Abstract

Second harmonic generation from the two-layer structure where a transition-metal dichalcogenide monolayer is put on a one-dimensional grating has been studied. This grating supports bound states in the continuum which have no leakage lying within the continuum of radiation modes, we can enhance the second harmonic generation from the transition-metal dichalcogenide monolayer by more than four orders of magnitude based on the critical field enhancement near the bound states in the continuum. In order to complete this calculation, the scattering matrix theory has been extended to include the nonlinear effect and the scattering matrix of a two-dimensional material including nonlinear terms; furthermore, two methods to observe the bound states in the continuum are considered, where one is tuning the thickness of the grating and the other is changing the incident angle of the electromagnetic wave. We have also discussed various modulation of the second harmonic generation enhancement by adjusting the azimuthal angle of the transition-metal dichalcogenide monolayer.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2017 (4)

G. T. Adamashvili and D. J. Kaup, “Optical surface breather in graphene,” Phys. Rev. A 95(5), 053801 (2017).
[Crossref]

H. Chen, V. Corboliou, A. S. Solntsev, D.-Y. Choi, M. A. Vincenti, D. de Ceglia, C. de Angelis, Y. Lu, and D. N. Neshev, “Enhanced second-harmonic generation from two-dimensional MoSe2 on a silicon waveguide,” Light Sci. Appl. 6(10), e17060 (2017).
[Crossref]

T. Wang and X. Zhang, “Improved third-order nonlinear effect in graphene based on bound states in the continuum,” Photon. Res. 5(6), 629–639 (2017).
[Crossref]

A. Aryshev, A. Potylitsyn, G. Naumenko, M. Shevelev, K. Lekomtsev, L. Sukhikh, P. Karataev, Y. Honda, N. Terunuma, and J. Urakawa, “Monochromaticity of coherent Smith-Purcell radiation from finite size grating,” Phys. Rev. Accel. Beams 20(2), 024701 (2017).
[Crossref]

2016 (4)

C. Blanchard, J. P. Hugonin, and C. Sauvan, “Fano resonances in photonic crystal slabs near bound states in the continuum,” Phys. Rev. B 94(15), 155303 (2016).
[Crossref]

M. Weismann and N. C. Panoiu, “Theoretical and computational analysis of second- and third- harmonic generation in periodically patterned graphene and transition-metal dichalcogenide monolayers,” Phys. Rev. B 94(3), 035435 (2016).
[Crossref]

S. Zhang and X. Zhang, “Strong second-harmonic generation from bilayer-graphene embedded in one-dimensional photonic crystals,” J. Opt. Soc. Am. B 33(3), 452–460 (2016).
[Crossref]

J. Niu, M. Luo, and Q. H. Liu, “Enhancement of graphene’s third-harmonic generation with localized surface plasmon resonance under optical/electro-optic Kerr effects,” J. Opt. Soc. Am. B 33(4), 615–621 (2016).
[Crossref]

2015 (3)

K. L. Seyler, J. R. Schaibley, P. Gong, P. Rivera, A. M. Jones, S. Wu, J. Yan, D. G. Mandrus, W. Yao, and X. Xu, “Electrical control of second-harmonic generation in a monolayer WSe2 transistor,” Nat. Nanotechnol. 10(5), 407–411 (2015).
[Crossref] [PubMed]

J. W. Yoon, S. H. Song, and R. Magnusson, “Critical field enhancement of asymptotic optical bound states in the continuum,” Sci. Rep. 5(1), 18301–18308 (2015).
[Crossref] [PubMed]

M. Zhang and X. Zhang, “Ultrasensitive optical absorption in graphene based on bound states in the continuum,” Sci. Rep. 5(1), 8266–82671 (2015).
[Crossref] [PubMed]

2014 (6)

F. Monticone and A. Alù, “Embedded Photonic Eigenvalues in 3D Nanostructures,” Phys. Rev. Lett. 112(21), 213903 (2014).
[Crossref]

Y. Yang, C. Peng, Y. Liang, Z. Li, and S. Noda, “Analytical Perspective for Bound States in the Continuum in Photonic Crystal Slabs,” Phys. Rev. Lett. 113(3), 037401 (2014).
[Crossref] [PubMed]

B. Zhen, C. W. Hsu, L. Lu, A. D. Stone, and M. Soljačić, “Topological Nature of Optical Bound States in the Continuum,” Phys. Rev. Lett. 113(25), 257401 (2014).
[Crossref] [PubMed]

M. A. Vincenti, D. de Ceglia, M. Grande, A. D’Orazio, and M. Scalora, “Third-harmonic generation in one-dimensional photonic crystal with graphene-based defect,” Phys. Rev. B 89(16), 165139 (2014).
[Crossref]

C. Janisch, Y. Wang, D. Ma, N. Mehta, A. L. Elías, N. Perea-López, M. Terrones, V. Crespi, and Z. Liu, “Extraordinary Second Harmonic Generation in Tungsten Disulfide Monolayers,” Sci. Rep. 4(1), 5530–5534 (2014).
[Crossref] [PubMed]

Y. Li, A. Chernikov, X. Zhang, A. Rigosi, H. M. Hill, A. M. van der Zande, D. A. Chenet, E.-M. Shih, J. Hone, and T. F. Heinz, “Measurement of the optical dielectric function of monolayer transition-metal dichalcogenides MoS2, MoSe2, WS2, and WSe2,” Phys. Rev. B 90(20), 205422 (2014).
[Crossref]

2013 (8)

L. M. Malard, T. V. Alencar, M. Ana Paula Barboza, K. F. Mak, and A. M. de Paula, “Observation of intense second harmonic generation from MoS2 atomic crystals,” Phys. Rev. B 87(20), 201401 (2013).
[Crossref]

Y. Li, Y. Rao, K. F. Mak, Y. You, S. Wang, C. R. Dean, and T. F. Heinz, “Probing Symmetry Properties of Few-Layer MoS2 and h-BN by Optical Second-Harmonic Generation,” Nano Lett. 13(7), 3329–3333 (2013).
[Crossref] [PubMed]

M. Xu, T. Liang, M. Shi, and H. Chen, “Graphene-Like Two-Dimensional Materials,” Chem. Rev. 113(5), 3766–3798 (2013).
[Crossref] [PubMed]

S. Z. Butler, S. M. Hollen, L. Cao, Y. Cui, J. A. Gupta, H. R. Gutiérrez, T. F. Heinz, S. S. Hong, J. Huang, A. F. Ismach, E. Johnston-Halperin, M. Kuno, V. V. Plashnitsa, R. D. Robinson, R. S. Ruoff, S. Salahuddin, J. Shan, L. Shi, M. G. Spencer, M. Terrones, W. Windl, and J. E. Goldberger, “Progress, Challenges, and Opportunities in Two-Dimensional Materials Beyond Graphene,” ACS Nano 7(4), 2898–2926 (2013).
[Crossref] [PubMed]

S.-Y. Hong, J. I. Dadap, N. Petrone, P.-C. Yeh, J. Hone, and R. M. Osgood., “Optical Third-Harmonic Generation in Graphene,” Phys. Rev. X 3(2), 021014 (2013).
[Crossref]

A. V. Gorbach, “Nonlinear graphene plasmonics: Amplitude equation for surface plamons,” Phys. Rev. A 87(1), 013830 (2013).
[Crossref]

C. W. Hsu, B. Zhen, J. Lee, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499(7457), 188–191 (2013).
[Crossref] [PubMed]

C. W. Hsu, B. Zhen, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Bloch surface eigenstates within the radiation continuum,” Light Sci. Appl. 2(7), e84–e89 (2013).
[Crossref]

2012 (3)

M. I. Molina, A. E. Miroshnichenko, and Y. S. Kivshar, “Surface bound states in the continuum,” Phys. Rev. Lett. 108(7), 070401 (2012).
[Crossref] [PubMed]

J. Lee, B. Zhen, S. L. Chua, W. Qiu, J. D. Joannopoulos, M. Soljačić, and O. Shapira, “Observation and Differentiation of Unique High-Q Optical Resonances Near Zero Wave Vector in Macroscopic Photonic Crystal Slabs,” Phys. Rev. Lett. 109(6), 067401 (2012).
[Crossref] [PubMed]

H. Zhang, S. Virally, Q. Bao, L. K. Ping, S. Massar, N. Godbout, and P. Kockaert, “Z-scan measurement of the nonlinear refractive index of graphene,” Opt. Lett. 37(11), 1856–1858 (2012).
[Crossref] [PubMed]

2011 (1)

Y. Plotnik, O. Peleg, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, and M. Segev, “Experimental Observation of Optical Bound States in the Continuum,” Phys. Rev. Lett. 107(18), 183901 (2011).
[Crossref] [PubMed]

2010 (2)

F. Bonaccorso, Z. Sun, T. Hasan, and A. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4(9), 611–622 (2010).
[Crossref]

R. F. Ndangali and S. V. Shabanov, “Electromagnetic bound states in the radiation continuum for periodic double arrays of subwavelength dielectric cylinders,” J. Math. Phys. 51(10), 102901 (2010).
[Crossref]

2008 (1)

E. N. Bulgakov and A. F. Sadreev, “Bound states in the continuum in photonic waveguides inspired by defects,” Phys. Rev. B 78(7), 075105 (2008).
[Crossref]

2007 (2)

Y. Zeng, Y. Fu, X. Chen, W. Lu, and H. Ågren, “Highly efficient generation of entangled photon pair by spontaneous parametric down conversion in defective photonic crystal,” J. Opt. Soc. Am. B 24(6), 1365–1368 (2007).
[Crossref]

M. Liscidini, A. Locatelli, L. C. Andreani, and C. De Angelis, “Maximum-Exponent Scaling Behavior of Optical Second-Harmonic Generation in Finite Multilayer Photonic Crystals,” Phys. Rev. Lett. 99(5), 053907 (2007).
[Crossref] [PubMed]

2003 (1)

Z. Y. Li and L. L. Lin, “Photonic band structures solved by a plane-wave-based transfer-matrix method,” Phys. Rev. E 67(4), 046607 (2003).
[Crossref] [PubMed]

2002 (1)

Y. Dumeige, I. Sagnes, P. Monnier, P. Vidakovic, I. Abram, C. Mériadec, and A. Levenson, “Phase-Matched Frequency Doubling at Photonic Band Edges: Efficiency Scaling as the Fifth Power of the Length,” Phys. Rev. Lett. 89(4), 043901 (2002).
[Crossref] [PubMed]

2001 (2)

C. De Angelis, F. Gringoli, M. Midrio, D. Modotto, J. S. Aitchison, and G. F. Nalesso, “Conversion efficiency for second-harmonic generation in photonic crystals,” J. Opt. Soc. Am. B 18(3), 348–351 (2001).
[Crossref]

A. H. Castro Neto, “Charge density wave, superconductivity, and anomalous metallic behavior in 2D transition metal dichalcogenides,” Phys. Rev. Lett. 86(19), 4382–4385 (2001).
[Crossref] [PubMed]

1998 (1)

N. Stefanou, V. Yannopapas, and A. Modinos, “Heterostructures of photonic crystals: frequency bands and transmission coefficients,” Comput. Phys. Commun. 13(113), 49–77 (1998).
[Crossref]

1992 (1)

1987 (1)

Abram, I.

Y. Dumeige, I. Sagnes, P. Monnier, P. Vidakovic, I. Abram, C. Mériadec, and A. Levenson, “Phase-Matched Frequency Doubling at Photonic Band Edges: Efficiency Scaling as the Fifth Power of the Length,” Phys. Rev. Lett. 89(4), 043901 (2002).
[Crossref] [PubMed]

Adamashvili, G. T.

G. T. Adamashvili and D. J. Kaup, “Optical surface breather in graphene,” Phys. Rev. A 95(5), 053801 (2017).
[Crossref]

Ågren, H.

Aitchison, J. S.

Alencar, T. V.

L. M. Malard, T. V. Alencar, M. Ana Paula Barboza, K. F. Mak, and A. M. de Paula, “Observation of intense second harmonic generation from MoS2 atomic crystals,” Phys. Rev. B 87(20), 201401 (2013).
[Crossref]

Alù, A.

F. Monticone and A. Alù, “Embedded Photonic Eigenvalues in 3D Nanostructures,” Phys. Rev. Lett. 112(21), 213903 (2014).
[Crossref]

Ana Paula Barboza, M.

L. M. Malard, T. V. Alencar, M. Ana Paula Barboza, K. F. Mak, and A. M. de Paula, “Observation of intense second harmonic generation from MoS2 atomic crystals,” Phys. Rev. B 87(20), 201401 (2013).
[Crossref]

Andreani, L. C.

M. Liscidini, A. Locatelli, L. C. Andreani, and C. De Angelis, “Maximum-Exponent Scaling Behavior of Optical Second-Harmonic Generation in Finite Multilayer Photonic Crystals,” Phys. Rev. Lett. 99(5), 053907 (2007).
[Crossref] [PubMed]

Aryshev, A.

A. Aryshev, A. Potylitsyn, G. Naumenko, M. Shevelev, K. Lekomtsev, L. Sukhikh, P. Karataev, Y. Honda, N. Terunuma, and J. Urakawa, “Monochromaticity of coherent Smith-Purcell radiation from finite size grating,” Phys. Rev. Accel. Beams 20(2), 024701 (2017).
[Crossref]

Bao, Q.

Blanchard, C.

C. Blanchard, J. P. Hugonin, and C. Sauvan, “Fano resonances in photonic crystal slabs near bound states in the continuum,” Phys. Rev. B 94(15), 155303 (2016).
[Crossref]

Bonaccorso, F.

F. Bonaccorso, Z. Sun, T. Hasan, and A. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4(9), 611–622 (2010).
[Crossref]

Bulgakov, E. N.

E. N. Bulgakov and A. F. Sadreev, “Bound states in the continuum in photonic waveguides inspired by defects,” Phys. Rev. B 78(7), 075105 (2008).
[Crossref]

Butler, S. Z.

S. Z. Butler, S. M. Hollen, L. Cao, Y. Cui, J. A. Gupta, H. R. Gutiérrez, T. F. Heinz, S. S. Hong, J. Huang, A. F. Ismach, E. Johnston-Halperin, M. Kuno, V. V. Plashnitsa, R. D. Robinson, R. S. Ruoff, S. Salahuddin, J. Shan, L. Shi, M. G. Spencer, M. Terrones, W. Windl, and J. E. Goldberger, “Progress, Challenges, and Opportunities in Two-Dimensional Materials Beyond Graphene,” ACS Nano 7(4), 2898–2926 (2013).
[Crossref] [PubMed]

Cao, L.

S. Z. Butler, S. M. Hollen, L. Cao, Y. Cui, J. A. Gupta, H. R. Gutiérrez, T. F. Heinz, S. S. Hong, J. Huang, A. F. Ismach, E. Johnston-Halperin, M. Kuno, V. V. Plashnitsa, R. D. Robinson, R. S. Ruoff, S. Salahuddin, J. Shan, L. Shi, M. G. Spencer, M. Terrones, W. Windl, and J. E. Goldberger, “Progress, Challenges, and Opportunities in Two-Dimensional Materials Beyond Graphene,” ACS Nano 7(4), 2898–2926 (2013).
[Crossref] [PubMed]

Castro Neto, A. H.

A. H. Castro Neto, “Charge density wave, superconductivity, and anomalous metallic behavior in 2D transition metal dichalcogenides,” Phys. Rev. Lett. 86(19), 4382–4385 (2001).
[Crossref] [PubMed]

Chen, H.

H. Chen, V. Corboliou, A. S. Solntsev, D.-Y. Choi, M. A. Vincenti, D. de Ceglia, C. de Angelis, Y. Lu, and D. N. Neshev, “Enhanced second-harmonic generation from two-dimensional MoSe2 on a silicon waveguide,” Light Sci. Appl. 6(10), e17060 (2017).
[Crossref]

M. Xu, T. Liang, M. Shi, and H. Chen, “Graphene-Like Two-Dimensional Materials,” Chem. Rev. 113(5), 3766–3798 (2013).
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Y. Li, A. Chernikov, X. Zhang, A. Rigosi, H. M. Hill, A. M. van der Zande, D. A. Chenet, E.-M. Shih, J. Hone, and T. F. Heinz, “Measurement of the optical dielectric function of monolayer transition-metal dichalcogenides MoS2, MoSe2, WS2, and WSe2,” Phys. Rev. B 90(20), 205422 (2014).
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Chernikov, A.

Y. Li, A. Chernikov, X. Zhang, A. Rigosi, H. M. Hill, A. M. van der Zande, D. A. Chenet, E.-M. Shih, J. Hone, and T. F. Heinz, “Measurement of the optical dielectric function of monolayer transition-metal dichalcogenides MoS2, MoSe2, WS2, and WSe2,” Phys. Rev. B 90(20), 205422 (2014).
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Choi, D.-Y.

H. Chen, V. Corboliou, A. S. Solntsev, D.-Y. Choi, M. A. Vincenti, D. de Ceglia, C. de Angelis, Y. Lu, and D. N. Neshev, “Enhanced second-harmonic generation from two-dimensional MoSe2 on a silicon waveguide,” Light Sci. Appl. 6(10), e17060 (2017).
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Chua, S. L.

C. W. Hsu, B. Zhen, J. Lee, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499(7457), 188–191 (2013).
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C. W. Hsu, B. Zhen, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Bloch surface eigenstates within the radiation continuum,” Light Sci. Appl. 2(7), e84–e89 (2013).
[Crossref]

J. Lee, B. Zhen, S. L. Chua, W. Qiu, J. D. Joannopoulos, M. Soljačić, and O. Shapira, “Observation and Differentiation of Unique High-Q Optical Resonances Near Zero Wave Vector in Macroscopic Photonic Crystal Slabs,” Phys. Rev. Lett. 109(6), 067401 (2012).
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Corboliou, V.

H. Chen, V. Corboliou, A. S. Solntsev, D.-Y. Choi, M. A. Vincenti, D. de Ceglia, C. de Angelis, Y. Lu, and D. N. Neshev, “Enhanced second-harmonic generation from two-dimensional MoSe2 on a silicon waveguide,” Light Sci. Appl. 6(10), e17060 (2017).
[Crossref]

Crespi, V.

C. Janisch, Y. Wang, D. Ma, N. Mehta, A. L. Elías, N. Perea-López, M. Terrones, V. Crespi, and Z. Liu, “Extraordinary Second Harmonic Generation in Tungsten Disulfide Monolayers,” Sci. Rep. 4(1), 5530–5534 (2014).
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Cui, Y.

S. Z. Butler, S. M. Hollen, L. Cao, Y. Cui, J. A. Gupta, H. R. Gutiérrez, T. F. Heinz, S. S. Hong, J. Huang, A. F. Ismach, E. Johnston-Halperin, M. Kuno, V. V. Plashnitsa, R. D. Robinson, R. S. Ruoff, S. Salahuddin, J. Shan, L. Shi, M. G. Spencer, M. Terrones, W. Windl, and J. E. Goldberger, “Progress, Challenges, and Opportunities in Two-Dimensional Materials Beyond Graphene,” ACS Nano 7(4), 2898–2926 (2013).
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D’Orazio, A.

M. A. Vincenti, D. de Ceglia, M. Grande, A. D’Orazio, and M. Scalora, “Third-harmonic generation in one-dimensional photonic crystal with graphene-based defect,” Phys. Rev. B 89(16), 165139 (2014).
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Dadap, J. I.

S.-Y. Hong, J. I. Dadap, N. Petrone, P.-C. Yeh, J. Hone, and R. M. Osgood., “Optical Third-Harmonic Generation in Graphene,” Phys. Rev. X 3(2), 021014 (2013).
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de Angelis, C.

H. Chen, V. Corboliou, A. S. Solntsev, D.-Y. Choi, M. A. Vincenti, D. de Ceglia, C. de Angelis, Y. Lu, and D. N. Neshev, “Enhanced second-harmonic generation from two-dimensional MoSe2 on a silicon waveguide,” Light Sci. Appl. 6(10), e17060 (2017).
[Crossref]

M. Liscidini, A. Locatelli, L. C. Andreani, and C. De Angelis, “Maximum-Exponent Scaling Behavior of Optical Second-Harmonic Generation in Finite Multilayer Photonic Crystals,” Phys. Rev. Lett. 99(5), 053907 (2007).
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C. De Angelis, F. Gringoli, M. Midrio, D. Modotto, J. S. Aitchison, and G. F. Nalesso, “Conversion efficiency for second-harmonic generation in photonic crystals,” J. Opt. Soc. Am. B 18(3), 348–351 (2001).
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H. Chen, V. Corboliou, A. S. Solntsev, D.-Y. Choi, M. A. Vincenti, D. de Ceglia, C. de Angelis, Y. Lu, and D. N. Neshev, “Enhanced second-harmonic generation from two-dimensional MoSe2 on a silicon waveguide,” Light Sci. Appl. 6(10), e17060 (2017).
[Crossref]

M. A. Vincenti, D. de Ceglia, M. Grande, A. D’Orazio, and M. Scalora, “Third-harmonic generation in one-dimensional photonic crystal with graphene-based defect,” Phys. Rev. B 89(16), 165139 (2014).
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de Paula, A. M.

L. M. Malard, T. V. Alencar, M. Ana Paula Barboza, K. F. Mak, and A. M. de Paula, “Observation of intense second harmonic generation from MoS2 atomic crystals,” Phys. Rev. B 87(20), 201401 (2013).
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Dean, C. R.

Y. Li, Y. Rao, K. F. Mak, Y. You, S. Wang, C. R. Dean, and T. F. Heinz, “Probing Symmetry Properties of Few-Layer MoS2 and h-BN by Optical Second-Harmonic Generation,” Nano Lett. 13(7), 3329–3333 (2013).
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Dreisow, F.

Y. Plotnik, O. Peleg, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, and M. Segev, “Experimental Observation of Optical Bound States in the Continuum,” Phys. Rev. Lett. 107(18), 183901 (2011).
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Dumeige, Y.

Y. Dumeige, I. Sagnes, P. Monnier, P. Vidakovic, I. Abram, C. Mériadec, and A. Levenson, “Phase-Matched Frequency Doubling at Photonic Band Edges: Efficiency Scaling as the Fifth Power of the Length,” Phys. Rev. Lett. 89(4), 043901 (2002).
[Crossref] [PubMed]

Elías, A. L.

C. Janisch, Y. Wang, D. Ma, N. Mehta, A. L. Elías, N. Perea-López, M. Terrones, V. Crespi, and Z. Liu, “Extraordinary Second Harmonic Generation in Tungsten Disulfide Monolayers,” Sci. Rep. 4(1), 5530–5534 (2014).
[Crossref] [PubMed]

Ferrari, A.

F. Bonaccorso, Z. Sun, T. Hasan, and A. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4(9), 611–622 (2010).
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Fu, Y.

Godbout, N.

Goldberger, J. E.

S. Z. Butler, S. M. Hollen, L. Cao, Y. Cui, J. A. Gupta, H. R. Gutiérrez, T. F. Heinz, S. S. Hong, J. Huang, A. F. Ismach, E. Johnston-Halperin, M. Kuno, V. V. Plashnitsa, R. D. Robinson, R. S. Ruoff, S. Salahuddin, J. Shan, L. Shi, M. G. Spencer, M. Terrones, W. Windl, and J. E. Goldberger, “Progress, Challenges, and Opportunities in Two-Dimensional Materials Beyond Graphene,” ACS Nano 7(4), 2898–2926 (2013).
[Crossref] [PubMed]

Gong, P.

K. L. Seyler, J. R. Schaibley, P. Gong, P. Rivera, A. M. Jones, S. Wu, J. Yan, D. G. Mandrus, W. Yao, and X. Xu, “Electrical control of second-harmonic generation in a monolayer WSe2 transistor,” Nat. Nanotechnol. 10(5), 407–411 (2015).
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Gorbach, A. V.

A. V. Gorbach, “Nonlinear graphene plasmonics: Amplitude equation for surface plamons,” Phys. Rev. A 87(1), 013830 (2013).
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Grande, M.

M. A. Vincenti, D. de Ceglia, M. Grande, A. D’Orazio, and M. Scalora, “Third-harmonic generation in one-dimensional photonic crystal with graphene-based defect,” Phys. Rev. B 89(16), 165139 (2014).
[Crossref]

Gringoli, F.

Gupta, J. A.

S. Z. Butler, S. M. Hollen, L. Cao, Y. Cui, J. A. Gupta, H. R. Gutiérrez, T. F. Heinz, S. S. Hong, J. Huang, A. F. Ismach, E. Johnston-Halperin, M. Kuno, V. V. Plashnitsa, R. D. Robinson, R. S. Ruoff, S. Salahuddin, J. Shan, L. Shi, M. G. Spencer, M. Terrones, W. Windl, and J. E. Goldberger, “Progress, Challenges, and Opportunities in Two-Dimensional Materials Beyond Graphene,” ACS Nano 7(4), 2898–2926 (2013).
[Crossref] [PubMed]

Gutiérrez, H. R.

S. Z. Butler, S. M. Hollen, L. Cao, Y. Cui, J. A. Gupta, H. R. Gutiérrez, T. F. Heinz, S. S. Hong, J. Huang, A. F. Ismach, E. Johnston-Halperin, M. Kuno, V. V. Plashnitsa, R. D. Robinson, R. S. Ruoff, S. Salahuddin, J. Shan, L. Shi, M. G. Spencer, M. Terrones, W. Windl, and J. E. Goldberger, “Progress, Challenges, and Opportunities in Two-Dimensional Materials Beyond Graphene,” ACS Nano 7(4), 2898–2926 (2013).
[Crossref] [PubMed]

Hasan, T.

F. Bonaccorso, Z. Sun, T. Hasan, and A. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4(9), 611–622 (2010).
[Crossref]

Heinrich, M.

Y. Plotnik, O. Peleg, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, and M. Segev, “Experimental Observation of Optical Bound States in the Continuum,” Phys. Rev. Lett. 107(18), 183901 (2011).
[Crossref] [PubMed]

Heinz, T. F.

Y. Li, A. Chernikov, X. Zhang, A. Rigosi, H. M. Hill, A. M. van der Zande, D. A. Chenet, E.-M. Shih, J. Hone, and T. F. Heinz, “Measurement of the optical dielectric function of monolayer transition-metal dichalcogenides MoS2, MoSe2, WS2, and WSe2,” Phys. Rev. B 90(20), 205422 (2014).
[Crossref]

S. Z. Butler, S. M. Hollen, L. Cao, Y. Cui, J. A. Gupta, H. R. Gutiérrez, T. F. Heinz, S. S. Hong, J. Huang, A. F. Ismach, E. Johnston-Halperin, M. Kuno, V. V. Plashnitsa, R. D. Robinson, R. S. Ruoff, S. Salahuddin, J. Shan, L. Shi, M. G. Spencer, M. Terrones, W. Windl, and J. E. Goldberger, “Progress, Challenges, and Opportunities in Two-Dimensional Materials Beyond Graphene,” ACS Nano 7(4), 2898–2926 (2013).
[Crossref] [PubMed]

Y. Li, Y. Rao, K. F. Mak, Y. You, S. Wang, C. R. Dean, and T. F. Heinz, “Probing Symmetry Properties of Few-Layer MoS2 and h-BN by Optical Second-Harmonic Generation,” Nano Lett. 13(7), 3329–3333 (2013).
[Crossref] [PubMed]

Hill, H. M.

Y. Li, A. Chernikov, X. Zhang, A. Rigosi, H. M. Hill, A. M. van der Zande, D. A. Chenet, E.-M. Shih, J. Hone, and T. F. Heinz, “Measurement of the optical dielectric function of monolayer transition-metal dichalcogenides MoS2, MoSe2, WS2, and WSe2,” Phys. Rev. B 90(20), 205422 (2014).
[Crossref]

Hollen, S. M.

S. Z. Butler, S. M. Hollen, L. Cao, Y. Cui, J. A. Gupta, H. R. Gutiérrez, T. F. Heinz, S. S. Hong, J. Huang, A. F. Ismach, E. Johnston-Halperin, M. Kuno, V. V. Plashnitsa, R. D. Robinson, R. S. Ruoff, S. Salahuddin, J. Shan, L. Shi, M. G. Spencer, M. Terrones, W. Windl, and J. E. Goldberger, “Progress, Challenges, and Opportunities in Two-Dimensional Materials Beyond Graphene,” ACS Nano 7(4), 2898–2926 (2013).
[Crossref] [PubMed]

Honda, Y.

A. Aryshev, A. Potylitsyn, G. Naumenko, M. Shevelev, K. Lekomtsev, L. Sukhikh, P. Karataev, Y. Honda, N. Terunuma, and J. Urakawa, “Monochromaticity of coherent Smith-Purcell radiation from finite size grating,” Phys. Rev. Accel. Beams 20(2), 024701 (2017).
[Crossref]

Hone, J.

Y. Li, A. Chernikov, X. Zhang, A. Rigosi, H. M. Hill, A. M. van der Zande, D. A. Chenet, E.-M. Shih, J. Hone, and T. F. Heinz, “Measurement of the optical dielectric function of monolayer transition-metal dichalcogenides MoS2, MoSe2, WS2, and WSe2,” Phys. Rev. B 90(20), 205422 (2014).
[Crossref]

S.-Y. Hong, J. I. Dadap, N. Petrone, P.-C. Yeh, J. Hone, and R. M. Osgood., “Optical Third-Harmonic Generation in Graphene,” Phys. Rev. X 3(2), 021014 (2013).
[Crossref]

Hong, S. S.

S. Z. Butler, S. M. Hollen, L. Cao, Y. Cui, J. A. Gupta, H. R. Gutiérrez, T. F. Heinz, S. S. Hong, J. Huang, A. F. Ismach, E. Johnston-Halperin, M. Kuno, V. V. Plashnitsa, R. D. Robinson, R. S. Ruoff, S. Salahuddin, J. Shan, L. Shi, M. G. Spencer, M. Terrones, W. Windl, and J. E. Goldberger, “Progress, Challenges, and Opportunities in Two-Dimensional Materials Beyond Graphene,” ACS Nano 7(4), 2898–2926 (2013).
[Crossref] [PubMed]

Hong, S.-Y.

S.-Y. Hong, J. I. Dadap, N. Petrone, P.-C. Yeh, J. Hone, and R. M. Osgood., “Optical Third-Harmonic Generation in Graphene,” Phys. Rev. X 3(2), 021014 (2013).
[Crossref]

Hsu, C. W.

B. Zhen, C. W. Hsu, L. Lu, A. D. Stone, and M. Soljačić, “Topological Nature of Optical Bound States in the Continuum,” Phys. Rev. Lett. 113(25), 257401 (2014).
[Crossref] [PubMed]

C. W. Hsu, B. Zhen, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Bloch surface eigenstates within the radiation continuum,” Light Sci. Appl. 2(7), e84–e89 (2013).
[Crossref]

C. W. Hsu, B. Zhen, J. Lee, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499(7457), 188–191 (2013).
[Crossref] [PubMed]

Huang, J.

S. Z. Butler, S. M. Hollen, L. Cao, Y. Cui, J. A. Gupta, H. R. Gutiérrez, T. F. Heinz, S. S. Hong, J. Huang, A. F. Ismach, E. Johnston-Halperin, M. Kuno, V. V. Plashnitsa, R. D. Robinson, R. S. Ruoff, S. Salahuddin, J. Shan, L. Shi, M. G. Spencer, M. Terrones, W. Windl, and J. E. Goldberger, “Progress, Challenges, and Opportunities in Two-Dimensional Materials Beyond Graphene,” ACS Nano 7(4), 2898–2926 (2013).
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Hugonin, J. P.

C. Blanchard, J. P. Hugonin, and C. Sauvan, “Fano resonances in photonic crystal slabs near bound states in the continuum,” Phys. Rev. B 94(15), 155303 (2016).
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Humlícek, J.

Ismach, A. F.

S. Z. Butler, S. M. Hollen, L. Cao, Y. Cui, J. A. Gupta, H. R. Gutiérrez, T. F. Heinz, S. S. Hong, J. Huang, A. F. Ismach, E. Johnston-Halperin, M. Kuno, V. V. Plashnitsa, R. D. Robinson, R. S. Ruoff, S. Salahuddin, J. Shan, L. Shi, M. G. Spencer, M. Terrones, W. Windl, and J. E. Goldberger, “Progress, Challenges, and Opportunities in Two-Dimensional Materials Beyond Graphene,” ACS Nano 7(4), 2898–2926 (2013).
[Crossref] [PubMed]

Janisch, C.

C. Janisch, Y. Wang, D. Ma, N. Mehta, A. L. Elías, N. Perea-López, M. Terrones, V. Crespi, and Z. Liu, “Extraordinary Second Harmonic Generation in Tungsten Disulfide Monolayers,” Sci. Rep. 4(1), 5530–5534 (2014).
[Crossref] [PubMed]

Joannopoulos, J. D.

C. W. Hsu, B. Zhen, J. Lee, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499(7457), 188–191 (2013).
[Crossref] [PubMed]

C. W. Hsu, B. Zhen, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Bloch surface eigenstates within the radiation continuum,” Light Sci. Appl. 2(7), e84–e89 (2013).
[Crossref]

J. Lee, B. Zhen, S. L. Chua, W. Qiu, J. D. Joannopoulos, M. Soljačić, and O. Shapira, “Observation and Differentiation of Unique High-Q Optical Resonances Near Zero Wave Vector in Macroscopic Photonic Crystal Slabs,” Phys. Rev. Lett. 109(6), 067401 (2012).
[Crossref] [PubMed]

Johnson, S. G.

C. W. Hsu, B. Zhen, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Bloch surface eigenstates within the radiation continuum,” Light Sci. Appl. 2(7), e84–e89 (2013).
[Crossref]

C. W. Hsu, B. Zhen, J. Lee, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499(7457), 188–191 (2013).
[Crossref] [PubMed]

Johnston-Halperin, E.

S. Z. Butler, S. M. Hollen, L. Cao, Y. Cui, J. A. Gupta, H. R. Gutiérrez, T. F. Heinz, S. S. Hong, J. Huang, A. F. Ismach, E. Johnston-Halperin, M. Kuno, V. V. Plashnitsa, R. D. Robinson, R. S. Ruoff, S. Salahuddin, J. Shan, L. Shi, M. G. Spencer, M. Terrones, W. Windl, and J. E. Goldberger, “Progress, Challenges, and Opportunities in Two-Dimensional Materials Beyond Graphene,” ACS Nano 7(4), 2898–2926 (2013).
[Crossref] [PubMed]

Jones, A. M.

K. L. Seyler, J. R. Schaibley, P. Gong, P. Rivera, A. M. Jones, S. Wu, J. Yan, D. G. Mandrus, W. Yao, and X. Xu, “Electrical control of second-harmonic generation in a monolayer WSe2 transistor,” Nat. Nanotechnol. 10(5), 407–411 (2015).
[Crossref] [PubMed]

Karataev, P.

A. Aryshev, A. Potylitsyn, G. Naumenko, M. Shevelev, K. Lekomtsev, L. Sukhikh, P. Karataev, Y. Honda, N. Terunuma, and J. Urakawa, “Monochromaticity of coherent Smith-Purcell radiation from finite size grating,” Phys. Rev. Accel. Beams 20(2), 024701 (2017).
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Kaup, D. J.

G. T. Adamashvili and D. J. Kaup, “Optical surface breather in graphene,” Phys. Rev. A 95(5), 053801 (2017).
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Kekoua, M. G.

Khoutsishvili, E. V.

Kivshar, Y. S.

M. I. Molina, A. E. Miroshnichenko, and Y. S. Kivshar, “Surface bound states in the continuum,” Phys. Rev. Lett. 108(7), 070401 (2012).
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Kockaert, P.

Kuno, M.

S. Z. Butler, S. M. Hollen, L. Cao, Y. Cui, J. A. Gupta, H. R. Gutiérrez, T. F. Heinz, S. S. Hong, J. Huang, A. F. Ismach, E. Johnston-Halperin, M. Kuno, V. V. Plashnitsa, R. D. Robinson, R. S. Ruoff, S. Salahuddin, J. Shan, L. Shi, M. G. Spencer, M. Terrones, W. Windl, and J. E. Goldberger, “Progress, Challenges, and Opportunities in Two-Dimensional Materials Beyond Graphene,” ACS Nano 7(4), 2898–2926 (2013).
[Crossref] [PubMed]

Lee, J.

C. W. Hsu, B. Zhen, J. Lee, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499(7457), 188–191 (2013).
[Crossref] [PubMed]

J. Lee, B. Zhen, S. L. Chua, W. Qiu, J. D. Joannopoulos, M. Soljačić, and O. Shapira, “Observation and Differentiation of Unique High-Q Optical Resonances Near Zero Wave Vector in Macroscopic Photonic Crystal Slabs,” Phys. Rev. Lett. 109(6), 067401 (2012).
[Crossref] [PubMed]

Lekomtsev, K.

A. Aryshev, A. Potylitsyn, G. Naumenko, M. Shevelev, K. Lekomtsev, L. Sukhikh, P. Karataev, Y. Honda, N. Terunuma, and J. Urakawa, “Monochromaticity of coherent Smith-Purcell radiation from finite size grating,” Phys. Rev. Accel. Beams 20(2), 024701 (2017).
[Crossref]

Levenson, A.

Y. Dumeige, I. Sagnes, P. Monnier, P. Vidakovic, I. Abram, C. Mériadec, and A. Levenson, “Phase-Matched Frequency Doubling at Photonic Band Edges: Efficiency Scaling as the Fifth Power of the Length,” Phys. Rev. Lett. 89(4), 043901 (2002).
[Crossref] [PubMed]

Li, Y.

Y. Li, A. Chernikov, X. Zhang, A. Rigosi, H. M. Hill, A. M. van der Zande, D. A. Chenet, E.-M. Shih, J. Hone, and T. F. Heinz, “Measurement of the optical dielectric function of monolayer transition-metal dichalcogenides MoS2, MoSe2, WS2, and WSe2,” Phys. Rev. B 90(20), 205422 (2014).
[Crossref]

Y. Li, Y. Rao, K. F. Mak, Y. You, S. Wang, C. R. Dean, and T. F. Heinz, “Probing Symmetry Properties of Few-Layer MoS2 and h-BN by Optical Second-Harmonic Generation,” Nano Lett. 13(7), 3329–3333 (2013).
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Y. Yang, C. Peng, Y. Liang, Z. Li, and S. Noda, “Analytical Perspective for Bound States in the Continuum in Photonic Crystal Slabs,” Phys. Rev. Lett. 113(3), 037401 (2014).
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Z. Y. Li and L. L. Lin, “Photonic band structures solved by a plane-wave-based transfer-matrix method,” Phys. Rev. E 67(4), 046607 (2003).
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M. Xu, T. Liang, M. Shi, and H. Chen, “Graphene-Like Two-Dimensional Materials,” Chem. Rev. 113(5), 3766–3798 (2013).
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Y. Yang, C. Peng, Y. Liang, Z. Li, and S. Noda, “Analytical Perspective for Bound States in the Continuum in Photonic Crystal Slabs,” Phys. Rev. Lett. 113(3), 037401 (2014).
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Lin, L. L.

Z. Y. Li and L. L. Lin, “Photonic band structures solved by a plane-wave-based transfer-matrix method,” Phys. Rev. E 67(4), 046607 (2003).
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Liscidini, M.

M. Liscidini, A. Locatelli, L. C. Andreani, and C. De Angelis, “Maximum-Exponent Scaling Behavior of Optical Second-Harmonic Generation in Finite Multilayer Photonic Crystals,” Phys. Rev. Lett. 99(5), 053907 (2007).
[Crossref] [PubMed]

Liu, Q. H.

Liu, Z.

C. Janisch, Y. Wang, D. Ma, N. Mehta, A. L. Elías, N. Perea-López, M. Terrones, V. Crespi, and Z. Liu, “Extraordinary Second Harmonic Generation in Tungsten Disulfide Monolayers,” Sci. Rep. 4(1), 5530–5534 (2014).
[Crossref] [PubMed]

Locatelli, A.

M. Liscidini, A. Locatelli, L. C. Andreani, and C. De Angelis, “Maximum-Exponent Scaling Behavior of Optical Second-Harmonic Generation in Finite Multilayer Photonic Crystals,” Phys. Rev. Lett. 99(5), 053907 (2007).
[Crossref] [PubMed]

Lu, L.

B. Zhen, C. W. Hsu, L. Lu, A. D. Stone, and M. Soljačić, “Topological Nature of Optical Bound States in the Continuum,” Phys. Rev. Lett. 113(25), 257401 (2014).
[Crossref] [PubMed]

Lu, W.

Lu, Y.

H. Chen, V. Corboliou, A. S. Solntsev, D.-Y. Choi, M. A. Vincenti, D. de Ceglia, C. de Angelis, Y. Lu, and D. N. Neshev, “Enhanced second-harmonic generation from two-dimensional MoSe2 on a silicon waveguide,” Light Sci. Appl. 6(10), e17060 (2017).
[Crossref]

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You, Y.

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J. Lee, B. Zhen, S. L. Chua, W. Qiu, J. D. Joannopoulos, M. Soljačić, and O. Shapira, “Observation and Differentiation of Unique High-Q Optical Resonances Near Zero Wave Vector in Macroscopic Photonic Crystal Slabs,” Phys. Rev. Lett. 109(6), 067401 (2012).
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ACS Nano (1)

S. Z. Butler, S. M. Hollen, L. Cao, Y. Cui, J. A. Gupta, H. R. Gutiérrez, T. F. Heinz, S. S. Hong, J. Huang, A. F. Ismach, E. Johnston-Halperin, M. Kuno, V. V. Plashnitsa, R. D. Robinson, R. S. Ruoff, S. Salahuddin, J. Shan, L. Shi, M. G. Spencer, M. Terrones, W. Windl, and J. E. Goldberger, “Progress, Challenges, and Opportunities in Two-Dimensional Materials Beyond Graphene,” ACS Nano 7(4), 2898–2926 (2013).
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M. Xu, T. Liang, M. Shi, and H. Chen, “Graphene-Like Two-Dimensional Materials,” Chem. Rev. 113(5), 3766–3798 (2013).
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R. F. Ndangali and S. V. Shabanov, “Electromagnetic bound states in the radiation continuum for periodic double arrays of subwavelength dielectric cylinders,” J. Math. Phys. 51(10), 102901 (2010).
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Nano Lett. (1)

Y. Li, Y. Rao, K. F. Mak, Y. You, S. Wang, C. R. Dean, and T. F. Heinz, “Probing Symmetry Properties of Few-Layer MoS2 and h-BN by Optical Second-Harmonic Generation,” Nano Lett. 13(7), 3329–3333 (2013).
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Nat. Nanotechnol. (1)

K. L. Seyler, J. R. Schaibley, P. Gong, P. Rivera, A. M. Jones, S. Wu, J. Yan, D. G. Mandrus, W. Yao, and X. Xu, “Electrical control of second-harmonic generation in a monolayer WSe2 transistor,” Nat. Nanotechnol. 10(5), 407–411 (2015).
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Figures (9)

Fig. 1
Fig. 1 Diagram of the TMDC monolayer-grating structure and the SH generation process (oblique view shown in (a) and side view presented in (b)). The lattice constant, width, thickness and relative permittivity of the grating are marked by a, w, d and ε, respectively, the grating is immersed in the air. When a plane wave with angular frequency ω is incident on the structure at the incident angle θ i , the transmitted and reflected SH fields are generated due to the second-order nonlinear effect of TMDC monolayer.
Fig. 2
Fig. 2 Transmission (a) and absorption (b) for the two layer structure as a function of the wavelength λ of the EM wave and thickness d of the grating. (c) Magnified absorption map for the square region in (b). (d)-(f) Corresponding absorption spectrums for different thicknesses of the grating d=312.8nm (d), d=321.71434nm (e) and d=331.2nm (f). The parameters are assumed as follows: a=460nm and w=0.6a.
Fig. 3
Fig. 3 (a) Maximal absorption near BIC as a function of the thickness of the grating. (b) Maximal field enhancement near BIC as a function of the thickness d of the grating. (c) The position spectrum of the maximal absorption (black line with black symbol) and maximal field enhancement (red line with red symbol) indicated by the wavelength λ of the EM wave and the thickness d of the grating. (d) Absorption spectrum at the maximum point in (a) d=325.87182nm. The other parameters are identical to those in Fig. 2.
Fig. 4
Fig. 4 SH generation enhancement ET of the transmitted field (a)-(d) and ER of the reflected field (e)-(h) as a function of the wavelength λ of the incident field for different thicknesses of the grating. The thicknesses of the grating are taken as follows: d=312.8nm for (a) and (e), d=321.71434nm for (b) and (f), d=325.87182nm for (c) and (g), and d=331.2nm for (d) and (h). The other parameters are identical to those in Fig. 2.
Fig. 5
Fig. 5 SH generation enhancement ET of the transmitted field (solid line) and ER of the reflected field (dashed line) as a function of the azimuthal angle of the WS 2 monolayer. The wavelength of the EM wave and thicknesses of the grating are taken as λ=1017.821nm and d=325.87182nm corresponding to the maximum point in Fig. 4(c).
Fig. 6
Fig. 6 (a) Absorption for two layer structure as a function of the wavelength λ and incident angle θ i of the EM wave. Maximal field enhancement (b) and maximal absorption (c) near BIC as a function of the incident angle θ i of the EM wave. (d) Absorption spectrum at the maximum point in (b) θ i =16.1723°. The parameters are assumed as follows: a=460nm, w=0.6a and d=1.0a.
Fig. 7
Fig. 7 SH generation enhancement ET of the transmitted field (a) and ER of the reflected field (b) as a function of the wavelength λ of the incident field at θ i =16.1723°. The other parameters are identical to those in Fig. 6.
Fig. 8
Fig. 8 Transmission and reflection of a plane wave from a TMDC monolayer at the SH frequency. The incident, transmitted and reflected waves are denoted by E i , E t and E r , the mediums up and down the interface are marked by the number 1 and 2.
Fig. 9
Fig. 9 Scattering of two waves from a TMDC monolayer. These two waves indicated by E 1 + and E 2 are incident from the up and down mediums, two waves E 2 + and E 1 are generated.

Equations (24)

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E k ( x,y,ω )= mn ( E kmn + ( ω )+ E kmn ( ω ) ) e i( k mnx ( ω )x+ k mny ( ω )y ) ,
E k ( x,y,2ω )= mn ( E kmn + ( 2ω )+ E kmn ( 2ω ) ) e i( k mnx ( 2ω )x+ k mny ( 2ω )y ) .
J ( 2 ) ( r ,2ω )=i d t 2ω ε 0 χ ( 2 ) : E 1 ( r ,ω ) E 1 ( r ,ω ).
J ( 2 ) ( r ,2ω )=i d t 2ω ε 0 χ ( 2 ) : mn ( pq E 1pq ( ω ) E 1mp,nq ( ω ) ) e i( k mnx ( 2ω )x+ k mny ( 2ω )y ) .
( E 1 + ( 2ω ) E 0 ( 2ω ) )=( Q 1 I ( 2ω ) Q 1 II ( 2ω ) Q 1 III ( 2ω ) Q 1 IV ( 2ω ) )( E 0 + ( 2ω ) E 1 ( 2ω ) )+( C 1 I ( 2ω ) C 1 II ( 2ω ) ),
( E 2 + ( 2ω ) E 1 ( 2ω ) )=( Q 2 I ( 2ω ) Q 2 II ( 2ω ) Q 2 III ( 2ω ) Q 2 IV ( 2ω ) )( E 1 + ( 2ω ) E 2 ( 2ω ) )+( C 2 I ( 2ω ) C 2 II ( 2ω ) ),
( E 2 + ( 2ω ) E 0 ( 2ω ) )=( Q 1,2 I ( 2ω ) Q 1,2 II ( 2ω ) Q 1,2 III ( 2ω ) Q 1,2 IV ( 2ω ) )( E 0 + ( 2ω ) E 2 ( 2ω ) )+( C 1,2 I ( 2ω ) C 1,2 II ( 2ω ) ),
C 1,2 I ( 2ω )= Q 2 I ( 2ω ) [ 1 Q 1 II ( 2ω ) Q 2 III ( 2ω ) ] 1 [ Q 1 II ( 2ω ) C 2 II ( 2ω )+ C 1 I ( 2ω ) ]+ C 2 I ( 2ω ),
C 1,2 II ( 2ω )= Q 1 IV ( 2ω ) [ 1 Q 2 III ( 2ω ) Q 1 II ( 2ω ) ] 1 [ Q 2 III ( 2ω ) C 1 I ( 2ω )+ C 2 II ( 2ω ) ]+ C 1 II ( 2ω ).
I t ( 2ω )= 1 2 ε 0 mn,i [ E 3mni + ( 2ω ) ] [ E 3mni + ( 2ω ) ] * ,
I r ( 2ω )= 1 2 ε 0 mn,i [ E 0mni ( 2ω ) ] [ E 0mni ( 2ω ) ] * ,
I i ( ω )= 1 2 ε 0 mn,i [ E 0mni + ( ω ) ] [ E 0mni + ( ω ) ] * .
ET= I t ( 2ω ) I 0t ( 2ω ) , ER= I r ( 2ω ) I 0r ( 2ω )
P x = ε 0 2 χ ( 2 ) sinϕcosϕ E x E x + ε 0 2 χ ( 2 ) ( sin 2 ϕ cos 2 ϕ ) E x E y + ε 0 2 χ ( 2 ) sinϕcosϕ E y E y , P y = ε 0 χ ( 2 ) ( sin 2 ϕ cos 2 ϕ ) E x E x + ε 0 χ ( 2 ) 4sinϕcosϕ E x E y + ε 0 χ ( 2 ) ( cos 2 ϕ sin 2 ϕ ) E y E y ,
{ E x t E x i E x r =0, H x t H x i H x r =σ E y + J y ( 2 ) , E y t E y i E y r =0, H y t H y i H y r =σ E x J x ( 2 ) ,
{ E x t = T x E x i + C tx , E y t = T y E y i + C ty , E x r = R x E x i + C rx , E y r = R y E y i + C ry .
{ T x = 2 q 2z / ε 2 q 1z / ε 1 + q 2z / ε 2 + q 1z q 2z σ/ ( ω ε 0 ε 1 ε 2 ) , T y = 2 q 1z q 1z + q 2z +ω μ 0 σ , R x = q 2z / ε 2 q 1z / ε 1 q 1z q 2z σ/ ( ω ε 0 ε 1 ε 2 ) q 1z / ε 1 + q 2z / ε 2 + q 1z q 2z σ/ ( ω ε 0 ε 1 ε 2 ) , R y = q 1z q 2z ω μ 0 σ q 1z + q 2z +ω μ 0 σ ,
{ C tx = J x ( 2 ) q 1z q 2z / ( ω ε 0 ε 1 ε 2 ) q 1z / ε 1 + q 2z / ε 2 + q 1z q 2z σ/ ( ω ε 0 ε 1 ε 2 ) , C ty = J y ( 2 ) ω μ 0 q 1z + q 2z +ω μ 0 σ , C rx = J x ( 2 ) q 1z q 2z / ( ω ε 0 ε 1 ε 2 ) q 1z / ε 1 + q 2z / ε 2 + q 1z q 2z σ/ ( ω ε 0 ε 1 ε 2 ) , C ry = J y ( 2 ) ω μ 0 q 1z + q 2z +ω μ 0 σ ,
( E x' t E y' t )=( cos 2 ϕ T x + sin 2 ϕ T y sinϕcosϕ( T x T y ) sinϕcosϕ( T x T y ) sin 2 ϕ T x + cos 2 ϕ T y )( E x' i E y' i )+( cosϕ C tx sinϕ C ty sinϕ C tx cosϕ C ty ),
( E 2 + E 1 )=( T ( 1,2 ) R ( 2,1 ) R ( 1,2 ) T ( 2,1 ) )( E 1 + E 2 )+( C 1 C 2 ),
T mn,pq ( i,j ) = δ mn,pq ( cos 2 ϕ mn T mnx ( i,j ) + sin 2 ϕ mn T mny ( i,j ) sin ϕ mn cos ϕ mn ( T mnx ( i,j ) T mny ( i,j ) ) sin ϕ mn cos ϕ mn ( T mnx ( i,j ) T mny ( i,j ) ) sin 2 ϕ mn T mnx ( i,j ) + cos 2 ϕ mn T mny ( i,j ) ),
R mn,pq ( i,j ) = δ mn,pq ( cos 2 ϕ mn R mnx ( i,j ) + sin 2 ϕ mn R mny ( i,j ) sin ϕ mn cos ϕ mn ( R mnx ( i,j ) R mny ( i,j ) ) sin ϕ mn cos ϕ mn ( R mnx ( i,j ) R mny ( i,j ) ) sin 2 ϕ mn R mnx ( i,j ) + cos 2 ϕ mn R mny ( i,j ) ),
C 1mn =( cos ϕ mn C mntx ( 1,2 ) sin ϕ mn C mnty ( 1,2 ) +cos ϕ mn C mnrx ( 2,1 ) sin ϕ mn C mnry ( 2,1 ) sin ϕ mn C mntx ( 1,2 ) +cos ϕ mn C mnty ( 1,2 ) +sin ϕ mn C mnrx ( 2,1 ) +cos ϕ mn C mnry ( 2,1 ) ),
C 2mn =( cos ϕ mn C mnrx ( 1,2 ) sin ϕ mn C mnry ( 1,2 ) +cos ϕ mn C mntx ( 2,1 ) sin ϕ mn C mnty ( 2,1 ) sin ϕ mn C mnrx ( 1,2 ) +cos ϕ mn C mnry ( 1,2 ) +sin ϕ mn C mntx ( 2,1 ) +cos ϕ mn C mnty ( 2,1 ) ),

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