application of sampling criterion on numerical diffraction from bacterial colonies

$: Application of sampling criterion on numerical diffraction from bacterial colonies$
Application of sampling criterion on numericaldiffraction from bacterial colonies

Euiwon Bae,1,* Nan Bai,1 and E. Daniel Hirleman2

1School of Mechanical Engineering, Purdue University, West Lafayette, Indiana 47907, USA2School of Engineering, University of California Merced, Merced, California 95343, USA

*Corresponding author: [email protected]

Received 29 June 2010; revised 13 January 2011; accepted 21 January 2011;posted 21 January 2011 (Doc. ID 130923); published 18 May 2011

Numerical diffraction from a bacterial colony was investigated from the viewpoint of applying the sam-pling criterion for both spatial and frequency domains. Once the morphology information of a bacterialcolony was given, the maximum diffraction angle was estimated to reveal the minimum and maximumlength of both the imaging and aperture domains. Scalar diffractionmodeling was applied to estimate thediffraction pattern, which provided that two phase functions were contributing to the phase modulation:chirp and Gaussian phase functions. Optimal sampling intervals for both phase functions were inves-tigated, and the effect of violating these conditions was demonstrated. Finally, the Fresnel approximationwas compared to the angular spectrum method for accuracy and applicability, which then revealed thatthe Fresnel approximation was valid for both large imaging distances and longer wavelengths. © 2011Optical Society of AmericaOCIS codes: 050.5080, 070.6120, 070.7345, 120.4570, 290.2558.

1. Introduction

The numerical modeling and digital computation oflight propagation has been studied in various fieldsof optics and their applications. For example, Cucheet al. introduced digital holography through the useof a numerical reconstruction rather than a physicalreconstruction of a recorded hologram [1,2]. In es-sence, the theory of scalar beam propagation origi-nated from the Rayleigh–Sommerfeld diffractiontheory, and it could be further simplified to Fresnelor Fraunhofer diffractions with appropriate assump-tions [3]. Recently, a novel bacterial colony identifica-tion methodology based upon unique diffractionpatterns has been introduced [4,5]. With this tool, ex-perimental results were compared with theoreticalpredictions based on modeling bacterial colonies asbiological spatial light modulators. These examplescommonly involved the numerical calculation of lightpropagation in space through various methods

depending on specific assumptions, i.e., the Fresnelapproximation, convolution, and angular spectrummethod. For an efficient computation of the propaga-tion, discrete Fourier transform (DFT) via the fastFourier transform (FFT) technique was typically im-plemented rather than direct integration. This inevi-tably involved sampling both in the spatial andfrequency domains, a step that requires a carefulunderstanding and selection of parameters to avoidunwanted artifacts mixed with the real signals origi-nating from the samples.

At this point we investigated the sampling criter-ion of the numerical light propagation calculationwith application to the diffraction patterns frombacterial colonies. A bacterial colony is a three-dimensional (3D) dome shaped structure severalhundred micrometers in diameter. We modeled theoptical interactions between the incident light andthe colony via amplitude and phase modulations ofthe incoming wave by the physical morphology of thecolony [4,6]. Therefore, to estimate the initial diffrac-tion characteristics of the bacterial colony, it is criti-cal that one understands the 3D colony morphology.

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2228 APPLIED OPTICS / Vol. 50, No. 15 / 20 May 2011

This was reported by several authors regarding somespecies of yeast and Escherichia coli that demon-strate a Gaussian-like profile (a flat escalated centerarea with tailing edge) [6–9]. The effect of morphol-ogy, whether it is from colony thickness or biomater-ial heterogeneity (refractive indices), was summedup to an effective phase modulation of the incidentbeam. Several authors from nematic liquid crystalresearch provided a simple method of estimating themaximum diffraction angle once the input voltage forthe molecule was given [10–14]. Similarly, once theshape of the colony was known, we could estimatethe maximum diffraction angle, which, in turn, de-fined the minimum two-dimensional (2D) imagingarea to capture the whole pattern. To accuratelymodel light propagation from colonies, it was criticalto understand the origin of the phase modulationthat was causing the diffraction pattern, i.e., thechirp and Gaussian phase functions. The samplingrequirement and the result of violating this conditionfor the chirp function have been studied in depth byvarious authors; conclusions have been drawn indi-cating that an optimal sample interval was requiredto avoid aliasing either in space or frequency do-mains [15–20]. In addition to these works, we haveinvestigated the sampling requirements for theGaussian phase function originating from the mor-phology of the bacterial colony. Since the Fresnel cal-culation is an approximation via simplifying thespherical phase function with a quadratic term, wehave compared its numerical calculation results withthose obtained from the angular spectrum methodand shown the consequences of nonideal sampling in-tervals for the diffraction calculations from bacterialcolonies.

2. Sampling of Computational Domain

A. Diffraction Modeling

To calculate the diffraction field from an aperture,there were several well-known numerical methods,which included the Fresnel approximation, convolu-tion, and angular spectrum methods. In this sectionwe start with bacterial colony morphology to intro-duce the computational requirements based on theFresnel approximation formula. As shown in Fig. 1,the amplitude and phase of the incident beam can beexpressed as [4,5,21–23]

Eaðxa; ya; z1Þ ¼ E0 exp�−ðX2

a þ X2aÞ

w2ðz1Þ�

× exp½jkz1� exp�jk

ðX2a þ X2

aÞ2Rðz1Þ

�; ð1Þ

where E0 is the on-axis field strength and the threeexponential terms account for the variations of theamplitude of the field, longitudinal phase, and radialphase, respectively. wðz1Þ and Rðz1Þ are the beamwidth and radius of curvature of the wavefront atthe colony plane (z ¼ z1), which are defined by

w2ðzÞ ¼ w20

�1þ

�zz0

�2�; RðzÞ ¼ z

�1þ

�z0z

�2�;

ð2Þ

where z0 is defined as the z location where the 1=e2

radius has expanded toffiffiffi2

ptimes of the beam waist

w0 . Then, we apply the Huygens–Fresnel principlein rectangular coordinates. Using Eq. (1), the electricfield Ei at the imaging plane can be expressed as

EiðXi;YiÞ ¼1jλ

ZZΣEaðXa;YaÞ exp½jkðϕðXa;YaÞÞ�

× exp�jkrairai

�cos θdXadYa; ð3Þ

where λ is the wavelength and rai is the distance fromthe aperture to the imaging plane. Based on the Fres-nel diffraction theory, assuming its validity criteriaare met, the diffracted field on the image planecan be described as

EiðXi;YiÞ ≈C1

ZZΣexp

�−ðX2

aþY2aÞ

w2ðz1Þ�exp

�jkðX2

aþY2aÞ

2Rðz1Þ�

×exp�jkðX2

aþY2aÞ

2z2

�×exp½jkϕðXa;YaÞ�

×exp½−i2πðf xXaþ f yYaÞ�dXadYa; ð4Þ

where C1 is a constant that is not related to theintegration variables and f x and f y are the spatialfrequencies. If we drop constant C1 and rearrangeEq. (4), we get [4,21]

Fig. 1. (Color online) Coordinate definition for modeling diffrac-tion from a bacterial colony. Colony is located at the aperture plane(Xa, Ya) with a half-domain size of La. The incident beam is aGaussian beam with beam waist radius of wb, and the colony isassumed to have a Gaussian shape with a diameter of D, a centerheight of H0, and refractive index of n. The imaging plane (Xi, Yi)is located at a distance of z with a maximum half-diffraction angleof θ=2 and a half-domain size of Li.

20 May 2011 / Vol. 50, No. 15 / APPLIED OPTICS 2229

EiðXi;YiÞ ≈ZZ

Σt exp½jϕr� exp½jϕc� exp½jϕg�

× exp½−i2πðf xXa þ f yYaÞ�dXadYa; ð5Þ

tðXa;YaÞ ¼ exp�−ðX2

a þ Y2aÞ

w2ðz1Þ�; ð6Þ

ϕrðXa;YaÞ ¼ k

�ðX2a þ Y2

aÞ2Rðz1Þ

�; ð7Þ

ϕcðXa;YaÞ ¼ k

�ðX2a þ Y2

aÞ2z2

�; ð8Þ

ϕgðXa;YaÞ ¼ kðn1 − 1ÞH0 exp�−ðX2

a þ Y2aÞ

w2b

�: ð9Þ

If we break down Eq. (5), the amplitude is modifiedwith the Gaussian profile Eq. (6), while the phase hasthree contributing terms: the first [Eq. (7)] is the ra-dial phase term originating from the incident beamexpression [Eq. (1)], the second term [Eq. (8)] is thequadratic phase from the Fresnel approximation,and the third is the Gaussian phase component[Eq. (9)]. Since the radial phase term of Eq. (7) wasrelatively small compared to the other two phaseterms, we only concentrated on the quadratic andGaussian phase terms for the rest of this paper. Next,the amplitude- and phase-modulated light wave pro-pagated to the imaging plane to generate colony-profile-dependent diffraction rings.

To address the sampling condition for accuratecomputation, we first defined the grids of the aper-ture and imaging planes (Fig. 1). To estimate theupper and lower bounds, we started with a bacterialcolony of diameter D. Since previous experimentshave shown that the aspect ratio (diameter/centerheight) of a bacterial colony was approximately 10∶1on average, we defined the center heightH0 based onthis estimation and modeled the colony profile as aGaussian shape. In this case, the maximum diffrac-tion angle played a critical role in defining the ima-ging domain since the computational domain had toinclude the largest scattering component to displaythe correct diffraction patterns. To provide an esti-mate, we applied the following formula [11,12]:

θ=2max ≅1k

�dΔΦdr

�max

; ð10Þ

whereΔΦ was the phase difference across the radialdirection from ϕg and k was the wavenumber. Since

H0 was known, Eq. (10) was estimated and yieldedthe minimum computation domain size requiredfor the imaging plane as

Li ≥ z tan−1ðθ=2Þmax; ð11Þ

where Li was the one-sided length of the imagingplane and z was the imaging distance from the aper-ture. Following the aperture and imaging plane rela-tionship [3], we estimated the minimum samplingfrequency required for the aperture plane via

FHS ≥Li

λz ; ð12Þ

where FHS was the half-sampling frequency for theaperture plane. When we defined the one-sidedlength of the aperture domain as La, Eq. (12) wasrelated to La as

La ≤NFHS

; ð13Þ

where N=2 was the number of samples for one side.Therefore, we could conclude that, for a given bacter-ial diameter D, the whole aperture domain lengthshould satisfy

2wz ≤ 2La ≤2NFHS

; ð14Þ

where 2wz was the incident beam diameter.

B. Sampling of Chirp and Gaussian Phase Functions

As shown in Eq. (5), the diffraction calculation from acolony consisted of two different phase functions. Thefirst was the chirp function contributed from thelight propagation across the imaging distance, andthe second was the Gaussian phase function arisingfrom the bacterial colony profile. The samplingrequirement for unaliased chirp function has beenreported as [20]

dxc ¼λz2La

; ð15Þ

where undersampling or oversampling in spacecoordinates generated computational artifacts. Inaddition, we derived the optimal sampling require-ment for the Gaussian phase function. Since theGaussian phase term [Eq. (9)] was computed inthe complex exponential form, the unaliased sam-pling condition was satisfied when

dxgjdΔΦdr

j ≤ π; ð16Þ

where dxg was the optimal sampling interval for theGaussian phase function. Further simplification ofEq. (16) resulted in


dxg ≤π��kðn1 − 1ÞH0

2xw2

bexp

�−

x2

w2b

��max

; ð17Þ

where n1 was the refractive index and wb was theGaussian beam radius for the bacterial colony.Therefore, given λ, z, and La, we could compute theoptimal sampling interval for the impulse responsefunction and the Gaussian phase function. Therefore,to avoid aliasing for both the chirp and Gaussianphase functions, we had to strategically designatedxc and dxg values. Since the bacterial colony shapewas a physical quantity that acts as a constraintparameter, we must first compute dxg as Eq. (17)and then set dxc as the same value. Another condi-tion that had to be considered was that all para-meters had to satisfy the whole period of the complexexponential terms of Eq. (1) to avoid spectral leak-age. Therefore, the following three equations haveto be satisfied for the optimal sampling and thecorrect periodicity:

dxg ¼λz2La

; ð18Þ

hðxÞ ¼ exp�ik

L2a

2z

�; ð19Þ

�2πλL2a

2z

�¼ 2πP; ð20Þ

where P is an integer.

3. Results

A. Effect of Colony Shape on the Computational Domain

To correlate colony shape with the optimal settingsfor the computational domain, Eqs. (5)–(14) were ap-plied to estimate the effect of such a shape on the dif-fraction calculation. Figure 2 displays a series of 2Ddiffraction patterns predicted for the given morphol-ogy parameters of a bacterial colony. Figure 2(a)shows the diffraction pattern from a D ¼ 0:35mmand H0 ¼ 0:035mm colony, in which aliasing oc-curred with high-frequency components folded backat the edge of the computational domain. Once thebacterial colony diameter was given, the maximumhalf-diffraction angle [Eq. (10)] was determined,which also resulted in the minimum imaging planesize (Li). This was again correlated to the maximumhalf-frequency of the aperture plane [Eq. (12)], which

Fig. 2. (Color online) Effect of colony morphology on the minimum value of the one-sided length Li of the imaging plane. (a) Aliasing casewhere the imaging plane is not large enough to capture the high-frequency terms. This can be avoided either via (b) decreasing La (withconstant N) or (c) decreasing dx.


determined the maximum aperture plane size[Eq. (13)]. According to Eq. (10), the maximum half-diffraction angle with z ¼ 12mm estimated that theminimum imaging plane size of 1:32mm, the mini-mum half-frequency band of 174 lines=mm, and themaximum aperture domain size of 0:73mm wereneeded to capture the largest angular diffractionrings without aliasing. As shown in Table 1, however,for case (a), all Li sim, FHSsim, and La sim did not satisfythe minimum and maximum values estimated pre-viously. Figures 2(b) and 2(c) display the same pat-tern while avoiding this aliasing problem. Thiswas possible either via decreasing La (the aperturedomain size) or via increasing N (equivalentlydecreasing dx).

B. Effect of Sampling Intervals

Since Fresnel diffraction was computed via DFT, wehave investigated the effect of different sampling in-

tervals on the magnitude and phase plots in both thespatial and frequency domains. Figures 3(a) and 3(b)show the Gaussian phase profile expðikΦgÞ) withundersampling in spatial and frequency domains,while Figs. 3(c) and 3(d) display the oversampledcase. Since the Gaussian phase function, kΦg, wascomputed via a complex exponential, the optimalsampling interval (dxg) was obtained via Eq. (17).When the Gaussian phase function was 25% and50% undersampled from dx0, the unwrapped phasefunction in the spatial domain did not follow the ori-ginal Gaussian profile [Fig. 3(a)], which in turn re-sulted in an aliasing in the phase plot of the DFT ofthe Gaussian profile in the frequency domain. How-ever, the oversampling case displayed no deviationfrom the optimal case in either spatial or frequencydomains.

According to Eq. (1), the diffraction calculation wasperformed via multiplying the chirp with the Gaus-sian phase function before DFT. Therefore, it wasworthwhile to investigate the influence of the sam-pling interval for both chirp and Gaussian phasefunctions. Figure 4 displays the magnitude and un-wrapped phase for chirp [Figs. 4(a) and 4(b)] andGaussian [Figs. 4(c) and 4(d)] phase functions. First,the plots of the DFT of the chirp phase functionshowed severe oscillations in magnitude for eitherundersampled or oversampled intervals, which con-firmed the previously published results [20]. The un-wrapped phase of the chirp function showed phase

Fig. 3. (Color online) Unwrapped phase function plots of the Gaussian phase profile and their DFT phase plots. (a) Effect of undersam-pling in space coordinates; (b) phase plot of the DFT of (a). (c) Oversampling in space coordinates; (d) phase plot of the DFT of (c).

Table 1. Simulation Parameters for Aliased and Normal Cases

Unit Case (a) Case (b) Case (c)

ðdΔΦdr Þmax rad=mm 1100 1100 1100

Li mm 1.32 1.32 1.32Li sim mm 0.97 1.38 1.94FHS line=mm 174 174 174FHS;SIK line=mm 127 182 255La mm 0.73 0.73 1.46La sim mm 1.00 0.70 1.00Nsim - 512 512 1024


deviation especially for the undersampled case.Second, the magnitude of the DFT of the Gaussianphase function for the undersampled case illustratedsignificant deviation from the optimal case, while theoversampled case was similar to the result with theoptimal sampling interval. The phase of the DFT ofthe Gaussian phase deviated from that of the opti-mal case.

C. Effect of Period

The effect of using the whole or partial periodic func-tion was formulated by Eqs. (18)–(20). While Eq. (18)stated that the sampling intervals of the Gaussianand chirp functions had to be identical for an optimalsampling, Eq. (20) stated that the physical para-meters (L, z, λ) had to be carefully designated suchthat the whole periodicity was utilized for the DFT,otherwise it would generate spectral leakage frommodeling the period differently. If Eqs. (18) and(20) were satisfied simultaneously, there was an in-teger P such that L and z are formulated as a func-tion of the Gaussian function sampling interval dxg,integer P, and wavelength λ. When this condition wasviolated, the ratio of L2

a=ðλ2zÞ was a noninteger thusgenerating artifacts in the DFT calculation. Figure 5displays the comparison of three ratios for L2

a=ðλ2zÞ:Poptimalð60Þ, P1ð60:46Þ, and P2ð60:70Þ. Figures 5(a)and 5(b) display the effect of periodicity on theDFT of the chirp phase function while Figs. 5(c)

and 5(d) show the effect on the Gaussian phase func-tion. For the chirp phase function, the cases P1 andP2 showed significant oscillations in magnitude butsimilar phase plots. For the Gaussian phase function,both the magnitude and phase plots were similar toeach other.

D. Comparison of Diffraction Patterns

To better understand the artifacts that can be intro-duced from the discussions above, we computed aseries of diffraction patterns with optimal and sub-optimal sampling intervals with both the Fresnelapproximation (FA) and angular spectrum (AS)methods. Figure 6 shows the results for commonparameters of La ¼ 1:6mm, z ¼ 12mm, H0 ¼0:025mm, and λ ¼ 0:635 μm. Figures 6(a)–6(d) dis-play the result for FA without Gaussian windows forundersampling (N ¼ 1024, dx ¼ 0:00313mm),optimal (N ¼ 1344, dx ¼ 0:00238mm), and oversam-pling (N ¼ 1800, dx ¼ 0:00178mm) in space coordi-nates. Figure 6(a) shows the undersampled casewhere the overestimated phase function dominatedthe imaging plane and generated a significantly dif-ferent pattern. The one-dimensional (1D) compari-son of Fig. 6(d) illustrates a severe oscillationbeyond the Xi ∼ 1mm location. Figure 6(b) illus-trates the optimal interval case, which shows thediffraction pattern from the colony along with diffrac-tions originating from the edge of the computational

Fig. 4. (Color online) (a), (b) Magnitude and phase plots of the chirp function and (c), (d) the Gaussian phase function with undersampling(1.25 dx0), optimal (dx0), and oversampling (0.75 dx0) cases.


domain. Figure 6(c) displays the result for the over-sampling case, which shows an effect of band cutoff inthe 1:5 ≤ Xi ≤ 2:2mm range in Fig. 6(d). Once we in-cluded the Gaussian apodization (window), all threecases were merged to the same diffraction pattern asshown in Figs. 6(e)–6(h).

The same cases were also computed with the ASmethod as shown in Fig. 7. Figures 7(a)–7(c) showthe effect of undersampling, optimal, and oversam-pling cases. Figure 7(a) shows the predicted diffrac-tion pattern that displays a larger diffraction anglethan the optimal case, while Fig. 7(c) shows a smallerone. Figure 7(d) displays the 1D comparison of theinverse Fourier transform (IFT) of the spatial propa-gation term (the chirp phase), which thereforeparallels the similar characteristics in Fig. 4(a).Corresponding 1D diffraction patterns were com-bined in Fig. 7(e), illustrating deviations dependenton the sample interval dx. The results from the FAshowed the similar patterns across a different num-ber of grid points, while the AS method showed sig-nificantly different diffraction patterns dependent onthe dx value. While the optimal dx for the AS methodshowed similar patterns with that computed fromthe FA, the undersampled case (b) generated aspatially larger diffraction pattern (equivalently pre-dicted higher spatial frequencies) while the over-sampled case showed a smaller diffraction pattern(equivalently predicted lower spatial frequencies).

To compare the effect of the imaging distance z, thetwo models were calculated with 6 ≤ z ≤ 20mm.Figure 8(a) displays the difference between thesetwo methods where the deviation was calculatedvia the summed difference of power Pdiff defined as

Pdiff ¼P jPFA � PASjP

PAS; ð21Þ

where PFA and PAS were the normalized powers forthe FA and AS methods for three different wave-lengths (solid curve, 500nm; dotted curve, 635nm;dashed curve, 800nm). Figure 8(b) displays the com-parison of the aperture and imaging domain size ofthe two methods. The AS (curves with circles) meth-od demonstrated the same domain size for both theaperture and imaging domains irrespective of z,while the FA can extend the imaging domain sizefor a larger distance without causing aliasing.

4. Discussions

With the recent findings of diffraction-based bacter-ial colony identification, computational light propa-gation had not been thoroughly linked between thecolony morphology and the required optical simula-tion parameters. First, a contribution of this work in-volved the analytical estimation of the maximumdiffraction angle from the bacterial colony diameter.Since the colony diameter was one of the most

Fig. 5. (Color online) (a), (b) Magnitude and phase plots of the chirp function and (c), (d) the Gaussian phase function with nonideal (P1,P2) and ideal (Poptimal) values.


pronounced physical traits during the growth of thebacteria, relating this parameter to the imaging do-main and subsequently to the aperture domain wasimportant to model light propagation without arti-facts. Assuming a Gaussian profile with an aspect ra-tio of 10∶1, we could estimate the appropriatesampling conditions [Eqs. (10)–(14)] for the apertureplane with N number of data points. If given compu-tational parameters violate Eqs. (5)–(8), aliasing is-sues could be avoided by either increasing N ordecreasing La.

A second contribution included the optimal sam-pling condition for the phase functions involved inmodeling light propagation from bacterial colonies.The FA of Eq. (5) clearly displayed two major phasecomponents contributing to the light propagation,which were the chirp function and the colony phasefunction (modeled as a Gaussian shape). Similar tothe optimal sampling requirement for the chirp func-tion, Eqs. (16) and (17) provided requirements forsampling the Gaussian phase function without alias-

ing. A careful investigation of the analytical expres-sion showed that the upper bound for the samplinginterval was governed by the maximum slope of thephase profile, which was directly correlated to the as-pect ratio of the colony shape. In other words, a nar-row and high colony (largeH0 and smallwb) requireda smaller sampling interval rather than a wide andlow colony (smallH0 and largewb) for the given aper-ture domain parameter La. This qualitative under-standing was supported by Eq. (17) quantitativelyfor the given simulation conditions.

Even though there are different formulationsavailable for successfully modeling light propaga-tion, most of these varied computations implementDFT with the FFT technique. This induced somesampling requirements in both spatial and frequencydomains, since the FA, AS method, and directconvolution require one, two, and three FFT opera-tions. If we modeled the experimental setup withoutconsidering the effect of parameters on the computa-tional domain, this could lead to computational

Fig. 6. Comparison of the predicted diffraction patterns via the FA with and without a Gaussian window. (a) N ¼ 1024, (b) N ¼ 1344,(c) N ¼ 1800 without the Gaussian window before DFT. (d) 1D plot across Yi ¼ 0 for (a)–(c). (f) N ¼ 1024, (g) N ¼ 1344, (h) N ¼ 1800 withthe Gaussian window before DFT. (e) 1D plot cross Yi ¼ 0 for (f)–(h). Common variables are La ¼ 1:6mm,H0 ¼ 0:025mm, and z ¼ 12mm.Optimal value [λz=ð2LaÞ] for the sampling interval dx is 0.00238 while (a) and (f) were computed with 0.00313 (undersampled in space), (b)and (g) with 0.00238, and (c) and (h) with 0.00178 (oversampled in space).


artifacts. Equation (19) represents a chirp function ina 1D form with the FA. The argument of Eq. (14) hadto satisfy the whole period requirement with givenphysical parameters La, λ, and z, where an integerPð¼ L2

a=ðλ2zÞÞ should exist. Figure 5 displays theconditions for modeling the whole period of the phasemodulation with P1 and P2, which were deviationsfrom the optimal value P. Even though the magni-tude of the chirp function showed oscillations overall the frequency range with more than 50% occur-ring at the higher frequencies, the phase of the chirpand the magnitude and phase of the Gaussian phasefunction did not vary depending on different Pvalues.

Figure 6 shows the results of the predicted diffrac-tion patterns from the FA with and without a Gaus-sian window. In physical measurements, a Gaussianlaser beam of ∼1mm was used, which validated theneed of the Gaussian intensity modulation in Eq. (5)before two phase terms. The effect of Gaussian apo-dization was clearly shown in Figs. 6(a)–6(d) wherethe undersampling (N ¼ 1024) case in Figs. 6(a) and

6(d) showed overshooting of intensity, while the over-sampling (N ¼ 1800) resulted in spatial frequencycutoff (Xi ≥ 1:5mm). The optimal sampling intervalhas shown diffraction patterns with additional dif-fractions generated from the incident light and thecomputational edges. As shown in Figs. 6(e)–6(h),this issue was resolved once the Gaussian apodiza-tion term [expð−x2=w2

z Þ] was multiplied into Eq. (5)before DFT, which also correctly modeled the actualphysical phenomenon. Figures 7(a)–7(c) show signif-icant deviations from the optimal interval of case (b).Undersampling generated oscillation in magnitude[Fig. 7(d)] for Xi ≥ 1mm while oversampling showedfluctuation and frequency cutoff for Xi ≥ 1:2mm,which resulted in different diffraction patterns[Fig. 7(e)]. This could be understood from the devia-tion of the two phase functions with sampling inter-vals above and below the optimal value as shown inFig. 4. Both the magnitude and phase of the chirpand Gaussian phase functions with nonoptimal sam-pling displayed amplitude oscillations or phase lag/lead compared to the plots with the optimal one.

Fig. 7. Predicted diffraction patterns via the AS method: (a) N ¼ 1024, (b) N ¼ 1344, (c) N ¼ 1800. Common variables are La ¼ 1:6mm,H0 ¼ 0:025mm, and z ¼ 12mm. Optimal value [λz=ð2LaÞ] for the sampling interval dx is 0.00238, while (a) was computed with 0.00313(undersampled in space), (b) with 0.00238, and (c) with 0.00178 (oversampled in space). (d) 1D plot of the IFT of the spatial propagationterm for three cases, which shows oscillations (N ¼ 1024) or cutoff (N ¼ 800) for higher-frequency components. (e) Resulting 1D diffractionpattern that shows peak locations at the imaging plane.


To access the accuracy and applicability of FA andAS, we compared the difference of the predicteddiffractions against variable distance z. The resultFig. 8(a) indicated that as z increased and λ de-creased, the difference between FA and ASincreased. For applicability, the FA method couldmodel the diffraction from a larger imaging distancez, while the AS method had to apply the same ima-ging domain size as the aperture domain, which wasinefficient for a larger z.

5. Conclusion

To conduct an effective light propagation modelingarising from bacterial colonies, we have investigatedconditions for diffraction modeling without aliasing

based on the physical morphology of the bacterial col-ony. The effect of the sampling interval for both thechirp and Gaussian phase functions was investigatedand the analytical expression for the optimal sam-pling interval was provided. The DFT operation,which is sensitive to the periodicity of the input spa-tial function, was also investigated for the aperturedomain size and colony-imaging plane distance to sa-tisfy the whole period requirement to avoid artifactsin the frequency domain. Finally, the above discus-sions were applied to a series of calculations ofdiffraction modeling both for the FA and the ASmethod, which provided for a growing fundamentalunderstanding of the sampling requirements forlight propagation models.

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Fig. 8. Comparison of FA and AS for different wavelengths.(a) Pdiff versus the imaging distance z. (b) Imaging domain size ver-sus distance z. For AS, the aperture domain and imaging domainsize remain the same, while FA can be increased.


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application of sampling criterion on numerical diffraction from bacterial colonies

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