Transcript Lecture 26

Design Realization
lecture 26
John Canny
11/25/03
Last time
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Reflection, Scattering
Refraction, TIR
Retro-reflection
Lenses
This time
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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
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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:
1 1 1
 
f
f1 f 2
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:
 2 
a b  1 
r2     Mr1  



 c d   y1 
 y2 
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:
1
M 
0


1
1

f
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:
1

M   2
 0


1 
1

f
Matrix method: translation
 Within a thick lens, direction does not change
but the intercept changes
 1
M  d

  2
0

1

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):
1 d
f 2

M
 d
 
1 1
d
 
f1 f 2 f1 f 2
d
1
f1




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 multielement 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 thickness, 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 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, antiglare plastic coatings.
 Holographic surface etching: provides tightlycontrolled 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
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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