Download Indirect and azimuthal compensators for elliptically birefringent media

A new phase difference compensation
method for elliptically birefringent media
Piotr Kurzynowski, Sławomir Drobczyński
Institute of Physics
Wrocław University of Technology
Poland
Scheme of presentation
 The literature background
 Compensators for linearly birefringent media
 Elliptically birefringent medium in the compensator
setup
 A phase plate eliminating the medium ellipticity
 Numerical calculations
 The measurement procedure
 Experimental results
 Conclusions
The literature background
 H.G. Jerrard, „Optical Compensators for Measurements of Elliptical Polarization”,
JOSA, Vol.38 (1948)
 H. De Senarmont, Ann. Chim. Phys.,Vol.73 (1840)
 P. Kurzynowski, „Senarmont compensator for elliptically birefringent media”, Opt.
Comm., Vol.171 (2000)
 J. Kobayashi, Y. Uesu, „A New Method and Apparatus ‘HAUP’ for Measuring
Simultaneously Optical Activity and Birefringence of Crystals. I. Principles and
Constructions”, J.Appl. Cryst., Vol.16 (1983)
 C.C. Montarou, T.K. Gaylord, „Two-wave-plate compensator for single-point
retardation measurements”, Appl. Opt., Vol.43 (2004)
 P.Kurzynowski, W.A. Woźniak, „Phase retardation measurement in simple and
reverse Senarmont compensators without calibrated quarter wave plates”, Optik,
Vol.113 (2002)
 M.A. Geday, W. Kaminsky, J.G. Lewis, A.M. Glazer, „Images of absolute
retardance using the rotating polarizer method”,
J. of Micr., Vol.198 (2000)
Direct compensators
for linearly birefringent media
C =-45
M f=45
-unknown
P P=0
A A=90
x-variable
Iout  x ,    1  cos x   
The phase shift compensation idea
for direct compensators
 A rule: the total phase shift introduced by two media is equal
to the difference phase shifts introduced by the medium M
and the compensator C, because  M  C  90
 Transversal compensators (e.g. the Wollastone one): for
some x0 co-ordinate axis
 x0    M   C x0   0
 Inclined compensators (e.g. the Ehringhause one):
for some inclination angle 0
 0    M   C 0   0
Azimuthal compensators
for linearly birefringent media
/4 =0
M f=45
 -unknown
P P=0
A A-variable
A0=90
Iout  ,    1  cos2   
The phase shift compensation idea for
azimuthal compensators
 A rule: the quarter wave plate transforms the polarization
state of the light after te medium M to the linearly one:
  2   A
Linearly birefringent medium
in the compensator setup
the Stokes vector V of the light after the medium M
1
0

  1  1
cos 2  cos 2  cos   0 cos 


V 
0
 sin 2  cos 2   0  0

 
 
sin
2

sin


 
 0 sin 
*
*
*
*
* 1
* 1
 
* 0
 
* 0
the light azimuth angle doesn’t change; the light ellipticity angle is
equal to the half of the phase shift  introduced by the medium M
Elliptically birefringent medium
in the compensator setup (1)
the Stokes vector V of the light after the medium M
1
0

 1

 0
cos 
cos 

V  
  sin  sin 2 f  0  sin  sin 2 f

 
sin

cos
2

f

 0 sin  cos 2 f
*
*
*
*
* 1
* 1
 
* 0

* 0
but
1

 1

 0
cos 

V  
 sin  sin 2 f  0

 
sin

cos
2

f

 0
0
0
1
0
0 cos 2 f
0 sin 2 f
0
 1 


0
 cos  
 sin 2 f   0 

cos 2 f   sin  
This is a rotation matrix R(2f) !
Elliptically birefringent medium
in the compensator setup (2)
hence
so


V  R 2 f  V


V  R  2 f  V
Elliptically birefringent medium
in the compensator setup (3)
 elimination of the f medium M ellipticity influence
-
the rotation matrix
 1  1
cos   0


 0  0

 
sin


 0
-
-
0
1
0
0
0 cos 2 f
0  sin 2 f
0
 1
 0
cos 

sin 2 f  0  sin  sin 2 f

cos 2 f  0 sin  cos 2 f
0
0
0 0 1
* * 1
 
* *  0 
 
* * 0
the rotation matrix a linearly birefringent medium C with the azimuth
angle =0° and introducing the phase shift 
so if the medium C is introduced in the setup, the light azimuth angle
doesn’t change if only =2·f
Proposed compensator setups
 for direct compensators:
 
 




 
P 0  M 45  C 0 ,    C  45 ,  x  A 90
 for azimuthal compensators:
 
 


P 0  M 45  C 0 ,   




0   A 
4

The direct compensation setup
A A=90
P P=0
M f=45
,f unknown
C =0
 -variable
C =-45
C-variable
The azimuthal compensation setup
/4 =0
P P=0
M f=45
,f unknown
C =0
 -variable
A A-variable
A0=90
The output light intensity distribution



I out  , f ;  ; X  1  cos   cos X  sin   sin X  cos 2 f   
where
or
generally
where
and
X C
for direct compensators
X  2 A for azimuthal ones .
Iout  X ; X 0   1  V  cos X  X 0 

tan X 0  tan   cos 2 f   


V 2  1  sin 2   sin 2 2 f   


Numerical calculations-the Wollastone compensator setup
 The normalized intensity distribution for i =  - 2f
1= 0 <2<3
Normalized intensity a
1
1
2
3
0,5
0
11,8
12
12,2
12,4
12,6
x [mm]
12,8
13
13,2
Numerical calculations-the Wollastone compensator setup
 The normalized intensity distribution for = - 2f0
1
Normalized intensitya
2
0,5
1
0
11,8
12
12,2
12,4
12,6
x [mm]
12,8
13
13,2
Numerical calculations-the Wollastone compensator setup
 The normalized intensity distribution for  =  - 2f=0
1
Normalized intensitya
2
0,5
1
0
11,8
12
12,2
12,4
x [mm]
12,6
12,8
13
13,2
The measurement procedure
 The direct compensators:
a) the ellipticity angle f measurement the inclined
(for example Ehringhause one) compensator C
action the fringe visibility maximizing  f
b) the absolute phase shift  measurement
two-wavelength or white-light analysis of
the intensity light distribution at the setup output
The Senarmont configuration
 Two or one compensating plates?
 




P 0  M 45 ,   C 0 ,   




0 ,90   A 
4

 A quarter wave plate action is from mathematically point of wiev a
rotation matrix R(90°)
 So symbolically
 




 4 0   C 0 ,90    
C 0 , 
 The new Senarmont setup configuration!
P 0   M 45   C 0 ,     A 
V  1  sin   cos    2 f 
2
2
2
V  1     90  2 f
Experimental results (1)
1
<
2
<
3
Experimental results (2)
160

Intensity
120

80

40
0
0
50
100
150
Pixel
200
250
300
Conclusions
 Due to the compensating plate C application there is
possibility to measure in compensators setups not only the
phase shift introduced by the medium but also its ellipticity
 The solution (,f) is univocal independently of medium
azimuth angle f sign (±45°) indeterminity
 A new (the last or latest?) Senarmont compensator setup has
been presented