## Abstract

I present an approach which determines analytically, for any desired output Gaussian pulse-envelope modulated chirped harmonic wave of arbitrary width at a given propagation distance in an absorptive, dispersive linear metamaterial medium acting as a compressor, the required, specific input pulse modulated wave. Employed asymptotic techniques yield the analytic description of the dynamical evolution of the appropriate input wave which is time-focused, and dispersion compensated upon propagation in a linear time-invariant system with a transfer function representing a chosen, physically realizable, Lorentz-type metamaterial. This approach is validated upon comparison with the results of different employed numerical experiments.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Mode-locking, initially in dye and later in solid-state lasers, coupled with compression and dispersion compensation schemes underpin efforts regarding the generation, amplification, shaping and characterization of femtosecond pulses from the near infrared to the visible spectral regions [1–10]; Carrier-Envelope Offset (CEO) stabilization techniques ushered these efforts in the single-cycle regime. Successive achievements, include shorter pulse durations, and increasingly higher energies, average powers, and pulse repetition rates [1,2]. Currently, intense, femtosecond, waveform-controlled few-cycle pulses are also central to the generation of (isolated or trains of) record-setting attosecond pulses in the soft X-ray (SXR) spectral region [2–4,9,11,12]; the branch of attosecond science emerged.

This stream of efforts belongs to a virtuous circle of new theoretical approaches coupled with constant experimental innovations that lead to the emergence of diverse applications. The latter encompass the interaction of high-intensity radiation with matter, spectroscopy, study of ultrafast processes, metrology, atmospheric propagation and information processing and transmission systems [1–17]. Physics, chemistry, material science, biology, and medicine, have already benefitted and a plethora of industrial applications are emerging.

Τhe theoretical treatment of such pulses is predominantly dependent upon variants of the Slowly-Varying-Envelope-Approximation (SVEA) with their inherent limitations especially in the single and sub-cycle pulse regimes [1–5,7,8,11,13–16,18–36]. An important theoretical alternative that circumvents these limitations is based upon asymptotic analysis and resulted in an accurate, uniformly valid description of the propagated field dynamics due to rapid rise/fall-time or exponentially varying input fields, in linear, causally dispersive media [1,7,29,37–40].

A research area which also attracts increasing scientific and industrial interest deals with metamaterials, which facilitate unprecedented, manipulation of the electric and magnetic fields and lead to unique properties not found in nature. Transformation optics, spatial cloacking, super-resolution imaging, sub-wavelength field localization, confinement and amplification in plasmonic resonators, antennae with superior properties, ultrafast pulse shaping and a host of other theoretical predictions have been experimentally verified and added impetus to related research efforts [6,7,10,16,17,21–23,25–27,35,41].

Here, metamaterials are utilized to compress and shape electromagnetic pulses to a desired chirped Gaussian waveform of arbitrary width. The presented approach employs asymptotic techniques and does not depend on the assumptions inherent in the SVEA-based analytic approaches. Therefore, it is valid from the quasi-monochromatic to the single-cycle regimes. The derived, intrinsically coupled, pair of input-output pulse waves manifests temporal-compression, in addition to dispersion-compensation. Therefore, metamaterials are shown here to enable temporal-focusing (and temporal-shift) as was reported for generic dispersive media [1,4–8,10,13–18,23,24,26–28,30–32,34–36], in addition to spatial-focusing achieved in the case of a super-lens [7,22,41]. The increased insight and control of the temporal dynamics of ultrashort (few-cycle) waveforms may then be applied in (ultra-broadband) radio-frequency photonics [4], temporal pulse-field characterization [3], and may be extended to optical frequencies in order to be utilized for isolated attosecond pulse generation [4], and to facilitate pulse (pre-)shaping applications [4,6,7,10,13,14,17].

## 2. Definition of the problem

Any orthogonal component of the electric or magnetic field, Hertz vector or vector potential and the scalar potential, in the half-space $z > 0$ occupied by a linear, homogeneous, isotropic, temporally dispersive, non-hysteretic medium, due to an input arbitrary plane wave, is given exactly by [1,3,6,7,13,15,18,22–24,29,37–40,42,43]

*c*the vacuum speed of light; the integration contour

*L*in the complex $\omega - $plane is the real frequency axis or any homotopic contour to this axis. For rapid rise/fall-time or exponentially-varying input pulses [1,4,6,7,13–18,20,24,26,28,29,32–40], absorption and dispersion lead to increasing distortion and even break-up of the output pulse wave, and eventually to the appearance of precursor fields, especially when the corresponding input pulse width approaches the single and eventually the sub-cycle regimes.

Consider that, a metamaterial medium occupies $z > 0$ and is characterized by the Lorentz-type, (effective) electric permittivity and (effective) magnetic permeability [21–23,25,41,42],

Herein, the generic problem of interest is to identify analytically the required input pulse modulated wave which upon propagation in this metamaterial generates, at a fixed, arbitrary distance $z = d$, a chosen, arbitrary, (waveform-controlled) Gaussian pulse-envelope modulated chirped harmonic wave with temporal behavior [7,35,40,42]

*C*; the constant phase $\psi $ is zero for a sine or ${\pi / 2}$ for a cosine wave.

## 3. Analytic determination of the input pulse modulated wave in a Lorentz-type metamaterial medium

The generic, metamaterial propagation scheme is illustrated in Fig. 1, and may be considered as a linear time-invariant system whose frequency domain output is ${A_{{m_{out}}}}({z,\omega } )= {H_m}({z,\omega } ){A_{{m_{in}}}}({z = 0,\omega } )$ [4,6,7,15,24,33,34], with ${H_m}({z,\omega } )$ denoting the metamaterial’s frequency response. The absorptive and dispersive effects due to propagation in the metamaterial may be compensated by a proper choice of the input pulse wave ${A_{{m_{in}}}}({z = 0,\omega } )$. In order to determine analytically the dynamical evolution of ${A_{{m_{in}}}}({z = 0,\omega } )$, consider the propagation scheme in Fig. 2 with ${A_{com{p_{out}}}}({z^{\prime},\omega } )= {H_{comp}}({z^{\prime},\omega } ){A_{com{p_{in}}}}({z^{\prime} = 0,\omega } )$, where ${H_{comp}}({z^{\prime},\omega } )$ denotes the frequency response of the input-stage compensation filter [4,7,14,33,34]. Also consider that,

Let, ${A_{com{p_{in}}}}({z^{\prime} = 0,\omega } )= g(\omega )$ be the Fourier-Laplace transform of the desired (output) chirped Gaussian pulse wave $g(t )$. The boundary conditions in Eq. (4) are satisfied provided that

The unified asymptotic approach [29,37–40,42] is invoked in order to evaluate the total propagated field in Eq. (4) (${A_{com{p_{out}}}}({z^{\prime},t} )$ at point $(\Gamma )$ in Fig. 2 or, equivalently, ${A_{{m_{in}}}}({z = 0,t} )$ at point $({\rm B} )$ in Fig. 1). Accordingly, the temporal behavior of ${A_{com{p_{out}}}}({z^{\prime},t} )$ is obtained from the exact, unified integral expression [42]

The application of the unified asymptotic approach entails the identification of the dynamics of the saddle points ${\omega _{S{P_{Ul}}}}(\theta )$, $l \in N$(where the index *l* is a physical number) as well as of the topography of the real and imaginary parts of ${\Phi _U}({\omega ,\theta } )$ in the complex $\omega - $plane, in order to assess the relevance and dominance of each saddle point. Cauchy’s residue theorem is then applied in order to deform continuously the original integration path *L* such that, at each $\theta - $value, it is a sum of sub-paths each passing only through a single relevant saddle point. The total propagated field ${A_{com{p_{out}}}}({z^{\prime},t} )= \sum\nolimits_{l = 1}^n {{A_{Ul}}({z^{\prime},t} )} $, $n \in N$, is a linear superposition of pulse components ${A_{Ul}}({z^{\prime},t} )$ each encapsulating the asymptotic contribution of a respective relevant saddle point ${\omega _{S{P_{Ul}}}}(\theta )$, whose dynamical evolution depends upon the input field ($2T$, ${\omega _c}$, *C*, ${t_0}$, $\psi $) and medium (${\omega _{ep}}$, ${\omega _{e0}}$, ${\gamma _e}$, ${\omega _{mp}}$, ${\omega _{m0}}$, ${\gamma _m}$) parameters and the propagation distance ($z^{\prime}$) [42]. Herein, the first two asymptotic series expansion terms are retained, and each ${A_{Ul}}({z^{\prime},t} )$ is given by

According to Eq. (9)-(10), the asymptotic analysis depends dynamically on the input field and medium parameters and the propagation distance and must be performed when any of them changes value. Moreover, ${\omega _{S{P_{Ul}}}}(\theta )$, $l \in N$ appears in the exponent in Eq. (9) and must be determined very accurately in order to obtain an accurate description of ${A_{Ul}}({z^{\prime},t} )$, and also of the ensuing ${A_{com{p_{out}}}}({z^{\prime},t} )$.

## 4. Results and discussion

The propagated field ${A_{com{p_{out}}}}({z^{\prime} = d,t} )$ is evaluated and depicted in Fig. 3(a) and Fig. 3(c) based upon the contribution of a single relevant saddle point in the unified asymptotic approach. Moreover, ${A_{com{p_{out}}}}({z^{\prime} = d,t} )$ is evaluated and depicted in Fig. 3(b) and Fig. 3(d) based upon an implementation of the inverse Laplace-transform [43]; this purely numerical approach has been thoroughly elaborated, calibrated upon comparison with respective results of additional numerical experiments and utilized previously in dispersive pulse propagation problems [29,37–40,43]. In particular, ${A_{com{p_{out}}}}({z^{\prime} = d,t} )$ corresponds to two cases of the input chirped Gaussian pulse wave ${A_{com{p_{in}}}}({z^{\prime} = 0,t} )= g(t )$. Case-1, in Fig. 3(a) and Fig. 3(b), and Case-2, in Fig. 3(c) and Fig. 3(d), both correspond to an input Gaussian pulse-envelope with $2T = 0.236 \times {10^{ - 9}}s$, centered at ${t_0} = 15T$ which modulates a sine wave ($\psi = 0$) with ${\omega _c} = 2\pi \times 21.156 \times {10^9}{s^{ - 1}}$ that is unchirped (${C^0} = 0.0{s^{ - 2}}$); these cases correspond to a five-cycle input pulse wave. However, the propagation distance in Case-1 is $z^{\prime} = 2.5 \times {10^{ - 3}}m$ while in Case-2 it is $z^{\prime} = 7.5 \times {10^{ - 3}}m$ in the same metamaterial medium considered in previous research [23,25,42] (${\omega _{ep}} = 56\pi \times {10^9}{s^{ - 1}}$, ${\omega _{e0}} = 0.0{s^{ - 1}}$, ${\gamma _e} = 1.6 \times {10^9}{s^{ - 1}}$, ${\omega _{mp}} = 49\pi \times {10^9}{s^{ - 1}}$, ${\omega _{m0}} = 42\pi \times {10^9}{s^{ - 1}}$, ${\gamma _m} = 4.0 \times {10^9}{s^{ - 1}}$).

In both cases, ${\omega _c}$ is chosen to lie at the absorption band center, in order to impose stringent requirements on the accuracy of the asymptotic analysis. According to Fig. 3, there is close agreement between the description of the propagated field afforded by the unified asymptotic approach and the corresponding prediction of the inverse Laplace numerical approach except at the rear end of the propagated pulse. As depicted in Fig. 4, the inclusion of the second expansion term in the propagated field expressions (9)-(10), results in improved agreement with respect to the situation when only the first dominant expansion term is retained.

In Fig. 3(a), the root mean square (RMS) width [1] of ${A_{com{p_{out}}}}({z^{\prime} = d,t} )$ at point $(\Gamma )$ is ${\sigma _1} = 0.464 \times {10^{ - 9}}s$ whereas the respective RMS width of $g(t )$ at point $(\Delta )$ is ${\sigma _0} = {T / 2} = 0.059 \times {10^{ - 9}}s$ and the broadening factor is ${{{\sigma _1}} / {{\sigma _0}}} = 7.864$. In Fig. 3(c), ${\sigma _2} = 0.823 \times {10^{ - 9}}s$ whereas, again, ${\sigma _0} = {T / 2} = 0.059 \times {10^{ - 9}}s$, and ${{{\sigma _2}} / {{\sigma _0}}} = 13.949$.

In Fig. 5, (a) numerical experiment employing the Finite Difference Time Domain (FDTD) method has been utilized to directly solve the differential form of Maxwell’s equations and determine the propagated field ${A_{{m_{out}}}}({z = d,t} )$ in the output stage of the metamaterial pulse compression scheme. Each of Fig. 5(a)–5(d) corresponds to an input ${A_{{m_{in}}}}({z = 0,t} )$, which is chosen to be the appropriate output pulse wave ${A_{com{p_{out}}}}({z^{\prime} = d,t} )$ in the respective Fig. 3(a)–3(d), as evaluated using both the unified asymptotic approach and the inverse Laplace numerical approach. The agreement between ${A_{{m_{out}}}}({z = d,t} )$ and the corresponding desired $g(t )$ is evident in Fig. 5(b) and Fig. 5(d) when ${A_{{m_{in}}}}({z = 0,t} )$ is taken to be the prediction of the latter numerical approach, thus validating the proposed metamaterial pulse compression scheme. Moreover, when ${A_{{m_{in}}}}({z = 0,t} )$ is derived from the unified asymptotic approach, a similar level of agreement is obtained in the five-cycle pulse wave cases depicted in Fig. 5(a) and Fig. 5(c) for the two increasing propagation distances. In order to improve the accuracy of the unified asymptotic description for the entire evolution of ${A_{{m_{out}}}}({z = d,t} )$, especially when $g(t )= {A_{{m_{out}}}}({z = d,t} )$ approaches the single-cycle regime, additional asymptotic series expansion terms need to be retained in the expressions (9)-(10).

According to the unified asymptotic approach, for chirped Gaussian-pulse propagation in a generic dielectric medium the envelope of a pulse component $|{{A_{Ul}}({z,t} )} |$ attains its stationary points when the general condition

In order to consider analytically the propagation of an input pulse wave with an increasingly shorter temporal-duration, the expressions in Eq. (2) must represent accurately the dielectric properties of a considered linear medium over a wide frequency range. The unified asymptotic approach [1–4,7,11,13–16,18–36], considers the medium’s absorptive and dispersive behavior without the assumptions inherent in all SVEA-based analytic approaches [1–4,7,11,13–16,18–36]; the asymptotic analysis that is employed here is applicable from the SVEA or quasi-monochromatic to the single-cycle regime. Its accuracy may be improved when additional asymptotic series expansion terms are retained.

Effectively, in order to generate a desired (waveform-controlled) output Gaussian pulse-envelope modulated chirped harmonic wave $g(t )= {A_{{m_{out}}}}({z = d,t} )$ at point $({\rm A} )$ in Fig. 1, the dynamical evolution of the input pulse wave ${A_{{m_{in}}}}({z = 0,t} )$ at point $({\rm B} )$ must exactly cancel-out the expected, accumulated, absorptive and dispersive impact due to propagation at a distance $z$ in the considered linear metamaterial; the unified asymptotic approach determines analytically the required dynamical evolution of ${A_{{m_{in}}}}({z = 0,t} )$. Adaptive (pre and/or post) shaping approaches, albeit purely experimental or numerical and applicable in the SVEA regime, have been proposed for pulse distortion compensation in optical fiber communication systems [4,7,10,13,14], in Ultra-Wideband antenna systems [4], or in epsilon-near-zero metamaterials [17]. Moreover, appropriately engineered ultrathin metasurfaces have been shown numerically to provide a limited level of compression/broadening of ultrashort optical pulses [6]. Here, a distinctively different analytic approach is shown to overcome the challenge of dispersion compensation over a large spectral range associated with ultrashort pulse generation relying on a (temporally) broad input pulse wave exhibiting the dynamics described by the unified asymptotic approach. This approach is applicable for any chosen set of parameter values describing the desired output pulse wave $g(t )$, the Lorentz-type metamaterial ${n_m}(\omega )$, and at any $z$, in the linear propagation regime. As depicted in Fig. 5(a) and Fig. 5(c), the same $g(t )$ is obtained at two increasing propagation depths in the same metamaterial, by properly choosing the respective dynamical evolution of ${A_{{m_{in}}}}({z = 0,t} ) = {A_{com{p_{out}}}}({z^{\prime} = d,t} )$.

Comparison of the input-output pulse wave pairs depicted in Fig. 1 and Fig. 3 with the respective pairs in Fig. 2 and Fig. 5 shows that at the output stage of the metamaterial pulse compression scheme the compression factor (i.e. the ratio of the RMS width of $g(t ) = {A_{{m_{out}}}}({z = d,t} )$ over the respective RMS width of ${A_{{m_{in}}}}({z = 0,t} ) = {A_{com{p_{out}}}}({z^{\prime} = d,t} )$) is the inverse of the corresponding pulse broadening factor associated with Fig. 3. Moreover, in the considered cases, the metamaterial practically eliminates the chirping present in the input pulse wave ${A_{{m_{in}}}}({z = 0,t} ) = {A_{com{p_{out}}}}({z^{\prime} = d,t} )$ resulting in an output pulse wave which oscillates at the desired carrier frequency ${\omega _c}$. Finally, a comparison of the respective input-output pulse wave pairs in Fig. 3 and Fig. 5 shows that the peak (amplitude) of the output pulse wave $g(t )$ occurs earlier in time than the time-instant when the peak amplitude of the respective input pulse wave ${A_{{m_{in}}}}({z = 0,t} ) = {A_{com{p_{out}}}}({z^{\prime} = d,t} )$ enters the Lorentz-type metamaterial, in accordance with previous observations in, passive and active, causally dispersive media [1,7,15,17,18,23,25,26,28,30–32,34–36]. Taken together, the input and output stages of the proposed pulse compression scheme perform a temporal-shift (here, a translation at the specific earlier time-instant), which does not violate causality and may be explained by the deformation of the pulse in the metamaterial medium.

## 5. Conclusions

An analytic approach, based upon linear system theory and asymptotic analysis, provides an accurate description of the dynamics of the required input pulse wave which, at a chosen propagation distance in a linear, Lorentz-type metamaterial, is compressed to a desired output (waveform-controlled) Gaussian modulated chirped harmonic wave of arbitrary width; the accuracy of this analytic approach is verified upon comparison with numerical experiments. The metamaterial propagation scheme is considered as a linear time-invariant system. The ensuing asymptotic analysis is critically dependent upon the unified phase function and its associated saddle points, which encompass the influence of the input pulse and the dispersive medium parameters as well as of the propagation distance. As more terms are retained in the asymptotic series expansion, the accuracy of the analytic description of the input pulse wave dynamics is increased. This analytic approach is applicable form the SVEA to the single-cycle regimes.

The dynamics of the input pulse wave cancel-out the absorptive and dispersive impact that is accumulated upon propagation in the chosen metamaterial medium yielding a time-focused output pulse wave with the desired characteristics. It is shown that, when the carrier frequency of the desired output pulse wave is chosen to lie at the center of the metamaterial’s absorption band, the derived, intrinsically coupled, pair of input-output pulse waves exhibits temporal-focusing and temporal-shift. Moreover, in this case the peak envelope velocity is negative, which does not violate relativistic causality and may be explained by the deformation of the pulse in the metamaterial medium. Furthermore, the same output pulse wave may be obtained at different propagation distances in the same metamaterial, by a proper choice of the input pulse wave. Finally, the metamaterial is also shown here to practically eliminate the chirping in the input pulse wave, yielding an unchirped Gaussian modulated harmonic wave of chosen carrier frequency. This asymptotic approach provides increased insight and control of the temporal dynamics of ultrashort (few-cycle) waveforms.

## Acknowledgments

The author gratefully acknowledges the assistance of Dr. Theodoros Zygiridis with the FDTD numerical experiment.

## Disclosures

The author declares no conflicts of interest.

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