A simple scheme to synthesize non-uniform patterns of polarization across the transverse section of a beam is proposed with the standard materials in an undergraduate optics laboratory. The experiment is based on the superposition of two orthogonally polarized fields obtained by using a Fresnel biprism and dichroic polarizers. Although no interference pattern appears in the superposition area, a non-uniformly totally polarized field is synthesized. Analytic expressions for the Jones vector and Stokes parameters of the output beam are calculated and, in the process, students can cement their knowledge about the representation of polarized light with these formalisms. The experimental polarization pattern is either obtained from intensity measurements with a CCD camera or measured directly with a commercial polarimeter modified with a pinhole. This experiment will help students discover an easy way to vary the state of polarization across the transverse section of a light field.

1.
M.
Born
and
E.
Wolf
,
Principles of Optics
, 6th (corrected) ed. (
Cambridge U.P.
,
Cambridge
,
1980
).
2.
F.
Gori
, “
Polarization basis for vortex beams
,”
J. Opt. Soc. Am. A
18
,
1612
1617
(
2001
).
3.
G.
Piquero
and
J.
Vargas-Balbuena
, “
Non-uniformly polarized beams across their transverse profiles: An introductory study for undergraduate optics courses
,”
Eur. J. Phys.
25
,
793
800
(
2004
).
4.
V.
Ramírez-Sánchez
,
G.
Piquero
, and
M.
Santarsiero
, “
Generation and characterization of spirally polarized fields
,”
J. Opt. A: Pure Appl. Opt.
11
,
085708
(
2009
).
5.
Q.
Zhan
, “
Cylindrical vector beams: from mathematical concepts to applications
,”
Adv. Opt. Photon.
1
,
1
57
(
2009
).
6.
T. G.
Brown
and
Q.
Zhan
, “
Focus issue: Unconventional polarization states of light
,”
Opt. Express
18
,
10775
10776
(
2010
).
7.
A. M.
Beckley
,
T. G.
Brown
, and
M. A.
Alonso
, “
Full Poincaré beams
,”
Opt. Express
18
,
10777
10785
(
2010
).
8.
E. J.
Galvez
,
S.
Khadka
,
W. H.
Schubert
, and
S.
Nomoto
, “
Poincaré-beam patterns produced by nonseparable superpositions of Laguerre-Gauss and polarization modes of light
,”
Appl. Opt.
51
,
2925
2934
(
2012
).
9.
J. A.
Jones
,
A. J.
D'Addario
,
B. L.
Rojec
,
G.
Milione
, and
E. J.
Galvez
, “
The Poincaré-sphere approach to polarization: Formalism and new labs with Poincaré beams
,”
Am. J. Phys.
84
,
822
835
(
2016
).
10.
B.
Pérez-García
,
C.
López-Mariscal
,
R. I.
Hernández-Aranda
, and
J. C.
Gutiérrez-Vega
, “
On-demand tailored vector beams
,”
Appl. Opt.
56
,
6967
6972
(
2017
).
11.
H.
Rubinsztein-Dunlop
,
A.
Forbes
,
M. V.
Berry
,
M. R.
Dennis
,
D. L.
Andrews
,
M.
Mansuripur
,
C.
Denz
,
C.
Alpmann
,
P.
Banzer
,
T.
Bauer
,
E.
Karimi
,
L.
Marrucci
,
M.
Padgett
,
M.
Ritsch-Marte
,
N. M.
Litchinitser
,
N. P.
Bigelow
,
C.
Rosales-Guzmán
,
A.
Belmonte
,
J. P.
Torres
,
T. W.
Neely
,
M.
Baker
,
R.
Gordon
,
A. B.
Stilgoe
,
J.
Romero
,
A. G.
White
,
R.
Fickler
,
A. E.
Willner
,
G.
Xie
,
B.
McMorran
, and
A. M.
Weiner
, “
Roadmap on structured light
,”
J. Opt.
19
,
013001
(
2017
).
12.
S. K.
Mohanty
,
K. D.
Rao
, and
P. K.
Gupta
, “
Optical trap with spatially varying polarization: Application in controlled orientation of birefringent microscopic particle(s)
,”
Appl. Phys. B
80
,
631
634
(
2005
).
13.
W.
Cui
,
F.
Song
,
F.
Song
,
D.
Ju
, and
S.
Liu
, “
Trapping metallic particles under resonant wavelength with 4π tight focusing of radially polarized beam
,”
Opt. Express
24
,
20062
20068
(
2016
).
14.
F.
Gori
, “
Measuring Stokes parameters by means of a polarization grating
,”
Opt. Lett.
24
,
584
586
(
1999
).
15.
O.
Arteaga
,
R.
Ossikovski
,
E.
Kuntman
,
M. A.
Kuntman
,
A.
Canillas
, and
E.
Garcia-Caurel
, “
Mueller matrix polarimetry on a Young's double-slit experiment analog
,”
Opt. Lett.
42
,
3900
3903
(
2017
).
16.
J. C. G.
de Sande
,
M.
Santarsiero
, and
G.
Piquero
, “
Spirally polarized beams for polarimetry measurements of deterministic and homogeneous samples
,”
Opt. Lasers Eng.
91
,
97
105
(
2017
).
17.
S.
Nolte
,
C.
Momma
,
G.
Kamlage
,
A.
Ostendorf
,
C.
Fallnich
,
F.
von Alvensleben
, and
H.
Welling
, “
Polarization effects in ultrashort-pulse laser drilling
,”
Appl. Phys. A
68
,
563
567
(
1999
).
18.
M.
Meier
,
V.
Romano
, and
T.
Feurer
, “
Material processing with pulsed radially and azimuthally polarized laser radiation
,”
Appl. Phys. A
86
,
329
334
(
2007
).
19.
S.
Matsusaka
,
Y.
Kozawa
, and
S.
Sato
, “
Micro-hole drilling by tightly focused vector beams
,”
Opt. Lett.
43
,
1542
1545
(
2018
).
20.
J.
Serna
and
G.
Piquero
, “
Beam moments and angular momentum in non-uniformly polarized beams
,”
Opt. Commun.
282
,
1973
1975
(
2009
).
21.
E.
Karimi
,
S.
Slussarenko
,
B.
Piccirillo
,
L.
Marrucci
, and
E.
Santamato
, “
Polarization-controlled evolution of light transverse modes and associated Pancharatnam geometric phase in orbital angular momentum
,”
Phys. Rev. A
81
,
053813
(
2010
).
22.
M.
Alonso
,
G.
Piquero
, and
J.
Serna
, “
Proposals for the generation of angular momentum from non-uniformly polarized beams
,”
Opt. Commun.
285
,
1631
1635
(
2012
).
23.
D.
Maluenda
,
A.
Carnicer
,
R.
Martínez-Herrero
,
I.
Juvells
, and
B.
Javidi
, “
Optical encryption using photon-counting polarimetric imaging
,”
Opt. Express
23
,
655
666
(
2015
).
24.
N. A.
Rubin
,
A.
Zaidi
,
M.
Juhl
,
R. P.
Li
,
J. B.
Mueller
,
R. C.
Devlin
,
K.
Leósson
, and
F.
Capasso
, “
Polarization state generation and measurement with a single metasurface
,”
Opt. Express
26
,
21455
21478
(
2018
).
25.
G.
Piquero
,
L.
Monroy
,
M.
Santarsiero
,
M.
Alonzo
, and
J. C. G.
de Sande
, “
Synthesis of full Poincaré beams by means of uniaxial crystals
,”
J. Opt.
20
,
065602
(
2018
).
26.
F.
Gori
,
M.
Santarsiero
, and
R.
Borghi
, “
Vector mode analysis of a Young interferometer
,”
Opt. Lett.
31
,
858
860
(
2006
).
27.
T.
Setälä
,
J.
Tervo
, and
A. T.
Friberg
, “
Stokes parameters and polarization contrasts in Young's interference experiment
,”
Opt. Lett.
31
,
2208
2210
(
2006
).
28.
A.
Luis
, “
Ray picture of polarization and coherence in a Young interferometer
,”
J. Opt. Soc. Am. A
23
,
2855
2860
(
2006
).
29.
M.
Santarsiero
, “
Polarization invariance in a Young interferometer
,”
J. Opt. Soc. Am. A
24
,
3493
3499
(
2007
).
30.
J.
Tervo
,
P.
Réfrégier
, and
A.
Roueff
, “
Minimum number of modulated Stokes parameters in Young's interference experiment
,”
J. Opt. A: Pure Appl. Opt.
10
,
055002
(
2008
).
31.
R.
Martínez-Herrero
and
P. M.
Mejías
, “
Maximizing Young's fringe visibility under unitary transformations for mean-square coherent light
,”
Opt. Express
17
,
603
610
(
2009
).
32.
Y.
Li
,
X.-L.
Wang
,
H.
Zhao
,
L.-J.
Kong
,
K.
Lou
,
B.
Gu
,
C.
Tu
, and
H.-T.
Wang
, “
Young's two-slit interference of vector light fields
,”
Opt. Lett.
37
,
1790
1792
(
2012
).
33.
M.
Alonzo
,
M.
Santarsiero
, and
F.
Gori
, “
Maximizing Young fringe visibility with a universal SU2 polarization gadget
,”
Opt. Lett.
43
,
2844
2847
(
2018
).
34.
H.
Partanen
,
B. J.
Hoenders
,
A. T.
Friberg
, and
T.
Setälä
, “
Young's interference experiment with electromagnetic narrowband light
,”
J. Opt. Soc. Am. A
35
,
1379
1384
(
2018
).
35.
W. R.
Mellen
, “
Interference of linearly polarized light with perpendicular polarizations
,”
Am. J. Phys.
30
,
772
772
(
1962
).
36.
R.
Hanau
, “
Interference of linearly polarized light with perpendicular polarizations
,”
Am. J. Phys.
31
,
303
304
(
1963
).
37.
J. L.
Hunt
and
G.
Karl
, “
Interference with polarized light beams
,”
Am. J. Phys.
38
,
1249
1250
(
1970
).
38.
D.
Pescetti
, “
Interference between elliptically polarized light beams
,”
Am. J. Phys.
40
,
735
740
(
1972
).
39.
M.
Henry
, “
Fresnel-arago laws for interference in polarized light: A demonstration experiment
,”
Am. J. Phys.
49
,
690
691
(
1981
).
40.
P.
Andrés
,
A.
Pons
, and
J.
Ojeda-Castañeda
, “
Young's experiment with polarized light: Properties and applications
,”
Am. J. Phys.
53
,
1085
1088
(
1985
).
41.
S.
Mallick
, “
Interference with polarized light
,”
Am. J. Phys.
41
,
583
584
(
1973
).
42.
J. L.
Ferguson
, “
A simple, bright demonstration of the interference of polarized light
,”
Am. J. Phys.
52
,
1141
1142
(
1984
).
43.
E. F.
Carr
and
J. P.
McClymer
, “
A laboratory experiment on interference of polarized light using a liquid crystal
,”
Am. J. Phys.
59
,
366
367
(
1991
).
44.
B. M.
Rodríguez-Lara
and
I.
Ricardez-Vargas
, “
Interference with polarized light beams: Generation of spatially varying polarization
,”
Am. J. Phys.
77
,
1135
1143
(
2009
).
45.
D.
Gossman
,
B.
Pérez-García
,
R. I.
Hernández-Aranda
, and
A.
Forbes
, “
Optical interference with digital holograms
,”
Am. J. Phys.
84
,
508
516
(
2016
).
46.
G. R.
Fowles
,
Introduction to Modern Optics
, 2nd ed. (
Holt, Rinehart, and Winston
,
New York
,
1975
).
47.
Y.
Gorodetski
,
G.
Biener
,
A.
Niv
,
V.
Kleiner
, and
E.
Hasman
, “
Space-variant polarization manipulation for far-field polarimetry by use of subwavelength dielectric gratings
,”
Opt. Lett.
30
,
2245
2247
(
2005
).
48.
J. C. G.
de Sande
,
M.
Santarsiero
,
G.
Piquero
, and
F.
Gori
, “
Longitudinal polarization periodicity of unpolarized light passing through a double wedge depolarizer
,”
Opt. Express
20
,
27348
27360
(
2012
).
49.
M.
Santarsiero
,
J. C. G.
de Sande
,
G.
Piquero
, and
F.
Gori
, “
Coherence-polarization properties of fields radiated from transversely periodic electromagnetic sources
,”
J. Opt.
15
,
055701
(
2013
).
50.
G.
Piquero
,
J. M.
Movilla
,
P. M.
Mejías
, and
R.
Martínez-herrero
, “
Degree of polarization of non-uniformly partially polarized beams: a proposal
,”
Opt. Quantum Electron.
31
,
223
226
(
1999
).
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