Abstract

A general approach for describing (1 + 1)-D subwavelength optical field whose waist is much smaller than the wavelength is presented. Exploiting the vectorial Rayleigh-Sommerfeld diffraction theory, a suitable expansion in the ratio between the beam waist and the wavelength allows us to prove the a (1+1)D highly nonparaxial field is generally the product of a cylindrical wave carrier and an envelope which is angularly slowly varying. We apply our general approach to the case of highly nonparaxial Hermite-Gaussian beams whose description is fully analytical.

© 2010 Optical Society of America

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  1. J. B. Pendry, "Negative Refraction Makes a Perfect Lens," Phys. Rev. Lett. 85, 3966 (2000).
    [CrossRef] [PubMed]
  2. J. Takahara, S. Yamagishi, H. Taki, A. Morimoto, and T. Kobayashi, "Guiding of a one-dimensional optical beam with nanometer diameter," Opt. Lett. 22, 475 (1997).
    [CrossRef] [PubMed]
  3. H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, "Beaming Light from a Subwavelength Aperture," Science 297, 820 (2002).
    [CrossRef] [PubMed]
  4. L. Verslegers, P.B. Catrysse, Z. Yu, J. S. White, E. S. Barnard, M. L. Brongersma, and S. Fan, "Planar Lenses Based on Nanoscale Slit Arrays in a Metallic Film," Nano Lett. 9, 235 (2009).
    [CrossRef]
  5. A. Ciattoni, B. Crosignani, and P. Di Porto, "Vectorial free-space optical propagation: a simple approach for generating all-order nonparaxial corrections," Opt. Commun. 1779 (2000).
    [CrossRef]
  6. A. Yu. Savchencko and B. Ya. Zel’dovich, "Wave propagation in a guiding structure: one step beyond the paraxial approximation," J. Opt. Soc. Am. B 13, 273 (1996).
    [CrossRef]
  7. A. Ciattoni, B. Crosignani, and P. Di Porto, "Vectorial analytical description of propagation of a highly nonparaxial beam," Opt. Commun. 202, 17 (2002).
    [CrossRef]
  8. Z. Mei and D. Zhao, "Nonparaxial analysis of vectorial Laguerre-Bessel-Gaussian beams," Opt. Express 15, 11942 (2007).
    [CrossRef] [PubMed]
  9. R. Martnez-Herrero, P. M. Mejas, and A. Carnicer, "Evanescent field of vectorial highly non-paraxial beams," Opt. Express 16, 2845 (2008).
    [CrossRef]
  10. P. Liu, B. Lü, and K. Duan, "Propagation of vectorial nonparaxial Gaussian beams through an annular aperture," Opt. Laser Technol. 38, 133 (2006).
    [CrossRef]
  11. E. Moreno, F. J. García-Vidal, and L. Martín-Moreno, "Enhanced transmission and beaming of light via photonic crystal surface modes," Phys. Rev. B 69, 121402 (2004).
    [CrossRef]
  12. S. K. Morrison and Y. S. Kivshar, "Engineering of directional emission from photonic-crystal waveguides," Appl. Phys. Lett. 86, 081110 (2005).
    [CrossRef]
  13. C. Liu, N. Chen, and C. Sheppard, "Nanoillumination based on self-focus and field enhancement inside a subwavelength metallic structure," Appl. Phys. Lett. 90, 011501 (2007).
    [CrossRef]
  14. V. I. Smirnov, A Course of Higher Mathematics, Vol. 4 (Pergamon, Oxford, 1975).

2009

L. Verslegers, P.B. Catrysse, Z. Yu, J. S. White, E. S. Barnard, M. L. Brongersma, and S. Fan, "Planar Lenses Based on Nanoscale Slit Arrays in a Metallic Film," Nano Lett. 9, 235 (2009).
[CrossRef]

2008

2007

Z. Mei and D. Zhao, "Nonparaxial analysis of vectorial Laguerre-Bessel-Gaussian beams," Opt. Express 15, 11942 (2007).
[CrossRef] [PubMed]

C. Liu, N. Chen, and C. Sheppard, "Nanoillumination based on self-focus and field enhancement inside a subwavelength metallic structure," Appl. Phys. Lett. 90, 011501 (2007).
[CrossRef]

2006

P. Liu, B. Lü, and K. Duan, "Propagation of vectorial nonparaxial Gaussian beams through an annular aperture," Opt. Laser Technol. 38, 133 (2006).
[CrossRef]

2005

S. K. Morrison and Y. S. Kivshar, "Engineering of directional emission from photonic-crystal waveguides," Appl. Phys. Lett. 86, 081110 (2005).
[CrossRef]

2004

E. Moreno, F. J. García-Vidal, and L. Martín-Moreno, "Enhanced transmission and beaming of light via photonic crystal surface modes," Phys. Rev. B 69, 121402 (2004).
[CrossRef]

2002

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, "Beaming Light from a Subwavelength Aperture," Science 297, 820 (2002).
[CrossRef] [PubMed]

A. Ciattoni, B. Crosignani, and P. Di Porto, "Vectorial analytical description of propagation of a highly nonparaxial beam," Opt. Commun. 202, 17 (2002).
[CrossRef]

2000

A. Ciattoni, B. Crosignani, and P. Di Porto, "Vectorial free-space optical propagation: a simple approach for generating all-order nonparaxial corrections," Opt. Commun. 1779 (2000).
[CrossRef]

J. B. Pendry, "Negative Refraction Makes a Perfect Lens," Phys. Rev. Lett. 85, 3966 (2000).
[CrossRef] [PubMed]

1997

1996

Barnard, E. S.

L. Verslegers, P.B. Catrysse, Z. Yu, J. S. White, E. S. Barnard, M. L. Brongersma, and S. Fan, "Planar Lenses Based on Nanoscale Slit Arrays in a Metallic Film," Nano Lett. 9, 235 (2009).
[CrossRef]

Brongersma, M. L.

L. Verslegers, P.B. Catrysse, Z. Yu, J. S. White, E. S. Barnard, M. L. Brongersma, and S. Fan, "Planar Lenses Based on Nanoscale Slit Arrays in a Metallic Film," Nano Lett. 9, 235 (2009).
[CrossRef]

Carnicer, A.

Catrysse, P.B.

L. Verslegers, P.B. Catrysse, Z. Yu, J. S. White, E. S. Barnard, M. L. Brongersma, and S. Fan, "Planar Lenses Based on Nanoscale Slit Arrays in a Metallic Film," Nano Lett. 9, 235 (2009).
[CrossRef]

Chen, N.

C. Liu, N. Chen, and C. Sheppard, "Nanoillumination based on self-focus and field enhancement inside a subwavelength metallic structure," Appl. Phys. Lett. 90, 011501 (2007).
[CrossRef]

Ciattoni, A.

A. Ciattoni, B. Crosignani, and P. Di Porto, "Vectorial analytical description of propagation of a highly nonparaxial beam," Opt. Commun. 202, 17 (2002).
[CrossRef]

A. Ciattoni, B. Crosignani, and P. Di Porto, "Vectorial free-space optical propagation: a simple approach for generating all-order nonparaxial corrections," Opt. Commun. 1779 (2000).
[CrossRef]

Crosignani, B.

A. Ciattoni, B. Crosignani, and P. Di Porto, "Vectorial analytical description of propagation of a highly nonparaxial beam," Opt. Commun. 202, 17 (2002).
[CrossRef]

A. Ciattoni, B. Crosignani, and P. Di Porto, "Vectorial free-space optical propagation: a simple approach for generating all-order nonparaxial corrections," Opt. Commun. 1779 (2000).
[CrossRef]

Degiron, A.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, "Beaming Light from a Subwavelength Aperture," Science 297, 820 (2002).
[CrossRef] [PubMed]

Devaux, E.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, "Beaming Light from a Subwavelength Aperture," Science 297, 820 (2002).
[CrossRef] [PubMed]

Di Porto, P.

A. Ciattoni, B. Crosignani, and P. Di Porto, "Vectorial analytical description of propagation of a highly nonparaxial beam," Opt. Commun. 202, 17 (2002).
[CrossRef]

A. Ciattoni, B. Crosignani, and P. Di Porto, "Vectorial free-space optical propagation: a simple approach for generating all-order nonparaxial corrections," Opt. Commun. 1779 (2000).
[CrossRef]

Duan, K.

P. Liu, B. Lü, and K. Duan, "Propagation of vectorial nonparaxial Gaussian beams through an annular aperture," Opt. Laser Technol. 38, 133 (2006).
[CrossRef]

Ebbesen, T. W.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, "Beaming Light from a Subwavelength Aperture," Science 297, 820 (2002).
[CrossRef] [PubMed]

Fan, S.

L. Verslegers, P.B. Catrysse, Z. Yu, J. S. White, E. S. Barnard, M. L. Brongersma, and S. Fan, "Planar Lenses Based on Nanoscale Slit Arrays in a Metallic Film," Nano Lett. 9, 235 (2009).
[CrossRef]

Garcia-Vidal, F. J.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, "Beaming Light from a Subwavelength Aperture," Science 297, 820 (2002).
[CrossRef] [PubMed]

García-Vidal, F. J.

E. Moreno, F. J. García-Vidal, and L. Martín-Moreno, "Enhanced transmission and beaming of light via photonic crystal surface modes," Phys. Rev. B 69, 121402 (2004).
[CrossRef]

Kivshar, Y. S.

S. K. Morrison and Y. S. Kivshar, "Engineering of directional emission from photonic-crystal waveguides," Appl. Phys. Lett. 86, 081110 (2005).
[CrossRef]

Kobayashi, T.

Lezec, H. J.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, "Beaming Light from a Subwavelength Aperture," Science 297, 820 (2002).
[CrossRef] [PubMed]

Linke, R. A.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, "Beaming Light from a Subwavelength Aperture," Science 297, 820 (2002).
[CrossRef] [PubMed]

Liu, C.

C. Liu, N. Chen, and C. Sheppard, "Nanoillumination based on self-focus and field enhancement inside a subwavelength metallic structure," Appl. Phys. Lett. 90, 011501 (2007).
[CrossRef]

Liu, P.

P. Liu, B. Lü, and K. Duan, "Propagation of vectorial nonparaxial Gaussian beams through an annular aperture," Opt. Laser Technol. 38, 133 (2006).
[CrossRef]

Lü, B.

P. Liu, B. Lü, and K. Duan, "Propagation of vectorial nonparaxial Gaussian beams through an annular aperture," Opt. Laser Technol. 38, 133 (2006).
[CrossRef]

Martin-Moreno, L.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, "Beaming Light from a Subwavelength Aperture," Science 297, 820 (2002).
[CrossRef] [PubMed]

Martín-Moreno, L.

E. Moreno, F. J. García-Vidal, and L. Martín-Moreno, "Enhanced transmission and beaming of light via photonic crystal surface modes," Phys. Rev. B 69, 121402 (2004).
[CrossRef]

Martnez-Herrero, R.

Mei, Z.

Mejas, P. M.

Moreno, E.

E. Moreno, F. J. García-Vidal, and L. Martín-Moreno, "Enhanced transmission and beaming of light via photonic crystal surface modes," Phys. Rev. B 69, 121402 (2004).
[CrossRef]

Morimoto, A.

Morrison, S. K.

S. K. Morrison and Y. S. Kivshar, "Engineering of directional emission from photonic-crystal waveguides," Appl. Phys. Lett. 86, 081110 (2005).
[CrossRef]

Pendry, J. B.

J. B. Pendry, "Negative Refraction Makes a Perfect Lens," Phys. Rev. Lett. 85, 3966 (2000).
[CrossRef] [PubMed]

Savchencko, A. Yu.

Sheppard, C.

C. Liu, N. Chen, and C. Sheppard, "Nanoillumination based on self-focus and field enhancement inside a subwavelength metallic structure," Appl. Phys. Lett. 90, 011501 (2007).
[CrossRef]

Takahara, J.

Taki, H.

Verslegers, L.

L. Verslegers, P.B. Catrysse, Z. Yu, J. S. White, E. S. Barnard, M. L. Brongersma, and S. Fan, "Planar Lenses Based on Nanoscale Slit Arrays in a Metallic Film," Nano Lett. 9, 235 (2009).
[CrossRef]

White, J. S.

L. Verslegers, P.B. Catrysse, Z. Yu, J. S. White, E. S. Barnard, M. L. Brongersma, and S. Fan, "Planar Lenses Based on Nanoscale Slit Arrays in a Metallic Film," Nano Lett. 9, 235 (2009).
[CrossRef]

Yamagishi, S.

Yu, Z.

L. Verslegers, P.B. Catrysse, Z. Yu, J. S. White, E. S. Barnard, M. L. Brongersma, and S. Fan, "Planar Lenses Based on Nanoscale Slit Arrays in a Metallic Film," Nano Lett. 9, 235 (2009).
[CrossRef]

Zel’dovich, B. Ya.

Zhao, D.

Appl. Phys. Lett.

S. K. Morrison and Y. S. Kivshar, "Engineering of directional emission from photonic-crystal waveguides," Appl. Phys. Lett. 86, 081110 (2005).
[CrossRef]

C. Liu, N. Chen, and C. Sheppard, "Nanoillumination based on self-focus and field enhancement inside a subwavelength metallic structure," Appl. Phys. Lett. 90, 011501 (2007).
[CrossRef]

J. Opt. Soc. Am. B

Nano Lett.

L. Verslegers, P.B. Catrysse, Z. Yu, J. S. White, E. S. Barnard, M. L. Brongersma, and S. Fan, "Planar Lenses Based on Nanoscale Slit Arrays in a Metallic Film," Nano Lett. 9, 235 (2009).
[CrossRef]

Opt. Commun.

A. Ciattoni, B. Crosignani, and P. Di Porto, "Vectorial free-space optical propagation: a simple approach for generating all-order nonparaxial corrections," Opt. Commun. 1779 (2000).
[CrossRef]

A. Ciattoni, B. Crosignani, and P. Di Porto, "Vectorial analytical description of propagation of a highly nonparaxial beam," Opt. Commun. 202, 17 (2002).
[CrossRef]

Opt. Express

Opt. Laser Technol.

P. Liu, B. Lü, and K. Duan, "Propagation of vectorial nonparaxial Gaussian beams through an annular aperture," Opt. Laser Technol. 38, 133 (2006).
[CrossRef]

Opt. Lett.

Phys. Rev. B

E. Moreno, F. J. García-Vidal, and L. Martín-Moreno, "Enhanced transmission and beaming of light via photonic crystal surface modes," Phys. Rev. B 69, 121402 (2004).
[CrossRef]

Phys. Rev. Lett.

J. B. Pendry, "Negative Refraction Makes a Perfect Lens," Phys. Rev. Lett. 85, 3966 (2000).
[CrossRef] [PubMed]

Science

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, "Beaming Light from a Subwavelength Aperture," Science 297, 820 (2002).
[CrossRef] [PubMed]

Other

V. I. Smirnov, A Course of Higher Mathematics, Vol. 4 (Pergamon, Oxford, 1975).

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Figures (5)

Fig. 1.
Fig. 1.

Modulus of the normalized field Fx (x,z)/max(∣Fx ∣) of the fundamental Gaussian field (m = 0 subplot (a)) and of the first order Hermite-Gaussian field (m = 1 subplot (b)) where λ= 0.5μm, λ 0 = 0, w = λ/10. The refractive index is n = 1.

Fig. 2.
Fig. 2.

Modulus of the normalized field Ex (x,z)/max(∣Ex ∣) (subplot (a)) and Ez (x,z)/max(∣Ez ∣) (subplot (b)) of the fundamental Gaussian field (m = 0) for λ = 0.5μm, λ 0 = 0, w = λ/10 and n = 1.

Fig. 3.
Fig. 3.

Modulus of the normalized field Ex (x,z)/max(∣Ex ∣) (subplot (a)) and Ez (x,z)/max(∣Ez ∣) (subplot (b)) of the first order Hermite-Gaussian field (m = 1) for λ = 0.5μm, λ 0 = 0, w = λ/10 and n = 1.

Fig. 4.
Fig. 4.

Normalized Poynting vector components Sx (x,z)/max(∣S∣) (subplot (a)) and Sz (x,z)/max(∣S∣) (subplot (b)) of the Gaussian field (m = 0) for λ = 0.5μm, λ 0 = 0, w = λ/10 and n = 1.

Fig. 5.
Fig. 5.

Normalized Poynting vector components Sx (x,z)/max(∣S∣) (subplot (a)) and Sz (x,z)/max(∣S∣) (subplot (b)) of the first order Hermite-Gaussian field (m = 1) for λ = 0.5μm, x 0 = 0, w = λ/10 and n = 1.

Equations (18)

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

E ( x , z ) = i 2 + dx z ( H 0 ( 1 ) ( kR ) ) E ( x , 0 ) ,
E z ( x , z ) = i 2 + dx x ( H 0 ( 1 ) ( kR ) ) E x ( x , 0 ) ,
F ( x , z ) = i 2 + dx H 0 ( 1 ) ( kR ) E ( x , 0 ) ,
E = F z , E z = F x x .
B y = k 2 F x ,
F = + dx 1 2 πkR exp [ i ( kR + π 4 ) ] E ( x , 0 ) .
F = exp [ i ( kr + π 4 ) ] 1 2 πkr G ( x , z ) ,
G ( x , z ) = + dx exp ( ikxx r + ik z 2 x 2 2 r 3 ) E ( x , 0 )
E ( x , 0 ) = E 0 H m ( x x 0 w ) exp [ ( x x 0 ) 2 2 w 2 ] e ̂ x
G ( x , z ) = E 0 e ̂ x w m m x 0 m + dx exp [ ikxx r + ik z 2 x 2 2 r 3 ( x x 0 ) 2 2 w 2 ]
G ( x , z ) = E 0 e ̂ x w m + 1 2 π P ( x , z ) m g ( x , z | x 0 ) x 0 m
g = exp [ w 2 2 r 2 P ( ikx r x 0 w 2 ) 2 x 0 2 2 w 2 ] , P = 1 ik w 2 z 2 r 3 .
G ( r , θ ) = e ̂ x E 0 w 2 π P exp [ k 2 w 2 2 P sin 2 θ ] ,
E x ( r , θ ) = E 0 kw cos θ Pkr exp [ ikr + i π 4 k 2 w 2 2 P sin 2 θ ] ×
× { ( i 1 2 kr ) k w 2 Pr [ sin 2 θ + i 2 kr ( 1 k 2 w 2 P sin 2 θ ) ( 3 sin 2 θ 1 ) ] } ,
E z ( r , θ ) = E 0 kw sin θ Pkr exp [ ikr + i π 4 k 2 w 2 2 P sin 2 θ ] ×
× { ( i 1 2 kr ) k w 2 Pr [ cos 2 θ + i 2 kr ( 1 k 2 w 2 P sin 2 θ ) ( 3 cos 2 θ ) ] } ,
S ( x , z ) = 1 2 μ 0 Re ( E × B * ) = k 2 2 ω μ 0 Im ( F x * F x x e ̂ x + F x * F x z e ̂ z ) .

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