Abstract

Diffraction of TM-polarized waves by a slit in a thick screen of infinite conductivity is treated. The case of an arbitrary incident beam wave is considered. We study the resonances that appear when the wavelength of the incident beam wave is larger than the slit width, i.e., the subwavelength regime where a one-mode model for the slit can be considered. High anomalous values (resonances) of the transmission coefficient, the angular diffracted energy, and the magnetic field within the slit are analyzed. A simple linear relationship to determine the resonant wavelengths is proposed. We show that the transmission coefficient, the normal diffracted energy, and the magnetic field within the cavity are linear functions of the resonant wavelength and the thickness of the screen. Additionally and surprisingly, we reveal that under certain conditions the incident beam wave via the diffraction can give a suppressed light transmission; i.e., a minimum in the transmission is obtained where a maximum is expected.

© 2007 Optical Society of America

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References

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  1. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature (London) 391, 667-668 (1998).
    [CrossRef]
  2. D. Van Labeke, F. I. Baida, and J.-M. Vigoureux, "A new structure for enhanced transmission through a two-dimensional metallic grating," J. Microsc. 213, 140-143 (2004).
    [CrossRef] [PubMed]
  3. E. Popov, M. Nevière, S. Enoch, and R. Reinisch, "Theory of light transmission through subwavelength periodic hole arrays," Phys. Rev. B 62, 16100-16108 (2000).
    [CrossRef]
  4. J. A. Porto, F. J. García-Vidal, and J. B. Pendry, "Transmission resonances on metallic gratings with very narrow slits," Phys. Rev. Lett. 83, 2845-2848 (1999).
    [CrossRef]
  5. Q. Cao and P. Lalanne, "Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits," Phys. Rev. Lett. 88, 057403-1-057403-4 (2002).
    [CrossRef]
  6. E. Popov, M. Nevière, J. Wenger, P. F. Lenne, H. Rigneault, and P. Chaumet, "Field enhancement in a single subwavelength aperture," J. Opt. Soc. Am. A 23, 2342-2348 (2006).
    [CrossRef]
  7. F. I. Baida, D. Van Labeke, and B. Guizal, "Enhanced confined light transmission by single subwavelength apertures in metallic films," Appl. Opt. 42, 6811-6815 (2003).
    [CrossRef] [PubMed]
  8. Y. Takakura, "Optical resonance in a narrow slit in a thick metallic screen," Phys. Rev. Lett. 86, 5601-5603 (2001).
    [CrossRef] [PubMed]
  9. F. Yang and J. R. Sambles, "Determination of microwave permitivities using a metallic slit," J. Phys. D 35, 3049-3051 (2002).
    [CrossRef]
  10. F. Yang and J. R. Sambles, "Determination of the microwave permitivities of nematic liquid crystals using a single-metallic slit technique," Appl. Phys. Lett. 81, 2047-2049 (2002).
    [CrossRef]
  11. L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, A. Degiron, and T. W. Ebbesen, "Theory of highly directional emission from a single subwavelength aperture surrounded by surface corrugations," Phys. Rev. Lett. 90, 167401-1-167401-4 (2003).
    [CrossRef]
  12. F. J. García-Vidal, H. J. Lezec, T. W. Ebbesen, and L. Martín-Moreno, "Multiple paths to enhance optical transmission through a single subwavelength slit," Phys. Rev. Lett. 90, 213901-1-213901-4 (2003).
    [CrossRef]
  13. J. Bravo-Abad, L. Martín-Moreno, and F. J. García-Vidal, "Transmission properties of a single metallic slit: from the subwavelength regime to the geometrical-optics limit," Phys. Rev. E 69, 026601-1-026601-6 (2004).
    [CrossRef]
  14. R. Gordon, "Light in a subwavelength slit in a metal: propagation and reflection," Phys. Rev. B 73, 153405-1-153405-3 (2006).
    [CrossRef]
  15. P. Lalanne, J. P. Hugonin, and J. C. Rodier, "Approximate model for surface-plasmon generation at slit apertures," J. Opt. Soc. Am. A 23, 1608-1615 (2006).
    [CrossRef]
  16. O. Mata-Mendez, J. Avendaño, and F. Chavez-Rivas, "Rigorous theory of the diffraction of Gaussian beams by finite gratings: TM polarization," J. Opt. Soc. Am. A 23, 1889-1895 (2006).
    [CrossRef]
  17. J. T. Foley and E. Wolf, "Note on the far field of a Gaussian beam," J. Opt. Soc. Am. 69, 761-764 (1979).
    [CrossRef]
  18. O. Mata-Mendez and F. Chavez-Rivas, "Diffraction of Hermite-Gaussian beams by a slit," J. Opt. Soc. Am. A 12, 2440-2445 (1995).
    [CrossRef]
  19. O. Mata-Mendez and F. Chavez-Rivas, "New property in the diffraction of Hermite-Gaussian beams by a finite grating in the scalar diffraction regime: constant-intensity angles in the far field when the beam center is displaced through the grating," J. Opt. Soc. Am. A 15, 2698-2704 (1998).
    [CrossRef]
  20. O. Mata-Mendez and F. Chavez-Rivas, "Diffraction of Gaussian and Hermite-Gaussian beams by finite gratings," J. Opt. Soc. Am. A 18, 537-545 (2001).
    [CrossRef]

2006

2004

J. Bravo-Abad, L. Martín-Moreno, and F. J. García-Vidal, "Transmission properties of a single metallic slit: from the subwavelength regime to the geometrical-optics limit," Phys. Rev. E 69, 026601-1-026601-6 (2004).
[CrossRef]

D. Van Labeke, F. I. Baida, and J.-M. Vigoureux, "A new structure for enhanced transmission through a two-dimensional metallic grating," J. Microsc. 213, 140-143 (2004).
[CrossRef] [PubMed]

2003

F. I. Baida, D. Van Labeke, and B. Guizal, "Enhanced confined light transmission by single subwavelength apertures in metallic films," Appl. Opt. 42, 6811-6815 (2003).
[CrossRef] [PubMed]

L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, A. Degiron, and T. W. Ebbesen, "Theory of highly directional emission from a single subwavelength aperture surrounded by surface corrugations," Phys. Rev. Lett. 90, 167401-1-167401-4 (2003).
[CrossRef]

F. J. García-Vidal, H. J. Lezec, T. W. Ebbesen, and L. Martín-Moreno, "Multiple paths to enhance optical transmission through a single subwavelength slit," Phys. Rev. Lett. 90, 213901-1-213901-4 (2003).
[CrossRef]

2002

F. Yang and J. R. Sambles, "Determination of microwave permitivities using a metallic slit," J. Phys. D 35, 3049-3051 (2002).
[CrossRef]

F. Yang and J. R. Sambles, "Determination of the microwave permitivities of nematic liquid crystals using a single-metallic slit technique," Appl. Phys. Lett. 81, 2047-2049 (2002).
[CrossRef]

Q. Cao and P. Lalanne, "Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits," Phys. Rev. Lett. 88, 057403-1-057403-4 (2002).
[CrossRef]

2001

Y. Takakura, "Optical resonance in a narrow slit in a thick metallic screen," Phys. Rev. Lett. 86, 5601-5603 (2001).
[CrossRef] [PubMed]

O. Mata-Mendez and F. Chavez-Rivas, "Diffraction of Gaussian and Hermite-Gaussian beams by finite gratings," J. Opt. Soc. Am. A 18, 537-545 (2001).
[CrossRef]

2000

E. Popov, M. Nevière, S. Enoch, and R. Reinisch, "Theory of light transmission through subwavelength periodic hole arrays," Phys. Rev. B 62, 16100-16108 (2000).
[CrossRef]

1999

J. A. Porto, F. J. García-Vidal, and J. B. Pendry, "Transmission resonances on metallic gratings with very narrow slits," Phys. Rev. Lett. 83, 2845-2848 (1999).
[CrossRef]

1998

1995

1979

Avendaño, J.

Baida, F. I.

D. Van Labeke, F. I. Baida, and J.-M. Vigoureux, "A new structure for enhanced transmission through a two-dimensional metallic grating," J. Microsc. 213, 140-143 (2004).
[CrossRef] [PubMed]

F. I. Baida, D. Van Labeke, and B. Guizal, "Enhanced confined light transmission by single subwavelength apertures in metallic films," Appl. Opt. 42, 6811-6815 (2003).
[CrossRef] [PubMed]

Bravo-Abad, J.

J. Bravo-Abad, L. Martín-Moreno, and F. J. García-Vidal, "Transmission properties of a single metallic slit: from the subwavelength regime to the geometrical-optics limit," Phys. Rev. E 69, 026601-1-026601-6 (2004).
[CrossRef]

Cao, Q.

Q. Cao and P. Lalanne, "Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits," Phys. Rev. Lett. 88, 057403-1-057403-4 (2002).
[CrossRef]

Chaumet, P.

Chavez-Rivas, F.

Degiron, A.

L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, A. Degiron, and T. W. Ebbesen, "Theory of highly directional emission from a single subwavelength aperture surrounded by surface corrugations," Phys. Rev. Lett. 90, 167401-1-167401-4 (2003).
[CrossRef]

Ebbesen, T. W.

L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, A. Degiron, and T. W. Ebbesen, "Theory of highly directional emission from a single subwavelength aperture surrounded by surface corrugations," Phys. Rev. Lett. 90, 167401-1-167401-4 (2003).
[CrossRef]

F. J. García-Vidal, H. J. Lezec, T. W. Ebbesen, and L. Martín-Moreno, "Multiple paths to enhance optical transmission through a single subwavelength slit," Phys. Rev. Lett. 90, 213901-1-213901-4 (2003).
[CrossRef]

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature (London) 391, 667-668 (1998).
[CrossRef]

Enoch, S.

E. Popov, M. Nevière, S. Enoch, and R. Reinisch, "Theory of light transmission through subwavelength periodic hole arrays," Phys. Rev. B 62, 16100-16108 (2000).
[CrossRef]

Foley, J. T.

García-Vidal, F. J.

J. Bravo-Abad, L. Martín-Moreno, and F. J. García-Vidal, "Transmission properties of a single metallic slit: from the subwavelength regime to the geometrical-optics limit," Phys. Rev. E 69, 026601-1-026601-6 (2004).
[CrossRef]

L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, A. Degiron, and T. W. Ebbesen, "Theory of highly directional emission from a single subwavelength aperture surrounded by surface corrugations," Phys. Rev. Lett. 90, 167401-1-167401-4 (2003).
[CrossRef]

F. J. García-Vidal, H. J. Lezec, T. W. Ebbesen, and L. Martín-Moreno, "Multiple paths to enhance optical transmission through a single subwavelength slit," Phys. Rev. Lett. 90, 213901-1-213901-4 (2003).
[CrossRef]

J. A. Porto, F. J. García-Vidal, and J. B. Pendry, "Transmission resonances on metallic gratings with very narrow slits," Phys. Rev. Lett. 83, 2845-2848 (1999).
[CrossRef]

Ghaemi, H. F.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature (London) 391, 667-668 (1998).
[CrossRef]

Gordon, R.

R. Gordon, "Light in a subwavelength slit in a metal: propagation and reflection," Phys. Rev. B 73, 153405-1-153405-3 (2006).
[CrossRef]

Guizal, B.

Hugonin, J. P.

Lalanne, P.

P. Lalanne, J. P. Hugonin, and J. C. Rodier, "Approximate model for surface-plasmon generation at slit apertures," J. Opt. Soc. Am. A 23, 1608-1615 (2006).
[CrossRef]

Q. Cao and P. Lalanne, "Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits," Phys. Rev. Lett. 88, 057403-1-057403-4 (2002).
[CrossRef]

Lenne, P. F.

Lezec, H. J.

F. J. García-Vidal, H. J. Lezec, T. W. Ebbesen, and L. Martín-Moreno, "Multiple paths to enhance optical transmission through a single subwavelength slit," Phys. Rev. Lett. 90, 213901-1-213901-4 (2003).
[CrossRef]

L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, A. Degiron, and T. W. Ebbesen, "Theory of highly directional emission from a single subwavelength aperture surrounded by surface corrugations," Phys. Rev. Lett. 90, 167401-1-167401-4 (2003).
[CrossRef]

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature (London) 391, 667-668 (1998).
[CrossRef]

Martín-Moreno, L.

J. Bravo-Abad, L. Martín-Moreno, and F. J. García-Vidal, "Transmission properties of a single metallic slit: from the subwavelength regime to the geometrical-optics limit," Phys. Rev. E 69, 026601-1-026601-6 (2004).
[CrossRef]

F. J. García-Vidal, H. J. Lezec, T. W. Ebbesen, and L. Martín-Moreno, "Multiple paths to enhance optical transmission through a single subwavelength slit," Phys. Rev. Lett. 90, 213901-1-213901-4 (2003).
[CrossRef]

L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, A. Degiron, and T. W. Ebbesen, "Theory of highly directional emission from a single subwavelength aperture surrounded by surface corrugations," Phys. Rev. Lett. 90, 167401-1-167401-4 (2003).
[CrossRef]

Mata-Mendez, O.

Nevière, M.

E. Popov, M. Nevière, J. Wenger, P. F. Lenne, H. Rigneault, and P. Chaumet, "Field enhancement in a single subwavelength aperture," J. Opt. Soc. Am. A 23, 2342-2348 (2006).
[CrossRef]

E. Popov, M. Nevière, S. Enoch, and R. Reinisch, "Theory of light transmission through subwavelength periodic hole arrays," Phys. Rev. B 62, 16100-16108 (2000).
[CrossRef]

Pendry, J. B.

J. A. Porto, F. J. García-Vidal, and J. B. Pendry, "Transmission resonances on metallic gratings with very narrow slits," Phys. Rev. Lett. 83, 2845-2848 (1999).
[CrossRef]

Popov, E.

E. Popov, M. Nevière, J. Wenger, P. F. Lenne, H. Rigneault, and P. Chaumet, "Field enhancement in a single subwavelength aperture," J. Opt. Soc. Am. A 23, 2342-2348 (2006).
[CrossRef]

E. Popov, M. Nevière, S. Enoch, and R. Reinisch, "Theory of light transmission through subwavelength periodic hole arrays," Phys. Rev. B 62, 16100-16108 (2000).
[CrossRef]

Porto, J. A.

J. A. Porto, F. J. García-Vidal, and J. B. Pendry, "Transmission resonances on metallic gratings with very narrow slits," Phys. Rev. Lett. 83, 2845-2848 (1999).
[CrossRef]

Reinisch, R.

E. Popov, M. Nevière, S. Enoch, and R. Reinisch, "Theory of light transmission through subwavelength periodic hole arrays," Phys. Rev. B 62, 16100-16108 (2000).
[CrossRef]

Rigneault, H.

Rodier, J. C.

Sambles, J. R.

F. Yang and J. R. Sambles, "Determination of the microwave permitivities of nematic liquid crystals using a single-metallic slit technique," Appl. Phys. Lett. 81, 2047-2049 (2002).
[CrossRef]

F. Yang and J. R. Sambles, "Determination of microwave permitivities using a metallic slit," J. Phys. D 35, 3049-3051 (2002).
[CrossRef]

Takakura, Y.

Y. Takakura, "Optical resonance in a narrow slit in a thick metallic screen," Phys. Rev. Lett. 86, 5601-5603 (2001).
[CrossRef] [PubMed]

Thio, T.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature (London) 391, 667-668 (1998).
[CrossRef]

Van Labeke, D.

D. Van Labeke, F. I. Baida, and J.-M. Vigoureux, "A new structure for enhanced transmission through a two-dimensional metallic grating," J. Microsc. 213, 140-143 (2004).
[CrossRef] [PubMed]

F. I. Baida, D. Van Labeke, and B. Guizal, "Enhanced confined light transmission by single subwavelength apertures in metallic films," Appl. Opt. 42, 6811-6815 (2003).
[CrossRef] [PubMed]

Vigoureux, J.-M.

D. Van Labeke, F. I. Baida, and J.-M. Vigoureux, "A new structure for enhanced transmission through a two-dimensional metallic grating," J. Microsc. 213, 140-143 (2004).
[CrossRef] [PubMed]

Wenger, J.

Wolf, E.

Wolff, P. A.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature (London) 391, 667-668 (1998).
[CrossRef]

Yang, F.

F. Yang and J. R. Sambles, "Determination of microwave permitivities using a metallic slit," J. Phys. D 35, 3049-3051 (2002).
[CrossRef]

F. Yang and J. R. Sambles, "Determination of the microwave permitivities of nematic liquid crystals using a single-metallic slit technique," Appl. Phys. Lett. 81, 2047-2049 (2002).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

F. Yang and J. R. Sambles, "Determination of the microwave permitivities of nematic liquid crystals using a single-metallic slit technique," Appl. Phys. Lett. 81, 2047-2049 (2002).
[CrossRef]

J. Microsc.

D. Van Labeke, F. I. Baida, and J.-M. Vigoureux, "A new structure for enhanced transmission through a two-dimensional metallic grating," J. Microsc. 213, 140-143 (2004).
[CrossRef] [PubMed]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Phys. D

F. Yang and J. R. Sambles, "Determination of microwave permitivities using a metallic slit," J. Phys. D 35, 3049-3051 (2002).
[CrossRef]

Nature (London)

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature (London) 391, 667-668 (1998).
[CrossRef]

Phys. Rev. B

E. Popov, M. Nevière, S. Enoch, and R. Reinisch, "Theory of light transmission through subwavelength periodic hole arrays," Phys. Rev. B 62, 16100-16108 (2000).
[CrossRef]

R. Gordon, "Light in a subwavelength slit in a metal: propagation and reflection," Phys. Rev. B 73, 153405-1-153405-3 (2006).
[CrossRef]

Phys. Rev. E

J. Bravo-Abad, L. Martín-Moreno, and F. J. García-Vidal, "Transmission properties of a single metallic slit: from the subwavelength regime to the geometrical-optics limit," Phys. Rev. E 69, 026601-1-026601-6 (2004).
[CrossRef]

Phys. Rev. Lett.

L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, A. Degiron, and T. W. Ebbesen, "Theory of highly directional emission from a single subwavelength aperture surrounded by surface corrugations," Phys. Rev. Lett. 90, 167401-1-167401-4 (2003).
[CrossRef]

F. J. García-Vidal, H. J. Lezec, T. W. Ebbesen, and L. Martín-Moreno, "Multiple paths to enhance optical transmission through a single subwavelength slit," Phys. Rev. Lett. 90, 213901-1-213901-4 (2003).
[CrossRef]

J. A. Porto, F. J. García-Vidal, and J. B. Pendry, "Transmission resonances on metallic gratings with very narrow slits," Phys. Rev. Lett. 83, 2845-2848 (1999).
[CrossRef]

Q. Cao and P. Lalanne, "Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits," Phys. Rev. Lett. 88, 057403-1-057403-4 (2002).
[CrossRef]

Y. Takakura, "Optical resonance in a narrow slit in a thick metallic screen," Phys. Rev. Lett. 86, 5601-5603 (2001).
[CrossRef] [PubMed]

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Figures (8)

Fig. 1
Fig. 1

Our configuration. A slit of width l in a thick planar screen of thickness h is parallel to the O z axis, i.e., perpendicular to the plane of the figure. The position of the incident Gaussian beam is fixed by the parameters b and c.

Fig. 2
Fig. 2

Ratio E d E d ( θ ) as a function of the wavelength for slit width l = 0.5 μ m , metallic screen thickness h = 8.0 μ m , and angles of diffraction θ = 10 ° , 30°, 50°, 70°.

Fig. 3
Fig. 3

Total diffracted energy E d versus the wavelength for an incident plane wave (Pw) and a Gaussian beam wave (Gw) with width L = 20 2 μ m , both at the angle of incidence θ i = 20 ° , for a slit width l = 0.5 μ m and metallic screen thickness h = 8.0 μ m .

Fig. 4
Fig. 4

Resonant wavelength as a function of the screen thickness h for slit width l = 0.5 μ m . Our Eq. (28) (dashed curves) and Eq. (9) of Takakura (dotted curves) are compared with a rigorous diffraction theory (solid curves).

Fig. 5
Fig. 5

Angular diffracted energy E d ( θ ) as a function of the wavelength for the angles of diffraction θ = 0 ° , 10°, 30°, 50°, 70°. The solid curve corresponds to θ = 0 ° , the dashed curves to the other angles of diffraction. We consider an incident Gaussian beam of width L = 20 2 μ m at the angle of incidence θ i = 20 ° , for slit width l = 0.5 μ m and screen thickness h = 8.0 μ m .

Fig. 6
Fig. 6

Diffracted energy E d (arbitrary units) as a function of the wavelength for an incident distorted beam wave (Dw) and a Gaussian beam wave (Gw). Same parameters as in Fig. 3.

Fig. 7
Fig. 7

Resonant transmission coefficient τ r e s as a function of the resonant wavelength λ n for a normally incident plane wave, a slit width l = 0.5 μ m , and several screen thickness. Equation (32) (dashed curves) is compared with the rigorous diffraction theory (solid curves).

Fig. 8
Fig. 8

Resonant transmission coefficient τ r e s as a function of the screen thickness for a normally incident plane wave, when the slit width is l = 0.5 μ m . Equation (33) (dashed curves) is compared with the rigorous diffraction theory (solid curves).

Equations (48)

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

2 H x 2 + 2 H y 2 + k 0 2 H = 0 ,
H ( x , y ) = 1 2 π H ̂ ( α , y ) exp ( i α x ) d α ,
H ̂ ( α , y ) = 1 2 π H ( x , y ) exp ( i α x ) d x .
H 1 ( x , y ) = 1 2 π k 0 k 0 A ( α ) e i ( α x β y ) d α + 1 2 π B ( α ) e i ( α x + β y ) d α ( for y > h 2 ) ,
H 2 ( x , y ) = 1 2 π C ( α ) e i ( α x β y ) d α ( for y < h 2 ) ,
H 3 ( x , y ) = [ a 0 cos ( k 0 y ) + b 0 sin ( k 0 y ) ] ϕ 0 ( x ) ( h 2 < y < h 2 ) .
ϕ 0 ( x ) = { 1 if 0 x l 0 elsewhere ,
ϕ 0 ( x ) , ϕ 0 ( x ) = ϕ 0 ( x ) ϕ 0 ( x ) * d x = l .
B ( α ) = A ( α ) exp ( i β h ) i β exp ( i β h 2 ) k 0 [ a 0 sin ( k 0 h 2 ) + b 0 cos ( k 0 h 2 ) ] ϕ ̂ 0 ( α ) ,
C ( α ) = i β exp ( i β h 2 ) k 0 [ a 0 sin ( k 0 h 2 ) + b 0 cos ( k 0 h 2 ) ] ϕ ̂ 0 ( α ) ,
ϕ ̂ 0 ( α ) = 2 π sin ( α l 2 ) α exp ( i α l 2 ) .
H ̂ 1 ( α , h 2 ) H ̂ 3 ( α , h 2 ) , ϕ ̂ 0 ( α ) = 0 ,
H ̂ 3 ( α , h 2 ) H ̂ 2 ( α , h 2 ) , ϕ ̂ 0 ( α ) = 0 ,
a 0 = S 0 k 0 sin ( k 0 h 2 ) I + l cos ( k 0 h 2 ) ,
b 0 = S 0 k 0 cos ( k 0 h 2 ) I + l sin ( k 0 h 2 ) ,
S 0 = A ( α ) exp ( i β h 2 ) , ϕ ̂ 0 ( α ) = 0 l E i ( x , h 2 ) d x
I = i β ϕ ̂ 0 ( α ) , ϕ ̂ 0 ( α ) = 2 i π sin 2 ( α l 2 ) α 2 β d α .
H i ( x , y = 0 ) = exp [ 2 ( x b ) 2 L 2 ] .
A ( α ) = L 2 q 2 ( θ i ) exp [ i ( α b + β c ) ] exp [ q 1 ( θ i ) 2 L 2 8 ] ,
E d = Im [ a 0 b 0 * ] k 0 l .
E d = k 0 2 l 2 S 0 2 Im I F 2 + ( l k 0 Im I ) 2 = 1 2 k 0 2 l 4 S 0 2 F 2 + l 6 k 0 2 4 ,
F = 1 2 ( l 2 k 0 2 I 2 ) sin ( k 0 h ) + k 0 l Re I cos ( k 0 h ) .
E d ( θ ) = k 0 2 l 2 S 0 2 F 2 + k 0 2 l 2 ( Im I ) 2 ϕ ̂ 0 ( k 0 sin θ ) 2 = 2 π k 0 2 l 2 S 0 2 F 2 + k 0 2 l 6 4 sin 2 ( k 0 sin θ l 2 ) k 0 2 sin 2 θ .
E d E d ( θ ) = Im I ϕ ̂ 0 ( k 0 sin θ ) 2 = π k 0 2 l 2 4 sin 2 θ sin 2 ( k 0 l sin θ 2 ) .
E d E d ( θ ) π .
E d ( θ ) = k 0 2 l 4 2 π S 0 2 F 2 + l 6 k 0 2 4 = constant ;
tan ( k 0 h ) + 2 k 0 l Re I l 2 k 0 2 I 2 = 0 .
λ n = 2.05 n h + a n l ,
n λ n = constant ( for n > 1 ) .
( E d ) r e s = S 0 2 Im I = 2 S 0 2 l 2 .
( E d ) r e s = 2 .
τ r e s = 1 π ( λ n l ) .
τ r e s = ( 2.05 n l π ) h + a n π ,
[ E d ( θ ) ] r e s = S 0 2 ( Im I ) 2 ϕ ̂ 0 ( k 0 sin θ ) 2 = 8 π S 0 2 l 4 sin 2 ( k 0 sin θ l 2 ) k 0 2 sin 2 θ ,
[ E d ( θ ) ] r e s = l 2 ( Im I ) 2 ϕ ̂ 0 ( k 0 sin θ ) 2 = 8 π l 2 sin 2 ( k 0 sin θ l 2 ) k 0 2 sin 2 θ .
[ E d ( 0 ° ) ] r e s = 2 π ,
[ E d ( 0 ° ) k 0 l ] r e s = ( 1 π 2 l ) λ n ,
[ E d ( 0 ° ) k 0 l ] r e s = ( 2.05 π 2 l n ) h + a π 2 n ,
H 3 ( x , y ) 2 = S 0 2 [ cos 2 ( k n y ) M 1 2 + sin 2 ( k n y ) M 2 2 ] ,
M 1 2 = [ k n sin ( k n h 2 ) Re I + l cos ( k n h 2 ) ] 2 + k n 2 sin 2 ( k n h 2 ) ( Im I ) 2 ,
M 2 2 = [ k n cos ( k n h 2 ) Re I + l sin ( k n h 2 ) ] 2 + k n 2 cos 2 ( k n h 2 ) ( Im I ) 2 .
λ n = 2.05 n h .
H 3 ( x , y ) 2 = { S 0 2 [ cos 2 ( k n y ) l 2 + sin 2 ( k n y ) k n 2 I 2 ] n even S 0 2 [ cos 2 ( k n y ) k n 2 I 2 + sin 2 ( k n y ) l 2 ] n odd ] .
H 3 ( x , h 2 ) 2 = H 3 ( x , h 2 ) 2 = S 0 2 l 2 ,
H 3 max 2 λ n 2 .
H 3 max 2 ( h n ) 2 .
H 3 max 2 = b ( 2.05 n ) 2 h 2 + b n ,
H 3 max 2 = a λ n 2 + b ,

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