Abstract

The usual differential method for solving the grating problem in the H|| case is shown to be unable to predict the efficiencies of blazed gratings in a reliable manner. Its predictions are compared with those obtained using the integral method developed by Maystre, a reliable method that has shown its validity over a wide range of applications. The efficiencies of sinusoidal gratings as a function of angle of incidence are calculated by both methods for two values of the groove-height-to-period ratio. For 0.05 (low modulations) both formalisms yield similar results, but for 0.2 only a qualitative agreement is observed. The differential method is shown to involve an approximation valid only for low-modulated surfaces, a fact that accounts for the observed discrepancies. As a self-consistency test, the fulfillment of the electromagnetic boundary conditions is checked by calculating the jumps of the field components, which should be continuous at the grating surface.

© 1987 Optical Society of America

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References

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  1. G. Cerutti-Maori, R. Petit, M. Cadilhac, “Etude numerique du champ diffracte par un reseau,”C. R. Acad. Sci. (Paris) 268B, 1060–1063 (1969).
  2. N. Neviere, P. Vincent, R. Petit, “Sur la theorie du reseau conducteur et ses applications a l’optique,” Nouv. Rev. Opt. 5, 65–77 (1974).
  3. P. Vincent, “Differential methods,” in Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics, R. Petit, ed. (Springer-Verlag, Berlin, 1980), p. 101.
    [Crossref]
  4. R. A. Depine, J. M. Simon, “Diffraction grating efficiencies: an exact differential algorithm valid for high conductivities,” Opt. Acta 30, 1273–1286 (1983).
    [Crossref]
  5. J. M. Simon, R. A. Depine, “Diffraction grating efficiencies: differential methods for H||case,” Optik 67, 145–153 (1984).
  6. D. Maystre, “A new general integral theory for dielectric coated gratings,”J. Opt. Soc. Am. 68, 490–495 (1978).
    [Crossref]
  7. D. Maystre, “Integral methods,” in Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics, R. Petit, ed. (Springer-Verlag, Berlin, 1980), p. 63.
    [Crossref]
  8. D. Maystre, M. Neviere, R. Petit, “Experimental verifications and applications of the theory,” in Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics, R. Petit, ed. (Springer-Verlag, Berlin, 1980), p. 159.
    [Crossref]

1984 (1)

J. M. Simon, R. A. Depine, “Diffraction grating efficiencies: differential methods for H||case,” Optik 67, 145–153 (1984).

1983 (1)

R. A. Depine, J. M. Simon, “Diffraction grating efficiencies: an exact differential algorithm valid for high conductivities,” Opt. Acta 30, 1273–1286 (1983).
[Crossref]

1978 (1)

1974 (1)

N. Neviere, P. Vincent, R. Petit, “Sur la theorie du reseau conducteur et ses applications a l’optique,” Nouv. Rev. Opt. 5, 65–77 (1974).

1969 (1)

G. Cerutti-Maori, R. Petit, M. Cadilhac, “Etude numerique du champ diffracte par un reseau,”C. R. Acad. Sci. (Paris) 268B, 1060–1063 (1969).

Cadilhac, M.

G. Cerutti-Maori, R. Petit, M. Cadilhac, “Etude numerique du champ diffracte par un reseau,”C. R. Acad. Sci. (Paris) 268B, 1060–1063 (1969).

Cerutti-Maori, G.

G. Cerutti-Maori, R. Petit, M. Cadilhac, “Etude numerique du champ diffracte par un reseau,”C. R. Acad. Sci. (Paris) 268B, 1060–1063 (1969).

Depine, R. A.

J. M. Simon, R. A. Depine, “Diffraction grating efficiencies: differential methods for H||case,” Optik 67, 145–153 (1984).

R. A. Depine, J. M. Simon, “Diffraction grating efficiencies: an exact differential algorithm valid for high conductivities,” Opt. Acta 30, 1273–1286 (1983).
[Crossref]

Maystre, D.

D. Maystre, “A new general integral theory for dielectric coated gratings,”J. Opt. Soc. Am. 68, 490–495 (1978).
[Crossref]

D. Maystre, “Integral methods,” in Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics, R. Petit, ed. (Springer-Verlag, Berlin, 1980), p. 63.
[Crossref]

D. Maystre, M. Neviere, R. Petit, “Experimental verifications and applications of the theory,” in Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics, R. Petit, ed. (Springer-Verlag, Berlin, 1980), p. 159.
[Crossref]

Neviere, M.

D. Maystre, M. Neviere, R. Petit, “Experimental verifications and applications of the theory,” in Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics, R. Petit, ed. (Springer-Verlag, Berlin, 1980), p. 159.
[Crossref]

Neviere, N.

N. Neviere, P. Vincent, R. Petit, “Sur la theorie du reseau conducteur et ses applications a l’optique,” Nouv. Rev. Opt. 5, 65–77 (1974).

Petit, R.

N. Neviere, P. Vincent, R. Petit, “Sur la theorie du reseau conducteur et ses applications a l’optique,” Nouv. Rev. Opt. 5, 65–77 (1974).

G. Cerutti-Maori, R. Petit, M. Cadilhac, “Etude numerique du champ diffracte par un reseau,”C. R. Acad. Sci. (Paris) 268B, 1060–1063 (1969).

D. Maystre, M. Neviere, R. Petit, “Experimental verifications and applications of the theory,” in Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics, R. Petit, ed. (Springer-Verlag, Berlin, 1980), p. 159.
[Crossref]

Simon, J. M.

J. M. Simon, R. A. Depine, “Diffraction grating efficiencies: differential methods for H||case,” Optik 67, 145–153 (1984).

R. A. Depine, J. M. Simon, “Diffraction grating efficiencies: an exact differential algorithm valid for high conductivities,” Opt. Acta 30, 1273–1286 (1983).
[Crossref]

Vincent, P.

N. Neviere, P. Vincent, R. Petit, “Sur la theorie du reseau conducteur et ses applications a l’optique,” Nouv. Rev. Opt. 5, 65–77 (1974).

P. Vincent, “Differential methods,” in Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics, R. Petit, ed. (Springer-Verlag, Berlin, 1980), p. 101.
[Crossref]

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

G. Cerutti-Maori, R. Petit, M. Cadilhac, “Etude numerique du champ diffracte par un reseau,”C. R. Acad. Sci. (Paris) 268B, 1060–1063 (1969).

J. Opt. Soc. Am. (1)

Nouv. Rev. Opt. (1)

N. Neviere, P. Vincent, R. Petit, “Sur la theorie du reseau conducteur et ses applications a l’optique,” Nouv. Rev. Opt. 5, 65–77 (1974).

Opt. Acta (1)

R. A. Depine, J. M. Simon, “Diffraction grating efficiencies: an exact differential algorithm valid for high conductivities,” Opt. Acta 30, 1273–1286 (1983).
[Crossref]

Optik (1)

J. M. Simon, R. A. Depine, “Diffraction grating efficiencies: differential methods for H||case,” Optik 67, 145–153 (1984).

Other (3)

P. Vincent, “Differential methods,” in Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics, R. Petit, ed. (Springer-Verlag, Berlin, 1980), p. 101.
[Crossref]

D. Maystre, “Integral methods,” in Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics, R. Petit, ed. (Springer-Verlag, Berlin, 1980), p. 63.
[Crossref]

D. Maystre, M. Neviere, R. Petit, “Experimental verifications and applications of the theory,” in Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics, R. Petit, ed. (Springer-Verlag, Berlin, 1980), p. 159.
[Crossref]

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Figures (6)

Fig. 1
Fig. 1

Presentation of the grating problem.

Fig. 2
Fig. 2

Zero-order efficiency for a sinusoidal grating with h/d = 0.05, = −5.28 + i1.48, and λ/d = 0.5. Solid curve, differential method; dashed curve, integral method.

Fig. 3
Fig. 3

−3-, −2-, and +1-order efficiencies for a sinusoidal grating with h/d = 0.2, = −5.28 + i1.48, and λ/d = 0.5. Solid curves, differential method; dashed curves, integral method.

Fig. 4
Fig. 4

−1- and zero-order efficiencies for a sinusoidal grating with h/d = 0.2, = −5.28 + i1.48, and λ/d = 0.5. Solid curves, differential method; dashed curves, integral method.

Fig. 5
Fig. 5

The jump of the tangential component of the field E along the groove surface of the grating considered in Fig. 2 and 3. The horizontal line indicates the greatest value of |Et|.

Fig. 6
Fig. 6

The jump of the normal component of the field D along the groove surface of the grating considered in Figs. 2 and 3. The horizontal line indicates the greatest value of |Dn|.

Equations (17)

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H ( x , y ) = f ( x , y ) e - i ω t z ^ ,
[ f ( x , y ) ( x , y ) ] + ω 2 c 2 f ( x , y ) = 0.
x [ 1 ( x , y ) f x ] + y [ 1 ( x , y ) f y ] + ω 2 c 2 f = 0.
f + [ x , g ( x ) ] = f - [ x , g ( x ) ] ,
f + [ x , g ( x ) ] n ^ = 1 f - [ x , g ( x ) ] n ^ .
f + y = f - y ( n y 2 + n x 2 ) + f - x n x n y ( 1 - 1 ) ,
f + x = f - x ( n x 2 + n y 2 ) + f - y n x n y ( 1 - 1 ) .
{ exp ( i γ n x ) } n = - : f ( x , y ) = n = - 0 f n ( y ) exp ( i γ n x ) ,
f ˜ ( x , y ) = 1 ( x , y ) f ( x , y ) y = n = f ˜ n ( y ) exp ( i γ n x ) ,
γ n = k sin θ + 2 π d n ,             k = w / c .
Δ E t = E t + - E t - , E t = 1 i k ( x , y ) ( n x f x + n y f y ) .
Δ E t = n x i k ( f + x - 1 f - x ) + n y i k ( f + y - 1 f - y ) ,
Δ E t = n x i k ( f + x - 1 f - x ) = n x i k - 1 f x .
Δ E t = n x k - 1 n γ n f n [ g ( x ) ] exp ( i γ n x ) .
Δ D n = n x ( 1 - ) i k n f ˜ n [ g ( x ) ] exp ( i γ n x )
Δ E x = 0 ,
Δ E y = Δ E t / n x .

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