1. 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 Coulomb-potential 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, lead to formation of $em$ transmission bands enclosed by forbidden frequency spectrum known as photonic bandgap (PBG) [3,4]. Over the last four decades, photonic crystals 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, photonic crystals 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 [6–11]. Bragg reflection based waveguiding configurations have been employed as polarization selection device in miniaturized optical sources [12,13]. Bragg-reflection-waveguides with a vacuum core have been forecasted as plausible hosts for future optical particle accelerators driven by 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 led to the formation of stable spatial-solitons in nonlinear Bragg-reflection-waveguides [15]. Recently, the bulk modes of a Bragg-reflection -waveguides or optical surface states in DBR have been found to be promising candidates for optical parametric processes and higher harmonic generation [16–18]. Chirped DBR systems have been introduced in the context of distributed-feedback-lasers for broadband output generation and this has been immensely successful strategy technologically [19,20]. In order to explore the propagation characteristics in a chirped system, the matrix method as well as the coupled-mode theory have been employed [19–21]. The chirped-DBR configurations have also been explored for devising group-velocity-compensating elements in broadband frequency conversion processes [22]. Optimally-designed chirped-DBR architectures have also been explored from the perspective of generation of slow-light and realizing light-trapping schemes [23–26].

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 ($\leq ~0.5~ps$) pulses [27]. 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 [28–31].

2. Adiabatic following in DBR

A periodic variation (with periodicity $\Lambda$) in dielectric constant ($\epsilon$) along $z$-direction in a medium could be expressed as [32]

In case of DBR, the coupling between a forward propagating mode ($A_p \equiv A_i$) and a backward propagating mode ($A_q \equiv A_r$) is of interest. In absence of any other intermodal interactions, the forward-backward mode coupling (assuming $m=1$) would be governed by [32–34],

The solution of Eqs. (5) and (6) for close-to-perfect phase-matched ($\Delta \beta ~\approx ~0$) situation leads to a strong coupling from a forward propagating mode ($|{i}\rangle$) to a backward propagating mode ($|{r}\rangle$) 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 Eqs. (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, $\vert n^{2}_1 - n^{2}_2 \vert$ is fixed and consequently, the width of PBG remains unchanged.

 

Fig. 1. Oblique incidence of $em$-wave on distributed-Bragg-reflector (DBR). $z$-direction represents the optical axis

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2.1 Adiabatic phase-matching

We consider the following transformation to a rotating frame given by

It is apparent that $\Delta k = 0$ along the entire DBR length for the central PBG frequency and $\Delta k \neq 0$ 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 $\Delta k$ from a large negative to a large positive value along the DBR length such that the sweep always remains much smaller than the coupling ($\kappa$). 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 [27,36]

2.2 Chirped DBR configuration for adiabatic mode-conversion

In order to adopt an adiabatic coupling scheme, we consider a DBR configuration with a linear chirp in the duty cycle of each unit cell i.e. thickness $d_{1M} = d_1+ M\delta$ and $d_{2M} = \Lambda - d_1 - M\delta$ define the thickness of layers $A$ and $B$ respectively in $M^{th}$ unit cell as shown in Fig. 2. The unit cell period ($\Lambda$), however remains unchanged. Here, $M = 0,1,2,3,\ldots,(N-1)$ where $N$ is the total number of unit cells in the chirped-DBR (C-DBR). This leads to a longitudinal variation in $\Delta k$ through a monotonic change in average refractive index of an unit cell which could be defined as $\bar{n} = \sqrt {\frac {d_{1M}{n_1}^{2} + d_{2M}{n_2}^{2}}{\Lambda }}$. It is worthwhile to note that the variation of $\bar{n}$ also manifests in the form $z$-dependence of $\kappa$. Using Eqs. (4), (7) and (8), the variation in $\Delta k$ and $\kappa$ for normal incidence ($\theta = 0$) would be expressed as [32]

For an arbitrarily chosen chirp-length of $\delta = 10~nm$, $d_1 = 10~nm$ and $N=39$, the variation in $\Delta k$ and $\kappa$ for the C-DBR is shown in Fig. 3(a). It is apparent that $\Delta \beta$ ($\approx 2\Delta k$) 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 ($\kappa$), on the other hand, reaches a maximum at the center of C-DBR geometry ($M~=~19$) and negligibly small at $z=0,L$. Figure 3(b) shows the variation of $|\tilde {\kappa } \frac {d\Delta \beta }{dz} - \Delta \beta \frac {d\tilde {\kappa }}{dz}|$ in $M^{th}$ unit cell of C-DBR. It is apparent that this is much smaller than $({\tilde {\kappa }}^{2}+\Delta \beta ^{2})^{3/2}$ at any point within the C-DBR. Therefore, the adiabaticity condition described by Eq. (14) is completely satisfied in case of C-DBR. It is interesting to note that the auto-resonant condition $i.e.~\big \vert \frac {\Delta \beta }{\tilde {\kappa }}\big \vert < 2$ is satisfied in a significant fraction of the C-DBR ($M\approx 10$ to $M\approx 26$) as shown in Fig. 3(c) [38].

 

Fig. 2. A schematic to describe the geometry of a chirped-DBR

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Fig. 3. a) shows the variation of $\Delta \beta$ and $\tilde {\kappa }$ in $M^{th}$ unit cell. b) shows the variation of LHS and RHS of the inequality given in Eq. (14) in $M^{th}$ unit cell. c) shows the variation of $\frac {\Delta \beta }{\tilde {\kappa }}$ as a function of unit cell no. ($M$) for depicting a significant fraction of DBR length satisfies the (condition for auto-resonance). d), e) and f) shows a comparison between the reflection spectrum of a normal-DBR (dashed green line) of $d_1 = d_2 = 200~nm$ with that for C-DBRs (solid red line) having $\delta =~10~nm$, $\delta =~5~nm$ & $\delta =~2.5~nm$ respectively.

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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. The material refractive indices $n_1 = 2.5$ (for $TiO_2$) and $n_2 = 1.5$ (for $SiO_2$) were assumed to be constant for the simulations. In order to compare, the reflection spectrum for a normal DBR (with $50\%$ duty-cycle and $\Lambda = 400~nm$) is also shown in 3(d). It is evident that there is $\approx ~40~THz$ increase in PBG for C-DBR as compared to the normal DBR. A normal DBR with arbitrary duty-cycle would have similar PBG but it could be more symmetrically positioned with respect to the PBG of C-DBR. Nevertheless, the PBG enhancement in case of C-DBR would still be $\approx ~40~THz$ with respect to the PBG for a normal DBR. It is interesting to note that the reflectivity drop at the band-edges is relatively smooth with complete suppression of small reflection resonances outside the PBG. An alteration in chirp length has a discernible impact on satisfying the adiabatic constraints as a consequence of change in $\Delta \beta$ and $\kappa$. 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 $\delta = 5~nm$ with $d_1 = 100~nm$ and $N=39$, the reflection spectrum is shown in Fig. 3(e) where the PBG shrinks to $\approx ~30~THz$. 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 $\delta = 2.5~nm$ ($d_1=150~nm$ and $N=39$), PBG for the C-DBR shrinks to $15~THz$ 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 $\kappa$ (see Eq. (16)). By reducing the chirp length (without changing the DBR length $L$), the minimum value of $\tilde {\kappa }$ (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 $\delta = 5~nm$ and $\delta = 2.5~nm$. However, reducing the chirp has a weak impact on $\frac {d\Delta \beta }{dz}$. Overall, adiabaticity conditions are partially satisfied for smaller chirp length and consequently, we obtain a smaller PBG. The C-DBR, in general, could be viewed as a geometry comprised of stacked group of normal DBRs with varying duty cycle. Accordingly, the PBG due to neighbouring unit cells in the C-DBR overlap partially with each other. From this perspective, a random variation in $d_{1M}$ and $d_{2M}$, say up to $10\%$ results in negligible change in overall PBG of C-DBR.

2.3 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 amplitudes $\tilde {a_i}$ and $\tilde {a_r}$ respectively. The parameters determining the geometrical path $\Delta$ (frequency detuning) and $\Omega$ (Rabi frequency) are replaced by $\Delta k$ (phase-mismatch) and $\kappa$ (coupling coefficient). Therefore, the dynamical trajectory on the Bloch-sphere for a perfectly phase-matched ($\Delta k = 0$) interaction in the DBR should resemble an on-resonance interaction in two-level atomic system. Due to the fact that we have one-to-one mapping between the respective parameters in a two-level atomic system with that for a DBR, we expect identical solutions for Eq. (13) in the context of resonant interaction as well as interaction under adiabatic constraints [27]. Consequently, the dynamics of state vector ($\vec {S}$) corresponding to the central PBG frequency (for which $\Delta k = \Delta \beta = 0$) of a normal DBR is given by $S_x = 0,~S_y = sin(\tilde {\kappa }z), ~S_z = -cos(\tilde {\kappa }z)$ [27]. The form of Stokes vector depict the truncation of DBR at a length $z = L_c$ such that $\tilde {\kappa }.L_c = \pi$ would yield a complete conversion of $|{\tilde {a_i}}\rangle$ to $|{\tilde {a_r}}\rangle$ at $L_c$. Figure 4(a) represent such an evolution where the state-vector ($\vec {S}$) traverses from south-pole to the north-pole (for $z=L_c$) along the great circle (red-curve). Here, the DBR has periodicity $\Lambda = 400~nm$, $d_1 = d_2 = \frac {\Lambda }{2}$ and the PBG central frequency is $\nu _c = 181.8~THz$ (where $\Delta k = 0$). It is extremely crucial to note that if the DBR length is i.e. $z~>~L_c$, there is a possibility of backconversion of optical power to the forward-propagating mode. However, from a practical point of view, $\vert \tilde {a_r}\vert ~\approx ~0$ at $z~\geq ~L_c$ due to the fact that a dominant fraction of $\tilde {a_i}$ has already been converted into $\tilde {a_r}$ for $z~<~L_c$ and $|{\tilde {a_r}}\rangle$ is propagating in $-$ve $z$-direction. Consequently, the backconversion to $|{\tilde {a_i}}\rangle$ is comparatively very small. Further, the conversion from $|{\tilde {a_i}}\rangle$ to $|{\tilde {a_r}}\rangle$ resumes for $z~\geq ~2L_c$ of the DBR. Therefore, there is a progressive conversion of optical power to the backward propagating mode which spans over several $L_c$. In such a scenario, complete conversion ($\eta = 1$) is said to be achieved asymptotically ($L \rightarrow \infty$) for $\Delta k = \Delta \beta = 0$. Any non-zero $\Delta k$ (analogous to the detuned interaction) would mean a smaller mode-conversion. In case of frequencies within the PBG (say $\nu = \frac {c}{\lambda } = 200~THz$), the dynamical trajectory would result in termination of the state-vector ($\vec {S}$) at some point on the surface of northern-hemisphere of the DBR Bloch sphere as shown in Fig. 4(b). In consistency with the aforementioned argument, there is also a progressive growth of power along the $z$. For frequencies outside the PBG ($\vert \Delta k\vert >> 0$) or in the transmission band (say $\nu = 300~THz$ or $\nu = 120~THz$), the dynamical trajectory traced by $\vec {S}$ terminates within the southern hemisphere which could be observed in Figs. 4(c) and (d) respectively. The conversion efficiency ($\eta = \frac {S_z+1}{2}$) as a function of propagation distance ($z$) in the normal DBR is shown in Fig. 5(a). Interestingly, the Stokes vector mentioned-above provides an exact description of conversion efficiency (or reflectivity) for all frequencies outside the PBG. It could be noted that $\eta < 1$ for all frequencies except central PBG frequency ($\nu _c = 181.8~THz$) which is consistent with the description in Fig. 4. In order to represent the dynamics of C-DBR, we seek the solutions to Eq. (13) under the constraints imposed by adiabatic following. When the angle ($\Gamma$) between the magnetic field analog and the state vector is very small, the exact solutions are given by [27],

Therefore, $\eta$ could be exactly unity if $\tilde {\kappa } = 0$ or $\Delta k \rightarrow \infty$ at $z = L$. From Fig. 3(a), it is apparent that the former condition is exactly satisfied in case of the designed C-DBR and consequently, we would have $\eta ~=~1$. Due to the fact that $\tilde {\kappa } = \tilde {\kappa }(z)$ and $\Delta \beta = \Delta \beta (z)$ (see Fig. 3(a)), $\vec {S}$ follows a spiralling trajectory from the south-pole to the north-pole. The geometric representation for a C-DBR is shown in Fig. 4 for frequencies within the PBG (Fig. 4(e,f)) as well as outside the PBG (Fig. 4(g,h)). Since, the parameters $\Delta k$ and $\kappa$ exhibit a slow longitudinal variation, $\vec {S}$ always remain perpendicular to $\vec {B}$ at each $z$. Figure 4(e) represents a dynamical trajectory of $\vec {S}$ (at $\nu _c=181.8~THz$) for the variation in $\Delta k$ (or $\Delta \beta$) shown in Fig. 3(a). In this case, $\vec {S}$ traverses from the south-pole to the north-pole through a spiralling trajectory and this results in complete conversion of $|{i}\rangle$ to $|{r}\rangle$ as shown in Fig. 5(b). For another frequency $\nu = 200~THz$ which is within the PBG, the dynamical trajectory appears similar but the spiralling trajectory is asymmetric with respect to the equator 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). In fact, the conversion efficiency for all the frequencies within the PBG of C-DBR reaches (nearly) unity by taking a different path on the DBR-Bloch sphere. On the other hand, $\vec {S}$ remains within the southern hemisphere for frequency outside the PBG (see Figs. 4(g) and (h)). The conversion efficiency for such frequencies remains negligible (close to zero) along the DBR length which could be observed in Fig. 5(b) (blue and red curves). In Fig. 5(c), we plotted the reflection spectrum for a normal-DBR of length $L~=~2~\mu m$ ($N~=~5$), $L~=~2.4~\mu m$ ($N~=~6$) and $L~=~2.8~\mu m$ ($N~=~7$). Similarly, the reflection spectrum for a C-DBR segment of length $L~=~2.8~\mu m$ is shown in Fig. 5(d). This C-DBR segment has been chosen such that it contains $7$-unit cells of layer thicknesses $d_{1p} = 170~nm+ p\delta$ and $d_{2p} = 230~nm - p\delta$ with $\delta = 10~nm$ and $p = 0,1,2,3,4,5,6$.

 

Fig. 4. Shows the evolution of state vector $\vec {S}$ on the Bloch sphere for the normal DBR at frequencies a) $\nu _c = 181.8~THz$ (central frequency) b) $\nu = 200~THz$ c) $\nu = 300~THz$ d) $\nu = 120~THz$ and C-DBR at frequencies (e) $\nu _c = 181.8~THz$ (central frequency) (f) $\nu = 200~THz$ (g) $\nu = 300~THz$ and (h) $\nu = 120~THz$ respectively.

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Fig. 5. a) and b) shows the variation of conversion efficiency as a function of propagation distance for normal DBR and C-DBR at frequencies used in Fig. 4. The black solid line represents the evolution of the central frequency of the PBG, the green solid lines for frequency that lie closer to the band-edges. The blue/purple solid lines correspond to frequencies that lie outside the PBG. c) shows the reflectivity spectrum for normal-DBR z = 2$\mu m$ (N=5), z = 2.4$\mu m$ (N=6) and z = 2.8$\mu m$ (N=7). d) shows the the reflectivity spectrum for C-DBR for z = 2.8$\mu m$ (N=7).

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2.4 Oblique incidence and angular dispersion

A detailed comparison of reflection spectrum and dispersion for a normal DBR and a C-DBR (with $\delta = 10~nm$) 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 = $\approx 60^{\circ }$ 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 $\geq ~50^{\circ }$, thereby leading to a high-reflection band extending from $150~THz$ to $750~THz$ ($1600~nm$ 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 $200-250~THz$, $310-400~THz$ and $490-550~THz$ 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 ($\approx ~60^{\circ }$). By taking material dispersion into consideration, we expect a small change ($\pm ~1^{\circ }$) in Brewster’s angle and hence, an insignificant spectral variation of the transmission spectrum. The PBGs tend to merge thereafter ($\geq ~65^{\circ }$) 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 $(\approx 150 - 750~THz)$ polarization filter for $\approx ~60^{\circ }$ AOI.

 

Fig. 6. a) and b) shows the variation reflection spectrum for TE and TM polarization respectively in a normal DBR ($\Lambda = 400~nm$ 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 $\delta =~10~nm$ and $d_1 = 10~nm$. All the cases, total number of units is $N=39$.

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3. 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 $\kappa$ in addition to the sweep in $\Delta k$ (or $\Delta \beta$). 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 [27]. 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 two- and three-dimensional photonic crystals with focus on applications such as sensing and enhanced nonlinear optical interactions.