## 1.

## Introduction

Pathogenic bacteria have caused numerous infectious diseases in host living organisms either by contact or food consumption. Recent outbreaks from numerous food sources such as tomatoes and peanut butter remind us of the vulnerability of the food supply chain to contamination with pathogenic bacteria. The gold standard to detect and isolate a bacterial species is to grow colonies in a petri dish. A bacteria colony is defined as a group of individual bacterium $(\sim {10}^{8})$ compactly packed together to form an entity different from a surrounding environment. A colony consists of not only the bacteria itself but also extracellular materials, which are secreted as the colony grows. This combined structure can be analyzed to provide distinguishable phenotypic characteristics with proper interrogation methods. For example, differences of color, size, and shape can be utilized as distinguishing traces of a certain bacteria species under certain growth conditions. However, a selective medium that allows only certain types of species to grow and substantially suppresses the growth of others limits applicability to a small number of bacterial species. Therefore, it is advantageous to use nonselective media, which can grow various species on the same media. When different bacteria species are grown in this general media, their phenotypic characteristics tend to be quite similar, which renders morphological differentiation challenging.

Recently, a group of researchers reported a novel method of applying a forward scattering technique to identify a bacterial colony with highly accurate classification and broad applicability. The proposed technology, called BARDOT (bacteria rapid detection using optical scattering technology), provided highly repeatable and distinguishable forward scattering patterns all the way down to strain levels for some of the test cases. The previous research provided the theoretical modeling using scalar diffraction theory,^{1, 2} experimental verification for time-resolved scattering,^{3} and application to the actual food samples.^{4} In life science, automated bacterial colony counting is a topic of interest and various image processing techniques have been suggested for optimal performance considering the optical response characteristics of the colony.^{5, 6, 7, 8, 9} Several authors have modeled the profile of the bacterial colony elevation as a convex shape with different radii of curvature,^{1} a thin film with a decreasing tailing edge,^{5} and a Gaussian profile.^{6} The results of our most recent study on the smaller bacterial colony profile indicate that a Gaussian profile is quite satisfactory for describing the actual profile of the bacterial colony.^{10} Therefore, we model a bacterial colony as an optical amplitude/phase modulator with a Gaussian cross section in our study.

Similar ring-type diffraction patterns were observed from previous research involving neumatic liquid crystals (LCs), where the patterns were due to the laser-induced self-phase modulation phenomenon.^{11, 12, 13, 14, 15, 16, 17, 18, 19, 20} LC molecules were rotated in proportion to the intensity of the incident laser, which results in refractive index variations with a Gaussian profile. When this laser passes through an LC film of several hundred micrometer thickness, the optical wavefront is modulated to create a Gaussian wavefront that is propagated to a screen to create several diffraction rings. Some researchers investigated the field curvature effect on the diffraction rings,^{16, 17} while others worked on the relationship between Fredericksz transitions and primary/secondary diffraction rings.^{11, 12, 13, 14, 20}

Although the fundamental idea was revealed, more quantitative analysis is required to understand the morphological parameters of the bacterial colony and their correlation to the forward scattering pattern. Since the wavefront after the bacterial colony combines the incoming wavefront with the depth-averaged effect of a phase modulator, we designate some important morphological parameters and analyze their correlations to the far-field scattering patterns. Section 2 introduces theoretical modeling of light colony interactions and experimental procedures. Section 3 provides the simulation results when diameter and colony height are varied independently as well as experimental measurements of scattering patterns, colony diameters, and profiles. Section 4 discusses the validation and correlation between the scattering pattern and colony morphology.

## 2.

## Materials and Methods

## 2.1.

### Modeling Light-Colony Interactions

## 2.1.1.

#### Derivation of diffraction modeling

We define the coordinate systems for the source, the colony, and the imaging coordinate as
${x}_{s},{y}_{s}$
;
${x}_{a},{y}_{a}$
; and
${x}_{i},{y}_{i}$
, respectively, as shown in Fig. 1
. Assuming a
${\mathrm{TEM}}_{00}$
mode of laser beam centered on the
$z$
axis, electric field on the aperture plane
${E}_{1}$
can be expressed as^{1}

## Eq. 1

$${E}_{1}({x}_{a},{y}_{a},{z}_{1})={E}_{0}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}[-\frac{({x}_{a}^{2}+{y}_{a}^{2})}{{w}^{2}\left({z}_{1}\right)}]\mathrm{exp}\left(ik{z}_{1}\right)\mathrm{exp}\left[ik\frac{({x}_{a}^{2}+{y}_{a}^{2})}{2R\left({z}_{1}\right)}\right],$$## Eq. 2

$${w}^{2}\left(z\right)={w}_{0}^{2}[1+{\left(\frac{z}{{z}_{0}}\right)}^{2}],\phantom{\rule{1em}{0ex}}R\left(z\right)=z[1+{\left(\frac{{z}_{0}}{z}\right)}^{2}],$$## Eq. 3

$${E}_{2}({x}_{i},{y}_{i})=\frac{1}{i\lambda}\underset{\Sigma}{\int \int}t({x}_{a},{y}_{a}){E}_{1}({x}_{a},{y}_{a})\mathrm{exp}\left\{ik\left[\varphi ({x}_{a},{y}_{a})\right]\right\}\mathrm{exp}\left(\frac{ik{r}_{ai}}{{r}_{ai}}\right)\mathrm{cos}\phantom{\rule{0.2em}{0ex}}\theta \mathrm{d}{x}_{a}\mathrm{d}{y}_{a},$$## Eq. 4

$${E}_{2}({x}_{i},{y}_{i})\approx {C}_{1}\underset{\Sigma}{\int \int}t({x}_{a},{y}_{a})\mathrm{exp}[-\frac{({x}_{a}^{2}+{y}_{a}^{2})}{{w}^{2}\left({z}_{1}\right)}]\mathrm{exp}\left[ik\frac{({x}_{a}^{2}+{y}_{a}^{2})}{2R\left({z}_{1}\right)}\right]\mathrm{exp}\left[ik\frac{({x}_{a}^{2}+{y}_{a}^{2})}{2{z}_{2}}\right]\times \mathrm{exp}\left[ik\varphi ({x}_{a},{y}_{a})\right]\mathrm{exp}[-i2\pi ({f}_{x}{x}_{a}+{f}_{y}{y}_{a})]\mathrm{d}{x}_{a}\mathrm{d}{y}_{a},$$## Eq. 5

$${C}_{1}=\frac{{E}_{0}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}\left(ik{n}_{2}{\Delta}_{2}\right)\mathrm{exp}\left[ik({z}_{1}+{z}_{2})\right]\mathrm{exp}\left[(ik\u22152{z}_{2})({x}_{i}^{2}+{y}_{i}^{2})\right]}{i\lambda {z}_{2}},$$^{1}

## 2.1.2.

#### Amplitude Component

Our scanning electron microscopy (SEM) observation revealed both microscopic and macroscopic details of how millions of individual bacteria were packed to form a colony.^{2} The results indicated that the individual bacterium tended to form a layered structure rather than a distribution of randomly oriented bacterium, which provided for us a ground on which to model the light-bacteria interactions using a layer model as a first-order approximation. We divided the bacterial colony profile as multiple layers of bacteria with a thickness of
$\Delta \phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}$
, considering the physical dimensions of the individual bacteria and extracellular materials. Although the reduction in transmitted light can originate from both reflections and absorptions, we assume the normal incident reflection is the major contributor of field loss to our layer model based on the interrogating wavelength and refractive index of a bacterium. With respect to this assumption, Fig. 2
shows the field attenuated for the
$k$
’th layer of bacterium, which can be modeled as

## Eq. 7

$$t({x}_{a},{y}_{a})=\frac{{E}_{\mathrm{out}}}{{E}_{0}}=(1-{r}_{1}){(1-{r}_{k})}^{2l}(1-{r}_{2}),$$## Eq. 9

$${r}_{k}=\left|\frac{{n}_{1}-{n}_{\mathrm{ec}}}{{n}_{1}+{n}_{\mathrm{ec}}}\right|,\phantom{\rule{1em}{0ex}}{r}_{1}=\left|\frac{{n}_{0}-{n}_{1}}{{n}_{0}+{n}_{1}}\right|,\phantom{\rule{1em}{0ex}}{r}_{2}=\left|\frac{{n}_{1}-{n}_{2}}{{n}_{1}+{n}_{2}}\right|,$$## 2.1.3.

#### Phase Component

The phase component of the diffraction integral is governed by the optical path length term

where ${H}_{0}$ and $H({x}_{a},{y}_{a})$ are the center thickness and a profile of the bacterial colony. In addition, we define the effective diameter $D$ of the Gaussian profile colony aswhere ${w}_{b}$ is the $1\u2215e$ radius of a Gaussian profile, and $F$ is the factor (1.6) multiplied to render a $1\u2215{e}^{3}$ radius. Equation 10 can be further rearranged when we assume the Gaussian profile of the bacterial colony as^{21}

## Eq. 12

$$\varphi ({x}_{a},{y}_{a})=({n}_{1}-1){H}_{0}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}[-\frac{({x}_{a}^{2}+{y}_{a}^{2})}{{w}_{b}^{2}}]+{H}_{0}.$$## Eq. 13

$${E}_{2}({x}_{i},{y}_{i})\approx {C}_{2}\underset{\Sigma}{\int \int}t({x}_{a},{y}_{a})\mathrm{exp}[-\frac{({x}_{a}^{2}+{y}_{a}^{2})}{{w}^{2}\left({z}_{1}\right)}]\mathrm{exp}\left[ik{\varphi}^{\prime}({x}_{a},{y}_{a})\right]\times \mathrm{exp}[-i2\pi ({f}_{x}{x}_{a}+{f}_{y}{y}_{a})]\mathrm{d}{x}_{a}\mathrm{d}{y}_{a},$$## Eq. 14

$${\varphi}^{\prime}({x}_{a},{y}_{a})=\left(\frac{{x}_{a}^{2}+{y}_{a}^{2}}{2}\right)[\frac{1}{R\left({z}_{1}\right)}+\frac{1}{{z}_{2}}]+({n}_{1}-1){H}_{0}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}[-\frac{({x}_{a}^{2}+{y}_{a}^{2})}{{w}_{b}^{2}}],$$Then, the intensity of the electric field (power) is computed via

where $c$ is the speed of light in a vacuum, and $\epsilon $ is the permittivity. We then computed the diffraction pattern at the far field according to the parameters defined in Table 1 .## Table 1

Common simulation parameters.

Source | ${Z}_{1}$ | 100 | mm |

$w$ | 0.5 | mm | |

$\lambda $ | 0.635 | $\mu \mathrm{m}$ | |

$P$ | 1 | mW | |

Colony | ${Z}_{2}$ | 30 | mm |

$\Delta $ | 2 | $\mu \mathrm{m}$ | |

${n}_{1}$ | 1.38 | — | |

${n}_{2}$ | 1.33 | — | |

${n}_{\mathrm{ec}}$ | 1.35 | — | |

Image | $N$ | $1024\times 1024$ | — |

## 2.2.

### Experiments

## 2.2.1.

#### Sample Preparation

*Salmonella montevideo* cultures were selected to be inoculated in BHI broth and incubated at
$37\phantom{\rule{0.2em}{0ex}}\xb0\mathrm{C}$
overnight. The culture was then diluted (10-fold dilution) in phosphate buffer saline (PBS). We spread
$100\phantom{\rule{0.3em}{0ex}}\mu \mathrm{l}$
of the appropriate concentration thoroughly on the BHI agar and cultivated at
$37\phantom{\rule{0.2em}{0ex}}\xb0\mathrm{C}$
in the incubator. At the specified growth time, the plate was taken to BARDOT for measurement, which were
$6.5\phantom{\rule{0.3em}{0ex}}\mathrm{h}$
(diameter around
$90\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}$
),
$7\phantom{\rule{0.3em}{0ex}}\mathrm{h}$
$\left(120\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}\right)$
,
$8\phantom{\rule{0.3em}{0ex}}\mathrm{h}$
$\left(160\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}\right)$
, and
$9\phantom{\rule{0.3em}{0ex}}\mathrm{h}$
$\left(300\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}\right)$
.

## 2.2.2.

#### Forward Scatterometer

A forward scatterometer [Fig. 1] was built using (1) a
$635\text{-}\mathrm{nm}$
diode laser (Coherent 0221-698-01 REV B, California) with
$1\phantom{\rule{0.3em}{0ex}}\mathrm{mW}$
of power and a
$1\text{-}\mathrm{mm}$
diameter; (2) a single biconvex lens (BK7 H32-717, Edmund Optics, New Jersey) with a focal length of
$150\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$
; (3) a monochromatic IEEE1394 CMOS array (PixeLINK, Ottawa, Ontario, Canada) with
$1280\times 1024$
resolution and
$6.7\times 6.7\phantom{\rule{0.3em}{0ex}}\mu {\mathrm{m}}^{2}$
for each pixel, used as the imaging sensor; and (4) an
$XY$
linear translation stage (two of 850-HS, Newport, New York) and a multiaxis closed-loop controller (ESP300, Newport, New York) to move the plate in the
$x\text{-}y$
plane to align the laser with the center of each selected bacterial colony. The biconvex lens was used to control the beam diameter similar to the bacterial colony diameter via controlling the beam waist within the Raleigh range of the focused Gaussian beam. The focusing beam diameter
$d$
is computed via^{22}

## 2.2.3.

#### Phase Contrast Microscopy and Confocal Displacement Meter

The diameters of the bacterial colonies were measured by a phase contrast microscope (Leica Microsystems, Illinois). The objective was
$\times 10$
and the phase contrast image mode was used to image and measure the diameter of the colonies. Even though the phase contrast microscope provided qualitative phase information, it did not correlate the image with the colony profile quantitatively. Therefore, a confocal displacement meter (Keyence LT9010, New Jersey) was integrated with the
$X\text{-}Y$
motorized stage to obtain the height profile of each individual bacterial colony [Fig. 1]. We tested conventional laser triangular sensors to measure the profile, but the complexity of the sample (diffuse reflectivity, transparency, and multiple surface reflections) resulted in an unreliable shape profile (data not shown). The CDM instrument implements a “confocal principle” in the laser triangulation technique to deal with these issues and provided highly reliable surface scanning method by using a
$670\text{-}\mathrm{nm}$
laser light source with the spot size of
$2\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}$
and the vertical reference distance of
$6\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$
and a scanning resolution up to^{23}
$0.01\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}$
.

## 3.

## Results

We theoretically investigated the effect of morphological variation of the bacterial colonies to diffraction patterns on the imaging plane and measured the actual bacterial colony morphologies via microscope and profilometer. To decouple the effect of morphological parameters, we computed the colony center thickness effect and the colony diameter effect separately.

## 3.1.

### Height Variation versus Diffraction Pattern

The diameter of the bacterial colony
$D$
was kept constant at
$1.024\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$
and the center thickness
${H}_{0}$
of the bacterial colony was modified from
$10\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}110\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}$
in
$5\text{-}\mu \mathrm{m}$
steps. The maximum thickness was adapted from the previously published experimental results from the confocal microscopy.^{1} The corresponding profile was modified with Eqs.
3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 and their respective diffraction patterns were displayed for radii of
$0\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}5\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$
due to their circularly symmetric patterns. Since the dynamic range of the diffraction pattern varied significantly, we plotted the logarithm of the intensity (in milliwatts per square millimeters). Figure 3
shows the result for
${H}_{0}$
for 30, 50, 70, and
$90\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}$
. The computational results indicated the following two observations:

1. When the ${H}_{0}$ value increased for a constant diameter, the maximum diffraction angle increased.

2. When the ${H}_{0}$ value increased for a constant diameter, the total number of rings also increased.

## 3.2.

### Diameter Variation versus Diffraction Pattern

In this study, we kept the ${H}_{0}$ value of the bacterial colony as $50\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}$ and investigated the effect of the $D$ variations when it was varied from $0.384\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}1.05\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$ in $10\text{-}\mu \mathrm{m}$ steps. The corresponding profile was also modified with Eqs. 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 and their respective diffraction patterns are displayed in Fig. 4 for the colony diameters of 0.384, 0.48, 0.704, and $0.96\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$ . The diffraction calculation provided the following understandings:

1. When the center thickness was kept constant, the maximum diffraction angle was inversely proportional to the colony diameter.

2. When the center thickness was kept constant, the total number of rings was also kept constant irrespective of the colony diameter variation.

## 3.3.

### Correlation between Morphology and Diffraction Patterns

To understand the correlation between the colony morphology and the diffraction patterns, we repeated the calculation for different $D$ (Sec. 3.1) and different ${H}_{0}$ (Sec. 3.2). Figure 5 shows the relationship between the ${H}_{0}$ and the total number of peaks (diffraction rings) when $D$ was varied for $D=0.768$ , 0.896, and $1.024\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$ . The result clearly indicated that the ${H}_{0}$ value was the major factor contributing to the total number of diffraction rings observed at the imaging plane irrespective of the $D$ value. Figure 5 indicates the relationship between the colony diameter and the total number of peaks, which remained relatively constant across the diameter but increased as the ${H}_{0}$ values increased, which confirmed the result from Fig. 5.

Figure 6 displays the phase [Fig. 6] and the first derivative of the phase [Fig. 6] for a diameter of $1.024\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$ and ${H}_{0}$ values of 30, 50, and $70\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}$ . Since the center thicknesses are different, the phase lag from $70\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}$ is approximately 2.5 times as great as that from a $30\text{-}\mu \mathrm{m}$ thickness, as shown in Fig. 6. The derivative of the phase in Fig. 6 for the three thicknesses shows the minima at the ${X}_{i}=0.23\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$ location since this is the deflection point of the phase in Fig. 6. The magnitude of the derivative of the wavefront for $70\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}$ shows the largest value, which corresponded to the half angle of $4.42\phantom{\rule{0.3em}{0ex}}\mathrm{deg}$ , while $30\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}$ shows the smallest value of $2.13\phantom{\rule{0.3em}{0ex}}\mathrm{deg}$ .

## 3.4.

### Scatterometer Measurement

Using the forward scatterometer, a series of scatterogram was aquired as the bacteria grew. The measurement was aquired as the incident laser beam waist was varied close to the colony diameter. The top row of Fig. 7 shows scatterograms from approximately 89, 119, and $144\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}$ with ${Z}_{2}$ set around $8\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$ . Meanwhile, the bottom row displays the result for the colony diameters of approximately 193, 253, and $302\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}$ . For these sets, ${Z}_{2}$ was set to approximately $12\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$ such that larger area of sensor was utilized to the resolve the multiple number of rings. The result clearly indicates the increase of the number of rings as the colony diameters increased, which were measured from phase contrast microscope.

## 3.5.

### Colony Profile Measurement

To accurately assess the
${H}_{0}$
, CDM was used to acqure the profile, which was operated on the profile mode with 220 data points with a
$5\text{-}\mu \mathrm{m}$
interval^{23} in
$Y$
direction. Figure 8
displays the colony profile measurement and corresponding diameter measured from the phase contrast microscope. Figure 8 consolidates the 1-D crossectional profile across the
${H}_{0}$
location for seven different colony diameters. The result indicates the following two obervations: (1) proporationality between
$D$
and
${H}_{0}$
, which is approximately 10:1 ratio, and (2) the tailing edge profile was clearly captured for the most of the colony profiless, which supports our assumption on the Gaussian colony profile. Figures 8 and 8 show the phase contrast image and corresponding 2-D profiles from CDM in peudocolors.

To provide quantitative validations between theory and experiment, correlations between three different approaches were investigated: estimation from nematic crystal research,^{12, 15} estimation from the proposed model, and the experimental result (Fig. 9
). The red square indicates the estimated ring counts [Fig. 9] and maximum diffraction angle [Fig. 9] from the
${H}_{0}$
measurement of the CDM, while the green triangle represents the those from the experiment (Fig. 7). Finally, the blue circle shows the estimation from the proposed diffraction model. In the experiment, the beam diameter was controlled to be similar to the colony diameter therefore Eq. 12 was applied to convert from
${H}_{0}$
to phase difference. To compare with our previous modeling effort, we fitted the measured colony profile with three different models and estimated their performance, which were Gaussian profile fit (G-fit), two-radius fit (TR-fit), and one-radius fit (OR-fit). The performance was measured by calculating the sum of the squared difference from the proposed model and the 1-D CDM profile across the peak location. Table 2
shows the results and indicates that the G-fit and TR-fit are comparable to each other and clearly capture the colony profile, while the OR-fit (single spherical profile) does not reflect the actual colony profile.

## Table 2

Comparison of the Gaussian (G), two-radius (TR), and one-radius (OR) fits to the measured colony profiles.

Diameter (μm) | Height (μm) | Σ(PM−Fit)2 | ||
---|---|---|---|---|

G-fit | TR-fit | OR-fit | ||

109 | 9.04 | 10.21 | 21.42 | 6.39 |

124 | 11.28 | 29.84 | 25.43 | 564.49 |

204 | 22.83 | 150.65 | 128.42 | 2181.03 |

285 | 26.95 | 209.81 | 263.18 | 4808.79 |

318 | 28.19 | 145.64 | 265.87 | 1387.36 |

367 | 33.64 | 275.18 | 341.80 | 3436.84 |

421 | 52.37 | 700.19 | 2993.44 | 14,350.91 |

## 4.

## Discussion

## 4.1.

### Maximum Diffraction Angle

According to the results in Sec. 3, the maximum diffraction angle and the total number of rings are closely related to the bacterial colony morphology. These two parameters are important since they are directly measurable from the experiment and thus provide the phenotypic characteristics of the bacterial colonies. First, the maximum half diffraction angle is the result of the interference of two or more wavevectors pointing at the same direction. According to Eq. 13, the Fresnel diffraction pattern is the fast Fourier transform (FFT) of the input wavefront right after the wave passes through the bacterial colony. This accumulates all the light-matter interactions such as absorption, transmission, and reflection, which results in an overall phase modulation. Since the bacterial colony is modeled as a Gaussian profile, the wavefront emerging from the bacterial colony also resembles this Gaussian profile. Coupled with the radial phase effect, the overall wavefront is determined by Eq. 14. Therefore, the maximum diffraction angle is governed by the direction of the wave vector, which is the normal vector to the emerging wavefront. This implies that the magnitude of the slope of the wavefront is the key factor to determining the maximum diffraction angle, a theory that was also suggested by other researchers using nematic crystals.^{12, 13, 14, 15, 16} Previous research from nematic LCs reported the equation for estimating the maximum half cone angle
$\theta \u2215{2}_{\mathrm{max}}$
as,

## Eq. 18

$$\theta \u2215{2}_{\mathrm{max}}\cong \frac{1}{k}{\left(\frac{\mathrm{d}\Delta \Phi}{\mathrm{d}r}\right)}_{\mathrm{max}},$$^{20}Figure 6 provides the derivative of the phase profile based on model profile of Fig. 6. The maximum half cone angle of the total number of rings (when $D=1.024$ ) for the series of ${H}_{0}$ for 0.03, 0.05, and $0.07\phantom{\rule{0.3em}{0ex}}\mathrm{mm}$ as 2.13, 3.21, and 4.42, which agrees with 1.91, 3.08, and 4.25 from Eq. 18.

## 4.2.

### Number of Diffraction Rings

The total number of rings is another parameter closely related to the colony morphology. As shown in Fig. 5, the diffraction rings are dependent on the center thickness and are not dependent on the the colony diameter. This is because the optical path difference (OPD) increases as the
${H}_{0}$
is increased. To help clarify this argument, we provide a schematic diagram in Fig. 10
, which represents the wavefront from the bacterial colony. Since the wavefront captures the colony profile, the emerging wavefront is also a Gaussian profile. Since the diffraction peaks and valleys are generated from the interference of two or more wave vectors, we can relate the phase lag
$\left(\Delta \Phi \right)$
to the number of rings. Since the wave vectors
${\mathbf{k}}_{1}$
and
${\mathbf{k}}_{2}$
will interfere in at the far field, their OPD determines whether we observe a peak or valley, depending on constructive and destructive interference. As shown in Fig. 10, this OPD is determined from
${\delta}_{1}$
and
${\delta}_{2}$
, which decreases when the bacterial colony profile gets flatter (
${H}_{0}$
is decreased). Therefore, we can infer that the increase of
${H}_{0}$
results in a larger OPD and creates a larger number of diffraction rings. This conclusion coincides with conclusions drawn from nematic crystal research,^{12, 15} where the number of rings
${N}_{\text{ring}}$
was estimated as

## 4.3.

### Experimental Verification

The proposed Guassian colony profile model was compared with experiment and other models of Eqs. 18, 19. Figure 9 shows the comparision of the number of rings versus the colony diameter up to the
$350\text{-}\mu \mathrm{m}$
range, where the predicted and measured ring counts agree well. Some differences are still observed, though especially for the smaller colonies where the bright central spot lowers the accuracy in the experimental ring counts due to the limited dynamic ranges of the CMOS sensor. Figure 9 displays a similar comparison for a maximum diffraction angle. Both the proposed model and Eq. 19 predict similar results and the experimental measurement matched within a reasonable range of angles. A possible error source could arise from the difference between the Gaussian fit and the real colony and the position measurement of
${Z}_{2}$
, since the scattering angle masurement is sensitive to the colony-image plane distance. The result from Table 2 clearly shows the possibility of using a Gaussian profile to analyze the optical characteristics of the colony. Although the TR-fit model showed comparable accuracy with the G-fit model (except for diameters larger than
$350\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}$
), the proposed model requires only the
${H}_{0}$
and
$D$
values, while the TR-fit requires iterative optimizations of three different spherical profiles to accurately model the profile. Thus, we can directly estimate the morphological characteristics of the bacterial colony based on observation of the diffraction pattern. This explanation can be applied to the previous results from the time-resolved scattering pattern analysis.^{3} Our theory argues that vertical growth velocity is approximately equal to lateral growth velocity in the lag and logarithmic bacterial growth phase, which results in a wide diffraction pattern since the colony is narrow and tall. On the other hand, once the colony reaches the stationary point, vertical growth is slowed compared to lateral growth; hence, the diffraction pattern tends to shrink since the colony shape becomes wider and shorter.

## 5.

## Conclusion

Scalar diffraction theory was applied to model a bacterial colony as optical amplitude and phase modulators. Assuming a Gaussian profile for the colony shape, two main morphological characteristics of center thickness and colony diameter were calculated to correlate with the corresponding diffraction pattern. The results indicated that the maximum diffraction angle is dependent on the magnitude of the slope of the wavefront emerging from the bacterial colony, while the total number of rings is dependent on the center thickness of the bacterial colony. These results verify a fast method of understanding the bacterial colony morphology, which can be related to the understanding and classification of bacterial species.

## Acknowledgments

This research was supported through a cooperative agreement with the Agricultural Research Service of the U.S. Department of Agriculture Project No. 1935-42000-035 and the Center for Food Safety and Engineering at Purdue University.

## References

*Serratia liquefaciens*,” J. Math. Biol., 40 (1), 27 –63 (2000). https://doi.org/10.1007/s002850050004 0303-6812 Google Scholar