Abstract

The standard method of matching boundary conditions at the interfaces of a multilayer plane dielectric stack is shown to be numerically unstable for the evanescent orders when a large number of layers is present. For an isolated dielectric stack with an incident propagating beam there is no need to calculate the evanescent orders; however, when a diffraction grating is buried under the stack there is mixing of orders, and it may be important to calculate the evanescent as well as the propagating orders. It is shown that the impedance formalism removes the numerical instability completely. This method may be coupled to either boundary integral or differential equation methods for the grating to provide the complete solution for the grating–stack system.

© 1990 Optical Society of America

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References

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  1. R. Petit, Ed., Electromagnetic Theory of Gratings (Springer-Verlag, New York, 1980).
    [CrossRef]
  2. L. C. Botten, “A New Formalism for Transmission Gratings,” Opt. Acta 25, 481–499 (1978).
    [CrossRef]
  3. J. Chandezon, M. T. Dupuis, G. Cornet, D. Maystre, “Multicoated Gratings: a Differential Formalism Applicable in the Entire Optical Region,” J. Opt. Soc. Am. 72, 839–846 (1982).
    [CrossRef]
  4. D. Maystre, “A New Theory for Multiprofile, Buried Gratings,” Opt. Commun. 26, 127–137 (1978).
    [CrossRef]
  5. L. M. Brekhovskikh, Waves in Layered Media (Academic, New York, 1960).

1982 (1)

1978 (2)

L. C. Botten, “A New Formalism for Transmission Gratings,” Opt. Acta 25, 481–499 (1978).
[CrossRef]

D. Maystre, “A New Theory for Multiprofile, Buried Gratings,” Opt. Commun. 26, 127–137 (1978).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Acta (1)

L. C. Botten, “A New Formalism for Transmission Gratings,” Opt. Acta 25, 481–499 (1978).
[CrossRef]

Opt. Commun. (1)

D. Maystre, “A New Theory for Multiprofile, Buried Gratings,” Opt. Commun. 26, 127–137 (1978).
[CrossRef]

Other (2)

L. M. Brekhovskikh, Waves in Layered Media (Academic, New York, 1960).

R. Petit, Ed., Electromagnetic Theory of Gratings (Springer-Verlag, New York, 1980).
[CrossRef]

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Figures (1)

Fig. 1
Fig. 1

Diffraction by a diffraction grating with multiple dielectric coating layers.

Equations (34)

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2 H + k 2 H = 0 ,
2 E + k 2 E = 0 ,
n × H 1 = n × H 2 ,
n × E 1 = n × E 2 ,
H ( j ) = n = - ( exp ( i α n x ) { A n ( j ) exp [ - i β n ( j ) y ] + B n ( j ) exp ( i β n y ) } ) ,
α n = k ( j ) sin θ ( j ) + n K ,
k ( j ) sin θ ( j ) = k ( N + 1 ) sin θ ( N + 1 ) by Snell ' s law ,
β n ( j ) = [ k ( j ) 2 - α n 2 ] 1 / 2 .
[ B n ( j ) A n ( j ) ] = 1 2 { [ 1 + u ( j ) ] b - ( j ) [ 1 - u ( j ) ] / b + ( j ) [ 1 - u ( j ) ] b + ( j ) [ 1 + u ( j ) ] / b - ( j ) } [ B n ( j - 1 ) A n ( j - 1 ) ] ,
u ( j ) = [ ν ( j ) / ν ( j - 1 ) ] 2 ,
b - ( j ) = exp { i [ β n ( j - 1 ) - β n ( j ) ] Δ ( j ) } ,
b + ( j ) = exp { i [ β n ( j - 1 ) + β n ( j ) ] Δ ( j ) } .
exp { ± i [ β ( j ) - β ( k ) ] j Δ ( j ) } .
Z E t / H t = E cos  θ / H in TM polarization ,
= E / H cos  θ in TE polarization .
H n ( j ) = exp ( i α n x ) ( A n ( j ) exp { - i β n ( j ) [ y - y ( j ) ] } + B n ( j ) exp { i β n [ y - y ( j ) ] } ) .
E n ( j ) = k ( j ) β n ( j ) [ ( j ) μ 0 ] 1 / 2 exp ( i α n x ) ( A n ( j ) exp { - i β n ( j ) [ y - y ( j ) ] } - B n ( j ) exp { i β n [ y - y ( j ) ] } ) .
Z n ( j ) = k ( j ) β n ( j ) [ μ 0 ( j ) ] 1 / 2 cos  θ ( j ) .
Z i n , n ( j ) E t , n ( j ) H t , n ( j ) | y = y ( j + 1 ) = Z n ( j ) A n ( j ) exp [ - i ϕ n ( j ) ] - B n ( j ) exp [ i ϕ n ( j ) ] A n ( j ) exp [ - i ϕ n ( j ) ] + B n ( j ) exp [ i ϕ n ( j ) ]
B n ( j ) A n ( j ) = Z n ( j ) - Z in , n ( j ) Z n ( j ) + Z in , n ( j ) exp [ - 2 i ϕ n ( j ) ] .
Z n ( j ) { A n ( j ) exp [ - i ϕ n ( j ) ] - B n ( j ) exp [ i ϕ n ( j ) ] } = Z n ( j + 1 ) [ A n ( j + 1 ) - B n ( j + 1 ) ]
A n ( j ) exp [ - i ϕ n ( j ) ] + B n ( j ) exp [ i ϕ n ( j ) ] = A n ( j + 1 ) + B n ( j + 1 ) .
A n ( j + 1 ) A n ( j ) = exp [ - i ϕ n ( j ) ] - Z n ( j ) Z n ( j + 1 ) Z n ( j + 1 ) + Z in , n ( j ) Z n ( j ) + Z in , n ( j ) ,
B n ( j + 1 ) B n ( j ) = exp [ i ϕ n ( j ) ] Z n ( j ) Z n ( j + 1 ) Z n ( j + 1 ) - Z in , n ( j ) Z n ( j ) - Z in , n ( j ) .
Z in , n ( j ) E t , n ( j + 1 ) H t , n ( j + 1 ) = Z n ( j + 1 ) [ A n ( j + 1 ) - B n ( j + 1 ) ] A n ( j + 1 ) + B n ( j + 1 ) .
Z in , n ( j - 1 ) = Z n ( j ) Z in , n ( j ) + i Z n ( j ) tan [ ϕ n ( j ) ] Z n ( j ) + i Z in , n ( j ) tan [ ϕ n ( j ) ] .
Z in , n ( j - 1 ) = Z in , n ( j ) .
Z i n , n ( N ) = Z n ( N + 1 ) - B n ( N + 1 ) exp [ i ϕ n ( N + 1 ) ] B n ( N + 1 ) exp [ i ϕ n ( N + 1 ) ] = - Z n ( N + 1 ) ,             n 0.
B n ( 1 ) = j R n j A j ( 1 ) ,
B n ( 1 ) = f n ( 1 ) A n ( 1 ) ,
f n ( 1 ) Z n ( 1 ) - Z in , n ( 1 ) Z n ( 1 ) + Z in , n ( 1 ) exp [ - 2 i ϕ n ( 1 ) ] .
k = - k 0 ( δ n k - R n k f k ) B k ( 1 ) A 0 ( 1 ) = R n 0 ,             n 0.
B 0 ( 1 ) A 0 ( 1 ) = f 0 ( 1 ) .
Z in , 0 ( 1 ) = Z 0 ( 1 ) 1 - [ B 0 ( 1 ) / A 0 ( 1 ) ] exp [ 2 i ϕ 0 ( 1 ) ] 1 + [ B 0 ( 1 ) / A 0 ( 1 ) ] exp [ 2 i ϕ 0 ( 1 ) ] .

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