Nonlocality‐Enabled Pulse Management in Epsilon‐Near‐Zero Metamaterials

Ultrashort optical pulses are integral to probing various physical, chemical, and biological phenomena and feature in a whole host of applications, not least in data communications. Super‐ and subluminal pulse propagation and dispersion management (DM) are two of the greatest challenges in producing or counteracting modifications of ultrashort optical pulses when precise control over pulse characteristics is required. Progress in modern photonics toward integrated solutions and applications has intensified this need for greater control of ultrafast pulses in nanoscale dimensions. Metamaterials, with their unique ability to provide designed optical properties, offer a new avenue for temporal pulse engineering. Here an epsilon‐near‐zero metamaterial is employed, exhibiting strong nonlocal (spatial dispersion) effects, to temporally shape optical pulses. The authors experimentally demonstrate, over a wide bandwidth of tens of THz, the ability to switch from sub to superluminal and further to “backward” pulse propagation (±c/20) in the same metamaterial device by simply controlling the angle of illumination. Both the amplitude and phase of a 10 ps pulse can be controlled through DM in this subwavelength device. Shaping ultrashort optical pulses with metamaterials promises to be advantageous in laser physics, optical communications, imaging, and spectroscopy applications using both integrated and free‐standing devices.


DOI: 10.1002/adma.202107023
analysis, spectroscopy, optoelectronics, and imaging but perhaps, most importantly, they also play a crucial role in modern optical communication systems and quantum technologies. In all these applications, the temporal pulse shaping is important to reveal dynamic response of the material or chemical systems. As the pulse propagates in a material, it is distorted due to the dispersion effects, affecting time resolution or information transmission. Therefore, there is a constant pursuit for solutions to control and re-shape light pulses and compensate their dispersion.
In recent years, the control of optical pulses has been achieved via the phenomena of fast and slow light. These terms refer to situations when the group velocity v g , the velocity of the peak of the optical pulse traveling through the material system, is considerably different from the vacuum speed of light c ( Figure S1, Supporting Information). [1,2] Experimental observation of faster-than-light propagation has been reported for media featuring absorption resonances [3] and bandgaps [4] in different types of photonic devices like, for example, tapered waveguides, [5] subwavelength holes, [6] or more recently, in semiconductor heterostructures operating in a super-radiant emission regime. [7] Furthermore, under certain extreme conditions, the peak of the transmitted pulse might emerge from the material medium before entering, so that the pulse envelope appears to propagate backward inside the material. This effect exists in sufficiently thick materials with negative group refractive index and results in a negative group delay (GD) and correspondingly a negative v g . Such backward propagating light was observed experimentally in resonant media, such as saturable absorbers [8] and erbium-doped optical fibers. [9] The superluminality of v g or backward propagation does not violate causality or Einstein's special relativity as even if the peak of the pulse moves at a superluminal velocity, the information content of the pulse travels no faster than the speed of light in vacuum [10] and the flow of the power is always from the entrance to the exit of the medium.
On the other extremity are the phenomena of subluminal propagation when the group velocity of the light pulse is much slower than c. Experimentally, light has been slowed down [11] to Ultrashort optical pulses are integral to probing various physical, chemical, and biological phenomena and feature in a whole host of applications, not least in data communications. Super-and subluminal pulse propagation and dispersion management (DM) are two of the greatest challenges in producing or counteracting modifications of ultrashort optical pulses when precise control over pulse characteristics is required. Progress in modern photonics toward integrated solutions and applications has intensified this need for greater control of ultrafast pulses in nanoscale dimensions. Metamaterials, with their unique ability to provide designed optical properties, offer a new avenue for temporal pulse engineering. Here an epsilon-near-zero metamaterial is employed, exhibiting strong nonlocal (spatial dispersion) effects, to temporally shape optical pulses. The authors experimentally demonstrate, over a wide bandwidth of tens of THz, the ability to switch from sub to superluminal and further to "backward" pulse propagation (±c/20) in the same metamaterial device by simply controlling the angle of illumination. Both the amplitude and phase of a 10 ps pulse can be controlled through DM in this subwavelength device. Shaping ultrashort optical pulses with metamaterials promises to be advantageous in laser physics, optical communications, imaging, and spectroscopy applications using both integrated and freestanding devices.

Introduction
Ultrashort optical pulses are central to many applications and techniques in the development of new materials, chemical 17 ms −1 or even stopped entirely [12] using ultracold atom clouds. This was obtained with the use of electromagnetically induced transparency (EIT), a quantum interference effect arising from the nonlinear coherent coupling of broad and narrow resonances. [13] Such coupling is manifested by a sharp transmission peak located within the absorption profile. Although the vital advantage of the described method is the substantial suppression of optical losses, the constraints come from the fact that the frequency of the incident light must match precisely the resonance frequency of the atoms. These obstacles were partially mitigated by introducing artificially created structures exhibiting the EIT-like effect, including, microresonators, [14,15] electric circuits, [16] and waveguides. [17] Slow and fast light can also be obtained at room temperature via the nonlinear process of coherent population oscillations (CPO). Similar to EIT, subluminal propagation in CPO is associated with a dip in the absorption spectrum. In this method, the dip is obtained when pump and signal beams of slightly different frequencies interact inside a saturable absorber. These interfering pump and signal beams produce a beat frequency, which the ground and excitedstate populations of the material oscillate at. The coupling between the pump beam and the oscillation of the population results in a reduction of the absorption of the signal beam and alters its pulse propagation speed. [8,18] It was also demonstrated that a high degree of control over the velocity of light could be achieved via stimulated Brillouin scattering (SBS). [19] In this process, through electrostriction, a high-frequency acoustic wave is induced in the material, causing a time-varying change in the refractive index. The nonlinear interaction between pump and signal beams and the acoustic wave gives rise to narrow isolated amplifying (Stokes) or absorbing (anti-Stokes) resonances, in the vicinity of which either slow or fast light can be achieved, respectively. [20] Above-mentioned velocity control processes (EIT, CPO, and SBS), are all nonlinear, and the time delay of the propagating optical pulse is proportional to the intensity of the pump pulse. However, these methods also share three problems. The first one is associated with limited bandwidth over which the slow light effect can be observed. For example, for a typical optical fiber, the Brillouin linewidth is only 20 to 50 MHz. [21] Second, switching between slow and fast light operation is either impossible or difficult. Lastly, all these techniques require complicated experimental configurations.
Another critical aspect of pulse propagation control is the influence of material dispersion on pulse properties and, thus, on the carried information content. Since a laser pulse is generally composed of different frequencies, chromatic dispersion leads to pulse broadening and chirping, which limits the bandwidths of, among others, telecommunication networks. This degradation of optical signals is accumulated over multiple devices, as the optical amplifiers and relays do not restore the output signal to its original state. [22] Therefore, dispersion management (DM) is a critical part in the process of designing modern optical systems. DM methods are not only applied to compensate unwanted dispersion-related effects, they can also serve to tailor and enhance dispersion in order to boost other desirable functionalities. For example, the sensitivity of many types of interferometers can be dramatically improved by placing highly dispersive material within one arm. [21] Another example comes from the field of mode-locked lasers, where, for optimum pulse generation, it is often beneficial to overcompensate the chromatic dispersion in order to utilize the regime of anomalous dispersion. [23] DM also becomes especially important in the contexts of slow and fast light phenomena, which relies mostly on the fast change of the refractive index with respect to the light frequency in the vicinity of the resonance.
Over the years, many different types of DM setups have been developed. The standard configurations rely either on the Michelson interferometer, [24] propagation through a dispersive bulk medium of varying thickness, [25] grating compressors, [26] chirped mirrors, [27] acousto-optic modulators, [28] spatial light modulators, [29] or fiber-based geometries. [30] However, the fundamental limits imposed by physics make these solutions difficult to implement in integrated photonics, slow or not dynamically tunable. A common problem in both velocity and DM in light pulses is the inflexibility of current devices to be dynamically tuned, and their bulky or complex nature. With the introduction of integrated photonic solutions and nanophotonic waveguides with high dispersion in modern technology, the requirements for pulse control on the nanoscale become increasingly important.
Metamaterials, with their unique and tailorable optical properties, offer the potential to solve some of these problems. Metamaterial structures derive their optical properties from their designed geometrical arrangement of subwavelength elements, rather than just from their constituent material properties. This has led to metamaterials being used in many applications to control the properties of light, from polarization [31] to focusing. [32] With this ability to design optical properties, it is not surprising that metamaterials and metasurfaces have attracted attention for slow and fast light applications and effective DM. Using metamaterials, it has been demonstrated that controlling the speed of light can be done by an analog of EIT either by using metallic structures in the near-infrared, [33,34] or split rings in the terahertz region. [35] A substantial difference in the speed of light can also be obtained in metamaterial waveguides. [36,37] Similarly, the problem of DM has been investigated theoretically in phase-engineered sheet metamaterials [38] and experimentally in left-handed transmission lines. [39] Still, these approaches require complex structures and once fabricated, their properties are often difficult to adapt.
In this paper, we propose and demonstrate the unique opportunities for controlling pulse dispersion provided by a nonlocal (spatial dispersion) response of an epsilon-near-zero (ENZ) metamaterial. Such structural nonlocality in metamaterials has already been shown to crucially affect nonlinearity and spontaneous emission in metamaterials. [40,41] We show, for the first time, that the nonlocal nanorod-based metamaterial operating in the ENZ regime has the ability to provide either slow (≈c/20), fast or backward (≈−c/20) propagation at the same wavelength, over a wide bandwidth of tens of THz in the visible spectral range, depending on the angle of incidence. The nanorods architecture also offers the ability to easily tune the operation of this self-assembled metamaterial into the NIR. [42] We show numerically and experimentally that switching between propagation regimes is obtained not by a nonlinear process, but simply by changing the angle of incidence of the pulses. Furthermore, the dispersion properties can also be tailored, with control over the sign and strength of the material second and third-order dispersion, up to 5 orders of magnitude higher than in bulk glass. Finally, we study the influence of the nanorod metamaterial on the amplitude and phase of 10 ps pulses and demonstrate that a large variety of different positive and negative pulse chirps can be obtained. All these tunable phenomena are contained in a single metamaterial device of subwavelength thickness suitable for integration in nanophotonic devices.

Physics of Pulse Management
Light pulses can be represented as a superposition of an infinite number of monochromatic plane waves: The interference of these weighted plane wave components determines the shape, width, and location of the pulse envelope in space and time. When a pulse propagates through the dispersive medium, the frequencydependent response of a material, manifested by its refractive index n, results in different phase velocities v p of each monochromatic component v p (ω) = c/n(ω). At the same time, the peak of the pulse, where the constituent Fourier components add up in phase, moves at the group velocity given by v g = c/n g (ω), where the group index n g (ω) and its dependence on the angular frequency ω and the phase refractive index n(ω) is In order to obtain slow-, fast-and backward-light phenomena, the group index must be significantly different from unity. This condition can be satisfied either by taking a medium with extreme values of phase refractive index n(ω), for example, close to zero, or by using a material with strong positive or negative dispersion. In the case of EIT, CPO, SBS, and in the nanorod metamaterial, sub and superluminal propagations are realized by the second approach. [21] Both the amplitude and the phase of the electric field need to be considered in order to obtain full information about the pulse. Modern telecommunications systems require not only distortion-free amplitudes of optical pulses but also highquality phases. This relates to the fact that the modulation of the optical phase may be used to correct chromatic dispersion and/or nonlinearities using digital backpropagation and for quadrature phase-shift keying information transmission. [43] Notably, the spectral phase ϕ(ω) plays a central role in determining the shape of short pulses. There are several quantities related to the spectral phase, that are often used to describe the evolution of a pulse shape. For a narrow band pulse, the spectral phase can be expanded in a Taylor series around the center pulse frequency ω 0 : where and ϕ (0) is known as the carrier-envelope phase or the absolute phase, ϕ (1) is the GD and describes the arrival time of various spectral components of the signal. It is also directly proportional to the group refractive index and is inversely proportional to the group velocity: GD = z·n g /c = z/v g , where z is the propagation distance, which, in our analysis, will often be equal to the thickness of the metamaterial sample. ϕ (2) is the GD dispersion (GDD) with positive (negative) values corresponding to normal (anomalous) chromatic dispersion. The GDD per unit length is the group velocity dispersion (GVD), GVD = ϕ (2) /z, which is convenient for comparing the performance of different DM devices. GDD and GVD are the parameters that describe dispersive temporal broadening and compression of ultrashort pulses. Since dispersion is also responsible for group velocity mismatch of different waves, nonlinear effects, such as parametric interactions or spectral broadenings, are strongly affected. [44] The higher-order terms in Equation (2) result in the distortion of the pulse from the original form. For example, strong third-order dispersion (TOD), ϕ (3) , which comes from the frequency dependence of the GDD, leads to appearance of pre-or post-pulses in the temporal domain. [25] In some applications exploiting ultrashort pulses, TOD and higher-order terms also have to be appropriately compensated. [45] Similarly, one can also describe the optical pulse in the time domain. We can define the instantaneous frequency, ν inst , which provides a measure of the frequency domain signal energy concentration as a function of time: Time dependence of the instantaneous frequency is related to the chirp of an optical pulse. A chirp means that each frequency experiences a different delay in time.
Both GD and instantaneous frequency can be used to describe the characteristics of a given pulse and give information on how to process the pulse in order to obtain a specific shape or width. Depending on the application, the chirp of the pulse can be either compensated, as happens in telecommunication systems, or further enhanced and modified to, for example, exploit physics of interactions of atoms or molecules. [46,47] In the following, we aim to demonstrate that a metamaterial composed of a gold nanorod matrix can be used for pulse management. As shown in the schematics in Figure 1a, such a metamaterial can act as the tunable optical element, which can introduce either positive or negative chirp of varying magnitude into the optical pulse.

Local and Nonlocal Effects in Nanorod Metamaterial
The nanorods metamaterial consists of an array of metallic rods oriented perpendicular to the substrate (Figure 1b). The optical response is governed by the interaction between cylindrical surface plasmons of closely spaced nanorods. [48] Such a metamaterial can be used to control fluorescence lifetime [49,41] to obtain enhanced nonlinear optical effects, [32] as well as in ultrasensitive sensors. [50,51] Many of these properties are linked with the strong anisotropy of the metamaterial with optical axis parallel to the nanorods [48] (Figure 1c): within the effective medium theory (EMT), eff e ff x y ε ε = components of the effective permittivity tensor are always positive while eff z ε typically changes its sign from positive to negative with the decreasing frequency. Accordingly, optical waves in the metamaterial have three dispersion regimes in different wavelength ranges: elliptic, epsilon-near-zero (ENZ), and hyperbolic. [52] The spectral response of the structure can be engineered and tuned by choosing the material of the rods, their length, diameter, a separation distance between them, and the material of the embedded matrix. [42] To illustrate the physics of pulse management in the gold nanorod metamaterial, we consider three different samples with different geometrical and material parameters and thus distinctive types of optical responses. The gold nanorods, were grown by direct electrodeposition into pores formed in thin-film anodized aluminum oxide (see Experimental Section for more details).
Comparison between the experimentally measured optical properties of the metamaterials and numerical simulations in either local EMT, where the sample is treated as a homogenous, anisotropic slab of material, or modeling using the full vectorial finite-difference time-domain method, where we take into account the presence of individual rods (see Experimental Section for details), allows us to understand the role of nonlocal effects for the samples with different parameters (Figure 2). The extinction spectra of Sample #1 exhibit two typical, polarization-dependent, well-defined features at around 540 nm due to the electric field component perpendicular to the nanorod axes and around 615 nm due to the field component along the nanorod axes. The longer-wavelength mode appears in the proximity of the ENZ region, where the component of the effective permittivity tensor in z-direction has values close to zero: Re( ) 0 eff z ε ≈ . The resonance associated with this ENZ regime is designed to be wide in order to provide broad operating bandwidth of the device. The local EMT and the full wave numerical modeling agree well between themselves and with the measured spectra, indicating that nonlocal effects are not important for the metamaterial with geometrical parameters of Sample #1.
Sample #2, which is composed of longer rods, reveals the presence of additional features, not described by the local EMT model, manifested by splitting of the ENZ-resonance (Figure 2e,f). Eigenmode analysis shows that the nanorod metamaterial actually supports two transverse-magnetic (TM) waves in this case. [53] When the coupling between the cylindrical surface plasmons of individual nanorods is sufficiently strong, these two TM waves can interfere. [54] This interference is extremely sensitive to the angle of incidence and leads to wavevector dependent dispersion in the metamaterial and is described as nonlocal, due to the effect the electric field component in one TM wave has on the permittivity experienced by the other TM wave in a distinct location. Consequently, the effective permittivity of the metamaterial depends not only on the frequency of light but also on the incident light wave-vector. This nonlocal effect is not visible in the extinction spectra calculated using the local effective medium theory (Figure 2d).
Since the nonlocal effects depend on the subwavelength structure of the metamaterial (the strength of coupling between the longitudinal plasmonic oscillations excited on individual metal rods), its characteristics can be tuned by varying geometrical parameters of the structure (see Supporting Information for details). Another critical factor influencing nonlocality is the quality of the metal used for the construction of the nanorods. In solution-derived gold, the mean freeelectron path can be restricted not only by electron-electron or electron-phonon interactions but also by the presence of internal structure or grain boundaries. [55] This leads to alteration of the value of the permittivity and as a consequence, changes the strength of nonlocal effects. [53] In Sample #3 (Figure 2h,i), the nonlocal effects are less pronounced in comparison to Sample #2, due to lower mean free electron path in gold for this sample ( Table 1)

Slow, Fast and Backward Light in Nanorod Metamaterial
Gold nanorod metamaterials are highly dispersive structures, with a spectral response that can be precisely designed. Together with the property that the strength of the extinction resonance can be varied by simple alteration of the illumination angle, in the context of Equation (1), such a platform should be the right candidate for a pulse velocity control device.
The parameter which is directly related to the motion of the pulse is the GD. In Figure 3, we present numerically modeled and experimentally measured GD values as functions of wavelength obtained for considered nanorod metamaterials. In Sample #1, which does not exhibit nonlocal effects, the GD follows the dependence known from typical resonance media, in which the presence of a region of anomalous dispersion allows superluminal or even negative delay. [3] In the nanorod metamaterial however, the delay values can be continuously modified by changing the angle of incidence. This feature results from the fact that the nature of this resonance does not originate from material absorption, but from the interaction of two TM waves. Therefore, the GD in Sample #1 monotonically decreases with a larger illumination angle and for 50 o (red line in Figure 3c) and at the wavelength of 618 nm, the GD is 10 fs smaller than at normal incidence (black line in Figure 3c). The measured GDs confirm that, for high angles of incidence (>45 o ), the nanorod   metamaterial not only supports superluminal pulse propagation but as indicated by the negative GD values, also backward pulse propagation. When the nonlocal effects become significant, the situation becomes far more intriguing. Initially, for Sample #2, we observe similar GD versus angle dependence as for sample #1. However, when the illumination angles are increased beyond 40 o , the change in GD values greatly accelerates, and values as low as −47 fs for a 525 nm thick sample were measured in the experiment (dark green line, Figure 3f). Such strong, negative delay indicates backward waves propagating with a group velocity of ≈−c/30. When the material exhibits negative group refractive index, the operation bandwidth is around 20 nm (≈14 THz). For illumination angles larger than 42 o , the extremum in GD instantaneously changes from dip to peak and reaches 23 fs what corresponds to ≈c/15. Full wave simulations (Figure 3e) confirm the existence of a GD singularity, present for an angle between 41.5 o and 42 o . They also indicate that even more extreme velocities can be obtained in the system (c/115 & −c/170) if only the angle of incidence is more precisely controlled. Moreover, unlike in the typical resonant systems, the maximum change of GD in the nonlocal metamaterials is not necessarily directly linked with the highest absorption values. This is seen by comparing GD from Figure 3f with corresponding extinction spectra from Figure 2f. The strongest absorption was measured for 50 o angle of incidence, whereas extreme GD values were obtained for 40 o and 45 o . If no nonlocal corrections are considered, the GD-angle dependence has a similar monotonic relation, as presented earlier for sample #1 (Figure 3a,d).
To our knowledge, this is the first experimental demonstration of the angle-of-incidence-induced switching between slow and fast light over tens of THz bandwidth, enabled by nonlocality in metamaterials. However, Sample #2 does not allow  the selection of arbitrary values of GD at the same wavelength as the nonlocality not only changes the direction of GD but also results in the significant shift of the position of the peaks with the angle of illumination (Figure 3e). The solution to the problem of nonlocality-induced shift of the peaks is to control the strength of the nonlocal effects in the nanorod metamaterial by optimizing the sample geometry and/or by influencing the electron mean free path in the metal. Sample #3, fabricated as a result of such a parameter optimization procedure, behaves similar to sample #2 in terms of exhibiting a GD singularity (Figure 3h,i), but this time the extreme positive and negative values of GD are achieved for almost exactly the same wavelength, corresponding to the center of the ENZ range. This allows the selection of any desired GD value between the two extremes by precise control of the angle of incidence.
Finally, we examine how the nanorod metamaterial will affect the propagation of different spectral components of a traveling optical pulse (Figure 4). Since the GVD is the derivative of the GD with respect to the angular frequency, similarly, as for the classical resonance materials, GVD has opposite signs around the extinction peak in the ENZ region. Again, unlike in absorbing bulk media, or even in the theoretically proposed sheet metamaterials, [38] in the nanorod metamaterials, the GVD value is not explicitly determined by the elemental material composition or fixed because of unchangeable sample geometry after final fabrication stage, but can be fluently altered with the illumination angle. Depending on the local or nonlocal sample type, the potential of the proposed metamaterial as a DM device is different. In sample #1 (Figure 4a-c), the selection of the sign of the dispersion comes with the choice of the operating wavelength. In practice it means that one needs two samples, optimized for different optical frequencies, to have the possibility of changing GVD in full positive and negative value range. It is for this reason that the advantage of nonlocal nanorod metamaterials becomes apparent. In sample #2 and especially sample #3, where the position of the resonant wavelength is fixed, it  is possible to obtain subluminal, superluminal, and negative delay for the same wavelength by simple choice of the angle of incidence (sample rotation). The GVD introduced by the metamaterial can have both positive and negative values occurring for a selected optical frequency. This property is confirmed both in the full vectorial FDTD simulations (Figure 4e,h) and in the experiment (Figure 4f,i), and it is unprecedented in a subwavelength thick device, as such versatility would currently require a complex optical setup and time-consuming alignment. Furthermore, the measured values of GVD, are almost 5 orders of magnitude larger than for the case of fused silica glass (brown curve in Figure 4a,d,g). Numerical modeling indicates that these values can be even higher if the angle of incidence is more precisely controlled (inset in Figure 4h).

Pulse Management in Nanorod Metamaterial
In practical applications for managing optical pulses, one must control simultaneously time delays as well as dispersion, which affects pulse shape, or, more generally, the information content of a spectral phase. In this context, the key parameter related with the performance of a DM device, is the bandwidth of operation. Strongly dispersive materials need to have the operational bandwidth much broader than the spectrum of the propagating pulse. If this condition is not fulfilled, the material will introduce strong, high order distortions in the pulse, which will be very difficult or impossible to compensate at the later stage. We analyzed the influence of the nanorod metamaterial on a 10 ps, bandwidth-limited light pulse typical in telecommunications. As a test sample, we have used sample #3, with strong presence of nonlocal effects. Illumination with TM polarized light at an angle of 42° corresponds to the backward light propagation (Figure 3h). From the dependence of the normalized intensity of the output pulse, as a function of time and wavelength (Figure 5a), one can obtain a −1.5 ps delay at a 649.05 nm central wavelength, with only small distortions in the pulse shape. This is possible because the assumed pulse has a much narrower spectral width (Δλ = 0.06 nm) than the FWHM associated with the ENZ-resonance of the GD peak (Δλ = 20 nm, inset in Figure 3h). The amount of total delay means that a 1 nm slice of the structure introduces an astonishing additional delay of ≈2.86 fs, with respect to the propagation time through the metamaterial at normal incidence.
As a final point, we discuss the capabilities of the nanorod metamaterial to operate as a DM device. Figure 5b shows a multiparameter plot of the instantaneous frequency of a 10 ps pulse after the propagation through sample #3, where in order to reflect the scale of characteristic quantities, we have plotted a corresponding normalized GVD and normalized TOD of the metamaterial together with an illustrative spectrum of 10 ps pulse and its intensity envelope in time. Starting from the situation when the incoming pulse has a central wavelength of 648.8 nm, the metamaterial introduces positive, almost linear chirp into an optical signal propagating through it. When the structure is illuminated with pulses with longer central wavelength, the time delays of particular spectral components become longer, and the strongest chirp is observed at 649 nm wavelength where GVD reaches maximum. At wavelengths closer to the center of the resonance, higher order dispersion effects start to influence the propagation of the optical pulse, bending the tails of the instantaneous frequency dependence. Gradually, they start to dominate over a second-order GVD and, in the very center of the ENZ region, the shape of the chirp is determined mainly by TOD. If the metamaterial was thicker or the pulses shorter, we should observe for this frequency additional oscillations after the main optical pulse. On the other side of the center of the resonance, we observe negative chirp: first the decrease of TOD and increase in GVD and next the lessening of all dispersion effects. It is important to note that for shorter pulses, the associated spectral broadening can lead to more complicated dispersion effects on the pulse temporal shape if different spectral components of the pulse propagate in different regimes (i.e., with different sign of GD) introducing strong chromatic dispersion.
To illustrate the variety of different pulse chirps which can be obtained using a nonlocal metamaterial, the instantaneous frequency of the output pulse is shown for different central frequencies of the pulse (Figure 5c). Having in mind that i) the spectral position of the resonance can be engineered and ii) by controlling the angle of incidence it is possible to control the strength and sign of dispersion and the type of propagation (by exploiting the nonlocal effects), it is possible to obtain all combinations of dispersion effects for a desired wavelength. To our knowledge, there are no existing nanoscale devices that demonstrate such versatility and adaptability in pulse management. On the other hand, conventional and dielectric metasurfacebased DM devices with similar functionality require multiple optical components unsuitable for integration. [45]

Conclusion
We have experimentally demonstrated a versatile pulse management platform comprised of a nanorod metamaterial structure exhibiting nonlocal effects. Near the ENZ frequency, where the role of nonlocal effects is especially pronounced due to the interference of the two excited TM modes in the metamaterial layer, [41,53] the ultra-strong dispersion of the ultrashort pulses is observed as the pulse can couple to both TM modes. Pulse propagation is strongly affected in the nonlocal regime as normal and additional TM waves have very different refractive indexes. [53] The geometrical parameters of the metamaterial can be adjusted in order to provide the desired dispersion characteristics for a given wavelength. The magnitude and sign of the dispersive effects can be altered by simply changing the angle of incidence of the signal pulse. We have demonstrated experimentally, the ability to obtain slow-(≈c/20) and fast-light (≈−c/20) beam propagation for the same wavelength, over a wide bandwidth of tens of THz in the visible spectral range. To our knowledge, such versatility has not been demonstrated in previous DM devices; and certainly none that are easily integrable into nanophotonic devices like the proposed metamaterial, with thicknesses of around 500 nm, exhibiting 5 orders of magnitude larger GVD than fused silica. The thickness and associated losses can be further reduced depending on the required range of parameters of pulses to be controlled. Since nonlinear optical effects are also enhanced in the ENZ regime, [32] the nonlinearity may offer an additional degree of freedom to manipulate the temporal and spectral properties of ultrashort pulses in such metamaterials. Natural ENZ materials may also be exploited if the suitable nonlocal spatial-dispersion operating regime can be achieved in the required wavelength range. The proposed approach not only has the promise to create new dispersion and pulse management devices for data communication at both the nano-and macro-scale, light buffers for optical packet switching, variable-sensitivity interferometry but also in the design and fabrication of new pulsed laser sources.

Experimental Section
Sample Fabrication: The nanorod metamaterial is an example of a self-organized metamaterial structure and is fabricated via an electrochemical process. [48] The geometry consists of an array of gold nanorods oriented perpendicular to a glass substrate and this lithography-free fabrication method can achieve samples in the cm 2 scale. These nanorods are embedded in an alumina template, which can be etched away to create a free-standing array of nanorods. In order to fabricate a sample, an aluminum film was sputtered via magnetron sputtering on a substrate comprised of a glass slide covered with a 15 nm thick adhesion layer of tantalum pentoxide and a 10 nm thick Au film acting as an electrodeposition electrode. The thickness of the aluminum determines the maximum length of the nanorods. Next, the aluminum was anodized in two steps using 0.3 m sulfuric and oxalic acid solution at 20 and 40 V respectively. The first anodization was carried out for 5 min. After it, the porous oxide layer was completely etched in a mixed solution of H 3 PO 4 (3.5%) and CrO3 (20 g L −1 ) at 70 °C leaving an ordered, patterned surface. The samples were then subjected to a second anodization step under the same conditions as in the first step to produce an alumina film with an array of highly ordered pores. The pores were subsequently etched in 30 mm NaOH to remove a barrier layer at the bottom of the pores, thus exposing the gold electrode, and to tune the diameter of the pores. The diameter and separation of these pores can be carefully tuned by controlling the parameters of the anodization process. Then gold was electrodeposited into the porous template using a non-cyanide solution. Depending on the application, the sample can be further annealed, to control the mean free electron path of electrons in gold and thus, its optical properties. [53] Numerical Modeling of the Optical Response of Metamaterial: The extinction spectra, as well as GDs and group velocity dispersions of the plasmonic metamaterial, are calculated using either the transfer matrix algorithm with implemented EMT or the full-vectorial finitedifference time-domain where the presence of individual gold rods is taken into account. In the first approach, the sample was treated as a homogenous, anisotropic slab of material with a diagonal anisotropic permittivity tensor. In the framework of Maxwell-Garnet approximation, the components of the tensor can be calculated as [56] ε ε ε ε ( )  (6) where N = πr 2 /p 2 is the concentration of the nanorods, defined via their radius r and the separation p in the square array, and ε Au and ε Al2O3 are the permittivities of gold and AAO, respectively. Using the transfer matrix method, for each angle of illumination and for each wavelength, the complex amplitude transmission coefficients t = |t|e iφ were calculated. They contain information about the optical extinction in the sample (−log(|t| 2 )). Furthermore, from the phase of the amplitude transmission coefficient, from Equation (4), one can also calculate GD and group velocity dispersion values.
In the second approach, the direct composite structure of the metamaterial is taken into account. The optical response was simulated using finite-difference time-domain algorithm (FDTD, Lumerical). The authors have considered the diameter and length of the nanorods, as well as their surface roughness (by introducing random distortions in the surface of rods) and realistic permittivity values. In the model, the structure was illuminated with various angles with the broadband pulse. Afterward, via the chirped z-transform, [57] the complex amplitude transmission coefficients have been calculated, GD, and group velocity dispersion values. To eliminate the problem with the spread in the injection angles associated with time domain simulations, an implementation of the broadband fixed angle source technique had been used. [58] Both in TMM and FDTD approaches, the authors have taken into account the variation of the mean free path of electrons in the electrodeposited gold due to granularity effects, which can be controlled via the sample annealing process. [48] Such restriction in electron movement is a consequence of the nanorod fabrication procedure and is linked to the presence of metal grain boundaries. The effective permittivity of the gold can be calculated via the equation [48] ε ε ω τ ω ωτ ωτ ( ) ( )( ) where L = 35.7 nm is the mean free path of the electrons in bulk, ω p = 13.7 × 10 15 Hz is the plasma frequency, τ = 2.53 × 10 −14 s is the relaxation time. The effective mean free path R restricted by the effects of the structure, was estimated from the extinction spectra measurements, individually for each of the studied samples.
Optical Characterization: The dispersion effects introduced by the metamaterial were measured using an adapted version of the spatial encoded arrangement for temporal analysis by dispersing a pair of light E-fields technique (SEA TADPOLE). [59] In the SEA TADPOLE, the information about the spectral-phase difference between two optical pulses, traveling through different optical paths, is coded in spatial fringes along the vertical axis of the spectrometer camera, while the light is spectrally resolved along the horizontal axis. In contrast to standard spectral interferometry, in the SEA TADPOLE the spectral resolution of the measurement is not reduced, and it is equal to the resolution of the spectrometer. In the setup ( Figure S1, Supporting Information) the light from the supercontinuum laser (Fianium) is divided by the beam-splitter into the reference beam and the probe beam. The probe beam passes through the sample and thus its phase carries the information about the dispersion effects introduced by the metamaterial. The presence of the delay line ensures that the probe and reference beam, crossing at a small angle at the entrance of an imaging spectrometer (ISOPLANE 320, Pixis 256), are overlapping both in time and space. The generated spatial fringes (insert in Figure S1, Supporting Information) can be described by the equation: ω ω ω ω ω θ ϕ ω ϕ ω ( )