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

These slides were prepared by Austin Roorda, except where otherwise noted. Full permission is granted to anyone who would like to use any or all of these slides for educational purposes.

Relationships between pupil size, refractive error and blur Geometrical Optics Relationships between pupil size, refractive error and blur Start with some of the most basic, but profound effects on image quality, which can be explained entirely with geometrical optics. These have to do with the relationships between pupil size, refractive error and blur. This is the simplest thing to learn, but you could argue that it’s the most serious cause for image quality degradation. After all, how many people are accurately refracted?

Optics of the eye: Depth of Focus When refractive errors are present, then large pupils experience more blur. 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

Courtesy of RA Applegate 7 mm pupil Bigger blur circle Courtesy of RA Applegate

Courtesy of RA Applegate 2 mm pupil Smaller blur circle Courtesy of RA Applegate

Role of Pupil Size and Defocus on Retinal Blur Demonstration Role of Pupil Size and Defocus on Retinal Blur When do you experience this? You discover this when you are giving a lecture, and you dim the lights. The people in the of the class suddenly have difficulty reading the board. Not because they developed a refractive error. But because their pupils have opened and the refractive error that they can deal with when the light are on produces an intolerable blur with large pupil. We should never forget these fundamental and simple relationships for image quality. We should not think that aberrations, PSFs, MTFs are so much more important. But we are at a point where the next levels in understanding image quality and how to improve it requires a different way of thinking about how light works. We have to think of light as a wave, and how it interferes and diffracts. 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 To start with, we will describe the wavefront. This is the one of the most fundamental and useful description of the optical properties of the eye, from which most of the image quality metrics can be derived.

What is the Wavefront? parallel beam = plane wavefront converging beam spherical wavefront So, what is the wavefront. Line that is perpendicular to all the rays. While it is a bit more abstract in the sense of understanding the light paths, it is simpler than rays because many rays can be represented by a single wavefront surface. Parallel beam = plane or flat wavefront Converging beam = spherical wavefront

What is the Wavefront? parallel beam ideal wavefront = plane wavefront defocused wavefront Now, consider if the light is converging to point in front of the image plane, then the wavefront takes a new shape. It is more curved, compared to the ideal wavefront that would be required to focus the light onto the image plane.

What is the Wavefront? parallel beam ideal wavefront = plane wavefront aberrated beam = irregular wavefront If the lens has aberrations ….

What is the Wavefront? diverging beam = spherical wavefront aberrated beam = irregular wavefront In reverse, a similar ray distortion take place except now that wave aberration is a distortion of an otherwise plane wave. ideal wavefront

The Wave Aberration

What is the Wave Aberration? diverging beam = spherical wavefront wave aberration The wave aberration is a measure of the difference between the ideal wavefront and the actual wavefront. You are able to choose whatever ideal wavefront you want, but you commonly choose the ideal wavefront as one that would focus the light to the image plane, or a plane. In this example, ideally the light will emerge as a perfect collimated beam, or parallel rays, so the ideal surface is a plane. Over a pupil, the wave aberration defines a surface, whose height indicates the difference form the ideal surface.

Wave Aberration of a Surface -3 -2 -1 1 2 3 Wavefront Aberration mm (right-left) mm (superior-inferior) Here, the different map is plotted across the whole pupil, and looks like a surface, or a contour. They can be plotted in a number of different ways.

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! When a parallel beam passes though an aperture, the light distribution does not simply take the shape of the aperture, like geometrical theory would predict. Because light interferes with itself, diffraction occurs and the light forms what is called a diffraction pattern. When the aperture is far from the screen, then one type of pattern, called a Fraunhofer pattern, is formed. A Fraunhofer diffraction pattern also forms at the focal point of a lens

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 Remember one important thing. Smaller apertures generate more diffraction. The closer the edges of the aperture are to each other, the more the perpendicular spread of light. This is counterintuitive but it is true.

circular aperture Airy Disc Because the circular aperture is rotationally symmetric, so is the diffraction pattern. At the focal point of a lens with a circular aperture, you do not get a point, you get an Airy disk pattern.

The Point Spread Function

The PSF is analogous to the Impulse Response Function in electronics. 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 q

As the pupil size gets larger, the Airy disc gets smaller. 2.5 2 1.5 separatrion between Airy disk peak and 1st min (minutes of arc 500 nm light) This shows the inverse relationship between pupil size and potential image quality. Larger pupils can resolve smaller objects. Recall that the human eye can only resolve about 60 c/deg 1 0.5 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 How bad is the wavefront aberration? Here is an example from a typical human eye.

Observe Your Own Point Spread Function Demonstration Observe Your Own Point Spread Function

Resolution

Unresolved point sources Rayleigh resolution limit Resolved Two points are resolved at the Rayleigh resolution limit when the peak of the Airy disc from one point is above the first minimum of the other. Therefore, the equation for the Rayleigh resolution limit is the same as is used for the size of the Airy disk. Resolved

uncorrected corrected AO image of binary star k-Peg on the 3.5-m telescope at the Starfire Optical Range About 1000 times better than the eye!

About 4500 times better than the eye! Keck telescope: (10 m reflector) About 4500 times better than the eye! Wainscott

Convolution

Convolution

Simulated Images 20/20 letters 20/40 letters

MTF Modulation Transfer Function

low medium high object: 100% contrast image contrast spatial frequency 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).

MTF: Cutoff Frequency 1 mm 1 2 mm 4 mm 6 mm 8 mm 0.5 50 100 150 200 Rule of thumb: cutoff frequency increases by ~30 c/d for each mm increase in pupil size 8 mm modulation transfer 0.5 50 100 150 200 250 300 spatial frequency (c/deg)

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

PTF Phase Transfer Function

low medium high object image 180 phase shift -180 spatial frequency

Relationships Between Wave Aberration, PSF and MTF

The PSF is the Fourier Transform (FT) of the pupil function The MTF is the real part of the FT of the PSF The PTF is the imaginary part of the FT of the PSF

Adaptive Optics Flattens the Wave Aberration AO OFF AO ON

Other Metrics to Define Imagine Quality

Strehl Ratio diffraction-limited PSF Hdl actual PSF Heye

Retinal Sampling

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

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

Nyquist Sampling Theorem

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

Photoreceptor Sampling = 2 x Spatial Frequency 1 I 1 I nearly 100% transmitted

Photoreceptor Sampling = Spatial Frequency 1 I 1 I nothing transmitted

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)