1. Introduction

Nonlinear mixing in optical devices is a well known means of providing phase-sensitive amplification and can be used to regenerate binary phase shift keyed (BPSK) signals [1–4]. Quadrature phase shift keyed (QPSK) signals can be regenerated by arranging two phase-sensitive amplifiers in parallel [5]. More recently, a phase-sensitive amplifier with a four-step phase transfer characteristic has been demonstrated that can regenerate QPSK signals in a single device [6,7]. Phase-sensitive frequency conversion of BPSK using nonlinear mixing has also been shown [8] and has the potential for greater phase discrimination because of the isolation of the output from the input signal.

In previous work, we found that it is also possible to convert the two orthogonal phase components of a signal simultaneously to separate output frequencies and thus convert a QPSK signal to two BPSK outputs [9]. The devices employed were semiconductor optical amplifiers (SOAs), which have been extensively studied as nonlinear mixing elements [10–13] and offer a number of advantages. They require only sub-milliwatt input powers, provide gain, are compact and are also readily integrated.

In this paper, we verify experimentally that SOAs can be efficient nonlinear mixing devices over a broad range of wavelengths and difference frequencies. We then show that there are two mechanisms by which phase-sensitive frequency conversion can be obtained. The first is that exploited in Croussore and Li’s experiment [8], but only the second, introduced in this work, allows more than one phase component to be converted to separate outputs simultaneously. Using a numerical model validated by comparison with the experimental results, we demonstrate a novel phase-to-polarization converter that generates a pair of BPSK outputs in orthogonal polarization states from a QPSK input signal. We also simulate a scheme using a four-tooth frequency comb similar to our earlier work, but with double the symbol rate and comb spacing, in order to show that SOA-based phase-sensitive signal processing is feasible with 40GBd QPSK data.

Potential applications of QPSK separation include demultiplexing data, replacing a complex receiver with a pair of simpler BPSK receivers and use as an input stage to an all-optical QPSK regenerator.

2. Experimental measurements

In order to assess the efficiency of nonlinear mixing in SOAs, devices were characterized over a range of wavelengths and difference frequencies. The measurements were carried out by injecting two continuous wave (CW) pump inputs, at wavelengths λ₁ and λ₂, into the SOA and observing the output spectrum. Nonlinear mixing between the two pump inputs produced additional frequency components spaced at multiples of the difference frequency. The output powers of the two first-order sidebands and the amplified pumps were recorded with an optical spectrum analyzer. Wavelength dependence was investigated by stepping λ₁ from 1530nm to 1585nm in 5nm increments. With λ₁ fixed at each wavelength step, λ₂ was swept from λ₁ + 0.08nm to λ₁ + 11nm, thus varying the difference frequency from ~10GHz to ~1.4THz. To speed up data collection, the experiment was largely automated using LabVIEW (Fig. 1).

 

Fig. 1 Experimental system (VOA: variable optical attenuator, RPC: remote polarization controller, PBS: polarization beam splitter, PC: polarization controller, PM: power meter, OSA: optical spectrum analyzer). The solid lines represent optical paths and the dotted lines are data and control paths.

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The bias current of the SOA was set to 400mA and two variable optical attenuators kept the power of each pump at a constant −10dBm at the input facet for all wavelength settings. Since the relative polarization of the pumps at the ends of their connecting fibers varied with tuning, two remote polarization controllers were used to ensure that both injected signals remained co-polarized for maximum nonlinear mixing efficiency. The pumps were combined and passed through a polarization beamsplitter. A downhill simplex algorithm adjusted the remote polarization controllers to minimize the power measured at the unused output port. A manual polarization controller set the polarization state at the SOA input for maximum gain.

In order to refer the measured powers to the SOA facets, the coupling efficiencies at the input and output of the SOA were measured. A CW source was connected to each fiber tail in turn and varied in power, and the photocurrent generated in the SOA with the PN junction at 0V bias was measured. With the assumption that all of the light reaching the active region stimulated photocurrent, the coupling efficiency is given by η = $ħ$ωI/eP, where $ħ$ω is the photon energy, I is the photocurrent, e is the charge of an electron, and P is the injected power.

The waveguide loss is one of the parameters used in the SOA model (described in the Appendix) and is equal to the loss of the SOA when the bias current is set to give a net material gain of 0dB. At this bias, which depends on input wavelength, the number of carriers generated by absorption balances the number removed by stimulated emission when averaged along the length of the device. There is, therefore, no change in junction voltage when the input power is varied. The bias for transparency was found by amplitude modulating the optical input at 1kHz and observing the phase reversal of the junction voltage variation when the transition point between net material gain and loss was reached. The waveguide loss was measured at a number of points across the gain band and showed little wavelength dependence.

3. Results

Sets of measurements were made on a number of SOAs using a range of wavelengths for the fixed pump. Here we concentrate on the SOA that was previously employed to demonstrate separation of the in-phase and quadrature phase components of a signal (a CIP NL-OEC-1550) [9]. When biased at 400mA, this device had a peak small-signal gain of 32dB at 1560nm and a saturated output power of 10.4dBm at 1550nm (all measurements referred to the SOA facets).

Nonlinear mixing results for a fixed-pump wavelength of 1555nm, the center wavelength in our previous experiment, are shown in Fig. 2, where the output pump powers and the powers of the two first-order sidebands have been plotted against the frequency difference between the pumps. Simulated results from the model described in the Appendix are also shown. For difference frequencies up to 50GHz there was a strong transfer of power from the blue (fixed) pump to the red (swept) pump and the red sideband was larger than the blue sideband. At higher frequencies, the sideband powers rolled off with increasing slope, but remained greater than the input pump powers (−10dBm) until the difference frequency reached 600GHz. The availability of frequency conversion gain up to such a high frequency is attributed to carrier temperature modulation.

 

Fig. 2 Pump and sideband powers for fixed-pump wavelength = 1555nm. Symbols are measurements and lines are model results.

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For difference frequencies up to 700GHz, the simulated sideband powers were within 1dB of the experimental measurements. However, above this frequency the roll-off of the measured sideband powers was more rapid than the 6dB/octave expected from an exponential decay of the carrier temperature with time (Eq. (16). Dispersion in the SOA did not appear to account for this discrepancy because calculations based on the ASE fringe spacing suggest that dispersion should not have significant effect for difference frequencies below 2THz.

The sideband powers were also plotted for fixed-pump wavelengths of 1530nm, 1565nm and 1585nm (Fig. 3), which were the shortest wavelength used, the value giving maximum sideband power and the longest wavelength used, respectively. For difference frequencies up to 400GHz, the frequency conversion process showed gain for all these wavelengths and the variation of sideband power with fixed-pump wavelength was less than 5dB, demonstrating that SOAs are well suited to the phase-sensitive frequency conversion applications described in the next section.

 

Fig. 3 Sideband powers for three fixed-pump wavelengths. Lines are to guide the eye.

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4. Principles of phase-sensitive frequency conversion

The nonlinear mixing between optical fields that occurs at each point along the length of an SOA can be regarded as a two-stage process. Firstly, interference between the fields gives rise to beats in the total power and the carrier density and temperature vary in response to the beats. Secondly, these variations cause phase and amplitude modulation of the fields, generating new sidebands [11,14,15]. We show in the following that there are consequently two mechanisms by which the amplitude of a frequency converted output can become sensitive to signal phase. Either the interference between the signal and some pump fields is arranged so that the power beats at a given frequency vary in amplitude with signal phase, or the output can be the sum of two or more sidebands whose relative phases depend on signal phase. Using the first mechanism, only one phase component of the signal can be converted to a new frequency. However, by using the second mechanism, multiple outputs dependent on different phase components of the signal are possible.

The beats between a signal, with electric field given by

 

Fig. 4 Examples of phase-sensitive frequency conversion schemes. (a) Croussore and Li’s scheme with two pumps and one probe. Both outputs follow the same phase component of the signal. (b) Alternative scheme with one pump and two probes. Probe phases set to select Δϕ signal component. (c) With a third probe, in-phase and quadrature outputs can be obtained simultaneously.

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However, it is also possible to use an inverse configuration with one pump and two probes, with frequencies ± Δω relative to the output frequency, ω₀ (Fig. 4(b)). Consider first a single probe with electric field V(t). The signal-pump beats (Eq. (3) cause both phase and amplitude modulation of the probe, and the first-order lower and upper sidebands are given by:

Thus the output of this configuration also shows signal-phase selectivity, being proportional to one phase component only (Δϕ) of the input signal. The desired signal phase component is selected by setting the phase difference between V₂(t) and V₁(t) to θ-2Δϕ, where θ is the phase difference (given by Eqs. (8) and (9)) for Δϕ = 0. (The effect of beating between the probes has been ignored in this analysis because it does not generate any sidebands at the output frequency, ω₀. It can, however, modify the probes themselves but the probe input amplitudes and phases can be adjusted to compensate.)

There is an important difference in the operating mechanism of the two configurations. In contrast to the Croussore and Li scheme, the amplitude of the power modulation in the configuration proposed here does not vary with signal phase. Instead, phase selectivity is obtained by forming the output from the vector sum of two sidebands. As a result, a further pair of probes can be introduced to generate a second output with a different dependence on signal phase (Fig. 4(c)). By setting Δϕ = 0 for one probe pair and Δϕ = -π/2 for the second, both the in-phase and quadrature components of the signal can be extracted simultaneously in a single device. For example, three probes can be used, i.e. two pairs sharing a common central probe, to generate the in-phase and quadrature outputs at different frequencies. Alternatively, the two probe pairs can share the same two frequencies but have orthogonal polarizations (Fig. 5). In this case, the in-phase and quadrature outputs also share the same frequency but occupy orthogonal polarization states: the device becomes a phase-to-polarization converter. The beats between the probes are negated with this configuration.

 

Fig. 5 Phase-to-polarization converter. Probes Pr1 and Pr2 have orthogonal polarization states. Their vertically and horizontally polarized components can be regarded as separate pairs of probes with the phase differences shown. The in-phase and quadrature outputs can be separated by a linear polarization splitter.

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More complex configurations are possible, such as the examples reported previously [9] and in the following section, where the signal is placed in one gap in a four-tooth frequency comb and orthogonal outputs are generated in the other two gaps. The signal must be phase locked to the comb. In these cases, the frequency comb teeth act as both pumps and probes and multiple mixing processes contribute to each output.

5. Simulation of phase-sensitive frequency conversion

Using the numerical model presented in the Appendix and verified by comparison with the measurements presented in Section 3, we simulated two different phase-sensitive frequency conversion schemes: firstly the orthogonal polarization scheme described in Section 4 and secondly a four-tooth frequency comb scheme similar to our earlier work [9] but with doubled comb spacing and symbol rate. In both cases, instead of a single SOA, a Mach-Zehnder interferometer (MZI) with an SOA in each arm [17] was employed in order to reject the amplified pumps and probes, thus reducing the demands placed on the bandpass filters used for the output signals.

5.1 Phase-to-polarization converter

The arrangement for the first scheme is shown in Fig. 6. The input signal, consisting of a train of raised cosine half-width RZ pulses at 1560nm with 19.4GBd QPSK modulation, was connected to one input port of the MZI and the pump and two probes (all CW) were connected to the other input. The difference frequency, Δf, between the pump and the signal was 40GHz (it need not be related to the symbol rate) and the pump and signal were co-polarized.

 

Fig. 6 Phase-to-polarization converter simulation. The MZI rejects the amplified pump and probes.

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The two probes were in counter-rotating circular polarization states, ensuring that the beats between their vertically polarized and horizontally polarized components were in antiphase (Fig. 5). Their optical frequencies were ± Δf relative to the output frequency, 193.4THz (1550nm). The output of the MZI was bandpass filtered and split into orthogonal linearly polarized components. Following numerical optimization of the pump and probe powers and the probe phases, these two output signals became replicas of the BPSK components of the input signal. The values of all the inputs are shown in Table 1. Constellation diagrams for the two outputs, on which the optimization target was based, are shown in Fig. 7, together with eye diagrams obtained by DPSK demodulation of the signals (to show that phase separation was obtained without coherent detection). Crosstalk between the two phase components of the input signal was evidently low. The mean output powers were 7dB greater than the mean input signal power (but filter and SOA coupling losses were not included in the simulation).

Table 1. Inputs and outputs for the phase-to-polarization converter

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Fig. 7 BPSK constellations of the phase-to-polarization converter outputs with eye diagrams after DPSK demodulation. The field components are in units of √(2W).

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Retaining the optimized pump and probe settings, the signal was replaced by a CW input with the same mean power and optical frequency. The phase of this CW input was stepped through the range -π to π radians and the two outputs were calculated for each step (Fig. 8). Each output is the sum of two modulation sidebands, one from each of the two probes (Section 4), and their vector addition is shown graphically with the input phase set to 0 and π/2 (Fig. 9). The in-phase and quadrature output powers varied in good approximation to the cos²ϕ_S and sin²ϕ_S responses that would indicate ideal resolution into orthogonal components, and there was strong phase regeneration. However, the minimum and maximum powers did not occur at exactly ϕ_S = 0 and π/2. It was generally found that optimization for a modulated signal did not give the best performance with CW inputs. The reverse was also true: optimization for CW inputs did not give the best signal performance. In this case, the rms error vector magnitude was doubled when the optimization was based on CW inputs.

 

Fig. 8 In-phase and quadrature outputs of the phase-to-polarization converter v. CW input phase with ideal cos²ϕ_S and sin²ϕ_S responses.

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Fig. 9 Vector components of the phase-to-polarization converter outputs. Vector a is the lower sideband of the higher frequency probe and vector b is the upper sideband of the lower frequency probe. The blue arrows show their summation for the CW input phase, ϕ_S = 0, and the red arrows correspond to ϕ_S = p/2. The dashed lines show the locus of the vector sum for all ϕ_S.

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5.2 Four-tooth frequency comb scheme

For this simulation, a four-tooth frequency comb with 160GHz spacing and center wavelength 1555nm was connected to input port 1 and the symbol rate of the QPSK signal connected to port 2 was increased to 40GBd in order to investigate phase-sensitive signal processing in SOAs at a higher data rate (Fig. 10). The signal carrier frequency was phase locked to the comb and set at the midpoint between the two highest frequency comb teeth, giving a signal-pump difference frequency, Δf, of 80GHz. The comb powers and phases (Table 2) were optimized to obtain BPSK outputs at the comb center and between the two lower frequency comb teeth (−2 Δf relative to the comb center). As in the previous simulation, the two outputs showed low crosstalk (Fig. 11) and gains of 4dB relative to the mean signal power despite the increased frequency spacing (not including filter and SOA coupling losses).

 

Fig. 10 Simulation of four-tooth frequency comb scheme. Frequencies shown are relative to the comb center frequency.

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Table 2. Inputs and outputs for the four-tooth comb scheme.

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Fig. 11 BPSK constellations from the four-tooth frequency comb scheme with eye diagrams after DPSK demodulation. The field components are in units of √ (2W).

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Simulations with a CW input stepped through a range of phase values again showed that comb settings for optimum signal performance and for optimum CW response did not exactly coincide (Fig. 12). Calculation of the constituent vectors of the two outputs revealed that the behavior of this scheme was rather more complex. The comb teeth beat with the signal and with each other to generate power beats at all integer multiples of the difference frequency from Δf to 6 Δf. Each output was the sum of five vectors and, in contrast to the previous scheme, each vector varied in amplitude as well as phase with signal phase (Fig. 13), indicating that both mechanisms of phase-sensitive frequency conversion identified in Section 4 were in operation.

 

Fig. 12 In-phase and quadrature outputs of the four-tooth frequency comb scheme v. CW input phase.

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Fig. 13 Vector components of the four-tooth frequency comb scheme outputs. Table 3 lists the vectors starting from the origin of the plots. Column 1 shows the input component from which each vector is derived. Colum 2 shows the relative frequency of the modulation sideband contributing to the in-phase output and, similarly, column 3 shows which sideband contributes to the quadrature output. Blue arrows show their summation for the CW input phase, ϕ_S = 0, and red arrows correspond to ϕ_S = π/2. Dashed lines show the locus of the vector sum for all ϕ_S.

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Table 3. Vector components shown in Fig. 13.

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6. Conclusions

We have shown experimentally that SOAs are efficient as nonlinear mixing devices over a broad wavelength range and provide frequency conversion gain for signal-pump difference frequencies that are sufficiently large for most telecommunications applications. There are two mechanisms by which frequency conversion can be made sensitive to signal phase: either the magnitude of the power beats can vary with the phase of the signal relative to the pump inputs or the output can be the superposition of more than one modulation product (both mechanisms can be combined). The second mechanism permits the generation of additional outputs proportional to different phase components of the signal.

We have proposed a novel scheme based on the second mechanism that converts the orthogonal phase components of a signal to orthogonally polarized outputs at a different frequency. Numerical simulations of this scheme show that a QPSK signal can be converted to a pair of BPSK outputs with low crosstalk. We have also demonstrated that SOAs can provide phase-sensitive frequency conversion at 40GBd with conversion gain by adapting our previously reported four-tooth frequency comb scheme. With this arrangement, the QPSK signal was converted into two BPSK outputs at different frequencies, each comprising multiple vector components.

Such schemes could be used to demultiplex data, simplify receivers or as part of an all-optical QPSK regenerator. It may also be advantageous to resolve higher order quadrature modulated amplitude (QAM) formats into their in-phase and quadrature components.

Appendix: numerical model of the SOA

The simulation results presented in this paper were obtained with a time-domain model similar to those widely used to model cross-phase and cross-gain modulation in SOAs [18–21]. With the SOA divided into a number of length elements, rate equations for the carrier density and carrier temperature were solved and a polynomial approximation to the gain spectrum was evaluated for each length element and time step. In order to model the nonlinear mixing between inputs which is of interest here, the input fields were summed and represented by a single equivalent-baseband optical field:

(11)$$E_{in}={\displaystyle \sum _k{E}_k\left(t\right)\exp i\left({\omega }_k-{\omega }_c\right)t},$$

where the E_k are the complex amplitudes and the ω_k are the frequencies of the component fields, and ω_c is a common carrier frequency. The mixing products generated by the nonlinear response of the SOA are obtained by taking the Fourier transform of the output field.

The growth in forward travelling optical power, P (P = EE*/2), is given by:

(12)$$\frac{\partial P}{\partial z}+\frac{\partial P}{v_g\partial t}=\left(\Gamma g_m-{\alpha }_L\right)P,$$

where v_g is the group velocity, Γ is the proportion of the power in the active region and α_L is the waveguide loss (measured as described in Section 2). g_m is the material gain, which varies with time, t, and position, z, and depends on wavelength, λ, according to:

(13)$$g_m=g_p\left[3{\left(\frac{{\lambda }_z-\lambda }{{\lambda }_z-{\lambda }_p}\right)}^2-2{\left(\frac{{\lambda }_z-\lambda }{{\lambda }_z-{\lambda }_p}\right)}^3\right],$$

where λ_z is the bandgap wavelength. In a simplification of the expression introduced by Leuthold et al. [22], λ_z is assumed to be constant to facilitate calculation of concatenated gain spectra and the amplified spontaneous emission (ASE). (In this work, the effects of ASE were negligible.) g_p and λ_p are the gain and wavelength at the peak of the gain spectrum, given by $g_p=a_0\left(N_{eff}-N_0\right)$and${\lambda }_p={\lambda }_0-b_0\left(N_{eff}-N_0\right)-b_1{\left(N_{eff}-N_0\right)}^2$, where a₀ is the differential gain, N₀ is the transparency carrier density and λ₀, b₀ and b₁ are constants.

Band modelling has shown that the effects of carrier heating in the center of the gain region can be approximated by a reduction in carrier density to an effective density, N_eff, given by:

(14)$$N_{eff}=N-\frac{\partial N_{eff}}{\partial T}\left(T-T_0\right),$$

where T is the temperature of the conduction band carriers and T₀ is the carrier temperature in the absence of optical power. The actual carrier density, N, evolves according to the standard rate equation:

(15)$$\frac{\partial N}{\partial t}=I_e-a_{s}N-b_s{N}^2-c_s{N}^3-\frac{g_m\Gamma P}{A\hslash \omega },$$

where I_e is the electron injection rate per unit volume; a_s, b_s and c_s are the spontaneous recombination coefficients; A is the cross-sectional area of the active region and $\hslash \omega$ is the photon energy. T is assumed to rise in proportion to the number of carriers removed by stimulated recombination and to decay exponentially:

(16)$$\frac{\partial T}{\partial t}=\frac{{\epsilon }_T{g}_m\Gamma P}{\sqrt{N}A\hslash \omega }-\frac{T-T_0}{{\tau }_{CH}},$$

where $\epsilon$_T is a constant and τ_CH is the cooling time constant. The root N term is introduced as an approximation to band modelling results.

The rate equations are solved using uniform length elements and time steps, with the interval between time samples equal to the optical transit time of a length element. When modelling nonlinear mixing, step sizes must be chosen that are short enough to sample all the important beat frequencies in the optical field. The phase changes induced on the optical field are related to the carrier density and temperature by:

(17)$$\frac{\partial \varphi }{\partial z}+\frac{\partial \varphi }{v_g\partial t}=\frac{\Gamma a_0}{2}\left[\left(N-N_0\right){\alpha }_{BF}-\left(T-T_0\right)\frac{\partial N_{eff}}{\partial T}{\alpha }_{CH}\right],$$

where α_BF and α_CH are the alpha factors for the band filling and carrier heating processes, respectively [16,23].

The unknown model parameters were obtained by comparing the outputs with experimental data from a CIP XN-OEC-1550, an SOA similar to the one used in this work. All measurements were made at the same bias current in order to avoid changes in junction temperature, which is not included in this model. Accurate fits were obtained both to a set of gain and ASE spectra measured with the device saturated by different levels of CW optical power and also to a set of pump-probe amplitude and phase responses measured in the region of the gain peak. The values for a₀, α_BF, $\epsilon$_T, τ_CH and α_CH were further optimized to improve the fit to the experimental results presented in Fig. 2. Results for difference frequencies above 700GHz were not included because the measured and modelled slopes diverged significantly in this region. The final set of parameter values used for the simulations in Section 5 is shown in Table 4.

Table 4. SOA model parameters.

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Acknowledgment

This work was funded by Science Foundation Ireland grant 06/IN/I969.

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