Abstract

Rigorous coupled-wave theory of diffraction by dielectric gratings is extended to cover E-mode polarization and losses. Unlike in the H-mode-polarization case, it is shown that, in the E-mode case, direct coupling exists between all diffracted orders rather than just between adjacent orders.

© 1983 Optical Society of America

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References

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  1. M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981).
    [CrossRef]
  2. C. L. Liu and J. W. S. Liu, Linear Systems Analysis (McGraw-Hill, New York, 1975).
  3. R. Magnusson and T. K. Gaylord, “Analysis of multiwave diffraction of thick gratings,” J. Opt. Soc. Am. 67, 1165–1170 (1977).
    [CrossRef]
  4. J. A. Kong, “Second-order coupled-mode equations for spatially periodic media,” J. Opt. Soc. Am. 67, 825–829 (1977).
    [CrossRef]
  5. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
    [CrossRef]
  6. M. G. Moharam and T. K. Gaylord, “Chain-matrix analysis of arbitrary-thickness dielectric reflection gratings,” J. Opt. Soc. Am. 72, 187–190 (1982).
    [CrossRef]

1982 (1)

1981 (1)

1977 (2)

1969 (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Diffraction efficiencies of forward-diffracted waves for a lossless ϕ = 120° slanted grating (120° angle from z axis to grating vector) for both H-mode and E-mode polarizations. The average permittivity inside and outside the grating is the same ( = 2.25). The angle of incidence θ′ = 42° is at the first Bragg angle (m = 1). For these conditions the i = −1 field is evanescent (cut off). The modulation is 1/0 = 0.120. The diffraction efficiencies for all diffracted waves not shown are less than 0.01. (a) Rigorously calculated results. The fields i = −4 to i = +5 were retained to achieve convergence in field amplitudes. (b) Multiwave first-order coupled-wave theory results, showing the effect of neglecting second derivatives and boundary effects. Notice that the diffraction efficiency of the i = −1 field is predicted to be as large as 9% even though this wave, in fact, is evanescent! The fields i = −4 to i = +5 were retained to achieve convergence in the field amplitudes. (c) Two-wave (i = 0, +1) second-order coupled-wave theory results showing the effect of neglecting higher-order waves.

Fig. 2
Fig. 2

Diffraction efficiencies of forward-diffracted and backward-diffracted waves for a lossless ϕ = 150° slanted grating for both H-mode and E-mode polarizations. The average permittivity inside and outside the grating is the same ( = 2.25). The angle of incidence θ′ = 20° is at the first Bragg angle (m = 1). The modulation is 1/0 = 0.330. The diffraction efficiencies for all diffracted waves not shown are less than 0.01. (a) Rigorously calculated results. The fields i = −4−+5 were retained to achieve convergence in field amplitudes. (b) Multiwave first-order coupled-wave theory results, showing the effect of neglecting second derivatives and boundary effects. The fields i = −4−+5 were retained to achieve convergence in field amplitudes. (c) Two-wave (i = 0, +1) second-order coupled-wave theory results, showing the effect of neglecting higher-order waves.

Fig. 3
Fig. 3

Rigorously calculated diffraction efficiencies of forward-diffracted waves for a lossy ϕ = 120° slanted grating for both H-mode and E-mode polarizations. The average conductivity is σ0 = 400 (ohm − m)−1. The angle of incidence θ′ = 20° is at the first Bragg angle. The modulation is 1/0 = 0.120, and the wavelength λ = 514.5 nm. The average permittivity outside the grating is the same as that inside ( = 2.25).

Equations (16)

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( x , z ) = ̂ 0 + 1 cos [ K ( x sin ϕ + z cos ϕ ) ] ,
̂ 0 = 0 j σ 0 / ω ɛ 0 ,
2 Ē + ( Ē ) + k 2 ( x , z ) Ē = 0 ,
2 H ¯ + × × H ¯ + k 2 ( x , z ) H ¯ = 0 ,
2 E + k 2 ( x , z ) E = 0 .
2 H ( ) H + k 2 ( x , z ) H = 0 .
E ( x , z ) = i = + S i ( z ) exp ( j σ ¯ i r ¯ ) ,
1 2 π 2 d 2 S i ( z ) d z 2 j 2 π [ ( ̂ 0 I sin θ ) 1 / 2 λ i cos ϕ Λ ] d S i ( z ) d z + 2 i ( m i ) Λ 2 S i ( z ) + 1 λ 2 [ S i + 1 ( z ) + S i 1 ( z ) ] = 0 ,
m 2 ( Λ / λ ) [ I 1 / 2 sin ϕ sin θ + ( ̂ 0 I sin 2 θ ) 1 / 2 cos ϕ ] .
( ) H = 1 sin ( K ¯ r ¯ ) 0 + 1 cos ( K ¯ r ¯ ) ( sin ϕ H x + cos ϕ H z ) 2 π Λ ,
1 sin ( K ¯ r ¯ ) 0 + 1 cos ( K ¯ r ¯ ) = j h = + A h exp ( j h K ¯ r ¯ ) ,
H ( x , z ) = i = + U i ( z ) exp ( j σ ¯ i . r ¯ ) ,
1 2 π 2 d 2 U i ( z ) d z 2 j 2 π [ ( ̂ 0 I sin θ ) 1 / 2 λ i cos ϕ Λ ] d U i ( z ) d z j cos ϕ π Λ h A h d U i h ( z ) d z + 2 i ( m i ) Λ 2 U i ( z ) + 1 λ 2 [ U i + 1 ( z ) + U i 1 ( z ) ] + 2 Λ 2 h ( i h m 2 ) × A h U i h ( z ) = 0 .
H x = ( j / ω μ ) z i = + S i ( z ) exp ( j σ ¯ i r ¯ ) .
1 ( x , z ) = h = + G h exp ( j h K ¯ r ¯ ) ,
E x = ( j / ω ɛ 0 ) i = + exp ( j σ ¯ i r ¯ ) × h = + G h [ d U i h ( z ) d z j ( σ ¯ i h z ̂ ) U i h ( z ) ] .