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SECTION 1

The principle of causality forbids the movement of any object at a speed exceeding that of light in a vacuum $(c)$. However, various theoretical and experimental studies demonstrated that faster-than-$c$ propagation of light pulses can be realized utilizing specific dispersion characteristics of materials in the vicinity of resonance frequencies. The speed mentioned here (and throughout this paper) is, of course, different from the signal velocity at which information travels, which is defined as that of the front of a square pulse and cannot exceed $c$ [1]. A gain doublet [1], [2], [3], electromagnetically induced absorption [4], coherent population oscillation [5], and negative-index metamaterials [6] have been studied to implement such desired dispersion properties. Recently, several researchers reported superluminal propagation of optical pulses through various waveguide structures using highly birefringent optical fibers [7], nonlinear processes [8], or the resonance of ring resonators [9]. In this paper, we would like to put forward a new strategy of superluminal transmission of optical wavepackets through waveguides. The key idea is to adopt an Airy wavepacket [10], [11], [12], [13], [14], [15], [16], [17], [18] as the initial field launched to a waveguide. Then, the launched Airy wavepacket becomes associated with a waveguide mode and forms a type of hybrid wavepacket. This resultant wavepacket propagates like an optical bullet without any spatial diffraction or temporal spreading [16]. Its effective propagation speed can be easily enhanced higher than $c$ by changing only the initial field configuration (by using an external modulation) without any changes in the physical aspects of the waveguide.

SECTION 2

Let us consider an electromagnetic wavepacket launched to a waveguide along the $+z$ direction (see Fig. 1). We will limit our attention only to 1-D waveguides in this paper. However, the extension to higher dimensional ones is quite straightforward. The wavepacket can be written as $\phi = A_{T}(t, z)g(y)\exp[j(\beta_{0} z - \omega_{0} t)]$, where $\omega_{0}$ is the carrier frequency of the wavepacket, and $g(y)$ describes the transverse profile of a waveguide mode with a wavenumber of $\beta_{0}$ at $\omega_{0}$. $\phi$ can be $E_{x}$ and $H_{x}$ field components for TE- and TM-polarized light, respectively. Assuming $A_{T}$ is a slowly varying function of $z$ (i.e., with the paraxial approximation) and including the dispersive property of the wavenumber $\beta$ up to the second order (note that the physics investigated in this paper does not change much when we include the third-order dispersive property of $\beta$ as was in [16]), we can easily show that $A_{T}(t, z)$ can be a nonspreading Airy wavepacket given by TeX Source $$\displaylines{A_{T}(\tau, z) = Ai\left[(-1)^{m}{\tau \over \tau_{0}} - \left({\beta_{2}z \over 2\tau_{0}^{2}}\right)^{2} - j{a\beta_{2}z \over \tau_{0}^{2}}\right]\exp\left[-j \left({(-1)^{m}\tau + a^{2}\tau_{0} \over 2\tau_{0}}{\beta_{2}z \over \tau_{0}^{2}} - {1 \over 12}\left({\beta_{2}z \over \tau_{0}^{2}}\right)^{3}\right)\right]\hfill\cr\hfill\times\ \exp\left[(-1)^{m}{a\tau \over \tau_{0}} - {a \over 2}\left({\beta_{2}z \over \tau_{0}^{2}}\right)^{2}\right]\quad\hbox{(1)}}$$ where $\tau_{0}$ is an arbitrary scaling factor, and $m$ is either 0 or 1 and related to the acceleration direction of the wavepacket (see (2), shown below, and the related discussion for details). $v_{g}\ [= (\partial\beta/\partial\omega\vert_{\omega_{0}})^{-1}]$ and $\beta_{2}\ [= \partial^{2}\beta/\partial\omega^{2}\vert_{\omega_{0}}]$ denote the group velocity and the group-velocity dispersion coefficient at $\omega_{0}$, respectively, and a transformation of $\tau = t - (z/v_{g})$, where $\tau$ represents time in a reference frame moving at $v_{g}$ has been introduced. We note that the above equation seems to be much different from [16 eq. (6)], which also describes a nonspreading Airy wavepacket, but if, in [16 eq. (6)], we write $\tau_{0}$ as $(-1)^{m}\tau_{0}$ (to explicitly consider both its positive and negative values) and set the parameter related to the third-order dispersive property $(b)$ equal to zero, we can immediately obtain this expression.

The initial field $A_{T}(t, 0)$ (at $z = 0$ we have $t = \tau$) is given as $Ai[(-1)^{m}t/\tau_{0}]\exp[(-1)^{m}(a^{\prime}t/ \tau_{0}\ - ja^{\prime\prime}t/\tau_{0})]$, where $a^{\prime}$ denotes the exponential apodization parameter which must satisfy $(-1)^{m}a^{\prime}\ >\ 0$ for the finite-energy (physically realizable) Airy wavepacket, and $a^{\prime\prime}$ is the phase modulation factor along the scaled temporal coordinate which should be nonnegative to avoid unphysical solutions. From the above discussion, we can see that a type of hybrid wavepacket, which is given by the product of a waveguide mode and a finite Airy wavepacket, can propagate through the waveguide.

Here, we would like to point out two unique features of the Airy wavepacket given by (1). One is that it undergoes neither spatial diffraction nor temporal spreading if $a^{\prime} = 0$. Due to nonzero values of $a^{\prime}$, the actual finite-energy Airy wavepacket diffracts or spreads a little bit. This means that the waveguide mode-Airy wavepacket suffers minimal diffraction or spreading, propagating through the waveguide like an optical bullet. The other is that during propagation along the $+z$ direction, its peak position shifts laterally along the $\tau$ direction. The peak of the Airy wavepacket follows the following spatiotemporal trajectory $(\tau_{p}, z_{p})$ which can be obtained by setting the real arguments of the Airy function in (1) to be zero TeX Source $$(-1)^{m}\tau_{p} = {a^{\prime\prime}\beta_{2}\over \tau_{0}}z_{p} + {1\over 2} {\beta_{2}^{2} \over 2\tau_{0}^{3}}z_{p}^{2}\eqno{\hbox{(2)}}$$ where we can find that, during propagation, the peak position shifts along the $+\tau$ or $-\tau$ direction, depending on whether $m$ is 0 or 1. These shifts along the positive and negative time coordinates in a moving frame can be interpreted as the slowdown and speedup of the wavepacket, respectively.

SECTION 3

From the second feature mentioned above, we can see that the Airy wavepacket is either accelerated (when $m = 1$) or decelerated (when $m = 0$) during propagation. Since the Airy part corresponds to the temporal envelope of the hybrid wavepacket, the transmission speed of the Airy wavepacket determines that of the waveguide mode-Airy hybrid one. As we want *fast* transmission of optical wavepackets, we will limit our attention to the case of $m = 1$. We have
TeX Source
$${\beta_{2}^{2} \over 2\tau_{0}^{3}}z_{p}^{2} + 2\left({a^{\prime \prime}\beta_{2} \over \tau_{0}} - {1 \over v_{g}} \right)z_{p} + 2t = 0\eqno{\hbox{(3)}}$$ and the trajectory $(t, z_{p})$ of the peak of the hybrid wavepacket can be written as
TeX Source
$$z_{p}(t) = {2\tau_{0}^{3} \over \beta_{2}^{2}} \left({1 - \alpha \over v_{g}} - \sqrt{\left({1 - \alpha \over v_{g}}\right)^{2} - {\beta_{2}^{2} t \over \tau_{0}^{3}}}\right)\eqno{\hbox{(4)}}$$ where $\alpha = a^{\prime \prime}\beta_{2}v_{g}/\tau_{0}$. By the differentiation with respect to $t$, we can obtain the speed of the hybrid wavepacket as
TeX Source
$$v_{p}(t) = \left[\left({1 - \alpha \over v_{g}}\right)^{2} - {\beta_{2}^{2} t \over \tau_{0}^{3}}\right]^{-1/2} = {v_{0} \over \sqrt{1 - t/T_{m}}}\eqno{\hbox{(5)}}$$ where $v_{0} = v_{g}/\vert 1 - \alpha\vert$, and $T_{m} = \tau_{0}^{3}/(\beta_{2}^{2}v_{0}^{2})$. Above results hold only when $t\ <\ T_{m}$. Otherwise, (3) does not have a root for a given time $t$, meaning that the temporal profile of the wavepacket no longer maintains its original peak or overall shape. Therefore, we can say that $T_{m}$ denotes the maximum time during which the hybrid wavepacket can be transported with minimal diffraction or spreading.

From the above equation, we can see that $1 - \alpha$ must be positive for the forward propagation of the wavepacket in the region of our interest ($z_{p} \geq 0$ because our initial field is assumed to be launched at $z = 0$). The initial launching speed of the hybrid wavepacket is given as $v_{0} = v_{g}/ (1 - \alpha)$, which can be easily changed by controlling $\alpha$ or $a^{\prime\prime}$. By making $\alpha$ close to 1, we can enhance the initial launching speed significantly. If we write $v_{g} = c/n_{g}$, we can obtain a superluminal wavepacket by making $\alpha\ >\ 1 - n_{g}^{-1}$, that is, $a^{\prime\prime}\ >\ \tau_{0}(n_{g} - 1)/(\beta_{2}c)$ when $\beta_{2}\ >\ 0$ or $a^{\prime\prime}\ < \tau_{0}(n_{g} - 1)/(\beta_{2}c)$ when $\beta_{2}\ <\ 0$. The latter condition cannot be satisfied since it requires $n_{g}\ <\ 1$ for a nonnegative $a^{\prime\prime}$. In addition, $v_{p}(t)$ is a monotonically increasing function with respect to $t$, that is, this hybrid wavepacket always moves faster than the initial velocity $v_{0}$ and speeds up with passing time. That is, the transmission speed of the hybrid wavepacket can be enhanced further during propagation, making its superluminal transmission a lot easier to realize: even if $v_{0}\ <\ c$, $v_{p}(t)$ becomes superluminal when $t\ >\ \tau_{0}^{3}[n_{g}^{2}(1 - \alpha)^{2} - 1]/(\beta_{2}^{2} c^{2})$.

One of the most important parameters in our scheme is $T_{m}$. If we put $T_{m} = N\tau_{0}$, we have $\tau_{0}^{2}(1 - \alpha)^{2} = N(\beta_{2}v_{g})^{2}$. Another crucial one is $\alpha$ which can be written as $\alpha = (a^{\prime\prime}/\tau_{0})\beta_{2}v_{g}$ and determines the initial launching speed of the wavepacket. Therefore, we can see that $\beta_{2}v_{g}$, whose value is determined by the waveguide structure (materials and geometry), plays a key role in our scheme. From above results, we can obtain $\tau_{0} = \sqrt{N}\beta_{2}v_{g}/(1 - \alpha)$ and $a^{\prime \prime} = \sqrt{N}\alpha/(1 - \alpha)$. If the waveguide gives us $\beta_{2}v_{g}$ and we select appropriate values of $\alpha$ and $N$, then we can determine two design parameters $\tau_{0}$ and $a^{\prime\prime}$ using the above relations.

SECTION 4

Let us examine the above discussions in more detail using a numerical example. We should mention that since we require a positive $\alpha$ (close to 1), $\beta_{2}$ must be positive as well. That is, anomalous dispersion $(\partial v_{g}/\partial\omega\ <\ 0)$ is necessary in our scheme. For this purpose, we adopted a metamaterial waveguide: a slab composed of a negative-index core between silica clads. The negative-index metamaterial was 0.55 $\mu\hbox{m}$ thick and its relative permittivity and permeability were given as $\varepsilon_{co} = -4$ and $\mu_{co} = -2$ at $\lambda_{0} = 1550\ \hbox{nm}$. We used the Sellmeier equation and the Drude model to take into account the dispersive property of silica and metamaterial layers, respectively. At $\omega_{0}\ (= 2\pi c/\lambda_{0})$, we obtained a backward guiding mode having $\beta_{0} = -1.6372 k_{0}\ (k_{0} = 2 \pi/\lambda_{0})$, $v_{g} = c/13.91$, and $\beta_{2} = 5.666 \times 10^{-21}$. In Fig. 2(a), we showed the intensity profiles of the hybrid wavepacket at the center of the negative-index core $(y = 0)$ for various longitudinal $(z)$ positions. We set $N = 10$, $\alpha = 0.95$, $a = 0.05 - j60.08$, and $\tau_{0} = 7.72\ \hbox{ps}$. From the results, we can easily find that the peak of the hybrid wavepacket was accelerated along the $-\tau$ direction. In addition, the wavepacket underwent little diffraction or spreading even after it propagated $2000\lambda_{0}$. In Fig. 2(b), we plotted these results again: in this case using the time coordinate $(t)$ instead of that in a moving frame $(\tau)$. Here, we can see that the hybrid wavepacket in its peak position traveled $2000\lambda_{0}$ in 7 picoseconds, resulting in an average speed of $1.48c$. If we note that the group velocity of the waveguide mode itself is only about $0.072c$, the transmission speed is enhanced significantly via the introduction of the Airy wavepacket and its association with the waveguide mode.

In Fig. 3, we plotted further the calculated speed of the hybrid wavepacket as it propagates through the negative-index-core slab. We can identify two mechanisms which enhance the transmission speed of the hybrid wavepacket. One is via the initial launching speed: by controlling $a^{\prime\prime}$ or $\alpha$ which means physically the additional modulation of the initial field profile. The initial field of the waveguide mode-Airy wavepacket at $z = 0$ can be written as $A_{T}(t, 0; a^{\prime\prime}) = A_{T}(t, 0; a^{\prime\prime} = 0)\ \times \exp(ja^{\prime\prime}t/\tau_{0})$. Therefore, we can discern that by modulating $A_{T}(t, 0; a^{\prime\prime} = 0)$, we can enhance the initial launching speed quite easily, which can be found in Fig. 3 by comparing the cases of $a^{\prime\prime} = 0$ and $a^{\prime\prime} = 60.08$. The other is due to the acceleration of the hybrid wavepacket along the $-\tau$ coordinate, which can also be identified in Fig. 3; as the hybrid wavepacket travels, its transmission speed increases.

SECTION 5

We showed that superluminal transmission of a dispersionless wavepacket can be realized via the introduction of the waveguide mode-Airy hybrid wavepacket. We have found two mechanisms which contribute to the faster-than- $c$ propagation of the hybrid wavepacket: i) the initial launching speed, which can be easily enhanced by changing the initial field configuration of the wavepacket, and ii) the acceleration feature of the Airy wavepacket along the time domain in a moving frame. We have considered a concrete example using a negative-index-core slab waveguide and proved superluminal transportation of the waveguide mode-Airy wavepacket through it numerically.

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