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19. Geometrical Optics
Geometrical light rays
Free space propagation
Lenses
Ray Matrices and Ray Vectors
Ray tracing
f-numbers
Cylindrical lenses
Two ways to describe the propagation of a light beam1. Ray optics
• easy to understand• easy to calculate• easy to find situations where it is wrong
2. Gaussian beams• less easy• more accurate
TODAY: ray optics
Ray Optics
We'll define "light rays" as directions in space, corresponding, roughly,to k-vectors of light waves.
axisinput
output
Each optical system will have an axis, and all light rays will be assumed to propagate at small angles to the axis. This is called the Paraxial Approximation.
The Optic Axis
A mirror deflects the optic axis into a new direction.
This ring has an optic axis that is rectangular.
Optic axis A ray propagating through this system
We define all rays relative to the relevant optic axis.
Rays xin, in
xout, out
its position, x
its slope,
Optical axis
optical ray
x
These parameters will change as the ray propagates through an optical system.
A light ray can be defined by two coordinates:
These are often written in vector form:
a ‘ray vector’
x
Ray Matrices
Ray matricescan describe both simple and complex systems.
These matrices are often called “ABCD Matrices.”
A BC D
Optical system ↔ 2x2 Ray matrix
in
in
x
out
out
x
The effect on a ray is determined by multiplying its ray vector by the appropriate ray matrices.
For many optical components, we can define 2x2 “ray matrices.”
Easiest example: rays in free space or a uniform medium
If xin and in are the position and slope at z = 0, and xout and out are the position and slope after propagating from z = 0 to z = z0, then:
0out in in
out in
x x z
xin, in
z = 0
xout out
z0
01
0 1out in
out in
x xz
Rewriting this expression in matrix notation:
(notice the small angle approximation: tan )
Ray Matrix for an Interface
At the interface, clearly:
xout = xin.
Now calculate out.
out = [n1 / n2]in
in
n1
out
n2
xin xout
1 2
1 00 /interfaceM
n n
Snell's Law says: n1 sin(in) = n2 sin(out)
which becomes for small angles: n1 in = n2 out
Ray matrix for a curved interface
1 2 1 2
1 0( / 1) / /
curvedinterface
Mn n R n n
Now the output angle depends on the input position, too.
n1 n2
12If the interface has
spherical curvature:
Ray matrices as derivativesSince the displacements and angles are assumed to be small, we can think in terms of partial derivatives.
As we have just seen, it’s easy to evaluate these derivatives for many situations.
out ioutout
i ni
n in nx x xx
x
out in ioutut
nin i
onx
x
out
in
angular magnification
out
in
xx
spatial magnification
out in
out inDB x
Cx A
For cascaded elements, multiply ray matrices
3 2 1out in
out in
x xM M M
Notice that the order looks opposite to what it should be. Order matters!
M1 M3M2in
in
x
out
out
x
An important optical element: a lensA lens is a dielectric (transparent, zero absorption) slab of material with two surfaces, at least one of which is curved.
It affects the paths of light rays because of refraction (i.e., Snell’s law) at the two surfaces.
different types of lenses:
Simple lens: comprised of just one piece of glass
Compound lens: comprised of two or more pieces of glass of different types, attached to each other.
A thin lens is just two curved interfaces.
1 2 1 2
1 0( / 1) / /
curvedinterface
Mn n R n n
We’ll neglect the glass in between (it’s a really thin lens!). And, we’ll take n1 = 1.
2 1 2 1
1 01 0
11 / 1 / 1/thin lens curved curvedinterface interface
M M Mn R n R n
n
n=1
R1 R2
n≠1n=1
2 1
1 0( 1)(1/ 1/ ) 1n R R
1 01/ 1f
This can be written:
This is called the “Lens-Maker’s Formula”
where: 1 2
1 1 11
nf R R
Ray matrix for a lens
The quantity f is the focal length of the lens. It’s the most important parameter describing a lens. It can be positive or negative.
1 0=
1/ 1
lensMf
If f > 0, the lens deflects rays toward the axis.
f > 0R1 > 0R2 < 0
If f < 0, the lens deflects rays away from the axis.
f < 0R1 < 0R2 > 0
1 2
1 1 11
nf R R
where
A lens focuses parallel rays to a point one focal length away.
1 1 00 1 1/ 1 0
out in
out
x f xf
f
f
A lens followed by propagation by one focal length:
Assume all input rays have in = 0
The opposite occurs if the arrows are all reversed. Rays diverging from a point are made parallel by the lens.
For all rays xout = 0!
00/1/ 1 0
in
in
f xx ff
At the focal plane, all rays converge to the z axis (xout = 0) independent of input position.
Lenses focus light!
The f-number, “f / #”, of a lens is the ratio of its focal length and its diameter.
f / # = f / d
f
f
d1
f / # = 1
f
f
d2
f / # = 2
Small f-number lenses collect more light but are harder to engineer.
The f-number of a lens
Near-sightedness (myopia)
In nearsightedness, a person can see nearby objects well, but has difficulty seeing distant objects. Objects focus before the retina. This is usually caused by an eye that is too long or a lens system that has too much focusing power.
Myopia is corrected with a negative-focal-length lens. This lens causes the light to diverge slightly before it enters the eye.
Near-sightedness
Real lens systems can be very complicated
The ability to describe a complicated lens system using a single 2-by-2 matrix is very valuable!
Example: a 13-element telephoto lens (Olympus)
Abberations: imperfections in lensesThere are numerous different kinds of lens abberations. Most can be corrected with clever lens design.
Rays which are far from the optic axis focus to a different point than rays close to the axis.
Off-axis rays focus to different points.
The focal length of a lens can depend on the wavelength of the light.
Abberations: imperfections in lenses
Astigmatism
Astigmatism is when the horizontal and vertical planes of a lens have different focal lengths.
This can happen when a lens is used in an off-axis configuration, as shown here:
It can also happen if the lens is not axially symmetric, so that the curvature is ellipsoidal rather than spherical. Roughly 1/3 of the population suffers from astigmatism of the cornea.
Cylindrical lensesA "spherical lens" focuses in both transverse directions.A "cylindrical lens" focuses in only one transverse direction.This is astigmatism on purpose!
Examples of cylindrical lenses:
When using cylindrical lenses, we must perform two separate Ray Matrix analyses, one for each transverse direction.
Ray tracing
Geometric ray tracing - the process of following a collection of rays through an optical system to characterize its performance.
This concept has been adapted by the computer graphics world for rendering of three-dimensional scenes. One must accurately account for reflection, refraction, and absorption.
