1. 3. Beam optics

2. 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

3. Gaussian beams
z0 : Rayleigh range.

4. 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

5. 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

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

7. 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 !

8. 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

9. Wavefront - bending
Wavefronts (= surfaces of constant phase) :

10. 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

11. Gaussian parameters : Relationships between parameters

13. 3.2 TRANSMISSION THROUGH OPTICAL COMPONENTS
A. Transmission Through a Thin Lens

14. B. Beam Shaping Beam Focusing
If a lens is placed at the waist of a Gaussian beam,
If (2 z0 ) >> f ,

15. (a) z and z’ :

16. Gaussian Beams higher order beams
Hermite-Gaussian
Bessel Beams

17. q(z) ?