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FZ_polar_4_miroirs

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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
•Only angular misalignment tilts dq x,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                        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
for 1mrad misalignment tilts

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 :

For 1mrad misalignment tilts
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
      1mrad
      tilts




                   2D

                                       12
                    X check
                  Low finesse
                       2D        Cavity
Eigen                            gain
vectors




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

                  S2,in  1
                   1mrad
                   tilts

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




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

                      S2,in  1

                        1mrad
                        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
                                    1mrad
                                    tilts leads
                                    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




                                       17

				
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