1. Introduction

Integrated Bragg gratings (IBGs) have become critical components for many optical applications such as optical communications [1–3], large-scale quantum photonic systems [4, 5], sensing [6, 7], etc. Recently, considerable efforts have been devoted to develop IBGs on silicon-on-insulator (SOI) platforms due to the compatibility with standard microelectronics fabrication processes and the technological advances in silicon photonics [8, 9]. Silicon IBGs have many advantages such as high extinction ratio, easy fabrication and low insertion loss. But the most distinguishing one is the high flexibility in their spectral responses, which can be freely tailored to offer different spectral functionalities [10, 11]. This feature is particularly appealing for applications where flexible control of spectral responses of the devices is of interest, such as optical signal processing [12, 13] and wavelength-division multiplexing (WDM) systems [14–16]. Spectral tailoring of Bragg gratings is essentially achieved by modulating coupling coefficient, κ, along the grating according to that required by the designed response, which is called apodization [17]. Thus, the apodization performance is of the utmost importance for spectral tailoring of IBGs.

Reliable and accurate apodization for silicon IBGs, however, still remains challenging compared with that in fiber Bragg gratings, with two main issues. The first one is that the control of κ in silicon IBGs usually suffers from a relatively lowprecision/resolution and dynamic range. This is because the modulation of κ in silicon IBGs is generally realized by modifying the physical waveguide grating structure, such as the corrugation width [18]. Due to the small dimension of the single-mode silicon strip waveguide, the resolution and precision of the κ control in practice are limited by the fabrication constraints, such as lithography resolution [19]. Also, the high index contrast of SOI waveguide places an inherent difficulty to achieve weak κ under fabrication limitations [20], which constrains the apodization dynamic range. Fortunately, the issue can be considerably alleviated by using different advanced apodization methods, such as duty-cycle [21, 22] and lateral misalignment [23, 24] modulation schemes. For example, it is suggested that the lateral misalignment modulation scheme can lead to an order of magnitude increase in the apodization resolution and precision compared with the conventional corrugation width modulation method [24].

The second apodization issue for silicon IBGs, which has received much less attention, is that the physical waveguide structure modifications for controlling κ in silicon IBGs can also introduce unwanted phase variations or noise in the gratings for some particular apodization schemes, including lateral misalignment and grating duty-cycle modulations [12]. Such apodization phase noise (APN) mainly comes from two sources. The first and primary one is the average effective refractive index variations along the waveguide grating due to the waveguide structure modifications [18, 22, 25]. These index changes will vary the local effective period of the grating and thus lead to phase variations. Also, unwanted phase changes can arise from feature position modulations involved in some apodization schemes for silicon IBGs, such as lateral misalignment modulations and similar methods with grating corrugation shifts [14, 24, 26]. This can be seen by noting that variations of the feature displacements with respect to the period centers can introduce phase shifts in the grating [11, 16]. APN due to these two factors can actually lead to significant distortions of the complex response of an apodized IBG, as will be shown in this paper. However, as yet, there has been little study to characterize the impact of APN on the responses of apodized IBGs, and to eliminate the APN to correct the distorted grating responses to be consistent with the designs. APN is difficult to be included inconventional coupled-mode theory (CMT) based modeling methods, where the modeled grating is identified by its coupling coefficient profile only while the physical grating structure is ignored [17].

In this work, we propose a powerful modeling tool for apodized silicon IBGs that can take into account the APN, which is performed by directly synthesizing the physical structure of the grating with the assistance of the transfer matrix method (TMM). By using this structure synthesis-based TMM (SS-TMM), we model a series of different silicon IBGs apodized by modulating lateral misalignment ($\mathrm{\Delta }L$) and duty-cycle (DC) and designed with different responses. The modeling results show that the responses of the apodized gratings due to the APN can be significantly distorted from the designs, especially when an elaborate spectral response is designed. Then, to address this issue, we propose a methodology to compensate/eliminate the APN of an apodized IBG to correct the distorted grating response. The original and APN-compensated silicon IBGs designed were fabricated and measured experimentally. The accuracy of the SS-TMM is examined by comparing the measured grating spectra with those predicted by the SS-TMM, with good agreement obtained for each case. Then, spectral corrections are demonstrated in Gaussian-apodized gratings based on $\mathrm{\Delta }L$- and DC-modulated IBGs and a square-shaped filter based on a $\mathrm{\Delta }L$-modulated IBG. Finally, we achieve a complex (amplitude & phase) spectral correction of a photonic Hilbert transformer developed on a $\mathrm{\Delta }L$-modulated IBG.

2. SS-TMM for spectral modeling of apodized silicon IBGs

The transfer matrix method (TMM) is a powerful tool for the characterization of nonuniform Bragg gratings with arbitrary apodization profiles [27]. The traditional TMM for Bragg gratings is based on CMT, in which the standard transfer matrix solution is derived from the CMT equations [17]. However, in the case of silicon IBGs where the apodization is typically achieved via grating structure modifications, the CMT-TMM, as we shall see later, will no longer be accurate. This is because, as mentioned before, in the field propagation calculation of CMT, each grating section is identified by its coupling coefficient while the physical structure of the grating is ignored. Thus, the model can only provide the ideal spectral response produced by a coupling coefficient profile. But it cannot take into account the structure modification-induced unexpected phase variations, which however can considerably impact the spectral response of the apodized silicon IBG.

Here, we propose the SS-TMM model for apodized IBGs which is operated by directly synthesizing the physical waveguide grating structures. This SS-TMM model, owing to its structure synthesis operation, can take into account any structure modification-induced phase variations, and thus is more accurate than conventional CMT-based modeling methods for apodized silicon IBGs. The proposed SS-TMM in the current work is used for characterizing the APN-impacted spectral responses of apodized IBGs to allow the APN distribution to be extracted for the subsequent APN compensation. However, it will also be useful for investigating impacts of different physical structure variations on the IBG spectra, such as fabrication imperfections in lithography process [28].

 

Fig. 1 (a) Schematic flow showing the process of the SS-TMM modeling. (b) The upper figure plots the normalized Gaussian apodization profile (blue, left axis) and the translated lateral misalignment-to-period ratio $\mathrm{\Delta }L/\mathrm{\Lambda }$ (red, right axis) along the grating; the bottom diagrams illustrate the grating structures at the different positions (1-3), whose locations are indicated in the upper plot. (c) Schematic illustrations of spatial sampling of the cell structure of a lateral misalignment-modulated IBG in different cases of (i) rectangular and (ii) sinusoidal grating shapes, where the bottom figures plot the corresponding $\mathrm{\Delta }W\left(k\right)$ profiles; $\mathrm{\Delta }W\left(k\right)$ is defined as the width variation of the kth segment from the unperturbed waveguide width [denoted as W in Fig. 1(c)]. (d) Illustration of the transfer matrices describing the wave propagation through an interface (left) and through a uniform section (right); the upper diagrams show the original grating structures while the bottom ones illustrate the equivalent multiplayer structures used in the SS-TMM modeling.

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2.1. Principle and implementation of the SS-TMM

We illustrate the implementation of the SS-TMM by modeling a silicon IBG apodized by modulating the lateral misalignment [represented as $\mathrm{\Delta }L$ in the bottom diagram of Fig. 1(b)]. The modulation of $\mathrm{\Delta }L$ can vary the interference between the two sides of the gratings, thereby controlling the coupling coefficient of the IBG [23]. The IBG is developed on a single-mode SOI strip waveguide with a cross section of 500 × 220 nm, and is designed for the fundamental quasi-TE mode. The grating period Λ is 316 nm, corresponding to a Bragg resonance of around 1550 nm. A Gaussian apodization profile [blue in the upper plot of Fig. 1(b)] is applied to the grating to minimize the spectral side-lobes. The total period number is 800, which translates to a grating length of ∼0.25 mm. The bottom schematic diagrams of Fig. 1(b) illustrate the $\mathrm{\Delta }L$-modulated grating structures at the different positions (1-3), whose locations are indicated in the upper plot of Fig. 1(b). The APN here can be expected to come from both the feature position shifts and the differences in the average effective refractive index for the different grating periods with different $\mathrm{\Delta }L$. Figure 1(a) is a schematic flow showing the process of SS-TMM modeling. The first step is to determine the whole IBG physical structure from the grating coupling coefficient and phase profiles, which are $\kappa \left(z\right)$ and ${\phi }_G\left(z\right)$ respectively. Note that there is no phase modulation required by the current Gaussian-apodized IBG, i.e., ${\phi }_G\left(z\right)=0$. Phase modulations of gratings are usually required when elaborate spectral responses are designed, as we shall see later. Several fundamental grating physical parameters are needed to be decided first, including the corrugation width [denoted as $\mathrm{\Delta }{W}_C$ in Fig. 1(c)], the grating period Λ (which has been chosen to be 316 nm), and the corrugation shape (rectangular or sinusoidal). Then, the whole grating structure can be mapped from the normalized apodization profile, ${\kappa }_n\left(z\right)$, based on the rule of the chosen apodization scheme. Here, ${\kappa }_n\left(z\right)$ is translated to a lateral misalignment as a function of the grating period number, $\mathrm{\Delta }L\left(i\right)$, through the following relationship [23]

The calculated $\mathrm{\Delta }L\left(i\right)$ profile is plotted as the red curve in the upper plot of Fig. 1(b).

Once the entire grating structure is determined, the grating structure is spatially sampled into many short uniform segments, as schematically illustrated in Fig. 1(c). Then, the width variation of each grating segment from the unperturbed waveguide width [W in Fig. 1(c)] can be obtained, which is defined as $\mathrm{\Delta }W\left(k\right)$, where k is the index number of the grating segment. Figures 1(c)-i and 1(c)-ii schematically illustrate the discretized cell structures of $\mathrm{\Delta }L$-modulated IBGs with rectangular and sinusoidal corrugation shapes, respectively, while the bottom plots show the corresponding $\mathrm{\Delta }W\left(k\right)$profiles. The sampling interval can be regarded as the fabrication resolution (or the grid size of the lithography) of the IBG. Since the effective refractive index of the waveguide is related to its width, the $\mathrm{\Delta }W\left(k\right)$ profile can be translated to the effective refractive index changes along the grating segments, $\mathrm{\Delta }{n}_{eff}\left(k\right)$. To obtain the relationship between$\mathrm{\Delta }W$ and $\mathrm{\Delta }{n}_{eff}$ that can be used in our SS-TMM, eigenmode analysis is first used to calculate the real $\mathrm{\Delta }{n}_{eff}$-versus-$\mathrm{\Delta }W$ relationship of the waveguide, and the scale of the obtained curve is then reduced by a factor of ∼2.3. This $\mathrm{\Delta }{n}_{eff}$-versus-$\mathrm{\Delta }W$ curve with a reduced scale is finally used in our SS-TMM for translating the $\mathrm{\Delta }W\left(k\right)$ into $\mathrm{\Delta }{n}_{eff}\left(k\right)$ profile. The reason for scaling down the curve is because the effective refractive index variations due to different waveguide widths here are not totally equivalent to the refractive index changes in a multi-layer structure, and the actual mode coupling strength here is smaller than that in a real multi-layer structure [12]. The scaling factor of ∼2.3 used here has been estimated based on our previous experimental results of SOI-based IBGs fabricated in EBL. It is worth noting that the relationship between $\mathrm{\Delta }W$ and $\mathrm{\Delta }{n}_{eff}$ can also be directly obtained from experimental results [8].

Once the $\mathrm{\Delta }{n}_{eff}\left(k\right)$ profile is obtained, the discretized IBG can be treated as an optical multilayer structure which consists of a stack of layers with different refractive indices. The spectral characterization of the IBG then can be regarded as the general problem of the wave propagation in a multi-layer structure [29]. Thus, the TMM used in optical multilayer structures now can be borrowed for characterizing the current discretized IBG, which will be elaborated on in the following.

The transfer matrix that relates the complex amplitudes of forward and backward propagating waves (defined as A and B respectively) is defined as

Two basic individual transfer matrices can be built: one is for wave propagation through an interface between neighboring grating segments [$T_{k\left(k+1\right)}$ in left of Fig. 1(d)], and another one is for wave propagation through a uniform grating segment [T_k in right of Fig. 1(d)]. These two basic transfer matrices can be derived using Fresnel equations to be [29]

The total transfer matrix for the wave propagation through the whole discretized IBG can be obtained by the multiplication of all individual transfer matrices.

This complete transfer matrix relates the incident and reflected waves at the input port with the incident and reflected waves at the output port for the grating. Therefore, it can be used to extract the grating reflection and transmission coefficients, defined as r and t, respectively, which will be [29]

By repeating the above TMM calculation for each wavelength, the reflection and transmission spectrum of the apodized IBG can be finally obtained. It is important to note that the dependence of the effective refractive index on wavelength, i.e., the waveguide dispersion, must be considered when repeating the calculation at different wavelengths.

 

Fig. 2 Comparison of the implementation process in conventional CMT-TMM and SS-TMM.

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Now, we can highlight the difference between CMT-TMM and SS-TMM. Figure 2 compares the implementation processes of the two modeling methods. Both of them start from using the target grating profiles, $\kappa \left(z\right)$ and ${\phi }_G\left(z\right)$. For CMT-TMM, it will first discretize the grating profiles that is necessary for the subsequent TMM calculation. Then, the transfer matrix calculation is performed directly based on the discretized grating profiles, where the basic transfer matrix of the mth profile segment is based on the general transfer solution of the coupled equations [30]:

It should be noted that although in principle it would be possible to correct the grating profiles to include the APN in the CMT-TMM modeling without a grating structure construction, such a correction would require additional empirical models for each different apodization method. For example, to include the APN coming from the average effective refractive index variation ($\mathrm{\Delta }{n}_{av}$), one needs to build an empirical relationship between $\mathrm{\Delta }{n}_{av}$ [or the Bragg wavelength shift] and the apodization physical parameters, which are $\mathrm{\Delta }L$ and the duty-cycle for the lateral misalignment and grating duty-cycle modulation methods [12], respectively. This calibration work is time-consuming and challenging in practice. Further, even if the $\mathrm{\Delta }{n}_{av}$ distribution along an apodized IBG is known and included in the grating profiles, the CMT-TMM modeling still does not take into account the second source of APN, namely the unexpected grating phase shifts arising from feature position changes. For the proposed SS-TMM, however, all the APN can be naturally taken into account by directly synthesizing the constructed grating structure without any additional calibration work, thus offering a more reliable and simpler modeling tool for various apodized IBGs. Moreover, the SS-TMM as a structure-aware modeling method also has useful applications that cannot be achieved in CMT-based modeling methods, such as investigating the impact of the fabrication resolution and the corrugation shape on the grating response, as we shall see later.

 

Fig. 3 (a) Calibrated model for $\mathrm{\Delta }{n}_{eff}$-versus-$\mathrm{\Delta }W$ used in the SS-TMM for IBGs developed on 500 × 220 nm SOI strip waveguides. (b) Reflection and transmission spectra of the $\mathrm{\Delta }L$-modulated Gaussian-apodized IBG calculated by the SS-TMM using a sampling interval of 6 nm. (c) Comparison of the calculated reflection spectra of the IBG using different sampling intervals. (d) Comparison of the calculated reflection spectra of the IBGs with different grating shapes of rectangular and sinusoidal under the same sampling interval of 6 nm. The black curves in (c)-(d) represent the ideal reflection spectrum calculated via the CMT-TMM.

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2.2. Spectral modeling of different apodized IBGs through the SS-TMM

In this section, we will perform spectral modeling of several different apodized IBGs to examine the SS-TMM model, and more importantly, to investigate the impact of APN on apodized IBG spectral responses. The $\mathrm{\Delta }L$-modulated Gaussian-apodized IBG designed above is first characterized. The grating shape is rectangular, and the corrugation width is 8 nm. We use Matlab to generate the vertices of the apodized IBG structure and perform the waveguide sampling to obtain the $\mathrm{\Delta }W\left(k\right)$ profile. Then, $\mathrm{\Delta }W\left(k\right)$ is converted to $\mathrm{\Delta }{n}_{eff}\left(k\right)$ based on the calibrated model for $\mathrm{\Delta }{n}_{eff}$-versus-$\mathrm{\Delta }W$ plotted in Fig. 3(a), which, as described before, is obtained from eigen mode analysis with the scale of the curve decreased by a factor of ∼2.3. Figure 3(b) shows the reflection and transmission spectra of the $\mathrm{\Delta }L$-modulated Gaussian-apodized IBG calculated by the SS-TMM using a sampling interval of 6 nm. Figure 3(c) compares the calculated reflection spectra of the IBG using different sampling intervals of 2 nm and 6 nm, where the ideal reflection spectrum obtained from the CMT-based TMM is also included for comparison. The grating spectra predicted by the SS-TMM using different sampling intervals are significantly broader than the ideal one. Such a spectral broadening is caused by the APN coming from both the average effective refractive index changes along the grating period due to the spatially varying $\mathrm{\Delta }L$ and the corrugation position shifts, leading to a chirp-like effect along the grating. One can also find that the use of different sampling intervals, which can be considered as different fabrication resolutions of the gratings, also results in small differences in the spectral results. The calculated spectrum for the larger sampling interval of 6 nm has higher ripple on the two sides of the spectrum than that with the smaller sampling interval of 2 nm. This indicates that a high fabrication resolution would decrease the spectral ripples of the IBG. Note that as the EBL writing resolution used in the fabrication will be 6 nm, the sampling intervals have been chosen to be 6 nm for all the SS-TMM calculation results shown in the below. Figure 3(d) compares the calculated reflection spectra of the Gaussian-apodized IBGs with rectangular and sinusoidal grating shapes under the same sampling interval of 6 nm. The spectrum of the sinusoidal IBG shows lower ripple compared with that of the rectangular one. This should be due to that sinusoidal grating profiles are less subject to the discretization issue than rectangular grating profiles. Note that a sinusoidal grating shape will also lead to a smaller κ and thus a lower grating reflectivity than a rectangular grating shape under the same corrugation width [8], as also indicated in Fig. 3(d).

 

Fig. 4 (a) Schematic illustration of the DC modulation in an IBG (left) and the relationship between the DC and grating strength (right). (b) Normalized apodization profile (blue, left axis) and the converted DC distribution along the grating (red, right axis). (c) Spectrum of the DC-modulated Gaussian-apodized IBG calculated by the SS-TMM (blue), and the ideal spectrum (black).

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Next, we use the SS-TMM model to characterize an IBG apodized via the duty-cycle (DC) modulation [22]. This apodization technique uses the coupling strength dependence on the DC, defined as the ratio of the grating groove width to the grating period, as schematically illustrated in Fig. 4(a). The relationship between the grating strength and the duty-cycle follows a sinusoidal function

For this apodization technique, the major source of APN can be expected to be the average effective refractive index variations between different grating periods with different DCs. The same Gaussian apodization profile as the previous IBG is used and is plotted in Fig. 4(b) (blue, left axis), where the DC distribution obtained based on Eq. (8) is also shown (red, right axis). The fundamental grating parameters, including the grating shape, Λ and $\mathrm{\Delta }{W}_C$ are also the same as those of the previous $\mathrm{\Delta }L$-modulated IBG. Figure 4(c) shows the SS-TMM modeling result and the ideal spectrum. Due to the effective refractive index variations arising from the spatially varying DC, the spectrum of the apodized IBG is considerably broadened and distorted from the ideal one.

 

Fig. 5 (a) Spectrum of the designed square filter, which has been modified to be physically realizable. (b) Grating coupling coefficient κ (upper) and phase φ_G (bottom) profiles required by the designed spectral response, calculated via LPA. The blue curves in (c) and (d) are the SS-TMM predicted spectra of the $\mathrm{\Delta }L$- and DC-modulated IBGs, respectively, where the ideal spectrum (black) calculated by the CMT-TMM is also included in each figure for comparison.

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Having characterized the Gaussian-apodized IBGs above, we now use the SS-TMM to model more complicated IBGs designed with elaborate spectral responses. We design a square filter on both $\mathrm{\Delta }L$- and DC-modulated silicon IBGs. The designed square filtering response is shown in Fig. 5(a), which has a flattop bandwidth of ∼4 nm. Note that the spectrum in Fig. 5(a) has been modified to be physically realizable, by truncating and time-shifting its temporal response to make it finite and casual. The layer peeling algorithm (LPA) [31] is then used to calculate the grating coupling coefficient profile, $\kappa \left(z\right)$, and phase profile, ${\phi }_G\left(z\right)$, required by the design. The obtained $\kappa \left(z\right)$ and ${\phi }_G\left(z\right)$ are shown as a function of the period number in Fig. 5(b). The total grating period number is 2486. The physical structures of the apodized IBGs are then created based on the $\kappa \left(z\right)$ and ${\phi }_G\left(z\right)$ profiles. The corrugation widths of the IBGs are 10 nm, which are decided according to the maximum value of the $\kappa \left(z\right)$ profile. The grating period is 316 nm, and the corrugation shapes are rectangular. The normalized $\kappa \left(z\right)$ profile is first converted to $\mathrm{\Delta }L$ and DC values as a function of the grating period number for the $\mathrm{\Delta }L$- and DC-modulated IBGs, respectively. The phase profile ${\phi }_G\left(z\right)$ is then applied into the gratings by inserting an additional waveguide section with a length of half the grating period in each π-phase shift position. By this way, the whole apodized IBG structures can be determined, and the SS-TMM can be performed on them to calculate their spectral responses. The spectral results obtained from the SS-TMM for the $\mathrm{\Delta }L$- and DC-modulated IBGs are shown in Figs. 5(c) and 5(d) respectively, with the ideal grating spectrum calculated via the CMT-TMM included in each figure for comparison. For both the two apodization techniques, the spectrum of the apodized IBG is largely broadened and distorted from the ideal one, indicating a failure of the spectral tailoring. This shows that the spectral distortion due to the APN can be even larger when an elaborate spectral response is designed.

 

Fig. 6 (a) Complex spectral response of the designed photonic Hilbert transformer, which has been modified to be physically realizable. (b) Grating coupling coefficient κ (upper) and phase φ_G (bottom) profiles required by the design. (c) Complex spectral response of the $\mathrm{\Delta }L$-modulated IBG calculated by the SS-TMM.

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Finally, we investigate the impact of APN on the complex (amplitude & phase) spectral response of an apodized IBG. We design and model a photonic Hilbert transformer based on a $\mathrm{\Delta }L$-modulated IBG for the investigation. A photonic Hilbert transformer is essentially a band pass filter with a discrete π-phase shift at the central wavelength in the phase response [24, 32]. The complex spectral response of the designed photonic Hilbert transformer is shown in Fig. 6(a), which has been modified to be physically realizable. The phase response has been wrapped into the range $\left(0,2\pi \right]$. The bandwidth of the transformer is ∼5.8 nm. A spectral notch exists at the central wavelength due to the π-phase jump [24]. The $\kappa \left(z\right)$ and ${\phi }_G\left(z\right)$ profiles required by the design are calculated by LPA and the results are shown in Fig. 6(b). The total grating period number is 3053. The whole physical structure of the $\mathrm{\Delta }L$-modulated IBG is created based on the $\kappa \left(z\right)$ and ${\phi }_G\left(z\right)$ profiles and the created grating structure is then modeled by the SS-TMM. The modeling result is shown in Fig. 6(c), where we can see that both the amplitude and phase responses are distorted due to the APN. The phase response is considerably deviated from the designed staircase-like behavior. Such a distorted complex spectral response may not provide a sufficient performance for photonic Hilbert transforming applications. These results show the considerable impact of APN on both the amplitude and phase responses of an apodized IBG.

 

Fig. 7 Design process of an IBG with the APN compensation included; the procedures enclosed by the red dashed line are for the extraction of the APN distribution. LPA: layer peeling algorithm [31].

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3. APN compensation and spectral correction

In this section, we describe our proposed methodology for the APN compensation and spectral correction of apodized IBGs. The whole design process of an IBG with the APN compensation included is shown in Fig. 7. The main concept of the methodology is to first extract the APN distribution along the apodized grating assisted by the SS-TMM, with the relevant procedures enclosed by the red dashed line in Fig. 7. Then, the IBG physical structure is modified accordingly to compensate the APN and thus to correct the spectral response. Next, we will illustrate how to perform the APN compensation for the previously designed square filter in Fig. 5 developed on a $\mathrm{\Delta }L$-modulated IBG.

3.1. APN distribution extraction

As mentioned before, we will first extract the APN distribution along the apodized grating. For this purpose, we create a temporary grating that is dedicated for the APN extraction. The structure of the temporary grating is determined based on only the ${\kappa }_n\left(z\right)$ profile required by the design while discarding the phase profile ${\phi }_G\left(z\right)$, i.e., setting the phase to be a constant along the grating. Such a temporary IBG will be modeled by the SS-TMM, and the obtained spectral result will be synthesized to retrieve the grating phase and thus the APN distribution. Note that the modeling of the temporary grating can also be performed by 3D finite-difference time-domain (3D-FDTD) calculations, which can be more accurate but also much more computationally intensive. The reason for discarding ${\phi }_G\left(z\right)$ when creating the temporary IBG is that the use of only ${\kappa }_n\left(z\right)$ is sufficient to extract the APN distribution from the spectral response. This is because the APN distribution along the grating is only related to the spatial distribution of the apodizaiton physical parameter [$\mathrm{\Delta }L\left(i\right)$ here] which is solely determined by the ${\kappa }_n\left(z\right)$ profile. Also, if ${\phi }_G\left(z\right)$ is imposed on the temporary grating, the grating phase profile synthesized from its spectral response will be superimposed by both the APN and ${\phi }_G\left(z\right)$, making it difficult for the APN extraction. Note that such a temporary grating with ${\phi }_G\left(z\right)$ set to be a constant is only used in the APN distribution calculation stage and will be discarded once the APN distribution is determined from its complex response. The IBG that is corrected and finally fabricated will be still apodized based on the original $\kappa \left(z\right)$ and ${\phi }_G\left(z\right)$ profiles required by the design. Also, note that in cases where there is no phase modulation required by the design, such as Gaussian-apodized gratings, the temporary grating will be equivalent to the designed grating.

 

Fig. 8 (a) Coupling coefficient (blue, left axis) and phase (red, right axis) profiles used for creating the temporary IBG structure that is dedicated for the APN extraction purpose. (b) Amplitude response of the temporary IBG calculated bythe SS-TMM. (c) Coupling coefficient (blue, left axis) and phase (red, right axis) profiles synthesized from the complex response of the temporary IBG using LPA. (d) Phase differences between neighboring periods.

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The amplitude response of the temporary grating calculated by the SS-TMM is shown in Fig. 8(b). Note that the wavelength range of the spectrum is as large as 70 nm to obtain a high spatial resolution of the spectral synthesis result using LPA [31]. As the SS-TMM model has taken the APN into account, it is possible to extract the APN distribution by synthesizing the calculated complex response of the temporary grating. Thus, we use LPA on the amplitude and phase (not shown) responses of the temporary grating to retrieve its coupling coefficient and phase (ϕ) profiles. One can expect that if there is no phase variation due to the apodization, the retrieved phase should be a constant along the grating as designed. However, as can be seen in Fig. 8(c), due to the existence of the APN, the retrieved grating phase now varies considerably along the grating. These unexpected phase variations along the grating thus represent the APN. To facilitate the following phase-to-structure translation, we calculate the phase difference between adjacent periods, i.e, $\mathrm{\Delta }\varphi \left(i\right)=\varphi \left(i\right)-\varphi \left(i-1\right)$. The obtained $\mathrm{\Delta }\varphi \left(i\right)$, shown in Fig. 8(d), will be regarded as the APN distribution and be used for the APN compensation process described next.

3.2. IBG structure corrections for the APN compensation

Now, with the APN distribution, $\mathrm{\Delta }\varphi \left(i\right)$, along the apodized IBG determined, we can introduce an additional phase profile along the IBG to compensate or eliminate the APN. The compensation phase as a function of the grating period number, $\mathrm{\Delta }{\varphi }_C\left(i\right)$, can be easily determined to be

The APN compensation task now becomes to translate this compensation phase profile into the modifications/corrections of the IBG physical structure. Such a phase difference profile along the grating can be interpreted by either of the two physical meanings. The first one is the first derivative of the grating phase, which corresponds to the effective spatial frequency of the grating index modulation profile, or the effective grating period. The second physical interpretation of $\mathrm{\Delta }{\varphi }_C\left(i\right)$ is discrete phase shifts between neighboring grating periods. These two interpretations will lead to different phase-to-structure translation schemes, which will be elaborated on in the following.

 

Fig. 9 Schematic illustrations of the correction of (a) grating period Λ and (b) distance between adjacent grating corrugations d according to different values of $\mathrm{\Delta }{\varphi }_C\left(i\right)$ in a uniform IBG.

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For the first interpretation, the compensation phase profile, $\mathrm{\Delta }{\varphi }_C\left(i\right)$, is considered as the variations of the effective grating periods, and thus should be converted to the corrections of the local periods. The period modification for the ith period, $\text{ΔΛ}\left(i\right)$, is related with $\mathrm{\Delta }{\varphi }_C\left(i\right)$ via

Figure 9(a) schematically illustrates the modification of the period according to different values of $\mathrm{\Delta }{\varphi }_C\left(i\right)$ in a uniform IBG.

In the second scheme, the phase difference profile is regarded as discrete phase shifts between adjacent grating periods. Such phase shifts will be translated to the corrugation position shifts, or equivalently, the variations of the distances between adjacent corrugations, denoted as $\mathrm{\Delta }d\left(i\right)$ for the distance modification between the $\left(i-1\right)$th and ith corrugations. $\mathrm{\Delta }d\left(i\right)$ is related to $\mathrm{\Delta }{\varphi }_C\left(i\right)$ via [11]

Figure 9(b) illustrates the modification of $d\left(i\right)$ in cases of different values of $\mathrm{\Delta }{\varphi }_C\left(i\right)$. Note that the original value of d is half the grating period (158 nm)

 

Fig. 10 (a) and (c) show the Λ and d correction profiles, respectively, for the designed square filter based on the $\mathrm{\Delta }L$-modulated IBG. (b) and (d) present the SS-TMM calculated spectra of the Λ- and d-corrected IBGs, respectively, with the ideal spectrum (black) included in each figure for comparison.

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Simulations are then performed to validate the APN compensation methodology. We correct the apodized IBG structure based on both the Λ and d correction schemes. Figures 10(a) and 10(c) show, respectively, the Λ and d correction profiles of the designed square grating filter, which are calculated from the compensation phase profile $\mathrm{\Delta }{\varphi }_C\left(i\right)$ using Eqs. (10) and (11), respectively. One may notice from Eqs. (10) and (11) and Figs. 10(a) and 10(c) that $\mathrm{\Delta }d\left(i\right)$ and $\text{ΔΛ}\left(i\right)$ profiles required by a certain $\mathrm{\Delta }{\varphi }_C\left(i\right)$ profile will be exactly the same. This will lead to the same total lengths of the IBGs corrected by the two different phase-to-structure mapping schemes, indicating the physical equivalence of the two schemes. The corrected IBG structures are then modeled by the SS-TMM to obtain their new spectral responses. Figures 10(b) and 10(d) shows the calculated spectral results of the Λ- and d-corrected IBGs, respectively, where the ideal grating spectrum (black) is also included for comparison. As can be seen, the spectral responses of both the two corrected IBGs now are in excellent agreement with the ideal one. This clearly shows that the APN has been nearly eliminated, validating our spectral correction methodology. Note that as period corrections are more straightforward and easier to implement in different apodization schemes, this correction scheme is chosen for the validation of our APN compensation methodology shown below.

 

Fig. 11 (a)-(c) are scanning electron microscope (SEM) images of the testing circuit of a grating and the structures of fabricated $\mathrm{\Delta }L$-modulated and DC-modulated silicon IBGs, respectively; BDC: broadband directional coupler [33].

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4. Experimental validation

To experimentally validate the SS-TMM model and the spectral correction methodology, we fabricated and tested the previously designed silicon IBGs. Both the original and APN-compensated gratings were fabricated and characterized for each design. The fabrication was performed based on electron-beam lithography, using a single etch process on an SOI wafer with 220 nm thick silicon on a 3 μm thick buried oxide layer. A 2 μm thick silicon dioxide cladding layer was deposited on the etched sample. The silicon waveguide width was 500 nm. The resolution grid of the lithography was 6 nm, and the minimum feature size/spacing was conservatively established as a design rule of 60 nm. The scanning electron microscope (SEM) images of the testing circuit of a grating and the structures of fabricated $\mathrm{\Delta }L$-modulated and DC-modulated IBGs are shown in Figs. 11(a)-11(c), respectively. Two full-etched sub-wavelength grating couplers (GCs) [34], fabricated on a 127 μm pitch, were used to couple light into and out of the chip from a polished polarization maintaining single-mode optical fiber array with a 127 μm fiber-to-fiber pitch (PLC Connections Inc.). A broadband directional coupler (BDC) [33] was placed between the GCs and the IBG to direct the reflected light back. In the current work, only the reflected signals were of interest and measured, and the transmission signals were terminated at the end of the IBGs via waveguide tapers. An optical vector analyzer (OVA, Luna Innovations) was used to measure the reflection spectra of the gratings. A time-domain filter provided by the OVA was used to eliminate the back reflections from the GCs, the BDC, and fiber interfaces. The waveguide between the directional coupler and the measured IBG was long enough (> 200 μm) to provide sufficient temporal spacing to filter out the unwanted reflections.

 

Fig. 12 (a)-(c) Experimental data of the $\mathrm{\Delta }L$-modulated Gaussain-apodized IBG. (a) Measured (blue) and SS-TMM predicted (red) spectra of the originally designed IBG. (b) Λ-correction profile of the IBG. (c) Measured spectrum of the Λ-corrected IBG (blue). (d)-(f) Experimental data of the DC-modulated Gaussain-apodized IBG. (d) Measured (blue) and SS-TMM predicted (red) spectra of the originally designed IBG. (e) Λ-correction profile of the IBG. (f) Measured spectrum of the Λ-corrected IBG (blue). The black curves in (a), (c), (d) and (f) are the ideal spectra for comparison.

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Table 1. Spectral parameters of the Gaussian-apodized IBGs

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4.1. Gaussian-apodized IBGs

We first show the experimental results of the Gaussian-apodized IBGs modulated by the $\mathrm{\Delta }L$ and DC which are designed based on Fig. 1(b) and Fig. 4(b), respectively. The corrugation widths of the IBGs, $\mathrm{\Delta }{W}_C$, are 8 nm. Figures 12(a) and 12(d) compare the measured (blue) and SS-TMM predicted (red) spectra for the original $\mathrm{\Delta }L$- and DC-modulated IBGs, respectively, with the ideal spectrum (black) included in each figure for comparison. As can be seen, for each apodization technique, the measured spectrum of the originally designed IBG due to the APN is largely deviated from the ideal one but is in close agreement with that predicted by the SS-TMM. This demonstrates the high reliability and accuracy of the SS-TMM and its capability to take into account the APN. The Λ-correction profiles for the $\mathrm{\Delta }L$- and DC-modulated IBGs are plotted in Figs. 12(b) and 12(e), respectively, which are obtained based on the procedures described in the last section. One may notice that the Λ-corrections are much smaller than the resolution grid of the lithography (6 nm). These period corrections below the lithography resolution, however, can be resolved in the actual fabrication. This can be explained if we consider the lithography as a spatial sampling process of the grating structure. Thus, these Λ variations, even much smaller than the sampling interval, can be still captured provided that the sampling frequency is sufficiently large compared with the spatial frequency of the grating structure.

Figures 12(c) and 12(f) shows the measured spectra of the Λ-corrected $\mathrm{\Delta }L$- and DC-modulated IBGs, respectively, together with the ideal spectra (black). It can be seen that the spectra of both the two corrected IBGs have been fully corrected, which now are in close agreement with the ideal one, validating our APN compensation and spectral correction methodology. Furthermore, the side-lobe suppression ratios (SLSRs) are 27.3 dB and 16.5 dB for the $\mathrm{\Delta }L$- and DC-modulated IBGs, respectively, which are remarkably high compared with those achieved in recently work [1, 18, 35]. This indicates that the APN has been almost exactly compensated with little phase error introduced. The lower SLSR of the corrected DC-modulated IBG compared with that of the corrected $\mathrm{\Delta }L$-modulated one should be due to the lower apodization dynamic range of the DC modulation scheme, which prevents the achievement of the entire Gaussian apodization profile into the grating. This can be seen by noting that realizing low κ for the DC modulation scheme requires small feature spacings in the grating structure [as illustrated in the diagrams of Fig. 4(a)], which is limited by the minimum achievable feature spacing in the fabrication. Hence, the feature spacings near both the ends of the DC-modulated grating would be too small to be resolved in the fabrication, which truncates the applied apodization profile and limits the maximum achievable SLSR. Due to the larger apodization dynamic range of the $\mathrm{\Delta }L$ modulation method, this apodization scheme has been chosen for implementing more complicated IBGs, as we shall see later. Table 1 summarizes the 10 dB and 20 dB bandwidths (${\text{BW}}_{10\text{dB}}$ and ${\text{BW}}_{20\text{dB}}$) and SLSRs of the original and corrected Gaussian-apodized IBGs, where the parameters of the ideal spectrum calculated from the CMT-TMM is also included for comparison.

 

Fig. 13 Experimental data of the square filter based on the $\mathrm{\Delta }L$-modulated IBG. (a) Measured (blue) and SS-TMM predicted (red) spectra of the originally designed IBG. (b) Λ-correction profile of the IBG. (c) Measured spectrum of the Λ-corrected IBG (blue). The black curves in (a) and (c) are the ideal spectra for comparison.

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Table 2. Spectral parameters of the square filters based on $\mathrm{\Delta }L$-modulated IBGs

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4.2. Square flat-top filters

Then, we show the measurement results of the square grating filters which are designed based on Fig. 5 and developed on $\mathrm{\Delta }L$-modulated IBGs. The W_C of the IBGs are 10 nm. Figure 13(a) shows the experimental (blue) and SS-TMM predicted (red) spectra of the original $\mathrm{\Delta }L$-modulated IBGs, with the ideal spectrum (black) included for comparison. Again, the measured spectrum of the originally designed IBG due to the APN is greatly distorted from the ideal one, but shows good agreement with that predicted by the SS-TMM. Figure 13(b) plots the Λ-correction profile for the IBG, while Fig. 13(c) shows the measured spectrum of the Λ-corrected IBG. After the APN compensation, the spectrum of the IBG has been corrected and now agrees well with the ideal one. This validates the capability of the methodology for correcting complicated IBGs designed with elaborate responses. Table 2 summarizes 5 dB bandwidths (${\text{BW}}_{5\text{dB}}$), rising bandwidths and SLSRs of the original and corrected IBGs, where those of the ideal spectral response are also included for comparison. The rising bandwidth here is defined as the bandwidth over which the reflection goes from -0.5 dB to -10 dB, and is used to measure the steepness at the edges of the square response. A smaller rising bandwidth means sharper spectral edges and thus a higher quality of the square filtering response. It can be seen from the table that the corrected IBG has a bandwidth closer to that of the ideal, a much smaller rising bandwidth and a much higher SLSR compared with those of the originally designed IBG. These results highlight that in addition to correcting the apodized IBG response to be closer to the design, the APN compensation can also lead to an improved spectral performance due to the elimination of the phase noise.

 

Fig. 14 (a) Measured (blue) and SS-TMM predicted (red) amplitude and phase responses of the photonic Hilbert transformer based on the original $\mathrm{\Delta }L$-modulated IBG. (b) Λ-correction profile of the IBG. (c) Measured complex response of the Λ-corrected IBG (blue), and the ideal response (black). (d) Building block used in the numerical analysis for achieving a single side-band filtering response. (e) Calculated transfer functions of the building block when using the experimental S parameters of the original (red) and corrected (blue) IBGs.

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4.3. Photonic Hilbert transformers

Finally, we demonstrate the correction of the complex spectral response of the previously designed photonic Hilbert transformer (Fig. 6) based on the $\mathrm{\Delta }L$-modulated IBG. The W_C of the IBG is 7 nm. The amplitude and phase spectral responses of the IBGs were measured using the OVA. The linear portions of the measured phase responses shown in Figs. 14(a) and 14(c) due to the length of the testing circuit have been removed. Also, all the phase responses presented in Fig. 14 have been wrapped into the range $\left(0,2\pi \right]$ and offset to have the same baseline for comparison purpose. The measured and SS-TMM predicted complex responses of the originally designed IBG are plotted in Fig. 14(a). Both the measured amplitude and phase responses exhibit good agreement with those predicted by the SS-TMM. Figure 14(b) shows the period correction profile along the IBG. Figure 14(c) compares the measured complex spectral response of the corrected IBG with the ideal response. We can see that both the amplitude and phase responses have been corrected and exhibit excellent agreement with the ideal one, which demonstrates the capability of the proposed methodology for correcting complex spectral responses of apodized IBGs.

The corrected complex response of the photonic Hilbert transformer will lead to a corresponding improvement in the related overall system performance. To verify this, we calculate the transfer function of a typical building block using photonic Hilbert transformer for achieving a single side-band (SSB) filter [36], as illustrated in Fig. 14(d). We use the experimental S parameters measured from the IBGs in the calculation. The building block has a MZI structure, with one arm consisting of an optical S-parameter operating at reflection mode, and the other having an optical delay and attenuator to match the phase and the intensity of the first branch, respectively. This configuration essentially exploits the π-phase shift between the left- and right-side wavelength bands of the photonic Hilbert transformer to realize a SSB filter. In particular, when the phase of the light traveling through the bottom arm is tuned to match to that of the left (right)-side wavelength band of the photonic Hilbert transformer, a constructive interference will occur on the left (right)-side band while a destructive interference will happen on the right (left)-side band due to the π-phase shift between them, which finally forms a SSB filtering behaviour for the transfer function of the system. Figure 14(e) shows the calculated transfer functions of the building block when using the experimental S parameters of the original (red) and corrected (blue) IBGs. We can see that the obtained SSB filtering response for the corrected IBG has a much higher extinction ratio (14.5 dB) compared with that of the original IBG (4.5 dB). This higher extinction ratio for the corrected IBG is mainly because of the higher quality of the grating phase response which is much closer to the desired staircase-like behaviour, thus leading to a better SSB filtering response. The result clearly shows that the APN compensation to correct the spectral response of an IBG will bring about an improvement in the related system performance.

5. Conclusion and future work

In summary, this work focuses on the characterization of the impacts of apodization phase noise (APN) on the spectral responses of apodized silicon IBGs, and proposes an APN compensation methodology to correct the APN-distorted grating responses. We propose the reliable and accurate SS-TMM for spectral modeling of apodized silicon IBGs that can take the APN into account, which is difficult for conventional CMT-based modeling methods. By using the SS-TMM to model different silicon IBGs, itis shown that APN can greatly distort both the amplitude and phase responses of an apodized IBG such that the responses are deviated significantly from the design. Then, we present a methodology to eliminate the APN so as to correct the distorted response, which is implemented by first extracting the APN distribution of the apodized IBG, and then correcting the grating structure accordingly to compensate the APN. The designed silicon IBGs were fabricated and experimentally characterized. The measured spectra for the originally designed IBGs are largely deviated from the designs due to APN but exhibit good agreement with those predicted by the SS-TMM. For the APN-compensated IBGs, their experimental spectra are successfully corrected and agree well with the designs. Finally, we demonstrate a complex spectral correction of an IBG-based photonic Hilbert transformer. For some applications, a perfect control of the grating response may be not critical and thus the APN compensation may not be essential, such as IBG-based sensors and high extinction ratio filters. However, there are many applications that will benefit from our work where the complex response of a designed grating is required to be consistent with the target one for achieving the required functionality. Those include all-optical signal processors, multi-channel bandpass filters in WDM networks, dispersion compensators/controllers in optical communications, etc.

The SS-TMM in the current work is for modeling silicon IBGs based on strip waveguides. There will however be benefits in some applications to develop IBGs on other silicon waveguides such as ridge and slot waveguides. For example, IBGs based on ridge waveguides, which typically have larger cross sections than those of strip waveguides, would be easier to have weak grating strengths to achieve narrowband filtering characteristics [37]; those based on slot waveguides, where the mode can be more sensitive to the refractive index changes of the cladding, would be preferred in sensing applications [6]. Thus, one direction of future research is to extend the SS-TMM model to IBGs based on those different silicon waveguides. The SS-TMM implementation in those cases should be similar to that used in current strip waveguide-based IBGs. However, the effective refractive index of each layer of the discretized grating will no longer be simply related to its waveguide width, which would need to be solved through other mathematical techniques, such as effective index method [38] and photonic band-structure calculations [12], or be extracted from experimental results. Another area of future research could include the manufacturing imperfections and variations in the SS-TMM model for a better spectral prediction. As an example, the grating structure to be modeled could include waveguide width variations and waveguide roughness, and be modified via a lithography model which can take lithography smoothing effects and minimum feature size/spacing into account [8]. After these, the virtually fabricated grating can be re-simulated by the SS-TMM to obtain a better prediction of the forthcoming experimental results.