© 2009 OSA/FiO/LS/AO/AIOM/COSI/LM/SRS 2009 a954_1.pdf JWB5.pdf JWB5.pdf
High Resolution Wavefront Sensing and Mirror Control for Vision Science by Quantitative Phase Imaging Alaster Meehan1,2, Phillip Bedggood 1, Brendan Allman2, Keith Nugent2, Andrew Metha1 1
2
Department of Optometry & Vision Sciences , School of Physics , the University of Melbourne, 3010, Australia Author e-mail address:
[email protected]
Abstract: Quantitative Phase Imaging displays attractive features for ocular wavefront aberrometry. An adaptive-optics mirror control algorithm for ophthalmoscopy is demonstrated that takes advantage of its superior lateral resolution and similar accuracy compared to HartmannShack systems. ©2009 Optical Society of America OCIS Codes: (100.5070) Phase retrieval, (110.1080) Active or adaptive optics
1. Introduction One of the most common wavefront sensors in the field of adaptive optics (AO) is the Hartmann-Shack (HS) wavefront sensor. The HS wavefront sensor is very reliable and accurate but is limited in resolution and range by the size and focal length of the lenslet array. It is sometimes desirable to have a higher resolution in a wavefront sensor particularly in the field of vision science where we may wish to study tear film effects, the precise effects of contact lenses and corneal damage that may occur during refractive surgery. High-resolution Adaptive Optics is also necessary to achieve enhanced imaging of the retina. Better retinal imaging can help us better understand the anatomy of a living eye and examine the effects of ageing and disease. In this paper we introduce the use of Quantitative Phase Imaging (QPI) as an alternative method for wavefront sensing. QPI can be used as a high resolution, fast and accurate wavefront sensor for adaptive optics in vision science. A new mirror control algorithm, useful for high-resolution wavefront sensors is also presented. 2. The QPI algorithm The QPI algorithm is based on a solution of the transport of intensity equation (TIE). The TIE explains the propagation of electromagnetic radiation and is primarily based on energy conservation. It is usefully expressed by a paraxial approximation that describes the energy flow in a particular direction represented by the z axis, x and y are coordinates perpendicular to energy flow. 2 π ∂I ( x , y ) = ∇ • (I ( x, y )∇ φ ( x, y ) ) λ ∂z
(1)
Here, λ is the wavelength of light, I(x,y) and φ (x,y) are the electromagnetic intensity and phase respectively, perpendicular to the direction of the energy flow, ∂I(x,y) is the derivative of intensity, it is calculated by taking the difference of two images slightly out of focus I1–I2. A full derivation of Equation 1, the TIE, is given by Teague [1]. A Fourier based solution to the phase term (Equation 2) was developed and described by Paganin and Nugent [2]. −1 φ x = − kF φ = φx + φy φ = − k F −1 y
kx 1 k F F − 1 x2 F ∂ z I ( x , y ) k r2 I ( x , y ) kr
(2)
ky k 1 F F − 1 y2 F ∂ z I ( x , y ) k r2 I ( x , y ) kr
Here F-1 denotes the inverse Fourier transform, F denotes forward Fourier transform and kx and ky are the Fourier variables where kr2 = kx2 + ky2. This is a direct calculation of the phase of the wavefront, unlike other curvature sensors that use an iterative process to solve the wavefront [3], hence it is much faster to compute. The nature of the algorithm is also highly separable with only FFTs and image multiplications where each pixel is independent. The separable nature QPI lends itself to parallel computing. For example, computing QPI on a Graphics Processing Unit (GPU) achieves a ~10-fold increase in speed and gets 10 Mega Pixels per second output.
© 2009 OSA/FiO/LS/AO/AIOM/COSI/LM/SRS 2009 a954_1.pdf JWB5.pdf JWB5.pdf
3. Comparisons between QPI and Hartmann-Shack for wavefront aberrometry We compared HS and QPI wavefront sensors simultaneously using beam-splitter in the optical path of our AO ophthalmoscope. We passed 655nm collimated laser light through trial lenses placed at the pupil plane, conjugate with an adaptive mirror, to present to the sensors a range of quantified wavefront profiles. The pupil diameter was 5mm. The QPI wavefront sensor consisted of two IMI tech IMB-147FT cameras and a beam-splitter at a defocus difference of 10mm. The HS sensor used a Pulnix TM-1020 Camera with lenslets spaced at 0.4mm and 24mm focal length. Cylinderical Lenses
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Fig. 1. QPI and HS results (derived dioptric power) for simultaneous wavefront measurement of spherical (left) and cylindrical (right) trial lenses placed in the pupil plane. The adaptive mirror was held flat for these measurements. Standard errors of the mean for these data are smaller than the symbol.
The test-retest repeatability was similar for both HS and QPI with all Zernike coefficients differing by less than 0.005µm (~λ/100). The range of lens powers measurable was ±20D for QPI compared to about ±16D with HS, corresponding to ±32µm dynamic range (~100 λ). Sensitivity and the dynamic range of QPI can be varied by changing the defocus. Overall the two wavefront sensors had similar performance with the exception of lateral resolution, an example of which can be seen in Fig 2.
(a) (b) (c) (d) Fig. 2. (a) A highly aberrated wavefront measured with QPI. (b) The same highly aberrated wavefront measured with HS. (c) A flat wavefront of about 0.02um RMS. Small dimples can be seen where the actuators are. (d) The same flat wavefront measured with HS. Note: All scales in µm. The HS images are produced from Zernike polynomials fitted to HS spot displacements.
The highly aberrated wavefronts are a result of ‘poking’ random actuators. These high level localised distortions resulted in some HS spots not being identified or severely distorted. These missing and distorted spots may explain the slightly higher peaks and troughs measured on the wavefront. QPI has no problems with distortions of this type. The enhanced resolution of the QPI wavefront sensor also revealed some small dimples on the DM surface where the actuators are positioned, these dimples are not do not appear in HS wavefront sensing. These dimples are still present when the mirror is switched off, and vary between 50nm and 100nm in depth. Inspecting the fine detail in deformable mirrors becomes possible with a high resolution wavefront sensor. 4. Adaptive Mirror Control Our AO mirror was a MIRAO52 from Imagine Eyes with 52 actuators. The resolution of the QPI wavefront sensor is limited by the pixel density (696x520; approximately 250,000 data points in a 5mm pupil). Mirror control algorithms for adaptive optics are based on relating wavefront measurements to ‘influence functions’, which can be in terms of Zernike polynomials [4], HS spot displacements [5], or any other measure of the result from ‘poking’ actuators.
© 2009 OSA/FiO/LS/AO/AIOM/COSI/LM/SRS 2009 a954_1.pdf JWB5.pdf JWB5.pdf
A QPI-based control algorithm using a 10th order Zernike reconstruction matrix resulted in ‘waffling’, a product of the Zernike representation under-sampling the wavefront [4]. To avoid this we have developed a new algorithm where the wavefront W was fitted directly with each of the actuator influence functions I. These influence functions are the result of QPI wavefront measurements by ‘poking’ each actuator in turn (Figure 3). However unlike Zernike polynomials the influence functions are not orthogonal. Hence, to get the actuator voltages vi, we form the matrix cross-product of the vector (Ii•W) and the inverse of an interaction matrix, A, describing how each influence function interacts with every other influence function:
v i = A−1 × [Ii •W ]
(3)
The interaction matrix A is n×n, where n is the number of actuators. Since it is square, a regular matrix inversion can used rather that a pseudo-inverse. A-1 is precomputed by finding the dot product of each influence function with every other influence function. (4) Ai , j = I i • I j
Fig. 3. A selection of influence functions for actuators i = 1, 24, 25, 30, 45
Tilt and tip were not corrected and as these terms do not affect image quality, but rather just shift the image. The tilt and tip components were removed from both the wavefront and influence functions. Due to the high resolution of the wavefront sensor there was no waffle. This algorithm has shown to be very stable and apart from tilt and tip there is no need to suppress any other modes so the mirror can be used to its full capability. Even with a control-loop gain close to 1 the algorithm is stable although a slightly lower RMS can be achieved with the gain set to around 0.4-0.5, but high values for the gain can be used for faster convergence. For a pupil diameter of 5mm we can achieve a RMS of 0.01~0.02 um. Our closed loop control operates at about 5Hz in Matlab™. This algorithm can be adapted for use with a HS wavefront sensor where the spot displacements are used as influence functions. The computational order of this algorithm is nm+m2 where n is the number of data points (spots or pixels) and m is the number of actuators on the mirror. Since we have many more wavefront data points than mirror actuators, the order can be approximated by nm. This is the same complexity as fitting Zernike polynomials when the number of Zernike polynomials is approximately equal to the number of actuators. 5. Conclusions As a method of wavefront sensing, QPI demonstrates abilities similar to the HS wavefront sensor but with a superior lateral resolution. Compared to other high-resolution wavefront sensors, such as curvature sensors, QPI is faster to compute. We have also developed a very effective mirror control algorithm that makes use of the high resolution. A higher resolution wavefront sensor may also bring benefits in monitoring and controlling the deformable mirrors used in adaptive optics. Sometimes the actuators that are used to adjust the mirrors shape can induce high order aberrations where they interact with the mirror surface that may not be accurately detected by a HS sensor. 6. References Teague, M.R., Deterministic phase retrieval: a Green’s function solution. J.Opt.Soc.Am, 1983. 73: p. 1434-1441. Paganin, D. and K.A. Nugent, Non-interferometric phase imaging using partially coherent light. Phys Rev Lett, 1998(80): p. 2586-2589. Roddier, F. and C. Roddier, Wavefront reconstruction using iterative Fourier transforms. Appl. Opt., 1991. 30: p. 1325. Porter, J., et al., Adaptive Optics for Vision Science. 2006: Wiley. Goncharov, A.V., et al., Laboratory MCAO test-bed for developing wavefront sensing concepts. Optics Express, 2005 / Vol. 13, No. 14 /. 13: p. 5580-5590. Acknowledgments: ARC Industry Linkage Grant Lp0454885 with Iatia Ltd, www.iatia.com.au, and a University of Melbourne Science Faculty Research and Development Grant (2008). 1. 2. 3. 4. 5.
