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Cantor diffractals are waves that have encountered a Cantor grating. In this paper, we report an important property of Cantor diffractals, namely that of redundancy. We observe that the Fraunhofer diffraction pattern comprises of several bands, each containing complete information about the fractal aperture. This redundancy allows for a faithful reconstruction of the Cantor grating by an inverse Fourier transformation of an arbitrary band.
© 2012 Optical Society of America
1. Introduction
Fractals are geometric objects with regular or stochastic hierarchic structure which persists at all levels of magnification. As a result, they exhibit self-similar and scale-invariant properties [1, 2]. They provide a description for many forms in Nature such as coastlines, trees, blood vessels, turbulence, dielectric breakdown, etc. where Euclidean geometry is inappropriate. A surprising feature of fractals is that their dimension is non-integer, greater than the topological dimension but less than the Haussdorff dimension [1, 3]. A coastline for example thus has a fractal dimension between 1 and 2.
Diffractals, as defined by Berry, are waves that have encountered a fractal aperture [4]. Their short-wavelength regime allows an exploration of the ever finer levels of the fractal structure. Since geometric optics is not applicable on these length scales, scattering methods hold a special appeal for the study of self-similar objects. Diffractals propagate in space and time to produce the diffraction pattern of the fractal aperture.
We report a novel property of diffractals, namely that of redundancy. Though well studied in the context of optical holography, its presence in diffractals has been unnoticed and unexplored thus far. Literally, redundancy indicates superfluous information that can be omitted without loss of meaning. In holograms, information about a recorded interference pattern of light generated from a reference beam and an object beam is spread all across [5,6]. As a consequence, the entire image can still be reconstructed from scratched, dirty or fragmented holograms without serious loss of information.
The Cantor grating is a typical example of a regular, rectangular fractal. The interaction of electromagnetic waves with it yields Cantor diffractals [7–9,12]. We first define the generator of the n-th generation Cantor grating and calculate the corresponding diffraction pattern. It consist of several “bands” which we define shortly. Next, we demonstrate redundancy. This premise is based on our observation that complete information of the fractal aperture is repeated in any and every band of the diffraction pattern. It can therefore be reconstructed from an arbitrary band by inverse propogation. Such a band excludes the zero spatial frequency component and does not have a significant energy content in it. Notwithstanding, this spatial filtering continues to yield a faithful reconstruction of the aperture and is not masked by edge-enhancement effects expected in Fourier optics [10,11]. We also observe that the band contains information about all the previous generations of the Cantor grating. Interestingly, these can be reconstructed from smaller and smaller nested-clips. And a window of width 32, in a band whose width is 3n, is sufficient to yield the fractal generator! So for example in the case n = 4, information in a tenth of the band suffices for this purpose.
The paper is organized as follows. In Section 2, we present the mathematical formulation required in the context of Cantor diffraction. Section 3 contains our primary results demonstrating redundancy in Cantor diffractals. A summary of our findings and their potential applicability is discussed in Section 4.
2. Mathematical formulation
Let Ro(x) = rect(ε0 = 2a, x = 0), where rect(w, x) is a rectangle function of width w placed symmetrically about point x. Defining Rn(x) = rect(εn = εn−1/3, x = 0), the first generation Cantor grating is then given by
where * denotes the convolution operation and with xi = ±2a/3. Generalizing Eq. (1) results in the n-th generation Cantor grating where with xi = ±2a/3 ± 2a/32 ± · · · ± 2a/3n.The corresponding diffraction pattern can be obtained by a Fourier transformation of Eq. (2) using the convolution theorem [11]. Its amplitude is given by
where f is the spatial frequency of the scattered wave having dimensions of inverse length and sinc(x) = sin(x)/x. Under coherent illumination, the diffraction pattern can also be presented in terms of the diffraction angle θ. The two presentations are related by the expression sinθ = λf, where λ is the wavelength of the incident radiation. For the purpose of our formulation however, the spatial frequency domain f is more appropriate. Subsequently, without loss of generality, we have normalized λ to 1. Scattering experiments however, measure the intensity It is simple algebra to check that the intensity scales as In(f) = (2/3)2 In−1 (f/3) [13–15]. The corresponding structure factor is given by: The fractal dimension D can be obtained from the integrated structure factor [12, 16, 17]:We now present some of the more familiar properties of fractals in general and Cantor fractal in particular. Figure 1(a) depicts the n = 4 Cantor grating generated from Eq. (2). We have chosen 2a = 1 here, and in all subsequent evaluations, for convenience. The corresponding diffracted field, obtained from Eq. (3), is shown in Fig. 1(b). The scaled intensity profiles for a few generations (n = 1,2,3 and 4) are shown in Fig. 1(c). They demonstrate self-similarity and scale-invariance of Cantor diffractals [18]. Figure 1(d) shows the structure factor, S(f) vs. f, for n = 4 on a log-log scale. The structure factor too is self-similar [26]. In Fig. 1(e), we plot the corresponding integrated structure factor, 〈S(f)〉 vs. f, for Δ = 1/3. The slope of the best fit line yields D ≈ 0.63 as expected for the Cantor grating [1, 2].
Fig. 1 (a) Cantor grating with n = 4, (b) corresponding diffraction pattern, (c) scaled intensity profiles for n = 1,2,3,4, (d) structure factor and (e) integrated structure factor.
3. Redundancy in Cantor Diffractals
In this section, we focus on establishing redundancy in Cantor diffractals and consequently in the diffraction profile. A schematic representation of the optical set-up for this purpose is presented in Fig. 2. In the “4F configuration” depicted, coherent illumination incident on Cantor grating G yields the Fraunhoffer diffraction pattern on the plane D. The lens L1 here performs the operation of the Fourier transformation. The aperture A placed in front allows only a part of the diffraction pattern to be incident on the lens L2. The passage through L2 results in yet another Fourier transformation thereby reconstructing the Cantor aperture on screen R.
Next, we explain the formation of bands and their significant features. Notice the well-separated regions with prominent amplitude variation in the diffraction pattern of Fig. 1(b). We call them bands. They are not unexpected and arise due to the zeros of the sinc function which occur at f = ±no3n; no = 1, 2,... (see Eq. (3)). The internal structure of the bands is a result of diffractal interference, mathematically represented by the product of cosines in Eq. (3). The distinctive features of the bands is their identical, self-similar and scale-invariant (internal) structure. Some attributes of the product which lead to the above properties are the following: (i) It contains information about interference of diffractals from all n generations of the Cantor grating. Therefore, information of all n length scales in the fractal grating is contained in the band. (ii) The zeros of the m-th cosine term are given by fm = ±(2mo + 1)3m/4; mo = 0, 1, 2,... As fm = 3fm−1, the zeros of orders m′ < m are nested in those of order m. Consequently, the periods of all the cosine terms are commensurate. These features are seen in Fig. 3, where we plot the cosine terms for the n = 4 case in the window f = [34,2×34] corresponding to the first secondary band on the right of the central band. The relevant portion of the sinc function and the resulting intensity profile are also shown for completeness.
The other important characteristic, viz., the fractal dimension, can be obtained from the integrated structure factor of any band. This is also evident in Fig. 1(d). A band therefore has complete information of the diffracting fractal aperture. We can thus expect an accurate reconstruction of the later by an inverse Fourier transformation of a fragment of the diffraction field, vis-a-vis the entire pattern in conventional gratings. We discuss these reconstructions for the Cantor grating next.
In the set of Figs. 4(a)–4(d), we show the reconstructed Cantor grating from portions of the first secondary band. We used the entire band to reconstruct order 4, and the energy content enclosed between zeros of cos(2f/34), cos(2f/33) and cos(2f/32) to obtain reconstructions of order 3, 2 and 1 respectively. These nested intervals are indicated by dashed lines in the intensity profile of Fig. 3. Notice that the generator of the Cantor grating can be reconstructed from just 1/10 th of the energy content in the band. We have tried such reconstructions from several bands. Reconstructions from distant bands are faint, but the information content is unaltered.
Fig. 4 Results using Virtual lab: Row 1 depicts the n = 4 cantor grating and its diffraction pattern. Row 2 and 3 show reconstructions and the corresponding intensity profiles.
In Fig. 5 we demonstrate the results of Fig. 4 using VirtualLab 4 software from LightTrans GmbH [20]. We wish to point out that VirtualLab software provides efficient simulations of optical phenomena with high physical accuracy. It is customarily used when laboratory experiments are unavailable. The top row depicts the Cantor grating of order 4 (left) and its diffraction pattern (right). The pictures in the next row are reconstructions akin to those in Figs. 4(a)–4(d) using VirtualLab. The corresponding intensity profiles are also shown in the row below. A point of concern could be the lack of exactness in the reconstruction of Cantor gratings. In this regard, it should be noted that the maximum intensity peaks in the reconstruction are correctly placed. This information is sufficient not only for the precise determination of fractal length scales, but also for the evaluation of the fractal dimension [1, 3].
4. Summary and conclusions
We conclude this paper by summarizing our main results. We find that a Cantor grating can be reconstructed from secondary bands of the Cantor diffraction pattern. Being far away from the zero-spatial frequency component, these bands have an insignificant energy content in them. Nevertheless, they provide a faithful reconstruction of the fractal grating. In fact the fractal generator and the fractal dimension can be obtained from yet smaller fragments of the secondary bands. The repetitive information or redundancy is absent in diffraction patterns generated by non-fractal apertures. Analogous reconstructions here, are strongly affected by edge-enhancement, a known evil aroused due to the exclusion of the zero frequency [11]. We attribute this novel distinction to the presence of many, but self-similar length scales which camouflage the effects of edge enhancement in reconstruction from bands. We emphasize that redundancy is a generic feature of diffractals, especially when they emanate from deterministic fractals [21].
Fractals are ubiquitous in nature. Their signature resides in diverse physical settings such as coastlines, snowflakes, growth phenomena, chaos, etc. [1, 3, 22, 23]. Diffraction provides a convenient tool to probe the properties of this fascinating geometry which persists at all length scales. Of late, the presence of fractal architecture in diffractive elements such as gratings, zone plates and lenses is receiving considerable attention [18, 22–24]. These are essential in image forming set-ups such as tomography, soft x-ray microscopy and lithography, optical tweezers, spatial light modulators and multifocal contact lenses to name a few. It has been proved that diffraction from fractal apertures yields images with improved depth fields. So far, the focus in this technologically important area has primarily been on obtaining sharp diffraction profiles. Though an inherent property, redundancy in fractal diffraction has remained unnoticed so far. This redundancy can be useful in many of the above applications, especially when restoration from partial information is required. For example, it can be of assistance in the identification and classification of the non-deterministic fractal morphologies which result in stochastic growth phenomena. Or in image reconstruction, when there is loss of information due to optical aberrations and disorder. We hope that this study provokes experiments on the utility of redundancy in these applications.
Acknowledgments
We thank S. Puri and K. Thyagarajan for useful suggestions. Financial support from CSIR grants 086(0951)/09/EMR-I and 03(1077)/06/EMR-II is also acknowledged.
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