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Polarisation effects in 4 mirrors cavities
•Introduction•Polarisation eigenmodes calculation•Numerical illustrations
F. Zomer LAL/OrsayPosipol 2008 Hiroshima 16-19 june
2
2D: bow-tie cavity 3D: tetrahedron cavity
L~500mm
h~100mm
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
L~500mm
h~100mm
V0
V0
3
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)
4
y yx
z
12
1 2
Planar mirror
Spherical mirror
Planar mirror
Spherical mirror
Example of a 3D cavity.
k1
k2
p1
s1
p2
s2
s2
k3
p2’
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.
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}
, 1i iN
i i+1 i i+1
i i+1 i i+1
s s p' s
s p p' p
, ,
, ' ,
| | 0, such
0 | |
s
p
is r s i s
ir p i pp
r e E ER R
E Er e
With s≠p when mirrors are misaligned !!!rs ≠ rp when incidence angle ≠ 0
V0
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1 41 4 34 3 23 2 12J R N R N R N R NTaking the mirror 1 basis as the reference basis one gets the Jones Matrix for a round trip
01
n
n
TJ
circulatingE 0V
And the electric field circulating inside the cavitywhere V0 is the incident polarisation vector in the s1,p1 basis
The 2 eigenvalues of J are ei = |eiexp(ii) and 1≠ a priori.
The 2 eigenvectors are noted ei . One gets
1
2
1
i
1
2
1
10
1
10
1
,
with the normalised eignevectors =
i i
ii
e e
e e
eU U T
e
U
circulatingE
p p
1 21 1
1 21
i
i
0
1
s e s
V
e
' e ' e
ee
e
is the round trip phase: =2L if the cavity is locked on one phase, e.g. the first one 1=2,
then 2=221
Transmission matrix
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2 1
2 1
1 11 10 0
( )
( )
0 0
1
2
1 11 1 1
2
10
1If :
10
1
10
1If :
10
1
i
i
eT T U U T
e
eT T U U T
e
e
e
circ
circ
E
E
V V
V V
1 2
2
0
1 0
e e V
e Ve
Experimentally one can lock on the maximum mode coupling, so that the circulating field inside the cavity is computed using a simple algorithm :
Numerical study : 2D and 3D•L=500mm, h=50mm or 100mm for a given V0 •Only angular misalignment tilts x,y = {-1,0,1} mrad or rad 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
Spherical mirror
Spherical mirror
Planar mirror
Planar mirror
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An example of a mirror misalignments configuration : 3D with 3D missalignments
Spherical mirror
Spherical mirror
planar mirror
planar mirror
9
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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Stokes parameters for the eigenvectors shown using thePoincaré sphère
Numerical examples of eigenvectors for 1mrad misalignment tilts
2D
S3=0
3 mirror coef. of reflexion consideredNlayer=16, 18 and 20
S3=1
3D
Circular polarisation Linear polarisationElliptical polarisation otherwise
S1
S2
S3
28 entries/plots(misalignments configurations)
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2D
3D
3, 1inS
0
1
2V =
i
2
For 1mrad misalignment tilts and
The circulating field is computed for :
Then the cavity gain is computed
gain = |Ecirculating|2 for |Ein|2=1
12
2D
3, 1inS
0
1
2V =
i
2
1mradtilts
3D
Stokes Parameters distributions
13
2, 1inS
0
1
2V =
1
2
1mradtilts
X checkLow finesse
2DEigenvectors
Cavitygain
Stokes parameters Stokes
parameters
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2, 1inS
0
1
2V =
1
2
1mradtilts
X-checklow finesse
3DCavitygain
Stokes parameters Stokes
parameters
Stokes parameters
15
3, 1inS
0
1
2V =
i
2
1radtilts leads to ~10% effecton the gainfor the highest finesseN=20
Numerical examples for U or Z 2D & 3D cavities (6reflexions for 1 cavity round-trip)
(proposed by KEK)
U 2D U 3D
Z 2D
‘closed orbits’ are always self retracinghighest sensitivity to misalignments viz bow-tie cavties
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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 …
17
U 2D L=500.0;h=150.0, ra=1.e-7, S3=1