July 2003 Chuck DiMarzio, Northeastern University 10351-8-1

ECEG105 & ECEU646 Optics for Engineers

Course NotesPart 8: Gaussian Beams

Prof. Charles A. DiMarzio

Northeastern University

Fall 2003

July 2003 Chuck DiMarzio, Northeastern University 10351-8-2

Some Solutions to the Wave Equation

• Plane Waves– Fourier Optics

• Spherical Waves– Spherical Harmonics; eg. In Mie Scattering

• Gaussian Waves– Hermite- and Laguerre- Gaussian Waves

July 2003 Chuck DiMarzio, Northeastern University 10351-8-3

The Spherical-Gaussian Beam • Gaussian Profile

Rayleigh Range

• Diameter

• Radius of Curvature

• Axial Irradiance

iyxiwyx eee

w

PU //

2

222222

2

0 1

b

zww

2

020

4

dwb

20

2

w

PE

z

bz

2

2

0 1

b

zdd

July 2003 Chuck DiMarzio, Northeastern University 10351-8-4

Size Scales of Gaussian BeamsP

EP

0.86P0.14E 0.95P

0.76P0.5E 0.5P

0.21P0.79E 0.5P

d

dd 59.02

2ln

d34.0

2/21 wre wrerf /2

July 2003 Chuck DiMarzio, Northeastern University 10351-8-5

Visualization of Gaussian Beam

z=0

w

Center ofCurvature

July 2003 Chuck DiMarzio, Northeastern University 10351-8-6

Parameters vs. Axial Distance

-5 0 50

1

2

3

4

5

z/b, Axial Distance

d/d

0, Bea

m D

iam

eter

-5 0 5-5

0

5

z/b, Axial Distance

/b,

Rad

ius

of C

urv

atu

re

z

bz

2

2

0 1

b

zdd

zd0

4

0d

z

z

b2

m4053 m4053

July 2003 Chuck DiMarzio, Northeastern University 10351-8-7

Complex Radius of Curvature

• Spherical Wave

• Gaussian Spherical Wave

222

2224zyxike

zyx

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222

2224ibzyxike

ibzyx

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July 2003 Chuck DiMarzio, Northeastern University 10351-8-8

Paraxial Approximation

222

2224

ibzyxike

ibzyx

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2

22

21

24

ibz

yxibzik

eibz

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222

222

22

24

yxibz

bkyx

ibz

zik

ibzik eeeibz

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July 2003 Chuck DiMarzio, Northeastern University 10351-8-9

Complex Radius of Curvature: Physical Results

22

2222

2222 22224

yxbz

bkyx

bz

zik

bzik eeebz

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'22

22

2222

4b

yxk

yxik

ibzik eeebz

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21

ibz

z

2'

1

ibz

b

b

z

bz 22

b

bzb

22

'

2

'w

b

222

222

22

24

yxibz

bkyx

ibz

zik

ibzik eeeibz

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July 2003 Chuck DiMarzio, Northeastern University 10351-8-10

Collins Chart

z

bz 22

b

bzb

22

'

ibzq '

11

b

i

q

z

b

constant

constant' b constantz

constantb

July 2003 Chuck DiMarzio, Northeastern University 10351-8-11

A Lens on the Collins Chart

'

11

b

i

q

z

b

in

constant' b

sin 'sout

outinf 111

fqq inout

111

out

July 2003 Chuck DiMarzio, Northeastern University 10351-8-12

Looking For Solutions on the Collins Chart (1)

-z1

You Can’t Focus a Beam of diameter d1

any Further Away than z1

b’=b’2

b’=b’1

You Can’t Keep a beam diameter less than d2

over a distance greater than.

z

July 2003 Chuck DiMarzio, Northeastern University 10351-8-13

Looking For Solutions on the Collins Chart (2)

b’=b’3 There may be 0, 1, or 2 solutions.

Watch out for your tie!

I want to put a beam waist at a distance z3 from a starting diameter of d3.

z

b

July 2003 Chuck DiMarzio, Northeastern University 10351-8-14

Making a Laser Cavity

Make the Mirror Curvatures Match Those of the Beam You Want.

July 2003 Chuck DiMarzio, Northeastern University 10351-8-15

Hermite-Gaussian Beams (1)• Expansion in Hermite

Gaussian Functions– Orthogonal Functions

• Infinite x,y

– Freedom to Choose w• Use Best Fit for Lowest

Mode

• Alternative– Laguerre Gaussians

• For Circular Symmetry

July 2003 Chuck DiMarzio, Northeastern University 10351-8-16

Hermite-Gaussian Beams (2)

• Possible Applications– Approximation to Real

Beams• Simple Propagation

– Description of Modes of Real Lasers

– Calculation of Losses at Square Apertures

July 2003 Chuck DiMarzio, Northeastern University 10351-8-17

Coefficients for HG Expansion

July 2003 Chuck DiMarzio, Northeastern University 10351-8-18

Propagation Problems

July 2003 Chuck DiMarzio, Northeastern University 10351-8-19

Uniform Circular Aperture

0 1 2 3 4 5 6-60

-50

-40

-30

-20

-10

0

Radial Distance

Nor

ma

lize

d I

rra

dia

nce

Original Function

1 term

8 terms

20terms

0 1 2 3 4 5 6-60

-50

-40

-30

-20

-10

0

Radial Distance

Nor

ma

lize

d I

rra

dia

nce

Far Field Diffraction

1 term

8 terms

20terms

1.22 /D

July 2003 Chuck DiMarzio, Northeastern University 10351-8-20

Sample Hermite Gaussian Beams0:0 0:1 0:3

1:0 1:1 1:3

2:0 2:1 2:3

5:0 5:1 5:3

(0:1)+i(1:0)=“Donut Mode”

Most lasers prefer rectangular modes because something breaks the circular symmetry. Note: Irradiance Images rendered with =0.5

from matlab program 10021.m

July 2003 Chuck DiMarzio, Northeastern University 10351-8-21

Losses at an Aperture (1)

g,GainAperture

E1

r1,mirror

r2,mirror

E2

Straight-Line Layout

E1E2

E1

E1 = E1gMr2gr1One round trip:

What is M?

July 2003 Chuck DiMarzio, Northeastern University 10351-8-22

Losses at an Aperture (2)

E1E2

E1

C1 = C1gMr2gr1One round trip:Now, g and M and maybe r are matrices. All but M are likely to be nearly diagonal.

Large Apertures: M is diagonalFinite Apertures: Diagonal elements become smaller, and off-diagonal elements become non-zero