Gaussian beams

1
Gaussian beams
slow variable
kzn?/c
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2-D Fourier Transform in space
Represent the field as a superposition of plane
waves
Solution
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2-D Gaussian shape
gaussian
Initial beam shape
w0-beam waist radius
Total Power must be conserved
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Fourier Transform of Gaussian beam
Fourier beam waist radius is 2/w0
z0kzw02/2? w02n/?
Diffraction (Raleigh) length
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Gaussian beam propagation
Apply inverse Fourier transform
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Propagation
Beam radius
Radius of curvature
Extra phase shift
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Beam parameters
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Diffraction angle
z0? w02n/?
zgtgtz0 2w(z)?2zw0/z02z?/n?w0z?d, where
diffraction angle is ?D2?/ n?w0
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Meaning of the radius of curvature
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Propagation through the thin lens
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Focusing
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q-parameter
qz-jz0
z0? w02n/?
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Matrices and Gaussian beam
qz-jz0
Free space
Thin lens
For an arbitrary matrix also
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Gaussian mode in the resonator
The mode must be reproduced after one round trip
ADlt2 or 0lt(AD2)/4lt1
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Stability
Symmetric cavity
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Gaussian beam parameters in the cavity
z0 1/Rc0
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Gaussian beam parameters in the cavity
At the waist
At the mirror
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Specific cavities
Symmetric confocal cavity dR
Symmetric concentric cavity d2R
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Specific cavities
Flat cavity
More typical cavity R2d
R
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Higher order modes
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Higher order modes
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Resonant Frequencies
(symmetric cavity)
Resonant condition
R
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Specific cavities transverse modes
Plane cavity R??
Symmetric confocal cavity dR
Symmetric concentric cavity d2R
More typical cavity R2d
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Diffraction loss
Fresnel number Na2/d?