Gaussian beams Flashcards
Why can plane and spherical waves nto be considered beams
Althought yes PW are unidirectional, they are infinite laterryly,
spherical waves are not unidirectional
why can one neglect 2nd order z derivs when simplyfing th ehemholtz eqn to the paraxial we
We assume the varions in the the amplitude wrt z with a distance of the order of a wavelength in the z direction are negligable
i.e. the amplitude of the wave changes very little in the direction of propogation over the propogation distance z that is on the order of a wavelength (it propogates too fast forwards along z to haev significan osicllation z changes in xy )
what is the spot size of a guassian beam
w is a spot of radius w if projected onto a secreen
this is a lateral distance from the z axis of the beam
where the intensity is a factor e^2 (7.3) smaller than its on axis peak intensitity
What are p(Z) AND q(Z) int ht eporposed solution gaussian form to paraxial WE
p(z) is in the e^(ip(z)) term
it is a phase factor which changes with z
this is the longitunial phase
q(z) is in the sigma spot
it is the width of the gaussian
indication of the radial spot size
q(z) is broken up into a real and imaginary part see eq what are they for
real: ;1/R(z) this is the curvature of the beam wavefront and related to the radial phase of the beam (hence R)
imag: lambda/(piw^2) relates to the size of the beam since it contains the beam rad w(z)
what is the z0
The rayleigh range
zo = piw0^2/lambda
what is the z0
-The rayleigh range is a measure of the length of the wait region where the spot size is smallest
-the distance (measured from the waist) over which the beam diverges by a swrt(2) factor
-i.e. at z = z0, w(z) = sqrt2 * w0
-this is thus th ept where the intensity is reuced to half and the beam area (transverse to z) is doubled
what is th w(z)
The beam radius as z changes
taken at 1/e^2 pts of the intensity distrib (or 1/e of E field)
eqn w(Z) = w0 sqrt(some factors )
beam is smallest at z = 0 with w0 = w(o) the beam waist
What happens at the beam waist
THis is at z=0 w(z=0) =w0
Here the beam has R= infty and if flat PW front
What hapens in the far field o fgauss beam
z > z0
the rad of curvature of the wavefront (R(z)) is equal to that of a equivalent pt source at z= 0
What hapens in the far field o fgauss beam
z > z0
the rad of curvature of the wavefront (R(z)) is equal to that of a equivalent pt source at z= 0
what is b
The confocal paramater
the depth of focus/diffraction length
full distnace between the sqrt2*w0 pts from the z = 0 x 2
b = 2z0
How does waist size affect the rayleigh range
a smaller w0 at the beam waist gives a greater div angel and thus a greater rate of growth with z of the spot size from w0. so z0 is smaller with smaller spot size
similar to aperture diffraction
the smaller the aperture diameter the greater the diffraction
How does wavelength affect gauss beams
Longer wavelengths have shorter rayleigh ranges (and thus greater divergence angles)
so shorter wavleength beams diverge less
