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Abstract
The high resolution of optical imaging and optogenetic stimulation in the deep tissue requires focusing light against strong scattering with high contrast. Digital optical phase conjugation (DOPC) has emerged recently as a promising solution for this requirement, because of its short latency. A digital micromirror device (DMD) in the implementation of DOPC enables a large number of modulation modes and a high speed of modulation both of which are important when dealing with a highly dynamic scattering medium. Here, we propose full-polarization DOPC (fpDOPC) in which two DMDs simultaneously modulate the two orthogonally polarized components of the optical field, respectively, to mitigate the effect of depolarization caused by strong scattering. We designed a simple system to overcome the difficulty of alignment encountered when modulating two polarized components independently. Our simulation and experiment showed that fpDOPC could generate a high-contrast focal spot, even though the polarization of light had been highly randomized by scattering. In comparison with the conventional method of modulating the polarization along a particular direction, fpDOPC can improve the peak to background ratio of the focal spot by a factor of two. This new technique has good potential in applications such as high-contrast light focusing in vivo.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
When light is propagating through a scattering medium, the inhomogeneity of the refractive index spatially randomizes the amplitude, phase, and polarization of the light so that the conventional optical components cannot focus the light deep within the medium. This phenomenon compromises the spatial resolution of optical imaging and optogenetic stimulation in the deep tissue [1,2], which however are critical for biological research. For example, neuroscientists want to image and stimulate each individual neuron in a circuit in the deep brain to completely understand the neural function [3].
Wavefront shaping (WS) is a technique manipulating the optical field so that it can compensate for the scattering distortion and form a focal spot deep within the medium [4,5]. To find an appropriate optical field compensating for the distortion, people, in general, have three options, including the iterative optimization [6–9], transmission matrix measurement [10–13], and optical phase conjugation (OPC) [14–19]. Among these three options, OPC has an obvious advantage in terms of speed, because it interferometrically measures an optical field in a few exposures, without the time-consuming repetitive measurements necessary for the iterative optimization and transmission matrix methods. Moreover, it has been demonstrated in the light focusing against thick scattering media, showing a potential for deep-tissue optical imaging, manipulation, and therapy [20].
The hardware modulating the optical field in OPC can be the photorefractive crystal [21–23], liquid crystal spatial light modulator (LC-SLM) [14,15,24–26], or digital micromirror device (DMD) [27–29]. The use of DMD has been common, when dealing with the highly dynamic scattering in the biological tissue, due to a large number of modulation modes, high phase-conjugate reflectivity, and high frame rate. The playback latency of OPC based on DMD can be as short as 5.3ms [27] which is good enough to stimulate genetically encoded photochromic guide stars in vivo inside a mouse tumor [29].
Here, we present a new method, called full-polarization digital OPC (fpDOPC), to improve the contrast of the focal spot. The conventional OPC method modulates only one polarized component in the optical field with a single DMD, which neglects the information carried in the orthogonally polarized component due to the depolarization in the scattering medium. In the presented method, we introduced the second DMD to modulate two orthogonally polarized components, simultaneously. Therefore, the full optical field was used appropriately and the peak to background ratio (PBR) of the focal spot was increased by a factor of two. We designed a special optical setup in the demonstration to measure the two polarized components simultaneously and retrieve their phase maps from their interference patterns with a reference beam. We developed a simple method to spatially align the two polarized components, modulated separately, by adding an auxiliary 4f module.
2. Methods
To illustrate how fpDOPC works, we define the incident optical field as ${{\bf E}^{(1)}}(x,y)$ and the field distorted by the scattering medium as ${{\bf E}^{(2)}}(x^{\prime},y^{\prime})$ that consists of the horizontally polarized component ${\bf E}_\textrm{H}^{(2)}(x^{\prime},y^{\prime})$ and vertically polarized component ${\bf E}_\textrm{V}^{(2)}(x^{\prime},y^{\prime})$ (Fig. 1). Two cameras CamH and CamV interferometrically image these two components, respectively. A program calculates the patterns of phase modulation from the images for two DMDs that modulate two polarized components ${\bf E}_\textrm{H}^{(3)}(x^{\prime},y^{\prime})$ and ${\bf E}_\textrm{V}^{(3)}(x^{\prime},y^{\prime})$ for the playback, respectively. After an appropriate recombination of ${\bf E}_\textrm{H}^{(3)}(x^{\prime},y^{\prime})$ and ${\bf E}_\textrm{V}^{(3)}(x^{\prime},y^{\prime})$, the modulation and scattering will largely cancel with each other, and we obtain the wanted optical field ${{\bf E}^{(4)}}(x,y)$ (Fig. 1). When the light propagates through the scattering medium, the scattering scrambles the polarization of the light. The modulation on the two polarized components induces focusing against the scattering. Therefore, we will end up with a twice-enhanced PBR for light focusing through a thick scattering medium.
Fig. 1. Schematic of fpDOPC. (a) Schematic of the recording step with interferometrically imaging the horizontally polarized component ${\bf E}_\textrm{H}^{(2)}(x^{\prime},y^{\prime})$ and vertically polarized component ${\bf E}_\textrm{V}^{(2)}(x^{\prime},y^{\prime})$ in the scattered light. (b) Schematic of playback step. Two DMDs modulate the light and generate ${\bf E}_\textrm{H}^{(3)}(x^{\prime},y^{\prime})$ and ${\bf E}_\textrm{V}^{(3)}(x^{\prime},y^{\prime})$ conjugate to ${\bf E}_\textrm{H}^{(2)}(x^{\prime},y^{\prime})$ and ${\bf E}_\textrm{V}^{(2)}(x^{\prime},y^{\prime})$, respectively. The modulation and scattering cancel with each other.
2.1 Recording process
Based on the random matrix model, the incident optical field ${{\bf E}^{(1)}}$ can be expressed as a generalized Jones vector
We represent the transmission matrix T describing the effect of scattering on the optical field as [25,30]
Then, the horizontally and vertically polarized components in the scattered optical field are obtained as
2.2 Playback process
We measure the phase maps for ${\bf E}_\textrm{H}^{(2)}$ and ${\bf E}_\textrm{V}^{(2)}$ as follows. ${\bf E}_\textrm{H}^{(2)}$ at pixel i is equal to $t_{i1}^{\textrm{HH}} + t_{i1}^{\textrm{HV}} = A_{i1}^\textrm{H}\exp (i\varphi _{i1}^\textrm{H})$ according to Eq. (4). Supposing $A_{\textrm{ref}}^\textrm{H}$ is the amplitude of ${{\bf E}_{\textrm{H\_ref}}}$ and much larger than $A_{i1}^\textrm{H}$, the measured intensity $I_i^\textrm{H}$ at pixel i can be approximated as
2.3 Simulation
According to the equations introduced above, we set N = 2000 and simulated the process of a 45-degree linear polarized light propagating through a strong scattering medium. The result (Fig. 2(a)) showed that fpDOPC compared with the conventional single-polarization digital OPC (spDOPC) increased the PBR of the focal spot by a factor of 2, because the energy in the other polarized component also contributed in the focal spot. We also analyzed how the degree of depolarization caused by scattering would influence the PBR in spDOPC and fpDOPC (Fig. 2(b)). Averaging over 1000 randomly generated transmission matrices at each α, we found that fpDOPC kept the value of PBR around 0.156N regardless of the degree of depolarization in the scattering medium. In contrast, the depolarization of scattering degraded the performance of spDOPC considerably, and the PBR dropped to its lowest level around 0.079N when α = 0.5. Thus, modulating two orthogonally polarized components simultaneously is desirable, especially when the scattering medium scrambles the polarization seriously.
Fig. 2. Simulation of fpDOPC. (a) Enhanced contrast of focal spot realized by fpDOPC. α = 0.5. (b) PBR realized by fpDOPC and spDOPC with respect to varying depolarization.
3. Experimental setup and characterization
Given the proof of fpDOPC in simulation, we next demonstrate fpDOPC in the experiment. Figure 3(a) shows the schematic of our fpDOPC system. A collimated beam generated from a continues-wave laser source (640 nm wavelength, RLK 40200 TS, Lasos Inc.) was split into three beams (Beams 1-3) by two variable attenuators, one was composed of HW1 and PBS1, and the other was composed of HW2 and PBS2. Between HW2 and PBS2, the beam diameter was expanded by the lenses L1 and L2 to match the size of the sample beam and the DMD chips (DLP7000, Texas Instruments).
Fig. 3. Schematic of the experimental setup. (a) Schematic of fpDOPC system. BS, beam splitter; PBS, polarizing beam splitter; L, lens; M, mirror; HW, half-wave plate; SM, scattering medium; Cam, camera. (b) Compartment for recording. (c) Compartment for playback. BS3, L4 and Cam1 are flipped for illustration.
In recording step (Fig. 3(b)), after scattered by the medium (#34-473, Edmund Optics, Inc.), the sample beam (Beam 1) polarized randomly and the reference beam (Beam 2) polarized at 45° would be merged at the beam splitter BS1. The downstream polarizing beam splitter PBS5 separated the vertically and horizontally polarized components in the interference, and the cameras Cam2 and Cam3 (acA1920-155um, Basler) captured the images of the two components, respectively. Following Eq. (6), we then retrieved the phase maps from the two images.
In playback step (Fig. 3(c)), a half-wave plate HW3 and a polarizing beam splitter PBS3 converted the playback beam (Beam3) to a horizontally polarized beam and vertically polarized beam with equal energy. Two DMDs (DMDV and DMDH) then modulated the two polarized beams, respectively, according to the calculated phase maps. A wave retarder (LCC1223-A, Thorlabs, Inc.) compensated the phase difference between the two modulated beams, and the playback beam recombined from them at PBS4 would be reflected by BS1 and the mirror M3 to the scattering medium. Since the modulation was to compensate the scattering, we placed BS3, a focusing lens L4, and a camera Cam1 (CMLN-13S2M-CS, Point Grey Research, Inc) to evaluate the performance of fpDOPC.
The curvature of DMD could result in errors in the modulation. To mitigate this effect, we measured the curvature of the two DMDs through the phase-shift method and corrected the modulation errors during the calculation of binary conjugate phase maps. The curvature of DMD was measured through phase-shifting interferometry. We added a reference beam to interfere with the optical field reflected from the DMD. A phase shifter generated 0, π/2, π, and 3π/2 phase shifts. Then, the phase error caused by the curvature as shown in the insets of Fig. 3 could be calculated from the four interference patterns. What’s more, how to precisely align the sample beam and two playback beams in the coordinates defined by the DMDs and cameras is a critical issue. To solve this problem, we introduced an auxiliary module (lenses L5 and L6 and mirror M4) (Fig. 4). We first loaded a checkboard pattern on DMDH and marked a label on M4. We then adjusted the positions of M4 and Cam3, with respect to the image of checkboard and label captured by Cam3, so that the three components DMDH, M4 and Cam3 were conjugate to one another. Furthermore, we loaded an array of points on DMDH and accordingly obtained a lookup table mapping the pixels between DMDH and Cam3. In the same way, we also aligned DMDV and Cam2.
Fig. 4. Schematic of system alignment and pixel matching. BS, beam splitter; PBS, polarizing beam splitter; L, lens; M, mirror; Cam, camera.
Finally, we demonstrated fpDOPC in the experiment. The medium with strong scattering in the demonstration could effectively depolarize the incident light. We tested this in a simple setup with a polarizer and an analyzer before and after the scattering medium, respectively (Fig. 5(a)). The polarizer selected the polarization of incident light and the analyzer helped measure the energy in each polarized component. According to the speckles captured by a camera, the mean of intensity of the horizontally polarized component was 35.2 and that of the vertically polarized component was 36.4, which confirmed the polarization coupling coefficient α ≈ 0.49.
Fig. 5. Performance of fpDOPC. (a) Measurement of depolarization caused by the scattering medium. (b) Images captured by Cam1 in Fig. 3 after modulating only the horizontally polarized component (left), only the vertically polarized component (middle), and both the two components (right). The dashed box highlights the focal spot. Each image at the bottom was normalized with respect to its own peak intensity. (c) Intensity profiles of the focal spots generated in three different ways.
With this medium and the setup shown in Fig. 3, we tested the performance of fpDOPC. As shown in Figs. 5(b) and (c), when we only modulated the light on a single polarization, the peak values of the focus were 62 and 58 with respect to the horizontal and vertical polarization modulation, respectively, and the corresponding PBR were 115 and 108. When we modulated the two polarized components, the peak value of the focus was 225, about four times that of single polarization modulation. The PBR of the focus was 210, about twice that of single polarization modulation. Thus, we found that fpDOPC in comparison with spDOPC could increase the contrast of focal spot.
4. Conclusion and discussion
We have developed fpDOPC for high-contrast light focusing against strong scattering. This method modulated two orthogonally polarized components simultaneously, and the additional polarized component contributed to the enhancement of contrast. We solved the challenge in the system alignment by introducing an auxiliary module and establishing unified coordinates.
There are many ways further improving the contrast and speed of fpDOPC. One factor that is constraining the contrast is the accuracy of modulation. DMDs in our experiment provided only binary modulation. By taking the super-pixel modulation strategy, DMDs can support more modulation states and recover a more accurate conjugate phase maps than the demonstrated case. In terms of speed, fpDOPC has already greatly benefitted from the high frame rate of DMD and OPC, showing an obvious advantage over the other wavefront shaping methods. To further accelerate fpDOPC to deal with a highly dynamic scattering medium, such as high energy density plasma, we can capture the images with a streak camera which is fast enough so that the DMD, with a custom embedded system processing and transferring information among devices, can be used at its highest frame rate to focus light against fast dynamic scattering.
Moreover, the iterative optimization and transmission matrix methods can potentially benefit from our technique, by introducing the modulation on the second polarized component. Focusing light within the scattering medium is also possible, after introducing an appropriate guide star. This upgrade is even more meaningful for the applications in biomedical imaging and optogenetics.
Funding
Shanghai Pujiang Program (20PJ1408700); Foundation of National Facility for Translational Medicine (Shanghai) (TMSK-2020-129); Natural Science Foundation of Shanxi Province (202103021224015).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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