1. Introduction

Many natural systems exhibit fractal properties [1]; in fact, scale invariance underlies many self-similar phenomena, from frost crystallization to animal colouration to stock-market pricing. In the realm of optics, the fractal anatomy of systems is widely associated with aggregated dielectric and metal colloids [2], crystals [3], and tissues [4]. Fractal systems are also observed in nonlinear optics [5, 6] and fractalized optical attributes can even efficiently characterize or enhance the response of materials [7–9]. Less well utilized are the features of diffractals, or the diffraction patterns of fractal signals [10–12]. Diffractals feature in methods of encrypting data [13] as a versatile approach to random-phase encoding [14]. In a departure from previously-mentioned work, we exploit fractals to provide spatial multiplexing in images or wide-area signal transmission.

Here, we show that the free-space propagation of diffractal-signal architectures provides algorithmic value and spatial multiplexing properties. Any arbitrary subsection of a diffractal contains sufficient information to recreate the original sparse signal that is transmitted with a specific fractal architecture. Both robustness and redundancy are identified numerically and experimentally in the transmission properties of diffractal gratings in [15, 16], whose ideas we advance in 2D. In a manner similar to compressive imaging [17, 18]—where sparse signals reveal greater information via the diffraction through structures—here the fractal structuring within the signal sparseness prevents the loss of information.

Like other applications of fractals in communication systems, the diffractal architecture exhibits trade-offs. Fractal antennas for the radio frequency and microwave regimes are known for being compact and versatile over wide spectral bands but are power intensive [19, 20]; fractal encoding algorithms enable image compression with higher-resolution at the expense of greater algorithmic complexity [21]; here, our research identifies that diffractal architectures prevent the loss of information but require greater signal preprocessing. This investigation extends our understanding of fractal structures in signal communications and may increase robustness and transmission rates of satellite, wireless, and interplanetary communication systems, i.e., networks that support a large number of roaming receivers.

The remainder of this report is organized as follows. First, we formalize the expression of a transmitted fractal signal, show that its far-field diffraction pattern or diffractal is also a fractal, and illustrate the reciprocal nature of fractals with the Sierpinski carpet. Second, we demonstrate the robust retrieval of a signal from a diffractal; the original signal is reconstructed even when the majority of the diffractal signal is blocked. Finally, we discuss the future applications for diffractal spatial multiplexing in free-space communication systems and conclude.

2. Theoretical description

2.1. Spatial multiplexing of the diffractal

We generate the fractal transmittance pattern from any base matrix via recursive iterations where the matrix is resized and convolved with itself repeatedly [22]. The base matrix B(x,y) adopts a general form,

Subsequently, the diffractal is the Fourier transform or the far-field of the transmittance function $\tilde{T}$ [11],

The Sierpinski carpet is one example of a fractal that is generated by this process and via the iterated substitution of a 3 × 3 base matrix of ones with removal of the center element:

The second substitution of Eq. (4) represents the mathematical expression for the base matrix B₀(x,y). In the case of the Sierpinski-carpet base-matrix elements, r = 3, and x_j and y_j are the perimeter coordinates of a 3 × 3 9-unit block centered at the origin, and (x_j,y_j) ∈ [(1,1),(1,0),(1,−1),(0,−1),(−1,−1),(−1,0),(−1,1),(0,1)]. Since each of the Dirac delta functions in B₀(x,y) yields a phase shift in the Fourier domain, i.e., $ℱ\{\delta (\alpha x-x_i)\}=e^{2\pi i_c{k}_x{x}_i}/\left|\alpha \right|$, its scaled Fourier transform at the i^th iteration of the Sierpinski carpet [Eq. (1)] becomes:

With each iteration, the spatial frequency components k_x,k_y of the diffractal increase by a factor of 3 and spread the diffractal across a 3-times wider k_x,k_y-range, which is evident in Eq. (5); the cutoff of ${\tilde{T}}_n$ scales in proportion with k_x,k_y ∝ rⁿ⁻¹.

We calculate the Sierpinski carpet T recursively [Eq. (2)]: a second-order fractal is generated from the Kronecker tensor product of the base matrix $\left[\begin{array}{ccc}1& 1& 1\\ 1& 0& 1\\ 1& 1& 1\end{array}\right]$ with itself; a third-order fractal is generated from the Kronecker tensor product of a second-order fractal and the same base matrix see Code 1, Ref. [23]. Sierpinski carpets of n = 1,3, and 5 are shown in Fig. 1(a) with the corresponding diffractals $\tilde{T}$ [Fig. 1(b)].

 

Fig. 1 (a) Fractalized signals (FS) of orders n = 1, 3, and 5, where the original signal OS is a 3×3 array of 1’s with a 0 in the center. (b) Corresponding Fourier transforms of signals, or diffractal signals (DS) on logarithm scale. (c) Reconstructed Fourier transforms (RFS) from the 1% black-outlined subset, the blocked diffractal signal (BDS). (d) Enlargement of a portion of the n = 5 diffractal, which illustrates similar patterns at different length scales. (e) The peak signal-to-noise ratio (PSNR) of RFS to FS for various fractal orders as a function of the fractional BDS size.

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As the fractal order increases, $\tilde{T}$ exhibits smaller self-similar features at higher k_x,k_y; the diffractal also exhibits a fractal architecture, as observed in other literature [10, 11]. Moreover, when the fractal order, n is large, an arbitrary subsection of $\tilde{T}$ closely resembles the whole, and the self-similarity is already apparent with n = 5 [Figs. 1(c) and 1(d)]. The iterated, self-similar, wide-spatial-frequency features contained in the diffractal ${\tilde{T}}_n$ [Eq. (3)] enable robust reconstruction of itself, which is described in the next subsection.

2.2. Robust reconstruction of a blocked diffractal signal

We now refer to the sparse matrix B₀ and transmitted signal T as the original and fractalized signal, OS and FS, respectively; the diffractal signal DS is $\tilde{T}$, or, the Fourier transform of FS; BDS refers to an off-axis subsection of DS that is filtered; a reconstructed fractalized signal RFS refers to the inverse-Fourier transform of BDS; a regenerated version of the original signal ROS interpolates RFS in order to obtain OS.

In Fig. 1(b), a subsection or BDS is outlined with a black square, which represents 1% of DS. The corresponding RFS from BDS are shown in Fig. 1(c). For n = 1, 3, RFS is ostensibly blank because BDS carries negligible power. In contrast, when n =5, RFS carries features that resemble FS. We refer to the capacity to reproduce FS with 1% of the off-axis area from DS as robust reconstruction.

Robust reconstruction—where RFS resembles FS—is possible even when the size of BDS is significantly reduced [15]. A corresponding increase in the FS fractal order n will yield RFS that resembles FS when BDS is arbitrarily small. With the example of the Sierpinski carpet, Fig. 1(a)–1(c), if the size of BDS is reduced to 1% of OS, then a comparable RFS is produced by increasing the fractal order of FS from n = 1 to n = 5.

Here, we also refer to robust reconstruction as the capacity to regenerate OS from RFS from a simple threshold function see Code 2, Ref. [24]. The threshold function of the Sierpinski carpet divides RFS into a 3 × 3 array (identical size as OS) and measures the intensity in each of the 9 elements. Above a certain threshold value, the element is assigned a value of 1, and otherwise assigned a value of 0. The 3 × 3 array that is processed with the threshold function, ROS, is theoretically identical to OS when OS is transmitted with the diffractal architecture.

The comparison between RFS and FS in peak signal-to-noise ratio (PSNR) for various fractal orders as a function of the fractional BDS size is shown in Fig. 1(e). The error bars represent the standard deviation from the mean value of 200 random locations of BDS. Since the size of RFS is reduced from FS, RFS is resized with nearest-neighbour interpolation. We provide 2 observations from Fig. 1(e): first, when BDS is greater than 0.01% of the fractional size of FS, larger fractal orders produce higher PSNR; second, as the fractional area of BDS increases the location of BDS becomes irrelevant since the standard deviation around the mean diminishes. For fractional BDS sizes smaller than 0.01% lower fractal orders yield a higher PSNR since the minimum size matrix needed to completely recreate FS is larger for higher fractal orders. For example the minimum size matrix needed to recreate FS of order 1 is 3 × 3, for FS of order 2 it is 9 × 9, etc.

We observe that ROS is generally identical to OS when BDS is above 0.01% and the fractal order is greater than 3, regardless of BDS location. A partial explanation for the robust reconstruction is that increasing-order FS carry arbitrarily-high k_x,k_y and enable arbitrarily-small BDS to carry the information of FS or OS. If we strictly limit OS to binary or Dirac-delta functions, then FS has no k_x,k_y cut-off and as n approaches infinity, fractalized features appear in FS without a k_x,k_y cut-off. In fact, the amplitude of the additional k_x,k_y gained from each iteration scales inversely with rⁿ⁻¹, which provides a detection limit only in practice; in theory, each iteration generates high-k_x,k_y copies of OS [Eq. (4)] that are spatially distributed from the origin.

Yet it is worth noting that the robust reconstruction is achieved because the diffractal architecture also couples k_x and k_y in iterated products [Eq. (3)]. As a result, RFS will resemble FS when either the BDS size or location changes. In a manner similar to spatial filtering, RFS will produce outlines of FS if n is not sufficiently large; however, unlike a high-pass spatial filter of a multi-scale random high-k_x,k_y pattern [17], a change in the location or the size of BDS will not distort the outline of RFS. Subsequently, the diffractal architecture provides superior performance over other algorithms that regenerate sparse data, OS [17,18].

3. Experimental results

We experimentally demonstrate the features of diffractals with a 4-f optical arrangement where the 2D Fourier transform of a collimated fractal image, DS, lies in the focal plane of an imaging lens [25]. The experimental setup that produces, filters and reconstructs FS is shown in Fig. 2(a) and experimentally reconstructed images, RFS, are shown in Fig. 2(b). An aperture of area approximately 0.8mm² blocks the majority of DS. The placement of the aperture is shifted roughly 4mm horizontally and 5mm vertically from the central point, in the focal plane of the lens. The SLM image has a resolution of 800×600 pixels (16mm×12mm).

 

Fig. 2 (a) Light from a λ = 632.8-nm wavelength laser is spatially filtered, expanded, and collimated, and illuminates the full area of a 800×600 pixel spatial light modulator (SLM). The 4-f system is composed of two 5-cm focal length lenses placed after the SLM. The first lens transforms the fractal signal (FS) and produces the diffractal signal (DS) at the focal plane. An aperture is placed off-center at the focal plane and only transmits a portion of the diffractal (BDS). The second lens reconstructs the SLM image with BDS (RFS). (b) The Sierpinski carpet hologram of order n= 5. (c) Experimentally-measured DS, where the red circle is the BDS utilized to (d) experimentally-reconstruct RFS.

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When we employ diffractal architectures of high orders (n >4), we observe the phenomenon that is shown numerically: the placement of the aperture in DS is irrelevant in order to reconstruct the original image. When the aperture is moved laterally in the focal plane, RFS maintains strong resemblance to FS. In fact, only the intensity of RFS diminishes as the aperture translates further from the DS center; the outline remains fixed as the aperture moves. In general the intensity profile of DS follows the inverse power law from the center of the diffractal, observed in other work [11, 12]. For smaller fractal orders, RFS resembles FS only when the aperture is placed within 0.8mm of the center, where the spatial frequency components are concentrated. The signal-to-noise of the experiment and CCD camera sensitivity limit effective reconstruction, while the number of SLM pixels limits the highest fractal order of FS.

4. Discussion of applications

Here we make the distinction that diffractals are specific fractal structures. Not all fractals enable robust signal communications and the recursive transformation employed to generate FS and DS in Eqs. (1) and (5) differ fundamentally. For example, both of the Fourier transform pairs FS and DS are fractals and carry iterated, self-similar features, but if we reverse their roles in our transmission system, the reconstruction will instead depend severely on the size and location of BDS. In fact, if in the example of the Sierpinski carpet, the center subsection becomes BDS, then the ROS will not resemble the OS, regardless of fractal order. The diffractal is unique from general fractals and self-similar scale-invariance alone is an insufficient precondition for our system of robust reconstruction and spatial multiplexing.

It may seem contradictory that higher-order fractals lead to more robust signal transmission since the finer structure of a higher-order fractal is itself harder to reconstruct. There are two perspectives of diffractals that explain the robust reconstruction. First, the self-similar structures of higher-order fractals have a greater spatial frequency range and finer detail, and subsequently smaller BDS carry sufficient information to reconstruct OS. Second, the higher-order diffractals carry higher spatial-frequency components where the k_x and k_y components are coupled, and subsequently the location of the subsection in DS is unimportant. Here we have shown that BDS of arbitrary size and location carry sufficient information to reconstruct OS but in practice, there exists clear limitations for the robust reconstruction even in the limit of infinite-order FS.

There exists a trade-off with the diffractal architecture between robust reconstruction and high bit rate; a greater bit rate is achieved with a larger base matrix, which can limit the maximum fractal order that is transmitted. For example, with the 3 × 3 or 9-element OS, there are 512 possible spatial bits, three of which are illustrated in Fig. 3(a)–3(c). A 4 × 4 base matrix requires ${(\frac{4}{3})}^{2n}$ more pixels than a 3 × 3 to achieve the same fractal order, n. With a limited SLM pixel resolution, there is a choice between the generation of higher-order fractals and the utility of higher spatial bits. If a higher bit rate reduces the fractal order then a larger BDS size will be required in order to achieve equivalent robustness.

 

Fig. 3 Three examples of 512 9-bit spatial patterns, associated with base matrices (a) $\left[\begin{array}{ccc}1& 0& 1\\ 1& 1& 1\\ 1& 0& 1\end{array}\right]$, (b) $\left[\begin{array}{ccc}1& 1& 0\\ 0& 1& 1\\ 0& 1& 1\end{array}\right]$, and (c) $\left[\begin{array}{ccc}1& 0& 0\\ 1& 1& 0\\ 1& 1& 1\end{array}\right]$. The fractal signals FS are shown with their corresponding experimentally-reconstructed fractal signals RFS from the experimental setup in Fig. 2(a). The lower-right inset shows the reconstructed original signal ROS. (d) Example application: transmitted fractal signal FS is received at a far-field distance as a diffractal signal DS, where a roaming set of receivers, with only a diffractal subsection BDS, which is inverse-Fourier transformed from the far-field propagation, reconstructs the original signal OS.

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If the trade-off between bit rate and robust reconstruction are mitigated, then the diffractal architecture could support a large number of roaming receivers with only one transmitter, such as wireless or satellite networks shown in Fig. 3(d). The self-similar properties of the diffractal architecture and their corresponding far-field pattern provide a method to reach a large number of receivers, possibly moving, without signal degradation. In fact, in such a system, the far-field propagation of the transmitted FS would lead to DS at the location of the receivers without the requirement of a lens. The self-similar properties of the diffractal architecture provide a method to reach a large number of receivers, possibly moving, without signal degradation. The processing times required in the calculation of FS from OS are not trivial and scale with r²ⁿ. The spatial multiplexing of even sparse signals may be employed to increase the bandwidth of data that is currently transmitted in the time domain. Secondly, the refresh rates of a spatial light modulator or similar adaptive-optics device present constraints on the maximum achieved bit rate, which requires further consideration.

5. Conclusion

We have shown that the diffractal architecture provides the beneficial features of spatial multiplexing and robust reconstruction of original sparse signals. We have centered our demonstrations with the Sierpinski carpet, a familiar fractal, although our results could have been demonstrated with 511 other patterns similarly imprinted with the diffractal architecture. Data that is transmitted with the diffractal architecture is highly robust to intermediate-obstacle signal blocks and diffractal subsections of arbitrary size and location carry sufficient information to regenerate the original signal without distorting outlines of its pattern. Our research illuminates potential applications when one transmitter sends data to a large number of moving receivers or through noisy media.