Pixilated spatial light modulators are efficient devices to shape the wavefront of a laser beam or to perform Fourier optical filtering. When conjugated with the back focal plane of a microscope objective, they allow an efficient redistribution of laser light energy. These intensity patterns are usually polluted by undesired spots so-called ghosts and zero-orders whose intensities depend on displayed patterns. In this work, we propose a model to account for these discrepancies and demonstrate the possibility to efficiently reduce the intensity of the zero-order up to 95%, the intensity of the ghost up to 96% and increase diffraction efficiency up to 44%. Our model suggests physical cross-talk between pixels and thus, filtering of addressed high spatial frequencies. The method implementation relies on simple preliminary characterization of the SLM and can be computed a priori with any phase profile. The performance of this method is demonstrated employing a Hamamatsu LCoS SLM X10468-02 with two-photon excitation of fluorescent Rhodamine layers.
©2012 Optical Society of America
Holographic phase modulation generated by means of Liquid Crystal Spatial Light Modulator (LC-SLM) , originally used for wavefront correction  and three-dimensional optical traps generation , has been recently also employed in biology for patterned photoactivation of caged compounds and optogenetics actuators [4–8]. Among them, parallel aligned nematic liquid crystal SLMs in particular, offer the possibility to imprint graded values of phase shifts between 0 and 2π to the wave-front of an impinging beam, allowing the experimentalist to create arbitrary programmable optics.
A typical implementation of such devices consists in conjugating them with the back focal plane of an objective lens. Without any phase modulation, light beam yields a diffraction limited spot at the focal point of the objective lens. A beam, phase-modulated by means of a SLM, diffracts energy towards different regions and thus allow intensity modulation in the vicinity of the focal point with an efficient redistribution of light energy. Iterative Gerchberg-Satxon-like algorithms [9, 10] provide a way to design appropriate phase patterns to arbitrarily shape the light distribution.
One of the most crucial aspects of SLMs is then their diffraction efficiency, which may be defined as the ratio between the light power redirected into the desired pattern and the light power in the focal point of the lens when the SLM is off.
Diffraction efficiency of an SLM can be affected by several factors. First, the squarely pixilated structure of SLMs constrains the diffracted pattern below a (sin(x)/x)2 envelope [4, 8]. Remaining power is then sent to higher diffraction orders. Second, diffraction efficiency is limited by the filling factor of phase actuators . Electrically addressed reflective LCoS-SLMs (Liquid Crystal on Silicon Spatial Light Modulators) currently offer the largest filling factors. Third, the imperfect anti-reflection coating of the front electrode of the SLM leaves a fraction of the impinging beam un-modulated. Both these two latter effects yield an un-modulated beam, focused by the lens into a so-called zero-order spot. Finally, in general, diffraction efficiency and light power in zero-order spot also depend on the specific displayed pattern either due to electronic limitations [12–14] or physical electric field averaging between electrodes [15–17].
Several strategies were developed to remove this unwanted component. It can be achieved by simply introducing a beam block in an intermediate image. This method is very efficient but introduces a blind-zone in the excitation field and the energy present in the zero order is lost. Alternatively, access to the central region can be restored by moving the zero-order axially [18, 19]. However, axial shifting of the pattern with a Fresnel lens on the SLM also deteriorates diffraction efficiency . Another strategy to eliminate the undiffracted light is to modify the hologram phase pattern by computing and superimposing an additional spot onto the zero-order spot, and to modulate its phase to get a destructive interference [21, 22]. Without any alteration of the optics architecture, this method elegantly reduces the undiffracted contribution. However, in this method, the modification of the hologram phase pattern requires a precise knowledge of the amplitude and phase of the zero-order beam which depends on the phase pattern itself and is not predictable a priori.
Here, we characterized a LCoS-SLM (Hamamatsu X10468-02) having 95% filling factor, optimized for near infrared (750nm-850nm) for which diffraction efficiency is observed to collapse more quickly than the expected (sin(x)/x)2 envelope. Moreover, depending on the displayed phase pattern, fraction of total power in the zero-order varies from ~2% up to ~40%. This effect is all the more critical when dealing with applications consisting in bi-photonic excitation because of the quadratic dependence of the absorption process on intensity. Further degradation of the diffraction efficiency and limitations of the applicability of the SLM are due to the presence of undesired spots symmetrically displaced in respect to the desired pattern, so called ghost spots , which we observed to range from ~2% up to ~12% depending on the phase pattern.
In this article we analyze the SLM pattern dependence and we propose a novel approach to both reject the zero-order component and enhance diffraction efficiency. To do so, we first measure diffraction efficiency and zero-order addressing simple grid patterns onto the SLM. We observe that the SLM behavior is equivalent to a low-spatial-frequency filtering of displayed phase pattern. On the basis of this model, systematic optimization methods are proposed and successfully tested with our SLM.
In the present paper a parallel aligned LCoS-SLM (Hamamatsu X10468-02) was characterized using a previously described setup . Briefly, the SLM was illuminated by a linearly polarized 800nm light beam delivered by a mode-locked Ti:Sapphire laser (MaiTai, Spectra Physics), previously expanded to cover the SLM surface (Fig. 1 ). The SLM was then imaged in the back focal plane of an objective lens (60x, W 0.9 NA, Olympus) through a telescope (f1 = 1000mm; f2 = 500mm) in order that the pupil diameter matches the SLM short axis.
Preliminary calibration experiments were performed by placing a photodiode power meter in the intermediate image plane (Fig. 1). We displayed horizontal and vertical binary phase grating profile on the SLM with different periods and gray level modulation depths. Power diffracted in orders 0, ± 1 and ± 2 in the vertical and horizontal directions were then measured by the power meter through an iris. Binary checkerboard patterns were also displayed on the SLM in order to measure the power in 0, ± 1 and ± 2 diffraction orders in the diagonal directions.
Next, diffraction efficiency measurements were performed when displaying 2D patterns (disks or arbitrary shapes) in the image plane, generated by an iterative Gershberg-Saxton like algorithm [9, 10], and laterally displayed at various distances from the focal point. Light power in the pattern and in the zero-order spot were measured in the optical configuration previously described.
Fluorescence images were carried out by exciting a spin-coated fluorescent layer of Rhodamine 6G in PMMA (1.6% w/v in Chloroform) in the focal plane of the objective lens. Fluorescence was collected by a second objective lens (60x, W 1.2 NA, Olympus) placed on the opposite side, filtered by a set of dichroic filters (Chroma Technology 640DCSPXR, HQ 535/50M and Chroma Technology 640DCSPXR) and imaged onto a CCD camera (CoolSNAP HQ2, Roper Scientific) through a collection tube lens (TL = 150mm). In the paper, fluorescence imaging was performed in the forward direction thanks to a second microscope objective lens, but this experimental configuration is not essential and the same results will be obtained by using the same objective for excitation and imaging.
3. SLM characterization and modeling
SLM experimental characterization
In order to study the spatial frequency dependence of the SLM diffraction efficiency, we addressed binary gratings and checkerboards of different modulation depths ranging from 0 to 255 gray level values and of different periods (2 and 4 pixels). The diffracted powers in orders 0, ± 1, ± 2 was measured with a power meter and normalized to the total power reflected on the SLM when off. The results obtained for spots produced by a binary phase grating of 2-pixels period are shown in Fig. 2(a) .
For a binary crenel-shaped phase grating, the theoretical dependence of the normalized power on the modulation depth φ, in the zero-order and in ± 1 orders is expected to be and, respectively  (Fig. 2(b), black lines). Then, when displaying a binary phase grating with a 2π-modulation depth (corresponding to a 212 gray level at 800 nm, according to the manufacturer lookup tables for our SLM) the power should be only in the zero order. Conversely, for a modulation depth of 106 gray value (corresponding to a π phase modulation depth) the amount of energy in the first orders should be maximum (~40.57% in each) and no light should be present in the zero order spot. In Fig. 2(a), an almost reverse experimental result is obtained: for a 106 gray level modulation depth, the fraction of light in the zero and first orders are ~41% and ~27%, respectively, while the minimum of the zero-order power (~2%) and the maximum of the ± 1-order (~36%) are reached for modulation depth of 203 and 165, respectively. Similar results, although less pronounced, were observed for 4-pixels period grids where, in both x and y directions, the minimum of the zero-order spot was obtained for a gray level of ~140 and the maximum of first orders were obtained for gray levels of ~120 (curves not shown). This discrepancy becomes negligible for larger periods, confirming values specified by the manufacturer.
One possible cause of such behavior can be related to the electronic of the device, in particular to a spatial-frequency dependent phase/gray-level conversion . However, this effect is not sufficient to account for our results as under this assumption, the minimum of the zero-order should still correspond to the maximum of the first order. As shown in Fig. 2(a) experimental measurements reveal a zero order minimum at ~ 203 and the maximum of power in the first order at~165.
This observation suggests that, when addressing such gratings, pixels do not diffract as perfect rectangular gates and that the effectively produced phase grating is not actually crenel-like but more sine-like shaped. Indeed, for a sine-like phase grating, the normalized power in the zero and firsts orders varies as , and , where and are Bessel functions of the first kind (Fig. 2(b), red lines). For such a phase profile, the minimum of zero order is obtained for a modulation depth value larger than the one that is required to get the maximum of the first order.
More precisely, by expanding the crenel phase pattern in Fourier series, an almost perfect matching with experimental diffraction efficiencies in orders 0, ± 1, ± 2 is obtained for a phase profile described as a sine function summation truncated at the fifth harmonic (Fig. 3 ).
We noted that experimental results for the ± 1 orders exhibit slightly different behaviors for vertical and horizontal grating directions (data not shown). In particular, the positive and negative orders induced by horizontal grids are symmetric (Fig. 2(a) and Fig. 3), while those generated by vertical grids presents a slight asymmetry which could not be attributed to the illumination geometry. This asymmetric diffraction pattern supposes an asymmetric phase pattern. This behavior may be related to the way pixels are addressed [19, 26].
SLM phenomenological modeling
Former observations suggest that the phase pattern addressed onto the SLM is filtered, and high spatial frequencies attenuated. This effect can be accounted by a previously proposed model [15–17] involving a fringing-field effect which affects the LC molecules orientation of adjacent electrodes inducing a broadening and smearing of the desired phase grating profile. Then, any pixilated phase pattern containing abrupt discontinuities in the pattern will be altered by this smoothing effect. Mathematically, we can model it by a convolution of the theoretical phase pattern (containing discontinuities) with a kernel smoothing function. In previous studies [15–17], for asymmetric displayed pattern (like a blazed grating), the kernel function was observed to be slightly asymmetric, suggesting a non-linear filtering effect. A rigorous numerical modeling of technical limitations of nematic LCoS-SLMs is beside the scope of this article. However, we demonstrate here that a phenomenological model based on experimental evidences which assume a linear filter is sufficient to practically balance the filtering of the SLM.
Given a theoretically expected phase pattern p(r), the phase pattern effectively displayed by the SLM, f(r) will be given by:
If p is a periodic function, f is also periodic and both can be expanded in Fourier series. P and F are then sums of Dirac’s functions.
We have previously seen that if p(r) is a crenel-like function then f(r) is a sine series truncated at the fifth harmonic. Therefore, assuming that g(r) does not depend on the phase pattern addressed onto the SLM, we can determine G(k) as the ratio between the Fourier transform of a sine series truncated at the fifth harmonic and the Fourier transform of a crenel-like function:Fig. 3(a), red dots). By interpolating these coefficients, it is possible to trace the filtering function G(k) for the all spatial frequencies of the displayed pattern (Fig. 3(a), black curve). In the following sections, we propose two strategies to numerically compensate for this filtering effect.
Rigorous Phase profile correction
As a first model we propose to correct the phase patterns displayed on the SLM by using the filter function, G(k) (Fig. 3, inset (a)) to compensate the distortions introduced by the SLM. That is, given a theoretical phase pattern, P(k), we generate the corrected one, Pcorr(k) as:
Unfortunately, this method is limited by the pixilation of the SLM. Indeed, compensating completely distortions introduced by spatial filtering would require to correct every spatial frequency present in the Fourier spectrum, P(k), of the desired phase profile. However, in practice only spatial frequencies lower than 1/(2a) (a being the pixel pitch) can be corrected. Higher frequencies correspond to inaccessible sub-pixel scales.
As a consequence the restoration of the SLM distortions and the suppression of the zero order are not fully achieved. To illustrate it, let us consider a 2-pixel periodic grating. If we follow the correction methods proposed in Eq. (4), the spectrum of the binary grating will be multiplied by the inverse of G(k). The first harmonic will then be multiplied by C(k1) = 1/G(k1), where k1 stands for the first harmonic. In the spatial domain, this means that the theoretical modulation depth of the binary grating function, φ, is multiplied by C(k1) = 1/G(k1):Fig. 3), then C(k1) = 1/0.62 = 1.61. Thus in particular for φ = 106, the corrected modulation depth is, φ’ = 171. Referring to Fig. 2(a), for such a modulation depth, the fraction of power in the zero order experimentally obtained is ~6%, decreased by 85% with respect to the uncorrected case which exhibit 41% of total power in the zero-order. However, we also notice that the minimum reachable power in the zero-order is ~2%, corresponding to 203 gray level modulation. This suggests that further optimizations could be potentially obtained with a total fraction of zero-order reduction of 95%.
Phase Profile correction: zero-order optimization
As a second method, we propose here to maximize zero-order suppression by optimizing the multiplicative coefficients C(k).
Specifically, in the case of a binary grating, in order to minimize the experimental zero-order, the modulation depth has to be enhanced by a factor C0(k1) given by the ratio between the phase value corresponding to the minimum of the experimental zero-order diffraction and 106, which represents the theoretical minimum:
For a grating with a lower spatial frequency, as already mentioned, the filtering effect is less pronounced and the experimental minimum of the zero-order diffraction is obtained for modulation depths closer to the theoretical value of 106 gray-level. Thus, in this case, lower multiplicative coefficients have to be applied. Thus, we calculated the correction coefficient (according to Eq. (6) C0(k1/2) for a 4-pixel-periodic grating for which the fundamental frequency is 1/4a. Higher harmonics (3/4a, 5/4a…) are not addressable. Phase patterns with larger periods contain more than one harmonic within the addressable compact support and render analysis more complicated.
In Fig. 4(a) we show multiplicative coefficients required to cancel the zero order when displaying 2-pixel and 4-pixel period gratings with vertical and horizontal orientations. Checkerboard patterns with 2-pixel and 4-pixel periods, which display diffraction orders along diagonals, were also measured in order to acquire information about diagonal spatial frequencies. For a 2-pixel periodic checkerboard, the experimental local minimum of the zero-order was observed to be for a gray level higher than 255, thus inaccessible. However, even if the C0 correction value cannot be precisely derived in this case, we can argue that it should be higher than 2.5.
Numerical fitting of these correction values provides C0(k), the correction function for any spatial frequencies k. Here, we fitted correcting values with a second order polynomial in both horizontal and vertical directions:
According to experimental data, we measured and used for zero-order suppression (Fig. 4(b)):
This function must then be multiplied in the frequency domain with the Fourier transform of the phase mask applied to the SLM to correct filtering of each spatial frequency. The final corrected phase mask to be displayed in order to suppress the zero-order beam is then given by:
We point out that the described algorithm is also suitable to maximize diffraction efficiency, rather than minimize zero-order beam component. In this case correction coefficients are obtained by considering gray levels for which diffraction efficiencies are maxima. Doing so we obtained as a correcting function:
In the next section we demonstrate the ability of this technique to reject the un-diffracted light component as well as the ghost image and to enhance the diffracted light which can be obtained.
4. Experimental results
Demonstration of SLM Correction efficiency
We demonstrated the efficiency of the proposed method with our LCoS-SLM (Hamamatsu X10468-02) by suppressing the zero-order spots in various illumination patterns. As demonstrated below, it also results in a reduction of ghost images and improvement of diffraction efficiency in locations of interest.
As previously explained, the Fourier transform of the desired phase profile was multiplied by the 2D correction map (C0 and C1 depending on the specific purpose) previously derived, to compensate the spatial smooth filtering effect. However, this numerical calculation, corresponding to a high spatial frequency enhancement, yields some phase patterns with required gray-level values larger than 255 or smaller than 0. Therefore, the corrected gray level maps were truncated for addressing to the SLM. Before truncation, mean value of the output gray profile was centered on 128 to make use of the full gray level dynamic of the SLM.
We first demonstrated the efficiency of the algorithm by producing circular disks of 10μm in diameters at various distances from the focal point. Corresponding phase profiles were computed thanks to a Gerchberg-Saxton iterative algorithm. In the present experimental configuration, maximum distance of 120µm in the image plane corresponds to the maximum spatial frequency of 1/2a on the SLM with an expected theoretical maximum diffraction efficiency of 40.57%.
In Fig. 5 , we measured the power in the zero order and in the diffracted spot before and after applying the C0(k) correction map as described in the previous section. Zero order removal appears to be efficient in every direction. Fraction of power in the zero order can be kept bellow 2% for spots displayed in horizontal and vertical directions, and almost always below 5% for spots in the diagonal direction, whilst uncorrected patterns left around 20% or 40% of total power un-diffracted. Relying on the proposed model, numerical simulations (data not shown) predict a better suppression of the zero order for the diagonal direction. The remaining un-diffracted power must then be attributed to saturation of the gray levels or polarization rotation in the LC matrix.
Moreover, as expected, the power in the diffracted disk was improved even though the filter is specifically optimized for zero-order suppression (Fig. 5(b)). Correction improves diffraction efficiency and partially redistributes power initially present in the zero-order spot towards the shape of interest. For the disk displayed in the horizontal direction at 115µm for instance, the 20% reduction of power initially in the zero order results in a 12% increase of power in the diffracted order (the remaining energy being transferred to other orders). This redistribution corresponds to a 44% increase of power in the disk of interest. Let us point out that, as already noted, the SLM behavior is slightly asymmetric when dealing with spots displayed along the horizontal direction. Powers collected in the zero order and in the disk shape are slightly different for spots displayed on either side.
Finally, ghost images arising from spatial filtering, in particular symmetrically with respect to the focal point, are also reduced (Fig. 6 ). In Fig. 6, we measured power reduction in the brightest ghosts, which appear symmetrically with respect to zero for disks displayed at position up to x = 60µm, and at position x-120µm for shapes displayed at x = 80µm and x = 100µm. For a disk displayed at 60microns in the horizontal and in the vertical direction, the power in the ghost image is decreased by 96% and 92%, respectively.
Compared to C0(k), the use of the C1(k) correction map exhibited analogous results, of course increasing diffraction efficiency but also slightly degrading zero-order suppression.
Two-photon excitation fluorescence
In order to demonstrate the benefits of the proposed correction in typical imaging experimental situations we tested its performances with two-photon excitation fluorescence. A thin layer labeled with Rhodamine 6G was illuminated with various shapes.
In Fig. 7 we show disks of 10µm in diameter placed at 60µm from the focal point, that is, at half way from maximum distance. At this distance, although power is mainly diffracted toward the disk (Fig. 5), the peak intensity of the zero-order is larger than the one generated by the 10µm disk of interest because of spreading of optical power on a much smaller surface. Results are enhanced by the quadratic dependence of two-photon excitation on impinging intensity. When compensating for spatial filtering by the SLM, zero-order contribution becomes negligible when the disk is displayed in the vertical and horizontal direction (Figs. 7(g) and 7(h)) and of the same order of magnitude when displayed along the diagonal (Fig. 7(i)). Actually, the relevant measurement is the total amount of photo-excited material in the volume of the sample which is obtained by integrating fluorescence intensity over a full axial stack. We measured that, for a spot displayed in the diagonal direction, the integrated amount of photo-excitation by the zero-order represents only 12% of the total amount of photo-excited material for a corrected phase pattern while it represents 64% when uncorrected. This means that although the zero-order spot is still visible in Fig. 7(f), its influence may be neglected.
In Fig. 8 the excitation light pattern is shaped to match the fluorescence profile of a previously stored fluorescence image of a dendrite to demonstrate the applicability to photo-activation with arbitrary shapes. We can see that the undiffracted light is almost completely suppressed by employing the proposed algorithm. Moreover, we also notice that pattern illumination is more homogeneous and therefore, is more in agreement with the algorithm convergence. To observe this effect, speckle pattern in the shapes was minimized. To achieve so, the target shape was simply discretized in an array of spots. Spaced multi-spots do not interfere in the focal plane and consequently, random intensity fluctuations are suppressed. However, this method gives the shapes a pixilated aspect which can be observed due to the higher resolution for imaging than for excitation. In the SLM plane, multi-spot excitation simply consisted in padding the phase pattern twice along both x and y directions within the pupil plane of the objective. This padding of the phase pattern on the SLM results in an inter-spot distance equal to twice minimal resolution at the sample plane. This numerical treatment allows us to appreciate an improvement in the energy distribution in the shined pattern when applying our correcting algorithm.
In the present paper we proposed a method to remove undesired ghosts and zero-orders introduced by a LCoS-SLM device applying a frequency based algorithm to any phase pattern.
The method is based on a preliminary characterization of the SLM diffraction efficiency obtained by a set of 6 binary phase gratings. Numerical fitting of experimental data suggests a model involving cross-talk between nearby pixels of the SLM, thus resulting in filtering of the highest spatial frequencies of the phase pattern displayed.
Relying on this model, we elaborated a simple algorithm able to correct for distortions introduced by the device. The correction is obtained by multiplying the Fourier transform of displayed phase patterns, by a 2D correction function, to enhance high spatial frequencies and partially balance for spatial filtering.
Experimental application of this method with arbitrary shapes generated by iterative Gershberg Saxton algorithm demonstrates a reduction of the power of the undesired zero-order beam up to ~90% in the centre of the image. Ghost images, corresponding to symmetric replica of the desired patterns are also reduced by this treatment. Furthermore, the reduction of light power in undesired spots results in an improvement of diffraction efficiency toward regions of interest.
The current analysis was done with an LCoS-SLM (Hamamatsu X10468-02) optimized for the infrared region from 750nm to 850nm. This wavelength range corresponds to the maximum of emission of pulsed femtoseconds Ti:Sa lasers and suits to many two photon-absorption experiments. These improvements are all the more important in bi-photonic excitation due to the quadratic dependence of excitation on intensity. Estimated reduction up to 99% of the excitation on the optical axis is obtained.
In the present article we mainly focused on a correction aiming at the complete rejection of the zero-order beam. If a beam block in an intermediate image plane can be placed to remove zero order, then optimization of diffraction efficiency, rather than zero-order suppression, might be more relevant.
Finally, it should be possible to extend this method to other SLM models working at different wavelength ranges in order to maximize their experimental performances.
The authors thank E. Papagiakoumou and O. Hernandez-Cubero for sharing the experimental setup and helping with it. V.E. acknowledges Human Frontier Science Program (RGP0013/2010) and Fondation pour la Recherche Médicale (FRM équipe).
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