## Abstract

We present a flexible approach to generate arbitrary vector beams with a trapezoid Sagnac interferometer. With the interferometer, the different orders of two orthogonally polarized beams from computer-generated holograms coincide with each other in Fourier spectrum domain, and coaxially combine into the vector beams. This approach provides convenient way to experimentally study the properties of vector beams with complex polarization.

©2012 Optical Society of America

## 1. Introduction

Optical vector fields, with spatially variant states of polarization, have attracted growing research interests over the past few years, among which the most noticeable are the cylindrically polarized vector beams [1]. Many intriguing phenomena of the tight focused vector beams are discovered due to the inhomogeneous polarization [2–9]. Based on the unique focusing properties [2–5] and novel angular momenta [6–9] associated with the vector beams, people has reported much works with the expectation of exploring new optical effects or phenomena both in scientific and engineering applications, such as surface plasmon excitations [10], superresolution [11], optical tweezers [12], and laser micromachining [13], etc. Up to now, a variety of generation methods of vector fields have been proposed [14–27], and can be mainly clarified into two types: direct and indirect manners. For direct methods, the designed optical elements, such as liquid crystal [14], subwavelength grating [15], quantized Pancharatnam-Berry phase optical elements [16], sectored half-wave plates [17], spiral phase retarder [18], etc., are used to directly convert the polarization of the laser beam to the cylindrical ones. While for indirect methods, the vector fields are composed by combining two traditionally polarized beams via interferometric approaches [19–26]. Among these methods, spatial light modulators are widely used for generating a variety of vector fields due to its tunability and versatility [22–27]. Especially in Refs [22,26], the authors proposed a theoretical description of an arbitrary vector beam, which can be decomposed into two circular polarized beams with opposite handedness generated from the ± 1 order components of a one- or two-dimensional holographic grating. Although they obtained perfect vector beams, there still exits a flaw that the experimental setup needs a particular grating to guarantee the coaxial interference of the two circular polarized beams. If the periods of the grating and the holographic grating are mismatched, the coaxiality is broken.

In this paper, we present an interfering approach for generating arbitrary vector beams with spatially variant polarization based on a trapezoid Sagnac interferometer, with which the coaxial adjustment of the two interfering beams can be easily achieved. The most attractive feature of Sagnac interferometer is that the two interfering beams share the same optical path and elements, so that the output is less affected by the external disturbances [21,24]. Here, we demonstrate two generation ways with one-dimensional (1D) and two-dimensional (2D) computer-generated holograms (CGHs), respectively.

## 2. Experimental setup

The experimental setup for generating vector beams is illustrated in Fig. 1
. A vertically polarized beam from Ar^{+} laser with wavelength of *λ* = 514.5nm is focused on a rotating diffuser (RD, rotating frosted glass plate) and then collimated, with its polarization state rotated by a half-wave (*λ*/2) plate. The RD is used to integrate the changing speckles over time and generate the partially coherent light field with homogeneous intensity distribution. Actually, the RD can be removed from the setup if a high quality of optical field is not required. The input collimated beam is split by a polarizing beam splitter (PBS) into two orthogonally polarized beams, i.e. the vertically (*s*-component) and horizontally (*p*-component) polarized ones. The ratio of the two components is adjusted to be 1:1 by rotating the half-wave plate. In Fig. 1, the PBS and three mirrors (M_{2}, M_{3} and M_{4}) compose a Sagnac interferometer, and the *s*- and *p*-components are combined by the PBS after passing through the same path *a*-*b*-*c*-*d*-*a*. The coaxiality of the two components is easily achieved by adjusting the angle of an arbitrary mirror. Once the coaxiality is achieved, it will be insensitive to the initial conditions of the input beam (such as the incident angle and the collimation degree). A CGH (see the inset of CGH in Fig. 1) is placed at the position *e*, which ensures that the optical paths of the two components after passing through the CGH are equal, i.e. *ad* + *de* = *ab* + *bc* + *ce*. This makes the path consisted of the PBS and the three mirrors has to be a trapezoid rather than a rectangle. If the position of the CGH mismatches the position *e*, an additional phase different *φ*_{0} between *s*- and *p*-components will be attached. As a result, the CGH position can be slightly adjusted around position *e* to control the phase different. After combined by the PBS, the ± 1 orders of the *s*- and *p*-components are filtered through the same open aperture F placed on the Fourier plane of L_{3} (the aperture positions for different CGHs are marked by the dotted circles in Fig. 1), and are respectively converted into the left- and right-hand circularly polarized beams by a quarter-wave (*λ*/4) plate (with its *c*-axis tilted 45° relative to the horizontal). The output field is detected by a CCD camera in the image plane of the CGH. It deserves special mention that, to make sure the optical paths of *s*- and *p*-components after the CGH being equal, the positions of the CGH and the lens L_{3} need to be adjusted until the CGH images on the CCD with the two components are both clear.

To facilitate the adjustment of CGH, we load the CGH on a transmissive spatial light modulator (SLM) with 1024 × 768 pixels in gray scale. It is specially noted that the SLM can change the polarization states of the *s*- and *p*-components. Although the two components could be changed back into the *s*- and *p*-polarizations via the PBS, the polarization conversions of the two components are not equivalent, finally leading to the distortion of the superposed polarizations. To avoid this distortion, two film polarizers are posted on both facets of the SLM, with the polarization axis at a 45° angle to the horizontal. As a result, the intensities of *s*- and *p*- components are both weakened 75%, and the generating efficiency of vector beam is greatly reduced. In general, a designed CGH etched on the glass or captured in the film is much more efficient.

## 3. Results and discussions

#### 3.1 Generating with one-dimensional computer-generated holograms (CGHs)

For a 1D CGH, the amplitude transmission function is depicted as Ref [22], i.e.

where*δ*is the additional phase distribution, and

*D*is the grating constant of the CGH.

For the same CGH, the *s*-component reads *t _{s} = t*, while the

*p*-component reads

*t*[1 + cos(2π

_{p}=*y*/

*D*+

*δ*

_{−})]/2, where

*δ*

_{−}=

*δ*(−

*x*,

*y*). After passing through the trapezoid Sagnac interferometer, the ± 1 orders of the

*s*-component just overlap that of the

*p*-component, respectively (see the inset of Fourier spectrum from 1D CGH in Fig. 1), with phase difference

*φ*

_{0}. Only one order, for instance the + 1 order [of which the phase distributions for the

*s*- and

*p*-components are exp(i

*δ*) and exp(i

*δ*

_{−}+ i

*φ*

_{0}), respectively], is allowed to pass through. After converted to circular polarizations with

*λ*/4 plate, the optical fields of the two components can be expressed as

*A*

_{0}is the constant factor. Then the superposed field is

*δ*of the CGH and adjusting the phase difference

*φ*

_{0}, arbitrary spatially variant polarization beams can be obtained. However, an unexpected phase distribution exp[i(

*δ*+

*δ*

_{−}+

*φ*

_{0})/2] is attached to the generated beam. To avoid the influence of additional phase distribution, the phase function is designed to satisfy

*δ*

_{−}= −

*δ*+

*δ*

_{0}, where

*δ*

_{0}is the additional constant phase. For example, when

*δ = mφ*+

*δ*

_{0}(where

*φ*is the azimuth angle, and

*m*is the topological charge),

*δ*

_{−}= −

*mφ*+

*m*π +

*δ*

_{0}, then a cylindrical vector beam [cos(

*mφ*-

*m*π/2-

*φ*

_{0}/2), sin (

*mφ*-

*m*π/2-

*φ*

_{0}/2)]

^{T}is formed.

Figure 2
shows the experimental results of four kinds of vector beams generated with designed 1D CGH, where Figs. 2(a,b) and 2(c,d) represent single- and double-modes, respectively, with the theoretical polarizations marked by arrowheads. For the single-modes of Figs. 2(a) and 2(b), we select the parameters as *δ = φ*, *φ*_{0} = -π, and *δ =* 2*φ*, *φ*_{0} = 0, respectively. While for the double-modes, there contain two types of polarization states separated by the circular line *r* = *r*_{0}/2 (*r* is the radial coordinate, *r*_{0} is the radius of the circular field). In Fig. 2(c), the parameters of CGH are *δ =* -*φ* and *φ* for inside- and outside-modes, respectively, and *φ*_{0} = -π. In Fig. 2(d), the parameters are *δ =* 2*φ* and *φ* for inside and outside modes, respectively, and *φ*_{0} = 0. The top of Fig. 2 shows the vector beams with homogeneous intensity distributions. Due to the singularity of the polarization, the center of each field presents a dark spot. Especially in the light fields of double-modes, the regions of singular polarizations present dark rings. To analyze the polarization of the generated fields, polarization analyzers (see Fig. 1) with the axis at 0° and 45° angle to the horizontal are employed, and the corresponding intensity distributions are depicted in the middle and bottom of Fig. 2, respectively. The results are in good agreement with the theoretical polarizations. Furthermore, if we replace the *λ*/4 plate with a *λ*/2 plate, and set the *c*-axis on a 22.5° angle relative to the horizontal, a special vector beam with hybrid polarization [cos(*mφ*-*m*π/2-*φ*_{0}/2),i sin (*mφ*-*m*π/2-*φ*_{0}/2)]^{T} can be obtained [25].

#### 3.2 Generating with two-dimensional CGHs

The trapezoid Sagnac interferometer can also be used to generate vector beams with variant polarization and phase distributions, as proposed in Ref [26]. For this purpose, we employ a 2D CGH with amplitude transmission function

*δ*

_{2−}(

*x*,

*y*) =

*δ*

_{2}(−

*x*,

*y*). The 2D CGH can be considered as the superposition of two orthogonal oblique 1D CGHs [see the top of Figs. 3(a) and 3(b), with the amplitude transmission function

*t*

_{1}

*=*{1 + cos[2π(

*x*+

*y*)/

*D*+

*δ*

_{1}]}/4 and

*t*

_{2}

*=*{1 + cos[2π(

*x*-

*y*)/

*D*+

*δ*

_{2−}]}/4, respectively]. In particular, the ± 1 orders of the Fourier spectrum from the 2D CGH can be considered as the superposition of the spectra from the two 1D CGHs. The Fourier spectra of the two oblique CGHs (with ± 1 orders of the

*s*- and

*p*-components denoted by ± 1

*and ± 1*

_{p}*, respectively) are illustrated in the bottoms of Figs. 3(a) and 3(b), respectively. It reveals that the*

_{s}*p*-polarized ± 1 order from the CGH

*t*

_{1}coincides with the

*s*-polarized ± 1 order from

*t*

_{2}. Here, the influence of the phase difference

*φ*

_{0}is identical to the case with 1D CGH. To simplify the discussion, we adjust the phase difference

*φ*

_{0}as a constant, e.g.

*φ*

_{0}= 0. Filtered by an open aperture in the Fourier plan, the + 1 orders of the

*s-*and

*p-*components from the 2D CGH [with the phase distributions exp(i

*δ*

_{1}) and exp(i

*δ*

_{2}), respectively] pass through and translate into left- and right-hand circularly polarized beams, i.e. the two components can be respectively expressed as

Then the superposed field is

Figures 3(c)-3(e) depict three types of vector beams generated by the 2D CGHs, where the top presents the light intensity distributions (with the local polarization directions marked by white arrowheads), the middle and bottom present the intensity distributions of light fields passing through the polarization analyzers with the horizontal and vertical axes, respectively. The double-mode vector field carrying a helical phase exp(i*φ*), as shown in Fig. 3(c), is generated with *δ*_{1} *=* 2*φ* + π/2, *δ*_{2} *=* −π/2 for the inside mode and *δ*_{1} *=* 3*φ*, *δ*_{2} *=* −*φ* for the outside mode. Since the phase structures of these vector beams have been verified [25], here we will no longer discuss them. Figures 3(d) and 3(e) depict two vector beams with the polarization varied with the radial coordinate *r*, and the parameters for CGHs are *δ*_{1} *= φ* + π, *δ*_{2} *=* 2.5π*r*/*r*_{0} (*r*_{0} is the radius of the circular field) and *δ*_{1} *=* 3π*r*/*r*_{0}, *δ*_{2} *=* −3π*r*/*r*_{0}, respectively. The polarization state in Fig. 3(d) has azimuthal and radial dependence, similar to the phase structure of the helical-conical beam, which presents peculiar focusing property [28]. In Fig. 3(e), the polarization state is radially varying, and can possibly carry a novel angular momentum [8]. The intensity distributions from the analyzers conform to the theoretically calculated polarizations.

In addition, we can also obtain the vector fields with the interference between two linear polarized beams. In the setup shown in Fig. 1, the *λ*/4 plate is removed, and an additional intensity modulation is attached to the 2D CGH, of which the amplitude transmission function is *t*(*x*,*y*) *=* {2 + *A*_{1}cos[2π(*x* + *y*)*/D* + *δ*_{1}] + *A*_{2}cos[2π(*x*−*y*)*/D* + *δ*_{2}]}/4. Then the superposed field is proportional to *A*_{1}exp(i*δ*_{1})[1,0]^{T} + *A*_{2}exp(i*δ*_{2})[0,1]^{T} *=* exp(i*δ*_{1})[*A*_{1},*A*_{2}exp(i*δ*_{2}-i*δ*_{1})]^{T}. It can describe an arbitrary (including linear, circular and elliptical) polarization distribution. For a radially polarized beam, *A*_{1} *=* cos*φ*, *A*_{2}exp(i*δ*_{2}-i*δ*_{1}) *=* sin*φ*. However, due to the effect of aperture filter, some spectral information of the CGH lost, and the intensity modulation (*A*_{1} and *A*_{2}) cannot be exactly reconstructed via the holography. As a result, the vector beams are difficult to be perfectly created. Especially with partially coherent light, the distortion of reconstructed light field is more intense. Thus, we remove the rotating diffuser, and obtain two vector beams as shown in Fig. 4
, where 4(a) and 4(b) correspond to the radial [see Fig. 2(a)] and radial-variant [see Fig. 3(e)] polarizations, respectively. The transmission functions of the CGH used in Figs. 4(a) and 4(b) are *t*(*x*,*y*) *=* {2 + cos*φ* cos[2π(*x* + *y*)*/D*] + sin*φ* cos[2π(*x*−*y*)*/D*]}/4 and *t*(*x*,*y*) *=* {2 + cos(3π*r*/*r*_{0})cos[2π(*x* + *y*)*/D*] + sin(3π*r*/*r*_{0})cos[2π(*x*−*y*)*/D*]}/4, respectively. From the analyzer, the intensity distributions basically meet the theoretical polarization, with some local regions ambiguous.

Here, we emphasize that the coincidence in the Fourier spectrum domain of different orders of two polarization components from CGHs merely happens in the trapezoid Sagnac interferometer rather than in the triangular one. In the interferometer, the amplitude transmission functions of the CGH respectively read by *s*- and *p*-components are mirror symmetric with each other. Besides, the wavefront of the beam will be flipped horizontally during each reflection. To make the outputs of *s*- and *p*-components have reversed wavefronts, an odd number of mirrors are needed. If, for instance, the trapezoid Sagnac interferometer is replaced by a triangular one, the output field from *s*- and *p*-components are exactly the same both for 1D and 2D CGHs, and the superposed field is always linearly polarized. Therefore, trapezoid Sagnac interferometer should be an optimized one for our generating method.

## 4. Conclusion

In summary, we have presented a flexible approach to generate arbitrary vector beams with a trapezoid Sagnac interferometer. With the interferometer, the different orders of two orthogonally polarized beams from CGHs coincide with each other in Fourier spectrum domain, and coaxially combine into the vector beams. This coincidence merely happens in the trapezoid Sagnac interferometer rather than in the triangular one. Furthermore, to improve the efficiency of vector beam generation, a designed CGH etched on the glass or captured in the film need to replace the SLM. This approach has the advantages of easy adjustment and good stability. It provides convenient way to experimentally study the properties of vector beams with complex polarization.

## Acknowledgments

This work was supported by the 973 Program (2012CB921900), the Natural Science Basic Research Plan in Shaanxi Province of China (2012JQ1017), the Northwestern Polytechnical University (NPU) Foundation for Fundamental Research (JC20120251), and the Technology Innovation Foundation of NPU (2011KJ01011).

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