Efficient implementation of a spatial light modulator as a diffractive optical microlens array in a digital Shack–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 the incoming wavefront into a number of segments and to measure the phase profile over the cross section of a given light beam. We customized a digital SHWS by encoding a spatial light modulator (SLM) with a diffractive optical lens (DOL) pattern to function as a diffractive optical microlens array. This SHWS can offer great flexibility for various applications. Through fast-Fourier-transform (FFT) analysis and experimental 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 microstructure of the SLM, we proposed three important strategies for the proper implementation of DOLs and DOL arrays with a SLM. Experiments demonstrated that these design rules were necessary and sufficient for generating 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) was invented by Roland Shack in the early 1970s. It was subsequently used in different applications, ranging from ophthalmology, astronomy, adaptive optics, and optical alignment to commercial optical testing.1 This sensing system uses a microlens array to dissect an incoming wavefront into a number of segments, leading to the creation of a focal spot by each lenslet on a CCD, as shown in Fig. 1. By measuring these focusing spots, one can calculate the phase distribution over the cross section of a given light beam. In our study we made the sensor component customized and programmable by using a spatial light modulator (SLM) to generate the diffractive microlens array. SLMs are pixelated liquid-crystal minidisplays that allow the implementation of a diffractive optical element by feeding the display with computer data in
real time or in a time-division-multiplexing scheme. These devices can modulate light spatially in amplitude and phase, so they act as dynamic optical elements. The optical function or information to be displayed can be taken from the optical design directly and can be transferred by means of a computer interface. In our experiment we used an LC2002 manufactured by HOLOEYE (see Ref. 2), a plug-andplay LCD device containing a Sony SVGA (800 ⫻ 600 pixels) LCD and driver electronics in a compact box. As far as we know, ours is the first study of the efficient encoding of a DOL pattern onto a SLM so that the latter can function as a dynamic microlens array. To make full use of SLMs as lenses, we investigated the performance of these generated DOLs both experimentally and theoretically. Three important design rules were concluded and are proposed in this paper. 2. Phase Function of a Diffractive Optical Lens
L. Zhao (
[email protected]), X. Li, and Z. P. Fang are with the Singapore Institute of Manufacturing Technology, 71 Nanyang Drive, Singapore 638075. N. Bai, L. S. Ong, and A. K. Asundi are with the School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore 639798. Received 16 June 2005; revised 5 August 2005; accepted 10 August 2005. 0003-6935/06/010090-05$15.00/0 © 2006 Optical Society of America 90
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Diffractive optics has emerged from holography and is making its way into industry. The areas of applications range from biotechnology via printing, material processing, sensing, and contactless testing to technical optics and optical metrology.3 Typical DOLs have multilevel microreliefs or continuous microreliefs4 (see Fig. 2). In contrast to conventional bulk optical lenses that derive their functionality from ei-
Fig. 1. Schematic of the SHWS.
ther refraction or reflection, DOLs derive their functionality from diffraction, which allows DOLs to be very thin.5,6 They therefore can offer significant savings in terms of mass and volume of optical systems. A diffractive lens on a planar substrate is described by a phase function. The phase function of a rotationally symmetric diffractive lens with an arbitrary profile can be of the form3 (r) ⫽ 2(a2r2 ⫹ a4r4 ⫹ · · · ),
(1)
where r is the radial coordinate in the plane of the diffractive lens. The optical power of the diffractive lens in the mth diffraction order is then given by 1兾f ⫽ ⫺2a20m,
(2)
where 0 is the design wavelength and f0 is the design focal length. 3. Efficient Sampling of the Continuous Phase Function of a Diffractive Optical Lens
Owing to the SLMs pixelated structure, a DOL with a continuous phase function, as shown in Fig. 3(a), cannot be implemented, and a pixelated and digitized DOL pattern can be generated, as shown in Fig. 3(b). The big square shown in Fig. 4 is a typical lenslet with diameter d, and each small square represents a pixel. Because one pixel can implement only one gray level, we use the center phase value of each pixel, which is calculated from the phase function, to represent the whole pixel.
Fig. 2. Diffractive lens with (a) continuous microreliefs and (b) multilevel microreliefs.
Fig. 3. (a) Continuous phase profile and (b) staircaselike phase profile of a DOL.
For the pixels that sit in the critical rings [such as those in the Q1 and Q2 solid rings in Fig. 5(a)], in principle, their phases should be zero for the last zone and should be 2 for the next zone, and they are defined by the DOL’s phase function. Also, corresponding to its location, each pixel has its own phase value that is calculated by the phase function. Therefore there are three methods for setting out the pixels located at the critical rings: (i) Set them black to correspond to a 0 phase level. (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 methods are shown in Fig. 5(b). Figure 6 gives their corresponding fast-Fourier-transform (FFT) spectra, and Fig. 7 depicts their corresponding focal spots. We can observe that under the same CCD settings, the DOL generated by method (i) has a much less noisy focusing spot and the smallest spot size among the three
Fig. 4. Typical lenslet with d ⫻ d pixels. 1 January 2006 兾 Vol. 45, No. 1 兾 APPLIED OPTICS
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Fig. 5. DOLs (d ⫽ 50 pixels, f ⫽ 75 mm) generated by three different sampling methods.
methods. We can conclude that the first sampling method generates DOLs the most efficiently. 4. Strategies for Efficient Generation of Diffractive Optical Lenses by a Spatial Light Modulator
Scalar theory predicts an ideal diffraction efficiency of 100% for DOLs with a continuous surface-relief profile as shown in Fig. 8. However, due to the SLM’s pixelated structure configuration, the generated DOLs can have only staircaselike phase profiles instead of continuous profiles, so a diffraction efficiency of 100% can never be achieved. Nevertheless, we try to achieve a better diffraction efficiency by increasing the number of phase levels in one zone. For a zone with eight phase levels, the corresponding local diffraction efficiency is already as high as 95%.3 Based on this, we proposed three requirements for an efficient DOL realized by a SLM.
Fig. 8. DOL continuous phase profile (a) side view and (b) top view.
A.
First-Zone Requirement
Our designed DOL is, practically speaking, a ring grating with gradually reduced pitches from the center to the edge of the lens. To achieve a diffraction efficiency as high as 95% and a quality spot image, we require that the first zone include at least eight pixels for eight phase levels of realization. The number of pixels in the first zone are N1 ⫽
⫽
冑1兾a2 b
冑2f b
,
(3)
where b is pixel size, f is focal length, and is wavelength. Since N1 should be larger than or at least equal to eight pixels, we get f ⱖ 64b2兾2.
(4)
Fig. 6. FFT spectra for the three DOLs shown in Fig. 5.
In our experiment we use an LC2002 with a pixel size of 32 m, and the minimum focal length to be efficiently achieved is 51.7 mm. B.
Fig. 7. Focusing spots for the three DOLs shown in Fig. 5. 92
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Effective Aperture Range
The focal length of a DOL determines the pixel number in each zone. The diameter determines the number of zones in one lenslet of fixed focal length. So the diameter of the lenslet should be larger than or equal to 2N1 pixels. As a result, the minimum diameter of
our designed DOLs should be dmin ⫽ 2冑2f.
(5)
The zone widths of a DOL, such as r1, (r2-r1), (r3r2), (r4-r3) as shown in Fig. 8, are gradually reduced from the center to the rim of the lens. So, as the diameter of the lens grows to some value, the width of that local zone will be too small to cover only one pixel. In this situation, the only single pixel to be completely covered in this zone will be set black according to our design algorithm, as one pixel can have only one gray level. This means that the phase levels and zone changes can no longer be implemented by pixel gray levels. As a result, the effective aperture of the lens cannot be extended further. Assuming that N is the number of pixels covered in the nth zone, we have
N⫽
n⫽
冑n ⫺ 冑n ⫺ 1 b
冑2f ⫽ 1,
1 b2 ⫹ ⫹ . 2b2 2 8f
⫽
632.8 ⫻ 10⫺9 32 ⫻ 10⫺6
⫽ 1.1331°. (6)
The maximum diameter of the DOLs should be 2f ⫹ b. b
(7)
So the effective and efficient aperture range is 2冑2f ⱕ d ⱕ
sin ⫽ m兾p,
⫽ 0.019775
f
dmax ⫽
Fig. 9. (a) SLM microstructure and (b) focal image of DOL (d ⫽ 50 pixels, f ⫽ 60 mm).
2f ⫹ b. b
(8)
C. Lenslet-Spacing Requirement
The SLM used in our project is a 600 ⫻ 800 pixel LCD panel; its microstructure is actually a twodimensional (2D) grating with a pitch size of 32 m, as shown in Fig. 9(a), and we can see that the fill ratios in the horizontal and vertical directions are different. As shown in Fig. 9(b), the bright dot in the center is the focusing spot of a DOL, and the other four surrounding dots with low intensity are noisy dots due to the diffraction of the SLM microstructure. We also observed that the two different horizontal and vertical fill ratios caused a stronger diffraction effect in the vertical direction than in the horizontal direction in our experiments. As a result, our investigation focused mainly on noisy dots in the vertical direction. From the diffraction theory for a grating,7 we can derive the following equation: p sin ⫽ m.
(9)
When m ⫽ 1 and ⫽ 632.8 m, we can calculate
(10)
This is the theoretical value of the diffraction angle of the first-order diffraction formed by a SLM grating structure. From Eq. (10) we can conclude that the diffraction angle is decided and fixed by the wavelength and the pitch size. As a result, the distance between the SLM’s first-order diffraction and the DOL’s focusing spot is determined by the focal length and is independent of the diameter of the generated DOLs. After carrying out some experiments, we found that the positions of these diffraction dots were really independent of the diameter of the DOLs but were closely related to their focal lengths. The correlation existing between them is that the distance between any two lenslets in a microlens array has to avoid S pixels, where S ⫽ s兾n 共n ⫽ 1, 2, 3, . . . 兲 and s ⫽ 70%关f兴, and [f] is a dimensionless figure and is equal to the numerical value of f, and f itself is in millimeters. The fixed ratio 70% was obtained through our experiments with the LC2002. Using the empirical 70% ratio to calculate the practical diffraction angle, we can get tan ⫽
0.7关f兴 ⫻ 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 assumption on the formation of these observed noise dots. In the SHWS if the undesired diffraction dot of one lenslet overlaps with the focusing spot of another lenslet, the centroid position of the spot cannot reflect the real 1 January 2006 兾 Vol. 45, No. 1 兾 APPLIED OPTICS
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in the same column that is 77 pixels, 39 pixels, and so on. Figure 10 is the image in focus of a 3 ⫻ 3 diffractive lens array with a 40 pixel diameter. In Fig. 10 bright spot lines 1, 2, and 3 are the focusing spot lines produced by the first, second, and third lines of the lenslet array, respectively, and the diffraction-related noisy dot lines 1⬘, 2⬘, and 3⬘ correspond to the first, second, and third lines of the lenslet array, respectively. We can see that cross talk almost occurs here owing to the improper pitch of the lenslet array. Although the four diffraction dots are always evenly positioned around the corresponding focusing spot in both directions, they still may restrict the dynamic range of the SHWS, since the wavefront tilt in one lenslet will cause its spot to move relative to the others and may overlap unwanted orders from the neighboring lenslets. 5. Conclusion
Fig. 10. Image captured on the focal plane of a DOL (40 pixels, 125 mm, 3 ⫻ 3) array. Cross talk almost occurs owing to the improper pitch of the lenslet array.
local wavefront information any more, and we call this cross talk. Once cross talk among lenses happens, the measurement accuracy and the performance of the SHWS will be degraded. However, we cannot eliminate cross talk through special patterning of DOL; instead, we can suppress its negative effect through proper spacing of the lenslets in the lens array. We need to ensure that the distance between two neighboring lenslets in a lens array is not close to S ⫽ f兾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
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In this paper we investigated how to efficiently adapt a SLM to function as a diffractive microlens array. We sampled a DOL’s continuous phase function with three different methods to deal with the transition pixels in the critical rings. The resulting digitized DOL patterns were compared, and the most efficient one was selected. Three important strategies for efficient sampling of DOLs generated by a SLM were summarized. Diffraction-related cross talk, which is found to be caused by the SLM’s microstructure and is independent of the designed DOLs, was studied as well. Although these diffraction-related noisy spots cannot be eliminated, we can take precautions when designing the lens array by setting the spacing between the lenslets properly. However, in real Shack– Hartmann wavefront measurements, the distances between neighboring centroids varies continuously and this restricts the measurement range. We will discuss this issue in a follow-up paper that is currently in preparation. References 1. 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 diffractive microlens fabrication by thin film deposition,” Appl. Opt. 29, 931–936 (1990). 6. W. C. Sweat, “Describing holographic and optical elements as lenses,” J. Opt. Soc. Am. 67, 803– 808 (1977). 7. E. Hecht, Optics, 2nd ed. (Addison-Wesley, 1990), pp. 424 – 426.
