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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
1. Introduction
Spirally polarized beams (SPB) have been receiving increasing attention since Gori introduced them as a polarization basis for vortex vector beams [1]. In his original paper, he highlighted the advantages of representing the polarization states of vortex beams in an orthogonal basis with angular dependence. This advantage was applied soon later to decompose vector propagation-invariant and rotating fields [2]. After the pioneering work [1], several articles were published studying the characteristics of the SPBs in the paraxial and non-paraxial regimes [2–8]. SPBs have been generated by using suitable superpositions of Hermite-Gaussian beams [3,4], computer-generated dielectric gratings [9,10], spatial light modulators in combination with waveplates [11], polarization converters and rotators [12], geometric phase manipulation through $q$–plates [13], optical fiber micro-axicons [14], and fiber-optic modes [15]. Some works studying the superposition of radially and azimuthally polarized beams have been reported, for example in [16] a matrix formalism is introduced to propagate SPBs, and in [17] the authors show how to construct spiral polarization distributions using a S-wave plate and apply them to signal processing. The propagation of light through uniaxial crystals to shape vector beams, including spirally polarized structures, was studied by Fadeyeva et al. [18] by analyzing the behavior of the scalar and vector singularities. More recently, the SVBs were applied to perform polarimetry measurements of deterministic and homogeneous samples by de Sande et al. [19,20].
One of main features of the SVBs is the fact that the streamlines of the vector field across the transverse plane are readily expressed in closed-form, thus providing a precise visual depiction of the spatial structure of the vector field. The streamlines are the family of curves that are instantaneously tangent to the electric field vector on the transverse plane. In particular, the streamlines of SPBs form logarithmic spirals and constitute continuous intermediate cases between purely radially polarized and purely azimuthally polarized beams. The streamline visualization technique is particularly useful for observation of the spatial distribution of 2-D and 3-D optical vector fields. The behavior of the streamlines allows to grasp the underlying nature of the physical process and understand the evolution of dynamical phenomena, such as the structure of polarization singularities and the energy flux line distributions. The streamline technique has been applied successfully in several optical studies, e.g. vortices in vector fields [21–23], Poynting distributions in nanoplasmonic devices [24,25], and propagating modes [26].
In this paper, (a) we generalize the model of the SPBs introduced by Gori [1] and studied by multiple authors [2–8]; (b) analyze in detail their propagation through arbitrary polarization optical systems described by complex Jones matrices, and (c) generate them experimentally using a technique based on a Sagnac interferometer. Usually, the literature refers to SPBs as those in which the electric field is linearly polarized at any point across the transverse section, but the polarization direction changes with the angular coordinate forming stable spiral streamlines invariant in propagation and time. Our generalization consists on superposing two suitable spiral beams with orthogonal polarization to generate spiral beams where now the electric field can be elliptically polarized at any point across the transverse section instead of linearly polarized. We will refer these beams to Generalized Spirally Polarized Beams (GSPBs) and their streamlines are varying on time. The GSPBs exhibit a more complex vector structure that allows to study the morphologies of vector singularities immersed in optical fields. The possibility of adjusting both polarization components may be applied in polarimetry measurements using spiral beams [19,20].
We concentrate our attention on describing the morphological transformation of the streamlines of the GSPBs upon its passage through a polarization system. We have found that the description of the passage of GSPBs through the polarization system is equivalent to the stability theory of autonomous systems of ordinary differential equations (ODEs). While the streamlines of the GSPBs exhibit a spiral geometry, the streamlines of the output field may display different shapes, such as saddles, nodes, ellipses, and stars. A major advantage of expressing he polarization states of vortex beams in the spiral basis is that it is possible to find closed expressions for the streamlines even after their transformation by the polarization system. Using a novel experimental technique [27], we have been able to reproduce in the laboratory each one of the different cases of GSPBs and record their corresponding characteristic streamline morphologies. We remark that the generation of vector beams can be realized applying a variety of techniques. In this work we have implemented an efficient arrangement based on a Sagnac interferometer which permit us to control and calibrate all the parameters involved very accurately.
2. Definition of the generalized spirally polarized beams
In this section we introduce the GSPBs and discuss their basic properties. We begin by describing the spiral polarization basis in order to establish notation.
2.1 Spiral polarization basis
At a definite position $\mathbf {r}=\left ( x,\;y\right ) = \left ( r\cos \theta ,\;r\sin \theta \right )$ on the transverse plane, the unit vectors $\left (\mathbf {\hat {u},\hat {v}}\right )$ of the spiral polarization basis are related to the polar (radial and azimuthal) unit vectors $(\mathbf {\hat {r}},\mathbf {\hat {\theta })}$, to the Cartesian vectors $\left ( \mathbf {\hat {x},\hat {y}}\right )$, and to the circular polarization (left- and right-handed) vectors $( \hat {{\mathbf {c}}}_L, \hat {{\mathbf {c}}}_R)$ as follows [1]:
In Fig. 1(a) we show the distribution of the spiral unit vectors $\left ( \mathbf {\hat {u} ,\hat {v}}\right )$ on the plane $\left ( x,\;y\right )$ for $\alpha =\pi /6$. Geometrically, $\alpha$ is the constant angle that the unit vector $\mathbf {\hat {u}}$ makes at each point with the radial direction $\mathbf {\hat {r}}$ and the vector $\mathbf {\hat {v}}$ with the azimuthal direction $\mathbf {\hat {\theta }}$. Similar to the polar vectors $(\mathbf {\hat {r}},\mathbf {\hat {\theta })}$, the orientation of the vectors $\left ( \mathbf {\hat {u},\hat {v}}\right )$ varies as a function of the angular position $\theta$, and the origin $\mathbf {r}=0$ is a singular point of the vector basis.
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 $\left ( x,\;y\right ) .$
The streamlines of the vector fields polarized along the vectors $\mathbf {\hat {u}}$ and $\mathbf {\hat {v}}$ can be derived in closed-form. In particular, they are the solutions of the first-order differential equations $\mathrm {d} y/\mathrm {d}x=u_{y}/u_{x}$ and $\mathrm {d}y/\mathrm {d}x=v_{y}/v_{x},$ where $u_{x},\;u_{y},\;v_{x},\;v_{y}$ are the Cartesian components of the vector fields $\mathbf {\hat {u}}$ and $\mathbf {\hat {v}}$ in Eq. (1). Integrating these equations and expressing the streamlines in polar coordinates we get
2.2 Definition of generalized spirally polarized beams
Using the definition of the spiral vector basis in Eq. (1), we now define a monochromatic GSPB as the superposition of a $\mathbf {\hat {u}}$-field with amplitude $\mathcal {U}$ and a $\mathbf {\hat {v}}$-field with amplitude $\mathcal {V}$, namely
We now derive the analytic expression for the streamlines of the GSPBs. To this end, we first express the field $\mathbf {E}_{uv}$ in the Cartesian basis, namely
An important point to note in Eq. (6) is that the amplitudes $U$ and $V$ oscillate harmonically with frequency $\omega .$ Therefore, from Eq. (5) we see that the pattern of streamlines of a GSPB will fluctuate with period $2\pi /\omega$. This fluctuation is similar to that of a circularly polarized conventional Gaussian beam, in which its straight streamlines rotate performing a full turn per period. In order to show this variation, in Fig. 2 we have plotted the polarization ellipses and the streamlines of GSPBs for several values of the amplitudes $\mathcal {U}$, $\mathcal {V}$, and parameter $\alpha$, at two different temporal phases $\omega t.$ Although the streamlines are fluctuating, note that the polarization states at each point keep constant in time. An animation of the first case shown in Fig. 2 is included in the supplementary material of this paper, see Visualization 1.
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.
From Eq. (5) we can identify the following cases of interest:
- 1. Pure spiral $\mathbf {\hat {u}}$- and $\mathbf {\hat {v}}$- polarized fields. The streamlines of the $\mathbf {\hat {u}}$- and the $\mathbf {\hat {v}}$-fields in Eqs. (2) are two special cases when either $\mathcal {U}$ or $\mathcal {V}$ are zero.
- 2. Azimuthal polarization. Twice in a cycle, the streamlines become circles centered at the origin when $U\cos \alpha -V\sin \alpha =0$, that is, when $\omega t = \arg \{ \mathcal {U}\cos \alpha - \mathcal {V}\sin \alpha \} \pm \pi /2$.
- 3. Radial polarization. Twice in a cycle, the streamlines become straight radial lines emerging from the origin when $U\sin \alpha +V\cos \alpha =0$, that is, when $\omega t = \arg \{ \mathcal {U}\sin \alpha + \mathcal {V}\cos \alpha \} \pm \pi /2$. The radial and circular streamlines appear with a phase difference of 90 degrees. In Fig. 2(c) we show the streamlines just at the time when the field is radially polarized.
Taking the components from Eq. (3), the third Stokes parameter is given by $S_3=2\textrm{Im} \left \{ E_{x}^{\ast }E_{y}\right \}=-2\textrm{Im} \left \{ \mathcal {UV}^{\ast }\right \}$. From this result we can identify two additional special cases:
- 4. Pure spiral linearly polarized field. If either $\mathcal {U}$ or $\mathcal {V}$ are zero, or $\arg \mathcal {U} = \arg \mathcal {V} + m\pi$, with $m=\{-1,0,1\},$ then $S_{3}=0$ and therefore the field at every point keeps linearly polarized along the tangent of the spiral streamline. In these cases, the time dependencies in Eq. (5) cancel out and the spiral streamlines are invariant in time.
- 5. Pure spiral circularly polarized field. If $\mathcal {U}=\pm i\mathcal {V}$, then $S_{1}=S_{2}=0$ and the field at every point of the transverse plane is circularly polarized. The sign of $S_{3}$ defines the sense of rotation of the electric field vector. We illustrate this case in Fig. 2(c).
Finally, we recall that polarization singularities are also customarily characterized by the Poincaré-Hopf index $\eta =\Delta \theta /(2\pi )$ associated with the winding (rotation) angle $\theta$ of the vectors surrounding the singularity in the vector field [21,22,29]. For the case of the GSPBs, the Poincaré-Hopf index is $\eta =+1$.
3. Transformation by polarization optical systems
The spiral morphology of the streamlines of a GSPB is invariant upon its propagation through homogeneous, isotropic media. In order to better understand this invariance, note that the GSPB in Eq. (4) may be decomposed into two orthogonal linearly polarized vortex modes with spatial structure $g_1\left ( r\right ) \exp \left ( \mathrm {i}\theta \right ) + g_2\left ( r\right ) \exp \left ( -\mathrm {i}\theta \right )$, where $g_{1,2}$ are functions which depend only on the radial coordinate $r$. Due to the invariance of the azimuthal factors $\exp (\pm \mathrm {i}\theta )$ on propagation, it is clear that when the GSPB passes through an isotropic ABCD system, diffraction affects only the radial dependence of the beam leading to a propagated field of the form $\mathbf {E}(x,\;y,\;z,\;t)=F(r,\;z)(\mathcal {U}\mathbf {\hat {u}}+\mathcal {V} \mathbf {\hat {v}})e^{-\mathrm {i}\omega t},$ where $F\left ( r,\;z\right )$ can be determined solving the known Collins diffraction integral through an ABCD system. We then conclude that the vector structure of the GSPBs remains unchanged under propagation through linear ABCD isotropic optical systems. However, the same does not hold for propagation in anisotropic media, where the orthogonal vector components experiment different phase retardations leading to a modification of the vector structure of the field. This effect is illustrated graphically in Fig. 3. In this section, we study the transformation of the field by anisotropic media and characterize the corresponding changes in the morphology of the field streamlines.
Fig. 3. The streamline morphology of a GSPB is transformed by its passage through an anisotropic polarization system.
3.1 Output field
We investigate the passage of the GSPBs through a polarization optical system characterized by the Jones matrix
Adjusting the values $J_{mn}$ in Eq. (7) allows one to model a variety of polarization devices, including for example, ideal and partial linear polarizers, wave plates, polarization rotators, or an arbitrary sequence in cascade of these devices. The matrix $\mathbf {J}$ accounts for the overall amplitude and phase factors introduced by the optical system.When the input spiral beam $\mathbf {E}_{xy}^{\mathrm {in}}$ in Eq. (4) passes through the system $\mathbf {J},$ the polarization state of the resulting output beam is $\mathbf {E}_{xy}^{\mathrm {out}}=\mathbf {JE}_{xy}^{\mathrm {in}}$. The expression of the output field reads as
3.2 Morphologies of the streamlines
The streamlines of the output field Eq. (9) are solutions of the first-order differential equation $\mathrm {d}y/\mathrm {d}x=\operatorname {Re} E_{y}^{\mathrm {out}}\left ( \mathbf {r}\right ) /\operatorname {Re} E_{x}^{\mathrm {out}}\left ( \mathbf {r}\right )$ that can be recast as a nonlinear autonomous system of ODEs of the form
The origin (0,0) is a singular point of the spiral basis. Therefore, a physically realizable field $\mathbf {E}$ must vanish at the origin, i.e. $R\left ( 0\right ) =0$. To this end, we assume that $R(r)$ is proportional to the radius near the origin, i.e. $R(r)\sim r,$ which means that we take the first term of the Taylor expansion of $R\left ( r\right )$ around $r=0$. This assumption is physically reasonable and has two important consequences: (a) it ensures that both $E_{x}$ and $E_{y}$ vanish at $\left ( 0,0\right )$ and thus the origin becomes an isolated critical point of the autonomous system, and (b) the nonlinear system in Eqs. (11) reduces to a linear system of the form
The matrix $\mathbf {Q}$ characterizes the morphologies of the streamlines of the field $\mathbf {E}^{\mathrm {out}}$. Let
The solution of the system in Eq. (12) subject to the initial conditions $x(0)=x_{0}$ and $y(0)=y_{0}$ is given as a linear superposition of eigenvectors $\left \vert \mathbf {q}_{1}\right \rangle$ and $\left \vert \mathbf {q}_{2}\right \rangle ,$ namely
From the theory of linear autonomous systems [30], it is known that the trace $\tau$ and the determinant $\Delta$ of the matrix $\mathbf {Q}$ determine the shapes of the curves $x\left ( \xi \right )$ and $y\left ( \xi \right ) .$ In Fig. 4(a) we show the streamline stability chart $\left ( \tau ,\Delta \right )$ to identify all possible scenarios, which are:
- • Node: Region $0\;<\;\Delta\;<\;\tau ^{2}/4$. In this region, the eigenvalues are real, distinct, and of the same sign ($\lambda _{1}\lambda _{2}\;>\;0$), and the eigenvectors $\left \vert q_{j}\right \rangle$ are real. Suppose for definiteness that $0\;<\;\lambda _{1}\;<\;\lambda _{2}.$ In the limit $\xi \rightarrow -\infty ,$ the streamlines become parallel to the eigenvector $\left \vert \mathbf {q}_{1}\right \rangle$ and tend to the origin, see Fig. 4(b). On the other hand, as $\xi \rightarrow \infty ,$ the streamlines tend to infinity and become parallel to the eigenvector $\left \vert \mathbf {q}_{2}\right \rangle$. An improper node occurs when the eigenvectors are parallel, i.e. $\left \vert \mathbf {q}_{1}\right \rangle =\left \vert \mathbf {q}_{2}\right \rangle ,$ see Fig. 4(c). In the nodal region the Poincaré-Hopf index is $\eta =+1$.
- • Star: Region $\Delta =\tau ^{2}/4$. The region corresponds to a point falling just on the parabola. In this region the eigenvalues degenerate, i.e. $\lambda _{1}=\lambda _{2}=\tau /2$, and the eigenvectors are real and equal, i.e. $\left \vert \mathbf {q}_{1}\right \rangle =\left \vert \mathbf {q}_{2}\right \rangle .$ The streamlines are straight lines emerging radially from the origin. For a star the Poincaré-Hopf index is $\eta =+1$.
- • Saddle: Region $\Delta\;<\;0$. In this region, the eigenvalues are real with opposite signs ($\lambda _{1}\lambda _{2}\;<\;0$), and the eigenvectors $\left \vert q_{j}\right \rangle$ are real. Suppose for definiteness that $\lambda _{1}\;<\;0\;<\;\lambda _{2}.$ As shown in Fig. 4(e), the eigenvectors $\left \vert \mathbf {q}_{1}\right \rangle$ and $\left \vert \mathbf {q} _{2}\right \rangle$ define the separatrices of the saddle which are the asymptotes of the streamlines that have the form of hyperbolae. For a saddle the Poincaré-Hopf index is $\eta =-1$.
- • Center: Region $\tau =0$ and $\Delta\;>\;0,$ i.e. the positive vertical axis. In this region, the eigenvalues are purely imaginary, i.e. $\lambda _{1,2}=\mp \mathrm {i}g=\mp \mathrm {i}\Delta ^{1/2}$ and the eigenvectors $\left \vert q_{j}\right \rangle$ are complex. The streamlines are ellipses centered at the origin with parametric equations$$\begin{bmatrix} x(\xi)\\ y(\xi) \end{bmatrix} _{\mathrm{center}}= \begin{bmatrix} x_0 \cos\left( {g\xi}\right) +\textrm{Im} \left\{ C_{1}q_{1x}-C_{2}q_{2x}\right\} \sin\left( {g\xi}\right) \\ y_0 \cos\left( {g\xi}\right) +\textrm{Im} \left\{ C_{1}q_{1y}-C_{2}q_{2y}\right\} \sin\left( {g\xi}\right) \end{bmatrix}.$$For the Center morphology the Poincaré-Hopf index is $\eta =+1$.
- • Spiral: Region $\Delta\;>\;\tau ^{2}/4$ and $\tau \neq 0$. In this region, the eigenvalues are conjugate complex i.e. $\lambda _{1,2} =h\mp \mathrm {i}g$, but not purely imaginary, and the eigenvectors $\left \vert q_{j}\right \rangle$ are complex with the general form $\left \vert \mathbf {q}_{1,2}\right \rangle =[a,\;b\mp \mathrm {i}c]^{T}$, where $T$ means transpose. The streamlines are skewed logarithmic spirals given by $[x(\xi ),\;y(\xi )]_{\mathrm {spiral}}^{T} = e^{h\xi }[x(\xi ),\;y(\xi )]_{\mathrm {center}}^{T}$, see Fig. 4(b). For a spiral configuration the Poincaré-Hopf index is $\eta =+1$.
- • Straight lines: Region $\Delta =0,$ i.e. the horizontal axis. If the determinant of the matrix $\mathbf {Q}$ vanishes, then it becomes singular, therefore one eigenvalue vanishes and the remaining one equals the trace, i.e. $\lambda _{1}=\tau ,\lambda _{2}=0,$ and the eigenvectors are real and different. The streamlines become straight lines parallel to the vector $\left \vert \mathbf {q}_{1}\right \rangle$. The special point at the origin $\left ( \tau ,\Delta \right ) =\left ( 0,0\right )$ leads to the degenerate case $\lambda _{1}=\lambda _{2}=0$ and identical and real eigenvectors $\left \vert \mathbf {q}_{1}\right \rangle =\left \vert \mathbf {q}_{2}\right \rangle .$ In this case, the streamlines are also parallel to $\left \vert \mathbf {q}_{1}\right \rangle$. For the straight lines the Poincaré-Hopf index vanishes, i.e. $\eta =0$.
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.
We finally remark that the theory presented in this subsection encounters strong analogies with the stability theory of autonomous systems of ordinary differential equations in other areas like in fluid mechanics [30], and also in the theory of polarization singularities in optics [21,22].
3.3 Time evolution of streamlines
The polarization states across the transverse plane of the output field $\mathbf {E}_{xy}^{\mathrm {out}}$ are determined by the Jones vector of Eq. (9) and are, of course, time-invariant. However, the same is not true on the streamlines because they are tangent to the instantaneous and local direction of the electric field $\mathbf {E}_{xy}^{\mathrm {out}}$ which is rotating with frequency $\omega$. Let us now turn our attention to how the streamlines evolve with time in a complete period $2\pi /\omega$ of the field. As we saw, the streamline patterns are fully characterized by the trace and determinant of the matrix $\mathbf {Q}$. Since the elements $Q_{mn}$ depend harmonically with time according to Eqs. (13), then $\tau$ and $\Delta$ are functions of time as well. Replacing $Q_{mn}$ in Eqs. (15) we get
From the above equations, it can be seen that the trace oscillates harmonically within the interval $\tau \leq |T|$ with frequency $\omega ,$ whereas the determinant oscillates with frequency $2\omega$ and maximum excursion $|D|$ around the mean value $\overline {\Delta }=\mathrm {Re}\{c_{1}c_{3}^{\ast }+c_{2}c_{4}^{\ast }\}/2$. Therefore the curve $\Delta \left ( \tau \right )$ on the plane $\left ( \tau ,\Delta \right )$ corresponds in general to a $2$:$1$ Lissajous curve with two lobes as shown in Fig. 5(a).
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.
The morphological variation of the streamlines during one field oscillation period can be traced by following the point $(\tau ,\Delta )$ on the stability chart as time advances. In Fig. 5 we show the polarization ellipses and the streamlines of the input and output fields at three different phase values $\omega t$ chosen specifically to illustrate the spiral, node, and saddle morphologies. The corresponding points on the curve $\Delta \left ( \tau \right )$ are also depicted in Fig. 5(a) for each case. In the example, the polarization system is a quarter-wave retarder with fast axis making an angle of $20{}^{\circ }$ respect to the horizontal. Comparing the plots at different times, we see that the polarization ellipses do not change with time, but the streamlines vary by virtue of the rotation of the electric vector a each point. A video showing the dynamic behavior of the streamlines is included in the supplementary material, see Visualization 2.
We have also plotted the distribution of the third Stokes parameter $S_{3}\left (x,\;y\right ) =2\operatorname {Im}\left ( E_{x}^{\ast }E_{y}\right )$ in Fig. 5(b). Notice how the cross section of the output beam is divided in four sectors where the handedness of the polarization ellipses is positive (green) or negative (yellow). The red straight lines correspond to the loci $S_{3}=0,$ i.e. L-lines where the polarization state is linear. In this particular example, the L-lines intersect perpendicularly, but for other combination of parameters, the L-lines intersect at different angles. Actually, if $K \equiv \operatorname {Im} \left \{ c_1^{*}c_4 + c_2^{*}c_3 \right \}$, $L \equiv \operatorname {Im} \left \{ c_1^{*}c_3 - c_2^{*}c_4 \right \}$, and $H \equiv \operatorname {Im} \left \{c_2^{*}c_3 - c_1^{*}c_4 \right \}$ then the angles of the two L-lines are given explicitly by $\theta _{\pm } = \arctan (L/K)/2 \pm \arccos (H/\sqrt {K^{2}+L^{2}})/2 - \alpha$.
From Eqs. (19) we can extract additional conclusions about the time evolution of the streamlines:
- • In a complete period the curve $\Delta \left ( \tau \right )$ goes through every region of the stability chart and crosses the axes $\tau =0$ and $\Delta =0$, as well as the parabola $\Delta =\tau ^{2}/4$. Therefore, in one period the streamlines exhibit each one of the possible morphologies, including spirals, saddles, nodes, stars, ellipses and straight lines. The parameters $c_j$ can be adjusted to raise or lower the value of $\overline {\Delta }$ in such a way that the graph falls entirely within the spiral or the saddle region, respectively.
- • $\tau (t)$ always crosses twice the vertical axis $\tau =0,$ therefore twice per cycle the streamlines become ellipses (if the cross point occurs for $\Delta\;>\;0$), or symmetric saddles (if the cross point occurs for $\Delta\;<\;0$). The condition $T=0$ implies that $\tau (t)=0,$ thus the curve $\Delta \left (\tau \right )$ oscillates on the vertical axis. In this case, the streamlines are always elliptical and/or symmetric saddles.
- • The Lissajous curve $\Delta \left ( \tau \right )$ becomes a parabola when $\arg D = 2\arg T + m\pi$, where $m=\{-1,0,1\}$. The parabola opens upward for $m=0$, and downward for $m=\pm 1$. One particular example of this case is when the input field is purely polarized in the $\mathbf {\hat {u}}$ direction, i.e. $\mathcal {U}=1, \mathcal {V}=0$, and the optical system is a linear retarder. In this case, both $T$ and $D$ become real numbers and thus $\arg D = \arg T = 0$.
- • The needed condition to have stationary streamlines is $\left \vert T\right \vert =\left \vert D\right \vert =0,$ which leads to a point on the vertical axis $\tau =0$ at $\Delta =\overline {\Delta }.$
4. Experimental generation of generalized spirally polarized beams
4.1 Experimental setup
To produce GSPBs in the laboratory we have used Laguerre–Gauss (LG) beams $ {\textrm {LG}}{}_\ell (r,\;\theta ,\;z=0) = (r/w_0)^{ {\left |{\ell }\right |}} \exp (-r^{2} / w_0^{2}) \exp (i\ell \theta )$, where $\ell$ is the winding number and $w_0$ is the beam waist. For the experiment we have set $w_0=1.5$ mm. LG beams are characterized by their doughnut-shaped transverse intensity profiles. For convenience, we rewrite the input vector field $ {\mathbf {E}}^{\mathrm {in}}$ using LG beams in the circular polarization basis $( \hat {{\mathbf {c}}}_L, \hat {{\mathbf {c}}}_R)$ as [Eq. (1)]
In order to study the streamlines of the GSPBs and their transformation after their propagation through an anisotropic medium, we have built the experimental setup shown in Fig. 6(a). The GSPB was generated by independently shaping each transverse circular component of the electric field, as we previously reported in [27]. The resulting field is then analyzed to determine its transverse intensity, phase and polarization state. In our experiment, a horizontally polarized HeNe laser serves as a scalar seed and is expanded and collimated. We directed this scalar beam to illuminate the entire screen of a Spatial Light Modulator (SLM Holoeye PLUTO VIS) and generated two scalar beams by independently addressing each half of the SLM screen, split in the horizontal direction. A blazed grating was added to the phase modulation (digital hologram) to allow spatial filtering of the desired beams. A half–wave plate (HWP) then sets the polarization state to +45 degrees on both beams and these are sent to an interferometer.
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.
We have used a Sagnac interferometer to perform the superposition of the ortogonal components due to its common path properties. Common path interferometers are inherently stable [31], in particular the Sagnac interferometer ensures a zero path difference between the interfering beams, and is insensitive to lateral displacements of the mirrors or the beam splitter. Our setup allows us to conveniently control the phase difference between the field components directly via the SLM, in turn providing with exquisite control of the polarization distribution of the generated beams.
A quarter–wave plate (QWP) was placed at the output of the interferometer to change the polarization basis from linear to circular polarization. The superposition of both circular components resulted in the desired GSPBs ideal for our study. The generated GSPB was then propagated through different anisotropic media, namely, half–wave plates, quarter–wave plates and a combination of them. Finally, we analyzed the resulting output beam.
In order to characterize the propagated GSPB, we measured the intensity and spatial phase distribution of each circularly polarized component of the beam. The intensity measurement of each component was carried out by simply projecting on the circular polarization basis. This was performed by placing a right–handed or left–handed circular polarizer in front of the CCD detector and then recording the field. On the other hand, to obtain the phase of each component, we used the technique described by Takeda [32]. This method is based on the interference pattern that arises from the superposition of the beam and a reference plane wave.
As shown in Fig. 6, the beam coming out of the laser is split into two copies, one of them is used as the reference wave while the other is employed for the generation of the GSPB. The neutral density filter (ND) is employed for matching the power of the reference beam to that of the GSPB coming out of the Sagnac interferometer, this is done in order to achieve maximum contrast in the interference pattern. The circular polarizer (CP2) is chosen to project the reference beam into a right or left circularly polarized state, depending on the field component one is trying to measure. It is important to remark that due to the time averaging performed by the CCD detector and its sensibility to the field intensity rather than the amplitude, this measurement scheme sets the time reference to $t=0$, or for instance any other time for which $\omega t$ is a multiple of $2\pi$. In this respect, Fig. 7 depicts the streamlines of the GSPB at a reference time $t=0$.
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.
The interference pattern is captured by a CCD camera (Thorlabs DCX) and later processed numerically. First, a numerical Fourier transform is performed. This spectrum features three bright regions, the 0–th and $\pm 1$ orders. The +1 order is numerically selected and shifted to the origin of the numerical window. Finally, a numerical inverse Fourier transform of the selected order is performed and the phase is extracted. In our experiment, we performed the previously described strategy for each circular component by interfering the unknown beam with a reference plane wave with a matching polarization state.
The information provided by the amplitude and phase of each circular component allows us to fully characterize the properties of the vector beam. For instance, we can calculate the polarization ellipses at each point on the transverse plane, or we can also compute the streamlines that correspond to the GSPB and their transformation after propagating through a polarizing system. The generation of the streamlines is obviously result of the post-processing of the intensity measurements.
4.2 Results and discussion
Figures 7(a)–7(c) show a GSPB with $\mathcal {U}=1/\sqrt {2}$, $\mathcal {V}=1/\sqrt {2}$ and $\alpha =0$. Using our instrument, we have measured the inhomogeneous polarization map shown in Fig. 7(a). Notice that this GSPB features local linear polarization states with varying orientations which collectively trace a spiral pattern across the transverse plane. This characteristic pattern is more evident in the streamlines depicted in Fig. 7(c), calculated from the intensity and phase measurements of each component (Fig. 7(b)). The color of the streamlines corresponds to the orientations of the polarization ellipse, whereas the thickness of the lines is proportional to the local intensity of the field. Now we turn our attention to the propagation of the GSPB through a polarization system comprised of a HWP and a QWP with their fact axis oriented at $0^{\circ }$ and $-22.5^{\circ }$ with respect to the $x$-axis, respectively, see Figs. 7(d)–7(f). The propagated GSPB features a symmetric saddle morphology (Fig. 7(f)), corresponding to the region $\Delta\;<\;0$ of the chart $(\tau , \Delta )$ shown in Fig. 4(a). This characteristic is evident in the streamlines, but not necessarily in the inhomogeneous polarization map depicted in Fig. 7(d). The presence of different elliptical polarization states makes it difficult to infer the morphology of the propagated GSPB.
Figures 8(a)–8(c) show a GSPB with $\mathcal {U}=\sqrt {3}/2$, $\mathcal {V}=\frac {1}{2}\exp (i\pi /3)$ and $\alpha =0$. This GSPB depicts a rich inhomogeneous polarization map (Fig. 8(a)), that locally features elliptical polarization states with varying orientations, tracing collectively a spiral pattern across the transverse plane. This behaviour also occurs in the streamlines depicted in Fig. 8(c), calculated from the intensity and phase measurements of each component (Fig. 8(b)). The propagation of the GSPB through a polarization system comprised of two QWPs with their fast axis oriented at $180^{\circ }$ and $36^{\circ }$, is shown in Figs. 8(d)–8(f). The propagated GSPB features a skewed spiral morphology (Fig. 8(f)), corresponding to the region $\Delta\;>\;\tau ^{2}/4$ of the chart $(\tau , \Delta )$ shown in Fig. 4(a). Finally, we mention that the fields shown in Fig. 7 and Fig. 8 can be considered as a special case of the Stokes vortex fields [33].
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.
5. Conclusions
In summary, we have studied theoretically and experimentally the transformation of generalized spirally polarized vector beams by anisotropic media with emphasis in the morphologies of the streamlines of the electric field. These morphologies can be understood from their equivalence to the phase portraits in the stability diagram of dynamical systems. We have found that in a complete temporal oscillation period the streamlines fluctuate periodically exhibiting saddles, spirals, nodes, centers, stars, and straight lines. We have characterized each one of these topological cases mathematically and experimentally with a novel technique to generate arbitrary vector beams accurately. The theory discussed in this paper can be generalized to beams with higher topological charges for the spiral basis.
Funding
Consejo Nacional de Ciencia y Tecnología (PN2016-3140); Instituto Tecnológico y de Estudios Superiores de Monterrey.
Acknowledgments
We acknowledge financial support from Tecnológico de Monterrey.
Disclosures
The authors declare no conflicts of interest.
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