## Ultra-compact photonic crystal waveguide spatial mode converter and its connection to the optical diode effect |

Optics Express, Vol. 20, Issue 27, pp. 28388-28397 (2012)

http://dx.doi.org/10.1364/OE.20.028388

Acrobat PDF (1023 KB)

### Abstract

We design an extremely compact photonic crystal waveguide spatial mode converter which converts the fundamental even mode to the higher order odd mode with nearly 100% efficiency. We adapt a previously developed design and optimization process that allows these types of devices to be designed in a matter of minutes. We also present an extremely compact optical diode device and clarify its general properties and its relation to spatial mode converters. Finally, we connect the results here to a general theory on the complexity of optical designs.

© 2012 OSA

## 1. Introduction

1. J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photon. **6**, 488–496 (2012). [CrossRef]

2. L. H. Gabrielli, D. Liu, S. G. Johnson, and M. Lipson, “On-chip transformation optics for multimode waveguide bends,” Nat. Commun. **3**, 1217 (2012). [CrossRef] [PubMed]

3. D. A. B. Miller, “All linear optical devices are mode converters,” Opt. Express **20**, 23985–23993 (2012). [CrossRef] [PubMed]

4. D. A. B. Miller, “How complicated must an optical component be?,” Submitted to J. Opt. Soc. Am. A. http://arxiv.org/abs/1209.5499.

5. V. P. Tzolov and M. Fontaine, “A passive polarization converter free of longitudinally-periodic structure,” Opt. Commun. **127**, 7–13 (1996). [CrossRef]

13. S. Y. Tseng and M. C. Wu, “Adiabatic mode conversion in multimode waveguides using computer-generated planar holograms,” IEEE Photon. Technol. Lett. **22**, 1211–1213 (2010). [CrossRef]

14. P. Sanchis, J. Marti, J. Blasco, A. Martinez, and A. Garcia, “Mode matching technique for highly efficient coupling between dielectric waveguides and planar photonic crystal circuits,” Opt. Express **10**, 1391–1397 (2002). [CrossRef] [PubMed]

15. I. L. Gheorma, S. Haas, and A. F. J. Levi, “Aperiodic nanophotonic design,” J. Appl. Phys. **95**, 1420–1426 (2004). [CrossRef]

17. V. Liu, Y. Jiao, D. A. B. Miller, and S. Fan, “Design methodology for compact photonic-crystal-based wavelength division multiplexers,” Opt. Lett. **36**, 591–593 (2011). [CrossRef] [PubMed]

18. G. G. Denisov, G. I. Kalynova, and D. I. Sobolev, “Method for synthesis of waveguide mode converters,” Radiophys. Quantum El. **47**, 615–620 (2004). [CrossRef]

19. M. Qiu, “Effective index method for heterostructure-slab-waveguide-based two-dimensional photonic crystals,” Appl. Phys. Lett. **81**, 1163–1165 (2002). [CrossRef]

20. Y. Li and J.-M. Jin, “A vector dual-primal finite element tearing and interconnecting method for solving 3-D large-scale electromagnetic problems,” IEEE Trans. Antennas Propag. **54**, 3000–3009 (2006). [CrossRef]

## 2. Mode converter

### 2.1. Design considerations based on general properties of the scattering matrix

*ε*= 12) of radius 0.2

*a*in air (

*ε*= 1) on a square lattice of lattice constant

*a*(Fig. 1). The fields are polarized with the electric field out-of-plane. The input and output waveguides (blue regions in Fig. 1) are formed by removing two lines of rods. These waveguides support two modes: a fundamental even-symmetric mode and a higher order odd-symmetric mode. Our aim is to design the coupler region (orange region in Fig. 1) between the waveguides that enables modal conversion.

### 2.2. Numerical design and optimization

*p*is the conversion efficiency at frequency

_{ω}*ω*. For the examples shown in the remainder of this paper, we evaluate

*J*by summing the conversion efficiency at a center frequency of 0.4025 × 2

*πc/a*and two neighboring frequencies 0.3% above and below.

*a*). Any structures encountered in the combinatorial search that show promising conversion behavior (

*J*≲ 0.5) are considered initial candidates for further optimization.

*J*in Eq. (1). The gradient of

*J*with respect to rod radius is straightforward to calculate by applying the adjoint variable method throughout the computational process [21

21. G. Veronis, R. W. Dutton, and S. Fan, “Method for sensitivity analysis of photonic crystal devices,” Opt. Lett. **29**, 2288–2290 (2004). [CrossRef] [PubMed]

22. Y. Jiao, S. Fan, and D. A. B. Miller, “Systematic photonic crystal device design: global and local optimization and sensitivity analysis,” IEEE J. Quantum Elect. **42**, 266–279 (2006). [CrossRef]

^{20}possible rod configurations, and a systematic enumeration of the entire space of about one million combinations is performed using an efficient simulation method tailored to the particular class of structures being analyzed [23]. The Dirichlet-to-Neumann (DtN) map is used to model each unique type of unit cell [24

24. Y. Huang and Y. Y. Lu, “Scattering from periodic arrays of cylinders by dirichlet-to-neumann maps,” J. Lightwave Technol. **24**, 3448–3453 (2006). [CrossRef]

25. Z. Hu and Y. Y. Lu, “Efficient analysis of photonic crystal devices by dirichlet-to-neumann maps,” Opt. Express **16**, 17383–17399 (2008). [CrossRef] [PubMed]

*Ax*=

*b*where

*x*represents the fields on the boundaries of the unit cells,

*b*represents the incident wave source, and

*A*is the system matrix formed from the DtN maps.

^{2}. On a local cluster with 80 cores, the entire combinatorial search of 2

^{20}structures at a single frequency can be completed in 10 minutes, while the subsequent gradient optimization usually takes no longer than 5 minutes on a personal computer.

### 2.3. Results

*r*= 0.2

*a*), and further optimization of rod radius did not produce substantial improvement in device performance. As shown in Fig. 3(a), the mode converter exhibits greater than 99% peak conversion efficiency from mode A to mode D with a relative frequency width of 0.176% at the 90% threshold. For a center wavelength at 1.55

*μ*m, this corresponds to a bandwidth of 2.73 nm. Note that the spectrum for the other three conversion processes (B to C, C to B, D to A) must all have the same spectrum, as required by inversion symmetry and reciprocity. Therefore when considering the scattering matrix of this device, one can immediately see that it performs as a nearly perfect mode converter. Quantitatively, the reflection (from a mode back into the same mode) and crosstalk (conversion efficiency to an unwanted mode) parameters at the operating frequency of

*ω*= 0.4025 × 2

*πc/a*are all lower than −45 dB, demonstrating excellent suppression of undesired behavior. Representative field patterns for the device for the A to D conversion and the B to C conversion are shown in Fig. 3(b). The entire coupler region occupies an area of 4 × 10 unit cells, which is about four wavelengths long and two wavelengths wide. To the best of our knowledge, this is the most compact design of a dielectric mode converter at the present time.

## 3. Optical diode

26. M. Greenberg and M. Orenstein, “Irreversible coupling by use of dissipative optics,” Opt. Lett. **29**, 451–453 (2004). [CrossRef] [PubMed]

30. C. Lu, X. Hu, H. Yang, and Q. Gong, “Ultrahigh-contrast and wideband nanoscale photonic crystal all-optical diode,” Opt. Lett. **36**, 4668–4670 (2011). [CrossRef] [PubMed]

31. S. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, Comment on “nonreciprocal light propagation in a silicon photonic circuit,” Science **335**, 38 (2012). [CrossRef] [PubMed]

*ω*= 0.4025× 2

*πc/a*exceed 97%, while the reflection and crosstalk parameters (the remaining mode conversions) are below −35 dB.

## 4. Design complexity

4. D. A. B. Miller, “How complicated must an optical component be?,” Submitted to J. Opt. Soc. Am. A. http://arxiv.org/abs/1209.5499.

4. D. A. B. Miller, “How complicated must an optical component be?,” Submitted to J. Opt. Soc. Am. A. http://arxiv.org/abs/1209.5499.

*M*= 4) and 4 output modes (

_{I}*M*= 4). The most general linear device relating input to output modes could therefore be represented by a 4 × 4 device matrix, which in this case is the same matrix as the most general 4×4 scattering matrix. That most general device matrix would require therefore 16 complex numbers, corresponding to 32 real numbers (a “complexity number”

_{O}*N*= 32 [4

_{D}*M*×

*M*unitary matrix, to specify the first column of the matrix requires 2

*M*− 1 real numbers (the “−1” arises because the sum of the modulus squared elements must equal 1, thus reducing the real degrees of freedom by 1 in the choice of these numbers). In general, for the second column of a unitary matrix, it must be orthogonal to the first column, which means both the real and imaginary parts of the inner product of these two columns must be zero, thereby reducing the number of degrees of freedom by 2, so that second column can only have 2

*M*− 3 real degrees of freedom to specify it. Additionally, because the matrix here is to be symmetric, the top element of this column must be the same as the second row element of the first column, reducing the degrees of freedom by a further 2, leaving 2

*M*− 5. For each successive column, if required, we reduce the degrees of freedom by a further 4. We only need to continue until we have specified half the columns (i.e.,

*M*/2); the other columns are then all known as a result of the matrix symmetry and the orthogonality of the columns. The total number of degrees of freedom (real numbers) to specify this unitary symmetric matrix is therefore Since we do not care about the phase of the outputs relative to the inputs, we have one less real constraint for each of the

*M*/2 columns we can specify, so the final complexity number for an arbitrary unitary symmetric operator with floating output phases is For our present devices, with

*M*= 4, we therefore have

*N*= 8. That is, with 8 real degrees of freedom in our design, we would have enough variables in principle to design any such device with 4 input and output modes, including, therefore, the specific ones we have chosen to design. Note in our designs we have used 20 binary numbers in the simplest designs (namely, the presence or absence of each of 20 rods). How we should relate the number of real numbers,

_{D}*N*, required to the number of binary numbers used is not quantitatively clear, but we note we have used only a moderately larger number of binary variables here than

_{D}*N*. This observation is similar to the behavior of the 3-mode-to-3-mode converter of [16

_{D}16. Y. Jiao, S. Fan, and D. A. B. Miller, “Demonstration of systematic photonic crystal device design and optimization by low-rank adjustments: an extremely compact mode separator,” Opt. Lett. **30**, 141–143 (2005). [CrossRef] [PubMed]

## 5. Conclusion

## Acknowledgments

## References and links

1. | J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photon. |

2. | L. H. Gabrielli, D. Liu, S. G. Johnson, and M. Lipson, “On-chip transformation optics for multimode waveguide bends,” Nat. Commun. |

3. | D. A. B. Miller, “All linear optical devices are mode converters,” Opt. Express |

4. | D. A. B. Miller, “How complicated must an optical component be?,” Submitted to J. Opt. Soc. Am. A. http://arxiv.org/abs/1209.5499. |

5. | V. P. Tzolov and M. Fontaine, “A passive polarization converter free of longitudinally-periodic structure,” Opt. Commun. |

6. | T. D. Happ, M. Kamp, and A. Forchel, “Photonic crystal tapers for ultracompact mode conversion,” Opt. Lett. |

7. | A. Talneau, P. Lalanne, M. Agio, and C. M. Soukoulis, “Low-reflection photonic-crystal taper for efficient coupling between guide sections of arbitrary widths,” Opt. Lett. |

8. | P. Sanchis, J. Marti, A. Garcia, A. Martinez, and J. Blasco, “High efficiency coupling technique for planar photonic crystal waveguides,” Electron. Lett. |

9. | P. Lalanne and A. Talneau, “Modal conversion with artificial materials for photonic-crystal waveguides,” Opt. Express |

10. | P. Bienstman, S. Assefa, S. G. Johnson, J. D. Joannopoulos, G. S. Petrich, and L. A. Kolodziejski, “Taper structures for coupling into photonic crystal slab waveguides,” J. Opt. Soc. Am. B |

11. | E. Khoo, A. Liu, and J. Wu, “Nonuniform photonic crystal taper for high-efficiency mode coupling,” Opt. Express |

12. | J. Castro, D. F. Geraghty, S. Honkanen, C. M. Greiner, D. Iazikov, and T. W. Mossberg, “Demonstration of mode conversion using anti-symmetric waveguide bragg gratings,” Opt. Express |

13. | S. Y. Tseng and M. C. Wu, “Adiabatic mode conversion in multimode waveguides using computer-generated planar holograms,” IEEE Photon. Technol. Lett. |

14. | P. Sanchis, J. Marti, J. Blasco, A. Martinez, and A. Garcia, “Mode matching technique for highly efficient coupling between dielectric waveguides and planar photonic crystal circuits,” Opt. Express |

15. | I. L. Gheorma, S. Haas, and A. F. J. Levi, “Aperiodic nanophotonic design,” J. Appl. Phys. |

16. | Y. Jiao, S. Fan, and D. A. B. Miller, “Demonstration of systematic photonic crystal device design and optimization by low-rank adjustments: an extremely compact mode separator,” Opt. Lett. |

17. | V. Liu, Y. Jiao, D. A. B. Miller, and S. Fan, “Design methodology for compact photonic-crystal-based wavelength division multiplexers,” Opt. Lett. |

18. | G. G. Denisov, G. I. Kalynova, and D. I. Sobolev, “Method for synthesis of waveguide mode converters,” Radiophys. Quantum El. |

19. | M. Qiu, “Effective index method for heterostructure-slab-waveguide-based two-dimensional photonic crystals,” Appl. Phys. Lett. |

20. | Y. Li and J.-M. Jin, “A vector dual-primal finite element tearing and interconnecting method for solving 3-D large-scale electromagnetic problems,” IEEE Trans. Antennas Propag. |

21. | G. Veronis, R. W. Dutton, and S. Fan, “Method for sensitivity analysis of photonic crystal devices,” Opt. Lett. |

22. | Y. Jiao, S. Fan, and D. A. B. Miller, “Systematic photonic crystal device design: global and local optimization and sensitivity analysis,” IEEE J. Quantum Elect. |

23. | V. Liu, D. A. B. Miller, and S. Fan, “Highly tailored computational electromagnetics methods for nanophotonic design and discovery,” Proc. IEEE (Accepted). |

24. | Y. Huang and Y. Y. Lu, “Scattering from periodic arrays of cylinders by dirichlet-to-neumann maps,” J. Lightwave Technol. |

25. | Z. Hu and Y. Y. Lu, “Efficient analysis of photonic crystal devices by dirichlet-to-neumann maps,” Opt. Express |

26. | M. Greenberg and M. Orenstein, “Irreversible coupling by use of dissipative optics,” Opt. Lett. |

27. | L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science |

28. | A. E. Serebryannikov, “One-way diffraction effects in photonic crystal gratings made of isotropic materials,” Phys. Rev. B |

29. | C. Wang, C.-Z. Zhou, and Z.-Y. Li, “On-chip optical diode based on silicon photonic crystal heterojunctions,” Opt. Express |

30. | C. Lu, X. Hu, H. Yang, and Q. Gong, “Ultrahigh-contrast and wideband nanoscale photonic crystal all-optical diode,” Opt. Lett. |

31. | S. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, Comment on “nonreciprocal light propagation in a silicon photonic circuit,” Science |

**OCIS Codes**

(130.2790) Integrated optics : Guided waves

(130.3120) Integrated optics : Integrated optics devices

(230.3120) Optical devices : Integrated optics devices

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: October 22, 2012

Revised Manuscript: November 19, 2012

Manuscript Accepted: November 23, 2012

Published: December 6, 2012

**Citation**

Victor Liu, David A. B. Miller, and Shanhui Fan, "Ultra-compact photonic crystal waveguide spatial mode converter and its connection to the optical diode effect," Opt. Express **20**, 28388-28397 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-27-28388

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### References

- J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photon.6, 488–496 (2012). [CrossRef]
- L. H. Gabrielli, D. Liu, S. G. Johnson, and M. Lipson, “On-chip transformation optics for multimode waveguide bends,” Nat. Commun.3, 1217 (2012). [CrossRef] [PubMed]
- D. A. B. Miller, “All linear optical devices are mode converters,” Opt. Express20, 23985–23993 (2012). [CrossRef] [PubMed]
- D. A. B. Miller, “How complicated must an optical component be?,” Submitted to J. Opt. Soc. Am. A. http://arxiv.org/abs/1209.5499 .
- V. P. Tzolov and M. Fontaine, “A passive polarization converter free of longitudinally-periodic structure,” Opt. Commun.127, 7–13 (1996). [CrossRef]
- T. D. Happ, M. Kamp, and A. Forchel, “Photonic crystal tapers for ultracompact mode conversion,” Opt. Lett.26, 1102–1104 (2001). [CrossRef]
- A. Talneau, P. Lalanne, M. Agio, and C. M. Soukoulis, “Low-reflection photonic-crystal taper for efficient coupling between guide sections of arbitrary widths,” Opt. Lett.27, 1522–1524 (2002). [CrossRef]
- P. Sanchis, J. Marti, A. Garcia, A. Martinez, and J. Blasco, “High efficiency coupling technique for planar photonic crystal waveguides,” Electron. Lett.38, 961–962 (2002). [CrossRef]
- P. Lalanne and A. Talneau, “Modal conversion with artificial materials for photonic-crystal waveguides,” Opt. Express10, 354–359 (2002). [CrossRef] [PubMed]
- P. Bienstman, S. Assefa, S. G. Johnson, J. D. Joannopoulos, G. S. Petrich, and L. A. Kolodziejski, “Taper structures for coupling into photonic crystal slab waveguides,” J. Opt. Soc. Am. B20, 1817–1821 (2003). [CrossRef]
- E. Khoo, A. Liu, and J. Wu, “Nonuniform photonic crystal taper for high-efficiency mode coupling,” Opt. Express13, 7748–7759 (2005). [CrossRef] [PubMed]
- J. Castro, D. F. Geraghty, S. Honkanen, C. M. Greiner, D. Iazikov, and T. W. Mossberg, “Demonstration of mode conversion using anti-symmetric waveguide bragg gratings,” Opt. Express13, 4180–4184 (2005). [CrossRef] [PubMed]
- S. Y. Tseng and M. C. Wu, “Adiabatic mode conversion in multimode waveguides using computer-generated planar holograms,” IEEE Photon. Technol. Lett.22, 1211–1213 (2010). [CrossRef]
- P. Sanchis, J. Marti, J. Blasco, A. Martinez, and A. Garcia, “Mode matching technique for highly efficient coupling between dielectric waveguides and planar photonic crystal circuits,” Opt. Express10, 1391–1397 (2002). [CrossRef] [PubMed]
- I. L. Gheorma, S. Haas, and A. F. J. Levi, “Aperiodic nanophotonic design,” J. Appl. Phys.95, 1420–1426 (2004). [CrossRef]
- Y. Jiao, S. Fan, and D. A. B. Miller, “Demonstration of systematic photonic crystal device design and optimization by low-rank adjustments: an extremely compact mode separator,” Opt. Lett.30, 141–143 (2005). [CrossRef] [PubMed]
- V. Liu, Y. Jiao, D. A. B. Miller, and S. Fan, “Design methodology for compact photonic-crystal-based wavelength division multiplexers,” Opt. Lett.36, 591–593 (2011). [CrossRef] [PubMed]
- G. G. Denisov, G. I. Kalynova, and D. I. Sobolev, “Method for synthesis of waveguide mode converters,” Radiophys. Quantum El.47, 615–620 (2004). [CrossRef]
- M. Qiu, “Effective index method for heterostructure-slab-waveguide-based two-dimensional photonic crystals,” Appl. Phys. Lett.81, 1163–1165 (2002). [CrossRef]
- Y. Li and J.-M. Jin, “A vector dual-primal finite element tearing and interconnecting method for solving 3-D large-scale electromagnetic problems,” IEEE Trans. Antennas Propag.54, 3000–3009 (2006). [CrossRef]
- G. Veronis, R. W. Dutton, and S. Fan, “Method for sensitivity analysis of photonic crystal devices,” Opt. Lett.29, 2288–2290 (2004). [CrossRef] [PubMed]
- Y. Jiao, S. Fan, and D. A. B. Miller, “Systematic photonic crystal device design: global and local optimization and sensitivity analysis,” IEEE J. Quantum Elect.42, 266–279 (2006). [CrossRef]
- V. Liu, D. A. B. Miller, and S. Fan, “Highly tailored computational electromagnetics methods for nanophotonic design and discovery,” Proc. IEEE (Accepted).
- Y. Huang and Y. Y. Lu, “Scattering from periodic arrays of cylinders by dirichlet-to-neumann maps,” J. Lightwave Technol.24, 3448–3453 (2006). [CrossRef]
- Z. Hu and Y. Y. Lu, “Efficient analysis of photonic crystal devices by dirichlet-to-neumann maps,” Opt. Express16, 17383–17399 (2008). [CrossRef] [PubMed]
- M. Greenberg and M. Orenstein, “Irreversible coupling by use of dissipative optics,” Opt. Lett.29, 451–453 (2004). [CrossRef] [PubMed]
- L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science333, 729–733 (2011). [CrossRef] [PubMed]
- A. E. Serebryannikov, “One-way diffraction effects in photonic crystal gratings made of isotropic materials,” Phys. Rev. B80, 155117 (2009). [CrossRef]
- C. Wang, C.-Z. Zhou, and Z.-Y. Li, “On-chip optical diode based on silicon photonic crystal heterojunctions,” Opt. Express19, 26948–26955 (2011). [CrossRef]
- C. Lu, X. Hu, H. Yang, and Q. Gong, “Ultrahigh-contrast and wideband nanoscale photonic crystal all-optical diode,” Opt. Lett.36, 4668–4670 (2011). [CrossRef] [PubMed]
- S. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, Comment on “nonreciprocal light propagation in a silicon photonic circuit,” Science335, 38 (2012). [CrossRef] [PubMed]

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