A Review of OpticsA 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.

Geometrical Optics

Relationships between pupil size, refractive error

and blur

Optics of the eye: Depth of Focus Optics of the eye: Depth of Focus

2 mm 4 mm 6 mm

2 mm 4 mm 6 mm

In focus

Focused in front of retina

Focused behind retina

Optics of the eye: Depth of Focus Optics of the eye: Depth of Focus

7 mm pupil

Bigger blurcircle

Courtesy of RA ApplegateCourtesy of RA Applegate

Smaller blurcircle

2 mm pupil

Courtesy of RA ApplegateCourtesy of RA Applegate

DemonstrationRole 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?What is the Wavefront?

converging beam=

spherical wavefront

parallel beam=

plane wavefront

What is the Wavefront?What is the Wavefront?ideal wavefrontparallel beam

=plane wavefront

defocused wavefront

What is the Wavefront?What is the Wavefront?parallel beam

=plane wavefront

aberrated beam=

irregular wavefront

ideal wavefront

What is the Wavefront?What is the Wavefront?

aberrated beam=

irregular wavefront

diverging beam=

spherical wavefront

ideal wavefront

The Wave Aberration

What is the What is the Wave AberrationWave Aberration??diverging beam

=spherical wavefront wave aberration

-3 -2 -1 0 1 2 3

-3

-2

-1

0

1

2

3

Wavefront Aberration

mm (right-left)m

m (

supe

rior

-infe

rior)

Wave Aberration of a SurfaceWave Aberration of a Surface

Diffraction

DiffractionDiffraction

“Any deviation of light rays from a rectilinear path which cannot be interpreted as reflection or refraction”

Sommerfeld, ~ 1894

Fraunhofer DiffractionFraunhofer 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 InterferenceDiffraction 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

Airy Disc

circular aperture

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.

Airy Disc

The Point Spread FunctionThe 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 DiskAiry Disk

1.22

a

angle subtended at the nodal point

wavelength of the light

pupil diameter

1.22

a

a

0

0.5

1

1.5

2

2.5

1 2 3 4 5 6 7 8

pupil diameter (mm)sepa

ratr

ion

betw

een

Airy

dis

k pe

ak a

nd 1

st m

in

(min

utes

of a

rc 5

00 n

m li

ght)

As the pupil size gets larger, the Airy disc gets smaller.

Point Spread Function vs. Pupil SizePoint Spread Function vs. Pupil Size

1 mm 2 mm 3 mm 4 mm

5 mm 6 mm 7 mm

Small PupilSmall Pupil

Larger pupilLarger pupil

1 mm 2 mm 3 mm 4 mm

5 mm 6 mm 7 mm

Point Spread Function vs. Pupil SizePerfect Eye

pupil images

followed by

psfs for changing pupil size

1 mm 2 mm 3 mm 4 mm

5 mm 6 mm 7 mm

Point Spread Function vs. Pupil SizeTypical Eye

DemonstrationObserve Your Own Point Spread Function

Resolution

Rayleigh resolution

limit

Unresolved point sources

Resolved

AO image of binary star k-Peg on the 3.5-m telescope at the Starfire Optical Range

uncorrected corrected

arc of seconds 064.05.3

1090022.122.1 9

min

a

About 1000 times better than the eye!

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

Wainscott

Convolution

( , ) ( , ) ( , )PSF x y O x y I x y

Convolution

Simulated Images

20/40 letters

20/20 letters

MTFModulation Transfer

Function

low medium high

object:100% contrast

image

spatial frequency

cont

rast

1

0

• 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 FrequencyMTF: Cutoff Frequency

0

0.5

1

0 50 100 150 200 250 300

1 mm2 mm4 mm6 mm8 mm

mo

du

lati

on

tra

nsf

er

spatial frequency (c/deg)

cut-off frequency

57.3cutoff

af

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

Effect of Defocus on the MTF

Charman and Jennings, 1976

450 nm

650 nm

PTFPhase Transfer

Function

object

image

spatial frequency

phas

e sh

ift 180

0

-180

low medium high

Relationships Between Wave Aberration,

PSF and MTF

2

( , ), ( , )

i W x y

i iPSF x y FT P x y e

, Amplitude ( , )x y i iMTF f f FT PSF x y

The PSF is the Fourier Transform (FT) of the pupil function

The MTF is the real part of the FT of the PSF

, Phase ( , )x y i iPTF f f FT PSF x y

The PTF is the imaginary part of the FT of the PSF

Adaptive Optics Flattens the Wave Aberration

AO ON

AO OFF

Other Metrics to Define Imagine Quality

Strehl RatioStrehl Ratio

diffraction-limited PSF

Hdl

Heye

actual PSF

Strehl Ratio = eye

dl

H

H

Retinal Sampling

Projected Image Sampled Image

5 arc minutes20/20 letter

Sampling by Foveal ConesSampling by Foveal Cones

5 arc minutes20/5 letter

Projected Image Sampled Image

Sampling by Foveal ConesSampling by Foveal Cones

Nyquist Sampling TheoremNyquist Sampling Theorem

Photoreceptor Sampling >> Spatial Frequency

I

0

1

I

0

1

nearly 100% transmitted

I

0

1

I

0

1

nearly 100% transmitted

Photoreceptor Sampling = 2 x Spatial Frequency

I

0

1

I

0

1

nothing transmitted

Photoreceptor Sampling = Spatial Frequency

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)