“Gaussian beam”

27. Gaussian Beam 27. Gaussian Beam

: Gaussian function

“Gaussian beam”

How to determine - Beam size W at z - beam waist

- beam radius

- divergence angle

Wave Equation

: Helmholtz equation

( wavenumber )

“The wave equation for monochromatic waves”

Now, let’s start with the wave equation in free-space

Paraxial Helmholtz equation Paraxial Helmholtz equation

ÎSlowly varying envelope approximation of the Helmholtz equation Î Paraxial Helmholtz equation.

: Helmholtz equation

: Consider a plane wave propagating in z-direction

: Slowly varying approximation

2 2

2

2 2

T

x y

⎛ ∇ ≡ ∂ + ∂ ⎞

⎜ ∂ ∂ ⎟

⎝ ⎠

One simple solution to the paraxial Helmholtz equation : paraboloidal waves

Another solution of the paraxial Helmholtz equation : Gaussian beams

A paraxial wave

by a complex envelope A(r) that is a slowly varying function of position:

The complex envelope A(r) must satisfy the paraxial Helmholtz equation

Gaussian beam Gaussian beam

W₀ : beam waist where,

Gaussian beam

2

0 0

z π W λ

⎛ = ⎞

⎜ ⎟

⎝ ⎠

z₀ : Rayleigh range

Gaussian beam

Gaussian beam

Intensity of Gaussian beam Intensity of Gaussian beam

The intensity is a Gaussian function of the radial distance ρ. Æ This is why the wave is called a Gaussian beam.

On the beam axis (ρ = 0)

At z = z₀ , I = I_o/2

Gaussian beam : Power Gaussian 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 ρ in the transverse plane at position z to the total power is

( ρ

= a )

Power ratio clipped by aperture

Beam radius Beam radius

At the Beam waist : Waist radius = W₀

Spot size = 2W₀

(divergence angle) (far-field)

Depth of Focus Depth of Focus

The axial distance within which the beam radius lies within a factor root(2) of its minimum value (i.e., its area lies within a factor of 2 of its minimum) is known as the depth of focus or confocal parameter

beam area at waist

= λ

A small spot size and a long depth of focus cannot be obtained simultaneously !

Depth of focus, Rayleigh range, and Beam waist

Gaussian parameters

: Relationships between parameters Gaussian parameters

: Relationships between parameters

Phase of the Gaussian beam Phase of the Gaussian beam

kz : the phase of a plane wave.

: a phase retardation ranging from - π/2 to - π/2 . : This phase retardation corresponds to an excess delay of the wavefront in comparison with a plane wave or a spherical wave

The total accumulated excess retardation as the wave travels from

Guoy effect

Wavefront - bending Wavefront - bending

Wavefronts (= surfaces of constant phase) :

Wavefronts near the focus Wavefronts near the focus

Wave fronts:

π/2 phase shift

relative to

spherical wave

TRANSMISSION THROUGH OPTICAL COMPONENTS TRANSMISSION THROUGH OPTICAL COMPONENTS

A. Transmission Through a Thin Lens

Gaussian beam relaying

Gaussian beam Focusing

If a lens is placed at the waist of a Gaussian beam,

If (2 z₀ ) >> f ,

Other Beams

higher order beams

Other Beams

higher order beams

Hermite-Gaussian

Bessel Beams