Abstract

The ability to both spatially and spectrally demultiplex wireless transmitters enables communication networks with higher spectral and energy efficiency. In practice, demultiplexing requires sub-millisecond latency to map the dynamics of the user space in real-time. Here, we present a system architecture, referred to as k-space imaging, which channelizes the radio frequency signals both spatially and spectrally through optical beamforming, where the latency is limited only by the speed of light traversing the optical components of the receiver. In this architecture, a phased antenna array samples radio signals, which are then coupled into electro-optic modulators (EOM) that coherently up-convert these signals to the optical domain, preserving their relative phases. The received signals, now optical sidebands, are transmitted in optical fibers of varying path lengths, which act as true time delays that yield frequency-dependent optical phases. The output facets of the optical fibers form a two-dimensional optical phased array in an arrangement preserving the phases generated by the angle of arrival (AoA) and the time-delay phases. Directing the beams emanating from the fibers through an optical lens produces a two-dimensional Fourier transform of the optical field at the fiber array. Accordingly, the optical beam formed at the back focal plane of the lens is steered based upon the phases, providing the angle of arrival and instantaneous frequency measurement (IFM), with latency determined by the speed of light over the optical path length. We present a numerical evaluation and experimental demonstration of this passive AoA- and frequency-detection capability.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Massive Multiple-Input Multiple-Output (mMIMO) systems have been implemented within fifth-generation (5G) mobile communication networks and are continued to be proposed for beyond-5G networks, with the aim of increasing spatial diversity and thereby improving the utilization of the available spectrum [1,2]. Spatial division multiple access (SDMA), a key technique in these 5G and proposed beyond 5G (B5G) networks afforded by phased-array technology, utilizes spatial beam forming to divide the user space into spatially isolated data channels [3], a spatial analog to frequency division multiple access (FDMA). Carrier frequencies transitioning from sub-6 GHz to those exceeding 27 GHz enables improved beam resolution for a given aperture size afforded by the shorter wavelengths of the millimeter waves, potentially allowing for a ten-fold increase in the expected data capacity from prior wireless network generations [4]. However, the reliance of these systems on pilot-based channel-state estimation entails the need to adequately address pilot contamination and the and the maintenance of channel orthogonality.

Pilot sequences, or training sequences, are signals used on the uplink to provide channel information to properly beamform a downlink transmission to the appropriate device. Pilot contamination, arising from the re-use of pilots, generates interference in the channel state information (CSI), which reduces the accuracy of the downlink beam forming. Additional challenges to obtaining CSI using the pilot sequence approach include the need for a transceiver calibration to realize channel reciprocity, timely acquisition of CSI in a highly dynamic radio frequency (RF) environment, and difficulty in scaling up to millimeter waves (mmW) [5]. Accordingly, there has been a great interest in blind channel estimation techniques to obtain channel state information free of pilot training sequences, which passively formulate a spatial-spectral mapping of the dynamic user beam space [6].

Microwave photonics holds key advantages to acquiring this mapping of the spatial-temporal spectrum through passive millimeter wave imaging [710]. Investigated approaches include computational interferometric imaging methods, such as optical hyperspectral imaging [11], which provide full three-dimensional mapping of the RF signal space [1214], in addition to techniques relying exclusively on RF-domain processing [15]. Common among these systems is the need for extensive calibration procedures and/or computational reconstruction, which leads to latency in obtaining CSI, and therefore limit the applicability of the techniques.

In this paper, we modify the system architecture discussed in [1214], which employs computational algorithms to reconstruct a three-dimensional Fourier transform of the received RF signal space, to a system that directly produces a two-dimensional Fourier transform on a photodetector array. The modification of the imaging modality allows us to alleviate latency constraints imposed by these computational algorithms, and produce the Fourier transform at the speed of light using a lens-based optical system. Figure 1 shows a generalized system architecture consisting of an RF phased antenna array, which couples directly to electro optic modulators (EOM) to produce optical sidebands with the phases of the RF signals preserved across the optical fiber array. Each of the EOMs outputs several optical fibers of varying lengths to produce frequency-dependent phase shifts, similar to an arrayed waveguide grating (AWG) [16,17]. The arrayed fiber gratings (AFG) associated with each antenna transports the up-converted signals to a two-dimensional lenslet array having a geometry matching the spatial-temporal aperture produced by the antenna-element spatial distribution along the x direction and the optical fiber lengths along the y direction, see Fig. 1. Combining the spatial antenna array, and the temporal optical fiber array in this specific manner results in passive phase shifting that steers the optical beam in response to the received RF-wave angle of arrival (AoA) and its frequency along two orthogonal axes. This beam is focused on a photodetector (PD) array by an optical lens and produces a direct Fourier transform of the angular and temporal spectrum, which is detected as an optical-power distribution at the PD array.

 figure: Fig. 1.

Fig. 1. Conceptual diagram of a photonic k-space imaging for resolving the angle of arrival and frequency of incident radio frequency wave fronts.

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Below, we present and discuss the numerical evaluation and experimental implementation of a microwave-photonic spectral imaging system where we demonstrate the use optical beamforming to resolve RF emitters by their angles of arrival and frequency. The paper is organized as follows: In Sec. 2 we discuss the principles underlying the system operation, and numerically evaluate the model we develop in Sec. 3. The experimental system configuration and results are discussed in detail in Sec. 4. Finally, the concluding remarks are presented in Sec. 5.

2. Operational principle

A source oscillating at frequency Ω and amplitude A produces a spherical wave centered at its position and expanding at the speed of light c. Receiver antenna located distance r away from the source captures the wave to produce time-dependent electrical signal, which after amplification may be expressed as

$$U = U(t )= A\frac{{{e^{j({\Omega t - Kr + \phi } )}}}}{r} + \textrm{c}\textrm{.c}\textrm{.},$$
where t is time, $K = {\Omega / c},$ $\phi $ is the RF phase offset, and ‘c.c.’ indicates the presence of a complex-conjugate term that makes U real. This signal modulates the phase of an optical carrier $a{\mkern 1mu} {e^{j\omega t}}$ to yield
$$u = a\textrm{exp} \left[ {j({{\omega_{}}t + \varphi } )+ \pi \frac{{U(t )}}{{{V_\pi }}}} \right] + \textrm{c}\textrm{.c}\textrm{.},$$
where a is the optical-beam amplitude, $\omega $ is the carrier frequency, $\varphi $ is the optical phase offset, and ${V_\pi }$ is the half-wave voltage of the modulator. For low modulation index where $|{U(t )} |\ll {V_p},$ u in (2) may be approximated using Taylor expansion of the exponential function to obtain
$$u = a{\mkern 1mu} {e^{j({\omega t + \varphi } )}}\left[ {1 + \pi \frac{{U(t )}}{{{V_\pi }}}} \right] + o\left( {\frac{U}{{{V_\pi }}}} \right) + \textrm{c}\textrm{.c} \simeq a{\mkern 1mu} {e^{j({\omega t + \varphi } )}}\left[ {1 + \pi \frac{A}{{{V_\pi }}}\frac{{{e^{j({\Omega t - Kr} )}}}}{r} + \textrm{c}\textrm{.c}\textrm{.}} \right] + \textrm{c}\textrm{.c}\textrm{.}$$
where the higher-order terms $o({{U / {{V_\pi }}}} )$ have been omitted in the last expression. Multiplying out the square bracket and rearranging the c.c. terms in (3) yields
$$\begin{aligned} u &= a{\mkern 1mu} {e^{j({\omega t + \varphi } )}} + \textrm{c}\textrm{.c}\textrm{.}\\ &\textrm{ + }\frac{\pi }{{{V_\pi }}}\frac{{aA}}{r}\;\textrm{exp} [{j({\omega - \Omega } )t + \varphi - \phi } ]\;\textrm{exp} ({jKr} )+ \textrm{c}\textrm{.c}\textrm{.}\\ &\textrm{ + }\frac{\pi }{{{V_\pi }}}\frac{{aA}}{r}\;\textrm{exp} [{j({\omega + \Omega } )t + \varphi + \phi } ]\;\textrm{exp} ({ - jKr} )+ \textrm{c}\textrm{.c}\textrm{.} \end{aligned}$$

The first term in (4) may be identified as the optical carrier, which contains no information about the modulating signal. The remaining two terms are the lower and upper sidebands, respectively, that include the effect of the modulation in the form of RF signal amplitude A, frequency $\Omega ,$ and phase offset $\phi .$ The sidebands also include the information on the distance from the source in the form of the phase term $\textrm{exp} ({ \pm jKr} ).$ For a single receiving antenna, this phase term enters (4) on the same footing as the RF phase $\phi $ or the optical phase $\varphi $ and as such it cannot be distinguished from the two. However, the system considered here comprises an array of antennas distributed at different physical locations to yield generally different position-dependent phases, which may be used to pin-point the source location. In addition, the modulator outputs are split and the beams are carried by optical fibers of different lengths, a configuration that effectively extends the spatial aperture to the time domain.

To analyze the system, we introduce index n enumerating the optical fibers in the array; notably, each fiber has an associated antenna, but due to modulator-output splitting, multiple fibers may be associated with a single antenna. Thus, multiple values of index n may correspond to the same antenna position. The latter are described using vectors ${\textbf r} + {{\textbf r}_n},$ where ${\textbf r}$ points from the source to the array reference point, for example to the center of the array, and ${{\textbf r}_n}$ is the vector from the array reference point to the nth antenna of the array. Similarly, we express time delays as $t + {t_n}$ where different values ${t_n}$ result from different optical-fiber lengths. Thus, at the end of the nth optical fiber, the signal amplitude of the upper sideband is

$${u_n} = \frac{\pi }{{{V_\pi }}}\frac{{aA}}{{|{{\textbf r} + {{\textbf r}_n}} |}}\;\textrm{exp} [{j({\omega + \Omega } )({t + {t_n}} )+ {\varphi_n} + \phi } ]\;\textrm{exp} ({ - jK|{{\textbf r} + {{\textbf r}_n}} |} )+ \textrm{c}\textrm{.c}.$$

For a distant source, $|{\textbf r} |\gg |{{{\textbf r}_n}} |$ and therefore $|{{\textbf r} + {{\textbf r}_n}} |$ may be replaced by $|{\textbf r} |= r$ in the overall amplitude factor. To evaluate the position-dependent phase term $\textrm{exp} ({ - jK|{{\textbf r} + {{\textbf r}_n}} |} )$ in (5), we first note that for a distant source, the spherical wave looks approximately like a plane wave with wavevector ${{\textbf K}_n} = K{{({{\textbf r} + {{\textbf r}_n}} )} / {|{{\textbf r} + {{\textbf r}_n}} |}}$ at the antenna position so that

$$\textrm{exp} ({ - jK|{{\textbf r} + {{\textbf r}_n}} |} )\simeq \textrm{exp} [{ - j{{\textbf K}_n} \cdot ({{\textbf r} + {{\textbf r}_n}} )} ].$$

Additionally,

$${{\textbf K}_n} = K\frac{{{\textbf r} + {{\textbf r}_n}}}{{|{{\textbf r} + {{\textbf r}_n}} |}} \simeq K\frac{{{\textbf r} + {{\textbf r}_n}}}{{|{\textbf r} |}} = K\left( {\frac{{\textbf r}}{{|{\textbf r} |}} + \frac{{{{\textbf r}_n}}}{{|{\textbf r} |}}} \right) \simeq K\frac{{\textbf r}}{{|{\textbf r} |}} \equiv {\textbf K},$$
where in the two approximations, we used the condition $|{\textbf r} |\gg |{{{\textbf r}_n}} |.$ Eq. (7) expresses the notion that for a distant source, all elements of the array see essentially the same incoming plane wave.

To evaluate the time-dependent phase term in (5), we note that the optical phase ${\varphi _n}$ may be freely adjusted in the system by applying a DC bias voltage. We use this freedom, to set its value so that $\omega {t_n} + {\varphi _n}$ is an integer multiple of $2\pi $ for each n, and as such may be omitted within the exponent.

Gathering these approximations, we may rewrite the upper-sideband signal as

$${u_n} = \left\{ {\frac{\pi }{{{V_\pi }}}\frac{{aA}}{r}\;\textrm{exp} [{j({\omega + \Omega } )t - j{\textbf K} \cdot {\textbf r} + j\phi } ]} \right\}\;\textrm{exp} [{ - j{\textbf K} \cdot {{\textbf r}_n}} ]\;\textrm{exp} [{j\Omega {t_n}} ]\; + \textrm{c}\textrm{.c}.$$

In (8), the complex amplitude of the signal in curly brackets $\{\ldots \}$ is the same for all elements of the array. The last two phase terms depend on the relative time delay ${t_n}$ due to different fiber lengths, and on the position of the antenna in the array ${{\textbf r}_n}.$

Consider now a situation where the sources are distributed along one direction. For example, the position of distant terrestrial sources is determined by the azimuth angle only as the elevation above or below horizon is mostly negligible. In this case, only the horizontal components ${K_X},\;{K_Y}$ of K appear in (8). As a result, the vertical component ${({{{\textbf r}_n}} )_Y} = {r_{nY}}$ of ${{\textbf r}_n}$ in (8) is irrelevant, and the antenna-position-dependent phase term becomes $\textrm{exp} [{ - j{K_X}{\mkern 1mu} {r_{nX}}} ]$, as shown in Fig. 2, for a planar antenna array oriented perpendicular to the Y axis.

 figure: Fig. 2.

Fig. 2. Illustration of the wavefront, K, projected onto the x-axis of the antenna array positions, rnX, with apparent incident angle θ. Note the apparent elevation angle is negligible.

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In the optical processor, the fiber-lenslet array is a scaled replica of the spatio-temporal array defined by the x-component of the antenna position ${({{{\textbf r}_n}} )_X} = {r_{nX}}$ and the time delay ${t_n},$ i.e.

$$({{x_n},{y_n}} )= ({s{\mkern 1mu} {r_n}_X,s^{\prime}{\mkern 1mu} {t_n}} ),$$
where xn, yn are the coordinates of the nth fiber in the array, whereas s and s’ are the horizontal and vertical scaling factors, respectively. The lenslet array is placed in the focal plane of an optical lens, which, in the paraxial approximation, performs a Fourier transformation of the optical field. As a result, the optical field strength U as a function of position $({u,v} )$ in the back focal plane of the lens is
$$\begin{aligned} U({u,v} )&= \frac{1}{{j\lambda f}}\sum\limits_n {{u_n}\textrm{exp} \left[ { - \frac{{2\pi j}}{{\lambda f}}({{x_n}{\mkern 1mu} u + {y_n}{\mkern 1mu} v} )} \right]} + \textrm{c}\textrm{.c}\textrm{.}\\ &= \frac{1}{{j\lambda f}}\{\ldots \}\sum\limits_n {\textrm{exp} [{ - j{K_X}{\mkern 1mu} {r_{nX}}} ]\;\textrm{exp} [{j\Omega {t_n}} ]\textrm{exp} \left[ { - \frac{{2\pi j}}{{\lambda f}}({{x_n}{\mkern 1mu} u + {y_n}{\mkern 1mu} v} )} \right]} + \textrm{c}\textrm{.c}\textrm{.} \end{aligned}$$
where the fiber outputs have been treated as approximately point sources in the front focal plane. Using (9) in (10) yields
$$U({u,v} )= \frac{1}{{j\lambda f}}\{\ldots \}\sum\limits_n {\textrm{exp} \left[ { - j\left( {{K_X} + \frac{{2\pi s}}{{\lambda f}}u} \right){\mkern 1mu} {r_{nX}}} \right]\;\textrm{exp} \left[ {j\left( {\Omega - \frac{{2\pi s^{\prime}}}{{\lambda f}}v} \right){t_n}} \right]} + \textrm{c}\textrm{.c}\textrm{.}$$

The sum in the total field distribution (11) contains only phase terms, which are complex numbers lying on a unit circle in the complex plane. The absolute value of the sum is maximized when all terms are in phase, which is the condition for the optical power density ${|{U({u,v} )} |^2}$ to reach a peak value. For arbitrary, irregular distributions of ${r_{nX}}$ and ${t_n},$ the in-phase condition is obtained when the terms in parentheses $(\ldots )$ of (11) vanish, i.e.,

$$\begin{array}{c} \left( {{K_X} + \frac{{2\pi s}}{{\lambda f}}u} \right) = 0 \Rightarrow u ={-} \frac{{\lambda f}}{{2\pi s}}{K_X}\\ \left( {\Omega - \frac{{2\pi s^{\prime}}}{{\lambda f}}v} \right) = 0 \Rightarrow v = \frac{{\lambda f}}{{2\pi s^{\prime}}}\Omega . \end{array}$$

Equations (12) show that the coordinates $({u,v} )$ of the optical-amplitude peak correspond to the spatial ${K_X}$ and spectral $\Omega $ frequencies of the incoming wave, with fixed scaling factors ${{\lambda f} / {2\pi s}}$ and ${{\lambda f} / {2\pi s^{\prime}}},$ respectively, set by the geometry of the system encoded in s, s’, and f, and by the optical wavelength $\lambda .$ The spatial frequency ${K_X}$ may be expressed in terms of the angle of incidence $\theta $ of the incoming wave on the antenna-array plane ${K_X} = K\sin (\theta ),$ so that the coordinates of the optical-amplitude peak become

$$\begin{array}{c} u ={-} \frac{{\lambda f}}{{2\pi s}}K\sin (\theta )= \frac{{\lambda f}}{{2\pi sc}}\Omega \sin (\theta )\\ v = \frac{{\lambda f}}{{2\pi s^{\prime}}}\Omega .. \end{array}$$

Equations (13) show that that the optical processor produces a bright spot at a position determined by the position of the transmitter relative to the antenna array encoded in the incidence angle $\theta ,$ and by the frequency of the incoming radiation $\Omega .$ Notably, the vertical component v of the bright-spot position depends only on the frequency, whereas its horizontal component u depends on both the frequency and on the angle of incidence. This dependence of the horizontal component on frequency is known as squint.

When multiple sources in the range of the receiver transmit at different frequencies and/or have different positions relative to the receiver array, multiple bright spots appear at the back focal plane of the lens, and the positions of the bright spots correspond to the positions and frequencies of the respective sources. Thus, an image is created instantaneously that depicts the spatial-spectral distribution of transmitters operating within the range of the receiving array.

Above, we analyzed the optical processor by treating fibers as point sources. More realistically, we represent the beams emanating from the fibers as Gaussian. Thus, following [20], the complex amplitude of the field at the output of the nth fiber is proportional to:

$${\left. {\frac{1}{{z - j{z_R}}}\textrm{exp} \left[ {jk\frac{{{{({{x_n} - x} )}^2} + {{({{y_n} - y} )}^2}}}{{2({z - j{z_R}} )}}} \right]} \right|_{z = 0}} = \frac{1}{{ - j{z_R}}}\textrm{exp} \left[ { - \frac{{{{({{x_n} - x} )}^2} + {{({{y_n} - y} )}^2}}}{{{w_0}^2}}} \right]$$
where zR is the Rayleigh range associated with the Gaussian beam waist, w0, as in
$$w_0^2 = \frac{{{z_R}\lambda }}{\pi }.$$

Accounting for the fiber positions in a way similar to one that led to (11), we arrive at the following expression for optical power density at the back focal plane of the Fourier lens:

$$\begin{aligned} {|{U({u,v} )} |^2} &= 2{\left( {\frac{\pi }{{{V_\pi }}}\frac{{aA}}{r}\frac{1}{{\lambda f}}} \right)^2}\textrm{exp} \left[ { - {{\left( {\frac{{2\pi {w_0}}}{{\lambda f}}} \right)}^2}({{u^2} + {v^2}} )} \right] \times \\ &\times {\left|{\sum\limits_n {\textrm{exp} \left[ { - j\left( {{K_X} + \frac{{2\pi s}}{{\lambda f}}u} \right){\mkern 1mu} {r_{nX}}} \right]\;\textrm{exp} \left[ {j\left( {\Omega - \frac{{2\pi s^{\prime}}}{{\lambda f}}v} \right){t_n}} \right]} \textrm{exp} ({ - j{\varphi_n}} )} \right|^2}, \end{aligned}$$
where we included an additional phase term φn that signifies the freedom to electronically steer the optical beam across the optical array by applying DC bias to the phase modulators. To find the instantaneous optical power detected by an element of the photodetector array, we integrate the intensity over the area A of a photodetector:
$$P = \int\limits_{\textrm{Area}} {{{|{U({u,v} )} |}^2}{d^2}x \simeq } {|{U({u,v} )} |^2}A.$$
In this section, we have shown how the two-dimensional Fourier transformation performed by the optical processor leads to RF beam steering and, as a result, yields an instantaneous image of the spatial and spectral frequencies of the incoming RF wave front. The next section details a numerical model, which evaluates the theoretical description using real system parameters.

3. Numerical evaluation

The general expressions derived in Section 2 are applied to a numerical model, which we use to investigate a two-dimensional spatio-temporal array geometry, referred to as the five-arm spiral. The spatio-temporal array comprises 30 elements arranged in a spiral, with six elements along five distinct arms, previously studied as a purely spatial imaging system in [18,19]. The geometry, representing the combined antenna positions and optical path lengths are compared with the spatial fiber array element positions in Fig. 3. This spiral geometry is chosen for the spatio-temporal array as the aperiodic structure results in the removal of grating lobes, the harmonic maxima arising from periodicity in the array structure, increasing the alias-free Field-of-View (FOV). As discussed in Section 2, the fiber array geometry is related to the spatio-temporal array by an offset scaling factor, s, which controls the effective beam size in the optical domain. The improved alias-free FOV allows for an arbitrary scaling of the fiber array to alter the optical beam size within the Gaussian beam profile defined by the beam waist without introducing grating lobes into the response, as will be further explored in this section. This geometry additionally allows for the re-use of custom optical components of the same geometry developed in previous studies [19]. The added field of view of the aperiodic five-arm spiral, as with other non-uniform array geometries such as optimized random arrays [20], comes with a trade-off in elevated side lobe levels. Alternative geometries should be considered in future studies to further optimize the side lobe levels in the imaging response increasing either the FOV or spatio-temporal resolutions of the system.

 figure: Fig. 3.

Fig. 3. (a) Spatial-temporal array geometry of the radio-frequency antennas and optical fiber lengths (Xn, Ln). (b) Spatial array geometry of the two-dimensional fiber array (xn, yn). (c) the resulting optical point spread function of the array geometry with an RF wavefront of incident angle 0° and temporal frequency of 33 GHz, in the middle of the band, which is steered electronically to the center of the detector plane.

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The RF antenna elements comprising the spatial array in Fig. 3(a) operate over a 14 GHz bandwidth from 26–40 GHz, covering the portion of the microwave spectrum known as Ka band. The temporal portion of the array shown in Fig. 3(a) contains a maximum fiber length difference of 13.5 cm. Accounting for the refractive index $\eta ,$ this equates to an effective optical path length difference of 20.25 cm. The n-th time delay can be calculated from the n-th fiber delay line length by the simple relation ${L_n}/c.$ We calculate the equivalent temporal Rayleigh resolution of the delay line aperture as the change in frequency corresponding to the maximum difference in optical path lengths, resulting in the minimum detectable phase shift, as:

$$\Delta f = \frac{c}{{\eta \Delta {L_{\max }}}};$$
given this relation, we have an expected 3 dB frequency resolution of 1.51 GHz. Similarly, the Rayleigh resolution of the spatial array is determined by the relation between the minimum resolvable angle $\Delta \theta $ and the minimum phase shift across the spatial aperture:
$$\Delta \theta = \frac{\lambda }{D},$$
where $D = 50.88\;\textrm{cm}$ is the diameter for the RF antenna array, which yields an angular resolution range of 0.8–1.4° over the RF bandwidth. The frequency of the laser oscillator is approximately 193 THz, corresponding to the telecommunications laser wavelength of 1550 nm. For simplicity, the antenna efficiencies and optical modulation indices are assumed uniform while the optical bandpass filter isolating the upper sideband is considered ideal.

The maximum dimensions of the fiber array in the X-Y plane, placed in the front focal plane and illustrated in Fig. 3(b), are both 1.8 mm. Along the X-axis, this size corresponds to 1/280 scaling of the antenna positions, whereas along the Y-axis, scaling by 1/75 of the optical-path-lengths applies. CCD array comprising 320 × 256 detectors with a pitch of 30 µm (FLIR SC2500) placed in the back focal plane detects the resulting distribution of light intensity. Adding the steering phase offset, ${\varphi _n}$ in (16), allows us to center the point spread function corresponding to the mid-band frequency, 33 GHz, on the photodetector array as in Fig. 3(c). As indicated by Eq. (12), the location of the optical beam along the v-axis of the optical detector plane is proportional to the temporal frequency of the received radio signal. To analyze the bandwidth of this frequency scanning, we evaluate the cross-sections of the individual point spread functions for fourteen separate sidebands at an interval of 1 GHz, covering the entire RF bandwidth, along the v-axis. These wavefronts are considered to arrive at broadside incidence relative to the antenna array, appearing centered upon the u-axis of the detector plane. For visualization, the cross-sections of each of the point spread functions are overlayed together in Fig. 4 and normalized relative to the mid-band point spread function at 33 GHz. For the sake of visualization, the span of the photodetector response is truncated to ±1.5 mm. The fiber lengths are specifically designed to confine the 14 GHz RF bandwidth within the 3-dB roll-off produced by the element gain. This does not wholly account for the spectral dependence of the system components such as the antenna directivity, amplifier gain and modulator efficiencies. As such, the true optical 3-dB FOV is expected to be narrower than that defined exclusively by the element gain.

 figure: Fig. 4.

Fig. 4. Numerically evaluated cross-section of optical intensity response at the u-v detector plane for an optical sideband offset by the received radio frequency swept from 26 to 40 GHz in 1 GHz increments. Note the frequencies are listed as the offset from the optical carrier.

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Since the array is aperiodic and does not produce grating lobes, the relative size of the optical fiber array can be increased to allow for improved power confinement of the steered beam. Scaling the spacings of the fiber array to a larger physical size resulting in a reduced roll-off over the desired RF bandwidth, while equalizing the frequency response closer to a flat-top band-pass. The effects of scaling the fiber array pitch on beam steering were evaluated for the fiber array between 100% of its original size up to 250% of its original size in 50% increments, as shown in Fig. 5. The increase in physical size results in an increase in the spatial frequency of the optical beams incident upon the photodetector array, scaling the array factor relative to the Gaussian element pattern. As a result, the fiber array shown in Fig. 5(d), corresponding to 250% scaling, has a fiber array with a maximum spacing of 5 mm, and allows the full 14 GHz bandwidth with only 10% variation in optical power density. Compare this variation to the nearly 50% decrease in power at the band edges for the original array size.

 figure: Fig. 5.

Fig. 5. Adjusting the relative size of the fiber array, shown in Fig. 3(b), from (a) original scaling, (b) 150% scaling, (c) 200% scaling and (d) 250% scaling. Note the same color mapping is used as in Fig. 4 relating the optical beam to corresponding sideband offset frequency.

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The array scaling constrains the field-of-view of the spatial response. However, since the optical path lengths are not scaled, the ratio of the beam spot size to temporal frequency is preserved. Thus, the reduced spatial FOV is of no consequence on the frequency-dependent beam steering, as the alias-free bandwidth of the spectral beam steering, or free spectral range, is maintained.

Now we consider the full two-dimensional beamforming property outlined by Eq. (12). This indicates that the position of the optical beam along the u-axis relates to the spatial frequency in addition to the temporal frequency scanning along the v-axis. From the detected positions of the optical beam on the 2D detector plane, we find the angle of arrival and measure the radio frequency. To illustrate this beamforming capability, the numerical case is evaluated in Fig. 6 below for beams steered with varying AoA and temporal frequencies in Fig. 6 below. In this evaluation, the two-dimensional beam power pattern is shown over the u-v detector plane. To further clarify the location of the beam produced by the novel beamforming architecture, the cross-sections of the main beam are projected along both axes of the detector plane. The cross-section of the u-axis is illustrated by the black line representing the spatial frequency of the main beam. The cross-section along the v-axis is presented in a similar manner as Fig. 4, which changes color with respect to the temporal frequency.

 figure: Fig. 6.

Fig. 6. Numerical evaluation of the two-dimensional beamforming technique shown for four separate cases of receiving from emitters at spatio-temporal locations of: (a) −5° and 27 GHz, (b) 5° and 27 GHz, (c) −5° and 33 GHz, (d) 5° and 33 GHz, (e) −5° and 39 GHz and (d) 5° and 39 GHz. Note the projection of the main beam on the u-axis corresponding to the emitter’s spatial location, while the projection on the v-axis corresponds to the temporal frequency.

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Additional evaluation of the two-dimensional beamforming is performed by assuming the superposition of fields from several emitters at varying AoA and operational frequencies. The evaluated source configurations are illustrated in Fig. 7 where columns (a) and (b) refer to different spatial placements of emitters, whereas rows (i) through (iii) refer to different frequency configurations. Accordingly, one of these emitters is placed broadside and scanned spectrally with 31 GHz in row (i), 33 GHz in row (ii) and 35 GHz in row (iii), whereas the other emitter is held at a constant frequency of 33 GHz and repositioned spatially at 1° in column (a), and 2.5° in column (b) off broadside.

 figure: Fig. 7.

Fig. 7. Numerical evaluation of two-dimensional spatial-spectral imaging capability with two incoherent RF emitters. Column (a): Emitter displaced by 1° from broadside at 33 GHz with secondary emitter positions of (i) Broadside at 31 GHz (ii) Broadside at 33 GHz, sources are not resolvable within this scene (iii) Broadside at 35 GHz. Column (b): Emitter displaced by 2.5° at 33 GHz with secondary emitter positions of (i) Broadside at 31 GHz (ii) Broadside at 33 GHz (iii) Broadside at 35 GHz. We note that aside form (a-ii), wherein both emitters are placed in the same spatial-spectral bin relative to the array resolution, each emitter is uniquely identifiable by its spatial-spectral location.

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It is to be noted that column (a) refers to a spatial separation which is not resolvable by the array, hence the blurring of the two emitters at the same frequency in (a-ii). This result demonstrates that ambiguity occurs only when emitters are placed at the same spatial and spectral locations.

4. Experimental results

Demonstration of the k-space imaging technique was performed with a system shown in Figs. 8 and 9(a) comprising thirty tapered-slot antenna elements in a phased array having an operational bandwidth from 26.5 to 40 GHz (Ka-band) and the FOV +/- 10°. Arranging the spatio-temporal array in a sparse two-dimensional five-arm spiral configuration, see Fig. 3, allows for the suppression of grating lobes while maximizing the resolution in the azimuth and frequency directions [19], which is 0.4° at the highest frequency, and 1.52 GHz. In this system, each antenna element within the array couples into a commercial electro-optic phase modulator (EOM) that is fed by a common laser source. In the EOM, the RF signal from the corresponding antenna element is up-converted to an optical sideband that is offset by the microwave frequency of the received signal from a 193 THz carrier. Following up-conversion, the signals are output to optical fibers that couple them into a micro-lens array matching the geometry of the five-arm spiral RF antenna array, as shown in Fig. 3. In order to obtain the desired wavelength-dependent beamforming, the optical fibers are spliced to precise optical path lengths, which match a scaled geometry of the antenna array spacing. Distributing the fiber length in this arrangement is critical for enabling spatial beamforming to maintain temporal coherence of the true time delay phases relative to the spatial coherence of the phases emanating from the spatial micro-lens array into free space, as explained in Sec. 2. Due to the antenna array’s geometry, the transmitting sources are maintained in the same elevation plane throughout the experiment to prevent additional location-dependent phase shift along the vertical direction, which would cause ambiguity in the frequency-elevation plane. However, this constraint may be alleviated by constructing a one-dimensional antenna array with varying fiber lengths routing into a two-dimensional lens array. Likewise, the constraint that emitters must be confined to the azimuth dimension may be alleviated by alterations in the system configurations potentially allowing for a direct imaging method to recover all three-dimensions in future investigations.

 figure: Fig. 8.

Fig. 8. Conceptual diagram of the experimental configuration of the two-dimensional five-arm spiral antenna array and the corresponding optical fiber array comprising thirty elements.

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 figure: Fig. 9.

Fig. 9. (a) Notional schematic diagram of the two-dimensional spatial-spectral imaging system. PBS: Polarizing beam splitter. BS: beam splitter. EDFA: Erbium-doped fiber amplifier. (b) 30-element phased array in five-arm spiral configuration. (c) Close-up of the 200-element fiber-lenslet array, of which only the inner 30 lenslets are coupled to optical fibers. (d) Corresponding optical image of the 30-element fiber-lenslet array as seen at the location of the phase-detection board. (e) Interference pattern generated by the 30-element optical phased array as seen at the CCD image-sensor location.

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This three-dimensional array is visualized in Fig. 8, which shows the two-dimensional antenna array with a corresponding optical path-length following each element that is proportional to the vertical position of the element (shown previously in Fig. 3(a)). This arrangement was intentional as the projection of the 2D antenna array onto a linear array would produce element spacing that results in mutual coupling between the antennas; on the other hand, preserving the geometry was critical for implementation as it allowed the use of our existing custom lensed fiber array.

The specified maximum length difference of the fiber lengths, roughly 13.1 cm, corresponds to a spectral resolution of 1.52 GHz and an optical time delay of 436 ps in fiber; considering that the total optical path length is 1 meter, the spatial-spectral Fourier transform is computed optically in just 3.7 nanoseconds. Similar system architectures have employed this arrayed fiber waveguide grating with spectral resolution engineered on the order of 50 MHz demonstrated in [21], provided a trade-off in the covered RF bandwidth; system architectures employing computational methods, such as those in [14,21], have demonstrated simultaneous high bandwidth and resolution with a concomitant trade-off in computational complexity.

Following the projection of the signals into free-space by the fiber lenslet array, the optical waves pass through a polarizing beam-splitter, which allows the incoming light through. Therein, the optical carriers and sidebands pass through a quarter-wave plate to circularize the polarization of the light, and then through distributed Bragg reflectors, which act as a band-pass filter rejecting the optical carrier and upper side band while selectively passing the lower side band. The reflected carrier and one sideband pass through the quarter-wave plate again to become linearly polarized at a right angle with respect to the original, incoming light. The PBS reflects this light and free-space optics re-images the original fiber bundle on a custom photo-detector array, arranged in the five-arm spiral geometry, where the signals are heterodyned with a reference beam from the common laser source. Heterodyning the imaged fiber bundle with the common source on individual photodetectors enables the monitoring of the optical phases in each element of the array [10]. The measured optical phase fluctuations are then compensated for through a feedback loop using a field-programmable gate array (FPGA) [10]. High fidelity coherence is achieved with phase variations of 2.5° rms on average [9].

Meanwhile, the lower sideband passes through the distributed Bragg reflector and follows toward an optical lens. Situated at the back focal plane of the lens is a short-wave infrared InGaAs CCD array with 256 × 320 pixels (FLIR SC2500), which is thus illuminated by the two-dimensional far-field diffraction pattern, Fig. 9(e).

Calibration of the phases for each channel prior to operation is performed to set the bandwidth covered on the CCD array; the center frequency is chosen at 33 GHz so as to cover the full bandwidth of 26.5–40 GHz (+/- 7 GHz). The calibration transmitter is a WR-28 Ka-band horn antenna with 20 dB directive gain, which is fed a signal from a programmable signal generator (Keysight E8267D PSG Vector Signal Generator). Operating this source at 10 dBm at four meters away from the receiver ensures mostly flat phase curvature across the entire array for the calibration procedure. Each modulator then has an offset phase added onto to the optical waves to steer the 33 GHz offset sideband to the center of the detector plane. Calibrating the optical path lengths allows us to correct for fluctuations in the spliced fiber lengths. This was performed by limiting element operation to a reference, chosen based upon the shortest propagation delay length, and a receiving element under test out of the entirety of the remaining 29-elements in the array. Isolating two elements enables monitoring of the optical fringes produced on the CCD, while sweeping the frequency of a calibration antenna with a CW tone over the receiving antenna bandwidth allows us to measure the relative optical fringe shift. Based on the numerically calculated phase shifts, we were able to correct the path lengths for each receiving element within a +/- 2.5 mm tolerance, i.e., within +/- 35% error from the shortest RF wavelength; further investigations may see improvement in the path length tolerances with the use of an optical backscatterer reflectometer to precisely measure the delay lengths. The measured optical array response taken after the calibration procedure is shown in Fig. 10(a), and compared with the numerically evaluated array response, with tolerances in optical path length accounted for, shown in Fig. 10(b).

 figure: Fig. 10.

Fig. 10. Comparison of the optical point spread functions in ADU counts for a source at 33 GHz produced by (a) the experimental system, and (b) a numerical evaluation with empirically derived phase variations from path-length mismatching accounted for. The experimental point spread function is intentionally saturated to raise the side-lobe distribution to detectable levels.

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The deviation of the measured side-lobe energy distribution from the expected values calculated in Sec. 3 is likely the result of tolerances in optical path lengths, and errors introduced to the phase steering of a center frequency on the CCD array.

Following the analysis corresponding to Fig. 4 in Sec. 3, cross-sections of the measured camera response were taken to illustrate the spatial mapping of the up-converted RF spectrum over an 11-GHz bandwidth. As visualized in Fig. 11, the optical beam is steered in 1 GHz increments, where the RF frequency listed corresponds to the offset of the optical sideband from the optical carrier. The band edges are neglected due to the test emitter’s (WR22 horn antenna) cut-off at 26.3 GHz paired with the gaussian-limited element factor in the point spread function reducing the amplitude-response relative to the side-lobe levels at the edges of the optical beam pattern. This variation in the observed amplitude is caused by differences in the LNA gain, modulator efficiency and array gain at different RF frequencies. Additional variations are due to residual phase errors causing changes in the level of constructive interference on the CCD array.

 figure: Fig. 11.

Fig. 11. Spatial distribution of frequency-steered optical sidebands along a center cross-section of the measured camera response. Steering is shown as a composite in 1 GHz steps for a 10-GHz bandwidth of 28–39 GHz.

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To demonstrate the spatial and spectral channelization afforded by the array geometry, a second transmitter (another WR28 Horn antenna) was placed at the same distance from the array as the initial transmitting source, as shown in Fig. 12. The experimental setup follows the numerical simulations presented in Section 3. The resulting images were compiled based on the physical locations and the independent frequency control of each transmitting source.

 figure: Fig. 12.

Fig. 12. Spatial-Spectral dispersion from two transmitters shown for: (a) transmitters positioned at 1 cm separation, (b) left transmitter displaced by 24 cm. The right-hand transmitter (red) is swept over frequencies of (i) 31 GHz (ii) 33 GHz (iii) 35 GHz, while the frequency of the left-hand transmitter (yellow) is maintained at 33 GHz. The vertical displacement of the sources in the CCD images corresponds directly to the frequency offset of the transmitters.

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Accordingly, the frequency of one of the sources (circled in yellow) is maintained at 33 GHz while the other source (circled in red) is swept in 2 GHz increments from 31 to 35 GHz. While the frequency-swept source (red) is kept stationary, the 33 GHz transmitter (yellow) is moved along the azimuth plane from 1 cm to 24 cm separation between the two horn antennas. The resulting real-time images produced by the spatial-spectral imaging system are shown in Fig. 12. Therein, column (a) refers to the transmitters positioned 1 cm apart, whereas column (b) refers to the left transmitter positioned 24 cm away from the right transmitter. Note that the different physical positions of the sources result in different lateral displacement of the bright spots in the captured images. Rows (i) through (iii) correspond to the progression of the frequency sweeping in the right source from 31 GHz shown in (i), 33 GHz in (ii), and 35 GHz (iii). Change in frequency results in a vertical displacement of the corresponding bright spot in the captured images. The spatio-temporal locations of the emitters may be estimated using several methods, most prominently used within this work is the method of peak detection to derive the frequency and angle from the known beamsteering relation outlined in Eq. (13). The calibration emitter was located at 33 GHz, and broadside relative to the array, corresponding to the red source pictured in both (a-ii) and (b-ii). Using the known calibration emitters spatial displacements on the CCD array, the angle of arrival of the yellow emitter in column (a) is estimated as 0.5°, while the location of this emitter in column (b) is estimated as 4.5°.As shown in Fig. 11, the position of an isolated emitter may be located below the diffraction resolution limit using this known beamsteering relation to an estimated 10× improvement. Future investigations may potentially benefit from alternative image processing and detection methods allowing noise-limited super-resolution of emitters.

5. Summary

In this paper, an experimental microwave-photonic system is presented, which combines the angular steering of a spatial phased antenna arrays and the frequency-steering of an arrayed waveguide grating to produce a real-time, two-dimensional spatial-spectral Fourier transform of received radio-frequency wave fronts. This beam-forming technique enables optical channelization of received RF signals based upon the angle of arrival and frequency for several transmitters operating simultaneously. Future implementations will use high-speed photodetectors to enable direct down-conversion and signal recovery in addition to the spatial-spectral channelization capability.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

1. L. Lu, G. Y. Li, A. L. Swindlehurst, A. Ashikhmin, and R. Zhang, “An Overview of Massive MIMO: Benefits and Challenges,” IEEE J. Sel. Top. Signal Process. 8(5), 742–758 (2014). [CrossRef]  

2. J. Brady, N. Behdad, and A. M. Sayeed, “Beamspace MIMO for Millimeter-Wave Communications: System Architecture, Modeling, Analysis, and Measurements,” IEEE Trans. Antennas Propagat. 61(7), 3814–3827 (2013). [CrossRef]  

3. Q. H. Spencer, A. L. Swindlehurst, and M. Haardt, “Zero-forcing methods for downlink spatial multiplexing in multiuser MIMO channels,” in IEEE Trans. on Signal Processing, 52(2), 461–471 (2004).

4. E. G. Larsson, O. Edfors, F. Tufvesson, and T. L. Marzetta, “Massive MIMO for next generation wireless systems,” in IEEE Communications Magazine, 52(2), 186–195 (2014).

5. O. Elijah, C. Y. Leow, A. R. Tharek, S. Nunoo, and S. Z. Iliya, “Mitigating pilot contamination in massive MIMO system — 5G: An overview,” 10th Asian Control Conference (ASCC), 1–6 (2015).

6. H. Q. Ngo and E. G. Larsson, “EVD-based channel estimation in multicell multiuser MIMO systems with very large antenna arrays,” 2012 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Kyoto, 3249–3252 (2012).

7. C. Schuetz, R. Martin, T. Dillon, P. Yao, D. Mackrides, C. Harrity, A. Zablocki, K. Shreve, J. Bonnett, P. Curt, and D. Prather, “Realization of a Video-Rate Distributed Aperture Millimeter-Wave Imaging System using Optical Upconversion,” Proc. SPIE 8715, 87150I (2013). [CrossRef]  

8. C. A. Schuetz, R. D. Martin, I. Biswas, M. S. Mirotznik, S. Shi, G. J. Schneider, J. Murakowski, and D. W. Prather, “Sparse aperture millimeter-wave imaging using optical detection and correlation techniques,” Proc. SPIE 6548, 65480B (2007). [CrossRef]  

9. R. D. Martin, S. Shi, Y. Zhang, A. Wright, P. Yao, K. P. Shreve, C. A. Schuetz, T. E. Dillon, D. G. Mackrides, C. E. Harrity, and D. W. Prather, “Video rate passive millimeter-wave imager utilizing optical upconversion with improved size, weight, and power,” Proc. SPIE 9462, 946209 (2015). [CrossRef]  

10. R. D. Martin, C. A. Schuetz, D. W. Prather, and T. E. Dillon, “United States Patent: 8159737 - Controlling the phase of optical carriers,” U.S.patent 8159737 (April 17, 2012).

11. N. A. Hagen and M. W. Kudenov, “Review of snapshot spectral imaging technologies,” Opt. Eng. 52(9), 090901 (2013). [CrossRef]  

12. J. Murakowski, G. J. Schneider, S. Shi, C. A. Schuetz, and D. W. Prather, “Photonic probing of radio waves for k-space tomography,” Opt. Express 25(14), 15746–15759 (2017). [CrossRef]  

13. D. D. Ross, C. J. Ryan, G. J. Schneider, J. Murakowski, and D. W. Prather, “Passive Three-Dimensional Spatial-Spectral Analysis Based on k-Space Tomography,” in IEEE Photonics Technology Letters, 30(9), 817–820 (2018).

14. D. D. Ross, J. Murakowski, C. J. Ryan, G. J. Schneider, and D. W. Prather, “Compressive k-Space Tomography,” in Journal of Lightwave Technology, 36(19), 4478–4485 (2018).

15. A. V. Diebold, M. F. Imani, T. Fromenteze, D. L. Marks, and D. R. Smith, “Passive microwave spectral imaging with dynamic metasurface apertures,” Optica 7(5), 527–536 (2020). [CrossRef]  

16. Y. Yoshikuni, “Semiconductor arrayed waveguide gratings for photonic integrated devices,” in IEEE Journal of Selected Topics in Quantum Electronics, 8(6), 1102–1114 (2002).

17. H. Takahashi, K. Oda, H. Toba, and Y. Inoue, “Transmission characteristics of arrayed waveguide N/spl times/N wavelength multiplexer,” in Journal of Lightwave Technology, 13(3), 447–455 (1995).

18. J. N. Mait, R. D. Martin, C. A. Schuetz, and D. W. Prather, “Millimeter wave image processing through point spread function engineering,” Proc. SPIE 7936, 79360K (2011). [CrossRef]  

19. D. G. Mackrides, C. A. Schuetz, R. D. Martin, T. E. Dillon, P. Yao, and D. W. Prather, “Progress toward a video-rate, passive millimeter-wave imager for brownout mitigation,” Proc. SPIE 8022, 802203 (2011). [CrossRef]  

20. Y. Lo, “A mathematical theory of antenna arrays with randomly spaced elements,” IEEE Trans. Antennas Propagat. 12(3), 257–268 (1964). [CrossRef]  

21. C. J. Ryan, W. L. Beardell, J. Murakowski, D. D. Ross, G. J. Schneider, and D. W. Prather, “Log-periodic temporal apertures for grating lobe suppression in k-space tomography,” Opt. Express 28(11), 15969–15983 (2020). [CrossRef]  

References

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  1. L. Lu, G. Y. Li, A. L. Swindlehurst, A. Ashikhmin, and R. Zhang, “An Overview of Massive MIMO: Benefits and Challenges,” IEEE J. Sel. Top. Signal Process. 8(5), 742–758 (2014).
    [Crossref]
  2. J. Brady, N. Behdad, and A. M. Sayeed, “Beamspace MIMO for Millimeter-Wave Communications: System Architecture, Modeling, Analysis, and Measurements,” IEEE Trans. Antennas Propagat. 61(7), 3814–3827 (2013).
    [Crossref]
  3. Q. H. Spencer, A. L. Swindlehurst, and M. Haardt, “Zero-forcing methods for downlink spatial multiplexing in multiuser MIMO channels,” in IEEE Trans. on Signal Processing, 52(2), 461–471 (2004).
  4. E. G. Larsson, O. Edfors, F. Tufvesson, and T. L. Marzetta, “Massive MIMO for next generation wireless systems,” in IEEE Communications Magazine, 52(2), 186–195 (2014).
  5. O. Elijah, C. Y. Leow, A. R. Tharek, S. Nunoo, and S. Z. Iliya, “Mitigating pilot contamination in massive MIMO system — 5G: An overview,” 10th Asian Control Conference (ASCC), 1–6 (2015).
  6. H. Q. Ngo and E. G. Larsson, “EVD-based channel estimation in multicell multiuser MIMO systems with very large antenna arrays,” 2012 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Kyoto, 3249–3252 (2012).
  7. C. Schuetz, R. Martin, T. Dillon, P. Yao, D. Mackrides, C. Harrity, A. Zablocki, K. Shreve, J. Bonnett, P. Curt, and D. Prather, “Realization of a Video-Rate Distributed Aperture Millimeter-Wave Imaging System using Optical Upconversion,” Proc. SPIE 8715, 87150I (2013).
    [Crossref]
  8. C. A. Schuetz, R. D. Martin, I. Biswas, M. S. Mirotznik, S. Shi, G. J. Schneider, J. Murakowski, and D. W. Prather, “Sparse aperture millimeter-wave imaging using optical detection and correlation techniques,” Proc. SPIE 6548, 65480B (2007).
    [Crossref]
  9. R. D. Martin, S. Shi, Y. Zhang, A. Wright, P. Yao, K. P. Shreve, C. A. Schuetz, T. E. Dillon, D. G. Mackrides, C. E. Harrity, and D. W. Prather, “Video rate passive millimeter-wave imager utilizing optical upconversion with improved size, weight, and power,” Proc. SPIE 9462, 946209 (2015).
    [Crossref]
  10. R. D. Martin, C. A. Schuetz, D. W. Prather, and T. E. Dillon, “United States Patent: 8159737 - Controlling the phase of optical carriers,” U.S.patent 8159737 (April 17, 2012).
  11. N. A. Hagen and M. W. Kudenov, “Review of snapshot spectral imaging technologies,” Opt. Eng. 52(9), 090901 (2013).
    [Crossref]
  12. J. Murakowski, G. J. Schneider, S. Shi, C. A. Schuetz, and D. W. Prather, “Photonic probing of radio waves for k-space tomography,” Opt. Express 25(14), 15746–15759 (2017).
    [Crossref]
  13. D. D. Ross, C. J. Ryan, G. J. Schneider, J. Murakowski, and D. W. Prather, “Passive Three-Dimensional Spatial-Spectral Analysis Based on k-Space Tomography,” in IEEE Photonics Technology Letters, 30(9), 817–820 (2018).
  14. D. D. Ross, J. Murakowski, C. J. Ryan, G. J. Schneider, and D. W. Prather, “Compressive k-Space Tomography,” in Journal of Lightwave Technology, 36(19), 4478–4485 (2018).
  15. A. V. Diebold, M. F. Imani, T. Fromenteze, D. L. Marks, and D. R. Smith, “Passive microwave spectral imaging with dynamic metasurface apertures,” Optica 7(5), 527–536 (2020).
    [Crossref]
  16. Y. Yoshikuni, “Semiconductor arrayed waveguide gratings for photonic integrated devices,” in IEEE Journal of Selected Topics in Quantum Electronics, 8(6), 1102–1114 (2002).
  17. H. Takahashi, K. Oda, H. Toba, and Y. Inoue, “Transmission characteristics of arrayed waveguide N/spl times/N wavelength multiplexer,” in Journal of Lightwave Technology, 13(3), 447–455 (1995).
  18. J. N. Mait, R. D. Martin, C. A. Schuetz, and D. W. Prather, “Millimeter wave image processing through point spread function engineering,” Proc. SPIE 7936, 79360K (2011).
    [Crossref]
  19. D. G. Mackrides, C. A. Schuetz, R. D. Martin, T. E. Dillon, P. Yao, and D. W. Prather, “Progress toward a video-rate, passive millimeter-wave imager for brownout mitigation,” Proc. SPIE 8022, 802203 (2011).
    [Crossref]
  20. Y. Lo, “A mathematical theory of antenna arrays with randomly spaced elements,” IEEE Trans. Antennas Propagat. 12(3), 257–268 (1964).
    [Crossref]
  21. C. J. Ryan, W. L. Beardell, J. Murakowski, D. D. Ross, G. J. Schneider, and D. W. Prather, “Log-periodic temporal apertures for grating lobe suppression in k-space tomography,” Opt. Express 28(11), 15969–15983 (2020).
    [Crossref]

2020 (2)

2017 (1)

2015 (1)

R. D. Martin, S. Shi, Y. Zhang, A. Wright, P. Yao, K. P. Shreve, C. A. Schuetz, T. E. Dillon, D. G. Mackrides, C. E. Harrity, and D. W. Prather, “Video rate passive millimeter-wave imager utilizing optical upconversion with improved size, weight, and power,” Proc. SPIE 9462, 946209 (2015).
[Crossref]

2014 (1)

L. Lu, G. Y. Li, A. L. Swindlehurst, A. Ashikhmin, and R. Zhang, “An Overview of Massive MIMO: Benefits and Challenges,” IEEE J. Sel. Top. Signal Process. 8(5), 742–758 (2014).
[Crossref]

2013 (3)

J. Brady, N. Behdad, and A. M. Sayeed, “Beamspace MIMO for Millimeter-Wave Communications: System Architecture, Modeling, Analysis, and Measurements,” IEEE Trans. Antennas Propagat. 61(7), 3814–3827 (2013).
[Crossref]

C. Schuetz, R. Martin, T. Dillon, P. Yao, D. Mackrides, C. Harrity, A. Zablocki, K. Shreve, J. Bonnett, P. Curt, and D. Prather, “Realization of a Video-Rate Distributed Aperture Millimeter-Wave Imaging System using Optical Upconversion,” Proc. SPIE 8715, 87150I (2013).
[Crossref]

N. A. Hagen and M. W. Kudenov, “Review of snapshot spectral imaging technologies,” Opt. Eng. 52(9), 090901 (2013).
[Crossref]

2011 (2)

J. N. Mait, R. D. Martin, C. A. Schuetz, and D. W. Prather, “Millimeter wave image processing through point spread function engineering,” Proc. SPIE 7936, 79360K (2011).
[Crossref]

D. G. Mackrides, C. A. Schuetz, R. D. Martin, T. E. Dillon, P. Yao, and D. W. Prather, “Progress toward a video-rate, passive millimeter-wave imager for brownout mitigation,” Proc. SPIE 8022, 802203 (2011).
[Crossref]

2007 (1)

C. A. Schuetz, R. D. Martin, I. Biswas, M. S. Mirotznik, S. Shi, G. J. Schneider, J. Murakowski, and D. W. Prather, “Sparse aperture millimeter-wave imaging using optical detection and correlation techniques,” Proc. SPIE 6548, 65480B (2007).
[Crossref]

1964 (1)

Y. Lo, “A mathematical theory of antenna arrays with randomly spaced elements,” IEEE Trans. Antennas Propagat. 12(3), 257–268 (1964).
[Crossref]

Ashikhmin, A.

L. Lu, G. Y. Li, A. L. Swindlehurst, A. Ashikhmin, and R. Zhang, “An Overview of Massive MIMO: Benefits and Challenges,” IEEE J. Sel. Top. Signal Process. 8(5), 742–758 (2014).
[Crossref]

Beardell, W. L.

Behdad, N.

J. Brady, N. Behdad, and A. M. Sayeed, “Beamspace MIMO for Millimeter-Wave Communications: System Architecture, Modeling, Analysis, and Measurements,” IEEE Trans. Antennas Propagat. 61(7), 3814–3827 (2013).
[Crossref]

Biswas, I.

C. A. Schuetz, R. D. Martin, I. Biswas, M. S. Mirotznik, S. Shi, G. J. Schneider, J. Murakowski, and D. W. Prather, “Sparse aperture millimeter-wave imaging using optical detection and correlation techniques,” Proc. SPIE 6548, 65480B (2007).
[Crossref]

Bonnett, J.

C. Schuetz, R. Martin, T. Dillon, P. Yao, D. Mackrides, C. Harrity, A. Zablocki, K. Shreve, J. Bonnett, P. Curt, and D. Prather, “Realization of a Video-Rate Distributed Aperture Millimeter-Wave Imaging System using Optical Upconversion,” Proc. SPIE 8715, 87150I (2013).
[Crossref]

Brady, J.

J. Brady, N. Behdad, and A. M. Sayeed, “Beamspace MIMO for Millimeter-Wave Communications: System Architecture, Modeling, Analysis, and Measurements,” IEEE Trans. Antennas Propagat. 61(7), 3814–3827 (2013).
[Crossref]

Curt, P.

C. Schuetz, R. Martin, T. Dillon, P. Yao, D. Mackrides, C. Harrity, A. Zablocki, K. Shreve, J. Bonnett, P. Curt, and D. Prather, “Realization of a Video-Rate Distributed Aperture Millimeter-Wave Imaging System using Optical Upconversion,” Proc. SPIE 8715, 87150I (2013).
[Crossref]

Diebold, A. V.

Dillon, T.

C. Schuetz, R. Martin, T. Dillon, P. Yao, D. Mackrides, C. Harrity, A. Zablocki, K. Shreve, J. Bonnett, P. Curt, and D. Prather, “Realization of a Video-Rate Distributed Aperture Millimeter-Wave Imaging System using Optical Upconversion,” Proc. SPIE 8715, 87150I (2013).
[Crossref]

Dillon, T. E.

R. D. Martin, S. Shi, Y. Zhang, A. Wright, P. Yao, K. P. Shreve, C. A. Schuetz, T. E. Dillon, D. G. Mackrides, C. E. Harrity, and D. W. Prather, “Video rate passive millimeter-wave imager utilizing optical upconversion with improved size, weight, and power,” Proc. SPIE 9462, 946209 (2015).
[Crossref]

D. G. Mackrides, C. A. Schuetz, R. D. Martin, T. E. Dillon, P. Yao, and D. W. Prather, “Progress toward a video-rate, passive millimeter-wave imager for brownout mitigation,” Proc. SPIE 8022, 802203 (2011).
[Crossref]

R. D. Martin, C. A. Schuetz, D. W. Prather, and T. E. Dillon, “United States Patent: 8159737 - Controlling the phase of optical carriers,” U.S.patent 8159737 (April 17, 2012).

Edfors, O.

E. G. Larsson, O. Edfors, F. Tufvesson, and T. L. Marzetta, “Massive MIMO for next generation wireless systems,” in IEEE Communications Magazine, 52(2), 186–195 (2014).

Elijah, O.

O. Elijah, C. Y. Leow, A. R. Tharek, S. Nunoo, and S. Z. Iliya, “Mitigating pilot contamination in massive MIMO system — 5G: An overview,” 10th Asian Control Conference (ASCC), 1–6 (2015).

Fromenteze, T.

Haardt, M.

Q. H. Spencer, A. L. Swindlehurst, and M. Haardt, “Zero-forcing methods for downlink spatial multiplexing in multiuser MIMO channels,” in IEEE Trans. on Signal Processing, 52(2), 461–471 (2004).

Hagen, N. A.

N. A. Hagen and M. W. Kudenov, “Review of snapshot spectral imaging technologies,” Opt. Eng. 52(9), 090901 (2013).
[Crossref]

Harrity, C.

C. Schuetz, R. Martin, T. Dillon, P. Yao, D. Mackrides, C. Harrity, A. Zablocki, K. Shreve, J. Bonnett, P. Curt, and D. Prather, “Realization of a Video-Rate Distributed Aperture Millimeter-Wave Imaging System using Optical Upconversion,” Proc. SPIE 8715, 87150I (2013).
[Crossref]

Harrity, C. E.

R. D. Martin, S. Shi, Y. Zhang, A. Wright, P. Yao, K. P. Shreve, C. A. Schuetz, T. E. Dillon, D. G. Mackrides, C. E. Harrity, and D. W. Prather, “Video rate passive millimeter-wave imager utilizing optical upconversion with improved size, weight, and power,” Proc. SPIE 9462, 946209 (2015).
[Crossref]

Iliya, S. Z.

O. Elijah, C. Y. Leow, A. R. Tharek, S. Nunoo, and S. Z. Iliya, “Mitigating pilot contamination in massive MIMO system — 5G: An overview,” 10th Asian Control Conference (ASCC), 1–6 (2015).

Imani, M. F.

Inoue, Y.

H. Takahashi, K. Oda, H. Toba, and Y. Inoue, “Transmission characteristics of arrayed waveguide N/spl times/N wavelength multiplexer,” in Journal of Lightwave Technology, 13(3), 447–455 (1995).

Kudenov, M. W.

N. A. Hagen and M. W. Kudenov, “Review of snapshot spectral imaging technologies,” Opt. Eng. 52(9), 090901 (2013).
[Crossref]

Larsson, E. G.

H. Q. Ngo and E. G. Larsson, “EVD-based channel estimation in multicell multiuser MIMO systems with very large antenna arrays,” 2012 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Kyoto, 3249–3252 (2012).

E. G. Larsson, O. Edfors, F. Tufvesson, and T. L. Marzetta, “Massive MIMO for next generation wireless systems,” in IEEE Communications Magazine, 52(2), 186–195 (2014).

Leow, C. Y.

O. Elijah, C. Y. Leow, A. R. Tharek, S. Nunoo, and S. Z. Iliya, “Mitigating pilot contamination in massive MIMO system — 5G: An overview,” 10th Asian Control Conference (ASCC), 1–6 (2015).

Li, G. Y.

L. Lu, G. Y. Li, A. L. Swindlehurst, A. Ashikhmin, and R. Zhang, “An Overview of Massive MIMO: Benefits and Challenges,” IEEE J. Sel. Top. Signal Process. 8(5), 742–758 (2014).
[Crossref]

Lo, Y.

Y. Lo, “A mathematical theory of antenna arrays with randomly spaced elements,” IEEE Trans. Antennas Propagat. 12(3), 257–268 (1964).
[Crossref]

Lu, L.

L. Lu, G. Y. Li, A. L. Swindlehurst, A. Ashikhmin, and R. Zhang, “An Overview of Massive MIMO: Benefits and Challenges,” IEEE J. Sel. Top. Signal Process. 8(5), 742–758 (2014).
[Crossref]

Mackrides, D.

C. Schuetz, R. Martin, T. Dillon, P. Yao, D. Mackrides, C. Harrity, A. Zablocki, K. Shreve, J. Bonnett, P. Curt, and D. Prather, “Realization of a Video-Rate Distributed Aperture Millimeter-Wave Imaging System using Optical Upconversion,” Proc. SPIE 8715, 87150I (2013).
[Crossref]

Mackrides, D. G.

R. D. Martin, S. Shi, Y. Zhang, A. Wright, P. Yao, K. P. Shreve, C. A. Schuetz, T. E. Dillon, D. G. Mackrides, C. E. Harrity, and D. W. Prather, “Video rate passive millimeter-wave imager utilizing optical upconversion with improved size, weight, and power,” Proc. SPIE 9462, 946209 (2015).
[Crossref]

D. G. Mackrides, C. A. Schuetz, R. D. Martin, T. E. Dillon, P. Yao, and D. W. Prather, “Progress toward a video-rate, passive millimeter-wave imager for brownout mitigation,” Proc. SPIE 8022, 802203 (2011).
[Crossref]

Mait, J. N.

J. N. Mait, R. D. Martin, C. A. Schuetz, and D. W. Prather, “Millimeter wave image processing through point spread function engineering,” Proc. SPIE 7936, 79360K (2011).
[Crossref]

Marks, D. L.

Martin, R.

C. Schuetz, R. Martin, T. Dillon, P. Yao, D. Mackrides, C. Harrity, A. Zablocki, K. Shreve, J. Bonnett, P. Curt, and D. Prather, “Realization of a Video-Rate Distributed Aperture Millimeter-Wave Imaging System using Optical Upconversion,” Proc. SPIE 8715, 87150I (2013).
[Crossref]

Martin, R. D.

R. D. Martin, S. Shi, Y. Zhang, A. Wright, P. Yao, K. P. Shreve, C. A. Schuetz, T. E. Dillon, D. G. Mackrides, C. E. Harrity, and D. W. Prather, “Video rate passive millimeter-wave imager utilizing optical upconversion with improved size, weight, and power,” Proc. SPIE 9462, 946209 (2015).
[Crossref]

J. N. Mait, R. D. Martin, C. A. Schuetz, and D. W. Prather, “Millimeter wave image processing through point spread function engineering,” Proc. SPIE 7936, 79360K (2011).
[Crossref]

D. G. Mackrides, C. A. Schuetz, R. D. Martin, T. E. Dillon, P. Yao, and D. W. Prather, “Progress toward a video-rate, passive millimeter-wave imager for brownout mitigation,” Proc. SPIE 8022, 802203 (2011).
[Crossref]

C. A. Schuetz, R. D. Martin, I. Biswas, M. S. Mirotznik, S. Shi, G. J. Schneider, J. Murakowski, and D. W. Prather, “Sparse aperture millimeter-wave imaging using optical detection and correlation techniques,” Proc. SPIE 6548, 65480B (2007).
[Crossref]

R. D. Martin, C. A. Schuetz, D. W. Prather, and T. E. Dillon, “United States Patent: 8159737 - Controlling the phase of optical carriers,” U.S.patent 8159737 (April 17, 2012).

Marzetta, T. L.

E. G. Larsson, O. Edfors, F. Tufvesson, and T. L. Marzetta, “Massive MIMO for next generation wireless systems,” in IEEE Communications Magazine, 52(2), 186–195 (2014).

Mirotznik, M. S.

C. A. Schuetz, R. D. Martin, I. Biswas, M. S. Mirotznik, S. Shi, G. J. Schneider, J. Murakowski, and D. W. Prather, “Sparse aperture millimeter-wave imaging using optical detection and correlation techniques,” Proc. SPIE 6548, 65480B (2007).
[Crossref]

Murakowski, J.

C. J. Ryan, W. L. Beardell, J. Murakowski, D. D. Ross, G. J. Schneider, and D. W. Prather, “Log-periodic temporal apertures for grating lobe suppression in k-space tomography,” Opt. Express 28(11), 15969–15983 (2020).
[Crossref]

J. Murakowski, G. J. Schneider, S. Shi, C. A. Schuetz, and D. W. Prather, “Photonic probing of radio waves for k-space tomography,” Opt. Express 25(14), 15746–15759 (2017).
[Crossref]

C. A. Schuetz, R. D. Martin, I. Biswas, M. S. Mirotznik, S. Shi, G. J. Schneider, J. Murakowski, and D. W. Prather, “Sparse aperture millimeter-wave imaging using optical detection and correlation techniques,” Proc. SPIE 6548, 65480B (2007).
[Crossref]

D. D. Ross, C. J. Ryan, G. J. Schneider, J. Murakowski, and D. W. Prather, “Passive Three-Dimensional Spatial-Spectral Analysis Based on k-Space Tomography,” in IEEE Photonics Technology Letters, 30(9), 817–820 (2018).

D. D. Ross, J. Murakowski, C. J. Ryan, G. J. Schneider, and D. W. Prather, “Compressive k-Space Tomography,” in Journal of Lightwave Technology, 36(19), 4478–4485 (2018).

Ngo, H. Q.

H. Q. Ngo and E. G. Larsson, “EVD-based channel estimation in multicell multiuser MIMO systems with very large antenna arrays,” 2012 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Kyoto, 3249–3252 (2012).

Nunoo, S.

O. Elijah, C. Y. Leow, A. R. Tharek, S. Nunoo, and S. Z. Iliya, “Mitigating pilot contamination in massive MIMO system — 5G: An overview,” 10th Asian Control Conference (ASCC), 1–6 (2015).

Oda, K.

H. Takahashi, K. Oda, H. Toba, and Y. Inoue, “Transmission characteristics of arrayed waveguide N/spl times/N wavelength multiplexer,” in Journal of Lightwave Technology, 13(3), 447–455 (1995).

Prather, D.

C. Schuetz, R. Martin, T. Dillon, P. Yao, D. Mackrides, C. Harrity, A. Zablocki, K. Shreve, J. Bonnett, P. Curt, and D. Prather, “Realization of a Video-Rate Distributed Aperture Millimeter-Wave Imaging System using Optical Upconversion,” Proc. SPIE 8715, 87150I (2013).
[Crossref]

Prather, D. W.

C. J. Ryan, W. L. Beardell, J. Murakowski, D. D. Ross, G. J. Schneider, and D. W. Prather, “Log-periodic temporal apertures for grating lobe suppression in k-space tomography,” Opt. Express 28(11), 15969–15983 (2020).
[Crossref]

J. Murakowski, G. J. Schneider, S. Shi, C. A. Schuetz, and D. W. Prather, “Photonic probing of radio waves for k-space tomography,” Opt. Express 25(14), 15746–15759 (2017).
[Crossref]

R. D. Martin, S. Shi, Y. Zhang, A. Wright, P. Yao, K. P. Shreve, C. A. Schuetz, T. E. Dillon, D. G. Mackrides, C. E. Harrity, and D. W. Prather, “Video rate passive millimeter-wave imager utilizing optical upconversion with improved size, weight, and power,” Proc. SPIE 9462, 946209 (2015).
[Crossref]

D. G. Mackrides, C. A. Schuetz, R. D. Martin, T. E. Dillon, P. Yao, and D. W. Prather, “Progress toward a video-rate, passive millimeter-wave imager for brownout mitigation,” Proc. SPIE 8022, 802203 (2011).
[Crossref]

J. N. Mait, R. D. Martin, C. A. Schuetz, and D. W. Prather, “Millimeter wave image processing through point spread function engineering,” Proc. SPIE 7936, 79360K (2011).
[Crossref]

C. A. Schuetz, R. D. Martin, I. Biswas, M. S. Mirotznik, S. Shi, G. J. Schneider, J. Murakowski, and D. W. Prather, “Sparse aperture millimeter-wave imaging using optical detection and correlation techniques,” Proc. SPIE 6548, 65480B (2007).
[Crossref]

D. D. Ross, C. J. Ryan, G. J. Schneider, J. Murakowski, and D. W. Prather, “Passive Three-Dimensional Spatial-Spectral Analysis Based on k-Space Tomography,” in IEEE Photonics Technology Letters, 30(9), 817–820 (2018).

R. D. Martin, C. A. Schuetz, D. W. Prather, and T. E. Dillon, “United States Patent: 8159737 - Controlling the phase of optical carriers,” U.S.patent 8159737 (April 17, 2012).

D. D. Ross, J. Murakowski, C. J. Ryan, G. J. Schneider, and D. W. Prather, “Compressive k-Space Tomography,” in Journal of Lightwave Technology, 36(19), 4478–4485 (2018).

Ross, D. D.

C. J. Ryan, W. L. Beardell, J. Murakowski, D. D. Ross, G. J. Schneider, and D. W. Prather, “Log-periodic temporal apertures for grating lobe suppression in k-space tomography,” Opt. Express 28(11), 15969–15983 (2020).
[Crossref]

D. D. Ross, J. Murakowski, C. J. Ryan, G. J. Schneider, and D. W. Prather, “Compressive k-Space Tomography,” in Journal of Lightwave Technology, 36(19), 4478–4485 (2018).

D. D. Ross, C. J. Ryan, G. J. Schneider, J. Murakowski, and D. W. Prather, “Passive Three-Dimensional Spatial-Spectral Analysis Based on k-Space Tomography,” in IEEE Photonics Technology Letters, 30(9), 817–820 (2018).

Ryan, C. J.

C. J. Ryan, W. L. Beardell, J. Murakowski, D. D. Ross, G. J. Schneider, and D. W. Prather, “Log-periodic temporal apertures for grating lobe suppression in k-space tomography,” Opt. Express 28(11), 15969–15983 (2020).
[Crossref]

D. D. Ross, C. J. Ryan, G. J. Schneider, J. Murakowski, and D. W. Prather, “Passive Three-Dimensional Spatial-Spectral Analysis Based on k-Space Tomography,” in IEEE Photonics Technology Letters, 30(9), 817–820 (2018).

D. D. Ross, J. Murakowski, C. J. Ryan, G. J. Schneider, and D. W. Prather, “Compressive k-Space Tomography,” in Journal of Lightwave Technology, 36(19), 4478–4485 (2018).

Sayeed, A. M.

J. Brady, N. Behdad, and A. M. Sayeed, “Beamspace MIMO for Millimeter-Wave Communications: System Architecture, Modeling, Analysis, and Measurements,” IEEE Trans. Antennas Propagat. 61(7), 3814–3827 (2013).
[Crossref]

Schneider, G. J.

C. J. Ryan, W. L. Beardell, J. Murakowski, D. D. Ross, G. J. Schneider, and D. W. Prather, “Log-periodic temporal apertures for grating lobe suppression in k-space tomography,” Opt. Express 28(11), 15969–15983 (2020).
[Crossref]

J. Murakowski, G. J. Schneider, S. Shi, C. A. Schuetz, and D. W. Prather, “Photonic probing of radio waves for k-space tomography,” Opt. Express 25(14), 15746–15759 (2017).
[Crossref]

C. A. Schuetz, R. D. Martin, I. Biswas, M. S. Mirotznik, S. Shi, G. J. Schneider, J. Murakowski, and D. W. Prather, “Sparse aperture millimeter-wave imaging using optical detection and correlation techniques,” Proc. SPIE 6548, 65480B (2007).
[Crossref]

D. D. Ross, J. Murakowski, C. J. Ryan, G. J. Schneider, and D. W. Prather, “Compressive k-Space Tomography,” in Journal of Lightwave Technology, 36(19), 4478–4485 (2018).

D. D. Ross, C. J. Ryan, G. J. Schneider, J. Murakowski, and D. W. Prather, “Passive Three-Dimensional Spatial-Spectral Analysis Based on k-Space Tomography,” in IEEE Photonics Technology Letters, 30(9), 817–820 (2018).

Schuetz, C.

C. Schuetz, R. Martin, T. Dillon, P. Yao, D. Mackrides, C. Harrity, A. Zablocki, K. Shreve, J. Bonnett, P. Curt, and D. Prather, “Realization of a Video-Rate Distributed Aperture Millimeter-Wave Imaging System using Optical Upconversion,” Proc. SPIE 8715, 87150I (2013).
[Crossref]

Schuetz, C. A.

J. Murakowski, G. J. Schneider, S. Shi, C. A. Schuetz, and D. W. Prather, “Photonic probing of radio waves for k-space tomography,” Opt. Express 25(14), 15746–15759 (2017).
[Crossref]

R. D. Martin, S. Shi, Y. Zhang, A. Wright, P. Yao, K. P. Shreve, C. A. Schuetz, T. E. Dillon, D. G. Mackrides, C. E. Harrity, and D. W. Prather, “Video rate passive millimeter-wave imager utilizing optical upconversion with improved size, weight, and power,” Proc. SPIE 9462, 946209 (2015).
[Crossref]

J. N. Mait, R. D. Martin, C. A. Schuetz, and D. W. Prather, “Millimeter wave image processing through point spread function engineering,” Proc. SPIE 7936, 79360K (2011).
[Crossref]

D. G. Mackrides, C. A. Schuetz, R. D. Martin, T. E. Dillon, P. Yao, and D. W. Prather, “Progress toward a video-rate, passive millimeter-wave imager for brownout mitigation,” Proc. SPIE 8022, 802203 (2011).
[Crossref]

C. A. Schuetz, R. D. Martin, I. Biswas, M. S. Mirotznik, S. Shi, G. J. Schneider, J. Murakowski, and D. W. Prather, “Sparse aperture millimeter-wave imaging using optical detection and correlation techniques,” Proc. SPIE 6548, 65480B (2007).
[Crossref]

R. D. Martin, C. A. Schuetz, D. W. Prather, and T. E. Dillon, “United States Patent: 8159737 - Controlling the phase of optical carriers,” U.S.patent 8159737 (April 17, 2012).

Shi, S.

J. Murakowski, G. J. Schneider, S. Shi, C. A. Schuetz, and D. W. Prather, “Photonic probing of radio waves for k-space tomography,” Opt. Express 25(14), 15746–15759 (2017).
[Crossref]

R. D. Martin, S. Shi, Y. Zhang, A. Wright, P. Yao, K. P. Shreve, C. A. Schuetz, T. E. Dillon, D. G. Mackrides, C. E. Harrity, and D. W. Prather, “Video rate passive millimeter-wave imager utilizing optical upconversion with improved size, weight, and power,” Proc. SPIE 9462, 946209 (2015).
[Crossref]

C. A. Schuetz, R. D. Martin, I. Biswas, M. S. Mirotznik, S. Shi, G. J. Schneider, J. Murakowski, and D. W. Prather, “Sparse aperture millimeter-wave imaging using optical detection and correlation techniques,” Proc. SPIE 6548, 65480B (2007).
[Crossref]

Shreve, K.

C. Schuetz, R. Martin, T. Dillon, P. Yao, D. Mackrides, C. Harrity, A. Zablocki, K. Shreve, J. Bonnett, P. Curt, and D. Prather, “Realization of a Video-Rate Distributed Aperture Millimeter-Wave Imaging System using Optical Upconversion,” Proc. SPIE 8715, 87150I (2013).
[Crossref]

Shreve, K. P.

R. D. Martin, S. Shi, Y. Zhang, A. Wright, P. Yao, K. P. Shreve, C. A. Schuetz, T. E. Dillon, D. G. Mackrides, C. E. Harrity, and D. W. Prather, “Video rate passive millimeter-wave imager utilizing optical upconversion with improved size, weight, and power,” Proc. SPIE 9462, 946209 (2015).
[Crossref]

Smith, D. R.

Spencer, Q. H.

Q. H. Spencer, A. L. Swindlehurst, and M. Haardt, “Zero-forcing methods for downlink spatial multiplexing in multiuser MIMO channels,” in IEEE Trans. on Signal Processing, 52(2), 461–471 (2004).

Swindlehurst, A. L.

L. Lu, G. Y. Li, A. L. Swindlehurst, A. Ashikhmin, and R. Zhang, “An Overview of Massive MIMO: Benefits and Challenges,” IEEE J. Sel. Top. Signal Process. 8(5), 742–758 (2014).
[Crossref]

Q. H. Spencer, A. L. Swindlehurst, and M. Haardt, “Zero-forcing methods for downlink spatial multiplexing in multiuser MIMO channels,” in IEEE Trans. on Signal Processing, 52(2), 461–471 (2004).

Takahashi, H.

H. Takahashi, K. Oda, H. Toba, and Y. Inoue, “Transmission characteristics of arrayed waveguide N/spl times/N wavelength multiplexer,” in Journal of Lightwave Technology, 13(3), 447–455 (1995).

Tharek, A. R.

O. Elijah, C. Y. Leow, A. R. Tharek, S. Nunoo, and S. Z. Iliya, “Mitigating pilot contamination in massive MIMO system — 5G: An overview,” 10th Asian Control Conference (ASCC), 1–6 (2015).

Toba, H.

H. Takahashi, K. Oda, H. Toba, and Y. Inoue, “Transmission characteristics of arrayed waveguide N/spl times/N wavelength multiplexer,” in Journal of Lightwave Technology, 13(3), 447–455 (1995).

Tufvesson, F.

E. G. Larsson, O. Edfors, F. Tufvesson, and T. L. Marzetta, “Massive MIMO for next generation wireless systems,” in IEEE Communications Magazine, 52(2), 186–195 (2014).

Wright, A.

R. D. Martin, S. Shi, Y. Zhang, A. Wright, P. Yao, K. P. Shreve, C. A. Schuetz, T. E. Dillon, D. G. Mackrides, C. E. Harrity, and D. W. Prather, “Video rate passive millimeter-wave imager utilizing optical upconversion with improved size, weight, and power,” Proc. SPIE 9462, 946209 (2015).
[Crossref]

Yao, P.

R. D. Martin, S. Shi, Y. Zhang, A. Wright, P. Yao, K. P. Shreve, C. A. Schuetz, T. E. Dillon, D. G. Mackrides, C. E. Harrity, and D. W. Prather, “Video rate passive millimeter-wave imager utilizing optical upconversion with improved size, weight, and power,” Proc. SPIE 9462, 946209 (2015).
[Crossref]

C. Schuetz, R. Martin, T. Dillon, P. Yao, D. Mackrides, C. Harrity, A. Zablocki, K. Shreve, J. Bonnett, P. Curt, and D. Prather, “Realization of a Video-Rate Distributed Aperture Millimeter-Wave Imaging System using Optical Upconversion,” Proc. SPIE 8715, 87150I (2013).
[Crossref]

D. G. Mackrides, C. A. Schuetz, R. D. Martin, T. E. Dillon, P. Yao, and D. W. Prather, “Progress toward a video-rate, passive millimeter-wave imager for brownout mitigation,” Proc. SPIE 8022, 802203 (2011).
[Crossref]

Yoshikuni, Y.

Y. Yoshikuni, “Semiconductor arrayed waveguide gratings for photonic integrated devices,” in IEEE Journal of Selected Topics in Quantum Electronics, 8(6), 1102–1114 (2002).

Zablocki, A.

C. Schuetz, R. Martin, T. Dillon, P. Yao, D. Mackrides, C. Harrity, A. Zablocki, K. Shreve, J. Bonnett, P. Curt, and D. Prather, “Realization of a Video-Rate Distributed Aperture Millimeter-Wave Imaging System using Optical Upconversion,” Proc. SPIE 8715, 87150I (2013).
[Crossref]

Zhang, R.

L. Lu, G. Y. Li, A. L. Swindlehurst, A. Ashikhmin, and R. Zhang, “An Overview of Massive MIMO: Benefits and Challenges,” IEEE J. Sel. Top. Signal Process. 8(5), 742–758 (2014).
[Crossref]

Zhang, Y.

R. D. Martin, S. Shi, Y. Zhang, A. Wright, P. Yao, K. P. Shreve, C. A. Schuetz, T. E. Dillon, D. G. Mackrides, C. E. Harrity, and D. W. Prather, “Video rate passive millimeter-wave imager utilizing optical upconversion with improved size, weight, and power,” Proc. SPIE 9462, 946209 (2015).
[Crossref]

IEEE J. Sel. Top. Signal Process. (1)

L. Lu, G. Y. Li, A. L. Swindlehurst, A. Ashikhmin, and R. Zhang, “An Overview of Massive MIMO: Benefits and Challenges,” IEEE J. Sel. Top. Signal Process. 8(5), 742–758 (2014).
[Crossref]

IEEE Trans. Antennas Propagat. (2)

J. Brady, N. Behdad, and A. M. Sayeed, “Beamspace MIMO for Millimeter-Wave Communications: System Architecture, Modeling, Analysis, and Measurements,” IEEE Trans. Antennas Propagat. 61(7), 3814–3827 (2013).
[Crossref]

Y. Lo, “A mathematical theory of antenna arrays with randomly spaced elements,” IEEE Trans. Antennas Propagat. 12(3), 257–268 (1964).
[Crossref]

Opt. Eng. (1)

N. A. Hagen and M. W. Kudenov, “Review of snapshot spectral imaging technologies,” Opt. Eng. 52(9), 090901 (2013).
[Crossref]

Opt. Express (2)

Optica (1)

Proc. SPIE (5)

J. N. Mait, R. D. Martin, C. A. Schuetz, and D. W. Prather, “Millimeter wave image processing through point spread function engineering,” Proc. SPIE 7936, 79360K (2011).
[Crossref]

D. G. Mackrides, C. A. Schuetz, R. D. Martin, T. E. Dillon, P. Yao, and D. W. Prather, “Progress toward a video-rate, passive millimeter-wave imager for brownout mitigation,” Proc. SPIE 8022, 802203 (2011).
[Crossref]

C. Schuetz, R. Martin, T. Dillon, P. Yao, D. Mackrides, C. Harrity, A. Zablocki, K. Shreve, J. Bonnett, P. Curt, and D. Prather, “Realization of a Video-Rate Distributed Aperture Millimeter-Wave Imaging System using Optical Upconversion,” Proc. SPIE 8715, 87150I (2013).
[Crossref]

C. A. Schuetz, R. D. Martin, I. Biswas, M. S. Mirotznik, S. Shi, G. J. Schneider, J. Murakowski, and D. W. Prather, “Sparse aperture millimeter-wave imaging using optical detection and correlation techniques,” Proc. SPIE 6548, 65480B (2007).
[Crossref]

R. D. Martin, S. Shi, Y. Zhang, A. Wright, P. Yao, K. P. Shreve, C. A. Schuetz, T. E. Dillon, D. G. Mackrides, C. E. Harrity, and D. W. Prather, “Video rate passive millimeter-wave imager utilizing optical upconversion with improved size, weight, and power,” Proc. SPIE 9462, 946209 (2015).
[Crossref]

Other (9)

R. D. Martin, C. A. Schuetz, D. W. Prather, and T. E. Dillon, “United States Patent: 8159737 - Controlling the phase of optical carriers,” U.S.patent 8159737 (April 17, 2012).

Q. H. Spencer, A. L. Swindlehurst, and M. Haardt, “Zero-forcing methods for downlink spatial multiplexing in multiuser MIMO channels,” in IEEE Trans. on Signal Processing, 52(2), 461–471 (2004).

E. G. Larsson, O. Edfors, F. Tufvesson, and T. L. Marzetta, “Massive MIMO for next generation wireless systems,” in IEEE Communications Magazine, 52(2), 186–195 (2014).

O. Elijah, C. Y. Leow, A. R. Tharek, S. Nunoo, and S. Z. Iliya, “Mitigating pilot contamination in massive MIMO system — 5G: An overview,” 10th Asian Control Conference (ASCC), 1–6 (2015).

H. Q. Ngo and E. G. Larsson, “EVD-based channel estimation in multicell multiuser MIMO systems with very large antenna arrays,” 2012 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Kyoto, 3249–3252 (2012).

Y. Yoshikuni, “Semiconductor arrayed waveguide gratings for photonic integrated devices,” in IEEE Journal of Selected Topics in Quantum Electronics, 8(6), 1102–1114 (2002).

H. Takahashi, K. Oda, H. Toba, and Y. Inoue, “Transmission characteristics of arrayed waveguide N/spl times/N wavelength multiplexer,” in Journal of Lightwave Technology, 13(3), 447–455 (1995).

D. D. Ross, C. J. Ryan, G. J. Schneider, J. Murakowski, and D. W. Prather, “Passive Three-Dimensional Spatial-Spectral Analysis Based on k-Space Tomography,” in IEEE Photonics Technology Letters, 30(9), 817–820 (2018).

D. D. Ross, J. Murakowski, C. J. Ryan, G. J. Schneider, and D. W. Prather, “Compressive k-Space Tomography,” in Journal of Lightwave Technology, 36(19), 4478–4485 (2018).

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (12)

Fig. 1.
Fig. 1. Conceptual diagram of a photonic k-space imaging for resolving the angle of arrival and frequency of incident radio frequency wave fronts.
Fig. 2.
Fig. 2. Illustration of the wavefront, K, projected onto the x-axis of the antenna array positions, rnX, with apparent incident angle θ. Note the apparent elevation angle is negligible.
Fig. 3.
Fig. 3. (a) Spatial-temporal array geometry of the radio-frequency antennas and optical fiber lengths (Xn, Ln). (b) Spatial array geometry of the two-dimensional fiber array (xn, yn). (c) the resulting optical point spread function of the array geometry with an RF wavefront of incident angle 0° and temporal frequency of 33 GHz, in the middle of the band, which is steered electronically to the center of the detector plane.
Fig. 4.
Fig. 4. Numerically evaluated cross-section of optical intensity response at the u-v detector plane for an optical sideband offset by the received radio frequency swept from 26 to 40 GHz in 1 GHz increments. Note the frequencies are listed as the offset from the optical carrier.
Fig. 5.
Fig. 5. Adjusting the relative size of the fiber array, shown in Fig. 3(b), from (a) original scaling, (b) 150% scaling, (c) 200% scaling and (d) 250% scaling. Note the same color mapping is used as in Fig. 4 relating the optical beam to corresponding sideband offset frequency.
Fig. 6.
Fig. 6. Numerical evaluation of the two-dimensional beamforming technique shown for four separate cases of receiving from emitters at spatio-temporal locations of: (a) −5° and 27 GHz, (b) 5° and 27 GHz, (c) −5° and 33 GHz, (d) 5° and 33 GHz, (e) −5° and 39 GHz and (d) 5° and 39 GHz. Note the projection of the main beam on the u-axis corresponding to the emitter’s spatial location, while the projection on the v-axis corresponds to the temporal frequency.
Fig. 7.
Fig. 7. Numerical evaluation of two-dimensional spatial-spectral imaging capability with two incoherent RF emitters. Column (a): Emitter displaced by 1° from broadside at 33 GHz with secondary emitter positions of (i) Broadside at 31 GHz (ii) Broadside at 33 GHz, sources are not resolvable within this scene (iii) Broadside at 35 GHz. Column (b): Emitter displaced by 2.5° at 33 GHz with secondary emitter positions of (i) Broadside at 31 GHz (ii) Broadside at 33 GHz (iii) Broadside at 35 GHz. We note that aside form (a-ii), wherein both emitters are placed in the same spatial-spectral bin relative to the array resolution, each emitter is uniquely identifiable by its spatial-spectral location.
Fig. 8.
Fig. 8. Conceptual diagram of the experimental configuration of the two-dimensional five-arm spiral antenna array and the corresponding optical fiber array comprising thirty elements.
Fig. 9.
Fig. 9. (a) Notional schematic diagram of the two-dimensional spatial-spectral imaging system. PBS: Polarizing beam splitter. BS: beam splitter. EDFA: Erbium-doped fiber amplifier. (b) 30-element phased array in five-arm spiral configuration. (c) Close-up of the 200-element fiber-lenslet array, of which only the inner 30 lenslets are coupled to optical fibers. (d) Corresponding optical image of the 30-element fiber-lenslet array as seen at the location of the phase-detection board. (e) Interference pattern generated by the 30-element optical phased array as seen at the CCD image-sensor location.
Fig. 10.
Fig. 10. Comparison of the optical point spread functions in ADU counts for a source at 33 GHz produced by (a) the experimental system, and (b) a numerical evaluation with empirically derived phase variations from path-length mismatching accounted for. The experimental point spread function is intentionally saturated to raise the side-lobe distribution to detectable levels.
Fig. 11.
Fig. 11. Spatial distribution of frequency-steered optical sidebands along a center cross-section of the measured camera response. Steering is shown as a composite in 1 GHz steps for a 10-GHz bandwidth of 28–39 GHz.
Fig. 12.
Fig. 12. Spatial-Spectral dispersion from two transmitters shown for: (a) transmitters positioned at 1 cm separation, (b) left transmitter displaced by 24 cm. The right-hand transmitter (red) is swept over frequencies of (i) 31 GHz (ii) 33 GHz (iii) 35 GHz, while the frequency of the left-hand transmitter (yellow) is maintained at 33 GHz. The vertical displacement of the sources in the CCD images corresponds directly to the frequency offset of the transmitters.

Equations (19)

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U = U ( t ) = A e j ( Ω t K r + ϕ ) r + c .c . ,
u = a exp [ j ( ω t + φ ) + π U ( t ) V π ] + c .c . ,
u = a e j ( ω t + φ ) [ 1 + π U ( t ) V π ] + o ( U V π ) + c .c a e j ( ω t + φ ) [ 1 + π A V π e j ( Ω t K r ) r + c .c . ] + c .c .
u = a e j ( ω t + φ ) + c .c .  +  π V π a A r exp [ j ( ω Ω ) t + φ ϕ ] exp ( j K r ) + c .c .  +  π V π a A r exp [ j ( ω + Ω ) t + φ + ϕ ] exp ( j K r ) + c .c .
u n = π V π a A | r + r n | exp [ j ( ω + Ω ) ( t + t n ) + φ n + ϕ ] exp ( j K | r + r n | ) + c .c .
exp ( j K | r + r n | ) exp [ j K n ( r + r n ) ] .
K n = K r + r n | r + r n | K r + r n | r | = K ( r | r | + r n | r | ) K r | r | K ,
u n = { π V π a A r exp [ j ( ω + Ω ) t j K r + j ϕ ] } exp [ j K r n ] exp [ j Ω t n ] + c .c .
( x n , y n ) = ( s r n X , s t n ) ,
U ( u , v ) = 1 j λ f n u n exp [ 2 π j λ f ( x n u + y n v ) ] + c .c . = 1 j λ f { } n exp [ j K X r n X ] exp [ j Ω t n ] exp [ 2 π j λ f ( x n u + y n v ) ] + c .c .
U ( u , v ) = 1 j λ f { } n exp [ j ( K X + 2 π s λ f u ) r n X ] exp [ j ( Ω 2 π s λ f v ) t n ] + c .c .
( K X + 2 π s λ f u ) = 0 u = λ f 2 π s K X ( Ω 2 π s λ f v ) = 0 v = λ f 2 π s Ω .
u = λ f 2 π s K sin ( θ ) = λ f 2 π s c Ω sin ( θ ) v = λ f 2 π s Ω . .
1 z j z R exp [ j k ( x n x ) 2 + ( y n y ) 2 2 ( z j z R ) ] | z = 0 = 1 j z R exp [ ( x n x ) 2 + ( y n y ) 2 w 0 2 ]
w 0 2 = z R λ π .
| U ( u , v ) | 2 = 2 ( π V π a A r 1 λ f ) 2 exp [ ( 2 π w 0 λ f ) 2 ( u 2 + v 2 ) ] × × | n exp [ j ( K X + 2 π s λ f u ) r n X ] exp [ j ( Ω 2 π s λ f v ) t n ] exp ( j φ n ) | 2 ,
P = Area | U ( u , v ) | 2 d 2 x | U ( u , v ) | 2 A .
Δ f = c η Δ L max ;
Δ θ = λ D ,