## Reduction of chromatic dispersion due to coupling for synchronized-router-based flat-passband filter using multiple-input arrayed waveguide grating

Optics Express, Vol. 17, Issue 24, pp. 22260-22270 (2009)

http://dx.doi.org/10.1364/OE.17.022260

Acrobat PDF (261 KB)

### Abstract

An approach to reducing the chromatic dispersion due to coupling between input waveguides before the input slab for a synchronized-router-based flat-passband filter using a multiple-input arrayed waveguide grating (AWG) is proposed. The proposed method uses phase compensation at the waveguide array of the AWG by correction of waveguide lengths. The characteristics of the flat-passband filter that consists of a multiple-input AWG combined with cascaded Mach-Zehnder interferometers (MZIs) are simulated using a theoretical model of the multiple-input AWG based on Fourier optics and the coupled-mode theory. The simulation result reveals that the chromatic dispersion within the passband can be significantly reduced by using phase compensation and additional dummy waveguides at the input just before the slab.

© 2009 OSA

## 1. Introduction

1. K. Okamoto and H. Yamada, “Arrayed-waveguide grating multiplexer with flat spectral response,” Opt. Lett. **20**(1), 43–45 (
1995). [CrossRef] [PubMed]

13. C. R. Doerr, M. A. Cappuzzo, E. Y. Chen, A. Wong-Foy, L. T. Gomez, and L. L. Buhl, “Wideband arrayed waveguide grating with three low-loss maxima per passband,” IEEE Photon. Technol. Lett. **18**(21), 2308–2310 (
2006). [CrossRef]

4. C. Dragone, “Efficient techniques for widening the passband of a wavelength router,” J. Lightwave Technol. **16**(10), 1895–1906 (
1998). [CrossRef]

8. C. R. Doerr, L. W. Stulz, R. Pafchek, and S. Shunk, “Compact and low-loss manner of waveguide grating router passband flattening and demonstration in a 64-channel blocker/multiplexer,” IEEE Photon. Technol. Lett. **14**(1), 56–58 (
2002). [CrossRef]

13. C. R. Doerr, M. A. Cappuzzo, E. Y. Chen, A. Wong-Foy, L. T. Gomez, and L. L. Buhl, “Wideband arrayed waveguide grating with three low-loss maxima per passband,” IEEE Photon. Technol. Lett. **18**(21), 2308–2310 (
2006). [CrossRef]

8. C. R. Doerr, L. W. Stulz, R. Pafchek, and S. Shunk, “Compact and low-loss manner of waveguide grating router passband flattening and demonstration in a 64-channel blocker/multiplexer,” IEEE Photon. Technol. Lett. **14**(1), 56–58 (
2002). [CrossRef]

12. C. R. Doerr, M. A. Cappuzzo, E. Y. Chen, A. Wong-Foy, and L. T. Gomez, “Low-loss rectangular-passband multiplexer consisting of a waveguide grating router synchronized to a three-arm interferometer,” IEEE Photon. Technol. Lett. **17**(11), 2334–2336 (
2005). [CrossRef]

13. C. R. Doerr, M. A. Cappuzzo, E. Y. Chen, A. Wong-Foy, L. T. Gomez, and L. L. Buhl, “Wideband arrayed waveguide grating with three low-loss maxima per passband,” IEEE Photon. Technol. Lett. **18**(21), 2308–2310 (
2006). [CrossRef]

14. K. Maru, T. Mizumoto, and H. Uetsuka, “Modeling of multi-input arrayed waveguide grating and its application to design of flat-passband response using cascaded Mach-Zehnder interferometers,” J. Lightwave Technol. **25**(2), 544–555 (
2007). [CrossRef]

16. C. Dragone, “Efficient N x N star couplers using Fourier optics,” J. Lightwave Technol. **7**(3), 479–489 (
1989). [CrossRef]

18. P. Muñoz, D. Pastor, and J. Capmany, “Modeling and design of arrayed waveguide gratings,” J. Lightwave Technol. **20**(4), 661–674 (
2002). [CrossRef]

19. K. Maru, T. Mizumoto, and H. Uetsuka, “Demonstration of flat-passband multi/demultiplexer using multi-input arrayed waveguide grating combined with cascaded Mach-Zehnder interferometers,” J. Lightwave Technol. **25**(8), 2187–2197 (
2007). [CrossRef]

20. K. Maru, T. Mizumoto, and H. Uetsuka, “Super-high-Δ silica-based flat-passband filter using AWG and cascaded Mach-Zehnder interferometers,” in *Proceedings of 12th Optoelectronics and Communications Conf./16th International Conf. on Integrated Optics and Optical Fiber Communication (OECC/IOOC**2007**)*, 12E4–3.

14. K. Maru, T. Mizumoto, and H. Uetsuka, “Modeling of multi-input arrayed waveguide grating and its application to design of flat-passband response using cascaded Mach-Zehnder interferometers,” J. Lightwave Technol. **25**(2), 544–555 (
2007). [CrossRef]

19. K. Maru, T. Mizumoto, and H. Uetsuka, “Demonstration of flat-passband multi/demultiplexer using multi-input arrayed waveguide grating combined with cascaded Mach-Zehnder interferometers,” J. Lightwave Technol. **25**(8), 2187–2197 (
2007). [CrossRef]

21. C. R. Doerr, L. W. Stulz, R. Pafchek, L. Gomez, M. Cappuzzo, A. Paunescu, E. Laskowski, L. Buhl, H. K. Kim, and S. Chandrasekhar, “An automatic 40-wavelength channelized equalizer,” IEEE Photon. Technol. Lett. **12**(9), 1195–1197 (
2000). [CrossRef]

## 2. Principle

### 2.1 Structure

*f*and a cascaded MZI structure [22

_{FSR}22. N. Takato, K. Jinguji, M. Yasu, H. Toba, and M. Kawachi, “Silica-based single-mode waveguides on silicon and their application to guided-wave optical interferometers,” J. Lightwave Technol. **6**(6), 1003–1010 (
1988). [CrossRef]

*M*input waveguides and a waveguide array consisting of 2

*I*+ 1 waveguides. Here, the length of the waveguides in the waveguide array is slightly changed for phase compensation from the normal design.

14. K. Maru, T. Mizumoto, and H. Uetsuka, “Modeling of multi-input arrayed waveguide grating and its application to design of flat-passband response using cascaded Mach-Zehnder interferometers,” J. Lightwave Technol. **25**(2), 544–555 (
2007). [CrossRef]

*M*= 4) to achieve a small chip size as well as sufficient flatness. The two slabs in the AWG have the same focal length

*z*. The waveguides in the array are connected to the edges of the two slabs with a waveguide interval of

*d*. The signals from a first-stage MZI are demultiplexed by second-stage MZIs by setting the FSR of the second-stage MZIs, Δ

*f*, to twice that of the first-stage one. The signals with four equally spaced frequencies

_{MZI}*f*

_{1}, …,

*f*

_{4}within one FSR of the second-stage MZIs are first divided by the first-stage MZI between the two groups

*f*

_{1},

*f*

_{3}and

*f*

_{2},

*f*

_{4}, and next divided by the second-stage MZIs into individual signals. The lower port of the upper second-stage MZI and the upper port of the lower second-stage MZI should cross each other so that the signals

*f*

_{1}, …,

*f*

_{4}are spatially arranged in this order at the input side of the AWG. To obtain an appropriate demultiplexing function, the channel spacing of the AWG should be the same value as the FSR of the final-stage MZIs, i.e., to Δ

*f*.

_{MZI}### 2.2 Phase compensation

**25**(2), 544–555 (
2007). [CrossRef]

*δ*. The field distribution at the edge of the input slab illuminated by the

*m*-th output of the cascaded MZI structure is expressed aswhere

*u*(

^{o}_{in}*x*) is the mode field without coupling,

*x*is the position of the

_{m}*m*-th input waveguide along the edge of the input slab, Δ

*x*is the constant interval between two adjacent input waveguides, and

*χ*is the relative amplitude of path-through light.

*i*-th waveguide (

*l*for phase compensation aswhere Δ

_{i}*L*is the constant difference in length between adjacent waveguides in the array and

*L*

^{0}is the length of the 0th waveguide without phase compensation. The phase delay of the

*i*-th waveguide for phase compensation,

*θ*, is expressed aswhere

_{i}*n*is the effective refractive index of the waveguides in the array and

_{a}*λ*

_{0}is the central wavelength. Here, the contribution of the phase compensation Θ(

*y*) is defined aswhere Δ

*y*is defined as Δ

*y*=

*λ*

_{0}

*z*/(

*n*) and

_{s}d*n*is the effective refractive index of the slab waveguide.

_{s}**25**(2), 544–555 (
2007). [CrossRef]

*n*-th output waveguide located at

*y*along the edge of the output slab,

_{n}*t*(

*y*;

_{n}*f*), is derived as the convolution between the transfer function without phase compensation and the contribution of the phase compensation Θ(

*y*) asandwhere

*D*(

_{N}*x*) is the Dirichlet kernel

*D*(

_{N}*x*) = sin(

*Nπx*)/sin(

*πx*) [25],

*E*(

*x*;

_{m}*f*) is the amplitude as a function of the optical frequency

*f*for the

*m*-th input waveguide connected at the position

*x*along the edge of the input slab,

_{m}*u*(

_{out}*y*) is the mode field function of the output waveguide at the interface to the output slab, and

*f*(

_{a}*x*) and

*f*(

_{b}*y*) are the images of the input-side and output-side mode field functions of a single waveguide in an array produced on the input and output edges of the slabs.

*g*

_{1}(

*y*) and

*g*

_{2}(

*y*) with a period of Δ

*y*, i.e.,When the mode field function,

*u*(

_{in}*y*), is sufficiently narrow so that the energy is substantially limited over the interval of the convolution, -Δ

*y*/2 <

*y*< Δ

*y*/2, the first convolution of the right hand in Eq. (5) is reduced towhere

*U*(

_{in}*id*) is the Fourier transform of

*u*(

_{in}*x*), i.e.,where

*k*is the wave number defined as

*k*= 2

*πn*

_{s}/λ_{0}. Using Eq. (1), when there is a coupling to adjacent input waveguides,

*U*(

_{in}*id*) is reduced towhere

*U*(

^{0}_{in}*id*) is the Fourier transform of

*u*(

^{o}_{in}*x*). From Eq. (10), the distortion in phase delay due to the coupling is regarded as tan

^{−1}[2

*δ*cos(2

*πi*Δ

*x*/Δ

*y*)/

*χ*]. Hence, to reduce the chromatic dispersion due to the coupling, the phase delay for phase compensation should bewhere

*θ’*is the constant. Therefore, to improve chromatic dispersion due to coupling characterized with the parameters

*δ*and

*χ*, the waveguides in the array should be lengthened by

*l*so as to give the phase change shown as Eq. (11).

_{i}## 3. Model for coupling before input slab

26. A. Yariv, “Coupled-mode theory for guided-wave optics,” J. Quantum Electron. **9**(9), 919–933 (
1973). [CrossRef]

19. K. Maru, T. Mizumoto, and H. Uetsuka, “Demonstration of flat-passband multi/demultiplexer using multi-input arrayed waveguide grating combined with cascaded Mach-Zehnder interferometers,” J. Lightwave Technol. **25**(8), 2187–2197 (
2007). [CrossRef]

*E*(

*x*;

_{m}*f*) be the amplitude of the

*m-*th input waveguide (

*E*(

*x*;

_{m}*f*) is given by [14

**25**(2), 544–555 (
2007). [CrossRef]

*n’*is the arbitrary output port number and

*T*is the

*M*x

*M*transfer matrix of input waveguides between cascaded MZIs and the input slab including coupling. Here, the uniform coupling between

*M*parallel and identical waveguides with a length

*L*is considered as a simple model of the coupling before the input slab. Under this simplification,

*T*is derived from a superposition of supermodes of the

*M*waveguides [27

27. E. Kapon, J. Katz, and A. Yariv, “Supermode analysis of phase-locked arrays of semiconductor lasers,” Opt. Lett. **9**(4), 125–127 (
1984). [CrossRef] [PubMed]

*β*, the coupling coefficient be

*κ*, and the transfer function

*T*=

*e*[

^{-jβL}*t*] (

_{ij}*t*for

_{ij}*M*= 4 is given as a function of

*κL*by [19

**25**(8), 2187–2197 (
2007). [CrossRef]

*π*/2 radians as the light is coupled to the next adjacent waveguide when

*κL*is small. When

*κL*is assumed to be small enough to satisfy the approximations

*T*can be approximated byThat is, the relative amplitude coupled to nearest adjacent waveguides

*t*(|

_{ij}*i*–

*j*| = 1) is approximately –

*jκL*. Hence, compared with Eq. (1), phase delay due to the coupling to nearest adjacent waveguides is compensated when

*x*

_{-1}and

*x*are the positions of the dummy waveguides along the edge of the input slab. In this case, the dimension of the transfer matrix

_{M}*T*becomes (

*M*+ 2) x

*M*. When

*κL*is assumed to be as small as that in the derivation of Eq. (16), the transfer matrix

*T*for

*M*= 4 with the dummy waveguides is given by

## 4. Simulation results and discussion

**25**(2), 544–555 (
2007). [CrossRef]

*I*+ 1 is defined as the number of waveguides in an array perfectly occupies one Brillouin zone [28

_{P}28. C. R. Doerr, M. Cappuzzo, E. Laskowski, A. Paunescu, L. Gomez, L. W. Stulz, and J. Gates, “Dynamic wavelength equalizer in silica using the single-filtered-arm interferometer,” IEEE Photon. Technol. Lett. **11**(5), 581–583 (
1999). [CrossRef]

*u*(

^{o}_{in}*y*) and

*u*(

_{out}*y*) aswhere

*w*is the spot size and

_{u}*η*is the coupling efficiency between

*u*(

^{o}_{in}*y*) and

*u*(

_{out}*y*).

*κL*without phase compensation. When coupling occurs, the chromatic dispersion is no longer zero and the maximum absolute value within the passband increases as

*κL*increases. Three valleys in the chromatic dispersion around the frequencies of the transition of light from one input waveguide to the next (i.e.,

*f/*Δ

*f*= 0, ± 0.25) are caused by the change in the phase of the light launched into the input slab due to the dependence of the amplitude of the coupled light on frequency. The transmittance around the passband also decreases as the coupling coefficient

_{MZI}*κL*increases.

*κL*= 0.2 and 0.4 correspond to the changes in the optical path lengths of about 0.19 and 0.38 μm at the wavelength of 1.55 μm, respectively. It implies that the fabrication resolution in optical path length of the order of a few tens of nanometers would be required. The transmittance and chromatic dispersion of the flat-passband filters with and without phase compensation for

*κL*= 0.2 and 0.4 are plotted in Figs. 4 and 5 . The chromatic dispersion within the passband can be significantly reduced by using phase compensation. The chromatic dispersion within the passband of ± 0.35 x Δ

*f*becomes –9.4 to 0.5 ps/nm by using phase compensation, whereas that is –19.7 to 5.4 ps/nm without phase compensation when

_{MZI}*κL*= 0.2. The chromatic dispersion near the edge of the passband is still larger than that around the center of the passband. This is because the coupling is asymmetrical when the light mainly propagates either side of the four input waveguides (i.e., the 0th or 3rd waveguide) and it leads to the deviation of the phase distortion from the compensated values. The amount of chromatic dispersion below 10 ps/nm may be allowable for some applications such as point-to-point communications with a bit rate below 10 Gb/s. However, when the filter is used for applications in which many multi/demultiplexers are cascaded (e.g., optical cross-connect), this amount of chromatic dispersion may be not allowable, especially, for high bit-rate signals such as 40 Gb/s.

*κL*= 0.2 and 0.4 are plotted in Figs. 7 and 8 . Compared with the result in Figs. 4 and 5, the chromatic dispersion around the edge of the passband can be further reduced to –0.3 to 0.1 ps/nm by using the additional dummy waveguides when

*κL*= 0.2.

*κL*values even if the phase is compensated because the approximations

**25**(8), 2187–2197 (
2007). [CrossRef]

*δ*and

*χ*in Eq. (11) as far as the coupling at this portion is small.

30. K. Takada, T. Tanaka, M. Abe, T. Yanagisawa, M. Ishii, and K. Okamoto, “Beam-adjustment-free crosstalk reduction in a 10 GHz-spaced arrayed-waveguide grating via photosensitivity under UV laser irradiation through metal mask,” Electron. Lett. **36**(1), 60–61 (
2000). [CrossRef]

## 5. Conclusion

*f*can be significantly reduced to –9.4 to 0.5 ps/nm by using phase compensation, whereas that is –19.7 to 5.4 ps/nm without phase compensation when

_{MZI}*κL*= 0.2. The chromatic dispersion can be further reduced to –0.3 to 0.1 ps/nm by using additional dummy waveguides.

## Acknowledgment

## References and links

1. | K. Okamoto and H. Yamada, “Arrayed-waveguide grating multiplexer with flat spectral response,” Opt. Lett. |

2. | M. R. Amersfoort, J. B. D. Soole, H. P. LeBlanc, N. C. Andreadakis, A. Rajhel, and C. Caneau, “Passband broadening of integrated arrayed waveguide filters using multimode interference couplers,” Electron. Lett. |

3. | K. Okamoto and A. Sugita, “Flat spectral response arrayed-waveguide grating multiplexer with parabolic waveguide horns,” Electron. Lett. |

4. | C. Dragone, “Efficient techniques for widening the passband of a wavelength router,” J. Lightwave Technol. |

5. | G. H. B. Thompson, R. Epworth, C. Rogers, S. Day, and S. Ojha, “An original low-loss and pass-band flattened SiO |

6. | T. Kamalakis and T. Sphicopoulos, “An efficient technique for the design of an arrayed-waveguide grating with flat spectral response,” J. Lightwave Technol. |

7. | J.-J. He, “Phase-dithered waveguide grating with flat passband and sharp transitions,” J. Select. Topics Quantum Electron. |

8. | C. R. Doerr, L. W. Stulz, R. Pafchek, and S. Shunk, “Compact and low-loss manner of waveguide grating router passband flattening and demonstration in a 64-channel blocker/multiplexer,” IEEE Photon. Technol. Lett. |

9. | M. Kohtoku, H. Takahashi, I. Kitoh, I. Shibata, Y. Inoue, and Y. Hibino, “Low-loss flat-top passband arrayed waveguide gratings realised by first-order mode assistance method,” Electron. Lett. |

10. | C. Dragone, “Theory of wavelength multiplexing with rectangular transfer functions,” J. Select. Topics Quantum Electron. |

11. | C. R. Doerr, R. Pafchek, and L. W. Stulz, “Integrated band demultiplexer using waveguide grating routers,” IEEE Photon. Technol. Lett. |

12. | C. R. Doerr, M. A. Cappuzzo, E. Y. Chen, A. Wong-Foy, and L. T. Gomez, “Low-loss rectangular-passband multiplexer consisting of a waveguide grating router synchronized to a three-arm interferometer,” IEEE Photon. Technol. Lett. |

13. | C. R. Doerr, M. A. Cappuzzo, E. Y. Chen, A. Wong-Foy, L. T. Gomez, and L. L. Buhl, “Wideband arrayed waveguide grating with three low-loss maxima per passband,” IEEE Photon. Technol. Lett. |

14. | K. Maru, T. Mizumoto, and H. Uetsuka, “Modeling of multi-input arrayed waveguide grating and its application to design of flat-passband response using cascaded Mach-Zehnder interferometers,” J. Lightwave Technol. |

15. | K. Maru, T. Mizumoto, and H. Uetsuka, “Flat-passband arrayed waveguide grating employing cascaded Mach-Zehnder interferometers,” in |

16. | C. Dragone, “Efficient N x N star couplers using Fourier optics,” J. Lightwave Technol. |

17. | H. Takenouchi, H. Tsuda, and T. Kurokawa, “Analysis of optical-signal processing using an arrayed-waveguide grating,” Opt. Express |

18. | P. Muñoz, D. Pastor, and J. Capmany, “Modeling and design of arrayed waveguide gratings,” J. Lightwave Technol. |

19. | K. Maru, T. Mizumoto, and H. Uetsuka, “Demonstration of flat-passband multi/demultiplexer using multi-input arrayed waveguide grating combined with cascaded Mach-Zehnder interferometers,” J. Lightwave Technol. |

20. | K. Maru, T. Mizumoto, and H. Uetsuka, “Super-high-Δ silica-based flat-passband filter using AWG and cascaded Mach-Zehnder interferometers,” in |

21. | C. R. Doerr, L. W. Stulz, R. Pafchek, L. Gomez, M. Cappuzzo, A. Paunescu, E. Laskowski, L. Buhl, H. K. Kim, and S. Chandrasekhar, “An automatic 40-wavelength channelized equalizer,” IEEE Photon. Technol. Lett. |

22. | N. Takato, K. Jinguji, M. Yasu, H. Toba, and M. Kawachi, “Silica-based single-mode waveguides on silicon and their application to guided-wave optical interferometers,” J. Lightwave Technol. |

23. | B. H. Verbeek, C. H. Henry, N. A. Olsson, K. J. Orlowsky, R. F. Kazarinov, and B. H. Johnson, “Integrated four-channel Mach-Zehnder multi/demultiplexer fabricated with phosphorous doped SiO |

24. | C. K. Madsen, and J. H. Zhao, |

25. | H. Dym, and H. P. McKean, |

26. | A. Yariv, “Coupled-mode theory for guided-wave optics,” J. Quantum Electron. |

27. | E. Kapon, J. Katz, and A. Yariv, “Supermode analysis of phase-locked arrays of semiconductor lasers,” Opt. Lett. |

28. | C. R. Doerr, M. Cappuzzo, E. Laskowski, A. Paunescu, L. Gomez, L. W. Stulz, and J. Gates, “Dynamic wavelength equalizer in silica using the single-filtered-arm interferometer,” IEEE Photon. Technol. Lett. |

29. | I. Kaminow, and T. Li, |

30. | K. Takada, T. Tanaka, M. Abe, T. Yanagisawa, M. Ishii, and K. Okamoto, “Beam-adjustment-free crosstalk reduction in a 10 GHz-spaced arrayed-waveguide grating via photosensitivity under UV laser irradiation through metal mask,” Electron. Lett. |

**OCIS Codes**

(130.0130) Integrated optics : Integrated optics

(230.7390) Optical devices : Waveguides, planar

(080.1238) Geometric optics : Array waveguide devices

(130.2755) Integrated optics : Glass waveguides

(130.7408) Integrated optics : Wavelength filtering devices

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: October 5, 2009

Revised Manuscript: November 6, 2009

Manuscript Accepted: November 17, 2009

Published: November 20, 2009

**Citation**

Koichi Maru and Yusaku Fujii, "Reduction of chromatic dispersion due to coupling for synchronized-router-based flat-passband filter using multiple-input arrayed waveguide grating," Opt. Express **17**, 22260-22270 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-24-22260

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### References

- K. Okamoto and H. Yamada, “Arrayed-waveguide grating multiplexer with flat spectral response,” Opt. Lett. 20(1), 43–45 (1995). [CrossRef] [PubMed]
- M. R. Amersfoort, J. B. D. Soole, H. P. LeBlanc, N. C. Andreadakis, A. Rajhel, and C. Caneau, “Passband broadening of integrated arrayed waveguide filters using multimode interference couplers,” Electron. Lett. 32(5), 449–451 (1996). [CrossRef]
- K. Okamoto and A. Sugita, “Flat spectral response arrayed-waveguide grating multiplexer with parabolic waveguide horns,” Electron. Lett. 32(18), 1661–1662 (1996). [CrossRef]
- C. Dragone, “Efficient techniques for widening the passband of a wavelength router,” J. Lightwave Technol. 16(10), 1895–1906 (1998). [CrossRef]
- G. H. B. Thompson, R. Epworth, C. Rogers, S. Day, and S. Ojha, “An original low-loss and pass-band flattened SiO2 on Si planar wavelength demultiplexer,” in Proceedings of Optical Fiber Communication Conference (OFC ’98), p. 77.
- T. Kamalakis and T. Sphicopoulos, “An efficient technique for the design of an arrayed-waveguide grating with flat spectral response,” J. Lightwave Technol. 19(11), 1716–1725 (2001). [CrossRef]
- J.-J. He, “Phase-dithered waveguide grating with flat passband and sharp transitions,” J. Select. Topics Quantum Electron. 8(6), 1186–1193 (2002). [CrossRef]
- C. R. Doerr, L. W. Stulz, R. Pafchek, and S. Shunk, “Compact and low-loss manner of waveguide grating router passband flattening and demonstration in a 64-channel blocker/multiplexer,” IEEE Photon. Technol. Lett. 14(1), 56–58 (2002). [CrossRef]
- M. Kohtoku, H. Takahashi, I. Kitoh, I. Shibata, Y. Inoue, and Y. Hibino, “Low-loss flat-top passband arrayed waveguide gratings realised by first-order mode assistance method,” Electron. Lett. 38(15), 792–794 (2002). [CrossRef]
- C. Dragone, “Theory of wavelength multiplexing with rectangular transfer functions,” J. Select. Topics Quantum Electron. 8(6), 1168–1178 (2002). [CrossRef]
- C. R. Doerr, R. Pafchek, and L. W. Stulz, “Integrated band demultiplexer using waveguide grating routers,” IEEE Photon. Technol. Lett. 15(8), 1088–1090 (2003). [CrossRef]
- C. R. Doerr, M. A. Cappuzzo, E. Y. Chen, A. Wong-Foy, and L. T. Gomez, “Low-loss rectangular-passband multiplexer consisting of a waveguide grating router synchronized to a three-arm interferometer,” IEEE Photon. Technol. Lett. 17(11), 2334–2336 (2005). [CrossRef]
- C. R. Doerr, M. A. Cappuzzo, E. Y. Chen, A. Wong-Foy, L. T. Gomez, and L. L. Buhl, “Wideband arrayed waveguide grating with three low-loss maxima per passband,” IEEE Photon. Technol. Lett. 18(21), 2308–2310 (2006). [CrossRef]
- K. Maru, T. Mizumoto, and H. Uetsuka, “Modeling of multi-input arrayed waveguide grating and its application to design of flat-passband response using cascaded Mach-Zehnder interferometers,” J. Lightwave Technol. 25(2), 544–555 (2007). [CrossRef]
- K. Maru, T. Mizumoto, and H. Uetsuka, “Flat-passband arrayed waveguide grating employing cascaded Mach-Zehnder interferometers,” in Proceedings of 11th Optoelectronics and Communications Conf. (OECC2006), 5B2–5.
- C. Dragone, “Efficient N x N star couplers using Fourier optics,” J. Lightwave Technol. 7(3), 479–489 (1989). [CrossRef]
- H. Takenouchi, H. Tsuda, and T. Kurokawa, “Analysis of optical-signal processing using an arrayed-waveguide grating,” Opt. Express 6(6), 124–135 (2000). [CrossRef] [PubMed]
- P. Muñoz, D. Pastor, and J. Capmany, “Modeling and design of arrayed waveguide gratings,” J. Lightwave Technol. 20(4), 661–674 (2002). [CrossRef]
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