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 [110]; 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 [24,9,11,12]; 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 [117]. 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 [15,7,8,11,1316,1836]. 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 [1,7,29,3740].

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 [6,7,10,16,17,2123,2527,35,41].

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 [1,48,10,1318,23,24,2628,3032,3436], 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 [1,3,6,7,13,15,18,2224,29,3740,42,43]

$$A({z,t} )= \frac{1}{{2\pi }}\int_L {f(\omega )} {e^{i[{k(\omega )z - \omega t} ]}}d\omega ,\textrm{ }\forall z > 0.$$
Here, $f({z = 0,\omega } )= f(\omega )$ is the Fourier-Laplace transform of the input wave, $k(\omega )= ({{\omega / c}} )n(\omega )$ the wave number, $n(\omega )$ the refractive index of the medium, and 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 [1,4,6,7,1318,20,24,26,28,29,3240], 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 [2123,25,41,42],

$${\varepsilon _m}(\omega )= 1 - \frac{{\omega _{ep}^2 - \omega _{e0}^2}}{{{\omega ^2} - \omega _{e0}^2 + i{\gamma _e}\omega }},\textrm{ and}{\bf }\textrm{ }{\mu _m}(\omega )= 1 - \frac{{\omega _{mp}^2 - \omega _{m0}^2}}{{{\omega ^2} - \omega _{m0}^2 + i{\gamma _m}\omega }}.$$
Here, ${\omega _{ep}}$, ${\omega _{e0}}$ and ${\gamma _e}$ respectively represent the electric plasma frequency, resonance frequency and phenomenological damping constant; their magnetic counterparts are ${\omega _{mp}}$, ${\omega _{m0}}$ and ${\gamma _m}$, respectively. The (effective) index of refraction for this physically realizable metamaterial is given by ${n_m}(\omega )= \sqrt {{\varepsilon _m}(\omega )} \sqrt {{\mu _m}(\omega )} $.

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]

$$g(t )= \exp \{{ - {{[{{{({t - {t_0}} )} / T}} ]}^2}} \}\sin [{{\omega_c}t + ({{C / 2}} ){t^2} + \psi } ].$$
Here, $2T$ is the full width of the Gaussian envelope centered at ${t_0} \gg 2T$ which modulates a harmonic wave of carrier frequency ${\omega _c}$ that is linearly chirped with a fixed, arbitrary chirp parameter 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,

$${A_{{m_{in}}}}({z = 0,\omega } )= {A_{com{p_{out}}}}({z^{\prime} = d,\omega } ),{\bf }\textrm{and}{\bf }{A_{{m_{out}}}}({z = d,\omega } )= {A_{com{p_{in}}}}({z^{\prime} = 0,\omega } ).$$

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

$${H_{comp}}({z^{\prime},\omega } )= \exp \{{i[{{k_{comp}}(\omega )z^{\prime}} ]} \},$$
$${H_m}({z,\omega } )= \exp \{{i[{{k_m}(\omega )z} ]} \},$$
$${k_{comp}}(\omega )={-} {k_m}(\omega )={-} ({{\omega / c}} ){n_m}(\omega ).$$

 

Fig. 1. Output stage of the metamaterial pulse compression scheme

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Fig. 2. Input stage of the metamaterial pulse compression scheme

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The unified asymptotic approach [29,3740,42] 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]

$${A_{com{p_{out}}}}({z^{\prime},t} )= \frac{1}{{2\pi }}{\rm{Re}} \left\{ {i\int_L {{U_U}\exp \left[ {\frac{{z^{\prime}}}{c}{\Phi _U}({\omega ,\theta } )} \right]d\omega } } \right\},{\bf }\forall z^{\prime} > 0,$$
where ${U_U} = [{{{({2T{\pi^{{1 / 2}}}} )} / {{{({4 + i2C{T^2}} )}^{{1 / 2}}}}}} ]\exp \{{ - [{({{{t_0^2} / {{T^2}}}} )+ i\psi } ]} \}$ is the frequency independent unified spectral amplitude, ${\Phi _U}({\omega ,\theta } )= \phi ({\omega ,\theta } )- \left\{ {{{\left[ {c{T^2}{{\left( {\omega - {\omega_c} - i\frac{{2{t_0}}}{{{T^2}}}} \right)}^2}} \right]} / {[{z^{\prime}({4 + i2C{T^2}} )} ]}}} \right\}$ is the unified phase with $\phi ({\omega ,\theta } )= \{{i\omega [{{n_{comp}}(\omega )- \theta } ]} \}= \{{i\omega [{ - {n_m}(\omega )- \theta } ]} \}$ denoting the classical phase and $\theta = {{({ct} )} / {z^{\prime}}}$ is a dimensionless space-time parameter; the terms ${\rm{Re}} \{{\cdot} \}$ and ${\mathop{\rm Im}\nolimits} \{{\cdot} \}$ denote, respectively, the real and imaginary parts of the quantity inside the brackets. Equations (7)-(8) describe an output pulse wave at $z^{\prime}$ due to the input pulse wave $g(t )$, which propagates in the negative $z^{\prime} - $ direction in the considered medium.

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

$${A_{Ul}}({z^{\prime},t} )= \frac{1}{{2\pi }}{\rm{Re}} {\left\{ {i{e^{\frac{{z^{\prime}}}{c}{\Phi _U}({\omega ,\theta } )}}{{\left( {\frac{{\pi c}}{{z^{\prime}}}} \right)}^{\frac{1}{2}}}{U_U}{{\left[ { - \frac{1}{2}\frac{{{d^2}{\Phi _U}({\omega ,\theta } )}}{{d{\omega^2}}}} \right]}^{ - \frac{1}{2}}}\left[ {\textrm{H}({\omega ,\theta } )+ O\left( {{{z^{\prime}}^{ - \frac{5}{2}}}} \right)} \right]} \right\}_{{\omega _{S{P_{Ul}}}}(\theta )}}$$
where the function $\textrm{H}({\omega ,\theta } )$ is defined as
$$\textrm{H}({\omega ,\theta } )= 1 - \frac{c}{{z^{\prime}}}\left\{ {\frac{{15}}{{72}}{{\left[ {\frac{{{d^3}{\Phi _U}({\omega ,\theta } )}}{{d{\omega^3}}}} \right]}^2}{{\left[ {\frac{{{d^2}{\Phi _U}({\omega ,\theta } )}}{{d{\omega^2}}}} \right]}^{ - 3}} - \frac{1}{8}\left( {\frac{{{d^4}{\Phi _U}({\omega ,\theta } )}}{{d{\omega^4}}}} \right){{\left( {\frac{{{d^2}{\Phi _U}({\omega ,\theta } )}}{{d{\omega^2}}}} \right)}^{ - 2}}} \right\}.$$

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 [29,3740,43]. 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] (${\omega _{ep}} = 56\pi \times {10^9}{s^{ - 1}}$, ${\omega _{e0}} = 0.0{s^{ - 1}}$, ${\gamma _e} = 1.6 \times {10^9}{s^{ - 1}}$, ${\omega _{mp}} = 49\pi \times {10^9}{s^{ - 1}}$, ${\omega _{m0}} = 42\pi \times {10^9}{s^{ - 1}}$, ${\gamma _m} = 4.0 \times {10^9}{s^{ - 1}}$).

 

Fig. 3. Dynamical evolution with $\theta $ of ${A_{com{p_{out}}}}({z^{\prime} = d,t} )$ in the input stage of the metamaterial pulse compression scheme, evaluated using both the unified asymptotic approach and the inverse Laplace numerical approach. (a) and (b) depict Case-1 and (c) and (d) Case-2. Moreover, (a) and (c) depict the description due to the unified asymptotic approach whereas (b) and (d) the prediction of the inverse Laplace numerical approach. In (a) and (c), the square denotes the ordered pair $({\theta _{U{A_{\max }}}},{A_{U{A_{\max }}}})$ and in (b) and (d) the corresponding ordered pair $({\theta _{H{A_{\max }}}},{A_{H{A_{\max }}}})$ of the space-time point when the total (absolute) maximum for the respective ${A_{com{p_{out}}}}({z^{\prime} = d,t} )$ occurs.

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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.

 

Fig. 4. Dynamical evolution with $\theta $ of the propagated field ${A_{com{p_{out}}}}({z^{\prime} = d,t} )$ and its associated envelope $\left| {{A_{com{p_{out}}}}({z^{\prime} = d,t} )} \right|$ in the input stage of the metamaterial pulse compression scheme, evaluated using the unified asymptotic approach, when one as well as when two asymptotic series expansion terms are retained in Eq. (9)-(10). (a) and (b) depict Case-1, whereas (c) and (d) Case-2. In (a) and (c) one asymptotic term is retained, whereas in (b) and (d) two asymptotic terms are retained in the respective propagated field expressions. In (a)-(d) the propagated field is depicted with a solid (black) line, its associated envelope is depicted with a solid (red) line and the respective (black) square denotes the ordered pair $({\theta _{U{A_{\max }}}},{A_{U{A_{\max }}}})$ of the space-time point when the total (absolute) maximum for the respective ${A_{com{p_{out}}}}({z^{\prime} = d,t} )$ occurs.

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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).

 

Fig. 5. Numerically determined, dynamical evolution with $\theta $ of ${A_{{m_{out}}}}({z = d,t} )$ in the output stage of the metamaterial pulse compression scheme, when ${A_{{m_{in}}}}({z = 0,t} ) = {A_{com{p_{out}}}}({z^{\prime} = d,t} )$. (a) and (b) correspond to Case-1 and (c) and (d) to Case-2. Moreover, (a) and (c) depict, superimposed upon the (desired) field $g(t )$ (blue continuous line), the numerically determined output pulse wave (red dashed line) when the input pulse wave coincides, respectively, with the field dynamics in Fig. 3(a) and Fig. 3(c). Furthermore, (b) and (d) depict, superimposed upon the (desired) field $g(t )$ (blue continuous line), the numerically determined output pulse wave (red dashed line) when the input pulse wave coincides, respectively, with the field dynamics in Fig. 3(b), and Fig. 3(d. In (a) and (c), the (red) square denotes the ordered pair $({\theta _{U{A_{\max }}}},{A_{U{A_{\max }}}})$ and in (b) and (d) the corresponding ordered pair $({\theta _{H{A_{\max }}}},{A_{H{A_{\max }}}})$ of the space-time point when the total (absolute) maximum of the respective ${A_{{m_{out}}}}({z = d,t} )$ occurs.

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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

$$\omega _{S{P_{Ul}}}^i(\theta )= \frac{d}{{dt}}\left\{ {\ln {{\left[ {\frac{{{{\left|{ - \frac{{{d^2}{\Phi _U}({\omega ,\theta } )}}{{d{\omega^2}}}} \right|}^{\frac{1}{2}}}}}{{|{{\rm H}({\omega ,\theta } )} |}}} \right]}_{{\omega_{S{P_{Ul}}}}(\theta )}}} \right\},$$
is satisfied [40]; here, $\omega _{S{P_{Ul}}}^i(\theta )= {\mathop{\rm Im}\nolimits} \{{{\omega_{S{P_{Ul}}}}(\theta )} \}$. Let the ordered pair $({{\theta_{UEn{v_{\max }}}},{A_{UEn{v_{\max }}}}} )$ denote the space-time point when the total maximum of $|{{A_{Ul}}({z,t} )} |$ occurs. For Case-1 in Fig. 3(a), the envelope peak $A_{UEn{v_{\max }}}^1$ of ${A_{com{p_{out}}}}({z^{\prime} = d,t} )$ is located at $\theta _{UEn{v_{\max }}}^1 = 220.216$; when determined from Eq. (11) it is located at $\theta _{UEn{v_{\max }}}^1 = 220.215$. For Case-2 in Fig. 3(c), $A_{UEn{v_{\max }}}^2$ is located at $\theta _{UEn{v_{\max }}}^2 = 118.147$; according to Eq. (11) at $\theta _{UEn{v_{\max }}}^2 = 118.146$. At the input stage of the metamaterial pulse compression scheme in Fig. 2, the velocity of the peak envelope of ${A_{com{p_{out}}}}({z^{\prime} = d,t} )$ propagating in the negative $z^{\prime} - $direction in the considered medium due to $g(t )$, is obtained by
$${\upsilon _{UEn{v_{comp}}}} = \frac{{ - d}}{{{t_{UEn{v_{\max }}}} - {t_0}}} ={-} \frac{c}{{{\theta _{UEn{v_{\max }}}} - ({{{c{t_0}} / d}} )}}.$$
Accordingly, for Case-1, $\upsilon _{UEn{v_{comp}}}^1 ={-} 0.131c$ and for Case-2 $\upsilon _{UEn{v_{comp}}}^2 ={-} 0.021c$. At the output stage of the metamaterial pulse compression scheme in Fig. 1, Eq. (1) together with the condition in Eq. (4) describe the dynamical evolution of the desired output pulse wave due to the input pulse wave ${A_{{m_{in}}}}({z = 0,t} )$, which propagates in the positive $z - $direction in the considered metamaterial. The velocity of the peak envelope of the propagated field is also given by Eq. (12) (the displacement in the numerator is positive but the time-difference in the denominator is the opposite). Therefore, in the two considered cases ${\upsilon _{UEn{v_m}}} = {\upsilon _{UEn{v_{comp}}}}$, and the combined action of absorption and dispersion on the spectral content of the input pulse wave ${A_{{m_{in}}}}({z = 0,t} )$ leads to the formation of an output pulse wave $g(t )$ whose envelope peak occurs prior to the occurrence of the envelope peak of the input pulse wave. The observed negative values of the peak envelope velocity when ${\omega _c}$ is situated at the center of the (metamaterial’s) absorption band, has been theoretically predicted and explained in accordance with relativistic causality, and has also been experimentally observed in, passive and active, dispersive media [1,7,15,18,23,25,26,28,3032,34,36]. Moreover, the analysis presented here is consistent with experimental and theoretical results in metamaterials in the microwave [2123,25,27,35] and the optical [6,16,17,26,41] regimes.

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 [14,7,11,1316,1836], considers the medium’s absorptive and dispersive behavior without the assumptions inherent in all SVEA-based analytic approaches [14,7,11,1316,1836]; 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 [1,7,15,17,18,23,25,26,28,3032,3436]. 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.

References

1. S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, Optics of Femtosecond Laser Pulses (AIP, New York, 1992), Chap. 1.

2. G. Steinmeyer, D. H. Sutter, L. Gallmann, N. Matuschek, and U. Keller, “Frontiers in Ultrashort Pulse Generation: Pushing the Limits in Linear and Nonlinear Optics,” Science 286(5444), 1507–1512 (1999). [CrossRef]  

3. I. A. Walmsley and C. Dorrer, “Characterization of ultrashort electromagnetic pulses,” Adv. Opt. Photonics 1(2), 308–437 (2009). [CrossRef]  

4. A. M. Weiner, “Ultrafast optical pulse shaping: A tutorial review,” Opt. Commun. 284(15), 3669–3692 (2011). [CrossRef]  

5. C. Ding, O. Korotkova, Y. Zhang, and L. Pan, “Cosine-Gaussian correlated Schell-model pulsed beams,” Opt. Express 22(1), 931–942 (2014). [CrossRef]  

6. E. Rahimi and K. Sendur, “Femtosecond pulse shaping by ultrathin plasmonic metasurfaces,” J. Opt. Soc. Am. B 33(2), A1–A7 (2016). [CrossRef]  

7. S. J. Orphanidis, Electromagnetic Waves and Antennas, 2016, https://www.ece.rutgers.edu/∼orfanidi/ewa/.

8. C. Ding, M. Koivurova, J. Turunen, and L. Pan, “Temporal self-splitting of optical pulses,” Phys. Rev. A 97(5), 053838 (2018). [CrossRef]  

9. M. Miranda, F. Silva, L. Neoričic, C. Guo, V. Pervak, M. Canhota, A. S. Silva, I. J. Sola, R. Romero, P. T. Guerreiro, A. L’Huillier, C. L. Arnold, and H. Crespo, “All-optical measurement of the complete waveform of octave-spanning ultrashort light pulses,” Opt. Lett. 44(2), 191–194 (2019). [CrossRef]  

10. S. Divitt, W. Zhu, C. Zhang, H. J. Lezec, and A. Agrawal, “Ultrafast Optical Pulse Shaping using Dielectric Metasurfaces,” Science 364(6443), 890–894 (2019). [CrossRef]  

11. M. Chini, K. Zhao, and Z. Chang, “The generation, characterization and applications of broadband isolated attosecond pulses,” Nat. Photonics 8(3), 178–186 (2014). [CrossRef]  

12. T. Gaumnitz, A. Jain, Y. Pertot, M. Huppert, I. Jordan, F. Ardana-Lamas, and H. J. Wörner, “Streaking of 43-attosecond soft-X-ray pulses generated by a passively CEP-stable mid-infrared driver,” Opt. Express 25(22), 27506–27518 (2017). [CrossRef]  

13. B. H. Kolner and M. Nazarathy, “Temporal imaging with a time lens,” Opt. Lett. 14(12), 630–632 (1989). [CrossRef]  

14. I. Jacobs and J. K. Shaw, “Optimal dispersion precompensation by pulse chirping,” Appl. Opt. 41(6), 1057–1062 (2002). [CrossRef]  

15. Y. Pinhasi, A. Yahalom, and G. A. Pinhasi, “Propagation analysis of ultrashort pulses in resonant dielectric media,” J. Opt. Soc. Am. B 26(12), 2404–2413 (2009). [CrossRef]  

16. B. Dastmalchi, P. Tassin, T. Koschny, and C. M. Soukoulis, “Strong group-velocity dispersion compensation with phase-engineered sheet metamaterials,” Phys. Rev. B 89(11), 115123 (2014). [CrossRef]  

17. P. Kelly and L. Kuznetsova, “Adaptive pre-shaping for ultrashort pulse control during propagation in AZO/ZnO multilayered metamaterial at the epsilon-near-zero spectral point,” OSA Continuum 3(2), 143–152 (2020). [CrossRef]  

18. C. G. B. Garrett and D. E. McCumber, “Propagation of a Gaussian Light Pulse through an Anomalous Dispersion Medium,” Phys. Rev. A 1(2), 305–313 (1970). [CrossRef]  

19. S. Szatmári, P. Simon, and M. Feuerhake, “Group-velocity-dispersion-compensated propagation of short pulses in dispersive media,” Opt. Lett. 21(15), 1156–1158 (1996). [CrossRef]  

20. T. Brabec and F. Krausz, “Nonlinear Optical Pulse Propagation in the Single-Cycle Regime,” Phys. Rev. Lett. 78(17), 3282–3285 (1997). [CrossRef]  

21. R. A. Shelby, D. R. Smith, S. C. Nemat-Nasser, and S. Schultz, “Microwave transmission through a two-dimensional, isotropic, left-handed metamaterial,” Appl. Phys. Lett. 78(4), 489–491 (2001). [CrossRef]  

22. R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E 64(5), 056625 (2001). [CrossRef]  

23. M. Mojahedi, K. J. Malloy, G. V. Eleftheriades, J. Woodley, and R. Y. Chiao, “Abnormal Wave Propagation in Passive Media,” IEEE J. Sel. Top. Quantum Electron. 9(1), 30–39 (2003). [CrossRef]  

24. H. Cao, A. Dogariu, and L. J. Wang, “Negative Group Delay and Pulse Compression in Superluminal Pulse Propagation,” IEEE J. Sel. Top. Quantum Electron. 9(1), 52–58 (2003). [CrossRef]  

25. J. F. Woodley and M. Mojahedi, “Negative group velocity and group delay in left-handed media,” Phys. Rev. E 70(4), 046603 (2004). [CrossRef]  

26. G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous Negative Phase and Group Velocity of Light in a Metamaterial,” Science 312(5775), 892–894 (2006). [CrossRef]  

27. J. Woodley and M. Mojahedi, “Backward wave propagation in left-handed media with isotropic and anisotropic permittivity tensors,” J. Opt. Soc. Am. B 23(11), 2377–2382 (2006). [CrossRef]  

28. G. M. Gehring, A. Schweinsberg, C. Barsi, N. Kostinski, and R. W. Boyd, “Observation of Backward Pulse Propagation Through a Medium with a Negative Group Velocity,” Science 312(5775), 895–897 (2006). [CrossRef]  

29. K. E. Oughstun, Electromagnetic and Optical Pulse Propagation 2: Temporal Pulse Dynamics in Dispersive, Attenuative Media (Springer Series in Optical Sciences, New York, 2009), Vol. 144.

30. R. W. Boyd, “Slow and fast light: fundamentals and applications,” J. Mod. Opt. 56(18-19), 1908–1915 (2009). [CrossRef]  

31. R. W. Boyd, “Material slow light and structural slow light: similarities and differences for nonlinear optics,” J. Opt. Soc. Am. B 28(12), A38–A43 (2011). [CrossRef]  

32. R. T. Glasser, U. Vogl, and P. D. Lett, “Demonstration of images with negative group velocities,” Opt. Express 20(13), 13702–13710 (2012). [CrossRef]  

33. Y. Xiao, D. N. Maywar, and G. P. Agrawal, “New approach to pulse propagation in nonlinear dispersive optical media,” J. Opt. Soc. Am. B 29(10), 2958–2963 (2012). [CrossRef]  

34. D. Ye, Y. Salamin, J. Huangfu, S. Qiao, G. Zheng, and L. Ran, “Observation of Wave Packet Distortion during a Negative-Group-Velocity Transmission,” Sci. Rep. 5(1), 8100 (2015). [CrossRef]  

35. A. F. Mota, A. Martins P, L. L. Bertarini, and B.-H. V. Borges, “Dispersion Management With Microwave Pulses in Metamaterials in the Negative Refraction Regime,” Antennas Wirel. Propag. Lett. 14, 1377–1380 (2015). [CrossRef]  

36. J. D. Swaim and R. T. Glasser, “Causality and information transfer in simultaneously slow- and fast-light media,” Opt. Express 25(20), 24376–24386 (2017). [CrossRef]  

37. K. E. Oughstun and C. M. Balictsis, “Gaussian Pulse Propagation in a Dispersive, Absorbing Dielectric,” Phys. Rev. Lett. 77(11), 2210–2213 (1996). [CrossRef]  

38. C. M. Balictsis and K. E. Oughstun, “Generalized asymptotic description of the propagated field dynamics in Gaussian pulse propagation in a linear, causally dispersive medium,” Phys. Rev. E 55(2), 1910–1921 (1997). [CrossRef]  

39. C. M. Balictsis, “Unified asymptotic description of Gaussian pulse propagation of arbitrary initial pulse width in a Lorentz-type gain medium,” Phys. Rev. E 87(1), 013304 (2013). [CrossRef]  

40. C. M. Balictsis, “Chirped dispersive pulse propagation of arbitrary initial width: a saddle point perspective,” J. Opt. Soc. Am. B 36(5), 1300–1312 (2019). [CrossRef]  

41. Y. Liu and X. Zhang, “Metamaterials: a new frontier of science and technology,” Chem. Soc. Rev. 40(5), 2494–2507 (2011). [CrossRef]  

42. C. M. Balictsis, “Asymptotic Analysis of Chirped Gaussian Pulse Propagation in a Metamaterial Medium,” in Proceedings of the 2019 International Conference on Electromagnetics in Advanced Applications (ICEAA), Granada, Spain, 2019, p. 530–533.

43. P. Wyns, D. P. Foty, and K. E. Oughstun, “Numerical analysis of the precursor fields in linear dispersive pulse propagation,” J. Opt. Soc. Am. A 6(9), 1421–1429 (1989). [CrossRef]  

References

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  1. S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, Optics of Femtosecond Laser Pulses (AIP, New York, 1992), Chap. 1.
  2. G. Steinmeyer, D. H. Sutter, L. Gallmann, N. Matuschek, and U. Keller, “Frontiers in Ultrashort Pulse Generation: Pushing the Limits in Linear and Nonlinear Optics,” Science 286(5444), 1507–1512 (1999).
    [Crossref]
  3. I. A. Walmsley and C. Dorrer, “Characterization of ultrashort electromagnetic pulses,” Adv. Opt. Photonics 1(2), 308–437 (2009).
    [Crossref]
  4. A. M. Weiner, “Ultrafast optical pulse shaping: A tutorial review,” Opt. Commun. 284(15), 3669–3692 (2011).
    [Crossref]
  5. C. Ding, O. Korotkova, Y. Zhang, and L. Pan, “Cosine-Gaussian correlated Schell-model pulsed beams,” Opt. Express 22(1), 931–942 (2014).
    [Crossref]
  6. E. Rahimi and K. Sendur, “Femtosecond pulse shaping by ultrathin plasmonic metasurfaces,” J. Opt. Soc. Am. B 33(2), A1–A7 (2016).
    [Crossref]
  7. S. J. Orphanidis, Electromagnetic Waves and Antennas, 2016, https://www.ece.rutgers.edu/∼orfanidi/ewa/ .
  8. C. Ding, M. Koivurova, J. Turunen, and L. Pan, “Temporal self-splitting of optical pulses,” Phys. Rev. A 97(5), 053838 (2018).
    [Crossref]
  9. M. Miranda, F. Silva, L. Neoričic, C. Guo, V. Pervak, M. Canhota, A. S. Silva, I. J. Sola, R. Romero, P. T. Guerreiro, A. L’Huillier, C. L. Arnold, and H. Crespo, “All-optical measurement of the complete waveform of octave-spanning ultrashort light pulses,” Opt. Lett. 44(2), 191–194 (2019).
    [Crossref]
  10. S. Divitt, W. Zhu, C. Zhang, H. J. Lezec, and A. Agrawal, “Ultrafast Optical Pulse Shaping using Dielectric Metasurfaces,” Science 364(6443), 890–894 (2019).
    [Crossref]
  11. M. Chini, K. Zhao, and Z. Chang, “The generation, characterization and applications of broadband isolated attosecond pulses,” Nat. Photonics 8(3), 178–186 (2014).
    [Crossref]
  12. T. Gaumnitz, A. Jain, Y. Pertot, M. Huppert, I. Jordan, F. Ardana-Lamas, and H. J. Wörner, “Streaking of 43-attosecond soft-X-ray pulses generated by a passively CEP-stable mid-infrared driver,” Opt. Express 25(22), 27506–27518 (2017).
    [Crossref]
  13. B. H. Kolner and M. Nazarathy, “Temporal imaging with a time lens,” Opt. Lett. 14(12), 630–632 (1989).
    [Crossref]
  14. I. Jacobs and J. K. Shaw, “Optimal dispersion precompensation by pulse chirping,” Appl. Opt. 41(6), 1057–1062 (2002).
    [Crossref]
  15. Y. Pinhasi, A. Yahalom, and G. A. Pinhasi, “Propagation analysis of ultrashort pulses in resonant dielectric media,” J. Opt. Soc. Am. B 26(12), 2404–2413 (2009).
    [Crossref]
  16. B. Dastmalchi, P. Tassin, T. Koschny, and C. M. Soukoulis, “Strong group-velocity dispersion compensation with phase-engineered sheet metamaterials,” Phys. Rev. B 89(11), 115123 (2014).
    [Crossref]
  17. P. Kelly and L. Kuznetsova, “Adaptive pre-shaping for ultrashort pulse control during propagation in AZO/ZnO multilayered metamaterial at the epsilon-near-zero spectral point,” OSA Continuum 3(2), 143–152 (2020).
    [Crossref]
  18. C. G. B. Garrett and D. E. McCumber, “Propagation of a Gaussian Light Pulse through an Anomalous Dispersion Medium,” Phys. Rev. A 1(2), 305–313 (1970).
    [Crossref]
  19. S. Szatmári, P. Simon, and M. Feuerhake, “Group-velocity-dispersion-compensated propagation of short pulses in dispersive media,” Opt. Lett. 21(15), 1156–1158 (1996).
    [Crossref]
  20. T. Brabec and F. Krausz, “Nonlinear Optical Pulse Propagation in the Single-Cycle Regime,” Phys. Rev. Lett. 78(17), 3282–3285 (1997).
    [Crossref]
  21. R. A. Shelby, D. R. Smith, S. C. Nemat-Nasser, and S. Schultz, “Microwave transmission through a two-dimensional, isotropic, left-handed metamaterial,” Appl. Phys. Lett. 78(4), 489–491 (2001).
    [Crossref]
  22. R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E 64(5), 056625 (2001).
    [Crossref]
  23. M. Mojahedi, K. J. Malloy, G. V. Eleftheriades, J. Woodley, and R. Y. Chiao, “Abnormal Wave Propagation in Passive Media,” IEEE J. Sel. Top. Quantum Electron. 9(1), 30–39 (2003).
    [Crossref]
  24. H. Cao, A. Dogariu, and L. J. Wang, “Negative Group Delay and Pulse Compression in Superluminal Pulse Propagation,” IEEE J. Sel. Top. Quantum Electron. 9(1), 52–58 (2003).
    [Crossref]
  25. J. F. Woodley and M. Mojahedi, “Negative group velocity and group delay in left-handed media,” Phys. Rev. E 70(4), 046603 (2004).
    [Crossref]
  26. G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous Negative Phase and Group Velocity of Light in a Metamaterial,” Science 312(5775), 892–894 (2006).
    [Crossref]
  27. J. Woodley and M. Mojahedi, “Backward wave propagation in left-handed media with isotropic and anisotropic permittivity tensors,” J. Opt. Soc. Am. B 23(11), 2377–2382 (2006).
    [Crossref]
  28. G. M. Gehring, A. Schweinsberg, C. Barsi, N. Kostinski, and R. W. Boyd, “Observation of Backward Pulse Propagation Through a Medium with a Negative Group Velocity,” Science 312(5775), 895–897 (2006).
    [Crossref]
  29. K. E. Oughstun, Electromagnetic and Optical Pulse Propagation 2: Temporal Pulse Dynamics in Dispersive, Attenuative Media (Springer Series in Optical Sciences, New York, 2009), Vol. 144.
  30. R. W. Boyd, “Slow and fast light: fundamentals and applications,” J. Mod. Opt. 56(18-19), 1908–1915 (2009).
    [Crossref]
  31. R. W. Boyd, “Material slow light and structural slow light: similarities and differences for nonlinear optics,” J. Opt. Soc. Am. B 28(12), A38–A43 (2011).
    [Crossref]
  32. R. T. Glasser, U. Vogl, and P. D. Lett, “Demonstration of images with negative group velocities,” Opt. Express 20(13), 13702–13710 (2012).
    [Crossref]
  33. Y. Xiao, D. N. Maywar, and G. P. Agrawal, “New approach to pulse propagation in nonlinear dispersive optical media,” J. Opt. Soc. Am. B 29(10), 2958–2963 (2012).
    [Crossref]
  34. D. Ye, Y. Salamin, J. Huangfu, S. Qiao, G. Zheng, and L. Ran, “Observation of Wave Packet Distortion during a Negative-Group-Velocity Transmission,” Sci. Rep. 5(1), 8100 (2015).
    [Crossref]
  35. A. F. Mota, A. Martins P, L. L. Bertarini, and B.-H. V. Borges, “Dispersion Management With Microwave Pulses in Metamaterials in the Negative Refraction Regime,” Antennas Wirel. Propag. Lett. 14, 1377–1380 (2015).
    [Crossref]
  36. J. D. Swaim and R. T. Glasser, “Causality and information transfer in simultaneously slow- and fast-light media,” Opt. Express 25(20), 24376–24386 (2017).
    [Crossref]
  37. K. E. Oughstun and C. M. Balictsis, “Gaussian Pulse Propagation in a Dispersive, Absorbing Dielectric,” Phys. Rev. Lett. 77(11), 2210–2213 (1996).
    [Crossref]
  38. C. M. Balictsis and K. E. Oughstun, “Generalized asymptotic description of the propagated field dynamics in Gaussian pulse propagation in a linear, causally dispersive medium,” Phys. Rev. E 55(2), 1910–1921 (1997).
    [Crossref]
  39. C. M. Balictsis, “Unified asymptotic description of Gaussian pulse propagation of arbitrary initial pulse width in a Lorentz-type gain medium,” Phys. Rev. E 87(1), 013304 (2013).
    [Crossref]
  40. C. M. Balictsis, “Chirped dispersive pulse propagation of arbitrary initial width: a saddle point perspective,” J. Opt. Soc. Am. B 36(5), 1300–1312 (2019).
    [Crossref]
  41. Y. Liu and X. Zhang, “Metamaterials: a new frontier of science and technology,” Chem. Soc. Rev. 40(5), 2494–2507 (2011).
    [Crossref]
  42. C. M. Balictsis, “Asymptotic Analysis of Chirped Gaussian Pulse Propagation in a Metamaterial Medium,” in Proceedings of the 2019 International Conference on Electromagnetics in Advanced Applications (ICEAA), Granada, Spain, 2019, p. 530–533.
  43. P. Wyns, D. P. Foty, and K. E. Oughstun, “Numerical analysis of the precursor fields in linear dispersive pulse propagation,” J. Opt. Soc. Am. A 6(9), 1421–1429 (1989).
    [Crossref]

2020 (1)

2019 (3)

2018 (1)

C. Ding, M. Koivurova, J. Turunen, and L. Pan, “Temporal self-splitting of optical pulses,” Phys. Rev. A 97(5), 053838 (2018).
[Crossref]

2017 (2)

2016 (1)

2015 (2)

D. Ye, Y. Salamin, J. Huangfu, S. Qiao, G. Zheng, and L. Ran, “Observation of Wave Packet Distortion during a Negative-Group-Velocity Transmission,” Sci. Rep. 5(1), 8100 (2015).
[Crossref]

A. F. Mota, A. Martins P, L. L. Bertarini, and B.-H. V. Borges, “Dispersion Management With Microwave Pulses in Metamaterials in the Negative Refraction Regime,” Antennas Wirel. Propag. Lett. 14, 1377–1380 (2015).
[Crossref]

2014 (3)

M. Chini, K. Zhao, and Z. Chang, “The generation, characterization and applications of broadband isolated attosecond pulses,” Nat. Photonics 8(3), 178–186 (2014).
[Crossref]

C. Ding, O. Korotkova, Y. Zhang, and L. Pan, “Cosine-Gaussian correlated Schell-model pulsed beams,” Opt. Express 22(1), 931–942 (2014).
[Crossref]

B. Dastmalchi, P. Tassin, T. Koschny, and C. M. Soukoulis, “Strong group-velocity dispersion compensation with phase-engineered sheet metamaterials,” Phys. Rev. B 89(11), 115123 (2014).
[Crossref]

2013 (1)

C. M. Balictsis, “Unified asymptotic description of Gaussian pulse propagation of arbitrary initial pulse width in a Lorentz-type gain medium,” Phys. Rev. E 87(1), 013304 (2013).
[Crossref]

2012 (2)

2011 (3)

R. W. Boyd, “Material slow light and structural slow light: similarities and differences for nonlinear optics,” J. Opt. Soc. Am. B 28(12), A38–A43 (2011).
[Crossref]

A. M. Weiner, “Ultrafast optical pulse shaping: A tutorial review,” Opt. Commun. 284(15), 3669–3692 (2011).
[Crossref]

Y. Liu and X. Zhang, “Metamaterials: a new frontier of science and technology,” Chem. Soc. Rev. 40(5), 2494–2507 (2011).
[Crossref]

2009 (3)

I. A. Walmsley and C. Dorrer, “Characterization of ultrashort electromagnetic pulses,” Adv. Opt. Photonics 1(2), 308–437 (2009).
[Crossref]

Y. Pinhasi, A. Yahalom, and G. A. Pinhasi, “Propagation analysis of ultrashort pulses in resonant dielectric media,” J. Opt. Soc. Am. B 26(12), 2404–2413 (2009).
[Crossref]

R. W. Boyd, “Slow and fast light: fundamentals and applications,” J. Mod. Opt. 56(18-19), 1908–1915 (2009).
[Crossref]

2006 (3)

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous Negative Phase and Group Velocity of Light in a Metamaterial,” Science 312(5775), 892–894 (2006).
[Crossref]

J. Woodley and M. Mojahedi, “Backward wave propagation in left-handed media with isotropic and anisotropic permittivity tensors,” J. Opt. Soc. Am. B 23(11), 2377–2382 (2006).
[Crossref]

G. M. Gehring, A. Schweinsberg, C. Barsi, N. Kostinski, and R. W. Boyd, “Observation of Backward Pulse Propagation Through a Medium with a Negative Group Velocity,” Science 312(5775), 895–897 (2006).
[Crossref]

2004 (1)

J. F. Woodley and M. Mojahedi, “Negative group velocity and group delay in left-handed media,” Phys. Rev. E 70(4), 046603 (2004).
[Crossref]

2003 (2)

M. Mojahedi, K. J. Malloy, G. V. Eleftheriades, J. Woodley, and R. Y. Chiao, “Abnormal Wave Propagation in Passive Media,” IEEE J. Sel. Top. Quantum Electron. 9(1), 30–39 (2003).
[Crossref]

H. Cao, A. Dogariu, and L. J. Wang, “Negative Group Delay and Pulse Compression in Superluminal Pulse Propagation,” IEEE J. Sel. Top. Quantum Electron. 9(1), 52–58 (2003).
[Crossref]

2002 (1)

2001 (2)

R. A. Shelby, D. R. Smith, S. C. Nemat-Nasser, and S. Schultz, “Microwave transmission through a two-dimensional, isotropic, left-handed metamaterial,” Appl. Phys. Lett. 78(4), 489–491 (2001).
[Crossref]

R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E 64(5), 056625 (2001).
[Crossref]

1999 (1)

G. Steinmeyer, D. H. Sutter, L. Gallmann, N. Matuschek, and U. Keller, “Frontiers in Ultrashort Pulse Generation: Pushing the Limits in Linear and Nonlinear Optics,” Science 286(5444), 1507–1512 (1999).
[Crossref]

1997 (2)

T. Brabec and F. Krausz, “Nonlinear Optical Pulse Propagation in the Single-Cycle Regime,” Phys. Rev. Lett. 78(17), 3282–3285 (1997).
[Crossref]

C. M. Balictsis and K. E. Oughstun, “Generalized asymptotic description of the propagated field dynamics in Gaussian pulse propagation in a linear, causally dispersive medium,” Phys. Rev. E 55(2), 1910–1921 (1997).
[Crossref]

1996 (2)

K. E. Oughstun and C. M. Balictsis, “Gaussian Pulse Propagation in a Dispersive, Absorbing Dielectric,” Phys. Rev. Lett. 77(11), 2210–2213 (1996).
[Crossref]

S. Szatmári, P. Simon, and M. Feuerhake, “Group-velocity-dispersion-compensated propagation of short pulses in dispersive media,” Opt. Lett. 21(15), 1156–1158 (1996).
[Crossref]

1989 (2)

1970 (1)

C. G. B. Garrett and D. E. McCumber, “Propagation of a Gaussian Light Pulse through an Anomalous Dispersion Medium,” Phys. Rev. A 1(2), 305–313 (1970).
[Crossref]

Agrawal, A.

S. Divitt, W. Zhu, C. Zhang, H. J. Lezec, and A. Agrawal, “Ultrafast Optical Pulse Shaping using Dielectric Metasurfaces,” Science 364(6443), 890–894 (2019).
[Crossref]

Agrawal, G. P.

Akhmanov, S. A.

S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, Optics of Femtosecond Laser Pulses (AIP, New York, 1992), Chap. 1.

Ardana-Lamas, F.

Arnold, C. L.

Balictsis, C. M.

C. M. Balictsis, “Chirped dispersive pulse propagation of arbitrary initial width: a saddle point perspective,” J. Opt. Soc. Am. B 36(5), 1300–1312 (2019).
[Crossref]

C. M. Balictsis, “Unified asymptotic description of Gaussian pulse propagation of arbitrary initial pulse width in a Lorentz-type gain medium,” Phys. Rev. E 87(1), 013304 (2013).
[Crossref]

C. M. Balictsis and K. E. Oughstun, “Generalized asymptotic description of the propagated field dynamics in Gaussian pulse propagation in a linear, causally dispersive medium,” Phys. Rev. E 55(2), 1910–1921 (1997).
[Crossref]

K. E. Oughstun and C. M. Balictsis, “Gaussian Pulse Propagation in a Dispersive, Absorbing Dielectric,” Phys. Rev. Lett. 77(11), 2210–2213 (1996).
[Crossref]

C. M. Balictsis, “Asymptotic Analysis of Chirped Gaussian Pulse Propagation in a Metamaterial Medium,” in Proceedings of the 2019 International Conference on Electromagnetics in Advanced Applications (ICEAA), Granada, Spain, 2019, p. 530–533.

Barsi, C.

G. M. Gehring, A. Schweinsberg, C. Barsi, N. Kostinski, and R. W. Boyd, “Observation of Backward Pulse Propagation Through a Medium with a Negative Group Velocity,” Science 312(5775), 895–897 (2006).
[Crossref]

Bertarini, L. L.

A. F. Mota, A. Martins P, L. L. Bertarini, and B.-H. V. Borges, “Dispersion Management With Microwave Pulses in Metamaterials in the Negative Refraction Regime,” Antennas Wirel. Propag. Lett. 14, 1377–1380 (2015).
[Crossref]

Borges, B.-H. V.

A. F. Mota, A. Martins P, L. L. Bertarini, and B.-H. V. Borges, “Dispersion Management With Microwave Pulses in Metamaterials in the Negative Refraction Regime,” Antennas Wirel. Propag. Lett. 14, 1377–1380 (2015).
[Crossref]

Boyd, R. W.

R. W. Boyd, “Material slow light and structural slow light: similarities and differences for nonlinear optics,” J. Opt. Soc. Am. B 28(12), A38–A43 (2011).
[Crossref]

R. W. Boyd, “Slow and fast light: fundamentals and applications,” J. Mod. Opt. 56(18-19), 1908–1915 (2009).
[Crossref]

G. M. Gehring, A. Schweinsberg, C. Barsi, N. Kostinski, and R. W. Boyd, “Observation of Backward Pulse Propagation Through a Medium with a Negative Group Velocity,” Science 312(5775), 895–897 (2006).
[Crossref]

Brabec, T.

T. Brabec and F. Krausz, “Nonlinear Optical Pulse Propagation in the Single-Cycle Regime,” Phys. Rev. Lett. 78(17), 3282–3285 (1997).
[Crossref]

Canhota, M.

Cao, H.

H. Cao, A. Dogariu, and L. J. Wang, “Negative Group Delay and Pulse Compression in Superluminal Pulse Propagation,” IEEE J. Sel. Top. Quantum Electron. 9(1), 52–58 (2003).
[Crossref]

Chang, Z.

M. Chini, K. Zhao, and Z. Chang, “The generation, characterization and applications of broadband isolated attosecond pulses,” Nat. Photonics 8(3), 178–186 (2014).
[Crossref]

Chiao, R. Y.

M. Mojahedi, K. J. Malloy, G. V. Eleftheriades, J. Woodley, and R. Y. Chiao, “Abnormal Wave Propagation in Passive Media,” IEEE J. Sel. Top. Quantum Electron. 9(1), 30–39 (2003).
[Crossref]

Chini, M.

M. Chini, K. Zhao, and Z. Chang, “The generation, characterization and applications of broadband isolated attosecond pulses,” Nat. Photonics 8(3), 178–186 (2014).
[Crossref]

Chirkin, A. S.

S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, Optics of Femtosecond Laser Pulses (AIP, New York, 1992), Chap. 1.

Crespo, H.

Dastmalchi, B.

B. Dastmalchi, P. Tassin, T. Koschny, and C. M. Soukoulis, “Strong group-velocity dispersion compensation with phase-engineered sheet metamaterials,” Phys. Rev. B 89(11), 115123 (2014).
[Crossref]

Ding, C.

C. Ding, M. Koivurova, J. Turunen, and L. Pan, “Temporal self-splitting of optical pulses,” Phys. Rev. A 97(5), 053838 (2018).
[Crossref]

C. Ding, O. Korotkova, Y. Zhang, and L. Pan, “Cosine-Gaussian correlated Schell-model pulsed beams,” Opt. Express 22(1), 931–942 (2014).
[Crossref]

Divitt, S.

S. Divitt, W. Zhu, C. Zhang, H. J. Lezec, and A. Agrawal, “Ultrafast Optical Pulse Shaping using Dielectric Metasurfaces,” Science 364(6443), 890–894 (2019).
[Crossref]

Dogariu, A.

H. Cao, A. Dogariu, and L. J. Wang, “Negative Group Delay and Pulse Compression in Superluminal Pulse Propagation,” IEEE J. Sel. Top. Quantum Electron. 9(1), 52–58 (2003).
[Crossref]

Dolling, G.

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous Negative Phase and Group Velocity of Light in a Metamaterial,” Science 312(5775), 892–894 (2006).
[Crossref]

Dorrer, C.

I. A. Walmsley and C. Dorrer, “Characterization of ultrashort electromagnetic pulses,” Adv. Opt. Photonics 1(2), 308–437 (2009).
[Crossref]

Eleftheriades, G. V.

M. Mojahedi, K. J. Malloy, G. V. Eleftheriades, J. Woodley, and R. Y. Chiao, “Abnormal Wave Propagation in Passive Media,” IEEE J. Sel. Top. Quantum Electron. 9(1), 30–39 (2003).
[Crossref]

Enkrich, C.

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous Negative Phase and Group Velocity of Light in a Metamaterial,” Science 312(5775), 892–894 (2006).
[Crossref]

Feuerhake, M.

Foty, D. P.

Gallmann, L.

G. Steinmeyer, D. H. Sutter, L. Gallmann, N. Matuschek, and U. Keller, “Frontiers in Ultrashort Pulse Generation: Pushing the Limits in Linear and Nonlinear Optics,” Science 286(5444), 1507–1512 (1999).
[Crossref]

Garrett, C. G. B.

C. G. B. Garrett and D. E. McCumber, “Propagation of a Gaussian Light Pulse through an Anomalous Dispersion Medium,” Phys. Rev. A 1(2), 305–313 (1970).
[Crossref]

Gaumnitz, T.

Gehring, G. M.

G. M. Gehring, A. Schweinsberg, C. Barsi, N. Kostinski, and R. W. Boyd, “Observation of Backward Pulse Propagation Through a Medium with a Negative Group Velocity,” Science 312(5775), 895–897 (2006).
[Crossref]

Glasser, R. T.

Guerreiro, P. T.

Guo, C.

Heyman, E.

R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E 64(5), 056625 (2001).
[Crossref]

Huangfu, J.

D. Ye, Y. Salamin, J. Huangfu, S. Qiao, G. Zheng, and L. Ran, “Observation of Wave Packet Distortion during a Negative-Group-Velocity Transmission,” Sci. Rep. 5(1), 8100 (2015).
[Crossref]

Huppert, M.

Jacobs, I.

Jain, A.

Jordan, I.

Keller, U.

G. Steinmeyer, D. H. Sutter, L. Gallmann, N. Matuschek, and U. Keller, “Frontiers in Ultrashort Pulse Generation: Pushing the Limits in Linear and Nonlinear Optics,” Science 286(5444), 1507–1512 (1999).
[Crossref]

Kelly, P.

Koivurova, M.

C. Ding, M. Koivurova, J. Turunen, and L. Pan, “Temporal self-splitting of optical pulses,” Phys. Rev. A 97(5), 053838 (2018).
[Crossref]

Kolner, B. H.

Korotkova, O.

Koschny, T.

B. Dastmalchi, P. Tassin, T. Koschny, and C. M. Soukoulis, “Strong group-velocity dispersion compensation with phase-engineered sheet metamaterials,” Phys. Rev. B 89(11), 115123 (2014).
[Crossref]

Kostinski, N.

G. M. Gehring, A. Schweinsberg, C. Barsi, N. Kostinski, and R. W. Boyd, “Observation of Backward Pulse Propagation Through a Medium with a Negative Group Velocity,” Science 312(5775), 895–897 (2006).
[Crossref]

Krausz, F.

T. Brabec and F. Krausz, “Nonlinear Optical Pulse Propagation in the Single-Cycle Regime,” Phys. Rev. Lett. 78(17), 3282–3285 (1997).
[Crossref]

Kuznetsova, L.

L’Huillier, A.

Lett, P. D.

Lezec, H. J.

S. Divitt, W. Zhu, C. Zhang, H. J. Lezec, and A. Agrawal, “Ultrafast Optical Pulse Shaping using Dielectric Metasurfaces,” Science 364(6443), 890–894 (2019).
[Crossref]

Linden, S.

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous Negative Phase and Group Velocity of Light in a Metamaterial,” Science 312(5775), 892–894 (2006).
[Crossref]

Liu, Y.

Y. Liu and X. Zhang, “Metamaterials: a new frontier of science and technology,” Chem. Soc. Rev. 40(5), 2494–2507 (2011).
[Crossref]

Malloy, K. J.

M. Mojahedi, K. J. Malloy, G. V. Eleftheriades, J. Woodley, and R. Y. Chiao, “Abnormal Wave Propagation in Passive Media,” IEEE J. Sel. Top. Quantum Electron. 9(1), 30–39 (2003).
[Crossref]

Martins P, A.

A. F. Mota, A. Martins P, L. L. Bertarini, and B.-H. V. Borges, “Dispersion Management With Microwave Pulses in Metamaterials in the Negative Refraction Regime,” Antennas Wirel. Propag. Lett. 14, 1377–1380 (2015).
[Crossref]

Matuschek, N.

G. Steinmeyer, D. H. Sutter, L. Gallmann, N. Matuschek, and U. Keller, “Frontiers in Ultrashort Pulse Generation: Pushing the Limits in Linear and Nonlinear Optics,” Science 286(5444), 1507–1512 (1999).
[Crossref]

Maywar, D. N.

McCumber, D. E.

C. G. B. Garrett and D. E. McCumber, “Propagation of a Gaussian Light Pulse through an Anomalous Dispersion Medium,” Phys. Rev. A 1(2), 305–313 (1970).
[Crossref]

Miranda, M.

Mojahedi, M.

J. Woodley and M. Mojahedi, “Backward wave propagation in left-handed media with isotropic and anisotropic permittivity tensors,” J. Opt. Soc. Am. B 23(11), 2377–2382 (2006).
[Crossref]

J. F. Woodley and M. Mojahedi, “Negative group velocity and group delay in left-handed media,” Phys. Rev. E 70(4), 046603 (2004).
[Crossref]

M. Mojahedi, K. J. Malloy, G. V. Eleftheriades, J. Woodley, and R. Y. Chiao, “Abnormal Wave Propagation in Passive Media,” IEEE J. Sel. Top. Quantum Electron. 9(1), 30–39 (2003).
[Crossref]

Mota, A. F.

A. F. Mota, A. Martins P, L. L. Bertarini, and B.-H. V. Borges, “Dispersion Management With Microwave Pulses in Metamaterials in the Negative Refraction Regime,” Antennas Wirel. Propag. Lett. 14, 1377–1380 (2015).
[Crossref]

Nazarathy, M.

Nemat-Nasser, S. C.

R. A. Shelby, D. R. Smith, S. C. Nemat-Nasser, and S. Schultz, “Microwave transmission through a two-dimensional, isotropic, left-handed metamaterial,” Appl. Phys. Lett. 78(4), 489–491 (2001).
[Crossref]

Neoricic, L.

Oughstun, K. E.

C. M. Balictsis and K. E. Oughstun, “Generalized asymptotic description of the propagated field dynamics in Gaussian pulse propagation in a linear, causally dispersive medium,” Phys. Rev. E 55(2), 1910–1921 (1997).
[Crossref]

K. E. Oughstun and C. M. Balictsis, “Gaussian Pulse Propagation in a Dispersive, Absorbing Dielectric,” Phys. Rev. Lett. 77(11), 2210–2213 (1996).
[Crossref]

P. Wyns, D. P. Foty, and K. E. Oughstun, “Numerical analysis of the precursor fields in linear dispersive pulse propagation,” J. Opt. Soc. Am. A 6(9), 1421–1429 (1989).
[Crossref]

K. E. Oughstun, Electromagnetic and Optical Pulse Propagation 2: Temporal Pulse Dynamics in Dispersive, Attenuative Media (Springer Series in Optical Sciences, New York, 2009), Vol. 144.

Pan, L.

C. Ding, M. Koivurova, J. Turunen, and L. Pan, “Temporal self-splitting of optical pulses,” Phys. Rev. A 97(5), 053838 (2018).
[Crossref]

C. Ding, O. Korotkova, Y. Zhang, and L. Pan, “Cosine-Gaussian correlated Schell-model pulsed beams,” Opt. Express 22(1), 931–942 (2014).
[Crossref]

Pertot, Y.

Pervak, V.

Pinhasi, G. A.

Pinhasi, Y.

Qiao, S.

D. Ye, Y. Salamin, J. Huangfu, S. Qiao, G. Zheng, and L. Ran, “Observation of Wave Packet Distortion during a Negative-Group-Velocity Transmission,” Sci. Rep. 5(1), 8100 (2015).
[Crossref]

Rahimi, E.

Ran, L.

D. Ye, Y. Salamin, J. Huangfu, S. Qiao, G. Zheng, and L. Ran, “Observation of Wave Packet Distortion during a Negative-Group-Velocity Transmission,” Sci. Rep. 5(1), 8100 (2015).
[Crossref]

Romero, R.

Salamin, Y.

D. Ye, Y. Salamin, J. Huangfu, S. Qiao, G. Zheng, and L. Ran, “Observation of Wave Packet Distortion during a Negative-Group-Velocity Transmission,” Sci. Rep. 5(1), 8100 (2015).
[Crossref]

Schultz, S.

R. A. Shelby, D. R. Smith, S. C. Nemat-Nasser, and S. Schultz, “Microwave transmission through a two-dimensional, isotropic, left-handed metamaterial,” Appl. Phys. Lett. 78(4), 489–491 (2001).
[Crossref]

Schweinsberg, A.

G. M. Gehring, A. Schweinsberg, C. Barsi, N. Kostinski, and R. W. Boyd, “Observation of Backward Pulse Propagation Through a Medium with a Negative Group Velocity,” Science 312(5775), 895–897 (2006).
[Crossref]

Sendur, K.

Shaw, J. K.

Shelby, R. A.

R. A. Shelby, D. R. Smith, S. C. Nemat-Nasser, and S. Schultz, “Microwave transmission through a two-dimensional, isotropic, left-handed metamaterial,” Appl. Phys. Lett. 78(4), 489–491 (2001).
[Crossref]

Silva, A. S.

Silva, F.

Simon, P.

Smith, D. R.

R. A. Shelby, D. R. Smith, S. C. Nemat-Nasser, and S. Schultz, “Microwave transmission through a two-dimensional, isotropic, left-handed metamaterial,” Appl. Phys. Lett. 78(4), 489–491 (2001).
[Crossref]

Sola, I. J.

Soukoulis, C. M.

B. Dastmalchi, P. Tassin, T. Koschny, and C. M. Soukoulis, “Strong group-velocity dispersion compensation with phase-engineered sheet metamaterials,” Phys. Rev. B 89(11), 115123 (2014).
[Crossref]

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous Negative Phase and Group Velocity of Light in a Metamaterial,” Science 312(5775), 892–894 (2006).
[Crossref]

Steinmeyer, G.

G. Steinmeyer, D. H. Sutter, L. Gallmann, N. Matuschek, and U. Keller, “Frontiers in Ultrashort Pulse Generation: Pushing the Limits in Linear and Nonlinear Optics,” Science 286(5444), 1507–1512 (1999).
[Crossref]

Sutter, D. H.

G. Steinmeyer, D. H. Sutter, L. Gallmann, N. Matuschek, and U. Keller, “Frontiers in Ultrashort Pulse Generation: Pushing the Limits in Linear and Nonlinear Optics,” Science 286(5444), 1507–1512 (1999).
[Crossref]

Swaim, J. D.

Szatmári, S.

Tassin, P.

B. Dastmalchi, P. Tassin, T. Koschny, and C. M. Soukoulis, “Strong group-velocity dispersion compensation with phase-engineered sheet metamaterials,” Phys. Rev. B 89(11), 115123 (2014).
[Crossref]

Turunen, J.

C. Ding, M. Koivurova, J. Turunen, and L. Pan, “Temporal self-splitting of optical pulses,” Phys. Rev. A 97(5), 053838 (2018).
[Crossref]

Vogl, U.

Vysloukh, V. A.

S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, Optics of Femtosecond Laser Pulses (AIP, New York, 1992), Chap. 1.

Walmsley, I. A.

I. A. Walmsley and C. Dorrer, “Characterization of ultrashort electromagnetic pulses,” Adv. Opt. Photonics 1(2), 308–437 (2009).
[Crossref]

Wang, L. J.

H. Cao, A. Dogariu, and L. J. Wang, “Negative Group Delay and Pulse Compression in Superluminal Pulse Propagation,” IEEE J. Sel. Top. Quantum Electron. 9(1), 52–58 (2003).
[Crossref]

Wegener, M.

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous Negative Phase and Group Velocity of Light in a Metamaterial,” Science 312(5775), 892–894 (2006).
[Crossref]

Weiner, A. M.

A. M. Weiner, “Ultrafast optical pulse shaping: A tutorial review,” Opt. Commun. 284(15), 3669–3692 (2011).
[Crossref]

Woodley, J.

J. Woodley and M. Mojahedi, “Backward wave propagation in left-handed media with isotropic and anisotropic permittivity tensors,” J. Opt. Soc. Am. B 23(11), 2377–2382 (2006).
[Crossref]

M. Mojahedi, K. J. Malloy, G. V. Eleftheriades, J. Woodley, and R. Y. Chiao, “Abnormal Wave Propagation in Passive Media,” IEEE J. Sel. Top. Quantum Electron. 9(1), 30–39 (2003).
[Crossref]

Woodley, J. F.

J. F. Woodley and M. Mojahedi, “Negative group velocity and group delay in left-handed media,” Phys. Rev. E 70(4), 046603 (2004).
[Crossref]

Wörner, H. J.

Wyns, P.

Xiao, Y.

Yahalom, A.

Ye, D.

D. Ye, Y. Salamin, J. Huangfu, S. Qiao, G. Zheng, and L. Ran, “Observation of Wave Packet Distortion during a Negative-Group-Velocity Transmission,” Sci. Rep. 5(1), 8100 (2015).
[Crossref]

Zhang, C.

S. Divitt, W. Zhu, C. Zhang, H. J. Lezec, and A. Agrawal, “Ultrafast Optical Pulse Shaping using Dielectric Metasurfaces,” Science 364(6443), 890–894 (2019).
[Crossref]

Zhang, X.

Y. Liu and X. Zhang, “Metamaterials: a new frontier of science and technology,” Chem. Soc. Rev. 40(5), 2494–2507 (2011).
[Crossref]

Zhang, Y.

Zhao, K.

M. Chini, K. Zhao, and Z. Chang, “The generation, characterization and applications of broadband isolated attosecond pulses,” Nat. Photonics 8(3), 178–186 (2014).
[Crossref]

Zheng, G.

D. Ye, Y. Salamin, J. Huangfu, S. Qiao, G. Zheng, and L. Ran, “Observation of Wave Packet Distortion during a Negative-Group-Velocity Transmission,” Sci. Rep. 5(1), 8100 (2015).
[Crossref]

Zhu, W.

S. Divitt, W. Zhu, C. Zhang, H. J. Lezec, and A. Agrawal, “Ultrafast Optical Pulse Shaping using Dielectric Metasurfaces,” Science 364(6443), 890–894 (2019).
[Crossref]

Ziolkowski, R. W.

R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E 64(5), 056625 (2001).
[Crossref]

Adv. Opt. Photonics (1)

I. A. Walmsley and C. Dorrer, “Characterization of ultrashort electromagnetic pulses,” Adv. Opt. Photonics 1(2), 308–437 (2009).
[Crossref]

Antennas Wirel. Propag. Lett. (1)

A. F. Mota, A. Martins P, L. L. Bertarini, and B.-H. V. Borges, “Dispersion Management With Microwave Pulses in Metamaterials in the Negative Refraction Regime,” Antennas Wirel. Propag. Lett. 14, 1377–1380 (2015).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

R. A. Shelby, D. R. Smith, S. C. Nemat-Nasser, and S. Schultz, “Microwave transmission through a two-dimensional, isotropic, left-handed metamaterial,” Appl. Phys. Lett. 78(4), 489–491 (2001).
[Crossref]

Chem. Soc. Rev. (1)

Y. Liu and X. Zhang, “Metamaterials: a new frontier of science and technology,” Chem. Soc. Rev. 40(5), 2494–2507 (2011).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (2)

M. Mojahedi, K. J. Malloy, G. V. Eleftheriades, J. Woodley, and R. Y. Chiao, “Abnormal Wave Propagation in Passive Media,” IEEE J. Sel. Top. Quantum Electron. 9(1), 30–39 (2003).
[Crossref]

H. Cao, A. Dogariu, and L. J. Wang, “Negative Group Delay and Pulse Compression in Superluminal Pulse Propagation,” IEEE J. Sel. Top. Quantum Electron. 9(1), 52–58 (2003).
[Crossref]

J. Mod. Opt. (1)

R. W. Boyd, “Slow and fast light: fundamentals and applications,” J. Mod. Opt. 56(18-19), 1908–1915 (2009).
[Crossref]

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (6)

Nat. Photonics (1)

M. Chini, K. Zhao, and Z. Chang, “The generation, characterization and applications of broadband isolated attosecond pulses,” Nat. Photonics 8(3), 178–186 (2014).
[Crossref]

Opt. Commun. (1)

A. M. Weiner, “Ultrafast optical pulse shaping: A tutorial review,” Opt. Commun. 284(15), 3669–3692 (2011).
[Crossref]

Opt. Express (4)

Opt. Lett. (3)

OSA Continuum (1)

Phys. Rev. A (2)

C. G. B. Garrett and D. E. McCumber, “Propagation of a Gaussian Light Pulse through an Anomalous Dispersion Medium,” Phys. Rev. A 1(2), 305–313 (1970).
[Crossref]

C. Ding, M. Koivurova, J. Turunen, and L. Pan, “Temporal self-splitting of optical pulses,” Phys. Rev. A 97(5), 053838 (2018).
[Crossref]

Phys. Rev. B (1)

B. Dastmalchi, P. Tassin, T. Koschny, and C. M. Soukoulis, “Strong group-velocity dispersion compensation with phase-engineered sheet metamaterials,” Phys. Rev. B 89(11), 115123 (2014).
[Crossref]

Phys. Rev. E (4)

C. M. Balictsis and K. E. Oughstun, “Generalized asymptotic description of the propagated field dynamics in Gaussian pulse propagation in a linear, causally dispersive medium,” Phys. Rev. E 55(2), 1910–1921 (1997).
[Crossref]

C. M. Balictsis, “Unified asymptotic description of Gaussian pulse propagation of arbitrary initial pulse width in a Lorentz-type gain medium,” Phys. Rev. E 87(1), 013304 (2013).
[Crossref]

R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E 64(5), 056625 (2001).
[Crossref]

J. F. Woodley and M. Mojahedi, “Negative group velocity and group delay in left-handed media,” Phys. Rev. E 70(4), 046603 (2004).
[Crossref]

Phys. Rev. Lett. (2)

K. E. Oughstun and C. M. Balictsis, “Gaussian Pulse Propagation in a Dispersive, Absorbing Dielectric,” Phys. Rev. Lett. 77(11), 2210–2213 (1996).
[Crossref]

T. Brabec and F. Krausz, “Nonlinear Optical Pulse Propagation in the Single-Cycle Regime,” Phys. Rev. Lett. 78(17), 3282–3285 (1997).
[Crossref]

Sci. Rep. (1)

D. Ye, Y. Salamin, J. Huangfu, S. Qiao, G. Zheng, and L. Ran, “Observation of Wave Packet Distortion during a Negative-Group-Velocity Transmission,” Sci. Rep. 5(1), 8100 (2015).
[Crossref]

Science (4)

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous Negative Phase and Group Velocity of Light in a Metamaterial,” Science 312(5775), 892–894 (2006).
[Crossref]

G. M. Gehring, A. Schweinsberg, C. Barsi, N. Kostinski, and R. W. Boyd, “Observation of Backward Pulse Propagation Through a Medium with a Negative Group Velocity,” Science 312(5775), 895–897 (2006).
[Crossref]

S. Divitt, W. Zhu, C. Zhang, H. J. Lezec, and A. Agrawal, “Ultrafast Optical Pulse Shaping using Dielectric Metasurfaces,” Science 364(6443), 890–894 (2019).
[Crossref]

G. Steinmeyer, D. H. Sutter, L. Gallmann, N. Matuschek, and U. Keller, “Frontiers in Ultrashort Pulse Generation: Pushing the Limits in Linear and Nonlinear Optics,” Science 286(5444), 1507–1512 (1999).
[Crossref]

Other (4)

S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, Optics of Femtosecond Laser Pulses (AIP, New York, 1992), Chap. 1.

S. J. Orphanidis, Electromagnetic Waves and Antennas, 2016, https://www.ece.rutgers.edu/∼orfanidi/ewa/ .

K. E. Oughstun, Electromagnetic and Optical Pulse Propagation 2: Temporal Pulse Dynamics in Dispersive, Attenuative Media (Springer Series in Optical Sciences, New York, 2009), Vol. 144.

C. M. Balictsis, “Asymptotic Analysis of Chirped Gaussian Pulse Propagation in a Metamaterial Medium,” in Proceedings of the 2019 International Conference on Electromagnetics in Advanced Applications (ICEAA), Granada, Spain, 2019, p. 530–533.

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Figures (5)

Fig. 1.
Fig. 1. Output stage of the metamaterial pulse compression scheme
Fig. 2.
Fig. 2. Input stage of the metamaterial pulse compression scheme
Fig. 3.
Fig. 3. Dynamical evolution with $\theta $ of ${A_{com{p_{out}}}}({z^{\prime} = d,t} )$ in the input stage of the metamaterial pulse compression scheme, evaluated using both the unified asymptotic approach and the inverse Laplace numerical approach. (a) and (b) depict Case-1 and (c) and (d) Case-2. Moreover, (a) and (c) depict the description due to the unified asymptotic approach whereas (b) and (d) the prediction of the inverse Laplace numerical approach. In (a) and (c), the square denotes the ordered pair $({\theta _{U{A_{\max }}}},{A_{U{A_{\max }}}})$ and in (b) and (d) the corresponding ordered pair $({\theta _{H{A_{\max }}}},{A_{H{A_{\max }}}})$ of the space-time point when the total (absolute) maximum for the respective ${A_{com{p_{out}}}}({z^{\prime} = d,t} )$ occurs.
Fig. 4.
Fig. 4. Dynamical evolution with $\theta $ of the propagated field ${A_{com{p_{out}}}}({z^{\prime} = d,t} )$ and its associated envelope $\left| {{A_{com{p_{out}}}}({z^{\prime} = d,t} )} \right|$ in the input stage of the metamaterial pulse compression scheme, evaluated using the unified asymptotic approach, when one as well as when two asymptotic series expansion terms are retained in Eq. (9)-(10). (a) and (b) depict Case-1, whereas (c) and (d) Case-2. In (a) and (c) one asymptotic term is retained, whereas in (b) and (d) two asymptotic terms are retained in the respective propagated field expressions. In (a)-(d) the propagated field is depicted with a solid (black) line, its associated envelope is depicted with a solid (red) line and the respective (black) square denotes the ordered pair $({\theta _{U{A_{\max }}}},{A_{U{A_{\max }}}})$ of the space-time point when the total (absolute) maximum for the respective ${A_{com{p_{out}}}}({z^{\prime} = d,t} )$ occurs.
Fig. 5.
Fig. 5. Numerically determined, dynamical evolution with $\theta $ of ${A_{{m_{out}}}}({z = d,t} )$ in the output stage of the metamaterial pulse compression scheme, when ${A_{{m_{in}}}}({z = 0,t} ) = {A_{com{p_{out}}}}({z^{\prime} = d,t} )$. (a) and (b) correspond to Case-1 and (c) and (d) to Case-2. Moreover, (a) and (c) depict, superimposed upon the (desired) field $g(t )$ (blue continuous line), the numerically determined output pulse wave (red dashed line) when the input pulse wave coincides, respectively, with the field dynamics in Fig. 3(a) and Fig. 3(c). Furthermore, (b) and (d) depict, superimposed upon the (desired) field $g(t )$ (blue continuous line), the numerically determined output pulse wave (red dashed line) when the input pulse wave coincides, respectively, with the field dynamics in Fig. 3(b), and Fig. 3(d. In (a) and (c), the (red) square denotes the ordered pair $({\theta _{U{A_{\max }}}},{A_{U{A_{\max }}}})$ and in (b) and (d) the corresponding ordered pair $({\theta _{H{A_{\max }}}},{A_{H{A_{\max }}}})$ of the space-time point when the total (absolute) maximum of the respective ${A_{{m_{out}}}}({z = d,t} )$ occurs.

Equations (12)

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A ( z , t ) = 1 2 π L f ( ω ) e i [ k ( ω ) z ω t ] d ω ,   z > 0.
ε m ( ω ) = 1 ω e p 2 ω e 0 2 ω 2 ω e 0 2 + i γ e ω ,  and   μ m ( ω ) = 1 ω m p 2 ω m 0 2 ω 2 ω m 0 2 + i γ m ω .
g ( t ) = exp { [ ( t t 0 ) / T ] 2 } sin [ ω c t + ( C / 2 ) t 2 + ψ ] .
A m i n ( z = 0 , ω ) = A c o m p o u t ( z = d , ω ) , and A m o u t ( z = d , ω ) = A c o m p i n ( z = 0 , ω ) .
H c o m p ( z , ω ) = exp { i [ k c o m p ( ω ) z ] } ,
H m ( z , ω ) = exp { i [ k m ( ω ) z ] } ,
k c o m p ( ω ) = k m ( ω ) = ( ω / c ) n m ( ω ) .
A c o m p o u t ( z , t ) = 1 2 π R e { i L U U exp [ z c Φ U ( ω , θ ) ] d ω } , z > 0 ,
A U l ( z , t ) = 1 2 π R e { i e z c Φ U ( ω , θ ) ( π c z ) 1 2 U U [ 1 2 d 2 Φ U ( ω , θ ) d ω 2 ] 1 2 [ H ( ω , θ ) + O ( z 5 2 ) ] } ω S P U l ( θ )
H ( ω , θ ) = 1 c z { 15 72 [ d 3 Φ U ( ω , θ ) d ω 3 ] 2 [ d 2 Φ U ( ω , θ ) d ω 2 ] 3 1 8 ( d 4 Φ U ( ω , θ ) d ω 4 ) ( d 2 Φ U ( ω , θ ) d ω 2 ) 2 } .
ω S P U l i ( θ ) = d d t { ln [ | d 2 Φ U ( ω , θ ) d ω 2 | 1 2 | H ( ω , θ ) | ] ω S P U l ( θ ) } ,
υ U E n v c o m p = d t U E n v max t 0 = c θ U E n v max ( c t 0 / d ) .

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