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 as¹

Eq. 1

$$E_1(x_a,y_a,z_1)=E_0\exp [-\frac{(x_a^2+y_a^2)}{w^2\left(z_1\right)}]\exp \left(ik{z}_1\right)\exp \left[ik\frac{(x_a^2+y_a^2)}{2R\left(z_1\right)}\right],$$

where $E_0$ is the on-axis field strength; three terms account for variations of amplitude of field, longitudinal phase, and radial phase, respectively; $x_a$ and $y_a$ are the coordinates in the aperture; and $w\left(z_1\right)$ and $R\left(z_1\right)$ are the beam waist and radius of the wavefront in the colony plane $(z=z_1)$ , which is defined by

Eq. 2

$$w^2\left(z\right)=w_0^2[1+{\left(\frac{z}{z_0}\right)}^2],R\left(z\right)=z[1+{\left(\frac{z_0}{z}\right)}^2],$$

where $z_0$ is defined as the $z$ location where $1∕e^2$ radius has expanded to $\sqrt{2}$ times of beam waist $w_0$ . Then, we apply the Huygens-Fresnel principle in rectangular coordinates. Using Eq. 1, the electric field $E_2$ at the imaging plane can be expressed as

Eq. 3

$$E_2(x_i,y_i)=\frac{1}{i\lambda }\underset{\Sigma }{\int \int }t(x_a,y_a)E_1(x_a,y_a)\exp \left\{ik\left[\varphi (x_a,y_a)\right]\right\}\exp \left(\frac{ik{r}_{ai}}{r_{ai}}\right)\cos \theta \mathrm{d}{x}_a\mathrm{d}{y}_a,$$

where $x_i$ and $y_i$ are points on the image plane, $\sum$ denotes the colony surface, $t(x_a,y_a)$ is the 2-D transmission coefficient, $\varphi (x_a,y_a)$ is the 2-D phase modulation factor, $r_{ai}$ is the distance from the aperture plane to the image plane, $\lambda$ is the wavelength, and $n_2$ and ${\Delta }_2$ represent the refractive index and thickness of Brain heart infusion (BHI) agar, respectively. Based on the scalar diffraction theory, the diffracted field on the image coordinates can be formulated via Fresnel approximation as

Eq. 4

$$E_2(x_i,y_i)\approx C_1\underset{\Sigma }{\int \int }t(x_a,y_a)\exp [-\frac{(x_a^2+y_a^2)}{w^2\left(z_1\right)}]\exp \left[ik\frac{(x_a^2+y_a^2)}{2R\left(z_1\right)}\right]\exp \left[ik\frac{(x_a^2+y_a^2)}{2z_2}\right]\times \exp \left[ik\varphi (x_a,y_a)\right]\exp [-i2\pi (f_x{x}_a+f_y{y}_a)]\mathrm{d}{x}_a\mathrm{d}{y}_a,$$

where $f_x$ and $f_y$ are defined as $f_x=x_i∕\lambda z_2$ and $f_y=y_i∕\lambda z_2$ . When Eq. 2 is rearranged, the constants and independent variables are consolidated as

Eq. 5

$$C_1=\frac{E_0\exp \left(ik{n}_2{\Delta }_2\right)\exp \left[ik(z_1+z_2)\right]\exp \left[(ik∕2z_2)(x_i^2+y_i^2)\right]}{i\lambda z_2},$$

where $k$ is the wave number. Detailed assumptions and the derivation are shown elsewhere.¹

Fig. 1

Experimental setup and definition of coordinate system: (a) forward scatterometer with laser diode $\left(635\mathrm{nm}\right)$ ; (b) confocal displacement measurement (CDM) system with 2-D rastering via $XY$ stage system; and (c) coordinate system for source, aperture, and image coordinates and the bacterial colony morphology fitted with Gaussian shape. Here $H_0$ is the center thickness, $w_b$ is the $1∕e$ radius, $n_1$ and $n_2$ are the refractive indices for colony and agar, and ${\Delta }_2$ represents the agar thickness.

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 $\Delta \mu \mathrm{m}$ , 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. 6

$$E_{k+1}=E_k(1-r_k),$$

where $r_k$ is the reflection coefficient for the $k$ ’th layer and is assumed to be constant for all the layers. Considering all the multiple layers in the colony, the overall 2-D transmission coefficient is defined as

Eq. 7

$$t(x_a,y_a)=\frac{E_{\mathrm{out}}}{E_0}=(1-r_1){(1-r_k)}^{2l}(1-r_2),$$

where the exponent $l$ is defined with the colony profile $H(x_a,y_a)$ , and the layer thickness $\Delta$ as

Eq. 8

$$l=H(x_a,y_a)∕\Delta .$$

The different reflection coefficients $r_k$ , $r_1$ , and $r_2$ are defined since $r_k$ represents the interbacterium reflection, while $r_1$ and $r_2$ model the reflection between air-bacterium and bacterium-agar. This can be modeled as

Eq. 9

$$r_k=\left|\frac{n_1-n_{\mathrm{ec}}}{n_1+n_{\mathrm{ec}}}\right|,r_1=\left|\frac{n_0-n_1}{n_0+n_1}\right|,r_2=\left|\frac{n_1-n_2}{n_1+n_2}\right|,$$

where $n_0$ , $n_1$ , $n_2$ , and $n_{\mathrm{ec}}$ are the refractive indices for air, bacteria, agar, and extracellular materials. Since our scalar diffraction theory does not consider the polarization effect, we simply model the attenuation from the thickness of the colony and take the absolute value of the reflection coefficients.

Fig. 2

Schematic diagram for amplitude modulation model via multiple layers, where the rectangular box and the shaded box represent bacteria and extracellular material: $E_k$ denotes the electric field incident on the $k\text{th}$ layer of bacteria exiting as $E_{k+1}$ . The loss is modeled via the reflection coefficient $r_k$ with the layer ( $\text{bacterium}+\text{extracellular}$ material) thickness as $\Delta$ .

2.2.

Experiments

3.1.

Height Variation versus Diffraction Pattern

The diameter of the bacterial colony $D$ was kept constant at $1.024\mathrm{mm}$ and the center thickness $H_0$ of the bacterial colony was modified from $10\text{to}110\mu \mathrm{m}$ 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 $0\text{to}5\mathrm{mm}$ 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 $90\mu \mathrm{m}$ . The computational results indicated the following two observations:

3.2.

Diameter Variation versus Diffraction Pattern

In this study, we kept the $H_0$ value of the bacterial colony as $50\mu \mathrm{m}$ and investigated the effect of the $D$ variations when it was varied from $0.384\text{to}1.05\mathrm{mm}$ 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 $0.96\mathrm{mm}$ . The diffraction calculation provided the following understandings:

Fig. 3

Forward scattering pattern for center thickness $\left(H_0\right)$ variation when diameter is kept constant: $H_0=\left(\mathrm{a}\right)$ 0.03, (b) 0.05, (c) 0.07, and (d) $0.09\mathrm{mm}$ . The number of peaks and maximum scattering angle increases as $H_0$ increases.

Fig. 4

Forward scattering pattern for colony diameter variation when $H_0$ is kept constant and $D$ is (a) 0.384, (b) 0.48, (c) 0.704, and (d) $0.96\mathrm{mm}$ . Note that the maximum scattering angle decreases as $D$ increases.

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 $1.024\mathrm{mm}$ . 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.

Fig. 5

Correlation of the number of rings when (a) colony thickness and (b) colony diameter were varied.

Figure 6 displays the phase [Fig. 6] and the first derivative of the phase [Fig. 6] for a diameter of $1.024\mathrm{mm}$ and $H_0$ values of 30, 50, and $70\mu \mathrm{m}$ . Since the center thicknesses are different, the phase lag from $70\mu \mathrm{m}$ 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 $X_i=0.23\mathrm{mm}$ location since this is the deflection point of the phase in Fig. 6. The magnitude of the derivative of the wavefront for $70\mu \mathrm{m}$ shows the largest value, which corresponded to the half angle of $4.42\mathrm{deg}$ , while $30\mu \mathrm{m}$ shows the smallest value of $2.13\mathrm{deg}$ .

Fig. 6

(a) Phase and (b) derivative of phase of the wavefront after the incident beam passes the bacterial colony when the center thickness is 0.03, 0.05, and $0.07\mathrm{mm}$ .

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 $144\mu \mathrm{m}$ with $Z_2$ set around $8\mathrm{mm}$ . Meanwhile, the bottom row displays the result for the colony diameters of approximately 193, 253, and $302\mu \mathrm{m}$ . For these sets, $Z_2$ was set to approximately $12\mathrm{mm}$ 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.

Fig. 7

Recorded forward scattering patterns against the colony diameter for Salmonella monteviedo. The diameter was varied from $89\text{to}302\mu \mathrm{m}$ .

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 $5\text{-}\mu \mathrm{m}$ interval²³ 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.

Fig. 8

Bacterial colony profile measurements: (a) the cross-sectional profile (vertical axis not to scale) for various size of bacterial colonies $\left(109\text{to}450\mu \mathrm{m}\right)$ , (b) the corresponding phase contrast images, and (c) the 2-D profile from CDM. (Note: second figure of (b) shows $187\mu \mathrm{m}$ horizontal diameter but the vertical diameter is measured to be $204\mu \mathrm{m}$ ).

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.

Fig. 9

Comparison of model and experimental result: (a) comparisons of the ring count versus colony diameter and (b) maximum half cone angle versus colony diameter. Red squares in (a) denote the estimated ring count from the $H_0$ measured from CDM using Eq. 16, while in (b) the derivative of phase is calculated from the profile and Eq. 17 was used for estimation. The green triangle represents the actual ring count from scatterograms (Fig. 7), while the blue circle shows the estimated ring count from the proposed model. (Color online only.)

Table 2

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

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 $\theta ∕2_{\max }$ as,

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

Fig. 10

Schematic diagram for understanding the interference pattern generated by the emerging wavefront: $\Delta \Phi$ represents the phase lag from the colony center thickness $H_0$ , ${\mathbf{k}}_1$ and ${\mathbf{k}}_2$ represent the wave vectors, and ${\delta }_1$ and ${\delta }_2$ represent the OPD difference of these two wave vectors.

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 $350\mu \mathrm{m}$ ), 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.