FM Detection from an Optical Chaotic Carrier

Jpn. J. Appl. Phys. Vol. 40 (2001) pp. 4549–4551 Part 1, No. 7, July 2001

2001 The Japan Society of Applied Physicsc

FM Detection from an Optical Chaotic Carrier

Shaw-Tzuu HUANG, Tsung-Min HWANG¹Cheng JUANG, Jonq JUANG²and Wen-Wei LIN³ Electronics Department, Ming Hsin College, Hsinchu, Taiwan, R.O.C.

1Department of Mathematics, National Normal University, Taipei, Taiwan, R.O.C.

2Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan, R.O.C.

3Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan, R.O.C.

(Received November 6, 2000; accepted for publication March 9, 2001)

To synchronize two identical chaotic systems with different initial conditions, a drive and response system model is constructed according to Pecora and Carroll’s theory. The drive and response systems become asymptotically synchronized if an FM signal is added to the drive system. The FM signal can be detected when the drive and response systems are strongly coupled.

KEYWORDS: synchronization, self-pulsating laser diodes, optical chaotic FM detection

1. Introduction

Pecora and Carroll have shown that a chaotic system (the drive system) can be synchronized with a separate chaotic system (the response system) provided that (1) the drive sys- tem transmits some of its state variable to the response sys- tem, and (2) the conditional Lyapunov exponents (CLE) as- sociated with the variational equation are negative.^{1, 2)} The ability to design synchronizing systems has opened up oppor- tunities for applications of chaos to secure communications.³⁾ In optical chaotic communication which has the potential for high-speed communication, various approaches have been used to achieve chaotic synchronization and transmission.^4–7) Goedegebuer et al. have shown the synchronization of two hyperchaotic laser diodes (two positive Lyapunov exponents) which are generated by the delayed feedback technique.⁴⁾ Sivaprakasam and Shore have reported a scheme of optical synchronization of two chaotic single-mode external-cavity semiconductor lasers.⁵⁾An erbium-doped fiber-ring laser has also been used to achieve chaotic synchronization and sig- nal transmission.⁶⁾ Spencer et al. have investigated the opti- cal chaotic synchronization of external-cavity vertical-cavity surface-emitting lasers based on a master-slave model.⁷⁾

In this study, a drive-response model using self-pulsating laser diodes has been adopted based on Pecora and Carroll’s theory.⁸⁾ Using the self-pulsating laser diodes, chaos can be easily achieved at a relatively low bias and small modulation depth compared to those in the case of the conventional laser diodes.⁹⁾ Synchronization can also be achieved for optical simplex and duplex transmissions. Optical chaotic FM de- tection is further investigated by asymptotic synchronization.

With an FM signal added to the drive system, the two sys- tems are no longer identical and become asymptotically syn- chronized. A strong coupling between the drive and response systems can lead to optical chaotic FM demodulation.

2. Formulation and Synchronization

The drive system described by a three-dimensional rate equation is given by¹⁰⁾

d N1

dt = F1(N1, N2, S) + a + b sin 2π f t + δ( ˆS − S), d N2

dt = F2(N1, N2, S), (1)

d S

dt = F3(N1, N2, S),

and the response system (ˆ) is given by d ˆN1

dt = ˆF1( ˆN1, ˆN2, ˆS) + ˆa + ˆb sin 2π ˆft + ˆδ(S − ˆS), d ˆN2

dt = ˆF2( ˆN1, ˆN2, ˆS), (2)

d ˆS

dt = ˆF3( ˆN1, ˆN2, ˆS),

where S is the photon density, N1 is the electron density in the active region, N2 is the electron density in the saturable absorption region, I = a + b sin 2π f t + δ( ˆS − S) is the in- jection current, andδ is the coupling coefficient. Note that I is normalized by a factor of eV1, where V1is the active layer volume. The nonlinear functions F1, F2, and F3 which de- scribe the self-pulsating laser diodes are written by¹⁰⁾

F1= −k₁ξ1

V₁ (N1− Ng1)S − N₁ τs

− N₁− N2

T₁₂ , F2= −k₂ξ2

V₂ (N2− Ng2)S − N₂ τs

− N₂− N1

T₂₁ , (3) F3= [k1ξ1(N1− Ng1) + k2ξ2(N2− Ng2) − Gth]S

+ CN1V1

τs

,

where τs is the carrier lifetime, ξ is the confinement fac- tor, Gth is the threshold gain level, T is the carrier time dif- fusion constant between the two layers, k is the linear ap- proximation constant for the gain curve, Ng is the transpar- ent level of electron density, and C is the coupling ratio be- tween the spontaneous field and lasing mode. The subscripts 1 and 2 describe each term in the active layer and absorp- tion layer, respectively. Table I lists all the parameters of the self-pulsating laser diode obtained from ref. 10 used in the simulation. When a = 30 mA is injected, the corresponding self-pulsating frequency f0is determined to be 2.28 GHz.

According to Pecora and Carroll’s theory, synchronization can only be achieved for all negative CLEs. CLEs are cal- culated from the real part of eigenvalues of A⁻¹ in the dif- ference system d E(t)/dt = AE(t), where EN₁ = N1− ˆN1, EN₂ = N2− ˆN2and ES = S − ˆS. If all of the CLEs are neg- ative (limt→∞E(t) = 0), the two identical systems will be

4549

4550 Jpn. J. Appl. Phys. Vol. 40 (2001) Pt. 1, No. 7 S.-T. HUANGet al.

Table I. Parameters used for the simulation of the self-pulsating laser diodes.

Parameters Value Unit

k1 3.08 × 10⁻¹² m³/S

k2 1.232 × 10⁻¹¹ m³/S

ξ1 0.2034 —

ξ2 0.1449 —

Ng1 1.4 × 10²⁴ m⁻³

Ng2 1.6 × 10²⁴ m⁻³

V1 72 µm³

V2 102.96 µm³

T12 2.65 ns

T21 4.452 ns

Gth 3.91 × 10¹¹ S⁻¹

C 1.573 × 10⁻²³ µm⁻³

τs 3 ns

Fig. 1. The first CLE of the drive-response system against ˆδ for δ = 0, f = 0.6 f0, b= 9 mA, and a = 30 mA.

synchronized independent of the initial conditions. Figure 1 shows the first CLE against ˆδ when δ = 0, f = 0.6 f0 and b= 9 mA. Note that the second and the third Lyapunov expo- nents are all smaller than zero. For ˆδ ≥ 43, there is no positive Lyapunov exponent and the synchronization is ensured. Syn- chronization can also be verified by directly solving eqs. (1) and (2). Figure 2 shows ˆS(t) versus S(t) for a weak coupling when ˆδ = 7, δ = 0, f = 0.6 f0, and b= 9 mA. Drive and re- sponse systems with closely correlated initial conditions have trajectories which quickly become uncorrelated. In this case, there is a positive CLE. However, if the gain stage of the response system is tuned to a strong coupling ( ˆδ = 700), syn- chronization occurs, as shown in Fig. 2.

3. FM Detection

In the process of synchronization, the FM signal is added to the injection current I = a + b sin (2π f t + θ(t)), where θ(t) = b sin 2πf t is the FM signal. In the response sys- tem, I = a + b sin 2π f t. Thus, the two chaotic systems are no longer identical. Ifθ(t) is small and the coupling is strong, the two chaotic systems become asymptotically syn- chronized. This asymptotic synchronization can be described

Fig. 2. ˆS(t) versus S(t) for ˆδ = 1 and 700, when δ = 0, f = 0.6 f0, and b= 9 mA in the drive-response system.

by

tlim→∞|S(t) − ˆS(t)| = lim

t→∞|ES(t)| = k

| | (δ + ˆδ)



, (4) where k is the proportional constant and is the difference term between the drive and response systems without the cou- pling. With the FM signal, is given by

= b(sin (2π f t + θ(t)) − sin 2π f t). (5) Note that without the FM signal, limt→∞|ES(t)| = 0. In the conventional FM technique, a coincidence detector ex- tracts the signal by multiplying two quadrature carriers, that is, cos 2π f t. Thus, let the output signal P(t) be

P(t) =

(δ + ˆδ)cos 2π f t

≈ b

2(δ + ˆδ)[sin θ(t) + sin θ(t) cos 2π2 f t]

≈ b

2(δ + ˆδ)[θ(t) + θ(t) cos 2π2 f t]

≈ bb

2(δ + ˆδ)[sin 2πf t + sin 2πf t cos 2π2 f t].

(6)

Jpn. J. Appl. Phys. Vol. 40 (2001) Pt. 1, No. 7 S.-T. HUANGet al. 4551

Fig. 3. Waveforms of the signal, transmitted chaotic light S(t), and the output signal P(t), where f/f = b/b = 10⁻³, ˆδ = 700, and δ = 0.

Fig. 4. Frequency spectrum of the output signal P(t) in log-plot.

Fig. 5. Signal-to-noise ratio as a function of ˆδ.

In the frequency domain, three peaks (f , 2 f ± f ) are shown from eq. (6). The harmonic frequencies (2 f ± f ) can be easily filtered out, leaving the FM signal. Equation 6 can then be used as a baseline principle for optical chaotic FM detection.

Figure 3 shows the waveforms of the signal, transmitted chaotic light S(t), and the output signal P(t), where f/f = b/b = 10⁻³, ˆδ = 700, and δ = 0. The carrier frequency f = 0.6 f0 = 1.368 GHz and the FM signal are at an in- termediate frequency (f = 1.368 MHz). The transmitted chaotic light S(t) with a complex and unpredictable pattern would ensure the secure transmission of the signal. The out- put signal P(t) shown in the figure clearly recovers an en- velope with a frequency of f . Within the envelope, there are high-frequency components corresponding to frequency 2 f ± f . Figure 4 shows the frequency spectrum of the P(t) (log-plot) forb/b = 10⁻³, ˆδ = 700, and δ = 0. Three dis- tinct peaks are found atf , 2 f ± f . This clearly indicates the validity of eq. (6).

Any other peak that is present as a direct result of the FM detection is considered to be noise, particularly, the small peak at f . If ˆδ becomes small, the noise peak will increase.

In order to characterize this noise effect, one can calculate the power ratio between the signal peak (at f ) and the noise peak (at f ). Figure 5 shows the signal-to-noise ratio as a function of ˆδ. As ˆδ decreases, the drive and response systems become less correlated. Therefore, noise level increases, and the turning point occurs at 150.

4. Conclusions

In conclusion, synchronization between two identical chaotic systems with different initial conditions (drive and re- sponse systems) can be achieved provided that the conditional Lyapunov exponents are all negative. By adding an FM signal to the drive system, the two systems become asymptotically synchronized. The FM signal can be recovered by multiply- ing two quadrature carriers.

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