Efficient high-charge Laguerre-Gaussian mode conversion by using a periscopic axicon mirror

S. J. Zhang; S. J. Zhang; H. B. Zhuo; H. B. Zhuo; Y. Yin; D. B. Zou; N. Zhao; N. Zhao; W. M. Zhou; W. M. Zhou

doi:10.1364/OE.452499

1. Introduction

The angular momentum of a light beam is of fundamental interest. It has a spin part associated with polarization and an orbital part associated with spatial distribution of the light field. Any optical vortex whose transverse field distribution contains the complex component exp(ilϕ) would have an orbital angular momentum (OAM) of l${\hbar}$ per photon, where ϕ is the azimuthal coordinate and integer l is the topological charge (or azimuthal mode index) [1]. Due to the spatial phase singularity on the propagation axis, the field here vanishes to satisfy the continuity condition, which results in a donut intensity distribution. Optical vortex beams have become a powerful optical tool in applications fields ranging from optical manipulation [2–4], optical communication [5,6], high-resolution microscopy [7] to laser process [7,8]. Especially, high-charge optical vortex beams can be applied in high-angular-momentum quantum entanglement [9] and high-precision spatial resolution up to the picometer range [10].

Among all vortex beam modes, the Laguerre-Gaussian (LG) beams, as the eigen solutions of the paraxial Helmholtz equation in cylindrical coordinate, are most widely investigated. LG beams can be effectively converted from Hermite-Gaussian (HG) beams via two cylindrical lenses [11], but it relies on the generation of HG beams first. By contrast, direct conversion from conventional fundamental laser beams is simpler and more reliable. The stack of wedge can transform the fundamental Gaussian beam into vortical beam which preserves its structure when it is focused by the circular aperture. However, the topological charge of this vortex is determined by the number of wedges, which impedes the scheme from generating vortex with rather high charge [12,13]. The spiral phase plates (SPP) [14,15] or spatial light modulators (SLM) [16] are the most widely-used spiral phase optics to shape the fundamental Gaussian beam into a Gaussian vortex (GV). In a GV, the spiral phase-adjusted term exp(iLϕ) is imposed into the field of fundamental Gaussian beam, where L represents the topological charge of these phase-adjusted optics. In low-order cases, the GV can be expanded into a distribution of LG modes centered around a dominant mode. It has been proven that for the lowest-order GV of L = 1, the dominant LG mode (of azimuthal mode index l = 1 and radial mode index p = 0) occupies 93.08% the total energy [17], and the purity of dominant mode can be further increased up to over 99% by spatial filtering other redundant modes [18]. However, in high-order cases, because the transverse intensity of GV deviates severely from that of LG mode, the LG components of a high-order GV become more and more complex as the order increases. It still remains a challenge to convert the fundamental Gaussian beam to an extremely high-order LG mode with satisfactory conversion efficiency and mode purity.

In this paper, we first study the mode decomposition of GV of different order. It is found that with the increase of the order of GV, the ratio of the dominant LG component significantly reduces. To improve the conversion efficiency from the fundamental Gaussian mode to a high-order LG mode with higher purity, we theoretically proposed a three-staged scheme based on the periscopic axicon mirror (PAM) and SPPs. The PAM is able to modulate the transverse intensity of the incident ring beam to better match the transverse intensity of the specified high-order LG beam. We quantitatively discussed the reflection of ring beam in PAM by using the modified Bessel Gaussian (MBG) beam to approximate the LG beam for the first time. The MBG beam is a kind of ring beam which keeps its topological shape unchanged when it is reflected by the PAM, only with its ring radius enlarged. We analytically calculated the optimal relationship of beam parameters to achieve the maximum match of transverse intensity between MBG beam and LG beam. Based on the calculation, the optimal parameters for the PAM are obtained. At last, particle-in-cell simulations are conducted to verify the correctness of our theoretical calculation and the effectiveness of the three-stage scheme. The simulation results show that it is accurate and appropriate to use MBG beam to approximate LG beam to study its reflection in PAM. Our three-stage scheme can efficiently convert the fundamental Gaussian mode into the high-order LG mode, and the optimal parameters is given theoretically, which can provide an effective reference for future experiments.

2. Generation of high-order LG modes

2.1 Mode decomposition of the Gaussian vortex

When a fundamental Gaussian beam passes through a SPP of order $L$, the fundamental Gaussian beam is transformed to the GV of order L

(1)$$G{V_L}({r,\phi ,0} )= L{G_{00}}({r,\phi ,0} )\exp ({iL\phi } ).$$

LG₀₀(r,ϕ,0) is the complex field of fundamental Gaussian beam at focal plane. The field of GV at any distance z can be deduced via Fresnel diffraction [19]. It is found that the beam’s transverse structure changes rapidly at the initial stage of propagation. The GV can be decomposed as a summation of LG modes which form the complete set of orthonormal bases. The normalized complex field of LG beam is

(2)$$\begin{aligned} L{G_{pl}}(r,\phi ,z) &= {C_{pl}}\frac{1}{{w(z)}}{\left[ {\frac{{\sqrt 2 r}}{{w(z)}}} \right]^{|l|}}\exp \left[ { - \frac{{{r^2}}}{{{w^2}(z)}}} \right]L _p^{|l|}\left[ {\frac{{2{r^2}}}{{{w^2}(z)}}} \right]\\ &\times \exp \left[ {\frac{{ik{r^2}}}{{2R(z)}}} \right]\exp [ - i\psi (z)]\exp (il\phi )\exp (ikz). \end{aligned}$$

Here, p is the radial mode index. ${C_{pl}} = \sqrt {2p!/\pi (p + |l|)!}$ is the normalization constant. $w(z) = {w_0}\sqrt {1 + {{({z/{z_r}} )}^2}}$ is the beam radius. $R(z )= z[{1 + {{({{{{z_r}} / z}} )}^2}} ]$ is the curvature radius of the spherical wavefront.$\psi (z )= ({2p + |l |+ 1} )\arctan ({{z / {{z_r}}}} )$ is the Gouy phase. w₀ is the waist radius at focus plane and ${z_r} = {{\pi w_0^2} / \lambda }$ is the Rayleigh length over which the beam radius changes significantly.$L _p^{|l |}$ is associated Laguerre polynomial. For single-ring modes of p = 0, the polynomial $L _0^{|l |} = 1$. Note that in this paper we use the sign convention where the phase shift kz contributes to a positive phase term exp(ikz). The other phase terms are in accordance with this convention.

At the focal plane, R(0) $\infty$, $\psi(0)=0$, so the field of LG beam is simplified to

(3)$$\begin{aligned} L{G_{pl}}(r,\phi ,0) &= {C_{pl}}\frac{1}{{{w_0}}}{\left( {\frac{{\sqrt 2 r}}{{{w_0}}}} \right)^{|l|}}\exp \left( { - \frac{{{r^2}}}{{w_0^2}}} \right)L _p^{|l|}\left( {\frac{{2{r^2}}}{{w_0^2}}} \right)\exp (il\phi )\\ &\equiv A_{pl}^{LG}({r;{w_0}} )\exp (il\phi ). \end{aligned}$$

Here, $A_{pl}^{LG}({r;{w_0}} )$ represents the field amplitude of LG beam at the focal plane. Equation (1) turns to be

(4)$$G{V_L}(r,\phi ,0) = A_{00}^{LG}({r;{w_0}} )\exp (iL\phi ).$$

Theoretically, the GV could be expanded as the summation of LG modes under arbitrary waist radius w₁. It is not physically compulsory to keep w₁ to be the same as w₀ of the incident beam. The conversion efficiency from the GV_L mode of waist radius w₀ to arbitrary ${LG _{{p_1}{l_1}}}$ mode of waist radius w₁ is given by the square of the inner product of their fields:

(5)$$\begin{aligned} \eta &= \left|{\left\langle {A_{{p_1}{l_1}}^{LG}({r;{w_1}} )\exp ({i{l_1}\phi } )| A_{00}^{LG}({r;{w_0}} )\exp (iL\phi )} \right\rangle } \right|^2\\ &= \left|{{{\int_0^\infty {A_{{p_1}{l_1}}^{LG}} }^\ast }({r;{w_1}} )A_{00}^{LG}({r;{w_0}} )rdr} \right|^2 {\left|{\int_0^{2\pi } {\exp } [{i({L - {l_1}} )\phi } ]d\phi } \right|^2} \\ &\equiv {\eta _r}{\eta _\phi }. \end{aligned}$$

The integral is split into two parts $\eta_r$ and $\eta_{\phi}$, which are related to the radial and azimuthal integrals, respectively. The azimuthal integral $\eta_{\phi}$ yields (2π)² only when l₁ = L, otherwise yields 0. Therefore, the decomposed LG modes must have the same topological charge as that of GV_L. The radial integral $\eta_{\phi}$ denotes the coupling of transverse intensity between $A_{00}^{LG}({r;{w_0}} )$ and $A_{{p_1}{l_1}}^{LG}({r;{w_1}} )$. Let l₁ = L, then the intensity coupling between GV_L and ${LG _{{p_1}L}}$ mode is given by

(6)$$\eta = {(2\pi )^2}{\left|{\frac{2}{\pi }\sqrt {\frac{{{p_1}!}}{{({{p_1} + |L|} )!}}} \frac{1}{{{w_0}{w_1}}}\int_0^\infty {{{\left( {\frac{{\sqrt 2 r}}{{{w_1}}}} \right)}^{|L|}}} \exp \left[ { - {r^2}\left( {\frac{1}{{w_0^2}} + \frac{1}{{w_1^2}}} \right)} \right]L_{{p_1}}^L\left( {\frac{{2{r^2}}}{{w_1^2}}} \right)rdr} \right|^2}.$$

According to Eq. (6), Fig. 1 gives the intensity coupling between GV_L mode and ${LG _{{p_1}L}}$ modes of different p₁ under different waist ratio w₁/w₀. Figure 1(a), 1(b), and 1(c) correspond to three cases of L = 1, L = 4 and L = 16, respectively. To achieve the maximal coupling to the dominant mode of p₁ = 0, the optimal waist ratio as the following is to be satisfied [20]

(7)$${{{w_1}} / {{w_0} = {1 / {\sqrt {|L |+ 1} }}}}.$$

Under this optimal waist radius, the GV could be well reconstructed with fewest LG modes. For the lowest-order GV of L = 1, the optimal ratio is w₁/w₀≈0.71 and the dominant LG₀₁ mode occupies 93.08% of the total energy. The LG₂₁ mode occupies 3.49%, and other modes are negligible. The energy proportion of p = 1 mode is always 0 at the optimal waist ratio. With the topological charge of GV increasing, the conversion efficiency to the dominant mode decrease, while the proportions of other high-order radial modes rise evidently. For example, when L = 6, the optimal ratio of waist radius is 0.24, and the corresponding energy proportion of the dominant mode dramatically drops to 42.82%. The LG₂ ₁₆ mode occupies 16.05% of the total energy, which is of the same order of magnitude comparing with the dominant mode and cannot be ignored. The far field diverging angle of LG modes is ${\theta _{pl}} = \left( {\sqrt {l + 2p + 1} {{{w_0}} / {{w_1}}}} \right){\theta _0}$, where ${\theta _0} = {{2{\lambda _0}} / {\pi {w_0}}}$ is the diverging angle of the fundamental Gaussian mode. With L increasing, the optimal w₀/w₁ becomes larger, so the dominant mode diverges faster.

Fig. 1. The dependence of conversion efficiency from the GV_L beam to different ${LG _{{p_1}L}}$ modes on ratio of waist radius w₁/w₀, with the topological charge of (a) L = 1, (b) L = 4, (c) L = 16, respectively. The grey dashed lines correspond to the optimal waist ratios at which the maximal conversion efficiency to the central mode of p₁ = 0 is achieved.

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Using spiral phase optics can introduce vortical phase, but it cannot adjust the transverse intensity of the beam. For the high-order case, due to the serious deviation between the transverse intensity of the GV and that of the high-order LG mode, it is impossible to efficiently convert the fundamental Gaussian beam into high-order LG mode with high purity. In order to solve this problem, the periscopic axicon mirror is used to modulate the transverse intensity of the incident beam.

2.2 Periscopic axicon mirror

Figure 2(a) depicts the phase and intensity modulation in our three-stage scheme. The scheme mainly uses two SPPs and a double periscope axicon. First, the SPP1 of L = 1 converts the fundamental Gaussian beam into the Gaussian vortex. An additional appropriately-sized aperture (not shown in the figure) can be used in Fourier frequency domain to remove the redundant components in the central region to further improve the purity of dominant LG₀₁ mode [18]. Then, the transverse intensity of LG₀₁ mode is modulated by the double periscopic axicon mirror. The left periscopic axicon mirror and the right one are oppositely aligned. Each periscope consists of an inner and an outer cone mirror. The inner cone mirror and the outer cone mirror have the same cone angle α = 45°, share the same axis, and have the reflection plane facing the opposite directions. The periscope transforms the ring beam with a smaller radius into a ring beam with a larger radius, or vice versa, depending on which side the beam comes from. Finally, the SPP2 transforms the output radius-adjusted ring beam into a high-order LG beam. The detailed structure of the SPP is shown in Fig. 2(b), which has a continuous spiral increase in thickness. Figure 2(c) gives the longitudinal slice of the double conical periscope. The two inner mirrors are completely symmetric. The outer mirror can slide freely along the axis, so the mirror distance D between the reflection surface of the inner and the outer mirror along radial direction is continuously adjustable.

Fig. 2. (a) The schematic diagram. The system is mainly composed of two SPPs and a double periscope axicon. (b) The detailed structure of the SPP with L = 4, where different colors represent different phase variations. (c) The longitudinal section of the double periscopic axicon. The inner mirrors are symmetric about z = 0 plane. The outer axicon mirror is movable longitudinally so as to adjust the mirror distance D.

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The light ray is originally incident at r = r₀, as the black dashed arrow shows in Fig. 2(c). After the light ray passes through the double periscopic axicon, the radial position becomes r = r₀ + ΔD. Here, ΔD = D₁ - D₂ is the difference of mirror distance between the left and the right periscope. The transverse intensity of the output beam from conical periscope varies with ΔD. The periscopic axicon with the optimal ΔD can modulate the incident low-order LG beam into a ring beam whose transverse intensity matches well with that of the desired high-order LG mode. To avoid that the inner mirror blocks the light beam reflected by the outer mirror of the left periscope, D₁ must exceed a certain value D’. If the optimal ΔD satisfies ΔD > D’, then the right periscope can be removed, in which case ΔD = D₁. Otherwise, the right periscope is used to assist the left one to adjust ΔD according to ΔD = D₁ - D₂.

2.3 Optimal parameter relationship between LG0 and MBG0

In order to study the reflection of the incident LG beam in the periscopic axicon mirror, we use the modified Bessel Gaussian (MBG) beam to approximate the LG beam. The MBG beam is also a kind of vortical ring beam. The m-order MBG beam can be analytically expressed as the equally-weighted superposition of infinite fundamental Gaussian beams whose axes are homogenously distributed on the cylindrical surface oriented in z direction. Each compositing beam at ϕ’ is loaded with a phase difference exp(imϕ’). Suppose the focal spots of these compositing beams are on the z = 0 plane, and the radius of the cylinder is R_d, then the field of MBG beam at the focal plane is derived by integrating over all the component beams along the azimuthal direction [21,22]:

(8)$$\begin{aligned} MB{G_{m,{R_d}}}(r,\phi ,0) &= C\int_0^{2\pi } {\exp } \left[ { - \frac{{R_d^2 + {r^2} - 2{R_d}r\cos ({{\phi^\prime } - \phi } )}}{{w_0^2}}} \right]\exp ({im{\phi^\prime }} )\frac{{d{\phi ^\prime }}}{{2\pi }}\\ &= {C_m}\frac{1}{{{w_0}}}\exp \left( { - \frac{{R_d^2}}{{w_0^2}}} \right)\exp \left( { - \frac{{{r^2}}}{{w_0^2}}} \right){I _m}\left( {\frac{{2{R_d}r}}{{w_0^2}}} \right)\exp (im\phi )\\ &\equiv A_m^{MBG}({r;{R_d},{w_0}} )\exp (im\phi ). \end{aligned}$$

Here, ${[{R_d^2 + {r^2} - 2{R_d}r\cos ({\phi^{\prime} - \phi } )} ]^{1/2}}$ is the distance between the compositing beam’s focal spot at (R_d, ϕ’) and the field point at (r, ϕ). I_m is the m-order modified Bessel function.${C_m} = \sqrt {{2 / \pi }} \exp ({{{R_d^2} / {2w_0^2}}} ){[{{I_m}({{{R_d^2} / {w_0^2}}} )} ]^{ - 1/2}}$ is the normalized constant.$A_m^{MBG}({r;{R_d},{w_0}} )$ represents the field amplitude at z = 0. According to Fresnel diffraction theory, the field at any longitudinal position is [22]

(9)$$\begin{aligned} MB{G_{m,{R_d}}}(r,\phi ,z) &= {C_m}\frac{1}{{{w_0}}}{q^2}\exp \left( { - {q^2}\frac{{R_d^2}}{{w_0^2}}} \right)\exp \left( { - {q^2}\frac{{{r^2}}}{{w_0^2}}} \right)\\ &\times {I _m}\left( {{q^2}\frac{{2{R_d}r}}{{w_0^2}}} \right)\exp (im\phi )\exp (ikz). \end{aligned}$$

The diffraction behavior of the beam is contained in parameter ${q^2} = {({1 + iz/{z_r}} )^{ - 1}}$.

Due to the unique superposition feature of MBG beam, its reflection in periscopic conical mirror is equivalent to that its component beams are reflected respectively. Owing to the annular symmetry, the output beam is still a MBG beam composed of these reflected component beams. For the left periscopic conical mirror, after the successive reflections by the inner mirror and the outer mirror, the radial position of the axes of the component beams changes from R_d to R_d + D₁. The corresponding input beam $\textrm{MB}{\textrm{G}_{m,{R_d}}}$ turns to be the output beam $\textrm{MB}{\textrm{G}_{m,{R_d} + {D_1}}}$, and the focal plane of the output beam falls behind that of the incident beam by a distance of D₁. The lag-behind of the focal plane intuitively originates from the radial propagation of MBG beam between the inner and outer mirror. This is different from propagation in thin optical elements. Similarly, for the double conical periscope, the output beam becomes $\textrm{MB}{\textrm{G}_{m,{R_d} + \varDelta D}}$, and the focal plane lags behind by a distance of D₁ + D₂. In order to avoid the diffraction of incident beam at the apex of the conical mirror, the parameter ${R_d}$ of the incident MBG beam should at least meet ${R_d} \ge \sqrt 2 {w_{MBG}}$, so that the intensity of the component beams at the axis of the conical mirror can be ignored.

The intensity coupling between the single-ring LG beam of order $\; l\; $ and the MBG beam of order $\; m\; $ at z = 0 plane is

(10)$$\eta = {(2\pi )^2}{\left|{\int_0^\infty {A_{0l}^{L{G^\ast }}} ({r;{w_{LG}}} )A_m^{MBG}({r;{R_d},{w_{MBG}}} )rdr} \right|^2}.$$

Take $A_{0l}^{LG}$ in Eq. (3) and $A_m^{MBG}$ in Eq. (8) into Eq. (10), then

(11)$$\begin{aligned} \eta &= {\left[ {2\pi \frac{{{C_{0l}}{C_m}}}{{{w_{LG}}{w_{MBG}}}}\exp \left( { - \frac{{R_d^2}}{{w_{MBG}^2}}} \right)} \right]^2}\\ &\times {\left|{\int_0^\infty {{{\left( {\frac{{\sqrt 2 r}}{{{w_{LG}}}}} \right)}^{|l|}}} \exp \left( { - \frac{{{r^2}}}{{w_{LG}^2}}} \right)\exp \left( { - \frac{{{r^2}}}{{w_{MBG}^2}}} \right){I_m}\left( {\frac{{2{R_d}r}}{{w_{MBG}^2}}} \right)rdr} \right|^2}. \end{aligned}$$

Here, w_LG and w_MBG represent the waist radius of LG beam and MBG beam, respectively. According to the integral

(12)$$\int_0^\infty {{x^{l + 1}}} \exp ({ - {a^2}{x^2}} ){I _m}(bx)dx = {2^{ - 1 - m}}{a^{ - 2 - l - m}}{b^m}\frac{{\Gamma \left( {\frac{{2 + l + m}}{2}} \right)}}{{\Gamma (1 + m)}}M \left( {\frac{{2 + l + m}}{2},1 + m,\frac{{{b^2}}}{{4{a^2}}}} \right),$$

the intensity coupling is analytically expressed as

(13)$$\begin{aligned} \eta &= {\left|{{2^{\frac{l}{2}}}\pi \frac{{\Gamma \left( {\frac{{2 + l + m}}{2}} \right)}}{{\Gamma (1 + m)}}{C_{0l}}{C_m}{{({w_{LG}^2 + w_{MBG}^2} )}^{\frac{{ - 2 - l - m}}{2}}}w_{LG}^{1 + m}w_{MBG}^{1 + l - m}R_d^m} \right|^2}\\ &\times {\left|{\exp \left( { - \frac{{R_d^2}}{{w_{MBG}^2}}} \right)M \left[ {\frac{{2 + l + m}}{2},1 + m,\frac{{R_d^2w_{LG}^2}}{{w_{MBG}^2({w_{LG}^2 + w_{MBG}^2} )}}} \right]} \right|^2}. \end{aligned}$$

Here $\Gamma (x )= \int_0^\infty {{t^{x - 1}}{e^{ - t}}dt}$ is Gamma function. $M ({a,b,x} )= \sum\nolimits_{k = 0}^\infty {{{{{(a )}_k}{x^k}} / {{{(b )}_k}}}k!}$ is Kummer confluent hypergeometric function, and ${(a )_k} = {{\Gamma ({a + k} )} / {\Gamma (a )}}$ is Pochhammer symbol.

According to Eq. (13), Fig. 3 gives the intensity coupling η between the single-ring LG beam with l = 1, l = 4, l = 16 and the MBG beam with m = 1 under different parameters R_d/w_MBG and w_LG/w_MBG. w_MBG is chosen as the normalization coefficient because it is invariant when MBG beam is reflected by the axicon periscope.

Fig. 3. The intensity coupling η between single-ring LG modes of (a) l = 1, (b) l = 4, (c) l = 16, and the MGB beam of order m = 1 under different R_d/w_MBG and w_LG/w_MBG. The cross points of the grey dashed lines represent the optimal parameters to achieve the highest coupling. The line plots in the bottom row are the transverse intensity profiles of the MBG beam (solid blue line) and the LG beam (red dashed line) under the optimal parameters given in each plot.

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First, we solve the optimal parameters R_d and w_MBG of the 1st-order MBG beam that are used to approximate the LG₀₁ beam. Calculating the limit of Eq. (9) under R_d→0, according to the following limit

(14)$$\mathop {\lim }\limits_{{R_d} \to 0} \frac{{{I _m}({2{R_d}r} )}}{{\sqrt {{I _m}({R_d^2} )} }} = \sqrt {\frac{1}{{|m |!}}} {\left( {\sqrt 2 r} \right)^{|m |}}$$

it can be proven that when R_d→0, the m-order MBG beam is identical to the single-ring Laguerre-Gaussian beam of order l = m, p = 0. This is consistent with the plot in Fig. 3(a) which shows that the intensity coupling equals 1.0 when R_d/w_MBG = 0 and w_LG/w_MBG = 1.0. However, these parameters do not satisfy the demand ${R_d} \ge \sqrt 2 {w_{MBG}}$. In Fig. 3(a), the maximal coupling in the region ${{{R_d}} / {{w_{MBG}}}} \ge \sqrt 2$ exactly locates on the line ${{{R_d}} / {{w_{MBG}}}} = \sqrt 2$. The numerical calculation shows that the maximal coupling on that line is η = 98.68%, and the corresponding parameter relationships (marked by the cross points of the grey dashed line) are

(15)$${{{w_{LG0}}} / {{w_{MBG0}}}} = 1.55,$$

(16)$${{{R_{d0}}} / {{w_{MBG0}}}} = \sqrt 2 .$$

Here, we use LG0 to denote the incident $\; \textrm{L}{\textrm{G}_{01}}\; $ beam, and use MBG0 to denote the MBG beam that is used to approximate LG0, as shown in Fig. 1(a). After MBG0 is reflected by the periscopic conical mirror, the outgoing beam is still a 1st-order MBG beam (denoted as MBG1), and its parameters satisfy:

(17)$${w_{MBG1}} = {w_{MBG0}},$$

(18)$${R_{d1}} = {R_{d0}} + \Delta D.$$

2.4 Optimal parameter relationship between MBG1 and LG1

Next, we solve the optimal coupling parameters relationship between the MBG1 beam and the high-order LG beam (denoted as LG1). As shown in Fig. 3(b) and 3(c), for the LG beam of order l = 4 and l = 16, the highest intensity couplings between the 1st-order MBG beam and the LG beam are 99.76% and 99.93%, respectively. As shown in Fig. 4(d), the coupling is infinitely close to 100% when l becomes increasingly larger. In this limiting case, the maximum transverse intensity of the LG beam and the radial position of the maximum are the same as that of the MBG beam. Based on these two criterions, the optimal R_d/w_MBG and w_LG/w_MBG achieving the maximal coupling are analytically solved in the following text.

Fig. 4. (a) the radial position of intensity maximum under different R_d for MBG beam. (b) the optimal R_d/w_MBG. (c) the optimal w_LG/w_MBG. (d) the maximal intensity coupling under the optimal parameters. For all these plots, the black solid lines represent the numerical results, and the red dashed line represents the theoretical asymptotic values given by the formula in each plot.

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For LG beam, take the partial derivative of Eq. (3) with respect to r and calculate the zeros, the radial position of intensity maximum is

(19)$$r_{\max }^{LG} = \sqrt {{l / 2}} {w_{LG}}.$$

The maximal field amplitude here is

(20)$$A_{0l}^{LG}({r_{\max }^{LG};{w_{LG}}} )= \sqrt {\frac{2}{\pi }} \sqrt {\frac{1}{{l!}}} \frac{1}{{{w_{LG}}}}{l^{\frac{l}{2}}}\exp \left( { - \frac{l}{2}} \right).$$

For MBG beam, the partial derivative of Eq. (8) with respect to r has no explicit expression of zeros. We can only calculate the approximation value for the case of large l. According Eq. (19), $r_{\max }^{MBG} = r_{\max }^{LG}$ is a large value, so the argument of ${I _1}({{{2{R_d}r_{\max }^{MBG}} / {w_0^2}}} )$ is also a large value. Taking the limit

(21)$${I _1}(x )\to \exp (x ){({2\pi x} )^{ - {1 / 2}}}$$

into Eq. (8), the radial position of intensity maximum for MBG beam can be explicitly derived to be

(22)$$r_{\max }^{MBG} = \left( {{R_d} + \sqrt {R_d^2 - w_{MBG}^2} } \right)/2 \to [{{R_d}/{w_{MBG}} - 1/({4{R_d}/{w_{MBG}}} )} ]{w_{MBG}}.$$

As shown in Fig. 4(a), it is in good consistence with the precise numerical value. At $r = r_{\max }^{MBG}$, using the limit (21) again, the maximal field amplitude of MBG beam is

(23)$$A_1^{MBG}({r_{\max }^{MBG};{R_d},{w_{MBG}}} )= \sqrt {\frac{2}{\pi }} \frac{1}{{{w_{MBG}}}}{\left( {2\pi \frac{{R_d^2}}{{w_{MBG}^2}}} \right)^{ - \frac{1}{4}}}{\left( {2 - \frac{{w_{MBG}^2}}{{2R_d^2}}} \right)^{ - \frac{1}{2}}}.$$

Based on $r_{\max }^{LG} = r_{\max }^{MBG}$ and $A_{0l}^{LG}({r_{\max }^{LG};{w_{LG}}} )= A_1^{MBG}({r_{\max }^{MBG};{R_d},{w_{MBG}}} )$, and using Stirling's approximation for large l

(24)$$l! \to \sqrt {2\pi l} {({l/e} )^l},$$

the optimal parameters to achieve the highest intensity coupling between the 1st-order MBG beam and the LG beam of order $\; l\; $ are derived to be

(25)$$\frac{{{R_{d1}}}}{{{w_{MBG1}}}} = \frac{{\sqrt l + \sqrt {l + 1} }}{2},$$

(26)$$\frac{{{w_{LG1}}}}{{{w_{MBG1}}}} = \sqrt 2 .$$

As shown in Fig. 4(b) and 4(c), the optimal parameters match well with the precise numerical results for large l.

According to Eq. (15)–18), (25) and (26), the optimal ΔD of periscopic axicon mirror to achieve highest conversion efficiency from LG₀₁ mode to high-order LG_0l mode is

(27)$$\varDelta D = {R_{d1}} - {R_{d0}} \approx \frac{\left( {\sqrt{l} + \sqrt{l + 1} - 2\sqrt{2}} \right){w_{LG0}}}{3.1},$$

and the waist radius of the output LG₀₁ beam with respect to the waist radius of LG₀₁ is

(28)$${w_{LG1}} \approx \frac{{\sqrt 2 {w_{LG0}}}}{{1.55}} \approx 0.91{w_{LG0}}.$$

From Fig. 3(c), it is shown that the parameter range in which the conversion efficiency is close to 1 exists in a long and narrow region. It indicates that the small deviation of ΔD from the optimal value would not significantly affect the effective conversion to high-order LG mode.

Under the optimal parameters, the conversion efficiency from the fundamental Gaussian mode to LG₀₁ mode is about 93.08%, and the conversion efficiency from LG₀₁ mode to high-order LG_0l mode is about 98.68%. The total conversion efficiency from the fundamental Gaussian beam to LG_0l mode is η = 93.08% × 98.68% = 91.85%. This conversion efficiency is much higher than the case without axicon periscope. The waist radius of LG_0l with respect to the waist radius of the fundament Gaussian beam is w_LG ≈ 0.91w_LG₀ ≈ 0.91 × 0.71w₀ = 0.64w₀. It is a constant value independent of order l. Compared with waist radius of the central mode of high-order Gaussian vortex in Eq. (7), the waist radius of output beam is larger after using the periscopic axicon mirror, so the divergency angle is smaller.

2.5 Numerical simulation

Among the three steps of our scheme, the validity of first step that transforms the fundamental Gaussian beam into LG₀₁ mode has been demonstrated via both the theory and experiment [18]. The third step that introduces spiral phase by means of SPP has also been widely proven. In our simulations, we are only concerned about the second step that whether LG₀₁ beam would propagate in the pcpectation. We use three-dimensional (3D) particle-in-cell (PIC) code EPOCH [23] to conduct the simulations. The particle-in-cell method is extensively used to study the interaction of laser and plasma, in which the standard finite-difference time-domain method (FDTD) is adopted to solve Maxwell’s equation. When the laser intensity is not enough to drive electrons to make relativistic motion, the nonlinear effect can be ignored, and the response of high-density plasma to laser is just like that of ordinary optical medium.

The incident LG₀₁ beam is set to be azimuthally polarized. The left periscopic axicon mirror is used in the simulation to reflect the incident pulse. The apex of the inner axicon mirror is at z = −25λ₀, on which plane the incident pulse is focused. The mirror distance between the inner mirror and outer mirror is D₁ = 40λ₀. The incident LG beams of two different waist radius are studied: one is of w_LG₀ = 4.0λ₀, and the other is of w_LG₀ = 8.0λ₀. According to Eq. (15)–(18), after being reflected by the axicon periscope, the parameters for the output 1st-order MBG beam are w_MBG₁ = 2.58λ₀, R_d₁ = 43.65λ₀ for the case of w_LG₀ = 4.0λ₀, and w_MBG₁ = 5.16λ₀, R_d₁ = 47.30λ₀ for the case of w_LG₀ = 8.0λ₀. The focal plane is at z = −65λ₀, delayed by D₁ with respect to the original focus plane. The transverse field E_ϕ and intensity distribution of the output beam at z = 0, which corresponds to the z = 65λ₀ plane in Eq. (9), are shown in Fig. 5. It can be seen that for both the two cases, the simulation results match well with the theoretical predictions. Especially for the case of w_LG₀ = 4.0λ₀, even though the diffraction is evident during the propagation, the diffraction details still fit well with the theory. It confirms that using the MBG beam to approximate the LG beam to study its reflection in the axicon mirror is effective.

Fig. 5. Field E_ϕ (on the left) and intensity distribution (on the right) of the output beams after the azimuthally polarized LG₀₁ beam is reflected by the axicon periscope. The plots on the top row are for the case of w_LG₀ = 4.0λ₀ and on the bottom row are for the case of w_LG₀ = 8.0λ₀. Plots (I) are simulation results. Plots (II) are theoretical results. Plots (III) are lines data extracted at $X=0$, where red dashed lines are theoretical results and black solid lines are simulation results. The data in all these plots are normalized by their own maximal absolute values.

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3. Conclusion

In this paper, we proposed a three-staged phase-modulated scheme to effectively generate high-order LG beam with high purity directly from the fundamental Gaussian beam. Compared with the scheme with only SPPs, an axicon periscope is added to make the transverse intensity of the output ring beam well match that of the desired high-order LG beam, so as to greatly improve the conversion efficiency and mode purity to the high-order LG mode. With the advent of nanofabrication in recent years, there are several techniques that manage to fabricate the high-order SPP with nano-level planeness. For example, the spiral phase mirrors of extremely high order can be produced by direct machining of the surface of an aluminum disk with an ultra-precision single-point diamond lathe [24]. Our simulations show that the accuracy of high-order phase modulation is acceptable at least at L = 32. The enlargement of the beam radius of output beam via PAM lowers the requirement of high spatial resolution at the exact center of SPP, which makes the experiment simpler and more feasible. The generated high-charge optical vortices can be applied in precise metrology or in high-angular-momentum quantum entanglement.

It is particularly pointed out that the PAM in our scheme is appliable to adjust the radius of the beam with near relativistic intensity. Even though the reflective mirror is ionized by the laser pulse instantaneously, as long the topological structure of the mirror is not destroyed, the plasma mirrors with high damage threshold are still able to specularly reflect the incident beam. This is an evident advantage of PAM over the commercially available diffractive beam expanders with low damage threshold.

Funding

National Natural Science Foundation of China (11774430, 11775202, 12075157, 12175310).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Efficient high-charge Laguerre-Gaussian mode conversion by using a periscopic axicon mirror

Abstract

1. Introduction

2. Generation of high-order LG modes

2.1 Mode decomposition of the Gaussian vortex

2.2 Periscopic axicon mirror

2.3 Optimal parameter relationship between LG0 and MBG0

2.4 Optimal parameter relationship between MBG1 and LG1

2.5 Numerical simulation

3. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (5)

Equations (28)

Optics Express