Laser Physics 15
15..
Passive optical resonators (cont.)
cont.)
Pál Maák
Atomic Physics Department
1
Passive optical resonators (rev.)
Characteristics of passive optical resonators
open (modes with loss), dimensions >>
Estimation of the resonator lifetime -
r
spectral bandwidth of the modes Q-factor
laser
1
r
2
r
Q
r
Types
Plane parallel resonator - approximate determination of the l,m,n
frequencies
- transverse amplitude distribution TEMlm and
loss
c
n
l ,m ,n 1
l ,m ,n
2L
2
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L=R
Passive optical resonators
z=0
z
Confocal resonator – normalized field distribution on the mirrors
Analytic solution when L>>a (a is the radius of the round mirror), cos
~ 1, r ~ L and N >>1 – lossless confocal resonator in paraxial
approximation.
U 2 P2
i
2
U1 P1 exp ikr 1 cos
r
1
dS1
Fresnel-Kirchoff diffraction integral
Normalized field distribution on the mirrors:
U lm x , y
H l H m exp
x2
y2
L
Hl,Hm are the Hermite polynomials, l and m are integer and show the
order of the polynomials:
H0 x 1, H1 x 2 x ,
H2 x
3
2 2x 2 1 ,
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L=R
Passive optical resonators
z
z=0
Confocal resonator - normalized field distribution on the mirrors (cont.)
The lowest order mode (TEM00) has a Gaussian distribution, the
normalized field distribution:
U 00 x , y
exp
L
x2
y 2 , H0
1.
The amplitude falls to the ratio of e-1 in the x or y direction when
1/ 2
L
ws
, ws is the radius of the laser spot on the mirror.
2
x2
y2
spot
Gaussian beam is characterized
by the beam radius and not with
the FWHM!
4
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L=R
Passive optical resonators
z=0
z
Confocal resonator – normalized field distribution at any place of the resonator
U x, y ,z
2x
2y
w0
Hl
Hm
e
w z
w z
w z
x2 y 2
w2 z
e
i kz 1 l m
z
e
longitudinal
amplitude factor
2 12
w z
w0 1
2z L
Rz
z1
L 2z ,
2
L=zc
transversal
phase factors
, w z
z
ik x 2 y 2 2 R z
0
w0
tg 1 2z L .
L2
12
, L
2 w 02
,
w0 is the minimal beam
radius or the beam waist at
z=0!
ws
2 w0
zc is the confocal parameter
zc=2 w02/ .
5
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Passive optical resonators
Confocal resonator – determination of
l,m,n frequencies by the longitudinal
phase factor
e i kz 1 l m
L
k
2
kL
4
1
1 l
1 l
m
c 2n
l ,m ,n
2
1 l
4L
n , k
m
Frequency difference of
consecutive longitudinal modes:
Frequency difference of
consecutive transversal modes:
6
1 l
2
c
.
e i kz 1 l m tg
1
2z L
,
z
tg 1 2z L
4
L
k
2
m tg 1
z
m tg
1
1
n
,
Degenerated modes!
n
m
l ,m ,n 1
l ,m 1,n
l ,m ,n
c
2L
l ,m ,n
c
4L
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Passive optical resonators
Confocal resonator – wavefronts
Because of the transversal phase factor the surfaces of z = constant are not
wavefronts
therefore the wavefronts are not plane!
e
i kz 1 l m
z
e
ik x 2 y 2 2R z
We can determine the surfaces of constant phase by neglecting (1+l+m) (z):
(x, y)
k x2 y 2
2R
wavefront
k z
k z0
R
P
R z0
7
0
z z0
z
The left side is the equation of a
paraboloid of revolution around the z axis
with a radius of R at z=z0!
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Passive optical resonators
Confocal resonator – wavefronts
The equation of a spherical surface from point P is:
x2+y2+(z+R-z0)2=R2
(x, y)
in case z-z0<<R
wavefront
(z+R-z0)2=R2+2R(z-z0)+(z-z0)2,
R
P
2
y
x2
y2
2R z z0
k x2 y 2
2R
k z
x
8
2
R
2
2R z z0
R
2
R z0
0
z z0
z
0
k z0
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Passive optical resonators
Confocal resonator – wavefronts
The surfaces of constant phase are ~ spheres, in the middle and in the
infinity the wavefronts are plane, on the mirrors the radius of curvature
of the wavefront is exactly equal with the geometrical radius of
curvature of the mirror!
9
2
Rz
z1
L 2z ,
z
0, R 0
,
z
, R
,
z
L
L
, R
2
2
L.
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Passive optical resonators
Confocal resonator – complex q parameter
The transversal part of the amplitude is:
U x, y ,z
2x
2y
w0
Hm
Hl
e
w z
w z
w z
U t ~ exp
k x2 y 2
i
2R
x2 y2
w2 z
e i kz 1 m l
z
e ik x
2
y 2 2R z
x2 y 2
w2
With the notation
1 1
i
q R
w2
Important: q is full complex when
the wavefront is plane!
k x2 y 2
i
Ut ~ exp
.
2q
q is the complex radius of the beam.
10
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Passive optical resonators
Confocal resonator – calculation of the loss (numerically)
11
The loss in the resonator with spherical mirrors is significantly smaller
in practice the resonators consist of spherical mirrors and the
radius of curvature of the mirrors is usually large compared to the
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length of the resonator.
Passive optical resonators
Gaussian modes – transversal mode patterns (solutions of the diffraction
integral at given symmetry)
Hermite-Gaussian modes
x, y symmetry
12
Laguerre-Gaussian modes
circular symmetry
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Passive optical resonators
General resonator
We can put a mirror with a suitable
radius of curvature in the place where
the wavefront bending is the same,
e.g. in place of wavefronts 1’ and 2’
and we will have a general resonator.
All stable resonator with at least one spherical mirror has its equivalent
confocal resonator.
R1
z1 1
L
z1 0
R1
13
z2
z1 z2
R2
zc
2z1
2
, R2
z2 1
zc
2z2
2
,
L
In the symmetrical resonator (R1=R2=R):
L
zc2 2R L L, mert z
2
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