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Phase-sensitive amplification of picosecond optical pulses was demonstrated using an SOA as the nonlinear medium inside a Sagnac interferometer. Ratios of maximum to minimum gain of more than 3:1 were experimentally measured. Numerical simulations using a semiconductor amplifier model are consistent with experiments.
©2008 Optical Society of America
1. Introduction
Optical phase sensitive amplification has been of interest for decades [1]. Current applications for this technology include the amplification and regeneration of novel optical modulation formats for telecommunications. Phase-sensitive amplifiers have been shown to provide high gain while adding less noise to the signal than an ideal laser amplifier [2], and used to regenerate DPSK formatted signals [3, 4]. Parametric amplifiers in general work as efficient switches that use a weak control signal to re-direct the power of a strong pump [5, 6]. Despite these demonstrations there remains to be implemented a compact and efficient waveguide implementation of a phase sensitive amplifier.
As summarized below, there have been numerous implementations and demonstrations of systems that have an amplification coefficient that is dependent on the phase of the input signal as defined with respect to some other optical beam. Not only have there been phase sensitive amplifiers built at different wavelengths and different geometries, but there have also been a number of different nonlinear optical processes and nonlinear materials used.
There is a long history of using second order nonlinearity (χ(2)) for parametric down-conversion (PDC). Parametric down conversion is the inverse process of second harmonic generation, and produces optical gain that is dependent on the phase relationship between the 2ω pump and ω signal beams [7, 8]. While very successful, this process has difficulties in waveguide geometries because it requires single-mode waveguides and phase matching for two very different wavelengths.
Third order nonlinear processes based on χ(3), can also be used for the implementation of phase sensitive amplification using four-wave mixing. This can be accomplished with a pair of pumps of slightly different frequencies to amplify a signal at a frequency exactly between them [9]. Another approach is to use the same frequency for the pump and signal, with the nonlinear medium placed inside a Mach-Zehnder or Sagnac interferometer [2, 10]. Because the wavelengths are closer together than for a second order process, the third order implementation is preferred for waveguide implementations. For these interferometers the phase sensitive gain in a nonlinear interferometer is approximately, (2πn2IL/λ)2, where n2 is the nonlinear index of refraction, I is the pump intensity, L is the interaction length, and λ is the wavelength [11]. However, most materials have either a (non-resonant) low value of the nonlinear coefficient, or high values of the coefficient that are associated with a resonance and strong linear absorption [7]. Weak nonlinearity has resulted in amplifiers that require high optical power or long interaction lengths. Some of the best results to date were obtained using highly nonlinear optical fibers that were typically ~1 km long. New materials with improved nonlinear coefficients will dictate advances in this approach.
In this work we investigated the phase sensitive gain in a semiconductor optical amplifier (SOA) that is placed in a Sagnac interferometer as illustrated on Fig. 1. The phase sensitive amplification is the result of gain saturation in the SOA, which can happen with relatively low input powers. The large linewidth enhancement factor in semiconductor materials results in a significant phase shift that is associated with the gain saturation in the SOA. While the optical arrangement is similar to that used to implement phase sensitive parametric gain in fiber based nonlinear loop mirrors (NOLM) [2, 10], the nonlinear process exploited here can not be described by the conventional third order coefficient χ(3). Instead a time dependent dynamic solution is required. We will present a theoretical analysis of this phase sensitive amplifier, experimental results, and show that an “effective” nonlinear index of refraction for our SOA is orders of magnitude larger than other modern materials.
Fig.1. . chematic of the nonlinear interferometer. Electric field strengths and devices are labeled to describe components of the simulation.
Placing an SOA in a Sagnac interferometer is not a new idea. In fact, the arrangement that is implemented here resembles a terahertz optical asymmetric de-multiplexer (TOAD) [12–19]. In a TOAD fast response times are obtained in spite of the relatively slow relaxation time of an SOA by intentionally locating the SOA asymmetrically in the loop and saturating the SOA with an intense pump pulse. The asymmetric layout causes the pulses that are propagating in opposite directions to experience different gains. This unbalance destroys the symmetry that is required for destructive interference at the output of the interferometer and exists only for the difference in the arrival times of the pulses at the SOA.
There are two major differences in the arrangement described here and a TOAD. First, the SOA is intentionally located symmetrically in the loop. This symmetry allows for strong pump pulses to be used without unbalancing the loop mirror. Secondly, the control signal that is injected into the output port is at the same wavelength as the strong pump. Because of the phase shifts inherent in the beam splitter, the clockwise propagating signal and pump pulses interfere constructively, while the counter-clockwise propagating signal and pump pulses will interfere destructively. This difference in interference creates the unbalance of the Sagnac interferometer that results in phase sensitive amplification of the control signal.
2. Numerical simulation
The performance of our nonlinear Sagnac interferometer was simulated using the phenomenological rate equations model for gain saturation in the SOA [20]. While more sophisticated solid state models exist, it has been shown that the phenomenological models are sufficient to model the effects experienced by the photons as long as the parameters are accurately determined [21]. Gain saturation in SOA’s has been successfully modeled using this approach [22].
Assuming that the clockwise and counter-clockwise fields are written in the form
The slowly varying envelope approximation is used to simplify the governing differential equations for the clockwise and counter-clockwise propagating pulses:
Where n is the average real component of the refractive index, c is the speed of light in vacuum and g(z,t) is the optical gain coefficient, which is a finite value in the SOA, and assumed to be zero outside of the SOA. The linewidth enhancement factor, αH, is included as a constant. There is no wavelength dependence in Eq. (2) and (3) because we are concerned with coherent nonlinear interactions, and we can therefore perform the simulations on a single wavelength. Since the wavelengths are identical, it is assumed that both the clockwise and counter-clockwise fields contribute to the gain saturation equally. Therefore the intensities additively contribute to the saturation.
Here, τc is the gain relaxation time constant, g0 is the small signal gain, and U sat is the gain saturation energy.
The simulated region consists of the entire SOA and a small section of the surrounding passive interferometer. Omitting most of the passive fiber optic propagation length minimizes calculation time, and does not introduce any error because only the SOA contributes to the variation of the interferometer’s output.
The dynamic non-linear behavior of the SOA is modeled by segmenting the simulated region into 1ps segments. The resulting spatial grid (δd) is non-uniform and can be determined by δd = cδt/n, where c is the speed of light, δt is the temporal grid size, and n is the index of refraction. As long as the segmentation volumes are small compared to the pulse envelope, co-propagating and counter-propagating pulses can be successfully modeled.
The input signals are introduced assuming a perfect 50:50 splitter. That is, the clockwise and counter-clockwise propagating pulses in the loop consist of half of the signal and pump fields:
Because the pump pulse (Ep) and the signal pulses (Es), are introduced from opposite inputs of the input coupler there is a phase difference between the ports of the splitter. Neglecting optical component loss inside the Sagnac interferometer, the electric field strength of the output pulse is
The accurate determination of the parameters needed for these simulations are imperative to perform even the most cursory comparison of simulation and experiment. The nonlinear parameters of the real SOA that was used for the experiments discussed here have been characterized elsewhere [23]. The small signal gain, g0, is 67cm-1, the gain recovery time, T1, is 400ps, saturation energy, Usat, is 0.05pJ, linewidth enhancement factor, αH, is 5, and the SOA is 1.2mm long. The first simulations explored the variation in output of the interferometer based on the relative delay of the two counter-propagating pulses at the SOA. This is required to adequately understand the correct tuning of the fiber section of the nonlinear interferometer. The calculated output with no signal pulse (Es=0) is shown in Fig. 2.
Fig. 2. Simulated output of the interferometer with only the pump pulses present. The labeled points correspond to the symmetric location of the SOA (A), a non-symmetric geometry (B), and the symmetry point where the pulses arrive at equal and alternating intervals at the SOA (C).
There are two symmetry points of importance that correspond to minima of the output. The first symmetry point (A) corresponds to the perfect symmetry of the interferometer where the pulses meet at the center of the SOA. Another minimum is observed at point C, where the pulses alternately pass through the SOA. At point B the interferometer is unbalanced because one of the pulses arrived before the other and experiencing a larger gain. For example, if the clockwise (CW) propagating pulse arrives at the SOA earlier than the counter-clockwise (CCW) propagating pulse it will saturate the SOA and experience a larger gain. When both pulses meet at the 3dB coupler the two pulses will not completely interfere resulting in an output of the interferometer. The output is at its maximum when the later pulse arrives exactly when the first is exiting. This corresponds with a completely saturated SOA with a maximum recovery time until the next pulse pair arrives.
It is interesting to compare our results with those published in reference 13. In particular, even though we are analyzing different experiments, our results shown in Fig 2 appear similar to Kang’s reported results for the case of a zero switching-window offset. The experiment analyzed in reference 13 explores the switching window for two weak probes that is created by a strong pump. In our device there are only two strong pumps that saturate the SOA. The common cancelation of the output in both results is the result of a dynamic balance of the interferometer. Unlike Kang’s work that requires a non-symmetric arrangement for optimal performance; we use the symmetrical configuration where the output is at a minimal as the operating point of highest interest.
Figure 3 shows the calculated pulse energy as a function of position for the counter-propagating pulses in the amplifier. For this figure, the input energy of each pulse is assumed to be 0.1 Usat. There are clearly three regions in Fig. 3: flat regions outside of the SOA, regions of exponential gain associated with unsaturated gain, and a region of linear gain associated with saturated gain in the SOA. Although the two pulses are being amplified in the same SOA it is apparent that each pulse experiences a heavily saturated second half of the amplifier. Therefore, the majority of the competition for gain in the SOA occurs during the first half of the device. This gain competition is the essence of the unbalancing that creates the phase-sensitive amplification of the nonlinear interferometer.
Fig. 3. Pulse energy of the clockwise solid (blue) and counter-clockwise dashed (red) propagating pulses in the simulation region. There are clearly three regions: flat regions outside of the SOA, region of exponential gain associated with unsaturated gain, and a region of linear gain associated with saturated gain in the SOA.
The simulation of phase sensitive amplification was completed with the addition of a weak control signal (7% of the pump power) with the pulse delay set for the symmetric arrival of the pulses at the SOA, which corresponds to point A in Fig. 2. The system gain that is calculated from the ratio of output power divided by the signal power is shown on the polar plot in Fig. 4. The non-circular shape of this plot quantifies the phase sensitive gain. One important parameter that characterizes the phase sensitive amplifier is the ratio of the maximum radius to the minimum radius of this figure. For the specific case illustrated on Fig. 4 this ratio is R=3:1.
Fig. 4. Simulated phase-sensitive gain (G) as a function of phase (φ) of the nonlinear Sagnac interferometer.
The ratio R depends on many parameters such as input powers, amplifier gain, recovery time and the line width enhancement factor αH. Fig. 5(a) shows the ratio R as a function of input pump power with signal power scaling as 7% of the pump power. Also shown on the graph is the gain at the optimal phase that rapidly drops at higher inputs.
Fig. 5. Plot of the contrast ratio (dashed-circles) and gain (solid-squares) at the optimal input phase as a function of input energy for an (a) unsaturated gain of 35 dB and (b) unsaturated gain of 69 dB.
The model shows that increasing the length of the SOA and keeping the total unsaturated gain (35 dB), has almost no effect on the curves shown on Fig. 5(a). Plot (a) uses the same coefficients for Figs. 2, 3 and 4. However, increasing the length by a factor of two, and keeping the gain per unit length constant (effectively doubling the unsaturated gain to 69 dB) enhances the phase sensitivity by an order of magnitude, as can be seen on Fig. 5(b). Increasing αH, and decreasing the gain recovery time also significantly increase R. The simulation results show that significant phase dependent gain is expected with pump powers in the milliwatt range using a 1 mm long SOA. This should be compared to power requirements ~1 Watt for 1km of the highly nonlinear fibers that are available today.
3. Experiment
The experimental set up to test the concept described above was constructed and is shown in Fig. 6. To minimize facet reflections inside the interferometer, the components inside of the Sagnac interferometer were fusion spliced with standard telecommunications components. Two polarization controllers were used to adjust the polarization in each arm. The delay line inside of the interferometer (optical delay #1) was used to adjust the relative arrival times of the clockwise and counter-clockwise pulses at the SOA. The optical delay outside of the interferometer (optical delay #2) is used to control the pulse overlap of the pump and signal pulses. The ports D2 and D4 provide monitoring signals for determining the relative phase and relative strength of the input pump and signal pulses. The ports D1 and D3 are used for monitoring the gain of the SOA.
Fig. 6. Experimental setup of the nonlinear interferometer comprising of a narrow band pass filter (BPF), optical power monitoring ports (D1-D4), polarization controllers (PC), the SOA and optical delay lines. Splitters are labeled by power ratios.
With only the pump pulses introduced into the interferometer, the couplers at ports D1 and D3 were used to independently monitor the optical power of the CW and CCW propagating pulses while optimizing the polarization for maximum gain in the SOA. Next, the pump pulses were used to examine the relative timing of the CW and CCW pulses at the SOA by adjusting the optical delay line #1. Figure 7 shows the output from the interferometer as the optical delay was varied from 0ps to 250ps.
Fig. 7. Measured output power of the nonlinear interferometer with only the pump pulses present. The labeled points correspond to the symmetric location of the SOA (A), a nonsymmetric geometry (B), and the symmetry point where the pulses arrive at equal and alternating intervals at the SOA (C).
Similar to the simulations shown in Fig. 2, a periodic pattern is observed with two minima in the output per period. The sharp minimum Point (A at 62ps) corresponds to pairs of pulses meeting in the middle of the SOA. The shallow minimum (Point C at 118ps) occurs when the CW and CCW pulses alternately pass through the SOA at equal intervals. Comparing the curve with the theoretical model predictions, shown on Fig. 2, we see several differences. The maxima are asymmetric about the dips. This is due to unbalanced losses in the two arms of the interferometer. This can be readily remedied in future implementations with a more careful selection of components. Also, the sharp minimum is somewhat narrower than that predicted by our simple model. The most likely explanation of this discrepancy is due to the oversimplified model for gain relaxation which does not include any fast (sub-picosecond) gain dynamics.
In order to test the concept of phase sensitive amplifications we used one common optical source to create both the strong pump and the weak control signal. As can be see in Fig. 6, the optical pump goes through a delay line and a phase shifter. By actively sweeping the input phase of the pump pulses the gain as a function of phase can be mapped to produce graphs like Fig. 4. The phase shifter consists of a 2-inch diameter piezoelectric cylinder with 5 meters of optical fiber coiled on the outside surface. The piezoelectric transducer stretches the fiber and produces a periodic phase variation between the signal and the pump when driven with a sinusoidal voltage. A phase shift of approximately 4π is achieved at a frequency of 90Hz.
Figure 8 shows typical experimental results observed with the pump and signal pulses injected into the interferometer. The optical delay is adjusted to set the interferometer to point A of the curve on Fig. 7. The lower trace (#1) is the output from port D2 showing the relative input phase variation between the probe and signal. The amplitude of the curve was multiplied by a factor of four to make it easier to observe the relative phase variation. The output of the interferometer is shown on upper trace (#2). This output of the interferometer shows strong dependence on the phase. It is evident on this graph that the output is varying at twice the frequency of the input phase variation. This doubling is the result of phase sensitive amplification in the Sagnac interferometer. Symmetrical phase sensitive gain is observed when the optical delay line of the interferometer is set to points A and C.
Fig. 8. Measured input (D2) and output of the nonlinear interferometer with both the pump and signal beams. The optical delay line is set to the symmetry point A. The amplitude of lower trace #1 was multiplied by a factor of four to make it appear bigger.
Fig. 9. Plots of the gain vs. relative phase from the experimental data and simulated data for the symmetry points A, B and C. Cartoons at the tops of the figures show relative timing of both pulse train relative to the SOA. Phase sensitive gain is quantified by the non-circular shape of these plots.
On the left sides of Figs. 9(a), (b) and (c) are polar plots of the experimentally measured gain as a function of the phase between the pump and signal, while simulation results are shown on the right side of each sub-figure. As seen in Fig. 9, the phase sensitive gain varies depending on the setting of the optical delay line #1. For setting A & C, there is phase sensitive gain, but when the optical delay line is at point B, the resulting circle indicates very little phase sensitive gain.
4. Discussion
The work presented here demonstrates phase sensitive amplification of optical pulses using an SOA inside a Sagnac interferometer with a weak control signal and strong pump pulse. This concept is similar to balanced homodyne detection where interfering a strong pump pulse with a weak signal in homodyne detection provides enhancement of the information from the weak signal that is transferred to electrical power in a pair of square law detector in homodyne detection. The fluctuations of the strong pump are cancelled out by balancing the electrical signals. Here, the interference produces variations of the gain in different spatial regions of the SOA which is transferred onto phase modulation of the pump beam. The fluctuations of the pump are cancelled out interferometrically in the optical domain.
One important concern in using an SOA for amplifying high-speed OOK data streams is pattern dependence, which is due to the gain relaxation time constant being comparable to the bit period. This should not be as much of a concern here as with some other nonlinear switches. Because there is always a strong pump pulse that saturates the SOA, while the control signal produces only small variations. The bidirectional use of the SOA is also expected to reduce pattern dependence effects because it more uniformly distributes the saturated gain throughout the SOA.
This phase sensitive amplifier can be particularly useful for regeneration of DPSK signals in which binary data is encoded in pulses that are 180 degrees out of phase with each other. The phase diagrams in Fig. 9 show that both symbols will experience equal gain. Small optical phase fluctuations will be reduced just like it was accomplished in the fiber based phase sensitive parametric amplifier [17]. Timing jitter of the signal pulses will also be reduced by the interaction with well clocked pump pulses.
One important feature of parametric amplifiers is the possibility of gain without additional amplified spontaneous emission (ASE) noise added to the signal. Clearly this is not expected in our system because of the ASE from the SOA. Noise properties of our phase sensitive amplifier are presently under investigation and their discussion is beyond the scope of this manuscript. We do expect that the noise figure here will be below that of the SOA as a direct optical amplifier. Also, it should be possible to operate the SOA with a wavelength that is close to or below the bandgap energy. In this region the gain will decrease, the saturation intensity will increase, but the linewidth enhancement factor will be significantly higher. These changes should cause the system to behave more like a Kerr medium, and amplification with a low noise figure should be possible. Of course, the operation in this region will increase the required pump power, but it should remain orders of magnitude below requirements for other nonlinear materials.
5. Conclusion
The described experiments represent a parametric amplifier designed to demonstrate the feasibility of phase sensitive gain. Phase sensitive gains with a ratio of more than 3:1 were experimentally measured. The experimental results and simulation predictions indicate reasonable agreement and show several paths that could result in significant performance improvements.
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