Abstract

The system of coupled-wave equations for electromagnetic calculations of lamellar gratings is transformed to a new operator–vector form. The numerical procedure is based on truncation of the transformed system and proves to be stable, to be free of ill conditioning, and to preserve the power-conservation requirement for a lossless dielectric with high accuracy. To execute the procedure a very compact matlab-based program is developed, and numerical simulations for thick intrinsic silicon gratings are performed. Zero-reflectance phenomena at normal incidence for both TE and TM polarizations are studied. The ratios of the grating dimensions to be wavelengths at which these anomalies occur are found numerically. It is shown that by keeping the period- and slot-width-to-wavelength ratios constant and by increasing the slot depth one can repeat the anomalies. An antiblazing property at oblique incidence is also considered. The connection with recent directional polarized-emission experiments on intrinsic silicon gratings is discussed.

© 1994 Optical Society of America

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References

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  1. D. Maystre, N. Neviere, R. Petit, “Experimental verification and applications of the theory,” in Electromagnetic Theory of Gratings, R. Petit, ed., Topics in Current Physics Series (Springer, New York, 1980), pp. 159–225.
    [CrossRef]
  2. N. Garcia, V. Celli, N. R. Hill, N. Cabrera, “Ill-conditioned matrices in the scattering of waves from hard corrugated surfaces,” Phys. Rev. B 18, 5184–5189 (1978).
    [CrossRef]
  3. K. Knop, “Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves,” J. Opt. Soc. Am. 68, 1206–1210 (1978).
    [CrossRef]
  4. P. J. Hesketh, J. N. Zemel, B. Gebhart, “Polarized spectral emittance from periodic micromachined surfaces. I. Doped silicon: the normal direction,” Phys. Rev. B 37, 10796–10802 (1988).
  5. P. J. Hesketh, J. N. Zemel, B. Gebhart, “Polarized spectral emittance from periodic micromachined surfaces. II. Doped silicon: the angular dependences,” Phys. Rev. B 37, 10803–10813 (1988).
    [CrossRef]
  6. T. K. Wang, J. N. Zemel, “Polarized spectral emittance from periodic micromachined surfaces—III. Undoped silicon: the normal direction in shallow lamellar gratings,” Infrared Physics 32, 477–488 (1991).
    [CrossRef]
  7. T. K. Wang, J. N. Zemel, “Polarized spectral emittance from periodic micromachined surfaces: IV. Undoped silicon: Normal direction in deep lamellar gratings,” Appl. Opt. 31, 732–736 (1992).
    [CrossRef] [PubMed]
  8. T. K. Wang, J. N. Zemel, “Polarized spectral emittance from periodic micromachined surfaces: V. Undoped silicon: angular measurement in shallow lamellar gratings,” Appl. Opt. 32, 2021–2027 (1993).
    [CrossRef] [PubMed]
  9. M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981);“Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. 72, 1385–1392 (1982).
    [CrossRef]
  10. D. E. Aspnes, “Optical functions of intrinsic silicon: table of refraction index, extinction coefficient and absorption coefficient vs energy (0 to 400 eV),” in Properties of Silicon, No. 4 of Emis Datareviews Series (Institution of Electrical Engineers, London, 1988), Sec. 2.6, pp. 72–79.
  11. E. Popov, L. Tsonev, D. Maystre, “Gratings–general properties of the Littrow mounting and energy flow distribution,” J. Mod. Opt. 36, 367–377 (1990).
    [CrossRef]

1993 (1)

1992 (1)

1991 (1)

T. K. Wang, J. N. Zemel, “Polarized spectral emittance from periodic micromachined surfaces—III. Undoped silicon: the normal direction in shallow lamellar gratings,” Infrared Physics 32, 477–488 (1991).
[CrossRef]

1990 (1)

E. Popov, L. Tsonev, D. Maystre, “Gratings–general properties of the Littrow mounting and energy flow distribution,” J. Mod. Opt. 36, 367–377 (1990).
[CrossRef]

1988 (2)

P. J. Hesketh, J. N. Zemel, B. Gebhart, “Polarized spectral emittance from periodic micromachined surfaces. I. Doped silicon: the normal direction,” Phys. Rev. B 37, 10796–10802 (1988).

P. J. Hesketh, J. N. Zemel, B. Gebhart, “Polarized spectral emittance from periodic micromachined surfaces. II. Doped silicon: the angular dependences,” Phys. Rev. B 37, 10803–10813 (1988).
[CrossRef]

1981 (1)

1978 (2)

N. Garcia, V. Celli, N. R. Hill, N. Cabrera, “Ill-conditioned matrices in the scattering of waves from hard corrugated surfaces,” Phys. Rev. B 18, 5184–5189 (1978).
[CrossRef]

K. Knop, “Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves,” J. Opt. Soc. Am. 68, 1206–1210 (1978).
[CrossRef]

Aspnes, D. E.

D. E. Aspnes, “Optical functions of intrinsic silicon: table of refraction index, extinction coefficient and absorption coefficient vs energy (0 to 400 eV),” in Properties of Silicon, No. 4 of Emis Datareviews Series (Institution of Electrical Engineers, London, 1988), Sec. 2.6, pp. 72–79.

Cabrera, N.

N. Garcia, V. Celli, N. R. Hill, N. Cabrera, “Ill-conditioned matrices in the scattering of waves from hard corrugated surfaces,” Phys. Rev. B 18, 5184–5189 (1978).
[CrossRef]

Celli, V.

N. Garcia, V. Celli, N. R. Hill, N. Cabrera, “Ill-conditioned matrices in the scattering of waves from hard corrugated surfaces,” Phys. Rev. B 18, 5184–5189 (1978).
[CrossRef]

Garcia, N.

N. Garcia, V. Celli, N. R. Hill, N. Cabrera, “Ill-conditioned matrices in the scattering of waves from hard corrugated surfaces,” Phys. Rev. B 18, 5184–5189 (1978).
[CrossRef]

Gaylord, T. K.

Gebhart, B.

P. J. Hesketh, J. N. Zemel, B. Gebhart, “Polarized spectral emittance from periodic micromachined surfaces. I. Doped silicon: the normal direction,” Phys. Rev. B 37, 10796–10802 (1988).

P. J. Hesketh, J. N. Zemel, B. Gebhart, “Polarized spectral emittance from periodic micromachined surfaces. II. Doped silicon: the angular dependences,” Phys. Rev. B 37, 10803–10813 (1988).
[CrossRef]

Hesketh, P. J.

P. J. Hesketh, J. N. Zemel, B. Gebhart, “Polarized spectral emittance from periodic micromachined surfaces. II. Doped silicon: the angular dependences,” Phys. Rev. B 37, 10803–10813 (1988).
[CrossRef]

P. J. Hesketh, J. N. Zemel, B. Gebhart, “Polarized spectral emittance from periodic micromachined surfaces. I. Doped silicon: the normal direction,” Phys. Rev. B 37, 10796–10802 (1988).

Hill, N. R.

N. Garcia, V. Celli, N. R. Hill, N. Cabrera, “Ill-conditioned matrices in the scattering of waves from hard corrugated surfaces,” Phys. Rev. B 18, 5184–5189 (1978).
[CrossRef]

Knop, K.

Maystre, D.

E. Popov, L. Tsonev, D. Maystre, “Gratings–general properties of the Littrow mounting and energy flow distribution,” J. Mod. Opt. 36, 367–377 (1990).
[CrossRef]

D. Maystre, N. Neviere, R. Petit, “Experimental verification and applications of the theory,” in Electromagnetic Theory of Gratings, R. Petit, ed., Topics in Current Physics Series (Springer, New York, 1980), pp. 159–225.
[CrossRef]

Moharam, M. G.

Neviere, N.

D. Maystre, N. Neviere, R. Petit, “Experimental verification and applications of the theory,” in Electromagnetic Theory of Gratings, R. Petit, ed., Topics in Current Physics Series (Springer, New York, 1980), pp. 159–225.
[CrossRef]

Petit, R.

D. Maystre, N. Neviere, R. Petit, “Experimental verification and applications of the theory,” in Electromagnetic Theory of Gratings, R. Petit, ed., Topics in Current Physics Series (Springer, New York, 1980), pp. 159–225.
[CrossRef]

Popov, E.

E. Popov, L. Tsonev, D. Maystre, “Gratings–general properties of the Littrow mounting and energy flow distribution,” J. Mod. Opt. 36, 367–377 (1990).
[CrossRef]

Tsonev, L.

E. Popov, L. Tsonev, D. Maystre, “Gratings–general properties of the Littrow mounting and energy flow distribution,” J. Mod. Opt. 36, 367–377 (1990).
[CrossRef]

Wang, T. K.

Zemel, J. N.

T. K. Wang, J. N. Zemel, “Polarized spectral emittance from periodic micromachined surfaces: V. Undoped silicon: angular measurement in shallow lamellar gratings,” Appl. Opt. 32, 2021–2027 (1993).
[CrossRef] [PubMed]

T. K. Wang, J. N. Zemel, “Polarized spectral emittance from periodic micromachined surfaces: IV. Undoped silicon: Normal direction in deep lamellar gratings,” Appl. Opt. 31, 732–736 (1992).
[CrossRef] [PubMed]

T. K. Wang, J. N. Zemel, “Polarized spectral emittance from periodic micromachined surfaces—III. Undoped silicon: the normal direction in shallow lamellar gratings,” Infrared Physics 32, 477–488 (1991).
[CrossRef]

P. J. Hesketh, J. N. Zemel, B. Gebhart, “Polarized spectral emittance from periodic micromachined surfaces. I. Doped silicon: the normal direction,” Phys. Rev. B 37, 10796–10802 (1988).

P. J. Hesketh, J. N. Zemel, B. Gebhart, “Polarized spectral emittance from periodic micromachined surfaces. II. Doped silicon: the angular dependences,” Phys. Rev. B 37, 10803–10813 (1988).
[CrossRef]

Appl. Opt. (2)

Infrared Physics (1)

T. K. Wang, J. N. Zemel, “Polarized spectral emittance from periodic micromachined surfaces—III. Undoped silicon: the normal direction in shallow lamellar gratings,” Infrared Physics 32, 477–488 (1991).
[CrossRef]

J. Mod. Opt. (1)

E. Popov, L. Tsonev, D. Maystre, “Gratings–general properties of the Littrow mounting and energy flow distribution,” J. Mod. Opt. 36, 367–377 (1990).
[CrossRef]

J. Opt. Soc. Am. (2)

Phys. Rev. B (3)

P. J. Hesketh, J. N. Zemel, B. Gebhart, “Polarized spectral emittance from periodic micromachined surfaces. I. Doped silicon: the normal direction,” Phys. Rev. B 37, 10796–10802 (1988).

P. J. Hesketh, J. N. Zemel, B. Gebhart, “Polarized spectral emittance from periodic micromachined surfaces. II. Doped silicon: the angular dependences,” Phys. Rev. B 37, 10803–10813 (1988).
[CrossRef]

N. Garcia, V. Celli, N. R. Hill, N. Cabrera, “Ill-conditioned matrices in the scattering of waves from hard corrugated surfaces,” Phys. Rev. B 18, 5184–5189 (1978).
[CrossRef]

Other (2)

D. Maystre, N. Neviere, R. Petit, “Experimental verification and applications of the theory,” in Electromagnetic Theory of Gratings, R. Petit, ed., Topics in Current Physics Series (Springer, New York, 1980), pp. 159–225.
[CrossRef]

D. E. Aspnes, “Optical functions of intrinsic silicon: table of refraction index, extinction coefficient and absorption coefficient vs energy (0 to 400 eV),” in Properties of Silicon, No. 4 of Emis Datareviews Series (Institution of Electrical Engineers, London, 1988), Sec. 2.6, pp. 72–79.

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Figures (7)

Fig. 1
Fig. 1

Fragment of a silicon lamellar-grating structure.

Fig. 2
Fig. 2

Spectral reflectances at normal incidence of a grating of Λ = 22 μm, w = 8.8 μm, and h = 2 μm. Curve 1, TE polarization; curve 2, TM polarization; dotted curve, reflection from a flat silicon surface.

Fig. 3
Fig. 3

Calculated spectral reflectances at normal incidence for two gratings reported in Ref. 6: Curve 1, Λ = 10 μm, w = 6.5 μm, and h = 3 μm, TE polarization; curve 2, Λ = 14 μm, w = 9.1 μm, and h = 2 μm, TM polarization.

Fig. 4
Fig. 4

TE polarization spectral efficiencies of a grating with Λ = 9.7379 μm, w = 6.2419 μm, and h = 2.9394 μm. Solid curves, reflectance (R) and 0 reflected order; dotted curve, ±1st reflected order. Zero reflectance at λ = 10.4953 μm.

Fig. 5
Fig. 5

TM polarization spectral efficiencies of a grating with Λ = 6.7284 μm, w = 4.8388 μm, and h = 2.9929 μm. Solid curves, reflectance (R) and 0 reflected order; dotted curve, ±1st reflected order. Zero reflectance at λ = 10.4953 μm.

Fig. 6
Fig. 6

Slot-depth dependencies of reflectance at λ = 10.4953 μm. Dotted curve, TE polarization, Λ/λ = 0.9278, w/λ = 0,5947, Dashed curve, TM polarization, Λ/λ = 0.6441, w/λ = 0.4610.

Fig. 7
Fig. 7

Incidence-angle dependence of TE reflectance for real dielectric constant gratings: ε(λ) = 11.7060, Λ/λ = 0.6653, w/λ = 0.4956, h/λ = 0.6980. Zero reflectance at θ = 30°.

Tables (2)

Tables Icon

Table 1 Calculated Dimension-to-Wavelength Ratios for Zero Reflectance in TE Polarization for Intrinsic Silicon Gratings

Tables Icon

Table 2 Calculated Dimension-to-Wavelength Ratios for Zero Reflectance in TM Polarization for intrinsic Silicon Gratings

Equations (31)

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E = ( 0 , Ψ , 0 ) , H = i k 0 ( z Ψ , 0 , x Ψ ) ,
( 2 x 2 + 2 z 2 ) Ψ + k 0 2 ε g ( x , z ) Ψ = 0 .
E = i ε g ( x , z ) k 0 ( z Ψ , 0 , x Ψ ) , H = ( 0 , Ψ , 0 ) ,
[ x 1 ε g ( x , z ) x + z 1 ε g ( x , z ) x ] Ψ + k 0 2 Ψ = 0 .
Ψ ( x + Λ , z ) = exp ( i k x Λ ) Ψ ( x , z ) ,
Ψ ( x , z ) = n = + Ψ n ( z ) exp ( i k 0 γ n x ) , γ n = sin θ + λ Λ n .
Ψ n I ( z ) = δ n , 0 exp ( i k 0 cos θ z ) + R n exp [ i k 0 ( 1 γ n 2 z ) 1 / 2 ] , z > 0 ,
Ψ n III ( z ) = T n exp [ i k 0 ( ε ( λ ) γ n 2 ) 1 / 2 ( z + h ) ] , z < h ,
d 2 d z 2 Ψ n II ( z ) + k 0 2 m = + A n m Ψ m II ( z ) = 0 , < n < + , 0 > z > h .
A n m = ε n m l = + τ n l γ l ( τ 1 ) l m γ m ,
ε n m = ε ( λ ) δ n , m [ ε ( λ ) 1 ] sin [ π ( n m ) w Λ ] π ( n m ) ,
σ ( λ ) = { 1 TE ε ( λ ) TM ,
d 2 d z 2 Ψ ( z ) + k 0 2 A Ψ ( z ) = 0 , 0 > z > h ,
Ψ ( z ) = exp ( i k 0 Q z ) X + exp ( i k 0 Q z ) Y , Q = ( A ) 1 / 2 ,
r n m = τ n m ( 1 γ m 2 ) 1 / 2 , t n m = τ n m σ ( λ ) ( ε ( λ ) γ m 2 ) 1 / 2 ;
S = ( Q + t ) 1 ( Q t ) ,
Γ ( h ) = [ I exp ( i k 0 Q h ) S exp ( i k 0 Q h ) ] 1 × [ I + exp ( i k 0 Q h ) S exp ( i k 0 Q h ) ] ,
R = [ I + Γ ( h ) Q 1 r ] 1 [ I Γ ( h ) Q 1 r ] Δ ,
T = ( Q + t ) 1 exp ( i k 0 Q h ) [ ( Q + r ) Δ + ( Q r ) R ] .
R = 1 cos θ n = + | R n | 2 Re ( 1 γ n 2 ) 1 / 2 = 1 cos θ Re ( R τ 1 r R ) ,
T = 1 cos θ n = + | T n | 2 Re [ 1 σ ( λ ) ( ε ( λ ) γ n 2 ) 1 / 2 ] = 1 cos θ Re ( T τ 1 t T ) .
ρ n = ( ε ( λ ) γ n 2 ) 1 / 2 σ ( λ ) ( 1 γ n 2 ) 1 / 2 ( ε ( λ ) γ n 2 ) 1 / 2 + σ ( λ ) ( 1 γ n 2 ) 1 / 2 , ϕ n = k 0 d ( ε ( λ ) γ n 2 ) 1 / 2 ,
t ¯ n m = t n m 1 ρ m exp ( 2 i ϕ m ) 1 + ρ m exp ( 2 i ϕ m )
c n m = δ n , m exp ( i ϕ m ) 1 + ρ m exp ( 2 i ϕ m ) × ( ε ( λ ) γ n 2 ) 1 / 2 ( ε ( λ ) γ n 2 ) 1 / 2 + σ ( λ ) ( 1 γ n 2 ) 1 / 2 ,
T = 2 c ( Q + t ¯ ) 1 exp ( i k 0 Q h ) [ ( Q + r ) Δ + ( Q r ) R ] ,
T = 1 cos θ Re ( T τ 1 r T ) .
N g = integer { Λ λ [ | sin θ | + n ( λ ) ] } .
R + T = 1 ,
R = R [ ε ( λ ) , λ , Λ , w , h , θ ] R [ ε ( λ ) , 1 , Λ λ , w λ , h λ , θ ] .
Λ λ = 0.8674 , w λ = 0.4204 , h λ = 0.1553 .
Λ λ = 0.6653 , w λ = 0.4956 , h λ = 0.6980 ,

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