Abstract

We develop a rigorous electromagnetic analysis to study the field diffracted by a metallic or dielectric grating made of rectangular rods, lying over a stratified media. The analysis generalizes the modal theory of perfectly conducting lamellar gratings combined with the R-matrix propagation algorithm in order to avoid numerical instabilities, thus allowing treatment of layers and gratings of arbitrary thickness. We then study the field map obtained under the conditions stated in a previous patent that described a process for casting on a support the faithful reproduction of a mask pierced with periodically distributed slits. We make a systematic study of the influence of the various parameters (incidence, mark–space ratio, groove spacing, groove depth, polarization, conductivity of the metal) on the field map below the mask. We discover a large tolerance over the parameters, close to the values stated in the patent. The result is that the setup described in the patent valid for periodic maps can be used for duplicating nonperiodic masks (e.g., linear Fresnel zone plates) as well as chirped gratings or gratings with nonrectilinear grooves.

© 1996 Optical Society of America

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  1. M. Okai, S. Tsuji, N. Chinone, T. Haruda, “Novel method to fabricate corrugation for a λ/4-shifted distributed feedback laser using a grating photomask,” Appl. Phys. Lett. 55, 415–417 (1989).
    [CrossRef]
  2. D. M. Tennant, T. L. Koch, P. P. Mulgrew, R. P. Gnall, F. Ostermeyer, J. M. Verdiell, “Characterization of near-field holography grating masks for optoelectronics fabricated by electron beam lithography,”J. Vac. Sci. Technol. B 10, 2530–2535 (1992).
    [CrossRef]
  3. T. E. Yeo, N. J. Phillips, S. J. Clements, S. Ojha, “Photopolymer replication—a new technique for 0/25 μm phase shifted DFB–LD grating manufacture?” Proc. IEE 379, 186–191 (1993).
  4. G. Pakulski, R. Moore, C. Maritan, F. Shepher, F. Fallahi, L. Templeton, G. Champion, “Fused silica masks for distributed feedback lasers,” Appl. Phys. Lett. 62, 222–224 (1993).
    [CrossRef]
  5. J. L. Roumiguières, M. Nevière, “Procédé pour reporter sur un support l’ombre fidéle d’un masque percé de fentes distribuées périodiquement, et application de ce procédé notamment en photolithogravure,” French patent792254 (1979); European patent80 401 211-0 (1980); Japanese patent1625035 (1980); U.S.A. patent4 389 094 (1983), Canadian patent1140373 (1983).
  6. J. L. Roumiguières, D. Maystre, R. Petit, “Diffraction par un réseau lamellaire infiniment conducteur en présence de dioptres parallèles aux lamelles,” Opt. Commun. 7, 402–405 (1973).
    [CrossRef]
  7. M. Nevière, “Sur un formalisme différentiel pour les problèmes de diffraction dans le domaine de résonance; applications à l’étude des réseaux optiques et de diverses structures périodiques,” thèse d’état A. O. 11556 (Université d’Aix-Marseille III, Centre de St. Jérôme, Marseille, France, 1979), Chap. III, pp. 84–94.
  8. R. Petit, “A tutorial introduction,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 1–52.
    [CrossRef]
  9. L. Li, “Multilayer modal theory for diffraction gratings of arbitrary profile, depth, and conductivity,” J. Opt. Soc. Am. A 10, 2581–2591 (1993).
    [CrossRef]
  10. F. Montiel, M. Nevière, “Differential theory of gratings: extension to deep gratings of arbitrary profile and permittivity through the R-matrix propagation algorithm,” J. Opt. Soc. Am. A 11, 3241–3250 (1994).
    [CrossRef]
  11. O. Parriaux, H. Vuilliomenet, P. Sixt, N. Cuny, “Spatial frequency bandwidth in the photolithographic transfer of submicron gratings,” Opt. Eng. (to be published).

1994

1993

L. Li, “Multilayer modal theory for diffraction gratings of arbitrary profile, depth, and conductivity,” J. Opt. Soc. Am. A 10, 2581–2591 (1993).
[CrossRef]

T. E. Yeo, N. J. Phillips, S. J. Clements, S. Ojha, “Photopolymer replication—a new technique for 0/25 μm phase shifted DFB–LD grating manufacture?” Proc. IEE 379, 186–191 (1993).

G. Pakulski, R. Moore, C. Maritan, F. Shepher, F. Fallahi, L. Templeton, G. Champion, “Fused silica masks for distributed feedback lasers,” Appl. Phys. Lett. 62, 222–224 (1993).
[CrossRef]

1992

D. M. Tennant, T. L. Koch, P. P. Mulgrew, R. P. Gnall, F. Ostermeyer, J. M. Verdiell, “Characterization of near-field holography grating masks for optoelectronics fabricated by electron beam lithography,”J. Vac. Sci. Technol. B 10, 2530–2535 (1992).
[CrossRef]

1989

M. Okai, S. Tsuji, N. Chinone, T. Haruda, “Novel method to fabricate corrugation for a λ/4-shifted distributed feedback laser using a grating photomask,” Appl. Phys. Lett. 55, 415–417 (1989).
[CrossRef]

1973

J. L. Roumiguières, D. Maystre, R. Petit, “Diffraction par un réseau lamellaire infiniment conducteur en présence de dioptres parallèles aux lamelles,” Opt. Commun. 7, 402–405 (1973).
[CrossRef]

Champion, G.

G. Pakulski, R. Moore, C. Maritan, F. Shepher, F. Fallahi, L. Templeton, G. Champion, “Fused silica masks for distributed feedback lasers,” Appl. Phys. Lett. 62, 222–224 (1993).
[CrossRef]

Chinone, N.

M. Okai, S. Tsuji, N. Chinone, T. Haruda, “Novel method to fabricate corrugation for a λ/4-shifted distributed feedback laser using a grating photomask,” Appl. Phys. Lett. 55, 415–417 (1989).
[CrossRef]

Clements, S. J.

T. E. Yeo, N. J. Phillips, S. J. Clements, S. Ojha, “Photopolymer replication—a new technique for 0/25 μm phase shifted DFB–LD grating manufacture?” Proc. IEE 379, 186–191 (1993).

Cuny, N.

O. Parriaux, H. Vuilliomenet, P. Sixt, N. Cuny, “Spatial frequency bandwidth in the photolithographic transfer of submicron gratings,” Opt. Eng. (to be published).

Fallahi, F.

G. Pakulski, R. Moore, C. Maritan, F. Shepher, F. Fallahi, L. Templeton, G. Champion, “Fused silica masks for distributed feedback lasers,” Appl. Phys. Lett. 62, 222–224 (1993).
[CrossRef]

Gnall, R. P.

D. M. Tennant, T. L. Koch, P. P. Mulgrew, R. P. Gnall, F. Ostermeyer, J. M. Verdiell, “Characterization of near-field holography grating masks for optoelectronics fabricated by electron beam lithography,”J. Vac. Sci. Technol. B 10, 2530–2535 (1992).
[CrossRef]

Haruda, T.

M. Okai, S. Tsuji, N. Chinone, T. Haruda, “Novel method to fabricate corrugation for a λ/4-shifted distributed feedback laser using a grating photomask,” Appl. Phys. Lett. 55, 415–417 (1989).
[CrossRef]

Koch, T. L.

D. M. Tennant, T. L. Koch, P. P. Mulgrew, R. P. Gnall, F. Ostermeyer, J. M. Verdiell, “Characterization of near-field holography grating masks for optoelectronics fabricated by electron beam lithography,”J. Vac. Sci. Technol. B 10, 2530–2535 (1992).
[CrossRef]

Li, L.

Maritan, C.

G. Pakulski, R. Moore, C. Maritan, F. Shepher, F. Fallahi, L. Templeton, G. Champion, “Fused silica masks for distributed feedback lasers,” Appl. Phys. Lett. 62, 222–224 (1993).
[CrossRef]

Maystre, D.

J. L. Roumiguières, D. Maystre, R. Petit, “Diffraction par un réseau lamellaire infiniment conducteur en présence de dioptres parallèles aux lamelles,” Opt. Commun. 7, 402–405 (1973).
[CrossRef]

Montiel, F.

Moore, R.

G. Pakulski, R. Moore, C. Maritan, F. Shepher, F. Fallahi, L. Templeton, G. Champion, “Fused silica masks for distributed feedback lasers,” Appl. Phys. Lett. 62, 222–224 (1993).
[CrossRef]

Mulgrew, P. P.

D. M. Tennant, T. L. Koch, P. P. Mulgrew, R. P. Gnall, F. Ostermeyer, J. M. Verdiell, “Characterization of near-field holography grating masks for optoelectronics fabricated by electron beam lithography,”J. Vac. Sci. Technol. B 10, 2530–2535 (1992).
[CrossRef]

Nevière, M.

F. Montiel, M. Nevière, “Differential theory of gratings: extension to deep gratings of arbitrary profile and permittivity through the R-matrix propagation algorithm,” J. Opt. Soc. Am. A 11, 3241–3250 (1994).
[CrossRef]

M. Nevière, “Sur un formalisme différentiel pour les problèmes de diffraction dans le domaine de résonance; applications à l’étude des réseaux optiques et de diverses structures périodiques,” thèse d’état A. O. 11556 (Université d’Aix-Marseille III, Centre de St. Jérôme, Marseille, France, 1979), Chap. III, pp. 84–94.

J. L. Roumiguières, M. Nevière, “Procédé pour reporter sur un support l’ombre fidéle d’un masque percé de fentes distribuées périodiquement, et application de ce procédé notamment en photolithogravure,” French patent792254 (1979); European patent80 401 211-0 (1980); Japanese patent1625035 (1980); U.S.A. patent4 389 094 (1983), Canadian patent1140373 (1983).

Ojha, S.

T. E. Yeo, N. J. Phillips, S. J. Clements, S. Ojha, “Photopolymer replication—a new technique for 0/25 μm phase shifted DFB–LD grating manufacture?” Proc. IEE 379, 186–191 (1993).

Okai, M.

M. Okai, S. Tsuji, N. Chinone, T. Haruda, “Novel method to fabricate corrugation for a λ/4-shifted distributed feedback laser using a grating photomask,” Appl. Phys. Lett. 55, 415–417 (1989).
[CrossRef]

Ostermeyer, F.

D. M. Tennant, T. L. Koch, P. P. Mulgrew, R. P. Gnall, F. Ostermeyer, J. M. Verdiell, “Characterization of near-field holography grating masks for optoelectronics fabricated by electron beam lithography,”J. Vac. Sci. Technol. B 10, 2530–2535 (1992).
[CrossRef]

Pakulski, G.

G. Pakulski, R. Moore, C. Maritan, F. Shepher, F. Fallahi, L. Templeton, G. Champion, “Fused silica masks for distributed feedback lasers,” Appl. Phys. Lett. 62, 222–224 (1993).
[CrossRef]

Parriaux, O.

O. Parriaux, H. Vuilliomenet, P. Sixt, N. Cuny, “Spatial frequency bandwidth in the photolithographic transfer of submicron gratings,” Opt. Eng. (to be published).

Petit, R.

J. L. Roumiguières, D. Maystre, R. Petit, “Diffraction par un réseau lamellaire infiniment conducteur en présence de dioptres parallèles aux lamelles,” Opt. Commun. 7, 402–405 (1973).
[CrossRef]

R. Petit, “A tutorial introduction,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 1–52.
[CrossRef]

Phillips, N. J.

T. E. Yeo, N. J. Phillips, S. J. Clements, S. Ojha, “Photopolymer replication—a new technique for 0/25 μm phase shifted DFB–LD grating manufacture?” Proc. IEE 379, 186–191 (1993).

Roumiguières, J. L.

J. L. Roumiguières, D. Maystre, R. Petit, “Diffraction par un réseau lamellaire infiniment conducteur en présence de dioptres parallèles aux lamelles,” Opt. Commun. 7, 402–405 (1973).
[CrossRef]

J. L. Roumiguières, M. Nevière, “Procédé pour reporter sur un support l’ombre fidéle d’un masque percé de fentes distribuées périodiquement, et application de ce procédé notamment en photolithogravure,” French patent792254 (1979); European patent80 401 211-0 (1980); Japanese patent1625035 (1980); U.S.A. patent4 389 094 (1983), Canadian patent1140373 (1983).

Shepher, F.

G. Pakulski, R. Moore, C. Maritan, F. Shepher, F. Fallahi, L. Templeton, G. Champion, “Fused silica masks for distributed feedback lasers,” Appl. Phys. Lett. 62, 222–224 (1993).
[CrossRef]

Sixt, P.

O. Parriaux, H. Vuilliomenet, P. Sixt, N. Cuny, “Spatial frequency bandwidth in the photolithographic transfer of submicron gratings,” Opt. Eng. (to be published).

Templeton, L.

G. Pakulski, R. Moore, C. Maritan, F. Shepher, F. Fallahi, L. Templeton, G. Champion, “Fused silica masks for distributed feedback lasers,” Appl. Phys. Lett. 62, 222–224 (1993).
[CrossRef]

Tennant, D. M.

D. M. Tennant, T. L. Koch, P. P. Mulgrew, R. P. Gnall, F. Ostermeyer, J. M. Verdiell, “Characterization of near-field holography grating masks for optoelectronics fabricated by electron beam lithography,”J. Vac. Sci. Technol. B 10, 2530–2535 (1992).
[CrossRef]

Tsuji, S.

M. Okai, S. Tsuji, N. Chinone, T. Haruda, “Novel method to fabricate corrugation for a λ/4-shifted distributed feedback laser using a grating photomask,” Appl. Phys. Lett. 55, 415–417 (1989).
[CrossRef]

Verdiell, J. M.

D. M. Tennant, T. L. Koch, P. P. Mulgrew, R. P. Gnall, F. Ostermeyer, J. M. Verdiell, “Characterization of near-field holography grating masks for optoelectronics fabricated by electron beam lithography,”J. Vac. Sci. Technol. B 10, 2530–2535 (1992).
[CrossRef]

Vuilliomenet, H.

O. Parriaux, H. Vuilliomenet, P. Sixt, N. Cuny, “Spatial frequency bandwidth in the photolithographic transfer of submicron gratings,” Opt. Eng. (to be published).

Yeo, T. E.

T. E. Yeo, N. J. Phillips, S. J. Clements, S. Ojha, “Photopolymer replication—a new technique for 0/25 μm phase shifted DFB–LD grating manufacture?” Proc. IEE 379, 186–191 (1993).

Appl. Phys. Lett.

M. Okai, S. Tsuji, N. Chinone, T. Haruda, “Novel method to fabricate corrugation for a λ/4-shifted distributed feedback laser using a grating photomask,” Appl. Phys. Lett. 55, 415–417 (1989).
[CrossRef]

G. Pakulski, R. Moore, C. Maritan, F. Shepher, F. Fallahi, L. Templeton, G. Champion, “Fused silica masks for distributed feedback lasers,” Appl. Phys. Lett. 62, 222–224 (1993).
[CrossRef]

J. Opt. Soc. Am. A

J. Vac. Sci. Technol. B

D. M. Tennant, T. L. Koch, P. P. Mulgrew, R. P. Gnall, F. Ostermeyer, J. M. Verdiell, “Characterization of near-field holography grating masks for optoelectronics fabricated by electron beam lithography,”J. Vac. Sci. Technol. B 10, 2530–2535 (1992).
[CrossRef]

Opt. Commun.

J. L. Roumiguières, D. Maystre, R. Petit, “Diffraction par un réseau lamellaire infiniment conducteur en présence de dioptres parallèles aux lamelles,” Opt. Commun. 7, 402–405 (1973).
[CrossRef]

Proc. IEE

T. E. Yeo, N. J. Phillips, S. J. Clements, S. Ojha, “Photopolymer replication—a new technique for 0/25 μm phase shifted DFB–LD grating manufacture?” Proc. IEE 379, 186–191 (1993).

Other

J. L. Roumiguières, M. Nevière, “Procédé pour reporter sur un support l’ombre fidéle d’un masque percé de fentes distribuées périodiquement, et application de ce procédé notamment en photolithogravure,” French patent792254 (1979); European patent80 401 211-0 (1980); Japanese patent1625035 (1980); U.S.A. patent4 389 094 (1983), Canadian patent1140373 (1983).

M. Nevière, “Sur un formalisme différentiel pour les problèmes de diffraction dans le domaine de résonance; applications à l’étude des réseaux optiques et de diverses structures périodiques,” thèse d’état A. O. 11556 (Université d’Aix-Marseille III, Centre de St. Jérôme, Marseille, France, 1979), Chap. III, pp. 84–94.

R. Petit, “A tutorial introduction,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 1–52.
[CrossRef]

O. Parriaux, H. Vuilliomenet, P. Sixt, N. Cuny, “Spatial frequency bandwidth in the photolithographic transfer of submicron gratings,” Opt. Eng. (to be published).

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Figures (13)

Fig. 1
Fig. 1

Schematic representation of a periodic mask. Medium 1 is generally glass, and medium 2 is generally air.

Fig. 2
Fig. 2

Field intensity maps for various negative ordinates for a chromium mask (d = 0.65 μm, h = 0.18 μm, a/d = 0.5) illuminated under first-order Littrow incidence; λ = 0.44 μm in vacuum. The various ordinates are printed as parameters (in air). ν1 = 1.5, ν2 = ν = 1.

Fig. 3
Fig. 3

Transmitted amplitudes in the zero and −1 orders below a mask illuminated near −1-order Littrow position as a function of incidence θ (in degrees). ν1 = 1.5, ν2 = ν = 1, h = 0.18 μm, a/d = 0.5, d = 0.4 μm, λ = 0.441 μm. (a) perfectly conducting mask, (b) chromium mask.

Fig. 4
Fig. 4

Transmitted amplitudes below the mask shown in Fig. 3(a) as a function of groove spacing d, with a/d = 0.5. We recall that θ = 21.61° in glass and that Littrow configuration occurs for d = 0.4 μm, λ = 0.441 μm.

Fig. 5
Fig. 5

Transmitted amplitudes below a chromium mask as a function of aspect ratio a/d; d = 0.5 μm, h = 0.2 μm, λ = 0.442 μm, ν1 = 1.5, ν2 = ν = 1; Littrow incidence.

Fig. 6
Fig. 6

Same as Fig. 4 but for different values of h: (a) h = 0.3 μm, (b) h = 0.05 μm.

Fig. 7
Fig. 7

Transmitted amplitudes below a chromium mask as a function of d for three different groove depths h. λ = 0.44 μm, ν1 = 1.5, ν2 = ν = 1, a/d = 0.5. (a) h = 0.30 μm, (b) h = 0.15 μm, (c) h = 0.05 μm.

Fig. 8
Fig. 8

Same device as in Fig. 7(a), but chromium is replaced by glass.

Fig. 9
Fig. 9

Same device as in Fig. 7(a), but the refractive index of the rods is 1.5 + i1. (a) h = 0.3 μm, (b) h = 0.2 μm, (c) h = 0.1 μm.

Fig. 10
Fig. 10

Transmitted amplitudes below a finite-conductivity mask (refractive index 1.5 + i1) as a function of incidence θ; d = 0.4 μm, h = 0.3 μm, λ = 0.44 μm, ν1 = 1.5, ν2 = ν = 1, a/d = 0.5. Littrow incidence is 21.61°.

Fig. 11
Fig. 11

First and −2 absolute efficiencies as a function of d for a mask with a/d = 0.5, λ = 0.44 μm, h = 0.2 μm, ν1 = 1.5, ν2 = ν = 1. (a) Infinite conductivity, (b) chromium.

Fig. 12
Fig. 12

Transmitted amplitudes below the mask as a function of incidence θ (in degrees); for a chromium mask with d = 0.4 μm, λ = 0.442 μm, ν1 = 1.5, ν2 = ν = 1. (a) h = 0.2 μm, infinite conductivity of rods; (b) h = 0.2 μm, chromium rods; (c) h = 0.1 μm, chromium rods; (d) h = 0.07 μm, chromium rods.

Fig. 13
Fig. 13

Transmitted amplitudes as a function of groove spacing d for a chromium mask with h = 0.1 μm, λ = 0.442 μm, ν1 = 1.5, ν2 = ν = 1. (a) Littrow angle for d = 0.3 μm, (b) Littrow angle for d = 0.6 μm.

Equations (45)

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E i ( x , y ) = exp [ i ( α x - β y ) ] ,
If y > h :             E 1 ( x , y ) = exp [ i ( α x - β y ) ] + m B m exp [ i ( α m x + β 1 , m y ) ] .
If y > 0 :             E 2 ( x , y ) = m T m exp [ i ( α m x - β 2 , m y ) ] ,
β i , m 2 = ( 2 π λ ν 1 ) 2 - α m 2 ,             Re ( β i , m ) + Im ( β i , m ) > 0 ,             i [ 1 , 2 ] .
E g ( x , y ) = m = 1 w n ( y ) G m ( x ) ,
μ m 2 = ( 2 π λ ν ) 2 - ( m π a ) 2 ,             Re ( μ m ) + Im ( μ m ) > 0.
F ( x ) = 1 , 0 < x < a , F ( x ) = 0 , a < x < d .
E 1 ( x , h ) = F ( x ) E g ( x , h ) ,
F ( x ) E 1 ( x , y ) y | y = h = F ( x ) E g ( x , y ) y | y = h ,
E 2 ( x , 0 ) = F ( x ) E g ( x , 0 ) ,
F ( x ) E 2 ( x , y ) y | y = 0 = F ( x ) E g ( x , y ) y | y = 0 .
γ m ( x ) = F ( x ) exp ( i α m x ) ,             ρ m ( x ) = F ( x ) G m ( x ) ,
exp ( - i β h ) exp ( i α x ) + m = - + B m exp ( i β 1 , m h ) exp ( i α m x ) = m = 1 + w m ( h ) ρ m ( x ) ,
- i β exp ( - i β h ) γ 0 ( x ) + m i β 1 , m B m exp ( i β 1 , m h ) γ m ( x ) = m = 1 w m y | y = h ρ m ( x ) ,
m = - + T m exp ( i α m x ) = m = 1 w m ( 0 ) ρ m ( x ) ,
m = - + ( - i β 2 , m ) T m γ m ( x ) = m = 1 w m ( y ) y | y = 0 ρ m ( x ) .
γ n , m = 1 d 0 d γ m ( x ) exp ( - i α n x ) d x ,
B ˜ n - m = 1 [ a m cos ( μ m h ) + b m sin ( μ m h ) ] ρ n , m = - exp ( - i β h ) δ n , 0 ,
m = - + i β 1 , m B ˜ m γ n , m - m = 1 μ m [ - a m sin ( μ m h ) + b m cos ( μ m h ) ] ρ n , m = i β exp ( - i β h ) γ n , 0 ,
T n = m = 1 ρ n , m a m = 0 ,
m = - + i β 2 , m T m γ n , m + m = 1 ρ n , m μ m b m = 0.
1 ν 1 2 H 1 ( x , y ) y | y = h = F ( x ) ( 1 ν 2 H g ( x , h ) y ) | y = h ,
F ( x ) H 1 ( x , h ) = F ( x ) H g ( x , h ) ,
1 ν 2 2 H 2 ( x , y ) y | y = 0 = F ( x ) ( 1 ν 2 H g ( x , h ) y ) | y = 0 , F ( x ) H 2 ( x , 0 ) = F ( x ) H g ( x , 0 ) .
i ν 1 2 β 1 , n B ˜ n - 1 ν 2 m = 1 μ m [ - a m sin ( μ m h ) + b m cos ( μ m h ) ] × ρ n , m = 1 ν 1 2 i β exp ( - i β h ) δ n , 0 ,
m = - + B ˜ m γ n , m - m = 1 [ a m cos ( μ m h ) + b m sin ( μ m h ) ] ρ n , m = - γ n , 0 exp ( - i β h ) ,
i ν 2 2 β 2 , n T n + 1 ν 2 m = 1 μ m b m ρ n , m = 0 ,
m T m γ n , m - m = 1 a m ρ n , m = 0.
Ψ 1 , m = w m ( 0 ) , Ψ 2 , m = w m ( h ) , Ψ 3 , m = w m y | y = 0 , Ψ 4 , m = w m y | y = h ,
[ Ψ 1 , m Ψ 2 , m ] = r [ Ψ 3 , m Ψ 4 , m ] .
w m ( 0 ) = a m ,
w m ( h ) = a m cos ( μ m h ) + b m sin ( μ m h ) ,
w m y | y = 0 = b m μ m ,
w m y | y = h = μ m [ - a m sin ( μ m h ) + b m cos ( μ m h ) ] .
w m ( 0 ) = 1 μ m tan ( μ m h ) w m y | y = 0 - 1 μ m sin ( μ m h ) w m y | y = h , w m ( h ) = 1 μ m sin ( μ m h ) w m y | y = 0 - 1 μ m tan ( μ m h ) w m y | y = h ,
r = [ 1 μ m tan ( μ m h ) - 1 μ m sin ( μ m h ) 1 μ m sin ( μ m h ) - 1 μ m tan ( μ m h ) ] .
γ n , m = 1 d 0 d d x γ m ( x ) exp ( - i α n x ) = 1 d 0 d d x f ( x ) exp ( i α m x ) exp ( - i α n x ) = 1 d 0 a d x exp [ i ( m - n ) K x ] .
If m = n :             γ n , m = a / d ; if n m :             γ n , m = 1 d K 1 i ( m - n ) × { exp [ i ( m - n ) K a ] - 1 } .
ρ n , m = 1 d 0 d d x ρ m ( x ) exp ( - i α n x ) = 1 d 0 d d x f ( x ) sin ( m π x a ) exp ( - i α n x ) = 1 d 0 a d x [ exp ( i m π x a ) - exp ( - i m π x a ) 2 i ] × exp ( - i α n x ) = 1 2 i d { 0 a d x exp [ i ( m π a - a n ) x ] - 0 a d x exp [ - i ( m π a + α n ) x ] } .
If m π / a - α n = 0 :             ρ n , m = a / 2 i d . If m π / a + α n = 0 :             ρ n , m = - a / 2 i d .
If ( m π a ) 2 - α n 2 0 :
ρ n , m = 1 2 i d ( [ 1 i ( m π / a - α n ) ] { exp [ i ( m π / a - α n ) a ] - 1 } - [ 1 - i ( m π / a + α n ) ] { exp [ - i ( m π / a + α n ) a ] - 1 } ) = 1 2 i d [ ( - 1 ) m exp ( - i α n x ) - 1 ] × [ 1 i ( m π / a - α n ) + 1 i ( m π / a + α n ) ] = 1 2 i d [ ( - 1 ) m exp ( - i α n a ) - 1 } { 2 i m ( π / a ) - [ ( m π / a ) 2 - α n 2 ] } = m π a d 1 α n 2 - ( m π / a ) 2 [ ( - 1 ) m exp ( - i α n a ) - 1 ] .
ρ n , m = 1 d 0 d d x ρ m ( x ) exp ( - i α n x ) = 1 d 0 d d x f ( x ) cos ( m π x / a ) exp ( - i α n x ) = 1 d 0 a d x [ exp ( i m π x a ) + exp ( - i m π x a ) 2 ] × exp ( - i α n x ) = 1 2 d { 0 a d x exp [ i ( m π / a - α n ) x ] + 0 a d x exp [ - i ( m π / a + α n ) x ] } .
If ( m π / a - α n ) ( m π / a + α n ) = 0 :             ρ n , m = a / 2 d . If ( m π / a ) 2 - α n 2 0 :
ρ n , m = 1 2 d ( 1 i ( m π / a - α n ) { exp [ i ( m π / a - α n ) a ] - 1 } + 1 - i ( m π / a + α n ) { exp [ - i ( m π / a + α n ) a ] - 1 } ) = 1 2 d [ ( - 1 ) m exp ( - i α n a ) - 1 ] × [ - 2 i α n ( m π / a ) 2 - α n 2 ] = + i α n d 1 α n 2 - ( m π / a ) 2 [ ( - 1 ) m exp ( - i α n a ) - 1 ] .

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