Abstract
The polarization of light can exhibit unusual features when singular optical beams are involved. In 3dimensional polarized random media the polarization orientation around singularities describe 1/2 or 3/2 Möbius strips. It has been predicted that if singular beams intersect noncollinearly in free space, the polarization ellipse rotates forming manyturn Möbius strips or twisted ribbons along closed loops around a central singularity. These polarization features are important because polarization is an aspect of light that mediate strong interactions with matter, with potential for new applications. We examined the noncollinear superposition of two unfocused paraxial light beams when one of them carried an optical vortex and the other one a uniform phase front, both in orthogonal states of circular polarization. It is known that these superpositions in 2dimensions produce spacevariant patterns of polarization. Relying on the symmetry of the problem, we extracted the 3dimensional patterns from projective measurements, and confirmed the formation of manyturn Möbius strips or twisted ribbons when the topological charge of one of the component beams was odd or even, respectively. The measurements agree well with the modelings and confirmed that these types of patterns occur at macroscopic length scales and in ordinary superposition situations.
Introduction
Electromagnetic waves can produce in 3dimensions (3d) unusual patterns of fields that are not immediately obvious. Superpositions of optical beams carrying scalar phase singularities, optical vortices, have been shown to produce knots in the line that follows the phase singularity in 3d^{1,2,3,4}. Vector fields can also produce unusual patterns. They may, for example, involve a traveling wave that when focused produces an axial component of the field. This is the case at the waist of a radially or azimutallypolarized beam^{5}. Because of this, they have long been discussed for use in chargedparticle accelerators^{6}. Radial and azimuthal vector beams are already a rarity in themselves because in their paraxial form they carry a polarization singularity: its center has all linearpolarization orientations. It is not a contradiction because, like with optical vortices, the intensity decreases asymptotically to zero at the singular point. These beams belong to a class of beams known as singular optical beams. Likewise, Poincaré beams can carry various types of polarization singularities. Among them, Cpoints, which have all orientations of the polarization ellipse, reached asymptotically from the polarization orientation of the surrounding field. The singular point in this case may have nonzero intensity because it has circular polarization, which is singular in orientation. The polarization field surrounding the Cpoint contains a disclination in the orientation of the polarization ellipse^{7,8,9,10,11,12,13,14}, and recent studies have investigated the full range of disclinations that can be produced with designer singular optical beams^{15,16,17}.
Evanescent fields are also situations that may give rise to unexpected arrangements of fields^{18,19}. Freund proposed that electromagnetic fields involving randomlypolarized fields contain unique features: the polarization ellipse describes Möbius strips of either 1/2 or 3/2 turns along closed paths surrounding a singular point^{20,21,22}. These features have been confirmed analytically by independent analysis^{23}. However, they appear on length scales of the order of the wavelength, which makes them hard to measure^{22}. Recently, Bauer et al. verified experimentally 3/2 and 5/2 Möbius strips appearing at the waist of a tightlyfocused Poincaré beam^{24}. They did so with the novel technique of Mie scattering nanointerferometry that they developed^{25}. A more recent prediction claimed that singular optical beams would produce 3d fields that describe manyturn Möbius strips at macroscopic length scales in the intersection of paraxial beams. These patterns would appear along a macroscopic closed path about the singular point^{26,27}. Moreover, crossed paraxial beams with opposite circular polarization produce twists in the polarization along any closed paths with any type of modes^{26,27}. The twists may be halfinteger, forming Möbius strips only when the paths surround a polarization singularity.
The previous measurements of these types of features used a novel scattering method of tightly focused beams^{25}. We made this determination using polarimetry. A particular challenge posed by this approach was that polarization projections cannot be used to measure the zcomponent of the field independently of the other components. However, we were able to derive it from projective measurements by relying on the simplified formalism of a symmetric geometric configuration. Using this method we were able to successfully reconstruct the 3d orientations of the polarization ellipse, as shown below.
Materials and Methods
Theory
The setup of the problem involves two paraxial beams of halfwidth w crossing in free space. The beams were in orthogonal states of circular polarization. One beam was a standard Gaussian beam and the other carried an optical vortex, which for ease of analysis was prepared in a singlyringed LaguerreGauss mode. This is depicted in Fig. 1(a). We analyzed the pattern in a reference frame (x, y, z), with zaxis coplanar with the propagation directions of the two beams, and bisecting them forming an angle \(\theta \) with each direction, as shown in the figure. We label the z = 0 plane as the “observing plane”. We can write the equations for the field at t = 0 in this plane as^{28}
where
and
with k being the wavenumber, \(\ell \) the topological charge, \({A}_{\ell }\) and \({A}_{0}\) mode normalization constants, and \({\delta }_{0}\) the relative phase between the two beams. We neglect the curvature of the wavefront.
Because the field in Eq. (1) has all three components in a nonfactorable form, the polarizationellipse field is 3dimensional. To qualitatively visualize the pattern that the ellipses make, let us assume that \(\theta \) is a small angle, so that the zcomponent of the field is small. We express the x and ycomponents of the field in terms of the right and left circular components: \({\hat{e}}_{R}={2}^{1/2}({\hat{e}}_{x}{\rm{i}}{\hat{e}}_{y})\) and \({\hat{e}}_{L}={2}^{1/2}({\hat{e}}_{x}+{\rm{i}}{\hat{e}}_{y})\). Neglecting the zcomponent, the orientation of the polarization ellipse is half the relative phase between the circular components, or approximately
where \(\varphi \) is the transverse angular coordinate. For \(\theta \) above 10^{−4} rad the variation in the xdependence of the phase is larger than the azimuthal phase variation and the polarization displays polarization fringes, as shown in the measurements of Fig. 1(b), for which \(\theta =3\cdot {10}^{4}\) rad^{28}. In these fringes the polarization ellipse rotates by half a turn from fringe to fringe. So, if we follow a closed path around the center of the intersection of the two beams (the origin), the polarization ellipse rotates by N half turns on the top side of the fringe pattern, and by \(N+\ell \) halfturns in the opposite sense on the bottom side of the path. This leads to a net \(\ell \) halfturns for a completed closed path.
Note also that in Eq. (1) the absolute value of the y and zcomponents of the field are proportional to each other. Figure 1(c) shows the calculated amplitude squared of the zcomponent of the field. Adjacent fringes are out of phase with each other. That is, at the nodes of the fringe pattern the instantaneous zcomponent of the field changes sign, implying that the polarization ellipse is changing its tilt relative to the plane of the picture. In following this zcomponent along the circular path we see that it undergoes an odd number of tilt flips. When we combine the twodimensional rotations with the zcomponent switching, we get that the semimajor axis of the polarization ellipse describes either a Möbius strip or a twisted ribbon.
We can quantify the twists the following way: the number of twists depends on the radius of the circular path r relative to the fringe spacing \(\lambda /(2\,\sin \,\theta )\) ^{27}, where λ is the wavelength of the light. The number of turns is then given by
where \(c=0\) for \(\ell \) even, and \(c=1\) for \(\ell \) odd. Note that a twisted ribbon also appears for \(\ell =0\), in the intersection of two plane waves. Indeed, the simple interference of noncollinear circularly polarized plane waves shows interesting features in its transverse momentum^{29}.
Projective Measurements
The next problem we faced was the determination of the field of the light pattern. One approach that works with tightlyfocused beams is to scatter the light with a small metallic sphere, and infer the field from the scattered pattern^{24,25}. In our case the optical beams are unfocused, a situation where scattering may be too weak to use as means of measurement. We opted for using projective measurements with a polarizer. The challenge is that such measurements do not measure the zcomponent of the field independently of the other components^{30}. However, as with polarimetry, one can obtain polarizationellipse parameters via projective measurements plus knowledge of the modes of the polarization of the incoming beams and their geometric configuration. One additional restriction that simplifies the algebra is the use of an observing plane with a normal that bisects the two propagationvector directions, as presented above in the setup of the problem.
To analyze the result of projective measurements we specify the field components of the two beams:
with respective propagation unit vectors:
The intensity of the transmitted light past a polarizer with transmission axis along unit vector \(\hat{p}\) is
where^{30,31}
is the unit vector orthogonal to the propagation direction along the plane that contains \(\hat{p}\). We consider four polarization projections: \({\hat{p}}_{V}={\hat{e}}_{y}\), \({\hat{p}}_{H}={\hat{e}}_{x}\), \({\hat{p}}_{D}=({\hat{e}}_{x}+{\hat{e}}_{y})/\sqrt{2}\) and \({\hat{p}}_{A}=({\hat{e}}_{x}{\hat{e}}_{y})/\sqrt{2}\), which correspond to vertical (V), horizontal (H), diagonal (+45 degrees; D), and antidiagonal (−45 degrees; A) directions, respectively. The field component along the ydirection can be extracted directly from the projective measurement along V: \({E}_{y}={I}_{V}^{1/2}\). However, we cannot separate the x and z components from the projections: \({I}_{H}={{E}_{x}}^{2}+{{E}_{z}}^{2}\). The field of Eq. (1) can be written as
which can be expressed to within an overall phase as
where \(\delta ={\delta }_{+}{\delta }_{}\). With the above equations we can establish relations between the amplitudes of the z and xcomponents of the field, and extract them from the V and H projective measurements: \({E}_{z}={E}_{y}\sin \,\theta ={I}_{V}^{1/2}\,\sin \,\theta \) and \({E}_{x}={({I}_{H}{I}_{V}{\sin }^{2}\theta )}^{1/2}\). The phase δ is obtained from the D and A projections:
where \(f(\theta )={(1+{\cos }^{2}\theta )}^{2}/(2\,{\cos }^{4}\theta {\cos }^{2}\,\theta +1)\). Thus, our measurement constitutes only a partial measurement of the fields because it uses a priori knowledge of the component beams to obtain the amplitudes of the x and zcomponents of the field. We also used the fringe spacing from the data to obtain an accurate value of \(\theta \). With this information we can reconstruct the field of Eq. (13). However, the expression only gives us an instantaneous value of the field. Our goal is to obtain the polarization ellipse. If we express the electric field in terms of the ellipse semiaxes, then it is given by
where \({\hat{e}}_{a}\) and \({\hat{e}}_{b}\) are unit vectors along the semimajor and semiminor axes of the ellipse, respectively, as shown in Fig. 2, and \({E}_{a}\) and \({E}_{b}\) are real coefficients. The phase γ is related to the “rectification phase” \(\beta ={\tan }^{1}[({E}_{b}/{E}_{a})\tan \,\gamma ]\), which is the angle that the instantaneous field vector makes with the semimajor axis of the ellipse^{9,32}, as shown in Fig. 2. The phase γ can be obtained directly from the field via the relation^{33}
Thus, per Eq. (15), the semimajor and semiminor axis vectors are respectively the real and imaginary components of the product of Eqs (13) and (16) ^{27,33}.
Apparatus
A schematic of the apparatus is shown in Fig. 3. Light from a HeliumNeon laser was spatially filtered, expanded and sent through a MachZehnder type interferometer where a spatial light modulator (SLM) was used as a folding mirror for both beams. Two panes of the SLM encoded spatial modes onto the light in firstorder diffraction, as shown in the insert to Fig. 3. A polarizer insured that the polarization after the SLM was in a vertical plane. A halfwave plate in the path of one of the beams flipped the polarization to horizontal. The beams were recombined forming an angle 2θ by a mirror and polarizing beam splitter. A quarterwave plate converted the polarization states to the circular states. The beams overlapped at a digital camera closely preceded by neutral density filters and a film or wiregrid polarizer. The angle θ was obtained by measuring the fringe density of the interference pattern when both modes were fundamental Gaussian. We also did full polarimetric analysis of the light.
We took data for various angles, with the largest limited only by the camera resolution. We also took data for various topological charges of one of the beams. Images taken after the above mentioned 4 polarizer orientations were used to extract the semimajor axis of the polarization ellipse of every point in the image plane.
We have become aware or recent work measuring ½ Möbius strip in a focused Poincaré beam in Bauer, T. et al.^{34}.
Results and Discussion
Figure 4(a) and (b) show the modeled and measured fringe patterns for \(\ell =1\) at a shallow angle of \(40\pm 1\) arc sec. We also show in pane (c) the modeled 2d view of the semimajor axis of the ellipse along a closed circle shown in pane (a), respectively, which describe a halfturn Möbius strip. Because we graph the semimajor axis vector (\({E}_{a}{\hat{e}}_{a}\)), the evolution of the vector after a closed path yields the initial and final vectors parallel to each other but pointing in opposite directions. In pane (d) of the same figure we show the same quantity as extracted from the data. The colourcoding is labeled at the bottom of the figure. It helps in visualizing the 3d orientation of the semimajor axis: redmagenta and blue/green are used when the vector describing the axis is above and below the observation plane, respectively. The redgreen and magentablue are when the vector has a component that points toward or away the center of the circle, respectively.
The comparison of the two is in excellent qualitative agreement. This agreement is remarkable given the imperfections of the measured optical beams, which suffer aberrations, mode deformations due to the lack of mode purity and unwanted contributions from light for other diffraction orders. The adjustable parameters in the modelings were the relative phase between the two beams \({\delta }_{0}\), angle \(\theta \), and the center of the pattern. The shortcoming is, of course, that we do not make independent measurement of the x and zcomponents of the field and have to rely on knowledge of the geometry of the problem. The method could become more widely applicable to situations where the exact geometry is not known if a method to do independent measurements of the x and zcomponents of the field is devised.
Note in Fig. 4(a) and (b) that the circle along which we do our calculation of the polarization ellipse barely goes over 12 fringes. In doing so, the Möbius strip carries only a half twist. As the circle covers more fringes, the polarization ellipse describes more twists^{27}. We verified this as well, as shown below. The largest angle θ that we investigated was of about one degree, which involved about 7 camera pixels from fringe to fringe, yielding a Möbius strip with about 161.5 turns for a circular path with \(r\simeq w\).
We experimented with other values of the topological charge, as shown in Fig. 5 for a few cases. It can be seen that for the same value of \(\theta =1.63\pm 0.03\) arc min, the cases for \(\ell =1\) and \(\ell =3\) [(a), (b) and (e), (f), respectively] produced Möbius strips of 5/2 and 11/2 twists, respectively, whereas the \(\ell =2\) case [(c) and (d)] involves a ribbon of 3 twists. The 3d views have different perspectives to better visualize the general character of the patterns. We added the vertical polarization projections to show the fringes that give rise to the modeled and measured reconstructions. We note that because the angle is shallow, the inplane (x, y) coordinates are not in the same scale as the outofplane zcoordinate.
Conclusions
In summary, we have confirmed the predicted patterns of twists in three dimensions that the polarization of the light describes when it is formed by the noncollinear interference of singularoptical beams. We confirmed that the polarization ellipse describes Möbius strips for \(\ell \) odd and twisted ribbons for \(\ell \) even^{27}. Should both beams carry optical vortices of charge \({\ell }_{1}\) and \({\ell }_{2}\) the number of twists would be determined by the parity of \(\ell ={\ell }_{1}{\ell }_{2}\) in Eq. (5).
We did these demonstrations at shallow angles to allow measurements to be done with a digital camera. The variations in the polarization are scalable via the angle formed by the propagation vectors of the two beams (2θ). Patterns formed at larger angles, where all components are of comparable magnitude could not be imaged because the polarization fringe density was smaller than current camera pixel sizes. This work shows that noncollinear paraxial beams can be used to produce 3d patterns that can be manipulated by adjusting the beam modes and their relative angle. If the medium were composed of molecules that interact strongly with polarization, these patterns could be used to manipulate them^{35,36}. Beyond the fundamental interest, studies of 3d polarization may have potential for adding polarization encoding to the storage of 3d information. In this work we investigated the most basic situation: two paraxial beams with one of them bearing an optical vortex. The addition of more beams and modes is likely to show new interesting effects, opening a new dimension of complex light.
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Acknowledgements
This work was supported by the National Science Foundation grant PHY1506321. We thank T. Bauer and W. Loeffler for useful discussions and S. Zhang for help.
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E.J.G. did the theory and analysis helped by I.D.; K.B. and J.J.Z. took the data and J.A.J. and B.K. contributed to the analysis and conceptual understanding of the problem.
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Galvez, E.J., Dutta, I., Beach, K. et al. Multitwist Möbius Strips and Twisted Ribbons in the Polarization of Paraxial Light Beams. Sci Rep 7, 13653 (2017). https://doi.org/10.1038/s41598017131991
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