2.1.

Principle and Experimental Setup

The experimental setup is depicted in Fig. 1. The signal pulses at ${\lambda }_s=1.064\mu \text{m}$ are delivered by a partially frequency-doubled Q-switched mode-locked Nd:YAG laser (PL2250 Series from EKSPLA). The nearly ${\mathrm{TEM}}_{00}$ IR pulses have a 20-ps duration (FWHM), a time-bandwidth product close to 1, and a 10-Hz repetition rate. The pump pulses at ${\lambda }_p={\lambda }_s/2=0.532\mu \text{m}$ are provided by partial frequency-doubling of the IR radiation in a potassium dihydrogen phosphate crystal. The remaining IR light is separated from the green light by a dichroic mirror and then attenuated and vertically polarized. The IR pulses illuminate a resolution target, whose image is formed inside the diffusing medium (biological tissues or translucent adhesive ribbon) with a magnification of $1/20$ by means of an aspherical lens with a focal lens of 8 mm and a numerical aperture of 0.5. During the imaging process, the diffusing medium can be moved either in the transverse plane or along the propagation axis. The output scattered light is then collected with a second identical aspherical lens that conjugates the image plane of the first lens with the input face of the nonlinear crystal. The total magnification of the imaging system between the object and the crystal is close to 1. The $5\times 5{\mathrm{mm}}^2$ wide and 2.5 mm long potassium titanyl phosphate (KTP) crystal from Cristal Laser SA is cut for degenerate type 2 collinear phase-matching¹⁹ and its 20 mrad phase-matching angular range is adapted to the numerical aperture of the imaging system. The neutral axes of the crystal are oriented respectively along the vertical and the horizontal directions. The pump pulse reflected by a dichroic mirror is mixed with the IR scattered signal wave in the KTP crystal. A small amount of the scattered light reflected by the dichroic mirror is used to directly observe the light transmitted through the diffusing medium. Direct imaging through the diffusing medium is performed with a charge-coupled device (CCD) camera (CCD2) placed in the image plane of the second aspherical lens.

Fig. 1

Experimental setup.

In a degenerate three-wave mixing interaction, the idler wave is generated at the same wavelength as the signal and with a transverse wave-vector opposite to that of the signal [Fig. 2(a)], giving forward phase-conjugation¹⁹ [Fig. 2(b)]. In this work, as in Ref. 15, the idler wave is back-reflected by the coated exit face of the crystal (high reflecting coating at 1.064 μm), resulting in reversing the longitudinal component of the idler wave vector that leads to full phase-conjugation [Fig. 2(c)]. The reflected waves (amplified signal and idler) back-propagate toward the diffusing medium. The PC image is formed by the idler wave at a symmetric position of the object with respect to the $50/50$ beam-splitter and recorded on a low-noise CCD camera (CCD1: Princeton Instruments, Spec-10 400B). A Glan–Taylor polarizer placed in front of CCD1 rejects the vertically polarized distorted signal wave and transmits the horizontally polarized idler wave carrying the PC image. We assume here that the PC image keeps its horizontal polarization while back-propagating in the diffusing medium. This nontrivial issue will be discussed in the next section. The signal amplification gain $G_s$, defined as the ratio between the signal pulse energies with and without amplification, has been measured, giving a value close to 2. Because the added power on the signal is, for identical signal-idler wavelengths, equal to the generated idler energy, the gain of phase conjugation $G_{\mathrm{PC}}$, defined as the ratio between the incident signal energy and the energy of the PC wave, is given by $G_{\mathrm{PC}}=(G_s-1)$.

Fig. 2

(a) k-wavevector diagram for noncolinear three-wave mixing. (b) Principle of forward three-wave mixing phase-conjugation imaging. (c) Full phase-conjugation by three-wave mixing. The horizontal and vertical arrows represent the crossed polarization directions of the signal and idler waves in a type 2 interaction. For the sake of clarity, the pump plane wave is not represented.

2.2.

Diffusing Media

In these experiments, two kinds of diffusing media are studied. The first one consists of a translucent adhesive ribbon fixed on a 1-mm-thick microscope slide. The value of the ribbon thickness is a few tens of microns and the light traversing the rough surface evolves in a fully developed speckle. From measurements of the size of the speckle halo, the mean size of the diffusing grains is estimated to be 1 μm. The degree of polarization of the scattered light is given by the ratio $I_{\Vert }/I_{\perp }$, where $I_{\Vert }$ and $I_{\perp }$ are, respectively, the energies of the scattered light, measured in the parallel and orthogonal polarization directions with respect to the polarization direction of the incident light. A ratio of $385\pm 35$ is measured while this ratio is 500 through the clean microscope slide. Thirty-five percent of loss due to scattering is also measured by comparing the energies transmitted through the imaging system with and without the diffusing medium. These results indicate that, with this kind of medium, most of the light is scattered in the forward direction and the scattering process is almost polarization independent. With such a thin scattering medium, direct imaging is possible with a good SNR when the object is imaged at the surface of the diffusing medium [Fig. 3(b)], because of the so-called shower curtain effect.²⁰ An SNR close to 2 is deduced from the peak value of the normalized covariance function ${\mathrm{cov}}_{\max }$ (Ref. 21) between the image obtained through the clean microscope slide [Fig. 3(a)] and the image obtained through the diffusing medium [Fig. 3(b)], such that

Fig. 3

Results of direct imaging results through different media: (a) The clean microscope slide. (b) and (c) One layer of translucent adhesive ribbon when the image of the object is formed on the diffusing medium and when the diffusing medium is placed before the image plane, respectively. (d) Two layers of adhesive ribbon on both faces of the microscope slide. (e) and (f) Biological tissues: 0.7-mm-thick Parma ham slice and 5-mm-thick chicken breast, respectively.

In the direct imaging configuration, the image resolution is limited by the size of the speckle grains. When the diffusing medium is placed before or after the image plane, the image of the object is completely blurred and the object can no more be recognized [Fig. 3(c)]. For the sake of comparison, a second layer of adhesive ribbon was fixed on the other face of the microscope slide. In this case, direct imaging is impossible even when the object is imaged on the first adhesive ribbon layer [Fig. 3(d)] and whatever be the position of the image plane with respect to the diffusing medium.

The second kind of diffusing media that we studied consists of two different biological tissues. The first one is formed with slices of Parma ham with thicknesses from 0.7 to 2.1 mm and the second one is a 5-mm-thick piece of chicken breast. The tissues are maintained between two microscope slides. With a standard scattering coefficient ${\mu }_s=100{\mathrm{cm}}^{-1}$ and an anisotropy coefficient $g=0.9$, the reduced scattering coefficient is ${\mu }_s^{\prime }={\mu }_s(1-g)=10{\mathrm{cm}}^{-1}$. Then, the thicknesses of the biological samples, expressed in number of reduced scattering mean free paths ($mfp\prime$), lie in the range 0.7 to 5 $mfp\prime$. We measured the degree of polarization and losses through the two kinds of biological tissues for the translucent adhesive ribbon as well. Average degrees of polarization of 20, 10, and 2 and losses of 50, 90, and 97% have been measured, respectively, for 0.7- and 2.1-mm-thick slices of Parma ham and a 5-mm-thick piece of chicken breast. It should be stressed that the studied biological tissues (consisting of muscle fibers and fats) are very inhomogeneous. Therefore, these measurements fluctuate with large amplitudes depending on the area crossed by the light. Finally, we verified that direct imaging is impossible through biological tissues [Fig. 3(e) and 3(f)], whatever be the position of the diffusing medium with respect to the intermediate image plane.

2.3.

Image Restoration by Three-Wave Mixing Phase-Conjugation

In this section, we present the results of image restoration by backward propagation of the PC idler wave through diffusing media. We have investigated the resolution limits in restored images as well as the consequences of a motion of the diffusing medium during the acquisition time. Figure 4(a) shows the limits of resolution in images formed by the backward PC wave propagating through the clean microscope slide. In this image, the line group $16{\mathrm{mm}}^{-1}$ of the USAF resolution chart is resolved. Since the magnification of the imaging system between the object and the type 2 KTP crystal is close to 1, it corresponds also to a resolution of $16{\mathrm{mm}}^{-1}$ in the crystal, which is in good agreement with the theoretical resolution given by the phase-matching conditions.²² Figure 4(b) shows the PC image obtained through one static layer of translucent adhesive ribbon positioned such that direct imaging is impossible. In the restored image, the resolution and the SNR are seriously degraded by the speckle noise. The $5.04{\mathrm{mm}}^{-1}$ line group can be recognized and an SNR equal to 2 is measured. As with a static diffusing medium, the speckle noise patterns are strongly correlated from one laser shot to another, and the SNR in PC images can be seriously improved if the diffusing medium moves during the acquisition time. Indeed, in this case, uncorrelated speckle noise patterns are summed by CCD1 during the acquisition time. This speckle averaging reduces the noise fluctuations and therefore improves the SNR of the PC images. In these experiments the acquisition time is 1 s, corresponding to the acquisition of 10 laser shots. Figure 4(c) and 4(d) clearly shows the SNR and resolution improvement when the diffusing medium is moving. These figures correspond to the PC images obtained when, respectively, the medium is moved along the transversal axis ($Y$) and along the propagation axis ($Z$). In both cases, the diffusing medium is translated with an amplitude of $\pm 50\mu \text{m}$ around its initial position during the exposure time of CCD1. In PC images, the $16{\mathrm{mm}}^{-1}$ line group of the USAF resolution chart is now resolved and an SNR equal to 4 is deduced from the normalized covariance function [Fig. 4(e)] calculated between Fig. 4(a) and 4(d).

Fig. 4

Resolution limits through translucent adhesive ribbon. (a) Phase-conjugated (PC) image through the clean microscope slide. (b) PC image through a static single layer of translucent adhesive ribbon. PC images when the diffusing medium is moved (c) along the transversal axis (Y) and (d) along the propagation axis (Z) during the acquisition time. White rectangles surround the $16{\mathrm{mm}}^{-1}$ line group. (e) Normalized covariance between PC images (a) and (d).

The same experiments were performed with a 0.7-mm-thick biological tissue. Figure 5 shows these results with a static medium and a medium in motion. Although the biological tissue is much thicker than the translucent adhesive ribbon, similar performances are obtained: the $14.24{\mathrm{mm}}^{-1}$ line group is resolved and an SNR equal to 2.6 is measured. We would like to stress that in the intermediate image plane, the resolution is $285{\mathrm{mm}}^{-1}$ according to the $1/20$ magnification of the first aspherical lens. Moreover, surprisingly, we observed that the restored image is linearly polarized in a crossed polarization direction with respect to the incident signal polarization direction. It shows that the process of image restoration by TWMPC does not require that the back-propagating PC wave has the same polarization state as the scattered signal wave emerging from the scattering medium. This particular point will be discussed in Sec. 3.

Fig. 5

Resolution limits through 0.7-mm-thick Parma ham slice: (a) PC image through the clean microscope slide. (b) PC image through static biological tissue. PC images when biological tissue is moved (c) along the transversal axis (Y) and (d) along the propagation axis (Z) during the acquisition time. White rectangles surround the $14.24{\mathrm{mm}}^{-1}$ line group. (e) Normalized covariance between PC images (a) and (d).

The last experiment consists of image restoration through biological tissues with different thicknesses. Parma ham slices of thicknesses 0.7, 1.4, and 2.1 mm and a 5-mm-thick piece of chicken breast were used. Figure 6 shows the PC images of the 1.41 and $2.83{\mathrm{mm}}^{-1}$ line groups of the USAF target obtained through the Parma ham and chicken breast samples, respectively. Figure 6(a) and 6(b) corresponds to the PC images of the two line groups obtained through the clean microscope slide. Figure 6(c) to 6(f) corresponds to the PC images obtained through static biological tissues with increasing thicknesses. Because the PC images are seriously degraded by the speckle noise, the initial object is more and more difficult to recognize when the thickness of the biological tissue increases. When biological tissues are moved during the acquisition time of CCD1, the SNR of PC images is improved significantly as in previous experiments [Fig. 6(g) to 6(j)]. While motion of biological media seriously improves the resolution of PC images, the SNR clearly decreases when the thickness of the biological tissues increases. Indeed, losses also increase with the thickness. Thus, the PC image is progressively drowned in the background noise, mainly because of the back-scattered light. Experimentally, increased losses are compensated partially by increasing the amplification gain of the nonlinear TWM process (i.e., by increasing the pump pulse energy).

Fig. 6

Image restoration through 0.7- to 5-mm-thick biological tissues. (a) and (b) reference PC images of 1.41 and $2.83{\mathrm{mm}}^{-1}$ line groups of the USAF obtained through the clean microscope slide. (c) to (f) PC images obtained through static biological media with 0.7-, 1.4-, and 2.1-mm-thick Parma ham slices, respectively, and 5-mm-thick piece of chicken breast. (g) to (j) PC images obtained through the same biological tissues moving along Y axis.