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Statistical analysis of geometrical imperfections from the images of 2D photonic crystals

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Abstract

High resolution images of planar photonic crystal (PC) optical components fabricated by e-beam lithography in various materials are analyzed to characterize statistical properties of common 2D geometrical imperfections. Our motivation is to attempt an intuitive, while rigorous statistical description of fabrication imperfections to provide a realistic input into theoretical modelling of PC device performance.

©2005 Optical Society of America

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Figures (7)

Fig. 1.
Fig. 1. (a) Image of a hole together with a detected edge. (b) Shape of a rugged edge is fitted with Fourier series in θ. Smooth curve is an M=1 circle fit. (c) On a scale <2nm hole edge can not be represented by a single valued analytical function rfitM (θ). (d) Edge roughness is self-similar on very different scales suggesting fractal description.
Fig. 2.
Fig. 2. (a,c)Probability density distribution of fit error for different number of angular momenta components M in a fit. (b) RMS of fit error decreases slowly as the number of angular momenta components M in a fit increases, suggesting that there is no simple coarse description of a feature shape. (c) RMS of fit error decreases dramatically when ellipticity M=2 of a feature is included in a fit, suggesting ellipticity as a dominant coarse parameter.
Fig. 3.
Fig. 3. (a) “Height to height” correlation function and (b) auto-correlation function of an edge deviation from smooth fits with M angular components.
Fig. 4.
Fig. 4. (a) Power spectral density (blue). Linear fit is over 2 decades starting from the largest length scale. (b) RMS of a fit error (blue). Linear fit spans the lowest angular momenta starting with M=1. (c) Power spectral density (blue). Linear fit is over 1 decade in the interval 30nm≳λ≳200nm (d) RMS of a fit error (blue). Linear fit is in the range 4<M≲40. In red are the statistical functions of a noise level due to finite resolution of an image.
Fig. 5.
Fig. 5. (a)PC lattice of holes with 2 missing rows. Vertices of a fitted perfectly periodic underlying lattice are shows as white dots. (b)PDDs of hole center deviations from the vertices of a perfect lattice along 2 principal directions (solids) together with Gaussian fits (dotted lines): perpendicular to the waveguide σ1 (blue), and parallel to the waveguide σ2 (red). (c) RMS deviations σ1,2 (along 2 principle directions) of hole centers from an underlying lattice against the number of features in a fit. Features in a fit are included one by one, row by row starting from the upper left corner of an image.
Fig. 6.
Fig. 6. (a) Uniform square PC lattice [26]. (b) σ1,2 as a function of the number of features in a fit. Distribution of feature centers around the vertices of an underlying perfect lattice is isotropic. (c) Triangular PC lattice with a waveguide and a bend [28]. (d) σ1,2 as a function of the number of features in a fit. Distribution of feature centers around the vertices of an underlying perfect lattice is anisotropic.
Fig. 7.
Fig. 7. (a) Image of a hole with a moderate contrast and a high noise level. Insert: histogram of pixel values. Hole edges are detected with: (b) tol=0.37 (c) tol=0.40 (d) tol=0.43

Tables (4)

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Table 1. Parameterization of features in Fig. 1(a), InP/InGaAsP/InP [22].

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Table 2. Parameterization of features in Fig. 2(c), Air/Si membranes [26].

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Table 3. Parameterization of features in Fig. 6(a), Air/Si membranes [26].

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Table 4. Ranges of statistical parameters over various PC lattices [21].

Equations (23)

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Q edge M = 1 N edge i = 1 N edge ( r fit M ( θ i ) r edge ( θ i ) ) 2 ,
r fit M ( θ i ) = R 0 + m = 2 M ( A m M Sin ( m θ i ) + B m M Cos ( m θ i ) ) .
R = R a v ± δ R a v R a v = 1 N f n = 1 N f R 0 n δ R a v 2 = 1 N f n = 1 N f ( R 0 n R a v ) 2
σ 2 ( M ) = n = 1 N f N edge n σ n 2 ( M ) n = 1 N f N edge n
[ C , Γ , S ] M ( λ ) = 1 N f n = 1 N f [ C , Γ , S ] n M ( λ ) .
δ r M = r fit n , M ( θ i ) r edge n ( θ i ) ,
P ( δ r M ) = 1 2 π σ ( M ) exp ( δ r M ) 2 2 σ 2 ( M ) .
δ n M ( θ i ) = r fit n , M ( θ i ) r edge n ( θ i ) ,
f ( θ + ε ) f ( θ ) ε H , ε 0 ,
C n M ( λ ) = ( δ n M ( θ + λ R 0 n ) δ n M ( θ ) ) 2 θ = 1 2 π 0 2 π d θ ( δ n M ( θ + λ R 0 n ) δ n M ( θ ) ) 2 .
C n M ( λ ) = 2 ( σ n 2 ( M ) Γ n M ( λ ) ) ,
Γ n M ( λ ) = δ n M ( θ + λ R 0 n ) δ n M ( θ ) θ δ n N ( θ ) θ 2 .
C n M ( λ ) λ 0 λ 2 H ; C n M ( λ ) λ > λ n c M 2 σ n 2 ( M ) Γ n M ( λ ) λ > λ n c M 0 ; ( σ n 2 ( M ) Γ n M ( λ ) ) λ 0 λ 2 H .
C n M ( λ ) = 2 σ n 2 ( M ) ( 1 exp ( ( λ λ n c M ) 2 H ) )
Γ n M ( λ ) = σ n 2 ( M ) exp ( ( λ λ n c M ) 2 H ) .
δ n 1 ( θ ) = r edge n ( θ ) R 0 n = m = 2 N edge n ( A m Sin ( m θ ) + B m Cos ( m θ ) ) ,
Γ n 1 ( λ ) = 1 2 m = 2 N edge n ( A m 2 + B m 2 ) cos ( m λ R 0 n ) .
S n 1 ( λ m ) = 1 2 π R 0 n 0 2 π R 0 n d λ ˜ Γ n 1 ( λ ˜ ) exp ( i 2 π λ m λ ˜ ) = 1 4 ( A m 2 + B m 2 ) ,
S n 1 ( λ m ) = 1 4 ( A m 2 + B m 2 ) λ m 0 λ m 1 + 2 H ,
δ n M ( θ ) = m = M + 1 N edge n ( A m Sin ( m θ ) + B m Cos ( m θ ) ) .
σ n 2 ( M ) = ( δ n M ( θ ) ) 2 θ = 1 2 m = M + 1 N edge n ( A m 2 + B m 2 ) M H .
Q lat = 1 N f n = 1 N f ( r ̅ 0 n j 1 n a ̅ 1 j 2 n a ̅ 2 ) 2 ,
P ( δ ̅ c ) = 1 2 π σ 1 σ 2 exp ( ( δ c x δ c y ) T R T ( 1 2 σ 1 2 0 0 1 2 σ 2 2 ) R ( δ c x δ c y ) ) ,
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