1. Introduction

Using optics to convey and process analog radio-frequency (RF) signals has long been recognized for its potential to overcome barriers associated with purely electronic approaches [1–9]. For example, the low propagation loss and negligible dispersion over broad bandwidth has been seen as the main advantage of employing RF over fiber, but the nonlinear transfer function that limits spur-free dynamic range (SFDR) and optical-power/modulation-frequency tradeoff that compromises noise figure have been considered the main drawbacks of the technology [10,11]. Whereas the published work on RF-photonic processing largely focuses on individual time-variable signals, it has been shown [12–18] that spatially coherent up-conversion of radio waves to optical domain, with the attendant shortening of the wavelength by several orders of magnitude, offers unique opportunities in the analog spatial processing of the RF waves in a small physical volume. Passive millimeter-wave imager [13–15] and imaging receiver [19] are two examples of systems that exploit the coherent photonic up-conversion of RF and free-space optical processing to achieve functionalities and performance that prove challenging using all-electronic approaches. However, in those systems, the role of optical fibers is limited to the conveying of coherently up-converted RF waves from antennas probing the electromagnetic field to the free-space optical processor where the actual functionality of the system is realized. To ensure proper image formation in the optical domain, the fibers are cut to identical length and the instantaneous bandwidth is restricted so as to limit squint, i.e., the chromatic dispersion that arises from the conversion of fixed true-time delays to phase differences.

Here, we extend the optical processing in the architectures of [13–18] to include the optical fibers by introducing unequal lengths of fibers conveying the modulated optical beams from the antennas to the free-space portion of the optical processor. As a result, the interference pattern produced by the outputs of such fibers contains correlations among signals having various time delays; this enables the detection of RF frequencies in addition to the angle of arrival (AoA) afforded by the spatial extent of the antenna array. The spectral resolution is proportional to the reciprocal of the range of fiber-length differences and the angular resolution is proportional to the reciprocal of the antenna-array physical size. The variation of fiber lengths that provides for the detection of temporal correlations, and the consequent ability to detect frequency, carries the cost of scrambling the image of the RF environment formed in the optical domain (i.e., breaking the direct, one-to-one spatial correspondence between the optical energy distribution and the AoA-s of emitters in the scene, as demonstrated in [13–15]). Thus, to recover the full information about the electromagnetic environment, captured by the antenna array, from the detected interference pattern, computational reconstruction employing tomography algorithms [20–33] is used. The functionality of the resulting opto-electronic system can be thought of as providing an instant Computed Tomography [34] (CT) scan of the electromagnetic environment.

This new capability is particularly timely in light of society’s increasing reliance on electromagnetic (EM) waves for wireless communication, sensing, and countless other pursuits. With the recent opening of frequency bands between 700 MHz and 71 GHz to cellular networks, and the incorporation of beam forming to increase network capacity in nascent 5G technologies [35–38], the complete spatial and spectral awareness of the electromagnetic environment becomes paramount to best take advantage of this finite resource for both practical and scientific ends. The k-space tomograph demonstrated here addresses this essential need to provide broadband awareness of the EM environment—to take an instant CT scan of radio waves. It does so by harnessing photonics for coherent probing of radio waves over an aperture extended in space and time, and by optical processing of the intercepted signals.

In the remainder of this article, we expound on the operation principle of the system in Sec. 2, which is followed by computer simulations in Sec. 3, and experimental results, in Sec. 4, that include the description of the physical implementation of the system. We summarize our findings in Sec. 5. The appendix, Sec. 6, details the mathematical foundations of the system operation, and Sec. 7 provides the description of visualizations available online.

2. Principle of operation

Radio waves filling the space around us can be represented as a superposition of plane waves arriving at the speed of light from various directions [39]. A plane wave is characterized by the direction of propagation and frequency that together form a four-vector (ω,k), where, in free space, the angular frequency ω is related to the wave propagation vector k according to ω = c|k|, where c is the speed of light. Given this relation, it is sufficient to specify a three-vector k, in addition to signal amplitude, to fully characterize a plane wave; such three-vectors are referred to as k-vectors, and their collection, as k-space. Thus, to fully specify the electromagnetic field that varies in both space and time, it is sufficient to provide the amplitudes of its plane-wave components, i.e., the amplitude distribution in k-space. Such k-space representation of the electromagnetic field is natural from the standpoint of a radio-frequency (RF) receiver that intercepts incoming waves. In this representation, an incoming plane wave shows as a bright spot in the three-dimensional k-space, see Visualization 1 and Sec. 7 for details.

To fully identify the incoming waves, a receiver must sense the field distribution over both space and time, and process this spatio-temporal information accordingly. The k-space tomograph investigated here does so by harnessing light for the collection of RF signals and their processing [13–15,18], see Fig. 1. The front end is a distributed array of antennas that probe the electromagnetic field at discrete spatial locations. Signals thus obtained are used to modulate an optical beam split among the antenna locations, where each antenna is coupled to a respective electro-optic modulator. The modulation imposes sidebands on the optical carrier fed into the modulators. The optical power in the sidebands is directly proportional to the RF power captured by the antennas, and the modulation process preserves phase information contained in the RF signal. With the optical carrier shared among all the antennas, the distributed modulation has the effect of coherently up-converting the radio wavefront received by the antennas to an optical wavefront carried by optical fibers.

 

Fig. 1 Conceptual diagram of a k-space tomograph.

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To provide for temporal sensing and processing, the optical fibers coming out of the modulators differ in length. This way, the spatial aperture formed by the antenna array is extended into temporal dimension, which thereby allows sensing the (temporal) spectrum of the incoming waves, along with sensing their angles of arrival afforded by the spatial extent of the antenna array. The fiber ends form an array, out of which light propagates in free space. Following carrier-wavelength and upper-wavelength sidebands suppression, the remaining modulation sidebands generate an interference pattern, captured by a CCD camera, that contains information about the incoming RF waves, Fig. 1. To recover this information, i.e. to reconstruct the distribution of waves in k-space, we note that the optical power distribution in the interference pattern is a linear function of the RF power distribution in k-space received at the antenna front end, since the power in the sidebands is directly proportional to the captured RF power. As such, the optical power $P_n$ detected by the n^th element of the CCD can be expressed in the following form (see Appendix for details):

Equation (1) has a form analogous to that encountered in computed tomography (CT) where an X-ray is attenuated as it passes through tissue by the integrated (summed) density of material in its path. With a sufficient number of differently oriented X-rays penetrating the volume of interest, the 3D distribution of density may be recovered computationally from the detected integrated densities obtained for all rays. Similarly, with a sufficient diversity of patterns in the weight matrix ($a_{nk}$) the reconstruction of the wave distribution $S$ in k-space from detected intensities $P_n$ is possible. Accordingly, algorithms used in real-space computed tomography are employed in k-space reconstruction, including the Kaczmarz method [20,40], also known as the algebraic reconstruction technique (ART), and its multiplicative varieties (MART) [21,23].

3. Numerical simulations

The ability to reconstruct the distribution of incoming waves in k-space is first tested numerically. For convenience in visualization, we limited the simulations to two dimensions, $k_x$ and $k_y$ and chose two octaves of bandwidth between 15 GHz and 60 GHz. Accordingly, we constructed a numerical model of a k-space tomograph with 180 antennas uniformly distributed around a 10-cm-wide circle, and assumed a random fiber-length variation within 10 cm, illustrated in Fig. 2.

 

Fig. 2 Notional configuration of the system used for numerical simulations of a k-space tomograph; inset shows an example of a weight map a_n corresponding to one of the photo-detectors in the detector array; the labeled equi-frequency contours in the weight map provide scale in the k-space.

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Such an antenna arrangement allows the detection of k-vectors in plane, i.e., the azimuth AoA and frequency, and serves to illustrate the flexibility in arranging antennas along arbitrary geometries. The specific choice of fiber-length random distribution has little effect on the quality of reconstruction, as long as the probability is set to uniform when generating the random fiber lengths. Light in our numerical model was captured with a 15,000-element photo-detector array. Each of the photo-detectors has a weight map $a_n$ associated with it—similar to the example weight map shown in the inset of Fig. 2.

Assuming the sources are far from the antenna array, the incoming RF radiation arrives at the receiver as approximately plane waves and is represented as distributions of Dirac delta functions in k-space. Panels (a), (b), (c) and (d) in Fig. 3 show several such distributions with different numbers of RF sources randomly scattered throughout k-space; the scenes contain 23, 43, 83, and 163 sources, respectively. The corresponding reconstructions obtained using Kaczmarz method [20,40] are shown in panels (a’), (b’), (c’) and (d’). Notably, the positions in k-space of all the sources in the scenes have been faithfully recovered. The reconstructed sizes of the sources are larger than the original point sources in the scene set-up. This dilation is the result of diffraction due to the finite size of the receiving aperture (10 cm) and the finite range of fiber-length distribution (10 cm) combined with a finite wavelength of the detected RF radiation. The spatial and spectral resolutions of such a system may be estimated by analyzing the granularity of the weight maps, such as that shown in the inset of Fig. 2, and, to the first order may be approximated by $\lambda /D$ for angular, and $1/\Delta t$ for spectral resolution, where $\lambda$ is the RF wavelength, D is the circle diameter, and $\Delta t$ is the maximum time-delay difference induced by the uneven fiber lengths. The reconstructed scenes also contain a background ‘noise’ pattern. This ‘noise’ pattern is an artifact of the iterative reconstruction process used in the simulations and may be further suppressed by increasing the number of iterations.

 

Fig. 3 Comparison of a distribution of sources in k-space and their tomographic reconstruction from Eq. (1). (a), (b), (c), (d) show scenes set up with various numbers of emitters distributed throughout k-space whereas (a’), (b’), (c’), (d’) show the corresponding reconstructions. These scenes contain 23, 43, 83, and 163 sources, respectively.

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4. Experimental testing

To test the concept of k-space tomography experimentally, we constructed a receiver with eight broadband antennas, 3.5 cm apart, coupled to electro-optic modulators—see Figs. 4 and 5 for details. Figure 4 shows a linear distribution of Vivaldi antennas forming the front end of the system, followed by low-noise amplifiers that boost the signals fed to electro-optic modulators. The optical inputs to the modulators are all derived from a common laser, indicated in Fig. 5, that is amplified using an erbium-doped fiber amplifier (EDFA) before splitting it 8 ways. The modulator outputs, carrying up-converted RF signals, proceed through a set of optical fiber segments that have been custom spliced so as to realize a desired set of regularly spaced total fiber path lengths, comprising a temporal aperture of 13.4 cm (difference between longest and shortest paths), equivalent to 19.6 cm in free space. The lengths of these segments were (in centimeters): 43.5, 37.8, 45.4, 40.9, 33.9, 35.9, 41.6, 47.3. These fiber sections next feed the up-converted RF signals through an array of phase shifters, Fig. 5. This custom-made, integrated LiNbO₃, 1x8 phase-shifter array is designed to provide low-speed phase corrections to compensate for phase variations induced as the signals travel through separate loose fibers, each subject to unequal vibrations and other environmental effects [16,17]. Finally, the outputs of the 8 phase-control modulators proceed through an array of fibers, whose end facets have been mounted in a 1x8 silicon V-groove array, from whence the up-converted signals propagate into the free-space optical processor. All fibers used in the system are polarization maintaining.

 

Fig. 4 Linear k-space tomograph constructed at the University of Delaware. Inset shows a simplified schematic diagram.

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Fig. 5 Schematic diagram of an 8-channel k-space tomograph. EDFA: erbium-doped fiber amplifier. PBS: polarizing beam splitter. BS: beam splitter. LNA: low-noise amplifier.

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In the free-space optical processor, a linear lenslet array collimates the outputs of the fibers, and the combined beam is directed to a polarizing beam-splitter cube oriented to pass it. Then, the beam passes through a quarter-wave plate oriented at 45° to circularize its polarization, which is followed by a set of filters designed to pass the lower-wavelength sideband while reflecting the optical carrier wavelength. A set of lenses relay the sideband beam to the sensor of an infrared camera that captures the resulting interference pattern. One-dimensional (1D) interferograms are generated from the 2D sensor output (WiDy SWIR 320U from New Imaging Technologies) by extracting in software only the region of the sensor illuminated by the interference pattern (a 174-column x 48-row sub-region of the array) and averaging over the rows.

The reflected carrier is directed back through the quarter-wave plate, whereupon it is converted back to linear polarization, but with the orientation rotated 90° with respect to the fiber outputs. Therefore, it is reflected by the polarizing beam splitter into a separate branch of the optical processor, see Fig. 5. It passes through a relay lens so positioned as to form an image of the fiber output facets. This image is combined on a linear photo-detector array with a reference beam derived from the primary laser using a 5% fiber tap prior to the 1x8 splitter described above. The resulting beat signal is used to track the phase drift in the 8 separate fiber paths. The feedback signal is calculated in real time using field-programmable gate array FPGA (Xilinx ML605), and delivered to the phase-control modulator array discussed previously. See Refs. [16,17] for further details on the operation of the phase-control feedback loop.

We calibrated the receiver by measuring the weight maps for the photodetectors over a bandwidth of 12-18 GHz and over ± 15° of azimuth angle at a resolution of 100 MHz and 0.5°, respectively. Figure 6(a) shows one of the 174 weight maps measured in this tomograph. Visualization 7 shows all captured weight maps as a function of pixel index in an animation; see also Sec. 7 for details. We then set up scenes comprising test sources transmitting at various frequencies and placed at various locations relative to the receiver to produce a variety of different electromagnetic waves incident on the antenna array. A typical photo-diode-array response, i.e., a distribution $P_n$ corresponding to an RF wave intercepted by the antenna array is shown in Fig. 6(b). Such a distribution served as input to Eq. (1), along with the measured weight matrix ($a_{nk}$) obtained from calibration, to reconstruct the distribution of incoming waves in k-space. Some of these reconstructions are shown in Figs. 6(c) though 6(f). For easy reference, the positions of the incoming waves in k-space, as set up by tuning the source and its physical placement relative to the receiver array, are indicated with a plus ‘ + ’ sign, with the sizes of the ‘ + ’ signs in Fig. 6(f) indicating the relative power of the sources. It can be ascertained from these figures that reasonable quality of reconstruction in terms of the recovery of the waves’ AoAs and frequencies is possible even with the considerably limited size of the array counting only eight elements. The presence of artifacts, including faint ‘phantom’ sources visible in the reconstruction, may be attributed to the small number of elements in the array and the regular distribution of the antennas in the array as well as regularly-spaced fiber lengths. The deviation of the reconstructed positions of sources in k-space from their set positions may be the result of a small drift of RF phases introduced by the front-end LNA-s between the calibration and the acquisition of data used in the reconstruction.

 

Fig. 6 Experimental testing of k-space tomography with a system of Figs. 4 and 5. (a) Weight map corresponding to one of the photodetectors in the tomograph of Figs. 4 and 5. (b) Distribution of light among photo-detectors in the array with a single RF plane wave at normal incidence and 16.7 GHz frequency. (c)-(f) Reconstruction of k-space distributions for various source distributions. The plus ‘ + ’ signs indicate the nominal positions and frequencies of sources as set up in the experiment. Sizes of the ‘ + ’ signs in (f) indicate the relative powers of the sources.

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5. Summary

We presented above a method of using spatial and temporal optical coherence to achieve a new imaging modality for a complete tomographic reconstruction of an electromagnetic environment in the RF portion of the spectrum. The k-space tomograph employs coherent optical up-conversion of electromagnetic waves sampled at discrete points in space, and temporal skewing afforded by fiber-length dispersion, to produce a spatio-temporal aperture that can sense both the frequency and AoA of incoming waves. Algebraic reconstruction techniques, commonly employed in computed tomography (CT), are then applied to reconstruct the distribution of incoming waves in k-space. Numerical simulations and experimental testing indicate the ability to simultaneously recover the frequencies and AoA-s of intercepted waves. As shown in Sections 3 and 4, the frequencies and AoA-s of multiple sources may be accurately recovered using the approach described here. As a result, we are able to image in the radio-spectrum of electromagnetic waves—to probe the space of radio waves and provide complete awareness of transmitters operating in the vicinity.

6 Appendix

The formation of the interference pattern captured by CCD can be expressed as follows:

(2)$$C_n=\frac{1}{\sqrt{N}}{\displaystyle \sum _{m=0}^{M-1}{B}_m{e}^{i\left(\omega t+{\theta }_{nm}+{\phi }_m\right)}}+\text{c}\text{.c}\text{.}$$

where $B_m$ is the amplitude of the field at the output of the m^th fiber, $C_n$ is the amplitude of the field at the n^th pixel of the CCD, ${\phi }_m$ is the (RF-modulated) phase of the optical beam in the m^th fiber, and ${\theta }_{nm}$ is the phase the optical beam picks up as it propagates in free space from m^th fiber to n^th pixel; it is assumed that there are M optical fibers, N sensing elements in the CCD array, and that the intensity of light coming out of each fiber is evenly distributed among the N sensors of the CCD. ‘c.c.’ denotes the complex conjugate of the preceding term.

Assume now that the scene consists of K point sources that emit radio waves incoherently. In case of phase modulation, the phases in the individual M channels are

(3)$${\phi }_m={\displaystyle \sum _{k=0}^{K-1}{\tilde{S}}_k}\cos [{\Omega }_k(t+t_m)+{\varphi }_{km}],$$

where ${\Omega }_k$ is the angular frequency of the k^th source, t is the time, $t_m$ is the time required for the signal in m^th channel to traverse the fiber from the modulator to the fiber array, and ${\varphi }_{km}$ is the phase picked up by the propagating wave on the way to the antenna; it depends on both the scene element, indexed by k, and on the channel, indexed by m, as well as on frequency ${\Omega }_k$. ${\tilde{S}}_k$ are the amplitudes of the signals scaled by the modulation efficiency of the modulators and the distance from the source to the receiving antenna. Identical modulation efficiency is assumed for modulators in all channels.

Substituting (3) into (2) and applying optical spectral filtering to extract the lower sideband (LSB) yields, after simple algebra and linear approximation, the following expression for the optical field amplitude at n^th pixel:

(4)$$C_n^{LSB}=\frac{i{e}^{i\omega t}}{2\sqrt{N}}{{\displaystyle \sum _{mk}{B}_m{\tilde{S}}_{k}e}}^{i[{\theta }_{nm}-{\varphi }_{km}-{\Omega }_k(t+t_m)]}+c.c.$$

Optical sensing elements detect the intensity ${\left|C_n^{LSB}\right|}^2$ rather than the amplitude of the field. Furthermore, they integrate, or average, this intensity over time to yield $〈{\left|C_n^{LSB}\right|}^2〉$ as the detected signal. Assuming that the integration time is much longer than the reciprocal of the modulation frequency ${\Omega }_k$, the optical power detected at the n^th pixel is found as

(5)$$〈{\left|C_n^{LSB}\right|}^2〉=\frac{1}{2N}{\displaystyle \sum _k{\left|{\displaystyle \sum _m{\tilde{S}}_k{B}_m{e}^{i\left({\theta }_{nm}-{\varphi }_{km}-{\Omega }_k{t}_m\right)}}\right|}^2}.$$

To account for finite size photo-detectors and their uneven illumination, Eq. (5) is generalized to

(6)$$〈{\left|C^{LSB}\left(x\right)\right|}^2〉=\frac{1}{2N}{\displaystyle \sum _k{\left|{\displaystyle \sum _m{\tilde{S}}_k{a}_m\left(x\right)B_m{e}^{-i\left({\varphi }_{km}+{\Omega }_k{t}_m\right)}}\right|}^2},$$

where $〈{\left|C_{}^{LSB}(x)\right|}^2〉$ is the linear (or areal) optical power density of lower sideband incident on the plane of photo-detector array at point x, and $a_m(x)$ is the amplitude density of the optical beam emanating from m^th fiber and incident at point x; note that the phase ${\theta }_{nm}$ of Eq. (5) is absorbed into the complex amplitude $a_m(x)$. For RF sources sufficiently far away from the antenna aperture, the waves incident on the antenna array are approximately plane waves. Therefore, phase ${\varphi }_{km}$ can be expressed as ${\varphi }_{km}=K_k\cdot X_m$ where $K_k$ is the wave-vector corresponding to a wave produced by source k, and $X_m$ is the position of m^th antenna in the array. For uncorrelated sources, Eq. (6) simplifies to

(7)$$〈{\left|C^{LSB}\left(x\right)\right|}^2〉=\frac{1}{2N}{\displaystyle \sum _k{\left|{\tilde{S}}_k\right|}^2{\left|{\displaystyle \sum _m{a}_m\left(x\right)B_m{e}^{-i\left(K_k\cdot X_m+{\Omega }_k{t}_m\right)}}\right|}^2}.$$

The power detected by the n^th photo-detector is obtained by integrating power density $〈{\left|C^{LSB}\left(x\right)\right|}^2〉$ of Eq. (7) over the photo-detector area. For a photo-detector sufficiently small that the power density changes little over its extent, the integration may be carried out by simply multiplying $〈{\left|C_{}^{LSB}(x)\right|}^2〉$ by the photo-detector area A:

(8)$$\begin{array}{c}{P}_n={\displaystyle \underset{\begin{array}{c}\text{P/D}\\ \text{area}\end{array}}{\int }〈{\left|C^{LSB}\left(x\right)\right|}^2〉}{d}^{2}x=〈{\left|C^{LSB}\left(x_n\right)\right|}^2〉A=\\ =\frac{A}{2N}{\displaystyle \sum _k{\left|{\tilde{S}}_k\right|}^2{\left|{\displaystyle \sum _m{a}_m\left(x_n\right)B_m{e}^{-i\left(K_k\cdot X_m+{\Omega }_k{t}_m\right)}}\right|}^2},\end{array}$$

where $x_n$ is the position of n^th photo-detector.

Expression (8) has the following form

(9)$$P_n={\displaystyle \sum _k{a}_{nk}{S}_k},$$

where $S_k~{\left|{\tilde{S}}_k\right|}^2$ is the power in the plane wave associated with the k^th source in the scene that is captured by the receiver, and

(10)$$a_{nk}=\frac{A}{2N}{\left|{\displaystyle \sum _m{a}_m\left(x_n\right)B_m{e}^{-i\left(K_k\cdot X_m+{\Omega }_k{t}_m\right)}}\right|}^2.$$

Equation (9) is a matrix equation relating the RF power distribution in k-space with the optical power detected by the photo-detector array. It is referred to as Eq. (1) in Section 2, and is the basis for operation of the k-space tomograph described therein.

For Gaussian beams emanating from the ends of the fibers, the complex amplitude of the optical field takes the following form

(11)$$a_m\left(x_n\right)=\frac{1}{z-i{z}_0}\exp \left[\frac{2\pi i}{\lambda }\frac{{\left|x_n-x_m\right|}^2}{2\left(z-i{z}_0\right)}\right],$$

where $x_m$ is the (lateral) position of the m^th fiber in the array, $x_n$is the (lateral) position of the n^th photo-detector, z is the axial distance from the end of the fiber array to the photo-detector array, $\lambda$is the optical wavelength, and $z_0$ is the Rayleigh distance that is related to the Gaussian-beam waist $w_0^{}$ according to

(12)$$w_0^2=\frac{z_0\lambda }{\pi }.$$

For z large, i.e., much greater than $z_0$ and $\left|x_m\right|$, or when using an optical system (a lens) to bring $z=\infty$ to a finite position, expression (11) for the complex Gaussian amplitude reduces to

(13)$$a_m\left(x_n\right)=\frac{1}{z-i{z}_0}\exp \left[\frac{2\pi i}{\lambda }\frac{{\left|x_n\right|}^2}{2\left(z-i{z}_0\right)}\right]\exp \left[-\frac{2\pi i}{\lambda }\frac{x_n\cdot x_m}{\left(z-i{z}_0\right)}\right],$$

which substituted into (8) yields

(14)$$\begin{array}{c}{P}_n=\frac{1}{2N}\frac{A}{z^2}\exp \left[-\frac{1}{2}{\left|\frac{2\pi }{\lambda }{w}_0\frac{x_n}{z}\right|}^2\right]\times \\ \times {\displaystyle \sum _k{\left|{\tilde{S}}_k\right|}^2{\left|{\displaystyle \sum _m{B}_m\exp \left[-i\left(\frac{2\pi }{\lambda }{x}_m\cdot \frac{x_n}{z}+K_k\cdot X_m+{\Omega }_k{t}_m\right)\right]}\right|}^2}.\end{array}$$

7. Visualizations

Figure 7, Fig. 8, Fig. 9, Fig. 10, and Fig. 11 are snapshots of animations available online.

 

Fig. 7 RF sources in k-space. Single-frame of an animation, Visualization 1, showing five transmitters, four frequency-hopping and one frequency-swept. Crowded frequency spectrum (see Visualization 1) shown in lower left pane, and the frequency variation makes it difficult to identify the sources. Discerning the angle of arrival, the azimuth and elevation in the upper left pane, allows for easy identification of the positions of the sources. K-space tomography provides full 3D spatial-spectral location of all transmitters, right pane.

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Fig. 8 k-space mapping for array of antennas coupled to fibers with identical lengths. Left side corresponds to a two-dimensional k-space weight map. On the right, antennas are coupled to optical fibers that form an array. A lens images light emanating from the fiber array onto a line of photo-detectors. Different photo-detectors receive different contributions from RF sources represented as points in k-space, see Visualization 2. The brighter the regions in k-space, the greater the contribution from the RF sources with the corresponding k-vectors to the signal detected at the photo-detector position of the bright star on the right.

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Fig. 9 Effect of changing fiber lengths on weight map. As the lengths of the fibers at the top of the array increase relative to the fibers at the bottom, the bright region in the k-space weight map deforms from a straight line to a hyperbola (see Visualization 3). For fibers with fixed lengths increasing linearly from the top to the bottom fiber, the bright region of high contribution in the weight map on the left shifts as the position of the photo-detector changes on the right (see Visualization 4). The bright curve retains the shape of a hyperbola.

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Fig. 10 Effect of randomizing fiber lengths on the weight map. The fibers change from equal lengths to a random length distribution (Visualization 5). As a result, the well-organized weight map containing a single straight bright line, and fainter parallel traces induced by antenna side-lobes, becomes discombobulated. Multiple bright regions scattered randomly throughout the k-space provide diverse contribution to the detection. For a fixed (pseudo‑)random distribution of fiber lengths (Visualization 6), each position of the photodetector (represented by the bright star on the right) corresponds to a different random distribution of contributions from the k-space as represented by the weight maps on the left. With sufficiently diverse collection of weight maps, the sources in k-space may be reconstructed computationally.

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Fig. 11 Weight maps measured experimentally. Left pane shows weight maps for different pixels of the photo-detector array, see Visualization 7. The animation of Visualization 7 is analogous to that of Visualization 6 obtained using computer simulations. As the photo-detector index is incremented from 1 to 174, the weight map changes as shown in the video of Visualization 7. Right pane shows the contribution of a single k-vector, indicated on the weight map with a white circle at frequency 16.7 GHz and 0.0-degrees incidence angle, to the different pixels of the photo-detector array.

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Funding

U.S. DoD.

Acknowledgment

We would like to acknowledge the support of this work by the U.S. Air Force and other U.S. Government agencies.

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