Generation of superposition modes by polarization-phase coupling in a cylindrical vector orbital angular momentum beam

Tae Moon Jeong; Sergei Bulanov; Sergei Bulanov; Sergei Bulanov; Wenchao Yan; Stefan Weber; Georg Korn

doi:10.1364/OSAC.2.002718

1. Introduction

The light field is described by three fundamental parameters: field strength, polarization, and phase. For conventional beams such as linearly-polarized (LP) and circularly-polarized (CP) beams, the polarization is homogeneous in space. In contrast, the polarization of the cylindrical vector (CV) beam (radially-polarized or azimuthally-polarized beam) is not homogeneous and has a vector symmetry about the beam axis [1]. The CV beam has a polarization singularity in the axis and can be generated by several techniques described elsewhere [2–4]. The focused field distribution has been widely studied in Ref. [5]. The polarization singularity of the CV beam is characterized by the Poincaré-Hopf (PH) index [6]. The PH index represents the number of polarization rotation in a closed curve around the polarization singularity. Thus, the PH index of the linearly-polarized beam is 0 and the PH index of the radially- or azimuthally-polarized beam is 1.

It has been demonstrated that helical phase shifts can impose the orbital angular momentum (OAM) on the beam [7,8]. The helical phase shift can be achieved using spiral phase-plates, computer-generated hologram methods, or other techniques as described in Ref. [8–11]. The OAM beam has a phase singularity and the singularity is characterized by the topological charge (TC), which corresponds to the number of phase rotation [0, 2π] around the singular point. Much attention has been paid to the use of OAM beams in emerging optics fields [12,13] and in high-intensity laser-matter interactions for generating x-ray vortex beams, electron vortex beams, or mega-gauss axial magnetic field [14–20].

Although LP and CP OAM [21] beams were mostly used in experiments, the generation and use of CV OAM beam [22] becomes more interesting. The CV OAM beam can be generated by passing a LP beam through a radial/azimuthal polarization plate (RAPP) and a spiral phase plate (SPP). The generation of CV OAM beam and mathematical description on the beam profile of CV OAM beam have been introduced in Ref. [22,23]. Because the CV OAM beam possesses the orientation-angle-dependent polarization and helical phase, it is conceived that these two parameters can be coupled in the beam through the polarization-phase coupling (P-P coupling) and a resultant electric field distribution will be determined by the P-P coupling. Due to the P-P coupling, interesting features in the field distribution are also expected, especially in the focal plane. However, it has not yet thoroughly understood how the P-P coupling modifies the focused field distribution of the CV OAM beam especially with arbitrary PH index and TC.

In this paper, we derive the analytical formulas describing focused electric field distribution of a CV OAM beam with arbitrary PH index and TC, and then examine the P-P coupling occurring in the CV OAM beam and how the P-P coupling modifies the focused electric field distribution. Our results show that the P-P coupling in the CV OAM beam induces Laguerre-Gaussian (LG) beam modes with different orders determined by PH index and TC, and the resultant electric field distribution is a superposition of these induced LG modes. Analytical expressions describing the resultant electric fields have been derived from the vector diffraction integrals and confirmed by numerical calculations. The benefit of using vector diffraction integrals is to provide information on focused electric fields for all polarization components, including transverse and longitudinal polarizations which are seriously considered under tight focusing conditions [24].

2. Focused electric field of CV OAM beam under the paraxial approximation

According to the theory of polarization optics [25], the polarization state of an electric field on the Poincaré sphere is expressed by

(1)$$\left[ {\begin{array}{c} {{\varphi_x}({2\theta ,2\phi } )}\\ {{\varphi_y}({2\theta ,2\phi } )} \end{array}} \right] = \left[ {\begin{array}{c} {\cos \theta \cos \phi - i\sin \theta \sin \phi }\\ {\cos \theta \sin \phi + i\sin \theta \cos \phi } \end{array}} \right].$$

Here, 2θ means the latitude in [-π/2, π/2] and 2ϕ the azimuthal angle in [0, 2π] on the Poincaré sphere [see Fig. 1(a)]. The functions φ_x and φ_y denote the x- and y-polarization states of the electric field, respectively. Let us assume an electric field with a polarization state (2θ and 2ϕ) on the Poincaré sphere passes through an RAPP and an SPP. Then, after the RAPP and the SPP, the polarization state of the electric field on a higher-order Poincaré sphere [26] is written as follows:

(2)$$\left[ {\begin{array}{c} {\varphi_x^{m,l}({2\theta ,2\phi } )}\\ {\varphi_y^{m,l}({2\theta ,2\phi } )} \end{array}} \right] = {e^{il\gamma }}T({m\gamma } )\left[ {\begin{array}{c} {\cos \theta \cos \phi - i\sin \theta \sin \phi }\\ {\cos \theta \sin \phi + i\sin \theta \cos \phi } \end{array}} \right].$$

Here, the superscript, l, in Eq. (2) means the TC of the helical phase and $T({m\gamma } )$ refers to a transformation matrix $\left[ {\begin{array}{cc} {\cos ({m\gamma } )}&{ - \sin ({m\gamma } )}\\ {\sin ({m\gamma } )}&{\cos ({m\gamma } )} \end{array}} \right]$ for the RAPP with a PH index of m. An LP electric field at a polarization state (θ=0 and ϕ=0) turns its polarization state into the radial polarization state of $\left[ {\begin{array}{c} {\cos (\gamma )}\\ {\sin (\gamma )} \end{array}} \right]$ when it passes through a RAPP with m = 1. By using the identities of $\cos ({m\gamma } )= {{({{e^{im\gamma }} + {e^{ - im\gamma }}} )} \mathord{\left/ {\vphantom {{({{e^{im\gamma }} + {e^{ - im\gamma }}} )} 2}} \right.} 2}$ and $\sin ({m\gamma } )= {{({{e^{im\gamma }} - {e^{ - im\gamma }}} )} \mathord{\left/ {\vphantom {{({{e^{im\gamma }} - {e^{ - im\gamma }}} )} {2i}}} \right.} {2i}}$, Eq. (2) can be rewritten as

(3)$$\begin{aligned} \left[ {\begin{array}{c} {\varphi_x^{m,l}({2\theta ,2\phi } )}\\ {\varphi_y^{m,l}({2\theta ,2\phi } )} \end{array}} \right] &= \frac{1}{2}\left( \begin{array}{l} \cos \theta \cos \phi - \\ i\sin \theta \sin \phi \end{array} \right)\left\{ {{e^{i({l + m} )\gamma }}\left[ {\begin{array}{c} 1\\ { - i} \end{array}} \right] + {e^{i({l - m} )\gamma }}\left[ {\begin{array}{c} 1\\ i \end{array}} \right]} \right\}\\ &+ i\frac{1}{2}\left( \begin{array}{l} \cos \theta \sin \phi + \\ i\sin \theta \cos \phi \end{array} \right)\left\{ {{e^{i({l + m} )\gamma }}\left[ {\begin{array}{c} 1\\ { - i} \end{array}} \right] - {e^{i({l - m} )\gamma }}\left[ {\begin{array}{c} 1\\ i \end{array}} \right]} \right\}. \end{aligned}$$

The function $\varphi _x^{m,l}$ and $\varphi _y^{m,l}$ in Eq. (3) are general expressions for polarization states when an LP electric field passes through an RAPP with a PH of m and an SPP with a TC of l.

Fig. 1. (a) Poincare sphere representing the polarization state (PS) of the beam. S1, S2, S3 mean the stoke parameters. (b) Radial polarization (blue arrows) and Laguerre-Gaussian intensity profile. (c) Helical phase shift dependent on the orientation angle γ. l is the topological charge.

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Equation (3) shows some interesting characteristics of the electric field that passes through the RAPP and the SPP. First, the P-P coupling occurs in the electric field, and two different azimuthal modes (l + m and l-m) are generated by the P-P coupling. Second, both real and imaginary parts of the electric field are the superposition states of sine or cosine modes [i.e., $\cos ({l + m} )\gamma \pm \cos ({l - m} )\gamma$ or $\sin ({l + m} )\gamma \pm \sin ({l - m} )\gamma$] of two coupled modes, l + m and l-m. As will be seen later, the superposition of two modes produced by the P-P coupling determines the electric field distribution in the focal plane. Third, the sine and cosine functions of the real and imaginary components [$\cos ({l + m} )\gamma \pm \cos ({l - m} )\gamma + i\{{\sin ({l + m} )\gamma \pm \sin ({l - m} )\gamma } \}$ or vice versa] of the electric field are responsible for the helical propagation of the electric field.

It is now interesting to examine the electric field distribution in the focal plane. An LP state was taken as an input polarization. Different and complex polarization states can be chosen by taking a different θ-ϕ combination. By taking θ=0 and ϕ=0 in Eq. (3) and changing the expression from the polarization-state representation to the field distribution representation, the x- and y-polarized electric fields are expressed as follows:

(4)$$\left[ {\begin{array}{c} {{E_x}({{x_S},{y_S},{z_S}} )}\\ {{E_y}({{x_S},{y_S},{z_S}} )} \end{array}} \right] = \frac{{{E_0}}}{2}{e^{ - \frac{{r_S^2}}{{2\omega _0^2}}}}\left\{ {{e^{i({l + m} )\gamma }}\left[ {\begin{array}{c} 1\\ { - i} \end{array}} \right] + {e^{i({l - m} )\gamma }}\left[ {\begin{array}{c} 1\\ i \end{array}} \right]} \right\}.$$

Here, ${E_x}({{x_S},{y_S},{z_S}} )$ and ${E_y}({{x_S},{y_S},{z_S}} )$ mean the electric fields at a source point (x_S, y_S, z_S) on a focusing optic system. For the mathematical simplicity, the y-component of the incident electric field, ${E_y}$, is taken to be zero in Eq. (4). As shown in Fig. 2(a), an LP Gaussian beam passes through the RAPP and the SPP, and it is focused by the focusing system. The Gaussian function, ${E_0}{e^{ - {{r_S^2} \mathord{\left/ {\vphantom {{r_S^2} {2\omega_0^2}}} \right.} {2\omega _0^2}}}}$, is multiplied to describe the Gaussian beam profile. Here, r_S is defined by $\sqrt {x_S^2 + y_S^2}$. The inclination and rotation angles to the source point are defined by $\vartheta$ and $\gamma$, respectively. Based on the vector diffraction theory, the x-polarization component of the focused electric field [27] at the observation position (x_P, y_P, z_P) near the focal region can be calculated as follows:

(5)$$\begin{array}{l} {E_x}^{\prime}({{x_P},{y_P},{z_P}} )= \\ - i\frac{{{E_0}}}{{2\pi }}\int_0^\alpha {\int_0^{2\pi } {\left[ \begin{array}{l} {e^{ - {{{{\sin }^2}\vartheta } \mathord{\left/ {\vphantom {{{{\sin }^2}\vartheta } {2{{\sin }^2}{\vartheta_0}}}} \right.} {2{{\sin }^2}{\vartheta_0}}}}}{e^{i({l + m} )\gamma }}{\cos^{1/2}}\vartheta \sin \vartheta \times \\ \{{\cos \vartheta + ({1 - \cos \vartheta } ){{\sin }^2}\gamma } \}{e^{ik{r_P}\cos \varepsilon }} \end{array} \right]d\vartheta } d\gamma } \\ - i\frac{{{E_0}}}{{2\pi }}\int_0^\alpha {\int_0^{2\pi } {\left[ \begin{array}{l} {e^{ - {{{{\sin }^2}\vartheta } \mathord{\left/ {\vphantom {{{{\sin }^2}\vartheta } {2{{\sin }^2}{\vartheta_0}}}} \right.} {2{{\sin }^2}{\vartheta_0}}}}}{e^{i({l - m} )\gamma }}{\cos^{1/2}}\vartheta \sin \vartheta \times \\ \{{\cos \vartheta + ({1 - \cos \vartheta } ){{\sin }^2}\gamma } \}{e^{ik{r_P}\cos \varepsilon }} \end{array} \right]d\vartheta } d\gamma } . \end{array}$$

Here, r_P (=$\sqrt {x_P^2 + y_P^2 + z_P^2}$) is the distance from the origin of the coordinate to the observation position. The cosine function, $\cos \varepsilon$, is defined by $\cos \vartheta \cos \chi + \sin \vartheta \sin \chi \cos ({\gamma - q} )$. The angles, χ and q, mean the inclination and the rotation angle to the observation position, respectively [see Fig. 2(b)]. The upper limit α in the integral is defined by the f-number (the ratio of focal length to the diameter of entrance pupil) of the focusing system. Under the paraxial approximation (${\vartheta _0} \ll \alpha$), the inclination angle $\vartheta$ approaches 0 and $\cos \vartheta$ becomes close to unity. The Gaussian function ${E_0}{e^{ - {{r_S^2} \mathord{\left/ {\vphantom {{r_S^2} {2\omega_0^2}}} \right.} {2\omega _0^2}}}}$ is then expressed as ${e^{ - {{{\vartheta ^2}} \mathord{\left/ {\vphantom {{{\vartheta^2}} {2\vartheta_0^2}}} \right.} {2\vartheta _0^2}}}}$ in terms of $\vartheta$ through r_S=${\rho _S}\sin {\vartheta _S}$ and ω₀=${\rho _S}\sin {\vartheta _0}$.

Fig. 2. Coordinates used when the electric field distributions are calculated near the focal plane. Definition of coordinates (a) for the source point and (b) for the observation point.

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Now, by using the integral identity of $\int_0^{2\pi } {{e^{in\gamma }}{e^{i\rho \cos ({\gamma - q} )}}d\gamma } = 2\pi {i^n}{e^{inq}}{J_n}(\rho )$ and taking the Taylor’s series expansion of sine function as $\sum\limits_{n = 0}^\infty {{{{{({ - 1} )}^n}{\vartheta ^{2n + 1}}} \mathord{\left/ {\vphantom {{{{({ - 1} )}^n}{\vartheta^{2n + 1}}} {({2n + 1} )!}}} \right.} {({2n + 1} )!}}}$, Eq. (5) becomes under the paraxial approximation,

(6)$$\begin{array}{l} {E_x}^{\prime}({{x_P},{y_P},{z_P}} )= \\ - \frac{{{i^{l + m + 1}}}}{{2\pi }}{E_0}{e^{ik{r_P}\cos \chi }}{e^{i({l + m} )q}}\sum\limits_{n = 0}^\infty {\frac{{{{({ - 1} )}^n}}}{{({2n + 1} )!}}} \int_0^\alpha {{e^{ - \frac{{{\vartheta ^2}}}{{2\vartheta _0^2}}}}{J_{l + m}}({k{r_P}\sin \chi \cdot \vartheta } ){\vartheta ^{2n + 1}}d\vartheta } \\ - \frac{{{i^{l - m + 1}}}}{{2\pi }}{E_0}{e^{ik{r_P}\cos \chi }}{e^{i({l - m} )q}}\sum\limits_{n = 0}^\infty {\frac{{{{({ - 1} )}^n}}}{{({2n + 1} )!}}} \int_0^\alpha {{e^{ - \frac{{{\vartheta ^2}}}{{2\vartheta _0^2}}}}{J_{l - m}}({k{r_P}\sin \chi \cdot \vartheta } ){\vartheta ^{2n + 1}}d\vartheta } . \end{array}$$

When integrating Eq. (6), the upper limit α can be approximated as infinity by assuming the contribution from the periphery of the Gaussian function to the field formation is small. This allows one to integrate Eq. (6) with the integral formula [28] of

(7)$$\int_0^\infty {{e^{ - {x^2}}}{x^{2\mu + \nu + 1}}{J_\nu }\left( {2x\sqrt z } \right)dx} = \frac{{\mu !}}{2}{e^{ - z}}{z^{{\nu \mathord{\left/ {\vphantom {\nu 2}} \right.} 2}}}L_\mu ^\nu (z ). $$

By replacing ${{{k^2}r_P^2{{\sin }^2}\chi } \mathord{\left/ {\vphantom {{{k^2}r_P^2{{\sin }^2}\chi } 4}} \right.} 4}$ with $R_P^2$, the resultant electric field distribution in the focal region has the form of a superposition of two Laguerre-Gaussian (LG) beam modes, $R_P^{|{l + m} |}{e^{ - R_P^2}}L_{n - {{|{l + m} |} \mathord{\left/ {\vphantom {{|{l + m} |} 2}} \right.} 2}}^{l + m}({R_P^2} )$ and $R_P^{|{l - m} |}{e^{ - R_P^2}}L_{n - {{|{l - m} |} \mathord{\left/ {\vphantom {{|{l - m} |} 2}} \right.} 2}}^{l - m}({R_P^2} )$ or simply $LG_{n - {{|{l + m} |} \mathord{\left/ {\vphantom {{|{l + m} |} 2}} \right.} 2}}^{l + m}$ and $LG_{n - {{|{l - m} |} \mathord{\left/ {\vphantom {{|{l - m} |} 2}} \right.} 2}}^{l - m}$. The expression $LG_\mu ^\nu$ means the LG mode with a radial order of μ and an azimuthal of ν. Thus, Eq. (6) becomes

(8)$$\begin{array}{l} {E_x}^{\prime}({{x_P},{y_P},{z_P}} )= \\ - \frac{{{i^{l + m + 1}}}}{{2\pi }}{E_0}{e^{ik{r_P}\cos \chi }}\sum\limits_{n = 0}^\infty {\frac{{{2^n}\vartheta _0^{2n + 2}{{({ - 1} )}^n}}}{{({2n + 1} )!}}} \left( {n - \frac{{l + m}}{2}} \right)!LG_{n - \frac{{|{l + m} |}}{2}}^{l + m}{e^{i({l + m} )q}}\\ - \frac{{{i^{l - m + 1}}}}{{2\pi }}{E_0}{e^{ik{r_P}\cos \chi }}\sum\limits_{n = 0}^\infty {\frac{{{2^n}\vartheta _0^{2n + 2}{{({ - 1} )}^n}}}{{({2n + 1} )!}}} \left( {n - \frac{{l - m}}{2}} \right)!LG_{n - \frac{{|{l - m} |}}{2}}^{l - m}{e^{i({l - m} )q}}. \end{array}$$

Here, the radial order $n - {{|{l \pm m} |} \mathord{\left/ {\vphantom {{|{l \pm m} |} 2}} \right.} 2}$ in Eq. (8) should be greater than zero. The P-P coupling expressed by $|{l \pm m} |$ is revealed by modifying the radial and azimuthal orders in the LG beam.

Because of ${\vartheta _0} \gg \vartheta _0^2 \gg \vartheta _0^3 \cdots$ under the paraxial approximation, the lowest modes are dominant in Eq. (8). When $|{l \pm m} |$ satisfies 2u (u = 0, 1, 2, ···), the lowest order mode becomes the Gaussian. Since the lowest-order radially- or azimuthally-polarized (RP or AP) beam has a PH index (m) of ±1, the condition of $|{l \pm m} |= 0$ can be satisfied when the TC (l) of the OAM beam is 1. Thus, the focused intensity of the lowest-order vector OAM beam is expected to be a Gaussian beam. When $|{l \pm m} |$ satisfies 2u + 1, the lowest mode can be calculated from the integral formula [28] of

(9)$$\int_0^\infty {\vartheta {e^{ - \alpha {\vartheta ^2}}}{J_{l \pm m}}({2{R_p}\vartheta } )d\vartheta } = \frac{{\sqrt \pi {R_p}}}{{4{\alpha ^{{3 \mathord{\left/ {\vphantom {3 2}} \right.} 2}}}}}{e^{ - \frac{{R_p^2}}{{2\alpha }}}}\left[ {{I_{\frac{{l \pm m}}{2} - \frac{1}{2}}}\left( {\frac{{R_p^2}}{{2\alpha }}} \right) - {I_{\frac{{l \pm m}}{2} + \frac{1}{2}}}\left( {\frac{{R_p^2}}{{2\alpha }}} \right)} \right],$$

with n = 0 in Eq. (6). Here, ${I_{{{({l \pm m} )} \mathord{\left/ {\vphantom {{({l \pm m} )} 2}} \right.} 2} \mp {1 \mathord{\left/ {\vphantom {1 2}} \right.} 2}}}({{{R_p^2} \mathord{\left/ {\vphantom {{R_p^2} {2\alpha }}} \right.} {2\alpha }}} )$ is the modified Bessel function of the first kind. The higher-order field distribution becomes a half-integer radial-order LG mode as discussed in Eq. (9) and Ref. [29]. The general expression for the half-integer Laguerre function can be found in Ref. [30].

The advantage of using vector diffraction integrals is to calculate the electric fields with the other transverse and longitudinal polarizations which are different from the input polarization. The strength of these field components is negligibly small under the normal focusing condition (f-number >>1), however it becomes significant to modify the overall intensity distribution under the tight focusing condition (f-number <1). The analytical expressions of the electric field distribution for y- and z-components can be obtained through the same approach using Eq. (2.26) in Ref. [27]. After the straightforward calculation, the resultant field distributions for y- and z-polarizations have the following forms with constant coefficients C_i,n and D_i,n (i = 1,2,3,4),

(10)$$\begin{array}{l} {E_y}^{\prime}({{x_P},{y_P},{z_P}} )\sim \\ {i^{l + m - 1}}{E_0}\left( {\sum\limits_n {{C_{1,n}} \cdot LG_{n - \frac{{|{l + m + 2} |}}{2}}^{l + m + 2}{e^{i({l + m + 2} )q}}} - \sum\limits_n {{C_{2,n}} \cdot LG_{n - \frac{{|{l + m - 2} |}}{2}}^{l + m - 2}{e^{i({l + m - 2} )q}}} } \right)\\ + {i^{l - m - 1}}{E_0}\left( {\sum\limits_n {{C_{3,n}} \cdot LG_{n - \frac{{|{l - m + 2} |}}{2}}^{l - m + 2}{e^{i({l - m + 2} )q}}} - \sum\limits_n {{C_{4,n}} \cdot LG_{n - \frac{{|{l - m - 2} |}}{2}}^{l - m - 2}{e^{i({l - m - 2} )q}}} } \right), \end{array}$$

and

(11)$$\begin{array}{l} {E_z}^{\prime}({{x_P},{y_P},{z_P}} )\sim \\ {i^{l + m - 1}}{E_0}\left( {\sum\limits_n {{D_{1,n}} \cdot LG_{n - \frac{{|{l + m + 1} |}}{2}}^{l + m + 1}{e^{i({l + m + 1} )q}}} - \sum\limits_n {{D_{2,n}} \cdot LG_{n - \frac{{|{l + m - 1} |}}{2}}^{l + m - 1}{e^{i({l + m - 1} )q}}} } \right)\\ + {i^{l - m - 1}}{E_0}\left( {\sum\limits_n {{D_{3,n}} \cdot LG_{n - \frac{{|{l - m + 1} |}}{2}}^{l - m + 1}{e^{i({l - m + 1} )q}}} - \sum\limits_n {{D_{4,n}} \cdot LG_{n - \frac{{|{l - m - 1} |}}{2}}^{l - m - 1}{e^{i({l - m - 1} )q}}} } \right). \end{array}$$

It should be noted that Eqs. (8), (10)–(11) are derived under the paraxial approximation. Thus, the exact electric field and intensity distribution under a tight focusing condition should be calculated directly by the vector diffraction integrals. However, one can imagine, by using Eqs. (8), (10)–(11), how all polarization components of the focused CV OAM beam changes under the tight focusing condition.

3. Numerical calculation results for focused CV OAM electric fields

The formation of a superposed LG mode has been confirmed by the numerical calculation. The vector diffraction integrals based on Stratton-Chu’s approach [31,32] were used to calculate all polarization components with the helical phase of input beam. In calculations, it is assumed that the x-polarized beam passing through RAPP and SPP is immediately focused by an ideal parabolic mirror. Figure 3(a) shows the calculated electric field and intensity distributions in the focal plane when a linearly- (x-) polarized (m = 0) OAM (l = 1) beam is focused. The focusing condition of f-number of 2 (NA: ∼0.24) is assumed in the calculation. Under the condition of m = 0 and l = 1, the lowest mode for x-polarization is explicitly expressed as ${R_p}{e^{ - {\vartheta _0}R_p^2}}[{{I_0}({{\vartheta_0}R_p^2} )- {I_1}({{\vartheta_0}R_p^2} )} ]{e^{iq}}$ from Eq. (9). The real and imaginary fields are generated by the spiral phase of ${e^{iq}}$ and the intensity distribution becomes annular. Figure 3(b) shows the calculated electric field and intensity distributions when the RP (m = 1) OAM (l = 1) beam is focused. In this case, from Eq. (8), the real and imaginary parts of the x-polarization component become $LG_0^2\sin 2q$ and $- ({{{\vartheta_0^2} \mathord{\left/ {\vphantom {{\vartheta_0^2} {2\pi }}} \right.} {2\pi }}} )LG_0^0 - ({{{\vartheta_0^4} \mathord{\left/ {\vphantom {{\vartheta_0^4} {6\pi }}} \right.} {6\pi }}} )LG_0^2\cos 2q$, respectively. Therefore, the real part is the LG sine mode and the imaginary part is the superposition of Gaussian and LG cosine modes. This is clearly seen by numerical simulation in Fig. 3(b). Contrary to the LP beam case, the $LG_0^0$ is the strongest field component for the RP vector OAM beam and the intensity distribution becomes a modified Gaussian shape as shown in Fig. 3(b).

Fig. 3. Electric field and Intensity distributions. (a) LP OAM and (b) RP OAM beams. The wavelength (λ) is 0.8 µm, and horizontal and vertical dimension of the calculation window is (12.2λ)×(12.2λ). Red color in the field means a positive value and blue means a negative value.

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The y- and z-polarization components are usually ignored under normal focusing conditions. However, the intensity for those components becomes crucial under tight focusing conditions and modifies the overall intensity distribution. The electric field and intensity distributions for the induced y- and z-polarizations can be analyzed in the same way with Eqs. (10) and (11). For an x-polarized LP (m = 0) OAM (l = 1) beam, the first and third azimuthal orders are induced for y-polarization due to the P-P coupling ($|{l \pm m + 2} |$ and $|{l \pm m - 2} |$), and the zeroth and second azimuthal orders are induced for z-polarization due to the P-P coupling ($|{l \pm m + 1} |$ and $|{l \pm m - 1} |$). The resultant electric fields are superpositions of those modes. For the RP (m = 1) OAM (l = 1) beam, the lowest orders for y-polarized field is determined by the P-P coupling that generate azimuthal orders ($|{l - m + 2} |$ and $|{l + m - 2} |$). But, the field strength of the y-component induced by ${E_x}$ from Eq. (10) is so weak as to be neglected when compared to the field strength directly generated by ${E_y}$ obtained using Eq. (5). This is why the y-polarization component in Fig. 3(b) shows the same shape as the x-polarization component. Finally, the azimuthal order of 1 becomes dominant for the z-polarized component, as shown in Eq. (11).

The generation of the superposition mode described in Eqs. (8), (10)–(11) can be understood graphically as shown in Fig. 4. The left side of Fig. 4 represents a superposition mode, and the right side does modes that create the superposition mode. Figure 4(a) shows the superposition of the zeroth and second azimuthal modes shown in Fig. 3. Figure 4(b) and 4(c) show the superposition of the first and third azimuthal modes but with different orientations. As shown in Fig. 4(b), the electric field for y-polarization of Fig. 3(a) is the superposition mode of $|{l - m} |= 1$ and $|{l + m} |= 3$ modes which are generated by P-P coupling. It is also confirmed that the electric field distribution of Fig. 4(c) can be also generated by superposing the first and third orders appearing in a RP (m = 1) OAM beam with l = 2. Thus, the electric field distributions of the focused vector OAM beam are well explained by the superposition of LG modes induced by the P-P coupling.

Fig. 4. Graphical calculation of superposed modes. The calculation window is (12.2λ)×(12.2λ) as in Fig. 3.

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It is also interesting to examine the propagation property of OAM beam near the focal plane. Figures 5 and 6 show how the electric field changes as the LP and RP OAM beams propagate near the focal plane. The f-number of 2 is used in the calculations by assuming the normal focusing condition. The field distribution at different locations is calculated by multiplying the propagation phase factor (${e^{ - i\omega t}}$) as in Ref. [32]. For an LP OAM beam, the real and imaginary components of x-polarization component show a typical helical propagation (see Fig. 5). However, in the case of a RP OAM beam, the real and imaginary parts show complex propagation characteristics due to the superposition of modes generated by the P-P coupling (see Fig. 6). The exact field description of the propagating OAM beams will be crucial for characterizing the laser-matter interactions using high-intensity OAM laser beams to generate secondary vortex beams or mega-gauss axial magnetic field. In general, an OAM beam with a different azimuthal order shows a different size of annular beam shape. This property can be used as an evidence for OAM exchange between photons and electrons. A reflected laser pulse changes its OAM state from $|{l \pm m} |$ to $|{l - 2 \pm m} |$ after laser-matter interaction. The exchange of OAM can be verified by monitoring the change in laser mode (size of annular shape) before and after interacting with electrons.

Fig. 5. Electric fields and intensities at different positions showing the propagation property of the linearly-polarized OAM beam with (a) a TC of 1 and (b) a TC of 2. The calculation window is (12.2λ)×(12.2λ) as in Fig. 3.

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Fig. 6. Electric fields and intensities at different positions showing the propagation property of the cylindrical vector OAM beam with (a) a TC of 1 and (b) a TC of 2. The calculation window is (12.2λ)×(12.2λ) as in Fig. 3.

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4. Conclusion

In conclusion, the analytical formulas describing the focused electric field distribution of CV OAM beams with an arbitrary PH index and TC have been derived and the P-P coupling of the focused CV OAM beam has been investigated. The P-P coupling occurs when focusing the CV OAM beam and generates coupling modes having different radial and azimuthal orders. The resultant electric field is determined by the superposition of these modes. Over the past few years, the use of OAM beams is of growing interest in laser-plasma interaction studies. Therefore, the results described in the paper will be useful in understanding physical mechanisms and focusing properties of secondary OAM beams generated during the high-intensity laser-matter interactions using the CV OAM beams.

Funding

European Regional Development Fund (CZ.02.1.01/0.0/0.0/15_003/0000449, CZ.02.1.01/0.0/0.0/16_019/0000789).

Acknowledgments

High Field Initiative (CZ.02.1.01/0.0/0.0/15_003/0000449) from European Regional Development Fund and Advanced research using high intensity laser produced photons and particles (CZ.02.1.01/0.0/0.0/16_019/0000789).

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Generation of superposition modes by polarization-phase coupling in a cylindrical vector orbital angular momentum beam

Abstract

1. Introduction

2. Focused electric field of CV OAM beam under the paraxial approximation

3. Numerical calculation results for focused CV OAM electric fields

4. Conclusion

Funding

Acknowledgments

References

Cited By

Figures (6)

Equations (11)

OSA Continuum