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Abstract
We present a heterogeneously coupled Si/SiO2/SiN waveguide structure that can achieve extremely high dispersions (> | ± 107| ps · nm−1km−1). A strong mode coupling between the Si and SiN waveguides introduces a normal dispersion to symmetric mode and an anomalous dispersion to anti-symmetric mode, and the large group velocity difference between the two waveguides results in such high dispersions. Geometric parameters of the structure control the peak dispersions and the central wavelength of the mode coupling, and these engineering capabilities are studied numerically. Analytical representations on the heterogeneously coupled waveguides are also introduced and these equations explain the effects of geometric parameters. This extremely dispersive waveguide scheme can be constructed with other material combinations as well and should be of interest in ultrafast signal processing and spectroscopic applications.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Integration of discrete photonic components on a chip can miniaturize a complicated and bulky optical system into a tiny photonic circuitry. This makes such a complicated system portable and allows us to use it outside of a laboratory, and this has advanced many optical applications such as a telecommunication [1–3], bio- and chemical-sensing [4,5], high-resolution spectroscopy [6,7], and light detection and ranging (LIDAR) [8, 9]. Photonic chips also can improve some functionalities and the robustness of a system, and their wafer-scale fabrication process can reduce the cost of individual chips. Recent advances in nanophotonics and nanotechnology keep revolutionizing photonic chip technologies; for example, all-dielectric metamaterials and plasmonic nanostructures have been used to further scale down the device footprints [10–12]; low-loss waveguides and high quality factor microresonators have been developed for Kerr frequency combs [13–15] and other nonlinear/quantum optical processes [16,17]; and, hybrid integrations of photonic chips with other atomic or ion-qubit systems have been implemented for laser stabilization and quantum information processing [18–21].
In an optical system of ultrafast pulse processing, dispersive photonic components that control the dispersion of a system play vital roles. Dispersions in a system determine the duration and chirp rate of an optical pulse, thus controlling dispersions is essential for various ultrashort pulse applications [22,23]. For example, in high bit-rate data communication networks, dispersions in the system induce a pulse broadening, which causes a signal overlapping, and this limits the capacity of data transmission. Thus, to avoid the signal overlapping, a typical optical communication system requires dispersion compensating components that can prevent the pulse broadening. Furthermore, wavelength dispersive components also can be used in various spectroscopy and sensing applications [24–26], nonlinear pulse compressor [27], mode-locked lasers [28], and supercontinuum generations [29–31].
Over the years, several photonic structures have been proposed to control the dispersions of the system. Such structures include chirped vertical [32] and Bragg [33,34] gratings, angled [35,36] or concentric [37,38] microresonators, asymmetric dual-core or slot waveguides [39–41], and coupled strip-slot waveguides [42–45]. Among these structures, many of them use a mode coupling approach to control the dispersions [37–45]; i.e., when the two asymmetric modes are close by and coupled together, the coupled supermodes try to avoid the mode crossing and this induces a normal dispersion to the symmetric mode and an anomalous dispersion to the anti-symmetric mode. These mode-coupling-induced dispersions can be achieved with various asymmetric modes: for example, guided modes with different widths of slots; with different bending radii; different orders, polarizations, or types of modes, etc. Such dispersion-engineered coupled modes are used in practice for different applications, for example, the coupled modes in a concentric microresonator were used to induce anomalous dispersion for a Kerr frequency comb generation [37].
In this paper, we present a heterogeneously coupled waveguide, the asymmetric structure of which is given by different material composites. The waveguide consists of a Si/SiO2/SiN multilayer stack and shows extremely high dispersions (> | ± 107| ps · nm−1km−1) due to the large group velocity difference between the Si and SiN waveguides. We analytically describe the dispersion profiles of the heterogeneously coupled waveguides and use these equations to fundamentally understand our structure. The dispersion peaks and the central wavelength of a mode-coupling can be engineered using the geometric parameters of the structure, and these engineering capabilities are studied numerically.
2. Mode coupling in heterogeneously coupled SiN/SiO2/Si waveguides
Figure 1(a) shows the schematic cross-section of the proposed waveguide structure that is heterogeneously stacked with Si (red), SiO2 (grey), and SiN (blue) layers. The parameters h1, h2, and h3 are the thicknesses of the Si, SiO2 and SiN layers, respectively, and w0 is the width of the waveguide. Such SiN/SiO2/Si multilayer structures have been explored previously for the development of hybrid optoelectronic platforms and can be fabricated through the vertical integration of SiN on top of silicon-on-insulator (SOI) wafer [46,47]. This approach starts with a standard SOI process first, followed by a spin coating of a flowable oxide and an annealing in the oxygen environment to form a flat SiO2 surface. Then, a low pressure chemical vapor deposition (LPCVD) can be employed to deposit a thick SiN film. Depositing a thick (> 900 nm) SiN film is not trivial due to the high film stress, and deep trenches are required to mitigate the high film stress and to avoid the cracking [13,48,49]. An electron-beam lithography can write a high-resolution pattern on a negative-tone hydrogen-silsesquioxane (HSQ) resist, and the SiN film can be dry-etched through an inductively coupled plasma etching to define the structure. Alternatively, such a hybrid structure also can be implemented through the post-bonding process of the two separate (SOI and SiN) wafers [50,51]; each wafer can be prepared separately with a thin intermediate SiO2 layer on top of Si and SiN structures, and then a hydrophilic fusion bonding of the intermediate SiO2 layers can be used to integrate them into one piece. On a SiN wafer, a set of vent channels needs to be added for a high-quality bonding, and these channels also relieve the film stress. In both approaches, a high temperature (> 1100°C) annealing process can be included to reduce the material loss of a SiN film by removing the residue N-H bonds [13,14,49]. (This step can be skipped for some applications that do not require a low-loss propagation).
Fig. 1 (a) Schematic cross-section of the heterogeneously coupled SiN/SiO2/Si waveguides with geometric parameters; h1, h2, and h3 are the heights of Si, SiO2, and SiN layers, respectively, and w0 is the width of the waveguide. (b) Normalized electric fields (Ey) of the coupled symmetric (Sym, upper) and anti-symmetric (Anti, lower) modes at the wavelengths of λ < λc, λ = λc, and λ > λc, where λc is the mode crossing wavelength. (c) Effective refractive indices (neff) of the symmetric (orange solid line) and anti-symmetric (blue solid line) modes. Orange and blue dashed lines are the neff of the fundamental TM modes at Si and SiN, respectively, when they are isolated without coupling. (d) Second-order dispersion and (e) Third-order dispersion (TOD) profiles of symmetric (orange) and anti-symmetric (blue) modes through the mode coupling. The black dashed line in (e) indicates the mode coupling wavelength (ω = ω0). Geometric parameters are h1 = 200 nm, h2 = 1200 nm, h3 = 1050 nm, and w0 = 1000 nm.
Figure 1(b) shows the numerically simulated mode profiles (normalized Ey) of the symmetric (Sym, upper figures) and anti-symmetric (Anti, lower figures) modes at different wavelengths λ; from left to right, wavelengths shorter (λ < λc), at (λ = λc), and longer (λ > λc) than the mode crossing wavelength λc. The commercially available eigenmode solver was used for the simulation [52], considering the material dispersions of Si, SiO2, and SiN. The geometric parameters are h1 = 200 nm, h2 = 1200 nm, h3 = 1050 nm, and w0 = 1000 nm. Throughout the paper, we restricted the waveguide width to w0 ≈ 1000 nm and our geometries require a thick (> 900 nm) SiN film for a mode coupling; however, in principle, we can achieve a similar mode coupling on a thinner SiN film with a larger w0. Figure 1(c) shows the simulated effective refractive indices (neff) of the coupled supermodes, and Figs. 1(d) and 1(e) are the corresponding second-order dispersions (Dλ) and third-order dispersions (TOD), respectively; orange and blue solid lines are the coupled symmetric and anti-symmetric modes, respectively. The dashed lines in Fig. 1(c) are the fundamental TM (TM0) modes at Si (orange) and SiN (blue) waveguides when they are isolated without mode coupling. Notice that, in Fig. 1(c), the effective indices of the coupled-modes (solid lines) exhibit a bending around the mode crossing wavelength (λc ≈ 1553 nm) of the two isolated modes (dashed lines). The neff bending in the coupled supermodes induces a normal dispersion to the symmetric mode and an anomalous dispersion to the anti-symmetric mode, and these are clearly shown in the dispersion curves of Fig. 1(d). In this regard, the mode coupling phenomenon provides an unique engineering capability for controlling dispersions, and this technique has been applied to achieve various dispersion profiles in waveguides [39–45] and microresonators [37, 38]. For an efficient and selective excitation of a symmetric or anti-symmetric mode, an adiabatic mode converter can be designed to smoothly convert a decoupled (or an isolated) mode into a desired symmetric or anti-symmetric mode [12,37,41]. Our goal in this paper is to achieve extremely high dispersions through such a mode coupling phenomenon, using the coupled Si and SiN waveguide modes, whose group velocity difference is very large with two different composite materials. With the current geometric parameters of Fig. 1, the peak dispersions are approximately ±2.3 × 106 ps·nm−1km−1 (Fig. 1(d)), but these values can be increased/decreased more/less than ±107 ps·nm−1km−1 with further optimization processes in the later section. As the second-order dispersion is made higher, the peak values of the TOD will also become higher; however, as shown in Fig. 1(e), the wavelengths of the peak TOD are different from the wavelength of the mode coupling (ω = ω0) and there would be a negligible TOD effect on the wavelengths of our interest.
3. Analytical representations of heterogeneously coupled waveguides
In this section, we discuss analytical representations of dispersion profiles in heterogeneously coupled waveguides. The propagation constant of a waveguide mode β(ω) can be represented with Taylor series expansions of angular frequency ω, and, assuming the third- and higher-order dispersions are negligible, we can approximate it as the following [22]:
where ν and Dω are the group velocity and group velocity dispersion (GVD), respectively, which are defined by, The ω0 is the angular frequency of the mode crossing point and all the derivatives are evaluated at ω = ω0. In our configuration (as in Fig. 1(a)), there are two waveguide modes β1 and β2 that correspond to the isolated TM0 modes in Si and SiN waveguides. When the two isolated modes β1 and β2 are nearby and coupled together, they form a symmetric (β+) and antisymmetric (β−) supermodes, whose propagation constants are written as the following [1]: The |κ| is the coupling coefficient between the two modes and it depends on the spacing between them (i.e., the thickness of the SiO2 spacer h2) and can be calculated by [1,2] where E1(x, y) and E2(x, y) are the normalized electric fields of each waveguide mode (isolated) and Δε(x, y) is the dielectric perturbation between the two waveguides. By inserting the approximated β1 and β2 of Eq. (1) into Eq. (4), and using the definition of the GVD in Eq. (3), the GVD of the coupled modes can be written as the following: where ω̃ = (ω − ω0) /δω, with the characteristic bandwidth of, Equations (6) and (7) can be approximated further by assuming Dω1 ≈ Dω2 = Dω0, and as follows [41], The peak dispersions (maximum/minimum ) can be obtained at ω = ω0 (ω̃ = 0) and can be written, where c and ng are the speed of light in a vacuum and the group index, respectively. Notice that, the peak values of are quadratically proportional to the group index difference between the two isolated modes. In our heterogeneous structure, the large group index difference (or group velocity difference) is introduced by using the two different sets of materials (Si and SiN) for the waveguides; this is the key idea behind using the SiN/SiO2/Si heterostructure for achieving the extremely high dispersions in the coupled supermodes.The dispersion (Dλ) also can be defined as a second derivative of neff with respect to the wavelength λ and is written as the following:
Note that the sign for the normal and anomalous dispersions are opposite for Dω and Dλ, i.e., Dω > 0 and Dλ < 0 indicate normal dispersions, while Dω < 0 and Dλ > 0 indicate anomalous dispersions. In the following, we plot Dλ that indicates positive values (Dλ > 0) for anomalous dispersion and negative values (Dλ < 0) for normal dispersion.Figure 2 shows the analytical evaluations of the mode coupling, using the equations addressed above. First, the coupling coefficient |κ| between the two waveguides (Si and SiN waveguides) is calculated in Fig. 2(a). The mode profiles of each waveguide are simulated numerically and the |κ| is extracted using Eq. (5). The |κ| is fairly constant for the given spectrum range. Then the dispersion profiles of the symmetric (orange) and anti-symmetric (blue) modes are plotted in Fig. 2(b) using Eqs. (6) and (7). Note that these dispersion profiles from analytical calculations show quite similar results as in Fig. 1(d), which are calculated by full numerical simulations. The full-width-half-maximum (FWHM) bandwidths of the dispersion curves are extracted by fitting the curves and plotted (blue line) in Fig. 2(c) as a function of |κ| (reversed). The bandwidth of the dispersion is almost proportional to the coupling coefficient |κ| and this corresponds with Eq. (9). The peak dispersion wavelengths are also plotted in the same figure, and they are identical at ≈ 1553 nm. The peak values (maximum/minimum) of the dispersions (anti-symmetric/symmetric) can be evaluated using Eq. (10), and they are presented in Fig. 2(d) as a function of |κ| (reversed). As suggested in Eq. (10), the peak dispersions are inversely proportional to |κ|. In a realistic coupled waveguide structure, this |κ| can be controlled by engineering the separation distance between the two coupled waveguides, i.e., the larger separation distance results in a lower |κ| and it is opposite for the smaller separation distance. Notice that the peak dispersions can be higher/lower than ±1 × 107 ps·nm−1km−1 with a coupling coefficient κ < 800 m−1. Again, these extremely high dispersion peaks are due to the large group velocity difference between the two waveguide modes (Si and SiN), which have the quadratic relation with the dispersion peaks (Eq. (10)).
Fig. 2 Analytical evaluations on the heterogeneously coupled waveguides. (a) Numerically calculated coupling coefficient |κ| between the Si and SiN waveguides. Equation (5) is used with the same geometric parameters as in Fig. 1. (b) Dispersion profiles of the symmetric (orange) and anti-symmetric (blue) modes using Eqs. (6) and (7), and the |κ| of (a). (c) The full-width-half-maximum (FWHM) bandwidth (blue) and the peak dispersion wavelength (orange) for different |κ|. (d) Dispersion peaks (maximum/minimum) of the symmetric (orange) and anti-symmetric (blue) modes as a function of |κ|.
4. Dispersion engineering with heterogeneously coupled waveguides
In previous sections, we have seen that the mode coupling in heterogeneously coupled waveguides can induce extremely high dispersions. Such phenomenon can be analyzed using the equations in Sec. 3, and these equations help us to understand the phenomena. To fully utilize its capability to achieve high dispersions and to control them, further optimization process is required. In the following, we will use the numerical simulations to further explore the engineering capability of the dispersion controls with geometric parameters.
As shown in Eq. (10), the peak values of the dispersions in a mode coupling depends on the group velocity difference and the the coupling coefficient between the two modes. The large group velocity difference can be given by choosing two different materials for constructing each waveguide mode, but controlling this group velocity difference with geometric parameters is limited as changing the geometries shifts the mode coupling wavelength as well. However, the coupling coefficient between the two waveguide modes can be engineered independently by the spacing between them, i.e., in our configuration, the height of SiO2 spacer h2. Equation (5) indicates that coupling coefficient |κ| is proportional to the modal overlap between the two modes; in other words, a lower coupling coefficient for a larger separation distance and it’s opposite for a smaller separation distance. To confirm this and to optimize h2, we conducted full modal simulations to track the peak dispersion values for different spacer thickness h2. Figure 3(a) shows the numerically simulated dispersion peaks (maximum/minimum) of the anti-symmetric (blue) and symmetric (orange) mode. For the simulation, the same geometric parameters as in Fig. 1 were used. Note that, peak dispersions over/under ±107 ps·nm−1km−1 are achieved for the spacer thickness of h2 > 1500 nm. This is also in fairly good agreement with Fig. 2(d), which shows the analytically calculated dispersion peaks using Eq. (10). However, there is a trade-off between the peak dispersion and the bandwidth, i.e., the bandwidth decreases as the peak dispersion increases. The blue curve in Fig. 3(b) shows the numerically calculated FWHM bandwidth of the mode coupling induced dispersions for different h2. We can observe the FWHM decreases as the h2 increases (|κ| decreases) that is also in good agreement with Eq. (9) and Fig. 2(c). The wavelength of the peak dispersion (orange curve) is also plotted in Fig. 3(b) and shows an almost fixed wavelength at ≈ 1553 nm. Thus, we can use the SiO2 spacer thickness to control the peak dispersions and the bandwidth of a mode coupling while fixing the central wavelength of it.
Fig. 3 Numerical analysis on the heterogeneously coupled waveguides. (a) Simulated dispersion peaks (blue: anti-symmetric, orange: symmetric) of the heterogeneously coupled waveguides with different heights of SiO2 spacer h2. Other parameters are the same as in Fig. 1. (b) Simulated full-width-half-maximum (FWHM) bandwidth (blue) and the peak dispersion wavelength (orange) as a function of h2. The wavelength of the peak dispersions is almost constant at ≈ 1553 nm.
The wavelength of peak dispersions or the mode coupling wavelength can be engineered by shifting the mode crossing point of the two (Si and SiN) waveguides. This can be done by changing the heights (h1 and h2) and the width (w0) of the waveguides. In our configuration, changing the waveguides’ heights is more effective in shifting the mode coupling wavelength as the two waveguides share the same width w0. To estimate the mode coupling wavelength with different combinations of waveguide heights, the neff of each Si and SiN waveguide are simulated when they are separated and isolated, and they are plotted in Fig. 4(a) with dashed and solid lines, respectively. Different colors represent different heights and the width is fixed at w0 = 1000 nm. In this plot, the wavelength of the crossing point between the dashed and solid lines indicates the mode coupling wavelength. For example, the orange dashed line (h1 = 200 nm) and the orange solid line (h3 = 1050 nm) shows the crossing point at λ ≈ 1553, which corresponds to the cases of Figs. 1–3. The neff of both Si and SiN waveguide modes increases for thicker waveguide heights, due to the higher modal confinement within the waveguide core. For the mode crossing point (wavelength), increasing the h1 shifts the mode coupling wavelength to the longer wavelengths, while increasing the h3 shifts the wavelength to shorter wavelengths; this opposite trend is because the SiN waveguide has a higher group velocity (slope in Fig. 4(a)) than the Si waveguide. To see this trend in detail, the mode crossing wavelengths between the two modes are plotted in Fig. 4(b); blue and orange lines are the tracked mode crossing wavelengths from Fig. 4(a) while changing the waveguide heights of Si (h1) and SiN (h3), respectively. For each case, the other waveguide height is fixed at its original design, i.e., h3 = 1050 nm for the blue line and h1 = 200 nm for the orange line. The full modal simulations on the coupled structure are also conducted for comparison and the mode coupling wavelengths from these simulations are plotted in Fig. 4(b); blue circles and orange crosses match well with their respective blue and orange lines, confirming our estimations on the central wavelengths of the mode coupling with Fig. 4(a). The degree of shift in the mode coupling wavelength is more drastic when changing the h1 than the h3 (Δλc/Δh1 ≈ 4.3 nm/nm and Δλc/Δh3 ≈ 0.68 nm/nm), as the index of Si is higher than that of the SiN; thus, the neff of the Si waveguide is more sensitive to the geometric variations. In other words, the height of Si (h1) can be used as a coarse tuning knob to control the mode coupling wavelength, while the height of SiN (h3) works as a fine tuning knob. In our case, h1 and h3 are chosen to be 200 nm and 1050 nm, respectively, targeting the central wavelength to be at telecommunication wavelength (≈ 1550 nm), but this wavelength can be shifted easily by choosing appropriate combinations of h1 and h3.
Fig. 4 (a) Simulated effective refractive indices (neff) of the isolated Si and SiN waveguides for different thicknesses (dashed lines: neff of Si waveguide for different h1, solid lines: neff of SiN waveguide for different h3). The mode coupling wavelength can be estimated by tracking the crossing point of dashed (neff of Si waveguide) and solid (neff of SiN waveguide) lines. (b) Mode crossing wavelength for different thicknesses. Blue and orange lines are the tracked mode crossing wavelengths from (a), for different h1 (while fixing the h3 = 1050 nm) and h3 (while fixing the h1 = 200 nm), respectively. Blue circles and orange crosses are the mode coupling wavelengths (or the wavelengths of peak dispersions) from the full modal simulations (other geometric parameters: h2 = 1500 nm and w0 = 1000 nm).
Changing the h1 and h3 will also effect the dispersion peaks of both anti-symmetric and symmetric modes, as they will change the group velocities of each mode. Figures 5(a) and 5(b) show the numerically simulated peak dispersions of the anti-symmetric (blue) and symmetric (orange) modes for different waveguide heights of Si (h1) and SiN (h3), respectively. For each simulation, the other waveguide heights are fixed at h3 = 1050 nm (Fig. 5(a)) and h1 = 200 nm (Fig. 5(b)), respectively, and other parameters are set to be h2 = 1500 nm and w0 = 1000 nm. Notice that, as the film thickness increases, both figures show slight changes in peak dispersions but opposite trends; i.e., increasing the h1 decreases the peak dispersions while increasing the h3 increases the peak dispersions. This is because the group index of Si waveguides is greater than that of SiN waveguides. Figures 5(c) and 5(d) show the numerically simulated group indices (ng = c/v) of Si and SiN waveguides, respectively, for different heights of each waveguide (different colors). Circles in each figure represent the mode coupling wavelengths (and corresponding group indices) at each thickness and the dashed black arrows guide the increase of waveguide heights. As noted, the group index of the Si waveguide is greater than that of the SiN waveguide; thus, the group index difference between the two waveguides decreases as the group index of the Si waveguide decreases, while it’s opposite for the SiN waveguide. Figure 5(c) clearly shows that the group index of the Si waveguide decreases as the h1 increases (thus, the group index difference decreases) and this corresponds with the trend in Fig. 5(a) as the peak dispersions are quadratically proportional to the group index difference (Eq. (10)). However, for the SiN waveguide, increasing the h3 decreases the group index of the SiN waveguide and increases the group index difference, thus increasing the peak dispersions as shown in Fig. 5(b).
Fig. 5 Numerically simulated peak dispersions of anti-symmetric (blue) and symmetric (orange) modes while varying (a) h1 (h3 is fixed at 1050 nm) and (b) h3 (h1 fixed at 200 nm). Other parameters are set to be h2 = 1500 nm and w0 = 1000 nm. Simulated group indices (ng) of isolated (c) Si and (d) SiN waveguides for different heights of h1 and h3, respectively. The black dashed arrow guides the increase of each thickness.
Changing the waveguide width w0 will also alter the neff of each waveguide (Si and SiN), thus it will also effect the mode coupling phenomena. However, the effect of the w0 variation will be lower than that of the height parameters, as both Si and SiN waveguides share the waveguide width; i.e., increasing the w0 will increase the neff of both waveguides at the same time. Figure 6(a) shows numerically simulated dispersion peaks (blue: anti-symmetric and orange: symmetric) for different waveguide widths w0. The peak dispersions are relatively constant for different w0, compared to other cases in Figs. 3(a), 5(a), and 5(b). The FWHM bandwidth (blue) and the central wavelength of the dispersion peaks (orange) are also plotted in Fig. 6(b), and show almost flat bandwidth and small variations in the wavelength of dispersion peaks (Δλc/Δw0 ≈ 0.05 nm/nm). Thus, w0 can be used as a finer tuning knob for engineering a mode coupling wavelength. Furthermore, this fine-shift in a mode coupling wavelength with w0 suggests that we can broaden the bandwidth of the dispersion peaks by tapering the waveguide width w0 through the propagation direction [53].
Fig. 6 (a) Peak dispersions of the anti-symmetric (blue) and symmetric (orange) modes for different waveguide width w0. (b) Full-width-half-maximum (FWHM) bandwidth (blue) and central wavelengths of the dispersion peaks (orange) for different w0. Other parameters are set to be h1 = 200 nm, h2 = 1500 nm and h3 = 1050 nm.
5. Conclusion
In summary, we have proposed a heterogeneously coupled waveguides scheme, which is composed of SiN/SiO2/Si asymmetric layers. Such a heterostructure introduces a large group velocity difference between the two waveguide modes, which results in extremely high dispersion peaks when they are coupled. The coupling strength between the two modes can be engineered to control the peak values and the bandwidth of the dispersions, and the peak dispersions > | ± 107| ps·nm−1km−1 are achieved by optimizing the SiO2 spacer thickness. The central wavelength of the peak dispersions also can be engineered covering a broadband at near-infrared, with an appropriate combination of Si and SiN waveguide heights. A fine-tuning on the mode coupling wavelength is also feasible by changing the waveguide width, and tapering the waveguide widths will increase the overall bandwidth of such extreme dispersions. In our configuration, we have used TM modes for the mode coupling; however, in principle, TE modes also can be used to achieve similar high dispersions with different geometric parameters and sensitivities. A similar concept can be applied to other types of heterostructures with different combinations of materials and the capability to achieve such extreme dispersions should be useful in diverse applications such as on-chip pulse compressors, dispersion compensators, and radio-frequency photonic filters.
Funding
This work was supported by a faculty startup fund from Texas Tech University.
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