## Principles and application of reduced beat length in MMI couplers

Optics Express, Vol. 14, Issue 19, pp. 8753-8764 (2006)

http://dx.doi.org/10.1364/OE.14.008753

Acrobat PDF (481 KB)

### Abstract

A unified method was proposed to reduce the beat length of a multimode interference (MMI) coupler. By properly adjusting the phase difference of the N-fold images, the mode evolution is changed to generate self-images at a much shorter distance. The effect of adjusting the phase difference can be regarded as dividing the original MMI coupler into multiple sub-MMI couplers. Such an effect can be applied for both symmetric- and paired-interference cases. We applied the principle to design compact optical splitters operating at dual wavelength bands. The simulation shows that excellent performance with reduced coupler length can be obtained for splitters operating at both 1.3 and 1.55 µm bands.

© 2006 Optical Society of America

## 1. Introduction

1. H. Sasaki, E. Shki, and N. Mikoshiba, “Propagation characteristics of optical guided wave in asymmetric branching waveguides,” IEEE J. Quantum Electron. **QE-17**, 1051–1058 (1981). [CrossRef]

4. R. Baets and P. E. Lagasse, “Calculation of radiation loss in integrated-optic tapers and Y-junctions,” Appl. Opt. **21**, 1972–1978 (1982). [CrossRef] [PubMed]

5. O. Mikami and S. Zembutsu, “Coupling-length adjustment for an optical direction coupler as a 2×2 switch,” Appl. Phys. Lett. **35**, 38–40 (1979). [CrossRef]

7. M. Rajarajan, B. M. A. Rahman, and K. T. V. Grattan, “A rigorous comparison of the performance of directional couplers with multimode interference devices,” J. Lightwave Technol. **17**, 243–248 (1999). [CrossRef]

8. 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. Y. Gottesman, E. V. K. Rao, and B. Dagens, “A novel design proposal to minimize reflections in deep-ridge multimode interference couplers,” IEEE Photon. Technol. Lett. **12**, 1662–1664 (2000). [CrossRef]

9. K.-C. Lin and W.-Y. Lee, “Guided-wave 1.3/1.55µm wavelength division multiplexer based on multimode interference,” Electron. Lett. **32**, 1259–1261 (1996). [CrossRef]

## 2. Operation principles

### 2.1 Field representation in MMI

13. O. Bryngdahl, “Image formation using self-imaging techniques,” J. Opt. Soc. Amer. **63**, 416–419 (1973). [CrossRef]

*c*denotes the expansion coefficient for the m-th eigenmode of which the field distribution is

_{m}*φ*and the propagation constant is

_{m}*β*. The coordinates of

_{m}*x*and

*z*stand for the transverse and longitudinal directions, respectively.

*β*can be expressed as:

_{m}*λ*

_{0}and

*n*denote the free-space wavelength and the refractive index of the core waveguide, respectively.

_{r}*W*is the effective width that includes the Goos-Hanschen shifts at the waveguide boundaries. The beat length

_{e}*L*is defined as [8

_{π}8. 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]

*φ*can be approximated as [14

_{m}14. 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]

### 2.2 Principles of reduced beat length

*z*=0) where two self images are formed. Assume that one of the two images is represented as Φ(

*x*,0), as given in Eq. (5). The two images can be treated as the point sources for generating the following wave propagation. Therefore, the field evolution can be regarded as the superposition of responses from the two point sources. The relative phases of the N-fold images can be derived for the general or symmetric-interference cases [16

16. M. Bachmann, P. A. Besse, and H. Melchior, “General self-imaging properties in N×N multimode interference couplers including phase relations,” Appl. Opt. **33**, 3905–3911 (1994). [CrossRef] [PubMed]

17. E. R. Thoen, L. A. Molter, and J. P. Donnelly, “Exact modal analysis and optimization of N×N×1 cascaded waveguide structures with multimode guiding sections,” IEEE J. Quantum Electron. **33**, 1299–1307 (1997). [CrossRef]

*x*=0, as shown in Fig. 1. A superposition of purely odd modes results in a null along the center of waveguide(

*x*=0), which is equivalent to the case that a mirror (or a perfect conductor) is placed along the center. The field evolution in the MMI becomes the propagation in two sub-MMI couplers with a half of width. This can be verified by setting

*x*′=

*x*-

*W*4and

_{e}*m*=2

*m*′+1. Eq. (5) becomes:

*πλ*

_{0}/4

*n*(

_{r}*W*′)

_{e}^{2}with

*W*′=

_{e}*W*/2. Eqs (6) and (8) prove that the net effect of applying the π-phase shift is to divide the original MMI into two isolated regions. Since the effective width is halved, the beat length becomes

_{e}*L*π

_{′}=

*L*

_{π}/4. The above principles can be applied to general types or restricted-interference types of MMI couplers. The phase-shift scheme can also be applied to the longitudinal positions where the

*N*-fold (

*N*>2) images are formed.

*N*=3, if phase changes are applied to the local images so that the adjacent images are out-of-phase, the resultant wave evolution can be treated as if the MMI is divided into three isolated regions with an effective width of

*W*′=

_{e}*W*/3. This can be easily verified for the case with three equally-spaced images. Assume that the images appear at

_{e}*x*=-

*W*/3, 0,

_{e}*W*/3, respectively. When phase changes are applied to the three images to provide relative phase shifts of π, 0, and, π, respectively, the resultant field can be expressed as:

_{e}*x*,0) with Eq. (1) and (4) for

*z*=0, it is easy to show that the expansion coefficient is not vanishing only if

*m*=2, 5, 8, … Let

*m*=3

*m*′+2, Eq. (9) can be rewritten as:

*m*=3

*m*′+2, the propagation constant can be written as:

*x*=-

*W*/6 and

_{e}*W*/6, respectively. The beat length is reduced to

_{e}*L′*=

_{π}*L*/9.

_{π}*N*=4 or above to show the same effects from the phase shifts. Therefore, with appropriate phase shifts on the

*N*-fold images, the effective width and beat length can be reduced to

*W*and

_{e}/N*L*

_{π}/N^{2}, respectively.

18. S. Nagai, G. Morishima, H. Inayoshi, and K. Utaka, “Multimode interference photonic switches (MIPS),” J. Lightwave Technol. **20**, 675–681 (2002). [CrossRef]

*L*is the length of the phase shift region and Δ

_{m}*n*is the change in the refractive index. The size and location of the phase shift region could affect the wave evolution in a MMI. In our design, the phase shift region is placed on the self-imaging spots. Therefore, the design is not sensitive to the size variation as long as the region covers the spot and does not overlap with other self-imaging spots [18

_{r}18. S. Nagai, G. Morishima, H. Inayoshi, and K. Utaka, “Multimode interference photonic switches (MIPS),” J. Lightwave Technol. **20**, 675–681 (2002). [CrossRef]

8. 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}for the single mode access waveguide and the refractive index difference between the core and the cladding is 0.75%.

## 3. Restricted interference cases

### 3.1. Modified symmetric interference

*c*=0 for

_{m}*m*=1, 3, 5…., and the periodic distance is

*L*=3

_{c}*L*[8

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

*π*to one image, the modal expansion is modeled as the superposition of two out-of-phase point sources. Thus, the field becomes the summation of purely odd modes. The resultant effective width is equal to one half of the original one and the periodic distance is reduced by a factor of 1/2

^{2}. In general, the periodic distance is reduced to

*L′*=

_{c}*L*when appropriate phase shifts are applied to the

_{c}/N^{2}*N*-fold images. Figure 2 depicts the light evolution along the MMI coupler for the standard symmetric-interference case and the ones with phase shifts on two- to five-fold images. The rules for applying the phase shifts are summarized in Table 1. It is clear that the effects of applying phase shifts on the

*N*-fold images are like adding mirrors to divide a MMI to

*N*sub-MMI couplers.

### 3.2. Modified paired-interference

*c*=0 for

_{m}*m*=2, 5, 8…; and the periodic distance is

*L*. By applying a

_{c}=L_{π}*π*/2-phase shift to one of the two-fold images and make them to have

*π*phase difference, the modified paired-interference can also be described by Eq. (6). The net effect looks as if the MMI is divided into two sub-MMI’s. The wave evolution in each sub-MMI possesses the property of paired-interference. When the same principle is applied to the

*N*-fold images,

*L′*.

_{c}/L_{c}/N^{2}*N*sub-MMI’s. The above schemes of applying phase shifts result in point sources with alternating 0 and

*π*phases. The out-of-phase sources generate anti-symmetric wave evolution patterns that vanish at the intermediate planes. Another interesting arrangement of the phases is to make the

*N*-fold images to have symmetric phases such that the paired-interference property is turned into the symmetric-interference case. This changes the self-imaging patterns and the periodic distance. A simulation example is shown in Fig. 3(e) by using the phase shifts listed in Table 2. The periodic distance is reduced to

*L*/4.

_{c}*N*-fold images can also be adjusted to include both symmetric- and paired-interference cases. Figure 3(f) depicts the wave evolution when the 4-fold images are adjusted to have two pairs of phases that have a phase difference of

*π*, as listed in Table 2. The results reveal that the periodic interval can be reduced to

*L*/16 though the effective width is only reduced to one half. From the above results, there are a variety of phase shift combinations to alter the periodic distance of a MMI coupler. Since the beat lengths for the symmetric-type and paired-type excitations are different, the proposed method offers flexibility in choosing the periodic distance for a MMI coupler.

_{c}## 4. Applications

*λ*

_{1}and λ

_{2}), the total length of the MMI coupler should satisfy:

*L*and

_{c,λ1}*L*are the beat lengths for the λ

_{c,λ2}_{1}and λ

_{2}wavelengths, respectively.

*q*and

_{1}*q*are integers with no common divisor with

_{2}*N*. Using a conventional MMI structure to realize a dual-band 1×N power splitter would typically require a long device, because the beat length is only weakly wavelength dependent. The difference between

*L*and

_{c,λ1}*L*is usually small, so the values

_{c,λ2}*q*and

_{1}*q*are large.

_{2}*L*. When the length of the first region

_{MMI}=L_{d1}+L_{d2}*L*satisfies:

_{d1}*N*-fold images of λ

_{1}light and the single image of λ

_{2}light will appear at the same z position.

*s*and

_{1}*s*are integers that have no common divisor with

_{2}*N*. Under such a condition, applying the phase shift on the

*N*-fold images for λ

_{1}except the single-image spot for λ

_{2}can affect only the phase difference of λ

_{1}light. Therefore, the

*N*-fold images of λ

_{1}light will repeat with a periodic distance of

*L*, while the periodic distance of λ

_{c,λ1/N}_{2}light remains unchanged. When the length of the second part

*L*obeys this relation:

_{d2}*N*-fold images of λ

_{1}light and the

*N*-fold images of λ

_{2}light will appear at the same z position. Under such a condition, 1×N power splitting can be achieved for both wavelengths.

_{1}and λ

_{2}are interchanged. Thus, the alternative expressions for

*L*and

_{d1}*L*are:

_{d2}*s*and

_{3}*s*are integers that have no common divisor with

_{4}*N*.

*PO*and

_{max}*PO*are the maximum and minimum powers at the output waveguides.

_{min}### 4.1 1×2 dual-band splitter

_{1}and λ

_{2}be 1.3 µm and 1.55 µm, respectively. For obtaining compact devices, the s-integers and the MMI width should be optimized. Though a smaller MMI width is desired to shorten the beat length, the width should be large enough to support as many modes as required for self-imaging. From numerical simulation, we limit the MMI width to

*W*≧38µm. Moreover, the solution for the shortest MMI does not necessarily occur at the smallest

_{M}*W*. After calculation, the optimal solution for each case is listed in Table 3.

_{M}_{2}and q

_{1}are 43 and 37, respectively, when

*W*=38µm. It requires a long MMI section (

_{M}*L*=45045 µm). By using the phase-shift scheme listed in Table 2, the shortest device is found while (

_{MMI}*s*)=(7, 3) and

_{3}, s_{4}*W*=45 µm. The MMI coupler is 15057 µm long under this condition. Figure 5(a) and (b) shows the light propagating pattern along the MMI coupler for the shortest solution. Power splitting can be achieved for the two wavelengths. Figure 6(a) illustrates the simulated imbalance and excess loss as a function of the deviation in device length (ΔL

_{MMI}_{MMI}). The solid and dashed lines show the performance for the TE and TM polarization, respectively. The imbalance is below 0.15 and 0.25 dB for the 1.3- and 1.55-µm wavelengths, respectively. The excess loss is below 0.1 dB for both wavelengths. Among the solutions listed in Table 3, the one with the shortest device has better performance in terms of the excess loss and imbalance.

20. J. Z. Huang, R. Scarmozzio, G. Nagy, M. J. Steel, and R. M. Osgood Jr., “Realization of a compact and single-mode optical passive polarization converter,” IEEE Photon. Technol. Lett. **12**, 317–319 (2000). [CrossRef]

19. M. Rajarajan, B. M. A. Rahman, T. Wongcharoen, and K. T. V. Grattan, “Accurate analysis of MMI devices with two-dimensional confinement,” J. Lightwave Technol. **14**, 2078–2084 (1996). [CrossRef]

### 4.2. 1×3 dual-band splitter

*q*)=(43, 37) and

_{2}, q_{1}*W*=38 µm, which gives a MMI length of 23100 µm for constructing a dual band splitter. By using the phase-shift scheme listed in Table 1, The best solution is obtained when (

_{MMI}*s*)=(7, 2) and

_{3}, s_{4}*W*=52 µm, which gives the shortest MMI coupler (

_{MMI}*L*=8802µm). The MMI with phase shifts can be much shorter than the conventional one. Figure 7 shows the power evolution along the MMI coupler for the two wavelength bands. The 1×3 power splitting function can be obtained for both wavelengths. Figure 8 illustrates the simulated imbalance and excess loss as a function of the device length (Δ

_{MMI}*L*) and wavelength. The imbalance is below 0.01 and 0.04 dB for the 1.3- and 1.55-µm wavelengths, respectively. The excess loss is below 0.06 dB for both wavelengths. The wavelength tolerance for less than 0.5dB of excess loss is larger than 30 nm. Again, from Table 4, the solution with the shortest device has better performance in terms of the excess loss and imbalance.

_{MMI}## 5. Conclusions

## Acknowledgments

## References and links

1. | H. Sasaki, E. Shki, and N. Mikoshiba, “Propagation characteristics of optical guided wave in asymmetric branching waveguides,” IEEE J. Quantum Electron. |

2. | M. Belanger, G. L. Yip, and M. Haruna, “Passive planar multibranch optical power divider: Some design considerations,” Appl. Opt. |

3. | M. Haruna and J. Koyama, “Electrooptic branching waveguide-switch and the application to 1×4 optical switching network,” J. Lightwave Technol. |

4. | R. Baets and P. E. Lagasse, “Calculation of radiation loss in integrated-optic tapers and Y-junctions,” Appl. Opt. |

5. | O. Mikami and S. Zembutsu, “Coupling-length adjustment for an optical direction coupler as a 2×2 switch,” Appl. Phys. Lett. |

6. | H. A. Haus and C. G. Fonstad, “Three waveguide couplers for improved sampling and filtering,” IEEE J. Quantum Electron. |

7. | M. Rajarajan, B. M. A. Rahman, and K. T. V. Grattan, “A rigorous comparison of the performance of directional couplers with multimode interference devices,” J. Lightwave Technol. |

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

9. | K.-C. Lin and W.-Y. Lee, “Guided-wave 1.3/1.55µm wavelength division multiplexer based on multimode interference,” Electron. Lett. |

10. | Y.-J. Lin and S.-L. Lee, “InP-based 1.3/1.55µm wavelength demultiplexer with multimode interference and chirped grating,” Opt. and Quantum Electron. |

11. | Y. Ma, S. Park, L. Wang, and S. T. Ho, “Ultracompact multimode interference 3-dB coupler with strong lateral confinement by deep dry etching,” IEEE Photon. Technol. Lett. |

12. | Y. Gottesman, E. V. K. Rao, and B. Dagens, “A novel design proposal to minimize reflections in deep-ridge multimode interference couplers,” IEEE Photon. Technol. Lett. |

13. | O. Bryngdahl, “Image formation using self-imaging techniques,” J. Opt. Soc. Amer. |

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

15. | J. M. Heaton, R. M. Jenkins, D. R. Wight, J. T. Parker, J. C. H. Birbeck, and K. P. Hilton, “Novel 1-to-N way integrated optical beam splitters using symmetric mode mixing in GaAs/AlGaAs multimode waveguides,” Appl. Phys. Lett. |

16. | M. Bachmann, P. A. Besse, and H. Melchior, “General self-imaging properties in N×N multimode interference couplers including phase relations,” Appl. Opt. |

17. | E. R. Thoen, L. A. Molter, and J. P. Donnelly, “Exact modal analysis and optimization of N×N×1 cascaded waveguide structures with multimode guiding sections,” IEEE J. Quantum Electron. |

18. | S. Nagai, G. Morishima, H. Inayoshi, and K. Utaka, “Multimode interference photonic switches (MIPS),” J. Lightwave Technol. |

19. | M. Rajarajan, B. M. A. Rahman, T. Wongcharoen, and K. T. V. Grattan, “Accurate analysis of MMI devices with two-dimensional confinement,” J. Lightwave Technol. |

20. | J. Z. Huang, R. Scarmozzio, G. Nagy, M. J. Steel, and R. M. Osgood Jr., “Realization of a compact and single-mode optical passive polarization converter,” IEEE Photon. Technol. Lett. |

**OCIS Codes**

(130.3120) Integrated optics : Integrated optics devices

(230.1360) Optical devices : Beam splitters

(230.7370) Optical devices : Waveguides

**ToC Category:**

Optical Devices

**History**

Original Manuscript: June 22, 2006

Revised Manuscript: September 7, 2006

Manuscript Accepted: September 10, 2006

Published: September 18, 2006

**Citation**

Lung-Wei Chung, San-Liang Lee, and Yen-Juei Lin, "Principles and application of reduced beat length
in MMI couplers," Opt. Express **14**, 8753-8764 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-19-8753

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### References

- H. Sasaki, E. Shki, and N. Mikoshiba, "Propagation characteristics of optical guided wave in asymmetric branching waveguides," IEEE J. Quantum Electron. 17, 1051-1058 (1981). [CrossRef]
- M. Belanger, G. L. Yip, and M. Haruna, "Passive planar multibranch optical power divider: Some design considerations," Appl. Opt. 22, 2283-2289 (1983). [CrossRef]
- M. Haruna and J. Koyama, "Electro-optic branching waveguide-switch and the application to 1 × 4 optical switching network," J. Lightwave Technol. 1, 233-247 (1983).
- R. Baets and P. E. Lagasse, "Calculation of radiation loss in integrated-optic tapers and Y-junctions," Appl. Opt. 21, 1972-1978 (1982). [CrossRef] [PubMed]
- O. Mikami and S. Zembutsu, "Coupling-length adjustment for an optical direction coupler as a 2 × 2 switch," Appl. Phys. Lett. 35, 38-40 (1979). [CrossRef]
- H. A. Haus and C. G. Fonstad, "Three waveguide couplers for improved sampling and filtering," IEEE J. Quantum Electron. 17, 2321-2325 (1981). [CrossRef]
- M. Rajarajan, B. M. A. Rahman, and K. T. V. Grattan, "A rigorous comparison of the performance of directional couplers with multimode interference devices," J. Lightwave Technol. 17, 243-248 (1999). [CrossRef]
- 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]
- K.-C. Lin and W.-Y. Lee, "Guided-wave 1.3/1.55μm wavelength division multiplexer based on multimode interference," Electron. Lett. 32, 1259-1261 (1996). [CrossRef]
- Y.-J. Lin and S.-L. Lee, "InP-based 1.3/1.55μm wavelength demultiplexer with multimode interference and chirped grating," Opt. and Quantum Electron. 34, 1201-1212 (2002). [CrossRef]
- Y. Ma, S. Park, L. Wang, and S. T. Ho, "Ultracompact multimode interference 3-dB coupler with strong lateral confinement by deep dry etching," IEEE Photon. Technol. Lett. 12, 492-494 (2000). [CrossRef]
- Y. Gottesman, E. V. K. Rao, and B. Dagens, "A novel design proposal to minimize reflections in deep-ridge multimode interference couplers," IEEE Photon. Technol. Lett. 12, 1662-1664 (2000). [CrossRef]
- O. Bryngdahl, "Image formation using self-imaging techniques," J. Opt. Soc. Amer. 63, 416-419 (1973). [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]
- J. M. Heaton, R. M. Jenkins, D. R. Wight, J. T. Parker, J. C. H. Birbeck, and K. P. Hilton, "Novel 1-to-N way integrated optical beam splitters using symmetric mode mixing in GaAs/AlGaAs multimode waveguides," Appl. Phys. Lett. 61, 1754-1756 (1992). [CrossRef]
- M. Bachmann, P. A. Besse, and H. Melchior, "General self-imaging properties in N×N multimode interference couplers including phase relations," Appl. Opt. 33, 3905-3911 (1994). [CrossRef] [PubMed]
- E. R. Thoen, L. A. Molter, and J. P. Donnelly, "Exact modal analysis and optimization of N×N×1 cascaded waveguide structures with multimode guiding sections," IEEE J. Quantum Electron. 33, 1299-1307 (1997). [CrossRef]
- S. Nagai, G. Morishima, H. Inayoshi, and K. Utaka, "Multimode interference photonic switches (MIPS)," J. Lightwave Technol. 20, 675-681 (2002). [CrossRef]
- M. Rajarajan, B. M. A. Rahman, T. Wongcharoen, and K. T. V. Grattan, "Accurate analysis of MMI devices with two-dimensional confinement," J. Lightwave Technol. 14, 2078-2084 (1996). [CrossRef]
- J. Z. Huang, R. Scarmozzio, G. Nagy, M. J. Steel, R. M. Osgood, and Jr., "Realization of a compact and single-mode optical passive polarization converter," IEEE Photonics Technol. Lett. 12, 317-319 (2000). [CrossRef]

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