Abstract

A major issue in the fabrication of integrated Bragg grating filters in highly confined waveguides is the average effective index fluctuations caused by waveguide dimension variations. Lateral variations are caused by the sidewall roughness created during the etching process while vertical variations are coming from the wafer silicon layer thickness non-uniformity. Grating spectral distortions are known to result solely from the low spatial frequency components of these variations. As a result, in this work, we present an experimental method to quantify such relevant spatial components by stitching a hundred high-resolution scanning electron microscope images. Additionally, we propose two techniques to reduce, in the design, the phase noise impact on integrated Bragg gratings without relying on fabrication process improvements. More specifically, we show that the use of hybrid multimode/singlemode waveguides reduce by more than one order of magnitude the effect of sidewall roughness on integrated Bragg gratings while we show that the fabrication of ultra-compact gratings in spiral waveguides mitigate the impact of the silicon layer thickness variations.

© 2013 OSA

1. Introduction

Due to its compatibility with CMOS processes, silicon photonics has emerged as an attractive solution for the fabrication of low-cost small-footprint photonic integrated circuits. Alongside these developments, fiber Bragg gratings (FBG) provide filters with versatile and precise spectral responses, but their footprint could be significantly reduced if integrated in the silicon-on-insulator (SOI) platform. Integrated Bragg gratings (IBGs) with complex spectral responses require long structures with tens of thousands of periods but, unfortunately, these long gratings are sensitive to phase noise [1]. The grating phase defines the local Bragg wavelength or, equivalently, the grating detuning coefficient whereas the grating strength refers to the coupling coefficient. In this paper, we therefore use the term “phase noise” to regroup perturbations that have an impact on the local Bragg wavelength of the grating [1]. Any perturbation to the waveguide geometry that changes its effective refractive index is thus a source of phase noise that can induce distortion in the grating spectral response. Recent work reported significant spectral distortions of IBGs in SOI due to waveguide dimension variations that perturb the mode effective index [1,2]. This effect is significant due to the high index contrast of SOI waveguides. Waveguide width variations are caused by the sidewall roughness (SWR) introduced by the lithography and etching processes whereas the height variations are caused by the silicon layer thinning and polishing processes, i.e. wafer height fluctuation (WHF). Since these phenomena modify the waveguide effective index, they consequently perturb the grating phase or equivalently introduce variations in the local Bragg wavelength along the waveguide.

The study of waveguide non-uniformities in SOI has recently received growing attention due to the associated performance degradation observed for many devices. The impact of SWR was mostly studied in the context of waveguide group delay [3], propagation losses [4]- [5] and spectral distortions in IBGs [1], while WHF was recently identified as a major concern in the reproducibility of micro-ring [6] and IBGs [7] resonances. Furthermore, much work was already dedicated to reduction of the SWR [811] as well as to the improvement of silicon thickness uniformity [12].

The modeling of SWR and its impact on IBG spectral responses was discussed in [1], but this work did not include the effect of WHF. Furthermore, the experimental characterization of those two random processes has never been done in the context of IBGs that are more sensitive to dimension variations with low spatial frequency components, in contrast to propagation loss that is more severely affected by the high spatial frequency components and less by the low frequency ones. Thus, the first objective of this paper is to present an experimental method to quantify the phase noise terms that affect the quality of IBG spectral responses. Those parameters can be used as inputs to a Bragg grating emulator in order to predict device yield [1]. The second objective is to demonstrate how the phase noise can be reduced by appropriate design, i.e. without relying on fabrication process improvements. It should be mentioned that propagation loss also affects IBGs spectral response. However this phenomenon can already be considered by adding a complex index of refraction in the coupled-mode equations.

The paper is organized as follows: section 2 presents the phase-noise model that links the SWR and WHF random processes to the Bragg wavelength fluctuations observed in IBGs. Previous studies have already addressed the characterization of the SWR random process using scanning electron microscope (SEM) images ([5,1315] to name a few). However, these previous reports did not characterize the low spatial frequency content of SWR, which is critical to predict spectral distortions of IBGs, because the required characterization length exceeds the field-of-view (FOV) of SEM images. Consequently, we propose in section 3 an experimental method to increase the length over which the waveguide is analyzed, thus enabling the characterization of the low spatial frequency content of SWR. We also discuss how to extract the parameters characterizing WHF from IBGs optical measurements. Finally, section 4 presents two techniques to improve IBG robustness to phase noise: the first one is to fabricate IBGs on wider waveguide to reduce the impact of SWR, while the second one proposes to fabricate ultra-compact gratings to mitigate the impact of the WHF, i.e. using spiral IBGs [16,17].

2. Phase-noise model

This section presents the theoretical background that links the parameters of the SWR and the WHF random processes to the standard deviation of the grating Bragg wavelength profile. These two processes are modeled by a normal distribution and an exponentially decaying autocorrelation function [1820]. In section III, we will show that this model provides a good fit to the SWR experimental results and we assume that this model could be extended to the WHF. The good correspondence between the model and the experimental results presented in the following sections confirms this hypothesis.

The spatial frequency content of a random process, X, is described by its power spectral density function, GX, which is the Fourier transform of its autocorrelation function, RX. For the random processes SWR and WHF, the autocorrelation function are defined by

RSWR(Δz)=limL1LL/2L/2Δx(z)Δx(z+Δz)dzRWHF(Δz)=limL1LL/2L/2Δy(z)Δy(z+Δz)dz
where Δx is the sidewall deviation from its mean position, Δy is the wafer height deviation from the average silicon thickness and Δz is the spatial shift along the waveguide axis. The axes are indicated on the SEM image of Fig. 1(a), which shows a typical top-down image of a photonic wire. Throughout this paper, many random processes will be discussed (i.e. the effective index, the Bragg wavelength, the sidewall roughness, the wafer height fluctuation, etc…). To simplify the notation, σX is going to refer to the standard deviation of the process. Similarly, Lc,X is going to refer to the autocorrelation length of X. We use the usual exponential function to model the autocorrelation function of the SWR and the WHF processes, which is given by [18]
RSWR(Δz)=σSWR2exp(|Δz|Lc,SWR)RWHF(Δz)=σWHF2exp(|Δz|Lc,WHF)
Resulting from Eq. (2), the power spectral density functions are given by
GSWR(fz)=2σSWR2Lc,SWR1+4π2Lc,SWR2fz2GWHF(fz)=2σWHF2Lc,WHF1+4π2Lc,WHF2fz2
where fz is the spatial frequency (along the z-axis). The parameters describing the sidewall roughness (σSWR and Lc,SWR) are measured in section 3.1 using SEM images. Due to experimental limitations, a combination of the parameters describing the wafer height fluctuation (σWHF and Lc,WHF) will be extracted using optical measurements in section 3.2.

 

Fig. 1 (a) Typical top-down SEM image of a photonic wire. (b) Typical intensity profile of a pixel column (taken at the position of the red line in (a)). (c) Typical top-down SEM image converted into a binary image with the threshold shown by the black line in (b). The red lines are the outer waveguide sidewalls. (d) Typical superposition of the waveguide width fluctuation from every image of a set of measurement. The thick blue line is the linear regression of the average waveguide width fluctuation per pixel (black curve).

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In order to analyze IBG phase noise, the model describing SWR and WHF must be linked to the variations of the local Bragg wavelength. Since the standard deviation of the Bragg wavelength is related to the effective index standard deviation, we write

σλB=2Λσn
where Λ is the grating period and σn is the effective index standard deviation. Considering the presence of SWR and WHF noise sources, σn is given by
σn=2CSWR2σSWR2+CWHF2σWHF2
where CSWR and CWHF are the proportionality constants that link the waveguide width and height variations to an effective index variation. Those two parameters are evaluated around the waveguide average dimension using a finite element mode solver.

In analogy to the phase modulation of a carrier that introduces sidebands around the carrier frequency, spatial frequency components of grating phase noise introduce spectral distortions located at specific optical frequency bands of the grating spectrum. As the spatial frequency of the phase noise increases, the distortions will appear in frequency bands located farther away from the grating resonance [1]. Consequently, the grating spectral response is not affected by high frequency noise components because their impact will be outside the relevant optical bandwidth. Furthermore, because experimental measurements have a limited dynamic range, the distortions induced by high frequency noise content are often below the noise floor of the detection system. Thus, since only low spatial frequencies have an impact on the grating spectral response, the noise can be filtered to consider only such frequencies lower than the cut-off frequency, fc, given by

fc=2ngλB(λBΔλ1)2ngΔλλB2,
where ng is the group index and Δλ is the spectral interval between the Bragg wavelength and the wavelength where the grating reflection becomes hidden by the background noise in the measurement. The derivation of Eq. (6) has been done in [1,21]. Since the integral of the power spectral density over the frequencies is, by definition, equal to the variance of the noise, the variance of the noise which has a significant impact on IBGs spectral response, noted with “~”, is defined by
σ˜2=fcfcGΔx(fz)dfz=4σ2Lcfc4π2Lc2fc2+1.
In the special case where the cut-off frequency is significantly smaller than (2πLc)−1, σ˜2can be well approximated by 4σ2Lcfc. The filtered variances are thus given by
σ˜SWR28ngLcΔλλB2σSWR2σ˜WHF28ngLcΔλλB2σWHF2
and the filtered Bragg wavelength standard deviation σ˜λBis obtained by using Eq. (8) in Eqs. (4) and (5), such as
σ˜λB=2Λσ˜n=2Λ2CSWR2σ˜SWR2+CWHF2σ˜WHF2
which results in
σ˜λB=8ΛngΔλλB(CSWR2Lc,SWRσSWR2+12CWHF2Lc,WHFσWHF2)
By using the parameters that describe the waveguide dimension fluctuations experimentally determined in section 3, Eq. (10) will be used in section 4 to compare the model to the experimentally retrieved Bragg wavelength standard deviations. Equation (10) must be used instead of Eq. (4) since the experimental measurements of the Bragg wavelength were low-pass filtered by the optical characterization process since the noise floor of the spectral measurement hides the higher frequency content of the noise.

3. Phase-noise characterization

As mentioned previously, the measurement of the low frequency content is critical for the modeling of IBG spectral responses. In section 3.1, the SWR parameters will be directly measured by stitching high-resolution images from a SEM. In section 3.2, the characterization of WHF will be done by optical measurements of IBGs. The parameters will be extracted using the results of section 3.1 and by reconstructing the grating structure using an integral layer peeling algorithm [22]. Through Eq. (9), the grating phase gives information on the WHF parameters but this method only allows the determination of the product σWHFLc,WHF1/2.Unfortunately, atomic force microscope (AFM) images could not be used to properly characterize the surface because, in order to measure low frequency components, the scan size should be made as large as possible. However, the mechanical properties of the piezoelectric element that moves the AFM probe near the surface introduce 2nd and 3rd order curvatures that are usually called “bow”. These phenomena increase with the scan size. As a result, the bow must be removed using a third order polynomial regression, which suppressed the low frequency contents that needs to be characterize.

3.1 Characterization of sidewall roughness

The high resolution images needed to resolve the SWR random process having a σSWR of typically a few nanometers can be obtained with SEM images [5,1315]. The algorithm used in this paper to determine the sidewall position follows closely the one presented in [14] and is summarized in section 3.1.1. However, those measurements require special care since high-resolution SEM pictures provide a relatively limited field of view (FOV). If the FOV is too small, Lc will be under-estimated. Consequently, to increase the FOV along the waveguide propagation axis without degrading the resolution, hundreds of successive images have been taken with an overlap between them of roughly half a FOV. An algorithm was developed to properly align them in order to obtain the waveguide width fluctuation over a length of ~100 µm. Section 3.1.2 presents the alignment algorithm while the retrieval of the SWR parameters is described in section 3.1.3.

Measurements reported in this section were made on uncorrelated straight SOI waveguides having nominal cross sections of 220 nm x 500 nm that were fabricated using 193 nm deep-UV photolithography (IMEC). The SEM was a Zeiss 1540XB CrossBeam. The picture magnification was 60 kX and the acceleration tension was 20 kV which results in a probe size of 1.1 nm. The FOV was about 1.9 μm along the waveguide axis.

3.1.1 Waveguide edge determination

The algorithm used to determine the positions of the waveguide edges is based on a signal-threshold analysis, i.e. the edges are determined using a threshold value on the signal itself, as opposed to a derivate-threshold analysis that determines the edges using a threshold on the derivative signal. The derivative-threshold analysis has been discarded due to its higher noise sensitivity [14].

As can be seen on Fig. 1(a), the waveguide is defined by two bright lines. The algorithm can be applied either on the inner part or the outer part of the lines or could locate the position of the maximal brightness. However, it has been shown [14] that the retrieved edge functions using those three positions are alike. In this work, we chose to analyze the SWR using the outer part of the bright lines.

The first step after the image has been transformed into an array of pixel is to determine the pixel size (1.86 nm in this paper; the picture length along the waveguide axis is 1024·1.86 nm = 1905 nm). The pixel size evaluation is done by the SEM and this value does not change from one picture to another. Afterward, since the image noise is white whereas the useful information is mostly composed of low frequencies, the images are low-pass filtered using a Gaussian filter having a size of 5 x 5 pixels [14]. Then, the average background intensity is subtracted to the filtered-picture and the intensity is normalized to the maximal intensity value of the waveguide edge. Consequently, the threshold value is a ratio smaller than 1 (0.2 in this paper as shown by the black line in Fig. 1(b)). Since the intensity might differ between the two waveguide walls, the normalization is done independently on both sides of the waveguide which explains the discontinuity at the center of the intensity profile shown in Fig. 1(b). At this point, the sidewall positions can be easily determined. Figure 1(c) shows the top-down SEM image converted into a binary image with the retrieved sidewall positions.

To conclude, pixel size up to 4 nm does not have significant impact on the edge position [14]. As a result, the resolution used for this measurement is sufficient for our purpose. Furthermore, the retrieved sidewall profile is not changing as a function of the threshold value [14], aside for a position shift (as long as the threshold is fixed to a value superior to the noise floor). However, since we are not interested by the absolute position of the sidewall but by its variation around its mean, as will be discussed in more details in the next subsection, the choice of the threshold has no impact on the waveguide width variation measurement.

3.1.2 Picture alignment

Before aligning the pictures, the systematic source of error coming from the tilt that exists between the sample surface and the electron beam must be considered. Ideally, the beam would be at normal incidence on the silicon substrate; however, a small residual tilt can have a major impact on the waveguide roughness analysis. The misalignment of the e-beam on the substrate can be separated in three components; a rotation around the x-axis, the y-axis and the z-axis.

Rotation around the y-axis is the most obvious since it simply rotates the waveguide in the picture. Although this effect is often corrected by doing a linear fit of the retrieved sidewall function and by rotating numerically the picture accordingly, this cannot be done when analysing low frequency components since this operation suppresses an important portion of these low frequencies. Consequently, instead of correcting the picture misalignment and analysing the two sidewalls independently, it is wiser to analyse the waveguide width fluctuation instead, which is defined by removing the average waveguide width (w0) to the waveguide width profile. This procedure cancels out the misalignment without influencing the retrieved SWR parameter since both sides of the waveguide are equally affected by this tilt and because the random processes affecting both sidewalls are identical. Therefore, the retrieved Lc parameter of the waveguide width fluctuation is the same as Lc,SWR while the retrieved σ should be divided by a factor √2 to obtain σSWR.

The rotation around the z-axis has a negligible impact since it simply reduce w0 in a similar manner on every picture. Once again, we are not interested in the absolute width value. As a result, this error source is not relevant.

However, the rotation around the x-axis is critical. This tilt has a dramatic impact on the measured waveguide width fluctuation since it systematically increases the width on one side of the picture and reduces it on the other side. Obviously, this effect is very small and, when one analyzes one SEM picture at a time, this effect can easily be neglected. However, in this work, since a hundred pictures are put together, this effect adds up and must be removed. To this end, we superimposed every waveguide width fluctuation measurement from the different images as a function of pixel number as shown on Fig. 1(d). Since a random process with zero-mean is involved, the waveguide width fluctuation averaged over every picture should be equal to zero for every pixel. However, as can be seen from this figure, there is a small residual slope (in blue) which would result in a waveguide width variation of ~0.6 nm/picture (or equivalently 0.315 nm/µm) if this effect is not removed. Obviously, the left side of the sample was closer to the e-beam source than the right side for this measurement. It is interesting to evaluate what would be the Bragg wavelength chirp of a uniform grating if one was fabricated on this nominally constant waveguide. A grating having a length of 100 µm would have a waveguide width varying by about 30 nm. Thus, considering the effective index change of a waveguide having a width varying from 480 nm to 510 nm, the associate Bragg wavelength chirp of a uniform Bragg grating would be of the order of ~20 nm/100 µm (i.e. 2000 nm/cm). However, such chirp ratio has never been measured on a nominally straight waveguide (i.e. narrowband Bragg gratings can be fabricated with this technology [2,7]). As a result, this slope is obviously a measurement artifact and has been removed on every picture. Finally, if the waveguide would have been designed in a tapered-shaped instead, this procedure would also have removed this linear width variation. However, since we are interested in the random process that affects the waveguide width, we can discard any source of systematic width variation.

The translation from one picture to another is about half a FOV. However, we must apply a correction to both w0 and the waveguide axis (z-axis), in order to stitch together the images and be able to extract the width fluctuation profile over many tens of microns. For the z-alignment, the retrieved waveguide width of the Mth image is scanned over the M - 1 image until the fluctuations are matched. The optimal alignment corresponds to the position where the root mean square error is minimal. This is necessary because the translation stage of the SEM has a precision of about ± 0.3 µm. Fortunately the presence of high frequency components in the waveguide width measurement eases the alignment process.

An error in w0 is introduced by the focus variations (working distance) from one picture to another. To illustrate this point, many pictures of a waveguide were taken at the same position but with the working distance being modified around the optimal value. Figure 2(a) shows the superposition of the different measurements of waveguide width fluctuations once w0 is removed. It can be noticed that the measured waveguide width fluctuations are not changing significantly, which means that a small error in the focus will not affect significantly the roughness analysis although, as shown in Fig. 2(b), w0 is strongly modified. The waveguide image remains clear on a small range of working distance, which corresponds to a variation of the measured average waveguide width of about 5 nm. Sensitivity of waveguide width measurements to focus adjustments makes this parameter hard to characterize experimentally. As a result, because of this error in waveguide width measurement, a correction must be introduced which is simply done by adjusting the width of the image M + 1 to the width of the Mth image. This procedure is not problematic since we are only interested in the waveguide width fluctuation. Once every images are properly aligned, the average width is removed which provides the waveguide width fluctuation.

 

Fig. 2 (a) Waveguide width fluctuation obtained from top-down SEM taken at the same location with different working distance. (b) Dependency of the average waveguide width caused by a variation of the working distance.

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The red, blue and green curves of Fig. 3(a) are typical results of waveguide width fluctuations retrieved from consecutive SEM measurements after a proper alignment. The good correspondence of the waveguide width fluctuations measured in overlapping regions gives good confidence in the precision of this technique. The black curve in Fig. 3(a) and (b) is the final profile that will be used from now on and was obtained by averaging the superimposed curves.

 

Fig. 3 (a) Typical overlap of the waveguide width fluctuation between three consecutive images (red, blue and green). The black curve is the averaged waveguide width as a function of position. (b) Typical waveguide width fluctuation profile as a function of position. The black line is the raw data (corresponding to the black line of (a)) while the blue line is a filtered version of the waveguide width profile. (c) Autocorrelation function of the waveguide width fluctuation of the filtered (blue) and unfiltered (black) curve of (b). The red curve is the decaying exponential autocorrelation fit. (d) Probability density functions of the sidewall roughness. The red curve is a Gaussian function.

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3.1.3 Extraction of sidewall roughness parameters

To obtain the parameters σSWR and LSWR, the autocorrelation of the waveguide width fluctuation function is calculated as displayed in Fig. 3(c). The amplitude at Δz = 0 is, by definition, the variance of the process (i.e. to obtain the SWR variance, this value must be divided by a factor 2). In the region Δz < 1 μm, the autocorrelation of the raw data experiences a very steep decay (identified by the red arrows). This region is associated to high frequency fluctuations and, since the high spatial frequency has no impact on IBG spectral response, the retrieved profile (in black on Fig. 3(b)) can be low-pass filtered with a cut-off frequency of ~0.7 μm−1 (in blue on Fig. 3(b)). This cut-off frequency has been optimized in order to suppress the steep decay around Δz = 0 while maintaining the other parts of the function unchanged as shown by the blue curve of Fig. 3(c). This part of the autocorrelation function that is removed can be obtained by analyzing a series of SEM images without the alignment procedure explained above. Those high frequency components are irrelevant for IBG analysis, but are critical to propagation loss calculations [5,13]. The noise spatial frequency range relevant for propagation loss is discussed in [20]. Inversely, the low frequency components of waveguide width fluctuations discussed in this paper are not important to characterize propagation loss.

The red curve of Fig. 3(c) shows the autocorrelation, after filtering, modeled by a decaying exponential function. It should be mentioned that the filtered part of the waveguide width fluctuation measurement (high spatial frequencies), by itself, can also be modeled by a decaying exponential function as mentioned in [13], but obviously with a smaller autocorrelation length (a few tens to a few hundreds of nm). If one is interested to consider both the high and the low spatial frequencies of the waveguide sidewall roughness, the autocorrelation could be parameterized by the summation of two decaying exponential functions, with their respective set of parameters, σSWR and LSWR.

The whole measurement procedure was repeated for seven photonic wires over a length of ~100 µm on three chips of the same wafer. The retrieved parameters are: σSWR = 1.8 nm ± 0.1 nm and LSWR = 14 200 nm ± 600 nm. The measurement length of 100 µm (i.e. 7x longer than the measured autocorrelation length) seems sufficient since it has been shown that the measurement length of a process to be characterized should be longer than the autocorrelation length by a factor in the range between three to eight [2325].

Finally, as mentioned in section 2, the SWR is modeled by a random process with a normal distribution. This assumption is widely used for convenience since this kind of process can be easily handled mathematically. However, to our knowledge, this point has never been proven experimentally. To verify this affirmation, we considered the seven waveguide width profiles retrieved and plot in Fig. 3(d) a histogram to retrieve the probability density function of the process. The red curve is a Gaussian function with the width defined by the variance of the process. The overlap between the two confirms indeed that a normal distribution is an appropriate model for this kind of random process.

3.2 Characterization of wafer height fluctuation

In this section, we use measurements of IBG spectral responses to estimate the phase-noise originating from WHFs. The IBGs samples were fabricated on multimode section of a hybrid multimode/singlemode waveguide [2,26] as shown on Fig. 4. Singlemode sections with adiabatic couplers are placed before and after the multimode section in order to predominantly excite the fundamental mode of the multimode section. As a result, the grating is behaving as if it were in a singlemode waveguide, but with less distortion. Input/output light coupling is achieved with grating couplers having etched depth of 70 nm in 10 μm wide waveguides. These couplers are followed by 600 μm long adiabatic tapers that lead to single-TE mode waveguides having a width of 500 nm and a length of 190 μm. Then two 125 μm-long adiabatic tapers connect the single-TE mode waveguides to the multimode waveguide (having either a width of 800 nm or 1200 nm) which contains 2 mm-long IBGs. The silicon layer thickness is 220 nm. The IBG corrugations have recesses of 30 nm and a mean grating period of 858.24 nm for the 1200 nm wide waveguides and recesses of 10 nm and a mean grating period of 872.97 nm for the 800 nm wide waveguides. Third order gratings were chosen due to the fabrication limitation of a 300 nm minimal pitch. The grating average duty-cycle, defined as the ratio of the corrugation width to the grating period, was of 25% to allow the presence of a significant third order resonance. Although the duty-cycle that optimizes the grating reflectivity is 50% for rectangular corrugations, due to the small feature size of the grating, the lithography and etching processes strongly smooth the corrugation shape [27]. As a result, a 50% duty-cycle with such corrugation shape would not result in significant third order grating reflection.

 

Fig. 4 Schematic of the IBGs.

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Characterization of the IBGs was performed by measuring their complex spectral responses using a commercial optical frequency domain reflectometer to which we removed the coupling losses. To eliminate the strong reflections at the end of the cleaved fibers, we performed time filtering of the grating temporal response [21]. The 2 mm-long single-mode waveguide sections on both ends of the gratings provided sufficient temporal separation to filter out the unwanted reflections. We then retrieved the profiles of the Bragg wavelength, λB(z), and of the index modulation amplitude, Δn(z), using an inverse scattering algorithm, namely the integral layer peeling algorithm proposed in [22]. The amplitude of a typical spectral response is shown in red in Fig. 5(a), while the retrieved grating profiles are displayed in Fig. 5(b) and (c). The maximal grating reflectivity was designed to be very small to ease the convergence of the grating reconstruction algorithm. The retrieved grating amplitude and phase profiles, after appropriate filtering, are used to calculate the reconstructed spectral responses, showed by the black curve in Fig. 5(a), using a standard transfer matrix solution of the coupled mode equations. We confirm the precision of the retrieved λB and Δn profiles by obtaining a good correspondence between the reconstructed spectra and the measured ones. More details on the data post-processing procedure can be found in [21].

 

Fig. 5 (a) Comparison of the experimental reflection spectrum of a typical straight grating on a 1200 nm wide waveguide (in red) with the reconstructed reflection spectrum (in black). Retrieved (b) λB and (c) Δn profiles, which are used to calculate the black curve of (a).

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Due to a Fourier transform relationship, the spatial resolution of the retrieved profiles depends on the measured optical bandwidth and, since IBGs are narrowband filters, most of the out-of-band information is noise. Consequently, the retrieved Bragg wavelength must be low-pass filtered [21] with a cut-off frequency given by Eq. (6). For the gratings studied in this paper and the one shown in Fig. 5, the noise level allowed us to characterize the grating over a spectral band of roughly Δλ = ± 1 nm around the resonances. As a result, a cut-off frequency of fc ≈3 000 m−1 has been used.

Using Eq. (10) with the sidewall roughness parameters obtained in section 3.1 and the value of σ˜λBexperimentally determined from measurements of IBGs fabricated on 1200 nm wide waveguides, the productσWHFLc,WHF1/2can be estimated. A total of 11 gratings were measured on four different chips from two different wafers. We obtained σ˜λB = 0.17 nm ± 0.01 nm resulting in σWHFLc,WHF1/2 = 6.6 x 10−13 m3/2 using the calculated values CWHF (3.6x10−3 nm−1) and CSWR (1.2x10−4 nm−1) for these waveguides. This result is significantly lower than the experimentally determined product 2σSWRLc,SWR1/2 = 1.4 x 10−11 m3/2 of the SWR. This means that, for a typical singlemode waveguide having a width of 500 nm, even though CWHF (4.0x10−3 nm−1) is about two times larger than CSWR (1.9x10−3 nm−1), SWR distortion is about one order of magnitude more damaging for IBGs spectral responses than WHF.

4. Phase-noise reduction techniques

Although an obvious way to reduce IBGs spectral distortions is to improve fabrication techniques to reduce the WHF, as discussed in [12], and the SWR, as discussed in [911]. In the latter case, it is unclear if the proposed approaches, which are appropriate to reduce propagation losses, are compatible with the fabrication of corrugation based IBGs. These techniques optimize either the lithography, etching or post-etch process conditions in order to reduce the propagation losses, which are mainly influenced by spatial frequencies near the propagation constant of the guided mode [18]. Since the noise spatial frequencies that those techniques aim to remove are close to the grating spatial frequency, the grating corrugations will also be strongly suppressed during the fabrication process. Consequently, in this section we propose two approaches to reduce the phase noise by optimizing the design. The first one reduces the impact of sidewall roughness on IBGs spectra while the second one reduces the impact of wafer height fluctuations.

4.1 Reduction of the impact of sidewall roughness

The impact of SWR and WHF on the IBG phase noise is shown in Fig. 6(a) as a function of w0. The curves were calculated for a waveguide with a 220 nm height using Eq. (10) with the experimentally determined parameter values for σSWR, LSWR and σWHFLc,WHF1/2, and using a finite element mode solver to obtain CSWR(w0)and CWHF(w0). In Fig. 6(a), the red and the blue curves represent the respective contributions of the WHF and SWR impact onσ˜λB, while the black curve shows their combined effect. Clearly, wider waveguide reduce considerably the IBG phase noise coming from sidewall roughness while leaving the impact of WHF unchanged. The cyan bars shows the value of σ˜λBdetermined from optical measurements of gratings in waveguides having width of 1200 nm (i.e. the results used to obtain the product σWHFLc,WHF1/2 in the previous section) and in waveguides having a width of 800 nm (a total of 14 gratings were measured on five different chips and two different wafers). The latter result is in good agreement with the theoretical curve, hence confirming the validity of this model. As mentioned previously, the 1200 nm wide waveguides had an averaged Bragg wavelength standard deviation of σ˜λB = 0.17 nm ± 0.01 nm while the 800 nm wide waveguides had a standard deviation of σ˜λB = 0.31 nm ± 0.05 nm.

 

Fig. 6 (a) Bragg wavelength standard deviation as a function of the waveguide width. The black curve contains both the impact of the sidewall roughness and the wafer height fluctuations, while the blue and red curve takes those to effect independently. The cyan lines are the optical measurement of 2 mm-long IBGs while the purple line is the optical measurement for spiral IBGs. (b) Schematic of spiral IBGs.

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Since the fundamental mode becomes more strongly guided as w0 increases, CSWR decreases rapidly. It will be reduced by one order of magnitude when the waveguide width increases from 500 nm to 1050 nm when considering a waveguide thickness of 220 nm. As a result, as w0 increases, the noise becomes dominated by WHF and the related Bragg wavelength standard deviation becomes

σ˜λB,WHF8ΛCWHFσWHFλBngΔλLc,WHF2
which corresponds to the minimal amount of noise that can be obtained using hybrid single-mode waveguides [2,26]. This approach allows a significant reduction of the phase noise but, as the calculations show, the improvement is limited by the level of WHF. Another motivation for designing IBGs in hybrid multimode/singlemode waveguides is the fact that the amount of backscattered light due to SWR is reduced as the waveguide width increases [28], i.e. when CSWR is reduced. Considering that, in future work, longer grating structures with weaker coupling coefficients could be required to achieve integrated optical filters with elaborate spectral responses. As a result, backscattering noise could become problematic for standard singlemode waveguides (500 nm wide).

4.2 Reduction of the impact of wafer height fluctuations

Considering that the phase noise that affects IBGs is composed of low frequency components and assuming that the autocorrelation length of the WHF is much longer than the SWR one, the fabrication of IBGs along a spiral as shown in Fig. 6(b) should reduce the WHF impact on the grating spectral response [16,17]. With proper tuning of the grating period to compensate for the effective index variations caused by the curvature, as discussed in [17], those IBGs are as flexible as straight grating, provide highly compact devices and are less affected by phase noise.

A total of nine spiral gratings on five different chips from the same wafer have been characterized [29]. The procedure to obtain the Bragg wavelength standard deviation is similar to the one described above for straight gratings. For 1200 nm wide waveguide and 30 nm corrugation recesses, we obtained a Bragg wavelength standard deviation of 0.12 ± 0.01 nm. This result is indicated by the purple dot in Fig. 6(a) and can be compared to the straight grating result. It should be mentioned that, unlike the straight gratings, a layer of silica has been put on the wafer containing the spiral gratings. As a result, the improvement showed in Fig. 6 (a) is probably underestimated due to the addition of a phase noise source caused by the possible presence of air-gaps between the waveguide and the silica layer. It should be mentioned that the passivation is causing a slight modification on CSWR and CWHF. The reduction of these parameters are responsible for about 10% of the measured improvement of the Bragg wavelength standard deviation obtained with passivated spiral gratings while the remaining 90% improvement is coming from the waveguide compactness.

To assess the impact of the SWR and the WHF reduction techniques on IBGs spectrum, a comparison is made in Fig. 7 between 2 mm-long straight gratings having widths of 800 nm and 1200 nm, with a spiral grating having a width of 1200 nm. The standard deviation of the Bragg wavelength of those gratings is 0.26 nm, 0.16 nm and 0.11 nm respectively, which makes them among the best of their category. The 3-dB bandwidths were respectively 0.22 nm (0.13 nm), 0.18 nm (0.13 nm) and 0.14 nm (0.12 nm). The value in parenthesis refers to the simulated 3-dB bandwidth of each design (presented in blue in Fig. 7). As can be seen from this figure, the use of wider straight waveguides improved the sidelobe suppression ratio by ~3 dB and brought the grating 3-dB bandwidth closer to the design by ~0.04 nm. Furthermore, the improvement from 0.16 nm to 0.11 nm of the Bragg wavelength standard deviation obtained by using 1200 nm wide spiral-IBGs had a significant impact on the grating spectrum leading to a symmetrical main lobe that corresponds closely to the design (in blue). The side-lobe suppression ratio has also been improved by ~3 dB compared to the straight 1200 nm wide IBGs and the first side-lobe corresponds more closely to the design. Figure 8 presents the superposition of every Bragg wavelength measurements for the three types of IBG described above. The improvement due to the two phase-noise reduction techniques is clearly illustrated although further improvement could be achieved with even more compact spiral waveguides. In this work, the IBG strength was deliberately designed to be low in order to ease the reconstruction of the Bragg wavelength profile along the grating length but the two approaches could easily be extended to stronger grating filters.

 

Fig. 7 Typical spectral response of 2 mm-long (a) straight grating on a 800 nm wide waveguide, (b) straight grating on a 1200 nm wide waveguide and (c) spiral grating on a 1200 nm wide waveguide. The blue curves are the designs while the black curves are the experimental results.

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Fig. 8 Superposition of every Bragg wavelength fluctuation measurements for (a) 800 nm straight waveguide, (b) 1200 nm straight waveguide and (c) 1200 nm spiral waveguide.

Download Full Size | PPT Slide | PDF

5. Conclusion

In this paper, we presented an improved technique to characterize SWR of silicon-on-insulator photonic waveguides and described how this phenomenon impacts the spectral response of IBGs. Many papers have discussed in length how to retrieve the sidewall roughness from SEM images. However, previously proposed technique could only be applied to single SEM image and, consequently, resulted in power spectral density described by an autocorrelation length of a few tens to a few hundreds of nm. In our work, we clearly show that there is also a significant portion of low frequency fluctuations. IBGs are predominantly influenced by the low spatial frequency content of the SWR whereas the high frequencies are relevant to characterize optical loss but are not sufficient to model and predict IBGs distortions. This paper clearly demonstrates that, in addition to its amplitude, the spatial frequency content of SWR is of critical importance for IBG-based devices. Furthermore, the impact of the WHF on IBGs spectral response has been modeled and quantified for the first time.

This paper also presented two techniques to improve IBG robustness to phase noise at the design step. The first one proposes the use of hybrid multimode/singlemode waveguides to reduce by more than one order of magnitude the effect of SWR on IBGs. The second one takes advantage of the fabrication of ultra-compact gratings in spiral waveguides to mitigate the impact of the silicon layer thickness variation.

These results are of importance because longer grating structures with weaker coupling coefficients are required in order to achieve integrated optical filters with elaborate spectral responses, both in amplitude and phase. Since longer gratings are more affected by phase noise, such demonstrations have been so far very limited. We believe that those phase-noise reduction techniques open the door to many new grating-based optical filter designs.

Acknowledgments

This project is part of the research program in Advanced photonics technologies for communications of the Canada research chair APTECS. The research was funded by NSERC, CMC Microsystems and NanoQuébec. The authors acknowledge Dr. Dan Deptuck for insightful discussions. The spiral gratings described in this paper were fabricated using the OpSIS service through IME A*STAR in Singapore.

References and links

1. A. D. Simard, N. Ayotte, Y. Painchaud, S. Bedard, and S. LaRochelle, “Impact of Sidewall Roughness on Integrated Bragg Gratings,” J. Lightwave Technol. 29(24), 3693–3704 (2011). [CrossRef]  

2. A. D. Simard, N. Belhadj, Y. Painchaud, and S. LaRochelle, “Apodized Silicon-on-Insulator Bragg Gratings,” IEEE Photon. Technol. Lett. 24(12), 1033–1035 (2012). [CrossRef]  

3. M. A. Schneider and S. Mookherjea, “Modeling Transmission Time of Silicon Nanophotonic Waveguides,” IEEE Photon. Technol. Lett. 24(16), 1418–1420 (2012). [CrossRef]  

4. K. K. Lee, D. R. Lim, H.-C. Luan, A. Agarwal, J. Foresi, and L. C. Kimerling, “Effect of size and roughness on light transmission in a Si/SiO2 waveguide: Experiments and model,” Appl. Phys. Lett. 77(11), 1617 (2000). [CrossRef]  

5. K. P. Yap, A. Delâge, B. Lamontagne, S. Janz, D.-X. Xu, J. Lapointe, P. Waldron, J. Schmid, P. Chow-Chong, E. Post, and B. Syrett, “Scattering loss measurement of SOI waveguides using 5X17 integrated optical star coupler,” in Conference Proceedings on Integrated Optoelectronic Devices, International Society for Optics and Photonics, 64770J (2007).

6. W. A. Zortman, M. R. Watts, and D. C. Trotter, “Determination of Wafer and Process Induced Resonant Frequency Variation in Silicon Microdisk-Resonators,” in Conference Proceedings on Integrated Photonics and Nanophotonics Research and Applications, paper IMC5 (2009).

7. X. Wang, W. Shi, H. Yun, S. Grist, N. A. F. Jaeger, and L. Chrostowski, “Narrow-band waveguide Bragg gratings on SOI wafers with CMOS-compatible fabrication process,” Opt. Express 20(14), 15547–15558 (2012). [CrossRef]   [PubMed]  

8. S. Sardo, F. Giacometti, S. Doneda, U. Colombo, M. D. Muri, A. Donghi, R. Morson, G. Mutinati, A. Nottola, M. Gentili, and M. C. Ubaldi, “Line edge roughness (LER) reduction strategy for SOI waveguides fabrication,” Microelectron. Eng. 85(5–6), 1210–1213 (2008). [CrossRef]  

9. D. K. Sparacin, S. J. Spector, and L. C. Kimerling, “Silicon Waveguide Sidewall Smoothing by Wet Chemical Oxidation,” J. Lightwave Technol. 23(8), 2455–2461 (2005). [CrossRef]  

10. J. Cai, Y. Wang, Y. Ishikawa, Y. Yamashita, Y. Kamiura, and K. Wada, “Hydrogen plasma treatment for Si waveguide smoothing,” in 8th IEEE International Conference on Group IV Photonics, 95–97, London, United Kingdom (2011). [CrossRef]  

11. K. K. Lee, D. R. Lim, L. C. Kimerling, J. Shin, and F. Cerrina, “Fabrication of ultralow-loss Si/SiO2 waveguides by roughness reduction,” Opt. Lett. 26(23), 1888–1890 (2001). [CrossRef]   [PubMed]  

12. S. K. Selvaraja, E. Rosseel, L. Fernandez, M. Tabat, W. Bogaerts, J. Hautala, and P. Absil, “SOI thickness uniformity improvement using corrective etching for silicon nano-photonic device,” in in 8th IEEE International Conference on Group IV Photonics London, United Kingdom, 71–73 (2011). [CrossRef]  

13. K. P. Yap, A. Delage, J. Lapointe, B. Lamontagne, J. H. Schmid, P. Waldron, B. A. Syrett, and S. Janz, “Correlation of Scattering Loss, Sidewall Roughness and Waveguide Width in Silicon-on-Insulator (SOI) Ridge Waveguides,” J. Lightwave Technol. 27(18), 3999–4008 (2009). [CrossRef]  

14. G. P. Patsis, V. Constantoudis, A. Tserepi, E. Gogolides, and G. Grozev, “Quantification of line-edge roughness of photoresists. I. A comparison between off-line and on-line analysis of top-down scanning electron microscopy images,” J. Vac. Sci. Technol. B 21(3), 1008–1018 (2003). [CrossRef]  

15. V. Constantoudis, G. P. Patsis, A. Tserepi, and E. Gogolides, “Quantification of line-edge roughness of photoresists. II. Scaling and fractal analysis and the best roughness descriptors,” J. Vac. Sci. Technol. B 21(3), 1019–1026 (2003). [CrossRef]  

16. A. D. Simard, Y. Painchaud, and S. LaRochelle, “Integrated Bragg Gratings in Curved Waveguides,” in the 23rd Annual Meeting of the Photonics Society Denver, USA, ThU3 (2010). [CrossRef]  

17. A. D. Simard, Y. Painchaud, and S. LaRochelle, “Integrated Bragg gratings in spiral waveguides,” Opt. Express 21(7), 8953–8963 (2013). [CrossRef]   [PubMed]  

18. F. Ladouceur, J. D. Love, and T. J. Senden, “Measurement of surface roughness in buried channel waveguides,” Electron. Lett. 28(14), 1321–1322 (1992). [CrossRef]  

19. T. Barwicz and H. A. Haus, “Three-Dimensional Analysis of Scattering Losses Due to Sidewall Roughness in Microphotonic Waveguides,” J. Lightwave Technol. 23(9), 2719–2732 (2005). [CrossRef]  

20. T. Barwicz and H. I. Smith, “Evolution of line-edge roughness during fabrication of high-index-contrast microphotonic devices,” J. Vac. Sci. Technol. B 21(6), 2892–2896 (2003). [CrossRef]  

21. A. D. Simard, Y. Painchaud, and S. LaRochelle, “Characterization of Integrated Bragg Grating Profiles,” in Conference Proceedings on Bragg Gratings, Photosensitivity, and Poling in Glass Waveguides, paper BM3D.7 (2012).

22. A. Rosenthal and M. Horowitz, “Inverse scattering algorithm for reconstructing strongly reflecting fiber Bragg gratings,” IEEE J. Quantum Electron. 39(8), 1018–1026 (2003). [CrossRef]  

23. V. Eswaran and S. B. Pope, “Direct numerical simulations of the turbulent mixing of a passive scalar,” Phys. Fluids 31(3), 506–520 (1988). [CrossRef]  

24. S. B. Pope, “Turbulent flows,” (Cambridge University Press, 2000).

25. P. L. O’Neill, D. Nicolaides, D. Honnery, and J. Soria, “Autocorrelation functions and the determination of integral length with reference to experimental and numerical data,” in 15th Australasian fluid mechanics conference, Sydney, Australia (2004).

26. S. Spector, M. W. Geis, D. Lennon, R. C. Williamson, and T. M. Lyszczarz, “Hybrid multi-mode/single-mode waveguides for low loss,” in Integrated Photonics Research, Optical Society of America (2004).

27. X. Wang, W. Shi, M. Hochberg, K. Adam, E. Schelew, J. F. Young, N. A. F. Jaeger, and L. Chrostowski, “Lithography simulation for the fabrication of silicon photonic devices with deep-ultraviolet lithography,” in 9th International Conference on Group IV Photonics (GFP), 288 –290 (2012). [CrossRef]  

28. F. Morichetti, A. Canciamilla, C. Ferrari, M. Torregiani, A. Melloni, and M. Martinelli, “Roughness Induced Backscattering in Optical Silicon Waveguides,” Phys. Rev. Lett. 104(3), 033902 (2010). [CrossRef]   [PubMed]  

29. T. Baehr-Jones, R. Ding, Y. Liu, A. Ayazi, T. Pinguet, N. C. Harris, M. Streshinsky, P. Lee, Y. Zhang, A. E.-J. Lim, T.-Y. Liow, S. H.-G. Teo, G.-Q. Lo, and M. Hochberg, “Ultralow drive voltage silicon traveling-wave modulator,” Opt. Express 20(11), 12014–12020 (2012). [CrossRef]   [PubMed]  

References

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  • |
  • |

  1. A. D. Simard, N. Ayotte, Y. Painchaud, S. Bedard, and S. LaRochelle, “Impact of Sidewall Roughness on Integrated Bragg Gratings,” J. Lightwave Technol.29(24), 3693–3704 (2011).
    [CrossRef]
  2. A. D. Simard, N. Belhadj, Y. Painchaud, and S. LaRochelle, “Apodized Silicon-on-Insulator Bragg Gratings,” IEEE Photon. Technol. Lett.24(12), 1033–1035 (2012).
    [CrossRef]
  3. M. A. Schneider and S. Mookherjea, “Modeling Transmission Time of Silicon Nanophotonic Waveguides,” IEEE Photon. Technol. Lett.24(16), 1418–1420 (2012).
    [CrossRef]
  4. K. K. Lee, D. R. Lim, H.-C. Luan, A. Agarwal, J. Foresi, and L. C. Kimerling, “Effect of size and roughness on light transmission in a Si/SiO2 waveguide: Experiments and model,” Appl. Phys. Lett.77(11), 1617 (2000).
    [CrossRef]
  5. K. P. Yap, A. Delâge, B. Lamontagne, S. Janz, D.-X. Xu, J. Lapointe, P. Waldron, J. Schmid, P. Chow-Chong, E. Post, and B. Syrett, “Scattering loss measurement of SOI waveguides using 5X17 integrated optical star coupler,” in Conference Proceedings on Integrated Optoelectronic Devices, International Society for Optics and Photonics, 64770J (2007).
  6. W. A. Zortman, M. R. Watts, and D. C. Trotter, “Determination of Wafer and Process Induced Resonant Frequency Variation in Silicon Microdisk-Resonators,” in Conference Proceedings on Integrated Photonics and Nanophotonics Research and Applications, paper IMC5 (2009).
  7. X. Wang, W. Shi, H. Yun, S. Grist, N. A. F. Jaeger, and L. Chrostowski, “Narrow-band waveguide Bragg gratings on SOI wafers with CMOS-compatible fabrication process,” Opt. Express20(14), 15547–15558 (2012).
    [CrossRef] [PubMed]
  8. S. Sardo, F. Giacometti, S. Doneda, U. Colombo, M. D. Muri, A. Donghi, R. Morson, G. Mutinati, A. Nottola, M. Gentili, and M. C. Ubaldi, “Line edge roughness (LER) reduction strategy for SOI waveguides fabrication,” Microelectron. Eng.85(5–6), 1210–1213 (2008).
    [CrossRef]
  9. D. K. Sparacin, S. J. Spector, and L. C. Kimerling, “Silicon Waveguide Sidewall Smoothing by Wet Chemical Oxidation,” J. Lightwave Technol.23(8), 2455–2461 (2005).
    [CrossRef]
  10. J. Cai, Y. Wang, Y. Ishikawa, Y. Yamashita, Y. Kamiura, and K. Wada, “Hydrogen plasma treatment for Si waveguide smoothing,” in 8th IEEE International Conference on Group IV Photonics, 95–97, London, United Kingdom (2011).
    [CrossRef]
  11. K. K. Lee, D. R. Lim, L. C. Kimerling, J. Shin, and F. Cerrina, “Fabrication of ultralow-loss Si/SiO2 waveguides by roughness reduction,” Opt. Lett.26(23), 1888–1890 (2001).
    [CrossRef] [PubMed]
  12. S. K. Selvaraja, E. Rosseel, L. Fernandez, M. Tabat, W. Bogaerts, J. Hautala, and P. Absil, “SOI thickness uniformity improvement using corrective etching for silicon nano-photonic device,” in in 8th IEEE International Conference on Group IV Photonics London, United Kingdom, 71–73 (2011).
    [CrossRef]
  13. K. P. Yap, A. Delage, J. Lapointe, B. Lamontagne, J. H. Schmid, P. Waldron, B. A. Syrett, and S. Janz, “Correlation of Scattering Loss, Sidewall Roughness and Waveguide Width in Silicon-on-Insulator (SOI) Ridge Waveguides,” J. Lightwave Technol.27(18), 3999–4008 (2009).
    [CrossRef]
  14. G. P. Patsis, V. Constantoudis, A. Tserepi, E. Gogolides, and G. Grozev, “Quantification of line-edge roughness of photoresists. I. A comparison between off-line and on-line analysis of top-down scanning electron microscopy images,” J. Vac. Sci. Technol. B21(3), 1008–1018 (2003).
    [CrossRef]
  15. V. Constantoudis, G. P. Patsis, A. Tserepi, and E. Gogolides, “Quantification of line-edge roughness of photoresists. II. Scaling and fractal analysis and the best roughness descriptors,” J. Vac. Sci. Technol. B21(3), 1019–1026 (2003).
    [CrossRef]
  16. A. D. Simard, Y. Painchaud, and S. LaRochelle, “Integrated Bragg Gratings in Curved Waveguides,” in the 23rd Annual Meeting of the Photonics Society Denver, USA, ThU3 (2010).
    [CrossRef]
  17. A. D. Simard, Y. Painchaud, and S. LaRochelle, “Integrated Bragg gratings in spiral waveguides,” Opt. Express21(7), 8953–8963 (2013).
    [CrossRef] [PubMed]
  18. F. Ladouceur, J. D. Love, and T. J. Senden, “Measurement of surface roughness in buried channel waveguides,” Electron. Lett.28(14), 1321–1322 (1992).
    [CrossRef]
  19. T. Barwicz and H. A. Haus, “Three-Dimensional Analysis of Scattering Losses Due to Sidewall Roughness in Microphotonic Waveguides,” J. Lightwave Technol.23(9), 2719–2732 (2005).
    [CrossRef]
  20. T. Barwicz and H. I. Smith, “Evolution of line-edge roughness during fabrication of high-index-contrast microphotonic devices,” J. Vac. Sci. Technol. B21(6), 2892–2896 (2003).
    [CrossRef]
  21. A. D. Simard, Y. Painchaud, and S. LaRochelle, “Characterization of Integrated Bragg Grating Profiles,” in Conference Proceedings on Bragg Gratings, Photosensitivity, and Poling in Glass Waveguides, paper BM3D.7 (2012).
  22. A. Rosenthal and M. Horowitz, “Inverse scattering algorithm for reconstructing strongly reflecting fiber Bragg gratings,” IEEE J. Quantum Electron.39(8), 1018–1026 (2003).
    [CrossRef]
  23. V. Eswaran and S. B. Pope, “Direct numerical simulations of the turbulent mixing of a passive scalar,” Phys. Fluids31(3), 506–520 (1988).
    [CrossRef]
  24. S. B. Pope, “Turbulent flows,” (Cambridge University Press, 2000).
  25. P. L. O’Neill, D. Nicolaides, D. Honnery, and J. Soria, “Autocorrelation functions and the determination of integral length with reference to experimental and numerical data,” in 15th Australasian fluid mechanics conference, Sydney, Australia (2004).
  26. S. Spector, M. W. Geis, D. Lennon, R. C. Williamson, and T. M. Lyszczarz, “Hybrid multi-mode/single-mode waveguides for low loss,” in Integrated Photonics Research, Optical Society of America (2004).
  27. X. Wang, W. Shi, M. Hochberg, K. Adam, E. Schelew, J. F. Young, N. A. F. Jaeger, and L. Chrostowski, “Lithography simulation for the fabrication of silicon photonic devices with deep-ultraviolet lithography,” in 9th International Conference on Group IV Photonics (GFP), 288 –290 (2012).
    [CrossRef]
  28. F. Morichetti, A. Canciamilla, C. Ferrari, M. Torregiani, A. Melloni, and M. Martinelli, “Roughness Induced Backscattering in Optical Silicon Waveguides,” Phys. Rev. Lett.104(3), 033902 (2010).
    [CrossRef] [PubMed]
  29. T. Baehr-Jones, R. Ding, Y. Liu, A. Ayazi, T. Pinguet, N. C. Harris, M. Streshinsky, P. Lee, Y. Zhang, A. E.-J. Lim, T.-Y. Liow, S. H.-G. Teo, G.-Q. Lo, and M. Hochberg, “Ultralow drive voltage silicon traveling-wave modulator,” Opt. Express20(11), 12014–12020 (2012).
    [CrossRef] [PubMed]

2013 (1)

2012 (4)

2011 (1)

2010 (1)

F. Morichetti, A. Canciamilla, C. Ferrari, M. Torregiani, A. Melloni, and M. Martinelli, “Roughness Induced Backscattering in Optical Silicon Waveguides,” Phys. Rev. Lett.104(3), 033902 (2010).
[CrossRef] [PubMed]

2009 (1)

2008 (1)

S. Sardo, F. Giacometti, S. Doneda, U. Colombo, M. D. Muri, A. Donghi, R. Morson, G. Mutinati, A. Nottola, M. Gentili, and M. C. Ubaldi, “Line edge roughness (LER) reduction strategy for SOI waveguides fabrication,” Microelectron. Eng.85(5–6), 1210–1213 (2008).
[CrossRef]

2005 (2)

2003 (4)

T. Barwicz and H. I. Smith, “Evolution of line-edge roughness during fabrication of high-index-contrast microphotonic devices,” J. Vac. Sci. Technol. B21(6), 2892–2896 (2003).
[CrossRef]

A. Rosenthal and M. Horowitz, “Inverse scattering algorithm for reconstructing strongly reflecting fiber Bragg gratings,” IEEE J. Quantum Electron.39(8), 1018–1026 (2003).
[CrossRef]

G. P. Patsis, V. Constantoudis, A. Tserepi, E. Gogolides, and G. Grozev, “Quantification of line-edge roughness of photoresists. I. A comparison between off-line and on-line analysis of top-down scanning electron microscopy images,” J. Vac. Sci. Technol. B21(3), 1008–1018 (2003).
[CrossRef]

V. Constantoudis, G. P. Patsis, A. Tserepi, and E. Gogolides, “Quantification of line-edge roughness of photoresists. II. Scaling and fractal analysis and the best roughness descriptors,” J. Vac. Sci. Technol. B21(3), 1019–1026 (2003).
[CrossRef]

2001 (1)

2000 (1)

K. K. Lee, D. R. Lim, H.-C. Luan, A. Agarwal, J. Foresi, and L. C. Kimerling, “Effect of size and roughness on light transmission in a Si/SiO2 waveguide: Experiments and model,” Appl. Phys. Lett.77(11), 1617 (2000).
[CrossRef]

1992 (1)

F. Ladouceur, J. D. Love, and T. J. Senden, “Measurement of surface roughness in buried channel waveguides,” Electron. Lett.28(14), 1321–1322 (1992).
[CrossRef]

1988 (1)

V. Eswaran and S. B. Pope, “Direct numerical simulations of the turbulent mixing of a passive scalar,” Phys. Fluids31(3), 506–520 (1988).
[CrossRef]

Agarwal, A.

K. K. Lee, D. R. Lim, H.-C. Luan, A. Agarwal, J. Foresi, and L. C. Kimerling, “Effect of size and roughness on light transmission in a Si/SiO2 waveguide: Experiments and model,” Appl. Phys. Lett.77(11), 1617 (2000).
[CrossRef]

Ayazi, A.

Ayotte, N.

Baehr-Jones, T.

Barwicz, T.

T. Barwicz and H. A. Haus, “Three-Dimensional Analysis of Scattering Losses Due to Sidewall Roughness in Microphotonic Waveguides,” J. Lightwave Technol.23(9), 2719–2732 (2005).
[CrossRef]

T. Barwicz and H. I. Smith, “Evolution of line-edge roughness during fabrication of high-index-contrast microphotonic devices,” J. Vac. Sci. Technol. B21(6), 2892–2896 (2003).
[CrossRef]

Bedard, S.

Belhadj, N.

A. D. Simard, N. Belhadj, Y. Painchaud, and S. LaRochelle, “Apodized Silicon-on-Insulator Bragg Gratings,” IEEE Photon. Technol. Lett.24(12), 1033–1035 (2012).
[CrossRef]

Canciamilla, A.

F. Morichetti, A. Canciamilla, C. Ferrari, M. Torregiani, A. Melloni, and M. Martinelli, “Roughness Induced Backscattering in Optical Silicon Waveguides,” Phys. Rev. Lett.104(3), 033902 (2010).
[CrossRef] [PubMed]

Cerrina, F.

Chrostowski, L.

Colombo, U.

S. Sardo, F. Giacometti, S. Doneda, U. Colombo, M. D. Muri, A. Donghi, R. Morson, G. Mutinati, A. Nottola, M. Gentili, and M. C. Ubaldi, “Line edge roughness (LER) reduction strategy for SOI waveguides fabrication,” Microelectron. Eng.85(5–6), 1210–1213 (2008).
[CrossRef]

Constantoudis, V.

G. P. Patsis, V. Constantoudis, A. Tserepi, E. Gogolides, and G. Grozev, “Quantification of line-edge roughness of photoresists. I. A comparison between off-line and on-line analysis of top-down scanning electron microscopy images,” J. Vac. Sci. Technol. B21(3), 1008–1018 (2003).
[CrossRef]

V. Constantoudis, G. P. Patsis, A. Tserepi, and E. Gogolides, “Quantification of line-edge roughness of photoresists. II. Scaling and fractal analysis and the best roughness descriptors,” J. Vac. Sci. Technol. B21(3), 1019–1026 (2003).
[CrossRef]

Delage, A.

Ding, R.

Doneda, S.

S. Sardo, F. Giacometti, S. Doneda, U. Colombo, M. D. Muri, A. Donghi, R. Morson, G. Mutinati, A. Nottola, M. Gentili, and M. C. Ubaldi, “Line edge roughness (LER) reduction strategy for SOI waveguides fabrication,” Microelectron. Eng.85(5–6), 1210–1213 (2008).
[CrossRef]

Donghi, A.

S. Sardo, F. Giacometti, S. Doneda, U. Colombo, M. D. Muri, A. Donghi, R. Morson, G. Mutinati, A. Nottola, M. Gentili, and M. C. Ubaldi, “Line edge roughness (LER) reduction strategy for SOI waveguides fabrication,” Microelectron. Eng.85(5–6), 1210–1213 (2008).
[CrossRef]

Eswaran, V.

V. Eswaran and S. B. Pope, “Direct numerical simulations of the turbulent mixing of a passive scalar,” Phys. Fluids31(3), 506–520 (1988).
[CrossRef]

Ferrari, C.

F. Morichetti, A. Canciamilla, C. Ferrari, M. Torregiani, A. Melloni, and M. Martinelli, “Roughness Induced Backscattering in Optical Silicon Waveguides,” Phys. Rev. Lett.104(3), 033902 (2010).
[CrossRef] [PubMed]

Foresi, J.

K. K. Lee, D. R. Lim, H.-C. Luan, A. Agarwal, J. Foresi, and L. C. Kimerling, “Effect of size and roughness on light transmission in a Si/SiO2 waveguide: Experiments and model,” Appl. Phys. Lett.77(11), 1617 (2000).
[CrossRef]

Gentili, M.

S. Sardo, F. Giacometti, S. Doneda, U. Colombo, M. D. Muri, A. Donghi, R. Morson, G. Mutinati, A. Nottola, M. Gentili, and M. C. Ubaldi, “Line edge roughness (LER) reduction strategy for SOI waveguides fabrication,” Microelectron. Eng.85(5–6), 1210–1213 (2008).
[CrossRef]

Giacometti, F.

S. Sardo, F. Giacometti, S. Doneda, U. Colombo, M. D. Muri, A. Donghi, R. Morson, G. Mutinati, A. Nottola, M. Gentili, and M. C. Ubaldi, “Line edge roughness (LER) reduction strategy for SOI waveguides fabrication,” Microelectron. Eng.85(5–6), 1210–1213 (2008).
[CrossRef]

Gogolides, E.

G. P. Patsis, V. Constantoudis, A. Tserepi, E. Gogolides, and G. Grozev, “Quantification of line-edge roughness of photoresists. I. A comparison between off-line and on-line analysis of top-down scanning electron microscopy images,” J. Vac. Sci. Technol. B21(3), 1008–1018 (2003).
[CrossRef]

V. Constantoudis, G. P. Patsis, A. Tserepi, and E. Gogolides, “Quantification of line-edge roughness of photoresists. II. Scaling and fractal analysis and the best roughness descriptors,” J. Vac. Sci. Technol. B21(3), 1019–1026 (2003).
[CrossRef]

Grist, S.

Grozev, G.

G. P. Patsis, V. Constantoudis, A. Tserepi, E. Gogolides, and G. Grozev, “Quantification of line-edge roughness of photoresists. I. A comparison between off-line and on-line analysis of top-down scanning electron microscopy images,” J. Vac. Sci. Technol. B21(3), 1008–1018 (2003).
[CrossRef]

Harris, N. C.

Haus, H. A.

Hochberg, M.

Horowitz, M.

A. Rosenthal and M. Horowitz, “Inverse scattering algorithm for reconstructing strongly reflecting fiber Bragg gratings,” IEEE J. Quantum Electron.39(8), 1018–1026 (2003).
[CrossRef]

Jaeger, N. A. F.

Janz, S.

Kimerling, L. C.

Ladouceur, F.

F. Ladouceur, J. D. Love, and T. J. Senden, “Measurement of surface roughness in buried channel waveguides,” Electron. Lett.28(14), 1321–1322 (1992).
[CrossRef]

Lamontagne, B.

Lapointe, J.

LaRochelle, S.

Lee, K. K.

K. K. Lee, D. R. Lim, L. C. Kimerling, J. Shin, and F. Cerrina, “Fabrication of ultralow-loss Si/SiO2 waveguides by roughness reduction,” Opt. Lett.26(23), 1888–1890 (2001).
[CrossRef] [PubMed]

K. K. Lee, D. R. Lim, H.-C. Luan, A. Agarwal, J. Foresi, and L. C. Kimerling, “Effect of size and roughness on light transmission in a Si/SiO2 waveguide: Experiments and model,” Appl. Phys. Lett.77(11), 1617 (2000).
[CrossRef]

Lee, P.

Lim, A. E.-J.

Lim, D. R.

K. K. Lee, D. R. Lim, L. C. Kimerling, J. Shin, and F. Cerrina, “Fabrication of ultralow-loss Si/SiO2 waveguides by roughness reduction,” Opt. Lett.26(23), 1888–1890 (2001).
[CrossRef] [PubMed]

K. K. Lee, D. R. Lim, H.-C. Luan, A. Agarwal, J. Foresi, and L. C. Kimerling, “Effect of size and roughness on light transmission in a Si/SiO2 waveguide: Experiments and model,” Appl. Phys. Lett.77(11), 1617 (2000).
[CrossRef]

Liow, T.-Y.

Liu, Y.

Lo, G.-Q.

Love, J. D.

F. Ladouceur, J. D. Love, and T. J. Senden, “Measurement of surface roughness in buried channel waveguides,” Electron. Lett.28(14), 1321–1322 (1992).
[CrossRef]

Luan, H.-C.

K. K. Lee, D. R. Lim, H.-C. Luan, A. Agarwal, J. Foresi, and L. C. Kimerling, “Effect of size and roughness on light transmission in a Si/SiO2 waveguide: Experiments and model,” Appl. Phys. Lett.77(11), 1617 (2000).
[CrossRef]

Martinelli, M.

F. Morichetti, A. Canciamilla, C. Ferrari, M. Torregiani, A. Melloni, and M. Martinelli, “Roughness Induced Backscattering in Optical Silicon Waveguides,” Phys. Rev. Lett.104(3), 033902 (2010).
[CrossRef] [PubMed]

Melloni, A.

F. Morichetti, A. Canciamilla, C. Ferrari, M. Torregiani, A. Melloni, and M. Martinelli, “Roughness Induced Backscattering in Optical Silicon Waveguides,” Phys. Rev. Lett.104(3), 033902 (2010).
[CrossRef] [PubMed]

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M. A. Schneider and S. Mookherjea, “Modeling Transmission Time of Silicon Nanophotonic Waveguides,” IEEE Photon. Technol. Lett.24(16), 1418–1420 (2012).
[CrossRef]

Morichetti, F.

F. Morichetti, A. Canciamilla, C. Ferrari, M. Torregiani, A. Melloni, and M. Martinelli, “Roughness Induced Backscattering in Optical Silicon Waveguides,” Phys. Rev. Lett.104(3), 033902 (2010).
[CrossRef] [PubMed]

Morson, R.

S. Sardo, F. Giacometti, S. Doneda, U. Colombo, M. D. Muri, A. Donghi, R. Morson, G. Mutinati, A. Nottola, M. Gentili, and M. C. Ubaldi, “Line edge roughness (LER) reduction strategy for SOI waveguides fabrication,” Microelectron. Eng.85(5–6), 1210–1213 (2008).
[CrossRef]

Muri, M. D.

S. Sardo, F. Giacometti, S. Doneda, U. Colombo, M. D. Muri, A. Donghi, R. Morson, G. Mutinati, A. Nottola, M. Gentili, and M. C. Ubaldi, “Line edge roughness (LER) reduction strategy for SOI waveguides fabrication,” Microelectron. Eng.85(5–6), 1210–1213 (2008).
[CrossRef]

Mutinati, G.

S. Sardo, F. Giacometti, S. Doneda, U. Colombo, M. D. Muri, A. Donghi, R. Morson, G. Mutinati, A. Nottola, M. Gentili, and M. C. Ubaldi, “Line edge roughness (LER) reduction strategy for SOI waveguides fabrication,” Microelectron. Eng.85(5–6), 1210–1213 (2008).
[CrossRef]

Nottola, A.

S. Sardo, F. Giacometti, S. Doneda, U. Colombo, M. D. Muri, A. Donghi, R. Morson, G. Mutinati, A. Nottola, M. Gentili, and M. C. Ubaldi, “Line edge roughness (LER) reduction strategy for SOI waveguides fabrication,” Microelectron. Eng.85(5–6), 1210–1213 (2008).
[CrossRef]

Painchaud, Y.

Patsis, G. P.

V. Constantoudis, G. P. Patsis, A. Tserepi, and E. Gogolides, “Quantification of line-edge roughness of photoresists. II. Scaling and fractal analysis and the best roughness descriptors,” J. Vac. Sci. Technol. B21(3), 1019–1026 (2003).
[CrossRef]

G. P. Patsis, V. Constantoudis, A. Tserepi, E. Gogolides, and G. Grozev, “Quantification of line-edge roughness of photoresists. I. A comparison between off-line and on-line analysis of top-down scanning electron microscopy images,” J. Vac. Sci. Technol. B21(3), 1008–1018 (2003).
[CrossRef]

Pinguet, T.

Pope, S. B.

V. Eswaran and S. B. Pope, “Direct numerical simulations of the turbulent mixing of a passive scalar,” Phys. Fluids31(3), 506–520 (1988).
[CrossRef]

Rosenthal, A.

A. Rosenthal and M. Horowitz, “Inverse scattering algorithm for reconstructing strongly reflecting fiber Bragg gratings,” IEEE J. Quantum Electron.39(8), 1018–1026 (2003).
[CrossRef]

Sardo, S.

S. Sardo, F. Giacometti, S. Doneda, U. Colombo, M. D. Muri, A. Donghi, R. Morson, G. Mutinati, A. Nottola, M. Gentili, and M. C. Ubaldi, “Line edge roughness (LER) reduction strategy for SOI waveguides fabrication,” Microelectron. Eng.85(5–6), 1210–1213 (2008).
[CrossRef]

Schmid, J. H.

Schneider, M. A.

M. A. Schneider and S. Mookherjea, “Modeling Transmission Time of Silicon Nanophotonic Waveguides,” IEEE Photon. Technol. Lett.24(16), 1418–1420 (2012).
[CrossRef]

Senden, T. J.

F. Ladouceur, J. D. Love, and T. J. Senden, “Measurement of surface roughness in buried channel waveguides,” Electron. Lett.28(14), 1321–1322 (1992).
[CrossRef]

Shi, W.

Shin, J.

Simard, A. D.

Smith, H. I.

T. Barwicz and H. I. Smith, “Evolution of line-edge roughness during fabrication of high-index-contrast microphotonic devices,” J. Vac. Sci. Technol. B21(6), 2892–2896 (2003).
[CrossRef]

Sparacin, D. K.

Spector, S. J.

Streshinsky, M.

Syrett, B. A.

Teo, S. H.-G.

Torregiani, M.

F. Morichetti, A. Canciamilla, C. Ferrari, M. Torregiani, A. Melloni, and M. Martinelli, “Roughness Induced Backscattering in Optical Silicon Waveguides,” Phys. Rev. Lett.104(3), 033902 (2010).
[CrossRef] [PubMed]

Tserepi, A.

V. Constantoudis, G. P. Patsis, A. Tserepi, and E. Gogolides, “Quantification of line-edge roughness of photoresists. II. Scaling and fractal analysis and the best roughness descriptors,” J. Vac. Sci. Technol. B21(3), 1019–1026 (2003).
[CrossRef]

G. P. Patsis, V. Constantoudis, A. Tserepi, E. Gogolides, and G. Grozev, “Quantification of line-edge roughness of photoresists. I. A comparison between off-line and on-line analysis of top-down scanning electron microscopy images,” J. Vac. Sci. Technol. B21(3), 1008–1018 (2003).
[CrossRef]

Ubaldi, M. C.

S. Sardo, F. Giacometti, S. Doneda, U. Colombo, M. D. Muri, A. Donghi, R. Morson, G. Mutinati, A. Nottola, M. Gentili, and M. C. Ubaldi, “Line edge roughness (LER) reduction strategy for SOI waveguides fabrication,” Microelectron. Eng.85(5–6), 1210–1213 (2008).
[CrossRef]

Waldron, P.

Wang, X.

Yap, K. P.

Yun, H.

Zhang, Y.

Appl. Phys. Lett. (1)

K. K. Lee, D. R. Lim, H.-C. Luan, A. Agarwal, J. Foresi, and L. C. Kimerling, “Effect of size and roughness on light transmission in a Si/SiO2 waveguide: Experiments and model,” Appl. Phys. Lett.77(11), 1617 (2000).
[CrossRef]

Electron. Lett. (1)

F. Ladouceur, J. D. Love, and T. J. Senden, “Measurement of surface roughness in buried channel waveguides,” Electron. Lett.28(14), 1321–1322 (1992).
[CrossRef]

IEEE J. Quantum Electron. (1)

A. Rosenthal and M. Horowitz, “Inverse scattering algorithm for reconstructing strongly reflecting fiber Bragg gratings,” IEEE J. Quantum Electron.39(8), 1018–1026 (2003).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

A. D. Simard, N. Belhadj, Y. Painchaud, and S. LaRochelle, “Apodized Silicon-on-Insulator Bragg Gratings,” IEEE Photon. Technol. Lett.24(12), 1033–1035 (2012).
[CrossRef]

M. A. Schneider and S. Mookherjea, “Modeling Transmission Time of Silicon Nanophotonic Waveguides,” IEEE Photon. Technol. Lett.24(16), 1418–1420 (2012).
[CrossRef]

J. Lightwave Technol. (4)

J. Vac. Sci. Technol. B (3)

T. Barwicz and H. I. Smith, “Evolution of line-edge roughness during fabrication of high-index-contrast microphotonic devices,” J. Vac. Sci. Technol. B21(6), 2892–2896 (2003).
[CrossRef]

G. P. Patsis, V. Constantoudis, A. Tserepi, E. Gogolides, and G. Grozev, “Quantification of line-edge roughness of photoresists. I. A comparison between off-line and on-line analysis of top-down scanning electron microscopy images,” J. Vac. Sci. Technol. B21(3), 1008–1018 (2003).
[CrossRef]

V. Constantoudis, G. P. Patsis, A. Tserepi, and E. Gogolides, “Quantification of line-edge roughness of photoresists. II. Scaling and fractal analysis and the best roughness descriptors,” J. Vac. Sci. Technol. B21(3), 1019–1026 (2003).
[CrossRef]

Microelectron. Eng. (1)

S. Sardo, F. Giacometti, S. Doneda, U. Colombo, M. D. Muri, A. Donghi, R. Morson, G. Mutinati, A. Nottola, M. Gentili, and M. C. Ubaldi, “Line edge roughness (LER) reduction strategy for SOI waveguides fabrication,” Microelectron. Eng.85(5–6), 1210–1213 (2008).
[CrossRef]

Opt. Express (3)

Opt. Lett. (1)

Phys. Fluids (1)

V. Eswaran and S. B. Pope, “Direct numerical simulations of the turbulent mixing of a passive scalar,” Phys. Fluids31(3), 506–520 (1988).
[CrossRef]

Phys. Rev. Lett. (1)

F. Morichetti, A. Canciamilla, C. Ferrari, M. Torregiani, A. Melloni, and M. Martinelli, “Roughness Induced Backscattering in Optical Silicon Waveguides,” Phys. Rev. Lett.104(3), 033902 (2010).
[CrossRef] [PubMed]

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P. L. O’Neill, D. Nicolaides, D. Honnery, and J. Soria, “Autocorrelation functions and the determination of integral length with reference to experimental and numerical data,” in 15th Australasian fluid mechanics conference, Sydney, Australia (2004).

S. Spector, M. W. Geis, D. Lennon, R. C. Williamson, and T. M. Lyszczarz, “Hybrid multi-mode/single-mode waveguides for low loss,” in Integrated Photonics Research, Optical Society of America (2004).

X. Wang, W. Shi, M. Hochberg, K. Adam, E. Schelew, J. F. Young, N. A. F. Jaeger, and L. Chrostowski, “Lithography simulation for the fabrication of silicon photonic devices with deep-ultraviolet lithography,” in 9th International Conference on Group IV Photonics (GFP), 288 –290 (2012).
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A. D. Simard, Y. Painchaud, and S. LaRochelle, “Characterization of Integrated Bragg Grating Profiles,” in Conference Proceedings on Bragg Gratings, Photosensitivity, and Poling in Glass Waveguides, paper BM3D.7 (2012).

S. K. Selvaraja, E. Rosseel, L. Fernandez, M. Tabat, W. Bogaerts, J. Hautala, and P. Absil, “SOI thickness uniformity improvement using corrective etching for silicon nano-photonic device,” in in 8th IEEE International Conference on Group IV Photonics London, United Kingdom, 71–73 (2011).
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A. D. Simard, Y. Painchaud, and S. LaRochelle, “Integrated Bragg Gratings in Curved Waveguides,” in the 23rd Annual Meeting of the Photonics Society Denver, USA, ThU3 (2010).
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K. P. Yap, A. Delâge, B. Lamontagne, S. Janz, D.-X. Xu, J. Lapointe, P. Waldron, J. Schmid, P. Chow-Chong, E. Post, and B. Syrett, “Scattering loss measurement of SOI waveguides using 5X17 integrated optical star coupler,” in Conference Proceedings on Integrated Optoelectronic Devices, International Society for Optics and Photonics, 64770J (2007).

W. A. Zortman, M. R. Watts, and D. C. Trotter, “Determination of Wafer and Process Induced Resonant Frequency Variation in Silicon Microdisk-Resonators,” in Conference Proceedings on Integrated Photonics and Nanophotonics Research and Applications, paper IMC5 (2009).

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Figures (8)

Fig. 1
Fig. 1

(a) Typical top-down SEM image of a photonic wire. (b) Typical intensity profile of a pixel column (taken at the position of the red line in (a)). (c) Typical top-down SEM image converted into a binary image with the threshold shown by the black line in (b). The red lines are the outer waveguide sidewalls. (d) Typical superposition of the waveguide width fluctuation from every image of a set of measurement. The thick blue line is the linear regression of the average waveguide width fluctuation per pixel (black curve).

Fig. 2
Fig. 2

(a) Waveguide width fluctuation obtained from top-down SEM taken at the same location with different working distance. (b) Dependency of the average waveguide width caused by a variation of the working distance.

Fig. 3
Fig. 3

(a) Typical overlap of the waveguide width fluctuation between three consecutive images (red, blue and green). The black curve is the averaged waveguide width as a function of position. (b) Typical waveguide width fluctuation profile as a function of position. The black line is the raw data (corresponding to the black line of (a)) while the blue line is a filtered version of the waveguide width profile. (c) Autocorrelation function of the waveguide width fluctuation of the filtered (blue) and unfiltered (black) curve of (b). The red curve is the decaying exponential autocorrelation fit. (d) Probability density functions of the sidewall roughness. The red curve is a Gaussian function.

Fig. 4
Fig. 4

Schematic of the IBGs.

Fig. 5
Fig. 5

(a) Comparison of the experimental reflection spectrum of a typical straight grating on a 1200 nm wide waveguide (in red) with the reconstructed reflection spectrum (in black). Retrieved (b) λB and (c) Δn profiles, which are used to calculate the black curve of (a).

Fig. 6
Fig. 6

(a) Bragg wavelength standard deviation as a function of the waveguide width. The black curve contains both the impact of the sidewall roughness and the wafer height fluctuations, while the blue and red curve takes those to effect independently. The cyan lines are the optical measurement of 2 mm-long IBGs while the purple line is the optical measurement for spiral IBGs. (b) Schematic of spiral IBGs.

Fig. 7
Fig. 7

Typical spectral response of 2 mm-long (a) straight grating on a 800 nm wide waveguide, (b) straight grating on a 1200 nm wide waveguide and (c) spiral grating on a 1200 nm wide waveguide. The blue curves are the designs while the black curves are the experimental results.

Fig. 8
Fig. 8

Superposition of every Bragg wavelength fluctuation measurements for (a) 800 nm straight waveguide, (b) 1200 nm straight waveguide and (c) 1200 nm spiral waveguide.

Equations (11)

Equations on this page are rendered with MathJax. Learn more.

R SWR ( Δz )= lim L 1 L L/2 L/2 Δx(z)Δx(z+Δz)dz R WHF ( Δz )= lim L 1 L L/2 L/2 Δy(z)Δy(z+Δz)dz
R SWR ( Δz )= σ SWR 2 exp( | Δz | L c,SWR ) R WHF ( Δz )= σ WHF 2 exp( | Δz | L c,WHF )
G SWR ( f z )= 2 σ SWR 2 L c,SWR 1+4 π 2 L c,SWR 2 f z 2 G WHF ( f z )= 2 σ WHF 2 L c,WHF 1+4 π 2 L c,WHF 2 f z 2
σ λ B =2Λ σ n
σ n = 2 C SWR 2 σ SWR 2 + C WHF 2 σ WHF 2
f c = 2 n g λ B ( λ B Δλ 1 ) 2 n g Δλ λ B 2 ,
σ ˜ 2 = f c f c G Δx ( f z )d f z = 4 σ 2 L c f c 4 π 2 L c 2 f c 2 +1 .
σ ˜ SWR 2 8 n g L c Δλ λ B 2 σ SWR 2 σ ˜ WHF 2 8 n g L c Δλ λ B 2 σ WHF 2
σ ˜ λ B =2Λ σ ˜ n =2Λ 2 C SWR 2 σ ˜ SWR 2 + C WHF 2 σ ˜ WHF 2
σ ˜ λ B = 8Λ n g Δλ λ B ( C SWR 2 L c,SWR σ SWR 2 + 1 2 C WHF 2 L c,WHF σ WHF 2 )
σ ˜ λ B,WHF 8Λ C WHF σ WHF λ B n g Δλ L c,WHF 2

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