## Abstract

We describe compression and expansion of the time–bandwidth product of signals and present tools to design optical data compression and expansion systems that solve bottlenecks in the real-time capture and generation of wideband data. Applications of this analog photonic transformation include more efficient ways to sample, digitize, and store optical data. Time–bandwidth engineering is enabled by the recently introduced Stretched Modulation (${\mathrm{S}}_{\mathrm{M}}$) Distribution function, a mathematical tool that describes the bandwidth and temporal duration of signals after arbitrary phase and amplitude transformations. We demonstrate design of time–bandwidth engineering systems in both near-field and far-field regimes that employ engineered group delay (GD), and we derive closed-form mathematical equations governing the operation of such systems. These equations identify an important criterion for the maximum curvature of warped GD that must be met to achieve time–bandwidth compression. We also show application of the ${\mathrm{S}}_{\mathrm{M}}$ Distribution to benchmark different GD profiles and to the analysis of tolerance to system nonidealities, such as GD ripples.

© 2014 Optical Society of America

## 1. INTRODUCTION

With the quantity of data growing exponentially, new approaches to data capture and compression are urgently needed. With respect to the field of optics, this predicament arises in fiber optic communication and in real-time optical instruments [1]. Such instruments are used in study of optical rogue waves [2–5], in ultrafast signal measurement [6–12], and ultrafast imaging [13–15]. Owing to their high measurement rate, real-time instruments produce a fire hose volume of data that overwhelms even the most advanced computers [16]. This necessitates innovations in data management and inline processing techniques. As a case study, consider a dispersive Fourier transform (DFT)-based real-time spectrometer or camera, such as the time stretch microscopy [13] that captures single-shot optical spectra or images at a 100 MHz frame rate with each frame containing 1000 samples. Assuming each sample is digitized with 10 bits of accuracy, such a system produces 1 terabit of data each second [16]. This is equivalent to capturing and storing $>10\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{h}$ of full-HD movies each second! Time–bandwidth compression can alleviate this problem.

The generation of waveforms with wide instantaneous bandwidth can be a bottleneck in radar, communication systems, and laboratory instruments. To this end, photonic arbitrary waveform generation that employs spectrum to time mapping has been proposed as a potential solution [17–19]. These systems are limited by the time bandwidth of the component used for spectrum encoding. Time–bandwidth expansion may improve the performance in these applications.

The Stretched Modulation (${\mathrm{S}}_{\mathrm{M}}$) Distribution is a recently introduced mathematical tool that describes the information bandwidth and the record length [20,21] of signal intensity after arbitrary transformations. It provides graphical and intuitive description of time–bandwidth transformation and can be used to benchmark different optical systems. It was recently used for the design and demonstration of the first optical real-time data compression system [22].

In this work, we present foundations of methods to manipulate the time bandwidth of optical signals using photonic platforms. We employ the ${\mathrm{S}}_{\mathrm{M}}$ Distribution to design time–bandwidth engineering (TBE) systems in both near-field and far-field regimes. Using the ${\mathrm{S}}_{\mathrm{M}}$ distribution we derive, for the first time to the best of our knowledge, the equations governing the TBE and identify an important criterion on the maximum curvature of warped group delay (GD) that must be met in order to achieve time–bandwidth compression. This criterion is then used to compare various GD profiles as it relates to their utility for TBE. We also show, for the first time to the best of our knowledge, that ${\mathrm{S}}_{\mathrm{M}}$ Distribution can be used to analyze the effect of system nonidealities in TBE systems. In particular, we study the effect of experimentally measured GD ripples of a chirped fiber Bragg grating fabricated for optical real-time data compression application on the bandwidth compression factor and time–bandwidth product (TBP). ${\mathrm{S}}_{\mathrm{M}}$ Distribution analysis of four optical systems indicate that the TBP in far-field operation is determined by GD ripples at lower frequencies, whereas, in the case of near-field operation, it is determined by ripples at higher frequencies.

## 2. STRETCH MODULATION DISTRIBUTION

The system block diagrams for general TPE systems are shown in Fig. 1. Input electric field ${E}_{i}(t)$ has a time duration of ${\mathrm{T}}_{\text{in}}$ and bandwidth of ${\mathrm{BW}}_{\text{in}}$. The transformation is described by phase kernel $\tilde{K}(\omega )$ and a nonlinear operation, a simple example of which is the field-to-intensity conversion ${E}_{i}(t)\xb7{E}_{i}{(t)}^{*}$. In the case of TBP compression [Fig. 1(a)], the transformation matches the signal bandwidth to that of the back-end digitizer (e.g., an oscilloscope or spectrometer), while at the same time minimizing the record length to avoid generation of redundant data. In the case of expansion, the transformation increases the modulation TBP of a generated signal. Since this transformation intentionally warps the signal, it can be referred to as the anamorphic transform, a metaphoric reference to the technique of anamorphism in graphic arts [16].

The ${\mathrm{S}}_{\mathrm{M}}$ Distribution is a complex-valued three-dimensional plot that allows one to design TBE systems. It describes the information bandwidth and duration of the signal intensity after a transformation that is mathematically described by the kernel of ${\mathrm{S}}_{\mathrm{M}}$. The Distribution can be written as [20,21]

Considering the Distribution as a cross correlation of the signal spectrum after transformation with its time-shifted waveform, the maximum absolute amount of time variable $t$ at which the ${\mathrm{S}}_{\mathrm{M}}(\omega ,t)$ function has non-zero values is the duration of output signal ${\mathrm{T}}_{\text{out}}$. This is given by the half-height of the plot in ${\mathrm{S}}_{\mathrm{M}}$ Distribution or half-extent in the temporal direction; see Fig. 2.

From Eq. (1), it can be shown that at $t=0$ the cross section of the Distribution is the output modulation (intensity) spectrum:

where FT is the Fourier transform. Thus, the width of the cross section at $t=0$ gives the output modulation bandwidth; see Fig. 2.The phase kernel $\tilde{K}(\omega )$, which is the transfer function of an optical system, allows one to engineer the TBP of the modulation envelope. Let the operation kernel be defined by a nonlinear phase operator:

The kernel’s phase profile $\beta (\omega )$ is characterized by the phase derivative, i.e., the GD profile $\tau (\omega )=d\beta (\omega )/d\omega $. The information of the TBP can be manipulated by operating on the signal’s complex field with a specific kernel followed by a nonlinear operation provided by the photo-detector.## 3. DESIGN AND ANALYSIS OF TBE SYSTEMS

To show how the Distribution can be used to design and analyze TBE systems, we consider systems in which the operation kernel represents a phase filter with different GD profiles. In practice, the filter operation can be implemented with a dispersive optical element with a designed GD profile. After dispersion, the signal can be in the far-field or near-field regime, depending on whether the stationary phase approximation is satisfied or not. The far field is achieved for a large amount of GD dispersion and/or when the signal has a very large bandwidth, leading to one-to-one mapping of frequency into time. Conversely, the near field refers to the regime prior to the stationary phase approximation being satisfied. We note that in the far field there is warped (nonlinear or nonuniform) frequency-to-time mapping (warped dispersive Fourier transform), whereas in the near field there is no one-to-one mapping. Therefore, the near-field case, on its own, is not a Fourier transform or frequency–time mapping for arbitrary signals.

We consider three types of GD profiles, namely, linear, sub-linear, and super-linear. Here the sub-linear and super-linear profiles are defined as functions that grow slower or faster than a linear function when its argument becomes very large. Figure 3 shows qualitative profiles of these three types of GD. Here we compare the ${\mathrm{S}}_{\mathrm{M}}$ Distribution plots associated with these three GDs. The input signal used for numerical simulations in this paper is shown in the time and spectral domains in Fig. 4. For this set of simulations, we have normalized the input signal to its maximum bandwidth. The input was designed to have a collection of different temporal features, such as coarse and fine temporal features, with different durations. It also includes both closely spaced as well as sparsely spaced features.

Figure 5 shows the ${\mathrm{S}}_{\mathrm{M}}$ Distribution of these three systems when operated in the near-field and far-field regimes. As shown in Fig. 5(a), when the GD has a linear profile, the Distribution is linearly tilted, resulting in a reduced modulation bandwidth. However, record length is proportionally increased. In this case, the TBP is either conserved (in the far-field regime) or expanded (in the near-field regime). However, if we cause a nonlinear tilt (i.e., a warp) having the shape shown in Fig. 5(b), corresponding to the sub-linear GD profile shown in Fig. 3, the bandwidth is reduced but the record length is not increased proportionally. In this case, one achieves TBP compression in both the near-field and far-field regimes. Finally, if the Distribution is warped in the manner shown in Fig. 5(c), corresponding to the super-linear GD profile shown in Fig. 3, the TBP is expanded in both the near-field and far-field regimes.

## 4. MATHEMATICAL FOUNDATIONS OF THE ${\mathsf{S}}_{\mathsf{M}}$ DISTRIBUTION

Here we calculate the TBP of a bandwidth-limited signal $E(t)$ after propagation through a filter or a dispersive optical element with GD response $\tau (\omega )=d\beta (\omega )/\mathrm{d}\omega $, where $\beta (\omega )$ is the phase profile of the system kernel. By substituting Eq. (3) into Eq. (1) and changing variables (${\omega}_{1}\to {\omega}_{1}-\omega /2$), the absolute value of the ${\mathrm{S}}_{\mathrm{M}}$ Distribution at the output can be represented as follows:

In deriving Eq. (5), we have used the assumption that the profile of the phase kernel, $\beta (\omega $), is an even function. For the case that the GD curvature, ${\tau}^{\prime \prime}(\omega )$, is sufficiently small, i.e., $|{\tau}^{\prime \prime}(\omega /2)|\ll 24\xb7|\tau (\omega /2)|/\mathrm{\Delta}{\omega}^{2}$, where $\mathrm{\Delta}\omega $ is the input signal complex-field bandwidth, we have

In this case, Eq. (4) is simplified to

We showed earlier using Eq. (1) that the duration of the output signal is given by half-extent of the ${\mathrm{S}}_{\mathrm{M}}$ Distribution. Thus, the following equation gives the output signal duration:

We have graphically shown how the output signal duration is related to the input signal and the system kernel. This relation is shown in Fig. 6. Mathematically described by Eq. (8), it is valid for both the near-field and far-field regimes. In the far field, Eq. (8) can be approximated by ${\mathrm{T}}_{\mathrm{out}}\approx 2\xb7\tau (\mathrm{\Delta}\omega /2)$.

As seen in Fig. 3 the modulation bandwidth, ${\mathrm{BW}}_{\text{out}}$ or $\mathrm{\Delta}{\omega}_{m}$, is the $t=0$ intercept of the ${\mathrm{S}}_{\mathrm{M}}$ plot. Mathematically, it is equivalent to finding the minimum frequency at which the input duration plus the GD shift is larger than zero:

This is the estimated output intensity bandwidth assuming that the input signal has infinite bandwidth. Because in a physical system the maximum output intensity bandwidth is limited by twice the input bandwidth, the $\mathrm{\Delta}{\omega}_{m}$ is the minimum amount of input signal bandwidth and the amount estimated for $\mathrm{\Delta}{\omega}_{m}$ with infinite input bandwidth given by Eq. (9):

The above conclusion assumes that the signal is broadband, comprising both fast and slow features, i.e., high and low instantaneous frequencies with respect to the bandwidth of the GD. Such a signal has redundancy in the time domain (see Fig. 4). If the signal has only high frequencies, then the effect of GD will be different than that described above. Also, as with any other data compression method, the maximum compression that can be achieved is signal dependent. In particular, it will depend on the amount of redundancy in the signal, a quantity that will be reflected in the probability distribution function of the signal instantaneous frequency.

To examine the output TBP, we have compared the calculated value from Eq. (11) to numerical simulations. The input optical signal is shown in the time and spectral domains in Fig. 4. The input signal has duration of 200 ps and complex-field bandwidth of 1 THz. Duration and the bandwidth here are defined as the 1% value (from peak). We have considered three systems with different types of GD profiles, i.e., with linear, sub-linear, and super-linear GD profiles shown in Fig. 7(a). The super-linear case has the same total GD as the linear case and the sub-linear case has the same slope as the linear case at the origin. The GD profiles of these functions are given by

We can identify three regions of operation spanning the near- to far-field regimes. For the case of linear GD, an example of ${\mathrm{S}}_{\mathrm{M}}$ Distribution in each region is shown in Fig. 8. Region 1 ($k<0.5$) corresponds to the case when $\mathrm{\Delta}{\omega}_{m}=2\xb7\mathrm{\Delta}\omega $. Since in this region the output bandwidth is fixed but the output duration is increased [given by Eq. (8)], the TBP is enlarged as shown in Fig. 8(a). However, in Regions 2 and 3, the TBP gets close to unity as dispersion is increased [see Figs. 7(b), 8(b), and 8(c)].

## 5. PHYSICAL IMPLEMENTATION

The system block diagram for a simple optical TBE system is shown in Fig. 9(a). The desired phase kernel profile for TBE can be obtained using a chirped fiber Bragg grating (CFBG). A CFBG offers great flexibility in dispersion profile and has low insertion loss. An example of a designed CFBG period profile along grating distance to demonstrate a sub-linear GD profile is shown in Fig. 9(b). This reflection CFBG is 100 cm long, its nominal period is 504.54 nm, and the effective refractive index of the fiber is 1.54. The measured GD profile of a fabricated CFBG using this CFBG design is shown in Fig. 9(c). CFBGs exhibit GD ripples, which are problematic. There are demonstrated techniques for mitigating the effect of ripples that can be employed in our technique to calibrate the GD ripples in post-processing analysis [12,23]. We note that the transformed signal has both phase and amplitude requiring complex field recovery before reconstruction in the digital domain [20–22].

Finally, we note that, if the carrier wavelength is moved away from the center of symmetry of the GD curve, a different characteristic for the TBE system is achieved. In particular, “sub-linear” and “super-linear” descriptions imply the carrier wavelength is at the center of symmetry. If the carrier is moved away from the center of symmetry, the sub-linear curve will no longer be sub-linear and the effect on the TBP will change. The ${\mathrm{S}}_{\mathrm{M}}$ Distribution can be used to study such effects.

## 6. ANALYSIS OF TOLERANCE TO SYSTEM NONIDEALITIES

The Distribution function can also be used to assess the impact of nonidealities in the performance of TBE. In a dispersive TBE system, nonidealities include GD ripple, polarization dependence, and temperature fluctuations. The ${\mathrm{S}}_{\mathrm{M}}$ Distribution visualizes the effect of nonidealities on information bandwidth and record length and provides insight into how to mitigate them.

In the examples studied here, we consider four systems with GD profiles shown in Fig. 10. We consider operation in the far field [Figs. 10(a) and 10(b)] as well as the near field [Figs. 10(c) and 10(d)]. In each regime we consider two cases: GD ripples of less than $\pm 4\%$ and $\pm 8\%$ of the maximum GD. The far-field GD profile with 4% ripples is the experimentally measured result for a fabricated grating that was recently used to demonstrate optical real-time data compression [22] [see Fig. 10(a)]. The grating was designed to compress the bandwidth of a signal with 1 THz bandwidth to 8 GHz so it can be digitized in real time, and at the same time to compress the record length and, hence, the digital data size. For the far-field example with 8% GD ripples [Fig. 10(b)], we have simulated a GD profile with the same profile but twice the ripple amplitude. For the near-field examples, Figs. 10(c) and 10(d), we have scaled down the measured profiles in Figs. 10(a) and 10(b) 20 times.

The time domain and spectrum of the input signal are shown in Fig. 4. Figure 11 shows the ${\mathrm{S}}_{\mathrm{M}}$ Distribution corresponding to the large GD profiles with 4% and 8% ripples. Since in both cases the system is operating in the far-field regime, the expected output duration is 10 ns which is the total system GD. The ${\mathrm{S}}_{\mathrm{M}}$ Distribution shows that systems with 4% and 8% ripples have similar output durations (10 ns), suggesting that the effect of GD ripples on the output signal duration is very small. The ${\mathrm{S}}_{\mathrm{M}}$ Distribution also shows that, in the case with 4% GD ripples, the bandwidth is compressed to the desired 8 GHz baseband bandwidth (16 GHz passband bandwidth). However, the case with 8% GD ripples results in 27 GHz bandwidth.

The ${\mathrm{S}}_{\mathrm{M}}$ Distribution suggests that, depending on its relative magnitude, GD ripples can dramatically impact the TBP in a TBE system. In the present example, limiting tolerances to 4% ensures the system reaches the desired 8 GHz output bandwidth. The Distributions depicted in Fig. 11 also show that the output bandwidth in the far-field regime is mostly related to the GD ripples around the carrier frequency. This can be seen by comparing the Distribution plots in Figs. 11(a) and 11(b) to their corresponding GD profiles in Fig. 10(a) and 10(b). The large ripple in the right side in Figs. 10(a) and 10(b) results in the new track seen in their ${\mathrm{S}}_{\mathrm{M}}$ plots in Fig. 11, as indicated with triangles.

The small ripple in the left side of the GD profiles in Figs. 10(a) and 10(b) results in the new track seen in their ${\mathrm{S}}_{\mathrm{M}}$ plots in Fig. 11, indicated with squares. These Distributions show that these two ripples do not contribute to the output bandwidth; however, the ripples around the carrier frequency ([see zoomed plots in Figs. 10(a) and 10(b)] dramatically impact the output bandwidth [see zoomed plots in Figs. 11(a) and 11(b)].

We have also investigated the use of ${\mathrm{S}}_{\mathrm{M}}$ plots to analyze the effect of ripples on the TBP in the near field. The near-field GD profiles with 4% and 8% GD ripples are shown in Figs 10(c) and 10(d). The corresponding ${\mathrm{S}}_{\mathrm{M}}$ Distributions are shown in Figs. 12(a) and 12(b). In this case the Distribution plots also show that effect of GD ripples on the output duration is small and the main impact is on the output modulation bandwidth. In particular, in the case with 4% ripples the bandwidth is compressed to 200 GHz (400 GHz intensity bandwidth), as opposed to 380 GHz in the 8% case. When comparing the ${\mathrm{S}}_{\mathrm{M}}$ plots in Figs. 12(a) and 12(b) to their corresponding GD profiles in Figs. 10(c) and 10(d), we find that the large GD ripples on the right side of the profile in Fig. 10(d) have resulted in a dramatic increase in the output bandwidth. In contrast to the far-field regime, the ${\mathrm{S}}_{\mathrm{M}}$ plots show that the modulation bandwidth in the near field is determined mainly by the GD ripples at higher frequencies.

## 7. RELATION TO ANAMORPHIC STRETCH TRANSFORM

The ${\mathrm{S}}_{\mathrm{M}}$ Distribution has been used to design an anamorphic stretch transform (AST), which is a new way of compressing analog and digitized data by selectively stretching and warping the signal in the frequency domain [20,21]. AST has been used to demonstrate the first analog optical real-time data compression instrument [22]. This analog compression is achieved in an open-loop fashion, without prior knowledge of the input waveform. The discrete implementation in the form of an algorithm has been applied to digital image compression, where it has shown superior performance compared to standard image compression techniques [24].

## 8. RELATION TO ANAMORPHIC TEMPORAL IMAGING

The ${\mathrm{S}}_{\mathrm{M}}$ Distribution has been recently used to design an analog optical transform, called anamorphic temporal imaging, for engineering the time bandwidth within a temporal imaging system. In temporal imaging systems [8,25,26], the TBP is not changed, whereas the anamorphic temporal imaging system can compress or expand the TBP [27,28].

When placed in front of a spectrometer, this technique enhances the spectral resolution $\delta \omega $ and the update rate by performing optical real-time data compression without losing information. Figure 13(a) shows a block diagram of anamorphic temporal imaging implementation. In this technique, the signal is warped in the time lens by mixing it with a local oscillator (LO) that has a nonlinear instantaneous frequency. Two power spectrum measurements are used for complex-field detection of the output signal [29]. ${\mathrm{S}}_{\mathrm{M}}$ Distribution is used to design the instantaneous frequency profile in the warped time lens that leads to time–bandwidth compression. Figure 3(b) shows the qualitative ${\mathrm{S}}_{\mathrm{M}}$ plots before and after an anamorphic temporal imaging system with an LO with a sub-linear instantaneous frequency profile. As seen, spectral resolution $\delta \omega $ is increased, but the bandwidth is not expanded proportionally, i.e., time–bandwidth compression.

## 9. RELATION TO OTHER TIME–FREQUENCY DISTRIBUTIONS

The ${\mathrm{S}}_{\mathrm{M}}$ Distribution belongs to general class of time–frequency distribution functions that includes short-time Fourier transform [30], Wigner distribution [31], and the ambiguity function [32]. Short-time Fourier transform and Wigner distribution analyze the time and frequency dependence of the signal’s electric field. In contrast, ${\mathrm{S}}_{\mathrm{M}}$ Distribution describes the signal’s modulation intensity envelope. The ambiguity function, used mainly in pulsed radar and sonar, is a two-dimensional function of time delay and Doppler frequency [32]. None of these functions are designed for or have been used for TBE. Unique to the ${\mathrm{S}}_{\mathrm{M}}$ Distribution, the built-in operation kernel enables the Distribution to show how the time duration and information bandwidth are affected when the signal is transformed by a nonlinear phase operation.

## 10. SUMMARY

We have described foundations of optical methods to engineer the time bandwidth of information-carrying signals. Here the time represents the record length and the bandwidth is the modulation, i.e., the information bandwidth. To design such systems, we use the ${\mathrm{S}}_{\mathrm{M}}$ Distribution, a mathematical tool that describes the bandwidth and temporal duration of the signal intensity (instantaneous power) before and after operation with a phase filter with an arbitrary GD. We provided closed-form mathematical expressions that describe how the signal’s duration and bandwidth are shaped by the GD characteristics. We also showed how the Distribution can provide visual insight and tolerances to nonidealities, such as GD ripples. Finally, the mathematical tools presented in this paper offer opportunities for follow-up work directed to detailed study of different implementations of the TBE system, including studying and benchmarking various GD profiles.

## FUNDING INFORMATION

Office of Naval Research (ONR) (N00014-13-1-0678).

## ACKNOWLEDGMENT

The program manager for the ONR MURI program funding was Dr. Steve Pappert.

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