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Department of Electrical and Computer Engineering, McMaster University, 1280 Main Street West,. Hamilton, L8S 4K1 ... ID 140496); published March 15, 2011.
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OPTICS LETTERS / Vol. 36, No. 6 / March 15, 2011

Complex coupled-mode theory for tapered optical waveguides Jianwei Mu* and Wei-Ping Huang Department of Electrical and Computer Engineering, McMaster University, 1280 Main Street West, Hamilton, L8S 4K1, Ontario, Canada *Corresponding author: [email protected] Received January 4, 2011; revised February 14, 2011; accepted February 15, 2011; posted February 16, 2011 (Doc. ID 140496); published March 15, 2011 A coupled-mode formulation based on complex local modes is developed for tapered and longitudinally varying optical waveguides. Different from the conventional coupled-mode theory that requires integration over the entire spectrum of radiation modes, the new formulation treats the radiation fields via discrete complex modes similarly to the guided modes. Accuracy, convergence, and scope of validity for the solutions of the complex coupled-mode equations are investigated in detail for a typical single-mode waveguide taper. It is demonstrated that the complex coupled-mode theory has overcome the difficulties of the conventional theory in simulation of radiation field effects while preserving the simplicity and intuitiveness of this popular method. © 2011 Optical Society of America OCIS codes: 130.0130, 000.4430.

Tapered waveguides are frequently used in various optical waveguide structures to facilitate efficient coupling and field profile conversion between devices with different cross-sectional geometries [1,2]. To minimize the transition loss due to the radiation field coupling, quasiadiabatic tapered waveguide structures are normally adopted, in which the geometry/index variations along the propagation direction are slow and gradual. Though the design criterion of adiabatic taper has been investigated and proposed [3,4], efficient and simple simulation techniques are still desirable to predict the impact of different geometrical/index parameters on the performance of given structures. Of the many modeling techniques [5–14], the coupled-mode theory (CMT) based on local modes is applied as a physically intuitive and mathematically simple tool for investigation of radiation loss [11–14]. The fact that the power loss results from coupling from the guided mode to the higher-order radiation modes means that the unbounded radiation modes must be considered in the formulations and solutions. Treating the radiation modes, however, is complicated due to their continuum nature. Recently, a complex coupled-mode theory based on the ideal modes of the global reference waveguide was developed as a unified approach for investigation of the guided and radiation mode coupling [15]. The accuracy, effectiveness, and versatility of the proposed method are validated through its applications to periodic optical structures based on slab, circular waveguides [16,17]. For certain waveguide structures with nonuniform perturbations along the propagation direction, local mode analysis is superior to ideal mode study in regard to efficiency and accuracy. We have detailed the capability and feasibility of the complex coupled mode based on ideal modes in past publications. However, it is uncertain whether the complex coupled-mode theory could be extended to nonuniform waveguide structures. In this Letter, the validity and efficiency of complex coupledmode theory based on local modes have been assessed and explored using examples of a tapered waveguide structure. It is should be noted that efficient design of 0146-9592/11/061026-03$15.00/0

a wide-angle taper is beyond the scope of this article and will not be discussed. In complex coupled-mode theory [15], the waveguide structures to be investigated are enclosed transversely by the perfectly matched layer (PML) and terminated by the perfectly reflecting boundary (PRB) condition. Accordingly, the conventional power orthogonality that uses the overlapping integrals with complex conjugate is no longer valid [15–17]. Instead, the following orthogonality relation is always held for both guided modes and complex modes: Z Z ðetm × htn Þ · ^zda ¼ 0:

ð1Þ

Despite the fact that an ideal taper waveguide structure is continuous, the staircase approximation is a good approximation for the evaluation of radiation loss. For z-variant structures with the refractive index profile defined by nðzÞ, the local modes at an arbitrary position are given by en ðr t ; zÞ ¼ ½etn ðr t ; zÞ þ ezn ðr t ; zÞ expð−jβn zÞ;

ð2aÞ

hn ðr t ; zÞ ¼ ½htn ðr t ; zÞ þ hzn ðr t ; zÞ expð−jβn zÞ:

ð2bÞ

The modes have been normalized with respect to power as 1 2

Z Z

ðetm × htm Þ · ^zda ¼ 1:

ð3Þ

We can decompose the field at an arbitrary point as Eðr t ; zÞ ¼ E t þ E z ;

ð4aÞ

Hðr t ; zÞ ¼ H t þ H z :

ð4bÞ

Expanding the fields with the local modes, we have © 2011 Optical Society of America

March 15, 2011 / Vol. 36, No. 6 / OPTICS LETTERS

E t ðr t ; zÞ ¼

X ðan ðzÞ þ bn ðzÞÞetn ðr t ; zÞ;

ð5aÞ

H t ðr t ; zÞ ¼

X ðan ðzÞ − bn ðzÞÞhtn ðr t ; zÞ;

ð5bÞ

E z ðr t ; zÞ ¼

X ðan ðzÞ − bn ðzÞÞezn ðr t ; zÞ;

ð5cÞ

H z ðr t ; zÞ ¼

X

ðan ðzÞ þ bn ðzÞÞhzn ðr t ; zÞ:

1027

ð5dÞ

Utilizing the orthogonality relation (1), and with some mathematical derivations, we obtain X X d am ¼ −jβm am − κ mn an − χ mn bn ; dz X X d bm ¼ jβm am − κ mn bn − χ mn an ; dz

ð6aÞ

Fig. 1. Geometry of a linear taper waveguide structure and its staircase approximation.

ð6bÞ

(Δz) along the propagation direction is determined by step height (Δd). The step height (Δd), on the other hand, is decided by the beam resolution. For the linear taper shown above, the refractive index profiles of adjacent sections are the same everywhere except in the vicinity of the boundary between the cladding and the core. The backward-coupling coefficient given by Eq. (9b) indicates that the backward coupling could be reasonably disregarded, because the phase-matching condition will not be satisfied for practical waveguide taper structures. Based on the analysis, we can simplify the coupling coefficients for TE modes as

where κ mn

Z Z 

1 ¼ 4N m

χ mn ¼

1 4N m

∂etn ∂h × htm þ etm × tn ∂z ∂z

Z Z 

∂etn ∂h × htm − etm × tn ∂z ∂z

 · ^zda;

ð7aÞ

· ^zda;

ð7bÞ



and Nm

1 ¼ 2

Z Z ðetm × htm Þ · ^zda:

ð8Þ

The coupling coefficients can be further simplified as the approaches used by Marcuse [12]: κmn ¼

1 ωε0 4N m βn − βm

χ mn ¼

1 ωε0 4N m βn þ βm

Z Z 

∂n2 e ·e ∂z tm tn

 · ^zda;

ð9aÞ

 Z Z  2 ∂n etm · etn · ^zda: ∂z

ð9bÞ

Note that the normalization factor N m and the coupling coefficients κ mn , χ mn are both complex, except for the guided modes. We consider a linear taper with angle θ, which was investigated in reference [11]. The structure is represented by dðzÞ ¼ din þ z tan θ:

κ mn ðzÞ ¼

ωε0 n2co − n2cl tan θ · ðetm · etn Þx¼d : 4N m βn − βm

ð11Þ

The computation parameters are: thickness of the substrate ds ¼ 55 μm; computation window W ¼ 70 μm, PML thickness 2:5 μm on both sides, and PML reflection 1e − 4. The power transmission coefficient obtained by solving the coupled equations [Eq. (6)] will be referred to the full CMT. For a weakly coupled waveguide structure in which the mutual coupling among the higher order modes is negligible, we may consider the coupling between the fundamental mode and higher order modes separately, e.g., instead of using Eq. (6), the coupled equations can be expressed as d a ¼ −jβ1 a1 − κ 1n an ; dz 1

ð12aÞ

ð10Þ

The structure geometry is shown in Fig. 1. The physical parameters are: refractive index of the substrate ns ¼ 1:515, refractive index of the core nco ¼ 1:517, refractive index of the cladding ncl ¼ 1:0, width of the input waveguide core din ¼ 5 μm, and width of the output waveguide core dout ¼ 10 μm. The working wavelength selected is λ ¼ 1:32 μm. Though the ideal taper is continuous, we can use staircase approximation to evaluate the radiation loss. The number of steps (M) and grid size

Fig. 2.

(Color online) Power transmission of a linear taper.

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OPTICS LETTERS / Vol. 36, No. 6 / March 15, 2011

Fig. 3. (Color online) Convergence behavior of grid size (Δd).

Fig. 4. (Color online) Convergence behavior of number of modes.

d a ¼ −jβn an − κn1 a1 : ð12bÞ dz n The transmission coefficient is approximated by XX an am P mn ; ð13Þ T ¼1−

In summary, we have developed and presented a general complex coupled-mode theory for tapered and longitudinally varying optical waveguides based on local mode expansion. The accuracy and the convergence behavior in respect to the mesh size and the mode numbers have been studied. It is found that both the full CMT and the reduced CMT agree well with the mode-matching method for small taper angles. For large taper angles, the full CMT overestimates the power transmission, while the reduced CMT leads to an underestimation. The full CMT converges with mode numbers and mesh sizes. Moreover, it is noted that the relative error is increased with the increase of taper angle.

n

where P mn ¼

1 4

Z

m

ðetm × htn þ etn × htm Þ · ^zda:

ð14Þ

In this case, the higher mode cross power is very weak and negligible, and we have X T ¼1− jan j2 : ð15Þ n

The transmission obtained through Eq. (13) will be referred to the reduced CMT. The power transmission of the linear taper of interest as a function of taper angle is shown in Fig. 2. Compared to the benchmark obtained from the mode-matching method, the results from full CMT overestimate the transmission; on the other hand, the results of the reduced CMT underestimate the power transmission. For a small taper angle, both the full CMT and the reduced CMT are good approximations. However, for a large angle, the full CMT is more accurate as the mutual coupling among higher modes becomes pronounced. As a staircase approximation is utilized to facilitate the computation model, we investigated the convergence of the full CMT with the grid size. The relative errors of power transmission for different grid sizes (Δd) are shown in Fig. 3. It is noted that continuity of the refractive index profile is implied with the framework of the complex coupled-mode method. Considering the staircase approximation used in this study, the accuracy is grid-size dependent and a smaller grid size is preferred to obtain reasonable results with the tradeoff of computational demands. We also studied the convergence of the full CMT with respect to the number of modes being used. It is observed from Fig. 4 that the relative error decreases with the number of modes. The accuracy deteriorates with the increase in the taper angle. It is concluded that, in order to reach convergence, at least 20 modes are required, in which a total of 19 complex modes are used.

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