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LIGHT AND THE RETINAL IMAGE: KEY POINTS Light travels in (more or less) straight lines: the pinhole camera’s inverted image Enlarging the pinhole leads to BLUR How a lens prevents blur: refraction reunites light rays by bending them Point-to-point projection from object to inverted image Refraction: which way is light bent? Slowing in glass: lifeguard analogy. The eye: retina, lens and cornea; fovea, periphery and blind spot Focus errors; distant vision and near vision Myopia, hypermetropria, emmetropia, accommodation; emmetropization Visual angle and image size: (in radians) = size/distance (in degrees) = (180/ ) * size/distance (minutes of arc) = 60 * (180/ ) * size/distance Point spread function: width is 1 minute in visual angle, or 5 microns (.005 mm) Sources of light spread making the image imperfect: focus error; chromatic aberration; other aberrations; diffraction Direct observation of the image: Helmholtz’s ophthalmoscope Quality of the image: spread is about 5 microns (1 minute of arc) Visual resolution limit: about 1 minute of arc or 30 cpd (for 20/20 vision) Can vision be perfected?? William’s magic mirror and laser surgery Aliasing through sampling by the photoreceptor mosaic: Nyquist limit (60cpd)

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A Review of Optics Austin Roorda, Ph.D. University of Houston College of Optometry

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(Most of) these slides were prepared by Austin Roorda, (UC Berkeley Optometry School) and used by permission.

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Geometrical Optics Relationships between pupil size, refractive error and blur

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Optics of the eye: Depth of Focus 2 mm4 mm6 mm

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2 mm4 mm6 mm In focus Focused in front of retina Focused behind retina Optics of the eye: Depth of Focus

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7 mm pupil Bigger blur circle Courtesy of RA Applegate

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Smaller blur circle 2 mm pupil Courtesy of RA Applegate

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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 made 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.

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Physical Optics The Wavefront

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What is the Wavefront? converging beam = spherical wavefront parallel beam = plane wavefront

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What is the Wavefront? ideal wavefront parallel beam = plane wavefront defocused wavefront

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What is the Wavefront? parallel beam = plane wavefront aberrated beam = irregular wavefront ideal wavefront

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What is the Wavefront? aberrated beam = irregular wavefront diverging beam = spherical wavefront ideal wavefront

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The Wave Aberration

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What is the Wave Aberration? diverging beam = spherical wavefront wave aberration

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Wave Aberration of a Surface

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Diffraction

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Diffraction “Any deviation of light rays from a rectilinear path which cannot be interpreted as reflection or refraction” Sommerfeld, ~ 1894

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Diffraction and Interference diffraction causes light to bend perpendicular to the direction of the diffracting edge interference due to the size of the aperture causes the diffracted light to have peaks and valleys

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rectangular aperture square aperture

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Airy Disc circular aperture

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The Point Spread Function

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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.

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Airy Disc 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.

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Airy Disk

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0 0.5 1 1.5 2 2.5 12345678 pupil diameter (mm) separatrion between Airy disk peak and 1 st min (minutes of arc 500 nm light) As the pupil size gets larger, the Airy disc gets smaller.

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Point Spread Function vs. Pupil Size 1 mm2 mm3 mm4 mm 5 mm6 mm7 mm

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Small Pupil Little spreading due to defocus or aberrations So diffraction is limiting

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Larger pupil: Less diffraction (not shown) But more blur and more aberrations

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Aberrations

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1 mm2 mm3 mm4 mm 5 mm6 mm7 mm Point Spread Function vs. Pupil Size Perfect Eye (Diffraction Limited)

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pupil images followed by psfs for changing pupil size 1 mm2 mm3 mm4 mm 5 mm6 mm7 mm Point Spread Function vs. Pupil Size Typical Eye with aberrations

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Demonstration Observe Your Own Point Spread Function

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Resolution

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Rayleigh resolution limit Unresolved point sources Resolved

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Keck telescope: (10 m reflector) About 4500 times better than the eye! “Pupil” is 10M: almost no diffraction Wainscott

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Compound eye: Each facet must be large to fight diffraction Many facets (pixels) needed to capture details

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Convolution with the PSF

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Convolution

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Simulated Images 20/40 letters 20/20 letters

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MTF Modulation Transfer Function

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low mediumhigh object: 100% contrast image spatial frequency contrast 1 0

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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).

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MTF: Cutoff Frequency 0 0.5 1 050100150200250300 1 mm 2 mm 4 mm 6 mm 8 mm modulation transfer spatial frequency (c/deg) cut-off frequency Rule of thumb: cutoff frequency increases by ~30 c/d for each mm increase in pupil size

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Effect of Defocus on the MTF Charman and Jennings, 1976 450 nm 650 nm

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Relationships Between Wave Aberration, PSF and MTF

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Retinal Sampling

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Projected Image Sampled Image 5 arc minutes 20/20 letter Sampling by Foveal Cones

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5 arc minutes 20/5 letter Projected Image Sampled Image Sampling by Foveal Cones

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Nyquist Sampling Theorem

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Photoreceptor Sampling >> Spatial Frequency I 0 1 I 0 1 nearly 100% transmitted

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I 0 1 I 0 1 Photoreceptor Sampling = 2 x Spatial Frequency

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I 0 1 I 0 1 nothing transmitted Photoreceptor Sampling = Spatial Frequency

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Nyquist theorem: The maximum spatial frequency that can be detected is equal to ½ of the sampling frequency. foveal cone spacing ~ 120 samples/deg maximum spatial frequency: 60 cycles/deg (20/10 or 6/3 acuity)

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