Gaussian Beams &

Resonators

Electro-Optics & Applications Prof. Elias N. Glytsis

School of Electrical & Computer Engineering National Technical University of Athens

08/11/2018

Gaussian Beams r

z θ

R(z1) R(z2)

w0

z1 z2

2 Prof. Elias N. Glytsis, School of ECE, NTUA

TEM00 TEM10 TEM11

TEM20 TEM02 TEM21

TEM30 TEM31 TEM32

TEM33 TEM40 TEM63

Gaussian Beams Patterns

Hermite Polynomials Generator

3 Prof. Elias N. Glytsis, School of ECE, NTUA

Hermite Polynomials (recursive)

Experimental Patterns of Gaussian Beams

From A. Yariv and P. Yeh, “Photonics” 6th Ed. Oxford University Press, 2007

4 Prof. Elias N. Glytsis, School of ECE, NTUA

TEM00 TEM10 TEM01

TEM11 TEM21

Gaussian Beams Patterns

5 Prof. Elias N. Glytsis, School of ECE, NTUA

Laguerre-Gaussian Beams

Laguerre Polynomials Generator

6 Prof. Elias N. Glytsis, School of ECE, NTUA

Laguerre Polynomials (recursive)

Laguerre-Gaussian Beams Patterns

7 Prof. Elias N. Glytsis, School of ECE, NTUA

Gaussian Beams and ABCD Law

qin qout

Input Plane

Output Plane

8 Prof. Elias N. Glytsis, School of ECE, NTUA

Prof. Elias N. Glytsis, School of ECE, NTUA 9

θi

θt

θi

A

B

C

D

E

t nf

n0

n0

Transmitted Waves

Reflected Waves

Fabry-Perot Interferometer

Prof. Elias N. Glytsis, School of ECE, NTUA 10

Fabry-Perot Interferometer

Prof. Elias N. Glytsis, School of ECE, NTUA 11

Symmetric Fabry-Perot Interferometer

νm νm+1 νm-1 ν

FSR

Resonant Frequencies

Prof. Elias N. Glytsis, School of ECE, NTUA 12

Symmetric Fabry-Perot Interferometer

Coefficient of Finesse

Prof. Elias N. Glytsis, School of ECE, NTUA 13

Symmetric Fabry-Perot Interferometer

δ

δ/k0

mλ01 mλ02

2πm 2πm

1

1/2

Δδ

Minimum Wavelength Separation

Prof. Elias N. Glytsis, School of ECE, NTUA 14

νm νm+1 νm-1 ν

FSR

Symmetric Fabry-Perot Interferometer

Resolving Power

Finesse

Prof. Elias N. Glytsis, School of ECE, NTUA 15

Asymmetric Fabry-Perot Interferometer with Gain or Loss

G = eγt

Gain Loss

Fabry-Perot Interferometer

Prof. Elias N. Glytsis, School of ECE, NTUA 16

Two-Mirror Laser Resonator

17 Prof. Elias N. Glytsis, School of ECE, NTUA

http://www.optique-ingenieur.org/en/courses/OPI_ang_M01_C03/co/Contenu_11.html

ℓ

Two-Mirror Laser Resonator Example

R1 = Infinite

R2 = 1m (R2>0)

18 Prof. Elias N. Glytsis, School of ECE, NTUA

Two-Mirror Laser Resonator Example

R1 = Infinite, R2 = 1m (>0), L = R2/2 = 0.5m

z0 = 0.5m, w0 = 317.35μm, z1 = 0, z2 = 0.5m

19 Prof. Elias N. Glytsis, School of ECE, NTUA

Two-Mirror Laser Resonator Example

R1 = 0.3m (R1>0)

R2 = 0.5m (R2>0)

20 Prof. Elias N. Glytsis, School of ECE, NTUA

Two-Mirror Laser Resonator Example

R1 = 0.3m (>0), R2 = 0.5m (>0), L = 0.6m R1 = 0.3m (>0), R2 = 0.5m (>0), L = 0.2m

z0 = 0.15m, w0 = 173.82μm, z1 = -0.15m, z2 = 0.45m

z0 = 0.15m, w0 = 173.82μm, z1 = -0.15m, z2 = 0.05m

21 Prof. Elias N. Glytsis, School of ECE, NTUA

Two-Mirror Laser Resonator Example

R1 = -0.3m (R1<0)

R2 = 0.5m (R2>0)

22 Prof. Elias N. Glytsis, School of ECE, NTUA

Two-Mirror Laser Resonator Example

R1 = -0.3m (<0), R2 = 0.5m (>0), L = 0.25m R1 = -0.3m (<0), R2 = 0.5m (>0), L = 0.5m

z0 = 0.1382m, w0 = 166.84μm, z1 = 0.2083m, z2 = 0.4583m

z0 = 0.0928m, w0 = 136.71μm, z1 = 0.0321m, z2 = 0.4821m

23 Prof. Elias N. Glytsis, School of ECE, NTUA

http://www.optique-ingenieur.org/en/courses/OPI_ang_M01_C03/co/Contenu_12.html

Different Laser Resonator Geometries

24 Prof. Elias N. Glytsis, School of ECE, NTUA

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Cavity Lifetime – Approximate Approach

Assume N round-trips in the cavity: ℓ

R1 R2

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Cavity Lifetime – Accurate Approach

Rate of decrease in intensity (or number of photons) in the cavity:

Survival Ratio:

Round-Trip Time:

ℓ

R1 R2