Abstract

Based on an exact solution of the pertinent boundary-value problem, a method is presented for finding the electromagnetic fields scattered or guided by lossy dielectric gratings having arbitrary profiles. This method unifies the treatment of both perpendicular (TE) and parallel (TM) polarizations by expressing the fields in terms of two coupled first-order differential equations. Their solution is obtained by resorting to difference equations in conjunction with the algorithm of Adams–Moulton, which easily leads to accurate results for a large variety of practical problems. To illustrate the application of this approach, quantitative results are presented for the scattering of plane waves by lossy corrugated structures and for the guiding of (leaky) surface waves by triangular gratings with symmetric or asymmetric profiles.

© 1980 Optical Society of America

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References

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  1. A. R. Neureuther and K. Zaki, “Numerical methods for the analysis of scattering from nonplanar and periodic structures,” Alta Freq. 38, 282–285 (1969).
  2. P. M. van den Berg, “Rigorous diffraction theory of optical reflection and transmission gratings,” Thesis, Delft University of Technolgy, The Netherlands, Report No. 1971–16, 1971 (unpublished).
  3. N. Nevière, R. Petit, and M. Cadilhac, “About the theory of optical grating coupler-waveguide systems.”Opt. Commun. 8, 113–117 (1973).
    [Crossref]
  4. M. Nevière, P. Vincent, R. Petit, and M. Cadilhac, “Systematic study of resonances of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
    [Crossref]
  5. M. Nevière, P. Vincent, and R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nouv. Rev. Opt. 5, 65–77 (1974).
    [Crossref]
  6. S. T. Peng, T. Tamir, and H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
    [Crossref]
  7. D. Marcuse, “Exact theory of TE-wave scattering from blazed dielectric gratings,” Bell Syst. Tech. J. 55, 1295–1317 (1976).
    [Crossref]
  8. D. E. Tremain and K. K. Mei, “Application of the unimoment method to scattering from periodic dielectric structures,” J. Opt. Soc. Am. 68, 775–783 (1978).
    [Crossref]
  9. A. Wirgin, “A new theoretical approach to scattering from a periodic interface,” Opt. Commun. 27, 189–194 (1978).
    [Crossref]
  10. D. Maystre, “A new general integral theory for dielectric coated gratings,” J. Opt. Soc. Am. 68, 490–495 (1978).
    [Crossref]
  11. K. C. Chang, “Surface-wave scattering by dielectric gratings with arbitrary profiles,” Ph.D. thesis, Polytechnic Institute of New York, Brooklyn, N.Y., 1979 (unpublished).
  12. W.-H. Lee and W. Streifer, “Radiation loss calculation for corrugated dielectric waveguides,” J. Opt. Soc. Am. 68, 1701–1707 (1978);J. Opt. Soc. Am. 69, 1671–1676 (1979).
    [Crossref]
  13. K. C. Chang and T. Tamir, “Simplified approach to surface-wave scattering by blazed dielectric gratings,” Appl. Opt. 19, 282–288 (1980).
    [Crossref] [PubMed]
  14. For surface waves, the sign is chosen as in the case of the scattering problem. For leaky waves, the same rule holds except that if n< 0 and βn> 0, the sign must usually be chosen so that the imaginary part of kzn(u) is negative. The reader should refer for a detailed discussion of this aspect to Antenna Theory, edited by R. E. Collin and P. J. Zucker, (McGraw-Hill, New York, 1969), Sec. 19.10, pp. 203–208.
  15. C. F. Gerald, Applied Numerical Analysis (Addison-Wesley, Reading, Mass., 1973), p. 124.
  16. Reference 15, p. 116.
  17. M. Nevière, (personal communication).
  18. D. Maystre and R. Petit, “Brewster incidence’ for metallic gratings,” Opt. Commun. 17, 196–200 (1976).
    [Crossref]
  19. E. G. Loewen and M. Nevière, “Dielectric coated gratings: A curious property,” Appl. Opt. 16, 3009–3011 (1977).
    [Crossref] [PubMed]
  20. V. Shah and T. Tamir, “Brewster phenomena in lossy structures,” Opt. Commun. 23, 113–117 (1977).
    [Crossref]
  21. T. Tamir and S. T. Peng, “Analysis and design of grating couplers,” Appl. Phys. 14, 235–254 (1977).
    [Crossref]

1980 (1)

1978 (4)

1977 (3)

E. G. Loewen and M. Nevière, “Dielectric coated gratings: A curious property,” Appl. Opt. 16, 3009–3011 (1977).
[Crossref] [PubMed]

V. Shah and T. Tamir, “Brewster phenomena in lossy structures,” Opt. Commun. 23, 113–117 (1977).
[Crossref]

T. Tamir and S. T. Peng, “Analysis and design of grating couplers,” Appl. Phys. 14, 235–254 (1977).
[Crossref]

1976 (2)

D. Maystre and R. Petit, “Brewster incidence’ for metallic gratings,” Opt. Commun. 17, 196–200 (1976).
[Crossref]

D. Marcuse, “Exact theory of TE-wave scattering from blazed dielectric gratings,” Bell Syst. Tech. J. 55, 1295–1317 (1976).
[Crossref]

1975 (1)

S. T. Peng, T. Tamir, and H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[Crossref]

1974 (1)

M. Nevière, P. Vincent, and R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nouv. Rev. Opt. 5, 65–77 (1974).
[Crossref]

1973 (2)

N. Nevière, R. Petit, and M. Cadilhac, “About the theory of optical grating coupler-waveguide systems.”Opt. Commun. 8, 113–117 (1973).
[Crossref]

M. Nevière, P. Vincent, R. Petit, and M. Cadilhac, “Systematic study of resonances of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
[Crossref]

1969 (1)

A. R. Neureuther and K. Zaki, “Numerical methods for the analysis of scattering from nonplanar and periodic structures,” Alta Freq. 38, 282–285 (1969).

Bertoni, H. L.

S. T. Peng, T. Tamir, and H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[Crossref]

Cadilhac, M.

M. Nevière, P. Vincent, R. Petit, and M. Cadilhac, “Systematic study of resonances of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
[Crossref]

N. Nevière, R. Petit, and M. Cadilhac, “About the theory of optical grating coupler-waveguide systems.”Opt. Commun. 8, 113–117 (1973).
[Crossref]

Chang, K. C.

K. C. Chang and T. Tamir, “Simplified approach to surface-wave scattering by blazed dielectric gratings,” Appl. Opt. 19, 282–288 (1980).
[Crossref] [PubMed]

K. C. Chang, “Surface-wave scattering by dielectric gratings with arbitrary profiles,” Ph.D. thesis, Polytechnic Institute of New York, Brooklyn, N.Y., 1979 (unpublished).

Gerald, C. F.

C. F. Gerald, Applied Numerical Analysis (Addison-Wesley, Reading, Mass., 1973), p. 124.

Lee, W.-H.

Loewen, E. G.

Marcuse, D.

D. Marcuse, “Exact theory of TE-wave scattering from blazed dielectric gratings,” Bell Syst. Tech. J. 55, 1295–1317 (1976).
[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 and R. Petit, “Brewster incidence’ for metallic gratings,” Opt. Commun. 17, 196–200 (1976).
[Crossref]

Mei, K. K.

Neureuther, A. R.

A. R. Neureuther and K. Zaki, “Numerical methods for the analysis of scattering from nonplanar and periodic structures,” Alta Freq. 38, 282–285 (1969).

Nevière, M.

E. G. Loewen and M. Nevière, “Dielectric coated gratings: A curious property,” Appl. Opt. 16, 3009–3011 (1977).
[Crossref] [PubMed]

M. Nevière, P. Vincent, and R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nouv. Rev. Opt. 5, 65–77 (1974).
[Crossref]

M. Nevière, P. Vincent, R. Petit, and M. Cadilhac, “Systematic study of resonances of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
[Crossref]

M. Nevière, (personal communication).

Nevière, N.

N. Nevière, R. Petit, and M. Cadilhac, “About the theory of optical grating coupler-waveguide systems.”Opt. Commun. 8, 113–117 (1973).
[Crossref]

Peng, S. T.

T. Tamir and S. T. Peng, “Analysis and design of grating couplers,” Appl. Phys. 14, 235–254 (1977).
[Crossref]

S. T. Peng, T. Tamir, and H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[Crossref]

Petit, R.

D. Maystre and R. Petit, “Brewster incidence’ for metallic gratings,” Opt. Commun. 17, 196–200 (1976).
[Crossref]

M. Nevière, P. Vincent, and R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nouv. Rev. Opt. 5, 65–77 (1974).
[Crossref]

N. Nevière, R. Petit, and M. Cadilhac, “About the theory of optical grating coupler-waveguide systems.”Opt. Commun. 8, 113–117 (1973).
[Crossref]

M. Nevière, P. Vincent, R. Petit, and M. Cadilhac, “Systematic study of resonances of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
[Crossref]

Shah, V.

V. Shah and T. Tamir, “Brewster phenomena in lossy structures,” Opt. Commun. 23, 113–117 (1977).
[Crossref]

Streifer, W.

Tamir, T.

K. C. Chang and T. Tamir, “Simplified approach to surface-wave scattering by blazed dielectric gratings,” Appl. Opt. 19, 282–288 (1980).
[Crossref] [PubMed]

V. Shah and T. Tamir, “Brewster phenomena in lossy structures,” Opt. Commun. 23, 113–117 (1977).
[Crossref]

T. Tamir and S. T. Peng, “Analysis and design of grating couplers,” Appl. Phys. 14, 235–254 (1977).
[Crossref]

S. T. Peng, T. Tamir, and H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[Crossref]

Tremain, D. E.

van den Berg, P. M.

P. M. van den Berg, “Rigorous diffraction theory of optical reflection and transmission gratings,” Thesis, Delft University of Technolgy, The Netherlands, Report No. 1971–16, 1971 (unpublished).

Vincent, P.

M. Nevière, P. Vincent, and R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nouv. Rev. Opt. 5, 65–77 (1974).
[Crossref]

M. Nevière, P. Vincent, R. Petit, and M. Cadilhac, “Systematic study of resonances of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
[Crossref]

Wirgin, A.

A. Wirgin, “A new theoretical approach to scattering from a periodic interface,” Opt. Commun. 27, 189–194 (1978).
[Crossref]

Zaki, K.

A. R. Neureuther and K. Zaki, “Numerical methods for the analysis of scattering from nonplanar and periodic structures,” Alta Freq. 38, 282–285 (1969).

Alta Freq. (1)

A. R. Neureuther and K. Zaki, “Numerical methods for the analysis of scattering from nonplanar and periodic structures,” Alta Freq. 38, 282–285 (1969).

Appl. Opt. (2)

Appl. Phys. (1)

T. Tamir and S. T. Peng, “Analysis and design of grating couplers,” Appl. Phys. 14, 235–254 (1977).
[Crossref]

Bell Syst. Tech. J. (1)

D. Marcuse, “Exact theory of TE-wave scattering from blazed dielectric gratings,” Bell Syst. Tech. J. 55, 1295–1317 (1976).
[Crossref]

IEEE Trans. Microwave Theory Tech. (1)

S. T. Peng, T. Tamir, and H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[Crossref]

J. Opt. Soc. Am. (3)

Nouv. Rev. Opt. (1)

M. Nevière, P. Vincent, and R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nouv. Rev. Opt. 5, 65–77 (1974).
[Crossref]

Opt. Commun. (5)

A. Wirgin, “A new theoretical approach to scattering from a periodic interface,” Opt. Commun. 27, 189–194 (1978).
[Crossref]

N. Nevière, R. Petit, and M. Cadilhac, “About the theory of optical grating coupler-waveguide systems.”Opt. Commun. 8, 113–117 (1973).
[Crossref]

M. Nevière, P. Vincent, R. Petit, and M. Cadilhac, “Systematic study of resonances of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
[Crossref]

D. Maystre and R. Petit, “Brewster incidence’ for metallic gratings,” Opt. Commun. 17, 196–200 (1976).
[Crossref]

V. Shah and T. Tamir, “Brewster phenomena in lossy structures,” Opt. Commun. 23, 113–117 (1977).
[Crossref]

Other (6)

K. C. Chang, “Surface-wave scattering by dielectric gratings with arbitrary profiles,” Ph.D. thesis, Polytechnic Institute of New York, Brooklyn, N.Y., 1979 (unpublished).

For surface waves, the sign is chosen as in the case of the scattering problem. For leaky waves, the same rule holds except that if n< 0 and βn> 0, the sign must usually be chosen so that the imaginary part of kzn(u) is negative. The reader should refer for a detailed discussion of this aspect to Antenna Theory, edited by R. E. Collin and P. J. Zucker, (McGraw-Hill, New York, 1969), Sec. 19.10, pp. 203–208.

C. F. Gerald, Applied Numerical Analysis (Addison-Wesley, Reading, Mass., 1973), p. 124.

Reference 15, p. 116.

M. Nevière, (personal communication).

P. M. van den Berg, “Rigorous diffraction theory of optical reflection and transmission gratings,” Thesis, Delft University of Technolgy, The Netherlands, Report No. 1971–16, 1971 (unpublished).

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Figures (6)

FIG. 1
FIG. 1

Geometry of the dielectric grating: (a) general profile, (b) special case with straight horizontal portions.

FIG. 2
FIG. 2

Scattering of a perpendicularly polarized plane wave incident on a lossy dielectric grating backed by a perfectly conducting plane: (a) relative magnitude |C0| of the specularly reflected field; and (b) relative magnitude | C−1| of the “resonant” field, as functions of θ for various values of the loss factor δf. The variations of |C0| and |C−1| around θ = 35° illustrate an absorption anomaly.

FIG. 3
FIG. 3

Leaky-wave guidance by a symmetric triangular grating for the TE0 mode, showing the variations of the radiation efficiency ηa and the normalized leakage factor αλ as functions of the normalized grating height tg/λ.

FIG. 4
FIG. 4

Leaky-wave guidance by the dielectric grating of Fig. 3, except that now the TM0 mode is shown.

FIG. 5
FIG. 5

Leaky-wave guidance by a dielectric grating as in Fig. 3, except that now the grating profile is asymmetric.

FIG. 6
FIG. 6

Leaky-wave guidance by a dielectric grating as in Fig. 3, except that now the TM0 mode is shown and the grating profile is asymmetric.

Tables (3)

Tables Icon

TABLE I Comparison of results for scattering by a sinusoidal interface, as described by Fig. 1 with a = 1, r = f = s = 4, θ = 45°, and d/λ = 1. Here P n ( u ) refers to power of propagating harmonics in air (u = a) or dielectric (u = s) while ΣP denotes the total scattered power.

Tables Icon

TABLE II Comparison of results for guiding by a sinusoidal grating, as described by Fig. 1 with a = 1,∊r = (1.62 + i10−3)2, f = 2.465, s = 2.295, tr = (0.48 − tg/2) μm, tf = 0.80 μm, d = 0.606 μm, and λ = 0.6328 μm. All β0 and α are in (μm)−1.

Tables Icon

TABLE III Comparison of results for guiding by a rectangular grating, as described by Fig. 1 with a = 1, r = f = 3, s = 2.3, and d/λ = 0.5. The width of the rectangular corrugation (along x) is d1 = d/2 and tr = 0.

Equations (61)

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F ( i ) = exp [ i k 0 a ( x sin θ z cos θ ) ] ,
F ( a ) ( x , z ) = F ( i ) ( x , z ) + n C n exp ( i k z n ( a ) z ) exp ( i β n x ) ( for z > t g ) ,
F ( r ) ( x , z ) = n [ B n exp ( i k z n ( r ) z ) + B n + exp ( i k z n ( r ) z ) ] exp ( i β n x ) ( for 0 > z > t r ) ,
F ( f ) ( x , z ) = n [ A n exp ( i k z n ( f ) z ) + A n + exp ( i k z n ( f ) z ) ] exp ( i β n x ) ( for t r > z > t f t r ) ,
F ( s ) ( x , z ) = n D n exp ( i k z n ( s ) z ) exp ( i β n x ) ( for z < t f t r ) ,
β n = k 0 a sin θ + ( 2 n π / d ) .
k z n ( u ) = ± ( k 0 2 u β n 2 ) 1 / 2 ,
F ( g ) ( x , z ) = n f n ( z ) exp ( i β n x ) ,
( x , z ) = n n ( z ) exp ( i 2 n π / d ) .
[ ( x , z ) ] 1 = n ξ n ( z ) exp ( i 2 n π / d ) .
T ( 0 ) = d f ( 0 ) d z + L ( 0 ) f ( 0 ) = o .
l m n ( 0 ) = i e 1 k z n ( r ) 1 r r n 1 + r r n δ m n ,
r r n = ( k z n ( r ) + e 2 k z n ( f ) ) r f n + ( k z n ( r ) e 2 k z n ( f ) ) exp ( 2 i k z n ( f ) t r ) ( k z n ( r ) e 2 k z n ( f ) ) r f n + ( k z n ( r ) + e 2 k z n ( f ) ) exp ( 2 i k z n ( f ) t r ) × exp [ 2 i k z n ( r ) t r ] ,
r f n = k z n ( f ) e 3 k z n ( s ) k z n ( f ) + e 3 k z n ( s ) exp [ 2 i k z n ( f ) ( t r + t f ) ] .
e 1 = { 1 ( for TE ) a / r ( for TM ) ,
e 2 = { 1 ( for TE ) r / f ( for TM ) ,
e 3 = { 1 ( for TE ) f / s ( for TM ) .
t m n ( z ) = { δ m n ( for TE ) a ξ n m ( z ) ( for TM ) .
T ( t g ) = d f ( t g ) d z + L ( g ) f ( t g ) = u ,
l m n ( g ) = i k z n ( a ) δ m n ,
u n = 2 i k z 0 ( a ) δ n 0 exp ( i k z 0 ( a ) t g ) .
d f ( z ) d z = Q ( z ) g ( z ) ,
d g ( z ) d z = P ( z ) f ( z ) ,
G ( g ) ( x , z ) = n g n ( z ) exp ( i β n x ) .
p m n ( z ) = { i ω 0 [ ( β n / k 0 ) 2 δ m n n m ( z ) ] ( for TE ) i ω μ 0 [ δ m n ( β m β n / k 0 2 ) ξ n m ( z ) ] ( for TM ) ,
q m n ( z ) = { i ω μ 0 δ m n ( for TE ) i ω 0 n m ( z ) ( for TM ) ,
f [ ( l + 1 ) h 1 ] = f ( l h 1 ) + h 1 Q ( l h 1 ) g ( l h 1 ) ,
g [ ( l + 1 ) h 1 ] = g ( l h 1 ) + h 1 P ( l h 1 ) f ( l h 1 ) × ( l = 0 , 1 , 2 , , 3 ν 1 2 , 3 ν 1 1 ) ,
f ( j ) = M j ( f ) f ( 0 ) ,
g ( j ) = M j ( g ) f ( 0 ) .
M j ( f ) = M j 1 ( f ) + h 24 [ 55 Q ( j 1 ) M j 1 ( g ) 59 Q ( j 2 ) M j 2 ( g ) + 37 Q ( j 3 ) M j 3 ( g ) 9 Q ( j 4 ) M j 4 ( g ) ] ,
M j ( g ) = M j 1 ( g ) + h 24 [ 55 P ( j 1 ) M j 1 ( f ) 59 P ( j 2 ) M j 2 ( f ) + 37 P ( j 3 ) M j 3 ( f ) 9 P ( j 4 ) M j 4 ( f ) ] .
M ¯ j ( f ) = M j 1 ( f ) + h 24 [ 9 Q ( j ) M j ( g ) + 19 Q ( j 1 ) M j 1 ( g ) 5 Q ( j 2 ) M j 2 ( g ) + Q ( j 3 ) M j 3 ( g ) ] ,
M ¯ j ( g ) = M j 1 ( g ) + h 24 [ 9 P ( j ) M j ( f ) + 19 P ( j 1 ) M j 1 ( f ) 5 P ( j 2 ) M j 2 ( f ) + P ( j 3 ) M j 3 ( f ) ] ,
Mf ( 0 ) = u ,
M = T ( ν ) Q ( ν ) M ν ( g ) + L ( g ) M ν ( f ) .
f ( t g ) = M ν ( f ) f ( 0 ) .
| M | = 0 ,
γ n = β n + i α ,
η a + η s = 1
P 0 ( a )
P 1 ( a )
P 0 ( s )
P 1 ( s )
P 1 ( s )
P 2 ( s )
P 0 ( a )
P 1 ( a )
P 0 ( s )
P 1 ( s )
P 1 ( s )
P 2 ( s )
D n exp [ i k z n ( s ) ( t r + t f ) ] = A n exp [ i k z n ( f ) ( t r + t f ) ] + A n + exp [ i k z n ( f ) ( t r + t f ) ]
k z n ( s ) D n exp [ i k z n ( s ) ( t r + t f ) ] = k z n ( f ) { A n exp [ i k z n ( f ) ( t r + t f ) ] A n + exp [ i k z n ( f ) ( t r + t f ) ] } .
d f n d z + i k z n ( r ) 1 r r n 1 + r r n f n ( 0 ) = 0 ,
d f n ( t g ) d z i k z n ( a ) f n ( t g ) = 2 i k z 0 ( a ) δ n 0 exp [ i k z 0 ( a ) t g ] ,
f n ( 0 ) = B n ( 1 + r r n ) ,
m = ξ n m ( 0 ) d f m ( 0 ) d z = B n ( i k z n ( r ) / r ) ( r r n 1 ) .
m = a ξ n m ( 0 ) d f m ( 0 ) d z + i k z n ( r ) ( a r ) ( 1 r r n 1 + r r n ) f n ( 0 ) = 0 ,
C n exp [ i k z n ( a ) t g ] δ n 0 exp [ i k z n ( a ) t g ] = f n ( t g )
i k z n ( a ) a C n exp [ i k z n ( a ) t g ] i k z 0 ( a ) a δ n 0 exp [ i k z 0 ( a ) t g ] = m = ξ n m ( t g ) d f m ( t g ) d z .