ECE 5616Curtis

Gaussian Beams and Lasers

• Gaussian Beams Introduction• Matrix Method• Equivalent Ray tracing• Example calculation• Laser Basics

ECE 5616Curtis

Gaussian BeamsSolution of scalar paraxial wave equation (Helmholtz equation) is aGaussian beam, given by:

Note that R(z) does not obey ray tracing sign convention. Unfortunately there’s no particularly good way to fix this.

ECE 5616Curtis

Gaussian BeamsMain points

Gaussian beam can be completely described once you know two things

1. wo beam waist, which is the point where the field is down 1/e compared to on axis, wavelength

2. Z=0, location of beam waist

Half apex angle for far field of aperture wo, about 86% of beam power is contained within this cone

ECE 5616Curtis

Gaussian Beams Detailed view, Rayleigh distance

Real part of E vs. radius and z

Measure of the convergence of the beam, smaller zo stronger convergence.The phase on the beam axis is retarded by π/4 relative to plane wave.

ECE 5616Curtis

Guassian Beams Detailed view, at beam center and far away

Real part of E vs. radius and z

Near beam center (z<< zo)Beam intensity is ~ uniform across wavefront and Guay phase is zeroIn other words it is like a plane wave.

Far from beam waist (z >> zo)Wave is ~ like a spherical wave (paraxially) Since R(z) ≈ z and w(z) ≈ woz/zo,but with extra phase of ζ(z) =π/2.

ECE 5616Curtis

Gaussian Beams Conversion formulas

ECE 5616Curtis

Gaussian Beam Parameter q(z)

ECE 5616Curtis

Gaussian Beam Parameter q(z)

ECE 5616Curtis

How does q change with transfer and refraction ?

ECE 5616Curtis

ABCD Matrix for Guassians

ECE 5616Curtis

ABCD Matrix for Guassians

ECE 5616Curtis

Matrix Method

Amnon Yariv, Optical Electronics

where

So can use them like before for multiple elements

ECE 5616Curtis

ExampleGaussian beam focusing

ECE 5616Curtis

ExampleGaussian beam focusing

ECE 5616Curtis

ExampleGaussian beam in lens waveguide/resonator

F=2/RFor mirrors

ECE 5616Curtis

ExampleGaussian beam in lens waveguide/resonator

F=2/RFor mirrors

For Gaussian beam confinement θ must be real. This yields the condition below for stable beam confinement or resonance…

This is the same condition as we derived for RAYS !!!

ECE 5616Curtis

Representation of Gaussian Beams by complex rays (1)

ECE 5616Curtis

Representation of Gaussian Beams by complex rays (1’)

ECE 5616Curtis

Representation of Gaussian Beams by complex rays (2)

ECE 5616Curtis

Representation of Gaussian Beams by complex rays (2’)

ECE 5616Curtis

Representation of Guassian Beams by complex rays (3)

Notes• In (1), second waist is at Fourier plane, as expected.

• In (2), second waist occurs before image plane, as expected.

• In (3), as distance to lens increases, waist moves to paraxial image plane

ECE 5616Curtis

Representation of Guassian Beams by complex rays (4)

Notes• In (1), waist is centered at zero (as expected of FT geometry)

• In (2), image is at -10 μm, expected from M=-1.

• This type of problem is not possible with the ABCD formalism.

ECE 5616Curtis

Do Guassian Beams Obey Paraxial Imaging ? (1/3)

Answer: Yes. The image is also a Gaussian E field distribution in amplitude andany point on the object down from the peak by some value, say 1/e for the pointw(z), will image to the point on the image down from the peak by the same value.Shown above only for real objects conjugate to real images ( -t f ).

ECE 5616Curtis

Do Guassian Beams Obey Paraxial Imaging ? (2/3)

Works for virtual objects as well…

ECE 5616Curtis

Do Guassian Beams Obey Paraxial Imaging ? (3/3)

Conclusion: All parts of the object Gaussian image correctly to the appropriate parts of the image Gaussian including both real and virtual objects and images.Corollary: If you apply paraxial imaging to the object Gaussian over all z, you generate the image Gaussian over all z. Gaussian beams obey paraxial imaging exactly.

ECE 5616Curtis

Example: Collimation Lens

ECE 5616Curtis

Laser BasicsAbsorption and Emission

ECE 5616Curtis

Laser basicsSpontaneous Emission and Lifetime

ECE 5616Curtis

Laser basicsSpontaneous and Stimulated Emission

ECE 5616Curtis

Laser basicsStimulated Emission versus Absorption

ECE 5616Curtis

Laser basicsPopulation Inversion

ECE 5616Curtis

Line Broadening: Homogeneous

ECE 5616Curtis

Line Broadening: Inhomogeneous

(gas) different velocities

ECE 5616Curtis

Laser basicsAmplification and Lasing

ECE 5616Curtis

Laser basicsCavity Modes

ECE 5616Curtis

Single versus Multimode

Single Mode Multimode Mode

ECE 5616Curtis

Spectral hole burning

ECE 5616Curtis

Spectral hole burning

ECE 5616Curtis

Multimode/Inhomogeneous

ECE 5616Curtis

Multimode Lasers

ECE 5616Curtis

Laser basics3 level vs 4 level systems

N2=No/2 + Nt/2

N2~ Nt

N1=No/2 - Nt/2

N1~0

t

o

level

level

NN

NN

2)()(

42

32 ≈

Can be large ~100

ECE 5616Curtis

Laser basicsSome simple laser equations

(Add in α which is passive loss in medium)

ECE 5616Curtis

Threshold InversionMinimum Power Required

)ln1()(

8212

2

rrlg

tnN spont

t −= αλυ

πυ

υΔ≈

)(1

og

)1( Rcnltc −

≈

Gain equals losses and then solve for ΔN

The cavity decay time (tc) assuming α=0, r1 ~ r2 is approximately

The threshold population is many times written using tc as

)(8

3

23

υυπgtc

tnN

c

spontt =

Power needed to do this is given by

24)(

tVhNP t

levels

υ=

LinewidthAlso 1/lw will give you minimum pulse length

ECE 5616Curtis

ExampleErbium Doped Fiber Amplifier: EDFA