1. INTRODUCTION

When a small particle, such as an aerosol particle, is introduced into a collimated laser beam, a portion of the beam will be scattered and may also be absorbed. One effect will be a net reduction of the amount of radiant power propagating in the beam downstream from the particle, and this reduction is extinction. It is thus common to state that extinction is the combined effect of absorption and scattering, and usually, it is quantified in terms of a cross section, the extinction cross section ${C^{{\rm ext}}}$ [1,2]. Indeed, extinction is fundamental; it is the conservation of energy in classical elastic electromagnetic scattering, where ${C^{{\rm ext}}}$ is equal to the sum of the particle’s absorption and (total) scattering cross sections, ${C^{{\rm abs}}}$ and ${C^{{\rm sca}}}$, respectively. In a loose sense, ${C^{{\rm ext}}}$ may be regarded as the shadow that a particle casts on a sensor facing the oncoming beam, although the shadow may not be well defined and dark depending on the particle size [2]. A careful examination of extinction in terms of the incident and scattered electromagnetic waves, and their associated energy-flow terms, reveals important facts. Extinction is not simply a blocking or deflection of the beam as may be thought [3]. Rather, the beam attenuation is a consequence of a subtle interference process [3–7]. The wave scattered by a particle interferes with the incident wave to redistribute the energy flow otherwise attributed to the beam in the particle’s absence. When understood in this manner, extinction is seen to be linked to holography [8], which provides a convenient method to measure ${C^{{\rm ext}}}$ under certain conditions [9].

 

Fig. 1. (a) Optical arrangement used to record digital in-line holograms of single, fixed, micro-particles at wavelengths $\lambda$ every 20 nm for $\lambda \in [440, 1040]\; {\rm nm}$, and (b)–(d) examples of the resulting data. The output of a supercontinuum laser is filtered by a monochromator to provide a wavelength-tunable beam propagating along the $z$-axis in (a). The beam is expanded and its profile is partly cleaned by a spatial filter consisting of lenses (L1) and (L2) and a pinhole (PH). Then, a single particle on an anti-reflection window (${W}$) is illuminated. Unscattered and scattered portions of the beam constitute the reference and object waves, respectively, which interfere across the sensor. In (b) is shown the reference measurement ${I^{{\rm ref}}}$, the raw hologram measurement ${I^{{\rm holo}}}$ in (c), and the contrast hologram $H_{{\exp}}^{{\rm con}}$ in (d). Note the particle’s interference fringes in (c) near the red arrow and the same in (d).

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Measurements of aerosol-particle extinction are important in many applications, such as quantifying the influence of aerosols on the Earth’s radiation budget, remote sensing including lidar [10,11], and visibility in urban environments [12]. In many cases, it is important to know an aerosol’s extinction over a spectrum of wavelength, $\lambda$. For example, the radiative impact of atmospheric aerosols occurs across, and indeed beyond, the visible spectrum spanning $\lambda \in (380{,}700)\; {\rm nm}$. Most conventional methods to measure aerosol extinction involve cavity ring-down (CRD), where an aerosol sample is introduced into an optical cavity [13–16]. Once calibrated, CRD provides an absolute measurement for the extinction within the cavity [13]. For an aerosol stream, the measurement originates from an ensemble of particles and it is thus difficult to associate the measurement to individual particle properties, such as particle size and shape. Consequently, supplementary analysis or assumptions are required in such measurements [13]. Combining a CRD system with an optical or electrodynamic trap permits the measurement of single particle extinction [15,16]. With the exception of spherical liquid droplets, however, it remains difficult to also know the particle size and shape [16].

The aim of this work is to explain and demonstrate a method to measure ${C^{{\rm ext}}}$ for fixed single particles across a broad spectrum by recording and analyzing an in-line digital hologram. The concept is based on [8,9] where it is shown that ${C^{{\rm ext}}}$ can be estimated by integrating such holograms; see also Raviasio et al. [17]. The method of [18] is incorporated as well to prevent issues with [8,9] that can arise with holograms containing significant noise. Using a supercontinuum laser, the measurements below yield ${C^{{\rm ext}}}$ for $\lambda \in [440,1040] \;{\rm nm}$ as well as images of the particles reconstructed from the same holograms. As such, a unique aspect of the method is that the measured ${C^{{\rm ext}}}$ is simultaneously accompanied by knowledge of the particle size and shape. The particles tested include a glass microsphere, volcanic ash, and an iron(III) oxide particle, all of which are in the ${\sim}30\,\,\unicode{x00B5}{\rm m}$ to 50 µm size range. This range is chosen because the images are then well resolved, which permits the use of the extinction paradox to provide an independent estimate for ${C^{{\rm ext}}}$, and thus, the method’s performance can be quantified across the spectrum. While not as accurate as CRD, the holographic method is comparatively simple to realize and provides particle morphology all from a single measurement. The method may thus find use in a variety of applications where extinction measurements of small particles are needed, e.g., in aerosol science, detection, and remote sensing.

2. METHOD

To describe the process to measure spectral extinction begins with Fig. 1(a), where the optical setup used to capture in-line holograms is shown. Here, a supercontinuum laser (Leukos, SMHP-40) emits a pulsed broadband beam with a spectrum of $\lambda \in [410, 2200]\;{\rm nm}$ and an approximate pulse length of 5 ps at a repetition rate of 40 MHz, and an average power of 4 W. The broadband beam, traveling along the $z$-axis, is then filtered by a Czerny-Turner type monochromator (Digikröm, CM110) outputting a quasi-monochromatic beam of wavelength $\lambda$ tunable in the range $\lambda \in [440, 875)\; {\rm nm}$ from the grating’s first diffraction order. By installing a removable long-pass filter (LPF: Semrock, LL01-514-25), the output range is slightly extended into the IR via the grating’s second diffraction order. To provide better continuity of the beam intensity ${I_{ o}}$ as $\lambda$ is varied, the filter is installed at $\lambda = 720\;{\rm nm} $. In all, this provides a tunable output of $\lambda \in [440, 1040]\; {\rm nm}$, albeit with significantly reduced intensity over the extended portion of the spectrum. The filtered beam is then expanded by a spatial filter consisting of lenses L1 ($f = 25\;{\rm mm} $) and L2 ($f = 50\;{\rm mm} $) and a pinhole (PH: 75 µm diameter), to a diameter of approximately 1 cm. The expanded, collimated beam then illuminates an anti-reflection coated fused quartz window (W) where a single particle resides on the window’s exit-face, i.e., the face farthest from the laser. The majority of the beam continues on to illuminate a monochrome image sensor (FLIR, Grasshopper) with $4096 \times 3000\;{\rm pixels}$ of size $p \times p = 3.45\,\,\unicode{x00B5}{\rm m} \times 3.45\,\,\unicode{x00B5}{\rm m}$. Meanwhile, a small portion scatters from the particle, which is also received by the sensor. These waves interfere and the sensor records the intensity of the interference pattern, constituting a so-called raw hologram ${I^{{\rm holo}}}$ at the wavelength selected by the monochromator. Note that other optical configurations are possible to perform DIH, e.g., see [19–21].

Shown in Fig. 1(b) is an example of the sensor’s output when no particle is present on $W$. The wavelength is set to $\lambda = 440\;{\rm nm} $ here. This measurement constitutes the so-called reference image, ${I^{{\rm ref}}}$, since it simply resolves the beam profile. Then, a single 50 µm diameter glass microsphere (ThermoFisher, 9000 series) is placed on $W$ and fringes are now present, which can be seen near the red arrow in Fig. 1(c) as a collection of nested rings. This measurement provides ${I^{{\rm holo}}}$. Finally, the difference ${I^{{\rm ref}}} - {I^{{\rm holo}}}$ is calculated resulting in the contrast hologram, $H_{{\exp}}^{{\rm con}}$, which is shown in Fig. 1(d). Note that in [22], the contrast is defined differently as $H_{{\exp}}^{{\rm con}} = ({I^{{\rm ref}}} - {I^{{\rm holo}}})/\def\LDeqbreak{}{I^{{\rm ref}}}$, i.e., it is normalized by the reference measurement. When one is interested in imaging particles only, there is little practical difference between these definitions. Here, however, the former definition is required since the contrast hologram should have dimensions of intensity for use in the extinction-extraction method of [9]. By scanning the monochromator in steps of $\Delta \lambda = 20\;{\rm nm} $, while keeping all other parameters of the experiment constant, a collection of holograms is made covering the spectrum $\lambda \in [440, 1040]\;{\rm nm}$.

Notice that to obtain $H_{{\exp}}^{{\rm con}}$ at different $\lambda$, it is necessary to repeat the measurement of ${I^{{\rm ref}}}$ and ${I^{{\rm holo}}}$ each time $\lambda$ changes. In doing so, it is important that the particle’s position in the beam does not significantly change due to the nonuniformity of ${I^{{\rm ref}}}$. This is achieved by use of two identical windows ${W}$ and a quick-release magnetic mount (Thor Labs, CP44F). The mount allows an optical element (the window) to be precisely positioned in the beam, easily removed, and then later replaced in the same position. One window is particle-free while the other contains the particle. For a given $\lambda$, the particle-free window is first installed in the mount and 10 measurements of ${I^{{\rm ref}}}$ are taken with constant laser and sensor settings. These are then averaged to suppress any minor fluctuations of the beam intensity. The window is then replaced by the particle-laden window and 10 more measurements are taken, and averaged, to constitute ${I^{{\rm holo}}}$. Then, $\lambda$ is changed by $\Delta \lambda$ and the process is repeated. In this way, there is no significant change in either the position of the particle or the distance $d$ between the particle and sensor. There is, however, a significant change in the overall beam intensity as $\lambda$ varies, which is typical of supercontinuum lasers; see, e.g., Fig. 1(a) in [23]. Fortunately, such variation is unimportant here because the particle is stationary and the sensor exposure time can be varied (for a given $\lambda$) as needed to resolve the reference and hologram images.

The contrast holograms are the main data-set used to both reconstruct an image of the particle and estimate the extinction cross section. Notice how the beam profile, ${I^{{\rm ref}}}$ in Fig. 1(b), is not uniform, or even smoothly varying across the image like one would expect for a Gaussian beam. This is largely due to aberrations introduced by the monochromator that remain following the spatial filter. Similar aberrations are present in ${I^{{\rm holo}}}$, and so, the aberrations largely subtract in $H_{{\exp}}^{{\rm con}}$ and the particle’s fringes become accentuated. In other words, despite the rather poor quality beam profile, good particle images can be reconstructed since the profile nonuniformity largely cancels in the contrast hologram. However, as explained in [18], there remains a sufficient degree of nonuniformity, i.e., noise, in $H_{{\exp}}^{{\rm con}}$ that the method of [9] to estimate ${C^{{\rm ext}}}$ would fail. Indeed, such noise is typical of many hologram measurements in practice and it is seen in all of the holograms taken here across the spectrum. Thus, it is necessary to first review the process of [18] used to reduce the noise in $H_{{\exp}}^{{\rm con}}$ before discussing the cross section results.

Close inspection of Fig. 1(d) reveals that the particle’s interference fringes are not exactly at the center of the image. That is because the particle does not reside at the center of the beam profile on W—recall Fig. 1(a)—although it is close to the center. The ${C^{{\rm ext}}}$ estimation method of [9] is simplest if the fringes do reside at the center of $H_{{\exp}}^{{\rm con}}$. Also, as explained in [22], the fringes are not resolved across the entire sensor area because of the sensor’s limited dynamic range. The area of the sensor not resolving fringes, which is the majority of its total area as can be seen in Fig. 1(d), then serves no useful purpose and will only burden later computations. Thus, the first task is to center and crop $H_{{\exp}}^{{\rm con}}$ so that the fringes appear at the center of a new $H_{{\exp}}^{{\rm con}}$ and its size (in pixels) is moderately larger than what is needed to retain the resolved fringes. Figure 2(a) shows the result of this process applied to Fig. 1(d). Now the particle’s fringes are more visible and are seen to oscillate between positive and negative pixel gray levels as indicated by the scale bar. Also notice the noise background.

 

Fig. 2. Demonstration of the noise-reduction method of [18] applied to the cropped and centered contrast hologram measured in Fig. 1 for $\lambda = 440\;{\rm nm} $. (a), (b) Hologram and the reconstructed particle image ${I^{{\rm img}}}$ with scale bars revealing the pixel gray-level variations. (c) Histogram of the pixel values of (b) where small (large) gray levels denote white (black) pixels. A mask $M$, shown in (d), is formed by binarizing the image using a threshold $\eta$ to zero all pixels with values less than $\eta$. Multiplication of $M$ with the complex-valued image amplitude ${E^{{\rm img}}}$ gives the cleaned image amplitude $E_{{\rm cln}}^{{\rm img}}$ of (e) where the background noise is now removed. Here, the complex-valued $E_{{\rm cln}}^{{\rm img}}$ is plotted with colors representing phase and color-saturation representing magnitude. (f) A cleaned contrast hologram $H_{{\rm cln}}^{{\rm con}}$ is then produced by evaluating the inverse Fresnel transform $2{\rm Re}\{{\rm Fr}_{- d}^{- 1}\{\ldots \}\}$ of the cleaned image amplitude.

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The first step to reduce the noise is to reconstruct an image ${I^{{\rm img}}}$ of the particle from the centered, cropped, contrast hologram. The reconstruction process is explained in many references and several approaches may be taken. Here, the Fresnel transform ${{\rm Fr}_z}\{\ldots \}$ is used, which essentially calculates the (scalar) diffraction of a plane wave through the hologram to form the focused particle image ${I^{{\rm img}}}$ in a parallel image-plane at $z = d$ as the absolute square of the complex-valued diffracted amplitude, i.e., ${I^{{\rm img}}} = |{E^{{\rm img}}}{|^2}$. More detail, including the code to perform this reconstruction, is given in the tutorial [22]. The focused image is shown in Fig. 2(b), where the 50 µm spherical particle is clearly resolved. Although it is not obvious in Fig. 2(b), there is significant noise present in the background of the image. Figure 2(c) shows a histogram of the pixel gray levels in ${I^{{\rm img}}}$. Noting the log scale in this plot, we see that the majority of pixels exhibit small gray levels, i.e., a light gray or white appearance in Fig. 2(b), while the pixels of the particle image are found at the largest gray levels. Thus, by defining a threshold $\eta$ in Fig. 2(c), it is possible to differentiate, approximately, between the particle image pixels and those that are mostly noise. In the case when there is only a single particle on ${W}$, it is found that the exact gray-level value for $\eta$ is not critical; a value approximately at the midpoint of the gray-level range can be selected.

Next, a mask $M$ is generated by assigning all pixels in ${I^{{\rm img}}}$ below $\eta$ a value of zero and those above $\eta$ a value of one; see Fig. 2(d). Taking the element-wise multiplication of the mask with the complex-valued image amplitude, $M \odot {E^{{\rm img}}}$, has the effect of erasing the background noise, resulting in the cleaned image amplitude $E_{{\rm cln}}^{{\rm img}}$ shown in Fig. 2(e). Lastly, the reconstruction process is inverted [18], i.e., by applying $2{\rm Re}\{{{\rm Fr}_{- d}^{- 1}\{\ldots \}} \}$ to $E_{{\rm cln}}^{{\rm img}}$, which generates the new, cleaned hologram $H_{{\rm cln}}^{{\rm con}}$ shown in Fig. 2(f). Comparing Fig. 2(a) with Fig. 2(f) shows that, indeed, the noise present in $H_{{\exp}}^{{\rm con}}$ is now removed and suitable for use in the method of [9] to estimate ${C^{{\rm ext}}}$.

The discussion so far concerns a hologram measurement at $\lambda = 440\;{\rm nm} $. Consider Fig. 3 for a sense of the performance of the method across the spectrum. The first row shows the $\lambda = 440\;{\rm nm} $ results again, except a slice through $H_{{\exp}}^{{\rm con}}$ and $H_{{\rm cln}}^{{\rm con}}$ along the $x$-axis is selected, indicated by the red line in Figs. 3(a) and 3(c). Also, Fig. 3(b) shows a zoomed-in view of the particle image magnitude $|{E^{{\rm img}}}|$. The magnitude better reveals the noise background surrounding the particle than does the image intensity $|{E^{{\rm img}}}{|^2}$ in Fig. 2(b). Figure 3(d) plots the pixel gray levels along the red line in $H_{{\exp}}^{{\rm con}}$ and $H_{{\rm cln}}^{{\rm con}}$. Comparing the measured hologram’s curve (in blue) to that of the cleaned hologram (in black) clearly demonstrates a significant degree of noise reduction.

 

Fig. 3. Performance of the noise-reduction method of Fig. 2 for the 50 µm glass microsphere at four wavelengths: $\lambda = 440\;{\rm nm} $, row (a); $\lambda = 640$, row (e); $\lambda = 840$, row (i); and $\lambda = 1040$, row (m). The plots in column (d) show the noisy oscillations of the measured contrast holograms $H_{{\exp}}^{{\rm con}}$ in blue and the resulting noise-reduced holograms $H_{{\rm cln}}^{{\rm con}}$ in black corresponding to the red lines in the associated holograms. Column (b) shows the image magnitude, $|{E^{{\rm img}}}|$, reconstructed from $H_{{\exp}}^{{\rm con}}$.

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Subsequent rows in Fig. 3 show the same results for other wavelengths for the same 50 µm spherical particle: $\lambda = 640\;{\rm nm} $ for Figs. 3(e)–3(h), $\lambda = 840\;{\rm nm} $ for Figs. 3(i)–3(l), and $\lambda = 1040\;{\rm nm} $ for Figs. 3(m)–3(p). Several features of the data are notable. First, compare the measured holograms as a function of increasing $\lambda$, i.e., going down the first column in Fig. 3. The nested-ring fringe pattern is seen to broaden as $\lambda$ increases. Such behavior is expected from the basic nature of scattering. That is, as $\lambda$ increases for a particle of a given size $R$, the size parameter $kR = 2\pi R/\lambda$ decreases, and the angular spacing between features in the particle’s scattered light will increase. Notice that the strength and structure of the noise in $H_{{\exp}}^{{\rm con}}$ varies with $\lambda$. One can also see a diagonal, linear fringe pattern at $\lambda = 840\;{\rm nm} $ and $\lambda = 1040\;{\rm nm} $, which is due to the long-pass filter installed at $\lambda = 720\;{\rm nm} $. The second column in Fig. 3, going down, shows how the particle’s image changes with increasing $\lambda$. Remember that this is the same particle in each image. Overall, there is an increase in the background noise with $\lambda$ accompanied by a decrease in the image resolution. The final two columns in Fig. 3 show that despite the overall degradation of both the measured hologram or the reconstructed image, the noise reduction method performs well. Even in the worst case with respect to noise, $\lambda = 1040\;{\rm nm} $, the method is able to produce a high quality hologram from a rather low signal-to-noise measurement, which is best seen in the plot of Fig. 3(p). With such results, it is straightforward to process the reduced-noise holograms, $H_{{\rm cln}}^{{\rm con}}$, to obtain estimates of ${C^{{\rm ext}}}$.

3. RESULTS

The measurements and methods above are applied to three particles of different characteristics: a 50 µm diameter glass sphere, an ash particle collected from the 2021 Tajogaite volcano eruption (Cumbre Vieja ridge) on La Palma island, and an iron(III) oxide (${{\rm Fe}_2}{{\rm O}_3}$) particle. The spherical particle is used to calibrate the method. That is, because the particle’s size and refractive index $m(\lambda)$ are known, Lorenz-Mie theory can be used to calculate [1] values for ${C^{{\rm ext}}}$ and verify that that the holographic images exhibit the correct particle size. The ash is chosen as an example of a unique particle with relevance to atmospheric remote sensing applications, while the iron(III) oxide particle is chosen as a loose representation of an absorbing mineral-dust particle.

Consider first Fig. 4(a) showing the cross section values obtained via [9] across the spectrum. These values are denoted $C_{{\rm holo}}^{{\rm ext}}$ and are shown in black with gray error bounds. The size of the error, ${\pm}10\%$, is based on the variability of the data in [9]. The solid red curve shows the Lorenz-Mie theory calculated cross section $C_{{\rm Mie}}^{{\rm ext}}$ using the open-source software MiePlot [24]. An error bound is also shown, but now is based on calculating $C_{{\rm Mie}}^{{\rm ext}}$ for different values of the particle diameter. The manufacturer certifies that the spheres are mono-disperse in size but with some variability, specifically, $D = 49.3 \pm 1.4\,\,\unicode{x00B5}{\rm m}$. Also shown is a straight, dashed, red line representing twice the area of the particle’s geometric cross section, ${C^{{\rm geo}}}$. This value represents the so-called extinction paradox. In short, when a particle is much larger than the wavelength, i.e., is in the geometric regime, ${C^{{\rm ext}}} \sim 2{C^{{\rm geo}}}$ independent of $m(\lambda)$ or $\lambda$ (provided $\lambda \ll D$ holds) [25]. Since the sphere here is 50 to 100 times the wavelength, the geometric regime is realized and the paradox can be used as a theoretical estimate for ${C^{{\rm ext}}}$.  Of course, doing so here is unnecessary since Lorenz-Mie theory is available. But, in the following, the paradox will be the only theoretical estimate possible for the other particles since Lorenz-Mie theory does not apply to non-spherical particles. Comparing $C_{{\rm Mie}}^{{\rm ext}}$ to $2{C^{{\rm geo}}}$ in Fig. 4(a) shows that the paradox estimates the true cross section well. Lorenz-Mie theory requires $m(\lambda)$, which is available in [26] and is plotted in the inset in Fig. 4(a). Note that although the real part of $m$ varies with $\lambda$, its imaginary part (absorption) is negligible over the spectral range.

 

Fig. 4. Hologram-estimated spectral extinction cross sections $C_{{\rm holo}}^{{\rm ext}}(\lambda)$ for three particles: 50 µm glass microsphere in (a), and volcanic ash and iron(III) oxide (${{\rm Fe}_2}{{\rm O}_3}$) particles in (b). The $C_{{\rm holo}}^{{\rm ext}}$ values are plotted in black along with theoretical estimates in red based on either Lorenz-Mie theory for the sphere, $C_{{\rm Mie}}^{{\rm ext}}$, or the geometric area of the image ${C^{{\rm ext}}} \simeq 2{C^{{\rm geo}}}$ for the other particles. The gray and red bands represent estimated error.

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Now consider Fig. 4(b) showing $C_{{\rm holo}}^{{\rm ext}}$ for the ash and iron(III) oxide particles (black curves). Again, each curve is given a ${\pm}10\%$ error bound. To provide some sense for whether the values are reasonable, the paradox estimate ${C^{{\rm ext}}} = 2{C^{{\rm geo}}}$ is shown (in red) with similar error bounds. Unlike the sphere, the particle sizes are not known a priori; they can, however, be measured from ${I^{{\rm img}}}$. Using a reconstructed image of the sphere, which is shown in the inset in Fig. 4(a), the sphere’s diameter in numbers of pixels ${N_{p}}$ is found to be ${N_{p}} \sim 17$. Multiplying by the pixel size $p$ provides a hologram-derived diameter ${D_{ h}} \sim 51\,\,\unicode{x00B5}{\rm m}$, which is in good agreement with the manufacturer-stated size range. Thus, to obtain ${C^{{\rm geo}}}$ for the ash and iron(III) oxide particles, one simply counts ${N_{p}}$ over the (entire) particle image and then multiplies by ${p^2}$. The values found are ${C^{{\rm geo}}} \sim 1150\,\,\unicode{x00B5}{\rm m}^2$ for ash and ${C^{{\rm geo}}} \sim 490\,\,\unicode{x00B5}{\rm m}^2$ for iron(III) oxide. For reference, the 50 µm scale bar in Fig. 4(a) applies to the holographic particle images in Fig. 4(b) as well.

One novelty of the experiment is that by measuring holograms at every 20 nm from $\lambda = 440\;{\rm nm} $ to $\lambda = 1040\;{\rm nm} $, it is possible to study the evolution of the reconstructed image across the visible-NIR spectrum. Although the sensor is monochrome, the image pixels can be assigned a false color based on $\lambda$. In Fig. 5(a), a color wheel is displayed with the reconstructed image at each $\lambda$ around the wheel’s circumference. Starting at the right side of Fig. 5(a) and going counterclockwise, the colors of each angular segment, or slice, of the wheel are selected to mimic the visual appearance of light at the $\lambda$ indicated, ranging from $\lambda \in [440, 700]\; {\rm nm}$. Beyond $\lambda = 700\;{\rm nm} $, in the NIR, the color becomes monochrome. Each angular segment is divided into 10 radial portions with varying saturation, or opacity. An opacity of 0%, corresponding to white, would indicate that the particle is fully transparent and no image forms. An opacity of 100% represents a completely opaque particle, and appears as a dark silhouette in the color associated with $\lambda$ for $\lambda \lt 700\;{\rm nm} $. Beyond $\lambda = 700\;{\rm nm} $, 100% opacity simply corresponds to black.

 

Fig. 5. Behavior of the false-color reconstructed images ${I^{{\rm img}}}$ of the particle in Fig. 4 across the spectrum. Each image thumbnail is reconstructed from $H_{{\rm cln}}^{{\rm con}}$ for the $\lambda$ indicated around the circumference of the color wheel and assigned a color. The color approximately conveys what the eye would perceive for that $\lambda$ while the saturation indicates the opacity of the image. In the NIR, the color is monochrome. Notice how different particle types exhibit opacity structure in their image, or a lack thereof, as $\lambda$ advances counterclockwise around the wheel.

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At first view, the particle images in Fig. 5(a) show the sphere with little apparent variation. However, closer inspection reveals that the internal portions of the image display some structure, in terms of opacity, as $\lambda$ changes. For example, compare the images for $\lambda = 1000\;{\rm nm} $ and $\lambda = 1020\;{\rm nm} $. The $\lambda = 1000\;{\rm nm} $ image shows a disk-like region at the center that is more opaque than the annular portion surrounding it. At $\lambda = 1020\;{\rm nm} $, the inverse is seen. Overall, the images across all $\lambda$ exhibit some degree of varying opacity, implying that light is not uniformly transmitted (or attenuated) through the particle, and a degree of transmission is observed in each case. At some level, this is expected because the sphere is glass, with excellent transmission properties across the spectrum, but the curved surface would still deviate the light.

The ash particle in Fig. 5(b) shows much the same qualitative behavior. The images are largely opaque for all $\lambda$, although some opacity structure can be discerned in cases. While the exact composition of the particle is unknown, it is likely a silicate mineral with (optically) absorbing compounds. That is, one would expect some degree of transparency, much like the glass sphere, except the opacity should vary more with $\lambda$ due to the absorbing components. Yet, it is difficult to claim a clear wavelength-dependent change in absorption due to the small particle size and lack of sufficient detail in the images. A significantly larger particle may however, show such behavior. Because the particle is nonspherical, Fig. 5(b) is also useful to appreciate how the image resolution is affected by an increase in $\lambda$. The best resolved image is for $\lambda = 440\;{\rm nm} $, and going around the figure counterclockwise, one sees that the resolution degrades. The sharp edges of the image become smoothed and slight distortions to the shape develop, becoming worse for the largest $\lambda$. This behavior is typical of imaging with DIH as explained, e.g., in Sec. 4.3 of [22].

Figure 5(c) shows the results for the iron(III) oxide particle. Unlike the others, here little opacity variation is seen with $\lambda$; the opacity is near 100% overall. One explanation could be that this material is known to be absorbing over the spectrum. According to [27], ${\rm Im}\{m \} = 1.1$ at $\lambda = 440\;{\rm nm} $ and drops to ${\rm Im}\{m \} = 0.012$ at $\lambda = 1040\;{\rm nm} $. While the latter value may seem comparatively small, realize that this particle is approximately 30 µm in size, and with such size, the absorption can manifest to a significant degree. That is, consider a simple exponential description [28] of light intensity $I$ at $\lambda$ through a thickness $\ell$ of material: $I(\ell ,\lambda)/{I_{ o}} \simeq \exp (- 4\pi {\rm Im}\{{m(\lambda)} \}\ell /\lambda)$. At $\lambda = 440\;{\rm nm} $, $I/{I_{ o}} = 0$ and at $\lambda = 1040\;{\rm nm} $, $I/{I_{o}} \simeq 0.01$, which is only 1% transmission of light. Thus, one would indeed expect a completely opaque appearance for all images here. Also notable in Fig. 5(c) is the impact of $\lambda$ on image resolution. Again, $\lambda = 440\;{\rm nm} $ represents the best-resolution image. By the point that $\lambda$ has increased into the NIR, the images show somewhat more degradation than the other particles. Likely, this is due to the smaller particle size.

As a final remark, Fig. 5 qualitatively explains some of the errors in $C_{{\rm holo}}^{{\rm ext}}$ in Fig. 4. For example, the trend in the $C_{{\rm holo}}^{{\rm ext}}$ values for the iron(III) oxide particle in Fig. 4(b) drops around $\lambda = 660\;{\rm nm} $. And in Fig. 5(c), the images appear to become slightly smaller due to the resolution degradation and this seems to persists for larger wavelength. Notice that the trend correlates with the decrease of $C_{{\rm holo}}^{{\rm ext}}$ with $\lambda$ away from the paradox estimate. While it is tempting to attribute the discrepancy between $C_{{\rm holo}}^{{\rm ext}}$ and the paradox estimate to this decrease in image size, Fig. 3 does not support that interpretation. That is, although a smaller image would produce a smaller mask $M$ in Fig. 2(d), the holograms $H_{{\rm cln}}^{{\rm con}}$ generated from the masked reconstructed image still agree well with the measured holograms, $H_{{\exp}}^{{\rm con}}$. Further investigation is needed to understand the origin for the drift of $C_{{\rm holo}}^{{\rm ext}}$ in Fig. 4 away from the paradox estimate.

4. DISCUSSION

In all, Fig. 4 demonstrates that the estimation of ${C^{{\rm ext}}}$ across the spectrum performs reasonably well. Overlap of the $C_{{\rm holo}}^{{\rm ext}}$ and theoretical values is seen for portions, but not all, of the spectrum. Aside from the factors already described, the source of the disagreement between $C_{{\rm holo}}^{{\rm ext}}$ and $C_{{\rm Mie}}^{{\rm ext}}$ (for the sphere) or the paradox estimate is difficult to isolate. One possibility not accounted for in Fig. 4 relates to the supercontinuum beam. Referring to Fig. 1(a), if the beam is not well collimated between the particle window (${ W}$) and the sensor, the extinction-extraction method of [9,18] may not yield ${C^{{\rm ext}}}$ accurately. For example, suppose the beam is slightly diverging over this distance. Then, the hologram pattern would be slightly magnified in size across the sensor, leading to an estimated ${C^{{\rm geo}}}$ greater than the true value. The inverse would be true as well, i.e., a converging beam would underestimate ${C^{{\rm geo}}}$. The possibility is relevant because the monochromator used slightly alters the input-beam characteristics as it scans the spectrum. Given that the hologram fringes also naturally expand (contract) as $\lambda$ increases (decreases) for a collimated beam—recall Fig. 3—it is difficult to deconvolute the effect from any that may be due to the monochromator. In future work, one could resolve such effects by slight translations of the beam-expander lenses in Fig. 1(a) or by replacement, or elimination, of the monochromator.

Whether or not the errors in Fig. 4 are acceptable would depend on the application, but certainly, the results are far from the accuracy expected from CRD measurements. However, the particle-ensemble nature of CRD measurements prevents knowledge of individual particle properties. Here, however, one has an image, and so can interpret $C_{{\rm holo}}^{{\rm ext}}$ with knowledge of the particle size and shape. Moreover, Fig. 5 suggests that the (spectral) particle images may give some sense for whether optical absorption is a significant contribution to the extinction. That is, the iron(III) oxide in Fig. 5(c) displays significantly more image opacity than the glass sphere in Fig. 5(a), suggesting that absorption is a more significant contribution to the extinction for ${{\rm Fe}_2}{{\rm O}_3}$ than it is for the sphere. Of course, knowing the compositions of the particles, that claim is obvious. But one may find use in applying the same argument to particles of unknown composition.

Next, consider the coherence properties of the supercontinuum laser. One may wonder how well holography could be done with such a light source simply based on the inverse relationship between the temporal coherence length ${L_{ c}}$ and the bandwidth $\Delta \lambda$ of the source, namely, ${L_{c}} \sim {\lambda ^2}/\Delta \lambda$ [29]. The formation of quality images in DIH requires the resolution of as many interference fringes as possible and with the supercontinuum source operating with $\Delta \lambda \sim 600\;{\rm nm} $, a coherence length of ${L_{c}} \lt 1\,\,\unicode{x00B5}{\rm m}$ may be expected. While the true coherence properties of supercontinuum lasers appear to be more complicated that this simple description, an ${L_{ c}}$ of such scale would imply difficulty to form holograms with sufficient fringes to image particles larger than ${L_{ c}}$. Of course, the results in, e.g., Fig. 3, demonstrate that this is not the case. The reason, we assess, is at least partly because the light is passed through the monochromator, which outputs a beam of far smaller bandwidth, usually several nanometers in scale. Thus, the light used to form the holograms is, in reality, sufficiently coherent and quality images can be obtained. Note that this also implies that one could apply the method here using conventional monochromatic lasers, where $C_{{\rm holo}}^{{\rm ext}}$ would be provided for a single $\lambda$, as demonstrated in [9].

One point to discuss concerns the twin image produced in DIH and the impact it could have on the images in Fig. 5. For any DIH measurement, the reconstruction process always produces two images of the particle. When one is well focused, the other is strongly blurred and appears as a diffuse system of rings surrounding the focused image. This twin image is caused by the reference and object waves in the measurement arrangement sharing the same optical axis, i.e., an on-axis arrangement. In cases where the particle size is a significant fraction of the beam diameter, the twin is usually detrimental and other approaches, such as an off-axis arrangement, are advised for imaging purposes. Here, however, the particles are far smaller, and in such cases, simply performing the reconstruction from the contrast hologram usually suppresses the twin’s effect strongly, e.g., see Fig. 12 in [22]. Nevertheless, the twin is not completely removed and one may wonder if it affects the formation of the mask $M$ in Fig. 2(d). If so, then erroneous results for $H_{{\rm cln}}^{{\rm con}}$ and $C_{{\rm holo}}^{{\rm ext}}$ may result. Such is not the case for particles of the size considered here. The clearest evidence for this can be seen in Fig. 3, fourth column, where the agreement between $H_{{\exp}}^{{\rm con}}$ and $H_{{\rm cln}}^{{\rm con}}$ in terms of the angular spacing of the fringes is excellent. If the twin were significant in the reconstruction, then the resulting mask would be larger than it should be. The consequence would be an $H_{{\rm cln}}^{{\rm con}}$ corresponding to a (falsely) larger particle, and thus, a mismatch between the fringe patterns in Fig. 3, which is not seen.

As a final remark, the results of this work could be viewed as somewhat trivial in that the particles are each sufficiently large that the paradox provides an estimate for ${C^{{\rm ext}}}$ well within 10% error; recall Fig. 4(a). The absence of wavelength dependence in the paradox (provided $\lambda \ll D$) means that one simply needs to reconstruct a particle’s image, determine ${C^{{\rm geo}}}$, and multiply the result by two to obtain a good estimate for ${C^{{\rm ext}}}$. In other words, for such large particles, the method of [9] is unnecessary. Much better would be to test the method above on particles roughly 1–5 µm in overall size, where ${C^{{\rm ext}}}$ is known to change strongly with $\lambda$ and where ${C^{{\rm ext}}}$ cannot be obtained directly from the particle image, thus necessitating the method of [9]. Indeed, it is shown via simulations in [9] that ${C^{{\rm ext}}}$ estimates with error less than 10% can be obtained for particles as small as $D \sim \lambda /3$, although this remains to be shown from measurements. With the experimental parameters in Fig. 1, e.g., $\lambda \in [{440, 1040}]\; {\rm nm}$, $p = 3.45\,\,\unicode{x00B5}{\rm m}$, etc., the image resolution is approximately $2p = 6.9\,\,\unicode{x00B5}{\rm m}$, and likely larger [22]. Thus, particles smaller than ${\sim}7\,\,\unicode{x00B5}{\rm m}$ would be completely unresolved and no paradox estimate for ${C^{{\rm ext}}}$ could be made since ${C^{{\rm geo}}}$ could not be found. We therefore focus on particles large enough to yield unambiguous ${C^{{\rm geo}}}$ values, thereby enabling a paradox estimate. In addition, particles 1–5 µm in size scatter light orders of magnitude less than the larger particles here. We find that with the equipment used, holograms for such small particles could not be resolved above the sensor noise-floor due to its limited dynamic range. A sensor with increased pixel size $p$ will feature a greater dynamic range, but of course, that simultaneously raises the resolution limit $2p$; see Sec. 4 in [22]. A solution may be to employ a telecentric lens between W and the sensor in Fig. 1 to magnify the hologram interference pattern and future work may consider this approach.