Miniaturization of 2 × 4 90-Degree Hybrid Optical Couplers

In this work we study the limits of miniaturization of a 90-degree hybrid coupler working in the L, C and S bands, with respect to a number of performance parameters aimed at its application for balanced detection. We investigate the main effects responsible for the degradation of the performance of the devices during miniaturization, and establish the minimal dimension that such devices can have without significant degradation for photonic applications such as balanced detection. The miniaturized device in InP generic technology has a footprint of only $2200\mu \text{m}^{2}$ , more than 5 times smaller than the conventional device used as reference. The scaling approach is based on the use of the number of propagating modes which are sustained in both the miniaturized MMI and port waveguides as scaling parameters. This approach allows us to generalize the miniaturization problem from a specific platform and offers a methodology which is flexible and transferable to multiple platforms. We tested the scaling methodology based on the number of modes in other platforms commonly used in integrated photonics, such as Si/SiO2(SOI), TriPleX or polymer platforms, obtaining comparable results and proving the universality of our approach, finally we performed a fabrication tolerance analysis of the miniaturized devices.


I. INTRODUCTION
M ULTI-MODE interferometer couplers (MMI) [1], [2], [3], are a fundamental type of photonic components, whose main function is to distribute the incoming light over several output ports, each with a well-defined power ratio and phase difference with respect to each other. The working principle is based on self-imaging caused by internal interference of the modes excited and propagating in it [2], [4], [5], [6]. They find an impressive number of applications such as splitters [7], couplers [8], multiplexers [9], demultiplexers [10], power combiners [11], Manuscript  filters [12], sensors [13] to name a few. 90-degree hybrid couplers are a subclass of MMIs, which finds applications in coherent detection systems. They are commonly used in telecommunications for demodulating optical signals with quadrature phase shift keyed (QPSK) modulation format and balanced detection [14], [15], [16], [17], [18], [19]. The structure of a typical 2 × 4 MMI hybrid coupler is shown in Fig 1. It consists of two input ports on the left, and four output ports on the right, the first input port, symmetrical to the output channel 1, is used for the received Signal (S), and the second, symmetrical to the output channel 3, is used for the Local Oscillator (LO) [20], [21]. Because of this configuration, this structure is also referred to 4 × 4 hybrid in the literature [21], [22], with reference to the 2 unused virtual ports opposite to the output channels. Other configurations of 90-degree couplers also exist, usually involving cascades of lower order MMIs [20], [23] or more exotic configurations [24].
In a properly designed hybrid the two signals mix inside the MMI producing equally distributed powers at the four output channels, and a phase difference between the signal and local oscillator ports at 0 • , ±90 • , ∓90 • and 180 • degrees (ignoring constant phase offsets), yielding the output fields as in In this configuration, the output channels 1 and 4 can be used as in-phase components and the output channels 2 and 3 as quadrature components for balanced detection, as shown schematically in Fig.1. The detection itself consists of four identical photodiodes which convert the light power coming from the output ports of the MMI into current which can then be detected by standard electronic instrumentation. The balanced detection allows for high S/N ratios as it supplies a superposition of signal and local oscillator, and also suppresses the dc component in detection and its drifts [14].
The optimization of the MMI to work as a proper 90-degree hybrid depends on both the widths w of the input ports, the separation d and the width W and length L of the MMI itself.
According to the self-imaging theory, an optimal hybrid can usually be realized for relatively long MMIs, often a few hundreds of microns. This might hinder the implementation in large scale integration systems which require devices to fit within a densely packed chip. It is thus important to implement strategies for the miniaturization of these devices, without compromising their performance and the quality of self-imaging. The miniaturization of an MMI presents three main challenges: the first is that a reduced size in the input waveguides decreases the confinement of the fundamental mode reducing the efficiency of the coupling with the modes propagating in the MMI. The second is that a reduction of the width of the MMI implies a reduced number of propagating modes inside the MMI itself which leads to reduced imaging quality. The third is that crosstalk between adjacent waveguides may induce significant errors if their separation is too small. The first target of this paper is to explore the limits of miniaturization of a working 2 × 4 MMI 90-degree hybrid coupler for a specific platform with respect to these challenges, establishing the minimal dimensions which do not affect the device performance for coherent receivers, and to explore the operational bandwidth of the miniaturized device. The second target is to show that the quality of self-imaging is directly related to the number of modes propagating in the input waveguides and, in the MMI, the latter can be used as a universal, platform-independent scaling unit, to switch between equally performing platforms. The novelty of the paper is that not only we present concrete dimensions for reduction of 90-degree hybrid couplers for a number of common platforms to operate in systems for coherent receivers with minimal footprint, but we also establish a method to translate the dimensions (and corresponding performance) from one platform to another by using the number of modes propagating in the input waveguides and MMI as a universal platform-independent scaling unit.

A. Performance Parameters
In order to test the quality of operation of our devices we establish four families of performance parameters [25], [24] which are dependent on the eight scattering parameters, S [1,2,3,4],[S,L O] defined as the ratio between the electric field propagating to one of the four output ports [1], [2], [3], [4] and the corresponding exciting signal in one of the two input ports [S, LO], and the corresponding power ratios which are defined as P 1) The Common Mode Rejection Ratio (CMRR) is a metric used to quantify the ability of the device to reject commonmode signals. In view of the applications for balanced photodetection we define it as a ratio of the photocurrents that will circulate on the receiving photodiodes and which are directly proportional to the detected power (supposing all the four photodiodes to be the same). It is thus expressed in electrical decibels [dBe] and defined for the in-phase and quadrature signals as The CMRR is supposed to be small so that all CMRR in dBe units are negative numbers, the more negative the better the performance. We take as performance parameter the highest value (which is the less negative, hence the worst) between the two.
2) The insertion loss (IL), expressed in decibels [dB], is defined as the ratio between the total power before and after the hybrid 3) The imbalances for the in-phase and quadrature signals, also expressed in [dB] are defined using the total power at the two ports respectively. The absolute value of the imbalance must be as small as possible, thus we take the highest (the worst) value between those two as criterion to evaluate the performance.
4) The phase errors, expressed in degrees [ • ], between the phases of the in-phase output channels, those in quadrature and between the two output components are defined as and the one between the two output components The higher the phase error in modulus, the lower the performance, thus we take the highest among these phase errors, as performance parameters.
We consider a miniaturized device as suitable for coherent receivers if both CMRR < −20dBe, |ε| < 5 • ,IL< 1dB and |Imb| < 1dB, at the operation wavelength. These conditions are stricter than those which normally apply for a 90-degree hybrid aimed at balanced detection [26].

B. Methodology of the Simulations
All the simulations are performed using an EigenMode Expansion Solver (EME) [27], with perfectly matching layers (PML) at all external interfaces. The number of modes used in simulations are always at least five more than the total number of propagating TE and TM modes. The input modes were all the propagating TE modes and the simulations were performed separately for Signal and Local Oscillator inputs. In order to evaluate the phase and power in a simple way, without loss of generality, the input power and phase for the Signal and Local Oscillator were the same in our simulations. In order to compare different platforms, we considered all the refractive indices as purely real (no losses), and constant over the wavelength band in consideration [1460nm -1625nm]. We verified the validity of our simulations by comparing the results obtained with EME to those obtained with FTDT [27] for identical small test structures. The results are comparable, but the computational time of FTDT is much longer than for EME, which remains our preferential choice for the simulations.

III. REFERENCE DESIGN A. Structure of the InP Platform Chosen as Reference
The cross section of the MMI layerstack for a deep-etched waveguide in the InP generic platform [28] is shown in figure  Fig 2a. it consists of a p-doped InP (refractive index n = 3.17 [29], [30]) top cladding of 1500nm, a InGaAsP (n = 3.38 [30]), and lower cladding of n-doped InP (n = 3.163 [29], [30]) of 200nm on the same substrate [5]. The device is surrounded by a layer of polyimide (n = 1.5) for potential integration to active components [28]. The thickness of the layer of polyimide (usually at least a couple of micron [28]) is enough for the propagating modes to be completely confined in the system.

B. Waveguide Dimensions
The width of the input and output waveguides was initially chosen to be d = 3.5µm in order to accommodate 3 modes, this dimension is large enough to guarantee an efficient coupling to the propagating modes in the MMI.

C. Positions of Input and Outputs Waveguide of MMI
The hybrid coupler shown in Fig. 1 has 4 outputs, q = 1, 2, 3, 4 whose position is determined by the theory of imaging [5], [6] as where N is the number of images reproduced at the output of the MMI (N = 4 for the hybrid), M is a positive integer without common divisor with N, and W is the width of the MMI. The distance between the waveguides is fixed to be d = 1µm in order to limit efficiently the crosstalk between waveguides. The distance between the centers of the neighboring waveguides then becomes x q − x q−1 = w + d = 4.5µm. For M = 1 the required phase conditions at the output channels for the Signal and Local Oscillator outputs in formula Cross sections for the waveguides analyzed in this work: a) InP in generic platform, which will be used to demonstrate the scaling, b) SOI structure, c) Polymer waveguide and d) TriPleX, on which we will apply the scaling properties, more details in section VIII. The vertical dimensions are represented in scale, and are not changed in this study, the variation of the dimensions of the waveguides is specified in the main text.
(1) are respected, and we find from (6) that the optimal nominal width for this configuration is W = 18µm.
However the self-image theory neglects the evanescent part of the fields, considering all modes perfectly confined in the MMI. In order to take this into account we have to calculate the effective width W M of the MMI, so that the TE modes are effectively present till W [2] Here n e f f is the effective refractive index of a planar infinite waveguide (n e f f = 3.282) and n 0 is the refractive index of the surrounding medium (n 0 = 1.5forthestructureinfig.1a) For the device in Fig. 1

D. Length of the MMI
According to the theory of self-imaging, N images show up at is the coupling length between the two lowest-order modes L c = π β 01 . For M = 1, W M = 18µm and n e f f = 3.2818 in our case translates as L = 686µm. However this is an approximate method which reduces the system from 3D to a 2D problem, leaving room for additional optimization. With accurate EME numerical simulations, we find the optimal length for a device of physical width W = 17.8µm to be L = 671µm, which takes into account also δW . These last ones are the dimensions for our reference device in the InP generic platform, which correspond to a footprint of around 12000 µm 2 .

E. Number of Modes in the MMI
In order to create a relation between the dimensions of MMIs and waveguides and the number of modes propagating in them, we use an effective index method approximation [31]. More specifically, the 3D structured waveguide is described as an adapted planar waveguide perpendicular to the real one whose core of dimension w is sandwiched between two infinite layers. The refractive index of the core of this planar waveguide is the effective refractive index n e f f of the original waveguide for the fundamental TE mode, and can be calculated numerically. The refractive index of the cladding n clad is the highest refractive index among those of the materials constituting the cladding of the original waveguide. The number of TE modes X can be extracted by adapting the calculation for a slab waveguide [32] The results of this relation are in agreement with EME simulations for the corresponding structures.
The expression (10) holds for both the modes in the input and output waveguides and it gives a linear dependence between the number of modes and the width of the device, which means that the width of the working devices can be associated to the number of modes.

IV. DEPENDENCE OF PERFORMANCE ON THE DIMENSION
OF THE PORT WAVEGUIDES We start by analyzing how the performance of the MMI changes by varying the width w of the waveguides at its ports [33].
To do so, we take the MMI at the reference size, and vary the width of all the waveguides by a factor α as in Fig. 3. This changes the efficiency of coupling to the modes propagating in the MMI.
In Fig. 4 we show how a scaling of all the waveguides at all ports affects the performance parameters. We scale all the waveguides from w = 3.5µm to 1.5µm (which means by applying a scaling factor α = 1 to 0.43, to the original waveguides), which corresponds to reducing the number of propagating modes from 3 (above 3µm) to 2 modes (above 1.75µm) to 1 mode (below 1.75µm).
For each performance parameter, we show the corresponding values for the in-phase (I, line in blue) and quadrature ports (Q, line in red), as well as the worst performance of the two (dark black line), which serves as the one chosen for the performance parameter. A dashed horizontal line, when present, represents the threshold for that performance parameter. All data below the dashed line in the figures are considered suitable for receiver applications.
It can be observed that all the performance parameters degrade when the waveguide shrinks, in particular we notice that when the waveguide has three modes (w > 3µm), it has similar performance to the reference; when it has two modes (w > 1.75µm) it has a slightly worse performance, but still below the threshold, while for narrower waveguides, at least Fig. 4. Evolution of the performance parameters a) CMRR, b) insertion loss, c) absolute imbalance and d) absolute phase error for the MMI of fixed dimension described in the text, on the width w of the port waveguides (and their correspondent scaling factor from the reference port), the dark black line, corresponding to the worst conditions, is the result for the corresponding performance parameter, the dashed horizontal line, is the threshold below which the device satisfies the requirements for balanced detection application. one of the four performance parameters is above threshold, which makes it unsuitable.
From this analysis we conclude that, for an equally sized MMI, a port waveguide can be miniaturized till 2 modes (w ≥ 2µm in the InP platform), without significant loss of performance.

V. DEPENDENCE OF PERFORMANCE ON THE MINIATURIZATION OF THE HYBRID
The next step is to study the evolution of the performance of the hybrid by miniaturizing the whole structure as shown in Fig. 5. Because of the geometry of the hybrid, a scaling of the width of the device cannot be independent from the scaling of the waveguides, which must be fully contained in it.
Thus we adopt the approach to scale the width of the waveguides w, their relative distance d and width of the MMI W by the same factor α, and to scale the length of the MMI L by α 2 in order to maintain the same proportions as the reference for image formation as in (9) This means that the footprint of the MMI scales with α 3 . In Fig. 6 we compare the performance parameters for the miniaturized device depending on the scaling factor α. Since the number Fig. 6. Evolution of the performance parameters for the fully scaled device on the scaling factor for the whole device α, the number of modes and the dimensions in the x direction are linearly proportional to it and can be used as alternative scaling factors (such as the port waveguide width w, shown here for direct comparison to the results in Fig. 4). of modes is linearly proportional to the width W, which scales proportionally to α, α represents also the ratio between the number of modes of the reference sample and of the miniaturized one.
We observe that for a scaling to w min = 2µm, corresponding to α = 0.57, all the four performance parameters are within the required specification range. This corresponds to a miniaturized device with dimensions W min = 10.2µm, L min = 219µm, w min = 2µm, d min = 570nm and a footprint of around 2200µm 2 which is more than 5 times less than that of the reference device.
For these miniaturization conditions, the port waveguide of width w min can sustain the propagation of 2 modes (fulfilling also the requirements of section IV), while the MMI of width W min allows the propagation of 11 modes. These are the number of modes, that we will use as scaling factors to transfer the miniaturization conditions to other platforms. Fig. 7 shows the power distribution in the reference and miniaturized MMI, it can be noticed that all the characteristic features for self imaging (such as images at L/2, 4 output images) which are observable in the reference MMI are clearly observable also in the miniaturised device.

VI. ANALYSIS OF CROSSTALK BETWEEN WAVEGUIDES
Crosstalk between waveguides, as shown in Fig. 8, is an important limitation to the performance of the devices, it affects mainly the output ports which are very close to each other. Crosstalk does not only create an imbalance of power with respect to the nominal conditions, but also creates an additional mixing of the output phases in the output channels themselves, seriously degrading their phase configuration.
Following the directional coupler analysis [31] we define the crosstalk power distributed in two waveguides for the fundamental modes as P 1,2 = 1 2 (1 ± cos βx) where x is the distance, and β = 2π λ n e f f _even − n e f f _odd is the difference in propagation constants of the normal modes associated to the even and odd sums of the fundamental modes in the coupled waveguide system. With simple trigonometric manipulation we extrapolate the fraction of the power transferred from one waveguide to another to be P 2 P 1 = tan 2 βx 2 . This is dependent from the refractive indices, geometry and separation of the waveguides. We consider the crosstalk to be negligible if the S 12 is below a threshold of -30dB for a length of at least x = 200µm, which is enough to allow the output waveguides to depart from the critical dimension with a sufficient large bending radius to minimize losses [28].
For the reference structure (w = 3.5µm, d = 1µm) the crosstalk is negligible with P 2 P 1 < −65d B at x = 200µm, while for the miniaturized device (w = 2µm, d = 571nm) we calculate P 2 P 1 = −36.7d B, which is still below the threshold.

VII. EXTENSION OF THE ANALYSIS TO S, C AND L BANDS
Once the optimal scaling parameters were obtained for the operational wavelength of λ = 1550nm, we extend the analysis to the S [1460nm -1530nm], C [1530nm -1565nm] and L [1565nm -1625nm] bands. In Fig. 9 we show a comparison of the four performance parameters for both the reference and the miniaturized device. It can be observed that the miniaturized device has comparable performance to the reference one, with a slightly worse performance in the C-band, but still well beyond the threshold, while the two devices are essentially equivalent in the L and S bands. The performance is overall very good with around 80nm of useful bandwidth, a value in line with other 90-degree couplers [24], [34].

VIII. EXTENSION TO OTHER PLATFORMS
In order to prove the validity of the scaling approach method we perform a comparison with other platforms commonly used in photonics. The approach is to calculate the effective refractive indices for these structures, and the corresponding relation with the number of propagating modes with respect to the width. Once these have been calculated, we can scale the device width W as and w and d accordingly, in order to have the same number of propagating modes, and scale the length accordingly as in (9). In other words we can transfer the optimized W dimension for the devices that we calculated for a platform (e.g. InP in this case) to a new platform using the relation The other platforms that were tested are shown in Figs 1 (b, c, d). They are all commonly used platforms and have slightly different waveguide shapes: 1) Si/SiO 2 SOI [3]: substrate SiO 2 (n = 1.445 [35]), 220nm thick core Si (n = 3.455), in air (n = 1), with n e f f = 2.835. In this platform the thin Si core waveguide lays on a SiO 2 substrate and is in the open air, without any casing, the high difference of refractive index of the Si core with respect to SiO 2 and air provides good more confinement, despite the tiny dimension of the waveguides. 2) SU8 polymer waveguide [36], [37]: substrate SiO 2 , 1.7µm thick core SU8 (n = 1.573), in air, with n e f f = 1.537. This platform is similar to the previous one, but the refractive index of the core is now comparable to those of SiO 2 and air thus the dimension of the waveguides must be larger for good confinement. 3) TriPleX [38]: 50nm thick core Si 3 N 4 (n = 2.463 [35]), embedded in SiO 2 , with n e f f = 1.496. Here the core Si 3 N 4 waveguide is totally embedded in SiO 2 , the SiO 2 is supposed to be thick enough that all modes are herein contained. As previously said, for an accurate comparison of the performance of the different devices we neglect the imaginary part of the refractive indices and consider them constant for the studied wavelengths.
To scale the devices from one platform to another we simply take the y-dimensions w, d, W of the miniaturized hybrid with the satisfactory performance and we scale them so that the corresponding device in the new platform has the same number of modes. In the present study the reference number of modes in MMI with respect to which we scale the devices is X = 11 as this proved to be the minimal one to maintain acceptable performance for balanced detection. Since X is linearly proportional to the width of the waveguide, this guarantees that also the corresponding scaled port waveguides of width w' have 2 propagating modes each, as it was for the InP platform, which ensures good coupling. In table I we show the effective refractive indices and the widths w and distances d of the port waveguides, after renormalization to have the same number of modes.
It is important to ensure that the distance between the waveguides at the output ports of the miniaturized device, is such that the crosstalk condition of power coupling of -30dB described in section VI is respected. This is not the case for the miniaturized SOI and TriPleX structures. In fact for the SOI structure (w = 0.697µm and d = 199nm) the power coupling amounts to 5.4dB while for TriPleX (w = 5µm and d = 1.429µm) to −19.8dB. These values are well above the threshold, and imply that the device will not be efficiently performant because of mode mixing at the output channels. In such cases, the miniaturized device is designed by choosing the width of the waveguides in order to have 2 propagating modes, which corresponds to a dimension w which ensures efficient coupling in the MMI, as described in section IV; then the distance d, for which the crosstalk between two waveguides of width w satisfy the threshold condition (< −30dB) is calculated, and finally one can proceed to calculate W and L similarly to what we did in section III-C for the reference device.
The dimensions for the adapted SOI and TriPleX devices are also shown in table I. In table II we show the main results for the miniaturized device at the operational wavelength of λ = 1550nm: width W, length L, footprint FP, and the four performance parameters. It can be seen that all the devices satisfy the threshold conditions for balanced detection.
In Fig. 10 we show a comparison of performance in the S, C and L bands between the miniaturized structure in the InP platform and the correspondent one in the SU8 platform. The structures are indeed similarly performing, with just a slightly lower performance for the SU8 platform concerning the imbalance and phase error. This can be further improved by applying tiny variations of W and L, similarly to what was done for the reference structure in section III-D.

IX. FABRICATION TOLERANCE
Finally we study what is the fabrication tolerance of the miniaturized devices. The studied structures can be fabricated, Fig. 10. Comparison of the performance parameters in the L, C and S bands, for the miniaturized hybrid in both the InP generic platform and the SU8 platform where the latter is obtained by mode translation without further optimization.
using standard clean room techniques, however due to the small dimensions of the miniaturized devices, some deviations from the nominal dimensions should be expected. By defining the structure using scanner, electron beam or optical lithography, followed by etching via ICP we have a variation of the critical dimensions of around ±20nm. ±60nm, and ±160nm respectively [39]. This means that in general a waveguide would be larger or narrower than its nominal width w by a random but constant value within the amount of the variation. In standard nanofabrication conditions we assume that these values are valid both in the x and y horizontal directions, so if there is a fabrication error δ f along the x-axis, the same error δ f will be also along the y-axis. The vertical component z and the refractive indices are more precise in generic technology [28], as growing techniques for alloys (eg. epitaxy) or oxides (eg. PECVD) are usually calibrated to deposit material with nanometric precision, and with stoichiometric resolution. In the case of the hybrids, in presence of a fabrication error of δ f ,we have , as the position of the center waveguides is usually well defined and with negligible errors with present technology, it should be noticed, that, since the fabrication error adds linearly to each direction, our study on fabrication tolerance is not differential as in other studies [2], [33], [40], but additive. The image formation suitable for an MMI of width W' will appear at In order for a device affected by fabrication errors to have a similar performance to a nominal device, it is thus necessary that this error in length defined as E L = 4n e f f (δ f 2 +2W δ f ) 3λ −δ f , might be less than a factor k of the resolution [2] of the images ρ = W m where W is the width of the nominal MMI, m here is the highest order of the propagating modes in it. As we pointed out in section III-E, m is proportional to W, so ρ is independent on the device size, we establish then a general condition for fabrication tolerance that E L < kρ where k is a proportionality factor of the resolution of the images, to be determined on the base of the accepted performance parameters of the corresponding device.
In fig. 11, we study how the performance parameters of the miniaturized InP coupler are affected by fabrication errors ranging from −200nm to 200nm for three wavelength of interest: 1530nm, 1550mn and 1570nm, covering the whole C band.
It can be noted that insertion loss and phase error are the most critical parameters, and that a fabrication error of +50nm actually provides a slightly better device, this might be due to the role of equation (7) for the miniaturized device as discussed in III-D and VIII.
As expected a shorter wavelength provides better performance for negative fabrication error, and the converse for longer wavelength, and this should be taken into account when designing the device with aim at a certain operation band. A fabrication error range of ±50nm with respect to the real optimum device of 50nm, guarantees a good performance for the whole C band. If instead we consider a single wavelength of operation (eg. 1550nm) the device shows a good performance for a fabrication error up to ±125nm. In the case of InP ρ = 0.93µm and E L = 2.82δ f 2 +56.61δ f from which we extrapolate k C−band = 3.05 and k 1550nm = 7.66.
From this we conclude that if a fabrication error δ f is such that E L < 3.05ρ the miniaturized device maintains an acceptable performance in the C band.

X. CONCLUSION
In conclusion, we have demonstrated that a standard 90-degree hybrid coupler in InP technology can be scaled from a footprint of 12000µm 2 to one of only 2200µm 2 maintaining a sufficient performance for applications of balanced detection. We compared the principal performance parameters for the miniaturized device and the reference one in the S, C and L bands, showing that they have similar performance over a range of wavelengths of 80nm. We then related the performance of the miniaturized devices to the number of modes propagating in the MMI and use this unit as a scaling factor to transfer the technology to other platforms, to obtain miniaturized devices with similar performance. Finally we studied the fabrication tolerance of the miniaturized devices. These results are very important for the application of 90-degree hybrid couplers in densely packed photonic circuits, where footprint and dimension are a major constraint.