Phase-sensitive amplifier-based optical conversion for direct detection of complex modulation format to bridge long-haul transmissions and short-reach interconnects

Jiabin Cui; Yuefeng Ji; Guo-Wei Lu; Hongxiang Wang; Min Zhang

doi:10.1364/OE.380482

1. Introduction

With the arrival of 5G era, amount of large data traffic applications emerge prominently [1]. 4K/8K video, Internet of Things and dense mobile social network have been gradually inseparable with people’s work and life. As the carrier of whole communication network, higher transmission capacities are in constant demand for optical networks. Various multiplexing methods [2], advanced modulation formats [3] and other emerging technologies are continually proposed and embody their advanced characteristics. Coherent detection and direct detection (DD) are the major enabling method for the long-haul transmissions and short-reach interconnects, respectively. Besides, modulation formats are another key means for implementing the high capacity transmissions. For long-haul transmissions, quadrature phase shift keying (QPSK) and quadrature amplitude modulation (QAM) signals are considered as the pivotal modulation formats. The QPSK is the standard format for the 100G wide area networks [4], and is also main candidate for the 400G long-haul networks standard format, along with 8QAM and 16QAM [5]. Moreover, because of its maturity and universality, QPSK is applied in some new-rising areas and scenarios, such as mode division multiplexing [6] and free-space optical communication [7]. For short-reach interconnects, being limited by the system cost and simplicity requirements, the intense modulation (IM) formats are the major adopted measures, such as on-off keying (OOK) and pulse amplitude modulation (PAM) signals [8]. Among them, PAM4 signal has been elected as the standard format for the 400G short-reach networks and is under research for amount future network scenarios, such as 50G passive optical networks (PONs) [9] and data center interconnects [10].

Nonlinear optics-based signal processing techniques have been a research hotpot for flexible handling optical vector signals. Different optical networks have their own proper transmission methods and modulation formats. QPSK and PAM4 are the main modulation formats in the long-haul transmissions and short-reach interconnects, respectively. For flexible controlling and efficient managing the multifarious optical networks, an optical format conversion node which can bridge the signals in different networks is an impressive work [11]. Further, the conversion node between long-haul and short-reach networks has drawn a great deal of attention for its practical meaning and various conversion node schemes have been proposed, such as 16QAM-to-PAM4 conversion [12,13], PAM4-to-QPSK conversion [14] and 64QAM-to-PAM8 conversion [15]. Focusing on the QPSK-to-PAM4 conversion, the node scheme proposed by [16] achieves the conversion while its converted PAM4 is not a proper normal PAM signal since it cannot be directly detected by a photo-detector (PD). The scheme in [17,18] realizes the conversion and gets a power ratio editable normal PAM4 signal while it changes the signal original wavelength and cannot flexible control the PAM4 constellation shape. The flexible controlling of PAM4 power ratio and constellation shaping are both meaningful. The uniform PAM4 signal with equal power interval is usually perceived as the optimal PAM4 and has the best transmission performance. While the non-uniform PAM4 signal also has its suited application scenarios and its many topics are under research. For example, amount researches report that the non-uniform PAM4 has potential applications in PON, comparing with uniform PAM4, it can increase the aggregated capacity [19,20], improve the receiver power budget [21] and receiver sensitivity [22] in PONs. Constellation shaping is an effective method to improve the signal transmission performance and enrich its application scenarios. The hybrid constellation shaping technology contains geometric constellation shaping and probabilistic constellation shaping [23]. These constellation shaping technologies play a key role both in IM/DD system [24] and coherent system [25]. Moreover, with the signal formats and baud rate becoming more complicated and higher, the generating methods of broadband advanced signals have to be abundant. Hierarchical modulation is one of hot technologies which are being researched to improve and enrich the optical modulation technology [26,27]. PAM4 signals can be used to aggregate high rate QAM signals and the constellation shape editable PAM4 signals can be applied into flexible generating geometric constellation shaped QAM signals.

The processing of optical vector signals in optical domain is always a critical issue in signal processing field. Phase-sensitive amplifier (PSA) technology is one of the effective measures to flexibly handle the optical vector. For an input signal, the amplification offered by PSA according to its relative phase to the PSA pumps [28]. PSA has been a hot topic in nonlinear optics-based signal process techniques and amount of related researches have been reported due to its specific phase-sensitive amplification characteristic. The reported nonlinear optical mediums employed to realize PSA include but not limited to high nonlinear fiber (HNLF) [29], periodically poled lithium niobate (PPLN) waveguide [30], quantum-dot semiconductor optical amplifier (QD-SOA) [31], and silicon waveguide [32]. The reported PSA-based applications mainly contain signal regeneration [33], format conversion [34], low-noise amplification [35], phase quantizing [36], phase-sensitive wavelength multicasting [37], and so on.

In this paper, a PSA-based optical conversion node scheme is proposed and verified by simulations. An input noisy 10G Baud QPSK signal is converted into a 10G Baud normal PAM4 signal. The constellation shape and power ratio of the converted PAM4 can be designed flexibly according to the applications requirements. Five kinds of uniform and non-uniform PAM4 signals are generated in this research to verify the constellation shaping functionality. The conversion node contains three parts: degenerate PSA-based quadrature de-multiplexer (QD), non-degenerate PSA-based bi-directional vector mover (VM), and flexible vector coupling. The detailed principle and transfer characteristics of these three parts are expressed by formula derivations and quantitative analysis. For the input QPSK with the input optical signal-to noise ratio (OSNR) range of (10 dB$\sim$30 dB), the constellations, eye diagrams, error vector magnitude (EVM), phase and amplitude impact factors (IFs) of the signals went through every stage are measured to indicate the system key performances. The QPSK and converted PAM4s bit error ratio (BER) performances are also performed. For the QPSK with the input OSNR of 15 dB, 20 dB and 25 dB, with the BER of ${10^{ - 3}}$, the receiver OSNR of converted uniform PAM4s is 16.6 dB, 16.2 dB and 16 dB, respectively. For the format conversion and constellation flexible shaping functions, the proposed node scheme has vast potential applications in the optical network intermediate node, advanced format signals generating and shaping.

2. Operating principle

As shown in Fig. 1, an optical conversion node scheme is proposed by this paper. For an input noisy QPSK signal, the node can convert it into a PAM4 signal which can be directly detected by a PD. Moreover, the constellation shape of the converted PAM4 is flexibly editable to address different application scenarios. The node system is constituted by three main parts: degenerate PSA-based QD, non-degenerate PSA-based bi-directional VM and flexible coupling part. The input QPSK is firstly quadrature de-multiplexed by the QD, and its in-phase (I) and quadrature (Q) parts can be viewed as two binary phase shift keying (BPSK) signals. Then BPSK-I and BPSK-Q are processed by the bi-directional VM, and two OOK signals generated after the vector moving processing. The flexible coupling part makes the OOK-I and OOK-Q add two-dimensionally, the PAM4 is generated finally. Because of the vector adding is controllable and two-dimensional, the constellation of converted PAM4 can be flexible shaping. In the following subsections, the principles of QD, VM and coupling process are introduced in detail. Besides, the generation of phase-locked pumps is a key issue for PSA realizing, so the phase-locking solution this paper adopted is also expressed. The node scheme in this paper is a basic model which realizes converting a noisy one-channel QPSK signal into a constellation editable PAM4 signal. The conversion researches oriented to higher-order formats and multiplexing technologies will be our future important research directions to make the node scheme more complete.

2.1 Quadrature de-multiplexer

The degenerate PSA-based QD is proposed by [38], its setup, spectrum and process constellations are shown in Fig. 2. The QD is a bi-directional degenerate PSA and a HNLF is employed to be the nonlinear medium in this paper. The input signal (S), phase-locked pumps (P1, P2) with the spectrum as Fig. 2(b) are injected into the QD. The electrical fields of the launched signals at the points ${A_1}$ and ${A_2}$ can be expressed as:

(1)$$\left[ {\begin{array}{c} {{A_1}}\\ {{A_2}} \end{array}} \right] = \left[ {\begin{array}{c} {{A_{p1}}\exp \left( {j{\varphi _{p1}}} \right) + {A_s}\exp \left( {j{\varphi _s}} \right)}\\ {{A_{p2}}\exp \left( {j{\varphi _{p2}}} \right)} \end{array}} \right]$$

The ${A_s}\exp \left ( {j{\varphi _s}} \right )$, ${A_{p1}}\exp \left ( {j{\varphi _{p1}}} \right )$and ${A_{p2}}\exp \left ( {j{\varphi _{p2}}} \right )$are the optical vector expressions of S, P1 and P2, respectively. The launched signals go through the coupler1, the signals of ${B_1}$ and ${B_2}$ are:

(2)$$\left[ {\begin{array}{c} {{B_1}}\\ {{B_2}} \end{array}} \right] = \sqrt {\frac{1}{2}} \left[ {\begin{array}{c} {{A_{p1}}\exp \left( {j{\varphi _{p1}}} \right) + {A_s}\exp \left( {j{\varphi _s}} \right) + j{A_{p2}}\exp \left( {j{\varphi _{p2}}} \right)}\\ {j{A_{p1}}\exp \left( {j{\varphi _{p1}}} \right) + j{A_s}\exp \left( {j{\varphi _s}} \right) + {A_{p2}}\exp \left( {j{\varphi _{p2}}} \right)} \end{array}} \right]$$

Then the signals occur four-wave mixing (FWM) effect in the HNLF, the electrical fields of output signals at ${C_1}$ and ${C_2}$ are:

(3)$$\left[ {\begin{array}{c} {{C_1}}\\ {{C_2}} \end{array}} \right] = \sqrt {\frac{G}{2}} \left[ {\begin{array}{c} {{A_{p1}}\exp \left( {j{\varphi _{p1}}} \right) + {A_s}\exp \left( {j{\varphi _s}} \right) + j{A_i}\exp \left( {j{\varphi _i}} \right) + j{A_{p2}}\exp \left( {j{\varphi _{p2}}} \right)}\\ {j{A_{p1}}\exp \left( {j{\varphi _{p1}}} \right) + j{A_s}\exp \left( {j{\varphi _s}} \right) + {A_i}\exp \left( {j{\varphi _i}} \right) + {A_{p2}}\exp \left( {j{\varphi _{p2}}} \right)} \end{array}} \right]$$

the $\sqrt G$ in Eq. (3) is the amplitude gain caused by FWM process, ${A_i}\exp \left ( {j{\varphi _i}} \right )$ is the idler (Idl) wave generated in the FWM process and its phase has the relationship of ${\varphi _i} = \pi /2 + {\varphi _{p1}} + {\varphi _{p2}} - {\varphi _s}$. Then these signals go through coupler1 again, the electrical fields of ${D_1}$ and ${D_2}$ are:

(4)$$\left[ {\begin{array}{c} {{D_1}}\\ {{D_2}} \end{array}} \right] = j\sqrt G \left[ {\begin{array}{c} {{A_{p1}}\exp \left( {j{\varphi _{p1}}} \right) + {A_s}\exp \left( {j{\varphi _s}} \right)}\\ {{A_i}\exp \left( {j{\varphi _i}} \right) + {A_{p2}}\exp \left( {j{\varphi _{p2}}} \right)} \end{array}} \right]$$

In this process, the S and Idl of degenerate PSA are separated in space by the QD. Then S and Idl are filtered out by optical band pass filter and coupled by coupler2, the signals at ${E_1}$ and ${E_2}$ are:

(5)$$\left[ {\begin{array}{c} {{E_1}}\\ {{E_2}} \end{array}} \right] = j\sqrt {\frac{G}{2}} \left[ {\begin{array}{c} {{A_s}\exp \left( {j{\varphi _s}} \right) + j{A_i}\exp \left( {j{\varphi _i}} \right)}\\ {j{A_s}\exp \left( {j{\varphi _s}} \right) + {A_i}\exp \left( {j{\varphi _i}} \right)} \end{array}} \right]$$

The ${\varphi _s}$ can be defined as ${\varphi _s} = {\varphi _{sc}} + {\varphi _{mn}}$, ${\varphi _{sc}}$ and ${\varphi _{mn}}$ are the carrier phase and information phase (including noise phase) of S. The Eq. (5) can be written as:

(6)$$\left[ {\begin{array}{*{20}{c}} {{E_1}}\\ {{E_2}} \end{array}} \right] = j\frac{{\sqrt G {A_s}\exp \left( {j{\varphi _{sc}}} \right)}}{{\sqrt 2 }}\left[ {\begin{array}{*{20}{c}} {\exp \left( {j{\varphi _{mn}}} \right){\rm{ + }}jm\exp \left( {j\left( {\theta - {\varphi _{mn}}} \right)} \right)}\\ {j\exp \left( {j{\varphi _{mn}}} \right){\rm{ + }}m\exp \left( {j\left( {\theta - {\varphi _{mn}}} \right)} \right)} \end{array}} \right] $$

where $\theta \textrm {{ = }}\pi /2 + {\varphi _{p1}} + {\varphi _{p2}} - 2{\varphi _{sc}}$, $m$ is the amplitude ratio of Idl and S. The $\theta$ is the relative phase of input S and two pumps, if the P1 and P2 are phase-locked pumps of S, is a constant value and can be controlled by adjusting pumps phase. $m$ is also can be controlled by changing the power ratio of input S and pumps. Here we suppose $\theta \textrm {{ = 3}}\pi \textrm {{/2}} \pm \textrm {{2}}\pi$ and $m\textrm {{ = }}1$, the Eq. (6) can be rewritten as:

(7)$$\left[ {\begin{array}{c} {{E_1}}\\ {{E_2}} \end{array}} \right] = j\sqrt {2G} {A_s}\exp \left( {j{\varphi _{sc}}} \right)\left[ {\begin{array}{c} {cos{\varphi _{mn}}}\\ { - \sin {\varphi _{mn}}} \end{array}} \right]$$

As expressed in Eq. (7), the I and Q components are get at ${E_1}$ and ${E_2}$. Moreover, the output information phase and gain of the QD output signals are:

(8)$${\varphi _I} = \left\{ {\begin{array}{cc} {2l\pi} ,&{- \pi /2 \le {\varphi _{mn}} \pm 2l\pi\;<\;\pi /2,}\\ {\left( {2l + 1} \right)\pi} ,&{ \pi /2 \le {\varphi _{mn}} \pm 2l\pi\;<\;3\pi /2.} \end{array}} \right.$$

(9)$${\varphi _Q} = \left\{ {\begin{array}{cc} {\left( {2l + 1} \right)\pi },& {0 \le {\varphi _{mn}} \pm 2l\pi\;<\;\pi ,}\\ {2l\pi },& {\pi \le {\varphi _{mn}} \pm 2l\pi\;<\;2\pi .} \end{array}} \right.$$

(10)$${G_{total}} = \left\{ {\begin{array}{c} {{G_I} = 10\log \left( {2G{{\cos }^2}{\varphi _{mn}}} \right),}\\ {{G_Q} = 10\log \left( {2G{{\sin }^2}{\varphi _{mn}}} \right).} \end{array}} \right.$$

where ${\varphi _I}$ and ${\varphi _Q}$ are the output information phase of I and Q components of S. ${G_I}$ and ${G_Q}$ are the total gain of the I and Q outputs comparing with input S, $l$ is an integer. From Eq. (8) and (9), we can observe that degenerate PSA has a well two-level phase quantizing property when $m\textrm {{ = }}1$. The Eq. (10) tells that the gain of PSA with two-level phase quantizing property is phase-sensitive and has a period of $\pi$.

After above derivations, the QD fulfills the optical vector quadrature de-multiplexing function. When input signal is a QPSK signal, the signals at ${E_1}$ and ${E_2}$ will be two BPSK tributaries which carries I and Q component information separately (BPSK-I, BPSK-Q).

Fig. 1. Conceptual graph of the conversion node system. N-PSA: non-degenerate PSA; Att: attenuator; $\Delta \varphi$: phase shift.

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Fig. 2. (a), (b) and (c) are the setup, spectrum and constellations of the degenerate PSA-based QD, respectively. cpl: coupler; BPF: band pass filter.

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2.2 Vector mover

After the QD part, two BPSK tributaries are generated. These two BPSKs need to be converted into two OOKs simultaneously by vector moving processing. [39] employs a non-degenerate PSA to realize a format conversion from BPSK to OOK while we found that a non-degenerate PSA can be designed to be an optical VM. The setup of non-degenerate PSA-based VM is shown in Fig. 3(a). Input signal (S) and two pumps (P1 and P0) are launched into the VM, their spectrum are shown as Fig. 3(b). The VM formula derivation process are as follows. According to [28], the non-degenerate PSA transfer function is:

(11)$$\left[ {\begin{array}{c} {{A_{so}}\exp \left( {j{\varphi _{so}}} \right)}\\ {{A_{io}}\exp \left( {j{\varphi _{io}}} \right)} \end{array}} \right] = \left[ {\begin{array}{cc} \mu & \nu \\ {{\nu ^*}} & {{\mu ^*}} \end{array}} \right]\left[ {\begin{array}{c} {{A_{si}}\exp \left( {j{\varphi _{si}}} \right)}\\ {{A_{ii}}\exp \left( { - j{\varphi _{ii}}} \right)} \end{array}} \right]$$

where ${A_{so}}\exp \left ( {j{\varphi _{so}}} \right )$ and ${A_{si}}\exp \left ( {j{\varphi _{si}}} \right )$ are the output and input signal electrical fields of VM, ${A_{io}}\exp \left ( {j{\varphi _{io}}} \right )$ and ${A_{ii}}\exp \left ( {j{\varphi _{ii}}} \right )$ are the output and input idler of the FWM in PSA, here the idler is P0. So the output S can be expressed as:

(12)$${A_{so}}\exp \left( {j{\varphi _{so}}} \right) = \mu {A_{si}}\exp \left( {j{\varphi _{si}}} \right) + \nu A_{p0}^*\exp \left( { - j{\varphi _{p0}}} \right)$$

where $\mu$ and $\nu$ are the transfer functions of non-degenerate PSA, they satisfy the auxiliary equation ${\left | \mu \right |^2} - {\left | \nu \right |^2} = 1$, their detailed expressions are:

(13)$$\left[ {\begin{array}{c} \mu \\ \nu \end{array}} \right] = \left[ {\begin{array}{c} {\cosh \left( {\kappa z} \right) + j\left( {\delta /\kappa } \right)\sinh \left( {\kappa z} \right)}\\ {j\left( {\gamma /\kappa } \right)\sinh \left( {\kappa z} \right)} \end{array}} \right]$$

where $z$ is the HNLF length, the growth rate $\kappa \textrm {{ = }}{\left ( {{{\left | \gamma \right |}^2} - {\delta ^2}} \right )^{1/2}}$, $\delta \textrm {{ = }}j\beta /2 + j\bar \gamma {P_{p1}}$, linear wave vector mismatch term $\beta \textrm {{ = }}{k_s} + {k_{p0}} - 2{k_{p1}}$, $k$ is wave vector; $\gamma \textrm {{ = }}2\bar \gamma {\left ( {{A_{p1}}\exp \left ( {j{\varphi _{p1}}} \right )} \right )^2}$, $\bar \gamma$ is the nonlinear coupling coefficient of HNLF, ${A_{p1}}\exp \left ( {j{\varphi _{p1}}} \right )$ is the electrical fields of P1. It can be concluded that ${\varphi _\gamma }\textrm {{ = 2}}{\varphi _{p1}}$.

When the FWM process fulfills the phase-matching condition, $\delta \textrm {{ = }}0$and $\kappa \textrm {{ = }}\left | \gamma \right |$, Eq. (13) can be expressed as:

(14)$$\left[ {\begin{array}{c} \mu \\ \nu \end{array}} \right] = \left[ {\begin{array}{c} {\cosh \left( {\left| \gamma \right|z} \right)}\\ {j\exp \left( {2j{\varphi _{p1}}} \right)\sinh \left( {\left| \gamma \right|z} \right)} \end{array}} \right]$$

Fig. 3. (a), (b) and (c) are the setup, spectrum and constellations of the VM; (d) and (e) are the setup and constellations of bi-directional VM proposed by this paper. BPF: band pass filter.

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With the Eq. (14), Eq. (12) can be rewritten as:

(15)$${A_{so}}\exp \left( {j{\varphi _{so}}} \right) = \cosh \left( {\left| \gamma \right|z} \right){A_{si}}\exp \left( {j{\varphi _{si}}} \right) + j\sinh \left( {\left| \gamma \right|z} \right)A_{p0}^*\exp \left( {j\left( {2{\varphi _{p1}} - {\varphi _{p0}}} \right)} \right)$$

When HNLF is certain, the $\cosh \left ( {\left | \gamma \right |z} \right )$ and $\sinh \left ( {\left | \gamma \right |z} \right )$ are constant real values. The ${\varphi _{si}}$ also can be expressed as ${\varphi _{si}}\textrm {{ = }}{\varphi _{sc}} + {\varphi _{mn}}$, then Eq. (15) can be expressed as:

(16)$${A_{so}}\exp \left( {j{\varphi _{so}}} \right) = \sqrt {{G_1}} {A_{si}}\exp \left( {j{\varphi _{sc}}} \right)\left( {\exp \left( {j{\varphi _{mn}}} \right) + M\exp \left( {j{\theta _1}} \right)} \right)$$

where $\sqrt {{G_1}} = \cosh \left ( {\left | \gamma \right |z} \right )$, the $\left | \gamma \right |$ and $z$ are effected by the length, nonlinear coupling coefficient of HNLF and power of P1. We define that ${\theta _1} = \pi /2 + 2{\varphi _{p1}} - {\varphi _{p0}} - {\varphi _{sc}}$ and $M = {{\sinh \left ( {\left | \gamma \right |z} \right )A_{p0}^*} \mathord {\left / {\vphantom {{\sinh \left ( {\left | \gamma \right |z} \right )A_{p0}^*} {\cosh \left ( {\left | \gamma \right |z} \right ){A_{si}}}}} \right. } {\cosh \left ( {\left | \gamma \right |z} \right ){A_{si}}}}$. If P1 and P0 are phase-locked pumps of S, ${\theta _1}$ is a constant value and can be adjusted by controlling the phase of pumps. By controlling the parameters of HNLF and power of P1 and P0, is also a controllable constant value. Suppose that ${\theta _1}\textrm {{ = }}2l\pi$, the Eq. (15) can be expressed as:

(17)$${A_{so}}\exp \left( {j{\varphi _{so}}} \right) = \sqrt {{G_1}} {A_{si}}\exp \left( {j{\varphi _{sc}}} \right)\left( {\exp \left( {j{\varphi _{mn}}} \right) + M} \right)$$

From Eq. (17), the output signal ${A_{so}}\exp \left ( {j{\varphi _{so}}} \right )$ can be viewed as the sum of input S and a coherent constant optical vector which means the VM realizes optical vector moving. The output signal information phase ${\varphi _{mno}}$, its first derivative over ${\varphi _{mn}}$ and total gain of VM are:

(18)$${\varphi _{mno}} = \arctan \left( {\frac{{\sin {\varphi _{mn}}}}{{M + \cos {\varphi _{mn}}}}} \right)$$

(19)$${\varphi _{mno}}' = 1/2,\quad M = 1,{\varphi _{mn}} \ne \left( {2l + 1} \right)\pi$$

(20)$${G_{total}} = 2\left( {M + \cos {\varphi _{mn}}} \right)$$

If the input S is a BPSK signal, the output signal can be an OOK signal when $M = 1$. After the QD process, a QPSK is de-multiplexed into two BPSK tributaries. In order to maintain the coherence between the system I and Q branches, we design the bi-directional VM as shown in Fig. 3(d). The bi-directional VM can process two input BPSK tributaries simultaneously within the same medium and the constellations of moving process are shown in Fig. 3(e). After the Bi-directional VM part, BPSK-I and BPSK-Q are converted into OOK-I and OOK-Q, respectively.

2.3 Phase-locked pumps generation

Phase-locked pumps are critical issues for PSA realizing process. In introduces of 2.1 and 2.2, the $\theta$ and ${\theta _1}$ are defined as the relative phase among pumps and signal carrier of degenerate and non-degenerate PSA, and both $\theta$ and ${\theta _1}$ need to be constant and controllable values. Here we employ optical frequency comb to generate the pumps and signal carrier. Optical frequency comb -based signal processing is a promising and hot topic in optical signal processing field, amount of signal processing schemes are proposed in recent years. Generally, optical frequency combs are generated by employing radio frequency (RF) signals to drive a phase modulator or a Mach-Zehnder modulator (MZM) to modulate a continuous wave (CW). In this paper, the scheme in [40] is employed to get a serious of waves with equal frequency interval and constant relative phase. The optical combs generation scheme is shown in Fig. 4(a), two RF signals are employed to drive the two branches of a MZM to modulate a CW signal. By adjusting the parameters of RFs and MZM bias, an optical frequency comb with flat spectrum is generated. The output signal electric field of MZM is:

(21)$${E_{out}} = \frac{1}{2}{E_{in}}\sum_{l = - \infty }^\infty {\left[ {\exp \left( {j\frac{\pi }{{{E_\pi }}}{V_1}\left( t \right)} \right) + \exp \left( {j\frac{\pi }{{{E_\pi }}}{V_2}\left( t \right)} \right)} \right]}$$

where ${E_{in}}$ and ${E_{out}}$ are the input and output signals of MZM, ${E_{in}}\textrm {{ = }}{E_c}\exp \left ( {j{\omega _0}t + j{\varphi _0}} \right )$ is input CW signal, ${E_c}$, ${\omega _0}$ and ${\varphi _0}$ are the amplitude, frequency and initial phase of ${E_{in}}$. ${E_\pi }$ is the half-wave voltage (RF) of MZM. ${V_1}\left ( t \right ) = {E_1}\cos \omega t$ and ${V_2}\left ( t \right ) = {E_2}\cos \omega t$ are the RF signals, ${E_1}$ and ${E_2}$ are their amplitude, $\omega$ is their frequency. The Eq. (21) can be expressed as:

(22)$${E_{out}} = \frac{1}{2}{E_c}\exp \left( {j{\omega _0}t + j{\varphi _0}} \right)\sum_{l = - \infty }^\infty {\left[ {\exp \left( {j\frac{\pi }{{{E_\pi }}}\left( {{E_1} + {E_2}} \right)\cos \omega t} \right)} \right]}$$

With the Jacobi-Anger identity:

(23)$$\exp \left( {jx\cos \varphi } \right) = \sum_{l = - \infty }^\infty {{j^n}{J_l}\left( x \right)} \exp \left( {jl\varphi } \right)$$

where the ${J_l}\left ( x \right )$ is the ${l^{th}}$ Bessel function of the first kind and it is a real value effected by $x$, we substitute Eq. (23) into Eq. (22):

(24)$${E_{out}} = \frac{1}{2}{E_c}\sum_{l = - \infty }^\infty {{j^l}\left[ {{J_l}\left( {\frac{\pi }{{{E_\pi }}}{E_1}} \right) + {J_l}\left( {\frac{\pi }{{{E_\pi }}}{E_2}} \right)} \right]} \exp \left( {j{\omega _0}t + j{\varphi _0} + jl\omega t} \right)$$

Fig. 4. (a) Phase-locked pumps generation setup; (b) Phase-locked pumps spectrum. CW: continuous wave; MZM: Mach-Zehnder modulator; RF: radio frequency signals; PF: programmable filter.

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Then we get the expressions of the optical frequency comb whose frequency interval and phase difference between two adjacent harmonics are $\omega$ and $\pi /2$. The MZM-based optical frequency comb is employed to be the signal carrier and phase-locked pumps of the PSAs in QD and VM. Here the MZM-based phase-locked pumps generation scheme is verified by simulations. As shown in Fig. 4(a), the laser is set to emit a CW signal with the frequency of 193.2 THz and the power of 100 mW. The half-wave voltage (RF) and half-wave voltage (Bias) of MZM are both 8.8-volt. The ${E_1}$ is set to be 5-volt and RF frequency is 50GHz, ${V_1}\left ( t \right ) = {V_2}\left ( t \right )$ and bias voltage on both branches are 0-volt. From the spectrum shown in Fig. 4(b), the output signal of the optical frequency comb is a serious of equal frequency interval CWs with flat spectrum which can be employed as the signal carrier and pumps in PSAs. The phase-locked pumps generation scheme employed in this paper is effected by the frequency and voltage of the driving RF signals. For processing the signals with ultra-high baud rate or free-running carriers, a more effective pumps generation scheme need to be investigated and employed.

2.4 Flexible coupling

After the QD and VM parts, the input QPSK signal are converted into two OOK signals which are to be added two-dimensionally to generate one PAM4 signal in the coupling process, the coupling process can be expressed as:

(25)$${A_{pam}}\exp \left( {j{\varphi _{pam}}} \right) = {A_{ook - I}}\exp \left( {j{\varphi _{ook - I}}} \right) + {A_{ook - Q}}\exp \left( {j{\varphi _{ook - Q}}} \right)$$

where ${A_{pam}}\exp \left ( {j{\varphi _{pam}}} \right )$, ${A_{ook - I}}\exp \left ( {j{\varphi _{ook - I}}} \right )$ and ${A_{ook - Q}}\exp \left ( {j{\varphi _{ook - Q}}} \right )$ are the electrical fields of PAM4, OOK-I and OOK-Q. To realize flexible coupling, an attenuator and a phase shifter can be placed on Q branch of the coupling part to control the power ratio and relative phase between the two OOKs as shown in Fig. 5(a). For direct detection, the PAM4 signal power is:

(26)$${P_{pam}} = {P_{ook - I}} + {P_{ook - Q}} + 2\sqrt {{P_{ook - I}}{P_{ook - Q}}} \cos \left( {{\varphi _{ook - I}} - {\varphi _{ook - Q}}} \right)$$

Equation (26) indicates that the PAM4 power ratio can be flexibly editable because of the flexible two-dimensional adding. Here we set ${\varphi _{ook - I}} - {\varphi _{ook - Q}} = \pi /2$ to simplify the PAM4 shaping process, now the PAM4 power is:

(27)$${P_{pam}} = {P_{ook - I}} + {P_{ook - Q}}$$

Fig. 5. (a) Setup of the flexible coupling part; (b) and (c) are the constellations and eye diagrams of the coupling process, respectively. Att: attenuator; $\Delta \varphi$: phase shift.

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The PAM4 generating process is shown in Fig. 5(b), the data corresponding relationships are given in Table 1. Setting the relative phase between two OOKs to be $\pi /2$ is not only to simply the PAM4 shaping process, also can reduce damages to the synchronism between two branches by employing a 50:50 coupler to replace the phase shifter in Q branch.

Table 1. Data corresponding relationships of the conversion node.

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Two OOKs are added vertically to generate a PAM4, whose power levels are: $0$, ${P_{ook - Q}}$, ${P_{ook - I}}$ and ${P_{ook - Q}}\textrm {{ + }}{P_{ook - I}}$. We distinguish the PAM4s by the power ratio of eyes in their eye diagrams as shown in Fig. 5.(c), which is ${P_{ook - Q}}:\left ( {{P_{ook - I}} - {P_{ook - Q}}} \right ):{P_{ook - Q}}$. This power ratio can be controlled by attenuating the power of OOK-Q, the parameters settings of the five examples in Fig. 5(c) are listed in Table 2.

Table 2. Coupling part settings for PAM4s with different power ratios.

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With the flexible two-dimensional coupling process, a power-ratio editable PAM4 can be generated by OOK-I and OOK-Q. The editable PAM4s can be employed in different transmission scenarios and generating constellation shaped advanced signals. In this paper, the original input QPSK is converted into five kinds of PAM4s, including PAM-111, PAM4-121, PAM4-131, PAM4-212 and PAM4-313. In the following analysis, the critical performances of these PAM4s are measured to indicate their different characteristics.

3. Simulations & discussions

The node system simulations are based on the software of VPItransmissionMaker. The proposed conversion are verified by simulations as shown in Fig. 6. A CW signal with frequency of 193.2 THz is split into two paths. The upper path is employed to generate phase-locked pumps, include the P1, P2 in QD and P0, P1 in VM. The parameters of input CW signal, MZM and RF signals in phase-locked pumps generation are the same with the descriptions in subsection 2.3. The lower path 10 mW CW signal is modulated by a 20 Gbps pseudo random bit sequence (PRBS) to generate an original QPSK signal. The CW signals are filtered and amplified to be P1 (frequency: 193.15 THz, power: 107mW) and P2 (frequency: 193.25 THz, power: 107mW) of QD. The QD contains HNLF1 (length: 700 m, nonlinear coefficient: 13.2 ${W^{ - 1}}k{m^{ - 1}}$), two couplers (50:50), two circulators (3 ports) and two band pass filters (center frequency: 193.2 THz, bandwidth: 20 GHz). The VM contains HNLF2 (length: 500 m, nonlinear coefficient: 13.2 ${W^{ - 1}}k{m^{ - 1}}$), two circulators (3 ports), two band pass filters (center frequency: 193.2 THz, bandwidth: 20 GHz) and one coupler (50:50). The attenuator in VM is employed to control the power ratio of converted PAM4 signals follow the rules in Table 2. Amplified spontaneous emission noise is added to the signal at the point A and F in Fig. 6 to control the input and receiver OSNR of the system. The signals constellations, eye diagrams, EVMs, IFs and BERs are depicted and measured to indicate the conversion node performances. The IF is defined as:

(28)$$IF = 10\log \left( {\frac{{de{v_{in}}}}{{de{v_{out}}}}} \right)$$

where $de{v_{out}}$ and $de{v_{in}}$ are the standard deviations of output and input signals. IF expresses the influences of signal processing on signal distribution characteristic in amplitude and phase. The constellations, eye diagrams, EVMs and IFs are measured by coherent receiving with ${2^{12}} = 4096$ signal symbols and BERs are estimated by direct detection with ${2^{14}} = 16384$ signal symbols propagated through the node. The curves of EVMs, IFs and BERs are performed with input QPSK carries variable input OSNR. The constellations and eye diagrams are depicted after normalization.

3.1 QD & VM transfer characteristics

Before the system quantitative analysis, the QD and VM transfer characteristics are firstly performed. A CW signal (frequency: 193.2 THz, power: 10 mW) with a modulation phase range of $\left ( { - 1.5\pi \sim 3.5\pi } \right )$ is launched into the QD. The test signal constellations before and after the QD are shown in Fig. 7(a), the input vector signal shows a circular constellation with full phase states. After the quadrature de-multiplexing process, the output I and Q vector signals show line constellations with only two phase states. The output phase-input phase and gain-input phase transfer curves of I and Q components are shown in Fig. 7(b) and 7(c). The phase transfer curves indicate that the phase two-level quantizing character of degenerate PSA which is expressed in Eq. (8) and (9). Gain transfer curves both have the amplification ratio of 43.4 dB demonstrate the phase-sensitive amplification of PSA and is consistent with Eq. (10).

Fig. 6. Conversion node simulation setup. CW: continuous wave; MZM: Mach-Zehnder modulator; RF: radio frequency signals; PF: programmable filter; IQ: IQ modulator; PRBS: pseudo random binary sequence; ASE: amplified spontaneous emission noise; BPF: band pass filter; Att: attenuator.

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Fig. 7. QD transfer characteristics.

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Then a CW signal (frequency: 193.2 THz, power: 5 mW) with a modulation phase range of $\left ( { - 1.5\pi \sim 3.5\pi } \right )$ is launched into the VM. In Fig. 8(a), the input vector signal shows a circular constellation and the output vector signal is also a circle while added by a rightward vector. The constellations before and after VM do a good job of verifying the vector moving functionality. The VM gain-input phase curve in Fig. 8(b) is consistent with Eq. (20) and has a period of which is different with QD. The VM realizes vector moving processing also by phase-sensitive amplification character of PSA and it has gain ratio of 46.9 dB. The phase transfer and output phase first derivative curves in Fig. 8(c) also follow the derivation in Eq. (18) and (19). The VM phase transfer curve shows a sawtooth style and the first derivative curve is identically equal to 0.5 when the input information phase is not . Figure 7 and Fig. 8 demonstrate that the QD and VM can work well and fulfil the quadrature de-multiplexing and vector moving processing.

Fig. 8. VM transfer characteristics.

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3.2 Conversion performances analysis

Then the noisy QPSK signal are launched into the node system, the constellations and eye diagrams of the signals at the points in Fig. 6 are shown in Fig. 9 and Fig. 10. The constellations and eye diagrams are depicted when the input OSNR of QPSK is 25 dB. The constellations of QD process are shown in Fig. 9(a), the input QPSK signal is de-multiplexed into BPSK-I and BPSK-Q, which carries the I and Q components information of QPSK separately. From their eye diagrams of Fig. 10(a), 10(c) and 10(d), it can be observed that the input typical QPSK is converted into two normal BPSKs. Figure 9(b) and 9(c) present the VM process that two BPSKs are moved to be two OOKs and the eye diagrams are shown in Fig. 10(e) and 10(f). We set 3dB attenuation to OOK-Q and generate a uniform PAM4 (PAM4-111) signal whose constellation and eye diagram are Fig. 9(d) and Fig. 10(b). In Fig. 9(d), the amplitude states of PAM4 are: 0, ${A_m}$, $\sqrt 2 {A_m}$and $\sqrt 3 {A_m}$ which means its power ratio of the eyes in eye diagram is 1:1:1. Here the ${A_m}$ is defined as a reference quantity. Moreover, the power efficiency of the QD and VM is also estimated which is defined as the power ratio of the output and input signals. When the node is injected into a QPSK with input OSNR of 25dB, and outputs a PAM4-111 signal, the power efficiency of QD and VM are 3.2% and 3.8%, respectively. The low power efficiency is mainly caused by the low nonlinear efficiency, optical filter, attenuator and coupler.

Fig. 9. Constellations of the conversion process.

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Fig. 10. Eye diagrams of the conversion process.

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The quantitative analysis of QD and VM process are shown in Fig. 11 for the QPSK with input OSNR range of (10 dB$\sim$30 dB). From the EVM versus input OSNR curves, the BPSK-I and BPSK-Q have similar EVM performances, so as OOK-I and OOK-Q. When the input OSNR lower than 22 dB, the EVMs of BPSKs are lower than QPSK which is caused by the PSA phase squeezing functionality. If the QPSK has high signal quality, the effect of PSA phase squeezing is not evident. Comparing the EVMs of BPSKs and OOKs, the curves indicate that the VM process generates extra noise within most input OSNR range. Moreover, the cross point of EVM curves of QPSK and OOKs presents at the input OSNR of 15 dB. IF represents the influence of signal processing on signal distribution in amplitude and phase fields and IF=0 dB is a critical point for the signal characteristics variation. Negative IFs means that the system deteriorates the signal standard deviation. Figure 11(b) shows the phase and amplitude IFs of QD, we can see that the QD narrows the signal phase standard deviation for phase squeezing functionality. While the signal amplitude distribution gets worse because of the phase-sensitive amplification. In Fig. 11(c), the VM amplitude and phase IF curves both varies within a small range which is smaller than 1 dB. The VM phase IFs are all negative which means the VM deteriorates the signal phase distribution around the input OSNR range. For amplitude distribution, VM slightly improves its standard deviation within the input OSNR range of (10 dB$\sim$18 dB), and deteriorates the standard deviation within the rest input OSNR range.

Fig. 11. EVMs, IFs of the QD and VM.

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Except for PAM4-111, other four kinds of PAM4s are generated by the node. Their constellations and eye diagrams are shown in Fig. 12 and Fig. 13. The PAM4s amplitude states are also marked in the constellations, the power ratio can be calculated as: PAM4-121, PAM4-131, PAM4-212 and PAM4-313 which can also be observed from the eye diagrams. The EVMs, IFs and BERs of five kinds of PAM4s and input QPSK are estimated and made comparisons to indicate the conversion node performance and signals quality.

Fig. 12. Constellations of non-uniform PAM4s.

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Fig. 13. Eye diagrams of non-uniform PAM4s.

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As shown in Fig. 14(a), PAM4s EVM performances are similar with OOKs, the PAM4s except PAM4-131 present lower EVMs comparing with the input QPSK when input OSNR is lower than 17 dB while performs higher EVMs when input OSNR is higher than 17 dB. The PAM4-131 performs higher EVMs than other PAM4s for the cross point with QPSK EVM curve presents at input OSNR of 14 dB. The node system IFs here are calculated with the standard deviation of converted PAM4s and input QPSK. From Fig. 14(b), the 0 dB point of the phase IF curves present at 15 dB (PAM4-131), 17 dB (PAM4-121, PAM4-212, PAM4-313) and 19 dB (PAM4-111). The PAM4-111 performs best phase IF performance, and PAM4-131 phase IF has a clear gap comparing with other PAM4s. The system amplitude IF curves are depicted in Fig. 14(c), it can be observed that the five PAM4s have similar IF performances, the 0 dB point of PAM4-111 curve presents at 17 dB, PAM4-131 presents at 15 dB and other PAM4s are between them. Taken together, PAM4-111 shows the best performances, the PAM4-131 shows the worst and other PAM4s are between them. There is a clear gap between the PAM4-131 and other four kinds of PAM4s performances.

Fig. 14. EVMs, IFs of QPSK and PAM4s.

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BER is a key indicator for evaluating the system and signal performances. In this paper, the BER versus receiver OSNR curves of converted PAM4s are depicted in Fig. 15. The BER curves are measured when the QPSK input OSNR is 15 dB, 20 dB and 25 dB. One QPSK BER curve are also estimated for the QPSK with the same noise to be a reference. From Fig. 15, with the BER of ${10^{ - 3}}$, PAM4-121 and PAM4-212 shows nearly the same receiver OSNR. The PAM4 curves can be divided into four groups: PAM4-111, PAM4-121 & PAM4-212, PAM4-313 and PAM4-131. Under the input OSNR of 15 dB and BER of ${10^{ - 3}}$, the receiver OSNR of PAM4-111, PAM4-121 & PAM4-212, PAM4-313 and PAM4-131 are 16.6 dB, 17.4 dB, 20.5 dB and 21.6 dB. Under the input OSNR of 20 dB and BER of ${10^{ - 3}}$, the receiver OSNR of PAM4-111, PAM4-121 & PAM4-212, PAM4-313 and PAM4-131 are 16.2 dB, 17 dB, 18 dB and 21 dB. Under the input OSNR of 25 dB and BER of ${10^{ - 3}}$, the receiver OSNR of PAM4-111, PAM4-121 & PAM4-212, PAM4-313 and PAM4-131 are 16 dB, 16.6 dB, 17.7 dB and 19.6 dB. The PAM4-313 and PAM4-131 have a clear gap in BER performances with other PAM4s when the QPSK input OSNR is 15 dB. With the improvement of input OSNR, the differences of PAM4s receiver OSNR gradually diminishes, especially the PAM4-313 signal. It can be observed from Fig. 15 that, with the BER of ${10^{ - 3}}$, the receiver OSNR of PAM4s have the rule as: $OSN{R_{111}}\;<\;OSN{R_{121\&212}}\;<\;OSN{R_{313}}\;<\;OSN{R_{131}}$, $OSN{R_x}$ is the receiver OSNR of PAM4-x. Totally, the successful receiving of converted PAM4s means that the node format conversion functionality works well and the flexible shaping for PAM4 is feasible.

Fig. 15. (a), (b) and (c) are the BER versus receiver OSNR curves with input OSNR of 15 dB, 20 dB and 25dB.

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4. Conclusion

In this paper, a PSA-based optical conversion node scheme for direct detection of complex modulation format is proposed and verified by simulations. A 10G Baud noisy QPSK signal is converted into a 10G Baud PAM4 signal which can be directly detected by a PD. In the PAM4 generating process, its power ratio and constellation shape can be editable flexibly according to different application scenarios. Five kinds of power ratio PAM4 signals are generated in this paper to verify the shaping functionality. The node is constructed by three parts: degenerate PSA-based QD, non-degenerate PSA-based bi-directional VM, and flexible vector coupling part. The principle and transfer characteristics of every part are introduced by equation derivations and quantitative analysis. The constellations, eye diagrams, EVMs, phase and amplitude IFs of the signals went through every part are measured to indicate the system key performances. The BER performances of converted PAM4s are also estimated. For the QPSK with input OSNR of 15 dB, 20 dB and 25 dB, with the BER of ${10^{ - 3}}$, the receiver OSNR of converted PAM4-111 is 16.6 dB, 16.2 dB and 16 dB. The format conversion and constellation shaping functionalities have great potential applications in bridging long-haul transmissions and short-reach interconnects, advanced format signals generations and shaping.

Funding

National Natural Science Foundation of China (61831003); National Basic Research Program of China (973 Program) (2018YFB1800802).

Disclosures

The authors declare that they have no conflicts of interest.

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QPSK $φ_{m n}$	BPSK-I $φ_{m n}^{I}$	BPSK-Q $φ_{m n}^{Q}$	OOK-I logic	OOK-Q logic	PAM4 level
$π / 4$	0	0	1	1	$A_{4}$
$3 π / 4$	$π$	0	0	1	$A_{2}$
$5 π / 4$	$π$	$π$	0	0	$A_{1}$
$7 π / 4$	0	$π$	1	0	$A_{3}$

$Δ φ$ in Q Branch	Attenuation in Q Branch	Power Ratio of PAM4
90 degree	3dB	1-1-1
90 degree	4.77dB	1-2-1
90 degree	6dB	1-3-1
90 degree	1.77dB	2-1-2
90 degree	1.25dB	3-1-3

QPSK $φ_{m n}$	BPSK-I $φ_{m n}^{I}$	BPSK-Q $φ_{m n}^{Q}$	OOK-I logic	OOK-Q logic	PAM4 level
$π / 4$	0	0	1	1	$A_{4}$
$3 π / 4$	$π$	0	0	1	$A_{2}$
$5 π / 4$	$π$	$π$	0	0	$A_{1}$
$7 π / 4$	0	$π$	1	0	$A_{3}$

$Δ φ$ in Q Branch	Attenuation in Q Branch	Power Ratio of PAM4
90 degree	3dB	1-1-1
90 degree	4.77dB	1-2-1
90 degree	6dB	1-3-1
90 degree	1.77dB	2-1-2
90 degree	1.25dB	3-1-3

Phase-sensitive amplifier-based optical conversion for direct detection of complex modulation format to bridge long-haul transmissions and short-reach interconnects

Abstract

1. Introduction

2. Operating principle

2.1 Quadrature de-multiplexer

2.2 Vector mover

2.3 Phase-locked pumps generation

2.4 Flexible coupling

3. Simulations & discussions

3.1 QD & VM transfer characteristics

3.2 Conversion performances analysis

4. Conclusion

Funding

Disclosures

References

Cited By

Figures (15)

Tables (2)

Equations (28)

Optics Express