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Abstract
Holographic techniques enable precise laser manipulation, but suffer from two considerable limitations: speckle and deterioration of axial distribution. Here, we propose a cylindrical quadratic phase (CQP) method with temporal focusing (TF) to generate speckle-free holographic illumination with high axial resolution. TF-CQP utilizes a superposed cylindrical phase as the initial guess to iteratively optimize phase hologram, realizing speckle-free holographic reconstruction on the target focal plane and eliminating secondary focus on the defocused planes. TF-CQP further disperses defocused beams symmetrically by a blazed grating, placed conjugate to the focal plane, which enhances axial confinement. Simulation and experimental results show that TF-CQP reconstructs speckle-free illumination with arbitrary shapes and <10 µm axial resolution. Compared to TF-GS (Gerchberg-Saxton algorithm), widely used in holographic optogenetics, TF-CQP shows increased uniformity of 200% and improved modulation efficiency of 32.33% for parallel holographic illumination, as well as a 10% increment in axial resolution.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Computer-generated hologram (CGH), combining holography and digital processing technology [1], has successfully synthesized the complex optical field with custom intensity distributions. Compared to other beam manipulation approaches (e.g., freeform surfaces [2] and diffractive optical elements [3]), CGH has reported superior performance in terms of flexibility [4] and dynamic modulation [5]. In particular, recent advances in ultrafast pulsed lasers have opened up a new avenue to CGH due to the capabilities of three-dimensional addressing and ultrahigh precision [6], becoming widely used in optical trapping [7,8], lithography [9] and optogenetics [10,11]. However, two considerable limitations, speckle noise and deterioration of axial distribution, restrain the precision of beam manipulation and decrease the spatial resolution of holographic illumination [12] in ultrafast pulsed laser-based CGH.
Speckle noise is generated by randomized phase (RAP) interference of the coherent light sources [13]. Over the last decades, both hardware and numerical solutions have been proposed for overcoming this problem [14]. The typical hardware method is multi-CGHs averaging [15–17], which depends on the different CGHs with noise diversity [15] to average out the speckle noise. However, it sacrifices time resolution and requires a fast-switching phase modulation device like a ferroelectric spatial light modulator (SLM). The amplitude reconstruction denoising method is another optical hardware method, which utilizes a low-pass filter to smooth speckle noise [18,19]. However, this method still fails to obtain a completely homogeneous intensity distribution [18] because the spatial frequency information of target illumination and speckles cannot be completely separated. Compared to optical hardware solutions, numerical reconstructed algorithms suppress speckle noise by reducing random phase, which is more conveniently integrated in the commonly used optical layout of CGHs. The traditional phase optimization method relies on iterative algorithms such as the widely employed Gerchberg-Saxton (GS) algorithm [20], which significantly enhances holographic modulation efficiency but suffers from severe speckle noise. The typical speckle-suppression phase optimization method is based on the Fidoc method [21], which partitions the output plane into the signal and the freedom region while only constraining amplitude modulus in the signal region. Fidoc performs well in speckle suppression, but suffers from a substantial loss in modulation efficiency due to lots of light scattered in the freedom region [22–24]. Recently, RAP-free methods have been proposed to fundamentally solve speckle noise by avoiding the random phase. In RAP-free methods for Fourier CGH, the quadratic phase is employed as the initial phase for the iterative Fourier transform loop (IFTL) and optimizes phase hologram with continuous distribution spectrum, which has succeeded in generating arbitrary speckle-free holographic illumination [25,26]. However, there are rare presentations about three-dimensional (3D) intensity distribution generated by quadratic phase methods in reported works. This is because the optimized phase hologram by the conventional spherical quadratic phase algorithm (SQP) leads beam to focus during axial transmission. As a result, the intensity of the secondary focus on defocused planes is higher than that of the reconstructed holographic illumination on the target projection plane, limiting the quadratic phase method applied in high-axial-resolution holographic applications.
Axial confinement is another important research direction for CGH, aiming to mitigate the degradation of axial resolution as the size of holographic illumination increases. The typical approach is based on random-phase encoding technologies, which leverages the orthogonality of high-dimensional random vectors to rapidly reduce the beam intensity on defocus planes [27,28]. But this approach cannot be combined with the quadratic phase method to confine axial distribution of speckle-free holographic illumination, because phase continuity will be destroyed by the random phase. By a trade-off restrict of amplitude and phase spectrum of the projection plane during the iteration, the stochastic gradient descent (SGD) method confines the 3D distribution of the holographic beam [29,30] but suffers a loss of modulation efficiency. The integration of temporal focusing (TF) and CGH can optimize axial resolution while maintaining intact reconstruction performance on the target projection plane [31,32]. The TF uses a blazed grating to disperse the short-pulsed laser beam on defocused planes; thereby, the greater dispersion power the grating has, the stronger axial confinement can be obtained. However, it is difficult to simultaneously realize high-axial-resolution and speckle-free holographic illumination by simply combining TF and quadratic phase method. Because even though TF disperses the secondary focused point into line distribution on defocused planes, the local intensity is still higher than that of 2D or 3D reconstructed holographic illumination on the target focal plane.
To solve this challenge, we propose a cylindrical quadratic phase (CQP) method with TF (TF-CQP) to generate speckle-free holographic illumination with high axial resolution. In the numerical reconstruction, CQP uses bandwidth-limited cylindrical initial phase and weighted iteration to generate the phase hologram with continuous distribution spectrum, eliminating random phase and thus completely suppressing speckles. Additionally, this algorithm discretizes the secondary focus on the defocused plane in the conventional SQP algorithm, transforming it into a line distribution. To further reduce the defocused beam intensity of line distribution, we integrate the TF module in the Fourier CGH system. The dispersion direction of the blazed grating in TF module is set to 45° with the generatrices of the cylindrical-like phase hologram, providing the same degree of beam dispersion on both sides of the objective focal plane. Numerical and experimental results show that with a femtosecond laser source, our proposed TF-CQP not only succeeds in generating speckle-free holographic illumination with micrometer-level axial resolution but also performs well in modulation efficiency. Compared with the traditional TF-GS widely used for optogenetic illumination, TF-CQP effectively reduces the undesired illumination on non-target neurons and improves the optical sectioning capability in optogenetic illumination.
2. Principle of TF-CQP
2.1 Cylindrical quadratic phase (CQP) algorithm
The basic optical layout for Fourier CGHs is shown in Fig. 1(a), which consists a Fourier transform (FT) lens, and the hologram plane and projection plane located in the front and back focal plane of this lens, respectively. CQP algorithm is proposed to design phase hologram to modulate the incident beam to be focused as the desired target illumination. CQP follows the general procedure of the conventional quadratic phase algorithm [33], including initial values setting and weighted IFTL. In the first step, CQP sets three initial values, the amplitude spectrum of incident Gaussian beam (uin), the target illumination patterns (uT), and the bandwidth-limited cylindrical initial phase ($\varphi _{\textrm{in},\; \textrm{CQP}}^{(1 )}$). The initial phase can be assumed as
where (x, y) are the coordinates of the hologram plane, cx and cy are the guessing cylindrical phase factors, and the superscript (1) represents that the current iteration number is 1. The approach to calculate the cylindrical phase factors is consistent with previously reported methods [25,33], which are briefly reviewed here. The beam bandwidth of the target illumination on the output plane isSpeckle suppression requires the input-plane effective spatial frequency to be between 0.5 to 2 times the bandwidth of the target illumination [25]. According to Eq. (2, 3), the range of initial cylindrical factors are
By stepping the initial cylindrical factors within the range shown in Eq. (4), different reconstructed illuminations are output by the weighted IFTL. The modulation efficiency and root-mean-square error are calculated as the cost functions. The best parameters of cx and cy are determined by the cost values, and then the optimal cylindrical initial phase can be obtained.
Fig. 1. CQP algorithm for speckle-free Fourier CGH. (a) Schematic diagram for Fourier CGH. f = f’, focal length of the FT lens. (b) Block diagram of the weighted IFTL in CQP algorithm. Symbol $ \odot $ is a Hadamard product, k is the iteration number and |û| means that the modulus of u is normalized using |û|=|u|/max(|u|). (c) Comparison of SQP (left) and CQP (right). Top panel, Phase holograms. Bottom panels, A 20-µm diameter circular spot is holographically reconstructed on the target projection plane (middle), and its intensity distributions on the defocused planes at ±5 µm away from the target focal plane. Scale bar, 15 µm.
In the second step, taking this bandwidth-limited cylindrical phase as initial phase to start a new weighted IFTL. In each iteration, dual FTs between the hologram plane and the projection plane are used to satisfy each plane’s amplitude constraint in each iteration, while the phase spectrums are remained (Fig. 1(b)). The final phase hologram with continuous distribution spectrum can be achieved when the iteration is converged.
We performed numerical simulations to test the speckle suppression and axial distribution of the proposed CQP algorithm. The computation parameters are as follows: the CGH is 512 × 512 pixels and the pixel size is 15 µm. The FT lens is set as a 40X Nikon objective (F = 5 mm). And the incident beam and the desired target illumination are set as a 4-mm-diameter Gaussian beam and a 20-µm-diameter circular spots, respectively. Simulation results show that compared to the conventional quadratic phase method, which uses the spherical phase as the initial phase (SQP), CQP generates the phase hologram no longer similar to the lens’s phase distribution, but with a cylindrical-like distribution (top panel in Fig. 1(c)). And the phase hologram via CQP discretizes the secondary focus on the defocused plane to line distribution (bottom panel in Fig. 1(c)), reducing the local intensity by about 52 times.
Two parameters, uniformity (U) and modulation efficiency (η), are used to evaluate reconstructed quality on the target focal plane:
where I and IT are reconstructed and target intensity distributions respectively, and the symbol S represents where the target beam locates at. CQP shows similar reconstructed quality as the previous SQP (ηCQP = 0.96, ηSQP =0.98; UCQP = 0.98, USQP =0.99). Therefore, the proposed CQP preliminarily reduces local beam intensity on defocused planes while generating high-modulation-efficiency and speckle-free holographic illumination on the focal plane, laying the groundwork for further axial confinement by TF.2.2 High axial resolution realized by TF-CQP
To further discretize the defocused line-distributed beams in CQP algorithm, we used the TF module to provide dispersion for defocused beams. The temporal focusing module comprises a telescope and a blazed grating placed at the focal plane of the two lenses, which temporally focused the reconstructed pattern at the objective focal plane [34]. The dispersion direction of the grating is set to 45° with respect to the two generatrices of the orthogonal cylindrical phase hologram. This configuration provides the same dispersion degree on both sides of the objective focal plane, controlling the maximum intensity distributed in three dimensions to be located on the target projection plane (Fig. 2).
Fig. 2. Optical design of TF-CQP. A phase modulation device, such as an SLM, is put on the hologram plane and performs the holographic phase modulation for generating the custom-designed illumination. The illumination pattern is then focused on a blazed grating for TF through lens L1. The first diffraction order is collimated by lens L2 and directed to the objective (OBJ). Finally, the temporally focused reconstructed result is projected on the focal plane of OBJ.
To analyze the optical sectioning ability of TF-CQP, we calculate the beam dispersion along the axial transmission. For femtosecond laser at central wavelength λ and pulse width Δτ, the beam radius expansion after TF is
where κ is the time-bandwidth product, f2 is the focal length of L2, c is the speed of light, and d is the groove density of the grating. At a distance Δz from the objective focal plane, the beam expansion caused by dispersion isTo test TF-CQP capabilities on speckle suppression and axial confinement, we perform the angular-spectrum transmission simulation to show the 3D distribution of reconstructed holographic illumination. The computation parameters for TF are as follows: the beam is set as 1064-nm laser with pulse width of 100 fs; the focal lengths of L1 and L2 are 100 mm and 200 mm respectively, and the groove density of the grating is 1200 l/mm.
Simulation results show that compared to the previous SQP method with TF (TF-SQP, shown in top panel in Fig. 3(a)), TF-CQP constrains the maximum illumination intensity distributed in three dimensions to be focused on the objective focal plane (top panel in Fig. 3(c)). And compared to the traditional TF-GS (top panel in Fig. 3(b)), TF-CQP suppresses speckle noise completely (Fig. 3(c)). From axial maximum-intensity projections, the dispersion along the x-direction reduces the local intensity on defocus planes (bottom right panel in Fig. 3(a-c)), in agreement with the mathematical analysis of Eq. (7). We further use axial resolution to quantify the optical sectioning ability, which is defined as the full width at half maximum (FWHM) of the average intensity value on the different planes of the optical stacks. And the average intensity value is calculated from,
Fig. 3. Simulation comparison of TF-SQP (a), TF-GS (b), and our proposed TF-CQP (c). Top panel, 3D intensity distribution of 20-µm-diameter circular spot reconstructed by the three methods. Red dashed section represents the objective focal plane. Bottom right panel, x-y intensity distribution on the objective focal plane, and x – z and y − z projections of the holographic illumination. Bottom right panel, axial profile of average intensity. Dashed line represents the axial position of the reconstructed illumination.
3. Experimental results
To further demonstrate the optical effects of our proposed algorithm, experiments are also performed. The schematic of the experimental setup is shown in Fig. 4. The laser at 1064 nm with an 80 MHz repetition rate and 100 fs pulse (ALCOR1064, Spark lasers) first passes through a pockels cell for power adjustment. Then, the beam is spatially filtered by two lenses (L1 and L2) and a pinhole to fill the SLM active area with Gaussian distribution. A half-wave plate (HWP) is set to optimize laser polarization for the maximum efficiency of the reflective SLM (Model P512-0785, Meadowlark Optics, 7.68 × 7.68 mm2 active area, 512 × 512 pixels). After modulation by the phase hologram on SLM, the zero-order beam is blocked by a beam stop (BS) and the speckle-free illumination pattern is focused on a blazed grating for TF through lens L3. The first diffraction order is collimated by lens L4 and directs to the objective (OBJ1, Nikon 40X). The illumination pattern on the focal plane of OBJ1 is made on a CCD camera, through a second opposite objective (OBJ2) and a tube lens (L5). In addition, since the signal-to-noise ratio (SNR) of images reflects the modulation efficiency, we use SNR to evaluate reconstructed efficiency in experiments.
We first conduct experimental validation on the 20-µm-diameter circular illumination. On the focal plane, TF-CQP generates circular holographic illumination with uniformity of 0.73 and SNR of 10.64, which are generally consistent with the traditional TF-SQP (Table 2). However, TF-SQP shows extremely poor axial projection, where the defocused beam intensity is higher than that of reconstructed illumination on the target focal plane (Fig. 5(a)). In contrast, the beam intensity generated by TF-CQP rapidly decreases when the beam is out of the target focus plane, with the axial projection images showing the maximum intensity distributed in three dimensions to be located at the target focal plane (Fig. 5(c)). TF-CQP also eliminates speckle noise shown in TF-GS (Fig. 5(b)), with an 18% improvement in axial resolution (Table 2).
Fig. 4. Scheme of optical setup. PC, pockels cell; P, pinhole; HWP, half-wave plate; BS, beam stop; BG, blazed grating; M, mirror; L, lens; OBJ, objective.
Fig. 5. Experimental results of 20-µm-diameter circular illumination generated by TF-SQP (a), TF-GS (b) and TF-CQP (c). Top panel, x-y intensity distribution on the objective focal plane, and x – z and y − z projections of the holographic illumination. Bottom panel, axial profile of average beam intensity. Dashed line represents the axial position of the reconstructed illumination.
Table 2. Performance Comparison of Various Methods for Different-shaped Holographic Illuminationa
To test the potential of TF-CQP applied in optogenetic illumination, we randomly select neuronal soma (Fig. 6(a)) and dendrite (Fig. 6(c)) as the target illumination pattern. Results show that the reconstruction performance of TF-CQP is pattern-independent. Compared to TF-SQP, TF-CQP limits the maximum illumination intensity on the target focal plane (Fig. 6(b), (d)) and shows similar SNR and uniformity for reconstructed illumination (Table 2). Compared to TF-GS, TF-CQP not only improves uniformity by 287.0% and 131.6% for reconstructed soma-shaped and axon-shaped illumination, respectively, but also improves the axial resolution of both illuminations by ∼10% (Fig. 6(e), (f), and Table 2).
Fig. 6. Comparison of TF-SQP, TF-GS and TF-CQP for generating soma-shaped and axon-shaped illumination. (a), (c) Cortical field of view image and the soma-target (a) and axon-target (c) selection. (b), (d) x - y intensity distribution on the objective focal plane, and x – z and y − z projections of reconstructed soma-shaped (b) and axon-shaped (d) illumination. (e), (f) Axial profile of average intensity in (b), (d).
We further test TF-CQP’s capability in holographic parallel illumination, where the target illumination patterns are randomly selected as several neuronal somata (Fig. 7(a)). Due to the weak axial confinement in the reconstruction of a single illumination pattern, TF-SQP is not included for comparison in this experiment. Reconstructed results indicate that TF-CQP generates parallel multi-cell illumination with SNR of 6.63 and uniformity of 0.84 (top panel in Fig. 7(c)), whereas the SNR and uniformity for TF-GS are only 5.01 and 0.04, respectively (top panel in Fig. 7(b)). Compared to TF-GS, TF-CQP increases SNR by 32.33% and uniformity by 200%, respectively. To demonstrate that the high axial resolution of TF-CQP can suppress the crosstalk of optogenetic illumination on neighboring non-target neurons, we magnify an individual reconstructed illumination pattern (red) and merge it with 3D neuron distribution close to the target neuron (green) (bottom right panel in Fig. 7(b), (c)). TF-GS shows undesired illumination on non-target neurons (bottom right panel in Fig. 7(b), and sections shown in bottom left panel in Fig. 7(b)). In contrast, TF-CQP effectively reduces the intensity of undesired illumination (bottom panel in Fig. 7(c)), with more than a factor of two (Fig. 7(d)), enabling precise illumination only on selected target neurons.
Fig. 7. Parallel illumination on selected several neurons. (a) Target illumination patterns (yellow dashed line) of multi-cells in 3D cortical field of view image. (b), (c) Top panel, reconstructed results of multi-cell illumination in (a), generated by TF-GS (b) and TF-GSSIP (c) respectively, with the intensities normalized to the maximum intensity in each case. Bottom right panel, merge images of an individual reconstructed illumination pattern (red box in top panel and color in red) and 3D distribution of neurons near the target neuron #1 (color in green). Bottom left panel, sections along the red dotted line in bottom right panel. (d) Intensity profiles along the direction of white arrows in bottom left panel in (b), (c), normalized to the maximum intensity of the TF-CQP.
4. Conclusion
In this paper, we proposed a TF-CQP method to confine the axial distribution of speckle-free holographic illumination with high modulation efficiency for pulsed laser-based CGH. This method optimizes CGH through both numerical reconstruction and optical configuration. In the process of calculating phase hologram, TF-CQP sets the initial phase for weighted IFTL as a superposed phase of two orthogonally placed cylindrical lenses with concavity and convexity characteristics, respectively. This process not only generates phase hologram with continuous distribution spectrum for speckle-free reconstruction, but also eliminates the secondary focus on the defocused plane in the conventional SQP method, discretizing it to line distribution. In the optical modification, TF-CQP uses the TF module to further disperse the defocused line-distributed beams, in which the dispersion direction of the blazed grating is designed as 45° to the generatrices of the cylindrical-like phase hologram.
Simulation and experimental results show our proposed TF-CQP achieves speckle-free holographic illumination and controlling the maximum intensity distributed in three dimensions to be precisely located on the target projection plane. In contrast to the TF-GS, commonly used in holographic optogenetics, TF-CQP not only offers notable enhancements in both uniformity and modulation efficiency, but also exhibits an ∼10% improvement in axial resolution, thereby effectively mitigating the impreciseness caused by weak axial confinement. The parallel capability of TF-CQP meets the requirements of optically regulating neural ensembles, showing potentials in exploring the function of neural ensembles. In combination with the MTF-CGH optical configuration [34], TF-CQP is expected to design and reconstruct speckle-free holographic illumination patterns addressing arbitrary 3D locations by a simple lens phase [37,38] or the phase designed by multi-plane GS algorithm [39].
Funding
STI 2030—Major Projects (2021ZD0200401); National Natural Science Foundation of China (61975178); Key R&D Program of Zhejiang Province (2021C03001); Zhejiang Provincial Outstanding Youth Science Foundation (LR22F050007); CAMS Innovation Fund for Medical Sciences (2019-I2M-5-057); Fundamental Research Funds for the Central Universities.
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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