Psych 221/EE362 Applied Vision and Imaging Systems Zernike Polynomials and Their Use in Describing the Wavefront Aberrations of the Human Eye Psych 221/EE362 Applied Vision and Imaging Systems Course Project, Winter 2003 Patrick Y. Maeda pmaeda@stanford.edu Stanford University

Introduction and Motivation Great interest in correcting higher order aberrations of the eye Laser eye surgery (PRK, LASIK) Currently, only defocus and astigmatism being corrected (2nd order aberrations) Improve vision better than 20/20 Correct problems caused or induced by current generation of laser surgery Imaging of the retina and other structures in the eye using adaptive optics Correction requires measurement of optical aberrations Defocus and astigmatism can be determined using sets of lenses Measurement of higher orders require more sophisticated techniques Measurement of the wavefront aberration with Shack-Hartmann Wavefront Sensor Mathematical description of the aberrations needed Accurate description of wave aberration function Accurate estimation of wave aberration function from measurement data

Project Outline Introduction/Motivation General Optical System Description Monochromatic Wavefront Aberrations PSF and MTF calculations Why Use Zernike Polynomials? Definition of Zernike Polynomials Describing Wave Aberrations using Zernike Polynomials Simulating the Effects of Wave Aberrations Wavefront Measurement and Data Fitting with Zernike Polynomials Conclusion, References, Source Code Appendix

Coordinate Systems

Reference Spherical Wavefront Wave Aberration Exit Pupil Image Plane Aberrated Wavefront Reference Spherical Wavefront Wave Aberration W(x,y) y z x The wavefront aberration, W(x,y), is the distance, in optical path length (product of the refractive index and path length), from the reference sphere to the wavefront in the exit pupil measured along the ray as a function of the transverse coordinates (x,y) of the ray intersection with the reference sphere. It is not the wavefront itself but it is the departure of the wavefront from the reference sphere.

Describing Optical Aberrations Optical system aberrations have historically been described, characterized, and catalogued by power series expansions Many optical systems have circular pupils Application of experimental results typically require data fitting It is, therefore, desirable to expand the wave aberration in terms of a complete set of basis functions that are orthogonal over the interior of a circle

Why Use Zernike Polynomials? Zernike polynomials form a complete set of functions or modes that are orthogonal over a circle of unit radius Convenient for serving as a set of basis functions Expressible in polar coordinates or Cartesian coordinates Scaled so that non-zero order modes have zero mean and unit variance Puts modes in a common reference frame for meaningful relative comparison Other power series descriptions are not orthogonal Wave aberrations in an optical system with a circular pupil accurately described by a weighted sum of Zernike polynomials The Orthonormal set of Zernike polynomials is recommended for describing wave aberration functions and for data fitting of experimental measurements for the eye7 Terms are normalized so that the coefficient of a particular term or mode is the RMS contribution of that term

Mathematical Formulae 3 factorial.m zernike.m

List of Zernike Polynomials 7, 9,10

Wave Aberration Description WaveAberration.m

Double-Index Zernike Polynomials Azimuthal Frequency, m -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 Radial Order, n 1 2 3 4 5 6 Common Names7 Piston Tilt Astigmatism (m=-2,2), Defocus(m=0) Coma (m=-1,1), Trefoil(m=-3,3) Spherical Aberration (m=0) Secondary Coma (m=-1,1) Secondary Spherical Aberration (m=0) ZernikePolynomial.m

Double-Index Zernike Polynomial PSFs Azimuthal Frequency, m -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 Radial Order, n 1 2 3 4 5 6 Common Names7 Piston Tilt Astigmatism (m=-2,2), Defocus(m=0) Coma (m=-1,1), Trefoil(m=-3,3) Spherical Aberration (m=0) Secondary Coma (m=-1,1) Secondary Spherical Aberration (m=0) ZernikePolynomialPSF.m

Double-Index Zernike Polynomial MTFs Azimuthal Frequency, m -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 Radial Order, n 1 2 3 4 5 6 Common Names7 Piston Tilt Astigmatism (m=-2,2), Defocus(m=0) Coma (m=-1,1), Trefoil(m=-3,3) Spherical Aberration (m=0) Secondary Coma (m=-1,1) Secondary Spherical Aberration (m=0) Pupil Diameter = 4 mm 0 to 50 cycles/degree = 570 nm RMS wavefront error = 0.2 ZernikePolynomialMTF.m MTFy MTFx

Double-Index Zernike Polynomial MTFs Azimuthal Frequency, m -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 Radial Order, n 1 2 3 4 5 6 Common Names7 Piston Tilt Astigmatism (m=-2,2), Defocus(m=0) Coma (m=-1,1), Trefoil(m=-3,3) Spherical Aberration (m=0) Secondary Coma (m=-1,1) Secondary Spherical Aberration (m=0) Pupil Diameter = 7.3 mm 0 to 50 cycles/degree = 570 nm RMS wavefront error = 0.2 ZernikePolynomialMTF.m MTFy MTFx

Simulation based on Human Eye Data WaveAberrationMTF.m WaveAberration.m WaveAberrationPSF.m

Measurement Setup Pupil Retina Iris Real Aberrated Wavefront Ideal Planar Wavefront y z x Incoming Light Beam

Shack-Hartmann Sensor Layout CCD Pupil Relay Optics PBS Light Source Lenslet Array

Shack-Hartmann Wavefront Sensor Lenslet Array Focal Length f y(x1, y1) y(x1, y2) y(x1, y3) y(x1, y4) Aberrated Wavefront

Data Fitting with Zernike Polynomials Equations (14) and (15) can be used to determine the Wj’s using Least-squares Estimation

Least-squares Estimation

Benefits of Orthogonality

Conclusions Zernike Polynomials well suited for Describing wave aberration functions of optical systems with circular pupils Estimation of wave aberration coefficients from wavefront measurements Able to integrate Psych 221 learning with material from optical systems and Fourier optics courses Linear systems theory make image formation and image quality evaluation straightforward Suggestions for future work Extend simulation to incorporate chromatic effects Investigate the how wave aberration changes with accommodation Conduct simulations on a wide set of patient data Simulate the higher order aberrations induced by the PRK and LASIK Research some of the new wavefront technologies like implantable lenses

References [1] MacRae, S. M., Krueger, R. R., Applegate, A. A., (2001), Customized Corneal Ablation, The Quest for SuperVision, Slack Incorporated. [2] Williams, D., Yoon, G. Y., Porter, J., Guirao, A., Hofer, H., Cox, I., (2000), “Visual Benefits of Correcting Higher Order Aberrations of the Eye,” Journal of Refractive Surgery, Vol. 16, September/October 2000, S554-S559. [3] Thibos, L., Applegate, R.A., Schweigerling, J.T., Webb, R., VSIA Standards Taskforce Members (2000), "Standards for Reporting the Optical Aberrations of Eyes," OSA Trends in Optics and Photonics Vol. 35, Vision Science and its Applications, Lakshminarayanan,V. (ed) (Optical Society of America, Washington, DC), pp: 232-244. [4] Goodman, J. W. (1968). Introduction to Fourier Optics. San Francisco: McGraw Hill [5] Gaskill, J. D. (1978). Linear Systems, Fourier Transforms, Optics. New York: Wiley [6] Fischer, R. E. (2000). Optical System Design. New York: McGraw Hill [7] Thibos, L. N.(1999), Handbook of Visual Optics, Draft Chapter on Standards for Reporting Aberrations of the Eye. http://research.opt.indiana.edu/Library/HVO/Handbook.html [8] Bracewell, R. N. (1986). The Fourier Transform and Its Applications. McGraw Hill [9] Mahajan, V. N. (1998). Optical Imaging and Aberrations, Part I Ray Geometrical Optics, SPIE Press [10] Liang, L., Grimm, B., Goelz, S., Bille, J., (1994), “Objective Measurement of Wave Aberrations of the Human Eye with the use of a Hartmann-Shack Wave-front Sensor,” J. Opt. Soc. Am. A, Vol. 11, No. 7, 1949-1957. [11] Liang, L., Williams, D. R., (1997), “Aberration and Retinal Image Quality of the Normal Human Eye,” J. Opt. Soc. Am. A, Vol. 14, No. 11, 2873-2883.

Appendix I Matlab Source Code Files: zernike.m ZernikePolynomial.m ZernikePolynomialPSF.m ZernikePolynomialMTF.m WaveAberration.m WaveAberrationPSF.m WaveAberrationMTF.m