Abstract

In this paper, we present extremely compact designs of both broadband mode converter and optical diode in linear rod-type photonic crystal (PhC) waveguide with functional region consisting of only 4 × 1 unit cells of perfect PhC. The dielectric distribution inside functional region are optimized by combining geometry projection method and method of moving asymptotes. Bidirectional mode converter realizes above 60% transmission efficiency within bandwidth 0.02c/a, where c and a represent light velocity and PhC lattice constant respectively. Optical diode achieves above 19 dB unidirectionality for even mode within bandwidth 0.01c/a. Moreover, the proposed designs have reasonable tolerance of rod boundary fluctuation. We expect the results will help developing recipes for future PhC devices in all-optical integrated circuits.

© 2015 Optical Society of America

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References

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2014 (2)

2013 (3)

D. Jalas, A. Petrov, M. Eich, W. Freude, S. H. Fan, Z. F. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is - and what is not - an optical isolator,” Nat. Photonics 7(8), 579–582 (2013).
[Crossref]

A. Khavasi, M. Rezaei, A. P. Fard, and K. Mehrany, “A heuristic approach to the realization of the wide-band optical diode effect in photonic crystal waveguides,” J. Opt. 15(7), 075501 (2013).
[Crossref]

S. Feng and Y. Wang, “Unidirectional reciprocal wavelength filters based on the square-lattice photonic crystal structures with the rectangular defects,” Opt. Express 21(1), 220–228 (2013).
[Crossref] [PubMed]

2012 (5)

C. Wang, X. L. Zhong, and Z. Y. Li, “Linear and passive silicon optical isolator,” Sci. Rep. 2, 674 (2012).
[PubMed]

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]

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. Express 20(21), 23985–23993 (2012).
[Crossref] [PubMed]

L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. Qi, “An all-silicon passive optical diode,” Science 335(6067), 447–450 (2012).
[Crossref] [PubMed]

2011 (2)

D. Wang, Z. Yu, Y. Liu, P. Lu, L. Han, H. Feng, X. Guo, and H. Ye, “The optimal structure of two dimensional photonic crystals with the large absolute band gap,” Opt. Express 19(20), 19346–19353 (2011).
[Crossref] [PubMed]

C. C. Lu, X. Y. Hu, Y. B. Zhang, Z. Q. Li, X. A. Xu, H. Yang, and Q. H. Gong, “Ultralow power all-optical diode in photonic crystal heterostructures with broken spatial inversion symmetry,” Appl. Phys. Lett. 99(5), 051107 (2011).
[Crossref]

2009 (1)

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljacić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461(7265), 772–775 (2009).
[Crossref] [PubMed]

2008 (2)

F. D. M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100(1), 013904 (2008).
[Crossref] [PubMed]

W. R. Frei, H. T. Johnson, and K. D. Choquette, “Optimization of a single defect photonic crystal laser cavity,” J. Appl. Phys. 103(3), 033102 (2008).
[Crossref]

2005 (1)

1987 (1)

K. Svanberg, “The method of moving asymptotes-a new method for structural optimization,” Int. J. Numer. Methods Eng. 24(2), 359–373 (1987).
[Crossref]

Baets, R.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. H. Fan, Z. F. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is - and what is not - an optical isolator,” Nat. Photonics 7(8), 579–582 (2013).
[Crossref]

Bulgakov, E. N.

Chen, G.

Chong, Y.

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljacić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461(7265), 772–775 (2009).
[Crossref] [PubMed]

Choquette, K. D.

W. R. Frei, H. T. Johnson, and K. D. Choquette, “Optimization of a single defect photonic crystal laser cavity,” J. Appl. Phys. 103(3), 033102 (2008).
[Crossref]

Ding, Y.

Doerr, C. R.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. H. Fan, Z. F. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is - and what is not - an optical isolator,” Nat. Photonics 7(8), 579–582 (2013).
[Crossref]

Eich, M.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. H. Fan, Z. F. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is - and what is not - an optical isolator,” Nat. Photonics 7(8), 579–582 (2013).
[Crossref]

Elesin, Y.

Fan, L.

L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. Qi, “An all-silicon passive optical diode,” Science 335(6067), 447–450 (2012).
[Crossref] [PubMed]

Fan, S.

Fan, S. H.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. H. Fan, Z. F. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is - and what is not - an optical isolator,” Nat. Photonics 7(8), 579–582 (2013).
[Crossref]

Fard, A. P.

A. Khavasi, M. Rezaei, A. P. Fard, and K. Mehrany, “A heuristic approach to the realization of the wide-band optical diode effect in photonic crystal waveguides,” J. Opt. 15(7), 075501 (2013).
[Crossref]

Feng, H.

Feng, S.

Frandsen, L. H.

Frei, W. R.

W. R. Frei, H. T. Johnson, and K. D. Choquette, “Optimization of a single defect photonic crystal laser cavity,” J. Appl. Phys. 103(3), 033102 (2008).
[Crossref]

Frellsen, L. F.

Freude, W.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. H. Fan, Z. F. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is - and what is not - an optical isolator,” Nat. Photonics 7(8), 579–582 (2013).
[Crossref]

Gabrielli, L. H.

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]

Gong, Q. H.

C. C. Lu, X. Y. Hu, Y. B. Zhang, Z. Q. Li, X. A. Xu, H. Yang, and Q. H. Gong, “Ultralow power all-optical diode in photonic crystal heterostructures with broken spatial inversion symmetry,” Appl. Phys. Lett. 99(5), 051107 (2011).
[Crossref]

Guo, X.

Haldane, F. D. M.

F. D. M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100(1), 013904 (2008).
[Crossref] [PubMed]

Han, L.

Hu, X. Y.

C. C. Lu, X. Y. Hu, Y. B. Zhang, Z. Q. Li, X. A. Xu, H. Yang, and Q. H. Gong, “Ultralow power all-optical diode in photonic crystal heterostructures with broken spatial inversion symmetry,” Appl. Phys. Lett. 99(5), 051107 (2011).
[Crossref]

Jalas, D.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. H. Fan, Z. F. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is - and what is not - an optical isolator,” Nat. Photonics 7(8), 579–582 (2013).
[Crossref]

Joannopoulos, J. D.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. H. Fan, Z. F. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is - and what is not - an optical isolator,” Nat. Photonics 7(8), 579–582 (2013).
[Crossref]

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljacić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461(7265), 772–775 (2009).
[Crossref] [PubMed]

Johnson, H. T.

W. R. Frei, H. T. Johnson, and K. D. Choquette, “Optimization of a single defect photonic crystal laser cavity,” J. Appl. Phys. 103(3), 033102 (2008).
[Crossref]

Johnson, S. G.

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]

Kang, J. U.

Khavasi, A.

A. Khavasi, M. Rezaei, A. P. Fard, and K. Mehrany, “A heuristic approach to the realization of the wide-band optical diode effect in photonic crystal waveguides,” J. Opt. 15(7), 075501 (2013).
[Crossref]

Li, Z. Q.

C. C. Lu, X. Y. Hu, Y. B. Zhang, Z. Q. Li, X. A. Xu, H. Yang, and Q. H. Gong, “Ultralow power all-optical diode in photonic crystal heterostructures with broken spatial inversion symmetry,” Appl. Phys. Lett. 99(5), 051107 (2011).
[Crossref]

Li, Z. Y.

C. Wang, X. L. Zhong, and Z. Y. Li, “Linear and passive silicon optical isolator,” Sci. Rep. 2, 674 (2012).
[PubMed]

Lipson, M.

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]

Liu, D.

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]

Liu, V.

Liu, Y.

Lu, C. C.

C. C. Lu, X. Y. Hu, Y. B. Zhang, Z. Q. Li, X. A. Xu, H. Yang, and Q. H. Gong, “Ultralow power all-optical diode in photonic crystal heterostructures with broken spatial inversion symmetry,” Appl. Phys. Lett. 99(5), 051107 (2011).
[Crossref]

Lu, P.

Mehrany, K.

A. Khavasi, M. Rezaei, A. P. Fard, and K. Mehrany, “A heuristic approach to the realization of the wide-band optical diode effect in photonic crystal waveguides,” J. Opt. 15(7), 075501 (2013).
[Crossref]

Melloni, A.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. H. Fan, Z. F. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is - and what is not - an optical isolator,” Nat. Photonics 7(8), 579–582 (2013).
[Crossref]

Miller, D. A. B.

Mitrovic, M.

Niu, B.

L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. Qi, “An all-silicon passive optical diode,” Science 335(6067), 447–450 (2012).
[Crossref] [PubMed]

Petrov, A.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. H. Fan, Z. F. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is - and what is not - an optical isolator,” Nat. Photonics 7(8), 579–582 (2013).
[Crossref]

Popovic, M.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. H. Fan, Z. F. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is - and what is not - an optical isolator,” Nat. Photonics 7(8), 579–582 (2013).
[Crossref]

Qi, M.

L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. Qi, “An all-silicon passive optical diode,” Science 335(6067), 447–450 (2012).
[Crossref] [PubMed]

Raghu, S.

F. D. M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100(1), 013904 (2008).
[Crossref] [PubMed]

Renner, H.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. H. Fan, Z. F. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is - and what is not - an optical isolator,” Nat. Photonics 7(8), 579–582 (2013).
[Crossref]

Rezaei, M.

A. Khavasi, M. Rezaei, A. P. Fard, and K. Mehrany, “A heuristic approach to the realization of the wide-band optical diode effect in photonic crystal waveguides,” J. Opt. 15(7), 075501 (2013).
[Crossref]

Sadreev, A. F.

Shen, H.

L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. Qi, “An all-silicon passive optical diode,” Science 335(6067), 447–450 (2012).
[Crossref] [PubMed]

Sigmund, O.

Soljacic, M.

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljacić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461(7265), 772–775 (2009).
[Crossref] [PubMed]

Svanberg, K.

K. Svanberg, “The method of moving asymptotes-a new method for structural optimization,” Int. J. Numer. Methods Eng. 24(2), 359–373 (1987).
[Crossref]

Vanwolleghem, M.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. H. Fan, Z. F. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is - and what is not - an optical isolator,” Nat. Photonics 7(8), 579–582 (2013).
[Crossref]

Varghese, L. T.

L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. Qi, “An all-silicon passive optical diode,” Science 335(6067), 447–450 (2012).
[Crossref] [PubMed]

Wang, C.

C. Wang, X. L. Zhong, and Z. Y. Li, “Linear and passive silicon optical isolator,” Sci. Rep. 2, 674 (2012).
[PubMed]

Wang, D.

Wang, J.

L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. Qi, “An all-silicon passive optical diode,” Science 335(6067), 447–450 (2012).
[Crossref] [PubMed]

Wang, Y.

Wang, Z.

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljacić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461(7265), 772–775 (2009).
[Crossref] [PubMed]

Weiner, A. M.

L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. Qi, “An all-silicon passive optical diode,” Science 335(6067), 447–450 (2012).
[Crossref] [PubMed]

Xu, X. A.

C. C. Lu, X. Y. Hu, Y. B. Zhang, Z. Q. Li, X. A. Xu, H. Yang, and Q. H. Gong, “Ultralow power all-optical diode in photonic crystal heterostructures with broken spatial inversion symmetry,” Appl. Phys. Lett. 99(5), 051107 (2011).
[Crossref]

Xuan, Y.

L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. Qi, “An all-silicon passive optical diode,” Science 335(6067), 447–450 (2012).
[Crossref] [PubMed]

Yang, H.

C. C. Lu, X. Y. Hu, Y. B. Zhang, Z. Q. Li, X. A. Xu, H. Yang, and Q. H. Gong, “Ultralow power all-optical diode in photonic crystal heterostructures with broken spatial inversion symmetry,” Appl. Phys. Lett. 99(5), 051107 (2011).
[Crossref]

Ye, H.

Yu, Z.

Yu, Z. F.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. H. Fan, Z. F. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is - and what is not - an optical isolator,” Nat. Photonics 7(8), 579–582 (2013).
[Crossref]

Yvind, K.

Zhang, Y. B.

C. C. Lu, X. Y. Hu, Y. B. Zhang, Z. Q. Li, X. A. Xu, H. Yang, and Q. H. Gong, “Ultralow power all-optical diode in photonic crystal heterostructures with broken spatial inversion symmetry,” Appl. Phys. Lett. 99(5), 051107 (2011).
[Crossref]

Zhong, X. L.

C. Wang, X. L. Zhong, and Z. Y. Li, “Linear and passive silicon optical isolator,” Sci. Rep. 2, 674 (2012).
[PubMed]

Appl. Phys. Lett. (1)

C. C. Lu, X. Y. Hu, Y. B. Zhang, Z. Q. Li, X. A. Xu, H. Yang, and Q. H. Gong, “Ultralow power all-optical diode in photonic crystal heterostructures with broken spatial inversion symmetry,” Appl. Phys. Lett. 99(5), 051107 (2011).
[Crossref]

Int. J. Numer. Methods Eng. (1)

K. Svanberg, “The method of moving asymptotes-a new method for structural optimization,” Int. J. Numer. Methods Eng. 24(2), 359–373 (1987).
[Crossref]

J. Appl. Phys. (1)

W. R. Frei, H. T. Johnson, and K. D. Choquette, “Optimization of a single defect photonic crystal laser cavity,” J. Appl. Phys. 103(3), 033102 (2008).
[Crossref]

J. Opt. (1)

A. Khavasi, M. Rezaei, A. P. Fard, and K. Mehrany, “A heuristic approach to the realization of the wide-band optical diode effect in photonic crystal waveguides,” J. Opt. 15(7), 075501 (2013).
[Crossref]

Nat. Commun. (1)

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]

Nat. Photonics (1)

D. Jalas, A. Petrov, M. Eich, W. Freude, S. H. Fan, Z. F. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is - and what is not - an optical isolator,” Nat. Photonics 7(8), 579–582 (2013).
[Crossref]

Nature (1)

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljacić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461(7265), 772–775 (2009).
[Crossref] [PubMed]

Opt. Express (5)

Opt. Lett. (2)

Phys. Rev. Lett. (1)

F. D. M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100(1), 013904 (2008).
[Crossref] [PubMed]

Sci. Rep. (1)

C. Wang, X. L. Zhong, and Z. Y. Li, “Linear and passive silicon optical isolator,” Sci. Rep. 2, 674 (2012).
[PubMed]

Science (1)

L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. Qi, “An all-silicon passive optical diode,” Science 335(6067), 447–450 (2012).
[Crossref] [PubMed]

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Figures (4)

Fig. 1
Fig. 1 (a) Schematic diagram of proposed two-dimensional PhC mode converter and optical diode . PhC is based on silicon rods (ɛ = 12) in air. (b) Dispersion curves of even mode and odd mode supported by W2 waveguide. (c) Schematic diagram of GPM: The rod shape is determined by intersection line of level plane and three dimensional surface formed by control points. FEM + MMA are adopted to adjust heights of control points to find the optimum solution.
Fig. 2
Fig. 2 (a) Optimized shape of rods inside functional region for broadband mode converter. (b) Spectrum of transmission efficiency and corresponding electric field profiles.
Fig. 3
Fig. 3 (a) Optimized shape of rods inside functional region for broadband optical diode. (b) Spectrum of unidirectionality and corresponding electric field profiles.
Fig. 4
Fig. 4 (a) Spectra of transmission efficiency of mode converters with perfect and 5% imperfect shape. (b) Spectra of unidirectionality of optical diode with perfect and 3% imperfect shapes.

Tables (2)

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Table 1 Deviations between optimized rods for mode converter and reference circular rods

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Table 2 Deviations between optimized rods for optical diode with reference circular rods

Equations (2)

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f obj (ω)= P1 P(ω)ds+ P3 P(ω)ds P2 P(ω)ds
F obj = 1 5 i=1 5 w i f obj ( ω i ) = 1 5 i=1 5 w i [ P1 P( ω i )ds+ P3 P( ω i )ds i=2,4,5,6 P i P( ω i )ds ]

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