Abstract

We performed spectroscopic ellipsometry measurements of the specular and off-specular reflections of a one-dimensional polymethyl methacrylate grating on a Au-coated glass substrate and analyzed the data with efficient theoretical modeling. Very good agreement is obtained between calculation and experiment for both specular and off-specular reflections. It is found that the resonance peak of the off-specular reflection is much sharper than specular reflection at a wavelength that matches the condition for exciting the interface (surface plasmon) mode. When different media are used as the ambient, we found that the off-specular reflection near the surface-plasmon resonance can give significant enhancement in detection sensitivity compared with the corresponding specular reflection. This study may open a new way for biosensing. To analyze the data, we also developed a method to compute quasi-photonic band structures for periodic structures with frequency-dependent complex dielectric constants.

© 2012 Optical Society of America

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N. Rotenberg, J. N. Caspers, and H. M. van Driel, “Tunable ultrafast control of plasmonic coupling to gold films,” Phys. Rev. B 80, 245420 (2009).
[CrossRef]

D. Schmidt, B. Booso, T. Hofmann, E. Schubert, A. Sarangan, and M. Schubert, “Monoclinic optical constants, birefringence, and dichroism of slanted titanium nanocolumns determined by generalized ellipsometry,” Appl. Phys. Lett. 94, 011914 (2009).
[CrossRef]

2008

Y.-C. Chang, S.-H. Hsu, P.-K. Wei, and Y. D. Kim, “Optical nanometrology of Au nanoparticles on a multilayer film,” Phys. Status Solidi C 5, 1194–1197 (2008).
[CrossRef]

S.-H. Hsu, E.-S. Liu, Y.-C. Chang, J. N. Hilfiker, Y. D. Kim, T. J. Kim, C.-J. Lin, and G.-R. Lin, “Characterization of Si nanorods by spectroscopic ellipsometry with efficient theoretical modeling,” Phys. Status Solidi A 205, 876–879 (2008).
[CrossRef]

Ju-Yi Lee, Teng-Ko Chou, and Hsueh-Ching Shih, “Polarization-interferometric surface-plasmon-resonance imaging system,” Opt. Lett. 33, 434–436 (2008).
[CrossRef]

S.-Y. Wu and H.-P. Ho, “Single-beam self-referenced phase-sensitive surface plasmon resonance sensor with high detection resolution,” Chin. Opt. Lett. 6, 176–178 (2008).
[CrossRef]

2006

2005

R. Naraoka and K. Kajikawa, “Phase detection of surface plasmon resonance using rotating analyzer method,” Sens. Actuators B 107, 952–956 (2005).
[CrossRef]

J. J. Wang, L. Chen, S. Kwan, F. Liu, and X. Deng, “Resonant grating filters as refractive index sensors for chemical and biological detections,” J. Vac. Sci. Technol. B 23, 3006–3010 (2005).
[CrossRef]

2004

H. Wormeester, E. S. Kooij, A. Mewe, S. Rekveld, and B. Poelsema, “Ellipsometric characterisation of heterogeneous 2D layers,” Thin Solid Films 455–456, 323–334 (2004).
[CrossRef]

B. Kaplan, T. Novikova, A. D. Martino, and B. Drévillon, “Characterization of bidimensional gratings by spectroscopic ellipsometry and angle-resolved Mueller polarimetry,” Appl. Opt. 43, 1233–1240 (2004).
[CrossRef]

I. R. Hooper and J. R. Sambles, “Differential ellipsometric surface plasmon resonance sensors with liquid crystal polarization modulators,” Appl. Phys. Lett. 85, 3017–3019 (2004).
[CrossRef]

2003

J. Yoon, G. Lee, S. H. Song, C.-H. Oh, and P.-S. Kim, “Surface-plasmon photonic band gaps in dielectric gratings on a flat metal surface,” J. Appl. Phys. 94, 123–129 (2003).
[CrossRef]

2002

S. W. Harmer and P. D. Townsend, “Wavelength selectivity of on-axis surface plasmon laser filters,” J. Phys. D 35, 2516–2519 (2002).
[CrossRef]

1999

B. P. Nelson, A. G. Frutos, J. M. Brockman, and R. M. Corn, “Near-infrared surface plasmon resonance measurements of ultrathin films. 1. Angle shift and SPR imaging experiments,” Anal. Chem. 71, 3928–3934 (1999).
[CrossRef]

J. Homola, S. S. Yee, and G. Gauglitz, “Surface plasmon resonance sensors: review,” Sens. Actuators B 54, 3–15(1999).
[CrossRef]

A. N. Grigorenko, P. I. Nikitin, and A. V. Kabashin, “Phase jumps and interferometric surface plasmon resonance imaging,” Appl. Phys. Lett. 75, 3917–3919 (1999).
[CrossRef]

1996

1995

1992

O. Solgaard, F. Ho, J. I. Thackara, and D. M. Bloom, “High frequency attenuated total internal reflection light modulator,” Appl. Phys. Lett. 61, 2500–2502 (1992).
[CrossRef]

1970

Bloom, D. M.

O. Solgaard, F. Ho, J. I. Thackara, and D. M. Bloom, “High frequency attenuated total internal reflection light modulator,” Appl. Phys. Lett. 61, 2500–2502 (1992).
[CrossRef]

Booso, B.

D. Schmidt, B. Booso, T. Hofmann, E. Schubert, A. Sarangan, and M. Schubert, “Monoclinic optical constants, birefringence, and dichroism of slanted titanium nanocolumns determined by generalized ellipsometry,” Appl. Phys. Lett. 94, 011914 (2009).
[CrossRef]

Brockman, J. M.

B. P. Nelson, A. G. Frutos, J. M. Brockman, and R. M. Corn, “Near-infrared surface plasmon resonance measurements of ultrathin films. 1. Angle shift and SPR imaging experiments,” Anal. Chem. 71, 3928–3934 (1999).
[CrossRef]

Caspers, J. N.

N. Rotenberg, J. N. Caspers, and H. M. van Driel, “Tunable ultrafast control of plasmonic coupling to gold films,” Phys. Rev. B 80, 245420 (2009).
[CrossRef]

Chang, Y. C.

Chang, Y.-C.

S.-H. Hsu, Y.-C. Chang, Y.-C. Chen, P.-K. Wei, and Y. D. Kim, “Optical metrology of randomly-distributed Au colloids on a multilayer film,” Opt. Express 18, 1310–1315 (2010).
[CrossRef]

Y.-C. Chang, S.-H. Hsu, P.-K. Wei, and Y. D. Kim, “Optical nanometrology of Au nanoparticles on a multilayer film,” Phys. Status Solidi C 5, 1194–1197 (2008).
[CrossRef]

S.-H. Hsu, E.-S. Liu, Y.-C. Chang, J. N. Hilfiker, Y. D. Kim, T. J. Kim, C.-J. Lin, and G.-R. Lin, “Characterization of Si nanorods by spectroscopic ellipsometry with efficient theoretical modeling,” Phys. Status Solidi A 205, 876–879 (2008).
[CrossRef]

Chen, L.

J. J. Wang, L. Chen, S. Kwan, F. Liu, and X. Deng, “Resonant grating filters as refractive index sensors for chemical and biological detections,” J. Vac. Sci. Technol. B 23, 3006–3010 (2005).
[CrossRef]

Chen, Y.-C.

Chou, Teng-Ko

Chu, H.

Corn, R. M.

B. P. Nelson, A. G. Frutos, J. M. Brockman, and R. M. Corn, “Near-infrared surface plasmon resonance measurements of ultrathin films. 1. Angle shift and SPR imaging experiments,” Anal. Chem. 71, 3928–3934 (1999).
[CrossRef]

Deng, X.

J. J. Wang, L. Chen, S. Kwan, F. Liu, and X. Deng, “Resonant grating filters as refractive index sensors for chemical and biological detections,” J. Vac. Sci. Technol. B 23, 3006–3010 (2005).
[CrossRef]

Drévillon, B.

Frutos, A. G.

B. P. Nelson, A. G. Frutos, J. M. Brockman, and R. M. Corn, “Near-infrared surface plasmon resonance measurements of ultrathin films. 1. Angle shift and SPR imaging experiments,” Anal. Chem. 71, 3928–3934 (1999).
[CrossRef]

Gauglitz, G.

J. Homola, S. S. Yee, and G. Gauglitz, “Surface plasmon resonance sensors: review,” Sens. Actuators B 54, 3–15(1999).
[CrossRef]

Gaylord, T. K.

Grann, E. B.

Grigorenko, A. N.

V. G. Kravets, F. Schedin, A. V. Kabashin, and A. N. Grigorenko, “Sensitivity of collective plasmon modes of gold nanoresonators to local environment,” Opt. Lett. 35, 956–958 (2010).
[CrossRef]

A. N. Grigorenko, P. I. Nikitin, and A. V. Kabashin, “Phase jumps and interferometric surface plasmon resonance imaging,” Appl. Phys. Lett. 75, 3917–3919 (1999).
[CrossRef]

Harmer, S. W.

S. W. Harmer and P. D. Townsend, “Wavelength selectivity of on-axis surface plasmon laser filters,” J. Phys. D 35, 2516–2519 (2002).
[CrossRef]

Haus, H. A.

H. A. Haus, Waves and Fields in Optoelectronics (Central Book Company, 1985).

Hilfiker, J. N.

S.-H. Hsu, E.-S. Liu, Y.-C. Chang, J. N. Hilfiker, Y. D. Kim, T. J. Kim, C.-J. Lin, and G.-R. Lin, “Characterization of Si nanorods by spectroscopic ellipsometry with efficient theoretical modeling,” Phys. Status Solidi A 205, 876–879 (2008).
[CrossRef]

Ho, F.

O. Solgaard, F. Ho, J. I. Thackara, and D. M. Bloom, “High frequency attenuated total internal reflection light modulator,” Appl. Phys. Lett. 61, 2500–2502 (1992).
[CrossRef]

Ho, H.-P.

Hofmann, T.

D. Schmidt, B. Booso, T. Hofmann, E. Schubert, A. Sarangan, and M. Schubert, “Monoclinic optical constants, birefringence, and dichroism of slanted titanium nanocolumns determined by generalized ellipsometry,” Appl. Phys. Lett. 94, 011914 (2009).
[CrossRef]

Homola, J.

J. Homola, S. S. Yee, and G. Gauglitz, “Surface plasmon resonance sensors: review,” Sens. Actuators B 54, 3–15(1999).
[CrossRef]

Hooper, I. R.

I. R. Hooper and J. R. Sambles, “Differential ellipsometric surface plasmon resonance sensors with liquid crystal polarization modulators,” Appl. Phys. Lett. 85, 3017–3019 (2004).
[CrossRef]

Hsu, S.-H.

S.-H. Hsu, Y.-C. Chang, Y.-C. Chen, P.-K. Wei, and Y. D. Kim, “Optical metrology of randomly-distributed Au colloids on a multilayer film,” Opt. Express 18, 1310–1315 (2010).
[CrossRef]

S.-H. Hsu, E.-S. Liu, Y.-C. Chang, J. N. Hilfiker, Y. D. Kim, T. J. Kim, C.-J. Lin, and G.-R. Lin, “Characterization of Si nanorods by spectroscopic ellipsometry with efficient theoretical modeling,” Phys. Status Solidi A 205, 876–879 (2008).
[CrossRef]

Y.-C. Chang, S.-H. Hsu, P.-K. Wei, and Y. D. Kim, “Optical nanometrology of Au nanoparticles on a multilayer film,” Phys. Status Solidi C 5, 1194–1197 (2008).
[CrossRef]

Kabashin, A. V.

V. G. Kravets, F. Schedin, A. V. Kabashin, and A. N. Grigorenko, “Sensitivity of collective plasmon modes of gold nanoresonators to local environment,” Opt. Lett. 35, 956–958 (2010).
[CrossRef]

A. N. Grigorenko, P. I. Nikitin, and A. V. Kabashin, “Phase jumps and interferometric surface plasmon resonance imaging,” Appl. Phys. Lett. 75, 3917–3919 (1999).
[CrossRef]

Kajikawa, K.

R. Naraoka and K. Kajikawa, “Phase detection of surface plasmon resonance using rotating analyzer method,” Sens. Actuators B 107, 952–956 (2005).
[CrossRef]

Kaplan, B.

Kathman, A. D.

D. C. O’Shea, T. J. Suleski, A. D. Kathman, and D. W. Prather, Diffractive Optics: Design, Fabrication, and Test (SPIE, 2004).

Kim, P.-S.

J. Yoon, G. Lee, S. H. Song, C.-H. Oh, and P.-S. Kim, “Surface-plasmon photonic band gaps in dielectric gratings on a flat metal surface,” J. Appl. Phys. 94, 123–129 (2003).
[CrossRef]

Kim, T. J.

S.-H. Hsu, E.-S. Liu, Y.-C. Chang, J. N. Hilfiker, Y. D. Kim, T. J. Kim, C.-J. Lin, and G.-R. Lin, “Characterization of Si nanorods by spectroscopic ellipsometry with efficient theoretical modeling,” Phys. Status Solidi A 205, 876–879 (2008).
[CrossRef]

Kim, Y. D.

S.-H. Hsu, Y.-C. Chang, Y.-C. Chen, P.-K. Wei, and Y. D. Kim, “Optical metrology of randomly-distributed Au colloids on a multilayer film,” Opt. Express 18, 1310–1315 (2010).
[CrossRef]

S.-H. Hsu, E.-S. Liu, Y.-C. Chang, J. N. Hilfiker, Y. D. Kim, T. J. Kim, C.-J. Lin, and G.-R. Lin, “Characterization of Si nanorods by spectroscopic ellipsometry with efficient theoretical modeling,” Phys. Status Solidi A 205, 876–879 (2008).
[CrossRef]

Y.-C. Chang, S.-H. Hsu, P.-K. Wei, and Y. D. Kim, “Optical nanometrology of Au nanoparticles on a multilayer film,” Phys. Status Solidi C 5, 1194–1197 (2008).
[CrossRef]

Kooij, E. S.

H. Wormeester, E. S. Kooij, A. Mewe, S. Rekveld, and B. Poelsema, “Ellipsometric characterisation of heterogeneous 2D layers,” Thin Solid Films 455–456, 323–334 (2004).
[CrossRef]

Kravets, V. G.

Kwan, S.

J. J. Wang, L. Chen, S. Kwan, F. Liu, and X. Deng, “Resonant grating filters as refractive index sensors for chemical and biological detections,” J. Vac. Sci. Technol. B 23, 3006–3010 (2005).
[CrossRef]

Lee, G.

J. Yoon, G. Lee, S. H. Song, C.-H. Oh, and P.-S. Kim, “Surface-plasmon photonic band gaps in dielectric gratings on a flat metal surface,” J. Appl. Phys. 94, 123–129 (2003).
[CrossRef]

Lee, Ju-Yi

Li, G.

Li, L.

Lin, C.-J.

S.-H. Hsu, E.-S. Liu, Y.-C. Chang, J. N. Hilfiker, Y. D. Kim, T. J. Kim, C.-J. Lin, and G.-R. Lin, “Characterization of Si nanorods by spectroscopic ellipsometry with efficient theoretical modeling,” Phys. Status Solidi A 205, 876–879 (2008).
[CrossRef]

Lin, G.-R.

S.-H. Hsu, E.-S. Liu, Y.-C. Chang, J. N. Hilfiker, Y. D. Kim, T. J. Kim, C.-J. Lin, and G.-R. Lin, “Characterization of Si nanorods by spectroscopic ellipsometry with efficient theoretical modeling,” Phys. Status Solidi A 205, 876–879 (2008).
[CrossRef]

Liu, E.-S.

S.-H. Hsu, E.-S. Liu, Y.-C. Chang, J. N. Hilfiker, Y. D. Kim, T. J. Kim, C.-J. Lin, and G.-R. Lin, “Characterization of Si nanorods by spectroscopic ellipsometry with efficient theoretical modeling,” Phys. Status Solidi A 205, 876–879 (2008).
[CrossRef]

Liu, F.

J. J. Wang, L. Chen, S. Kwan, F. Liu, and X. Deng, “Resonant grating filters as refractive index sensors for chemical and biological detections,” J. Vac. Sci. Technol. B 23, 3006–3010 (2005).
[CrossRef]

Martino, A. D.

Mewe, A.

H. Wormeester, E. S. Kooij, A. Mewe, S. Rekveld, and B. Poelsema, “Ellipsometric characterisation of heterogeneous 2D layers,” Thin Solid Films 455–456, 323–334 (2004).
[CrossRef]

Moharam, M. G.

Naraoka, R.

R. Naraoka and K. Kajikawa, “Phase detection of surface plasmon resonance using rotating analyzer method,” Sens. Actuators B 107, 952–956 (2005).
[CrossRef]

Nelson, B. P.

B. P. Nelson, A. G. Frutos, J. M. Brockman, and R. M. Corn, “Near-infrared surface plasmon resonance measurements of ultrathin films. 1. Angle shift and SPR imaging experiments,” Anal. Chem. 71, 3928–3934 (1999).
[CrossRef]

Nikitin, P. I.

A. N. Grigorenko, P. I. Nikitin, and A. V. Kabashin, “Phase jumps and interferometric surface plasmon resonance imaging,” Appl. Phys. Lett. 75, 3917–3919 (1999).
[CrossRef]

Novikova, T.

O’Shea, D. C.

D. C. O’Shea, T. J. Suleski, A. D. Kathman, and D. W. Prather, Diffractive Optics: Design, Fabrication, and Test (SPIE, 2004).

Oh, C.-H.

J. Yoon, G. Lee, S. H. Song, C.-H. Oh, and P.-S. Kim, “Surface-plasmon photonic band gaps in dielectric gratings on a flat metal surface,” J. Appl. Phys. 94, 123–129 (2003).
[CrossRef]

Opsal, J.

Palik, E. D.

E. D. Palik, Handbook of Optical Constants of Solids(Academic, 1985).

Poelsema, B.

H. Wormeester, E. S. Kooij, A. Mewe, S. Rekveld, and B. Poelsema, “Ellipsometric characterisation of heterogeneous 2D layers,” Thin Solid Films 455–456, 323–334 (2004).
[CrossRef]

Pommet, D. A.

Prather, D. W.

D. C. O’Shea, T. J. Suleski, A. D. Kathman, and D. W. Prather, Diffractive Optics: Design, Fabrication, and Test (SPIE, 2004).

Raether, H.

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer, 1988).

Rekveld, S.

H. Wormeester, E. S. Kooij, A. Mewe, S. Rekveld, and B. Poelsema, “Ellipsometric characterisation of heterogeneous 2D layers,” Thin Solid Films 455–456, 323–334 (2004).
[CrossRef]

Rotenberg, N.

N. Rotenberg, J. N. Caspers, and H. M. van Driel, “Tunable ultrafast control of plasmonic coupling to gold films,” Phys. Rev. B 80, 245420 (2009).
[CrossRef]

Sambles, J. R.

I. R. Hooper and J. R. Sambles, “Differential ellipsometric surface plasmon resonance sensors with liquid crystal polarization modulators,” Appl. Phys. Lett. 85, 3017–3019 (2004).
[CrossRef]

Sarangan, A.

D. Schmidt, B. Booso, T. Hofmann, E. Schubert, A. Sarangan, and M. Schubert, “Monoclinic optical constants, birefringence, and dichroism of slanted titanium nanocolumns determined by generalized ellipsometry,” Appl. Phys. Lett. 94, 011914 (2009).
[CrossRef]

Schedin, F.

Schmidt, D.

D. Schmidt, B. Booso, T. Hofmann, E. Schubert, A. Sarangan, and M. Schubert, “Monoclinic optical constants, birefringence, and dichroism of slanted titanium nanocolumns determined by generalized ellipsometry,” Appl. Phys. Lett. 94, 011914 (2009).
[CrossRef]

Schubert, E.

D. Schmidt, B. Booso, T. Hofmann, E. Schubert, A. Sarangan, and M. Schubert, “Monoclinic optical constants, birefringence, and dichroism of slanted titanium nanocolumns determined by generalized ellipsometry,” Appl. Phys. Lett. 94, 011914 (2009).
[CrossRef]

Schubert, M.

D. Schmidt, B. Booso, T. Hofmann, E. Schubert, A. Sarangan, and M. Schubert, “Monoclinic optical constants, birefringence, and dichroism of slanted titanium nanocolumns determined by generalized ellipsometry,” Appl. Phys. Lett. 94, 011914 (2009).
[CrossRef]

Shih, Hsueh-Ching

Solgaard, O.

O. Solgaard, F. Ho, J. I. Thackara, and D. M. Bloom, “High frequency attenuated total internal reflection light modulator,” Appl. Phys. Lett. 61, 2500–2502 (1992).
[CrossRef]

Song, S. H.

J. Yoon, G. Lee, S. H. Song, C.-H. Oh, and P.-S. Kim, “Surface-plasmon photonic band gaps in dielectric gratings on a flat metal surface,” J. Appl. Phys. 94, 123–129 (2003).
[CrossRef]

Suleski, T. J.

D. C. O’Shea, T. J. Suleski, A. D. Kathman, and D. W. Prather, Diffractive Optics: Design, Fabrication, and Test (SPIE, 2004).

Thackara, J. I.

O. Solgaard, F. Ho, J. I. Thackara, and D. M. Bloom, “High frequency attenuated total internal reflection light modulator,” Appl. Phys. Lett. 61, 2500–2502 (1992).
[CrossRef]

Tien, P. K.

Townsend, P. D.

S. W. Harmer and P. D. Townsend, “Wavelength selectivity of on-axis surface plasmon laser filters,” J. Phys. D 35, 2516–2519 (2002).
[CrossRef]

Ulrich, R.

van Driel, H. M.

N. Rotenberg, J. N. Caspers, and H. M. van Driel, “Tunable ultrafast control of plasmonic coupling to gold films,” Phys. Rev. B 80, 245420 (2009).
[CrossRef]

Wang, J. J.

J. J. Wang, L. Chen, S. Kwan, F. Liu, and X. Deng, “Resonant grating filters as refractive index sensors for chemical and biological detections,” J. Vac. Sci. Technol. B 23, 3006–3010 (2005).
[CrossRef]

Wei, P.-K.

S.-H. Hsu, Y.-C. Chang, Y.-C. Chen, P.-K. Wei, and Y. D. Kim, “Optical metrology of randomly-distributed Au colloids on a multilayer film,” Opt. Express 18, 1310–1315 (2010).
[CrossRef]

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Figures (6)

Fig. 1.
Fig. 1.

(a) Construction of angles for higher order reflections from grating. (b) Schematic datagram of 1D PMMA gratings on Au-coated glass substrate, (c) Optical image of PMMA grating on Au-coated glass substrate with different periods (d) SEM image of PMMA grating with period of 600 nm.

Fig. 2.
Fig. 2.

SE measurements and model calculations of 1D PMMA grating samples (a) for specular reflection with Λ = 600 nm , (b) for specular reflection with Λ = 900 nm , (c) for ( 1 ) order off-specular reflection with Λ = 600 nm , and (d) for ( 1 ) order off-specular reflection with Λ = 900 nm .

Fig. 3.
Fig. 3.

Angle-dependent reflectance spectra (ARS) for off-specular reflection and band structure diagram of the PMMA grating for TM mode. (a) ARS with Λ = 600 nm . (b) Minimum | q | plot with Λ = 600 nm . (c) ARS with Λ = 900 nm . (d) Minimum | q | plot with Λ = 900 nm . The dashed line in each plot shows the relation ω / c = k x / sin θ i at an incident angle of θ i = 60 ° .

Fig. 4.
Fig. 4.

(a)  | H y | 2 distribution within one pitch for a PMMA grating structure at 1.38 eV for θ i = 60 ° . (b) Calculated s -polarized specular reflectance ( R s ) spectra (black dashed curve), s -polarized off-specular reflectance ( R s ) spectra (brown dashed curve), p -polarized specular reflectance ( R p ) spectra (cyan stars), p -polarized off-specular reflectance ( R p ) spectra (pink solid curve), p -polarized specular phase ( Φ p ) spectra (open circles) and p -polarized off-specular phase ( Φ p ) spectra (blue solid curve) for PMMA grating with period 900 nm at θ i = 60 ° .

Fig. 5.
Fig. 5.

Model calculations of SE parameters for light scattering from a 1D PMMA grating with 600 nm period for an incident angle of 60° with ambient being either water or salty water when the reflection is detected at (a) specular angle and (b) ( 1 ) order off-specular angle near 0°.

Fig. 6.
Fig. 6.

Model calculations of SE parameters for light scattering from a 1D PMMA grating with 900 nm period for an incident angle of 60° with ambient being either water or salty water when the reflection is detected at (a) specular angle and (b) ( 1 ) order off-specular angle near 0°.

Equations (9)

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× ε 1 × H = ( 2 / t 2 ) H = k 0 2 H ,
H = 0 or k z H z = ( K x H x + K y H y ) ,
q j 2 = k z 2 , z 2 Ψ = L j ( z ) Ψ q j 2 Ψ ,
L j ( z ) = ( ε j k 0 2 K x 2 ε j K y ε j 1 K y ε j K y ε j 1 K x K x K y ε j K x ε j 1 K y K y K x ε j k 0 2 K y 2 ε j K x ε j 1 K x ) .
( E y E x ) = ε 1 [ 1 + ( K x 2 K x K y K y K x K y 2 ) k z 2 ] k z ( H x H y ) D ( z ) k z Ψ .
Ψ = j S j e Q j z f j + j S j e Q j z R j f j ,
R j = e Q j d j ( M j + M j + R j + 1 ) ( M j + + M j R j + 1 ) 1 e Q j d j ,
R n = ( E n r × × E n r * ) e z ( E i × × E i * ) e z = ( E n r × k n × E n r * ) e z ( E i × k 0 × E i * ) e z = k n z | E n r | 2 k 0 z | E i | 2 = k n z k 0 z | r n | 2 ,
q 2 = k m x 2 n eff 2 k 0 2 = 0 or k 0 = 2 π λ = | m 2 π Λ / ( n eff ± sin θ i ) | ; m is an integer .

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