Solid State Communications 151 (2011) 579–581

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Solid State Communications

journal homepage: www.elsevier.com/locate/ssc

Reply to comment on ‘‘Negative refraction in 1D photonic crystals’’Ragini Srivastava a,∗, Khem B. Thapa b, Shyam Pati a, S.P. Ojha c

a Department of Physics, Udai Pratap Autonomous College, Varanasi, 221002, Indiab Department of Physics, U.I. E.T, C. S. J. M. University, Kanpur-208024, Indiac C. C. S. University, Meerut- 200005, India

a r t i c l e i n f o

Article history:Received 13 December 2010Accepted 21 December 2010by M.S. SkolnickAvailable online 21 January 2011

Keywords:A. Photonic crystalD. Negative refractionD. Group velocity

a b s t r a c t

I refute the criticisms of Srivastava et al. (2008) [13] raised in Bénédicto et al. [14].© 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Photonic crystals are artificial materials which show anextraordinary strong nonlinear dispersion at wavelengths close tothe bandgap. Under certain conditions, they abnormally refractthe light as if they had a negative refractive index, which isreferred to as negative refraction effect [1,2]. Carefully designedphotonic crystals are used for obtaining negative refraction and thetransmission of energy is possible when thewave comes in a rangeof frequencies to form energy packets [1,3]. In photonic crystals,light travels as Bloch waves, which travel through the crystalswith a definite propagation direction, despite of the presence ofscattering. In the long wavelength regime, the average direction ofenergy propagation can be shown to be the same as the directionof group velocity [4].

We know that neither a negative group velocity nor a negativeeffective index is a prerequisite for negative refraction. In fact, thelowest band, everywherewithin the first Brillouin zone, ismeaningthat the group velocity is never opposite to the phase velocity. Inthis sense, we are operating in a regime of positive effective index.In fact, photonic crystal is behaving much like a uniform, ‘‘right-handed’’ medium with hyperbolic dispersion relation [5].

Refraction-like behavior could be expected to occur in pho-tonic crystals that exhibit negative refraction for certain lattice pa-rameters and the simple one-dimensional Kronig–Penney modelprovided an exactly soluble example of a photonic crystal withnegative refraction [6].

DOI of original article: 10.1016/j.ssc.2010.12.026.∗ Corresponding author. Tel.: +91 9839927479.

E-mail address: ragini_upc@yahoo.co.in (R. Srivastava).

0038-1098/$ – see front matter© 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.ssc.2010.12.029

2. Theoretical analysis

We know that the decouple equation for the electromagneticwave is

∇⃗2E +

ω2

c2εrµr E⃗ = 0 and ∇⃗

2H⃗ +ω2

c2εrµr H⃗ = 0. (1)

For homogeneous and isotropic media, the solution of Eq. (1);

E(r, t) = E(r)e−i(ωt+K⃗ .r⃗) and H(r, t) = H(r)e−i(ωt+K⃗ .r⃗) (2)

where E(r) and H(r) are constants and are the amplitudes and ωis angular frequency. The K is magnitude of wave vector which isgiven by

K = |k⃗| = ω√

εµ =ω

cn. (3)

This equation is called the dispersion relation of the field. If ‘ωt −

k⃗.r⃗ = constant’ for a plain (surface of constant phase) normal tothewave vector k⃗ at any instant time t , then this surface of constantphases is called wave front (eikonal) [7]. The phase velocity of thewave is given by

vp =ω

k(ω). (4)

For the homogeneous and isotropic periodic dielectric stack,k⃗ is the dispersion relation of photonic crystals, i.e., K⃗(ω). Thisdispersion relation shows the radiation modes of the photoniccrystal. The wave function could also be calculated. In additionto the eigen frequency and eigen function, there are severalparameters that characterize the radiational waves; one of themis the wave velocity. In contrast to the case particles for which

580 R. Srivastava et al. / Solid State Communications 151 (2011) 579–581

Fig. 1. Group velocity Vs normalized frequency for a = 0.75 and b = 0.25 andrefractive index of air/ZnSe is n1 = 1 and n2 = 2.3 at angle = 45.

the velocity has single meaning; waves have three differentkinds of velocities, i.e., the phase velocity, group velocity andthe energy velocity. These velocities are equal to each other inuniform materials with dielectric contrasts real and independentof frequency. The effective index of photonic band gap materialshas been calculated by Dowling and Bowden [7] using the phasevelocity. So that effective phase index is given by

neff(p) =cvp

. (5)

The phase velocity is defined as the velocity of the propagationof an equi-phase surface. This velocity has a definite meaning,for example, for plane waves and spherical waves for whichequi-phase surface can be defined without ambiguity. In thephotonic crystals, however, the equi-phase surface cannot bedefined rigorously, since its eigen function is a superposition ofplane waves. This means that the phase velocity cannot be definedappropriately in the photonic crystals. Hence, the calculation ofthe group velocity is an essential task for understanding of theiroptical properties. The group velocity of the radiation modes hasvery important role in light propagation and optical response inphotonic crystals. The group velocity and effective group indexin one-dimensional photonic crystal, which is given by [8–10]as

vg =dω

dk(ω)(6)

and neff(g) =cvg

. (7)

Sowehave considered the group velocity and effective group indexeffective using Eqs. (6) and (7) respectively.

Recently, Wang et al. [11] have experimentally observedthe superluminal propagation at negative group velocity in C60polymethyl methacrylate (PMMA) firstly. The largest time ad-vancement of 7.61 ms was obtained at 1 mg/cm3 of sample con-centration; the corresponding group velocity was −0.657 m/s.

3. Result and discussion

Asmentioned above, the group velocity governs the energy flowof a light beam: when Vgx > 0, the wave propagates along thepositive direction of x axis, and the beam refracts to the other sideof the normal according to the Snell law;Vgx = 0 is associatedwiththe normal propagation; but Vgx < 0 means that the transmittingbeam will bend to the ‘wrong’ side, that is the same side as theincident beam, which means the negative refraction if the beampasses through the PC.

From Fig. 1, it is clear that at the bandgap edge there is stronggroup velocity dispersion. As frequency increase, Vgx decreasesfrom a positive to negative value and falls sharply to the negativeminimum at the edge of band and then jumps to the positive

Fig. 2. Transmittance Vs normalized frequency for a = 0.75 and b = 0.25 andrefractive index of air/ZnSe is n1 = 1 and n2 = 2.3 at angle = 45.

Fig. 3. Phase velocity Vs normalized frequency for a = 0.75 and b = 0.25 andrefractive index of air/ZnSe is n1 = 1 and n2 = 2.3 at angle = 45.

maximum. Afterwards Vgx decreases again and so on. At Vgx < 0the energy flowmay tend to the negative direction of the x-axis, sonegative refraction phenomenon may occur at some frequency inthe bandgap. At normal incidence Vgx always equals zero.

Fig. 2 shows that in the first and third bandgap, the wavesare completely reflected and do not pass through the crystalmeans that the transmittance is zero but in the second and fourthbandgaps, the transmittance is not zero, which means part of thewave can pass. Hence, there exists a negative group velocity in thelow frequency edge of the second bandgap, at the same time; thetransmittance is not zero, so the transmissionwavewill bend to thenegative direction of the x-axis, which is the negative refractionphenomenon. While in the first and third bandgap, some groupvelocity is negative, so there is no transmission of energy andthe negative refraction cannot occur. Therefore, we can concludethat negative refraction may occur in the frequency near the lowfrequency edge of the second and fourth bandgap because of thestrong group velocity dispersion.

According to the Fig. 3, phase velocity is infinite in the secondand fourth bandgaps because the wave is evanescent, the wavenumber is complex and only the imaginary part exists. In theory,the evanescent wave decays to zero in infinite photonic crystal,so the phase of the wave does not exist therefore phase velocityhas no meaning [12]. In the first and third bandgaps, the wavenumber is complex, but the real part is not zero, so the phasevelocity exists, and the phase of the wave varies because of thereflection at interface boundary, the waves are all reflected andcannot pass through photonic crystal. Therefore, we can concludethat negative refraction may occur in the frequency near the lowfrequency edge of the second and fourth bandgap because of thestrong group velocity dispersion.

4. Conclusions

In conclusion, we can say that the negative refraction occurin such type of materials near the low frequency edge of thesecond and fourth bandgaps in 1D photonic crystals for an obliqueincidence of wave.

R. Srivastava et al. / Solid State Communications 151 (2011) 579–581 581

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