Information Theoretic Framework for the Analysis of a Slow-Light Delay Device Mark A. Neifeld, 1,2 and Myungjun Lee 1, * 1 Department of Electrical Co

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1 To be published in Journal of the Optical Society of America B: Title: {FEATURE: SL} Information Theoretic Framework for the Analysis of a Slow-Light Delay Device Authors: Mark Neifeld and Myungjun Lee Accepted: 16 July 2008 Posted: 22 July 2008 Doc. ID: 94969

2 Information Theoretic Framework for the Analysis of a Slow-Light Delay Device Mark A. Neifeld, 1,2 and Myungjun Lee 1, * 1 Department of Electrical Computer Engineering, University of Arizona, Tucson, AZ , USA 2 College of Optical Sciences, University of Arizona, Tucson, AZ, , USA * Corresponding author: mjlee76@ .arizona.edu We present a framework for the information theoretic analysis of slow light devices. We employ a model in which the device input is a binary-valued data sequence and the device output is considered within a window of finite-duration. We use the mutual information between these two quantities to measure information content. This approach enables the information theoretic definitions of delay and throughput. We use our new framework to analyze a delay device based on stimulated Brillouin scattering (SBS) and find good agreement with previous SBS delay bounds Optical Society of America OCIS codes: , , , Introduction Tunable all-optical delay is expected to play an important role within all-optical networks, enabling efficient all-optical data buffering, synchronization and retiming functions. For 1

3 this reason, a substantial number of novel slow-light delay devices have attracted recent interest for their potential application in future all-optical communication and signal processing systems. A wide variety of novel physical processes have been investigated towards the realization of all-optical delay (see e.g. the recent review in [1]). Many of these approaches, including engineered resonant structures (i.e., micro-rings and/or photonic bandgap structures) [2-4], wavelength conversion/dispersion [5,6], and stimulated gain and absorption processes (e.g., Brillouin or Raman) [7-9], have produced promising room-temperature experimental demonstrations. This impressive success has motivated a renewed interest in the fundamental limits of such slow light devices. Several notable analyses of slow-light delay bounds have been recently undertaken. For example, recent work by Khurgin analyzes tunable buffers based on optical amplifiers [10]. This analysis concludes that although gain broadening may provide increased bandwidth, delay-bandwidth product remains quite modest in these systems. Related work by Tucker et. al., investigates a broader class of linear devices and employs a wide range of performance metrics based on the physical size of the stored bit including delaybandwidth product and storage capacity [11,12]. A more recent analysis from Miller further generalizes the performance of linear delay structures. Using a powerful form of modal analysis he derives an upper bound on the delay-bandwidth product that can be achieved for any one-dimensional device structure [13,14]. Despite these important recent contributions, still lacking is an information theoretic framework that would enable formal information-based definitions of delay and capacity for slow light optical components and systems. 2

4 In this paper we have endeavored to answer two practically important questions. (1) Given a slow light device and a typical signaling scheme how does the fidelity of the output, measured using the Shannon mutual information, depend upon the parameters of the device? (2) When is the largest fraction of the input information available at the device output? Note that the answer to question (1) is not simply the Shannon channel capacity nor is the answer to question (2) simply the delay spread. In what follows we will present an information theoretic formulation that provides the answers to these two questions and we will demonstrate the application of this new formulation to a slow light device based on stimulated Brillouin scattering (SBS). 2. Information Theoretic Delay Information theory is central to nearly all forms of modern communication and storage. Especially important within optical communications applications is the concept of channel capacity, which is frequently used to analyze the quality of physical links, the efficiency of novel modulation and coding schemes, and the effectiveness of candidate receiver architectures. The Shannon information capacity of a communication channel is defined as the maximum rate at which error-free data may be conveyed between a transmitter and a receiver [15,16]. Note that the information theoretic definition of capacity is not simply the time-bandwidth product of the channel. Formally, the capacity (C) is defined as the supremum of the mutual information between the channel input and 3

5 output: C = sup[i(x;y)], where X represents the channel input, Y represents the channel output, I(X;Y) is the mutual information between these two random variables, and the sup[ ] operation involves maximizing I(X;Y) over all possible modulation, signaling, and coding schemes. It is important to note that this definition of channel capacity is asymptotic, relying on codes of arbitrary length and complexity. Unfortunately, because it is an asymptotic quantity, capacity is an inconvenient metric for the evaluation of delay systems in which finite delays are of primary interest. In order to overcome this limitation we have elected to use the mutual information directly. The mutual information I(X;Y), measures the information about the data X that is contained in the measured signal Y, for a specific choice of input/output signaling protocol. This approach will therefore enable us to constrain the input modulation format as well as the input and output signal durations, thus providing a natural method for quantifying information theoretic delay and throughput. In general the mutual information can be expressed as I(X;Y) = H(X) H(X Y), where H(X) is the entropy of the input and H(X Y) is the entropy of the input conditioned on the output. Physically we may interpret H(X) as quantifying the number of possible input states. So for a n-dimensional binary-valued sequence the number of states would be M=2 n, resulting in an entropy of H(X) = log 2 (M) = n bits. Similarly, H(X Y) quantifies the number of possible input states that might have been transmitted, consistent with the measurement Y. Note that if the channel output is a unique (i.e., one-to-one) and deterministic function of the channel input, then H(X Y) = log 2 (1) = 0 and I(X;Y) = n bits. 4

6 This is the physical reason why a many-to-one channel and/or noise will increase H(X Y) thus reducing I(X;Y). Our formalism is based on the channel model depicted in Fig. 1. We assume that the channel input sequence X is binary-valued and that it is used to modulate via on-off keying (OOK), a train of truncated Gaussian pulses (GP). Return-to-zero modulation is used with a bit period of T 0 and a Gaussian pulse width of T pulse =T 0 /2 (pulse width at 1/e 2 intensity). The delay device is represented by the operator H SL. Although this operator may in general be nonlinear, our current study limits H SL to a linear operator represented by a channel matrix which operates on a sampled representation of the continuous-time output from the modulator. We also assume that the measurement is corrupted by noise. In general the form of the noise probability density function (PDF) will depend upon the specific device physics. Here we employ a zero-mean additive white Gaussian noise (AWGN) assumption for simplicity. Because our approach involves the incorporation of this noise via Monte Carlo sampling, extension to alternate noise statistics will be straightforward. Note that even though the channel input X is discrete (i.e., the logical data to be stored), the channel output Y is continuous (i.e., a delayed and distorted train of pulses). Using this channel model the mutual information may be computed explicitly according to: I(X; Y) = M i = 1 p(x i) log 2 p(x i) + M i= 1 p(x i, Y) log 2 p(xi Y)dY (1) 5

7 = n + M i= 1 p(x i)p(y x i) log 2 p(y x i)p(x i) dy, p(x j)p(y x j) M j= 1 (2) where p(x i ) is the prior probability of a specific n-bit input sequence x i, p(x i,y) is the joint PDF of x i and Y, p(y x i ) is the PDF of Y conditioned on x i and M = 2 n is the number of possible n-bit input sequences. We will assume independent and equiprobable bits so that p(x i ) = (1/2) n. The AWGN assumption implies that the probability of Y conditioned on a specific input x i is given by the Gaussian PDF: p(y x i ) exp Y H x i, 2 n L 2 SL (2πσ ) 2σ where σ represents the standard deviation of the noise and L represents the number of simulation samples used to represent a single truncated GP. In order to numerically compute I(X;Y) from these expressions, we consider all possible input sequences of which there are M. The sums in Eqs. (1) and (2) are therefore exhaustive. The required integrals over Y are solved using a Monte Carlo technique with importance sampling of the joint and conditional densities. Note that importance sampling is a popular simulation technique to reduce the variance of a given simulation estimator, and thus achieving reliable results while decreasing simulation time [17]. For the results reported here we have obtained reliable data using fewer than 10 6 noise samples per input sequence. (3) 6

8 Let us examine the application of this formalism to an ideal delay device. For this example we use T 0 = 40ns. We select H SL to be a convolution matrix whose corresponding impulse response is simply h SL (t) = δ(t-t D ), where δ denotes the Dirac delta function and T D is the desired value of time delay. The value of delay was selected to be T D = 270ns in this example. In order to acknowledge the finite signal durations and delay values that are of interest in practical systems we impose the window structure depicted in Fig. 2(a). This figure shows an example 6-bit sequence of pulses that will serve as an input to the delay device. The input signal is contained within an input window of duration T W. Also shown is the corresponding output signal for the case of an ideal distortion-free delay device with σ = 0. Two candidate output windows (dotted and solid) at two different values of window offset are shown in Fig. 2(b). For a specific choice of candidate output window our information theoretic analysis of such a delay system involves computing the mutual information between X and only that part of Y contained in the output window. Repeating this analysis for many different output window positions we obtain the mutual information as a function of window offset as shown in Fig. 2(b). Note that we may interpret the peak of this graph as quantifying the amount of information that can be transmitted through H SL (6-bits in this case); whereas, the location of this peak provides an information-theoretic measure of delay. We will refer to the peak-height as the information throughput (IT) of the delay device and the peak location as the information delay (ID) of the device. 3. Information Analysis of SBS Slow Light Delay 7

9 The SBS process provides a convenient mechanism through which a weak probe beam may experience gain upon counter-propagating with a strong pump beam in an optical fiber [7,8]. This interaction is mediated via an acoustic mode in the fiber and requires that the pump (f p ) and probe (f s ) frequencies satisfy the Brillouin shift f p -f s = F B, where F B ~ 12 GHz in typical fibers. When this condition is met, a signal modulated onto the probe beam at the fiber input E(f, 0), will be amplified at the fiber output according to E(f,L) = E(f,0)exp(i k(f) L), where f represents frequency, L denotes the fiber length, and k(f) is a Lorentzian profile given by k(f) γgl =, 4 π(f f p + FB + iγ) where γ = 25MHz is the SBS half-width half-maximum (HWHM) linewidth. The linecenter gain experienced by the probe beam is determined by the strength of the pump beam according to g = g 0 P 0 /A, where g 0 is the SBS gain coefficient, P 0 is the (monochromatic) pump power and A is the fiber mode area. Note that the function k(f) also has a real part which gives rise to dispersion and the associated group delay T g which can be tuned via the gain according to [18,19] (4) T g gl. 4π γ (5) 8

10 This gain-dependent group delay is the basis for using SBS as a tunable delay device. Note that Eq. (5) represents the maximum achievable group delay (i.e., upper bound of T g ) at the line-center [18]. We apply our information theoretic framework to this SBS delay system by setting H SL equal to the linear operator associated with the SBS transfer function H(f) = exp(i k(f) L). This is accomplished by making H SL a convolution matrix corresponding to the impulse response h SL (t) = F -1 [H(f)], where F -1 [] denotes the inverse Fourier transform operator. Figure 3 presents an example input/output pair for this choice of H SL. Figure 3(a) shows a 6-bit input pulse sequence (T 0 = 40ns) in which we have normalized the pulse heights to one arbitrary unit of optical power. Figure 3(b) shows the corresponding output signal for the case gl=5. We note the large gain (i.e., line-center gain exp(5) = 148) and substantial distortion associated with the signal depicted in Fig. 3(b). Because large values of exp(gl) may complicate our interpretation of fundamental trends, we also consider the normalized signal shown in Fig. 3(c). This signal has been scaled to have energy (i.e., time-integrated power) equal to that of the input signal shown in Fig. 3(a). There is physical justification for considering such a normalized output signal. For example, system-level constraints might require delay components with unity gain, in which case a variable absorber would be required to follow the SBS gain component. Alternately, the signal to noise ratio (SNR) of the output signal may be limited by the input signal SNR, in which case this normalization will insure constant signal strength relative to σ, facilitating the use of the model shown in Fig. 1. Note that the peaks in Fig. 3(c) are no longer achieving one unit of power. This is because distortion arising from propagation 9

11 through H SL has resulted in energy leaking into the nulls as well as energy leaking outside the window. Although for this value of gl the SBS distortion does not result in a manyto-one channel mapping, this energy leakage will have a deleterious impact on measurement SNR and an associated reduction in information content. Figure 4 presents the results of computing mutual information as a function of window offset for several cases of interest using the energy normalization described above. Figure 4(a) shows the mutual information for a case of moderate noise strength with σ = 0.45, and several values of gl. We note from this data two expected trends: increasing gl results in (a) increasing distortion and a resulting decrease in the information throughput and (b) increasing group delay and a corresponding increase in information delay. Note that the case gl = 1 experiences negligible distortion and so the SBS system achieves IT ~ 6 bits as expected. Figure 4(b) presents these mutual information curves for three different values of noise strength (σ = 0.45, 0.59, 0.71) and gl=10. From this data we note that IT decreases with increasing σ as expected. Note also that there does not appear to be a strong relationship between σ and ID. It is convenient to extract the IT and ID values from curves like those shown in Fig. 4. The resulting data will enable us to summarize the information theoretic delay and associated information throughput for SBS slow light devices as a function of gl. Such a presentation will facilitate comparison with other studies of slow light delay via SBS. The results of this process are shown in Fig. 5. The left-hand axis in Fig. 5(a) corresponds to the solid curves, and presents IT as a function of gl for two values of noise strength σ = 10

12 0.45 and σ = The right-hand axis in Fig. 5(a) corresponds to the dashed curve, and presents ID as a function of gl. As expected we find that increasing gl results in increased ID and decreased IT. In Fig. 5(b) we compare three different methods for computing delay: (a) the group delay T g [18,19], (b) the eye-opening based delay T e [20,21], and (c) the information-based analysis presented here. The eye-opening based delay T e is measured when the eye-opening instant of the output pulse sequence is the maximum. We note good agreement among these various methods. We also note that these previous authors report distortion metrics (i.e., eye-opening and pulse broadening) that are monotonic with gl, in qualitative agreement with the IT trends shown in Fig. 5(a). Note that because we have used T 0 = 40ns < 120ns [18], we observe that T g is larger than both ID and T e as expected. For smaller values of T 0, the maximum achievable delay will be smaller than T g [18]. Based on the results shown in Fig. 5(a) we may conclude that with T 0 =40ns and σ = 0.45, a SBS delay device can utilize a gain of gl = 6 without sacrificing more than 10% of the input signal information (i.e., IT > 5.4 bits). This IT constraint therefore results in a maximum IT-constrained delay of 14ns for sequences comprising T 0 =40ns. Repeating this exercise for many values of T 0, we obtain the IT-constrained delay results shown in Fig. 6. In this figure the solid curve represents the fractional ID (= ID / T pulse ) versus normalized bandwidth (B N = 1 /2γT 0 ) subject to two constraints. The first constraint limits gl 10 to avoid the onset of nonlinear amplifier behavior and the second constraint limits gl to insure that IT > 5.4 bits. The choice of 5.4 bits is based on the (somewhat arbitrary) desire to retain > 90% of the transmitted information. We 11

13 observe that for small values of bandwidth, the fractional ID is constrained primarily by the nonlinearity-limit and for large values of bandwidth the fractional ID is constrained primarily by the IT-limit. We note that where these two constraints are balanced, an optimum value of ID is obtained. For the case presented here we find that the maximum IT-constrained fractional delay is at an optimum bandwidth of B N = 0.4 (i.e., optimum T 0 = 50ns). The dashed curve in Fig. 6(a) represents analogous data for eyeopening-constrained fractional delay (= T e / T pulse ) [20]. This data is based on the same maximum gain constraint gl 10 and an eye-opening (EO) distortion constraint of EO > We observe similar trends from the EO-constrained data, with a maximum EOconstrained fractional delay of at the optimum B N of 0.4. The corresponding optimal gain results for both cases are shown in Fig. 6b. The design curves presented in Figs. 6(c) and 6(d) represents bounds on the performance of SBS delay devices subject to real-world operating constraints on output signal quality measured via either eye-opening or the newly presented information theoretic metric IT. It is interesting to compare our information theoretic results with the previously published bounds from Khurgin [10] and Miller [14]. Such a comparison is possible because all of these analyses utilize input sequences that employ a common definition of input bits (i.e., on/off signals in distinct time-domain slots). The comparison is complicated however, by the fact that these analyses do not utilize a common definition of output bits. Specifically, both Khurgin and Miller retain the distinct time slots definition for output bits; whereas, the information theoretic analysis employs a very different number of distinguishable signals definition. 12

14 This disconnect between definitions suggests that the following results should be interpreted with some caution. For a Lorentzian gain profile in the slow-light medium, Khurgin provides an upper bound of the delay-bandwidth product that depends on the peak gain [10]. His expression for fractional delay is approximately T d / T pulse (G/50) 2/3, where the peak gain G[dB] = 10 log 10 [exp(gl)] and T d is the delay [10]. For gl=10, his upper limit is approximately 0.94 and this value agrees with our maximum delay results (both T e / T pulse = and ID / T pulse = 0.933) at B N = 0.4, as shown in Fig. 6(a). Miller also gives a delay-bandwidth product bound based on the relative variation of the dielectric constant. To accommodate the delay definition used in this paper (i.e, pulse delay versus bit delay), we can rewrite his equation (9) as T d / T pulse (4/3 1/2 ) n avg (n avg -n min )L/λc, where n avg is the average refractive index, n min is the minimum refractive index at any position or frequency of interest, and λc is the carrier frequency [14]. For gl=10, n avg =1.43, and L= 1km, the effective variation of the refractive index n avg -n min is approximately 6.2* that is induced in the optical fiber as a result of the SBS process, resulting a delaybandwidth bound less than This value is somewhat larger than the Khurgin and IT results; however, it is important to recall that the Miller result presumes arbitrary use of the refractive index resources. Recall that the results presented above were predicated on the use of output signal energy normalization as discussed in conjunction with Fig. 3(c). Although this normalization strategy is physically meaningful in some circumstances, we note that un-normalized signals (e.g., Fig. 3b) are perhaps more common. For example the un-normalized model 13

15 would be relevant to SBS systems that are not input SNR limited and/or are not required to employ some form of compensation to achieve unity gain. In this case we observe that because the output signal strength is exponential in gl, even small values of gl provide extremely good noise tolerance. For this reason we see that even for very large values of noise strength, IT increases with gl. This is shown in Fig 7(a), where we plot mutual information as a function of window offset for σ = 10 and several values of gl. Note that for gl = 1 the output SNR is approximately exp(1)/10 = This is a small value of SNR, resulting in an IT value of only 0.35 bit. For a larger gain value of gl = 4, we find a peak output SNR of approximately 5.5, resulting in IT ~ 6 bits. Note that the effect of pulse distortion is entirely compensated by the increased signal strength associated with larger gl. This observation points to the inadequacy of distortion as a measure of information. It is important to realize that the only factor causing information to be lost is the potential indistinguishability of output signals deriving from different input data sequences. Although the distortion of these high-gain SBS systems can be severe, in cases for which noise is small, the distorted output signals remain distinguishable and therefore retain nearly all of the input information. It is interesting to note that for values of gl 4 the mutual information curves in Fig. 7(a) exhibit obvious peaks, resulting in an unambiguous definition for ID. In contrast with this behavior, consider the case of gl=8 for which IT = 6 bits is achieved for a large range of window offset. This requires a modified definition of ID. Specifically we will define ID as the largest value of window offset at which I(X;Y) reaches 0.99*IT. This definition accommodates an important observation from the previous paragraph: 14

16 information can be extracted from distinguishable signals regardless of their degree of distortion. A summary of these un-normalized results is shown in Fig. 7(b). In this figure we plot IT (left axis) and ID (right axis) as functions of gl. We observe that IT saturates for small values of gl, indicating that the corresponding output signals have high SNR, even for the very large noise strength of σ = 10. We also observe delay values larger than those seen in the normalized case. This is because high SNR enables the use of window offsets positioned beyond the center of the output signal. In these cases we may actually exploit signal distortion to facilitate distinguishing output signals using energy in their tails. Although the current study has been limited to AWGN, it is interesting to consider how other signal- and/or gain-dependent noise sources (e.g., spontaneous SBS noise) might impact these results. With the energy normalization that we have employed thus far we may speculate: (1) IT will decrease faster with increasing gl as a result of additional (e.g., ASE) noise sources and (2) because we find ID essentially independent of noise strength we expect ID will be largely unaffected by these additional noise sources. Note that for cases in which energy normalization is nonphysical, we show below that IT can saturate with increasing gl. In the presence of gain-dependent noise we would expect a rollover in this behavior for some value of gl, beyond which IT once again decreases. The results presented thus far have utilized 6-bit input sequences. We have limited our current efforts to 6-bit inputs in order to enable the presentation of a more complete 15

17 design study. It is important to note that this is a limitation not of the formalism, but of our current computational abilities with regard to the Monte Carlo integration of Eq. (2). The results that we have presented will be approximately independent of the sequence length neglecting boundary effects. With the goal of demonstrating a point design using a longer 10-bit input sequence, we repeat the exercise represented by the data in Fig. 5(a). Specifically we compute the IT and ID values obtained from propagating all possible 10-bit input pulse sequences with T 0 = 40ns and σ = 0.45 through the SBS transfer function. The results are shown in Fig. 8. As expected we obtain ID results that are nearly identical to those obtained in the 6-bit case; whereas, the 10-bit IT data is nearly 1.67 (=10/6) times larger than the 6-bit case. 4. Conclusions: We have presented a novel framework that facilitates the application of information theoretic metrics to the analysis of slow light optical delay devices and systems. We have employed a simple model which describes the propagation of binary-valued input data through a slow light system, generating a continuous-valued signal at the system output. By using a numerical technique for estimating the mutual information between the input data and the output signal within a window of finite-duration, we have shown how information theoretic measures of delay and throughput may be computed. This new framework is completely general and can be applied to arbitrary modulation (e.g., nonbinary and/or complex-valued) and coding schemes. It can be applied to a wide variety of 16

18 slow light devices and systems and can be used to describe linear or nonlinear physical systems subject to one or more arbitrary stochastic noise sources. As an initial application of this framework we have analyzed a simple all-optical delay system based on SBS. The information theoretic trends which result from our analysis are both physically sensible and in good agreement with previous analyses of SBS delay systems. 5. Acknowledgements: We gratefully acknowledge the financial support of the DARPA DSO Slow-Light Program. 6. References [1] E. Parra and J. R. Lowell, Toward Applications of Slow Light Technology, OPN, 41-45, (November, 2007). [2] G. Lenz, B. J. Eggleton, C.K. Madsen, and R. E. Slusher, Optical Delay Lines Based on Optical Filters, IEEE J. Quant. Elect. 37, (2001). [3] J. E. Heebner, V. Wong, A. Schweinsberg, R. W. Boyd, and D. J. Jackson, Optical Transmission Characterisitics of Fiber Ring Resonators, IEEE J. Quant. Elect (2004). 17

19 [4] J. Mok, C. M. Sterke, and B. J. Eggleton, Delay-tunable gap-soliton-based slow-light system, Opt. Exp. 14, (2006). [5] Y. Okawachi, J. E. Sharping, C. Xu, and A. L. Gaeta, Large tunable optical delays via self-phase modulation and dispersion, Opt. Exp. 14, (2006) [6] R. Pant, M. D. Stenner, and M. A. Neifeld, Limitations of self-phase modulation based tunable delay system for all-optical buffer design, submitted to App. Opt. [7] R. W. Boyd, D. J. Gauthier, and A. L. Gaeta, Slow Light: From basics to future prospects, Photon. Spectra 40, (2006). [8] M. González Herráez, K. Y. Song, and L. Thévenaz, Arbitrary-bandwidth Brillouin slow light in optical fibers, Opt. Exp. 14, (2006) [9] D. Dahan and G. Eisenstein, Tunable all optical delay via slow and fast light propagation in a Raman assisted fiber optical parametric amplifier: a route to all optical buffering, Opt. Exp. 13, (2005). [10] J. B. Khurgin, Performance limits of delay lines based on optical amplifiers, Opt. Lett. 31, (2006). [11] R. S. Tucker, P.C. Ku, and C. J. Chang-Hasnain, Slow-Light Optical Buffers: Capabilities and Fundamental Limitations, J. Light. Tech. 23, (2005). [12] R. S. Tucker, The role of optics and electronics in high-capacity routers, J. Light. Tech. 24, (2006). [13] D. A. B. Miller, Fundamental limit for optical components, J. Opt. Soc. Am. B 24, A1-18 (2007). [14] D. A. B. Miller, Fundamental limit to linear one-dimensional slow light structures, Phys. Rev. Lett. 99, (2007). 18

20 [15] C. E. Shannon, A mathematical theory of communication, Bell System Technical Journal 27, (July 1948). [16] C. E. Shannon, A mathematical theory of communication, Bell System Technical Journal 27, (October 1948). [17] C. P. Robert and G. Casella, Monte Carlo Statistical Methods, (Springer, 1999), pp [18] Z. Zhu, D. J. Gauthier, Y. Okawachi, J. E. Sharping, A. L. Gaeta, R. W. Boyd, and A. E. Willner, Numerical study of all-optical slow-light devices via stimulated Brillouin scattering in an optical fiber, J. Opt. Soc. Am. B 22, (2005). [19] Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, Tunable all-optical delays via Brillouin slow light in an optical fiber, Phys. Rev. Lett. 94, (2007). [20] M. Lee, R. Pant, M. D. Stenner, and M. A. Neifeld, SBS gain-based slow light system with a Fabry-Perot resonator, Opt. Comm. 281, (2008). [21] R. Pant, M. D. Stenner, M. A. Neifeld, Z. Shi, R. W. Boyd, and D. J. Gauthier, Maximizing the opening of eye diagrams for slow-light systems, App. Opt. 46, (2007). 19

21 AWGN X OOK GP H SL + Y Figure 1: Block diagram representing the information theoretic model of a slow light delay device. X is a binary-valued discrete input sequence, the OOK GP block uses X to modulate via on-off keying, a sequence of truncated Gaussian pulses, H SL is the slow light operator, AWGN represents additive white Gaussian noise, and Y is the continuous-valued output signal. 20

22 1 (a) WO2 WO1 T w OW1 OW2 T w Power (a.u.) Time (ns) 6 (b) I(X;Y) bits Window offset (ns) Figure 2: Application of the information theoretic analysis to an ideal delay device. (a) Example signals at the input (dashed) and output (solid) of the slow light operator. The input window along with two candidate output windows (solid and dotted) are show. (b) Mutual information as a function of output window offset. Note that WO represents the window offset and OW represents the output window. 21

23 Power (a.u.) (a) Time (ns) Power (a.u.) (b) Power (a.u.) Time (ns) (c) Time(ns) Figure 3: Example signals used in the information theoretic analysis of SBS delay. (a) 6-bit modulated signal used as input to H SL, (b) output of H SL for gl=5, and (c) the signal shown in (b) normalized to have total energy equal to that of (a). 22

24 6 5 (a) gl=1 gl=5 gl= (b) σ=0.71 σ=0.59 σ=0.45 I(X:Y) (bits) I(X;Y) (bits) ID Window Offset (ns) Window offset (ns) Figure 4: Mutual information versus window offset for some example SBS systems. (a) Mutual information for σ = 0.45 and three values of gl=1, 5, 10. (b) Mutual information for gl = 10 and three values of σ =0.45, 0.59,

25 IT (bits) σ = 0.45 σ = 0.71 (a) ID (ns) 10 Delay (ns) T g ID T e (b) Gain (gl) Gain (gl) Figure 5: Summary of information theoretic results for SBS. (a) Information throughput (IT) on the left axis (solid curves) and information delay (ID) on the right axis (dashed curve). (b) Delay results computed using three different methods (ID, T g, and T e ). 0 24

26 Fractional delay (a) gl (b) B N B N 6 (c) 1 (d) IT (bits) B N EO B N Figure 6: (a) Maximum delay versus bandwidth for SBS slow light devices under real-world operating constraints on maximum gain (gl 10) and output signal fidelity: solid curve corresponds to constraint on IT > 5.4 bits and dashed curve corresponds to constraint on eye-opening > (b) Optimal gain. (c) ITconstraint (solid) and IT-limit (dotted). (d) EO-constraint (dashed) and EO-limit (dotted). 25

27 I(X;Y) (bits) (a) gl=1 gl=2 gl=3 gl=4 gl=8 IT (bits) (b) ID (ns) Window offset (ns) Gain (gl) Figure 7: Mutual information versus window offset for some example SBS systems. This data is obtained from un-normalized SBS output signals. (a) Mutual information for σ = 10 and five values of gl. (b) Summary of information theoretic results. Information throughput (IT) on left axis (solid curve) and information delay (ID) right axis (dashed curve). 26

28 10 (a) 30 (b) 9 25 IT (bits) ID (ns) Gain (gl) Gain (gl) Figure 8: Comparison of information theoretic results for 6-bit (dashed curve with circle) and 10-bit (solid curve) input sequences for σ = (a) Information throughput (IT) and (b) information delay (ID). 27