Adaptive Control in an Adaptive Optics Experiment

Adaptive Control in an Adaptive Optics Experiment with Simulated Turbulence-induced Optical Wavefronts Salman Monirabbasi, Steve Gibson Mechanical and Aerospace Engineering University of California, Los Angeles 90095-1597 ABSTRACT This paper presents results from an adaptive optics experiment in which an adaptive control loop augments a classical adaptive optics feedback loop. A membrane deformable mirror is used for wavefront correction, and a set of frequency-weighted modes based on the actuator geometry are used to define the control channels for the adaptive controller. In the adaptive optics experiment, the wavefront sensor in the control loop is a three step phase shifting self-referencing interferometer. The corrected laser beam is imaged by a diagnostic CCD camera. The effect of atmospheric turbulence is simulated in the experiment by a sequence of wavefronts that is generated by a WaveTrain adaptive optics model and added to the laser beam by a spatial light modulator. The experimental results show the improved closed-loop wavefront errors and diagnostic images produced by the adaptive control loop as compared to the classical adaptive optics loop. Keywords: Adaptive optics, adaptive control, turbulence

1. INTRODUCTION In adaptive optics (AO), deformable mirrors are used to correct wavefront errors in optical systems. The commands to the deformable mirror (DM) are generated by a control loop that feeds back an estimate of the wavefront error, usually obtained from a wavefront sensor.1–8 Classical adaptive optics loops are based on classical feedback control methods, so that the controllers are linear with fixed gains. In recent years, adaptive control algorithms based on adaptive estimation of optimal reconstructor matrices have been proposed9–14 to improve the performance of adaptive optics systems in applications with strong, time-varying atmospheric turbulence. Adaptive controllers have two properties that classical AO loops lack: the capability to compensate for loop latency by predicting wavefront error, and the capability to identify optimal gains in real time as the wavefront statistics change. This paper presents an experimental implementation of an adaptive control scheme based on a multichannel recursive least-squares (RLS) lattice filter.15 The adaptive controller has been used in simulations of high energy laser systems,16 and a quasi-adaptive experimental implementation was presented previously.17 To introduce wavefront errors representative of the effects of atmospheric turbulence, a spatial light modulator in the current experiment adds to the laser beam a wavefront sequence generated by the WaveTrain simulation package for high energy laser systems. The wavefront errors are corrected by an adaptive optics system using a deformable mirror as the control actuator and a self-referencing interferometer to measure the closed-loop wavefront errors. Section 2 of this paper describes the experiment setup and the main optical components. Section 3 describes the disturbance commands to the spatial light modulator, including some details of the WaveTrain simulation that produced the two wavefront sequences used; also, Section 3 describes the parameterization of the commands to the deformable mirror. Section 4 discusses the use of the self-referencing interferometer as the wavefront sensor and the two-dimensional phase unwrapping algorithm used to obtain the final wavefront measurements. Section 5 describes the estimation of the poke matrix mapping actuator commands to wavefront sensor measurements, and Section 6 presents the classical AO loop and the adaptive control loop. Then Section 7 presents experimental results that illustrate how the adaptive control loop improves the performance of the AO system and how the adaptive loop tracks a change in the nature of the wavefront statistics. Email: [email protected], [email protected] This work was supported by the U.S Office of Naval Research and the High Energy Laser Joint Technology Office under Grant N00014 07-1-1063. Advanced Wavefront Control: Methods, Devices, and Applications VII, edited by Richard A. Carreras, Troy A. Rhoadarmer, David C. Dayton, Proc. of SPIE Vol. 7466, 746608 · © 2009 SPIE · CCC code: 0277-786X/09/$18 · doi: 10.1117/12.826197

Proc. of SPIE Vol. 7466 746608-1

Figure 1. Photograph of the experiment.

2. DESCRIPTION OF THE EXPERIMENT The experimental setup consists of many optical components which are aligned carefully for accurate results. Figure 1 shows a photograph of the experiment, and Figure 2 shows a corresponding diagram. The diagram is not to scale. The main components of the experiment are the laser source, a spatial light modulator, a self-referencing interferometer, a deformable mirror and a scoring camera. Laser Source. The laser source is a Helium-Neon point laser (λ = 634nm) with a linearly polarized output beam. A gas laser is used to satisfy the required coherence length for interferometry. A linear polarizer (Polarizer 1 in Figure 2) after the laser source adjusts the beam intensity. Then a beam expander, which consists of a 10× objective lens and a 25μm pinhole, produces a diverging outgoing beam. Spatial Light Modulator. A Spatial Light Modulator (SLM) is used in this experiment to induce the simulated atmospheric-turbulence aberrations on the wavefront. A reflective Nematic Liquid Crystal (NLC) phaseonly SLM18, 19 from Boulder Nonlinear Systems, Inc. (BNS) is used. The diverging beam from the beam expander passes through Polarizer 2 in Figure 2, which aligns the beam’s polarization with the SLM’s active axis. The non-polarizing Beam Splitter 1 then directs the beam to the SLM. Self-referencing Interferometer. A three-step phase-shifting self-referencing Mach-Zehnder interferometer (SRI)1, 8 is used as the wavefront sensor. The components of the interferometer are shown in Figure 2. The first element of the inteferometer is the polarizing Beam Splitter 2, which produces a reference beam and a signal beam. The reference beam passes through Lens 1 for collimation and then the LCD phase shifter adds three separate phase shifts in succession during each sampling interval for the SRI. Lenses 3 and 4 expand the beam so that a smooth portion of the reference wavefront can be interfered with the signal beam at Beam Splitter 5. The signal beam travels from Beam Splitter 2 to Beam Splitter 3 to the deformable mirror (DM), then out through Beam Splitter 3 and the collimating Lens 2 and to Beam splitter 4, which sends part of the signal beam to the target camera (i.e., scoring camera) and part of the signal beam to Beam Splitter 5 where the signal and reference beams interfere. The resulting beam passes through Polarizer 4, which adjusts the intensities of the two beams to yield the best contrast for the fringe patterns. Then the wavefront sensor camera captures the fringe patterns on a 58 × 58 pixel array for processing as described in Section 4.


Figure 2. Schematic diagram of the experiment.

Deformable Mirror. The control actuator is the deformable mirror (DM), which is in the path of the signal beam as shown in Figure 2. This DM, a one-inch membrane mirror from Active Optical Systems (AOS), has 31 actuators. Target Camera. A CCD camera is used to measure the corrected signal beam’s intensity distribution on the camera’s focal plane, which represents the target in a directed energy application. The goal of the experiment is to maximize intensity on the center of the target. The non-polarizing Beam Splitter 4 in Figure 2 directs part of the corrected signal beam to the target camera. An intensity filter scales down the intensity so that it does not saturate the CCD camera, while maintaining the shape of the intensity profile.

3. COMMANDS TO SLM AND DM 3.1 Disturbance Commands The wavefront errors put on the spatial light modulator were generated with the BLAT01 model in WaveTrain, a program for high-fidelity simulation of wave propagation through atmospheric turbulence. The BLAT01 model contains all the elements of a typical high energy laser system, including an extended turbulence path, a complete adaptive optics system on an airborne platform and a moving target. However, to generate the wavefront sequences used for this paper, only the track loop was closed in the WaveTrain simulation to remove tilts from the wavefront sequences. Hence the wavefront errors had minimal first-order tilt. Table 3 lists the parameters for the simulation used to generate the wavefronts for the experiment described in this paper. As Table 3 indicates, wavefront sequences were generated for two turbulence models, the only difference being that the platform and target velocities for Turbulence 2 are double those for Turbulence 1. These


increased velocities increase the Greenwood frequency for Turbulence 2, thereby increasing the bandwidth of the wavefront errors that the control loops need to correct. Although the experiment runs at a slow and not very constant sample rate due to limitations of the hardware used, the spatial and temporal statistics of the open-loop and closed-loop wavefront sequences are determined by the 2500 Hz frame rate in the WaveTrain simulation. Hence the performance of the control loops should be the same as if the experiment were running at the constant 2500 Hz frame rate. Table 1. Parameters of the Wavetrain simulation.

Parameter

Turbulence 1

Turbulence 2

(0,50,0) (50,0,0)

(0,100,0) (100,0,0)

Platform velocity Target velocity Parameter

Turbulence 1 and 2

Wind velocity HEL path length HEL platform altitude Target altitude Turbulence strength Number of phase screens Control and sensor frame rate HEL wavelength Beacon wavelength

(0,0,0) 20000 m 2755 m 1231 m Clear I 20 2500 Hz 1315 nm 1315 nm

3.2 Control Commands A command voltage vi to the ith actuator generates an electric field that produces a DM displacement (i.e., phase shift) di at the actuator, with displacement and command related by di = avi2

(1)

where a is a constant. The electronic driver for the deformable mirror provides eight-bit resolution for the DM commands, which take integer values between 0 and 255. For the control loops, the command voltages are parameterized as vb = 180 2552 /2 ci + vb2 , (2) vi = round where ci is the mathematical control command and vb is an applied bias voltage. The commands ci can take values in the interval −vb2 ≤ ci ≤ vb2 . The motivation for (2) is that, if the quantization error due to the rounding is neglected, then (1) and (2) yield (3) di = a(ci + vb2 ), which is an affine relationship between control commands and DM displacements. The bias voltage vb , which is the same for all actuators, produces a focus shape on the DM.

4. WAVEFRONT SENSING AND PHASE UNWRAPPING 4.1 Wavefront Sensing One common class of wavefront sensors is self-referencing interferometers.1, 20–22 The experiment described in this paper uses an amplitude splitting Mach-Zehnder self-referencing interferometer.1, 8 In self referencinginterferometers (SRI) the beam is split into two beams that travel different paths and are later interfered with each other to create the fringe patterns sent to the wavefront sensor. In an amplitude splitting Mach-Zehnder interferometer, both beams carry all the phase information. In order to use a Mach-Zehnder interferometer as


a real time wavefront sensor, the reference beam transmits through a phase shifter which creates three known phase shifts during each measurement and a CCD camera should capture the fringe patterns in real time. To reconstruct the phase, it is assumed that the portion of the reference beam (which is expanded by Lenses 3 and 4 in Figure 2) used to interfere with the signal beam can be approximated as a plane wave. Under this assumption, it can be shown1 that the phase of the beam can be given by θ − φ0 = arctan{

2I(π/2) − I(0) − I(π) }. I(0) − I(π)

(4)

In this equation θ is the wavefront phase, φ0 is a constant phase due to the optical path difference of the reference and signal beam, and I is the intensity of the fringe patterns captured by the wavefront sensor CCD. As was mentioned before a three-step phase-shifting interferometry is used so at each step a constant phase shift is added to the reference beam while an intensity image of the fringe patterns is stored. The phase shifts used in this experiment are 0, π/2 and π. For instance, I(π) is the fringe pattern captured when the phase shift on the reference beam is π. Equation (4) can be solved for all points in the interference aperture and as a result the phase of the signal beam can be calculated. The main issue with solving (4) is that the solution will have a range of (−π, π) and so the result is a wrapped wavefront which must be unwrapped before it can be used in the control loops. This is done using two-dimensional phase unwrapping.

4.2 Phase Unwrapping In recent years there has been an increasing demand for accurate and robust two dimensional phase unwrapping algorithms. Some common applications requiring phase unwrapping are synthetic aperture radar (SAR), magnetic resonance imaging (MRI), acoustic imaging and adaptive optics among other applications.23 A number of methods have been developed for fast and accurate phase unwrapping.24, 25 In this experiment the Goldstein Branch Cut Algorithm is used.23 This algorithm involves three simple steps. In the first step, the wrapped phase profile is searched for residues and their coordinates are stored. In the second step, these residues are connected with branch cuts to create neutral lines or clusters of residues. Finally, a line integral is calculated over the gradient profile of the wrapped phase, avoiding the branch cuts until the whole array is unwrapped. Detailed discussion of this algorithm can be found in [23, Chapter 4].

5. LEAST-SQUARES IDENTIFICATION OF THE POKE MATRIX The control loops are based on the following model relating the control commands to measured wavefront: y − yb = Γ c

(5)

where y is the wavefront vector (i.e., phase), yb is the output bias, c is the vector containing the actuator commands ci , and Γ is the poke matrix, which has dimensions 582 × 31. The bias yb , which is predominantly focus, is the net phase due to the actuator command bias vb , the bias on the SLM, and any other biases from lenses, slight misalignments, etc. The output bias yb was measured with only the bias voltage vb applied to the actuators. All of the optical components were aligned to minimize yb . Then, to identify the poke matrix, a random command sequence with 220 frames (i.e., command vectors c) was sent to the DM, with no control loop closed, to generate an output sequence of wavefronts. Then Γ was chosen to minimize the norm of the fit-to-data error vector eΓ = y − yb − Γ c.

(6)

This identification reduces to 582 independent least-squares problems for the individual rows of Γ . Figure 3 shows the relative error norm eΓ (t)/y(t) where t is the frame number. The mean value of this ratio is 0.04.


Fit-to-data Error for Identification of Γ

0.1 0.09 0.08

|| eΓ (t) || / || y(t) ||

0.07 0.06 0.05 0.04 0.03 0.02 0.01

0

50

100 150 t = frame number

200

250

Figure 3. Relative fit-to-data error for 1 ≤ t ≤ 220.

6. ADAPTIVE OPTICS AND ADAPTIVE CONTROL 6.1 The Adaptive Optics Problem The block diagram in Figure 4 shows the signals and control loops for the adaptive optics problem. The vector c contains the commands to the deformable mirror, the vector y represents the measured wavefront (after phase unwrapping), and w represents the sequence of wavefront errors added to the beam by the spatial light modulator. The linear model of the relationship between input and output signals is y = yb + w − Γ c .

(7)

The classical adaptive optics (AO) loop shown in Figure 4 consists of a low-pass digital filter with gain K = 0.01 and pole α = 0.985, the modal transformation matrix V and the least-squares reconstructor E0 . (When α = 1, the low-pass filter is an integrator.) The values of K and α were chosen to maximize the error-rejection bandwidth of the classical AO loop without amplifying high-frequency disturbance and noise. As indicated by the z −1 block in Figure 4, the AO system has a one-frame loop latency. Since tilt was removed from the wavefront sequences generated by WaveTrain for the experiments discussed in this paper, there is no need for a track loop. The columns of V represent a set of frequency-weighted deformable mirror modes that are orthogonal with respect to the actuator geometry.16, 17 All of the modes have zero tilt. Both the classical AO loop and the adaptive control loop use the DM modes as control channels. The reconstructor matrix E0 is the pseudo inverse of Γ V . Since the poke matrix Γ has linearly independent columns, E0 Γ V = I,

(8)

which implies that the modal control channels are uncoupled in the linear model of the system when only the classical AO loop is closed.

6.2 Adaptive Control The adaptive control loop is shown inside the dashed box in Figure 4. As shown in this block diagram, the adaptive controller is a feedback controller but it’s structure is based on a feedforward disturbance-rejection design with the adaptive feedforward filter L(z). The tuning signal e and input signal r for the adaptive controller are given, respectively, by e = E0 y ,

r = G(z)u − e ,


(9)

yb vV

SLM

w − d? - d?cDM SRI

y - −1 z

Classical AO Loop z Kz − α d 6 u Adaptive Control Loop

E0

- G(z) Z } Z L(z)

− e r ? d Z Z

Figure 4. Block diagram of the digital control loops for adaptive optics.

where G(z) is a model of the transfer function from the adaptive control signal u to the tuning signal e with only the classical AO loop closed. Since the modal control channels are uncoupled for G(z), this multi-inputmulti-output (MIMO) transfer function is diagonal with each diagonal term given by G(z) =

−K . z+K −α

(10)

The adaptive controller in the experiments reported here uses an fourth-order finite-impulse-response (FIR) lattice filter L(z) with a recursive-least-squares (RLS) adaptation algorithm.15 This adaptive filter derives numerical stability and efficiency from an orthogonalization of the data channels. Theoretically, an IIR filter should produce optimal steady performance for stationary turbulence statistics; however, in practice an FIR filter performs more robustly, and FIR orders higher than four have not yielded significant improvements in performance. The adaptive filter L(z) identifies gains to minimize the mean-square value of the tuning signal e over space and time. The vector signal e contains the projections of the measured closed-loop wavefront error onto the DM modes used in the control loops. The adaptive control loop couples the modal control channels, but the fact that channels are uncouple for the transfer function G(z) has the following important consequence for the adaptive controller: the problem of computing the optimal gains for the MIMO filter L(z) reduces to a set of independent RLS problems for the rows of L(z).

7. EXPERIMENT RESULTS The main purpose of the experiments reported here is to investigate the performance of the adaptive control loop. However, an initial step was to determine the number of frequency-weighted DM modes to be used in both the adaptive control loop and the classical AO loop. A series of experiments showed that using 15 DM modes in the control loops gave the best performance in most cases. Table 2 shows the average values for wavefront error reduction and target intensity improvement for different numbers of modes when the classical AO loop (the top loop in Figure 4) was closed. For Table 2, the Turbulence 1 wavefront sequence described in Section 3 was used. That the performance decreases with more than 15 modes likely indicates that there is significant measurement noise and/or reconstruction error associated with the modes that have higher spatial frequencies. For all subsequent results, 15 DM modes were used in both the classical AO loop and the adaptive control loop.


Table 2. Performance of the classical AO loop using different number of modes of the deformable mirror.

Number of DM modes

Wavefront error RMS reduction (%)

Target intensity improvement (%)

1 2 5 10 15 20 31

24.75 31.43 42.84 44.26 46.13 44.78 45.07

22.79 30.01 41.92 43.75 44.89 38.28 40.68

Figures 5–8 and Tables 4–5 show results from three experiments. One experiment was open-loop (i.e., neither control loop was closed). In the second experiment, only the classical AO loop was closed. In the third experiment, the adaptive control loop was closed after 1000 learning steps (frames); only the classical AO loop was closed during the first 1000 time steps, while the adaptive filter estimated optimal gains for the adaptive controller. The results for the second and third experiments are almost identical for the first 1000 steps, but not quite because ambient light and air vary slightly from one experiment to the next. In each of the three experiments, the WaveTrain-generated wavefront sequences were added to the laser beam by the SLM. The Turbulence 1 sequence was used during the first 2500 time steps, and the Turbulence 2 sequence was used during the last 1500 time steps to investigate the capability of the adaptive loop to adapt to changing turbulence. The magnitudes of the phases in the Turbulence 2 sequences were increased by 50% from the phases generated by WaveTrain to increase the challenge to the control loops. Figure 5 shows the RMS wavefront errors in units of wavelength for the three experiments, and Figure 6 shows the average intensity for the five pixels in a diamond pattern at the center of the target camera’s focal plane. The results of the open-loop experiment, the experiment with classical AO loop only, and with the adaptive control loop are plotted in red, green and blue, respectively. When the Turbulence model suddenly changes after time step 2500, all three intensity plots in Figure 6 decrease, but the curve produced by the adaptive controller continues to be higher than that produced by the classical AO loop alone. Table 4 shows the RMS values over space and time of wavefront error calculated at steady state for each of the three experiments. The steady state segments are chosen as time steps 1100 through 2400 for Turbulence 1 and time steps 2600 through 3900 for Turbulence 2. The reduced wavefront error produced by the adaptive control loop translates to an improved intensity distribution on the target camera, as indicated by the intensity plots in Figure 6. Table 3 shows steady-state RMS values (over time, in CCD counts) of the intensities at the center of the target camera focal plane. Figure 7 shows the average intensity profiles on the target for the first turbulence sequence (open-loop on the left, classical AO loop in the center and the adaptive loop on the right). Figure 8 shows the corresponding plots for the second turbulence sequence. Table 5 summarizes the peak values from these two figures. These improvements are important because they indicate that the adaptive loop increases the mean and total energy at the center of the target.

8. CONCLUSIONS The paper has presented an experimental implementation of adaptive control in adaptive optics. By predicting the wavefront error, the multichannel adaptive control loop, which augments a classical AO loop, reduces wavefront error and sharpens the intensity distribution on a target camera. The experimental results show that the adaptive loop also tracks changing wavefront characteristics.


Target Intensity (Center Diamond)

RMS Wavefront Error

45

0.35 Open Loop Classical AO Loop Adaptive Loop

40

0.3

CCD Camera Count

RMS Wavefront Error / λ

35 0.25

0.2

30

25

20 0.15

Adaptive Loop Classical AO Loop Open Loop

15

0.1

0

500

1000

1500

2000 2500 Time Steps

3000

3500

4000

10

Figure 5. RMS wavefront error.

Open Loop

Classical AO Loop 80

80

70

70

70

60

60

50

50

40

40

40

30

30

30

20

20

20

10

10

10

0

0 0

20

40

40 20

1500

Open Loop

75

0

20

40

3000

Classical AO Loop

3500

4000

0

20

40

Figure 7. Target average intensity profile for Turbulence 1.

Adaptive Loop

70

70

60

60

60

50

50

40

40

40

30

30

20

20

20

10

10

10

23

0

40 20

2000 2500 Time Steps

70

30

25

40 20

1000

60

51

50

0

500

Figure 6. Target center diamond intensity.

Adaptive Loop

80

0

46

50

0

0 40 20

0

20

40

58

40 20

0

20

40

40 20

0

20

40

Figure 8. Target average intensity profile for Turbulence 2.


Table 3. RMS values of the target intensity for each of the turbulence sequences for steady state. The steady state part for Turbulence 1 is selected to be between time steps 1100 and 2400 and for Turbulence 2 to be between time steps 2600 and 3900.

Experiment Type Open Loop Classical AO Loop Adaptive Loop

Turbulence 1

Turbulence 2

Center Pixel

Center Diamond

Center Pixel

Center Dimaond

25.7 51.4 75.6

19.0 27.8 35.8

23.0 44.7 56.1

17.7 25.2 31.7

Table 4. RMS values of the RMS wavefront error for each of the turbulence sequences for steady state. The steady state part for Turbulence 1 is selected to be between time steps 1100 and 2400 and for Turbulence 2 to be between time steps 2600 and 3900.

Experiment Type

Turbulence 1

Turbulence 2

Open Loop Classical AO Loop Adaptive Loop

0.254 0.192 0.142

0.274 0.234 0.182

Table 5. Peak values of the target mean intensity profiles for each of the turbulence sequences for steady state. The steady state part for Turbulence 1 is selected to be between time steps 1100 and 2400 and for Turbulence 2 to be between time steps 2600 and 3900.

Experiment Type

Turbulence 1

Turbulence 2

Open Loop Classical AO Loop Adaptive Loop

25 51 75

23 46 58

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