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Abstract
One of the main challenges in the wavefront shaping technique is to enable controllable light propagation through scattering media. However, the scanning of the focus generated by wavefront shaping is limited to a small range determined by the optical memory effect (ME). Here, we propose and demonstrate efficiently scanning a focus behind scattering media beyond the ME region using the wavefront tilting and re-optimization (WFT&RO) method. After scanning an initial focus to a desired position by wavefront tilting, our approach utilizes the scanned focus at a new position as the “guide star” to do wavefront re-optimization, which can not only enhance the intensity of the focus to the value before scanning but also accelerate the optimization speed. Repeat such a process, we can theoretically fast scan the focus to any position beyond the ME region while maintaining a relatively uniform intensity. We experimentally demonstrate the power of the method by scanning a focus with uniform intensity values through an optical diffuser within a range that is at least 5 folds larger than the ME region. Additionally, for the case of two cascaded optical diffusers, the scanning range achieved is at least 7 folds larger than the ME region. Our method holds promising implications for applications such as imaging through media, where the ability to control light through scattering media is crucial.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Wavefront shaping (WFS) technology has demonstrated its effectiveness in focusing and transmitting light through scattering media [1–4]. When light passes through scattering media, such as biological tissues or clouds, the uneven distribution of the refractive index causes wavefront distortion and the formation of speckle patterns [5]. However, by manipulating the incident light's wavefront, WFS can focus light to a diffraction-limited spot behind or within the scattering media [6,7]. This capability has sparked considerable interest among researchers and has promoted many new applications such as optical trapping [8], phototherapy [9], and optogenetics [10]. The increasing breadth of applications places high demands on the ability of wavefront shaping technology to control and transmit light through scattering media, one significant challenge is achieving controllable scanning of the focus across the entire field of view.
Indeed, measuring the full transmission matrix (TM) [11–14] of the scattering media provides a method for achieving customized focus throughout the full field of view. However, measuring the full TM is computationally complex and memory consuming. Alternatively, a subset of the full TM can be measured for scanning a focus through the scattering media [15], but the process of obtaining multiple optimized phase masks remains time-consuming. Of course, the measurement of the full TM can also be further accelerated by using fast digital micromirror devices (DMD) [16,17]. Another approach involves using optical phase conjugation (OPC) [18–23] or feedback-based iterative optimization [24–30] to obtain optimized wavefronts for each desired focus position, but this approach is also not fast enough. To mitigate the need for multiple optimizations, one can initially obtain the focus through wavefront shaping and subsequently scan it by introducing a phase ramp or physically tilting the wavefront [31–33]. However, this method is still limited to a small scanning range defined by the optical memory effect (ME) [34,35], and the intensity of the focus decreases significantly as the scanning distance increases.
In this work, we propose and demonstrate efficiently scanning a focus behind the scattering media beyond the ME region using the wavefront tilt and re-optimization (WFT&RO) method. This method eliminates the need for additional scanning devices and achieves efficient focus scanning solely by superimposing the phase ramp on the spatial light modulator (SLM). After obtaining the initial focus, the utilization of WFT enables rapid scanning of the focus at the frame rate of the SLM, which is three to four orders of magnitude faster than optimization from scratch at each point. During the wavefront re-optimization process, the focus at the new position acts as a ‘guide star’, facilitating the rapid increase of the intensity to the value before scanning. More importantly, by repeating the WFT&RO process, we can rapidly scan the focus to a range that is at least 5 folds larger than the ME region while maintaining a relatively uniform intensity value throughout the scanning region. This method opens up new possibilities for controlling light propagation through scattering media beyond the ME region and provides an effective solution for applications that require customized focus over a large range such as optical microscopy [36], super-resolution imaging [37], and large field of view imaging [38].
2. Principle
To implement the WFT&RO technique, an initial focus is required behind scattering media, which can be achieved by utilizing feedback-based iterative optimization with a genetic algorithm (GA) [26]. Its basic components include an initial population, a cost function, a crossover process, and random mutations in the offspring, as shown in Fig. 1(a). The GA begins with a population of N randomly generated phase masks. Subsequently, the individuals in the population are ranked based on a cost function, which quantifies the peak-to-background ratio (PBR). A crossover operation is then applied to the candidates with higher measurement values, generating a new offspring. To introduce diversity, random mutations occur by altering a specified number of elements in each phase mask of the offspring. Finally, the individuals with the lowest measured values are removed to proceed to the next iteration. Through the iterative modification of the input light field's wavefront using the above process, the GA seeks to enhance the PBR, which results in an improvement in focusing quality.
Fig. 1. Principle of WFT&RO method with the genetic algorithm (GA). (a) The flowchart of GA. (b) Principle of WFT&RO. (b1) Focusing scattered light behind scattering media by iterative wavefront optimization. (b2) Scanning the focus within the ME region by wavefront tilting (WFT). (b3) Improving the focusing quality of the scanned focus by wavefront re-optimization (WFRO).
Through iterative wavefront optimization, the incident field ${E_n}({{x_n},{y_n}} )$ is optimized and transmitted to the surface of the scattering media, resulting in the generation of a sharp focus, as shown in Fig. 1(b1). The scattering model can be expressed as a linear relationship between the complex field at the incident field and the output field [6]:
To achieve 2-D scanning of the focus, wavefront tilting (WFT) is employed by superimposing a phase ramp on the spatial light modulator (SLM), as shown in Fig. 1(b2). For simplicity, we concentrate solely on scanning the focus along the X-axis direction. To shift the focus towards X-axis by a distance $\Delta x$, a phase ramp $\Delta \varphi = \Delta {k_x}\cdot {x_n}$ is introduced to the incident field ${E_n}({{x_n},{y_n}} )$. As a result, the output field ${E_m}({{x_m},{y_m}} )$ exhibits the same tilt $\Delta {k_x}$. The relationship between $\Delta x$ and $\Delta {k_x}$ can be mathematically expressed as:
where d is the Euclidean distance from the scattering plane to the observation plane. In the context of 2-D scanning of the focus, we can further describe the WFT method as:The WFT method enables scanning the focus within the ME region at the frame rate of the SLM. However, as the scanning distance $\sqrt {{{(\Delta x)}^2} + {{(\Delta y)}^2}} $ approaches the edge of the ME region, the intensity of the scanned focus tends to decrease due to the decorrelation of the scattering media. To overcome this limitation, the wavefront re-optimization (WFRO) technique is then employed to maintain the focusing quality of the scanned focus, as illustrated in Fig. 1(b3). By utilizing the previous incident field ${E_n}({{x_n},{y_n}} )$ with a phase ramp $\Delta \varphi $ as input to the GA with an appropriate mutation rate, the scanned focus serves as a “guide star” for the re-optimization process. As a result, a new output field $E_m^{\prime}({x_m^{\prime},y_m^{\prime}} )$ is obtained, where the focus intensity is re-optimized to the value before scanning.
In addition, by performing several rounds of the WFT&RO process, we can fast scan the focus beyond the ME region with a relatively uniform intensity, which allows for the expansion of the scanning range beyond the conventional limits of the ME.
3. Experimental setup
As a proof-of-concept demonstration, we built the experimental setup as depicted in Fig. 2. A continuous wave laser (1125P, Lumentum) with a wavelength of 632.8 nm is expanded on the center of a phase-only SLM (FSLM-2K70-VIS, CAS MICROSTAR), which has a resolution of $1920 \times 1080$, a single pixel size of 8 μm. The SLM phase pattern is then conjugated onto the surface of the SC (DG10-120, ground glass diffuser, Thorlabs) after passing through a 4f system. Before the optimization process, a blazed grating phase is loaded onto the SLM to ensure that the first diffraction order has sufficient optical power for making a focus, an aperture is placed on the focal plane of L3 and L4 to filter other diffraction orders. Note that the other diffraction orders are not shown apart from the zeroth and the first diffraction order. A CMOS camera ($1600 \times 1200$, 4.5 μm, UI-3250-M_GL, IDS) is placed behind the SC on the observation plane to provide feedback signals for the optimization process. In the experiments, the SLM central pixels in size $1080 \times 1080$ were grouped into large segments to create $108 \times 108$ input modulation segments as one phase mask. The PBR can be expressed as: $PBR = {I_F}/{I_B}$, where ${I_F}$ is the intensity of the focus and ${I_B}$ is the mean intensity of the overall background. Here, we take the mean intensity of the central $5 \times 5$ pixels around the interested position as ${I_F}$, and the mean intensity of all $1600 \times 1200$ pixels as ${I_B}$.
Fig. 2. Schematic of the experimental setup. P: polarizer; A: aperture; SC: scattering media; L1, L2, L3, and L4: lenses with focal lengths of 50, 150, 150, and 50 mm; SLM: spatial light modulator.
4. Results
To prove our concept, we first demonstrated the WFT&RO for scanning and re-optimizing a focus behind scattering media within the ME region. The initial focus at the center of the observation plane is obtained by feedback-based wavefront shaping, whose intensity pattern is shown in Fig. 3(d), in which the PBR finally reaches 83. Its corresponding phase mask is shown in Fig. 3(a). Once the initial phase mask is determined, scanning the focus to various positions within the ME region becomes straightforward thanks to WFT. Next, we obtained the phase mask(Fig. 3(b)) to scan the initial focus by combining a phase ramp with the initial phase mask. The resulting intensity pattern of the scanned focus is presented in Fig. 3(e), and it exhibits a displacement of 50 pixels along the X-axis from the initial focus (Fig. 3(g)). However, due to the substantial scanning distance, the peak intensity of the scanned focus is reduced to 61% of its initial value (Fig. 3(g) and Fig. 3(h)). To enhance the quality of the scanned focus, we employed WFRO to obtain a re-optimization(RO) phase mask (Fig. 3(c)). The RO phase mask was then used to generate a RO focus as illustrated in Fig. 3(f). Importantly, after wavefront re-optimization, the peak intensity of the RO focus increases up to the same level as the initial focus, as observed in Fig. 3(g) and Fig. 3(h). The results demonstrate the effectiveness of the WFT&RO approach in scanning and re-optimizing the focus while maintaining a high-intensity level.
Fig. 3. Experimental results of WFT&RO behind scattering media within the ME region. (a-c) SLM phase masks for the initial focus, the scanned focus, and the RO focus. (d-f) The observation plane images of the initial focus, the scanned focus, and the RO focus. These images highlight the center $120 \times 120$ pixels of the original camera images. The colored arrows in the observation plane images indicate the lines for sectional intensity curves. Intensity profile of the initial focus, the scanned focus, and the RO focus along (g) X-axis and (h) Y-axis. The inset is a local amplification of (h). Scale bars: 10 camera pixels, equivalent to 45 µm.
To further demonstrate the practical efficiency of the WFT&RO technique, we conducted experiments with different step lengths and analyzed their impact on the WFRO process. The intensity patterns of the scanned focus at different positions before and after WFRO are shown in Fig. 4(a). Before WFRO, the intensity of the scanned focus tends to decrease as the WFT process progresses. However, through the WFRO technique, we successfully re-optimized the intensity of the scanned focus, enhancing their focusing quality, as detailed in Fig. 4(b). The performance of the WFRO technique with different step lengths is further illustrated in Fig. 4(c). After 1000 iterations, the final contrast achieved in the original optimization is 83. The results indicate that different numbers of iterations are required to achieve the same contrast as the original optimization for different step lengths: on average 256, 836, 437, 672, and 833 iterations for step lengths of 10, 20, 30, 40, and 50 pixels, respectively. In a word, at a position, WFRO offers a time saving of no less than 16.4% when contrasted with the time required for optimization from scratch.
Fig. 4. Experimental demonstration of scanning a focus behind scattering media by WFT&RO with different step lengths. (a) The intensity patterns of the scanned focus at different positions before and after WFRO, respectively. (b) The intensity profiles of the scanned focus before and after WFRO, respectively. (c) Variation of the contrast along with the number of iterations under various step lengths. The number in the parenthesis refers to the step length.
Despite the slower increase of contrast when the step length is set to a small value of 10 or 20 pixels, the strong initial contrast achieved after the WFT process with a small displacement often renders the WFRO unnecessary in many cases. However, it is worth noting that our re-optimization method still proves successful in optimizing the focusing quality after a certain number of iterations, even in such cases. On the other hand, as the step length increased to 30, 40, and 50 pixels, the number of iterations required for the WFRO process also increased accordingly, which indicates that larger step lengths introduce more iterations to achieve the desired contrast. These findings highlight the importance of selecting an appropriate step length in the WFT&RO process to balance the speed of scanning and the efficiency of optimization.
Undoubtedly, our approach brings efficiency gains when compared to the extensive iteration required in traditional optimization processes. In contrast to performing 100 rounds of optimizations from scratch to achieve focusing along 100 different positions, our method can start with an initial optimization to generate an initial focus, proceed to employ WFT to obtain 49 scanned focuses, then execute a WFRO process with a step length of 50 pixels to generate a RO focus, and ultimately use WFT to obtain remaining scanned 49 focuses. Notably, the WFT operations involved in acquiring the 98 scanned focuses require a time equivalent to a mere 0.2% of the time taken by a single optimization process. Additionally, the WFRO process, executed with a step length of 50 pixels, saves at least 16.4% of the time compared to an optimization from scratch. Consequently, in this scenario, compared to optimizing from scratch at each position, our WFT&RO method only requires about 1.84% of the time consumed by the former. This underscores the efficiency enhancement our method offers in scanning a focus through scattering media.
Having verified the efficiency and effectiveness of WFT&RO for scanning and re-optimizing a focus, further exploration was conducted to assess its capability beyond the ME limits. In Fig. 5, we present a comparative analysis between the WFT&RO and the conventional optimization from scratch method. We set a step length of 50 pixels for the WFRO process and the optimization from scratch process to obtain several discrete focuses (Fig. 5(a)), and the remaining focus was obtained through WFT operations (Fig. 5(b)). Interestingly, the WFT&RO expands the scanning of the focus to a range of 350 pixels, which is at least 5 folds larger than the ME region, as illustrated in Visualization 1. Furthermore, the WFRO significantly accelerates the optimization speed, leading to higher focusing intensity levels within the same number of iterations. Throughout the entire scanning range, the mean intensity of the scanned focus attained through the WFRO was 35.4% higher compared to that achieved through optimization from scratch. Evidently, our WFT&RO technique for focus scanning behind scattering media demonstrates the ability to expand the scanning range significantly beyond the confines of the ME region. Moreover, it achieves this with enhanced intensity within the same number of iterations compared to the conventional optimization from scratch method.
Fig. 5. The comparison of the WFT&RO and optimization from scratch. (a) The intensity profiles of WFRO and optimization from scratch at a range of 350 pixels. The ME region of the diffuser is about 60 pixels. The number of iterations was all set to 200, the step length was all set to 50 pixels. (b) The measured intensity of the scanned focus as a function of the scanning distance. The scanned focus is obtained by the WFT operation after WFRO and optimization from scratch, respectively. More details can be seen in Visualization 2.
Regardless of scattering properties and the range size of the memory effect of the scattering media, WFT & RO technology can be applied in principle as long as effective focusing behind the media can be performed. In the following, we also show the experiment using two cascaded optical diffusers (DW110-1500D, ground glass diffuser, LBTEK) as samples, where the distance of the two scattering surfaces is 2 mm and the ME region is much smaller and is about 8 pixels as can be noted in Fig. 6. Within the entire scanning range, the mean intensity of the scanned focus obtained by the WFT&RO is 19.2% higher than that achieved through optimization from scratch with WFT, as shown in Fig. 6. The results underscore the enduring effectiveness of the WFRO approach in accelerating optimization speed even under the influence of complex scattering effects. In addition, despite the strong scattering effect and the reduced ME region, our WFT&RO method still proves to scan a focus within the range that is at least 7 folds larger than the ME region.
Fig. 6. The experimental results for scanning a focus behind two cascaded optical diffusers by the WFT operation after WFRO and optimization from scratch, respectively. The number of iterations was all set to 200, the step length was all set to 8 pixels.
5. Discussion
In this work, we introduced a WFT&RO technique as an efficient method for scanning a focus behind the scattering media beyond the conventional ME region. By performing several rounds of WFT&RO, we successfully achieved a scanning range of at least 5 folds larger than the ME region (Fig. 5). Furthermore, we experimentally demonstrated the effectiveness and flexibility of scanning a focus beyond the ME region in our approach, even in challenging scenarios with two cascaded optical diffusers, achieved a scanning range that is at least 7 folds larger than the ME region (Fig. 6). The extended scanning range overcomes the limitations of the scanning methods based on ME [32,33,36], enlarging the available field of view for applications such as scattering imaging.
It is important to highlight the trade-off inherent in our WFT&RO technique, which involves balancing the mean intensity of the scanned focus against the total number of iterations. This trade-off is concisely summarized in Table 1. To illustrate, let's consider a scanning range of 300 pixels after an original optimization from a random speckle, 9, 7, and 5 WFRO executions are required for the step lengths of 30, 40, and 50 pixels, respectively. This finding underscores a key principle: employing a larger step length facilitates coverage of the same scanning range with fewer iterations. Conversely, opting for a smaller step length yields a higher mean intensity of the scanned focus. This trade-off provides flexibility in selecting the appropriate step length based on the specific requirements of the application. For example, in scenarios like fluorescence imaging through scattering media, where the fluorescence excitation power exerts a significant influence on imaging quality, a higher focusing intensity becomes imperative. To elaborate, even though employing a step length of 30 pixels requires nearly an additional 800 iterations to cover the same scanning range as a 50 pixels step length, it achieves a 45.9% higher focusing intensity than the latter. This heightened intensity could translate to an elevated signal-to-noise ratio for imaging, thus enhancing image quality.
Table 1. The trade-off between the mean intensity and the total number of iterations.
Indeed, the WFT&RO technique theoretically allows scanning a focus to any position on the observation plane if we disregard the intensity decrease caused by the large displacement from the initial focus position. However, in our experiment, as the scanning range increases to a distance longer than 350 pixels, re-optimizing the focus to its original value becomes challenging due to the reduced available photon budget [39]. Alternatively, we could increase the laser excitation power to achieve a higher focus intensity at a larger scanning range.
We concentrate here on the expansion of the scanning range and the reduction of the optimization time during the scanning process. However, it's worth noting that the potential acceleration of our technique can be further harnessed through faster optimization algorithms [30,40,41] or advanced hardware configurations [42–44]. While our implementation utilizes a phase-only SLM operating at a frame rate of 60 frames per second, the optimization time can be systematically reduced as long as it preserves the previous phase mode through the WFT operation. Recent advancements have introduced modulation techniques based on DMD with modulation speeds exceeding 22.7 kHz. Leveraging DMD-based holographic modulation techniques, such as the LEE method [45] or superpixel method [46], offers the potential for faster wavefront modulation, which might further promote our method’s application in dynamic scattering media such as biological tissues. Notably, in biological tissues, the memory effect is diminished. The recent advancements in utilizing DMD for high-speed wavefront shaping present a compelling prospect, with the capability to achieve light focusing within a matter of milliseconds. If this technique was employed to establish continuous 50 focuses along 50 pixels in biological tissues with the memory effect region limited to 10 pixels, assuming the optimization time for a single focus is 33.8 ms achieved in [16], it would take a total of 1.69s to complete the construction of these focuses. However, our method has the potential to accelerate this process. After obtaining the initial focus, the establishment of these 50 focuses could be achieved through a combination of the WFRO process, which would generate 4 RO focuses, and the WFT operation, which would obtain 45 scanned focuses. Given that WFRO saves at least 16.4% of the time taken by optimization from scratch, a single WFRO process would take about 28.3 ms. The WFT process for acquiring the 45 scanned focuses could be performed at the frame rate of the DMD, taking approximately 2 ms in total. Taking these considerations into account, the utilization of WFT&RO would enable the scanning of these 50 focal points in a mere 149 ms, yielding a 91.2% time savings compared to optimization from scratch at each point. In addition, the time saved by the WFRO technique holds the potential to reduce the duration that light interacts with biological tissues, thereby mitigating potential photo-damage effects. Moreover, there is a report of projecting a focus through various dynamic scattering samples within a few microseconds using a 350 kHz modulator [44], which might further enhance the light manipulation ability of our method in biological tissues.
In this work, we operate under the assumption that both sides of the sample are accessible. However, for scenarios where access to the back of the sample is restricted, alternative approaches such as invasive “guide star” can be employed for focusing purposes. Moreover, recent developments in WFS [47–50] make it possible to form an optical focus non-invasively through scattering media. Our WFT&RO technique might hold substantial promise for integrating the non-invasive focusing to enable the customized focus throughout a large field of view in a non-invasive way. Interestingly, previous advancements in focus scanning have been applied for imaging through scattering media [32], although the field of view remains limited by the memory effect. We believe that our work has the potential to create the required lighting modes through scattering media beyond the memory effect, thereby utilizing this controllable lighting to further achieve non-invasive large field-of-view imaging through scattering media [38].
6. Conclusion
In summary, we have demonstrated that WFT&RO offers an efficient approach for scanning a focus with relatively uniform intensity values behind the scattering media beyond the ME region. After scanning the initial focus to the desired position by WFT, our method subsequently utilizes the scanned focus as a “guide star” for WFRO, resulting in the restoration of the focusing quality to its original value and acceleration of the optimization speed. The WFT&RO technique is not theoretically limited by the ME, although maintaining a certain focal intensity over a larger scanning range becomes challenging due to the reduction in the available photon budget. Our experimental results show that WFT&RO can rapidly scan a focus with uniform intensity values within a range that is at least 5 folds larger than the ME region. In addition, we experimentally achieved a scanning range that is at least 7 folds larger than the ME region through two cascaded optical diffusers while maintaining a relatively uniform focus intensity. The technique effectively mitigates the intensity degradation and limited range encountered in the scanning approaches based on ME, thereby expanding the applicability of the method in various optical systems and imaging setups.
Funding
National Natural Science Foundation of China (62275188); International Science and Technology Cooperation Program of Shanxi Province (202104041101009); Natural Science Foundation of Shanxi Province (202103021223091, 20210302123169).
Disclosures
The authors declare no conflicts of interest.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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