27 gaussian beams
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27. Gaussian Beam
27. Gaussian Beam
: Gaussian function
Gaussian beam
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How to determine
- Beam size W at z- beam waist- beam radius- divergence angle
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Wave Equation
: Helmholtz equation
( wavenumber )
The wave equation for monochromatic waves
Now, lets start with the wave equation in free-space
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Paraxial Helmholtz equationParaxial Helmholtz equation
Slowly varying envelope approximation of the Helmholtz equation
Paraxial Helmholtz equation.
: Helmholtz equation
: Consider a plane wave propagating in z-direction
: Slowly varying approximation
2 2
2
2 2Tx y
+
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One simple solution to the paraxial Helmholtz equation : paraboloidal waves
Another solution of the paraxial Helmholtz equation : Gaussian beams
A paraxial wave is a plane wave e-jkz modulatedby a complex envelopeA(r) that is a slowly varying function of position:
The complex envelopeA(r) must satisfy the paraxial Helmholtz equation
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Gaussian beamGaussian beam
W0 : beam waist
where,
Gaussian beam
2
0 0z W
=
z0 : Rayleigh range
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Gaussian beamGaussian beam
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Intensity of Gaussian beamIntensity of Gaussian beam
The intensity is a Gaussian functionof the radial distance .
This is why the wave is called a Gaussian beam.
On the beam axis ( = 0)
At z = z0 , I = Io/2
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Gaussian beam : PowerGaussian beam : Power
The result is independent of z, as expected.The beam power is one-half the peak intensity times the beam area.
The ratio of the power carried within a circle of radius in the transverseplane at position z to the total power is
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( )0 a =
Power ratio clipped by aperture
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Beam radiusBeam radius
At the Beam waist :
Waist radius =W0Spot size =2W0
(divergence angle)(far-field)
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Depth of FocusDepth of Focus
The axial distance within which the beam radius lies within a factor root(2) ofits minimum value (i.e., its area lies within a factor of 2 of its minimum) isknown as the depth of focus or confocal parameter
beam area at waist
=
A small spot size and a long depth of focus cannot be obtained simultaneously !
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Depth of focus, Rayleigh range, and Beam waist
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Gaussian parameters
: Relationships between parameters
Gaussian parameters
: Relationships between parameters
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Phase of the Gaussian beamPhase of the Gaussian beam
kz : the phase of a plane wave.
: aphase retardation
ranging from - /2 to - /2 .
: This phase retardation corresponds to an excessdelay of the wavefront in comparison with a planewave or a spherical wave
The total accumulated excess retardation as the wave travels from
Guoy effect
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Wavefront - bendingWavefront - bending
Wavefronts (= surfaces of constant phase) :
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Wavefronts near the focusWavefronts near the focus
Wave fronts:
/2 phase shiftrelative to
spherical wave
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TRANSMISSION THROUGH OPTICAL COMPONENTSTRANSMISSION THROUGH OPTICAL COMPONENTS
A. Transmission Through a Thin Lens
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Gaussian beam relaying
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Gaussian beam Focusing
If a lens is placed at the waist of a Gaussian beam,
If (2 z0 ) >> f ,
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Other Beamshigher order beams
Other Beamshigher order beams
Hermite-GaussianBessel Beams
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