Abstract

We present a generalization of the known spirally polarized beams (SPBs) which we will call generalized spirally polarized beams (GSPBs). We characterize in detail both theoretically and experimentally the streamline morphologies of the GSPBs and their transformation by arbitrary polarization optical systems described by complex Jones matrices. We find that the description of the passage of GSPBs through a polarization system is equivalent to the stability theory of autonomous systems of ordinary differential equations. While the streamlines of the GSPB exhibit a spiral geometry, the streamlines of the output field may exhibit spirals, saddles, nodes, ellipses, and stars as well. Using a novel experimental technique based on a Sagnac interferometer, we have been able to generate in the laboratory each one of the different cases of GSPBs and record their corresponding characteristic streamline morphologies.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2019 (2)

N. A. Ochoa, “A unifying approach for the vectorial Rayleigh-Sommerfeld diffraction integrals,” Opt. Commun. 448, 104–110 (2019).
[Crossref]

S. K. Pal and P. Senthilkumaran, “Synthesis of Stokes vortices,” Opt. Lett. 44(1), 130–133 (2019).
[Crossref]

2018 (2)

2017 (4)

2016 (1)

2012 (3)

2011 (2)

2010 (1)

2009 (1)

V. Ramírez-Sánchez, G. Piquero, and M. Santarsiero, “Generation and characterization of spirally polarized fields,” J. Opt. A: Pure Appl. Opt. 11(8), 085708 (2009).
[Crossref]

2008 (2)

B. Hao and J. Leger, “Numerical aperture invariant focus shaping using spirally polarized beams,” Opt. Commun. 281(8), 1924–1928 (2008).
[Crossref]

V. Ramírez-Sánchez and G. Piquero, “The beam quality parameter of spirally polarized beams,” J. Opt. A: Pure Appl. Opt. 10(12), 125004 (2008).
[Crossref]

2007 (1)

2006 (2)

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006).
[Crossref]

F. Flossmann, U. T. Schwarz, M. Maier, and M. R. Dennis, “Stokes parameters in the unfolding of an optical vortex through a birefringent crystal,” Opt. Express 14(23), 11402–11411 (2006).
[Crossref]

2005 (1)

2004 (1)

2003 (1)

2002 (4)

2001 (2)

I. Freund, “Polarization flowers,” Opt. Commun. 199(1-4), 47–63 (2001).
[Crossref]

F. Gori, “Polarization basis for vortex beams,” J. Opt. Soc. Am. A 18(7), 1612–1617 (2001).
[Crossref]

1982 (1)

1963 (1)

Ala-Nissila, T.

Alonso, M. A.

Biener, G.

Bomzon, Z.

Borghi, R.

Boyce, W. E.

W. E. Boyce, R. C. DiPrima, and D. B. Meade, Elementary differential equations and boundary value problems, vol. 9 (Wiley, 1992).

Brasselet, E.

Brown, T. G.

Campos-Martínez, J.

de Sande, J. C. G.

J. C. G. de Sande, G. Piquero, and M. Santarsiero, “Polarimetry with azimuthally polarized light,” Opt. Commun. 410, 961–965 (2018).
[Crossref]

J. C. G. de Sande, M. Santarsiero, and G. Piquero, “Spirally polarized beams for polarimetry measurements of deterministic and homogeneous samples,” Opt. Laser Eng. 91, 97–105 (2017).
[Crossref]

J. C. G. de Sande, G. Piquero, and C. Teijeiro, “Polarization changes at Lyot depolarizer output for different types of input beams,” J. Opt. Soc. Am. A 29(3), 278–284 (2012).
[Crossref]

Dennis, M. R.

Desyatnikov, A. S.

DiPrima, R. C.

W. E. Boyce, R. C. DiPrima, and D. B. Meade, Elementary differential equations and boundary value problems, vol. 9 (Wiley, 1992).

Dyson, J.

Fadeyeva, T. A.

Flossmann, F.

Freund, I.

I. Freund, M. S. Soskin, and A. I. Mokhun, “Elliptic critical points in paraxial optical fields,” Opt. Commun. 208(4-6), 223–253 (2002).
[Crossref]

I. Freund, “Polarization flowers,” Opt. Commun. 199(1-4), 47–63 (2001).
[Crossref]

Gori, F.

Gutiérrez-Vega, J. C.

Hao, B.

B. Hao and J. Leger, “Numerical aperture invariant focus shaping using spirally polarized beams,” Opt. Commun. 281(8), 1924–1928 (2008).
[Crossref]

Hasman, E.

Hernandez-Aranda, R. I.

Ina, H.

Izdebskaya, Y. V.

Kivshar, Y. S.

Kleiner, V.

Kobayashi, S.

Krishna, C. H.

Krolikowski, W.

Leger, J.

B. Hao and J. Leger, “Numerical aperture invariant focus shaping using spirally polarized beams,” Opt. Commun. 281(8), 1924–1928 (2008).
[Crossref]

Leger, J. R.

López-Mariscal, C.

Maier, M.

Manzo, C.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006).
[Crossref]

Marrucci, L.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006).
[Crossref]

Matsuo, S.

Meade, D. B.

W. E. Boyce, R. C. DiPrima, and D. B. Meade, Elementary differential equations and boundary value problems, vol. 9 (Wiley, 1992).

Miret, J. J.

Miret-Artés, S.

Mokhun, A. I.

I. Freund, M. S. Soskin, and A. I. Mokhun, “Elliptic critical points in paraxial optical fields,” Opt. Commun. 208(4-6), 223–253 (2002).
[Crossref]

Neshev, D. N.

Niv, A.

Ochoa, N. A.

N. A. Ochoa, “A unifying approach for the vectorial Rayleigh-Sommerfeld diffraction integrals,” Opt. Commun. 448, 104–110 (2019).
[Crossref]

Pääkkönen, P.

Pal, S. K.

Paparo, D.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006).
[Crossref]

Pastor, D.

Perez-Garcia, B.

Philip, G. M.

Piquero, G.

J. C. G. de Sande, G. Piquero, and M. Santarsiero, “Polarimetry with azimuthally polarized light,” Opt. Commun. 410, 961–965 (2018).
[Crossref]

J. C. G. de Sande, M. Santarsiero, and G. Piquero, “Spirally polarized beams for polarimetry measurements of deterministic and homogeneous samples,” Opt. Laser Eng. 91, 97–105 (2017).
[Crossref]

J. C. G. de Sande, G. Piquero, and C. Teijeiro, “Polarization changes at Lyot depolarizer output for different types of input beams,” J. Opt. Soc. Am. A 29(3), 278–284 (2012).
[Crossref]

V. Ramírez-Sánchez, G. Piquero, and M. Santarsiero, “Generation and characterization of spirally polarized fields,” J. Opt. A: Pure Appl. Opt. 11(8), 085708 (2009).
[Crossref]

V. Ramírez-Sánchez and G. Piquero, “The beam quality parameter of spirally polarized beams,” J. Opt. A: Pure Appl. Opt. 10(12), 125004 (2008).
[Crossref]

Ram, B. S. B.

Ramírez-Sánchez, V.

V. Ramírez-Sánchez, G. Piquero, and M. Santarsiero, “Generation and characterization of spirally polarized fields,” J. Opt. A: Pure Appl. Opt. 11(8), 085708 (2009).
[Crossref]

V. Ramírez-Sánchez and G. Piquero, “The beam quality parameter of spirally polarized beams,” J. Opt. A: Pure Appl. Opt. 10(12), 125004 (2008).
[Crossref]

Roy, S.

Ruchi,

Santarsiero, M.

J. C. G. de Sande, G. Piquero, and M. Santarsiero, “Polarimetry with azimuthally polarized light,” Opt. Commun. 410, 961–965 (2018).
[Crossref]

J. C. G. de Sande, M. Santarsiero, and G. Piquero, “Spirally polarized beams for polarimetry measurements of deterministic and homogeneous samples,” Opt. Laser Eng. 91, 97–105 (2017).
[Crossref]

V. Ramírez-Sánchez, G. Piquero, and M. Santarsiero, “Generation and characterization of spirally polarized fields,” J. Opt. A: Pure Appl. Opt. 11(8), 085708 (2009).
[Crossref]

R. Borghi, M. Santarsiero, and M. A. Alonso, “Highly focused spirally polarized beams,” J. Opt. Soc. Am. A 22(7), 1420–1431 (2005).
[Crossref]

R. Borghi and M. Santarsiero, “Nonparaxial propagation of spirally polarized optical beams,” J. Opt. Soc. Am. A 21(10), 2029–2037 (2004).
[Crossref]

Sanz, A. S.

Schwarz, U. T.

Senthilkumaran, P.

Sharma, A.

Shvedov, V. G.

Sihvola, A.

Soskin, M. S.

I. Freund, M. S. Soskin, and A. I. Mokhun, “Elliptic critical points in paraxial optical fields,” Opt. Commun. 208(4-6), 223–253 (2002).
[Crossref]

Spilman, A. K.

Takeda, M.

Teijeiro, C.

Tervo, J.

Turunen, J.

Tzarouchis, D. C.

Vahimaa, P.

Viswanathan, N. K.

Volyar, A. V.

Ylä-Oijala, P.

Zapata-Rodríguez, C. J.

Zhan, Q.

Appl. Opt. (5)

J. Opt. A: Pure Appl. Opt. (2)

V. Ramírez-Sánchez and G. Piquero, “The beam quality parameter of spirally polarized beams,” J. Opt. A: Pure Appl. Opt. 10(12), 125004 (2008).
[Crossref]

V. Ramírez-Sánchez, G. Piquero, and M. Santarsiero, “Generation and characterization of spirally polarized fields,” J. Opt. A: Pure Appl. Opt. 11(8), 085708 (2009).
[Crossref]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (5)

J. Opt. Soc. Am. B (1)

Opt. Commun. (5)

B. Hao and J. Leger, “Numerical aperture invariant focus shaping using spirally polarized beams,” Opt. Commun. 281(8), 1924–1928 (2008).
[Crossref]

J. C. G. de Sande, G. Piquero, and M. Santarsiero, “Polarimetry with azimuthally polarized light,” Opt. Commun. 410, 961–965 (2018).
[Crossref]

I. Freund, “Polarization flowers,” Opt. Commun. 199(1-4), 47–63 (2001).
[Crossref]

I. Freund, M. S. Soskin, and A. I. Mokhun, “Elliptic critical points in paraxial optical fields,” Opt. Commun. 208(4-6), 223–253 (2002).
[Crossref]

N. A. Ochoa, “A unifying approach for the vectorial Rayleigh-Sommerfeld diffraction integrals,” Opt. Commun. 448, 104–110 (2019).
[Crossref]

Opt. Express (6)

Opt. Laser Eng. (1)

J. C. G. de Sande, M. Santarsiero, and G. Piquero, “Spirally polarized beams for polarimetry measurements of deterministic and homogeneous samples,” Opt. Laser Eng. 91, 97–105 (2017).
[Crossref]

Opt. Lett. (4)

Phys. Rev. Lett. (1)

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006).
[Crossref]

Other (1)

W. E. Boyce, R. C. DiPrima, and D. B. Meade, Elementary differential equations and boundary value problems, vol. 9 (Wiley, 1992).

Supplementary Material (2)

NameDescription
» Visualization 1       Dynamical behavior of the streamlines of a GSPB.
» Visualization 2       Dynamical behavior of the streamlines of a GSPB and their transformation by the polarization device.

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Figures (8)

Fig. 1.
Fig. 1. (a) Spiral orthonormal polarization basis $\left ( \mathbf {\hat {u},\hat {v}}\right )$ for $\alpha =\pi /6$. Vector $\mathbf {\hat {u}}$ is shown in red, and $\mathbf {\hat {v}}$ in blue. (b) $u$- and $v$-lines intersect orthogonally forming spiral streamlines on the plane $\left ( x,\;y\right ) .$
Fig. 2.
Fig. 2. Polarization maps and spiral streamlines of GSPBs for $\omega t=0$ and $\omega t=2\pi /3$ with (a) $\alpha =\pi /6$, $\mathcal {U}=1$, $\mathcal {V}=1\exp \left ( \mathrm {i}\pi /3\right )$. (b) $\alpha =-\pi /4$, $\mathcal {U}=1$, $\mathcal {V}=0$.$7\exp \left (-\mathrm {i}5\pi /12\right )$. (c) $\alpha =\pi /6,$ $\mathcal {U}=\mathrm {i},$ $\mathcal {V}=1$. The size of the ellipses is proportional to the amplitude of the electric field and the green and yellow colors mean positive and negative polarization handedness, respectively. The arrows on the streamlines show the instantaneous direction of the tangent electric field. The Poincaré-Hopf index of the GSPBs is $\eta =+1$. The gray background is proportional to the beam intensity, i.e. $R^{2}(r) \propto r^{2} \exp (-2r^{2}/w_0^{2})$ where $w_0=\sqrt {2}$ in arbitrary units. The size of each image is $3.6w_0 \times 3.6w_0$. An animation of the case (a) is included in the supplementary material, see Visualization 1.
Fig. 3.
Fig. 3. The streamline morphology of a GSPB is transformed by its passage through an anisotropic polarization system.
Fig. 4.
Fig. 4. (a) Chart $\left ( \tau ,\Delta \right )$ for the streamline morphologies. (b) Node point. The streamlines tend to the eigenvectors $\left \vert \mathbf {q}_{j}\right \rangle .$ (c) Improper node occurs when eigenvectors are degenerate $\left \vert \mathbf {q}_{1}\right \rangle =\left \vert \mathbf {q}_{2}\right \rangle .$ (d) Skewed spiral occurs for region $\Delta\;>\;\tau ^{2}/4.$ (e) Saddle point. The streamlines approach the eigenvectors $\left \vert \mathbf {q}_{j}\right \rangle$ asymptotically.
Fig. 5.
Fig. 5. Input GSPB with parameters $\alpha =\pi /6,$ $\mathcal {U}=1,$ $\mathcal {V}=1\exp \left ( \mathrm {i}\pi /12\right )$. The anisotropic device is a quarter-wave retarder with its fast axis at an angle of $20{{}^{\circ }}$ respect to the $x$ axis. (a) Curve $\Delta \left ( \tau \right )$ plotted on the streamline stability chart. (b) Distribution of the third Stokes parameter $S_{3}=2\operatorname {Im}\left ( E_{x}^{\ast }E_{y}\right )$ across the transverse plane. (c)–(e) Streamlines and polarization maps of the input and output beam for three different phases $\omega t$ illustrating different streamline morphologies. Red lines correspond to L-lines, i.e. points where the polarization is linear. These L-lines are invariant on time. A video showing the dynamic behavior of the streamlines is included in the supplementary material, see Visualization 2.
Fig. 6.
Fig. 6. (a) Experimental setup. Laser, HeNe Laser source; BE, beam expander; SLM, spatial light modulator; L1,L2, lenses; SF1,SF2, spatial-filters; HWP, half-wave plate; PBS, polarizing beam splitter; M1–M7 mirrors; BS, non–polarizing beam splitter; QWP, quarter-wave plate; Anisotropic medium, wave plates; CP1, CP2, right handed and left handed circular polarizers; CCD, camera. (b) CGH displayed on the SLM.
Fig. 7.
Fig. 7. (a)–(c) Experimental generation of a GSPB with amplitudes $\mathcal {U} = 1/\sqrt {2}$ and $\mathcal {V} = 1/\sqrt {2}$, and $\alpha = 0$. (a) Polarization map. (b) Decomposition of the vector field into two scalar fields left and right circularly polarized. (c) Spiral streamlines. Different colors correspond to varying inclinations of the polarization ellipse. (d)–(f) Experimental field after a polarization system comprised of a HWP and a QWP oriented at $0^{\circ }$ and $-22.5^{\circ }$ with respect to the $x$-axis, respectively. (d) Polarization map. (e) Circularly polarized components. (f) Streamlines showing a symmetric saddle. Size of the square images is $3\times 3$ mm.
Fig. 8.
Fig. 8. (a)–(c) Experimental generation of a GSPB with amplitudes $\mathcal {U}=\sqrt {3}/2$, $\mathcal {V}=\frac {1}{2}\exp (i\pi /3)$ and $\alpha =0$. (d)–(f) Experimental field after a polarization system comprised of two QWPs oriented at $180^{\circ }$ and $36^{\circ }$ respect to the $x$-axis, respectively.

Equations (23)

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[ u ^ v ^ ] = [ cos α sin α sin α cos α ] [ r ^ θ ^ ] = [ cos Θ sin Θ sin Θ cos Θ ] [ x ^ y ^ ] = [ e i Θ e i Θ i e i Θ i e i Θ ] [ c ^ L c ^ R ] ,
r u ( θ ) = r 0 exp [ ( θ θ 0 ) / tan α ] , r v ( θ ) = r 0 exp [ ( θ θ 0 ) tan α ] ,
E u v ( r , z = 0 , t ) = [ E u E v ] = [ U V ] R ( r ) e i ω t ,
E x y = [ E x E y ] = [ cos Θ sin Θ sin Θ cos Θ ] E u v = [ U cos Θ V sin Θ U sin Θ + V cos Θ ] R ( r ) e i ω t .
r E ( θ , t ) = r 0 exp [ ( θ θ 0 ) U cos α V sin α U sin α + V cos α ] ,
U ( t ) Re { U e i ω t } , V ( t ) Re { V e i ω t } ,
J = [ J 11 J 12 J 21 J 22 ] .
E x y o u t = J E x y i n = [ J 11 J 12 J 21 J 22 ] [ cos Θ sin Θ sin Θ cos Θ ] [ U V ] R ( r ) e i ω t ,
E x y o u t = [ c 1 cos Θ c 2 sin Θ c 3 sin Θ + c 4 cos Θ ] R ( r ) e i ω t ,
c 1 U J 11 + V J 12 , c 2 V J 11 U J 12 , c 3 U J 22 V J 21 , c 4 U J 21 + V J 22 .
d x d ξ = Re E x o u t = Re { ( c 1 cos Θ c 2 sin Θ ) R ( r ) e i ω t } ,
d y d ξ = Re E y o u t = Re { ( c 3 sin Θ + c 4 cos Θ ) R ( r ) e i ω t } ,
d d ξ [ x y ] = Q [ x y ] = [ Q 11 Q 12 Q 21 Q 22 ] [ x y ] ,
Q 11 Re { ( c 1 cos α c 2 sin α ) e i ω t } , Q 12 Re { ( c 2 cos α c 1 sin α ) e i ω t } ,
Q 21 Re { ( c 4 cos α + c 3 sin α ) e i ω t } , Q 22 Re { ( c 3 cos α c 4 sin α ) e i ω t } .
λ 1 , 2 = τ 2 ± τ 2 4 Δ , | q 1 = [ q 1 x q 1 y ] , | q 2 = [ q 2 x q 2 y ] ,
τ = t r   Q = Q 11 + Q 22 , Δ = det Q = Q 11 Q 22 Q 12 Q 21 ,
[ x ( ξ ) y ( ξ ) ] = C 1 e λ 1 ξ | q 1 + C 2 e λ 2 ξ | q 2 ,
C 1 = q 2 y x 0 q 2 x y 0 q 1 x q 2 y q 1 y q 2 x , C 2 = q 1 y x 0 + q 1 x y 0 q 1 x q 2 y q 1 y q 2 x .
[ x ( ξ ) y ( ξ ) ] c e n t e r = [ x 0 cos ( g ξ ) + Im { C 1 q 1 x C 2 q 2 x } sin ( g ξ ) y 0 cos ( g ξ ) + Im { C 1 q 1 y C 2 q 2 y } sin ( g ξ ) ] .
τ ( t ) = | T | cos ( ω t arg T ) , Δ ( t ) = | D | cos ( 2 ω t arg D ) + R e { c 1 c 3 + c 2 c 4 } / 2 ,
T ( c 1 + c 3 ) cos α ( c 2 + c 4 ) sin α , D c 1 c 3 + c 2 c 4 2 = ( U 2 + V 2 ) det J 2 .
E i n ( r , θ ) = ( U i V ) e i α LG 1 ( r , θ ) c ^ L + ( U + i V ) e i α LG 1 ( r , θ ) c ^ R ,

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