## Abstract

A coherent receiver based on a 120° downconverter architecture, inherited from previous approaches at the microwave and optical fields, is proposed, analyzed, numerically evaluated and compared to the conventional 90° downconverter alternative. It is shown that, due to its superior calibration procedure, the new downconverter architecture allows full compensation of the imbalances in its optical front-end thus leading to an extended dynamic range and a broader operating bandwidth than its 90° counterpart. Simulation results from monolithically integrated downconverters show that our approach can be an interesting alternative to support efficient modulation schemes such as M-QAM that is being studied as potential candidate for the next generation of optical communication systems.

©2012 Optical Society of America

## 1. Introduction

Optical coherent systems have significant advantages as they enable high spectral efficiency via complex modulation formats as M-ary phase shift keying (M-PSK) or quadrature amplitude modulation (M-QAM), high capacity from wavelength division multiplexing (WDM) and electronic compensation of transmission impairments. Polarization multiplexed QPSK has been recently proposed as the transmission modulation for 100Gbps Dense-WDM networks by the Optical Internetworking Forum [1], and higher order QAM formats are being foreseen as a viable alternative to increase a step further system transmission capacity and efficiency. For example, in the frame of the MIRTHE project, 16-QAM monolithically integrated transmitter and receivers for 400Gbps are being pursued [2] and several groups are investigating the possibilities of high density 64-512 QAM for optical communications in different scenarios [3]. These high order modulation formats benefit from symbol rate minimization thus increasing tolerance to chromatic and polarization dispersion, allowing lower speed electronic devices to be used, and reducing power consumption. Besides that, recent advances in broadband optical amplification technologies would allow to take full advantage of the S- C- and Extended L-band to cover a 1490-1620 nm optical bandwidth [4, 5]. Hence, it is clear that there exist a potential interest in developing high performance receivers capable of demodulating complex modulation signals in wide wavelength spans with the highest quality.

The vast majority of the optical coherent receivers in the bibliography make use of the well-known 90° hybrid based downconverter, a phase diversity coherent receiver which was initially proposed in the eighties [6] and which has recently received much attention as it is an important piece of hardware in the optical front end [7]. These hybrids can be fabricated in bulk optics [8] or in integrated optics technology. The later allows photoreceivers to be integrated in the same chip [9,10], making it a cost-efficient solution appropriate for commercial applications. Obviously 90° hybrid optical downconverters cannot be designed to have ideal performance in a wide wavelength span, so real hardware suffers from different kinds of unbalances (mainly measured in practice through common mode rejection ratio, CMRR) which usually worsens at the ends of the operative band. Digital signal processing algorithms following photodetection are designed to mitigate this non-ideal performance. This is for example the case of Gram-Schmidt orthogonalization procedure (GSOP) which is used for 90° hybrid quadrature imbalance compensation [11]. However, not all the effects due to 90° hybrid unbalances can be compensated by this kind of approach. Effectively, as it has been shown previously [12], hybrid imbalances introduce two types of distortion on the received signal constellation: linear and nonlinear; and as will be shown in the following Section 4.3 GSOP algorithms only compensates for the linear type of distortion.

Fortunately, other types of phase diversity downconverters exist which are not based on the 90° hybrid. This is the case of the 120° phase diversity receiver, a type of the well-known multiport radiofrequency receiver [13], which has been recently used as a wide bandwidth, hardware-impairment tolerant receiver at microwave frequencies [14]. This kind of 120° downconverters, although not very frequent, have also been reported for optical communications by making use of 3x3 fiber couplers: 680 Mbps Amplitude-Shift Keying demodulation was already demonstrated in the eighties [6], and 1.4 Gbps 4-QAM has been recently reported based on this approach [15].

In this work we propose a 120° monolithically integrated downconverter based on a 2x3 multimode interference coupler (MMI). The performance of this approach will be analyzed and compared with the traditional 90° hybrid receiver architecture for different spectrally-efficient modulation schemes (e.g. 64-QAM and 256-QAM). It will be demonstrated that the proposed architecture offers the same noise performance that the classical 90° hybrid counterpart. Furthermore, results will show that the proposed 120° coherent receiver exhibits lower distortion of received signal constellation, an improved dynamic range and a remarkable broadband wavelength response (providing complete S-C-L band coverage). This superior performance (compared with the classic 90° approach) is mainly due to the calibration strategy: a simple and linear process which almost perfectly compensates receptor impairments, so high tolerance to fabrication errors can be also expected. In authors' opinion, the 120° downconverter proposed here is an attractive alternative to the 90° hybrid approach for the next generation of coherent optical receivers which will be required in the short time.

This paper is organized as follows; standard 90° and proposed 120° downconverter architectures are shown in Section 2 and 3 respectively. There, analytical bit error rate (BER) estimation is carried out for an ideal hardware including noise sources. In Section 4, via numerical simulation, the performance degradation due to hardware non-idealities is carried out for each scheme, and the superior benefits of our proposal due to its superior calibration procedure are demonstrated. Finally Section 5 provides the main conclusions of this paper.

## 2. 90° Hybrid downconverter

#### 2.1 General setup

Figure 1 shows the schematic of the standard 90° downconverter, with a 90° hybrid and photodiodes in balanced configuration, for receiving in-phase and quadrature signal components [10]. After obtaining electrical signal components, they are digitized in two analog-to-digital converter (ADC) and combined to be processed in the digital signal processor (DSP). Assuming polarization control or polarization diversity, the electric fields of the received signal (neglecting transmission impairments) and the LO laser can be written in terms of their complex envelopes as

For homodyne detection the angular frequency ω_{o} of signal and LO is assumed to be the same, so that any frequency or phase mismatch between them would be corrected digitally afterwards in the ideal DSP block.

Considering ${P}_{s}={\left|{\tilde{e}}_{s}\right|}^{2}$and ${P}_{LO}={\left|{\tilde{e}}_{LO}\right|}^{2}$ as signal and LO power respectively, slowly varying complex signal envelope can be expressed as

where*n*represents the complex amplified stimulated emission (ASE) noise introduced by the optical amplification along transmission, being modeled as an additive white Gaussian noise (AWGN) with single-sided power spectral density

_{ASE}*N*. On the other hand,

_{ASE}*I*and

*Q*are the normalized in-phase and quadrature components of the incoming modulated signal.

The signal and LO are combined in the optical 2x4 90° hybrid, with scattering parameters *S _{ik}* defined between its six ports, and detected at the four output ports through photodiodes with responsivities

*R*. Then, neglecting the noise contribution, the output photocurrents can be calculated as

_{i}Defining the received symbol${\Gamma}^{s}={\tilde{e}}_{s}/{\tilde{e}}_{LO}$, in the complex plane, and the 90° hybrid centers as the ratio ${q}_{i}=-{S}_{i2}/S{}_{i1}$, Eq. (4) can be rewritten as

Output photocurrents *i _{I}* and

*i*, for in-phase an quadrature components, can be now obtained differentially from balanced detection, as shown in Fig. 1, and be described in matrix form as

_{Q}Table 1
shows the parameters introduced in Eq. (6). Three terms can be identified at right-hand side of Eq. (6): DC offset term (*α*), non-linear rectified wave distortion term (*γ*) and linear axes transformation terms (*u,v*). As it was demonstrated in [12], and it is repeated here for the sake of completeness, in the receiver performance the effect of DC offset and linear terms have a simple interpretation consisting of a translation *α* of the origin of coordinates, followed by a rotation and imbalance of reference axes as shown in Fig. 2(a)
. On the other hand, the non-linear rectified wave distortion term introduces a non-linear error proportional to the *P _{s}/P_{LO}* ratio that can severely distort the receiver constellation, as shown in Fig. 2(b), and whose effect will be considerable for spectrally-efficient high-order M-QAM modulation.

#### 2.2 Analytical BER estimation including noise sources for ideal hardware implementation

For a perfectly balanced receiver (photodiodes of same responsivity *R = R _{i}* and ideal 90° hybrid with scattering parameters {

*S*}), and in absence of noise, it can be easily obtained that

_{31}= S_{41}= S_{51}= S_{61}= S_{32}= ½, S_{42}= -½, S_{52}= j/2, S_{62}= -j/2*α = 0*,

*γ = 0*and $u=-jv=R\sqrt{{P}_{LO}{P}_{s}}$, so Eq. (6) becomes

Therefore, the recovered complex signal can be defined as *i _{s} = i_{I} + ji_{Q}*, whose power can be calculated as

*<|i*(where < > denotes the ensemble average operator).

_{s}|^{2}>We now consider all relevant noise sources at reception: ASE noise, shot noise at the photodetectors, relative intensity noise (RIN) from signal and LO, and electronic trans impedance amplifier (TIA) noise. As shown in the Appendix, it is observed that for a perfectly balanced receiver ASE-signal beat noise and RIN from LO and signal are cancelled out. Besides TIA noise can usually be neglected, and in phase $\u3008{i}_{noise-I}^{2}\u3009$ and quadrature $\u3008{i}_{noise-Q}^{2}\u3009$ noise components can be calculated as in Eq. (A9), which is repeated here for convenience.

*B*is the effective noise bandwidth and

_{e}*q*is the electron’s charge. Then the inverse of the signal-to-noise ratio per symbol (SNR) can be written in the following closed form

If we assume M-QAM modulation with homodyne reception under AWGN noise sources, the bit error rate (BER) is given as a function of SNR (where *erfc* means the complementary error function) from [16].

Attending to Eq. (10), in the common situation of high LO power, there will be the expected limited shot-noise transmission under low signal power. This is easily appreciable in Fig. 3
, where BER performance versus signal power is depicted for different LO power, under 56 Gbps 64-QAM and 256-QAM transmission, when ASE noise is adjusted for a BER = 10^{−4} in absence of internal receiver noise (OSNR equals to 24.3 dB and 30.2 dB respectively).

## 3. 120° Coupler downconverter

#### 3.1 Setup

Figure 4
shows the scheme of the 120° downconverter, with a 2x3 120° optical coupler and three photodiodes. Photocurrent outputs are digitized in three ADC and processed in the DSP following a simple linear transformation to obtain in-phase and quadrature components. From a similar analysis to that performed in Section 2.1, the signal and LO waves, described by Eq. (1) and (2), will be combined now in the optical five-port, with scattering parameters *S _{ik}* defined between their ports, being the photocurrents generated at the output ports

Defining the five-port centers ${q}_{i}=-{S}_{i2}/S{}_{i1}$, Eq. (11) can be rewritten as

In the complex plane, Eq. (13) corresponds to three circles centered in *q _{i}* (ideally 120° apart) with radius depending on the received symbol. As it was shown in [17], after the work in [18], three power readings are enough to establish a linear set of equations to uniquely obtain I-Q signal components under arbitrary hardware unbalances of phase diversity circuit. Consequently, note that the only two power measurements obtained from the standard 90° downconverter (see Fig. 1 and Eq. (6) in Section 2.1), cannot cancel nonlinear constellation distortion (induced by the hardware unbalances of the phase diversity circuit) by linear means.

Equation (12) can be expressed as Eq. (5), or equivalently in matrix form as

Table 2
shows the parameters introduced in Eq. (14). Three parameters describe the photocurrents obtained at each of the three output ports: DC offset real parameter (*α*), signal power-dependent real parameter (*γ*) and LO-signal power-dependent complex parameter (*u*).

Attending to the three linear equations formulated in Eq. (14), in-phase and quadrature demodulated components can be easily described in terms of the three photocurrents measurements *i _{i}*.

*A*,

_{Ii}*A*,

_{Qi}*C*and

_{I}*C*can be determined from a simple calibration procedure, solving the linear equation system obtained from Eq. (15) when transmitting four known symbols [17].

_{Q}#### 3.2 Analytical BER estimation including noise sources for ideal hardware implementation

For a perfectly balanced 120° downconverter (photodiodes of a same responsivity *R = R _{i}* and an ideal 120° coupler with scattering parameter {${S}_{31}={S}_{41}={S}_{51}={S}_{42}=1/\sqrt{3}$;${S}_{32}={S}_{52}^{*}={e}^{j120\xba}/\sqrt{3}$}), and in the absence of noise, from Eq. (14) we obtain

Therefore, for this ideal situation in-phase and quadrature components, *i _{I}* and

*i*, can be easily obtained from the output photocurrents as

_{Q}Noise analysis can be carried out in a similar way as for the ideal 90° hybrid. This is detailed in the Appendix, where it is concluded from Eq. (A13) that in-phase and quadrature noise components for the ideal 120° coupler are the same as those obtained for the 90° hybrid from Eq. (9).

Thus it is easily shown that an ideal 120° coherent downconverter responds to the same SNR expression Eq. (10) obtained for an ideal 90° hybrid coherent downconverter. Therefore, this novel proposal does not suffer from additional penalty under ideal hardware realization (same dynamic range restrictions apply) and thus Fig. 3, originally obtained for the ideal 90° downconverter, is also valid for the ideal 120° one.

From this section we conclude that 90° hybrid and 120° coupler downconverter performances are the same in an ideal situation however, as it will be shown in the following sections, in presence of hardware impairments and under high *P _{s}/P_{LO}* ratios, the 120° architecture clearly outperforms the classical 90° approach due to its superior correction procedure.

## 4. Performance comparison of monolithically integrated 90° hybrid and 120° coupler based downconverters

Monolithically integrated downconverters for coherent optical communications are being targeted in several technologies as InP and SoI [19] as a mean to reduce complexity and cost of recently demanded 100 Gbps per wavelength optical communication systems. Almost all of the proposals are based on 90° downconverters achieved through a proper combination of 2x2 or 2x4 Multimode Interference Couplers (MMIs). In this section we will study, through simulation, the performance of two monolithically integrated receivers in InP: a classical 90°, and a novel 120° MMI based downconverters. The performance degradation, in terms of expected BER, of 90° and 120° downconverters due to hardware unbalances (2x4 and 2x3 MMIs phase diversity circuits and photodiode responsivity) will be compared to the ideal performance for perfectly balanced hardware described by Eqs. (8)-(11) (see Fig. 3).

#### 4.1. Monolithically integrated downconverters

The monolithically integrated downconverters, including phase diversity passive structures and p-i-n photodiodes, have been designed in InP technology similar to the one employed in [9, 10]. The passive circuitries (i.e. interconnection waveguides, tapers, crossings and MMI structures) are based on InP Rib waveguides with core thickness and etch depth around 1 μm and 0.4 μm respectively.

The 90° downconverter is based on a well-known architecture (see Fig. 1) already reported in [9]: a 2x4 MMI is monolithically integrated with four waveguide p-i-n photodiodes. Subtraction of these photocurrents in linear differential TIAs yield the two analog baseband signals *i _{I}(t)* and

*i*which are then digitized from two ADC and further corrected by well-known algorithms to get the received symbols. On the other hand, the 120° downconverter is based on a 2x3 MMI coupler monolithically integrated with three waveguide p-i-n photodiodes followed by TIAs (see Fig. 4). These three signals are then digitized from three ADC and finally baseband symbols are obtained from them after a calibration procedure.

_{Q}(t)MMIs have been carefully designed to minimize amplitude and phase imbalances for the in-plane (TE) polarization wave (notice we are assuming polarization control or a polarization diversity scenario) at its central wavelength. Although the explanation of the design process is out of the scope of this paper, it must be pointed out that it basically consists in the calculation of the geometry of the multimode region (i.e. MMI width and length) and the optimal positions and widths of the access waveguides. A detailed description of MMIs design process can be found in [20,21].

Figure 5 shows the simulated wavelength response (amplitude and phase imbalance) of the designed 2x4 MMI (90° hybrid). We have used FIMMWAVE, a full-vectorial electromagnetic simulator package, to obtain these results. A very low imbalance is observed within a small bandwidth around the central wavelength, thus confirming that the MMI has been properly designed since it completely covers the C-band (1530 to 1565 nm). Nevertheless, the device performance rapidly deteriorates for wavelengths far away from this central region. Imbalance in the lower limit of the S-band (1460 nm) and in the upper limit of the L-band (1625 nm) is too high, so the device cannot be used to cover these bands without an adequate calibration. The same conclusions can be obtained from Fig. 6 , where the simulated wavelength behavior of the 2x3 MMI (120° coupler) has been shown. It must be noticed that, although the 2x3 MMI exhibits a better bandwidth performance than the 2x4 MMI, this fact can be predicted using basic MMI theory [22] and therefore it is not due to a better design.

#### 4.2 Simulation scenario

In this section, main details of the setup adopted in the numerical simulation are shown. For the transmitter, we will assume here a conventional IQ-modulator for M-QAM, consisting on an electrical driver and an optical modulator [23]. Two orthogonal carriers from a LO are amplitude modulated by two Mach-Zehnder Modulators (MZM), which are biased and driven by electrical data signals, and combined to form optical signal. The pulse-shape modulator corresponds to the impulse response of a 5th order Bessel low-pass filter with electrical bandwidth 0.7*·R _{s}*.

The optical channel will be modeled as an AWGN channel degraded by ASE noise. The ASE noise level at the receiver input will be fixed from Eq. (11) to assure the minimum SNR needed for a BER = 10^{−4} (typical threshold when forward error correction is employed) in a noiseless ideal coherent receiver.

In the receiver side, it will be assumed there is not any other channel interfering on the incoming optical signal. We will consider ideal behavior of TIAs, ADCs and DSP blocks, and so we will focus on receiver non-idealities coming only from unbalanced 2x4 and 2x3 MMI couplers (as described in previous subsection). Photodiodes are modeled with in ideal square-law detector, a typical mean responsivity of 0.89 A/W and a CMRR of −22 dB (the parameter *CMRR = |R _{1}-R_{2}|/(R_{1} + R_{2})* describing the small responsivity imbalance between any pair of photodiodes

*R*and

_{1}*R*). Three different noise sources have been included in the receiver with reasonable levels used in practice [9]: RIN noise from signal and LO (

_{2}*RIN = −150 dB/Hz*), electronic TIA noise (${\alpha}_{TIA}=20pA/\sqrt{Hz}$), and photodiode shot noise (theoretical expression Eq. (A1). Also, in the numerical simulations, bit rate will be fixed to 56 Gbps (enabling 100Gbps Ethernet under Polarization multiplexing), and an ideal matched filter (effective noise bandwidth

*B*equal to the symbol rate

_{e}*R*) is considered in the DSP block, to achieve optimum noise limited performance for a perfectly balanced situation. Degraded BER from hardware non-idealities and electrical noise sources will be estimated for each setup from Monte-Carlo numerical simulations, transmitting 10

_{s}^{6}symbols of the chosen M-QAM.

#### 4.3. Performance of the 90° coherent receiver in the C and L bands

It is a well-known fact that hardware unbalances, existing in any realistic 90° downconverter, will cause a non-negligible distortion in received constellation (see Eq. (6) and Fig. 2), so these detrimental effects must be compensated for by a proper DSP correction algorithm. The standard procedure to correct phase and amplitude imbalances in a 90° hybrid coherent receiver is the GSOP method [11] which is based in a Gram-Schmidt orthogonalization. This algorithm has been implemented as an ideal DSP block at the receiver in Fig. 1.

Figure 7
shows 64-QAM and 256-QAM BER performance for the simulated 90° receiver, after GSOP compensation, for different LO and received signal powers and for two different wavelengths, λ = 1.54 μm (top) and λ = 1.62 μm (bottom), corresponding to central and upper L-band limit wavelengths, respectively. Simulations at the central wavelength, where minimum hardware unbalances exist, show a shot noise limited performance at low signal power levels (similar to the one already reported in Fig. 3 for a perfectly balanced hardware) but also show that receiver performance is degraded at high signal power levels. This effect, theoretically explained in [12], is due to the non-linear rectified wave distortion term which introduces a non-linear error proportional to the *P _{s}/P_{LO}* ratio which, apparently, is not being removed by the GSOP algorithm. This effect worsens for denser constellations as 256-QAM, where it is observed that, for a typical LO power of 10 dBm, the signal dynamic range (measured at the 1 dB equivalent SNR penalty) is reduced to one decade. As also seen at the bottom of Fig. 7, this behavior worsens for the upper limit of the L-band due to the increase in the downconverter unbalances. In this case, it is observed that no reception is possible for 256-QAM without significant performance degradation.

To gain insight into the causes of receiver degradation at high received signal levels, the 64-QAM constellation, after GSOP compensation, is plotted in Fig. 8
for two different *P _{s}/P_{LO}* ratios: −20 dB and −5 dB. This situation corresponds to the wavelength at the limit of the L-band where hardware unbalances are more appreciable. In the Fig. 8(a), corresponding to a

*P*ratio of −20 dB, the rectified wave distortion term (

_{s}/P_{LO}*γ*) of Eq. (6) can be considered negligible. Thus constellation distortion due to hardware unbalances is almost linear (as was illustrated in Fig. 2a) and GSOP algorithm can remove all impairments. On the other hand, in Fig. 8(b), corresponding to a

*P*ratio of −5 dB, the rectified wave distortion term (

_{s}/P_{LO}*γ*) of Eq. (6) has an important effect which creates a nonlinear distortion of received constellation, as was shown in Fig. 2(b). Being a linear orthogonalization algorithm, the GSOP is unable to remove this effect, yielding the distorted results of Fig. 8(b) which are responsible of the BER increase for high power levels.

#### 4.4 Performance of the 120° coherent receiver in the C and L bands

Following the calibration procedure summarized in Section 3.1 by Eq. (15) for 120° coherent receiver, in-phase and quadrature signal components can be easily obtained from the three diode photocurrents by a simple linear transformation which compensates all hardware non-idealities due to the 120° coupler and photodiode responsivity unbalances. Attending to the computational complexity, signal components for the 120° downconverter are calculated from a pair of linear equations depending on the three output photocurrents, instead of depending on the two differential output photocurrents for the 90° downconverter according to GSOP procedure. Therefore only two additional multiplications and additions are required per symbol, so a superior performance under hardware impairments is obtained with a slightly higher computational complexity.

This can be seen in Fig. 9
, where the results of previous section are repeated for the 120° receiver. Comparison of Fig. 9 with the previously obtained one for the 90° downconverter is straightforward. It is clearly noticed that, contrary to the 90° case, and despite dealing with a non-ideal hardware clearly exhibiting amplitude and phase imbalances (see Fig. 6), after calibration the 120° receiver shows a BER performance that is virtually identical to the ideal performance of the perfectly balanced receiver previously shown in Fig. 3. Furthermore this behavior is observed not only for the central operation wavelength, where unbalances are somehow limited, but also for the wavelength at the limit of L-band where deviation of the 120° coupler from its ideal performance is much more evident. To make this fact clearer, in Fig. 10
the demodulated constellation after calibration is plotted for the 120° receiver at the limit of the L-band. Comparing these results with those in Fig. 8, it can be clearly seen that the 120° receiver calibration can completely remove all linear and nonlinear constellation distortion due to hardware unbalances, and this removal is almost independent of the *P _{s}/P_{LO}* ratio.

Finally, comparison of dynamic performance of both types of receiver is plotted in Fig. 11
, where transmitted and demodulated I-Q plane and eye diagrams corresponding to a 16-QAM constellation are plotted for different *P _{s}/P_{LO}* ratios at the limit of the L-band for a LO power of 10 dBm. These results show again the better performance of 120° receiver.

#### 4.5 Ultra broad-band operation

As the proposed 120° downconverter exhibits a superior performance at both the central operation wavelength and the upper limit of the L-band, it is reasonable to ask ourselves whether the 120° receiver can be made to operate over an even greater wavelength span where much higher imbalances are observed (see Fig. 6). In doing so, we will not only show the possibilities of providing complete S-C-L band coverage with a simple and affordable receiver in monolithic technology, but we will also provide some indirect evidence it can be achieved with relaxed hardware constraints. This can have a dramatic influence on receiver sensibilities to fabrication tolerances and thus can greatly improve fabrication yield of practical integrated receivers.

Figure 12
shows the estimated BER for the 90° and 120° receivers for the complete wavelength span of 1.4-1.7μm for different *P _{s}/P_{LO}* ratios and two different modulation scenarios: 64-QAM and 256-QAM. Again the performance of 90° receiver followed by GSOP algorithm is compared with 120° receiver using the simple calibration linear algorithm described in Eq. (15). It is clearly shown that 90° receiver performance is only reasonable good for the center wavelength (where 90° hybrid has been designed and thus minimum hardware unbalances occur) and under low

*P*ratios which, as already shown, minimize the nonlinear signal constellation distortion which GSOP algorithm is unable to compensate for. Thus great SNR penalty is introduced by the 90° receiver far away from the designed wavelength or if the low

_{s}/P_{LO}*P*ratios assumption does not hold. On the contrary, due to its superior calibration procedure, the 120° receiver is able to almost completely compensate the detrimental effects due to hardware unbalances occurring outside the designed wavelength range, and this compensation can be done for a wide variety of

_{s}/P_{LO}*P*ratios (the inset of Figs. 12.c) and d) show that wideband operation can be achieved even for a low signal and LO power of 0 dBm).

_{s}/P_{LO}## 5. Conclusion

A novel 120° monolithically integrated downconverter, based on a 2x3 MMI, has been presented and compared to conventional 90° downconverter, based on a 2x4 MMI, for spectrally-efficient modulation schemes (e.g. 64-QAM and 256-QAM). Analytical BER estimation for a perfectly balanced situation has been compared to numerical BER performance considering amplitude and phase imbalances existing in real integrated phase diversity circuits and photodiodes. The 120° receiver has exhibited a superior performance by means of a simple linear calibration strategy which corrects hardware imbalances. As a consequence, it is expected that fabrication tolerances, that typically degrade the 90° downconverter performance, can be easily overcome in the proposed 120° downconverter. This can be used beneficially for both, relaxing fabrication tolerances of 120° coupler and increasing this way its fabrication yield. Results have shown that the 120° downconverter offers lower distortion of received signal constellation (as non-linear constellation distortion is fully compensated), better dynamic range and wide-band operation. Therefore, in the actual scenario where broadband optical amplification allows to operate on S-C-L bands and spectrally-efficient high-order M-QAM are potential candidates for the next generation optical communication systems, 120° proposal reveals itself as an attractive alternative to the 90° hybrid approach.

## Appendix A: Noise-induced degradation

Here, noise-induced degradation is evaluated in the 90° y 120° downconverter. The following noise sources will be considered and modeled as AWGN.

- • Optical ASE noise, see Eq. (3), that will manifest after mixing with signal and LO at the output ports through ASE-LO and ASE-signal beat noise terms.
- • LO and signal electrical shot noise. If a single photodiode received the LO and signal, it would generate an electrical shot noise proportional to its mean photocurrent <i>. So, the noise power contribution in an electrical receiver bandwidth
*B*, considering_{e}*q*the electron’s charge and R the responsitivity of the photodiode, would correspond toUnder an ideal realization, the received LO and signal will be equally split between the

*k*= 4 output ports of the 90° hybrid and k = 3 output ports of the 120° coupler. In the following*i*will represent the statistically independent shot noise in each photodiode, proportional to its mean photocurrent_{shot i}*<i*_{i}>. - • Relative intensity noise (RIN) from LO and signal. The statistically dependent electrical noise power contribution at each of the
*k*photodiodes will be proportional to the square of their mean photocurrent*<i*through the RIN parameter (typically described in dB/Hz) as_{i}> - • Trans impedance amplifier (TIA) noise. The amplification of in-phase and quadrature photocurrents in the 90° hybrid coherent receiver (after balanced detection), or output photocurrents in the 120° coherent receiver, will generate a current noise contribution,
*i*/_{TIAI}*i*and_{TIAQ}*i*respectively, of power_{TIAi}where

*α*corresponds to the noise current density, described in A/√Hz._{TIA} - • Thermal noise or dark current can be neglected.

*A.1. Noise-induced degradation in 90° hybrid coherent receiver*

From Eq. (6), considering ideal and perfect balanced hardware realization, we get the resulting photocurrents *i _{I}* and

*i*,

_{Q}Note that common mode noise, as ASE-signal beat noise or RIN from LO and signal, will be cancelled under ideal hardware. However ASE-LO beat noise, *i _{ASEI-LO}* and

*i*, appreciable from the noise contribution in the first term of Eqs. (A6), will remain. The power noise contribution of this latter will be

_{ASEQ-LO}The total power noise contribution to in-phase and quadrature signal components can be now easily expressed from Eqs. (A6), neglecting TIA noise, as

Finally, substituting Eq. (A2) and Eq. (A7) into Eq. (A8) yields (under low TIA noise assumption),

*A.2. Noise-induced degradation in 120° coherent receiver*

From Eq. (14), the output photocurrents including optical and electrical noise sources can be expressed as

ASE-LO and ASE-signal beat noise contributions are now appreciable on the third and fourth term respectively of Eqs. (A10). In addition to that, on each of the three output photocurrents, there will be an identical RIN noise contribution attending to Eqs. (16) and (A3).

Under ideal hardware realization, *i _{I}* and

*i*can be obtained from the algebraic relation derived in Eq. (17).

_{Q}Noise sources contribution are included on the last term of Eqs. (A11) from

As it happened in the ideal 90° hybrid coherent receiver, common mode noise as RIN from LO and signal or ASE-signal beat noise will be once again cancelled under ideal performance. From Eqs. (A2) and (A12) the total power noise contribution to in-phase and quadrature components then becomes (under low TIA noise assumption),

## Acknowledgments

The authors gratefully acknowledge the design support from Robert Halir and Sebastián Romero-García, and the fruitful discussion with Oriol Bertran-Pardo. This work has been partially funded under Andalusian Regional Ministry of Science Innovation and Business project P09-TIC-5268, Spanish Ministry of Science and Innovation project TEC2009-10152 and EU 7th Framework Programme project MIRTHE ICT-2009-5 no. 257980.

## References and links

**1. **Optical Internetworking Forum (OIF), “100G ultra long haul DWDM framework document,” document OIF-FD-100G-DWDM-01.0 (June 2009), http://www.oiforum.com/public/impagreements.html.

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