Efficient implementation of a spatial light modulator as adiffractive optical microlens array in a digitalShack–Hartmann wavefront sensor

Liping Zhao, Nan Bai, Xiang Li, Lin Seng Ong, Zhong Ping Fang, and Anand Krishna Asundi

A traditional Shack–Hartmann wavefront sensor (SHWS) uses a physical microlens array to sample theincoming wavefront into a number of segments and to measure the phase profile over the cross sectionof a given light beam. We customized a digital SHWS by encoding a spatial light modulator (SLM) witha diffractive optical lens (DOL) pattern to function as a diffractive optical microlens array. This SHWScan offer great flexibility for various applications. Through fast-Fourier-transform (FFT) analysis andexperimental investigation, we studied three sampling methods to generate the digitized DOL pattern,and we compared the results. By analyzing the diffraction efficiency of the DOL and the microstructureof the SLM, we proposed three important strategies for the proper implementation of DOLs and DOLarrays with a SLM. Experiments demonstrated that these design rules were necessary and sufficient forgenerating an efficient DOL and DOL array with a SLM. © 2006 Optical Society of America

OCIS codes: 220.3620, 230.6120, 260.1960, 120.4640.

1. Introduction

The Shack–Hartmann wavefront sensor (SHWS) wasinvented by Roland Shack in the early 1970s. It wassubsequently used in different applications, rangingfrom ophthalmology, astronomy, adaptive optics, andoptical alignment to commercial optical testing.1 Thissensing system uses a microlens array to dissect anincoming wavefront into a number of segments, lead-ing to the creation of a focal spot by each lenslet on aCCD, as shown in Fig. 1. By measuring these focusingspots, one can calculate the phase distribution overthe cross section of a given light beam. In our studywe made the sensor component customized and pro-grammable by using a spatial light modulator (SLM)to generate the diffractive microlens array.

SLMs are pixelated liquid-crystal minidisplaysthat allow the implementation of a diffractive opticalelement by feeding the display with computer data in

real time or in a time-division-multiplexing scheme.These devices can modulate light spatially in ampli-tude and phase, so they act as dynamic optical ele-ments. The optical function or information to bedisplayed can be taken from the optical design di-rectly and can be transferred by means of a computerinterface. In our experiment we used an LC2002manufactured by HOLOEYE (see Ref. 2), a plug-and-play LCD device containing a Sony SVGA (800� 600 pixels) LCD and driver electronics in a com-pact box.

As far as we know, ours is the first study of theefficient encoding of a DOL pattern onto a SLM sothat the latter can function as a dynamic microlensarray. To make full use of SLMs as lenses, we inves-tigated the performance of these generated DOLsboth experimentally and theoretically. Three impor-tant design rules were concluded and are proposed inthis paper.

2. Phase Function of a Diffractive Optical Lens

Diffractive optics has emerged from holography andis making its way into industry. The areas of appli-cations range from biotechnology via printing, mate-rial processing, sensing, and contactless testing totechnical optics and optical metrology.3 Typical DOLshave multilevel microreliefs or continuous microre-liefs4 (see Fig. 2). In contrast to conventional bulkoptical lenses that derive their functionality from ei-

L. Zhao (lpzhao@Simtech.a-star.edu.sg), X. Li, and Z. P. Fangare with the Singapore Institute of Manufacturing Technology, 71Nanyang Drive, Singapore 638075. N. Bai, L. S. Ong, and A. K.Asundi are with the School of Mechanical and Aerospace Engi-neering, Nanyang Technological University, Singapore 639798.

Received 16 June 2005; revised 5 August 2005; accepted 10August 2005.

0003-6935/06/010090-05$15.00/0© 2006 Optical Society of America

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ther refraction or reflection, DOLs derive their func-tionality from diffraction, which allows DOLs to bevery thin.5,6 They therefore can offer significant sav-ings in terms of mass and volume of optical systems.

A diffractive lens on a planar substrate is describedby a phase function. The phase function of a rotation-ally symmetric diffractive lens with an arbitrary pro-file can be of the form3

�(r) � 2�(a2r2 � a4r

4 � · · · ), (1)

where r is the radial coordinate in the plane of thediffractive lens. The optical power of the diffractivelens in the mth diffraction order is then given by

1�f � �2a2�0m, (2)

where �0 is the design wavelength and f0 is the designfocal length.

3. Efficient Sampling of the Continuous PhaseFunction of a Diffractive Optical Lens

Owing to the SLMs pixelated structure, a DOL witha continuous phase function, as shown in Fig. 3(a),cannot be implemented, and a pixelated and digitizedDOL pattern can be generated, as shown in Fig. 3(b).

The big square shown in Fig. 4 is a typical lensletwith diameter d, and each small square represents apixel. Because one pixel can implement only one graylevel, we use the center phase value of each pixel,which is calculated from the phase function, to rep-resent the whole pixel.

For the pixels that sit in the critical rings [such asthose in the Q1 and Q2 solid rings in Fig. 5(a)], inprinciple, their phases should be zero for the last zoneand should be 2� for the next zone, and they aredefined by the DOL’s phase function. Also, corre-sponding to its location, each pixel has its own phasevalue that is calculated by the phase function. There-fore there are three methods for setting out the pixelslocated at the critical rings:

(i) Set them black to correspond to a 0 phaselevel.

(ii) Use their center-own-point phase value.(iii) Set them white to correspond to a 2� phase

level.

DOL patterns generated by these three methodsare shown in Fig. 5(b). Figure 6 gives their corre-sponding fast-Fourier-transform (FFT) spectra, andFig. 7 depicts their corresponding focal spots. We canobserve that under the same CCD settings, the DOLgenerated by method (i) has a much less noisy focus-ing spot and the smallest spot size among the three

Fig. 1. Schematic of the SHWS.

Fig. 2. Diffractive lens with (a) continuous microreliefs and (b)multilevel microreliefs.

Fig. 3. (a) Continuous phase profile and (b) staircaselike phaseprofile of a DOL.

Fig. 4. Typical lenslet with d � d pixels.

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methods. We can conclude that the first samplingmethod generates DOLs the most efficiently.

4. Strategies for Efficient Generation of DiffractiveOptical Lenses by a Spatial Light Modulator

Scalar theory predicts an ideal diffraction efficiencyof 100% for DOLs with a continuous surface-reliefprofile as shown in Fig. 8. However, due to the SLM’spixelated structure configuration, the generatedDOLs can have only staircaselike phase profiles in-stead of continuous profiles, so a diffraction efficiencyof 100% can never be achieved. Nevertheless, we tryto achieve a better diffraction efficiency by increasingthe number of phase levels in one zone. For a zonewith eight phase levels, the corresponding local dif-fraction efficiency is already as high as 95%.3 Basedon this, we proposed three requirements for an effi-cient DOL realized by a SLM.

A. First-Zone Requirement

Our designed DOL is, practically speaking, a ringgrating with gradually reduced pitches from the cen-ter to the edge of the lens. To achieve a diffractionefficiency as high as 95% and a quality spot image, werequire that the first zone include at least eight pixelsfor eight phase levels of realization.

The number of pixels in the first zone are

N1 ��1�a2

b

��2f�

b , (3)

where b is pixel size, f is focal length, and � is wave-length. Since N1 should be larger than or at leastequal to eight pixels, we get

f � 64b2�2�. (4)

In our experiment we use an LC2002 with a pixel sizeof 32 m, and the minimum focal length to be effi-ciently achieved is 51.7 mm.

B. Effective Aperture Range

The focal length of a DOL determines the pixel num-ber in each zone. The diameter determines the num-ber of zones in one lenslet of fixed focal length. So thediameter of the lenslet should be larger than or equalto 2N1 pixels. As a result, the minimum diameter of

Fig. 5. DOLs (d � 50 pixels, f � 75 mm) generated by threedifferent sampling methods.

Fig. 6. FFT spectra for the three DOLs shown in Fig. 5.

Fig. 7. Focusing spots for the three DOLs shown in Fig. 5.

Fig. 8. DOL continuous phase profile (a) side view and (b) topview.

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our designed DOLs should be

dmin � 2�2f�. (5)

The zone widths of a DOL, such as r1, (r2-r1), (r3-r2), (r4-r3) as shown in Fig. 8, are gradually reducedfrom the center to the rim of the lens. So, as thediameter of the lens grows to some value, the width ofthat local zone will be too small to cover only onepixel. In this situation, the only single pixel to becompletely covered in this zone will be set black ac-cording to our design algorithm, as one pixel can haveonly one gray level. This means that the phase levelsand zone changes can no longer be implemented bypixel gray levels. As a result, the effective aperture ofthe lens cannot be extended further.

Assuming that N is the number of pixels covered inthe nth zone, we have

N ��n � �n � 1

b�2f� � 1,

n �f�

2b2 �12 �

b2

8f�. (6)

The maximum diameter of the DOLs should be

dmax �2f�b � b. (7)

So the effective and efficient aperture range is

2�2f� d 2f�b � b. (8)

C. Lenslet-Spacing Requirement

The SLM used in our project is a 600 � 800 pixel LCDpanel; its microstructure is actually a two-dimensional (2D) grating with a pitch size of 32 m,as shown in Fig. 9(a), and we can see that the fillratios in the horizontal and vertical directions aredifferent. As shown in Fig. 9(b), the bright dot in thecenter is the focusing spot of a DOL, and the otherfour surrounding dots with low intensity are noisydots due to the diffraction of the SLM microstructure.We also observed that the two different horizontaland vertical fill ratios caused a stronger diffractioneffect in the vertical direction than in the horizontaldirection in our experiments. As a result, our inves-tigation focused mainly on noisy dots in the verticaldirection.

From the diffraction theory for a grating,7 we canderive the following equation:

p sin � � m�. (9)

When m � 1 and � � 632.8 m, we can calculate

sin � � m��p,

�632.8 � 10�9

32 � 10�6

� 0.019775

� � 1.1331°. (10)

This is the theoretical value of the diffraction angleof the first-order diffraction formed by a SLM gratingstructure. From Eq. (10) we can conclude that thediffraction angle is decided and fixed by the wave-length and the pitch size. As a result, the distancebetween the SLM’s first-order diffraction and theDOL’s focusing spot is determined by the focal lengthand is independent of the diameter of the generatedDOLs.

After carrying out some experiments, we foundthat the positions of these diffraction dots were reallyindependent of the diameter of the DOLs but wereclosely related to their focal lengths. The correlationexisting between them is that the distance betweenany two lenslets in a microlens array has to avoidS pixels, where S � s�n �n � 1, 2, 3, . . . � and s �70%�f�, and [f] is a dimensionless figure and is equalto the numerical value of f, and f itself is in millime-ters. The fixed ratio 70% was obtained through ourexperiments with the LC2002.

Using the empirical 70% ratio to calculate the prac-tical diffraction angle, we can get

tan � �0.7�f� � 32 � 10�6

f � 10�3

� 0.0224

� � 1.2832°. (11)

Now we can see that these two values in Eqs. (10)and (11) are very close, which validates our assump-tion on the formation of these observed noise dots. Inthe SHWS if the undesired diffraction dot of one len-slet overlaps with the focusing spot of another lenslet,the centroid position of the spot cannot reflect the real

Fig. 9. (a) SLM microstructure and (b) focal image of DOL (d� 50 pixels, f � 60 mm).

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local wavefront information any more, and we callthis cross talk. Once cross talk among lenses hap-pens, the measurement accuracy and the perfor-mance of the SHWS will be degraded. However, wecannot eliminate cross talk through special pattern-ing of DOL; instead, we can suppress its negativeeffect through proper spacing of the lenslets in thelens array. We need to ensure that the distance be-tween two neighboring lenslets in a lens array is notclose to

S � �f�np, (12)

where n � 1, 2, 3, . . . .Consider a lens array with a 125 mm focal length as

an example. We should avoid a lens design with d� 77 pixels and with spacing between any two lenses

in the same column that is 77 pixels, 39 pixels, and soon. Figure 10 is the image in focus of a 3 � 3 diffrac-tive lens array with a 40 pixel diameter. In Fig. 10bright spot lines 1, 2, and 3 are the focusing spot linesproduced by the first, second, and third lines of thelenslet array, respectively, and the diffraction-relatednoisy dot lines 1�, 2�, and 3� correspond to the first,second, and third lines of the lenslet array, respec-tively. We can see that cross talk almost occurs hereowing to the improper pitch of the lenslet array. Al-though the four diffraction dots are always evenlypositioned around the corresponding focusing spot inboth directions, they still may restrict the dynamicrange of the SHWS, since the wavefront tilt in onelenslet will cause its spot to move relative to theothers and may overlap unwanted orders from theneighboring lenslets.

5. Conclusion

In this paper we investigated how to efficiently adapta SLM to function as a diffractive microlens array. Wesampled a DOL’s continuous phase function with threedifferent methods to deal with the transition pixels inthe critical rings. The resulting digitized DOL patternswere compared, and the most efficient one was se-lected. Three important strategies for efficient sam-pling of DOLs generated by a SLM were summarized.Diffraction-related cross talk, which is found to becaused by the SLM’s microstructure and is indepen-dent of the designed DOLs, was studied as well.

Although these diffraction-related noisy spots can-not be eliminated, we can take precautions when de-signing the lens array by setting the spacing betweenthe lenslets properly. However, in real Shack–Hartmann wavefront measurements, the distancesbetween neighboring centroids varies continuouslyand this restricts the measurement range. We willdiscuss this issue in a follow-up paper that is cur-rently in preparation.

References1. B. C. Platt and R. Shack, “History and principles of Shack–

Hartmann wavefront sensing,” J. Refract. Surg. 17, 573–577(2001).

2. See http://holoeye.com/ (accessed on 31 May 2005).3. H. P. Herzig, Micro-optics, Elements, Systems and Applications

(Taylor & Francis, 1997), pp. 19–23.4. J. R. Leger and M. P. Griswold, “Binary-optics miniature Talbot

cavities for laser beam addition,” Appl. Phys. Lett. 56, 4–6(1990).

5. J. Jahns and S. J. Walker, “Two-dimensional array of diffractivemicrolens fabrication by thin film deposition,” Appl. Opt. 29,931–936 (1990).

6. W. C. Sweat, “Describing holographic and optical elements aslenses,” J. Opt. Soc. Am. 67, 803–808 (1977).

7. E. Hecht, Optics, 2nd ed. (Addison-Wesley, 1990), pp. 424–426.

Fig. 10. Image captured on the focal plane of a DOL (40 pixels,125 mm, 3 � 3) array. Cross talk almost occurs owing to theimproper pitch of the lenslet array.

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