## Abstract

A tunable microstructure optical fiber for different orbital angular momentum states generation is proposed and investigated by simulation. The microstructure optical fiber is composed of a high refractive index ring and a hollow core surrounded by four small air holes. The background material of the microstructure fiber is pure silica. The hollow core and the surrounded four small air holes are infiltrated by optical functional material whose refractive index can be modulated via physical parameters, leading to the conversion between circular polarized fundamental mode and different orbital angular momentum states at tunable operating wavelengths. A theoretical model is established and the coupling mechanism is systematically analyzed and investigated based on coupled mode theory. The fiber length can be designed specifically to reach the maximum coupling efficiency for every OAM mode respectively, and can also be fixed at a certain value for several OAM modes generation under tunable refractive index conditions. The proposed fiber coupler is flexible and compact, making it a good candidate for tunable OAM generation and sensing systems.

© 2015 Optical Society of America

## 1. Introduction

Orbital angular momentum (OAM) beams, which carry OAM of $L\cdot \hslash $ per photon, have found a variety of applications in optical tweezers and atom manipulation [1–3 ], nano-scale microscopy, as well as optical communications [4–6 ]. Research on optical vortex beams began in free space while fiber was not available at that time, and has been extensively studied for decades [7,8 ]. In 1998, optical fiber which can transmit OAM modes was first discussed by Alexeyev [9], and then the corresponding fiber OAM research boom followed in the last decade. Specially designed OAM fibers (which can support OAM modes and usually possess a high refractive index ring in fiber core) [10–12 ] and different approaches to generate OAM states in fiber have been proposed and demonstrated. Twisted special fibers [13–16 ], microbend fiber gratings [17] and dual-grating based on acoustic-optic interaction [18] are demonstrated to realize the excitation of OAM states. Novel compact OAM couplers with a simple structure utilized different kinds of Schott glass were designed by Y. Yan and his associates to realize the excitation of higher order OAM modes [19]. However, the high refractive index of Schott glass and the square core make it unsuitable for directly spliced with conventional fibers. Furthermore, with a fixed structure, the wavelength of the excited OAM mode in the above-mentioned fiber-based OAM coupler is also fixed, and only one kind of OAM mode can be generated, which brings difficulty to the development of highly capacity and integrated all-fiber OAM communication system.

In this paper, we design and show a new kind of microstructure fiber for different OAM modes generation at tunable operating wavelengths. The microstructure fiber is composed of a high refractive index ring and a hollow core surrounded by four small air holes. The background material is pure silica. The hollow core and the surrounded four small air holes are infiltrated by optical functional material whose refractive indices can be modulated via physical parameters, allowing for the coupling between circularly polarized fundamental mode and different higher order OAM modes at tunable resonant wavelengths. We build a theoretical model, systematically analyze and explain the physical mechanism of mode couplings in this fiber based on coupled mode theory. The corresponding refractive index (RI) matching conditions, 3 dB spectral bandwidths, the tuning rates of resonant wavelengths, and the coupling lengths for different OAM modes are investigated in detail. The fiber length can be designed specifically to reach the maximum coupling efficiency for every OAM mode respectively, and can also be fixed at a certain value for several OAM modes generation under tunable RI conditions.

## 2. Fiber structure

The diameter of the designed microstructure fiber is 125 μm, and the background material of the microstructure fiber is made of pure silica. This fiber is composed of a high refractive index ring and a hollow core surrounded by four small air holes. The diameters of the hollow core and four small air holes are 4 μm and 2 μm, respectively. The ring size is also fixed with an inner radius of 5 μm and an outer radius of 6.5 μm, and the refractive index (RI) difference between the ring and background silica is 0.08, as shown in Fig. 1 .

The hollow core and the surrounded four small air holes can be infiltrated by optical functional materials whose refractive indices are modulated by physical parameters, such as temperature, voltage, magnetic field, light intensity. The infiltrated material changes the dispersion curves of the fiber modes, allowing for the coupling between the input circularly polarized fundamental mode and the higher-order OAM modes in the fiber ring. Moreover, by regulating the RI of the functional material, this fiber can realize the coupling between circularly polarized fundamental mode and different higher order OAM modes at tunable operating wavelengths. The detailed analysis and calculations are described in the next section.

## 3. Theoretical model and results

To analyze the mechanism of mode conversion in the proposed optical fiber, we divide this fiber structure into two parts, as shown in Fig. 2 . Part ① is composed of four satellite holes, and part ② is a symmetric multilayered fiber which consists of a circular fiber core and a high refractive index ring. Without part ①, the fundamental core mode and the higher order mode in the high refractive index ring are orthogonal to each other in the circularly symmetric multilayered fiber. While in the proposed fiber, the surrounded four cylinders break the circular symmetry of the multilayered fiber and act as spatially dependent perturbation for this fiber, resulting in the mode conversion between fundamental core mode and the corresponding higher order modes in the high refractive index ring within a certain wavelength range.

When the incident light propagates from the single mode fiber to the proposed fiber core, the electric fields of the fundamental core mode A and the high order mode B in the ring can be expressed as:

Where ${E}_{A}(x,y)$and ${E}_{B}(x,y)$are normalized mode fields, ${\beta}_{A}$and ${\beta}_{B}$ are the mode-propagation constants, and the coefficients $A(z)$ and $B(z)$ which varies with $z$represent the amplitudes of the optical field.The existence of the four satellite cylinders will induce a spatially dependent perturbation which can be represented by a perturbing polarization $\Delta P(r)$ to the circularly symmetric multilayered fiber, leading to the energy exchange between fundamental core mode and higher order modes in the high index ring.

According to the coupled-mode theory [20], the coefficients $A(z)$ and $B(z)$ satisfy the following conditions:

Thus the coefficients $A(z)$ and $B(z)$ of the two resonant modes can be obtained:

The mode coupling coefficient $\kappa $ is given by:The mode self-coupling terms can be removed by expressing the coefficients $A(z)$ and $B(z)$ as $A(z)=\tilde{A}(z){e}^{i{\kappa}_{AA}z}$ and $B(z)=\tilde{B}(z){e}^{i{\kappa}_{BB}z}$. So the coupled mode equations can be simplified to:

Where $\delta $ is the phase mismatching coefficient, $\delta =\left({\beta}_{B}\text{+}{\kappa}_{BB}-{\beta}_{A}-{\kappa}_{AA}\right)/2$.We suppose that the light power is only launched into the fundamental mode in fiber core at the initial position, and the initial value are $\tilde{A}(0)=1$ and $\tilde{B}(0)=0$. On this condition, the coupled mode equation can be solved as:

When we divide the designed fiber into two parts and consider the proposed structure as a symmetric multilayered fiber affected by perturbation, the mode coupling mechanism of the two modes can be solved by the coupled-mode theory. While we consider the proposed fiber as a whole waveguide, the total optical field of this fiber behaves as a pair of supermodes. As we have discussed above, Eqs. (11) and (12) show the variation of the mode field amplitudes for the fundamental core mode and high order mode in the high index ring. So the total mode field in the proposed fiber can be obtained as the combination of the two mode fields:

Next, we employ a commercial finite element code (Comsol) to investigate the dispersion relation of the resonant normal modes and the corresponding supermodes in this fiber. The refractive index of background silica is set to be 1.444 at 1550 nm and its material dispersion is considered by the Sellmeier equation [21]. We fill the five air holes with refractive index matching fluid which is produced by Cargille Laboratories Inc as a sample, whose refractive index is 1.473 for 1550 nm at 25°C and has a thermal-optic coefficient of −0.0004 refractive index unit per centigrade (RIU/°C). Thus the RI of the fluid can be regulated and controlled by external temperature. The material dispersion of the liquid is also fitted by the Cauchy equation [22].

We first built the whole fiber structure model in COMSOL and accurately
calculated the dispersion curves of supermodes in the proposed fiber,
which are shown in the left of Fig.
3
(hyperbolic lines). The strongest mode coupling occurs at the waist
of the hyperbolic curves ($\delta =0$), and the wavelength of the waist point is
also defined as the coupling wavelength ${\lambda}_{c}$. The two pairs of supermodes in the insets
are the superposition of HE_{11}&EH_{51} modes (inset
(1)) and HE_{11}&HE_{71} modes (inset (2)). The four
dispersion curves of supermodes in the insets exhibit a repulsion
phenomenon in spectral branches, which is also defined as the
avoided-crossing effect in microstructure fiber [23]. Accordingly, the modal energy distributions of
supermodes formed from the resonant modes EH_{51} and
HE_{71} at the coupling wavelengths are shown in the middle of
Fig. 3. To ensure the resonant
wavelengths are near 1550 nm, we adjust the external temperature at 14°C
so that the RI of the fluid will change to 1.477617 at 1550 nm. The
strongest mode coupling occurs at the two waist points (point I and point
II) of the three dispersion curves: ${\lambda}_{EH51}\text{=}1547.04$nm for HE_{11} and EH_{51}
modes and ${\lambda}_{HE71}\text{=}1550$nm for HE_{11} and HE_{71}
modes.

Next, we built the symmetric multilayered fiber model in COMSOL and
calculated the new propagation constants of the resonant modes
HE_{11}, HE_{71} and EH_{51} based on their mode
self-coupling coefficients obtained by Eq. (8), as shown in the right of Fig. 3. There exist two crossing points in the
dispersion curves where the strongest mode couplings will occur. However,
the wavelength locations of the crossing points don’t accurately match the
waist of the supermodes. This slight deviation is mainly because the
calculations of the self-coupling coefficients based on the coupled-mode
theory do not consider the change of mode field distribution after the
perturbation. In fact, the introduction of the four cylinders has obvious
influence on the distribution of mode fields, especially the fundamental
core mode. Thus the self-coupling coefficients can only represent the
approximate modifications of new propagation constants.

From Eq. (15) we can indicate that if the fiber length satisfies the maximum power transfer condition, the 3 dB spectral bandwidth range can be defined as:

Due to the relationship: ${\kappa}_{AB}^{*}={\kappa}_{BA}$in the lossless fiber, Eq. (20) can be simplified to $\left|\delta \right|\le \left|{\kappa}_{AB}\right|$. Within this range, the two modes are well coupled, and the 3 dB spectral bandwidth can also be calculated from this equation. Moreover, $\delta =0$for perfect phase matching is achieved at the coupling wavelength. Therefore, the effective index differences between the two supermodes in the insets at the coupling wavelengths match the condition: $\Delta {n}_{\delta =0}\text{=}\lambda \left|{\kappa}_{AB}\right|/\pi $. So the coupling length of the corresponding resonance is determined by the waist of the hyperbolic curves: ${l}_{c}=\lambda /2\Delta {n}_{\delta =0}$. This principle also applies to other resonant modes, and the detailed 3 dB spectral bandwidth and coupling length for each mode is calculated and listed later in this paper.Similarly, the coupling between HE_{11} mode and higher order modes
HE_{91} and EH_{71} can also be realized by altering the
RI of the fluid to be 1.452817 at 75°C. The dispersion relationships for
the corresponding supermodes are shown at the left of Fig. 4
. Similarly, as shown in the right of Fig. 4, the dispersion curves of the resonant modes
obtained via the mode-coupling theory also have a slight deviation
compared to that of the supermodes. According to the dispersion curve of
supermodes, the strongest mode-coupling occurs at wavelengths
${\lambda}_{EH71}\text{=}1553.4$nm for HE_{11} and EH_{71}
modes and ${\lambda}_{HE91}\text{=}1549.86$ nm for HE_{11} and HE_{91}
modes, respectively.

In optical fibers, OAM modes exist as $\frac{\pi}{2}$-phase-shifted linear combinations of vector modes $H{E}_{L+1,m}$ or $E{H}_{L-1,m}$ [24]. This theorem can be expressed in equations [25]:

Where $L$ represents the topological charge, which means the number of 2π phase shifts along the circle around the beam axis [7], and $m$ is the number of concentric rings in the intensity profile of the modes. The superscript sign in $OA{M}_{{}_{\pm L,m}}^{\pm}$ or $OA{M}_{{}_{\pm L,m}}^{\mp}$ denotes the direction of the circular polarization, and $\pm L$ indicates the direction of the wave front rotation.

The fiber we designed can convert $H{E}_{11}^{even}$ and $H{E}_{11}^{odd}$ mode to different higher order modes simultaneously in different conditions at a specific wavelength. Take Fig. 3 as an example, if we input circularly polarized mode $H{E}_{11}^{even}\pm iH{E}_{11}^{odd}$ into the fiber core, the output mode in the high-index ring will be $OA{M}_{{}_{\pm 61}}^{\pm}=H{E}_{71}^{even}\pm iH{E}_{71}^{odd}$ with an orbital angular momentum of $\pm 6\hslash $ per photon at the wavelength of 1550.00 nm and $OA{M}_{{}_{\pm 61}}^{\mp}=E{H}_{51}^{even}\pm iE{H}_{51}^{odd}$ at the wavelength of 1547.04 nm. Thus the proposed fiber can realize the conversion between circular polarized fundamental mode and different higher-order orbital angular momentum fiber modes.

If we continuously increase the refractive index of the fluid, more
lower-order eigenmodes will be excited respectively. However, the
corresponding RI is beyond the variation range of the fluid mentioned
above at work temperature. So we change the filled refractive index
matching fluid to another type of the same series whose RI is 1.500 for
1550 nm at 25°C and has the same thermal-optic coefficient. We calculated
the excitation of four lower-order eigenmodes including EH_{31},
HE_{51}, EH_{11} and HE_{31} modes under this
fluid material respectively, and the detailed generation information is
shown below. Therefore, if the input light is circularly polarized mode
$H{E}_{11}^{even}\pm iH{E}_{11}^{odd}$, another four different vector modes
$OA{M}_{\pm 41}^{\pm}=H{E}_{51}^{even}\pm iH{E}_{51}^{odd}$,$OA{M}_{\pm 41}^{\mp}=E{H}_{31}^{even}\pm iE{H}_{31}^{odd}$,$OA{M}_{\pm 21}^{\mp}=E{H}_{11}^{even}\pm iE{H}_{11}^{odd}$and $OA{M}_{\pm 21}^{\pm}=H{E}_{31}^{even}\pm iH{E}_{31}^{odd}$ will be generated at different temperature
conditions.

With the RI variation of the filled material, not only different fiber
modes can be excited, but also the resonant wavelengths for each mode can
be controlled. We alter the RI of the corresponding fluid material for
each mode at a step of ${10}^{-4}$, and calculate the resonant wavelength
shifts for different resonant modes. Figure 5
shows the variations of the coupling wavelengths for different
modes as RI changes, and Table
1
lists the detailed regulation rates of the resonant
wavelengths for different fiber modes. The maximum tuning rate (−55683.8
nm/RIU) of coupling wavelength is 111 nm (from 1603 nm to 1492 nm) within
0.002 RI fluctuations for HE_{31} mode, and the minimum rate
(−8149.6 nm/RIU) is 16 nm (from 1539.5 nm to 1555.5 nm) within 0.002 RI
fluctuations for HE_{91} mode. The wavelength tuning rate is
related to the RI change coefficient of the filled material. For the fluid
with a negative thermal-optic coefficient of −0.0004 RIU/°C utilized in
this paper, the maximum and minimum temperature-tuning rates for
HE_{31} mode and HE_{91} mode are 22.3 nm/°C and 3.38
nm/°C, respectively.

According to the above formula and the simulation results, we calculated
and summarized a detailed table to list the generation information of the
eight eigenmodes around the resonant wavelength of 1550 nm, including the
corresponding RI of the filled material, the resonant wavelengths, 3 dB
spectral bandwidths and coupling lengths, respectively. As shown in Table 2
, different modes have different coupling lengths, and the
mode coupling efficiency can extend 100% if the fiber length is just the
odd times of the coupling length for a certain mode without consideration
of transmission loss at the coupling wavelength in theory. Thus the fiber
length can be designed specifically for every OAM mode. The minimum fiber
length is 1.39 mm for the generation of HE_{51} mode, and the
maximum fiber length can be designed to be 33.80 mm to generate
HE_{71} mode. So this microstructure fiber can be applied as a
wavelength tunable broadband high efficiency mode coupler for a single OAM
mode generation.

Moreover, we can also unify the fiber length at a certain value for generating several OAM modes. For example, if we fill the air holes with an index matching fluid with RI of 1.500 for 1550 nm at 25°C, as has been simulated above, the output will be different OAM modes: $OA{M}_{\pm 41}^{\pm}$mode (42.0°C), $OA{M}_{\pm 41}^{\mp}$mode (43.52°C), $OA{M}_{\pm 21}^{\mp}$mode (21.64°C) and $OA{M}_{\pm 21}^{\pm}$ mode (18.3°C) at the same resonant wavelength of 1550 nm. Thus we can fix the fiber length at 7.2 mm to respectively generate these eight OAM modes under different temperatures, and the energy conversion efficiencies are 92.2% ($OA{M}_{\pm 41}^{\pm}$), 96.2% ($OA{M}_{\pm 41}^{\mp}$), 61.1% ($OA{M}_{\pm 21}^{\mp}$) and 61.1% ($OA{M}_{\pm 21}^{\pm}$) for the eight OAM modes at 1550 nm, respectively. The actual number of the excited OAM modes is dependent on the RI adjustable range of the material filled in the air holes. With a wider RI adjustable range, more OAM modes can be excited with different RI conditions.

The proposed fiber device is controllable and flexible, for the functional material infiltrated in the hollow core and four small air holes can be any optical material whose refractive index matches the corresponding resonant conditions and can be modulated via physical parameters. For example, the refractive index matching fluid with a negative thermal-optic coefficient (−0.0004 RIU/°C) we have used in this paper, the magnetic fluid whose RI can be modulated by external magnetic field [26], the photoinduced azo-polymer mixture with a tunable RI which is dependent on light irradiation [27], and so on.

## 4. Conclusion

A flexible and compact microstructure fiber for different orbital angular momentum states generation at tunable operating wavelengths has been proposed and investigated by simulation. The fiber is composed of a high refractive index ring and a hollow core surrounded by four small air holes, and the background material is pure silica, which is more compatible to be integrated with conventional optical fibers. The hollow core and the surrounded four small air holes are infiltrated by optical functional material whose refractive index can be modulated by physical parameters, leading to the coupling between circularly polarized fundamental mode and different higher order OAM modes at tunable operating wavelengths. We build a theoretical model and systematically analyze the coupling mechanism based on coupled mode theory. The dispersion relations for different resonant eigenmodes and supermodes are investigated. And the corresponding refractive index matching conditions, 3 dB spectral bandwidths, the tuning rates of resonant wavelengths and the coupling lengths for different excited modes are calculated and listed in this paper. The fiber length can be designed specifically to reach the maximum coupling efficiency for every OAM mode respectively, and can also be fixed at a certain value for several OAM modes generation under tunable RI conditions. The proposed microstructure fiber is expected to be applied in tunable OAM generation and sensing systems.

## Acknowledgment

This work was supported by the National Key Basic Research and Development Program of China (Grant No. 2011CB301701), the National Natural Science Foundation of China (Grant Nos. 61322510, 11174154 and 11174155), the Tianjin Natural Science Foundation (Grant No. 14JCZDJC31300), and the National Undergraduate Training Program for Innovation and Entrepreneurship of China (Grant No. 201410055072).

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