## Electro-optic phase chaos systems with an internal variable and a digital key |

Optics Express, Vol. 20, Issue 23, pp. 25333-25344 (2012)

http://dx.doi.org/10.1364/OE.20.025333

Acrobat PDF (1183 KB)

### Abstract

We consider an electro-optic phase chaos system with two feedback loops organized in a parallel configuration such that the dynamics of one of the loops remains internal. We show that this configuration intrinsically conceals in the transmitted variable the internal delay times, which are critical for decoding. The scheme also allows for the inclusion, in a very efficient way, of a digital key generated as a long pseudorandom binary sequence. A single digital key can operate both in the internal and transmitted variables leading to a large sensitivity of the synchronization to a key-mismatch. The combination of intrinsic delay time concealment and digital key selectivity provides the basis for a large enhancement of the confidentiality in chaos-based communications.

© 2012 OSA

## 1. Introduction

1. S. Donati and C.R. Mirasso, “Feature section on optical chaos and applications to cryptography,” IEEE J. Quantum Electron. **38**, 1138–1184 (2002). [CrossRef]

2. R. Lavrov, M. Peil, M. Jacquot, L. Larger, V. Udaltsov, and J. Dudley, “Electro-optic delay oscillator with nonlocal nonlinearity: Optical phase dynamics, chaos, and synchronization,” Phys. Rev. E **80**, 026207/1–9 (2009). [CrossRef]

3. J. P. Goedgebuer, L. Larger, and H. Porte, “Optical cryptosystem based on synchronization of hyperchaos generated by a delayed feedback tunable laser diode,” Phys. Rev. Lett. **80**, 2249–2252 (1998). [CrossRef]

4. L. Larger, J. P. Goedgebuer, and F. Delorme, “Optical encryption system using hyperchaos generated by an optoelectronic wavelength oscillator,” Phys. Rev. E **57**, 6618–6624 (1998). [CrossRef]

5. A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C.R. Mirasso, L. Pesquera, and K.A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature **438**, 343–346 (2005). [CrossRef] [PubMed]

6. R. Lavrov, M. Jacquot, and L. Larger, “Nonlocal nonlinear electro-optic Phase dynamics demonstrating 10*Gbs/s* chaos communications,” IEEE J. Quantum Electron. **46**, 1430–1435 (2010). [CrossRef]

7. X. Li, W. Pan, B. Luo, and D. Ma, “Mismatch robustness and security of chaotic optical communications based on injection-locking chaos synchronization,” IEEE J. Quantum Electron. **42**, 953–960 (2006). [CrossRef]

9. V.S. Udaltsov, J.-P. Goedgebuer, L. Larger, J.-B. Cuenot, Pascal Levy, and W.T. Rhodes, “Cracking chaos-based encryption systems ruled by nonlinear time delay differential equations,” Phys. Lett. A **308**, 54–60 (2003). [CrossRef]

10. S. Ortín, J. Gutiérrez, L. Pesquera, and H. Vasquez, “Nonlinear dynamics extraction for time-delay systems using modular neural networks synchronization and prediction,” Physica A **351**, 133–141 (2005). [CrossRef]

2. R. Lavrov, M. Peil, M. Jacquot, L. Larger, V. Udaltsov, and J. Dudley, “Electro-optic delay oscillator with nonlocal nonlinearity: Optical phase dynamics, chaos, and synchronization,” Phys. Rev. E **80**, 026207/1–9 (2009). [CrossRef]

11. C. R. Mirasso, P. Colet, and P. García-Fernández, “Synchronization of chaotic semicondcutor lasers: Application to encoded communications,” Phot. Tech. Lett. **8**, 299–301 (1996). [CrossRef]

16. V. Z. Tronciu, C. Mirasso, P. Colet, M. Hamacher, M. Benedetti, V. Vercesi, and V. Annovazzi-Lodi, “Chaos generation and synchronization using an integrated source with an air gap,” IEEE J. Quantum Electron. **46**, 1840–1846 (2010). [CrossRef]

10. S. Ortín, J. Gutiérrez, L. Pesquera, and H. Vasquez, “Nonlinear dynamics extraction for time-delay systems using modular neural networks synchronization and prediction,” Physica A **351**, 133–141 (2005). [CrossRef]

17. R. Hegger, M. J. Bünner, H. Kantz, and A. Giaquinta, “Identifying and modeling delay feedback systems,” Phys. Rev. Lett. **81**, 558–561 (1998). [CrossRef]

19. L. Zunino, M. C. Soriano, I. Fischer, O. A. Rosso, and C. R. Mirasso, “Permutation-information-theory approach to unveil delay dynamics from time-series analysis,” Phys. Rev. E **82**, 046212/1–9 (2010). [CrossRef]

20. D. Rontani, A. Locquet, M. Sciamanna, and D. S. Citrin, “Loss of time-delay signature in the chaotic output of a semiconductor laser with optical feedback,” Opt. Lett. **32**, 2960 (2007). [CrossRef] [PubMed]

21. D. Rontani, A. Locquet, M. Sciamanna, D. S. Citrin, and S. Ortin, “Time-delay identification in a chaotic semiconductor laser with optical feedback: A dynamical point of view,” IEEE J. Quantum Electron. **45**, 879–891 (2009). [CrossRef]

22. R. M. Nguimdo, M. C. Soriano, and P. Colet, “Role of the phase in the identification of delay time in semiconductor lasers with optical feedback,” Opt. Lett. **36**, 4332–4334 (2011). [CrossRef] [PubMed]

23. R. M. Nguimdo, G. Verschaffelt, J. Danckaert, and G. Van der Sande, “Loss of time-delay signature in chaotic semiconductor ring lasers,” Opt. Lett. **37**, 2541–2544 (2012). [CrossRef] [PubMed]

2. R. Lavrov, M. Peil, M. Jacquot, L. Larger, V. Udaltsov, and J. Dudley, “Electro-optic delay oscillator with nonlocal nonlinearity: Optical phase dynamics, chaos, and synchronization,” Phys. Rev. E **80**, 026207/1–9 (2009). [CrossRef]

12. L. Larger, J. Goedgebuer, and V. Udaltsov, “Ikeda-based nonlinear delayed dynamics for application to secure optical transmission systems using chaos,” Comptes Rendus Physique **5**, 669–681 (2004). [CrossRef]

14. R. M. Nguimdo, P. Colet, and C. R. Mirasso, “Electro-optic delay devices with double feedback,” IEEE J. Quantum Electron. **46**, 1436–1443 (2010). [CrossRef]

24. R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. **16**, 347 (1980) [CrossRef]

**80**, 026207/1–9 (2009). [CrossRef]

25. J. Hizanidis, S. Deligiannidis, A. Bogris, and D. Syvridis, “Enhancement of chaos encryption potential by combining all-optical and electrooptical chaos generators,” IEEE J. Quantum Electron. **46**, 1642–1649 (2010). [CrossRef]

26. H. C. Wang, K. P. Ho, H. K. Chen, and H. C. Lu, J. Lightw. Technol. “Phase and amplitude responses of narrow-band optical filter measured by microwave network analyzer,” J. Lightwave Technol. **24**, 5075 (2006) [CrossRef]

27. L. Zimmermann, K. Voigt, G. Winzer, K. Petermann, and C. M. Weinert, “*C*-band optical 90°-hybrids based on silicon-on-insulator 4 × 4 waveguide couplers,” IEEE Photon. Technol. Lett. **21** (3), 143 (2009). [CrossRef]

22. R. M. Nguimdo, M. C. Soriano, and P. Colet, “Role of the phase in the identification of delay time in semiconductor lasers with optical feedback,” Opt. Lett. **36**, 4332–4334 (2011). [CrossRef] [PubMed]

**80**, 026207/1–9 (2009). [CrossRef]

28. R. M. Nguimdo, P. Colet, L. Larger, and L. Pesquera, “Digital key for chaos communication performing time delay concealment,” Phys. Rev. Lett. **107**, 034103/1–4 (2011). [CrossRef]

28. R. M. Nguimdo, P. Colet, L. Larger, and L. Pesquera, “Digital key for chaos communication performing time delay concealment,” Phys. Rev. Lett. **107**, 034103/1–4 (2011). [CrossRef]

*T*, and an imbalanced Mach-Zehnder interferometer with differential delay

_{i}*δT*. The Mach-Zehnder interferometer transforms in a nonlinear way the phase variations into intensity variations, which are finally detected by a photodiode. The electrical output of the photodiode after amplification is the input for the phase modulation of the other chain. Therefore the two delay chains operate in a serial configuration, and can be viewed as part of an overall delay loop. In this configuration time-delay concealment occurs only when the digital key is present and operates at a bit rate above a threshold given by the differential delay time of the chain in which the key is introduced.

_{i}28. R. M. Nguimdo, P. Colet, L. Larger, and L. Pesquera, “Digital key for chaos communication performing time delay concealment,” Phys. Rev. Lett. **107**, 034103/1–4 (2011). [CrossRef]

**107**, 034103/1–4 (2011). [CrossRef]

## 2. System

*i*= 1, 2 refer to a given loop. An electro-optic phase modulator (PM

_{1}) seeded by a continuous-wave (CW) telecom semiconductor laser (SL) is phase-modulated by a voltage proportional to

*x*

_{1}(

*t*). The output of PM

_{1}is then split into two parts. One part is sent to the receiver while the second part is successively phase modulated by a voltage proportional to

*x*

_{2}(

*t*) and by the digital key

*R*(

*t*), generated as a Pseudo-Random Bit Sequence (PRBS). After the double phase modulation the resulting optical signal is divided into two parts. Each part is fed to a fiber delay line which delays the signal by a time

*T*and then fed to an Mach-Zehnder interferometer (MZI

_{i}*) with imbalance time*

_{i}*δT*, which converts phase variations into intensity variations. The intensity variations are detected by a photodiode (PD

_{i}*) and amplified by an RF driver with an effective gain G*

_{i}*. The output of each amplifier, proportional to*

_{i}*x*, is applied to the respective RF electrode of PM

_{i}*to close the loop*

_{i}*i*. The message

*m*(

*t*) is encoded as an additional phase modulation using another PM placed in between the SL and PM

_{1}(as shown in Fig. 1) or alternatively just after PM

_{1}and prior to the split of the signal to be transmitted to the receiver. At this point we would like to note the following points: first, only the output of PM

_{1}is transmitted to the receiver, so loop 2 can be considered as internal. Second, a total of four phase modulations (two chaotic proportional to

*x*

_{1}(

*t*) and

*x*

_{2}(

*t*) + pseudorandom + message) are successively applied to the optical signal delivered at the SL output before its undergoes phase-to-intensity conversion. Third, this system requires less components than the previous one [28

**107**, 034103/1–4 (2011). [CrossRef]

**80**, 026207/1–9 (2009). [CrossRef]

6. R. Lavrov, M. Jacquot, and L. Larger, “Nonlocal nonlinear electro-optic Phase dynamics demonstrating 10*Gbs/s* chaos communications,” IEEE J. Quantum Electron. **46**, 1430–1435 (2010). [CrossRef]

**107**, 034103/1–4 (2011). [CrossRef]

*V*(

_{i}*t*) and proceeding as in [2

**80**, 026207/1–9 (2009). [CrossRef]

29. R. M. Nguimdo, R. Lavrov, P. Colet, M. Jacquot, Y. K. Chembo, and L. Larger, “Effect of fiber dispersion on broadband chaos communications implemented by electro-optic nonlinear delay phase dynamics,” J. Lightwave Technol. **28**, 2688–2616 (2010). [CrossRef]

*x*(

_{i}*t*) =

*πV*(

_{i}*t*)/(2

*V*) where

_{π,i}*V*is the half-wave voltage of the modulator PM

_{π,i}*where*

_{i}*du*/

_{i}*dt*=

*x*, Δ(

_{i}*F*)

_{t0}=

*F*(

*t*−

*t*

_{0}) −

*F*(

*t*−

*t*

_{0}−

*δt*

_{0}) and

*ϕ*is the static offset phase of MZI

_{i}*. For numerical simulations, we consider the key physical parameters arbitrary chosen, within the range of experimentally accessible values [2*

_{i}**80**, 026207/1–9 (2009). [CrossRef]

**107**, 034103/1–4 (2011). [CrossRef]

*T*

_{1}= 15 ns and

*T*

_{2}= 17 ns,

*τ*

_{1}= 20 ps,

*τ*

_{2}= 12.2 ps,

*θ*

_{1}= 1.6

*μ*s,

*θ*

_{2}= 1.6

*μ*s,

*δT*

_{1}= 510 ps,

*δT*

_{2}= 400 ps,

*ϕ*

_{1}=

*π*/4,

*ϕ*

_{2}=

*π*/8,

*G*

_{1}= 5 and

*G*

_{2}= 3. These parameters have been used for the original setup in [2

**80**, 026207/1–9 (2009). [CrossRef]

6. R. Lavrov, M. Jacquot, and L. Larger, “Nonlocal nonlinear electro-optic Phase dynamics demonstrating 10*Gbs/s* chaos communications,” IEEE J. Quantum Electron. **46**, 1430–1435 (2010). [CrossRef]

## 3. Delay Time Concealment

*C*(

*s*), delayed mutual information (DMI), extrema statistics and filling factor [17

17. R. Hegger, M. J. Bünner, H. Kantz, and A. Giaquinta, “Identifying and modeling delay feedback systems,” Phys. Rev. Lett. **81**, 558–561 (1998). [CrossRef]

19. L. Zunino, M. C. Soriano, I. Fischer, O. A. Rosso, and C. R. Mirasso, “Permutation-information-theory approach to unveil delay dynamics from time-series analysis,” Phys. Rev. E **82**, 046212/1–9 (2010). [CrossRef]

*C*(

*s*) and DMI are robust to noise perturbations and therefore are suitable to crack the time delay in practical situations. For a time series

*x*(

*t*),

*C*(

*s*) is defined as where

*x*(

_{s}*t*) =

*x*(

*t*−

*s*) and 〈...〉 stands for the time average. The DMI measures the information on

*x*(

*t*) that can be obtained by observing

*x*(

_{s}*t*) where

*p*(

*x*(

*t*)) is the probability distribution function of

*x*(

*t*) while

*p*(

*x*(

*t*),

*x*(

_{s}*t*)) is the joint probability distribution function.

*m*= 0). The relevant delay times for the model are

*T*

_{1},

*T*

_{1}+

*δT*

_{1},

*T*

_{2}and

*T*

_{2}+

*δT*

_{2}. Figure 2 displays the autocorrelation (a) and the DMI (b) without (solid line) and with a PRBS of amplitude

*π*/2 at 3 Gb/s (dashed line), computed from a long series for

*x*

_{1}(

*t*). Without PRBS, two relevant peaks are found both in the autocorrelation and in the DMI at delay times

*T*

_{1}and

*T*

_{1}+

*δT*

_{1}as expected. What is more relevant is that no peak is found around the internal loop delay time positions,

*T*

_{2}and

*T*

_{2}+

*δT*

_{2}. We have also checked that using the time distribution extrema and the filling factor methods these delay time signatures remain concealed. Therefore the system fully conceals the internal loop delay times even without digital key.

*j*

^{2}= −1, Δ̄(

*F*)

_{δt0}=

*F*(

*t*) −

*F*(

*t*−

*δt*

_{0}) and

**FT**{

*z*} stands for the Fourier transform of

*z*. For

*δT*

_{1}=

*δT*

_{2}and

*ϕ*

_{1}=

*ϕ*

_{2}, it turns out that Equation (5) establishes a linear relationship between

*x*

_{1}and

*x*

_{2}. Consequently information on the internal variable dynamics can be easily retrieved from the transmitted variable

*x*

_{1}(

*t*) and therefore for

*δT*

_{1}=

*δT*

_{2}one should expect that none of the time delays is concealed. And as shown in Eq. (5) this is certainly the case even if

*T*

_{1}is different from

*T*

_{2}. In fact, even considering different values for the offset phases,

*ϕ*

_{1}≠

*ϕ*

_{2}, we have numerically found that the delay times can be identified if

*δT*

_{1}=

*δT*

_{2}. The numerical results for the autocorrelation and the DMI for

*δT*

_{1}=

*δT*

_{2}= 400 ps,

*ϕ*

_{1}=

*π*/4 and

*ϕ*

_{2}=

*π*/8 computed from the transmitted variable

*x*

_{1}are shown in Fig. 3. For this specific case, we found that the maximum of the cross-correlation between

*x*

_{1}and

*x*

_{2}takes place at

*T*

_{2}−

*T*

_{1}(as predicted) and is quite large, 0.7. Peaks at

*T*

_{2}and

*T*

_{2}+

*δT*

_{2}are apparent. Clear peaks also appear at

*T*

_{2}−

*T*

_{1}(out of the figure range). In fact, while typically the delay time signature is reduced when increasing the overall loop gain (which increases the complexity of the chaos), for

*δT*

_{1}=

*δT*

_{2}, the delay time can always be identified even for

*G*

_{1}=

*G*

_{2}= 15, way beyond experimental limits.

*ξ*| increases the peak sizes both in C(s) and DMI(s) decrease, achieving full concealment for a mismatch greater than 20%. In particular, the delay time signature is completely lost in

*C*(

*s*) already at a 10% mismatch in correspondence with the decay time of the autocorrelation function while the DMI decays even faster to a residual value, which, although small, remains distinguishable all the way up to 20% mismatch. The reason for this larger range of detection capability is that mutual information measures the relationship between variables beyond a linear correlation. In any case, for a differential delay mismatch above 20% not even mutual information is capable of finding traces of the internal delay times in the transmitted signal.

*T*

_{1}and

*T*

_{1}+

*δT*

_{1}. As shown in Fig. 2 the addition of PRBS successfully conceals them for the autocorrelation function [Fig. 2(a)] but not for the DMI [Fig. 2(b)] (although the size of the peaks is significantly reduced). The fact that the PRBS does not completely suppress these peaks can be understood as follows. Without PRBS the size of the peaks signaling

*T*

_{1}and

*T*

_{1}+

*δT*

_{1}is stronger than in the case of the serial configuration for the same parameters [28

**107**, 034103/1–4 (2011). [CrossRef]

*x*

_{1}(

*t*) and its delayed version for the parallel configuration is stronger than for the serial one. The effective amplitude of the chaos driving the nonlinear term in Eq. (1) can be twice as large as that of the serial configuration since the signal delivered by the SL is successively modulated by

*x*

_{1}and

*x*

_{2}. Since the mixing of the PRBS and the chaos is less balanced the delay time is not concealed. Despite that, the PRBS remains efficiently masked by the chaos as the cross-correlation between

*x*

_{1}(

*t*) and

*R*(

*t*) is of the order of 10

^{−3}.

## 4. Synchronization

_{1}output, so

*x*

_{2}(

*t*) has to be generated at the receiver, through an internal closed loop. This makes the receiver to operate in semi-closed loop, which is known to be very sensitive to synchronization. The quality of the synchronization depends on several factors, including the coupling strength, parameter mismatch, noise, degradation due to fiber propagation effects. The latter has been considered in [29

29. R. M. Nguimdo, R. Lavrov, P. Colet, M. Jacquot, Y. K. Chembo, and L. Larger, “Effect of fiber dispersion on broadband chaos communications implemented by electro-optic nonlinear delay phase dynamics,” J. Lightwave Technol. **28**, 2688–2616 (2010). [CrossRef]

30. A. Argyris, E. Grivas, M. Hamacher, A. Bogris, and D. Syvridis, “Chaos-on-a-chip secures data transmission in optical fiber links,” Opt. Express , vol. **18**, 5188–5198, 2010. [CrossRef] [PubMed]

**107**, 034103/1–4 (2011). [CrossRef]

*dv*/

_{i}*dt*=

*y*. Since the message is encoded in the phase it has to be demodulated. This is done using a standard differential phase shift keying demodulator consisting in an MZI with an imbalance delay time

_{i}*δT*and a photodetector [28

_{m}**107**, 034103/1–4 (2011). [CrossRef]

*m*′ is obtained from

*P*. In the ideal case of perfect synchronization

*y*

_{1}=

*x*

_{1}and

*m*′ reproduces the original message

*m*.

*G*

_{2}= 0 in Eq. (6),

*y*

_{2}(

*t*) decays to zero after a characteristic time

**80**, 026207/1–9 (2009). [CrossRef]

**107**, 034103/1–4 (2011). [CrossRef]

*G*

_{2}= 0, and disregarding the message

*m*(

*t*) = 0, we gradually increase

*G*

_{2}in order to investigate the range of

*G*

_{2}for which synchronization is possible. This can be done estimating the largest conditional Lyapunov exponent (LCLE) [31

31. K. Pyragas, “Synchronization of coupled time-delay systems: Analytical estimations,” Phys. Rev. E **58**, 3067–3071 (1998). [CrossRef]

*δ*∈ 𝕃 (where 𝕃 is a suitable space function) constructed in the interval [

*t*−

*T*,

*t*]. Since the system has four different delay times,

*T*

_{1},

*T*

_{1}+

*δT*

_{1},

*T*

_{2},

*T*

_{2}+

*δT*

_{2}, we should consider the largest one

*T*=

_{D}*T*

_{2}+

*δT*

_{2}. Defining

*δ*=

_{i}*y*(

_{i}*t*) −

*x*(

_{i}*t*) the LCLE defined in [31

31. K. Pyragas, “Synchronization of coupled time-delay systems: Analytical estimations,” Phys. Rev. E **58**, 3067–3071 (1998). [CrossRef]

*λ*< 0. By subtracting Eq. (1) from (6) and linearizing for

_{L}*δ*, one obtains where

_{i}*dε*/

_{i}*dt*=

*δ*. Thus

_{i}*δ*

_{1}(

*t*) to be used in Eq. (8) can be obtained by numerical integration of Eqs. (1) and (9). Note that

*λ*depends implicitly on the feedback strengths

_{L}*G*

_{1}and

*G*

_{2}. Synchronization between the external variables

*x*

_{1}(

*t*) and

*y*

_{1}(

*t*) is only possible if internal variables do synchronize first, i.e.

*δ*

_{2}(

*t*) = 0. Once

*δ*

_{2}= 0 the dynamics of

*δ*

_{1}decays to zero as

*λ*when synchronization takes place Figure 6(a) displays the LCLE as a function of

_{L}*G*

_{2}for

*G*

_{1}= 5 which corresponds to a relatively high gain in the external loop. Stable synchronization is found for

*G*

_{2}for which

*x*

_{1}(

*t*) and

*y*

_{1}(

*t*) synchronize, i.e.

*δ*

_{1}(

*t*) or

*δ*

_{2}(

*t*) grows in time and therefore

*λ*becomes positive indicating desynchronization between the emitter and receiver. We have found that even setting

_{L}*R*= 0, the range of values for

*G*

_{2}for which synchronization takes place remains the same. Similar values for the synchronization threshold

*G*

_{1}provided

*G*

_{1}> 3. Therefore in what follows we will consider only

*x*

_{1}(

*t*) and

*y*

_{1}(

*t*) can also be evaluated through the root-mean square synchronization error

*σ*in logarithmic scale as function of

*G*

_{2}. As expected from the LCLE analysis there is perfect synchronization (

*σ*< 10

^{−13}corresponding to the numerical accuracy) up to

*G*

_{2}, the synchronization rapidly degrades as indicated by an error of order 1.

## 5. Effect of the PRBS on Synchronization

*R*′ ≠

*R*the dynamics of

*δ*(

_{i}*t*) are given by These equations indicate that for

*R*′ ≠

*R*synchronization is degraded both on the internal and the transmitted variables since neither

*δ*

_{2}nor

*δ*

_{1}decay to zero.

*σ*as a function of PRBS mismatch for different values of the internal loop gain

*G*

_{2}while Fig. 7(b) shows the bit error rate (BER) of the recovered message. For

*G*

_{2}= 0, there is no internal variable and therefore synchronization degradation relies on the effect of the PRBS on the transmitted variable. The synchronization error grows faster with the mismatch and just a mismatch fraction

*η*= 1% in the PRBS leads to a synchronization error of 25% which corresponds to a quite poor synchronization. The BER grows linearly with the PRBS mismatch. The results obtained for

*G*

_{2}= 0 coincide with those obtained in the serial configuration when both loops have a relatively large gain,

*G*

_{1}=

*G*

_{2}= 5 [28

**107**, 034103/1–4 (2011). [CrossRef]

*G*

_{2}, the degradation becomes stronger both in synchronization error and BER. As an illustration, for

*G*

_{2}= 3 the degradation for

*η*= 0.4% (i.e ≈ 131 mismatched bits in the receiver PRBS for a key 2

^{15}= 32768 bit long) is equivalent to that obtained for

*η*= 2% (i.e ≈ 655 mismatched PRBS bits) when

*G*

_{2}= 0. In other words, the PRBS mismatch sensitivity for

*G*

_{2}= 3 is 5 times larger than that obtained in the serial configuration with

*G*

_{2}= 5 [28

**107**, 034103/1–4 (2011). [CrossRef]

*δT*, the effect of the PRBS is largely reduced. This is because at those low bit rates

_{i}*R*(

*t*) and

*R*(

*t*−

*δT*) have the same value most of the time. The same happens for

_{i}*R*′(

*t*) and

*R*′(

*t*−

*δT*). Therefore Δ(

_{i}*R*′ −

*R*)

_{Ti}=

*R*′(

*t*−

*T*) −

_{i}*R*′(

*t*−

*T*−

_{i}*δT*) −

_{i}*R*(

*t*−

*T*) +

_{i}*R*(

*t*−

*T*−

_{i}*δT*) vanishes even if

_{i}*R*and

*R*′ are different.

## 6. Conclusions

*G*

_{2}≈ 3.2. We have shown that the nonlinear dynamics of the system allows for a decorrelation between the internal variables and the transmitted signal so that the system intrinsically conceals the internal delay times. This was not possible in the serial configuration introduced before [28

**107**, 034103/1–4 (2011). [CrossRef]

**107**, 034103/1–4 (2011). [CrossRef]

## Acknowledgments

## References and links

1. | S. Donati and C.R. Mirasso, “Feature section on optical chaos and applications to cryptography,” IEEE J. Quantum Electron. |

2. | R. Lavrov, M. Peil, M. Jacquot, L. Larger, V. Udaltsov, and J. Dudley, “Electro-optic delay oscillator with nonlocal nonlinearity: Optical phase dynamics, chaos, and synchronization,” Phys. Rev. E |

3. | J. P. Goedgebuer, L. Larger, and H. Porte, “Optical cryptosystem based on synchronization of hyperchaos generated by a delayed feedback tunable laser diode,” Phys. Rev. Lett. |

4. | L. Larger, J. P. Goedgebuer, and F. Delorme, “Optical encryption system using hyperchaos generated by an optoelectronic wavelength oscillator,” Phys. Rev. E |

5. | A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C.R. Mirasso, L. Pesquera, and K.A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature |

6. | R. Lavrov, M. Jacquot, and L. Larger, “Nonlocal nonlinear electro-optic Phase dynamics demonstrating 10 |

7. | X. Li, W. Pan, B. Luo, and D. Ma, “Mismatch robustness and security of chaotic optical communications based on injection-locking chaos synchronization,” IEEE J. Quantum Electron. |

8. | Y. Chembo Kouomou, P. Colet, N. Gastaud, and L. Larger, “Effect of parameter mismatch on the synchronization of semiconductor lasers with electrooptical feedback,” Phys. Rev. E |

9. | V.S. Udaltsov, J.-P. Goedgebuer, L. Larger, J.-B. Cuenot, Pascal Levy, and W.T. Rhodes, “Cracking chaos-based encryption systems ruled by nonlinear time delay differential equations,” Phys. Lett. A |

10. | S. Ortín, J. Gutiérrez, L. Pesquera, and H. Vasquez, “Nonlinear dynamics extraction for time-delay systems using modular neural networks synchronization and prediction,” Physica A |

11. | C. R. Mirasso, P. Colet, and P. García-Fernández, “Synchronization of chaotic semicondcutor lasers: Application to encoded communications,” Phot. Tech. Lett. |

12. | L. Larger, J. Goedgebuer, and V. Udaltsov, “Ikeda-based nonlinear delayed dynamics for application to secure optical transmission systems using chaos,” Comptes Rendus Physique |

13. | M. C. Soriano, P. Colet, and C. R. Mirasso, “Security implications of open- and closed-loop receivers in all-optical chaos-based communications,” IEEE Photon. Technol. Lett. |

14. | R. M. Nguimdo, P. Colet, and C. R. Mirasso, “Electro-optic delay devices with double feedback,” IEEE J. Quantum Electron. |

15. | U. Leonora, M. Santagiustina, and V. Annovazzi-Lodi, “Enhancing chaotic communication performances by Manchester coding”, IEEE Phot. Tech. Lett. |

16. | V. Z. Tronciu, C. Mirasso, P. Colet, M. Hamacher, M. Benedetti, V. Vercesi, and V. Annovazzi-Lodi, “Chaos generation and synchronization using an integrated source with an air gap,” IEEE J. Quantum Electron. |

17. | R. Hegger, M. J. Bünner, H. Kantz, and A. Giaquinta, “Identifying and modeling delay feedback systems,” Phys. Rev. Lett. |

18. | M.D. Prokhorov, V.I. Ponomarenko, A.S. Karavaev, and B.P. Bezruchko, “Reconstruction of time-delayed feedback systems from time series,” Physica D |

19. | L. Zunino, M. C. Soriano, I. Fischer, O. A. Rosso, and C. R. Mirasso, “Permutation-information-theory approach to unveil delay dynamics from time-series analysis,” Phys. Rev. E |

20. | D. Rontani, A. Locquet, M. Sciamanna, and D. S. Citrin, “Loss of time-delay signature in the chaotic output of a semiconductor laser with optical feedback,” Opt. Lett. |

21. | D. Rontani, A. Locquet, M. Sciamanna, D. S. Citrin, and S. Ortin, “Time-delay identification in a chaotic semiconductor laser with optical feedback: A dynamical point of view,” IEEE J. Quantum Electron. |

22. | R. M. Nguimdo, M. C. Soriano, and P. Colet, “Role of the phase in the identification of delay time in semiconductor lasers with optical feedback,” Opt. Lett. |

23. | R. M. Nguimdo, G. Verschaffelt, J. Danckaert, and G. Van der Sande, “Loss of time-delay signature in chaotic semiconductor ring lasers,” Opt. Lett. |

24. | R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. |

25. | J. Hizanidis, S. Deligiannidis, A. Bogris, and D. Syvridis, “Enhancement of chaos encryption potential by combining all-optical and electrooptical chaos generators,” IEEE J. Quantum Electron. |

26. | H. C. Wang, K. P. Ho, H. K. Chen, and H. C. Lu, J. Lightw. Technol. “Phase and amplitude responses of narrow-band optical filter measured by microwave network analyzer,” J. Lightwave Technol. |

27. | L. Zimmermann, K. Voigt, G. Winzer, K. Petermann, and C. M. Weinert, “ |

28. | R. M. Nguimdo, P. Colet, L. Larger, and L. Pesquera, “Digital key for chaos communication performing time delay concealment,” Phys. Rev. Lett. |

29. | R. M. Nguimdo, R. Lavrov, P. Colet, M. Jacquot, Y. K. Chembo, and L. Larger, “Effect of fiber dispersion on broadband chaos communications implemented by electro-optic nonlinear delay phase dynamics,” J. Lightwave Technol. |

30. | A. Argyris, E. Grivas, M. Hamacher, A. Bogris, and D. Syvridis, “Chaos-on-a-chip secures data transmission in optical fiber links,” Opt. Express , vol. |

31. | K. Pyragas, “Synchronization of coupled time-delay systems: Analytical estimations,” Phys. Rev. E |

**OCIS Codes**

(060.0060) Fiber optics and optical communications : Fiber optics and optical communications

(060.5060) Fiber optics and optical communications : Phase modulation

(140.1540) Lasers and laser optics : Chaos

(250.0250) Optoelectronics : Optoelectronics

(250.4390) Optoelectronics : Nonlinear optics, integrated optics

**ToC Category:**

Optoelectronics

**History**

Original Manuscript: May 25, 2012

Revised Manuscript: July 13, 2012

Manuscript Accepted: August 12, 2012

Published: October 23, 2012

**Citation**

Romain Modeste Nguimdo and Pere Colet, "Electro-optic phase chaos systems with an internal variable and a digital key," Opt. Express **20**, 25333-25344 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-23-25333

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### References

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- Y. Chembo Kouomou, P. Colet, N. Gastaud, and L. Larger, “Effect of parameter mismatch on the synchronization of semiconductor lasers with electrooptical feedback,” Phys. Rev. E69, 056226/1–15 (2004).
- V.S. Udaltsov, J.-P. Goedgebuer, L. Larger, J.-B. Cuenot, Pascal Levy, and W.T. Rhodes, “Cracking chaos-based encryption systems ruled by nonlinear time delay differential equations,” Phys. Lett. A308, 54–60 (2003). [CrossRef]
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- C. R. Mirasso, P. Colet, and P. García-Fernández, “Synchronization of chaotic semicondcutor lasers: Application to encoded communications,” Phot. Tech. Lett.8, 299–301 (1996). [CrossRef]
- L. Larger, J. Goedgebuer, and V. Udaltsov, “Ikeda-based nonlinear delayed dynamics for application to secure optical transmission systems using chaos,” Comptes Rendus Physique5, 669–681 (2004). [CrossRef]
- M. C. Soriano, P. Colet, and C. R. Mirasso, “Security implications of open- and closed-loop receivers in all-optical chaos-based communications,” IEEE Photon. Technol. Lett.21, 426–428 (2009). [CrossRef]
- R. M. Nguimdo, P. Colet, and C. R. Mirasso, “Electro-optic delay devices with double feedback,” IEEE J. Quantum Electron.46, 1436–1443 (2010). [CrossRef]
- U. Leonora, M. Santagiustina, and V. Annovazzi-Lodi, “Enhancing chaotic communication performances by Manchester coding”, IEEE Phot. Tech. Lett.20, 401–403 (2008). [CrossRef]
- V. Z. Tronciu, C. Mirasso, P. Colet, M. Hamacher, M. Benedetti, V. Vercesi, and V. Annovazzi-Lodi, “Chaos generation and synchronization using an integrated source with an air gap,” IEEE J. Quantum Electron.46, 1840–1846 (2010). [CrossRef]
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- R. M. Nguimdo, G. Verschaffelt, J. Danckaert, and G. Van der Sande, “Loss of time-delay signature in chaotic semiconductor ring lasers,” Opt. Lett.37, 2541–2544 (2012). [CrossRef] [PubMed]
- R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron.16, 347 (1980) [CrossRef]
- J. Hizanidis, S. Deligiannidis, A. Bogris, and D. Syvridis, “Enhancement of chaos encryption potential by combining all-optical and electrooptical chaos generators,” IEEE J. Quantum Electron.46, 1642–1649 (2010). [CrossRef]
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- L. Zimmermann, K. Voigt, G. Winzer, K. Petermann, and C. M. Weinert, “C-band optical 90°-hybrids based on silicon-on-insulator 4 × 4 waveguide couplers,” IEEE Photon. Technol. Lett.21 (3), 143 (2009). [CrossRef]
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- R. M. Nguimdo, R. Lavrov, P. Colet, M. Jacquot, Y. K. Chembo, and L. Larger, “Effect of fiber dispersion on broadband chaos communications implemented by electro-optic nonlinear delay phase dynamics,” J. Lightwave Technol.28, 2688–2616 (2010). [CrossRef]
- A. Argyris, E. Grivas, M. Hamacher, A. Bogris, and D. Syvridis, “Chaos-on-a-chip secures data transmission in optical fiber links,” Opt. Express, vol. 18, 5188–5198, 2010. [CrossRef] [PubMed]
- K. Pyragas, “Synchronization of coupled time-delay systems: Analytical estimations,” Phys. Rev. E58, 3067–3071 (1998). [CrossRef]

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