## Abstract

The Fibonacci grating (FbG) is an archetypal example of aperiodicity and self-similarity. While aperiodicity distinguishes it from a fractal, self-similarity identifies it with a fractal. Our paper investigates the outcome of these complementary features on the FbG diffraction profile (FbGDP). We find that the FbGDP has unique characteristics (e.g., no reduction in intensity with increasing generations), in addition to fractal signatures (e.g., a non-integer fractal dimension). These make the Fibonacci architecture potentially useful in image forming devices and other emerging technologies.

© 2014 Optical Society of America

Mathematically, numbers in the Fibonacci sequence can be obtained by the recurrence relation ${F}_{n}={F}_{n-1}+{F}_{n-2}$ with seed values ${F}_{0}=1$ and ${F}_{1}=1$ [1,2]. In the limit $n\to \infty $, the ratio between successive numbers converges to the *golden ratio* $\varphi =(1+\sqrt{5})/2$. The Fibonacci sequence can also be constructed by a *replacement rule*: $0\to 01$ and $1\to 0$, starting with 1. The first few generations of the Fibonacci sequence are therefore: ${f}_{0}=1$, ${f}_{1}=0$, ${f}_{2}=01$, ${f}_{3}=010$, ${f}_{4}=01001$, ${f}_{5}=01001010$, and so forth. The number of elements in a generation is the Fibonacci number corresponding to it, i.e., $|{f}_{n}|={F}_{n}$. Further, the ratio of the number of 1’s to 0’s is again the golden ratio. As this number is irrational, no finite portion of the Fibonacci sequence will have exactly the same ratio of 1 to 0 as the infinite sequence. As a consequence, repeating a finite sequence indefinitely cannot produce the infinite Fibonacci sequence. Such a sequence, which is predictable but not periodic, is referred to as aperiodic or quasi-periodic.

Fibonacci diffracting elements (FbDE) (e.g., gratings, lenses, zone plates, *etc.*) can be created from a Fibonacci sequence by associating 1 with a transparency of length ${l}_{1}$ and 0 with an opacity of length ${l}_{2}$. Sometimes, they appear naturally in physical systems. A striking example are the icosahedral phases in metallic alloys or *quasi-crystals* [3,4]. In this case, the Fibonacci lattice provides a prototypical model to explain the five-fold symmetry in the diffraction pattern and their unusual electronic properties and energy spectrum. Of late, Fibonacci lenses are being used for generating arrays of optical vortices to trap micro-particles and set them into rotations to drive optical pumps [5,6]. Further, fiber Bragg gratings with Fibonacci sequences have been found to transform evanescent waves into propagating waves for far-field super-resolution imaging [7,8]. Thus, FbDE are finding a large number of new applications in diverse areas of technological importance.

In a related context, fractal diffracting elements have been more studied in the recent past [9–12]. They are being suggested as excellent replacements for conventionally used *regular* diffractive elements in image forming devices, especially under polychromatic, wide-band illumination [13–16]. Fractals exhibit self-similarity as well as scale-invariance and are characterized by a fractal dimension. The FbDE, on the other hand, are quasi-periodic and self-similar. They cannot be categorized as fractals because of the lack of scale-invariance in their structure. But fractal signatures are expected because of the element of self-similarity. It is therefore intriguing to probe the similarities and differences between FbDE and their fractal counterparts to mainstream emerging technologies based on them. There are many questions of relevance in this context. For example, is there a fractal dimension associated with FbDE? How similar is their diffraction pattern to *diffractals* or electromagnetic waves emanated by fractal apertures? Do FbDE have improved imaging capabilities? In this paper, we provide answers to some of these questions for the *Fibonacci grating* (FbG)—an archetypal example of aperiodicity and self-similarity.

The main results in our paper are as follows. We found that the FbG diffraction pattern (FbGDP) has many features of diffractals. It was comprised of self-similar bands, each containing complete information about the FbG. This *redundancy* can therefore be used for reconstruction of the FbG by an inverse Fourier transformation (FT) of a secondary band, although it excludes the zero-frequency component and contains an insignificant energy content in it. But this redundancy was limited as compared to fractals, where reconstructions are possible even from small fragments of the band [11]. The integrated structure factor calculated from the FbGDP yielded a fractal dimension of ${d}_{f}\approx 0.88$. Further, like in diffractals, the FbGDP also showed robustness to malformations or disorder in the aperiodic aperture. For higher generations, the fractal dimension was unaffected even if approximately 50% of the transparencies and opacities had an incorrect placement in the Fibonacci sequence. A significant distinction from fractal diffraction was that the intensity of bands did not decrease for higher generations. This, in conjunction with robustness, is consequential for optical devices based on the Fibonacci architecture. For example, the inherent (manufacturing) defects in higher order FbG will still provide intense images. In fact, our study suggest that the FbG is better suited for polychromatic imaging than its fractal counterparts. We emphasize that our analyses are generic and applicable to a variety of aperiodic diffracting elements. Further, the distinction between aperiodic and fractal optical elements has been fuzzy in the literature. Our observations demarcate with precision the two classes of diffracting elements that hold promise in image forming devices.

Let ${R}_{0}(x)=\text{rect}({\u03f5}_{0}=2a)$ be a rectangle function of width ${\u03f5}_{0}$ placed symmetrically about point $x$. Let ${R}_{n}(x)=\text{rect}({\u03f5}_{n}=2a/{F}_{n})$ where ${F}_{n}$ is the Fibonacci number corresponding to the $n$-th level. The first few generations ${G}_{n}(x)$ of the FbG can be written as

A few generations of the FbG using Eq. (2) are depicted in Fig. 1. The corresponding diffraction pattern can be obtained by taking the FT of the grating function in Eq. (2) and using the convolution theorem. Its amplitude is given by:

Scattering experiments, however, measure the intensity ${I}_{n}(f)={|{A}_{n}(f)|}^{2}$. It is easy to check that the relation as ${I}_{n}(f)={I}_{n-1}(f/\varphi )$ *scales* the envelope sinc functions in these profiles. This is depicted in Fig. 3 where we plot the scaled intensity for orders 10, 11, and 12. It should be noted that the $y$ axis does not require a scale factor as the intensity is unaffected by increasing orders. Observe that the internal structure exhibits self-similarity and only a *partial* scale-invariance. The latter improves for higher orders of the FbG. On the other hand, diffractals from fractal gratings of different orders scale perfectly by a dilation and contraction defined by ${I}_{n}(f)={(2/3)}^{2}{I}_{n-1}(f/3)$ [10,11]. But the issue of reduced intensity for higher generations is a deterrent in many applications requiring fractal diffracting elements [11,12].

The term $\sum _{k=1}^{{F}_{n-2}}{e}^{-i2\pi f{x}_{k}}$ in Eq. (3) yields the structure factor $S(f)$. The fractal dimension ${d}_{f}$ can be obtained from the integrated structure factor $\overline{S(f)}=[{\int}_{f-\mathrm{\Delta}}^{f}\mathrm{d}qS(q)]/\mathrm{\Delta}\propto {f}^{-{d}_{f}}$ [19–21]. In Fig. 4, we plot the corresponding integrated structure factor, $\overline{S(f)}$ versus $f$, evaluated from the first secondary band. The slope of the best fit line yields ${d}_{f}\approx 0.88$ and is consistent with earlier evaluations [22,23].

After identifying the fractal dimension and self-similarity in the FbGDP, we probed the property of redundancy observed in diffractals [11,12]. In an earlier paper, we found Cantor diffractals comprised of scale-invariant, self-similar bands each containing complete information about the fractal aperture [11]. This redundancy allowed the reconstruction of the Cantor grating by an inverse FT of an *arbitrary* band. Such spatial filtering was not masked by edge enhancement effects as expected in Fourier optics [24]. We also observed that the band contained information about *all* the previous generations of the Cantor grating. The fractal generator could hence be obtained from small fragments of the arbitrary band [11].

We investigated redundancy in the FbGDP experimentally using a “4F configuration” depicted schematically in Fig. 5. Here, coherent plane wave illumination of G yields the Fraunhoffer diffraction pattern on the plane $D$. Lens ${\mathrm{L}}_{1}$ performs the operation of FT. The aperture A placed after allows only a part of the diffraction pattern to be incident on the lens ${\mathrm{L}}_{2}$. The passage through ${\mathrm{L}}_{2}$ results in yet another FT thereby reconstructing the FbG on screen R. Figure 6 depicts (top row) the FbG of orders 6 and 7, (middle row) reconstruction from the entire diffraction profile, and (bottom row) reconstruction from the first secondary band. The reconstructions from the secondary bands were excellent, and as predicted, the intensity did not diminish with increasing orders. Inverse FT of smaller clips of the secondary bands, however, did not yield the FbG. Thus, the FbGDP exhibits *limited* redundancy, unlike its fractal counterpart due to the lack of scale-invariance. Checks of redundancy can also be done numerically. We repeated the above experiment by evaluating the FT numerically using Matlab and reconfirmed our observations.

Another significant property of diffractals is that of robustness to disorder $D$ in the grating. For Cantor diffractals, we obtained the correct fractal dimension even when half the transparencies and opacities in the grating were the wrong way ($D\sim 50\%$). Further, the fractal generator could also be obtained unambiguously by inverse propagation of a fragment of an arbitrary band. A query regarding the robustness of the FbGDP is also very relevant because of its technological implications. The question we address is: how much disorder in the FbG can be accommodated by the FbGDP to still yield the correct fractal dimension? The disorder was introduced by blocking and opening a fraction of randomly chosen transparencies and opacities. Figure 7 depicts a few realizations of the FbG with $D\sim 50\%$.

Diffraction pattern from disordered FbG also exhibits redundancy. Therefore, the fractal dimension ${d}_{f}$ can be obtained from an arbitrary band. To obtain reliable numerics, it is essential to average data over several realizations of disorder. All the data that we present has been averaged over 40 disorder realizations. Table 1 presents $\u3008{d}_{f}\u3009$ calculated from the first secondary band for different values of disorder $D$ and order $n$. The angular brackets $\u3008\cdots \u3009$ indicate an averaging over disorder realizations. These data indicate that ${d}_{f}$ was correct to the first significant digit of the un-disordered value even if $D$ was as large as 50% for higher $n$. A query regarding the stability of FbGDP to perturbations was also made by Grushina *et al.* [18]. The authors did not provide a quantitative answer to this question.

Summarizing, we have studied diffraction from the aperiodic FbG. The FbG diffraction pattern (FbGDP) is comprised of self-similar bands, each containing complete information about the FbG. A significant number in the context of Fibonacci architecture is the *golden ratio* $\varphi =(1+\sqrt{5})/2$. Its presence is ubiquitous and has important implications. For instance, the ratio of the light-transmitting and opaque regions in the FbG approaches the golden ratio as the generation $n\to \infty $. As a result, the intensity of the diffraction profile remains unaltered even for higher generations of the grating (this is in sharp contrast to fractal diffraction where field intensity reduces to zero in just a few generations). Further, the positions of consecutive local maxima in a band are related by the golden ratio. The locations of zeros of sinc functions (or bandwidths) in consecutive generations $n$ are also in accordance with the golden ratio. The intensity profiles are therefore self-similar and can be scaled by the relation ${I}_{n}(f)={I}_{n-1}(f/\varphi )$, where $f$ is the spatial frequency of the scattered wave.

The presence of self-similarity introduces fractal signatures. The FbG was characterized by a *fractal dimension* ${d}_{f}\approx 0.88$ evaluated from the structure factor component of the diffraction pattern. The FbGDP also exhibits the property of *redundancy* characteristic of diffractals or waves diffracted by a fractal grating. It is therefore possible to obtain the FbG by an inverse FT of an arbitrary band, although it excludes the zero spatial frequency component and has insignificant energy content in it. Like diffractals, the FbGDP also shows *robustness* to disorder in the FbG. For higher generations, the fractal dimension is marginally affected even if approximately 50% of the transparencies and opacities are placed wrongly. Having demonstrated the properties of constant intensity, redundancy, and robustness in the FbGDP, our conclusion is that higher generation FbG are excellent candidates for emerging technologies based on polychromatic, wide-band imaging.

RV acknowledges financial support from CSIR, India.

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