## Abstract

We present a versatile, non-interferometric method for generating vector fields and vector beams which can produce all the states of polarization represented on a higher-order Poincar*é* sphere. The versatility and non-interferometric nature of this method is expected to enable exploration of various exotic properties of vector fields and vector beams. To illustrate this, we study the propagation properties of some vector fields and find that, in general, propagation alters both their intensity and polarization distribution, and more interestingly, converts some vector fields into vector beams. In the article, we also suggest a modified Jones vector formalism to represent vector fields and vector beams.

© 2012 Optical Society of America

## 1. Introduction

A Bessel beam “heals” itself when obstructed by an obstacle [1]; an Airy beam “bends” as it propagates [2]; an optical-vortex beam rotates micron-sized particles when focused [3]. These seemingly strange properties result from judiciously tailoring the phase and the amplitude of an optical field. Similarly, optical vector fields [4], which exhibit a (spatially) nonuniform polarization distribution, can be engineered to either achieve specific tasks or exhibit exotic properties. In particular, vector fields that are beam-like solutions to the paraxial wave equation, referred to as vector beams, have been the subject of much research activity because of their importance to fields ranging from high-resolution microscopy [5–8], surface chirality determination [9], real-time polarimetry [10] to facilitating light-matter interaction at the nanoscale [8, 11, 12]. The most common types of vector beams are radial and azimuthal, where the electric-field vectors align radially and azimuthally about the beam axis, respectively. It is well known that the radial vector beam results in a significant on-axis longitudinal electric-field component upon strong focusing [5], while the azimuthal vector beam has an analogous effect for the magnetic-field component. Unfortunately, unlike the aforementioned Bessel and Airy beams, the exploration of the properties and potential applications of vector fields has been slow. This is primarily due to the lack of a convenient method of generation that satisfies the requirements of versatility in generating a diverse set of vector fields and stability. For example, the vast majority of generation techniques employ interferometry, which often is hampered by phase stability [13–15]. The handful of non-interferometric approaches [16, 17] (even the commercial systems), while stable, are often limited by versatility. A practical example where a versatile and stable vector field generator would be useful is in the creation of the flat top beam, which has been shown to be useful to a diverse set of applications requiring uniform illumination ranging from material processing [18] to improving the stability of the interferometers used to detect gravitational waves [19]. A flat top beam can be created by linearly combining the radial and azimuthal vector beams [18], at the expense of using interferometers.

In this paper, we present a novel approach to generating vector fields and vector beams that is both versatile and stable. We take advantage of the convenience of the method to generate several example vector fields, and to study the basic propagation properties of some of them. In particular, we demonstrate both numerically and experimentally that, in general, the intensity and polarization state of vector fields evolves with propagation, which may be of use in various fields including micro- and nanofabrication; we show that some of these fields become beam-like either upon traveling a sufficiently long distance or from conventional spatial filtering. We conclude by showing that our technique generates all the states of polarization represented on a recently proposed higher-order Poincar*é* sphere (HOPS) [20], adapted to describe vector fields.

## 2. Theory and experimental setup

To understand how we achieve versatile and stable vector-field generation, we begin by recalling a basic fact about nematic liquid crystal spatial light modulators (NLC-SLM). That is, the initial polarization state of an optical beam is altered by an NLC-SLM if the beam’s initial polarization makes a projection on both the fast and slow axes of the NLC-SLM [21]. Thus, for a well characterized NLC-SLM, this property can be exploited by the appropriate optical setup to effect the desired change in the polarization. In our scheme, the liquid crystal molecules of the NLC-SLM are aligned vertically with respect to the laboratory reference frame (the common reference henceforth used in this paper). Changing the control voltage applied to each pixel of the SLM changes the path length traversed by the vertically polarized component of an input optical field, and thus the polarization at each point of the field corresponding to a particular pixel would change accordingly; the orthogonal component is not affected by the applied voltage.

Figure 1 illustrates the concept of vector field generation. We begin by restricting ourselves to using an input beam that is right circularly polarized (RCP); the effect of an input beam with arbitrary polarization will later be analyzed. For an RCP input beam, the output polarization state in the Jones vector representation [22], can be written as a 2 × 1 column vector [(*cos* Φ/2) *sin*(Φ/2)]^{⊤} [23], where Φ is the phase retardation imparted by the SLM, and the first and second elements of the vectors represent components of the field along the horizontal (*x̂*) and vertical (*ŷ*) axes. This vector describes the output polarization at a point in the resultant field’s cross-section. Note that using the Jones calculus to describe the polarization distribution across the field cross-section requires some care. First, unlike the standard case of an optical field with homogeneous (uniform) polarization distribution, the application of the Jones calculus to a field with inhomogeneous polarization requires that the Jones vector be modified to be a function of position. Second, the phase difference between different points in a field’s cross-section must also be considered. With these constraints, the effect of the phase plate upon the beam can be represented as

*φ*is the azimuth angle on the beam, and

*ρ*is the angle the boundary of the phase transition on the phase plate makes with respect to

*x̂*. The term

*sgn*(

*ρ*−

*φ*)

*sgn*(

*ρ*+

*π*−

*φ*) ensures that the beam sees a phase step for azimuth angles between

*ρ*and

*ρ*+

*π*only. It should be noted that the phase term is a function of the position and cannot be discarded. To obtain a desired polarization distribution one has to carefully choose the spatial distribution of the retardation and the orientation of the phase plate. For a linear polarization distribution that possesses radial symmetry (such as in the case with radial and azimuthal vector beams), one can use a retardation value given by (2

*φ*+ 2

*α*)

*mod*2

*π*, where

*α*is the orientation of the polarization at

*φ*= 0. Hence, the Jones vector of the field at the output of the quarter-wave plate (QWP) can be written as

In this description, a polarization distribution identical to a radial beam can be obtained for *α* = 0 and that for an azimuthal beam can be obtained for
$\alpha =\genfrac{}{}{0.1ex}{}{\pi}{2}$. This approach can describe any state of linear polarization. To generate polarization states possessing ellipticity the QWP in Fig. 1 can be removed. Under this condition, the Jones vector in Eq. (2) becomes complex

The optical field obtained at the output of the QWP are vector fields. Some of these fields can be converted to vector beams using spatial filtering or long distance propagation as shown in the next section.

## 3. Results and discussion

In Fig. 2 we show the versatility of our system by generating a variety of vector fields (shown in red). A polarization analyzer (Thorlabs, LPNIR050-MP), followed by a CCD camera (Watec, WAT-902H), is placed at the output of the QWP (the analyzer and CCD are not shown in the figure). Figures 2(a)–2(c) represents three single-mode fields, i.e., those possessing uniform radial symmetry in polarization from the field center to the periphery, whereas Figs. 2(d)–2(f) represents three hybrid fields with radial distribution at the center (core). We observed that both the intensity and polarization distribution of the vector fields in Fig. 2 evolve with propagation, thus re-iterating that the fields at the output of the QWP are not beam-like. Note that the QWP at the output is used to obtain all the states except for that shown in Fig. 2(c) which was obtained by removing the QWP and setting *α* to 0. The top row in the figure shows the respective phase image used for the generation of each vector field.

To understand vector fields’ propagation properties, we carried out numerical simulations using the angular spectrum representation [11, 12] as well as experiments. Specifically, we chose two types of fields: a single-mode field with an azimuthal polarization distribution, Fig. 2(b), which we call quasi-azimuthal, and a hybrid-mode field with a radial polarization distribution at the core and an azimuthal polarization distribution at the periphery, Fig. 2(e), which we call radial-azimuthal. Both the total intensity and the projection through a 135° orientation of the analyzer (measured relative to horizontal axis) at propagation distances of 0, 10, 100, and 250 cm were recorded as shown in Figs. 3(a) and 3(b), column i. In column ii of Figs. 3(a) and 3(b), we show the corresponding simulations for both fields at these distances. For these simulations we assume that the phase plate imparts both a phase and an amplitude error of 0.95*e ^{i}^{π}*

^{/6}in the vicinity of the phase step. In Figs. 3(a) and 3(b), column iii, we show results of the simulation assuming an ideal phase plate with no error. As can be seen from Fig. 3(a), upon propagation both the quasi-azimuthal and radial-azimuthal fields converge to a doughnut shape. The simulated values and the experimentally observed data are in good qualitative agreement. Further, we observe that the polarization distribution for both of these fields resembles that of the conventional azimuthal vector beam after propagation, with the accuracy of the approximation improving with increasing distance. This is confirmed in Fig. 4(a) for a simulation of the behavior of both field types for 30 m of propagation, where the left and right columns show the results for the quasi-azimuthal and radial-azimuthal fields, respectively. The top row shows the total intensity, and the bottom row shows the results of projection through an analyzer at 135°. The conversion to a beam-like solution after long distance propagation is interesting and results from the portion of the field that does not satisfy the beam-like property diffracting more strongly away from the optical axis compared to the portion of the field satisfying the beam like property. The fact that the radial-azimuthal field converts to an azimuthal beam upon propagation is also interesting. To appreciate the reason behind this we used the angular spectrum approach and found that the portion of this field near its geometric center has a larger divergence than the portion closer to the periphery which signifies that the center of the field is lost due to diffraction leaving behind a polarization distribution that resembles that of its peripheral region. The conversion of the vector fields into conventional vector beams upon propagation is an interesting and important behavior. Most vector beam generation techniques employ the use of conventional spatial filtering to achieve beam-like solutions. Our approach provides an alternative. Figure 4(b) shows the effect of conventional spatial filtering on the quasi-radial beam. In the figure, column i and ii represent the experimental and simulated results taking non-ideality of the phase plate in account, respectively. Top row in each case shows total intensity of the beam whereas bottom row represents the projection of the beam through 135° oriented analyzer. The experimental results are in good agreement with the simulations. It is expected that a phase plate with graded index profile will reduce the diffraction resulting from the phase step, thereby leading to better results. In general, the deviation from the ideal cases is due to both pixelation and the finite phase resolution of the NLC-SLM, but, as can be seen from Figs. 3 and 4(b), reasonable approximations can be experimentally achieved.

It has recently been shown that a HOPS representation could be used to describe vector beams [20]. We adapt this to describe those vector fields which possess polarization symmetries similar to that of the vector beams by noting that the representation used in [20] only specifies the polarization distribution and not the intensity distribution. Next, we investigate the type of beams that can be generated by our setup. At first, we only consider an RCP input beam to the NLC-SLM. Expanding Eq. (2) and collecting similar terms for *sin*(*α*) and *cos*(*α*) we get *J* (*φ*) = *cos*(*α*) *V _{l}* +

*sin*(

*α*)

*H*, where

_{l}*V*=

_{l}*cos*(

*φ*)

*x̂*+

*sin*(

*φ*)

*ŷ*, and

*H*= −

_{l}*sin*(

*φ*)

*x̂*+

*cos*(

*φ*)

*ŷ*are the bases with topological charge

*l*= 1, as shown in Eqs. (4) and (5) in [20]. In the orthonormal circular polarization basis [20] we can represent the above field as

*J*(

*φ*) = [

*sin*(

*α*) −

*icos*(

*α*)]/2

*R*+ [

_{l}*sin*(

*α*) +

*icos*(

*α*)]/2

*L*, which, according to Eqs. (8)–(11) in [20], results in four Stokes vector elements of the form ${S}_{0}^{1}=1/2$, ${S}_{1}^{1}=\mathit{cos}(\alpha )/2$, ${S}_{2}^{1}=\mathit{sin}(\alpha )/2$, and ${S}_{3}^{1}=0$. These values relate to the coordinates for the azimuth angle $2\theta ={\mathit{tan}}^{-1}\left({S}_{2}^{1}/{S}_{1}^{1}\right)=\alpha $ and the latitude angle $2\phi ={\mathit{tan}}^{-1}\left({S}_{2}^{1}/{S}_{0}^{1}\right)=0$ on the HOPS. This indicates that for an RCP input to our system, the output states are limited to the equator of the HOPS. However, if the QWP in the outupt arm is removed, points outside the equator can be generated. This can be seen by starting with Eq. (3) and performing an analysis similar to the one followed for Eq. (2). Here, we add the caveat that Eqs. (12) and (13) in [20] use definitions of azimuth and latitude angles which are interchanged when compared to the generally used definitions in polarization optics [22, 24].

_{l}The results described in this paper have been for an RCP input beam. Now, we analyze the effect of an elliptically polarized input beam of the form *x̂* + *ibŷ*, where *b* is a complex coefficient relating to the weights of the orthogonal polarization components [25]. Such a beam can be generated by using a polarizer followed by an arbitrarily oriented QWP. Upon reflection from the NLC-SLM we obtain an output polarization proportional to [*e ^{i}*

^{Φ}

*ibe*

^{−}

^{i}^{Φ}]

^{⊤}; this beam, when passed through a QWP with its fast axis at 135°, results in an output polarization proportional to [

*e*

^{i}^{Φ}+

*be*

^{−}

*−*

^{iϕ}*ie*+

^{iϕ}*ibe*

^{−}

^{i}^{Φ}]

^{⊤}. For insight, we choose to rearrange this output polarization to the form

*bR*+

_{l}*L*, and observe that this resembles the equation of a general vector beam in a basis of right and left-hand circular polarization, as shown in Eq. (1) in [20]. It has been shown that this equation represents any point on the HOPS with a topological charge of 1 [20], and thus we conclude that our setup can also generate all such states. It is worth highlighting that

_{l}*V*and

_{l}*H*are functions that describe the radial and an azimuthal vector beams, respectively. Since any state represented in the

_{l}*R*and

_{l}*L*basis can be represented in the

_{l}*V*and

_{l}*H*basis, all states on the HOPS result from a weighted superposition of radial and azimuthal vector fields.

_{l}## 4. Conclusion

We have presented a non-interferometric technique to generate arbitrary vector fields and vector beams possessing topological charge of *l* = ±1. Our approach is versatile and can generate any polarization distribution that corresponds to a coordinate on the higher-order Poincar*é* sphere, with basis functions of radial and azimuthal polarization. Fields with a higher topological charge can be generated by using phase plates with more segments. The stability resulting from our approach is expected to facilitate the use of vector beams in quantitative studies in metrology of anisotropic materials [10], microscopy [5], nanophotonics [11], and atomic physics [26]. As an illustrative example of the usefulness of the technique we analyzed the effect of propagation on some vector fields and interestingly found that, in general, these fields change polarization with propagation–a property of potential interest for systems requiring spatio-polarization-encoded activation for specific reaction process such as in cross-linking of polarization sensitive polymers for three dimensional fabrication. The experiments also showed that some vector fields convert to vector beams upon propagation which presents an alternative to the conventional spatial filtering in vector beam generation.

## Acknowledgments

The authors acknowledge Dr. Scott Kim and Lattice Electro Optics for custom fabrication of the phase plate. This work was supported in part by the University of Illinois at Urbana-Champaign and by the NSF CAREER award ( NSF DBI 09-54155).

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