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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 9 — Apr. 25, 2011
  • pp: 8102–8107
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Analysis and engineering of chromatic dispersion in silicon waveguide bends and ring resonators

Lin Zhang, Yang Yue, Raymond G. Beausoleil, and Alan E. Willner  »View Author Affiliations


Optics Express, Vol. 19, Issue 9, pp. 8102-8107 (2011)
http://dx.doi.org/10.1364/OE.19.008102


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Abstract

We analyze chromatic dispersion in tightly curved silicon strip and slot waveguides with high index contrast. It is found that the dispersion profile is changed dramatically at both polarization states, when bending radius is reduced to a few microns. Zero-dispersion wavelength may shift by more than 220 nm, which raises a critical issue in design and optimization of micro-resonator-based devices for nonlinear applications. We propose a slot structure to tailor in-cavity dispersion and obtain spectral lines with the standard deviation of frequency-dependent free spectral range of the slot-waveguide resonator made 460 times smaller than that of a strip-waveguide resonator, making it suitable for on-chip octave-spanning frequency comb generation in mid-infrared wavelength range.

© 2011 OSA

1. Introduction

Chromatic dispersion in optical cavities can be quite important for micro-resonator-based nonlinear applications [12

12. P. Del'Haye, O. Arcizet, M. L. Gorodetsky, R. Holzwarth, and T. J. Kippenberg, “Frequency comb assisted diode laser spectroscopy for measurement of microcavity dispersion,” Nat. Photonics 3(9), 529–533 (2009). [CrossRef]

15

15. J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photonics 4(1), 37–40 (2010). [CrossRef]

], e.g., on-chip frequency comb generation enabled by cascaded four wave mixing with a low pump power. Flat and low in-cavity dispersion produces almost equidistant resonance peaks, which are desired not only for reducing pump power but also for improving measurement precision in comb-based frequency metrology. However, highly nonlinear waveguides or photonic wires typically have strong index contrast and waveguide dispersion, making it difficult to obtain equally spaced resonance peaks over a wide band. In-cavity dispersion of high-index-contrast micro-resonators needs to be engineered.

In this paper, we analyze chromatic dispersion in tightly curved silicon waveguides. It is shown that both strip and slot waveguides would have a dramatic dispersion change when the bending radius is reduced to a few microns. Both polarization states are examined, and they exhibit quite different sensitivity of dispersion to waveguide bending. The shift of zero-dispersion wavelength (ZDW) in these silicon waveguides could be up to 220 nm when the bending radius is 2 or 3 μm. Dispersion value at a given wavelength can have a change up to 1000 ps/(nm·km) for a different bending radius. This indicates that the dispersion change in bent waveguides is an important issue to be considered in device design. We also show that in-cavity dispersion in silicon microring resonators can be flattened, using a horizontal slot structure, over a 563-nm wavelength range from 1566 to 2129 nm. Free spectral range (FSR) of the resonator as a function of wavelength has a standard deviation reduced to 0.023 GHz, 460 times smaller than that produced by a strip waveguide resonator.

2. Dispersion in circular waveguide bends

When the waveguide is bent with a structural bending radius R0 that is measured from the center of rotation to the waveguide center, the guided optical mode is extended outwards, as shown in Fig. 1
Fig. 1 A horizontal slot waveguide has a low-index slot between upper and lower high-index layers. When the upper layer and the slot are removed (Hu = 0, Hs = 0), the waveguide becomes a strip waveguide. In waveguide bends and micro-resonators, traveling optical mode feels an effective bending radius, Reff, that is greater than structural bending radius, R0.
, and feels an “effective” bending radius Reff that is greater than R0. The increased path length of the guided mode is dependent on wavelength, which induces an additional dispersion. The effective bending radius Reff also depends on waveguide geometry, especially waveguide width. To use the waveguide bends and ring resonators for nonlinear optics applications, one would have a ZDW close to a wavelength range of interest, near 1550 nm. From previously reported results [16

16. L. Yin, Q. Lin, and G. P. Agrawal, “Dispersion tailoring and soliton propagation in silicon waveguides,” Opt. Lett. 31(9), 1295–1297 (2006). [CrossRef] [PubMed]

,17

17. J. I. Dadap, N. C. Panoiu, X. Chen, I.-W. Hsieh, X. Liu, C.-Y. Chou, E. Dulkeith, S. J. McNab, F. Xia, W. M. J. Green, L. Sekaric, Y. A. Vlasov, and R. M. Osgood Jr., “Nonlinear-optical phase modification in dispersion-engineered Si photonic wires,” Opt. Express 16(2), 1280–1299 (2008). [CrossRef] [PubMed]

], it can be seen that the waveguide size has to be relatively large. For silicon strip waveguides, we choose height Hl = 400 nm, and waveguide width W = 800 nm. For silicon horizontal slot waveguides, we set the heights Hu and Hl of two silicon layers to be 240 nm, and slot height Hs = 40 nm, width W = 500 nm. In both cases, 2-μm-thick SiO2 is considered above the silicon substrate, and top cladding is air.

Dispersion curves in the waveguide bends are shown in Figs. 2(c) and 2(d). For the quasi-TM mode, the anomalous waveguide dispersion, induced by light confinement, is strong enough to over-balance the normal material dispersion in silicon, which causes a ZDW at a wavelength close to 1300 nm. Since the strip waveguide with a large width produces a relatively loose light confinement in horizontal direction, the quasi-TE mode has a smaller waveguide dispersion compared to the quasi-TM mode and has ZDWs at longer wavelengths. When R0 is reduced from 4.5 to 2 μm, the dispersion curves are shifted to longer wavelengths for both polarization states. At a given wavelength, dispersion value changes rapidly. For example, at 1600 nm wavelength, the quasi-TE mode has a dispersion value increasing from −77 to 660 ps/(nm·km), while the dispersion changes from 1674 to 2614 ps/(nm·km) for the quasi-TM mode. Such a large dispersion change may greatly affect nonlinear interaction efficiency of optical waves traveling inside a ring resonator [14

14. A. C. Turner, M. A. Foster, A. L. Gaeta, and M. Lipson, “Ultra-low power parametric frequency conversion in a silicon microring resonator,” Opt. Express 16(7), 4881–4887 (2008). [CrossRef] [PubMed]

].

The position of ZDW in a dispersion curve is critical in nonlinear parametric processes. Figures 2(e) and 2(f) show that the ZDW of the curved waveguides shifts relative to that of a straight waveguide, as a function of R0 for different waveguide widths. With a width of 800 nm, the ZDW of the quasi-TE mode is blue-shifted first as R0 is reduced from 32 to 8 μm and then rapidly red-shifted by >220 nm, from −47 to 175 nm, as R0 is reduced from 8 to 2 μm. All results are relative to the ZDW in the straight waveguide at 1448 nm.

Bending-induced dispersion change in a horizontal slot waveguide is examined, with the structural parameters given above. Figure 3(a)
Fig. 3 Dispersion change induced by waveguide bending in horizontal slot waveguides. (a) effective banding radius, (b) dispersion profile and (c) ZDW shift.
shows that the change in Reff is relatively small as compared to the strip waveguides, because of a small waveguide width of 500 nm. At R0 = 3 μm, (Reff - R0) for the slot mode decreases from 44 to 21 nm as wavelength is changed from 1300 to 1800 nm. As a result, dispersion profiles are quite tolerant to waveguide bending in Fig. 3(b). We vary the slot height from 30 to 50 nm, when keeping other parameters the same, and plot the ZDW shift versus R0 in Fig. 3(c). The ZDW is blue-shifted as R0 decreases.

3. Flattening of in-cavity dispersion

It was shown above that curving waveguides may cause a large change in dispersion inside a cavity, which provides a potentially effective method to shift ZDWs, enabling more freedom in device design, but dispersion flatness does not change significantly. For ultra-wideband nonlinear applications, flat and low in-cavity dispersion over a wide wavelength range would be beneficial to produce equidistant resonance frequencies and increase efficiency of cascaded four wave mixing and frequency comb generation in micro-cavities. We propose a horizontal slot waveguide that forms a cavity with 460 times smaller standard deviation of FSRs than a silicon strip waveguide with the same size and without a slot layer. Structural parameters are: upper silicon height Hu = 286 nm, silica slot height Hs = 32 nm, lower silicon height Hl = 510 nm, and waveguide width W = 500 nm. Quasi-TM mode in the straight slot waveguide has a flat dispersion of ± 15 ps/(nm·km) over a 563-nm wavelength range, from 1566 to 2129 nm. The flattened dispersion results from an anti-crossing effect when the guided mode transits from a strip mode at short wavelength to a slot-like mode at long wavelength, which causes a slightly negative waveguide dispersion [20

20. L. Zhang, Y. Yue, R. G. Beausoleil, and A. E. Willner, “Flattened dispersion in silicon slot waveguides,” Opt. Express 18(19), 20529–20534 (2010). [CrossRef] [PubMed]

] to balance the convex dispersion in strip waveguides without a slot structure. Figure 4(a)
Fig. 4 (a) Dispersion inside the cavity is changed as R0 is reduced from 16 to 3 μm. Both averaged value and flatness of the dispersion profiles are varied. (b) FSR's standard deviation changes rapidly with R0.
shows in-cavity dispersion profiles as R0 is reduced from ∞ to 3 μm. Little change in the dispersion is seen for R0 >11 μm, but the average dispersion shifts by >100 ps/(nm·km) as R0 is from 6 to 3 μm, since in this range more optical field extends outwards, which causes strong wavelength dependence of the effective index.

One can obtain the resonance frequencies that satisfy 2πReff(fm)neff(fm)fm = mc and also the FSR as a function of frequency. Figure 4(b) shows the standard deviation of the FSRs that changes with R0 for a single-ring resonator with the proposed slot structure. Due to the flat and low in-cavity dispersion, the FSR is nearly constant with a standard deviation of 0.023 GHz for R0 = 32 μm from 1566 to 2129 nm wavelength. The averaged FSR is 376.3 GHz, and the averaged resonance linewidth is 2.97 GHz. In contrast, if the slot structure is taken out, a microring resonator (its bending radius is 32 μm) formed by a silicon strip waveguide with a height of 510 nm and a width of 500 nm produces a fast-changing FSR over the same wavelength band. The FSR standard deviation is 10.53 GHz, which is 460 times larger, and the linewidth is 2.79 GHz. In both cases, we assume 2 dB/cm waveguide loss. To illustrate the uniformity of the resonance frequencies, we plot a frequency-domain “eye-diagram”. As shown in Fig. 5
Fig. 5 Spectral response of a single-ring resonator is sliced with a frequency spacing equal to the averaged FSR. All the spectrum pieces are plotted together to show the uniformity of the resonance peaks. Proposed slot-waveguide resonator has well-aligned resonance peaks over a 563-nm wavelength band, in contrast with a strip-waveguide resonator.
, the frequency axis is discretized with an interval equal to the averaged FSR, and these spectrum pieces are plotted on top of each other. If the FSR changes with frequency, the resonance peaks will be shifted relative to each other in the “eye-diagram”. As shown in Fig. 5, the resonance peaks produced by the strip-waveguide resonator drift by multiple resonance linewidths over a 100-nm bandwidth. The slot-waveguide resonator generates the well-aligned peaks in the “eye-diagrams” over a 563-nm bandwidth. This great improvement of FSR uniformity is enabled by engineering in-cavity dispersion with the proposed slot structure, allowing a large number of equidistant spectral lines determined by cascaded four wave mixing to be on resonance, which would help increase nonlinear efficiency and obtain octave-spanning spectral coverage of a frequency comb generated on a chip.

4. Conclusion

We have analyzed the dispersion change induced by tight waveguide bends in high-index-contrast silicon waveguides. Both strip and slot waveguides are simulated with a bending radius as small as a few microns. The dispersion is changed dramatically, which could raise an important issue in the design of micro-resonator devices for cavity-enhanced nonlinear optics applications. By engineering the in-cavity dispersion using a slot structure, we have improved uniformity of resonance peaks in frequency response of a silicon micro-cavity, which would be beneficial to generate a mid-infrared frequency comb on a chip.

Acknowledgments

The authors would thank Drs. Mark Oxborrow and Qiang Lin for helpful discussions. This work is supported by the Defense Advanced Research Projects Agency (DARPA) (contract # HR0011-09-C-0124) and the HP Laboratories.

References and links

1.

E. A. Marcatili, “Bends in optical dielectric guides,” Bell Syst. Tech. J. 48, 2103–2132 (1969).

2.

H. F. Taylor, “Losses at corner bends in dielectric waveguides,” Appl. Opt. 16(3), 711–716 (1977). [CrossRef] [PubMed]

3.

E.-G. Neumann, “Curved dielectric optical waveguides with reduced transition losses,” IEE Proc. Microwaves, Antennas Propag. 129, 278–280 (1982).

4.

R. Espinola, R. Ahmad, F. Pizzuto, M. Steel, and R. Osgood, “A study of high-index-contrast 90 degree waveguide bend structures,” Opt. Express 8(9), 517–528 (2001). [CrossRef] [PubMed]

5.

C. Xudong, C. Hafner, R. Vahldieck, and F. Robin, “Sharp trench waveguide bends in dual mode operation with ultra-small photonic crystals for suppressing radiation,” Opt. Express 14(10), 4351–4356 (2006). [CrossRef] [PubMed]

6.

A. Sakai, G. Hara, and T. Baba, “Propagation characteristics of ultrahigh-Δ optical waveguide on silicon on-insulator substrate,” Jpn. J. Appl. Phys. 40(Part 2, No. 4B), L383–L385 (2001). [CrossRef]

7.

A. Sakai, T. Fukazawa, and T. Baba, “Estimation of polarization crosstalk at a micro-bend in Si-photonic wire waveguide,” J. Lightwave Technol. 22(2), 520–525 (2004). [CrossRef]

8.

Y. Vlasov and S. McNab, “Losses in single-mode silicon-on-insulator strip waveguides and bends,” Opt. Express 12(8), 1622–1631 (2004). [CrossRef] [PubMed]

9.

P. A. Anderson, B. S. Schmidt, and M. Lipson, “High confinement in silicon slot waveguides with sharp bends,” Opt. Express 14(20), 9197–9202 (2006). [CrossRef] [PubMed]

10.

Q. Xu, D. Fattal, and R. G. Beausoleil, “Silicon microring resonators with 1.5-microm radius,” Opt. Express 16(6), 4309–4315 (2008). [CrossRef] [PubMed]

11.

Z. Sheng, D. Dai, and S. He, “Comparative study of losses in ultrasharp silicon-on-insulator nanowire bends,” IEEE J. Sel. Top. Quantum Electron. 15(5), 1406–1412 (2009). [CrossRef]

12.

P. Del'Haye, O. Arcizet, M. L. Gorodetsky, R. Holzwarth, and T. J. Kippenberg, “Frequency comb assisted diode laser spectroscopy for measurement of microcavity dispersion,” Nat. Photonics 3(9), 529–533 (2009). [CrossRef]

13.

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450(7173), 1214–1217 (2007). [CrossRef] [PubMed]

14.

A. C. Turner, M. A. Foster, A. L. Gaeta, and M. Lipson, “Ultra-low power parametric frequency conversion in a silicon microring resonator,” Opt. Express 16(7), 4881–4887 (2008). [CrossRef] [PubMed]

15.

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photonics 4(1), 37–40 (2010). [CrossRef]

16.

L. Yin, Q. Lin, and G. P. Agrawal, “Dispersion tailoring and soliton propagation in silicon waveguides,” Opt. Lett. 31(9), 1295–1297 (2006). [CrossRef] [PubMed]

17.

J. I. Dadap, N. C. Panoiu, X. Chen, I.-W. Hsieh, X. Liu, C.-Y. Chou, E. Dulkeith, S. J. McNab, F. Xia, W. M. J. Green, L. Sekaric, Y. A. Vlasov, and R. M. Osgood Jr., “Nonlinear-optical phase modification in dispersion-engineered Si photonic wires,” Opt. Express 16(2), 1280–1299 (2008). [CrossRef] [PubMed]

18.

M. Oxborrow, “Traceable 2-D finite-element simulation of the whispering-gallery modes of axisymmetric electromagnetic resonators,” IEEE Trans. Microw. Theory Tech. 55(6), 1209–1218 (2007). [CrossRef]

19.

L. Zhang, Y. Yue, Y. Xiao-Li, J. Wang, R. G. Beausoleil, and A. E. Willner, “Flat and low dispersion in highly nonlinear slot waveguides,” Opt. Express 18(12), 13187–13193 (2010). [CrossRef] [PubMed]

20.

L. Zhang, Y. Yue, R. G. Beausoleil, and A. E. Willner, “Flattened dispersion in silicon slot waveguides,” Opt. Express 18(19), 20529–20534 (2010). [CrossRef] [PubMed]

OCIS Codes
(130.3060) Integrated optics : Infrared
(130.3120) Integrated optics : Integrated optics devices
(260.2030) Physical optics : Dispersion
(130.3990) Integrated optics : Micro-optical devices

ToC Category:
Integrated Optics

History
Original Manuscript: March 16, 2011
Revised Manuscript: April 11, 2011
Manuscript Accepted: April 11, 2011
Published: April 13, 2011

Citation
Lin Zhang, Yang Yue, Raymond G. Beausoleil, and Alan E. Willner, "Analysis and engineering of chromatic dispersion in silicon waveguide bends and ring resonators," Opt. Express 19, 8102-8107 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-9-8102


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References

  1. E. A. Marcatili, “Bends in optical dielectric guides,” Bell Syst. Tech. J. 48, 2103–2132 (1969).
  2. H. F. Taylor, “Losses at corner bends in dielectric waveguides,” Appl. Opt. 16(3), 711–716 (1977). [CrossRef] [PubMed]
  3. E.-G. Neumann, “Curved dielectric optical waveguides with reduced transition losses,” IEE Proc. Microwaves, Antennas Propag. 129, 278–280 (1982).
  4. R. Espinola, R. Ahmad, F. Pizzuto, M. Steel, and R. Osgood, “A study of high-index-contrast 90 degree waveguide bend structures,” Opt. Express 8(9), 517–528 (2001). [CrossRef] [PubMed]
  5. C. Xudong, C. Hafner, R. Vahldieck, and F. Robin, “Sharp trench waveguide bends in dual mode operation with ultra-small photonic crystals for suppressing radiation,” Opt. Express 14(10), 4351–4356 (2006). [CrossRef] [PubMed]
  6. A. Sakai, G. Hara, and T. Baba, “Propagation characteristics of ultrahigh-Δ optical waveguide on silicon on-insulator substrate,” Jpn. J. Appl. Phys. 40(Part 2, No. 4B), L383–L385 (2001). [CrossRef]
  7. A. Sakai, T. Fukazawa, and T. Baba, “Estimation of polarization crosstalk at a micro-bend in Si-photonic wire waveguide,” J. Lightwave Technol. 22(2), 520–525 (2004). [CrossRef]
  8. Y. Vlasov and S. McNab, “Losses in single-mode silicon-on-insulator strip waveguides and bends,” Opt. Express 12(8), 1622–1631 (2004). [CrossRef] [PubMed]
  9. P. A. Anderson, B. S. Schmidt, and M. Lipson, “High confinement in silicon slot waveguides with sharp bends,” Opt. Express 14(20), 9197–9202 (2006). [CrossRef] [PubMed]
  10. Q. Xu, D. Fattal, and R. G. Beausoleil, “Silicon microring resonators with 1.5-microm radius,” Opt. Express 16(6), 4309–4315 (2008). [CrossRef] [PubMed]
  11. Z. Sheng, D. Dai, and S. He, “Comparative study of losses in ultrasharp silicon-on-insulator nanowire bends,” IEEE J. Sel. Top. Quantum Electron. 15(5), 1406–1412 (2009). [CrossRef]
  12. P. Del'Haye, O. Arcizet, M. L. Gorodetsky, R. Holzwarth, and T. J. Kippenberg, “Frequency comb assisted diode laser spectroscopy for measurement of microcavity dispersion,” Nat. Photonics 3(9), 529–533 (2009). [CrossRef]
  13. P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450(7173), 1214–1217 (2007). [CrossRef] [PubMed]
  14. A. C. Turner, M. A. Foster, A. L. Gaeta, and M. Lipson, “Ultra-low power parametric frequency conversion in a silicon microring resonator,” Opt. Express 16(7), 4881–4887 (2008). [CrossRef] [PubMed]
  15. J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photonics 4(1), 37–40 (2010). [CrossRef]
  16. L. Yin, Q. Lin, and G. P. Agrawal, “Dispersion tailoring and soliton propagation in silicon waveguides,” Opt. Lett. 31(9), 1295–1297 (2006). [CrossRef] [PubMed]
  17. J. I. Dadap, N. C. Panoiu, X. Chen, I.-W. Hsieh, X. Liu, C.-Y. Chou, E. Dulkeith, S. J. McNab, F. Xia, W. M. J. Green, L. Sekaric, Y. A. Vlasov, and R. M. Osgood., “Nonlinear-optical phase modification in dispersion-engineered Si photonic wires,” Opt. Express 16(2), 1280–1299 (2008). [CrossRef] [PubMed]
  18. M. Oxborrow, “Traceable 2-D finite-element simulation of the whispering-gallery modes of axisymmetric electromagnetic resonators,” IEEE Trans. Microw. Theory Tech. 55(6), 1209–1218 (2007). [CrossRef]
  19. L. Zhang, Y. Yue, Y. Xiao-Li, J. Wang, R. G. Beausoleil, and A. E. Willner, “Flat and low dispersion in highly nonlinear slot waveguides,” Opt. Express 18(12), 13187–13193 (2010). [CrossRef] [PubMed]
  20. L. Zhang, Y. Yue, R. G. Beausoleil, and A. E. Willner, “Flattened dispersion in silicon slot waveguides,” Opt. Express 18(19), 20529–20534 (2010). [CrossRef] [PubMed]

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