Figure 2.4: Emitter setup of basic electro-optic intensity chaos generator, adapted from [103].
One of the systems belonging to the suitable class of chaotic systems able to develop high complexity was proposed by Goedgebuer et al.
in 2002 [102]. This system uses a nonlinear delay feedback loop illuminated by a CW semiconduc-tor laser (see Fig. 2.4). The nonlinearity is im-plemented through a lithium niobate (LiNbO)3 Mach-Zehnder modulator, which is a customized integrated optics telecom device. This architec-ture has many advantages for optical communi-cations. Through having good stability and con-trollability in real conditions, it also has great
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architectural flexibility so that some components can be replaced to change differ-ent parameters (bandwidth, noise, efficiency etc..) or even to modify the architecture (additional delays, transformations etc..). Furthermore, it can be easily designed in matched pair. The system is composed of:
• A CW semiconductor laser (SL) delivering a constant power P0 =hνI (where h stands for the Planck constant, ν is the photon emission frequency and I ∝
|E(t)|2 the photon number).
• A Mach-Zehnder modulator (MZM): The light coming from the SL is evenly split into the two arms of the MZM and interferes at its output. The refractive index of one arm is modulated by the output voltage of an electronic driver.
The applied voltage has two components: a constant or DC component VB that allows to select the operating point of the modulator; and a radio-frequency (RF) component V(t) which is used to generate the chaos. The complex enve-lope of the electric field at the MZM output can be written as
E(t) = 1
where VπRF and VπDc stand for the RF wave and the bias electrode half-wave, respectively andE0 is the amplitude of the SL output. The optical output power is given by
• A fiber delay line used to delay the optical signal in time. The fiber is assumed not to be dispersive (independent to the frequency of delayed signal) so that the delay time T is given by as T =L/Vg, where Lis the fiber length while Vg
is the group velocity.
• An amplified diode with sensitivity S to detect the optical signal and convert it into an electrical signal,
• A RF driver whose output modulates the MZM and closes the delay loop. The RF driver is naturally a filter which can be low-pass, high-pass or band-pass of any order. Table 2.1 overviews the main fundamental 1st order filters.
Here, we assume that the RF driver behaves as a first-order bandpass linear filter with gain G0. Thus the system can be described by the RF output voltage as
1 + τ
Filter kind Representation in time domain Bandwidth Low-pass 1st order x(t) +τdx(t)dt [0, fH = 1/(2πτ)]
high-pass 1st order 1θRt
t0x(t′)dt′ [fB = 1/(2πθ),∞] band-pass 1st order 1 + τθ
x(t) +τdx(t)dt +1θRt
t0x(t′)dt′ [fB, fH] Table 2.1: Filters and their corresponding equations.
0 1 2 3 4 5
G -2
0 2
Amplitude
Figure 2.5: Bifurcation diagram showing signal amplitude distribution in dependence on feedback gainGfor parametersτ = 20 ps,θ= 1.6µs,G= 5,T= 30 ns andφ=π/4.
whereη0 accounts for overall losses. Since we are going to consider systems for which τ ≪θ, the term τ /θ can be neglected as compared to 1.
The message m(t) is added within the chaotic carrier as shown in Fig. 2.4.
In practice, this is achieved through another SL having the same wavelength but with orthogonal polarization in order to prevent for the interference between light beams of the chaotic optical intensity carrier and the optical intensity binary data.
For security issues, fast polarization scrambling is required to prevent from a simple polarization splitting attack on the transmission line. For the modeling purposes, we introducex(t) =πV(t)/(2VπRF),u=Rt
t0x(t′)dt′so that the system can be described by
x+τdx dt +1
θu = G
αmm(t−T) + cos2[x(t−T) +φ] , (2.6) du
dt = x, (2.7)
where the parameters are the fast cutoff time scale τ, the slow cutoff response time θ, the offset phase φ = πVB/(2VπDC) and the normalized electro-optical loop gain G = πSG0P0η0/2VπRF while αm is the ratio between the message and the carrier light beam. For high values of the bifurcation parameter G, the interplay between
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nonlinearity and delay generates a chaotic output. Typical parameters used to induce strong chaotic behavior in this system are τ = 20 ps, θ = 1.6 µs, G = 5, T = 30 ns and φ=π/4. Figure 2.5 displays the bifurcation diagram of such system as a function of the feedback gain G. It can be seen that for large value of G, the system is highly chaotic. Its Lyapunov dimension has found to be greater than 1000 for T = 30 ns and G= 5 [104]. With such large dimension, one may expect that it will be computationally very complex to reconstruct the high-dimensional attractor from the time series.
To decrypt the encoded message a receiver is built similarly to the emitter using an open loop configuration as shown in Fig. 2.6. The signal coming from the emitter is split in two parts. One part is used to drive the nonlinear processing branch that, in suitable conditions, regenerates the chaotic carrier without message x′(t). The receiver can be described by
x′+τ′dx′
In a back-to-back configuration, x′(t) synchronizes with x(t) for identical pa-rameters. As shown in Fig. 2.6 the output of the receiver MZM is then detected by the photodiode PD− and combined with the second part of the transmitted signal detected by PD+. Therefore, the message is obtained by canceling the chaos in the combiner. The output of the combiner reads
m′(t)∝G
αmm(t) + cos2[x(t) +φ] +G′cos2[x(t) +φ′]. (2.10)
Figure 2.6: Receiver setup of basic electro-optic intensity chaos generator, adapted from [103] .
The cancellation of the message is eas-ily achieved experimentally operating the MZM with aπ−shifted static phase in the receiver (e.g.
φ′ =φ+π/2). Alternatively this could have been done using balanced photodiodes and exchang-ing their inputs or usexchang-ing an inverted amplifier at the receiver. Note that for small mismatch, ac-ceptable synchronization quality is still achieved [105]. Moreover, practical demonstrations have shown that even when the emitter and the re-ceiver are located far one from another, good synchronization quality (with BER of the order 10−7 for a message at 3 Gb/s) is obtained after compensating for the fiber losses and its disper-sion effects [74].