Scalable Multi-Core Dual-Polarization Coherent Receiver Using a Metasurface Optical Hybrid

The space-division multiplexed (SDM) coherent transmission using a multi-core fiber (MCF) is a promising technology for further increasing the capacity of optical communications and interconnects. For broader application of SDM systems, dual-polarization (DP) coherent receivers that can be directly coupled to MCFs without bulky fan-in/fan-out (FIFO) devices and polarization beam splitters (PBSs) are desirable. Unlike intensity-modulation direct-detection (IMDD) receivers, however, scaling a DP coherent receiver to multiple spatial channels in a compact form factor is not straightforward. Here, we propose and demonstrate a compact and scalable surface-normal multi-core DP coherent receiver using an ultrathin dielectric metasurface (MS). The MS is composed of silicon nanoposts on quartz, which are judiciously designed to function as the DP optical hybrids and focusing lenses for all spatial channels. Using the fabricated device, simultaneous homodyne detection of DP 64-ary quadrature-amplitude-modulation signals from a four-core fiber is demonstrated with an error vector magnitude of less than 3% for all spatial/polarization channels from 1530 to 1570 nm. The demonstrated receiver can be assembled into a compact module to enable low-cost SDM coherent receivers without bulky PBS and FIFO devices.


I. INTRODUCTION
O VER the past few decades, numerous key technologies have been introduced in optical communications to address the continuously growing data traffic [1], [2].In the 2010 s, dual-polarization (DP) digital coherent systems were put into practical use, which have substantially expanded the transmission capacity by fully utilizing optical phase and polarization degrees of freedom, in addition to optical intensity employed in intensity-modulation and direct-detection (IMDD) systems.Digital coherent systems, which have initially become dominant in the backbone and metro networks, are now expanding to shorter-reach networks, such as datacenter interconnects and optical access [3], [4], [5].
Another emerging technology to expand the transmission capacity is the space-division multiplexing (SDM) that utilizes multiple spatial modes of a multi-core fiber (MCF) or a multi-mode fiber (MMF) as an additional transmission degree of freedom [6], [7], [8].In particular, use of uncoupled MCFs is expected in the near-future practical systems since they do not require power-hungry multi-input multi-output (MIMO) processing to compensate for complex modal coupling among spatial channels [9], [10].SDM systems using dual-core uncoupled MCF have recently been deployed in submarine links, which will be in service in a few years [11].We can expect that such SDM systems would extend to shorter-reach networks, where compact and cost-effective multi-core DP coherent transceivers would be desired.
In order to handle multi-core DP coherent signals in a scalable manner, integrated transceivers that can be coupled directly to MCFs without bulky fan-in/fan-out (FIFO) devices and polarization beam splitters (PBSs) would be ideal.For IMDD systems, multi-core integrated transceivers using vertical-cavity surface-emitting lasers and surface-illuminating photodetector (PD) arrays have been demonstrated [12].Unlike IMDD format, however, dense 2D array of coherent receivers are challenging to realize due to their inherent complexity.As a result, a coherent receiver that can directly receive multi-core DP coherent signals from an MCF has never been demonstrated to our knowledge.
A straightforward approach to scale conventional integrated waveguide-based receivers to multiple 2D spatial channels is to use arrayed grating couplers (GCs).Indeed, single-polarization signals from an MCF can be efficiently coupled to multiple single-mode waveguides using fan-shaped GCs [13], [14], [15].In a DP coherent receiver, however, both polarization signals need to be coupled to multiple waveguides simultaneously, which makes use of GCs challenging.A polarization-separating (PS) GC that guides xand y-polarization components to separate waveguides in different directions has been explored [16], [17], [18].However, PSGCs generally require a large footprint and/or complicated routing to two independent waveguides per each PSGS, which make these devices not easily scalable to a dense 2D coherent receiver array.To this end, we have recently proposed an alternative approach of realizing a scalable multicore DP coherent receiver using a dielectric metasurface (MS) and a high-speed PD array [19].The MS consists of an array of silicon (Si) nanoposts, which were designed judiciously to focus signal light from each core to five PDs with precisely controlled polarization-dependent phases.By mixing with a local oscillator (LO) light, IQ signals of both polarizations were retrieved from five photocurrent signals without using FIFO or PBS.The designed MS was fabricated on a compact silicon-onquartz (SOQ) chip to demonstrate simultaneous demodulation of four DP coherent signals from a four-core step-index MCF at 1550-nm wavelength.
In this article, we provide a rigorous analysis of the receiver sensitivity of the proposed receiver and derive an explicit condition for the MS to achieve the highest sensitivity.Furthermore, detailed design methodology of MS, comprehensive experimental results using an SMF, and additional experimental investigations on broad wavelength operation from 1530 to 1570 nm are provided.
Throughout this paper, we employ following mathematical notations: c(t) denotes the expected value of a scalar function c(t), v is the Euclidean norm of a vector v, M T , M † , M F , and tr(M) are the transpose matrix, the adjoint matrix, the Frobenius norm, and the trace of a matrix M, respectively.

II. DEVICE CONCEPT
The configuration of the proposed SDM DP coherent receiver is shown in Fig. 1.Fig. 1(a) illustrates the case where a singlechannel DP coherent signal is coupled from an SMF [20].Signal light emitted from an SMF is incident on the MS and is focused with equal intensity at five points forming a regular pentagon on the surface of a PD array.Optical phases at respective focal points are designed to maximize the detection sensitivity as we will discuss in the next subsection.In an optimal case, the phases of adjacent focusing points differ by 2π/5 and 4π/5 for x and y polarizations, respectively, as shown in Fig. 1(b).Then, collimated LO light with 45 • linear polarization is combined and irradiated onto the same PD array, so that signal-LO beat components are detected.The nonlinear signal-signal beat noise can be eliminated effectively by taking the difference between the photocurrents of adjacent PDs.Subsequently, we obtain four real-valued electrical signals, from which the four distinct elements of DP coherent signal (i.e., in-phase and quadrature signals of both polarization components) can be retrieved through 4 × 4 linear operation by digital signal processing (DSP).
We should note that this device relies on the same principle as the waveguide-based coherent receiver with five PDs, demonstrated by the authors [21].Compared with conventional DP coherent receivers, which require eight PDs (or four balanced PDs) [22], [23], [24], the number of PDs in our receiver is reduced to five.As a result, we can minimize the footprint of the PD array.Moreover, owing to its surface-normal configuration, the proposed scheme can naturally be scaled to receive spatially parallelized coherent signals.Since the MS acts as a focusing lens, multiple signal beams from different input positions are focused to five points at shifted positions.This feature enables simultaneous detection of multiple signals from all cores when an MCF is placed at the input as shown in Fig. 1(c).Importantly, this can be accomplished without modifying the MS or other optical Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
components.Consequently, this configuration offers inherent scalability with respect to the number of channels, enabling compact implementation of SDM digital coherent receivers.

A. Theory
Here, we present an analytical model of a generalized surfacenormal DP coherent receiver comprising N PDs.To describe the electric field distributions of the xand y-polarization components on the PD array surface for the signal and LO lightwaves, respectively, we use Jones vectors denoted as where ) represent the normalized complex IQ amplitudes for both polarizations.Since the electric field at the PD array surface is described as , where ω c is the carrier angular frequency, the photocurrent I n (n = 1, . . ., N) of n-th PD is described as where R is the responsivity of PD.The first term, P bn , denotes the LO-LO beat component, which is spatially integrated within the entire detection area of n-th PD and is written as Similarly, P a1n and P a2n in the second term in (2) represent the signal-signal beat components for the x and y polarizations, respectively, and are defined as Finally, the third term in (2) represents the signal-LO beat component using Q n and A, which are defined as where To eliminate the signal-signal beat component term that are nonlinear with respect to A, differential photodetection is generally required.This operation is mathematically written as where S denotes the four real-valued signal components, C is a 4 × N matrix representing the differentiation operation, and In order to remove the signal-signal beat components, C must satisfy the following conditions: When these conditions are satisfied, S is expressed by inserting (2) to (8) as where we define In (10), the first term inside the parenthesis represents the signal-LO beat components, whereas the second term is a constant vector representing the remaining LO-LO beat components after the differentiation.If the matrix CQ is regular, the desired signal component A can be obtained from (10) as

B. Sensitivity Analysis
In general, the receiver sensitivity of a coherent receiver is ultimately limited by the shot noise.We consider a case where the detected PD current I meas includes noise components ΔI ≡ [ΔI 1 , . . ., ΔI n ] T and is written as Assuming that LO light intensity is large compared with the signal, variance of the shot noise current can be expressed as [22] where q is the elementary charge and W is the detection bandwidth.We can calculate the variance of the difference between the decoded IQ signal A retrieved from I meas and the actual IQ signal A (see Appendix A for details) as where Σ L is a N × N diagonal matrix consisting of the square roots of the LO light intensity expressed as We now define a dimensionless parameter K as Since K is proportional to the normalized noise ε, it is useful in comparing the receiver sensitivity of different configurations.
In the following, we evaluate the quantitative receiver sensitivity assuming that the incident signal power is normalized, so that |a 1 | 2 dxdy = |a 2 | 2 dxdy = 1, and the LO power is sufficiently large so that E LO 2 dxdy = P LO 1. First, we consider a conventional DP coherent receiver with a 90 • hybrid (N = 8) under ideal conditions.We assume that the xand y-polarization components of the signal and LO light are split and focused to n = 1, . . ., 4 and n = 5, . . ., 8, respectively, with equal power and precise phase relationship.Then, the signal and LO power detected by each PD can be expressed as Considering the phase shift of 90 • hybrid, H kn and Q are written as where |H| is defined as The matrix C for the four balanced PDs is expressed as By inserting parameters of ( 19), (22), and ( 24) to (18), we obtain Next, we investigate the sensitivity of our proposed configuration with five PDs (N = 5) shown in Fig. 1.Assuming that the power transmittance of the beam splitter (BS) is T , signal and LO power at each PD can be written as Since half of the LO power interferes with each polarization of the signal in this case, |H kn | can be written as Then, H kn and Q are derived as = |H| 1 e i2π/5 e i4π/5 e −i4π/5 e −i2π/5 , (28) Since we take the difference between the photocurrents of adjacent PDs, C is written as Note that the LO-LO beat component is also eliminated by the current differentiation in this case, leading to P ΔLO = 0. Inserting ( 26), (30), and (31) to (18), we obtain From (32), we see that K approaches 16 by increasing T , provided that sufficient LO power is input to the receiver.Consequently, we can conclude that the theoretical sensitivity limit of the proposed configuration is half of the conventional polarization-diversity 90 • -hybrid configuration given by ( 25).This 3-dB penalty originates from the fact that both polarization components of the signal are mixed with the LO light and received together in our five-PD scheme.This is in contrast to the conventional polarization-diversity configuration with eight PDs, where orthogonal polarization components of the signal are first demultiplexed, then combined with the LO light having the same polarization state, and finally detected by separate PDs for respective polarizations.In the proposed method, therefore, only half of the detected signal in average has the same polarization state as the LO light and can interfere to generate the signal-LO beat components, resulting in a 3-dB penalty.
Although the theoretical 3-dB reduction in the sensitivity is obviously a drawback of our five-PD configuration, we should note that fiber-to-chip coupling losses at GCs and insertion losses at the waveguide-based 90 • optical hybrids, which are required in conventional DP coherent receivers, can easily add up to exceed 3 dB in practice [23], [24].These losses should be eliminated or relaxed in our surface-normal receiver.In addition, Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.the elimination of PBS and reduction of the number of PDs from eight to five in our scheme would offer a substantial advantage in minimizing the device footprint and complexity.Besides, in a system where an optical preamplifier is deployed before the receiver, the amplified-spontaneous-emission (ASE) noise becomes dominant.In such a case, the proposed scheme does not have any sensitivity penalty compared with conventional polarization-diversity-based DP coherent receivers.
Finally, we should note that the minimum number of PDs to demodulate a DP coherent signal by linear operations in general is five.Through a numerical parameter search, it is confirmed that our proposed scheme with the phase distributions shown in Fig. 1(b) exhibits the theoretical maximum sensitivity among all the possible phase configurations using five PDs.

III. DESIGN AND FABRICATION OF METASURFACE
To experimentally demonstrate the proposed device, we designed and fabricated an MS composed of Si nanoposts on SiO 2 .The MS was designed to focus light from each core into five different spots with precise phase relationship as derived in the previous section.Fig. 2 illustrates the setup, where a fiber output is placed at a distance of l 1 from the MS and focused at a distance of l 2 from the MS output.Assuming an input fiber with the mode field diameter (MFD) of d, the diameter of each focused spot would be (l 2 /l 1 )d.Then, when an MCF with the core spacing of Λ is deployed, convolutions of these five spots would appear at the focal plane with a distance of (l 2 /l 1 )Λ.
In this work, a four-core step-index MCF [25] with d = 10.4 μm and Λ = 40 μm at a wavelength of 1550 nm was employed.The diameter of the focused spot was set to 31.2 μm by setting l 1 = 2 mm and l 2 = 6 mm.The diameter of entire MS region was set to 600 μm, so that more than 99% of the light emitted from the fiber was captured.To simplify the design of MS, a highly symmetric structure is preferable.Thus, the focal points were arranged in a regular pentagon, as depicted in Fig. 2(b).The size of the regular pentagon was determined to maximize the spacing between the focal points, resulting in a spacing of p = 45.8 μm.Assuming Gaussian beam approximation, we estimated that 98.7% of the total optical power of each focused spot was within the diameter of p, indicating sufficiently low crosstalk to adjacent PDs.
The MS was designed based on the methodology demonstrated previously [26], [27].As shown in Fig. 3(a), Si elliptical nanoposts with 1.05-μm height on a SiO 2 substrate were employed as meta-atoms and arranged in a square lattice.By adjusting the diameters (D x , D y ) of each meta-atom, arbitrary phase shifts (φ x , φ y ) for xand y-polarization components could be applied.Using the rigorous coupled-wave analysis (RCWA), a look-up table (LUT) was first generated, which indicated the required (D x , D y ) to achieve desired (φ x , φ y ) [28].Assuming the lattice constant L of 0.9 μm and the ranges of D x and D y from 200 nm to 840 nm, we successfully constructed LUT that gives arbitrary φ x and φ y with an average transmission as high as 94%.
The desired phase shift characteristics of MS were then calculated.The target phase profiles at the input and output of the MS were obtained through forward propagation from the input fiber mode as well as backward propagation from the ideal electric field at the PD array surface.By taking the difference of them, required phase profiles for xand y-polarization components were obtained.Since dielectric MS can control only the phase, electric field profiles on the PD array generally deviate from the ideal ones, causing, for example, intensity variations between each focal point.Nonetheless, owing to the high rotational symmetry in the complex amplitude distribution in the designed layout, such errors were minimized.Using these phase shift profiles and LUT, the distribution of meta-atom dimensions was derived as shown in Fig. 3(b).
To confirm the validity of the designed MS, rigorous simulation using the finite-difference time-domain (FDTD) method was performed.Fig. 3(c) shows the intensity and phase profiles of electric field (E x , E y ) at the PD plane for xand y-polarized inputs.The white line denotes the assumed photosensitive area of each PD, with a diameter of 40 μm.We can confirm that the light is focused at the five PDs with precise phase differences as designed.The total loss from incident light to PD is 3.1 dB and 3.7 dB for x and y polarizations, respectively.The intensity difference between different PDs is less than 0.2 dB, while the crosstalk from the neighboring input cores is suppressed below -16 dB for both polarization components.From (18), the receiver sensitivity penalty from the theoretical limit is derived to be 3.9 dB.Furthermore, it is numerically confirmed that the variation of sensitivity using our MS is less than 0.2 dB within the C-band, which is comparable or smaller than the loss variation of conventional waveguide-based 90 • hybrids using multimode interference couplers [29].
The primary sources of the loss compared to the ideal case are attributed to two factors.First, since a lossless dielectric MS can only manipulate the optical phase, the actual wavefront generated by MS may not match perfectly with the ideal profile, which generally has a nonuniform intensity distribution.This difference is estimated to account for ∼1 dB of loss.By employing dual MS configuration, it is possible to control both the intensity and phase distributions [30], which should reduce the penalty caused by such intensity mismatching.The second factor is the interaction between adjacent meta-atoms, which is estimated to contribute to ∼2 dB of loss.While the LUT generated through RCWA assumes an infinite array of meta-atoms with identical dimensions, the actual designed MS consists of spatially varying meta-atoms, which results in phase errors.We can, therefore, expect that these losses can be eliminated to some extent by employing more sophisticated design algorithms that take into account complex interactions between neighboring meta-atoms [31], [32], [33].
The designed MS was fabricated using a silicon-on-quartz (SOQ) substrate with 1.05-μm-thick crystalline Si layer.The meta-atom patterns were formed by electron-beam lithography with ZEP520 A resist, followed by inductively coupled plasma reactive-ion etching (ICP-RIE) using SF 6 and C 4 F 8 gases (Bosch process).Microscope and scanning-electron-microscope (SEM) images of the fabricated MS with a diameter of 600 μm are shown in Fig. 4. By optimizing fabrication conditions, the undesired scalloping effect on the sidewalls induced by the Bosch process [34] was minimized.Besides, high sidewall verticality exceeding 88 • and good in-plane uniformity were obtained.To evaluate the yield of our MS, we fabricated 20 devices on different chips and confirmed that the deviation of meta-atom dimensions was generally within ±10 nm, indicating high yield and reproducibility of our fabrication process.

IV. CHARACTERIZATION OF METASURFACE
The fabricated MS was evaluated by measuring the intensity and phase characteristics at the focal plane.Fig. 5(a) illustrates the experimental setup.Continuous-wave (CW) light at a wavelength of 1550 nm was emitted from an SMF, focused at P1' plane, and incident on MS.The polarization state was controlled using a quarter-wave plate (QWP), a half-wave plate (HWP), and a polarizer.The focal plane P2 at the output of MS was magnified using a 4-f system comprising two lenses, f 3 and f 4 , and then observed with an InGaAs camera (Hamamatsu C10633-13, 320 × 256 pixels).In order to measure phase profiles, a reference path was placed by splitting the collimated beam and combined with the MS output to form an interferometer.We employed the digital off-axis holography method to extract phase information from the camera image [35], [36].To measure the focusing efficiency, each focusing point was selected by an iris and the transmitted power was detected by an optical power meter.
Fig. 5(b) shows the complex amplitude distributions obtained for xand y-polarization inputs.We can confirm that the light is focused precisely onto five points at the vertices of regular pentagon for both polarizations as designed.The optical phase at each focal point, averaged over the photosensitive area of each PD (indicated by white circles in Fig. 3(c)), is plotted for xand y-polarization inputs in Fig. 5(c).The average and worst phase errors from the ideal case are 0.018π and 0.047π, respectively.Fig. 5(d) shows the measured focusing efficiencies to five points.The average efficiency to each of these five focal point is −11.8 dB (i.e., 4.8 dB below the intrinsic splitting loss of 1/5), so that the total insertion loss of our device is derived to be 4.8 dB.The intensity variations among the five focal points are less than 1.1 dB for both polarizations.The efficiencies obtained by the FDTD simulation are also plotted by the dashed lines for comparison.The ∼1.5-dB difference between the experiment and the simulations is attributed to imperfections in fabrication.6) was used as a reference light or LO; we extracted the light from only one core by using an iris and collimated to an InGaAs camera.The other path (the upper path in Fig. 6) was used as a signal.As  were stabilized.Note that the modulation speed was limited by the read-out time of the camera and the response time of VRs in this experiment.A real-valued 5 × 2 MIMO adaptive equalizer by a decision-driven least-mean-square (DD-LMS) algorithm [37], [38] was applied to obtain IQ components and to compensate for any polarization variation inside MCF and other fluctuations in the setup.Prior to the experiment, the adaptive equalizer was trained using transmitted symbols.Then, no knowledge about the transmitted data was used during the measurement to retrieve IQ signals.

V. MULTI-CORE COHERENT DETECTION EXPERIMENT
First, complex optical field distributions on the focal plane of MS measured at 1550-nm wavelength without IQ modulation are shown in Fig. 7(a) for both polarizations.Once again, the relative optical phases are extracted from these distributions for 20 focal points and plotted in Fig. 7(b).Now, we see that the signals from the four cores are focused on 20 spots with the same relative optical phases as in the single-core case shown in Fig. 5(b).The average and worst phase errors compared with the ideal case for the 20 focusing spots are 0.074π and 0.26π, respectively.Finally, Fig. 8 shows the retrieved constellations of xand y-polarization components from all four cores at wavelengths of 1530, 1550, and 1570 nm.Here, an averaged value of approximately 500 symbols is shown for each constellation.The error vector magnitude (EVM) derived from Fig. 8 is plotted for each wavelength in Fig. 9.The EVM is less than 3% for all wavelengths, cores, and polarization states, demonstrating that high-accuracy homodyne detection is achieved across the entire C-band.

VI. CONCLUSION
We have proposed and experimentally demonstrated a spatially scalable multi-core DP coherent receiver based on a dielectric MS.The proposed receiver realizes all the necessary functionalities of optical hybrids and focusing lenses for multiple channels using a single dielectric MS.Rigorous analytical model was introduced to derive explicit conditions for MS to achieve the highest receiver sensitivity.We designed and fabricated MS composed of Si nanoposts array on SiO 2 to demonstrate simultaneous detection of four DP 64QAM signals from a four-core step-index MCF.The EVM was less than 3% for all spatial/polarization channels from 1530 to 1570 nm.While the speed was limited by the InGaAs camera in this work, use of high-speed PD arrays [39], [40], [41] can easily achieve >100 Gb/s bandwidth per single spatial/polarization channel, enabling >Tb/s total data rate for four-core DP coherent signals.
Our designed MS can simply replace micro-optic lenses used in conventional receivers and can be inserted between an MCF and a PD array separated by less than 1 cm.The entire receiver can therefore be assembled into a compact module equivalent to current commercial single-channel coherent receivers [42], enabling low-cost and practical SDM coherent receivers without PBS and FIFO devices.
Finally, we should note that the scalable coherent receiver demonstrated in this work would be useful not only in optical communications and interconnects, but also in various other emerging applications.For example, frequency-modulated continuous-wave (FMCW) light detection and ranging (LiDAR) systems utilize the coherent nature of lightwaves to achieve high-sensitivity three-dimensional (3D) sensing.Compact and large-scale coherent receiver arrays are desired to achieve highspeed and high-resolution 3D imaging without using bulky and fragile mechanical beam steerers [43].In the field of optical neural networks, on the other hand, scalable coherent receiver arrays would be useful in performing large-scale multiplyaccumulate operations in the optical domain at ultra-low latency and computational energy [44], [45].We, therefore, believe that this work paves the way for a wide range of applications that Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
utilize the spatial parallelism of coherent light, including optical communication, sensing, and computing.

APPENDIX A DERIVATION OF SENSITIVITY FORMULA
Based on (13) and (14), The difference between the received symbol A and the actual symbol A is expressed as To quantitatively evaluate the sensitivity of receivers with different configurations, we derive the expression for the variance ε defined as Since the shot noise expressed by ( 15) is superimposed on each PD current, the noise vector ΔI follows an N -dimensional Gaussian distribution N N as It is known that a general N -dimensional vector X ≡ [X 1 , . . ., X N ] T ∼ N N (µ, Σ) has a following property about a transformation by a N × N matrix M [46]: Applying ( 35) and ( 36) to (33), the distribution of A − A can be written as (37) Next, the variance of the norm of X is calculated as where X n is the n-th element of X, which follows the onedimensional Gaussian distribution.When the mean and the standard deviation of X n are defined as μ n and σ n , the second order moment X 2 n can be calculated using the moment generating function M X n (t) = exp(μ n t + σ 2 n t 2 /2) as Equations ( 38) and (39) yield Applying (37) and ( 40) to (34), ε is expressed as Here, we used the Frobenius norm defined as

Fig. 1 .
Fig. 1.Proposed spatially scalable DP coherent receiver.(a) Schematic of the receiver when a single signal is input from an SMF.(b) Spatial distributions of signal and LO lightwaves on the PD array plane (a) when a single signal is input from an SMF and (c) when multiple signals are input from a 19-core MCF.In both configurations, the same MS can be used.

Fig. 2 .
Fig. 2. Functionalities of the designed MS.(a) Light propagation from each core.MFD: mode field diameter.(b) Intensity pattern on the PD array surface when a four-core MCF (Λ = 40 µm) is placed at the input.The light from each core forms a regular pentagon.

Fig. 3 .
Fig. 3. Design and numerical analysis of MS.(a) Schematic of the MS composed of Si elliptical nanoposts on SiO 2 , having the x and y diameters of (D x , D y ) and the lattice constant of L. (b) Designed distributions of D x and D y for L = 0.9 µm.(c) Electric field distribution on the PD array obtained by FDTD simulation.

Fig. 5 .
Fig. 5. Phase measurement of fabricated MS at 1550-nm wavelength for single-channel input.(a) Experimental setup.LD: laser diode, VOA: variable optical attenuator, SMF: single-mode fiber, QWP: quarter-wave plate, HWP: half-wave plate, Pol.: polarizer, f 1 = f 2 = 50 mm, f 3 = 4 mm, f 4 = 200 mm.(b) Electric field distribution at the PD array for x and y polarization inputs.(c) Relative optical phases of x and y components at each focal point.(d) Focusing efficiency at each focal point.Solid lines: measurement, dashed lines: FDTD simulation.

Fig. 6 .
Fig. 6.Experimental setup of four-channel DP coherent detection using the fabricated MS and a step-index four-core MCF.TLD: tunable laser diode, FC: fiber collimator, VR: liquid-crystal-based variable retarder, f 1 = f 2 = 50 mm, f 3 = 10 mm, f 4 = 200 mm, f 5 = 40 mm, f 6 = 125 mm, f 7 = 200 mm.The components in the broken-line box are used to emulate an IQ modulator.Blue arrows indicate the angle of light polarization, the slow axes of the VRs, or the orientations of the polarizers.

Fig. 6
Fig. 6 shows the experimental setup to demonstrate simultaneous homodyne detection of four-channel DP coherent signals from an MCF.CW light from a wavelength-tunable laser (Santec TSL-510) was irradiated to the input facet of a four-core stepindex MCF (cladding diameter = 125 μm, MFD = 10.4 μm at λ = 1550 nm, Λ = 40 μm) through a focusing lens.Output light from MCF was converted to 45 • linear polarization and split into two paths.One path (the lower path in Fig.6) was used as a reference light or LO; we extracted the light from only one core by using an iris and collimated to an InGaAs camera.The other path (the upper path in Fig.6) was used as a signal.As

Fig. 7 .
Fig. 7. Phase measurement of MS for four-core inputs at 1550-nm wavelength.(a) Electric field distribution at the PD array for x and y polarization inputs.(b) Relative optical phases of x and y components at each focal point for four cores.

Fig. 9 .
Fig. 9. Measured wavelength dependence of EVM in derived from measured data in Fig. 8.