Negative refraction of light in Bragg mirrors made of porous silicon

Physics Letters A 339 (2005) 387–392 www.elsevier.com/locate/pla

Negative refraction of light in Bragg mirrors made of porous silicon A.V. Kavokin a,∗ , G. Malpuech a , I. Shelykh a,b a LASMEA, UMR 6602, CNRS, Université Blaise-Pascal, 24, av. des Landais, 63177 Aubiere, France b St. Petersburg State Polytechnical University, ul. Polytekhnicheskaya 29, 195251 St. Petersburg, Russia

Received 11 February 2005; accepted 21 February 2005 Available online 21 March 2005 Communicated by V.M. Agranovich

Abstract In allowed photonic bands of one-dimensional photonic crystals (Bragg mirrors) light modes propagate with negative effective masses at certain frequencies. We demonstrate theoretically that this effect allows for the negative refraction of the visible light. We propose a structure made of two porous silicon Bragg mirrors with one rotated by 90◦ with respect to the other. This structure that may serve as a Veselago lens with a focal distance of the order of 10−5 m.  2005 Elsevier B.V. All rights reserved.

Negative refraction of electromagnetic waves is widely discussed now as a tool for fabrication of the shortfocus Veselago lenses [1]. While much progress in realization of such lenses working in the microwave region has been achieved [2,3], creation of Veselago lenses for visible light remains an unsolved problem up to now. A few interesting ideas have been recently proposed for achievement of the negative refraction for visible light. The potentiality of metamaterials [4–6], plasmonic modes [7], photonic crystals [8,9] for this purpose has been discussed in the literature. Experimental testing of the proposed structures is not an easy task, as fabrication of most of them requires a huge technological effort. Here we propose a simple system suitable for realization of negative refraction and application in the Veselago lenses. We apply the same principle as Ref. [9], i.e., use a photonic structure having a negative in-plane effective mass for the light. In contrast to Ref. [9] we do not operate with two-dimensional photonic crystals but propose a simpler system based on distributed Bragg reflectors (DBR). Technology of fabrication of high quality DBR is very well developed now. We specifically address the porous silicon structures as this material allows for the rapid growth of thick multilayer structures with a high contrast of the refractive indices governed by the degree of porosity. * Corresponding author.

E-mail address: [email protected] (A.V. Kavokin). 0375-9601/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.02.077

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Fig. 1. (a) Scheme of the photonic structure composed by two Bragg mirrors we consider for realization of the Veselago lens. (b) Solid line shows the photonic band structure for the light propagating in one mirror in the direction perpendicularly to the layers. Dashed line shows light dispersion within the second photonic subband in the plane parallel to the mirror layers.

Fig. 1 shows schematically the structure we consider. It consists of two DBRs having the same layer thicknesses (a, b) and refractive indices (n1 , n2 , respectively) but being oriented in perpendicular directions. The parameters we used in the calculation correspond to realistic porous silicon structures [10]. We took na = 1.4, nb = 2.4, na a = 456.5 nm, nb b = 913 nm. Using the transfer matrix techniques one can calculate the dispersion of the photonic bands [11] in an infinite periodic structure from cos k(a + b) =

t11 (ω, kz ) + t22 (ω, kz ) , 2

(1)

where k and kz are normal-to-plane and in-plane components of the pseudowavevector of light in the structure, t11 and t22 are elements of the transfer matrix T across its period, which can be calculated as a product of the transfer matrices through a and b layers: T = Ta Tb . The expressions for Ta and Tb are given in Ref. [11]. The dispersion of photonic modes of such structure in the growth direction is shown in Fig. 1(b) by solid lines. The difference between the optical thicknesses of the layers a and b opens the photonic gap at k = 0 which would not be the case in a balanced DBR satisfying na a = nb b. The second allowed band ends at 843 meV and is characterized by a negative parabolicity above 750 meV. The in-plane dispersion is also parabolic close to the band edge but is characterized by a positive effective mass as shown by the dashed line in Fig. 1(b). Expanding (1) into Taylor series, one can easily obtain the effective mass of light in the Bragg mirrors in x- (normal to layers) and z- (parallel

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to layers) directions: mTE x

= mTM x

TE/TM mz

    
 a(n2a + n2b ) ωj nb b ω j na a h¯ sin cos bnb + = 2nb c c (a + b)2 c       2 2 b(na + nb ) ωj na a ω j nb b sin cos , + ana + 2na c c

  
   a(n2a + n2b ) ω j na a ωj nb b h¯ ωj cos bnb + sin = 2 2nb c c c       b(n2a + n2b ) ω j na a ω j nb b sin cos + ana + 2na c c
      2 2 a(na + nb ) ωj nb b ω j na a b + × sin cos nb c c 2n2a nb       2 2 b(na + nb ) ω j na a ω j nb b a + sin cos + na c c 2na n2b     ωj nb b −1 ωj na a (na + nb )2 (na − nb )2 c sin ∓ sin , c c 2n3a n3b ωj

(2)

(3)

where the signs “−” and “+” in the denominator of (3) correspond to the TE- and TM-polarized light, ωj is a position of the band edge, which is given by the following equation:           ωna a ωnb b 1 na ωnb b nb ωna a sin cos − sin . + 1 = cos (4) c c 2 nb na c c For our choice of parameters, mx = −5 × 10−37 kg and mz = 2.3 × 10−36 kg. To obtain negative refraction we use the fact that if in a given direction the effective mass is negative, the corresponding components of group and phase velocities of light have different signs. Hence, in an isotropic medium characterized by a negative mass, group and phase velocities are exactly opposite. The medium we consider is strongly anisotropic, so that the effective masses have different signs in in-plane and normal-to-plane directions. We shall further refer to such media as left-handed (LHM). In the following we analyze refraction of light at the boundary between conventional media characterized by a linear dispersion ω = ck/neff (further referred to as the right-handed media, RHM) and LHM. Using the continuity of the electric and magnetic fields at the boundary, one can easily obtain the expressions for the refraction angle ϕr and reflection coefficient R: sin2 ϕr =  1−

mx  mz

sin2 ϕ0 sin2 ϕ0 +

2m2x c2 (ω−ω0 ) h¯ neff mz ω2

    neff ω sin ϕ0 − 2mz c2 (ω−ω0 ) −  h¯ R =   2  0)  neff ω sin ϕ0 + 2mz c (ω−ω − h¯

,

mz n2eff ω2 mx mz n2eff ω2 mx

(5) 2  sin ϕ0   ,   sin2 ϕ0  2

(6)

where ϕ0 is an angle of the incidence, ω0 is a frequency of the top of the second subband, ω < ω0 and (ω0 − ω)/ω0  1. The latter condition limits the validity of the effective mass approximation we use. Note that expressions (5), (6) are valid for the boundary between two DBRs rotated by 90◦ with respect to one other shown in Fig. 1(a). This is because light propagating in-plane of a DBR within the lowest allowed photonic subband has

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Fig. 2. Propagation of a light beam in the two first mirrors. The propagation takes place in the (x, z)-plane. The color plot shows the intensity of the electric field which allows to see the modulations whose period corresponds to the wave-length of light at the center of the propagating wave-packet. The arrows show the wave vector directions. In this geometry the left medium is LHM and the right medium is RHM.

a linear dispersion. Such a light has a zero normal-to-plane component of the wavevector, neff =

na a + nb b . a+b

(7)

Fig. 2 shows the intensity of the electric field of the wave-packet of light centered at the energy of 827 meV (λ = 1.55 µm) and propagating in the (x, z)-plane of our structure. The calculation has been performed using the scattering state technique described in Ref. [12]. The left medium is RHM for light propagating in the (x, z)-plane and has a negative mass in the y-direction. The right medium is LHM in the (x, z)-plane and RHM in the (x, y)plane. One can see that inside the left medium the wave-packet moves parallel to the wave-vector direction in this case. At the boundary, each of plane waves composing our wave-packet is weakly positively refracted at the interface whereas the wave-packet itself changes its propagation direction by almost 90◦ showing a pronounced negative refraction. Beyond the interface, in the LHM, the wave-packet now moves almost perpendicularly to the wavevector. This image is very similar to one obtained in Ref. [9] for a two-dimensional photonic crystal. An advantage of our structure consists in its extreme simplicity which allows for cheap and rapid fabrication of samples. Also advantageous is a possibility to calculate analytically the effective masses for photonic modes with use of Eqs. (2)–(4), while to obtain effective masses of photons in photonic crystals one should perform numerical calculations, in general. On a larger spatial scale, propagation of the light beams in our model structure is shown in Figs. 3, 4. The negative refraction and focalization of light in the (x, y)-plane of the structure under consideration is clearly seen in Fig. 3(a), (b) for two different incidence angles. For this propagation plane, the left mirror is LHM and the right mirror is RHM. Fig. 4(a), (b) shows propagation of light pulses in the (x, z)-plane of our model structure. For this propagation plane, the left mirror is RHM and the right mirror is LHM. Due to the symmetry of the structure, the focal distances for light propagating in (x, y)- and (x, z)-planes are exactly the same, so that the degree of astigmatism in our lens in expected to be extremely low. On the other hand, the focal distance is dependent on the incidence angle, as one can see comparing images in Fig. 3(a), (b). That is why the aberrations in our Veselago lens

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Fig. 3. Propagation of light pulses in a two-Bragg mirror Veselago lens in the (x, y)-plane (a), (b) at different incidence angles (20◦ and 25◦ , respectively). The brightness of color in increases monotonically with the intensity of light.

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Fig. 4. Propagation of light pulses in a two-Bragg mirror Veselago lens in the (x, z)-plane (a), (b) at different incidence angles (20◦ and 25◦ , respectively). The brightness of color in increases monotonically with the intensity of light.

are indeed important. Anyway, for all angles, the focal distance is extremely short (of the order of tens of microns). The lens we propose is acting correctly only in a relatively narrow frequency range given by the overlap of the allowed photonic bands in all mirrors (of 90 meV width in our model structure). The working interval of these lenses depends on the position of stop-bands in the DBR which is governed by the layer thicknesses and can be varied in large limits. Thus, the lens composed by multiple pairs of Bragg mirrors with different layer thicknesses could be operating for all the visible spectrum. Here we considered a single couple of DBRs which is a building block of such a full-spectrum Veselago lens. In conclusion, we propose an extremely simple photonic structure that exhibits negative refraction of visible light. We suggest porous silicon as a material mostly adapted for fabrication of periodic photonic structures which allow for realization of the short-focus Veselago lenses.

Acknowledgements This work has been supported by the Marie-Curie RTN “Clermont2” (contract MRTN-CT-2003-503677).

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