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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 11 — Jun. 2, 2014
  • pp: 13280–13287
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Pulse compression and broadening by reflection from a moving front of a photonic crystal

Elena A. Ulchenko, Dirk Jalas, Alexander Yu. Petrov, Michel Castellanos Muñoz, Slawa Lang, and Manfred Eich  »View Author Affiliations


Optics Express, Vol. 22, Issue 11, pp. 13280-13287 (2014)
http://dx.doi.org/10.1364/OE.22.013280


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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

1. H. E. Ives, “The Doppler effect from moving mirrors,” J. Opt. Soc. Am. 30(6), 255–257 (1940). [CrossRef]

], propagating dielectric interfaces [2

2. C. S. Tsai and B. A. Auld, “Wave interactions with moving boundaries,” J. Appl. Phys. 38(5), 2106–2115 (1967). [CrossRef]

], and ionization fronts [3

3. M. Lampe, E. Ott, and J. H. Walker, “Interaction of electromagnetic waves with a moving ionization front,” Phys. Fluids 21(1), 42–54 (1978). [CrossRef]

6

6. C. H. Lai, R. Liou, T. C. Katsouleas, P. Muggli, R. Brogle, C. Joshi, and W. B. Mori, “Demonstration of microwave generation from a static field by a relativistic ionization front in a capacitor array,” Phys. Rev. Lett. 77(23), 4764–4767 (1996). [CrossRef] [PubMed]

] 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 (c0+v)/(c0v). 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

7. V. I. Semenova, “Reflection of electromagnetic waves from an ionization front,” Radiophys. Quantum Electron. 10(8), 599–604 (1967). [CrossRef]

].

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

8. M. Kando, Y. Fukuda, A. S. Pirozhkov, J. Ma, I. Daito, L.-M. Chen, T. Zh. Esirkepov, K. Ogura, T. Homma, Y. Hayashi, H. Kotaki, A. Sagisaka, M. Mori, J. K. Koga, H. Daido, S. V. Bulanov, T. Kimura, Y. Kato, and T. Tajima, “Demonstration of laser-frequency upshift by electron-density modulations in a plasma wakefield,” Phys. Rev. Lett. 99(13), 135001 (2007). [CrossRef] [PubMed]

]. 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

9. D. Kiefer, M. Yeung, T. Dzelzainis, P. S. Foster, S. G. Rykovanov, C. L. S. Lewis, R. S. Marjoribanks, H. Ruhl, D. Habs, J. Schreiber, M. Zepf, and B. Dromey, “Relativistic electron mirrors from nanoscale foils for coherent frequency upshift to the extreme ultraviolet,” Nat. Commun. 4, 1763 (2013). [CrossRef] [PubMed]

]. However, the electron density still was not sufficient to achieve a perfect mirror reflection.

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

14. F. Biancalana, A. Amann, A. V. Uskov, and E. P. O’Reilly, “Dynamics of light propagation in spatiotemporal dielectric structures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(4), 046607 (2007). [CrossRef] [PubMed]

,15

15. A. B. Shvartsburg, “Optics of nonstationary media,” Phys.- Usp. 48(8), 797–823 (2005). [CrossRef]

] 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.

×H=dDdt=εdEdt+Edεdt
(1)

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

16. M. D. Thomson, S. M. Tzanova, and H. G. Roskos, “Terahertz frequency upconversion via relativistic Doppler reflection from a photoinduced plasma front in a solid-state medium,” Phys. Rev. B 87(8), 085203 (2013). [CrossRef]

,17

17. X. Liu and D. McNamara, “The use of the FDTD method for electromagnetic analysis in the presence of independently time-varying media,” Int. J. Infrared Millim. Waves 28(9), 759–778 (2007). [CrossRef]

]. 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 dε/dt with σ. 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

18. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2000).

]. 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

4. C. H. Lai, T. C. Katsouleas, W. B. Mori, and D. Whittum, “Frequency upshifting by an ionization front in a magnetized plasma,” IEEE Trans. Plasma Sci. 21(1), 45–52 (1993). [CrossRef]

]: ωitkir=ωretkrer, where r=vft, vf is front velocity, ωi, ωre are frequencies of incident and reflected waves, ki,kre 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 ω(4)=(ω/c,k) and r(4)=(ct,r). In the frame moving with the front the frequency is conserved and front position can be fixed at r=0, thus the phase ω(4)r(4) is continuous. Due to invariance of dot-multiplication to linear coordinate transformation, it follows that relation ωi(4)r(4)=ωre(4)r(4) is true in any frame. Then, the same is true for the stationary frame. Phase continuity can be written as:

ωre=ωi+(kreki)vf
(2a)
ΔωΔk=ωreωikreki=vf
(2b)

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. ωre,kre are obtained from these intersection points. Phase continuity line (2a) has the slope of the front velocity vf [4

4. C. H. Lai, T. C. Katsouleas, W. B. Mori, and D. Whittum, “Frequency upshifting by an ionization front in a magnetized plasma,” IEEE Trans. Plasma Sci. 21(1), 45–52 (1993). [CrossRef]

]. 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 tok+G, where G is the reciprocal vector equal to an integer multiple of 2π/a and a is the lattice constant, the phase continuity line is duplicated by the G vector ω=ωi+(kki+G)vf. 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 n1d1=n2d2, where n1 and n2 are the refractive indices and d1, d2 are the thicknesses of dielectric layers. The refractive index niof the surrounding material where incident and reflected waves are propagating is selected such that n1d1+n1d2=(d1+d2)ni. 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 Δε(x) is described by the following function:

ε(x,t)=εi+Δε(x)(111+exp(γ(xvft)))
(3)

The front is moving with velocity vf. The geometric extension of the front is defined by the parameter γ. 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)
Fig. 1 The front of photonic crystal is moving against the propagation of the incident pulse. (a) Dispersion relation for the surrounding homogeneous dielectric medium where both the incident (red line) and reflected (green line) waves propagate. The red dot corresponds to the incident wave and the green dots to the reflected waves. The gray lines correspond to phase continuity condition with slope vf=0.1c0 which appears multiple due to the series of allowed photonic crystal reciprocal lattice vectors. (b) Frequency dependent amplitude of the reflected pulse for a moving sharp front with γ=20μm1. The frequencies of the reflected wave correspond to the points of intersections between the dispersion relation of the surrounding medium and the phase continuity lines. (c) Frequency dependent amplitude of the reflected wave for a moving smooth front with a width of γ=1μm1. Due to the adiabatic transition, only a single frequency band is observed.
. 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 f0 = 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 ni=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 (γ=20μm-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 Δf=ff0=f02vf(c/nivf)=208THz, where f’ is the Doppler shifted frequency. Figure 1(c) demonstrates the simulation results for the smooth front with an extension of γ=1μm-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 2π/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 +2π/a.

Figure 2
Fig. 2 Schematic representation of spectral broadening when the front is moving against the propagation of the incident wave. The input signal (in red) has a narrow frequency range which lies inside the band gap of the photonic crystal. The reflected wave shows an altered frequency range (in green).
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 velocityvf. 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 (1+vf/vre)/(1vf/vre) 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 π/a point. From the graphical representation we can see that the center frequency of the pulse is not changed due to reflection.

Figure 3
Fig. 3 Schematic representation of spectral compression when the direction of the front motion coincides with the propagation direction of the incident wave. The input signal (in red) has a wide frequency range which lies inside the photonic band gap. The reflected wave has a narrowed frequency range (in green). The dotted lines depict the adiabatic change of the photonic bands in case of a smooth moving front.
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 t0 the incident wave strikes the interface it experiences low contrast of refractive indexes. At the time t1>t0 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
Fig. 4 (a) FDTD simulation of frequency range broadening (green) with respect to the incident spectral width (red). The front velocity is equal to 0.1c0. (b) FDTD simulation of frequency range compression (green) with respect to the incident spectral width (red). The front velocity is equal to 0.1c0.
. The front velocity for bandwidth broadening is equal to 0.1c0. 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 0.1c0 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

19. J. H. Wülbern, S. Prorok, J. Hampe, A. Petrov, M. Eich, J. Luo, A. K.-Y. Jen, M. Jenett, and A. Jacob, “40 GHz electro-optic modulation in hybrid silicon-organic slotted photonic crystal waveguides,” Opt. Lett. 35(16), 2753–2755 (2010). [CrossRef] [PubMed]

22

22. X. Gan, R.-J. Shiue, Y. Gao, K. F. Mak, X. Yao, L. Li, A. Szep, D. Walker Jr, J. Hone, T. F. Heinz, and D. Englund, “High-contrast electrooptic modulation of a photonic crystal nanocavity by electrical gating of graphene,” Nano Lett. 13(2), 691–696 (2013). [CrossRef] [PubMed]

].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

23. H. Lira, Z. Yu, S. Fan, and M. Lipson, “Electrically driven nonreciprocity induced by interband photonic transition on a silicon chip,” Phys. Rev. Lett. 109(3), 033901 (2012). [CrossRef] [PubMed]

] 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

24. M. Castellanos Muñoz, A. Y. Petrov, L. O’Faolain, J. Li, T. F. Krauss, and M. Eich, “Optically induced indirect photonic transitions in a slow light photonic crystal waveguide,” Phys. Rev. Lett. 112(5), 053904 (2014). [CrossRef] [PubMed]

]. 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 λ/20, where λ is the wavelength of the reflected wave. The results were then compared with the theoretical prediction. According to Tsai and Auld [2

2. C. S. Tsai and B. A. Auld, “Wave interactions with moving boundaries,” J. Appl. Phys. 38(5), 2106–2115 (1967). [CrossRef]

], 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).
EreEi=1+vf/vre1vf/vreR
(4)
where vf is the front velocity, vre is the velocity of incident and reflected waves in the medium, R is Fresnel amplitude reflection coefficient.

In Fig. 5
Fig. 5 Relations between electric field amplitudes of reflected and incident waves for different velocities of the front. Solid line represents theoretical relation of Eq. (4). Results obtained using conventional FDTD algorithm are marked with squares, results given by modified FDTD algorithm are depicted with circles. For these calculations the refractive index is varied from 3.5 to 4.5 with a sharp front.
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 dε/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”.

References and links

1.

H. E. Ives, “The Doppler effect from moving mirrors,” J. Opt. Soc. Am. 30(6), 255–257 (1940). [CrossRef]

2.

C. S. Tsai and B. A. Auld, “Wave interactions with moving boundaries,” J. Appl. Phys. 38(5), 2106–2115 (1967). [CrossRef]

3.

M. Lampe, E. Ott, and J. H. Walker, “Interaction of electromagnetic waves with a moving ionization front,” Phys. Fluids 21(1), 42–54 (1978). [CrossRef]

4.

C. H. Lai, T. C. Katsouleas, W. B. Mori, and D. Whittum, “Frequency upshifting by an ionization front in a magnetized plasma,” IEEE Trans. Plasma Sci. 21(1), 45–52 (1993). [CrossRef]

5.

W. B. Mori, T. Katsouleas, J. M. Dawson, and C. H. Lai, “Conversion of DC fields in a capacitor array to radiation by a relativistic ionization front,” Phys. Rev. Lett. 74(4), 542–545 (1995). [CrossRef] [PubMed]

6.

C. H. Lai, R. Liou, T. C. Katsouleas, P. Muggli, R. Brogle, C. Joshi, and W. B. Mori, “Demonstration of microwave generation from a static field by a relativistic ionization front in a capacitor array,” Phys. Rev. Lett. 77(23), 4764–4767 (1996). [CrossRef] [PubMed]

7.

V. I. Semenova, “Reflection of electromagnetic waves from an ionization front,” Radiophys. Quantum Electron. 10(8), 599–604 (1967). [CrossRef]

8.

M. Kando, Y. Fukuda, A. S. Pirozhkov, J. Ma, I. Daito, L.-M. Chen, T. Zh. Esirkepov, K. Ogura, T. Homma, Y. Hayashi, H. Kotaki, A. Sagisaka, M. Mori, J. K. Koga, H. Daido, S. V. Bulanov, T. Kimura, Y. Kato, and T. Tajima, “Demonstration of laser-frequency upshift by electron-density modulations in a plasma wakefield,” Phys. Rev. Lett. 99(13), 135001 (2007). [CrossRef] [PubMed]

9.

D. Kiefer, M. Yeung, T. Dzelzainis, P. S. Foster, S. G. Rykovanov, C. L. S. Lewis, R. S. Marjoribanks, H. Ruhl, D. Habs, J. Schreiber, M. Zepf, and B. Dromey, “Relativistic electron mirrors from nanoscale foils for coherent frequency upshift to the extreme ultraviolet,” Nat. Commun. 4, 1763 (2013). [CrossRef] [PubMed]

10.

E. J. Reed, M. Soljacić, and J. D. Joannopoulos, “Reversed Doppler effect in photonic crystals,” Phys. Rev. Lett. 91(13), 133901 (2003). [CrossRef] [PubMed]

11.

E. J. Reed, M. Soljacić, and J. D. Joannopoulos, “Color of shock waves in photonic crystals,” Phys. Rev. Lett. 90(20), 203904 (2003). [CrossRef] [PubMed]

12.

E. J. Reed, M. Soljacic, M. Ibanescu, and J. D. Joannopoulos, “Reversed and anomalous Doppler effects in photonic crystals and other time-dependent periodic media,” J. Comput. Aided Mater. Design 12, 1–15 (2005).

13.

N. Seddon and T. Bearpark, “Observation of the inverse Doppler effect,” Science 302(5650), 1537–1540 (2003). [CrossRef] [PubMed]

14.

F. Biancalana, A. Amann, A. V. Uskov, and E. P. O’Reilly, “Dynamics of light propagation in spatiotemporal dielectric structures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(4), 046607 (2007). [CrossRef] [PubMed]

15.

A. B. Shvartsburg, “Optics of nonstationary media,” Phys.- Usp. 48(8), 797–823 (2005). [CrossRef]

16.

M. D. Thomson, S. M. Tzanova, and H. G. Roskos, “Terahertz frequency upconversion via relativistic Doppler reflection from a photoinduced plasma front in a solid-state medium,” Phys. Rev. B 87(8), 085203 (2013). [CrossRef]

17.

X. Liu and D. McNamara, “The use of the FDTD method for electromagnetic analysis in the presence of independently time-varying media,” Int. J. Infrared Millim. Waves 28(9), 759–778 (2007). [CrossRef]

18.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2000).

19.

J. H. Wülbern, S. Prorok, J. Hampe, A. Petrov, M. Eich, J. Luo, A. K.-Y. Jen, M. Jenett, and A. Jacob, “40 GHz electro-optic modulation in hybrid silicon-organic slotted photonic crystal waveguides,” Opt. Lett. 35(16), 2753–2755 (2010). [CrossRef] [PubMed]

20.

X. Zhang, A. Hosseini, S. Chakravarty, J. Luo, A. K. Jen, and R. T. Chen, “Wide optical spectrum range, subvolt, compact modulator based on an electro-optic polymer refilled silicon slot photonic crystal waveguide,” Opt. Lett. 38(22), 4931–4934 (2013). [CrossRef] [PubMed]

21.

X. Zhang, A. Hosseini, X. Lin, H. Subbaraman, and R. Chen, “Polymer-based hybrid-integrated photonic devices for silicon on-chip modulation and board-level optical interconnects,” IEEE J. Sel. Top. Quantum Electron. 19(6), 196–210 (2013). [CrossRef]

22.

X. Gan, R.-J. Shiue, Y. Gao, K. F. Mak, X. Yao, L. Li, A. Szep, D. Walker Jr, J. Hone, T. F. Heinz, and D. Englund, “High-contrast electrooptic modulation of a photonic crystal nanocavity by electrical gating of graphene,” Nano Lett. 13(2), 691–696 (2013). [CrossRef] [PubMed]

23.

H. Lira, Z. Yu, S. Fan, and M. Lipson, “Electrically driven nonreciprocity induced by interband photonic transition on a silicon chip,” Phys. Rev. Lett. 109(3), 033901 (2012). [CrossRef] [PubMed]

24.

M. Castellanos Muñoz, A. Y. Petrov, L. O’Faolain, J. Li, T. F. Krauss, and M. Eich, “Optically induced indirect photonic transitions in a slow light photonic crystal waveguide,” Phys. Rev. Lett. 112(5), 053904 (2014). [CrossRef] [PubMed]

OCIS Codes
(060.2630) Fiber optics and optical communications : Frequency modulation
(050.5298) Diffraction and gratings : Photonic crystals
(130.7405) Integrated optics : Wavelength conversion devices

ToC Category:
Photonic Crystals

History
Original Manuscript: March 5, 2014
Revised Manuscript: April 13, 2014
Manuscript Accepted: April 26, 2014
Published: May 27, 2014

Citation
Elena A. Ulchenko, Dirk Jalas, Alexander Yu. Petrov, Michel Castellanos Muñoz, Slawa Lang, and Manfred Eich, "Pulse compression and broadening by reflection from a moving front of a photonic crystal," Opt. Express 22, 13280-13287 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-11-13280


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References

  1. H. E. Ives, “The Doppler effect from moving mirrors,” J. Opt. Soc. Am. 30(6), 255–257 (1940). [CrossRef]
  2. C. S. Tsai, B. A. Auld, “Wave interactions with moving boundaries,” J. Appl. Phys. 38(5), 2106–2115 (1967). [CrossRef]
  3. M. Lampe, E. Ott, J. H. Walker, “Interaction of electromagnetic waves with a moving ionization front,” Phys. Fluids 21(1), 42–54 (1978). [CrossRef]
  4. C. H. Lai, T. C. Katsouleas, W. B. Mori, D. Whittum, “Frequency upshifting by an ionization front in a magnetized plasma,” IEEE Trans. Plasma Sci. 21(1), 45–52 (1993). [CrossRef]
  5. W. B. Mori, T. Katsouleas, J. M. Dawson, C. H. Lai, “Conversion of DC fields in a capacitor array to radiation by a relativistic ionization front,” Phys. Rev. Lett. 74(4), 542–545 (1995). [CrossRef] [PubMed]
  6. C. H. Lai, R. Liou, T. C. Katsouleas, P. Muggli, R. Brogle, C. Joshi, W. B. Mori, “Demonstration of microwave generation from a static field by a relativistic ionization front in a capacitor array,” Phys. Rev. Lett. 77(23), 4764–4767 (1996). [CrossRef] [PubMed]
  7. V. I. Semenova, “Reflection of electromagnetic waves from an ionization front,” Radiophys. Quantum Electron. 10(8), 599–604 (1967). [CrossRef]
  8. M. Kando, Y. Fukuda, A. S. Pirozhkov, J. Ma, I. Daito, L.-M. Chen, T. Zh. Esirkepov, K. Ogura, T. Homma, Y. Hayashi, H. Kotaki, A. Sagisaka, M. Mori, J. K. Koga, H. Daido, S. V. Bulanov, T. Kimura, Y. Kato, T. Tajima, “Demonstration of laser-frequency upshift by electron-density modulations in a plasma wakefield,” Phys. Rev. Lett. 99(13), 135001 (2007). [CrossRef] [PubMed]
  9. D. Kiefer, M. Yeung, T. Dzelzainis, P. S. Foster, S. G. Rykovanov, C. L. S. Lewis, R. S. Marjoribanks, H. Ruhl, D. Habs, J. Schreiber, M. Zepf, B. Dromey, “Relativistic electron mirrors from nanoscale foils for coherent frequency upshift to the extreme ultraviolet,” Nat. Commun. 4, 1763 (2013). [CrossRef] [PubMed]
  10. E. J. Reed, M. Soljacić, J. D. Joannopoulos, “Reversed Doppler effect in photonic crystals,” Phys. Rev. Lett. 91(13), 133901 (2003). [CrossRef] [PubMed]
  11. E. J. Reed, M. Soljacić, J. D. Joannopoulos, “Color of shock waves in photonic crystals,” Phys. Rev. Lett. 90(20), 203904 (2003). [CrossRef] [PubMed]
  12. E. J. Reed, M. Soljacic, M. Ibanescu, J. D. Joannopoulos, “Reversed and anomalous Doppler effects in photonic crystals and other time-dependent periodic media,” J. Comput. Aided Mater. Design 12, 1–15 (2005).
  13. N. Seddon, T. Bearpark, “Observation of the inverse Doppler effect,” Science 302(5650), 1537–1540 (2003). [CrossRef] [PubMed]
  14. F. Biancalana, A. Amann, A. V. Uskov, E. P. O’Reilly, “Dynamics of light propagation in spatiotemporal dielectric structures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(4), 046607 (2007). [CrossRef] [PubMed]
  15. A. B. Shvartsburg, “Optics of nonstationary media,” Phys.- Usp. 48(8), 797–823 (2005). [CrossRef]
  16. M. D. Thomson, S. M. Tzanova, H. G. Roskos, “Terahertz frequency upconversion via relativistic Doppler reflection from a photoinduced plasma front in a solid-state medium,” Phys. Rev. B 87(8), 085203 (2013). [CrossRef]
  17. X. Liu, D. McNamara, “The use of the FDTD method for electromagnetic analysis in the presence of independently time-varying media,” Int. J. Infrared Millim. Waves 28(9), 759–778 (2007). [CrossRef]
  18. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2000).
  19. J. H. Wülbern, S. Prorok, J. Hampe, A. Petrov, M. Eich, J. Luo, A. K.-Y. Jen, M. Jenett, A. Jacob, “40 GHz electro-optic modulation in hybrid silicon-organic slotted photonic crystal waveguides,” Opt. Lett. 35(16), 2753–2755 (2010). [CrossRef] [PubMed]
  20. X. Zhang, A. Hosseini, S. Chakravarty, J. Luo, A. K. Jen, R. T. Chen, “Wide optical spectrum range, subvolt, compact modulator based on an electro-optic polymer refilled silicon slot photonic crystal waveguide,” Opt. Lett. 38(22), 4931–4934 (2013). [CrossRef] [PubMed]
  21. X. Zhang, A. Hosseini, X. Lin, H. Subbaraman, R. Chen, “Polymer-based hybrid-integrated photonic devices for silicon on-chip modulation and board-level optical interconnects,” IEEE J. Sel. Top. Quantum Electron. 19(6), 196–210 (2013). [CrossRef]
  22. X. Gan, R.-J. Shiue, Y. Gao, K. F. Mak, X. Yao, L. Li, A. Szep, D. Walker, J. Hone, T. F. Heinz, D. Englund, “High-contrast electrooptic modulation of a photonic crystal nanocavity by electrical gating of graphene,” Nano Lett. 13(2), 691–696 (2013). [CrossRef] [PubMed]
  23. H. Lira, Z. Yu, S. Fan, M. Lipson, “Electrically driven nonreciprocity induced by interband photonic transition on a silicon chip,” Phys. Rev. Lett. 109(3), 033901 (2012). [CrossRef] [PubMed]
  24. M. Castellanos Muñoz, A. Y. Petrov, L. O’Faolain, J. Li, T. F. Krauss, M. Eich, “Optically induced indirect photonic transitions in a slow light photonic crystal waveguide,” Phys. Rev. Lett. 112(5), 053904 (2014). [CrossRef] [PubMed]

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