Non-invasive correction of thermally induced wavefront aberrations of spatial light modulator in holographic projection

Jan Bolek; Michal Makowski

doi:10.1364/OE.27.010193

1. Introduction

Spatial light modulators (SLM) are devices capable of modifying optical fields in terms of phase, amplitude, or both. They have been proven useful in applications such as holographic projection [1], holographic displays [2], optical trapping [3–5], mode conversion [6], correlation for microscopy [7], spatial filtering [8] and many others [9]. With recent advances in the technology of phase-only reflective SLMs, such as increase of their resolution and fill factor, as well as reduction of their pixel size, it has become possible to demonstrate high-resolution holographic projectors with high efficiency due to phase-only modulation [10].

However, within all applications mentioned before, one expects an SLM to modulate light only in a deserved manner, without induction of any other phase disturbances, i.e. wavefront aberrations. In case of the reflective SLMs there exists a curvature of the backplane of the SLM, which induces aberrations in the modulated light field, usually measured with interferometers [9, 11]. As a result, there is observed a significant decrease of the image quality and resolution of projected images [12] or loss of trapping force in case of the optical tweezers [13,14]. Spatially varying phase response is another important cause of aberrations induced by the SLM [13,15–18]. That may be caused by non-uniform liquid crystal cell thickness or spatially varying distribution of the electric field which reorients the liquid crystal molecules.

Thanks to the ability to modulate phase of the light in the SLM plane, it is possible to correct aberrations present in the optical setup by changing phase in a certain way, so that the effective aberration phase is canceled. The problem then becomes focused on finding such phase distributions. Various method has been proposed, many based on finding a set of Zernike polynomials coefficients that minimize a given merit function [12,19–21]. In this case algorithms tend to be complicated and they may require constant observation and feedback of the images of the PSFs (point spread functions) using a camera. Moreover, the result is restricted only to the space spanned by the chosen set of Zernike polynomials, thus some high-order aberrations may be missing from the results. Methods based on speckle field analysis were also proposed [22], but resulting phase maps are noisy and exhibit optical vortices, which are unacceptable for application in holographic projection, as they increase the coherent noise. Methods for correcting aberrations induced by non-uniform phase response over the surface of the SLM has been proposed, including altering the look-up tables (LUTs) of the phase responses in sub-regions of the SLM [13,18] or remapping the displayed phase masks [17].

Another important issue regarding application of an SLM is a susceptibility of the backplane of the SLM to deform under varying temperature conditions [19,23]. Heating of the SLM may be caused by variable external temperature, power dissipated by the electronics of the SLM or, in case of higher powers of the light source, absorption of the light incident of the SLM. The backplane reflective surface, along the other layers of the LCoS (liquid crystal on silicon) device, experience mechanical deformations with changes of temperature due to thermal expansion. This effect has to be accounted for in order to achieve the best performance and resolution of the optical setup. This effect was described earlier in [23], but the method proposed there based on use of an interferometer, which complicates setup too much to use in practical, miniaturized applications. The effect of the thermal dependency on the phase modulation curve and the effect of spatially varying phase response of the SLM may be also considered [15–18]. Non-uniform light intensity distribution illuminating the SLM is an important factor here [18], as it can affect the phase response locally in the areas of greater light intensity, hence inducing more optical aberrations.

In this paper we present a non-invasive method of determination of the phase distribution of said aberrations based on acquisition of defocused PSF spots [24,25] and previously proposed phase retrieval algorithm [26–28], which is realized in a simple optical setup, as opposed to an interferometric setup [9,11,22,23]. Here we use it for the characterization of the influence of temperature on the aberrations of our particular SLM. We also show results of the wavefront correction in projection of diffusive Fourier-type computer-generated holograms (CGHs). We prove the accuracy and usefulness and additionally show the necessity of performing the periodic aberration corrections in order to achieve the maximum possible projection resolution under variable temperature and stress of the SLM. We also show an excellent pixel-in-pixel agreement of diffraction-limited projected fields with numerical simulations, which allows practically error-free experimental projection of intensity fields.

2. Proposed aberrations measurement methodology

2.1. Aberration correction method

The proposed method of retrieving aberrations of an optical field in the SLM plane involves the capture of three images of focused light field PSFs, each in a slightly different distance. This provides information about the light field not only in a plane perpendicular to the propagation direction, but also along the optical axis, thus giving the information about three dimensional nature of the beam. From the physical point of view, the evolution of the beam intensity pattern measured at three distances gives an additional partial information about the phase distribution of the beam, which cannot be unambiguously derived from a single measurement. Similar methods were described earlier but in our case we avoid using complicated algorithms which result only in a finite set Zernike or other polynomial set coefficients [20, 23] instead of a full wavefront function or moving the camera or a sample (in case of a microscopic setup) along the optical axis [20]. The main computational cost of our method consists of the Fast Fourier Transform, which are implemented very efficiently on graphical processing units (GPU) and even for large matrices can be performed in a few milliseconds [29].

In our method, in order to avoid the necessity to physically move the camera sensor used for capturing the PSFs which greatly complicates the setup [30], we used the inherent ability of a phase-only SLM to apply additional defocus, which virtually changes the distance of the observed light field [21]. By comparing the Fresnel integral of a light field observed at a distance z from the input plane:

u_{o} (x_{o}, y_{o}; z) = \frac{1}{i λ z} e^{i k (z + \frac{x_{o}^{2} + y_{o}^{2}}{2 z})} \iint_{\pm \infty} u_{i} (x_{i}, y_{i}; 0) e^{\frac{i k}{2 z} (x_{i}^{2} + y_{i}^{2})} e^{- \frac{i k}{2} (x_{i} x_{o} + y_{i} y_{o})} d x_{i} d y_{i}

with an equation in which there has been added a convergent spherical wave of radius q, with

p = {(\frac{1}{z} - \frac{1}{q})}^{- 1}

being the virtual observation distance:

u_{o}^{'} (x_{o}, y_{o}; z) = \frac{1}{i λ z} e^{i k (z + \frac{x_{o}^{2} + y_{o}^{2}}{2 z})} \iint_{\pm \infty} u_{i}^{'} (x_{i}, y_{i}; 0) e^{\frac{i k}{2 p} (x_{i}^{2} + y_{i}^{2})} e^{- \frac{i k}{p} (x_{i} \frac{x_{o}}{\frac{z}{p}} + y_{i} \frac{y_{o}}{\frac{z}{p}})} d x_{i} d y_{i}

one can see that in the latter case the effective propagation distance (EPD) is changed from z to p and the output coordinates are scaled by a factor of

\frac{z}{p}

, thus performing the light field capture involves capturing the PSFs and scaling the image by a factor dependent on ratio of the distance to the camera z to the effective propagation distance p. The EPDs in which the PSFs are captured are chosen to obtain the images of the focused beam within its waist, behind it and in front of it. This ensures that the optimal amount of information is gathered for the phase retrieval process.

An iterative algorithm, based on Gerchberg-Saxton algorithm [24,27,31,32], depicted in Fig. 1, is then used to retrieve phase in the SLM plane. The planes positions of the planes along the Z axis are as follows: z₀ = 0 for the SLM plane and z₁, z₂ and z₃ for the consecutive PSF planes. Prop(z_j − z_i) denotes the propagation operations from the plane positioned at distance z_i to the plane at z_j. The order of the propagations are denoted above the corresponding propagation. The algorithm starts with propagation of the light field consisting of the measured amplitude of the first measured PSF to the plane of the second PSF. Then the measured amplitude of the second PSF is imposed on the field, keeping the phase unchanged. The resulting light field is propagated to the third PSF plane, where the measured amplitude of the third PSF is imposed. The light field is propagated back to the second and the the first planes of the PSFs where the corresponding measured amplitudes are again being imposed. Please note that the used PSF amplitudes are measured only once, before executing the computations.

Fig. 1 Flowchart of the algorithm.

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Next, the light field is propagated back to the SLM plane, where the amplitude reflected from the SLM, i.e. a constant amplitude over the SLM aperture is imposed. The presence of the constant amplitude in the optical setup is assured by use of the pinhole-based spatial filter and a lens forming a quasi-planar wavefront. The aperture of the SLM is determined by its active rectangular area with dimensions depending on the utilized SLM. The incident beam illuminates the whole active area of the SLM.

Then the process is repeated with light field propagated again to the first, second and third PSF with imposition of the respective measured PSF amplitudes and back to the SLM plane with imposition of the previously described amplitude in the SLM plane. The process is executed until no significant improvement in the iterated light field is achieved, i.e. a stagnation has occurred. The occurrence of the stagnation can be determined by observation of the change of a chosen merit function. When the change has decreased below a chosen threshold value the iteration loop is terminated. For our purpose the merit function was a measure of the uniformness of the amplitude in the SLM plane, i.e. ratio of the standard deviation of the amplitude to its average value. Termination threshold of the merit function was set to 0.02. This merit function has performed well in our research, but other functions can be chosen, e.g. RMS differences between the amplitudes or unwrapped phases of the light field in the consecutive iterations.

The computations were performed using a PC computer equipped with GeForce GTX 1070 GPU using CUDA platform which enables low processing time. On average, one iteration takes 0.17 seconds, so depending on the number of iterations, normally between 5 and 10, the total computing time adds up to 1–2 seconds.

3. Experimental setup

The temperature control of the SLM was performed with use of a Peltier module, which provided ability for both cooling and heating the panel with respect to the ambient temperature. The scheme of the temperature controller attached to the SLM panel is shown in Fig. 2(a).

Fig. 2 a) Temperature controller of the SLM. M – the SLM panel, T – temperature sensor, S – aluminum spacer, P – Peltier module, I – thermal isolator, H – heatsink, R – SLM panel holder. b) Experimental setup: P – polarizer, λ/2 – half-wave plate, SF – spatial filter, L – lens, BS – 50/50 beamsplitter, SLM – spatial light modulator with TC – temperature controller.

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The temperature was controlled with accuracy of 0.5°C, measured with a digital temperature sensor attached directly to the metal backplane of the SLM. The Peltier module with a heatsink were mounted delicately so they would not induce additional aberrations due to any mechanical stress. Good thermal conductivity was ensured by use of a thermal grease. The absence of the induced stress was evaluated by comparing aberrations before and after installation of the Peltier module. Our measurement were performed in a laboratory conditions, hence high rigidity of the assembly was not crucial. The front frame of the SLM was also thermally isolated from the ambient. It must be noted however that in practice one may be unable to avoid induction of any stress due to a necessity of rigidly mounting the SLM in an optical setup. This stress may change with temperature, but resulting aberrations are just part of the total aberrations induces by the SLM.

The optical setup realizing holographic projection of 2-D images is shown in Fig. 2(b). He-Ne laser is used with a polarizer and a half-wave plate places in the beam path to ensure correct polarization orientation necessary for the SLM to obtain optimal phase-only modulation. Plane wave, formed by a spatial filter and a lens, illuminates the HoloEye PLUTO SLM with pixel pitch of 8 μm and rectangular aperture with resolution of 1920 by 1080 pixels, corresponding to physical dimensions of 15.36 mm by 8.64 mm, at a right angle after being reflected by a beamsplitter. The beam focused by a converging lens displayed on the SLM propagates to a camera fixed at a distance of 1 m away from the SLM and is captured on the CMOS matrix of the Basler puA2500-14um camera.

4. Experimental results

The temperature was continuously changing from 18°C to 34°C as the SLM was heating up and the measurements were performed for each 1°C step. Six measurements were performed for each temperature step. The EPD distances varied by 4 mm for each of the six measurements to account for possible uncertainty caused by selection of the capture distances.

The measurements resulted in series of phase masks, six for each temperature. The examples of the retrieved phase masks are shown in Fig. 3 for temperature increment of 2°C. The residual piston and tilt terms were removed as they do not introduce any substantial information. Change of the deformation is very significant and is clearly seen in the provided figure.

Fig. 3 Retrieved aberration masks for different temperatures ( Visualization 1). The gray levels on the image correspond to phase retardation.

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Figure 4 depicts the accuracy of the method in terms of deviation between masks obtained for each temperature with varying EPD of PSFs capture. For each temperature, an average depth mask was computed from each of the unwrapped masks. Then six RMS values were computed between the averaged mask and each of the six measurements. The average RMS deviation, over all 102 measurements for all temperatures, is equal to 0.013λ, which proves good accuracy of the presented method.

Fig. 4 RMS deviation from averaged depth mask for each temperature. The blue points show, for the given temperature, RMS values between the averaged phase mask each of the six measurements with varying EPS distance The red points show average deviation for each temperature and the red line shows average deviation for all the measurements.

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Note that it was necessary to change the base EPD distances at which the PSFs were captured in order to account for change of focal length with temperature. The dependence was assumed to be approximately linear, changing from 89.3 cm for 18°C to 98.4 cm for 34°C. This is caused by the fact that the effective focal length induced by the SLM, mostly associated with defocus Zernike term Z_2,0 varies strongly with temperature, which is shown in the next subchapter.

4.1. Temperature influence on the Zernike coefficients of aberrations

The iteratively retrieved masks were decomposed into the Zernike polynomials up to fourth radial orders by fitting into a modified, orthogonalized set based on the Zernike polynomials and later converted into the Zernike coefficients (see Appendix). The graphs illustrating the dependence are shown in Fig. 5.

Fig. 5 Temperature dependency of the Zernike coefficients of the retrieved masks, up to 4th radial order. The irrelevant piston and tilt terms are omitted.

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Mean values and standard deviations were calculated for each temperature to estimate measured value and corresponding uncertainty. The values of the coefficients manifest low deviations in comparison with their value. The largest contribution to the wave front error comes from the defocus term, which effectively changes focal length of the SLM, i.e. moves the focal point closer. The curvature of the light reflected by the SLM, relating to the effective curvature of the device, was calculated with use of the Zernike coefficients containing quadratic radial term, i.e. defocus, primary astigmatisms and primary spherical aberration (see Appendix) and the relation is shown in the graph in Fig. 6. The fitted approximate change of curvature with temperature was estimated to be $- 0.257 \frac{m}{° C}$ . With the base curvature of about 10 m for 20°C the change of the curvature can significantly affect the effective optical power of the setup.

Fig. 6 Temperature dependency on the curvature of the light reflected from the SLM.

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4.2. Reconstructed PSFs

Figure 7(a) presents averaged cross-sections of the PSFs reconstructed from the SLM with corrected aberrations. The amplitude, i.e. square root of the intensity is shown, instead of intensity, for better visualization of the weaker side lobes. The theoretical function $A (ν_{x}, ν_{y}) = \sqrt{I (ν_{x}, ν_{y})} = A_{0} | sinc (a ν_{x}) sinc (b ν_{y}) |$ , i.e. the Fourier transform of the rectangular aperture of the PLUTO SLM, is fitted to the presented data. The standard deviation for every point of the graph, shown by two black, dashed lines, has been included as well. Figure 7(b) shows histogram of normalized RMS errors between each of the captured PSFs after correction and the function fitted from the averaged cross-sections shown in Fig. 7(a). The RMS were normalized by dividing their values by the maximum amplitude value of the theoretical PSFs. The reference point for RMS is the theoretically assumed PSF of the perfectly flat SLM in the given configuration. Most of the RMS values fall into range between 0.023 and 0.035 for the X axis and 0.009 to 0.025 for the Y axis. The connection between the RMS values and the PSF shape is also shown by giving the examples of the worst, median and best PSFs in terms of the RMS error. This exemplary images are shown as a difference between the given measured and the theoretical, fitted PSFs. Along the X axis only one of the 102 measurements possesses significantly larger value of the RMS connected to asymmetry of the side lobes. This again proves high accuracy of the proposed aberration retrieval method because the aberration-limited PSF can be obtained only with very low residual aberrations present in the setup. The consistent values of the RMS values of the captured PSF after correction, shown in the histogram in Fig. 7(b), indicate that the aberrations are kept low in the whole temperature range.

Fig. 7 (a) Cross-sections of the averaged PSF with theoretical fit; (b) Histogram of RMS values, as well as differential images showing measured PSFs with the smallest, median and the largest RMS values. The measured PSFs are subtracted from theoretical, fitted sinc function. The PSFs were normalized to maximum value of 1 and the values of the differences correspond to this maximum value.

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4.3. Reconstruction of Fourier holograms

The retrieved phase masks were used to perform a test of a holographic projection with corrected aberrations in the same experimental setup. The projection was performed by displaying on the SLM a Fourier hologram of a mosaic of the USAF-1951 test pattern, shown in Fig. 8(a). The chosen test image consist of groups of lines with periods decreasing down to smallest possible details that can be theoretically resolved, i.e. two-pixel lines seen in the groups (8,1–3) of Fig. 8(c). The phase-only hologram was diffusive, i.e. computed using an initial random phase distribution, which creates speckle noise in the image plane.

Fig. 8 The input image (a), shown magnified in b) and c). The central part of the reconstruction of Fourier holograms in the simulation (d) and in the experiment with proper correction applied (e), showing the diffraction-limited image resolution. The red circles highlight the high agreement of positions of the speckle grains in the simulation and in the experiment.

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The phase mask displayed on the SLM consisted of the Fourier hologram, a Fourier-transforming lens and the aberration correction phase mask obtained by negating the retrieved aberration phase mask: φ_SLM = φ_holo + φ_lens − φ_aberr. The hologram is reconstructed at focal plane of the displayed lens, i.e. the plane of the camera sensor. Figure 8(d) shows the reconstruction of the hologram in simulation, i.e. the ideal, aberration-free conditions. Figure 8(e) shows the reconstruction of the hologram with correction applied for the actual SLM temperature, here 25°C. With the aberrations corrected the diffraction-limited resolution was achieved as the groups corresponding to two-pixel line pairs of a two megapixel (Full HD) hologram, are clearly distinguishable (horizontal groups [8, 1–3] and vertical groups [7, 2–3]), as well as the digits. It is important to note that the speckle pattern in the experimental reconstruction, observed in Fig. 8(e), corresponds precisely to the simulation shown in Fig. 8(d), even up to the position of speckle noise spots. This underlines the precision and importance of the performed aberration correction.

Figure 9 shows reconstructions of the Fourier hologram in the experiment with varying temperature of the SLM. Applying a correction only for the defocus term, corresponding to spherical curvature of the SLM while neglecting the higher-order aberrations, leads to loss of resolution, as shown in Fig. 9(a). This effect is less prominent in case of the temperature value of 25°C when comparing to 34°C, because values of higher-order aberrations are smaller in the first case, as can be deduced from Fig. 5. Figure 9(b) also shows the effect of degradation of the reconstruction quality when the temperature of the SLM is altered without changing the correction mask. Here, the correction mask is obtained when the SLM has temperature of 25°C. The quality of the image is significantly lower when the temperature raises by 1°C. Further increase of the temperature by 2°C and 3°C causes a loss of half of the resolution in the as the smallest groups become unresolvable. In the endpoint cases of 18°C and 34°C, corresponding to change of −7°C and +9°C respectively, the whole central part of the image becomes completely illegible. However, the quality and full resolution is fully restored in every of the presented cases after applying a proper correction mask obtained for the actual temperature, see Fig. 9(c). Consistency of the effects of the corrections is also shown in Fig. 10 by comparing values of the RMS errors between the simulated reconstructions shown in Fig. 8(d) and experimental reconstructions shown in Figs. 9(b) and 9(c). This proves that the method presented in this paper enables one to successfully correct the aberrations of the SLM in the broad range of temperatures. Theoretically, based on the extrapolation of the linear fit in Fig. 6, the SLM is capable of adding optical power, sufficient for displaying the corrective phase masks used in the presented method, with temperatures down to −20°C, which obviously exceeds the operating range of the used liquid crystal.

Fig. 9 The central part of the experimental projections from holograms with different thermal phase correction applied: a) comparison of full correction of aberrations vs correction of defocus alone, i.e. quadratic phase; b) deterioration of projection quality with a growing mismatch of actual temperature and fixed phase correction mask; c) dynamic thermal correction adapted to the actual temperature allows diffraction-limited imaging in a wide range of SLM temperatures (proposed method).

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Fig. 10 RMS errors between simulation and experimental reconstructions for different temperatures in two cases: when the aberrations are corrected for the actual temperature (red bars) and for temperature of 25°C, regardless of the actual temperature (blue bars).

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5. Conclusions

We examined the quasi-real-time thermal dependence of the aberrations induced by the SLM. It is a significant factor and even a change as small as 1°C can cause a significant decrease of the performance and the effective resolution in holographic image projection. Therefore in practical setup the temperature of the SLM has to be constantly monitored and either kept at the same value without changing the correction mask, or the aberrations have to by dynamically corrected with varying temperature. Otherwise, one must anticipate a significant loss in image quality in case of a projection or imaging using an SLM or loss in trapping force in case of application in an optical tweezers.

As a solution we have demonstrated the feasibility of previously proposed multi-defocus phase retrieval method for a specific case of precise evaluation and cancellation of thermally-variable aberrations in an SLM-based holographic projection setup. The method is non-invasive in that it requires only little modification of the optical setup, i.e. one additional fixed camera, which can be seamlessly installed on the side using a beam sampler without obstructing the main projection beam. The proposed method has allowed us to experimentally demonstrate diffraction-limited holographic 2-D projections with very high fidelity to numerical simulations, which could facilitate further progress e.g. in experimental validation and feedback loop in iterative optimization of CGHs. It is also important to note that although we show results only for one particular model and piece of the SLM device, it is possible to apply the suggested methodology to determine and correct aberrations of other modulators or SLMs combined with an arbitrary set of thermally-sensitive optical elements.

Appendix

Orthogonalization of the Zernike polynomials for a non-circular domain

The Zernike polynomials are useful tool for quantitative description of optical aberrations. However, their property of orthogonality is preserved only on a circular domain. In order to use them for decomposition of aberrations on a rectangular aperture of an SLM they must be converted to an orthogonal basis [33]. Here we present method used for analysis of the data in our research.

The orthogonalized set of polynomials is derived using a method called Gram-Schmidt orthogonalization. The process of orthogonalization goes as follows:

\begin{array}{l} 𝒱_{1} & = & 𝒵_{1} \\ 𝒱_{2} & = & 𝒵_{2} - \frac{< 𝒵_{2}, 𝒱_{1} >}{< 𝒱_{1}, 𝒱_{1} >} 𝒱_{1} \\ 𝒱_{3} & = & 𝒵_{3} - \frac{< 𝒵_{3}, 𝒱_{1} >}{< 𝒱_{1}, 𝒱_{1} >} 𝒱_{1} - \frac{< 𝒵_{3}, 𝒱_{2} >}{< 𝒱_{2}, 𝒱_{2} >} 𝒱_{2} \\ \dots \end{array}

The first vector of the new set {𝒱_i} is taken from the old set {𝒵_i} (here the Zernike polynomials) without any changes. The second vector is calculated by subtracting a projection (scalar product) of the second old basis vector on the first new basis vector to remove any non-orthogonal part of the previous vector. In then next step, for the third vector, projection of the third vector from old basis on the first and second of the new vectors are subtracted. The process continues for the next vectors any any vector can be computed using the general formula:

𝒱_{i} = 𝒵_{i} - \sum_{j = 1}^{i - 1} \frac{< 𝒵_{i}, 𝒱_{j} >}{< 𝒱_{j}, 𝒱_{j} >} 𝒱_{j}

For convenience the vectors can be normalized to unity, becoming an orthonormal set {𝒱̃_i}.

{\tilde{𝒱}}_{i} = \frac{𝒱_{i}}{\sqrt{< 𝒱_{i}, 𝒱_{i} >}}

The coefficients {

c_{i}^{𝒱}

} of decomposition of any function in the {𝒱̃_i} basis are computed as follows with use of an scalar product:

c_{i}^{𝒱} = < f, {\tilde{𝒱}}_{i} > = \sum_{k = 1}^{N} \sum_{l = 1}^{M} f_{[k, l]} {\tilde{𝒱}}_{i [k, l]}

(where k and l are indices associated with X and Y axes and N and M are number of pixels in the respective axes) and the function is a linear combination of the basis vectors and the computed coefficients:

f_{[k, l]} = \sum_{i} c_{i}^{𝒱} {\tilde{𝒱}}_{i [k, l]]}

Knowing the coefficients { $c_{i}^{𝒱}$ } in the orthogonalized basis one can compute coefficients in the old, nonorthogonal basis {𝒵_i}:

f_{[k, l]} = \sum_{i} c_{i}^{𝒵} 𝒵_{i [k, l]]}

The equation for the decomposition can be written in the matrix form in the orthogonal basis { $c_{i}^{𝒱}$ } represented by a unit matrix:

f = [{\tilde{𝒱}}_{1} | {\tilde{𝒱}}_{2} | {\tilde{𝒱}}_{3} | \dots | {\tilde{𝒱}}_{n}] [\begin{matrix} c_{1}^{𝒱} \\ c_{2}^{𝒱} \\ c_{3}^{𝒱} \\ ⋮ \\ c_{n}^{𝒱} \end{matrix}] = [\begin{matrix} 1 & 0 & 0 & \dots & 0 \\ 0 & 1 & 0 & \dots & 0 \\ 0 & 0 & 1 & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & 0 & \dots & 1 \end{matrix}] [\begin{matrix} c_{1}^{𝒱} \\ c_{2}^{𝒱} \\ c_{3}^{𝒱} \\ ⋮ \\ c_{n}^{𝒱} \end{matrix}] = \hat{I} {\vec{C}}^{𝒱}

For the nonorthogonal basis {𝒵_i} the decomposition is given as follows:

f = [𝒵_{1} | 𝒵_{2} | \dots | 𝒵_{n}] [\begin{matrix} c_{1}^{𝒵} \\ c_{2}^{𝒵} \\ ⋮ \\ c_{n}^{𝒵} \end{matrix}] = \hat{𝒵} {\vec{C}}^{𝒵}

Matrix 𝒵̂ can be derived from the Eq. (4) defining the connection between vectors 𝒱_i and 𝒵_i. In the matrix form:

\hat{I} = \hat{𝒵} - \hat{I} \hat{A}

where

\hat{A}

is an upper-triangular matrix of scalar products. The items of matrix

\hat{A}

are defined as:

a_{j i} = {\begin{array}{l} \frac{< 𝒵_{i}, 𝒱_{j} >}{< 𝒱_{j}, 𝒱_{j} >} & j < i \\ 0 & j \geq i \end{array}

and the matrix 𝒵̂ will be equal to:

\hat{𝒵} = \hat{I} + \hat{A} = [\begin{matrix} 1 & a_{12} & a_{13} & \dots & a_{1 n} \\ 0 & 1 & a_{23} & \dots & a_{2 n} \\ 0 & 0 & 1 & \dots & a_{3 n} \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & 0 & \dots & 1 \end{matrix}]

Equations (7) and (8) must be equal, so:

(\hat{I} + \hat{A}) {\vec{C}}^{𝒵} = {\vec{C}}^{𝒱} m

The above equation is an linear equation for the unknown coefficients {

c_{i}^{𝒵}

}. It can be easily solved in a recurrent manner:

\begin{array}{l} c_{n}^{𝒵} & = & c_{n}^{𝒱} \\ c_{n - 1}^{𝒵} & = & c_{n - 1}^{𝒱} - a_{n - 1, n} c_{n}^{𝒱} \\ c_{n - 2}^{𝒵} & = & c_{n - 2}^{𝒱} - a_{n - 2, n - 1} c_{n - 1}^{𝒱} - a_{n - 2, n} c_{n}^{𝒱} \\ \dots \end{array}

In general:

c_{i}^{𝒵} = c_{i}^{𝒱} - \sum_{j = 1 + 1}^{n} a_{i j} c_{j}^{𝒱}

Quadratic phase coefficient and the effective curvature of the wavefront

Zernike coefficients with radial order up to 4th containing the quadratic term ρ² are defocus c_2,0, oblique astigmatism c_2,−2, vertical astigmatism c_2,2, and primary spherical aberration c_4,0:

\begin{array}{l} c_{2, 0} = \sqrt{3} (2 ρ^{2} - 1) \\ c_{2, - 2} = \sqrt{6} ρ^{2} sin 2 θ \\ c_{2, 2} = \sqrt{6} ρ^{2} cos 2 θ \\ c_{4, 0} = \sqrt{5} (ρ^{4} - 6 ρ^{2} + 1) \end{array}

By extracting the terms with ρ² the effective coefficient corresponding to quadratic radial term can be obtained:

c_{ρ^{2}} = 2 \sqrt{3} c_{2, 0} - \sqrt{6} \sqrt{c_{2, - 2}^{2} + c_{2, 2}^{2}} - 6 \sqrt{5} c_{4, 0}

By comparing the expression describing wavefront by the coefficient c_ρ² and quadratic wavefront with a radius of f :

2 π c_{ρ^{2}} ρ^{2} = \frac{π}{λ f} {(R ρ)}^{2}

(where ρ is coordinate defined on a unit circle and R is real radius of the aperture) one can calculate the curvature of the wavefront knowing the coefficient c_ρ² :

f = \frac{R^{2}}{2 λ c_{ρ^{2}}}

where R is a radius of the circle on which the decomposition into the Zernike polynomials was performed.

Funding

Foundation for Polish Science (TEAM TECH/2016-3/18,POIR.04.04.00-00-3DD9/16-00).

References

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Non-invasive correction of thermally induced wavefront aberrations of spatial light modulator in holographic projection

Abstract

1. Introduction

2. Proposed aberrations measurement methodology

2.1. Aberration correction method

3. Experimental setup

4. Experimental results

4.1. Temperature influence on the Zernike coefficients of aberrations

4.2. Reconstructed PSFs

4.3. Reconstruction of Fourier holograms

5. Conclusions

Appendix

Orthogonalization of the Zernike polynomials for a non-circular domain

Quadratic phase coefficient and the effective curvature of the wavefront

Funding

References

Supplementary Material (1)

Cited By

Figures (10)

Equations (20)

Optics Express