Misalignment effects of the Shack–Hartmann sensor

Johannes Pfund, Norbert Lindlein, and Johannes Schwider

The Shack–Hartmann sensor uses a microlens array and a CCD camera for wave-front measurements.To obtain wave-front measurements with high accuracy, an accurate relative alignment of both isessential. The different states of misalignment of the Shack–Hartmann sensor are divided into groupsand are treated theoretically and experimentally. Their effect on the accuracy of wave-front measure-ments is evaluated. In addition, a practical method for proper alignment of the Shack–Hartmann sensoris proposed. © 1998 Optical Society of America

OCIS codes: 010.7350, 130.6010, 220.1140.

1. Introduction

The use of microlens arrays instead of a simple arrayof holes in an opaque screen has improved the Hart-mann test considerably.1–6 The principle of theShack–Hartmann sensor is shown in Fig. 1. Thewave front W to be measured is sampled by a micro-lens array. In the focal plane ~Dz 5 f ! of the micro-lens array one obtains a spot array that is detected bya CCD camera. If the dislocations of the spot posi-tions ~Dx, Dy! are measured with respect to the ref-erence points, a discrete two-dimensional field ofpartial derivatives of W can be calculated from Eq.~1!. In the case of a so-called internal reference, thereference points are the crossing points of the localoptical axis with the CCD plane. Here p and q in-dicate the rows and columns of the subapertureswithin the array:

$=W%pq 5 51]

]x]

]y2W6

pq

51f

3 HSDxDyDJpq

. (1)

One way to estimate W from the discrete field ofpartial derivatives is calculation of the expansion co-efficients of a polynomial in the xy or the Nijboer–

The authors are with Lehrstuhl fur Optik, UniversitatErlangen-Nurnberg, Physikalisches Institut, D-91058 Erlangen,Staudtstrasse 7yB2, Germany.

Received 10 June 1997; revised manuscript received 5 Septem-ber 1997.

0003-6935y98y010022-06$10.00y0© 1998 Optical Society of America

22 APPLIED OPTICS y Vol. 37, No. 1 y 1 January 1998

Zernike expansion. The common procedure forcalculating a xy polynomial,

Wxy 5 (i50

G

(j50

j#i

aijxjyi2j, (2)

of degree G is by least-squares fitting.7 Here wehave chosen a one-dimensional order for the mono-mials xjyi2j,

xjyi2j3 Pk~x, y!, (3)

and for the values of the discrete field of derivatives,

S]W]x Dpq

3 S]W]x Dm

, S]W]y Dpq

3 S]W]y Dm

. (4)

N is the number of expansion coefficients, and M isthe number of points of the measured field of deriv-atives at positions ~xm, ym! with m [ $1, . . . , M%.

The calculation of the minima,

]G~a!

]aj5 0, (5)

of the merit function

G~a! 5 (r51

M HF]Wxy~xr, yr!

]x2 S]W

]x DrG2

1 F]Wxy~xr, yr!

]y2 S]W

]y DrG2J (6)

leads to a set of linear equations:

! 3 a 5 b, (7)

where

Amn 5 (r51

M F ]

]xPm~xr, yr! 3

]

]xPn~xr, yr!

1]

]yPm~xr, yr! 3

]

]yPn~xr, yr!G , (8)

bn 5 (r51

M FS]W]x Dr

3]

]xPn~xr, yr! 1 S]W

]y Dr

3]

]yPn~xr, yr!G . (9)

The vector a contains the expansion coefficients thatwe are looking for. It can be calculated with stan-dard methods for solving a set of linear equations.

In general, the measured lateral spot positions @i.e.,the crossing point of the chief ray with the focal planeof a particular microlens ~p, q!# are dependent on therelative adjustment of the microlens array and theCCD plane. In Section 2 we deal with the effects ofdifferent types of misalignment, and in Section 4 wedescribe an accurate alignment method for theShack–Hartmann sensor.

2. Analysis of the Misalignments

The severity of misalignments between the CCD andthe lens array is discussed. Since we measure the

Fig. 1. Principle of the Shack–Hartmann sensor.

gradients, Wx and Wy, of a scalar wave front, themixed second derivatives should fulfill the condition

]

]xWy 5

]

]yWx. (10)

We can check our discussion to determine whetherthe considered misalignment violates this conditionand infer from this the misalignment that is the mostdisturbing. One further implication is the fact thatthe gradient of W should be unique, which is a con-dition excluding wave-front warping. The latter is aself-evident assumption from a mathematical point ofview.

If the microlens array is assumed to be fixed rela-tive to the coordinate system of the wave front, thesystem consisting of the CCD and the microlens arrayhas six degrees of freedom ~Fig. 2!. If the micro-lenses and the CCD camera are misaligned with re-spect to each other, errors are generated in thepolynomial of Eq. ~2!. The degrees of freedom can bedivided into four groups, two of them being transla-tion groups and two being rotation groups:

~a! Lateral translation ~x and y!.~b! Axial translation ~z!.~c! Rotation about the x and y axis ~a, b!.~d! Rotation about the z axis ~g!.

Translations are represented by linear coordinatetransforms of the form

r3 r* 5 r 1 dr, (11)

while the mathematical tool for performing rotationsare rotation matrices:

r3 r* 5 $a 3 $b 3 $g 3 r. (12)

A. Lateral Translation of the CCD ~Tilt Error!

All spots are displaced from their reference points bythe same displacement vector:

DrWdx,dy5 SDxW 1 dx

DyW 1 dyD . (13)

Fig. 2. Coordinate system of the Shack–Hartmann sensor.

1 January 1998 y Vol. 37, No. 1 y APPLIED OPTICS 23

Continuation leads to the differential equations

]W~dx,dy!

]x5

]W]x

1dxf

,

]W~dx,dy!

]y5

]W]y

1dyf

, (14)

which can be analytically integrated to

W~dx,dy! 5 W 11f

3 ~xdx 1 ydy!, (15)

because Eq. ~10! is fulfilled. Thus a lateral misalign-ment of the CCD camera feigns a tilt.

B. Axial Translation of the CCD ~Sensitivity Error!

In this case the focal length f has to be replaced by

f3 f9 5 f 1 dz (16)

~see Fig. 3!.Considerations similar to those in Subsection 2.A

yield an aberration

Wdz 5 S1 1dzf D 3 W. (17)

Fig. 3. Axial displacement of the CCD plane by dz . 0 amplifyingthe original wave aberrations.

This misalignment type keeps the shape of the cal-culated wave front from being affected but makesrecalibrations necessary.

C. Rotations about the x and y Axis

The differential equations for this case can be derivedas follows: The coordinate system of the rotatedCCD plane is calculated by multiplication of the co-ordinate vectors with rotation matrices. By calcu-lating the crosspoints of the chief ray of eachsubaperture with the rotated CCD plane, the localspot positions ~Dx, Dy! can be calculated. Fromthese the analytical form of the field of derivativesunder misalignment ~c! is obtained.

Provided the measured wave-front aberrations Ware not too large and assuming only one elementaryrotation around the x or the y axis, the followingdifferential equations follow for the case of rotationsaround the y axis:

Wb, x 5xf

3 S 1cos b

2 1D 11

cos b3

]W]x

, (18a)

Wb,y 5]W]y

. (18b)

For a nonplanar wave front this field of derivativescannot be integrated to the wanted scalar functionWb, because Eq. ~10! is violated for Wb:

]

]xWb, y 2

]

]yWb, x 5

d

]x]W]y

3 S1 21

cos bD< 2

]

]x]W]y

3b2

2Þ 0 if b Þ 0. (19)

Here we have used the fact that W is a scalar func-tion, and thus Eq. ~10! is fulfilled for W.

Fig. 4. Effect of misalignment type ~c!.

24 APPLIED OPTICS y Vol. 37, No. 1 y 1 January 1998

D. Rotation around the z Axis

If rW denotes the spot coordinates and rrp denotes thecoordinates of the reference points in the CCD plane,the spot positions in the local coordinate system of asubaperture are given by

DrW 5 rW 2 rrp. (20)

Because a rotation of the coordinate system of theCCD can be described by a rotation of the spot field inthe opposite direction, the transformation of the spotpositions,

DrWg5 rWg

2 rrp 5 Scos gsin g

2sin gcos gD3 rW 2 rrp

5 F~xrp 1 DxW! 3 cos g 2 ~yrp 1 DyW! 3 sin g 2 xrp

~xrp 1 DxW! 3 sin g 1 ~yrp 1 DyW! 3 cos g 2 yrpG , (21)

leads to the differential equations

Wg,x 5 Sxf 1]W]xD3 cos g 2 Syf 1

]W]yD3 sin g 2

xy

, (22a)

Wg,y 5 Sxf 1]W]xD3 sin g 1 Syf 1

]W]yD3 cos g 2

yf

. (22b)

These are not conservative as well because

]

]xWg, y 2

]

]yWg,x 5 1 sin g 3 S2f 1

]2W]x2 1

]2W]y2D

Þ 0 if g Þ 0. (23)

3. Consequences of the Nonintegrability ofMisalignments ~c! and ~d!

If misalignments ~c! and ~d! are not eliminated, themeasured fields of Eqs. ~18! and ~22! are not integra-ble. However, our integration algorithm ~see Sec-tion 1! will construct a solution vector a, which is thebest approximation in the sense of least-squares fit-ting. The effect of misalignment ~d! is illustrated byexperimental results below.

The Shack–Hartmann sensor is illuminated with aplanar wave front that is not necessarily parallel tothe plane of the microlens array. If small g rotations

Fig. 5. Misalignment type ~d!, mean fit error G.

are carried out, the mean fitting error G, which wedefine as

G 51M

3 (l51

M HF]Wxy~xl, yl!

]x2 S]W

]x DlG2

1 F]Wxy~xl, yl!

]y2 S]W

]y DlG2J1y2

, (24)

increases linearly with the angle of rotation ~Fig. 5!because of the nonzero rotation of the discrete field ofderivatives, which prevents an integral function Wxyfrom existing. The peak-to-valley ~PV! value of thexy polynomial is growing linearly with misalignmentg ~Fig. 6!, which is mainly caused by numerical fittingerrors at the rim of the spot field. A plot of the xypolynomial, calculated under misalignment of g 52.67°, is shown in Fig. 7. The variations in the wave-front aberrations at the rim are caused by an increas-ing dislocation of the spots, which is growing in theradial direction from the axis of rotation. Similarerrors occur if other wave fronts are measured.

4. Alignment Techniques

A. Lateral Translation of the CCD ~Tilt Error!

The lateral translations are not really misalignmentsbecause they can be removed easily. This can be

Fig. 6. Misalignment type ~d!, PV value of the polynomial.

Fig. 7. Misalignment ~d!, plot of polynomial ~g 5 2.667°!, PV 52.669l.

1 January 1998 y Vol. 37, No. 1 y APPLIED OPTICS 25

done by calculating the mean value of the local spotpositions and adding this value to each spot position.

B. Axial Translation of the CCD ~Sensitivity Error!

This misalignment can be eliminated by measuring areference wave front incident onto the microlens ar-ray ~e.g., a spherical wave front!, recalculating theeffective focal length feff from the polynomial repre-sentation, and substituting f 3 feff in Eq. ~1!.

The alignment can be done by illuminating themicrolens array with a known wave front ~i.e., spher-ical! and changing the z position of the CCD until thecalculated representation of Eq. ~2! matches the orig-inal wave front, i.e., reproduces the radius of curva-ture.

C. Rotations around the x and y Axis

In the case of a nearly plane wave front incident ontothe microlens array, from Eqs. ~18! an angle can becalculated that would be detectable assuming a spot-finding accuracy Dxmin:

]Wplane,b

]x5

xf

3 S 1cos b

2 1D 11

cos b

3]Wplane

]x<

xf

3 S 1cos b

2 1Df Dx 5 x 3 S 1

cos b2 1D

fDx5Dxmin

bmax 5 arccosxmax

Dxmin 1 xmax. (25)

With the rotation axis positioned in the center of theCCD chip ~8.3 mm 3 6.4 mm! the maximum distanceof a reference point from the rotation axis is xmax 54.15 mm, and with Dxmin 5 0.11 mm the maximalacceptable angle due to a misalignment of type ~c!would be bmax 5 0.42°. Such a tolerance is easilymet by adjusting the reflections originating from theCCD and the microlens array on the optical axis.

D. Rotation around the z Axis

Because the z position of the spots is unchanged byrotation of the microlens array around the z axis, theeffect of a misalignment of type ~d! can be removed by

Fig. 8. Sum of the gray-scale values of the rows of the CCD frameat the correct g alignment.

26 APPLIED OPTICS y Vol. 37, No. 1 y 1 January 1998

measurement of a reference wave front ~i.e., a planarwave front!.

If the misalignment is not eliminated by a refer-ence, an accurate alignment is desirable. With Eq.~22b! and the approximations DrW ' 0 ~a nearly pla-nar wave front!, cos g ' 1 and sin g ' g ~smallmisalignments!, the minimum detectable angle ofmisalignment d would be theoretically

DyWg5 ~xrp 1 DxW! 3 sin g 1 ~yrp 1 DyW!

3 cos g 2 yrp < xrp 3 g 1 yrp

3 1 2 yrp 5 xrp 3 g,

f gmin 5DyWg

xrp,max5

0.11 mm4.4 mm

5 2.5 3 1025 rad. (26)

It should be emphasized that such an accuracy doesnot have to be achieved practically because the effectof such a small misalignment onto the wave-frontmeasurements may be neglected. The followingtechnique is sufficient for the practical use of theShack–Hartmann sensor. One measure for the ac-curacy of the g alignment is the mean fit error, whichobviously should be minimized. Another techniqueis to calculate the sum of the gray-scale values of allrows and columns. As shown in Figs. 8 and 9, theresult peaks are located at those rows and columnswhere the spots are situated. The aim is to maxi-mize the height and to minimize the width of thepeaks. Figure 8 shows the row sums of a correctlyaligned measurement and Fig. 9 those of a badlyaligned measurement.

5. Conclusion

In this paper we have calculated the partial deriva-tives of wave-front measurements in the presence ofdifferent types of misalignment. In the case of mis-alignments ~a! and ~b! the analytical form of the re-constructed polynomial could be calculated @Eqs. ~15!and ~17!#. For the case of misalignments ~c! and ~d!we have shown that the fields of derivatives ~18! and~22! are no longer integrable.

Summarizing the above discussion there are bigdifferences between the effects of the different mis-alignments on the achievable final accuracy. Themost severe errors are caused by violations of the

Fig. 9. Sum of the gray-scale values of the rows of the CCD frameat a misalignment of g 5 0.667°.

integrability conditions by rotations of the lens arrayin relation to the CCD camera. But discussion andmeasurement have shown that the proposed align-ment criteria are sufficient to keep these effects belowthe level of the spot-finding accuracy of an approxi-mately 1⁄100-pixel distance.

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