Edge scan wave-front analyzer for low order aberrations

Edge scan wave-front analyzer for low order aberrations

Julian S. Nichols and Dennis C. Duneman

The principles and concepts of a wave-front analyzer based on spot edge scanning techniques and rising andfalling illuminance slope comparisons are reviewed. An analyzer implementing these techniques and de-

signed to be used with an adaptive optics element for the real time correction of low order aberrations is de-scribed. An evaluation of the tilt output channels of the analyzer shows a linear operating range exceeding+500 !lrad, frequency response exceeding 250 Hz, and a noise equivalent tilt angle of 4 jtrad. For the focus

and astigmatism channels, the analyzer output is linear over an - 2.5-3 -ttm range of aberration input for adiffraction-limited (DL) spot. The instrument gain falls from -8.5 V/4m for a 1 X DL spot to -4 V/Mm for

a 6 x DL spot. The noise equivalent aberration increases from 0.025 Mtm for a 1 X DL spot to -0.05 /Am for

a 6 X DL spot.

1. Introduction

Barakat and Houston' developed analytical rela-tionships among the point spread function, the linespread function, the cumulative line spread function,and the system optical transfer function. Emphasis wason a system with a circular aperture and aberrationswith rotational symmetry. Titian2 expanded this workto show that the transfer function can be derived fromthe edge response function.

Holmes3 et al. confirmed experimentally that theedge response function can be used to derive the opticaltransfer function for specified instances of rotationallysymmetric aberrations.

Barakat and Houston4 have also explored the rela-tionship between the transfer function and the totalilluminance for incoherent radiation. They point outthat the total illuminance of an Airy system will showthe fastest rise of all possible systems. A system witha transfer function degraded with induced aberrationswill show a correspondingly slower rise. Several ex-amples showing the reduced illuminance rise with in-creased rotationally symmetric aberrations are calcu-lated and plotted.

A scanning wave-front analyzer has been built todetect low order aberrations-two axes of tilt (P2, P3),defocus (P4), and two axes of astigmatism (P5, P6)-which implement many ideas explored in Refs. 1-4. InSec. II, the concepts underlying the operation of theanalyzer are discussed. In Sec. III, the analyzer is de-scribed. Sec. IV the laboratory setup used to evaluate

The authors are with U. S. Air Force Weapon Laboratory (AFSC),

Kirtland Air Force Base, New Mexico 87117..Received 2 May 1983.

the analyzer is described. The experimental results arepresented in Sec. V followed by concluding remarks inSec. VI.

11. Principles and Concepts

The implementation chosen for the analyzer includesa dual chopper wheel on a common shaft. The systemfocal plane for a condition of no defocus lies midwaybetween the two choppers. They are identical but arerotationally offset so that one uncovers the beam, whilethe second covers the beam. We thus have two partialedge scans of the beam-one by the ahead-of-focuschopper and one by the behind-focus chopper. Thusthe output signal from the cooled detector shows a (il-luminance) rise corresponding to the beam at theahead-of-focus position, while the (illuminance) fallcorresponds to the beam at the behind-focus posi-tion.

To clarify the concept, let us assume that the beambeing analyzed is nominally a Gaussian, which, whenfocused midway between the two chopper wheels, isdescribed by

I = Io exp(-x2/2cr2). (1)

(This 1-D model is adequate for this discussion. It caneasily be extrapolated to two dimensions.)

At any plane removed from the focal plane the imageis defocused. To the first order the beam can be de-scribed by

7 = +/ exp(- 2/2u2), (2)

where

Il1 = exp[-(27rNx) 2 ],

Io(3)

2836 APPLIED OPTICS / Vol. 22, No. 18 / 15 September 1983

and N. is the number of waves of the optical path dif-ference (OPD) at the sensing wavelength correspondingto the condition of defocus.

Differentiation of Eq. (2) shows that the maximumslope of the beam occurs at x = o- and is given by

- = 0.24 I,-dxmax U2

(4)

If the chopper wheels are located at a position cor-responding to a defocus of 0.164Nx, I, = 0.3461o, and themaximum slope at the wheel positions for the conditionof beam focus midway between the two wheel positionsis, from Eq. (4),

P4 P P6 VOLTAGE DIRECTION FRONT FRRO'A

COMPONENT OF SCAN FRN

V0 - I I 0CHANNEL A

CHANNEL B *

V45 C I i -CHANNEL CI 0

CHANNEL A

- = 0.084 -.dx a

CHANNEL (5)

Since the two choppers are equally spaced from thenominal focal plane, the leading edge slope will equalthe trailing edge slope. An algorithm which subtractsthe trailing edge slope from the leading edge slope yieldsa zero signal for this condition.

If a system induced focus aberration of 0.1X is intro-duced so as to move the beam focus away from thenominal focal plane toward the behind-focus chopper,the illuminance slope at chopper 1 is decreased whilethat at chopper 2 is increased:

dXwheel = 0.2 Io exp[-(27r X 0.264)21 = 0.015 Io (6)d~wel 0.24 Io2

d = 2 Io exp[-(2ir X 0.064)2 = 0.204 Io (7)dxwheel 2 0,2U

In this case the slope subtraction algorithm yieldsa signal proportional to (Io/o-2) (0.015-0.204) =-0.189(Io/a 2).

If a system induced focus aberration of equal mag-nitude but of opposite sign were introduced, the algo-rithm would yield a signal proportional to +0.189Io/.2.The conditions required for generating a null sensingsignal for a servo drive are met.

Equation (4) shows that the illuminance rise for thehypothetical gaussian beam is a function of both in-tensity I, and the standard deviation . Introductionof focus aberration decreases I, [Eq. (3)]. Introducinga source larger than diffraction-limited has the effectof increasing a. From Eq. (5), we see that increasing -results in a decreased slope. From Eqs. (6) and (7) wesee that applying the slope differencing algorithm forthe case of an increased o- yields a result which is de-creased in absolute value for the same magnitude offocus aberration. Thus for this analyzer configuration,an extended source, i.e., larger than diffraction-limited,results in a decreased sensitivity, or scale factor, of theanalyzer.

Ill. Analyzer Design

The discussion in the previous section shows thefeasibility of using the illuminance rise (the edge re-sponse function), two chopper wheels, and a single de-tector to detect focus aberrations and develop a nullingservo signal. To expand the capability of the analyzerto detect two axes of astigmatism (in the Zernike for-

Fig. 1. Detector layout sketch and channel voltage output depen-dence on aberration components.

mutation), in addition to defocus, the incoming beamis split into three beams. Each is chopped and sent toa detector. The detectors are located equidistant fromthe center of the axis of the chopper wheels at 0, 45, and90° locations (see Fig. 1).5 Keeping in mind that theprocessing algorithm essentially subtracts the illumi-nance fall slope generated by behind-focus chopperfrom the illuminance rise slope generated by theahead-of-focus chopper, one can confirm the matrixshown in Fig. 1. This matrix shows the dependence ofthe signal out of each of the three channels on the threeaberrations P4, P5, P6 to within a scale factor k4 , k5 , k6 .Inverting the matrix in Fig. 1 yields an expression forthe aberrations P4, P5, P6 in terms of the channelvoltages Vo, Vgo, and V 4 5 :

P4 = - (Vo + V9o);2k4

P5 = -(Vo - V90);2k5(8)

P6 = - [1/2 (VO + V9o) - 45].k6

Thus the low order aberrations P4, P5, P6 can be ex-pressed as algebraic sums of the voltages derived bysubtracting the behind-focus edge response function(ERF) from the ahead-of-focus ERF in each detectorchannel.

For clarity, the edge response function has been usedin the discussion above. The signal processing algo-rithm actually implemented in the analyzer is com-monly known as the S4 image sharpness6 algorithmgiven by

2A =f 9tsdt (9)

where s is the illuminance rise slope and t is time.The analyzer processing implements this function,

with finite differences rather than derivatives, to match

15 September 1983 / Vol. 22, No. 18 / APPLIED OPTICS 2837

Fig. 2. Scanning wave-front analyzer.

CHANNEL ADETECTORCHANNEL CDETECTORCHANNEL DETECTOR

CONDENSERS

Fig. 3. Optical schematic of analyzer.

the sharpness of the ahead-of-focus and behind-focusedge response functions. 5 A signal is generated pro-portional to

fIA21't~ 2 ,I2ERF 2 YR2A = SV 'T') dt - T2 ) dt (10)

leading edge or trailing edge or

ahead of focus behind focus

Two tilt channels P2, P 3 are also included in the an-alyzer. The implementation of these functions is astraightforward phase comparison. The phase of thechopped signal out of channel A (V90 ) is compared withthe signal from a chopped reference light emitting diodebeam to derive an azimuth tilt signal. In a similarmanner elevation tilt information is derived fromChannelB (V0 ).

A photograph of the analyzer is shown in Fig. 2; anoptical schematic is shown in Fig. 3.

IV. Experimental

A. Tilt Channels

The Barnes collimator model 6-105 is designed toproduce a 30-cm diam collimated IR wave front froma point source. The optical train includes two orthog-onal mirrors, the position of which can be controlledwith an external electrical signal. To calibrate the tiltchannels, P2 and P3, of the analyzer the sensor wasaligned to look into the collimator, while calibrated in-crements of tilt angle were introduced into the orthog-onal mirrors of the collimator. The point source wasa pinhole-either 0.1- or 0.03-cm diam-illuminated bya quartz encased tungsten filament lamp.

For the frequency response evaluation of the tiltchannels, the orthogonal mirrors were driven by anexternal signal generator, while the tilt channel outputsof the analyzer were monitored. The upper frequencylimit is set by the frequency response of the orthogonalmirrors-3 dB down at -240 Hz with a 1350 phaselag.

B. Aberration Channels

The laboratory setup shown in Fig. 4 was used toevaluate the focus P4 and two axes of astigmatism, P5and P6, channels of the analyzer. The critical elementhere was the edge actuated deformable mirror (DM).The performance of DM was described in Ref. 7. TheBarnes collimator and the He-Ne laser were used forinitial alignment only. The 3-5-ynm source was a pin-hole illuminated by a quartz enclosed tungsten filamentlamp. Analyzer performance data were obtained usingfour different pinhole sizes ranging from diffraction-limited (1 X DL) to -6 X DL. The beam expander(BX) produced a 10-cm diam collimated beam, whichis the beam diameter for which the DM and the analyzerwere designed.

The experimental procedure consisted of (1) ad-justing P4, P5, P6 electrical inputs to the DM to removeresidual aberrations so that the aberration outputchannels of the interferometer were zeroed, (2) insertingknown increments of aberration-either P4, P5, orP6-into the DM and monitor the output of the ana-lyzer. To evaluate cross-coupling between aberrationchannels, all three output channels of the analyzer weremonitored even if only a single aberration was imposedon the mirror.

PINHOLE

LAMP

Fig. 4. Laboratory optical layout.


TVI

----------- 11 TM

14

12 X-X TILT VOLTAGE10- O-Y TILT VOLTAGE

>6/

-2

-2

-4

.0

-a

.10

.12

-14-500 -400 .300 -200 -100 0 100 200 300 400 500

X TILT (QRAWIANS)

Fig. 5. Tilt channel static response.

Extraneous and excessive tilt inputs to the analyzerarising from residual misalignment, atmospheric tur-bulence, laboratory vibrations, etc. were canceled byclosing the analyzer tilt output channels around a dy-namic tilt mirror (TM2).

V. Results

A. Tilt Channels

A typical static tilt response curve of the analyzer overthe range of 500-Arad input is shown in Fig. 5. Theresponse is relatively independent of the pinhole sizeand the source intensity. Pinholes of 0.1- and 0.03-cmdiam were used. Combinations of these two pinholesizes and two lamp sources of different intensitiesyielded SNR variation from 12 to 150. The analyzershowed a gain or scale factor of G = 0.025 V/,urad. Wecan define a noise equivalent tilt angle (NETA) to be

NETA = (Vnoise/G).

For the analyzer in the test configuration of this eval-uation NETA _ 4,4rad.

The open loop frequency response of the analyzer tiltchannels exceeds 250 Hz for the 3-dB down point. Thefrequency response of the calibrating instrument, theBarnes collimator, proved to be the limiting element.

B. Aberration Channels

The open loop response of each of the three channels(P4, P5, P6) of the analyzer for each of four pinholediameters (0.076, 0.1588, 0.3175, and 0.4763 cm) wasdetermined. The results are shown in Fig. 6 for defocusP4 and in Figs. 7 and 8 for the two axes of astigmatismP5, P6.

It is important to reemphasize that the analyzer wasdesigned to be used in a closed loop system with anadaptive optic element for real time correction of sensedwave-front aberrations. In this sense the analyzer is anull sensing instrument. However, the data in Figs. 6,7, and 8 were taken open loop, and the analyzer wasdriven significantly off-null.

Parameters of interest for the present evaluationincluded

(a) linearity, the range of input wave-front aberra-tion, in microns, over which the open loop response ofthe analyzer is linear;

(b) gain, analyzer voltage output, over the linearrange, for unit aberration input;

(c) coupling, a measure of the analyzer output in aspecific aberration output channel with a different ab-erration input (i.e., the output on the P5 or P6 channelonly when P4 aberrations are applied).

(d) Noise equivalent aberration (NEAb), the mini-mum aberration input which can be detected in thenoise environment existing in the experimental setup.The NEAb is given by

NEAb = (Vnoise/gain)

and is system dependent.These parameters are summarized in Table I for the

focus channel and Tables II and III for the two astig-matism channels.

In an earlier section a decrease of instrument gain asthe object size increased was anticipated. This be-havior is confirmed in Fig. 9 in which the aberrationchannel gain is plotted vs the object spot size.

In Fig. 10, the NEAb for each of the three aberrationchannels is plotted vs spot size.

VI. Summary

The concept of using an edge scan of an image spotto detect and measure low order optical aberrations hasbeen demonstrated in the analyzer described here.Two axes of tilt (P2, P3), focus (P4), and two axes ofastigmatism (P5, P6) are implemented.

The tilt channels are implemented using a conven-tional phase comparison technique in which thechopped signal is compared to a fixed reference signal.The tilt channels are shown to be linear over a range ofat least 500 urad with a frequency response in excessof 250 Hz and a noise equivalent tilt angle of -4,urad.

The focus and two axes of astigmatism channels areimplemented by comparing the rising illuminance slopeahead of nominal focus position with the falling illu-minance slope behind the nominal focus position. Animage sharpness algorithm is used to make this com-parison.

The focus channel output is linear with aberrationinput over an 2.3-,um range which changes little as theobject spot size varies from diffraction-limited (DL) to6 X DL. The gain of the focus channel over the samerange of spot size decreases from 8.1 for DL to 3.8 for 6X DL, which is consistent with previous analyticalpredictions. Coupling to other channels is independentof spot size and is generally in the 15-20% range over thelinear range. When used in a closed loop, null seekingmode, this coupling is expected to be significantly re-duced. The NEAb for the system setup in this evalu-ation shows a variation from 0.025 um for a DL spot to0.053 m for a 6 X DL spot.


Performance of the astigmatism channels are gen-erally consistent with the focus channels (compareTables II and III with Table I). One notable exceptionis that the linear range of the astigmatism channels in-creases as the spot size increases while the focus channellinear range does not.

Generally any limitations in the analyzer (NEAb,cross-channel coupling, gain, etc.) which were encoun-

+10

.076

tered were due to the electronic processing and not tothe sensor. The three cooled detectors' implementationrequires three channels of parallel processing. Foraccurate operation these three channels must be bal-anced. An imbalance in one or more channels will yieldinaccurate results and increase channel cross-coupling.Since the processing is implemented in analog circuitry,keeping the channels balanced is a difficult task. The

+10 -

+5-

I-0

~.1.II_ .

.5

-lok

I-

C

C0

.5

P4

P5

.P6

-10 P4

…------ P5._._._;_.. P6

+2.56 0 -2.56INPUT WAVEFRONT ABERRATION ( m)

(a)

.3175 cm PINHOLE

- P4- P5

._._._._ P6

I-C)

U)

C3

+2.56 0 -2.56P4 INPUT WAVEFRONT ABERRATION ( m)

(b)

E 4763 cm PINHOLE+10

+51-

0

.5

-10 _

P4

+2.56 0 -2.56

INPUT WAVEFRONT ABERRATION ( m) P4

(C)

Fig. 6. Focus P4 channel response to defocus aberration:0.476-cm source.

+2.56 0 -2.56INPUT WAVEFRONT ABERRATION ( m)

(d)

(a) 0.076-; (b) 0.159-; (c) 0.318-; (d)


P4

+10

I-

C a

-10- ____ P4 …-- --P5._.---- .P 6

.I

l w E E -

+5~

------------------

Il

l l

automatic gain control (AGC) circuitry proved to be areal problem. Residual noise in the input stages of thechannel resulted in a dc offset in the output of the AGCcircuitry. This feature created significant problems inkeeping the channels balanced.

Control loops from the aberration channels of theanalyzer to a deformable mirror have been closed suc-cessfully in the laboratory. In this case the channel

+10

C

fa--JC3

+5

0

5

-lo

.076 cm PINHOLEi

I

II I

I

I -

II

/

J ___I

,8 .__

cross-coupling has not created severe problems since inall cases the aberrations were canceled and the analyzeroutputs zeroed before the loops were closed. In a lessbenign environment in which the outputs of the ana-lyzer cannot be zeroed before loop closure one mightexpect to encounter instabilities resulting from channelcross-coupling.

The linear portions of the analyzer response curves

+10

a.-

I-C2

+5.

01

-5

- P4. P5

P6

I I I

.1588 cm PINHOLEIIIIIIIIII

IIII.1 I I

IIII

IIIIII -

III

- P4____ P5

...... P6

+2.33 0 -2.33

P5 INPUT WAVEFRONT ABERRATION ( m)

(a)

:.3175 cm PINHOLE X

P4P5P6

+2.33 0 -2.33P5 INPUT WAVEFRONT ABERRATION (m)

(b)

+101

Ia-C3U)

I-CDC.

+5

0

.5

-10

IIII

III

II

III

IIIII

II

III

_1/

K .4763 cm PINHOLE -,

II

II

III

/

III

,' PSE _ ___. P!

IfI -PI.

-I~~~~~~P


(c)


(d)

Fig. 7. Astigmatism P5 channel response to P5 aberration: (a) 0.076-; (b) 0.159-; (c) 0.318-;(d) 0.476-cm source.


+10

+5

a

8.-

CD

U)ba-C

-5

. s

* X t Z. .l

._.

-10 _

45

(Figs. 6-8) which are discussed in this paper extend overonly a small range of input aberrations around the zeroaberration point. For larger aberrations the curves turnover resulting in a double valued function of voltageversus optical path difference. For large OPD both the

+10 I

+5a-C3

Caa--5a

.5

-10 F

.076 cm PINHOLE /

iI

ii

i

i

/

1i ._

voltage output and the slopes of the volts vs aberrationcurves decrease. An attempt to use the analyzer to closea loop under conditions of large OPD could result in avery unresponsive, low gain, low bandwidth instru-ment.

+101

a-

Ca

U)a.--j

C0

+51

0

.5

- P4,__ P5 -101

.I5UU cm PINHULE it

/

/

II

i

i

l !I ____

. ... __

P4

R5RE;

-2.22 0 +2.22

P6 INPUT WAVEFRONT ABERRATION (m)

(a)

-2.22 0 +2.22

P6 INPUT WAVEFRONT ABERRATION (jim)

(b)

.3175 cm PINHOLEiii

i

/ -I

!I

II

- -

i --------- P.!, .I.

a-C

a.-C

+5

0

-5

-10 _

.4763 cm PINHOLE

~1

i

11-1/ '-_%_

"I i~~~~~-"I~~~~~~,,' ,. /,,#" .,~~~I,

f1J1

-2.22 0 +2.22P6 INPUT WAVEFRONT ABERRATION (jim)

(c)

-2.22 0 +2.22

P6 INPUT WAVEFRONT ABERRATION (m)

(d)

Fig. 8. Astigmatism P6 channel response to P6 aberration: (a) 0.076-; (b) 0.159-; (c) 0.318-;

(d) 0.476-cm source.


+10

+5

0

C

C

-5

-10I

___--- P4…--- - P5

._._._._._ P6

. _ _ _ _ I I

--- - . _... _. -

+10o

4

I

Table 1. Focus P4 Channel Response Parameters

Pinhole size0.076 0.159 0.318 0.476cm cm cm cm

Linearity (m) 2.3 2.3 2.3 2.3Gain (V/Am) 8.1 8.1 5.7 3.8Coupling (%) 15 22 19 17.7NEAb (m) 0.025 0.025 0.037 0.053

Table II. Astigmatism P5 Channel Response Parameters


Linearity (m) 3 - 4.4 4.7Gain (V/Am) 8.6 8.7 6.4 4.9Coupling (%) 12 9.5 7 7.5NEAb (m) 0.023 0.023 0.031 0.041

Table Ill. Astigmatism P6 Channel Response Parameters


Linearity (m) 2.66 3.1 - 4.88Gain (V/Mm) 9.9 8.9 6.2 4.3Coupling (%) 25 18 30 35NEAb (m) 0.02 0.02 0.032 0.047

8

6

4

2

0

.05 -

E .04

z .03P4

-- P5

.021

0

I I I

.0794 .1588 cm .3175

TARGET SIZE

.4763 cm

Fig. 9. Analyzer gain variation with source diameter.

.~~~~- -- P 4

* *~~~~~~~-----_. P6

I~~~~~~~~~~~~~~~

IXDL 2XDLTARGET SIZE

4XDL

Fig. 10. NEAb vs source diameter.

References1. R. Barakat and A. Houston, J. Opt. Soc. Am. 54, 768 (1964).2. B. Tatian, J. Opt. Soc. Am. 55, 1014 (1965).3. D. Holmes et al., Air Force Weapons Laboratory Technical Report

AFWL-TR-76-268 (1977).4. R. Barakat and A. Houston, J. Opt. Soc. Am. 53, 1244 (1963).5. C. Neufeld et al., AFWL TR-81-72 (1982), Vol. 1, contract

F29601-77-C-0062.6. A. Erteza, Appl. Opt., 15, 877 (1976).7. J. Nichols and D. Duneman, Opt. Eng. 22, 366 (1983).


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6XDL

Edge scan wave-front analyzer for low order aberrations

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Transcript of Edge scan wave-front analyzer for low order aberrations