Geometric representation of adiabatic distributed-Bragg-reflectors and broadening the photonic bandgap

Adiabatic following has been an widely-employed technique for achieving near-complete population transfer in a `two-level' quantum mechanical system. The theoretical basis, however, could be generalized to a broad class of systems exhibiting SU(2) symmetry. In the present work, we present an analogy of population transfer dynamics of two level atomic system with that of light propagation in a classical `one-dimensional' photonic crystal, commonly known as distributed-Bragg-reflector (DBR). This formalism facilitates in adapting the idea of adiabatic following, more precisely the rapid adiabatic passage (RAP) which is usually encountered in a broad class of quantum-mechanical systems. We present a chirped DBR configuration in which the adiabatic constraints are satisfied by virtue of optimally chirping the DBR. The reflection spectrum of the configuration exhibit broadening of photonic bandgap (PBG) in addition to a varying degree of suppression of sharp reflection peaks in the transmission band. The intermodal coupling between counter-propagating modes as well as their phase-mismatch, for the DBR configuration, exhibits a longitudinal variation which is usually observed in `Allen-Eberly' scheme of adiabatic population transfer in two-level atomic systems.


Introduction
The propagation characteristics of electromagnetic (em) waves in a periodically stratified dielectric medium closely resemble the features exhibited by the matter waves in crystalline solids [1,2]. In crystals, the periodic Coulombpotential leads to formation of continuous energy bands separated by forbidden energy spectrum, also known as bandgaps. In resemblance, analogous periodic photonic architectures, commonly termed as photonic crystals (PCs), lead to formation of em transmission bands enclosed by forbidden frequency spectrum known as photonic bandgap (PBG) [3,4]. Over the last four decades, PCs has evolved as backbone of many technological advancements which essentially hinge upon manipulating the spatial and spectral characteristics of light beam [5]. In one-dimension, the PCs are also known as distributed-Bragg-reflector (DBR) and they have turned out to be primary ingredients for devising reflectors/anti-reflectors, spectral and spatial filtering elements and creating efficient sensing platforms which include those involving plasmonic interactions [6,7,8,9,10,11]. Bragg reflection based waveguiding configurations have been employed as polarization selection device in miniaturized optical sources [12,13]. Bragg-reflection-waveguides (BRWs) with a vacuum core have been forecasted as plausible hosts for future optical particle accelerators driven by arXiv:2108.05227v1 [physics.optics] 10 Aug 2021 extremely high power lasers [14]. On the other hand, the possibility of field confinement and therefore, reducing the em interaction volume to sub-wavelength scale has lead to the formation of stable spatial-solitons in nonlinear BRWs [15]. Recently, the bulk modes of a BRWs or optical surface states in DBR have been found to be promising candidates for optical parametric processes and higher harmonic generation [16,17,18]. As mentioned before, a DBR is primarily characterized by PBG whose magnitude is primarily dictated by the refractive index contrast of the DBR constituents and the location in a spectral band is governed by the thickness of constituent layers [5]. Therefore, for a given pair of material forming the DBR, the magnitude of PBG is unique and fixed. Within the limit of optical transparency for the constituent DBR materials, we present a formalism to broaden the PBG (and suppress the transmission band) by utilizing the concept of adiabatic coupling to a backward propagating mode from a forward propagating mode. Therefore, we draw an equivalence of the coupled-wave equations in a DBR with that encountered while describing the dynamics of quantum two-level atomic systems. Subsequently, we propose plausible DBR configurations for adopting the adiabatic following in such systems which leads to broadening of PBG spectrum. The formalism could be translated to any spectral band and PBG broadening is limited by the restriction imposed by material transparency only. Interestingly, this idea provides an viable platform to tailor the backscattered phase from such systems as well.
Adiabatic following, also known as rapid adiabatic passage (RAP) has been an well-established technique for realizing near 100% population transfer in a two-level atomic systems using optimum ultrashort (≤ 0.5 ps) pulses [19]. In fact, it is quite straightforward to ascertain that the RAP mechanism could be adopted in any equivalent system exhibiting special unitary (2) or SU(2) symmetry. For example, RAP provides an efficient platform for ultrashort pulse frequency conversion as well as tailoring spatial characteristics of optical beams in optical nonlinear medium [20,21,22,23].

Adiabatic-following in DBR
A periodic variation (with periodicity Λ) in dielectric constant ( ) along z-direction in a medium could be expressed as [24] (x, y, z) = (x, y) + ∆ (x, y, z) where ∆ (x, y, z) defines the periodic modulation along z. This dielectric perturbation gives rise to the possibility of intermodal interactions. Let us consider any two modes, namely |p and |q propagating through the medium described by dielectric function in equation (1). Assuming the modes are propagating along z-direction, the em-field corresponding to the modes (in the paraxial limit) are given by |p = E p (x, y)A p (z)e i(ωt−βpz) and |q = E q (x, y)A q (z)e i(ωt−βqz) . Under the slowly varying approximation, the evolution of z-dependent mode-amplitude (A p ) of the p th -mode due to the presence of q th -mode (with amplitude A q ) is given by [24] dA q dz where β p and β q are the normal components of wavevector k p and k q respectively. κ pq (m) defines the magnitude of coupling coefficient which couples the modes through the m th Fourier component of dielectric function equation (1). The m th coupling coefficient is expressed as where m is the m th component of Fourier-series expansion of (see equation (1)). The coupling between the interacting modes is maximum when the longitudinal phase-matching condition is exactly satisfied, In case of DBR, the coupling between a forward propagating mode (A p ≡ A i ) and a backward propagating mode (A q ≡ A r ) is of interest. In absence of any other intermodal interactions, the forward-backward mode coupling (assuming m = 1) would be governed by [24,25,26], where we have dropped the indices from κ and ∆β = β r − β i − 2π Λ . In case the DBR constituent materials are isotropic, the dielectric function (equation (1)) is a purely scalar and consequently, transverse-electric (T E) to transverse-magnetic (T M ) mode-coupling (and vice-versa) is forbidden. It is worth noting that the phase-matching condition for coupling the forward propagating mode (|i ) to a backward propagating mode (|r ) or vice-versa is contra-directional in nature and therefore, Eq. (4) could be expressed as ∆β = 2β − 2π Λ = 0 where β i = −β r = β [24]. As shown in Fig.1, the phase-matching condition for the contra-directional coupling process is modified to ∆β = 2β cos θ − 2π Λ = 0 in case of oblique incidence (θ with respect to z-axis),. For a simple case when the two layers of the DBR (with refractive indices n 1 and n 2 ) share the same thickness i.e. d 1 = d 2 , the coupling coefficient takes a simplified form given by [24], The solution of equations (5) and (6) for close-to-perfect phase-matched (∆β ≈ 0) situation leads to a strong coupling from a forward propagating mode (|i ) to a backward propagating mode (|r ) for broad range of frequencies. This is realized for all the frequencies within the PBG and consequently, we obtain a strong reflection band. It is apparent from equations (7) (or (8)) that the contrast in refractive index (or dielectric constant) is the primary factor determining the strength of coupling and hence, the sharpness of band edges. For a given choice of DBR constituents, |n 2 1 − n 2 2 | is fixed and consequently, the width of PBG remains unchanged.

2
. The complex amplitudesã i andã r represent the power contained in the forward and backward propagating modes in the rotated reference frame. It is worthwhile to note that |ã i | 2 = |A i | 2 and |ã r | 2 = |A r | 2 . The substitution of equation (9) and (10) into equations (5) and (6) yields which is an equivalent representation of time-dependent Schrödinger equation in optics. In fact, if we define a state Ψ = ã r a i then −i d dz |Ψ =Ĥ |Ψ where time-dependence is replaced by the z-dependence and the Hamiltonian H = σ. B. Here, σ = (σ x , σ y , σ z ) are the Pauli-spin matrices and B = (κ, 0, ∆k) represents a fictitious magnetic field. The Hamiltonian has a close resemblance with that representing the dynamics of a spin-1/2 particle in an external magnetic field where |i , |r are the equivalent to the spin-up (|↑ ) and spin-down (|↓ ) states [27]. In order to represent a particular pseudo-spin state, the equivalent Stokes parameters could be given by S j = σ j (j ≡ x, y, z) and the evolution of the state would be dictated by where˙ S ≡ ∂ S ∂z ). The states and their evolution described by equations (12) and (13) determine the evolution dynamics of the forward and backward propagating modes. In other words, equations (12) and (13) could be mapped onto an equivalent Bloch-sphere where |i (spin-up) and |r (spin-down) states are located the south-pole and north-pole respectively. Any point of the DBR Bloch-sphere represents a particular superposition of eigenstates |i and |r . Sec. 2.2 discusses the geometric representation in detail for specific cases of evolution of these states. The fictitious magnetic-field ( B) constitutes a parameter space for exerting a torque on the system which leads to precession of the state vector ( S) about B with a frequency | B| = |κ| 2 + ∆k 2 . It is worthwhile to point out that the value S z (equation (12)) determines the mode-conversion (from the forward-to-backward or vice-versa) factor. The state S = [0, 0, −1] represents a situation where the incident beam contains all the optical power i.e.ã i = 1 and the state S = [0, 0, 1] represents the case when all the optical power is present in the reflected beam (ã r = 1). The conversion efficiency to the reflected beam is expressed as η = Sz+1 2 as shown in Fig. 5. The dynamics, here, is primarily controlled by the parameters ∆k and κ which are essentially equivalent to factors detuning (∆) and Rabi-frequency (Ω) respectively which are commonly encountered in a two-level atomic system. In a perfectly phase-matched case (∆k = 0), which is satisfied for the central PBG frequency, a gradual semi-circular rotation of the state-vector ( S) from south-pole to the north-pole takes place on DBR Bloch-sphere which implies a complete transfer of optical power from the incident beam to the reflected beam in the DBR. In order not to obscure the salient features of the equivalent dynamics, we have ignored the role played by absorption loss (α) in the dielectric materials. However, it would be worthwhile to point out that the role of absorption loss is equivalent to that played by semi-phenomenological decay constant (T 2 ) in a two-level atomic system [19]. For ∆k = 0, S traces a trajectory on the DBR Bloch-sphere about the axisn = κx+∆kẑ √ κ 2 +∆k 2 . This implies that ∆k = 0 leads to transportation of state-vector S to a point in northern-hemisphere. This point may be near the north-pole but it will never be the north-pole of the equivalent DBR Bloch-sphere. Nevertheless, S z still positive and the reflectivity is large (not maximum). The frequencies for which S z remains positive forms a band of frequencies constituting the PBG. However, a sufficiently large ∆k results in an such that S is transported to a point which lies on the southern hemisphere of DBR Bloch-sphere (assuming S starts precessing about B from south-pole). Therefore, S z would be negative after propagation through the DBR. This corresponds to frequencies outside the PBG (or within the transmission band) where reflectivity is much smaller than the PBG. In fact, the equator of DBR Bloch-sphere distinguishes the PBG (northern hemisphere) from the transmission (pass) band (southern hemisphere) with respect to the trajectory of S.
It is apparent that ∆k = 0 along the entire DBR length for the central PBG frequency and ∆k = 0 for all other frequencies. The aforementioned analogy to a two-level atomic system, thus, allows us to adopt the Rapid Adiabatic Passage (RAP) mechanism for broadening the PBG spectrum and suppressing the transmission (pass) bands in the DBR. Consequently, we propose DBR configurations which leads to slow variation in ∆k from a large negative to a large positive value along the DBR length such that the sweep always remains much smaller than the coupling (κ). This would ensure nearly complete transfer of optical power from a forward propagating to a backward propagating mode across a frequency band extending much beyond the conventional PBG of a DBR. The RAP condition is expressed as [19,28] κ∆k − ∆kκ << (|κ| 2 + ∆k 2 ) which essentially implies that the rate of change of polar angle Φ = tan −1 ( κ ∆k ) (angle between B and z axis) during the evolution is much slower with the frequency of rotation (= √ κ 2 + ∆k 2 ) for S about B. If κ is assumed to be constant (along z), the adiabaticity condition appears as d∆k dz << (κ 2 +∆k 2 ) 3/2 κ . Also, in order to achieve near-complete optical power transfer, it is essential that the two states (modes) |i and |r are decoupled at the entry (z = 0) and exit (z = L) faces of the DBR. Alternately, this is mathematically expressed as | ∆β κ | >> 2 at z = 0, L which is equivalent to satisfying the condition of autoresonance in 'two-wave' interaction system [29,30]. In this case, autoresonance essentially ensures that the counter-propagating modes remain phase-locked when the parameters of the Hamiltonian undergo an adiabatic change. This manifests into a near-complete transfer of optical power from the forward to the backward propagating mode.

Chirped DBR configuration for adiabatic mode-conversion
It is worthwhile to note that the variation ofn also manifests in the form z-dependence of κ. Using equations (4), (7) and (8), the variation in ∆k and κ for normal incidence (θ = 0) would be expressed as [24] For an arbitrarily chosen chirp-length of δ = 10 nm, d 1 = 10 nm and N = 39, the variation in ∆k and κ for the chirped-DBR (C-DBR) is shown in Fig. 3(a). It is apparent that ∆β (= 2∆k) varies symmetrically from a large negative (at z = 0 or M = 0) to a large positive value (at z = L or M = 38). The coupling coefficient (κ), on the other hand, reaches a maximum at the center of C-DBR geometry (M = 20) and negligibly small at z = 0, L. Figure  3(b) shows the variation of κ d∆β dz − ∆β dκ dz in M th unit cell of C-DBR. It is apparent that this is much smaller than (κ 2 + ∆β 2 ) 3/2 at any point within the C-DBR. Therefore, the adiabaticity condition described by equation (14) is completely satisfied in case of C-DBR. It is interesting to note that the auto-resonant condition i.e. ∆β κ < 2 is satisfied in a significant fraction of the C-DBR (M ≈ 10 to M ≈ 28) as shown in Fig. 3(c). This essentially implies that the counter-propagating modes (at a certain frequency) would be phase-locked in the entire interaction region if they are phase-matched (∆β = 0) in any one unit cell of C-DBR.
The impact of closely satisfying the adiabatic constraints have a profound impact on reflection spectrum shown in Fig. 3(d). In order to obtain the reflection spectrum, we consider a cross-section as shown in Fig. 2 where N = 39 is considered. A broadband plane-wave is incident on the C-DBR from z = 0. The reflection spectrum was obtained using finite element method (FEM) (wave-optics module, COMSOL Multiphysics). The periodic boundary condition is imposed along the transverse direction and a mesh-size of 5 nm is considered for the simulation. Also, we ignore the material dispersion in the present case and assumed n 1 = 2.5 (A ≡ T iO 2 ) and n 2 = 1.5 (B ≡ SiO 2 ) across the entire spectrum. In order to compare, the reflection spectrum for a normal DBR (no-chirping and Λ = 400 nm) is also shown in 3(d). It is evident that there is ≈ 240 nm increase in PBG for C-DBR as compared to the normal DBR. Also, the reflectivity drop at the band-edges is relatively smooth with complete suppression of small reflection resonances outside the PBG. In fact, any alteration in structural parameters has an impact ∆β and κ and consequently, the adiabatic constraints are not satisfied closely. This is expected to reduce the PBG in reflection spectrum along with appearance of sharp transmission resonance (outside PBG). For example, when the chirp-length changes to δ = 5 nm with d 1 = 100 nm and N = 39, the reflection spectrum is shown in Fig. 3(e) where the PBG shrinks to ≈ 182 nm. It is also worth noting that the reflection spectrum is accompanied by oscillating side-bands with sharp transmission resonances on both ends of the PBG. On reducing the chirp length further to δ = 2.5 nm (d 1 = 150 nm and N = 39), PBG for the C-DBR shrinks to 100 nm and discernibly sharper transmission resonances on both ends of PBG which is similar to that exhibited by normal DBR. The underlying cause behind this observation could be traced to the variation of κ (see equation (16)). By reducing the chirp length (without changing the DBR length L), the minimum value of κ (at z = 0) and z = L) increases. Therefore, the forward and backward propagating modes are not completely decoupled at the ends of C-DBR when δ = 5 nm and δ = 2.5 nm. However, reducing the chirp has a weak impact on d∆β dz . Overall, adiabaticity conditions are partially satisfied for smaller chirp length and consequently, we obtain a smaller PBG.

Geometric representation of propagation characteristics in DBR
The analogy between a two-level atomic system and propagation dynamics in a DBR is apparent from Eqs. (12) and (13) where the time-evolution has been replaced by evolution along z-axis. The ground-state and excited state populations are analogues to complex amplitudesã i andã r respectively. The parameters determining the geometrical path ∆ (frequency detuning) and Ω (Rabi frequency) are replaced by ∆k (phase-mismatch) and κ (coupling coefficient). Therefore, the dynamical trajectory on the Bloch-sphere for a perfectly phase-matched (∆k = 0) interaction in the DBR should resemble an on-resonance interaction in two-level atomic system. Consequently, a π-pulse (κ.z = π) at the resonance is equivalent to exactly satisfying the Bragg's condition which manifests into complete conversion ofã i intoã r resulting in reflectivity R = 1. On an equivalent Bloch-sphere, Fig. 4(a) represent such an evolution where the state-vector ( S) traverses from south-pole to the north-pole along the great circle (red-curve). Here, the DBR has periodicity Λ = 400 nm, d 1 = d 2 = Λ 2 and the PBG central wavelength is λ c = 1650 nm (where ∆k = 0). Any non-zero ∆k (analogous to detuned interaction) would mean a smaller mode-conversion. In case of frequencies within the PBG (say λ = 1500 nm), the dynamical trajectory would result in termination of the state-vector at some point on the surface of northern-hemisphere of the DBR Bloch sphere as shown in Fig. 4(b). Interestingly, for frequencies outside the PBG (|∆k| >> 0) or in the transmission band (say λ = 1000 nm or λ = 2500 nm), the dynamical trajectory traced by S terminates within the southern hemisphere which could be observed in Figs. 4(c) and (d) respectively. The conversion efficiency (η = Sz+1 2 ) as a function of propagating distance (z) for all the aforementioned frequencies in normal DBR is shown in Fig.5(a). It could be noted that η < 1 for all wavelengths except central PBG wavelength (λ c = 1650 nm) which is consistent with the description in Fig. 4.
The geometric representation for a C-DBR is shown in Fig. 4 for frequencies within the PBG (Fig. 4e,f) as well as outside the PBG (Fig. 4g,h). Since, the parameters ∆k and κ exhibit a slow longitudinal variation, S always remain perpendicular to B at each z. Therefore, S follows a spiralling trajectory after beginning its journey from the south-pole. Figure 4(e) represents a dynamical trajectory of S (at λ c = 1650 nm) for the variation in ∆k (or ∆β) shown in Fig.  3(a). In this case, S traverses from the south-pole to the north-pole through a spiralling trajectory and this results in complete complete conversion of |i to |r as shown in Fig. 5(b). For another wavelength λ = 1500 nm which is within the PBG, the dynamical trajectory appears similar but it exhibits asymmetrically distributed spirals (with respect to the equator) while travelling from the south-pole to the north-pole of DBR Bloch sphere (see Fig. 4(b). In this case, the conversion efficiency reaches unity through a different path as shown in Fig. 5(b). On the other hand, S remains within the southern hemisphere for wavelength outside the PBG (see Figs. 4(g) and (h)).

Oblique incidence and angular dispersion
A detailed comparison of reflection spectrum and dispersion for a normal DBR and a C-DBR (with δ = 10 nm) is elucidated in Fig. 6(a)-(d). The dependence of reflection spectrum on angle-of-incidence (AOI) for TE and TM polarization in a normal-DBR is shown in Fig. 6(a) and (b) respectively. The oblique incidence essentially leads to a smaller value of normal component of the wavevector and consequently, the PBGs shift to higher frequencies with increasing angle of incidence. As expected, the PBGs tend to broaden for the TE polarization on oblique incidence. On the other hand, the low frequency PBGs for TM polarization tends to reduce up to an AOI = ≈ 60 • and increases thereafter. The drop in PBG for TM polarization is essentially due to the Brewster's effect at the interface of high and low-index layers. Figure 6(c) and (d) represent the reflection spectrum for the TE and TM polarization respectively for C-DBR. A comparison between Fig. 6(a) and (c) reveals that the C-DBR exhibits appreciably broad PBGs with very narrow transmission bands separating them. At oblique incidence, the PBG in C-DBR broadens further and shifts to higher frequencies. This is accompanied by shrinking of transmission bands. In fact, the PBGs tend to overlap for AOI ≥ 50 • , thereby leading to a high-reflection band extending from 150 T Hz to 750 T Hz (1600 nm band). Such broad PBGs are usually not found in normal DBR, even with very wide refractive index contrast. It is also interesting to note that the reflection spectrum (in Fig. 6(c)) exhibits three omnidirectional PBGs which are located in 160 − 250 T Hz, 310 − 400 T Hz and 490 − 550 T Hz frequency range. The PBG, in this case (for normal as well as oblique incidence) is limited by the material transparency window and could be extended further (on both spectral ends) through suitable choice of materials. A similar comparison between Fig. 6(b) and (d) depict broadened PBG in case of TM polarization in C-DBR and narrow transmission bands. However, Fig. 6(d) exhibits a sharp transmission resonance at the Brewster's angle (≈ 60 • ). The PBGs tend to merge thereafter (≥ 65 • ) giving rise to a broad high reflection band. A comparison between Fig. 6(c) and (d)reveals that the C-DBR could be employed as a broadband (≈ 150 − 750 T Hz) polarization filter for ≈ 60 • AOI. Figure 6: a) and b) shows the variation reflection spectrum for TE and TM polarization respectively in a normal DBR (Λ = 400 nm and d 1 = d 2 ) as a function of angle of incidence. c) and d) shows the reflection spectrum for TE and TM polarization respectively in C-DBR as a function of angle of incidence. The C-DBR parameters are δ = 10 nm and d 1 = 10 nm. All the cases, total number of units is N = 39 .

Conclusion
In conclusion, we present an approach to understand the propagation characteristics of modes in a DBR using general techniques adopted in a wide variety of systems exhibiting SU(2) dynamical symmetry. The coupled-mode equations describing the forward and backward propagating modes in a DBR could be represented in the form of a single optical Bloch-equation where the evolution of state-vector depicts the dynamical behaviour of the wave propagation. This provides a platform to draw an analogy with a two-level atomic system and consequently, adopt a formalism for adiabatic evolution in DBR based configurations. In order to realize conditions imposed by adiabatic constraints, C-DBR configurations have been investigated in detail. The DBR variants exhibit enhancement of PBG along with varying degree of suppression of sharp transmission resonances in the reflection spectrum. The impact of alteration in physical parameters of the DBR is explored in detail. It is worth pointing out that the C-DBR configuration involves a discernible longitudinal variation in mode-coupling coefficient κ in addition to the sweep in ∆k (or ∆β). Conventionally, such adiabatic process have a close resemblance with the Allen-Eberly scheme defined in the context of population-transfer in two-level atomic systems [19]. Novel DBR designs could further be explored which satisfy the adiabatic constraints. An interesting extension of this proposal would be to investigate the evolution of geometric-phase in such DBR configurations and the possibility to control the backscattered (reflection) phase through suitable DBR design. This promises to provide an unique and flexible platform for tailoring the spatial features of an optical beam using adiabatic DBR configurations. Nevertheless, a natural extension of this proposal would be to explore the viability of this formalism in twoand three-dimensional photonic crystals with focus on applications such as sensing and enhanced nonlinear optical interactions.

Disclosures
The authors declare that there are no conflicts of interest related to this article.