A Field Trial of Multi-Homodyne Coherent Detection Over Multi-Core Fiber for Encryption and Steganography

The concept of multi-homodyne coherent detection is used for stealth and secured optical communications, demonstrating a transmission rate of 20 Gbps under negative (in dB) optical signal-to-noise ratio (OSNR), in a deployed multi-core fiber (MCF) network. A 10 GBd QPSK signal is assigned with a bandwidth of multi-hundreds GHz, optically encoded with spectral phase mask, and transmitted below ASE noise in the MCF channel. To enable such transmission scheme, a semiconductor based gain-switched laser (GSL) frequency comb, used as both the signal carrier and local oscillator (LO), are then mixed in intradyne coherent receiver. The ultra-short time-domain overlapping between the pulsed signal and the pulsed LO allows the authorized user to collect the information, while discriminating most of the noise. Consequently, the electrical SNR is dramatically improved compared to the optical SNR, resulting in optical processing gain. Nevertheless, ultra-short pulses mixing is rather susceptible to a chromatic dispersion (CD), which imposes a major challenge in this spread-spectrum transmission technique. In this contribution, the effect of CD in multi-homodyne coherent detection is analyzed in an analytic model and in experiments. The model suggests a modified CD transfer function, comprised of the optical sampling of a conventional CD transfer function. In addition, the SNR penalty associated with the CD effect is quantified, as well as its dependency in propagation length and overall GSL's bandwidth. To circumvent the devastating effect of the CD, multi-core fibers are proposed. Two different weakly-coupled cores of a homogeneous MCF allow the signal and the LO to co-propagate through a very similar dispersive channel. A field trial in a deployed 6.29 km MCF network has validated an encrypted and stealthy 20 Gbps coherent transmission, at −2 dB OSNR.


I. INTRODUCTION
T HE concept of multi-homodyne coherent detection (MHCD) [1], also referred to as coherent pulse detection (CPD), is a coherent detection of a modulated pulsed signal performed by mixing it with a synchronized pulsed local oscillator (LO). Coherent optical code-division multiple access (Co-OCDMA) [2], [3], [4], [5], [6], [7], [8], [9] employs MHCD to assign different users a lower bandwidth information over a widely modulated bandwidth. Alternatively, the bandwidth redundancy can be utilized to discriminate random noise from a pseudo random signal, which in turn, enable detection of events in noisy environments [10]. In addition, in some applications [11], by using pulsed laser at the transmitter side, the LO wavelength stability can be relaxed to the range of few nanometers, thus giving rise to an LO based on uncooled distributed-feedback (DFB) laser. In a previously published work [1], we suggested a physical layer security scheme based on MHCD that enables reliable detection under negative (in dB) OSNR conditions, while exploiting the spread spectrum transmission into optical processing gain. Though MHCD is a promising approach for next-generation optical communications, it encompasses considerable engineering challenges. The first is the synchronization between two remote mode-locked lasers (MLLs). The second is the enhanced susceptibility to physical impairments that have to be mitigated in the optical domain, such as the chromatic dispersion (CD).
In optical coherent communication receivers, CD is commonly compensated by digital signal processing (DSP). Since the local oscillator (LO) and the signal are both continuous waves (CWs), the CD effect is pronounced as an all-pass filter, thus, can be eliminated in post DSP with minimal penalty [12]. In MHCD, however, where both the signal carrier and the LO are trains of ultra-short pulses, CD impairs the detection performance considerably and permanently, i.e. it cannot be mitigated once the opto-electronic conversion has been carried out. The CD-related SNR penalty eliminates the optical processing gain after only several kilometers, as showed in [13]. In the latter, numerical simulations have supported this assumption, while some thorough theoretical background was yet to be provided. In this paper, CD is introduced in the analytic model of MHCD and extensively examined. A closed-form expression for the SNR as a function of the CD parameters is derived and validated by comparison with simulation results. Furthermore, it is shown here, that transmitting the LO and the signal through similar dispersive media, yields a drastic reduction of the impact of CD.
Independently of CD, deviation in the GSL's free-spectral range (FSR), which is directly related to optical sampling error, can cause a major performance degradation. However, the desired generation of a synchronized LO GSL in the receiver-side is a challenging task which requires a complex phase-locked loop (PLL) electro-optic circuits. Thus, another merit of cotransmitting the signal carrier and the LO, is obviating the PLL and the synchronization of the LO. The two aforementioned challenges associated with MHCD: copping with chromatic dispersion and LO synchronization, can both be resolved by co-transmitting the LO with the signal. Multi-core fibers (MCFs) allow appropriate solution, as they dense more fiber cores, thus allow LO transmission without scarifying additional fibers. Moreover, the very similar dispersive properties of the different cores ensure optimal performance for MHCD, as the transmitted LO and carrier accumulates similar phase over same spectral bandwidth. Such configuration does not affect the confidentiality of the system, as the encryption key is the optical code (OC) used for spectral phase encoding, while any other parameters, such as the spectral properties of the laser, are anyhow assumed to be known for the Eavesdropper.
In this work it is analytically shown, and verified by numerical simulations, that the case of co-propagating LO is highly tolerant to chromatic dispersion. Subsequently, this concept was further tested in a field trial, using a deployed uncoupled-core MCFs. Two separate optical signals were transmitted in parallel via two different cores of the MCF: In one core, the QPSK signal was transmitted at varying OSNR values, while in the second core an unmodulated branch of the same MSL was transmitted, and, in turn, being used as a remote reference oscillator, replacing the functionality of a conventional LO. Results show successful transmission of single polarization QPSK at 20 Gpbs. This scheme may be extended to higher order modulation formats, as the optical processing gain depends exclusively on the number of modes. Yet, higher order of modulation has higher OSNR requirement. Furthermore, the proposed system does not limit the transmission distance, as long as chromatic dispersion is compensated optically, and as long as the PMD related differential group delay is limited to a fraction of a symbol unit interval. An error-free transmission after pre-FEC BER of 1E-3 is achieved under negative −2 dB OSNR conditions. A small discrepancy between optical back-to-back and field transmission is observed, due to chromatic dispersion imbalance between the different cores. The field trial was conducted in the world's first testbed for space-division multiplexed transmission, using multi-core fibers deployed in the city of L'Aquila, in Italy [14].

II. MULTI-HOMODYNE COHERENT DETECTION
The MHCD is a unique coherent detection scheme, incorporating pulsed signal-carrier and a pulsed LO, both generated by a gain-switched laser source. A conceptual block diagram of a MHCD system in dispersive media, aided with time and frequency-domain illustrations, is provided in Fig.  1. It is important to note that MHCD based security system is additionally comprised of spectral phase encoding and noise loading [1]. Here, however, to analyze chromatic dispersion effect, these two steps can be ignored, and therefore are not discussed. The simulation's results presented in this figure use a set of parameters, that were chosen according to the hardware of the experiment to be described later in this paper.
In the system shown in Fig. 1, GSL with 10 GHz FSR and 11 equal power tones is simulated, modulating a 10 GBaud QPSK signal. Inset (a) shows the LO and the signal at the output of the transmitter, in the optical spectrum domain. Once modulated, the signal's power spectral density (PSD) becomes wide flat-top shaped, with a total bandwidth of approximately 110 GHz. The total bandwidth is calculated according to −3 dB level. Additional bandwidth broadening should be attributed to non-optimal Nyquist pulse shaping and the roll-off factor of the SRRC. The flat spectral shape is enabled by digital pulse shaping and by matching the GSL's FSR to the baud. The transmitter filter was selected to maintain flat spectral response in the optical domain, thus accommodate the signal stealthiness.
Both the signal and the LO then propagate in a standard single-mode fiber (SSMF). The fiber is modeled here as a purely dispersive channel with dispersion coefficient D and dispersion slope S. Additionally, Fig. 1(b) plots the pulse train amplitude in the time domain. Even being modulated, the QPSK signal's envelope remains a train of equal-height pulses, as its amplitude is constant for all of the alphabet symbols and minor ISI is introduced with digital pulse shaping using square-root raised-cosine (SRRC) filter. For the theoretical analysis, the spectral aliasing that results from non-optimal Nyquist shaping, i.e. the use of non-zero SRRC roll-off factor, is neglected. In practice, the trade-off between the spectral aliasing and Tx limitation needs to be considered and optimized.
As the link distance increases, the pulse spreading becomes more noticeable. For example, the transmitted pulses (red solid . The analytic signal at various points is highlighted, with notations that follows the formalism provided in [1]. Gain-switched laser (GSL) with eleven pre-equalized tones is used as both the signal carrier (blue dashed line) and LO (red solid line), as shown in Inset (a). The GSL's modulated output is amplified using EDFA, which generates 15 dB OSNR, and transmitted through standard single mode fiber (SSMF) with a linear chromatic dispersion. The time-domain pulses, before mixing in the coherent receiver, are plotted in Inset (b). For convenience, noise is removed in Inset (b) so that the pulses broadening due to 1 km (blue dashed) and 6 km (black dash-dotted) of SSMF, can be easily noticed, compared to the original pulse train (red solid line). The detected signals' power spectral densities (PSDs) are plotted in Inset (c), after MHCD and digital filtering, for propagation length of 15 km. Simulation result, plotted with dashed blue line, agrees well with the analytical model prediction (solid red). These two differ significantly from the transmitted PSD (dash-dotted black line) due to CD effect in MHCD system. line) attain a rather flattened envelope (i.e., pulse spreading) after only 6 km of fiber (black dash-dotted line), due to CD and substantial ISI. Each of the 11 tones intercepts the CD transfer function in its corresponding wavelength. Thus, different phase accumulation per mode exempts the phase-locking condition. This behaviour well describes both the LO and the signal, therefore, while being mixed in the coherent receiver, the temporal overlapping is maximal, allowing optimal SNR conditions. Inset (c) presents the spectrum of the sampled electrical signal, after multi-homodyne coherent detection and post digital filtering, which is matched to the transmitter filter. A low-pass filter is introduced by MHCD interacting with the CD passband channel, as shown by both simulation (blue dashed line) and by solving the analytical model (red solid line), which is developed later in this paper. A good agreement between the model and the simulations is demonstrated, up to some discrepancy at low amplitude level, due to the numerical noise floor of the simulation. The resultant filter deviates from the flat-top shape of the original square root raised cosine (SRRC) Tx filter, with a smooth descent and sharp spectral notches at roughly ±4.7 GHZ. This behaviour demonstrates the main affect of the CD in MHCD system and stresses the difference from the case of traditional coherent system, where the latter uses CW laser sources, and CD is pronounced as an all-pass filter, and thus can be fixed by receiver DSP.

III. COMPLEX BASEBAND EQUIVALENT CHANNEL FOR MHCD
Following a qualitative description, in the following section a rigorous derivation of the complex baseband equivalent channel is provided, supporting the general MHCD theory. In order to evaluate the MHCD detection performance, e.g. SNR, it is useful to address the signal term and the noise term separately [1].
Since the signal and the noise are uncorrelated, the SNR can be computed as follows where E s and E n are the power of the signal and the noise, respectively [15]. The derivation of E s relies on the theorem of a stochastic process (the data) undergoing a linear filter (the complex baseband equivalent channel). The complex baseband equivalent channel is derived from the proposed model, which describes the interaction of the MHCD system with a general passband optical channel. In particular, we study the case where the passband optical channel is governed by the CD transfer function. To highlight the CD effect, all the electrical and optical components are assumed to be ideal and the system to be noiseless. The derivation of the complex baseband equivalent channel addresses the diagram depicted in Fig. 1. The multi-tone signal carrier considered here, is a GSL with FSR δ f ≥ W , central optical frequency f c , a total power of P GSL and weighting coefficients A m , which define the amplitude relation between its M tones. To prevent aliasing, it is provided that δ f ≥ W , where W is the modulated data finite bandwidth. Therefore, assuming an odd number of tones, the GSL E-field can be expressed as where P GSL is the associated average power of the GSL's E-field, thus it is necessary to demand a normalized energy, i.e. Our goal, similar to other contributions [16], is to find the complex baseband equivalent channel that concludes the passband system. The analysis of the complex baseband equivalent channel is derived in the frequency domain, using the convolution theorem. The Fourier transform of (2) is expressed as a sum of weighted and frequency shifted Dirac's delta functions δ(·). A continuous-time input data signal x(t) with a Fourier transform X(f ) modulates the GSL, resulting in the transmitted data-carrying E-field i.e., M replicas of the Tx signal are shifted over the laser's bandwidth. Furthermore, E GSL (t) is split, so that the associated power of the signal carrier, and the LO E-fields are P c and P LO , respectively.
The signal E-field, E tx (t), and the LO E-field, E LO (t), are then transmitted through possibly different passband complex channels, which are denoted as h s (t) and h LO (t), with Fourier transforms H s (f ) and H LO (f ), respectively. Next, the optical intradyne coherent receiver detects the beating product of where * represents the convolution operator. Starting form the information signal's path (the lower branch), once it passes through a linear channel, the received signal E-field takes the following form On the LO path (the upper branch), the laser experiences the channel H LO (f ). Since the LO model is a finite train of spectral impulses, as dictated by Eqs. 2 and 3, the LO channel is optically sampled, resulting in the frequency domain signal To keep the passband filters lossless, it is necessary to define another pair of received powers P rx and P LO, rx . These two constants reflect the fiber losses which are introduced by the signal and the LO channels, respectively. It is therefore implied in (6) that the LO samples its passband channel in a set of M discrete frequencies given by f c + nδ f . Both the LO and the signal fields are then coherently mixed and detected, to generate the RF signal [17] where the photodiodes' responsivities are assumed to be equal, and denoted as R. In addition, the non-informative DC terms are discarded following the balanced pairs of photodiodes. Based on (7), the frequency-domain representation of the received electrical signal is provided as Using Eqs. (4), (5) and (8), the received electrical signal is expressed in double summation form, as follows To illustrate the fundamental mechanism of the MHCD, R(f ) is plotted in Fig. 2. For this purpose, a simplified setup is considered, with M = 11, δf = 10 GHz, equal-power GSL modes and ideal channels H While the transmitted electrical bandwidth of x(t) is limited to W , the detected electrical signal, r(t), has a 2M -times wider bandwidth. Fig. 2 depicts the entire spectrum of the detected electrical signal R(f ), spans over 210 GHz, resulting from the convolution of two 110 Ghz combs. The coherent addition effect, enabled by the MHCD system, aggregates M replicas to the baseband, M − 1 replicas to the baseband adjacent channels, and M − m replicas to ±m channels' PSD. This effect is well demonstrated in Fig. 2, with a triangular envelope of the electrical spectrum (triangular in a linear scale), demonstrating a 21 dB = 20 log(11) difference between the baseband aggregated PSD to the marginal channels, obtained for 21 GSL modes.
The best SNR is achieved for the the baseband channel [1], therefore R(f ) is filtered with a LPF, P (f ), that is matched to the pulse shape, yieldingX(f ) = R(f )P (f ). Due to this filtering, only the baseband-folded replicas, those that satisfy m = n in (9), contribute to the baseband channel. The electrical signal after the filter is therefore expressed by a single summation This expression is in a form of a signal that is multiplied by a channel. Thus, the complex baseband equivalent channel is The last results allows one to compute a complex baseband equivalent channel for any given linear passband channel. In the following sections the dispersive and the non-dispersive passband channels will be analyzed using the result of (11).

IV. CHROMATIC DISPERSION TRANSFER FUNCTION
A simplified, yet fairly accurate, analytic derivation of the CD transfer function, at location L, is presented below: where higher orders (k > 3) of β are negligible for small signal bandwidth. This will be used later on, to obtain, what shall be defined as the "folded CD transfer function". We use the specification of a standard single-mode fiber to derive an approximation for the chromatic dispersion transfer function over the wavelength range of interest (C-band). To obtain the constants D, S, and V g which defines [β k ] k=3 k=0 in (12), we consider an explicit expression for the dispersion coefficients, found in Corning SMF-28 TM product specifications as [19] where λ 0 = 1310.3 nm and S 0 = 0.086 ps/(nm 2 · km) are the zero dispersion wavelength and the zero dispersion slope, respectively. Although λ 0 is within the o-band, the analytical D specs (λ) holds for a wide wavelength range 1200 < λ < 1600 nm. Therefore, (13) can be used to derive a linearization of D specs (λ) in the operating wavelength range over the c-band. The Taylor series expansion for D(λ) around λ c = 1550 nm, the center wavelength of the c-band, is provided by: Fig. 3. Chromatic dispersion coefficient versus wavelength. The linear approximation, D 1st order (λ), plotted in dashed line, is compared to the textbook data [18]. The inset plots the same data as the main axis, over a wider range of telecommunication windows (1270-1620 nm), while the c-band is denoted with a green rectangle.
Considering the group velocity, In Eqs. (14) and (15) we obtained the three coefficients needed for the linearized CD transfer function of (12). The results of this model are plotted in Fig. 3. In this figure, we compare D s (λ), D 1st order (λ), and we also plot data from textbook [18] D textbook (λ).

V. MHCD IN NONDISPERSIVE CHANNEL
To asses the effect of chromatic dispersion in MHCD system, a reference nondispersive channel is examined. As a figure of merit, the signal's power E s , and the spectral shape of the complex baseband equivalent filter, H BB (f ), will be compared between the nondispersive, and the dispersive channel. The latter will be developed in Section VI. For the nondispersive channel, i.e. back-to-back transmission, it is required to eliminate the CD effect, thus the channels' response h s (t) and h LO (t) are both considered as δ(t), inducing a unity frequency response over all frequency range, i.e.
The input signal is assumed to be a linearly modulated signal with i.i.d. symbols from the set {x n } N n=1 , with a zero mean, and variance of σ 2 x . p(t) is the fundamental pulse waveform, whose Fourier transform we denoted as P (f ) with a finite bandwidth W , and the signal spectral density is [15] The complex baseband equivalent filter, which was developed in (11) for a general passband channel, obtains the following reduced form In this case the equivalent baseband filter obviously reduces to unity.
It is necessary to relate between the received power and the passband channel, via the complex baseband equivalent filter. Therefore, following two linear filters [15], H BB (f ) and P (f ), the PSD of the detected signalx(t) iŝ Consequently, the average power of the signal is It is shown that the detected power in the case of a non-dispersive channel (19) entirely exploits the power of the received signal and the LO. Note that in this analysis we assumed arbitrary small roll-off factor of the SRRC filter, thus the effect of spectral broadening of the base-band signal beyond the FSR is neglected. For specific pulse shaping filters, the performance degradation due to aliasing or low-pass filtering should be considered. These detection performances are identical to the case of conventional coherent detection, where both the signal carrier and the LO are CW sources, holding the powers of P rx and P LO,rx , respectively. Moreover, this result does not depend on the spectral shape of the GSL, as long as the signal carrier and the LO hold the same one. In the dispersive channel, however, it will be shown that the maximal power is no longer obtained, as dispersion impairs the coherent addition mechanism.

VI. MHCD IN DISPERSIVE CHANNEL
While in Section V the chromatic dispersion was excluded from the model, here we apply the dispersive channel derived in Section IV, to investigate the power spectral density of the detected baseband signal in MHCD system. Additional implications of the inclusive model are the related SNR degradation and the temporal accuracy required for the synchronization circuits. Alongside with the dispersive channel model, two LO schemes are analyzed: (1) the LO is generated at the Rx side, and (2) the LO is generated at the Tx side, and co-propagates with the signal to the Rx. These two system configurations are summarized in Table I, as well as the trivial scheme, where dedicated hardware compensates the CD. Compared to the configuration where LO is generated at the Rx side, co-propagating LO provides a remedy to the devastating effect of CD in MHCD, as will be shown in the analysis below.
First, we discuss the case where both the signal and the LO experience the same dispersive channel (the second configuration in Table I). Consequently, the CD transfer function, introduced in (12), is applicable for both signal and LO branches, i.e. H s (f ) = H LO (f ) = H CD (f ). It is shown, that while the CD optical passband channel is an all-pass filter, the complex baseband equivalent filter in the MHCD scenario damages the amplitude as well as the phase of the baseband electrical signal. Thus, using Eqs. (11) and (12), the complex baseband equivalent Both the contributions of the co-propagating LO and signal are denoted with a corresponding label in the last expression. Considering the nondispersive channel, which allows a maximum utilization of the power carried by the M tones of the GSL, as shown in (17). With the dispersive channels, however, the phase matching is degraded, thus only partial coherent summation occurs which, in turn, degrades the signal power, and distorts its PSD. What can be further studied from (20), is an M -summation over product of two exponential functions, forming the baseband filter. In this product, one exponents originates from the LO contribution, and the other from the signal contribution. Essentially, the resulting baseband filter consists of the optical passband channel "folded" into the baseband M times, each time it is shifted by δf . Thus, the LO contribution is reflected as a constant phase (frequency independent). On the contrary, the dispersed-signal contribution can be considered as polynomial phase, with the order of the CD Taylor expansion (K). For an GSL with a total bandwidth below 5 THz and dispersion of a standard SMF, the third order of K and higher are rather negligible. To further convey this mechanism, Fig. 4 plots the LO and the signal phase contributions, focusing the quadratic term. The two curves corresponding the two exponential arguments of (20), when taking only k = 2. To remind that the linear term induces a linear time shift, and third order (and above) are negligible. The LO phase, is presented as zero-order hold, as it retains a constant phase per the entire replica of which is about to fold to the baseband. Due to the FSR matching condition, the crossing of the accumulated quadratic phase is exactly in the center of each replica. This condition guarantees that each exponential generated in the MHCD will encompass a phase that is approximately symmetrical around the origin.
A simplified version of (20) can be obtained for a constant dispersion (K = 2), and a flat optical spectrum, i.e. A m = A ∀ m. In this case, the complex baseband equivalent filter gets a simplified form of (21) This filter has an amplitude of A 2 M at f = 0 and lower amplitudes for every f = 0, and in any rate it is bounded by A 2 M , the maximal value of the all-pass filter in the nondispersive case. Moreover, (21) predicts the spectral notches, upon the nulling of its numerator. The frequencies of the notches are given by f q = q/(±2πβ 2 Lδ f M ), where q is an integer. This makes sense, as each of the following quantities increases the system susceptibility to chromatic dispersion effect: the propagation length (L), the GSL's FSR (δ f ) and the number of modulated tones (M ). For example, if using R b = 10 GBd, δ f = 10 GHz, M = 21, and L = 12 km, the first notch appears at 4 GHz, as can be seen in Fig. 5.
Similarly to the derivation of (16), the PSD of the received signal, in the presence of dispersive channel, is obtained and the signal power is provided by integrating over the signal's bandwidth Examining (23) reveals that it is bounded by the received signal power in non-dispersive channel (19). The term |H D BB (f )| 2 , which is provided in (20), follows the format | m |A m | 2 e jφ(f ) | 2 , where φ(f ) may represent any arbitrary function determined by the system's constants. This invites using the triangle inequality: where the last equality stems from the normalization requirement of (17). From the last expression, |H D BB (f )| 2 ≤ 1∀f and it is therefore shown that the total detected power in dispersive channel (23) is always lower than the power of the non-dispersive channel (19).
The quantities of (22) are plotted in Fig. 5 with both numerical simulations results, and solution of analytical model. The numerical simulations of data transmission were conducted for the cases where no CD compensation was applied (configurations 1 and 2 of Table I). In order to get a reasonable memory size of processed vectors, the 193.414 THz center frequency (equivalent to 1550 nm) of the GSL carriers, was shifted to the baseband. The 10 GBd QPSK symbols are sampled with high sample-persymbol ratio, to simulate passband "optical" transmission. Fig. 5 shows a very good agreement between analytical model and the simulation, as the blue-dashed, and the solid red curves are well correlated. Only the first spectral notch is pronounced for this parameters set, while the anticipated second and the third are highly attenuated due to the pulse-shaping filter P (f ) roll-off factor of 0.2.
The SNR obtained at the MHCD receiver, versus fiber length L is plotted in Fig. 6 for various number of GSL tones, and for both cases of Rx-side generated LO (MHCD configuration 1), and LO that is co-propagating with the signal through the same dispersive channel (MHCD configuration 2). This figure manifests the main advantage of the co-propagating LO configuration in terms of resilience to chromatic dispersion. A considerable SNR degradation is observed as the number of tones (M ) or the propagation length (L) increases. However, in the case of co-propagating LO, this degradation is certainly tolerable for most of MHCD-based use-cases. For example, considering an Fig. 6. The CD effect on the detected SNR vs. the fiber length and for various number of GSL tones, is plotted. The system initial SNR (optical back-to-back), when maximal optical processing gain is achieved, is 18.2 dB, an is denoted with an horizontal dashed line. Vertical dashed line represent the SNR at 6.29 km of fiber length, emphasizing the predicated results in the experimental part of this paper. Fig. 7. The SNR susceptibility for the signal-LO temporal misalignment in terms of symbol unit interval (UI), for M = 11 tones GSL. Both LO that is generated at the Rx side (a) and co-propagating LO (b) are simulated. System initial SNR, at optimal signal-LO temporal alignment, is 18 dB. 11 tones GSL (the red curves), a 1 dB of SNR degradation is observed after less than 2 km when LO is generated locally, at the Rx side. On the other hand, in the case of co-propagating LO, 1 dB of SNR penalty are only introduced after more than 10 km. Moreover, up to 5 km almost no penalty occurs in the co-propagating LO scenario (few tenths of dB). From link design perspective, if considering data center interconnect (DCI) link of 80 km length, the LO and the signal can be both compensated with anywhere between 70 km to 90 km of DCF/DCM. In Figs. 5 and 6, it is assumed that the sig.-LO mixing occurs with a perfect temporal alignment. However, it is important to examine the SNR sensitivity to such temporal sig.-LO misalignment, which inevitably exists due to temperature and mechanical variations. Fig. 7 shows a plot of the SNR vs. sig.-LO misalignment, measured in fractions of unit interval (UI), for 11 tones GSL, for two different uncompensated propagation lengths, and for the two system configurations discussed here.
Since CD effects the MHDC performances, the absolute SNR value is degraded with the increase of uncompensated fiber length, however resulting in an improved resilience to sig.-LO misalignment. In both panels of Fig. 7, the blue curves represent back-to-back measurements, while the red curves represent measurements of uncompensated fiber link that causes 1 dB penalty for an ideal sig.-LO misalignment (the y-axis crossing). In other words, the solid blue curves, which represents back-to-back SNR measurements, start from an higher SNR value (18.2 dB), compared to the dashed red curves which start from lower value (17.2 dB), as they represent some amount of uncompensated fiber link (1.5 km in config. 1 -left, and 12 km in config. 3 -right). Nevertheless, the dashed red curves exhibits moderate descending compared to the solid blue curves, thus indicating better tolerance to sig.-LO misalignment. Both configurations presented in Fig. 7(a) and (b) show similar behaviour. However, a careful quantitative comparison can be attained by measuring the temporal sig.-LO misalignment that introduces additional 3 dB of SNR penalty. Such measurements reveal a second advantage for the co-propagating LO configuration.
In these two panels of Fig. 7, the dashed red curves represent a condition in which dispersion causes 1 dB of SNR penalty with ideal temporal alignment. This condition is set with either 1.5 km of uncompensated length in the Rx-LO configuration or 12 km of uncompensated length in the co-propagating LO configuration. Additional SNR penalty is introduced as shifting rightward, until, for instance, the SNR curve drops with 3 dB compared to an ideal sig.-LO alignment. Therefore the points that are marked with red circle, stand for total 4 dB of SNR penalty, as they are a result of: r Rx LO configuration, ∼1.5 km uncompensated fiber and ∼ ±5.1% UI sig.-LO temporal misalignment (shown in Fig. 7(a)).
r Co-propagating LO configuration for over ∼12 km, and ∼ ±7.5% UI misalignment (shown in Fig. 7(b)). It is therefore concluded that co-propagating LO is more resilience to sig.-LO misalignment compared to the Rx-LO configuration.

VII. EXPERIMENT
The experimental setup is depicted in Fig. 8. At the transmitter-side, the data bits are mapped into QPSK symbols, upsampled and digitally shaped using an SRRC filter with roll-off factor of 0.2. The QPSK symbols stream feeds a Mach-Zehnder IQ modulator (I/Q-MZM) which modulates all the GSL tones at once, at 10 Gbaud. Note that the modulated signals' bandwidth in each replica exceeds 10 GHz due to non-zero roll-off factor and deviation from optimal Nyquist pulse shaping. The SRRC parameters were optimized based on receiver SNR criterion considering the effect of the spectral aliasing, and transmitter limitations. Transmission of 20 Gbps is demonstrated here, while polarization diversity transmitter and receiver can readily double the data rate. A liquid crystal on silicon (LCoS) based SLM is then used to determine the number of GSL tones (M ) and equalize the spectrum of the GSL. The spectral phase mask is used to encode the replicated signal by assigning each of its spectral replicas a one or more random phase terms, equivalent to a random time delay. The resulting time-domain signal, previously a train of ultra-short pulses, is smeared according to the pattern of the mask, to gain stealthiness in the time domain and to prevent the coherent addition at the unauthorized user's receiver. In the system described here, the spectral phase mask is comprised of 10 GHz spectral bins, corresponding to the FSR of the GSL and the Baud rate of the signal. The phase mask was further tuned to aligned the centers of the mask's bins with the center of each replica. The phase elements are a set of pseudorandom variables {φ β } M β=1 , where φ β is the spectral phase term associated with the β-th spectral bin of the mask, uniformly distributed within the range [0, 2π], i.e. φ β ∼ U (0, 2π). The phase elements are shared between the transmitter and the receiver, where the latter uses the key to decode the signal by applying the conjugate phase mask.
Following the phase mask encoding, a noise loading mechanism, comprised of a variable optical attenuator (VOA) and an erbium-doped fiber amplifier (EDFA) adjusts various, possibly negative, OSNR levels. Here we defined OSNR as the power ratio of the signal PSD over the noise PSD, per the same bandwidth. The stealth signal is transmitted to one of the cores of the MCF, while a second core is used for the LO transmission. The LO is split from the native un-modulated GSL source with 50/50 PM-coupler.
At the receiver, to maximize the temporal overlapping between the LO and signal, an optical delay line (ODL) is used. Two polarization controllers (PCs) are then applied to align both polarization states of the signal and the LO before they are mixed at the coherent receiver. The product of the multi-homodyne coherent detection results in an electrical baseband signal that is being digitized by the analog-to-digital converter (ADC). Following, a Rx DSP is applied: first, filtering the baseband term (to eliminate high-order interference terms), second, running a conventional coherent detection chain, similarly to [20]. This chain includes a front-end correction (spur cancellation), time synchronization and deskew of the transmitted sequence, IQ imbalance correction, carrier phase estimation, matched filter which downsamples the signal before being sliced and QAMdemapped.

A. Gain-Switched Laser Based Frequency Comb Generation
The gain-switched laser source, used in this work, is based on an externally injected gain switched laser (EI-GSL) [21]. In comparison to other optical frequency combs generation techniques, EI-GSL based on direct modulation is simple, cost efficient and flexible. Some other comb generation techniques, such as Kerr microresonators and electro-optic (EO) modulation [22] can be considered as well for MHCD. The external injection from a single low linewidth master tunable laser (TL) into the gain switched slave laser enables the EI-GSL source to offer low linewidth [23], tunability of the central emission wavelength [24], a high degree of phase correlation between all of the comb's tones [25], and a tunable and stable free spectral range [26]. It is worth noting that this configuration is suitable for photonic integration [27]. A shortcoming associated with this comb generation technique is the limited number of generated comb tones (directly related to the bandwidth of the laser).
The EI-GSL optical frequency comb generation schematic was previously explained [1]. A commercially available Fabry-Perot (FP) laser (slave) used in our experiment is a 200 μm long device encased in a high speed butterfly package. The threshold current of the FP was around 8 mA at 25 • C and the small signal modulation bandwidth was measured to be around 11 GHz, when biased at 60 mA. Gain switching of this FP laser was achieved by driving the laser diode with a large amplified sinusoidal signal (24 dBm @ 10 GHz) in conjunction with a dc bias current of 60 mA, while the laser was temperature controlled at 25 • C. A semiconductor based TL, with a linewidth of ∼300 kHz and an output power of −4 dBm, was used to inject light via a polarization maintaining circulator into the gain switched laser. However, the number of lines generated (12 lines, within 3 dB from spectral peak) was low due to the limited bandwidth of the FP laser used. Hence, the generated comb was expanded by passing it through a 20 Gb/s phase modulator biased at 2.75V π and driven by the same RF signal as used for the comb generation. The expanded EI-GSL comb spectrum, as illustrated in Fig. 9(b), yielded 20 clearly resolved phase correlated optical tones, with each of the tones offset by an integer multiple of the drive frequency (10 GHz). The corresponding optical pulse train is shown in Fig. 9(d) with a pulse width of ∼18.5 ps. The measured pulse width is limited by the oscilloscope temporal response (15.4 ps) convolved with the natural width of the laser's pulse shape of 20 modes equalized comb (5 ps pulse width). Typical transmitted signal with 20 equalized modes, at 4 dB optical SNR, is shown in Fig. 9(c). The optical spectrum measurements were obtained with high-resolution 10 MHz interferometric optical spectrum analyzer. The spectral flatness of the equalized modes is roughly 2 dB, and it is related to the spectral resolution of the SLM device. This technique was further used to determine 5, 10, and 20 modes transmission, with different levels of optical SNR.

B. Multi-Core Fiber Network
The multi-core fiber network is an optical test-bed infrastructure built in 2019 in the city of L'Aquila [14], [28]. A single jelly-filled loose-tube cable, with an outer diameter of 6 mm and a total length of 6.29 km, has been deployed in a multiservice underground tunnel with both ends accessible from the same location. The cable accommodates three different kinds of MCFs for a total of 18 strands: twelve are randomly coupled four-core MCFs (RC-4CF), four are uncoupled four-core MCFs (UC-4CF), and two are uncoupled eight-core MCFs (UC-8CF). The first two kinds (RC-4CF and UC-4CF) are optimized for the C-band window (1550 nm), while the latter (UC-8CF) is optimized for the O-band window (1310 nm). In the experiment presented here, the uncoupled UC-4CF were used.

VIII. RESULTS AND DISCUSSION
To assess the model presented in this paper, three sets of a multi-homodyne coherent detection transmission experiments were conducted, and presented in Fig. 10. In the first set, a baseline optical back-to-back (OB2B) non-dispersive transmission was taken as a reference, using 5 and 10 GSL's tones (blue lines). In the second set, in order to validate the theoretical model developed in Section VI above, a dispersive propagation in two cores of 6.29 km MCF was tested using 5 and 10 tones (yellow lines). Finally, a third set of measurements was dedicated to demonstrate the MHCD system with security applications, i.e. optical encoding is applied over the transmitted signal, while both the authorized and the unauthorized receivers are used to decode the signal (red lines, Fig. 10(b)). The first two sets of experiments, focusing back-to-back vs. MCF co-propagating transmission, are summarized in Fig. 10(a). The third one, focusing the security of the phase mask, is presented in in Fig.  10(b).
In Fig. 10(a), the SNR is plotted vs. optical SNR. Both the number of GSL's tones, and the optical noise level were accurately determined with the setup depicted in Fig. 8. As a figure of merit, the SNR was extracted from the acquired MHCD signal, after applying a conventional coherent DSP receiver. Below 7 dB SNR, the system performance are mainly limited by the optical noise, thus rather linear curves are observed. However, while approaching 12 dB, the SNR curves begin to saturate due to the rise of signal-dependent electro-optical impairments. This is in agreement with the MHCD theoretical model which is based on the optical processing gain defined in optical noise environment, which dominates below 7 dB SNR [1]. Considering the optical Fig. 10. In (a), the SNR vs. optical SNR of a 10 Gbaud single-polarization QPSK signal detected by the authorized user for 5, and 10 GSL's tones is presented. The optical back-to-back (OBTB) measurements of 5 and 10 tones are plotted in blue, and the multi-core fibers transmission over 6.29 km, for 5 and 10 tones, in yellow. In (b), the SNR detected by the unauthorized user, for an arbitrary phase mask vs. the authorized user, both for 20 tones in OB2B, are plotted in red. In both subfigures, the BER thresholds are shown, to indicate detectable measurements. back-to-back curves, one can see that SNR improves proportionally with the number of GSL's tones [1]. The absolute SNR values reflect good agreement with the theory, up to 2 dB gap, potentially due to the non-ideal sig.-LO temporal alignment, which was tuned with a manual optical delay line, and the limited flatness of the GSL spectrum. In Fig. 10(b), the security of the system is demonstrated, while comparing the authorized user SNR (upper curve) versus the unauthorized (lower curve). The phase mask used by the unauthorized is chosen randomly, evoking incoherent addition of the signal [1]. The vertical gap in this subfigure implies for the performance discrimination, allowing MHCD system to work in "error-free" condition, while the unauthorized user is limited to 6-8 dB lower SNR.
The case of MCF transmission is shown with two yellow lines. The results are comparable with the simulations (depicted in Fig.  6) as the SNR difference between 10 and 11 tones is negligible. Transmission of 5 and 10 GSL's tones was successfully demonstrated, and the latter yielded optical processing gain, which allowed "error-free" transmission in 0 dB optical SNR. A comparison between OB2B and MCF SNR (blue versus yellow curves) suggests some degradation, predicted by the model of MHCD in dispersive channel (formulated by (23) and plotted in Fig. 6). For 5 tones 10 GHz FSR and 6.29 km uncompensated length there is 0.4 dB average SNR penalty (one vertical division in average, in Fig. 10(a)). For 10 tones, 10 GHz FSR and 6.29 km uncompensated length introduces 0.8 dB average SNR penalty (two vertical divisions in average). The latter agrees with the intersection of the two dashed lines in Fig. 6, around 0.4 dB above the red solid line, up to a discrepancy of about 0.4 dB. This gap can be explained as the two different MCF cores do not match perfectly with their dispersion properties. Therefore, the performance model of MHCD in dispersive channel, developed in this paper, well describes the experimental results, for co-propagating signal and LO.

IX. CONCLUSION
Multi-core fibers have proven to enable MHCD transmission by solving key hardware-complicated mechanism. Here, significant relaxation of two critical hardware challenges was demonstrated, including pulse locking between the LO and the received signal and chromatic dispersion compensation. Furthermore, stealthy and encrypted transmission was demonstrated and quantitatively analyzed. The proposed scheme is based on co-transmission of both the LO and the optical signal over separated cores of the MCF. A shortcoming associated with this technique, is the losses experienced by the LO, and the need to control its incident polarization at the receiver. Theoretical modeling, simulations, and "real world" experiments over deployed multi-core fibers network in a metropolitan environment have validated the proposed method. He is currently a Lecturer with the School of Electronic Engineering, DCU, Principal Investigator for Science Foundation Ireland and Enterprise Ireland, and the Director of the Photonics Systems and Sensing Laboratory. He has authored or coauthored more than 250 articles in internationally peer reviewed journals and conferences and also a Holder of seven international patents. His main research interests include elastic optical networks, photonic sensing, spectrally efficient modulation formats, and radio-over-fibre distribution systems.

Luca Potì biography not available at the time of publication.
Dan Sadot received the B.Sc., M.Sc., and Ph.D. degrees in electrical and computer engineering (Summa Cum Laude) from Ben Gurion University, Beersheba, Israel. He is currently a Professor with the School of ECE, Ben Gurion University. He was a Postdoc with Stanford University, Stanford, CA, USA, where his studies were supported by the Clore, Rothchild, and Fulbright scholarships. He was the Chairman of the ECE Department, Ben Gurion University during 2007-2013. He has authored or coauthired more than 200 papers in peer reviewed journals and conference proceedings, and holds more than 35 patents. He was the Founder and CTO of four startup companies: TeraCross, Xlight Photonics, MultiPhy, and CyberRidge.