1. Introduction

The propagation of fast- and slow-light pulses has been the subject of intense study recently, largely for its potential application in all-optical networking [1]. Most of the effort in this field has been devoted to studying and designing propagation media and devices for slowing (or speeding up) light pulses. As a result of these studies, we now have a broad and ever increasing array of slow-light techniques including fiber-based gain systems [2], photonic techniques [3], mixed linear/nonlinear approaches [4], and many others.

 

Fig. 1. (a) The detection model considered herein. A pulse is generated within some temporal window (the input window) and detected within a corresponding output window. Only energy falling within the output window contributes to the identification of the bit. (b) The discrete linear representation of pulse propagation. The submatrix H _s describes the pulse in the output window in terms of the input pulse. The matrices H _i1 and H _i2 similarly describe the interference, the energy that falls outside the output window.

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In this paper, we present a complimentary study of pulse design. We present a technique for generating pulse shapes designed to be reliably detected in a given temporal window after propagating through a slow-light medium. In addition to providing good pulse designs for a given medium, this approach also raises the possibility of tuning delay by adjusting pulse parameters rather than medium parameters. Finally, the techniques and results described herein lead to questions emerging within the community about the meaning and definition of delay in communications systems; should we think of delay as a property of media, pulses, detection, or some combination thereof [5]?

2. Slow light pulse design

In order to effectively design pulses for a communication system, one must know the medium properties as well as the detection scheme. We consider here a detection scheme in which the measured energy falling within some temporal window is used to determine whether or not a pulse is present, as depicted in Fig. 1(a). In the limit that the output window duration t_wo goes to zero, this energy measurement becomes a simple measurement of the instantaneous output power. Given this measurement scheme and simple binary intensity modulation (e.g., on-off keying), the data is detected by distinguishing zeros (low energy) from ones (high energy) using a simple threshold. We constrain input pulses only in that they have unit energy, all of which falls inside an input window. The mathematical framework developed below does not require that the input and output windows have equal duration (t_wi=t_wo), but they do for all examples presented herein.

If the chief obstacle in distinguishing zeros from ones is noise, then the best strategy is to choose input pulse shapes such that as much energy as possible falls within the output window, thus maximizing the signal-to-noise ratio (SNR). For sampled linear systems, one can model propagation as

where f is the input pulse, g is the output pulse, and H is a Toeplitz matrix whose first column is the impulse response of the system (beginning at time zero) and first row (after the first element) is all zeros. The values of both f and g correspond to the electric field sampled at a vector of times t. Given this framework, a temporal input window is mapped to a temporal output window by H _s, the “signal” submatrix of H shown in Fig. 1(b).

For the goal of maximizing the SNR of the output pulses, we must simply choose f to maximize the 2-norm of g _w, the part of g in the output window. The well-known solution to such a problem is to calculate the singular value decomposition (SVD) of the matrix H _s=UDV ^†. The ideal input pulse is then given by the column of V corresponding to the largest singular value. Examples of such pulses are presented in the next section.

While the approach described above does indeed maximize the output SNR, in doing so it completely ignores the pulse energy falling outside the output window. In practical communication systems, pulse sequences are used, in which neighboring bits are equally likely to be zero or one. Undesired power from those neighboring bits will be integrated in the output window and reduce signal fidelity. This inter-symbol interference (ISI) must also be managed. The contribution to the interference part of the output pulse—the power falling outside the output window—is controlled by the interference submatrices H _i1 and H _i2. We can then define the signal-to-noise-plus-interference ratio (SNIR) as

σ ² is the variance of the noise on the output pulse energy measurement, and the input pulse has been normalized to unit energy. We can optimize this SNIR by differentiating with respect to f and setting the result to zero.

This last result is a generalized eigenvalue equation of the form Ax=λ Bx in which the SNIR is the eigenvalue. Tools for solving such problems are readily available. Also, in the limit that σ ²→∞, this becomes the SNR-optimal solution provided by the SVD. Physically, this corresponds to the dominance of noise in the SNIR denominator.

3. Results and discussion

Figure 2 shows the results of applying the above design techniques to an example system consisting of a single Lorentzian gain line [6, 7]. The input pulse intensity, output intensity, and input power spectra are presented. Results are shown for the SNR-optimal and SNIR-optimal pulses, along with impulse (delta function) and square pulses for comparison. The square pulse has no free parameters, spanning the full input window. The impulse is selected to occur inside the input window at the moment that maximizes the output SNR. This also maximizes the SNIR because the total output energy and pulse shape do not depend on the impulse position within the input window. For all temporal plots, the pulses have been normalized to unit height whereas the power spectra have been normalized to unit area (power). For computational and presentation simplicity, all pulse propagation excludes the pure linear dispersion from propagation through free space, which enter the problem as a simple temporal shift. As a result, any difference in shape or temporal location between an input pulse and corresponding output pulse arises entirely from the gain. If there were no gain (gL=0) then the plotted input and output pulses would have identical shape and no delay.

 

Fig. 2. a) Input pulse intensity for several candidate pulse shapes: SNIR-optimal (solid), SNR-optimal (dashed), square (dot-dashed), and impulse (dotted). The light vertical lines bound the input window. b) Output pulses, normalized to unit height. The light vertical lines bound the output window. c) Power spectra for SNIR- and SNR-optimal pulses and the square magnitude of the medium transfer function (solid line with squares).

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For this example, the Lorentzian gain exponent is gL=20 and the input and output window widths are given by t_wi=t_wo=t_w=10×(2π/γ) where γ is the Lorentzian half-width at half-maximum (HWHM). The window offset, the amount by which the output window lags the input window, is t ₀=18:4×(2π/γ). The noise level σ ² is defined at the output and is given by exp(gL)/σ ²=70 dB, which would imply a 70 dB SNR if the unit-normalized input pulse were able to achieve the full continuous-wave line-center gain of exp(gL) and have all its energy fall within the output window. However, because the actual pulses place power away from the gain center, they do not experience the full gain and do not achieve this 70 dB upper bound on the SNR and SNIR. This method of defining σ ² is convenient because it requires only knowledge of the output noise level without regard for the origin of the noise; it can come from the amplifier, detector, etc. Also, by defining the noise level in terms of the gain, we can fairly compare performance at different gain values, as discussed below. Herein, all comparisons in dB refer to optical pulse energy. The pulse design process takes these system parameters (t_w, t ₀, gL, and σ ²) as inputs. The specific values used here have been chosen because they represent reasonable physical choices which have been demonstrated experimentally [2]. Because the SNR-optimal and SNIR-optimal pulses are identical in the high-noise limit (as discussed below), we have chosen a small noise value to illustrate their differences.

By comparing the temporal pulses and their spectra, we see that the SNIR-optimal pulse most effectively places a large fraction of its energy inside the output window, thereby reducing the ISI. Although not visible on the height-normalized plots, the SNR-optimal pulse achieves the most energy inside the output window, but without regard for the energy falling outside. The energy within the output window is 6.8×10³, 5.2×10⁷, 4.5×10⁵, and 3.9×10⁷ times the input energy for the SNIR-optimal, SNR-optimal, impulse, and square pulses respectively. The total output energy for each of these is 7.5×10³, 2.0×10⁸, 8.0×10⁵, and 2.6×10⁸ times the input energy. The resulting fractions of energy inside the output window are 90%, 26%, 57%, and 15%.

Each of the optimal pulses is similar to one of the standard reference pulses. The SNR-optimal pulse achieves high energy inside the output window by creating large gain overall. This is precisely what the square pulse does by using low bandwidth. The SNR-optimal pulse does outperform the square pulse slightly because it also temporally shifts the energy to place as much as possible inside the output window. For the SNIR-optimal pulse, minimizing energy outside the window also matters. Using a narrow input pulse helps produce a narrow output pulse. Therefore, the SNIR-optimal pulse is qualitatively similar to the impulse. However, its slight differences result in significant improvement in the SNIR. For example, the lobed spectrum yields similar shape and delay to the impulse, but with higher gain for noise tolerance. The lobed spectrum leads to the visible oscillation at the difference frequency of the two lobes. The spectral bandwidth will narrow in the presence of high noise to achieve higher gain, as shown below.

Note that by traditional measures [2, 8], the SNIR-optimal pulse exhibits the most “distortion” of all cases, both in the sense that the output pulse is “ugly” and in that the output pulse has a very different shape from the input pulse. In this model of pulse propagation, we have abandoned the desire for input and output pulses to have the same shape, and care only that the output pulses are reliably detectable. Rather than designing media to minimize the change in pulse shape, the process described herein designs pulses so that they are well shaped by the medium.

 

Fig. 3. Eye diagrams for the pulses from Fig. 2. The shaded region in each eye diagram indicates the “open” region.

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Figure 3 contains several eye diagrams using the pulses from Fig. 2(b) to demonstrate their suitability for communication. However, one must recall that the pulses have been designed for detection by integrating the pulse energy in the output window, not by thresholding the instantaneous pulse power. The shape of the eye diagram depends on the pulse shape only and does not include noise. Each curve corresponds to a single binary sequence and is formed by incoherently adding the pulse energy from adjacent bits, with zero energy for a zero-bit. In each pulse type the shaded region indicates the “open” region of the eye within the output window. Here, we see that the SNIR-optimal pulse is clearly superior in the sense of addressing ISI. The other pulses all have reasonable open regions, but most of it is not contained within the output window, and would therefore provide better SNIR for smaller window offset, as will be shown below. Clearly the amount of energy in the output window depends on the neighboring bits and is therefore pattern-dependent. However, the simple ISI model used here is valid in the statistical sense if the bits are independent and identically distributed.

The results presented in Figs. 2 and 3 are for a single window offset. Figures 4 (a) and (b) show how the optimal SNR and SNIR vary with window offset. All other parameters have been kept the same: gL=20, t_w=10×(2π/γ), and σ ²=exp(gL)/(70 dB). As expected, the SNR-optimal pulses achieve the greatest SNR for all window offsets, followed closely by the square pulse. By contrast, the SNIR-optimal pulses yield dramatically better SNIR than all other pulse types for almost all values of window offset.

While the SNR-optimal pulse provides little performance benefit over the square pulse, it is interesting both as a high-noise limit of the SNIR-optimal pulse (as discussed below) and as a mathematical stepping stone to the SNIR-optimal pulse. By contrast, the SNIR-optimal pulse provides substantial benefit over other standard pulse shapes, yielding either approximately 10 dB improvement over the impulse for the same window offset or an increased window offset of about 0.3 times the window width in the limit of large window offset. For small offsets, the improvement is even greater.

These plots can be divided into three regions of window offset. For mid-range offset, roughly 5 to 15, all pulses perform reasonably well because the “natural” delay of the system, defined by the peak or centroid of the impulse response, is approximately 9. For comparison, the group delay in this medium is t_g=gL/(2γ)=10. The central region of natural delay also explains the flat region in the SNR and SNIR of the impulse pulse; the input pulse is shifted within the input window in order to follow the output window. For smaller window offset, SNIR-optimal pulses can yield good SNIR by placing power in spectral regions which experience little delay. This effectively achieves the needed small pulse delay despite a fixed medium with large natural delay. Finally, in the large-offset domain, the SNR and SNIR of pulses degrade with the same asymptotic behavior. In this domain, pulses change very little as the window offset varies and performance decreases as the output window moves into the tails of the pulses.

 

Fig. 4. a) SNR of SNIR-optimal (solid), SNR-optimal (dashed), square (dot-dashed), and impulse (dotted) pulses vs window offset for an example gain/window-size/noise. b) SNIR of the same pulses. c) Centroid (with dots) and peak (without) delay vs window offset.

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Figure 4(c) shows traditional delay metrics (the delays of the pulse peak peak and centroid) as functions of window offset. Note that these metrics are presented only for reference and have no impact on the performance of the detection model used here. For the purpose of communication, the window offset completely defines the transmission/detection timing, and the SNR/SNIR determine the detection fidelity. Both impulse and square pulse delays are presented, but they have nearly identical and constant values given by the locations of the peak and centroid of the impulse response because the impulse response is nearly symmetric [6]. These represent the natural delay values for the medium mentioned above. The fluctuations in the SNIR-optimal delays (and SNR) come from its slight oscillatory shape. The counter-intuitive decrease in the SNR-optimal pulse delay as window offset increases is because the input pulse shape changes for each window offset; for very small window offsets, the peak and centroid of the input pulse occur early in the input window. Both input and output peak (and centroid) occur later with increasing window offset, but the shift occurs faster for the input pulse.

This methodology leads to an interesting distinction between window offset and pulse delay. Window offset is purely a property of the transmission and detection mechanism and not a function of the pulses or medium. One can set the window offset to any value, although the quality of the result (SNR or SNIR) may suffer for extreme choices. By contrast, pulse delay (as measured by the peak, centroid, edges, etc.) is purely a property of the pulse and medium. For example, the square pulse has fixed centroid delay, but it can be used in a communication channel with any window offset. That said, natural delay and window offset are clearly related—best performance is achieved when the window offset is chosen to be near the natural delay, although we’ve seen that proper pulse choice can dramatically affect such results.

4. Variation of pulse shape and performance

The technique described above yields optimal pulses (in the sense of maximizing SNR or SNIR) for arbitrary linear media. In this section, we demonstrate how the pulse shape, SNR, and SNIR are affected by varying system properties including window width, noise, and gain. These three parameters are representative of the type of system constraints commonly encountered in the design of slow-light systems. Window width (relative to the medium linewidth) is a property of the transmission/detection scheme directly related to bit-rate. Varying the gain allows us to demonstrate system performance for different media. Finally, all systems have noise and in this case we are particularly interested in the relationship between noise and ISI.

Figure 5 shows several input and output pulses, as well as spectra for varying window width. In all cases the input and output window widths are equal, although this is not required. All other parameters are identical to those used in Fig. 2. The narrow-window limit dramatically illustrates the difference between the SNR-optimal and SNIR-optimal pulses. Although both input pulses are forced to be narrow and therefore have larger bandwidth, the SNR-optimal solution still places most of the pulse power near the carrier frequency (zero frequency) where there is maximum gain. By contrast, the SNIR-optimal pulse puts power in the spectral wings and achieves much larger peak and centroid delay.

 

Fig. 5. Input pulses (a), output pulses (b), and power spectra (c) of SNIR-optimal (solid) and SNR-optimal (dashed) pulses for several different window widths. From top to bottom, the window widths are {14,10,6,2}×(2π/γ). All other parameters are the same as for Fig. 2. The boxes indicate the input and output windows.

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Fig. 6. SNR (a) and SNIR (b) vs window offset for different values of gain. The plots show both SNIR-optimal (solid) and SNR-optimal (dashed) pulses. The symbols correspond to gain with gL taking on the values 10 (plus), 20 (dot), 30 (asterisk). All other values are identical to those used in Fig. 2.

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Figure 5 illustrates an important distinction between pulse bandwidth, which we define as the full-width at half-maximum (FWHM) of the pulse power spectrum, and the “window bandwidth,” given by 2π/t_w. Even for large temporal windows, it may be advantageous to use high-bandwidth pulses. That is, the optimal pulses may have large time-bandwidth products.

Figures 6 (a) and (b) show optimal SNIR and SNR as functions of the window offset for three different gain values; gL=10, 20, and 30. All other parameters are identical to those used in Fig. 2. We see here that lower-gain systems perform better for small window-offsets and high-gain systems perform better for large window-offsets. This further emphasizes the importance of the medium’s natural delay. For a simple Lorentzian gain line, the group delay is given by t_g=gL/(2γ)=gL/2, which corresponds roughly with the peaks of the curves in Fig. 6 for the SNR-optimal pulses. Like the SNR-optimal pulses, the SNIR-optimal pulses perform better at small (large) window offsets for low (high) gain. However, even for high gain, the SNIR-optimal pulses perform well at small window offsets. This is because for offsets smaller than the window width, high-bandwidth pulses can be designed which experience little modification (and delay) because the bulk of the power is placed in the spectral wings of the medium.

Figure 7 shows several pulses designed for varying noise levels. The SNR-optimal pulse design process does not depend on noise, so the corresponding pulse (dashed) does not change. However, the SNIR-optimal pulse balances the contributions from noise and ISI, and so varies with noise. For both input and output pulses, the pulses with edges farther to the right have been designed for smaller noise. The line with dots is the pulse from Fig. 2. As we now see, that is the “low noise” domain in that further reducing the noise has little impact on the pulse design. As the noise increases, the ISI contribution to SNIR becomes less important and the SNIR-optimal pulses eventually become identical to the SNR-optimal pulses in the “high noise” domain. Figure 7 shows SNIR-optimal pulse design over the full range of noise domains; at low noise, the SNIR-optimal pulses are designed to reduce ISI, whereas at high noise they attempt to increase the power in the output window, overcoming the noise by sacrificing ISI.

 

Fig. 7. Input pulses (a), output pulses (b), and power spectra (c) for SNIR-optimal (solid) and SNR-optimal (dashed) pulses for different values of noise. There is only one SNR-optimal pulse because it does not depend on noise. With decreasing noise, both input and output SNIR-optimal pulses shift to the right. The corresponding power spectra get wider. The SNR values are 0, 10, 30, 70, and 90 dB, as defined in the text. The line with dots is the SNIR-optimal pulse used in Fig. 2.

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5. Conclusions

The technique presented in this paper allows for the creation of optimal pulses in the sense of SNIR and SNR for this window-based transmission/detection scheme. There are several implications of this approach. For example, this technique facilitates joint design of the entire slow-light system, creating both media and pulses for optimal performance. It also opens the possibility of controlling delay by designing pulses for different output windows, and letting the detector choose the detection synchronize to the incoming pulse sequence. However, because current communications schemes rely on simple uniform (and ideally, unchanging) pulse shape, such devices cannot simply be dropped into existing networks; they would require translation hardware and would therefore be most likely to find application in optical local area networks or other isolated environments. Perhaps a more near-term application of these techniques is the design of a single pulse shape to be robust with respect to medium-controlled delay. That is, to maintain good detection properties as the medium is changed to adjust the delay.

Finally, these results suggest a different approach to determining delay; rather than measuring the delay of a pulse, choose a window offset or more general “detection delay” and quantify the signal fidelity given that choice. The latter approach reflects actual communication-oriented performance rather than an arbitrary measure based on a specific pulse shape.

This work was supported by the DARPA/DSO Slow Light program.