A large number of degrees of freedom are required to produce a high quality focus through random scattering media. Previous demonstrations based on spatial phase modulations suffer from either a slow speed or a small number of degrees of freedom. In this work, a high speed wavefront determination technique based on spatial frequency domain wavefront modulations is proposed and experimentally demonstrated, which is capable of providing both a high operation speed and a large number of degrees of freedom. The technique was employed to focus light through a strongly scattering medium and the entire wavefront was determined in 400 milliseconds, ~three orders of magnitude faster than the previous report.
©2011 Optical Society of America
The advance of optical imaging technologies has revolutionized a broad range of biomedical research fields [1–6]. Over the past decade, new techniques have been developed to provide unprecedented spatial resolution and imaging speed [2, 7]. However, the optical imaging depth in biological tissues remains very limited . Many research fields such as development biology  and neuron science  would benefit greatly from the technologies that can provide deeper imaging depth for functional imaging, in particular fluorescence imaging.
Elastic scattering is the dominant factor limiting the optical imaging depth in tissues. Take gray matter as an example, at 800 nm the scattering coefficient is 77 cm−1 and the absorption coefficient is 0.2 cm−1 . If there is a way to suppress scattering, the optical imaging depth could be greatly improved. Despite the apparent randomness, scattering is a deterministic process. A properly engineered wave can propagate inside scattering media and form a focus, a well understood phenomenon in the time reversal and optical phase conjugation (OPC) studies [10–16]. The same principle has also been employed in the research field of adaptive optics for astronomy and biomedical applications [17–22]. How to use the established techniques for fluorescence imaging in highly scattering media is not apparent. OPC often requires a coherent signal from the target, which is not the case in fluorescence imaging. Conventional adaptive optics methods require a wavefront sensor to detect the wavefront of the target signal . For a broadband light source such as a fluorophore, the emitted light at different wavelength propagates along different path and thus has a different wavefront . Restricting the detection bandwidth can alleviate the problem at the severe cost of losing a large percentage of signals.
A more practical solution is to directly modulate the input optical wavefront while detecting the target signal, and to determine the correct wavefront by analyzing the relation between the input modulation and the detected output modulation [23–25]. In such a scheme, the emitted signal at all wavelengths can be used to determine the correct input wavefront. This scheme was originally employed at the Hughes Research Laboratory in 1970’s to help focus a laser beam onto a remote target through air turbulence and was named Coherent Optical Adaptive Technique (COAT) [23, 24]. Compared to atmospheric turbulence, the refractive index variation in biological tissues can cause much severer wavefront distortion. Recently, Mosk’s team has successfully employed a similar scheme to form an optical focus through a strongly scattering medium . In such experiments, the focus peak to background ratio is linearly proportional to the number of degrees of freedom . To achieve a high quality focus, a spatial light modulator (SLM) was employed to provide a large number of degrees of freedom. A data acquisition time of ~one second was used to determine the phase value of one degree of freedom .
For applications on biological tissues, acquisition time on the order of one millisecond (ms) per degree of freedom is desired [26, 27]. Deformable mirrors can provide a high modulation speed. However the degrees of freedom are rather limited. A phase-only SLM can provide about one million degrees of freedom at a much lower modulation speed. In this work, I present a novel method, capable of providing as many degrees of freedom as a SLM with a data acquisition time of one ms per degree of freedom. The method was employed to focus light through a random scattering medium with a 400 ms total data acquisition time, ~three orders of magnitude faster than the previous report .
The essence of a COAT system is to phase modulate different input spatial modes while detecting the output signal from the target. To greatly improve the operation speed, the experiment requires a device that can provide fast phase modulation and can access a large number of spatial modes very quickly. To meet these two requirements, a pair of scanning Galvanometer mirrors was used to quickly visit different modes in the spatial frequency domain or k space, and a frequency shifted reference beam was provided for a heterodyne detection. The wavefront profile was first determined in k space and then transformed to the spatial domain. The spatial phase profile was displayed on a SLM to focus light onto the target. In such a design, the number of degrees of freedom is limited by the number of pixels on the SLM and the experiment speed is determined by the scanning mirror speed.
The optical setup is shown in Fig. 1 . A continuous wave 785 nm laser was used in the experiment. To generate two laser beams with a tunable frequency difference, two acousto-optic modulators (AOM) were used to provide two frequency shifted beams. For a compact design, the two AOMs were placed in serial and both AOMs were aligned to optimize the diffraction for 50MHz acoustic waves. The 0th order beam was blocked. One of the two frequency shifted beams was directed to a Galvanometer mirror (X mirror), rotating in the horizontal direction. The X mirror was then relayed to the other Galvanometer mirror (Y mirror), rotating in the vertical direction. Three folding mirrors were used to fold the laser beam back to the horizontal plane, as shown in the dashed box in Fig. 1. The Y mirror was relayed to the rear pupil plane of a NA 0.5 20x objective. The other frequency shifted beam was directed to a phase-only reflective SLM (Holoeye, PLUTO, 1920 x 1080 pixels). The SLM was also relayed to the rear pupil plane of the 20x objective. In such a way, the X mirror, the Y mirror and the SLM operated on the same plane. All the three optical relay systems were achieved with 4f lens pairs. The two frequency shifted beams were combined by a 50:50 beam splitter before entering the 20x objective lens. A 1.6 mm thick glass diffuser was placed in front of the 20x objective as the random scattering medium. The transmitted light was collected by a NA 0.9 100x objective and imaged onto an 8-bit CCD camera (CCD1) and a photodiode detector with 100 kHz bandwidth. A 50 micron diameter pinhole was mounted on the photodiode so that the photodiode collected the light from a 500 nm diameter area on the focal plane of the 20x objective. A second 8-bit CCD camera (CCD2) equipped with a camera lens was used to measure the interference pattern of the two frequency shifted beams.
2.2 operation principle
The principle of the experiment is schematically shown in Fig. 2 . A two-channel function generator controlled the frequency shifting of the XY mirror beam (50 MHz) and the SLM beam (50.05 MHz). The frequency difference of the two beams was therefore 50 kHz. During the measurement, the XY mirrors were programmed to visit a matrix of k space positions. At the end of the measurement, the 50 kHz frequency component at each k space position was analyzed to yield the phase and amplitude information. The information was then transformed to the spatial domain. The phase profile in the spatial domain was displayed on the SLM to focus light onto the target.
2.3 system calibration
Four requirements need to be met before running the experiment. Firstly, the rotation axis of XY mirrors and the rotation center need to be determined on the SLM coordinate. Secondly, the quantitative relation between the XY mirrors’ rotation angle and the SLM phase slope needs to be measured. Thirdly, The SLM beam and the XY mirror beam should have identical spatial phase profile because the XY mirror beam was used in the wavefront measurement and the determined phase profile was eventually carried by the SLM beam. Fourthly, the beating frequency of the two beams needs to be locked to the data acquisition device’s clock.
For the first requirement, the relayed image of the SLM and the XY mirrors were imaged by a camera lens onto CCD2. The two AOMs were driven by a two-channel function generator, both running at 50MHz (no beating). The interference between the SLM beam and the XY beam was observed on CCD2. The X mirror was first rotated slowly, causing a variation of the interference pattern except the line on the rotation axis. The line of the stationary interference pattern was the X mirror’s rotation axis. In the same way, the Y mirror’s rotation axis was also determined. The intersection of the two rotation axes was the rotation center.
For the second requirement, the beam rotations caused by the XY mirrors and the SLM phase slopes need to be quantitatively compared. This task can be done with either CCD1 or CCD2. To use CCD1, the glass diffuser was removed from the setup and the focus of the 20x objective was imaged onto CCD1. Mirror rotations and SLM phase slopes can both move the focus on CCD1. Comparing the measured movement could yield the relation between the mirror rotation and the SLM phase slope. Alternatively, CCD2 can be employed to measure the interference pattern variation caused by the mirror rotation and the SLM phase slope, which could also provide the required information.
For the third requirement, the X and Y mirrors were offset from their center positions to perform a digital off-axis holography. The spatial phase difference between the XY mirror beam and the SLM beam was determined through Fourier transforms. Alternatively, a phase-shifting holography can be performed by setting the XY mirrors to their center positions and beating the two beams at a rate a quarter of the CCD2 frame rate. Four consecutive frames can be used to determine the phase profile.
For the fourth requirement, the data acquisition card (NI PCI 6110) used in the experiment was configured to output a 10 MHz pulse train to the reference input of the function generator to tightly synchronize the timing of the data acquisition and the beating of the two beams.
3. Experimental results
To test if the system can indeed focus light through a random scattering sample, a 1.6 mm thick glass diffuser was used as the scattering sample and the transmitted light within a 500 nm diameter area was detected by a photodiode as the target signal. The XY mirrors were programmed to rotate from −0.12° to 0.12° in 20 steps such that 400 k space positions were visited during the measurement. The time spent at each position is 1 ms. In the first 400 μs, the mirror position was in transition from its previous position to its current position. In the latter 600 μs, the mirror position was stable and the feedback signals of the XY mirrors and the target signal within the 600 μs were recorded. All the signals were digitized at 1 MHz. The recorded raw data were shown in Fig. 3 . (a). The center area of Fig. 3. (a) is magnified and shown in Fig. 3. (b). The 50 kHz oscillation is clearly seen.
The raw data was divided into 400 arrays. Each data array was Fourier transformed and the amplitude and phase information of the 50 kHz signal was extracted and transformed to the spatial domain. The phase profile was displayed on the SLM. At this point, a bright round focus was observed on CCD1. The observed focus and the phase pattern displayed on the SLM are shown in Fig. 4 . (a) and (b) respectively. Without the compensation phase pattern, the transmission through the scattering sample was a random speckle, as shown in Fig. 4. (c). The FWHM of the formed focus was 0.82 ± 0.01 μm observed by the NA 0.9 objective. Assuming a 0.48 μm detection point spread function (PSF) of the NA 0.9 objective, the actual FWHM of the formed focus is estimated to be 0.66 ± 0.01 μm. The ideal focus FWHM of the NA 0.5 objective is 0.84 μm. The focus formed through scattering media is smaller than the ideal value is a well known effect in time reversal and phase conjugation studies [12, 13], which is beneficial for imaging applications.
In summary, a high speed wavefront determination method is proposed and experimentally demonstrated. Its principle is based on the Coherent Optical Adaptive Technique. Different from the methods based on direct spatial phase modulations, the reported method employed Galvanometer mirrors to quickly visit different modes in k space and a frequency shifted reference beam for a high speed phase modulation. Compared to existing techniques, the reported method can provide both a high operation speed and a large number of degrees of freedom. In the current design, the operation speed is limited by the scanning mirror speed and the maximum number of degrees of freedom is limited by the SLM pixel number. In this demonstration, 400 spatial modes in k space were visited and the determined phase profile was displayed on the SLM. Depending on the scattering property of the media, more (up to 1920 x 1080) or less number of degrees of freedom can be used to optimize the focus quality and the operation speed.
Using a stepwise position scanning, the method achieves an operation speed of one ms (400 μs transition time + 600 μs recording time) per spatial mode, ~three orders of magnitude faster than the previous report. Using a continuous position scanning and a faster position scanner such as resonant scanning mirrors, polygon mirror scanners, or acousto-optic deflectors, the operation speed can be potentially increased by at least one order of magnitude. It is anticipated that the reported technique will find a broad range of applications in biomedical deep tissue imaging.
The author thanks Na Ji, Eric Betzig, and Mats Gustafsson for helpful discussions and thanks Karel Svoboda for providing the scanning mirrors. This work is supported by Howard Hughes Medical Institute.
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