Analysis of First-Order Gratings in Silicon Photonic Waveguides

A simple thin film effective index analysis for first-order gratings in Si photonic waveguides is shown to provide highly accurate results for reflected and transmitted power spectrums as long as the waveguide remains single mode and non-radiating. A cover layer can be added to the grating region of a Si photonic waveguide to increase the strength of the grating, modify transition losses from the input waveguide to the grating waveguide region, and/or modify the width of the reflectivity spectrum. For a given grating period, the peak reflection and spectral width of the reflectivity decrease as the duty cycle is decreased or increased from ∼50%. For both radiating and multimode structures, the coupling between all modes, power radiated towards the superstrate (upwards), power radiated downwards (substrate) and transmitted power analyzed by Floquet-Bloch, Eigenmode Expansion and Finite Difference Time Domain methods show excellent agreement. Coupling coefficients calculated using analytic formulas are shown to be accurate only for shallow grating depths.


I. INTRODUCTION
N UMEROUS studies [1], [2], [3], [4] have described grating outcouplers in silicon photonic waveguides in the vicinity of the second-order Bragg condition, where the longitudinal propagation constant β (β = 2π/λ g ) is equal to the grating wave vector K (K = 2π/Λ), where λ g is the wavelength of the propagating mode in the waveguide (λ g = λ o /n eff ), n eff is the real part of the complex effective index of the propagating mode, λ o is the free space wavelength, and Λ is the period of the grating. This paper considers the reflective properties of first-order gratings in the extended vicinity of 2β = K. In cases where the waveguide Manuscript  is single mode and non-radiating, a simple, easy to understand thin-film effective index approach [5], [6], [7] using coupled mode theory is highly accurate. Such single mode, non-radiating waveguides occur not only in silicon photonics, but dominate III-V optical devices such as semiconductor lasers, particularly the single-frequency telecom and moderate power devices. The discussion in this paper is limited to waveguides with one-dimensional cross sections. However, many waveguides with two dimensional cross sections can be analyzed as two one-dimensional waveguides [8], [9]. A result of such analyses is that the longitudinal propagation constant β, used in the calculation of the grating period is modified. As an example, numerous single frequency semiconductor lasers use a ridge structure for lateral confinement with a horizontal (y axis) index step on the order 10 −2 to 10 −3 compared to a vertical (x axis) index step of multiples of 10 −1 . Such a disparity in index contrast in the two directions result in the two-dimensional propagation constant β being only very slightly reduced from the one-dimensional propagation constant. For two-dimensional waveguides with such index disparities, the one-dimensional analyses presented herein are nearly identical to the two-dimensional analysis. The calculation of the exact grating period (Λ = π/β = λ o /(2n eff )) for such two-dimensional waveguides only requires using an effective index approach [8], [9] to obtain the modified two dimensional propagation constant.
Analyses of radiating and/or multimode structures herein use Floquet-Bloch [10], Eigenmode Expansion (EME) [11], Finite Difference Time Domain (FDTD) [12] methods, since coupling can occur between modes and power can radiate towards the superstrate (upwards), and substrate (downwards) in addition to being transmitted and reflected. For the non-radiating, single mode waveguides considered in this paper, Floquet Bloch and EME approaches have excellent agreement with the slab effective index method even with only one grating period and the FDTD approach shows very good agreement with all approaches for gratings that contain more than 5 periods.
The Floquet-Bloch approach is fast (executes in seconds), free and is the only method of the three that calculates the complex modal effective index. Our Floquet-Bloch software is presently limited to waveguides with one-dimensional cross sections. The EME method requires judicious choice of boundary conditions, grid size and number of spatial harmonics to obtain accurate solutions with typical execution times of many tens of minutes. conditions and grid size and execution times can exceed multiple hours for the structures considered in this paper. Both ModProp (EME) and Fullwave (FDTD) have professional user interfaces with detailed user manuals and can treat two-dimensional waveguides. However, commercial versions of EME and FDTD software are relatively expensive. Fig. 1(a) shows a generic Si photonic waveguide with a SiO 2 substrate with an index of 1.5, a Si core with an index of 3.5 and a thickness of 0.22 μm, and a SiO 2 superstrate with an index of 1.5. Fig. 1(b) and 1(c) show the same Si photonic waveguide modified with a thin amorphous Si cover layer over both regions ( Fig. 1(b)) and only over the grating region ( Fig. 1(c)). The grating grooves contain a low-index material such as SiO 2 (or a spin-on glass polymer) with an assumed index of 1.5. Fig. 1(d) shows an equivalent thin-film model of the waveguides in Fig. 1(a), 1(b), and 1(c), where n 1 and n 2 are the corresponding effective indices of the groove and tooth regions of either waveguide.
In many instances, a narrow reflectivity spectrum (generally associated with a long, shallow grating) is desired to select a single wavelength or a narrow range of wavelengths in distributed reflector lasers [13], [14], [15], [16], [17]. At other times, a broad reflectivity spectrum (generally associated with a short grating length) is desired to provide reflectivity over a wide wavelength range, allowing laser operation over a large temperature range, as in the case of VCSELs [18]. Fig. 2(a) shows the reflectivity over a wide wavelength range and duty cycle range for the simple "equivalent" thin film model ( Fig. 1(d)) of the Si photonic waveguide in Fig. 1(a) designed for a peak reflectivity at a wavelength of 1.55 μm at first-order. The n = 2 nd , 3 rd , 4 th and 5 th grating orders (corresponding to 2β = nK) are located at wavelengths of ∼0.87, 0.60, 0.46 and 0.37 μm. Although this "slab effective index" method (Figs. 1 and 2) is accurate for layered coatings, it does not allow for radiation that occurs at higher Bragg orders in waveguides or for radiation occurring c) comparison of the reflection at the first Bragg peak for a 50% duty cycle grating calculated using the Floquet-Bloch, slab effective index, FDTD and EME methods for κ·L = 1; and d) comparison of the reflection at the first Bragg peak for a 50% duty cycle grating calculated using the slab effective index, Floquet-Bloch, EME and FDTD methods for one grating period (κ·L = 0.0319, L = 0.2738 μm), three grating periods (κ·L = 0.0957, L = 0.8214 μm) and 5 grating periods (κ·L = 0.1595, L = 1.369 μm).
The tilt of the reflectivity spectrum with respect to wavelength in the expanded top view of Fig. 2(b) of the n = 1 order in Fig. 2(a) is the result of the dependence of the average permittivity of the grating layer on the duty cycle (duty cycle = Λ 1 / Λ as shown in Fig. 1). A comparison of the reflectivity spectrum curve for the waveguide in Fig. 1(a) with a peak reflection wavelength of 1.55 μm and a duty cycle of 50%, as shown in Fig. 2(c), shows excellent agreement between the simple slab effective index, Floquet-Bloch, EME, and FDTD methods for a grating with 40 periods (L = 10.952 μm). The Floquet-Bloch and EME methods shows good agreement with the slab effective index calculation for only one grating period and the FDTD approach shows very good agreement with all approaches for gratings that contain more than 5 periods. (Fig. 2(d)).
The modal coupling coefficients for sinusoidal and rectangular gratings based on the coupled mode theory are given by [27] κ sin corresponds to a sinusoidal boundary and κ rect corresponds to a rectangular boundary. The ratio of (κ rect /κ sin ) for a 50% duty cycle is 4/π, Δn eff = n 2 − n 1, n 2 is the effective index of the tooth section of the grating region, and n 1 is the effective index of the groove section of the grating region. n 1 and n 2 are the index values used in the planar thin-film calculations, which is referred to as the slab effective index method. Unless stated otherwise, for all figures in this paper, the product of the grating depth and grating length L is normalized such that πΔn eff /(2λ 0 ) · L = 1, or equivalently, κ sin L = 1 [27].
The coupling coefficient can be enhanced [25] by applying a cover layer over the grating, as can be seen from the more general formula for the coupling strength κ pq between a forward propagating mode (mode p) and a backward propagating mode (mode q) of a first-order grating of depth "t g " formed in an optical waveguide.
Equation (2) shows that the strength of the coupling coefficient increases as both 1) the fraction of the mode power in the grating "layer" (the integral term in (2)) increases and 2) as the difference between the relative permittivities (ε rel = n 2 ) on either side of the grating increases. The angular frequency of the radiation is ω, permittivity of free space is ε 0 , and b m is the Fourier coefficient corresponding to the first-order grating period of the grating profile. In addition to increasing the coupling coefficient κ by adding a cover layer ( Fig. 1(b) and 1(c)) of appropriate thickness, losses can be reduced by minimizing the field mismatch between the input waveguide mode and the waveguide mode in the grating region. The grating confinement factor Γg is defined as follows: where E g is the electric field in the grating waveguide. Based on our calculations comparing Floquet-Bloch, FDTD, and EME methods to a thin-film slab effective index method based on coupled mode theory for both Si photonic and III-V waveguides, the simple slab effective index method provides accurate results for reflection, transmission, and corresponding spectral widths for wavelengths around the first Bragg if the waveguide remains single mode and non-radiating, as shown in Fig. 2(c).

A. Modal Overlaps At Interfaces
Radiation loss and reflectivity at a step discontinuity transition in an optical waveguide increase in part as the term (1 -Γ i ) increases, where Γ i is a normalized intensity overlap integral [28] of the fields on either side of the discontinuity. E w denotes the electric field of the input waveguide. Fig. 3 shows these fields for the structures shown in Figs. 1(a) and 1(b) for a grating depth of 25 nm and cover layers of 55 nm. Table I summarizes the associated normalized overlap integral Γ i (no cover layers in Fig. 1(a), cover layers in both regions ( Fig. 1(b)), and for no cover layer on the input waveguide, but a cover layer in the grating region (Fig. 1(c)). The corresponding index and electric field profiles for these cases are shown in Fig. 3. For these cases, a cover layer thickness of 55 nm was chosen to provide an increase in Γ g without making the waveguide multimode or having a larger percentage of the mode in the cover layer than in the core region. For such shallow gratings, there is little difference in Γ i . Even for deeper gratings (e.g., 100 nm), because of the strong confinement of Si photonic waveguides, Γ i is close to one (Table II), which shows the optimum cover-layer thickness to maximize Γ i for a duty cycle of 50%. The modal overlap integrals in Tables I and II assume a 50% duty cycle in calculating the average permittivity of the grating layers and were evaluated using WAVEGUIDE III [29].
The variation of the normalized intensity overlap integral as a function of duty cycle for several structures (see insets) with and without 35 nm thick cover layers and 25 nm grating depths are shown in Fig. 4(a). The optimum cover layer thicknesses and normalized intensity overlap integrals for one of the structures are shown in Fig. 4(b) as a function of duty cycle.

B. Complex Effective Indices
Numerical methods using the Floquet-Bloch approach have long been used to analyze gratings in optical waveguides [10], [23], [24], [25] by matching the boundary conditions of the electromagnetic field at every layer interface contained in the grating region (and at every layer interface in the complete optical waveguide) to obtain a detailed and accurate solution for the grating strength and the distance over which a large fraction of the waveguide light is reflected, transmitted, and radiated by the grating. Assuming that modes propagate in the positive z direction as exp(jωt-γz), the electric field is written as where γ = α + jβ is the complex propagation constant of the mode and F(x, z) = F(x, z + Λ) is a periodic function expandable in a Fourier series.
where ψ n (x) is the transverse variation of the n th space harmonic. The longitudinal propagation constant γ n of a space harmonic is given by where β is equal to 2π/λ g (λ g is the wavelength of the field variation along the z-axis of the waveguide), and K = 2π/Λ where Λ is the grating period. The field attenuation coefficient α (2α is the power attenuation constant) is related to the amount of reflected light for a given grating length. The function F(x, z) modulates the propagating mode near the Bragg conditions. The accuracy of this numerical approach is limited only by the number of terms (space harmonics) in (6) included in the calculation. In practice, as the number of space harmonics increases, the values of both β and α approach Fig. 5. a) Real parts (TE 0 (blue) and TE 1 (pink)) and imaginary parts (TE 0 (red) and TE 1 (dashed blue)) of the complex effective indices and b) percent of total power (green), reflected power (red), transmitted power (blue), upward coupled power (yellow) and downward coupled power (purple) for T E 0 -T E 0 and T E 0 -T E 1 coupling for a conventional Si photonics waveguide ( Fig. 1(a)) with a grating depth of 25 nm.
the limiting values, and do not change with further increases in the number of space harmonics. A minimum of five space harmonics, from −2 to +2, were used in all the Floquet-Bloch calculations performed in this study.
Unlike a thin-film stack ( Fig. 1(d)), waveguides can support multiple modes, each with a corresponding β, α and effective complex index n eff = (β + jα)/k o . Plots of the real and imaginary parts of the effective indices are shown for waveguides in Fig. 5(a) (grating depth of 25 nm, no cover layer ( Fig. 1(a)), Fig. 6(a) (grating depth 25 nm, 55 nm cover layer ( Fig. 1(b)), and Fig. 7(a) (a single-mode DBR grating waveguide in a III-V compound (grating depth of 7.68 nm, and a 0.3 μm core layer thickness with refractive indices the same as a Si photonic waveguide, except that the substrate has an index of 3.2).
The first-order modal coupling coefficient κ for any grating region in the waveguide ( Fig. 1(a)-(c)) is the maximum value of α (κ = α peak ). Table III shows a comparison of κ calculated by the analytic perturbation formula [30], [31], [32], [33] κ = κ pq (column 3) and the slab effective index method κ = κ rect Fig. 6. a) Real parts (TE 0 (blue) and TE 1 (pink)) and imaginary parts (TE 0 (red) and TE 1 (dashed blue)) of the complex effective indices; b) percent of total power (green), reflected power (red), transmitted power (blue), upward coupled power (yellow) and downward coupled power (purple) for T E 0 -T E 0 and T E 0 -T E 1 coupling; c) percent of total power (green), reflected power (red), transmitted power (blue), upward coupled power (yellow) and downward coupled power (purple) for T E 0 -T E 1 and T E 1 -T E 1 coupling; and d) reflected power comparison for T E 0 − T E 0 , T E 0 − T E 1 and T E 1 − T E 1 coupling; The orange line correspond to a TE 1 input and the blue line corresponds to a TE 0 input. All plots are for a Si photonics waveguide with a grating depth of 25 nm and a cover layer of 55 nm. (column 5) with the numerically exact Floquet-Bloch method κ = α peak (column 2).
The slab effective index method has excellent agreement (<2%) with the coupling coefficient κ calculated by Floquet Bloch at the first Bragg condition for even deep gratings, but because the analytic formulas are based on perturbation theories that only include the first term in a Taylor series expansion, the agreement is reasonable (<10%) only if the grating depth is less than ∼20% of the silicon photonic waveguide core thickness. Revision of the analytic formulas to include higher order terms in the derivation would increase the agreement with the exact coupling coefficient α peak for deeper grating depths.
If the waveguide supports multiple modes with propagation constants β p and β q , the same grating wavector K couples the p-p, p-q, and q-q modes for wavelengths that satisfy [30]:  Fig. 5(a)) and at λ ο ∼ 1.28 μm (Fig. 6(a)) for TE 0 -TE 1 , and at λ ο ∼ 1.1 μm (Fig. 6(a)) for TE 1 -TE 1 . The imaginary part of the effective index α/k o is zero outside of the TE o -TE o and TE o -TE 1 resonances, except for wavelengths shorter than the TE o -TE 1 resonance in Figs. 5(a) and Fig. 6(a) (see the vertical dashed line at λ ο ∼ 1.21 μm). For these shorter wavelengths, the grating couples light out of the waveguide, as shown in Figs. 5(b) and 6(b), which indicates the amount of light reflected (red), transmitted (blue) and outcoupled downwards towards the substrate (purple) and upward towards the superstrate (yellow). The green line is the sum of the reflected, transmitted, and outcoupled light. The deviation of the sum of the powers from 100% is an indication of numerical accuracy.
The transmission, reflection, upward radiation, and downward radiation calculations in Fig. 6(b) assume an incident TE 0 mode, and those in Fig. 6(c) assume an incident TE 1 mode. Fig. 6(d) shows the agreement between the TE o -TE 1 coupling calculated using both assumptions.
In Fig. 7(a), the III-V waveguide remains single mode at wavelengths corresponding to the second Bragg resonance peak, which occurs at a wavelength of ∼0.775 μm. However, the grating structure in Fig. 7(a) begins radiating, mainly into the substrate, at wavelengths shorter than 1.54 μm.

C. Reflection, Reflectivity Spectral Width and Peak Reflection Wavelength
Comparisons of the peak reflection, full width at half maximum (FWHM) of the reflection spectrum, and the shift of the peak reflection wavelength for III-V DBR and DFB waveguides as a function of duty cycle, calculated by both the Floquet Bloch (solid lines) and the slab effective index method (dashed lines), are shown in Fig. 8, showing excellent agreement between the two methods. In both cases, the thickness of the center layer is 0.3 microns, the grating depth is 7.68 nm and the grating lengths are 295 μm (DBR) and 202 μm (DFB). In agreement with Fig. 1, the maximum reflectivity occurs at a ∼50% duty cycle, where the spectral width is also a maximum. The wavelength corresponding to the reflection peak increases with duty cycle because the average effective permittivity of the grating layer, and therefore, the overall modal effective index increases with duty cycle.
The core-substrate index step of III-V waveguides [34] is in the 0.3 to 0.5 range compared to the large refractive index steps (∼2) between the central core region and the superstrate and substrate of silicon photonic waveguides. Unlike the tightly bound modes of silicon photonic waveguides, increasing the grating depth in III-V waveguides eventually can result in a decreasing coupling coefficient κ and the modes in the grating waveguide regions can become cut-off as they transition from bound to leaky modes.
The reflection spectra of Si photonic waveguides with (red) and without (blue) a 55 nm cover layer are shown in Fig. 9 for a grating depth of 25 nm and grating lengths of a) 11 μm and b) 2 μm. The addition of the cover layer increases the reflected power by ∼20%, increases the spectral width by 15%  for the 11 μm long grating, and increases the reflected power by ∼100%, and decreases the spectral width by ∼7% for the 2 μm long grating. For a Si photonics waveguide with a grating depth of 2.5 nm, changing the grating duty cycle from 10% to about 50% changes the maximum reflection by 81%, the reflection spectral width by 30% and the peak reflection wavelength by ∼0.1% for a grating Fig. 10. a) Reflection, reflection spectral width and peak wavelength shift for a grating length of 119 μm as a function of duty cycle; b) required grating length, reflection spectral width and peak wavelength shift for a fixed reflection peak of 95% as a function of duty cycle; and c) required grating length, reflection spectral width and peak wavelength shift for a fixed reflectivity of 30%. as a function of duty cycle. All cases are for a Si photonics waveguide with a grating depth of 2.5 nm. Calculations done by SEI. Fig. 11. Required grating length, reflection spectral width and peak wavelength shift for a fixed reflectivity of 30% for a Si photonics waveguide with a grating depth of 25 nm as a function of duty cycle. Calculations done by SEI. Fig. 12. a) Reflection, reflection spectral width and peak wavelength shift for a grating length of 11μm as a function of duty cycle for a Si photonics waveguide with (solid line) and without (dashed line) a 55 nm cover layer; and b) required grating length, reflection spectral width and peak wavelength shift for a fixed reflection peak of 95% as a function of duty cycle for a Si photonics waveguide with (solid line) and without (dashed line) a 55 nm cover layer. All cases are for a Si photonics waveguide with a grating depth of 25 nm and grating period of 0.2740 μm. Calculations done by SEI. length of 119 μm (Fig. 10(a)). To obtain a fixed reflectivity of 95%, the required grating length varies from 666 μm to 204 μm and the reflection spectral width varies from 1.0 to 3.1 nm as the duty cycle changes from 10% to 50%. The peak wavelength shifts from 1.549 to 1.551 μm as the duty cycle changes from 10% to 90% (Fig. 10(b)). For a fixed reflectivity of 30%, the required grating length varies from 186 μm to 58 μm and the reflection spectral width varies from 1.8 to 6.0 nm as the duty cycle changes from 10% to 50%. The peak wavelength shifts from 1.549 to 1.551 μm as the duty cycle changes from 10% to 90% (Fig. 10(c)).
For a Si photonic waveguide with a grating depth of 25 nm, changing the grating duty cycle from 10% to 50% for a fixed reflectivity of 30% changes the grating length from 18 μm to 5 μm and the reflection spectral width from 20 to 65 nm. The peak wavelength changes from 1.538 to 1.567 μm as the duty cycle changes from 10% to 90% (Fig. 11).
For a Si photonic waveguide without a cover layer and with a grating depth of 25 nm, changing the grating duty cycle from 10% to 50% changes the maximum reflection by 60% and the reflection spectral width by 56%. The peak wavelength changes by 1.9% for a grating length of 11 μm as grating duty cycle changes from 10% to 90%. If the same photonic waveguide has a cover layer, changing the grating duty cycle from 10% to 50% increases the peak reflection by 75%, reflection spectral width by 56%. The peak wavelength increases by 2.5% as the grating duty cycle changes from 10% to 90%. (Fig. 12(a)).
To obtain a fixed reflectivity of 95% for a Si photonic waveguide without a cover layer with a grating depth of 25 nm, the required grating length varies from 66 to 19 μm the reflection spectral width varies from 11 to 35 nm as the duty cycle changes from 10% to 50%. The peak wavelength shifts from 1.538 to 1.567 μm as the duty cycle changes from 10% to 90% ( Fig. 12(b)). If the same photonic waveguide has a cover layer, changing the grating duty cycle from 10% to 50% changes the required grating length required for 95% reflectivity from 47 μm to 14 μm and the reflection spectral width from 14 to 46 nm. The peak wavelength changes from 1.533 to 1.573 μm as the duty cycle changes from 10% to 90% (Fig. 12(b)).

III. CONCLUSION
A simple thin-film analysis of first-order gratings in Si photonic waveguides provides highly accurate results for reflected and transmitted power spectra as long as the waveguide remains single mode and non-radiating. The differences in the coupling coefficients, reflectivities, and reflectivity spectral widths calculated by the slab effective index method, FDTD, EME and Floquet-Bloch methods are very small (e.g., ∼0.1% for first-order gratings on Si photonic waveguides.) However, commonly used analytic formulas for gratings only provide accurate values of coupling coefficients in silicon photonic waveguides for grating depths of less than ∼20% of the core thickness. Additionally, the Floquet Bloch, slab effective index and EME methods show near-exact agreement even for only one grating period, and the FDTD method has excellent agreement once the grating has more than 5 grating periods (Fig. 2(d)).
A cover layer can be added to the grating region of a Si photonic waveguide to increase the strength of the grating and the width of the reflectivity spectrum. For a given grating period, the peak reflection and reflectivity spectral width decreases as the duty cycle decreased or increased from approximately 50%.
The modes of Si photonic waveguides are tightly bound because of the large refractive index steps (∼2) between the central core region and the superstrate and substrate, unlike III-V waveguides in which the core-substrate index step is in the 0.3 to 0.5 range. As a result, increasing the grating depth in III-V waveguides can result in a decreasing coupling coefficient κ, or cause the modes in the grating waveguide regions to be cut-off.
To reduce the coupling coefficients and obtain narrower reflection spectral widths, the regions between the grating teeth could be filled with higher-index materials such as Si 7 N 3 [35] or higher-index polymer materials.
The introduction of a higher-order mode in Si photonic waveguides at λ ∼ 1.25 μm (Fig. 5) and λ ∼ 1.3 μm (Fig. 6) provides a lower wavelength limit for WDM applications and may be a limit in designing silicon photonic waveguides [36].