Tunable arbitrary unitary transformer based on multiple sections of multicore fibers with phase control

Junhe Zhou; Jianjie Wu; Qinsong Hu

doi:10.1364/OE.26.003020

1. Introduction

Mode division multiplexing (MDM) has been viewed as a competitive candidate to boost the transmission capacity for the next generation optical communication systems [1,2]. Mode multiplexers/demultiplexers (MUX/DEMUXs) [3–10] and mode switches are the key components within the network, which have attracted significant attentions. Ways to realize the mode MUX/DEMUX include the effective index matching approach [3–6] and the interferometric approach [7–9].

Perfect mode conversion and switching without any loss can be regarded as optical unitary transforms, and this concept has already been implemented to realize mode MUX/DEMUXs [10]. Due to the intrinsic merit of the unitary transform based approach [10], the corresponding mode MUX/DEMUXs have much lower mode dependent loss (MDL) and insertion loss (IL) in comparison with those by other approaches. In addition to the applications in the MDM systems, unitary transforms have been widely implemented in other areas of optics, such as quantum information processing [11,12]. Hence, a tunable arbitrary unitary transformer, which can achieve mode conversion and switching in any multimode waveguides and can realize arbitrary interference for the photons, is of particular interest to the research community.

Realization of a tunable arbitrary unitary transformer is nontrivial. The most widely applied approach is called the multi-plane method, which converts the spatial fields through Fourier transforms by the lenses and uses the holograms to alter the phases between the two lenses. In order to reduce the system complexity and the insertion loss, G. Labroille et. al. [12] proposed a reflection and feedback approach to realize the mode MUX-DEMUX via the multi-plane concept. The pioneering work has paved the way for the realization of an arbitrary optical unitary transformer; however, it is a specifically designed and fixed transformer, which cannot adapt to different waveguide structures or switch the modes. Although the spatial light modulator (SLM) [13] can be used to tune the phases, it is not cost effective for the communication system implementation. To achieve cost-effective tuning, R. Tang [14] proposed a tunable arbitrary unitary transformer based on the multimode interference coupler via the multi-plane concept. Despite these innovations, there is still a requirement for the fiber based tunable arbitrary optical unitary transformers, because the free space optics based [10,13] or the waveguide based devices [14] have high insertion loss when they are implemented in the fiber related applications. Additionally, the phase tuning algorithm is also worth investigating, as the random optimization methods in [13,14] might be time consuming during system applications. Another popular approach is to use cascaded tunable couplers and phase shifters [15–17], which offers more degrees of tuning. It, however, requires quite a large amount of tunable couplers and may have relatively large loss when the cascading stage increases.

In this work, we have proposed a novel fiber based approach to realize the tunable arbitrary unitary transformer. Multiple sections of multi-core fibers (MCFs) are used along with the phase controllers embedded before and after them. The MCFs have their cores closely aligned and the cores couple to each other. The input signal is therefore transformed by the MCF. With the aid of the phase controllers, it is possible to mimic the multi-plane concept. A simple yet efficient method is proposed to tune the phase shifters. The numerical simulation verifies the proposed concept and a tunable mode MUX/DEMUX for a multimode fiber is designed and simulated based on it.

2. Theory

2.1 Transformation for each section of the MCFs

There are two types of MCFs. The first one has the cores isolated with respect to each other and the second one has the core coupled to each other. While most of the applications for the MCF transmission systems use the former type, we implement the latter in this work.

Without loss of generality, the cores are assumed to be aligned in a circular array [18]. The amplitudes of waves in the cores obey the following coupled equation during the propagation.

\begin{array}{l} \frac{d a_{1}}{d z} = i β a_{1} + i C a_{N} + i C a_{2} \\ \frac{d a_{i}}{d z} = i β a_{i} + i C a_{i - 1} + i C a_{i + 1} \\ \frac{d a_{N}}{d z} = i β a_{1} + i C a_{N - 1} + i C a_{1} \end{array}

where a_i is the amplitude of wave in the ith core, N the total number of the cores and C the coupling coefficient between the neighboring cores. Rewriting Eq. (1) in the matrix form, we have

\begin{array}{l} \frac{d a}{d z} = i β I a + i M a \\ M = (\begin{matrix} 0 & C & 0 & \dots & 0 & C \\ C & 0 & C & 0 & \dots & 0 \\ 0 & C & 0 & C & ⋱ & ⋮ \\ ⋮ & ⋱ & ⋱ & ⋱ & ⋱ & 0 \\ 0 & \dots & ⋱ & ⋱ & ⋱ & C \\ C & 0 & \dots & 0 & C & 0 \end{matrix}) \end{array}

Matrix M is a circulant matrix and can be decomposed by [12]

\begin{array}{l} M = F (\begin{matrix} λ_{1} \\ λ_{2} \\ ⋱ \\ λ_{N} \end{matrix}) F^{H} \\ F_{m n} = \frac{1}{\sqrt{N}} \exp (- j \frac{2 π}{N} (m - 1) (n - 1)) \\ λ_{k} = 2 C \cos (\frac{2 π (k - 1)}{N}) \end{array}

where F is the Fourier transform matrix, and hence, the solution of Eq. (2) is [18]

\begin{array}{l} a (z) = e^{j β z} F D F^{H} a (0) \\ D (z) = (\begin{matrix} \exp (λ_{1} z) \\ \exp (λ_{2} z) \\ ⋱ \\ \exp (λ_{N} z) \end{matrix}) \end{array}

2.2 Arbitrary unitary transformer

Assuming K sections of multi-core fibers are connected in a cascaded way, and the length of the k^th section is L_k. To realize an arbitrary unitary transformer, we mimic the multi-plane concept and insert the phase controllers before each section of the MCF. The phase controller array before the k^th section will have its contribution as a diagonal matrix θ^(k). Therefore, the input and output vectors are related to each other by the following expression.

\begin{array}{l} a_{o u t} = e^{j β \sum_{k} L_{k}} \prod_{k} (F D (L_{k}) F^{H} θ^{(k)}) a_{i n} \\ θ^{(k)} = (\begin{matrix} \exp (j θ_{1}^{(k)}) \\ \exp (j θ_{2}^{(k)}) \\ ⋱ \\ \exp (j θ_{N}^{(k)}) \end{matrix}) \end{array}

From Eq. (5), it can be seen that the transfer matrix of the proposed structure resembles the transfer matrix for the multi-plane method, which has one phase shifter matrix between the two Fourier transform matrices. To increase the degrees of control, one may add one more phase shifter array after the last section of the MCF. The detailed and rigorous mathematical proof in the appendix shows that an arbitrary unitary transformer can be realized by the proposed structure. Although the proof in the appendix indicates that quite a large number of MCF couplers and phase shifters are required, numerical test verifies that the proposed structure can realize arbitrary unitary transforms provided that the number of phase controllers together with the matrix elements constrains is larger than the number of the unknowns in the unitary matrix. When the proposed structure is implemented to realize the mode MUX/DEMUXs, a transfer matrix should be included in Eq. (5), which will associate the amplitudes of the modes in the multimode fiber with the amplitudes of the waves in the MCF cores.

It is worth noting that it is not mandatory to use the circular array. Other core array alignment method can also be used, e.g the linearly aligned cores can also realize the arbitrary unitary transformer with the aid of phase shifters.

2.3 Simple algorithm to realize the optimal phase setup

It is not easy to find the optimal control parameters for the phase shifters as the number of the variables to be optimized is quite large. The optimization problem is neither linear nor quadratic and hence the efficient optimization methods, such as linear programming and quadratic programming, cannot be implemented. Random optimization approaches, such as the simulated annealing algorithm [13, 14] and the genetic algorithm can be used to find the optimal phase setup. They are, however, difficult to be used for phase tuning in the practical systems due to the significant computational complexity. The gradient search can also be used to find the optimal solution. It is, however, also computationally intensive when the number of the variables is large.

Here, we propose a simple approach to optimize the phase setup by sequentially tuning the phase shifters in the device. Defining the cost function to be minimized as

E = {\sum_{m, n} | T_{m n} - O_{m n} |}^{2}

where T_mn is the element of the transfer matrix of the device T, O_mn the element of the objective unitary matrix O. Each time, one optimizes one of the phase shifters while keeping other phase shifters unchanged and obtains the minimized cost function E_k. It can be proved that E_k will converge to a specific value, because we have

0 < \dots E_{k + 1} \leq E_{k} \leq E_{k - 1} \dots

Therefore, E_k is a sequence which is descending its value but with a lower bound, and it will converge to its limit. We may use that limit as the optimal setup for the phase shifters. To avoid being trapped in the local optima, the order of the phase shifters are randomized during each round of optimization, e.g. one may optimize phase shifter 2, phase shifter 1, phase shifter 3.... during the first round of optimization while a different optimization order is adopted during the next round of optimization. It is worth mentioning that the proposed method could have its final performance worse than that of the global optimization method such as the simulated annealing method. However, it is a trade-off between efficiency and performance.

When we are trying to realize the devices like power splitters, mode MUX/DEMUXs and mode switches based on the arbitrary unitary transformers, there will be no specific requirement on the phases of the matrix elements. In such cases, the cost function in Eq. (6) can be defined as

E = {\sum_{m, n} | | T_{m n} | - | O_{m n} | |}^{2}

3. Results and discussions

Numerical simulations have been performed to verify the proposed device and the optimization algorithm. First of all, we verify the effectiveness of the proposed approach to realize arbitrary unitary transforms. Six sections of four-core MCFs were used with five arrays of phase shifters embedded between each two of them. The first section and the last section of MCF were connected with two arrays of phase shifters, and hence, totally seven arrays of phase shifters were used. The MCFs were assumed to have the coupling strength and their length product CL = 0.35π (One may have a different CL product if all of the elements within the transfer matrix are kept non-zero). The phase shifters were tuned sequentially to find the optimal setup. First of all, we assumed that the target matrix to be realized is the well known DFT matrix with its elements illustrated in Table 1. By tuning the phase shifters with the tuning resolution of 1 degree in the proposed device, it is possible to realize the transfer matrix shown in Table 2, and it can be seen from the two tables that the elements of the two matrices are extremely close. The mean square error (MSE) between the two matrices is 1.73 × 10⁻⁵. Secondly, to test the robustness of the device, we chose a purely random 4X4 matrix, which is shown in Table 3. Again, by tuning the phase controllers, it is possible to obtain a transfer matrix, which is very close to the target matrix (shown in Table 4). The mean square error between the two matrices is 1.85 × 10⁻⁴.

Table 1. The target matrix: the 4X4 DFT matrix

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Table 2. The realized transfer matrix with Table 1 as the target

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Table 3. The target matrix: a 4X4 random matrix

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Table 4. The realized transfer matrix with Table 3 as the target

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To investigate the capability of the arbitrary unitary transformer with the randomly selected target matrix, we calculated the MSE between the unitary matrix realized by the transformer and the random target unitary matrix. The probability density function (PDF) of MSE with respect to 1000 random realizations is plotted in Fig. 1, and the MSE interval to evaluate the PDF is 5 × 10⁻⁵. It can be seen from the figure that the PDF reaches 0% when the MSE is 3 × 10⁻³ with the peak of the PDF located below the MSE value of 0.5 × 10⁻³. The statistical results illustrate that the proposed structure can achieve arbitrary unitary transforms with high accuracy.

Fig. 1 Probability density function (PDF) of MSE with respect to 1000 random realizations. The MSE interval to evaluate the PDF is 5 × 10⁻⁵.

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Next, we implemented the proposed concept to design a tunable MUX/DEMUX based on the cascaded MCF structure. The MUX/DEMUX was designed for a three-mode fiber. The input modes for the multimode fiber were converted to the modes in the MCF by a photonic lantern [19, 20]. The converter schematic is shown in Fig. 1, and the tapered multi-core fiber structure is used [20]. The multimode fiber has the core index as n₀ and the cladding index as n₁. The photonic lantern has the outer cladding index as n₀, the inner cladding index as n₁ and the core index as n₂. n₀ = 1.435, n₁ = 1.44 and n₂ = 1.445 respectively [19]. The signal wavelength is 1550nm. The taper length for the photonic lantern is 20mm, and the inner cladding radius will gradually increase from 6μm to 35μm. The multimode fiber has the core radius of 6μm, and the single mode cores at the output of lantern have the core radii of 4μm, which has been specially designed to guarantee single mode and triple mode (with two modes being degenerated) operation. The core pitch at the output of the lantern is 41.6μm, which can reduce the coupling between the cores to a negligible level. The schematic of the photonic lantern is shown in Fig. 2. The proposed geometrical parameters can be easily implemented in real fiber/photonic lantern fabrications [19, 21, 22].

Fig. 2 Schematic of the photonic lantern for multimode to single mode conversion.

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With the photonic lantern, it is possible to convert the modes in the multimode fiber to the modes in the MCF, which is shown in Fig. 3. From the BPM simulation results in Fig. 3, it is possible to evaluate a matrix B, which relates the amplitudes of the multimode fiber modes to the amplitudes in the MCF cores. The rest part of the MUX/DEMUX, which was realized by the cascaded MCFs with phase shifters, would try to achieve a transfer matrix G, so that GB will be a matrix with only one none zero element for each column/row. We have

G = \prod_{k} K θ^{(k)}

where K is the transfer matrix of the coupler. In such a way, each multimode fiber mode will be converted to one core mode of the MCF.

Fig. 3 The converted pattern from the modes in the multimode (a) LP01 (b) LP11a and (c) LP11b to the modes in the multi-core single mode fibers (d,e,f).

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The MCF based couplers used in the tunable MUX are shown in Fig. 4. The cores of the MCF have the index of 1.445 and the cladding has the index of the 1.44. The core radius is 4μm. At the input and the output of the section, there are two tapers, which gradually increase/decrease the core pitch from 20.8μm to 41.6μm. The lengths of the two tapers are 20mm. In the middle of the MCF based coupler, there is a strong coupling region. In the region, the core pitch remains 20.8μm and the length of it is 17mm. With input from each port of the MCF, the coupler will have the corresponding output pattern, which is shown in Fig. 5. Similarly, we evaluated the transfer matrix K of the coupler from the data in Fig. 5.

Fig. 4 Schematic of the multi-core fiber based three-port coupler.

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Fig. 5 The output pattern of the MCF based coupler with the input from (a) port 1 (b) port 2 and (c) port 3.

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The overall structure of the tunable mode MUX/DEMUX is shown in Fig. 6. It consists of one photonic lantern, four MCF based couplers, and four arrays of the phase shifters. Each array has three phase shifters. It is worth noting that the relative phase of the input/output modes does not impact unitary transform in this mode MUX-DEMUX application, and therefore, we have removed the phase shifters after the output MCF couplers. The reason to choose such a set-up is as follows. A 3×3 unitary transform has 13 unknowns, and it has 6 matrix elements constrains (3 constrains to guarantee the uniformity of the rows and 3 constrains to guarantee the orthogonality between the rows). Hence, the minimum number of phase controller will be 7. In the phase controller scheme, only relative phase matters. Therefore, a phase array has only two effective phase controllers. From this aspect, we must use 4 phase shifter arrays (8 effective phase controllers) and 5 MCFs.

Fig. 6 The overall structure of the tunable mode MUX/DEMUX.

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In the simulation, the photonic lantern and the MCF based couplers were simulated by the BPM using the structural parameters described in the previous paragraphs. The phase shifters were simulated by varying the index in the middle of the single mode waveguides which connects these components. In practice, the phase shifters can be realized by several ways. One choice is to use the integrated phase shifters, such as the LiNbO3 phase shifters, which are small albeit with a larger loss when connecting with the fibers. Fiber-based device has lower loss and larger size. Generally, it is well known that fiber-based phase shifters can be realized by PZT [23] and are commercially available. However, such fiber-based device may induce phase instabilities in terms of temperature variations. Phase errors, as discussed in latter part of the section, can reduce the mode extinction ratio of the proposed device. Careful environmental temperature management should be applied in this case. Another method to realize the fiber-based phase shifters is to coat the fiber with special materials which have the electro-optic or thermal-optic effect, e.g. Graphene [24]. The device will be smaller, but more expensive. Hence, a trade-off should be considered while selecting the proper phase shifters in terms of size, phase tuning stability, insertion loss and cost.

The overall transfer matrix, which associates the amplitudes of the multimode fiber modes with the amplitudes of the MCF cores, is

T = (\prod_{k = 1}^{4} K θ^{(k)}) B

Before conducting the full scale simulation, we evaluated the transfer matrix of the photonic lantern B and the transfer matrix of the MCF based coupler K from the simulated data, and optimize the phase shifter values using the sequential optimization algorithm described in the previous section so that abs(T) = [1 0 0; 0 1 0; 0 0 1] (mode MUX/DEMUX), abs(T) = [0 0 1; 0 1 0; 1 0 0] (switch the outputs for mode 1 and mode 3) or abs(T) equals other target matrices. Since the phase term at the output of the mode MUX/DEMUX does not impact the performance, we used the absolute value of the transfer matrix T as the optimization target to approach the target matrix O.

With the obtained phases, they are used in the BPM for the final simulations. First of all, we set the target matrix O = [1 0 0; 0 1 0; 0 0 1], which means the mode LP01, LP11a and LP11b will be de-multiplexed to core 1, core 2 and core 3 of the MCF. Figure 7 illustrates the output pattern at the output of each MCF-based coupler. The fourth coupler output is the output of the tunable MUX. It can be seen that perfect de-multiplexing is achieved. Without considering the extra loss, such as the splicing loss and the losses induced by the phase shifters, the performance of the device is as follows: over 35 dB mode extinction ratio, less than 0.3dB insertion loss, and less than 0.1dB MDL. After that, we changed the target matrix O as [0 0 1; 0 1 0; 1 0 0], and the corresponding output pattern is shown in Fig. 8. Clearly, the output cores for mode LP01 and LP11b have switched in comparison with Fig. 7. The switching has been realized by changing the phase shifter values. The performance of the device during the switching status is as follows: over 35 dB mode extinction ratio, less than 0.3dB insertion loss, and less than 0.1dB MDL. Further investigation shows that proper adjustment of the phase shifters can lead to switching of the output cores for two arbitrarily selected modes without performance degradation.

Fig. 7 The output of each MCF based coupler with the target transfer matrix (O) = [1 0 0; 0 1 0; 0 0 1], the x and y axis units are μm.

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Fig. 8 The output of each MCF based coupler with the target transfer matrix (O) = [0 0 1; 0 1 0; 1 0 0], the x and y axis units are μm.

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The operation bandwidth of the proposed device is investigated. It is found that when the center signal wavelength varies +/−15nm, +/−10nm, and +/−5nm, the insertion loss and the MDL do not change much, but the worst mode extinction ratio changes to 17dB, 20dB, and 26dB respectively. From the system application perspective, the mode extinction ratio remains at an acceptable level within the 30nm-wide C band.

Afterwards, robustness analysis of the proposed device with respect to different parameters is performed. The lengths of the tapers and the coupling regime for the photonic lantern and the MCF-based couplers change by +/−2mm, and no performance degradation for the mode MUX/DEUMX is observed. This can be understood by the fact that the phase shifters can adjust their values to adapt to those parameters variations.

The unbalanced coupling between the cores is also considered, which can be attributed to the core pitch and core diameter variations [26, 27]. The cores in the MCF vary their center positions by +/−0.5μm and their diameters by +/−0.1μm, and no significant performance degradation has been observed. Furthermore, instead of the MCF coupler proposed in this work, we replace it with a realistic 3×3 fiber coupler with its transfer matrix experimentally measured in [28]. The realistic coupler is lossy and has its power division unbalanced among the three output ports, which makes the transfer matrix non-unitary. With the implementation of the realistic fiber coupler, it is found that the mode extinction ratio is over 29dB, the insertion loss is less than 1.6dB, and the MDL is less than 0.4dB.

Next, we investigate the polarization dependence of the device. If 0.1dB PDL and 5 degree polarization dependent phase errors are assumed in the simulation, the mode extinction ratio maintains over 25dB, which is still acceptable from the practical point of view. Certainly, the implementation of fiber couplers made of polarization maintaining fibers [29] along with polarization controllers before the couplers can result in a better performance in terms of polarization stability.

Finally, we added the core dependent insertion loss induced by photonic lantern the phase shifters and the splicing process, which will impact the unitary property of the transfer matrix. It was reported that in [22] the core dependent loss is less than 0.15dB while the insertion loss in addition to the core dependent loss is less than 0.25dB (or 0.4 dB maximum insertion loss). We adopt this value and 0.5dB core dependent insertion loss was assumed for each stage to account for the contribution from the MCF coupler, the phase shifters and the splicing process. It is found that the worst mode extinction ratio decreases to 26dB, the insertion loss increases to 1.7dB, and the MDL increases to 0.8dB respectively. The performance of the proposed device is still quite stable with the introduction of the core dependent loss, however, the power balance problem should be taken care of while one wants to improve the device performance.

4. Summary

In summary, we have proposed a fiber-based all-optical unitary transformer which can be implemented in the MDM systems and other areas of optics. The proposed device is based on multiple sections of MCFs with the phase controllers embedded before and after each section. A simple but efficient algorithm is proposed to find the optimal phase setup for the phase shifters. Numerical simulation verifies the proposed device and its application as a tunable mode MUX/DEMUX for the few mode fibers.

Appendix A The proof that an arbitrary unitary transformer can be realized based on the proposed structure

In this appendix, the strict mathematical proof is provided to demonstrate that an arbitrary unitary transformer can be realized by the proposed structure given in the paper.

To give the strict proof, we have to state the two theorems which have been rigorously proved in [25] and prove a lemma.

Theorem I [25]: An arbitrary unitary n×n matrix (the group of U(n, C)) can be realized by cascaded multiplication of the matrix T(i,j) (the group of U_ij) and diagonal matrices (the group of D(n,C)), where matrix T(i,j) is a unitary matrix in the form of

T (i, j) = (\begin{matrix} 1 \\ 1 \\ ⋱ \\ \cos θ & \dots & \sin θ \\ ⋮ & 1 & ⋮ \\ - \sin θ & \dots & \cos θ \\ ⋱ \\ 1 \end{matrix})

Theorem II [25]: H is a subgroup of U(n,C) which contains the diagonal matrix group D(n,C). If there exist a matrix A in H, with two of its elements a_ik≠0 and a_jk≠0 (with i≠j) for some k, we have

H \geq U_{i j}

Lemma:

The transfer matrices of the circularly aligned MCF couplers and the transfer matrices of the phase shifter arrays, as well as their matrix product will form a subgroup of U(n,C).

Proof:

First of all, it is easy to see that all of the matrices within the set are unitary matrices. Also, it is straight forward to show that any diagonal matrices realized by the phase shifters have their inverse matrices to be contained within the set. Therefore, to prove the above matrix set is a subgroup of U(n,C), we need to prove that if we have one MCF coupler transfer matrix A in the set, we will has its inverse matrix A⁻¹ to be contained in the set as well.

Afterwards, it can be easily proved that the transfer matrix A is a symmetrical matrix based on Eq. (2-4). We may choose a diagonal matrix C₁ realized by the phase shifters such that

\begin{array}{l} C_{1} = (\begin{matrix} e^{j θ_{1}} \\ e^{j θ_{2}} \\ ⋱ \\ e^{j θ_{n}} \end{matrix}) \\ θ_{m} = - 2 \times a n g l e (A (1, m)) \end{array}

where angle indicates the phase of a complex variable. It can be found that the matrix product AC₁A has its first row and first column to be with only one non-zero element, i.e.

A C_{1} A = (\begin{matrix} 1 & 0 & \dots & 0 \\ 0 & E_{11} & E_{1, n - 1} \\ ⋮ \\ 0 & E_{n - 1, 1} & E_{n - 1, n - 1} \end{matrix})

Since A and C₁ are symmetrical and unitary, it can be concluded the block matrix E is also symmetrical and unitary. Following the above steps, it is possible to find a diagonal matrix C₂ which is realized by the phase shifters, such that

\begin{array}{l} A C_{1} A C_{2} A C_{1} A = (\begin{matrix} 1 & 0 & \dots & 0 \\ 0 & E_{11} & E_{1, n - 1} \\ ⋮ \\ 0 & E_{n - 1, 1} & E_{n - 1, n - 1} \end{matrix}) C_{2} (\begin{matrix} 1 & 0 & \dots & 0 \\ 0 & E_{11} & E_{1, n - 1} \\ ⋮ \\ 0 & E_{n - 1, 1} & E_{n - 1, n - 1} \end{matrix}) \\ = (\begin{matrix} 1 & 0 & \dots & \dots & 0 \\ 0 & 1 & 0 & \dots & 0 \\ ⋮ & 0 & F_{11} & F_{1, n - 2} \\ ⋮ & ⋮ \\ 0 & 0 & F_{n - 2, 1} & F_{n - 2, n - 2} \end{matrix}) \end{array}

where the (n-2)X(n-2) block matrix F is still symmetrical and unitary. Hence, we may find the diagonal matrix C₃...C_n sequentially, such that

A C_{1} A \dots A C_{n - 1} A C_{n} A C_{n - 1} A \dots A C_{1} A = I

where I is the identity matrix. Hence, we have

A^{- 1} = C_{1} A \dots A C_{n - 1} A C_{n} A C_{n - 1} A \dots A C_{1} A

i.e. A⁻¹ can be realized by the matrix product of the transfer matrix A and the diagonal matrices C₁ to C_n. Hence, A⁻¹ is contained in the set. Therefore, the set will form a subgroup of U(n,C).

End of proof.

Based on the above two theorems and the lemma, it is possible to prove that our proposed structure can realize an arbitrary unitary transformer.

Proof:

From Theorem I, since we have the tunable phase shifter arrays to realize the arbitrary diagonal matrices, it can be inferred that the key to realize an arbitrary unitary matrix is to realize an arbitrary matrix T(i,j) in the group of U_ij.

From Theorem II, it can be inferred that to realize an arbitrary matrix T(i,j) in the group of U_ij, the key is to find a subgroup H, which contains a matrix A with the non zero elements of a_ik and a_jk.

Therefore, we propose to use the subgroup H, which has been discussed in the lemma. Clearly, such a subgroup H contains the diagonal matrix group D(n,C), and the transfer matrix A of the MCF coupler, which can be obtained by Eq. (4):

\begin{array}{l} A (z) = F D (z) F^{H} \\ D (z) = (\begin{matrix} \exp (λ_{1} z) \\ \exp (λ_{2} z) \\ ⋱ \\ \exp (λ_{N} z) \end{matrix}) \end{array}

where z is the length of the MCF coupler. As discussed before, A is a symmetrical matrix.

From (A8), we have

A_{i j} (z) = \sum_{l} F_{i l} D_{l l} (z) F_{j l}^{*}

Clearly, A_ij(z) is a non-zero function of z. Identifying the zeros of A_ij(z) and do not choose those values for z, we have A_ij(z)≠0. Since A(z) is symmetrical, A_ji(z) ≠0 if we have avoided the zeros of A_ij(z). Such a matrix A(z) contains two non-zero elements, i.e. A_ij and A_ji. Thus, the proposed subgroup H contains matrix T(i,j).

We may further design a different MCF coupler, whose transfer matrix M contains the non-zero elements M_i'j' and M_j'i' to find the other matrix T(i',j') to be contained in the subgroup as well. In this way, an arbitrary matrix in the form of T(i,j) (with i, j being a integer from 1 to n and i≠j) can be included in the subgroup H. By theorem I, it is known that an arbitrary unitary matrix can be realized by the cascaded multiplication of the matrices within the subgroup.

Therefore, it is strictly proved that an arbitrary unitary transformer can be realized by the proposed structure given in the paper.

End of proof.

From the discussion above, it can be inferred that the lengths of the MCF couplers should be chosen carefully to ensure more non-zero elements within the transfer matrix. If one wants to use identical MCF couplers, the length of the MCF couplers should be chosen so that no zero elements appear in the transfer matrix. Furthermore, the absolute values of the elements should be as close as possible.

We may choose the CL value to minimize the difference between the maximum absolute value of the elements and the minimum absolute value within the transfer matrix.

For example, to realize an arbitrary transformer in U(2,C), we choose the coupling strength and the length product CL= 0.25π, so that we have the transfer matrix as

N_{o p t} (2) = (\begin{matrix} \frac{1}{\sqrt{2}} & j \frac{1}{\sqrt{2}} \\ j \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{matrix})

In Eq. (15) all of the elements within the matrix have equal absolute values.

To realize an arbitrary transformer in U(3, C), one may choose the optimal CL as 2/9π, so that the transfer matrix have equal absolute values for the elements of the transfer matrix.

N_{o p t} (3) = \frac{1}{\sqrt{3}} (\begin{matrix} \exp (- j \frac{1}{18} π) & \exp (j \frac{11}{18} π) & \exp (j \frac{11}{18} π) \\ \exp (j \frac{11}{18} π) & \exp (- j \frac{1}{18} π) & \exp (j \frac{11}{18} π) \\ \exp (j \frac{11}{18} π) & \exp (j \frac{11}{18} π) & \exp (- j \frac{1}{18} π) \end{matrix})

To realize an arbitrary transformer in U(4, C), one may choose the optimal CL as 0.25π,, so that the transfer matrix have equal absolute values for the elements of the transfer matrix.

N_{o p t} (4) = (\begin{matrix} 0.5 & 0.5 i & - 0.5 & 0.5 i \\ 0.5 i & 0.5 & 0.5 i & - 0.5 \\ - 0.5 & 0.5 i & 0.5 & 0.5 i \\ 0.5 i & - 0.5 & 0.5 i & 0.5 \end{matrix})

Appendix B Realization of a 3×3 arbitrary unitary transformer

In this appendix, we discuss about the possible number of T(i,j) matrices (indicated by Eq. (11)) required to realize a 3×3 arbitrary unitary transformer. We will prove the following theorem to show that only three T(i,j) matrices are required:

Theorem III: An arbitrary unitary matrix U can be decomposed into the product of three unitary block matrices A⁽¹⁾, D, A⁽²⁾, i.e.

\begin{array}{l} U = A^{(1)} D A^{(2)} \\ A^{(1)} = (\begin{matrix} a_{11}^{(1)} & a_{12}^{(1)} \\ a_{21}^{(1)} & a_{22}^{(1)} \\ a_{33}^{(1)} \end{matrix}), A^{(2)} = (\begin{matrix} a_{11}^{(2)} & a_{12}^{(2)} \\ a_{21}^{(2)} & a_{22}^{(2)} \\ a_{33}^{(2)} \end{matrix}) \\ D = (\begin{matrix} d_{11} \\ d_{22} & d_{23} \\ d_{32} & d_{33} \end{matrix}) \end{array}

Proof:

Considering an arbitrary unitary matrix U, we may find a block unitary matrix C such that their product results in a unitary matrix V whose element on the upper right corner V₁₃ is zero, i.e.

\begin{array}{l} U = (\begin{matrix} u_{11} & u_{12} & u_{13} \\ u_{21} & u_{22} & u_{23} \\ u_{31} & u_{32} & u_{33} \end{matrix}) \\ C = (\begin{matrix} c_{11} & c_{12} & 0 \\ c_{21} & c_{22} & 0 \\ 0 & 0 & c_{33} \end{matrix}) \\ c_{11} = \frac{- u_{23}}{\sqrt{{| u_{13} |}^{2} + {| u_{23} |}^{2}}}, c_{12} = \frac{u_{13}}{\sqrt{{| u_{13} |}^{2} + {| u_{23} |}^{2}}}, c_{21} = - c_{12}^{*}, c_{22} = c_{11}^{*}, c_{33} = e^{j η} \\ V = C U = (\begin{matrix} v_{11} & v_{12} & 0 \\ v_{21} & v_{22} & v_{23} \\ v_{31} & v_{32} & v_{33} \end{matrix}) \end{array}

Since V is unitary, the rows of the matrix are orthogonal to each other, we have

\begin{array}{l} v_{11}^{*} v_{21} + v_{12}^{*} v_{22} = 0 \\ v_{11}^{*} v_{31} + v_{12}^{*} v_{32} = 0 \end{array}

With the matrix V, it is possible to find a block matrix B, whose elements fulfill the following:

\begin{array}{l} B = (\begin{matrix} b_{11} & b_{12} \\ b_{21} & b_{22} \\ b_{33} \end{matrix}) \\ b_{11} = \frac{v_{11}^{*}}{\sqrt{{| v_{11} |}^{2} + {| v_{12} |}^{2}}}, b_{21} = \frac{v_{12}^{*}}{\sqrt{{| v_{11} |}^{2} + {| v_{12} |}^{2}}}, b_{12} = - b_{21}^{*}, b_{22} = b_{11}^{*}, b_{33} = e^{j ζ} \end{array}

The product of VB will result in a matrix D, which has two of its elements on the bottom left corner to be zero:

D = V B = (\begin{matrix} v_{11} & v_{12} & 0 \\ v_{21} & v_{22} & v_{23} \\ v_{31} & v_{32} & v_{33} \end{matrix}) (\begin{matrix} b_{11} & b_{12} \\ b_{21} & b_{22} \\ b_{33} \end{matrix}) = (\begin{matrix} d_{11} & d_{12} & d_{13} \\ 0 & d_{22} & d_{23} \\ 0 & d_{32} & d_{33} \end{matrix})

Since D is unitary, we have

\begin{array}{l} | d_{11} | = 1 \\ d_{12} = 0 \\ d_{13} = 0 \end{array}

Hence, we have matrix U to be decomposed into:

\begin{array}{l} U = C^{- 1} D B^{- 1} \\ = (\begin{matrix} c_{11}^{*} & c_{21}^{*} & 0 \\ c_{12}^{*} & c_{22}^{*} & 0 \\ 0 & 0 & c_{33}^{*} \end{matrix}) (\begin{matrix} d_{11} & 0 & 0 \\ 0 & d_{22} & d_{23} \\ 0 & d_{32} & d_{33} \end{matrix}) (\begin{matrix} b_{11}^{*} & b_{21}^{*} \\ b_{12}^{*} & b_{22}^{*} \\ b_{33}^{*} \end{matrix}) \end{array}

Clearly, Eq. (29) is in accordance with Eq. (23).

End of proof.

Combining the lemma and the theorems I II and III, and using the procedures indicated in [25], it is possible to determine the maximum number of MCF couplers and phase-shifters required to realize an arbitrary 3×3 unitary transform.

Funding

National Natural Science Foundation of China (61775168); Natural Science Foundation of Shanghai (16ZR1438600).

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0.4948-0.0035i	0.5011 + 0.0000i	0.5010 + 0.0007i	0.5031-0.0006i
0.5006 + 0.0055i	0.0028-0.5057i	−0.4956-0.0041i	−0.0010 + 0.4981i
0.5027 + 0.0004i	−0.4964 + 0.0037i	0.5015-0.0057i	−0.4994-0.0007i
0.5019 + 0.0013i	0.0047 + 0.4968i	−0.5019-0.0008i	0.0009-0.4995i

−0.3146 + 0.4636i	−0.0736 + 0.4996i	0.0432 + 0.4144i	−0.2334 + 0.4505i
0.2468-0.3336i	0.4868-0.0783i	−0.5222 + 0.1347i	−0.2915 + 0.4570i
−0.5596 + 0.2379i	0.3275 + 0.0331i	−0.4397-0.4080i	0.4026 + 0.0073i
−0.1821-0.3332i	0.4617 + 0.4247i	0.0981 + 0.4075i	0.0630-0.5316i

−0.3138 + 0.4434i	−0.0764 + 0.5194i	0.0478 + 0.4191i	−0.2305 + 0.4452i
0.2439-0.3360i	0.4881-0.0755i	−0.5218 + 0.1353i	−0.2928 + 0.4554i
−0.5775 + 0.2387i	0.3318 + 0.0307i	−0.4359-0.4036i	0.3814 + 0.0005i
−0.1762-0.3331i	0.4548 + 0.4031i	0.0915 + 0.4125i	0.0555-0.5541i

0.4948-0.0035i	0.5011 + 0.0000i	0.5010 + 0.0007i	0.5031-0.0006i
0.5006 + 0.0055i	0.0028-0.5057i	−0.4956-0.0041i	−0.0010 + 0.4981i
0.5027 + 0.0004i	−0.4964 + 0.0037i	0.5015-0.0057i	−0.4994-0.0007i
0.5019 + 0.0013i	0.0047 + 0.4968i	−0.5019-0.0008i	0.0009-0.4995i

−0.3146 + 0.4636i	−0.0736 + 0.4996i	0.0432 + 0.4144i	−0.2334 + 0.4505i
0.2468-0.3336i	0.4868-0.0783i	−0.5222 + 0.1347i	−0.2915 + 0.4570i
−0.5596 + 0.2379i	0.3275 + 0.0331i	−0.4397-0.4080i	0.4026 + 0.0073i
−0.1821-0.3332i	0.4617 + 0.4247i	0.0981 + 0.4075i	0.0630-0.5316i

Tunable arbitrary unitary transformer based on multiple sections of multicore fibers with phase control

Abstract

1. Introduction

2. Theory

2.1 Transformation for each section of the MCFs

2.2 Arbitrary unitary transformer

2.3 Simple algorithm to realize the optimal phase setup

3. Results and discussions

4. Summary

Appendix A The proof that an arbitrary unitary transformer can be realized based on the proposed structure

Appendix B Realization of a 3×3 arbitrary unitary transformer

Funding

References and links

Cited By

Figures (8)

Tables (4)

Equations (29)

Optics Express

0.5	0.5	0.5	0.5
0.5	−0.5i	−0.5	0.5i
0.5	−0.5	0.5	−0.5
0.5	0.5i	−0.5	0.5i