ReviewReview ofof BasicBasic PrinciplesPrinciples inin Optics,Optics, WavefrontWavefront andand WavefrontWavefront ErrorError Austin Roorda, Ph.D. University of California, Berkeley Google my name to find copies of these slides for free use and distribution Geometrical Optics Relationships between pupil size, refractive error and blur Optics of the eye: Depth of Focus 2 mm 4 mm 6 mm Optics of the eye: Depth of Focus Focused behind retina In focus Focused in front of retina 2 mm 4 mm 6 mm Demonstration Role of Pupil Size and Defocus on Retinal Blur Draw a cross like this one on a page. Hold it so close that is it completely out of focus, then squint. You should see the horizontal line become clear. The line becomes clear because you have used your eyelids to make your effective pupil size smaller, thereby reducing the blur due to defocus on the retina image. Only the horizontal line appears clear because you have only reduced the blur in the horizontal direction. ComputationComputation ofof GeometricalGeometrical BlurBlur SizeSize blur[mrad][]blur[minutes]3.44[]DpupilsizemmDpupilsizemm=×=×× where D is the defocus in diopters ApplicationApplication ofof BlurBlur EquationEquation • 1 D defocus, 8 mm pupil produces 27.52 minute blur size ~ 0.5 degrees Physical Optics The Wavefront WhatWhat isis thethe Wavefront?Wavefront? parallel beam = converging beam plane wavefront = spherical wavefront WhatWhat isis thethe Wavefront?Wavefront? parallel beam ideal wavefront = plane wavefront defocused wavefront WhatWhat isis thethe Wavefront?Wavefront? parallel beam ideal wavefront = plane wavefront aberrated beam = irregular wavefront WhatWhat isis thethe Wavefront?Wavefront? diverging beam = aberrated beam spherical wavefront = irregular wavefront ideal wavefront The Wave Aberration WhatWhat isis thethe WaveWave AberrationAberration?? diverging beam = spherical wavefront wave aberration WaveWave Aberration:Aberration: DefocusDefocus Wavefront Aberration 3 2 1 0 -1 mm (superior-inferior) -2 -3 -3 -2 -1 0 1 2 3 mm (right-left) WaveWave Aberration:Aberration: ComaComa Wavefront Aberration 3 2 1 0 -1 mm (superior-inferior) -2 -3 -3 -2 -1 0 1 2 3 mm (right-left) WaveWave Aberration:Aberration: AllAll TermsTerms Wavefront Aberration 3 2 1 0 -1 mm (superior-inferior) -2 -3 -3 -2 -1 0 1 2 3 mm (right-left) Zernike Polynomials WaveWave AberrationAberration ContourContour MapMap 2 1.5 1 0.5 0 -0.5 -1 mm (superior-inferior) -1.5 -2 -2.5 -2 -1 0 1 2 mm (right-left) BreakdownBreakdown ofof ZernikeZernike TermsTerms Coefficient value (microns) -0.5 0 0.5 1 1.5 2 1 2 3 astig. nd 4 defocus 2 order 5 astig. 6 trefoil 7 coma rd 8 coma 3 order 9 trefoil 10 11 12 spherical aberration 4th order Zernike term 13 14 15 16 17 5th order 18 19 20 The Point Spread Function The Point Spread Function, or PSF, is the image that an optical system forms of a point source. The point source is the most fundamental object, and forms the basis for any complex object. The PSF is analogous to the Impulse Response Function in electronics. TheThe PointPoint SpreadSpread FunctionFunction The PSF for a perfect optical system is the Airy disc, which is the Fraunhofer diffraction pattern for a circular pupil. Airy Disc AiryAiry DiskDisk 1.22aλθ⋅= angle subtended at the nodal point wavelength of the light pupil diameteraθλ≡≡≡ θ As the pupil size gets larger, the Airy disc gets smaller. angle subtended at the nodal point wavelength of the light pupil diameter1.22aaθλλθ≡≡≡⋅= 2.5 2 (minutes) 1.5 radius 1 Disk 0.5 Airy 0 PSF 1 2 3 4 5 6 7 8 pupil diameter (mm) PointPoint SpreadSpread FunctionFunction vs.vs. PupilPupil SizeSize 1 mm 2 mm 3 mm 4 mm 5 mm 6 mm 7 mm Small Pupil PointPoint SpreadSpread FunctionFunction vs.vs. PupilPupil SizeSize 1 mm 2 mm 3 mm 4 mm 5 mm 6 mm 7 mm Perfect Eye Typical Eye Larger pupil Resolution Unresolved point sources Rayleigh resolution limit Resolved As the pupil size gets larger, the Airy disc gets smaller. minmin angle subtended at the nodal point wavelength of the light pupil diameter1.22aaθλλθ≡≡≡⋅= 2.5 2 (minutes) 1.5 radius 1 Disk 0.5 Airy 0 PSF 1 2 3 4 5 6 7 8 pupil diameter (mm) Keck telescope: (10 m reflector) About 4500 times better than the eye! Convolution Convolution (,) (,) (,)PSFxyOxyIxy⊗= SimulatedSimulated ImagesImages 20/20 letters 20/40 letters MTF Modulation Transfer Function low medium high object: 100% contrast image 1 contrast 0 spatial frequency MTF:MTF: CutoffCutoff FrequencyFrequency cut-off frequency 57.3cutoffafλ=⋅ 1 mm 1 2 mm 4 mm 6 mm Rule of thumb: cutoff 8 mm frequency increases by ~30 c/d for each mm 0.5 increase in pupil size modulation transfer 0 0 50 100 150 200 250 300 spatial frequency (c/deg) ModulationModulation TransferTransfer FunctionFunction 0.8 0.6 0.4 0.2 vertical spatial horizontal spatial -100 0 100 frequency (c/d) c/deg frequency (c/d) PTF Phase Transfer Function low medium high object image 180 0 phase shift -180 spatial frequency PhasePhase TransferTransfer FunctionFunction • Contains information about asymmetry in the PSF • Contains information about contrast reversals (spurious resolution) Relationships Between Wave Aberration, PSF and MTF The PSF is the Fourier Transform (FT) of the pupil function ()2(,),(,)iWxyiiPSFxyFTPxyeπλ−= The MTF is the amplitude component of the FT of the PSF (){},(,)xyiiMTFffAmplitudeFTPSFxy= The PTF is the phase component of the FT of the PSF (){},(,)xyiiPTFffPhaseFTPSFxy= The OTF (MTF and PTF) can also be computed as the autocorrelation of the pupil function Wavefront Aberration Point Spread Function 0.5 0 -0.5 -2 -1 0 1 2 -200 -100 0 100 200 mm (right-left) arcsec Modulation Transfer Function Phase Transfer Function 0.8 0.5 0.6 0 0.4 -0.5 0.2 -100 0 100 -100 0 100 c/deg c/deg Wavefront Aberration Point Spread Function 0.5 0 -0.5 -2 -1 0 1 2 -200 -100 0 100 200 mm (right-left) arcsec Modulation Transfer Function Phase Transfer Function 150 0.8 100 50 0.6 0 0.4 -50 0.2 -100 -150 -100 0 100 -100 0 100 c/deg c/deg Wavefront Aberration Point Spread Function 1.5 1 0.5 0 -0.5 -2 -1 0 1 2 -1000 -500 0 500 1000 mm (right-left) arcsec Modulation Transfer Function Phase Transfer Function 150 0.8 100 50 0.6 0 0.4 -50 0.2 -100 -150 -100 0 100 -100 0 100 c/deg c/deg Conventional Metrics to Define Imagine Quality RootRoot MeanMean SquareSquare ()()()()()21,, pupil area, wave aberration, average wave aberrationRMSWxyWxydxdyAAWxyWxy=−−−−∫∫ RootRoot MeanMean Square:Square: AdvantageAdvantage ofof UsingUsing ZernikesZernikes toto RepresentRepresent thethe WavefrontWavefront ()()()()222220212223.......RMSZZZZ−−=+++ …… trefoil term defocus term astigmatism term astigmatism term StrehlStrehl RatioRatio diffraction-limited PSF Strehl Ratio = eyedlHH Hdl actual PSF Heye ModulationModulation TransferTransfer FunctionFunction 1 0.9 20/20 20/10 0.8 0.7 0.6 Area under the MTF 0.5 contrast 0.4 0.3 0.2 0.1 0 0 50 100 150 spatial frequency (c/deg) MetricsMetrics toto DefineDefine ImageImage QualityQuality Other Metrics Campbell,C.E. (2004). Improving visual function diagnostic metrics with the use of higher-order aberration information from the eye. J.Refract.Surg. 20, S495-S503 Cheng,X., Bradley,A., Hong,X., & Thibos,L. (2003). Relationship between refractive error and monochromatic aberrations of the eye. Optom.Vis.Sci. 80, 43-49. Cheng,X., Bradley,A., & Thibos,L.N. (2004). Predicting subjective judgment of best focus with objective image quality metrics. J.Vis. 4, 310-321. Guirao,A. & Williams,D.R. (2003). A method to predict refractive errors from wave aberration data. Optom.Vis.Sci. 80, 36-42. Marsack,J.D., Thibos,L.N., & Applegate,R.A. (2003). Scalar metrics of optical quality derived from wave aberrations predict visual performanc. J.Vis. 4, 322-328. Sarver,E.J. & Applegate,R.A. (2004). The importance of the phase transfer function to visual function and visual quality metrics. J.Refract.Surg. 20, S504-S507 TypicalTypical ValuesValues forfor WaveWave AberrationAberration Strehl Ratio • Strehl ratios are about 5% for a 5 mm pupil that has been corrected for defocus and astigmatism. • Strehl ratios for small (~ 1 mm) pupils approach 1, but the image quality is poor due to diffraction. TypicalTypical ValuesValues forfor WaveWave AberrationAberration Population Statistics trefoil coma coma trefoil spherical aberration TypicalTypical ValuesValues forfor WaveWave AberrationAberration Change in aberrations with pupil size 1.2 Shack Hartmann Methods Iglesias et al, 1998 Other Methods Navarro et al, 1998 1 Liang et al, 1994 Liang and Williams, 1997 Liang et al, 1997 (microns) 0.8 Walsh et al, 1984 He et al, 1999 Calver et al, 1999 0.6 Calver et al, 1999 Porter et al., 2001 He et al, 2002 aberration 0.4 He et al, 2002 Xu et al, 2003 Paquin et al, 2002 wave 0.2 Paquin et al, 2002 Carkeet et al, 2002 rms Cheng et al, 2004 0 0 1 2 3 4 5 6 7 8 9 pupil size (mm) TypicalTypical ValuesValues forfor WaveWave AberrationAberration Change in aberrations with age Monochromatic Aberrations as a Function of Age, from Childhood to Advanced Age Isabelle Brunette,1 Juan M. Bueno,2 Mireille Parent,1,3 Habib Hamam,3 and Pierre Simonet3 Other Optical Factors that Degrade Image Quality Retinal Sampling SamplingSampling byby FovealFoveal ConesCones Projected Image Sampled Image 20/20 letter 5 arc minutes SamplingSampling byby FovealFoveal ConesCones Projected Image Sampled Image 20/5 letter 5 arc minutes NyquistNyquist SamplingSampling TheoremTheorem Photoreceptor Sampling >> Spatial Frequency 1 I 0 1 I 0 nearly 100% transmitted Photoreceptor Sampling = 2 x Spatial Frequency 1 I 0 1 I 0 nearly 100% transmitted Photoreceptor Sampling = Spatial Frequency 1 I 0 1 I 0 nothing transmitted Nyquist theorem: The maximum spatial frequency that can be detected is equal to _ of the sampling frequency.