Imaging through random media using low-coherence optical heterodyning A. Schmidt, R. Corey, and P. Saulnier Department of Physics, Gustavus Adolphus College Saint Peter, MN 56082 ABSTRACT Optical heterodyning is a detection scheme that allows the preferential detection of ballistic photons transmitted through a random media, enabling the imaging of absorbing structures within. We introduce a normalized cross-correlation function as a quantitative measure of image quality and employ it to investigate the effect of source coherence length on the ability of optical heterodyning to detect these absorbing structures. The ability to image, in transmission, is found to be greatly enhanced using a reduced coherence length source. The effect on image quality of the transport mean-freepath length is quantified and found to depend on source coherence length. The necessity of considering scatterer particle-particle correlations is also presented. PACS numbers: 42.30.Wb, 42.25.Bs, 42.62.Be, 42.68.Ay 1
Using non-ionizing optical wavelength radiation to image structures within human tissue possesses many benefits and challenges. Multiply scattered light quickly dominates the ballistic component of the transmitted intensity as the scattering coefficient, the number of extinction mean-free-path lengths per unit sample length increases. This rapid attenuation of the ballistic component of transmitted light is embodied in Beer's law [1],, where is the ballistically transmitted intensity, is the incident intensity, is the sample thickness, and is the extinction mean-freepath length. Temporal imaging techniques all rely on the fact that the ballistic light will be the first light to arrive at the detection apparatus while the multiply scattered component will be greatly delayed, providing the necessary rejection. Various time-of-flight detection schemes have been used including streak camera [2], electronic [3], nonlinear gating [4, 5], coherent temporal gating [6], as well as electronic [7] and photographic [8] pulsed holography. In contrast to time-resolved methods, spatially resolved techniques rely on directional selectivity to suppress the diffuse component of the transmitted light. Spatially resolved techniques include optical heterodyning [9] and confocal imaging [10]. While effective, these spatial techniques may have difficulty rejecting light that has been multiply scattered back into the ballistic direction. In this Letter we use a low coherence CW transmission optical heterodyning technique [11] to quantitatively examine the degradation of image quality with increasing scattering coefficient. Additionally, the effects of source coherence length and transport mean-free-path length on image quality are also quantified. Finally, we illustrate the necessity of using a Mie cross-section that has been corrected for scatterer particleparticle correlations when determining the merit of an imaging system. Figure 1 illustrates the modified Mach-Zehnder experimental arrangement. Two laser sources were used to determine the effect of source coherence length on image quality. The long coherence length source was a HeNe laser (633 nm) possessing a coherence length of 0.4 m, while the short coherence length source was a diode laser (780 nm) with a 300 m coherence length. In each case, the intensity incident upon the 2 sample cell was 0.75 mwcm. 2
AOM1 80.1 MHz AOM2 80 MHz BE1 CC BS1 SC.. BS2 PD1 BE2 PH2 PH1 Lock-in Reference Laser TS1 Lock-in Signal Figure 1: The modified Mach-Zehnder interferometer used to image absorbing structures within a random media, in transmission, via optical heterodyning. A diode laser (780 nm) and HeNe laser (633 nm) were used along with other major components; BS1, BS2, beam splitters; AOM1, AOM2, acousto-optic modulators; BE1, BE2, beam expanders; TS1, TS2, 1D and 2D translation stages; PH1, PH2, 800 m and 400 m pinholes; PD1, PD2, photo-diode detectors; and SC, CC, sample and compensating cells. PD2 TS2 The light in the reference arm of the interferometer was Doppler shifted by using two acousto-optic modulators (AOMs). The first modulator upshifted the frequency by 80.1 MHz while the second modulator downshifted it by 80 MHz. Beam expanders were used to enlarge the beam to a size slightly larger than the 1.0 cm wide sample cell so as to provide an unobstructed path for light to reach the lock-in reference detector. A compensating cell was used to match the optical path length of the ballistic light traversing the sample cell. The random media used as diffusely scattering samples were monodisperse suspensions of polystyrene spheres in water, 0.997 m and 0.107 m diameter spheres. The mean-free-path lengths were varied over more than a decade by successively diluting a stock solution to obtain a volume fraction range of 0-0.08 % for the large spheres and 0-5.0 % for the small spheres. These vastly different volume fraction ranges produced nearly identical scattering coefficients due to a large discrepancy in Mie scattering crosssections between the two sphere sizes. 3
Both two-dimensional and one-dimensional absorbing masks were imaged. The two-dimensional mask used had a donut-like absorption profile while the onedimensional mask produced a square-wave absorption profile with a 4 mm period. The four edged one-dimensional mask was used primarily to speed accumulation of data. The detector aperture, used in all scans, was 400 m while the sample-to-detector distance was 10 cm. The one-dimensional mask was used for the quantitative study with data being collected every 200 m. One of the primary goals of this Letter is to quantify the effect that various experimental parameters have on the ability of optical heterodyning to image absorbing structures within a random media. Image cross-correlation was employed. The discrete cross-correlation is defined as (1) where are pixel indices and are image intensity at specific pixel locations. A reference intensity profile,, is obtained by imaging the absorbing maskobject through clear water. This profile is correlated with all other sample images obtained through suspensions with increased scattering coefficients. The maximum correlation value, which occurs when the two images are perfectly overlapped, is a measure of the image quality. We define a normalized correlation function where max is the maximum value of the correlation between the sample and reference image and max is the maximum value of the correlation between the reference image and itself. Thus, an image of comparable quality to that obtained when no scatterers are present will possess a normalized correlation value of one. As stated above, the extinction mean-free-path length is the light scattering parameter that indicates the spatial rate of attenuation of ballistic photons while the transport mean-free-path length is the average distance needed to randomize the photon direction. In a random media with max max many low angle snake-like photon trajectories may exist, possibly impacting the ability to image absorbing structures within [12]. The influence of transport mean-free-path length on image quality within an optical heterodyning experiment was investigated by viewing absorbing structures through random media with 10 (0.997 m spheres) and 1 (0.107 m spheres). Judicious choice of volume fraction range for the two sphere sizes was made so that identical, and hence optical thicknesses L, were probed. The influence of on image quality was found to depend on the source coherence length. Figure 2 summarizes 4
the results for the short coherence length source. No dependence on was observed as both sphere sizes exhibited identical rate of image degradation with the image finally being lost at L 22, corresponding to a ballistic attenuation of 22. The necessity of properly accounting for scatterer particle-particle correlations is also illustrated by Fig. 2. If the number of spherical scatterers per cubic wavelength becomes large the scattering path lengths, calculated within the independent scatterer approximation are significantly underestimated. This underestimation leads to the curve represented by the diamond markers in Fig. 2 (the 0.997 m spheres being virtually unaffected owing to their small volume fraction). The independent scatterer approximation yields the scattering path lengths as 1 and 1 where is the scatterer number density and are the extinction and transport cross-sections calculated from Mie theory [13]. If this method of calculating the scattering path lengths were used an erroneous conclusion would be drawn from Fig. 2. The incorporation of scatterer particle-particle correlations requires that the Mie cross-sections be modified by the inclusion of a structure factor. The Percus-Yevick [14] structure factor for a hard sphere potential [15] works well [16] and yields the curve given by the circular markers. L / l for 0997µ. m 0.0 0.5 1.0 1.5 2.0 2.5 3.0 L / l for 0107µ. m 0 5 10 15 20 25 30 Normalized Correlation 1.0 0.8 0.6 0.4 0107. µm 0997. µm 0107. µm w / uncorrelated l 0 5 10 15 20 25 30 L / l Figure 2: Image quality expressed as a normalized correlation function versus optical thickness L, where L 1 cm is the sample thickness and is the extinction mean-free- 5
path length. Also shown are scales that indicate the number of transport mean-free-path lengths across the sample, L/, for the two scatterer diameters 0.997 m and 0.107 m. Except for the diamond markers all mean-free-path lengths, account for particle-particle correlations by inclusion of the hard sphere Percus-Yevick structure factor into the Mie cross-sections. Data obtained using a 300 m coherence length source. The above experiment was repeated, under identical conditions, with the long coherence length source. The results are presented in Fig. 3. The first observation is that there is a dramatic difference in the optical thickness through which images can be successfully obtained. An imaging limit of L 8 is obtained for the 0.997 m spheres and L 11 for the 0.107 m spheres compared with the L 22 previously obtained for each. The rate of image degradation is also significantly different than that obtained with the short coherence length source. 1.2 L / l for 0997µ. m 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 L / l for 0107µ. m 0 2 4 6 8 10 12 14 Normalized Correlation 1.0 0.8 0.6 0.4 0107. µm 0997. µm See Optics Letters 20, 404 (1995) for this inset. 0.2 0 2 4 6 8 10 12 14 L / l Figure 3: Image quality expressed as a normalized correlation function versus optical thickness L, where L 1 cm is the sample thickness and is the extinction mean-freepath length. Also shown are scales that indicate the number of transport mean-free-path lengths across the sample, L/, for the two scatterer diameters 0.997 m and 0.107 m. Data obtained using a 0.4 m coherence length source. Inset shows the 6
temporal distribution of transmitted photons for the two scatterer sizes, each with L 10. Finally, a dependence of image quality on using the long coherence length source is noted in Fig. 3. Two scattering solutions each possessing identical extinction mean-free-path lengths L 10, but differing transport mean-free-path lengths L 1 and L 10, yield a normalized correlation value of 0.2 and 0.6 respectively. It is perhaps somewhat surprising that the sample with the longest transport mean-free-path length proved to be the most difficult one to image through. A qualitative explanation for this fact can be illustrated with the aid of the inset in Fig. 3, which shows the temporal distribution of light diffusely transmitted through slabs with L 1, 10 respectively [17]. Phase randomization, due to multiple scattering, will lead to fewer of these photons contributing to the heterodyne signal, nevertheless, only diffuse light that is within the coherence time of the laser may contribute a heterodyne signal. Furthermore, the visibility of these interference fringes will be determined by the photon arrival time expressed as a fraction of the coherence time of the laser source. It is clear from the inset that the sample with L 1 has more photons of higher visibility available to degrade the image, as observed. The short coherence length laser has a coherence time of 1 psec, virtually eliminating any contribution to the image from diffusely scattered light and thus any dependence on. We are pleased to acknowledge support from a William and Flora Hewlett Foundation Award of Research Corporation and the donors of The Petroleum Research Fund, administered by the American Chemical Society, for support of this work. [1] H. C. van de Hulst, Light Scattering by Small Particles (Dover Publications, Inc., New York, NY, 1981), p. 401. [2] K. M. Yoo, B. B. Das, and R. R. Alfano, Opt. Lett. 17, 958 (1992). [3] S. Andersson-Engels, R. Berg, S. Svanberg, and O. Jarlman, Opt. Lett. 15, 1179 (1990). [4] K. M. Yoo, Q. Xing, and R. R. Alfano, Opt. Lett. 16, 1019 (1991). [5] M. D. Duncan, R. Mahon, L. L. Tankersley, and J. Reintjes, Opt. Lett. 16, 1868 (1991). [6] M. R. Hee, J. A. Izatt, J. M. Jacobson, J. G. Fujimoto, and E. A. Swanson, Opt. Lett. 18, 950 (1993). 7
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