Abstract

A rigorous modal theory for the diffraction of Gaussian beams from N equally spaced slits (finite grating) in a planar perfectly conducting thin screen is presented. The case of normal incidence and TE polarization state is considered; i.e., the electric field is parallel to the slits. The characteristics of the far-field diffraction patterns, the transmission coefficient, and the normally diffracted energy as a function of several optogeometrical parameters are analyzed within the so-called vectorial region, where the polarization effects are important. The diffraction pattern of an aperiodic grating is also considered. In addition, one diffraction property known to be valid in the scalar region is generalized to the vectorial region: the existence of constant-intensity angles in the far field when the incident beam wave is scanned along the N slits. The classical grating equation is tested for incident Gaussian beams under several conditions.

© 2003 Optical Society of America

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References

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  1. E. Loewen, E. Popov, Diffraction Gratings and Applications (Marcel Dekker, New York, 1997).
  2. C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys. 17, 35–99 (1954).
    [CrossRef]
  3. K. Hongo, “Diffraction of an electromagnetic plane wave by a thick slit,” IEEE Trans. Antennas Propag. AP-26, 494–499 (1978).
    [CrossRef]
  4. T. Otsuki, “Diffraction by two parallel slits in a plane,” J. Math. Phys. 19, 911–915 (1978).
    [CrossRef]
  5. T. Otsuki, “Reexamination of diffraction problem of a slit by a method of Fourier orthogonal functions transformation,” J. Phys. Soc. Jpn. 41, 2046–2051 (1976).
    [CrossRef]
  6. B. K. Sachdeva, R. A. Hurd, “Diffraction by multiple slits at the interface between two different media,” Can. J. Phys. 53, 1013–1021 (1975).
    [CrossRef]
  7. A. S. Zil’bergleit, “Diffraction of electromagnetic waves by an ideal plate with an even number of symmetrically placed slits,” Sov. Phys. Tech. Phys. 20, 292–295 (1975).
  8. T. Otsuki, “Diffraction by multiple slits,” J. Opt. Soc. Am. A 7, 646–652 (1990).
    [CrossRef]
  9. H. A. Kalhor, “Diffraction of electromagnetic waves by plane metallic gratings,” J. Opt. Soc. Am. 68, 1202–1205 (1978).
    [CrossRef]
  10. J. J. Stamnes, H. A. Eide, “Exact and approximate solutions for focusing of two-dimensional waves. I. Theory,” J. Opt. Soc. Am. A 15, 1285–1291 (1998).
    [CrossRef]
  11. M. Wirgin, “Influence de l’épaisseur de l’écran sur la diffraction par une fente,” C. R. Acad. Sci. Paris 270, 1457–1460 (1970).
  12. H. Henke, H. Fruchting, “Irradiation in a slotted half space and diffraction by a slit in a thick screen,” Nachrichtentech. 29, 401–405 (1976).
  13. J. L. Roumiguières, D. Maystre, R. Petit, M. Cadilhac, “Etude de la diffraction par une fente pratiquée dans un écran infiniment conducteur d’épaisseur quelconque,” Opt. Commun. 9, 402–405 (1973).
  14. F. L. Neerhoff, G. Mur, “Diffraction of a plane electromagnetic wave by a slit in a thick screen placed between two different media,” Appl. Sci. Res. 28, 73–88 (1973).
  15. K. Hongo, “A method of evaluating the near diffracted field,” IEEE Trans. Antennas Propag. AP-28, 409–412 (1980).
    [CrossRef]
  16. D. Marcuse, Light Transmission Optics (Van NostrandReinhold, New York, 1982).
  17. H. S. Tan, “On Kirchhoff’s theory in non-planar scalar diffraction,” Proc. Phys. Soc. Jpn. 91, 768–773 (1967).
    [CrossRef]
  18. O. Mata-Mendez, “Diffraction and beam-diameter measurement of Gaussian beams at optical and microwave frequencies,” Opt. Lett. 16, 1629–1631 (1991).
    [CrossRef] [PubMed]
  19. O. Mata-Mendez, F. Chavez-Rivas, “Diffraction of Gaussian and Hermite–Gaussian beams by finite gratings,” J. Opt. Soc. Am. A 18, 537–545 (2001).
    [CrossRef]
  20. O. Mata-Mendez, M. Cadilhac, R. Petit, “Diffraction of a two-dimensional electromagnetic beam wave by a thick slit pierced in a perfectly conducting screen,” J. Opt. Soc. Am. 73, 328–331 (1983).
    [CrossRef]
  21. O. Mata-Mendez, F. Chavez-Rivas, “Diffraction of Hermite–Gaussian beams by a slit,” J. Opt. Soc. Am. A 12, 2440–2445 (1995).
    [CrossRef]
  22. G. A. Suedan, E. V. Jull, “Two-dimensional beam diffraction by a half-plane and wide slit,” IEEE Trans. Antennas Propag. AP-35, 1077–1082 (1987).
    [CrossRef]
  23. R. A. Depine, D. C. Skigin, “Multilayer modal method for diffraction from dielectric inhomogeneous apertures,” J. Opt. Soc. Am. A 15, 675–683 (1998).
    [CrossRef]
  24. D. C. Skigin, R. A. Depine, “Scattering by lossy inhomogeneous apertures in thick metallic screens”, J. Opt. Soc. Am. A 15, 2089–2096 (1998).
    [CrossRef]
  25. O. Mata-Mendez, 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]
  26. T. Otsuki, “Diffraction by multiple slits,” J. Opt. Soc. Am. A 7, 646–652 (1990).
    [CrossRef]
  27. B. Guizal, D. Felbacq, “Electromagnetic beam diffraction by a finite strip grating,” Opt. Commun. 165, 1–6 (1999).
    [CrossRef]
  28. Em. E. Kriezis, P. K. Pandelakis, A. G. Papagiannakis, “Diffraction of a Gaussian beam from a periodic planar screen,” J. Opt. Soc. Am. A 11, 630–636 (1994).
    [CrossRef]
  29. J.-I. Lee, C.-H. Lee, Y.-S. Lee, Y.-K. Cho, “Diffraction of a Gaussian wave by finite periodic slots in a parallel-plate waveguide,” IEICE Trans. Commun. E84–B, 95–99 (2001).
  30. J. S. Uppal, P. K. Gupta, R. G. Harrison, “Aperiodic ruling for the measurement of Gaussian laser beam diameters,” Opt. Lett. 14, 683–685 (1989).
    [CrossRef] [PubMed]

2001 (2)

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

J.-I. Lee, C.-H. Lee, Y.-S. Lee, Y.-K. Cho, “Diffraction of a Gaussian wave by finite periodic slots in a parallel-plate waveguide,” IEICE Trans. Commun. E84–B, 95–99 (2001).

1999 (1)

B. Guizal, D. Felbacq, “Electromagnetic beam diffraction by a finite strip grating,” Opt. Commun. 165, 1–6 (1999).
[CrossRef]

1998 (4)

1995 (1)

1994 (1)

1991 (1)

1990 (2)

1989 (1)

1987 (1)

G. A. Suedan, E. V. Jull, “Two-dimensional beam diffraction by a half-plane and wide slit,” IEEE Trans. Antennas Propag. AP-35, 1077–1082 (1987).
[CrossRef]

1983 (1)

1980 (1)

K. Hongo, “A method of evaluating the near diffracted field,” IEEE Trans. Antennas Propag. AP-28, 409–412 (1980).
[CrossRef]

1978 (3)

H. A. Kalhor, “Diffraction of electromagnetic waves by plane metallic gratings,” J. Opt. Soc. Am. 68, 1202–1205 (1978).
[CrossRef]

K. Hongo, “Diffraction of an electromagnetic plane wave by a thick slit,” IEEE Trans. Antennas Propag. AP-26, 494–499 (1978).
[CrossRef]

T. Otsuki, “Diffraction by two parallel slits in a plane,” J. Math. Phys. 19, 911–915 (1978).
[CrossRef]

1976 (2)

T. Otsuki, “Reexamination of diffraction problem of a slit by a method of Fourier orthogonal functions transformation,” J. Phys. Soc. Jpn. 41, 2046–2051 (1976).
[CrossRef]

H. Henke, H. Fruchting, “Irradiation in a slotted half space and diffraction by a slit in a thick screen,” Nachrichtentech. 29, 401–405 (1976).

1975 (2)

B. K. Sachdeva, R. A. Hurd, “Diffraction by multiple slits at the interface between two different media,” Can. J. Phys. 53, 1013–1021 (1975).
[CrossRef]

A. S. Zil’bergleit, “Diffraction of electromagnetic waves by an ideal plate with an even number of symmetrically placed slits,” Sov. Phys. Tech. Phys. 20, 292–295 (1975).

1973 (2)

J. L. Roumiguières, D. Maystre, R. Petit, M. Cadilhac, “Etude de la diffraction par une fente pratiquée dans un écran infiniment conducteur d’épaisseur quelconque,” Opt. Commun. 9, 402–405 (1973).

F. L. Neerhoff, G. Mur, “Diffraction of a plane electromagnetic wave by a slit in a thick screen placed between two different media,” Appl. Sci. Res. 28, 73–88 (1973).

1970 (1)

M. Wirgin, “Influence de l’épaisseur de l’écran sur la diffraction par une fente,” C. R. Acad. Sci. Paris 270, 1457–1460 (1970).

1967 (1)

H. S. Tan, “On Kirchhoff’s theory in non-planar scalar diffraction,” Proc. Phys. Soc. Jpn. 91, 768–773 (1967).
[CrossRef]

1954 (1)

C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys. 17, 35–99 (1954).
[CrossRef]

Bouwkamp, C. J.

C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys. 17, 35–99 (1954).
[CrossRef]

Cadilhac, M.

O. Mata-Mendez, M. Cadilhac, R. Petit, “Diffraction of a two-dimensional electromagnetic beam wave by a thick slit pierced in a perfectly conducting screen,” J. Opt. Soc. Am. 73, 328–331 (1983).
[CrossRef]

J. L. Roumiguières, D. Maystre, R. Petit, M. Cadilhac, “Etude de la diffraction par une fente pratiquée dans un écran infiniment conducteur d’épaisseur quelconque,” Opt. Commun. 9, 402–405 (1973).

Chavez-Rivas, F.

Cho, Y.-K.

J.-I. Lee, C.-H. Lee, Y.-S. Lee, Y.-K. Cho, “Diffraction of a Gaussian wave by finite periodic slots in a parallel-plate waveguide,” IEICE Trans. Commun. E84–B, 95–99 (2001).

Depine, R. A.

Eide, H. A.

Felbacq, D.

B. Guizal, D. Felbacq, “Electromagnetic beam diffraction by a finite strip grating,” Opt. Commun. 165, 1–6 (1999).
[CrossRef]

Fruchting, H.

H. Henke, H. Fruchting, “Irradiation in a slotted half space and diffraction by a slit in a thick screen,” Nachrichtentech. 29, 401–405 (1976).

Guizal, B.

B. Guizal, D. Felbacq, “Electromagnetic beam diffraction by a finite strip grating,” Opt. Commun. 165, 1–6 (1999).
[CrossRef]

Gupta, P. K.

Harrison, R. G.

Henke, H.

H. Henke, H. Fruchting, “Irradiation in a slotted half space and diffraction by a slit in a thick screen,” Nachrichtentech. 29, 401–405 (1976).

Hongo, K.

K. Hongo, “A method of evaluating the near diffracted field,” IEEE Trans. Antennas Propag. AP-28, 409–412 (1980).
[CrossRef]

K. Hongo, “Diffraction of an electromagnetic plane wave by a thick slit,” IEEE Trans. Antennas Propag. AP-26, 494–499 (1978).
[CrossRef]

Hurd, R. A.

B. K. Sachdeva, R. A. Hurd, “Diffraction by multiple slits at the interface between two different media,” Can. J. Phys. 53, 1013–1021 (1975).
[CrossRef]

Jull, E. V.

G. A. Suedan, E. V. Jull, “Two-dimensional beam diffraction by a half-plane and wide slit,” IEEE Trans. Antennas Propag. AP-35, 1077–1082 (1987).
[CrossRef]

Kalhor, H. A.

Kriezis, Em. E.

Lee, C.-H.

J.-I. Lee, C.-H. Lee, Y.-S. Lee, Y.-K. Cho, “Diffraction of a Gaussian wave by finite periodic slots in a parallel-plate waveguide,” IEICE Trans. Commun. E84–B, 95–99 (2001).

Lee, J.-I.

J.-I. Lee, C.-H. Lee, Y.-S. Lee, Y.-K. Cho, “Diffraction of a Gaussian wave by finite periodic slots in a parallel-plate waveguide,” IEICE Trans. Commun. E84–B, 95–99 (2001).

Lee, Y.-S.

J.-I. Lee, C.-H. Lee, Y.-S. Lee, Y.-K. Cho, “Diffraction of a Gaussian wave by finite periodic slots in a parallel-plate waveguide,” IEICE Trans. Commun. E84–B, 95–99 (2001).

Loewen, E.

E. Loewen, E. Popov, Diffraction Gratings and Applications (Marcel Dekker, New York, 1997).

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van NostrandReinhold, New York, 1982).

Mata-Mendez, O.

Maystre, D.

J. L. Roumiguières, D. Maystre, R. Petit, M. Cadilhac, “Etude de la diffraction par une fente pratiquée dans un écran infiniment conducteur d’épaisseur quelconque,” Opt. Commun. 9, 402–405 (1973).

Mur, G.

F. L. Neerhoff, G. Mur, “Diffraction of a plane electromagnetic wave by a slit in a thick screen placed between two different media,” Appl. Sci. Res. 28, 73–88 (1973).

Neerhoff, F. L.

F. L. Neerhoff, G. Mur, “Diffraction of a plane electromagnetic wave by a slit in a thick screen placed between two different media,” Appl. Sci. Res. 28, 73–88 (1973).

Otsuki, T.

T. Otsuki, “Diffraction by multiple slits,” J. Opt. Soc. Am. A 7, 646–652 (1990).
[CrossRef]

T. Otsuki, “Diffraction by multiple slits,” J. Opt. Soc. Am. A 7, 646–652 (1990).
[CrossRef]

T. Otsuki, “Diffraction by two parallel slits in a plane,” J. Math. Phys. 19, 911–915 (1978).
[CrossRef]

T. Otsuki, “Reexamination of diffraction problem of a slit by a method of Fourier orthogonal functions transformation,” J. Phys. Soc. Jpn. 41, 2046–2051 (1976).
[CrossRef]

Pandelakis, P. K.

Papagiannakis, A. G.

Petit, R.

O. Mata-Mendez, M. Cadilhac, R. Petit, “Diffraction of a two-dimensional electromagnetic beam wave by a thick slit pierced in a perfectly conducting screen,” J. Opt. Soc. Am. 73, 328–331 (1983).
[CrossRef]

J. L. Roumiguières, D. Maystre, R. Petit, M. Cadilhac, “Etude de la diffraction par une fente pratiquée dans un écran infiniment conducteur d’épaisseur quelconque,” Opt. Commun. 9, 402–405 (1973).

Popov, E.

E. Loewen, E. Popov, Diffraction Gratings and Applications (Marcel Dekker, New York, 1997).

Roumiguières, J. L.

J. L. Roumiguières, D. Maystre, R. Petit, M. Cadilhac, “Etude de la diffraction par une fente pratiquée dans un écran infiniment conducteur d’épaisseur quelconque,” Opt. Commun. 9, 402–405 (1973).

Sachdeva, B. K.

B. K. Sachdeva, R. A. Hurd, “Diffraction by multiple slits at the interface between two different media,” Can. J. Phys. 53, 1013–1021 (1975).
[CrossRef]

Skigin, D. C.

Stamnes, J. J.

Suedan, G. A.

G. A. Suedan, E. V. Jull, “Two-dimensional beam diffraction by a half-plane and wide slit,” IEEE Trans. Antennas Propag. AP-35, 1077–1082 (1987).
[CrossRef]

Tan, H. S.

H. S. Tan, “On Kirchhoff’s theory in non-planar scalar diffraction,” Proc. Phys. Soc. Jpn. 91, 768–773 (1967).
[CrossRef]

Uppal, J. S.

Wirgin, M.

M. Wirgin, “Influence de l’épaisseur de l’écran sur la diffraction par une fente,” C. R. Acad. Sci. Paris 270, 1457–1460 (1970).

Zil’bergleit, A. S.

A. S. Zil’bergleit, “Diffraction of electromagnetic waves by an ideal plate with an even number of symmetrically placed slits,” Sov. Phys. Tech. Phys. 20, 292–295 (1975).

Appl. Sci. Res. (1)

F. L. Neerhoff, G. Mur, “Diffraction of a plane electromagnetic wave by a slit in a thick screen placed between two different media,” Appl. Sci. Res. 28, 73–88 (1973).

C. R. Acad. Sci. Paris (1)

M. Wirgin, “Influence de l’épaisseur de l’écran sur la diffraction par une fente,” C. R. Acad. Sci. Paris 270, 1457–1460 (1970).

Can. J. Phys. (1)

B. K. Sachdeva, R. A. Hurd, “Diffraction by multiple slits at the interface between two different media,” Can. J. Phys. 53, 1013–1021 (1975).
[CrossRef]

IEEE Trans. Antennas Propag. (3)

K. Hongo, “Diffraction of an electromagnetic plane wave by a thick slit,” IEEE Trans. Antennas Propag. AP-26, 494–499 (1978).
[CrossRef]

K. Hongo, “A method of evaluating the near diffracted field,” IEEE Trans. Antennas Propag. AP-28, 409–412 (1980).
[CrossRef]

G. A. Suedan, E. V. Jull, “Two-dimensional beam diffraction by a half-plane and wide slit,” IEEE Trans. Antennas Propag. AP-35, 1077–1082 (1987).
[CrossRef]

IEICE Trans. Commun. (1)

J.-I. Lee, C.-H. Lee, Y.-S. Lee, Y.-K. Cho, “Diffraction of a Gaussian wave by finite periodic slots in a parallel-plate waveguide,” IEICE Trans. Commun. E84–B, 95–99 (2001).

J. Math. Phys. (1)

T. Otsuki, “Diffraction by two parallel slits in a plane,” J. Math. Phys. 19, 911–915 (1978).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (9)

J. Phys. Soc. Jpn. (1)

T. Otsuki, “Reexamination of diffraction problem of a slit by a method of Fourier orthogonal functions transformation,” J. Phys. Soc. Jpn. 41, 2046–2051 (1976).
[CrossRef]

Nachrichtentech. (1)

H. Henke, H. Fruchting, “Irradiation in a slotted half space and diffraction by a slit in a thick screen,” Nachrichtentech. 29, 401–405 (1976).

Opt. Commun. (2)

J. L. Roumiguières, D. Maystre, R. Petit, M. Cadilhac, “Etude de la diffraction par une fente pratiquée dans un écran infiniment conducteur d’épaisseur quelconque,” Opt. Commun. 9, 402–405 (1973).

B. Guizal, D. Felbacq, “Electromagnetic beam diffraction by a finite strip grating,” Opt. Commun. 165, 1–6 (1999).
[CrossRef]

Opt. Lett. (2)

Proc. Phys. Soc. Jpn. (1)

H. S. Tan, “On Kirchhoff’s theory in non-planar scalar diffraction,” Proc. Phys. Soc. Jpn. 91, 768–773 (1967).
[CrossRef]

Rep. Prog. Phys. (1)

C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys. 17, 35–99 (1954).
[CrossRef]

Sov. Phys. Tech. Phys. (1)

A. S. Zil’bergleit, “Diffraction of electromagnetic waves by an ideal plate with an even number of symmetrically placed slits,” Sov. Phys. Tech. Phys. 20, 292–295 (1975).

Other (2)

E. Loewen, E. Popov, Diffraction Gratings and Applications (Marcel Dekker, New York, 1997).

D. Marcuse, Light Transmission Optics (Van NostrandReinhold, New York, 1982).

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Figures (11)

Fig. 1
Fig. 1

Our configuration composed of N slits of width l, separation d, and period D = l + d in an infinitely thin planar screen. The slits are parallel to the Oz axis. The position of the normally incident Gaussian beam is fixed by the parameter b. The angle θ is measured with respect to the screen normal.

Fig. 2
Fig. 2

Plot of the square root of the intensity diffracted for six slits of width l = 2 and period D = 4 for a normally incident plane wave of wave number k 0 = 2.0 . Fair agreement between Otsuki’s theory (solid curve) and our rigorous modal theory (dashed curve) is found.

Fig. 3
Fig. 3

Diffraction patterns of a finite grating with N = 10 , and D / l = 2.0 for a normally incident Gaussian beam with L / l = 20 / 2 , b / l = 9.5 , and wavelengths (a) λ / l = 0.1 , (b) λ / l = 0.8 , (c) λ / l = 1.5 , and (d) λ / l = 2.5 .

Fig. 4
Fig. 4

(a) Intensity diffracted in the normal direction normalized to the total incident energy ( E / I 0 ) and (b) transmission coefficient τ as a function of the wavelength λ / l for finite gratings composed of 2, 3, 4, 5, and 8 slits of period D / l = 1.5 and for a normally incident Gaussian beam with width L / l = 5 / 2 .

Fig. 5
Fig. 5

Transmission coefficient τ as a function of the normalized beam width L / l for finite gratings with N = 2 , 3, 4, 5, 6, 7 and period D / l = 1.5 for a normally incident Gaussian beam with wavelength λ / l = 0.90 whose beam waist is located at the middle of each finite grating.

Fig. 6
Fig. 6

Intensity diffracted in the normal direction normalized to the total incident energy ( E / I 0 ) as a function of the beam width L / l . Parameters are set as in Fig. 5.

Fig. 7
Fig. 7

Diffraction patterns from a finite grating when N = 15 , D / l = 1.5 , and l = 1 for a normally incident Gaussian beam of wavelength λ / l = 0.9 , beam position b / l = 11 , and beam widths (a) L / l = 7 / 2 , (b) L / l = 10 / 2 , (c) L / l = 45 / 2 2 , and (d) L / l = 5000 / 2 .

Fig. 8
Fig. 8

(a) Intensity diffracted in the normal direction normalized to the total incident energy ( E / I 0 ) and (b) transmission coefficient τ versus normalized beam position b / l for a finite grating with N = 5 and period D / l = 1.5 for a normally incident Gaussian beam with wavelength λ / l = 0.90 , beam width L / l = 50 / 2 , and beam position b / l = 3.5 .

Fig. 9
Fig. 9

Coupling between slits. Transmission coefficient τ as a function of normalized separation d / l between slits for a finite grating with N = 5 for a normally incident Gaussian beam with wavelength λ / l = 0.90 , beam width L / l = 500 / 2 , and beam position b / l = 0.5 .

Fig. 10
Fig. 10

Diffracted intensity normalized to the total incident energy [ I ( θ ) / I 0 ] for a finite aperiodic grating with N = 6 , period D / l = 1.5 , and metal separation d / l = 0.5 , where the mid-separation d / l = 0.5 has been replaced by d / l = 0.1 , 0.45, and 5.0, when L / l = 20 / 2 , λ / l = 0.9 , and the spot is located at the middle of the finite aperiodic grating.

Fig. 11
Fig. 11

Diffracted intensity normalized to the incident energy [ I ( θ ) / I 0 ] for a finite grating with five slits of period D / l = 1.5 , d / l = 0.5 , when L / l = 10 / 2 , and λ / l = 0.90 . Constant-intensity angles are pointed out by arrows.

Equations (35)

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2 E x 2 + 2 E y 2 + k 0 2 E = 0 ,
E ( x ,   y ) = 1 2 π - E ˆ ( α ,   y ) exp ( i α x ) d α ,
E ˆ ( α ,   y ) = 1 2 π - E ( x ,   y ) exp ( - i α x ) d α .
E ^ 1 ( α ,   y ) = A ( α ) exp ( - i β y ) + B ( α ) exp ( i β y )
( for y > 0 )
E ^ 2 ( α ,   y ) = C ( α ) exp ( - i β y ) + D ( α ) exp ( i β y )
( for y < 0 ) ,
E 1 ( x ,   y ) = 1 2 π - k 0 k 0 A ( α ) exp [ i ( α x - β y ) ] d α + 1 2 π - B ( α ) exp [ i ( α x + β y ) ] d α
( for y > 0 ) ,
E 2 ( x ,   y ) = 1 2 π - C ( α ) exp [ i ( α x - β y ) ] d α
( for y < 0 ) .
E 3 ( x ) = n = 1 a n 1 ϕ n 1 ( x ) + n = 1 a n 2 ϕ n 2 ( x ) +     + n = 1 a nN ϕ nN ( x ) ,
ϕ np = sin ( x - ( p - 1 ) D )   n π l if ( p - 1 ) D x l + ( p - 1 ) D 0 otherwise ,
ϕ np ,   ϕ mq = - ϕ np ϕ mq * d x = l 2   δ nm δ pq ,
A ( α ) + B ( α ) = n = 1 a n 1 ϕ ^ n 1 ( α ) + n = 1 a n 2 ϕ ^ n 2 ( α ) +     + n = 1 a nN ϕ ^ nN ( α ) ,
C ( α ) = n = 1 a n 1 ϕ ^ n 1 ( α ) + n = 1 a n 2 ϕ ^ n 2 ( α ) +     + n = 1 a nN ϕ ^ nN ( α ) ,
E 1 y   ( x ,   0 ) - E 2 y   ( x ,   0 ) ,   ϕ mq ( x ) = 0 .
E ^ 1 y   ( α ,   0 ) - E ^ 2 y   ( α ,   0 ) ,   ϕ ^ mq ( α ) = 0 ,
n = 1 a n 1 β ϕ ^ n 1 ,   ϕ ^ mq + n = 1 a n 2 β ϕ ^ n 2 ,   ϕ ^ mq +    
+ n = 1 a nN β ϕ ^ nN ,   ϕ ^ mq = β A ( α ) ,   ϕ ^ mq ,
- E ^ 1 y   ( α ,   0 ) ϕ ^ mq * ( α ) d α = - i - β ( α ) A ( α ) ϕ ^ mq * ( α ) d α + i - β ( α ) B ( α ) ϕ ^ mq * ( α ) d α .
E ^ i y   ( α ,   0 ) = - i β A ( α ) ,
E ^ 1 y   ( α ,   0 ) - E ^ i y   ( α ,   0 ) ,   ϕ ^ mq = i β ( α ) B ( α ) ,   ϕ ^ mq .
β ( α ) B ( α ) ,   ϕ ^ mq = 0 .
E 1 y   ( x ,   0 ) = E i y   ( x ,   0 ) ,
τ = - k 0 k 0 β ( α ) | C ( α ) | 2 d α - k 0 k 0 β ( α ) | A ( α ) | 2 d α
ρ = - k 0 k 0 β ( α ) | B ( α ) | 2 d α - k 0 k 0 β ( α ) | A ( α ) | 2 d α ,
ρ + τ = 1 .
I ( θ ) = k 0 2 2 μ 0 ω cos 2   θ | E ^ 3 ( k 0 sin   θ ,   0 ) | 2 .
E i ( x ,   y = 0 ) = exp - 2 ( x - b ) 2 L 2 ,
A ( α ) = L 2 exp ( - i α b ) exp - α 2 L 2 8 .
( L / l ) max = N   D 2 ;
E ( b ) = 0.72873   exp [ - 3.9375 ( b - 3.5 ) 2 / L 2 ]
τ ( b ) = 0.142201   exp [ - 3.8875 ( b - 3.5 ) 2 / L 2 ] .
E = 0.9224   N τ λ ,

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