Abstract

In this paper, we report large area four-wave mixing microscopy studies on silicon-on-insulator based partially etched two-dimensional zero-contrast gratings. The zero-contrast gratings offer an additional degree of freedom for the design of spectral resonances by varying the etch depth of the grating structures. This is leveraged by designing signal resonance at 1580 nm, operating in the sub-wavelength, zeroth-order diffraction region and pump fixed at 1040 nm operating in the higher order diffraction region. The zero-contrast gratings are fabricated on standard 220 nm silicon-on-insulator substrates with etch depth chosen as 140 nm. The fabricated structures are characterized to measure the linear transmission and nonlinear four-wave mixing performance. Multi-spectral four-wave mixing images acquired across the grating structures for varying input signal wavelength show maximum enhancement of four-wave mixing signal at 1575 nm, with the on-grating four-wave mixing signal enhanced by ∼ 450 times when compared to the un-patterned film. Zero-contrast gratings present a promising platform for realizing sub-wavelength scale nanostructured surfaces for nonlinear wave-mixing applications using standard silicon-on-insulator substrates. Such structures can find potential applications in wavelength conversion across widely separated wavelength bands and as substrates for nonlinear frequency conversion.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2019 (1)

R. Colom, L. Xu, L. Marini, F. Bedu, I. Ozerov, T. Begou, J. Lumeau, A. E. Miroshnishenko, D. Neshev, B. T. Kuhlmey, S. Palomba, and N. Bonod, “Enhanced Four-Wave Mixing in Doubly Resonant Si Nanoresonators,” ACS Photonics 6(5), 1295–1301 (2019).
[Crossref]

2018 (1)

2017 (1)

G. Grinblat, Y. Li, M. P. Nielsen, R. F. Oulton, and S. A. Maier, “Degenerate Four-Wave Mixing in a Multiresonant Germanium Nanodisk,” ACS Photonics 4(9), 2144–2149 (2017).
[Crossref]

2016 (2)

S. V. Makarov, A. N. Tsypkin, T. A. Voytova, V. A. Milichko, I. S. Mukhin, A. V. Yulin, S. E. Putilin, M. A. Baranov, A. E. Krasnok, I. A. Morozov, and P. A. Belov, “Self-adjusted all-dielectric metasurfaces for deep ultraviolet femtosecond pulse generation,” Nanoscale 8(41), 17809–17814 (2016).
[Crossref]

A. I. Kuznetsov, A. E. Miroshnichenko, M. L. Brongersma, Y. S. Kivshar, and B. Lukyanchuk, “Optically resonant dielectric nanostructures,” Science 354(6314), aag2472 (2016).
[Crossref]

2015 (6)

J. Butet, P.-F. Brevet, and O. J. F. Martin, “Optical Second Harmonic Generation in Plasmonic Nanostructures: From Fundamental Principles to Advanced Applications,” ACS Nano 9(11), 10545–10562 (2015).
[Crossref]

M. R. Shcherbakov, P. P. Vabishchevich, A. S. Shorokhov, K. E. Chong, D.-Y. Choi, I. Staude, A. E. Miroshnichenko, D. N. Neshev, A. A. Fedyanin, and Y. S. Kivshar, “Ultrafast All-Optical Switching with Magnetic Resonances in Nonlinear Dielectric Nanostructures,” Nano Lett. 15(10), 6985–6990 (2015).
[Crossref]

T. Sun, W. Yang, and C. J. Chang-Hasnain, “Surface-normal coupled four-wave mixing in a high contrast gratings resonator,” Opt. Express 23(23), 29565–29572 (2015).
[Crossref]

Y. Yang, W. Wang, A. Boulesbaa, I. I. Kravchenko, D. P. Briggs, A. Puretzky, D. Geohegan, and J. Valentine, “Nonlinear Fano-Resonant Dielectric Metasurfaces,” Nano Lett. 15(11), 7388–7393 (2015).
[Crossref]

M. R. Shcherbakov, A. S. Shorokhov, D. N. Neshev, B. Hopkins, I. Staude, E. V. Melik-Gaykazyan, A. A. Ezhov, A. E. Miroshnichenko, I. Brener, A. A. Fedyanin, and Y. S. Kivshar, “Nonlinear Interference and Tailorable Third-Harmonic Generation from Dielectric Oligomers,” ACS Photonics 2(5), 578–582 (2015).
[Crossref]

P. Qiao, L. Zhu, W. Cho Chew, and C. J. Chang-Hasnain, “Theory and design of two-dimensional high-contrast-grating phased arrays,” Opt. Express 23(19), 24508–24524 (2015).
[Crossref]

2014 (3)

R. Magnusson, “Wideband reflectors with zero-contrast gratings,” Opt. Express 39(15), 4337–4340 (2014).
[Crossref]

H. Aouani, M. Rahmani, M. Navarro-Cía, and S. A. Maier, “Third-harmonic-upconversion enhancement from a single semiconductor nanoparticle coupled to a plasmonic antenna,” Nat. Nanotechnol. 9(4), 290–294 (2014).
[Crossref]

M. R. Shcherbakov, D. N. Neshev, B. Hopkins, A. S. Shorokhov, I. Staude, E. V. Melik-Gaykazyan, M. Decker, A. A. Ezhov, A. E. Miroshnichenko, I. Brener, A. A. Fedyanin, and Y. S. Kivshar, “Enhanced Third-Harmonic Generation in Silicon Nanoparticles Driven by Magnetic Response,” Nano Lett. 14(11), 6488–6492 (2014).
[Crossref]

2013 (1)

Y. Zhang, F. Wen, Y.-R. Zhen, P. Nordlander, and N. J. Halas, “Coherent Fano resonances in a plasmonic nanocluster enhance optical four-wave mixing,” Proc. Natl. Acad. Sci. U. S. A. 110(23), 9215–9219 (2013).
[Crossref]

2012 (2)

J. Butet, I. Russier-Antoine, C. Jonin, N. Lascoux, E. Benichou, and P.-F. Brevet, “Sensing with Multipolar Second Harmonic Generation from Spherical Metallic Nanoparticles,” Nano Lett. 12(3), 1697–1701 (2012).
[Crossref]

C. J. Chang-Hasnain and W. Yang, “High-contrast gratings for integrated optoelectronics,” Adv. Opt. Photonics 4(3), 379–440 (2012).
[Crossref]

2011 (1)

Y. Wang, C.-Y. Lin, A. Nikolaenko, V. Raghunathan, and E. O. Potma, “Four-wave mixing microscopy of nanostructures,” Adv. Opt. Photonics 3(1), 1–52 (2011).
[Crossref]

2006 (1)

2005 (1)

1993 (2)

W. Y. Ching and M.-Z. Huang, “Calculation of optical excitation in cubic semiconductors. III. Third harmonic generation,” Phys. Rev. B 47(15), 9479–9491 (1993).
[Crossref]

S. S. Wang and R. Magnusson, “Theory and applications of guided-mode resonance filters,” Appl. Opt. 32(14), 2606–2613 (1993).
[Crossref]

1969 (1)

J. J. Wynne, “Optical Third-Order Mixing in GaAs, Ge, Si, and InAs,” Phys. Rev. 178(3), 1295–1303 (1969).
[Crossref]

Aouani, H.

H. Aouani, M. Rahmani, M. Navarro-Cía, and S. A. Maier, “Third-harmonic-upconversion enhancement from a single semiconductor nanoparticle coupled to a plasmonic antenna,” Nat. Nanotechnol. 9(4), 290–294 (2014).
[Crossref]

Baranov, M. A.

S. V. Makarov, A. N. Tsypkin, T. A. Voytova, V. A. Milichko, I. S. Mukhin, A. V. Yulin, S. E. Putilin, M. A. Baranov, A. E. Krasnok, I. A. Morozov, and P. A. Belov, “Self-adjusted all-dielectric metasurfaces for deep ultraviolet femtosecond pulse generation,” Nanoscale 8(41), 17809–17814 (2016).
[Crossref]

Bedu, F.

R. Colom, L. Xu, L. Marini, F. Bedu, I. Ozerov, T. Begou, J. Lumeau, A. E. Miroshnishenko, D. Neshev, B. T. Kuhlmey, S. Palomba, and N. Bonod, “Enhanced Four-Wave Mixing in Doubly Resonant Si Nanoresonators,” ACS Photonics 6(5), 1295–1301 (2019).
[Crossref]

Begou, T.

R. Colom, L. Xu, L. Marini, F. Bedu, I. Ozerov, T. Begou, J. Lumeau, A. E. Miroshnishenko, D. Neshev, B. T. Kuhlmey, S. Palomba, and N. Bonod, “Enhanced Four-Wave Mixing in Doubly Resonant Si Nanoresonators,” ACS Photonics 6(5), 1295–1301 (2019).
[Crossref]

Belov, P. A.

S. V. Makarov, A. N. Tsypkin, T. A. Voytova, V. A. Milichko, I. S. Mukhin, A. V. Yulin, S. E. Putilin, M. A. Baranov, A. E. Krasnok, I. A. Morozov, and P. A. Belov, “Self-adjusted all-dielectric metasurfaces for deep ultraviolet femtosecond pulse generation,” Nanoscale 8(41), 17809–17814 (2016).
[Crossref]

Benichou, E.

J. Butet, I. Russier-Antoine, C. Jonin, N. Lascoux, E. Benichou, and P.-F. Brevet, “Sensing with Multipolar Second Harmonic Generation from Spherical Metallic Nanoparticles,” Nano Lett. 12(3), 1697–1701 (2012).
[Crossref]

Biswas, R.

Bonod, N.

R. Colom, L. Xu, L. Marini, F. Bedu, I. Ozerov, T. Begou, J. Lumeau, A. E. Miroshnishenko, D. Neshev, B. T. Kuhlmey, S. Palomba, and N. Bonod, “Enhanced Four-Wave Mixing in Doubly Resonant Si Nanoresonators,” ACS Photonics 6(5), 1295–1301 (2019).
[Crossref]

Boulesbaa, A.

Y. Yang, W. Wang, A. Boulesbaa, I. I. Kravchenko, D. P. Briggs, A. Puretzky, D. Geohegan, and J. Valentine, “Nonlinear Fano-Resonant Dielectric Metasurfaces,” Nano Lett. 15(11), 7388–7393 (2015).
[Crossref]

Brener, I.

M. R. Shcherbakov, A. S. Shorokhov, D. N. Neshev, B. Hopkins, I. Staude, E. V. Melik-Gaykazyan, A. A. Ezhov, A. E. Miroshnichenko, I. Brener, A. A. Fedyanin, and Y. S. Kivshar, “Nonlinear Interference and Tailorable Third-Harmonic Generation from Dielectric Oligomers,” ACS Photonics 2(5), 578–582 (2015).
[Crossref]

M. R. Shcherbakov, D. N. Neshev, B. Hopkins, A. S. Shorokhov, I. Staude, E. V. Melik-Gaykazyan, M. Decker, A. A. Ezhov, A. E. Miroshnichenko, I. Brener, A. A. Fedyanin, and Y. S. Kivshar, “Enhanced Third-Harmonic Generation in Silicon Nanoparticles Driven by Magnetic Response,” Nano Lett. 14(11), 6488–6492 (2014).
[Crossref]

Brevet, P.-F.

J. Butet, P.-F. Brevet, and O. J. F. Martin, “Optical Second Harmonic Generation in Plasmonic Nanostructures: From Fundamental Principles to Advanced Applications,” ACS Nano 9(11), 10545–10562 (2015).
[Crossref]

J. Butet, I. Russier-Antoine, C. Jonin, N. Lascoux, E. Benichou, and P.-F. Brevet, “Sensing with Multipolar Second Harmonic Generation from Spherical Metallic Nanoparticles,” Nano Lett. 12(3), 1697–1701 (2012).
[Crossref]

Briggs, D. P.

Y. Yang, W. Wang, A. Boulesbaa, I. I. Kravchenko, D. P. Briggs, A. Puretzky, D. Geohegan, and J. Valentine, “Nonlinear Fano-Resonant Dielectric Metasurfaces,” Nano Lett. 15(11), 7388–7393 (2015).
[Crossref]

Brongersma, M. L.

A. I. Kuznetsov, A. E. Miroshnichenko, M. L. Brongersma, Y. S. Kivshar, and B. Lukyanchuk, “Optically resonant dielectric nanostructures,” Science 354(6314), aag2472 (2016).
[Crossref]

Butet, J.

J. Butet, P.-F. Brevet, and O. J. F. Martin, “Optical Second Harmonic Generation in Plasmonic Nanostructures: From Fundamental Principles to Advanced Applications,” ACS Nano 9(11), 10545–10562 (2015).
[Crossref]

J. Butet, I. Russier-Antoine, C. Jonin, N. Lascoux, E. Benichou, and P.-F. Brevet, “Sensing with Multipolar Second Harmonic Generation from Spherical Metallic Nanoparticles,” Nano Lett. 12(3), 1697–1701 (2012).
[Crossref]

Chang-Hasnain, C. J.

Ching, W. Y.

W. Y. Ching and M.-Z. Huang, “Calculation of optical excitation in cubic semiconductors. III. Third harmonic generation,” Phys. Rev. B 47(15), 9479–9491 (1993).
[Crossref]

Cho Chew, W.

Choi, D.-Y.

M. R. Shcherbakov, P. P. Vabishchevich, A. S. Shorokhov, K. E. Chong, D.-Y. Choi, I. Staude, A. E. Miroshnichenko, D. N. Neshev, A. A. Fedyanin, and Y. S. Kivshar, “Ultrafast All-Optical Switching with Magnetic Resonances in Nonlinear Dielectric Nanostructures,” Nano Lett. 15(10), 6985–6990 (2015).
[Crossref]

Chong, K. E.

M. R. Shcherbakov, P. P. Vabishchevich, A. S. Shorokhov, K. E. Chong, D.-Y. Choi, I. Staude, A. E. Miroshnichenko, D. N. Neshev, A. A. Fedyanin, and Y. S. Kivshar, “Ultrafast All-Optical Switching with Magnetic Resonances in Nonlinear Dielectric Nanostructures,” Nano Lett. 15(10), 6985–6990 (2015).
[Crossref]

Claps, R.

Colom, R.

R. Colom, L. Xu, L. Marini, F. Bedu, I. Ozerov, T. Begou, J. Lumeau, A. E. Miroshnishenko, D. Neshev, B. T. Kuhlmey, S. Palomba, and N. Bonod, “Enhanced Four-Wave Mixing in Doubly Resonant Si Nanoresonators,” ACS Photonics 6(5), 1295–1301 (2019).
[Crossref]

Decker, M.

M. R. Shcherbakov, D. N. Neshev, B. Hopkins, A. S. Shorokhov, I. Staude, E. V. Melik-Gaykazyan, M. Decker, A. A. Ezhov, A. E. Miroshnichenko, I. Brener, A. A. Fedyanin, and Y. S. Kivshar, “Enhanced Third-Harmonic Generation in Silicon Nanoparticles Driven by Magnetic Response,” Nano Lett. 14(11), 6488–6492 (2014).
[Crossref]

Deka, J.

Dimitropoulos, D.

Evans, C. L.

Ezhov, A. A.

M. R. Shcherbakov, A. S. Shorokhov, D. N. Neshev, B. Hopkins, I. Staude, E. V. Melik-Gaykazyan, A. A. Ezhov, A. E. Miroshnichenko, I. Brener, A. A. Fedyanin, and Y. S. Kivshar, “Nonlinear Interference and Tailorable Third-Harmonic Generation from Dielectric Oligomers,” ACS Photonics 2(5), 578–582 (2015).
[Crossref]

M. R. Shcherbakov, D. N. Neshev, B. Hopkins, A. S. Shorokhov, I. Staude, E. V. Melik-Gaykazyan, M. Decker, A. A. Ezhov, A. E. Miroshnichenko, I. Brener, A. A. Fedyanin, and Y. S. Kivshar, “Enhanced Third-Harmonic Generation in Silicon Nanoparticles Driven by Magnetic Response,” Nano Lett. 14(11), 6488–6492 (2014).
[Crossref]

Fedyanin, A. A.

M. R. Shcherbakov, P. P. Vabishchevich, A. S. Shorokhov, K. E. Chong, D.-Y. Choi, I. Staude, A. E. Miroshnichenko, D. N. Neshev, A. A. Fedyanin, and Y. S. Kivshar, “Ultrafast All-Optical Switching with Magnetic Resonances in Nonlinear Dielectric Nanostructures,” Nano Lett. 15(10), 6985–6990 (2015).
[Crossref]

M. R. Shcherbakov, A. S. Shorokhov, D. N. Neshev, B. Hopkins, I. Staude, E. V. Melik-Gaykazyan, A. A. Ezhov, A. E. Miroshnichenko, I. Brener, A. A. Fedyanin, and Y. S. Kivshar, “Nonlinear Interference and Tailorable Third-Harmonic Generation from Dielectric Oligomers,” ACS Photonics 2(5), 578–582 (2015).
[Crossref]

M. R. Shcherbakov, D. N. Neshev, B. Hopkins, A. S. Shorokhov, I. Staude, E. V. Melik-Gaykazyan, M. Decker, A. A. Ezhov, A. E. Miroshnichenko, I. Brener, A. A. Fedyanin, and Y. S. Kivshar, “Enhanced Third-Harmonic Generation in Silicon Nanoparticles Driven by Magnetic Response,” Nano Lett. 14(11), 6488–6492 (2014).
[Crossref]

Geohegan, D.

Y. Yang, W. Wang, A. Boulesbaa, I. I. Kravchenko, D. P. Briggs, A. Puretzky, D. Geohegan, and J. Valentine, “Nonlinear Fano-Resonant Dielectric Metasurfaces,” Nano Lett. 15(11), 7388–7393 (2015).
[Crossref]

Grinblat, G.

G. Grinblat, Y. Li, M. P. Nielsen, R. F. Oulton, and S. A. Maier, “Degenerate Four-Wave Mixing in a Multiresonant Germanium Nanodisk,” ACS Photonics 4(9), 2144–2149 (2017).
[Crossref]

Halas, N. J.

Y. Zhang, F. Wen, Y.-R. Zhen, P. Nordlander, and N. J. Halas, “Coherent Fano resonances in a plasmonic nanocluster enhance optical four-wave mixing,” Proc. Natl. Acad. Sci. U. S. A. 110(23), 9215–9219 (2013).
[Crossref]

Hopkins, B.

M. R. Shcherbakov, A. S. Shorokhov, D. N. Neshev, B. Hopkins, I. Staude, E. V. Melik-Gaykazyan, A. A. Ezhov, A. E. Miroshnichenko, I. Brener, A. A. Fedyanin, and Y. S. Kivshar, “Nonlinear Interference and Tailorable Third-Harmonic Generation from Dielectric Oligomers,” ACS Photonics 2(5), 578–582 (2015).
[Crossref]

M. R. Shcherbakov, D. N. Neshev, B. Hopkins, A. S. Shorokhov, I. Staude, E. V. Melik-Gaykazyan, M. Decker, A. A. Ezhov, A. E. Miroshnichenko, I. Brener, A. A. Fedyanin, and Y. S. Kivshar, “Enhanced Third-Harmonic Generation in Silicon Nanoparticles Driven by Magnetic Response,” Nano Lett. 14(11), 6488–6492 (2014).
[Crossref]

Huang, M.-Z.

W. Y. Ching and M.-Z. Huang, “Calculation of optical excitation in cubic semiconductors. III. Third harmonic generation,” Phys. Rev. B 47(15), 9479–9491 (1993).
[Crossref]

Jalali, B.

Jha, K. K.

Jonin, C.

J. Butet, I. Russier-Antoine, C. Jonin, N. Lascoux, E. Benichou, and P.-F. Brevet, “Sensing with Multipolar Second Harmonic Generation from Spherical Metallic Nanoparticles,” Nano Lett. 12(3), 1697–1701 (2012).
[Crossref]

Kivshar, Y. S.

A. I. Kuznetsov, A. E. Miroshnichenko, M. L. Brongersma, Y. S. Kivshar, and B. Lukyanchuk, “Optically resonant dielectric nanostructures,” Science 354(6314), aag2472 (2016).
[Crossref]

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M. R. Shcherbakov, D. N. Neshev, B. Hopkins, A. S. Shorokhov, I. Staude, E. V. Melik-Gaykazyan, M. Decker, A. A. Ezhov, A. E. Miroshnichenko, I. Brener, A. A. Fedyanin, and Y. S. Kivshar, “Enhanced Third-Harmonic Generation in Silicon Nanoparticles Driven by Magnetic Response,” Nano Lett. 14(11), 6488–6492 (2014).
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Tsypkin, A. N.

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S. V. Makarov, A. N. Tsypkin, T. A. Voytova, V. A. Milichko, I. S. Mukhin, A. V. Yulin, S. E. Putilin, M. A. Baranov, A. E. Krasnok, I. A. Morozov, and P. A. Belov, “Self-adjusted all-dielectric metasurfaces for deep ultraviolet femtosecond pulse generation,” Nanoscale 8(41), 17809–17814 (2016).
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ACS Nano (1)

J. Butet, P.-F. Brevet, and O. J. F. Martin, “Optical Second Harmonic Generation in Plasmonic Nanostructures: From Fundamental Principles to Advanced Applications,” ACS Nano 9(11), 10545–10562 (2015).
[Crossref]

ACS Photonics (3)

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Lumerical FDTD: https://www.lumerical.com/products/fdtd/

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Figures (8)

Fig. 1.
Fig. 1. (a) Top-view and (b) side-view of the ZCG structures with the Cartesian axis shown. (c) Transmission contour as a function of the incident wavelength and varying etch depth, H-h for the ZCG. (d) The line profile of the transmission spectrum shown for three different etch depths and wavelength ranges of 1 to 1.1 µm and 1.4 to 1.8 µm.
Fig. 2.
Fig. 2. Cross sectional field profiles at: (a) pump field – 1040 nm, (b) signal resonance field – 1580 nm and (c) nonlinear polarization at 775 nm. X, Y, Z components and total absolute value are shown in the figures with the Z-X cross-sectional profiled. The dimension of the ZCG used for these simulations are: Λ = 1.2 µm, D = 660 nm, H = 220 nm and h = 80 nm. The multiplication factor used to get the magnitude of the fields to the same range, as indicated by the colorbar below the respective columns is shown in the figures.
Fig. 3.
Fig. 3. The transmission contour shown as a function of the signal wavelength (x-axis) and angle-of-incidence (y-axis) with respect to the normal. The dimension of the ZCG used for these simulations are: Λ = 1200 nm, D = 660 nm, H = 220 nm and h = 80 nm.
Fig. 4.
Fig. 4. (a) SEM image, (b) AFM image of the fabricated ZCG structures with field of view of 7 × 5 µm and 10 × 10 µm respectively. (c) Transmission spectrum obtained for the fabricated ZCG structure. Blue curve (left axis) corresponds to experiments and the black curve (right axis) corresponds to simulations. The fabricated device dimensions are: Λ = 1200 nm, D = 660 nm and etch depth = 140 nm.
Fig. 5.
Fig. 5. The schematic of the experimental setup used for linear transmission and nonlinear four-wave mixing experiments.
Fig. 6.
Fig. 6. (a) Power dependence of the FWM signal (measured as PMT voltage) as a function of the input signal and pump power level. (b) The dependence of FWM signal (shown as PMT voltage) as a function of the input signal polarization rotated in the XY plane. The experimental data is shown by blue circles and simulated fit to the data is the maximum nonlinear polarization intensity – normalized to the peak experimental data. The black and red simulated fit corresponds to two different nonlinear susceptibility ratios, r = 0.75 and 0.5 respectively.
Fig. 7.
Fig. 7. FWM microscopy images as a function of time delay, Δt between the input pump and signal laser pulses. The image field-of-view is of size 100 × 100 µm.
Fig. 8.
Fig. 8. (a) Four-wave mixing microscopy image (shown as PMT voltage) at 1575 nm signal wavelength with large field-of-view of 300 × 300 µm. Multi-spectral FWM microscopy images acquired at input signal wavelengths of: (b) 1550 nm, (c) 1560 nm, (d) 1575 nm, (e) 1590 nm, (f) 1600 nm and (g) 1620 nm. Field-of-view for the multispectral imaging is 100 × 100 µm. (h) Comparison of average FWM signals both on- and off- gratings as a function of the input signal wavelengths used in multispectral studies. Maximum resonant enhancement of 450 is observed.

Equations (3)

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P F W M x ( 3 ) = ε o ( χ x x x x ( 3 ) E p x 2 E s x + 2 χ x x y y ( 3 ) E p x E p y E s y + 2 χ x x y y ( 3 ) E p x E p z E s z + χ x y y x ( 3 ) E p y 2 E s x + χ x y y x ( 3 ) E p z 2 E s x )
P F W M y ( 3 ) = ε o ( χ x x x x ( 3 ) E p y 2 E s y + 2 χ x x y y ( 3 ) E p y E p x E s x + 2 χ x x y y ( 3 ) E p y E p z E s z + χ x y y x ( 3 ) E p x 2 E s y + χ x y y x ( 3 ) E p z 2 E s y )
P F W M z ( 3 ) = ε o ( χ x x x x ( 3 ) E p z 2 E s z + 2 χ x x y y ( 3 ) E p z E p y E s y + 2 χ x x y y ( 3 ) E p z E p x E s x + χ x y y x ( 3 ) E p y 2 E s z + χ x y y x ( 3 ) E p x 2 E s z )

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