Abstract

We propose a family of exact solutions of Maxwell's equations to model some aspects of the imaging process involved in the scanning surface plasmon microscope (SSPM). More precisely, we compute the SSPM response of a spherical nanoparticle immobilized close to a thin gold layer and illuminated by a tightly focused spot. We discuss the influence of parameters such as the defocus and the width of the gold layer on the image contrast. We show that this microscopy combines a subwavelength spatial resolution together with high sensitivity to small changes in dielectric properties on the nanoparticle.

© 2010 Optical Society of America

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  1. M. G. Somekh, “Surface plasmon and surface wave microscopy,” in Optical Imaging and Microscopy, Vol. 87 of Springer Series in Optical Sciences, P.Torok and F. J.Kao, eds. (Springer Verlag, 2003), pp. 275-307.
  2. E. M. Yeatman and E. A. Ash, “Surface plasmon microscopy,” Electron. Lett. 23, 1091-1092 (1987).
    [CrossRef]
  3. B. Rothenhausler and W. Knoll, “Surface-plasmon microscopy,” Nature 332, 615-617 (1988).
    [CrossRef]
  4. C. E. H. Berger, R. P. H. Kooyman, and J. Greve, “Resolution in surface plasmon microscopy,” Rev. Sci. Instrum. 65, 2829-2836 (1994).
    [CrossRef]
  5. H. Kano, S. Mizuguchi, and S. Kawata, “Excitation of surface-plasmon polaritons by a focused laser beam,” J. Opt. Soc. Am. B 15, 1381-1386 (1998).
    [CrossRef]
  6. M. G. Somekh, S. Liu, and T. S. Velinov, “Optical V(z) for high-resolution 2π surface plasmon microscopy,” Opt. Lett. 25, 823-825 (2000).
    [CrossRef]
  7. A. N. Grigorenko, A. A. Beloglazov, P. I. Nikitin, C. Kuhne, G. Steiner, and R. Salzer, “Dark-field surface plasmon resonance microscopy,” Opt. Commun. 174, 151-155 (2000).
    [CrossRef]
  8. L. Berguiga, S.-J. Zhang, F. Argoul, and J. Elezgaray, “High-resolution surface-plasmon imaging in air and in water: V(z) curve and operating conditions,” Opt. Lett. 32, 509-511 (2007).
    [CrossRef] [PubMed]
  9. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253, 358-379 (1959).
    [CrossRef]
  10. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901-233904 (2003).
    [CrossRef] [PubMed]
  11. Q. Zhan, “Evanescent Bessel beam generation via surface plasmon resonance excitation by a radially polarized beam,” Opt. Lett. 31, 1726-1728 (2006).
    [CrossRef] [PubMed]
  12. Z. Zhu, M. G. Somekh, and M. M. Steven, “Behavior of localized surface plasmon near focus,” Opt. Commun. 207, 113-119 (2002).
    [CrossRef]
  13. A. Doicu, Y. A. Eremin, and T. Wriedt, “Convergence of the T-matrix method for light scattering from a particle on or near a surface,” Opt. Commun. 159, 266-277 (1999).
    [CrossRef]
  14. G. Videen, “Light scattering from a sphere behind a surface,” J. Opt. Soc. Am. A 10, 110-117 (1993).
    [CrossRef]
  15. M. G. Somekh, S. Liu, T. S. Velinov, and C. W. See, “High-resolution scanning surface-plasmon microscopy,” Appl. Opt. 39, 6279-6287 (2000).
    [CrossRef]
  16. C. J. R. Sheppard and T. Wilson, “Effects of high angles of convergence on V(z) in the scanning acoustic microscope,” Appl. Phys. Lett. 38, 858-859 (1981).
    [CrossRef]
  17. G. Lévêque and O. J. F. Martin, “Optical interactions in a plasmonic particle coupled to a metallic film,” Opt. Express 14, 9971-9981 (2006).
    [CrossRef] [PubMed]

2007 (1)

2006 (2)

2003 (1)

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901-233904 (2003).
[CrossRef] [PubMed]

2002 (1)

Z. Zhu, M. G. Somekh, and M. M. Steven, “Behavior of localized surface plasmon near focus,” Opt. Commun. 207, 113-119 (2002).
[CrossRef]

2000 (3)

1999 (1)

A. Doicu, Y. A. Eremin, and T. Wriedt, “Convergence of the T-matrix method for light scattering from a particle on or near a surface,” Opt. Commun. 159, 266-277 (1999).
[CrossRef]

1998 (1)

1994 (1)

C. E. H. Berger, R. P. H. Kooyman, and J. Greve, “Resolution in surface plasmon microscopy,” Rev. Sci. Instrum. 65, 2829-2836 (1994).
[CrossRef]

1993 (1)

1988 (1)

B. Rothenhausler and W. Knoll, “Surface-plasmon microscopy,” Nature 332, 615-617 (1988).
[CrossRef]

1987 (1)

E. M. Yeatman and E. A. Ash, “Surface plasmon microscopy,” Electron. Lett. 23, 1091-1092 (1987).
[CrossRef]

1981 (1)

C. J. R. Sheppard and T. Wilson, “Effects of high angles of convergence on V(z) in the scanning acoustic microscope,” Appl. Phys. Lett. 38, 858-859 (1981).
[CrossRef]

1959 (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253, 358-379 (1959).
[CrossRef]

Argoul, F.

Ash, E. A.

E. M. Yeatman and E. A. Ash, “Surface plasmon microscopy,” Electron. Lett. 23, 1091-1092 (1987).
[CrossRef]

Beloglazov, A. A.

A. N. Grigorenko, A. A. Beloglazov, P. I. Nikitin, C. Kuhne, G. Steiner, and R. Salzer, “Dark-field surface plasmon resonance microscopy,” Opt. Commun. 174, 151-155 (2000).
[CrossRef]

Berger, C. E. H.

C. E. H. Berger, R. P. H. Kooyman, and J. Greve, “Resolution in surface plasmon microscopy,” Rev. Sci. Instrum. 65, 2829-2836 (1994).
[CrossRef]

Berguiga, L.

Doicu, A.

A. Doicu, Y. A. Eremin, and T. Wriedt, “Convergence of the T-matrix method for light scattering from a particle on or near a surface,” Opt. Commun. 159, 266-277 (1999).
[CrossRef]

Dorn, R.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901-233904 (2003).
[CrossRef] [PubMed]

Elezgaray, J.

Eremin, Y. A.

A. Doicu, Y. A. Eremin, and T. Wriedt, “Convergence of the T-matrix method for light scattering from a particle on or near a surface,” Opt. Commun. 159, 266-277 (1999).
[CrossRef]

Greve, J.

C. E. H. Berger, R. P. H. Kooyman, and J. Greve, “Resolution in surface plasmon microscopy,” Rev. Sci. Instrum. 65, 2829-2836 (1994).
[CrossRef]

Grigorenko, A. N.

A. N. Grigorenko, A. A. Beloglazov, P. I. Nikitin, C. Kuhne, G. Steiner, and R. Salzer, “Dark-field surface plasmon resonance microscopy,” Opt. Commun. 174, 151-155 (2000).
[CrossRef]

Kano, H.

Kawata, S.

Knoll, W.

B. Rothenhausler and W. Knoll, “Surface-plasmon microscopy,” Nature 332, 615-617 (1988).
[CrossRef]

Kooyman, R. P. H.

C. E. H. Berger, R. P. H. Kooyman, and J. Greve, “Resolution in surface plasmon microscopy,” Rev. Sci. Instrum. 65, 2829-2836 (1994).
[CrossRef]

Kuhne, C.

A. N. Grigorenko, A. A. Beloglazov, P. I. Nikitin, C. Kuhne, G. Steiner, and R. Salzer, “Dark-field surface plasmon resonance microscopy,” Opt. Commun. 174, 151-155 (2000).
[CrossRef]

Leuchs, G.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901-233904 (2003).
[CrossRef] [PubMed]

Lévêque, G.

Liu, S.

Martin, O. J. F.

Mizuguchi, S.

Nikitin, P. I.

A. N. Grigorenko, A. A. Beloglazov, P. I. Nikitin, C. Kuhne, G. Steiner, and R. Salzer, “Dark-field surface plasmon resonance microscopy,” Opt. Commun. 174, 151-155 (2000).
[CrossRef]

Quabis, S.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901-233904 (2003).
[CrossRef] [PubMed]

Richards, B.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253, 358-379 (1959).
[CrossRef]

Rothenhausler, B.

B. Rothenhausler and W. Knoll, “Surface-plasmon microscopy,” Nature 332, 615-617 (1988).
[CrossRef]

Salzer, R.

A. N. Grigorenko, A. A. Beloglazov, P. I. Nikitin, C. Kuhne, G. Steiner, and R. Salzer, “Dark-field surface plasmon resonance microscopy,” Opt. Commun. 174, 151-155 (2000).
[CrossRef]

See, C. W.

Sheppard, C. J. R.

C. J. R. Sheppard and T. Wilson, “Effects of high angles of convergence on V(z) in the scanning acoustic microscope,” Appl. Phys. Lett. 38, 858-859 (1981).
[CrossRef]

Somekh, M. G.

Z. Zhu, M. G. Somekh, and M. M. Steven, “Behavior of localized surface plasmon near focus,” Opt. Commun. 207, 113-119 (2002).
[CrossRef]

M. G. Somekh, S. Liu, T. S. Velinov, and C. W. See, “High-resolution scanning surface-plasmon microscopy,” Appl. Opt. 39, 6279-6287 (2000).
[CrossRef]

M. G. Somekh, S. Liu, and T. S. Velinov, “Optical V(z) for high-resolution 2π surface plasmon microscopy,” Opt. Lett. 25, 823-825 (2000).
[CrossRef]

M. G. Somekh, “Surface plasmon and surface wave microscopy,” in Optical Imaging and Microscopy, Vol. 87 of Springer Series in Optical Sciences, P.Torok and F. J.Kao, eds. (Springer Verlag, 2003), pp. 275-307.

Steiner, G.

A. N. Grigorenko, A. A. Beloglazov, P. I. Nikitin, C. Kuhne, G. Steiner, and R. Salzer, “Dark-field surface plasmon resonance microscopy,” Opt. Commun. 174, 151-155 (2000).
[CrossRef]

Steven, M. M.

Z. Zhu, M. G. Somekh, and M. M. Steven, “Behavior of localized surface plasmon near focus,” Opt. Commun. 207, 113-119 (2002).
[CrossRef]

Velinov, T. S.

Videen, G.

Wilson, T.

C. J. R. Sheppard and T. Wilson, “Effects of high angles of convergence on V(z) in the scanning acoustic microscope,” Appl. Phys. Lett. 38, 858-859 (1981).
[CrossRef]

Wolf, E.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253, 358-379 (1959).
[CrossRef]

Wriedt, T.

A. Doicu, Y. A. Eremin, and T. Wriedt, “Convergence of the T-matrix method for light scattering from a particle on or near a surface,” Opt. Commun. 159, 266-277 (1999).
[CrossRef]

Yeatman, E. M.

E. M. Yeatman and E. A. Ash, “Surface plasmon microscopy,” Electron. Lett. 23, 1091-1092 (1987).
[CrossRef]

Zhan, Q.

Zhang, S.-J.

Zhu, Z.

Z. Zhu, M. G. Somekh, and M. M. Steven, “Behavior of localized surface plasmon near focus,” Opt. Commun. 207, 113-119 (2002).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

C. J. R. Sheppard and T. Wilson, “Effects of high angles of convergence on V(z) in the scanning acoustic microscope,” Appl. Phys. Lett. 38, 858-859 (1981).
[CrossRef]

Electron. Lett. (1)

E. M. Yeatman and E. A. Ash, “Surface plasmon microscopy,” Electron. Lett. 23, 1091-1092 (1987).
[CrossRef]

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

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

Nature (1)

B. Rothenhausler and W. Knoll, “Surface-plasmon microscopy,” Nature 332, 615-617 (1988).
[CrossRef]

Opt. Commun. (3)

A. N. Grigorenko, A. A. Beloglazov, P. I. Nikitin, C. Kuhne, G. Steiner, and R. Salzer, “Dark-field surface plasmon resonance microscopy,” Opt. Commun. 174, 151-155 (2000).
[CrossRef]

Z. Zhu, M. G. Somekh, and M. M. Steven, “Behavior of localized surface plasmon near focus,” Opt. Commun. 207, 113-119 (2002).
[CrossRef]

A. Doicu, Y. A. Eremin, and T. Wriedt, “Convergence of the T-matrix method for light scattering from a particle on or near a surface,” Opt. Commun. 159, 266-277 (1999).
[CrossRef]

Opt. Express (1)

Opt. Lett. (3)

Phys. Rev. Lett. (1)

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901-233904 (2003).
[CrossRef] [PubMed]

Proc. R. Soc. London, Ser. A (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253, 358-379 (1959).
[CrossRef]

Rev. Sci. Instrum. (1)

C. E. H. Berger, R. P. H. Kooyman, and J. Greve, “Resolution in surface plasmon microscopy,” Rev. Sci. Instrum. 65, 2829-2836 (1994).
[CrossRef]

Other (1)

M. G. Somekh, “Surface plasmon and surface wave microscopy,” in Optical Imaging and Microscopy, Vol. 87 of Springer Series in Optical Sciences, P.Torok and F. J.Kao, eds. (Springer Verlag, 2003), pp. 275-307.

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Figures (10)

Fig. 1
Fig. 1

Schematic representation of the system: a spherical bead (index n 3 ) in a medium with optical index n 2 is illuminated by a focused beam, incoming from the objective immersed in an adapting oil medium (index n 0 ), going through a gold layer (index n 1 ), and finally reaching the bead. The position of the focus pointed with the letter O does not necessarily coincide with the center of the bead.

Fig. 2
Fig. 2

Plot of the modulus of the incident electric field with radial incident polarization. (a) z focus = 0.8 μ m , (b) z focus = 0 μ m , (c) z focus = 0.8 μ m , (d), (e), and (f) are zoomed views of (a), (b), and (c) in the vicinity of the axis ( x = 0 , y = 0 ). To get some insight into the variations of the field, in plots (d), (e), and (f) we actually display min ( | E | , 1.5 ) . The gold layer is a horizontal slab 0.05 μ m < z < 0 μ m .

Fig. 3
Fig. 3

Plot of the modulus of the incident electric field below the gold layer, with radial polarization, y = 0 , and x focus = y focus = 0 . (a), (b) The focus is 0.85 μ m below the gold layer; (c), (d) the focus is 0.05 μ m below the gold layer. In (b) and (d), the gold layer has been removed. Dashed (respectively solid) curves correspond to the field computed using 0 ° < θ < 72 ° (respectively 41 ° < θ < 42 ° ).

Fig. 4
Fig. 4

Modulus of the electric field for a bead of radius 0.05 μ m placed at 1 nm distance from the sample (gold or glass) surface. The incident field is radially polarized. (a) Gold sample surface, gold bead. (b) Glass sample surface, gold bead. (c) Gold sample surface, dielectric bead. (d) Glass sample surface, dielectric bead. The color coding is different for the dielectric and gold beads. δ F z = 0.8 μ m .

Fig. 5
Fig. 5

Modulus of the electric field for a bead of radius 0.05 μ m placed at 1 nm distance from the sample (gold or glass) surface. The incident field is radially polarized. The data in this figure correspond to sections of Fig. 4. (a), (b) Section at x = 0 ; (c), (d) section at z = 0 . Solid (respectively dashed) curves correspond to a gold (respectively glass with n 1 = n 0 ) sample surface. (a), (c) Gold bead ( ϵ 3 = 12 + i 1.2 ) ; (b), (d) dielectric bead ( ϵ 3 = 2.56 ) .

Fig. 6
Fig. 6

SSPM optical response v ( 0 , δ F z = 0.8 ) versus the radius of the bead, for a gold bead (respectively dielectric ϵ 3 = 2.56 bead, dashed curve) placed at 1 nm distance from a gold sample surface. The incident spot is centered on the bead, and the defocus is δ F z = 0.8 μ m . Gold layer thickness, 0.05 μ m .

Fig. 7
Fig. 7

v ( ρ b F , δ F z = 0.8 ) computed for one bead, R = 0.01 μ m , made of gold (continuous curve) or dielectric ( ϵ 3 = 2.56 ) (dashed curve). The data have been rescaled so that the maxima at ρ b F = 0 coincide.

Fig. 8
Fig. 8

Complex plane (phase versus radius) plot SSPM optical response v ( ρ b F = 0 , δ F z = 0.8 ) gold (hexagons) and dielectric ( n = 1.6 ) (squares) spherical beads of different sizes. (a) In the presence of a gold layer of width = 0.05 μ m . (b) The gold layer is replaced by glass. The radius of the particles ranges from 4 to 40 nm with a step of 2 nm . The symbol size is correlated to the radius of particles. The horizontal axis is logarithmic.

Fig. 9
Fig. 9

SSPM optical response as a function of the defocus δ F z . (a) | v ( 0 , δ F z ) | in the presence of a gold bead. (b) Unperturbed V ( z ) .

Fig. 10
Fig. 10

Maximum of the modulus of the normalized optical response max δ F z | v ( ρ b F = 0 , δ F z ) | as a function of the thickness of the gold layer (micrometers). Continuous (respectively dashed) graphs are such that the incoming rays are limited to 40 ° < θ < 50 ° (respectively 0 ° < θ < 60 ° ). (a) Gold bead, R = 0.01 μ . (b) Dielectric ( n 3 = 1.6 ) bead, same size.

Equations (39)

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E inc ( X ) = d Ω ( θ , φ ) cos ( θ ) E after ( θ , φ ) e i K 0 X e i K 0 F ,
K 0 = k n 0 ( sin θ cos φ , sin θ sin φ , cos θ ) ,
E before = E 0 [ P ρ ( φ ) e ̂ ρ + P φ ( φ ) e ̂ φ ] .
E after = E 0 [ P ρ ( φ ) e ̂ θ + P φ ( φ ) e ̂ φ ] .
E inc r ( X ) = E 0 cos ( θ ) [ P ρ ( φ ) e ̂ θ r P ( θ ) + P φ e ̂ φ r S ( θ ) ] e i K 0 ( X F ) e i k 0 z ( z 2 z 1 + 2 F z ) d Ω .
E inc t ( X ) = n 0 n 2 cos ( θ ) e θ ˆ ( θ , φ ) t P ( θ ) e i K X e i K 0 F d Ω ( θ , φ ) ,
n 2 sin ( θ ) = n 0 sin ( θ ) ,
K = 2 π λ n 2 ( sin ( θ ) cos ( φ ) , sin ( θ ) sin ( φ ) , cos ( θ ) ) .
0 2 π e i n φ e i x cos ( φ φ 0 ) d φ = 2 π i | n | J | n | ( x ) e i n φ 0 .
E inc t ( X ) = 2 π n 0 n 2 cos ( θ ) ( i cos ( φ x ) cos ( θ ) J 1 ( r k 0 sin ( θ ) ) i sin ( φ x ) cos ( θ ) J 1 ( r k 0 sin ( θ ) ) sin ( θ ) J 0 ( r k 0 sin ( θ ) ) ) t P ( θ ) e i k cos ( θ ) z e i k 0 F z sin θ d θ ,
( e θ ˆ e φ ˆ ) e i K X = n , m 4 i n D n , m [ ( i m π n | m | ( θ ) τ n | m | ( θ ) ) M n , m 1 ( K X ) + ( i τ n | m | ( θ ) m π n | m | ) N n , m 1 ( K X ) ] e i m φ ,
E inc t ( X ) = n > 0 m = n n 4 i n + 1 D n , m [ a n , m M n , m 1 ( k n 2 X ) + b n , m N n , m 1 ( k n 2 X ) ] ,
a n , m = n 0 n 2 cos ( θ ) t P ( θ ) m π n | m | ( θ ) e i K 0 F e i m φ d Ω ( θ , φ ) ,
b n , m = n 0 n 2 cos ( θ ) t P ( θ ) τ n | m | ( θ ) e i K 0 F e i m φ d Ω ( θ , φ ) .
a n , m = 2 π n 0 n 2 ( i | m | ) * cos ( θ ) sin ( θ ) t P ( θ ) m π n | m | ( θ ) e i k n 0 F cos ( θ ) cos ( θ F ) J | m | ( k n 0 F sin ( θ ) sin ( θ F ) ) e i m φ F d θ ,
b n , m = 2 π n 0 n 2 ( i | m | ) * cos ( θ ) sin ( θ ) t P ( θ ) τ n | m | ( θ ) e i k n 0 F cos ( θ ) cos ( θ F ) J | m | ( k n 0 F sin ( θ ) sin ( θ F ) ) e i m φ F d θ ,
F = ( F sin ( θ F ) cos ( φ F ) , F sin ( θ F ) sin ( φ F ) , F cos ( θ F ) ) .
E sc ( X ) = n , m e n , m M n , m 3 ( n 2 k X ) + f n , m N n , m 3 ( n 2 k X ) ,
( M n m 3 R ( X ) N n m 3 R ( X ) ) = n , m ( R n m n m M M R n m n m M N R n m n m N M R n m n m N N ) ( M n m 1 ( X ) N n m 1 ( X ) ) .
E sc R = n , m n , m e n , m ( R n m n m M M M n , m 1 + R n m n m M N N n , m 1 ) + f n , m ( R n m n m N M M n , m 1 + R n m n m N N N n , m 1 ) .
( M n m 3 ( X ) N n m 3 ( X ) ) = 0 2 π d α C + d β e i m α 2 π i n + 1 e i K X [ ( i τ n | m | ( β ) i m π n | m | ( β ) ) e ̂ α + ( m π n | m | ( β ) τ n | m | ( β ) ) e ̂ β ] sin ( β ) ,
e ̂ β = ( cos ( β ) cos ( α ) , cos ( β ) sin ( α ) , sin ( β ) ) ,
e ̂ α = ( sin ( α ) , cos ( α ) , 0 ) ,
K = ( sin ( β ) cos ( α ) , sin ( β ) sin ( α ) , cos ( β ) ) .
( M n m 3 R ( X ) N n m 3 R ( X ) ) = 0 2 π d α C d β e i m α 2 π i n + 1 sin ( β ) e i K X [ ( i τ n | m | ( π β ) i m π n | m | ( π β ) ) r S b ( π β ) e ̂ α + ( m π n | m | ( π β ) τ n | m | ( π β ) ) r P b ( π β ) e ̂ β ] ,
R n , m , n , m M M = δ m m i n n C [ τ n | m | ( β π ) r S b ( β π ) τ n | m | ( β ) + m 2 π n | m | ( β π ) r P b ( β π ) π n | m | ( β ) ] sin ( β ) d β ,
e n , m = T n , m M , M ( a n , m + n , m R n , m , n , m M M e n , m + n , m R n , m , n , m N M f n , m ) ,
f n , m = T n , m N , N ( b n , m + n , m R n , m , n , m N M e n , m + n , m R n , m , n , m N N f n , m ) .
E inside ( X ) = n , m c n , m M n , m 1 ( k 3 X ) + d n , m N n , m 1 ( k 3 X ) .
E , 0 ( θ , φ , r ) = 2 π i r k 0 e i k 0 r e 2 i k 0 ( δ F z ) r P ( θ ) cos θ e ̂ θ .
M n , m 3 ( i ) n e i k r i k r e i m φ ( i m π n | m | e ̂ θ τ n | m | e ̂ φ ) ,
N n , m 3 ( i ) n + 1 e i k r i k r e i m φ ( i m π n | m | e ̂ φ τ n | m | e ̂ θ ) .
δ E ( r , θ , φ ) e i K 0 X fb r e ̂ θ n , m ( i ) n + 1 e i k r i k r e i m φ t P b ( θ ) [ m π n | m | e n , m τ n | m | f n , m ] + e ̂ φ n , m ( i ) n + 1 e i k r i k r e i m φ t S b ( θ ) [ i τ n | m | e n , m i m π n | m | f n , m ] .
V ( X b , F ) = { [ E , 0 + δ E ] e ̂ θ P ρ ( φ ) [ E , 0 + δ E ] e ̂ φ P φ ( φ ) } R obj sin ( θ ) cos ( θ ) d θ d φ .
V ( X b , F ) = [ E , 0 + δ E ] e ̂ θ R obj sin ( θ ) cos ( θ ) d θ d φ .
I 0 = max F z | E , 0 e ̂ θ R obj sin ( θ ) cos ( θ ) d θ d φ | .
f 1 , 0 2 3 ( R λ ) 3 n 3 2 n 2 2 n 3 2 + 2 n 2 2 J 0 ( k n 0 F sin ( θ p ) sin ( θ F ) ) ,
f 1 , 1 = f 1 , 1 * 2 3 ( R λ ) 3 n 3 2 n 2 2 n 3 2 + 2 n 2 2 J 1 ( k n 0 F sin ( θ p ) sin ( θ F ) ) ,
δ z = λ 2 n 0 ( 1 cos θ P ) .

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