OAM states generation/detection based on the multimode interference effect in a ring core fiber

Junhe Zhou

doi:10.1364/OE.23.010247

1. Introduction

Optical Orbital angular momentum (OAM) multiplexing [1–11] has recently emerged as a promising candidate for spatial division multiplexing (SDM) in optical communication systems. The technique can be either implemented in free space [1–7] or ring core fibers [8–11].

The bottleneck to implement this technique lies on the generation and detection of OAM states either inside ring core fibers or in free space. In free space, OAM states can be generated via a spiral phase plate (SPP) [1, 3], a dove prism [4], or a ring resonator [5]. In ring core fibers, the generation of OAM states can be achieved by the effective index matching and coherent combining method [9], spin angular momentum (SAM) to OAM transference method [10] and the method of combining a circular array of coherent sources [11]. The last approach can be scaled to generate a large number of OAM modes and is easy to be integrated with the multi-core fibers and the planar lightwave circuits (PLCs) [6, 7]. The technology can also be extended to generate OAM states in free space [6, 7]. The principle for this method is as follows. The coherent sources are equally distributed on a circle, and they have the phase difference of 2πl/N between the neighboring ones, where l denotes the OAM charge number, N the total number of coherent sources within the circular array. Therefore, the phase distribution within the array will be 2πnl/N, where n stands for the n^th waveguide within the array. Such a phase distribution matches the spatial sampling of the azimuth phase of OAM state l. When such coherent sources are emitted into a piece of ring core fiber [11] or into free space [6, 7], OAM state l is generated.

This technique is promising. However, it requires pre-coding of the multiple coherent sources with discrete Fourier transform (DFT) [11]. Since the coherent optical communication system has already implemented complex DSP algorithms, such a requirement increases the digital signal processing (DSP) effort and makes DSP chip design a bottleneck in system design.

In this paper, a novel approach to generate/detect OAM states is proposed. The idea is based on the multimode interference (MMI) effect, which has already been proposed to realize the mode multiplexers/de-multiplexers (MUXs/DEMUXs) in planar waveguides [12, 13], rectangular waveguides [14] and coupled waveguide arrays [15]. It is well known that MMI effect occurs in planar waveguides and can convert a single input image into multiple duplicated images with different phases. In this work, we reveal that MMI effect also exists in ring core fibers and a comprehensive analytical theory is presented. The MMI effect inside a ring core fiber can also converts one input image into multiple duplicated output images with different phases. If the phases are adjusted by a phase shifter array, the phase distribution of 2πnl/N can be realized, where n stands for the n^th output port, and l an integer dependent on the number of input port. Therefore, by a ring shape MMI coupler and an array of fixed phase shifters at the output, one may generate different OAM states by inject the signal at different input ports. If the device is used inversely, OAM states detection can be realized. In comparison with other OAM MUXs/DEMUXs, the proposed device is compatible with the current optical fiber technology. And it can generate multiple OAM states and can be easily scaled to generate a large number of OAM states.

2. MMI Theory in ring core fibers

To start with, the MMI theory inside a ring core fiber will be derived. The mode field inside the ring core fiber can be expressed as [11]

{OAM}_{l, p} = R_{p} (r) \exp (j l φ) \exp (j β_{l, p} z)

where p stands for the number of peaks in the radial direction, l the OAM charge number in the azimuth domain, β_l,p the propagation constant in the z direction. It should be noted that the radial field distribution R_p(r) actually depends on l. However, if the radius of the ring is much larger than its width, it can be assumed to be independent of l. The ring can be designed in such a way that it is single mode in the radial direction, and hence p = 1 [11]

{OAM}_{l} = R_{1} (r) \exp (j l φ) \exp (j β_{l} z)

Rewriting the radial direction distribution as

R (r) = R_{1} (r)

and substituting Eq. (2) into the Helmholtz equation in polar coordinates, we have

\frac{1}{r} \frac{\partial}{\partial r} (r \frac{\partial R (r)}{\partial r}) - \frac{l^{2}}{r^{2}} R (r) + (k^{2} - β_{l}^{2}) R (r) = 0

where k = nk₀, n the refractive index of the ring core, k₀ the free space wave number. When the radius of the ring a is relatively larger than the width of the ring, we have r≈a within the ring. Therefore, inside the ring, Eq. (4) can be approximated by

R^{″} (r) + \frac{1}{a} R^{'} (r) + (k^{2} - β_{l}^{2} - \frac{l^{2}}{a^{2}}) R (r) = 0

Defining

\frac{R^{″} (r) + \frac{1}{a} R^{'} (r)}{R (r)} = k_{r}^{2}

The propagation constants for the modes can be calculated as

\begin{array}{l} β_{l} = \sqrt{k^{2} - (\frac{l^{2}}{a^{2}} + k_{r}^{2})} \\ \approx k (1 - \frac{\frac{l^{2}}{a^{2}} + k_{r}^{2}}{2 k^{2}}) \end{array}

The optical field inside the ring can be expressed as the superposition of the OAM modes. Neglecting the common phase term in Eq. (7), the optical field can be expressed as

E (r, φ, z) = \sum_{l} R (r) c_{l} \exp (j l φ) \exp (- \frac{l^{2} z}{2 k a^{2}})

where c_l is the coefficient of each OAM mode. The MMI effect which converts one input image into N output images will take place when the length of the ring core fiber is

L_{N} = \frac{2 k a^{2} π}{N}

In this case, Eq. (8) becomes

E (r, φ, L_{N}) = R (r) \sum_{l} c_{l} \exp (j l φ) \exp (- j \frac{l^{2} π}{N})

Assuming there N equally spaced inputs are coupled into the ring core fiber at z = 0, and the input field distribution in the azimuth domain is composed by the ideal Dirac functions (the non-Dirac-function case will be considered later), we have the field at z = 0 as

R (r) \sum_{m = 0}^{N - 1} a_{m} δ (φ - \frac{2 m π}{N})

where a_m is the coefficient of each input. By comparing Eq. (8) and Eq. (11), one may obtain the OAM mode coefficient c_l as

c_{l} = \frac{1}{2 π} \sum_{m = 0}^{N - 1} a_{m} \exp (- j \frac{2 m l π}{N})

There will be N output images whose coefficients are the linear superposition of the input coefficients and the relationship is derived as follows.

At z = L_N, the optical field can be calculated by substituting Eq. (12) into Eq. (10). Letting l = nN + k, where n is an integer, k an integer between 0 and N-1, one has

E (r, φ, L_{N}) = R (r) \frac{1}{2 π} \sum_{n} \exp (j n N φ) \sum_{k = 0}^{N - 1} \sum_{m = 0}^{N - 1} a_{m} \exp (- j \frac{2 m k π}{N}) \exp (- j \frac{k^{2} π}{N}) \exp (j k φ)

Using the following equalities

\begin{array}{l} \frac{N}{2 π} \sum_{n} \exp (j n N φ) = \sum_{n = 0}^{N - 1} δ (φ - \frac{2 π n}{N}) φ \subset [0, 2 π) \\ \sum_{k = 0}^{N - 1} \exp (- j \frac{k^{2} π}{N}) = \sqrt{N} \exp (- j \frac{π}{4}) \end{array}

one can reformulate Eq. (13) as

E (r, φ, L_{N}) = R (r) \frac{\exp (- j \frac{π}{4})}{\sqrt{N}} \sum_{n = 0}^{N - 1} (\sum_{m = 0}^{N - 1} a_{m} \exp (j \frac{m^{2} π}{N}) \exp (- j \frac{2 m n π}{N}) \exp (j \frac{n^{2} π}{N})) δ (φ - \frac{2 π n}{N})

It can be clearly seen from Eq. (15) that, with N equally spaced inputs at z = 0, there will be N equally spaced outputs at z = L_N, whose coefficients will be the linear superposition of the input coefficients.

Now, let's consider that the input image has an azimuth distribution f(φ). The input optical field can be expressed as

\begin{array}{l} E (r, φ, 0) \\ = R (r) (f (φ) \otimes \sum_{m = 0}^{N - 1} a_{m} δ (φ - \frac{2 m π}{N})) \\ = R (r) \sum_{m = 0}^{N - 1} a_{m} f (φ - \frac{2 m π}{N}) \end{array}

where

\otimes

denotes convolution. Due to the fact that the MMI process is a linear process, the output can be calculated as

\begin{array}{l} E (r, φ, L_{N}) = R (r) \sum_{n = 0}^{N - 1} b_{n} f (φ - \frac{2 π n}{N}) \\ b_{n} = \frac{\exp (- j \frac{π}{4})}{\sqrt{N}} \sum_{m = 0}^{N - 1} a_{m} \exp (j \frac{m^{2} π}{N}) \exp (- j \frac{2 m n π}{N}) \exp (j \frac{n^{2} π}{N}) \end{array}

Therefore, N equally spaced input images with an azimuth distribution will result in N equally spaced output images with the same azimuth distribution. Only the input and output coefficients will change accordingly. The input coefficient a_m (m = 0,...,N-1) and the output confinement b_n (n = 0,..., N-1) can be written in the vector form as a and b, which are related by

b = T a

while the transfer matrix T has the element as

\begin{array}{l} T_{m n} = \frac{\exp (- j \frac{π}{4})}{\sqrt{N}} \exp (j \frac{m^{2} π}{N}) \exp (- j \frac{2 m n π}{N}) \exp (j \frac{n^{2} π}{N}) \\ m = 0, \dots, N - 1 \\ n = 0, \dots, N - 1 \end{array}

3. MMI effect based OAM MUX and DEMUX

Following the analysis in the previous section, if the n^th output port is added with a phase shifter with the phase shift of

\exp (- j \frac{n^{2} π}{N})

Equation (18) will be modified as

\begin{array}{l} b = T^{'} a \\ T^{'} = D T \\ D = (\begin{matrix} \exp (- j \frac{0^{2} π}{N}) \\ ⋱ \\ \exp (- j \frac{{(N - 2)}^{2} π}{N}) \\ \exp (- j \frac{{(N - 1)}^{2} π}{N}) \end{matrix}) \end{array}

The element of T' will be

{T^{'}}_{m n} = \frac{\exp (- j \frac{π}{4})}{\sqrt{N}} \exp (- j \frac{2 m n π}{N}) \exp (j \frac{m^{2} π}{N})

Hence, if the m^th input port is injected with an optical wave, the phase distribution at the output will be -2πmn/N if the common phase term is neglected. Such a phase distribution can be used to generate OAM state -m. If the device is placed inversely, it can help to distinguish OAM state -m for the receiving signals. Henceforth, an OAM MUX/DEMUX can be realized with a piece of ring core fiber with the length of L_N, and an array of phase shifters with the constant phase shift of -n²π/N.

5. Numerical simulations

Numerical simulations based on the beam propagation method (BPM) have been performed to verify the proposed formulas and the OAM MUX/DEMXU proposed in section 4.

A ring core fiber with similar parameters as [11] is used as the MMI device. The ring core has the background index of 1.46 and the index difference between the core and the cladding is 1.2%. The refractive index difference for core with respect to the cladding is changed to 1.2% instead of 0.12% in [11], so as to sustain more modes in the azimuth direction to enable the MMI process (there are over 45 modes in this design). As illustrated in the theory, infinite number of azimuth modes will result in perfect MMI. Hence, the more the azimuth modes are, the better the performance is. It is worth mentioning that the proposed device still works in the weakly guiding regime, and therefore, the polarization effect [16, 17] is negligible. The ring has the inner radius as 16 μm and the outer radius as 22 μm, which are the same as [11]. There are eight ports equally distributed at the input and the output. These input/output ports are connected with single mode fibers with the radii of 3 μm, so that the width of the ring core fiber will match the diameters of the single mode fibers. The signal wavelength is 1550nm, and the length of the ring core fiber is calculated by Eq. (9) and adjusted as 1748μm after the optimizations during the simulations. The output ports are assumed to be connected with phase shifters before output optical field is used for OAM states generation. The cross section of the device is illustrated in Fig. 1(a) and the conceptual illustration of the device is in Fig. 1(b).

Fig. 1 (a) Cross section of the device with input/output port labeling (b) the conceptual illustration of the device as a mode MUX/DEMUX.

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The phase shifters with the phase shift of -n²π/N can be realized in many ways, such as the bending array waveguides, the waveguides with different refractive indexes [18] or external phase shifter arrays [19]. In this work, waveguides with different refractive indexes [18] which can provide the corresponding phase shift are used in the simulations. The radii of these waveguides are also 3μm.

Firstly, generation of N images at the output of the MMI device with respect to one input image is illustrated. The amplitude and the phase distributions at the output of the ring shape MMI coupler are illustrated in Fig. 2 when the input signals are injected into the 0th(m = 0) and the 1st(m = 1) input ports. The amplitudes in the figures are normalized with unit peak amplitude for clearer demonstration (so are the amplitudes in other figures.) It can be seen from the figures that one input image has been transformed into eight duplicated images at the output of the device. The phase distributions match the theoretical predictions in Eq. (19) if the common phase term is discarded. Quantitative evaluation of the phases is presented in the following paragraphs.

Fig. 2 (a, c) the amplitude and phase distribution when the input signal is at the 0th port. (b, d) the amplitude and phase distribution when the input signal is at the 1st port.

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Afterwards, the output signals are passed through a phase shifter array, which is composed by waveguides with different refractive indexes. The phases at the center of the eight images before and after phase adjustment are illustrated in Table 1 and Table 2 when the input signals are injected into the 0th (m = 0) and the 1st (m = 1) input ports. In the tables, it is clearly shown that the phase of each image matches the ideal phase distribution -2πmn/N after phase adjustment by the phase shifter array. Hence, these images can help to generate OAM state 0 and OAM state −1 either inside a ring core fiber or in free space.

Table 1,. Phase distributions of the duplicated images before and after phase adjustment when the input signal is injected into the 0th input port (m = 0)

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Table 2,. Phase distributions of the duplicated images before and after phase adjustment when the input signal is injected into the 1st input port (m = 1)

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To demonstrate the OAM state generation capability, after phase adjustment by the phase shifter array, the output field of the MMI device is injected into a ring core fiber with exactly the same parameters as the one in [11], i. e. with the background refractive index as 1.46 and the index difference between the core and the cladding as 0.12%. The inner radius and the outer radius remain unchanged as 16 μm and 22 μm [11]. As revealed in [11], such a ring core fiber will be able to sustain OAM states −3 to 3 in the core region. Higher order OAM modes will dissipate into the cladding [11]. Hence, when the optical signals are injected into the 0th-3rd (m = 0-3), and 5th to 7th (m = 5-7) input ports, the output field of the proposed MUX/DEMUX can stimulate the OAM state −3 to 3 inside the ring core fiber. Injecting the signal into the 4th input port of the device will result in the generation of OAM state 4 (combined with OAM state −4), which will vanish after long-distance propagation in the ring core fiber.

The OAM states generated in the ring core fiber mentioned above are illustrated in Fig. 3-Fig. 4. The corresponding signals are injected into the 0th-3rd ports (m = 0-3) of the ring shape MMI coupler. Phases are adjusted at the output of the MMI device by the phase shifter array, and then the optical waves are injected into the ring core fiber with length of 8cm. Higher order modes attenuate significantly along the propagation length (8cm) and only OAM states from −3 to 3 can remain inside the core. As shown in Fig. 3 and Fig. 4, OAM states 0 to −3 have been generated inside the core region when the signals are injected into the 0th-3rd input ports (m = 0 to 3) of the device, and thus validate its mode generation capability.

Fig. 3 OAM modes generated inside the ring core fiber (a, c) amplitude and phase of OAM state 0 generated by injecting the signal into the 0th input port of the OAM MUX/DEMUX. (b, d) amplitude and phase of OAM state −1 generated by injecting the signal into the 1st input port of the OAM MUX/DEMUX.

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Fig. 4 OAM modes generated inside the ring core fiber (a, c) amplitude and phase of OAM state −2 generated by injecting the signal into the 2nd input port of the OAM MUX/DEMUX. (b, d) amplitude and phase of OAM state −3 generated by injecting the signal into the 3rd input port of the OAM MUX/DEMUX.

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To demonstrate the OAM state generation capability in free space, the output field is emitted into free space and propagates for 1cm. Since the optical field tends to spread during the free space propagation, the required simulation ranges in the x direction and the y direction are quite large. Due to the high computational cost, the optical pattern is calculated according to the Fresnel diffraction theory in stead of the BPM. The output amplitude and the phase of the generated OAM states in free space are illustrated in Fig. 5 as a proof of concept. It can be discovered from the figures that OAM state 0 and OAM state −1 have been generated if the input signals are injected into the 0th (m = 0) and the 1st (m = 1) input ports of the MUX/DEMUX. Thus, the MUX/DEMUX can be applied to generate OAM states in free space.

Fig. 5 OAM modes generated in free space (a, c) amplitude and phase of OAM state 0 generated by injecting the signal into the 0th input port of the OAM MUX/DEMUX. (b, d) amplitude and phase of OAM state −1 generated by injecting the signal into the 1st input port of the OAM MUX/DEMUX.

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After the demonstration of OAM states generation using the device, OAM states detection will be demonstrated. The device is inversely used by injecting multiple inputs with different phases. These input signals are the spatial samples of the OAM states and the sampling process is completed by an array of circularly distributed waveguides. These waveguides have the radii of 3μm. Afterwards, the sampled signals will pass through an array of phase shifters which can adjust the phase according to Eq. (20) and then undergo the MMI process inside the ring core fiber.

All of the parameters for the MMI device and the phase shifters remain unchanged in the de-multiplexing process. With loss of generality, OAM states inside a ring core fiber with the background index of 1.46 and the index difference of 1.2% between the core and the cladding are used in the simulations of de-multiplexing. The reason to chose the above ring core fiber instead of the one in [11] is because the ring core fiber in [11] cannot sustain OAM state 4(−4) inside it.

OAM states −3 to 3 and OAM state 4 are spatially sampled and injected inversely into the proposed MUX/DEMUX. The output results are depicted in Fig. 6 and Fig. 7. With different OAM states at the input, the output optical field will concentrate at different output ports, and the output port number will match the input OAM state. OAM states 0 −1, −2, −3, and 4(−4) (m = 0 to 4) will have the corresponding output at the ports 0, 1, 2, 3 and 4, while OAM states 3, 2, 1 (m = 5 to7) will have the output at the ports 5, 6, and 7.

Fig. 6 OAM state detection using the device (a-d) OAM states 0, −1, −2, and −3 (m = 0, 1, 2, and 3) are spatially sampled and passed onto the device, and figures show the field patterns at the output of the device.

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Fig. 7 OAM state detection using the device (a-d) OAM states 4, 3, 2, and 1 (m = 4, 5, 6, and 7) are spatially sampled and passed onto the device, and the figures show the field patterns at the output of the device.

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As illustrated in the figures, the OAM states are de-multiplexed by the proposed device successfully. The mode extinction ratio is defined as optical power at the corresponding output port over the power at other output ports, and it remains as high as 29dB. This means the crosstalk from one mode to the others is below −29dB.

Finally, fabrication error tolerance analysis is conducted. Since OAM state generation and detection are realized on the same device, we evaluate the fabrication error impact on the OAM state detection as an illustration. When the length of the ring core fiber has the error of 2%, which is around 34μm, the mode extinction ratio will degrade to 25dB. When one of the phase shifter has the phase error of 20 degree, the mode extinction ratio will degrade to 24dB. Change of the radius of the ring core fiber can be proportionally related to the length variation through Eq. (9), which indicates that 1% variation in the average radius of the ring is equivalent to 2% variation in the length. From these simulations, it can be concluded that the device is quite robust with respect to the geometrical variations. However, phase error will degrade the performance quite significantly. Therefore, it is recommended that tunable phase shifter array should be used at the output of the device.

The operation wavelength range of the device can be estimated as follows. According to Eq. (9), the deviation of the wavelength can be proportionally related to deviation of the length. Therefore, with +/− 2% variation on the signal wavelength (+/− 30nm) around 1550nm, the DEMUX will maintain the mode extinction ratio of 25dB.

Last but not the least, the impact of random variation of the radius of the ring is investigated. The inner ring and the outer ring radii are assumed to vary randomly but with average radii unchanged. The standard deviations of the radii are assumed to be 0.5μm, which are quite large from the fiber technological point of view. In such an extreme case, the mode extinction ratio will remain 26dB. Therefore, the structure is quite robust with respect to random geometrical variations.

6. Conclusion

In summary, we have proposed a novel OAM MUX/DEMUX based on the MMI effect in a ring core fiber. A comprehensive theoretical framework for the MMI process inside ring core fibers is derived. Based on the theory, it is discovered that the OAM MUX/DEMUX can be realized with a piece of ring core fiber and an array of fixed phase shifters. Detailed numerical simulations are provided to verify the proposed theory and the functionality of the device. The novel MUX/DEMUX is compatible with the current fiber technology and can be both applicable to the OAM states generation/detection inside ring core fibers and in free space.

Acknowledgment

This work is partially supported by the National Science Foundation of China (Grant No. 61201068).

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port No.	phase at the output of the MMI coupler	Phase after passing the phase shifters
0	0.0	0.0
1	22.0	−0.5
2	86.8	−3.2
3	197.2	−5.3
4	−7.7	−7.7
5	197.0	−5.5
6	86.9	−3.1
7	22.4	−0.1

port No.	phase at the output of the MMI coupler	phase after passing the phase shifters	ideal phase shift -2πmn/N
0	0.0	0.0	0
1	−21.6	−44.1	−45
2	0.0	−90.0	−90
3	65.8	−136.7	−135
4	174.7	−185.3	−180
5	−29.3	−231.8	−225
6	174.6	−275.4	−270
7	65.9	−316.6	−315

port No.	phase at the output of the MMI coupler	Phase after passing the phase shifters
0	0.0	0.0
1	22.0	−0.5
2	86.8	−3.2
3	197.2	−5.3
4	−7.7	−7.7
5	197.0	−5.5
6	86.9	−3.1
7	22.4	−0.1

port No.	phase at the output of the MMI coupler	phase after passing the phase shifters	ideal phase shift -2πmn/N
0	0.0	0.0	0
1	−21.6	−44.1	−45
2	0.0	−90.0	−90
3	65.8	−136.7	−135
4	174.7	−185.3	−180
5	−29.3	−231.8	−225
6	174.6	−275.4	−270
7	65.9	−316.6	−315

OAM states generation/detection based on the multimode interference effect in a ring core fiber

Abstract

1. Introduction

2. MMI Theory in ring core fibers

3. MMI effect based OAM MUX and DEMUX

5. Numerical simulations

6. Conclusion

Acknowledgment

References and links

Cited By

Figures (7)

Tables (2)

Equations (22)

Optics Express