Abstract

Generalized Mie theory is employed to study the reflection and transmission properties of finite-sized spherical arrays of nanoparticles based on aperiodic geometries. To simulate realistic experimental conditions, a circular aperture is used to create an incident field with a finite beamwidth. The diffracted fields from the circular aperture are expanded in terms of vector spherical wave harmonics, which are then employed to derive the scattered fields using the generalized Mie theory. Expansion of diffracted fields in terms of spherical harmonics also led to new analytical expressions for two important integrals involving Bessel, associated Legendre, and trigonometric functions, which arise in electromagnetic diffraction problems. Subsequently, generalized scattering parameters were defined in terms of far-field specular energy fluxes. To verify the results, the method was applied to a truncated periodic array of spherical gold nanoparticles for which the generalized scattering parameters were compared and found to agree with the scattering parameters obtained for an infinite planar structure subject to periodic boundary conditions. The method was then applied to an aperiodic array of gold nanoparticles based on a Penrose geometry, and the lowest-order photonic resonances were observed in the predicted regions. Furthermore, it was shown that by proper scaling, the photonic resonances can be strategically placed in the plasmonic region of the gold, where they are enhanced due to strong coupling between the plasmonic and photonic modes.

© 2013 Optical Society of America

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2012 (1)

C. Bauer, G. Kobiela, and H. Giessen, “2D quasiperiodic plasmonic crystals,” Sci. Rep. 2, 1–6 (2012).
[CrossRef]

2010 (2)

M. A. Yurkin, D. D. Kanter, and A. G. Hoekstra, “Accuracy of the discrete dipole approximation for simulation of optical properties of gold nanoparticles,” J. Nanophoton. 4, 041585 (2010).
[CrossRef]

S. V. Boriskina, S. Y. K. Lee, J. J. Amsden, F. G. Omenetto, and L. D. Negro, “Formation of colorimetric fingerprints on nano-patterned deterministic aperiodic surfaces,” Opt. Express 18, 14568–14576 (2010).
[CrossRef]

2009 (1)

2008 (1)

A. Gopinath, S. V. Boriskina, N. Feng, B. M. Reinhard, and L. D. Negro, “Photonic-plasmonic scattering resonances in deterministic aperiodic structures,” Nano Lett. 8, 2423–2431 (2008).
[CrossRef]

2007 (2)

P. J. Cregg and P. Svedlindh, “Comment on ‘analytical results for a Bessel function times Legendre polynomials class integrals’,” J. Phys. A Math. Theor. 40, 14029–14031 (2007).
[CrossRef]

T. Matsui, A. Agrawal, A. Nahata, and Z. V. Vardeny, “Transmission resonances through aperiodic arrays of subwavelength apertures,” Nature 446, 517–521 (2007).
[CrossRef]

2006 (1)

A. A. R. Neves, L. A. Padilha, A. Fontes, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, “Analytical results for a Bessel function times Legendre polynomials class integrals,” J. Phys. A 39, L293–L296 (2006).
[CrossRef]

2003 (1)

T. Nieminen, H. Rubinsztein-Dunlop, and N. Heckenberg, “Multipole expansion of strongly focussed laser beams,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 1005–1017 (2003).
[CrossRef]

2000 (1)

M. E. Zoorob, M. D. B. Charlton, G. J. Parker, J. J. Baumberg, and M. C. Netti, “Complete photonic bandgaps in 12-fold symmetric quasicrystals,” Nature 404, 740–743 (2000).
[CrossRef]

1999 (2)

S. S. M. Cheng, L. M. Li, C. T. Chan, and Z. Q. Zhang, “Defect and transmission properties of two-dimensional quasiperiodic photonic band-gap systems,” Phys. Rev. B 59, 4091–4099(1999).
[CrossRef]

C. Jin, B. Cheng, B. Man, Z. Li, and D. Zhang, “Band gap and wave guiding effect in a quasiperiodic photonic crystal,” Appl. Phys. Lett. 75, 1848–1850 (1999).
[CrossRef]

1998 (3)

Y. L. Xu and R. T. Wang, “Electromagnetic scattering by an aggregate of spheres: theoretical and experimental study of the amplitude scattering matrix,” Phys. Rev. E 58, 3931–3948 (1998).
[CrossRef]

Y. L. Xu, “Efficient evaluation of vector translation coefficients in multiparticle light-scattering theories,” J. Comput. Phys. 139, 137–165 (1998).
[CrossRef]

L. M. Li, Z.-Q. Zhang, and X. Zhang, “Transmission and absorption properties of two-dimensional metallic photonic-band-gap materials,” Phys. Rev. E 58, 15589–15594 (1998).

1997 (3)

1996 (1)

Y. L. Xu, “Fast evaluation of Gaunt coefficients,” Math. Comput. 65, 1601–1612 (1996).
[CrossRef]

1995 (1)

1962 (1)

O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Quart. Appl. Math. 20, 33–40 (1962).

1961 (1)

S. Stein, “Addition theorems for spherical wave functions,” Quart. Appl. Math. 19, 15–24 (1961).

1929 (1)

J. A. Gaunt, “The triplets of helium,” Phil. Trans. R. Soc. A 228, 151–196 (1929).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, 1972).

Agrawal, A.

T. Matsui, A. Agrawal, A. Nahata, and Z. V. Vardeny, “Transmission resonances through aperiodic arrays of subwavelength apertures,” Nature 446, 517–521 (2007).
[CrossRef]

Amsden, J. J.

Arfken, G.

G. Arfken, Mathematical Methods for Physicists, 3rd ed.(Academic, 1985).

Barbosa, L. C.

A. A. R. Neves, L. A. Padilha, A. Fontes, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, “Analytical results for a Bessel function times Legendre polynomials class integrals,” J. Phys. A 39, L293–L296 (2006).
[CrossRef]

Bauer, C.

C. Bauer, G. Kobiela, and H. Giessen, “2D quasiperiodic plasmonic crystals,” Sci. Rep. 2, 1–6 (2012).
[CrossRef]

Baumberg, J. J.

M. E. Zoorob, M. D. B. Charlton, G. J. Parker, J. J. Baumberg, and M. C. Netti, “Complete photonic bandgaps in 12-fold symmetric quasicrystals,” Nature 404, 740–743 (2000).
[CrossRef]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-VCH, 2004).

Boriskina, S. V.

Cai, W.

W. Cai and V. Shalaev, Optical Metamaterials (Springer, 2010).

Cesar, C. L.

A. A. R. Neves, L. A. Padilha, A. Fontes, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, “Analytical results for a Bessel function times Legendre polynomials class integrals,” J. Phys. A 39, L293–L296 (2006).
[CrossRef]

Chan, C. T.

S. S. M. Cheng, L. M. Li, C. T. Chan, and Z. Q. Zhang, “Defect and transmission properties of two-dimensional quasiperiodic photonic band-gap systems,” Phys. Rev. B 59, 4091–4099(1999).
[CrossRef]

Charlton, M. D. B.

M. E. Zoorob, M. D. B. Charlton, G. J. Parker, J. J. Baumberg, and M. C. Netti, “Complete photonic bandgaps in 12-fold symmetric quasicrystals,” Nature 404, 740–743 (2000).
[CrossRef]

Cheng, B.

C. Jin, B. Cheng, B. Man, Z. Li, and D. Zhang, “Band gap and wave guiding effect in a quasiperiodic photonic crystal,” Appl. Phys. Lett. 75, 1848–1850 (1999).
[CrossRef]

Cheng, S. S. M.

S. S. M. Cheng, L. M. Li, C. T. Chan, and Z. Q. Zhang, “Defect and transmission properties of two-dimensional quasiperiodic photonic band-gap systems,” Phys. Rev. B 59, 4091–4099(1999).
[CrossRef]

Cregg, P. J.

P. J. Cregg and P. Svedlindh, “Comment on ‘analytical results for a Bessel function times Legendre polynomials class integrals’,” J. Phys. A Math. Theor. 40, 14029–14031 (2007).
[CrossRef]

Cruz, C. H. B.

A. A. R. Neves, L. A. Padilha, A. Fontes, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, “Analytical results for a Bessel function times Legendre polynomials class integrals,” J. Phys. A 39, L293–L296 (2006).
[CrossRef]

Cruzan, O. R.

O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Quart. Appl. Math. 20, 33–40 (1962).

Deloudi, S.

W. Steurer and S. Deloudi, Crystallography of Quasicrystals: Concepts, Methods and Structures (Springer, 2009).

Doicu, A.

Feng, N.

A. Gopinath, S. V. Boriskina, N. Feng, B. M. Reinhard, and L. D. Negro, “Photonic-plasmonic scattering resonances in deterministic aperiodic structures,” Nano Lett. 8, 2423–2431 (2008).
[CrossRef]

Fontes, A.

A. A. R. Neves, L. A. Padilha, A. Fontes, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, “Analytical results for a Bessel function times Legendre polynomials class integrals,” J. Phys. A 39, L293–L296 (2006).
[CrossRef]

Gaunt, J. A.

J. A. Gaunt, “The triplets of helium,” Phil. Trans. R. Soc. A 228, 151–196 (1929).
[CrossRef]

Giessen, H.

C. Bauer, G. Kobiela, and H. Giessen, “2D quasiperiodic plasmonic crystals,” Sci. Rep. 2, 1–6 (2012).
[CrossRef]

Gopinath, A.

A. Gopinath, S. V. Boriskina, B. M. Reinhard, and L. D. Negro, “Deterministic aperiodic arrays of metal nanoparticles for surface-enhanced Raman scattering (SERS),” Opt. Express 17, 3741–3753 (2009).
[CrossRef]

A. Gopinath, S. V. Boriskina, N. Feng, B. M. Reinhard, and L. D. Negro, “Photonic-plasmonic scattering resonances in deterministic aperiodic structures,” Nano Lett. 8, 2423–2431 (2008).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed. (Academic, 2000).

Heckenberg, N.

T. Nieminen, H. Rubinsztein-Dunlop, and N. Heckenberg, “Multipole expansion of strongly focussed laser beams,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 1005–1017 (2003).
[CrossRef]

Hoekstra, A. G.

M. A. Yurkin, D. D. Kanter, and A. G. Hoekstra, “Accuracy of the discrete dipole approximation for simulation of optical properties of gold nanoparticles,” J. Nanophoton. 4, 041585 (2010).
[CrossRef]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-VCH, 2004).

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, 1975).

Jin, C.

C. Jin, B. Cheng, B. Man, Z. Li, and D. Zhang, “Band gap and wave guiding effect in a quasiperiodic photonic crystal,” Appl. Phys. Lett. 75, 1848–1850 (1999).
[CrossRef]

Kanter, D. D.

M. A. Yurkin, D. D. Kanter, and A. G. Hoekstra, “Accuracy of the discrete dipole approximation for simulation of optical properties of gold nanoparticles,” J. Nanophoton. 4, 041585 (2010).
[CrossRef]

Kobiela, G.

C. Bauer, G. Kobiela, and H. Giessen, “2D quasiperiodic plasmonic crystals,” Sci. Rep. 2, 1–6 (2012).
[CrossRef]

Lee, S. Y. K.

Li, L. M.

S. S. M. Cheng, L. M. Li, C. T. Chan, and Z. Q. Zhang, “Defect and transmission properties of two-dimensional quasiperiodic photonic band-gap systems,” Phys. Rev. B 59, 4091–4099(1999).
[CrossRef]

L. M. Li, Z.-Q. Zhang, and X. Zhang, “Transmission and absorption properties of two-dimensional metallic photonic-band-gap materials,” Phys. Rev. E 58, 15589–15594 (1998).

Li, Z.

C. Jin, B. Cheng, B. Man, Z. Li, and D. Zhang, “Band gap and wave guiding effect in a quasiperiodic photonic crystal,” Appl. Phys. Lett. 75, 1848–1850 (1999).
[CrossRef]

Man, B.

C. Jin, B. Cheng, B. Man, Z. Li, and D. Zhang, “Band gap and wave guiding effect in a quasiperiodic photonic crystal,” Appl. Phys. Lett. 75, 1848–1850 (1999).
[CrossRef]

Matsui, T.

T. Matsui, A. Agrawal, A. Nahata, and Z. V. Vardeny, “Transmission resonances through aperiodic arrays of subwavelength apertures,” Nature 446, 517–521 (2007).
[CrossRef]

Nahata, A.

T. Matsui, A. Agrawal, A. Nahata, and Z. V. Vardeny, “Transmission resonances through aperiodic arrays of subwavelength apertures,” Nature 446, 517–521 (2007).
[CrossRef]

Negro, L. D.

Netti, M. C.

M. E. Zoorob, M. D. B. Charlton, G. J. Parker, J. J. Baumberg, and M. C. Netti, “Complete photonic bandgaps in 12-fold symmetric quasicrystals,” Nature 404, 740–743 (2000).
[CrossRef]

Neves, A. A. R.

A. A. R. Neves, L. A. Padilha, A. Fontes, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, “Analytical results for a Bessel function times Legendre polynomials class integrals,” J. Phys. A 39, L293–L296 (2006).
[CrossRef]

Nieminen, T.

T. Nieminen, H. Rubinsztein-Dunlop, and N. Heckenberg, “Multipole expansion of strongly focussed laser beams,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 1005–1017 (2003).
[CrossRef]

Omenetto, F. G.

Padilha, L. A.

A. A. R. Neves, L. A. Padilha, A. Fontes, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, “Analytical results for a Bessel function times Legendre polynomials class integrals,” J. Phys. A 39, L293–L296 (2006).
[CrossRef]

Parker, G. J.

M. E. Zoorob, M. D. B. Charlton, G. J. Parker, J. J. Baumberg, and M. C. Netti, “Complete photonic bandgaps in 12-fold symmetric quasicrystals,” Nature 404, 740–743 (2000).
[CrossRef]

Reinhard, B. M.

A. Gopinath, S. V. Boriskina, B. M. Reinhard, and L. D. Negro, “Deterministic aperiodic arrays of metal nanoparticles for surface-enhanced Raman scattering (SERS),” Opt. Express 17, 3741–3753 (2009).
[CrossRef]

A. Gopinath, S. V. Boriskina, N. Feng, B. M. Reinhard, and L. D. Negro, “Photonic-plasmonic scattering resonances in deterministic aperiodic structures,” Nano Lett. 8, 2423–2431 (2008).
[CrossRef]

Rodriguez, E.

A. A. R. Neves, L. A. Padilha, A. Fontes, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, “Analytical results for a Bessel function times Legendre polynomials class integrals,” J. Phys. A 39, L293–L296 (2006).
[CrossRef]

Rubinsztein-Dunlop, H.

T. Nieminen, H. Rubinsztein-Dunlop, and N. Heckenberg, “Multipole expansion of strongly focussed laser beams,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 1005–1017 (2003).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed. (Academic, 2000).

Senechal, M.

M. Senechal, Quasicrystals and Geometry (Cambridge University, 1996).

Shalaev, V.

W. Cai and V. Shalaev, Optical Metamaterials (Springer, 2010).

Stegun, I. A.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, 1972).

Stein, S.

S. Stein, “Addition theorems for spherical wave functions,” Quart. Appl. Math. 19, 15–24 (1961).

Steurer, W.

W. Steurer and S. Deloudi, Crystallography of Quasicrystals: Concepts, Methods and Structures (Springer, 2009).

Svedlindh, P.

P. J. Cregg and P. Svedlindh, “Comment on ‘analytical results for a Bessel function times Legendre polynomials class integrals’,” J. Phys. A Math. Theor. 40, 14029–14031 (2007).
[CrossRef]

Vardeny, Z. V.

T. Matsui, A. Agrawal, A. Nahata, and Z. V. Vardeny, “Transmission resonances through aperiodic arrays of subwavelength apertures,” Nature 446, 517–521 (2007).
[CrossRef]

Wang, R. T.

Y. L. Xu and R. T. Wang, “Electromagnetic scattering by an aggregate of spheres: theoretical and experimental study of the amplitude scattering matrix,” Phys. Rev. E 58, 3931–3948 (1998).
[CrossRef]

Wriedt, T.

Xu, Y. L.

Y. L. Xu, “Efficient evaluation of vector translation coefficients in multiparticle light-scattering theories,” J. Comput. Phys. 139, 137–165 (1998).
[CrossRef]

Y. L. Xu and R. T. Wang, “Electromagnetic scattering by an aggregate of spheres: theoretical and experimental study of the amplitude scattering matrix,” Phys. Rev. E 58, 3931–3948 (1998).
[CrossRef]

Y. L. Xu, “Electromagnetic scattering by an aggregate of spheres: far field,” Appl. Opt. 36, 9496–9508 (1997).
[CrossRef]

Y. L. Xu, “Fast evaluation of Gaunt coefficients: recursive approach,” J. Comput. Appl. Math. 85, 53–65 (1997).
[CrossRef]

Y. L. Xu, “Fast evaluation of Gaunt coefficients,” Math. Comput. 65, 1601–1612 (1996).
[CrossRef]

Y. L. Xu, “Electromagnetic scattering by an aggregate of spheres,” Appl. Opt. 34, 4573–4588 (1995).
[CrossRef]

Yurkin, M. A.

M. A. Yurkin, D. D. Kanter, and A. G. Hoekstra, “Accuracy of the discrete dipole approximation for simulation of optical properties of gold nanoparticles,” J. Nanophoton. 4, 041585 (2010).
[CrossRef]

Zhang, D.

C. Jin, B. Cheng, B. Man, Z. Li, and D. Zhang, “Band gap and wave guiding effect in a quasiperiodic photonic crystal,” Appl. Phys. Lett. 75, 1848–1850 (1999).
[CrossRef]

Zhang, X.

L. M. Li, Z.-Q. Zhang, and X. Zhang, “Transmission and absorption properties of two-dimensional metallic photonic-band-gap materials,” Phys. Rev. E 58, 15589–15594 (1998).

Zhang, Z. Q.

S. S. M. Cheng, L. M. Li, C. T. Chan, and Z. Q. Zhang, “Defect and transmission properties of two-dimensional quasiperiodic photonic band-gap systems,” Phys. Rev. B 59, 4091–4099(1999).
[CrossRef]

Zhang, Z.-Q.

L. M. Li, Z.-Q. Zhang, and X. Zhang, “Transmission and absorption properties of two-dimensional metallic photonic-band-gap materials,” Phys. Rev. E 58, 15589–15594 (1998).

Zoorob, M. E.

M. E. Zoorob, M. D. B. Charlton, G. J. Parker, J. J. Baumberg, and M. C. Netti, “Complete photonic bandgaps in 12-fold symmetric quasicrystals,” Nature 404, 740–743 (2000).
[CrossRef]

Appl. Opt. (3)

Appl. Phys. Lett. (1)

C. Jin, B. Cheng, B. Man, Z. Li, and D. Zhang, “Band gap and wave guiding effect in a quasiperiodic photonic crystal,” Appl. Phys. Lett. 75, 1848–1850 (1999).
[CrossRef]

J. Comput. Appl. Math. (1)

Y. L. Xu, “Fast evaluation of Gaunt coefficients: recursive approach,” J. Comput. Appl. Math. 85, 53–65 (1997).
[CrossRef]

J. Comput. Phys. (1)

Y. L. Xu, “Efficient evaluation of vector translation coefficients in multiparticle light-scattering theories,” J. Comput. Phys. 139, 137–165 (1998).
[CrossRef]

J. Nanophoton. (1)

M. A. Yurkin, D. D. Kanter, and A. G. Hoekstra, “Accuracy of the discrete dipole approximation for simulation of optical properties of gold nanoparticles,” J. Nanophoton. 4, 041585 (2010).
[CrossRef]

J. Phys. A (1)

A. A. R. Neves, L. A. Padilha, A. Fontes, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, “Analytical results for a Bessel function times Legendre polynomials class integrals,” J. Phys. A 39, L293–L296 (2006).
[CrossRef]

J. Phys. A Math. Theor. (1)

P. J. Cregg and P. Svedlindh, “Comment on ‘analytical results for a Bessel function times Legendre polynomials class integrals’,” J. Phys. A Math. Theor. 40, 14029–14031 (2007).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer (1)

T. Nieminen, H. Rubinsztein-Dunlop, and N. Heckenberg, “Multipole expansion of strongly focussed laser beams,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 1005–1017 (2003).
[CrossRef]

Math. Comput. (1)

Y. L. Xu, “Fast evaluation of Gaunt coefficients,” Math. Comput. 65, 1601–1612 (1996).
[CrossRef]

Nano Lett. (1)

A. Gopinath, S. V. Boriskina, N. Feng, B. M. Reinhard, and L. D. Negro, “Photonic-plasmonic scattering resonances in deterministic aperiodic structures,” Nano Lett. 8, 2423–2431 (2008).
[CrossRef]

Nature (2)

T. Matsui, A. Agrawal, A. Nahata, and Z. V. Vardeny, “Transmission resonances through aperiodic arrays of subwavelength apertures,” Nature 446, 517–521 (2007).
[CrossRef]

M. E. Zoorob, M. D. B. Charlton, G. J. Parker, J. J. Baumberg, and M. C. Netti, “Complete photonic bandgaps in 12-fold symmetric quasicrystals,” Nature 404, 740–743 (2000).
[CrossRef]

Opt. Express (2)

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[CrossRef]

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[CrossRef]

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Figures (6)

Fig. 1.
Fig. 1.

Geometry of an aperiodic AB tiling.

Fig. 2.
Fig. 2.

Normalized scattered field magnitude (dB) in the plane of the AB aperiodic array illuminated by a plane wave.

Fig. 3.
Fig. 3.

Normalized scattered field magnitude (dB) in the plane of the AB aperiodic array illuminated by circular aperture diffracted waves.

Fig. 4.
Fig. 4.

Scattering response of infinite (solid lines) and finite (dashed lines) periodic gold arrays obtained by CST MICROWAVE STUDIO and GMT with a finite beamwidth calculated using Eqs. (34) and (37).

Fig. 5.
Fig. 5.

Geometry of a Penrose aperiodic tiling composed of narrow (yellow) and wide (green) rhombi.

Fig. 6.
Fig. 6.

Scattering response of two finite aperiodic Penrose gold arrays with different tile sides (540 and 630 nm) obtained using GMT with a finite incident beamwidth produced by a circular aperture of radius a=2λ placed at a distance of 17 μm from the array. Values of T and R were calculated from Eqs. (34) and (37), respectively.

Equations (61)

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Eθ=iakeiρJ1(kasinθ)cosϕρsinθ,Eϕ=iakeiρJ1(kasinθ)sinϕcosθρsinθ,
Einc=n=1m=nniEmn[pmnNmn(1)+qmnMmn(1)].
Nmn(1)={r^n(n+1)Pnm(cosθ)jn(ρ)ρ+[θ^τmn(cosθ)+ϕ^iπmn(cosθ)]ψn(ρ)ρ}eimϕ,Mmn(1)=[θ^iπmn(cosθ)ϕ^τmn(cosθ)]jn(ρ)eimϕ,Emn=in[(2n+1)(nm)!n(n+1)(n+m)!]1/2,
πmn(cosθ)=msinθPnm(cosθ),τmn(cosθ)=ddθPnm(cosθ).
Pnm(x)=(1x2)m/2dmdxmPn(x),
qmn=i02π0πEinc·Mmn(1)*sinθdθdϕdρEmn02π0π|Mmn(1)|2sinθdθdϕdρ,
pmn=i02π0πEinc·Nmn(1)*sinθdθdϕdρEmn02π0π|Nmn(1)|2sinθdθdϕdρ.
jn(ρ)jm(ρ)dρ=π2n+1δnm,
Eθ=iakJ1(kasinθ)cosϕ[j1(ρ)+ij2(ρ)]sinθ,Eϕ=iakJ1(kasinθ)sinϕcosθ[j1(ρ)+ij2(ρ)]sinθ.
0π/2(J1(kasinθ)[π1n+cosθτ1n])dθ.
0π/2(J1(kasinθ)[π1n+cosθτ1n])dθ=n(n+1)Ωn(ka)+Ξn(ka).
02π0π|M1n(1)(ρ,θ,ϕ)|2sinθdθdϕ=4πn2(n+1)22n+1jn2(ρ).
q1nin2n(n+1)2n+1jn2(ρ)=akijn(ρ)[j1(ρ)+ij2(ρ)][n(n+1)Ωn(ka)+Ξn(ka)].
limρj1(ρ)=in1jn(ρ)ifnis odd,limρj2(ρ)=in2jn(ρ)ifnis even.
q1n=ak2n+1{n(n+1)Ωn(ka)+Ξn(ka)}2n(n+1).
0π/2(J1(kasinθ)[τ1n+cosθπ1n])dθ.
0π/2(J1(kasinθ)[τ1n+cosθπ1n])dθ=n(n+1)Ψn(ka).
p1n=ak2n+1Ψn(ka)4.
pmnj=exp(ikZj)pmn,qmnj=exp(ikZj)qmn.
pmnj=ν=1μ=ννEμνEmn[pμνAmnμν+qμνBmnμν],qmnj=ν=1μ=ννEμνEmn[pμνBmnμν+qμνAmnμν],
a(m,n,μ,ν,p)=(2p+1)2(pmμ)!(p+m+μ)!11Pnm(x)Pνμ(x)Ppm+μ(x)dx.
ESj=n=1Nmaxm=nniEmn[amnjNmn(3)+bmnjMmn(3)],
EIj=n=1Nmaxm=nniEmn[dmnjNmn(1)+cmnjMmn(1)],
ES(ρ,θ,ϕ)=n=1Nmaxm=nniEmn[amnNmn(3)+bmnMmn(3)],
amn=j=1Lν=1μ=νν[aμνjAmnμν+bμνjBmnμν],bmn=j=1Lν=1μ=νν[aμνjBmnμν+bμνjAmnμν].
NW=x+4x1/3+2,
ϵ(ω)=ϵωp2ω2+iΓω
θαθ0,D=rαcos(θα)
amn=j=1Lexp(ikΔj)amnj,bmn=j=1Lexp(ikΔj)bmnj,
ES(ρ,0,0)=ieiρρn=1Nmax2n+1[a1n+b1n]θ^.
Einc(ρ,0,0)=i(ak)2eiρ2ρθ^.
ETotal(ρ,0,0)=ieiρρ((ak)22+n=1Nmax2n+1[a1n+b1n]).
T=|ETotal(ρ,0,0)Einc(ρ,0,0)|2.
T=|12(ak)2n=1Nmax2n+1[a1n+b1n]|2.
R=|ES(ρ,π,0)Einc(ρ,0,0)|2.
ES(ρ,π,0)=ieiρρn=1Nmax(1)n2n+1[a1nb1n]θ^.
R=|2(ak)2n=1Nmax(1)n2n+1[a1nb1n]|2.
J1(kasinθ)[π1n+cosθτ1n]=J1(kasinθ)[Pn1(cosθ)sinθ+cosθdPn1(cosθ)dθ].
ddθ(sinθdPndθ)=n(n+1)sinθPn.
Pn1sinθ+cosθdPn1dθ=n(n+1)cosθPn+sinθPn1.
n(n+1)0π/2J1(kasinθ)cosθPndθ+0π/2J1(kasinθ)sinθPn1dθ.
Ωn(ka)0π/2J1(kasinθ)cosθPndθ,
Ξn(ka)0π/2J1(kasinθ)sinθPn1dθ.
cosθPn=12nt=0n2(1)t(2n2t)!(cosθ)(n2t+1)t!(nt)!(n2t)!,
0π/2(cosθ)(n2t+1)J1(kasinθ)dθ.
0π/2Jμ(zsinθ)(sinθ)1μ(cosθ)2ν+1dθ=s(μ+ν,νμ+1)(z)2μ1zν+1Γ(μ),
sμ,ν(z)=π2[Yν(z)0zzμJν(z)dzJν(z)0zzμYν(z)dz],
0π/2cosθ(n2t+1)J1(kasinθ)dθ=s(1t+n2,n2t)(ka)(ka)(1t+n2).
sν+1,ν(z)=π2[Yν(z)0zzν+1Jν(z)dzJν(z)0zzν+1Yν(z)dz].
xp+1Zp(x)dx=xp+1Zp+1(x),
sν+1,ν(z)=zν2νΓ(ν+1)Jν(z).
Ωn(ka)=1ka12nt=0n2(1)t(2n2t)!t!(nt)!(n2t)!Γ(1t+n2)2(n2t)J(n2t)(ka)(ka)(1t+n2).
J1(kasinθ)sin2θ2nt=0n12(1)t(2n2t)!(cosθ)(n2t1)t!(nt)!(n2t1)!.
0π/2Jμ(αsinθ)(sinθ)μ+1(cosθ)2ρ+1dθ=2ρΓ(ρ+1)α(ρ1)J(ρ+μ+1)(α).
0π/2J1(kasinθ)(sin2θ)(cosθ)(n2t1)dθ=2(n2t1)Γ(n2t)(ka)(tn2)J(n2t+1)(ka).
Ξn(ka)=12nt=0n12(1)t(2n2t)!2(n2t1)Γ(n2t)(ka)(tn2)J(n2t+1)(ka)t!(nt)!(n2t1)!.
0π/2J1(kasinθ)sinθ(dPn1dθ+cosθPn1sinθ)sinθdθ.
(dPn1dθ+cosθPn1sinθ)sinθ=ddθ(sinθPn1).
n(n+1)0π/2J1(kasinθ)Pndθn(n+1)Ψn(ka),
0π/2J1(kasinθ)PndθΨn(ka).
Ψn(ka)=1ka12nt=0n2(1)t(2n2t)!t!(nt)!(n2t)!Γ(n+12t)2(n12t)J(n12t)(ka)(ka)(n+12t).

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