Optics of Wavefront Austin Roorda, Ph.D. University of Houston College of Optometry
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
7 mm pupil Bigger blur circle Courtesy of RA Applegate
2 mm pupil Smaller blur circle Courtesy of RA Applegate
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.
Physical Optics The Wavefront
What is the Wavefront? parallel beam = plane wavefront converging beam = spherical wavefront
What is the Wavefront? parallel beam = plane wavefront ideal wavefront defocused wavefront
What is the Wavefront? parallel beam = plane wavefront ideal wavefront aberrated beam = irregular wavefront
What is the Wavefront? diverging beam = spherical wavefront aberrated beam = irregular wavefront ideal wavefront
The Wave Aberration
What is the Wave Aberration? diverging beam = spherical wavefront wave aberration
Wave Aberration: Defocus mm (superior-inferior) 3 2 1 0-1 -2-3 Wavefront Aberration -3-2 -1 0 1 2 3 mm (right-left)
Wave Aberration: Astigmatism mm (superior-inferior) 3 2 1 0-1 -2-3 Wavefront Aberration -3-2 -1 0 1 2 3 mm (right-left)
Wave Aberration: Coma 3 Wavefront Aberration mm (superior-inferior) 2 1 0-1 -2-3 -3-2 -1 0 1 2 3 mm (right-left)
Wave Aberration: All Terms mm (superior-inferior) 3 2 1 0-1 -2-3 Wavefront Aberration -3-2 -1 0 1 2 3 mm (right-left)
Zernike Polynomials
Wave Aberration Contour Map 2 1.5 mm (superior-inferior) 1 0.5 0-0.5-1 -1.5-2 -2.5-2 -1 0 1 2 mm (right-left)
Breakdown of Zernike Terms Zernike term -0.5 0 0.5 1 1.5 2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Coefficient value (microns) astig. defocus astig. trefoil coma coma trefoil spherical aberration 2 nd order 3 rd order 4 th order 5 th order
Diffraction
Diffraction Any deviation of light rays from a rectilinear path which cannot be interpreted as reflection or refraction Sommerfeld, ~ 1894
Fraunhofer Diffraction Also called far-field diffraction Occurs when the screen is held far from the aperture. Occurs at the focal point of a lens!
Diffraction and Interference diffraction causes light to bend perpendicular to the direction of the diffracting edge interference causes the diffracted light to have peaks and valleys
rectangular aperture square aperture
circular aperture Airy Disc
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.
The Point Spread Function The PSF for a perfect optical system is the Airy disc, which is the Fraunhofer diffraction pattern for a circular pupil. Airy Disc
1.22alq = Airy Disk angle subtended at the nodal point wave q
As the pupil size gets larger, the Airy disc gets smaller. angle subtended at the nodal point wave PSF Airy Disk radius (minutes) 2.5 2 1.5 1 0.5 0 1 2 3 4 5 6 7 8 pupil diameter (mm)
Point Spread Function vs. Pupil Size 1 mm 2 mm 3 mm 4 mm 5 mm 6 mm 7 mm
Small Pupil
Larger pupil
Point Spread Function vs. Pupil Size Perfect Eye 1 mm 2 mm 3 mm 4 mm 5 mm 6 mm 7 mm
Point Spread Function vs. Pupil Size Typical Eye 1 mm 2 mm 3 mm 4 mm pupil images followed by psfs for changing pupil size 5 mm 6 mm 7 mm
Demonstration Observe Your Own Point Spread Function
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 wa PSF Airy Disk radius (minutes) 2.5 2 1.5 1 0.5 0 1 2 3 4 5 6 7 8 pupil diameter (mm)
uncorrected corrected AO image of binary star k-peg on the 3.5-m telescope at the Starfire Optical Range q -9 1.22 l 1.22 900 10 = 0.064 seconds of a 3.5 About 1000 times better than the eye! min = = arc
Keck telescope: (10 m reflector) About 4500 times better than the eye!
Convolution
Convolution (,) (,) (,)PSFxyOxyIxyƒ=
Simulated Images 20/20 letters 20/40 letters
MTF Modulation Transfer Function
low medium high object: 100% contrast image contrast 1 0 spatial frequency
The modulation transfer function (MTF) indicates the ability of an optical system to reproduce (transfer) various levels of detail (spatial frequencies) from the object to the image. Its units are the ratio of image contrast over the object contrast as a function of spatial frequency. It is the optical contribution to the contrast sensitivity function (CSF).
modulation transfer 1 0.5 0 MTF: Cutoff Frequency 1 mm 2 mm 4 mm 6 mm 8 mm 0 50 100 150 200 250 300 spatial frequency (c/deg) cut-off frequency 57.3cutoffafl= Rule of thumb: cutoff frequency increases by ~30 c/d for each mm increase in pupil size
PTF Phase Transfer Function
low medium high object image phase shift 180 0-180 spatial frequency
Relationships Between Wave Aberration, PSF and MTF
()2(,),(,)iWxyiiPSFxyFTPxyepl-Ï =Ì Ó The PSF is the Fourier Transform (FT) of the pupil function The MTF is the amplitude component of the FT of the PSF (){},(,)xyiimtfffamplitudeftpsfxy=è Î The PTF is the phase component of the FT of the PSF (){},(,)xyiiptfffphaseftpsfxy=è Î
Adaptive Optics Flattens the Wave Aberration AO OFF AO ON
Conventional Metrics to Define Imagine Quality
Root Mean Square Root Mean Square ()()()()()21,, pupil area, wave aberratio
Root Mean Square: Advantage of Using Zernikes to Represent the Wavefront ()()()()222220212223...RMSZZZZ--=+++ astigmatism term defocus term astigmatism term trefoil term
diffraction-limited PSF Strehl Ratio Strehl Ratio = eyedlhh H dl actual PSF H eye
Modulation Transfer Function contrast 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 20/20 20/10 Area under the MTF 0 50 100 150 spatial frequency (c/deg)
Other Metrics Volume under the MTF Image Plane Metrics (Strehl ratio) vs Pupil Plane Metrics (eg RMS)
Other Optical Factors that Degrade Image Quality
Retinal Sampling
Sampling by Foveal Cones Projected Image Sampled Image 20/20 letter 5 arc minutes
Sampling by Foveal Cones Projected Image Sampled Image 20/5 letter 5 arc minutes
Nyquist Sampling Theorem
1 Photoreceptor Sampling >> Spatial Frequency I 0 1 I 0 nearly 100% transmitted
1 Photoreceptor Sampling = 2 x Spatial Frequency I 0 1 I 0 nearly 100% transmitted
1 Photoreceptor Sampling = Spatial Frequency I 0 1 I 0 nothing transmitted
Nyquist theorem: The maximum spatial frequency that can be detected is equal to _ of the sampling frequency. foveal cone spacing ~ 120 samples/deg maximum spatial frequency: 60 cycles/deg (20/10 or 6/3 acuity)
Thankyou!