Klein tunneling of light in fiber Bragg gratings

A photonic analogue of Klein tunneling (KT), i.e. of the exotic property of relativistic electrons to pass a large repulsive and sharp potential step, is proposed for pulse propagation in a nonuniform fiber Bragg grating with an embedded chirped region. KT can be simply observed as the opening of a transmission window inside the grating stop band, provided that the impressed chirp is realized over a length of the order of the analogue of the Compton wavelength.


I. INTRODUCTION
A remarkable prediction of the Dirac equation is that a below-barrier electron can pass a large repulsive and sharp potential step without the exponential damping expected for a non-relativistic particle. Such a transparency effect, originally predicted by Klein [1] and referred to as Klein tunneling (KT), arises from the existence of negative-energy solutions of the Dirac equation and requires a potential step height ∆V of the order of twice the rest energy mc 2 of the electron [2]. Relativistic tunneling across a smooth potential step, which describes the more physical situation of a constant electric field E in a finite region of space of length l, was subsequently studied by Sauter [3]. Sauter showed that to observe barrier transparency the potential increase ∆V ≃ eEl should occur over a distance l of the order or smaller than the Compton wavelength λ C =h/(mc), the transmission probability rapidly decaying toward zero for a smoother potential increase [2][3][4]. The required field corresponds to the critical field for e + e − pair production in vacuum, and its value is extremely strong making the observation of relativistic KT for electrons very challenging. Therefore, growing efforts have been devoted to find experimentally accessible systems to investigate analogs of relativistic KT [5]. Recently, great interest has suscitated the proposal [6] and first experimental evidences [7,8] of KT for non-relativistic electrons in graphene, which behave like massless Dirac fermions. On the other hand, optics has offered on many occasions a test bed to investigate the dynamical aspects embodied in a wide variety of coherent quantum phenomena (see, for instance, [9] and references therein). In optics, several proposals of KT analogs have been suggested as well, including light propagation in deformed honeycomb photonic lattices [10] whose band structure is similar to the one of graphene [11,12], light refraction at the interface between positive-index and negative-index media [13], spatial light propagation in binary waveguide arrays [14], and stationary light pulses in an atomic ensemble with electromagnetically induced transparency [15]. The experimental implementations of such schemes, however, might be a nontrivial matter, and an experimental observation of KT for photons is still lacking. On the other hand, multilayer and Bragg dielectric structures, such as fiber Bragg gratings (FBGs), are rather simple photonic devices with flexible design that have been successfully demonstrated to provide an accessible laboratory tool to investigate photonic analogues of non-relativistic tunneling phenomena [16][17][18]. Here it is shown that an optical analogue of KT can be achieved in a nonuniform FBG composed by two periodic sections linked by a chirped section which mimics an external potential step in the Dirac equation. Such a FBG-based system might be considered the simplest system proposed so far in order to observe Klein tunneling in any optical system.
where n 0 is the effective mode index in absence of the grating, ∆n ≪ n 0 is the peak index change of the grating, Λ is the nominal period of the grating defining the reference frequency ω B = πc/(Λn 0 ) of Bragg scattering, c is the speed of light in vacuum, and m(z ′ ), 2φ(z ′ ) describe the slow variation, as compared to the scale of Λ, of normalized amplitude and phase, respectively, of the index modulation. Note that the local spatial frequency of the grating is k(z ′ ) = 2π/Λ+2(dφ/dz ′ ), so that the local chirp rate is C = dk/dz ′ = 2(d 2 φ/dz ′2 ).
The periodic index modulation leads to Bragg scattering between two counterpropagating waves at frequencies close to ω B . By letting for the electric field in the fiber, where k B = π/Λ, the envelopes ϕ 1 and ϕ 2 of counterpropagating waves satisfy the coupled-mode equations [19] i and v g ∼ c/n 0 is the group velocity at the Bragg frequency.
The analogy between pulse propagation in the FBG and the Dirac equation in presence of an electrostatic field is at best captured by introducing the dimensionless variables z = z ′ /Z and τ = t/T , with characteristic spatial and time scales Z = 2n 0 /(k B ∆n) and T = Z/v g , and the new envelopes In this way, Eqs.(1-2) can be cast in the Dirac form for the spinor wave function ψ = (ψ 1 , ψ 2 ) T , where V (z) = (dφ/dz) and σ 1,3 are the Pauli matrices, defined by  It should be noted that, as compared to other photonic analogues of KT recently proposed in Refs. [10,14] and based on spatial light propagation in periodic photonic structures, the phenomenon of KT occurring in FBGs and discussed in the following section involves the temporal (rather than the spatial) light dynamics and can be therefore simply investigated in the frequency domain by spectrally-resolved transmission measurements.

III. KLEIN TUNNELING
To realize the analogue of KT, let us first assume that the optical pulse propagates in a region of the grating where m(z) is uniform and equal to one, and let us assume a chirp profile that mimics a step potential with an increase from V = 0 to V = ∆V which occurs over a length l (see Fig.1). Since for the Dirac equation (3) written in dimensionless units the Compton length is λ C = 1 and the rest energy is mc 2 = 1, according to Sauter's analysis KT is expected to be observable for l smaller than ∼ 1 and for a potential height ∆V larger than 2 [2][3][4]. The process of KT and tunneling inhibition for a smooth potential step can be simply explained by a graphical analysis of the space-energy diagrams (z, Ω) of the one-dimensional Dirac equation [2], which are shown in Figs z > 0 region after crossing a forbidden energy region, indicated by the bold segment AB in Fig.1(b), which vanishes for a sharp potential step [l = 0, see Fig.1(a)]. According to Sauter's analysis [2,3], the tunneling probability is appreciable provided that l is smaller than ∼ 1. In the FBG context, the energy diagrams of Fig.1 are equivalent to the bandreflection diagrams introduced by Poladian for a graphical analysis of nonuniform gratings [20], where the energy Ω represents the frequency detuning of the incoming wave from the Bragg frequency ω B . The Sauter's condition l <∼ 1 for KT can be derived following the analysis of Ref. [20] by computation of the transmittance of the effective grating associated to the evanescent region AB shown in Fig.1(b) (see Sec.V.A of Ref. [20]). In the previous discussion, we assumed m(z) = 1, however for a grating with finite spatial extent one has m(z) → 0 as z → ±∞. To inject and to eject the optical pulse into the m(z) = 1 grating region around z = 0, an input and an output apodization sections can be introduced, which adiabatically convert the input and output wave packets from the m(z) = 0 regions into the m(z) = 1 grating region (see Fig.1). Therefore, the general structure of the FBG that realizes a photonic analogue of relativistic tunneling across a potential step consists of five sections, as shown in Fig.1(c Such nonuniform FBG structures should be realizable with current FBG technology based on UV continuous laser writing [21]. It should be finally noticed that, as in an experiment the tracing of pulse evolution in the grating (Fig.2) can be a nontrivial task, the signatures of KT can be simply obtained from standard spectral transmission measurements of the grating. In fact, for a given value of ∆V > 2 and according to the band diagram of Fig.1(a), in the KT regime a transmission window at 1 < Ω < ∆V − 1 [the segment AB in Fig.1(a)], embedded into the two gaps ∆V − 1 < Ω < ∆V + 1 and −1 < Ω < 1 [the segments BC and AD in Fig.1(a)] should be observed in the transmission spectrum, the suppression of KT for a smooth potential corresponding to the lowering of such a transmission window. This is clearly shown in Fig.3

IV. CONCLUSIONS
In conclusion, a photonic analogue of Klein tunneling based on pulse propagation in nonuniform fiber Bragg gratings has been proposed. As compared to other photonic analogues of KT recently proposed in Refs. [10,14] and based on spatial light propagation in periodic photonic structures, the phenomenon of KT in FBGs suggested in this work can be simply observed in the frequency domain as the opening of a transmission window inside the grating stop band, provided that the impressed chirp is realized over a length of the order of the analogue of the Compton wavelength. Such a FBG-based system might be thus considered to be the simplest optical analogue proposed so far to observe KT.