Freeform engineered disordered metalenses for super-resolution imaging and communication

Mohammad Haghtalab; Safieddin Safavi-Naeini

doi:10.1364/OE.26.009749

1. Introduction

Diffraction limits the maximum resolution, which can be achieved to the order of the wavelength of light in conventional imaging systems. This is due to the limited capabilities in collecting the scattered high spatial-frequency components from the illuminated samples which render the reconstruction of sub-wavelength features impossible. Numerous attempts have been made to develop methods known as super-resolution techniques to overcome this limitation and push the boundaries of imaging capabilities into sub-diffractional regime. These attempts encompass the engineering of structured lenses, structured illuminating sources, and the development of computational algorithms which in combination with the proposed techniques have led to unprecedented achievements in the optical imaging systems.

Capturing and amplifying the evanescent components of scattered fields, metamaterial-based structures break the diffraction barriers and achieve resolutions beyond those imposed by the diffraction of light [1, 2]. For these kinds of structures, however, the super-resolution features are satisfied only for a fixed transverse plane near the structure. In addition, the limitations imposed by practical fabrication techniques and material losses and imperfections restrict their effective implementation.

Although some techniques achieve high resolution imaging by exploiting the nonlinear effects, the need for labeling or possessing nonlinear properties confines the application of these methods to certain groups of optical samples [3, 4]. Employing computational methods along with the optical imaging techniques such as structured illumination, on the other hand, provides new possibilities in extracting the sub-resolution information of samples. These breakthroughs are made possible through increasing the effective numerical aperture of the optical systems employing numerical algorithms like machine learning and without significant hardware modification [5, 6]. Relying on computational techniques, lens-less imaging systems provide a new scheme for optical imaging. The spatial coding realized by metallic nanostructured surfaces, for instance, has been used in combination with optimization algorithms to detect objects in on-chip lens-less imaging applications [7]. Despite using the visible light for imaging, the weak scattering signals of radiating objects, due to interaction with nanostructured surface, limit the maximum resolution which can be achieved to micro-scale ranges. The spectral content of radiated fields has also been used in other studies for post-numerical analyses. Examples include spectral encoding of spatial frequency, for the characterization of nanoscale structures, and interferometric spectroscopy used to quantify the statistics of refractive index fluctuations of random dielectric medium [8–10]. However, there are simplifying physical assumptions for the samples being imaged in these approaches, limiting their applications to certain type of materials such as weakly scattering objects.

In a recent study, it has been shown that using speckle correlation, high quality diffraction-limited imaging can be achieved. The study has been conducted for both lens-less imaging systems and those with illuminated samples accompanied by scattering layer(s) [11–13]. This technique is based upon the observation that the speckle images formed by different radiating point sources have minimal correlation over the image plane. Calculating the autocorrelation of measured intensity gives the same result as if an ideal, aberration-free lens is employed to image the object. Phase retrieval algorithms can be used, in the next step, to obtain the object's image [14, 15]. Making use of scattering properties of highly scattering media to achieve higher resolution has also been proposed by other studies. For example, a time-reversal (TR) based technique shows that in a rich scattering medium and at the presence of highly scattering microstructures, the information capacity of a communication system can be increased while the receiver antennas are located at sub-wavelength distances from each other [16]. The requirement for having a rich scattering medium limits the application and the maximum resolution can be achieved by this type of methods. While these techniques are performed under the assumption that sufficient number of measurements is recorded over the interested region, it has been shown that, when combined with a suitable optimization algorithm, scattering medium can also improve the resolution of “sparse” images with fewer number of samples over the image plane [17]. In a similar approach, using optimization algorithms, the combination of broadband illuminating sources with resonant structures and/or disordered medium is employed to realize high resolution imaging [18–20]. Introducing a novel approach based on engineered scattering medium which is capable of imaging beyond the diffraction limit and investigating the fundamental limitations imposed by various physical parameters of imaging system are clearly needed. Our goal is directed at such an approach in this paper.

Random systems exhibiting highly transmitting channels, known as “open channels”, have attracted considerable attention for creating new possibilities for a wide range of applications from biomedical imaging to energy-efficient ambient lighting and optical information systems [21–24]. While their main goal has been to compensate for the effects of multiple-scattering through controlling the incident wave, and consequently, to increase the transmitted power through such media, our focus, in this work, is on taking advantage of scattering effects in electrically thin scattering structures to increase the “number” of transmitting channels which are capable of carrying information to the far-field region. This is done through decomposing signals into orthogonal channels (modes) and maximizing the transmitted power of the highest order “mode” to the receiver side. In this paper, we reveal the possibility of extracting sub-wavelength information of illuminating sources from their far-field image distribution. To this end, we employ the singular value decomposition (SVD) technique to extract the information of all possible transmitting channels. Our approach is based on modifying the higher orders transmission channels by introducing disorder-based engineered all-dielectric lens structures in front of the illuminated samples or transmitting sources. The presented method paves the way towards realizing structures with remarkably enhanced scattering features. This is done through following a systematic design approach where we exploit the high degrees of freedom inherent in disordered strongly scattering media. Our results demonstrate a significant improvement in achieved resolutions (very small fraction of the wavelength), and computational complexity, compared to similar approaches reported previously, which require using techniques like the employment of self-assembled nanolenses and large processing time for sub-wavelength sensing [25].

The introduced scattering structures are composed of dielectric scatterers and combine two major features. The first feature is the enhancement of the transmitted power of the highest order mode signals to the far field region due to the interaction of incident fields emitted from the illuminating sources with the dielectric objects. The illuminated dielectric objects re-radiate the incident electromagnetic waves which, in combination with the primary fields emitted from the sources, can lead to an average enhancement of the received power at the far field region. As demonstrated, the maximal transmission through lossless random structures can be achieved by minimizing the reflection [26]. In the present work, however, we directly maximize the transmitted power of interested “modes” emitted from the source region by optimizing the configuration of dielectric objects. The second feature is the reduction of the correlation between the radiated signals emitted from adjacent point sources which can be enhanced by exploiting the disorder realized by sub-wavelength dielectric inclusions. The point spread functions (PSF) of the proposed structured lenses possess modified space-variant property leading to significant reduction of correlation between the images formed by point sources located at sub-wavelength distances from each other. Based on the PSFs of radiating point sources and using the singular value decomposition technique, orthogonal basis vectors (modes) on both source (object) and image vector spaces can be obtained. The orthogonal basis vectors (modes) which we refer to as the orthogonal transmission channels can be used to expand the single-shot measured image to resolve the sub-diffractional features of objects. Moreover, these orthogonal basis vectors can be used to combine information signals at the transmitter and restore the information from the received signal at the receiver side. The improvement achieved for the resolution due to the scattering medium and the enhancement of the information capacity are studied and discussed.

This paper is organized as follows. In Section 2, we present the analysis and design procedure of the proposed structures. The proposed method is applicable to any type of scattering medium, however, without loss of generality, in this paper we investigate scattering media composed of dielectric wires in more details. In Section 3, we apply our proposed method to design structures with modified scattering features and investigate how the presence of different designed structures affects the scattering properties of illuminating sources. The resultant improvements of features such as the resolution, and the maximum number of resolvable point sources, in a given imaging or communication system with a known signal to noise ratio (SNR) and numerical aperture (NA), are studied. The last Section is the conclusion.

2. Analysis and design of scattering features of disordered all dielectric medium

In this Section, the analysis and design of a disordered all-dielectric medium are explained. We consider a general configuration shown in Fig. 1. Dielectric cylinders with refractive index $= \sqrt{ϵ_{d}}$ and radius a are confined within a rectangular region. The structure is illuminated by a collection of $N$ point sources located over the source (object) region at a specific distance from the scattering medium. The transmitted fields are measured (amplitude and phase) at $M$ points over the image region which is located at a far distance from both the sources and the scattering structure.

Fig. 1 The geometry of problem. The radiated fields emitted from N point sources inside the source region propagate through the engineered scattering structure which is composed of a plurality of dielectric wires with radius “a” and dielectric constant of ε_d distributed inside a rectangular region with side length of t_d and l_d. The field measurement is performed at M points over the image region which is located at far-field region. In this work, we only consider the case where the point sources and measurement samples are located on two lines with length of l_o and l_i, located at d_o and d_i from the scattering medium, respectively.

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2.1 The analysis

We employ a full-vectorial approach to rigorously solve the Maxwell’s equations and calculate the scattered field from the dielectric objects at an arbitrary location. To obtain the total electromagnetic field at any point in space, the incident, internal and scattered fields are expanded in terms of cylindrical harmonics for each dielectric cylinder and the boundary conditions are imposed on the surface of wires [27]. Assuming that the dielectric wires are infinite in y-direction, for a TM incident wave we can formulate the electromagnetic problem in terms of the y component of the electric field. The scattered field by a dielectric wire due to a unity amplitude plane wave incident at an angle φ_i can be written as:

E^{y} = \sum_{m = - \infty}^{+ \infty} j^{- m} C_{m} H_{m}^{(2)} (k_{0} ρ) e^{j m (φ - φ_{i})},

where C_m are the mode coefficients for the scattered fields and can be expressed as [28]:

C_{m} = \frac{\frac{η_{0}}{η_{d}} J_{m} (k_{0} a) {J^{'}}_{m} (k_{d} a) - {J^{'}}_{m} (k_{0} a) J_{m} (k_{d} a)}{J_{m} (k_{d} a) H'_{m}^{(2)} (k_{0} a) - \frac{η_{0}}{η_{d}} {J^{'}}_{m} (k_{d} a) H_{m}^{(2)} (k_{0} a)} .

Assuming that the radius of wires are much smaller than the wavelength of the fields in the free space and the dielectric medium, higher order modes can be neglected and only C₀ is used to describe the scattered fields. If we have “

L

” number of dielectric wires, the incident waves exciting the k^th dielectric wire can be written as:

E_{k, i n c}^{y} = E_{e x t e r n a l s o u r c e s}^{y} (r_{k}) + \sum_{\begin{matrix} l = 1 \\ l \neq k \end{matrix}}^{L} E_{l}^{y} (r_{k}) = \sum_{n = 1}^{N} s_{n} H_{0}^{(2)} (k_{0} | r_{k} - r_{n} |) + \sum_{\begin{matrix} l = 1 \\ l \neq k \end{matrix}}^{L} E_{l}^{y} (r_{k}),

where

E_{l}^{y}

denotes the y component of the scattered field from the l^th dielectric wire,

r_{k}

is the location of the k^th dielectric wire,

r_{n}

is that of the n^th point source, and

s_{n}

represents the excitation coefficient of the n^th point source (

s_{n} = - ω μ s'_{n} / 4

). Using Eq. (1),

E_{k}^{y}

can be described by a linear combination of both incident fields due to external sources and the scattered fields by other dielectric wires. A system of “

L

” equations can be formed and solved for the unknown

E_{k}^{y}

(k = 1, 2,…, L). Once the scattered field by each wire is obtained, the total field can be computed at desired points in space.

Considering $N$ radiating point sources over the object plane and $M$ samples (here we assume $M \geq N$ ) over the image plane [Fig. 1], we can construct the transmission matrix as:

[E_{i m a g e}] = [t] [E_{s o u r c e}],

where [E_source] is a

N \times 1

matrix presenting the complex (real and imaginary) excitation coefficients (s_n) of point sources, and [t] is an

M \times N

matrix where

t_{m n}

relates the field transmission from the n^th radiating point source to the m^th sample over the image plane where the measured fields are expressed as [E_image]. Decomposing the transmission matrix, [t], gives:

[t] = [U] Σ {[V]}^{T} .

Here [V] = [V₁|V₂ |…|V_N] and [U] = [U₁|U₂|…|U_M] are

N \times N

and

M \times M

unitary matrices, respectively. The column vectors of [V] ([U]) are called the source (image) eigenchannels (functions).

Σ

is an

M \times N

non-negative real diagonal matrix containing the eigenchannel transmission coefficients,

σ_{n} = \sqrt{τ_{n}}

, where τ_n are the eigenvalues of [t]^T[t] and σ₁ > σ₂ > σ₃ >… >σ_N. Multiplying [E_source] by [V]^T maps any arbitrary distribution of excitation coefficients of point sources to eigenchannel vectors which are the columns of [V]. The n^th column represents one transmission channel and provides the required complex (amplitude and phase) excitation coefficients for each radiating point source so when combined and the radiated fields are transmitted through the structure and the medium, after measured at the

M

pre-specified points, gives the corresponding eigenfunction over the image plane (the n^th column of [U]) with transmittance of τ_n. The fact that the column vectors of [V] and [U] are orthonormal can be used to extract source information ([E_source]) from the image signal ([E_image]). For the

N

point sources and

M

samples over the image plane, [E_image] can be written as:

[E_{i m a g e}] = \sum_{n = 1}^{N} σ_{n} U_{n} (V_{n}^{T} [E_{s o u r c e}]) = \sum_{n = 1}^{N} σ_{n} U_{n} E_{source}^{n},

where V_n (U_n) represents the n^th column vector of [V] ([U]) and

V_{n}^{T} [E_{s o u r c e}] (= E_{source}^{n})

is the projection of the source signals onto the basis vectors

V_{n}

, which can be used to expand the source excitation coefficients as:

[E_{s o u r c e}] = \sum_{n = 1}^{N} (V_{n}^{T} [E_{s o u r c e}]) V_{n} = \sum_{n = 1}^{N} E_{source}^{n} V_{n} .

Multiplying both sides of Eq. (6) by

U_{n}^{T} / σ_{n}

and using the orthonormality of

U_{n}

,

E_{source}^{n}

can be extracted. In the next step, source distribution can be computed using the expansion of Eq. (7).

As it is clear, the larger the value of σ_n is, the more signal power of the n^th mode will be transferred through the corresponding eigenchannel and received by the detectors. Our goal, in this paper, is to design scattering structures to maximize the transmittance of the highest order eigenchannels (modes). This is to increase the number of channels with acceptable power level for a receiver with a known signal to noise ratio (SNR) to detect. As the exact extraction of source signals is limited by the receiver ability to detect the weakest signals which correspond to the lowest transmittances in the SVD formulation [Eq. (5)], by increasing the transmittance of the higher order modes, for a given level of noise power or signal to noise ratio (SNR) at the receiver, we increase the number of modes which can be detected. The more number of modes can be detected, the more detailed information can be extracted from the received signals. In imaging application, this means the enhancement of resolution by detecting higher order signals emitted from illuminating sources which are associated with the fine features of the object being imaged. Various parameters affecting the transmittance include the number of point sources and the distance between them, the number of samples over the image plane and the distance between them, the distance between the object plane and the image plane, and the scattering property of the medium the radiated signals are propagating through. The latter include the property of both the scattering structure and the free space. In this paper, we propose a new approach to design scattering structures in such a way that when they interact with the electromagnetic fields emitted from the point sources or illuminated object they increase the transmittance of the highest order mode. Before that, in this Section, we investigate how other parameters can affect the transmission of highest order signals through a given scattering medium.

As an example, we consider a structure consisting of 340 dielectric wires (ε_d = 4.61, a = 0.02λ) [Fig. 2]. We change the number of point sources (N) distributed uniformly over a given region (l_o) on source (object) plane and for each case we take M equally spaced samples over a given region (l_i) on the image plane. The distances between the source (object) plane and the structure and between that to image plane are denoted by d_o and d_i, respectively. Using Eq. (5) the transmittance of the last mode (the N^th mode) is calculated. As mentioned earlier, this is an indication of the resolution can be achieved in a given imaging or communication system. Larger transmittance is associated with higher SNR and accordingly higher resolution for the corresponding mode.

Fig. 2 The location map of a randomly configured structure composed of 340 dielectric wires.

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First, we set $d_{i} = 20 λ$ , $d_{o} = 0.16 λ$ , and $l_{i} = 400 λ .$ Changing $l_{o}$ , the transmittance (τ_N) is calculated as a function of N (number of point sources) and M (number of samples). As expected, with the same number of point sources (N), the transmittance increases as the distance between the point sources becomes larger [Fig. 3]. This is mainly due to the fact that the images obtained by different radiating sources located at farther distances are less correlated than those located at closer distances; thereby higher transmittance in SVD formulation is obtained. As the number of samples over the image plane increases, higher level of signal power is expected. However, this dependency is not linear and the rate of power change becomes very small in larger sampling points. The size of image region, where the samples are taken from, also affects the measured transmittance [Fig. 4]. With fewer numbers of samples, smaller sampling regions near the source region show higher transmittance. By increasing the number of sample points, however, larger sampling areas (over the image region) are superior in terms of transmittance. This confirms that the interaction of the incident field with the scattering structure results in diffracted fields which contain information over larger sampling region on the image plane. We define the numerical aperture as the angle seen from the centre of the source region to the sampling region over the image plane [Fig. 1]. For two identical systems, with a fixed numerical aperture, the changes in the distance between image and source (object) planes, while the image plane is in the far distance from the scattering medium, results in inversely proportional to distance (r) changes in the highest order mode transmittance [Fig. 5] verifying that the measurements are performed at the far-field regions where the evanescent fields are largely absent. The highest order modes contain information about sub-diffractional features of radiating sources. The presence of scattering structure can lead to the enhancement of the transmittance of these modes and our ability to resolve sub-wavelength information of the illuminated objects or radiating sources. In the next Section, we investigate the effect of scattering structures on the enhancement of the higher order modes transmittances. We propose a new approach in designing all dielectric scattering medium and demonstrate how they can lead to the enhancement of sub-wavelength information transfer. Although the proposed method is applied to scattering media composed of dielectric wires, it can be generalized and used, also, to design scattering structures composed of three dimensional dielectric scatterers.

Fig. 3 The highest order transmittance (dB) as a function of N and M (M ≥ N) and d_i = 20 λ, l_i = 400 λ. For (a) l_o = 0.25 λ, (b) l_o = 0.5 λ, (c) l_o = λ, and (d) l_o = 2 λ.

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Fig. 4 The highest order transmittance (dB) as a function of N and M (M ≥ N) for d_i = 20 λ, l_o = 2 λ. For (a) l_i = 20 λ, (b) l_i = 40 λ, (c) l_i = 400 λ, and (d) l_i = 800 λ.

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Fig. 5 The highest order transmittance (dB) as a function of N and M (M ≥ N) and l_o = 2 λ. For (a) d = d_i + d_o = 20 λ, li = 40 λ, and (b) d = d_i + d_o = 200 λ, l_i = 400 λ. The two cases, (a) and (b), represent the results for a same numerical aperture.

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2.2 Engineered disordered all dielectric medium

As mentioned in the previous Section, scattering medium can enhance the transmission of high resolution information to the far-field region. In this Section, we investigate this in more details and propose a new design approach for engineering the scattering properties of medium. This is done through designing structures composed of dielectric wires which are located within a confined specified region. The proposed devices provide a rich scattering medium maximizing the transfer of waves associated with the highest order radiating modes of the illuminated objects/ radiating sources to the receiver side located at the far-field region. It is important to note that the optimization problem we define here, like most problems in topological designs, is not convex which typically means that there are several different local maxima (minima), or non-unique solutions. The proposed methods known as topology optimization are based on the method of gradient descent where initial solutions are iteratively optimized to obtain a local maximum (minimum). On the other hand, due to the large number of variables, most global optimization methods cannot handle this type of problems efficiently [29]. In this paper we propose and employ a new stochastic-based optimization approach which is capable to search for the global solutions based on defined constraints and design parameters.

In this work, we assume the physical characteristics of dielectric wires including the radius and the dielectric constant (refractive index) are known. The designed structures, which consist of a plurality of identical dielectric wires, are confined within a rectangular region with known values for the length and the thickness. The optimization problem is defined to maximize the highest order mode transmittance for a set of radiating point sources located at known locations over the source (object) plane. This can be achieved through introducing dielectric wires at proper locations within the specified region by following our proposed procedure explained in this Section.

If we have a set of point sources with known positions radiating in free space, introducing an individual dielectric wire perturbs the scattered fields in the far field region. Depending on the position of the scattering object (a dielectric wire in this work) with respect to the radiating sources, the correlation between the radiated fields of adjacent point sources may differ by certain values. As mentioned previously, it is desired to have lower values for the correlation which is equivalent to higher values for the transmittance of higher order modes [See the SVD formulation, Eq. (5)]. Our proposed algorithm is based on finding a proper location for adding one dielectric wire so that for a given scattering medium, it maximizes the power of highest order radiating modes over the far-field region. Adding dielectric wires is continued until the transmittance of highest order radiating mode cannot be further enhanced or the preset maximum number of dielectric wires is reached. Controlling the design procedure is done through defining a number of parameters. The most important one is called the transmittance threshold. This determines the criterion for adding one new dielectric wire at a new location randomly selected by a computer simulator. The numerical value for this threshold is set using the statistical property of structure in a given physical configuration. For a given imaging or communication system, with known physical parameters and locations for radiating point sources, the average value of the highest order transmittance is calculated for different numbers of dielectric wires distributed randomly inside the specified rectangular region. We change the number of dielectric wires from one to an arbitrary maximum number (n_d = 1, 2, 3,…, N_d), and for each case the average value of transmittance is calculated for a large number of randomly configured dielectric wires generated inside the specified region. The result, which indicates the statistical property of structure, describes the average changes in the transmittance in terms of the number of dielectric wires and provides the required information about the effect of adding one dielectric wire to an existing scattering structure with $n_{d}$ dielectric wires. This can be understood from the slope of the graph at a point which corresponds to the number of dielectric wires (n_d) in an existing scattering medium. Depending on the number of dielectric wires, the initial value for the transmittance threshold, which is the key parameter in our design procedure, is chosen to be an integer multiple of the slope calculated at the corresponding point on the graph. Two other parameters used in the algorithm are denoted by M_t and N_t. M_t is used to control the maximum number of iterations in finding a proper location to add one new dielectric wire to an existing configuration. N_t determines the period over which the value of transmittance threshold is modified in order to improve the efficiency of algorithm in selecting the best locations for dielectric wires leading to an overall remarkable enhancement of the transmittance of the highest order radiating mode at the end of design procedure.

The flowchart shown in Fig. 14 (in the Appendix) describes our proposed algorithm for designing a dielectric medium with enhanced scattering properties. It starts with an initial configuration of dielectric wires located inside the region which provides a relatively good choice between many candidate configurations with the same number of dielectric wires. It is obtained either by running a large number of simulations generating random configurations and choosing the best one among them or choosing a known configuration which has been obtained previously. Depending on the initial number of dielectric wires in the primary configuration, the transmittance threshold is set to be significantly larger than the average change of transmittance due to adding one dielectric wire to the existing configuration. This can be done using the statistical property obtained previously by extracting the slope at a proper point. After choosing an initial configuration of dielectric wires, where the locations are shown by {(x, y)} the highest order transmittance (which is also the cost function denoted by C) for a given locations of point sources is calculated. The parameters including the transmittance threshold, M_t, and N_t are initialized to proper values. A new set of temporary dielectric wire locations is defined which is represented by {(x, y)_t}. This set includes the candidate locations to add one new dielectric wire to the existing configuration denoted by {(x, y)}. Adding one dielectric wire at each location which is saved in {(x, y)_t}, will change the transmittance to a new value which is recorded in a new set denoted by {C_t}. The design procedure starts by adding one temporary (test) dielectric wire at a random location (x_r, y_r) inside the specified region. The transmittance of the new configuration is calculated and is saved in C_r. If the enhancement in C is larger than the current transmittance threshold, the new location and corresponding transmittance are saved in {(x, y)_t} and {C_t}, respectively. This step is repeated for M_tmax times after which the best location in{(x, y)_t}, corresponding to the maximum transmittance saved in{C_t}, is chosen for the location of one new dielectric wire to be added to the current configuration ({(x, y)}). This procedure is repeated for N_tmax times when a maximum number of N_tmax dielectric wires can be added to the initial configuration. Depending on whether the number of dielectric wires has been changed (increased) or none of the random locations were selected for adding dielectric wires to the previous configuration, the value of transmittance threshold will be updated. If any dielectric wire has been added to the structure, the threshold is increased by some coefficient which can be set by user. This makes more stringent condition for adding new dielectric wires in the next iterations. This is continued until no dielectric wire can be added to enhance the transmittance greater than the updated threshold. At this point, we will gradually decrease the threshold (through one or successive iterations) so that at least one dielectric wire can be added to the current configuration. Although this can lead to a temporary, typically small reduction (depending on the threshold value and the current configuration), in the transmittance in a situation that we have a close to optimum structure (near one local maximum of the cost function), the new configuration, which includes the latest dielectric wire, can be improved (in the next steps) to obtain a more optimum configuration than the best configuration obtained so far in the previous steps. This process can be repeated until the maximum transmittance for the highest order radiating signal illuminating the designed structure is obtained, or the user can stop the process for any other reason such as reaching to the preset number of dielectric wires (the complexity of structure). More information on the proposed algorithm is provided in the Appendix.

Now, we consider the application of the proposed algorithm in a few examples. The effectiveness of our approach in designing the engineered structures can be seen through comparison with those which have periodic or totally random configurations.

3. Numerical results

3.1 Design examples

We consider a rectangular region like the one shown in Fig. 1, with the length sides of t_d = 0.34λ and l_d = 5.85λ. The region is going to be populated with dielectric wires having a radius of $0.02 λ$ and refractive index of 3. They are illuminated by point sources with known locations over the source (object) plane. The purpose is to design a structure composed of sufficient number of dielectric wires within the rectangular region to significantly increase the transmission of the highest order modes of the source signals to the far-field region. In the present examples we assume that the distance between the source (object) plane and the rectangular region is about one over sixth of the wavelength of illuminating sources (d_o = 0.16λ) and that between the rectangular region and the image plane is twenty times the wavelength (d_i = 20λ). The field is sampled over a region with length of l_i = 400λ and the total number of sample points is two hundred (M = 200). Two groups of sources will be considered here. The first group is composed of ten (N = 10) equally-spaced point sources distributed over a region with length of l_o = 0.58λ. In the second group, we increase the number of point sources to fifteen (N = 15) distributed over the same region. The design procedure starts with calculating the average transmittance of the structure for n_d = 1, 2, 3,…, 900 number of dielectric wires. To this end, for each n_d, we run a large number of simulations with random configurations and take the average of the calculated highest order transmittance for each simulation. The calculated values give an estimate of how much, on average, the highest order mode transmittance is increased if we add one new dielectric wire to a given configuration which consists of a total of n_d dielectric wires. As mentioned before, the slope of the resulting graph is used to set the initial value for the transmittance threshold in the optimization process. Figure 6 demonstrates the results for the two groups of sources.

Fig. 6 (a) and (b) show the highest order transmittance (dB) of 10 and 15 point sources, respectively, distributed uniformly on l_o = 0.58λ when illuminating random scattering structures. The horizontal axis shows the number of dielectric wires in the structure which are randomly configured in a rectangular region of side lengths t_d = 0.34λ and l_d = 5.85λ. The calculations are performed for a large number of random configurations, each denoted by a number on the vertical axis (number of simulations). (c) and (d) demonstrate the average value of the transmittance calculated in (a) and (b), respectively.

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The next step is to initialize the parameters in the algorithm and to define an initial configuration of dielectric wires as the starting point in the process of optimization. We set M_tmax = 20, N_tmax = 5 and the transmittance threshold to be 10 times larger than the average value it increases when one dielectric wire is added to an existing initial configuration (this value can be obtained from the graphs shown in Fig. 6). For a given configuration, and transmittance threshold, the algorithm chooses the best location for adding one new dielectric wire among a maximum M_tmax ( = 20) randomly selected locations which also improve the highest order transmittance by at least the value set by the transmittance threshold . It repeats this process for N_tmax ( = 5) times. Therefore, a maximum number of N_tmax ( = 5) dielectric wires can be added. If no wire is added, the threshold will be decreased by a factor of two, i.e. if the threshold is positive the new threshold becomes the multiplication of the previous one by half and if the threshold is negative, the updated threshold is the previous one multiplied by two. Otherwise, if at least one dielectric wire has been added, the threshold will be increased by the same factor, i.e. if the threshold is positive, the updated threshold becomes two times the previous one and if the threshold is negative, the updated threshold becomes the multiplication of the current threshold by half. As can be seen, the value of threshold can become very close to zero at some point. In this case, we define a crossover value which we set to be the initial threshold multiplied by 1e-7. If in the process of updating, the value of threshold becomes smaller than this value, the algorithm changes the sign of the new threshold. This way both positive and negative values for the threshold can be obtained (more detailed information about the algorithm is provided in the Appendix).

There are different options to set the initial configuration of dielectric wires. One can choose to start with an arbitrary number (n_d) of dielectric wires. For this, a large number of simulations can be performed, at each simulation, n_d numbers of dielectric wires are randomly distributed inside the region and the best one which has the largest highest order transmittance among others can be chosen as the initial configuration. To highlight the efficiency of our design method, we also consider starting with a periodic configuration. For this purpose, we calculate the transmittance of the highest order mode for all periodic configurations which can be defined within the specified rectangular region [Fig. 7]. We choose the best periodic configuration with the highest transmittance as the initial configuration for each group of point sources.

Fig. 7 The highest order transmittance (dB) for (a) 10 point sources and (b) 15 point sources distributed uniformly on l_o = 0.58λ when illuminating scattering structures populated uniformly by periodic configuration of dielectric wires. The horizontal and vertical axis shows the number of dielectric wires at each column and row, respectively.

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For the case of 10 point sources, the best periodic configuration is that of 9 $\times$ 41 and for the case of 15 point sources the periodic configuration of 9 $\times$ 63 gives the best highest order transmittance. Following the steps of the proposed algorithm which was described previously, we finally obtain structures composed of dielectric wires arranged in such a way that maximize the highest order transmittance of radiating sources. Figures 8(a) and 8(b) show the final configurations for two cases of ten and fifteen point sources, respectively. Although the structures have been designed for particular source locations, we characterize the final structures in terms of the highest order transmittance for different number of point sources distributed on different line scales. As compared to the free space where no scattering medium is present, the disorder-based engineered structures not only can significantly improve the information transmission of the particular sets of source locations they have been designed for, but also they can lead to considerable enhancement of the transmission of highest order signals for other combinations of point sources [(III) of Fig. (8)]. This is important to mention that depending on the configuration of point sources, the maximum achievable enhancement varies. In other words, although the achieved enhancements for other source configurations might be larger than the maximum enhancement we achieved for the particular set of point sources, they do not necessarily correspond to their optimum values which can be reached through the optimization process. To comply with the state-of-the-art fabrication techniques [30, 31], here we have assumed multilayer structures where the freedom in choice of dielectric wire locations is limited in one axis (z axis) while there is no limit to the freedom for the placement of dielectric wires in the horizontal axis (x axis). Nevertheless, there exist more regular types of structures with predefined locations for the dielectric wires in both axes which have less number of degrees of freedom as compared to our generally defined configuration. Various global optimization techniques such as genetic algorithm or simulated annealing [32] can be employed for the design of regular structures similar to works done previously for optical nanoantenna structures [23, 33].

Fig. 8 The highest order transmittance (dB) for disordered metalenses designed for 10 [(a), (c), (e)] and 15 [(b), (d), (f)] point sources distributed uniformly over l_o = 0.58λ. (I) The highest order transmittance for N = 1,2,...,30 and 0.003λ<l_o<4.57λ when there is no scattering medium introduced. (II) The highest order transmittance when the engineered scattering medium composed of 692, 888, 530, 520, 507, and 512 dielectric wires (ε_r = 9, a = 0.02λ), for (a) – (f), respectively, is introduced in front of the radiating point sources. (III) The enhancement achieved for the transmittance of the highest order radiated fields. The black square shows the target point in our design where an enhancement of about 78 dB, 100 dB, 80 dB, 90 dB, 78 dB, and 85 dB, for (a) – (f), respectively, is achieved. tan (φ) indicates the corresponding resolution, which is near 0.064λ [(a), (c), (e)] and 0.041λ [(b), (d), (f)] for 10 and 15 point sources, respectively.

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Now we investigate multilayer structures where the initial configurations are not periodic. Starting with a randomly configured 200 dielectric wires featuring the best transmittance among hundreds of randomly generated configurations, we obtain structures shown in Figs. 8(c) and 8(d). If we let the algorithm put the dielectric wires anywhere along both axes (x and z axis), we can obtain completely disordered structures [Figs. 8(e) and 8(f)], although they might be more complicated in terms of fabrication [34]. Comparing the results indicates that designing structures with more general configurations can lead to the almost the same level of enhancement with less number of scatterers in the structures [Figs. 8(c)-8(f)]. Any point on the 2D plots of Fig. 8 represents a particular number of point sources uniformly distributed over a specific length on the source (object) plane. The tangent of the angle ( $φ$ ) formed by the line connecting any point on these plots to the origin indicates the resolution we require to successfully image all the corresponding point sources. The color of these plots [(I) and (II)] shows the transmittance of the highest order mode signal emitted from corresponding point sources which determines the required SNR of the receiver. As can be seen, by introducing the engineered scattering medium, we can obtain the same level of transmittance with larger number of point sources distributed over the same length on the source plane. In other words, the points pertaining to the same color on (II) make smaller angle (higher resolution) than corresponding points on (I).

Thus far, we assumed the designed metalenses are composed of dielectric inclusions with negligible loss. However, concerning the practical situations, it is worth to investigate how loss would affect the scattering features of designed structures shown in Fig. 8. Assuming that the structures are illuminated by monochromatic radiating sources, the imaginary part of refractive index ( $\sqrt{ϵ_{d}} = n - j ĸ$ ) is changed from zero to 4, i.e. $n = 3,$ $0 \leq ĸ \leq 4$ , and for each value, the transmittance of all modes are computed [see Fig. 9]. In addition, we calculate the average of transmittances of all modes at the presence of designed structures with respect to that of propagating through the free space (relative average transmittance). As expected and is evident from Fig. 9, by increasing the loss of dielectric wires the transmittances of all modes will decrease. However, the rate of changes is not the same for different modes in various configurations. In general, for small values of loss, i.e. $0 \leq ĸ < ĸ' \approx 0.5$ , the transmittances of modes drops faster as compared to higher values of loss. Comparison between the results of engineered structures for ten and fifteen point sources (the left and right columns of plots in Fig. 9, respectively) reveals the higher susceptibility of the former’s modes transmittances to the changes of loss. In other words, the transmittances of the modes associated with ten point sources (including the highest order mode (10^th mode) which determines the ultimate resolution in our method) drop faster as loss increases in this range ( $0 \leq ĸ \leq 4$ ). For large values of $ĸ$ (not shown here), however, the transmittances approach to constant values. It can also be seen from the Fig. 9 that, as loss increases, after some threshold, the rate of changes of relative average transmittances becomes significantly small and approaches a constant value which is smaller in structures with larger number of dielectric wires.

Fig. 9 The transmittance (dB) of radiating modes versus the extinction coefficient (ĸ) of dielectric constant (the horizontal axis) for n = 3. (a), (c), and (e) correspond to the structures of Figs. 8(a), 8(c), and 8(e), respectively, designed for the ten point sources (ten modes), while (b), (d), and (f) correspond to those of Figs. 8(b), 8(d), and 8(f), respectively, designed for the fifteen point sources (fifteen modes). The black solid lines are the relative average transmittances (dB) for each configuration, the value of which can be read from the right vertical axes in the plots (a)-(f).

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In the next Section, we consider one of the designed structures and show how they can be used to enhance the transmission of information in a given imaging/communication system.

3.2 Information enhancement using modified orthogonal transmission channels

In this Section, we explain how the engineered scattering structures improve the transfer of information in an imaging/communication system. Regarding the communication systems, modulated radiating point sources (defined through excitation coefficients; s_n) can be used to transmit information signals. Similarly, from the imaging systems perspective, the scattered fields from illuminated objects can be expanded in terms of radiating point sources with excitation coefficients related to unknown object(s) physical features [35]. In either case, the problem is formulated as described by Eq. (3) and our goal is to retrieve the unknown excitation coefficients from the measured signals over the far-field region. As an example, we consider the designed structure of Fig. 8(c) which is optimized for the ten point sources located over the described source (object) plane. According to the SVD formulation [Eq. (5)], any distribution for the excitation coefficients of ten point sources can be described as a linear combination of ten orthogonal basis vectors (the columns of [V]). In other words, the orthogonal source basis vectors can be used to construct any distribution for the point source excitation coefficients. As mentioned previously, the main role of the designed scattering medium is to maximize the highest order transmittance of the modified orthogonal source basis vectors due to the presence of scattering medium. The modified orthogonal basis vectors, which we refer to as modified transmission channels, can be used to excite the point sources for efficient information transfer in the communication systems. Each vector of the source basis vectors corresponds to one particular vector on the receiver side (one column of [U]). Similar to the point source expansion in terms of orthogonal vectors on the transmitter side, these vectors (the columns of [U]) can be used to construct any field distribution due to the source excitation, over the image plane. The highest order transmittance, which is maximized at the presence of the designed structure, corresponds to the transmission of the source basis vector with the highest spatial frequency which is also related to the high resolution information of radiating sources/illuminated object(s). Figures 10 and 11 demonstrate the source basis vectors and the associated eigenvalues (square root of transmittance), before and after the scattering medium is introduced, respectively. As can be seen, the imaginary parts of the source basis vectors assume nonzero values when the scattering medium is introduced and become larger for higher order signals. This reveals that by applying proper phase shifts between source signals, the multiple scattering events that signals experience as they propagate through the scattering medium are compensated, leading to an improved transmission of sub-wavelength information to the far-field region. Similar phenomenon has been observed and exploited to achieve perfect optical focusing through disordered structures [36]. It is important to mention that the improvement in resolution comes at the expense of decreased “average total” transmittance which for the designed scattering medium is about 55% of that of the free space medium. This is attributed to the multiple scattering occurring inside the structure also causing some reflection from the medium. Figure 12 shows the total electric field distribution after exciting the highest order mode for both free space and the designed scattering structure. As it is clear, the scattering medium significantly improves the signal level at the far field region. Depending on the noise level, or SNR, the accurate extraction of the source information is limited by the power of the highest order mode which is the weakest component of the transmitted signals. According to Eq. (6), as mentioned before, each component can be obtained using the orthogonality property of basis vectors. However, the added noise is not cancelled in this process and has the maximum effect on reducing the SNR of the weakest component of the signal.

Fig. 10 (a) Orthogonal source basis vectors for 10 numbers of point sources distributed uniformly over l_o = 0.58λ and d_o = 0.16λ, when there is no scattering medium introduced. The field measurements are performed at $M = 200$ points over the image plane of l_i = 400λ and d_i = 20 λ. (b) The corresponding orthogonal basis vectors over the image plane.

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Fig. 11 (a) The modified orthogonal source basis vectors for 10 numbers of point sources distributed uniformly over l_o = 0.58λ and d_o = 0.16λ, when the scattering medium of Fig. 8(c) is introduced. The field measurements are performed at M = 200 points over the image plane of l_i = 400λ and d_i = 20 λ. (b) The corresponding modified orthogonal basis vectors over the image plane.

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Fig. 12 The electric field distribution (absolute value) (dB) for exciting (a) the highest order mode in free space and (b) the modified highest order mode at the presence of designed scattering structure [Fig. 8(c)].

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As an example we consider ten random source signals (N = 10) with excitation coefficients ( $s_{n}, n = 1, 2, \dots, 10$ ) lying on the perimeter of circle $r = (1 / \sqrt{10})$ over the complex (amplitude and phase) plane. To extract the source signals from the measured signal in Eq. (6), we discard terms with transmittance level lower than the noise level of the receiver. We add white Gaussian noise (WGN) with the SNR changing from 50 dB to 180 dB and take the average of root mean square (RMS) errors between hundreds of source distributions and the corresponding retrieved images. Comparing the results for propagation through the free space and the engineered scattering medium, we can observe the significant improvement in restoring the source signals [Fig. 13]. By introducing the engineered scattering structure of Fig. 8(c), with the same level of SNR, the source signals can be extracted more accurately. As can be seen from Fig. 13, even for lower SNRs the engineered structure reduces the RMS error. However, the significant improvement takes place once the SNR becomes close to or larger than 100 dB [blue curve in Fig. 13]. The reason can be understood by comparing the transmittances of highest order modes in Fig. 10 and Fig. 11. The highest order transmittance of the modified source basis vectors, which is about −96.1 dB, determines the required minimum SNR to accurately retrieve the information signals. The same procedure can be done for the case of seven point sources (N = 7) distributed uniformly over the same region, with excitation coefficients ( $s_{n}, n = 1, 2, \dots, 7$ ) lying on the perimeter of circle $r = (1 / \sqrt{7})$ over the complex (amplitude and phase) plane. The modified highest order transmittance for seven point sources is about −67.5 dB which set the minimum SNR for accurate extraction of information signals [green curve in Fig. 13]. From the communication point of view, if we were to send ten pieces of information signals through the ten point sources radiating in free space and located at sub-wavelength distances (0.064 $λ$ in this case), higher order modes get strongly attenuated as they propagate towards the receiver. Unless we have a very large value for the SNR, which is not practically possible, we are not able to restore all the source signals from the received signal. However, for the same level of SNR, the engineered scattering medium is capable to enhance the level of transmitted power of higher order modes so that the number of received signals (channels) increases. Depending on the SNR, the modified orthogonal transmission channels with acceptable level of transmittance can be used to modulate and combine more information signals which can be restored with minimum error. For example, if the SNR level at the receiver side is 70 dB (or 100 dB), according to the level of transmittances shown in Fig. 11, the first seven (or ten) independent modified basis vectors can be used to modulate information signals. For digital binary communication, that means a realization of $2^{7}$ (or $2^{10}$ ) pieces of information, which is a two(eight)-fold enhancement as compared to the propagation through the free space, where the first six (or seven) channels of acceptable transmittance levels, which is equivalent to the realization of $2^{6}$ (or $2^{7}$ ) pieces of information, can be chosen to modulate signals.

Fig. 13 The root mean square error versus the SNR of receiver for propagation through both free space (red and black) and engineered scattering structure (blue and green) of Fig. 8(c). The results show the average values taken over hundreds of random source distributions for both 10 point sources and 7 point sources distributed uniformly over l_o = 0.58λ and d_o = 0.16λ. The field measurements are performed at M = 200 points over the image plane of l_i = 400λ and d_i = 20 λ.

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4. Conclusion and perspective

We proposed a new approach for enhancing the transmission of high resolution signals emitted from point sources located at ultra sub-wavelength distances which leads to an enhanced imaging resolution and information capacity in communication systems. Our method is based on employing disordered engineered scattering structures. The high degrees of freedom inherent in such disordered medium enable us to come up with structures which have superior functionality as compared to their periodic and random counterparts. We demonstrated that the disorder in such structures can be exploited to significantly minimize the correlation between the radiated signals emitted from adjacent point sources at ultra sub-wavelength distances. An SVD technique is used to increase the transmittance of the highest order signals emitted from a given source region. For this goal, we applied our proposed global optimization method to design scattering structures with high degrees of freedom. Comparison with completely random and periodic configuration verifies the effectiveness of our method. We studied one particular example of ten point sources located at sub-wavelength distances from each other where the designed scattering medium significantly improves the level of accuracy in restoring the source signals for a given level of SNR at the receiver. Although the analysis and design procedure were performed for 2D structures and scattering medium with specific physical properties for scattering inclusions (dielectric wires with specific diameters and dielectric constants), the proposed approach can be applied to more general 3D structures with more stringent physical constraints which comply with available fabrication techniques and obtain even more improvements by introducing stronger scattering inclusions (higher dielectric constant) within the structure. Moreover, as it is shown, significant improvement on resolution can be achieved for other configuration of point sources or at different locations for image plane which, depending on the receiver SNR level, can further enhance the effectiveness of the method.

5 Appendix: Freeform engineered disordered metalenes: the algorithm

The flowchart shown in Fig. 14 summarizes the described algorithm for designing engineered disordered all-dielectric medium. It consists of a few main steps:

Fig. 14 The flowchart of the proposed algorithm used for engineering the scattering features of the dielectric medium.

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1. Initialization
- a. An initial configuration for dielectric inclusions (dielectric wires in this work) is chosen. The initial configuration features the best value for the cost function (the highest order transmittance in this work) among hundreds of random configurations with the same number of scatterers.
- b. The transmittance threshold is initialized. The value should be larger than the expected (average) value that the transmittance will be increased due to adding one new dielectric inclusion to the existing scattering structure which consists of $n_{d}$ dielectric inclusions.
- c. The controlling parameters, M_t and N_t, are set to one. M_tmax determines the maximum number of tries to find the best location for adding one new dielectric inclusion. N_tmax determines the maximum number of iterations before the transmittance threshold is updated. Each iteration consists of M_tmax tries to add one dielectric inclusion to the existing configuration. In addition, N_tmax represents the maximum number of dielectric inclusions which can be added where all satisfy the condition of increasing the cost function at least by the transmittance threshold, which is constant during the N_tmax iterations.
- d. New sets are defined for the temporary locations which are candidates for the location of one new dielectric inclusion, and the corresponding enhancements in the transmittance, as {(x, y)_t} and {C_t}, respectively.
2. Random Location
One random location for adding one temporary new dielectric inclusion is chosen within the structure. The location and resultant transmittance in the new configuration are denoted by (x_r, y_r), and C_r, respectively.
3. Temporary Selection Criterion
If the new transmittance is larger than the previous one at least by the current value of the transmittance threshold, (x_r, y_r) and C_r will be added to {(x, y)_t} and {C_t}, respectively. M_t is increased by one and the algorithm starts from step 2, unless M_t is equal to M_tmax.
4. Final Selection
After steps 2 and 3 are repeated for M_tmax times, the initial (current) configuration is updated by adding one new dielectric inclusion (if any) at the best location in {(x, y)_t}, which is associated with the maximum value in {C_t}. The two sets, {(x, y)_t} and {C_t}, are reset and M_t is set to one. N_t is increased by one and the algorithm starts from step 2, unless N_t is equal to N_tmax.
5. Updating Threshold
The transmittance threshold is updated after N_tmax repetitions of steps 2 and 3, where a maximum number of N_tmax dielectric inclusions could be added to the initial configuration. However, depending on the configuration or the value of transmittance threshold, the number of dielectric inclusions that have been added to the initial configuration varies. Two cases are, first, at least one dielectric inclusion has been added and, second, no location has been selected in previous steps to add one new dielectric inclusion. Depending on the case, the transmittance threshold is updated as follows:
$T h r e s h o l d_{n e w} = γ \times s i g n (| γ | - C r o s s o v e r v a l u e),^{} γ = (n - 1 + α^{β}) \times T h r e s h o l d_{o l d}$
where
$\begin{array}{l} α = (\frac{1}{n} + 1 - n), \\ β = \frac{1 \pm s i g n (T h r e s h o l d_{o l d})}{2}, (+ (-) : N o (a t l e a s t o n e) d i e l e c t r i c i n c l u s i o n a d d e d) . \end{array}$
$n$ is a coefficient which is used to update the threshold by multiplying the current threshold by $1 / n$ or $n$ which depends on the situation (two cases outlined before). The numerical value of this parameter affects how the configuration of dielectric inclusions is updated and how long it will take the algorithm to reach to the final configuration. Larger value may accelerate the process of optimization, however, the efficiency of algorithm in finding the best configuration might be negatively affected. This is due to the fact that the value of threshold becomes negative before sufficient number of iterations has been tried to find better locations for new dielectric inclusions maximizing the transmittance by larger values. On the other hand, smaller coefficients lead to longer process, because of the longer time it takes the algorithm to update the threshold to lower values to let one new dielectric inclusion be added to the configuration. This happens in a situation where the configuration represents a local maximum (minimum) in the solution space. In this work we assume a constant value for $n$ . The $C r o s s o v e r v a l u e$ is chosen to be a very small number (close to zero). As the threshold becomes smaller than this value, its sign will change. This avoids getting stuck on local maximum (minimum) and allows the algorithm to search for global maximum (minimum) in the solution space. This is done through accepting new dielectric inclusions into the locally optimum configuration making the cost function (the transmittance in this work) slightly lower (or higher for minimizing cost function) temporarily. However, according to Eq. (8), threshold will be updated, afterwards, in such a way that the algorithm looks for locations to add new dielectric inclusions in the new configuration which ultimately leads to the highest enhancement in the transmittance obtained so far.
6. Stopping Criterion
The previous steps will be repeated until a desired value for the cost function is obtained or the user stops the process for any other reason such as reaching to the preset number of dielectric inclusions.

Funding

This work has been supported by the National Science and Engineering Research Council (NSERC) of Canada, Black-Berry (formerly RIM), and C-COM Satellite Systems.

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Freeform engineered disordered metalenses for super-resolution imaging and communication

Abstract

1. Introduction

2. Analysis and design of scattering features of disordered all dielectric medium

2.1 The analysis

2.2 Engineered disordered all dielectric medium

3. Numerical results

3.1 Design examples

3.2 Information enhancement using modified orthogonal transmission channels

4. Conclusion and perspective

5 Appendix: Freeform engineered disordered metalenes: the algorithm

Funding

References and links

Cited By

Figures (14)

Equations (9)

Optics Express