Abstract

We present the design, analysis, and experimental characterization of a novel integrated add-drop filter capable of filtering simultaneously two independent channels that is based on a contra-directional grating assisted coupler with two different periods. The device performance is explained using Fourier analysis and confirmed with numerical simulations using the eigenmode expansion method. The devices were fabricated using electron-beam lithography on a silicon-on-insulator wafer with a 220 nm thick device layer. The Fourier analysis, simulations and experimental results are in agreement and show that the drop port response of the two-period configuration is the superposition of the drop port responses of two single-period gratings. Therefore, the output channels at drop port can be designed independently and can have different bandwidths.

© 2016 Optical Society of America

1. Introduction

Integrated optical filters are one of the essential building blocks of optical circuits and networks. Filters based on integrated Bragg gratings can provide a high extinction ratio with free spectral range (FSR)-free performance which is useful in many applications [1]. Silicon-on-Insulator (SOI) integrated Bragg gratings have been implemented by periodically modulating the side walls of the waveguide core [2,3], or the cladding of the waveguide [4]. In applications that require filtering multiple wavelength channels, it is possible to cascade multiple filters to isolate the different channels. However, cascaded structures are more sensitive to non-uniformity in the fabrication process [5]. Moreover, cascading filters leads to large device size when numerous ports are required. Therefore, a filter that simultaneously selects multiple channels would be more compact and less sensitive to fabrication variations. In [6], two separate wavelength channels are reflected by a single integrated Bragg grating by modulating the sidewalls of a silicon waveguide with two different periods. This method can be further extended by periodically modulating the sidewalls of the slab and core region of a ridge waveguide [7] to obtain a grating with four different periods. However, a circulator is required at the input port in order to extract the drop port response of an integrated Bragg grating. This limits the use of integrated Bragg gratings, since CMOS compatible integrated circulators are not available. In [8], an alternative approach is proposed that eliminates the need for a circulator by connecting two identical Bragg gratings to a 3-dB multimode interference coupler (MMI). If the two Bragg gratings are identical and the MMI coupler has sufficient bandwidth, the reflected signal from the two gratings add destructively with each other at the input port because of the 90 degree phase difference between the two outputs of the MMI coupler, and consequently light is directed to the other port of the MMI coupler where the reflected signals add up constructively. However, small fabrication errors in the MMI and variations between the Bragg gratings can cause reflections towards the input.

Contra-directional grating assisted couplers (CDGACs) are well suited to replace integrated Bragg gratings in integrated optical devices because their drop response propagates through a waveguide that is different from the input waveguide, and therefore there is no need to use a circulator at the input [9–11]. Moreover CDGACs, like Bragg gratings, can provide FSR-free performance and a high extinction ratio at the output. A SOI wavelength demultiplexer using CDGACs has been previously demonstrated in [12]. In this paper, we propose a CDGAC that can simultaneously filter two non-contiguous wavelength channels. We also demonstrate that the spectral properties of each filtered channel, such as their bandwidth and central wavelength, can be tailored independently. The proposed configuration enables the implementation of more compact devices. Moreover, the Fourier analysis of the configuration gives a thorough insight into the performance of the device and can be used to understand the behavior of similar configurations. The concepts demonstrated in this paper can have a significant impact on the device density required to perform advanced filtering functions since they enable more complex responses while requiring the same area as CDGACs previously reported in the literature.

2. Configuration

A CDGAC consists of two dissimilar waveguides in close proximity. Because of the difference between the waveguides, there is no significant power exchanged between them through evanescent coupling. However, by implementing a periodic perturbation, it is possible to couple light travelling in one direction in one waveguide to the opposite propagation direction in the second waveguide in a frequency-selective manner [11]. Figure 1(a) depicts the top view of a CDGAC. The gratings can be implemented on only one or both of the waveguides. The corrugations on the sidewalls for each period consists of a recessed and a protruded section of ΔW/2which adds up to the corrugation width of ΔW. The central wavelength of the drop signal of a CDGAC depends on the period as well as the average effective index of the grating. In order for light of wavelength λ0to be reflectively coupled from one waveguide to the other, the period of the grating (Λ0) must satisfy the following equation [14]:

β1(λ0)+β2(λ0)=2mπΛ0
where β1(λ0) and β2(λ0) are the propagation constants of the two waveguides at the wavelength λ0, and m (an integer) is the grating order. In addition to the contra-directional coupling mechanism, at certain wavelengths the light in each waveguide is reflected in the same waveguide, which results in intra-waveguide coupling. In order to reduce intra-waveguide coupling, which causes back reflections, the gratings on the side walls of each waveguide must be out of phase with each other [12,13], as shown in Fig. 1.

 

Fig. 1 (a) Top view of a uniform CDGAC with a grating only on one of the waveguides; (b) Top view of a uniform two-period CDGAC.

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The fabricated GACs have a 220 nm thick silicon core with a 450 nm width for the narrow waveguide (W1as depicted in Fig. 1) and a 600 nm width for the wide waveguide (W2in Fig. 1.). The separation between waveguides (g in Fig. 1.) is 100 nm and the device is cladded with silicon dioxide. ΔW1andΔW2are the widths of the grating modulation whereas the period of the grating on the narrow waveguide is Λ1 and is Λ2on the wide waveguide. Figure 1(b) depicts the top view of the two-period CDGAC configuration which consists of two different grating periods in contrast to single-period CDGACs which have a single-period grating. Two-period CDGACs exhibit a two channel drop port response. The Fourier analysis presented in the next section explains the behaviour of the two-period configuration in comparison with the conventional single-period structure.

3. Fourier analysis

The two-period CDGAC configuration depicted in Fig. 1 (b) consists of two gratings, with profiles a(z) and b(z), that are periodic along the length of the structure (in the z direction) with periods of Λ1, andΛ2. The modulation width of the grating on the narrow waveguide (ΔW1), and of the grating on the wide waveguide (ΔW2) can be different or similar. For the case of an integrated Bragg grating with two different grating periods on the two waveguide side walls, it was demonstrated that the spectral properties of the multi-period grating can be considered as the superposition of the spectral properties of the two corresponding gratings [6]. In this manuscript, in order to investigate the effect of having two different gratings acting simultaneously on the single CDGAC, we model the structure as a single-period grating with a period of Λ3=Λ1×Λ2(Λ1,Λ2), where (Λ1,Λ2)is the greatest common divisor of the two periods. In other words, Λ3 is the super-period of the two-period configuration. Two coupling mechanisms are present in CDGACs. The intra-waveguide coupling coefficient (ki) represents the coupling of a mode in one waveguide with an identical but oppositely propagating mode in the same waveguide. This is the same coupling mechanism as in Bragg gratings. The wavelength at which intra-waveguide coupling happens in a two-period CDGAC can be calculated with Eqs. (2) and (3).

2neff1ave(2πλ)=m2πΛ3
2neff2ave(2πλ)=m2πΛ3

In Eqs. (2) and (3), neff1aveand neff2aveare the average effective indexes of the two interacting modes of the structure in the narrow and wide waveguide respectively, Λ3is the super period of the grating, and m is an integer number. The other coupling mechanism is inter-waveguide coupling, which is the coupling between a mode of one waveguide to another mode in the other waveguide that propagates in the opposite direction. This occurs at wavelengths defined by:

(neff1ave+neff2ave)(2πλ)=m2πΛ3

The longitudinal phase match condition defined by Eq. (4) can be simplified as neff1ave+neff2ave=mλΛ3. Therefore, the wavelengths for which the phase match condition is satisfied can be found from the intersections of the neff1ave+neff2aveline and the mλΛ3lines when these quantities are plotted as a function of wavelength.

Figure 2. depicts the phase match condition for the fabricated structure with periods of 312 nm and 318 nm, and grating widths of ΔW1 = 50 nm and ΔW2 = 50 nm. The dashed lines show the mλΛ3conditions, while the solid line represents neff1ave+neff2aveas a function of wavelength that was extracted from the measurement results of the fabricated two-period CDGAC. The crossing of the dashed lines and of the neff1ave+neff2aveline defines the wavelengths at which coupling occurs. The orders of the dashed lines are listed on the right side of the graph.

 

Fig. 2 Longitudinal phase match condition for the two-period configuration with Λ1 = 312 nm, Λ2 = 318 nm, ΔW1 = 50 nm, and ΔW2 = 50 nm.

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According to Fig. 2., within the wavelength range from 1520 to 1580 nm, the orders of 52, 53, and 54 of the super-period grating cross the neff1ave+neff2aveline, which means that the phase match condition is satisfied at three wavelengths within this range. However, in order for coupling to take place, the coupling coefficient between the two modes must be non-zero at the phase match wavelengths. According to coupled mode theory, the magnitude of the coupling coefficient is proportional to the mthorder Fourier coefficient of the periodic perturbation, where m is the order at which the phase match condition is satisfied [14]. Therefore, to explain the performance of two-period CDGACs, one can investigate the Fourier coefficients of the grating at the phase match wavelengths.

The two-period grating profile c(z) can be seen as the summation of the individual grating profiles, a(z) and b(z), as demonstrated in Eq. (5) Since there is a finite number of periods in each grating, it can be concluded that the periodic functions a(z) and b(z) are multiplied by a window (p1(z), andp2(z) respectively). The width of the window determines the number of periods of the grating such that the width ofp1(z) isΝp1Λ1 and p2(z) has a width of Νp2Λ2. The shape of the window is defined by the apodization function of the grating.

c(z)=a(z)p1(z)+b(z)p2(z)

Assuming that both periodic functions are multiplied by identical windows, i.e. p1(z)=p2(z)=p(z), it can be concluded that Νp1Λ1=Νp2Λ2=Νp3Λ3. Since a(z) and b(z) are periodic functions, they can be defined in the Fourier domain as A(ω)=k=0,1...Akδ(ωk2πΛ1) and B(ω)=k=0,1...Bkδ(ωk2πΛ2), whereAkandBk are the Fourier series coefficients of the periodic functions a(z) and b(z), k is the Fourier coefficient order, and δ(ω) is the Dirac delta function. Considering P(ω) as the Fourier transform of p(z), the Fourier transform of c(z) can be expressed as follows:

C(ω)=(A(ω)+B(ω))*P(ω)=kAkP(ωk2πΛ1)+kBkP(ωk2πΛ2)
ω1=2πΛ1=Λ2(Λ1,Λ2)×2πΛ3=Λ2(Λ1,Λ2)×ω3=l1ω3
ω2=2πΛ2=Λ1(Λ1,Λ2)×2πΛ3=Λ1(Λ1,Λ2)×ω3=l2ω3
C(ω)=kAkP(ωkl1ω3)+kBkP(ωkl2ω3)

Therefore, the Fourier coefficients of the two-period configuration can be calculated using the following equation:

Cm=C(ωm=mω3)=kAkP((mkl1)ω3)+kBkP((mkl2)ω3)

To simplify the analysis, we consider a uniform grating for which P(ω)is a sinc function:

Cm=kAkejω3(mkl1)sinc((mkl1)Νp3)+kBkejω3(mkl2)sinc((mkl2)Νp3)

The contra-directional coupling coefficient (kc) between the two waveguides is calculated as follows [10]:

kc=ωε04E1ΔεmE2dxdy
where Δεm=CmΔn2 is themthorder Fourier coefficient of the periodic dielectric perturbation, and Ei(i=1,2) is the electric field in the corresponding waveguide.

Assuming the same CDGAC as described above, with Λ1 = 312 nm,Λ2 = 318 nm, and Νp3 = 20 which means that Νp1 = 1060, and Νp2 = 1040, the ratio of the Fourier coefficients of order m normalized to the zeroth order Fourier coefficient (i.e. Cm/C0) is calculated and presented in Fig. 3. According to the results, the Fourier coefficient are not zero for m = 52 and m = 53, but are equal to zero for all other orders that satisfy the phase match condition along the neff1+neff2line depicted in Fig. 2. Therefore, within the C and L-band, the two-period configuration provides only two drop channels that are centered on the wavelengths 1550 nm and 1573 nm. The Fourier coefficients of the two-period configuration are respectivelyC52=B1 and C53=A1, which means that the properties of the first channel (at 1550 nm) only depends on the characteristics of the grating with the period of Λ1 = 312 nm, and likewise those of the second channel (at 1573 nm) are only derived from the grating with the period ofΛ2 = 318 nm. Therefore, the response of the two channels can be designed independently by individually engineering the single-period gratings. The Fourier analysis allows the contribution of each single-period grating in the overall response of two-period CDGAC to be identified. However, it is important to mention that the high orders of 52 and 53 of the super-period configuration do not describe the order of the individual gratings. As shown by the fact that C52=B1 and C53=A1, both single-period gratings of two-period configuration are of the first diffraction order and the high grating orders are only related to the super-period model of two-period grating.

 

Fig. 3 Fourier coefficients of the two-period configuration with Λ1 = 312 nm and Λ2 = 318 nm having Νp3 = 20 super-periods.

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4. Numerical simulation results

In the previous section, Fourier analysis was used to demonstrate that the drop port response of the two-period configuration has two drop channels corresponding to the two periods of the grating. Using the eigen mode expansion method (EME) solver from Lumerical Solutions Inc., we simulated the two-period CDGAC. The simulated CDGAC has a narrow waveguide with a width of 450 nm (W1as depicted in Fig. 1.) and a wide waveguide with a width of 600 nm (W2 in Fig. 1.). The gap between waveguides (g in Fig. 1.) is 100 nm. We chose the modulation width to be ΔW1=ΔW2 = 50 nm for the two-period configuration. The period of the grating on the narrow waveguide is 312 nm, whereas the period on the wide waveguide is 318 nm. For simplicity, the simulation was performed on uniform gratings. The simulation results for the two-period configuration are presented in Fig. 4. The drop port of the two-period configuration has two channels: the first channel is centered at 1541.3 nm and has a 3 dB bandwidth of 6.94 nm, while the central wavelength of second channel is at 1560.2 nm and has a 3 dB bandwidth of 6.84 nm. The simulation results demonstrate that, as long as the wavelength ranges of the drop channels of the two gratings of the two-period CDGAC do not overlap with each other, the drop port response consists of two separate drop channels.

 

Fig. 4 Simulation results of a uniform two-period CDGAC with ΔW1=ΔW2 = 50 nm, Λ1 = 312 nm, Λ2 = 318 nm, and Νp3 = 20.

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5. Experimental results

The devices were fabricated using e-beam lithography on a silicon-on-insulator wafer with a 220 nm thick silicon device layer and a buried oxide thickness of 3.0 μm. Patterning was done using the negative resist HSQ and an oxide cladding between 2 to 3 μm was deposited using plasma enhanced chemical vapor deposition after the pattern was etched with reactive ion etching. To reduce the level of the side-lobe of drop port response, the modulation width of the gratings was apodized in a Gaussian manner [12]:

ΔW=ΔW0ea(x0.5NpNp)2
where ΔW0is the maximum grating width, a determines the strength of apodization, Νpis the number of periods, and x represents each period of the grating. However, the reduction in side-lobes provided by Gaussian apodization varies on each side of the passband [15], which explains the asymmetry in the side-lobes visible on the spectra presented below. As mentioned above, the side-etched gratings on the walls of each waveguide are out of phase with each other. In addition, based on the results from [12], both gratings were apodized with a = 10. In this case, in Eq. (10), P(ω)must be substituted by the Fourier transform of the truncated Gaussian function.

The devices were interrogated using grating couplers. A tunable laser was used to measure the output power of the device at different wavelengths. In order to eliminate the spectral response of the grating coupler from the results, the output spectrum of a straight waveguide was measured using the same grating coupler design that was used for the devices. Then the output spectrum of the devices was normalized to the spectrum of the straight waveguide.

The measured amplitude of the normalized through and drop port response of a two-period CDGAC with ΔW1=ΔW2 = 50 nm, and periods of Λ1 = 312 nm and Λ2 = 318 nm, a grating apodization factor of 10, and Np1=Np2=1300 is shown in Fig. 5 (a). The measurement results show two drop channels at 1555.9 nm and 1574.6 nm with 3 dB bandwidths of 6.99 nm for the first channel and of 7.53 nm for the second channel. The first channel of the drop port response of the two-period CDGAC is associated to the grating with a period of 312 nm, whereas the second channel is associated with the grating with a period of 318 nm. In order to compare the performance of the two-period CDGAC with single-period CDGACs, single-period CDGACs were also fabricated and characterized. It is important to mention that even in the ideal case where there is no fabrication error or difference between the two-period CDGAC and corresponding single-period CDGACs, the central wavelength and bandwidth of the drop channels of the two-period configuration and single-period structures would not be exactly the same. The reason is that the average effective index of the two-period CDGAC is larger than the corresponding single-period CDGAC. Based on our simulations, for the given configuration, the difference in the average effective index can lead to up to 0.13% increase in the central wavelength and up to 0.2% increase in the 3 dB bandwidth for the two-period configuration in comparison to the corresponding single-period gratings. Moreover, fabrication errors can cause larger variations between the responses of the single-period and two-period gratings.

 

Fig. 5 Through and drop port response of (a) two-period CDGAC with ΔW1=ΔW2 = 50 nm, Λ1 = 312 nm, Λ2 = 318 nm, and a = 10, (b) a single-period CDGAC with ΔW1 = 50 nm, ΔW2 = 0 nm, Λ = 312 nm, and a = 10, (c) a single-period CDGAC with ΔW1 = 0 nm, ΔW2 = 50 nm, Λ = 318 nm, and a = 10.

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Figure 5 (b) depicts the amplitude of the through and drop port responses of a single-period CDGAC with gratings only on the narrow waveguide with ΔW1 = 50 nm, a period of Λ = 312 nm, and an apodization strength of a = 10. The dropped channel has a central wavelength of 1551.5 nm, which is 4.4 nm (0.3%) smaller than the central wavelengths of the first drop channel of two-period configuration. In addition to the aforementioned inherent average effective index difference between two-period and single period gratings, the additional difference between the central wavelengths of the single period structure and that of the first channel of the two-period structure can be due to variation in the thickness of the core of the waveguide. A 3 nm reduction in the thickness of the core of the waveguide results in a decrease of about 2.6 nm in the central wavelength of the CDGAC. This adds up with the 2 nm central wavelength change due to average effective index difference. The 3 dB bandwidth of the single period CDGAC is 6.83 nm, which is 2% smaller than the two-period configuration. A slight change of less than 2 nm of the grating width can lead to this change in 3 dB bandwidth.

The measurement results of the single-period CDGAC with gratings on the wide waveguide is shown in Fig. 5 (c). The maximum modulation width of this configuration is 50 nm, the grating apodization factor is 10, and the period of the grating is 318 nm. According to the measurement results, the 3 dB bandwidth is 8.85 nm, which is 17% larger than the bandwidth of the second drop channel of the two-period configuration. A 6 nm increase in the grating width would cause a 15% increase in the 3 dB bandwidth of the response but not a significant change in the central wavelength. The central wavelength of the drop channel is at 1573.3 nm which is 0.08% smaller than the central wavelength of the second channel of the two-period grating. Therefore, the responses of the single-period CDGACs match the one of the two-period CDGAC within the fabrication tolerance when the impact of the variation of the average refractive index is considered. The difference in the insertion loss of Figs. 5. (a) and (b) is due to fabrication imperfection of input grating couplers of device in Fig. 5 (b) for which light could not couple in and out of the device as efficiently as Fig. 5 (a).

6. Bandwidth tailoring

It is possible to modify the grating modulation width to tailor the 3 dB bandwidths of the drop port channels to achieve identical or different bandwidths. Figure 6. depicts the amplitude of the drop and through port response of a two-period CDGAC with 3 dB bandwidths of 4.7 nm for the first channel and of 2.5 nm for the second channel. The grating with the shorter period has a maximum grating width of ΔW2 = 50 nm, whereas the grating with the longer period has a maximum grating width of ΔW2 = 24 nm. The grating with the smaller width has a smaller coupling coefficient, and therefore, it has a smaller 3 dB drop channel bandwidth.

 

Fig. 6 Transmission and drop port response of a two-period CDGAC with ΔW1 = 24 nm, ΔW2 = 50 nm, Λ1 = 318 nm, Λ2 = 312 nm, and a = 10.

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7. Conclusion

We proposed a novel integrated add-drop filter with two independent drop channels. This structure is based on a single contra-directional grating assisted coupler with two different grating periods. The device was analysed using Fourier analysis, numerical simulations, and experimental measurements to demonstrate that the overall response is the superposition of two individual grating responses. This device can reduce the footprint of two channel add-drop filter since only a single structure is required instead of cascading two add-drop filters. Moreover, the bandwidth and central wavelengths of the channels can be designed separately.

Acknowledgments

Fabrication support was provided via the Natural Sciences and Engineering Research Council of Canada (NSERC) Silicon Electronic-Photonic Integrated Circuits (SiEPIC) Program. The passive devices were fabricated by Richard Bojko at the University of Washington Nanofabrication Facility, which is part of the National Science Foundation’s National Nanotechnology Infrastructure Network (NNIN). The authors would like to thank CMC (Canada Microsystems Corporation) for their help and support.

References

1. A. D. Simard, N. Belhadj, Y. Painchaud, and S. LaRochelle, “Apodized silicon-on-insulator Bragg gratings,” IEEE Photonics Technol. Lett. 24(12), 1033–1035 (2012). [CrossRef]  

2. X. Wang, W. Shi, R. Vafaei, N. A. F. Jaeger, and L. Chrostowski, “Uniform and sampled Bragg gratings in SOI strip waveguides with sidewall corrugations,” IEEE Photonics Technol. Lett. 23(5), 290–292 (2011).

3. G. Jiang, R. Chen, Q. Zhou, J. Yang, M. Wang, and X. Jiang, “Slab-modulated sidewall Bragg gratings in silicon-on-insulator ridge waveguides,” IEEE Photonics Technol. Lett. 23(1), 6–8 (2011).

4. D. T. H. Tan, K. Ikeda, and Y. Fainman, “Cladding-modulated Bragg gratings in silicon waveguides,” Opt. Lett. 34(9), 1357–1359 (2009). [CrossRef]   [PubMed]  

5. L. Chrostowski, X. Wang, J. Flueckiger, Y. Wu, Y. Wang, and S. Talebi Fard, “Impact of fabrication non-uniformity on chip-scale silicon photonic integrated circuits,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2014), paper Th2A.37. [CrossRef]  

6. M. W. Puckett, F. Vallini, A. Grieco, and Y. Fainman, “Multichannel Bragg gratings in silicon waveguides with asymmetric sidewall modulation,” Opt. Lett. 40(3), 379–382 (2015). [CrossRef]   [PubMed]  

7. X. Wang, H. Yun, N. A. F. Jaeger, and L. Chrostowski, “Multi-period Bragg gratings in silicon waveguides,” in Proceedings of IEEE Photonics Conference (IEEE 2013), pp. 442–443.

8. A. D. Simard and S. LaRochelle, “Complex apodized Bragg grating filters without circulators in silicon-on-insulator,” Opt. Express 23(13), 16662–16675 (2015). [CrossRef]   [PubMed]  

9. W. Shi, X. Wang, C. Lin, H. Yun, Y. Liu, T. Baehr-Jones, M. Hochberg, N. A. F. Jaeger, and L. Chrostowski, “Silicon photonic grating-assisted, contra-directional couplers,” Opt. Express 21(3), 3633–3650 (2013). [CrossRef]   [PubMed]  

10. H. Qiu, G. Jiang, T. Hu, H. Shao, P. Yu, J. Yang, and X. Jiang, “FSR-free add-drop filter based on silicon grating-assisted contradirectional couplers,” Opt. Lett. 38(1), 1–3 (2013). [CrossRef]   [PubMed]  

11. D. Marcuse, “Directional couplers made of nonidentical asymmetric slabs part II: grating-assisted couplers,” J. Lightwave Technol. 5(2), 268–273 (1987). [CrossRef]  

12. W. Shi, H. Yun, C. Lin, X. Wang, J. Flueckiger, N. Jaeger, and L. Chrostowski, “Silicon CWDM demultiplexers using contra-directional couplers,” in Proceedings of CLEO, OSA Technical Digest (online) (Optical Society of America, 2013), paper CTu3F.5. [CrossRef]  

13. X. Wang, Y. Wang, J. Flueckiger, R. Bojko, A. Liu, A. Reid, J. Pond, N. A. Jaeger, and L. Chrostowski, “Precise control of the coupling coefficient through destructive interference in silicon waveguide Bragg gratings,” Opt. Lett. 39(19), 5519–5522 (2014). [CrossRef]   [PubMed]  

14. A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications (Oxford University, 2007).

15. T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15(8), 1277–1294 (1997). [CrossRef]  

References

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  1. A. D. Simard, N. Belhadj, Y. Painchaud, and S. LaRochelle, “Apodized silicon-on-insulator Bragg gratings,” IEEE Photonics Technol. Lett. 24(12), 1033–1035 (2012).
    [Crossref]
  2. X. Wang, W. Shi, R. Vafaei, N. A. F. Jaeger, and L. Chrostowski, “Uniform and sampled Bragg gratings in SOI strip waveguides with sidewall corrugations,” IEEE Photonics Technol. Lett. 23(5), 290–292 (2011).
  3. G. Jiang, R. Chen, Q. Zhou, J. Yang, M. Wang, and X. Jiang, “Slab-modulated sidewall Bragg gratings in silicon-on-insulator ridge waveguides,” IEEE Photonics Technol. Lett. 23(1), 6–8 (2011).
  4. D. T. H. Tan, K. Ikeda, and Y. Fainman, “Cladding-modulated Bragg gratings in silicon waveguides,” Opt. Lett. 34(9), 1357–1359 (2009).
    [Crossref] [PubMed]
  5. L. Chrostowski, X. Wang, J. Flueckiger, Y. Wu, Y. Wang, and S. Talebi Fard, “Impact of fabrication non-uniformity on chip-scale silicon photonic integrated circuits,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2014), paper Th2A.37.
    [Crossref]
  6. M. W. Puckett, F. Vallini, A. Grieco, and Y. Fainman, “Multichannel Bragg gratings in silicon waveguides with asymmetric sidewall modulation,” Opt. Lett. 40(3), 379–382 (2015).
    [Crossref] [PubMed]
  7. X. Wang, H. Yun, N. A. F. Jaeger, and L. Chrostowski, “Multi-period Bragg gratings in silicon waveguides,” in Proceedings of IEEE Photonics Conference (IEEE 2013), pp. 442–443.
  8. A. D. Simard and S. LaRochelle, “Complex apodized Bragg grating filters without circulators in silicon-on-insulator,” Opt. Express 23(13), 16662–16675 (2015).
    [Crossref] [PubMed]
  9. W. Shi, X. Wang, C. Lin, H. Yun, Y. Liu, T. Baehr-Jones, M. Hochberg, N. A. F. Jaeger, and L. Chrostowski, “Silicon photonic grating-assisted, contra-directional couplers,” Opt. Express 21(3), 3633–3650 (2013).
    [Crossref] [PubMed]
  10. H. Qiu, G. Jiang, T. Hu, H. Shao, P. Yu, J. Yang, and X. Jiang, “FSR-free add-drop filter based on silicon grating-assisted contradirectional couplers,” Opt. Lett. 38(1), 1–3 (2013).
    [Crossref] [PubMed]
  11. D. Marcuse, “Directional couplers made of nonidentical asymmetric slabs part II: grating-assisted couplers,” J. Lightwave Technol. 5(2), 268–273 (1987).
    [Crossref]
  12. W. Shi, H. Yun, C. Lin, X. Wang, J. Flueckiger, N. Jaeger, and L. Chrostowski, “Silicon CWDM demultiplexers using contra-directional couplers,” in Proceedings of CLEO, OSA Technical Digest (online) (Optical Society of America, 2013), paper CTu3F.5.
    [Crossref]
  13. X. Wang, Y. Wang, J. Flueckiger, R. Bojko, A. Liu, A. Reid, J. Pond, N. A. Jaeger, and L. Chrostowski, “Precise control of the coupling coefficient through destructive interference in silicon waveguide Bragg gratings,” Opt. Lett. 39(19), 5519–5522 (2014).
    [Crossref] [PubMed]
  14. A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications (Oxford University, 2007).
  15. T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15(8), 1277–1294 (1997).
    [Crossref]

2015 (2)

2014 (1)

2013 (2)

2012 (1)

A. D. Simard, N. Belhadj, Y. Painchaud, and S. LaRochelle, “Apodized silicon-on-insulator Bragg gratings,” IEEE Photonics Technol. Lett. 24(12), 1033–1035 (2012).
[Crossref]

2011 (2)

X. Wang, W. Shi, R. Vafaei, N. A. F. Jaeger, and L. Chrostowski, “Uniform and sampled Bragg gratings in SOI strip waveguides with sidewall corrugations,” IEEE Photonics Technol. Lett. 23(5), 290–292 (2011).

G. Jiang, R. Chen, Q. Zhou, J. Yang, M. Wang, and X. Jiang, “Slab-modulated sidewall Bragg gratings in silicon-on-insulator ridge waveguides,” IEEE Photonics Technol. Lett. 23(1), 6–8 (2011).

2009 (1)

1997 (1)

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15(8), 1277–1294 (1997).
[Crossref]

1987 (1)

D. Marcuse, “Directional couplers made of nonidentical asymmetric slabs part II: grating-assisted couplers,” J. Lightwave Technol. 5(2), 268–273 (1987).
[Crossref]

Baehr-Jones, T.

Belhadj, N.

A. D. Simard, N. Belhadj, Y. Painchaud, and S. LaRochelle, “Apodized silicon-on-insulator Bragg gratings,” IEEE Photonics Technol. Lett. 24(12), 1033–1035 (2012).
[Crossref]

Bojko, R.

Chen, R.

G. Jiang, R. Chen, Q. Zhou, J. Yang, M. Wang, and X. Jiang, “Slab-modulated sidewall Bragg gratings in silicon-on-insulator ridge waveguides,” IEEE Photonics Technol. Lett. 23(1), 6–8 (2011).

Chrostowski, L.

Erdogan, T.

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15(8), 1277–1294 (1997).
[Crossref]

Fainman, Y.

Flueckiger, J.

Grieco, A.

Hochberg, M.

Hu, T.

Ikeda, K.

Jaeger, N. A.

Jaeger, N. A. F.

W. Shi, X. Wang, C. Lin, H. Yun, Y. Liu, T. Baehr-Jones, M. Hochberg, N. A. F. Jaeger, and L. Chrostowski, “Silicon photonic grating-assisted, contra-directional couplers,” Opt. Express 21(3), 3633–3650 (2013).
[Crossref] [PubMed]

X. Wang, W. Shi, R. Vafaei, N. A. F. Jaeger, and L. Chrostowski, “Uniform and sampled Bragg gratings in SOI strip waveguides with sidewall corrugations,” IEEE Photonics Technol. Lett. 23(5), 290–292 (2011).

Jiang, G.

H. Qiu, G. Jiang, T. Hu, H. Shao, P. Yu, J. Yang, and X. Jiang, “FSR-free add-drop filter based on silicon grating-assisted contradirectional couplers,” Opt. Lett. 38(1), 1–3 (2013).
[Crossref] [PubMed]

G. Jiang, R. Chen, Q. Zhou, J. Yang, M. Wang, and X. Jiang, “Slab-modulated sidewall Bragg gratings in silicon-on-insulator ridge waveguides,” IEEE Photonics Technol. Lett. 23(1), 6–8 (2011).

Jiang, X.

H. Qiu, G. Jiang, T. Hu, H. Shao, P. Yu, J. Yang, and X. Jiang, “FSR-free add-drop filter based on silicon grating-assisted contradirectional couplers,” Opt. Lett. 38(1), 1–3 (2013).
[Crossref] [PubMed]

G. Jiang, R. Chen, Q. Zhou, J. Yang, M. Wang, and X. Jiang, “Slab-modulated sidewall Bragg gratings in silicon-on-insulator ridge waveguides,” IEEE Photonics Technol. Lett. 23(1), 6–8 (2011).

LaRochelle, S.

A. D. Simard and S. LaRochelle, “Complex apodized Bragg grating filters without circulators in silicon-on-insulator,” Opt. Express 23(13), 16662–16675 (2015).
[Crossref] [PubMed]

A. D. Simard, N. Belhadj, Y. Painchaud, and S. LaRochelle, “Apodized silicon-on-insulator Bragg gratings,” IEEE Photonics Technol. Lett. 24(12), 1033–1035 (2012).
[Crossref]

Lin, C.

Liu, A.

Liu, Y.

Marcuse, D.

D. Marcuse, “Directional couplers made of nonidentical asymmetric slabs part II: grating-assisted couplers,” J. Lightwave Technol. 5(2), 268–273 (1987).
[Crossref]

Painchaud, Y.

A. D. Simard, N. Belhadj, Y. Painchaud, and S. LaRochelle, “Apodized silicon-on-insulator Bragg gratings,” IEEE Photonics Technol. Lett. 24(12), 1033–1035 (2012).
[Crossref]

Pond, J.

Puckett, M. W.

Qiu, H.

Reid, A.

Shao, H.

Shi, W.

W. Shi, X. Wang, C. Lin, H. Yun, Y. Liu, T. Baehr-Jones, M. Hochberg, N. A. F. Jaeger, and L. Chrostowski, “Silicon photonic grating-assisted, contra-directional couplers,” Opt. Express 21(3), 3633–3650 (2013).
[Crossref] [PubMed]

X. Wang, W. Shi, R. Vafaei, N. A. F. Jaeger, and L. Chrostowski, “Uniform and sampled Bragg gratings in SOI strip waveguides with sidewall corrugations,” IEEE Photonics Technol. Lett. 23(5), 290–292 (2011).

Simard, A. D.

A. D. Simard and S. LaRochelle, “Complex apodized Bragg grating filters without circulators in silicon-on-insulator,” Opt. Express 23(13), 16662–16675 (2015).
[Crossref] [PubMed]

A. D. Simard, N. Belhadj, Y. Painchaud, and S. LaRochelle, “Apodized silicon-on-insulator Bragg gratings,” IEEE Photonics Technol. Lett. 24(12), 1033–1035 (2012).
[Crossref]

Tan, D. T. H.

Vafaei, R.

X. Wang, W. Shi, R. Vafaei, N. A. F. Jaeger, and L. Chrostowski, “Uniform and sampled Bragg gratings in SOI strip waveguides with sidewall corrugations,” IEEE Photonics Technol. Lett. 23(5), 290–292 (2011).

Vallini, F.

Wang, M.

G. Jiang, R. Chen, Q. Zhou, J. Yang, M. Wang, and X. Jiang, “Slab-modulated sidewall Bragg gratings in silicon-on-insulator ridge waveguides,” IEEE Photonics Technol. Lett. 23(1), 6–8 (2011).

Wang, X.

Wang, Y.

Yang, J.

H. Qiu, G. Jiang, T. Hu, H. Shao, P. Yu, J. Yang, and X. Jiang, “FSR-free add-drop filter based on silicon grating-assisted contradirectional couplers,” Opt. Lett. 38(1), 1–3 (2013).
[Crossref] [PubMed]

G. Jiang, R. Chen, Q. Zhou, J. Yang, M. Wang, and X. Jiang, “Slab-modulated sidewall Bragg gratings in silicon-on-insulator ridge waveguides,” IEEE Photonics Technol. Lett. 23(1), 6–8 (2011).

Yu, P.

Yun, H.

Zhou, Q.

G. Jiang, R. Chen, Q. Zhou, J. Yang, M. Wang, and X. Jiang, “Slab-modulated sidewall Bragg gratings in silicon-on-insulator ridge waveguides,” IEEE Photonics Technol. Lett. 23(1), 6–8 (2011).

IEEE Photonics Technol. Lett. (3)

A. D. Simard, N. Belhadj, Y. Painchaud, and S. LaRochelle, “Apodized silicon-on-insulator Bragg gratings,” IEEE Photonics Technol. Lett. 24(12), 1033–1035 (2012).
[Crossref]

X. Wang, W. Shi, R. Vafaei, N. A. F. Jaeger, and L. Chrostowski, “Uniform and sampled Bragg gratings in SOI strip waveguides with sidewall corrugations,” IEEE Photonics Technol. Lett. 23(5), 290–292 (2011).

G. Jiang, R. Chen, Q. Zhou, J. Yang, M. Wang, and X. Jiang, “Slab-modulated sidewall Bragg gratings in silicon-on-insulator ridge waveguides,” IEEE Photonics Technol. Lett. 23(1), 6–8 (2011).

J. Lightwave Technol. (2)

D. Marcuse, “Directional couplers made of nonidentical asymmetric slabs part II: grating-assisted couplers,” J. Lightwave Technol. 5(2), 268–273 (1987).
[Crossref]

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15(8), 1277–1294 (1997).
[Crossref]

Opt. Express (2)

Opt. Lett. (4)

Other (4)

A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications (Oxford University, 2007).

W. Shi, H. Yun, C. Lin, X. Wang, J. Flueckiger, N. Jaeger, and L. Chrostowski, “Silicon CWDM demultiplexers using contra-directional couplers,” in Proceedings of CLEO, OSA Technical Digest (online) (Optical Society of America, 2013), paper CTu3F.5.
[Crossref]

L. Chrostowski, X. Wang, J. Flueckiger, Y. Wu, Y. Wang, and S. Talebi Fard, “Impact of fabrication non-uniformity on chip-scale silicon photonic integrated circuits,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2014), paper Th2A.37.
[Crossref]

X. Wang, H. Yun, N. A. F. Jaeger, and L. Chrostowski, “Multi-period Bragg gratings in silicon waveguides,” in Proceedings of IEEE Photonics Conference (IEEE 2013), pp. 442–443.

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Figures (6)

Fig. 1
Fig. 1 (a) Top view of a uniform CDGAC with a grating only on one of the waveguides; (b) Top view of a uniform two-period CDGAC.
Fig. 2
Fig. 2 Longitudinal phase match condition for the two-period configuration with Λ 1 = 312 nm, Λ 2 = 318 nm, Δ W 1 = 50 nm, and Δ W 2 = 50 nm.
Fig. 3
Fig. 3 Fourier coefficients of the two-period configuration with Λ 1 = 312 nm and Λ 2 = 318 nm having Ν p 3 = 20 super-periods.
Fig. 4
Fig. 4 Simulation results of a uniform two-period CDGAC with Δ W 1 = Δ W 2 = 50 nm, Λ 1 = 312 nm, Λ 2 = 318 nm, and Ν p 3 = 20.
Fig. 5
Fig. 5 Through and drop port response of (a) two-period CDGAC with Δ W 1 = Δ W 2 = 50 nm, Λ 1 = 312 nm, Λ 2 = 318 nm, and a = 10, (b) a single-period CDGAC with Δ W 1 = 50 nm, Δ W 2 = 0 nm, Λ = 312 nm, and a = 10, (c) a single-period CDGAC with Δ W 1 = 0 nm, Δ W 2 = 50 nm, Λ = 318 nm, and a = 10.
Fig. 6
Fig. 6 Transmission and drop port response of a two-period CDGAC with Δ W 1 = 24 nm, Δ W 2 = 50 nm, Λ 1 = 318 nm, Λ 2 = 312 nm, and a = 10.

Equations (13)

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

β 1 ( λ 0 ) + β 2 ( λ 0 ) = 2 m π Λ 0
2 n e f f 1 a v e ( 2 π λ ) = m 2 π Λ 3
2 n e f f 2 a v e ( 2 π λ ) = m 2 π Λ 3
( n e f f 1 a v e + n e f f 2 a v e ) ( 2 π λ ) = m 2 π Λ 3
c ( z ) = a ( z ) p 1 ( z ) + b ( z ) p 2 ( z )
C ( ω ) = ( A ( ω ) + B ( ω ) ) * P ( ω ) = k A k P ( ω k 2 π Λ 1 ) + k B k P ( ω k 2 π Λ 2 )
ω 1 = 2 π Λ 1 = Λ 2 ( Λ 1 , Λ 2 ) × 2 π Λ 3 = Λ 2 ( Λ 1 , Λ 2 ) × ω 3 = l 1 ω 3
ω 2 = 2 π Λ 2 = Λ 1 ( Λ 1 , Λ 2 ) × 2 π Λ 3 = Λ 1 ( Λ 1 , Λ 2 ) × ω 3 = l 2 ω 3
C ( ω ) = k A k P ( ω k l 1 ω 3 ) + k B k P ( ω k l 2 ω 3 )
C m = C ( ω m = m ω 3 ) = k A k P ( ( m k l 1 ) ω 3 ) + k B k P ( ( m k l 2 ) ω 3 )
C m = k A k e j ω 3 ( m k l 1 ) sin c ( ( m k l 1 ) Ν p 3 ) + k B k e j ω 3 ( m k l 2 ) sin c ( ( m k l 2 ) Ν p 3 )
k c = ω ε 0 4 E 1 Δ ε m E 2 d x d y
Δ W = Δ W 0 e a ( x 0.5 N p N p ) 2

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