24 guassian beams and lasers - university of colorado …ecee.colorado.edu/~ecen5616/webmaterial/24...

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Gaussian Beams and Lasers

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

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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.

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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

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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.

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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.

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Gaussian Beams Conversion formulas

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Gaussian Beam Parameter q(z)

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Gaussian Beam Parameter q(z)

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How does q change with transfer and refraction ?

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ABCD Matrix for Guassians

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ABCD Matrix for Guassians

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Matrix Method

Amnon Yariv, Optical Electronics

where

So can use them like before for multiple elements

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ExampleGaussian beam focusing

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ExampleGaussian beam focusing

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ExampleGaussian beam in lens waveguide/resonator

F=2/RFor mirrors

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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 !!!

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Representation of Gaussian Beams by complex rays (1)

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Representation of Gaussian Beams by complex rays (1’)

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Representation of Gaussian Beams by complex rays (2)

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Representation of Gaussian Beams by complex rays (2’)

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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

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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.

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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 ).

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

Works for virtual objects as well…

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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.

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Example: Collimation Lens

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Laser BasicsAbsorption and Emission

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Laser basicsSpontaneous Emission and Lifetime

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Laser basicsSpontaneous and Stimulated Emission

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Laser basicsStimulated Emission versus Absorption

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Laser basicsPopulation Inversion

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Line Broadening: Homogeneous

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Line Broadening: Inhomogeneous

(gas) different velocities

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Laser basicsAmplification and Lasing

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Laser basicsCavity Modes

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Single versus Multimode

Single Mode Multimode Mode

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Spectral hole burning

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Spectral hole burning

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Multimode/Inhomogeneous

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Multimode Lasers

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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

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Laser basicsSome simple laser equations

(Add in α which is passive loss in medium)

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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

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ExampleErbium Doped Fiber Amplifier: EDFA