1. Introduction

Controlling light propagation through scattering media is of fundamental interest in optics and critical for applications in biomedical imaging and materials inspection [1–20]. Recently there has been an increasing interest in wavefront control techniques for focusing through turbid media [2–19]. These techniques rely on the deterministic nature of multiple scattering to shape the incident wavefront and pre-compensate for the scattering effects of light propagation in the material and beyond. Iterative methods divide the light incident on a scattering sample into N spatial input modes [2–4]. The optimal phase of each mode is measured and set to create a focus on the opposing side of the scattering material. Other iterative techniques optimize the input modes in parallel, thus increasing the speed at which the focus is formed [4–6]. Other techniques measure the transmission matrix through the scattering material [7,8]. With the transmission matrix the relationship between the input modes and output modes through the sample is understood quantitatively and phase masks can be calculated to focus to any mode in the output plane. Optical or digital phase conjugation have also been used to record the scattered field and return a focusing beam through the turbid media [9,10]. Many of these focusing techniques have already been employed in imaging through scattering media demonstrations [11–14].

An application of particular interest in this area of research is imaging in biological materials. The imaging depth into biological materials is limited by scattering, which could be compensated via wavefront control. However, living biological materials have speckle decorrelation times on the millisecond timescale [15–17]. This fast rate of change makes most current methods of focusing through turbid media too slow, because of switching rate limitations imposed by the wavefront modulation device. Most current methods use liquid crystal spatial light modulators (LC-SLM) for phase-only wavefront modulation [2–7], which is more efficient for creating a focus than amplitude only modulation [18]. The LC-SLMs switching speed is limited by the rate at which the liquid crystals can align in the device. Typically this is in the 10s of Hz: much slower than the kHz rate required for the millisecond timescale of biological tissues. Thus, new high-speed techniques for optimizing phase masks are required to implement focusing through biological samples. A first step in this direction has been implemented recently in a system that uses a scanned laser beam for wavefront determination and an LC-SLM for the final phase modulation [19]. The system achieved wavefront determination in 400ms. Here we introduce a new high-speed phase mask optimization technique, which utilizes off-axis binary-amplitude computer-generated holography implemented on a deformable mirror device (DMD) [21] and demonstrate wavefront determination about one order of magnitude faster than the prior state of the art. Furthermore, the transfer matrix approach provides a general and thorough characterization of the scattering medium that not only allows for the focusing on a given point in space but also enables the determination of wavefronts for other optical processing applications [7,13,20].

2. Binary amplitude holography

We introduce the computer-generated holographic technique to simultaneously implement phase-only control of the wavefront using the high switching speed of DMDs. Phase-only wavefront modulation has in principle a theoretical enhancement five times higher than binary amplitude modulation for the same number of input modes [2,18]. Considering that biological samples have a limited timeframe for focusing in the 10s of milliseconds and that commercially available DMDs have a maximum full-image frame rate of 22.7 kHz, the possible number of optimized modes available under these conditions is limited. For this reason we combine the 1024x768 array of binary pixels into far fewer modes for phase-only modulation through off-axis binary amplitude holography. As a result, we are able to simultaneously take advantage of the high efficiency of phase modes and high frame rate of DMDs.

Binary amplitude off-axis holography is a well known method for generation of uniform-amplitude phase-modulated images [22]. We generate an amplitude hologram, t(x,y) using the Lee method [22]. An off-axis reference wave encodes the desired phase distribution, φ(x,y):

This technique can be extended to phase and amplitude modulation by selecting thresholds that give the appropriate relative amplitude of t(x,y) [22]. However, the available amplitude levels are limited by the number of pixels per period which must be relatively small to allow sufficient separation between diffraction orders. The desired wavefront is produced in the −1st diffraction order, thus an aperture spatial filter placed in the Fourier plane of the hologram around the −1st diffraction order blocks out all other diffraction orders. After another Fourier transforming lens the image is created with uniform intensity and phase variation determined by φ(x,y). Figure 1b illustrates a Lee hologram, h(x,y), corresponding to the phase distribution shown in Fig. 1a.

 

Fig. 1 (a) An example of a phase distribution, φ(x,y), for a single Hadamard basis element surrounded by a phase reference for transmission matrix measurement. (b) The binary amplitude Lee hologram, h(x,y), which encodes the phase distribution shown in (a).

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3. Transmission matrix focusing

The transmission matrix focusing method [7] was selected for wavefront determination, because it uses a set of predefined phase masks and requires only three measurements per input mode. Using a predefined set of phase masks minimizes data transfer time between the DMD and the computer allowing the DMD to display all preloaded images at a maximum frame rate.

The observed transmission matrix (K_obs) is measured by calculating the complex field response for a set of given input basis modes. The Hadamard basis set is selected, because it can be represented as a phase basis with uniform amplitude. The complex field response is measured by interference between the Hadamard basis element and known phase reference beams. To minimize the number of measurements made, we use a three-phase method to recover the complex field [23,24], instead of the four-phase method [7]. Each Hadamard basis element interferes with phase references of 0, π/2, and π displayed on the frame of each function as shown in Fig. 1a and detected after propagation through the scattering medium. Furthermore, due to CCD frame rate limitations, we use a photodetector for high speed intensity measurements. This simplifies the transmission matrix measured into an Nx1 matrix, defined as the response of N input modes to a single output mode. The observed transmission matrix response for Hadamard basis element n is calculated with the intensity measurements at the output mode from all three phase references ($I^0,I^{\pi /2},$and $I^{\pi }$) according to:

Once the observed transmission matrix is measured, the appropriate phase mask for creating a focus at the single output mode is calculated:

The three-phase reference transmission matrix measurement method can find a suitable phase mask after measurements of 3*N predefined phase masks, providing a 25 percent improvement in speed over the 4*N measurements previously used [7]. Numerical simulations show that, with experimental level noise, the overall signal to background enhancement is comparable to the enhancement achieved with four-phase references.

4. System

A collimated and expanded 532 nm laser illuminates the DMD (DLP Discovery Kit D4100), which consists of an array of 1024x768 mirrors, as shown in Fig. 2 . Each mirror is individually controlled to two angular positions, which are used to encode the binary amplitude Lee hologram. For our purposes we use N = 256 or 1024 inputs to a single output mode defined by the photodetector. To implement the transmission matrix measurement method with the DMD we generate 768 binary amplitude holograms for N = 256, or 3072 holograms for N = 1024. Figure 1a shows an example phase distribution, with the centered Hadamard basis element surrounded by the reference. Figure 1b shows the resulting Lee hologram that generates the phase distribution in Fig. 1a. The experimental diffraction efficiency of the holograms with the DMD was 6-10% of the incident power. All holograms are loaded onto the DMD memory, which in conjunction with high speed software, allows for DMD control at maximum frame rate.

 

Fig. 2 Diagram illustrating the experimental apparatus. A collimated 532nm laser beam is encoded with a Lee binary amplitude hologram by the DMD. The iris (I) passes the −1st diffraction order, which contains the hologram information, and blocks all other orders. The uniform amplitude, phase variation image is created at the back aperture of the 20X objective and focused onto the scattering sample. A 100X objective images a plane about a millimeter behind the scattering sample. This image is relayed to a pinhole placed before a photodetector. The photodetector signal is sent to a USB oscilloscope for A/D conversion before being processed by a PC to create a binary amplitude mask. A CCD and beamsplitter capture the focal spot image. DMD: deformable mirror device. f_A, f_B, and f_t: lenses. I: iris. S: scattering sample. P: polarizer. BS: beamsplitter. PH: pinhole. PD: photodetector.

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A Fourier transforming lens is placed one focal length away from the DMD. An iris placed after this lens in the Fourier plane blocks all diffraction orders, except for the −1st diffraction order, where the phase mask information is encoded. The −1st order light is then propagated through another Fourier transforming lens, which images the phase mask at the back aperture of a 20X (NA = 0.5) objective lens that focuses the beam onto the scattering sample. A 100X (NA = 0.75) objective images a plane ~1 mm behind the scattering sample onto a 50 μm pinhole placed before a photodetector. The back objective and the pinhole size are selected to match the pinhole to the speckle size of the light scattered by the sample. The photodetector voltage is digitized and sent to the computer, where it is used to calculate the transmission matrix through the scattering material to the single output mode. A LabVIEW program controls all system computation and synchronization. By using a photodetector the intensity measurement is oversampled in time and an average value is used for the intensity measure to filter noise. A non-polarizing beamsplitter placed after the tube lens and before the pinhole creates a second image plane on a CCD array for imaging the focus spot.

5. Experiment

Using 120-grit ground glass (Edmund Optics, NT83-381) as the scattering medium we test the off-axis binary amplitude hologram focusing system. Each binary amplitude hologram is displayed on the DMD for 22 μs. With a switching time of 22 μs the total period for each mask is 44 μs (22.7 kHz). Thus, for N = 256, all 768 measurements for transmission matrix determination occur in 33.8 ms. The photodetector signal is digitized and sent to the computer where the average intensity value for each measurement is used to calculate the transmission matrix of the system. Using the transmission matrix of the N input modes mapped to the single output mode, the phase conjugate mask is calculated and used to maximize the intensity of the light at the photodetector. The enhancement of the focus is calculated using the focus image taken with the CCD.

Using this system we have demonstrated signal enhancements over the background intensity level of 164 and 454 with N = 256 and N = 1024 respectively. The values are comparable to enhancements obtained using a phase-only liquid crystal spatial light modulator [2,7]. Interestingly, the intensity enhancement does not scale with N as predicted by theory in the ideal case, which is likely a result of phase hologram degradation for higher N due to the limited degrees of freedom of the DMD [22]. Figure 3a illustrates how the output mode intensity varies with each binary amplitude hologram. This is the sample data corresponding to the first 25 Hadamard basis elements and their three phase references. Figure 3b shows an example of a focus spot created through the scattering sample with 1024 modes, with signal enhancement of 450 over the background level, while Fig. 3c shows the intensity distribution with a single Hadamard basis element and reference phase hologram on the DMD to illustrate the speckle field without optimization. With either 256 or 1024 input modes a focus spot with FWHM of 1.0 μm is created, which is comparable to previous reports with the same scattering sample [5,19].

 

Fig. 3 (a) First 3.3 ms of digitized sampling data from photodetector showing the intensity of the first 25 Hadamard basis modes interfered with three phase references. (b) Focus spot with enhancement of 450 created after optimization with 1024 modes. (c) Speckle field without the optimized phase mask.

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6. Focusing through dynamic turbid media

To explore the implication of dynamic changes of the transmission matrix in focusing through multiply scattering turbid materials with the off-axis binary amplitude technique, the algorithm was run with temporally dynamic turbid media of various persistence times. The persistence time of a turbid sample is the time over which the speckle pattern remains stable [15]. For this experiment a 785 nm laser was used with the previously described system with N = 256 input modes. The samples were prepared using varying amounts of Gelatin and water to create semifluid materials with Intralipid added as a light scatterer. These were 1 mm thick and had mean free path of ~200 μm. Given that the system run time between the start of measuring the TM to when the optimized mask displays is ~300 ms (Fig. 4f ), speckle decorrelation times greater than 300 ms were investigated, namely 350 ms, 650 ms, and 850 ms. The measured average peak enhancements were 28, 54, and 69, respectively (Fig. 4a-c).

 

Fig. 4 Enhancement of the focus spot versus time with temporally dynamic turbid samples of persistence times: (a) 350 ms (Media 1), (b) 650 ms (Media 2), and (c) 850 ms (Media 3). (d) Speckle field before optimization. (e) Focal spot after wavefront optimization through turbid sample with T_p = 850ms. (f) The timing of the system: Measure transmission matrix (TM): 34ms (red). Transfer data to computer, compute new mask, and transfer data to DMD and project: 270 ms (yellow). Display optimized binary amplitude mask: 200 ms (green).

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Figure 4f shows the timing of the system. The first 34 ms have the lowest enhancement and correspond to the measurement time of the TM when the DMD mirrors are moving quickly and less light arrives at the focal plane. This is followed by a 270 ms interval with measured enhancements of one. This corresponds to the time spent transferring the measurement data between the USB oscilloscope and computer, followed by transmission of the optimized focusing TM to the DMD. During this time the DMD has the last mask from the TM measurement displayed which creates a random speckle at the output. Note that as an alternative, one could place the previous optimal mask on the DMD instead, in order to keep a high enhancement. The last part of the cycle, the 200 ms high enhancement, corresponds to the time when the optimized wavefront is encoded with the DMD. The enhancement can be seen to decrease during this time due to the short speckle decorrelation time. This process continuously repeats creating a bright focus twice per second.

7. Discussion and conclusion

We have demonstrated high speed wavefront optimization for focusing through turbid media using a DMD with off-axis binary amplitude holography for phase control and the transmission matrix method adapted to the task. We measured the transmission matrix using a three-phase method and a single photodetector to decrease the total measurement time. With this approach we demonstrated an order of magnitude improvement in measurement speed over the current fastest wavefront determination method [19] and three orders of magnitude improvement over LC-SLM methods [2]. We also demonstrated focusing through temporally dynamic turbid materials with persistence times similar to the system focusing time. The current system focusing time is limited by the transfer of data from the USB oscilloscope to the computer and to the DMD board and does not represent a fundamental limit. Current state of the art custom electronics could improve the data transfer time by at least one order of magnitude.

Based on its speed, the binary amplitude holographic technique could find application in sensing and imaging of biological materials. By measuring seven basis modes per millisecond this method should have enough speed to overcome the fast speckle decorrelation times of biological samples and generate enough focusing power for a variety of biomedical sensing and imaging applications.