Design Realization lecture 26

John Canny

11/25/03

Last time

Reflection, Scattering Refraction, TIR Retro-reflection Lenses

This time

Lenses reviewed: convex spherical lenses. Ray diagrams. Real and virtual images. More on lenses. Concave and aspheric lenses. Fresnel optics:

Lenses: spherical and aspheric Lenticular arrays Prisms

Refraction – ray representation

In terms of rays, light bends toward the normal in the slower material.

Refractive indices

Water is approximately 1.33 Normal glass and acrylic plastic is about 1.5 Polycarbonate is about 1.56 Highest optical plastic index is 1.66 Bismuth glass is over 2 Diamond is 2.42

Lenses If light comes from a point source that is

further away than the focal length, it will focus to another point on the other side.

Lenses

When there are two focal points f1 , f2 (sometimes called conjugates), then they satisfy:

21

111

fff

Ray diagrams – real & virtual images

Tracing a pair of rays from the top and bottom of the object allows us to find the orientation and size of an image. The pair of rays from a point converge at some

distance from the lens, defining the image distance. One pair of rays are usually straight ray through the

axis of the lens.

Real images

An object further than the focal length away from the lens forms a convergent real image.

Virtual images

An object closer than the focal length forms a virtual image on the same side of the lens.

Virtual images

Virtual images can be created with concave lenses, which are smaller than the object.

Spherical Lenses

If a thin lens consists of spherical surfaces with radii r1 and r2, then the focal length satisfies

1/f = ( - 1) (1/r1 - 1/r2)

this is known as the “lens-maker’s formula”.

Thick Lenses

The above approximations apply to “thin” lenses. Thick lenses use different approximations (based on paraxial rays).

Principal planes and Gullstrands equation are used to compute focal length etc. See:http://hyperphysics.phy-astr.gsu.edu

Thick Lenses

The above approximations apply to “thin” lenses. Thick lenses use different approximations (based on paraxial rays).

Principal planes and Gullstrand’s equation are used to compute focal length etc. See:http://hyperphysics.phy-astr.gsu.edu

The matrix method can also be used:

Matrix method

Lens effects can be approximated with 2D matrices. r1 = incoming ray, r2 = outgoing.

Let r = (, y) be a ray, where is its angle from horizontal, and y is its vertical coordinate.

A lens can be represented as a matrix M:

1

11

2

22 ydc

baMr

yr

Matrix method: thin lens example

Rays through the origin do not change direction, so a = 1.

Rays through the origin do not change y-value, so c = 0.

Assume the lens is at the origin, so intercept does not change, d = 1.

If incoming angle = 0, outgoing rays converge at the focal length, so b = -1/f.

Matrix method: thin lens example

Thin lens matrix is:

10

1 1

fM

Matrix method: half-lens example

For the transition from air to glass on the entry side of the lens, the incoming ray angle is weakened by the refractive index ratio, so:

10

1

2

1fM

Matrix method: translation

Within a thick lens, direction does not change but the intercept changes

1

01

2dM

Thick lens matrix

We derive the thick-lens matrix by multiplying two half-lenses with a translation in between. The result is (d is lens thickness):

11

212

1

1

1

21

f

dd

ff

d

fff

d

M

Spherical aberration

Cylindrical lenses do not converge to a point – outer rays converge closer:

Multi-element lenses Are used to reduce aberration.

Aspheric lenses

Lens shape generated to provide better convergence between two conjugates (focal points) at specified distances.

Used to replace multi-element lenses. Increasingly popular.

Parabolic and elliptical mirrors

Curved mirrors provide very similar performance to lenses.

A parabolic mirror perfectly focuses parallel light to a point.

Parabolic and elliptical mirrors

Elliptical mirrors have two focal points, and focus light from one to the other.

A pair of parabolic mirrors also does this.

Fresnel lenses

Thin lenses are accurate but provide weak magnification. Thick lenses provide power but increase aberration.

Much of the aberration in thick lenses comes from the thick glass (not from the surfaces).

Fresnel lenses provide magnification without thickness.

Fresnel lenses

Remove the thick-ness, but preservepower.

Some artifacts areintroduced, but are invisible for large viewing areas(e.g. diplays).

Fresnel lenses

Fresnel lenses have no “thickness”, and simplify analysis for spherical and aspheric lenses.

In particular, aspheric lens equations can be written in closed form.

Two conjugates are needed because the lens equation is exact.

Fresnel lenses

Fresnel lenses can be made with high precision and low cost from optical plastics by pressure molding.

They are available in arbitrarily large sizes from custom manufacturers – and off the shelf up to about 5’ x 3’.

Fresnel grooves/inch may be 100 or more. Better for display than for imaging.

Lenticular arrays

Many lenses printed on one sheet. Simplest version: array of cylindrical lenses. Used to budget 3D vision:

Lenticular arrays

Simplest version: array of cylindrical lenses.

Lenticular arrays

Lenticular screens are rated in LPI for lines per inch. Typical range is 40-60 LPI, at about $10 per square foot.

Budget color printers can achieve 4800 dpi.

At 40 LPI that gives 120 images in approx 60 viewing range, or 0.5 per image.

Lenticular stereograms

By interleaving images from views of a scene spaced by 0.5, you can achieve a good 3D image.

At 1m viewing distance, 0.5 translates to 1cm spacing between images.

Eye spacing is about 6 cm.

Diffusers

Diffusers spread collimated (parallel) light over a specified range of angles.

Can control viewing angle for a display. Controls sense of “presence” in partitioned

spaces.

Geometric diffusers Arrays of tiny lenses (lenticular arrays).

Can be cylindrical (diffusion in one direction only), used in rear-projection screens.

Surface etching. Using in shower glass, anti-glare plastic coatings.

Holographic surface etching: provides tightly-controlled diffusion envelope.

Low-quality surface finish(!) on plastics gives diffusion effect.

Geometric diffusers Arrays of tiny lenses (lenticular arrays).

Can be cylindrical (diffusion in one direction only), used in rear-projection screens.

Surface etching. Using in shower glass, anti-glare plastic coatings.

Holographic surface etching: provides tightly-controlled diffusion envelope.

use a material with diffusing properties: E.g. small spheres in refractive material

Fresnel prisms

Similar idea to lenses. Remove the thickness of the prism and stagger the surface facets.

Useful for bending light over a large area, e.g. for deflecting daylight.

Also used for vision correction.

Summary

Ray diagrams. Real and virtual images. More on lenses. Concave and aspheric lenses. Parabolic and elliptical mirrors. Fresnel optics:

Lenses: spherical and aspheric Lenticular arrays Prisms