## Variable splitting ratio 2×2 MMI couplers using multimode waveguide holograms

Optics Express, Vol. 15, Issue 14, pp. 9015-9021 (2007)

http://dx.doi.org/10.1364/OE.15.009015

Acrobat PDF (405 KB)

### Abstract

Variable power splitting ratio 2×2 MMI couplers using multi-mode waveguide holograms are analyzed. Theoretical analysis shows that variable splitting ratios can be obtained with surface relief holograms on MMI couplers with fixed dimensions. Devices with paired-imaging lengths are designed on a silicon-on-insulator (SOI) platform. Beam propagation simulations are used to verify a matrix theory analysis and to investigate proposed device performance. Fabrication tolerance of the proposed device is also analyzed.

© 2007 Optical Society of America

## 1. Introduction

1. 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]

2. P. A. Besse, M. Bachmann, H. Melchior, L. B. Soldano, and M. K. Smit, “Optical bandwidth and fabrication tolerances of multimode interference couplers,” J. Lightwave Technol. **12**, 1001–1009 (1994). [CrossRef]

3. M. Bachmann, P. A. Besse, and H. Melchior, “Overlapping-image multimode interference couplers with a reduced number of self-images for uniform and nonuniform power splitting,” Appl. Opt. **34**, 6898–6910 (1995). [CrossRef] [PubMed]

4. P. A. Besse, E. Gini, M. Bachmann, and H. Melchior, “New 2×2 and 1×3 multimode interference couplers with free selection of power splitting ratios,” J. Lightwave Technol. **14**, 2286–2293 (1996). [CrossRef]

5. D. S. Levy, Y. M. Li, R. Scarmozzino, and R. M. Osgood Jr. , “A multimode interference-based variable power splitter in GaAs-AlGaAs,” IEEE Photon. Technol. Lett. **9**, 1373–1375 (1997). [CrossRef]

6. D. J. Y. Feng, T. S. Lay, and T. Y. Chang, “Waveguide couplers with new power splitting ratios made possible by cascading of short multimode interference sections,” Opt. Express **15**, 1588–1593 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-4-1588 [CrossRef] [PubMed]

7. S.-Y. Tseng, Y. Kim, C. J. K. Richardson, and J. Goldhar, “Implementation of discrete unitary transformations by multimode waveguide holograms,” Appl. Opt. **45**, 4864–4872 (2006). [CrossRef] [PubMed]

7. S.-Y. Tseng, Y. Kim, C. J. K. Richardson, and J. Goldhar, “Implementation of discrete unitary transformations by multimode waveguide holograms,” Appl. Opt. **45**, 4864–4872 (2006). [CrossRef] [PubMed]

8. H. Wei, J. Yu, Z. Liu, X. Zhang, W. Shi, and C. Fang, “Fabrication of 4×4 tapered MMI coupler with large cross section,” IEEE Photon. Technol. Lett. **13**, 466–468 (2001). [CrossRef]

## 2. Matrix theory of hologram in MMI power splitters at paired-imaging length

*L*of the multimode section is at half the beat length,

*L*

_{π}[1

1. 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]

*n*is the core effective index,

_{c}*W*is the effective width [1

_{e}1. 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]

*λ*is the wavelength. By placing the access waveguides at ±

*W*/6 from the center of the multimode section, selected modes are excited to produce a paired-image at the 3dB length of

_{e}*L*

_{π}/2 [1

**13**, 615– 627 (1995). [CrossRef]

*n*at different parts of the multimode section. Magnitude of ∆

*n*can be varied by changing the depth of the surface relief patterns [7

7. S.-Y. Tseng, Y. Kim, C. J. K. Richardson, and J. Goldhar, “Implementation of discrete unitary transformations by multimode waveguide holograms,” Appl. Opt. **45**, 4864–4872 (2006). [CrossRef] [PubMed]

**45**, 4864–4872 (2006). [CrossRef] [PubMed]

9. J. M. Heaton and R. M. Jenkins, “General matrix theory of self-imaging in multimode interference (MMI) couplers,” IEEE Photon. Technol. Lett. **11**, 212–214 (1999). [CrossRef]

*i*th and the

*j*th eigenmodes corresponds to the

*ij*th element in matrix

**K**, which relates input vector

**V**

_{i}and output vector

**V**

_{o}through unitary matrix-vector multiplication as

*L*is the length of the hologram.

_{H}**e**

_{a}=[1,0]

^{T}and

**e**

_{b}=[0,1]

^{T}can be used to represent the complex electric field amplitudes in the input waveguides. When there is no hologram, the transfer matrix

**T**representing the MMI is

**K**is a Hermitian matrix representing the hologram as

## 3. Device design

10. K. Solehmainen, M. Kapulainen, M. Harjanne, and T. Aalto, “Adiabatic and multimode interference couplers on silicon-on-insulator,” IEEE Photon. Technol. Lett. **18**, 2287–2289 (2006). [CrossRef]

11. T. Tsuchizawa, K. Yamada, H. Fukuda, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, “Microphotonics devices based on silicon microfabrication technology,” IEEE J. Sel. Top. Quantum Electron. **11**, 232–240 (2005). [CrossRef]

*H*is 3

*μ*m, the slab height

*h*is 2μm, the buried oxide layer thickness

*d*is 1

*μ*m, the width of access waveguides

*W*is 4

_{a}*μ*m, the width of the multimode section

*W*is 24

_{mmi}*μ*m, the length

*L*of the multimode section is 1080

*μ*m, the centers of access waveguides are placed ±4.3

*μ*m from the center of the multimode section for paired-imaging. The placement of access waveguides is chosen by approximating the effective width

*W*of MMI by taking into account the lateral penetration of the fundamental mode into the cladding as in Ref. [1

_{e}**13**, 615– 627 (1995). [CrossRef]

*n*=1.0(air),

_{a}*n*=3.6, and

_{Si}*n*=1.46. The device is optimized for 1550nm input wavelength and TE polarization.

_{ox}12. G. R. Hadley, “Wide-angle beam propagation using Pade approximant operators,” Opt. Lett **17**, 1426–1428 (1992). [CrossRef] [PubMed]

13. K. Kawano and T. Kiton, *Introduction to Optical Waveguide Analysis* (Wiley, New York, 2001). [CrossRef]

14. Optical Modesolver, Photonics Research Laboratory, University of Maryland, MD (2006), http://www.photonics.umd.edu/software/modesolver.zip

## 4. Device simulation

*μm*× 1

*μm*pixels in the simulation.

### 4.1. Variable power splitting ratio

*n*using an input wavelength of 1550nm. The normalized output power in the bar and cross ports are plotted as a function of index modulation in Fig. 3. The two curves in the same figure are theoretical fits with Eq.(8) and Eq.(9). Excellent agreement between the matrix theory and WA-BPM simulations can be seen in the figure. An effective index modulation of-0.002 is required for a full swing of the power splitting ratio at the output ports. Using the effective index method, it is found that a ∆

*n*of-0.002 can be obtained by surface relief patterns etched into the multimode section of the MMI with a depth of approximately 300nm.

### 4.2. Wavelength dependence of MMI power splitter with hologram

*R*as the ratio between bar port output intensity and cross port output intensity. Using power splitters designed and optimized for operation at 1550nm, we vary the input wavelengths from 1500nm to 1600nm in increments of 10nm to investigate wavelength dependence of these devices. We define the difference between R and the designed value at 1550nm,

*R*

_{1550}, as ∆

*R*. These variations for a 50:50 MMI splitter without holograms, along with three hologram-loaded MMIs with index modulations of-0.0005,-0.0010,-0.0015 are normalized against the designed value

*R*

_{1550}and shown in Fig. 4. It can be seen that the spectral variations in the power splitting ratios of hologram-loaded MMIs are comparable to that of the MMI splitter without holograms. We also plot the normalized total transmittance of these devices as a function of wavelength in Fig. 5. Transmittance decrease as wavelength is detuned from the design wavelength. As expected, excess loss is observed when index modulation is increased. Overall, the devices exhibit excellent power splitting performance and high transmittance over a wide bandwidth of 100nm.

## 5. Fabrication tolerance

## 6. Summary

*N*×

*N*power splitters or interconnects with new splitting ratios.

## Acknowledgments

## References and links

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

2. | P. A. Besse, M. Bachmann, H. Melchior, L. B. Soldano, and M. K. Smit, “Optical bandwidth and fabrication tolerances of multimode interference couplers,” J. Lightwave Technol. |

3. | M. Bachmann, P. A. Besse, and H. Melchior, “Overlapping-image multimode interference couplers with a reduced number of self-images for uniform and nonuniform power splitting,” Appl. Opt. |

4. | P. A. Besse, E. Gini, M. Bachmann, and H. Melchior, “New 2×2 and 1×3 multimode interference couplers with free selection of power splitting ratios,” J. Lightwave Technol. |

5. | D. S. Levy, Y. M. Li, R. Scarmozzino, and R. M. Osgood Jr. , “A multimode interference-based variable power splitter in GaAs-AlGaAs,” IEEE Photon. Technol. Lett. |

6. | D. J. Y. Feng, T. S. Lay, and T. Y. Chang, “Waveguide couplers with new power splitting ratios made possible by cascading of short multimode interference sections,” Opt. Express |

7. | S.-Y. Tseng, Y. Kim, C. J. K. Richardson, and J. Goldhar, “Implementation of discrete unitary transformations by multimode waveguide holograms,” Appl. Opt. |

8. | H. Wei, J. Yu, Z. Liu, X. Zhang, W. Shi, and C. Fang, “Fabrication of 4×4 tapered MMI coupler with large cross section,” IEEE Photon. Technol. Lett. |

9. | J. M. Heaton and R. M. Jenkins, “General matrix theory of self-imaging in multimode interference (MMI) couplers,” IEEE Photon. Technol. Lett. |

10. | K. Solehmainen, M. Kapulainen, M. Harjanne, and T. Aalto, “Adiabatic and multimode interference couplers on silicon-on-insulator,” IEEE Photon. Technol. Lett. |

11. | T. Tsuchizawa, K. Yamada, H. Fukuda, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, “Microphotonics devices based on silicon microfabrication technology,” IEEE J. Sel. Top. Quantum Electron. |

12. | G. R. Hadley, “Wide-angle beam propagation using Pade approximant operators,” Opt. Lett |

13. | K. Kawano and T. Kiton, |

14. | Optical Modesolver, Photonics Research Laboratory, University of Maryland, MD (2006), http://www.photonics.umd.edu/software/modesolver.zip |

**OCIS Codes**

(090.1760) Holography : Computer holography

(130.2790) Integrated optics : Guided waves

(130.3120) Integrated optics : Integrated optics devices

**ToC Category:**

Holography

**History**

Original Manuscript: May 2, 2007

Revised Manuscript: June 28, 2007

Manuscript Accepted: June 29, 2007

Published: July 6, 2007

**Citation**

Shuo-Yen Tseng, Canek Fuentes-Hernandez, Daniel Owens, and Bernard Kippelen, "Variable splitting ratio 2 × 2 MMI couplers using multimode waveguide holograms," Opt. Express **15**, 9015-9021 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-14-9015

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### References

- 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]
- P. A. Besse, M. Bachmann, H. Melchior, L. B. Soldano, and M. K. Smit, "Optical bandwidth and fabrication tolerances of multimode interference couplers," J. Lightwave Technol. 12, 1001-1009 (1994). [CrossRef]
- M. Bachmann, P. A. Besse, and H. Melchior, "Overlapping-image multimode interference couplers with a reduced number of self-images for uniform and nonuniform power splitting," Appl. Opt. 34, 6898-6910 (1995). [CrossRef] [PubMed]
- P. A. Besse, E. Gini, M. Bachmann, and H. Melchior, "New 2×2 and 1×3 multimode interference couplers with free selection of power splitting ratios," J. Lightwave Technol. 14, 2286-2293 (1996). [CrossRef]
- D. S. Levy, Y.M. Li, R. Scarmozzino, R.M. OsgoodJr., "A multimode interference-based variable power splitter in GaAs-AlGaAs," IEEE Photon. Technol. Lett. 9, 1373-1375 (1997). [CrossRef]
- D. J. Y. Feng, T. S. Lay, and T. Y. Chang, "Waveguide couplers with new power splitting ratios made possible by cascading of short multimode interference sections," Opt. Express 15, 1588-1593 (2007),://www.opticsinfobase.org/abstract.cfm?URI=oe-15-4-1588> [CrossRef] [PubMed]
- S.-Y. Tseng, Y. Kim, C. J. K. Richardson, and J. Goldhar, "Implementation of discrete unitary transformations by multimode waveguide holograms," Appl. Opt. 45, 4864-4872 (2006). [CrossRef] [PubMed]
- H. Wei, J. Yu, Z. Liu, X. Zhang, W. Shi, and C. Fang, "Fabrication of 4×4 tapered MMI coupler with large cross section," IEEE Photon. Technol. Lett. 13, 466-468 (2001). [CrossRef]
- J. M. Heaton and R. M. Jenkins, "General matrix theory of self-imaging in multimode interference (MMI) couplers," IEEE Photon. Technol. Lett. 11, 212-214 (1999). [CrossRef]
- K. Solehmainen, M. Kapulainen, M. Harjanne, and T. Aalto, "Adiabatic and multimode interference couplers on silicon-on-insulator," IEEE Photon. Technol. Lett. 18, 2287-2289 (2006). [CrossRef]
- T. Tsuchizawa, K. Yamada, H. Fukuda, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, "Microphotonics devices based on silicon microfabrication technology," IEEE J. Sel. Top. Quantum Electron. 11, 232-240 (2005). [CrossRef]
- G. R. Hadley, "Wide-angle beam propagation using Pade approximant operators," Opt. Lett 17, 1426-1428 (1992). [CrossRef] [PubMed]
- K. Kawano and T. Kiton, Introduction to Optical Waveguide Analysis (Wiley, New York, 2001). [CrossRef]
- Optical Modesolver, Photonics Research Laboratory, University of Maryland, MD (2006), http://www.photonics.umd.edu/software/modesolver.zip

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