## Topology optimized mode conversion in a photonic crystal waveguide fabricated in silicon-on-insulator material |

Optics Express, Vol. 22, Issue 7, pp. 8525-8532 (2014)

http://dx.doi.org/10.1364/OE.22.008525

Acrobat PDF (1718 KB)

### Abstract

We have designed and for the first time experimentally verified a topology optimized mode converter with a footprint of ~6.3 μm × ~3.6 μm which converts the fundamental even mode to the higher order odd mode of a dispersion engineered photonic crystal waveguide. 2D and 3D topology optimization is utilized and both schemes result in designs theoretically showing an extinction ratio larger than 21 dB. The 3D optimized design has an experimentally estimated insertion loss lower than ~2 dB in an ~43 nm bandwidth. The mode conversion is experimentally confirmed in this wavelength range by recording mode profiles using vertical grating couplers and an infrared camera. The experimentally determined extinction ratio is > 12 dB and is believed to be limited by the spatial resolution of our setup.

© 2014 Optical Society of America

## 1. Introduction

1. Y. Huang, G. Xu, and S.-T. Ho, “An ultracompact optical mode order converter,” IEEE Photon. Technol. Lett. **18**(21), 2281–2283 (2006). [CrossRef]

7. J. Lu and J. Vučković, “Objective-first design of high-efficiency, small-footprint couplers between arbitrary nanophotonic waveguide modes,” Opt. Express **20**(7), 7221–7236 (2012). [CrossRef] [PubMed]

1. Y. Huang, G. Xu, and S.-T. Ho, “An ultracompact optical mode order converter,” IEEE Photon. Technol. Lett. **18**(21), 2281–2283 (2006). [CrossRef]

4. B. T. Lee and S. Y. Shin, “Mode-order converter in a multimode waveguide,” Opt. Lett. **28**(18), 1660–1662 (2003). [CrossRef] [PubMed]

5. G. Chen and J. U. Kang, “Waveguide mode converter based on two-dimensional photonic crystals,” Opt. Lett. **30**(13), 1656–1658 (2005). [CrossRef] [PubMed]

7. J. Lu and J. Vučković, “Objective-first design of high-efficiency, small-footprint couplers between arbitrary nanophotonic waveguide modes,” Opt. Express **20**(7), 7221–7236 (2012). [CrossRef] [PubMed]

4. B. T. Lee and S. Y. Shin, “Mode-order converter in a multimode waveguide,” Opt. Lett. **28**(18), 1660–1662 (2003). [CrossRef] [PubMed]

6. V. Liu, D. A. B. Miller, and S. Fan, “Ultra-compact photonic crystal waveguide spatial mode converter and its connection to the optical diode effect,” Opt. Express **20**(27), 28388–28397 (2012). [CrossRef] [PubMed]

7. J. Lu and J. Vučković, “Objective-first design of high-efficiency, small-footprint couplers between arbitrary nanophotonic waveguide modes,” Opt. Express **20**(7), 7221–7236 (2012). [CrossRef] [PubMed]

8. M. P. Bendsøe and N. Kikuchi, “Generating optimal topologies in structural design using a homogenization method,” Comput. Methods Appl. Mech. Eng. **71**(2), 197–224 (1988). [CrossRef]

9. P. I. Borel, A. Harpøth, L. H. Frandsen, M. Kristensen, P. Shi, J. S. Jensen, and O. Sigmund, “Topology optimization and fabrication of photonic crystal structures,” Opt. Express **12**(9), 1996–2001 (2004). [CrossRef] [PubMed]

10. J. S. Jensen and O. Sigmund, “Topology optimization for nano-photonics,” Laser Photon. Rev. **5**(2), 308–321 (2011), doi:. [CrossRef]

_{0}) mode and the first higher order odd mode (TE

_{1}) in a photonic crystal waveguide (PhCW) fabricated in silicon-on-insulator (SOI) material. Mode conversion is obtained by applying topology optimization to the PhCW and demonstrates the practical feasibility of using TO in order to realize ultra-compact mode conversion in PICs. Furthermore, TO is used to optimize the coupling efficiency of the TE

_{1}mode from the PhCW to a photonic wire (PhW).

## 2. Design, optimization, and modelling

### 2.1 Dispersion engineering the photonic crystal waveguide

_{2}= 0.8Λ. Consequently, the calculated [11

11. S. Johnson and J. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express **8**(3), 173–190 (2001). [CrossRef] [PubMed]

_{1}mode (cyan cross) is made monotonic, hence, more practical to work with, and has near-linear dispersion (blue circle) in an ~45 nm wavelength region (grey) centered around ~0.273 Λ/λ. The linear regime of the TE

_{0}mode (pink cross and red circle) is left unchanged. The possibility to alter the dispersion properties of a PhCW mode by perturbing the waveguide design is well-known [12

12. L. H. Frandsen, A. V. Lavrinenko, J. Fage-Pedersen, and P. I. Borel, “Photonic crystal waveguides with semi-slow light and tailored dispersion properties,” Opt. Express **14**(20), 9444–9450 (2006). [CrossRef] [PubMed]

13. J. Li, T. P. White, L. O’Faolain, A. Gomez-Iglesias, and T. F. Krauss, “Systematic design of flat band slow light in photonic crystal waveguides,” Opt. Express **16**(9), 6227–6232 (2008). [CrossRef] [PubMed]

_{1}mode relies on the different spatial mode distributions of the TE

_{0}and TE

_{1}mode as shown in Fig. 1(c). Because the TE

_{1}mode extends deeper into the cladding of the PhCW compared to the tightly confined TE

_{0}mode, only the TE

_{1}mode will be affected by changes in the second row.

### 2.2 Topology optimization of the mode converter and the output coupling interface

_{0}mode of the PhCW into the TE

_{1}mode in the ~45 nm wavelength region located in the index-guided regime around 0.273 Λ/λ. Figure 2(a) shows the PhCW used as a basis for the TO which is comprised of a 450 nm wide and single-mode input PhW, a 6.3 μm long dispersion engineered PhCW (c.f. previous discussion), and an 1150 nm wide output PhW supporting

*more*than two modes. In the TO, the width of the output PhW was widened compared to a 750 nm wide PhW supporting

*only*two modes in order to reduce any ravaging reflections back into the PhCW caused by an improper coupling interface between the PhCW and the output PhW. This was done well-knowing that the output coupling interface during optimization was not optimal from a transmission point of view. However, low transmission into the PhW had no impact on the TO as the objectives for the optimization were located inside the PhCW as described in the following and did not depend on the transmission to the PhW.

14. Y. Elesin, B. S. Lazarov, J. S. Jensen, and O. Sigmund, “Design of robust and efficient photonic switches using topology optimization,” Photon. Nanostruct. Fund. Appl. **10**(1), 153–165 (2012). [CrossRef]

15. Y. Elesin, B. S. Lazarov, J. S. Jensen, and O. Sigmund, “Time domain topology optimization of 3D nanophotonic devices,” Photon. Nanostruct. Fund. Appl. **12**(1), 23–33 (2014). [CrossRef]

_{1}mode. The design domain (green) in Fig. 2(a) is ~2.5 μm × ~2.5 μm and was arbitrarily chosen to be a set of circular and/or elliptical regions around the 3 innermost rows of holes in order to prevent dramatic changes in the design and to avoid appearance of isolated silicon islands that would be incompatible with e.g. a membrane configuration. Although we could have chosen a larger and/or a continuous design domain, numerical experiments showed that relatively small and isolated design areas were enough to yield the targeted mode conversion. Had the conversion efficiency been unsatisfactory, the design areas could have been changed in order to introduce more freedom in the TO. After having obtained efficient mode conversion in the PhCW at position #1, we address the output coupling interface to a 750 nm wide and double-mode PhW by doing a separate TO on the design shown in Fig. 2(b). Here, the TE

_{1}mode is excited inside the PhCW and TO is performed using the design domain sketched in green in Fig. 2(b). Here, the objective is to optimize the transmission of the TE

_{1}mode from the PhCW to position #2 (dashed red) in the output PhW. The final mode converter design is obtained by combining the optimized converter and the optimized output coupling designs into one PhCW design.

11. S. Johnson and J. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express **8**(3), 173–190 (2001). [CrossRef] [PubMed]

_{Si}= 3.418 and has a silica buffer with index n

_{SiO2}= 1.444 below and air above. The TE

_{0}and TE

_{1}modes are partly located above the silica-line in the ~45 nm wavelength region of interest, thus, both modes may leak into the silica buffer layer. However, due to the short length of the PhCW the out-of-plane leakage-loss for the generic PhCW is expected to be negligible. The design domain extends vertically through the silicon slab but designed structures are constrained to be uniform in the vertical direction to make the optimized designs feasible for fabrication with dry etching techniques. In all optimizations, a ~280 nm spectrally broad (full-width at half-maximum) Gaussian-shaped pulse is used as optical input centered on ~1540 nm. Figures 2(c) and 2(d) show the 2D and 3D topology optimized (TopOpt) designs, respectively. The final designs of the mode converters have been obtained after ~200 TO iteration steps whereas coupling interface designs have reached convergence after only ~25 iteration steps. By comparing the 2D and 3D optimized structures some features are seen to be similar and both optimizations involve intricate tuning of the individual photonic crystal holes. As intuitively expected from the symmetric intensity profile of the TE

_{1}mode, the TopOpt coupling interfaces are (close to being) symmetric around the axis of propagation, even though such symmetry was not enforced explicitly.

### 2.3 Modelling the topology optimized structures

_{0}light input into the 3D TopOpt (blue circle) mode converter normalized to the transmission spectrum of the TE

_{0}mode propagating through a PhW. Mode conversion of the TE

_{0}to the TE

_{1}mode is clearly observed as illustrated in Fig. 3(b) showing the propagation of the orthogonal H-field (H

_{z}) at 1545 nm through the 3D TopOpt mode converter from Fig. 2(d). As seen, the mode conversion is obtained by a non-trivial twisting of the TE

_{0}mode, hence, intuitively explaining the asymmetric structure of the converter design. The inset of Fig. 3(a) shows the output power flux recorded at 1545 nm in the PhW as a function of the relative position across the PhW with 0 nm being the PhW center. The flux reveals a TE

_{1}/TE

_{0}mode extinction ratio better than 21 dB. The 3D TopOpt mode converter has an ~43 nm operational bandwidth from ~1523 nm to ~1565 nm showing lower than 2 dB insertion loss compared to a PhW and better than ~16 dB TE

_{1}/TE

_{0}mode extinction ratio. Figure 3(a) also shows the 3D FDTD-calculated transmission spectrum for the 2D TopOpt (red square) mode converter and clearly shows a lower transmission than the 3D TopOpt. This is in accordance with previous investigations [15

15. Y. Elesin, B. S. Lazarov, J. S. Jensen, and O. Sigmund, “Time domain topology optimization of 3D nanophotonic devices,” Photon. Nanostruct. Fund. Appl. **12**(1), 23–33 (2014). [CrossRef]

_{0}mode is not converted and, naturally, the butt coupling interface gives a higher transmission (of the TE

_{0}mode) than the topology optimized interface. The lowest insertion loss of the mode converter is 0.90 dB at 1545 nm. This loss includes coupling loss for the TE

_{0}mode at the standard input interface and we believe the insertion loss of the component may be decreased ~0.2-0.3 dB by applying topology optimization to the input coupling interface as well.

## 3. Experimental results

### 3.1 Fabrication

_{6}and C

_{4}F

_{8}gases to transfer the patterns into the top 340 nm silicon layer of a SOI wafer having a 3000 nm buffer layer of silica. Inversely tapered silicon ridge- and partly overlapping ~3 μm × 3 μm SU-8 polymer-waveguides are used to improve coupling to/from tapered and lensed single-mode fibers [16

16. M. Pu, L. Liu, H. Ou, K. Yvind, and J. M. Hvam, “Ultra-low-loss inverted taper coupler for silicon-on-insulator ridge waveguide,” Opt. Commun. **283**(19), 3678–3682 (2010). [CrossRef]

### 3.2 Characterization of the 3D topology optimized mode converter

_{0}-TE

_{1}(X1) and TE

_{0}-TE

_{1}-TE

_{0}(X2) mode converters were fabricated. The latter was realized by mirroring the X1 converter design subsequently after the TE

_{0}to TE

_{1}conversion. Figure 4(b) shows a SEM image of the fabricated X2 mode converter overlaid with the 3D FDTD-calculated H

_{z}-field at ~1530 nm demonstrating nice TE

_{0}-TE

_{1}-TE

_{0}conversion and the feasibility of mirroring and/or cascading TopOpt devices. Only the transmission spectrum for the X2 mode converter was recorded because no efficient tapered few-mode fiber supporting the TE

_{1}mode was available to us. The transmission spectrum for TE polarized light propagating through the X2 converter was recorded on an optical spectrum analyzer using a tunable laser source from 1480 nm-1580 nm. In the measurement, light is output from the X2 mode converter through a 350 nm wide and 1.6 mm long single-mode PhW to ensure that only a pure TE

_{0}mode is collected with the single-mode tapered and lensed fibers used.

17. Y. Ding, H. Ou, J. Xu, M. Xiong, and C. Peucheret, “On-chip mode multiplexer based on a single grating coupler,” in *Proceedings of IEEE Photonics Conference* (Institute of Electrical and Electronics Engineers, 2012), pp. 707–708. [CrossRef]

18. B. Wohlfeil, S. Burger, C. Stamatiadis, J. Pomplun, F. Schmidt, L. Zimmermann, and K. Petermann, “Numerical simulation of grating couplers for mode multiplexed systems,” Proc. SPIE **8988**, 89880K (2014). [CrossRef]

_{0}-TE

_{1}mode conversion in a broad range from ~1480 nm to ~1545 nm and the TE

_{0}-TE

_{1}-TE

_{0}conversion is confirmed. Figure 6(c) shows intensity line scans across the mode profiles for the X1 mode converter as a function of the relative position in the grating with 0 μm being the center of the grating. The mode profiles have been recorded at 1500 nm (red square), 1530 nm (orange triangle), and 1543 nm (green circle) using an objective with high numerical aperture (0.9) and 2 mm focal length (magnification x100). Experimentally, we find a TE

_{1}/TE

_{0}extinction ratio around 12 dB in the bandwidth of operation. However, we see that the node of the TE

_{1}profile (at relative position 0 μm in Fig. 6(c), i.e. at the center of the grating) is only resolved by a single point, thus, the highest possible extinction ratio we can measure is limited by the spatial resolution of the setup being the size of the grating coupler, the magnification of the microscope objective, and the pixel pitch of the detector (30 μm). Therefore, by utilizing a larger grating and/or a detector array with smaller pitch we will be able to resolve the node better and measure a more accurate extinction ratio, which we anticipate to increase. As the designs are reversible and characterized in a linear power regime, we can estimate the insertion loss of the TE

_{0}-TE

_{1}converter by halving the measured insertion loss of the TE

_{0}-TE

_{1}-TE

_{0}converter. The maximum error by doing so estimated from the 12 dB extinction error is < 0.25 dB. From Fig. 5 the 3D TopOpt X1 (X2) mode converter is experimentally estimated to have an insertion loss < ~2 dB (~4 dB) in a bandwidth from ~1500 nm to ~1543 nm corresponding to the linear and monotonic regime in Fig. 1. In the bandwidth from ~1510 nm to ~1540 nm the insertion loss is lower than ~1 dB (~2 dB).

## 4. Conclusion

## Acknowledgments

## References and links

1. | Y. Huang, G. Xu, and S.-T. Ho, “An ultracompact optical mode order converter,” IEEE Photon. Technol. Lett. |

2. | L. Luo, L. H. Gabrielli, and M. Lipson, “On-chip mode-division multiplexer,” in |

3. | Y. Ding, J. Xu, F. Da Ros, B. Huang, H. Ou, and C. Peucheret, “On-chip two-mode division multiplexing using tapered directional coupler-based mode multiplexer and demultiplexer,” Opt. Express |

4. | B. T. Lee and S. Y. Shin, “Mode-order converter in a multimode waveguide,” Opt. Lett. |

5. | G. Chen and J. U. Kang, “Waveguide mode converter based on two-dimensional photonic crystals,” Opt. Lett. |

6. | V. Liu, D. A. B. Miller, and S. Fan, “Ultra-compact photonic crystal waveguide spatial mode converter and its connection to the optical diode effect,” Opt. Express |

7. | J. Lu and J. Vučković, “Objective-first design of high-efficiency, small-footprint couplers between arbitrary nanophotonic waveguide modes,” Opt. Express |

8. | M. P. Bendsøe and N. Kikuchi, “Generating optimal topologies in structural design using a homogenization method,” Comput. Methods Appl. Mech. Eng. |

9. | P. I. Borel, A. Harpøth, L. H. Frandsen, M. Kristensen, P. Shi, J. S. Jensen, and O. Sigmund, “Topology optimization and fabrication of photonic crystal structures,” Opt. Express |

10. | J. S. Jensen and O. Sigmund, “Topology optimization for nano-photonics,” Laser Photon. Rev. |

11. | S. Johnson and J. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express |

12. | L. H. Frandsen, A. V. Lavrinenko, J. Fage-Pedersen, and P. I. Borel, “Photonic crystal waveguides with semi-slow light and tailored dispersion properties,” Opt. Express |

13. | J. Li, T. P. White, L. O’Faolain, A. Gomez-Iglesias, and T. F. Krauss, “Systematic design of flat band slow light in photonic crystal waveguides,” Opt. Express |

14. | Y. Elesin, B. S. Lazarov, J. S. Jensen, and O. Sigmund, “Design of robust and efficient photonic switches using topology optimization,” Photon. Nanostruct. Fund. Appl. |

15. | Y. Elesin, B. S. Lazarov, J. S. Jensen, and O. Sigmund, “Time domain topology optimization of 3D nanophotonic devices,” Photon. Nanostruct. Fund. Appl. |

16. | M. Pu, L. Liu, H. Ou, K. Yvind, and J. M. Hvam, “Ultra-low-loss inverted taper coupler for silicon-on-insulator ridge waveguide,” Opt. Commun. |

17. | Y. Ding, H. Ou, J. Xu, M. Xiong, and C. Peucheret, “On-chip mode multiplexer based on a single grating coupler,” in |

18. | B. Wohlfeil, S. Burger, C. Stamatiadis, J. Pomplun, F. Schmidt, L. Zimmermann, and K. Petermann, “Numerical simulation of grating couplers for mode multiplexed systems,” Proc. SPIE |

**OCIS Codes**

(000.3860) General : Mathematical methods in physics

(000.4430) General : Numerical approximation and analysis

(030.4070) Coherence and statistical optics : Modes

(130.3120) Integrated optics : Integrated optics devices

(130.5296) Integrated optics : Photonic crystal waveguides

(050.6624) Diffraction and gratings : Subwavelength structures

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: January 31, 2014

Revised Manuscript: March 24, 2014

Manuscript Accepted: March 27, 2014

Published: April 2, 2014

**Citation**

Lars H. Frandsen, Yuriy Elesin, Louise F. Frellsen, Miranda Mitrovic, Yunhong Ding, Ole Sigmund, and Kresten Yvind, "Topology optimized mode conversion in a photonic crystal waveguide fabricated in silicon-on-insulator material," Opt. Express **22**, 8525-8532 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-7-8525

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### References

- Y. Huang, G. Xu, S.-T. Ho, “An ultracompact optical mode order converter,” IEEE Photon. Technol. Lett. 18(21), 2281–2283 (2006). [CrossRef]
- L. Luo, L. H. Gabrielli, and M. Lipson, “On-chip mode-division multiplexer,” in CLEO: 2013, OSA Technical Digest (online) (Optical Society of America, 2013), paper CTh1C.6.
- Y. Ding, J. Xu, F. Da Ros, B. Huang, H. Ou, C. Peucheret, “On-chip two-mode division multiplexing using tapered directional coupler-based mode multiplexer and demultiplexer,” Opt. Express 21(8), 10376–10382 (2013). [CrossRef] [PubMed]
- B. T. Lee, S. Y. Shin, “Mode-order converter in a multimode waveguide,” Opt. Lett. 28(18), 1660–1662 (2003). [CrossRef] [PubMed]
- G. Chen, J. U. Kang, “Waveguide mode converter based on two-dimensional photonic crystals,” Opt. Lett. 30(13), 1656–1658 (2005). [CrossRef] [PubMed]
- V. Liu, D. A. B. Miller, S. Fan, “Ultra-compact photonic crystal waveguide spatial mode converter and its connection to the optical diode effect,” Opt. Express 20(27), 28388–28397 (2012). [CrossRef] [PubMed]
- J. Lu, J. Vučković, “Objective-first design of high-efficiency, small-footprint couplers between arbitrary nanophotonic waveguide modes,” Opt. Express 20(7), 7221–7236 (2012). [CrossRef] [PubMed]
- M. P. Bendsøe, N. Kikuchi, “Generating optimal topologies in structural design using a homogenization method,” Comput. Methods Appl. Mech. Eng. 71(2), 197–224 (1988). [CrossRef]
- P. I. Borel, A. Harpøth, L. H. Frandsen, M. Kristensen, P. Shi, J. S. Jensen, O. Sigmund, “Topology optimization and fabrication of photonic crystal structures,” Opt. Express 12(9), 1996–2001 (2004). [CrossRef] [PubMed]
- J. S. Jensen, O. Sigmund, “Topology optimization for nano-photonics,” Laser Photon. Rev. 5(2), 308–321 (2011), doi:. [CrossRef]
- S. Johnson, J. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8(3), 173–190 (2001). [CrossRef] [PubMed]
- L. H. Frandsen, A. V. Lavrinenko, J. Fage-Pedersen, P. I. Borel, “Photonic crystal waveguides with semi-slow light and tailored dispersion properties,” Opt. Express 14(20), 9444–9450 (2006). [CrossRef] [PubMed]
- J. Li, T. P. White, L. O’Faolain, A. Gomez-Iglesias, T. F. Krauss, “Systematic design of flat band slow light in photonic crystal waveguides,” Opt. Express 16(9), 6227–6232 (2008). [CrossRef] [PubMed]
- Y. Elesin, B. S. Lazarov, J. S. Jensen, O. Sigmund, “Design of robust and efficient photonic switches using topology optimization,” Photon. Nanostruct. Fund. Appl. 10(1), 153–165 (2012). [CrossRef]
- Y. Elesin, B. S. Lazarov, J. S. Jensen, O. Sigmund, “Time domain topology optimization of 3D nanophotonic devices,” Photon. Nanostruct. Fund. Appl. 12(1), 23–33 (2014). [CrossRef]
- M. Pu, L. Liu, H. Ou, K. Yvind, J. M. Hvam, “Ultra-low-loss inverted taper coupler for silicon-on-insulator ridge waveguide,” Opt. Commun. 283(19), 3678–3682 (2010). [CrossRef]
- Y. Ding, H. Ou, J. Xu, M. Xiong, and C. Peucheret, “On-chip mode multiplexer based on a single grating coupler,” in Proceedings of IEEE Photonics Conference (Institute of Electrical and Electronics Engineers, 2012), pp. 707–708. [CrossRef]
- B. Wohlfeil, S. Burger, C. Stamatiadis, J. Pomplun, F. Schmidt, L. Zimmermann, K. Petermann, “Numerical simulation of grating couplers for mode multiplexed systems,” Proc. SPIE 8988, 89880K (2014). [CrossRef]

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