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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

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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

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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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y yx

z

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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

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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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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

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2D

3, 1inS

0

1

2V =

i

2

1mradtilts

3D

Stokes Parameters distributions

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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

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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 …

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U 2D L=500.0;h=150.0, ra=1.e-7, S3=1