## Compact multimode interference couplers with arbitrary power splitting ratio

Optics Express, Vol. 16, Issue 10, pp. 7175-7180 (2008)

http://dx.doi.org/10.1364/OE.16.007175

Acrobat PDF (430 KB)

### Abstract

We show that it is possible to obtain 2×2 waveguide couplers with arbitrary power splitting ratios by interconnecting a pair of unequal-width waveguides as the phase-tuning section into the middle of two short MMI sections. These couplers have simple geometry and low loss. They offer valuable new possibilities for designing waveguide-based photonic integrated circuits.

© 2008 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]

4. S. Matsuo, Y. Yoshikuni, T. Segawa, Y. Ohiso, and H. Okamoto, “A widely tunable optical filter using ladder-type structure,” IEEE Photonics Technol. Lett. **15**, 1114–1116 (2003). [CrossRef]

5. V. M. Menon, W. Tong, C. Li, F. Xia, I. Glesk, P. R. Prucnal, and S. R. Forrest, “All-optical wavelength conversion using a regrowth-free monolithically integrated Sagnac interferometer,” IEEE Photonics Technol. Lett. **15**, 254–256 (2003). [CrossRef]

6. 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]

7. 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]

8. J. Leuthold and C. H. Joyner, “Multimode interference couplers with tunable power splitting ratios,” J. Lightwave Technol. **19**, 700–707 (2001). [CrossRef]

9. N. S. Lagali, M. R. Paiam, and R. I. MacDonald, “Theory of variable-ratio power splitters using multimode interference couplers,” IEEE Photonics Technol. Lett. **11**, 665–667 (1999). [CrossRef]

10. H. Ohe, H. Shimizu, and Y. Nakano, “InGaAlAs multiple-quantum-well optical phase modulators based on carrier depletion,” IEEE Photonics Technol. Lett. **19**, 1816–1818 (2007). [CrossRef]

11. Q. Lai, M. Bachmann, W. Hunziker, P. A. Besse, and H. Melchior, “Arbitrary ratio power splitters using angled silica on silicon multimode interference couplers,” Electron. Lett. **32**, 1576–1577 (1996). [CrossRef]

12. T. Saida, A. Himeno, M. Okuno, A. Sugita, and K. Okamoto, “Silica-based 2×2 multimode interference coupler with arbitrary power splitting ratio,” Electron. Lett. , **35**, 2031–2033 (1999). [CrossRef]

6. 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]

13. S. Y. Tseng, C. Fuentes-Hernandez, D. Owens, and B. Kippelen, “Variable splitting ratio 2×2 MMI couplers using multimode waveguide holograms,” Opt. Express , **15**, 9015–9021 (2007). [CrossRef] [PubMed]

14. 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). [CrossRef] [PubMed]

## 2. Design of 2×2 couplers with arbitrary power splitting ratio

_{PS}but of unequal waveguide widths (

*w*+δ

_{a}and

*w*+δ

_{b}). Waveguides “

*a*” and “

*b*” are located at the same (bar)- and opposite (cross)-side to the input waveguide, respectively. The MMI sections to be discussed in this paper are all 2×2 waveguide couplers. For the MMI sections, we set the input/output waveguide pairs of all the devices to have the same center-to-center separation

*s*. This distance

*s*will serve as the common unit of length scaling for the MMI sections. In order to minimize the size of the couplers,

*s*is generally chosen to be as short as the lithographic process would permit. The beat length (

*L*) between the two lowest guided modes of the same polarization in the MMI waveguide is given according to the standard mode propagation analysis (MPA) by [1

_{π}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]

7. 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]

*n*is the effective refractive index of the slab waveguide from which the MMI waveguide of effective width

_{s}*W*is formed,

_{e}*λ*is the vacuum wavelength, and

_{0}*r*≡

*W*/

_{e}*s*. Equation (1) holds for both TE and TM polarizations [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]

^{+}-InP substrate is assumed to have a 1.79 µm-thick InAlAs upper p-type cladding, a 0.14 µm-thick core layer containing triple quantum wells with the absorption edge at wavelength =1.48 µm, and a 0.14 µm-thick InAlAs lower n-type cladding. The basic layer structure in the waveguide ridge is shown schematically in Fig. 1(b). The side walls of the MMI section are assumed to be etched down to the InP substrate to keep the number of confined modes large, to reduce phase error of the confined modes, and to suppress polarization sensitivity. The

*n*of the TE fundamental slab mode is 3.2147 at λ

_{s}_{0}=1.55 µm (the base wavelength in this study). In order to match the low-order modes in 2-D simulation to those in 3-D simulation, the equivalent effective refractive index of the lateral cladding is set to 2.3. Considering the resolution and limitation of our photolithographic process, the width of the access waveguides

*w*is assumed to be 2.2 µm for only existing fundamental mode inside the input/output access waveguides and their separation is set to be

*s*=5 µm. The following four relatively short MMI sections with K≥0.5 have been considered as the building blocks.

*r*=1.44), we have

*W*=

_{e}*s*+

*w*=1.44s=7.2 µm. For the given values of

*s*and

*w*, this is the minimum practical effective width for a low loss MMI [1

**13**, 615–627 (1995). [CrossRef]

*L*is (3/2)

_{A}*L*=3.1104

_{πA}*As*

^{2}=215 µm according to MPA. For MMI-B (K=0.5,

*r*=3), we have

*W*=3

_{e}*s*=15 µm and

*L*=(1/2)

_{B}*L*=4.5

_{πB}*As*

^{2}=311 µm. The access waveguides for MMI-B are located exactly at ±

*W*/6 from the centerline of the multimode waveguide. For MMI-C (K=0.85,

_{e}*r*=2), we have

*W*=2

_{e}*s*=10 µm and

*L*=(3/4)

_{C}*L*=3

_{πC}*As*

^{2}=207 µm. The access waveguides are located exactly at ±

*W*/4 from the centerline of the MMI-C waveguide. For MMI-D (K=0.72,

_{e}*r*=2.5), we have

*W*=2.5

_{e}*s*=12.5 µm,

*L*=(3/5)

_{D}*L*=3.75

_{πD}*As*

^{2}=259 µm. The geometric centers of the access-waveguide pairs at two ends are offset in opposite directions by ±0.25

*s*from the center of the MMI-D waveguide.

7. 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]

*β*is the phase constant for the fundamental mode in the MMI section U, and

_{U}*L*is the length of that MMI section. The ± signs in the subscripts are significant only for MMI-D in which the access waveguides are not symmetrically placed. The asymmetry leads to two cross optical paths of unequal lengths. According to the 3-D BPM simulations [15], the total insertion losses of these four basic MMIs are in the range of 4 to 5%. Since all four basic MMI sections are 2×2 couplers with the same

_{U}*s*value, they can be simply aligned and cascaded. Using 3-D EigenMode Expansion (EME) method [16] to check the mode coupling efficiency for those interconnecting junctions as indicated in Fig. 1(a) by circular dot-line, the results show that only about 1% mode-mismatch loss is presented when one phase-shifter waveguide width, either “a” or “b”, is extended/reduced to 2.4/2.0 µm from 2.2 µm. Thus the phase-shifter waveguides can be considered as lossless, and their transfer functions are expressed as

*β*and

_{a}*β*are the propagation constant of fundamental mode for waveguide “

_{b}*a*” and “

*b*”, respectively; and L

_{PS}is the length of phase-shifting section. The transfer matrix of this cascaded device can be obtained by the following matrix multiplication.

*HX*

^{U(PS)V}

_{±}and

*HB*

^{U(PS)V}

_{±}are respectively the transfer functions to the cross output port and the bar output port for the cascaded device. The K-value is given by

**When L**=

_{PS}**0**, the MMI sections are actually butt-jointed without any interconnecting waveguides. We have examined the results of all possible combinations using Eq. (3) and verified their consistency with the results of 3-D BPM simulations. The resultant K-values are: K

^{AB}=0, K

^{AC}=0.15 (0.146), K

^{AD}=0.07 (0.075), K

^{BC}=0.85 (0.854), K

^{BD}=0.93 (0.925), K

^{CD}=0.64 (0.643), and K

^{DD}=0.80 (0.80). The alternative values given between parentheses are the MPA values. Among these, 0.07, 0.93, 0.64, and 0.80 are new K values that are so far not known to be possible obtained by using a constant-width MMI [7

**34**, 6898–6910 (1995). [CrossRef] [PubMed]

8. J. Leuthold and C. H. Joyner, “Multimode interference couplers with tunable power splitting ratios,” J. Lightwave Technol. **19**, 700–707 (2001). [CrossRef]

14. 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). [CrossRef] [PubMed]

*W*=1.94

_{e}*s*(MMI-E); it had been presented with a detailed discussion in [17

17. D. J. Y. Feng, T. S. Lay, and T. Y. Chang, “Compact 2×2 Couplers for Unequal Splitting of Power Obtained by Cascading of Short MMI Sections,” in *Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference*, OSA Technical Digest Series (CD) (Optical Society of America, 2007), paper JThA25, http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2007-JThA25

*As*according to MPA.

^{2}**When L**and the waveguide width in the phase-tuning section is the same as those of input/output access waveguides,

_{PS}≠0*i.e.*δ

_{a}=δ

_{b}=0, seven 2×2 cascaded MMI devices to be applied as the starting-point to achieve power splitter with freely chosen K-values are shown by simulated 2-D field maps in Fig. 3. According to 3-D BPM simulation, the resultant Kvalues of these seven cases are still the same as those of butt-jointed cascaded ones. The total insertion loss is 4% (0.17 dB) for A+A, 8% (0.36 dB) for A+B, A+C, and A+D, and 12% (0.55 dB) for B+D, C+D, and D+D.

**L**), continuous variations of L

_{PS}≠0 and δ_{a}≠δ_{b}_{PS}following with a series of consequent phase differences can induce a predictable change in K-values. In order to realize the shortest power splitter for each given K-value, we firstly analyze the MMI devices by using Eq. (3) and summarize the calculated results in Fig. 4 by solid-line to give a whole view about K-values vs. the total length of devices. Fig. 4(a) is for the cases of δ

_{a}=0.1/0.2 µm and δ

_{b}=0, while Fig. 4(b) is for the cases of δ

_{a}=0 and δ

_{b}=0.1/0.2 µm. The horizontal dash lines help us to catch the needed device length for each starting point simultaneously corresponding to those summarized in Fig. 2. Moreover, the 2-D EME simulation results, marked by open circles (δ

_{a,b}=0.1 µm) and triangles (δ

_{a,b}=0.2 µm), are also consistent with those calculation results from MPA and matrix multiplication of Eq. (3).

_{PS}is linearly extended to a π-phase shifting at L

_{PS}=621 and 328 µm for δ

_{a(b)}/δ

_{b(a)}=0.1/0 and 0.2/0 µm, respectively. For clarity, each coupler shown Fig. 4 is in range of its best application region. In the case of

*β*>

_{a}*β*as demonstrated in Fig. 4(a), the K-value for the A+D configuration starts decreasing and then increasing as the total length increases. On the other hand, a reverse trend for the K-value is observed in the B+D and C+D configurations. However, in case of

_{b}*β*<

_{a}*β*as demonstrated in Fig. 4(b), all K-values can be fine adjusted in either up or down without any saturated point before reaching a π-phase shifting. This result is caused by the asymmetric cross-state transfer function of MMI-D as indicated in Table I. In brief, the case of

_{b}*β*<

_{a}*β*offers us a chance to obtain relative shorter MMI-based couplers with K-values in those three special continuously tunable range of K=0.07-0.15 (A+D), 0.93-0.8 (B+D), and 0.64-0.5 (C+D).

_{b}## 4. Conclusions

## Acknowledgement

## 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. | D. G. Rabus, M. Hamacher, U. Troppenz, and H. Heidrich, “Optical filters based on ring resonators with integrated semiconductor optical amplifiers in GaInAsP-InP,” IEEE J. Sel. Topics Quantum Electron. |

3. | S. Suzuki, K. Oda, and Y. Hibino, “Integrated-optic double-ring resonators with a wide free spectral range of 100 GHz,” J. Lightwave Technol. |

4. | S. Matsuo, Y. Yoshikuni, T. Segawa, Y. Ohiso, and H. Okamoto, “A widely tunable optical filter using ladder-type structure,” IEEE Photonics Technol. Lett. |

5. | V. M. Menon, W. Tong, C. Li, F. Xia, I. Glesk, P. R. Prucnal, and S. R. Forrest, “All-optical wavelength conversion using a regrowth-free monolithically integrated Sagnac interferometer,” IEEE Photonics Technol. Lett. |

6. | 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. |

7. | 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. |

8. | J. Leuthold and C. H. Joyner, “Multimode interference couplers with tunable power splitting ratios,” J. Lightwave Technol. |

9. | N. S. Lagali, M. R. Paiam, and R. I. MacDonald, “Theory of variable-ratio power splitters using multimode interference couplers,” IEEE Photonics Technol. Lett. |

10. | H. Ohe, H. Shimizu, and Y. Nakano, “InGaAlAs multiple-quantum-well optical phase modulators based on carrier depletion,” IEEE Photonics Technol. Lett. |

11. | Q. Lai, M. Bachmann, W. Hunziker, P. A. Besse, and H. Melchior, “Arbitrary ratio power splitters using angled silica on silicon multimode interference couplers,” Electron. Lett. |

12. | T. Saida, A. Himeno, M. Okuno, A. Sugita, and K. Okamoto, “Silica-based 2×2 multimode interference coupler with arbitrary power splitting ratio,” Electron. Lett. , |

13. | S. Y. Tseng, C. Fuentes-Hernandez, D. Owens, and B. Kippelen, “Variable splitting ratio 2×2 MMI couplers using multimode waveguide holograms,” Opt. Express , |

14. | 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. | BeamPROP, version 5.1, RSoft Inc., NY (2005). |

16. | FimmProp, version 4.3, Photon Design, Oxford, UK (2004). |

17. | D. J. Y. Feng, T. S. Lay, and T. Y. Chang, “Compact 2×2 Couplers for Unequal Splitting of Power Obtained by Cascading of Short MMI Sections,” in |

**OCIS Codes**

(130.2790) Integrated optics : Guided waves

(130.3120) Integrated optics : Integrated optics devices

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: March 6, 2008

Revised Manuscript: May 1, 2008

Manuscript Accepted: May 1, 2008

Published: May 2, 2008

**Citation**

David J. Y. Feng and T. S. Lay, "Compact multimode interference couplers with arbitrary power splitting ratio," Opt. Express **16**, 7175-7180 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-10-7175

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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]
- D. G. Rabus, M. Hamacher, U. Troppenz, and H. Heidrich, "Optical filters based on ring resonators with integrated semiconductor optical amplifiers in GaInAsP-InP," IEEE J. Sel. Top. Quantum Electron. 8, 1405-1411 (2002). [CrossRef]
- S. Suzuki, K. Oda, and Y. Hibino, "Integrated-optic double-ring resonators with a wide free spectral range of 100 GHz," J. Lightwave Technol. 13, 1766-1770 (1995). [CrossRef]
- S. Matsuo, Y. Yoshikuni, T. Segawa, Y. Ohiso, and H. Okamoto, "A widely tunable optical filter using ladder-type structure," IEEE Photonics Technol. Lett. 15, 1114-1116 (2003). [CrossRef]
- V. M. Menon, W. Tong, C. Li, F. Xia, I. Glesk, P. R. Prucnal, and S. R. Forrest, "All-optical wavelength conversion using a regrowth-free monolithically integrated Sagnac interferometer," IEEE Photon. Technol. Lett. 15, 254-256 (2003). [CrossRef]
- P. A. Besse, E. Gini, M. Bachmann and H. Melchior, "New 2x2 and 1x3 Multimode Interference Couplers with Free Selection of Power Splitting Ratios," J. Lightwave Technol. 14, 2286-2293 (1996). [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]
- J. Leuthold and C. H. Joyner, "Multimode interference couplers with tunable power splitting ratios," J. Lightwave Technol. 19, 700-707 (2001). [CrossRef]
- N. S. Lagali, M. R. Paiam, and R. I. MacDonald, "Theory of variable-ratio power splitters using multimode interference couplers," IEEE Photon. Technol. Lett. 11, 665-667 (1999). [CrossRef]
- H. Ohe, H. Shimizu, and Y. Nakano, "InGaAlAs multiple-quantum-well optical phase modulators based on carrier depletion," IEEE Photon. Technol. Lett. 19, 1816-1818 (2007). [CrossRef]
- Q. Lai, M. Bachmann, W. Hunziker, P. A. Besse, and H. Melchior, "Arbitrary ratio power splitters using angled silica on silicon multimode interference couplers," Electron. Lett. 32, 1576-1577 (1996). [CrossRef]
- T. Saida, A. Himeno, M. Okuno, A. Sugita, and K. Okamoto, "Silica-based 2x2 multimode interference coupler with arbitrary power splitting ratio," Electron. Lett. 35, 2031-2033 (1999). [CrossRef]
- S. Y. Tseng, C. Fuentes-Hernandez, D. Owens, and B. Kippelen, "Variable splitting ratio 2�?2 MMI couplers using multimode waveguide holograms," Opt. Express 15, 9015-9021 (2007). [CrossRef] [PubMed]
- 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). [CrossRef] [PubMed]
- BeamPROP, version 5.1, RSoft Inc., NY (2005).
- FimmProp, version 4.3, Photon Design, Oxford, UK (2004).
- D. J. Y. Feng, T. S. Lay, and T. Y. Chang, "Compact 2x2 Couplers for Unequal Splitting of Power Obtained by Cascading of Short MMI Sections," in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, OSA Technical Digest Series (CD) (Optical Society of America, 2007), paper JThA25, http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2007-JThA25

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