Contact:

Ulrike Blumröder, e-mail: ulrike.blumroeder@uni-jena.de

Tobias Ullsperger, e-mail: tobias.ullsperger@uni-jena.de

Felix Zimmermann, e-mail: felix.zimmermann@uni-jena.de

Last edition: Ulrike Blumröder, March 2013

Adaptive Optics

Experimental Optics

Contents1 Overview 3

2 Safety issues 4

3 Theoretical Background 53.1 Shack-Hartmann Wavefront Sensor . . . . . . . . . . . . . . . . . . . . . . 53.2 Aberration Theory using Zernike Polynomials . . . . . . . . . . . . . . . . 53.3 Deformable Mirror . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.4 Closed-Loop Adaptive Optic Systems . . . . . . . . . . . . . . . . . . . . 8

4 Setup and equipment 94.1 Adjusting the SHWS and measuring the wavefront . . . . . . . . . . . . . 94.2 Evaluating the wavefront . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.3 Adjusting an optical system using the SHWS . . . . . . . . . . . . . . . . 9

5 Goals of the experimental work 115.1 Alignment of the Shack-Hartmann wavefront sensor . . . . . . . . . . . . . 115.2 Wavefront analysis with a Shack-Hartmann wavefront sensor . . . . . . . . 115.3 Aberration compensation in a microscope using a deformable mirror . . . . 125.4 Optimization of a telescope using wavefront analysis . . . . . . . . . . . . 12

A Prelimary Questions 14

B Final Questions 15

C Amplitude of the "defocus" term 16

D Initializing the software and setup 17

E Reading advice 19

Adaptive optics

1 OverviewAdaptive optics is widely used in optical systems like telescopes and microscopes for dy-namic correction of wavefront distortions [1, 2]. Currently adaptive optical components alsobecome important for high power lasers, beam and pulse shaping as well as compensationof distortion caused by nonlinear processes. The first part of this exercise is to learn howto characterize a beam by using a Shack-Hartmann Wavefront Sensor (SHWS) and how toexpress the deviations of an ideal beam in terms of Zernike polynomial. In the second part,a Micro-machined Membrane Deformable Mirror (MMDM) is used to correct for variousaberrations in a microscope, thus increasing image quality and resolution [1]. A typicalsetup can be seen in Fig. 1, which consists of a deformable membrane mirror (a) and aShack-Hartmann-wavefront sensor (b). The microscope unit (c) is vertically mounted.

a)

b)

c)

Figure 1: The adaptive optics setup used for aberration control in a microscope consisting of a de-formable membrane mirror (a) and a Shack-Hartmann-wavefront sensor (b). The microscope unit(c) is vertically mounted.

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Adaptive optics

2 Safety issuesEye hazardThe laser used emits light at 633 nm with a power of less than 1 mW. However, the manu-facturer classified it according to DIN IEC 60825-1 is classified as an 3R laser.

• Never look directly at the beam or its reflections or point it towards other people.

• Please, wear your laser safety goggles all the time.

• Remove all reflecting objects attached to your hands / wrist (e.g. rings, watches etc.)

• Never insert or remove an optical element from the rail unless the laser beam isblocked either mechanically or by shutting down the power supply.

• Never tilt elements of the setup or alignment discs such that the light reflected fromits surface may be directed towards you or your classmates.

Electrical hazardThe operation of the MMDM requires the use of a high voltage (HV) source. Do not switchon the HV supply if the output is not connected to MMDM-controller. Do not disconnectthe serial cable while the Power supply is in operation. Wait at least a minute after the powersupply is off before removing any cables.

Chemical hazardAcetone and its vapors are toxic. Use the minimal required quantity of acetone while clean-ing the optical elements. Do not sniff the vapors of the acetone for prolonged periods.Avoid contact with skin or eyes. If accidental contact happens, wash the affected area withabundant cold water. Do not hesitate to ask for assistance if pain persists.

General advice• Never touch the reflective surface of the deformable mirror! Never try to clean

it. Doing so will destroy the most valuable part of the setup.

• Try not to touch the surfaces of other mirrors, microscope objectives, lenses andbeam-splitters. Otherwise it is cleaning time!

• Do not use force when mounting or dismounting elements of the setup. Usually prob-lems can be avoided by unscrewing the element first.

• Think before altering the setup. Usually a new alignment of the setup takes verylong time. Therefore, better keep your hands off the microscope unit and the laseralignment mirrors.

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Adaptive optics

3 Theoretical BackgroundThe two key components of an adaptive optics setup include a device to measure the shape ofan incident wavefront and a dynamic method or correcting its aberrations. Most commonly,a Shack- Hartmann Wavefront Sensor and a deformable mirror are used [1]

3.1 Shack-Hartmann Wavefront SensorA Shack-Hartmann Wavefront Sensor (SHWS) comprises a microlens (lenslet) array and aCCD positioned at the focal length F of the microlenses. An ideal plane wavefront incidenton the microlens array will create a spot pattern on the CCD that matches the arrangementof the microlens array ideal grid. In the case of an aberrated wave, as shown in Fig. 2,individual spots in the pattern on the CCD array will be displaced due to a local tilt ofthe wavefront at the location of the corresponding microlens. By measuring the relativedisplacement ∆x of each of the points in the spot pattern from the ideal grid, the angleof the wavefront incident at each microlens can be determined to be ϑ = tan−1(∆x/F).Similarly, the phase difference Θ ≈ k(∆x)2/(2F) can be obtained, where k = 2π/λ is thewavenumber of the incident light of wavelength λ. Based on a grid of these values, theincident wavefront can be reconstructed using appropriate interpolation algorithms [3, 4].This wavefront information can then be expressed in terms of Zernike Polynomials, wherethe corresponding coefficients express the degree of aberration[5, 6].

4

(2)

(3)

Here, is defined from 0 to 1 , n is the radial order of the polynomial, and m is the azimuthal frequency, and is 1 for m

0.2 The first several Zernike Polynomials derived from these equations, as well as their corresponding aberration, can be seen in Table 1.

Any arbitrary wavefront can be decomposed into a weighted sum of individual Zernike Polynomials

where is the coefficient for the Zernike Polynomial . This decomposition method is utilized by adaptive optics to measure individual aberrations in an arbitrary wavefront. 3. Adaptive Optic E lements

The two key components of an adaptive optics setup include something measure the shape of an incident wavefront and a method or correcting its aberrations. In this case, a Shack-Hartmann Wavefront Sensor and a deformable mirror were used.

3.1 Shack-Hartmann Wavefront Sensor

A Shack-Hartmann Wavefront Sensor (SHWS) is made up of a microlens (lenslet) array positioned the focal length of the microlenses away from a CCD. An ideal plane wavefront incident on the microlens array will create a spot pattern on the CCD that matches the arrangement of the microlens array ideal grid. In the case of an aberrated wave, as shown in Figure 1, individual spots in the pattern on the CCD array will be displaced due to local tilt in the wavefront at the location of the corresponding microlens.

Table 1 F irst several Zernike Polynomials and corresponding aber rations 3

F igure 1 Operation of a Shack-Hartmann Wavefront Sensor4

Figure 2: Schematic of a Shack-Hartmann wavefront sensor (SHWS)

3.2 Aberration Theory using Zernike PolynomialsAberrations appear in any optical system because of deviations from the ideal paraxial ap-proximation, where the angle of an incident ray cannot be considered "small". Other sourcesinclude imperfections on optical surfaces, or distortion in the optical path such as atmo-spheric turbulence. The result is that rays incident on an optical element are not focused to

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Adaptive optics

the same location, causing an increase in the focal spot size and a decrease in image quality.Historically, aberrations have been described as a power series, where each term in the se-ries represents a particular type of aberration. Low order terms in the series are commonlyreferred to as Seidel aberrations, including spherical aberration, coma, and astigmatism.However, there are disadvantages in using a power series description of aberrations; they donot form a complete set and are not orthogonal over the unit circle. In order to overcomethose limitations Zernike polynomials were introduced [5, 6]. Usually Zernike polynomialsare expressed in radial coordinates (r, θ) of the unit circle, where θ is the azimuthal angle andr is the radius, expressed in units of the aperture radius 1. Because they form a complete,orthogonal basis, any wavefront W(r, θ) can be expressed as a weighted sum

W(r, θ) =

k∑n

n∑m=−n

Wmn Zm

n (1)

of the Zernike polynomials

Zmn (r, θ) =

{m > 0 : R−m

n (r) sin(−mθ)m ≥ 0 : Rm

n (r) cos(mθ)

}. (2)

The radial part of the polynomial Rmn (r) can be calculated with

Rmn (r) =

√2(n + 1)1 + δm0

n−m2∑

s=0

(−1)s(n − s)!

s!(

n+m2 − s

)!(

n−m2 − s

)!rn−2s, (3)

where δm0 is 1 for m = 0 and zero for any other m. Like the terms of the power seriesexpansion, each Zernike polynomial corresponds to a specific type of aberration. The firstseveral Zernike polynomials, as well as their corresponding aberration, are listed in Fig. 3.After a wavefront is decomposed into Zernike terms (Eq. 1), the amplitudes Wm

n provides aquantitative measure of the degree of aberration.

3.3 Deformable MirrorIf the shape of the aberrated wavefront is known, a deformable mirror can be adjusted toreshape it into an ideal plane wave [1, 2]. The used deformable mirror consisted of anoptically flat membrane suspended above a grid of many electronic actuators. By applyinga voltage to an individual actuator, the membrane will become electrically attracted to itresulting in a local deformation. By varying the voltages of all the actuators, the membranecan be reshaped to correct for arbitrary aberrations, as it is seen in Fig. 4.

1Therefore ranging from zero at the center and 1 at the boundary of the aperture.

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Adaptive optics

Figure 3: Table of the first 22 Zernike polynoms

Figure 4: Schematic of a micromachined deformable mirro (MMDM)

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Adaptive optics

3.4 Closed-Loop Adaptive Optic SystemsBoth the SHWS and deformable mirror are used in conjunction to form a closed-loop adap-tive system which is capable of measuring and dynamically correct aberrations [1, 2]. Forexample, in an adaptive optic setup used in ground-based astronomy, turbulences in the at-mosphere are random and permanently changing. In order to compensate for them, a SHWSis continuously measuring the aberrations in the wavefront, which can then be used to de-termine the appropriate shape of the deformable mirror, as shown in Fig. 5 for a telescopesetup. This is an iterative process, so as to adapt to new and different aberrations inducedby the atmosphere in order to maintain image quality over long exposure times. The sameiterative method is used in other applications as well in order to minimize wavefront aber-rations.

Figure 5: Setup of an astronomical telescope that incorporates adaptive optics to compensate foratmospheric turbulence

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Adaptive optics

4 Setup and equipmentThe principal setup is shown in Fig.1 and more detailed in Fig. 9, Fig. 11 and Fig. 10.Both SHWS and MMDM-controller are connected to the computer via USB. The MMDM-controller itself is connected to the MMDM-voltage supply via serial cables. The laserdiode can be switched on at its voltage supply. Throughout the experiment, the commer-cial software package Frontsurfer is used to measure with the SHWS and to compensate foraberrations with the MMDM. Not only does it include all the drivers, but also provides pow-erful tools to evaluate the measured wavefronts as well as a feedback system for wavefrontcorrection.

4.1 Adjusting the SHWS and measuring the wavefrontIn Fig. 6 it is demonstrated how to obtain a measurement of the wavefront. If the folder-button in the menu bar is pressed, an image of the pattern incident on the CCD-chip is taken.However, it is better to use the preview mode (camera button) to adjust the SHWS, since itcontinuously displays the current image of the SHWS taken by the CCD-chip 2. Make surethat the spot is well centered on the micro lens array and that as many lenses as possibleare illuminated. Then dim down the light using the filter wheel, until only a homogenous,hexagonal array of spots is visible. The better the adjustment of the SHWS is, the better anyfeedback loop will work. A good example is shown in the small window in Fig. 6, a ratherbad one in the big window.

4.2 Evaluating the wavefrontAfter a good image from the SHWS is obtained and the preview mode is stopped, the wave-front can be evaluated. Fig. 7 shows a typical screenshot. All the analysis is done at once bypressing the cyan arrow button in the menu bar. Alternatively all the evaluation tools can beaccessed from the measure menu. The function report displays the amplitudes of Zernikepolynomials. All data can be saved to .bmp, .ps, .txt and .csv formatted files. The latterone can be imported by MS Excel. Save every measurement separately and do not use theoption save everything, since this causes the program to crash.

4.3 Adjusting an optical system using the SHWSFrontsurfer can also perform a continuous wavefront analysis. This comes in handy foraligning optical systems. The loop modus can be accessed in the processing menu. It con-tinuously displays the first Zernike polynomials, which correspond to the Seidel aberrations.However, this tool does not help for greatly misaligned systems (Fig. 8), since the SHWSneed a reasonably collimated beam in order to measure correctly.

2It takes a few second to start the plugin after the button is pressed, so be patient. Be also patient whenrestarting the plugin (not the program) after a few minutes

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Adaptive optics

Figure 6: Adjusting the SHWS using the preview mode of the software.

Figure 7: Evaluating the wavefront using Frontsurfer’s various tools.

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Adaptive optics

Figure 8: The loop modus of Frontsurfer allows for continuously monitoring the lower Zernike poly-nomials, thus aligning for aberrations.

5 Goals of the experimental work

5.1 Alignment of the Shack-Hartmann wavefront sensorMeasure the wavefront without any additional optical elements. Observe the influence ofdifferent filters on the measured wavefront. What problems might occur? If your alignmentis correct the measured wavefront can be regarded as an undistorted reference. Discuss yourresults.

5.2 Wavefront analysis with a Shack-Hartmann wavefront sensor• Focus the beam of the laser diode with a lens (L1, focal length between 35 and

100 mm) onto the SHWS (Fig. 9). Use the preview modus of Frontsurfer to alignthe SHWS, dim the light with the filter wheel (F).

• Use Frontsufers wavefront analysis to determine the Zernike terms. Which one arepredominant and why?

• Evaluate the wavefront in terms of Zernike polynomials using the report function atten different lens positions. Measure the distance between the lens and the wavefrontsensor.

• Plot the amplitude of the defocus term against the displacement of the lens. Ex-perimentally validate the maximum tilt angle and radius of curvature. Compare theobtained results with the theoretical ones (see preliminary questions).

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Adaptive optics

BS L1

SHWS

F

Laser

Figure 9: Basic setup to do wavefront measurement with a Shack-Hartmann wavefront sensor(SHWS), the laser light is focused with (L1) in order to obtain a curved wavefront. Beamsplitter(BS) and filter wheel (F) are used to dim down the light.

5.3 Aberration compensation in a microscope using a deformablemirror

Switch on the voltage supply of the MMDM. Make sure that the calibration file of themirror is loaded (see Fig. 14). Then set all attenuators to zero using the set values functionof the mirror menu. Calibrate the mirror using the function calibrate. First try to improvethe beam quality of the setup with the feedback function of Frontsurfer without any glassslides. Document your improvements with the analysis tools provided by the program. Setthe values back to zero.

• Insert a carrier slide (S) in between the microscope objectives (O1, O2). What aber-rations does it cause? Measure the aberrations for one to five slides.

• Repeat the experiment, but correct for aberrations after each slide. Save the settings ofthe mirror (Mirror, Set Values Menu) each time. How many slides can the deformablemirror correct for?

5.4 Optimization of a telescope using wavefront analysis• Build a telescope with the lenses (L1, L2) given by the assistant. Mount one of the

lenses on the stage with the micrometer screw. What will be the magnification of yourtelescope? What orientation of the plano-convex lenses should be chosen to build upthe telescope?

• First, do the coarse alignment by collimating the beam using a piece of card board ora wall. In order to improve the quality of collimation, use a long optical path.

• Check the quality of your alignment by measuring the wavefront.

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Adaptive optics

BS

L1L2

M3

MMDM

SHWS

C L5 F

O2S

O1

M2

Laser

M1

Figure 10: Setup of the microscope with the adaptive mirror (MMDM)

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Adaptive optics

• If the aberrations are small you can do a fine alignment with the help of the micrometerscrew and the loop function of Frontsurfer. Document the quality of your alignmentin terms of Zernike amplitudes using the wavefront analysis tool.

C L3 BS L2

SHWS

F L1

LaserTelescope

Figure 11: Setup to measure and optimize aberrations of a telescope.

A Prelimary Questions• The microlens array used, has 127 microlenses. They have a pitch of 300 µm and

a focal length of 18 mm. Compute the maximal wavefront tilt ϑ that is measurablewithout focusing into the square of the adjacent microlens. The illuminating laserhas a wavelength of λ = 632 nm. To what phase difference Θ does the wavefront tiltcorrespond?

• Estimate the minimum wavefront tilt that is measurable. The pixel size of the CCD is9.9x9.9 µm2.

• The hexagonal microlens array has an average aperture radius of a = 1.9 mm. Giventhe maximal wavefront tilt angle ϑ from the first question, what is the minimal curva-ture radius the SHWS is able to measure?

• The MMDM has an aperture of 2a = 15 mm, the maximal deformation the centralattenuator can achieve is d = 9.5 µm. If used as a spherical mirror, what is the minimalfocal length f that can be achieved?

• How are the Zernike-Polynoms defined? Which aberration do they describe?

Keywordsray optics, telescope, microscope, image quality, resolution, aberrations, Strehl ratio, waveoptics, Gaussian optics, wave front analysis, point-spread-function, Shack-Hartmann wavesensor, Seidel aberrations, Zernike polynomials, adaptive mirrors.

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Adaptive optics

B Final Questions• For which beams SHWS and MMDM work best and why?

• How do specific aberrations affect the image quality / resolution / point spread func-tion?

• Which are the predominant aberrations and how can they best be compensated for?Which aberrations should be better compensated within the static setup?

• Where are the limits of dynamic wavefront correction?

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Adaptive optics

C Amplitude of the "defocus" termThe following is a derivation of the relation between a wavefront’s radius of curvature andthe coefficient of the Defocus Zernike term. A wavefront with radius of curvature R, enteringan aperture of radius a will have a phase difference from the optical axis at the edge of theaperture of ∆ (Fig. 12).

Figure 12: Geometry of a spherical wave with radius of curvature R

∆ = R −√

R2 − a2 (4)

The Zernike Polynomial corresponding to defocus is given by

Z02 = W0

2

√3(2ρ2 − 1) (5)

which has zeroes at ± 1√

2, and where W0

2 is a scaling coefficient. Normalizing Equation 1to an aperture of unit radius, in terms of wave numbers, and with zeroes at ± 1

√2

yields

∆ =aλ

√(R

a

)2

− ρ2 −

√(Ra

)2

−

(1√

2

)2 (6)

which can be simplified to

∆ =Rλ

√

1 −(ρaR

)2−

√1 −

(a√

2R

)2 (7)

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Adaptive optics

A Taylor series expansion of the square root terms gives

∆ =Rλ

1 − 12

(ρaR

)2− 1 +

12

(a√

2R

)2 (8)

From here, the equation can be readily simplified to

∆ = −a2

4λR

(2ρ2 − 1

)(9)

Multiply and divide by√

3 to give the same form as Equation 2

∆ = −a2

4√

3λR

√3(2ρ2 − 1

)(10)

Which leaves

W02 = −

a2

4√

3λR(11)

D Initializing the software and setupBefore starting any software, make sure that the SHWS is connected to the computer. Thesmall diode at the back of the SHWS should light up in red. If you use the MMDM checkthat the controller unit is connected (big red diode) as well and that the voltage converteris switched on (small blue diode and a noticeable cooler fan). After each program start orrestart, the configuration files of both SHWS have to be loaded. All necessary configurationfiles are .txt-files and are stored in the directory Eigene Dateien. Fig. 13 and Fig. 14show how this can be done. Additionally, for the SHWS small deviations due to irregu-larities of the micro lens array can be corrected by subtracting a number of Zernike termsfrom the measured wavefront. The configuration file extractterms.txt can be loaded ina similar fashion as the others by choosing Options, Extract terms from the menu.

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Adaptive optics

Figure 13: Loading the configuration file of the Shack-Hartmann wavefront sensor

Figure 14: Loading the configuration file of the deformable mirror.

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Adaptive optics

References[1] M Booth, M Neil, R Juskaitis, and T Wilson. Adaptive aberration correction in a con-

focal microscope. Proceedings of the National Academy of Sciences, Dec 2002.

[2] M Booth, M Neil, and T Wilson. Aberration correction for confocal imaging inrefractive-index-mismatched media. Journal of Microscopy, Dec 1998.

[3] D. L Fried. Least-square fitting a wave-front distortion estimate to an array of phase-difference measurements. Optical Society of America, 67:370, Mar 1977.

[4] R. H Hudgin. Wave-front reconstruction for compensated imaging. Optical Society ofAmerica, 67:375, Mar 1977.

[5] Virendra N Mahajan. Zernike circle polynomials and optical aberrations of systemswith circular pupils. Appl. Opt., 33:8121, Dec 1994.

[6] J. Y Wang and D. E Silva. Wave-front interpretation with zernike polynomials. AppliedOptics, 19:1510, May 1980.

E Reading adviceIn order to be prepared for this exercise, one should have grasped the fundamentals of classicbeam and Gaussian optics. One should know the basics of how telescopes and microscopesmagnify. Furthermore, the concepts of collimation and focusing should be known as theyare outlined in most undergraduate textbooks.

• Eugene Hecht: "‘Optics"’, 5 Geometric optics, especially, 5.7.5 The Microscope,5.7.7 The telecope

• Demtröder: "‘Experimentalphysik 2"’, 9.5.4 Linsensysteme, 9.5.6 Linsenfehler, 11.2Vergrößernde optische Instrumente, 12.3.2 Adaptive Optik

• Saleh Teich, "‘Fundamentals of photonics"’, Chapter 3: Beam Optics, especially sec-tions 3.1 and 3.2

In order to prepare for adaptive optics itself,

• Eugene Hecht, "‘Optics"’, 5.8.1 Adaptive optics, 6.3 Aberrations

• Definition of Zernike polynoms and their corresponding aberrations are given in [5, 6].

• The references [1, 2] are current research papers about the use of adaptive optics inmicroscopes. Reference [2] is interesting in context of the last part of the experiments.

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Adaptive optics

• The mathematical concepts to interpolate the wavefront using the SHWS measure-ments are given in [3, 4]. They go beyond the scope of this exercise.

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