## Analysis and design of arrayed waveguide gratings with MMI couplers

Optics Express, Vol. 9, Issue 7, pp. 328-338 (2001)

http://dx.doi.org/10.1364/OE.9.000328

Acrobat PDF (511 KB)

### Abstract

We present an extension of the AWG model and design procedure described in [

© Optical Society of America

## 1 Introduction

2. H. Takenouchi, H. Tsuda, and T. Kurokawa, “Analysis of optical-signal processing using an arrayed-waveguide grating,” Opt. Express **6**124–135 (2000), http://www.opticsexpress.org/oearchive/source/19103.htm [CrossRef] [PubMed]

3. K. Okamoto and A. Sugita, “Flat spectral response arrayed-waveguide grating multiplexer with parabolic waveguide horns,” Electron. Lett. **32**, 1661–1662 (1996). [CrossRef]

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

5. M.K. Smit and C. van Dam, “PHASAR-based WDM-devices: Principles, design and applications,” J. Sel. Top. Quant. Electron. **2**, 236–250 (1996). [CrossRef]

6. J. Soole e.a., “Use of multimode interference couplers to broaden the passband of wavelength-dispersive integrated WDM filters,” IEEE Photon. Technol. Lett. **8**, 1340–1342 (1996). [CrossRef]

## 2 AWG Theoretical Model Review

9. C. Dragone e.a., “Efficient N×N star couplers using Fourier optics,” J. Light. Technol. **7**, 479–489 (1989). [CrossRef]

10. C. Dragone e.a., “Efficient multichannel integrated optics star coupler on silicon,” IEEE Photon. Technol. Lett. **1**, 241–243 (1989). [CrossRef]

11. C. Dragone, C. Edwards, and R. Kistler, “Integrated optics N×N multiplexer on silicon,” IEEE Photon. Technol. Lett. **3**, 896–898 (1991). [CrossRef]

*l*, with respect to that preceeding it. The inset on the upper left corner of Fig. 1 shows the waveguide layout with its corresponding parameters, the waveguide width,

*W*, the gap between waveguides,

_{x}*G*, and the waveguide spacing,

_{x}*d*, where

_{x}*x*=

*i*,

*w*,

*o*, corresponding to IW’s, AW’s and OW’s respectively. The inset on the upper right side of the figure, shows the FPR’s layout. It consists of two sets of waveguides, the IW’s (OW’s) and the AW’s. The AW’s are positioned over a circumference of radius

*L*, which is called the focal length, whose center is located in the central IW (OW), CIW (COW). The rest of the IW’s are located over a circumference of diameter

_{f}*L*, called the

_{f}*Rowland circle*[5

5. M.K. Smit and C. van Dam, “PHASAR-based WDM-devices: Principles, design and applications,” J. Sel. Top. Quant. Electron. **2**, 236–250 (1996). [CrossRef]

2. H. Takenouchi, H. Tsuda, and T. Kurokawa, “Analysis of optical-signal processing using an arrayed-waveguide grating,” Opt. Express **6**124–135 (2000), http://www.opticsexpress.org/oearchive/source/19103.htm [CrossRef] [PubMed]

5. M.K. Smit and C. van Dam, “PHASAR-based WDM-devices: Principles, design and applications,” J. Sel. Top. Quant. Electron. **2**, 236–250 (1996). [CrossRef]

7. H. Takahashi e.a., “Transmission characterisitics of arrayed waveguide N×N wavelength multiplexer,” J. Light. Tech. **13**, 447–455 (1995). [CrossRef]

2. H. Takenouchi, H. Tsuda, and T. Kurokawa, “Analysis of optical-signal processing using an arrayed-waveguide grating,” Opt. Express **6**124–135 (2000), http://www.opticsexpress.org/oearchive/source/19103.htm [CrossRef] [PubMed]

*α*is:

*c*the light speed,

*L*the focal length of the FPR’s,

_{f}*n*its refractive index and

_{s}*ν*

_{0}the central design frequency of the AWG, i.e., the frequency corresponding to the center of the passing band from the CIW to the central output waveguide, COW. The total field distribution for the arrayed waveguides can be derived from the summation of the fundamental modes in the waveguides, each one weighted by a factor corresponding to the overlap integral between Eq. (2) and the following expression[13]:

*ω*the modal field radius of the AW’s. For a set of

_{g}*N*the field distribution over

*x*

_{1}can be rewritten as follows:

*δ*(

_{ω}*x*

_{1}) is a summation of delta function

*l*, is set to an integer multiple,

*m*, of the design wavelength in the waveguides:

*m*is known as the grating order, and

*n*

_{c}is the refraction index in the waveguides. The value of Δ

*l*ensures that the light wave from the CIW (

*p*=0), focuses on to the central output waveguide, COW (

*q*=0), at the design frequency

*ν*

_{0}. The constant length increment between consecutive waveguides, is incorporated into Eq. (5) to yield the field distribution over

*x*

_{2}:

*ϕ*(

*x*

_{2},

*ν*) is defined as:

*x*

_{3}in front of the OW’s, the Fourier transform of Eq. (9) is used to yield:

*γ*is the

*frequency-spatial dispersion parameter*, FSDP, relating the temporal frequencies of the input waveform to the spatial position at the output plane:

*B*(

_{g}*x*

_{3}) is the Fourier transform of the modal field expression in the AW’s:

*f*(

_{M}*x*

_{3}) is the Fourier transform of the truncated Gaussian function, corresponding to the first two terms at the left hand side of Eq. (9):

*B*(

_{g}*x*

_{3}). The baseline temporal delay of the waveforms travelling through the AWG is incorporated by

*ψ*(

*ν*). The summation term on the right hand side of the equation, is responsible of the shape of the passing bands, and the spatial repetition of the response over the focal plane of the second FPR, depending both on the space coordinate

*x*

_{3}and the frequency. The spatial repetition period of the response is called

*Spatial Free Spectral Range*, SFSR:

*Frequency Free Spectral Range*is the frequency difference between two adjacent spatial diffraction orders that makes them focus to the same point in the

*x*

_{3}plane [1]:

*ν*

_{FSR}_{,0}is the FSR for the AWG design frequency,

*ν*

_{0}. This expression illustrates how for a given diffraction order,

*r*, different frequencies focus to different points on the

*x*

_{3}plane. For

*ν*=

*ν*

_{0}, the order

*r*=

*m*is focused to the COW, as pointed previously.

*x*

_{3}, described by Eq. (18):

*q*is the OW number,

*d*

_{o}is the OW spacing and

*β*

_{o}(

*x*

_{3}) is the OW mode profile similar to Eq. (1). This expression corresponds to the field transmission coefficiente from the CIW, number 0, to an arbitrary OW,

*q*. It is possible to derive an expression for an arbitrary pair of IW-OW,

*t*(

_{p,q}*ν*) [1].

## 3 IW’s with MMI couplers

6. J. Soole e.a., “Use of multimode interference couplers to broaden the passband of wavelength-dispersive integrated WDM filters,” IEEE Photon. Technol. Lett. **8**, 1340–1342 (1996). [CrossRef]

*L*:

_{m}_{0}and ζ

_{1}are the effective indices of the fundamental and first order MMI coupler modes respectively. With this configuration, a center fed Gaussian field like the one in Eq. (1), is converted into a double Gaussian one, whose normalized power expression is [6

6. J. Soole e.a., “Use of multimode interference couplers to broaden the passband of wavelength-dispersive integrated WDM filters,” IEEE Photon. Technol. Lett. **8**, 1340–1342 (1996). [CrossRef]

*x*

_{3}, Eq. (12), can be easily modified using the shifting properties of the Fourier transform:

## 4 Modified Design Equations

### 4.1 Channel Bandpass 3 dB Bandwidth

*x*

_{3}, Eq. (23), to the Gaussian functions inside

*f*(

_{M}*x*

_{3}). Eq. (19) with

*q*=0, for the transmission between the CIW-COW pair, can be used to derive the expression for the channel bandpass 3 dB bandwidth:

*x*

_{0}, and the ouput plane,

*x*

_{3}, with identical input and output waveguides [6

**8**, 1340–1342 (1996). [CrossRef]

*x*=1.6173, and hence, with Δ

*ν*=Δ

*ν′*/2, the expression for the bandwidth is:

_{bw}### 4.2 Cross Talk

*β*(

_{i}*u*)=

_{n}*F*{

*B*(

_{i}*x*

_{1n})}, where

*B*(

_{i}*x*

_{1n}) and

*β*(

_{o}*u*) are the illumination and the OW mode profile respectively, for the normalized coordinates. For the case of MMI couplers, the modified illumination from Eq. (22) must be used instead of

_{n}*B*(

_{i}*x*

_{1n}). It is possible to write a similar normalized expression in this case, assuming that

*ω*=

_{i}*ω*and Δ

_{o}*x*=2

_{m}*ω*:

_{i}*σ*. The interpretation of this result is analog to the one corresponding to the ordinary AWG. Low uniformity of the illumination, Eq. (33), using small values for σ, yields low cross talk values. In general, reducing σ lowers the side lobes of

_{o}*β′*(

_{i}*u*), but also widens its main lobe. For small values of

_{n}*σ*, the cross talk value shows no dependence on σ, since despite the side lobes are lower, significant energy is coupled the main lobe of

_{o}*β′*(

_{i}*u*).

_{n}*σ*=5 in Fig. 4, the lower

_{o}*σ*, the lower the cross talk. A reduction of

*σ*can be attained for instance increasing the number of AW’s,

*N*, in Eq. (31), but it is not the only way due to the complex dependencies among the AWG parameters, as described in the design procedure elaborated in [1].

## 5 MMI-AWG Simulation Results

*n*=1.4529 and

_{s}*n*

_{c}=1.453 respectively, the normalized waveguide frequency,

*V*=3, the waveguide width,

*W*=14

_{x}*µm*, the number of AW’s,

*N*=128 and the shortest waveguide length,

*l*

_{0}=26mm. The module and delay CIW-COW responses obtained for the MMI-based AWG are ploted in Fig. 6. The desired HLR are well matched after simulation as Table 1 states.

## 6 Conclusions

## Acknowledgements

## References and links

1. | P. Muñoz, D. Pastor, and J. Capmany, “Modeling and designing arrayed waveguide gratings,” J. Light. Tech. (Submitted) |

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

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. Light. Tech. |

5. | M.K. Smit and C. van Dam, “PHASAR-based WDM-devices: Principles, design and applications,” J. Sel. Top. Quant. Electron. |

6. | J. Soole e.a., “Use of multimode interference couplers to broaden the passband of wavelength-dispersive integrated WDM filters,” IEEE Photon. Technol. Lett. |

7. | H. Takahashi e.a., “Transmission characterisitics of arrayed waveguide N×N wavelength multiplexer,” J. Light. Tech. |

8. | F. Pizzato, G. Perrone, and I. Montroset, “Arrayed waveguide grating demultiplexers: a new efficient numerical analysis approach,” in |

9. | C. Dragone e.a., “Efficient N×N star couplers using Fourier optics,” J. Light. Technol. |

10. | C. Dragone e.a., “Efficient multichannel integrated optics star coupler on silicon,” IEEE Photon. Technol. Lett. |

11. | C. Dragone, C. Edwards, and R. Kistler, “Integrated optics N×N multiplexer on silicon,” IEEE Photon. Technol. Lett. |

12. | G.P. Agrawal, |

13. | A.W. Snyder and J.D. Love, |

14. | L.B. Soldano and E.C. Pennings, “Optical multi-mode interference devices based on self-imaging: Principles and applications,” J. Light. Technol. |

**OCIS Codes**

(060.4230) Fiber optics and optical communications : Multiplexing

(070.2580) Fourier optics and signal processing : Paraxial wave optics

(070.6110) Fourier optics and signal processing : Spatial filtering

(230.1950) Optical devices : Diffraction gratings

(230.7390) Optical devices : Waveguides, planar

(350.2460) Other areas of optics : Filters, interference

**ToC Category:**

Research Papers

**History**

Original Manuscript: July 20, 2001

Published: September 24, 2001

**Citation**

Pascual Munoz, Daniel Pastor, and Jose Capmany, "Analysis and design of arrayed waveguide gratings with MMI couplers," Opt. Express **9**, 328-338 (2001)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-9-7-328

Sort: Journal | Reset

### References

- P. Munoz, D. Pastor and J. Capmany, "Modeling and designing arrayed waveguide gratings," J. Light. Tech. (Submitted).
- H. Takenouchi, H. Tsuda and T. Kurokawa, "Analysis of optical-signal processing using an arrayed-waveguide grating," Opt. Express 6, 124-135 (2000), http://www.opticsexpress.org/oearchive/source/19103.htm [CrossRef] [PubMed]
- K. Okamoto and A. Sugita, "Flat spectral response arrayed-waveguide grating multiplexer with parabolic waveguide horns," Electron. Lett. 32, 1661-1662 (1996). [CrossRef]
- C. Dragone, "Efficient techniques for widening the passband of a wavelength router," J. Light. Tech. 16, 1895-1906 (1998). [CrossRef]
- M. K. Smit and C. van Dam, "PHASAR-based WDM-devices: Principles, design and applications," J. Sel. Top. Quant. Electron. 2, 236-250 (1996). [CrossRef]
- J. Soole e.a., "Use of multimode interference couplers to broaden the passband of wavelength-dispersive integrated WDM filters," IEEE Photon. Technol. Lett. 8, 1340-1342 (1996). [CrossRef]
- H. Takahashi e.a., "Transmission characterisitics of arrayed waveguide N x N wavelength multiplexer," J. Light. Tech. 13, 447-455 (1995). [CrossRef]
- F. Pizzato, G. Perrone and I. Montroset, "Arrayed waveguide grating demultiplexers: a new efficient numerical analysis approach," in Silicon-based Optoelectronics, Houghton, D.C., Fitzgerald, E. A., eds., Proc. SPIE 3630, 198-206, 1999.
- C. Dragone e.a., "Efficient N x N star couplers using Fourier optics," J. Light. Technol. 7, 479-489 (1989). [CrossRef]
- C. Dragone e.a., "Efficient multichannel integrated optics star coupler on silicon," IEEE Photon. Technol. Lett. 1, 241-243 (1989). [CrossRef]
- C. Dragone, C. Edwards and R. Kistler, "Integrated optics N x N multiplexer on silicon," IEEE Photon. Technol. Lett. 3, 896-898 (1991). [CrossRef]
- G. P. Agrawal, Fiber-Optic Communications Systems, (John Wiley and Sons, New York, 1997).
- A. W. Snyder and J. D. Love, Optical Waveguide Theory, (Chapman & Hall, New York, 1983).
- L. B. Soldano and E. C. Pennings, "Optical multi-mode interference devices based on self-imaging: Principles and applications," J. Light. Technol. 13, 615-627 (1995). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.