1

ECE 645 – Optics and Photonics

Lecture 02 – Geometrical Optics

• HMY 645

• Lecture 02

• Spring Semester 2015

Stavros IezekielDepartment of Electrical and

Computer Engineering

University of Cyprus

iezekiel@ucy.ac.cy

THE RAY PICTURE OF LIGHT

2

3

A hierarchy of optics models

Geometrical optics

D >> λ

Wave optics

D ≈ λ

EM theory

D ≈ λ

Quantum electrodynamics (QED)

(Feynman)Increasing

“accuracy”

“Ease” of

use

• If λ → 0, wave optics → geometrical optics

4

• D ≈ λ

• Noticeable diffraction, so must use wave theory

• D >> λ

• Negligible diffraction; we can approximate behaviour using

rays to describe light (geometrical optics)

http://ngsir.netfirms.com/applets/diffraction/2X/Diffraction.htm

5

• The rays show us the direction that the light takes

6

Postulates of geometrical optics

1. Rays are perpendicular to the wavefronts. They show the direction the light

travels.

2. An optical medium is described by a refractive index n.

v

cn =

n = refractive index

c = speed of light in a vacuum (=2.998 × 108 ms-1)

v = speed of light in the optical medium

(1)

7

The optical path length (OPL) is the distance that light would travel in a vacuum during

the same time that it takes to travel a distance d in a medium with refractive index n.

c

OPL

v

dt ==

∫=B

A

dsn )(OPL r

In a non-uniform medium, the refractive index is a function of position:

ds

A

B

r = position vector = (x,y,z)

nddv

c== .OPL (2)

8

3. Fermat’s principle states that:

An optical ray travelling between points A and B will take a path so that the time

(or the OPL) between them is an extremum compared to adjacent paths. (An

extremum means the rate of change is zero.) For most cases, it is a minimum:

Rays of light travel along the path of least time

© M

IT ∫Γ

= dsn )(OPL r

Γ is chosen to minimise the

value of this integral compared

to other paths.

9

Example: Reflection from a mirror

A B

1 2 345

x

Mirror

x1 x2 x3 x4 x5

x

tAB(x)

tAB(x)MIN

x4

10

Example: Refraction in uniform media

n = n1

n = n2

A

B

θ1

θ2

h1

h2

x

w

variable

X

OPL =

AX n1 + XB n2

11

n = n1

n = n2

A

B

θ1

θ2

h1

h2

x

w

variable

X

[ ] 2

2

22

22

11

21

hxwc

n

xhc

n

c

nXBAXn

c

OPLtAB

+−+

+=

+=

=

22

1

1sinxh

x

+=θ

[ ] 2

2

22sin

hxw

xw

+−

−=θ

12

Solve for:

0=dx

dtAB

[ ] 2

2

2

2

22

1

1 )(

hxwc

nxw

xhc

xn

dx

dtAB

+−

−−

+=

[ ] 2

2

2

2

22

1

1 )(0

hxwc

nxw

xhc

xn

dx

dtAB

+−

−=

+⇒=

2211 sinsin θθ nn = Snell’s law (3)

13

Another example of the principle of least time: collect all the light that comes outof a point P, and collect it back at another point P'.

P P'

Optical Black Box

This is called focusing

14

How can this be possible? Make the paths take the same time.

In this picture, light near the top spends less time in the glass (which slows it), so:

tPQP

/ = tPRP

/ = tPP

/

This is called a focusing lens

P P'

R

Q

n1

n2

15

Geometrical Optics: The Paraxial Approximation

We define light rays as directions in space.

We will not worry about the phase.

Each optical system will have an axis, and all light rays will be assumed to

propagate at small angles to it. This is called the Paraxial Approximation.

axis

16

The Optical Axis

A mirror deflects the optic axis into a new direction.

This ring laser has an optic axis that scans out a rectangle.

Optical axis A ray propagating

through this system

We define all rays relative to the relevant optical axis.

17

The Ray Vector

A light ray can be defined by two co-ordinates

xin, θθθθin

xout, θθθθout

its position, x

its slope, θ

Optical axis

x

θ

These parameters define a ray vector,

which will change with distance and as

the ray propagates through optics.

x

θ

18

Ray Matrices

For many optical components, we can define 2 x 2 ray matrices.

An element’s effect on a ray is found by multiplying its ray vector.

Ray matrices

can describe

simple and complex

systems.

These matrices are often called ABCD Matrices.

in

in

x

θ

A B

C D

Optical system ↔ 2 x 2 Ray matrix

out

out

x

θ

19

Ray matrices as derivatives

out in

out inD

B x

C

x A

θ θ

=

out

in

θ

θ

∂

∂

out

in

x

x

∂

∂

out

inx

θ∂

∂

out

in

x

θ

∂

∂

angular

magnification

spatial

magnification

out ioutout

i n

i

n i

n nx xxx

xθ

θ= +

∂

∂

∂

∂

out in ioutut

nin i

onx

xθ

θ

θθ

θ= +

∂

∂

∂

∂

Since the displacements and angles are assumed

to be small, we can think in terms of partial

derivatives

(4)

We can write these

equations in matrix form.

20

For cascaded elements, we simply multiply ray matrices.

3 2 1 3 2 1

out in in

out in in

x x xO O O O O O

θ θ θ

= =

O1 O3O2

in

in

x

θ

out

out

x

θ

(5)

21

Ray matrix for free space or a medium

If xin and θin are the position and slope upon entering, let xout and θout be the

position and slope after propagating from z = 0 to z.

out in in

out in

x x z θ

θ θ

= +

=xin, θin

z = 0

xout θout

z 1

0 1

out in

out in

x xz

θ θ

=

Rewriting these expressions in

matrix notation:

1=

0 1space

zO

(6)

22

Ray Matrix for an Interface

At the interface, clearly:

xout = xin.

Now calculate θout

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

which becomes for small angles: n1 θin = n2 θout ⇒ θout = [n1 / n2] θin

θin

n1

θout

n2

xin xout

1 2

1 0

0 /interfaceO

n n

=

(7)

23

Ray matrix for a curved interface

At the interface, again:

xout = xin.

To calculate θout, we must

calculate θ1 and θ2.

If θs is the surface slope at

the height xin, then

θ1 = θin+ θs and θ2 = θout+ θs

If R is the surface radius of curvature, the surface z coordinate will be:

2 2 2 21

2

21

2

1 ( / ) 1 ( / )

( / )

in in in

in

z R R x R R x R R R x R

x R

= − − = − − ≈ − −

=/s in

in

dzx R

dxθ⇒ ≈ ≈

θin

n1

θout

n2

xin = xout

θ1θ2

θs

R

z

θs

z = 0

24

1 2 1 2

1 0

( / 1) / /curvedinterface

On n R n n

= −

1 2( / )( / ) /

out in in inn n x R x Rθ θ⇒ ≈ + −

1 2 1 2( / ) ( / 1) /out in inn n n n x Rθ θ⇒ ≈ + −

Now the output angle depends on the input position, too.

Snell's Law: n1 θ1 = n2 θ2

1 2( / ) ( / )

in in out inn x R n x Rθ θ⇒ + ≈ +

θ1 = θin+ xin / R and θ2 = θout+ xin / R

n1 n2

θ1θ2

(8)

25

A thin lens is just two curved interfaces.

1 2 1 2

1 0

( / 1) / /curvedinterface

On n R n n

= −

We neglect the glass in between (it is a very

thin lens!), and we take n1 = 1.

2 1 2 1

1 0 1 0

( 1) / (1/ 1) / 1/thin lens curved curved

interface interface

O O On R n n R n

= = − −

2 1 2 1

1 0 1 0

( 1) / (1/ 1) / (1/ ) ( 1) / (1 ) / 1n R n n R n n n R n R

= = − + − − + −

n=1

R1 R2

n≠1

n=1

2 1

1 0

( 1)(1/ 1/ ) 1n R R

= − −

1 0

1/ 1f

−

This can be written:

1 21/ ( 1)(1/ 1/ )f n R R= − − The Lens-Maker’s Formulawhere:

26

Ray matrix for a lens

The quantity, f, is the focal length of the lens. It is the most important

parameter of a lens. It can be positive or negative.

1 0=

-1/ 1lens

Of

If f > 0, the lens deflects

rays toward the axis.

f > 0

If f < 0, the lens deflects

rays away from the axis.

f < 0

1 21/ ( 1)(1/ 1/ )f n R R= − −

R1 > 0

R2 < 0

R1 < 0

R2 > 0

(9)

GEOMETRICAL OPTICS ANALYSIS OF

OPTICAL FIBRES

27 28

Optical Fibres: Basic Structure

• A circular core of refractive index n1 is surrounded by cladding with a slightly lower value of

refractive index (n2 < n1).

• Light is confined to the core of the fibre by total internal reflection – TIR at the core-cladding

interface.

CORE CLADDING BUFFER COATING

Not to scale!

29

• Snell’s law: 2211 sinsin φφ nn =

Refraction occurs when: φ1 < φC

Note: n2 < n1

n1

n2Normalline Refracted

ray

Reflected ray

Incident ray

Material boundary

θ1

θ2

θ1

φ1

φ2

2211 coscos θθ nn =

(10)

(11)

30

• At the critical angle, φ 1 = φ C :

211 sin nn =φ

Note: n2 < n1

n1

n2

Incident

ray

Material

boundary

φ1 = φC

φ2 = π/2

( )12

1sin nnC

−=φ

Ray propagates

along the material

boundary

(12)

31

2211 sinsin φφ nn =

• What happens when φ1 > φ C ?

1

2

12 sinsin φφ

n

n=⇒

Now, sin φ2 ≤ 1, hence:

1sin 1

2

1 ≤φn

n

1

21sin

n

n≤∴ φ

• Since n2 < n1, this implies that although the maximum value (“mathematically”)

for sin φ1 is unity, in this case it is limited to: 0 ≤ sin φ1 ≤ n2/n1.

32

• Beyond the critical angle, φ 1 > φ C :

Note: n2 < n1

n1

n2

Incident

ray

Material

boundary

φ1

φ2 = π/2

φ1

TIR is the physical principle upon which optical waveguides, and specifically optical

fibres, are based.

Total internal

reflection (TIR)

Totally

internally

reflected

ray

33

θ1 = θC

n1

n2

θ1

θ1 > θCθ1 < θC

n1

n2θ2

θ1

n1

n2

θ1

θ2=900

Summary

Total internal reflection (TIR) is the mechanism with which light propagates in

optical fibres.

It also has other applications (e.g. prisms).

Refraction Critical angle Total internal reflection

34

Does not propagate; φ < critical angle at core-cladding interface

Propagates through repeated TIR at core-cladding interface;θ0 denotes the acceptance angle.

Propagation in an ideal step-index fibre

Airn0

Cladding

Core

Cladding

n1

n2

n2

n1 > n2 > n0

θ0

θφ

Reflected ray

Refracted

ray

35

Numerical aperture (NA) in a step-index fibre

What is the maximum acceptance angle θ0?

Apply Snell’s law:

θθ sinsin 100 nn =21 sin nn C =φφC

θ

θ0

Cladding n2

Core n1

Air n0

2πφθ =+ C

36

2

2

2

1

2

1

21

2

1

1

1

100

1

sin1

cos

)2(sin

sinsin

nn

n

nn

n

n

n

nn

C

C

C

−=

−=

−=

=

−=

=

φ

φ

φπ

θθ

This equation defines the numerical aperture (NA) of a step-index fibre:

21 sin nn C =φφC

θ

θ0

Cladding n2

Core n1

2πφθ =+ C

2

2

2

100 sinNA nnn −== θ (13)

37

Basic types of optical fibre

• Single-mode step-index (monomode)

• Multimode step-index

• Multimode graded index

• Dispersion shifted and dispersion flattened fibre

38

Refractive index profile Fibre cross sections &

ray paths

Typical

dimensions

125 μm cladding

8 - 12 μm

core

n

n1n2

Step-index single mode

125 - 400 μm cladding

50 - 200 μm

coren

n1n2

Step-index multimode

125 – 140 μm cladding

50 - 100 μm

coren

n1n2

Graded index multimode

39

Single-mode fibres

• advantage: low dispersion

• disadvantages:

1) small numerical aperture (NA)

2) high tolerance connectors required

3) difficult to couple LED light into them

• used with laser diodes and for relatively long links.

125 μm cladding

8 - 12 μm

core

n

n1n2

Step-index single mode

n1n2

40

Multimode step-index fibres

• advantages:

1) large NA

2) low tolerance connectors

3) easy to couple LED light into them

• disadvantage: intermodal dispersion due to

different group velocities of the many modes

present

• used with LEDs and for “short” links, e.g.

LANs (local area networks)

125 - 400 μm cladding

50 - 200 μm

coren

n1n2

Step-index multimode

n1n2

41

Graded index fibres

• Multimode step index fibres cause dispersion because high order modes travel further than low

order modes in multimode fibre (causing broadening of the transmitted light pulse)

• This is minimised by using graded index fibres, where the refractive index n decreases smoothly

with increasing radius

125 – 140 μm cladding

50 - 100 μm

coren

n1n2

Graded index multimode

42

Dispersion

• If attenuation was the only source of signal degradation, it would not be too bad, due to

the existence of optical amplifiers:

• Unfortunately, real fibres also have dispersion ....

1 00

1 00

FIBRE

Input bit stream Output bit stream:

attenuated

due to optical losses in fibre

1 00

Original bit stream

OPTICAL

AMPLIFIER

43

p (t)p(t - τ)

pIN (t)

t

pOUT (t)

tτ

tτ

tτ

Attenuation onlyReduction in pulse energy

Attenuation & dispersionReduction in pulse energy

• Pulse spreading

Fibre

No change in

pulse shape

44

• We can (normally) consider the fibre to be a linear system, with an impulse

response as shown:

tt

h(t)

σ

δ (t)

pin(t)

pin(t)pout(t)

pout(t)

pin(t) = δ (t), hence pout(t) = h(t)

t

t = mean arrival time

σ = rms pulse spread

45

• Consider the output pulse:

σ

t

pout(t)

t

dttpE out∫∞

∞−

= )(

• Energy content

• E = area under pulse

dttptE

t out∫∞

∞−

= )(1

• Mean time of pulse arrival

FWHM = ∆τ

( )

22

22

)(1

)(1

tdttptE

dttpttE

out

out

−=

−=

∫

∫∞

∞−

∞

∞−

σ

• σ is root mean square spread of

pulse around mean arrival time

• It gives a measure of the dispersion

• An alternative measure is the

full width at half maximum (FWHM)

46

• If a pulse with an rms pulse width of σ1 is applied to a fibre, then the output

pulse spread will be given by:

h(t)

tm2

σ2

pout(t)

tm1

σ1

pin(t)

σ

22

1

2

2 σσσ +=

47

Dispersion leads to pulse spreading and overlapping:

48

49

Physical Causes of Dispersion

There are two major types of dispersion in optical fibres:

• Intermodal

- only occurs in multimode (MM) fibres, not in

single mode (SM)

- dominant source of dispersion for MM fibres

• Intramodal

- occurs in both SM & MM fibres

- dominates in single-mode (SM) fibres

- consists of:

• Material dispersion

• Waveguide dispersion

50

Intermodal Dispersion

• Light is transmitted along a multimode optical fibre by several modes (ray

paths).

• The distances travelled by the various paths are different, and hence the

transit times through the fibre also differ.

• A pulse of light, even if it is monochromatic, will have a spread of delays

and the received pulse will have a wider FWHM.

n1

n2

21

3

nO

n

Multimode step

index fibre. Ray

paths are different

so that rays arrive at

different times

input

t

+

+

=

t

t

output

• Consider a simple example, in which input light is launched into a fibre

and then propagates along three modes:

52

• Consider worst case scenario for step-index MM fibre:

A B

n2

n1

longest

pathshortest

path

φc

Intermodal dispersion in step-index multimode fibres

53

φc

LS

LL

sin φc = n2/n1 {Snell’s law for critical angle}

• Also, LS = LL sin φc = LL n2/n1

• Both rays have same velocity: v = c/n1

• τS = LS /v = LSn1/c

• τL = LL n1/c = LS . (n12/cn2)

• Hence δτ = τL - τS = (LSn1/c).{n1/n2 - 1}

• Consider time delay between longest and shortest paths through the fibre:

• So the pulse spread per unit length for intermodal dispersion in a

multimode fibre is:

−=

2

211

n

nn

c

n

L

τδ(14)

54

n1

n2

21

3

nO

n1

2

1

3

n

n2

OO' O''

n2

23

• Intermodal dispersion can be minimised by using graded index fibre:

Multimode step

index fibre. Ray

paths are different

so that rays arrive at

different times

Graded index fibre.

Ray paths are

different

but so are the

velocities

along the paths so

that all rays arrive

at approximately

the same time

55

GEOMETRICAL OPTICS ANALYSIS OF

LASER RESONATORS

56

57

Laser – Light Amplification by Stimulated Emission

• Three “ingredients” are needed for a laser:

1. Medium with optical gain

2. Pump (optical or electrical)3. Optical resonator

58

Types of Optical Resonator

Ring resonator

Planar mirror resonator

59

Spherical mirror resonator

zOptical

axis

R1 R2

d

Radius of curvature

60

Stability of optical resonators

Two flat mirrors, the flat-flat

laser cavity, is difficult to align

and maintain aligned.

Two concave curved mirrors, the

usually stable laser cavity, is

generally easy to align and

maintain aligned.

Two convex mirrors, the unstable

laser cavity, is impossible to

align!

Optical gain medium

http://webphysics.davidson.edu/applets/optics4/default.html

61

We will now examine the stability of a spherical mirror resonator

Define one round-trip as: going through the cavity twice, and being reflected

twice (once by the left mirror, once by the right mirror)

z

0

1

2

x0

x1

x2

d

62

• Unit cell

• Equivalent to one round trip in the

optical cavity

• Continually repeated, infinitely

We can model multiple reflections in the previous optical cavity with an

equivalent system of lenses spaced equally apart:

d

mm - 1 m + 1

z

63

z

d

R1 R2

z

f1 f2

CAVITY

MODEL OF CAVITY FOR

HALF A ROUND-TRIP

− 11

01

1f

− 11

01

2f

− 12

01

1R

− 12

01

2R

By inspection:

11

21

Rf=

22

21

Rf=

64

z

d

R1 R2

d

z

f1 f2 f1 f2 f1

CAVITY

MODEL OF CAVITY

A

A

B

B C

C

D

D

E

E

65

DC

BA

DC

BA

DC

BA

0

0

θ

x

DC

BA

DC

BA

1

1

θ

x

m

mx

θ

+

+

1

1

m

mx

θ

1 2 m-1 m+1m

=

0

0

θθ

x

DC

BAxm

m

m

A cascade of identical optical components

=

+

+

m

m

m

m x

DC

BAx

θθ 1

1

(22)

(23)

66

mmm

mmm

DCx

BAxx

θθ

θ

+=

+=

+

+

1

1

B

Axx

B

Axx

mmm

mmm

121

1

+++

+

−=⇒

−=

θ

θ

mmmm DCx

B

Axxθ+=

− ++ 12

B

ADxDxCx

B

Axx mmm

mm −+=

− +++ 112

67

( ) ( ) mmm xBCADxDAx −−+= ++ 12

mmm xOxOx ][Det ][ Trace 12 −= ++

DAO +=][ Trace

BCADO −=][Det

(24)

In many cases, the determinant of the ABCD matrix is one.

(25)

68

If we know the initial offset from the optical axis (x0) and the initial angle (θ0),

we can use the iterative equation (24) to find where the ray of light is at fixed points

(solid dots):

z

x

0 1 2 3 4……….

z

x

0 1 2 3 4……….

Unstable: ray eventually leaves system

Stable and periodic: ray never leaves system

Unit cell

69

DC

BA

DC

BA

DC

BA

0

0

θ

x

DC

BA

DC

BA

1

1

θ

x

m

mx

θ

+

+

1

1

m

mx

θ

1 2 m-1 m+1m

d

mm - 1 m + 1

z

For our model of the spherical mirror cavity, we have:

Need to

calculate

ABCD matrix

for this unit

cell

70

+−−+−−

−−

=

−−

−−=

−

−=

+

+

21

2

122121

2

2

2

2211

211

1

21

11

21

11

1

11

1

10

1

11

01

10

1

11

01

ff

d

f

d

f

d

ff

d

ff

f

dd

f

d

x

f

d

f

d

f

d

f

d

xd

f

d

f

x

m

m

m

m

m

m

θ

θθ(26)

71

+−−+−−

−−

=

21

2

122121

2

2

2

21

11

21

ff

d

f

d

f

d

ff

d

ff

f

dd

f

d

DC

BA

To calculate determinant, consider equation (26) on previous slide:

(27)

−

−=

10

1

11

01

10

1

11

01

21

d

f

d

fDC

BA

Det = 1 Det = 1 Det = 1 Det = 1

1][Det =−=∴ BCADO

72

( ) mmm xxDAx −+= ++ 12

So the iterative equation that we use to calculate the ray path now becomes

simpler:

Let:

( )DAb +=2

02 12 =+−∴ ++ mmm xbxx

(28)

(29)

Assume that the solution to (29) has the following form:

θjm

m exx 0= (30)

If we substitute (30) into (29) we get:

( ) 0122

0 =+− θθθ jjjmbeeex (31)

73

0122 =+−∴ θθ jj

bee

(32)21 bjbe

j −±=θ

Solve this quadratic in ejθ to get:

b

21 bj −

1

θje

So the general solution to (29) if b is real is:

θθ jmjm

m exexx−+= *

00

We can also write this using trigonometric functions, for example:

)sin(max αθ += mxxm

(33)

(34)

74

this analysis will be OK provided:

Since:2

1 bjbej −±=θ

012 ≥− b

1≤⇒ b

From the definition in (28),

12

≤+ DA

(35)

(36)

Condition for stability

75

12

1 ≤+

≤−DA

In other words:

If we add 1 to this equation and then divide by two, we get:

14

20 ≤

++≤

DA

Condition for stability

(37)

This version of the stability condition is easier to use for our equivalent lens system.

From (27):

21

2

12

21

ff

d

f

d

f

dD +−−=

2

1f

dA −=

76

−

−=

+−−=

+−−=

++

21

21

2

21

21

2

12

21

21

4221

224

4

1

4

2

f

d

f

d

ff

d

f

d

f

d

ff

d

f

d

f

dDA

Let:

−=

i

if

dg

21

(38)

(39)

77

• The condition for stability can now be written as:

10 21 ≤≤ gg (40)

−=

1

12

1f

dg

−=

2

22

1f

dg

78

−=

1

12

1f

dg

−=

2

22

1f

dg

1 2 3 4

-4 -3 -2 -1

-1

-2

-3

-4

1

2

3

4

STABLE REGION

421

dff ==

∞±== 21 ff

221

dff ==

79

ii Rf

21=

To relate our equivalent lens model with the actual optical cavity, remember:

−=

−=∴

ii

iR

d

f

dg 1

21

80

−=

1

1 1R

dg

−=

2

2 1R

dg

1 2 3 4

-4 -3 -2 -1

-1

-2

-3

-4

1

2

3

4

STABLE REGION

221

dRR ==

∞±== 21 RR

dRR == 21

∞=

=

2

1

R

dR

dR

R

=

∞=

2

1

htt

p:/

/co

rd.o

rg/c

m/l

eo

t/co

urs

e0

1_

mo

d0

7/m

od

01

-07

fra

me

.htm

http://perg.phys.ksu.edu/vqm/laserweb/java/cavity/cavstabe.htm

81

Closing Remarks

• Ray optics (geometrical optics) is a useful tool, especially when combined with

ABCD matrices

• It can be used for both analysis and design of optical systems

• Here, we have used it to analyse ray propagation in optical fibres and to look at

the stability of laser resonators

• However, the ray optics approach has limitations:

• It depends on the optical component size being bigger than the wavelength

• It is not useful for looking at phenomena such as diffraction

• It tells us nothing about the spectral content (i.e. wavelength) of the light

or other important properties like the state of polarisation.