Abstract

We investigate the interaction of highly focused linearly polarized optical beams with a metal knife-edge both theoretically and experimentally. A high numerical aperture objective focusses beams of various wavelengths onto samples of different sub-wavelength thicknesses made of several opaque and pure materials. The standard evaluation of the experimental data shows material and sample dependent spatial shifts of the reconstructed intensity distribution, where the orientation of the electric field with respect to the edge plays an important role. A deeper understanding of the interaction between the knife-edge and the incoming highly focused beam is gained in our theoretical model by considering eigenmodes of the metal-insulator-metal structure. We achieve good qualitative agreement of our numerical simulations with the experimental findings.

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  1. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
    [CrossRef]
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    [CrossRef] [PubMed]
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  4. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
    [CrossRef] [PubMed]
  5. G. Leuchs and S. Quabis, “Tailored polarization patterns for performance optimization of optical devices,” J. Mod. Opt. 53, 787–797 (2006).
    [CrossRef]
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    [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  29. O. Mata-Mendez, J. Avendano, and F. Chavez-Rivas, “Rigorous theory of the diffraction of Gaussian beams by finite gratings: TM polarization,” J. Opt. Soc. Am. A 23, 1889–1896 (2006).
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2009 (2)

Ş. E. Kocabaş, G. Veronis, D. A. B. Miller, and S. Fan, “Modal analysis and coupling in metal-insulator-metal wavequides,” Phys. Rev. B 79, 035120 (2009).
[CrossRef]

M. A. de Araujo, R. Silva, E. de Lima, D. P. Pereira, and P. C. de Oliveira, “Measurement of Gaussian laser beam radius using the knife-edge technique: improvement on data analysis,” Appl. Opt. 48, 393–396 (2009).

2007 (4)

2006 (5)

2005 (1)

S. A. Maier and H. A. Atwater, “Plasmonics: Localization and guiding of electromagnetic energy in metal/dielectric structures,” Appl. Phys. 98, 011101 (2005).
[CrossRef]

2003 (4)

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, 1917–1926 (2003).

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

M. J. Weber, Handbook of optical materials (CRC Press, 2003).

J. Sumaya-Martinez, O. Mata-Mendez, and F. Chavez-Rivas, “Rigorous theory of the diffraction of Gaussian beams by finite gratings: TE polarization,” J. Opt. Soc. Am. A 20, 827–835 (2003).
[CrossRef]

2000 (2)

K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical vector beams,” Opt. Express 7, 77–87 (2000).
[CrossRef] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[CrossRef]

1991 (1)

1987 (1)

A. Roberts and R. C. McPhedran, “Power losses in highly conducting lamellar gratings,” J. Mod. Opt 34, 511–538 (1987).
[CrossRef]

1985 (2)

1984 (1)

1983 (2)

1981 (1)

1977 (1)

1971 (1)

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. A 253, 358 – 379 (1959).
[CrossRef]

1955 (1)

A. E. Karbowiak, “Theory of imperfect waveguides: the effect of wall impedance,” Proc. IEEE Part B: Radio Electron. Engin. 102, 698 – 708 (1955).

1931 (1)

D. Karabacak, T. Kouh, C. C. Huang, and K. L. Ekinci, “Optical knife-edge technique for nanomechanical displacement detection,” Appl. Phys. Lett. 88, 193122 (2006).

Abtahi, A.

Arnaud, J. A.

Atwater, H. A.

J. A. Dionne, L. A. Sweatlock, and H. A. Atwater, “Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73, 035407 (2006).
[CrossRef]

S. A. Maier and H. A. Atwater, “Plasmonics: Localization and guiding of electromagnetic energy in metal/dielectric structures,” Appl. Phys. 98, 011101 (2005).
[CrossRef]

Avendano, J.

Bilger, H. R.

Brost, G.

Brown, T. G.

Cadilhac, M.

Chavez-Rivas, F.

Chiu, Y.

de Araujo, M. A.

de la Claviere, B.

de Lima, E.

de Oliveira, P. C.

Dionne, J. A.

J. A. Dionne, L. A. Sweatlock, and H. A. Atwater, “Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73, 035407 (2006).
[CrossRef]

Dorn, R.

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

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, 1917–1926 (2003).

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[CrossRef]

Eberler, M.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[CrossRef]

Ekinci, K. L.

D. Karabacak, T. Kouh, C. C. Huang, and K. L. Ekinci, “Optical knife-edge technique for nanomechanical displacement detection,” Appl. Phys. Lett. 88, 193122 (2006).

Fan, S.

Ş. E. Kocabaş, G. Veronis, D. A. B. Miller, and S. Fan, “Modal analysis and coupling in metal-insulator-metal wavequides,” Phys. Rev. B 79, 035120 (2009).
[CrossRef]

Firester, A. H.

Franke, E. A.

Franke, J. M.

Garetz, B. A.

Gentili, M.

Glöckl, O.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[CrossRef]

Gorkunov, M.

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

Habib, T.

Heller, M. E.

Horn, P. D.

Huang, C. C.

D. Karabacak, T. Kouh, C. C. Huang, and K. L. Ekinci, “Optical knife-edge technique for nanomechanical displacement detection,” Appl. Phys. Lett. 88, 193122 (2006).

Hubbard, W. M.

Hugonin, J. P.

Karabacak, D.

D. Karabacak, T. Kouh, C. C. Huang, and K. L. Ekinci, “Optical knife-edge technique for nanomechanical displacement detection,” Appl. Phys. Lett. 88, 193122 (2006).

Karbowiak, A. E.

A. E. Karbowiak, “Theory of imperfect waveguides: the effect of wall impedance,” Proc. IEEE Part B: Radio Electron. Engin. 102, 698 – 708 (1955).

Khosrofian, J. M.

Kocabas, S. E.

Ş. E. Kocabaş, G. Veronis, D. A. B. Miller, and S. Fan, “Modal analysis and coupling in metal-insulator-metal wavequides,” Phys. Rev. B 79, 035120 (2009).
[CrossRef]

Kouh, T.

D. Karabacak, T. Kouh, C. C. Huang, and K. L. Ekinci, “Optical knife-edge technique for nanomechanical displacement detection,” Appl. Phys. Lett. 88, 193122 (2006).

Lalanne, P.

Leuchs, G.

G. Leuchs and S. Quabis, “Tailored polarization patterns for performance optimization of optical devices,” J. Mod. Opt. 53, 787–797 (2006).
[CrossRef]

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, 1917–1926 (2003).

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

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[CrossRef]

Maier, S. A.

S. A. Maier and H. A. Atwater, “Plasmonics: Localization and guiding of electromagnetic energy in metal/dielectric structures,” Appl. Phys. 98, 011101 (2005).
[CrossRef]

Mandeville, G. D.

Mansuripur, M.

Mata Mendez, O.

Mata-Mendez, O.

McCally, R. L.

McPhedran, R. C.

A. Roberts and R. C. McPhedran, “Power losses in highly conducting lamellar gratings,” J. Mod. Opt 34, 511–538 (1987).
[CrossRef]

Miller, D. A. B.

Ş. E. Kocabaş, G. Veronis, D. A. B. Miller, and S. Fan, “Modal analysis and coupling in metal-insulator-metal wavequides,” Phys. Rev. B 79, 035120 (2009).
[CrossRef]

Moloney, J. V.

Pan, J.-H.

Pereira, D. P.

Petit, R.

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.

G. Leuchs and S. Quabis, “Tailored polarization patterns for performance optimization of optical devices,” J. Mod. Opt. 53, 787–797 (2006).
[CrossRef]

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, 1917–1926 (2003).

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

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[CrossRef]

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. A 253, 358 – 379 (1959).
[CrossRef]

Riza, N. A.

Roberts, A.

A. Roberts and R. C. McPhedran, “Power losses in highly conducting lamellar gratings,” J. Mod. Opt 34, 511–538 (1987).
[CrossRef]

Rodier, J. C.

Schneider, M. B.

Sheng, P.

Silva, R.

Sturman, B.

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

Sumaya-Martinez, J.

Sweatlock, L. A.

J. A. Dionne, L. A. Sweatlock, and H. A. Atwater, “Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73, 035407 (2006).
[CrossRef]

Umul, Y. Z.

Veronis, G.

Ş. E. Kocabaş, G. Veronis, D. A. B. Miller, and S. Fan, “Modal analysis and coupling in metal-insulator-metal wavequides,” Phys. Rev. B 79, 035120 (2009).
[CrossRef]

Webb, W. W.

Weber, M. J.

M. J. Weber, Handbook of optical materials (CRC Press, 2003).

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. A 253, 358 – 379 (1959).
[CrossRef]

Xie, Y.

Youngworth, K. S.

Zakharian, A. R.

Appl. Opt. (9)

J. A. Arnaud, W. M. Hubbard, G. D. Mandeville, B. de la Claviere, E. A. Franke, and J. M. Franke, “Technique for fast measurement of Gaussian laser beam parameters,” Appl. Opt. 10, 2775–2776 (1971).
[PubMed]

A. H. Firester, M. E. Heller, and P. Sheng, “Knife-edge scanning measurements of subwavelength focussed light beams,” Appl. Opt. 16, 1971–1974 (1977).
[CrossRef] [PubMed]

M. B. Schneider and W. W. Webb, “Measurement of submicron laser beam radii,” Appl. Opt. 20, 1382–1388 (1981).
[CrossRef] [PubMed]

R. L. McCally, “Measurement of Gaussian beam parameters,” Appl. Opt. 23, 2227–2227 (1984).
[CrossRef] [PubMed]

J. M. Khosrofian and B. A. Garetz, “Measurement of a Gaussian laser beam diameter through the direct inversion of knife-edge data,” Appl. Opt. 22, 3406–3410 (1983).
[CrossRef] [PubMed]

G. Brost, P. D. Horn, and A. Abtahi, “Convenient spatial profiling of pulsed laser beams,” Appl. Opt. 24, 38–40 (1985).
[CrossRef] [PubMed]

H. R. Bilger and T. Habib, “Knife-edge scanning of an astigmatic Gaussian beam,” Appl. Opt. 24, 686–690 (1985).
[CrossRef] [PubMed]

M. A. de Araujo, R. Silva, E. de Lima, D. P. Pereira, and P. C. de Oliveira, “Measurement of Gaussian laser beam radius using the knife-edge technique: improvement on data analysis,” Appl. Opt. 48, 393–396 (2009).

M. Gentili and N. A. Riza, “Wide-aperture no-moving-parts optical beam profiler using liquid-crystal displays,” Appl. Opt. 46, 506–512 (2007).
[CrossRef] [PubMed]

Appl. Phys. (1)

S. A. Maier and H. A. Atwater, “Plasmonics: Localization and guiding of electromagnetic energy in metal/dielectric structures,” Appl. Phys. 98, 011101 (2005).
[CrossRef]

Appl. Phys. Lett. (1)

D. Karabacak, T. Kouh, C. C. Huang, and K. L. Ekinci, “Optical knife-edge technique for nanomechanical displacement detection,” Appl. Phys. Lett. 88, 193122 (2006).

J. Mod. Opt (1)

A. Roberts and R. C. McPhedran, “Power losses in highly conducting lamellar gratings,” J. Mod. Opt 34, 511–538 (1987).
[CrossRef]

J. Mod. Opt. (2)

G. Leuchs and S. Quabis, “Tailored polarization patterns for performance optimization of optical devices,” J. Mod. Opt. 53, 787–797 (2006).
[CrossRef]

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, 1917–1926 (2003).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

Opt. Commun. (1)

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[CrossRef]

Opt. Express (3)

Opt. Lett. (1)

Phys. Rev. B (3)

J. A. Dionne, L. A. Sweatlock, and H. A. Atwater, “Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73, 035407 (2006).
[CrossRef]

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

Ş. E. Kocabaş, G. Veronis, D. A. B. Miller, and S. Fan, “Modal analysis and coupling in metal-insulator-metal wavequides,” Phys. Rev. B 79, 035120 (2009).
[CrossRef]

Phys. Rev. Lett. (1)

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

Proc. IEEE Part B: Radio Electron. Engin. (1)

A. E. Karbowiak, “Theory of imperfect waveguides: the effect of wall impedance,” Proc. IEEE Part B: Radio Electron. Engin. 102, 698 – 708 (1955).

Proc. R. Soc. 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. A 253, 358 – 379 (1959).
[CrossRef]

Other (1)

M. J. Weber, Handbook of optical materials (CRC Press, 2003).

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Figures (8)

Fig. 1
Fig. 1

Electron-micrographs of one of the gold samples investigated in the experiments. The knife-edge width and film thickness were determined by performing cuts with a focused ion beam (FIB) machine. The width of the investigated knife-edge is d = 2.0 μm, the slit width is l = 2.0 μm.

Fig. 2
Fig. 2

Schematic depiction of the knife-edge method for a two-dimensional beam (a,b). Typical beam profiling data (c) and their derivatives (d). The state of polarization always refers to the orientation of the electric field.

Fig. 3
Fig. 3

Distance between the peaks ds , dp versus wavelength λ derived from the experimental data for the s- and p-polarizations. The actual knife-edge width measured by SEM is shown by the gray bar (a). Difference of the peak positions of the photocurrent’s derivative ds dp versus wavelength λ for various Au samples (b). Difference of the peak positions ds dp of the photocurrent’s derivative versus sample height h for Au samples at various wavelengths (c). Difference of the peak positions ds dp of the photocurrent’s derivative versus wavelength λ for Ti and Ni samples (d). The colored lines represent results of numerical simulations. The dash-dotted lines in (b) represent a situation with noise artificially added to the photocurrent and smoothed with the same filter as in the experiment. The colored points represent experimental results.

Fig. 4
Fig. 4

Difference in the peak positions ds dp of the photocurrent’s derivative vs wavelength λ for various Au samples of 130 (a) and 70 (b) nm thickness. (T) means tempered samples (30 sec. 400C°). (S) means a sample with a standing-alone structure to investigate a possible influence of the opposite wall on the knife-edge measurement. In this case other structures are missing (a,b). Theoretical difference of the peak positions of the photocurrent’s derivative (ds dp ) versus slit width l for Au sample (h = 130 nm) at various wavelengths (c).

Fig. 5
Fig. 5

Dependence of the ratios wp /λ (a), ws /λ (b) on the wavelength λ for Au samples of varying thickness. Dependence of the ratios ws /λ, ws /λ on the on the wavelength λ for Ni (c) and Ti (d) samples of comparable thickness of the opaque film h = 130 nm. The colored curves represent results of numerical simulations, the colored points - of experimental measurements. Parameters of the model are w 0 p /λ = 0.9w 0 s /λ = 0.55. The dashed lines represent the FWHM’s of the squared electric field estimated from the Debye integrals [23].

Fig. 6
Fig. 6

Schematic depiction of a single plane wave component k = (±kx , kz ) impinging on a knife-edge. Here k 1 = (kx , kz ), k 2 = (−kx , kz ) and r = (x, z) (a). Sketch of the considered structure (b).

Fig. 7
Fig. 7

Absolute value of the electric field |E|/|E 0| distribution on one of the l = 2 μm wide slit walls for TE (a,c) and TM (b,d) radiation at λ = 532 nm (a,b) and 780 nm (c,d). The center of the incident beam is at x = 0. The Au knife-edge is situated at x < 0 and the beam propagates in negative z-direction. The parameters of the model are w 0 p = 0.9w 0 s = 0.55λ.

Fig. 8
Fig. 8

a) Fraction of energy transmitted by plasmons Tp /T as a function of the beam displacement x 0 for various samples of the same thickness h = 200 nm and width l = 2 μ m at λ = 532 nm (solid lines) and λ = 780 (dashed lines). b–d) Dependence of the angular spectral width Δkx of the transmitted signal’s spectra Strans on the beam displacement x 0. Thickness of the Au film is h = 200 nm, the slit width is l = 2 μm, the wavelengths are λ = 532 nm (b), 633 nm (c), 780 nm (d). The spectral FWHM Δkx is determined by fitting a Gaussian function S fit ( k x , x 0 ) = S 0 exp [ 2 ln 2 ( k x k x 0 ) 2 / Δ k x 2 ] to the spectral distribution Strans (kx , x 0). For comparison the black line represents the same dependence for an perfect knife-edge. Parameters of the theory are w 0 p = 0.9w 0 s = 0.55λ.

Equations (29)

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P = 0 S z ( x , x 0 ) d x ,
T x 0 = exp ( x 0 2 d 0 2 ) 2 x 0 d 0 2 exp ( x 0 2 d 0 2 ) Re υ κ k
( 2 x 2 + 2 z 2 + k ε 2 ) { E ( x , z ) H ( x , z ) } = 0 , k ε = ω ε c 0 ,
H x = β e c 0 ω Z 0 E y , H z = i c 0 ω Z 0 E y x , E x = E z = H y = 0.
E x = β m ω ε 0 ε ( x ) H y , E z = i ω ε 0 ε ( x ) H y x , H x = H z = E y = 0.
{ E y , ν ( x ) H y , ν ( x ) } = e ± i β ( e , m ) Z { C 1 e γ 2 x , γ 2 = β ( e , m ) 2 k 0 2 ε 2 , if x [ , 0 ] C 2 e γ 1 x + C 3 e γ 1 x , γ 1 = β ( e , m ) 2 k 0 2 ε 1 , if x [ 0 , l ] C 4 e γ 2 ( x l ) , γ 2 = β ( e , m ) 2 k 0 2 ε 2 , if x [ l , ]
e 2 γ 1 l = ( γ 1 γ 2 γ 1 + γ 2 ) 2 for TE , e 2 γ 1 l = ( γ 1 ε 2 γ 2 γ 1 ε 2 + γ 2 ) 2 for TM
π ν ( x ) = ( 1 ( 1 ) ν e γ 1 l ) e γ 2 x , if x [ , 0 ] π ν ( x ) = e γ 1 x ( 1 ) ν e γ 1 ( x l ) , if x [ 0 , l ] π ν ( x ) = ( e γ 1 l ( 1 ) ν ) e γ 2 ( x l ) , i f x [ l , ] ,
E y ( x ) , ν H x ( y ) , μ dx = δ ν μ G ν ν ,
N ν 2 = υ ( x ) π ν ( x ) π ν * ( x ) dx ,
N ν 2 = 1 + e 2 γ 1 l ( 1 ) ν 2 e γ 1 l cos γ 1 l γ 2 ( 1 ) ν 2 υ e γ 1 l sin γ 1 l γ 1 + υ e 2 γ 1 l 1 γ 1 .
Π ν ( k x ) = Φ ν ( k x ) + Ψ ν ( k x ) , Φ ν ( k x ) = [ 1 ( 1 ) ν e γ 1 * l ] γ 2 * + k x i ( 1 ) ν e i k x l ( γ 2 * k x i ) 2 π N ν ( γ 2 * 2 + k x 2 ) , Ψ ν ( k x ) = ( γ 1 * + k x i ) ( e ( γ 1 * i k x ) l 1 ) ( 1 ) ν ( γ 1 * k x i ) ( e γ 1 * l e i k x l ) 2 π N ν ( γ 1 * 2 + k x 2 ) .
F I , III ( x , z ) = 1 2 π S ( k x , z ) exp ( i k x x ) dk x ,
S ( k x , z ) = S in ( k x ) exp [ i β 1 ( k x ) z ] + S re f ( k x ) exp [ i β 1 ( k x ) z ] .
F G ( x , z = 0 ) = exp [ ( x x 0 ) 2 / W 0 2 ] ,
S in ( k x ) = W 0 2 exp ( i k x x 0 ) exp ( k x 2 W 0 2 / 4 ) .
S ( k x , z ) = S trans ( k x ) exp [ i β 3 ( k x ) z ] ,
F II ( x , z ) = ν = 1 [ a ν e i β ν z + b ν e i β ν ( z + h ) ] π ν ( x ) = ν = 1 F ν ( z ) π ν ( x ) ,
P beam = P trans + P re f + P abs ,
R = k 01 k 01 β 1 ( k x ) | S re f ( k x ) | 2 d k x k 01 k 01 β 1 ( k x ) | S in ( k x ) | 2 d k x , T = Re ( υ 3 ) k 03 k 03 β 3 ( k x ) | S trans ( k x ) | 2 d k x Re ( υ 1 ) k 01 k 01 β 1 ( k x ) | S in ( k x ) | 2 d k x ,
S re f ( k x ) = 1 i β 1 ( k x ) μ F μ z ( 0 ) Θ μ ( k x ) + S in ( k x ) , 2 S ν = μ [ i F μ z ( 0 ) J μ ν ( 1 ) + F μ ( 0 ) G μ ν ] ,
S trans ( k x ) = exp [ i β 3 ( k x ) h ] μ F μ ( h ) ϒ μ ( k x ) , 0 = μ [ i F μ z ( h ) J μ ν ( 3 ) F μ ( h ) G μ ν ] ,
Θ μ ( k x ) = υ 2 υ 1 Φ μ ( k x ) + Ψ μ ( k x ) , J μ ν ( 1 ) = β 1 1 ( k x ) Θ μ ( k x ) Π ν ( k x ) d k x , S ν = S in ( k in ) Π ν ( k x ) d k x , ϒ μ ( k x ) = υ 1 υ 3 Θ μ ( k x ) , J μ ν ( 3 ) = β 3 1 ( k x ) ϒ μ ( k x ) Π ν ( k x ) d k x , G μ ν = δ μ ν ϒ ν ( k x ) Π ν ( k x )
F μ ( 0 ) = a μ + b μ e i β μ h , F μ z ( 0 ) = i β μ ( a μ + b μ e i β μ h ) , F μ ( h ) = a μ e i β μ h + b μ , F μ z ( h ) = i β μ ( a μ e i β μ h + b μ ) .
2 S ν = μ ( G μ ν + β μ J μ ν ( 1 ) ) a μ + e i β μ h ( G μ ν β μ J μ ν ( 1 ) ) b μ , 0 = μ e i β μ h ( G μ ν β μ J μ ν ( 3 ) ) a μ + ( G μ ν + β μ J μ ν ( 3 ) ) b μ .
2 S = N a ( 1 ) a + N b ( 1 ) b , 0 = N a ( 2 ) a + N b ( 2 ) b ,
N a ( 1 ) = diag ( β μ ) J 1 + G , N b ( 1 ) = diag ( e i β μ h ) [ G diag ( β μ ) J 1 ] , N a ( 2 ) = diag ( e i β μ h ) [ diag ( β μ ) J 3 G ] , N b ( 2 ) = diag ( β μ ) J 3 G .
T = Re υ 3 Re υ 1 P beam Re μ ν i F μ z ( h ) F μ * ( h ) H μ ν ( 2 ) ,
H μ ν ( 2 ) = k 03 k 03 ϒ μ ( k x ) Π ν * ( k x ) d k x .

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