LIGHT AND THE RETINAL IMAGE: KEY POINTS Light travels in (more or less) straight lines: the pinhole cameras 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: Helmholtzs 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?? Williams magic mirror and laser surgery Aliasing through sampling by the photoreceptor mosaic: Nyquist limit (60cpd) A Review of Optics Austin Roorda, Ph.D. University of Houston College of Optometry

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

Physical Optics The Wavefront What is the Wavefront? parallel beam = plane wavefront

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

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

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

diverging beam = spherical wavefront ideal wavefront aberrated beam = irregular wavefront

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

Wave Aberration of a Surface Wavefront Aberration mm (superior-inferior) 3 2 1

0 -1 -2 -3 -3 -2 -1

0 1 mm (right-left) 2 3 Diffraction

Diffraction Any deviation of light rays from a rectilinear path which cannot be interpreted as reflection or refraction Sommerfeld, ~ 1894 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 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 Airy Disk 1.22 a

separatrion between Airy disk peak and 1st min (minutes of arc 500 nm light) As the pupil size gets larger, the Airy disc gets smaller. 1.22

a 2.5 angle subtended at the nodal point 2

wavelength of the light 1.5 a pupil diameter 1 0.5 0

1 2 3 4 5

pupil diameter (mm) 6 7 8

Point Spread Function vs. Pupil Size 1 mm 5 mm 2 mm 3 mm

6 mm 4 mm 7 mm Small Pupil Little spreading due to defocus or aberrations So diffraction is limiting

Larger pupil: Less diffraction (not shown) But more blur and more aberrations Aberrations Pupil Size Perfect Eye (Diffraction

1 mm 2 mm 3 mm 4 mm Limited) 5 mm 6 mm

7 mm Point Spread Function vs. Pupil Size Typical Eye 1 mm 2 mmwith aberrations

3 mm 4 mm pupil images followed by 5 mm psfs for changing pupil size 6 mm

7 mm Demonstration Observe Your Own Point Spread Function Resolution Unresolved point sources

Rayleigh resolution limit Resolved Keck telescope: (10 m reflector)

About 4500 times better than the eye! Pupil is 10M: almost no diffraction Wainscott Compound eye: Each facet must

be large to fight diffraction Many facets (pixels) needed to capture details Convolution with the PSF Convolution

PSF ( x, y ) O( x, y ) I ( x, y ) Simulated Images 20/20 letters 20/40 letters MTF Modulation Transfer

Function low medium object: 100% contrast

contrast image 1 0 spatial frequency

high 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).

MTF: Cutoff Frequency cut-off frequency 1 mm 2 mm 4 mm 6 mm 8 mm

modulation transfer 1 0.5 f cutoff

a 57.3 Rule of thumb: cutoff frequency increases by ~30 c/d for each mm increase in pupil size

0 0 50 100 150 200 250 spatial frequency (c/deg) 300 Effect of Defocus on the MTF

450 nm 650 nm Charman and Jennings, 1976 Relationships Between Wave Aberration, PSF and MTF

Retinal Sampling Sampling by Foveal Cones Projected Image 20/20 letter Sampled Image

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

5 arc minutes Nyquist Sampling Theorem 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. foveal cone spacing ~ 120 samples/deg maximum spatial frequency: 60 cycles/deg (20/10 or 6/3 acuity)