Semi-vector iterative method for modes of high-index-contrast nanoscale waveguides

K. Gehlot; A. Sharma

doi:10.1364/OE.21.009807

1. Introduction

Recent technological advancements in complementary metal oxide semiconductor (CMOS) compatible silicon (Si) photonics have generated a great interest in Silicon-on-insulator (SOI) photonic structures [1]. Small feature size, high optical field confinement and low bending loss of SOI waveguides permit to realize ultra-dense photonic integrated circuits on a single Si chip [2].

Several numerical and approximate methods have been developed to model and characterize such kind of waveguides. Rigorous numerical methods, such as the vector finite element method (VFEM) [3] give accurate results, but are computationally intensive. Hence there have always been efforts to develop simple and efficient approximate methods which can give fairly accurate results with less computational efforts.

The first approximate method of Marcatili [4] and subsequent analysis based on effective index approach [5, 6] have been successful in finding modal characteristics of low-index contrast waveguides. For modes close to cut-off, these approach do not give good results. This limitation was overcome by more efficient iterative methods such as the weighted index (WI) method [7] and the Vopt method [8], which are based on the variational principle. However, these methods based on scalar analysis cannot be directly applied to high-index contrast waveguides operating in the single mode regime.

Effective index method for vector modes (VEIM) proposed in [9] offers very simple modal analysis for such waveguides. It has been shown earlier [10] that for scalar modes the iterative process of the Vopt method gives higher accuracy than the effective-index method. To make use of this advantage, we have developed an iterative method using a procedure similar to the Vopt method to find approximate solution of the semi-vector wave equation. This method is named as the semi-vector optimal variational method (SV-Vopt). Accuracy of this method is tested by comparison with reference results obtained from the VFEM [3] and using the VEIM [9]. It is found that the SV-Vopt method gives very accurate results for vector modes over the practical region of single mode operation. The method, being iterative in nature, has a built-in procedure to improve the accuracy- a feature that is not available with the VEIM. In general, this method also can be applied to waveguides with complex refractive indices, such as lossy or plasmonic waveguides. The scope and limitations of the SV-Vopt method in case of such waveguides are still under investigation and would be reported elsewhere.

2. Semi-vector Vopt (SV-Vopt) method

This method is developed to find modal solution of the semi-vector wave equation by using an approach similar to that of the Vopt method. Although an example of only an Si strip waveguide is presented in this paper in section 3, this method can be applied to more general type of waveguides with piecewise constant or continuously varying refractive indices.

The vector modes of rectangular core waveguides are hybrid, and in general all the components of E and H-field are non-zero. In the semi-vector approach, we consider two types of mode: quasi-TE (or E^x) and quasi-TM (or E^y) mode. For the E^x mode propagating in the z-direction, we consider the mode field E of the form (E_x, 0, E_z)exp(iωt − iβz). For E^x mode, this reduces full vector wave equation to following semi-vector wave equation

\frac{\partial}{\partial x} {\frac{1}{n^{2}} \frac{\partial (n^{2} E_{x})}{\partial x}} + \frac{\partial^{2} E_{x}}{\partial y^{2}} + (k^{2} n^{2} - β^{2}) E_{x} = 0

An integral form of the semi-vector wave equation is obtained by assuming a field, separable in two transverse directions, namely the width and the depth directions. For the E^x mode, let us consider a separable field of form E_x(x, y) = χ(x)ϕ(y). In due course we will show that in SV-Vopt analysis χ(x) and ϕ(y) arise naturally as TM and TE mode fields of some planar index waveguides represented by index distributions

n_{x}^{2} (x)

and

n_{y}^{2} (y)

, respectively. We use an approach similar to that of scalar Vopt method to obtain the best values of

n_{x}^{2} (x)

and

n_{y}^{2} (y)

. Following this, first, we substitute the separable field, E_x(x, y) = χ(x)ϕ(y) in Eq. (1), multiply it by

n_{x}^{2} χ ϕ

and integrate,

β^{2} = k^{2} \int \int n^{2} n_{x}^{2} χ^{2} ϕ^{2} d x d y + \int \int n_{x}^{2} χ \frac{\partial}{\partial x} {\frac{1}{n^{2}} \frac{\partial (n^{2} χ)}{\partial x}} ϕ^{2} d x d y + \int ϕ \frac{d^{2} ϕ}{d y^{2}} d y

where the following normalization is used:

\int n_{x}^{2} χ^{2} (x) d x = 1 = \int ϕ^{2} (y) d y

Unless otherwise specified, all the integrals are evaluated over −∞ to +∞. Next we add and subtract two terms in Eq. (2) and rearrange as follows

\begin{array}{l} β^{2} & = & k^{2} \int {(n_{x}^{2} χ)}^{2} d x + \int n_{x}^{2} χ \frac{d}{d x} {\frac{1}{n_{x}^{2}} \frac{d (n_{x}^{2} χ)}{d x}} d x + \int ϕ \frac{d^{2} ϕ}{d y^{2}} d y \\ + k^{2} \int [\int (n^{2} - n_{x}^{2}) n_{x}^{2} χ^{2} d x + \frac{1}{k^{2}} \int n_{x}^{2} χ \frac{\partial}{\partial x} {χ \frac{\partial ln (n^{2} / n_{x}^{2})}{\partial x}} d x] ϕ^{2} d y \end{array}

In the RHS of Eq. (4), the first two terms have been added and the same are subtracted in the last term. Thus Eqs. (2) and (4) are, in fact, identical. The first two terms can be recognized as an integral form of the wave equation for the TM mode of the planar index profile

n_{x}^{2} (x)

and is thus equal to the square of the propagation constant,

β_{x}^{2}

, of its mode. Similarly, the last two terms are integral form of the wave equation for the TE mode of the planar index profile,

n_{y}^{2} (y)

and is equal to the square of the propagation constant,

β_{y}^{2}

. The index profile

n_{y}^{2} (y)

is obtained from the term in the square bracket in Eq. (4), which after simplification is expressed as follows:

n_{y}^{2} (y) = \int (n^{2} - n_{x}^{2}) n_{x}^{2} χ^{2} d x - \frac{1}{k^{2}} \int χ \frac{d (n_{x}^{2} χ)}{d x} \frac{\partial ln (n^{2} / n_{x}^{2})}{\partial x} d x

In a similar manner, we can add and subtract a term containing ϕ(y) in Eq. (2) and rearrange it to obtain

\begin{array}{l} β^{2} & = & k^{2} \int n_{y}^{2} ϕ^{2} d x + \int ϕ \frac{d^{2} ϕ}{d y^{2}} d y + \int n_{x}^{2} χ \frac{d}{d x} {\frac{1}{n_{x}^{2}} \frac{d (n_{x}^{2} χ)}{d x}} d x \\ + k^{2} \int [\int (n^{2} - n_{y}^{2}) ϕ^{2} d y + \frac{1}{k^{2}} \int n_{x}^{2} χ \frac{\partial}{\partial x} {χ \frac{\partial ln (n^{2} / n_{x}^{2})}{\partial x}} d x] n_{x}^{2} χ^{2} d x \end{array}

The last two terms on the RHS of Eq. (6) give the integral form of the wave equation for the TM mode of the index profile,

n_{x}^{2} (x)

which after simplification can now be defined as

n_{x}^{2} (x) = \int (n^{2} - n_{y}^{2}) ϕ^{2} d y - \frac{1}{k^{2}} \int [\int χ \frac{\partial (n_{x}^{2} χ)}{\partial x} \frac{\partial ln (n^{2} / n_{x}^{2})}{\partial x} d x] ϕ^{2} d y

The Eq. (7) represents an integral equation for

n_{x}^{2} (x)

and can be solved iteratively by starting with the

n_{x}^{2} (x)

of the previous SV-Vopt iteration cycle. Together Eqs. (5) and (7) form an iterative process, which starts with assuming an initial approximation for

n_{x}^{2} (x)

and uses Eqs. (5) and (7) to obtain new improved value of

n_{x}^{2} (x)

at the end of one cycle. These cycles are continued till convergence is achieved. In the converged state the x-component of the E-field for the E^x mode is E_x = χ(x)ϕ(y), and the propagation constant, β is obtained as:

β^{2} = β_{x}^{2} + β_{y}^{2}

Similar set of equations and procedure can be derived for E^y mode by merely exchanging terms

n_{x}^{2} (x)

, χ(x) with

n_{y}^{2} (y)

, ϕ(y). In most cases, the convergence is reached in 2 or 3 cycles. Whether one starts with an initial approximations for

n_{x}^{2} (x)

or

n_{y}^{2} (y)

, the two coupled equation in iterative process are same as Eqs. (5) and (7).

3. Modal analysis of Si strip waveguide

To demonstrate the efficiency and accuracy of method developed in the previous section, we present a detailed modal analysis of Si strip waveguide and show comparison with the results of the numerically rigorous VFEM analysis. The Si strip waveguide considered here consists of a rectangular Si core of extremely small cross-section surrounded by a silica cladding. Silica buffer of thickness as large as 1.5 micron is chosen to optically isolate waveguide core and avoid substrate leakage loss. For the present analysis the height of waveguide core is kept constant at 260 nm and the width is varied over a wide range. At the operating wavelength of 1.55 μm, Si and silica refractive indices are taken as 3.5 and 1.5, respectively. For the SV-Vopt, a planar waveguide with the width same as that of the core of the strip waveguide and Si and silica as core and cladding, respectively, is taken as initial approximation for $n_{x}^{2} (x)$ . Subsequently $n_{y}^{2} (y)$ and improved $n_{x}^{2} (x)$ are obtained using Eqs. (5) and (7). Converged results are obtained by iterating the procedure few times. For the present case, 2–3 iterations are enough to give converged results. Propagation constant and five component field are obtained from the results of the final iteration.

3.1. Effective mode index

In Fig. 1, variation of the effective index (n_eff = β/k) with waveguide core width (W) has been shown for $E_{11}^{x}$ , $E_{21}^{x}$ , $E_{31}^{x}$ and $E_{41}^{x}$ modes. The analysis of SV-Vopt method and VEIM [9] is applied to present structure and the results are plotted for comparison along with rigorous numerical results of the vector FEM analysis from [3]. The SV-Vopt analysis gives very accurate results for the fundamental mode, even near its cut-off. For higher order modes small error is found for near cut-off modes. From the inset plot in Fig. 1, it can be observed that the SV-Vopt method gives better estimate of n_eff than the VEIM. The SV-Vopt method is a good approximate method for waveguides operating in the single mode regime.

Fig. 1 Variation of effective index, n_eff with core width, W. The inset shows the variation of error in n_eff of fundamental mode with W for SV-Vopt and VEIM results. Error is defined as difference in n_eff obtained by SV-Vopt/VEIM and VFEM.

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3.2. Effective mode area

Field profiles obtained from the SV-Vopt method can be used to evaluate various other quantities of interest. One important parameter is effective mode area (A_eff), which defined as follows:

A_{eff} = \frac{{(\int \int {| E_{t} |}^{2} d x d y)}^{2}}{\int \int {| E_{t} |}^{4} d x d y}

where E_t = (E_x, E_y). Since the SV-Vopt approximates E_x = χ(x)ϕ(y) and E_y = 0 for Ex mode, E_t = (E_x, 0) is used here. Variation of the effective mode area, A_eff with core width, W, is shown in Fig. 2. Core height is kept constant at 260nm. It can be observed that value of A_eff is very large near the mode cut-off and attains a minimum value at some higher core width. For the fundamental mode, again the SV-Vopt results show very good agreement with the vector FEM results at all widths. This shows that, the SV-Vopt method not only gives very good estimate of propagation constant, but it also gives analytical expression for field profile with reasonable accuracy. Since the method is basically built around the fundamental mode (as is its scalar counterpart Vopt), accuracy for higher order modes is not as good, however, results can be used for making reasonable estimates.

Fig. 2 Variation of effective mode area, A_eff with core width, W.

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3.3. Power confinement factor

Another important parameter is the confinement factor, which gives an estimate of the power confined in various regions of the waveguide. For fields normalized to unit power, the power confinement factor for the core of the waveguide is defined as:

Γ_{core} = {\int \int}_{core} {E^{*} \times H}_{z} d x d y

where the integration is evaluated over the core of the waveguide. Since it is desirable to have waveguide with high power confinement in its core, we show variation of power confinement in core, Γ_core with core width in Fig. 3. Core height is kept constant at 260 nm. For higher order modes, there is significant error in estimation of confinement factor near the mode cutoff. However, for fundamental mode, the SV-Vopt results show very good agreement with the vector FEM results at all widths. It can be noted that Γ_core increases with increase in core width and saturates after a certain core width. An optimum core width can be chosen to have high power confinement, small effective mode area and small waveguide dimensions.

Fig. 3 Variation of power confinement factor in core, Γ_core with core width, W.

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3.4. Modal birefringence

Finally, we show that the SV-Vopt method gives equally good results for the E^y mode. From the propagation constants obtained for fundamental E^x and E^y modes, accurate estimation of birefringence can also be calculated. Birefringence is defined as difference in n_eff of the $E_{11}^{x}$ and $E_{11}^{y}$ modes. The SV-Vopt results and the reference FEM results for effective index, n_eff, and birefringence are plotted in Fig. 4. It can be seen that SOI strip waveguides show very large birefringence.

Fig. 4 Variation of n_eff for $E_{11}^{x}$ and $E_{11}^{y}$ mode (left scale) and modal birefringence (right scale) with core width, W.

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4. Conclusion

A new method, namely, the SV-Vopt for the analysis of the vector modes of general high contrast rectangular-core waveguides is proposed. It is an improvement over the scalar Vopt method. This method offers immense simplification over rigorous analysis, and can be used on wide variety of waveguide geometries. Various quantities of interest such as, the propagation constant, the effective mode area, the power confinement factor and the modal birefringence have been obtained with good accuracy using SV-Vopt for the fundamental mode.

Acknowledgments

One of the authors (Kanchan Gehlot) would like to thank the Council of Scientific and Industrial Research (CSIR), India, for providing Senior Research Fellowship.

References and links

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Semi-vector iterative method for modes of high-index-contrast nanoscale waveguides

Abstract

1. Introduction

2. Semi-vector Vopt (SV-Vopt) method

3. Modal analysis of Si strip waveguide

3.1. Effective mode index

3.2. Effective mode area

3.3. Power confinement factor

3.4. Modal birefringence

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (4)

Equations (10)

Optics Express