Abstract

The performance of the knife-edge method as a beam profiling technique for tightly focused light beams depends on several parameters, such as the material and height of the knife-pad as well as the polarization and wavelength of the focused light beam under study. Here we demonstrate that the choice of the substrate the knife-pads are fabricated on has a crucial influence on the reconstructed beam projections as well. We employ an analytical model for the interaction of the knife-pad with the beam and report good agreement between our numerical and experimental results. Moreover, we simplify the analytical model and demonstrate, in which way the underlying physical effects lead to the apparent polarization dependent beam shifts and changes of the beamwidth for different substrate materials and heights of the knife-pad.

© 2016 Optical Society of America

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References

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    [Crossref]
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  24. COMPONENTS LASER Germany GmbH, Si-Diode PDB-C601-1.
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2015 (1)

F. Huang, V. A. Tamma, Z. Mardy, J. Burdett, and H. Kumar Wickramasinghe, “Imaging nanoscale electromagnetic near-field distributions using optical forces,” Sci. Rep. 5, 10610 (2015).
[Crossref] [PubMed]

2014 (2)

T. Bauer, S. Orlov, U. Peschel, P. Banzer, and G. Leuchs, “Nanointerferometric amplitude and phase reconstruction of tightly focused vector beams,” Nature Photon. 8, 23–27 (2014).
[Crossref]

B. le Feber, N. Rotenberg, D. M. Beggs, and L. Kuipers, “Simultaneous measurement of nanoscale electric and magnetic optical fields,” Nature Photon. 8, 43–46 (2014).
[Crossref]

2013 (2)

2011 (1)

2010 (7)

2009 (2)

M. Burresi, R. J. P. Engelen, A. Opheij, D. van Oosten, D. Mori, T. Baba, and L. Kuipers, “Observation of polarization singularities at the nanoscale,” Phys. Rev. Lett. 102, 033902 (2009).
[Crossref] [PubMed]

M. W. Knight, Y. Wu, J. B. Lassiter, P. Nordlander, and N. J. Halas, “Substrates matter: influence of an adjacent dielectric on an individual plasmonic nanoparticle,” Nano Lett. 9(5), 2188–2192 (2009).
[Crossref] [PubMed]

2007 (1)

B. Sturman, E. Podivilov, and M. Gorkunov, “Eigenmodes for metal-dielectric light-transmitting nanostructures,” Phys. Rev. B 76, 125104 (2007).
[Crossref]

2006 (1)

2005 (1)

H. L. Offerhaus, B. van den Bergen, M. Escalante, F. B. Segerink, J. P. Korterik, and N. F. van Hulst, “Creating focused plasmons by noncollinear phasematching on functional gratings,” Nano Lett. 5(11), 2144–2148 (2005).
[Crossref] [PubMed]

2003 (2)

R. Dorn, S. Quabis, and G. Leuchs, “The focus of light - linear polarization breaks the rotational symmetry of the focal spot,” J. Mod. Opt. 50(12), 1917–1926 (2003).

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref] [PubMed]

2002 (1)

2001 (1)

N. Huse, A. Schönle, and S. W. Hell, “Z-polarized confocal microscopy,” J. Biomed. Opt. 6(3), 273–276 (2001).
[Crossref] [PubMed]

1981 (1)

1977 (1)

1971 (1)

1959 (2)

E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. London Ser. A 253, 349–357 (1959).
[Crossref]

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

1909 (1)

P. Debye, “Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie,” Annalen der Physik 335(14), 755–776 (1909).
[Crossref]

Agio, M.

Aizpurua, J.

M. Schnell, A. Garcia-Etxarri, J. Alkorta, J. Aizpurua, and R. Hillenbrand, “Phase-resolved mapping of the near-field vector and polarization state in nanoscale antenna gaps,” Nano Lett. 10(9), 3524–3528 (2010).
[Crossref] [PubMed]

Alkorta, J.

M. Schnell, A. Garcia-Etxarri, J. Alkorta, J. Aizpurua, and R. Hillenbrand, “Phase-resolved mapping of the near-field vector and polarization state in nanoscale antenna gaps,” Nano Lett. 10(9), 3524–3528 (2010).
[Crossref] [PubMed]

Arnaud, J. A.

Auguié, B.

B. Auguié, X. M. Bendaña, W. L. Barnes, and F. Javier Garcia de Abajo, “Diffractive arrays of gold nanoparticles near an interface: critical role of the substrate,” Phys. Rev. B 82(15), 155447 (2010).
[Crossref]

Baba, T.

M. Burresi, R. J. P. Engelen, A. Opheij, D. van Oosten, D. Mori, T. Baba, and L. Kuipers, “Observation of polarization singularities at the nanoscale,” Phys. Rev. Lett. 102, 033902 (2009).
[Crossref] [PubMed]

Banzer, P.

Barnes, W. L.

B. Auguié, X. M. Bendaña, W. L. Barnes, and F. Javier Garcia de Abajo, “Diffractive arrays of gold nanoparticles near an interface: critical role of the substrate,” Phys. Rev. B 82(15), 155447 (2010).
[Crossref]

Bauer, T.

T. Bauer, S. Orlov, U. Peschel, P. Banzer, and G. Leuchs, “Nanointerferometric amplitude and phase reconstruction of tightly focused vector beams,” Nature Photon. 8, 23–27 (2014).
[Crossref]

Beggs, D. M.

B. le Feber, N. Rotenberg, D. M. Beggs, and L. Kuipers, “Simultaneous measurement of nanoscale electric and magnetic optical fields,” Nature Photon. 8, 43–46 (2014).
[Crossref]

Bendaña, X. M.

B. Auguié, X. M. Bendaña, W. L. Barnes, and F. Javier Garcia de Abajo, “Diffractive arrays of gold nanoparticles near an interface: critical role of the substrate,” Phys. Rev. B 82(15), 155447 (2010).
[Crossref]

Biss, D. P.

Boer-Duchemin, E.

Brown, T. G.

Burdett, J.

F. Huang, V. A. Tamma, Z. Mardy, J. Burdett, and H. Kumar Wickramasinghe, “Imaging nanoscale electromagnetic near-field distributions using optical forces,” Sci. Rep. 5, 10610 (2015).
[Crossref] [PubMed]

Burr, G.

Burresi, M.

M. Burresi, R. J. P. Engelen, A. Opheij, D. van Oosten, D. Mori, T. Baba, and L. Kuipers, “Observation of polarization singularities at the nanoscale,” Phys. Rev. Lett. 102, 033902 (2009).
[Crossref] [PubMed]

Charraut, D.

Chen, X.-W.

Comtet, G.

de la Claviere, B.

Debye, P.

P. Debye, “Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie,” Annalen der Physik 335(14), 755–776 (1909).
[Crossref]

Dorn, R.

R. Dorn, S. Quabis, and G. Leuchs, “The focus of light - linear polarization breaks the rotational symmetry of the focal spot,” J. Mod. Opt. 50(12), 1917–1926 (2003).

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref] [PubMed]

Dujardin, G.

Engelen, R. J. P.

M. Burresi, R. J. P. Engelen, A. Opheij, D. van Oosten, D. Mori, T. Baba, and L. Kuipers, “Observation of polarization singularities at the nanoscale,” Phys. Rev. Lett. 102, 033902 (2009).
[Crossref] [PubMed]

Escalante, M.

H. L. Offerhaus, B. van den Bergen, M. Escalante, F. B. Segerink, J. P. Korterik, and N. F. van Hulst, “Creating focused plasmons by noncollinear phasematching on functional gratings,” Nano Lett. 5(11), 2144–2148 (2005).
[Crossref] [PubMed]

Firester, A. H.

Franke, E. A.

Franke, J. M.

Garcia-Etxarri, A.

M. Schnell, A. Garcia-Etxarri, J. Alkorta, J. Aizpurua, and R. Hillenbrand, “Phase-resolved mapping of the near-field vector and polarization state in nanoscale antenna gaps,” Nano Lett. 10(9), 3524–3528 (2010).
[Crossref] [PubMed]

Gorkunov, M.

B. Sturman, E. Podivilov, and M. Gorkunov, “Eigenmodes for metal-dielectric light-transmitting nanostructures,” Phys. Rev. B 76, 125104 (2007).
[Crossref]

Grosjean, T.

Gruber, C.

Halas, N. J.

M. W. Knight, Y. Wu, J. B. Lassiter, P. Nordlander, and N. J. Halas, “Substrates matter: influence of an adjacent dielectric on an individual plasmonic nanoparticle,” Nano Lett. 9(5), 2188–2192 (2009).
[Crossref] [PubMed]

Hell, S. W.

N. Huse, A. Schönle, and S. W. Hell, “Z-polarized confocal microscopy,” J. Biomed. Opt. 6(3), 273–276 (2001).
[Crossref] [PubMed]

Heller, M. E.

Hillenbrand, R.

M. Schnell, A. Garcia-Etxarri, J. Alkorta, J. Aizpurua, and R. Hillenbrand, “Phase-resolved mapping of the near-field vector and polarization state in nanoscale antenna gaps,” Nano Lett. 10(9), 3524–3528 (2010).
[Crossref] [PubMed]

Hohenau, A.

Huang, F.

F. Huang, V. A. Tamma, Z. Mardy, J. Burdett, and H. Kumar Wickramasinghe, “Imaging nanoscale electromagnetic near-field distributions using optical forces,” Sci. Rep. 5, 10610 (2015).
[Crossref] [PubMed]

Hubbard, W. M.

Huber, C.

Huse, N.

N. Huse, A. Schönle, and S. W. Hell, “Z-polarized confocal microscopy,” J. Biomed. Opt. 6(3), 273–276 (2001).
[Crossref] [PubMed]

Ibrahim, I.

Javier Garcia de Abajo, F.

B. Auguié, X. M. Bendaña, W. L. Barnes, and F. Javier Garcia de Abajo, “Diffractive arrays of gold nanoparticles near an interface: critical role of the substrate,” Phys. Rev. B 82(15), 155447 (2010).
[Crossref]

Jeppesen, C.

C. Jeppesen, N. A. Mortensen, and A. Kristensen, “The effect of Ti and ITO adhesion layers on gold split-ring resonators,” Appl. Phys. Lett. 97, 263103 (2010).
[Crossref]

Kindler, J.

Knight, M. W.

M. W. Knight, Y. Wu, J. B. Lassiter, P. Nordlander, and N. J. Halas, “Substrates matter: influence of an adjacent dielectric on an individual plasmonic nanoparticle,” Nano Lett. 9(5), 2188–2192 (2009).
[Crossref] [PubMed]

Korterik, J. P.

H. L. Offerhaus, B. van den Bergen, M. Escalante, F. B. Segerink, J. P. Korterik, and N. F. van Hulst, “Creating focused plasmons by noncollinear phasematching on functional gratings,” Nano Lett. 5(11), 2144–2148 (2005).
[Crossref] [PubMed]

Krenn, J.

Kristensen, A.

C. Jeppesen, N. A. Mortensen, and A. Kristensen, “The effect of Ti and ITO adhesion layers on gold split-ring resonators,” Appl. Phys. Lett. 97, 263103 (2010).
[Crossref]

Kuipers, L.

B. le Feber, N. Rotenberg, D. M. Beggs, and L. Kuipers, “Simultaneous measurement of nanoscale electric and magnetic optical fields,” Nature Photon. 8, 43–46 (2014).
[Crossref]

M. Burresi, R. J. P. Engelen, A. Opheij, D. van Oosten, D. Mori, T. Baba, and L. Kuipers, “Observation of polarization singularities at the nanoscale,” Phys. Rev. Lett. 102, 033902 (2009).
[Crossref] [PubMed]

Kumar Wickramasinghe, H.

F. Huang, V. A. Tamma, Z. Mardy, J. Burdett, and H. Kumar Wickramasinghe, “Imaging nanoscale electromagnetic near-field distributions using optical forces,” Sci. Rep. 5, 10610 (2015).
[Crossref] [PubMed]

Lassiter, J. B.

M. W. Knight, Y. Wu, J. B. Lassiter, P. Nordlander, and N. J. Halas, “Substrates matter: influence of an adjacent dielectric on an individual plasmonic nanoparticle,” Nano Lett. 9(5), 2188–2192 (2009).
[Crossref] [PubMed]

le Feber, B.

B. le Feber, N. Rotenberg, D. M. Beggs, and L. Kuipers, “Simultaneous measurement of nanoscale electric and magnetic optical fields,” Nature Photon. 8, 43–46 (2014).
[Crossref]

Le Moal, E.

Leuchs, G.

Maier, S. A.

S. A. Maier, Plasmonics: fundamentals and applications (Springer-Verlag, Berlin, 2007).

Mandeville, G. D.

Marchenko, P.

Mardy, Z.

F. Huang, V. A. Tamma, Z. Mardy, J. Burdett, and H. Kumar Wickramasinghe, “Imaging nanoscale electromagnetic near-field distributions using optical forces,” Sci. Rep. 5, 10610 (2015).
[Crossref] [PubMed]

Mivelle, M.

Mori, D.

M. Burresi, R. J. P. Engelen, A. Opheij, D. van Oosten, D. Mori, T. Baba, and L. Kuipers, “Observation of polarization singularities at the nanoscale,” Phys. Rev. Lett. 102, 033902 (2009).
[Crossref] [PubMed]

Mortensen, N. A.

C. Jeppesen, N. A. Mortensen, and A. Kristensen, “The effect of Ti and ITO adhesion layers on gold split-ring resonators,” Appl. Phys. Lett. 97, 263103 (2010).
[Crossref]

Nordlander, P.

M. W. Knight, Y. Wu, J. B. Lassiter, P. Nordlander, and N. J. Halas, “Substrates matter: influence of an adjacent dielectric on an individual plasmonic nanoparticle,” Nano Lett. 9(5), 2188–2192 (2009).
[Crossref] [PubMed]

Nugent, K. A.

Offerhaus, H. L.

H. L. Offerhaus, B. van den Bergen, M. Escalante, F. B. Segerink, J. P. Korterik, and N. F. van Hulst, “Creating focused plasmons by noncollinear phasematching on functional gratings,” Nano Lett. 5(11), 2144–2148 (2005).
[Crossref] [PubMed]

Opheij, A.

M. Burresi, R. J. P. Engelen, A. Opheij, D. van Oosten, D. Mori, T. Baba, and L. Kuipers, “Observation of polarization singularities at the nanoscale,” Phys. Rev. Lett. 102, 033902 (2009).
[Crossref] [PubMed]

Orlov, S.

Peschel, U.

Podivilov, E.

B. Sturman, E. Podivilov, and M. Gorkunov, “Eigenmodes for metal-dielectric light-transmitting nanostructures,” Phys. Rev. B 76, 125104 (2007).
[Crossref]

Quabis, S.

Rhodes, S. K.

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. Lond. A 253, 358–379 (1959).
[Crossref]

Roberts, A.

Rogez, B.

Rotenberg, N.

B. le Feber, N. Rotenberg, D. M. Beggs, and L. Kuipers, “Simultaneous measurement of nanoscale electric and magnetic optical fields,” Nature Photon. 8, 43–46 (2014).
[Crossref]

Rowland, S. W.

S. W. Rowland, “Computer implementation of image reconstruction formulas,” in Image reconstructions from projections, G. T. Herman, ed. (Springer-Verlag, Berlin, 1979), pp. 978.

Sandoghdar, V.

Schneider, M. B.

Schnell, M.

M. Schnell, A. Garcia-Etxarri, J. Alkorta, J. Aizpurua, and R. Hillenbrand, “Phase-resolved mapping of the near-field vector and polarization state in nanoscale antenna gaps,” Nano Lett. 10(9), 3524–3528 (2010).
[Crossref] [PubMed]

Schönle, A.

N. Huse, A. Schönle, and S. W. Hell, “Z-polarized confocal microscopy,” J. Biomed. Opt. 6(3), 273–276 (2001).
[Crossref] [PubMed]

Segerink, F. B.

H. L. Offerhaus, B. van den Bergen, M. Escalante, F. B. Segerink, J. P. Korterik, and N. F. van Hulst, “Creating focused plasmons by noncollinear phasematching on functional gratings,” Nano Lett. 5(11), 2144–2148 (2005).
[Crossref] [PubMed]

Sheng, P.

Sturman, B.

B. Sturman, E. Podivilov, and M. Gorkunov, “Eigenmodes for metal-dielectric light-transmitting nanostructures,” Phys. Rev. B 76, 125104 (2007).
[Crossref]

Suarez, M.

Tamma, V. A.

F. Huang, V. A. Tamma, Z. Mardy, J. Burdett, and H. Kumar Wickramasinghe, “Imaging nanoscale electromagnetic near-field distributions using optical forces,” Sci. Rep. 5, 10610 (2015).
[Crossref] [PubMed]

van den Bergen, B.

H. L. Offerhaus, B. van den Bergen, M. Escalante, F. B. Segerink, J. P. Korterik, and N. F. van Hulst, “Creating focused plasmons by noncollinear phasematching on functional gratings,” Nano Lett. 5(11), 2144–2148 (2005).
[Crossref] [PubMed]

van Hulst, N. F.

H. L. Offerhaus, B. van den Bergen, M. Escalante, F. B. Segerink, J. P. Korterik, and N. F. van Hulst, “Creating focused plasmons by noncollinear phasematching on functional gratings,” Nano Lett. 5(11), 2144–2148 (2005).
[Crossref] [PubMed]

van Oosten, D.

M. Burresi, R. J. P. Engelen, A. Opheij, D. van Oosten, D. Mori, T. Baba, and L. Kuipers, “Observation of polarization singularities at the nanoscale,” Phys. Rev. Lett. 102, 033902 (2009).
[Crossref] [PubMed]

Wang, T.

Webb, W. W.

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. Lond. A 253, 358–379 (1959).
[Crossref]

E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. London Ser. A 253, 349–357 (1959).
[Crossref]

Wu, Y.

M. W. Knight, Y. Wu, J. B. Lassiter, P. Nordlander, and N. J. Halas, “Substrates matter: influence of an adjacent dielectric on an individual plasmonic nanoparticle,” Nano Lett. 9(5), 2188–2192 (2009).
[Crossref] [PubMed]

Youngworth, K. S.

Zhang, Y.

Annalen der Physik (1)

P. Debye, “Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie,” Annalen der Physik 335(14), 755–776 (1909).
[Crossref]

Appl. Opt. (4)

Appl. Phys. Lett. (1)

C. Jeppesen, N. A. Mortensen, and A. Kristensen, “The effect of Ti and ITO adhesion layers on gold split-ring resonators,” Appl. Phys. Lett. 97, 263103 (2010).
[Crossref]

J. Biomed. Opt. (1)

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Other (3)

COMPONENTS LASER Germany GmbH, Si-Diode PDB-C601-1.

S. W. Rowland, “Computer implementation of image reconstruction formulas,” in Image reconstructions from projections, G. T. Herman, ed. (Springer-Verlag, Berlin, 1979), pp. 978.

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Figures (6)

Fig. 1
Fig. 1 Schematic sketch of the samples and the setup. The knife-pads are directly fabricated on GaAs- or Si-photodiodes (a) or manufactured on glass substrates (BK7) that are put on a Si-photodiode (size of the area: 6 × 6 μm) with a thin layer of immersion oil in between (b). A linearly polarized Gaussian beam is focused on the samples by a high numerical aperture objective (NA 0.9). The samples are mounted on a holder that can be moved by a 3D-piezostage with nanometer accuracy (c). For the knife-edge measurements, the sample is moved through the focal spot and the transmitted light is detected by a photodiode underneath (d,e).
Fig. 2
Fig. 2 Typical beam profiling data (a) and corresponding derivatives (b). The state of polarization (s: perpendicular; p: parallel) always refers to the orientation of the electric field.
Fig. 3
Fig. 3 Experimental and theoretical results plotted versus the wavelength for different substrate materials (GaAs-, Si-Diode and BK7 substrate) for Au knife-pads with a height h = 130 nm (a,c,e) and h = 70 nm (b,c,d). Shift dsdp between the maxima of the differentiated photocurrent curves (a, b). Reconstructed beam size of the focal spot for s- and p-polarized light Wp (c,d) and Ws (e,f) (each normalized to the wavelength). The beam size in the focal plane calculated by vectorial diffraction theory is plotted in (b–f) as well (black line).
Fig. 4
Fig. 4 Schematic depiction of an exemplary reconstructed beam profile (black), the corresponding electric energy density (red), its first (blue) and second derivative (green) for knife-pads interacting with the incident field via the local electric field (1,2) or via local gradients (3,4) (a). Dependence of the absolute beam shift from one edge d s , p = ( d s , p d ) / 2 (green dotted curve, n = 1) and the reconstructed beamwidth w s , p r (blue curve for n = 1 and red curve for n = 2) on the coefficients Cn, for ws,p = 1 as focal beam diameter.
Fig. 5
Fig. 5 Sketch of the considered structure containing the medium with dielectric constant ε1, the knife-pad (dielectric constant ε2) and the substrate (dielectric constant ε3). Schematic illustration of a single plane wave component k = (±kx,kz) impinging on the metal pad and being transmitted into the substrate through it or directly (a). Schematic depiction of a single plane wave component (k1 = (kx,kz), k2’= (−kx,kz)) impinging on the edge and r = (x,z). A plasmonic mode is also schematically depicted (b).
Fig. 6
Fig. 6 Experimentally measured beam profiles for p-polarization and two wavelengths (λ = 700 nm, 535 nm). The measurements have been performed using a knife-pad with a height of 70 nm placed on a BK7 glass substrate (a). Numerically calculated derivative of the photocurrent ∂P/∂x0 (red), expected beam profile UE(x0) (black) and a term representing distortions (∂P/∂x0 −UE(x0)) to the beam profile (green) for one particular wavelength λ = 535 nm (b).

Equations (19)

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P = P 0 d y 0 I ( x + x 0 , y , z = 0 ) d x ,
P P 0 x 0 = C 0 U E ( x 0 ) + n = 1 C n n U E ( x 0 ) x 0 n .
P P 0 x 0 = C 0 exp ( x 0 2 / w s , p 2 ) 2 x 0 C 1 w s , p 2 exp ( x 0 2 / w s , p 2 ) 2 ( w s , p 2 2 x 0 2 ) C 2 w s , p 4 exp ( x 0 2 / w s , p 2 )
P = P 0 d x d k x U ^ E ( k x , x 0 ) T ^ ( k x ) e i k x x .
P = P 0 d x d k x U ^ E ( k x , x 0 ) T 1 ( k x ) e i k x x + P 0 0 d x d k x U ^ E ( k x , x 0 ) T 2 ( k x ) e i k x x .
T 1 , 2 ( k x ) = T 10 , 20 + n = 1 k x n n ! n T 1 , 2 ( k x ) k x n | k x = 0 , T 10 T 1 ( k x = 0 ) , T 20 = T 2 ( k x = 0 ) .
P = P 0 x 0 d x [ T 10 U E ( x ) + n = 1 A n n U E ( x ) x n ] + P 0 x 0 d x [ T 20 U E ( x ) + n = 1 B n n U E ( x ) x n ] ,
P P 0 x 0 = ( T 10 T 20 ) U E ( x 0 ) + n = 1 ( A n B n ) n U E ( x 0 ) x 0 n .
C 0 = T 10 T 20 , C n = δ n , 2 m ( A n B n )
P t r a n s x 0 = T 10 U E ( x 0 ) x 0 Re υ κ k ,
P r e f = 0 d k x 0 h k x k 2 k x 2 d x U ^ E ( k x , x 0 ) R ( k 2 k x 2 ) k 2 k x 2 T 2 ( k x ) e i k x x = 0 d k x U ^ E ( k x , x 0 ) R ( k 2 k x 2 ) k 2 k x 2 T 2 ( k x ) i h k x 2 e k 2 k x 2 1 i k x ,
C 0 = 0 , C n = 1 i n n ! n k x n { T 2 ( k x ) R ( k 2 k x 2 ) k i k x k 2 k x 2 [ exp ( i h k x 2 k 2 k x 2 ) 1 ] } k x = 0 .
L = ε 2 0 d x E b ( x + x 0 ) exp ( i κ 2 x ) + ε 1 0 d x E b ( x + x 0 ) exp ( i κ 1 x ) ,
β = k ε 1 ε 2 ε 2 + ε 1 , κ 1 = k 2 ε 1 β 2 , κ 2 = k 2 ε 2 β 2 .
L = ε 2 d k x i E ^ b ( k x ) k x + κ 2 exp ( i k x x 0 ) + ε 1 d k x i E ^ b ( k x ) k x κ 1 exp ( i k x x 0 ) ,
L = n = 1 c n n 1 E b ( x 0 ) x 0 n 1 , c n = 1 i n 1 [ ε 2 ( κ 2 ) n + 1 ε 1 ( κ 1 ) n + 1 ] .
C n = m = 0 n c m c n m n ! m ! ( n m ) ! , for n > 1.
L = ε 1 w s , p π / 2 exp [ κ 1 4 ( κ 1 w s , p 2 4 i x 0 ) ] [ 1 + erf ( i κ 1 w s , p / 2 + x 0 / w s , p ) ] + ε 2 w s , p π / 2 exp [ κ 2 4 ( κ 2 w s , p 2 4 i x 0 ) ] [ 1 + erf ( i κ 2 w s , p / 2 + x 0 / w s , p ) ] ,
L ε 1 w s , p π e ( κ 1 w s , p 2 ) 2 e i κ 1 x 0 .

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