Abstract

Stokes polarimetry is a mature topic in optics, most commonly performed to extract the polarization structure of optical fields for a range of diverse applications. For historical reasons, most Stokes polarimetry approaches are based on static optical polarization components that must be manually adjusted, prohibiting automated, real-time analysis of fast changing fields. Here we provide a tutorial on performing Stokes polarimetry in an all-digital approach, exploiting a modern optical toolkit based on liquid-crystal-on-silicon spatial light modulators and digital micromirror devices. We explain in a tutorial fashion how to implement two digital approaches, based on these two devices, for extracting Stokes parameters in a fast, cheap, and dynamic manner. After outlining the core concepts, we demonstrate their applicability to the modern topic of structured light, and highlight some common experimental issues. In particular, we illustrate how digital Stokes polarimetry can be used to measure key optical parameters such as the state of polarization, degree of vectorness, and intra-modal phase of complex light fields.

© 2020 Optical Society of America

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2020 (3)

Q. Hu, Y. Dai, C. He, and M. J. Booth, “Arbitrary vectorial state conversion using liquid crystal spatial light modulators,” Opt. Commun. 459, 125028 (2020).
[Crossref]

A. Manthalkar, I. Nape, N. T. Bordbar, C. Rosales-Guzmán, S. Bhattacharya, A. Forbes, and A. Dudley, “All-digital Stokes polarimetry with a digital micromirror device,” Opt. Lett. 45, 2319–2322 (2020).
[Crossref]

B. Zhao, X.-B. Hu, V. Rodrguez-Fajardo, A. Forbes, W. Gao, Z.-H. Zhu, and C. Rosales-Guzmán, “Determining the non-separability of vector modes with digital micromirror devices,” Appl. Phys. Lett. 116, 091101 (2020).
[Crossref]

2019 (10)

A. Selyem, C. Rosales-Guzmán, S. Croke, A. Forbes, and S. Franke-Arnold, “Basis-independent tomography and nonseparability witnesses of pure complex vectorial light fields by Stokes projections,” Phys. Rev. A 100, 063842 (2019).
[Crossref]

E. Toninelli, B. Ndagano, A. Vallés, B. Sephton, I. Nape, A. Ambrosio, F. Capasso, M. J. Padgett, and A. Forbes, “Concepts in quantum state tomography and classical implementation with intense light: a tutorial,” Adv. Opt. Photon. 11, 67–134 (2019).
[Crossref]

X.-B. Hu, B. Zhao, Z.-H. Zhu, W. Gao, and C. Rosales-Guzmán, “In situ detection of a cooperative target’s longitudinal and angular speed using structured light,” Opt. Lett. 44, 3070–3073 (2019).
[Crossref]

B. Zhao, X.-B. Hu, V. Rodrguez-Fajardo, Z.-H. Zhu, W. Gao, A. Forbes, and C. Rosales-Guzmán, “Real-time Stokes polarimetry using a digital micromirror device,” Opt. Express 27, 31087–31093 (2019).
[Crossref]

M. A. Cox, E. Toninelli, L. Cheng, M. J. Padgett, and A. Forbes, “A high-speed, wavelength invariant, single-pixel wavefront sensor with a digital micromirror device,” IEEE Access 7, 85860–85866 (2019).
[Crossref]

A. Forbes and I. Nape, “Quantum mechanics with patterns of light: progress in high dimensional and multidimensional entanglement with structured light,” AVS Quantum Sci. 1, 011701 (2019).
[Crossref]

S. Shibata, N. Hagen, S. Kawabata, and Y. Otani, “Compact and high-speed Stokes polarimeter using three-way polarization-preserving beam splitters,” Appl. Opt. 58, 5644–5649 (2019).
[Crossref]

G. Lazarev, P.-J. Chen, J. Strauss, N. Fontaine, and A. Forbes, “Beyond the display: phase-only liquid crystal on silicon devices and their applications in photonics,” Opt. Express 27, 16206–16249 (2019).
[Crossref]

S. Scholes, R. Kara, J. Pinnell, V. Rodríguez-Fajardo, and A. Forbes, “Structured light with digital micromirror devices: a guide to best practice,” Opt. Eng. 59, 1–12 (2019).
[Crossref]

Y. Shen, X. Wang, Z. Xie, C. Min, X. Fu, Q. Liu, M. Gong, and X. Yuan, “Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities,” Light Sci. Appl. 8, 90 (2019).
[Crossref]

2018 (9)

A. Peña and M. F. Andersen, “Complete polarization and phase control with a single spatial light modulator for the generation of complex light fields,” Laser Phys. 28, 076201 (2018).
[Crossref]

E. Arbabi, S. M. Kamali, A. Arbabi, and A. Faraon, “Full-Stokes imaging polarimetry using dielectric metasurfaces,” ACS Photon. 5, 3132–3140 (2018).
[Crossref]

J. Chen, C. Wan, and Q. Zhan, “Vectorial optical fields: recent advances and future prospects,” Sci. Bull. 63(1), 54–74 (2018).
[Crossref]

C. Rosales-Guzmán, B. Ndagano, and A. Forbes, “A review of complex vector light fields and their applications,” J. Opt. 20, 123001 (2018).
[Crossref]

E. Otte, C. Alpmann, and C. Denz, “Polarization singularity explosions in tailored light fields,” Laser Photon. Rev. 12, 1700200 (2018).
[Crossref]

E. Otte and C. Denz, “Sculpting complex polarization singularity networks,” Opt. Lett. 43, 5821–5824 (2018).
[Crossref]

S. Shin, K. Lee, Z. Yaqoob, P. T. So, and Y. Park, “Reference-free polarization-sensitive quantitative phase imaging using single-point optical phase conjugation,” Opt. Express 26, 26858–26865 (2018).
[Crossref]

S. Liu, S. Qi, Y. Zhang, P. Li, D. Wu, L. Han, and J. Zhao, “Highly efficient generation of arbitrary vector beams with tunable polarization, phase, and amplitude,” Photon. Res. 6, 228–233 (2018).
[Crossref]

H. Larocque, D. Sugic, D. Mortimer, A. J. Taylor, R. Fickler, R. W. Boyd, M. R. Dennis, and E. Karimi, “Reconstructing the topology of optical polarization knots,” Nat. Phys. 14, 1079–1082 (2018).
[Crossref]

2017 (4)

B. Perez-Garcia, C. López-Mariscal, R. I. Hernandez-Aranda, and J. C. Gutiérrez-Vega, “On-demand tailored vector beams,” Appl. Opt. 56, 6967–6972 (2017).
[Crossref]

E. J. Galvez, I. Dutta, K. Beach, J. J. Zeosky, J. A. Jones, and B. Khajavi, “Multitwist Möbius strips and twisted ribbons in the polarization of paraxial light beams,” Sci. Rep. 7, 13653 (2017).
[Crossref]

S. Turtaev, I. T. Leite, K. J. Mitchell, M. J. Padgett, D. B. Phillips, and T. Čižmár, “Comparison of nematic liquid-crystal and DMD based spatial light modulation in complex photonics,” Opt. Express 25, 29874–29884 (2017).
[Crossref]

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).
[Crossref]

2016 (6)

2015 (3)

T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Boyd, and G. Leuchs, “Observation of optical polarization Möbius strips,” Science 347, 964–966 (2015).
[Crossref]

M. McLaren, T. Konrad, and A. Forbes, “Measuring the nonseparability of vector vortex beams,” Phys. Rev. A 92, 023833 (2015).
[Crossref]

Y.-X. Ren, R.-D. Lu, and L. Gong, “Tailoring light with a digital micromirror device,” Ann. Phys. 527, 447–470 (2015).
[Crossref]

2014 (1)

2013 (2)

2012 (4)

2010 (2)

2009 (2)

R. Komanduri and M. Escuti, “High efficiency reflective liquid crystal polarization gratings,” Appl. Phys. Lett. 95, 091106 (2009).
[Crossref]

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1, 1–57 (2009).
[Crossref]

2007 (1)

B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, “Measuring the Stokes polarization parameters,” Am. J. Phys. 75, 163–168 (2007).
[Crossref]

2006 (1)

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
[Crossref]

2004 (1)

E. Garcia-Caurel, A. De Martino, and B. Drevillon, “Spectroscopic Mueller polarimeter based on liquid crystal devices,” Thin Solid Films 455, 120–123 (2004).
[Crossref]

2002 (1)

A. Vaziri, G. Weihs, and A. Zeilinger, “Superpositions of the orbital angular momentum for applications in quantum experiments,” J. Opt. B 4, S47 (2002).
[Crossref]

2001 (2)

W. K. Wootters, “Entanglement of formation and concurrence,” Quantum Inf. Comput. 1, 27–44 (2001).

T. M. Kreis, P. Aswendt, and R. Höfling, “Hologram reconstruction using a digital micromirror device,” Opt. Eng. 40, 926–933 (2001).
[Crossref]

2000 (1)

J. M. Bueno, “Polarimetry using liquid-crystal variable retarders: theory and calibration,” J. Opt. A 2, 216 (2000).
[Crossref]

1994 (1)

J. B. Sampsell, “Digital micromirror device and its application to projection displays,” J. Vac. Sci. Technol. B 12, 3242–3246 (1994).
[Crossref]

1985 (1)

1982 (1)

R. Azzam, “Division-of-amplitude photopolarimeter (DOAP) for the simultaneous measurement of all four stokes parameters of light,” Opt. Acta 29, 685–689 (1982).
[Crossref]

1980 (2)

E. Collett, “Determination of the ellipsometric characteristics of optical surfaces using nanosecond laser pulses,” Surf. Sci. 96, 156–167 (1980).
[Crossref]

P. Hauge, “Recent developments in instrumentation in ellipsometry,” Surf. Sci. 96, 108–140 (1980).
[Crossref]

1977 (1)

1974 (2)

H. Berry, L. Curtis, D. Ellis, and R. Schectman, “Anisotropy in the beam-foil light source,” Phys. Rev. Lett. 32, 751 (1974).
[Crossref]

W.-H. Lee, “Binary synthetic holograms,” Appl. Opt. 13, 1677–1682 (1974).
[Crossref]

1971 (1)

D. Clarke and J. Grainger, “Polarized light and optical measurement,” Am. J. Phys. 40, 1055–1056 (1971).
[Crossref]

1968 (1)

E. Collett, “The description of polarization in classical physics,” Am. J. Phys. 36, 713–725 (1968).
[Crossref]

1954 (2)

E. Wolf, “Optics in terms of observable quantities,” Nuovo. Cim. 12, 884–888 (1954).
[Crossref]

W. H. McMaster, “Polarization and the Stokes parameters,” Am. J. Phys. 22, 351–362 (1954).
[Crossref]

1851 (1)

G. G. Stokes, “On the composition and resolution of streams of polarized light from different sources,” Trans. Cambridge Philos. Soc. 9, 399 (1851).

Alfano, R. R.

Alpmann, C.

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Figures (13)

Fig. 1.
Fig. 1. Conceptual schematics of (a) the conventional Stokes polarimetry measurements, together with the digital adaptations, implementing (b) a LCoS-SLM and (c) a DMD. QWP, quarter-wave plate; LP, linear polarizer; SLM, liquid-crystal-on-silicon (LCoS) spatial light modulator; PG, polarization grating; DMD, digital micromirror device; BS, 50:50 beam splitter.
Fig. 2.
Fig. 2. Experimental diagrams illustrating the arrangement of the optical components used to acquire Stokes parameters for (a) conventional Stokes polarimetry and the digital approaches, implementing (b) a LCoS-SLM and (c) a DMD. The representative holograms encoded onto the LCoS-SLM and DMD screens are provided as insets. HeNe, helium neon laser; QP, ${q}$-plate; M, mirror.
Fig. 3.
Fig. 3. Conventional Stokes polarimetry. Left: the generated field of interest, on which the Stokes polarimetry is performed [in this case described by Eq. (32)]. Middle: measured intensity profiles of the Stokes projections. Right: the corresponding calculated Stokes parameters (min = 0 and max = 1 for ${S_0}$, ${S_3}$, while min = −1 and max = 1 for ${S_1}$, ${S_2}$). Theoretical simulations are given as insets.
Fig. 4.
Fig. 4. Digital LCoS-SLM Stokes polarimetry. Left: the generated field of interest, on which the Stokes polarimetry is performed [in this case described by a superposition of the two equations in Eq. (34)]. Middle: measured intensity profiles of the Stokes projections. Right: the corresponding calculated Stokes parameters (min = 0 and max = 1 for ${S_0}$, ${S_1}$, while min = −1 and max = 1 for ${S_2}$, ${S_3}$). Theoretical simulations are given as insets.
Fig. 5.
Fig. 5. Diagram depicting the alignment of orthogonally polarized components through a 50:50 BS. As the misalignment in the beam paths improves (left to right), so the number of fringes decreases.
Fig. 6.
Fig. 6. Diagram showing the independent phase modulation of orthogonal polarization components (top two rows for DMD transmission functions, ${T_A}$ and ${T_B}$) and simulated interference of the components through a 50:50 BS (bottom row).
Fig. 7.
Fig. 7. Digital DMD Stokes polarimetry. Left: the generated field of interest, on which the Stokes polarimetry is performed [in this case described by Eq. (34)]. Middle: measured intensity profiles of the Stokes projections. Right: the corresponding calculated Stokes parameters (min = 0 and max = 1 for ${S_0}$, ${S_3}$, while min = −1 and max = 1 for ${S_1}$, ${S_2}$). Theoretical simulations are given as insets.
Fig. 8.
Fig. 8. Digital DMD Stokes polarimetry with curvature. Left: the generated field of interest, on which the Stokes polarimetry is performed [in this case described by Eq. (32)]. Middle: measured intensity profiles of the Stokes projections. Right: the corresponding calculated Stokes parameters (min = 0 and max = 1 for ${S_0}$, ${S_3}$, while min = −1 and max = 1 for ${S_1}$, ${S_2}$). Theoretical simulations are given as insets. Note the spiral lobes in ${I_D},{I_H},{S_1}$, and ${S_2}$.
Fig. 9.
Fig. 9. Diagram showing the relationship between the Cartesian co-ordinates of the Poincaré sphere and the ellipticity and orientation angles of the polarization ellipse.
Fig. 10.
Fig. 10. Reconstructed polarization structures (top row) for the conventional and digital DMD techniques, together with their reconstructed intra-modal phases (bottom row); the DMD results labeled as “imaged” were acquired by implementing the ${4f}$ imaging system discussed in Section 4.C. The ellipses are colored according to the intensity value of the optical field.
Fig. 11.
Fig. 11. Diagram showing the curvature developed in the SoP and intra-modal phase, $\delta$, of a TE vector vortex beam due to propagation over a distance $z = [0,(1.8 \times {10^{- 3}}){z_R}]$.
Fig. 12.
Fig. 12. Experimentally extracted intermodal phase profiles of a propagating vector vortex beam ($\ell = 1$), described by Eq. (34), illustrating the spiraling nature of the phase distribution.
Fig. 13.
Fig. 13. Plot of the measured vectorness (black data points) versus the QWP angle [positioned before the $q$-plate in Figs. 2(a) and 2(c)]. The blue, dashed curve denotes the theoretical prediction.

Tables (1)

Tables Icon

Table 1. Jones Matrix Parameters Imparted by the DMD to Acquire Stokes Intensity Measurements

Equations (51)

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E x ( t ) = E 0 x cos ( ω t + δ x ) ,
E y ( t ) = E 0 y cos ( ω t + δ y ) ,
E x 2 ( t ) E 0 x 2 + E y 2 ( t ) E 0 y 2 2 E x ( t ) E y ( t ) E 0 x E 0 y cos ( δ ) = sin 2 ( δ ) ,
E 0 x 2 cos 2 ( ω t δ x ) E 0 x 2 + E 0 y 2 cos 2 ( ω t δ y ) E 0 y 2 2 E 0 x E 0 y cos ( ω t δ x ) cos ( ω t δ y ) E 0 x E 0 y cos ( δ ) = sin 2 ( δ ) ,
E i ( t ) E j ( t ) = lim T 1 T 0 T E i ( t ) E j ( t ) d t   i , j = x , y .
cos 2 ( ω t δ x ) = cos 2 ( ω t δ y ) = 1 2 ,
cos ( ω t δ x ) cos ( ω t δ y ) = 1 2 cos ( δ ) ,
( E 0 x 2 + E 0 y 2 ) 2 ( E 0 x 2 E 0 y 2 ) 2 ( 2 E 0 x E 0 y cos ( δ ) ) 2 = ( 2 E 0 x E 0 y sin ( δ ) ) 2 .
S 0 = E 0 x 2 + E 0 y 2 , S 1 = E 0 x 2 E 0 y 2 , S 2 = 2 E 0 x E 0 y cos ( δ ) , S 3 = 2 E 0 x E 0 y sin ( δ ) .
S 0 2 = S 1 2 + S 2 2 + S 3 2 .
S 0 2 S 1 2 + S 2 2 + S 3 2 .
E x ( t ) = E 0 x e i ( ω t + δ x ) = E x e i ω t ,
E y ( t ) = E 0 y e i ( ω t + δ y ) = E y e i ω t .
S 0 = E x E x + E y E y , S 1 = E x E x E y E y , S 2 = E x E y + E y E x , S 3 = i ( E x E y E y E x ) .
S 0 = I H + I V , S 1 = I H I V , S 2 = I D I A , S 3 = I R I L ,
E D , A ( t ) = 1 2 ( E H ( t ) ± E V ( t ) ) , E L , R ( t ) = 1 2 ( E H ( t ) ± i E V ( t ) ) .
E x cos ( θ ) + E y sin ( θ ) ,
I ( θ ) = E x E x cos 2 ( θ ) + E y E y sin 2 ( θ ) + E x E y cos ( θ ) sin ( θ ) + E x E y cos ( θ ) sin ( θ ) .
I ( 0 ) = I H , I ( 90 ) = I V , I ( 45 ) = I D , I ( 135 ) = I A .
E x e i ϕ 2 cos ( θ ) + E y e i ϕ 2 sin ( θ ) ,
I ( θ , ϕ ) = E x E x cos 2 ( θ ) + E y E y sin 2 ( θ ) + E x E y e i ϕ cos ( θ ) sin ( θ ) + E x E y e i ϕ cos ( θ ) sin ( θ ) .
I ( 45 , 90 ) = I R , I ( 135 , 90 ) = I L .
cos 2 ( θ ) = 1 + cos ( 2 θ ) 2 , sin 2 ( θ ) = 1 cos ( 2 θ ) 2 , cos ( θ ) sin ( θ ) = sin ( 2 θ ) 2 .
I ( θ , ϕ ) = 1 2 [ S 0 + S 1 cos ( 2 θ ) + S 2 cos ( ϕ ) sin ( 2 θ ) + S 3 sin ( ϕ ) sin ( 2 θ ) ] .
S 0 = I R + I L , S 1 = 2 I H S 0 , S 2 = 2 I D S 0 , S 3 = I R I L ,
U ( r ) = U A ( r ) | R + U B ( r ) | L PG | U A ( r ) | e i δ A | R & | U B ( r ) | e i δ B | L ,
H A ( r ) = 1 2 + 1 2 s i g n ( cos ( 2 π ( G x x + G y y ) + π Φ A ( r ) ) cos ( π A A ( r ) ) ) ,
H B ( r ) = 1 2 + 1 2 s i g n ( cos ( 2 π ( G x x + G y y ) + π Φ B ( r ) ) cos ( π A B ( r ) ) ) .
T A = e i c A a n d T B = e i c B ,
U ( r ) = U A ( r ) + U B ( r ) = U A ( r ) e i φ A T A + U B ( r ) e i φ B T B ,
1 2 ( 1 1 1 1 ) B S M a t r i x ( A e i c A 0 0 B e i c B ) D M D M a t r i x 1 2 ( U A U B ) I n p u t F i e l d = 1 2 ( A e i c A B e i c B A e i c A B e i c B ) J o n e s M a t r i x o f S y s t e m ( U A U B ) = ( 1 2 ( A e i c A U A + B e i c B U B ) 1 2 ( A e i c A U A B e i c B U B ) ) B S O u t p u t s .
T E ( r ) = L G 0 1 ( r ) | R + L G 0 1 ( r ) | L .
L G p l ( r , ϕ , z ) = w 0 w ( z ) 2 p ! π ( | l | + p ) ! ( 2 r w ( z ) ) | l | L p l ( 2 r 2 w ( z ) 2 ) × e i ( 2 p + | l | + 1 ) ξ ( z ) e r 2 w ( z ) 2 e i k r 2 2 R ( z ) e i l ϕ ,
U ( r , ϕ ) = exp ( r 2 / ω 0 ) a n d U ( r , ϕ ) = exp ( r 2 / ω 0 ) exp ( i δ ( r , ϕ ) ) .
S 0 = S 1 2 + S 2 2 + S 3 2 ,
S 1 = S 0 cos 2 χ cos 2 ψ ,
S 2 = S 0 cos 2 χ sin 2 ψ ,
S 3 = S 0 sin 2 χ .
S 1 S 0 cos 2 χ cos 2 ψ = S 2 S 0 cos 2 χ sin 2 ψ ψ = 1 2 tan 1 ( S 2 S 1 ) .
S 1 2 + S 2 2 = S 0 2 cos 2 2 χ ( cos 2 2 ψ + sin 2 2 ψ ) χ = 1 2 tan 1 ( S 3 S 1 2 + S 2 2 ) .
H C P = { R f o r S 3 > 0 , L f o r S 3 < 0 .
δ ( r , ϕ ) = a r c t a n ( S 3 ( r , ϕ ) S 2 ( r , ϕ ) ) .
| Ψ = a | u ¯ R | R + 1 a | u ¯ L | L .
ρ = ( a | u ¯ R u ¯ R | a ( 1 a ) | u ¯ R u ¯ L | a ( 1 a ) | u ¯ L u ¯ R | ( 1 a ) | u ¯ L u ¯ L | ) .
C = m a x { 0 , λ 1 , λ 2 , λ 3 , λ 4 } ,
ρ ( σ 3 σ 3 ) ρ ( σ 3 σ 3 ) ,
Ψ = b | u + | R + c | u | R + d | u + | L + f | u | L ,
C ( Ψ ) = 2 | b f c d | .
| u ¯ i = d x d y | u ¯ i ( r ) | e i ϕ i ( r ) | x , y ,
C ( | Ψ ) = 2 u ¯ R | u ¯ R u ¯ L | u ¯ L | u ¯ R | u ¯ L | 2 .
C = 1 S 1 2 S 0 2 S 2 2 S 0 2 S 3 2 S 0 2 = 1 i = 1 3 S i 2 S 0 2 .

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