3. Beam optics

3-1. The Gaussian beam 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 One simple solution to the paraxial Helmholtz equation : paraboloidal waves Another solution of the paraxial Helmholtz equation : Gaussian beams

Gaussian beams z0 : Rayleigh range.

Gaussian beam : Intensity The intensity is a Gaussian function of the radial distance r.  This is why the wave is called a Gaussian beam. On the beam axis (r = 0) At z = z0 , I = Io/2

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 r in the transverse plane at position z to the total power is

Beam radius At the Beam waist : Waist radius = W0 Spot size = 2W0 (divergence angle)

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 (twice the Rayleigh range) A small spot size and a long depth of focus cannot be obtained simultaneously !

Phase of the Gaussian beam kz : the phase of a plane wave. : a phase retardation ranging from - p/2 to - p/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 Wavefronts (= surfaces of constant phase) :

wave fronts near the focus Wave fronts: p/2 phase shift relative to spherical wave wave fronts near the focus Changes in wavefront radius with propagation distance Radius of curvature

Gaussian parameters : Relationships between parameters

3.2 TRANSMISSION THROUGH OPTICAL COMPONENTS A. Transmission Through a Thin Lens

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 Hermite-Gaussian Bessel Beams

q(z) ?