The photonic polarization processor is composed of a 2D grating and four MZIs, as shown in Figure 1A. The 2D grating splits the two orthogonal components (defined as *x* and *y* polarizations) of the input light into different waveguide branches with the same TE mode. Both MZI 1 and MZI 4 can perform any arbitrary 2×2 unitary matrix transformation [40], [41], [42], [43], [44], [45]. The combination of MZIs 2 and 3 can perform an arbitrary 2×2 diagonal matrix transformation. The four MZIs constitute a complete network that can implement an arbitrary transformation matrix based on singular value decomposition [45], [46]. The light is coupled to the fiber array from the chip with TE gratings. By designing the transmission matrix, the chip is reconfigured to achieve three different functions.

Figure 1: Photonic polarization processing circuits.

(A) Detailed structure of the polarization processor chip. (B) Micrograph of fabricated chip. (C) Transmission spectra of the chip for random input SOPs and applied voltages. (D) Transmission spectra of the 2D grating. (E) Imparted phase depending on the applied voltage for the tested MZI structure.

Usually, the SOPs of two polarization channels are orthogonal and set as *x* and *y* polarizations. While crosstalk between different channels will be introduced both in the optical transmission link and in the mode multiplexer/demultiplexer, the SOPs of two channels will be changed or even become non-orthogonal. Assume that the Jones matrices of two channels in the receiving end are *P* and *Q*, respectively, given by

$$P={[{p}_{x},\text{\hspace{0.17em}}{p}_{y}]}^{\text{T}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}Q={[{q}_{x},\text{\hspace{0.17em}}{q}_{y}]}^{\text{T}}.$$(1)

In order to separate the two channels, a transformation matrix is needed to meet

$$M[P,\text{\hspace{0.17em}}Q]=\Lambda .$$(2)

Here, ᴧ=[*A*_{1}, *A*_{2}] is a diagonal matrix, representing the optical field distribution in the output ports (Ports 2 and 3) for two channels. The transformation matrix is then given by

$$M={[P,\text{\hspace{0.17em}}Q]}^{-1}\Lambda .$$(3)

From the above analysis, the *P*-polarized component (Channel 1) of the input light will emerge from Port 2 and the *Q*-polarized one (Channel 2) will emerge from Port 3. Similarly, the output ports can be switched by rotating the diagonal matrix by 90°. It proves that our chip is able to separate two arbitrary polarization-based channels in theory, which can be used to descramble the polarization-based channels.

Meanwhile, our chip can also transform an arbitrary polarization into a fixed known one and transform a fixed known polarization into an arbitrary one in reverse, acting as a polarization controller. The transformation matrix is given by

$$M={\left[P,\text{\hspace{0.17em}}{P}_{\perp}\right]}^{-1}\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}{[{P}_{\perp},\text{\hspace{0.17em}}P]}^{-1}.$$(4)

Here, *P*_{⊥} is the cross-polarization of *P*. When *P*-polarized light is incident on the 2D grating, only one output port will excite the light. And in the reverse, arbitrarily polarized light can be generated and emerge from the 2D grating when the light is incident on one of the output ports.

Furthermore, the four ports can output different polarization information, which can be used to measure the SOP of light, making the chip as a division-of-space polarization analyzer. The Stokes parameters *S*=[*S*_{0}, *S*_{1}, *S*_{2}, *S*_{3}] can be obtained by [30]

$$S=TI,$$(5)

where *I*=[*I*_{1}, *I*_{2}, *I*_{3}, *I*_{4}] is the measured optical power in the four output ports. *T* is a 4×4 matrix dependent on the internal parameters of chip. The four intensities of the different polarized components can be equivalently obtained at different times by applying four sets of direct current (DC) voltage signals on the phase shifters, making the chip as a division-of-time polarization analyzer.

The key point to configure these polarization functions is how to load a targeted transformation matrix on the chip. Here we employ a numerical gradient descent algorithm modified from deep learning [46], [47] to optimize this issue. According to the different purposes of our processor, a suitable and special cost function (CF) should be first defined. Then the only training target is to make the defined CF maximum using the numerical gradient descent algorithm. Theoretically, training needs to combine forward and backward propagation methods, similar to deep learning. Forward propagation is used to calculate the output as the data for the next iteration, and then backward propagation aims to estimate the errors and find the gradient descent. This training algorithm is also called the gradient descent algorithm, which is a common method for training artificial neural networks (ANNs). In our design, the optical chip can output automatically and timely provided the input is set. And the gradient descent can be alternatively measured by fine-tuning each parameter. So in our design, no backward propagation is needed and forward propagation can be implemented by the chip itself at the speed of light. Furthermore, the chip can be regarded as a “black box”. That means the internal structure of the chip is transparent to the users. The full training process is as follows:

Initialization: all the adjustable parameters *θ*(*i*=1, 2, …) are set randomly. Here, *θ*_{i} is the carried phase on the corresponding phase shifter.

Tuning each parameter: set *θ*_{1} to *θ*_{1}+Δ*θ* temporarily.

If CF(*θ*_{1}+Δ*θ*)≥CF(*θ*_{1}), replace *θ*_{1} with *θ*_{1}+Δ*θ*; else, replace *θ*_{1} with *θ*_{1}−Δ*θ*.

Repeat Step 2 for all adjustable parameters one by one.

Repeat Steps 2 and 3 until the CF is converged or reach the target value.

For the polarization MIMO descrambler, the CF is defined independently of the channel by

$$\text{CF}=\frac{\left|{A}_{1}\u2022{A}_{\mathrm{exp}1}\right|}{\left|{A}_{1}\right|\left|{A}_{\mathrm{exp}1}\right|}\frac{\left|{A}_{2}\u2022{A}_{\mathrm{exp}2}\right|}{\left|{A}_{2}\right|\left|{A}_{\mathrm{exp}2}\right|}.$$(6)

The operation “•” means the scalar product of two vectors. *A*_{exp}_{n} (n=1, 2) is the measured output power distribution in Ports 2 and 3 when only Channel n is open.

Similarly, the CF of the polarization controller is defined by

$$\text{CF}=\frac{\left|\text{Pow}\u2022{\text{Pow}}_{\mathrm{exp}}\right|}{\left|\text{Pow}\right|\left|{\text{Pow}}_{\mathrm{exp}}\right|}.$$(7)

Here, Pow is the desired output power distribution in four ports and Pow_{exp} is the measured one. For example, Pow can be set as [0, 1, 0, 0] if we want to make all the light output from Port 2.

CF ranges from 0 to 1, where CF=0 means that the experimental results are completely inconsistent with the targeted results and CF=1 means that they are completely consistent. Our training target is to make CF as close to 1 as possible. To guarantee power efficiency, the first eigenvalue of the transmission matrix is fixed to 1; namely the phase difference of two inner arms of the second MZI in Figure 1A is always equal to *π* and there is no light output from Port 1. In the following, the chip is reconfigured to achieve three different functions.

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