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Abstract
Time-of-flight (ToF) non-line-of-sight (NLoS) imaging reconstructs images of scenes with light that have undergone diffuse reflections. While, in the past, ToF light propagation and reconstruction methods have been described using their own inverse methods, it has recently been shown that ToF light transport can be described as the propagation of a wave, allowing it to be modeled by the same methods that are applied for direct imaging with electromagnetic or sound waves. This wave of fluctuating optical irradiance is called the phasor field ($\mathcal {P}$-field) wave. Here, we perform a series of experiments to show the wave-like behavior of this $\mathcal {P}$-field wave. We design a $\mathcal {P}$-field source and detector and use them to demonstrate interference of $\mathcal {P}$-field waves in a double slit experiment, as well as mutually-independent focusing and imaging of $\mathcal {P}$-field waves and their optical carrier. Besides establishing the properties of $\mathcal {P}$-field waves, our work demonstrates that imaging of ToF signals is possible without any computation enabling fast and energy-efficient NLoS imaging systems.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
While optical imaging yields great spatial resolution because of the short wavelength, it fails when there is no direct line of sight from the optical detector to the objects to be imaged. However, by utilizing ultrafast pulsed lasers and ultrafast detectors, the time-of-flight of the returned photons can be used to see around corners using diffuse reflections (see Fig. 1). This concept, first experimentally demonstrated by Velten et al. [1] and often referred to as non-line-of-sight (NLoS) imaging, has since attracted significant interest from researchers especially in the fields of optics, computer vision and computational imaging.
Fig. 1. Around the corner imaging with data obtained for different laser/camera position combinations.
Diffuse scattering from visible objects such as a relay wall enables photons to penetrate into spatial regions not directly visible from the detector. After possibly multiple reflections in the scene, some photons make their way back to the relay wall and from there to the detector. The photons recorded at multiple detector focus positions have different travel times depending on the scene geometry. A knowledge of the travel times of all photons allows for reconstructing a hidden space by (filtered) backprojection methods [1–8] or by various other techniques [9–11].
Recently, Reza et al. [12] introduced the theoretical and physical concept of phasor fields ($\mathcal {P}$-fields) which are described as slow temporal fluctuations in optical irradiance, and show that propagation of $\mathcal {P}$-fields between surfaces is described by the Rayleigh-Sommerfeld diffraction (RSD) integral that describes the propagation of all waves. Furthermore, [13] builds on the $\mathcal {P}$-field model in [12] to demonstrate its use in occlusion-aided imaging and [14] has expanded the theoretical framework of $\mathcal {P}$-field imaging. Moreover, Liu et al. [15] demonstrated a $\mathcal {P}$-field-based NLoS reconstruction approach which treats $\mathcal {P}$-field propagation analogously to the propagation of Electric fields (E-fields) which describe a conventional line-of-sight (LoS) imaging scenario as a sum of E-field contributions using the Huygens’ integral. Liu et al. consider the diffuse relay wall in an around-the-corner NLoS imaging scenario as a virtual aperture of a virtual $\mathcal {P}$-field camera that looks directly at the hidden scene. The scene is illuminated with short laser pulses and even though these illumination pulses are comprised of several modulation frequencies (i.e., Fourier components), individual $\mathcal {P}$-field waves can be extracted from the delta function time response via a computational post-processing filtering step. Whilst a vast majority of NLoS imaging techniques are based on pulsed light sources, sinusoidally-modulated continuous wave (CW) sources are more commonly used for 3D ranging in direct line-of-sight (LOS) scenarios and in distributed media [16–19].
In this paper, we perform a series of experiments that demonstrate a wave-like behavior of $\mathcal {P}$-fields as summarized in Fig. 2. Contrary to [15] - where $\mathcal {P}$-field is treated as a post data acquisition signal processing virtual quantity - in this paper we show that it can very well be used as a real quantity (which is the modulation signal that modulates a continuous wave optical carrier) for imaging purposes. The set of experiments we present in this paper show $\mathcal {P}$-field interference, focusing and imaging phenomena exhibited by other types of waves. A knowledge of $\mathcal {P}$-field properties holds great promise in the development of real-time ToF NLoS imaging methods and systems. For example, NLoS imaging systems can be developed that rely on directly measuring $\mathcal {P}$-fields instead of slow and computationally intense post-processing of recorded data such as the one proposed by Kadambi et al. in [20]. With an understanding of $\mathcal {P}$-field properties, real-time $\mathcal {P}$-field imaging systems can be designed where the evolution of a $\mathcal {P}$-field focal spot, $\mathcal {P}$-field detected power etc. can be directly measured for NLoS tracking and imaging applications.
The experiments which we perform, analyze and discuss in the paper are:
- 1. A double-slit experiment using two diffusers as $\mathcal {P}$-field apertures. The $\mathcal {P}$-field contributions from each diffuser interfere as conventional E-field waves do. In (Sec. 4.1), we demonstrate a $\mathcal {P}$-field fringe pattern which is the result of wave-like $\mathcal {P}$-field interference.
- 2. An experiment with a curved diffuser acting as a focusing mirror. We show in Sec. 4.2 that we can create a focused image of a $\mathcal {P}$-field source despite no focus of the optical carrier.
- 3. An optical system realized with large Fresnel lenses to demonstrate that an optical carrier can be focused without $\mathcal {P}$-field focus.
Next, in Sec. 2, we briefly recall the $\mathcal {P}$-field theory from [12] before proceeding to discuss our experimental setups, results and the $\mathcal {P}$-field properties we verify in light of our experimental results in Sec. 4.
2. Theoretical background of $\mathcal {P}$-fields
2.1 Introduction to $\mathcal {P}$-fields
Let us first provide a brief introduction to the notion of the $\mathcal {P}$-field introduced in [12,21]. It is well-known that the Huygens’ integral is a solution to the scalar wave equation and it describes the Electric field (E-field) at a location $(x,y)$ in a detection plane $\Sigma$ denoted by $z = Z$ as a sum of E-field spherical wavelet contributions from all locations $(x',y')$ of a specular surface $\mathcal {A}$ at $z = 0$. In the context of imaging, the Huygens’ integral
Fig. 3. An aperture plane $\mathcal {A}$ and an observation plane $\Sigma$ separated by a distance $Z$.
For a diffuse surface $\mathcal {A}$ and an optical signal which is amplitude modulated by a $\mathcal {P}$-field signal
of frequency $\Omega$, we demonstrated in [12] that the magnitude of the $\mathcal {P}$-field sum $[\mathcal {P}_{0,\Omega }(x,y)]_{\mathrm {Sum}}$ in a detection plane $\Sigma$ is described by as a sum of $\mathcal {P}$-field contributions $\mathcal {P}(r)$ from $\mathcal {A}$ in a Huygens’-like formulation as;As is seen from Eq. (3), we observe that $\mathcal {P}$-field contributions from a diffuse surface $\mathcal {A}$ add analogously to how E-fields from a specular surface add as is described by the Huygens’ integral in Eq. (1). The summation in Eq. (3) holds true when the condition $\lambda _{\mathrm E} \ll |\gamma | \ll \lambda _{\mathrm P}$ for the maximum roughness $\gamma$ of the diffuse surface is satisfied. We now verify the wave-like properties of $\mathcal {P}$-fields akin to the behavior of E-fields.
3. A $\mathcal {P}$-field detector and a $\mathcal {P}$-field source
Here, we define what is referred to as a $\mathcal {P}$-field detector and a $\mathcal {P}$-field source throughout the paper without further elaboration.
3.1 A $\mathcal {P}$-field source
A $\mathcal {P}$-field source refers to an amplitude modulated optical source as is shown in Fig. 4(a). The slowly-varying envelope of the average optical irradiance $\langle I(t) \rangle$ is the $\mathcal {P}$-field. In our experiments, we use a laser diode (LD) with 50 mW average power at an E-field wavelength $\lambda _{\mathrm E}$ of 520 nm. Light from the LD is amplitude modulated at a 1 GHz frequency ($\lambda _{\mathrm P}\thinspace \textrm{of} \thinspace {30}{\textrm {cm}}$) using an electro-optic modulator. Further details on the operation and the actual implementation of a $\mathcal {P}$-field source are provided in Appendix A.
Fig. 4. Overview of the operation of (a) a $\mathcal {P}$-field source, and (b) a $\mathcal {P}$-field detector.
3.2 A $\mathcal {P}$-field detector
A $\mathcal {P}$-field detector measures the amplitude of the modulation envelope of modulated optical irradiance. The $\mathcal {P}$-field detection process is shown in Fig. 4(b). The $\mathcal {P}$-field detector provides a single scalar $\mathcal {P}$-field amplitude value $\mathcal {P}_{0,\Omega }$ at each detection location analogously to a conventional E-field detector which records a single scalar measurement of the time-average optical irradiance. If the $\mathcal {P}$-field at location $(x,y)$ is given by
4. Experiments
4.1 Experiment 1: $\mathcal {P}$-field double slit interference experiment – measuring a $\mathcal {P}$-field fringe pattern
Our first experiment is analogous to a classical double slit experiment but in the realm of $\mathcal {P}$-fields. The two slits in a classical double slit experiment are replaced by two identical optical diffusers $D_1$ and $D_2$ separated by a distance $D_{\mathrm S}$, as is shown in Fig. 5(a).
Fig. 5. Shown is (a) the schematic and (b) the actual experimental setup for measuring a $\mathcal {P}$-field fringe through a diffuse aperture.
The detection plane $\Sigma$ is located at a distance $z = Z$ from the plane $\mathcal {A}$ containing the two diffusers. Each diffuser randomizes the optical phase of incident photons while having a minimal effect on the phase of $\left [\mathcal {P}_{0,\Omega }\right ]_1$ and $\left [\mathcal {P}_{0,\Omega }\right ]_2$; the $\mathcal {P}$-field contributions from $D_1$ and $D_2$, respectively. These $\mathcal {P}$-field contributions interfere like waves. Hence, this setup enacts a $\mathcal {P}$-field double slit experiment where a $\mathcal {P}$-field detector measures the sum of $\left [\mathcal {P}_{0,\Omega }\right ]_1$ and $\left [\mathcal {P}_{0,\Omega }\right ]_2$ analogous to the interference of two E-field contributions in a classical double slit experiment.
A 50:50 Beam Splitter (BS) splits the incident collimated Gaussian beam from a $\mathcal {P}$-field source into two identical collimated beams, each with exactly half the original beam power. We refer to these two beams as ’Beam 1’ and ’Beam 2’ and these beams remain collimated until reaching diffusers $D_1$ and $D_2$. Beam 1 and Beam 2 propagate through path lengths $L_1$ and $L_2$ before respective incidence at $D_1$ and $D_2$. The experiment was set up such that $L_1 = L_2$. The expected magnitude $[\mathcal {P}_{0,\Omega }(x)]_{\mathrm {Sum}}$ of the sum of the two $\mathcal {P}$-field contributions – depending on the detector location $x \in \Sigma$ (with $x=0$ denoting the central equidistant location from $D_1$ and $D_2$) – is given by
For actual measurements, we translate the $\mathcal {P}$-field detector along the $x$-direction and measure the normalized value $[\mathcal {P}_{\mathrm {Norm}}(x)]_{\mathrm M}$ of $\left |[\mathcal {P}_{0,\Omega }(x)]_{\mathrm {Sum}}\right |$ (where the subscript ’M’ denotes measured values).
We implemented the setup shown in Fig. 5(a). The actual experimental setup is presented in Fig. 5(b). Two identical diffusers from the Newport 20DKIT-C3 light shaping diffusers kit were used as $D_1$ and $D_2$. The distances $Z$ and $D_{\mathrm S}$ were set to 50 cm and 36 cm respectively. The diffusers provide a Gaussian-like illumination at the detection plane which were separately measured to have approximate effective $1/e^2$ beam radii of $w_{01} = {35.6}{\textrm {cm}}$ and $w_{02} = {35.7}\,{\textrm{cm}}$ at $\Sigma$. The obliquity factor is accounted for by measuring the $1/e^2$ waists of the Gaussian-like contributions along the x-direction and the subsequent normalization of the measured $\mathcal {P}$-field sum. Therefore the obliquity factor is embedded within our measurements and is fully accounted for.
Moreover, a correction factor has to be applied to data recorded at each measurement location $x \in \Sigma$ to account for a non-uniform illumination from the diffusers at $\Sigma$. We obtain a $\mathcal {P}$-field fringe pattern after applying this correction factor to measured data at each measurement location $x\in \Sigma$. This location-dependent correction factor $C(x)$, computed at every scan location, is given by
Fig. 6. Plots of (a) theoretically expected $[\mathcal {P}_{\mathrm {Norm}}(x)]_{\mathrm T}$ values and corresponding experimentally measured $\mathcal {P}$-field normalized sum $[\mathcal {P}_{\mathrm {Norm}}(x)]_{\mathrm M}$ values for different $\mathcal {P}$-field detector positions $x$ (b) Theoretical and experimental $\mathcal {P}$-field sums $\left [\mathcal {P}_{\mathrm {Norm}}(x)\right ]_{\mathrm {M-Uniform}}$ corrected for hypothetical uniform optical irradiance contributions from $D_1$ and $D_2$ at $\Sigma$.
It has to be noted that laser speckle is unavoidably present in this experiment as well as the next experiment. The effect of laser speckle in our measurements was not significant enough (as we did not register) to warrant the use of corrective measures. This is owing to the active $\mathcal {P}$-field detector area as well as inherent vibrations in the measurement setup which contribute in speckle reduction in a manner similar to how speckle is reduced in systems by vibrating polymers such as [22,23]. Had we encountered significant speckle in our measurements, we would also have used laser speckle reducers similar to [22,23].
4.2 Experiment 2: $\mathcal {P}$-field focusing with a $\mathcal {P}$-field focusing mirror – $\mathcal {P}$-field focus despite no optical focus
In this experiment, we demonstrate that $\mathcal {P}$-fields can be focused with the aid of a $\mathcal {P}$-field curved mirror. The operation of a $\mathcal {P}$-field lens is analogous to a conventional lens that achieves constructive interference of E-fields from a point object at the location of its focal point and destructive interference elsewhere. The $\mathcal {P}$-field focusing mirror can be realized using a diffuse surface with a curvature radius $R$ to obtain a desired focal length and a diameter $W$ which has to be larger than $\lambda _{\mathrm P}$.
We demonstrate that the $\mathcal {P}$-field focusing mirror focuses $\mathcal {P}$-fields without obtaining an optical focus and that such $\mathcal {P}$-field imaging follows the well-known lens imaging equation
where $D_{\mathrm {Obj}}$ and $D_{\mathrm {Img}}$ are the distances of the point light source and the image plane from the center of the $\mathcal {P}$-field focusing mirror respectively.Figure 7(a) illustrates the schematic of $\mathcal {P}$-field focusing using a curved diffuser, while a picture of the experiment is shown in Fig. 7(b). The $\mathcal {P}$-field source produces a diverging beam with a divergence angle $\alpha$ set to illuminate the curved diffuser completely.
Fig. 7. Shown here is (a) the proposed setup to obtain and measure a $\mathcal {P}$-field focus spot despite no optical (E-field) focus, and (b) its experimental implementation in the laboratory.
The magnitude of the $\mathcal {P}$-field sum $\left |[\mathcal {P}_{0,\Omega }(x,y)]_{\mathrm {Sum}}\right |$ is recorded by the $\mathcal {P}$-field detector at each image plane ($\Sigma$) location $(x,y)$ which can be expressed as a sum of all contributions from the virtual $\mathcal {P}$-field source point $S$. It can be expressed from Eq. (3) in terms of the divergence half-angle $\alpha$ of light contributions emanating from $S$ as
We perform simulations to demonstrate this $\mathcal {P}$-field focusing behavior with a $\mathcal {P}$-field focusing mirror where we summed all $\mathcal {P}$-field contributions from a point light source and computed the magnitude of this sum at every location in the image plane (using (Eq. (10)). The simulated $\mathcal {P}$-field spot in the image plane is plotted in Fig. 8(a).
Fig. 8. Images of (a) theoretically expected $\mathcal {P}$-field focal spot at the image plane, (b) experimentally measured $\mathcal {P}$-field focal spot at the image plane, and (c) experimentally measured optical irradiance in the image plane using a slow photo-detector
We use a $\mathcal {P}$-field focusing mirror with radius of curvature value of $R = {50}\,{\textrm{cm}}$ (i.e., a focal length of $f_{\mathrm D} = {25}\,{\textrm{cm}}$) and a diameter of $W = {82}\,{\textrm{cm}}$. Amplitude-modulated coherent light diverges rapidly from the $\mathcal {P}$-field source. To ensure complete diffuser illumination with a minimal number of photons outside the curved region of the diffuser, we use a spherical concave lens $L_{\mathrm {Div}}$ of focal length $f_{\mathrm {Div}}$ to widen the beam to illuminate entire $\mathcal {P}$-field mirror. To achieve this, a concave lens $L_{\mathrm {Div}}$ with a focal length $f_{\mathrm {Div}} = {-30}\,{\textrm{mm}}$ was placed at a distance of $D_{\mathrm {Lens}} = {5}\,{\textrm{cm}}$ from the fiber tip. The $\mathcal {P}$-field mirror was placed at a distance $D_{\mathrm {Obj}}$ from the virtual source point $S$ behind $L_{\mathrm {Div}}$ while $D_{\mathrm {Img}}$ was determined from Eq. (9) and known values of $D_{\mathrm {Obj}}$ and $R$. The distances $D_{\mathrm {Obj}}$ and $D_{\mathrm {Img}}$ were set to ${69.3}{\textrm {cm}}$ and ${43}\,{\textrm{cm}}$ respectively whereas distances $D_S$ and $D_I$ were set to ${40}\,{\textrm{cm}}$ and ${22.5}\,{\textrm{cm}}$ respectively (refer to Fig. 7(a)). A photograph of our actual experimental setup is provided in Fig. 7(b).
To measure $\left |[\mathcal {P}_{0,\Omega }(x,y)]_{\mathrm {Sum}}\right |$ at each scan location $(x,y)$, a $\mathcal {P}$-field detector scan was performed as a step-wise horizontal and vertical scan with a step size of 1 cm and a grid of 39$\times$39 scanning points. We plot the measured $\left |[\mathcal {P}_{0,\Omega }(x,y)]_{\mathrm {Sum}}\right |$ values for all scan locations in Fig. 8(b). We observe a $\mathcal {P}$-field focal spot with a peak in the center and a radially-symmetric amplitude drop-off. The $\mathcal {P}$-field focal spot is only slightly wider than the simulation results but this can be due to slight out-of-focus location of the $\mathcal {P}$-field image plane $\Sigma$ or an optical irradiance amplitude drop-off in the actual experiments.
The average irradiance was also measured simultaneously to the $\mathcal {P}$-field measurements at each scan location with a slow photo-detector. This average irradiance, normalized to the highest measured value, is plotted in Fig. 8(c). We observe that the $\mathcal {P}$-field focus observed with a $\mathcal {P}$-field detector is independent of the average irradiance distribution which drops off by a $1/|r|^2$ factor and no E-field focus.
4.3 Experiment 3: Imaging through Fresnel lenses – sharp optical focus with a variable $\mathcal {P}$-field focus
This experiment demonstrates the formation of a focused optical (E-field) image spot of a $\mathcal {P}$-field source while the $\mathcal {P}$-field at this optical focus is cancelled due to destructive $\mathcal {P}$-field interference to show that we can obtain E-field focus without $\mathcal {P}$-field focus.
Our experimental setup is shown in Fig. 9(a) where we implement an optical system (with four Fresnel lenses) which creates a diffraction-limited point-like image $S'$ in the image plane $\Sigma$ of a $\mathcal {P}$-field source $S$. While $\mathcal {P}$-field cancellation can be achieved with the use of a single-lens imaging system - for the shortest $\mathcal {P}$-field wavelength of $\lambda _{\mathrm P} = {30}\,{\textrm{cm}}$ attainable in our laboratory - it would have required a prohibitively large lens diameter to demonstrate maximum $\mathcal {P}$-field cancellation.
Fig. 9. Shown here is (a) the setup to demonstrate E-field focus with varying $\mathcal {P}$-field focus, and (b) an experimental setup in the laboratory.
The four identical Fresnel lenses in the setup are labeled as FL$_1$, FL$_2$, FL$_3$ and FL$_4$, each with a focal length of $f_{\mathrm {Fres}}$ and a radius $R_{\mathrm {Fres}}$. The necessity for using Fresnel lenses instead of four conventional spherical lenses and the role of the iris in achieving this are discussed in detail in Appendix B. Moreover, conditions on the Fresnel lens focal length $f_{\mathrm {Fres}}$ and radius $R_{\mathrm {Fres}}$ to obtain maximum possible $\mathcal {P}$-field interference are also discussed in Appendix B. Two spherical lenses SL$_1$ of focal length $f_1$ and SL$_2$ of focal length $f_2$ and a tunable aperture iris are used to adjust the divergence angle $\alpha$ of our $\mathcal {P}$-field source and re-focus it at location $S$.
From $S$, the beam passes through the series of four Fresnel lenses and forms an optical image spot $S'$ at a distance $f_{\mathrm {Fres}}$ from FL$_4$. The $\mathcal {P}$-field detector is positional at $S'$. The separation distance between FL$_2$ and FL$_3$ is set to $2f_{\mathrm {Fres}}$. For a Fresnel lens illumination radius of $R_{\mathrm {Illum}}$, the magnitude of the sum $\left |[\mathcal {P}_{0,\Omega }(R_{\mathrm {Illum}})]_{\mathrm {Sum}}\right |$ of all $\mathcal {P}$-field contributions $\mathcal {P}(r')$, detected by the $\mathcal {P}$-field detector at $S'$, can be simply expressed as
The $\mathcal {P}$-field interference sum $\left |[\mathcal {P}_{0,\Omega }(R_{\mathrm {Illum}})]_{\mathrm {Sum}}\right |$ was measured for various $R_{\mathrm {Illum}}$ settings. Satisfying the condition in (S7) for $\lambda _P = {30}\,{\textrm{cm}}$ required using Fresnel lenses with small f-numbers. We used four Fresnel lenses CP1300-1100 from FresnelFactory.com, each with $R_{\mathrm {Fres}} = {550}\,{\textrm{mm}}$ and $f_{\mathrm {Fres}} = {1300}\,{\textrm{mm}}$.
We used $f_1 = {50}\,{\textrm{mm}}$ for 1-inch SL$_1$. Instead of using a single lens SL$_2$, we replaced it with a lens pair comprising of two 1-inch spherical lenses with a focal lengths of 30 mm and 50 mm separated by a distance of 7 mm. The resulting Gaussian beam irradiance distribution at FL$_1$ is measured separately with a $1/e^2$ waist radius of $w_0 \approx {84}\,{\textrm{cm}}$ with iris completely open. The laboratory experimental setup is shown in Fig. 9(b).
We plot the theoretically normalized expected values of $\left |[\mathcal {P}_{0,\Omega }(R_{\mathrm {Illum}})]_{\mathrm {Sum}}\right |$. Refer to Appendix B for details on the theoretical estimate of $\left |[\mathcal {P}_{0,\Omega }(R_{\mathrm {Illum}})]_{\mathrm {Sum}}\right |$. Also in Fig. 10(a), we plot the measured values of $\left |[\mathcal {P}_{0,\Omega }(R_{\mathrm {Illum}})]_{\mathrm {Sum}}\right |$ normalized to the highest recorded value in the measurement dataset. Each data point in Fig. 10(a) was recorded for a different $R_{\mathrm {Illum}}$ value. At each $R_{\mathrm {Iris}}$ setting, the null-to-null $R_{\mathrm {Illum}}$ on FL$_1$ was measured with a long measuring scale while $\left |[\mathcal {P}_{0,\Omega }(R_{\mathrm {Illum}})]_{\mathrm {Sum}}\right |$ was recorded with the $\mathcal {P}$-field detector.
Fig. 10. Plot showing (a) comparison between expected normalized theoretical $\mathcal {P}$-field interference $[\mathcal {P}_{0,\Omega }(R_{\mathrm {Illum}})]_{\mathrm {Sum}}$ for different FL$_1$ illumination radius $R_{\mathrm {Illum}}$ values and experimentally measured data, and (b) a comparison of $\left [P_{\mathrm {DC-Norm}}(R_{\mathrm {Illum}})\right ]_{\mathrm {Theoretical}}$ to experimentally measured values of $P_{\mathrm {DC-Norm}}(R_{\mathrm {Illum}})$.
We observe an excellent agreement between the theoretical curve and the experimental data in Fig. 10(a) – despite Fresnel lenses imparting significant aberrations to propagating wavefronts. The maximum and minimum experimentally measured $\left |[\mathcal {P}_{0,\Omega }(R_{\mathrm {Illum}})]_{\mathrm {Sum}}\right |$ values are observed at approximately the same $R_{\mathrm {Illum}}$ values as predicted theoretically. Also, the overall trend of the measured data follows closely the theoretical predictions.
In order to demonstrate that the expected and measured trend of $\mathcal {P}$-field measurements with changing $R_{\mathrm {Illum}}$ does not simply follow the E-field trend, we also measure the average optical power $P_{\mathrm {DC}}(R_{\mathrm {Illum}})$ at $S'$ for different $R_{\mathrm {Illum}}$ settings using a very slowly integrating photo-detector. As shown in Fig. 9(a), these simultaneous measurements of $P_{\mathrm {DC}}$ were recorded by splitting the optical beam with a 50:50 power beam splitter BS immediately before $S'$ and measuring the average optical power with the slow photo-detector. These optical power measurements were also normalized to the maximum recorded value $P_{\mathrm {DC-Max}}$ (obtained when FL$_1$ was fully illuminated) of $P_{\mathrm {DC}}(R_{\mathrm {Illum}})$ to yield $P_{\mathrm {DC-Norm}}(R_{\mathrm {Illum}})$.
Computed $P_{\mathrm {DC-Norm}}(R_{\mathrm {Illum}})$ values are plotted in Fig. 10(b) which also includes a plot of the theoretically expected normalized average optical power values $\left [P_{\mathrm {DC-Norm}}(R_{\mathrm {Illum}})\right ]_{\mathrm {Theoretical}}$ for different values of $R_{\mathrm {Illum}}$. $\left [P_{\mathrm {DC-Norm}}(R_{\mathrm {Illum}})\right ]_{\mathrm {Theoretical}}$ is computed from the Gaussian FL$_1$ irradiance illumination function as
5. Applications of $\mathcal {P}$-fields in imaging
5.1 The use of $\mathcal {P}$-fields as a virtual wave for NLoS imaging
In Fig. 11(a), we depict an LoS imaging system that comprises a light source and a camera. The most fundamental operation of this imaging system is to image a point-like object. The source emits a monochromatic wave that travels to the object, passing through a lens, whose goal is to refocus the electromagnetic wave onto the object (we assume that the object is located at the focal spot of the lens). After light interacts with the object, it travels to the camera, once again passing through another lens which refocuses the diffuse light onto the receiver.
Fig. 11. Comparison of (a) Conventional E-field-based LoS imaging and (b) $\mathcal {P}$-field-based NLoS imaging.
The $\mathcal {P}$-field virtual wave approach can be used to describe any time-based lens-less imaging system, such as the one presented in [25] where the objective is to capture an image of a hidden scene via a diffuse reflection at a relay wall. NLoS hardware can project $\mathcal {P}$-field waves onto the relay wall and detect light intensity envelope contributions reflected from the hidden object and reflected again at the relay wall. The wall thus becomes the aperture of a holographic $\mathcal {P}$-field system where the detection system can be modeled as a virtual LoS $\mathcal {P}$-field camera. The method has recently been applied to the reconstruction of hidden scenes and initial results presented by Liu et.al. [15] are extremely promising for $\mathcal {P}$-field based NLoS imaging.
5.2 NLoS imaging applications through direct measurements of $\mathcal {P}$-fields
A common drawback of NLoS approaches using an amplitude modulated CW source is the large computational overhead required for spatial reconstruction from the acquired measurements [26,27]. This is due to the fact that these techniques deploy complex mathematical models and try to reconstruct images through computationally-intense inverse methods.
By contrast, we present physically measurable $\mathcal {P}$-fields to design exclusively $\mathcal {P}$-field imaging systems that use $\mathcal {P}$-field components (such as $\mathcal {P}$-field lenses, $\mathcal {P}$-field mirrors etc.) which can capture images with minimal or no digital computation.
In this work we demonstrate the focusing of a $\mathcal {P}$-field wave using a curved diffuser. In practical NLoS applications, this method of $\mathcal {P}$-field imaging would involve a large optical loss and possibly require a large optical setup (the dimensions of which can be reduced if higher $\mathcal {P}$-field frequency values are used). Contrary to NLoS imaging using virtual $\mathcal {P}$-fields [15] where back-projection of measurement data takes care of providing correct time shifts (effectively creating a virtual lens at the plane of the relay wall), a $\mathcal {P}$-field-based NLoS imaging scheme with direct modulation of optical carrier would require a delay-and-addition operation which is performed by a curved diffuser. This delay-and-addition operation could be more efficiently performed by an analog shifting electronic circuit in a setup where a $\mathcal {P}$-field light source can be focused on points of the relay wall and the phasor field contributions from the hidden scene that return to the wall are imaged onto an array of $\mathcal {P}$-field detectors. The detected $\mathcal {P}$-field signals can then be delayed and added electronically to create a $\mathcal {P}$-field image of the scene.
For imaging applications, the $\mathcal {P}$-field amplitude and frequency can be adjusted to operate in the traditional wireless communication cellphone bands which would allow for deploying existing wireless receiver hardware technology as a $\mathcal {P}$-field detector for low-cost $\mathcal {P}$-field imaging solutions. The resolution of the proposed imaging method shall depend on the $\mathcal {P}$-field wavelength $\lambda _P$ and increasing imaging resolution shall require reducing $\mathcal {P}$-field wavelength (i.e. increasing $\mathcal {P}$-field frequency). It may not be practical to increase $\mathcal {P}$-field frequency beyond few tens of GHz due to limitations in available hardware, which shall limit the imaging resolution to the millimeter scale.
6. Conclusion
In this paper, we provide experimental validation of the wave-like properties of the amplitude modulation envelope of an optical carrier which we refer to as phasor field (or $\mathcal {P}$-field). We show that these $\mathcal {P}$-field contributions exhibit wave-like properties of their own and the sum of $\mathcal {P}$-field contributions from a diffuse surface can be expressed in a formulation which is analogous to the Rayleigh-Sommerfeld diffraction integral which explains E-field summation and forms the basis of LoS imaging. This analogy between imaging with E-fields and $\mathcal {P}$-fields from a surface also allows us to model an NLoS imaging scenario as a virtual camera looking directly at the hidden scene through the diffuse relay wall which acts as a virtual lens.
In our experiments, we show that it is possible to achieve (1) a $\mathcal {P}$-field fringe interference pattern from a diffuse double slit aperture analogous to the Young’s double slit experiment for E-fields, (2) $\mathcal {P}$-field focusing with a $\mathcal {P}$-field lens in the absence of an E-field focus and 3) E-field (optical) focus with independently adjustable $\mathcal {P}$-field focus. Through our experiments, we also demonstrate the potential of using measurable $\mathcal {P}$-fields in low-cost real-time NLoS tracking and imaging applications.
Appendix A: Realizing a $\mathcal {P}$-field source and a detector
The details of the implementation of a $\mathcal {P}$-field source and a $\mathcal {P}$-field detector for our experiments are presented here. The same $\mathcal {P}$-field source and detector are used for all three experiments.
Implementation of a $\mathcal {P}$-field source
The objective of a $\mathcal {P}$-field source is to produce amplitude modulated light. The $\mathcal {P}$-field source consists of a traditional quasi-monochromatic light source such as laser beam produced by a laser diode amplitude-modulated by a signal of choice which we refer to as the $\mathcal {P}$-field signal. The $\mathcal {P}$-field envelope is imparted to an unmodulated optical signal (produced by the light source) by means of conventional direct modulation devices such as electro-optic or acousto-optic modulators. The light then carries this $\mathcal {P}$-field amplitude and phase information upon propagation. Figure 12(a) shows a quasi-monochromatic light wave produced by a light source such as a laser source which is directly modulated by a modulator. The peak amplitude of the irradiance envelope $\mathcal {P}_{0,\Omega }$ depends on the slope of the current versus optical power ($P_{\mathrm {Opt}}$) modulation transfer curve. The result is a modulated optical signal emanated by the $\mathcal {P}$-field source.
For the actual implementation of a $\mathcal {P}$-field for our experiments, we simply used a pigtailed laser diode mounted on a Thorlabs CLD1010LP laser diode controller with an in-built electro-optic (EO) modulator. The desired $\mathcal {P}$-field signal of 1 GHz frequency was applied directly to the EO modulator input port to obtain the modulated optical signal.
Implementation of a $\mathcal {P}$-field detector
In our experiments, a $\mathcal {P}$-field detector was used to directly detect the peak amplitude of a temporally slowly-varying envelope of optical irradiance. The concept of a $\mathcal {P}$-field detector is shown in Fig. 12(b), where a $\mathcal {P}$-field detector is depicted as a two-step process. First, the envelope of an amplitude modulated optical carrier is retrieved by an AC-coupled (avalanche) photo-detector (A)PD. An AC-coupled APD is used to provide signed modulation envelopes by removing the DC part of the modulation envelope. Also in order to avoid distorting the modulation envelope, an APD with an electrical bandwidth larger than the bandwidth of the modulating envelope has to be used.
Once the modulation envelope is retrieved by the APD, the signal is passed on to the second stage of the $\mathcal {P}$-field detector - namely the envelope peak detector. The aim of the peak detector is to simply read out, in real time, the peak value $\mathcal {P}_{0,\Omega }$ of the retrieved envelope. There are different ways a peak detector can achieve this. One simple way of measuring signal peak value is to use a simple diode circuit shown in Fig. 12(b) which first smooths the oscillating input $\mathcal {P}$-field signal amplitude and the quasi-stationary DC value is measured across a resister.
The peak detection of the retrieved modulating envelope is analogous to E-field detection where the temporal E-field variations are too fast to be detected by any photo-detector. Any photo-detector is only able to measure a time-averaged optical irradiance $\langle I(x,y) \rangle = |E(x,y)|^2$ – a scalar quantity at each location $(x,y)$ in a detection plane $\Sigma$.
For all our experiments, a $\mathcal {P}$-field detector was implemented by first retrieving the $\mathcal {P}$-field signal with a Menlo Systems APD210 photo-detector (with an electrical bandwidth $\Delta f = {1.6}{\textrm{GHz}} > \Omega /2\pi$) and then reading the $\mathcal {P}$-field peak value with an Agilent CXA N9000A RF spectrum analyzer.
Appendix B: Details and discussions on experiment 3
Deploying Fresnel lenses instead of conventional spherical lenses
In this section, we explain the use of Fresnel lenses instead of conventional spherical lenses in Experiment 3. Conventional spherical lenses image by counterbalancing the optical path lengths of different rays emanating from a given point source/object. These lenses do so by forcing rays which propagate lesser in air (in comparison to other rays) to pass through the thicker part of the lens and vice versa. Hence, at a particular location, the phases accumulated by all rays emitted by the point source are equal and interfere constructively. This location is called the image point and a collection of such points lie in a space called the image plane.
If four conventional spherical lenses were to be used in the setup of Fig. 9(a), the number of phase cycles added to each of the E-field as well as the $\mathcal {P}$-fields contributions from the point source $S$ would be equal at $S'$ regardless of whether we speak of rays propagating along the optical axis, marginal rays, or any other rays propagating through the system. All $\mathcal {P}$-field contributions – with one $\mathcal {P}$-field contribution associated with each ray – would also experience a simultaneous constructive interference at the point of optical focus $S'$ (E-field focus) owing to the optical path balancing property of spherical lenses. Therefore, for any ideal imaging system that uses conventional spherical lenses (including the classical 4-f imaging system), the location of highest $\mathcal {P}$-field constructive interference coincides with the location of the optical (E-field) image point $S'$.
Because our aim was to demonstrate optical interference independent of $\mathcal {P}$-field interference, we chose not to use conventional spherical lenses but Fresnel lenses instead. Contrary to imaging through optical path balancing, Fresnel lenses apply the correct modulo $2\pi$ phase to each E-field contribution to achieve optical focus instead of fully balancing optical paths. This residual phase balancing allows E-field contributions to constructively interfere even if each E-field contribution accumulates a different integer number $N_{\mathrm {m}}$ of complete $2\pi$ phase cycles. The total phase shift $\phi _m$ accumulated for the $m^{th}$ ray involving $N_m$ integer cycles of $2\pi$ phase accumulation and a residual phase $\phi$ is given by
while the residual phase $\phi$ for each of the rays remains uniform regardless of the ray index value $m$. Constructive E-field interference is possible in this manner due to the modulo $2\pi$ nature of time-harmonic E-fields. As a consequence of applying a correct residual phase shift to all propagating rays instead of aiming for a full-on optical path balancing, Fresnel lenses are much thinner than conventional spherical lenses.Owing to the small average thickness $\langle \mathrm {T} \rangle$ of Fresnel lenses, it is possible to set the $\mathcal {P}$-field wavelength such that $\langle \mathrm {T} \rangle \ll \lambda _{\mathrm P}$. In this case, the Fresnel lenses have a negligible effect on the phase accumulated by each $\mathcal {P}$-field contribution, and the phase accumulated by each $\mathcal {P}$-field contribution is mostly due to propagation between $S$ and $S'$. On the contrary, as $\lambda _{\mathrm E} \ll \lambda _{\mathrm P}$, E-field residual phase balancing results in an optical focus. In other words, phase accumulation for each individual $\mathcal {P}$-field contribution at $S' \in \Sigma$ is mostly due to propagation between $S$ and $S'$ with almost negligible phase contribution from the Fresnel lenses. This results in a dissimilar $\mathcal {P}$-field phase accumulation for each contribution and the degree of $\mathcal {P}$-field constructive interference is consequently independent of the complete E-field constructive interference. We exploit this fundamental property of Fresnel lenses to observe varying $\mathcal {P}$-field interference with a fixed E-field constructive interference.
Using an iris between SL$_1$ and SL$_2$ to observe full range of $\mathcal {P}$-field interference behavior
To demonstrate a comprehensive $\mathcal {P}$-field interference behavior which is independent of the omnipresent E-field focus, a mechanically tunable iris was placed between SL$_1$ and SL$_2$ – centered at the propagating collimated beam present in that location as is shown in Fig. 13. The iris is introduced to alter the illumination radius $R$ at FL$_1$. The collimated beam passes through the optical iris and the area of the circular optical illumination of each of the Fresnel lenses is controlled by changing the radius of the iris opening. The relationship between iris clear aperture radius $R_{\mathrm {Iris}}$ and Fresnel lens illumination radius $R_{\mathrm {Illum}}$ is simply given by the magnification introduced by lens pair SL$_2$ and FL$_1$ expressed in terms of their respective focal lengths $f_2$ and $f_{\mathrm {Fres}}$ as
The iris is chosen such that for less than or equal to the fully open setting, it allows the collimated beam between SL$_1$ and SL$_2$ to propagate through unchopped resulting in a subsequent unchopped identical illumination of radius $R_{\mathrm {Fres}}$ at each Fresnel lens. Furthermore, from Eq. (14) for $R_{\mathrm {Illum}} = R_{\mathrm {Fres}}$, the focal length $f_2$ of SL$_2$ is can be calculated for full illumination of all of the Fresnel lenses with the iris fully open. With the iris present in the setup, the sum of $\mathcal {P}$-field contributions $[\mathcal {P}_{0,\Omega }(R)]_{\mathrm {Sum}}$ that is measured by a $\mathcal {P}$-field detector is obviously a function of $R_{\mathrm {Illum}}$ corresponding to an iris opening radius of $R_{\mathrm {Iris}}$. $[\mathcal {P}_{0,\Omega }(R_{\mathrm {Illum}})]_{\mathrm {Sum}}$ is estimated in the following section and expressed as Eq. (17) described in the next section.Theoretical estimate of the $\mathcal {P}$-field sum in experiment 3
In order to obtain the theoretical plot in Fig. 10, we have to determine the cumulative $\mathcal {P}$-field amplitude of each contribution with a unique phase and sum all these $\mathcal {P}$-field contributions over the illuminated FL$_1$ area to determine the $\mathcal {P}$-field interference sum $\left |[\mathcal {P}_{0,\Omega }(R_{\mathrm {Illum}})]_{\mathrm {Sum}}\right |$. A single $\mathcal {P}$-field contribution with a specific peak amplitude $\mathcal {P}_{0,\Omega }(r')$ and phase $\phi (r')$ is produced by modulated photons located in a thin ring of radius $r'$ centered about the center of FL$_1$. From Fig. 14, the magnitude $\mathcal {P}_{0,\Omega }(r')$ of each $\mathcal {P}$-field contribution is determined by the number of photons present within the radial ring of radius $\approx r'$ and thickness $\Delta r'$ and its phase $\phi (r')$ relative to the $\mathcal {P}$-field component that propagates along the shortest possible physical distance (i.e., along the optical axis) is given by
where $\Delta D$ is the path length difference to the ray propagating along the optical axis. Substituting for $\Delta D$ in Eq. (15), we obtainFig. 14. Theoretically determining $\left |[\mathcal {P}_{0,\Omega }(R_{\mathrm {Illum}})]_{\mathrm {Sum}}\right |$ for Experiment 3.
In Eqs. (15) and (16), a coefficient value of ’4’ denotes the number of Fresnel lenses deployed in the imaging system. Moreover, for an identical and radially symmetric partial illumination of all Fresnel lenses with a radius of illumination $R_{\mathrm {Illum}}$, the resulting $\mathcal {P}$-field sum $[\mathcal {P}_{0,\Omega }(R_{\mathrm {Illum}})]_{\mathrm {Sum}}$ can be calculated from (10) and expressed as
Relationship between the Fresnel lens focal length and diameter for measuring full range of $\mathcal {P}$-field interference behavior
To obtain this comprehensive full range of constructive and destructive $\mathcal {P}$-field interference at $S'$, four identical Fresnel lenses each with a radius $R_{\mathrm {Fres}}$ were chosen such that for a completely open iris (i.e. complete Fresnel lens illumination), the maximum path length difference $(\Delta D)_{\mathrm {Max}}$ between $\mathcal {P}$-field contributions with the shortest and longest propagation paths $D_{\mathrm {Max}}$ and $D_{\mathrm {Max}}$ between $S$ and $S'$ was greater than or equal to the $\mathcal {P}$-field wavelength $\lambda _{\mathrm P}$ i.e.,
This ensures that each $\mathcal {P}$-field amplitude contribution $\mathcal {P}_{0,\Omega }(r')$ with a respective accumulated propagation phase $\phi (r')$ between $S$ and $S'$ in the range $0 \leq \phi (r') \leq \pi$ has a conjugate $\mathcal {P}$-field contribution with an accumulated phase in the range $\pi \leq \phi (r') \leq 2\pi$ to destructively interfere with it. This choice of $R_{\mathrm {Fres}}$ ensures that we are able to observe the largest possible constructive and destructive interference of $\mathcal {P}$-field contributions as well as all intermediate interference states for different iris settings. In other words, from Eq. (16),Funding
Defense Advanced Research Projects Agency (HR0011-16-C-0025); National Aeronautics and Space Administration (NNX15AQ29G); Office of Naval Research (N00014-15-1-2652).
Acknowledgments
The authors would like to acknowledge Mohit Gupta and his “Wision Lab” at the University of Wisconsin – Madison for lending us some laboratory equipment which was used in our experimental setups.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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