Abstract

This paper puts forward a subwavelength grating for highly sensitive refractive index (RI) sensing. The light-coupling condition of the grating covered by the liquid to be detected is sensitive to changes in RI of the liquid. The influence of the grating period and thickness on the coupling is studied. At the large angle of incidence, it is found that the effective RI of the grating slab is varied with the incidence angle, from which the coupling originating from the guided-mode resonance (GMR) impacted by such variation is revealed. Incidence angle is scanned at a fixed wavelength of 623.8 nm, and the calculated results indicate that the sensor sensitivity is mainly dependent on the period, while the resolution is controlled by the thickness. After the period and thickness have been optimized in sequence, an averaged sensitivity of 249.7°/RIU and resolution ranging from 0.1° to 0.3° are obtained; those of the grating sensor are better than those of the surface plasmon resonance (SPR) sensor, which has high sensitivity supported by a substrate with low RI.

© 2020 Optical Society of America

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2018 (3)

H. Hafez, K. Y. Hwan, and M. Robert, “Fiber-facet-integrated guided-mode resonance filters and sensors: experimental realization,” Opt. Lett. 43, 358–361 (2018).
[Crossref]

W. Zhou, K. Li, Y. Wei, P. Hao, M. Chi, Y. Liu, and Y. Wu, “Ultrasensitive label-free optical microfiber coupler biosensor for detection of cardiac troponin I based on interference turning point effect,” Biosens. Bioelectron. 106, 99–104 (2018).
[Crossref]

D. Kita, J. Michon, S. G. Johnson, and J. Hu, “Are slot and sub-wavelength grating waveguides better than strip waveguides for sensing?” Optica 5, 1046–1054 (2018).
[Crossref]

2017 (1)

P. K. Sahoo, S. Sarkar, and J. Joseph, “High sensitivity guided-mode-resonance optical sensor employing phase detection,” Sci. Rep. 7, 7607 (2017).
[Crossref]

2016 (5)

2013 (1)

E. Sader and A. Sayyed-Ahmad, “Design of an optical water pollution sensor using a single-layer guided-mode resonance filter,” Photon. Sens. 3, 224–230 (2013).
[Crossref]

2011 (1)

2010 (1)

A. Szeghalmi, E. B. Kley, and M. Knez, “Theoretical and experimental analysis of the sensitivity of guided mode resonance sensors,” J. Phys. Chem. C 114, 21150–21157 (2010).
[Crossref]

2009 (1)

2005 (1)

S. Tian, Z. Wang, W. Li, Y. Wu, and L. Chen, “Resonant excitation analysis of sub-wavelength dielectric grating,” J. Opt. A 8, 62–66 (2005).
[Crossref]

2004 (1)

H. V. Hsieh, Z. A. Pfeiffer, T. J. Amiss, D. B. Sherman, and J. B. Pitner, “Direct detection of glucose by surface plasmon resonance with bacterial glucose/galactose-binding protein,” Biosens. Bioelectron. 19, 653–660 (2004).
[Crossref]

2003 (1)

J. Homola, "Present and future of surface plasmon resonance biosensors,” Anal. Bioanal. Chem. 377, 528–539 (2003).
[Crossref]

1999 (1)

R. Magnusson, D. Shin, Z. Liu, S. Tibuleac, and K. Alavi, “Guided-mode resonance effects in thin-film diffractive optics and their applications,” Proc. SPIE 3729, 212–221 (1999).
[Crossref]

1997 (1)

Abdulhalim, I.

M. Abutoama and I. Abdulhalim, “Self-referenced biosensor based on thin dielectric grating combined with thin metal film,” IEEE J. Sel. Top. Quantum Electron. 23, 72–80 (2016).
[Crossref]

Abutoama, M.

M. Abutoama and I. Abdulhalim, “Self-referenced biosensor based on thin dielectric grating combined with thin metal film,” IEEE J. Sel. Top. Quantum Electron. 23, 72–80 (2016).
[Crossref]

Alavi, K.

R. Magnusson, D. Shin, Z. Liu, S. Tibuleac, and K. Alavi, “Guided-mode resonance effects in thin-film diffractive optics and their applications,” Proc. SPIE 3729, 212–221 (1999).
[Crossref]

Amiss, T. J.

H. V. Hsieh, Z. A. Pfeiffer, T. J. Amiss, D. B. Sherman, and J. B. Pitner, “Direct detection of glucose by surface plasmon resonance with bacterial glucose/galactose-binding protein,” Biosens. Bioelectron. 19, 653–660 (2004).
[Crossref]

Chakravarty, S.

Chen, L.

S. Tian, Z. Wang, W. Li, Y. Wu, and L. Chen, “Resonant excitation analysis of sub-wavelength dielectric grating,” J. Opt. A 8, 62–66 (2005).
[Crossref]

Chen, R. T.

Chi, M.

W. Zhou, K. Li, Y. Wei, P. Hao, M. Chi, Y. Liu, and Y. Wu, “Ultrasensitive label-free optical microfiber coupler biosensor for detection of cardiac troponin I based on interference turning point effect,” Biosens. Bioelectron. 106, 99–104 (2018).
[Crossref]

W. Zhou, K. Li, Y. Wei, P. Hao, M. Chi, Y. Liu, Y. Wu, Y. Hai, L. Huang, X. Xu, S. Chakravarty, and R. T. Chen, “Unique surface sensing property and enhanced sensitivity in microring resonator biosensors based on subwavelength grating waveguides,” Opt. Express 24, 29724–29733 (2016).
[Crossref]

Fan, S.

Friesem, A. A.

García, J.

Glasberg, S.

Hafez, H.

Hai, Y.

Hao, P.

W. Zhou, K. Li, Y. Wei, P. Hao, M. Chi, Y. Liu, and Y. Wu, “Ultrasensitive label-free optical microfiber coupler biosensor for detection of cardiac troponin I based on interference turning point effect,” Biosens. Bioelectron. 106, 99–104 (2018).
[Crossref]

W. Zhou, K. Li, Y. Wei, P. Hao, M. Chi, Y. Liu, Y. Wu, Y. Hai, L. Huang, X. Xu, S. Chakravarty, and R. T. Chen, “Unique surface sensing property and enhanced sensitivity in microring resonator biosensors based on subwavelength grating waveguides,” Opt. Express 24, 29724–29733 (2016).
[Crossref]

Ho, H. P.

H. P. Ho, S. Y. Wu, S. K. Kong, S. Zeng, and K. T. Yong, “SPR biosensors,” in Handbook of Photonics for Biomedical Engineering (Springer, 2017).

Homola, J.

J. Homola, "Present and future of surface plasmon resonance biosensors,” Anal. Bioanal. Chem. 377, 528–539 (2003).
[Crossref]

Hsieh, H. V.

H. V. Hsieh, Z. A. Pfeiffer, T. J. Amiss, D. B. Sherman, and J. B. Pitner, “Direct detection of glucose by surface plasmon resonance with bacterial glucose/galactose-binding protein,” Biosens. Bioelectron. 19, 653–660 (2004).
[Crossref]

Hu, J.

Huang, L.

Hwan, K. Y.

Janz, S.

Johnson, S. G.

Joseph, J.

P. K. Sahoo, S. Sarkar, and J. Joseph, “High sensitivity guided-mode-resonance optical sensor employing phase detection,” Sci. Rep. 7, 7607 (2017).
[Crossref]

Kikuta, H.

Kita, D.

Kley, E. B.

A. Szeghalmi, E. B. Kley, and M. Knez, “Theoretical and experimental analysis of the sensitivity of guided mode resonance sensors,” J. Phys. Chem. C 114, 21150–21157 (2010).
[Crossref]

Knez, M.

A. Szeghalmi, E. B. Kley, and M. Knez, “Theoretical and experimental analysis of the sensitivity of guided mode resonance sensors,” J. Phys. Chem. C 114, 21150–21157 (2010).
[Crossref]

Kong, S. K.

H. P. Ho, S. Y. Wu, S. K. Kong, S. Zeng, and K. T. Yong, “SPR biosensors,” in Handbook of Photonics for Biomedical Engineering (Springer, 2017).

Kuo, W. K.

Li, K.

W. Zhou, K. Li, Y. Wei, P. Hao, M. Chi, Y. Liu, and Y. Wu, “Ultrasensitive label-free optical microfiber coupler biosensor for detection of cardiac troponin I based on interference turning point effect,” Biosens. Bioelectron. 106, 99–104 (2018).
[Crossref]

W. Zhou, K. Li, Y. Wei, P. Hao, M. Chi, Y. Liu, Y. Wu, Y. Hai, L. Huang, X. Xu, S. Chakravarty, and R. T. Chen, “Unique surface sensing property and enhanced sensitivity in microring resonator biosensors based on subwavelength grating waveguides,” Opt. Express 24, 29724–29733 (2016).
[Crossref]

Li, W.

S. Tian, Z. Wang, W. Li, Y. Wu, and L. Chen, “Resonant excitation analysis of sub-wavelength dielectric grating,” J. Opt. A 8, 62–66 (2005).
[Crossref]

Li, X. F.

X. F. Li, W. Peng, Y.-L. Zhao, Q. Wang, and J.-L. Wei, “A subwavelength metal-grating assisted sensor of Kretschmann style for investigating the sample with high refractive index,” Chin. Phys. B 25, 037303 (2016).
[Crossref]

Lin, P.-Z.

Liu, Y.

W. Zhou, K. Li, Y. Wei, P. Hao, M. Chi, Y. Liu, and Y. Wu, “Ultrasensitive label-free optical microfiber coupler biosensor for detection of cardiac troponin I based on interference turning point effect,” Biosens. Bioelectron. 106, 99–104 (2018).
[Crossref]

W. Zhou, K. Li, Y. Wei, P. Hao, M. Chi, Y. Liu, Y. Wu, Y. Hai, L. Huang, X. Xu, S. Chakravarty, and R. T. Chen, “Unique surface sensing property and enhanced sensitivity in microring resonator biosensors based on subwavelength grating waveguides,” Opt. Express 24, 29724–29733 (2016).
[Crossref]

Liu, Z.

R. Magnusson, D. Shin, Z. Liu, S. Tibuleac, and K. Alavi, “Guided-mode resonance effects in thin-film diffractive optics and their applications,” Proc. SPIE 3729, 212–221 (1999).
[Crossref]

Magnusson, R.

R. Magnusson, D. Shin, Z. Liu, S. Tibuleac, and K. Alavi, “Guided-mode resonance effects in thin-film diffractive optics and their applications,” Proc. SPIE 3729, 212–221 (1999).
[Crossref]

R. Magnusson, “Guided-mode resonance sensors employing angular, spectral, modal, and polarization diversity for high-precision sensing in compact formats,” U.S. patent10,274,432 (April30, 2019).

Mattelin, M. A.

M. A. Mattelin, G. Van Steenberge, and J. Missinne, “An imprinted polymer-based guided mode resonance grating sensor,” in Society of Photo-optical Instrumentation Engineers Conference Series (2018).

Michon, J.

Missinne, J.

M. A. Mattelin, G. Van Steenberge, and J. Missinne, “An imprinted polymer-based guided mode resonance grating sensor,” in Society of Photo-optical Instrumentation Engineers Conference Series (2018).

Mizutani, A.

Nataraj, C. T.

C. T. Nataraj, G. R. Prashanth, and S. Talabattula, “Guided mode resonance grating based optical bio-sensor with enhanced bulk sensitivity,” in Advanced Photonics 2016, OSA Technical Digest (Optical Society of America, 2016), paper SeW1F.4.

Peng, W.

X. F. Li, W. Peng, Y.-L. Zhao, Q. Wang, and J.-L. Wei, “A subwavelength metal-grating assisted sensor of Kretschmann style for investigating the sample with high refractive index,” Chin. Phys. B 25, 037303 (2016).
[Crossref]

Pfeiffer, Z. A.

H. V. Hsieh, Z. A. Pfeiffer, T. J. Amiss, D. B. Sherman, and J. B. Pitner, “Direct detection of glucose by surface plasmon resonance with bacterial glucose/galactose-binding protein,” Biosens. Bioelectron. 19, 653–660 (2004).
[Crossref]

Pitner, J. B.

H. V. Hsieh, Z. A. Pfeiffer, T. J. Amiss, D. B. Sherman, and J. B. Pitner, “Direct detection of glucose by surface plasmon resonance with bacterial glucose/galactose-binding protein,” Biosens. Bioelectron. 19, 653–660 (2004).
[Crossref]

Prashanth, G. R.

C. T. Nataraj, G. R. Prashanth, and S. Talabattula, “Guided mode resonance grating based optical bio-sensor with enhanced bulk sensitivity,” in Advanced Photonics 2016, OSA Technical Digest (Optical Society of America, 2016), paper SeW1F.4.

Robert, M.

Rosenblatt, D.

Sader, E.

E. Sader and A. Sayyed-Ahmad, “Design of an optical water pollution sensor using a single-layer guided-mode resonance filter,” Photon. Sens. 3, 224–230 (2013).
[Crossref]

Sahoo, P. K.

P. K. Sahoo, S. Sarkar, and J. Joseph, “High sensitivity guided-mode-resonance optical sensor employing phase detection,” Sci. Rep. 7, 7607 (2017).
[Crossref]

Sarkar, S.

P. K. Sahoo, S. Sarkar, and J. Joseph, “High sensitivity guided-mode-resonance optical sensor employing phase detection,” Sci. Rep. 7, 7607 (2017).
[Crossref]

Sayyed-Ahmad, A.

E. Sader and A. Sayyed-Ahmad, “Design of an optical water pollution sensor using a single-layer guided-mode resonance filter,” Photon. Sens. 3, 224–230 (2013).
[Crossref]

Schmid, J. H.

Sharon, A.

Sherman, D. B.

H. V. Hsieh, Z. A. Pfeiffer, T. J. Amiss, D. B. Sherman, and J. B. Pitner, “Direct detection of glucose by surface plasmon resonance with bacterial glucose/galactose-binding protein,” Biosens. Bioelectron. 19, 653–660 (2004).
[Crossref]

Shin, D.

R. Magnusson, D. Shin, Z. Liu, S. Tibuleac, and K. Alavi, “Guided-mode resonance effects in thin-film diffractive optics and their applications,” Proc. SPIE 3729, 212–221 (1999).
[Crossref]

Sinclair, W.

Syu, S.-H.

Szeghalmi, A.

A. Szeghalmi, E. B. Kley, and M. Knez, “Theoretical and experimental analysis of the sensitivity of guided mode resonance sensors,” J. Phys. Chem. C 114, 21150–21157 (2010).
[Crossref]

Talabattula, S.

C. T. Nataraj, G. R. Prashanth, and S. Talabattula, “Guided mode resonance grating based optical bio-sensor with enhanced bulk sensitivity,” in Advanced Photonics 2016, OSA Technical Digest (Optical Society of America, 2016), paper SeW1F.4.

Tian, S.

S. Tian, Z. Wang, W. Li, Y. Wu, and L. Chen, “Resonant excitation analysis of sub-wavelength dielectric grating,” J. Opt. A 8, 62–66 (2005).
[Crossref]

Tibuleac, S.

R. Magnusson, D. Shin, Z. Liu, S. Tibuleac, and K. Alavi, “Guided-mode resonance effects in thin-film diffractive optics and their applications,” Proc. SPIE 3729, 212–221 (1999).
[Crossref]

Urakawa, S.

Van Steenberge, G.

M. A. Mattelin, G. Van Steenberge, and J. Missinne, “An imprinted polymer-based guided mode resonance grating sensor,” in Society of Photo-optical Instrumentation Engineers Conference Series (2018).

Wang, Q.

X. F. Li, W. Peng, Y.-L. Zhao, Q. Wang, and J.-L. Wei, “A subwavelength metal-grating assisted sensor of Kretschmann style for investigating the sample with high refractive index,” Chin. Phys. B 25, 037303 (2016).
[Crossref]

Wang, Z.

S. Tian, Z. Wang, W. Li, Y. Wu, and L. Chen, “Resonant excitation analysis of sub-wavelength dielectric grating,” J. Opt. A 8, 62–66 (2005).
[Crossref]

Wei, J.-L.

X. F. Li, W. Peng, Y.-L. Zhao, Q. Wang, and J.-L. Wei, “A subwavelength metal-grating assisted sensor of Kretschmann style for investigating the sample with high refractive index,” Chin. Phys. B 25, 037303 (2016).
[Crossref]

Wei, Y.

W. Zhou, K. Li, Y. Wei, P. Hao, M. Chi, Y. Liu, and Y. Wu, “Ultrasensitive label-free optical microfiber coupler biosensor for detection of cardiac troponin I based on interference turning point effect,” Biosens. Bioelectron. 106, 99–104 (2018).
[Crossref]

W. Zhou, K. Li, Y. Wei, P. Hao, M. Chi, Y. Liu, Y. Wu, Y. Hai, L. Huang, X. Xu, S. Chakravarty, and R. T. Chen, “Unique surface sensing property and enhanced sensitivity in microring resonator biosensors based on subwavelength grating waveguides,” Opt. Express 24, 29724–29733 (2016).
[Crossref]

Wu, S. Y.

H. P. Ho, S. Y. Wu, S. K. Kong, S. Zeng, and K. T. Yong, “SPR biosensors,” in Handbook of Photonics for Biomedical Engineering (Springer, 2017).

Wu, Y.

W. Zhou, K. Li, Y. Wei, P. Hao, M. Chi, Y. Liu, and Y. Wu, “Ultrasensitive label-free optical microfiber coupler biosensor for detection of cardiac troponin I based on interference turning point effect,” Biosens. Bioelectron. 106, 99–104 (2018).
[Crossref]

W. Zhou, K. Li, Y. Wei, P. Hao, M. Chi, Y. Liu, Y. Wu, Y. Hai, L. Huang, X. Xu, S. Chakravarty, and R. T. Chen, “Unique surface sensing property and enhanced sensitivity in microring resonator biosensors based on subwavelength grating waveguides,” Opt. Express 24, 29724–29733 (2016).
[Crossref]

S. Tian, Z. Wang, W. Li, Y. Wu, and L. Chen, “Resonant excitation analysis of sub-wavelength dielectric grating,” J. Opt. A 8, 62–66 (2005).
[Crossref]

Xu, D.-X.

Xu, X.

Yong, K. T.

H. P. Ho, S. Y. Wu, S. K. Kong, S. Zeng, and K. T. Yong, “SPR biosensors,” in Handbook of Photonics for Biomedical Engineering (Springer, 2017).

Yu, H. H.

Yu, Z.

Zeng, S.

H. P. Ho, S. Y. Wu, S. K. Kong, S. Zeng, and K. T. Yong, “SPR biosensors,” in Handbook of Photonics for Biomedical Engineering (Springer, 2017).

Zhao, Y.-L.

X. F. Li, W. Peng, Y.-L. Zhao, Q. Wang, and J.-L. Wei, “A subwavelength metal-grating assisted sensor of Kretschmann style for investigating the sample with high refractive index,” Chin. Phys. B 25, 037303 (2016).
[Crossref]

Zhou, W.

W. Zhou, K. Li, Y. Wei, P. Hao, M. Chi, Y. Liu, and Y. Wu, “Ultrasensitive label-free optical microfiber coupler biosensor for detection of cardiac troponin I based on interference turning point effect,” Biosens. Bioelectron. 106, 99–104 (2018).
[Crossref]

W. Zhou, K. Li, Y. Wei, P. Hao, M. Chi, Y. Liu, Y. Wu, Y. Hai, L. Huang, X. Xu, S. Chakravarty, and R. T. Chen, “Unique surface sensing property and enhanced sensitivity in microring resonator biosensors based on subwavelength grating waveguides,” Opt. Express 24, 29724–29733 (2016).
[Crossref]

Anal. Bioanal. Chem. (1)

J. Homola, "Present and future of surface plasmon resonance biosensors,” Anal. Bioanal. Chem. 377, 528–539 (2003).
[Crossref]

Appl. Opt. (2)

Biosens. Bioelectron. (2)

W. Zhou, K. Li, Y. Wei, P. Hao, M. Chi, Y. Liu, and Y. Wu, “Ultrasensitive label-free optical microfiber coupler biosensor for detection of cardiac troponin I based on interference turning point effect,” Biosens. Bioelectron. 106, 99–104 (2018).
[Crossref]

H. V. Hsieh, Z. A. Pfeiffer, T. J. Amiss, D. B. Sherman, and J. B. Pitner, “Direct detection of glucose by surface plasmon resonance with bacterial glucose/galactose-binding protein,” Biosens. Bioelectron. 19, 653–660 (2004).
[Crossref]

Chin. Phys. B (1)

X. F. Li, W. Peng, Y.-L. Zhao, Q. Wang, and J.-L. Wei, “A subwavelength metal-grating assisted sensor of Kretschmann style for investigating the sample with high refractive index,” Chin. Phys. B 25, 037303 (2016).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

M. Abutoama and I. Abdulhalim, “Self-referenced biosensor based on thin dielectric grating combined with thin metal film,” IEEE J. Sel. Top. Quantum Electron. 23, 72–80 (2016).
[Crossref]

J. Opt. A (1)

S. Tian, Z. Wang, W. Li, Y. Wu, and L. Chen, “Resonant excitation analysis of sub-wavelength dielectric grating,” J. Opt. A 8, 62–66 (2005).
[Crossref]

J. Opt. Soc. Am. A (1)

J. Phys. Chem. C (1)

A. Szeghalmi, E. B. Kley, and M. Knez, “Theoretical and experimental analysis of the sensitivity of guided mode resonance sensors,” J. Phys. Chem. C 114, 21150–21157 (2010).
[Crossref]

Opt. Express (3)

Opt. Lett. (1)

Optica (1)

Photon. Sens. (1)

E. Sader and A. Sayyed-Ahmad, “Design of an optical water pollution sensor using a single-layer guided-mode resonance filter,” Photon. Sens. 3, 224–230 (2013).
[Crossref]

Proc. SPIE (1)

R. Magnusson, D. Shin, Z. Liu, S. Tibuleac, and K. Alavi, “Guided-mode resonance effects in thin-film diffractive optics and their applications,” Proc. SPIE 3729, 212–221 (1999).
[Crossref]

Sci. Rep. (1)

P. K. Sahoo, S. Sarkar, and J. Joseph, “High sensitivity guided-mode-resonance optical sensor employing phase detection,” Sci. Rep. 7, 7607 (2017).
[Crossref]

Other (4)

R. Magnusson, “Guided-mode resonance sensors employing angular, spectral, modal, and polarization diversity for high-precision sensing in compact formats,” U.S. patent10,274,432 (April30, 2019).

H. P. Ho, S. Y. Wu, S. K. Kong, S. Zeng, and K. T. Yong, “SPR biosensors,” in Handbook of Photonics for Biomedical Engineering (Springer, 2017).

C. T. Nataraj, G. R. Prashanth, and S. Talabattula, “Guided mode resonance grating based optical bio-sensor with enhanced bulk sensitivity,” in Advanced Photonics 2016, OSA Technical Digest (Optical Society of America, 2016), paper SeW1F.4.

M. A. Mattelin, G. Van Steenberge, and J. Missinne, “An imprinted polymer-based guided mode resonance grating sensor,” in Society of Photo-optical Instrumentation Engineers Conference Series (2018).

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Figures (9)

Fig. 1.
Fig. 1. Schematic of highly sensitive RI sensor designed with subwavelength grating.
Fig. 2.
Fig. 2. Comparison between RGW and SPR sensor.
Fig. 3.
Fig. 3. Dependence of the angular sensitivity on the grating period and thickness given by (a) $f = {0.5}$ and (b) $P = {226}\;{\rm nm}$.
Fig. 4.
Fig. 4. Influence of incidence angle and grating thickness on GMR is shown by calculating the reflectance.
Fig. 5.
Fig. 5. Relationship between the grating thickness, effective RI of grating waveguide, and resonant angle of incidence.
Fig. 6.
Fig. 6. Reflectance as a function of incidence angle given by the different grating thicknesses of ${h_g} = {530}\;{\rm and}$ 550 nm.
Fig. 7.
Fig. 7. Influence of period of thin-film interference on incidence angle for excited and non-excited GMR.
Fig. 8.
Fig. 8. With the change of grating thickness, the resonances in thin-film interference and GMR are shown. Letters A, B, and C mark the peak positions of GMR.
Fig. 9.
Fig. 9. Resonant angle of incidence versus RI of samples to be detected.

Equations (8)

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{ λ 0 P n sin θ 0 = n e k 0 n e = β
Δ θ 0 = Δ n sin θ 0 + Δ n e n cos θ 0 .
{ A h g k 0 = m π + arctan ( B A ) + arctan ( C A ) , m = 0 , 1 , 2 , . . . A = n s l a b 2 n e 2 , B = n e 2 n 2 , C = n e 2 n s 2 ,
n s l a b = [ f n g 2 + ( 1 f ) n 2 + π 2 3 ( 1 f ) 2 ( n g 2 n 2 ) 2 ( P λ 0 ) 2 ] 1 / 2 .
{ S = Δ θ 0 Δ n tan θ 0 n + ( K 1 + ( 1 f ) K 2 ) n e cos θ 0 K 1 = h g + ( A + C ) ( B 2 + A C ) / ( ( B 2 + A 2 ) ( B 2 + C 2 ) ) h g + ( A + C ) / A C K 2 = h g + B 2 / ( A ( B 2 + A 2 ) ) h g + ( A + C ) / A C .
{ d P d θ 0 = P 2 λ 0 n cos θ 0 d n e d θ 0 = n cos θ 0 .
Λ = λ 0 2 n s l a b 1 ( n sin θ 0 / n s l a b ) 2 ) ,
Λ G M R = λ 0 2 n s l a b 2 ( n sin θ 0 λ 0 / P ) 2 )

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