## Rigorous modal analysis of silicon strip nanoscale waveguides

Optics Express, Vol. 18, Issue 8, pp. 8528-8539 (2010)

http://dx.doi.org/10.1364/OE.18.008528

Acrobat PDF (2081 KB)

### Abstract

A full-vectorial **H**-field Finite Element Method has been used for the rigorous modal analysis of silicon strip waveguides. The spatial variation of the full-vectorial **H** and **E**-fields are also discussed in details and further, the Poynting vector is also presented. The modal area, hybridness, single mode operation and birefringence are also described for such silicon strip waveguides.

© 2010 OSA

## 1. Introduction

_{3}) and Indium Phosphide (InP) that are complicated to process and unless their critical material properties, such as their excellent electro-optic properties or emissive capabilities are explicitly needed, are excessively expensive to produce. Recent research [1

1. M. Lipson, “Guiding, Modulating, and Emitting Light on Silicon-Challenges and Opportunities,” J. Lightwave Technol. **23**(12), 4222–4238 (2005). [CrossRef]

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

3. D. Dai and S. He, “Ultrasmall overlapped arrayed-waveguide grating based on Si nanowire waveguides for dense wavelength division multiplexing,” IEEE J. Sel. Top. Quantum Electron. **12**(6), 1301–1305 (2006). [CrossRef]

4. L. Liao, D. Samara-Rubio, M. Morse, A. Liu, D. Hodge, D. Rubin, U. D. Keil, and T. Franck, “High speed silicon Mach-Zehnder modulator,” Opt. Express **13**(8), 3129–3135 (2005). [CrossRef] [PubMed]

5. O. Boyraz and B. Jalali, “Demonstration of a silicon Raman laser,” Opt. Express **12**(21), 5269–5273 (2004). [CrossRef] [PubMed]

6. L. Yin, Q. Lin, and G. P. Agrawal, “Soliton fission and supercontinuum generation in silicon waveguides,” Opt. Lett. **32**(4), 391–393 (2007). [CrossRef] [PubMed]

_{z}has a large effect on the waveguide nonlinearity.

**H**-field finite-element method (VFEM) [7

7. B. M. A. Rahman and J. B. Davies, “Finite element solution of integrated optical waveguides,” J. Lightwave Technol. **2**(5), 682–688 (1984). [CrossRef]

**E**) is also derived from the vector

**H**-field obtained to characterize modal properties of such a waveguides.

## 2. Theory

**H**-Field formulation has been established as one of the most accurate and numerically efficient approaches to obtaining the modal field profiles and propagation constants of the fundamental and higher-order quasi-TE and TM modes. The full-vectorial formulation is based on the minimization of the following functional in terms of the nodal values of the full

**H**-field vector,where

**H**is the full vectorial magnetic field, * denotes a complex conjugate and transpose, ω

^{2}is the eigenvalue where ω is the angular frequency of the wave and ε and µ are the permittivity and permeability, respectively. The penalty function approach has also been incorporated to impose divergence-free condition of the magnetic field to reduce the appearance of spurious modes and

*p*is the dimensionless penalty parameter. It has been shown [8

8. B. M. A. Rahman and J. B. Davies, “Penalty Function Improvement of Waveguide Solution by Finite Elements,” IEEE Trans. Microw. Theory Tech. **32**(8), 922–928 (1984). [CrossRef]

*p*’ value of the order of 1/n

_{g}

^{2}would give a satisfactory result, where n

_{g}is the refractive index of the core. This full-vectorial FEM modal solution may also be used to determine the beat lengths between the fundamental and the higher order modes and also the polarization beat length between the two quasi-TE and TM polarized fundamental modes.

## 3. Results

### 3.1 Waveguide structure

_{2}) or air. This waveguide can be fabricated by using a SOI (Silicon-On-Insulator) wafer on a Si substrate. A resist mask can be used on the surface of the Si layer and the Si waveguide core may be formed by etching down to the SiO

_{2}buffer layer by use of an Inductively Coupled Plasma (ICP) dry etcher. The Si core can then either be buried under a thick SiO

_{2}over-layer or surrounded with air. In this study, the thickness of the core waveguide is taken as 260 nm and that of the lower SiO

_{2}buffer layer as 1.50 μm. The refractive index of the rectangular Si core at 1.55 μm wavelength is taken as 3.50. The refractive indices for the SiO

_{2}cladding or air cladding are taken as 1.50 and 1.0, respectively.

### 3.2 Modal solution

**H**-field based VFEM is used to obtain the modal solutions of such a waveguide. For this study, due to the availability of two-fold symmetry of the waveguide structure, only a quarter of the structure is considered, in which more than 80,000 irregular sized first order triangular elements have been employed to represent the waveguide structure. It takes about 2 minutes cpu time on a dual-core Pentium processor computer running solaris platform.

_{e}) with the width (W) for the fundamental H

_{y}

^{11}and higher order H

_{y}

^{21}, H

_{y}

^{31}and H

_{y}

^{41}quasi-TE modes are presented for the SiO

_{2}-cladding. The effective index, n

_{e}, of a given mode is a normalized propagation parameter, which can be defined by n

_{e}= β

_{0}/k

_{0}, where β

_{0}is the propagation constant of that mode and k

_{0}is the free space wavenumber defined as k

_{0}= ω(ε

_{0}μ

_{0})

^{1/2}= 2π/λ. It can be observed that when the width of the waveguide is large in comparison to the height of the structure, the waveguide supports many modes and n

_{e}is closer to the refractive index of Si. As W, decreases, n

_{e}also decreases and gets closer to the refractive index of SiO

_{2}where the mode reaches its cut-off. It is shown here that the cut-off widths for the H

_{y}

^{41}, H

_{y}

^{31}, H

_{y}

^{21}and H

_{y}

^{11}modes are 1000 nm, 700 nm, 400 nm and 200 nm, respectively. This simulation data generated suggest that the single-mode operation occurs when the waveguide width lies approximately between 200 nm to 400 nm when the operating wavelength, λ, is 1550 nm with a height of 260 nm. Accuracy of the solution for the H

_{y}

^{11}mode was identified by using Aitken extrapolation [9

9. B. M. A. Rahman and J. B. Davies, “Vector-H finite element solution of GaAs/GaAlAs rib waveguides,” Optoelectronics, IEE Proceedings J **132**, 349–353 (1985). [CrossRef]

### 3.3 Effective area

_{eff}) with the waveguide width for different quasi-TE modes. Following the second moment of intensity distribution (recommended by ISO Standard 11146), the definition of the effective area [10] can be given by:where

*E*is the transverse electric field vector and the integration is carried out over the whole cross-section of the waveguide, Ω.

_{t}_{eff}reduces to a minimum value and any further reduction of W, results in the sudden increases of the A

_{eff}as the mode approaches its cut-off. The minimum effective area, A

_{min}, for the H

_{y}

^{11}mode is obtained as 0.0956 μm

^{2}when the value of W = 320 nm, which means that the mode is more confined when the effective area is a minimum. Further rigorous simulations were carried out for the quasi-TE H

_{y}

^{21}, H

_{y}

^{31}and H

_{y}

^{41}modes. It was found that the widths for the minimum A

_{eff}values for the H

_{y}

^{21}, H

_{y}

^{31}and H

_{y}

^{41}modes are 650 nm, 1000 nm and 1300 nm, respectively. It can be noted that all the A

_{eff}values are very similar when the waveguide width, W, is large but it is only slightly higher for the higher order modes.

### 3.4 Power confinement

_{y}

^{11}, H

_{y}

^{21}, H

_{y}

^{31}and H

_{y}

^{41}modes. The confinement factor in any particular area normalized to the total power, which is obtained by integrating the Poynting vector, from the

**H**-and

**E**-fields as given below:

_{Si}would be close to 1.0. However, it can be noted that, the maximum power confinement in this case is closer to 0.85, because the height of the core was restricted to 260 nm. If the height of the core also becomes larger, then the power confinement in the Si core could approach 1.0. It can be observed here that as the width is reduced, the power confinement in the Si core also reduces. It can be also observed that although, for a wider waveguide, the power confinement for all the four modes shown here are similar, but for a narrower waveguide, the power confinements for the higher order modes are smaller.

_{e}, the effective area, A

_{eff}and the power confinement in the SiO

_{2}cladding region, Γ

_{SiO2}with the width, W, of the fundamental H

_{y}

^{11}mode are shown together in Fig. 4 , for their comparison. It can be observed that for the single mode operation, in the case of a 260 nm thick waveguide, the value of the width should lie between 200 nm and 400 nm and this may be chosen optimally to be around 320 nm, when the spot-size is also the smallest. In this case, the power confinement in SiO

_{2}is 0.31 and the remaining power will be in the Si core. From this figure, it can be seen that the power confinement range in SiO

_{2}for single mode operation range could be between 0.89 and 0.20 for those width values varying from 200 nm to 400 nm respectively.

### 3.5 Modal **H**-field profiles

_{y}field component is dominant, and H

_{x}and H

_{z}are the non-dominant components. The dominant H

_{y}field component of the H

_{y}

^{11}mode is shown as an inset in Fig. 5 for the waveguide width, W = 300 nm and height, H = 260 nm.

_{2}cladding region. To illustrate its variation more clearly, the variation of the H

_{y}field along the Y-axis is also shown by a dashed line in Fig. 5. In this case, the H

_{y}field is monotonically decreasing along the Y-axis and when y = 0.13 μm (as H = 260 nm), at the boundary between the Si and SiO

_{2}, the magnitude of the H

_{y}field is 60% of its maximum value. The variations of the H

_{y}field along the X-axis is also shown by a solid line, which also decreases monotonically but with a rapid reduction inside the Si core until at the boundary interface, following which its slope reduces in the SiO

_{2}cladding region. At the interface, (at x = 0.15 μm, as W = 300 nm) the magnitude of the H

_{y}field is only about 16% of its maximum value, which is significantly lower than field values at the upper and lower interfaces. In its contour plot it is clearly visible that the modal confinement in the horizontal direction is much stronger. The effective index, n

_{e}, of the H

_{y}

^{11}mode was found to be 2.00711 with its effective area A

_{eff}, equal to 0.098 μm

^{2}, when W = 300 nm.

_{z}field component of the H

_{y}

^{11}mode is shown as an inset in Fig. 7 . The maximum magnitude of H

_{z}is found to be 41% of the maximum H

_{y}field and it is significantly higher than the non-dominant of H

_{x}field. The H

_{z}field is zero along the X-axis and it can be observed that the maximum intensity occurs at the two horizontal interfaces between the Si and SiO

_{2}. This is because the value of H

_{z}is proportional to the derivative ∂E

_{x}/∂y and therefore the H

_{z}value is shown to peak in the y-direction. It can also be noted that all the three

**H**-field components of the H

_{y}

^{11}mode are continuous across the Si/SiO

_{2}interfaces.

### 3.6 Hybridness

_{y}

^{11}mode, the H

_{y}component is the dominant and H

_{x}and H

_{z}each are non-dominant. The variations of the modal hybridness with the width for the H

_{y}

^{11}, H

_{y}

^{21}, H

_{y}

^{31}and H

_{y}

^{41}modes are shown in Fig. 8 . The modal hybridness in this case is defined as the ratio of the maximum value of non-dominant H

_{x}field to the maximum value of dominant H

_{y}field. This is an important parameter to calculate polarization cross-talk [11

11. N. Somasiri and B. M. A. Rahman, “Polarization crosstalk in high index contrast planar silica waveguides with slanted sidewalls,” J. Lightwave Technol. **21**(1), 54–60 (2003). [CrossRef]

12. B. M. A. Rahman, S. S. A. Obayya, N. Somasiri, M. Rajarajan, K. T. V. Grattan, and H. A. El-Mikathi, “Design and characterization of compact single-section passive polarization rotator,” J. Lightwave Technol. **19**(4), 512–519 (2001). [CrossRef]

_{y}

^{11}mode its hybridness is shown by a solid line in Fig. 8. This is low when the width is large, and as W reduces, the hybridness reaches its maximum, then slowly reduces as the fundamental mode approaches its cut-off region when the A

_{eff}also increases. The maximum hybridness for the fundamental H

_{y}

^{11}mode is found to be 0.19 at W = 320 nm when its effective area was also the smallest. Furthermore, the behavior for the higher order modes (H

_{y}

^{21}, H

_{y}

^{31}and H

_{y}

^{41}) are shown to be similar to that of the fundamental mode, in which, as the width decreases, the hybridness increases until each higher order modes approaches its cut-off regions.

_{z}field, the hybridness can also be defined as the ratio of H

_{z}/H

_{y}, and this parameter is a function of the width, is shown in Fig. 9 . It can be observed that the hybridness increases as the width increases. It is also shown that the H

_{z}hybridness of the fundamental H

_{y}

^{11}mode is significantly larger than its H

_{y}hybridness. The H

_{z}hybridness of the fundamental and higher order modes follows a similar behavior, in which, they all decreases as width decreases. In this case, with W = 300 nm and H = 260 nm, the Si core waveguide is asymmetric. However, it was observed that in the case of when height varied with a fixed width, the H

_{z}hybridness increases as the waveguide height decreases (but this is not shown in here).

### 3.7 Modal **E**-field profiles

**H**-field are obtained, Maxwell’s ∇ ×

**H**equation is used to calculate the three components of the

**E**-field vector. The fundamental quasi-TE H

_{y}

^{11}mode contains all the three components of the electric field, E

_{x}, E

_{y}and E

_{z}: the E

_{x}field is dominant, and the E

_{y}and E

_{z}field components are non-dominant. Variations of the dominant E

_{x}field along the X and Y-axes for the fundamental H

_{y}

^{11}mode are shown in Fig. 10 for the situation when W = 300 nm and H = 260 nm. The variation of the E

_{x}field along the Y-axis, shown here as a dashed-line, reduces monotonically from the center of the waveguide core and is continuous at the interface between Si/SiO

_{2}, as required by the boundary condition. The SiO

_{2}and Si interfaces are shown by two vertical dashed lines. However, the E

_{x}field along the X-axis, shown here as a solid line, reduces more quickly in the core and at the Si/SiO

_{2}interface increases abruptly with a step change in the ratio of (3.5/1.5)

^{2}= 5.44 in the SiO

_{2}region. Therefore the magnitude of the E

_{x}field in the SiO

_{2}region can be significantly higher than that in the core region and this behavior is also shown in the 3D-contour of E

_{x}field, this being shown as an inset in Fig. 10.

_{y}field is illustrated here as an inset in Fig. 11 . From the E

_{y}contour, it can be observed that the

**E**-field is zero along the X and Y-axes. So the variations of E

_{y}inside the core area, but slightly away from the boundary interfaces are shown in Fig. 11. The variations of E

_{y}along the Y-direction at x = 0.1495 µm is shown by a dashed-line and the variations of E

_{y}along the X-direction at y = 0.1296 µm is shown further as a solid line. It can also be seen that the E

_{y}field is small inside the core and increases along the Y-direction. There is a step change noticeable at the SiO

_{2}and Si interface in the Y-direction, with E

_{y}being normal to this interface and seen as discontinuous here. On the other hand, although E

_{y}also peaks at the Si/SiO

_{2}interface along the X-direction, but this was more gradual. It can be noted that while the E

_{x}field was more concentrated in the core region (shown in Fig. 10), the E

_{y}field is predominantly present around the Si/SiO

_{2}interfaces, as can be seen from Fig. 11.

_{z}field component of the H

_{y}

^{11}mode is shown as an inset in Fig. 12 . It can be noted that E

_{z}is zero along the Y-axis. The variation of the E

_{z}field along the X-axis is also shown here. It can be observed that the maximum value of the longitudinal E

_{z}field is 80% of the maximum E

_{x}at the Si/SiO

_{2}interface and then is noted to reduce gradually in the SiO

_{2}region. Such a high longitudinal field can be used in the beam shaping, but this can also influence its polarization properties.

### 3.8 Poynting vector profiles

**E**and

**H**fields, the Poynting Vector (S

_{z}) may be calculated. The contour of the S

_{z}intensity profile is shown as an inset in Fig. 13 , where the core of the rectangular Si waveguide is also outlined by dashed lines. The variations of S

_{z}along the X and Y-axes are also shown here. Variations of the S

_{z}along the Y-axis are shown by a dashed-line, which clearly indicates that it reduces monotonically and continuous at the interface and extends more along the Y-axis. However, the variation of the S

_{z}along the X-axis is shown by solid line which reveals a small discontinuity step at the Si/SiO

_{2}interface and it decays more quickly in the Si core. It was observed, (but not shown here), that for a smaller waveguide width, the S

_{z}discontinuity step can be even larger with a significant S

_{z}field extending into the SiO

_{2}cladding in the horizontal X-direction.

_{y}

^{11}and H

_{x}

^{11}modes are shown in Fig. 14. In this study, mostly the effective index for the quasi-TE H

_{y}

^{11}mode is larger than the quasi-TM H

_{x}

^{11}mode: however when the waveguide’s width became smaller, the difference between the effective indices of the H

_{y}

^{11}and H

_{x}

^{11}modes becomes smaller, and they are equal when the waveguide becomes a square nano-scale waveguide with W = H = 260 nm.

_{2}cladding but be in air and the variations of their effective indices and the effective areas with the width are rigorously investigated. Here, the air-clad structure has only got one-fold symmetry, so half of the waveguide is used in the simulation. Figure 15 shows the variations of effective indices and effective areas for the fundamental H

_{y}

^{11}mode with the width for both air and SiO

_{2}cladding. It can be observed that when the waveguide is covered with SiO

_{2}both the effective indices and the effective areas are slightly higher than that with the air-cladding.

_{2}-cladding due to a stronger refractive index contrast between the Si core and upper cladding. It can be noted that the modal birefringence difference is larger when the height of the waveguide is smaller as in this case the waveguide asymmetry is increased. It is also shown here that the birefringence is zero when the waveguide’s width and height are the same.

## 4. Conclusion

**E**and

**H**-fields are shown, together with the S

_{z}profile. The modal hybridness, the power confinement and the modal birefringence for silica and air-clad Si nanowires are also shown here. It should be noted that the performance of these highly birefringence guided-wave devices are polarization dependent. However, in the presence of slanted side walls and bends, there may be additional polarization conversion that occurs due to the presence of higher magnitude of the non-dominant field components and this can only be studied by using a rigorous full-vectorial approach, as was done in this work.

## References and links

1. | M. Lipson, “Guiding, Modulating, and Emitting Light on Silicon-Challenges and Opportunities,” J. Lightwave Technol. |

2. | Y. Vlasov and S. J. McNab, “Losses in single-mode silicon-on-insulator strip waveguides and bends,” Opt. Express |

3. | D. Dai and S. He, “Ultrasmall overlapped arrayed-waveguide grating based on Si nanowire waveguides for dense wavelength division multiplexing,” IEEE J. Sel. Top. Quantum Electron. |

4. | L. Liao, D. Samara-Rubio, M. Morse, A. Liu, D. Hodge, D. Rubin, U. D. Keil, and T. Franck, “High speed silicon Mach-Zehnder modulator,” Opt. Express |

5. | O. Boyraz and B. Jalali, “Demonstration of a silicon Raman laser,” Opt. Express |

6. | L. Yin, Q. Lin, and G. P. Agrawal, “Soliton fission and supercontinuum generation in silicon waveguides,” Opt. Lett. |

7. | B. M. A. Rahman and J. B. Davies, “Finite element solution of integrated optical waveguides,” J. Lightwave Technol. |

8. | B. M. A. Rahman and J. B. Davies, “Penalty Function Improvement of Waveguide Solution by Finite Elements,” IEEE Trans. Microw. Theory Tech. |

9. | B. M. A. Rahman and J. B. Davies, “Vector-H finite element solution of GaAs/GaAlAs rib waveguides,” Optoelectronics, IEE Proceedings J |

10. | ISO 11146, Laser and laser related equipment – Test methods for laser beam widths, divergence and beam propagation ratios, International Organization for Standardization, Geneva, Switzerland, 2005. |

11. | N. Somasiri and B. M. A. Rahman, “Polarization crosstalk in high index contrast planar silica waveguides with slanted sidewalls,” J. Lightwave Technol. |

12. | B. M. A. Rahman, S. S. A. Obayya, N. Somasiri, M. Rajarajan, K. T. V. Grattan, and H. A. El-Mikathi, “Design and characterization of compact single-section passive polarization rotator,” J. Lightwave Technol. |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(040.6040) Detectors : Silicon

(230.7370) Optical devices : Waveguides

(350.4238) Other areas of optics : Nanophotonics and photonic crystals

**ToC Category:**

Physical Optics

**History**

Original Manuscript: December 24, 2009

Revised Manuscript: March 11, 2010

Manuscript Accepted: March 28, 2010

Published: April 8, 2010

**Citation**

D. M. H. Leung, N. Kejalakshmy, B. M. A. Rahman, and K. T. V. Grattan, "Rigorous modal analysis of silicon strip nanoscale waveguides," Opt. Express **18**, 8528-8539 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-8-8528

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### References

- M. Lipson, “Guiding, Modulating, and Emitting Light on Silicon-Challenges and Opportunities,” J. Lightwave Technol. 23(12), 4222–4238 (2005). [CrossRef]
- Y. Vlasov and S. J. McNab, “Losses in single-mode silicon-on-insulator strip waveguides and bends,” Opt. Express 12(8), 1622–1631 (2004). [CrossRef] [PubMed]
- D. Dai and S. He, “Ultrasmall overlapped arrayed-waveguide grating based on Si nanowire waveguides for dense wavelength division multiplexing,” IEEE J. Sel. Top. Quantum Electron. 12(6), 1301–1305 (2006). [CrossRef]
- L. Liao, D. Samara-Rubio, M. Morse, A. Liu, D. Hodge, D. Rubin, U. D. Keil, and T. Franck, “High speed silicon Mach-Zehnder modulator,” Opt. Express 13(8), 3129–3135 (2005). [CrossRef] [PubMed]
- O. Boyraz and B. Jalali, “Demonstration of a silicon Raman laser,” Opt. Express 12(21), 5269–5273 (2004). [CrossRef] [PubMed]
- L. Yin, Q. Lin, and G. P. Agrawal, “Soliton fission and supercontinuum generation in silicon waveguides,” Opt. Lett. 32(4), 391–393 (2007). [CrossRef] [PubMed]
- B. M. A. Rahman and J. B. Davies, “Finite element solution of integrated optical waveguides,” J. Lightwave Technol. 2(5), 682–688 (1984). [CrossRef]
- B. M. A. Rahman and J. B. Davies, “Penalty Function Improvement of Waveguide Solution by Finite Elements,” IEEE Trans. Microw. Theory Tech. 32(8), 922–928 (1984). [CrossRef]
- B. M. A. Rahman and J. B. Davies, “Vector-H finite element solution of GaAs/GaAlAs rib waveguides,” Optoelectronics, IEE Proceedings J 132, 349–353 (1985). [CrossRef]
- ISO 11146, Laser and laser related equipment – Test methods for laser beam widths, divergence and beam propagation ratios, International Organization for Standardization, Geneva, Switzerland, 2005.
- N. Somasiri and B. M. A. Rahman, “Polarization crosstalk in high index contrast planar silica waveguides with slanted sidewalls,” J. Lightwave Technol. 21(1), 54–60 (2003). [CrossRef]
- B. M. A. Rahman, S. S. A. Obayya, N. Somasiri, M. Rajarajan, K. T. V. Grattan, and H. A. El-Mikathi, “Design and characterization of compact single-section passive polarization rotator,” J. Lightwave Technol. 19(4), 512–519 (2001). [CrossRef]

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