## Abstract

We develop a method for generating focused vector beams with circular polarization at any transverse plane. Based on the Richards-Wolf vector model, we derive analytical expressions to describe the propagation of these set of beams near the focal area. Since the polarization and the amplitude of the input beam are not uniform, an interferometric system capable of generating spatially-variant polarized beams has to be used. In particular, this wavefront is manipulated by means of spatial light modulators displaying computer generated holograms and subsequently focused using a high numerical aperture objective lens. Experimental results using a *NA* = 0.85 system are provided: irradiance and Stokes images of the focused field at different planes near the focal plane are presented and compared with those obtained by numerical simulation.

© 2014 Optical Society of America

## 1. Introduction

The propagation of electromagnetic field distributions generated at the focal region has been extensively investigated in the last years [1–8]. Non-paraxial fields have demonstrated very useful in many fields for instance in high-resolution microscopy, particle trapping, high-density recording, tomography, electron acceleration, nonlinear optics, and optical tweezers [9]. Beam shaping in the focal area of a high numerical aperture objective lens requires a careful design of the input wavefront. In particular, full control of the complex amplitude and polarization distributions of the paraxial input field is required to generate focused fields adapted to the requirements of a specific problem [10]. Interestingly, several authors described inverse methods to find the pupil function from a predetermined field distribution in the focal area [11, 12]. Light shaping can be accomplished by using an optical setup able to generate beams with arbitrary polarization and shape distributions at a given plane. This is usually carried out by means of interferometric systems in combination with spatial light modulators and digital holography [13–21]. The objective of this paper is to present a method for designing focused fields with transverse circular polarization at any plane. Among many others applications, circularly polarized tight focused beams are useful in resolution improvement [22–24], third harmonic generation-based microscopy [25–27], plasmonics and nano-optics applications [28, 29], optical activity and chemical related problems [30,31] or, conical refraction [32].

Using the Richards-Wolf vector diffraction theory, we derive analytical expressions to describe the propagation of these set of beams near the focal area. Complex amplitude and polarization of the input beam are manipulated by means of spatial light modulators (SLM) displaying computer generated holograms. Numerical calculations and experimental results are compared and analyzed. Accordingly, the paper is organized as follows: in section 2 we derive the equations for describing circularly-polarized focused fields at any transverse plane. The experimental setup and the holographic procedure required to synthesize the beam are reviewed in section 3. Experimental results including irradiance images and polarization analysis are presented in section 4. Finally, the main conclusions are summarized in section 5.

## 2. Circularly-polarized highly focused beams

The electromagnetic field in the focal area of a high numerical aperture objective lens that obeys the sine condition is described by the Richards-Wolf vector equation [33]

*A*is a constant,

*r*,

*ϕ*and

*z*are the coordinates in the focal area, and angles

*φ*and

*θ*are the coordinates at the exit pupil; note that

*θ*

_{0}is the semi-aperture angle. Functions

*f*

_{1}(

*θ*,

*φ*) and

*f*

_{2}(

*θ*,

*φ*) are the azimuthal and radial components of incident field respectively,

**E**(

_{S}*θ*,

*φ*) = (

*E*,

_{Sx}*E*, 0) is the input beam considered transverse and the dot stands for the inner product. Vectors

_{Sy}**e**and ${\mathbf{e}}_{\mathbf{2}}^{\mathbf{i}}$ are unit vectors in the radial and azimuthal directions whereas ${\mathbf{e}}_{\mathbf{2}}^{\mathbf{o}}$ is the projection of ${\mathbf{e}}_{\mathbf{2}}^{\mathbf{i}}$ on the convergent wavefront surface, as shown in Fig. 1. This figure shows the geometrical variables used throughout this paper at different reference surfaces. Vectors

_{1}**e**, ${\mathbf{e}}_{\mathbf{2}}^{\mathbf{i}}$ and ${\mathbf{e}}_{\mathbf{2}}^{\mathbf{o}}$ are given by

_{1}To analyze the polarization structure of **E**(*r*, *ϕ*, *z*), an alternative base of mutually perpendicular unit vectors **u**_{+}, **u**_{−} and **u _{z}** is used

**E**(

*r*,

*ϕ*,

*z*) =

*E*

_{+}(

*r*,

*ϕ*,

*z*)

**u**

_{+}+

*E*

_{−}(

*r*,

*ϕ*,

*z*)

**u**

_{−}+

*E*(

_{z}*r*,

*ϕ*,

*z*)

**u**. According to Eq. (1), components (

_{z}*E*

_{+},

*E*

_{−},

*E*) read

_{z}*E*represent the right (+) and left (−) circular content of the transverse field at the vicinity of the focus plane and

_{±}*E*is the magnitude of the longitudinal component. Since our goal is to generate a focused field whose transverse component is circularly polarized at any plane

_{z}*z*, either

*E*

_{+}or

*E*

_{−}has to be zero. This condition is fulfilled when

*f*

_{1}(

*θ*,

*φ*) = ±

*i*cos

*θ f*

_{2}(

*θ*,

*φ*), which is equivalent to where,

*g*(

*θ*,

*φ*) is an arbitrary function. Additional characteristics of the global field can be obtained by choosing a suitable function

*g*(

*θ*,

*φ*). For example, to obtain a non-zero longitudinal component at the axis implies that

*g*(

*θ*,

*φ*) =

*g*(

*θ*).

In what follows and without loss of generality we choose the plus sign, i.e. we deal with right handed circularly polarized fields. For this kind of incident beams the transverse and longitudinal components of **E**(*r*, *ϕ*, *z*) become

*φ*, field components

*E*

_{+}and

*E*take the form

_{z}*J*

_{0}(

*x*) and

*J*

_{1}(

*x*) are the first kind Bessel functions of order 0 and 1 respectively. Interestingly,

*E*

_{+}presents topological charge

*e*

^{−iϕ}and

*|E*

_{+}|

^{2}and |

*E*|

_{z}^{2}show circular symmetry.

Figure 2 show irradiance maps for (a) the transverse component |*E*_{+}|^{2}, (b) the longitudinal component |*E _{z}*|

^{2}and (c) the total field

*I*= |

*E*

_{+}|

^{2}+|

*E*|

_{z}^{2}when a microscope objective

*NA*= 0.85 (

*θ*

_{0}≈ 1 rad) is used. The illumination is assumed to be Gaussian i.e. $g(\theta )=\text{exp}{\left(-\frac{1}{{f}_{0}}\frac{\text{sin}\theta}{\text{sin}{\theta}_{0}}\right)}^{2}$ and the filling factor

*f*

_{0}is set to 1. Notice that |

*E*|

_{z}^{2}presents high values at

*z*= 0 and drops very fast out of the focal plane. On the other hand, |

*E*

_{+}|

^{2}= 0 at

*r*= 0 at any plane

*z*. Figure 2(d) shows the profiles of the total irradiance

*I*at different distances from the focal plane (

*z*= 0,

*z*= −3

*λ*,

*z*= −5

*λ*and

*z*= −7

*λ*).

## 3. Synthesis of beam E_{S}

_{S}

Figure 3 depicts an experimental setup based on a Mach-Zehnder interferometer able to generate arbitrary spatially-variant polarized focused beams. An extended explanation on how this procedure can be used to generate beams with arbitrary polarization and shape can be found in [17]. A linearly polarized input beam **E*** _{in}* is split into two beams by means of polarizing beam splitter PBS

_{1}. Reflected by mirrors M

_{1}or M

_{2}the split beam (

*E*

_{in}_{1}or

*E*

_{in}_{2}) passes through wave plates HWP and QWP which rotate the oscillating plane and set the modulator to the required desired modulation curve. Then, light passes through a translucent SLM (Holoeye HEO 0017) displaying cell-based double-pixel holograms to encode complex transmittances

*C*(

_{x}*x*,

*y*) and

*C*(

_{y}*x*,

*y*) [34].

Precise alignment of the different optical components is required, especially a good match between the corresponding pixels of the two SLMs. This is carried out during the set up procedure by displaying the same distribution on SLM_{1} and SLM_{2} and imaging these scenes on camera 1. Note that both displays are controlled independently. Then, the scene displayed on one of the screens is shifted until a perfect match with the other one is accomplished. Shift values are used later to adapt the holograms displayed on both SLMs.

These beams are subsequently recombined by means of polarizing beam splitter PBS_{2} and fed into a *4f* system. A spatial filter removes higher-order terms whereas allowing pass the synthesized field **E*** _{S}*. The irradiance of this beam can be observed by means of camera 1. Afterward,

**E**

*is focused by means of a high numerical aperture microscope objective (MO)*

_{S}*NA*= 0.85. The beam in the focal area is reflected on a glass surface and imaged on camera 2. Polarization analysis is carried out by placing a polarizer (and a quarter-wave plate if required) next to camera 2.

According to Eqs. (2) and (6), the synthesized beam **E*** _{S}* has to be

**e**and

_{x}**e**are orthogonal Cartesian unit vectors as shown in Fig. 1. In order to synthesize

_{y}**E**the following complex valued distributions are coded on each SLM It is assumed that the radius of the entrance pupil is set to 1 and

_{S}*ρ*= sin

*θ*is the radial distance from the optical axis at the entrance pupil plane (see Fig. 1).

Figure 4 is a polar diagram displaying the values of the complex plane accessible by the codification method (gray small dots) and the set of physically accessible values by modulator SLM_{1} (red dots). A certain value *C* can be accessed as a combination of phasors *M _{L}* and

*M*, that belong to the modulation curve, and

_{R}*E*and

_{L}*E*, that are diffracted off-axis and removed by the spatial filter at the focal plane of lens L

_{R}_{1}. As shown in this Figure, not all values of the complex plane are accessed by the encoding procedure. This drawback could be overcome using a light source with a shorter wavelength to improve the modulation response of the SLM [35]. However, if the subset of accessible values

*C*within the circle of transmittance

*T*= 0.3 is used (see inset in Fig. 4), almost any complex transmittance can be generated. Non accessible values are approximated to the closest one belonging to the subset.

## 4. Experimental results

As explained in the previous section, the synthesized beam **E _{S}** is focused by means of the objective lens and subsequently reflected on the cover slip, back-propagated through the objective and imaged on camera 2 aided by lens L

_{2}. The cover slip (observation plane) is mounted on a stage that enables to modify the observation distance

*z*. Figure 5 (first row) shows the irradiance

*I*at

*z*= −3.5

*λ*, −5

*λ*and −7

*λ*. Distance

*z*is estimated by comparing the angular average of the experimental images with the numerical evaluation of

*I*and |

*E*

_{+}|

^{2}(Eqs. (8a) and (8b)). These curves are presented in the second row of Fig. 5. Notice that the irradiance at the focal plane

*z*= 0 is not analyzed due to lack of accuracy along the z-axis and insufficient resolution of the camera. Furthermore, the sudden increase in irradiance around the focal plane complicates the analysis, because camera is saturated.

To analyze the polarization of the focused beam a measure of the Stokes parameters has been carried out. These parameters are obtained according to

where*I*(

*α*,

*β*) stands for the recorded intensity when a polarizer is set at an angle

*α*with respect to the

*x*direction in front of camera 2;

*β*is the retardation between the

*x*and

*y*directions [36]. Retardation

*β*=

*π*/2 is accomplished by using also a quarter wave plate. Once the Stokes parameters are found in each point of the beam, the polarization map can be generated. Figure 6 show the Stokes images

*S*

_{0},

*S*

_{1},

*S*

_{2},

*S*

_{3}for the focused field at

*z*= −3.5

*λ*. Notice that the values of images

*S*

_{1}and

*S*

_{2}are very close to zero, whereas the high values present in

*S*

_{3}demonstrates that the field is circularly polarized.

In order to provide global parameters to describe the polarization of the whole beam, the following cumulative values S* _{i}* are introduced:

*k*,

*l*) are the indexes of the pixels of the Stokes image. Table 1 shows the values of S

_{1}, S

_{2}, S

_{3}for the three positions of the observation plane considered. A clear circular character of the beam along the

*z*-axis is recognized since ${\text{S}}_{3}^{2}\gg {\text{S}}_{1}^{2}+{\text{S}}_{2}^{2}$ at any transverse plane

*z*.

## 5. Conclusions

In this paper, a method for generating highly focused beams with circular polarization at any plane is presented. Using the Richards-Wolf diffraction formalism analytical expressions have been developed to design such fields. The analysis of the field in the focal area shows that the irradiance of the longitudinal component present very high values. The use of an interferometric setup for generating beams with arbitrary polarization combined with the use of digital holography techniques has enabled the experimental generation of such beams. Satisfactory practical results have been obtained showing a good agreement between theoretical predictions and the experimental behavior of the beam.

## Acknowledgments

This work was funded in part by the Spanish Ministerio de Ciencia e Innovación, project FIS2010-17543.

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