## Abstract

Conventional methods of compensating for self-distortion in liquid-crystal-on-silicon spatial light modulators (LCOS-SLM) are based on aberration correction, where the wavefront of the incident beam is modulated to compensate for aberrations caused by the imperfect optical flatness of the LCOS-SLM surface. However, the phase distribution of an LCOS-SLM varies with changes in ambient temperature and requires additional correction. We report a novel phase compensation method under temperature-varying conditions based on an orthonormal Legendre series expansion of the phase distribution. We investigated the temperature dependency by controlling the ambient temperature with an incubator and successfully corrected for self-distortion in a temperature range of 20 °C to 50 °C. Our approach has the potential to be adopted in tight-focusing applications which require wavefront modulation with very high accuracy.

© 2014 Optical Society of America

## 1. Introduction

Active wavefront modulation technology [1–3] using liquid-crystal-on-silicon spatial light modulators (LCOS-SLMs) is expected to be a key technology in optical systems such as laser processing [4–10], adaptive optics [11,12], microscope applications [13,14], optical manipulation [15–18], and advanced beam generation technologies [19,20]. Occasionally, these applications require high phase-modulation stability, and the imperfect optical flatness of LCOS-SLMs becomes more critical and may cause serious aberrations in the system. We previously proposed an effective method to compensate for the distortion caused by the imperfect flatness of an LCOS-SLM by displaying a compensation phase pattern obtained from interferometry [3].

The ambient temperature of LCOS-SLMs can vary under certain circumstances, i.e. equipped inside systems for field use or long-term operations. However, this compensation method can only be applied at the ambient temperature at which the compensation phase pattern was generated. Thermal stabilization with a Peltier cooling unit or heater may be one solution to this issue, but this involves problems such as an increase the size of the system and mechanical vibrations. In addition, direct cooling/heating of an LCOS-SLM has the possibility of causing a thermal gradient inside the device, which may cause additional irreversible aberrations and serious damage to the liquid crystal in the long term.

In the work described in this paper, we first analyzed the effects of ambient temperature on the optical flatness of LCOS-SLMs by using an orthonormal Legendre polynomial expansion [21, 22] of the corresponding phase map. We found several Legendre coefficients that follow quadratic functions of ambient temperature. This prompted us to propose an algorithm for correcting the temperature dependency by displaying a phase pattern using two simple steps: an initializing step and a temperature correction step. Reconstructed phase patterns taking account of the temperature were applied successfully in the temperature range of 20 °C to 50 °C, giving an optical flatness of *<* *λ*/10.

This paper is organized as follows. In Section 2, we introduce the interferometric method for achieving optical flatness of an LCOS-SLM. In Section 3, we analyze how the LCOS-SLM flatness depends on changes in ambient temperature. Section 4 explains the proposed self-compensation method in which the ambient temperature is taken into account and shows the experimental results obtained.

## 2. Interferometer setup and self-compensation of an LCOS-SLM

The optical flatness of reflecting devices, such as mirrors and LCOS-SLMs, can be easily measured by interferometry. Self-compensation phase patterns are defined as patterns in which the obtained phase distributions are inverted. A convenient and intuitive method to confirm the effects of distortion is to observe the intensity distribution of focal spots formed by focusing output light with a single lens, considered as Airy diffraction patterns.

#### 2.1. Experimental setup for optical flatness measurement

The basic idea for measuring the optical flatness of a reflective optical element is based on a Michelson interferometer configuration (Fig. 1). A continuous-wave, 633 nm-wavelength He-Ne laser beam was expanded to *ϕ*10mm in diameter after spatial filtering and collimation, producing a top-hat shaped intensity profile. Note that the polarization direction of the linearly polarized beam from the light source was set parallel to the director of the liquid crystal, which is usually in the horizontal direction. The incident beam enters the Michelson interferometer, where it is divided into two beams with a beam-splitter, and these two beams are incident on the reference mirror and the LCOS-SLM (Hamamatsu Photonics K.K., X10468-01), respectively. To distinguish the ambient temperature effects of LCOS-SLMs from other optical components, the LCOS-SLM was set inside an incubator (Mitsubishi, CN-40A 40L) having a transparent window to allow the beam to enter. Beams reflected from the mirror and LCOS-SLM were combined again at the beam-splitter to generate interference fringe patterns, which were relayed via a 4f telecentric lens system to a CMOS camera (Lumenera, LU135M-IO-WOIR). To increase the accuracy of temperature regulation, temperature IC sensors with an accuracy of ±0.2 °C were placed at several positions, such as on the front surface of the LCOS-SLM and on the inside and outside of the incubator. The temperatures at the device surface and inside the incubator were kept the same throughout the experiment.

The Fourier fringe analysis method [23] was adopted to reconstruct the phase distribution of the observed fringe pattern. Single-shot interference fringes of a tilted wavefront are described by

where*f*and

_{x}*f*are the spatial-carrier frequencies,

_{y}*ϕ*(

*x*,

*y*) expresses the desired phase distribution, and

*a*(

*x*,

*y*) and

*b*(

*x*,

*y*) represent unwanted irradiation variations caused by the experimental system. By taking the two dimensional Fourier transform of Eq. (1), the Fourier spectra will be separated by the spatial-carrier frequency $\sqrt{{f}_{x}^{2}+{f}_{y}^{2}}$. This spatial-carrier frequency must be below the Nyquist frequency of a camera. Set a signal window at the spectra on the spatial-carrier frequency, impose zero signal on the non-windowed spectra, and then shift $-\sqrt{{f}_{x}^{2}+{f}_{y}^{2}}$ to the origin of the Fourier spectra. From the inverse Fourier Transform of Fourier spectra and calculate arctangent of real and imaginary part, one can discriminated

*ϕ*(

*x*,

*y*) from

*a*(

*x*,

*y*) and

*b*(

*x*,

*y*). This non-contour fringe pattern analysis is effective for low spatial frequency or real-time analysis.

Figure 2 shows an example of the reconstructed phase distribution of the LCOS-SLM. A complex phase distribution with peak-to-valley values of the distribution exceeding 2*π* was observed. This directly appears as aberrations in the optical system, and some kind of compensation has to be performed.

#### 2.2. Self-compensation procedure

A major advantage of wavefront modulation is that it allows the use of a wrapped phase representation, which can cover a wide effective phase range of the device. Since the LCOS-SLM is a phase modulator, the phase modulation property is represented by

where the modulation term*δϕ*(

*x*,

*y*;

*V*) depends on both the pixel position and the control voltage, while

*ϕ*(

*x*,

*y*; 0) is a constant offset term given at each pixel position. Distortion compensation is achieved by modulating the incident wavefront

*ϕ*(

*x*,

*y*; 0) and inverting the measured phase distortion patterns obtained by interferometry.

The effects of phase distortion can be confirmed by observing the Fraunhofer diffraction pattern focused with an aplanatic lens. This Airy diffraction pattern, characterized by a circular aperture, is written as

*J*

_{1}is a first-order Bessel function of the first kind,

*λ*is the wavelength,

*k*= 2

*π/λ*is the wavenumber,

*f*is the focal length of the lens, and

*ω*is the diameter of the circular aperture. In cross-sections of the observed Airy diffraction pattern, the width of the central lobe, which is referred to as the resolving power (given by the Rayleigh criterion, or diffraction limit) can increase drastically depending on the aberrations in the system, i.e. blurring. This expression can be applied in a straightforward manner to describe the effect of aberrations qualitatively.

The resolving power depends on the focal length of the focusing lens and typically is on the order of micrometers. We focused the beam from the LCOS-SLM with a 400mm focal length lens (Sigma Koki, DLB-30-400PM) and magnified the focal spot with an objective lens (x10) to observe the intensity distribution in detail (Fig. 3). Figure 3 shows the focused beam pattern obtained without operating the LCOS-SLM, the pattern obtained after self-compensation, and cross-sectional profiles taken through the center of the patterns, respectively. It is obvious from a comparison of the beam patterns that the self compensation was effective. The profiles in Fig. 3(c) reflect the structure of the beam patterns, showing a symmetrical pattern after the distortion compensation, which indicated that the result meets the Rayleigh criterion. The ideal resolving power given by the Rayleigh criterion is 30.89*μ*m, and the diameter of the observed focal spot, after taking account of the magnification of the objective lens, was 31.1 ± 0.2*μ*m in this configuration, which shows good agreement.

## 3. Temperature dependency analysis of LCOS-SLM optical flatness

Transformation of the 2D phase distribution to a numerical representation is important for analyzing the effects of temperature. Typically, Zernike polynomials [1] are effective for analyzing a wavefront distribution in polar coordinates, but in the case of rectangular coordinates, Legendre polynomials may be more suitable [21,22]. In this section, we consider the effectiveness of a Legendre polynomial expansion and reconstruction, and we analyze the dependency of the LCOS-SLM’s optical flatness on changes in ambient temperature.

#### 3.1. Optical flatness analysis based on Legendre polynomial expansion

Legendre polynomials *P _{n}*(

*x*) [24] are the solution of Legendre’s differential equation,

If an arbitrary function *f* (*x*) is integrable in [−1,1] and its series expansion satisfies

*a*is given by

_{n}Additionally, extending the concept of Legendre polynomials to two dimensions, they are orthogonal in [−1,1] and satisfy

In the following, we refer to the pixel at position*x*horizontally and position

*y*vertically as simply the (

*x*,

*y*) pixel. Rewriting Eq. (8) into a straightforward expression for the LCOS-SLM, we have

*X*and

*Y*are sampling numbers in the

*x*and

*y*directions, respectively, which may be the number of pixels in the LCOS-SLM, and

*f*(

*x*,

*y*) is the phase value at (

*x*,

*y*). Legendre coefficients

*a*are calculated by substituting the phase distribution obtained by interferometry measurements.

_{nm}A Legendre polynomial expansion to 10-th order and reconstruction of the phase map showed good agreement (Fig. 4). We compared the intensity profiles of the focused patterns obtained with the original phase map, the reconstructed correction phase map, and the reference mirror, as shown in Fig. 5. All of the results meet the Rayleigh criterion, indicating that the Legendre expansion is a valid approach for this analysis. Note that this mathematical process constrains the spatial frequency of the reconstructed phase distribution, corresponding to the selected order for the calculation. There is a trade-off between the computational cost and the accuracy, since the cost increases exponentially with the order. High phase accuracy will be necessary for the analysis described in the next section, and an expansion up to 10-th order may be a suitable compromise.

#### 3.2. Temperature dependency of LCOS-SLM optical flatness

Thermal changes in the LCOS-SLM cause physical distortions associated with heat accumulation inside the system due to the high-power laser source, ambient temperature changes, or heat generated by other electronics. Figure 6 demonstrates the focused beam pattern at different ambient temperatures. In both cases, self-compensation was used with a common phase pattern, measured at a reference temperature *T _{ref}* = 27 °C. Note that the phase retrieval method can only correct the phase at a particular temperature, and the temperature dependency is considerable.

To extract meaningful information from the experiment, the temperature dependency of the phase difference as the temperature deviated from *T _{ref}* was obtained by changing the temperature of the incubator,

*T*. Single-shot interference fringes were observed while varying

*T*from 20 °C to 50 °C.

Since the Legendre polynomials of 10-th order have 65 orthonormal coefficients, which is too many for practical analysis, we focused attention on low order Legendre coefficients in the temperature range used (Fig. 7). The error bars in the results correspond to the standard deviation of five independent analyses with a single device. Quadratic components of Legendre coefficients *a*_{2,0} and *a*_{0,2} in the *x* and *y* directions showed comparatively high dependency on the temperature. This is probably due to the effect of differences in thermal expansion coefficient in the multilayer structure of the LCOS-SLM (Fig. 8). The silicon backplane of the LCOS-SLM serves as a reflecting surface, and the optical flatness of this surface directly affects the accuracy of wavefront modulation. However, this backplane is directly connected to the glass substrates by spacers for encapsulating the liquid crystal. The temperature expansion coefficients of the silicon backplane and the glass substrates are typically 2.6*μ*m · m^{−1} · K^{−1} and 3.2*μ*m · m^{−1} · K^{−1}, respectively. Although the difference between these values is small from the viewpoint of materials science, this difference can cause a difference of several *λ* in the optical phase, making it impossible to achieve an ideal wavefront. This mismatch between the layers may have resulted in a large defocus-like aberration, deforming the focused beam patterns depending on the temperature.

Note that differences in the temperature dependency of the Legendre coefficients between individual devices are typically small, as indicated by a comparison of the coefficients of three different LCOS-SLM devices shown in Fig. 9. There is still deviations of ±0.2[*λ*] between the devices at several temperatures, and we assumed that the computational uncertainty arises from the definition of the calculation domain, i.e., a unit square, in the Legendre series expansion. This indicates that the origin of the temperature-dependent distortions is away from the center of the LCOS-SLM, and might be caused by manufacturing errors between the devices.

## 4. Self distortion correction of LCOS-SLM under temperature-varying conditions

The Legendre coefficients were well-fitted to quadratic functions, via the least squares method, showing saturating trends of the temperature dependency (Fig. 7). This indicates the possibility of performing self-compensation simply by modulating the quadratic components according to the ambient temperature of the LCOS-SLM. In this section, we propose a phase correction method for recovering the distortion of the LCOS-SLM by self-correction, while taking account of the ambient temperature. The proposed algorithm is divided into two steps, an initializing step and a temperature correction step, to reduce the computational cost of real-time compensation.

#### 4.1. The flow of the temperature compensation process

The initial step of this process is to separate the phase distribution into temperature dependent components and independent residual components. We denote the initial phase distribution of the LCOS-SLM as *ϕ _{ref}* (

*x*,

*y*;

*T*), obtained with the interferometer at temperature

_{ref}*T*, the quadratic components of the Legendre coefficients

_{ref}*a*

_{2,0}and

*a*

_{0,2}as

*ϕ*(

_{quad}*x*,

*y*;

*T*), and residuals as

_{ref}*ϕ*(

_{residual}*x*,

*y*). Expanding

*ϕ*(

_{ref}*x*,

*y*;

*T*) with Legendre polynomials of (

_{ref}*n*,

*m*) = (0, 2) and (

*n*,

*m*) = (2, 0) only yields

*ϕ*(

_{quad}*x*,

*y*;

*T*). The components

_{ref}*ϕ*(

_{quad}*x*,

*y*;

*T*) can be removed from

_{ref}*ϕ*(

_{ref}*x*,

*y*;

*T*) due to the orthonormal properties, giving

_{ref}*ϕ*(

_{residual}*x*,

*y*).

*ϕ*(

_{residual}*x*,

*y*) will be necessary for calculating the phase in the next step and should be stored in a buffer memory of the computer.

As described in the previous section, the temperature dependency of *a*_{2,0}(*T*) and *a*_{0,2}(*T*) are determined by fitted the quadratic equations (Fig. 7), given by

The coefficients of the quadratic equations, *A*, *B*, and *C* may be predictable from the thermal expansion coefficients of the LCOS-SLM structure, and they are given by the fitted experimental results in this paper. Note that these coefficients of the fitted polynomial approximation will vary according to the multilayer structure of the device, since these factors are dependent on the thermal expansion behavior. This may occur when other structures have to be considered, such as the properties of the dielectric mirror, the thicknesses of the layers, or the coupling strength of the spacers.

Legendre coefficients *a*_{2,0}(*T*) and *a*_{0,2}(*T*) are calculated by substituting the monitored temperature *T*, followed by reconstruction of the phase distribution *ϕ _{quad}*(

*x*,

*y*;

*T*). Since the the temperature-correction process is based on an orthonormal operation, the temperature-corrected

*ϕ*(

_{quad}*x*,

*y*;

*T*) and

*ϕ*(

_{residual}*x*,

*y*) can be summed based on the composite phase modulo concept, as follows:

*ϕ*(

_{SLM}*x*,

*y*;

*T*) is the desired temperature-corrected phase map.

#### 4.2. Experimental results and discussions

The phase reconstructed with the temperature correction method at 50 °C matched the phase distribution obtained by the interferometric measurement, as shown in Fig. 10. There are differences between the results, which arise from temperature-dependent components that are not considered in the correction, such as *a*_{1,2} and *a*_{3,0}. In the following, the root-mean-square (RMS) errors were compared to quantify the difference between the two phases. The RMS error expresses the wavefront variance as the mean value of the wavefront over the measured area, i.e., the effective area of the LCOS-SLM.

RMS errors are given by

*ϕ*(

_{int}*x*,

*y*;

*T*) denotes the phase obtained with the interferometer at ambient temperature

*T*.

Figure 11(a) shows the RMS results of the temperature dependence with the self-compensation method. The black and gray dotted lines in the figure indicate *λ*/10 and *λ*/14, respectively, which are the target wavefront variances for attaining the diffraction limit. The temperature dependence of the RMS with a self-compensation phase pattern prepared at the reference temperature *T _{ref}* = 27 °C increased at both lower and higher temperatures. Note that the RMS error of the conventional method without considering the temperature might be effective only in the range

*T*± 3 °C for precise wavefront modulation applications, which is sufficient for room-temperature operation.

_{ref}In contrast, it is interesting that the evolution of the RMS error by correcting only a single coefficient, *a*_{2,0}, can suppress the optical flatness to near *λ*/10. Obviously, such correction is insufficient when the ambient temperature departs from *T _{ref}*, indicating that calibration with

*a*

_{0,2}is necessary. Figure 12 shows the intensity profiles of the focused patterns by correcting

*a*

_{2,0}and

*a*

_{0,2}in different ambient temperature. All of the results matches very well, meeting the Rayleigh criterion, directly indicates the appropriateness of the proposed method in experimental uses. However, there are several temperatures where the optical flatness is <

*λ*/14 by calibrating with only two coefficients. As shown in Fig. 11(b), this condition is met by limiting the calculation area to 600 × 600, which corresponds to the actual operation area since the incident laser beam profile is typically Gaussian or circular top-hatted. There is still some residual error when compensating with multiple coefficients (Fig. 11), and these are considered to arise from phase measurement errors in the analysis of non-contour Fourier fringes when separating the Fourier spectra on the fringe carrier. In other words, the computation domain used for the inverse Fourier transform greatly affects the accuracy of the correction. It may be useful to adopt other interferometric analysis methods, such as the phase-shifting method [25], to achieve more accurate compensation of the ambient temperature.

## 5. Summary and conclusion

In summary, we have reported a simple self-compensation method for correcting the distortion of an LCOS-SLM taking account of the ambient temperature. The method is based on an orthonormal Legendre expansion of a phase map obtained with an interferometer. We achieved a wavefront variance of < *λ*/10 in the temperature range 20 °C to 50 °C. This will be effective in high-accuracy wavefront modulation applications. Additionally, the concept of this method is extensible to the thermal behavior of other optical devices, such as lenses and mirrors, which have the possibility of causing unexpected aberrations.

It is known that the decay in liquid crystal birefringence is related to temperature [26]. This characteristic directly affects the diffraction efficiency and phase modulation accuracy of the device. Additional analysis and correction of this effect will be performed in a future study, in combination with our approach described here.

## Acknowledgments

The authors are grateful to A. Hiruma (President), T. Hara (Director), N. Mukohzaka, Y. Ohtake, T. Ando, H. Tanaka and N. Fukuchi of Hamamatsu Photonics for their numerous encouragements throughout this work. We also thank M. Oyaidu, M. Matsunaga, S. Takimoto, N. Matsumoto and S. Matsuda for their helpful support in the experiments.

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