2.1.

Temporal Response of Integrating Spheres

We assume a cavity as a hollow structure in which a collimated laser pulse can enter through an opening. This light is reflected multiple times ($n$-bounce path) with an efficiency depending on the surface reflection coefficient and eventually leaves the cavity through an opening in the direction of the sensing system. Therefore, a cavity acts similar to an integrating sphere.

Due to the multiple reflection processes, the exiting light intensity distribution is spatially and angular homogenized and temporally stretched. As given in Eq. (1), the temporal response $I_{\text{exit}}$ of an integrating sphere equals a convolution of the input light pulse $I_{\text{input}}$ and its temporal transfer function $T$. In Eq. (2), $T$ is defined as an exponential decay with a time constant $\tau$ given by the diameter $Ø$ of the integrating sphere, the speed of light $c$, and the average surface reflectivity $\overline{\rho }\in [\mathrm{0,1}]$:³¹

Due to the multireflections, a short laser pulse is stretched in a relatively long pulse. This stretching effect is increased with larger diameter $Ø$ (longer traveling paths) and higher reflectivity $\overline{\rho }$ (more bounces). We show hereafter that this empiric equation is valid only for pulse length equivalent to or greater than the sphere diameter. At shorter pulse lengths, the signals from different multiple reflection paths mix less and can be distinguished by high-resolution measurements as additional modulations of the temporal signal amplitude.

2.2.

Simulation of the Temporal Response by Transient Rendering

The temporal response of cavities can be simulated, e.g., by Monte Carlo simulations³² or transient rendering,³³ taking into account the light paths and the interaction of light with different surface facets. Light is partially reflected, expressed by the surface reflectivity $\rho$, and can bounce off the cavity multiple times before it reaches the receiver area, i.e., the opening port of the cavity. Multibounce light arrives temporally later at the receiver due to its longer traveling path. Both effects, multibounce and traveling path lengths, lead to a certain temporal stretching of the impulse excitation.

In our first simulation, we use a transient rendering software³³ that is capable of calculating the temporal response of a scene due to certain illumination and sensing conditions. The scene has to be described as a set of mesh structures that can have any geometrical shape. Each mesh is defined by its vertex points (defining the area), surface orientation, and optical properties. The rendering software calculates the photon propagation within the scene even with multiple reflections. At each surface, the reflection is calculated taking into account a selected surface model (e.g., Lambertian surface). Further, for each photon, the path length is calculated, and the transient count rate and the number of bounces are recorded.

In Fig. 2, we show some results for three different cavities: (a) sphere, (b) hemisphere, and (c) model of a real cave.²⁹ In each scenario, S indicates the spot of laser irradiation, and P is the sensing area where radiation is reflected toward the sensor. We assume illumination with a Gaussian laser pulse. In the diagrams, we show the temporal return of photons with different numbers of bounces $n$ (colored lines) and the total response (dashed) of the structures. The dotted line shows the total response when the structure is illuminated by a longer laser pulse.

Fig. 2

Numerical simulation of temporal response of (a) a sphere, (b) a hemisphere, and (c) a cave model.²⁹

In our simulation, we observe photons with $n$-bounces, $n\in [\mathrm{2,7}]$. A single bounce ($n=1$) return is not visible as we assume a bistatic configuration and the impact point is outside the sensor field of view. Further, we observe an oscillation at the onset of the returning signal due to the return of photons with a low number of bounces $n$ ($n\in [\mathrm{2,3},4]$). At larger times, the signatures start to mix and form an exponential decay as expected in the empirical model; see Eq. (1).

For a spherical cavity [Fig. 2(a)], we distinguish between photons with $n=2$, 3, and 4 bounces. For the hemisphere [Fig. 2(b)], we observe a similar behavior except that we do not observe a signature for $n=2$ due to the ideal flat surface neglecting quasi-evanescent radiation transfer. In contrast, in later experiments, we do observe this type of signature in most cases. Finally, for the cave model, we observe a behavior close to the hemisphere model except the fact that we observe an additional 2-bounce signature due to the rough surface structure.

2.3.

Simulation of the Radiant Transfer by Random Sampling

In the second simulation, we adopted a Monte Carlo approach³² to evaluate the radiant transfer between two surface elements. In our simulation, we consider only spherical and hemispherical cavities due to their known and simple geometries to estimate the signal peak position for different multibounce paths. We assume a cavitiy in the shape of a unit sphere with radius $r=1$ and the center in the origin of the coordinate system. Then, we randomly select two points on the cavity surface using a random sampling process ($s\stackrel{S}{\leftarrow }[-\mathrm{1,1}]$) to determine for each sampling point a set of azimuth and elevation angles with $\varphi =2\pi s$ and $\theta =\arccos (s)$.

Further, in the hemisphere model, we distinguish between the hemisphere dome and the base. In both cases, the azimuth angle is determined as aforementioned, but we limit the sampling of second parameter to the upper half of the sphere. For the hemisphere base, we set the heigth $z=0$ and randomly sample the radius weighted by $r=\sqrt{s}$.

We randomly sample two points on the cavity surface and evaluate the path length (i.e., analogous to the response time) and the probability of the exchange of optical energy due to the radiant transfer.³⁴ Here, we evaluate the orientation of the emitting and receiving surface elements to calculate their effective size and estimate the amount of radiation transferred. Our findings are summarized in Fig. 3.

Fig. 3

Analysis models consider different multibounce light paths in (a) spherical and (b) hemispherical cavities. We calculate the probability of radiant transfer (c) by random sampling of two points and obtain the probability function (d) of the photon path length to estimate the time response of the cavity. We distinguish paths with involvement of different surface patches: sphere–sphere (blue), hemisphere–hemisphere (orange), and base–hemisphere (green) paths.

As depicted in Figs. 3(a) and 3(b), light enters the structure through an opening and is reflected multiple times before it exits toward the sensor. For each $n$-bounce path, the returning peak position is denoted by $t_n$ [with $n$ indicating the number of internal reflections (bounces)]. Eventually, we observe light that is reflected at the opening rim to return a signal at $t_0$ without reflections inside the cavity.

Before the photon reaches sensing area P, we identify some significant photon paths. In the spherical model, the radiant energy is transferred between two points on the spherical hull (sphere–sphere), whereas in the hemispherical model, the energy is transferred between two points on the hemispheric dome (hemisphere–hemisphere) or between the dome and the base plate (hemisphere–base). Initially, we neglect radiant exchange within the base plate (base–base) because no energy can be transferred within an ideal plane $({\lim }_{{\theta }_i\to \pi /2}\cos ({\theta }_i)\to 0)$.

Our random sampling approach evaluates the flight path (i.e., the $L_2$-norm distance) and radiation transfer³⁴ (i.e., the amount of transferred energy by randomly selecting a pair of two points or surface elements, $\delta A_i$). Figure 3(d) shows the probability of a photon traveling a certain distance within the cavity hull. For each range, the radiant transfer weighs the probability of the exchange process, as illustrated in Fig. 3(c). The radiant transfer takes into account the face orientation (${\theta }_i$) and, thus, the effective size of the surface elements.

For the sphere–sphere exchange (blue), it can be clearly stated that the most probable distance is the sphere diameter, $D_{\mathrm{sph}-\mathrm{sph}}={Ø}_{\mathrm{sph}}$. In the hemisphere case, we obtain a most probable distances of $D_{\text{hemi}-\text{hemi}}\approx 1.7r$ for the hemisphere–hemisphere exchange (orange) and $D_{\text{hemi}-\text{base}}\approx 1r$ for the exchange with the base plate (green), with $r\equiv h_{\text{hemi}}$ being the hemisphere height.

2.4.

Path Length Model for Spherical and Hemispherical Cavities

Let us evaluate the path length in the cavity models. In the sphere model, as illustrated in Fig. 3(a), a small portion of light is scattered or diffracted while propagating toward the cavity. Therefore, we expect to observe a first signature at $t_0$ due to a reflection of light at opening O. Within the sphere, first, the light passes the cavity by distance $D_{\mathrm{OS}}$ to illuminate point S. From this point, the light can take various paths to reach P, the sensing area, and leaves the cavity opening toward the detector.

Due to the aforementioned analysis, most probable paths form the oscillations. A 2-bounce path is identified as the direct interaction of S and P, distance $D_{\mathrm{SP}}$ to form a signature at $t_2$. In a 3-bounce path, the light most likely interacts with face ${\mathrm{S}}^{\prime }$ opposing S, $D_{{\mathrm{SS}}^{\prime }}={Ø}_{\mathrm{sph}}$, and reaches P via $D_{{\mathrm{S}}^{\prime }\mathrm{P}}$. Then, (4-bounce path) the light passes two times the diameter and so on. Each path with $n$ bounces lead to a most probable time of arrival $t_n$ as summarized in Eqs. (3)–(5).

Path length model for spherical cavities:

Although we do not know the exact values of $D_{\mathrm{OS}}$, $D_{\mathrm{SP}}$, and $D_{{\mathrm{S}}^{\prime }\mathrm{P}}$, we determine the sphere diameter ${Ø}_{\mathrm{sph}}$ from the time of arrival difference between $t_2$ and $t_4$; see Eq. (6).

Further, we developed a similar model for hemisphere cavities, as illustrated in Fig. 3(b). Again, we name the distances from the opening to the illumination point as $D_{\mathrm{OS}}$ and the distance to the sensing area as $D_{\mathrm{SP}}$. Similar to above, we distinguish the radiant transfer between the base and hemisphere dome ($D_{\text{hemi}-\text{base}}\approx h_{\text{hemi}}$) and interhemisphere dome ($D_{\text{hemi}-\text{hemi}}\approx 1.7h_{\text{hemi}}$).

The hemisphere model is summarized in Eqs. (7)–(10). In contrast to the sphere model, we find two 4-bounce paths, $t_4$ and $t_{4^{\prime }}$, defining two different path lengths. We assume the second to be longer so that we distinguish both paths.

Similar to the sphere model, we determine the hemisphere height $h_{\text{hemi}}$ from the time of arrival difference between $t_2$ and $t_4$ [Eq. (11)].

Path length model for hemispherical cavities:

Both models, sphere [Eq. (6)] and hemisphere [Eq. (11)], give the same value. Later in the analysis, we therefore discuss only the target size and use sphere diameter ${Ø}_{\mathrm{sph}}$ and hemisphere height $h_{\text{hemi}}$ equally, depending on the shape of the target. Further, an alternative model to determine the hemisphere dimensions is given in Eq. (12), using the multiphoton path that includes a radiant transfer within the hemisphere dome [see Eq. (10)].

3.1.

Experimental Setup

Experimental investigations were carried out with a time-correlated single photon-counting setup consisting of a pulsed laser source (Picoquant LDH-P-C-640B, 80 ps, 640 nm, 2.5 MHz) and a photon-counting camera with a $32\times 32$ single photon-counting avalanche diode (SPAD) array (Photon Force PF32), as shown in Fig. 4. The camera has a sampling resolution of 56 ps per time bin with a 10 bit sampling depth (1024 time bins). Thus, the camera can sample a time window of 57.3 ns. Overall, the setup has a temporal resolution of about 150 ps (i.e., a path length of 4.5 cm) due to the instrument response function (IRF). Further, ambient light is filtered by a single bandpass filter (SBF) with a spectral width of about 20 nm.

Fig. 4

Our setup consists of a SPAD camera (C) and a collimated pulsed laser source (L). In a bistatic configuration (baseline $d_{\mathrm{LC}}$), the laser illuminates a point inside the cavity at a distance $D_{\text{range}}$. The camera captures reflected light and filters ambient light by a SBF.

The camera and laser were used in a bistatic configuration and were separated by a base line $d_{\mathrm{LC}}=30\mathrm{cm}$. The cavity investigated was placed at a distance of $D_{\text{range}}=200\mathrm{cm}$. The collimated laser illuminates a point within the cavity through an opening outside the sensor’s field of view. The light propagates within the cavity, exits at the opening, and is eventually projected by a lens onto the SPAD sensor array. For every photon event, the camera records the event time and position on the array. The field of view of the camera was chosen to match the opening port of the investigated cavities, and for the analysis, the signal was averaged over the whole image. Illumination and sensing paths cross and use the same opening port.

Each sensor element records the time for every detection event. In our experiences, we use 500.000 measurement cycles to enable statistical analysis. The datasets are converted to histograms to obtain a transient signal, i.e., the count rate for every time bin within the sampling window. Thus, for every measurement, we obtain a multidimensional dataset giving the count rate for each sensor element (position $u$, $v$) and time bin ($t_j$). The data structure is illustrated in Video 1 in the Supplementary Material; see Fig. 5. The still image [Fig. 5(a)] shows a frame of the count rate at a certain instant ($u,v$-cross-section at $t_j$) and [Fig. 5(b)] the mean transient signal indicating main signal features. Within the first frames ($t_j<7\mathrm{ns}$), we observe the laser pulse propagating toward the cavity (light in flight) and the reflection at the cavity rim ($t_0$). Later, the signatures of multibounce photons ($t_2$, $t_3$, and $t_4$) are visible before the exponential decay of the signal amplitude ($t_j>16\mathrm{ns}$).

Fig. 5

Visualization of experimental data as a still image of Video 1 in the Supplementary Material. (a) The frame illustrates a slice through the histogram data and gives the count rate distribution at a give instant. (b) The diagram shows the mean transient signal and indicates significant signatures (Video 1, MP4, 1 MB [URL: https://doi.org/10.1117/1.JARS.17.024503.s1]).

3.2.

Experimental Investigation of Cavities

We studied a set of eight cavities, as depicted in Fig. 6, with different shapes (sphere, hemisphere, and polygonal), sizes ($Ø\in [15\mathrm{cm},3\mathrm{m}]$) and surface materials [spectralon, expanded polystyrene (EPS), white paper, 3D printed nylon, and fabric]. The three sphere targets have different sizes (${Ø}_{\mathrm{sph}}\in [\mathrm{15,46},50\mathrm{cm}]$) and materials (spectralon and EPS), and the EPS hemisphere has a height of $h_{\text{hemi}}=23\mathrm{cm}$. The polygonal cavities are a cube and a dodecahedron, both made of white paper.

Fig. 6

Images of examined cavities with different dimensions, materials, and shapes: (a) sphere 1, (b) sphere 2, (c) sphere 3, (d) hemisphere, (e) cube, (f) dodecahedron, (g) cave, and (h) room.

Finally, we investigate two use cases. First, a 3D printed cave model made of selective laser sintered (SLS) nylon. This cavity has an irregular surface (nonconvex roughness of about 1 cm) that can be roughly described as a hemisphere. The base resembles an oval with a depth and width of $30\mathrm{cm}\times 40\mathrm{cm}$, and the height of the hemisphere is about $15\pm 0.5\mathrm{cm}$. The printed material has a smooth, diffusely reflective surface with high reflectivity ($\overline{\rho }\approx 0.9$). In contrast, the surfaces of natural caves have a more complex texture with a lower reflectivity. Both surface properties could change the signal amplitude but should not affect the round-trip time used in our analysis.

As a second use case, we consider a room size cavity with a volume of $3\mathrm{m}\times 3\mathrm{m}\times 2\mathrm{m}$ to demonstrate the up-scaling capability of our method. The room target was built of large white paper and aluminum compound plates coated with a diffuse reflective paint. In addition, we covered the floor with a white curtain fabric. All cavity parameters are summarized in Table 1.

Table 1

Shape, material, and dimensions of the cavities investigated.

Figure 7 shows the transient signatures recorded at the different targets. In all cases, we observe oscillations due to multibounce photon paths and at least determine the position of the first peaks ($t_0$, $t_2$, $t_3$, and $t_4$) using a multipeak detection algorithm. These values were used in our path length model for spherical and hemispherical cavities [Eqs. (6) and (11)] to estimate the cavity dimensions, sphere diameter ${Ø}_{\mathrm{sph}}$, or hemisphere height $h_{\text{hemi}}$, respectively.

Fig. 7

Transient signatures recorded at different targets: (a) small spectralon sphere, (b) large spectralon sphere, (c) EPS sphere, (d) EPS hemispheres, (e) paper cube, (f) paper dodecahedron, (g) 3D printed cave model, and (h) room.