## Geometrical optimization of the transmission and dispersion properties of arrayed waveguide gratings using two stigmatic point mountings

Optics Express, Vol. 11, Issue 19, pp. 2425-2432 (2003)

http://dx.doi.org/10.1364/OE.11.002425

Acrobat PDF (128 KB)

### Abstract

In this paper, the procedure to optimize flat-top Arrayed Waveguide Grating (AWG) devices in terms of transmission and dispersion properties is presented. The systematic procedure consists on the stigmatization and minimization of the Light Path Function (LPF) used in classic planar spectrograph theory. The resulting geometry arrangement for the Arrayed Waveguides (AW) and the Output Waveguides (OW) is not the classical Rowland mounting, but an arbitrary geometry arrangement. Simulation using previous published enhanced modeling show how this geometry reduces the passband ripple, asymmetry and dispersion, in a design example.

© 2003 Optical Society of America

## 1. Introduction

1. M.K. Smit and C. van Dam, “PHASAR-Based WDM-Devices: Principles, Design and Applications,” J. Sel. Top. Quantum 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/abstract.cfm?URI=OPEX-6-6-124 [CrossRef] [PubMed]

3. Y. Yoshikuni, “Semiconductor ArrayedWaveguide Gratings for Photonic Integrated Devices,” J. Sel. Top. Quantum Electron. **8**, 1102–1114 (2002). [CrossRef]

4. H. Takahashi, S. Suzuki, and I. Nishi, “Wavelength multiplexer based on SiO_{2}-Ta_{2}O_{5} arrayed-waveguide grating,” J. Lightwave Technol. **12**, 989–995 (1994). [CrossRef]

5. H. Takahashi, H. Toba, and Y. Inoue, “Multiwavelength ring laser composed of EDFAs and an arrayed-waveguide wavelength multiplexer,” Electron. Lett. **30**, 44–45 (1994). [CrossRef]

7. H. Takahashi, K. Oda, H. Toba, and Y. Inoue, “Transmission characteristics of arrayed waveguide N×N wavelength multiplexer,” J. Lightwave Technol. **13**447–455 (1995). [CrossRef]

8. C. Dragone, “Efficient N×N star couplers using Fourier Optics,” J. Lightwave Technol. **7**, 479–489 (1989). [CrossRef]

*L*. This introduces a frequency dependent phase change to the light traveling through the waveguides. The latter, combined with the second SC, splits the input frequencies over the second SC output, where additional waveguides, the output waveguides (OWs), are laid to collect the desired frequencies.

10. B. Soole e.a., “Use of multimode interference couplers to broaden the passband of wavelength-dispersive integrated WDM filters,” Phot. Tech. Lett. **8**, 1340–1342 (1996). [CrossRef]

11. L.B. Soldano and E.C.M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. **13**, 615–627 (1995). [CrossRef]

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

## 2. Light path function vectorial minimization

*Light Path Function*, LPF, [9], which tights together basic properties of the light as phase, wavelength and constructive interference, allowing for a simple, but yet powerfull, analysis of imaging systems as the AWG. Consider the layout of Fig. 1. The LPF is defined as:

*n*(

_{s}*λ*) and

*n*(

_{c}*λ*) are the refractive index in the slabs or Free Propagation Regions, FPR, and the arrayed waveguides index, AW, respectiveley. The fist three terms in the equation take into account the propagation length difference through FPR

_{1}, AWs and FPR

_{2}. This difference must be an integer number of times the wavelength considered for it to interfere construtively in the ouput point, that is,

*D*. Hence,

*G*(

*x*) takes an integer value for the AW at (

*x, z*(

_{G}*x*)).

*G*(

*x*) is known as the

*grating function*of the grating-like device. Abreviations can be introduced for the terms in the expression, yielding:

*F*(

_{i}*x*) represents the path difference on the input side of the AWG. Δ

*L*(

*x*) is the length increment along the curved section between the center AW, laid from

*O*to

_{I}*O*, and an arbitrary waveguide, laid from

*P*and

_{I}*P*. Finally,

*F*(

_{d}*x*) is the path difference at the output side. In Fig. 1,

*z*(

_{G}*x*) represents the

*grating line*of the device, which for simplicity will be assumed the same for the input and output part of the device. The

*focal line*of the device is represented as

*z*(

_{f}*x*).

*x*is considered, but a set of points

**x**(a vector), then Eq. (2) can be rewritten in a vectorial way:

**x**and

**z**

_{G}(

*x*). The solution for a particular output point

*D*, will yield a geometrical arrangement

**x**and

**z**

_{G}(

*x*), for a given wavelength

*λ*that cancels the LPF, i.e.,

_{D}**F**(

**x**)=0. The output points for which the LPF cancels, are known as

*stigmatic points*of the waveguide mounting [9]. Notice that on a first approach, there are three unkown geometrical variables or functions,

**x**,

**z**

_{G}(

*x*) and Δ

**L**(

**x**). Hence, and as proposed in [13

13. D. Wang, G. Jin, Y. Yan, and M. Wu, “Aberration theory of arrayed waveguide grating,” J. Lightwave Technol. **19**, 279–284 (2001). [CrossRef]

**L**(

**x**) must be kept linear. Therefore, only two unknowns are left in the problem, which are

**x**and

**z**

_{G}(

*x*), and only two stigmatic points must be given as initial conditions for Eq. (3). Consequently, the procedure to find the mounting, both at the AW side and the focal line side, must be divided into two steps:

### Step 1: AWs position

**F**(

**x**)=0, with two stigmatic points as initial conditions, is given by:

**x**and

**z**

_{G}(

*x*). The input data needed for the problem is:

- Two stigmatic points
*D*=(_{k}*x*) and their associated wavelength_{d,k}, z_{d,k}*λ*for_{k}*k*=1, 2. - The refractive indexes of the slab and arrayed waveguides,
*n*(_{s}*λ*) and*n*(_{c}*λ*) respectively. - The position of the input waveguide, IW, which will be assumed to be centered and at a distance
*R*from*O*(see Fig. 1)._{I} - The groove function
**G**(**x**), which is a vector of integers ranging from -*N*/2 to*N*/2-1 (with N even). - The waveguide length increment function, Δ
**L**(**x**).

**x**and

**z**

_{G}(

*x*), satisfying the stigmatic points, the groove function and the waveguide length increment function Δ

**L**(

**x**).

*d*is the AW spacing. The waveguide increment function is fixed to:

_{w}*λ*

_{0}=

*c/f*

_{0}is the AWG design wavelength. This preserves the focusing properties of the AWG, as mentioned before.

### Step 2: OWs positions

**x**,

**z**

_{G}(

*x*)), the OWs positions can be calculated solving the following equation:

*k*, (

*x*) for which the pass band center is

_{k}, z_{k}*f*=

_{k}*c/λ*is found. The initial condition is based, as mentioned before, on the Rowland mounting. The AWG physical lengths and parameters used as the starting point can be derived using any design methodology. For the example presented below, the methodology from [14

_{k}14. P. Mũnoz, D. Pastor, and J. Capmany, “Modeling and design of arrayed waveguide gratings,”, J. Lightwave Technol. **20**, 661–674 (2002). [CrossRef]

15. P. Mũnoz, D. Pastor, and J. Capmany, “Analysis and design of arrayed waveguide gratings with MMI couplers,” Opt. Express **9**, 328–338 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-7-328 [CrossRef]

## 3. Results and discussion

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

15. P. Mũnoz, D. Pastor, and J. Capmany, “Analysis and design of arrayed waveguide gratings with MMI couplers,” Opt. Express **9**, 328–338 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-7-328 [CrossRef]

*f*

_{0}=194 THz (

*f*

_{0}=

*c/λ*

_{0}, see equation 7), slab length

*R*=45804.120

*µ*m, AW spacing

*d*=10.44

_{w}*µ*m, OW spacing

*d*=19.75

_{ow}*µ*m, number of AW

*N*=1000, diffraction order

*m*=32, path length increment at the design frequency Δ

*L*(

*f*

_{0})=34.0562 µm and slab refractive index

*n*(λ

_{s}_{0})=1.4305. A 14 microns mode from a parabolic horn is used to make the device flat top as in [12

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

*W*=6

*µ*m width, with

*n*=1.4551 and

_{core}*n*=1.4451. The waveguides modes are found using a mode solver [17

_{cladding}17. M. Hammer, “WMM mode solver. Numerical simulation of rectangular integrated optical waveguides,” University of Twente, Faculty of Mathematical Sciences. http://www.physik.uni-osnabrueck.de/theophys/

18. C.D. Lee e.a., “The role of photomask resolution on the performance of arrayed-waveguide grating devices,” J. Lightwave Technol. **19**, 1726–1733 (2001). [CrossRef]

19. P. Mũnoz, D. Pastor, J. Capmany, and S. Sales, “Analytical and Numerical Analysis of Phase and Amplitude Errors in the Performance of Arrayed Waveguide Gratings,” J. Sel. Top. Quantum Electron. **8**, 1130–1141 (2002). [CrossRef]

1. M.K. Smit and C. van Dam, “PHASAR-Based WDM-Devices: Principles, Design and Applications,” J. Sel. Top. Quantum Electron. **2**, 236–250 (1996). [CrossRef]

1. M.K. Smit and C. van Dam, “PHASAR-Based WDM-Devices: Principles, Design and Applications,” J. Sel. Top. Quantum Electron. **2**, 236–250 (1996). [CrossRef]

*R*, centered at (0,R) on both slab couplers (see Fig. 1), while the obtained grating line is not symmetric. This is shown in Fig. 2, where the short AWs (smaller numbers) are further away from the (0,R) point, while the long AWs are nearer to the mentioned point. Conversely, on the focal line side, the focal line obtained is between 10 and 20 microns appart from the Rowland circle.

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

## 4. Conclusions

## Acknowledgements

## Footnotes

1 | In fact asymmetry should not be too large in the ideal design, but only after fabrication when errors in the FPR come into place. However, this is out of the scope of this paper. |

## References and links

1. | M.K. Smit and C. van Dam, “PHASAR-Based WDM-Devices: Principles, Design and Applications,” J. Sel. Top. Quantum Electron. |

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

3. | Y. Yoshikuni, “Semiconductor ArrayedWaveguide Gratings for Photonic Integrated Devices,” J. Sel. Top. Quantum Electron. |

4. | H. Takahashi, S. Suzuki, and I. Nishi, “Wavelength multiplexer based on SiO |

5. | H. Takahashi, H. Toba, and Y. Inoue, “Multiwavelength ring laser composed of EDFAs and an arrayed-waveguide wavelength multiplexer,” Electron. Lett. |

6. | D. Huang, T. Chin, and Y. Lai, “Arrayed waveguide grating DWDM interleaver,” Proc. OFC , |

7. | H. Takahashi, K. Oda, H. Toba, and Y. Inoue, “Transmission characteristics of arrayed waveguide N×N wavelength multiplexer,” J. Lightwave Technol. |

8. | C. Dragone, “Efficient N×N star couplers using Fourier Optics,” J. Lightwave Technol. |

9. | R. März, |

10. | B. Soole e.a., “Use of multimode interference couplers to broaden the passband of wavelength-dispersive integrated WDM filters,” Phot. Tech. Lett. |

11. | L.B. Soldano and E.C.M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. |

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

13. | D. Wang, G. Jin, Y. Yan, and M. Wu, “Aberration theory of arrayed waveguide grating,” J. Lightwave Technol. |

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

15. | P. Mũnoz, D. Pastor, and J. Capmany, “Analysis and design of arrayed waveguide gratings with MMI couplers,” Opt. Express |

16. | ITU-T G.692 Rec. “Optical interfaces for multichannel systems with optical amplifiers,” (1998). |

17. | M. Hammer, “WMM mode solver. Numerical simulation of rectangular integrated optical waveguides,” University of Twente, Faculty of Mathematical Sciences. http://www.physik.uni-osnabrueck.de/theophys/ |

18. | C.D. Lee e.a., “The role of photomask resolution on the performance of arrayed-waveguide grating devices,” J. Lightwave Technol. |

19. | P. Mũnoz, D. Pastor, J. Capmany, and S. Sales, “Analytical and Numerical Analysis of Phase and Amplitude Errors in the Performance of Arrayed Waveguide Gratings,” J. Sel. Top. Quantum Electron. |

20. | J.W. Goodman, |

**OCIS Codes**

(110.5100) Imaging systems : Phased-array imaging systems

(220.1000) Optical design and fabrication : Aberration compensation

(220.2740) Optical design and fabrication : Geometric optical design

(230.1150) Optical devices : All-optical devices

(230.1980) Optical devices : Diffusers

(230.7390) Optical devices : Waveguides, planar

(230.7400) Optical devices : Waveguides, slab

**ToC Category:**

Research Papers

**History**

Original Manuscript: August 4, 2003

Revised Manuscript: September 10, 2003

Published: September 22, 2003

**Citation**

P. Muñoz, D. Pastor, J. Capmany, and A. Martínez, "Geometrical optimization of the transmission and dispersion properties of arrayed waveguide gratings using two stigmatic point mountings," Opt. Express **11**, 2425-2432 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-19-2425

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### References

- M.K. Smit, C. van Dam, �??PHASAR-Based WDM-Devices: Principles, Design and Applications,�?? J. Sel. Top. Quantum Electron. 2, 236-250 (1996). [CrossRef]
- H. Takenouchi, H. Tsuda and T. Kurokawa, �??Analysis of optical-signal processing using an arrayed-waveguide grating,�?? Opt. Express 6 124-135 (2000), <a href= "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-6-6-124">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-6-6-124</a>. [CrossRef] [PubMed]
- Y. Yoshikuni, �??Semiconductor Arrayed Waveguide Gratings for Photonic Integrated Devices,�?? J. Sel. Top. Quantum Electron. 8, 1102-1114 (2002). [CrossRef]
- H. Takahashi, S. Suzuki, I. Nishi, �??Wavelength multiplexer based on SiO2-Ta2O5 arrayed-waveguide grating,�?? J. Lightwave Technol. 12, 989-995 (1994). [CrossRef]
- H. Takahashi, H. Toba, Y. Inoue, �??Multiwavelength ring laser composed of EDFAs and an arrayed-waveguide wavelength multiplexer,�?? Electron. Lett. 30, 44-45 (1994). [CrossRef]
- D. Huang, T. Chin, Y. Lai, �??Arrayed waveguide grating DWDM interleaver,�?? Proc. OFC, 3 WDD80 1-3 (2001).
- H. Takahashi, K. Oda, H. Toba, Y. Inoue, �??Transmission characteristics of arrayed waveguide N x N wavelength multiplexer,�?? J. Lightwave Technol. 13 447-455 (1995). [CrossRef]
- C. Dragone, �??Efficient N x N star couplers using Fourier Optics,�?? J. Lightwave Technol. 7, 479-489 (1989). [CrossRef]
- R. März, Integrated optics: design & modeling, (Artech House, 1995), Chap. 8.
- B. Soole e.a., �??Use of multimode interference couplers to broaden the passband of wavelength-dispersive integrated WDM filters,�?? Phot. Tech. Lett. 8, 1340-1342 (1996). [CrossRef]
- L.B. Soldano, E.C.M. Pennings, �??Optical multi-mode interference devices based on self-imaging: principles and applications,�?? J. Lightwave Technol. 13, 615-627 (1995). [CrossRef]
- K. Okamoto, A. Sugita, �??Flat spectral response arrayed-waveguide grating multiplexer with parabolic waveguide horns,�?? Electron. Lett. 32, 1661-1662 (1996). [CrossRef]
- D. Wang, G. Jin, Y. Yan and M. Wu, �??Aberration theory of arrayed waveguide grating,�?? J. Lightwave Technol. 19, 279-284 (2001). [CrossRef]
- P. Muñoz, D. Pastor, J. Capmany, �??Modeling and design of arrayed waveguide gratings,�?? J. Lightwave Technol. 20, 661-674 (2002). [CrossRef]
- P. Muñoz, D. Pastor, J. Capmany, �??Analysis and design of arrayed waveguide gratings with MMI couplers,�?? Opt. Express 9, 328-338 (2001), <a href= "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-7-328">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-7-328</a>. [CrossRef]
- ITU-T G.692 Rec. �??Optical interfaces for multichannel systems with optical amplifiers,�?? (1998).
- M. Hammer, �??WMM mode solver. Numerical simulation of rectangular integrated optical waveguides,�?? University of Twente, Faculty of Mathematical Sciences. <a href= "http://www.physik.uni-osnabrueck.de/theophys">http://www.physik.uni-osnabrueck.de/theophys</a>.
- C.D. Lee e.a., �??The role of photomask resolution on the performance of arrayed-waveguide grating devices,�?? J. Lightwave Technol. 19, 1726-1733 (2001). [CrossRef]
- P. Muñoz, D. Pastor, J. Capmany, S. Sales, �??Analytical and Numerical Analysis of Phase and Amplitude Errors in the Performance of Arrayed Waveguide Gratings,�?? J. Sel. Top. Quantum Electron. 8, 1130-1141 (2002). [CrossRef]
- J.W. Goodman, Introduction to Fourier Optics, (McGraw-Hill, 1994), Chaps. 4 and 5.

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