Higher order gaussian beams

Question 1

What is TEMmn
what is R(z) and w(z) in mn!=0 and TEM00

Answer 1

Transverse electro magnetic
mn is the orhter of the TEM mode

R(Z) and w(Z) are independent of m and n and therefore higher order Gauss beams propogate similarly to TEM00
they remain of the same order, described by the same q param if transformed by a normal ABCD matrix

Question 2

Difference between TEM00 and TEMmn in terms of waists

Answer 2

Since for a TEMmn the waist and divergence angle (th) depend on m and n and so the waist and div can be different in x direction

typically higher order -> larger waist
higher order beams extract from the gain (away from main 00 order?)

Question 3

Since higher order gaussians extract form the gain how can you minimise their negativ eeffects on a TEM00 mode

Answer 3

Discriminate away hiher orders using an aperture to cut out their larger waists
since TEMmn will typically have larger waists than 00

Question 4

TEM mode solns

Answer 4

QHOsc in both x and y directions having known soluns
hermite poly in cartesian
laguerre pol in cylindrical
mulitplies by gaussian functions and a longitundinal phase

Question 5

how is the pahse of the gauss beam changed from tem00

Answer 5

TEM00 theta(k) = kz - tan-1(z/z0)
now theta(k,m,n) = kz - (m+n+1) tain-1(z/z0)
so the phase of the beam is modified in higher order modes
just like the vq -> vqmn
w - > wmn, THETA (div) -> theta_mn

or pl if cylidnrical coords instead of mn

Question 6

Real beams are amde of what kind of modes

Answer 6

Arb mixtures of higher order modes
beams cannot be measured by normal waist etc but still want to know
-woM the beam waist measured at pts of 1/e^2 drop in I profile
-beam div ThM angle at a distance za from laser with za»z0

Question 7

What is M

Answer 7

Beam quality factor M^2 definined as the ratio of the measured beam divergence of the multimode beam and the beam div calc for an equiv zero order beam with the same beam waist as the multimode beam

M^2 = 1 corresp to eprfect gauss TEM00 beam, IRL m^2>1 .
Perf gauss beam with M^2 =1 can reallybe focused all the way down to the diffraction limit

Question 8

When do you have to start taking diffr effects into account

Answer 8

When w2 or w2 spprox a
when th emirror size is approx the spot size sine then 1/e^2 of the beam is lost
aka when Nfres near 1
for a resonator this means unstable and g1g2 outside of 0-1 rang

Question 9

What do you do when you need to calc diffr

Answer 9

Treat each pt in the beam acc Hugens principle as a source of a spherical wavelet and sum up all contribs on the plane youw ant with a Fresnel diffraction integral where you approx r = z

Question 10

What happens with Frauenhofer diffraction by an aperture

Answer 10

perfoming the Fresnel diffraction integral reults in a Bessel function in both E and Intensity .

The on axis intensity is given by an Airy pattern with nodes
the spot size is approx double the TEM00 Gaussian mode

Question 11

what happens in reality when people calculate diffraction integrals in resonators across multiple modes including possble diffr losses

Answer 11

Numerical iterativ emode calculasions so that after many (300) bounses beteen end mirrors that the higher order modes diffract the most. and so the lowest order survices.
IRL the Guassian mode tem00 diverges the lest and so ends up surviving even when we include diffration effects