Abstract

The correlation length for sidewall roughness in planar waveguides is comparable to the period of Bragg gratings written in such structures and thus might influence the performance and the spectral properties of such gratings. Using a coupled-mode formalism, we calculate the effect of roughness or inhomogeneity for an arbitrary grating and present specific results for uniform and phase-shifted gratings. The broad spectral characteristics of most gratings are insensitive to roughness. However, narrow spectral features (such as transmission resonances) that rely on interference effects are affected by the presence of roughness.

© 1997 Optical Society of America

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References

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  1. F. Ladouceur, J. D. Love, and T. J. Senden, “Effect of side wall roughness in buried channel waveguides,” IEE Proc. J 141, 242–248 (1994).
  2. F. Ladouceur and L. Poladian, “Surface roughness and back-scattering,” Opt. Lett. 21, 1833–1835 (1996).
    [CrossRef] [PubMed]
  3. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).
  4. F. Ladouceur and J. D. Love, Silica-Based Buried Channel Waveguides and Devices (Chapman & Hall, London, 1995), Chap. 8.
  5. J. E. Sipe, L. Poladian, and C. Martijn de Sterke, “Propagation through nonuniform grating structures,” J. Opt. Soc. Am. A 11, 1307–1320 (1994).
    [CrossRef]

1996 (1)

1994 (2)

J. E. Sipe, L. Poladian, and C. Martijn de Sterke, “Propagation through nonuniform grating structures,” J. Opt. Soc. Am. A 11, 1307–1320 (1994).
[CrossRef]

F. Ladouceur, J. D. Love, and T. J. Senden, “Effect of side wall roughness in buried channel waveguides,” IEE Proc. J 141, 242–248 (1994).

Ladouceur, F.

F. Ladouceur and L. Poladian, “Surface roughness and back-scattering,” Opt. Lett. 21, 1833–1835 (1996).
[CrossRef] [PubMed]

F. Ladouceur, J. D. Love, and T. J. Senden, “Effect of side wall roughness in buried channel waveguides,” IEE Proc. J 141, 242–248 (1994).

Love, J. D.

F. Ladouceur, J. D. Love, and T. J. Senden, “Effect of side wall roughness in buried channel waveguides,” IEE Proc. J 141, 242–248 (1994).

Martijn de Sterke, C.

Poladian, L.

Senden, T. J.

F. Ladouceur, J. D. Love, and T. J. Senden, “Effect of side wall roughness in buried channel waveguides,” IEE Proc. J 141, 242–248 (1994).

Sipe, J. E.

IEE Proc. J (1)

F. Ladouceur, J. D. Love, and T. J. Senden, “Effect of side wall roughness in buried channel waveguides,” IEE Proc. J 141, 242–248 (1994).

J. Opt. Soc. Am. A (1)

Opt. Lett. (1)

Other (2)

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

F. Ladouceur and J. D. Love, Silica-Based Buried Channel Waveguides and Devices (Chapman & Hall, London, 1995), Chap. 8.

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Figures (4)

Fig. 1
Fig. 1

Schematic representation of typical surface roughness along the vertical core–cladding interface of an etched silica-based waveguide. The correlation length of the surface roughness is Lc. The periodic shading indicates the variations in the refractive index that produce a Bragg grating with period d. Note that d and Lc are comparable in magnitude.

Fig. 2
Fig. 2

Spectral dependence of roughness perturbation for a uniform grating of strength κL=5.

Fig. 3
Fig. 3

Reflectance including roughness perturbation for a uniform grating of strength κL=50 shown in closeup near the first Fabry–Pérot resonance.

Fig. 4
Fig. 4

Reflectance including roughness perturbation for a phase-shifted grating of strength κL=10 shown in a closeup near the central transmission resonance. The dashed curve is for the unperturbed grating, and the solid curve includes the effects of roughness.

Equations (38)

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C(z-z)=f(z)f(z).
C(z)=δf2 exp(-|z|/Lc),
Γ(z-z)=s(z)s(z)=exp(-|z-z|/Lc).
ia1(z)+δa1(z)+κa2(z)=σ exp[-i(2π/d)z]s(z)a2(z),
-ia2(z)+δa2(z)+κa1(z)=σ exp[+i(2π/d)z]s(z)a1(z),
E(x, z)={a1(z)exp[i(π/d)z]+a2(z)exp[-i(π/d)z]}ψ(x),
ai(0)(z)+σai(1)(z)+σ2ai(2)(z)+,
a1(n+1)(z)a2(n+1)(z)=0Ls(z)G11(z, z)G12(z, z)G21(z, z)G22(z, z)×exp[-i(π/d)z]a2(n)(z)exp[+i(π/d)z]a1(n)(z)dz.
r=a2(0)=r(0)+σ0Ls(z)F(z)dz+σ20L0Ls(z)s(z)J(z, z)dzdz+O(σ3),
R=R(0)+σ2 Re 0L0LΓ(z-z)[F(z)F(z)*+2r(0)*J(z, z)]dzdz+O(σ3),
R=R(0)+σ2 Re 0L0LΓ(z-z)×{A(z, z)exp[i(2π/d)(z-z)]+B(z, z)exp[i(2π/d)(z+z)]}dzdz+O(σ3),
RR(0)+σ2 Re -LLΓ(y)0LA(z, z)exp[i(2π/d)y]+B(z, z)exp[i(4π/d)z]dzdy.
RR(0)+σ2S(2kB)LA(δ),
A(δ)=1L0LA(z, z)dz=1LRe 0L(1-R)[Pi(z)2+P2(z)2]+4r*2a12(z)a22(z)-2r*a1(z)a2(z)[P1(z)+P2(z)]1-Rdz
S(k)=2Lc1+k2Lc2.
RR(0)+2σ2LcL1+4kB2Lc2A(δ).
2σ2LcL1+4kB2Lc2σ2LLc,
(σ/κ)2(κL)2(Lc/L).
A(δ)=14(α2 cosh2 αL+δ2 sinh2 αL)3(3κ4-4α4)δ2+κ4(2α2+3κ2)cosh2 αL+2κ4(2α2-3κ2)×cosh4 αL-δ2κ2(4α2+9κ2)sinh 2αL2αL+κ4(9κ2-8α2)cosh2 αLsinh 2αL2αL,
RR(0)+σ2LS(2kB),
RR(0)+σ2L4 cosh4(κL)4-cosh 2κL+sinh 2κL2κL×S(2kB)R(0)-2σ2L exp(-2κL)S(2kB).
RR(0)+σ2L48(1+κ2L2)3(48+32κ2L2-13κ4L4-9κ6L6)S(2kB)R(0)-9σ2L48S(2kB).
Rσ2L1-κLnπ2+34κLnπ4S(2kB)34σ2LκLnπ4S(2kB).
Rσ2LS(2kB)exp(2κL)32κL.
(σ/κ)2(κL)2(Lc/L)
K(z)=f(z)U2W2ρ3β(1+W),
U=ρ(k2nco2-β2)1/2,
W=ρ(β2-k2ncl2)1/2,
σ=δfU2W2ρ3β(1+W).
Gjk(z, z)=-i1-R(0)[θ(z-z)aj(0)(z)ak(0)(z)*+θ(z-z)a3-j(0)(z)*a3-k(0)(z)-r(0)*aj(0)(z)a3-k(0)(z)],
F(z)=G21(0, z)exp[-i(2π/d)z]a2(0)(z)+G22(0, z)exp[+i(2π/d)z]a1(0)(z)
=-i{a1(0)(z)2 exp[+i(2π/d)z]+a2(0)(z)2 exp[-i(2π/d)z]},
J(z, z)=G21(0, z)G21(z, z)exp[-i(2π/d)×(z+z)]a2(0)(z)+G21(0, z)G22(z, z)×exp[-i(2π/d)(z-z)]a1(0)(z)+G22(0, z)G11(z, z)×exp[+i(2π/d)(z-z)]a2(0)(z)+G22(0, z)G12(z, z)×exp[+i(2π/d)(z+z)]a1(0)(z)
=-i{a2(0)(z)a2(0)(z)G21(z, z)exp[-i(2π/d)×(z+z)]+a2(0)(z)a1(0)(z)G22(z, z)×exp[-i(2π/d)(z-z)]+a1(0)(z)a2(0)(z)×G11(z, z)exp[+i(2π/d)(z-z)]+a1(0)(z)a1(0)(z)G12(z, z)×exp[+i(2π/d)(z+z)]}.
T=n¯cImddδln(r),
T=T(0)+σ2n¯cImddδ0L0LΓ(z-z)×J(z, z)r(0)-F(z)F(z)2r(0)2dzdz+O(σ3),
TT(0)+σ2S(2kB)LC(δ),
C(δ)=1Ln¯cImddδ0L(1+2R)a12(z)a22(z)-ra1(z)a2(z)[P1(z)+P2(z)]r2(1-R)dz.

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