Super cosh-Gauss nonuniformly correlated radially polarized beam and its propagation characteristics

Xinlei Zhu; Jiayi Yu; Fei Wang; Yahong Chen; Yangjian Cai; Yangjian Cai; Yangjian Cai

doi:10.1364/OE.468349

1. Introduction

Polarization is an intrinsic characteristic of a light field, which describes the amplitude and phase difference between two orthogonal electric field components, perpendicular to propagation axis in paraxial region. Over the past two decades, light beams with spatially varying polarization states (known as vector beams) have received considerable attention owing to their rich physical connotations and important applications in optical communications, optical manipulations, and material processing [1–3]. Radially polarized (RP) beam is perhaps one of the best-known vector beams [4]. The state of polarization (SOP) across beam cross section is linearly polarized, but the polarization direction at any point coincides with the radial direction. Due to such peculiar polarization distribution, the tightly focused RP beam will produce a strong longitude electric field beyond Gaussian beam diffraction limit, which has potential applications in super-resolution microscopy, trapping metallic particles, and optical guiding [5–8]. The RP beam was also found the much stronger ability to resist the turbulence-induced scintillation than that of a Gaussian beam, making it an ideal source from free-space optical communications [9].

In addition to polarization, spatial coherence is another salient property of the light field, describing the correlation of the optical fields at two or more spatial points. In 1994, James revealed that the polarization and spatial coherence of light beams are interacted during propagation [10], i.e., the coherence induces the changes of polarization state. Gori proposed 2 × 2 beam coherence polarization matrix in space time domain to characterize coherence and polarization of partially coherence vector beams (PCVBs) simultaneously [11]. Later, Wolf established the unified theory of coherence and polarization by applying the cross spectral density (CSD) matrix in space frequency domain [12]. Due to the pioneer work of Gori and Wolf, much progress has been achieved in the generation, propagation, and applications of the PCVBs [13–22]. By decreasing the spatial coherence of the RP beam, named partially coherent RP beam, it was found that the beam is de-polarized under propagation, but the SOP remains unchanged [23,24]. In addition, the partially coherent RP beam is more resistant to the turbulence disturbs compared to its coherent counterpart [25,26]. Recently, the control of coherence structure of partially coherent light has found a variety of applications [27–29]. Through modulating the coherence structure of the partially coherent RP beam, it not only provides a convenient way to shape the intensity profiled of the longitude field in tightly focused region [30], but also exhibits robust propagation properties, immune to opaque apertures and scattering objects in propagation path [31]. Hence, the spatial coherence gives a new freedom to manipulate the RP beam. Recently, the study of the partially coherent RP beam was extended to the beam with non-uniformly coherence states [32,33].

In this study, we introduce a new kind of PCVBs with radial polarization, named super cosh-Gauss nonuniformly correlated radially polarized (SCNRP) beam. The propagation characteristics including the spectral density, degree of polarization (DOP), and SOP in the source plane and during free-space propagation are investigated. The beam exhibits the self-focusing property and the invariance of DOP and SOP on propagation. We carry out an experiment to nearly real-time generate the SCNRP beam with controllable beam parameters. The intensity distribution and polarization properties of the generated beam are experimentally examined.

2. Theory

2.1 Theoretical model for a super cosh-Gauss correlated nonuniformly radially polarized beam

In this section, we first theoretically construct a kind of PCVBs, named double-H PCVBs whose kernel function is composed of the summation of two exponential functions. Then, as a subclass of double-H PCVBs, the SCNRP beam is introduced.

Consider a quasi-monochromatic, stochastic electromagnetic beam-like field, propagating along z-axis in the half-space (z > 0). According to the unified theory of optical coherence and polarization [12], the second-order statistics of such PCVB in the source plane (z = 0) may be characterized by 2×2 CSD matrix in space frequency domain

(1)$$\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over W} ({{{\mathbf r}_1},{{\mathbf r}_2}} )= \left[ {\begin{array}{cc} {{W_{xx}}({{{\mathbf r}_1},{{\mathbf r}_2}} )}&{{W_{xy}}({{{\mathbf r}_1},{{\mathbf r}_2}} )}\\ {{W_{yx}}({{{\mathbf r}_1},{{\mathbf r}_2}} )}&{{W_{yy}}({{{\mathbf r}_1},{{\mathbf r}_2}} )} \end{array}} \right],$$

with elements

(2)$${W_{\alpha \beta }}({{{\mathbf r}_1},{{\mathbf r}_2}} )= \left\langle {E_\alpha^\ast ({{{\mathbf r}_1}} ){E_\beta }({{{\mathbf r}_2}} )} \right\rangle ,\begin{array}{cc} {}&{({\alpha ,\beta = x,y} )} \end{array}$$

where r_i= (x_i, y_i), (i =1, 2) are two transverse position vectors in the source plane. E_x and E_y denote two mutually orthogonal components of the electric field realizations. The asterisk and angle brackets represent the complex conjugate and ensemble average over the field fluctuations, respectively. To guarantee a genuine CSD matrix, the necessary and sufficient condition [34,35] is that the elements can be written as the following integral formula

(3)$${W_{\alpha \beta }}({{{\mathbf r}_1},{{\mathbf r}_2}} )= \int_{ - \infty }^\infty {{p_{\alpha \beta }}(v )H_\alpha ^\ast ({{{\mathbf r}_1},v} ){H_\beta }({{{\mathbf r}_2},v} )dv} ,$$

where p_αβ(v) are the elements of the weighting matrix, i.e.,

(4)$$\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over p} (v )= \left[ {\begin{array}{cc} {{p_{xx}}(v )}&{{p_{xy}}(v )}\\ {{p_{yx}}(v )}&{{p_{yy}}(v )} \end{array}} \right],$$

and they must satisfy the inequalities:

(5)$${p_{xx}}(v )\ge 0,{\kern 7pt} {p_{yy}}(v )\ge 0, {\kern 7pt}Det[\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over p} (v)] \ge 0,$$

where Det stands for the determinant of the matrix. H_x_(y)(r, v) in Eq. (3) is an arbitrary kernel function of x(y) component. Note that v in Eqs. (3)–(5) is the 1D variable. On the basis of Eqs. (3)-(5), various kinds of PCVBs can be constructed by elaborate design of the kernel functions and weighting matrix. In the past studies, the choice of the kernel functions was almost limited to the form: ${H_{x(y )}}({{\mathbf r},v} )\propto \exp [{ikg({\mathbf r} )v} ]$, where g(r) is an arbitrary real function and k is a wavenumber of light. Here, we take H_x_(y)(r, v) to be a new form which is the summation of two conjugate phase functions (two single kernel functions):

(6)$${H_{x(y )}}({{\mathbf r},v} )= {\tau _{x(y )}}({\mathbf r} )\{{\exp [{ikg({\mathbf r} )v} ]+ \exp [{ - ikg({\mathbf r} )v} ]} \},$$

τ_x_(y)(r) relates the deterministic complex amplitude of x(y) component. Substituting Eqs. (4) and (6) into Eq. (3) and integrating over v, the elements of the CSD matrix yield

(7)$${W_{\alpha \beta }}({{{\mathbf r}_1},{{\mathbf r}_2}} )= \tau _\alpha ^ \ast ({{{\mathbf r}_1}} ){\tau _\beta }({{{\mathbf r}_2}} ){{P}_{\alpha \beta }}({{{\mathbf r}_1},{{\mathbf r}_2}} ),$$

where the correlation function P_αβ(r₁, r₂) is the superposition of four Fourier transforms of p_αβ(v) with different arguments, given by

(8)$$\begin{aligned} {P_{\alpha \beta }}({{{\mathbf r}_1},{{\mathbf r}_2}} )&= {{\tilde{p}}_{\alpha \beta }}[{g({{{\mathbf r}_2}} )- g({{{\mathbf r}_1}} )} ]+ {{\tilde{p}}_{\alpha \beta }}[{ - g({{{\mathbf r}_2}} )+ g({{{\mathbf r}_1}} )} ]\\ &\textrm{ + }{{\tilde{p}}_{\alpha \beta }}[{g({{{\mathbf r}_2}} )+ g({{{\mathbf r}_1}} )} ]\textrm{ + }{{\tilde{p}}_{\alpha \beta }}[{ - g({{{\mathbf r}_2}} )- g({{{\mathbf r}_1}} )} ]. \end{aligned}$$

${\tilde{p}_{\alpha \beta }}$ (α, β = x, y) in Eq. (8) denotes the Fourier transform of p_αβ. We term the PCVBs having correlation function in Eq. (8) as the double-H PCVBs. It is shown that the correlation function in general does not solely depend on the separation of two position vectors |r₂-r₁|, implying that the double-H PCVBs belong to a class of beams with spatially varying coherence states. Thereby, the degree of coherence (DOC) depends on a special reference point one chooses. By selecting complex amplitude τ_x_(y)(r), the function g(r) and weighting matrix, a wide class of the double-H PCVBs with prescribed DOC and SOP can be constructed theoretically.

As an example, we now pay attention to a particular subclass of the double-H PCVBs, named as SCNRP beam. The SOP of such beam is radially polarized, and the correlation function of each element is of super cosh-Gauss function. In this case, the complex amplitudes are chosen to be

(9)$${\tau _\alpha }({\mathbf r} )= \frac{\alpha }{{2{\omega _0}}}\exp \left( {\frac{{ - |{\mathbf r}{{\mathbf |}^2}}}{{2\omega_0^2}}} \right),(\alpha = x,y)$$

where ω₀ is the beam width. The elements of the weighting matrix are all Gaussian functions but with different widths

(10)$${p_{\alpha \beta }}(v )= \frac{{{B_{\alpha \beta }}}}{{\sqrt {2\pi a_{\alpha \beta }^2} }}\exp \left( { - \frac{{{v^2}}}{{2a_{\alpha \beta }^2}}} \right),(\alpha ,\beta = x,y)$$

where a_αβ is a positive constant. B_xx= B_yy= 1, B_xy= B_yx* are the complex correlation coefficients between the x and y components of the electric field, independent of the spatial position. Substitution Eqs. (9) and (10) into Eq. (3), and taking g(r) = |r|^m with m being an integer number, we obtain the expression for the CSD matrix’s elements of the SCNRP beam in the source plane

(11)$$\begin{aligned} {W_{\alpha \beta }}({{\mathbf r}_1},{{\mathbf r}_2}) &= \frac{{{\alpha _1}{\beta _2}{B_{\alpha \beta }}}}{{\omega _0^2}}\exp \left( { - \frac{{{{|{{{\mathbf r}_1}} |}^2} + {{|{{{\mathbf r}_2}} |}^2}}}{{2\omega_0^2}}} \right)\cosh \left( {\frac{{{{|{{{\mathbf r}_1}} |}^m}{{|{{{\mathbf r}_2}} |}^m}}}{{2\delta_{\alpha \beta }^{2m}}}} \right)\\ &\textrm{ } \times \exp \left( { - \frac{{{{|{{{\mathbf r}_1}} |}^{2m}} + {{|{{{\mathbf r}_2}} |}^{2m}}}}{{4\delta_{\alpha \beta }^{2m}}}} \right),(\alpha ,\beta = x,y) \end{aligned}$$

where cosh is the Hyperbolic cosine function. ${\delta _{\alpha \beta }} = \sqrt[{2m}]{{{1 / {2{k^2}a_{\alpha \beta }^2}}}}$ is a measure of spatial coherence of the beam. It follows from Eq. (11) that the correlation function is of super cosh-Gauss function if the index m is larger than 2. Nevertheless, the SOP in the source plane is closely related the correlation coefficient B_xy or B_yx and the coherence width δ_xy. In order to meet the condition that the source’s SOP is radially polarized, two additional conditions must be fulfilled: (1) the SOP at any point across the beam source is linearly polarized, (2) the orientation angle of the polarization ellipse at any point is θ(x, y) = arctan(y / x). By applying the above conditions, the parameters should satisfy

{B_{xy}} = {B_{yx}} = 1, \;\;{\delta _{xx}} = {\delta _{yy}} = {\delta _{yx}} = {\delta _g}.

Actually, the real-valued correlation coefficient implies the SOP is linearly polarized at any point across the beam source. The complete correlation between x and y components of the electric field, i.e., |B_xy|=|B_yx|=1 and the same coherence widths in x-x, y-y and x-y components ensure the orientation angle of the polarization being radial direction.

There are some definitions of the DOC about the PCVBs [36–39]. Here, we adopt the definition introduced by Tervo et al. [36], given by

(13)$${\mu ^2}({{{\mathbf r}_1},{{\mathbf r}_2}} )= \frac{{\textrm{Tr[}{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over W} }^\dagger }({{{\mathbf r}_1},{{\mathbf r}_2}} )\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over W} ({{{\mathbf r}_1},{{\mathbf r}_2}} )]}}{{\textrm{Tr[}\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over W} ({{{\mathbf r}_1},{{\mathbf r}_1}} )\textrm{]Tr[}\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over W} ({{{\mathbf r}_2},{{\mathbf r}_2}} )]}},$$

where the Tr and dagger denotes the trace of a matrix and the Hermitian adjoint, respectively. On substituting Eq. (11) into Eq. (13), the expression for the DOC of the SCNRP beam in the source plane is

(14)$${\mu ^2}({{{\mathbf r}_1},{{\mathbf r}_2}} )= \frac{{{{\cosh }^2}({{{{{|{{{\mathbf r}_1}} |}^m}{{|{{{\mathbf r}_2}} |}^m}} / {2\delta_g^{2m}}}} )}}{{\cosh ({{{|{{{\mathbf r}_1}} |}^{2m}}/2\delta_g^{2m}} )\cosh ({{{|{{{\mathbf r}_2}} |}^{2m}}/2\delta_g^{2m}} )}}.$$

Figure 1(a)-(d) presents the variation of DOC μ(x₁, 0, x₂, 0) as a function of (x₁ / δ_g, x₂ / δ_g) at y = 0 axis for different indices m. From Eq. (14), it readily finds that the beam is completely coherent at a pair of spatial points (x, 0) and (-x, 0), i.e., μ = 1.0. The results also can be seen from Fig. 1(a)-(c) at the anti-diagonal lines. Besides, the two points near x = 0 are almost completely coherent. However, as the index m increases, the two points near x and –x suddenly become incoherent if they leave the region near |x / δ_g| < 1. In Fig. 1(e)-(h), we plot the density distribution of the DOC when one point is fixed at r₂= 0. In this case, another point from near r₂ = 0 region is fully correlated with the reference point. The coherence pattern becomes more flat with the increase of index m. The coherence area under this circumstance is about πr² / δ_g² from m = 4. When the reference point moves out the central coherence area, the behavior is quite different. Figure 1(i)-(l) shows the density plots of the DOC when the reference point is at r₂ = (1.5δ_g, 1.5δ_g). Only the two points located at the same radial positions are completely coherent, while it drops fast in the case of m > 2 if one point leaves that radial positions. We may have to a conclusion that in the central region, the beam are highly coherence whereas outside this region, it is only perfectly coherent at the points from the ring with same radius.

Fig. 1. (a)-(d) Density plots of the DOC μ(x₁, 0, x₂, 0) at two points at y = 0 axis with different indices m. The distribution of DOC in the beam source plane with different indices m for different reference points (e)-(h) r₂ = 0, (i)-(l) r₂ = (1.5δ_g, 1.5δ_g).

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The spectral density of the SCNRP beam in the source plane is given by

(15)$$\begin{aligned} S({\mathbf r}) &= {W_{xx}}({\mathbf r},{\mathbf r}) + {W_{yy}}({\mathbf r},{\mathbf r})\\ &\textrm{ } = \frac{{{{|{\mathbf r} |}^2}}}{{2\omega _0^2}}\exp \left( { - \frac{{{{|{\mathbf r} |}^2}}}{{\omega_0^2}}} \right)\left[ {1 + \exp \left( { - \frac{{{{|{\mathbf r} |}^{2m}}}}{{\delta_g^{2m}}}} \right)} \right]. \end{aligned}$$

The source’s spectral density is closely pertinent to the spatial coherence width and the index m. When the spatial coherence width tends to be infinite (complete coherence), the spectral density reduces to that of the RP source, as expected. While for the case of the incoherent limit (${\delta _g} \to 0$), the spectral density also reduces to that of the RP source. This behavior is quite different from other kinds of partially coherent beams with spatially varying coherence states. Their spectral density in the source plane is independent of the coherence width. Figure 2 shows the normalized spectral density of the SCNRP beam in the source with different coherence widths and indices m. In the case of δ_g = 0.6 mm, the spectral density for m = 1 is similar to that of the RP beam. However, as the index m increases, the effect of δ_g gradually appears. The intensity near the dark hole is enhanced, and a bright ring is formed. When the coherence width decreases to 0.2 mm [see in Fig. 2(e)-(h)], the spectral density is nearly independent of m and δ_g, reducing to that of the RP source.

Fig. 2. Density plot of the normalized spectral density of the SCNRP beam in the source plane for different coherence widths and indices m.

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The degree of polarization (DOP) measures the ratio of the intensity of the completely polarization portion to the total intensity at a spatial point r. According to [20], the DOP of the SCNRP source can be evaluated by the following formula

(16)$$P({\mathbf r}) = \sqrt {1 - \frac{{4\textrm{Det[}\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over W} \textrm{(}{\mathbf r}\textrm{,}{\mathbf r}\textrm{)]}}}{{{{\{ \textrm{Tr}[\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over W} \textrm{(}{\mathbf r}\textrm{,}{\mathbf r}\textrm{)}]\} }^2}}}} = 1.$$

Equation (16) indicates that the DOP of the SNCRP source is completely polarized.

2.2 Propagation of the SCNRP beam in free space

Within the validity of the paraxial approximation, the CSD matrix’s elements of a PCVB propagating from the source plane (z = 0) to the half space (z >0) can be treated by the Huygens-Fresnel integrals [40]

(17)$$W_{\alpha \beta }^z({{{\mathbf{\rho} }_1},{{\mathbf{\rho} }_2}} )= \int {\int {{W_{\alpha \beta }}({{{\mathbf r}_1},{{\mathbf r}_2}} ){G^ \ast }({{{\mathbf r}_1},{{\mathbf{\rho} }_1}} )} } G({{{\mathbf r}_2},{{\mathbf{\rho} }_2}} )d{\mathbf r}_1^2d{\mathbf r}_2^2,(\alpha ,\beta = x,y)$$

where $G({{\mathbf r},{\mathbf{\rho} }} )={-} i\exp [{ik{{({{\mathbf r} - {\mathbf{\rho} }} )}^2}/2z} ]/\lambda z$ is the impulse response between the input plane and the output plane. It is hard to find the analytical expression for the CSD matrix if we substitute Eq. (11) into Eq. (17). Nevertheless, there are several methods such as coherent-mode/pseudo-mode decomposition [41–43], Fourier transform method [44], and the use of elementary function [45] to evaluate Eq. (17) numerically. Here, we use the pseudo-mode decomposition and fast Fourier transform (FFT) algorithm to numerically study the propagation properties of the SCNRP beam in free space. At first, the elements of the CSD matrix in the source plane is represented by the following discrete form via Eq. (3), as an approximation

(18)$${W_{\alpha \beta }}({{{\mathbf r}_1},{{\mathbf r}_2}} )= \sum\limits_{n = 1}^N {A_{n\alpha }^ \ast ({{{\mathbf r}_1}} )} {A_{n\beta }}({{{\mathbf r}_2}} ),(\alpha ,\beta = x,y)$$

where ${A_{n\alpha (\beta )}}({\mathbf r} )= \Delta {v^{{1 / 2}}}{p_{\alpha \beta }}({{v_n}} ){H_{\alpha (\beta )}}({{\mathbf r},{v_n}} )$, (n = 1, 2, 3,…N) represent pseudo-modes with mode weights Δv^1/2p_αβ(v_n). Δv is the interval of the discrete integral equation; N is the number of the discrete modes. In Eq. (18), the elements can be considered as the incoherent superposition of spatially coherent but mutually uncorrelated modes A_nα_(β). In such a decomposition manner, the modes in general are not orthogonal each other, hence, it is called pseudo-mode decomposition in some literatures [46,47]. Owing to that p_αβ(v_n) is Gaussian, it decays as |v_n| increases and tends to zero for $|{{v_n}} |\to \infty$. As a consequence, finite number of modes could represent the theoretical model accurately. On substituting Eq. (18) into Eq. (17) and after some mathematical operations, we obtain the expression for the elements in the output plane

(19)$$W_{\alpha \beta }^z({{{\mathbf{\rho} }_1},{{\mathbf{\rho} }_2}} )= \sum\limits_{n = 1}^N {\Theta _{n\alpha }^\ast ({{\mathbf{\rho} }_1}){\Theta _{n\beta }}({{\mathbf{\rho} }_2})} ,$$

with

(20)$$\Theta ({\mathbf{\rho} }) = \frac{{ - i}}{{\lambda z}}\exp \left( {\frac{{ik{{\mathbf{\rho} }^2}}}{{2z}}} \right)F[{{h_{n\alpha }}({\mathbf r})} ]\left\{ {\frac{{\mathbf{\rho} }}{{\lambda z}}} \right\},(\alpha = x,y)$$

where ${h_{n\alpha }}\textrm{(}{\mathbf r}\textrm{) = }{A_{n\alpha }}({\mathbf r} )\exp({{{i\pi {{\mathbf r}^2}} / {\lambda z}}} )$, F denotes the Fourier transform of the function h_nα. Equations (19) and (20) allow one to calculate the second-order statistics of the SCNRP beam at the propagation distance z with the help of the FFT algorithm. In the numerical calculation, the modes are sampled in the interval $|v |\le 3\sqrt 2 a$, and the parameter v_n is chosen to be ${v_n} = 3\sqrt 2 a[2(n - 1)/N - 1]$, (n = 1,2,..N). It is found that the involved number of modes N = 150 is accurately enough to represent the theoretical models.

Figure 3 illustrates the numerical results of the normalized spectral density of the SCNRP beam with index m = 1 and ω₀ = 0.7 mm at several propagation distances. The coherence widths at the first, second, and third rows are 0.6 mm, 0.2 mm, and 0.1 mm, respectively. It is shown that the coherence width has a significant influence on the evolution of the spectral density. When δ_g = 0.6 mm, the spectral density remains doughnut shape unchanged in 1.8 m propagation distance. This behavior is similar to that of the RP beam which is the completely coherent limit of the SCNRP beam. As the coherence width decreases, the beam displays the self-focusing characteristics. The size of ring gradually decreases and becomes the smallest at z = 1. 2 m, as shown in the second row of Fig. 3. In the case of δ_g = 0.1 mm, the self-focusing is much stronger compared to that of δ_g = 0.2 mm. The spectral density drops to a small bright ring at z = 0.5 m but there is a halo at the outskirt of the ring. The size of the bright ring then gradually increases whereas the halo gradually diminishes for the further propagation.

Fig. 3. Numerical results of the normalized spectral density of the SCNRP beam with m = 1 and ω₀ = 0.7 mm for different coherence widths in free-space propagation.

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To investigate the influence of the index m on the evolution of the spectral density, Fig. 4 presents the numerical results of the normalized spectral density of the SCNRP beam with δ_g = 0.6 mm and ω₀ = 0.7 mm for different m upon free-space propagation. The SOP across the beam section (denoted by blue lines) is also plotted. One can see that the SOP of the beam during propagation always keep the radial polarization, irrespective of the index m. Nevertheless, the index m intimately relates to the self-focusing behavior. Figure 5(a) shows the variation of the intensity maxima (normalized by the intensity maxima in the source plane) with the propagation distance for different m. The larger the m is, the shorter the focal length and the stronger the focusing ability. In addition, the focusing ability becomes stronger with the decrease of the coherence width shown in Fig. 5(b).

Fig. 4. Evolution of the spectral density and the SOP (denoted by blue lines) of the SCNRP beam with propagation distance for different indices m.

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Fig. 5. Variation of the normalized intensity maxima with propagation distance for (a) different indices m and (b) different coherence widths.

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Figure 6 presents the changes of the DOP at the point ρ = (0.02 mm, 0.02 mm) with the propagation distance. For comparison, the result of the partially coherent RP beam [24] is also plotted (solid curves). The DOP of the SCNRP beam with different indices m at that point keeps the value 1 and unchanged upon propagation, implying that the beam is still completely polarized. In fact, the SCNRP beam across the beam section at any propagation distance is completely polarized, independent of the index m and coherence width. For examples, the density plots of the DOP across beam section for m = 3 at four selected different propagation distances are illustrated in the inset of Fig. 5. This phenomenon is quite different from that of the partially coherent RP beam. The beam is de-polarized during propagation and finally degenerates to an unpolarized beam.

Fig. 6. The DOP of the SCNRP beam with different indices m and partially coherent RP beam (solid curves) at the point ρ = (0.02 mm, 0.02 mm) versus propagation distance z.

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4. Experimental generation of the SCNRP beam and its propagation properties

In this section, an experimental setup involving a radial polarization converter (RPC) and a digital micro-mirror device (DMD) is established to generate the SCNRP beam. The spectral density and polarization behavior during free-space propagation are examined experimentally.

Figure 7 shows the experimental setup for the generation of the SCNRP beam. A linearly polarized Gaussian beam (λ = 532nm) generated by a ND:YAG laser passes through a neutral density filter (NDF) and is expanded by a beam expander (BE). A RPC (ARCoptix company) is placed on the beam propagation path to convert the linearly polarized beam to the radially polarized beam. After the RPC, the electric field of the beam can be approximately expressed as

(21)$${\mathbf E}({\mathbf r} )= \left( \begin{array}{l} {E_x}({\mathbf r})\\ {E_y}({\mathbf r}) \end{array} \right) = \frac{1}{{2{\omega _0}}}\exp \left( {\frac{{ - |{\mathbf r}{{\mathbf |}^2}}}{{2\omega_0^2}}} \right)\left( \begin{array}{l} x\\ y \end{array} \right).$$

The generated radially polarized beam passes through an intensity beam splitter (BS), arriving at a DMD (ViALUX GmbH company, V-9501). The DMD screen has 1920×1080 micro mirrors (pixels) with each pixel pitch 10.8µm×10.8µm. The amplitude of the reflected light from the DMD is modulated by the micro-mirror array controlled by the computer program. In our experiment, an sine amplitude grating with its transmission function being ${t_n}({\mathbf r}) = 0.5 + 0.5\cos [2\pi {f_0}x + kg({\mathbf r}){v_n}]$ is loaded on the DMD, where f₀ is the spatial frequency along x-direction and kg(r)v_n is a spatial varying phase described in Eq. (6). Owing to that the DMD is an polarization insensitive device, the +1 and -1 diffraction orders reflected from the grating are still radially polarized, but they will carry the phase exp[ikg(r)v_n] and exp[-ikg(r)v_n], respectively. Hence, the electric fields of +1 and -1 diffraction orders can be written as

(22)$${{\mathbf E}_ \pm }({\mathbf r} )= \frac{1}{{2{\omega _0}}}\exp \left( {\frac{{ - |{\mathbf r}{{\mathbf |}^2}}}{{2\omega_0^2}}} \right)\exp [{\pm} ikg({\mathbf r}){v_n}]\left( \begin{array}{l} x\\ y \end{array} \right).$$

Fig. 7. Experimental setup for the generation of SCNRP beams. NDF: neutral density filter; BE: beam expander; RPC: radial polarization converter; BS: intensity beam splitter; DMD: micro-mirror device. AF: amplitude filter.

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The diffraction light from the DMD is then reflected by the BS, entering a 4f optical system consisting of Lens₁ and Lens₂ with focal lengths both 250 mm. In the rear focal plane of Lens₁, an amplitude filter (AF) is used to block other unwanted diffraction orders, and only allows the +1 and -1 diffraction orders to pass through. The two diffraction orders are then combined into a single beam by a Ronchi-grating located in the rear focal plane of Lens₂. The CSD matrix of the output beam after the Ronchi-grating takes the form

(23)$$\begin{aligned} {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over W} }_s}({{{\mathbf r}_1},{{\mathbf r}_2}} )&= ({{\mathbf E}_ +^\ast ({{\mathbf r}_1}) + {\mathbf E}_ -^\ast ({{\mathbf r}_1})} ){({{{\mathbf E}_ + }({{\mathbf r}_2}) + {{\mathbf E}_ - }({{\mathbf r}_2})} )^T}\\ &\textrm{ } = \left[ {\begin{array}{cc} {H_x^\ast ({{\mathbf r}_1},{v_n}){H_x}({{\mathbf r}_2},{v_n})}&{H_x^\ast ({{\mathbf r}_1},{v_n}){H_y}({{\mathbf r}_2},{v_n})}\\ {{H_x}({{\mathbf r}_2},{v_n})H_y^\ast ({{\mathbf r}_1},{v_n})}&{H_y^\ast ({{\mathbf r}_1},{v_n}){H_y}({{\mathbf r}_2},{v_n})} \end{array}} \right], \end{aligned}$$

where the superscript T denotes the transpose of a matrix. H_x_(y) are shown in Eq. (6). Compared Eq. (23) to Eqs. (3) and (18), one readily finds that the generate beam is just a single vector mode of the CSD matrix of the SCNRP beam if one set the g(r) function to be |r|^m. In order to synthesis of the real SCNRP beam, a series of amplitude gratings with different parameters v_n is first synthesized and is saved in the computer memory. The DMD displays the amplitude gratings randomly, but the appearance probability of the amplitude gratings is proportional to the mode weights $\Delta {v^{{1 / 2}}}{p_{\alpha \beta }}({{v_n}} )$, (α, β = x, y). By averaging over the modes with different v_n, the SCNRP is synthesized in the output plane. A CCD is located after the Ronchi-grating to measure the average intensity distribution of the generated beam. In fact, the refresh rate of the DMD can be up to 17kHz. Thus, the SCNRP beam can be generated nearly in real time. Finally, we emphasize that our experimental setup also can be used to synthesize other kinds of double-H vector beams by adjusting the function g(r), without changing physical layout of the setup.

The experimental results of the normalized intensity distribution of the SCNRP beam with different indices m after the Ronchi-grating are illustrated in Fig. 8. In the experiment, the total 150 discrete models are used. The other beam parameters are ω₀ = 0.7 mm, δ_g = 0.6 mm which is prescribed in discrete models. Compared the source’s intensity distributions to those theoretical results shown in Fig. 4(a), (f) and (k), the beam profiles between them seem to be a large deviation. The main reason we find is that the theoretical model used in Eq. (23) is somewhat different from the result in the experiment. There exists a mode mismatch when the RPC convert the linearly polarized Gaussian beam to the RP beam. Despite this, the evolution properties of the intensity distribution on propagation are reasonable consistent with those of theoretical predictions shown in Fig. 4. The beam with larger index m displays the strong self-focusing and the focal length is shorten with the increase of m. The results demonstrate that our experiment setup for the generation of the SCNRP beam is effective and flexible.

Fig. 8. Experimental results for the intensity distribution of the SCNRP beam for different indices m at several propagation distances.

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To examine the polarization of the propagating beam qualitatively, a polarizer is inserted before the CCD to observe the changes of the intensity distribution by rotating the polarizer. The transmission axis of the polarizer and the x-axis forms the angle θ. The first row in Fig. 9 shows the experimental results of the intensity distribution with m = 4 at propagation distance z = 0.5 m. For comparison, the corresponding theoretical results are shown in the second row. It can be seen that the SOP beam remain the radial polarization. Our other experimental results (no shown here) also demonstrate that the SOP of the beam keeps invariant during propagation, irrespectively of beam parameters ω₀, δ_g and m.

Fig. 9. (a) The normalized intensity distribution of the SCNRP beam with m = 4 at z = 0.5 m. (b)-(e) the changes of intensity distribution after passing through a polarizer with different angle θ. The second row are the corresponding theoretical results. The white arrows denote the transmission axis of the polarizer.

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5. Conclusion

As a summary, we have introduced a kind of nonuniformly correlated PCVBs, named SCNRP beam, as a subclass of double-H PCVBs whose kernel function is the summation of two conjugated phase functions. Such the SCNRP beam has initial radial polarization, and the beam is nearly completely coherent in the beam center region, while out of that region, only the two points with same radial position are perfectly coherent, and becomes partially coherent or even incoherent from different radial positions. With the help of the pseudo-mode decomposition and FFT algorithm, the spectral density and the polarization characteristics of the beam during free-space propagation are investigated numerically. It is found that the DOP and the SOP of the SCNRP beam keep invariant on propagation, i.e., the beam is still completely polarized and shows radial polarization. In addition, the beam shows the self-focusing propagation features. The strength of the focusing ability and focal length can be controlled by the beam index m and the spatial coherence width. We also establish an efficient coaxial interference optical system to generate the SCNRP beam nearly in real time by means of a DMD with fast modulation speed. The peculiar propagation properties of the generated beam are demonstrated in experiment. Our theoretical and experimental results suggest a new approach for tailoring the coherence state of the PCVBs and it will find applications in beam shaping, optical trapping, and others.

Funding

National Key Research and Development. Project of China (2019YFA0705000); National Natural Science Foundation of China (11874046, 11904247, 11974218, 12004218, 12192254); the Innovation Group of Jinan (2018GXRC010); the Local Science and Technology Development Project of the Central Government (YDZX20203700001766).

Disclosures

The authors declare no conflicts of interest.

Data availability

No data were generated or analyzed in the presented research.

References

1. Q. Zhan, Vectorial optical fields: Fundamentals and applications, (World Scientific2013).

2. X. Xie, Y. Chen, K. Yang, and J. Zhou, “Harnessing the point-spread function for high-resolution far-field optical microscopy,” Phys. Rev. Lett. 113(26), 263901 (2014). [CrossRef]

3. M. Meier, V. Romano, and T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys. A 86(3), 329–334 (2007). [CrossRef]

4. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009). [CrossRef]

5. S. Ramachandran, P. Kristensen, and M. F. Yan, “Generation and propagation of radially polarized beams in optical fibers,” Opt. Lett. 34(16), 2525–2527 (2009). [CrossRef]

6. Z. Nie, G. Shi, D. Li, X. Zhang, Y. Wang, and Y. Song, “Tight focusing of a radially polarized Laguerre–Bessel–Gaussian beam and its application to manipulation of two types of particles,” Phys. Lett. A 379(9), 857–863 (2015). [CrossRef]

7. C. Min, Z. Shen, J. Shen, Y. Zhang, H. Fang, G. Yuan, L. Du, S. Zhu, T. Lei, and X. Yuan, “Focused plasmonic trapping of metallic particles,” Nat. Commun. 4(1), 1–7 (2013). [CrossRef]

8. Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12(15), 3377–3382 (2004). [CrossRef]

9. W. Cheng, J. W. Haus, and Q. Zhan, “Propagation of vector vortex beams through a turbulent atmosphere,” Opt. Express 17(20), 17829–17836 (2009). [CrossRef]

10. D. F. James, “Change of polarization of light beams on propagation in free space,” J. Opt. Soc. Am. A 11(5), 1641–1643 (1994). [CrossRef]

11. F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23(4), 241–243 (1998). [CrossRef]

12. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5-6), 263–267 (2003). [CrossRef]

13. I. Vidal, E. J. Fonseca, and J. M. Hickmann, “Light polarization control during free-space propagation using coherence,” Phys. Rev. A 84(3), 033836 (2011). [CrossRef]

14. O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. 281(9), 2342–2348 (2008). [CrossRef]

15. M. Hyde, S. Bose-Pillai, D. G. Voelz, and X. Xiao, “Generation of Vector partially coherent optical sources using phase-only spatial light modulator,” Phys. Rev. Appl. 6(6), 064030 (2016). [CrossRef]

16. T. Li, D. Li, X. Zhang, K. Huang, and X. Lu, “Partially coherent radially polarized circular Airy beam,” Opt. Lett. 45(16), 4547–4550 (2020). [CrossRef]

17. X. Zhu, J. Yu, F Wang, Y Chen, Y Cai, and O Korotkova, “Synthesis of vector nonuniformly correlated light beams by a single digital mirror device,” Opt. Lett. 46(12), 2996–2999 (2021). [CrossRef]

18. R. Martínez-Herrero, D. Maluenda, I. Juvells, and A. Carnicer, “Effect of linear polarizer on the behavior of partially coherent and partially polarized highly focused field,” Opt. Lett. 43(14), 3445–3448 (2018). [CrossRef]

19. Y. Liu, Y. Chen, F. Wang, Y. Cai, C. Liang, and O. Korotkova, “Robust far-field imaging by spatial coherence engineering,” Opto-Electron. Adv. 4(12), 210027 (2021). [CrossRef]

20. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

21. O. Korotkova, Theoretical statistical optics (World Scientific, 2021).

22. Z. Mei and O. Korotkova, “Electromagnetic cosine-Gaussian Schell-model beams in free space and atmospheric turbulence,” Opt. Express 21(22), 27246–27259 (2013). [CrossRef]

23. F. Wang, Y. Cai, Y. Dong, and O. Korotkova, “Experimental generation of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 100(5), 051108 (2012). [CrossRef]

24. G. Wu, F. Wang, and Y. Cai, “Coherence and polarization properties of a radially polarized beam with variable spatial coherence,” Opt. Express 108(4), 891–895 (2012). [CrossRef]

25. F. Wang, X. Liu, L. Liu, Y. Yuan, and Y. Cai, “Experimental study of the scintillation index of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 103(9), 091102 (2013). [CrossRef]

26. J. Yu, Y. Huang, F. Wang, X. Liu, G. Gbur, and Y. Cai, “Scintillation properties of a partially coherent vector beam with vortex phase in turbulent atmosphere,” Opt. Express 27(19), 26676–26688 (2019). [CrossRef]

27. Y. Liu, X. Zhang, Z. Dong, D. Peng, Y. Chen, F. Wang, and Y. Cai, “Robust far-field optical image transmission with structured random light beams,” Phys. Rev. Appl. 17(2), 024043 (2022). [CrossRef]

28. Y. Chen, F. Wang, and Y. Cai, “Partially coherent light beam shaping via complex spatial coherence structure engineering,” Adv. Phys. 7, 2009742 (2022). [CrossRef]

29. D. Peng, Z. Huang, Y. Liu, Y. Chen, F. Wang, S. A. Ponomarenko, and Y. Cai, “Optical coherence encryption with structured random light,” PhotoniX 2(1), 6 (2021). [CrossRef]

30. C. Ping, C. Liang, F. Wang, and Y. Cai, “Radially polarized multi-Gaussian Schell-model beam and its tight focusing properties,” Opt. Express 25(26), 32475–32490 (2017). [CrossRef]

31. F. Wang, Y. Chen, X. Liu, Y. Cai, and S. A. Ponomarenko, “Self-reconstruction of partially coherent light beams scattered by opaque obstacles,” Opt. Express 24(21), 23735–23746 (2016). [CrossRef]

32. J. Yu, X. Zhu, S. Lin, F. Wang, G. Gbur, and Y. Cai, “Vector partially coherent beams with prescribed non-uniform correlation structure,” Opt. Lett. 45(13), 3824–3827 (2020). [CrossRef]

33. S. Lin, J. Wu, Y. Xu, X. Zhu, G. Gbur, Y. Cai, and J. Yu, “Analysis and experimental demonstration of propagation features of radially polarized specific non-uniformly correlated beams,” Opt. Lett. 47(2), 305–308 (2022). [CrossRef]

34. R. Martínez-Herrero, P. M. Mejias, and F. Gori, “Genuine cross-spectral densities and pseudo-modal expansions,” Opt. Lett. 34(9), 1399–1401 (2009). [CrossRef]

35. F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A: Pure Appl. Opt. 11(8), 085706 (2020). [CrossRef]

36. J. Tervo, T. Setälä, and A. T. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11(10), 1137–1143 (2003). [CrossRef]

37. T. Setälä, J. Tervo, and A. T. Friberg, “Complete electromagnetic coherence in the space–frequency domain,” Opt. Lett. 29(4), 328–330 (2004). [CrossRef]

38. F. Gori, M. Santarsiero, and R. Borghi, “Maximizing Young's fringe visibility through reversible optical transformations,” Opt. Lett. 32(6), 588–590 (2007). [CrossRef]

39. R. Martínez-Herrero and P. M. Mejías, “Maximum visibility under unitary transformations in two-pinhole interference for electromagnetic fields,” Opt. Lett. 32(11), 1471–1473 (2007). [CrossRef]

40. M. Born and E. Wolf, Principles of Optics, (Cambridge University Press, 1999).

41. B. E. Saleh and M. Rabbani, “Simulation of partially coherent imagery in the space and frequency domains and by modal expansion,” Appl. Opt. 21(15), 2770–2777 (1982). [CrossRef]

42. A. Starikov and E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and of their radiation fields,” J. Opt. Soc. Am. 72(7), 923–928 (1982). [CrossRef]

43. Y. Gu and G. Gbur, “Scintillation of pseudo-Bessel correlated beams in atmospheric turbulence,” J. Opt. Soc. Am. A 27(12), 2621–2629 (2010). [CrossRef]

44. F. Wang and O. Korotkova, “Convolution approach for beam propagation in random media,” Opt. Lett. 41(7), 1546–1549 (2016). [CrossRef]

45. A. Burvall, A. Smit, and C. Dainty, “Elementary functions: propagation of partially coherent light,” J. Opt. Soc. Am. A 26(7), 1721–1729 (2009). [CrossRef]

46. S. Yang, S. A. Ponomarenko, and Z. D. Chen, “Coherent pseudo-mode decomposition of a new partially coherent source class,” Opt. Lett. 40(13), 3081–3084 (2015). [CrossRef]

47. F. Gori and R. Martínez-Herrero, “Reproducing Kernel Hilbert spaces for wave optics: tutorial,” J. Opt. Soc. Am. A 38(5), 737–748 (2021). [CrossRef]

Super cosh-Gauss nonuniformly correlated radially polarized beam and its propagation characteristics

Abstract

1. Introduction

2. Theory

2.1 Theoretical model for a super cosh-Gauss correlated nonuniformly radially polarized beam

2.2 Propagation of the SCNRP beam in free space

4. Experimental generation of the SCNRP beam and its propagation properties

5. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (9)

Equations (23)

Optics Express