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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References

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Appl. Phys. Lett.

K.-C. Kwan, X. Zhang, et al. �??Effects due to disorder on photonic crystal-based waveguides,�?? Appl. Phys. Lett. 82, 4414-4416 (2003)
[CrossRef]

M.L. Povinelli, S.G. Johnson, et al. �??Effect of a photonic band gap on scattering from waveguide disorder,�?? Appl. Phys. Lett. 84, 3639-3641 (2004)
[CrossRef]

A. Talneau, M. Mulot, et al. �??Compound cavity measurement of transmission and reflection of a tapered singleline photonic-crystal waveguide,�?? Appl. Phys. Lett. 82, 2577-2579 (2003)
[CrossRef]

H. Altuga and J. Vuckovic, �??Two-dimensional coupled photonic crystal resonator arrays,�?? Appl. Phys. Lett. 84, 161-163 (2004)
[CrossRef]

M. Augustin, H.-J. Fuchs, et al. �??High transmission and single-mode operation in low-index-contrast photonic crystal waveguide devices,�?? Appl. Phys. Lett. 84, (2004)
[CrossRef]

J. App. Phys.

C. Monat, C. Seassal, et al. �??Two-dimensional hexagonal-shaped microcavities formed in a two-dimensional photonic crystal on an InP membrane,�?? J. App. Phys. 93, 23-31 (2003)
[CrossRef]

J. Appl. Phys.

S. Fan, P.R. Villeneuve, and J.D. Joannopoulos, �??Theoretical investigation of fabrication-related disorder on the properties of photonic crystals,�?? J. Appl. Phys. 78, 1415-1418 (1995)
[CrossRef]

B.C. Guptaa, and Z. Yeb, �??Disorder effects on the imaging of a negative refractive lens made by arrays of dielectric cylinders,�?? J. Appl. Phys. 94, 2173-2176 (2003)
[CrossRef]

J. Opt. A: Pure Appl. Opt.

N.A. Mortensen, M.D. Nielsen, et al. �??Small-core photonic crystal fibres with weakly disordered air-hole claddings,�?? J. Opt. A: Pure Appl. Opt. 6, 221223 (2004)
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

J. Vac. Sci. Technol.

M.Mulot, S.Anand, et al. �??Low-loss InP-based photonic crystal waveguides etched with Ar/Cl2 chemically assisted ion beam etching,�?? J. Vac. Sci. Technol. B21, 900-903 (2003).

J. Vac. Sci. Technol. B

A. Xing, M. Davanco, et al. �??Fabrication of InP-based two-dimensional photonic crystal membrane,�?? J. Vac. Sci. Technol. B 22, 70-73 (2004)
[CrossRef]

Nanotechnology

A.G. Martyn, D. Hermann, et al. �??Defect computations in photonic crystals: a solid state theoretical approach,�?? Nanotechnology 14, 177183 (2003)

Opt. Exp.

P.E. Barclay, K. Srinivasan, et al. �??Efficient input and output fiber coupling to a photonic crystal waveguide,�?? Opt. Lett. 29, 697-699 (2004)
[CrossRef] [PubMed]

Opt. Lett.

Photomechanics, Topics Appl. Phys.

D. J. Whitehouse, �??Surface Characterization and Roughness Measurement in Engineering,�?? Photomechanics, Topics Appl. Phys. 77, 413461 (2000)

Phys. Rev. B

V. Yannopapas, A. Modinos, and N. Stefanou, �??Anderson localization of light in inverted opals,�?? Phys. Rev. B 68, 193205 (2003)
[CrossRef]

M.A. Kaliteevski, J.M. Martinez, et al. �??Disorder-induced modification of the transmission of light in a two- dimensional photonic crystal, �?? Phys. Rev. B 66, 113101 (2002)
[CrossRef]

E. Lidorikis, M.M. Sigalas, et al. �??Gap deformation and classical wave localization in disordered two-dimensional photonic-band-gap materials,�?? Phys. Rev. B 61, 13458-13464 (2000)
[CrossRef]

S. Lan, K. Kanamoto, et al. �??Similar role of waveguide bends in photonic crystal circuits and disordered defects in coupled cavity waveguides: An intrinsic problem in realizing photonic crystal circuits,�?? Phys. Rev. B 67, 115208 (2003)
[CrossRef]

Phys. Rev. E

T. N. Langtry, A.A. Asatryan, et al. �??Effects of disorder in two-dimensional photonic crystal waveguides,�?? Phys. Rev. E 68, 026611 (2003)
[CrossRef]

A.A. Asatryan, P.A. Robinson, et al. �??Effects of geometric and refractive index disorder on wave propagation in two-dimensional photonic crystals,�?? Phys. Rev. E 62, 5711-5720 (2000)
[CrossRef]

M. Skorobogatiy, �??Modelling the impact of imperfections in high index-contrast photonic waveguides,�?? Phys. Rev. E 70, 46609 (2004)
[CrossRef]

Polym. Eng. Sci.

I. Arino, U. Kleist, et al. �??Surface Texture Characterization of Injection-Molded Pigmented Plastics,�?? Polym. Eng. Sci. 44, 1615-1626 (2004)
[CrossRef]

Proc. Instn. Mech. Engrs. Part J

D. J. Whitehouse, �??Some theoretical aspects of structure functions, fractal parameters and related subjects,�?? Proc. Instn. Mech. Engrs. Part J 215, 207-210 (2001)
[CrossRef]

Other

Developed software and analysed PC images are available at <a href="http://www.photonics.phys.polymtl.ca/PolyFIT/">http://www.photonics.phys.polymtl.ca/PolyFIT/</a>

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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)

Tables Icon

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

Tables Icon

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

Tables Icon

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

Tables Icon

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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