Abstract

We present a new formalism for the diffraction of an electromagnetic plane wave by a multicoated grating. Its basic feature lies in the use of a coordinate system that maps all the interfaces onto parallel planes. Using Maxwell’s equations in this new system leads to a linear system of differential equations with constant coefficients whose solution is obtained through the calculation of the eigenvalues and eigenvectors of a matrix in each medium. Through classical criteria, our numerical results have been found generally to be accurate to within 1%. The serious numerical difficulties encountered by the previous differential formalism for highly conducting metallic gratings completely disappear, whatever the optical region. Furthermore, our computer code provides accurate results for metallic gratings covered by many modulated dielectric coatings or for highly modulated gratings. We give two kinds of applications. The first concerns the use of dielectric coatings on a modulated metallic substrate to minimize the absorption of energy. Conversely, the second describes the use of highly modulated metallic gratings to increase this absorption.

© 1982 Optical Society of America

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References

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  1. R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980): R. Petit, Chap. 1, pp. 1–52; D. Maystre, Chap. 3, pp. 63–100; P. Vincent, Chap. 4, pp. 101–121.
    [CrossRef]
  2. J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. (Paris) 11, 235–241 (1980).
    [CrossRef]
  3. E. J. Post, Formal structure of Electromagnetics (North-Holland, Amsterdam, 1962), pp. 144–159.
  4. D. Maystre, J. P. Laude, P. Gacoin, D. Lepère, and J. P. Priou, “Gratings for tunable lasers: using multidielectric coatings to improve their efficiency,” Appl. Opt. 19, 3099–3102 (1980).
    [CrossRef] [PubMed]
  5. D. Maystre, M. Nevière, and R. Petit, in Electromagnetic Theory Of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 6, pp. 159–225.
    [CrossRef]
  6. J. Chandezon, “Les equatians de Maxwell sous forme covariante. Application à l’étude de la propagation dans les guides periodques et à la diffraction par les reseaux,” Ph.D. Thesis (Clermont-Ferrand University, Aubiere, France, 1979).
  7. In this usual mounting, there is a certain value of n for which the n th-diffracted wave and the incident wave are propagating in opposite directions, so βn= β0.
  8. D. Maystre, “A new general integral theory for dielectric coated gratings,” J. Opt. Soc. Am. 68, 490–495 (1978).
    [CrossRef]
  9. M. Nevière, P. Vincent, and R. Petit, “Theory of conducting gratings and their applications to optics,” Nouv. Rev. Opt. 5, 65–77 (1974).
    [CrossRef]
  10. G. H. Derrick, R. C. McPhedran, D. Maystre, and M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–51 (1979).
    [CrossRef]
  11. R. C. McPhedran, G. H. Derrick, and L. C. Botten, in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 7, pp. 227–276.
    [CrossRef]
  12. P. B. Clapham and M. C. Hutley, “Reduction of lens reflexion by the ‘moth eye’ principle,” Nature 244, 281–282 (1973).
    [CrossRef]
  13. D. Maystre, M. Cadilhac, and J. Chandezon, “Gratings: a phenomenological approach and its applications, perfect blazing in a non-zero deviation mounting,” Opt. Acta 28, 457–470 (1981).
    [CrossRef]

1981 (1)

D. Maystre, M. Cadilhac, and J. Chandezon, “Gratings: a phenomenological approach and its applications, perfect blazing in a non-zero deviation mounting,” Opt. Acta 28, 457–470 (1981).
[CrossRef]

1980 (2)

J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. (Paris) 11, 235–241 (1980).
[CrossRef]

D. Maystre, J. P. Laude, P. Gacoin, D. Lepère, and J. P. Priou, “Gratings for tunable lasers: using multidielectric coatings to improve their efficiency,” Appl. Opt. 19, 3099–3102 (1980).
[CrossRef] [PubMed]

1979 (1)

G. H. Derrick, R. C. McPhedran, D. Maystre, and M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–51 (1979).
[CrossRef]

1978 (1)

1974 (1)

M. Nevière, P. Vincent, and R. Petit, “Theory of conducting gratings and their applications to optics,” Nouv. Rev. Opt. 5, 65–77 (1974).
[CrossRef]

1973 (1)

P. B. Clapham and M. C. Hutley, “Reduction of lens reflexion by the ‘moth eye’ principle,” Nature 244, 281–282 (1973).
[CrossRef]

Botten, L. C.

R. C. McPhedran, G. H. Derrick, and L. C. Botten, in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 7, pp. 227–276.
[CrossRef]

Cadilhac, M.

D. Maystre, M. Cadilhac, and J. Chandezon, “Gratings: a phenomenological approach and its applications, perfect blazing in a non-zero deviation mounting,” Opt. Acta 28, 457–470 (1981).
[CrossRef]

Chandezon, J.

D. Maystre, M. Cadilhac, and J. Chandezon, “Gratings: a phenomenological approach and its applications, perfect blazing in a non-zero deviation mounting,” Opt. Acta 28, 457–470 (1981).
[CrossRef]

J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. (Paris) 11, 235–241 (1980).
[CrossRef]

J. Chandezon, “Les equatians de Maxwell sous forme covariante. Application à l’étude de la propagation dans les guides periodques et à la diffraction par les reseaux,” Ph.D. Thesis (Clermont-Ferrand University, Aubiere, France, 1979).

Clapham, P. B.

P. B. Clapham and M. C. Hutley, “Reduction of lens reflexion by the ‘moth eye’ principle,” Nature 244, 281–282 (1973).
[CrossRef]

Derrick, G. H.

G. H. Derrick, R. C. McPhedran, D. Maystre, and M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–51 (1979).
[CrossRef]

R. C. McPhedran, G. H. Derrick, and L. C. Botten, in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 7, pp. 227–276.
[CrossRef]

Gacoin, P.

Hutley, M. C.

P. B. Clapham and M. C. Hutley, “Reduction of lens reflexion by the ‘moth eye’ principle,” Nature 244, 281–282 (1973).
[CrossRef]

Laude, J. P.

Lepère, D.

Maystre, D.

D. Maystre, M. Cadilhac, and J. Chandezon, “Gratings: a phenomenological approach and its applications, perfect blazing in a non-zero deviation mounting,” Opt. Acta 28, 457–470 (1981).
[CrossRef]

J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. (Paris) 11, 235–241 (1980).
[CrossRef]

D. Maystre, J. P. Laude, P. Gacoin, D. Lepère, and J. P. Priou, “Gratings for tunable lasers: using multidielectric coatings to improve their efficiency,” Appl. Opt. 19, 3099–3102 (1980).
[CrossRef] [PubMed]

G. H. Derrick, R. C. McPhedran, D. Maystre, and M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–51 (1979).
[CrossRef]

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

D. Maystre, M. Nevière, and R. Petit, in Electromagnetic Theory Of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 6, pp. 159–225.
[CrossRef]

McPhedran, R. C.

G. H. Derrick, R. C. McPhedran, D. Maystre, and M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–51 (1979).
[CrossRef]

R. C. McPhedran, G. H. Derrick, and L. C. Botten, in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 7, pp. 227–276.
[CrossRef]

Nevière, M.

G. H. Derrick, R. C. McPhedran, D. Maystre, and M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–51 (1979).
[CrossRef]

M. Nevière, P. Vincent, and R. Petit, “Theory of conducting gratings and their applications to optics,” Nouv. Rev. Opt. 5, 65–77 (1974).
[CrossRef]

D. Maystre, M. Nevière, and R. Petit, in Electromagnetic Theory Of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 6, pp. 159–225.
[CrossRef]

Petit, R.

M. Nevière, P. Vincent, and R. Petit, “Theory of conducting gratings and their applications to optics,” Nouv. Rev. Opt. 5, 65–77 (1974).
[CrossRef]

D. Maystre, M. Nevière, and R. Petit, in Electromagnetic Theory Of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 6, pp. 159–225.
[CrossRef]

Post, E. J.

E. J. Post, Formal structure of Electromagnetics (North-Holland, Amsterdam, 1962), pp. 144–159.

Priou, J. P.

Raoult, G.

J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. (Paris) 11, 235–241 (1980).
[CrossRef]

Vincent, P.

M. Nevière, P. Vincent, and R. Petit, “Theory of conducting gratings and their applications to optics,” Nouv. Rev. Opt. 5, 65–77 (1974).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. (1)

G. H. Derrick, R. C. McPhedran, D. Maystre, and M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–51 (1979).
[CrossRef]

J. Opt. (Paris) (1)

J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. (Paris) 11, 235–241 (1980).
[CrossRef]

J. Opt. Soc. Am. (1)

Nature (1)

P. B. Clapham and M. C. Hutley, “Reduction of lens reflexion by the ‘moth eye’ principle,” Nature 244, 281–282 (1973).
[CrossRef]

Nouv. Rev. Opt. (1)

M. Nevière, P. Vincent, and R. Petit, “Theory of conducting gratings and their applications to optics,” Nouv. Rev. Opt. 5, 65–77 (1974).
[CrossRef]

Opt. Acta (1)

D. Maystre, M. Cadilhac, and J. Chandezon, “Gratings: a phenomenological approach and its applications, perfect blazing in a non-zero deviation mounting,” Opt. Acta 28, 457–470 (1981).
[CrossRef]

Other (6)

R. C. McPhedran, G. H. Derrick, and L. C. Botten, in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 7, pp. 227–276.
[CrossRef]

R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980): R. Petit, Chap. 1, pp. 1–52; D. Maystre, Chap. 3, pp. 63–100; P. Vincent, Chap. 4, pp. 101–121.
[CrossRef]

E. J. Post, Formal structure of Electromagnetics (North-Holland, Amsterdam, 1962), pp. 144–159.

D. Maystre, M. Nevière, and R. Petit, in Electromagnetic Theory Of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 6, pp. 159–225.
[CrossRef]

J. Chandezon, “Les equatians de Maxwell sous forme covariante. Application à l’étude de la propagation dans les guides periodques et à la diffraction par les reseaux,” Ph.D. Thesis (Clermont-Ferrand University, Aubiere, France, 1979).

In this usual mounting, there is a certain value of n for which the n th-diffracted wave and the incident wave are propagating in opposite directions, so βn= β0.

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Figures (6)

Fig. 1
Fig. 1

Presentation of the problem and notation for a total number of layers Q = 3.

Fig. 2
Fig. 2

Efficiency curves in a Littrow mount of a sinusoidal aluminum grating covered by 0 (for G0), two (G1), four (G2), six (G3), or eight (G4) stacks of λ/4 dielectric layers. The dashed line corresponds to the total efficiency (0 order and −1 order) and the solid line to the efficiency in the −1 order.

Fig. 3
Fig. 3

Efficiency in Littrow configuration and for TE polarization of an aluminum sinusoidal grating versus the groove-depth to groove-spacing ratio h/d. The dashed line represents the total efficiency and the solid line the −1-order efficiency.

Fig. 4
Fig. 4

The same as Fig. 3 but for TM polarization.

Fig. 5
Fig. 5

Efficiency in a non-Littrow configuration and for TE polarization of an aluminum grating versus the groove-depth to groove-spacing ratio h/d. The dashed line represents the total efficiency and the solid line the −1-order efficiency.

Fig. 6
Fig. 6

The same as Fig. 5 but for TM polarization.

Tables (1)

Tables Icon

Table 1 Comparison of the Values of βn (for n ≥ 0) and rqQ+1 Obtained for a Sinusoidal Grating of Groove Depth h = 4 μm and Groove Spacing d = 18 μm Illuminated under Normal Incidence with the Wavelength λ = 10 μm, the Truncation Being Made with N = 10

Equations (49)

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F i = exp ( i k x sin θ - i k y cos θ ) ,
F d = n B n exp [ i ( α n x + β n y ) ] ,
α n = k sin θ + n K ,
β n = ( k 2 - α n 2 ) 1 / 2             if             n U , β n = i ( α n 2 - k 2 ) 1 / 2             if             n U ,
sin θ n = sin θ + n λ / d .
F a d = n U B n exp [ i ( α n x + β n y ) ] ;
F d = F a d + F e d
E n = B n B n * cos θ n / cos θ ,
u = y - a ( x ) ,
F i = exp ( - i k u cos θ ) exp [ - i k a ( x ) cos θ ] exp ( i k x sin θ ) .
F i = m L m ( β 0 ) exp [ i ( α m x - β 0 u ) ] ,
L m ( t ) = 1 d 0 d exp { - i [ a ( x ) t + m K x ] } d x .
F a d = n U m B n L m - n ( - β n ) exp [ i ( α m x + β n u ) ] .
F = E z , G = k Z 0 H x for TE polarization , F = Z 0 H z , G = - k ν j 2 E x for TM polarization ,
F u = a 1 + a 2 F x + i 1 + a 2 G ,
G u = x ( i 1 + a 2 F x ) + i ν j 2 k 2 F + x ( a 1 + a 2 G ) ,
ψ = ( F G ) = ( E z k Z 0 H x ) for TE polarization , ψ = ( F G / ν j 2 ) = ( Z 0 H z - k E x ) for TM polarization ,
C ( x ) = 1 1 + a 2 ,             D ( x ) = a 1 + a 2 ,
C ( x ) = p C p exp ( i p K x ) ,
D ( x ) = p D p exp ( i p K x ) .
F = m F m ( u ) exp ( i α m x ) ,
G = m G m ( u ) exp ( i α m x ) , ψ = m ψ m ( u ) exp ( i α m x ) ,
- i F m u = p ( α p D m - p F p + C m - p G p ) ,
- i G m u = p [ ( - α m α p C m - p + k 2 ν j 2 δ m p ) F p + α m D m - p G p ] ,
γ N = ( F - N , F - N + 1 , F N , G - N , G - N + 1 , , G N )             for TE polarization ,
γ N = [ F - N , , F N , G - N / ν 2 ( u ) , , G N / ν 2 ( u ) ]             for TM polarization ,
- i d ξ d u = { [ R ( u ) ] - 1 T ( u ) R ( u ) } ξ ,
( A B C D ) ,
A m , n = α n D m - n , B m , n = C m - n , C m , n = - α m α n C m - n + k 2 ν j 2 ( u ) δ m n , D m , n = α m D m - n .
( R 1 0 0 R 2 ) ,
ξ ( u ) = q = 1 ξ q j exp ( i r q j u ) ,
ξ q j = b q j ( R - 1 V q j ) .
ξ = M j ϕ j ( u ) b j ,
M j ϕ j ( u j ) b j = M j + 1 ϕ j + 1 ( u j ) b j + 1 .
ϕ j ( u j ) [ ϕ j ( u j - 1 ) ] - 1 = ϕ j ( u j - u j - 1 ) = ϕ j ( e j ) , ϕ Q + 1 ( u Q ) = Q + 1 ( 0 ) = 1 ,
M Q + 1 b Q + 1 = [ M Q ϕ Q ( e Q ) ( M Q ) - 1 ] [ M j ϕ j ( e j ) ( M j ) - 1 ] [ M 1 ϕ 1 ( e 1 ) ( M 1 ) - 1 ] M 0 ϕ 0 ( - e ) b 0 .
M 0 b 0 = M 1 b 1 .
H b 0 = M Q + 1 b Q + 1 ,
H = H Q H Q - 1 H 1 M 0 ϕ 0 ( - e ) .
Im ( r q 0 ) < 0 ,             or             Im ( r q 0 ) = 0             and             Re ( r q 0 ) < 0.
ξ = M Q + 1 ϕ Q + 1 ( u ) b Q + 1 + exp ( - i β 0 u ) 1 + M ϕ ( u ) B ,
( 1 1 )
1 m = L m ( β 0 ) , 1 m = - ( β 0 - m K α 0 β 0 ) L m ( β 0 ) ,
( M M ) ,
M m n = L m - n ( - β n ) , M m n = [ β n - ( m - n ) K α n / β n ] L m - n ( - β n ) .
H b 0 = M Q + 1 b Q + 1 + 1 + M B .
Im ( r q Q + 1 ) > 0.
E - 1 = sin 2 [ ρ ( h ) ] ,
E - 1 = E sin 2 [ ρ ( h ) ] ,