Accurate theoretical and experimental characterization of optical grating coupler

Faezeh Fesharaki; Nadir Hossain; Sebastien Vigne; Mohamed Chaker; Ke Wu

doi:10.1364/OE.24.021027

1. Introduction

Grating coupler, a region on top of or below a waveguide where there is a grating, is a structure in which guided-mode resonance takes place. Once designed accurately, resonance happens for specific combinations of incident angles and light frequencies and allows the grating to couple light into a guided mode of the waveguide. They have prominent applications in integrated optics to efficient butt-coupling of light from a single mode fiber (~125μm diameter) to a thin-film planar waveguide (dimensions around ~1μm or less). The power distribution diffracted by a grating depends on the properties of electromagnetic fields guided by the structure [1], which necessitates a rigorous electromagnetic method in order to simulate, understand and design the grating coupler.

Numerous theoretical studies of grating couplers have been conducted based on both exact and approximate formulations [1–5]. However, the precedent methods based on an exact treatment of the pertinent boundary-value problem are complex and cumbersome to use [1–4]. On the other hand, traditional perturbation method, although simplifying the problem, has limitations and complications for treating general geometry cases, and is not accurate for grating with a small number of periods [6]. In the eigenmode expansion [2] and coupled-wave [4,5,7,8] approaches, the system of coupled differential equations may be large and difficult to solve [2]. In addition, some of the solution algorithms are unstable for thick gratings and may become inefficient for computation, as mentioned in references [2,5,8]. Two-dimensional (2D) modeling of the grating is quite common and widely used; although faster than 3D simulation, because of the simplifications in considering the whole radiated beam in a horizontal plane and also neglecting the radiation side lobes, it leads to an inaccurate result. Therefore, the real coupling efficiency, obtained with a three-dimensional (3D) configuration, will be lower than that of 2D model-based simulations [9]. Moreover, with the conventional design and simulation technique, it is difficult to transfer easily the existing well-known design of Silicon-on-Insulator (SOI) grating coupler to another material system such as III-V [10], which is essential for integration with active optoelectronic components [11].

In this paper, a 3D simulation of the grating coupler is performed. This approach enables a rigorous and general investigation of this periodic structure. Using the vector formulation of Maxwell’s electromagnetic equations with a commercial software package on the basis of a full-wave finite-element method (FEM), the power distribution diffracted by a grating is resolved and the grating efficiency is examined. A parametric study is performed to determine the dependence of the wave leakage on structural geometry, operating wavelength, and polarization as well as incident angle of the light. In addition, since the complex propagation constant (γ = $α$ + jβ) cannot be obtained directly from the simulation, we have used a numerical calibration method to extract this critical parameter. The numerical calibration method is a well-established technique for the study of any guided or leaky wave structures [12]. This method allows the analysis of optical periodic structures with a very large number of periods with minimal numerical problems. Such devices are of considerable technical importance [13]. The grating coupler is fabricated using electron-beam lithography (EBL), followed by plasma etching and measured for the validation of our simulation results.

2. Method of analysis

The grating coupler is an open guided-wave structure in which the gradual wave leakage along the propagation is used advantageously to couple the light to the fiber optic. The wave leakage is related to the wave guidance and is fundamentally characterized by the complex propagation constant, which depends on many parameters, including the structural geometry, polarization, the incident angle of light, material, groove spacing, operating frequency, and the guided mode.

The physical structure of the grating coupler is considered a leaky waveguide with a length L along which the leakage occurs. The propagation characteristic of the leaky mode in the longitudinal (z) direction is determined by phase constant β and leakage constant α, where α is a measure of the power leaked (radiated) per unit length. The theoretical determination of β and α, although difficult, is essential to any systematic design procedure. Once β and α are known for the considered wavelength, the required principal features such as coupling angle and coupling efficiency will follow in sequence.

In a grating coupler the fundamental mode is slow compared to the free space: the phase constant of the fundamental mode is larger than the free space wavenumber or $β > k_{0} = 2 π / λ$ [8] where λ is the wavelength in the free space. According to Floquet’s theorem [13], the grating periodicity introduces an infinite number of space harmonics, which are characterized by phase constant β_n:

β_{n} = β_{0} + \frac{2 n π}{d} .

where d is the period of the grating, and

β_{0}

is the phase constant of the fundamental mode of the waveguide without grating. The grating is designed in such a way that the first space harmonic (

n = - 1

) with phase constant

β_{- 1} = β_{0}

- 2

π / d

becomes fast (

β_{- 1} < k_{0}

). The beam can be created in either the forward or backward direction; however, the range in the forward quadrant is usually limited by the appearance of the

n = - 2

beam [14]. Design is carried out based on the criteria introduced by [6] which are:

n_{e f f} - \frac{λ}{d} < n_{a i r} .

and

\frac{2 λ}{d} - n_{e f f} < n_{s u b s t r a t e} .

in which n_eff is the refractive index of the optical waveguide, n_air is the air refractive index, and n_substrate is the substrate refractive index. Through three-dimensional simulations and using near and far field calculations, the field distribution and radiation patterns are monitored. A radiation boundary is used to simulate the grating coupler, which is an open problem. Radiation boundary allows waves to radiate infinitely far into space. Model exterior has been selected as HFSS-IE domain; in this case, conformal radiation volumes are used advantageously and the overall finite element solution domain is reduced, resulting in more efficient simulation for electrically large open boundary problems. The solution frequency is selected according to the operating wavelength.

To obtain the complex propagation constant γ = α + jβ, the numerical calibration technique [15] is used to extract the phase and attenuation constants. To perform the numerical calibration, the periodic structure of the grating coupler structure is simulated as a two-port waveguide with different lengths or different period numbers. The number of periods is optional; to speed up the simulation, a few periods are used. In this case, the whole structure consists of an input tapered section, grating section, and an output tapered section. The transmission matrix T can be written as a cascade matrix in the form of:

T = T_{i n} T_{g r a t i n g} T_{o u t} .

where

T_{g r a t i n g}

is a diagonal matrix

T_{g r a t i n g} = [\begin{matrix} e^{- γL} & 0 \\ 0 & e^{γL} \end{matrix}] .

in which L is the grating length. Once two different lengths (i and j) are simulated, T_i and T_j are obtained and the equations can be combined as:

T^{i j} T_{i n} = T_{i n} T_{g r a t i n g}^{i j} .

where

T^{i j} = T^{i} {(T^{j})}^{(- 1)}

and

T_{g r a t i n g}^{i j} = T_{g r a t i n g}^{i} {(T_{g r a t i n g}^{j})}^{- 1}

. Since

T_{g r a t i n g}^{i j}

is a diagonal matrix, the eigenvalues of

T^{i j}

are the diagonal elements of

T_{g r a t i n g}^{i j}

matrix. This feature is used to derive the unknown propagation constant of grating coupler:

e^{γ_{i} Δ L} = λ_{i} .

where

λ_{i}

is the eigenvalue of

T^{i j}

, and

Δ L

is the difference of two simulated gratings of differing lengths. Accurate complex propagation constants are therefore obtained from the parameters generated in two simulations. The extraction of propagation constant and attenuation from HFSS simulation and eigenvalues of the transmission matrix have been detailed in [12]. Since the proposed grating structure is uniform, the beam direction can be obtained from:

\sin θ_{m} \approx \frac{β}{k_{0}}

in which

θ_{m}

is the angle of the maximum of the beam. The larger value of α involves a larger leakage rate and results in a greater beamwidth; on the contrary, the lower value of attenuation constant α causes a narrower beam. When the length of the grating is finite and the attenuation constant is small, the beamwidth is determined primarily by the grating length and the value of α mainly affects the grating efficiency. The beamwidth can be calculated by

Δ θ \approx \frac{1}{(L / λ) c o s θ_{m}}

Knowing the complex propagation constant γ, the radiation or power pattern R(θ) is approximated by [14]:

R (θ) = \frac{c o s^{2} θ}{{(\frac{α}{k_{0}})}^{2} + {(\frac{β}{k 0} - s i n θ)}^{2}}

This equation is obtained from the Fourier transform of the grating filed distribution considering that the grating length is infinite and the grating field distribution is exponentially decaying. According to this equation, a grating coupler with an infinite length is ideally free of any side lobe radiation. However, in reality, the length of the grating is finite and therefore the efficiency is less and possesses side lobes. To improve the performance and efficiency in relation with side lobes, the structure must be designed in such a way that the value of α decreases slowly along the geometrical length of the grating, while the value of β remains constant so that the radiation from all parts of the grating direct to the same direction. As an example, a kind of tapered grating structure can be designed in which the width of the grating decreases along its length. The advantage of using numerical calibration and Eq. (10) is that for any structure and any length, the radiation pattern may be obtained systematically with a quick simulation. This allows the analysis of optical periodic structures with a very large number of periods with minimal numerical problems.

3. Simulations and calculations

A grating design based on SOI wafer was optimized for operation at λ = 1550nm wavelength. According to last section, once the material system is selected, the design procedure is as follows:

1. The refractive index of silicon (n_Si = 3.48) and silicon oxide (n_SiO2 = 1.46) are verified experimentally on the commercial wafer.
2. Using the finite difference beam propagation method of Optiwave [17], the width of the SOI strip waveguide is chosen to provide single mode propagation.
3. The cross-section of the waveguide is simulated using 2D finite element method solver of HFSS [15] and the effective refractive index of the fundamental mode (n_eff) is extracted.
4. From Eqs. (2) and (3), an estimated value for the grating period d is derived, which is used as an initial value in the 3D simulations.
5. The initial value of etching depth h is chosen around 0.1d to have a flat efficiency over the spectrum [18].
6. Starting from these initial values, a parametric setup sweep analysis is used to find the optimized value for grating period d and etching depth h.

For the commercially available SOI wafer with a 260nm silicon layer and 2 µm buried oxide, the waveguide width is selected equal to 500nm confirmed by Optiwave simulation to assure the single mode propagation. The fiber mode is approximately a Gaussian profile with a beam diameter of 10.5μm, and to maximize the overlap, the grating width is chosen 25 times larger than the width of the waveguide (12.5μm), and a total number of 16 to 18 repetitions is used. Figure 1 shows the complete 3D model of the SOI grating-waveguide devices that are used in this paper. In the simulations, we use half of this symmetrical structure to calculate the coupling efficiency from the waveguide to fiber, which is the same as the coupling from fiber to waveguide [9]. As illustrated in Fig. 2, the optical waveguide is excited using wave port excitation and the wave is propagating through the waveguide and taper. Field distributions and radiation patterns are monitored to achieve the desired coupling of the grating coupler.

Fig. 1 Complete three-dimensional model of the SOI grating-waveguide device is considered in this paper in both simulation and fabrication. In the simulation, half of the structure with shorter taper and waveguide length is simulated.

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Fig. 2 3D Simulated structure. Half of the device is simulated using the wave port excitation of the waveguide and the radiated beam is monitored.

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The length of the taper is optimized to 650μm using beam propagation method to minimize the radiation loss. However, in the 3D simulation, a much shorter length is used to speed up the simulation. To avoid any error in the analysis because of this estimation, the return loss of the wave port is observed; provided the matching condition is preserved, return loss is close to zero and trimming the taper length does not affect the normalized simulation result.

Figure 3 shows the 2D radiation pattern of the grating coupler with 70nm etching depth and three different periods. Part of the wave is radiated toward the substrate and part is a coupling beam radiating toward the fiber. The coupling angle varies with changing the grating period. The 530nm period provides the negative coupling angle around 7°; by increasing the period to 580nm and 600nm, the coupling angle is changed to positive angles around 7° and 13°. The ratio of the desired coupling beam to the whole radiated stream determines the coupling efficiency.

Fig. 3 The simulated radiation pattern of the proposed SOI grating coupler optimized to work at 1550nm wavelength for the different grating periods. SOI wafer (260nm silicon layer and 2µm buried oxide), waveguide width 500nm, grating width 12.5μm, etching depth 70nm, and taper length 650μm. Coupling angle is changed with the grating period.

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Figure 3 also shows the calculated radiation pattern for the proposed grating coupler with 580nm grating period using the numerical calibration (NC) method. For that, the simulation result of two different lengths (2 periods and 4 periods) is used to extract the complex propagation constant. The real and imaginary parts of the S-parameters are derived from HFSS simulation for two different lengths (∆L). These data are used to calculate the transmission matrix (T), whose eigenvalues are used to extract the propagation constant and attenuation. For the proposed structure, phase constant β = 6.1074*10⁵rad/m, and attenuation constant α = −1.0810*10⁵ np/m is derived, from which the radiation pattern is calculated. The slight difference in the radiation patterns of the simulation and theoretical model is observed. The reason is that in the calculation, the radiation pattern is obtained by taking the Fourier transform of the aperture field distribution, assuming infinite aperture length. However, in the simulation, the actual limited length of the grating structure is simulated. The Fourier transform relationship between field distribution and radiation is exact; however, since the electric field is not exact, radiation always involves an approximation. This formula is based on an infinite length and the result does not have side lobe radiation. In general, for a larger dimension, the beamwidth or angular bandwidth is smaller [19]. Figure 3 shows that it has been also confirmed with simulation. The grating coupler with 600nm grating period has been simulated using a total of 35 periods. As shown in the figure, the beamwidth is narrower compare to the original one with 18 periods.

From the radiation patterns, the radiated power to the substrate is identified and the efficiency of the grating coupler is estimated. Figure 4 shows the coupling efficiency versus incident angle for the proposed SOI grating coupler with different coupling angle. With a slight increase in the period, coupling angle is increased, keeping roughly the same efficiency, around 41%. The dimensions are optimized to obtain a compromise between angular bandwidth and efficiency. Deeper etch-depth can increase the efficiency, but in the same time period must be optimized to couple light toward the desired angle. Beamwidth is determined through Eq. (9). Considering that 90% of the power is radiated through the length, then:

\frac{L}{λ_{0}} \approx \frac{0.18}{α / K_{0}}

Therefore,

Δ θ \propto α / K_{0}

. As a result, decreasing loss would result in decreasing beamwidth (α↓⇒∆θ↓). It suggests that narrower beam can be created by smaller α value. On the other hand, side lobes are inevitable because the structure is finite along z. When we change the etch-depth to change the α, β would also be modified slightly. However, because we do not want β to change because it affects the angle of coupling, the period must be altered in the same time. In this work, using a specified material system, we have two parameters to play with: period and etch-depth. Therefore, it will include a two-step process: slightly increasing etch-depth while playing with period, to maintaining β constant to keep the angle of maximum radiation the same. In this work, the optimization of the structure is accomplished by simulating different periods and etching depths with an aim to obtain maximum efficiency over coupling angle from 4° to 14°.

Fig. 4 Simulated efficiency versus incident angles for the proposed SOI grating coupler at 1550nm wavelength for the different grating period. SOI wafer (260nm silicon layer and 2µm buried oxide), waveguide width 500nm, grating width 12.5μm, taper length 650μm, and etching depth 70nm.

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Figure 5 shows the simulated efficiency versus incident angle for the proposed SOI grating coupler at 1550nm wavelength with periods of 580nm and 600nm. For 70nm etching depth, more than 40% coupling efficiency is expected over the desired coupling angle. Figure 6 shows the variation of the simulated efficiency with wavelength for the proposed SOI grating coupler with a 580nm grating period and 70nm etching depth. According to Eq. (4), because the phase constant β changes with wavelength, the maximum coupling angle is also changed with wavelength. In this structure, by increasing the wavelength, coupling angle is decreasing and the maximum efficiency is expected at 1550nm wavelength.

Fig. 5 Simulated efficiency versus incident angle for the proposed SOI grating coupler at 1550nm wavelength for different grating etching depth. SOI wafer (260nm silicon layer and 2 µm buried oxide), waveguide width 500nm, grating width 12.5μm, taper length 650μm. a) grating period = 580nm, b) grating period = 600nm.

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Fig. 6 Simulated efficiency versus incident angle for the proposed SOI grating coupler for different wavelengths. SOI wafer (260nm silicon layer and 2 µm buried oxide), waveguide width 500nm, grating width 12.5 μm, taper length 650 μm, grating period 580nm, etching depth 70nm.

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4. Fabrication and measurement

Electron Beam Lithography (EBL) (Vistec VB6 UHR-EWF) and Inductively Coupled Plasma – Reactive Ion Etching (ICP-RIE) (Oxford PlasmaLab System100) were used for the fabrication of our device based on SOI substrate with a 260nm silicon device layer and a 2µm buried oxide (BOX) layer. Because the fiber couplers require a different etch depth than the other optical structures, they have to be fabricated in a separate process step. Two sets of alignment marks for these two-level microfabrication steps were first patterned by EBL in positive electron beam resist ZEP520a and then transferred into the top Si layer using an SF6/C4F8 ICP etch chemistry. The optical end-point detection system stops the etching process when the BOX is reached, to maximize the contrast during subsequent EBL exposures.

The sample was then cleaned, the same positive resist was spin coated, and the grating couplers were patterned by EBL using the first set of alignment marks. Grating couplers with 70nm etch depth were then obtained using the same ICP etching recipe described above. The sample was then cleaned again, and the same positive resist was spin coated. For the definition of the waveguide, the second set of alignment marks was used to obtain exposure by EBL 3 µm–wide trenches along the side of the waveguides, which were then transferred to the underlying top Si device layer (260nm), all the way down to the BOX layer, using the same SF6/C4F8 ICP-RIE process. The fabrication process was completed after removing the resist using oxygen plasma. Figure 7 above shows the atomic-force microscopy (AFM) scan of a grating coupler on one of our devices.

Fig. 7 AFM scan of a grating coupler.

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A grating design involving SOI wafer—with a 260nm silicon layer and 2µm buried oxide—was optimized for operation at 1550nm wavelength. The design is also able efficiently to couple the light over a broad range of wavelengths. Two different periods were studied, resulting in two slightly different coupling angles. Test structures with grating couplers and tapered strip waveguides were fabricated. They are characterized by various coupling angles over 100nm wavelength spectrum. The fabricated grating couplers show a coupling efficiency higher than 35% over 50nm wavelength bandwidth. The selection of period and etch depth of the proposed SOI grating coupler is done with consideration to achieving the maximum bandwidth. According to the theory, to attain wider bandwidth, a material system with larger refractive index must be used, as this value is the dominant factor in determining the maximum bandwidth. For lower value of n_eff, n = −2 beam begins propagating in a closer operating wavelength [20]. The optimized coupling angles are 7.5° and 12° for the 580nm and 600nm grating periods, respectively, in agreement with the full-wave simulation values of 7° and 13°. The corresponding coupling efficiencies were 42% and 38%. Figure 8 shows the measurement result for the 580nm grating period versus different coupling angle and the comparison with simulation. The discrepancy with simulation is less than 12% and also the curve trends are exactly the same. The discrepancy, using 2D eigenmode expansion method with perfectly matched layer boundary conditions, has been reported more than 34% and also 54% [21] and more than 20% in case of 2D finite-difference time-domain [22]. This deviation can be due to an aberration of the fabricated structure from the targeted structure. This excellent agreement among theory, simulation, and experiment fully validates the design method.

Fig. 8 Measured efficiency of SOI grating coupler (260nm silicon layer and 2 µm buried oxide, 580nm grating period and 70nm etching depth) at 1550nm wavelength and comparison with the simulation

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5. Conclusion

Three-dimensional full-wave analysis of an out-of-plane grating coupler is conducted and guided wave and leakage characteristics are studied in detail. The radiation pattern is obtained from the simulations, also using numerical calibration method. Coupling efficiency is derived from the radiation pattern. SOI grating coupler design is optimized to obtain a maximum coupling efficiency at operating 1550nm wavelength and over 4° to 14° degree coupling angle. The optimized grating couplers are fabricated using electron-beam lithography technique and plasma etching, and characterized. An excellent agreement between simulations and measurements is perceived, thereby validating the demonstrated method.

Funding

Natural Sciences and Engineering Research Council of Canada (NSERC), Discovery Grants Program, RGPIN-122137-2012.

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Accurate theoretical and experimental characterization of optical grating coupler

Abstract

1. Introduction

2. Method of analysis

3. Simulations and calculations

4. Fabrication and measurement

5. Conclusion

Funding

References and links

Cited By

Figures (8)

Equations (11)

Optics Express