Transcript Light polarisation and related instabilities in 4 mirror cavities.

Polarisation effects in 4
mirrors cavities
•Introduction
•Polarisation eigenmodes
calculation
•Numerical illustrations
F. Zomer LAL/Orsay
Posipol 2008 Hiroshima 16-19 june
1
3D: tetrahedron
cavity
2D: bow-tie cavity
V0
h~100mm
h~100mm
L~500mm
L~500mm
V0
V0 = the electric vector of the incident laser beam,
What is the degree of polarisation inside the resonator ?
Answer: ~the same if the cavity is perfectly aligned
different is the cavity is misaligned
numerical estimation of the polarisation effects is case
of unavoidable mirrors missalignments
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Calculations (with Matlab)
• First step : optical axis calculation
– ‘fundamental closed orbit’ determined using iteratively
Fermat’s Principal  Matlab numerical precision
reached
• Second step
– For a given set of mirror misalignments
• The reflection coefficients of each mirror are computed as a
function of the number of layers (SiO2/Ta2O5)
– From the first step the incidence angles and the mirror
normal directions are determined
– The multilayer formula of Hetch’s book (Optics) are then
used assuming perfect lambda/4 thicknesses when the
cavity is aligned.
• Third step
– The Jones matrix for a round trip is computed
following Gyro laser and non planar laser standard
techniques (paraxial approximation)
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Planar mirror
y
V0
x
y
P1
k1
p2
S1
Planar mirror
p1
k2
P2
s1
s2
z
p2’
k3
s2
Spherical mirror
S2
Spherical mirror
ni is the normal vector of mirror i
We have si=ni×ki+1/|| ni×ki+1||
and pi=ki×si/|| ki×si||,
pi’=ki+1×si/|| ki+1×si||,
where ki and ki+1 are the
wave vectors incident and
reflected by the mirror i.
Example of a 3D cavity.
Denoting by
• Ri the reflection matrix of the mirror i
• Ni,i+1 the matrix which describes the change of the basis {si,p’i,ki+1}
to the basis {si+1,pi+1,ki+1}
 | rs | eis
R
 0


 Er , s 
 Ei , s 
 , such 
  R

i p

E
E
| rp | e 
 r, p' 
 i, p 
0
With s≠p when mirrors are misaligned !!!
rs ≠ rp when incidence angle ≠ 0
 s i  s i+1 p'i  s i+1 
Ni ,i 1  

s

p
p'

p
 i i+1
i
i+1 
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J  R1 N 41R4 N34 R3 N 23 R2 N12
Taking the mirror 1 basis as the reference
basis one gets the Jones Matrix
for a round trip
And the electric field circulating inside the cavity
where V0 is the incident polarisation vector in
the s1,p1 basis
Ecirculating
  n
  J  T1V0
 n 0 
Transmission matrix
The 2 eigenvalues of J are ei = |ei|exp(ifi) and f1≠f2 a priori.
The 2 eigenvectors are noted ei . One gets
1

 1  e eif1 ei
1
Ecirculating  U 

0


 s 1  e1 s 1  e 2 
,
U 
 p '  e1 p '  e 2 
 1
1

0
1
1  e2 eif2 ei
with the normalised eignevectors e i =
ei
ei


 U 1T V
1 0



 is the round trip phase:
=2pn L
if the cavity is locked on
one phase,
e.g. the first one
f1=2p,
then
f2=2p f2f1
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Experimentally one can lock on the maximum mode coupling, so that the
circulating field inside the cavity is computed using a simple algorithm :
 1

0
 1 e

1
 U 1T V
If e1  T1V0  e 2  T1V0 : Ecirc  U 
1 0


1
 0
i (f2 f1 ) 
1

e
e
2


1


0
 1  e e  i (f2 f1 )

1
 U 1T V
If e 2  T1V0  e1  T1V0 : Ecirc  U 
1 0

1 
0


1

e
2 

Numerical study : 2D and 3D
•L=500mm, h=50mm or 100mm for a given V0
•Only angular misalignment tilts dqx,dqy = {-1,0,1} mrad or mrad
with respect to perfect aligned cavity
•38=6561 geometrical configurations (it takes ~2mn on my laptop)
•Stokes parameters for the eigenvectors and circulating field
computed for each configuration  histograming
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An example of a mirror misalignments configuration :
2D with 3D misalignments
Planar mirror
Spherical mirror
Planar mirror
Spherical mirror
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An example of a mirror misalignments configuration :
3D with 3D missalignments
planar
mirror
Spherical
mirror
planar
mirror
Spherical
mirror
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Results are the following:
For the eigen polarisation
•2D cavity : eigenvectors are linear for low mirror reflectivity
and elliptical at high reflect.
•3D cavity : eigenvectors are circular for any mirror reflectivities
Eigenvectors unstables for 2D cavity at high finesse
 eigen polarisation state unstable
For the circulating field
•In 2D the finesse acts as a bifurcation parameter for the polarisation state
of the circulating field
 The vector coupling between incident and circulating beam is unstable
 the
circulating power is unstable
•In 3D the circulating field is always circular at high finesse because only
one of the two eigenstates resonates !!!
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Numerical examples of eigenvectors
for 1mrad misalignment tilts
Stokes parameters for the
eigenvectors shown using the
Poincaré sphère
S3
3D
28 entries/plots
(misalignments
configurations)
S3=1
S1
S2
q0,p  Circular polarisation
qp/2  Linear polarisation
Elliptical polarisation otherwise
3 mirror coef. of reflexion considered
Nlayer=16, 18 and 20
2D
S3=0
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The circulating field is computed for :
For 1mrad misalignment tilts
and


V0 = 



3D
1 
2 
 S3,in  1
i 

2
Then the cavity gain is computed
gain = |Ecirculating|2 for |Ein|2=1
2D
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Stokes Parameters distributions
 1 
 2

V0 = 
 i 


 2
 S3,in  1
3D
1mrad
tilts
2D
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X check
Low finesse
2D
Eigen
vectors


V0 = 



Cavity
gain
1 
2 
1 

2
S 2,in  1
1mrad
tilts
Stokes
parameters
Stokes
parameters
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X-check
low finesse
3D
Cavity
gain


V0 = 



1 
2 
1 

2
Stokes
parameters
S 2,in  1
1mrad
tilts
Stokes
parameters
Stokes
parameters
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Numerical examples for U or Z 2D & 3D cavities (6reflexions for 1 cavity round-trip)
Z 2D
(proposed by KEK)
 1 
 2

V0 = 
 i 


 2
 S3,in  1
U 2D
1mrad
tilts leads
to ~10% effect
on the gain
for the
highest
finesse
N=20
‘closed orbits’ are
always self retracing
highest sensitivity to
misalignments viz
bow-tie cavties
U 3D
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Summary
• Simple numerical estimate of the effects of mirror
misalignments on the polarisation modes of 4 mirrors
cavity
– 2D cavity
• Instability of the polarisation of the eigen modes
Instability of the polarisation mode matching
between the incident and circulating fields
 power instability growing with the cavity finesse
– 3D cavity
• Eigen modes allways circular
• Power stable
– Z or U type cavities (4 mirrors & 6 reflexions) behave like 2D
bow-tie cavities with highest sensitivity to misalignments
• Most likely because the optical axis is self retracing
• Experimental verification requested …
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U 2D L=500.0;h=150.0, ra=1.e-7, S3=1
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