## Tuning of zero group velocity dispersion in infiltrated vertical silicon slot waveguides |

Optics Express, Vol. 21, Issue 2, pp. 1741-1750 (2013)

http://dx.doi.org/10.1364/OE.21.001741

Acrobat PDF (1103 KB)

### Abstract

In this work the design of Si / hybrid waveguides which contain a vertical infiltrated slot is studied. The case of slots infiltrated with a *χ*^{(3)} nonlinear material of relatively high refractive index (e.g. chalcogenide glasses) is specifically discussed. An optimized waveguide geometry with periodic refractive index modulation, a nonlinear figure of merit > 1 and minimum effective mode cross section is presented. Introducing a periodic refractive index variation along the waveguide allows the adjustment of the group velocity dispersion (GVD). Choosing the period accordingly, the phase matching condition for degenerate four wave mixing (GVD = 0) can be fulfilled at virtually any desired frequency and independently from the fixed optimized waveguide cross section.

© 2013 OSA

## 1. Introduction

*n*= 3.5) and the oxide cladding (

_{Si}*n*

_{SiO2}≈ 1.45) and air (

*n*= 1), respectively, enables a high confinement of light inside such waveguides. This enables the fabrication of small footprint devices. However, for many applications active devices exhibiting a nonlinear optical behavior are needed. Although Si shows large

_{Air}*n*

_{2}-values (

*n*

_{2}= 6 · 10

^{−18}m

^{2}/W [1

1. M. R. Lamont, C. M. de Sterke, and B. J. Eggleton, “Dispersion engineering of highly nonlinear As_{2}S_{3} waveguides for parametric gain and wavelength conversion,” Opt. Express **15**, 9458–9463 (2007). [CrossRef] [PubMed]

_{2}[2

2. T. Liang and H. Tsang, “Nonlinear absorption and Raman scattering in silicon-on-insulator optical waveguides,” IEEE J. Sel. Top. Quantum Electron. **10**, 1149 – 1153 (2004). [CrossRef]

*β*= 6.7 · 10

^{−12}m/W [2

2. T. Liang and H. Tsang, “Nonlinear absorption and Raman scattering in silicon-on-insulator optical waveguides,” IEEE J. Sel. Top. Quantum Electron. **10**, 1149 – 1153 (2004). [CrossRef]

*n*

_{2}/

*λβ*). For Silicon the FOM is approximately 0.6 at

*λ*= 1550 nm. Generally a FOM > 1 is necessary for efficient nonlinear optical processes [4

4. V. Mizrahi, K. W. DeLong, G. I. Stegeman, M. A. Saifi, and M. J. Andrejco, “Two-photon absorption as a limitation to all-optical switching,” Opt. Lett. **14**, 1140–1142 (1989). [CrossRef] [PubMed]

5. C. Koos, L. Jacome, C. Poulton, J. Leuthold, and W. Freude, “Nonlinear silicon-on-insulator waveguides for all-optical signal processing,” Opt. Express **15**, 5976–5990 (2007). [CrossRef] [PubMed]

_{2}S

_{3}[6

6. M. Asobe, T. Kanamori, K. Naganuma, H. Itoh, and T. Kaino, “Third order nonlinear spectroscopy in As_{2}S_{3} chalcogenide glass fibers,” J. Appl. Phys. **77**, 5518–5523 (1995). [CrossRef]

7. 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**, 13187–13193 (2010). [CrossRef] [PubMed]

8. 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]

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

11. P. Muellner, M. Wellenzohn, and R. Hainberger, “Nonlinearity of optimized silicon photonic slot waveguides,” Opt. Express **17**, 9282–9287 (2009). [CrossRef] [PubMed]

5. C. Koos, L. Jacome, C. Poulton, J. Leuthold, and W. Freude, “Nonlinear silicon-on-insulator waveguides for all-optical signal processing,” Opt. Express **15**, 5976–5990 (2007). [CrossRef] [PubMed]

_{2}S

_{3}[13

13. C. Tsay, E. Mujagi, C. K. Madsen, C. F. Gmachl, and C. B. Arnold, “Mid-infrared characterization of solution-processed As_{2}S_{3} chalcogenide glass waveguides,” Opt. Express **18**, 15523–15530 (2010). [CrossRef] [PubMed]

15. G. C. Chern, “Spin-coated amorphous chalcogenide films,” J. Appl. Phys. **53**, 6979 (1982). [CrossRef]

4. V. Mizrahi, K. W. DeLong, G. I. Stegeman, M. A. Saifi, and M. J. Andrejco, “Two-photon absorption as a limitation to all-optical switching,” Opt. Lett. **14**, 1140–1142 (1989). [CrossRef] [PubMed]

11. P. Muellner, M. Wellenzohn, and R. Hainberger, “Nonlinearity of optimized silicon photonic slot waveguides,” Opt. Express **17**, 9282–9287 (2009). [CrossRef] [PubMed]

16. Y. Yue, L. Zhang, J. Wang, R. G. Beausoleil, and A. E. Willner, “Highly efficient nonlinearity reduction in silicon-on-insulator waveguides using vertical slots,” Opt. Express **18**, 22061 (2010). [CrossRef] [PubMed]

## 2. Optimized waveguide cross section

### 2.1. Basic waveguide geometry and numerical model

*w*with a vertical centered slot. A layer of height

_{wg}*h*of an optical nonlinear material is situated on top of the substrate and is infiltrated into the slot. The half space above the waveguide is filled with air. Throughout the paper we consider the chalcogenide glass As

_{wg}_{2}S

_{3}as the infiltrated/coating nonlinear material. It is one of the most common chalcogenide glasses. Its refractive index of approx 2.34 is still quite low compared to other chalcogenide glasses and the linear absorption nearly vanishes around a wavelength of 1500 nm - our spectral range of interest. To calculate the TE-like mode frequencies and field distributions of the waveguide modes, the commercial finite element solver COMSOL was used [18

18. “www.comsol.com”.

**e**,

**h**) the complex third order nonlinear parameter

*γ*of the waveguide is calculated [19

19. 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**, 2298–2318 (2009). [CrossRef] [PubMed]

*n*

_{2}and nonlinear absorption coefficients

*β*are in general anisotropic [20

20. 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. Osgood Jr., “Nonlinear-optical phase modification in dispersion-engineered si photonic wires,” Opt. Express **16**, 1280–1299 (2008). [CrossRef] [PubMed]

21. Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: modeling and applications,” Opt. Express **15**, 16604–16644 (2007). [CrossRef] [PubMed]

*n*

_{2}and

*β*which are listed in Table 1.

*γ*measures the strength of the 3rd order nonlinear interactions in wave guiding structures corresponding to

*χ*

^{(3)}in bulk materials [21

21. Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: modeling and applications,” Opt. Express **15**, 16604–16644 (2007). [CrossRef] [PubMed]

*γ*with FOM = Re{

*γ*}/4

*π*Im{

*γ*}.

*A*

_{eff}, |

**e**|

^{4}is integrated over the areas filled with As

_{2}S

_{3}(

*D*

_{As2S3}). Thus,

*A*

_{eff}becomes minimal if the concentration of the field is maximized inside the strongly nonlinear optical material As

_{2}S

_{3}leading to waveguide geometries with compact mode volumes.

*A*

_{eff}the width and the height of the waveguides were varied while a fixed slot width of 100 nm was assumed. It has been reported that a smaller slot width gives rise to stronger field enhancement and thus a higher nonlinear optical FOM [5

5. C. Koos, L. Jacome, C. Poulton, J. Leuthold, and W. Freude, “Nonlinear silicon-on-insulator waveguides for all-optical signal processing,” Opt. Express **15**, 5976–5990 (2007). [CrossRef] [PubMed]

11. P. Muellner, M. Wellenzohn, and R. Hainberger, “Nonlinearity of optimized silicon photonic slot waveguides,” Opt. Express **17**, 9282–9287 (2009). [CrossRef] [PubMed]

_{2}S

_{3}.

### 2.2. Results

*w*> 550 nm. For these wide waveguides a large part of the mode is propagating within the two silicon rails at both sides of the slot, so that the low FOM of silicon starts to govern the FOM of the whole waveguide. On the other hand for waveguide widths approaching 300 nm a large part of the mode propagates inside the slot or leaks out beyond the silicon rails into the adjacent As

_{wg}_{2}S

_{3}slab regions. For these geometries the highest FOMs of about 3 to 4 are expected (the FOM of bulk Silicon and bulk As

_{2}S

_{3}differ by a factor of 100). Finally, for ultra-narrow waveguide widths below 200 nm the waveguide modes become unbound since the radiation can couple to propagating modes inside the neighboring As

_{2}S

_{3}slab regions. The modes within the infinitely extended As

_{2}S

_{3}slab regions left and right of the waveguide therefore determine the ”light cone” for the waveguide modes.

*A*

_{eff}for a varying waveguide width

*w*and height

_{wg}*h*. There is a maximum of 1/

_{wg}*A*

_{eff}(corresponds to a minimum of

*A*

_{eff}) at

*w*= 420 nm and

_{wg}*h*= 210 nm. The mode field distribution of the geometry with minimized

_{wg}*A*

_{eff}is shown in Fig. 1(b). Although most of the mode field intensity is confined in the slot a significant part of the mode field is already propagating in silicon giving rise to two photon absorption. Therefore the overall FOM for this geometry has a moderate value of 1.3.

## 3. Manipulation of the group velocity dispersion

*χ*

^{(3)}, are of interest here. Next we will specifically investigate the phase matching conditions for the degenerate four wave mixing process and how these can be fulfilled for the mentioned infiltrated slot waveguides.

*ω*and

_{p}*ω*interact to form a new wave

_{s}*ω*. For an efficient frequency conversion the frequencies and phases of the three interacting waves have to be matched according to:

_{i}*k*(

_{i}*ω*) and

_{i}*k*(

_{s}*ω*) in (4) can be developed into Taylor series around

_{s}*ω*.

_{p}*d*

^{2}

*k*/

*dω*

^{2}, vanishes if the phase matching condition is fulfilled. Silicon as well as As

_{2}S

_{3}and standard waveguides show normal dispersion, i.e., the phase index is larger for shorter wavelengths. One way to compensate this behavior is to use the anomalous dispersion of silica in the wavelength region around 1500 nm. Waveguides are then designed in such way that for longer wavelength (when the wave leaks out into the surrounding material, i.e. silica) its anomalous dispersion compensates the waveguide dispersion [23

23. 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. 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**, 13187–13193 (2010). [CrossRef] [PubMed]

9. 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]

24. 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]

25. L. Zhang, Y. Yue, Y. Xiao-Li, R. G. Beausoleil, and A. E. Willner, “Highly dispersive slot waveguides,” Opt. Express **17**, 7095–7101 (2009). [CrossRef] [PubMed]

*a*≫

*λ*

_{0}/

*n*) periodic modulation of the effective mode index along the propagation direction, with a periodicity

*a*. This adds a negative contribution of 2

*πm*/

*a*(

*m*= 1, 2, 3,...) to the phase matching condition (4). If properly designed, this allows the fulfillment of Eq. (4) in many conditions [26

26. J. B. Driscoll, N. Ophir, R. R. Grote, J. I. Dadap, N. C. Panoiu, K. Bergman, and R. M. Osgood, “Width-modulation of Si photonic wires for quasi-phase-matching of four-wave-mixing: experimental and theoretical demonstration,” Opt. Express **20**, 9227–9242 (2012). [CrossRef] [PubMed]

*a*<

*λ*

_{0}/

*n*. This refractive index modulations causes a 1-D photonic bandgap, which leads to additional band bending in its vicinity. This band bending yields an anomalous contribution to the GVD that can be used to achieve zero GVD. In practice, such a structure could be achieved by exploiting the photorefractive effect of As

_{2}S

_{3}. It is widely reported that the refractive index of chalcogenide glasses can be changed by illumination with above bandgap radiation. In the case of As

_{2}S

_{3}the refractive index can be altered up to 0.1 by exposure to visible or UV light [27

27. R. Todorov, D. Tsankov, J. Pirov, and K. Petkov, “Structure and optical properties of thin As_{2}S_{3} In_{2}S_{3} films,” J. Phys. D: Appl. Phys. **44**, 305401 (2011). [CrossRef]

### 3.1. Numerical Model

*n*

_{1}≈ 2.3 and

*n*

_{2}=

*n*

_{1}+ 0.05 is depicted in Fig. 3(a). The index

*n*1 corresponds to that of unexposed As

_{2}S

_{3}calculated by Sellmeier eq. with parameters given in Table 1. The offset of 0.05 was arbitrarily chosen as half of the maximum refractive index change reported in [27

27. R. Todorov, D. Tsankov, J. Pirov, and K. Petkov, “Structure and optical properties of thin As_{2}S_{3} In_{2}S_{3} films,” J. Phys. D: Appl. Phys. **44**, 305401 (2011). [CrossRef]

_{2}S

_{3}-layer is the same as the height of the waveguide and the upper half space above the waveguide is assumed to be air. Such a structure might be fabricated by periodic exposure of an already deposited As

_{2}S

_{3}film. The eigenfrequencys

*ω*(

*k*) of such a periodic system for given

*k*= 2

*π*/

*λ*-vectors were calculated in a 3D geometry using periodic boundary conditions in the propagation direction

*z*[28]. The finite element software COMSOL [18

18. “www.comsol.com”.

*n*

_{1}of the materials for different eigenfrequencies into account. The cross section of the waveguide geometry corresponds to the optimum condition with minimum

*A*

_{eff}as described above. For a one-dimensional periodic index modulation a band gap around

*λ*

_{0}/2

*n*=

*a*is expected [29]. For a design wavelength of

*λ*

_{0}= 1550 nm this leads to

*a*≈ 330 nm. To test the flexibility of our strategy we calculated the eigenfrequencies for different lattice constants

*a*.

### 3.2. Results

*a*= 380 nm is shown in Fig. 3(b). It shows the waveguide mode which is folded back, forming a bandgap at the Brillouin zone edge. Above the gap the guided modes of the second band extend up to the shaded area where the waveguide modes are able to couple to the As

_{2}S

_{3}-slab modes left and right of the waveguide and become lossy (light cone). To compensate the usual normal dispersion of the material and the waveguide mode the band bending of the second band around the upper band edge will be exploited. Only there the GVD vanishes.

*a*= 300 nm, 320 nm,...,400 nm are shown on Fig. 4(a). There a clear zero crossing is observed, which shifts to longer wavelengths for larger lattice constants. A |GVD| < 0.5 ps/(nm m) is achieved within a wavelength range of Δ

*λ*≈ 10 nm. This graph also represents the huge flexibility for tuning the GVD = 0 frequency to the desired spectral region. By simply changing the period of the index modulation the condition GVD = 0 can be shifted to nearly any wavelength within the near infra-red. The cross section of the waveguide can be left unchanged. In this way the method used here allows to choose the cross section of the waveguide in such a way to optimize parameters like effective mode area or FOM and then adjust the period of the index modulation to achieve the phase matching condition (GVD = 0) at the desired wavelength.

*βL*| <

*π*/2 from [21

21. Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: modeling and applications,” Opt. Express **15**, 16604–16644 (2007). [CrossRef] [PubMed]

*β*= 2

*k*(

_{p}*ω*) −

_{p}*k*(

_{i}*ω*) −

_{i}*k*(

_{s}*ω*) depends on the length of the waveguide

_{i}*L*. Assuming waveguide lengths between 1 and 5 mm the corresponding bandwidths were determined from the original dispersion curves and are presented in Fig. 4(b). As can be seen the bandwidth is well above 40 nm in any case and decreases more slowly with increasing waveguide length allowing even longer waveguides for narrow band four wave mixing.

*h*where the As

_{wg}_{2}S

_{3}layer extends by 20 nm to 100 nm above the waveguide. This is a realistic scenario when a fabricated silicon slot waveguide is infiltrated afterward with the chalcogenide glass and no further leveling or polishing is performed. The results of these investigations are shown in Fig. 4(c). The zero point of the GVD parameter shifts to longer wavelengths for thicker cladding layer. This is expected as the field of the mode extends a bit upward out of the slot and the mode ”sees” now still the relatively high refractive index of the added As

_{2}S

_{3}with Δ

*h*. As expected the impact of Δ

_{wg}*h*on the shift of the GVD = 0 frequency levels off for larger Δ

_{wg}*h*as the mode decays quickly above the slot. However, care has to be taken that the As

_{wg}_{2}S

_{3}-layer does not become too thick, as then the frequency of the As

_{2}S

_{3}-slab modes decreases considerably and the shaded area of the “radiative modes” in Fig. 3(b) extends close to the bandgap. Ultimately, the GVD = 0 frequency would be shifted into the leaky mode regime. This places a limit to the overall allowed As

_{2}S

_{3}layer thickness in the area between 500 nm to 1 micron depending on waveguide cross section parameters and contrast of index modulation.

*β*= 2

_{NL}*P*{

_{p}Re*γ*} induced by the Kerr effect was neglected. To estimate the influence of this phase shift we calculated the phase mismatch Δ

*β*= Δ

*β*− (2

_{NL}*k*(

_{p}*ω*) −

_{p}*k*(

_{i}*ω*− Δ

_{p}*ω*) −

*k*(

_{s}*ω*+ Δ

_{p}*ω*)) for Δ

*ω*= 2

*π*1.2 · 10

^{12}Hz ≙ Δ

*λ*≈ 10nm and

*Re*{

*γ*} = 90(Wm)

^{−1}at a pump power

*P*= 20dBm [21

_{p}**15**, 16604–16644 (2007). [CrossRef] [PubMed]

26. J. B. Driscoll, N. Ophir, R. R. Grote, J. I. Dadap, N. C. Panoiu, K. Bergman, and R. M. Osgood, “Width-modulation of Si photonic wires for quasi-phase-matching of four-wave-mixing: experimental and theoretical demonstration,” Opt. Express **20**, 9227–9242 (2012). [CrossRef] [PubMed]

*β*= 0 shifts by 2.5 nm towards the red from the original GVD = 0 wavelength. This demonstrates that the introduced band bending due to the photonic band gap is strong enough to compensate for linear and nonlinear dispersion characteristics. Even smaller index contrasts below Δ

*n*= 0.1 are sufficient to compensate for the material and waveguide dispersion. Furthermore the idea to use a periodic index modulation to achieve the phase matching for the degenerated four wave mixing process is not limited to photo-refractive materials. A periodic modulation of waveguide width or height or slot width will have a similar effect and will create a photonic bandgap with the required band bending as it was already reported for ”nanobeams” [30

30. P. B. Deotare, M. W. McCutcheon, I. W. Frank, M. Khan, and M. Lonar, “High quality factor photonic crystal nanobeam cavities,” Appl. Phys. Lett. **94**, 121106–121106–3 (2009). [CrossRef]

*λ*= 50 nm. In this respect our work has a certain similarity to the work on flat band slow light reported by Li et al. [31

31. J. Li, T. P. White, L. O’Faolain, A. Gomez-Iglesias, and T. F. Krauss, “Systematic design of flat band slow light in photonic crystal waveguides,” Opt. Express **16**, 6227–6232 (2008). [CrossRef] [PubMed]

*λ*−

_{i}*λ*) > 200 nm) the quasi phase matching strategy investigated by Driscoll et al. [26

_{p}26. J. B. Driscoll, N. Ophir, R. R. Grote, J. I. Dadap, N. C. Panoiu, K. Bergman, and R. M. Osgood, “Width-modulation of Si photonic wires for quasi-phase-matching of four-wave-mixing: experimental and theoretical demonstration,” Opt. Express **20**, 9227–9242 (2012). [CrossRef] [PubMed]

## 4. Conclusion

_{2}S

_{3}waveguides with optimized waveguide cross section and tailored group velocity dispersion is presented. This method is based on a refractive index modulation along the waveguide and allows to independently optimize the effective mode area

*A*

_{eff}and the GVD. An FOM of about 1.3 and GVD parameter < 0.5 ps/(nm m) was achieved for a wavelength range of Δ

*λ*= 10 nm. The bandwidth for degenerate four wave mixing processes amounts to Δ

*λ*≈ 50 nm for waveguide lengths of a few mm.

## 5. Appendix

*B*and

_{i}*C*as in Table 1 has been used to calculate the refractive indices of the materials.

_{i}## Acknowledgment

^{®}project 03Z2HN12.

## References and links

1. | M. R. Lamont, C. M. de Sterke, and B. J. Eggleton, “Dispersion engineering of highly nonlinear As |

2. | T. Liang and H. Tsang, “Nonlinear absorption and Raman scattering in silicon-on-insulator optical waveguides,” IEEE J. Sel. Top. Quantum Electron. |

3. | G. P. Agrawal, |

4. | V. Mizrahi, K. W. DeLong, G. I. Stegeman, M. A. Saifi, and M. J. Andrejco, “Two-photon absorption as a limitation to all-optical switching,” Opt. Lett. |

5. | C. Koos, L. Jacome, C. Poulton, J. Leuthold, and W. Freude, “Nonlinear silicon-on-insulator waveguides for all-optical signal processing,” Opt. Express |

6. | M. Asobe, T. Kanamori, K. Naganuma, H. Itoh, and T. Kaino, “Third order nonlinear spectroscopy in As |

7. | 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 |

8. | 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. |

9. | L. Zhang, Y. Yue, R. G. Beausoleil, and A. E. Willner, “Flattened dispersion in silicon slot waveguides,” Opt. Express |

10. | S. Mas, J. Caraquitena, J. V. Galn, P. Sanchis, and J. Mart, “Tailoring the dispersion behavior of silicon nanophotonic slot waveguides,” Opt. Express |

11. | P. Muellner, M. Wellenzohn, and R. Hainberger, “Nonlinearity of optimized silicon photonic slot waveguides,” Opt. Express |

12. | G. Agrawal, |

13. | C. Tsay, E. Mujagi, C. K. Madsen, C. F. Gmachl, and C. B. Arnold, “Mid-infrared characterization of solution-processed As |

14. | C. Tsay, Y. Zha, and C. B. Arnold, “Solution-processed chalcogenide glass for integrated single-mode mid-infrared waveguides,” Opt. Express |

15. | G. C. Chern, “Spin-coated amorphous chalcogenide films,” J. Appl. Phys. |

16. | Y. Yue, L. Zhang, J. Wang, R. G. Beausoleil, and A. E. Willner, “Highly efficient nonlinearity reduction in silicon-on-insulator waveguides using vertical slots,” Opt. Express |

17. | P. Muellner, “Fundamental characteristics of the soi slot waveguide structure,” Ph.D. thesis, Faculty of Physics, University of Vienna (2010). |

18. | |

19. | 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 |

20. | 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. Osgood Jr., “Nonlinear-optical phase modification in dispersion-engineered si photonic wires,” Opt. Express |

21. | Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: modeling and applications,” Opt. Express |

22. | I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Effective mode area and its optimization in silicon-nanocrystal waveguides,” Opt. Lett. |

23. | 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 |

24. | 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 |

25. | L. Zhang, Y. Yue, Y. Xiao-Li, R. G. Beausoleil, and A. E. Willner, “Highly dispersive slot waveguides,” Opt. Express |

26. | J. B. Driscoll, N. Ophir, R. R. Grote, J. I. Dadap, N. C. Panoiu, K. Bergman, and R. M. Osgood, “Width-modulation of Si photonic wires for quasi-phase-matching of four-wave-mixing: experimental and theoretical demonstration,” Opt. Express |

27. | R. Todorov, D. Tsankov, J. Pirov, and K. Petkov, “Structure and optical properties of thin As |

28. | A. von Rhein, S. Greulich-Weber, and R. B. Wehrspohn, “Multiphysics software gazes into photonic crystals,” Physics Best pp. 38–39 (2007). |

29. | J. D. Joannopoulos and J. N. Winn, |

30. | P. B. Deotare, M. W. McCutcheon, I. W. Frank, M. Khan, and M. Lonar, “High quality factor photonic crystal nanobeam cavities,” Appl. Phys. Lett. |

31. | J. Li, T. P. White, L. O’Faolain, A. Gomez-Iglesias, and T. F. Krauss, “Systematic design of flat band slow light in photonic crystal waveguides,” Opt. Express |

32. | M. Bass, C. DeCusatis, J. Enoch, G. Li, V. N. Mahajan, E. V. Stryland, and C. MacDonald, |

**OCIS Codes**

(190.4380) Nonlinear optics : Nonlinear optics, four-wave mixing

(190.4390) Nonlinear optics : Nonlinear optics, integrated optics

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: October 12, 2012

Revised Manuscript: November 22, 2012

Manuscript Accepted: November 24, 2012

Published: January 16, 2013

**Citation**

Peter W. Nolte, Christian Bohley, and Jörg Schilling, "Tuning of zero group velocity dispersion in infiltrated vertical silicon slot waveguides," Opt. Express **21**, 1741-1750 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-2-1741

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