In this work, we present an experimental method to characterize variable optical retarders, which can have linear or non-linear behavior of the retardance variation. A theoretical analysis of such is presented using a combination of Stokes vectors and Mueller matrixes for three different optical retarders. A straightforward method for phase unwrapping, or reconstructing the original phase from the measured retardance, is proposed that yields high-accuracy results. This work can be used in an undergraduate optics lab to help students understand the concepts of retardance and its control and also how variable retardance devices work.

1.
E.
Hecht
and
A.
Zajac
,
Optics
(
Addison-Wesley
,
Reading, MA
,
1974
), Chap. 8.
2.
F. Y.
Yap
, “
Some more simple laser experiments for the undergraduate laboratory
,”
Am. J. Phys.
37
,
1204
1206
(
1969
).
3.
B. S.
Perkalskis
and
J. R.
Freeman
, “
Demonstrating Fresnel's model for optical activity
,”
Am. J. Phys.
66
,
452
453
(
1998
).
4.
P. R.
Camp
, “
Demonstrating optical birefringence—A modification of Schneider's experiment
,”
Am. J. Phys.
65
,
672
674
(
1997
).
5.
W. S.
Bickel
and
W. M.
Bailey
, “
Stokes vectors, Mueller matrices, and polarized scattered light
,”
Am. J. Phys.
53
,
468
478
(
1985
).
6.
D.
Goldstein
,
Polarized Light
, 2nd ed. (
Marcel Dekker, Inc.
,
New York
,
2003
), Chaps. 1, 27, and 28.
7.
O. G.
Rodríguez-Herrera
,
D.
Lara
, and
C.
Dainty
, “
Far-field polarization-based sensitivity to sub-resolution displacements of a sub-resolution scatterer in tightly focused fields
,”
Opt. Express
18
(
6
),
5609
5628
(
2010
).
8.
D.
Lara
and
C.
Dainty
, “
Axially resolved complete Mueller matrix confocal microscopy
,”
Appl. Opt.
45
(
9
),
1917
1930
(
2006
).
9.
E.
Collet
,
Polarized Light for Scientists and Engineers
, 1st ed. (
The PolaWave Group
,
New Jersey
,
2012
).
10.
G.
Martínez-Ponce
,
C.
Solano
, and
C.
Pérez-Barrios
, “
Hybrid complete Mueller polarimeter based on phase modulators
,”
Opt. Lasers Eng.
49
,
723
728
(
2011
).
11.
R. L.
Heredero
,
N.
Uribe-Patarroyo
,
T.
Belenguer
,
G.
Ramos
,
A.
Sánchez
,
M.
Reina
,
V. Martínez
Pillet
, and
A.
Álvarez-Herrero
, “
Liquid-crystal variable retarders for aerospace polarimetry applications
,”
Appl. Opt.
46
(
5
),
689
698
(
2007
).
12.
P.
Terrier
,
J. M.
Charbois
, and
V.
Devlaminck
, “
Fast-axis orientation dependence on driving voltage for a Stokes polarimeter based on concrete liquid-crystal variable retarders
,”
Appl. Opt.
49
(
22
),
4278
4283
(
2010
).
13.
J.
Delgado-Aguillón
,
J.
Garduño-Mejía
,
J. M.
López-Téllez
,
N. C.
Bruce
,
M.
Rosete-Aguilar
,
C. J.
Román-Moreno
, and
R.
Ortega-Martínez
, “
Direct inversion methods for spectral amplitude modulation of femtosecond pulses
,”
Rev. Sci. Instrum.
85
(
4
),
043105
(
2014
).
14.
M. A.
Schofield
and
Y.
Zhu
, “
Fast phase unwrapping algorithm for interferometric applications
,”
Opt Lett.
28
(
14
),
1194
1196
(
2003
).
15.
R. M.
Goldstein
,
H. A.
Zebker
, and
C. L.
Werner
, “
Satellite radar interferometry: Two-dimensional phase unwrapping
,”
Radio Sci.
23
,
713
720
, doi: (
1988
).
16.
S.
Chavez
,
Q. S.
Xiang
, and
A.
Li
, “
Understanding phase maps in MRI: A new cutline phase unwrapping method
,”
IEEE Trans. Med. Imag.
21
(
8
),
966
977
(
2002
).
17.
L.
Aiello
,
D.
Riccio
,
P.
Ferraro
,
S.
Grilli
,
L.
Sansone
,
G.
Coppola
,
S. D.
Nicola
, and
A.
Finizio
, “
Green's formulation for robust phase unwrapping in digital holography
,”
Opt. Lasers Eng.
45
,
750
755
(
2007
).
18.
K.
Itoh
, “
Analysis of the phase unwrapping algorithm
,”
Appl. Opt.
21
(
14
),
2470
(
1982
).
19.
S. A.
Karout
,
M. A.
Gdeisat
,
D. R.
Burton
, and
M. J.
Lalor
, “
Two-dimensional phase unwrapping using a hybrid genetic algorithm
,”
Appl. Opt.
46
(
5
),
730
743
(
2007
).
20.
M.
Arevalillo-Herráez
,
M. A.
Gdeisat
, and
D. R.
Burton
, “
Hybrid robust and fast algorithm for three-dimensional phase unwrapping
,”
Appl. Opt.
48
(
32
),
6313
6323
(
2009
).
21.
See <http://www.ljmu.ac.uk/GERI/90202.htm/> for more information about the phase unwrapping problem.
22.
D. C.
Ghiglia
and
M. D.
Pritt
,
Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software
(
John Wiley & Sons
,
New York
,
1998
).
23.
See <www.meadowlark.com/> for more information on the fabrication and operation of the Liquid-Crystal Variable Retarders.
24.
See <www.cri-inc.com/> for more information on the fabrication and operation of the Amplitude Spatial-Light-Modulator.
25.
See <www.thorlabs.com/> for more information on the fabrication and operation of the Soleil-Babinet Compensator.
26.
J. M.
López-Téllez
and
N. C.
Bruce
, “
Stokes polarimetry using analysis of the nonlinear voltage-retardance relationship for liquid-crystal variable retarders
,”
Rev. Sci. Instrum.
85
(
3
),
033104
(
2014
).
27.
J. M.
López-Téllez
and
N. C.
Bruce
, “
Mueller polarimeter using analysis of the nonlinear voltage-retardance relationship for liquid-crystal variable retarders
,”
Appl. Opt.
53
(
24
),
5359
5366
(
2014
).
28.
H. R.
Phillip
,
Handbook of optical constants of solids
, edited by
E. D.
Palik
(
Academic Press London
,
1985
), p.
719
.
29.
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