Abstract

The extremely asymmetrical scattering (EAS) of bulk and guided electromagnetic waves in nonuniform periodic Bragg arrays with steplike variations of the grating amplitude is analyzed theoretically by means of a recently developed approach based on allowance for the diffractional divergence of the scattered wave. Arrays of finite and infinite widths are investigated. It is shown that, for thin nonuniform arrays, EAS has the same pattern as for uniform arrays with mean grating amplitude. On the contrary, for wide nonuniform arrays, the scattered wave amplitudes are well determined by local values of the grating amplitude. In this case, the energy of the scattered wave is shown to concentrate mainly in regions with smaller grating amplitude. The sensitivity of EAS to small imperfections of periodic arrays is investigated theoretically. The physical explanation of the observed effects is based on the diffractional divergence of the scattered wave.

© 1999 Optical Society of America

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References

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  1. Z. G. Pinsker, Dynamical Scattering of X-Rays in Crystals (Springer-Verlag, Berlin, 1978).
    [CrossRef]
  2. S. Kishino, “Anomalous transmission in Bragg-case diffraction of x-rays,” J. Phys. Soc. Jpn. 31, 1168–1173 (1971).
    [CrossRef]
  3. S. Kishino, A. Noda, K. Kohra, “Anomalous enhancement of transmitted intensity of diffraction of x-rays from a single crystal,” J. Phys. Soc. Jpn. 33, 158–166 (1972).
    [CrossRef]
  4. T. Bedynska, “On x-ray diffraction in an extremely asymmetric case,” Phys. Status Solidi A 19, 365–372 (1973).
    [CrossRef]
  5. T. Bedynska, “X-ray diffraction in the range of two diffracted beams for extremely asymmetric case,” Phys. Status Solidi A 25, 405–411 (1974).
    [CrossRef]
  6. A. V. Andreyev, “X-ray optics of surfaces (reflection and diffraction at grazing incident angles),” Sov. Phys. Usp. 28, 70–93 (1985), and references therein.
    [CrossRef]
  7. M. P. Bakhturin, L. A. Chernzatonskii, D. K. Gramotnev, “Planar optical waveguides coupled by means of Bragg scattering,” Appl. Opt. 34, 2692–2703 (1995).
    [CrossRef] [PubMed]
  8. D. K. Gramotnev, “Extremely asymmetrical scattering of Rayleigh waves in periodic groove arrays,” Phys. Lett. A 200, 184–190 (1995).
    [CrossRef]
  9. D. K. Gramotnev, “A new method of analysis of extremely asymmetrical scattering of waves in periodic Bragg arrays,” J. Phys. D 30, 2056–2062 (1997).
    [CrossRef]
  10. D. K. Gramotnev, “Extremely asymmetrical scattering of slab modes in periodic Bragg arrays,” Opt. Lett. 22, 1053–1055 (1997).
    [CrossRef] [PubMed]
  11. Yu. V. Gulyaev, V. V. Plesski, “Propagation of acoustic surface waves in periodic structures,” Sov. Phys. Usp. 32, 51–74 (1989).
    [CrossRef]
  12. G. I. Stegeman, D. Sarid, J. J. Burke, D. G. Hall, “Scattering of guided waves by surface periodic grating for arbitrary angles of incidence: perturbation field theory and implications to normal-mode analysis,” J. Opt. Soc. Am. 71, 1497–1507 (1981).
    [CrossRef]
  13. E. Popov, L. Mashev, “Analysis of mode coupling in planar optical waveguides,” Opt. Acta. 32, 265–280 (1985).
    [CrossRef]
  14. L. A. Weller-Brophy, D. G. Hall, “Local normal mode analysis of guided mode interactions with waveguide gratings,” J. Lightwave Technol. 6, 1069–1082 (1988).
    [CrossRef]
  15. D. G. Hall, “Coupled mode theory for corrugated optical waveguides,” Opt. Lett. 15, 619–621 (1990).
    [CrossRef]
  16. F. Ouellette, “Dispersion cancellation using linearly chirped Bragg grating filters in optical waveguides,” Opt. Lett. 12, 847–849 (1987).
    [CrossRef] [PubMed]
  17. C. M. de Sterke, J. E. Sipe, “Launching of gap solitons in nonuniform gratings,” Opt. Lett. 18, 269–271 (1993).
    [CrossRef] [PubMed]
  18. N. G. R. Broderick, C. M. de Sterke, “Analysis of nonuniform gratings,” Phys. Rev. E 52, 4458–4464 (1995).
    [CrossRef]

1997 (2)

D. K. Gramotnev, “A new method of analysis of extremely asymmetrical scattering of waves in periodic Bragg arrays,” J. Phys. D 30, 2056–2062 (1997).
[CrossRef]

D. K. Gramotnev, “Extremely asymmetrical scattering of slab modes in periodic Bragg arrays,” Opt. Lett. 22, 1053–1055 (1997).
[CrossRef] [PubMed]

1995 (3)

M. P. Bakhturin, L. A. Chernzatonskii, D. K. Gramotnev, “Planar optical waveguides coupled by means of Bragg scattering,” Appl. Opt. 34, 2692–2703 (1995).
[CrossRef] [PubMed]

D. K. Gramotnev, “Extremely asymmetrical scattering of Rayleigh waves in periodic groove arrays,” Phys. Lett. A 200, 184–190 (1995).
[CrossRef]

N. G. R. Broderick, C. M. de Sterke, “Analysis of nonuniform gratings,” Phys. Rev. E 52, 4458–4464 (1995).
[CrossRef]

1993 (1)

1990 (1)

1989 (1)

Yu. V. Gulyaev, V. V. Plesski, “Propagation of acoustic surface waves in periodic structures,” Sov. Phys. Usp. 32, 51–74 (1989).
[CrossRef]

1988 (1)

L. A. Weller-Brophy, D. G. Hall, “Local normal mode analysis of guided mode interactions with waveguide gratings,” J. Lightwave Technol. 6, 1069–1082 (1988).
[CrossRef]

1987 (1)

1985 (2)

A. V. Andreyev, “X-ray optics of surfaces (reflection and diffraction at grazing incident angles),” Sov. Phys. Usp. 28, 70–93 (1985), and references therein.
[CrossRef]

E. Popov, L. Mashev, “Analysis of mode coupling in planar optical waveguides,” Opt. Acta. 32, 265–280 (1985).
[CrossRef]

1981 (1)

1974 (1)

T. Bedynska, “X-ray diffraction in the range of two diffracted beams for extremely asymmetric case,” Phys. Status Solidi A 25, 405–411 (1974).
[CrossRef]

1973 (1)

T. Bedynska, “On x-ray diffraction in an extremely asymmetric case,” Phys. Status Solidi A 19, 365–372 (1973).
[CrossRef]

1972 (1)

S. Kishino, A. Noda, K. Kohra, “Anomalous enhancement of transmitted intensity of diffraction of x-rays from a single crystal,” J. Phys. Soc. Jpn. 33, 158–166 (1972).
[CrossRef]

1971 (1)

S. Kishino, “Anomalous transmission in Bragg-case diffraction of x-rays,” J. Phys. Soc. Jpn. 31, 1168–1173 (1971).
[CrossRef]

Andreyev, A. V.

A. V. Andreyev, “X-ray optics of surfaces (reflection and diffraction at grazing incident angles),” Sov. Phys. Usp. 28, 70–93 (1985), and references therein.
[CrossRef]

Bakhturin, M. P.

Bedynska, T.

T. Bedynska, “X-ray diffraction in the range of two diffracted beams for extremely asymmetric case,” Phys. Status Solidi A 25, 405–411 (1974).
[CrossRef]

T. Bedynska, “On x-ray diffraction in an extremely asymmetric case,” Phys. Status Solidi A 19, 365–372 (1973).
[CrossRef]

Broderick, N. G. R.

N. G. R. Broderick, C. M. de Sterke, “Analysis of nonuniform gratings,” Phys. Rev. E 52, 4458–4464 (1995).
[CrossRef]

Burke, J. J.

Chernzatonskii, L. A.

de Sterke, C. M.

N. G. R. Broderick, C. M. de Sterke, “Analysis of nonuniform gratings,” Phys. Rev. E 52, 4458–4464 (1995).
[CrossRef]

C. M. de Sterke, J. E. Sipe, “Launching of gap solitons in nonuniform gratings,” Opt. Lett. 18, 269–271 (1993).
[CrossRef] [PubMed]

Gramotnev, D. K.

D. K. Gramotnev, “A new method of analysis of extremely asymmetrical scattering of waves in periodic Bragg arrays,” J. Phys. D 30, 2056–2062 (1997).
[CrossRef]

D. K. Gramotnev, “Extremely asymmetrical scattering of slab modes in periodic Bragg arrays,” Opt. Lett. 22, 1053–1055 (1997).
[CrossRef] [PubMed]

M. P. Bakhturin, L. A. Chernzatonskii, D. K. Gramotnev, “Planar optical waveguides coupled by means of Bragg scattering,” Appl. Opt. 34, 2692–2703 (1995).
[CrossRef] [PubMed]

D. K. Gramotnev, “Extremely asymmetrical scattering of Rayleigh waves in periodic groove arrays,” Phys. Lett. A 200, 184–190 (1995).
[CrossRef]

Gulyaev, Yu. V.

Yu. V. Gulyaev, V. V. Plesski, “Propagation of acoustic surface waves in periodic structures,” Sov. Phys. Usp. 32, 51–74 (1989).
[CrossRef]

Hall, D. G.

Kishino, S.

S. Kishino, A. Noda, K. Kohra, “Anomalous enhancement of transmitted intensity of diffraction of x-rays from a single crystal,” J. Phys. Soc. Jpn. 33, 158–166 (1972).
[CrossRef]

S. Kishino, “Anomalous transmission in Bragg-case diffraction of x-rays,” J. Phys. Soc. Jpn. 31, 1168–1173 (1971).
[CrossRef]

Kohra, K.

S. Kishino, A. Noda, K. Kohra, “Anomalous enhancement of transmitted intensity of diffraction of x-rays from a single crystal,” J. Phys. Soc. Jpn. 33, 158–166 (1972).
[CrossRef]

Mashev, L.

E. Popov, L. Mashev, “Analysis of mode coupling in planar optical waveguides,” Opt. Acta. 32, 265–280 (1985).
[CrossRef]

Noda, A.

S. Kishino, A. Noda, K. Kohra, “Anomalous enhancement of transmitted intensity of diffraction of x-rays from a single crystal,” J. Phys. Soc. Jpn. 33, 158–166 (1972).
[CrossRef]

Ouellette, F.

Pinsker, Z. G.

Z. G. Pinsker, Dynamical Scattering of X-Rays in Crystals (Springer-Verlag, Berlin, 1978).
[CrossRef]

Plesski, V. V.

Yu. V. Gulyaev, V. V. Plesski, “Propagation of acoustic surface waves in periodic structures,” Sov. Phys. Usp. 32, 51–74 (1989).
[CrossRef]

Popov, E.

E. Popov, L. Mashev, “Analysis of mode coupling in planar optical waveguides,” Opt. Acta. 32, 265–280 (1985).
[CrossRef]

Sarid, D.

Sipe, J. E.

Stegeman, G. I.

Weller-Brophy, L. A.

L. A. Weller-Brophy, D. G. Hall, “Local normal mode analysis of guided mode interactions with waveguide gratings,” J. Lightwave Technol. 6, 1069–1082 (1988).
[CrossRef]

Appl. Opt. (1)

J. Lightwave Technol. (1)

L. A. Weller-Brophy, D. G. Hall, “Local normal mode analysis of guided mode interactions with waveguide gratings,” J. Lightwave Technol. 6, 1069–1082 (1988).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Phys. D (1)

D. K. Gramotnev, “A new method of analysis of extremely asymmetrical scattering of waves in periodic Bragg arrays,” J. Phys. D 30, 2056–2062 (1997).
[CrossRef]

J. Phys. Soc. Jpn. (2)

S. Kishino, “Anomalous transmission in Bragg-case diffraction of x-rays,” J. Phys. Soc. Jpn. 31, 1168–1173 (1971).
[CrossRef]

S. Kishino, A. Noda, K. Kohra, “Anomalous enhancement of transmitted intensity of diffraction of x-rays from a single crystal,” J. Phys. Soc. Jpn. 33, 158–166 (1972).
[CrossRef]

Opt. Acta. (1)

E. Popov, L. Mashev, “Analysis of mode coupling in planar optical waveguides,” Opt. Acta. 32, 265–280 (1985).
[CrossRef]

Opt. Lett. (4)

Phys. Lett. A (1)

D. K. Gramotnev, “Extremely asymmetrical scattering of Rayleigh waves in periodic groove arrays,” Phys. Lett. A 200, 184–190 (1995).
[CrossRef]

Phys. Rev. E (1)

N. G. R. Broderick, C. M. de Sterke, “Analysis of nonuniform gratings,” Phys. Rev. E 52, 4458–4464 (1995).
[CrossRef]

Phys. Status Solidi A (2)

T. Bedynska, “On x-ray diffraction in an extremely asymmetric case,” Phys. Status Solidi A 19, 365–372 (1973).
[CrossRef]

T. Bedynska, “X-ray diffraction in the range of two diffracted beams for extremely asymmetric case,” Phys. Status Solidi A 25, 405–411 (1974).
[CrossRef]

Sov. Phys. Usp. (2)

A. V. Andreyev, “X-ray optics of surfaces (reflection and diffraction at grazing incident angles),” Sov. Phys. Usp. 28, 70–93 (1985), and references therein.
[CrossRef]

Yu. V. Gulyaev, V. V. Plesski, “Propagation of acoustic surface waves in periodic structures,” Sov. Phys. Usp. 32, 51–74 (1989).
[CrossRef]

Other (1)

Z. G. Pinsker, Dynamical Scattering of X-Rays in Crystals (Springer-Verlag, Berlin, 1978).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

EAS in a nonuniform periodic array with steplike variation in the grating amplitude at x = L 1. The x 0 axis is parallel to the reciprocal lattice vector of the array. The wave vector of the scattered wave k1 is parallel to the array boundaries.

Fig. 2
Fig. 2

Dependences of the relative scattered wave amplitude |E 1/E 00| at the front boundary (i.e., at x = 0) of the semi-infinite (L 2 = +∞) nonuniform array with ε = 5, ε2 = 5 × 10-2, θ0 = π/4, and λ = 1 µm on grating amplitude ε1 in the first array (i.e., at 0 < x < L 1), and L 1 = 3 × 10-3 cm (curve 1), L 1 = 2 × 10-3 cm (curve 2), L 1 = 10-3 cm (curve 3), and L 1 = 5 × 10-4 cm (curve 4).

Fig. 3
Fig. 3

Dependences of |E 1(x)/E 00| in semi-infinite arrays with L 1 = 2 × 10-3 cm and L = +∞ on the x coordinate. Curve 1, the uniform array with ε1 = ε2 = 3 × 10-3. Curve 2, the nonuniform array with ε1 = 3 × 10-3 for 0 < x < L 1 and ε2 = 5 × 10-2 for x > L 1. Curve 3, the uniform array with ε1 = ε2 = 5 × 10-2. The angle of incidence is θ0 = π/4, λ = 1 µm, and ε = 5.

Fig. 4
Fig. 4

Dependences of |E 1(x)/E 00| in finite arrays with L 1 = 2 × 10-3 cm, L = 4 × 10-3 cm, θ0 = π/4, λ = 1 µm, and ε = 5 on the x coordinate. Curve 1, the uniform array with ε1 = ε2 = 3 × 10-3. Curve 2, the nonuniform array with ε1 = 3 × 10-3 for 0 < x < L 1 and ε2 = 5 × 10-2 for L 1 < x < L. Curve 3, the uniform array with ε1 = ε2 = 5 × 10-2.

Fig. 5
Fig. 5

Dependences of |E 1(x)/E 00| in finite arrays with L 1 = 10-3 cm, L = 2 × 10-3 cm, θ0 = π/4, λ = 1 µm, and ε = 5 on the x coordinate. Curve 1, the uniform array with ε1 = ε2 = 3 × 10-3. Curve 2, the nonuniform array with ε1 = 3 × 10-3 for 0 < x < L 1 and ε2 = 5 × 10-2 for L 1 < x < L. Curve 3, the uniform array with ε1 = ε2 = 5 × 10-2. Curve 4, the uniform array with the average grating amplitude (ε1 L 1 + ε2 L 2)/L = 2.65 × 10-2).

Fig. 6
Fig. 6

Illustration of the effect of small steplike variations of the grating amplitude on EAS in a finite array with L = 4 × 10-3 cm, θ0 = π/4, λ = 1 µm, and ε = 5. Curve 1, the uniform array with ε1 = ε2 = 4 × 10-3. Curve 2, the nonuniform array with L 1 = 2 × 10-3 cm and grating amplitudes ε1 = 4 × 10-3 for 0 < x < L 1 and ε2 = 5 × 10-3 for L 1 < x < L. Curve 3, the uniform array with ε1 = ε2 = 5 × 10-3.

Fig. 7
Fig. 7

Dependences |E 1(x)/E 00| in finite arrays with L 1 = 10-3 cm, L = 2 × 10-3 cm, and small variations of the grating amplitude. Curve 1, the uniform array with ε1 = ε2 = 4 × 10-3. Curve 2, the nonuniform array with grating amplitudes ε1 = 4 × 10-3 for 0 < x < L 1 and ε2 = 5 × 10-3 for L 1 < x < L. Curve 3, the uniform array with ε1 = ε2 = 5 × 10-3. Curve 4, the uniform array with the average value of the grating amplitude (ε1 L 1 + ε2 L 2)/L = 4.5 × 10-3. The angle of incidence is θ0 = π/4, λ = 1 µm, and ε = 5.

Fig. 8
Fig. 8

Dependences |E 1(x)/E 00| in finite arrays with L 1 = 1.5 × 10-3 cm, L = 3 × 10-3 cm, and small variations of the grating amplitude. Curve 1, the uniform array with ε1 = ε2 = 4 × 10-3. Curve 2, the nonuniform array with the grating amplitudes ε1 = 4 × 10-3 for 0 < x < L 1 and ε2 = 5 × 10-3 for L 1 < x < L. Curve 3, the uniform array with ε1 = ε2 = 5 × 10-3. Curve 4, the uniform array with the average value of the grating amplitude (ε1 L 1 + ε2 L 2)/L = 4.5 × 10-3. The angle of incidence is θ0 = π/4, λ = 1 µm, and ε = 5.

Equations (44)

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εs=ε+ε1 expiQxx+iQyy+c.c.0<x<L1,εs=ε+ε2 expiQxx+iQyy+c.c.L1<x<L,εs=εx<0 or x>L,
|ε1,2|/ε  1.
k1-k0=-Q,
Ex=E0xexpik0xx+ik0yy-iωt+E1xexpik0y-iωt,
dE1/dx0=iΓ0jE0,
dE0/dx0=iΓ1jE1,
Γ0j=-Γ1j*=-εj*ω2/2c2k0 cosη,
dE1=-iΓ0jE0 sinη-θ0dy.
d2E1x/dx2+K0jE0x=0,
dE0x/dx-iK1jE1x=0,
K0j=-2k1Γ0j sinη-θ0,  K1j=Γ1j cosη/cosθ0.
dE0x/dx=0,  d2E1x/dx2=0.
E0x=D1 expiγ1x+D2 exp-γ1x31/2+i/2+D3 expγ1x31/2-i/2,
E1x=C1 expiγ1x+C2 exp-γ1x31/2+i/2+C3 expγ1x31/2-i/2
E0x=G1 expiγ2x-L1+G2 exp-γ2x-L1×31/2+i/2,
E1x=F1 expiγ2x-L1+F2 exp-γ2x-L1×31/2+i/2,
γj=K0jK1j1/3 j=1, 2.
E0x=E00,  E1x=A1,
E0x=G1 expiγ2x+G2 exp-γ2x31/2+i/2+G3 expγ2x31/2-i/2,
E1x=F1 expiγ2x+F2 exp-γ2x31/2+i/2+F3 expγ2x31/2-i/2.
E0|x>L=E01,  E1|x>L=A2,
E0x=D1 expiγ1x+D2 exp-γ1x31/2+i/2,
E1x=C1 expiγ1x+C2 exp-γ1x31/2+i/2,
D1=E0031/2+i/121/2,  D2=E0031/2-i/121/2,
|Δε/ε10|  1,
γ1γ10+2γ10Reε10Δε*/3|ε10|2γ10+Δγ,
ΔE1E10Δε*/ε10-4Reε10Δε*/3ε102+xΔγiC10a0-31/2+iC20b0/2,
xΔλ  1.
ΔE1/E10Δε*/ε10-4Reε10Δε*/3ε102.
ΔE1/E10-Δε/3ε10;
|E1x|2-|E10x|2-2Δε|E10x|2/3ε10-31/2xΔγ|C20|2 exp-31/2xγ10-31/2xΔγ exp-31/2xγ10/2Re1-i31/2C10C20* exp3ixγ10/2.
|E1x|2-|E10x|2/|E10x|2-2Δε/3ε10.
L  γ10-1.
|E1x|2-|E10x|2/|E10x|2-2Δε/ε10.
ξ=d+ξ1fx0 0<x<L1,ξ=d+ξ2fx0 L1<x<L,
|ξ1,2|Q/2π  1.
fx0=p=-+ fp expipQxx+ipQyy=p=-+ fp expipQx0,
k1-k0=-pQ
E0|x=0=E00, dE1/dxx=0=0, dE1/dxx=L1-0=dE1/dxx=L1+0, E1|x=0+0=A1,   E1|x=L1-0=E1|x=L1+0,
C1=K01D1/γ12, C2=-K01D21-i31/2/2γ12, C3=-K01D31+i31/2/2γ12, F1=K02G1/γ22, F2=-K02G21-i31/2/2γ22.
D1=2i3-1/2E00bi31/2-1+di31/2+1σ+ρ2b+di31/2+1-b1+i31/2+di31/2+1/D0, D2=2E00a3+i31/2+2id31/2σ-ρa3-i31/2-2id31/2+2id31/2-2ia31/2/3D0, D3=2E00i31/2-2b-a1+i31/2σ+ρ2a+b1+i31/2-a1-i31/2+b1-i31/2/3D0, G1=3-i31/2E00abi31/2-1+2ad-bd1+i31/2/p+2bd+2ab+2adσ+ab1-i31/2-bd1-i31/2+2ad/3σρD0, G2=2iE00ab2ρ-i31/2-1+ad2ρ-1+i31/2+2bdρ+1σ+ρabi31/2-1+2bd-ad1+i31/2/31/2σD0, D0=-1+i31/2[-a1+i31/2+2d-b1-i31/2σ+2bp+ai31/2-1+2ρ-b1+i31/2+2dp+1,
a=expiγ1L1, b=exp-γ1L131/2+i/2, d=expγ1L131/2-i/2, σ=ε2*γ1/ε1*γ2, ρ=γ2/γ1,
E0x=E00,  E1x=A1,
A1=0.5K01γ1-22D1-D21-i31/2-D31+i31/2

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