1. 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
Stanford University

2. 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

3. 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

4. Coordinate Systems

5. 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.

6. 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

7. 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

8. Mathematical Formulae 3
factorial.m
zernike.m

9. List of Zernike Polynomials 7, 9,10

10. Wave Aberration Description
WaveAberration.m

11. Double-Index Zernike Polynomials
Azimuthal Frequency, m
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

12. Double-Index Zernike Polynomial PSFs
Azimuthal Frequency, m
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

13. Double-Index Zernike Polynomial MTFs
Azimuthal Frequency, m
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

14. Double-Index Zernike Polynomial MTFs
Azimuthal Frequency, m
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

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

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

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

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

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

20. Least-squares Estimation

21. Benefits of Orthogonality

22. 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

23. 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:
[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.
[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,
[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,

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