## Abstract

Previously, the effect of pulse bandwidth compression or broadening was observed in reflection from a moving front together with the Doppler shift. In this letter, an approach is presented, which alters pulse bandwidth without change in the central frequency. It occurs when light is reflected from a moving front of an otherwise stationary photonic crystal. This means that the photonic crystal lattice as such is stationary and only its boundary to the environment is moving, thus extruding (or shortening) the photonic crystal medium. The compression (broadening) factor depends on the front velocity and is the same as for the conventional Doppler shift. Complete reflection and transformation of the pulse can be achieved even with weak refractive index contrast, what makes the approach experimentally viable.

© 2014 Optical Society of America

## 1. Introduction

Reflection of an electromagnetic wave from a moving interface has been of interest for many years. Doppler effects occurring in the case of moving mirrors [1], propagating dielectric interfaces [2], and ionization fronts [3–6] have been theoretically and experimentally investigated by many authors. The frequency and bandwidth of an electromagnetic wave reflected from a relativistically moving mirror against the incident wave are increased by a factor $\left({c}_{0}+v\right)/\left({c}_{0}-v\right)$. In analogy with the moving mirror, a relativistically moving ionization front imposes the same Doppler up-shift and pulse compression upon the reflected wave [7].

For the realization of a relativistically moving mirror several techniques can be employed. The simplest method is to use a moving dielectric interface. However, depending on the technique employed to generate this interface, the expected difference of refractive indexes is in the order of 10^{−3}. Therefore, the reflection efficiency is very small. Another technique consists in using a plasma mirror. But the shorter the wavelength, the more energetically demanding is the realization of an effective plasma mirror due to the requirements of high concentration of free carriers. Experimentally, it was shown that a relativistically counter propagating plasma produced by a high power laser can cause a frequency multiplication factor exceeding 50 [8]. Experiments of Kiefer, et al. demonstrated the creation of relativistic electron bunches as reflectors for visible light by `blowing out' electrons from a freestanding, nanometre-scale thin foil with a high intensity laser pulse [9]. However, the electron density still was not sufficient to achieve a perfect mirror reflection.

Alternatively, strong reflection can be achieved from a moving front of a photonic crystal. A fundamentally different physical phenomenon of the inverse Doppler effect was presented in photonic crystals [10–12] and in periodic electrical transmission lines [13]. The previous works have concentrated on the shift of the center frequency in systems with non-relativistically moving fronts. We on the contrary concentrate on the bandwidth compression and broadening in systems with relativistically moving fronts. For this purpose we develop a graphical representation of the signal transformation involving the band diagram of a photonic crystal. The effect of adiabatic transformation described here directly follows from this representation. Also the condition for the light transmission into the photonic crystal is simple to obtain. Efficient reflection and large bandwidth variations without shift of the center frequency is predicted. FDTD simulations support the predictions of this model.

We considered the reflection of a wave from a moving front of a Bragg stack which moves with velocities close to the speed of light. We observed pulse broadening in the case of identical directions of the incident wave and the moving front and pulse compression when the front moves against the incident wave. This phenomenon cannot be explained by the traditional theory of Doppler effect, as it happens without change of the center frequency. Here we use the phase continuity condition and the photonic band diagram to explain the effect. A rigorous analytic solution of Maxwell's equations using methods developed for nonstationary media [14,15] are not required in this case.

## 2. Simulation methods

To explore the phenomenon one dimensional simulations with the finite-difference time-domain method (FDTD) were performed. In the FDTD algorithm the material parameters are assumed to be time-dependent, i.e. the second component on the right side in the Eq. (1) is not zero.

Time varying media are often simulated by the change of permittivity in the first component on the right side and still neglecting the second part [16,17]. It leads to reliable results for slowly varying parameters. However, for fast changes of permittivity, its time derivative should be considered in the simulations. The time derivative of the permittivity in the equation is mathematically equivalent to conductivity if we exchange $d\epsilon /dt$ with $\sigma $. In order to consider a time derivative of the permittivity we extended the FDTD algorithm in a similar way as the conventional conductivity is taken into account [18]. See the appendix of this article for a comparison between conventional and modified FDTD algorithm.

## 3. Phase continuity

To fulfil continuity of the electromagnetic fields at the moving dielectric front the condition of phase continuity at the front is required [4]: ${\omega}_{i}t-{k}_{i}r={\omega}_{re}t-{k}_{re}r$, where $r={v}_{f}t$, ${v}_{f}$ is front velocity, ${\omega}_{i}$, ${\omega}_{re}$ are frequencies of incident and reflected waves, ${k}_{i},{k}_{re}$ are wave numbers of incident and reflected waves. This relation can be derived as follows. Four vectors of frequency and coordinate in Minkowski space are ${\omega}^{(4)}=\left(\omega /c,k\right)$ and ${r}^{(4)}=\left(ct,r\right)$. In the frame moving with the front the frequency is conserved and front position can be fixed at $r=0$, thus the phase ${\omega}^{(4)}{r}^{(4)}$ is continuous. Due to invariance of dot-multiplication to linear coordinate transformation, it follows that relation ${\omega}_{i}^{(4)}{r}^{(4)}={\omega}_{re}^{(4)}{r}^{(4)}$ is true in any frame. Then, the same is true for the stationary frame. Phase continuity can be written as:

Thus, after interaction with the front only the modes exist that fulfill the condition (2a) which means that only frequencies can be observed for which the mode dispersion function intersects with the phase continuity line. ${\omega}_{re},{k}_{re}$ are obtained from these intersection points. Phase continuity line (2a) has the slope of the front velocity ${v}_{f}$ [4]. In addition, these intersections predict the frequencies of transmission. The intersection point belongs to transmission if the group velocity of the mode is larger than the front velocity.

We extend now the condition developed for homogeneous media to photonic crystals. Due to the fact that the Bloch wave in the photonic crystal contains wave vectors equal to$k+G$, where $G$ is the reciprocal vector equal to an integer multiple of $2\pi /a$ and $a$ is the lattice constant, the phase continuity line is duplicated by the $G$ vector $\omega ={\omega}_{i}+\left(k-{k}_{i}+G\right){v}_{f}$. In other words, the Bloch wave can be presented as a sum of discrete plane waves, each of them fulfilling the phase continuity condition. It also leads to the fact that in the frame moving with the front multiple frequencies will be generated.

The configuration of the Bragg stack is chosen to maximize the normal-incidence band gap. This is satisfied by the quarter-wave stack condition ${n}_{1}{d}_{1}={n}_{2}{d}_{2}$, where ${n}_{1}$ and ${n}_{2}$ are the refractive indices and ${d}_{1}$, ${d}_{2}$ are the thicknesses of dielectric layers. The refractive index ${n}_{i}$of the surrounding material where incident and reflected waves are propagating is selected such that ${n}_{1}{d}_{1}+{n}_{1}{d}_{2}=\left({d}_{1}+{d}_{2}\right){n}_{i}$. In this case, the dispersion line of the incident plane waves crosses the center of the band gap. This condition also maximizes the bandwidth available for pulse compression and broadening. The dispersion of the constituent materials is not taken into account.

A moving front can be described by the time varying permittivity. Mathematically the moving front of stationary one dimensional photonic crystal with periodic permittivity $\Delta \epsilon \left(x\right)$ is described by the following function:

The front is moving with velocity ${v}_{f}$. The geometric extension of the front is defined by the parameter $\gamma $. Two cases exhibiting different geometric extension of the front were considered. In the case of a sharp front the reflected wave contains equally spaced frequency bands, as it can be seen in Fig. 1(a). The frequency bands in the reflected wave correspond to the points where the phase continuity lines cut the dispersion line accountable for the backwards propagating wave. If the front width is much larger than the wavelength, an adiabatic transition takes place which slowly converts the incident mode into the reflected mode. No satellite frequencies appear in this case (Fig. 1(c)).

The presented theoretical predictions were confirmed by FDTD simulations. We have chosen the incident wavelength of light at 1.55 µm what corresponds to *f _{0}* = 193.5 THz in frequency. The values of refractive index vary from 3.07 to 4.07, the refractive index of the surrounding medium is ${n}_{i}=3.5$, which corresponds to silicon. Such a large variation of refractive indexes is used to reduce the simulation time. However, the conclusion drawn is also valid for cases of realistic figures of refractive index changes. The lattice constant a = 0.22 µm is chosen to open the band gap at the incident frequency.

Figure 1(a) depicts the dispersion relation of the incident and reflected waves and its intersections with phase continuity lines. Multiple phase continuity lines with the period of $G$ are shown. The front is propagating against the incident wave, thus the phase continuity lines have negative slope. Figure 1(b) shows the results for the sharp front ($\gamma =20\text{\hspace{0.17em}}{\text{\mu m}}^{\text{-1}}$) where equally spaced frequency bands with decreasing amplitudes are observed. The frequency bands correspond to intersections of the phase continuity lines with the reflected wave dispersion line. Higher order modes are excited due to the nonadiabatic transition. Still most of the energy is reflected without frequency shift. This can be explained by the fact that a photonic crystal interface is always smoothed by the finite penetration depth of the wave at the band gap frequency. If the front velocity is one tenth of the speed of light in vacuum, the frequency period is $\Delta f={f}^{\prime}-{f}_{0}={f}_{0}\cdot 2{v}_{f}(c/{n}_{i}-{v}_{f})=$$208\text{\hspace{0.17em}}\text{THz}$, where *f’* is the Doppler shifted frequency. Figure 1(c) demonstrates the simulation results for the smooth front with an extension of $\gamma =1\text{\hspace{0.17em}}{\text{\mu m}}^{\text{-1}}$, where no additional frequency bands are observed. As the reflection occurs from a continuously rising contrast of dielectric permittivity, the transition to the reflected state takes place adiabatically and the efficiency with which other frequency bands are excited is negligible.

## 4. Pulse compression and broadening

Now we will consider an adiabatic case and will look at the bandwidth modulation. We have reflection in the first band, thus, the phase continuity line shifted by $-2\pi /a$ defines the reflection frequency. To demonstrate graphically what happens with the bandwidth we have now shifted the dispersion line of the reflected wave to the right by $+2\pi /a$.

Figure 2 is the schematic representation of the frequency range broadening or, correspondingly, pulse length compression. The center frequency of the incident wave lies at the center of the photonic band gap. The front is moving towards the wave with the velocity${v}_{f}$. Thus the incident wave's spectral range is transferred from the red dispersion line with the positive slope to the green one with a negative slope along both gray phase continuity lines defined by the front velocity. The change of the frequency width is determined by the intersections between the dispersion line of the reflected wave and the phase continuity lines. From geometrical considerations the same broadening factor $\left(1+{v}_{f}/{v}_{re}\right)/\left(1-{v}_{f}/{v}_{re}\right)$ is obtained as by normal Doppler shift but in our case no central frequency shift is observed. The reflected bandwidth can be even larger than the band gap width as long as the phase continuity line does not intersect with the photonic crystal dispersion line. Otherwise photonic crystal modes will be excited and transmission into the photonic crystal is expected. We had shifted the dispersion line, thus, the reflected wave vector appears localized around the $-\pi /a$ point. From the graphical representation we can see that the center frequency of the pulse is not changed due to reflection.

Figure 3 demonstrates the case when the incident wave and the front propagate in the same direction. The input bandwidth can be wider than the band gap as long as the phase continuity lines do not cut the photonic crystal dispersion function. The reflection frequency range is narrowed. Here we will explain the effect of adiabatic transition. The dotted lines represent the band gap opening as the interaction of the smooth front of the photonic crystal and the incident wave gradually increases. If at time ${t}_{0}$ the incident wave strikes the interface it experiences low contrast of refractive indexes. At the time ${t}_{1}>{t}_{0}$ the wave has ‘penetrated’ further inside the photonic crystal hence the perceived index contrast has increased, therefore the band gap became wider. Thus, the incident wave undergoes an adiabatic change of its frequency components and wave vector spectrum, following the intersection point of the phase continuity lines with the two branches of the photonic crystal dispersion function. At some point in time the signal wave group velocity becomes smaller than the front velocity. This is a point of maximal signal penetration into the photonic crystal. Afterwards the signal reduces its velocity further on, reverses it and reflects back into the incident medium, now the perceived index contrast decreasing. The frequency spectrum is thus varying continuously by going through intermediate stages on the dotted lines.

The compression (broadening) factor entirely depends on the front velocity and on the condition that the phase continuity line does not cut the dispersion function of the bulk photonic crystal behind its front i.e. when the penetrating wave interacts with the full index contrast of the photonic crystal. FDTD simulations for broadening and compression of the bandwidth are presented in Fig. 4. The front velocity for bandwidth broadening is equal to $-0.1{c}_{0}$. The broadening factor obtained in the simulations is 2.07, which is close to the theoretical factor of 2.10. The front velocity for bandwidth compression is equal to $0.1{c}_{0}$ which results in the compression factor of 1/2.11. This is also very close to the theoretical value of 1/2.10. No center frequency shift is observed as predicted by the model. Gaussian shape of the frequency distribution is also conserved.

The effect discussed here does not depend on the refractive index contrast of the photonic crystal. With decrease of the index contrast the band gap shrinks which implies a narrowing of the incident spectral range. The observed phenomenon can be obtained for a realistic value of refractive index difference of 10^{−3}. In this case, arbitrary compression and broadening of the input bandwidths around 35 GHz is possible.

The experimental realization of the proposed model of a growing periodic structure with velocities close to the speed of light in a medium can be achieved by exploiting the electro-optical effect [19–22].The electro-optical modulator can realize the ideal case of symmetrical photonic crystal if a bias is used to achieve positive and negative refractive index change. It should be structured periodically similar to [23] but with a periodicity of 250 nm. Alternatively a nonlinear optical pulse propagating in a photonic crystal waveguide can be used to generate a front similar to our work in the reference [24]. The effect is present for any dispersion relations close to an anti-crossing point of modes with opposite group velocities. The adiabatic transition can be obtained by increasing the interaction of the modes gradually.

## 5. Conclusion

This work introduces a new method to vary the pulse duration and bandwidth without an accompanying frequency shift of the reflected pulse. A graphical representation is developed that explained frequency and wave vector variations of the signal pulse. Based on theoretical considerations the moving front can broaden and compress pulses with different factors depending on the velocity of the front. The theoretical predictions were confirmed by FDTD simulations. The effect can be demonstrated even with low refractive index contrasts obtained via the electro-optical effect or nonlinear pulse propagation.

## Appendix

To prove the reliability of the modified FDTD algorithm we considered a moving dielectric front and analyzed the dependency of the reflected wave amplitude on the front velocity. The value of discretization in these simulations was taken to be $\lambda /20$, where $\lambda $ is the wavelength of the reflected wave. The results were then compared with the theoretical prediction. According to Tsai and Auld [2], for a moving dielectric interface in a stationary medium, the relation between the amplitudes of reflected and incident waves is given by the Eq. (4).

where ${v}_{f}$ is the front velocity, ${v}_{re}$ is the velocity of incident and reflected waves in the medium, $R$ is Fresnel amplitude reflection coefficient.In Fig. 5 a comparison between the reflection coefficients obtained with conventional and modified FDTD algorithms, and the theoretical relation of Eq. (4) is presented. The results obtained in simulations with the modified algorithm correspond to the theoretical dependence. It should be mentioned that even the unmodified algorithm, where $d\epsilon /dt$ is not taken into account, predicts the correct Doppler shift. But the reflection coefficient is wrong in this case.

## Acknowledgments

This publication was supported by the German Research Foundation (DFG) under Grant EI 391/13-2 and the Hamburg University of Technology (TUHH) in the funding program “Open Access Publishing”.

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