† Corresponding author. E-mail:
Project supported by the National Key Research and Development Program of China (Grant No. 2017YFB1104500), the Natural Science Foundation of Beijing (Grant No. 7182091), and the National Natural Science Foundation of China (Grant No. 21627813).
This paper puts forward for the first time a combined transmission matrix (TM) method to measure the monochromatic TM of scattering media without a reference beam. This method can be named a sequential semi-definite programming method which combines the sequential algorithm and the semi-definite programming method. Firstly, each part of the TM is calculated respectively in proper sequence. Then every part of TM is combined to form a complete TM in accordance with a certain rule. The phase modulation of the incident light is achieved by using a high speed digital mirror device with the superpixel method. We have experimentally demonstrated that the incident light field is focused at the target through scattering media using the measured TM to optimize the wavefront of the incident light. Compared with the semi-definite programming method, our method takes less computational time and occupies less memory space. The sequential semi-definite programming method shows potential applications in imaging through biological tissues.
Wave propagation in complex media is a fundamental problem in acoustic, electromagnetism, and optics.[1] Transmission matrix (TM) is the basic model to describe wave propagation within a medium. In optics, controlling light propagation in scattering media has a wide range of application, such as biological imaging, information security, and ghost imaging.[2–4] In 2007, Vellekoop et al. put forward a method of wavefront-shaping for the first time. They focused the incident beam through scattering media with iterative optimization algorithm.[5] There are many algorithms to optimize the wavefront, such as genetic algorithm, sequential algorithm, and particle swarm optimization algorithm.[6–8] All of these algorithms require feedback to optimize the wavefront. In 2010, Popoff et al. proposed another method for wavefront-shaping.[9] They reconstructed an original image from the speckle field by measuring the monochromatic TM of multiple scattering media. In this TM measurement method, feedback regulation was not a necessity. However, the traditional TM measurement methods, such as four phase method[10] and Lee holographic method,[11] needed to measure the output complex field by interfering with the reference light field, which made the experimental setup more complex. In recent years, many phase retrieval algorithms were applied to measure TM of multiple scattering media, such as prGAMP, prSAMP, prVBEM, prVAMP, GS, WF, PhaseLift, and PhaseMax.[12] But the author used a high-throughput computing (HTC) cluster to compute TM of multiple scattering that personal computer could not complete. In 2017, N’Gom et al. measured the TM with the semi-definite programming (SDP) method,[13] which simplified the experimental setup because there was no reference beam and personal computer could complete the work. However, with the increase of TM elements, the method required more calculation time and consumed more memory space. For example, with the element number of 36, the method took 10.4 minutes and consumed 2-GB memory space. However, with the element number of 49, the method took 106 minutes and consumed 40 GB memory space. In consequence, as the number of transmission matrix elements increased, the calculation time and the memory space increased exponentially. Therefore, the computational complexity of this method was a great challenge for personal computers when the number of elements was large.
In this paper, we propose for the first time a combined transmission matrix method to measure the monochromatic TM of scattering media without a reference beam, i.e., the sequential semi-definite programming (SSDP) method. In this SSDP method, we first divide the TM into multiple parts and calculate each part in proper sequence. Then we assemble all parts together and obtain a complete TM after a corrective optimization. Therefore, the SSDP method can greatly reduce the calculation time and memory. This method also provides a new experimental idea to measure large TM of scattering medium because it can reduce the calculation time and memory. In the current experiments, we use digital micromirror device (DMD) to modulate the phase of the incident wavefront with the superpixel method.[14,15] The SSDP method is used to compute the TM of the scattering medium by recording only the intensity information of the output light field. Then based on this TM, we compute the optimal wavefront for focusing the incident light through the scattering medium. Our method has greatly improved the speed of light focusing compared with the SDP method.
We use DMD as spatial light modulator, which is an excellent electronic equipment for controlling light fields. It owns millions of switchable micromirror units and has an extremely fast refresh rate up to 23 kHz. However, the DMD can only achieve spatial light binary amplitude modulation in general and we need phase modulation to get random phase mask in the SSDP method. Thus, we use the superpixel method to achieve spatial light phase modulation on the DMD.[14,15] The superpixel method, which is simple and highly robust, takes full advantage of the characteristics of DMD. In this method, the adjacent m × m micromirrors are grouped into single surperpixel and each superpixel defines a complex field in the target plane. After a plane light beam illuminates DMD surface with an angle, the reflected light is divided into many diffraction orders. In the +1 or −1 order, the phase is modulated by superpixel-based DMD. A 4 × 4 superpixel is shown in Fig.
The process of TM measurement by the SSDP method is shown in Fig.
The input field and the output field are related by the N × N TM of the scattering medium
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As the number of transmission matrix elements increases, the calculation time and the memory space increase exponentially. Therefore, computing costs much time and memory space when the number of segment of input field is large. We use the following method to reduce the calculation time and memory space. First, we divide Tn into several parts. In order to minimize the calculation time and memory space, the number of parts is always the square root of the number of Tn elements. If the square root is not an integer, we choose an integer that is just greater than the value of the square root. Second, we calculate each part of Tn separately. Third, every part of Tn combines to form a complete TM. As shown in Fig.
As Step 1 shown in Fig.
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![](cpb_27_8_084201/cpb_27_8_084201_ieqn1.gif)
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![](cpb_27_8_084201/cpb_27_8_084201_ieqn4.gif)
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The above problem can be solved by using the SDP method[13] and the PhaseCut algorithm.[16] Equation (
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![](cpb_27_8_084201/cpb_27_8_084201_ieqn7.gif)
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For Step 2–Step 16 shown in Fig.
Though the maximal target |bn(i)| on the output field can be obtained by each individual optimal incident field
First, randomly selected phase shifts (
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![](cpb_27_8_084201/cpb_27_8_084201_ieqn13.gif)
![](cpb_27_8_084201/cpb_27_8_084201_ieqn14.gif)
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We also solve the above problem using the SDP method[13] and the PhaseCut algorithm,[16] such as Eq. (
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The experimental setup is shown in Fig.
We calculate a TM with 256 elements. First of all, we divide the TM into 4 × 4 parts and each part has 4 × 4 elements. In the same way, we divide the DMD into 4 × 4 parts and each part has 4 × 4 segments. Then each part of TM is measured in proper sequence as written in the Subsection
![]() | Fig. 4. (color online) (a) The intensity at the target for 766 measurements on the CCD camera. (b) The target field before optimization. (c) The target field after optimization. |
We also perform some experiments about different numbers of TM elements. We use the SDP method to calculate TM when N = 9, 16, 25, 36, and 49, and use SSDP method to calculate TM when N = 36, 81, 144, 256, and 400. The SSDP method and the SDP method are compared in four aspects. Due to the limitation of computer performance, we can only calculate the N = 49 TM with the SDP method. Each data point is the average of ten experiments. As shown in Fig.
In this paper, we put forward a combined transmission matrix method to measure the monochromatic TM of scattering media for the first time, i.e., the SSDP method. The calculation speed of the SSDP method is investigated and compared with the SDP method in experiment. We find that the calculation speed of the SSDP method is much faster than that of the SDP by comparing the time consumed. In addition, the SSDP method requires less computer memory. The method also provides a new experimental idea to measure large TM of scattering medium. In our experiment, by combining the SSDP method and the superpixel method, we have realized rapid light focusing through the scattering medium. The phase of the input light field is modulated by a DMD (a pure amplitude modulation device) with the superpixel method. Using the measured TM, the optimal mask is computed. We believe that the SSDP method requires less calculation time when we use high-performance computer clusters. The SSDP method has shown the potential for imaging or light delivery through biological tissues.
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