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Abstract
Parity-time symmetric Bragg gratings produce unidirectional reflection around the exceptional point. We propose and explore directional coupling of gain and loss modulated waveguide Bragg gratings operating at around 880 nm with long-range surface plasmon polaritons. Step-in-width modulation of a Ag stripe supporting long-range plasmons combined with a periodic modulation of the cladding were used to balance the real and imaginary index perturbation of the gratings. IR140 dye molecules in solvent forms a portion of the uppercladding, providing gain under optical pumping. We investigate directional coupling between a pair of parity-time symmetric waveguide Bragg gratings operating near their exceptional point, arranged in various configurations - duplicate, duplicate-shifted and duplicate-flipped. We also investigate coupling to a bus waveguide and to a conventional waveguide Bragg grating. Unidirectional multi-wavelength reflection and coupled supermode conversion are predicted.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Parity-time (PT) symmetry has emerged as a remarkable and promising research area in photonics, yielding new design paradigms and devices exhibiting unusual behaviour such as single-sided Bragg reflectors and vortex lasers [1,2]. The eigenvalues of a non-Hermitian Hamiltonian H will in general be complex, except when H commutes with the parity-time (PT) operator [H, PT] 0, in which case its eigenvalues are real and H is said to be PT symmetric or PT unbroken. If H is not PT symmetric then it is termed PT broken. The location in parameter space that separates the PT symmetric state from the PT broken state is termed the exceptional point, and at this location the eigenvalues of H are degenerate and its eigenvectors are parallel [3]. PT symmetry concepts and the splitting or merging of eigenvalues in the real or complex domains have inspired numerous applications and been applied, e.g., to micro-rings [4–6], directional couplers [7–9], and Bragg gratings [10] to realize functions such as generating vortex beams, unidirectional reflection, and coherent perfect absorption.
Consider a PT symmetric Bragg grating (PTSBG) with a modulated real and imaginary index following:
A single PTSBG incorporated within a waveguide provides symmetric transmission and asymmetric reflection. Incorporating PTSBGs within a directional waveguide coupling system enriches the system, leading to interesting performance characteristics. Previous work along this thread includes a theoretical study of directional coupling between gain and loss waveguides or gratings [14,15], along with PTSBG-assisted directional couplers [16], and a switch designed based on a PTSBG coupled to a bus waveguide [17].
In this paper, we consider waveguide PTSBGs operating with long-range surface plasmon-polaritons (LRSPPs) [18], coupled to a similar PTSBG (arranged in various configurations), a nearby bus waveguide, and a nearby conventional Bragg grating. By applying a step-in-width configuration, the real effective index can be modulated. Moreover, the LRSPP mode on the metal stripe is naturally lossy, whereas dye molecules integrated in the structure provide gain, which together are used to modulate the imaginary effective index. The structures of interest are described in Section 2. Computational results, their analysis and discussion are given in Section 3. Concluding remarks are given in Section 4.
2. Parity-time symmetric Bragg grating (PTSBG)
A PTSBG requires precisely-balanced modulated perturbations of the real and imaginary parts of the effective index for operation at the exceptional point. Figures 1(a) – 1(c) show the scheme investigated, which consists of an active LRSPP metal stripe waveguide patterned as a step-in-width structure. The specific geometry of the waveguide segments was determined via modal analysis (Fig. 2) and the grating response using the eigenmode expansion (EME) method at wavelengths near λ = 880 nm.
Fig. 1. (a) Isometric sketch of the proposed PTSBG structure; (b) top and (c) side views. (d) Effective index of the LRSPP along the grating structure with different colors representing different slices of the unit cell: neff1 = nave + Δnr + Δni; neff2 = nave - Δnr + Δni; neff3 = nave - Δnr - Δni and neff4 = nave + Δnr - Δni. (e) Transmittance and reflectance spectra for 500 periods (TMM calculation - dashed line) and 700 periods (EME simulation - solid line); R – right, L – left. Transmittance R to R implies reflectance for incidence from the right side (similarly for transmittance L to L). LRSPP mode profile shown in inset.
Fig. 2. Effective index of the LRSPP mode as function of: (a) width of the Ag stripe (hAg = 40 nm, hPMMA = 0 nm); (b) thickness of the Ag stripe (wAg = 1 µm, hPMMA = 0 nm); and (c) thickness of the PMMA cover layer (wAg = 1 µm, hAg = 40 nm). The solid curves correspond to the fundamental LRSPP (ssb°) whereas the dotted cuves in (a) pertain to the first high-order LRSPP (asb°) [24].
The grating has a period of $\mathrm{\Lambda }$ = 290 nm and 500 - 700 unit cells. The Ag stripe is selected as 40 nm thick on 1 nm of Cr (adhesion) on a BK7 wafer, to support LRSPPs. The Ag stripe has four steps in width per unit cell, w1 = 1.04 µm, w2 = 0.96 µm, w3 = 1.42 µm and w4 = 1.55 µm, each step being of 25% duty cycle, to produce the required real effective index modulation. The 300 nm thick PMMA cover layer is patterned following the same period as the Ag grating ($\mathrm{\Lambda }$ = 290 nm) but with a duty cycle of 50%, forming uppercladding segments on the Ag stripe portions of width w3 and w4. BK7 and PMMA were chosen as the substrate and upper cladding materials because they have a similar refractive index (nBK7 = 1.5093 [19], nPMMA = 1.498 [20]), thereby forming an almost symmetric waveguide.
IR140 dye molecules dissolved in solvent form the remainder of the upper cladding, infilling the grooved regions between PMMA segments. A gain of ∼360 cm−1 and a refractive index of 1.45 can be obtained from IR140 dissolved in a mixture of dimethyl sulfoxide and ethylene glycol near λ = 880 nm [21]. Here we propose the use of cinnamaldehyde as a replacement for ethylene glycol to achieve a higher index of 1.509 in order to match the index of BK7, and (conservatively) assume 250 cm−1 of material gain.
The grating segments covered by PMMA support a lossy LRSPP mode whereas the segments covered by the dye solution support an amplified LRSPP mode. As shown in Fig. 1(d), the perturbation of the real and imaginary effective index are matched and shifted by a quarter period. The computed average effective index of the LRSPP in this structure is amplifying (nave = (neff1 + neff2 + neff3 + neff4) / 4 = 1.5185 + j0.0054), implying that the gratings investigated could be used as distributed feedback lasers. The computed LRSPP mode profile shown in inset to Fig. 1(e) is clearly a highly symmetric LRSPP mode.
Figure 1(e) shows the transmission and reflection spectra of this design. The dashed lines are the calculation results based on the transfer matrix method (TMM) applied to an equivalent dielectric slice model of the structure [22], and the solid lines are the simulation results from the eigenmode expansion method (EME) [23]. The EME method considers scattering loss due to perturbation and mode mismatch along a grating, thereby providing a more accurate representation of the expected performance when the perturbation strength increases. Bidirectional transmission and unidirectional reflection (single-sided reflection from the left input) and are observed from these results, confirming operation at the EP. For a PTSBG, the average imaginary effective index (nave,im) can change the transmission and reflection levels. If nave,im > 0 as in this design, the transmission and reflection are amplified, whereas they will be unity if nave,im = 0 and lossy if nave,im < 0.
Figures 2(a) and 2(b) show the effective index of the LRSPP mode versus the width and thickness of the Ag stripe. Single (long-range) mode propagation requires a Ag stripe width of less than 2 µm. Control over the gain of the mode is provided by the PMMA cover layer as shown in Fig. 2(c). The two sharp peaks in Im{neff} in Figs. 2(a) and 2(b) are due to perturbations in coupling between edge and corner SPPs that constitute the LRSPP on a metal stripe, as observed previously in a similar structure [25]. By adjusting the thickness and width of the Ag stripe and the thickness of the PMMA layer, the effective index of the LRSPP can be perturbed such that a grating design operates at the exceptional point, as summarised in Fig. 1.
3. Directional coupling with parity-time symmetric Bragg gratings (PTSBGs)
In this section, we investigate in detail directional coupling involving one or two PTSBGs of the design shown in Fig. 1, to determine their behaviour and explore the possibility of applications using this system. We consider the input and output modes to be the symmetric and asymmetric supermodes on coupled uniform waveguide segments bounding the PTSBGs. The single waveguide input behavior can be understood by superposition of the symmetric and asymmetric supermodes at the input. In all figures that follow, ‘A’ and ‘S’ designate the asymmetric and symmetric supermodes, respectively, while ‘L’ and ‘R’ designate left and right input ports (if the latter are omitted then the response is the same regardless of which port is taken as the input). For all configurations investigated, the left and right reference planes, where the reflectance and transmittance are computed, are identified as ‘L’ and ‘R’ – see Figs. 3(a) to 7(a), and these quantities are defined in the usual way as ratios of mode power.
Fig. 3. (a) Top view of coupled duplicate PTSBGs. (b) Transmittance and (c) reflectance spectra. Mode profiles of the symmetric and asymmetric coupled modes (supermodes) are shown as the top right and bottom left insets to Part (b), respectively. Top view fragments of the distribution of the electric field magnitude for (d) incident symmetric supermode at the ‘LS to LS’ peak of Part (c), and (e) incident asymmetric supermode at the ‘LA to LA’ peak of Part (c).
3.1 Coupling involving duplicate and duplicate/shifted PTSBGs
A coupled system involving two PTSBGs is sketched in top view in Fig. 3(a), following the colour scheme of Fig. 1(d). In this system, a second PTSBG of identical design is coupled laterally to the first. The PTSBGs are each connected in cascade to input and output waveguide segments of the same design as used to implement the segment of neff4. These input and output waveguides are coupled and support LRSPP symmetric and asymmetric coupled modes (supermodes). The center-to-center separation between PTSBGs and between waveguides is set to 1.5 µm to produce a difference in the effective index of the coupled supermodes while ensuring that the asymmetric supermode remains guided.
The spectral responses of this system are plotted in Figs. 3(b) and 3(c). The behaviour of the supermodes follow intuitively the behaviour shown in Fig. 1(e): Bidirectional transmission and unidirectional reflection (single-sided reflection from the left input). Each supermode has a different behavior due to the slight difference in their complex effective index (mode profiles of the symmetric and asymmetric coupled supermodes are shown as insets to Fig. 3(b)). This slight difference causes a difference in the transmission responses observed in Fig. 3(b), and splits the center reflection wavelength as observed in Fig. 3(c). Also, the reflection from the right side is not exactly zero because in the coupled system, the two supermodes cannot be simultaneously located on the EP, so both are shifted slightly to PT broken states.
Figures 3(d) and 3(e) show in top view fragments of the distribution of the electric field magnitude at the ‘LS to LS’ and ‘LA to LA’ peaks of Fig. 3(c). Due to the large simulation volume of the 3D plasmonics model, these distributions were calculated for a 2D effective index equivalent model using Lumerical EME. The symmetric and asymmetric supermodes are clearly observed in Figs. 3(d) and 3(e), respectively.
A coupled system where the second PTSBG is shifted to the left by Λ/2 relative to the first PTSBG is sketched in Fig. 4(a). This system produces a very different behavior compared to the un-shifted system of Fig. 3(a). The transmission responses are now similar for both supermodes, as shown in Fig. 4(b). The structure still produces single-sided reflection from the left but now supermode conversion occurs, as shown in Fig. 4(c): The incident symmetric supermode emerges upon reflection as the asymmetric supermode, and vice versa. A single reflection peak is observed indicating that conversion occurs at the same wavelength for both supermodes. Indeed, the reflection response in this case (Fig. 4(c)) is identical with the reflection response of an isolated PTSBG (Fig. 1(e)).
Fig. 4. (a) Top view of coupled duplicate and shifted PTSBGs. (b) Transmittance and (c) reflectance spectra. Top view fragments of the distribution of the electric field magnitude for (d) incident symmetric supermode at the ‘LS to LA’ peak of Part (c), and (e) incident asymmetric supermode at the ‘LA to LS’ peak of Part (c).
Supermode conversion is caused by shifting the second PTSBG which introduces a phase shift in the reflection coefficient of this grating relative to the first. Consider an infinite periodic structure which supports a Floquet-Bloch mode having as a solution to the wave equation a field distribution of the following form [26]:
where $\psi (z )$ represents an electromagnetic field component distributed along the propagation direction z, k is the Bloch wavenumber, and ${u_k}(z )$ is a periodic function of the same period $\mathrm{\Lambda }$ as the effective index modulation. Two identical gratings support the same Bloch mode. However, if one of these gratings is shifted relative to the other by $\mathrm{\Delta \Lambda }$, the Bloch modes of the two gratings will differ in phase by $\mathrm{\Delta }\theta = 2\pi \Delta \mathrm{\Lambda }/\mathrm{\Lambda }$.In the present case $\mathrm{\Delta \Lambda }$ = Λ/2 which implies Δθ = π. Thus, the phase of the reflection coefficient looking into the second grating (in isolation) differs by π relative to the first grating (in isolation), thereby forcing supermode conversion (S ↔ A) at the input of the coupled system (Fig. 4(c)). Conversion involves both supermodes propagating with equal weight into the PTSBGs, which averages their effective index, such that reflection occurs at a center wavelength. Figures 4(d) and 4(e) show in top view fragments of the distribution of the electric field magnitude at the ‘LS to LA’ and ‘LA to LS’ peak of Fig. 4(c) – they are similar, and each can be viewed as equal-weight superpositions of the symmetric and asymmetric supermodes.
Any other shift in period would lead to a phase difference between 0 and ${\pm} $ causing three reflection peaks to occur - two peaks associated with the reflection of each supermode, and one associated with mode conversion. A quarter-period shift ($\mathrm{\Delta \Lambda }$ = Λ/4, Δθ = π/2) leads to three peaks of similar magnitude.
3.2 Coupling involving duplicate and flipped PTSBGs
A coupled system involving two PTSBGs of identical design but where the second is flipped (end-to-end) relative to the first is sketched in top view in Fig. 5(a). Combining PTSBGs that individually produce a unidirectional reflection (Fig. 1) in this manner yields a coupled system where the supermodes exhibit bidirectional reflection along with supermode conversion.
Fig. 5. (a) Top view of coupled duplicate and flipped PTSBGs. (b) Transmittance and (c) reflectance spectra.
Figures 5(b) and 5(c) give the computed transmittance and reflectance spectra, which are observed to be fully bidirectional (identical regardless of which port is taken as input). Figure 5(b) shows bidirectional transmission spectra similar to the previous cases. In this design, three reflection peaks are observed in Fig. 5(c), with the middle peak associated with supermode conversion, which occurs at the same wavelength as in the previous case (Fig. 4(c)). Mode conversion occurs because flipping introduces a phase difference in the reflection coefficients of the PTSBGs (in isolation). Shifting the flipped PTSBG relative to the first changes the phase relationship and the reflection level associated with mode conversion, but the latter remains at the same wavelength.
3.3 Coupling involving one PTSBG and a bus waveguide
A directional coupler where one PTSBG is coupled to a bus waveguide is shown in Fig. 6(a). Figures 6(b) and 6(c) show the transmittance and reflectance spectra computed for this case. Figure 6(b) shows bidirectional transmittance spectra similar to the other cases. Three unidirectional reflection peaks of similar level are evident in Fig. 6(c). Interestingly, and counter-intuitively, this structure also produces mode conversion upon reflection at the same center wavelength as the response of an isolated PTSBG (Fig. 1(e)), between the reflection peaks associated with the two supermodes, even though the bus waveguide produces minimal reflection of its own (in isolation) at the location of the coupled PTSBG. The bus waveguide, being longitudinally invariant, supports mode propagation in both longitudinal directions with arbitrary phases. Therefore both unconverted and converted coupled modes are valid modal solutions at the left input of the coupled system. Moreover, a bus waveguide has an index equal to the average index of two PTSBGs with a half period shift. The three reflection peaks in Fig. 6(c) can be seen as the superposition of the reflection peaks in Figs. 3(c) and 4(c).
Fig. 6. (a) Top view of a PTSBG coupled to a bus waveguide. (b) Transmittance and (c) reflectance spectra. Top view fragment of the distribution of the electric field magnitude for (d) incident asymmetric supermode at the intersection of the ‘LA to LA’ and ‘LA to LS’ curves (marked by a green circle) of Part (c), along with expanded views of the (e) input and (f) output regions.
The electric field profile of this system is similar to the two coupled systems discussed in Section 3.1 at the three reflection peaks. Figures 6(d), 6(e) and 6(f) show the electric field distribution at the intersection of the ‘LA to LA’ and ‘LA to LS’ curves (marked with a green circle). This field distribution can be viewed as superpositions of the symmetric and asymmetric supermodes with different weightings along the propagation direction.
3.4 Coupling involving one PTSBG and a conventional Bragg grating
A directional coupler where one PTSBG is coupled to a conventional Bragg grating of the same period is shown in Fig. 7(a). This system involves coupling between a unidirectional and a bidirectional grating.
Fig. 7. (a) Top view of a PTSBG coupled to a conventional Bragg grating of the same period. (b) Transmittance and (c) reflectance spectra.
Figure 7(b) shows bidirectional transmission spectra similar to the other cases. However, the reflection spectra of Fig. 7(c) reveal a different performance. There are still three strong reflection peaks for the left-side input, including one with mode conversion. However, a stronger mode conversion reflection peak arises for the right-side input. Perhaps unsurprisingly, coupling a PTSBG operating at the EP with a conventional grating yields a system that has less unidirectionality than the cases considered in Sections 3.1 and 3.3.
4. Concluding remarks
A PTSBG operating at its exceptional point and based on LRSPPs was proposed, and its use in directional coupling systems was analysed. Five coupling systems were considered: duplicate PTSBGs, duplicate and shifted PTSBGs, duplicate and flipped PTSBGs, one PTSBG coupled to a nearby bus waveguide, and one PTSBG coupled to a nearby conventional Bragg grating. All coupling configurations produce bi-directional supermode responses in transmission and unidirectional supermode responses in reflection except for the duplicate and flipped case. Unidirectional supermode conversion (symmetric to asymmetric and vice versa) can also be produced upon reflection in some cases, with the conversion strength adjusted by shifting the period between duplicate and coupled PTSBGs.
The material gain was adjusted such that the average effective index of the LRSPP in the isolated PTSBG, and of the LRSPP supermodes in the coupled systems, are amplifying. The material gain can easily be adjusted by altering the dye concentration or adjusting the optical pump intensity. Transmittance and reflectance levels can be greater than unity and increase with the number of unit cells, suggesting amplification and lasing applications.
The structures proposed and investigated are feasible in light of previous experimental work on similar LRSPP devices, including Bragg gratings and passive integrated optical structures [22,24], amplifiers [20,21], and DFB lasers [27] involving IR140.
Generally, unidirectional wavelength filters [11], switches [17] and PT symmetric lasers [2,13,28] are potential applications. A dual-wavelength single-sided DFB supermode laser would be an interesting application of the structures investigated here. Changing the separation in the directional couplers changes the effective index of the supermodes propagating thereon, leading to changes in the wavelength of the reflection peaks - the separation could thus be used as an additional design parameter to adjust the lasing wavelengths.
Funding
Natural Sciences and Engineering Research Council of Canada.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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