A new phase difference compensation method for elliptically birefringent media
Piotr Kurzynowski, Sławomir DrobczyńskiInstitute of Physics
Wrocław University of TechnologyPoland
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
P P=0
A A=90
M f=45
-unknown
xxoutI cos1,
C =-45
x-variable
The phase shift compensation ideafor 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
Transversal compensators (e.g. the Wollastone one): for some x0 co-ordinate axis
Inclined compensators (e.g. the Ehringhause one):for some inclination angle 0
000 xx CM
000 CM
90 CM
Azimuthal compensators for linearly birefringent media
P P=0
/4 =0
A A-variableA0=90
M f=45
-unknown
2cos1,outI
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:
A 2
Linearly birefringent medium in the compensator setup
the Stokes vector V of the light after the medium M
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
0011
**sin0**00**cos0**01
sin0
cos1
2sin2cos2sin2cos2cos
1
V
Elliptically birefringent medium in the compensator setup (1)
the Stokes vector V of the light after the medium M
but
This is a rotation matrix R(2f) !
0011
**2cossin0**2sinsin0**cos0**01
2cossin2sinsin
cos1
f
f
f
fV
sin0
cos1
2cos2sin002sin2cos00
00100001
2cossin2sinsin
cos1
ff
ff
f
fV
Elliptically birefringent medium in the compensator setup (2)
hence
so
VRV f 2
VRV f 2
Elliptically birefringent medium in the compensator setup (3)
elimination of the f medium M ellipticity influence- the rotation matrix
- 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
0011
**2cossin0**2sinsin0**cos00001
2cos2sin002sin2cos0000100001
sin0
cos1
f
f
ff
ff
Proposed compensator setups
for direct compensators:
for azimuthal compensators:
90,45,0450 ACCMP x
ACMP 04
,0450
The direct compensation setup
M f=45
,f unknownP P=0
C =0
-variable
C =-45
C-variable
A A=90
The azimuthal compensation setup
M f=45
,f unknownP P=0
C =0
-variable
/4 =0
A A-variableA0=90
The output light intensity distribution
where for direct compensators
or for azimuthal ones .
generally
where
and
ffout XXXI 2cossinsincoscos1;;,
CX
AX 2
00 cos1; XXVXXIout
fX 2costantan 0
fV 2sinsin1 222
Numerical calculations--the Wollastone compensator setup
The normalized intensity distribution for i = - 2f
1= 0 <2<3
0
0,5
1
11,8 12 12,2 12,4 12,6 12,8 13 13,2
x [mm]
Nor
mal
ized
int
ensi
ty a
2
1
3
Numerical calculations--the Wollastone compensator setup
The normalized intensity distribution for = - 2f0
0
0,5
1
11,8 12 12,2 12,4 12,6 12,8 13 13,2
x [mm]
Nor
mal
ized
int
ensi
ty a
1
2
Numerical calculations--the Wollastone compensator setup
The normalized intensity distribution for = - 2f=0
0
0,5
1
11,8 12 12,2 12,4 12,6 12,8 13 13,2
x [mm]
Nor
mal
ized
int
ensi
ty a
1
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?
A quarter wave plate action is from mathematically point of wiev a rotation matrix R(90°)
So symbolically
The new Senarmont setup configuration!
ACMP 90,04
,0,450
90,004
,0 CC
ACMP ,0450
fV 2901 fV 2cossin1 222
Experimental results (1)
1 < 2 < 3
Experimental results (2)
0
40
80
120
160
0 50 100 150 200 250 300
Pixel
Inte
nsity
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
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