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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 13, Iss. 7 — Apr. 4, 2005
  • pp: 2487–2502
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Statistical analysis of geometrical imperfections from the images of 2D photonic crystals

Maksim Skorobogatiy, Guillaume Bégin, and Anne Talneau  »View Author Affiliations


Optics Express, Vol. 13, Issue 7, pp. 2487-2502 (2005)
http://dx.doi.org/10.1364/OPEX.13.002487


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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

1. Introduction

Manufacturing imperfections and tight tolerances in photonic crystal (PC) structures present great challenge on the road of transferring this promising technology into the domain of commercial applications. Much work has been done to study an impact of imperfections on the performance of PCs [1

1. K.Y. Bliokh, Y.P. Bliokh, and V. D. Freilikher, “Resonances in one-dimensional disordered systems: localization of energy and resonant transmission,” J. Opt. Soc. Am. B 21, 113–120 (2004). [CrossRef]

17

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

]. It was established quite generally that small randomness in PC geometry and/or material constants leads to an overall reduction in a band gap size, as well as an increased back scattering and radiation loss, while stronger randomness can lead to appearance of localized impurity states. In the majority of theoretical studies various simplified models of randomness are assumed. Such models are typically chosen for simplicity of parameterization of a particular type of randomness, or because some modelling methods could only handle certain types of geometries. In 1D PC multilayers [1

1. K.Y. Bliokh, Y.P. Bliokh, and V. D. Freilikher, “Resonances in one-dimensional disordered systems: localization of energy and resonant transmission,” J. Opt. Soc. Am. B 21, 113–120 (2004). [CrossRef]

, 2

2. V.M. Apalkov, M.E. Raikh, and B. Shapiro, “Almost localized photon modes in continuous and discrete models of disordered media,” J. Opt. Soc. Am. B 21, 132–140 (2004). [CrossRef]

] one typically considers disorder in the thickness and value of a dielectric constant of individual layers. In 2D planar PCs and microstructured fibers [3

3. E. Lidorikis and 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]

13

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

] one frequently considers random displacement of features from an underlying ideal lattice, disorder in a feature size (radius of a hole, for example), disorder in the refractive index, distortion of feature shapes (ellipticity), as well as roughness of walls [14

14. W. Bogaerts, P. Bienstman, and R. Baets, “Scattering at sidewall roughness in photonic crystal slabs,” Opt. Lett. 28, 689–691 (2003). [CrossRef] [PubMed]

, 15

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

] which is sometimes modelled by protrusions of some average characteristic hight and width. In 3D PCs derived from lithographical techniques and opals [16

16. 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]

, 17

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

] additional imperfections are stacking faults, and surface roughness. In all these calculations disorder parameters are scanned from small to large and conclusions are drawn about their relative impacts. Propagation parameters can be sensitive functions of disorder parameters. For example, power in the back scattered modes from wall roughness in a planar TIR waveguide scales quadratically with roughness hight (perturbation theory) and is a very sensitive function of a roughness correlation length. Thus, for a rigorous comparison of theoretical estimates with experimental observations one has to be precise about the types and statistical importance of realistic imperfections.

The goal of this paper is to understand which are the statistical parameters of importance when describing disorder in PC lattices, and then to characterize such parameters quantitatively by analyzing high resolution experimental images of 2D planar slab PCs. We find that three intuitive sets of parameters are necessary to create a comprehensive statistical model of PC imperfections. First set of parameters describe coarse properties of features such as radius, ellipticity and other low angular momenta components in a feature shape. Such coarse variations of a shape can be either deliberately designed or result from an imperfect manufacturing process. Another set of parameters describe roughness of feature edges on a nanometer scale, that is wall roughness, which is ultimately determined by the random physical processes of electron scattering in a resist, resist development and etching. A final set of parameters describes deviations of feature centers from ideal periodic lattice.

When interpreting Scanning Electron Microscopy (SEM) images one has to always keep in mind that SEM image is a convolution of an imperfect fabrication (see Appendix) and an imperfect SEM capture. Reconstruction of actual dielectric profiles from SEM images can be a very non-trivial task far beyond the scope of this paper. In the following we apply our statistical analysis to SEM images assuming that they represent true dielectric profiles. Developed statistical formulation is, however, general and can be applied to any images.

The paper in organized as follows. We first characterize coarse variations and wall roughness in individual features from which the periodic lattice is constructed. Next, we characterize an imperfect lattice formed by individual features. Finally, we evaluate errors in deduced statistical parameters due to finite image resolution. We demonstrate our approach by analyzing various high resolution images of 2D planar PCs manufactured by e-beam lithography and detailed in the following publications: InP/InGaAsP/InP [22

22. A. Talneau and M. Mulot, et al. “Compound cavity measurement of transmission and reflection of a tapered single-line photonic-crystal waveguide,” Appl. Phys. Lett. 82, 2577–2579 (2003). [CrossRef]

, 23

23. M. Mulot and 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).

], Air/InP membranes [24

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

, 25

25. C. Monat and 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]

], Air/Si membranes [26

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

, 27

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

], and SiO 2/Nb 2 O 5/SiO 2 [28

28. M. Augustin and 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]

].

2. Statistical description of feature edges

In this section we introduce a parameterization model to characterize feature shapes. It is a general property of statistical fitting for the error to decrease with the number of parameters included in the model. The challenge is to define as small as possible set of parameters and yet to capture physically significant variations in a feature shape. Here we present a “common-sense” criterion to decide on a minimal number of parameters required to decompose the shape into a regular curve plus edge roughness noise. In what follows we adapt statistical methods developed for characterization of rheology of complex surfaces [18

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

, 19

19. 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]

, 20

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

] to describe fabrication imperfections in planar PCs.

2.1. Fitting a single feature edge

First, object recognition algorithm [21

21. Developed software and analysed PC images are available at http://www.photonics.phys.polymtl.ca/PolyFIT/

] (see discussion in section 4) is used to extract circular features and their edges Fig. 1(a). We start with images of highest resolution having few features an example of which is Fig. 1, [22

22. A. Talneau and M. Mulot, et al. “Compound cavity measurement of transmission and reflection of a tapered single-line photonic-crystal waveguide,” Appl. Phys. Lett. 82, 2577–2579 (2003). [CrossRef]

] with resolution of 0.46nm. Edge of each feature is then fitted to extract coordinates of its center, radius, ellipticity and higher order Fourier components in the edge shape Fig. 1(b). Particularly, we define an edge objective function

QedgeM=1Nedgei=1Nedge(rfitM(θi)redge(θi))2,
(1)

where redge (θi ) is a distance from a feature center (X 0,Y 0) to an edge point (Xi,Yi ),

rfitM(θi)=R0+m=2M(AmMSin(mθi)+BmMCos(mθi)).
(2)

Feature radius in (2) is an m=0 term, implicit m=1 terms correspond to the feature center coordinates (X 0,Y 0), while m=2 expansion coefficients in (2) define feature ellipticity as δel=(A2M)2+(B2M)2 with an angle of a major axis defined by cos(2θel )=B2Mel . For a given M there are (1+2M) fit parameters. We fit these parameters by minimizing an objective function (1) (finding zeros of its derivatives) using standard multidimensional Newton method, with a root mean square (RMS) of a fit error defined as σ(M)=QedgeM. In what follows M≥1, where for M=1 only the feature radius and coordinates of its center are fitted.

When more Fourier components are used in a fit, fit error σ(M) monotonically decreases, and the change in the individual fit coefficients becomes smaller than sub-nanometer image resolution for even modest values of M<10. Thus, for example, for a single hole in Fig. 1

M R 0(nm) X 0(nm) Y 0(nm) δel (nm) θel (Deg)σ(M)(nm)
1123.17342.89467.88  2.97
2123.06342.88467.832.0912.622.58
8123.00342.76467.662.1712.481.46
20122.99342.74467.672.1612.481.03

Note that, in general, assumption that feature edge r edge (θi ) can be fitted with a single valued analytical curve rfitM (θ) might not be true on a small enough scale (in a particular case of Fig. 1(c) this scale is below 2nm) where rough feature edge is fractal-like. For all the analyzed pictures we find that analytical form (2) is applicable on a larger than a nanometer scale.

If there are n=[1,Nf ], Nf >1 features in the image each containing Nedgen edge points, their shapes are first fitted individually. Then, all the relevant parameters and correlation functions are averaged over the features. For example, if R0n, σn2 (M), [C,Γ,S]nM (λ) are the radius, variance and correlation functions of a feature n, then their averaged counterparts are defined as

R=Rav±δRavRav=1Nfn=1NfR0nδRav2=1Nfn=1Nf(R0nRav)2
(3)
σ2(M)=n=1NfNedgenσn2(M)n=1NfNedgen
(4)
[C,Γ,S]M(λ)=1Nfn=1Nf[C,Γ,S]nM(λ).
(5)

2.2. Coarse parameters defining feature edges

The goal of this section is to define coarse parameters that characterize feature edges “globally” such as radius, ellipticity, quadruple contribution, etc. and to establish their relative relevance. Particularly, we consider statistics of deviations of feature edges from the corresponding smooth fits when only a small number of low angular components are included in a fit. We define random variable describing the fit error as

δrM=rfitn,M(θi)redgen(θi),
(6)

where n=[1,Nf ], i=[1,Nedgen ]. Note that <δrM >=0 as it is proportional to the derivative of (1) with respect to a feature radius. Variance of δrM is given by (4). In what follows, we find that for large enough values of M distribution of the remaining edge roughness δrM can always be fitted by a Gaussian probability density distribution

P(δrM)=12πσ(M)exp(δrM)22σ2(M).
(7)
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.

In Figs. 2(c,d) we analyze an image with resolution 2.45nm of a PC lattice [26

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

] where a pattern of elliptical features of graded radii was written to form a central vertical waveguide. In Fig. 2(c) we present probability density distribution (PDD) of δrM as a function of the number of angular momenta components M included in a fit.We observe that image data (solid curves) and Gaussian distribution (dotted curves) with mean 0 and variance σ2(M) defined by (4) match well for M≥2 indicating that after including ellipticity, the error of a fit is mostly random and normally distributed. From Fig. 2(d) it is clear that for this structure ellipticity and to a lesser extent quadruple contributions are important when describing feature shapes. That is, from Fig. 2(d) we observe that inclusion of ellipticity M=2 reduces fit error by a factor of 4, inclusion of M=3 does not change fit error substantially, while inclusion of a quadruple M=4 reduces the error further by almost a factor of 2. As the number of angular components in a fit increases beyond M=4 we observe a slow decrease (power law as established later) of RMS of fit error. Statistical model of coarse parameters defining feature edges can, thus, be specified by an average radius of an underlying circle R=178.46±7.82nm, feature ellipticity δel =22.11±1.87nm, direction of an ellipse major axis θel =87.3±1.5o, and a RMS deviation of an edge from an elliptical fit σ(2)=3.42nm. Note that averaged over features ellipticity has a very small deviation from its mean, and a well-defined direction of an ellipse major axis, thus signifying that in this system ellipticity is an intrinsic property of the features.

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.

In general, relevance of coarse parameters can be judged from dependence of a fit error on the number of included angular components M. Typically, we observe that coarse parameters corresponding to the first several angular components M≤10 are of major importance, and their inclusion leads to a considerable reduction in the fit error (Fig. 2(d), 1≤M≤4). Inclusion of higher order moments leads to a slow decrease of the fit error (Fig. 2(b), M>1 ; Fig. 2(d), M>4) signifying an onset of noise-like edge roughness. After differentiating coarse variations from noise-like edge roughness we can now complete statistical description of feature edges by specifying the parameters of edge roughness.

2.3. Characterization of edge roughness

Model of disorder or roughness requires a model for the statistics of the correlation functions. While simple analytical forms of the correlation functions, such as exponential or Gaussian, are frequently used due to their integrability, we aim to derive the proper statistical forms directly from analysis of the images. Introduction of fractal dimensions allows us to develop a “family” of possible statistical distribution functions to describe roughness of features in PC lattices.

The most straightforward way of describing feature edge roughness is by considering statistical properties of a fit error function, which for a feature n is defined as

δnM(θi)=rfitn,M(θi)redgen(θi),
(8)

where n=[1,Nf ], i=[1,Nedgen ]. Values of δnM (θ) for θθi are interpolated. In our studies we used “nearest neighbor”,“linear”, and “cubic” interpolation schemes with almost identical final values of statistical parameters. While interpolation of roughness, in general, is not trivial, in our case on a several pixel scale feature edges are continuous and relatively smooth curves as they correspond to physical boundaries, thus allowing us to perform interpolation in a consistent way. In what follows, for a feature n we consider interpolated values δnM (θ) on a uniform mesh θ=[0,2π-2π/Nedgen ] with Nedgen points, which allows a straightforward use of FFT transforms.

We now consider spectral and fractal properties of roughness, which present an alternative description to the angular momenta parameter approach of the preceding section. Fractal curves are scale-invariant structures, having a similar shape independent of the scale of observation. In practice, fractal stability should cover at least two decades in order to be unambiguously identified [20

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

]. In the case of 2D PCs, even for the images with the highest sub-nanometer resolution, self-similar behavior of edge roughness extended only for a maximum of two decades in spatial wavelength. Nevertheless, fractal methodology seems to be useful for our purposes as fractal exponents inferred from spectral and fractal analysis are consistent with each other.

To introduce fractal dimension we consider Lipschitz function f (θ) having the property

f(θ+ε)f(θ)εH,ε0,
(9)

where exponent H is called Lipschitz-Holder or Hurst exponent. When H=0, f (θ) is discontinuous, while if H=1, f (θ) is differentiable. For 0<H<1, function f (θ) is continuous but not differentiable and is known as fractal.

In order to perform a fractal analysis a “height to height” correlation function is introduced. For individual features it is defined as

CnM(λ)=(δnM(θ+λR0n)δnM(θ))2θ=12π02πdθ(δnM(θ+λR0n)δnM(θ))2.
(10)

Assuming that δnM (θ) is a fractal curve with Hurst exponent H, from definition (10) it follows that when λ→0, CnM (λ) ∝ λ2H . By explicit squaring of an integrand in (10) and after minor manipulations we write

CnM(λ)=2(σn2(M)ΓnM(λ)),
(11)

where autocorrelation function is defined as

ΓnM(λ)=δnM(θ+λR0n)δnM(θ)θδnN(θ)θ2.
(12)

For large enough values of λ that exceed noise correlation length λ>λncM , ΓnM (λ)→0, and consequently, CnM (λ)→2σn2 (M). In summary, asymptotics of spectral functions are

CnM(λ)λ0λ2H;CnM(λ)λ>λncM2σn2(M)ΓnM(λ)λ>λncM0;(σn2(M)ΓnM(λ))λ0λ2H.
(13)

We find that for all the images of 2D PCs analyzed the following parameterizations (consistent with asymptotics (13)) of functions CnM (λ) and ΓnM (λ) can be used to describe statistics of noise-like edge deviation from a smooth fit

CnM(λ)=2σn2(M)(1exp((λλncM)2H))
ΓnM(λ)=σn2(M)exp((λλncM)2H).
(14)
Fig. 3. (a) “Height to height” correlation function and (b) auto-correlation function of an edge deviation from smooth fits with M angular components.

These parameterizations work especially well when λλncM , while for λλncM oscillating features persist due to aliasing effects. Note, from their definitions on a periodic domain [C,Γ](λ)=[C,Γ](-λ)=[C,Γ](2πR0n -λ), therefore we will only consider 0≤λπR0n . Moreover, for asymptotics (13) to hold correlation length of fit error has to be λncMπR0n .

In Fig. 3(a) “height to height” correlation function of an edge deviation from a smooth fit with M angular components is presented as a function of a spatial wavelengthλ. From (13) it follows that we can extract Hurst exponent of edge roughness by fitting a straight line to CnM (λ) plotted on a log-log scale when λλncM . For wavelengths larger than correlation wavelength λ≲λncM , CnM (λ) approaches a constant value. To get a reliable fit of Hurst exponent one typically needs fractal behavior to persist over several decades of λ. In all the high resolution images that we analyzed, fractal behavior persisted over one to two decades, thus making determination of Hurst exponents from scaling ofCnM (λ) somewhat imprecise. Thus, for example, from Fig. 3(a) Hurst exponent for M=1 curve is H=0.5 when 2nm<λ<20nm interval is considered, while H=0.43 when curve is fitted over the 2nm<λ<90nm interval. The upper value of this interval, however exceeds correlation length λc1 =35nm and the fit underestimates the value of a Hurst exponent. As we pointed out earlier, on a smallest scale 0.46nm<λ<2nm our description of an edge as a single valued curve of θ is not valid any longer (see Fig. 1(c)) and this region can not be used in a fit. Note, that Hurst exponent of the remaining roughness is almost constant for the values M=(1,2,4,8) somewhat decreasing from H=0.5 to H=0.45 for largerM’s. Correlation length can be determined from Fig. 3(a) using parameterization (14) from which it follows that CnMMcn )=2σn2 (M)(e-1)/e. Thus, for M=1,2,4,8 the values of correlation lengths are λncM =35nm,22nm,11.2nm,6.4nm. We notice that λncM is a decreasing function of the number M of angular components in a fit.

In Fig. 3(b) auto-correlation function of edge deviation from a smooth fit with M angular components is presented as a function of spatial wavelength λ. Correlation length can be also determined from Fig. 3(b) using parameterization (14) from which it follows that ΓnM (λncM )=σn2 (M)/e. Both “height to height” and autocorrelation functions give very similar values of correlation lengths. To demonstrate that we plot in dotted lines parameterizations of autocorrelation function (14) with H=0.43,0.5, and correlation lengths deduced from asymptotics of “height to height” correlation function, and observe a good fit. Remaining oscillatory features for λλncM are due to aliasing effectss.

An alternativeway of extracting Hurst exponents of a fractal data is using spectral techniques. First, we use a small number of angular components M to fit feature center coordinates X0n,Y0n and radius R0n by minimizing edge objective function (1). These parameters converge rapidly as M increases, and we found that M=8 gave a reliable fit for all the analyzed images. Given feature center coordinates and a radius we consider deviation δn1 (θ) of an edge from a circle, interpolate it onto a uniform grid with the same number of Nedgen points as in a discretized image of an edge θ=(0:2π/Nedgen : 2π (1-1/Nedgen ), and use Fourier representation (2)

δn1(θ)=redgen(θ)R0n=m=2Nedgen(AmSin(mθ)+BmCos(mθ)),
(15)

where coefficients Am and Bm can now be efficiently computed using standard FFT. Substituting expansion (15) into (12) we get the following expression of the autocorrelation function

Γn1(λ)=12m=2Nedgen(Am2+Bm2)cos(mλR0n).
(16)

Next, we define power spectral density function Sn1m) of edge deviation from a circle as

Sn1(λm)=12πR0n02πR0ndλ˜Γn1(λ˜)exp(i2πλmλ˜)=14(Am2+Bm2),
(17)

where λm =2πR0n/m. Alternatively, using parameterization (14) and performing integration (17) we arrive to the following scaling relation

Sn1(λm)=14(Am2+Bm2)λm0λm1+2H,
(18)

from which one can extract Hurst exponent by plotting Sn (λm ) versus λm on a log-log scale.

Another spectral method that can be used to find Hurst exponent of a fractal data involves plotting on a log-log scale RMS of edge deviation from a smooth fit σn(M) as a function of the number of angular components M in a fit (same plot as Fig.2(b,d) but for all the values of M=(2:Nedgen )). In principle, to calculate σn(M) we have to first solve minimization problem (1) that includes (1+2M) fit parameters, which for even moderate values of M>10 becomes time consuming. A considerably faster way to evaluate σn(M) even for large M is to assume that coefficients AmM,BmM in expansion (2) are independent of M. Then, in the same way as in calculating power spectral density we first find all the expansion coefficients in (15) using FFT. Then, δnM (θ) can be expressed via expansion coefficients (2) as

δnM(θ)=m=M+1Nedgen(AmSin(mθ)+BmCos(mθ)).
(19)

Taking into account scaling (18) we find that for M large enough the following holds

σn2(M)=(δnM(θ))2θ=12m=M+1Nedgen(Am2+Bm2)MH.
(20)

Finally, when several features are present in the image we extract Hurst exponents from the averaged statistical functions (4).

In solid blue lines in Fig. 4(a,b) we present power spectral density and RMS of fit error for the same PC as in Fig. 1(a). Spectral density in Fig. 4(a) can be well fitted by a straight line over 2 decades starting from the largest length scale justifying our earlier finding that features in this PC can be characterized simply by an average radius and remaining roughness around a circular fit. Because only one feature is considered in this image the data is somewhat noisy and a range of Hurst exponents HS =0.45-0.6 is possible. RMS of fit error in Fig. 4(b) can be fitted by a straight line starting from the lowest angular momentum M=1 all the way to M=300 with a Hurst exponent Hσ =0.45. In solid red lines we present spectral functions of estimated discretization noise due to finite image resolution (for more discussion see section 4). In solid blue lines in Fig. 4(c,d) we present power spectral density and RMS of fit error for the same PC as in Fig. 2(c). Power spectral density in Fig. 4(c) can be well fitted by a straight line over 1 decade in the interval 30nm≲λ≲200nm giving an estimate of the Hurst exponent HS =0.3. RMS of fit error in Fig. 4(d) can be fitted by a straight line for angular momenta M > 4 all the way to M=40 with a Hurst exponent Hσ =0.28. As it was established earlier, ellipticity and a quadruple component are important contributions in the shape of the features, which is also clearly visible from Fig. 2(d), where power dependence of a spectral function is clearly observed only for M>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.

Following discussions of this section we now present in Tables 1,2 several parameterizations of features in PCs considered in Figs. 1(a), 2(c). Parameters Rav, δel, θel, Am, Bm -the radius of an underlying circle, ellipticity and other higher order moments of importance, are the coarse parameters of feature shapes averaged over features, whileσ (M), λcM , HC,S,σ describe statistical properties of the remaining roughness (edge deviation from a coarse fit).

3. Statistical description of feature lattices

In this section, we investigate variations in the feature positions from an ideal periodic lattice.

In Fig. 5(a) we present an image with resolution 5.54nm containing a PC lattice of 437 holes and a waveguide made of two rows of missing holes [22

22. A. Talneau and M. Mulot, et al. “Compound cavity measurement of transmission and reflection of a tapered single-line photonic-crystal waveguide,” Appl. Phys. Lett. 82, 2577–2579 (2003). [CrossRef]

]. At first, coordinates of the hole centers r¯0n =(X0n,Y0n ) are found by minimizing objective function (2) for various values of M. It was found that statistics of deviations of the hole centers from an underlying perfect lattice is not sensitive to a particular choice of M, and in what follows we choose M=3. Parameters of an underlying perfect lattice are then found by minimizing lattice objective function

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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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.
Qlat=1Nfn=1Nf(r̅0nj1na̅1j2na̅2)2,
(21)

where ā1,2 are the basis vectors of an underlying perfect lattice, and j1,2n are the integer lattice coordinates of an n’s hole center. It is relatively straightforward to find 2N f integer coordinates in (21) analytically given a reasonable approximation to the basis vectors, thus leaving 4 continuous parameters in ā1,2 to be fitted. As before, we perform a fit with multidimensional Newton method by minimizing the value of a fit variance function Qlat . In Fig. 5(a) vertices of a fitted perfectly periodic underlying lattice are shows as white dots.

We now define a 2D random variable δ¯c=r¯0nj1nā1j2nā2 which we assume to be 2D Gaussian distributed with PDD

P(δ̅c)=12πσ1σ2exp((δcxδcy)TRT(12σ120012σ22)R(δcxδcy)),
(22)
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.

where R=(cos(θ)sin(θ)sin(θ)cos(θ)) is a 2D rotation matrix, and σ1,2 are the variances along the two principle directions. One can deduce statistical parameters σ1,2,θ by using the following averages of a 2D Gaussian random variable, <δcxδcx >=σ12 cos2 (θ)+σ22 sin2 (θ), <δcy δcy >=σ12 sin2 (θ)+σ22 cos2 (θ), <δcxδcy >=2cos(θ)sin(θ)(σ12σ22 ).

In Fig. 5(b) we plot PDD of δ¯c along the two principle directions (θ=-1.6°) from the lattice fit (solid lines) and a corresponding Gaussian distribution (dotted lines). We find that a 2D distribution of feature center displacements from the vertices of an underlying perfect lattice indeed appears to be Gaussian and is highly anisotropic. The RMS of the hole center deviations from a perfect lattice is twice as large σ1=6.4nm in the direction perpendicular to the waveguide than in the parallel direction σ2=2.9nm. In Fig. 5(c) we investigate in more details the source of such an anisotropy. RMS deviations σ1,2 (along 2 principle directions) of hole centers from an underlying lattice are plotted against the number of features in a fit. Features in consecutive fits are added one by one, row by row starting from the upper left corner of a structure Fig. 5(a). At leat two rows are needed to fit both basis vectors. When only few features are included in a fit, parameters σ1,2 grow rapidly with each included feature, finally “saturating” when 50 features are included (2 rows). We observe, quite generally, that at least 10–20 features in each row are needed to determine parameters of a Gaussian fit reliably. When more then 5 layers are included in the fit, and while approaching a waveguide region, hole center deviations from an underlying lattice become highly anisotropic. As we have mentioned earlier, such an anisotropy can appear for non rectangular lattices. From analysis of various PC images we observe that anisotropy in the deviations of feature centers from an underlying perfect lattice is predominantly observed in the structures with symmetry breaking elements such as waveguides, bends, etc. For PC lattices with waveguides, for example, we find that frequentlyσ12, where σ1 is RMS of feature center deviations in the direction perpendicular to a waveguide, while σ2 is RMS of deviations in the direction parallel to a waveguide. Uniform rectangular PC lattices without functional elements are typically isotropic σ12. Most likely physical reason for such an anisotropy being varying along a non-uniform PC lattice e-beam proximity effects due to non-uniform local environments (see Appendix).

Image with resolution 1.63nm of a uniform square PC lattice containing 204 holes [26

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

] is presented in Fig. 6(a). Dependence of RMS parameters σ 1, σ2 along two principal directions for increasing number of features in a fit is plotted in Fig. 6(b). Features in consecutive fits are added one by one, column by column starting from the upper left corner of a structure. One observes that σ12~1.6nm for any number of features in a fit, “saturating” to their stationary values after two rows (~20 features) are included. In Table 3 we present a complete statistical model of feature shapes and feature center distribution from ideal lattice for Fig. 6(a).

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

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4. Discretized images and associated errors

In our edge detection algorithm, we first sort all the image pixels into two categories the ones that belong inside of a hole and the ones that belong to a substrate. We work with normalized grayscale images where the value of each pixel pix is between 0 (black-hole) and 1 (white-substrate). First, a 3–5 pixel convolution (“smoothing”) is applied to each image to reduce noise. To sort the pixels we compare their values to a threshold parameter tol, if pix<tol we consider a pixel to be in the hole and assign it a value 0, while in the opposite case we assign it a value 1. As different images have different contrast and noise levels to find a reasonable value of a tol parameter we use a histogram of pixel values. In Fig. 7(a) we show an image with resolution res=1.25nm of a hole [24

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

] and a histogram of pixel values in the insert. This image features a moderate index contrast (pixel value ratio for the two maxima corresponding to white and black is 0.6:0.1) and a relatively high noise (region 0.2-0.4 of substantial number of pixels with values in between two maxima). For edge detection we try several values of a threshold parameter tol=0.37,0.40,0.43 around the local minima of a histogram in between two maxima (see insert). Once all the pixels are sorted into two groups (with values 0 or 1) hole edge is detected using a standard method of image convolution with a Sobel-like 3×3 matrix. In Figs. 7(b,c,d) we present edges detected by using three different threshold parameters. As seen from the images, detected edges are somewhat different from each other.

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

An important question is how sensitive are the statistical parameters characterizing feature imperfections with respect to variations in tol. Detailed simulations show that for a Δtol/tol~15% change in a threshold parameter, statistical parameters only vary as ΔRav/Rav ~2%, Δδel /δel ~12%, Δσ (2)/σ (2)<1%, ΔHS,σ /HS,σ ~15%, Δσ1,21,2~5%. Note that for this image, the parameter affected most by variations in tol is a Hurst exponent of a remaining wall roughness. This is generally expected in the case of a noisy image where detected roughness on the smallest case (several nm) is strongly affected by image noise, and hence a value of a tol parameter. Thus, when the same sensitivity analysis is repeated for a higher resolution, lower noise image Fig. 1(a) we find that for a~10% change in a tol parameter Hurst exponents changes only by a few percents. We conclude that while there is indeed an uncertainty associated with a choice of a threshold parameter, the resultant values of the statistical parameters are weakly sensitive functions of tol given a good quality image. Moreover, these uncertainties can be further reduced by lowering the image noise level and increasing contrast.

Next, we estimate the level of discretization noise due to finite resolution of an image (red curves in Fig. 4). Particularly, given an image we first fit its coarse parameters R 0,X 0,Y 0,Am,Bm for a certain small number of angular momenta m=[2,M], M=3-8. Then, we consider an analytical curve rfitM (θ) with thus found fitted parameters (2) centered around X 0=0,Y 0=0. Next, we introduce a uniform mesh with the same resolution as that of an analyzed image and project an analytical edge onto a discrete mesh to get a discretized approximation of an edge rmeshM (θ). Then, we take a difference between an analytical curve and its discretized version δnoise (θi )=(rmeshM (θi )-rfitM (θi )) to estimate the level and statistics of a nose due to image discretization (red curves in Fig. 4). For all the images analyzed we find that RMS of δnoise (θ) is on the order of 0.4·res in the worst case, contributing most to the uncertainties in a wall roughness parameter σ (M). Moreover, when a discretized edge is fitted by minimizing (1), the error in the coarse parameters due to discretization is even smaller. Thus, when comparing the center coordinates X0mesh,Y0mesh by fitting a discretized edge with a circle and the exact center coordinates X0=0,Y0=0, we find that their maximum discrepancy over all the images and features did not exceed 0.1·res. Thus, uncertainties in the parameters σ1,2 of RMS deviation of feature centers from a perfect lattice are also at most 0.1·res.

5. Discussion and conclusions

We find that at least three sets of parameters are necessary to create a minimal statistical model of 2D disorder in PC lattices. First set of parameters describes coarse properties of individual shapes persistent over all features such as radius, ellipticity and other low angular momenta, among which radius is the most important. Another set of parameters describes higher angular momenta plus random edge roughness by a set of correlation functions (14) with parameters λcM , σ (M) and H corresponding to correlation length, standard deviation and Hurst exponent. Typically, unless written deliberately, we find that even low angular momenta components (such as ellipticity) are not persistent from one feature to another and can be simply described as part of a random edge roughness. A final set of parameters describes deviations of feature centers from an ideal periodic lattice in terms of a 2D Gaussian distribution parameterized by two principal directions and two variances σ1,2 along such directions. For PC lattices with symmetry breaking elements such as waveguides, bends, etc. we find that due to non-uniform e-beam proximity effects feature position disorder is frequently anisotropic σ1≫σ2.

Findings of this paper are based on the analysis of over 30 high resolution pictures of the “typical” e-beam written structures of various material combinations. In Table 4 we present the values (within 2 standard deviations from the averages) of various statistical parameters averaged over all analyzed images with resolutions 0.46nm-6nm. A somewhat surprising finding is that despite all the different material combinations from which these PC lattices are made a relatively narrow distribution of statistical parameters characterizing disorder is found.

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

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Appendix

We demonstrate physical processes responsible for disorder in PC lattices on the example of a particular fabrication process using direct e-beam writing in InP/InGaAsP/InP materials [22

22. A. Talneau and M. Mulot, et al. “Compound cavity measurement of transmission and reflection of a tapered single-line photonic-crystal waveguide,” Appl. Phys. Lett. 82, 2577–2579 (2003). [CrossRef]

].

One starts with a semiconductor multilayer where top InP layer is 200nm thick, followed by a 450nm thick optically guiding GaInAsP layer, then a 2µm thick buffer InP layer on the bottom, and, finally, an InP substrate. Processing steps are as follows: first, on the top of an InP layer one deposits a 250nm thick SiO 2 layer, then on the top of a SiO2 layer one deposits a 300nm thick PMMA polymer layer. After that, circular features are developed in PMMA with e-beam writing. The short penetration length of electrons precludes the use of a solid SiO 2 substrate as a mask directly. Next, SiO2 layer is dry etched with CHF3 using the PMMA layer as a mask. Finally, PMMA is removed and SiO2 layer is used as a mask for chemically assisted ion beam etching [23

23. M. Mulot and 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).

] of holes. Resulting holes are typically etched 3–4µm deep. The resultant structure is the etched PC lattice where roughness presents an accumulated effect of several fabrication steps: PMMA development by e-beam writing, SiO2 etching and semiconductor multilayer etching. Overall, it seems that the resultant roughness is less a function of a particular material combination but rather the details of a fabrication process. For example, an alternative process to create a PC lattice resulting in higher roughness would be a so-called lift-off process. In this process one deposits a PMMA layer directly on the top of an InP layer. Then, the complementary of holes are developed by e-beam writing and a continuous metallic layer is deposited. When washed by a solvent remainingPMMA dissolves by leaving a metallic mask for the holes while somewhat tearing the metallic mask layer near the hole edges.

A typical e-beam writing strategy for planar PC lattices is direct writing : a beam of electrons of a given diameter, moves pixel by pixel in x-y directions with a step size as small as 2.5nm. A feature boundary is typically coded as a polygon of many vertices (18 in this case). This polygon is then subdivided by software into elementary shapes such as rectangles and triangles for further exposure. If lattice is not rectangular than it becomes impossible to resolve exactly non-integer coordinates of the feature centers, which could introduce larger deviations of feature centers from an ideally periodic lattice along certain spatial directions. As the electrons penetrate into the resist material a considerable number of them experience large angle scattering leading to backscattering, thus causing additional exposure in the resist and what is called the electron beam proximity effect. Roughness introduced by e-beam proximity effects in the PMMA resist is theoretically estimated to be on the order of 3–10nm, however the measured roughness is typically smaller. If no software to compensate for proximity effects is used then exposure conditions for points in different local environments will be different. This can result in measurable distortions for the features located near the symmetry breaking features such as corners, waveguides, bends, resonators, etc. by comparison with features inside of a uniform periodic lattice. Finally, distortions in the shape of a feature (such as ellipticity of a hole) could also come from a non optimal setting of an imaging SEM in a form of a “residual astigmatism”, which could also make roughness look erroneously larger along certain spatial directions.

Acknowledgments

We would like to thank various experimental research groups and individuals including K. Srinivasan, P. Barclay, O. Painter, M. Augustin, A. Xing, E.L. Hu, C. Seassal, J. Vuckovic, and T. Charvolin who have expressed their interest in our work and contributed high resolution images. We would also like to acknowledgeY. Vlasov, S. Mingaleev and our referees for fruitful discussions and comments.

References and links

1.

K.Y. Bliokh, Y.P. Bliokh, and V. D. Freilikher, “Resonances in one-dimensional disordered systems: localization of energy and resonant transmission,” J. Opt. Soc. Am. B 21, 113–120 (2004). [CrossRef]

2.

V.M. Apalkov, M.E. Raikh, and B. Shapiro, “Almost localized photon modes in continuous and discrete models of disordered media,” J. Opt. Soc. Am. B 21, 132–140 (2004). [CrossRef]

3.

E. Lidorikis and 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]

4.

A.A. Asatryan and 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]

5.

M.A. Kaliteevski and 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]

6.

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

7.

B.Z. Steinberg, A. Boag, and R. Lisitsin, “Sensitivity analysis of narrowband photonic crystal filters and waveguides to structure variations and inaccuracy,” J. Opt. Soc. Am. A 20, 138 (2003). [CrossRef]

8.

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]

9.

S. Lan and 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]

10.

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

11.

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

12.

N.A. Mortensen and 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]

13.

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

14.

W. Bogaerts, P. Bienstman, and R. Baets, “Scattering at sidewall roughness in photonic crystal slabs,” Opt. Lett. 28, 689–691 (2003). [CrossRef] [PubMed]

15.

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

16.

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]

17.

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

18.

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

19.

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]

20.

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

21.

Developed software and analysed PC images are available at http://www.photonics.phys.polymtl.ca/PolyFIT/

22.

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

23.

M. Mulot and 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).

24.

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

25.

C. Monat and 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]

26.

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

27.

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

28.

M. Augustin and 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]

OCIS Codes
(030.5770) Coherence and statistical optics : Roughness
(100.2960) Image processing : Image analysis
(100.5010) Image processing : Pattern recognition
(130.1750) Integrated optics : Components
(130.3120) Integrated optics : Integrated optics devices

ToC Category:
Research Papers

History
Original Manuscript: January 31, 2005
Revised Manuscript: March 17, 2005
Published: April 4, 2005

Citation
Maksim Skorobogatiy, Guillaume Bégin, and Anne Talneau, "Statistical analysis of geometrical imperfections from the images of 2D photonic crystals," Opt. Express 13, 2487-2502 (2005)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-7-2487


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References

  1. K.Y. Bliokh, Y.P. Bliokh, and V. D. Freilikher, �??Resonances in one-dimensional disordered systems: localization of energy and resonant transmission,�?? J. Opt. Soc. Am. B 21, 113-120 (2004) [CrossRef]
  2. V.M. Apalkov, M.E. Raikh, and B. Shapiro, �??Almost localized photon modes in continuous and discrete models of disordered media,�?? J. Opt. Soc. Am. B 21, 132-140 (2004) [CrossRef]
  3. 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]
  4. 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]
  5. 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]
  6. K.-C. Kwan, X. Zhang, et al. �??Effects due to disorder on photonic crystal-based waveguides,�?? Appl. Phys. Lett. 82, 4414-4416 (2003) [CrossRef]
  7. B.Z. Steinberg, A. Boag, and R. Lisitsin, �??Sensitivity analysis of narrowband photonic crystal filters and waveguides to structure variations and inaccuracy,�?? J. Opt. Soc. Am. A 20, 138 (2003). [CrossRef]
  8. 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]
  9. 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]
  10. T. N. Langtry, A.A. Asatryan, et al. �??Effects of disorder in two-dimensional photonic crystal waveguides,�?? Phys. Rev. E 68, 026611 (2003) [CrossRef]
  11. A.G. Martyn, D. Hermann, et al. �??Defect computations in photonic crystals: a solid state theoretical approach,�?? Nanotechnology 14, 177183 (2003)
  12. 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]
  13. M. Skorobogatiy, �??Modelling the impact of imperfections in high index-contrast photonic waveguides,�?? Phys. Rev. E 70, 46609 (2004) [CrossRef]
  14. W. Bogaerts, P. Bienstman, and R. Baets, �??Scattering at sidewall roughness in photonic crystal slabs,�?? Opt. Lett. 28, 689-691 (2003) [CrossRef] [PubMed]
  15. 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]
  16. 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]
  17. V. Yannopapas, A. Modinos, and N. Stefanou, �??Anderson localization of light in inverted opals,�?? Phys. Rev. B 68, 193205 (2003) [CrossRef]
  18. D. J. Whitehouse, �??Surface Characterization and Roughness Measurement in Engineering,�?? Photomechanics, Topics Appl. Phys. 77, 413461 (2000)
  19. 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]
  20. I. Arino, U. Kleist, et al. �??Surface Texture Characterization of Injection-Molded Pigmented Plastics,�?? Polym. Eng. Sci. 44, 1615-1626 (2004) [CrossRef]
  21. 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>
  22. 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]
  23. 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).
  24. 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]
  25. 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]
  26. 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]
  27. H. Altuga and J. Vuckovic, �??Two-dimensional coupled photonic crystal resonator arrays,�?? Appl. Phys. Lett. 84, 161-163 (2004) [CrossRef]
  28. 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]

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