8.3.1 Equal amplitude and in-phase.
First we examine the case where both incoming waves have equal amplitude and are in phase. Along the left and bottom boundaries, we stipulate that the voltages are
given by
V1,j(t) =Asin(2πf t) (8.14a)
Vi,1(t) =Asin(2πf t) (8.14b)
For this experiment, we use nonlinear capacitors modeled by 8.12 with parameters b = 0.5 and VM = 1.9. Then, with the numerical setup given in Section 8.2, we find that the two incoming waves collide to produce one single outgoing wave of amplitude AR. By repeating the experiment with different values ofA, we study the dependence of the resonance amplitude AR on the incoming amplitude A. The resulting data is shown in Figure 8.5.
Based on this data, we make the following observations:
1. The ratio AR/Ameasures the efficiency of the combining; for linear combining, the output amplitude AR = 2A always. Since our lattice itself was nonlinear, we expected all incoming waves to combine nonlinearly, regardless of their am- plitude A. In our experiments, we found AR/A > 2 always, confirming this expectation.
2. When the voltages in the lattice are small (much less than unity), we may be in the KP regime, but whenAR ≥1V, it is clear that the weakly nonlinear KP theory no longer applies. At A = 0.3V, for example, we are able to produce an output signal that has more than six times the amplitude (AR > 6A) of the input. Note that for a range of inputs, 0.25V < A < 0.4V, the discrete resonance exceeds the AR = 4A bound established for the KP equation.
3. We also expected a saturation effect due to the form of nonlinearity we assumed in the capacitor model (8.12). For V > VM, the capacitance of each Cij is independent of V. A signal for which V > VM everywhere in space (e.g., a DC biased signal) would not see the nonlinearity of the lattice at all; for such a signal, linear dynamics prevail. However, the signals we deal with in this chapter, do not have any DC bias—they oscillate about zero with some
amplitude A. If A > VM, then some portion of the signal will experience only linear dynamics.
For this experiment in particular, VM = 1.9, so we do not expect nonlinear combination to work as efficiently when AR ≈ 1.9. This explains the drop in the ratio AR/A as A increases beyond A = 0.3. For A = 0.4, we are already producing an output signal with AR = 1.92. Increasing the input voltage to A = 0.5 has no effect on the output voltage—at this point, the nonlinearity of the capacitors has been saturated. In Figure 8.5, this corresponds to the part of the graph that is relatively flat.
The net effect of this circuit is to combine and convert the sinusoidal inputs into a soliton-like pulse, as shown in Figure 8.6.
We did not record in Figure 8.5 the wave vectors of the output signals. It is clear that the input signals have wave vectors k1 = (1,0) and k2 = (0,1). Recall from (8.6) that for KP soliton interactions, input signals with these wave vectors would combine to form an output signal with wave vector k3 = (1,1), which is precisely what we observed in all of the experiments we performed to generate Figure 8.5.
8.3.2 Unequal Amplitude and In-Phase
Next we consider the effect of varying the amplitude of the input signals, while keeping these signals in phase. Along the left and bottom boundaries, we prescribe:
V1,j(t) =ALsin(2πf t) (8.15a) Vi,1(t) =ABsin(2πf t). (8.15b) Just as for the equal amplitude case, the capacitor model is (8.12) with b = 0.5 and VM = 1.9. All other parameters are given in Section 8.2. Using repeated numerical simulations, we compute the amplitude of the output signal AR for various cases of input signal amplitudes AL and AB. The data is recorded in Table 8.1.
Homogeneity of the lattice (8.11) ensures that the table is symmetric across its
0.4 0.3 0.25 0.2 0.15 0.1 0.4 1.92 1.90 1.90 1.89 1.78 1.73 0.3 1.85 1.39 1.11 0.930 0.799
0.25 1.02 0.845 0.713 0.599
0.2 0.690 0.563 0.467
0.15 0.460 0.368
0.1 0.280
Table 8.1: AmplitudeARof the outgoing resonant pulse that forms from two incoming sinusoids of amplitude AL and AB. All amplitudes are in Volts.
0.4 0.3 0.25 0.2 0.15 0.1 0.4 2.40 2.71 2.92 3.15 3.24 3.45 0.3 3.08 2.53 2.23 2.07 2.00
0.25 2.04 1.88 1.78 1.71
0.2 1.73 1.61 1.56
0.15 1.53 1.47
0.1 1.40
Table 8.2: Efficiency AR/(AL+AB) (or the ratio of outgoing amplitude to the sum of incoming amplitudes) as a function of AL and AB. All amplitudes are in Volts.
diagonal, so only half the entries are shown. In all cases, it is clear that the efficiency ratio AR/(AL+AB)> 1, so the wave interaction is always nonlinear, just as in the equal-amplitude case. However, a closer inspection of this ratio, as shown in Table 8.2, reveals an interesting effect. WhenAL<0.4, the efficiency increases asAB increases.
However, whenAL = 0.4, the efficiency increases asAB decreases. In future work, we shall investigate theoretically the source of this phenomenon.
Other features of the unequal-amplitude case are the same as the equal-amplitude case. Let us specifically mention that input signals colliding at a right angle yield an output pulse that travels at a 45-degree angle. The net effect of the circuit is to generate concentrated soliton-like pulses from sinusoidal inputs.
8.3.3 Equal amplitude but out-of-phase
Finally we consider the effect of varying the interaction angle of the two inputs, while keeping their amplitudes fixed and equal:
V1,j(t) = 0.25 sin(2πf t+j∆ϕ) (8.16a)
Vi,1(t) = 0.25 sin(2πf t). (8.16b)
Each choice of the phase shift ∆ϕcorresponds to a certain interaction angleθbetween the two incoming waves. As an example, the trivial choice of ∆ϕ= 0 corresponds to an interaction angle of θ=π/2. In general,
cosθ = λ h
∆ϕ
2π , (8.17)
wherehis the distance between two nodes (i,1) and (i+ 1,1) andλis the wavelength of the incoming signal. In practice‡, for an input frequency of 10 GHz, the nonlinear lattice described in Section 8.2 has a ratio λ/h ∼ 22.4. We consider the collision at an angle θ of two waves that begin as sinusoidal forcing (8.16) with phase shift ∆ϕ.
From each collision, a resonant outgoing wave with amplitude AR results. We plot the outgoing amplitude AR as a function of θ in Figure 8.7.
Note that the efficiency ratio in this case is simply AR/(0.25 + 0.25) = 2AR. Therefore, it is clear that the resonant collision of maximum efficiency occurs when the interaction angle is 90 degrees. However, if the interaction angle is decreased to 82 degrees, the amplitude of the output signal is decreased only by 7%. For practical applications, even if the interaction angle is not precisely 90 degrees, we still see nonlinear resonant combining.
‡The ratio λ/h ∼ 22.4 was established through direct numerical simulation of the nonlinear lattice.