july 2003 chuck dimarzio, northeastern university 10351-8-1 eceg105 & eceu646 optics for...
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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
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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
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July 2003 Chuck DiMarzio, Northeastern University 10351-8-8
Paraxial Approximation
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22
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July 2003 Chuck DiMarzio, Northeastern University 10351-8-9
Complex Radius of Curvature: Physical Results
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2'
1
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22
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2
'w
b
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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