Abstract

Based on the phased-shifted interference between supermodes, a novel method that can directly convert LP01 mode to orbital angular momentum (OAM) mode in a dual-ring microstructure optical fiber is proposed. In this fiber, the resonance between even and odd HE11 modes in inner ring and higher order mode in outer ring will form two pairs of supermodes, and the intensities and phases of the complete superposition mode fields for the involved supermodes created by the resonance at different wavelengths and propagating lengths are investigated and exhibited in this paper. We demonstrate that OAM mode can be generated from π/2-phase-shifted linear combinations of supermodes, and the phase difference of the even and odd higher order eigenmodes can accumulate to π/2 during the coupling process, which is defined as “phase-shifted” conversion. We build a complete theoretical model and systematically analyze the phase-shifted coupling mechanism, and the design principle and optimization method of this fiber are also illustrated in detail. The proposed microstructure fiber is compact, and the OAM mode conversion method is simple and flexible, which could provide a new approach to generate OAM states.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Orbital angular momentum beams, which are characterized by a doughnut intensity profile and helical phase front, have drawn immense research interests in the recent past [1,2]. Due to the unique intrinsic spatial orthogonal property, OAM modes have found promising applications in large-capacity fiber optical communications [35]. In various fiber-based OAM applications, the generation of multiple OAM states is always highly desirable. Different approaches to generate OAM states in optical fiber have been proposed and demonstrated, such as helical or twisted special fibers and gratings [612], polarization controller or pressure assistant phase control methods [1317], and fused biconical fiber tapers [1821]. The OAM states are generated through the twisted structures or phase-control devices in the aforementioned approaches. In twisted structures, the helical perturbation of refractive index brings forth the coupling between normal fiber mode and OAM mode. However, the helical index-modulation is not easy to replicate due to the fabrication process. OAM modes can also be generated with the assistant of phase control devices: polarization controllers (PCs) or pressure slabs [1321]. The stress and the PCs change the phase velocities of the two orthogonal components of OAM modes. By adjusting the pressure or the PCs appropriately, the two orthogonal modes can achieve a π/2 phase difference. Acousto-optics gratings which constitute a low loss high purity technique for OAM generation have also been reported [10]. However, due to the phase-control devices and acousto-optic modulation equipment, the experiment setups in the related approaches are relatively complex. Novel compact all-fiber OAM couplers which can convert circularly polarized HE11 mode ($HE_{11}^{even} \pm iHE_{11}^{odd}$) to different OAM modes are also proposed [22,23], while the generation is seriously affected by the polarization state of incident light. Other special polarization-maintaining fibers which can convert LPm1 modes to $OA{M_{ {\pm} m}}$ modes are reported [24,25], and the π/2 phase difference between even and odd LPm1 modes comes from the large effective refractive index difference between two orthogonally decomposed LPm1 modes. Thus, this kind of polarization-maintaining fiber can only generate the same order linearly polarized $OA{M_{ {\pm} m}}$ modes, and the incident light is confined to LPm1 modes. Hence, compact and simple all-fiber OAM devices which offer the potential for direct generation of OAM modes in optical fiber from a regular fiber-mode input are still highly desirable and deeply significant.

In this paper, we propose a novel method to directly convert LP01 mode to OAM mode in a dual-ring microstructure optical fiber based on the phased-shifted interference between supermodes. The proposed microstructure optical fiber is composed of two high refractive index rings and two axisymmetric rods between the inner and outer rings. The LP01 mode in inner ring can be coupled to higher order modes in outer ring under certain conditions, and the resonance between even and odd HE11 modes and higher order mode will form two pairs of supermodes. We demonstrate that OAM mode can be generated from π/2-phase-shifted linear combinations of supermodes, and the phase difference of the even and odd higher order eigenmodes can accumulate to π/2 during the coupling process, which is defined as “phase-shifted” conversion. The effective refractive indices for different resonant eigenmodes and the corresponding supermodes are investigated, and the intensities and phases of the complete superposition mode fields for the involved supermodes created by the resonance between HE11 and higher order modes at different wavelengths and propagating lengths are investigated and exhibited in this paper. We build a complete theoretical model and systematically analyze the phase-shifted coupling mechanism, and the design principle and optimization method of this fiber are also illustrated in detail. The proposed microstructure fiber is compact, and the OAM mode conversion method is simple and flexible, which could provide a new approach to generate OAM states.

2. Principle and structure

The OAM generation in the proposed fiber is based on the phased-shifted interference between supermodes. The proposed fiber can be core-aligned and spliced with the single mode fiber. In single mode fiber, linearly polarized LP01 mode consists of two degenerate orthogonal modes: $HE_{11}^{even}$ and $HE_{11}^{odd}$ mode. When LP01 mode propagates from single mode fiber to the proposed fiber, the two HE11 modes can be coupled to higher order HE or EH modes under certain conditions. Here we take HE mode as an example to systematically analyze and explain the phase-shifted coupling mechanism in the proposed fiber. As each of the HE mode has two-fold orthogonal degeneracy, $HE_{11}^{even}$ (or $HE_{11}^{odd}$) mode can only be coupled to $HE_{mn}^{even}$ (or $HE_{mn}^{odd}$) mode, and the coupling mechanism between the two pair of orthogonal modes are similar.

When the incident light propagates from the single mode fiber to the proposed fiber core, according to the coupled-mode theory [26], the electric fields of the $HE_{11}^{even}$ mode and higher order $HE_{mn}^{even}$ mode can be expressed as:

$$E_{11}^{even}(r) = \left( {\cos {\gamma_{even}}z - \frac{{i{\delta_{even}}}}{{{\gamma_{even}}}}\sin {\gamma_{even}}z} \right)\exp (i{\delta _{even}}z)E_{11}^{even}(x,y)\exp [i(\beta _{11}^{even} + \kappa _{11}^{even})z,$$
$$E_{mn}^{even}(r) = \left( {\frac{{i{\kappa_{even}}}}{{{\gamma_{even}}}}\sin {\gamma_{even}}z} \right)\exp ( - i{\delta _{even}}z)E_{mn}^{even}(x,y)\exp [i(\beta _{mn}^{even} + \kappa _{mn}^{even})z.$$
Where $E_{11}^{even}(x,y)$ and $E_{mn}^{even}(x,y)$ are normalized mode fields, $\beta _{11}^{even}$ and $\beta _{mn}^{even}$ are the mode propagation constants of $HE_{11}^{even}$ and $HE_{mn}^{even}$ mode, $\kappa _{11}^{even}$ and $\kappa _{mn}^{even}$ are the self-coupling coefficients, ${\kappa _{even}}$ represents the mode coupling coefficient, and ${\delta _{even}}$ is the phase mismatching coefficient, ${\delta _{even}} = (\beta _{11}^{even} + \kappa _{11}^{even} - \beta _{mn}^{even} - \kappa _{mn}^{even})/2$, ${\gamma _{even}} = \sqrt {{\kappa _{even}}\kappa _{even}^\ast{+} \delta _{even}^2}$.

Thus, the complete field profile across the proposed fiber can be obtained as a combination of the two mode fields:

$${E_{even}}(r) = E_{11}^{even}(r) + E_{mn}^{even}(r) = {E_A}(x,y)\exp (i{\beta _A}z) + {E_B}(x,y)\exp (i{\beta _B}z).$$
Where
$${E_A}(x,y) = \frac{{({\gamma _{even}} - {\delta _{even}})E_{11}^{even}(x,y) + {\kappa _{even}}E_{mn}^{even}(x,y)}}{{2{\gamma _{even}}}},$$
$${E_B}(x,y) = \frac{{({\gamma _{even}} + {\delta _{even}})E_{11}^{even}(x,y) - {\kappa _{even}}E_{mn}^{even}(x,y)}}{{2{\gamma _{even}}}},$$
and $\overline {{\beta _{even}}} = (\beta _{11}^{even} + \kappa _{11}^{even} + \beta _{mn}^{even} + \kappa _{mn}^{even})/2$, ${\beta _A} = \overline {{\beta _{even}}} + {\gamma _{even}}$, ${\beta _B} = \overline {{\beta _{even}}} - {\gamma _{even}}$. ${E_A}(x,y)$ and ${E_B}(x,y)$ are independent of z. Equation (3) show that the total field ${E_{even}}(r)$ is a linear combination of two independent normal mode fields ${E_A}(x,y)$ and ${E_B}(x,y)$ with different propagation constants ${\beta _A}$ and ${\beta _B}$, which are also known as the supermodes in this fiber [23]. If we consider the inner ring and the outer ring as two different optical waveguides, the HE11 modes and the HEmn modes are approximate normal modes for the inner and outer rings respectively, and the variation of the mode field amplitudes can be expressed as the coupling between HE11 and HEmn modes. But for the entire dual-ring fiber, the supermodes are exact the normal mode solutions of this fiber, and the variation of the mode field amplitudes in the inner ring and outer ring can also be expressed as the interference of this pair of supermodes.

Based on coupled-mode theory, the maximum energy transfer between $HE_{11}^{even}$ and $HE_{mn}^{even}$ mode occurs at the coupling length:

$${l_0} = \frac{\pi }{{2{\gamma _{even}}}} = \frac{\pi }{{{\beta _A} - {\beta _B}}}.$$
And the complete energy transfer will happen at a particular optical frequency (defined as the coupling wavelength ${\lambda _c}$) when the phase matching condition is satisfied: ${\delta _{even}} = 0$. Thus, after propagating a distance of $L = (2n + 1) \cdot {l_0}$, the total mode field in the proposed fiber at the coupling wavelength ${\lambda _c}$ can be expressed as:
$${E_{even}}(r) = \frac{{{\gamma _{even}}E_{11}^{even}(x,y)}}{{2{\gamma _{even}}}}({e^{i{\beta _A}L}} + {e^{i{\beta _B}L}}) + \frac{{{\kappa _{even}}E_{mn}^{even}(x,y)}}{{2{\gamma _{even}}}}({e^{i{\beta _A}L}} - {e^{i{\beta _B}L}}) = E_{mn}^{even}(x,y){e^{i{\beta _A}L}}.$$

Similarly, the variation of the electric fields for $HE_{11}^{odd}$ and $HE_{mn}^{odd}$ can also be expressed as the interference of supermodes, as shown in Fig. 1:

$${E_{odd}}(r) = E_{11}^{odd}(r) + E_{mn}^{odd}(r) = {E_C}(x,y)\exp (i{\beta _C}z) + {E_D}(x,y)\exp (i{\beta _D}z).$$
Where
$${E_C}(x,y) = \frac{{({\gamma _{odd}} - {\delta _{odd}})E_{11}^{odd}(x,y) + {\kappa _{odd}}E_{mn}^{odd}(x,y)}}{{2{\gamma _{odd}}}},$$
$${E_D}(x,y) = \frac{{({\gamma _{odd}} + {\delta _{odd}})E_{11}^{odd}(x,y) - {\kappa _{odd}}E_{mn}^{odd}(x,y)}}{{2{\gamma _{odd}}}},$$
and $\overline {{\beta _{odd}}} = (\beta _{11}^{odd} + \kappa _{11}^{odd} + \beta _{mn}^{odd} + \kappa _{mn}^{odd})/2$, ${\beta _C} = \overline {{\beta _{odd}}} + {\gamma _{odd}}$, ${\beta _D} = \overline {{\beta _{odd}}} - {\gamma _{odd}}$. Also, after propagating a distance of $L = \frac{{(2m + 1)\pi }}{{{\beta _C} - {\beta _D}}}$, the superposed electric field pattern for $HE_{11}^{odd}$ and $HE_{mn}^{odd}$ at the coupling wavelength is $HE_{mn}^{odd}$ mode:
$${E_{odd}}(r) = {E_C}(x,y){e^{i{\beta _C}L}} + {E_D}(x,y){e^{i{\beta _D}L}} = E_{mn}^{odd}(x,y){e^{i{\beta _C}L}}.$$

 figure: Fig. 1.

Fig. 1. Principle of mode conversion between LP01 mode and OAM mode in the proposed fiber. The modal energy distributions of HE11 modes, HE71 modes, and the corresponding supermodes are shown as example images in this figure.

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Therefore, when $L{P_{01}} = HE_{11}^{even} + HE_{11}^{odd}$ mode propagates from single mode fiber to the proposed fiber, the complete field profile can be expressed as:

$$E(r) = {E_A}(x,y){e^{i{\beta _A}z}} + {E_B}(x,y){e^{i{\beta _B}z}} + {E_C}(x,y){e^{i{\beta _C}z}} + {E_D}(x,y){e^{i{\beta _D}z}}.$$

If the phase matching conditions for even and odd modes are satisfied at the same wavelength ${\lambda _c}$, and the propagation constants for each mode at ${\lambda _c}$ satisfy:

$$L = \frac{{(2n + 1)\pi }}{{{\beta _A} - {\beta _B}}}\textrm{ = }\frac{{(2m + 1)\pi }}{{{\beta _C} - {\beta _D}}}\textrm{ = }\frac{{(2p \pm \frac{1}{2})\pi }}{{{\beta _A} - {\beta _C}}} = \frac{{(2q \pm \frac{1}{2})\pi }}{{{\beta _B} - {\beta _D}}},$$
where $n$, $m$, $p$ and q can be any positive integers. Here we define Eq. (13) as “phase-shifted coupling condition”. After propagating a distance of L, the complete mode field in the proposed fiber at the coupling wavelength ${\lambda _c}$ can be calculated:
$$E(r) = E_{mn}^{even}(x,y){e^{i{\beta _A}L}} + E_{mn}^{odd}(x,y){e^{i{\beta _C}L}} = [{E_{mn}^{odd}(x,y) \pm iE_{mn}^{even}(x,y)} ]\cdot {e^{i{\beta _C}L}}.$$
Equation (14) indicates that the complete mode field $E(r)$ can be described as $OAM_{ {\pm} m - 1,n}^ \pm$ mode.

As analyzed above, the phase difference of the even and odd $H{E_{mn}}$ modes can accumulate to π/2 during the coupling process, and we define this phenomenon as “phase-shifted” conversion. The conclusion also indicate that OAM mode can be generated from π/2-phase-shifted linear combinations of supermodes, which could provide a new approach to generate OAM states.

The phase-shifted coupling mechanism and the variation of the electric fields between $H{E_{\textrm{11}}}$ and $E{H_{mn}}$ mode are similar, and LP01 mode can also be converted into $OAM_{ {\pm} m + 1,n}^ \mp$ mode through $E{H_{mn}}$ mode when its phase-shifted coupling condition is satisfied.

Based on the phase-shifted coupling mechanism analyzed above, we specially design and optimize a microstructure optical fiber, whose phase matching conditions for even and odd modes are satisfied at the same wavelength and the propagation constants for each coupled-mode meet the phase-shifted coupling condition, as shown in Fig. 2.

 figure: Fig. 2.

Fig. 2. Cross section of the proposed fiber.

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The proposed microstructure optical fiber is composed of two high refractive index rings and two axisymmetric rods between the inner and outer rings. The background material of this fiber is made of pure silica, which is more compatible to be integrated with conventional optical fibers. As we know, the fundamental mode and higher order modes are orthogonal to each other in circularly symmetric fibers. The two axisymmetric high refractive index rods are specially designed to break the circular symmetry of the dual-ring fiber and act as spatially dependent perturbation for this fiber so that the LP01 mode in the inner ring can be coupled to the higher order modes in the outer ring.

In the design process, the physical parameters of the inner and outer rings are adjusted firstly to ensure that the effective refractive indices (ERI) of the inner LP01 mode and the corresponding outer higher order mode are close to each other. The coupling coefficients and conversion properties between fundamental mode and higher order mode are closely related to the physical parameters of the two axisymmetric rods. The physical parameters n3 and Rhole of the axisymmetric rods are related to the symmetry of the proposed fiber. The cross section of this fiber is close to circular symmetry when Rhole is small and n3 approaches the background refractive index. When n3 gets higher and the physical size Rhole is bigger, the cross section of this fiber is close to axial symmetry, and the corresponding coupling coefficients become larger, so LP01 mode in inner ring can be coupled to higher order mode more easily at a relatively wide wavelength range. However, LP01 mode could be converted to EH and HE modes simultaneously at the same wavelength when the cross section tends to axial symmetry. Thus, the physical parameters n3 and Rhole should be adjusted carefully to ensure the coupling wavelengths of EH and HE modes are not in the same region.

Next, we have optimized the position y of the axisymmetric rods. As we illustrated before, $HE_{11}^{even}$ (or $HE_{11}^{odd}$) mode can only be coupled to $HE_{mn}^{even}$ (or $HE_{mn}^{odd}$) mode, and we should ensure that the phase matching conditions for even and odd modes are satisfied at the same wavelength. Figure 3 shows the optimization method to unify the coupling wavelengths for even and odd modes, and we also take HE modes as an example here. In the proposed fiber, the ERI of fundamental mode always decreases more slowly than higher order modes as wavelength increases, and the coupling wavelengths for even and odd modes can be calculated respectively through a commercial finite element code (COMSOL). Figures 3(a) and (b) indicates that ERI difference between HE11 modes is too large if the coupling wavelength for even mode is shorter than odd mode, while the ERI difference in HEmn modes should be reduced when the coupling wavelength for even mode is longer than odd mode.

 figure: Fig. 3.

Fig. 3. The optimization method to unify the coupling wavelengths for even and odd modes, and we suppose the ERI for even mode is larger than odd mode. (a) The coupling wavelength for even mode is shorter than odd mode. (b) The coupling wavelength for even mode is longer than odd mode. (c) The coupling wavelengths for odd and even modes are the same.

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The ERI difference between even and odd modes depends on the position y of the two axisymmetric rods. The ERI difference between $HE_{11}^{even}$ and $HE_{11}^{odd}$ mode gets larger if the axisymmetric rods are close to inner ring, while the ERI difference between $HE_{mn}^{even}$ and $HE_{mn}^{odd}$ mode becomes larger if the axisymmetric rods are close to outer ring. Therefore, we can adjust the position y to ensure that the phase matching conditions for even and odd modes are satisfied at the same wavelength.

Finally, we calculate the propagation constants for the involved four supermodes (${\beta _A}$, ${\beta _B}$, ${\beta _C}$ and ${\beta _D}$) and try to find an appropriate constant L to satisfy the phase-shifted coupling condition.

Based on the design principle and optimization method illustrated above, we have designed a series of fiber structure parameters for the generation of different OAM modes, as shown in Table 1 (n0 represents the refractive index of pure silica).

Tables Icon

Table 1. Structural parameters for the generation of different orders

The structure parameters to generate every OAM mode in each group are not unique, and Table 1 only shows some typical examples of the design. The radius of the inner ring in Table 1 is close to the fiber core of single mode fiber, which makes it more compatible to be integrated with conventional optical fibers. The materials n1, n2, and n3 can also be replaced by functional material whose refractive index can be modulated.

3. Results and properties

The mode analysis and the propagation properties in the proposed fiber are investigated by full-vector finite-element method. We first built the proposed fiber structure model according to Table 1 and accurately calculated the dispersion curves of each involved supermode. Figure 4 shows the dispersion curves and modal energy distributions of the four supermodes (${\beta _A}$, ${\beta _B}$, ${\beta _C}$ and ${\beta _D}$) created by the resonances between HE11 and HE51 modes. The phase matching conditions for even and odd modes are both satisfied at 1550.3 nm. To verify the possibility to generate OAM mode through the proposed phase-shifted conversion method, the intensities and phases of the superpositions of different supermodes are investigated.

 figure: Fig. 4.

Fig. 4. Dispersion curves and modal energy distributions of the four supermodes created by the resonance between HE11 and HE51 modes.

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As we illustrated before, the energy exchange between $HE_{11}^{even}$ and higher order even mode can be expressed as the interference of supermode A and B, so after propagating a distance of $L = \pi /({\beta _A} - {\beta _B})$, the superposition mode field of A and B can be expressed as:

$${E_{even}}(r) = {E_A}(x,y){e^{i{\beta _A}L}} + {E_B}(x,y){e^{i{\beta _B}L}} = [{{E_B}(x,y) - {E_A}(x,y)} ]{e^{i{\beta _B}L}}.$$
The intensity and phase of ${E_{even}}(r)$ at 1550.3 nm are shown in Fig. 5(d), and we can also obtain the superposition mode field of C and D after propagating a distance of $L = \pi /({\beta _C} - {\beta _D})$:
$${E_{odd}}(r) = {E_C}(x,y){e^{i{\beta _C}L}} + {E_D}(x,y){e^{i{\beta _D}L}} = [{{E_D}(x,y) - {E_C}(x,y)} ]{e^{i{\beta _D}L}},$$
whose intensity and phase are shown in Fig. 5(b).

 figure: Fig. 5.

Fig. 5. The intensities and phases of the superpositions of different supermodes created by the resonance between HE11 and HE51 modes at 1550.3 nm.

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If we can find a constant L which satisfies the condition: $L = \frac{\pi }{{{\beta _A} - {\beta _B}}}\textrm{ = }\frac{\pi }{{{\beta _C} - {\beta _D}}}\textrm{ = }\frac{{\pi /2}}{{{\beta _B} - {\beta _D}}}$, and after propagating L, the superposition mode field of the whole four supermodes at 1550.3 nm can be expressed as:

$$E(r) = [{{E_D}(x,y) - {E_C}(x,y) + i{E_B}(x,y) - i{E_A}(x,y)} ]{e^{i{\beta _D}L}}.$$
The intensity and phase of $E(r)$ are shown in Fig. 5(e), which present a doughnut profile and the $\exp ({\pm} 4i\phi )$ phase dependence.

We also simulated the intensities and phases of superposition mode field of ${E_D}(x,y) + i{E_B}(x,y)$ and ${E_C}(x,y) + i{E_A}(x,y)$, as shown in Figs. 5(a) and (c). The results also indicate that OAM mode could be generated from π/2-phase-shifted linear combinations of supermodes, which could provide a new approach to generate OAM states.

Next, we calculated the propagation constants for the involved four supermodes created by the resonance between HE11 and HE51 modes at 1550.3 nm and try to find an appropriate length L. Table 2 shows the matching coefficients under different transmission lengths according to Eq. (13). Meanwhile, the complete superposition mode fields at 1550.3 nm are simulated by Eq. (12) with the corresponding different propagation distances, as shown in Fig. 6. We calculated the mode purity for the generated OAM mode in outer ring based on Fourier series expansion method [27]. The purity of OAM41 mode generated in outer ring reaches the highest value 99.02% at the propagation length of 273 mm. However, the optical intensity in outer ring is relatively low at 273 mm. Considering both the purity and energy conversion efficiency, we think 268 mm is an appropriate propagation length to meet the phase-shifted coupling condition. This result indicate that the proposed fiber can convert LP01 mode to OAM41 mode based on the theoretical model.

 figure: Fig. 6.

Fig. 6. The intensity profiles, phase distributions, purities and azimuthal phase variations for the generated OAM41 modes in outer ring at 1550.3 nm with different propagation distances.

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Tables Icon

Table 2. The matching coefficients under different transmission lengths

Moreover, we investigate the influence of wavelength deviation on the performance of OAM mode conversion. The coupling wavelength between HE11 and HE51 mode is 1550.3 nm, where the complete energy transfer can happen. When the wavelength deviates from 1550.3 nm, the OAM mode conversion will be affected. The energy transfers between the two pairs of even and odd modes are incomplete, and the transfer ratios could be different.

On the other hand, the propagation constants of the four supermodes vary with wavelength, so the corresponding matching coefficients and the appropriate propagation length will also change. Figures 7(a)-(c) show the intensity profiles, phase distributions, purities and azimuthal phase variations of the generated OAM mode in outer ring at different wavelengths under the same propagation distance of 268 mm. The purity of OAM mode decreases fast and the generated mode in outer ring tends to become HE51 mode when the wavelength is away from 1550.3 nm. We also calculated the intensity profiles and phase distributions of the superposition mode fields of $E(r) = {E_D}(x,y) - {E_C}(x,y) + i{E_B}(x,y) - i{E_A}(x,y)$ at the corresponding different wavelengths, as shown in Figs. 7(d) and (e). The generated mode in outer ring of $E(r)$ still presents a doughnut profile and the $\exp ({\pm} 4i\phi )$ phase dependence in a certain wavelength range. This result indicates that the matching coefficients and the propagation length can no longer satisfy the OAM mode coupling condition, and we can still obtain OAM mode if the propagation length can be adjusted accordingly in a certain wavelength range.

 figure: Fig. 7.

Fig. 7. (a)-(c) The intensity profiles, phase distributions, purities and azimuthal phase variations of the generated OAM41 mode in outer ring at different wavelengths under the same propagation distance of 268 mm; (d)-(e) The intensity profiles and phase distributions of the superposition mode fields $E(r) = {E_D}(x,y) - {E_C}(x,y) + i{E_B}(x,y) - i{E_A}(x,y)$ at the corresponding different wavelengths.

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However, when the wavelength is far away from 1550.3 nm, the coupling between HE11 and HE51 modes becomes weak, and the energy transfer ratios decrease fast and could be different for odd and even modes. As a result, the purities of the generated OAM modes in outer ring are very low even if the phase-shifted coupling condition is satisfied, and the energy transfer ratios become the leading factor to affect the generation of OAM mode.

Next, we built several theoretical models to generate different OAM modes according to the physical parameters in Table 1. We calculated each group of supermodes and coupling wavelengths. Considering both the purity and energy conversion efficiency, the corresponding appropriate propagation lengths and matching coefficients of each model are proposed, as shown in Table 3.

Tables Icon

Table 3. The matching coefficients for different resonance modes

The complete superposition mode fields at the coupling wavelengths are simulated by Eq. (12) with different propagation distances, and the OAM mode purities and azimuthal phase variations for the generated modes in outer ring are also calculated, as shown in Figs. 812. $OAM_{ {\pm} \textrm{61}}^ \pm$, $OAM_{ {\pm} \textrm{81}}^ \pm$, $OAM_{ {\pm} \textrm{41}}^ \mp$, $OAM_{ {\pm} \textrm{61}}^ \mp$ and $OAM_{ {\pm} \textrm{81}}^ \mp$ modes can be generated respectively in the proposed fiber after propagating a certain distance at the corresponding coupling wavelength. The physical parameters in each group can also be further optimized to achieve better performance based on the proposed phase-shifted coupling mechanism and design principle. This result indicates that the proposed method could convert LP01 mode to different OAM modes through the interference of supermodes, which provides a new approach to generate OAM states.

 figure: Fig. 8.

Fig. 8. The intensity profiles, phase distributions, purities and azimuthal phase variations for the generated OAM61 modes in outer ring at different propagation distances. The OAM modes are generated from HE71 mode.

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 figure: Fig. 9.

Fig. 9. The intensity profiles, phase distributions, purities and azimuthal phase variations for the generated OAM81 modes in outer ring at different propagation distances (HE91 mode).

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 figure: Fig. 10.

Fig. 10. The intensity profiles, phase distributions, purities and azimuthal phase variations for the generated OAM41 modes in outer ring at different propagation distances (EH31 mode).

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 figure: Fig. 11.

Fig. 11. The intensity profiles, phase distributions, purities and azimuthal phase variations for the generated OAM61 modes in outer ring at different propagation distances (EH51 mode).

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 figure: Fig. 12.

Fig. 12. The intensity profiles, phase distributions, purities and azimuthal phase variations for the generated OAM81 modes in outer ring at different propagation distances (EH71 mode).

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The proposed fiber device is controllable and flexible, for the physical parameters to generate different OAM modes are not unique. Based on the proposed phase-shifted coupling mechanism and design principle, the structure and parameters of this fiber can be adjusted accordingly to realize the generation of different OAM modes, and the coupling wavelength and conversion length can also be modified. Moreover, the materials of the high refractive index rings and axisymmetric rods can also be replaced by optical functional materials whose refractive indices can be modulated by physical parameters. The proposed OAM mode conversion method and fiber structure have advantages of simplicity, compactness and flexibility, which have potential applications in fiber-based OAM generation and conversion systems.

4. Conclusion

In conclusion, we proposed a simple and flexible microstructure optical fiber to convert LP01 mode directly to orbital angular momentum mode. The proposed microstructure optical fiber is composed of two high refractive index rings and two axisymmetric rods between the inner and outer rings. The LP01 mode in inner ring can be coupled to higher order modes in outer ring under certain conditions, and the phase difference of the even and odd higher order eigenmodes can accumulate to π/2 during the coupling process, which is defined as “phase-shifted” conversion. We build a theoretical model and systematically analyze the phase-shifted coupling mechanism and illustrate the design principle and optimization method. The effective refractive indices for different resonant eigenmodes and the corresponding supermodes are investigated, and the intensities and phases of the complete superposition mode fields for the involved four supermodes created by the resonance between HE11 and higher order modes at different wavelengths and propagating lengths are investigated and exhibited in this paper. We also demonstrated that OAM mode can be generated from π/2-phase-shifted linear combinations of supermodes, which could provide a new approach to generate OAM states. The proposed microstructure fiber is compact, and the OAM mode conversion method is simple, which are expected to be applied in fiber-based OAM systems.

Funding

National Natural Science Foundation of China (11704283, 11804250, U1509207); Natural Science Foundation of Tianjin City (19JCQNJC01500, 18JCQNJC71300); National Key Research and Development Program of China (2018YFB1305200).

Disclosures

The authors declare no conflicts of interest.

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5. S. Ramachandran and P. Kristensen, “Optical vortices in fiber,” Nanophotonics 2(5-6), 455–474 (2013). [CrossRef]  

6. C. N. Alexeyev, T. A. Fadeyeva, B. P. Lapin, and M. A. Yavorsky, “Generation and conversion of optical vortices in long-period twisted elliptical fibers,” Appl. Opt. 51(10), C193–C197 (2012). [CrossRef]  

7. X. M. Xi, T. Weiss, G. K. L. Wong, F. Biancalana, S. M. Barnett, M. J. Padgett, and P. J. St. Russell, “Optical Activity in Twisted Solid-Core Photonic Crystal Fibers,” Phys. Rev. Lett. 110(14), 143903 (2013). [CrossRef]  

8. L. Li, S. Zhu, J. Li, X. Shao, A. Galvanauskas, and X. Ma, “All-in-fiber method of generating orbital angular momentum with helically symmetric fibers,” Appl. Opt. 57(28), 8182–8186 (2018). [CrossRef]  

9. C. Fu, S. Liu, Z. Bai, J. He, C. Liao, Y. Wang, Z. Li, Y. Zhang, K. Yang, B. Yu, and Y. Wang, “Orbital Angular Momentum Mode Converter Based on Helical Long Period Fiber Grating Inscribed by Hydrogen–Oxygen Flame,” J. Lightwave Technol. 36(9), 1683–1688 (2018). [CrossRef]  

10. P. Z. Dashti, F. Alhassen, and H. P. Lee, “Observation of Orbital Angular Momentum Transfer between Acoustic and Optical Vortices in Optical Fiber,” Phys. Rev. Lett. 96(4), 043604 (2006). [CrossRef]  

11. C. Fu, S. Liu, Y. Wang, Z. Bai, J. He, C. Liao, Y. Zhang, F. Zhang, B. Yu, S. Gao, Z. Li, and Y. Wang, “High-order orbital angular momentum mode generator based on twisted photonic crystal fiber,” Opt. Lett. 43(8), 1786–1789 (2018). [CrossRef]  

12. C. Zhu, P. Wang, H. Zhao, R. Mizushima, S. Ishikami, and H. Li, “DC-Sampled Helical Fiber Grating and its Application to Multi-Channel OAM Generator,” IEEE Photonics Technol. Lett. 31(17), 1445–1448 (2019). [CrossRef]  

13. S. Li, Z. Xu, R. Zhao, L. Shen, C. Du, and J. Wang, “Generation of Orbital Angular Momentum Beam Using Fiber-to-Fiber Butt Coupling,” IEEE Photonics J. 10(4), 1–7 (2018). [CrossRef]  

14. S. Li, Q. Mo, X. Hu, C. Du, and J. Wang, “Controllable all-fiber orbital angular momentum mode converter,” Opt. Lett. 40(18), 4376–4379 (2015). [CrossRef]  

15. Y. Jiang, G. Ren, Y. Lian, B. Zhu, W. Jin, and S. Jian, “Tunable orbital angular momentum generation in optical fibers,” Opt. Lett. 41(15), 3535–3538 (2016). [CrossRef]  

16. S. Wu, Y. Li, L. Feng, X. Zeng, W. Li, J. Qiu, Y. Zuo, X. Hong, H. Yu, R. Chen, I. P. Giles, and J. Wu, “Continuously tunable orbital angular momentum generation controlled by input linear polarization,” Opt. Lett. 43(9), 2130–2133 (2018). [CrossRef]  

17. C. Zhang, F. Pang, H. Liu, L. Chen, J. Yang, J. Wen, and T. Wang, “Highly efficient excitation of LP01 mode in ring-core fibers by tapering for optimizing OAM generation,” Chin. Opt. Lett. 18(2), 020602 (2020). [CrossRef]  

18. X. Heng, J. Gan, Z. Zhang, J. Li, M. Li, H. Zhao, Q. Qian, S. Xu, and Z. Yang, “All-fiber stable orbital angular momentum beam generation and propagation,” Opt. Express 26(13), 17429–17436 (2018). [CrossRef]  

19. Y. Jiang, G. Ren, Y. Shen, Y. Xu, W. Jin, Y. Wu, W. Jian, and S. Jian, “Two-dimensional tunable orbital angular momentum generation using a vortex fiber,” Opt. Lett. 42(23), 5014–5017 (2017). [CrossRef]  

20. T. Wang, F. Wang, F. Shi, F. Pang, S. Huang, T. Wang, and X. Zeng, “Generation of Femtosecond Optical Vortex Beams in All-Fiber Mode-Locked Fiber Laser Using Mode Selective Coupler,” J. Lightwave Technol. 35(11), 2161–2166 (2017). [CrossRef]  

21. Y. Huang, F. Shi, T. Wang, X. Liu, X. Zeng, F. Pang, T. Wang, and P. Zhou, “High-order mode Yb-doped fiber lasers based on mode-selective couplers,” Opt. Express 26(15), 19171–19181 (2018). [CrossRef]  

22. G. Yin, C. Liang, I. P. Ikechukwu, M. Deng, L. Shi, Q. Fu, T. Zhu, and L. Zhang, “Orbital angular momentum generation in two-mode fiber, based on the modal interference principle,” Opt. Lett. 44(4), 999–1002 (2019). [CrossRef]  

23. W. Huang, Y. Liu, Z. Wang, W. Zhang, M. Luo, X. Liu, J. Guo, B. Liu, and L. Lin, “Generation and excitation of different orbital angular momentum states in a tunable microstructure optical fiber,” Opt. Express 23(26), 33741–33752 (2015). [CrossRef]  

24. Y. Han, Y. Liu, W. Huang, Z. Wang, J. Guo, and M. Luo, “Generation of linearly polarized orbital angular momentum modes in a side-hole ring fiber with tunable topology numbers,” Opt. Express 24(15), 17272–17284 (2016). [CrossRef]  

25. X. Zeng, Q. Li, Q. Mo, W. Li, Y. Tian, Z. Liu, and J. Wu, “Experimental Investigation of LP11 Mode to OAM Conversion in Few Mode-Polarization Maintaining Fiber and the Usage for All Fiber OAM Generator,” IEEE Photonics J. 8(4), 1–7 (2016). [CrossRef]  

26. J. Liu, Photonic Devices (Cambridge University, 2005), Chap. 4.

27. S. Li and J. Wang, “Supermode fiber for orbital angular momentum (OAM) transmission,” Opt. Express 23(14), 18736–18745 (2015). [CrossRef]  

References

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  1. M. J. Padgett, “Orbital angular momentum 25 years on [Invited],” Opt. Express 25(10), 11265–11274 (2017).
    [Crossref]
  2. A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photonics 3(2), 161–204 (2011).
    [Crossref]
  3. L. Zhu and J. Wang, “A review of multiple optical vortices generation: methods and applications,” Front. Optoelectron. 12(1), 52–68 (2019).
    [Crossref]
  4. A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
    [Crossref]
  5. S. Ramachandran and P. Kristensen, “Optical vortices in fiber,” Nanophotonics 2(5-6), 455–474 (2013).
    [Crossref]
  6. C. N. Alexeyev, T. A. Fadeyeva, B. P. Lapin, and M. A. Yavorsky, “Generation and conversion of optical vortices in long-period twisted elliptical fibers,” Appl. Opt. 51(10), C193–C197 (2012).
    [Crossref]
  7. X. M. Xi, T. Weiss, G. K. L. Wong, F. Biancalana, S. M. Barnett, M. J. Padgett, and P. J. St. Russell, “Optical Activity in Twisted Solid-Core Photonic Crystal Fibers,” Phys. Rev. Lett. 110(14), 143903 (2013).
    [Crossref]
  8. L. Li, S. Zhu, J. Li, X. Shao, A. Galvanauskas, and X. Ma, “All-in-fiber method of generating orbital angular momentum with helically symmetric fibers,” Appl. Opt. 57(28), 8182–8186 (2018).
    [Crossref]
  9. C. Fu, S. Liu, Z. Bai, J. He, C. Liao, Y. Wang, Z. Li, Y. Zhang, K. Yang, B. Yu, and Y. Wang, “Orbital Angular Momentum Mode Converter Based on Helical Long Period Fiber Grating Inscribed by Hydrogen–Oxygen Flame,” J. Lightwave Technol. 36(9), 1683–1688 (2018).
    [Crossref]
  10. P. Z. Dashti, F. Alhassen, and H. P. Lee, “Observation of Orbital Angular Momentum Transfer between Acoustic and Optical Vortices in Optical Fiber,” Phys. Rev. Lett. 96(4), 043604 (2006).
    [Crossref]
  11. C. Fu, S. Liu, Y. Wang, Z. Bai, J. He, C. Liao, Y. Zhang, F. Zhang, B. Yu, S. Gao, Z. Li, and Y. Wang, “High-order orbital angular momentum mode generator based on twisted photonic crystal fiber,” Opt. Lett. 43(8), 1786–1789 (2018).
    [Crossref]
  12. C. Zhu, P. Wang, H. Zhao, R. Mizushima, S. Ishikami, and H. Li, “DC-Sampled Helical Fiber Grating and its Application to Multi-Channel OAM Generator,” IEEE Photonics Technol. Lett. 31(17), 1445–1448 (2019).
    [Crossref]
  13. S. Li, Z. Xu, R. Zhao, L. Shen, C. Du, and J. Wang, “Generation of Orbital Angular Momentum Beam Using Fiber-to-Fiber Butt Coupling,” IEEE Photonics J. 10(4), 1–7 (2018).
    [Crossref]
  14. S. Li, Q. Mo, X. Hu, C. Du, and J. Wang, “Controllable all-fiber orbital angular momentum mode converter,” Opt. Lett. 40(18), 4376–4379 (2015).
    [Crossref]
  15. Y. Jiang, G. Ren, Y. Lian, B. Zhu, W. Jin, and S. Jian, “Tunable orbital angular momentum generation in optical fibers,” Opt. Lett. 41(15), 3535–3538 (2016).
    [Crossref]
  16. S. Wu, Y. Li, L. Feng, X. Zeng, W. Li, J. Qiu, Y. Zuo, X. Hong, H. Yu, R. Chen, I. P. Giles, and J. Wu, “Continuously tunable orbital angular momentum generation controlled by input linear polarization,” Opt. Lett. 43(9), 2130–2133 (2018).
    [Crossref]
  17. C. Zhang, F. Pang, H. Liu, L. Chen, J. Yang, J. Wen, and T. Wang, “Highly efficient excitation of LP01 mode in ring-core fibers by tapering for optimizing OAM generation,” Chin. Opt. Lett. 18(2), 020602 (2020).
    [Crossref]
  18. X. Heng, J. Gan, Z. Zhang, J. Li, M. Li, H. Zhao, Q. Qian, S. Xu, and Z. Yang, “All-fiber stable orbital angular momentum beam generation and propagation,” Opt. Express 26(13), 17429–17436 (2018).
    [Crossref]
  19. Y. Jiang, G. Ren, Y. Shen, Y. Xu, W. Jin, Y. Wu, W. Jian, and S. Jian, “Two-dimensional tunable orbital angular momentum generation using a vortex fiber,” Opt. Lett. 42(23), 5014–5017 (2017).
    [Crossref]
  20. T. Wang, F. Wang, F. Shi, F. Pang, S. Huang, T. Wang, and X. Zeng, “Generation of Femtosecond Optical Vortex Beams in All-Fiber Mode-Locked Fiber Laser Using Mode Selective Coupler,” J. Lightwave Technol. 35(11), 2161–2166 (2017).
    [Crossref]
  21. Y. Huang, F. Shi, T. Wang, X. Liu, X. Zeng, F. Pang, T. Wang, and P. Zhou, “High-order mode Yb-doped fiber lasers based on mode-selective couplers,” Opt. Express 26(15), 19171–19181 (2018).
    [Crossref]
  22. G. Yin, C. Liang, I. P. Ikechukwu, M. Deng, L. Shi, Q. Fu, T. Zhu, and L. Zhang, “Orbital angular momentum generation in two-mode fiber, based on the modal interference principle,” Opt. Lett. 44(4), 999–1002 (2019).
    [Crossref]
  23. W. Huang, Y. Liu, Z. Wang, W. Zhang, M. Luo, X. Liu, J. Guo, B. Liu, and L. Lin, “Generation and excitation of different orbital angular momentum states in a tunable microstructure optical fiber,” Opt. Express 23(26), 33741–33752 (2015).
    [Crossref]
  24. Y. Han, Y. Liu, W. Huang, Z. Wang, J. Guo, and M. Luo, “Generation of linearly polarized orbital angular momentum modes in a side-hole ring fiber with tunable topology numbers,” Opt. Express 24(15), 17272–17284 (2016).
    [Crossref]
  25. X. Zeng, Q. Li, Q. Mo, W. Li, Y. Tian, Z. Liu, and J. Wu, “Experimental Investigation of LP11 Mode to OAM Conversion in Few Mode-Polarization Maintaining Fiber and the Usage for All Fiber OAM Generator,” IEEE Photonics J. 8(4), 1–7 (2016).
    [Crossref]
  26. J. Liu, Photonic Devices (Cambridge University, 2005), Chap. 4.
  27. S. Li and J. Wang, “Supermode fiber for orbital angular momentum (OAM) transmission,” Opt. Express 23(14), 18736–18745 (2015).
    [Crossref]

2020 (1)

2019 (3)

C. Zhu, P. Wang, H. Zhao, R. Mizushima, S. Ishikami, and H. Li, “DC-Sampled Helical Fiber Grating and its Application to Multi-Channel OAM Generator,” IEEE Photonics Technol. Lett. 31(17), 1445–1448 (2019).
[Crossref]

L. Zhu and J. Wang, “A review of multiple optical vortices generation: methods and applications,” Front. Optoelectron. 12(1), 52–68 (2019).
[Crossref]

G. Yin, C. Liang, I. P. Ikechukwu, M. Deng, L. Shi, Q. Fu, T. Zhu, and L. Zhang, “Orbital angular momentum generation in two-mode fiber, based on the modal interference principle,” Opt. Lett. 44(4), 999–1002 (2019).
[Crossref]

2018 (7)

S. Wu, Y. Li, L. Feng, X. Zeng, W. Li, J. Qiu, Y. Zuo, X. Hong, H. Yu, R. Chen, I. P. Giles, and J. Wu, “Continuously tunable orbital angular momentum generation controlled by input linear polarization,” Opt. Lett. 43(9), 2130–2133 (2018).
[Crossref]

Y. Huang, F. Shi, T. Wang, X. Liu, X. Zeng, F. Pang, T. Wang, and P. Zhou, “High-order mode Yb-doped fiber lasers based on mode-selective couplers,” Opt. Express 26(15), 19171–19181 (2018).
[Crossref]

L. Li, S. Zhu, J. Li, X. Shao, A. Galvanauskas, and X. Ma, “All-in-fiber method of generating orbital angular momentum with helically symmetric fibers,” Appl. Opt. 57(28), 8182–8186 (2018).
[Crossref]

C. Fu, S. Liu, Z. Bai, J. He, C. Liao, Y. Wang, Z. Li, Y. Zhang, K. Yang, B. Yu, and Y. Wang, “Orbital Angular Momentum Mode Converter Based on Helical Long Period Fiber Grating Inscribed by Hydrogen–Oxygen Flame,” J. Lightwave Technol. 36(9), 1683–1688 (2018).
[Crossref]

S. Li, Z. Xu, R. Zhao, L. Shen, C. Du, and J. Wang, “Generation of Orbital Angular Momentum Beam Using Fiber-to-Fiber Butt Coupling,” IEEE Photonics J. 10(4), 1–7 (2018).
[Crossref]

C. Fu, S. Liu, Y. Wang, Z. Bai, J. He, C. Liao, Y. Zhang, F. Zhang, B. Yu, S. Gao, Z. Li, and Y. Wang, “High-order orbital angular momentum mode generator based on twisted photonic crystal fiber,” Opt. Lett. 43(8), 1786–1789 (2018).
[Crossref]

X. Heng, J. Gan, Z. Zhang, J. Li, M. Li, H. Zhao, Q. Qian, S. Xu, and Z. Yang, “All-fiber stable orbital angular momentum beam generation and propagation,” Opt. Express 26(13), 17429–17436 (2018).
[Crossref]

2017 (3)

2016 (3)

2015 (4)

2013 (2)

X. M. Xi, T. Weiss, G. K. L. Wong, F. Biancalana, S. M. Barnett, M. J. Padgett, and P. J. St. Russell, “Optical Activity in Twisted Solid-Core Photonic Crystal Fibers,” Phys. Rev. Lett. 110(14), 143903 (2013).
[Crossref]

S. Ramachandran and P. Kristensen, “Optical vortices in fiber,” Nanophotonics 2(5-6), 455–474 (2013).
[Crossref]

2012 (1)

2011 (1)

A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photonics 3(2), 161–204 (2011).
[Crossref]

2006 (1)

P. Z. Dashti, F. Alhassen, and H. P. Lee, “Observation of Orbital Angular Momentum Transfer between Acoustic and Optical Vortices in Optical Fiber,” Phys. Rev. Lett. 96(4), 043604 (2006).
[Crossref]

Ahmed, N.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Alexeyev, C. N.

Alhassen, F.

P. Z. Dashti, F. Alhassen, and H. P. Lee, “Observation of Orbital Angular Momentum Transfer between Acoustic and Optical Vortices in Optical Fiber,” Phys. Rev. Lett. 96(4), 043604 (2006).
[Crossref]

Ashrafi, N.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Ashrafi, S.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Bai, Z.

Bao, C.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Barnett, S. M.

X. M. Xi, T. Weiss, G. K. L. Wong, F. Biancalana, S. M. Barnett, M. J. Padgett, and P. J. St. Russell, “Optical Activity in Twisted Solid-Core Photonic Crystal Fibers,” Phys. Rev. Lett. 110(14), 143903 (2013).
[Crossref]

Biancalana, F.

X. M. Xi, T. Weiss, G. K. L. Wong, F. Biancalana, S. M. Barnett, M. J. Padgett, and P. J. St. Russell, “Optical Activity in Twisted Solid-Core Photonic Crystal Fibers,” Phys. Rev. Lett. 110(14), 143903 (2013).
[Crossref]

Cao, Y.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Chen, L.

Chen, R.

Dashti, P. Z.

P. Z. Dashti, F. Alhassen, and H. P. Lee, “Observation of Orbital Angular Momentum Transfer between Acoustic and Optical Vortices in Optical Fiber,” Phys. Rev. Lett. 96(4), 043604 (2006).
[Crossref]

Deng, M.

Du, C.

S. Li, Z. Xu, R. Zhao, L. Shen, C. Du, and J. Wang, “Generation of Orbital Angular Momentum Beam Using Fiber-to-Fiber Butt Coupling,” IEEE Photonics J. 10(4), 1–7 (2018).
[Crossref]

S. Li, Q. Mo, X. Hu, C. Du, and J. Wang, “Controllable all-fiber orbital angular momentum mode converter,” Opt. Lett. 40(18), 4376–4379 (2015).
[Crossref]

Fadeyeva, T. A.

Feng, L.

Fu, C.

Fu, Q.

Galvanauskas, A.

Gan, J.

Gao, S.

Giles, I. P.

Guo, J.

Han, Y.

He, J.

Heng, X.

Hong, X.

Hu, X.

Huang, H.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Huang, S.

Huang, W.

Huang, Y.

Ikechukwu, I. P.

Ishikami, S.

C. Zhu, P. Wang, H. Zhao, R. Mizushima, S. Ishikami, and H. Li, “DC-Sampled Helical Fiber Grating and its Application to Multi-Channel OAM Generator,” IEEE Photonics Technol. Lett. 31(17), 1445–1448 (2019).
[Crossref]

Jian, S.

Jian, W.

Jiang, Y.

Jin, W.

Kristensen, P.

S. Ramachandran and P. Kristensen, “Optical vortices in fiber,” Nanophotonics 2(5-6), 455–474 (2013).
[Crossref]

Lapin, B. P.

Lavery, M. P. J.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Lee, H. P.

P. Z. Dashti, F. Alhassen, and H. P. Lee, “Observation of Orbital Angular Momentum Transfer between Acoustic and Optical Vortices in Optical Fiber,” Phys. Rev. Lett. 96(4), 043604 (2006).
[Crossref]

Li, H.

C. Zhu, P. Wang, H. Zhao, R. Mizushima, S. Ishikami, and H. Li, “DC-Sampled Helical Fiber Grating and its Application to Multi-Channel OAM Generator,” IEEE Photonics Technol. Lett. 31(17), 1445–1448 (2019).
[Crossref]

Li, J.

Li, L.

L. Li, S. Zhu, J. Li, X. Shao, A. Galvanauskas, and X. Ma, “All-in-fiber method of generating orbital angular momentum with helically symmetric fibers,” Appl. Opt. 57(28), 8182–8186 (2018).
[Crossref]

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Li, M.

Li, Q.

X. Zeng, Q. Li, Q. Mo, W. Li, Y. Tian, Z. Liu, and J. Wu, “Experimental Investigation of LP11 Mode to OAM Conversion in Few Mode-Polarization Maintaining Fiber and the Usage for All Fiber OAM Generator,” IEEE Photonics J. 8(4), 1–7 (2016).
[Crossref]

Li, S.

Li, W.

S. Wu, Y. Li, L. Feng, X. Zeng, W. Li, J. Qiu, Y. Zuo, X. Hong, H. Yu, R. Chen, I. P. Giles, and J. Wu, “Continuously tunable orbital angular momentum generation controlled by input linear polarization,” Opt. Lett. 43(9), 2130–2133 (2018).
[Crossref]

X. Zeng, Q. Li, Q. Mo, W. Li, Y. Tian, Z. Liu, and J. Wu, “Experimental Investigation of LP11 Mode to OAM Conversion in Few Mode-Polarization Maintaining Fiber and the Usage for All Fiber OAM Generator,” IEEE Photonics J. 8(4), 1–7 (2016).
[Crossref]

Li, Y.

Li, Z.

Lian, Y.

Liang, C.

Liao, C.

Lin, L.

Liu, B.

Liu, H.

Liu, J.

J. Liu, Photonic Devices (Cambridge University, 2005), Chap. 4.

Liu, S.

Liu, X.

Liu, Y.

Liu, Z.

X. Zeng, Q. Li, Q. Mo, W. Li, Y. Tian, Z. Liu, and J. Wu, “Experimental Investigation of LP11 Mode to OAM Conversion in Few Mode-Polarization Maintaining Fiber and the Usage for All Fiber OAM Generator,” IEEE Photonics J. 8(4), 1–7 (2016).
[Crossref]

Luo, M.

Ma, X.

Mizushima, R.

C. Zhu, P. Wang, H. Zhao, R. Mizushima, S. Ishikami, and H. Li, “DC-Sampled Helical Fiber Grating and its Application to Multi-Channel OAM Generator,” IEEE Photonics Technol. Lett. 31(17), 1445–1448 (2019).
[Crossref]

Mo, Q.

X. Zeng, Q. Li, Q. Mo, W. Li, Y. Tian, Z. Liu, and J. Wu, “Experimental Investigation of LP11 Mode to OAM Conversion in Few Mode-Polarization Maintaining Fiber and the Usage for All Fiber OAM Generator,” IEEE Photonics J. 8(4), 1–7 (2016).
[Crossref]

S. Li, Q. Mo, X. Hu, C. Du, and J. Wang, “Controllable all-fiber orbital angular momentum mode converter,” Opt. Lett. 40(18), 4376–4379 (2015).
[Crossref]

Molisch, A. F.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Padgett, M. J.

M. J. Padgett, “Orbital angular momentum 25 years on [Invited],” Opt. Express 25(10), 11265–11274 (2017).
[Crossref]

X. M. Xi, T. Weiss, G. K. L. Wong, F. Biancalana, S. M. Barnett, M. J. Padgett, and P. J. St. Russell, “Optical Activity in Twisted Solid-Core Photonic Crystal Fibers,” Phys. Rev. Lett. 110(14), 143903 (2013).
[Crossref]

A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photonics 3(2), 161–204 (2011).
[Crossref]

Pang, F.

Qian, Q.

Qiu, J.

Ramachandran, S.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

S. Ramachandran and P. Kristensen, “Optical vortices in fiber,” Nanophotonics 2(5-6), 455–474 (2013).
[Crossref]

Ren, G.

Ren, Y.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Shao, X.

Shen, L.

S. Li, Z. Xu, R. Zhao, L. Shen, C. Du, and J. Wang, “Generation of Orbital Angular Momentum Beam Using Fiber-to-Fiber Butt Coupling,” IEEE Photonics J. 10(4), 1–7 (2018).
[Crossref]

Shen, Y.

Shi, F.

Shi, L.

St. Russell, P. J.

X. M. Xi, T. Weiss, G. K. L. Wong, F. Biancalana, S. M. Barnett, M. J. Padgett, and P. J. St. Russell, “Optical Activity in Twisted Solid-Core Photonic Crystal Fibers,” Phys. Rev. Lett. 110(14), 143903 (2013).
[Crossref]

Tian, Y.

X. Zeng, Q. Li, Q. Mo, W. Li, Y. Tian, Z. Liu, and J. Wu, “Experimental Investigation of LP11 Mode to OAM Conversion in Few Mode-Polarization Maintaining Fiber and the Usage for All Fiber OAM Generator,” IEEE Photonics J. 8(4), 1–7 (2016).
[Crossref]

Tur, M.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Wang, F.

Wang, J.

L. Zhu and J. Wang, “A review of multiple optical vortices generation: methods and applications,” Front. Optoelectron. 12(1), 52–68 (2019).
[Crossref]

S. Li, Z. Xu, R. Zhao, L. Shen, C. Du, and J. Wang, “Generation of Orbital Angular Momentum Beam Using Fiber-to-Fiber Butt Coupling,” IEEE Photonics J. 10(4), 1–7 (2018).
[Crossref]

S. Li, Q. Mo, X. Hu, C. Du, and J. Wang, “Controllable all-fiber orbital angular momentum mode converter,” Opt. Lett. 40(18), 4376–4379 (2015).
[Crossref]

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

S. Li and J. Wang, “Supermode fiber for orbital angular momentum (OAM) transmission,” Opt. Express 23(14), 18736–18745 (2015).
[Crossref]

Wang, P.

C. Zhu, P. Wang, H. Zhao, R. Mizushima, S. Ishikami, and H. Li, “DC-Sampled Helical Fiber Grating and its Application to Multi-Channel OAM Generator,” IEEE Photonics Technol. Lett. 31(17), 1445–1448 (2019).
[Crossref]

Wang, T.

Wang, Y.

Wang, Z.

Weiss, T.

X. M. Xi, T. Weiss, G. K. L. Wong, F. Biancalana, S. M. Barnett, M. J. Padgett, and P. J. St. Russell, “Optical Activity in Twisted Solid-Core Photonic Crystal Fibers,” Phys. Rev. Lett. 110(14), 143903 (2013).
[Crossref]

Wen, J.

Willner, A. E.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Wong, G. K. L.

X. M. Xi, T. Weiss, G. K. L. Wong, F. Biancalana, S. M. Barnett, M. J. Padgett, and P. J. St. Russell, “Optical Activity in Twisted Solid-Core Photonic Crystal Fibers,” Phys. Rev. Lett. 110(14), 143903 (2013).
[Crossref]

Wu, J.

S. Wu, Y. Li, L. Feng, X. Zeng, W. Li, J. Qiu, Y. Zuo, X. Hong, H. Yu, R. Chen, I. P. Giles, and J. Wu, “Continuously tunable orbital angular momentum generation controlled by input linear polarization,” Opt. Lett. 43(9), 2130–2133 (2018).
[Crossref]

X. Zeng, Q. Li, Q. Mo, W. Li, Y. Tian, Z. Liu, and J. Wu, “Experimental Investigation of LP11 Mode to OAM Conversion in Few Mode-Polarization Maintaining Fiber and the Usage for All Fiber OAM Generator,” IEEE Photonics J. 8(4), 1–7 (2016).
[Crossref]

Wu, S.

Wu, Y.

Xi, X. M.

X. M. Xi, T. Weiss, G. K. L. Wong, F. Biancalana, S. M. Barnett, M. J. Padgett, and P. J. St. Russell, “Optical Activity in Twisted Solid-Core Photonic Crystal Fibers,” Phys. Rev. Lett. 110(14), 143903 (2013).
[Crossref]

Xie, G.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Xu, S.

Xu, Y.

Xu, Z.

S. Li, Z. Xu, R. Zhao, L. Shen, C. Du, and J. Wang, “Generation of Orbital Angular Momentum Beam Using Fiber-to-Fiber Butt Coupling,” IEEE Photonics J. 10(4), 1–7 (2018).
[Crossref]

Yan, Y.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Yang, J.

Yang, K.

Yang, Z.

Yao, A. M.

A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photonics 3(2), 161–204 (2011).
[Crossref]

Yavorsky, M. A.

Yin, G.

Yu, B.

Yu, H.

Zeng, X.

Zhang, C.

Zhang, F.

Zhang, L.

Zhang, W.

Zhang, Y.

Zhang, Z.

Zhao, H.

C. Zhu, P. Wang, H. Zhao, R. Mizushima, S. Ishikami, and H. Li, “DC-Sampled Helical Fiber Grating and its Application to Multi-Channel OAM Generator,” IEEE Photonics Technol. Lett. 31(17), 1445–1448 (2019).
[Crossref]

X. Heng, J. Gan, Z. Zhang, J. Li, M. Li, H. Zhao, Q. Qian, S. Xu, and Z. Yang, “All-fiber stable orbital angular momentum beam generation and propagation,” Opt. Express 26(13), 17429–17436 (2018).
[Crossref]

Zhao, R.

S. Li, Z. Xu, R. Zhao, L. Shen, C. Du, and J. Wang, “Generation of Orbital Angular Momentum Beam Using Fiber-to-Fiber Butt Coupling,” IEEE Photonics J. 10(4), 1–7 (2018).
[Crossref]

Zhao, Z.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Zhou, P.

Zhu, B.

Zhu, C.

C. Zhu, P. Wang, H. Zhao, R. Mizushima, S. Ishikami, and H. Li, “DC-Sampled Helical Fiber Grating and its Application to Multi-Channel OAM Generator,” IEEE Photonics Technol. Lett. 31(17), 1445–1448 (2019).
[Crossref]

Zhu, L.

L. Zhu and J. Wang, “A review of multiple optical vortices generation: methods and applications,” Front. Optoelectron. 12(1), 52–68 (2019).
[Crossref]

Zhu, S.

Zhu, T.

Zuo, Y.

Adv. Opt. Photonics (2)

A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photonics 3(2), 161–204 (2011).
[Crossref]

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Appl. Opt. (2)

Chin. Opt. Lett. (1)

Front. Optoelectron. (1)

L. Zhu and J. Wang, “A review of multiple optical vortices generation: methods and applications,” Front. Optoelectron. 12(1), 52–68 (2019).
[Crossref]

IEEE Photonics J. (2)

S. Li, Z. Xu, R. Zhao, L. Shen, C. Du, and J. Wang, “Generation of Orbital Angular Momentum Beam Using Fiber-to-Fiber Butt Coupling,” IEEE Photonics J. 10(4), 1–7 (2018).
[Crossref]

X. Zeng, Q. Li, Q. Mo, W. Li, Y. Tian, Z. Liu, and J. Wu, “Experimental Investigation of LP11 Mode to OAM Conversion in Few Mode-Polarization Maintaining Fiber and the Usage for All Fiber OAM Generator,” IEEE Photonics J. 8(4), 1–7 (2016).
[Crossref]

IEEE Photonics Technol. Lett. (1)

C. Zhu, P. Wang, H. Zhao, R. Mizushima, S. Ishikami, and H. Li, “DC-Sampled Helical Fiber Grating and its Application to Multi-Channel OAM Generator,” IEEE Photonics Technol. Lett. 31(17), 1445–1448 (2019).
[Crossref]

J. Lightwave Technol. (2)

Nanophotonics (1)

S. Ramachandran and P. Kristensen, “Optical vortices in fiber,” Nanophotonics 2(5-6), 455–474 (2013).
[Crossref]

Opt. Express (6)

Opt. Lett. (6)

Phys. Rev. Lett. (2)

P. Z. Dashti, F. Alhassen, and H. P. Lee, “Observation of Orbital Angular Momentum Transfer between Acoustic and Optical Vortices in Optical Fiber,” Phys. Rev. Lett. 96(4), 043604 (2006).
[Crossref]

X. M. Xi, T. Weiss, G. K. L. Wong, F. Biancalana, S. M. Barnett, M. J. Padgett, and P. J. St. Russell, “Optical Activity in Twisted Solid-Core Photonic Crystal Fibers,” Phys. Rev. Lett. 110(14), 143903 (2013).
[Crossref]

Other (1)

J. Liu, Photonic Devices (Cambridge University, 2005), Chap. 4.

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Figures (12)

Fig. 1.
Fig. 1. Principle of mode conversion between LP01 mode and OAM mode in the proposed fiber. The modal energy distributions of HE11 modes, HE71 modes, and the corresponding supermodes are shown as example images in this figure.
Fig. 2.
Fig. 2. Cross section of the proposed fiber.
Fig. 3.
Fig. 3. The optimization method to unify the coupling wavelengths for even and odd modes, and we suppose the ERI for even mode is larger than odd mode. (a) The coupling wavelength for even mode is shorter than odd mode. (b) The coupling wavelength for even mode is longer than odd mode. (c) The coupling wavelengths for odd and even modes are the same.
Fig. 4.
Fig. 4. Dispersion curves and modal energy distributions of the four supermodes created by the resonance between HE11 and HE51 modes.
Fig. 5.
Fig. 5. The intensities and phases of the superpositions of different supermodes created by the resonance between HE11 and HE51 modes at 1550.3 nm.
Fig. 6.
Fig. 6. The intensity profiles, phase distributions, purities and azimuthal phase variations for the generated OAM41 modes in outer ring at 1550.3 nm with different propagation distances.
Fig. 7.
Fig. 7. (a)-(c) The intensity profiles, phase distributions, purities and azimuthal phase variations of the generated OAM41 mode in outer ring at different wavelengths under the same propagation distance of 268 mm; (d)-(e) The intensity profiles and phase distributions of the superposition mode fields $E(r) = {E_D}(x,y) - {E_C}(x,y) + i{E_B}(x,y) - i{E_A}(x,y)$ at the corresponding different wavelengths.
Fig. 8.
Fig. 8. The intensity profiles, phase distributions, purities and azimuthal phase variations for the generated OAM61 modes in outer ring at different propagation distances. The OAM modes are generated from HE71 mode.
Fig. 9.
Fig. 9. The intensity profiles, phase distributions, purities and azimuthal phase variations for the generated OAM81 modes in outer ring at different propagation distances (HE91 mode).
Fig. 10.
Fig. 10. The intensity profiles, phase distributions, purities and azimuthal phase variations for the generated OAM41 modes in outer ring at different propagation distances (EH31 mode).
Fig. 11.
Fig. 11. The intensity profiles, phase distributions, purities and azimuthal phase variations for the generated OAM61 modes in outer ring at different propagation distances (EH51 mode).
Fig. 12.
Fig. 12. The intensity profiles, phase distributions, purities and azimuthal phase variations for the generated OAM81 modes in outer ring at different propagation distances (EH71 mode).

Tables (3)

Tables Icon

Table 1. Structural parameters for the generation of different orders

Tables Icon

Table 2. The matching coefficients under different transmission lengths

Tables Icon

Table 3. The matching coefficients for different resonance modes

Equations (17)

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E 11 e v e n ( r ) = ( cos γ e v e n z i δ e v e n γ e v e n sin γ e v e n z ) exp ( i δ e v e n z ) E 11 e v e n ( x , y ) exp [ i ( β 11 e v e n + κ 11 e v e n ) z ,
E m n e v e n ( r ) = ( i κ e v e n γ e v e n sin γ e v e n z ) exp ( i δ e v e n z ) E m n e v e n ( x , y ) exp [ i ( β m n e v e n + κ m n e v e n ) z .
E e v e n ( r ) = E 11 e v e n ( r ) + E m n e v e n ( r ) = E A ( x , y ) exp ( i β A z ) + E B ( x , y ) exp ( i β B z ) .
E A ( x , y ) = ( γ e v e n δ e v e n ) E 11 e v e n ( x , y ) + κ e v e n E m n e v e n ( x , y ) 2 γ e v e n ,
E B ( x , y ) = ( γ e v e n + δ e v e n ) E 11 e v e n ( x , y ) κ e v e n E m n e v e n ( x , y ) 2 γ e v e n ,
l 0 = π 2 γ e v e n = π β A β B .
E e v e n ( r ) = γ e v e n E 11 e v e n ( x , y ) 2 γ e v e n ( e i β A L + e i β B L ) + κ e v e n E m n e v e n ( x , y ) 2 γ e v e n ( e i β A L e i β B L ) = E m n e v e n ( x , y ) e i β A L .
E o d d ( r ) = E 11 o d d ( r ) + E m n o d d ( r ) = E C ( x , y ) exp ( i β C z ) + E D ( x , y ) exp ( i β D z ) .
E C ( x , y ) = ( γ o d d δ o d d ) E 11 o d d ( x , y ) + κ o d d E m n o d d ( x , y ) 2 γ o d d ,
E D ( x , y ) = ( γ o d d + δ o d d ) E 11 o d d ( x , y ) κ o d d E m n o d d ( x , y ) 2 γ o d d ,
E o d d ( r ) = E C ( x , y ) e i β C L + E D ( x , y ) e i β D L = E m n o d d ( x , y ) e i β C L .
E ( r ) = E A ( x , y ) e i β A z + E B ( x , y ) e i β B z + E C ( x , y ) e i β C z + E D ( x , y ) e i β D z .
L = ( 2 n + 1 ) π β A β B  =  ( 2 m + 1 ) π β C β D  =  ( 2 p ± 1 2 ) π β A β C = ( 2 q ± 1 2 ) π β B β D ,
E ( r ) = E m n e v e n ( x , y ) e i β A L + E m n o d d ( x , y ) e i β C L = [ E m n o d d ( x , y ) ± i E m n e v e n ( x , y ) ] e i β C L .
E e v e n ( r ) = E A ( x , y ) e i β A L + E B ( x , y ) e i β B L = [ E B ( x , y ) E A ( x , y ) ] e i β B L .
E o d d ( r ) = E C ( x , y ) e i β C L + E D ( x , y ) e i β D L = [ E D ( x , y ) E C ( x , y ) ] e i β D L ,
E ( r ) = [ E D ( x , y ) E C ( x , y ) + i E B ( x , y ) i E A ( x , y ) ] e i β D L .

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