## Abstract

Multi-segment liquid crystal spatial light modulators have received much attention recently for use as high-precision wavefront control devices for use in astronomical and non-astronomical applications. They act much like piston only segmented deformable mirrors. In this paper we investigate the use of these devices in conjunction with a Shack-Hartmann wave-front sensor. Previous investigators have considered Zernike modal control algorithms. In this paper we consider a zonal algorithm in order to take advantage of high speed matrix multiply hardware which we have in hand.

©1997 Optical Society of America

## 1. Introduction

Multi-segment spatial light modulators (SLM) have recently received much attention for use in adaptive optics applications. [1,2,3] These devices have the advantage that they are compact, light weight, and less expensive than deformable mirrors. They can also be used in transmission, which is desirable for some applications. These attributes make them useful in many applications including those not associated with astronomy. At present the devices are somewhat slow and have a complicated temporal behavior [3]. It is envisioned that their speed will be increased considerably in the near future. The maximum phase delay of the device is related to its thickness, however there is a tradeoff between thickness and speed. The throw of the device described in this paper is limited to about one wave in the near infrared.

When used with polarized light, the device can be made to act like a multi-segment piston-only deformable mirror. In order to form a closed loop adaptive optics system, the device must be used in conjunction with a wavefront sensor. Although a number of different types of wave-front sensors exist, the Shack-Hartmann sensor has the advantage that it is simple, easy to build and can be operated at a high temporal bandwidth when only the centroids of the sub-aperture spots are used to determine the average wave-front tilt across the sub-aperture.

Because the SLM can only perform piston wave-front correction, it can introduce waffle modes into the wavefront that are not observable by the Shack-Hartmann sensor. In addition, the multiple segments may introduce high order diffraction patterns into the spots produced by the individual Shack-Hartmann sub-apertures. These effects limit the operable configurations of the SLM/wave-front sensor combination.

The purpose of this paper is to discuss experiments and analysis using the Meadowlark hexagonal 127 element liquid crystal array in conjunction with a hexagonal Shack-Hartmann wave-front sensor. These experiments were performed in an attempt to understand and quantify the effects associated with this combination.

## 2. Higher order diffraction modes and waffle

Because the SLM acts like a piston only adaptive optic we must be concerned with the diffraction effects associated with the interference of light passing through adjacent SLM segments. Figure 1 shows a computer simulated one-dimensional cross-section of a Shack-Hartmann spot produced when the phase delays of two adjacent segments are displaced with respect to each other. We can see from the plots that for small phase differences each Shack-Hartmann lenslet will produce a well defined spot. However as the relative displacement increases, the intensity of one of the side lobes increases. As the displacement reaches one half wave, the spot breaks in two. This effect cannot be detected by a simple centroid operation.

Although high order aberrations can be calculated in the Shack-Hartmann spots, most wave-front sensors only work with the spot centroids. As the side-lobes in the diffraction pattern become significant, the error in the estimation of the centroid location increases. Figure 2 shows the percent error in the centroid calculations as a function of SLM segment displacement. It is seen that as the displacement approaches one half wave, the error approaches one hundred percent due to the bi-modal spot. This error introduces an ambiguity into the wave-front measurements that can lead to waffle modes developing in the SLM segments when we operate it as a closed-loop adaptive optic. In order to mitigate this problem, the commands applied to the SLM segments must be constrained such that the relative inter-segment displacements are less than one-half wave.

## 3. Zonal control of the spatial light modulator

The use of the SLM in a closed loop adaptive optics configuration, has been demonstrated in several places using a zernike modal control algorithm in conjunction with a Shack-Hartmann wave-front sensor [1,2]. In the references sited between ten and twenty zernike modes were corrected using the SLM.

Large aperture adaptive optics designed to work in the visible typically encounter turbulence with D/r0 greater than ten. In order to obtain near full order correction for such systems, a controller employing several hundred zernike modes would be required. In such instances zonal control algorithms, using a multi-input multi-output (MIMO) control algorithm are often used [4] in order to take advantage of high speed matrix multiply hardware. In this paper we investigate the use of a zonal control algorithm with the SLM similar to one used with a continuous face hexagonal deformable mirror.

Figure 3 shows the arrangement of Shack-Hartmann sub-apertures with respect to SLM segments that we used in our study. Each Shack-Hartmann sub-aperture is centered at the vertex of three SLM segments. One particular set of segments is shown at the right side of the full array. As discussed in the previous section, the piston only segments do not provide a smooth tilt across the Shack-Hartmann sub-aperture as a continuous face sheet deformable mirror would.

The effect of the phase discontinuity is illustrated in Figure 4 which shows an image of several Shack-Hartmann spots with varying amounts of phase piston displacement on a single SLM segment. Figures 4a and 4b show that for small displacements of the segment, the three surrounding spots move outward as desired. Figure 4c shows that as the segment displacement approaches one-half wave, the three spots become bi-modal. The side-lobes from these three spots coalesce to form a spot at the center of the SLM segment location. This effect causes errors in the centroid calculation for each of the three Shack-Hartmann spots and is detrimental to control system performance. The effect limits the relative displacement that can be allowed between two adjacent SLM segments.

In the remainder of this section we describe a zonal algorithm to provide closed loop MIMO control of the SLM. In this approach a control matrix is derived, that when multiplied by the Shack Hartmann centroid error measurement vector, produces control commands to the SLM This differs from a modal control algorithm in that the wave front measurements are not projected onto Zernike modes. The matrix based control algorithm allows us to use matrix multiply hardware which we have in hand.

We assume that for small displacements of adjacent segments, we can treat the control problem in the same way as for a continuous mirror. Referring to the three segment basic cell shown in the right hand side of Figure 3, an approximate relationship between the piston in the SLM segments and the spot displacement seen by the Shack-Hartmann sensor can be given as follows.

We then follow the standard least squares MIMO control approach [4] and put equations (1) and (2) in matrix form.

where Φ is the Shack-Hartmann phase gradient measurement vector, and X is the SLM segment displacement vector.

The matrix Γ is given by

The MIMO control matrix, H, is then formed using the pseudo inverse operation.

The H matrix is incorporated in a summation control algorithm, which attempts to drive the wave-front centroid errors to zero. The commands to the SLM segments can be described by:

Where Φ_{0} is a set of nominal centroid offsets obtained from a wave-front sensor calibration. The term *C _{k}* represents SLM segment commands at time step k, and g
represents the loop gain.

The closed loop feedback control system is shown in the following block diagram.

## 4. Experimental results

In this section we describe results of experimentation with the SLM configured in the zonal feedback control loop discussed previously. Figure 6 shows the optical layout of the experiment.

We first show results with a static aberration. Approximately one wave peak-to-valley of defocus and astigmatism was applied to the wave-front using a plate aberrator. The open and closed loop focal plane spots are shown in Figure 7. The open loop Strehl ratio was approximately .16 while the closed loop was .25.

Next we show open and closed loop performance with a series of dynamic aberrations. The dynamic aberrations were produced by allowing air to circulate in front of the static aberrator plate. The series is contained in an animation sequence which may be downloaded.

The animation represents about one-half second of data at one hundred frames per second. It shows that the closed loop configuration was able to correct small fluctuations in the aberration with a band-width of several hertz. Further study is required to determine exactly the closed loop disturbance rejection band-width of the system.

## 5. Conclusions

The liquid crystal spatial light modulator is a compact low cost alternative to deformable mirrors for adaptive optics systems. When used with polarized light, it acts like a piston only segmented deformable mirror. High order aberrations, produced by the discontinuities at the segment boundaries, can have an adverse effect when used with a Hartmann sensor. Special care must be exercised to mitigate this effect.

Zernike modal controllers have been proposed for use with this mirror in a closed loop feedback adaptive optics configuration. Such controllers have been shown to work well when only a few zernike modes are required to provide sufficient correction. However when working with a visible adaptive optics system, with D/r0 values greater than 10, several hundred Zernike modes would be required to achieve near full order correction. The computational complexity associated with fitting such a large number of zernike modes makes a zonal control algorithm attractive. We have in hand high-speed matrix multiply hardware for implementing real-time zonal control algorithms. For this reason we have been investigating the application of zonal control to the SLM.

The experimental results show that under the constraints discussed, a zonal control algorithm, similar to one used with a continuous face sheet hexagonal deformable mirror, can be used to control the SLM with a closed loop band-width of several hertz. The experimentation used one Shack-Hartmann sub-aperture per SLM segment. The Shack-Hartmann sub-apertures were off-set from the SLM segments by half a row and half a column. This allowed control of all 127 degrees of freedom possible with the corrector. The disturbance wave-front had only about one wave peak-to-valley distortion. Thus the displacements between SLM segments required to correct the wave-front were small and we did not have problems with the Shack-Hartmann spots breaking up. In addition the SLM segments have a maximum throw of only a little more than one wave, so the system can’t correct for larger distortions. In future experiments we plan to investigate a double pass configuration to increase the throw of each segment.

## References and links

**1. **G.D. Love, “Wavefront correction and production of Zernike modes with a liquid crystal SLM”, Appl. Opt. , **36**, 1517–1524 (1997). [CrossRef] [PubMed]

**2. **J. Gourlay, G.D. Love, P.M. Birch, R.M. Sharples, and A. Purvis, “A real-time closed-loop liquid crystal adaptive optics system: first results”, Opt. Commun. **137**, 17–21 (1997). [CrossRef]

**3. **A. Kudryashov, J. Gonglewski, S. Browne, and R. Highland, “Liquid crystal phase modulator for adaptive optics. Temporal performance characterization”, Opt. Commun. **141**, 247–253 (1997). [CrossRef]

**4. **W. Wild, E. Kibblewhite, and R. Vuilleumier, “Sparse matrix wave-front estimators for adaptive-optics system for large ground-based telescopes,” Opt. Lett. **20** (9), 995–957 (1995). [CrossRef]