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 f 1, f 2 (sometimes called conjugates), then they satisfy:

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 r 1 and r 2, then the focal length satisfies 1/f = (  - 1) (1/r 1 - 1/r 2 ) 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:

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:  The matrix method can also be used:

Matrix method  Lens effects can be approximated with 2D matrices. r 1 = incoming ray, r 2 = 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:

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:

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:

Matrix method: translation  Within a thick lens, direction does not change but the intercept changes

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):

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 preserve power.  Some artifacts are introduced, 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 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