Gauassian beams in resonators Flashcards
How does a gaussian beam propogate in space
iN VACUUM:
beam spot size and rad of curv change with propogation
the Guassian spherical wave form is ALWAYS maintained
used ABCD matrices since they tell you how q evoles adn that tells you how the rad of curvature and spot size evolves through an optical system
How can you calc that a gauss beam can be focussed to a very small spot
Start with abcs matrix for a thin lens and a prop dist d AFTER the lens
a.k.a beam at a pt after the lens (not yet sub in f)
then equate real and im parts of qf (from ABCD) to solve for R(d) and w^2(d)
Note that the waist of the beam occurs when R(z) = infty and sub in 0 for deno of R deno and solve for distance d
finally plug this d into the beam waist ormula you derived and you see thatnew beam waist is very small (w0_after lens)
what is a resonator mode
a steady SPATIAL pattern of intensity (and hence of the field) inside the resonator that does not change in time corresponding to succesive reflections off the mirrors
What is the condition for a filed to be a mode of resonator
It must have exactly the same value on the imaginary plane after a round trip as before (when it reaches same pt) and this msut be true regardelss of where the plane is chosen inside the resonator
i.e. q param must not change in a round trip through the resonator
ABCD cond for reosnating modes
q(z) no longer just f but must hold for all z planes chosen in the resonator
q(z) = (Aq(z) +B)/(Cq(z)+D)
i.e. qf= qi for all q(z) in a resonator
these values will change for diff z but must still hold for all z in order ot have a gaussian beam mode
What is the mirror conditions in a resonator to produce gaussian beams
The magnitude of the beam rad of curv must stay the same before and after a mirror reflection to avoid a change of phase front by the mirror
Require gaussian modes to be stable between spherical mirrors (the eg here)
the rad of curv of gaussian mode R(z`) at each mirror = magnitude of rad of curvature of the mirror’
if this is not the case then the mirror will change the magnitude of the beam radius upon reflection
What is the condition on phase of gaussian mode in a resonator
phase is given by: theta(z) = kz - tan-1(z/z0)
for a mode we enforce that the phase cna only chang eby an integer multiple of 2pi for a round trip
the(z2)-theta(z1) = 2pi q , q in ints(same subscript as Vq)
remember k must be in here cuase k det how the wave repeats itself in space (k -> lambda -> wave freq)
what does the condition on the phase of gaussian mode produce
THe allowed longitunial mode resonsance frequencies Vq
so ALL modes could exists but Vq tells you which modes RESONATE and thus will survive after a few passes (lasers require many passes)
What does Vg typically look like for gaussian modes in resonator
v1q depends on c, L, and the g params
for all g1,g2 we see that the mode spacing delta nu is c/2L which is the smae as we saw in mode locking
so the mode SPACING is UNAFFECTED by mirror geometry
Special case where g1g2 = 1 we get
vq = q (c/2L)
the same as normal modes, so mirror geometry DOES AFFECT the MODE POSITIONS (shifting them left and right)
these are the longitudinal modes
How is gaussian beams related to ray optics
If you want a gauss beam with finite x section and pliug into the w0 vlaues you find you must enforce g1g2 in [0,1]
this si the same as ray optics
How are gaussian beams affected by diffraction effects in a resonator
If a mirror is too small and allows spillover at the mirror edges diffraction effects and losses occur so we enforce
a»_space; w1, a» w2
(the mirror radius has to be much larger the the spot rad of the gaussian at both mirrors respectively)
plug in form of w1, w2 and you get
a»_space; sqrt{\lambdaL/pi} -> Nf
What is the frenel number
Nf the fresenel num of the resontor must be»_space;1 to minimise diffraction losses
seee formula dep on a , lambda, L where lambda is gotten from the allowed longitunial freq (det by the mirror curvatures)
lower bound of fresnel number (»1) is NB for good design
How can you minimise difr losses
-mirrors sufficnelty large than Nfresnel is »_space; 1
- use curved mirrors since these refocus the beam after each transit resulting in smaller intensity near mirror edges
so a confocal resonator looses less light than a plane // resonator
- make sure the mirror dia d>3w accoridng to rule of thumb for diffracting gaussians
for Nfres approx 1 the power looses due to diffraction are large
