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We discuss an all-optical DPSK wavelength conversion scheme comprising a delay-interferometer demodulation stage followed by a Mach-Zehnder interferometer, the arms of which are formed by nonlinear waveguides. If operated properly, the configuration shows regenerative behaviour. This is true for nonlinear waveguides with a dominant cross-gain nonlinearity (e. g., for an electro-absorption amplitude modulator) as well as for the case of a dominant cross-phase nonlinearity (e. g., for Kerr effect based phase modulator). In addition, we show that nonlinear materials exhibiting cross-gain modulation properties can provide a binary phase response so far only known from the transfer functions of digital electronics.
©2009 Optical Society of America
1 Introduction
All-optical signal regenerators and wavelength converters for differential phase-shift keying (DPSK) are of high interest in point-to-point and next generation meshed transparent networks. At low cost they allow larger link lengths by regenerating degraded signals, or simply by avoiding wavelength blocking in cross-connects and routers [1]. All-optical DPSK signal regeneration has been shown for bulk semiconductor optical amplifiers (SOA) in both Sagnac [2] and Mach-Zehnder interferometer (MZI) configurations [3] [4]. In these experiments, the dominant mechanism was cross-phase modulation (XPM). Highly nonlinear fibers (HNLF) have also been used in an interferometric configuration to regenerate DPSK signals [5] [6].In the future new highly nonlinear silicon waveguides could provide the necessary nonlinearty [7]. Important for the success of all these experiments was the use of nonlinear materials with a dominant XPM nonlinearity in an interferometric configuration. On the other hand, four-wave mixing (FWM) in HNLF can be implemented to reduce amplitude and phase noise [8]-[11]. It has been also noticed in [8] [9] that amplitude-only regeneration can effectively reduce the nonlinear phase noise (Gordon-Mollenauer phase noise). In most recent publications [11] [12], a promising method for phase-regenerative wavelength conversion was demonstrated by using phase-sensitive amplifiers.
However, many nonlinear materials exhibit only weak XPM or FWM, but show a dominant cross-gain modulation (XGM) nonlinearity. And in fact, XGM properties could be more favourable due to easier fabrication and better availability. Such XGM materials are typically used as amplitude modulators. Good examples are electro-absorption modulators [13], which exhibit a negligibly small phase-amplitude coupling or αH-factor (Henry’s αH-factor [14]). A small αH -factor is indicative for a small refractive index change and thus for a small XPM effect with respect to a given change of the charge carrier concentration. Wavelength converter based on saturated EAM has been demonstrated with regenerative properties in [15]. Another example for nonlinear amplitude modulators are quantum dot (QD)-SOA, which have attracted considerable attention due to their ultra fast carrier recovery dynamics [16]-[18]. QD-SOAs are also known to have a low αH-factor [19]- [21], if operated in the proper regime. In fact, by choosing the operating point carefully, the αH-factor in QD-SOA is close to zero when the bias is set just above [19] or well above the transparency point [20] [21]. Recent experimental studies of QD-SOA [22] [23] show that QD-SOA in a high carrier-injection regime are dominated by XGM effect.
In this work we show that all-optical DPSK wavelength converters with regenerative properties can be designed by relying solely on nonlinear elements (NLE) with XGM or XPM characteristics and any combination of the two. The case with XGM nonlinear elements is new and therefore deserves special attention. We compare the regenerative strength of conventional XPM schemes with the newly introduced XGM arrangement.
This paper is organized as follows. In Section 2 and 3 we first describe the interferometric DPSK wavelength converter (DPSK-WC) configuration and specify the associated transfer matrix. In Section 4 the usability of nonlinear materials with both large and vanishing αH-factor is discussed. The regenerative potential for the various materials is analyzed in Section 5. In greater detail we report on the regenerative effect on both phase and amplitude for an output signal behind a regenerative DPSK wavelength converter. The following observations will be discussed:
- • For an ideal DPSK-WC exploiting XPM nonlinearities only, input phase noise may be completely suppressed, while input amplitude noise is reduced due to the sinusoidal amplitude transmission behavior in MZI configurations. This is true as long as the amplitude fluctuations are not too large. For large amplitude fluctuations the phase noise can no longer be completely suppressed.
- • For an ideal DPSK-WC exploiting XGM nonlinearities only, perfect phase regeneration is achieved for any phase and amplitude noise of the input signal, because there are no XPM effects. If the XGM nonlinear elements are operated in a strong saturation regime, also the output amplitude noise can be reduced. Due to the inherently ideal phase regeneration after going through one DPSK-WC stage, amplitude regeneration can be even further improved by cascading another DPSK-WC stage employing XGM nonlinearities.
2 Configuration and operation principle
2.1. Configuration of the interferometric DPSK wavelength converter
The general DPSK wavelength conversion system is shown in Fig. 1 . This setup comprises a DPSK transmitter, followed by the interferometric DPSK wavelength converter and a balanced receiver with differential encoding. The wavelength converter consists of a delay interferometer (DI) stage and a MZI stage with nonlinear elements in the two arms, and an optical source with a target wavelength λ cnv to which the DPSK signal at λ in will be converted.
Fig. 1 DPSK wavelength converter schematic with nonlinear elements (NLE) in the arms of a Mach-Zehnder interferometer (MZI). Symbols () represents 2x2 couplers of the interferometers. An DPSK signal with electric field E in at a wavelength λ in is demodulated by a delay interferometer (DI, time delay difference Δt equals bit period T b) resulting in an OOK signal E up and an inverse OOK signal E low,. An electric field E cnv at the new “converted” wavelength λ cnv passes both NLE resulting in the fields E cnv,up and jE cnv,low in the upper and lower MZI arms. At the difference output port ∆, a filter selects the “converted” wavelength λ cnv resulting in a signal E Δ. A schematic representation of E Δ is sketched along with E cnv,up and E cnv,low for the cases (a) α Η ≠ 0 (mostly XPM, and XGM) and (b) α Η = 0 (XGM only). The optical output signal at port ∆ has been converted to PSK format, and the balanced receiver Rx requires differential encoding for recovering the original data.
Interferometric configurations usually work well with materials, which offer a high nonlinear change of the real refractive index (phase modulators). All-optical interferometric DPSK-WC that exploit XPM have already been demonstrated in [3] and [4].
Yet, it is more difficult to see that the very same configuration can perform efficient signal regeneration also with XGM nonlinear elements (amplitude modulators). As pointed out earlier, such nonlinear elements (NLE) are for instance QD-SOAs and electro-absorption modulators (EAMs). It is the goal of this work to quantitatively predict the signal regeneration performance of DPSK-WC schemes with both types of NLE.
2.2. Classification of nonlinear elements
The nonlinear elements under consideration are described by a complex refractive index , which has real and imaginary parts nr and ni, respectively. We first assume that, for a sufficiently narrow spectral range of interest around the carrier wavelength λ cnv (frequency f cnv), the complex refractive index at a position z within a NLE is constant and represented by
For our choice of harmonic time dependence exp(j 2π f t) and for propagation into positive z-direction, gain is described by ni < 0.The complex refractive index Eq. (1) depends on the total optical intensity, so that it changes nonlinearly in the presence of an input signal at wavelength λ in. The refractive index change is a consequence of, e.g., a gain saturation due to carrier depletion in an active medium, or of an (usually very small) absorption change due to third-order nonlinear interactions in a Kerr-type medium. We define Δnr(t) and Δni(t) to be the respective perturbations of the real and imaginary refractive indices from their unperturbed values given in Eq. (1). Assuming that Δnr(t) and Δni(t) are slowly varying functions of time with respect to the optical period 1/ f cnv [24] [25], we find the time-dependent complex refractive index and its real and imaginary parts nr(t) and ni(t) at wavelength λ cnv,
We further assume that Δnr(t) and Δni(t) have the same time dependence. In this case Δnr(t) and Δni(t) can be related by a constant α H-factorConcretely, in the case of an SOA for instance, an α H, SOA-factor is defined for differential changes of nr and ni with respect to a differential change in the free carrier number N [14],For a Kerr-type medium, a corresponding α H, Kerr-factor will be introduced in Section 3.1.The complex refractive index is relevant for describing signal propagation through a NLE. The signal propagation is given by an amplitude transmission T NLE relating the output and input signals of the NLE between the points z = L and z = 0, respectively. As the signal propagates and its power varies along the propagation direction z, the power dependence of the refractive index leads to a position dependent refractive index . The amplitude transmission with modulus |T NLE| and phase φ becomes
where k 0 = 2π f cnv / c 0 is the wave number and c 0 is the vacuum speed of light. The variable t is the time in a retarded time frame. A power gain requires |T NLE|2 >1.It is customary to define a parameter h(t), termed as the logarithmic power transmission,
The logarithmic power transmission h(t) is related to the single pass gain G(t) = exp[h(t)]. If a weak input signal at f cnv with cw power P cnv leads to a logarithmic transmission h cnv and a phase shift φ cnv, then a second strong signal launched into the NLE causes both a nonlinear gain change (XGM) and a nonlinear phase shift (XPM) for the cw signal. For this reason, the second strong signal is called “control signal”. The logarithmic power transmission change Δh(t) and the nonlinear phase shift Δφ (t) are
If we assume an initial phase shift φ cnv = 0, Eq. (5) can be rewritten with the help of Eq. (7),Note that Eq. (8) is an implicit expression for the intensity-dependent nonlinear amplitude transmission through the NLE.Instead of solving the amplitude transmission Eq. (8) iteratively, we first regard the NLE as a lumped device and regard nr (t,z), ni (t,z) and Δnr (t,z), Δni (t,z) in the integrals Eq. (6) and (7) as spatially constant. We further assume that Δnr(t) and Δni(t) have identical time dependency as is implied in Eq. (3). Yet, we extend this assumption to any position inside of the NLE. This allows us to eliminate the phase relation Δφ (t) by means of a constant α H-factor,
Thus we can rewrite the amplitude transmission function Eq. (8) for a NLE,The nonlinear phase change is thus related to a gain change by means of the α H-factor.With these definitions, we now classify nonlinear elements by the magnitude of the αH-factor. A NLE with a large α H mostly produces a phase modulation; in our case we are interested in XPM. In the extreme case where αH → ∞, the device can be understood as an ideal phase modulator. A NLE with a small or close to zero αH mostly causes a gain modulation, and here our interest is in XGM. If αH = 0, the NLE acts as an ideal gain modulator. Devices with dominant XGM effect are, e. g., QD-SOA or EAMs. A medium with a NLE characterized by 0 < α H < ∞ (e. g., a bulk SOA) will experience both XGM and XPM.
2.3. Operation principle of interferometric DPSK wavelength converter with various NLE
The operation principle of the XPM or XGM-based DPSK-WC is visualized in Fig. 1. An incoming DPSK formatted signal E in at λ in is transformed by the DI stage into an on-off keying (OOK) and an inverted OOK data stream. These on-off signals change the refractive index and the gain of the NLE in the two MZI arms. The two NLE in the MZI arms are assumed to have same gain and α H-factors. The amplitude transmission functions of the MZI arms are then approximated by Eq. (10).
A clock or a cw signal E cnv at a wavelength λ cnv is launched in the MZI and distributed to the two arms of the MZI. The two signals in the respective arms are denoted by E cnv,up and jE cnv,low. Note that a phase factor j will be added to the signal that couples into the cross output of the coupler. The relative phase and amplitude between E cnv,up and jE cnv,low is then controlled by the mark and space bits of the OOK and the inverse OOK via the NLE, similar to the push-pull operation of electrically controlled MZI modulators used in DPSK transmitters [1]. The modulated E cnv,up and jE cnv,low signals are recombined at the destructive (difference) output port Δ of the MZI. A filter centred at λ cnv selects the wavelength-converted signal E Δ at λ cnv.
We had already classified the NLE in terms of the α H-factor at end of Section 2.2. Figures 1(a) and 1(b) show schematically how the two amplitudes E cnv,up and E cnv,low in the MZI upper and lower arm evolve with an OOK input signal launched to the respective MZI arms for the two cases with α H = 0 and α H >> 0.
For α H = 0 we first assume that a logical “1” corresponding to a DPSK state “−1” induces a very strong gain suppression without any nonlinearly induced phase shift in the upper arm at λ in And there is no gain suppression in the lower arm. The filtered difference output signal at λ cnv is then E Δ = E cnv,up – E cnv,low < 0. If the gain is reduced in the lower arm by an inverted logical “0”, corresponding to a DPSK state “1”, we have the corresponding relation E Δ = E cnv,up – E cnv,low > 0. Thus, each logical “0” results in a signal with an intensity |E Δ|2 and an optical phase 0, while each logical “1” generates a signal with the same intensity |E Δ|2 and an optical phase π. This is illustrated in Fig. 1(b) for a signal inducing an intermediate gain reduction. This situation would hold true for, e. g., a QD-SOA or for an EAM.
For 0 < α H < ∞ (e. g., for a bulk SOA), both XPM and XGM are effective. The amplitude of the converted signal in the respective arm is both suppressed due to XGM, but also its phase is flipped due to XPM − if the input OOK signal is sufficiently strong to induce a π phase shift. This situation is illustrated in Fig. 1(a). The newly generated signal at the output is actually a PSK signal, where E Δ > 0 is for a PSK state “1” and E Δ < 0 is for a PSK state “−1”.
We therefore find, that the generation of the PSK signal does not depend on the αH -factor of the NLE, be it αH = 0 or 0 < αH < ∞. While the outcome is same, the physics behind the two NLE media is quite different and thus needs a more detailed discussion. For recovering the original signal from the PSK signal, an electrical encoder needs to be added [4]. There must be an identical number of delay interferometer stages and differential encoders in the transmission link. The two elements, however, can be arranged in any order.
3 Modeling
Our model focuses on describing the quasi-static amplitude transfer characteristics of generic nonlinear elements within the interferometric DPSK wavelength conversion configuration rather than on the specific temporal dynamic effects in the device. This means that we treat nonlinear effects as being “instantaneous”. As a consequence, we limit ourselves to bitrates such that the time constants of the respective nonlinear effects are negligible on the scale of the bit period. For larger bitrates, pattern dependence will degrade the NLE performance. While successful experiments with normal SOA are demonstrated at 40Gbit/s in [3] [4], Ultrafast response of the gain nonlinearity in QD-SOA [16]- [18] may enable operations at even higher bit rate.
To work out the specific differences between parametric or Kerr-type and non-parametric media (such as SOAs) it will be useful to have one formalism that describes the amplitude and phase characteristics of all NLE. It will be argued, that the two parameters P eq (the equivalent saturation power) and the αH-factor are sufficient to describe the media within first order. We will then use these two parameters to derive the DPSK wavelength converter transfer function for arbitrary NLE.
For clarity’s sake we further neglect noise generated by the NLE. Especially when using SOA, this noise would degrade the optical signal-to-noise ratio (OSNR). However, as we will show in the beginning Section 5.2.2, in view of other impairments this degradation is not significantly different for SOA wavelength converters of either XGM or XPM type.
3.1. Modeling of gain and phase effects in various nonlinear elements
Equation (10) provides the amplitude transmission function T NLE for a generic NLE. For determining T NLE in Eq. (10) we need finding Δh for the respective media. We therefore now discuss the logarithmic gain change for various important media.
(A) Gain and phase change in an SOA. A signal at wavelength λ cnv (frequency f cnv, converting signal in Fig. 1), is launched into the SOA and experiences a gain G cnv along the SOA. In addition, a strong input control signal, e.g. one of the OOK signals in Fig. 1, enters the SOA and leads to a gain suppression. We assume that the powers of the converting signal E cnv (power P cnv) and the power of the control signals E up or E low, (power Ps(t)) in the NLE simply add, i.e. we assume that the respective coherence times and the frequency difference of converting signal and control signals do not create additional beating terms. Thus the logarithmic power transmission of the cw signal h cnv and h(t) for the cases with and without control power Ps(t) can be described by [24],
The quantity h 0 is the unsaturated logarithmic power transmission and h 0 = g 0 L is given by the unsaturated material gain g 0 of, e. g., an SOA. For the sake of convenience, P sat is defined as the input saturation power. If a cw power P cnv = P sat ln(2)/(h 0 −ln(2)) is applied, the power gain G cnv = exp(h cnv) decreases by 3 dB with respect to G 0 = exp(h 0). From Eq. (11), the logarithmic power transmission change ∆h takes the form,From Eq. (9) and (12), the phase change is(B) Modeling of loss and phase change in a Kerr medium. In such a medium, the converting signal at f cnv experiences a power loss, i. e., a (possibly very weak) absorption. This linear power loss is usually described by a linear absorption coefficient α 0 [25]. The unsaturated logarithmic power transmission h 0 is given as h 0 = −α 0 L. If a sufficiently strong control signal, e.g. one of the OOK signals in Fig. 1, enters the Kerr-nonlinear element, the absorption changes with the input power due to coherent third-order nonlinear interaction. This interaction is described by a nonlinear power absorption coefficient α 2, also denoted as two photon absorption (TPA) coefficient [25]. The logarithmic power transmission of the converting signal, h cnv and h(t) for the cases with and without control signal Ps(t), can be written asThe factor 2 in front of P s(t) stems from collecting various third-order nonlinear interaction terms, which belong to the same frequency [25].To find a formulation similar to Eq. (12), we introduce an effective power P eff,
where A eff is the effective area for third-order nonlinear interaction. Equation (14) then becomesThe logarithmic power transmission change ∆h is now given byThe effective power P eff in a Kerr medium is interpreted as follows. If P cnv = P eff ln(2), then the absorption is increased by 3 dB, i. e., the effective power “gain” is decreased by 3 dB. The effective power is very large as a rule, because the nonlinear power absorption coefficient α 2 is usually very small.In the presence of a control signal P s, the nonlinear refractive index change Δnr = 2n 2 I is proportional to the intensity I via the nonlinear index coefficient n 2 [25]. The factor 2 again comes from the collection of third-order nonlinear interaction terms for XPM. Note that the refractive index change Δnr also varies with higher order nonlinear terms [25], which have been neglected. The phase shift due to XPM is now given as
Since α 2 is usually small in Kerr media like quartz glass, the αH, Kerr-factor is then very large. As an example, an As-Se chalcogenide HNLF was reported to have αH, Kerr ≈29 [26],.(C) Generalized saturation power andαH-factor. With respect to the gain suppression, P eff is equivalent to the saturation power P sat in an SOA. For convenience, we subsequently define an equivalent saturation power P eq by
Also we introduce the generalized αH-factor for an SOA-type and Kerr-type medium as3.2. Transmission function of a DPSK wavelength converter configuration
A transmission function formalism is used to describe the amplitude transmission behaviour of the DPSK-WC configuration. The overall transmission function needs to describe both the delay interferometer as well as the MZI with the nonlinear elements on the arms. We will discuss the two transmission matrices below.
The following notation will be used. The input DSPK signal centred at frequency f in is expressed by E in(t)exp(j 2π f in t). The quantity E in(t) = A in(t)exp[jΦin(t)] is the complex envelope, where A in(t) = |E in(t)| is the amplitude and Φin(t) represents the phase. The power of the signal is thus given by,
where ε 0 is the permittivity of free space and n 0 is the refractive index of the medium at low power. A eff in the amplifier is the mode cross-section with respect to the mode confinement in the waveguide [24]. A eff in the fibre is the effective area for third-order nonlinear interaction [25]. The input is a DPSK signal, whose phase difference between two consecutive bits carries the information. This phase difference is then expressed asFor a logical “0” we have ΔΦin = 0, and ΔΦin = ± π for a logical “1”.(A) Impulse response matrix for a delay interferometer. Following [27] we write the generalized impulse response of the delay-interferometer (DI) for the output fields E up, E low and the input field E in as assigned in Fig. 1 as
The first and third matrices under the integral of Eq. (23) describe the symmetric input and output directional couplers, where a phase factor j modifies a signal that couples into the cross output. The Dirac functions in the middle matrix are the impulse responses of the “long” (upper) and the “short” (lower) arm with the group delay difference Δt of the DI in Fig. 1. A multiplicative phase factor exp(j 2π f in τ) modifies the amplitude transmission in the “long” arm taking into account the phase shift of the optical carrier. The time delay Δt of the “long” arm is adjusted such that Δt approximates the bit period T b, and f in Δt is an integer number.Solutions of Eq. (23) with Δt = T b lead us to the output signals behind the DI,
At the outputs of the two 2x2 MZI couplers in front of NLE1,2 (coupler symbol “” in Fig. 1), the two signals Eq. (24) are superimposed to the converting signal E cnv (as specified in Fig. 1) and guided to the corresponding NLE in the upper and lower arms of the MZI.(B) Amplitude transmission function of the MZI with a generic NLE. The amplitude transmission function T(t) below relates the output signal E Δ of the destructive output port Δ of the MZI with the input signal E cnv, see Fig. 1. The transmission function of the MZI comprises the NLE defined in Eq. (10) on its arms. Depending on the NLE media another logarithmic power transmission h cnv and logarithmic power transmission change ∆h up/low will be needed, such as given by Eqs. (11), (12), (16) and (17). The αH–factor in Eq. (10) is defined by Eq. (9). This leads us to the general MZI amplitude transmission function T(t),
In this transmission function γ is the coupler link loss.In the next section we discuss the amplitude transmission function Eq. (25) and show to what extent regeneration is to be expected.
4 Regenerative properties for various nonlinear elements
The extent of the regenerative behaviour depends on the characteristics of the MZI and of the nonlinear elements. Not only the strength of the nonlinearity is of high interest, but also the question whether the changes of the real or the imaginary part of the complex refractive index dominate. In Kerr-effect elements, for instance in a quartz glass fibre and in the presence of a strong control signal, the change of real part of the refractive index dominates over the absorption change. On the other hand, in EAM and QD-SOA devices, absorption dominates over a change of the real part of the refractive index. In bulk SOAs, both the real and imaginary parts of the complex refractive index contribute significantly.
We now discuss the power response as well as the phase response of DPSK wavelength converters for various nonlinear elements, Fig. 2 .
Fig. 2 Power and phase responses of DPSK wavelength converters as a function of input DPSK signal power (left column, (I)) and phase difference ∆Φ (right column, (II)) between consecutive DPSK bits at the input. The power of the input DPSK signal is normalized to a reference input power P in,ref, which is explained in the paragraph before Eq. (26), while the power response is normalized to a value γ2exp(h cnv)/4. The phase difference ∆Φin is specified in Eq. (22). The MZI in the DPSK wavelength converter comprises (A) ideal amplitude modulators, (B) modulators showing amplitude and phase modulation (here an ordinary bulk SOA with αH = 8 and 0 < P eq < ∞), and (C) ideal phase modulators. The labels “1” and “−1” denote the respective ideal DPSK states. The corresponding optimum operating points are marked with short arrows pointing to filled circles (●), whose input powers are used to generate the plots in the right column (II).
The upper, middle and lower rows of Fig. 2 show the power and phase responses for a MZI formed (A) by pure amplitude modulators (αH = 0), (B) by modulators showing amplitude and phase modulation, and (C) by ideal phase modulators (αH → ∞), respectively. The left column of Fig. 2, marked with (I), shows the power and phase responses with increasing input DPSK signal power. The right column of Fig. 2, marked with (II), displays the power and phase response if the relative phase difference ∆Φ between two consecutive DPSK bits at the input increases from –π to π.
An important quantity influencing the power and phase responses is the equivalent saturation power P eq, defined in Eq. (19). For generating the plots in Fig. 2, we used a moderate non-zero equivalent saturation power in case (A) and (B), which is 20 dB below the equivalent saturation power P eq used in case (C).
In order to facilitate comparison between different concepts, we normalize all input powers P in (horizontal axes of the subfigures in the left column of Fig. 2) to a common reference power level P in,ref. This reference power level P in,ref is the power required to induce a π-phase shift onto an input signal P cnv = 0 dBm launched in SOAs with αH = 8, with P eq = 10 dBm (i.e. 0 < P eq < ∞) and G 0 = 30 dB, see Fig. 2(B). The resulting output powers in Fig. 2 have also been normalized with respect to the maximum output power γ2exp(h cnv)/4 of P cnv. All plots in Fig. 2(II) have been generated with the input powers at the respective optimum operating points, marked with short arrows pointing to filled circles (●) in Fig. 2(I).
The specific choice of αH = 8 for the SOAs is based on modeling and experimental results [28]. However, in contrast to this reference, in this work the time-dependence of the αH-factor has been dropped, as we assume (see beginning of Section 3) that the time constants of the respective nonlinear effects are negligible on the scale of the bit period under consideration.
We now discuss the three cases A to C in Fig. 2.
Figure 2(A): NLE with pure amplitude modulation (αH = 0). With αH = 0 the amplitude transmission function T(t) in Eq. (25) simplifies to
The conversion efficiency is best for highest input powers. The phase of the converted signal is not changed when increasing the power of input DPSK signal, since αH is virtually zero.The plot in Fig. 2(A,II) shows how the output signal will undergo a transition from amplitude “−1” to amplitude “+1” and again to amplitude “−1”, if the relative phase difference ΔΦ between two consecutive DPSK input signals of power P in ≈6P in,ref varies from −π to 0 and from 0 to + π. In fact, the converted signal at the output will have a phase with either 0 or π. The plot also shows the output powers if the input signals phases varies between −π and π.
Figure 2(A,I) shows the phases (green and blue) and output powers (black) of the converted signals, for consecutive input signals when the input power varies but has an ideal 0 or π phase difference between consecutive signals (indicted by “1” and “−1”). We first notice, that the output power converges towards 0.5 the stronger the input signals. This indicates amplitude regenerative behaviour if operated in strong saturation; a typical operating point is marked with a filled circle (●). This regenerative behaviour is similar to that of a limiting amplifier [17]. Also, the two flat phase responses in Fig. 2(A,I) and (A,II) indicate an almost ideal phase regenerative behaviour for DPSK signal phases. Yet, while the phases in these wavelength converters are reset, the power is not. For strong phase fluctuations one will introduce large power fluctuations as can be learnt from the power response in Fig. 2(A,II).
Figure 2(B): NLE with both amplitude and phase modulation (e.g. αH ≈8). In this case, the converting signal experiences both XGM and XPM in the NLE, and the transmission function from Eq. (25) provides the power and phase responses. This is a typical situation encountered with normal bulk SOAs. The transmission functions of Fig. 2(B,I) and (B,II) show that these wavelength converters with this type of NLE potentially could offer power and phase regeneration. The power and phase responses in Fig. 2(B,I) show oscillations with increasing signal power. These oscillations will appear if the nonlinear phase shift exceeds several π. Furthermore, one observes a damping of the transmission peaks of the converted signal. This is due to saturation effects in the nonlinear elements.
Figure 2(C): Pure phase modulation (αH → ∞). In this case, the quantity effective power P eq derived in Eq. (15) is large (P eq→ ∞). NLE such as an organic material [12] or HNLF indeed hardly show any XGM. The converting signal experiences dominantly XPM. An ideal MZI comprising ideal phase modulators can also perform DPSK wavelength conversion. This kind of DPSK signal regenerator based on phase-sensitive amplifiers has been recently discussed in [5] [6].
With pure phase modulators, the transmission function in Eq. (25) simplifies to
The power and phase responses of this type of DPSK wavelength converter are shown in Fig. 2(C,I) and (C,II). For generating the plots in Fig. 2(C), we used an effective power P eff which is 20 dB above that used in case (B), and we assumed an αH-factor αH = 500.From the power response in Fig. 2(C,II), one can expect a good amplitude regeneration, since the operating power P in,ref for generating Fig. 2(C,II) is at the optimum operating point. However, the phase of the converted signal linearly depends on the input power, Fig. 2(C,I). In fact, the phase response in Fig. 2(C,II) will vary considerably for different operating points.
5 Noise suppression properties with various nonlinear elements
5.1. Constellation maps
In this section we discuss the noise suppression potential of the DPSK wavelength converter for various nonlinear elements. The different NLE have been classified in terms of αH -factor and effective power. Best conversion efficiencies are obtained with schemes employing high αH-factors, as shown in Fig. 2(C). Yet, schemes with a high αH-factor are also very prone to power fluctuations. In fact, the transmission function for a small αH-factor becomes flatter, resulting in a more favourable regenerative behaviour around the two DPSK states. Unfortunately, this regenerative behaviour most often comes at the price of higher input powers that are required to reach the optimum operation point of low-αH devices.
In order to determine the regenerative effects on signals with amplitude and phase perturbation, we plot constellation diagrams that show how a particular phase-amplitude constellation space (Fig. 3(a) ) is mapped during all-optical wavelength conversion into a new constellation space (Fig. 3(b)).
Fig. 3 Constellation diagrams. (a) Input DPSK signal (b) Output wavelength-converted PSK signals for various αH-factors, corresponding to cases (A), (B) and (C) in Fig. 2. Areas in (a) are mapped onto the corresponding constellation areas in (b) having same shading. The amplitude is normalized to the amplitude value at the operating point.
The shaded areas in Fig. 3(a) indicate signals with perturbations. For a better visualization of the perturbations, we assume that the perturbations have been added to the optimum operating points. The optimum operating points correspond to the symbols marked with filled circles (●) in Fig. 2. We define the phasors of the optimum operating points as E in,0 = A 0 inexp(jΦ0 in), where A 0 in is the amplitude and Φ0 in is the absolute phase of the optimum operating point. Note that the phase Φ0 in is either 0 or π. The perturbation which adds to the phasor E in,0 is defined by δE in = δA exp(jδΦ), where δA is the amplitude perturbation and δΦ is the absolute phase perturbation. Thus, the phasor of the perturbed signal E in is
In generating the shaded areas in Fig. 3(a), we used equally distributed values of the relative amplitude perturbation δA/A 0 in within an interval 0% to 25%, 50% and 75%, respectively, and equally distributed phase perturbation δΦ within [−π, π].The three constellation diagrams in Fig. 3(b) are the constellation spaces after wavelength conversion for the different devices with an αH-factor of 0, 8 and 500. The amplitude of the subfigures in Fig. 3(b) is now normalized to the amplitude value of the output signal at the operating point. For the calculations in Fig. 3(b) we have used the same effective (or saturation) powers as for the corresponding calculations performed in column II of Fig. 2(A, B, C).
Generally, the plots in Fig. 3(b) show that the shaded areas after conversion are lying much closer to the ideal states “1” and “−1”, and thus indicate regenerative performance. While a medium with a high αH-factor leads to better conversion efficiency, as shown in Fig. 2(C), it does not necessarily provide better performance. As seen in the subfigures of Fig. 3(b), both amplitude and phase perturbations are visually more confined near the ideal “1” and “−1” states for the device with αH = 8 compared to the device with αH →∞. The figure on the left of Fig. 3(b) shows an ideal phase regeneration for devices with αH = 0..
5.2. Mathematical description of noise suppression properties
While above discussions based on transmission responses and constellation maps explore the output transmission function for a deviation of the input signal from its ideal form, we now discuss the noise suppression property of the wavelength converters when using different types of NLE. In the following discussion, we assume that input amplitude and phase fluctuations can be described by uncorrelated Gaussian noise. For the amplitude the Gaussian assumption is reasonable because optical line amplifiers are involved, For the phase noise this means the assumption of a Lorentzian laser source with a full-width at half-maximum (FWHM) linewidth ∆f in. The linewidth is related to the variance of the Gaussian-distributed differential phase noise (not the absolute phase noise) by (σ ph in)2 = 2π∆f in Tb [29] [30], where Tb is the width of a bit slot. While a Gaussian may not reflect the true probability function of the input phase of an in-line wavelength converter, it serves well for the purpose of discussing the regeneration performance
The input DPSK signal E in(t) is sampled in the middle of each bit slot. The n-th sample is now indexed by a subscript “n” as
The amplitude consists of an unperturbed term A 0 in and a noise term δAn in. To characterize the amplitude noise, we use a quantity , denoting the relative amplitude error, and a quantity for the standard deviation of this relative amplitude error. The phase difference between two DPSK bits ΔΦn in is also a random variable, which comprises a signal phase term ΔΦn,s in = {0, ± π} according to the logical information {0, 1} and a phase noise term δΦn in,The phase error δΦn in is characterized by its standard deviation σph in.In the following we discuss logical “1” states only. This can be done without loss of generality, and the noise suppression property of the logical “0” state is identical to that of the logical “1” state.
As the wavelength converter has two stages, we first show how the noise is transferred from the input DPSK signal to the OOK signal after the DI, and we then discuss how the noise is redistributed and suppressed by the MZI.
5.2.1 Noise after delay-interferometer stage
The DI stage of the converter maps a DPSK input signal onto an OOK and an inverted OOK signal. During this reformatting operation, amplitude and phase noise of the input DPSK signal are mapped onto amplitude noise in the two respective OOK signals. The two OOK signals are complementary and depend on the logical state of the input DPSK signal. For instance, the OOK signals from the upper and lower output ports of the DI are “1” (presence of a pulse) and “0” (absence of a pulse) for a logical state “1” of the input DPSK signal, and vice versa for a logical state “0”.
The powers in both arms are obtained by substituting the modulus of the DI output amplitude Eq. (24) in Eq. (21). For a logical “1” state, the modulus squared of the amplitudes are
The quantity in Eqs. (31) and (32) is defined as an average of two consecutive relative amplitude errors,We assume uncorrelated Gaussian amplitude deviations δAn +1 in and δAn in, the sum of which has also a Gaussian probability density function (pdf), but with a standard deviation compared to the standard deviation of the relative input amplitude error in Eq. (29).From Eqs. (31) and (32) the power variations δP up and δP low of the OOK signals in front of the NLE in the upper and lower arms of the MZI can also be calculated. They are obtained by neglecting the quadratic terms in Eqs. (31) and (32), and by normalizing the power variation to a power level Ps, 0 = (A 0 in)2 of a noiseless OOK pulse before the NLE,
5.2.2 Noise after MZI stage
The output signal after the MZI is a PSK signal. It is therefore of interest to discuss both amplitude and phase errors at the output. The n-th sample of the output signal is denoted as
where An out and Φn out are the amplitude and phase of the output signal. In the noiseless case, amplitude and phase of the output signal are denoted by A 0 out and Φ0 out. The amplitude and phase errors can be calculated through the transmission function of the MZI in Eq. (25). We write the MZI transmission function between the noiseless input and output signals as T 0 = E 0 out/E 0 in, and that between the n-th input and output signals as Tn = En out/En in, respectively. The relative output amplitude error then becomesSubsequently, we will discuss the cases of ideal amplitude modulators (αH = 0) and of ideal phase modulators (αH →∞) in the MZI arms. These two wavelength converters based on different ideal modulators predict the performance limits of amplitude and phase noise suppression. The problem then reduces to finding the respective T 0 and Tn for these two types of nonlinear elements.As mentioned in the beginning of Section 3, noise generated by the NLE has been neglected. Otherwise, a comparison of the generic properties of NLE like HNLF or SOA would be obscured by their widely different noise properties.
If ASE noise from SOA would have been taken into account, the optical signal-to-noise ratio (OSNR) after the MZI stage would deteriorate. We show in the following that the OSNR due to NLE-noise is about 3 dB worse for XGM converters than for XPM converters:
For an estimation of the influence of the NLE-generated ASE noise, we neglect any input signal noise and regard only the NLE-generated ASE noise. We assume that the NLE are saturated such that in the case of XGM the output signal extinction ratio is (virtually) infinite. The OSNR would be then determined by the ratio of output signal power PS and output noise P N of one single SOA, OSNRXGM = PS / P N. For XPM, and assuming comparable operating conditions, on the other hand, the coherent addition of (nearly) equal fields in both arms leads to a four-fold output signal power 4PS, while the (virtually equal) noise powers of both NLE add up to 2P N. Therefore, the OSNR for XPM-dominated wavelength conversion would be better by a factor of 2, OSNRXPM = 2 OSNRXGM = 2 PS / P N. If the XGM extinction ratio is not infinite but 20 dB, as long as the time constants of the respective nonlinear effects are negligible on the scale of the bit period e.g. shown in [16], the OSNRXPM / OSNRXGM factor would increase from 2 (3dB) to 2.4 (3.9dB).
Now we neglect the ASE noise from SOA and discuss the influence of the amplitude and phase noise on the SNR. For comparison, we assume Gaussian-distributed amplitude and phase noise. Without loss of generality, the SNR can be regarded to be reciprocally proportional to the square of the standard deviation of the relative amplitude errors defined in Eqs. (29) and (36). For instance, an increase of the standard deviation of the relative amplitude error from 0.1 to 0.1414 leads to a SNR decrease of 3dB. In addition the phase noise also leads to a SNR penalty. E.g., as can be calculated from [30], there is a SNR penalty of 3dB for a standard deviation of the phase noise of 0.19.
Thus, in view of other impairments, NLE noise plays only a secondary role when comparing XGM and XPM effects. Nevertheless, the NLE-generated ASE noise in the XGM case may be included as a penalty on the evaluated standard deviation of the output amplitude noise, to be seen in Fig. 4 . But, this penalty only comes into role when the output amplitude noise is not big.
Fig. 4 Noise suppression of the cascaded XGM-based (αH = 0) DPSK wavelength converter. (a) Schematic 2-stage XGM-based (αH = 0) wavelength converters. Note that the wavelengths of the two converting signals λ cnv,1 and λ cnv,2 are not necessary the same. (b) and (c) show simulation results of the standard deviations of the output amplitude noise after the first and second wavelength converter stages, while the bottom axes are the standard deviation of the input amplitude noise in front of the first stage. The standard deviations of the input phase noise σph in before the first stage are indicated in the legends. Dash-dotted lines are the borders of the regeneration regions.
(A) Noise suppression after an ideal amplitude modulator (αH = 0) based MZI
In Eq. (26) we derived the transmission function for αH = 0. This transmission function takes on positive or negative values only, depending on the difference between the logarithmic power transmission changes Δh up−Δh low, see Eq. (26). This actually corresponds to a perfect phase response as depicted in Fig. 2(A). That is, the output phase only takes on the values zero or π, but no other values in-between.
Now we give the transmission functions T 0 and Tn respectively for αH = 0 and for the logical “1” state. Without any noise, we have from Eq. (12)
Substituting Eq. (37) into Eq. (26) results in the MZI transmission function T 0 for αH = 0,Under the influence of noise, the respective values of Δh up and Δh low will change. If noise is included, the respective perturbations of the logarithmic transmissions of the upper and lower arms are denoted as δh up and δh low, respectively. They can be calculated by substituting δP up and δP low from Eq. (34) in Eq. (12) and Eq. (37), respectively,Note that the power perturbations δP low and δP up are neglected in the denominators of Eq. (39) . This assumption is generally valid since the power perturbations δP low and δP up are usually smaller than the power of the converting signal and the OOK control signals. Indeed, if these power perturbations are added in the denominators of Eq. (39), the respective values of the logarithmic transmission perturbations δh up and δh low, are smaller, which results in less amplitude/phase errors at the output. Equation (26) then provide the transmission function Tn of the n-th output signal En out Substituting Eqs. (38) and (40) into Eq. (36), finally provides the relative amplitude errorWith Eqs. (34) and (39), Eq. (41) shows that the output amplitude error contains the contributions from the input amplitude error and the input phase error , while the output phase error is virtually zero due to αH = 0. When the NLE in the upper arm (e. g., a QD-SOA) sees an OOK pulse, it becomes strongly saturated, while the other NLE in the lower arm is less saturated. So, the change of the logarithmic transmission Δh up in the strongly saturated NLE (P s,0 >> P cnv) approaches –h cnv according to Eq. (37), and exp(–Δh up/2) ≈exp(h cnv /2) >>1. Also, δh low >> δh up holds. Thus, from Eq. (41), we see that the amplitude error at the output signal mainly comes from the power perturbation in the lower arm δh low.An optimum performance of a DPSK-WC using QD-SOAs requires a proper choice of the input signal power Ps, 0 and of the converting signal power P cnv. In order to keep the noise low, Eq. (41), one needs minimizing δh up and especially δh low. The quantity δh low is small as long as δP low is small. A small δP low in Eq. (34) means a small Ps, 0 at a given input relative error. On the other hand, to achieve a good conversion efficiency (i. e., a large T 0) at the operating point, a high Ps ,0 is also needed to saturate QD-SOAs. This can be also seen from Eqs. (37) and (38): A large Ps ,0 leads to a small exp(Δh up /2) and thus to a large T 0. So the value Ps ,0 needs optimization. From Eq. (39), δh low is also small as long as the converting power P cnv is high. However, a high P cnv also suppresses the single pass gain and gives a smaller exp(h cnv /2), with other QD-SOA parameters fixed. Consequently, from Eq. (38), a large P cnv leads to a smaller eye opening of the converted signal. So an optimum exists for P cnv, too.
We performed system simulations with OptSim from RSoft, where the NLE model given in Section 3.1 is implemented by a MATLAB interface. A noisy NRZ-DPSK signal is generated with a PRBS sequences of 212 −1 at 40 Gbits/s. The phase noise is changed by varying the full-width at half-maximum (FWHM) linewidth ∆f in of the Lorentzian laser source. The linewidth is related to the variance of the Gaussian-distributed differntial phase noise by (σ ph in)2 = 2π∆f in Tb [29] [30], where Tb is the width of a bit slot. The amplitude noise is generated in OptSim by a Gaussian-distributed noise generator, and is added with proper normalization to the amplitude of the DPSK signal.
The parameters of the NLE under consideration are: Small-signal single-pass gain G 0 = 30 dB, saturation power P sat = 10 dBm, cw converting signal power (before MZI in Fig. 1) is 6 dBm, and optimized power of the input DPSK signal (before DI) P in = 15 dBm.
The amplitude-regeneration capability is measured in terms of the standard deviation of the relative amplitude error (with respect to the mean signal amplitude) in the centre of the bit slot [5]. This is a relevant measure, but does not imply the assumption of a Gaussian pdf, because saturation effects change the shape of the output pdf.
The amplitude regeneration in the case of cascaded wavelength converters is also of interest, where cascading means adding a second DPSK-WC stage with the same configuration (DI plus MZI) after the first stage, as depicted in Fig. 4(a).
For different noise loaded input signals, we depicted the standard deviation of the relative amplitude error Eq. (36) after the first stage, in Fig. 4(b), and after the second stage, in Fig. 4(c). The bottom axes of Figs. 4(b) and 4(c) are the standard deviations of the relative input amplitude error (Eq. (29)) before the first stage. The standard deviations of the input phase noise σph in (in radians) before the first stage are indicated in the legends in Figs. 4(b) and 4(c).
For such an ideal amplitude-modulator based wavelength converter, the amplitude regeneration can be achieved if the input phase noise is not too large. In Fig. 4(b) we observe that for weak amplitude noise the amplitude regeneration is limited by the input phase noise. This is because the input phase noise is transferred to the amplitude noise through the DI stage, see the terms sin2(/2) for the power perturbations δP up and δP low in Eq. (34). As the input amplitude noise is growing and exceeds a certain value, the input amplitude noise is becoming the dominant noise source. This amplitude regeneration could be totally destroyed for high input phase noise, even if the input amplitude noise is zero, as seen on the blue and green lines for in Fig. 4(b). But the amplitude regeneration can be enhanced if a second stage with the same configuration is cascaded, see the downshifted curves for the output amplitude noise in Fig. 4(c). This noise-suppression enhancement is due to the ideal phase regeneration in the first stage, i. e., due to the noiseless phase at the input of the second stage. It should be noted that the legends of Fig. 4(c) are the standard deviations of the input phase noise σph in before the first stage.
(B) Noise Suppression for an ideal phase modulator based MZI (αH → ∞)
In the case of ideal phase modulators, αH in Eq. (20) approaches infinity, and P eff → ∞ from Eq. (15), but αH /P eff = 4k 0 n 2 /(α 0 A eff) is a constant. The transmission function from Eq. (27) behaves as a linear MZI.
Now, we give the transmission functions T 0 for the logical “1” state without noise. We apply Eq. (17) for the two arms in the MZI and substitute them into Eq. (27),
The power Ps ,0 of the OOK pulse is assumed to be noiseless. From Fig. 2(C), the optimum operating point will be chosen so that the phase shift of the converting signal on the upper arm isand the transmission function in Eq. (27) becomesNext we are interested in the transmission function Tn for the logical “1” state with noise. By inserting the power perturbations with respect to P s,0 from Eq. (34) in Eq. (17), we have the respective perturbations in the logarithmic transmissionWith Eq. (45) and ∆h up = –2π /αH, Eq. (27) becomesSubstituting Eqs. (44) and (46) in Eq. (36), we now obtain the relative output amplitude error and the output phase error.From Eq. (47), first, it is interesting to see that the output phase error comprises only the input amplitude error. The output phase error follows a similar distribution compared to the input amplitude error, see Eqs. (33) and (47). Any transfer of the input phase error to an output phase error is totally suppressed. This is because the gain fluctuations δh low and δh up due to an input phase error are compensated, see Eq. (45). The second observation when inspecting Eq. (47) is that the conversion from the input phase noise to output amplitude noise is also suppressed by the nonlinear cosine function. The distribution of the output amplitude error is modified by the wavelength conversion. For two consecutive bits with a larger amplitude error in-between, the output amplitude is smaller than for the noiseless case. Also, if the perfect interference condition as assumed in Eq. (43) is not satisfied, the output amplitude is smaller than that in the noiseless case. So, the output amplitude is close to but never exceeds the output amplitude in the noiseless case, as can be seen in Fig. 3(b). As a result, the distribution function of the output amplitude error changes too.For the scheme Fig. 1, an RZ-DPSK signal modulated at 40 Gbits/s was used. Simulations were performed by setting αH = 500. Also, the effective power P eff = 27.5 dBm, the input DPSK signal (before DI) P in = 1.5 dBm, and the clock converting signal P cnv = 0 dBm was used. The remaining parameters are identical with those from before.
The conversion from the input amplitude or phase noise to the output amplitude or phase noise, respectively, has been computed numerically, and the results are depicted in Fig. 5 . Figure 5(a) shows the standard deviation of the output amplitude noise (vertical axis) versus the standard deviation of the input amplitude noise (horizontal axis), where the standard deviation of the input phase noise σph in varies for 0 and 0.25 radians. Figure 5(b) shows the standard deviation of the output phase noise σph out (left-axis) versus the standard deviation of the input phase noise σph in (bottom-axis), where the standard deviation of the input amplitude noise is indicated in the legend.
Fig. 5 Noise suppression property of an ideal XPM-based (αH –> ∞) DPSK wavelength converter. (a) Standard deviation of the output amplitude noiseversus standard deviation of input amplitude noisewhen the standard deviation of the input phase noise σph in is 0 or 0.25 radians. (b) Standard deviation of output phase noise σph out versus standard deviation of input phase noise σph in when varying the standard deviation of the input amplitude noisein an interval [0, 0.25]. Dash-dotted lines are the borders of the regeneration regions.
By inspecting Fig. 5, confirms the former conclusion that the output amplitude noise is suppressed with respect to the input amplitude/phase noise, Fig. 5(a), and that the output phase noise depends linearly on the input amplitude noise, Fig. 5(b). Amplitude regeneration is achieved in a large range of input amplitude noise, Fig. 5(a). However, as can be seen in Fig. 5(b), the output phase noise is beyond the region of regeneration (dash-dotted line in Fig. 5(b)) for not too large an input amplitude noise. It should also be emphasized that simultaneous phase and amplitude regeneration can be achieved in the case of small input amplitude noise.
6 Conclusion
An all-optical DPSK wavelength converter has potential for significant signal regeneration. The configuration comprises a delay-interferometer and a following Mach-Zehnder interferometer with nonlinear elements (NLE) in both arms. The wavelength converter can be implemented with NLE based on cross-gain modulation (XGM) or cross-phase modulation (XPM). If XGM dominates as is the case in, e. g., QD-SOA or EAM, strong phase regeneration can be realized for a very week (or even vanishing) XPM effect. Simultaneous amplitude regeneration can also be achieved, if the NLE are operated in a strong saturation regime. If XPM dominates as is true for, e. g., nonlinear organic Kerr materials or highly nonlinear fibers, significant amplitude regeneration is predicted. Phase regeneration can be achieved as long as input amplitude noise is not too large. We finally find that signal regeneration of XGM-based wave-length converters can be improved by cascading two stages.
Acknowledgements
This work was supported by the Center for Functional Nanostructures (CFN) of the Deutsche Forschungsgemeinschaft (DFG) within projects A3.1 and A4.4, and by the European project Euro-Fos (NoE 224402).
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