3-1. The Gaussian beam 3-1. The Gaussian beam

3. Beam optics

3. Beam optics

3-1. The Gaussian beam 3-1. The Gaussian beam

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 modulated 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 beams Gaussian beams

z₀ : Rayleigh range.

Gaussian beam : Intensity Gaussian beam : Intensity

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

Beam radius Beam radius

At the Beam waist : Waist radius = W₀ Spot size = 2W₀

(divergence angle)

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 !

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) :

wave fronts near the focus wave fronts near the focus

Radius of curvature

Wave fronts:

π/2 phase shift relative to spherical wave

Changes in wavefront radius with propagation distance

Gaussian parameters

: Relationships between parameters Gaussian parameters

: Relationships between parameters

q(z) ?

q(z) ?

3.2 TRANSMISSION THROUGH OPTICAL COMPONENTS 3.2 TRANSMISSION THROUGH OPTICAL COMPONENTS

A. Transmission Through a Thin Lens

B. Beam Shaping B. Beam Shaping

Beam Focusing

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

If (2 z₀ ) >> f ,

(a) z and z’ :

Gaussian Beams

higher order beams

Gaussian Beams

higher order beams

Hermite-Gaussian

Bessel Beams