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SECTION 1

Introduction

The principle of causality forbids the movement of any object at a speed exceeding that of light in a vacuum Formula$(c)$. However, various theoretical and experimental studies demonstrated that faster-than-Formula$c$ 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 Formula$c$ [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 Formula$c$ by changing only the initial field configuration (by using an external modulation) without any changes in the physical aspects of the waveguide.

SECTION 2

Waveguide Mode-Airy Wavepackets

Let us consider an electromagnetic wavepacket launched to a waveguide along the Formula$+z$ 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 Formula$\phi = A_{T}(t, z)g(y)\exp[j(\beta_{0} z - \omega_{0} t)]$, where Formula$\omega_{0}$ is the carrier frequency of the wavepacket, and Formula$g(y)$ describes the transverse profile of a waveguide mode with a wavenumber of Formula$\beta_{0}$ at Formula$\omega_{0}$. Formula$\phi$ can be Formula$E_{x}$ and Formula$H_{x}$ field components for TE- and TM-polarized light, respectively. Assuming Formula$A_{T}$ is a slowly varying function of Formula$z$ (i.e., with the paraxial approximation) and including the dispersive property of the wavenumber Formula$\beta$ 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 Formula$\beta$ as was in [16]), we can easily show that Formula$A_{T}(t, z)$ can be a nonspreading Airy wavepacket given by Formula 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 Formula$\tau_{0}$ is an arbitrary scaling factor, and Formula$m$ is either 0 or 1 and related to the acceleration direction of the wavepacket (see (2), shown below, and the related discussion for details). Formula$v_{g}\ [= (\partial\beta/\partial\omega\vert_{\omega_{0}})^{-1}]$ and Formula$\beta_{2}\ [= \partial^{2}\beta/\partial\omega^{2}\vert_{\omega_{0}}]$ denote the group velocity and the group-velocity dispersion coefficient at Formula$\omega_{0}$, respectively, and a transformation of Formula$\tau = t - (z/v_{g})$, where Formula$\tau$ represents time in a reference frame moving at Formula$v_{g}$ 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 Formula$\tau_{0}$ as Formula$(-1)^{m}\tau_{0}$ (to explicitly consider both its positive and negative values) and set the parameter related to the third-order dispersive property Formula$(b)$ equal to zero, we can immediately obtain this expression.

Figure 1
Fig. 1. Concept of a superluminal waveguide mode-Airy wavepacket. Formula$g(y)$ and Formula$A_{T}(t, z)$ denote the transverse profile of the waveguide mode at the carrier frequency Formula$\omega_{0}$ and the conventional Formula$(1 + 1)D$ Airy wavepacket profile, respectively.

The initial field Formula$A_{T}(t, 0)$ (at Formula$z = 0$ we have Formula$t = \tau$) is given as Formula$Ai[(-1)^{m}t/\tau_{0}]\exp[(-1)^{m}(a^{\prime}t/ \tau_{0}\ - ja^{\prime\prime}t/\tau_{0})]$, where Formula$a^{\prime}$ denotes the exponential apodization parameter which must satisfy Formula$(-1)^{m}a^{\prime}\ >\ 0$ for the finite-energy (physically realizable) Airy wavepacket, and Formula$a^{\prime\prime}$ 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 Formula$a^{\prime} = 0$. Due to nonzero values of Formula$a^{\prime}$, 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 Formula$+z$ direction, its peak position shifts laterally along the Formula$\tau$ direction. The peak of the Airy wavepacket follows the following spatiotemporal trajectory Formula$(\tau_{p}, z_{p})$ which can be obtained by setting the real arguments of the Airy function in (1) to be zero Formula 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 Formula$+\tau$ or Formula$-\tau$ direction, depending on whether Formula$m$ 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

Superluminal Transmission of Hybrid Wavepackets

From the second feature mentioned above, we can see that the Airy wavepacket is either accelerated (when Formula$m = 1$) or decelerated (when Formula$m = 0$) 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 Formula$m = 1$. We have Formula 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 Formula$(t, z_{p})$ of the peak of the hybrid wavepacket can be written as Formula 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 Formula$\alpha = a^{\prime \prime}\beta_{2}v_{g}/\tau_{0}$. By the differentiation with respect to Formula$t$, we can obtain the speed of the hybrid wavepacket as Formula 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 Formula$v_{0} = v_{g}/\vert 1 - \alpha\vert$, and Formula$T_{m} = \tau_{0}^{3}/(\beta_{2}^{2}v_{0}^{2})$. Above results hold only when Formula$t\ <\ T_{m}$. Otherwise, (3) does not have a root for a given time Formula$t$, meaning that the temporal profile of the wavepacket no longer maintains its original peak or overall shape. Therefore, we can say that Formula$T_{m}$ 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 Formula$1 - \alpha$ must be positive for the forward propagation of the wavepacket in the region of our interest (Formula$z_{p} \geq 0$ because our initial field is assumed to be launched at Formula$z = 0$). The initial launching speed of the hybrid wavepacket is given as Formula$v_{0} = v_{g}/ (1 - \alpha)$, which can be easily changed by controlling Formula$\alpha$ or Formula$a^{\prime\prime}$. By making Formula$\alpha$ close to 1, we can enhance the initial launching speed significantly. If we write Formula$v_{g} = c/n_{g}$, we can obtain a superluminal wavepacket by making Formula$\alpha\ >\ 1 - n_{g}^{-1}$, that is, Formula$a^{\prime\prime}\ >\ \tau_{0}(n_{g} - 1)/(\beta_{2}c)$ when Formula$\beta_{2}\ >\ 0$ or Formula$a^{\prime\prime}\ < \tau_{0}(n_{g} - 1)/(\beta_{2}c)$ when Formula$\beta_{2}\ <\ 0$. The latter condition cannot be satisfied since it requires Formula$n_{g}\ <\ 1$ for a nonnegative Formula$a^{\prime\prime}$. In addition, Formula$v_{p}(t)$ is a monotonically increasing function with respect to Formula$t$, that is, this hybrid wavepacket always moves faster than the initial velocity Formula$v_{0}$ 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 Formula$v_{0}\ <\ c$, Formula$v_{p}(t)$ becomes superluminal when Formula$t\ >\ \tau_{0}^{3}[n_{g}^{2}(1 - \alpha)^{2} - 1]/(\beta_{2}^{2} c^{2})$.

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

SECTION 4

Numerical Study Using a Metamaterial Waveguide

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

Figure 2
Fig. 2. (a) Propagating profiles (at Formula$z = 0$, Formula$400\lambda_{0}$, Formula$800 \lambda_{0}$, Formula$1200\lambda_{0}$, Formula$1600\lambda_{0}$, and Formula$2000 \lambda_{0}$) of the hybrid wavepacket at the center of the waveguide Formula$(y = 0)$ from the view point in which the transverse direction indicates time in a moving frame Formula$(\tau)$. The acceleration along the Formula$-\tau$ direction can be interpreted as a speedup of the wavepacket. (b) Propagating profiles (at Formula$z = 0$ and Formula$2000\lambda_{0}$) from the view point in which the transverse direction indicates time Formula$(t)$. The average speed of the hybrid wavepacket in its peak position is increased as high as Formula$1.48c$, although Formula$v_{g}$ of the waveguide mode is only about Formula$0.072c$. Two additional factors contribute to this speedup: the initial launching speed of the hybrid wavepacket and its sequential acceleration along the time domain in a moving frame (see Fig. 3 for details).

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

Figure 3
Fig. 3. Speedup of the hybrid wavepacket by increasing Formula$a^{\prime\prime}$ or Formula$\alpha$, i.e., by enhancing the initial launching speed (compare the cases of Formula$a^{\prime\prime} = 0$ and Formula$a^{\prime\prime} = 60.08$) and due to its acceleration as it propagates through the waveguide.
SECTION 5

Conclusion

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- Formula$c$ 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.

Footnotes

This work was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science, and Technology under Grant 2011-0011360. Corresponding author: K.-Y. Kim (e-mail: kykim@sejong.ac.kr).

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Kyoung-Youm Kim

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