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New design of optical waveguides is presented to synthesize Bragg grating waveguides without inducing birefringence. In the design, waveguide core has a thin trench on the top surface of the core and width of the trench is modulated concurrently with width modulation of the waveguide core. Effective refractive index profiles in Bragg grating waveguides are obtained by a differential inverse scattering algorithm and converted to waveguide width profiles by using the new core design. This procedure allows design of non-birefringent Bragg grating waveguides on planar substrate. The method is applied to the design of chromatic dispersion compensators. For characterization of Bragg grating waveguides, analysis based on short-time Fourier transform of Bragg grating response waveforms is presented.
©2010 Optical Society of America
1. Introduction
Bragg gratings have been of interest due to its applications in optical filters such as chromatic dispersion compensators and optical add/drop multiplexers in dense wavelength-division multiplexing (DWDM) optical fiber communication. Bragg grating filters in DWDM communication are designed with multiple Bragg gratings superimposed on each other to form an optical waveguide, with data being transmitted in multiple wavelength channels for DWDM communication. Each layer of the superimposed Bragg grating is designed to have a stopband of Bragg reflection in a respective wavelength channel. As a result of superimposition of multiple Bragg gratings, a non-uniform modulation profile of an effective refractive index is imprinted into the optical waveguide [1,2].
Bragg gratings have been formed in high-index contrast optical waveguides on planar substrates by periodically modulating the width or thickness of the waveguide core corresponding to the grating periods for Bragg reflection [3]. Bragg grating waveguides on planar substrates are useful for size reduction in optical filters and integration in low-loss planar lightwave circuits [4,5]. Bragg grating waveguides have been fabricated on wafers such as silicon and indium phosphide [6–9]. Multi-channel Bragg grating waveguides with non-uniform modulation have been fabricated and they exhibited a low reflectance side lobe spectrum [9]. The multi-channel Bragg grating waveguides fabricated on planar substrates have not been designed for non-birefringence, and as such it caused a difference between the effective refractive indices in transverse-electric (TE)-like and transverse-magnetic (TM)-like modes. Peak splitting between the stop-bands in TE-like and TM-like modes has been observed due to the birefringence [6,7].
Birefringence in optical waveguides causes separation of propagating optical pulses in time domain and induces cross-talk in data transmission [10]. Bragg gratings, therefore, must be synthesized by using non-birefringent waveguides to avoid data transmission errors. A method to eliminate birefringence in an optical waveguide, in which Bragg gratings are not formed, is simply to adjust the core width and core thickness to equalize effective refractive indices in the TE-like and TM-like modes [11]. In the case of a multi-channel Bragg grating waveguide having non-uniform modulation profile of effective refractive index, width and thickness of the waveguide must be adjusted concurrently over the entire range of effective refractive index modulation in superimposed Bragg gratings. In planar fabrication processes, width modulation in the Bragg gratings is achieved by lithography [4,5]. Thickness modulation is, on the other hand, controlled by depth in etching. Precise depth modulation allowing non-uniform effective index modulation in Bragg gratings is not a task achievable by the fabrication processes for planar lightwave circuits. For optical filters in coarse wavelength-division multiplexing (CDWM) application, non-birefringence in Bragg gratings was achieved by using a dual-core waveguide without concurrent modulation of core width and core thickness [12]. In the dual-core waveguide, four Bragg gratings, which produce Bragg reflection at four CDWM channels, were concatenated in series. In the DWDM, on the other hand, wavelength channels up to 50 or more are used, and a Bragg grating waveguide will be significantly long if Bragg gratings are concatenated. Superposition of Bragg gratings is thus essential to the DWDM filters.
Here, we present a new non-birefringent optical waveguide design to equalize effective refractive indices in the TE-like and TM-like modes for a range covering effective refractive index modulation in superimposed Bragg gratings. The waveguide design is exploited to synthesize Bragg grating waveguides for single-channel and multiple-channel chromatic dispersion compensators. Effective refractive index profiles of Bragg grating waveguides are obtained by a differential inverse scattering algorithm based on Gel'fand-Levitan-Marchenko (GLM) equations [13,14]. Differential inverse scattering algorithms have an advantage in reconstructing an effective refractive index profile, which is unique to a required optical spectrum, without iteration. Effective refractive index profiles are converted to waveguide width profiles which give rise to non-birefringent Bragg grating waveguides. Analysis based on grating spectrogram is presented for characterization of Bragg grating waveguides. Effective refractive index profiles of Bragg grating waveguides are utilized as response waveforms of the waveguides, and short-time Fourier transform of the response waveforms provide grating spectrograms.
2. Non-birefringent waveguide
To accommodate multi-channel Bragg gratings to a planar waveguide without inducing birefringence, a new waveguide having a schematic cross-section as shown in Fig. 1 is proposed. The core is made of silicon nitride and surrounded by silicon-dioxide cladding. Silicon nitride is a suitable material to serve as a loss-loss core for high-index contrast waveguides [15]. The core has a trench on its top with trench width w t and trench depth t t. Core width measured between the side walls is denoted as w c and core thickness measured between the top and bottom surfaces as t c. Change in w c affects the effective refractive index in TE-like polarization more strongly than that in w t. On the other hand, the latter affects the effective refractive index in TM-like polarization more strongly than the former. Therefore, the effective refractive indices in both polarization modes can be made to offset each other with a suitable choice of w t and w c.
Trench and core widths w t and w c are varied so as to produce effective refractive index profiles in multi-channel Bragg grating waveguides under consideration. Mode calculation described below provides conversion curves from effective refractive index to the trench and core widths. The trench depth and core thicknesses, on the other hand, are fixed to unique design values, respectively to be 0.1μm and 1.4μm. Both the top clad thickness and the bottom clad thickness are 1μm. Width of side clad on each side of the core is about 1.5μm. Refractive indices of silicon nitride and silicon dioxide are 2.05 and 1.45, respectively. Fabrication of such Bragg grating waveguides is feasible by deposition, optical lithography and etching processes for low-loss planar lightwave circuits in CMOS-compatible technology [15].
Effective refractive indices of the lowest-order TE-like and TM-like modes are calculated by the film mode matching simulation [16]. By changing w t and w c concurrently, effective refractive indices of the two modes are adjusted to a desired refractive index. This adjustment procedure is repeated in a range of effective refractive index between 1.92969 and 1.94046 with a residual ratio of less than 1.3x10−4. Trench width w t and core width w c after the adjustment are plotted with respect to effective refractive index n eff in Fig. 2 . These curves provide the conversion rule, by which w t and w c are uniquely determined for n eff, and are exploited for the synthesis of non-birefringent Bragg grating waveguides described below.
Fig. 2 Conversion curves from effective refractive index to waveguide parameters: trench width w t (red curve) and core width w c (blue curve).
3. Bragg grating synthesis and analysis
Bragg grating waveguides presented here are designed according to the following procedures:
- (i) The reflectance spectrum required for the Bragg grating waveguide is specified in terms of field magnitude and group delay. The required spectrum is transformed to a transient impulsive response by inverse Fourier transformation.
- (ii) Effective refractive index n eff of the Bragg grating waveguide is derived as a function of coordinate z along the waveguide by a differential inverse scattering algorithm based on GLM equations for forward and backward waves propagating in the Bragg grating waveguide [13,14]. In the inverse scattering algorithm, the coordinate z is treated as a dimension equivalent to time, and the transient impulse response obtained by inverse Fourier transform of Bragg grating reflectance spectrum is solved in time-reversal order to synthesize effective refractive index profile n eff(z) in the waveguide.
- (iii) The effective refractive index profile is transformed to waveguide width profiles w t(z) and w c(z) using the conversion curves in Fig. 2.
A single-channel chromatic dispersion compensator is designed and analyzed first as a simple example of waveguide design following steps (i) to (iii). Design and analysis of a Bragg grating for a multiple-channel chromatic dispersion compensator are presented next to demonstrate a waveguide having a high number of superposed Bragg gratings can be designed using the abovementioned procedures.
3.1 Single-channel chromatic dispersion compensator
The reflectance spectrum required for a Bragg grating waveguide for single-channel chromatic dispersion compensation is plotted in Fig. 3 (required). Absolute field reflectance is scaled in the left vertical axis, and group delay in the right vertical axis, respectively. Spectral phase of the required spectrum is obtained by frequency integral of the group delay spectrum. The field reflectance is normalized to unity. The reflectance channel is centered at 1590.41nm. The center wavelength is in channel L85 of 100GHz-spaced ITU grid. Reflectance bandwidth is 0.68nm (80GHz). The field reflectance in the reflection channel is 0.9 which implies that power reflectance is 0.81. The waveguide is designed for chromatic dispersion compensation in 30km dispersion-shifted fiber (DSF) which yields negative chromatic dispersion in L-band with a dispersion parameter of 2.9ps/nm/km at a wavelength of 1590nm. Thus, the group delay in the reflectance band of the required spectrum decreases linearly with wavelength, exhibiting a positive chromatic dispersion of −88ps/nm.
Fig. 3 Reflectance spectra required to single-channel chromatic dispersion compensator (required) and calculated using coupled mode equations for synthesized Bragg grating waveguide (realized).
The spectrum (realized) in the right-hand side of Fig. 3 is a reflectance spectrum calculated by coupled mode equations for the Bragg grating waveguide which has effective refractive index profile n eff(z) synthesized by the inverse scattering algorithm as explained below. The edges on both sides of the reflectance band in the realized spectrum are rounded since the length of the waveguide solved in the inverse scattering calculation is not infinite. The reflectance is lower than that in the required spectrum and there are ripples in the reflectance band, since the waveguide solved by the inverse scattering algorithm is truncated at both ends.
Effective refractive index profile n eff(z) is synthesized from the required spectrum by the inverse scattering algorithm, and then transformed to waveguide width profiles w t(z) and w c(z) using the conversion curves in Fig. 2 as plotted in Fig. 4 . The widths w t and w c oscillate along the coordinate z concurrently with each other as shown in the insets in Fig. 4. A local grating period in both w t(z) and w c(z) at a position in z provides a Bragg reflection wavelength at the position. The oscillation amplitudes in w t(z) and w c(z) are non-uniform and yield apodization envelopes inherent to the square reflectance band with no side lobes in the required spectrum. The number of sampling points in the width profiles is increased ten times of that in the inverse scattering solution by Whittaker-Shannon interpolation to reconstruct sinusoidal grating profiles as shown in the insets in Fig. 4 [17]. The sampling pitch in Fig. 4 is 0.02 in the unit of grating period corresponding to the center wavelength in the required spectrum.
Fig. 4 Profiles of waveguide widths w t (red dots) and w c (blue dots) with respect to coordinate z in a single-channel dispersion compensator. Insets: w t and w c in expanded z scale around 5.105mm.
The chromatic dispersion compensation is realized by the wavelength-dependent group delay in Fig. 3, and chirping is observed in the Bragg grating period in the width profiles in Fig. 4. To verify the chirping, it is useful to apply an analysis method based on short-time Fourier transform called as spectrogram [18]. Time-frequency representation based on spectrogram was incorporated into the design algorithm using Wigner-Ville distribution function for the synthesis of Bragg gratings [19]. Here, the spectrogram method is applied to the characterization of Bragg gratings. Effective refractive index profile n eff(z) of a Bragg grating waveguide under characterization is decomposed into average refractive index n av ( = 1.93508) and oscillation component Δn eff(z). The latter is taken as a response waveform of the Bragg grating waveguide under transformation of coordinate z into propagation time t p using the relation, z = ct p/(2n av). A factor of 2 in the transformation denotes double path in reflection and c the speed of light. Grating spectrogram S is defined by Eq. (1) as a function of grating frequency ν g and delay time τ d. Gate function g for short-time Fourier transform has a Gaussian shape with a full width at half maximum ~10ps.
Grating spectrogram is plotted in Fig. 5 as a function of delay time and grating wavelength. Grating wavelength is defined as 2n av Λ g using local Bragg grating period Λ g. The local grating period is obtained by c/(2n av ν g). Spectrogram amplitude is color-coded and normalized with respect to its maximum. For comparison with the spectrogram, Δn eff(z) is also plotted as a function of coordinate z in Fig. 5. The coordinate is given in the right vertical axis. The full-scale delay time in the left vertical axis is equivalent to the full length of the waveguide (~9.9mm) in the right vertical axis. The high distribution of the spectrogram amplitude shifts from longer to shorter local Bragg grating period with increasing delay time, and with increasing distance from the input end of the waveguide. This is a visual evidence of chirping in the local Bragg grating period. Such spectrogram analysis is particularly useful in the case to be described next, where a high number of Bragg gratings are superimposed in a waveguide resulting in a complicated effective refractive index profile.
Fig. 5 Grating spectrogram (left) and oscillation component Δn eff vs waveguide coordinate (right) for single-channel chromatic dispersion compensator.
3.2 Multi-channel chromatic dispersion compensator
A Bragg grating waveguide is designed for multi-channel chromatic dispersion compensation in 40km DSF. Dispersion slope of DSF, which is 0.07ps/nm2/km in L-band, is also taken into account. Group-delay spectrum required for the chromatic dispersion compensation is plotted in the bottom graph (required) in Fig. 6 . The dispersion slope produces larger group delay in a channel located at longer wavelength. Field reflectance is 0.9 and is flat over the entire wavelength range as shown in Fig. 6. The spectrum at the top of Fig. 6 is a group-delay spectrum (realized) obtained by coupled mode equations with effective refractive index profile synthesized by the inverse scattering algorithm.
Fig. 6 Group-delay spectra required to design of 50-channel chromatic dispersion compensator waveguide (required) and calculated for synthesized Bragg grating waveguide (realized).
Waveguide width profiles for the 50-channel chromatic dispersion compensator are obtained from synthesized effective refractive index profile according to the steps described above and plotted in Fig. 7 . The maximum oscillation amplitudes in w t and w c in Fig. 7 are more than ten times larger than those for the single-channel chromatic dispersion compensator in Fig. 4, since Bragg gratings of 50 channels are superimposed in this waveguide. The envelopes of the profiles have more complicated shapes than those in Fig. 4, although w t(z) and w c(z) oscillate with local Bragg grating periods as in Fig. 4. It is not possible to resolve the Bragg gratings in the width profiles into superimposed chirped Bragg gratings. Spectrogram analysis is necessary to decompose the Bragg grating profile into 50 channels of chirped Bragg gratings.
Fig. 7 Profile of waveguide width w t (red dots) and w c (blue dots) with respect to coordinate z for 50-channel dispersion compensator.
The Bragg grating waveguides will be fabricated by CMOS-based wafer processes. In fabrication, there will be variation in waveguide width and thickness over a wafer. Such intra-wafer variation can be suppressed by dose control in optical lithography. Tolerance for the variation in waveguide width, for example, is estimated as +/−10nm, which causes birefringence nearly same as the residual birefringence in the waveguide design. The residual birefringence is 1.3x10−4 in ratio as described in Section 2.
Grating spectrogram described in the analysis of the single-channel Bragg grating waveguide is applied to the analysis of the 50-channel Bragg grating waveguide. Grating spectrogram and Δn eff are plotted in Fig. 8 . The spectrogram has a comb-like amplitude distribution, which is decomposed into 50 channels in an array along the grating wavelength 2n av Λ g (Fig. 8). The 50 channels in grating wavelength occur at the same wavelengths as those in the required group-delay spectrum in Fig. 6. Intra-channel distribution of the spectrogram amplitude shows chirping as shown in Fig. 5, and is elongated along the output end of the waveguide for a channel centered at a longer wavelength. These characteristics reflect chromatic dispersion and dispersion slope characteristics over the 50 channels specified in the required spectrum. The grating spectrogram allows us to confirm that the effective index profile and hence the width profiles of the waveguide are composed of 50 chirped Bragg gratings superimposed in the waveguide.
Fig. 8 Grating spectrogram (left) and oscillation component vs waveguide coordinate (right) for 50-channel chromatic dispersion compensator.
4. Conclusion
New design of Bragg grating waveguides has been applied to single-channel and multiple-channel chromatic dispersion compensators in DWDM communication. The design of Bragg grating waveguides is based on the design of non-birefringent waveguide and Bragg grating synthesis by a differential inverse scattering algorithm. In the waveguide designed, the core has a trench on its top and width of the trench is modulated concurrently with modulation of core width to avoid birefringence. Effective refractive index profiles synthesized by the inverse scattering algorithm have been converted to profiles of the trench width and the core width. Analysis of Bragg gratings based on spectrogram has been presented and shown to be useful in the display of channeled spectral characteristics of a high number of Bragg gratings superimposed in a Bragg grating waveguide. The design and analysis described here can be applied to other optical devices such as add/drop optical filters.
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