# FZ_polar_4_miroirs by wuzhengqin

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```									              Polarisation effects in 4
mirrors cavities

•Introduction
•Polarisation eigenmodes
calculation
•Numerical illustrations

F. Zomer LAL/Orsay                               1
Posipol 2008 Hiroshima 16-19 june
2D: bow-tie cavity                         3D: tetrahedron
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       2
of unavoidable mirrors missalignments
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)
3
Planar mirror

y                    x                             y             Planar mirror
p1
V0         P1                                           P2
k1
p2                                                                    ni is the normal vector of mirror i
s1
p2’
z                     We have si=ni×ki+1/|| ni×ki+1||
S1         k2        s2                  k3                                           and pi=ki×si/|| ki×si||,
S2
pi’=ki+1×si/|| ki+1×si||,
s2
Spherical mirror          where ki and ki+1 are the
wave vectors incident and
Spherical mirror
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                   0               Er , s        Ei , s                                s i  s i+1 p'i  s i+1 
R                                       , such             R                           Ni ,i 1                           
 0                               i p
| rp | e             Er , p '      Ei , p                                 s i  pi+1 p'i  pi+1 
                                                                                                       
With s≠p when mirrors are misaligned !!!                                                                                4
rs ≠ rp when incidence angle ≠ 0
Taking the mirror 1 basis as the reference
basis one gets the Jones Matrix                                J  R1 N 41 R4 N34 R3 N 23 R2 N12
for a round trip

  n
  J  T1V0
And the electric field circulating inside the cavity
where V0 is the incident polarisation vector in                       Ecirculating
the s1,p1 basis                                                                         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          0                       is the round trip phase:
U                                      U 1T V             =2pn L
1
Ecirculating                                                  1 0
                        1                       if the cavity is locked on
        0                           
                  1  e2 eif2 ei   
one phase,
e.g. the first one
 s 1  e1 s 1  e 2                                                     f1=2p,
U                     ,                                          then
 p '  e1 p '  e 2 
 1         1                                                        f2=2p  f2f1
ei
with the normalised eignevectors e i =                                                           5
ei
Experimentally one can lock on the maximum mode coupling, so that the
circulating field inside the cavity is computed using a simple algorithm :

 1                                
 1 e                   0         
If e1  T1V0  e 2  T1V0 : Ecirc   U                                    U 1T V
1
1 0
                     1            
 0                                
              1  e2 ei (f2 f1 ) 
        1                          
 1  e e i (f2 f1 )         0    
If e 2  T1V0  e1  T1V0 : Ecirc   U                                     U 1T V
1
1 0
                               1 
        0                          
                            1  e2 

Numerical study : 2D and 3D
•L=500mm, h=50mm or 100mm for a given V0
with respect to perfect aligned cavity
•38=6561 geometrical configurations (it takes ~2mn on my laptop)
•Stokes parameters for the eigenvectors and circulating field                        6
computed for each configuration  histograming
An example of a mirror misalignments configuration :
2D with 3D misalignments

Planar mirror

Spherical mirror

Planar mirror

Spherical mirror                     7
An example of a mirror misalignments configuration :
3D with 3D missalignments

planar
mirror

Spherical
mirror

planar
mirror

Spherical
mirror

8
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
9
one of the two eigenstates resonates !!!
Numerical examples of eigenvectors

Stokes parameters for the                         3D
eigenvectors shown using the
Poincaré sphère
28 entries/plots
S3                (misalignments
configurations)

S3=1
S2
S1

2D
q0,p  Circular polarisation
qp/2  Linear polarisation
Elliptical polarisation otherwise        S3=0

3 mirror coef. of reflexion considered
Nlayer=16, 18 and 20
10
The circulating field is computed for :

and                                            3D

     1 
      2
V0 =          S3,in  1
     i 
       
      2

Then the cavity gain is computed
2D
gain = |Ecirculating|2 for |Ein|2=1

11
Stokes Parameters distributions

 1 
 2
V0 =      
 i 
     
 2
 S3,in  1
3D
tilts

2D

12
X check
Low finesse
2D        Cavity
Eigen                            gain
vectors

   1 
   2
V0 =      
   1 
     
   2

S2,in  1
tilts

Stokes
parameters                       Stokes
parameters
13
X-check
low finesse
3D
Cavity
gain

   1    Stokes
   2
parameters
V0 =      
   1 
     
   2

S2,in  1

tilts
Stokes
parameters                   Stokes
parameters
14
Numerical examples for U or Z 2D & 3D cavities (6reflexions for 1 cavity round-trip)

(proposed by KEK)                                           Z 2D
 1 
 2
V0 =      
 i 
     
 2
 S3,in  1
to ~10% effect
U 2D                    on the gain                U 3D
for the
highest
finesse
N=20

‘closed orbits’ are
always self retracing
highest sensitivity to
misalignments viz                     15
bow-tie cavties
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 …
16
U 2D L=500.0;h=150.0, ra=1.e-7, S3=1

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