## Abstract

In this paper, we introduce a new kind of vector multi-Gaussian Schell-model (MGSM) beam named radially polarized MGSM beam as an extension of recently introduced scalar MGSM beam [Opt. Lett. **37**, 2970 (2012)], and we obtain the realizability conditions of such beam. The tight focusing properties of a radially polarized MGSM beam passing through a high numerical aperture objective lens are investigated numerically based on the vectorial diffraction theory. It is interesting to find that the intensity distributions of both transverse and longitudinal fields near the focus display flat-top beam profiles due to the special correlation functions of the incident beam. Our results clearly show that engineering the structures of the correlation functions of a radially polarized partially coherent beam provides a novel way for shaping both transverse and longitudinal fields distributions, which have potential applications in particle acceleration and laser machining. Finally, we report experimental generation of a radially polarized MGSM beam with the help of a spatial light modulator and a radial polarization converter.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Study of tight focusing properties of a vector beam passing through a high numerical aperture objective lens is an important research topic in several branches of optics, and the polarization properties of the incident beam has a strong influence on the distributions of the three-dimensional electric and magnetic fields near the focal region [1–4]. Radially polarized beam and azimuthal polarized beam are typical kinds of vector beams with spatially non-uniform states of polarization. When a radially polarized beam is tightly focused by a high numerical aperture (NA) objective lens, a strong longitudinal electric field will be produced near the focus, and the focused beam spot size is much smaller than that of a focused linearly polarized beam. These unique features are useful for imaging of molecules, mapping the orientation of a single molecule, trapping metallic Rayleigh particles, particle acceleration, microscopy, material processing, high-resolution metrology, lithography and free-space optical communications [4–13]. When an azimuthally polarized beam is tightly focused, a strong magnetic field on the optical axis will be generated but the electric field exhibits a doughnut profile with pure azimuthal polarization near the focal region, which is useful for atom guiding [1, 4] and enhancement of Fano resonance in a gold spilt-ring resonator nanoarc structure [14]. By engineering polarization proportion properties of the incident beam, one can produce electric/magnetic field distributions with special patterns near the focal region, which are useful in selectively switching on and off electric/magnetic multipole resonances in single dielectric nanoparticles [15]. In addition, other interesting tightly focused field patterns such as multiple focal spots, needle-like focus and optical chain also can be achieved [16–18]. Besides the polarization, the phase also affects the electric and magnetic field distributions near the focus. Through manipulating the polarization and phase of the incident beam simultaneously, various special focused field patterns can be achieved [19–23]. For example, Wang *et. al* derived a purely longitudinal electric field by focusing a radially polarized beam with phase modulation [19]. Yuan *et. al* reported generation of a nondiffracting transversally polarized light by focusing an azimuthally polarized beam through a multibelt spiral phase hologram [20]. As one of the intrinsic attributes, spatial coherence plays an important role in determining the beam’s focusing properties and the performance in imaging system [24, 25]. The effects of spatial coherence on the intensity distribution and the polarization properties of the tightly focused partially coherent beams were investigated in [26–30], and it was found that the transverse field distribution can be shaped by varying the initial coherence width. However, the correlation functions (i.e., degrees of coherence) of the partially coherent beams in previous literatures on tight focusing are confined to Gaussian functions.

In recent years, partially coherent beams with prescribed correlation functions have been a subject of intense investigation since Gori and collaborators discussed the sufficient conditions for devising physically realizable correlation functions [31–56]. By manipulating the correlation functions of the partially coherent beams in the source plane, one can obtain any pre-established beam profile in the far field, which are useful in material processing, optical trapping, free-space optical communications, optical imaging, and information transfer [31–48]. One intriguing feature induced by correlation function is that partially coherent beam displays strong self-healing ability when scattered by opaque obstacles, e.g., both the beam profile and the state of polarization can be recovered [49]. Furthermore, the information of the orbital angular momentum of a partially coherent vortex beam is hidden in the correlation function in the far field [50], which also displays self-healing ability [51]. Multi-Gaussian Schell-model (MGSM) beam is a typical kind of partially coherent beam with prescribed correlation function introduced by Sahin and Korotkova [38, 39], whose degree of coherence is expressed as a sum of suitably weighted Gaussian functions with different widths and sign-alternating coefficients. Owing to its special correlation function, the far-field beam profile generated by a MGSM beam displays flat-topped shape, which is useful for material thermal processing. The MGSM beam has lower scintillation compared to the conventional partially coherent beam [Gaussian Schell-model (GSM) beam] upon propagation in turbulent atmosphere [52–55], which makes it attractive for free-space optical communications. Vector MGSM beam with spatially uniform state of polarization named electromagnetic MGSM beam was introduced in [56], and the changes of polarization and intensity pattern upon propagation in free space and in turbulent atmosphere were analyzed in detail.

In this paper, we introduce a new kind of vector MGSM beam with spatially non-uniform state of polarization named radially polarized MGSM beam, which combines the properties of non-uniform polarization and special correlation functions together, and we discuss the realizability conditions for such beam. We also explore tight focusing properties of a radially polarized MGSM beam passing through a high NA objective lens numerically, and we find that both transverse and longitudinal field patterns can be shaped by engineering the structures of the correlation functions of the incident beam. At last, we report experimental generation of a radially polarized MGSM beam.

## 2. Radially polarized multi-Gaussian Schell-model beam and its realizability conditions

According to the unified theory of optical coherence and polarization [57], the statistical properties of a partially coherent vector beam-like field, propagating along *z*-axis, can be characterized by the cross-spectral density (CSD) matrix $\overleftrightarrow{W}\left({r}_{\text{1}}\text{,}{r}_{\text{2}}\text{,}\omega ,z\right)$ in the space-frequency domain, where *ω* denotes the angular frequency, **r**_{1} and **r**_{2} are two arbitrary position vectors in the plane of *z*, perpendicular to the propagation axis. In the source plane (*z* = 0), the elements of the CSD matrix are expressed as

_{${E}_{x}\left(r\right)$}and ${E}_{y}\left(r\right)$ represent two orthogonally stochastic electric field components along

*x*and

*y*axis, respectively. The asterisks stand for the complex conjugate and the angular brackets denote an ensemble average. For simplicity, the dependence of the CSD matrix and other derived quantities on

*ω*will be omitted throughout the paper.

Assume that the vector source is a Schell-model source whose degree of coherence (DOC) only depends on the difference of two spatial position vectors. Under this circumstance, the elements of the CSD matrix can be written as

*α*=

*x*,

*y*) denotes

*x-x*or

*y-y*component of the spectral density,${\mu}_{\alpha \beta}$is the correlation function between

*α*and

*β*components of the electric field.

Furthermore, we suppose that the beam source displays radial polarization and its correlation functions are expressed in terms of multi-Gaussian functions. In this situation, the elements of the CSD matrix take the following forms

*M*is the number of Gaussian functions, $\left(\begin{array}{c}M\\ m\end{array}\right)$ stands for binomial coefficients, ${w}_{0}$ is the beam width of the source, ${\delta}_{xx}$, ${\delta}_{yy}$${\delta}_{xy}$are the coherence widths of the correlation functions of

*x*-

*x*,

*y*-

*y*and

*x*-

*y*components, respectively. ${B}_{\alpha \beta}=\left|{B}_{\alpha \beta}\right|\mathrm{exp}\left(i{\varphi}_{\alpha \beta}\right)$ is the complex correlation coefficient between $\alpha $ and $\beta $ component of the electric field with ${\varphi}_{\alpha \beta}$ being the phase difference. The beam source whose CSD matrix elements are given by Eq. (3) is termed as radially polarized MGSM source. When the parameter

*M*equals 1, the beam source reduces to a radially polarized Gaussian Schell-model (GSM) source.

Now, let us discuss the realizability conditions of a radially polarized MGSM source. First, the CSD matrix must be Hermitian conjugate from its definition shown in Eq. (1) [25]. Hence, the following conditions must be met

Second, to be a physically realizable CSD matrix [32], it is fulfilled to have a representation of the following integral form

The functions ${H}_{x}\left(r,v\right)$, ${H}_{y}\left(r,v\right)$ and ${p}_{\alpha \beta}\left(v\right)$ for a radially polarized MGSM source are given as follows

**v**. Thus, the first and the second inequalities in Eq. (8) are always true. Substituting Eq. (11) into the third inequality in Eq. (8), we obtain

Due to fact that the elements of the weighting matrix are all monotonically decreasing functions via **v**^{2}, by setting **v** = 0, we obtain the following inequality easily

*M*. Let us recall such condition for a GSM beam (

*M*= 1) derived by Gori et al. [58], given by

Besides the conditions shown in Eqs. (5) and (17), a radially polarized beam source should obey two additional restrictions on its parameters [59]

(a) Any point in the source plane is linearly polarized, which means that the minor semi-axes of the polarization ellipse equals to zero.

(b) The orientation angle of the polarization at any point in the source plane should satisfies $\theta \left(x,y\right)=\mathrm{arctan}\left(y/x\right)$.

According to [60], the orientation angle *θ* and ellipticity *ε* of the polarization ellipse in source plane can be expressed in terms of the elements of the CSD matrix, i.e.,

*B*. By applying the conditions (a) and (b), one can obtainIt follows from Eq. (23) that ${B}_{xy}={B}_{yx}=1$. Under this condition and combining Eq. (17), one can easily obtain ${\delta}_{xy}\text{=}{\delta}_{xx}\text{=}{\delta}_{yy}={\delta}_{\text{0}}$. Finally, the realibality conditions of a radially polarized MGSM beam can be simplified as

_{xy}The degree of coherence of a vector partially coherent beam is defined as [61]

## 3. Tight focusing properties of a radially polarized multi-Gaussian Schell-model beam passing through a high NA objective lens

The most interesting property of a scalar MGSM beam is that its initial Gaussian beam profile will evolve into a flat-topped beam profile in the far field in free space or in the focal plane when it was focused by a thin lens [38, 39]. In [38, 39], only transverse field distribution is discussed because the longitudinal field distribution can be ignored. What will happen when a radially polarized MGSM beam is focused by a high NA object lens? In this section, we will study the tight focusing properties of a radially polarized MGSM beam, and discuss the influence of correlation functions of the incident beam on the transverse and longitudinal fields distributions.

Let us refer to the scheme of tight focusing of a light beam focused by a high NA objective lens as shown in Fig. 1. According to the Richards–Wolf vectorial diffraction integral, the vectorial electric field in the focal region of a light beam passing through a high NA objective lens can be expressed as follows [1–4]

*r*,

*φ*,

*z*are the cylindrical coordinates of an observation point near the focal region, $\varphi $ is the azimuthal angle of an incident beam,

*θ*is the NA angle, and

*α*is the maximal NA angle given by the formula $\alpha =\mathrm{arcsin}\text{\hspace{0.17em}}{N}_{NA}/{n}_{1}$ with

*N*and

_{NA}*n*

_{1}being the NA number and the refractive index of the surrounding medium, ${k}_{1}=2\pi {n}_{1}/\lambda $ is the wave number with $\lambda $ being the wavelength of the incident beam,

*f*is the focal length of the lens, ${l}_{x}$ and ${l}_{y}$are the pupil apodization functions in the plane of aperture surface and are derived by setting $x=f\mathrm{sin}\theta \mathrm{cos}\varphi \text{\hspace{0.17em}}$and $y=f\mathrm{sin}\theta \mathrm{sin}\varphi \text{\hspace{0.17em}}$ in ${E}_{x}(x,y)$ and ${E}_{y}(x,y)$ of the incident beam, respectively.

Suppose that the incident beam is a radially polarized MGSM beam proposed in section 2, after passing through a high NA objective lens, the statistical properties of the field near the focal region can be characterized by the $3\times 3$ CSD matrix, and the elements is expressed

Applying Eqs. (3), (4), (27) and (28), we can obtain the integral formulae for the elements of the CSD matrix of a tightly focused radially polarized MGSM beam in the focal region. In this paper, we focus on the average intensity distribution of the focused radial polarized MGSM beam in the focal region, therefore, we only give the expressions for the diagonal elements of the CSD matrix in the focal plane, i.e.,

To explore the tight focusing properties of a radially polarized MGSM beam passing through a high NA objective lens, we carry out numerical calculations using above derived formulae. In the following numerical examples, we set *n*_{1} = 1, *N _{NA}* = 0.95,

*f*= 3.0mm,

*ω*

_{0}= 10mm and λ = 632.8nm. We calculate in Figs. 2 and 3 the contour graphs of the total intensity

*I*, transverse intensity

_{total}*I*and longitudinal intensity

_{tra}*I*of the tightly focused radially polarized MGSM beam in the focal plane for different values of beam index

_{z}*M*and coherence width

*δ*

_{0}. The intensity distributions in each row shown in Figs. 2 and 3 are normalized by the maximum value of the total intensity distribution, and the white solid line in Figs. 2 and 3 denote the cross line of the intensity distribution at

*ρ*= 0. One finds from Fig. 2 that the intensity distributions

_{y}*I*,

_{total}*I*and

_{tra}*I*in the transverse plane keep rotational symmetries irrespectively of the beam index

_{z}*M*. The reason is that the intensity distribution, the correlation functions and the polarization distribution of the incident beam all are of circular symmetries. When

*M*= 1,

*I*,

_{total}*I*and

_{tra}*I*all display quasi-Gaussian profiles due to the fact that the correlation functions of the incident beam are of Gaussian distributions [see Figs. 2(a)-2(c)]. It is interesting to find that with the increase of the beam index

_{z}*M*,

*I*,

_{total}*I*and

_{tra}*I*all gradually turn to flat-topped beam profiles [see in Figs. 2(h) and 2(i)], which is due to the effects of the special correlation functions of the incident beam. One can expect to produce other particular longitudinal intensity distributions such as doughnut and lattice profiles through manipulating the correlation functions of the incident beam. The ratio of the energy carried by the longitudinal field to the total energy of the beam in our case is about 50%. From Fig. 3, one sees that the focused intensity distributions are also controlled by the coherence width

_{z}*δ*

_{0}of the incident beam, e.g., with the increase of

*δ*

_{0}, the longitudinal and transverse intensity distributions gradually lose their flat-topped profiles. For the case of

*δ*

_{0}= 1.5mm, the longitudinal intensity distribution still remains a flat-topped profile, while the transverse intensity distribution degenerates to a single-peak profile [see Figs. 3(b) and 3(c)], which implies that the longitudinal field is more resistant to variance of

*δ*

_{0}. When

*δ*

_{0}= 10mm, the distributions of

*I*,

_{total}*I*and

_{tra}*I*are almost the same as those of a tightly focused coherent radially polarized beam [1–4]. Combining Fig. 2 and Fig. 3, we may come to the conclusion that to achieve a longitudinal field with flat-top profile, large beam order

_{z}*M*and small coherence width

*δ*

_{0}are required. Anyway, our numerical results clearly show that both transverse and longitudinal field distributions of a tightly focused radially polarized partially coherent beam can be shaped through manipulating the structures of its correlation functions, which have potential applications in particle acceleration and laser machining. On the other hand, it is known that the phase distribution of a tightly focused vector beam in the focal plane plays an important role from the point view of the light-matter interaction [15], and our numerical results (not presented here to save space) have also revealed that the phase distribution of a tightly focused radially polarized MGSM beam was almost not affected by its correlation functions. More information about the phase distribution of a tightly focused vector beam can be found in [15].

Now, we study the influences of the beam parameters *M* and *δ*_{0} on the behavior of the longitudinal field in the *ρ _{x}-z* plane near the focal region. We calculate in Fig. 4 the contour graph of the longitudinal intensity

*I*of the focused radial polarized MGSM beam in

_{z}*x*-

*z*plane for different values of

*M*and

*δ*

_{0}and the cross lines of the longitudinal intensity distribution at different distances. As illustrated in Fig. 4, the beam parameters

*M*and

*δ*

_{0}seriously affect the spot size of the longitudinal field in the

*ρ*plane. When

_{x}-z*δ*

_{0}= 0.5mm, the depth of focus (DOF) defined as the FHWM of its intensity profile in the longitudinal direction is calculated to be 2.40

*λ*, 3.04

*λ*and 3.74

*λ*for

*M*= 1, 3 and 10, respectively, which means the DOF increases with the increase of the beam index

*M*. The FHWM of the intensity distribution in

*ρ*direction at

_{x}*z*= 0 is calculated to be 2.56

*λ*, 3.44

*λ*and 4.40

*λ*for

*M*= 1, 3 and 10, respectively, implying that the ability for focusing in the transverse plane is weakened as

*M*increases. The cross line of the longitudinal intensity distribution for

*M*= 10 and

*δ*

_{0}= 0.5mm shows that in the plane z = 0.5λ, the beam profile in the transverse plane can still keep flat-topped shape, but in the plane z = λ, the beam profile degenerates to a Gaussian-like shape [see Fig. 4(f)]. As

*δ*

_{0}increases [see Figs. 4(g)-4(i)], both the longitudinal spot size in

*ρ*direction (transverse direction) and that in

_{x}*z*direction decrease, and the intensity distribution in the transverse plane at

*z*= 0 no longer exhibits flat-topped profile as we discussed in Fig. 3. Note that the spot size in

*ρ*direction becomes much smaller than that in

_{x}*z*direction for larger

*δ*

_{0}. Hence, the focal spot in the

*ρ*-

_{x}*z*plane turns to an elliptical shape [see Fig. 4(i)].

What’s more, our numerical results (not present here) also show that when the NA of the lens is low, the longitudinal field near the focus can be neglected, while the transverse field near the focus still can be shaped through engineering the structures of the correlation functions of the radial polarized MGSM beam and it also displays flat-topped beam profile by choosing suitable beam parameters *M* and *δ*_{0}.

## 4. Experimental generation of a radially polarized multi-Gaussian Schell-model beam

In this section, we report experimental generation of a radially polarized MGSM beam. In general, there are several outer-cavity methods for generating a partially coherent beam with prescribed correlations function. The first one is based on the Fourier transform of the intensity distribution involving the dynamic diffuser, i.e., rotating ground grass disk [43]. The second one resorts to random phase screens with designed correlation function with the help of spatial light modulator (SLM) [47]. The third one relies on a deformable mirror to synthesize a partially coherent beam with prescribed correlation function [48]. Here, we adopt the second method to generate a radially polarized MGSM beam with the help of a SLM and a radial polarization converter (RPC) [62], and the experiment setup is shown in Fig. 5. A horizontally polarized beam generated by a ND:YAG laser (λ = 532nm) first goes through a beam expander (BE), then reflected by a beam splitter (BS). The reflected beam from the BS goes towards a SLM, which act as random phase screens controlled by the personal computer. The approach for synthesizing random phase screens with designed correlation function is described in [47]. When we load the random phase screens whose correlation function is of multi-Gaussian functions, the first-order diffraction pattern reflected from the SLM can be regarded as a linearly polarized MGSM beam. After passing through the BS, the diffraction light from the SLM goes through a 4*f*-optical system comprised of two thin lenses (L_{1} and L_{2}) and a circular aperture (CA). Here the CA which is located in the rear focal plane L_{1} (front focal plane of L_{2}) is used to block the unwanted light and other order diffraction patterns except the first-order diffraction pattern. In order to convert the linearly polarized beam to a radially polarized beam, a half-wave plate (HWP) and a RPC are placed in the rear focal plane of L_{2} which is used to rotate the linear polarization to the vertical direction and convert linearly polarized beam to a radially polarized beam, respectively. The output beam from the RPC can be considered as a radially polarized MGSM beam. A charge-coupled device (CCD) is used to measure the intensity distribution or the squared modulus of the degree of coherence of the generated radially polarized MGSM beam. The principle and the procedure for measuring the degree of coherence by a CCD can be found in [34, 35].

Figure 6(a) shows our experimental result of the intensity distribution of the generated radially polarized MGSM beam with *M* = 10 captured by the CCD just behind the RPC. As expected, the intensity distribution displays a doughnut beam shape due to that the state of the polarization in the central point of the beam is undefined, i.e., polarization singularity. The beam width ${w}_{0}$ obtained from the experiment data is about 0.12mm. To verify that the generated beam displays radial polarization, we insert a linear polarizer between the RPC and the CCD, and Figs. 6(b)-6(d) show the experiment result of the intensity distribution captured by the CCD for different transmission axes of the polarizer. One finds from Fig. 6 that the generated beam indeed displays radial polarization. We want to point out that the beam profile of the generated radially polarized MGSM beam just after the RPC keeps almost unchanged if we change the beam parameter *M*. Note that the intensity distribution of the generated beam shown in Fig. 6(a) is somewhat inhomogeneous, which was caused by two factors. One is the protecting glass window located in front of the photo-sensitive area of the CCD, which leads to the interference fringes shown in Fig. 6(a) due to the interference effects between the front and rear surface of the protecting glass window, while this factor doesn’t affect the quality of the focal spot when the generated beam is tightly focused by a high NA lens. Another is the variable phase shifter embedded in the RPC. Such phase shifter has a π-phase jump line that coincides with the *x*-axis. Thus the intensity near the *x*-axis region is slightly weaker than the top or the low region of the beam. Figure 7 shows our experimental result of the squared modulus of the degree of coherence ${\left|\mu \left({r}_{1}-{r}_{2}\right)\right|}^{2}$of the generated radially polarized MGSM beam with *M* = 1 and *M* = 10 just after the RPC. The coherence width *δ*_{0} obtained from the experiment data for *M* = 1 or *M* = 10 is about 0.17mm. To show the influence of the correlation functions on the focusing properties, we also measure the focused intensity distribution of the generated radially polarized MGSM beam focused by a thin lens with focal length *f* = 15cm in the focal plane with *M* = 1 and *M* = 10, and the results are shown Fig. 8. In this case, the longitudinal field can be neglected, and Fig. 8 only shows the transverse field intensity distribution in the focal plane. As expected, the generated radially polarized MGSM beam with *M* = 1 displays quasi-Gaussian beam profile in the focal plane, while the generated radially polarized MGSM beam with *M* = 10 displays flat-topped beam profile in the focal plane.

## 5. Summary

We have introduced a new kind of vector MGSM beam named radially polarized MGSM beam as a natural extension of scalar MGSM beam and discussed its realizability conditions. Through exploring the tight focusing properties of the radially polarized MGSM beam passing through a high NA objective lens, we have found that one can shape both the transverse and longitudinal field distributions near the focus through varying the structures of the correlation functions. It is known that there are several methods in literatures for shaping the transverse filed distribution of a tightly focused beam, while our results have clearly show that engineering the structures of the correlation functions of a radially polarized partially coherent beam provides a novel way for shaping both transverse and longitudinal field distributions, which have potential applications in particle acceleration and laser machining. Furthermore, we have reported experimental generation of a radially polarized MGSM beam.

## Funding

National Natural Science Foundation of China (NSFC) (91750201&11474213); National Natural Science Fund for Distinguished Young Scholar (11525418); Project of the Priority Academic Program Development of Jiangsu Higher Education Institutions; Qing Lan Project of Jiangsu Province; Postgraduate Research & Practice Innovation Program of Jiangsu Province (KYCX17_2024).

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