## Abstract

Goniometry and optical scatter imaging have been used for optical determination of particle size based upon optical scattering. Polystyrene microspheres in suspension serve as a standard for system validation purposes. The design and calibration of a digital Fourier holographic microscope (DFHM) are reported. Of crucial importance is the appropriate scaling of scattering angle space in the conjugate Fourier plane. A detailed description of this calibration process is described. Spatial filtering of the acquired digital hologram to use photons scattered within a restricted angular range produces an image. A pair of images, one using photons narrowly scattered within 8 − 15° (*LNA*), and one using photons broadly scattered within 8 − 39° (*HNA*), are produced. An image based on the ratio of these two images, *OSIR* = *HNA/LNA*, following Boustany et al. (2002), yields a 2D Optical Scatter Image (OSI) whose contrast is based on the angular dependence of photon scattering and is sensitive to the microsphere size, especially in the 0.5−1.0*µm* range. Goniometric results are also given for polystyrene microspheres in suspension as additional proof of principle for particle sizing via the DFHM.

© 2016 Optical Society of America

## 1. Introduction

The ability to effectively determine the size of particles at and below the diffraction limit of resolution has drawn extensive attention and efforts over the last decade. An effective, minimally invasive means of optically sizing such particles comes via fluorescence and super resolution microscopy, both of which entail the use of exogenous fluorescent labels [1–4].

Scattering based microscopy serves as a truly non-invasive alternative to fluorescent microscopy. Particle structure, relative to its optical environment, results in scatter functions that directly correlate to particle size and geometries [5, 6]. Using Mie theory as a first-order approximation to particle geometries, Boustany et. al. developed Optical Scatter Imaging (OSI), which has had success in sizing polystyrene microspheres in solution and mitochondria in the intracellular environment [7–9]. This imaging technique maps the scattered light to a conjugate Fourier plane, where it undergoes spatial filtering before being imaged to a CCD camera. Spatial filtering is either achieved via physical alterations of an iris or use of a digital micromirror device. In either context, the images collected at the imaging plane correspond to a given spatial filter geometry. Since the metric used for particle sizing, the Optical Scatter Imaging Ratio (OSIR), is derived by taking the pixel-by-pixel ratio between two different spatially filtered images, measurements taken in this manner are bound by the geometry of the aperture employed for each respective image. The use of additional spatial filter geometries would require subsequent alterations to the spatial filter and the capture of subsequent images. Ultimately, particle sizing via OSI is achieved via the OSIR, a quantitative metric. Via OSI, resolution is sacrificed for a truly non-invasive and quantitative means of particle sizing.

As an alternative to this approach, we use a 4-f optical system to map the Fourier plane (contained within the housing of the objective) to and captured via a CCD camera at a conjugate Fourier plane. Having collected the unfiltered Fourier transform, and therefore the spatial distribution of scattered light, we are able to then use any desired spatial filtering geometry digitally via postprocessing. After spatially filtering the acquired scattered field during postprocessing, we then use a two-dimensional Fourier transform to recreate the filtered sample at the image plane. However, we are only capturing the intensity of the scattered field at the conjugate Fourier plane. To truly map the sample to the image plane via a Fourier transform requires information of both the scattered field’s amplitude and phase. By introducing a reference beam at the conjugate Fourier plane, we make use of digital holographic principles in order to recover phase information for the scattered field via interference between the sample and reference arms [10–12].

Digital Fourier holographic microscopy (DFHM) has been explained extensively as applied in reflectance mode [13–15]. The use of reflectance geometries generally implies the detection of obtuse scattering angles. We are interested in acute angle scattering in order to ultimately discriminate between tissue structures or intracellular objects, such as nuclei and mitochondria. Based upon their size and relative indices of refraction, Mie theory predicts such structures will have dominant acute scattering angles. As such, we have built the DFHM in transmission mode in order to capture acute angles of scattering and enhance the scattered signal detected over that achieved in reflectance mode.

Such a device has been briefly reported in the literature, but was not explored to its fullest [16]. In that work, the scattered fields from two known particle sizes were measured in the conjugate Fourier plane and then used to create a digital mask for spatial filtering. The mask was then used to spatially filter the scattered field collected from a sample consisting of a mixture of the two particle sizes. Essentially, that work presupposes known scattered fields a posteriori in order to distinguish between particle sizes in its application. In addition, such a mask would be limited to distinguishing between the two distinct particle sizes in their surroundings. The devised OSIR technique is more flexible, having the ability to distinguish between particles sizes over a large, continuous range [7–9].

This work goes beyond the previously reported DFHM systems by incorporating the full range of functionality afforded via the OSIR method over angles of acute, forward scattering. This process begins with a characterization of the DFHM by determining the scattering angle space at the conjugate Fourier plane. Once the plane of the CCD has been translated to scattering angle space, a series of spatial filters for scattering angles and geometries can be applied digitally to a single hologram during postprocessing. In this sense, the work presented here also gives a more flexible application than the original OSI systems, which had a one-to-one correspondence between image collection and spatial filters employed.

This work will begin with a physical description of the DFHM and the calibration process used to determine the scattering angle space at the CCD camera, which is positioned at the conjugate Fourier plane. Focus is then shifted towards the direct application of particle sizing for polystyrene microspheres in suspension. Functionality of the DFHM for particle sizing is validated with both goniometric and OSI results.

## 2. System design and implementation

The purpose of the DFHM is to map the Fourier transform, and therefore the scattered field, of an object to a conjugate Fourier plane. At this point, an image of the sample plane can be reconstructed via a two-dimensional inverse Fourier transform. Spatial filtering can be enacted digitally prior to image reconstruction in order to aid in particle sizing.

The system depicted in Fig. 1 begins with a laser (Melles Griot, 25-LGP-193-249, ≤ 5*mW*) with *λ* = 543.5*nm*. The laser light is then expanded to roughly 8*mm* in diameter and cleaned via a telescopic system (T) consisting of a 3-axis spatial filter (Newport 900 with 10X objective lens and Newport 900 PH-50 50*µm* pinhole) in sequence with a bi-convex lens (*f* = 100*mm*). Upon leaving the telescopic system, the beam then encounters a 50/50 beam splitter (BS), separating the beam into the sample and reference arms for interferrometry. Both the sample and reference arms have neutral density filters (ND) and electronic shutters (SH) in order to control the intensity and automated detection of each arm, respectively.

The collimated beam transmitted through the beam splitter is incident upon a sample. From the sample plane, the incident light will follow one of two paths, depending upon each photons’ encounter at the sample plane. Working with sparse, optically thin samples, the majority of the light incident upon the sample plane will transmit unimpeded and travel directly to the objective (O). Since the ballistic photons will enter the objective as collimated light, they will be mapped to a central maximum at the Fourier plane (F) within the objective.

Scattered photons will diverge from the sample plane in accordance with scattering theory. For particles sized on the order of the wavelength of incident light, Mie theory is used as a first order approximation to the scattering functions that result [5,6]. Photons scattered by the sample diverge from the small field of view of the objective lens, which approximately collimates these photons. While unscattered photons are focused by the lens to the center of the Fourier plane, the scattered photons are distributed in the Fourier plane. The DFHM takes advantage of this asymmetry in order to separate the scattered light from directly transmitted light, much in the fashion of dark field imaging [17]. Ultimately, the ability to remove the dominant, background signal due to directly transmitted photons will enhance the signal of the scattered photons.

A 4-f optical system (L1 and L2) is used to then map the scattered field at the Fourier plane, which is housed within the objective and therefore inaccessible, to a conjugate Fourier plane for imaging via a CCD camera (CCD1, Point Grey, Grasshoppper USB3, GS3-U3-28S4M-C, 2.8MP, 14bit, 1928×1448, *δx* = 3.69*µm*). As a result, the directly transmitted light will be focused at the center of the CCD as an unscattered DC component. Conversely, the scattered light will be mapped to some angular distribution of scattering angles at the conjugate Fourier plane. This scattered field is represented as a two-dimensional optical Fourier transform.

The CCD camera however only measures the intensity of the incident field. In order to properly take the two-dimensional inverse Fourier transform of the scattered field to recreate the image plane computationally, we need have information of the phase of the field. This phase is recovered via the introduction of interference between the sample and reference beams at the conjugate Fourier plane, where the reference arm is redirected towards the CCD via two mirrors (M) and an additional beam splitter prior to the CCD. Recovery of phase from the sample field via holography requires acquisition of intensities from the scattered (*I _{f}*) and reference (

*I*) fields individually, along with the resulting intensity from the superposition of the two fields (

_{r}*I*).

_{H}Multiplying the superposition of the sample and reference fields at the conjugate Fourier plane (CCD1) by its complex conjugate, interference between the sample and reference arms will result in the hologram intensity at the conjugate Fourier plane

where*f*and

*r*represent the amplitudes of the sample and reference fields, respectively, such that

*I*= |

_{f}*f*|

^{2}and

*I*= |

_{r}*r*|

^{2}correspond to their respective intensities [10–12]. The factor 2

*f rcosϕ*results from the cross terms in multiplying the superposition of the fields by its complex conjugate, along with the definition of the cosine function in terms of complex exponentials based on Euler’s formula. As a result, the overall phase

*ϕ*is really the difference between the phase of the reference and sample fields at a given pixel, Capturing and then subtracting the individual sample and reference intensities from Eq. (1) then returns the phase information of the hologram,

*k*and

_{x}*k*.

_{y}When imaging dynamic samples that evolve over time, the time lapse between the acquisition of intensities from the superposition of scattered and reference fields (*I _{H}*) and sample field alone (

*I*) leaves the possibility of motion or sample variations between the acquisitions, which could lead to artifacts in the reconstruction process. As an alternative to acquiring a separate sample intensity, we make use of the approximation

_{f}*cosϕ*(

*k*) is small and that

_{x},k_{y}*I*≫

_{r}*I*[12]. The latter requirement is typically met in digital holography, in that the reference signal can be used to amplify that of the sample via Eq. (3). Such is the case in the application of this system. Overall, the validity of both of these assumptions was tested via holographic reconstructions for this system with and without the application of Eq. (4). Reconstructions with and without the use of Eq. (4) were found to be equivalent for a static sample. Therefore these conditions are satisfied and the use of Eq. (4) is permitted in generating the phase information of the scattered light Eq. (3), thereby reducing the number of images acquired to those of the hologram itself Eq. (1) and of the reference alone. The reconstruction of phase from the sample field Eq. (3) then can be determined via

_{f}Given that the Fourier transform of a random distribution of particles will not necessarily be obvious, there is difficulty in knowing when exactly the sample is positioned at the working distance from the objective. As such, we make use of the portion of the sample arm that is redirected via the beam splitter and use a third lens (L3) to optically take the inverse Fourier transform, thereby mapping the scattered light to the image plane at a second CCD camera (CCD2, Point Grey, Firefly MV, FFMV-03M2M-CS, 640×480, *δx* = 6*µm*). This returns a bright field image which can be used to guide sample positioning and alignment.

Images collected by CCD2 are not used in data analysis and are solely for the purpose of alignment and visual confirmation. The mirrors in the reference arm are aligned so the reference beam is not normally incident on CCD1 at the conjugate Fourier plane, as is the case in off-axis holography. The alignment of these mirrors, along with the mirror and lens in the imaging arm of CCD2, redirect and focus the reference beam away from CCD2, thereby keeping the reference beam from imposing on the bright field image of the sample at CCD2.

Timing and acquisitions of both CCD cameras and electronic shutters are controlled via LabVIEW (National Instruments, Corp., Austin, TX), with the field of view from both cameras displayed in real-time on the front end GUI. Additional camera controls such as gain and exposure are also controlled via the GUI.

Upon acquisition of the hologram and reference intensities, and the application of Eq. (5) in order to regain phase information from the sample, we are ready to impart digital spatial filtering on Eq. (5) as an application of OSI. Alternatively, the intensity of the sample field measured at the conjugate Fourier plane (*I _{f}*) is equivalent to the scattering function

*F*(

*θ,φ*), and can be used directly for goniometry and comparison to the scattering functions predicted via Mie theory for known particles in suspension [5, 6].

## 3. Calibration

Before we can apply the DFHM towards either goniometry or OSI, we first must determine the scattering angle space generated at the conjugate Fourier plane. Ultimately, we want to map the scattering function *F*(*θ,φ*) to the two-dimensional, *x − y* pixel space of the CCD located at the conjugate Fourier plane. To do this, we make use of the diffraction pattern created by a holographic film transmission diffraction grating of known line spacing placed at the sample plane of the objective. With a beam normally incident on the grating, a central max will be created at the center of the CCD along with symmetric orders of diffraction to both sides. Positions of the diffraction maxima on the CCD are used along with the diffraction equation in order to calibrate the angle space on the CCD.

This process becomes a bit more convoluted however, in that the DFHM uses an oil-immersion objective (100X Olympus Plan Fluorite Oil Immersion Objective, ∞-corrected, 1.30 NA, 0.20 mm WD). In addition, samples of interest for this paper are polystyrene microspheres, *n _{ps}* = 1.59, suspended in Aqua-Poly/Mount mounting media,

*n*= 1.457 (Polysciences, Inc, Warrington, PA) between a glass slide and coverslip. Mie theory predicts the scattered field from spherical particles based upon their size and relative index of refraction with respect to their surroundings. However, upon traversing the media-glass and glass-immersion oil interfaces, refraction will have the effect of narrowing the angles of scattering as captured by the objective lens. This geometry is illustrated in Fig. 2 with a diffraction grating in place at the sample plane. The refraction effects reduce to a media-immersion oil interface via the application of Snell’s Law to this geometry,

_{med}*θ*corresponds to the scattering angle in mounting media. The refractive index of immersion oil

_{med}*n*= 1.518 is greater than that of the mounting media, such that angles of scattered light collected by the objective will be less than those predicted via Mie theory [18]. When imaging biological samples, such as cells in culture, the aqueous surrounding media will lead to a more dramatic scaling effect.

_{oil}In order to mimic the effects of refraction on samples when imaging diffraction patterns for calibration purposes, the diffraction grating is layered with mounting media and a glass coverslip. Immersion oil couples the glass coverslip to the objective. By bringing the mounting media into direct contact with the diffraction grating, we must also scale the predicted diffraction angles by the wavelength of light in the medium,

where*m*represents the

*m*order diffraction peak and

^{th}−*d*the spacing between lines on the diffraction grating. When using a diffraction grating,

*θ*is used in place of the more generic

_{m}*θ*variable to emphasize discrete angles of constructive interference due to the diffraction grating.

_{med}Having accounted for the effects of mounting media on diffraction angles and refraction from the overall mounting media-immersion oil interface, the diffraction pattern acquired at the conjugate Fourier plane is ready for scaling. A line scan is taken across the diffraction pattern in order to bring out the positioning of the diffraction peaks. Given the predicted angles of *m ^{th}*-ordered diffraction, along with their measured positions on the CCD, a pixels per degree scaling factor can be determined at the conjugate Fourier plane. For a 500 lines per millimeter transmission diffraction grating coupled to the coverslip via mounting media, a 18.6

*pix/*° scaling factor is determined at the conjugate Fourier plane [18].

The maximum angle of scattering accepted via the DFHM is 38.9° when using the Aqua-Poly/Mount mounting media [18]. Using water as the medium, such as when imaging cells *in vitro*, a similar scaling factor of 19.3*pix/*° and maximum angle of scattering accepted via the DFHM of 37.5° result [18].

The distribution of observed diffraction patterns confirm that a linear relationship between pixel and angle space at the conjugate Fourier plane is legitimate [18]. Having established the correlation between pixel and angle space at the conjugate Fourier plane, we are ready to use the DFHM for particle sizing via goniometry and OSI.

## 4. Experimental results

The scattered field as measured at the conjugate Fourier plane of the DFHM is equivalent to the scattering function *F*(*θ,φ*), which is symmetric with respect to the azimuthal angle *φ*. The scattering angle *θ* essentially projects onto the conjugate Fourier plane as what would normally be considered a radial dimension. Therefore, if we integrate the scattering function around the azimuthal angle *φ*, the one dimensional scattering function results

*rdφ*with

*θdφ*based upon the argument that the scattering angle

*θ*is projected along the “radial direction” of the conjugate Fourier plane. This geometry is depicted in Fig. 3.

Polystyrene microspheres with diameters from 0.36*µm* to 2.90*µm* are mounted separately in the Aqua-Poly/Mount mounting media. For goniometric purposes, the reference arm of the DFHM is blocked such that only the sample arm is detected at the conjugate Fourier plane—this simple goniometry experiment does not require a holographic reconstruction of the sample. A blank slide with only the Aqua-Poly/Mount media is also imaged and subtracted from the sample data in order to remove any background signal. The resulting two-dimensional scattering function is then integrated to return the one-dimensional scattering function via Eq. (8). In order to account for variations in particle numbers imaged between samples (and therefore the overall intensity of the measured scattering function), Eq. (8) is then normalized for each particle diameter. The results are then plotted against scattering functions predicted via Mie theory in Fig. 4 [19].

While the goniometric results are in good agreement with Mie theory, the use of OSI returns the OSIR as a predictive metric for quantitative particle sizing and has direct application beyond the model of polystyrene microspheres [7–9]. Some image processing is required in order to translate the hologram captured via the DFHM, Eq. (1), into a spatially filtered image. The first step has already been outlined, picking out the phase of the scattered field via Eq. (5), which represents the phase distribution of the Fourier transform of the sample field. Taking the inverse Fourier transform of Eq. (5) returns the scattered field

including twin images along with a central DC component. The twin images are attributed to the complex exponential terms that led to the 2*f rcosϕ*factor from Eq. (1). In classic holography, where film was used to record the phase of the hologram, these twin images resulted in a pair of real and virtual images that were created downstream and upstream from the holographic film, respectively.

The angle *θ _{ref}* at which the reference arm is introduced to the CCD at the conjugate Fourier plane works to impart a carrier frequency on the hologram, thereby separating the twin images from the DC component [10–12]. This effectively removes the DC component, or unscattered light, from the image, serving as a type of dark-field imaging and replacing the need for the central beam block employed in the original OSI system designs.

One of the two twin images can then be digitally selected and isolated from the remainder of the hologram for further analysis. We define this selected region of the field as *E*′(*x,y,z* = 0), such that it’s Fourier transform is defined as
${\mathcal{E}}^{\prime}({\overrightarrow{k}}_{x},{\overrightarrow{k}}_{y},0)$. The selected twin image will not be in focus initially and will need to be numerically propagated to an image plane, much as the optical inverse Fourier transform needs to propagate spatially before reaching a focal plane for imaging.

Numerical reconstruction can be explained starting with the Rayleigh-Sommerfield Solution to the Huygen’s-Kirchoff Principle

*z*= 0 and $E(\overrightarrow{\rho},z)$ represents the field downstream at a reconstructed depth

*z*[10, 20, 21]. The

*x*− and

*y*− dependence has been grouped into the vector ${\overrightarrow{\rho}}_{o}$ in the corresponding

*z−*plane. The remaining terms are those of the “impulse response”, By Huygen’s Principle, the incident field $E({\overrightarrow{\rho}}_{o},0)$ can be divided into infinitesimal units, each acting as an individual point, or secondary, source [22]. Observation of the field downstream is then the result of the interference between the waves from each of these secondary sources. Integrating over the

*x−y−*plane then is equivalent to summing over each of these infinitesimal, outwardly propagating waves in order to reconstruct the resulting field downstream at depth

*z*. Each secondary source is then a point source with an outgoing spherical wave, represented in Eq. (11) by the factor $\frac{{e}^{ikR}}{R}$. In addition, the cos

*θ*term in Eq. (11) accounts for the direction of propagation from each secondary source and the remaining coefficient $\frac{-ik}{2\pi}$ accounts for the time variation and phase of the wave at the

*z−*plane in comparison to that of the incident,

*z*= 0 plane [20].

Applying the Paraxial (or small angle) approximations cos*θ* → 1 and *R* → *z* to the impulse response, Eq. (11), we are able to relax the 1*/R* and cos*θ* dependence of the function. Taking a Taylor expansion of the exponential term in Eq. (11), the radial dependence can be reduced to a simpler *z*–dependence [21]

Making the notation changes *ρ*^{2} = *x*^{2} + *y*^{2} and
${\rho}_{o}^{2}={\xi}^{2}+{\eta}^{2}$ in the propagated and originating planes, respectively, the propagated field becomes

All together, Eq. (13) can be interpreted as a convolution of the incident field and propagation term in order to determine the reconstructed wave downstream from its originating plane,

By the Convolution Theorem of Fourier Theory, the Fourier transform of a convolution is equivalent to the product of the Fourier transforms of the individual functions being convolved [10]*x−y−*plane, and Employing the Convolution Theorem, Eq. (16), thereby eliminates the need for a convolution integral, Eq. (10), in favor of simply multiplying two Fourier transforms together. For the present application then, this process simplifies to

*z−*plane by substituting Eq. (21) into Eq. (20)

The entire holographic reconstruction process described by Eq. (9) through Eq. (22) is depicted in Fig. 5. This figure uses the scattered field from a sample of 2.90*µm* microspheres suspended in Aqua-Poly/Mount, beginning with a scattered field equivalent to Eq. (5).

Once the twin image *E*′(*x,y,z* = 0) is isolated from Eq. (9), we then follow by taking its Fourier transform, Eq. (18), for numerical propagation via Eq. (22). Since we are again working in Fourier space at this point, the appropriate time to employ digital spatial filtering for OSI purposes comes just after numerical propagation to the appropriate plane. The scattering function is spatially filtered and analyzed via optical scatter imaging using angles *θ _{block}* = 8°,

*θ*= 15° and

_{min}*θ*= 39°. Heterodyne filtering removes the DC component of the image, as well as the first 2° of scatter in the current system. When imaging cells

_{max}*in vitro*, setting

*θ*= 8° blocks nuclear scattering while accepting broader scattering from mitochondria. Future studies using a 2° − 8° spatial filter will study the scattering of nuclei.

_{block}Borrowing from Boustany’s notation, we let
${\mathcal{E}}_{HNA}({\overrightarrow{k}}_{\perp},0)$ and
${\mathcal{E}}_{LNA}({\overrightarrow{k}}_{\perp},0)$ represent the high numerical aperture and low numerical aperture filtered fields respectively, where the former emphasizes broadly scattered light and the later narrow scattered light [7]. The LNA and HNA digital masks used for spatial filtering are depicted in Figs. 6(a) and 6 (b), respectively. The inverse Fourier transform of these filtered fields are then taken in order to return to the image plane, giving *E _{HNA}*(

*x,y,z*) and

*E*(

_{LNA}*x,y,z*), Fig. 6(c and d) respectively. The OSI can then be determined by taking a pixel-by-pixel ratio between the intensities of the inverse Fourier transforms of these two propagated fields fields,

The resulting OSI can be noisy due to division of the HNA image Fig. 6(d) by the LNA image Fig. 6(c) when pixel values in the LNA image are at or near zero. In order to avoid this issue, the reconstructed hologram Fig. 6(e) is used to create a binary mask (using bwconncomp in MatLab) in order to isolate the OSIR of isolated particles (using labelmatrix and regionprops in conjunction with bwconncomp in MatLab). In this fashion, noise is eliminated from the Optical Scatter Image, as seen in Fig. 6(f).

The OSIR metric used for quantitative particle sizing is then represented by pixel values of the OSI. In order to judge the validity of this method, we take the average value of the OSIR from a sample of known polystyrene particles of the same size suspended in the mounting media. The average OSIR values recovered in this fashion closely match those predicted via Mie theory, as seen in Fig. 7.

## 5. Discussion

The critical step in recovering accurate particle sizing lies in the proper calibration of the angle to pixel space of the CCD located at the conjugate Fourier plane of the DFHM. This holds true for both goniomety and OSI experiments. For instance, without accounting for the refraction across the sample holder, as depicted in Fig. 2, the goniometric curves will be shifted from those predicted via Mie theory. When applied to OSI, if the measured angles of scattering aren’t similarly compensated, the scattering function *F*(*θ,φ*) will also be distorted from that predicted via Mie theory, such that the spatially filtered field will not be accurately quantified.

Goniometric results show excellent correspondence between predicted scattering functions and those measured via the CCD at the conjugate Fourier plane. These results are very accurate both in magnitude and in picking out the particular oscillations of the respective scattering functions.

OSIR results also follow the trend predicted for the metric based on Mie theory. For polystyrene microspheres suspended in mounting media, the OSIR falls off with increasing particle diameter. The greatest dynamic range of the OSIR curve lies below the 1*µm* particle diameter, giving the greatest sensitivity for detection of and differentiating between small particles.

Looking at the OSIR results generated via the DFHM system in more detail, the experimental resolution limit of the system is in the range of ~ 0.5*µm*. Judging by the numerical aperture of the objective however, we anticipate that the system should measure out to a maximum angle of 55.3° (when using the polystyrene microsphere mounting media) and have resolution down to 0.255*µm*. At present the system does not meet these theoretical limits, having a maximum angle of acceptance at the CCD of 38.9°. The lower range of angles accepted at the CCD is a result of the final lens in the system, prior to the conjugate Fourier plane, which over magnifies the Fourier transform, causing it to overfill the CCD at present. Replacing that *f* = 300*mm* biconvex lens with a new lens with a focal length less than 168*mm* would allow the capture of the maximum angle of 55.3° as predicted via the numerical aperture of the objective. A *f* = 150*mm* bi-convex lens would adequately map the full range of angles to the CCD, while not causing too great a loss of resolution by underfilling the CCD.

An additional note should be mentioned with respect to digital spatial filtering of the scattered field, as well as the isolation of the desired twin image. Each time we take a cut of the digital image or Fourier transform, we impart hard edges to the spatial (frequency) distribution. Upon enacting on such a cut or filtered digital representation with the Fourier (or inverse Fourier) transform will result in high frequency ringing. In order to limit this ringing effect, apodization can be used to smooth out the edges, thereby reducing or removing the high frequency noise that would otherwise result [10]. Common tools used for smoothing the spatial frequency window include the Hamming and Hann windows [24]. As an alternative, we use a modified version of the error function (erf in MatLab) in order to create a spatial frequency window which can be adjusted for width and the sharpness with which it drops off.

While the use of a smoothing spatial frequency window helps to remove high noise ringing upon Fourier (and inverse Fourier) filtering, it also has the effect of attenuating the signal at the edges of the sample or frequency space. As an alternative to the use of smoothing spatial frequency windows, images can be reconstructed with the hard edges and the resulting high frequency noise can be cleaned up via image thresholding and morphological operations [24]. Use of these alternative tools comes with their own set of caveats, which generally reduce again to a loss of information. A combination of both of these techniques has been employed in generation of the OSI results presented here. Boustany’s original OSI work gets around the noise generated by taking the pixel-by-pixel ratio of two images by binning pixels into larger pixel groups. Each bin is then averaged in order to determine the local average OSIR within the image [7–9].

## 6. Conclusions

The design and calibration of the digital Fourier holographic microscope in transmission are described. The calibration of angle to pixel space on the CCD camera, which accounts for refraction across any medium-coverslip-immersion oil interfaces, is paramount for the ability of the system to operate effectively as a metrology tool. Even small effects such as refraction across the medium-coverslip-immersion oil interfaces can shift the resulting data and has been accounted for here. The DFHM in transmission mode is then employed for particle sizing of polystyrene microspheres suspended in mounting media via goniometry and optical scatter imaging as proof of principle experiments.

Using the transmission mode of the DFHM allows for detection of scattering angles as small as ~ 2°, affording the ability to better discriminate between particle sizes than in the reflectance mode when scattering signals are weak. This is because Mie theory predicts small objects will scatter strongest at acute angles. Having to use the reflectance mode for the DFHM forces measurements generally forces scattering measurements out to obtuse angles, thereby lowering the scattered intensity by multiple orders of magnitude. This effect can be seen even within the range of detection of the DFHM in transmission mode presented here, where scattering intensities can drop three to four orders of magnitude over the first ~40° of scattering, as can be seen in Fig. 4. While scattering intensities do rebound for backscattered signals, particles of biological interest will still have their strongest scattering signals at acute scattering angles.

While a linear pixel-to-angle of scattering relationship across the CCD at the conjugate Fourier plane would be a natural assumption, it is not obvious that the relationship couldn’t vary otherwise. A linear pixel-to-angle of scattering relationship is demonstrated at the CCD, adding to the confidence in the pixel-to-scattering angle scaling at the conjugate Fourier plane. Having appropriately calibrated the DFHM system and demonstrated its effectiveness in metrology via both goniometry and OSI, the DFHM is ready for more novel biological particle sizing applications such as looking at nuclear or mitochondrial scattering in cells *in vitro* or collagen structure in thin tissue slices.

## Acknowledgments

The authors thank S. Alexandrov and N. Boustany for technical advice with respect to the DFHM and OSI, respectively.

## References and links

**1. **L. Peyser, A. Vinson, A. Bartko, and R. Dickson, “Photoactivated fluorescence from individual silver nanoclusters,” Science **291**, 103–106 (2001). [CrossRef] [PubMed]

**2. **J. Lichtman and J. Conchello, “Fluorescence microscopy,” Nat. Meth. **2**, 910–919 (2005). [CrossRef]

**3. **B. Giepmans, S. Adams, M. Ellisman, and R. Tsien, “The fluorescent toolbox for assessing protein location and function,” Science **312**, 217–224 (2006). [CrossRef] [PubMed]

**4. **B. Huang, M. Bates, and X. Zhuang, “Super resolution fluorescence microscopy,” Annu. Rev. Biochem. **78**, 993–1016 (2009). [CrossRef]

**5. **H. van de Hulst, *Light Scattering by Small Particles* (Dover, 1981).

**6. **C. Bohren and D. Huffman, *Absorption and Scattering of Light by Small Particles* (Wiley-VCH, 2004).

**7. **N. Boustany, S. Kuo, and N. Thakor, “Optical scatter imaging: subcellular morphometry in situ with fourier filtering,” Opt. Lett. **26**, 1063–1065 (2001). [CrossRef]

**8. **N. Boustany, R. Drezek, and N. Thakor, “Calcium-induced alterations in mitochondrial morphology quantified in situ with optical scatter imaging,” Biophys. J. **83**, 1697–1700 (2002). [CrossRef]

**9. **N. Boustany, Y. Tsai, B. Pfister, W. Joiner, and G. Oyler, “BCL − x_{L} – dependent light scattering by apoptotic cells,” Biophys. J. **87**, 4163–4171 (2004). [CrossRef] [PubMed]

**10. **J. Goodman, *Introduction to Fourier Optics*, 3rd ed. (Roberts and Company, 2005).

**11. **P. Picart and J. Li, *Digital Holography* (John Wiley and Sons, Inc., 2012).

**12. **M. Kim, *Digital Holographic Microscopy: Principles, Techniques, and Applications*, in Springer Series in Optical Sciences (Springer, 2011). [CrossRef]

**13. **S. Alexandrov, T. Hillman, and D. Sampson, “Spatially resolved fourier holographic light scattering angular spectroscopy,” Opt. Lett. **30**, 3305–3307 (2005). [CrossRef]

**14. **S. Alexandrov, T. Hillman, T. Gutzler, M. Same, and D. Sampson, “Particle sizing with spatially-resolved fourier-holographic light scattering angular spectroscopy,” in *Multimodal Biomedical Imaging*, F. Azar and D. Metaxas, eds. (SPIE, 2006). [CrossRef]

**15. **T. Hillman, S. Alexandrov, T. Gutzler, and D. Sampson, “Microscopic particle discrimination using spatially-resolved fourier-holographic light scattering angular spectroscopy,” Opt. Exp. **14**, 11088–11102 (2006). [CrossRef]

**16. **K. Seet, P. Blazkiewicz, P. Meredith, and A. Zvyagin, “Optical scatter imaging using digital fourier microscopy,” J. Phys. D: Appl. Phys. **38**, 3590–3598 (2005). [CrossRef]

**17. **P. Prasad, *Introduction to Biophotonics* (John Wiley and Sons, Inc., 2003). [CrossRef]

**18. **V. Rossi, “Digital fourier holographic microscopy and potential applications towards the design of photodynamic therapy of osteosarcoma,” Ph.D. Dissertation, Oregon State University, Department of Physics, 301 Weniger Hall, Corvallis, OR 97331 (2015).

**19. **S. Prahl, and OMLC, “Mie scattering calculator,” (2012).

**20. **D. Nolte, “Bioanalysis: advanced materials, methods, and devices,” *Optical Interferrometry for Biology and Medicine* (Springer, 2012). [CrossRef]

**21. **J. Mertz, *Introduction to Optical Microscopy* (Roberts and Company, 2010).

**22. **E. Hecht, *Optics*, 4th ed. (Addison-Wesley, 2002).

**23. **D. Voelz, “Computational Fourier Optics: a Matlab Tutorial,” *Tutorial Texts in Optical Engineering* (SPIE Press, 2011).

**24. **R. Gonzalez and R. Woods, *Digital Image Processing*, 3rd ed. (Prentice Hall, 2007).