Pulse propagation in finite linear one-dimensional periodic structures

Pulse Propagation in Finite Linear One-Dimensional PeriodicStructures

Guido Torrese and Henry P. Schriemer

University of Ottawa, Centre for Research in PhotonicsSchool of Information Technology and Engineering

800 King Edward Ave., Ottawa, ON, Canada, K1N 6N5

M. Cada

Dalhousie University, Department of Electrical and Computer Engineering1360 Barrington St., Halifax, NS, Canada, B3J 2X4

ABSTRACT

The optical propagation of a pulse through one dimensional finite gratings and photonic crystals is discussed.In the case of shallow gratings the light transport properties are derived within the frame of the Coupled ModeTheory, while the Transfer Matrix Method is used for investigating photonic crystal (PC) structures. The so-called superluminal tunneling of wave-packets through the band-gap region is investigated, and the dwell timeis computed. Because of the analyticity of the wave equation, the Einstein causality principle is not violated,although the group velocity can exceed the speed of the light in vacuum. We show that the dwell time is apropagation phenomenon and not a quasi-static process in which the incident pulse envelope modulates theamplitude of an exponentially decaying standing wave. Nevertheless, the group velocity cannot be always usedto compute the transmission group delay, because the latter does not represents the time spent by the energy topropagate through the band-gap.

Keywords: Photonic crystal, superluminality, group velocity, energy velocity, tunneling

1. INTRODUCTION

In the last few years there has been much interest in periodic microstructures characterized by a gap in theelectromagnetic spectrum, particularly those where wave propagation becomes forbidden in all directions andpolarization states across some frequency band.1, 2 A clear understanding of the relationship between wavetransport and the band (or stop) gap is required to drive innovation. Despite its apparent triviality, this is asubject which is perennially revisited whenever a novel system or experimental realization is found. At its heartlie the physical interpretations of the various transport velocities so ably introduced by Sommerfeld and Brillouinin the early part of the previous century.3 As shown by Steinberg et al.,4 waves traveling through a photoniccrystal excited within the band-gap appear to be delayed less than waves propagating over the same distancein vacuum. Similar results were obtained by Spielmann et al.,5 who measured the superluminal propagation offemtosecond optical pulses. By using correlation techniques, they obtained a negative delay of 6 fs through adistributed Bragg reflector (DBR). Larger advances in time have been demonstrated in a series of experimentsat the microwave regime.6, 7 The attraction of this subject devolves from the apparent superluminality of wavetransport through the gap, and the relation to relativistic causality. As late as 1992, it could be stated withsome frankness that the physical mechanism responsible for superluminal time delay was obscure and that thescientific community was not in consensus on this subject.8 While the situation has improved markedly in theintervening decade, this sophistication of insight has not been universally acknowledged, and the field still suffersfrom an astonishing lack of clarity in the popular view. In part, this has been due to the paucity of illustrationaccessible to scientific popularizers and illustrators. In this paper, we address this deficiency by considering thepropagation of an optical pulse through a one dimensional finite periodic structure. We directly compare the

Contact author: [email protected]

Photonics North 2004: Optical Components and Devices, edited by John C. Armitage,Simon Fafard, Roger A. Lessard, George A. Lampropoulos, Proceedings of SPIE Vol. 5577

(SPIE, Bellingham, WA, 2004) 0277-786X/04/$15 · doi: 10.1117/12.567367

568

pulse that has transited the barrier with the equivalent pulse that transits an equal distance of free space. In sodoing, we find that the so-called ”superluminal” pulse is always slower than the ”vacuum” pulse, thus explicitlydemonstrating causality.

We consider a ”barrier” of modest strength, and of familiar form, namely a one-dimensional periodic structure,or grating. This finite thickness ”Bragg stack” is a canonical λ/4 structure formed by alternating high (NH =1.66) and low (NL = 1.45) refractive index layers. We presume the materials to be isotropic, non-absorbing, andnon-dispersive and for illustration use a free-space wavelength of λ0 = 1000 nm. In section 2, we develop anexact mathematical formalism, suitable for addressing gratings of any strength, and based on the transfer matrixmethod (TMM). We follow this in section 3 with a review of the commonly applied approximate approach, basedon the coupled mode theory (CMT), which is suitable only in the limit of weak gratings. A comparison betweenresults obtained by these two approaches is done in Section 4, and causality is demonstrated.

2. PULSE PROPAGATION IN THE TMM FORMALISM - AN EXACT APPROACH

The central concept behind the paradoxical descriptions of superluminality is the concept of the transit time,often described as the time a wave packet spends interacting with the barrier potential. The definitions of thistransit (or dwell) time are many,9, 10 and include complex quantities.11 The literature on barrier traversalproceeds from two streams: those emphasizing the particle character of the wave packet, and those the wavecharacter, with the various ”tunneling” times in some manner described in terms of the conventional phase, groupand energy velocities. In most circumstances the velocity of the wave packet is given by the group velocity, whichrepresents the velocity of the dominant frequency component. In lossless uniform dielectric materials, phase,group and energy velocities are equal because different frequency components do not change phase with respectto one another. Conversely, in stratified media, for wavelengths matching the characteristic length scale of thecrystal, wave packets still suffer from dispersion even when the dielectric constant is frequency independent.For finite one dimensional photonic crystals, the dispersion can be taken into account by means of an effectiverefractive index,12 while the group velocity can be calculated as a function of a complex transmission coefficient.13

For linear causal media, it can be shown that the peak amplitude of a wave packet is delayed by a time

τg =∂φ(ω)

∂ω, (1)

where φ(ω) is the transmission phase and ω is the angular frequency. The time τg is called group delay and it isrelated to the group velocity by

vg =L

τg, (2)

with L the length of the system. For modes that are purely propagating waves, the wave vector k is real andthe derivative of the transmission phase, given by φ(ω) = kL, is

τg =∂φ(ω)

∂ω=

L∂ω∂k

. (3)

When combining equations (2) and (3), the group velocity results to be a frequency dependent quantity given bythe derivative of the angular frequency with respect to the wave vector. In other word, according to this definition,the group velocity is given by the slope of the dispersion curve. Because of the frequency dependence of thegroup velocity, different spectral components disperse during the propagation and do not arrive simultaneouslyat the structure end-face. The time delay td experienced by a pulse propagating through the structure can thenbe defined as

τd = τ0

(c

vg− 1

), (4)

with τ0 the time spent by the light to cross a distance equal to L at the velocity c in vacuum. Equation (4) canbe easily obtained by dividing both side of (3) by c, then computing the difference τd = τg − τ0. For frequenciestuned at the middle of the band-gap the dominant electromagnetic mode will be strongly attenuated. Becausedecaying modes are characterized by no phase variation, any phase change will be independent of the length

Proc. of SPIE Vol. 5577 569

of the barrier. Hence, the time for such a wave packet to transit a barrier will be invariant of its length; thishas been termed the Hartman effect. This phenomenon, although counterintuitive, is not at all paradoxical asclaimed by Winful,14 and we demonstrate this below.

We use the TMM formalism to determine the electric and magnetic field distribution associated with thepropagation of a Gaussian pulse through a one-dimensional (1D) finite photonic crystal (PC). Pulse propagationin a charge free dielectric material is governed by Maxwell’s equations, and we define E(z,t) as the electric fieldtraveling along the z direction, µ is the permeability, while ε is the material permittivity. Because we are dealingwith semiconductor materials, in the rest of this paper we assume µ = µ0, while the dielectric constant ε is aquantity depending on the position. The field E(z,t) is related to its spectral components through the Fourierrepresentation

E(z, t) =12π

∫ +∞

−∞E(z, ω)e+ωtdω, (5)

where the spectral amplitude E(z, ω) satisfies the time harmonic Maxwell’s equations with k(ω) = n(ω)ω/cdefining a complex propagation constant, with n(z, ω) the refractive index and c the speed of light in the vacuum.For an incident electric field linearly polarized in the x-direction, and normally incident to the system interface,the field within each layer may be given as the sum of forward and backward propagating field components:

E(z, ω) = E+(z, ω) + E−(z, ω), (6)

withE±(z, ω) = E±

0 (z)e∓k(ω)z. (7)

Without loss of generality, the electric field is taken to be directed such that E(z, ω) = E(z, ω)x, and hence theorthogonal magnetic field H(z, ω) = H(z, ω)y is obtained form the Maxwell’s equations as

H(z, ω) =1η[E+(z, ω) − E−(z, ω)], (8)

with the impedance η =√

µ/ε. Once the forward and backward electric field components are known, the electricand magnetic fields can be computed by using equations (6) and (8), respectively. This permits the fields ateach point z to be determined for each frequency ω. We now consider two different cases corresponding to thepropagation of the forward and the backward fields from one interface to the next and through an interface. Theelectric field components describing propagation in the uniform medium, of length Li and bounded by interfacesi and i + 1, are given by the propagation matrix

(E+

i

E−i

)=

1ti

(ekiLi 0

0 e−kiLi

)(E+

i+1

E−i+1

), (9)

with ki the transverse component of the wave vector for the ith-layer. Relating forward and backward fieldcomponents between layers is done by the imposition of the usual boundary conditions and yields the matchingmatrix (

E+i

E−i

)=

1tij

(1 rij

rij 1

)(E+

j

E−j

), (10)

where rij and tij are the Fresnel reflection and transmission coefficients at the interface between the media iand j, respectively. The forward and backward electric field components within the system can be determinedby multiplying matching and propagation matrices under the presumption of a forward propagating field ofunit measure incident on the leftmost interface, and no (backward) propagating field incident on the rightmostinterface. For example, when considering a single slab, we can write

E+1 =

1t12t23

(ek1L1 + r12r23e

−k1L1)E

′+2 (11)

E−1 =

1t12t23

(r12e

k1L1 + r23e−k1L1

)E

′+2 , (12)

570 Proc. of SPIE Vol. 5577

with E′+2 the forward propagating field in the most-right semi-infinite medium. When defining the wave reflection

and transmission coefficient as

T (ω) =E+(L, ω)E+(0, ω)

(13)

and

Γ(ω) =E−(0, ω)E+(0, ω)

, (14)

respectively, with 0 and L the coordinates corresponding to the input and output end of the system, a simplerecursive formula for the reflection coefficient may then be written as

Γi =ri,i+1 + Γi+1e

−2kiLi

1 + ri,i+1Γi+1e−2kiLi. (15)

Similarly, the transmission coefficient T obeys to the relation

Ti =ti,i+1Ti+1e

−kiLi

1 + ri,i+1Γi+1e−2kiLi. (16)

Equations (15) and (16) have been used to determine the reflection and transmission coefficients at each pointwithin the photonic crystal, and for each frequency. The net component of the field reflected from the system,and the net component transmitted, are then extracted at the end of the recursive process by using equations(13) and (14).

We consider a phase-modulated Gaussian light pulse defined by the incident electric field

E+1 (z = 0, t) = E0e

− 12 (

t−t0σ )2eωc(t−t0)eβ(t−t0)

2, (17)

where E0 is the peak amplitude, σ is the characteristic half-width at 1/e intensity point, ωc is the carrier angularfrequency, β is the frequency chirp parameter, and t0 is the time corresponding to maximum of the optical pulse.When applying the Fourier transform we have

E+1 (z = 0, ω) =

E0σ√

2π√q

e−(2t0qω+σ2(ω−ωc)2)

2q , (18)

with q = + 2βσ2. Equation (18) shows that in the frequency domain the incident pulse remains Gaussian, butit is characterized by a complex amplitude. The total electric field propagating in the crystal at the position zmay be computed by using the Fourier representation (5), with the electric field spectral amplitude given by alinear superposition of forward and backward traveling waves. If the spectral distribution is fairly sharply peakedaround the carrier frequency ωc, the wave vector can be expanded in Taylor series around the value ωc:

k(ω) = k(ωc) +(

∂k

∂ω

)ω=ωc

(ω − ωc) (19)

and the integral (5) performed. In this quasi-monochromatic approximation the group velocity

vg =(

∂ω

∂k

)ω=ωc

(20)

represents the velocity of a pulse that travels without significant shape distortion. For pulses for which |ω−ωc| �ωc, the quasi-monochromatic approximation fails and the different spectral components of the optical pulse travelinside the photonic crystal at different velocities. In this case dispersion-induced pulse broadening cannot beneglected and optical pulses no longer remain Gaussian on propagation and develop a tail with an oscillatorystructure.


Following the formulation of Bendickson,13 we extract the density of states from the reflection coefficient T,expressed in terms of its real and imaginary parts x(ω) and y(ω). Since ρ(ω), the total density of states (DOS),equals the the reciprocal of the global group velocity ∂ω/∂k, it is easy to show that

ρ(ω) =1L

y′x − x

′y

x2 + y2, (21)

where the prime denotes differentiation with respect to ω. The group delay based on a CW approach can then becomputed using equation (3). We defer discussion of the results until after we describe the CMT approximation,below.

3. PULSE PROPAGATION IN THE CMT APPROXIMATION

The CMT is typically employed in describing the response of weak, low refractive index contrast gratings, suchas those found in fiber Bragg grating structures. It has, however, also been applied to somewhat deeper gratingsrepresenting stronger barriers. The structure under consideration in this paper represents a barrier of modeststrength, suitable for assessing the failure modes of the CMT. Because of the periodic modulation of the refractiveindex along the longitudinal direction, forward and backward traveling waves are coupled through the structure.The CMT approximation assumes a uniform reference medium of refractive index n0, and treats the refractiveindex variation as a perturbation. In uniform periodic grating, the refractive index along the length variesperiodically as n0 + n1 cos(2πz/Λ), where n1 is the modulation depth, and Λ the grating period. Forward andbackward pulses with slowly varying envelopes E+(z, t) and E−(z, t) satisfy the following equations15

∂E+

∂z− 1

v

∂E+

∂t= δE− (22)

∂E−

∂z+

1v

∂E−

∂t= −δE+, (23)

where δ = πn1/λB is the coupling coefficient, λB the Bragg wavelength, and v the phase velocity in the un-perturbed medium. Equations (22) and (23) can be combined to yield the Klein-Gordon Equation (KGE), thesolution of which is obtained by a Green’s function approach.16 Once forward and backward electric fieldcomponents have been determined, the complex transmission and reflection coefficients are given by equations(13) and (14). In the case of vanishing coupling (δ = 0), the two equations become decoupled, yielding thetrivial solution E(z, ω) = E+(0, ω)e−kz + E−(0, ω)ekz. This represents a pair of independent plane waves withconstant amplitude propagating along the optical axis in two opposite directions. The solution of the coupledmode equations (22) and (23) can be written in matrix form as

(E+(L)E−(L)

)=

(cosh(Lγ) + ω

vγ sinh(Lγ) δγ sinh(Lγ)

− δγ sinh(Lγ) cosh(Lγ) − ω

vγ sinh(Lγ)

)(E+(0)E−(0)

), (24)

where γ2 = δ2 − (ω/v)2. In this way the shallow grating is described by a transfer matrix relating the forwardand backward field components at one end of the structure to the forward and backward field components at theother end. By utilizing the above description the transmission and reflection coefficients can be easily written as

T (ω) =1

cosh(Lγ) − ωvγ sinh(Lγ)

(25)

and

Γ(ω) =δγ sinh(Lγ)

cosh(Lγ) − ωvγ sinh(Lγ)

. (26)

Equations (25) and (26) have been obtained by substituting the field components E+(L), E+(0), and E−(0)into equations (13) and (14) and by imposing the condition E−(L) = 0. The latter implies, as with the TMM,


that there is no backward propagating wave input at the rightmost end of the structure. The group delay canfound by differentiation as per equation (1), given the transmission phase, which is

φ(ω) = arctan(

ω

vγtanh(Lγ)

). (27)

Doing so, we have

τg =sinh(2Lγ) − 2Lγ

δ2 (ωv )2

vγ[1 + cosh(2Lγ) − 2δ2 (ω

v )2]. (28)

4. NUMERICAL RESULTS

We now consider the results of our computations, for the structures in question, which we vary in length, thatis, in the number of layers. The structures are symmetric, the layer number being odd. Figure 1 shows thenormalized group velocity computed as the inverse of the density of states versus the normalized frequency.

0.9 1.0 1.10

1

2

3

4

5 17 layers 41 layers 101 layers 141 layers

Nor

mal

ized

Gro

up V

eloc

ity

Normalized Frequency

Figure 1. The group velocity vg(ω), normalized with respect to the velocity of light in vacuum, for multilayers ofincreasing layer number. The vertical dotted line corresponds to the normalized frequency at the middle of the band-gap.

The different curves correspond to structures characterized by a different number of layers, while the verticaldotted line indicates the normalized frequency corresponding to the middle of the band-gap. We first considerthe frequency range corresponding to the band-gap region. As the structure becomes thicker, the group velocitybegins to increase, eventually exceeding the speed of light in vacuum. On the other hand, in short structures,the scattering is insufficient to generate superluminality. As suggested by equation (20), the group velocity canbe unbounded in the limit of infinite thick crystals, because the derivative ∂k/∂ω vanishes when the field decay


constant is slowly varying with respect to the angular frequency. In other words, for frequencies within the bandgap, there are no propagating states, and the system response is described by tunneling. Outside the band gapregion, the response can be described in terms of propagating Bloch states, and the interpretation of the groupvelocity is consistent with that provided by an effective medium approach.

Similar considerations can be done when considering the group delay. For very spectrally narrow pulses, thegroup delay can be computed by using equation (3) without introducing a significant error. As shown in Figure 2,in the CW limit (solid line), the group delay is almost independent of the thickness of the crystal. This behavior

0 10 20 30

5

10

15

20

25

Gro

up d

elay

tim

e (f

s)

Barrier width (microns)

DOS Pulse TMM

CMT

Figure 2. The group delay obtained from the DOS (CW limit, solid line) is compared to that extracted from the pulsetransit time. Solid dots have been obtained using the exact TMM formalism, while solid squares represent the valuesobtained within the frame of the CMT approximation.

is consistent with the increase of the group velocity depicted in Figure 1. We note, in anticipation, that evenif the group velocity becomes superluminal within the band gap region, the absolute amount of light that exitsthe crystal is quite small. Consequently, in real experiments, it can be difficult to observe the velocity behaviorshown in Figure 1 without amplifying the transmitted light.

For spectrally broader pulses, such as the ones considered here, we need an alternate method for determiningthe group delay. The transit time of the peak of the field envelope is asymptotically equivalent to the CWapproach, and is the experimentally accessible quantity, being the measurement procedure usually followed. Weconsidered a spectrally moderate optical pulse with a characteristic half-width σ = 60π/(

√2ω0), with ω0 the

frequency corresponding to the middle of the band-gap. The frequency content of this incident pulse is shown,in Figure 3, in relation to the reflectance of a 41-layer structure; note that all frequency components of the pulsefall effectively within the stop gap. The time origin has been taken in such a way that the incident optical pulseenvelope reaches its maximum at the crystal input surface, i.e. t0 = 0, and no chirp has been considered (β = 0).We extract the group delay from the peak field amplitudes of the transmitted fields, using both the TMM and


0.9 1.0 1.1

-15

-10

-5

0

Ref

lect

ance

(dB

)

Normalized Frequency

Barrier (41 layers) TMM CMT

Pulse Am

plitude (dB)

Figure 3. The reflectance computed by using the CMT (dashed line) and the TMM (solid line) are plotted versus thenormalized frequency for a 41 layers barrier. The shadow pulse at the middle of the band-gap represents the electric fieldenvelope of the input Gaussian pulse.

CMT approach. As can be seen from Figure 2, results obtained by using the TMM formalism (solid dots) agreequite well with the CW approach - as they should for an exact theory. The CMT values (solid squares) divergefrom CW limit as one approaches the Hartman plateau. The simulations were not pushed to larger samplevalues because the TMM approach becomes numerically unstable at large sample thicknesses (about 100 layers,or 30 microns in our case). The small difference between results obtained using the CW approach (solid line)and those obtained using the TMM (solid dots) are attributable to group velocity dispersion. If group velocitydispersion, and all higher order dispersive effects, are not exactly zero, the velocity of the pulse peak does notrepresent the true velocity of the nominal centre frequency. Again, as shown in Figure 2, the difference betweenCW approach and TMM becomes more evident as the length structure increases since dispersion broadens thepulses during their propagation inside the crystal. We can thus conclude that only in the limit of spectrallynarrow incident pulses, where the stationary phase approximation is valid, will the group delay as computed viathe peak of the pulse coincide with that obtained by calculating the derivative of the transmission phase withrespect to the angular frequency.

When the length of the crystal increases, so does the scattering; the grating stop band narrows, while thereflectance tends to unity for all frequencies within the stop gap. The pulse propagation through such a gratingis illustrated in Figure 4. It shows the comparison between a pulse transmitted through a 101-layer barrier andthe same pulse propagating over the same distance in vacuum (dashed line). The solid curve refers to resultsobtained by using the Transfer Matrix Method, while results from the Coupled Mode Theory are represented bythe dot dashed curve. The top panel of Figure 4 shows the field envelopes normalized to unity. For both CMTand TMM, the peak of the pulse envelope appears at the output sooner than the incident pulse traveling the


-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.40.00

0.01

0.02

0.03

0.04

0.05

Tru

e A

mpl

itude

s

Time (ps)

0

1

Vacuum Barrier TMM

CMT N

orm

aliz

ed

Am

plitu

des

Figure 4. Comparison between the electric field envelope of a pulse exiting the endface of a 101-layer and one travelingover the same distance in vacuum (dashed line). The amplitude of the pulse propagating through the crystal has beencomputed by using both TMM (solid line) and CMT (dot dashed line). The top panel shows all data rescaled to unity,providing the false impression of noncausal transport, while the bottom panel maintains the true pulse amplitudes,demonstrating causality.

same distance in vacuum. However, one must not jump to the conclusion that the pulse propagating throughthe barrier is tunneled at superluminal group velocity, as is so frequently done. The proper view is to comparethese pulses on precisely equal footing, that is, without any rescaling. The bottom panel of Figure 4 shows thepulses when avoiding any such amplification. In this case, no part of the transmitted pulse preceeds the envelopeof the pulse propagating in vacuum. This explicitly demonstrates (or validates) causality. Because the CMT isan approximate theory the reduction in pulse amplitude is underestimated, but it too remains wholly causal atall times.

As previously alluded to, the scattering contributions increase with the number of interfaces, and hencethe transit time will monotonically increase with sample size. This situation is illustrated in Figure 5, whichshows similar data for pulses traversing barriers of increasing thicknesses for comparison. When consideringweaker barriers (top row: 11,17, and 31 layers), the pulse envelope peak is not found to the left of the onetransiting the vacuum. This is evident when referring back to Figure 2, which shows that the Hartmann plateauhas not yet been attained. Conversely, for barriers of increasing strength (i.e., length), the group delay doeseffectively become independent of sample thickness. Pulses crossing such barriers are shown in the bottom panelof Figure 5. For samples of increasing thickness, the pulse envelope peak shifted further and further to the left(i.e., increasing “superluminality”, see 1). Simultaneously, however, the distance between the rising edge of thevacuum-traversing pulse and that of the sample-traversing also increases, regardless of amplitude “set point”used for the reference. That such seemingly disparate behaviours are possible is due to the extremely rapidreduction in transmitted amplitude with increasing sample thickness. The construction of superluminal groupvelocities is a result of very poor nominclature, and the application of an analysis procedure to a regime where


0.0

0.5

1.0

11 layers

T

rue

Am

plitu

de

0.0

0.5

1.017 layers

0.0

0.5

1.031 layers

-0.2 0 0.2 0.40

0.1

0.2

0.3 71layers

Time (ps)

-0.2 0 0.2 0.40

0.01

0.02 101layers

Time (ps)

-0.2 0 0.2 0.40

0.001

0.002

0.003141layers

Time (ps)

Figure 5. Comparison between the electric field envelope of a pulse exits at the sample end (solid line) and one travelingover the same distance in vacuum (dashed line). The results have been obtained for different layer numbers by using theTMM.

such has no validity. No violation of causality actually occurs.

ACKNOWLEDGMENTS

One of the authors, G. Torrese, would like to acknowledge the pleasure of discussions with J. Taylor. This workwas supported by the Canada Research Chairs program and the National Capital Institute of Telecommunications(NCIT).

REFERENCES1. E. Yablonovitch, “Inhibited spontaneous emission in solid state physics and electronics,” Phys.Rev. Lett.

58, pp. 2059–2062, 1987.2. S. John, “Strong localization of photonics in certain disordered dielectric superlattices,” Phys.Rev. Lett. 58,

pp. 2486–2480, 1987.3. L. Brillouin, Wave Propagation and Group Velocity, Academic Press, NY, 1960.4. A. Steinberg, P. Kwiat, and R. Chiao, “Measuremets of the single-photon tunneling time,” Phys. Rev. Lett.

71, pp. 708–711, 1993.5. C. Spielmann, R. Szipocs, A. Stingl, and F. Krauzs, “Tunneling of optical pulses through photonic band-

gaps,” Phys. Rev. Lett. 73, pp. 2308–2311, 1994.6. A. Enders and G. Nimtz, “Photonic-tunneling experiments,” Phys. Rev. B 47, pp. 9605–9609, 1993.7. G. Nimtz and W. Heitmann, “Superluminal photonic tunneling and quantum-electronics,” Prog. Quantum

Electron. 21, pp. 81–108, 1997.


8. T. Martin and R. Landauer, “Time delay of evanescent electromagnetic waves and the analogy to particletunneling,” Phys. Rev. A 45, pp. 2611–2617, 1992.

9. E. Hauge and J. Stovneng, “Tunneling times: a critical review,” Rev. Mod. Phys. 61, pp. 917–936, 1989.10. M. Buttiker and R. Landauer, “Traversal time for tunneling,” Phys. Rev. Lett. 49, pp. 1739–1742, 1982.11. D. Sokolovski and L. Baskin, “Traversal time in quantum scattering,” Phys. Rev. A 36, pp. 4604–4611,

1987.12. M. Centini, C. Sibilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. Bloemer, C. Bowden, and I. Nefedov,

“Dispersive properties of finite, one-dimensional photonic band gap structures: Applications to nonlinearquadratic interactions,” Phys. Rev. E 60, pp. 4891–4898, 1999.

13. J. Bendickson, J. Dowling, and M. Scalora, “Analytic expression for the electromagnetic mode density infinite, one-dimensional, photonic band-gap structures,” Phys. Rev. E 53, pp. 4107–4121, 1996.

14. H. Winful, “Nature of superluminal barrier tunneling,” Phys. Rev. Lett. 90, p. 023901, 2003.15. R. Kashyap, Fiber Bragg Gratings, Academic Press, San Diego, 1999.16. D. Hall, “A comment on the coupled-mode equations in guided-wave optics,” Optics Communications 82,

pp. 453–455, 1991.


Pulse propagation in finite linear one-dimensional periodic structures

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