PART I MICROLENS ARRAY
Chapter 4 Self-assembled microlens on top of light emitting diodes using hydrophilic
5.4 Wavefront Sensor Computing Algorithm and Measurement Results79
The Shack-Hartmann wavefront sensor utilizes a microlens array to divide and reconstruct the incident wavefront. The incident light is divided into a number of Table 5-2 Methods for increasing the focal length of MLA.
Methods Advantage Disadvantage
Refractive index difference Significant increment (6X longer) Double layer (2nd PDMS coating is easy) Increase refractive index, n Single layer process Slightly (~1X, Depends on material
property)
Increase diameter Single layer process Results in thick photoresist and spin coating problems
spots and the measured wavefront focal spots provided by each lenslet. Each local wavefront slope corresponds to a wavefront distortion. The relationship between incoming wavefront slope θ, spot displacement Δd, and focal length of the lenslet f can be described by Equation (1.6) [7]: measurements. We chose modal reconstruction in our calculation. The wavefront is described in terms of functions that have analytic derivatives in the modal reconstruction method. The measured slope data was then fit to the derivative of these functions, allowing a direct determination of the wavefront from the fitting coefficients.
Therefore, it is desirable to expand the wavefront aberration in terms of a complete set of basis functions that are orthogonal, such as Zernike polynomials Zk(x,y) [10]:
where ωk represents the Zernike coefficient. So we can write the local wavefront slopes as in Equation (1.8):
1 calculate the unknown Zernike coefficient, ωk by the inverse and the transpose of above matrix. By the matrix operation, we could get each of the corresponding Zernike polynomial wavefront aberration coefficients and take them into Equation (1.7). The working principle of the Shack-Hartmann wavefront sensor included the recognition of the aperture size/lens focal length, the detection of the image spots for reference beam and distorted beam, the calculation of the spot center and the displacement, the calculation of the Zernike coefficients from the wavefront, and the reconstruction of wavefront.
In order to measure the wavefront aberration, we integrated our fabricated MLA with an image sensor to construct a Shack-Hartmann wavefront sensor. We show the whole experiment setup in Figure 5-5. The laser light source from optical fiber is collimated by an aspherical lens screwed in front of the fiber connector. The collimated laser produces a reference wavefront which passes through our microlens array and results in separated focal spots on the detector. In this set up, we chose our fabricated hexagonal shaped MLA to focus the light on sensor because it not only has a higher fill factor, but also higher reconstruction precision.
To evaluate our SHWS performance, testing samples such as PAL (progressive
laser light source and the SHWS.
For the measurement, we first took pictures of focal spots without aberration from the test samples as the reference points, as shown in Figure 5-6(a), and then a second picture of focal spots with aberration by putting the test lens between the laser light
Figure 5-5 (a)The experiment setup for measuring a wavefront aberration of testing lens (PAL) by using our fabricated MLA and SHWS. (b) The schematic diagram.
spots precisely, a hexagon mask is defined to cover only one point in every calculation.
We calculated 37 wavefront slopes of focal spots taken by the CMOS (complementary metal–oxide–semiconductor) image sensor. Therefore, our system was computed by four orders of Zernike coefficients with 10 aberration terms and each centroid position was calculated by the Labview® program. The comparison between the reference focal spots and the aberrated focal spots was shown in Figure 5-6(c), two previous pictures stacking, where red rectangles are the detected reference spots position and green rectangles were the detected aberrated spots position. The displacement was then calculated with the equations listed above.
The wavefront aberration generated by the testing lens was measured by both our SHWS and a commercial wavefront sensor from OKO Tech (UI2210-m, UEye, NL).
The lenslet array in the commercial wavefront sensor is in hexagonal configuration and the pitch size is 300 m. To evaluate the performance of double layer microlens arrays, we compared the measurement result between our SHWS and the commercial one. First,
Figure 5-6 (a) Focal spots image of the reference wavefront, (b) focal spots image of the aberrated wavefront, and (c) detection picture of image spots for reference beam and distorted beams.
aberration caused by titled incidence wavefront through PCX where rays pass through the lens at an angle to the axis θ. Figure 5-7 shows the measurement results by a three-dimensional phase map and its contour.
Figure 5-7 Wavefront measurement results of (a) PAL and (b) PCX by our SHWS
Because the reference and the distorted beams pass through exactly the same optical path in this system, the aberrations are eliminated by the reference beam. Thus, the wavefront of the optical objectives can be measured precisely. The displacements of the reference spots and the aberrated spots are detected by a CMOS image sensor and processed by the Labview® program on a personal computer. The calculation of Zernike coefficients and wavefront reconstruction are performed by Matlab®.
The wavefront measurement accuracy of our SHWS was evaluated by a comparative measurement with the commercial wavefront sensor. Unlike our SHWS that calculated 37 wavefront slopes, the commercial SHWS automatically chose the brightest centroid spots to calculate aberrations. The number of spots found is variable;
it depends on two parameters- the intensity threshold (with respect to the intensity of the brightest spot) and the upper limit of maximum number of spots. The commercial program searches for spots in the reference pattern and builds bounding boxes around them. Then, spots of the main pattern are searched within the bounding boxes calculated from the reference pattern. Our SHWS apply the same technique; however, the calculated number of spots is fixed to 37.
As shown in Figure 5-7(a), from a review of the coefficient and the contour map, we noticed that the astigmatism aberration dominates in which light in one plane (the
“plane of the paper” or meridional plane) focused at different location from the orthogonal plane. The root-mean-square (RMS) value of the wavefront aberration calculated by the commercial SHWS is 0.336λ (λ = 630 nm) while the RMS value
nevertheless the coma aberration still dominates as expected and the plot below shows the expected wavefront taken by the commercial SHWS, and. The peak to valley difference is 0.438λ and the RMS wavefront difference is 0.036λ. The commercial SHWS calculated 19 wavefront slopes. Table 5-3 shows the comparison of wavefront measurement results between commercial SHWS and our SHWS.
We believe that high sensitive wavefront sensors could be used in measuring dynamic deformation of microstructures under high operation frequency [11]. There, however, is a tradeoff between dynamic range and sensitivity according to the number of microlenses and the focal length [12]. A small-size, long-focal-length MLA could increase the measurement accuracy associated with Equation (1.6) [9]. A longer focal length will provide high sensitivity in determining the average slope across each lenslet under a given wavefront [13]. Nevertheless, a microlens array with a shorter focal length will have greater dynamic range at the cost of reduced sensitivity. According to Equation (1.3), the focal length of microlens array is related to the radius of curvature of the lens. In order to produce a short or long focal length MLA, the size of the lenslet
Table 5-3 The comparison of wavefront measurement by our SHWS and commercial SHWS.
simply modifying the refractive index difference without changing the MLA size. The master mold could be reused to replicate new optical transparent polymers and different MLA materials could be applied, which results in different focal length MLAs. Hence, we could build distinctive characteristic SHWS according to different applications.
In order to verify the feasibility of this technique, we also fabricated an SU-8 (n = 1.59) polymer MLA which has shorter focal length of 2.6 mm. Figure 5-8 shows the wavefront measurement compared between the previous built SHWS with long focal length and the new SHWS with SU-8 MLA as the lenslet component. Both evaluated wavefronts were dominated by defocus aberration which corresponds to the parabola-shaped optical path difference between two wavefronts that was tangent at the vertices and had different curvature radii. The peak-to-valley difference between long focal length MLA SHWS and shorter focal length MLA SHWS is 0.433λ. The RMS wavefront of the long focal length MLA SHWS is λ/20 less than the shorter focal length MLA. In other word, long focal length MLA SHWS has higher accuracy and sensitivity than the expected 1.17λ wavefront in rms. As a result, another advantage of the proposed microlens is that we can make different MLAs without changing lens curvature or pitch size. That means that the spatial resolution can be kept the same while the sensitivity is changed.
5.5 Conclusions
A simple and easy method to extend the focal length of MLAs was experimentally demonstrated. A covered PDMS layer was room-temperature cured upon the MLA.
Because of the reduced refractive index difference, the light is less refracted and this results in a long focal length. We used the UV-resin microlens to implement the MLA and other materials, such as SU-8, can be employed with the same concept, resulting in
Figure 5-8 Wavefront measurement results of defocus by using (a) long focal length (UV-resin) and (b) shorter focal length (SU-8) MLA as our SHWS lenslet
component.
master mold by simply modifying the refractive index of the microlens material. In other words, we can achieve different MLAs without changing the curvature or pitch distance. The method in this work also provides a significant increment in the focal length and the development of long focal length MLAs has been successful both using UV-resin and SU-8 as the MLA material.
The fabricated MLAs were also compatible with a CMOS image sensor in constructing wavefront sensors. The wavefront measurement accuracy of our wavefront sensor was evaluated by a comparative measurement using a commercial wavefront sensor, which was less than λ/25 in RMS difference. On the other hand, we also compared a long focal length MLA and a shorter focal length MLA produced by the same method with different refractive index difference. The wavefront measurement result verifies that the long focal length MLA has higher accuracy and sensitivity and its RMS is λ/20 better than that of the shorter focal length MLA. Hence, our MLA fabrication method has high reproducibility and could be implemented to build SHWS with different specifications. Spatial resolution remains the same while sensitivity is changed. The proposed double layer microlens was proven to be suitable for wavefront sensing applications.
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