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 beam

1. 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, θin

xout, θout

� its position, x

� its slope, θ

Optical axis

x

θ

These two parameters will change as the

ray propagates through an optical system.

A light ray can be defined by two coordinates:

We write these two quantities in vector form:

a ‘ray vector’

x

θ

Ray Matrices

Ray matrices

can describe both

simple and complex

systems.

These matrices are often called “ABCD Matrices.”

A B

C D

Optical system ↔ 2x2 Ray matrix

in

in

x

θ

out

out

x

θ

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.”

Multiplying a Matrix by a Vector

A B

C D

x

y

+ =

Ax By

+Cx Dy

Dot product: rows of first with columns of second:

Easiest example: rays in free space or a

uniform medium

If xin and θin are the position and slope at z = 0, and xout and θout are the position and slope after propagating from z = 0 to z = z0, then:

0out in in

out in

x x z= +

=

θ

θ θ

xin, θin

z = 0

xout θout

z0

01

0 1

out in

out in

x xz =

θ θRewriting this expression in matrix notation:

(notice the small angle approximation: tan θ ≈ θ )

Ray Matrix for an Interface

At the interface, clearly:

xout = xin.

Now calculate θout.

⇒ θout = [n1 / n2]θin

θin

n1

θout

n2

xin xout

1 2

1 0

0 /interfaceM

n n

=

Snell's Law says: n1 sin(θin) = n2 sin(θout)

which becomes for small angles: n1 θin = n2 θout

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

θ1 θ2If the interface has spherical curvature:

(radius of curvature = R)

Ray matrices as derivatives

Since 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 n

i

n i

n nx xxx

xθ

θ= + ∂

∂∂∂

out in ioutut

nin i

on

xx

θθθ

θ θ= + ∂∂

∂∂

out

in

θθ

∂∂

angular

magnification

out

in

x

x

∂∂

spatial

magnification

out in

out inD

B x

C

x A

θ θ

=

For cascaded elements, multiply ray matrices

3 2 1

out in

out in

x xM M M

θ θ

=

Notice that the order looks opposite to what it should be.

Order matters!

M1 M3M2in

in

x

θ

out

out

x

θ

An important optical element: a lens

A lens is a dielectric (transparent, ideally with 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≠1

n=1

2 1

1 0

1 1( 1) 1

= − −

thin lensM

nR R

1 0

1/ 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 > 0

R2 < 0

If f < 0, the lens deflects

rays away from the axis.

f < 0R1 < 0

R2 > 0

( )1 2

1 1 11

= − −

nf R R

where

A lens focuses parallel rays to a point one focal length away.

1 1 0

0 1 1/ 1 0

= −

out in

out

x f x

fθ

f

f

A lens followed by propagation by one focal length:

Assume all input rays have θin = 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 x

x 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 lenses

There 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 lenses

A "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.