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

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

0

0

1

1

**sin0

**00

**cos0

**01

sin

0

cos

1

2sin

2cos2sin

2cos2cos

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) !

0

0

1

1

**2cossin0

**2sinsin0

**cos0

**01

2cossin

2sinsin

cos

1

f

f

f

fV

sin

0

cos

1

2cos2sin00

2sin2cos00

0010

0001

2cossin

2sinsin

cos

1

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

0

0

1

1

**2cossin0

**2sinsin0

**cos0

0001

2cos2sin00

2sin2cos00

0010

0001

sin

0

cos

1

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

nsit

y

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