Low-crosstalk orbital angular momentum fiber coupler design

Zhishen Zhang; Jiulin Gan; Xiaobo Heng; Muqiao Li; Jiong Li; Shanhui Xu; Zhongmin Yang

doi:10.1364/OE.25.011200

1. Introduction

In recent years, the orbital angular momentum (OAM) beam has attracted much attention because of its great potentiality for scaling the transmission capacity of the classical [1–5] and quantum communication system [6]. The data transmission based on multiple light beams with different value of OAM has been successfully demonstrated in both free space [1–3] and fiber networks [4, 5]. The fiber system is more attractive than the bulk optics system due to the considerable flexibility and reliability. However, there are still some challenges in the OAM fiber communication system [7, 8]. For example, the OAM beams are hardly manipulated in the OAM fiber. There is inevitable high insertion loss when the OAM beams are coupled between the spatial light components and the OAM fiber. So it is of great important significance to design the fiber devices for manipulating the OAM beams.

The basic OAM fiber devices include OAM mode generators [9–12], couplers [13–15], filters [16], and multiplexers [17] and so on. The focus here is the coupler, which splits the OAM beam from one channel to multiple channels and is the one of the most common components in optical fiber links. Two OAM fiber coupler designs have been proposed with the ring fiber [13] and the few-mode fiber [14]. However, the crosstalk is ignored in both designs. The crosstalk is induced because of that other modes will be excited during the coupling process. Due to the non-cylindrical symmetry of the refractive index distribution of the coupling region, firstly, the horizontal and vertical polarization components of the OAM mode have different propagation constants and coupling coefficients, which will excite other modes besides the input OAM mode. Secondly, the asynchronous coupling feature of the even eigenmode and odd eigenmode will inverse the topological charge of the input OAM mode [14]. The crosstalk will result in transmission errors and increase complexity of the decoding module in communication system. So the crosstalk level should also be the key indicator for the OAM coupler. Furthermore, above two coupler designs are fixed structure. Such structure is inherently limited to specific coupling ratio and will sensitive to optical wavelength.

Toward the problems mentioned, we proposed a tunable OAM coupler design based on two separate and parallel microfibers. This OAM coupler design adopts the mode match method [18], which is appropriate for finding the optimum trade-off between the crosstalk and the coupling length. The crosstalk less than −20 dB is achieved by properly setting the center-to-center distance between microfibers. The designed coupler could achieve tunable coupling efficiency over the whole C-band, which is a prospective component for OAM fiber communication system.

2. Theoretical model

As shown in Fig. 1, the tunable OAM coupler consists of two separate and parallel microfibers, which are mounted in the specially designed box with the center-to-center distance d between two microfibers and the overlapping length L. The microfiber structure is submerged in the refractive index liquid for achieving the tunable characteristic and better protection the coupler from the external disturbance. Compared with traditional fused-type coupler, the major difference is that the coupling region is still two separate microfibers instead of one single 8-shaped waveguide, which is conducive to the low crosstalk level of the OAM coupler.

Fig. 1 Schematic diagram of the tunable OAM coupler coupler; insets are the refractive index profile of each part. The red arrows represent the direction of the OAM beam travel.

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Since the tunable OAM coupler is composed of microfibers, the weak-guidance approximation and the perturbation method no longer apply. So the coupled mode theory, supermode theory, and beam propagation method are unsuitable for calculating the coupling interaction in this coupler. The mode match method [18] is selected because it is a universal and efficient method for both the strong and the weak coupling situations. This method divides the coupling model into several parts and uses the interference of local eigenmodes in each part to calculate the electromagnetic field. So the designed coupler is divided into three parts: the input port, the coupling region, and the output ports, as shown in Fig. 1. Each part is assumed as the uniform and lossless waveguide. The local eigenmodes are calculated by the finite element method with the corresponding refractive index (RI) profile, as shown in the insets of the Fig. 1. The electric fields in three parts are expressed as the following:

E_{1} = \sum_{i = 1}^{N_{1}} a_{i} \cdot e_{1 i} \cdot \exp (j β_{1 i} z)

E_{2} = \sum_{i = 1}^{N_{2}} b_{i} \cdot e_{2 i} \cdot \exp (j β_{2 i} z)

E_{3} = \sum_{i = 1}^{N_{3}} c_{i} \cdot e_{3 i} \cdot \exp (j β_{3 i} z)

where

E

is the electric field,

e

is the normalized local eigenmode,

β

and

N

are the propagation constant and the number of the local eigenmodes in each part, the subscrip

1

,

2

, and

3

represent the parameter of three parts, respectively.

a_{i}

,

b_{i}

, and

c_{i}

are the expansion coefficients of the i-th local eigenmodes in three parts, respectively.

In order to simplify the calculation, the length of the input port and output port are set to be zero. So the output optical field of the input port is

E_{1} = \sum_{p = 1}^{N_{1}} a_{p} \cdot e_{1 p}

Using the Maxwell equations at the interface between two adjacent waveguides, the tangential electric field should be continuous. So the input optical field of the coupling region should be

E_{2_t} = E_{1_t}

where the subscript

_t

represents the tangential electric field.

From Eq. (3) and the orthogonality of the local eigenmodes, we have

b_{i} = \frac{\iint E_{2} \cdot e_{2 i}^{*} d s}{\iint e_{2 i} \cdot e_{2 i}^{*} d s}

By substituting Eqs. (4) and (5) in Eq. (6),

b_{i} = \frac{\iint (\sum_{p = 1}^{N_{1}} a_{p} \cdot e_{1 p}_{_t}) \cdot e_{2 i_t}^{*} d s}{\iint e_{2 i_t} \cdot e_{2 i_t}^{*} d s}

Here, the parameter $Γ_{mode 1_mode 2}$ was defined to estimate the overlapping rate between two modes.

Γ_{mode 1_mode 2} = \frac{\iint e_{mode 1} \cdot e_{mode 2}^{*} d s}{\iint e_{mode 2} \cdot e_{mode 2}^{*} d s}

So $b_{i}$ is expressed as

b_{i} = \sum_{p = 1}^{N_{1}} a_{p} \cdot Γ_{1 p_2 i}

The light in the coupling region is considered as the interference of the local eigenmodes. So the output optical field of the coupling region is

E_{2} = \sum_{i = 1}^{N_{2}} b_{i} \cdot e_{2 i} \cdot \exp (j β_{2 i} L)

where

L

is the length of the coupling region.

At the interface between the coupling region and the output port, the procedures are the same as Eqs. (5)-(9).

E_{3_t} = E_{2_t}

c_{q} = \frac{\iint (\sum_{i = 1}^{N_{2}} b_{i} \cdot e_{2 i_t} \cdot \exp (j β_{2 i} L)) \cdot e_{3 q_t}^{*} d s}{\iint e_{3 q_t} \cdot e_{3 q_t}^{*} d s} = \sum_{i = 1}^{N_{2}} b_{i} \cdot \exp (j β_{2 i} L) \cdot Γ_{2 i_3 q}

By substituting Eq. (9) in Eq. (12),

c_{q} = \sum_{i = 1}^{N_{2}} \sum_{p = 1}^{N_{1}} a_{p} \cdot Γ_{1 p_2 i} \cdot \exp (j β_{2 i} L) \cdot Γ_{2 i_3 q}

So when the p-th local eigenmode is launched in the input port, the coupling efficiency of the q-th local eigenmode in the output port is

η_{p q} = \frac{| \iint c_{q} \cdot e_{3 q} \cdot c_{q}^{*} \cdot e_{3 q}^{*} d s |}{| \iint a_{p} \cdot e_{1 p} \cdot a_{p}^{*} \cdot e_{1 p}^{*} d s |} = \frac{| \iint c_{q} \cdot c_{q}^{*} d s |}{| \iint a_{p} \cdot a_{p}^{*} d s |}

Usually, the coupling results of the OAM modes can be obtained by the superposition of the coupling results of the corresponding vector modes in fiber. It is inconvenient that both the power and the phase have to be dealt with. In fact, due to the orthogonality, the OAM modes can directly act as the eigenstates of the microfiber in the input/output ports. So the calculations of the coupling OAM modes are simplified and became faster and more convenient by using the mode matching method.

Considering that only the p-th OAM mode is launched in the input port,

a_{i} = δ_{i p}

η_{p q} = | [\sum_{i = 1}^{N_{2}} Γ_{1 p_2 i} \cdot \exp (j β_{2 i} L) \cdot Γ_{2 i_3 q}] \cdot {[\sum_{i = 1}^{N_{2}} Γ_{1 p_2 i} \cdot \exp (j β_{2 i} L) \cdot Γ_{2 i_3 q}]}^{*} |

Based on Eq. (16), the relationship between the coupling efficiency and the length can be very complex and has no periodicity. However, some physical rules still could be concluded from Eq. (16). For example, the maximum coupling efficiency would be achieved at a shorter coupling length with larger propagation constant difference between the excited local eigenmodes in the coupling region.

In our coupler design, two microfibers are identical. According to symmetry, the local eigenmodes in the input port and the output port are the same relative to the local eigenmodes in the coupling region, i.e., $Γ_{2 i_3 q} = Γ_{1 q_2 i}$ .

η_{p q} = | [\sum_{i = 1}^{N_{2}} Γ_{1 p_2 i} \cdot Γ_{1 q_2 i} \cdot \exp (j β_{2 i} L)] \cdot {[\sum_{i = 1}^{N_{2}} Γ_{1 p_2 i} \cdot Γ_{1 q_2 i} \cdot \exp (j β_{2 i} L)]}^{*} |

Based on Eq. (17), when $Γ_{1 p_2 i} \cdot Γ_{1 q_2 i} \neq 0$ , $η_{p q} \neq 0$ ( $p \neq q$ ), the new mode will be excited and the crosstalk is induced during the coupling process. The crosstalk will decrease with decreasing the value of $Γ_{1 p_2 i} \cdot Γ_{1 q_2 i}$ . Here, $Γ_{1 p_2 i} \cdot Γ_{1 q_2 i} = 0$ means that no common local eigenmodes are excited by different input mode, which can be achieved under the weak coupling situations. So the crosstalk will decrease with decreasing the coupling strength between two microfibers, which is related with the radius and the refractive index of microfiber, the refractive index of liquid, the interval between two microfibers and the input wavelength. In our paper, the crosstalk is improved noticeably by increasing the interval between two microfibers, which is a simple and effective way to decrease the coupling strength. More detailed discussions are shown in Section 3.

3. Results and discussion

In the following part, we design a low crosstalk OAM coupler for the first order OAM mode, which is represented as $〈 \pm 1, \pm 1 〉$ and $〈 + 1, + 1 〉 = H E_{21}^{e} + j \cdot H E_{21}^{o}$ . The symbol $〈 \pm 1, \pm 1 〉$ represents the left (right) circularly polarized OAM mode with topological charge $+ 1$ ( $- 1$ ). The symbol $H E_{21}^{}$ represents the second order vector mode in the microfiber. There are two orthogonal field distributions for the $H E_{21}^{}$ mode, which are represented with the superscript e and o, respectively. The refractive index liquid (the Cargille Laboratories) is selected to be RI = 1.2962 for 1550 nm at 25 °C and has a thermal-optic coefficient of $- 3.33 \times 10^{- 4}$ refractive index unit per centigrade (RIU/°C) and a boiling point of 250 °C. The RI of the liquid can be regarded as a constant over the C-band (1520-1580 nm). The material of microfiber is set to be the silicon dioxide and the dependence of the RI of the microfiber on the wavelength is calculated by the Sellmeier equation [19]. In order to shorten coupling length, the diameter of microfiber is optimized to be as small as possible until $H E_{21}^{}$ mode is cutoff at the C-band. Based on the calculation of the finite element method, the diameter of the microfiber is set to be 3.1 μm. The $〈 \pm 1, \pm 1 〉$ mode can propagate steadily in both the straight and bending microfiber, as shown in Appendix.

In order to obtain a high OAM mode purity and high maximum coupling efficiency, it is important to control the center-to-center distance d between two microfibers. Figure 2 shows the mode distributions, phase distributions and the coupling efficiency of $〈 + 1, + 1 〉$ mode at 1550 nm and room temperature (25 °C) in the cross output port with the different interval d. When two microfibers are close together ( $d = 3.1 μ m$ , i.e., the traditional 8-shaped waveguide), the input $〈 + 1, + 1 〉$ mode is greatly deteriorated. As show in Fig. 2(a), the phase distributions at different coupling length $L$ are no longer the spiral shape, which means that other modes are excited under such strong coupling state. More specifically, $T E_{01}^{}$ mode, $T M_{01}^{}$ mode and $〈 - 1, - 1 〉$ mode will be excited, which lowered the mode purity and coupling efficiency of the $〈 + 1, + 1 〉$ mode and induced the crosstalk, as shown in Fig. 2(b). However, when two microfibers are set with a proper interval (e.g., $d = 6.35 μ m$ ), the mode distributions and phase distributions at the both output ports of the different coupling length are all clear to recognize as the $〈 + 1, + 1 〉$ mode, as shown in Fig. 2(c). Figure 2(d) shows that the maximum coupling efficiency 97.59% with the crosstalk −20.8 dB is achieved at $L = 2.315 c m$ . The crosstalk is improved noticeably by increasing the interval between two microfibers. Because the proportion of the input mode field distribution in the adjacent microfiber is greatly lowered and the coupling effect between two microfibers is weaken. As shown in Fig. 2(e), the greater the interval, the higher mode purity and maximum coupling efficiency, and the longer the coupling length. There is a critical value of d for balancing the performance and the size of the coupler. Typically, the critical value of d is about twice the diameter of the microfiber, which makes the lowest purity of the $〈 + 1, + 1 〉$ mode is 99% and the crosstalk level is lower than −20 dB. This crosstalk value should be perfectly adequate for the mostly OAM fiber systems. The critical value of d is smaller while the evanescent wave of the input mode is more concentrated. So the critical value of d will decrease with increasing the refractive index of microfiber, decreasing the refractive index of liquid, and decreasing input wavelength. For example, at room temperature (25 °C), the calculated critical value of d for this $〈 + 1, + 1 〉$ mode coupler at 1520 nm and 1580 nm are 6.1 μm and 6.6 μm, respectively. Incidentally, only increasing the diameter of microfiber is not enough to decrease the coupling strength between two microfibers for achieving the low crosstalk level.

Fig. 2 When $〈 + 1, + 1 〉$ mode is launched at 1550 nm: (a) the mode distributions and phase distributions at the different coupling length with $d = 3.1 μ m$ ; (b) the coupling efficiencies of modes in the cross output port with $d = 3.1 μ m$ ; (c) and (d) are the coupling results with $d = 6.35 μ m$ ; (e) the coupling efficiency of $〈 + 1, + 1 〉$ mode in the cross output port with the different interval $d$ between two microfibers.

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Figure 3(a) shows the wavelength dependence of the OAM couple with a fix center-to-center distance ( $d = 6.6 μ m$ ) and a fix overlapping length ( $L = 3.535 c m$ ), where $〈 + 1, + 1 〉$ mode achieves the maximum coupling efficiency 99.19% with the crosstalk −24.9 dB. This coupler only achieves high coupling efficiency (less than 1% energy in the collinear output port) in 1545-1560 nm, which is not enough for the needs of the C-band communication. The bandwidth is difficult to expand by adjusting the microfiber’s diameter carefully. When the radius of the microfiber is comparable to the wavelength, the coupling efficiency, which is closely related to the mode distributions and the mode effective index, is highly sensitive to optical wavelength. Here, we propose to change the refractive index of liquid for tuning the operating wavelength and expanding the operating bandwidth. As shown in Fig. 3(b), the coupling curve of the operating wavelength 1580 nm (RI = 1.2878) and 1520 nm (RI = 1.3042) is almost overlapped with that of 1550 nm (RI = 1.2962). Also, this tuning method can achieve different coupling ratio. Figure 4(a) shows that the 1550 nm $〈 + 1, + 1 〉$ coupler ( $L = 3.885 c m$ ) achieves the coupling ratio 10:90, 50:50, and 70:30 when the liquid RI is 1.2837, 1.2778, and 1.2748, respectively. By using the thermosensitive liquid (the Cargille Laboratories), the $〈 + 1, + 1 〉$ coupler with $d = 6.1 μ m$ can achieve different coupling efficiency over the wholeC-band at $L = 3.885 c m$ when the temperatures is tuning from 25 to 150 °C, as shown in Fig. 4(b).

Fig. 3 (a) The wavelength dependence of $〈 + 1, + 1 〉$ mode in the output ports (at 3.535 cm); (b) the coupling efficiency of $〈 + 1, + 1 〉$ mode in the cross output port with (the refractive index of liquid, the input wavelength).

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Fig. 4 (a) The different coupling ratios achieved at 3.885 cm; (b) the tunable coupling efficiency over the C-band within 25-150 °C.

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From the results mentioned above, a tunable $〈 + 1, + 1 〉$ mode fiber coupler with adjustable coupling ratio over the whole C-band is effectively designed. It is worth nothing that this coupler is also suitable for the coupling of $〈 - 1, - 1 〉$ mode. Compared with $〈 + 1, + 1 〉$ mode, the synthesis of $〈 - 1, - 1 〉$ mode only has a different sign, i.e., $〈 - 1, - 1 〉 = H E_{21}^{e} - j \cdot H E_{21}^{o}$ . So the coupled results of $〈 - 1, - 1 〉$ mode and $〈 + 1, + 1 〉$ mode are the similar linear combination of the same eigenmodes in output ports and only have a different sign, which means that the coupling results of $〈 - 1, - 1 〉$ mode in the output ports are exactly the same as Fig. 3 and Fig. 4.

Based on the similar designing process, this tunable fiber coupler can be extended to higher order OAM modes with thicker microfiber. By proper setting the critical value of d, high OAM mode purity and high maximum coupling efficiency can also be achieved. The coupling length may be increased because of the coupling interaction is decreased with the larger diameter and distance of microfibers. Fortunately, such problem can be alleviated by increasing the refractive index of microfiber, which can optimize the diameter and distance of microfibers to the smaller values.

4. Conclusion

In conclusion, the tunable OAM fiber coupler designs have been proposed and simulated for $〈 \pm 1, \pm 1 〉$ mode. The designed coupler consists of two separate and parallel microfibers. Based on the mode match method, low crosstalk (< −20 dB) is achieved by optimizing the center-to-center distance d between the microfibers. Over the whole C-band, arbitrary coupling ratio can be achieved by adjusting the temperatures proper. The designed coupler can be potentially used for the OAM optical communication system and expand the application of OAM mode in fiber lasers, sensors and other various fiber systems.

Appendix

The OAM mode can propagate steadily in both the straight microfiber and the bending microfiber with proper curvature radius ( $\geq 1 m m$ ). Here is an example: the diameter of the microfiber is set to be 3.1 μm with the refractive index 1.444 at 1550 nm, and the refractive index liquid (the Cargille Laboratories) is selected to be RI=1.2962 for 1550 nm at 25 °C. For the bending microfiber, the radius of curvature is 1 mm. The effective refractive indices of the eigenmodes in the straight and bending microfibers are shown in Table 1. For both the straight and bending microfiber, the refractive index difference between $H E_{21}^{e}$ and $H E_{21}^{o}$ is less than $1 \times 10^{- 6}$ , and the minimum refractive index difference of different modes in microfiber is larger than $1 \times 10^{- 4}$ . Furthermore, the mode distribution and phase distribution of the $〈 + 1, + 1 〉$ mode in both the straight and bending microfiber are shown in Fig. 5. So the OAM mode can propagate steadily in both the straight and bending microfiber.

Table 1. The effective refractive indices of the eigenmodes in the microfiber.

View Table

Fig. 5 The mode distributions and phase distributions of the $〈 + 1, + 1 〉$ mode in the straight microfiber (a) and (b), and the bending microbiber (c) and (d).

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Funding

National Natural Science Foundation of China (NSFC) (61575064, U1609219); Science and Technology Project of Guangdong (2015B090926010); Tip-top Scientific and Technical Innovative Youth Talents of Guangdong special support program (2015TQ01X322); Fundamental Research Funds for Central Universities (2015ZP019); High-level Personnel Special Support Program of Guangdong Province (2014TX01C087); the National Science Fund for Distinguished Young Scholars of China (61325024).

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	$H E_{11}^{e}$	$H E_{11}^{o}$	$T E_{01}$	$H E_{21}^{e}$	$H E_{21}^{o}$	$T E_{01}$
$N_{eff}$ (straight)	1.410420	1.410420	1.363190	1.359627	1.359627	1.359399
$N_{eff}$ (bending)	1.410437	1.410436	1.363192	1.359629	1.359629	1.359406

Low-crosstalk orbital angular momentum fiber coupler design

Abstract

1. Introduction

2. Theoretical model

3. Results and discussion

4. Conclusion

Appendix

Funding

References and links

Cited By

Figures (5)

Tables (1)

Equations (17)

Optics Express