Abstract

A non-homogeneous composite guided-mode resonant (GMR) filter structure is proposed that avoids the multi-mode resonance effect and increases resonant wavelength tuning range. The composite filter structure is engineered using a combination of a varied-line-spacing (VLS) grating layer with a wedge-shaped waveguide layer. The grating is fabricated by holographic interference lithography (IL), while the wedge-shaped layer is fabricated using masked ion beam etching (MIBE) technology. The resonant wavelength has been observed to vary as a function of the spatial position on the structure. In the fabricated structure, over a length of 30 mm, the grating period increment is measured to be 149.2 nm, whereas the increment of the waveguide film thickness is approximately 100 nm. Experimental results show that a primary reflectance peak is achieved spanning a wavelength range of 805.8-1119.0 nm. The device is designed using the rigorous coupled-wave analysis (RCWA) method, and the proposed device is toward the practical application of GMR filters.

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

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References

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2017 (1)

T. Sang, Y. Wang, J. Li, J. Zhou, W. Jiang, J. Wang, and G. Chen, “Bandwidth tunable guided-mode resonance filter using contact coupled gratings at oblique incidence,” Opt. Commun. 382, 138–143 (2017).
[Crossref]

2016 (4)

2014 (1)

W. Chen, K. D. Long, H. Yu, Y. Tan, J. S. Choi, B. A. Harley, and B. T. Cunningham, “Enhanced live cell imaging via photonic crystal enhanced fluorescence microscopy,” Analyst (Lond.) 139(22), 5954–5963 (2014).
[Crossref] [PubMed]

2013 (1)

M. J. Uddin and R. Magnusson, “Guided-mode resonant thermo-optic tunable filters,” IEEE Photonics Technol. Lett. 25(15), 1412–1415 (2013).
[Crossref]

2012 (4)

M. J. Uddin and R. Magnusson, “Efficient guided-mode-resonant tunable color filters,” IEEE Photonics Technol. Lett. 24(17), 1552–1554 (2012).
[Crossref]

S. Foland, B. Swedlove, H. Nguyen, and J. B. Lee, “One-dimensional nanograting-based guided-mode resonance pressure sensor,” J. Microelectromech. Syst. 21(5), 1117–1123 (2012).
[Crossref]

H. A. Lin and C. S. Huang, “Linear variable filter based on a gradient grating period guided-mode resonance filter,” IEEE Photonics Technol. Lett. 28(9), 1042–1045 (2012).

R. Magnusson, “Spectrally dense comb-like filters fashioned with thick guided-mode resonant gratings,” Opt. Lett. 37(18), 3792–3794 (2012).
[Crossref] [PubMed]

2008 (2)

2007 (2)

A. S. P. Chang, K. J. Morton, H. Tan, P. F. Murphy, W. Wu, and S. Y. Chou, “Tunable liquid crystal-resonant grating filter fabricated by nano-imprint lithography,” IEEE Photonics Technol. Lett. 19(19), 1457–1459 (2007).
[Crossref]

F. Yang, G. Yen, and B. T. Cunningham, “Voltage-tuned resonant reflectance optical filter for visible wavelengths fabricated by nanoreplica molding,” Appl. Phys. Lett. 90(26), 261109 (2007).
[Crossref]

2006 (3)

P. Kiesel, O. Schmidt, S. Mohta, N. Johnson, and S. Malzer, “Compact, low-cost, and high-resolution interrogation unit for optical sensors,” Appl. Phys. Lett. 89(20), 201113 (2006).
[Crossref]

D. W. Dobbs and B. T. Cunningham, “Optically tunable guided-mode resonance filter,” Appl. Opt. 45(28), 7286–7293 (2006).
[Crossref] [PubMed]

D. W. Dobbs, I. Gershkovich, and B. T. Cunningham, “Fabrication of a graded-wavelength guided-mode resonance filter photonic crystal,” Appl. Phys. Lett. 89(12), 123113 (2006).
[Crossref]

2002 (1)

B. T. Cunningham, P. Li, B. Lin, and J. Pepper, “Colorimetric resonant reflection as a direct biochemical assay technique,” Sens. Actuators B Chem. 81(2), 316–328 (2002).
[Crossref]

2001 (1)

1998 (1)

1997 (1)

1993 (1)

Asquini, R.

Avrutsky, I.

Beccherelli, R.

Caputo, R.

Chang, A. S. P.

A. S. P. Chang, K. J. Morton, H. Tan, P. F. Murphy, W. Wu, and S. Y. Chou, “Tunable liquid crystal-resonant grating filter fabricated by nano-imprint lithography,” IEEE Photonics Technol. Lett. 19(19), 1457–1459 (2007).
[Crossref]

Chen, G.

T. Sang, Y. Wang, J. Li, J. Zhou, W. Jiang, J. Wang, and G. Chen, “Bandwidth tunable guided-mode resonance filter using contact coupled gratings at oblique incidence,” Opt. Commun. 382, 138–143 (2017).
[Crossref]

Chen, W.

W. Chen, K. D. Long, H. Yu, Y. Tan, J. S. Choi, B. A. Harley, and B. T. Cunningham, “Enhanced live cell imaging via photonic crystal enhanced fluorescence microscopy,” Analyst (Lond.) 139(22), 5954–5963 (2014).
[Crossref] [PubMed]

Choi, J. S.

W. Chen, K. D. Long, H. Yu, Y. Tan, J. S. Choi, B. A. Harley, and B. T. Cunningham, “Enhanced live cell imaging via photonic crystal enhanced fluorescence microscopy,” Analyst (Lond.) 139(22), 5954–5963 (2014).
[Crossref] [PubMed]

Chou, S. Y.

A. S. P. Chang, K. J. Morton, H. Tan, P. F. Murphy, W. Wu, and S. Y. Chou, “Tunable liquid crystal-resonant grating filter fabricated by nano-imprint lithography,” IEEE Photonics Technol. Lett. 19(19), 1457–1459 (2007).
[Crossref]

Cunningham, B. T.

W. Chen, K. D. Long, H. Yu, Y. Tan, J. S. Choi, B. A. Harley, and B. T. Cunningham, “Enhanced live cell imaging via photonic crystal enhanced fluorescence microscopy,” Analyst (Lond.) 139(22), 5954–5963 (2014).
[Crossref] [PubMed]

F. Yang, G. Yen, G. Rasigade, J. A. N. T. Soares, and B. T. Cunningham, “Optically tuned resonant optical reflectance filter,” Appl. Phys. Lett. 92(9), 091115 (2008).
[Crossref]

F. Yang, G. Yen, and B. T. Cunningham, “Voltage-tuned resonant reflectance optical filter for visible wavelengths fabricated by nanoreplica molding,” Appl. Phys. Lett. 90(26), 261109 (2007).
[Crossref]

D. W. Dobbs and B. T. Cunningham, “Optically tunable guided-mode resonance filter,” Appl. Opt. 45(28), 7286–7293 (2006).
[Crossref] [PubMed]

D. W. Dobbs, I. Gershkovich, and B. T. Cunningham, “Fabrication of a graded-wavelength guided-mode resonance filter photonic crystal,” Appl. Phys. Lett. 89(12), 123113 (2006).
[Crossref]

B. T. Cunningham, P. Li, B. Lin, and J. Pepper, “Colorimetric resonant reflection as a direct biochemical assay technique,” Sens. Actuators B Chem. 81(2), 316–328 (2002).
[Crossref]

d’Alessandro, A.

Dai, B.

De Sio, L.

Dobbs, D. W.

D. W. Dobbs and B. T. Cunningham, “Optically tunable guided-mode resonance filter,” Appl. Opt. 45(28), 7286–7293 (2006).
[Crossref] [PubMed]

D. W. Dobbs, I. Gershkovich, and B. T. Cunningham, “Fabrication of a graded-wavelength guided-mode resonance filter photonic crystal,” Appl. Phys. Lett. 89(12), 123113 (2006).
[Crossref]

Donisi, D.

Fang, C.

Foland, S.

S. Foland, B. Swedlove, H. Nguyen, and J. B. Lee, “One-dimensional nanograting-based guided-mode resonance pressure sensor,” J. Microelectromech. Syst. 21(5), 1117–1123 (2012).
[Crossref]

Gershkovich, I.

D. W. Dobbs, I. Gershkovich, and B. T. Cunningham, “Fabrication of a graded-wavelength guided-mode resonance filter photonic crystal,” Appl. Phys. Lett. 89(12), 123113 (2006).
[Crossref]

Harley, B. A.

W. Chen, K. D. Long, H. Yu, Y. Tan, J. S. Choi, B. A. Harley, and B. T. Cunningham, “Enhanced live cell imaging via photonic crystal enhanced fluorescence microscopy,” Analyst (Lond.) 139(22), 5954–5963 (2014).
[Crossref] [PubMed]

Hillier, A. C.

L. Liu, H. A. Khan, J. Li, A. C. Hillier, and M. Lu, “A strain-tunable nanoimprint lithography for linear variable photonic crystal filters,” Nanotechnology 27(29), 295301 (2016).
[Crossref] [PubMed]

Hong, R.

Huang, C. S.

H. A. Lin and C. S. Huang, “Linear variable filter based on a gradient grating period guided-mode resonance filter,” IEEE Photonics Technol. Lett. 28(9), 1042–1045 (2012).

Jiang, W.

T. Sang, Y. Wang, J. Li, J. Zhou, W. Jiang, J. Wang, and G. Chen, “Bandwidth tunable guided-mode resonance filter using contact coupled gratings at oblique incidence,” Opt. Commun. 382, 138–143 (2017).
[Crossref]

Johnson, N.

P. Kiesel, O. Schmidt, S. Mohta, N. Johnson, and S. Malzer, “Compact, low-cost, and high-resolution interrogation unit for optical sensors,” Appl. Phys. Lett. 89(20), 201113 (2006).
[Crossref]

Khan, H. A.

L. Liu, H. A. Khan, J. Li, A. C. Hillier, and M. Lu, “A strain-tunable nanoimprint lithography for linear variable photonic crystal filters,” Nanotechnology 27(29), 295301 (2016).
[Crossref] [PubMed]

Kiesel, P.

P. Kiesel, O. Schmidt, S. Mohta, N. Johnson, and S. Malzer, “Compact, low-cost, and high-resolution interrogation unit for optical sensors,” Appl. Phys. Lett. 89(20), 201113 (2006).
[Crossref]

Koike, M.

Lee, J. B.

S. Foland, B. Swedlove, H. Nguyen, and J. B. Lee, “One-dimensional nanograting-based guided-mode resonance pressure sensor,” J. Microelectromech. Syst. 21(5), 1117–1123 (2012).
[Crossref]

Li, J.

T. Sang, Y. Wang, J. Li, J. Zhou, W. Jiang, J. Wang, and G. Chen, “Bandwidth tunable guided-mode resonance filter using contact coupled gratings at oblique incidence,” Opt. Commun. 382, 138–143 (2017).
[Crossref]

L. Liu, H. A. Khan, J. Li, A. C. Hillier, and M. Lu, “A strain-tunable nanoimprint lithography for linear variable photonic crystal filters,” Nanotechnology 27(29), 295301 (2016).
[Crossref] [PubMed]

Li, P.

B. T. Cunningham, P. Li, B. Lin, and J. Pepper, “Colorimetric resonant reflection as a direct biochemical assay technique,” Sens. Actuators B Chem. 81(2), 316–328 (2002).
[Crossref]

Li, Z.

Lin, B.

B. T. Cunningham, P. Li, B. Lin, and J. Pepper, “Colorimetric resonant reflection as a direct biochemical assay technique,” Sens. Actuators B Chem. 81(2), 316–328 (2002).
[Crossref]

Lin, H. A.

H. A. Lin and C. S. Huang, “Linear variable filter based on a gradient grating period guided-mode resonance filter,” IEEE Photonics Technol. Lett. 28(9), 1042–1045 (2012).

Liu, L.

L. Liu, H. A. Khan, J. Li, A. C. Hillier, and M. Lu, “A strain-tunable nanoimprint lithography for linear variable photonic crystal filters,” Nanotechnology 27(29), 295301 (2016).
[Crossref] [PubMed]

Liu, Y.

Liu, Z. S.

Long, K. D.

W. Chen, K. D. Long, H. Yu, Y. Tan, J. S. Choi, B. A. Harley, and B. T. Cunningham, “Enhanced live cell imaging via photonic crystal enhanced fluorescence microscopy,” Analyst (Lond.) 139(22), 5954–5963 (2014).
[Crossref] [PubMed]

Lu, M.

L. Liu, H. A. Khan, J. Li, A. C. Hillier, and M. Lu, “A strain-tunable nanoimprint lithography for linear variable photonic crystal filters,” Nanotechnology 27(29), 295301 (2016).
[Crossref] [PubMed]

Magnusson, R.

Malzer, S.

P. Kiesel, O. Schmidt, S. Mohta, N. Johnson, and S. Malzer, “Compact, low-cost, and high-resolution interrogation unit for optical sensors,” Appl. Phys. Lett. 89(20), 201113 (2006).
[Crossref]

Mohta, S.

P. Kiesel, O. Schmidt, S. Mohta, N. Johnson, and S. Malzer, “Compact, low-cost, and high-resolution interrogation unit for optical sensors,” Appl. Phys. Lett. 89(20), 201113 (2006).
[Crossref]

Morton, K. J.

A. S. P. Chang, K. J. Morton, H. Tan, P. F. Murphy, W. Wu, and S. Y. Chou, “Tunable liquid crystal-resonant grating filter fabricated by nano-imprint lithography,” IEEE Photonics Technol. Lett. 19(19), 1457–1459 (2007).
[Crossref]

Murphy, P. F.

A. S. P. Chang, K. J. Morton, H. Tan, P. F. Murphy, W. Wu, and S. Y. Chou, “Tunable liquid crystal-resonant grating filter fabricated by nano-imprint lithography,” IEEE Photonics Technol. Lett. 19(19), 1457–1459 (2007).
[Crossref]

Namioka, T.

Nguyen, H.

S. Foland, B. Swedlove, H. Nguyen, and J. B. Lee, “One-dimensional nanograting-based guided-mode resonance pressure sensor,” J. Microelectromech. Syst. 21(5), 1117–1123 (2012).
[Crossref]

Pepper, J.

B. T. Cunningham, P. Li, B. Lin, and J. Pepper, “Colorimetric resonant reflection as a direct biochemical assay technique,” Sens. Actuators B Chem. 81(2), 316–328 (2002).
[Crossref]

Qian, L.

Rabady, R.

Rasigade, G.

F. Yang, G. Yen, G. Rasigade, J. A. N. T. Soares, and B. T. Cunningham, “Optically tuned resonant optical reflectance filter,” Appl. Phys. Lett. 92(9), 091115 (2008).
[Crossref]

Sang, T.

T. Sang, Y. Wang, J. Li, J. Zhou, W. Jiang, J. Wang, and G. Chen, “Bandwidth tunable guided-mode resonance filter using contact coupled gratings at oblique incidence,” Opt. Commun. 382, 138–143 (2017).
[Crossref]

Schmidt, O.

P. Kiesel, O. Schmidt, S. Mohta, N. Johnson, and S. Malzer, “Compact, low-cost, and high-resolution interrogation unit for optical sensors,” Appl. Phys. Lett. 89(20), 201113 (2006).
[Crossref]

Sheng, B.

Shin, D.

Soares, J. A. N. T.

F. Yang, G. Yen, G. Rasigade, J. A. N. T. Soares, and B. T. Cunningham, “Optically tuned resonant optical reflectance filter,” Appl. Phys. Lett. 92(9), 091115 (2008).
[Crossref]

Swedlove, B.

S. Foland, B. Swedlove, H. Nguyen, and J. B. Lee, “One-dimensional nanograting-based guided-mode resonance pressure sensor,” J. Microelectromech. Syst. 21(5), 1117–1123 (2012).
[Crossref]

Tan, H.

A. S. P. Chang, K. J. Morton, H. Tan, P. F. Murphy, W. Wu, and S. Y. Chou, “Tunable liquid crystal-resonant grating filter fabricated by nano-imprint lithography,” IEEE Photonics Technol. Lett. 19(19), 1457–1459 (2007).
[Crossref]

Tan, Y.

W. Chen, K. D. Long, H. Yu, Y. Tan, J. S. Choi, B. A. Harley, and B. T. Cunningham, “Enhanced live cell imaging via photonic crystal enhanced fluorescence microscopy,” Analyst (Lond.) 139(22), 5954–5963 (2014).
[Crossref] [PubMed]

Tao, C.

Tibuleac, S.

Uddin, M. J.

M. J. Uddin and R. Magnusson, “Guided-mode resonant thermo-optic tunable filters,” IEEE Photonics Technol. Lett. 25(15), 1412–1415 (2013).
[Crossref]

M. J. Uddin and R. Magnusson, “Efficient guided-mode-resonant tunable color filters,” IEEE Photonics Technol. Lett. 24(17), 1552–1554 (2012).
[Crossref]

Umeton, C.

Wang, J.

T. Sang, Y. Wang, J. Li, J. Zhou, W. Jiang, J. Wang, and G. Chen, “Bandwidth tunable guided-mode resonance filter using contact coupled gratings at oblique incidence,” Opt. Commun. 382, 138–143 (2017).
[Crossref]

Wang, Q.

Wang, S. S.

Wang, Y.

T. Sang, Y. Wang, J. Li, J. Zhou, W. Jiang, J. Wang, and G. Chen, “Bandwidth tunable guided-mode resonance filter using contact coupled gratings at oblique incidence,” Opt. Commun. 382, 138–143 (2017).
[Crossref]

Wu, W.

A. S. P. Chang, K. J. Morton, H. Tan, P. F. Murphy, W. Wu, and S. Y. Chou, “Tunable liquid crystal-resonant grating filter fabricated by nano-imprint lithography,” IEEE Photonics Technol. Lett. 19(19), 1457–1459 (2007).
[Crossref]

Xu, L.

Yang, F.

F. Yang, G. Yen, G. Rasigade, J. A. N. T. Soares, and B. T. Cunningham, “Optically tuned resonant optical reflectance filter,” Appl. Phys. Lett. 92(9), 091115 (2008).
[Crossref]

F. Yang, G. Yen, and B. T. Cunningham, “Voltage-tuned resonant reflectance optical filter for visible wavelengths fabricated by nanoreplica molding,” Appl. Phys. Lett. 90(26), 261109 (2007).
[Crossref]

Yen, G.

F. Yang, G. Yen, G. Rasigade, J. A. N. T. Soares, and B. T. Cunningham, “Optically tuned resonant optical reflectance filter,” Appl. Phys. Lett. 92(9), 091115 (2008).
[Crossref]

F. Yang, G. Yen, and B. T. Cunningham, “Voltage-tuned resonant reflectance optical filter for visible wavelengths fabricated by nanoreplica molding,” Appl. Phys. Lett. 90(26), 261109 (2007).
[Crossref]

Young, P. P.

Yu, H.

W. Chen, K. D. Long, H. Yu, Y. Tan, J. S. Choi, B. A. Harley, and B. T. Cunningham, “Enhanced live cell imaging via photonic crystal enhanced fluorescence microscopy,” Analyst (Lond.) 139(22), 5954–5963 (2014).
[Crossref] [PubMed]

Zahid, A.

Zhang, D.

Zhang, H.

Zheng, G.

Zhou, J.

T. Sang, Y. Wang, J. Li, J. Zhou, W. Jiang, J. Wang, and G. Chen, “Bandwidth tunable guided-mode resonance filter using contact coupled gratings at oblique incidence,” Opt. Commun. 382, 138–143 (2017).
[Crossref]

Zhuang, S.

Analyst (Lond.) (1)

W. Chen, K. D. Long, H. Yu, Y. Tan, J. S. Choi, B. A. Harley, and B. T. Cunningham, “Enhanced live cell imaging via photonic crystal enhanced fluorescence microscopy,” Analyst (Lond.) 139(22), 5954–5963 (2014).
[Crossref] [PubMed]

Appl. Opt. (3)

Appl. Phys. Lett. (4)

P. Kiesel, O. Schmidt, S. Mohta, N. Johnson, and S. Malzer, “Compact, low-cost, and high-resolution interrogation unit for optical sensors,” Appl. Phys. Lett. 89(20), 201113 (2006).
[Crossref]

D. W. Dobbs, I. Gershkovich, and B. T. Cunningham, “Fabrication of a graded-wavelength guided-mode resonance filter photonic crystal,” Appl. Phys. Lett. 89(12), 123113 (2006).
[Crossref]

F. Yang, G. Yen, and B. T. Cunningham, “Voltage-tuned resonant reflectance optical filter for visible wavelengths fabricated by nanoreplica molding,” Appl. Phys. Lett. 90(26), 261109 (2007).
[Crossref]

F. Yang, G. Yen, G. Rasigade, J. A. N. T. Soares, and B. T. Cunningham, “Optically tuned resonant optical reflectance filter,” Appl. Phys. Lett. 92(9), 091115 (2008).
[Crossref]

IEEE Photonics Technol. Lett. (4)

H. A. Lin and C. S. Huang, “Linear variable filter based on a gradient grating period guided-mode resonance filter,” IEEE Photonics Technol. Lett. 28(9), 1042–1045 (2012).

M. J. Uddin and R. Magnusson, “Guided-mode resonant thermo-optic tunable filters,” IEEE Photonics Technol. Lett. 25(15), 1412–1415 (2013).
[Crossref]

M. J. Uddin and R. Magnusson, “Efficient guided-mode-resonant tunable color filters,” IEEE Photonics Technol. Lett. 24(17), 1552–1554 (2012).
[Crossref]

A. S. P. Chang, K. J. Morton, H. Tan, P. F. Murphy, W. Wu, and S. Y. Chou, “Tunable liquid crystal-resonant grating filter fabricated by nano-imprint lithography,” IEEE Photonics Technol. Lett. 19(19), 1457–1459 (2007).
[Crossref]

J. Microelectromech. Syst. (1)

S. Foland, B. Swedlove, H. Nguyen, and J. B. Lee, “One-dimensional nanograting-based guided-mode resonance pressure sensor,” J. Microelectromech. Syst. 21(5), 1117–1123 (2012).
[Crossref]

Nanotechnology (1)

L. Liu, H. A. Khan, J. Li, A. C. Hillier, and M. Lu, “A strain-tunable nanoimprint lithography for linear variable photonic crystal filters,” Nanotechnology 27(29), 295301 (2016).
[Crossref] [PubMed]

Opt. Commun. (1)

T. Sang, Y. Wang, J. Li, J. Zhou, W. Jiang, J. Wang, and G. Chen, “Bandwidth tunable guided-mode resonance filter using contact coupled gratings at oblique incidence,” Opt. Commun. 382, 138–143 (2017).
[Crossref]

Opt. Express (1)

Opt. Lett. (6)

Sens. Actuators B Chem. (1)

B. T. Cunningham, P. Li, B. Lin, and J. Pepper, “Colorimetric resonant reflection as a direct biochemical assay technique,” Sens. Actuators B Chem. 81(2), 316–328 (2002).
[Crossref]

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Figures (7)

Fig. 1
Fig. 1 (a) Configuration and parameters of the composite GMR structure. (b) Schematic of the device fabrication process.
Fig. 2
Fig. 2 Fabrication mechanism of (a) wedge-shape film and (b) VLS grating.
Fig. 3
Fig. 3 (a) Calculated VLS grating period (blue line) and wedge-shape film thickness (red line) corresponding to spatial position, xi, as it varies from −15 mm to 15 mm. (b) Simulated GMR spectra under different grating periods. (c) Simulated reflect spectra for the proposed GMR filter at 7 locations spanning 30 mm with increments of 5 mm. (d) Simulated GMR wavelength for three kind structures.
Fig. 4
Fig. 4 (a) Calculated wedge-shape waveguide layer thickness difference and corresponding resonant wavelength shift, as functions of the angle φ. (b) Calculated grating period increment and corresponding resonant wavelength shift, as functions of the distances D.
Fig. 5
Fig. 5 Photos showing (a) the fabricated wedge-shaped TiO2 film and (b) final device with non-homogeneous composite structure.
Fig. 6
Fig. 6 (a) Top SEM and AFM views for the VLS grating at two positions of −15 mm and 15 mm. The grating period ang film thickness are ΛL = 609.5 nm, dL = 230 nm and ΛR = 460.3 nm, dL = 129.4 nm respectively. (b) Measured grating periods as a function of lateral position. (c) Measured TiO2 film thicknesses as a function of lateral position.
Fig. 7
Fig. 7 (a) Measured reflectance spectra for 7 different positions. (b) The relationship between resonant wavelength and the illumination positions. Black dots show the experimental measured results. Solid line shows the curve fitting.

Tables (1)

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Table 1 Reflection spectra information at different positions

Equations (2)

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Δ d = 2 N v a h s tan ( φ / 2 ) w a ,
Λ = λ ( n 1 ) x i D ( n 1 ) R + 2 sin ( θ 2 2 ) ,

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