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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