Lab Report 6: Adaptive Optics

LIN PEI-YING, BAIG JOVERIA

Abstract:

The objective of this experiment was to understand the principle of adaptive optics whereby,

the performance of optical systems is improved by compensating the effect of rapid distortions.

The wave front is analyzed and compensated with the adjustments of a deformable mirror as

described later in the sections.

Introduction:

The principle of adaptive optics is extensively used in astronomical telescopes and laser

communication systems. . It is used to

remove the effects of atmospheric

distortion, in microscopy, optical

fabrication and in retinal imaging

systems to reduce optical aberrations.

Adaptive optics works by measuring the

distortions in a wavefront and

compensating for them with a device

that corrects those errors such as

a deformable mirror or a liquid

crystal array. A typical deformable mirror used for such corrections is shown in Figure 1.

Theory:

When the light enters the earth's atmosphere, it is affected by turbulence which leads to

distortion. Hence, in order to image the light efficiently with a high resolution, it becomes

important to compensate for these distortions in real time.

Fried Parameter:

The Fried parameter gives an estimate of the optical transmission through the atmosphere in

the presence of inhomogenieties in the refractive index. These inhomogeneities can be due to

variation in temperature or density. It is defined as the diameter of a circular area over which

the root mean square wavefront aberration is equal to 1 radians in the atmosphere and is given

in units of length.

Figure 1: Deformable mirror used for correction

The setup used for adaptive optics is shown in Figure 2:

Figure 2: Experimental setup used for adaptive optics

The setup comprises of a HASO-32 used for wavefront sensing which comprises of a lenslet

array and a CCD matrix. It also comprises of a deformable mirror MIRAO. The working of each

of them are described below.

Shack-hartmann wavefront sensor:

Shack hartmann sensor is an optical component used to analyze and characterize the wavefront.

It comprises of a lens let array each of them with the same focal length. Depending on the

displacement of the image from the center of each pixel in a CCD, the tilt of the wave-front is

calculated. Hence the entire wave-front can be approximated in this way. The working of a

shack hartmann sensor is shown in Figure 3.

Figure 3: Shack hartmann sensor principle

Deformable Mirror:

The deformable mirror is a magnetic mirror with small magnets attached to the back of the mirror that

are pushed or pulled by the solenoids. By modifying the current on each actuator, the magnetic strength

of the solenoids are varied and there by producing a deformation in the mirror that can be used to

compensate for the distortions. The principle of a deformable magnetic mirror is shown in Figure 4.

Figure 4: Principle of a deformable magnetic mirror

Mathematical Background:

In order to compute the compensation required for the turbulence, it becomes important to analyze

mathematically the matrices that relate the displacement of the mirror to the effect on the image

obtained

Calculation of Interaction Matrix and Command Matrices:

The interaction matrix is the matrix that relates how the image changes by changing the current

provided on each actuator of the magnetic mirror. When the magnetic mirror is provided with variable

current on a solenoid, the effect on the image is recorded. This is repeated for all the actuators. Hence, a

matrix is obtained with the displacement on the lenslet array measured for each of the N lens lets for

the K actuators. Hence, this becomes an N x K matrix.

𝑁1:

𝑁𝑛=

𝐼11 ⋯ 𝐼𝑘1

⋮ ⋱ ⋮𝐼1𝑛 ⋯ 𝐼𝑘𝑛

𝐽1:𝐽𝑘

Command Matrix

The problem posed in this experiment is the inverse problem of the one described above. In order to

compensate for the distortion, it is important to calculate that in order to produce a given change in the

image, how much current should be supplied to each actuator. Hence, in order to analyze this problem,

single value decomposition is used. The interaction matrix can be decomposed in the following manner:

IM = U·W·VT

Where :

U orthogonal real matrix n x k,

W diagonal real matrix k x k,

V orthogonal real matrix k x k (The T symbol indicates the transposition).

n= number of lenslets

k= number of actuators

Hence, the command matrix which is the pseudo-inverse of the interaction matrix is given by:

P = V·W-1·UT

where P is a n x k matrix

Experiment

Study and setting afocal system

It was verified that the beam is collimated by the first lens, and then the HASO (Sensor) measurement of

a diverging wavefront. By slightly rotating the deformable mirror, it is seen that the image spots move,

but the signal on the CCD did not change, because it adds a lens L3 to combine the deformable mirror

and the wavefront analyzer.

Figure 1: Schematic of afocal assembly.

Study of the deformable mirror

The CASAO software starts in parallel the HASO software which allows to make standard calculations on

the wavefront measurement: Zernike polynomials decomposition, PSF and MTF calculation.

Figure 2: residual wavefront as a reference

The failure of the residual wavefront deformation of the deformable mirror happens when all actuators

are set to zero. This is not a problem since the adaptive optics is a servo system. We checked the

linearity of the default of the wavefront can be observed that the RMS deformation changes in

proportion to the variation of the value the actuator. For example, the actuator 30, a = 0.102 = 0.558

RMS. A = 0.2 RMS = 1.1, A = -0.1 RMS = 0.635 A = -0.2, RMS = 1.172.

One can measure the limitation due to computation software. When the deformation exceeds this

limitation, the software cannot calculate the image. This distortion becomes a hole. Also, we can verify

that the response in terms of linear front is a combination linear control actuators. The mirror does not

present hysteresis. Defects linearity and hysteresis are not troublesome for use in a closed loop since

the closed loop can correct defects according to feedback. On the other hand, they are inconvenient for

the open loop.

Construction of the interaction matrix

This part concerns the construction of the interaction matrix. To determine the voltage applying to the

mirror in order to compensate the displacement of the image spots of the micro-lenses due to a

disturbance of the wavefront, the procedure in reverse: it applies known voltage on each actuator, and

then it is calculated and stored in the order matrix interaction. CASAO, the software can automatically

perform this procedure.

Figure 3: the experimental results after driving the actuators individually. It was a relationship of

displacement related to the matrix interaction

Calculate the control matrix

The enslavement of the mirror involves solving the inverse problem: we measure vector displacement, caused by a wavefront disturbed; one must then compute the voltages to be applied to the deformable mirror to compensate for the best travel.

Because a beam with spherical wavefront is a divergent beam incident on the CCD, and if there is no

source of turbulence, the beam will be perfectly spherical. To use a null set point, it is proposed to add

another lens after L3 to turn the spherical wavefront beam into a collimated beam. The defects are the

spots on CCD image may move easily in a tilt of the deformable mirror.

We can observe the image spots of HASO camera. By calculating the positions of the image spots, we

can get a displacement vector. After multiplying it by the command matrix and the chosen gain, we can

also get a new calculated control voltage which can be applied on deformable mirrors in order to correct

the wavefront. We can see the difference if we apply feedback loop on our system in the following

Figure 4:

Before correction After correction

The Strehl ratio is 0.953. The PSF we got is like following Figure 5

The shape of the MTF is like following Figure 6:

The effect of a coma: If we increase the coma value, we can observe the distortion increases, too.

coma angle 45 (value 0)

coma angle 45 (value 0.119)

coma angle 45 (value 0.214)

Image spot

Wavefront

The effect of a defocus: If we defocus becomes stronger, we can observe the larger distortion.

Defocus (value 0)

Defocus (value 0.206)

Defocus (value 0.317)

Image spot

Wavefront

The effect of a tilt: If we tilt stronger, the image spot and wavefront don’t change much, instead the

position changes.

tilt (value 0)

tilt (value 0.452)

tilt (value 0.992)

Image spot

Wavefront

Because in the real time, the main problem is that it takes time in the feedback loop to correct the

wavefront by the deformable mirrors. Therefore, even the motor is running very slowly, the correction

will never be perfect.

After we use a microscope slide with a thin layer of silver, we can observe a realistic starry night sky is

created. The image is blurry and fluctuated with the disturbance. Then we start the controller and the

image improvement, i.e. the image is sharper. The bright spot is the rotation center. The quality of

correction of a perturbation on an extended object closed to the center is better.

Conclusion

Adaptive optics is a technique which corrects deformation in real time scale and non-predictive of a

wavefront with a deformable mirror. This lab work has to understand the main adaptive optical systems

and try to understand the method of construction of the interaction matrix to find the defect of the

wavefront caused by turbulence and offset by the deformable mirror. Adaptive optics gives excellent

results in astronomy, this technique is also imposed in the field of high-power lasers to correct the

wavefront of the beam and compensate the optical distortion, mirrors and crystals subjected to high

thermal stresses. Adaptive optics has developed in the field of ophthalmic optics, especially to improve

the resolution of images of the retina, as what we saw in the Laboratoire Aimé Cotton. New applications

are being developed especially for the microscopy and in the field of x-ray. It is likely that this technique

will continue to progress in all areas of the optical instrumental as the cost of spatial phase modulators

and wavefront analyzer decrease.