Abstract

Focusing light onto nanostructures thanks to spherical lenses is a first step in enhancing the field and is widely used in applications. Nonetheless, the electromagnetic response of such nanostructures, which have subwavelength patterns, to a focused beam cannot be described by the simple ray tracing formalism. Here, we present a method for computing the response to a focused beam, based on the B-spline modal method adapted to nanostructures in conical mounting. The eigenmodes are computed in each layer for both polarizations and are then combined for the computation of scattering matrices. The simulation of a Gaussian focused beam is obtained thanks to a truncated decomposition into plane waves computed on a single period, which limits the computation burden.

© 2014 Optical Society of America

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    [CrossRef]
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2014 (1)

2013 (2)

M. Walz, T. Zebrowski, J. Küchenmeister, and K. Busch, “B-spline modal method: a polynomial approach compared to the Fourier modal method,” Opt. Express 21, 14683–14697 (2013).
[CrossRef]

K. Edee, I. Fenniche, G. Granet, and B. Guizal, “Modal method based on subsectional Gegenbauer polynomial expansion for lamellar gratings: weighting function, convergence and stability,” Prog. Electromagn. Res. 133, 17–35 (2013).
[CrossRef]

2011 (3)

P. Bouchon, F. Pardo, B. Portier, L. Ferlazzo, P. Ghenuche, G. Dagher, C. Dupuis, N. Bardou, R. Haïdar, and J.-L. Pelouard, “Total funneling of light in high aspect ratio plasmonic nanoresonators,” Appl. Phys. Lett. 98, 191109 (2011).
[CrossRef]

F. Pardo, P. Bouchon, R. Haïdar, and J.-L. Pelouard, “Light funneling mechanism explained by magnetoelectric interference,” Phys. Rev. Lett. 107, 093902 (2011).
[CrossRef]

K. Edee, “Modal method based on subsectional Gegenbauer polynomial expansion for lamellar gratings,” J. Opt. Soc. Am. A 28, 2006–2013 (2011).
[CrossRef]

2010 (3)

P. Bouchon, F. Pardo, R. Haïdar, and J.-L. Pelouard, “Fast modal method for subwavelength gratings based on B-splines formulation,” J. Opt. Soc. Am. A 27, 696–702 (2010).
[CrossRef]

H. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9, 205–213 (2010).
[CrossRef]

J. Schuller, E. Barnard, W. Cai, Y. Jun, J. White, and M. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9, 193–204 (2010).
[CrossRef]

2009 (1)

2005 (1)

K. Edee, P. Schiavone, and G. Granet, “Analysis of defect in extreme UV lithography mask using a modal method based on nodal B-spline expansion,” Jpn. J. Appl. Phys. 44, 6458–6462 (2005).
[CrossRef]

1996 (2)

1995 (1)

1993 (1)

Atwater, H.

H. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9, 205–213 (2010).
[CrossRef]

Bardou, N.

P. Bouchon, F. Pardo, B. Portier, L. Ferlazzo, P. Ghenuche, G. Dagher, C. Dupuis, N. Bardou, R. Haïdar, and J.-L. Pelouard, “Total funneling of light in high aspect ratio plasmonic nanoresonators,” Appl. Phys. Lett. 98, 191109 (2011).
[CrossRef]

Barnard, E.

J. Schuller, E. Barnard, W. Cai, Y. Jun, J. White, and M. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9, 193–204 (2010).
[CrossRef]

Bloemer, M. J.

Bouchon, P.

F. Pardo, P. Bouchon, R. Haïdar, and J.-L. Pelouard, “Light funneling mechanism explained by magnetoelectric interference,” Phys. Rev. Lett. 107, 093902 (2011).
[CrossRef]

P. Bouchon, F. Pardo, B. Portier, L. Ferlazzo, P. Ghenuche, G. Dagher, C. Dupuis, N. Bardou, R. Haïdar, and J.-L. Pelouard, “Total funneling of light in high aspect ratio plasmonic nanoresonators,” Appl. Phys. Lett. 98, 191109 (2011).
[CrossRef]

P. Bouchon, F. Pardo, R. Haïdar, and J.-L. Pelouard, “Fast modal method for subwavelength gratings based on B-splines formulation,” J. Opt. Soc. Am. A 27, 696–702 (2010).
[CrossRef]

Brongersma, M.

J. Schuller, E. Barnard, W. Cai, Y. Jun, J. White, and M. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9, 193–204 (2010).
[CrossRef]

Busch, K.

Cai, W.

J. Schuller, E. Barnard, W. Cai, Y. Jun, J. White, and M. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9, 193–204 (2010).
[CrossRef]

D’Aguanno, G.

Dagher, G.

P. Bouchon, F. Pardo, B. Portier, L. Ferlazzo, P. Ghenuche, G. Dagher, C. Dupuis, N. Bardou, R. Haïdar, and J.-L. Pelouard, “Total funneling of light in high aspect ratio plasmonic nanoresonators,” Appl. Phys. Lett. 98, 191109 (2011).
[CrossRef]

Dupuis, C.

P. Bouchon, F. Pardo, B. Portier, L. Ferlazzo, P. Ghenuche, G. Dagher, C. Dupuis, N. Bardou, R. Haïdar, and J.-L. Pelouard, “Total funneling of light in high aspect ratio plasmonic nanoresonators,” Appl. Phys. Lett. 98, 191109 (2011).
[CrossRef]

Edee, K.

K. Edee, I. Fenniche, G. Granet, and B. Guizal, “Modal method based on subsectional Gegenbauer polynomial expansion for lamellar gratings: weighting function, convergence and stability,” Prog. Electromagn. Res. 133, 17–35 (2013).
[CrossRef]

K. Edee, “Modal method based on subsectional Gegenbauer polynomial expansion for lamellar gratings,” J. Opt. Soc. Am. A 28, 2006–2013 (2011).
[CrossRef]

K. Edee, P. Schiavone, and G. Granet, “Analysis of defect in extreme UV lithography mask using a modal method based on nodal B-spline expansion,” Jpn. J. Appl. Phys. 44, 6458–6462 (2005).
[CrossRef]

Fenniche, I.

K. Edee, I. Fenniche, G. Granet, and B. Guizal, “Modal method based on subsectional Gegenbauer polynomial expansion for lamellar gratings: weighting function, convergence and stability,” Prog. Electromagn. Res. 133, 17–35 (2013).
[CrossRef]

Ferlazzo, L.

P. Bouchon, F. Pardo, B. Portier, L. Ferlazzo, P. Ghenuche, G. Dagher, C. Dupuis, N. Bardou, R. Haïdar, and J.-L. Pelouard, “Total funneling of light in high aspect ratio plasmonic nanoresonators,” Appl. Phys. Lett. 98, 191109 (2011).
[CrossRef]

Gaylord, T.

Ghenuche, P.

P. Bouchon, F. Pardo, B. Portier, L. Ferlazzo, P. Ghenuche, G. Dagher, C. Dupuis, N. Bardou, R. Haïdar, and J.-L. Pelouard, “Total funneling of light in high aspect ratio plasmonic nanoresonators,” Appl. Phys. Lett. 98, 191109 (2011).
[CrossRef]

Ghosh, G.

E. Palik and G. Ghosh, Handbook of Optical Constants of Solids (Academic, 1985).

Granet, G.

G. Granet, “Efficient implementation of B-spline modal method for lamellar gratings,” J. Opt. Soc. Am. A 31, 332–337 (2014).
[CrossRef]

K. Edee, I. Fenniche, G. Granet, and B. Guizal, “Modal method based on subsectional Gegenbauer polynomial expansion for lamellar gratings: weighting function, convergence and stability,” Prog. Electromagn. Res. 133, 17–35 (2013).
[CrossRef]

K. Edee, P. Schiavone, and G. Granet, “Analysis of defect in extreme UV lithography mask using a modal method based on nodal B-spline expansion,” Jpn. J. Appl. Phys. 44, 6458–6462 (2005).
[CrossRef]

Grann, E. B.

Guizal, B.

K. Edee, I. Fenniche, G. Granet, and B. Guizal, “Modal method based on subsectional Gegenbauer polynomial expansion for lamellar gratings: weighting function, convergence and stability,” Prog. Electromagn. Res. 133, 17–35 (2013).
[CrossRef]

Haggans, C.

Haïdar, R.

P. Bouchon, F. Pardo, B. Portier, L. Ferlazzo, P. Ghenuche, G. Dagher, C. Dupuis, N. Bardou, R. Haïdar, and J.-L. Pelouard, “Total funneling of light in high aspect ratio plasmonic nanoresonators,” Appl. Phys. Lett. 98, 191109 (2011).
[CrossRef]

F. Pardo, P. Bouchon, R. Haïdar, and J.-L. Pelouard, “Light funneling mechanism explained by magnetoelectric interference,” Phys. Rev. Lett. 107, 093902 (2011).
[CrossRef]

P. Bouchon, F. Pardo, R. Haïdar, and J.-L. Pelouard, “Fast modal method for subwavelength gratings based on B-splines formulation,” J. Opt. Soc. Am. A 27, 696–702 (2010).
[CrossRef]

Jun, Y.

J. Schuller, E. Barnard, W. Cai, Y. Jun, J. White, and M. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9, 193–204 (2010).
[CrossRef]

Küchenmeister, J.

Lalanne, P.

Li, L.

Mattiucci, N.

Moharam, M.

Morris, G.

Palik, E.

E. Palik and G. Ghosh, Handbook of Optical Constants of Solids (Academic, 1985).

Pardo, F.

F. Pardo, P. Bouchon, R. Haïdar, and J.-L. Pelouard, “Light funneling mechanism explained by magnetoelectric interference,” Phys. Rev. Lett. 107, 093902 (2011).
[CrossRef]

P. Bouchon, F. Pardo, B. Portier, L. Ferlazzo, P. Ghenuche, G. Dagher, C. Dupuis, N. Bardou, R. Haïdar, and J.-L. Pelouard, “Total funneling of light in high aspect ratio plasmonic nanoresonators,” Appl. Phys. Lett. 98, 191109 (2011).
[CrossRef]

P. Bouchon, F. Pardo, R. Haïdar, and J.-L. Pelouard, “Fast modal method for subwavelength gratings based on B-splines formulation,” J. Opt. Soc. Am. A 27, 696–702 (2010).
[CrossRef]

Pelouard, J.-L.

P. Bouchon, F. Pardo, B. Portier, L. Ferlazzo, P. Ghenuche, G. Dagher, C. Dupuis, N. Bardou, R. Haïdar, and J.-L. Pelouard, “Total funneling of light in high aspect ratio plasmonic nanoresonators,” Appl. Phys. Lett. 98, 191109 (2011).
[CrossRef]

F. Pardo, P. Bouchon, R. Haïdar, and J.-L. Pelouard, “Light funneling mechanism explained by magnetoelectric interference,” Phys. Rev. Lett. 107, 093902 (2011).
[CrossRef]

P. Bouchon, F. Pardo, R. Haïdar, and J.-L. Pelouard, “Fast modal method for subwavelength gratings based on B-splines formulation,” J. Opt. Soc. Am. A 27, 696–702 (2010).
[CrossRef]

Polman, A.

H. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9, 205–213 (2010).
[CrossRef]

Pommet, D. A.

Portier, B.

P. Bouchon, F. Pardo, B. Portier, L. Ferlazzo, P. Ghenuche, G. Dagher, C. Dupuis, N. Bardou, R. Haïdar, and J.-L. Pelouard, “Total funneling of light in high aspect ratio plasmonic nanoresonators,” Appl. Phys. Lett. 98, 191109 (2011).
[CrossRef]

Scalora, M.

Schiavone, P.

K. Edee, P. Schiavone, and G. Granet, “Analysis of defect in extreme UV lithography mask using a modal method based on nodal B-spline expansion,” Jpn. J. Appl. Phys. 44, 6458–6462 (2005).
[CrossRef]

Schuller, J.

J. Schuller, E. Barnard, W. Cai, Y. Jun, J. White, and M. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9, 193–204 (2010).
[CrossRef]

Sibilia, C.

Walz, M.

White, J.

J. Schuller, E. Barnard, W. Cai, Y. Jun, J. White, and M. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9, 193–204 (2010).
[CrossRef]

Zebrowski, T.

Appl. Phys. Lett. (1)

P. Bouchon, F. Pardo, B. Portier, L. Ferlazzo, P. Ghenuche, G. Dagher, C. Dupuis, N. Bardou, R. Haïdar, and J.-L. Pelouard, “Total funneling of light in high aspect ratio plasmonic nanoresonators,” Appl. Phys. Lett. 98, 191109 (2011).
[CrossRef]

J. Opt. Soc. Am. A (7)

Jpn. J. Appl. Phys. (1)

K. Edee, P. Schiavone, and G. Granet, “Analysis of defect in extreme UV lithography mask using a modal method based on nodal B-spline expansion,” Jpn. J. Appl. Phys. 44, 6458–6462 (2005).
[CrossRef]

Nat. Mater. (2)

H. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9, 205–213 (2010).
[CrossRef]

J. Schuller, E. Barnard, W. Cai, Y. Jun, J. White, and M. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9, 193–204 (2010).
[CrossRef]

Opt. Express (2)

Phys. Rev. Lett. (1)

F. Pardo, P. Bouchon, R. Haïdar, and J.-L. Pelouard, “Light funneling mechanism explained by magnetoelectric interference,” Phys. Rev. Lett. 107, 093902 (2011).
[CrossRef]

Prog. Electromagn. Res. (1)

K. Edee, I. Fenniche, G. Granet, and B. Guizal, “Modal method based on subsectional Gegenbauer polynomial expansion for lamellar gratings: weighting function, convergence and stability,” Prog. Electromagn. Res. 133, 17–35 (2013).
[CrossRef]

Other (1)

E. Palik and G. Ghosh, Handbook of Optical Constants of Solids (Academic, 1985).

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Figures (5)

Fig. 1.
Fig. 1.

Grating system geometry in conical mounting. Regions I and III are semi-infinite homogeneous media, and region II is the structured layer.

Fig. 2.
Fig. 2.

Absolute error of the zeroth order of diffraction for light (λ=1μm) impinging on a silver grating, with w=0.5μm and d=h=1μm in conical mounting (θ=30°, φ=30°, and ψ=45°).

Fig. 3.
Fig. 3.

Conical absorption of a grating of grooves at λ=10μm. The structure considered is the grating system shown in Fig. 1, for grooves of height h=2μm, width w=150μm, and period d=5μm.

Fig. 4.
Fig. 4.

(a) Situation described by a nonperiodic beam but with a finite support: a single focal spot exists and (b) the focusing is periodized, allowing us to describe the situation by a discrete sum of plane waves.

Fig. 5.
Fig. 5.

(a) Computed incident field intensity on the structure, (b) field intensity in the structure for an (xz) cross section, (c) field enhancement inside the groove at different altitudes for an (xy) cross section [(c.1) z=0μm surface of the grating, (c.2) z=1μm middle height of the groove], (d) detail of the field intensity enhancement in the groove at its center (x=0μm) plotted along the y axis at the two different altitudes. The red cross on (c.1) highlights the position of the focal spot (x=y=z=0).

Equations (22)

Equations on this page are rendered with MathJax. Learn more.

Hx=0,
Hy(x,y,z)=nHyn(x)exp(ikznz+ikyyiωt),
Hz(x,y,z)=nHzn(x)exp(ikznz+ikyyiωt).
Hzn=kykznHyn.
iωByxEz+ikzEx=0.
Dx=ky2+kz2ωkzHy,
Dz=iωxHy.
μyHy+1k02x(1εzxHy)=(ky2+kz2k02)Hyϵx.
(μyN+1k021εzx2N)ΓQ=(ky2+kz2k02)1ϵxNΓQ,
(ϵyN+1k021μzx2N)ΓQ=(ky2+kz2k02)1μxNΓQ.
H1+H1S11=H2S21.
E1E1S11=E2S21.
S11=(12(1+H11H2E21E1)1)
S21=E21E1(1S11).
E0(x,y)=R2E˜0(kx,ky)eikxx+ikyydkxdky.
E˜0(kx,ky)=14π2R2E0(x,y)eikxxikyydxdy.
E0per(x,y)=p,qZE^0per(p,q)e2iπxpDx+2iπyqDy
E^0per(p,q)=1DxDySE0(x,y)e2iπxpDx2iπyqDydxdy.
E^0per(p,q)=Ce(p,q)Eince(p,q)+Cm(p,q)Eincm(p,q),
E0per(x,y)=p,qα{e,m}Cα(p,q)Eincα(p,q)e2iπxpDx+2iπyqDy.
Etot(x,y,z)=p,qα{e,m}Cα(p,q)Etotα(p,q)(x,z)e2iπxpDx+2iπyqDy,
E˜0(kx,ky)=W0xW0y2exp(kx2W0x2+ky2W0y24)u0(kx,ky),

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