylbenzene spheres (SPI Supplies) at a concentration o approximately particles per milliliter. The lens L 4 ( /1.15, 17.5-mm ocal length) images the exit surace o the turbid medium onto the sampling slit o the wave-ront sensor. The sensor is composed o a mm-wide sampling slit, ollowed by an /5.6, 60-mm-ocal-length transorm lens (L 5 ), a rotatable Glan Thompson prism polarizer, and a thermoelectrically cooled CCD array, which digitizes to 14 bits and has 460 pixel columns, separated by 29 m each. The scattered requency ( s) range (bandwidth) o this system is approximately ( 200, 200) mm 1. Both lenses, L 4 and L 5, are corrected or imaging with unit magniication. Fig. 5. Optical system or measurement o ield angular correlation: TP, Thompson prism; ES, electronic shutter system; WP, halwave plate; SA, sampling aperture; GT, Glan Thompson polarizing prism.

6 Brian G. Hoover Vol. 16, No. 5/May 1999/J. Opt. Soc. Am. A 1045 For alignment an incoherent source, e.g., a luorescent tube lamp, is placed on the entrance side o the tank, while a bar-pattern transparency is inserted on the exit surace o the turbid medium, providing a calibrated, high-contrast, incoherent pattern. The imaging lens L 4 is adjusted to achieve proper magniication o the bar pattern (M 2 or these measurements). The wave-ront sensor is then moved into place so that the image is accurately ocused onto the sampling slit. The intensity distribution recorded by the wave-ront sensor is highly sensitive to deocusing error; thereore the operation o L 4 is essential, as it is impossible to place the sampling slit exactly on the exit surace o the turbid medium, which coincides with the interior surace o the tank glass. The wave-ront sensor output under the incoherent source is recorded as the bar pattern is translated north south in Fig. 5. The orm o the intensity distribution obtained by sampling an incoherent ield is invariant or a properly aligned, well-ocused sensor, a result that may be veriied by application o Hopkins s ormula. 11 The bar pattern is B. Data Analysis Reerring to Eq. (22), the theory suggests that a sample o the scattered wave taken at an unspeciied sampling coordinate x yields a linear combination o J and J, the real and imaginary parts o the ield angular correlation unction. The task or data analysis is thereore to solve a system o data equations or the basis unctions J and J. Here a solution method is presented that is relatively ast and easy to implement on a personal computer. The intent is to establish experimental support or the theory rather than to strive or high-precision measurements. A more powerul solution method is being developed, involving optimizations and alternate algorithms that are, however, beyond the scope o this paper. The incoherent component is irst subtracted rom each sample intensity distribution, as prescribed in Subsection 2.B. A numerical deconvolution is perormed with the unction L( ) o Eq. (20), and the remaining coherent component o each distribution is normalized and itted with a polynomial by use o the least-squares method. Cubic polynomials (o 3) were used with good results. The normalized ield angular correlation unction is deined as i, i, s J i, i, s sinc 2 w i h s1, t; i1 2 h s2, t; i2 2. (24) 1/2 oriented parallel to the sampling slit, and the laser intererence ringes are made parallel to the bar pattern to complete the alignment o the sensor. Both incident plane waves were s polarized or the measurements, with one propagating parallel to the optical axis ( i1 0), normal to the tank sides, and the other angularly shited by 2 i. The ield angular correlation was measured in s-polarized scattered light or requencies i 0.286, 0.145, and mm 1, corresponding to spatial ringe periods 1.73, 3.42, and 6.13 mm. Faint intererence ringes o spatial period are visible on the exit surace o the turbid medium i the ield angular correlation is nonzero. The scattered wave may be sampled by translating the entire wave-ront sensor, or, equivalently, the ringe pattern may be scrolled by translating the second beam splitter o the Mach Zehnder intererometer, typically by a ew tenths o a micrometer per sample. I the coordinate o the sampling slit o the sensor relative to the spatial ringe pattern is known precisely, it is possible to obtain the ield angular correlation unction rom two samples o the ringe pattern, as indicated by Eq. (22). However, usually the path length dierence between the two incident waves cannot be strictly controlled, oten because o thermal drit, and thus the location o x 0 in Eq. (22) is not precisely known in the laboratory rame. It is thereore necessary to acquire a number o samples o the scattered wave at dierent values o x and solve a system o equations o the orm o Eq. (22). Between 20 and 30 samples were acquired or each requency i, with a camera integration time o 10 s per sample. According to Eq. (22), the data polynomials should orm a two-dimensional subset in the (o 1)-dimensional polynomial space. The standard inner product on the space o polynomial coeicient vectors is used to perorm a Gram Schmidt orthogonalization o the data polynomials. Each pair o data polynomials deines a plane, or which the mean residual, i.e., the data polynomial vector projection that is orthogonal to the plane spanned by the basis vectors, is computed over the data set. The plane with the minimum mean residual is chosen as the solution plane. Next the projections o the data polynomial vectors onto the orthonormal Gram Schmidt basis vectors are plotted on x, y axes. I the basis vectors happen to correspond to and, the real and imaginary parts o the normalized ield angular correlation unction, respectively, then according to Eq. (22) the data projections should lie on a unit circle. However, the basis vectors generally are linear combinations o the vectors corresponding to and ; hence in general the data projections should lie on an ellipse. An ellipse is itted to the data projections by minimizing the variance o s, the sum o the distances rom a data projection to the two oci o the ellipse. Figure 6 shows the data projections or two o the data sets. An elliptical locus o data projections indicates that the data unctions are in the orm o Eq. (22). The vectors corresponding to and may then be constructed rom the Gram Schmidt basis vectors through the orthogonal rotation and scaling operations that are required or transorming the ellipse into a unit circle. Implicit in this construction is the assumption that the vectors corresponding to and are orthogonal under the chosen inner product. This assumption is justiied by the physical argument that the ield angular correlation unction should be approximately symmetric about s 0 or the small incident angles that were examined

7 1046 J. Opt. Soc. Am. A/Vol. 16, No. 5/May 1999 Brian G. Hoover the turbid medium, as a unction o the scattering angle s, is proportional to V i, i, s i, i, s 2 i, i, s 2 1/2. (25) The odd symmetry o causes the phase o the ringe pattern to shit with the scattering angle, an eect that can be observed by placing a iducial mark on the exit surace and inspecting the ringe pattern rom dierent angles. Fig. 6. Data polynomial vector projections onto their respective solution planes, or 1.73 mm (crosses) and 3.42 mm (circles). Elliptical loci are predicted by the theory. Fig. 7. Experimental results. (a) Real and (b) imaginary parts o the normalized ield angular correlation unction at three spatial periods (mm) or transmission through a microsphere suspension. The mean incidence angle is small: i i. ( i 0). Since the data are generally ound to be nonsymmetric, as evidenced in Fig. 4, must have even symmetry and odd symmetry, or vice versa, which implies orthogonality under the chosen inner product. This argument should not hold as i increases, and a more general solution may have to be sought. The results o the ield angular correlation measurements are given in Fig. 7. The correlation is ound to be nearly symmetric about s 0. The method o solution described above does not distinguish between and, so there is ambiguity over which curve represents the real part and which represents the imaginary part o the correlation unction. On the basis o previous results, 5 the imaginary part is expected to be the smaller component. The visibility o the ringe pattern on the exit surace o 4. DISCUSSION Two eatures o the results o Fig. 7 are especially noteworthy. First, within the range o parameters examined, the magnitude o the ield angular correlation, which is proportional to the ringe visibility, is a nonlinear unction o the ringe period. Second, the ield angular correlation is in general a complex unction o the scattering angle. Both o these properties could be important in the utilization o coherent transport eects or characterization and imaging in inhomogeneous media. A particular example is the eect o a complex ield angular correlation unction on the ensemble-average wave-ront sensor imaging technique that was introduced in Re. 7. This technique is based on the premise that the original phase derivative o a scattered coherent wave is recoverable as the centroid o the ensemble average intensity distribution recorded by the wave-ront sensor. The wave composed o a pair o interering plane waves has a constant phase derivative, provided that the plane wave amplitudes are equal, but it has been ound in the measurements in this paper that a complex ield angular correlation unction implies that the centroid o the wave-ront sensor distribution shits as the corresponding scattered wave is sampled, that is, that the sensor distribution can change orm without a change in phase derivative. In conclusion, the technique presented in this paper should prove useul or (a) testing theoretical models o ield angular correlation and (b) applications utilizing coherent transport, i the relevant correlation unctions can be cataloged or the media o interest. APPENDIX A: PROOF OF THE NECESSARY CONDITION FOR THE EXISTENCE OF FIELD ANGULAR CORRELATION The condition necessary or the existence o ield angular correlation, expressed by Eq. (4), has been derived by several dierent methods in the literature. 4 6 The proo presented here relies only on the most basic mathematics. The condition will be established or an elementary diuser consisting o two scattering points conined to a line. The separation x o the scattering points is a random variable. The necessary condition derived or this elementary diuser will obviously remain valid as the dimension and the number o scattering elements are increased, since the addition o random parameters will not introduce additional correlation in the scattered light. As shown in Fig. 8, consider two correlated plane waves incident on the elementary diuser at angles i1 and i2. The ield angular correlation unction measures the cor-

8 Brian G. Hoover Vol. 16, No. 5/May 1999/J. Opt. Soc. Am. A 1047 d 1 d x d 2 d x, (A6) or, rom Eq. (A2), s1 i1 s2 i2, (A7) which is the condition expressed by Eq. (4). Fig. 8. Geometrical construction or proving the necessary condition or nonzero ield angular correlation. relation between the light scattered rom angle i1 to angle s1 and the light scattered rom angle i2 to angle s2. The two points are assumed to scatter isotropically. The upper point is chosen as the phase reerence and thus contributes the electric ield E 1 in the ar ield. For the realization o the elementary diuser speciied by separation x o the two points, the wave scattered rom i into s is described by where E 1 exp j, (A1) k OPD k sin i sin s x 2 i s x, (A2) where OPD stands or optical path dierence. With this inormation the ield angular correlation unction or the elementary diuser can be calculated as J s1, s2 ; i1, i2 E 1 E 2 * 1 exp j 1 1 exp j 2. (A3) The unction E 1 E 2 * is a random phasor. Because E 1 E 2 * is an average over an ininite number o such phasors, correlation may exist only i all o the E 1 E 2 * lie along the same line in the complex plane. This is equivalent to saying that the phase o E 1 E 2 * must be invariant over the ensemble variable ( x) i correlation is to exist. The mathematical expression o this condition is d tan 1 Im E 1E 2 * d x Re E 1 E 2 * or, equivalently, by using Eq. (A1), d Im E 1E 2 * d x Re E 1 E 2 * d d x 0, (A4) sin 1 sin 2 sin cos 1 cos 2 cos (A5) Perorming the indicated dierentiation reveals the solution o Eq. (A5) to be APPENDIX B: DERIVATION OF MUTUAL INTENSITY IN REAL SPACE FROM MUTUAL INTENSITY IN FOURIER SPACE Equation (8) can be veriied by simply substituting or J ( 1, 2 ) and perorming the indicated integration. The mutual intensity in Fourier space is deined as J 1, 2 U s 1, t U* s 2, t u s s 1, t u* s s 2, t exp 2 j 2 s 2 1 s 1 ds 1 ds 2, (B1) where u s (s, t) is the scattered ield, o polarization, at point s on the exit surace o the scattering medium at time t. Inserting this expression into the right-hand side o Eq. (8) gives J 1, 2 exp 2 j 1 x 1 2 x 2 d 1 d 2 ( u s s 1, t u* s s 2, t exp 2 j s 2 x 2 2 s 1 x 1 1 )d 1 d 2 ds 1 ds 2 u s s 1, t u* s s 2, t s 2 x 2 s 1 x 1 ds 1 ds 2 u s x 1, t u* s x 2, t J x 1, x 2. (B2) Equation (8) can be derived rom irst principles by solving the problem o the propagation o mutual intensity rom the exit surace to the ar ield. The details can be ound in a standard advanced text. 11 ACKNOWLEDGMENTS A sizable acknowledgment is due to Emmett Leith or support and technical advice. Also deserving credit are David Dilworth or technical support, Shawn Grannell and Joaquin Lopez or their laboratory assistance, Mayer Landau or analysis support, and Don Gillespie, El Don Engineering, or technical advice and or providing the El Don ES-100 Electronic Shutter System. This research was supported by the National Science Foundation under grant NSF-ECS and by the U.S. Army Research Oice under grant DAAH Address correspondence to Brian G. Hoover at the location on the title page or by phone, ; ax, ; or , hoover@engin.umich.edu.

9 1048 J. Opt. Soc. Am. A/Vol. 16, No. 5/May 1999 Brian G. Hoover REFERENCES 1. J. A. Moon, P. R. Battle, M. Bashkansky, R. Mahon, M. D. Duncan, and J. Reintjes, Achievable spatial resolution o time-resolved transillumination imaging systems which utilize multiply scattered light, Phys. Rev. E 53, (1996). 2. See J. S. Preston, Retro-relexion by diusing suraces, Nature (London) 213, (1967) and reerences therein. 3. K. M. Watson, Multiple scattering o electromagnetic waves in an underdense plasma, J. Math. Phys. 10, (1969). 4. D. Léger and J. C. Perrin, Real-time measurement o surace roughness by correlation o speckle patterns, J. Opt. Soc. Am. 66, (1976). 5. T. R. Michel and K. A. O Donnell, Angular correlation unctions o amplitudes scattered rom a one-dimensional, perectly conducting rough surace, J. Opt. Soc. Am. A 9, (1992). 6. Shechao Feng, C. Kane, P. A. Lee, and A. D. Stone, Correlations and luctuations o coherent wave transmission through disordered media, Phys. Rev. Lett. 61, (1988); Shechao Feng, Novel correlations and luctuations in speckle patterns, in Scattering and Localization o Classical Waves in Random Media, Ping Sheng, ed., Vol. 8 o World Scientiic Series on Directions in Condensed Matter Physics (World Scientiic, Singapore, 1990). 7. E. N. Leith, B. G. Hoover, D. S. Dilworth, and P. P. Naulleau, Ensemble-averaged Shack Hartmann waveront sensing or imaging through turbid media, Appl. Opt. 37, (1998). 8. D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, Localization o light in a disordered medium, Nature (London) 390, (1997). 9. J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap E. W. Marchand and E. Wol, Angular correlation and the ar-zone behavior o partially coherent ields, J. Opt. Soc. Am. 62, (1972). 11. M. Born and E. Wol, Principles o Optics, 6th ed. (Pergamon, Oxord, 1980), Chap. 10.