Abstract
A photonic analogue of Klein tunneling (KT), that is, 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.
1. 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 nonrelativistic 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 of the order of twice the rest energy of the electron [2]. Relativistic tunneling across a smooth potential step, which describes the more physical situation of a constant electric field in a finite region of space of length , was subsequently studied by Sauter [3]. Sauter showed that to observe barrier transparency, the potential increase should occur over a distance of the order or smaller than the Compton wavelength , the transmission probability rapidly decaying toward zero for a smoother potential increase [2–4]. The required field corresponds to the critical field for 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 [16–18]. 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.
2. Quantum-Optical Analogy
The starting point of our analysis if provided by a standard model of light propagation in an FBG with a longitudinal refractive index , where is the effective mode index in absence of the grating, is the peak index change of the grating, is the nominal period of the grating defining the reference frequency of Bragg scattering, is the speed of light in vacuum, and , describe the slow variation, as compared to the scale of , of normalized amplitude and phase, respectively, of the index modulation. Note that the local spatial frequency of the grating is , so that the local chirp rate is . The periodic index modulation leads to Bragg scattering between two counterpropagating waves at frequencies close to . By letting for the electric field in the fiber, where , the envelopes and of counterpropagating waves satisfy the coupled-mode equations [19] where and 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 and , with characteristic spatial and time scales and , and the new envelopes . In this way, (1) can be cast in the Dirac form for the spinor wave function , where and are the Pauli matrices, defined by
In its present form, (2) is formally analogous to the one-dimensional Dirac equation with in presence of an external electrostatic potential , playing the role of a dimensionless (and generally space-dependent) rest mass (see, for instance, [2, 4]). As is wellknown, a nonvanishing mass is responsible for the existence of a forbidden energy region, which separates the positive- and negative-energy branches of the massive Dirac equation. The optical analogue of the forbidden energy region is precisely the photonic stop band of the periodic grating. As the refractive index modulation of the grating, that is, the mass term in the Dirac equation (2), is decreased, the stop band region shrinks and the limit of a massless Dirac equation (similar to the one describing the dynamics of electrons in graphene near a Dirac point) is attained. The additional external potential in (2), related to the chirp of the grating according to , changes the local position of the forbidden energy region. Therefore, pulse propagation in an FBG with a suitably designed chirp profile can be used to mimic the relativistic tunneling of a wave packet in a potential step . It should be noted that, as compared to other photonic analogues of KT recently proposed in [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.
3. Klein Tunneling
To realize the analogue of KT, let us first assume that the optical pulse propagates in a region of the grating where is uniform and equal to one, and let us assume a chirp profile that mimics a step potential with an increase from to which occurs over a length (see Figure 1). Since for the Dirac equation (2) written in dimensionless units the Compton length is and the rest energy is ; according to Sauter's analysis KT is expected to be observable for smaller than and for a potential height larger than [2–4]. 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 of the one-dimensional Dirac equation [2], which are shown in Figures 1(a) and 1(b) for a sharp and for a smooth potential step, respectively. For the sake of clearness, in the figures the potential has been chosen to yield a nonvanishing and constant chirp rate over a length ; different forms for the potential step, such as the profile considered in the seminal work by Sauter [3], can be assumed as well without changing the main results.
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The space-energy diagrams of Figures 1(a) and 1(b) schematically show the behavior of the energy spectrum of (2) versus , which is composed by two branches—the electron and positron energy branches of the Dirac equation—separated by a gap of width and centered along the curve . The gap regions are visualized in the diagrams by the shaded areas. A wave packet (optical pulse) in the electron branch with an initial mean energy () coming from tunnels into the region after crossing a forbidden energy region, indicated by the bold segment AB in Figure 1(b). The forbidden gap region AB vanishes for a sharp potential step (, see Figure 1(a)). According to Sauter's analysis [2, 3], the tunneling probability is appreciable provided that is smaller than . In the FBG context, the energy diagrams of Figure 1 are equivalent to the band-reflection diagrams introduced by Poladian for a graphical analysis of nonuniform gratings [20], where the energy represents the frequency detuning of the incoming wave from the Bragg frequency . The Sauter's condition for KT can be derived following the analysis of [20] by computation of the transmittance of the effective grating associated to the evanescent region AB shown in Figure 1(b) (see Section of [20]). In the previous discussion, we assumed , however for a grating with finite spatial extent one has as . To inject and to eject the optical pulse into the grating region around , an input and an output apodization sections can be introduced, which adiabatically convert the input and output wave packets from the regions into the grating region (see Figure 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 Figure 1(c): two boundary apodization sections (regions I and V), and two uniform sections (regions II and IV) separated by a central chirped section of length (region III). In Figures 2(a)–2(c) typical examples of pulse tunneling across the potential step are presented, showing KT for a sharp potential step (Figure 2(a)) and inhibition of tunneling as the step gets smooth (Figures 2(b) and 2(c)). The figures depict the temporal evolution of —which is proportional to the field intensity averaged in time over a few optical cycles and in space over a few wavelengths—as obtained by numerical analysis of (1) for a grating length of with a quarter-cosine apodization profile (see Figure 2(d)), , and for a few values of . A forward-propagating Gaussian pulse of mean energy coming from and of duration (FWHM in intensity) has been assumed as an initial condition. For typical parameter values , and nm, which apply to FBGs used in optical communications, the spatial and temporal scales in Figure 2 are mm and ps, respectively. Hence, in physical units the grating length is cm whereas the optical analogue of the Compton length is mm. 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 (Figure 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 and according to the band diagram of Figure 1(a), in the KT regime a transmission window at (the segment AB in Figure 1(a)), embedded into the two gaps and (the segments BC and AD in Figure 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 Figure 3, where the spectral transmittance of the FBGs corresponding to the simulations of Figures 2(a), 2(b), and 2(c) are depicted. Note that, as the length of the chirped region is increased (from Figures 3(a) to 3(c)), the transmission window embedded in the two adjacent gaps disappears, which is the signature of KT inhibition.
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4. 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 [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.
Acknowledgment
The author acknowledges financial support by the Italian MIUR (Grant no. PRIN-2008-YCAAK, Project “Analogie ottico-quantistiche in strutture fotoniche a guida d'onda”).