Digital holography is a versatile method to image small particles. In its simplest form, often called digital in-line holography (DIH), a collimated laser beam illuminates a particle, and the interference pattern of scattered and unscattered light, called the object and reference wave, respectively, is recorded by an image sensor facing the oncoming beam. Examples of DIH are now commonplace in biology, atmospheric science, and industry, among others [1]. A well-suited application is the characterization of aerosol particles because multiple particle images may be obtained from a single hologram measurement in a contact-free manner using a simple optical arrangement. Notably, the locations of particles in the beam need not be known, or tightly controlled, because particles at different locations can be later re-focused computationally during the image generation [1].

Yet, aerosols pose a unique challenge for DIH. Because the particles are inherently in motion, a pulsed laser is usually employed to (stroboscopically) freeze the motion, and thus, only a single hologram can be captured [2]. Consequently, the variety of image-resolution enhancement methods commonly employed in DIH, such as pixel super-resolution [3–5], cannot be implemented on aerosols as multiple holograms of an unchanging particle configuration are needed. In practice, the image resolution $\delta$ obtained remains relatively poor, being constrained by the Nyquist–Shannon sampling theorem to $\delta _{\text {Ny}}=2p$, where $p$ is the sensor’s pixel size [1]. With commercial sensors possessing pixels typically >2 µm, aerosol particles smaller than $\sim$4 µm are not typically resolved with DIH methods. As such, particles in the so-called PM-2.5 or fine-mode size range are not subject to characterization by such methods. There are thus value efforts to improve the resolution of DIH for aerosols and, in particular, do so in a manner that preserves the flexibility and simplicity of conventional methods. This work describes a method capable of imaging $1$ µm free-flowing aerosol particles. Good imaging performance is realized using a simpler optical arrangement than comparable work. Importantly, the images obtained are not filtered, binarized, or otherwise enhanced, which is a common practice in many holography studies. In doing so, the results provide a genuine illustration of the imaging performance.

The approach taken is inspired by Spuler [6], where a bi-telecentric lens system in an aircraft instrument magnifies and relays the hologram patterns of cloud particles onto a sensor in a sealed wing-mounted canister. The telecentric system in [6] consists of seven lenses with a net magnification of $2.5\times$, yielding an image resolution of $4.7$ µm. In contrast, two lenses are used here, enabling a comparatively simpler design but with a smaller sensing region than in [6].

Bi-telecentricity is defined by the system’s entrance and exit pupils being at infinity; it is an afocal system [7]. A collimated beam entering the system will exit collimated, but with a beam diameter increased by the system magnification, $M$. An ideal system will form a constant-sized image of an object independent of the axial position of that object. The image appears orthographic in perspective in contrast to conventional lens systems that produce images of different sizes for different axial positions. Bi-telecentric systems are well-suited for DIH as they can uniformly magnify the hologram fringe patterns of particles within a volume almost independent of the particles’ positions.

Figure 1 depicts the experiment conducted. An aerosol is generated by the system in Fig. 1(a) where $\text {N}_{2}$ gas is supplied at a rate of 2 $\text {L}/\text {min}$ to a nebulizer (Aero Mist) to aerosolize size-calibrated polystyrene latex (PSL) microspheres (Polysciences, Inc., 64030-15) from suspension in (miliQ) water. The aerosol is drawn under negative pressure into a cell where the particles freely flow through a laser beam. A mass-flow controller (Alicat Scientific, MC-5SLPM-TFT/5M) between the cell’s output port and the vacuum pump sets the flow through the cell to 0.5 LPM.

 

Fig. 1. Arrangement used to image $1$ µm diameter, free-flowing, PSL aerosol particles with DIH. In (a) is the system to produce the aerosol. In (b) is the optical arrangement (not to scale) to capture the holograms via a bi-telecentric lens system. The inset (c) provides details of the sensing region in the aerosol cell. See the text for further explanation.

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Shown in Fig. 1(b) is the optical arrangement used to collect holograms. A laser (Coherent, OBIS-LX-405) emits pulses with a wavelength $\lambda =402$ nm and a pulse length $\tau \simeq 300$ ns with an average power $P_{\text {ave}}=100$ mW at a 7 Hz repetition rate. The collimated beam diameter from the laser is $\sim$1 mm, and the beam is directed to the aerosol cell to illuminate the flowing particles. Note that the Rayleigh range from the laser aperture is $\sim$1.25 m, which is much greater than the optical arrangement. The cell is an aluminum block with a 25.4 mm diameter bore sealed by two optical windows (W). Stainless steel tubes with an outer diameter (OD) and inner diameter (ID) of 2 mm and 1 mm, respectively, pass through the block to deliver and remove the aerosol. Figure 1(c) shows a view of the portion of the cell where the particles exit the inlet tube, travel in free-space approximately 2 mm and through the laser beam at $\sim$10 m/s, and are then removed by the outlet tube. The coordinate system shown in Fig. 1(b) is defined by the laser beam, which propagates along the $z$ axis; the aerosol flow is then along the negative $y$ axis.

Following the cell are two lenses, L1 (Thorlabs, THR254-040-A-ML) and L2 (Thorlabs, LA1172-A), with focal lengths $f_{1}=40$ mm and $f_{2}=400$ mm, respectively. Lens L1 is positioned a distance $f_{1}+d$ from the aerosol flow, where $d$ is an adjustable distance ranging from 0 to 5 mm. Lens L2 is positioned a distance $f_{1}+f_{2}$ from L1. In this configuration, i.e., with the lenses separated by the sum of their focal lengths, a simple bi-telecentric systems is realized. The telecentricity is verified with a shearing interferometer (Thorlabs, SI100) to ensure that, upon small adjustments to the lens positions, a collimated beam entering the system exits it collimated. The exiting beam diameter is $\sim$10 mm, which gives $M\sim$10 in agreement with a theoretical transverse magnification of $M_{\text {T}}=-f_{2}/f_{1}$. A 10-bit monochrome CMOS image sensor (Aluvium, 1800U-1240m) guarded by a $405\pm 3$ nm interference bandpass filter (F: Thorlabs, FBH405-3) is placed a distance $d_{\text {sen}}< f_{2}$ from L2 facing the oncoming beam. The sensor has square pixels of size of $p=1.85$ µm with an array of $4024\times 3036$ pixels, which is cropped in later analysis to a square array of $N\times N$ pixels where $N=3036$.

For more details, Fig. 1(c) shows an illuminated particle at a distance $d$ from a point $P$. The chief ray for $P$ is also shown (dashed), which can then be traced to the sensor in Fig. 1(b). The ray is parallel to the optical axis on both the object (cell) and image (sensor) side of the lens system, which is a characteristic of bi-telecentricity. Also shown in Fig. 1(c) is a plane, $S_{\text {eff}}$, passing through the beam at $P$ where the particle’s scattered wave is shown spreading across it. Thus, there is an interference pattern between the scattered and unscattered light in $S_{\text {eff}}$. The lenses magnify and relay this pattern onto the surface of the sensor, $S_{\text {sen}}$. Another way to view the effect of the lens system is given in Fig. 2. It is as if the sensor’s pixel array $S_{\text {sen}}$ with a pixel size $p$ becomes a virtual sensor at $S_{\text {eff}}$ a distance $d$ from a particle with an effective pixel size $p_{\text {eff}}=p/M\simeq 0.185$ µm. So, $\delta _{\text {Ny}}=2p_{\text {eff}}=0.37$ µm, which is well below the diffraction limit discussed later. As such, the image resolution $\delta$ will be constrained by the diffraction limit rather than the Nyquist–Shannon limit.

 

Fig. 2. Bi-telecentric system creating an effective pixel array.

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An illustration of the system’s telecentricity is presented in Fig. 3. Here, the cell is temporarily replaced by a 1951 USAF target (Thorlabs, R1DS1P). With the target in the beam at $S_{\text {eff}}$, i.e., at $d=0$, the lens system forms a focused image on the sensor; see Fig. 3(a). A 35 µm diameter red circle, equivalent to $35 \mathrm{\mu}\textrm{m}/p_{\text {eff}}=189$ image pixels across, is shown on the group-6 element-2 square to verify that the length is correctly measured in the analysis. Displacing the target along the negative $z$ axis a distance of $d\simeq 5$ mm from $S_{\text {eff}}$ means that a hologram now appears on the sensor. If the lens system has acceptable telecentricity, reconstruction of the image from this hologram should yield same-sized image features as seen for $d=0$, which is the case in Fig. 3(b). Note that Fig. 3 should not be considered an assessment of the system’s image resolution because the image in Fig. 3(b) is not reconstructed from a contrast hologram [1]. The aerosol particle images presented later are reconstructed from contrast holograms and exhibit better resolution. Further discussion of the image resolution is given later.

 

Fig. 3. Demonstration of the system’s telecentricity. A 1951 USAF target is placed in the sensing region of Fig. 1(b) at $S_{\text {eff}}$ for $d=0$ mm in (a) and $d\simeq 5$ mm in (b).

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Figure 4 shows the DIH images of $1$ µm diameter PSL aerosol particles. To obtain a hologram, an exposure of the sensor to a single laser pulse is first made without aerosol in the cell. This resolves the beam intensity profile and is called the reference measurement, $I^{\text {ref}}$. Then, aerosol is allowed into the cell and exposures are taken for each pulse emitted, at a rate of 7 Hz. A digital delay generator (Stanford Research Systems, DG645) is used to trigger and synchronize the laser and sensor for this purpose. Each exposure constitutes a so-called raw hologram $I^{\text {holo}}$, which will consist primarily of the beam profile with the possibility of relatively weak intensity fringe patterns due to any particles that may be present during the exposure.

 

Fig. 4. Holographic imaging of free-flowing $1$ µm diameter PSL aerosol particles. In (a) is a contrast hologram. In (b) is the image reconstructed for the red square in (a). The arrow identifies the twin image artifact. In (c) are 10 other examples of particle images. The red circles are $1$ µm in diameter.

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Because of the weak intensity of the patterns, it is necessary to form a contrast hologram $H^{\text {con}}=I^{\text {ref}}-I^{\text {holo}}$ to subtract away noise and the main features of the unscattered beam [1]. An example of $H^{\text {con}}$ is presented in Fig. 4(a). A particle’s pattern manifests as a cluster of nested rings, and several such clusters can be seen indicating the presence of multiple particles. The hologram in Fig. 4(a) is then processed by the Fresnel diffraction integral to reconstruct an image. The reconstruction process is reviewed well by many, see, e.g., [1] where the code is given to implement it along with a description for how to focus the image. The result is shown in Fig. 4(b), corresponding to the particle producing the fringes in the red square in Fig. 4(a). A $1$ µm diameter (red) circle is shown in Fig. 4(b) for the scale, which is clearly the same size as the particle image. Figure 4(c) presents an array of 10 other particle images reconstructed from different contrast holograms.

In DIH, a persistent image artifact is the twin image, which is a blurred mirror image of the particle appearing as halo of rings around the focused image [1]. Because the particles here are much smaller than the beam, image reconstruction from a contrast hologram will strongly suppress, but not eliminate, the artifact. The twin artifact is identified by the red arrow in Fig. 4(b), and it can also be seen in each image in Fig. 4(c). Although no filtering or enhancement is applied to these images, there are methods to further suppress the artifact; see [8].

Many holograms may be collected and not each one will contain a particle due to the erratic nature of aerosol generation, flow, and the limited frame rate of the sensor. In all, 105 contrast holograms are found to contain one or more particles. While the PSL microspheres used are marketed as monodisperse, they nevertheless have a degree of polydispersity. The manufacturer states a size of $0.95$–$1.05$ µm in diameter, and thus, some variability of the particle size should be present in the DIH images. By reconstructing a focused image of a particle and fitting a circle to the image perimeter [akin to Fig. 4(b)], the particle diameter $D$ is measured by the circle’s diameter in multiples of $p_{\text {eff}}$. There is thus an uncertainty of $\pm p_{\text {eff}}\simeq 0.19$ µm in $D$. The reconstruction process also provides the particle location in the beam (see [1]).

Figure 5(a) shows a histogram of the measured $D$. The mean diameter and standard deviation of all particles observed are $\langle D_{\text {a}} \rangle$ and $\sigma _{\text {a}}$, respectively, and the distribution is shown in gray. Of these, some are imaged with or without the aerosol passed through the diffusion dryer in Fig. 1(a). The dryer should remove any water that may coat the particles. The mean and standard deviation of particles processed through the dryer are $\langle D_{\text {d}} \rangle$ and $\sigma _{\text {d}}$, and the distribution is shown in blue; those without the dryer are $\langle D_{\text {nd}} \rangle$ and $\sigma _{\text {nd}}$ and likewise are shown in red. Dried particles show $\langle D_{\text {d}} \rangle =1.08\pm 0.19$ µm, slightly smaller than the no-dryer case where $\langle D_{\text {nd}} \rangle =1.15\pm 0.19$ µm. The larger $D$ values may be due to water coating. Note that the distributions in Fig. 5(a) should not be interpreted as exhibiting multiple modes due to the (statistically) small number of particles observed.

 

Fig. 5. Measured aerosol particle diameter $D$ in (a) and the spatial distribution in (b) for all $1$ µm PSL particles observed. Blue and red in (a) are the size distribution with and without the use of the diffusion dryer, while gray is the total distribution.

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Figure 5(b) shows the sensing region with spheres placed at the particle locations determined in the reconstruction of all 105 holograms. The sphere-symbol sizes are not to scale and the aerosol flow direction is indicated. Note how the particles appear at random locations, which is expected for an aerosol and proof that particles are not deposited on the cell windows. While the $x$ and $y$ axes of the region range $500$ µm, the $z$ axis ranges 3 mm and is compressed to aid the visualization. The volume of the sensing region is then estimated as $\sim$0.75 $\text {mm}^{3}$.

To give a broader sense for the method’s performance, Fig. 6 presents DIH imaging of a ragweed pollen (Ambrosia trifida) aerosol. Dried pollen grains (Stallergenes Greer) are drawn into the cell in Fig. 1(a) at the inlet tube under negative pressure from a particle-laden filter, and holograms are collected in the same manner as above. Figure 6(a) shows the image of a single aerosolized grain. In dried form, the grains are approximately spherical, 15–25 µm in diameter with a complex surface of spines that are 1–2 µm at their base and narrow to sub-micrometer points [9]. In Fig. 6(a), surface roughness is clearly visible on a sphere-like grain. Figure 6(b) shows a zoomed view where the roughness features are consistent with the expected size of the spines at their base. In Fig. 6(c), a scanning electron microscope (SEM) image of a representative grain is shown. The similarity of the grain surface features in Figs. 6(b) and 6(c) are illustrative of the DIH method’s ability to resolve micron-scale features on particles more complex, and larger, than the PSL microspheres.

 

Fig. 6. Holographic imaging of a single, dried, aerosolized ragweed pollen grain, (a) and (b). In (c) is a SEM image of a similar grain for comparison with a 2 µm (red) scale bar.

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The theoretical resolution limits that apply to Fig. 1 can be estimated. The ultimate constraint is the diffraction limit $\delta _{\text {NA}}=\lambda /(2\text {NA})$, where NA is the numerical aperture [1]. The NA of the telecentric system is defined by L1 in Fig. 1(b) as $\sin \left [\tan ^{-1}\left (D_{\text {L}}/2f\right )\right ]\simeq D_{\text {L}}/2f_{1}$, where $D_{\text {L}}$ is the lens diameter. Meanwhile, the NA associated with the holography, i.e., for $S_{\text {eff}}$, is $\text {NA}\simeq N p_{\text {eff}}/2d$. Estimating $d\sim$1 mm from Fig. 5(b), the diffraction limits for either the lens system or $S_{\text {eff}}$ are $\delta _{\text {NA}}\sim$0.63 µm and $\delta _{\text {NA}}\sim$0.72 µm, respectively. While these figures are approximate, they are at least consistent with the ability above to resolve the 1 µm particles, i.e., the particles are larger than these fundamental limits. Note that another reason the aerosol particle images exhibit better resolution than the USAF target in Fig. 3(b) is because the target produces a far more complex and intense interference pattern than a particle. Consequently, a portion of the target hologram’s finest fringes becomes averaged by the pixels, degrading the resolution to $\delta \simeq 2.5$ µm for the group-7 element-5 set resolved in Fig. 3(b), i.e., in addition to the target image not being reconstructed from a contrast hologram.

This work is not the first to achieve $1$ µm image resolution of aerosol particles. Perhaps the most relevant example is David et al. [10] where $\delta \simeq 0.77$ µm is achieved by optically trapping a particle and using a narrowband diode laser for the holography. The laser operates in continuous-wave mode, which requires the trapping to prevent particle motion from affecting the hologram. In that sense, the particle is not a free-flowing aerosol. Moreover, the optical arrangement uses diverging beam illumination, which restricts the sensing volume over which one can achieve such resolution. Also relevant is the work by Guildenbecher et al. [11] which employs a formidable object-space telecentric system with nine lenses and achieves $\delta \simeq 2$ µm resolution for aerosols in the test chamber.