Extraction of Signals from Chaotic Laser Data

John B. Geddes, Kevin M. Short, Kelly Black
1999 Physical Review Letters  
Several experimental groups have demonstrated communication with chaotic lasers. We analyze data collected from a message-modulated erbium-doped fiber-ring laser (provided by VanWiggeren and Roy). We show that the transmitted signal is dominated by convolution of the message with the response function of the laser. A simple model based on the topology of the laser allows us to recover a hidden message. While prior estimates indicate that the laser dynamics are high dimensional, we show that
more » ... four parameters are required, each of which can be recovered from the transmitted signal alone. PACS numbers: 05.45.Vx, 05.45.Tp, 89.70. + c A great deal of interest has been generated by the potential to use lasers running in a chaotic regime as the carriers of information in a secure chaotic communication scheme. Several researchers have been working to develop laser transmitter and receiver pairs which synchronize [1] [2] [3] [4] . These systems may represent a significant breakthrough for chaotic communication schemes, since they can provide for data transfer rates that are greater than 100 Mbits͞sec. One such laser system was developed by VanWiggeren and Roy [2-4], using an erbium-doped fiber-ring laser (EDFRL). The authors were able to generate a highdimensional chaotic waveform by message modulating the laser. They also introduced an additional delay loop in the hope of enhancing the privacy of the transmission. Message recovery was achieved by using an appropriately tuned open-loop receiver. This experiment is novel in two important aspectsthe communication rate (100 Mbits͞sec) is high and the dynamics appear to be high dimensional (.10) [3] . Previous attempts at secure chaotic communication have been shown to be susceptible to attacks based on nonlinear dynamic (NLD) forecasting [5] [6] [7] [8] and return map analysis [9] . However, it has been suggested [10] that transmitting a higher-dimensional waveform may make this difficult, and we will show that an entirely new approach was required in order to extract the message in this case. In this paper, we report on our investigation of the dynamics and security features of the EDFRL. In order to evaluate the security of the EDFRL, VanWiggeren and Roy provided us with a number of experimental data sets. The application of NLD forecasting was ineffective on this data possibly because subsequent research showed that the signal does not appear to be chaotic on the time scale of the message. Indeed, we will demonstrate that the transmitted signal is dominated by a convolution of the message bit stream with the response function of the laser. This response function is determined by the topology of the laser, i.e., the delay loops and amplification factors. We further show that these parameters are recoverable from the transmitted signal alone, and that first-order estimates of these parameters provide for excellent message recovery. Underlying model.-Our underlying model assumes that the message-modulated EDFRL can effectively be described in terms of a double echo-feedback loop (see Fig. 1 ). The transmitter consists of an inner loop with round-trip time T 1 , and an outer loop with round-trip time T 2 . The inner and outer loops are shared for part of the time. There are erbium-doped fiber amplifiers (EDFA1 and EDFA2) in both the shared inner loop and outer loop. These partly make up for the losses imposed by absorption and loop junctions. We assume that the amplifiers are linear and that a and b represent the net effects of amplification and attenuation in the inner and outer loops, respectively. We will ignore any underlying carrier, and focus on the linear response of the laser to the message bit stream. We assume that the transmitted intensity, I t ͑t͒, is dominated by a convolution of the message bit stream I m ͑t͒ with the response function of the laser, H͑t͒. Consider the evolution of a unit impulse which begins to circulate the laser at t 0. After amplification by EDFA1, this solitary "bit" splits at loop junction A. The inner-loop pulse continues to propagate, while the outer loop pulse is reamplified by EDFA2 before rejoining the inner loop (at loop junction B) as an "echo" of the original pulse. Since the transmitter is a closed-loop system, these pulses are continuously reamplified and resplit, thereby generating multiple echoes of the original pulse. In this way, the effect of a single bit persists for many roundtrips; this is analogous to feedback between a microphone and an amplifier in an echo chamber. It is the topology of the laser that dictates the form of the response function. After one round-trip, a single pulse generated at t 0 will produce two first generation echoes, the first at T 1 with amplitude a, and the second at T 2 with amplitude b. After another round-trip, the former echo will produce second-generation echoes at 2T 1 with amplitude a 2 , and at T 1 1 T 2 with amplitude ab. The latter echo will likewise split and produce second-generation echoes at T 1 1 T 2 with amplitude ab, and at 2T 2 with amplitude b 2 . The pulses at each subsequent generation will again 0031-9007͞99͞83(25)͞5389(4)$15.00
doi:10.1103/physrevlett.83.5389 fatcat:fwr7v6573fcw7ocsnchcptxmra