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Compound Opticsand PrismsTuesday, 9/26/2006

Physics 158Peter Beyersdorf

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Class Outline

Lens and mirror systems

Aberrations

Stops

Prisms

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Aberrations

Chromatic

Monochromatic

Spherical

Coma

Astigmatism

Field Curvature

Distortion

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Chromatic Aberration

Due to dispersion in glass lenses, different wavelengths will be bent by different amounts and be focus at different locations.

This can be compensated with a doublet using a suitable pair of materials for the two lenses

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Spherical AberrationsDeviation of a lens from the ideal shape of a cartesian oval (particularly great at large radial distances) gives rise to different focal points for rays at different radial positions

Transverse spread of a focused spot is called transverse spherical aberration (TSA)

Longitudinal spread of a focused spot is called Longitudinal spherical aberration (LSA)

The circle of least confusion is the location of minimal spot size

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Coma

Abberation from image “plane” not actually being plane

Off axis light creates a comet-like blurring of the image

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Astigmatism

Distortions of off-axis rays due to the difference in incident geometry of the sagittal and meridional rays

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Field Curvature

Aberration from object “plane” not being planar

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Distortion

Aberration from the positional dependance on the transverse magnification of a lens

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undistorted barrel distortionpincushion distortion

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Lens and Mirror Systems

Consider a system of multiple imaging elements

The image from one element can be the object for another

Multiple elements may be necessary to minimize certain aberrations (i.e. by compensating for the aberrations of one optic with those of another)

Various apertures in the system will limit the field of view and angular acceptance of the system

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Stops and Pupils

The element that ultimately limits how much light enters an optical system is the aperture stop for that system

The element that limits the field of the image is called the field stop

The image of the aperture stop seen from the object (at the optical axis) is called the entrance pupil

The image of the aperture stop seen from the image location (at the optical axis) is called the exit pupil

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Aperture Stops and Pupils

Find the aperture stop, entrance and exit pupil for the following compound system as seen from the object 80mm before the first lens

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x=-80 mm

x=0 mmΦ=50 mmf=100 mm

x=75 mmΦ=25 mm

x=100 mmΦ=25 mmf=50 mm

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Aperture Stops and Pupils

Find the aperture stop, entrance and exit pupil for the following compound system as seen from the object 80mm before the first lens

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++x=-80 mm

x=0 mmΦ=50 mmf=100 mm

x=75 mmΦ=25 mm

x=100 mmΦ=25 mmf=50 mm

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Ent.Pupil

x=∞

ExitPupil

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Numerical Aperture

The amount of light a lens collects is a function of its numerical aperture, approximately the ratio of its diameter to its focal length

The inverse quantity is called the f/# of a lens

A high numerical aperture is said to be a “fast lens”. Why?

A camera lens has a series of F/#s that form a geometric series (f/1, f/1.4, f/2, f/2.8, f/4 …) why?

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f/# =f

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NA = n0 sin !accept !D

f f Dθaccept

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Numerical Aperture

What are the benefits of having a fast lens (high numerical aperture)

Lots of light collection → bright image

Tightly focused spot (i.e. less diffraction)

What are the benefits of having a slow lens (low numerical aperture)

Reduced spherical aberation

Good depth of field

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Prisms

Prisms have a wide range of different applications in optics.

index measurement devicesretro-reflectorsoptical retarderswavelength based separatorsperiscopesbeam deflectorsbeam splitters

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From Snell’s law

and from symmetry

so that

giving

from inspection

Minimum Deviation Angle

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δ

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The deviation angle δ is

and so

giving

and from Snell’s law

This is a common way to measure very accurately the index of refraction of a material

Minimum Deviation Angle

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δ

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Prism Examples

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Littrow prism

Brewster’s angle at the entrance is n=tan θi1. For glass n≈1.5 so θi1≈56°

This type of prism is used in laser cavities as wavelength selection devices.

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Brewster prism

Two Littrow prisms back-to-back can be used as a transmissive wavelength selective device.

For glass of n=1.5, α=68°

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Prism Examples

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Direct vision prism

This is a color separator that produces an undeviated ray for a certain wavelength

Typically the center piece is flint glass and the two outer components are crown glass

Constant deviation prism

The Pellin-Broca prismThis prism has a geometry such that an outgoing ray exits at 90° relative to the incoming ray.

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60°30°

90°

60°

30°45°

45°

90°

It is a single block of glass but can be viewed a three separate triangles consisting of two 30°-60° triangles joined by a 45°-45° triangle.

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More Prism Examples

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The right angle prism

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The Amici prism

It works like a right-angle prism, but the roof-top shape at the hypotenuse adds a reversion to the image

Total internal reflection reflects 100% of the light and introduces a reversion to the image

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More Prism Examples

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The Porro prism

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Two TIR reflections. Image reversion from right-angle prism is cancelled. Hence output has the same handedness as the input

Dove prism

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If we rotate the prism, the image rotates at twice the rate

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More Prism Examples

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The Penta prism

Causes a 90°. deviation without affecting the orientation of the image

Corner cube

Uses three bounce to reflect light back into the direction it came from

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More Prism Examples

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Rhomboid prism

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A special case of the rhomboid prism is the Fresnel rhomb, which is used to generate a phase shift of the orthogonal polarizations to produce circularly polarized light for 45°linearly polarized input light. Two successive rhombs produce a half-waveplate effect .

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More Prism Examples

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Double Porro prism

Used in binoculars for image erection.

Prism beam expander

For a single prism the magnification is

When θti=α we get maximum magnification

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Example Problem

Show that in a prism beam expander, the magnification along one axis of the image is

while along the other axis the magnification is unity

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Example Problem

Show that in a prism beam expander, the magnification along one axis of the image is

while along the other axis the magnification is unity

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Summary

Real world imaging systems suffer from various aberrations

Finite lens sizes lead to limits on the amount and angle of light collected

Compound lenses can be used to minimize distortions of improve the amount of angle of light collected.

Prisms perform a variety of imaging functions

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