Abstract

We propose double-cladded and multi-cladded waveguide structures that enable flexible engineering of group-velocity dispersion over multi-octave spanning spectral range, allowing to produce intriguing dispersion characteristics that oscillate between normal and anomalous dispersion in a sinusoidal-like fashion with controllable magnitude, which will be very useful for nonlinear photonic applications. The proposed waveguides are easy to fabricate in practice, which we expect to have great potential for broad applications in nonlinear photonics, quantum optics, and optical communication.

© 2016 Optical Society of America

1. Introduction

Group-velocity dispersion (GVD) of photonic devices underlies many nonlinear optical phenomena, such as optical parametric amplification/oscillation [1], photonic signal processing [1], soliton formation and fission [1], supercontinuum generation [2], Kerr frequency comb generation [3], etc., that exhibit great potential for broad applications. In the past two decades, significant efforts have been devoted to engineer GVD in a variety of device structures and material platforms, from photonic crystal fibers [4] to chip-scale nanophotonic waveguides/resonators [5–23]. Although many approaches are able to tailor the dispersion curve to various spectral regions, the capability of engineering high-order dispersion are generally limited. It was shown recently that slotted waveguide structures exhibit great potential for flexible dispersion engineering [11–14,17–21]. However, current designs require extremely stringent precision of device nanofabrication and heterogeneous material etching, which imposes significant challenge on device fabrication in practice. On the other hand, a majority of current dispersion engineering focuses on producing flat dispersion profile over a broad band [9,11,12,14–23]. However, group-velocity matching is essential for many nonlinear optical phenomena [2,24–26], which cannot be achieved with a flat profile of constant GVD. For example, recent studies [26] showed that oscillatory GVD between normal and anomalous dispersion leads to multi-band group-index matching, which enables the generation of intriguing multicolor solitons. In this paper, we propose simple double-cladded and multi-cladded waveguide structures that are not only easy to fabricate, but also offer significant flexibility in dispersion engineering, allowing us to realize intriguing GVD characteristics over multi-octave spectral range. The proposed waveguide structures and dispersion characteristics would be of great potential for broad applications in nonlinear photonics, quantum optics, and optical communication.

2. Result and discussion

The proposed double-cladded waveguide structure is schematically shown in Fig. 1. It consists of a waveguide core, with a width of W and a height of H, sitting on a substrate, coated with two cladding layers with thickness of D1 and D2, respectively. The refractive indices are arranged such that nc > ns and n1 < (nc, n2), where nc and ns are those of the waveguide core and substrate, respectively, and n1 and n2 are those of the sandwiched and top cladding layers, respectively (see Fig. 1(b)). The condition nc > ns is simply to confine the optical mode inside the waveguide. The second condition n1 < (nc, n2) is to take advantage of the peculiar waveguide dispersion offered by the double-cladding wave guidance. As we will show below, this simple double-cladded waveguide structure exhibits significant advantage for dispersion engineering in comparison with conventional waveguides [5–23], allowing to realize desired intriguing dispersion characteristics over multi-octave spectral range. In particular, the proposed waveguide is easy to fabricate, since it requires only one etching step to define the waveguide core, and the two cladding layers can be coated with high precision of layer thickness via a certain deposition method such as chemical vapor deposition (CVD), atomic layer deposition (ALD), etc. The material of the top cladding layer can be either same as or different from that of the waveguide core, depending on applications, as long as it can be deposited conveniently.

 figure: Fig. 1

Fig. 1 (a) Schematic of the proposed double-cladded waveguide structure. (b) The cross section of the waveguide, where W and H are the width and height of the waveguide core, and D1 and D1 are the thicknesses of Cladding 1 and 2, respectively.

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To show the potential of the proposed waveguide for dispersion engineering, we use silicon carbide (SiC) as an example material platform in the following. The example waveguide consists of a SiC waveguide core sitting on a sapphire substrate, coated with a layer of aluminum oxide and a layer of SiC on the top, which is illustrated in the inset of Fig. 2(a). The dispersion of the waveguide is simulated by the finite element method, with the material dispersion of all layers taken into account [27,28]. As shown in Fig. 2(a), appropriate waveguide dimension is able to produce waveguide dispersion to compensate the material dispersion of SiC, resulting in group index nearly constant over an octave-spanning spectrum (Fig. 2(b)) for the fundamental quasi-transverse electric (quasi-TE) mode with the electric field dominantly lying in the device plane. Consequently, the waveguide mode exhibits small GVD value oscillating between ±0.005 ps2/m over nearly an octave spectral range from 1.6 μm to 3.1 μm (Fig. 2(c)), with four zero-dispersion wavelengths (ZDWLs) located at 1.651, 2.043, 2.647 and 3.053 μm, respectively. This magnitude of GVD is more than an order of magnitude smaller than conventional slotted waveguides but over a much broader spectral range [11–14, 17–21], clearly showing the advantage of the proposed waveguide structure. The oscillatory GVD results in oscillatory group-index profile, as shown in Fig. 2(b). As a result, the group index can be matched over multiple spectral regions, which will be very useful for resonant nonlinear optical interactions [2,24–26].

 figure: Fig. 2

Fig. 2 (a) Group velocity dispersion of the fundamental quasi-TE mode (blue) of the double-cladded waveguide schematically shown in the inset. The SiC waveguide core has a width of W = 1.597 μm and a height of H = 665 nm. The Cladding 1 is aluminum oxide with a thickness of D1 = 210 nm and the Cladding 2 is SiC with a thickness of D2 = 195 nm. To show the effect of waveguide dispersion, the red curve shows the case without material dispersion, where the refractive indices of materials are set as nSiC = 2.5686 and nAl2O3 = 1.7462. The green curve shows the material dispersion of SiC material. (b) Group index for the fundamental quasi-TE mode of the waveguide. (c) Detailed group-velocity dispersion of the waveguide, which is the same as the blue curve in (a). The insets show the optical mode field profiles at three wavelengths indicated by the arrows. (d) Blue curves: Energy ratios of the fundamental quasi-TE mode in the waveguide core, Cladding 1, and Cladding 2, respectively. Red curve: The effective mode area of the waveguide. See Ref. [29,30] for the detailed expression of the effective mode area.

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The double cladding structure primarily assists in the wave confinement to produce desired waveguide dispersion. It introduces only minor perturbation to the energy distribution of the field mode. As shown in Fig. 2(d) (solid blue curve), the fundamental quasi-TE mode is primarily confined inside the waveguide core, with a fraction of mode energy greater than 69% over the entire octave spectral range up to 3.2 μm. Only small fractions of energy are distributed in the two cladding layers, as indicated by the blue dashed and dotted curves of Fig. 2(d). The overall effective mode area increases roughly quadratically with optical wavelength, which is a typical signature of a normal optical waveguide. Moreover, as indicated by the optical mode field profiles given in the insets of Fig. 2(c), the fundamental quasi-TE mode exhibits great mode overlap for optical wavelengths covering a broad spectral range. These elegant features of mode confinement and overlap are very useful for many nonlinear photonic applications that might involve nonlinear interactions among optical waves with far separate wavelengths.

The double-cladded waveguide exhibits great capability of flexible dispersion engineering. For example, purely normal or anomalous GVD with nearly constant values can be obtained over very broad band with appropriate device dimensions. As shown in the black curve of Fig. 3(a), purely anomalous GVD with a value around −0.051 ps2/m and a variation magnitude of only ±0.003 ps2/m can be obtained over a broad spectral range from 1.8 to 2.9 μm, with waveguide dimensions of W = 1.517 μm, H = 697 nm, D1= 210 nm, and D2 = 195 nm. By changing the dimensions of the waveguide core to W = 1.677 μm and H = 637 nm while keeping the thicknesses of the two cladding layers unchanged, we are able to shift the whole dispersion curve to normal dispersion region with a value around 0.048 ps2/m and a variation magnitude of only ±0.010 ps2/m, over a broader spectral range from 1.4 μm to 3.2 μm (red curve in Fig. 3(a)). These types of dispersion characteristics would be very useful for broadband dispersion compensation, nonlinear photonics, and photonic signal processing.

 figure: Fig. 3

Fig. 3 (a) Examples of purely normal and anomalous GVD over broad band. The waveguide geometry of green line is same as Fig. 2 (a). Transferring from the anomalous GVD to normal GVD can be obtained by varying the width and height of the waveguide core simultaneously. Black line: W = 1.517 μm, H = 697 nm. Red line: W = 1.677 μm, H = 637 nm. Blue line: W = 1.757 μm, H = 613 nm. For all cases, the thicknesses of the two cladding layers remain as D1= 210 nm and D2 = 195 nm, respectively. (b) Group velocity dispersion of the fundamental quasi-TM mode, for a double-cladded waveguide with the following geometry: W = 768 nm, H = 1.420 μm, D1 = 303 nm, and D2 = 252 nm. The insets show the optical mode field profile at three wavelengths indicated by the arrows.

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The double-cladded waveguide also provides slotted-type waveguide confinement along the vertical direction similar to conventional slotted waveguides [11–14,17–21], which would offer flexible dispersion for the quasi-transverse magnetic (quasi-TM) guided modes with electric field dominantly along the vertical direction. One example is shown in Fig. 3(b), where the fundamental quasi-TM mode exhibits oscillatory GVD over a broad band of 1.0–4.2 μm, with a magnitude of ∼ ±0.1 ps2/m. In general, the quasi-TM modes are more sensitive to the waveguide confinement along the vertical direction, while the quasi-TE modes are more impacted by the waveguide confinement at the two sides of the waveguide. To obtain similar dispersion characteristics, the quasi-TE modes generally require less thickness of the waveguide core than the quasi-TM modes, which is more favorable for device fabrication in practice. Therefore, we focus on the fundamental quasi-TE modes in the following discussion.

In practice, due to fabrication imperfection, there generally exists a certain geometry variation of fabricated devices. It is thus important to investigate the sensitivity of device GVD to the variation of device geometry. Figure 4 shows the dependence of GVD on small variations of waveguide geometry in various dimensions, for the waveguide example given in Fig. 2(c). As shown in Fig. 4(a), the width variation of the waveguide core by ∼30 nm perturbs little to the GVD at wavelength below 2 μm where the guided mode dominantly resides inside the waveguide core, but changes GVD from −0.005 to −0.028 ps2/m at longer wavelengths around 3 μm where the guided mode becomes more sensitive to the waveguide core width. The overall dispersion curves, however, still show similar oscillatory features with four ZDWLs. Similar features are observed when the thickness of the sandwiched cladding layer varies, where GVD at wavelength below 2 μm is perturbed by small amount. In this case, however, GVD at longer wavelengths is changed fairly significantly since where the wave guidance is considerably impacted by the cladding layer. On the other hand, the height variation of the waveguide core (Fig. 4(b)) shifts the whole GVD curve and tilts it by a certain amount since such geometry perturbation is experienced by the guided mode at various wavelengths over a broad band due to the small waveguide height, a feature similar to conventional strip waveguides [5–10]. For a same reason, GVD behaves in a similar fashion when the thickness of the top cladding layer changes, as shown in Fig. 4(d). These studies indicate that the GVD of the double-cladded waveguide tends to be more sensitive to the layer thickness variation than that of the waveguide width. In particular, the thickness of the sandwiched cladding layer needs to be precisely controlled to obtained desired dispersion characteristics. This can be done easily for the double-cladded waveguide since the layer can be coated with very high thickness precision by certain deposition method such as ALD. In contrast, conventional slotted waveguides [12–14,19–21] rely on small etched slots for dispersion engineering, which is challenging to control in practice.

 figure: Fig. 4

Fig. 4 Dependence of GVD on the variations of various dimensions of the double-cladded waveguide example given in Fig. 2. (a) Impact of the width variation of the SiC waveguide core. (b) Impact of the height variation of the SiC waveguide core. (c) Impact of the thickness variation of Cladding 1 – the aluminum oxide cladding layer. (d) Impact of the thickness variation of Cladding 2 – the SiC cladding layer.

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Since the cladding layers can be conveniently deposited in practice, the idea of double-cladded waveguide can be extended to multi-cladded waveguides, which would allow even more degrees of freedom for dispersion engineering. The device structure is schematically shown in Fig. 5. In general, the refractive indices of the waveguide layers should be arranged alternately as {nc, n2, n4, ...} > {ns, n1, n3, ...} so as to take advantage of the intriguing waveguide dispersion. The number of cladding layers employed would depend on specific application. In such a multi-cladded waveguide, a guided wave at a short wavelength would be primarily confined inside the waveguide core, resulting in GVD dominantly determined by the waveguide core. When the optical wavelength increases, the guided wave experiences more wave guidance from the cladding layers, resulting in GVD considerably impacted by the cladding layers. The longer the optical wavelength, the more impacted the outer cladding layers impose on device dispersion. Therefore, the multi-cladded waveguide structure offers a large degree of freedom for broadband dispersion engineering.

 figure: Fig. 5

Fig. 5 Schematic of a multi-cladded waveguide structure.

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Figure 6 shows two examples. By adding one more cladding layer of aluminum oxide to form a three-cladded waveguide, we are able to achieve a GVD profile oscillating in a sinusoidal-like fashion, with five ZDWLs and a GVD magnitude of as small as ±0.017 ps2/m over a broad band from 1.5 to 4.6 μm (Fig. 6(a)). Interestingly, for a three-cladded waveguide, by changing the waveguide geometry, we can obtain a similar sinusoidal-like oscillatory GVD, but now with six ZDWLs over a very broadband spectrum from 1.4 to 5.7 μm covering nearly two octave spectrum, with a similar GVD magnitude of ±0.018 ps2/m (Fig. 6(b)). These types of intriguing dispersion characteristics would offer great opportunities for realizing many intriguing nonlinear optical phenomena and functionalities that cannot be achieved in devices with conventional dispersion properties. For example, our recent study [26] shows that devices with GVD characteristics shown in Fig. 6 supports the generation of peculiar multi-color solitons that are of great potential for broad applications in frequency metrology, optical frequency synthesis, and optical spectroscopy.

 figure: Fig. 6

Fig. 6 (a) Group-velocity dispersion with 5 ZDWLs, for a three-cladded waveguide. The SiC waveguide core has a width of W = 875 nm and a height of H = 1.865 μm. The Cladding 1 (Al2O3), Cladding 2 (SiC), and the top Cladding 3 (Al2O3) have thicknesses of D1= 125 nm, D2 = 195 nm, and D3= 1048 nm, respectively. (b) Group-velocity dispersion with 6 ZDWLs, for a three-cladded waveguide. The SiC waveguide core has a width of W = 884 nm and a height of H = 1.720 μm. The Cladding 1 (Al2O3), Cladding 2 (SiC), and the top Cladding 3 (Al2O3) have thicknesses of D1= 122 nm, D2 = 200 nm, and D3= 995 nm, respectively. The insets in (a) and (b) show the optical mode field profiles at the wavelengths indicated by the arrows.

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Note that, to focus on the essential physical phenomena, our simulations here assumed double-cladded and multi-cladded waveguide structures with ideal rectangular shape for simplicity. In practice, real fabricated devices might exhibit rounded corners and edges on the deposited cladding layers and the plasma-etched waveguide core might exhibit a certain slant angle on the sidewalls. They would impact the exact values of group-velocity dispersion of the waveguide, which need to be taken into account when one models real fabricated devices in practice. With the significant capability and flexibility of dispersion engineering offered by the proposed waveguide structure, these perturbations can be compensated easily with appropriate design of waveguide dimensions. Detailed analysis will be left for future exploration.

3. Conclusion

We have proposed new waveguide structures, double cladded waveguide and multi-cladded waveguide with appropriate refractive index arrangement, that are not only easy to fabricate in practice, but also exhibit very flexible dispersion engineering capability to obtain intriguing dispersion characteristics over multi-octave spectral region that have never been demonstrated before. Although the discussions above used SiC and aluminum oxide as material examples to show the related dispersion engineering capability and characteristics, the idea is universal and can be applied to other material platforms since the waveguide dispersion of the proposed waveguides can be flexibly designed to compensate material dispersion. Our detailed analysis shows that similar dispersion characteristics can be obtained with many other device materials such as silicon, silicon nitride, etc, as long as the refractive indices of cladding layers are arranged properly. Therefore, we expect the proposed waveguide structures and dispersion characteristics would be of great potential for broad applications in nonlinear photonics, quantum optics, optical communication, and photonic signal processing.

Funding

Defense Advanced Research Projects Agency SCOUT program (DARPA) (W31P4Q-15-1-0007) from AMRDEC.

Acknowledgments

The authors thank Dr. Lin Zhang for helpful discussion.

References and links

1. G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007).

2. J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135 (2006). [CrossRef]  

3. T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science 332, 555–559 (2011). [CrossRef]   [PubMed]  

4. P. St. J. Russell, “Photonic-crystal fibers,” J. Lightwave Technol. 24, 4729–4749 (2006). [CrossRef]  

5. E. Dulkeith, F. Xia, L. Schares, W. M. J. Green, and Y. A. Vlasov, “Group index and group velocity dispersion in silicon-on-insulator photonic wires,” Opt. Express 14, 3853–3863 (2006). [CrossRef]   [PubMed]  

6. A. C. Turner, C. Manolatou, B. S. Schmidt, M. Lipson, M. A. Foster, J. E. Sharping, and A. L. Gaeta, “Tailored anomalous group-velocity dispersion in silicon channel waveguides,” Opt. Express 14, 4357–4362 (2006). [CrossRef]   [PubMed]  

7. A. Saynatjoki, M. Molot, J. Ahopelto, and H. Lipsanen, “Dispersion engineering of photonic crystal waveguides with ring-shaped holes,” Opt. Express 15, 8323–8328 (2007). [CrossRef]   [PubMed]  

8. M. R. E. Lamont, C. M. de Stkerke, and B. J. Eggleton, “Dispersion engineering of highly nonlinear As2S3 waveguides for parametric gain and wavelength conversion,” Opt. Express 15, 9458–9463 (2007). [CrossRef]   [PubMed]  

9. X. Liu, W. M. J. Green, X. Chen, I.-W. Hsieh, J. I. Dadap, Y. A. Vlasov, and R. M. Osgood Jr, “Conformal dielectric overlayers for engineering dispersion and effective nonlinearity of silicon nanophotonic wires,” Opt. Lett. 33, 2889–2891 (2008). [CrossRef]   [PubMed]  

10. M. Ebnali-Heidari, C. Grillet, C. Monat, and B. J. Eggleton, “Dispersion engineering of slow light photonic crystal waveguides using microfluidic infiltration,” Opt. Express 17, 1628–1635 (2009). [CrossRef]   [PubMed]  

11. L. Zhang, Y. Yue, R. G. Beausoleil, and A. E. Willner, “Flattened dispersion in silicon slot waveguides,” Opt. Express 18, 20529–20534 (2010). [CrossRef]   [PubMed]  

12. S. Mas, J. Caraquitena, J. V. Galan, P. Sanchis, and J. Marti, “Tailoring the dispersion behavior of silicon nanophotonic slot waveguides,” Opt. Express 18, 20839–20844 (2010). [CrossRef]   [PubMed]  

13. C. E. de Nobriga, G. D. Hobbs, W. J. Wadsworth, J. C. Knight, D. V. Skryabin, A. Samarelli, M. Sorel, and R. M. De La Rue, “Supermode dispersion and waveguide-to-slot mode transition in arrays of silicon-on-insulator waveguides,” Opt. Lett. 35, 3925–3927 (2010). [CrossRef]   [PubMed]  

14. Q. Liu, S. Gao, Z. Li, Y. Xie, and S. He, “Dispersion engineering of a silicon-nanocrystal-based slot waveguide for broadband wavelength conversion,” Appl. Opt. 50, 1260–1265 (2011). [CrossRef]   [PubMed]  

15. M. Erdmanis, L. Karvonen, M. R. Saleem, M. Ruoho, V. Pale, A. Tervonen, S. Honkanen, and I. Tittonen, “ALD-assisted multiorder dispersion engineering of nanophotonic strip waveguides,” J. Lightwave Technol. 30, 2488–2493 (2012). [CrossRef]  

16. J. Riemensberger, K. Hartinger, T. Herr, V. Brasch, R. Holzwarth, and T. J. Kippenberg, “Dispersion engineering of thick high-Q silicon nitride ring-resonators via atomic layer deposition,” Opt. Express 20, 27661–27669 (2012). [CrossRef]   [PubMed]  

17. H. Ryu, J. Kim, Y. M. Jhon, S. Lee, and N. Park, “Effect of index contrasts in the wide spectral-range control of slot waveguide dispersion,” Opt. Express 20, 13189–13194 (2012). [CrossRef]   [PubMed]  

18. L. Zhang, Q. Lin, Y. Yue, Y. Yan, R. G. Beausoleil, and A. E. Willner, “Silicon waveguide with four zero-dispersion wavelengths and its application in on-chip octave-spanning supercontinuum generation,” Opt. Express 20, 1685–1690 (2012). [CrossRef]   [PubMed]  

19. M. Zhu, H. Liu, X. Li, N. Huang, Q. Sun, J. Wen, and Z. Wang, “Ultrabroadband falt dispersion tailoring of dual-slot silicon waveguides,” Opt. Express 20, 15899–15907 (2012). [CrossRef]   [PubMed]  

20. P. W. Nolte, C. Bohley, and J. Schilling, “Tuning of zero group velocity dispersion in infiltrated vertical silicon slot waveguides,” Opt. Express 21, 1741–1750 (2013). [CrossRef]   [PubMed]  

21. Z. Jafari and F. Emami, “Strip/slot hybrid arsenic tri-sulfide waveguide with ultra-flat and low dispersion profile over an ultra-wide bandwidth,” Opt. Lett. 38, 3082–3085 (2013). [CrossRef]   [PubMed]  

22. H. Jung, M. Poot, and H. X. Tang, “In-resonator variation of waveguide cross-sections for dispersion control of aluminum nitride micro-rings,” Opt. Express 23, 30634–30640 (2015). [CrossRef]   [PubMed]  

23. K. Y. Yang, K. Beha, D. C. Cole, X. Yi, P. Del’Haye, H. Lee, J. Li, D. Y. Oh, S. A. Diddams, S. B. Papp, and K. J. Vahala, “Broadband dispersion-engineered microresonator on a chip,” Nat. Photonics 10, 316–320 (2016). [CrossRef]  

24. D. V. Skryabin and A. V. Gorbach, “Looking at a soliton through the prism of optical supercontinuum,” Rev. Mod. Phys. 82, 1287 (2010). [CrossRef]  

25. P. Colman, S. Combrié, G. Lehoucq, A. de Rossi, and S. Trillo, “Blue self-frequency shift of slow solitons and radiation locking in a line-defect waveguide,” Phys. Rev. Lett. 109, 093901 (2012). [CrossRef]   [PubMed]  

26. R. Luo, H. Liang, and Q. Lin, “Multicolor cavity soliton,” Opt. Express 24, 16777–16787 (2016). [CrossRef]   [PubMed]  

27. W. J. Tropf and M. E. Thomas, “Infrared refractive index and thermo-optic coefficient measurement at APL,” Johns Hopkins APL Tech. Digest 19, 293–298 (1998).

28. I. Malitson, “Refraction and dispersion of synthetic sapphire,” J. Opt. Soc. Am. 52, 1377–1379 (1962). [CrossRef]  

29. S. Afshar V. and T. M. Monro, “A full vectorial model for pulse propagation in emerging waveguides with subwavelength structures part I: Kerr nonlinearity,” Opt. Express 17, 2298 (2009). [CrossRef]   [PubMed]  

30. The definition of the effective mode area given in Eq. (41) is employed here to calculate the red curve shown in Fig. 2(d).

References

  • View by:

  1. G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007).
  2. J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135 (2006).
    [Crossref]
  3. T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science 332, 555–559 (2011).
    [Crossref] [PubMed]
  4. P. St. J. Russell, “Photonic-crystal fibers,” J. Lightwave Technol. 24, 4729–4749 (2006).
    [Crossref]
  5. E. Dulkeith, F. Xia, L. Schares, W. M. J. Green, and Y. A. Vlasov, “Group index and group velocity dispersion in silicon-on-insulator photonic wires,” Opt. Express 14, 3853–3863 (2006).
    [Crossref] [PubMed]
  6. A. C. Turner, C. Manolatou, B. S. Schmidt, M. Lipson, M. A. Foster, J. E. Sharping, and A. L. Gaeta, “Tailored anomalous group-velocity dispersion in silicon channel waveguides,” Opt. Express 14, 4357–4362 (2006).
    [Crossref] [PubMed]
  7. A. Saynatjoki, M. Molot, J. Ahopelto, and H. Lipsanen, “Dispersion engineering of photonic crystal waveguides with ring-shaped holes,” Opt. Express 15, 8323–8328 (2007).
    [Crossref] [PubMed]
  8. M. R. E. Lamont, C. M. de Stkerke, and B. J. Eggleton, “Dispersion engineering of highly nonlinear As2S3 waveguides for parametric gain and wavelength conversion,” Opt. Express 15, 9458–9463 (2007).
    [Crossref] [PubMed]
  9. X. Liu, W. M. J. Green, X. Chen, I.-W. Hsieh, J. I. Dadap, Y. A. Vlasov, and R. M. Osgood, “Conformal dielectric overlayers for engineering dispersion and effective nonlinearity of silicon nanophotonic wires,” Opt. Lett. 33, 2889–2891 (2008).
    [Crossref] [PubMed]
  10. M. Ebnali-Heidari, C. Grillet, C. Monat, and B. J. Eggleton, “Dispersion engineering of slow light photonic crystal waveguides using microfluidic infiltration,” Opt. Express 17, 1628–1635 (2009).
    [Crossref] [PubMed]
  11. L. Zhang, Y. Yue, R. G. Beausoleil, and A. E. Willner, “Flattened dispersion in silicon slot waveguides,” Opt. Express 18, 20529–20534 (2010).
    [Crossref] [PubMed]
  12. S. Mas, J. Caraquitena, J. V. Galan, P. Sanchis, and J. Marti, “Tailoring the dispersion behavior of silicon nanophotonic slot waveguides,” Opt. Express 18, 20839–20844 (2010).
    [Crossref] [PubMed]
  13. C. E. de Nobriga, G. D. Hobbs, W. J. Wadsworth, J. C. Knight, D. V. Skryabin, A. Samarelli, M. Sorel, and R. M. De La Rue, “Supermode dispersion and waveguide-to-slot mode transition in arrays of silicon-on-insulator waveguides,” Opt. Lett. 35, 3925–3927 (2010).
    [Crossref] [PubMed]
  14. Q. Liu, S. Gao, Z. Li, Y. Xie, and S. He, “Dispersion engineering of a silicon-nanocrystal-based slot waveguide for broadband wavelength conversion,” Appl. Opt. 50, 1260–1265 (2011).
    [Crossref] [PubMed]
  15. M. Erdmanis, L. Karvonen, M. R. Saleem, M. Ruoho, V. Pale, A. Tervonen, S. Honkanen, and I. Tittonen, “ALD-assisted multiorder dispersion engineering of nanophotonic strip waveguides,” J. Lightwave Technol. 30, 2488–2493 (2012).
    [Crossref]
  16. J. Riemensberger, K. Hartinger, T. Herr, V. Brasch, R. Holzwarth, and T. J. Kippenberg, “Dispersion engineering of thick high-Q silicon nitride ring-resonators via atomic layer deposition,” Opt. Express 20, 27661–27669 (2012).
    [Crossref] [PubMed]
  17. H. Ryu, J. Kim, Y. M. Jhon, S. Lee, and N. Park, “Effect of index contrasts in the wide spectral-range control of slot waveguide dispersion,” Opt. Express 20, 13189–13194 (2012).
    [Crossref] [PubMed]
  18. L. Zhang, Q. Lin, Y. Yue, Y. Yan, R. G. Beausoleil, and A. E. Willner, “Silicon waveguide with four zero-dispersion wavelengths and its application in on-chip octave-spanning supercontinuum generation,” Opt. Express 20, 1685–1690 (2012).
    [Crossref] [PubMed]
  19. M. Zhu, H. Liu, X. Li, N. Huang, Q. Sun, J. Wen, and Z. Wang, “Ultrabroadband falt dispersion tailoring of dual-slot silicon waveguides,” Opt. Express 20, 15899–15907 (2012).
    [Crossref] [PubMed]
  20. P. W. Nolte, C. Bohley, and J. Schilling, “Tuning of zero group velocity dispersion in infiltrated vertical silicon slot waveguides,” Opt. Express 21, 1741–1750 (2013).
    [Crossref] [PubMed]
  21. Z. Jafari and F. Emami, “Strip/slot hybrid arsenic tri-sulfide waveguide with ultra-flat and low dispersion profile over an ultra-wide bandwidth,” Opt. Lett. 38, 3082–3085 (2013).
    [Crossref] [PubMed]
  22. H. Jung, M. Poot, and H. X. Tang, “In-resonator variation of waveguide cross-sections for dispersion control of aluminum nitride micro-rings,” Opt. Express 23, 30634–30640 (2015).
    [Crossref] [PubMed]
  23. K. Y. Yang, K. Beha, D. C. Cole, X. Yi, P. Del’Haye, H. Lee, J. Li, D. Y. Oh, S. A. Diddams, S. B. Papp, and K. J. Vahala, “Broadband dispersion-engineered microresonator on a chip,” Nat. Photonics 10, 316–320 (2016).
    [Crossref]
  24. D. V. Skryabin and A. V. Gorbach, “Looking at a soliton through the prism of optical supercontinuum,” Rev. Mod. Phys. 82, 1287 (2010).
    [Crossref]
  25. P. Colman, S. Combrié, G. Lehoucq, A. de Rossi, and S. Trillo, “Blue self-frequency shift of slow solitons and radiation locking in a line-defect waveguide,” Phys. Rev. Lett. 109, 093901 (2012).
    [Crossref] [PubMed]
  26. R. Luo, H. Liang, and Q. Lin, “Multicolor cavity soliton,” Opt. Express 24, 16777–16787 (2016).
    [Crossref] [PubMed]
  27. W. J. Tropf and M. E. Thomas, “Infrared refractive index and thermo-optic coefficient measurement at APL,” Johns Hopkins APL Tech. Digest 19, 293–298 (1998).
  28. I. Malitson, “Refraction and dispersion of synthetic sapphire,” J. Opt. Soc. Am. 52, 1377–1379 (1962).
    [Crossref]
  29. S. Afshar V. and T. M. Monro, “A full vectorial model for pulse propagation in emerging waveguides with subwavelength structures part I: Kerr nonlinearity,” Opt. Express 17, 2298 (2009).
    [Crossref] [PubMed]
  30. The definition of the effective mode area given in Eq. (41) is employed here to calculate the red curve shown in Fig. 2(d).

2016 (2)

K. Y. Yang, K. Beha, D. C. Cole, X. Yi, P. Del’Haye, H. Lee, J. Li, D. Y. Oh, S. A. Diddams, S. B. Papp, and K. J. Vahala, “Broadband dispersion-engineered microresonator on a chip,” Nat. Photonics 10, 316–320 (2016).
[Crossref]

R. Luo, H. Liang, and Q. Lin, “Multicolor cavity soliton,” Opt. Express 24, 16777–16787 (2016).
[Crossref] [PubMed]

2015 (1)

2013 (2)

2012 (6)

2011 (2)

2010 (4)

2009 (2)

2008 (1)

2007 (2)

2006 (4)

1998 (1)

W. J. Tropf and M. E. Thomas, “Infrared refractive index and thermo-optic coefficient measurement at APL,” Johns Hopkins APL Tech. Digest 19, 293–298 (1998).

1962 (1)

Afshar V., S.

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007).

Ahopelto, J.

Beausoleil, R. G.

Beha, K.

K. Y. Yang, K. Beha, D. C. Cole, X. Yi, P. Del’Haye, H. Lee, J. Li, D. Y. Oh, S. A. Diddams, S. B. Papp, and K. J. Vahala, “Broadband dispersion-engineered microresonator on a chip,” Nat. Photonics 10, 316–320 (2016).
[Crossref]

Bohley, C.

Brasch, V.

Caraquitena, J.

Chen, X.

Coen, S.

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135 (2006).
[Crossref]

Cole, D. C.

K. Y. Yang, K. Beha, D. C. Cole, X. Yi, P. Del’Haye, H. Lee, J. Li, D. Y. Oh, S. A. Diddams, S. B. Papp, and K. J. Vahala, “Broadband dispersion-engineered microresonator on a chip,” Nat. Photonics 10, 316–320 (2016).
[Crossref]

Colman, P.

P. Colman, S. Combrié, G. Lehoucq, A. de Rossi, and S. Trillo, “Blue self-frequency shift of slow solitons and radiation locking in a line-defect waveguide,” Phys. Rev. Lett. 109, 093901 (2012).
[Crossref] [PubMed]

Combrié, S.

P. Colman, S. Combrié, G. Lehoucq, A. de Rossi, and S. Trillo, “Blue self-frequency shift of slow solitons and radiation locking in a line-defect waveguide,” Phys. Rev. Lett. 109, 093901 (2012).
[Crossref] [PubMed]

Dadap, J. I.

De La Rue, R. M.

de Nobriga, C. E.

de Rossi, A.

P. Colman, S. Combrié, G. Lehoucq, A. de Rossi, and S. Trillo, “Blue self-frequency shift of slow solitons and radiation locking in a line-defect waveguide,” Phys. Rev. Lett. 109, 093901 (2012).
[Crossref] [PubMed]

de Stkerke, C. M.

Del’Haye, P.

K. Y. Yang, K. Beha, D. C. Cole, X. Yi, P. Del’Haye, H. Lee, J. Li, D. Y. Oh, S. A. Diddams, S. B. Papp, and K. J. Vahala, “Broadband dispersion-engineered microresonator on a chip,” Nat. Photonics 10, 316–320 (2016).
[Crossref]

Diddams, S. A.

K. Y. Yang, K. Beha, D. C. Cole, X. Yi, P. Del’Haye, H. Lee, J. Li, D. Y. Oh, S. A. Diddams, S. B. Papp, and K. J. Vahala, “Broadband dispersion-engineered microresonator on a chip,” Nat. Photonics 10, 316–320 (2016).
[Crossref]

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science 332, 555–559 (2011).
[Crossref] [PubMed]

Dudley, J. M.

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135 (2006).
[Crossref]

Dulkeith, E.

Ebnali-Heidari, M.

Eggleton, B. J.

Emami, F.

Erdmanis, M.

Foster, M. A.

Gaeta, A. L.

Galan, J. V.

Gao, S.

Genty, G.

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135 (2006).
[Crossref]

Gorbach, A. V.

D. V. Skryabin and A. V. Gorbach, “Looking at a soliton through the prism of optical supercontinuum,” Rev. Mod. Phys. 82, 1287 (2010).
[Crossref]

Green, W. M. J.

Grillet, C.

Hartinger, K.

He, S.

Herr, T.

Hobbs, G. D.

Holzwarth, R.

Honkanen, S.

Hsieh, I.-W.

Huang, N.

Jafari, Z.

Jhon, Y. M.

Jung, H.

Karvonen, L.

Kim, J.

Kippenberg, T. J.

Knight, J. C.

Lamont, M. R. E.

Lee, H.

K. Y. Yang, K. Beha, D. C. Cole, X. Yi, P. Del’Haye, H. Lee, J. Li, D. Y. Oh, S. A. Diddams, S. B. Papp, and K. J. Vahala, “Broadband dispersion-engineered microresonator on a chip,” Nat. Photonics 10, 316–320 (2016).
[Crossref]

Lee, S.

Lehoucq, G.

P. Colman, S. Combrié, G. Lehoucq, A. de Rossi, and S. Trillo, “Blue self-frequency shift of slow solitons and radiation locking in a line-defect waveguide,” Phys. Rev. Lett. 109, 093901 (2012).
[Crossref] [PubMed]

Li, J.

K. Y. Yang, K. Beha, D. C. Cole, X. Yi, P. Del’Haye, H. Lee, J. Li, D. Y. Oh, S. A. Diddams, S. B. Papp, and K. J. Vahala, “Broadband dispersion-engineered microresonator on a chip,” Nat. Photonics 10, 316–320 (2016).
[Crossref]

Li, X.

Li, Z.

Liang, H.

Lin, Q.

Lipsanen, H.

Lipson, M.

Liu, H.

Liu, Q.

Liu, X.

Luo, R.

Malitson, I.

Manolatou, C.

Marti, J.

Mas, S.

Molot, M.

Monat, C.

Monro, T. M.

Nolte, P. W.

Oh, D. Y.

K. Y. Yang, K. Beha, D. C. Cole, X. Yi, P. Del’Haye, H. Lee, J. Li, D. Y. Oh, S. A. Diddams, S. B. Papp, and K. J. Vahala, “Broadband dispersion-engineered microresonator on a chip,” Nat. Photonics 10, 316–320 (2016).
[Crossref]

Osgood, R. M.

Pale, V.

Papp, S. B.

K. Y. Yang, K. Beha, D. C. Cole, X. Yi, P. Del’Haye, H. Lee, J. Li, D. Y. Oh, S. A. Diddams, S. B. Papp, and K. J. Vahala, “Broadband dispersion-engineered microresonator on a chip,” Nat. Photonics 10, 316–320 (2016).
[Crossref]

Park, N.

Poot, M.

Riemensberger, J.

Ruoho, M.

Russell, P. St. J.

Ryu, H.

Saleem, M. R.

Samarelli, A.

Sanchis, P.

Saynatjoki, A.

Schares, L.

Schilling, J.

Schmidt, B. S.

Sharping, J. E.

Skryabin, D. V.

Sorel, M.

Sun, Q.

Tang, H. X.

Tervonen, A.

Thomas, M. E.

W. J. Tropf and M. E. Thomas, “Infrared refractive index and thermo-optic coefficient measurement at APL,” Johns Hopkins APL Tech. Digest 19, 293–298 (1998).

Tittonen, I.

Trillo, S.

P. Colman, S. Combrié, G. Lehoucq, A. de Rossi, and S. Trillo, “Blue self-frequency shift of slow solitons and radiation locking in a line-defect waveguide,” Phys. Rev. Lett. 109, 093901 (2012).
[Crossref] [PubMed]

Tropf, W. J.

W. J. Tropf and M. E. Thomas, “Infrared refractive index and thermo-optic coefficient measurement at APL,” Johns Hopkins APL Tech. Digest 19, 293–298 (1998).

Turner, A. C.

Vahala, K. J.

K. Y. Yang, K. Beha, D. C. Cole, X. Yi, P. Del’Haye, H. Lee, J. Li, D. Y. Oh, S. A. Diddams, S. B. Papp, and K. J. Vahala, “Broadband dispersion-engineered microresonator on a chip,” Nat. Photonics 10, 316–320 (2016).
[Crossref]

Vlasov, Y. A.

Wadsworth, W. J.

Wang, Z.

Wen, J.

Willner, A. E.

Xia, F.

Xie, Y.

Yan, Y.

Yang, K. Y.

K. Y. Yang, K. Beha, D. C. Cole, X. Yi, P. Del’Haye, H. Lee, J. Li, D. Y. Oh, S. A. Diddams, S. B. Papp, and K. J. Vahala, “Broadband dispersion-engineered microresonator on a chip,” Nat. Photonics 10, 316–320 (2016).
[Crossref]

Yi, X.

K. Y. Yang, K. Beha, D. C. Cole, X. Yi, P. Del’Haye, H. Lee, J. Li, D. Y. Oh, S. A. Diddams, S. B. Papp, and K. J. Vahala, “Broadband dispersion-engineered microresonator on a chip,” Nat. Photonics 10, 316–320 (2016).
[Crossref]

Yue, Y.

Zhang, L.

Zhu, M.

Appl. Opt. (1)

J. Lightwave Technol. (2)

J. Opt. Soc. Am. (1)

Johns Hopkins APL Tech. Digest (1)

W. J. Tropf and M. E. Thomas, “Infrared refractive index and thermo-optic coefficient measurement at APL,” Johns Hopkins APL Tech. Digest 19, 293–298 (1998).

Nat. Photonics (1)

K. Y. Yang, K. Beha, D. C. Cole, X. Yi, P. Del’Haye, H. Lee, J. Li, D. Y. Oh, S. A. Diddams, S. B. Papp, and K. J. Vahala, “Broadband dispersion-engineered microresonator on a chip,” Nat. Photonics 10, 316–320 (2016).
[Crossref]

Opt. Express (15)

R. Luo, H. Liang, and Q. Lin, “Multicolor cavity soliton,” Opt. Express 24, 16777–16787 (2016).
[Crossref] [PubMed]

H. Jung, M. Poot, and H. X. Tang, “In-resonator variation of waveguide cross-sections for dispersion control of aluminum nitride micro-rings,” Opt. Express 23, 30634–30640 (2015).
[Crossref] [PubMed]

S. Afshar V. and T. M. Monro, “A full vectorial model for pulse propagation in emerging waveguides with subwavelength structures part I: Kerr nonlinearity,” Opt. Express 17, 2298 (2009).
[Crossref] [PubMed]

E. Dulkeith, F. Xia, L. Schares, W. M. J. Green, and Y. A. Vlasov, “Group index and group velocity dispersion in silicon-on-insulator photonic wires,” Opt. Express 14, 3853–3863 (2006).
[Crossref] [PubMed]

A. C. Turner, C. Manolatou, B. S. Schmidt, M. Lipson, M. A. Foster, J. E. Sharping, and A. L. Gaeta, “Tailored anomalous group-velocity dispersion in silicon channel waveguides,” Opt. Express 14, 4357–4362 (2006).
[Crossref] [PubMed]

A. Saynatjoki, M. Molot, J. Ahopelto, and H. Lipsanen, “Dispersion engineering of photonic crystal waveguides with ring-shaped holes,” Opt. Express 15, 8323–8328 (2007).
[Crossref] [PubMed]

M. R. E. Lamont, C. M. de Stkerke, and B. J. Eggleton, “Dispersion engineering of highly nonlinear As2S3 waveguides for parametric gain and wavelength conversion,” Opt. Express 15, 9458–9463 (2007).
[Crossref] [PubMed]

M. Ebnali-Heidari, C. Grillet, C. Monat, and B. J. Eggleton, “Dispersion engineering of slow light photonic crystal waveguides using microfluidic infiltration,” Opt. Express 17, 1628–1635 (2009).
[Crossref] [PubMed]

L. Zhang, Y. Yue, R. G. Beausoleil, and A. E. Willner, “Flattened dispersion in silicon slot waveguides,” Opt. Express 18, 20529–20534 (2010).
[Crossref] [PubMed]

S. Mas, J. Caraquitena, J. V. Galan, P. Sanchis, and J. Marti, “Tailoring the dispersion behavior of silicon nanophotonic slot waveguides,” Opt. Express 18, 20839–20844 (2010).
[Crossref] [PubMed]

J. Riemensberger, K. Hartinger, T. Herr, V. Brasch, R. Holzwarth, and T. J. Kippenberg, “Dispersion engineering of thick high-Q silicon nitride ring-resonators via atomic layer deposition,” Opt. Express 20, 27661–27669 (2012).
[Crossref] [PubMed]

H. Ryu, J. Kim, Y. M. Jhon, S. Lee, and N. Park, “Effect of index contrasts in the wide spectral-range control of slot waveguide dispersion,” Opt. Express 20, 13189–13194 (2012).
[Crossref] [PubMed]

L. Zhang, Q. Lin, Y. Yue, Y. Yan, R. G. Beausoleil, and A. E. Willner, “Silicon waveguide with four zero-dispersion wavelengths and its application in on-chip octave-spanning supercontinuum generation,” Opt. Express 20, 1685–1690 (2012).
[Crossref] [PubMed]

M. Zhu, H. Liu, X. Li, N. Huang, Q. Sun, J. Wen, and Z. Wang, “Ultrabroadband falt dispersion tailoring of dual-slot silicon waveguides,” Opt. Express 20, 15899–15907 (2012).
[Crossref] [PubMed]

P. W. Nolte, C. Bohley, and J. Schilling, “Tuning of zero group velocity dispersion in infiltrated vertical silicon slot waveguides,” Opt. Express 21, 1741–1750 (2013).
[Crossref] [PubMed]

Opt. Lett. (3)

Phys. Rev. Lett. (1)

P. Colman, S. Combrié, G. Lehoucq, A. de Rossi, and S. Trillo, “Blue self-frequency shift of slow solitons and radiation locking in a line-defect waveguide,” Phys. Rev. Lett. 109, 093901 (2012).
[Crossref] [PubMed]

Rev. Mod. Phys. (2)

D. V. Skryabin and A. V. Gorbach, “Looking at a soliton through the prism of optical supercontinuum,” Rev. Mod. Phys. 82, 1287 (2010).
[Crossref]

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135 (2006).
[Crossref]

Science (1)

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science 332, 555–559 (2011).
[Crossref] [PubMed]

Other (2)

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007).

The definition of the effective mode area given in Eq. (41) is employed here to calculate the red curve shown in Fig. 2(d).

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Figures (6)

Fig. 1
Fig. 1 (a) Schematic of the proposed double-cladded waveguide structure. (b) The cross section of the waveguide, where W and H are the width and height of the waveguide core, and D1 and D1 are the thicknesses of Cladding 1 and 2, respectively.
Fig. 2
Fig. 2 (a) Group velocity dispersion of the fundamental quasi-TE mode (blue) of the double-cladded waveguide schematically shown in the inset. The SiC waveguide core has a width of W = 1.597 μm and a height of H = 665 nm. The Cladding 1 is aluminum oxide with a thickness of D1 = 210 nm and the Cladding 2 is SiC with a thickness of D2 = 195 nm. To show the effect of waveguide dispersion, the red curve shows the case without material dispersion, where the refractive indices of materials are set as nSiC = 2.5686 and nAl2O3 = 1.7462. The green curve shows the material dispersion of SiC material. (b) Group index for the fundamental quasi-TE mode of the waveguide. (c) Detailed group-velocity dispersion of the waveguide, which is the same as the blue curve in (a). The insets show the optical mode field profiles at three wavelengths indicated by the arrows. (d) Blue curves: Energy ratios of the fundamental quasi-TE mode in the waveguide core, Cladding 1, and Cladding 2, respectively. Red curve: The effective mode area of the waveguide. See Ref. [29,30] for the detailed expression of the effective mode area.
Fig. 3
Fig. 3 (a) Examples of purely normal and anomalous GVD over broad band. The waveguide geometry of green line is same as Fig. 2 (a). Transferring from the anomalous GVD to normal GVD can be obtained by varying the width and height of the waveguide core simultaneously. Black line: W = 1.517 μm, H = 697 nm. Red line: W = 1.677 μm, H = 637 nm. Blue line: W = 1.757 μm, H = 613 nm. For all cases, the thicknesses of the two cladding layers remain as D1= 210 nm and D2 = 195 nm, respectively. (b) Group velocity dispersion of the fundamental quasi-TM mode, for a double-cladded waveguide with the following geometry: W = 768 nm, H = 1.420 μm, D1 = 303 nm, and D2 = 252 nm. The insets show the optical mode field profile at three wavelengths indicated by the arrows.
Fig. 4
Fig. 4 Dependence of GVD on the variations of various dimensions of the double-cladded waveguide example given in Fig. 2. (a) Impact of the width variation of the SiC waveguide core. (b) Impact of the height variation of the SiC waveguide core. (c) Impact of the thickness variation of Cladding 1 – the aluminum oxide cladding layer. (d) Impact of the thickness variation of Cladding 2 – the SiC cladding layer.
Fig. 5
Fig. 5 Schematic of a multi-cladded waveguide structure.
Fig. 6
Fig. 6 (a) Group-velocity dispersion with 5 ZDWLs, for a three-cladded waveguide. The SiC waveguide core has a width of W = 875 nm and a height of H = 1.865 μm. The Cladding 1 (Al2O3), Cladding 2 (SiC), and the top Cladding 3 (Al2O3) have thicknesses of D1= 125 nm, D2 = 195 nm, and D3= 1048 nm, respectively. (b) Group-velocity dispersion with 6 ZDWLs, for a three-cladded waveguide. The SiC waveguide core has a width of W = 884 nm and a height of H = 1.720 μm. The Cladding 1 (Al2O3), Cladding 2 (SiC), and the top Cladding 3 (Al2O3) have thicknesses of D1= 122 nm, D2 = 200 nm, and D3= 995 nm, respectively. The insets in (a) and (b) show the optical mode field profiles at the wavelengths indicated by the arrows.

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