Abstract

Numerical diffraction from a bacterial colony was investigated from the viewpoint of applying the sampling criterion for both spatial and frequency domains. Once the morphology information of a bacterial colony was given, the maximum diffraction angle was estimated to reveal the minimum and maximum length of both the imaging and aperture domains. Scalar diffraction modeling was applied to estimate the diffraction 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 investigated, and the effect of violating these conditions was demonstrated. Finally, the Fresnel approximation was compared to the angular spectrum method for accuracy and applicability, which then revealed that the Fresnel approximation was valid for both large imaging distances and longer wavelengths.

© 2011 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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2010 (1)

E. Bae, N. Bai, A. Aroonnual, J. P. Robinson, A. K. Bhunia, and E. D. Hirleman, “Modeling light propagation through bacterial colonies and its correlation with forward scattering patterns,” J. Biomed. Opt. 15, 045001 (2010).
[CrossRef] [PubMed]

2009 (1)

2008 (1)

E. Bae, P. P. Banada, K. Huff, A. K. Bhunia, J. P. Robinson, and E. D. Hirleman, “Analysis of time-resolved scattering from macroscale bacterial colonies,” J. Biomed. Opt. 13, 014010(2008).
[CrossRef] [PubMed]

2007 (2)

2006 (1)

A. Brodzik, “On the Fourier transform of finite chirps,” IEEE Signal Process. Lett. 13, 541–544 (2006).
[CrossRef]

2004 (2)

L. Onural, “Some mathematical properties of the uniformly sampled quadratic phase function and associated issues in digital Fresnel diffraction simulations,” Opt. Eng. 43, 2557–2563 (2004).
[CrossRef]

A. Stern and B. Javidi, “Analysis of practical sampling and reconstruction from Fresnel fields,” Opt. Eng. 43, 239–250 (2004).
[CrossRef]

2000 (1)

1999 (2)

1998 (1)

S. H. Chen, J. Y. Fan, and J. J. Wu, “Novel diffraction phenomenon with the first order Fredericksz transition in a nematic liquid crystal film,” J. Appl. Phys. 83, 1337–1340 (1998).
[CrossRef]

1992 (1)

R. P. Pan, S. M. Chen, and C. L. Pan, “A quantitative study of the far-field laser-induced ring pattern from nematic liquid-crystal films,” Chin. J. Phys. 30, 457–466 (1992).

1988 (2)

1987 (1)

1984 (1)

1981 (1)

M. W. A. Lewis and J. W. T. Wimpenny, “The influence of nutrition and temperature on the growth of colonies of Escherichia coli K12,” Can. J. Microbiol. 27, 679–684 (1981).
[CrossRef] [PubMed]

1979 (1)

J. W. Wimpenny, “Growth and form of bacterial colonies,” J. Gen. Microbiol. 114, 483–486 (1979).
[PubMed]

Aroonnual, A.

E. Bae, N. Bai, A. Aroonnual, J. P. Robinson, A. K. Bhunia, and E. D. Hirleman, “Modeling light propagation through bacterial colonies and its correlation with forward scattering patterns,” J. Biomed. Opt. 15, 045001 (2010).
[CrossRef] [PubMed]

N. Bai, E. Bae, A. Aroonnual, A. K. Bhunia, J. P. Robinson, and E. D. Hirleman, “Development of a real-time system of monitoring bacterial colony growth and registering the forward-scattering pattern,” in Sensing for Agriculture and Food Quality and Safety (SPIE, 2009), pp. 73150Z–73158.

Bae, E.

E. Bae, N. Bai, A. Aroonnual, J. P. Robinson, A. K. Bhunia, and E. D. Hirleman, “Modeling light propagation through bacterial colonies and its correlation with forward scattering patterns,” J. Biomed. Opt. 15, 045001 (2010).
[CrossRef] [PubMed]

E. Bae, P. P. Banada, K. Huff, A. K. Bhunia, J. P. Robinson, and E. D. Hirleman, “Analysis of time-resolved scattering from macroscale bacterial colonies,” J. Biomed. Opt. 13, 014010(2008).
[CrossRef] [PubMed]

E. Bae, P. P. Banada, K. Huff, A. K. Bhunia, J. P. Robinson, and E. D. Hirleman, “Biophysical modeling of forward scattering from bacterial colonies using scalar diffraction theory,” Appl. Opt. 46, 3639–3648 (2007).
[CrossRef] [PubMed]

N. Bai, E. Bae, A. Aroonnual, A. K. Bhunia, J. P. Robinson, and E. D. Hirleman, “Development of a real-time system of monitoring bacterial colony growth and registering the forward-scattering pattern,” in Sensing for Agriculture and Food Quality and Safety (SPIE, 2009), pp. 73150Z–73158.

Bai, N.

E. Bae, N. Bai, A. Aroonnual, J. P. Robinson, A. K. Bhunia, and E. D. Hirleman, “Modeling light propagation through bacterial colonies and its correlation with forward scattering patterns,” J. Biomed. Opt. 15, 045001 (2010).
[CrossRef] [PubMed]

N. Bai, E. Bae, A. Aroonnual, A. K. Bhunia, J. P. Robinson, and E. D. Hirleman, “Development of a real-time system of monitoring bacterial colony growth and registering the forward-scattering pattern,” in Sensing for Agriculture and Food Quality and Safety (SPIE, 2009), pp. 73150Z–73158.

Banada, P. P.

E. Bae, P. P. Banada, K. Huff, A. K. Bhunia, J. P. Robinson, and E. D. Hirleman, “Analysis of time-resolved scattering from macroscale bacterial colonies,” J. Biomed. Opt. 13, 014010(2008).
[CrossRef] [PubMed]

E. Bae, P. P. Banada, K. Huff, A. K. Bhunia, J. P. Robinson, and E. D. Hirleman, “Biophysical modeling of forward scattering from bacterial colonies using scalar diffraction theory,” Appl. Opt. 46, 3639–3648 (2007).
[CrossRef] [PubMed]

Bevilacqua, F.

Bhunia, A. K.

E. Bae, N. Bai, A. Aroonnual, J. P. Robinson, A. K. Bhunia, and E. D. Hirleman, “Modeling light propagation through bacterial colonies and its correlation with forward scattering patterns,” J. Biomed. Opt. 15, 045001 (2010).
[CrossRef] [PubMed]

E. Bae, P. P. Banada, K. Huff, A. K. Bhunia, J. P. Robinson, and E. D. Hirleman, “Analysis of time-resolved scattering from macroscale bacterial colonies,” J. Biomed. Opt. 13, 014010(2008).
[CrossRef] [PubMed]

E. Bae, P. P. Banada, K. Huff, A. K. Bhunia, J. P. Robinson, and E. D. Hirleman, “Biophysical modeling of forward scattering from bacterial colonies using scalar diffraction theory,” Appl. Opt. 46, 3639–3648 (2007).
[CrossRef] [PubMed]

N. Bai, E. Bae, A. Aroonnual, A. K. Bhunia, J. P. Robinson, and E. D. Hirleman, “Development of a real-time system of monitoring bacterial colony growth and registering the forward-scattering pattern,” in Sensing for Agriculture and Food Quality and Safety (SPIE, 2009), pp. 73150Z–73158.

Bloisi, F.

Brodzik, A.

A. Brodzik, “On the Fourier transform of finite chirps,” IEEE Signal Process. Lett. 13, 541–544 (2006).
[CrossRef]

Bungay, H. R.

R. S. Kamath and H. R. Bungay, “Growth of yeast colonies on solid media,” J. Gen. Microbiol. 134, 3061–3069 (1988).
[PubMed]

Chen, S. H.

S. H. Chen, J. Y. Fan, and J. J. Wu, “Novel diffraction phenomenon with the first order Fredericksz transition in a nematic liquid crystal film,” J. Appl. Phys. 83, 1337–1340 (1998).
[CrossRef]

Chen, S. M.

R. P. Pan, S. M. Chen, and C. L. Pan, “A quantitative study of the far-field laser-induced ring pattern from nematic liquid-crystal films,” Chin. J. Phys. 30, 457–466 (1992).

Cipparrone, G.

Cuche, E.

Depeursinge, C.

Fan, J. Y.

S. H. Chen, J. Y. Fan, and J. J. Wu, “Novel diffraction phenomenon with the first order Fredericksz transition in a nematic liquid crystal film,” J. Appl. Phys. 83, 1337–1340 (1998).
[CrossRef]

Finn, G. M.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

Hirleman, E. D.

E. Bae, N. Bai, A. Aroonnual, J. P. Robinson, A. K. Bhunia, and E. D. Hirleman, “Modeling light propagation through bacterial colonies and its correlation with forward scattering patterns,” J. Biomed. Opt. 15, 045001 (2010).
[CrossRef] [PubMed]

E. Bae, P. P. Banada, K. Huff, A. K. Bhunia, J. P. Robinson, and E. D. Hirleman, “Analysis of time-resolved scattering from macroscale bacterial colonies,” J. Biomed. Opt. 13, 014010(2008).
[CrossRef] [PubMed]

E. Bae, P. P. Banada, K. Huff, A. K. Bhunia, J. P. Robinson, and E. D. Hirleman, “Biophysical modeling of forward scattering from bacterial colonies using scalar diffraction theory,” Appl. Opt. 46, 3639–3648 (2007).
[CrossRef] [PubMed]

N. Bai, E. Bae, A. Aroonnual, A. K. Bhunia, J. P. Robinson, and E. D. Hirleman, “Development of a real-time system of monitoring bacterial colony growth and registering the forward-scattering pattern,” in Sensing for Agriculture and Food Quality and Safety (SPIE, 2009), pp. 73150Z–73158.

Hou, J. Y.

Huff, K.

E. Bae, P. P. Banada, K. Huff, A. K. Bhunia, J. P. Robinson, and E. D. Hirleman, “Analysis of time-resolved scattering from macroscale bacterial colonies,” J. Biomed. Opt. 13, 014010(2008).
[CrossRef] [PubMed]

E. Bae, P. P. Banada, K. Huff, A. K. Bhunia, J. P. Robinson, and E. D. Hirleman, “Biophysical modeling of forward scattering from bacterial colonies using scalar diffraction theory,” Appl. Opt. 46, 3639–3648 (2007).
[CrossRef] [PubMed]

Javidi, B.

A. Stern and B. Javidi, “Analysis of practical sampling and reconstruction from Fresnel fields,” Opt. Eng. 43, 239–250 (2004).
[CrossRef]

Kamath, R. S.

R. S. Kamath and H. R. Bungay, “Growth of yeast colonies on solid media,” J. Gen. Microbiol. 134, 3061–3069 (1988).
[PubMed]

Khoo, I. C.

Lewis, M. W. A.

M. W. A. Lewis and J. W. T. Wimpenny, “The influence of nutrition and temperature on the growth of colonies of Escherichia coli K12,” Can. J. Microbiol. 27, 679–684 (1981).
[CrossRef] [PubMed]

Liu, T. H.

Marquet, P.

Michael, R. R.

Onural, L.

L. Onural, “Exact analysis of the effects of sampling of the scalar diffraction field,” J. Opt. Soc. Am. A 24, 359–367 (2007).
[CrossRef]

L. Onural, “Some mathematical properties of the uniformly sampled quadratic phase function and associated issues in digital Fresnel diffraction simulations,” Opt. Eng. 43, 2557–2563 (2004).
[CrossRef]

L. Onural, “Sampling of the diffraction field,” Appl. Opt. 39, 5929–5935 (2000).
[CrossRef]

Pan, C. L.

R. P. Pan, S. M. Chen, and C. L. Pan, “A quantitative study of the far-field laser-induced ring pattern from nematic liquid-crystal films,” Chin. J. Phys. 30, 457–466 (1992).

Pan, R. P.

R. P. Pan, S. M. Chen, and C. L. Pan, “A quantitative study of the far-field laser-induced ring pattern from nematic liquid-crystal films,” Chin. J. Phys. 30, 457–466 (1992).

Robinson, J. P.

E. Bae, N. Bai, A. Aroonnual, J. P. Robinson, A. K. Bhunia, and E. D. Hirleman, “Modeling light propagation through bacterial colonies and its correlation with forward scattering patterns,” J. Biomed. Opt. 15, 045001 (2010).
[CrossRef] [PubMed]

E. Bae, P. P. Banada, K. Huff, A. K. Bhunia, J. P. Robinson, and E. D. Hirleman, “Analysis of time-resolved scattering from macroscale bacterial colonies,” J. Biomed. Opt. 13, 014010(2008).
[CrossRef] [PubMed]

E. Bae, P. P. Banada, K. Huff, A. K. Bhunia, J. P. Robinson, and E. D. Hirleman, “Biophysical modeling of forward scattering from bacterial colonies using scalar diffraction theory,” Appl. Opt. 46, 3639–3648 (2007).
[CrossRef] [PubMed]

N. Bai, E. Bae, A. Aroonnual, A. K. Bhunia, J. P. Robinson, and E. D. Hirleman, “Development of a real-time system of monitoring bacterial colony growth and registering the forward-scattering pattern,” in Sensing for Agriculture and Food Quality and Safety (SPIE, 2009), pp. 73150Z–73158.

Roggemann, M. C.

Santamato, E.

Shen, Y. R.

Siegman, A. E.

A. E. Siegman, Lasers (University Science, 1986).

Simoni, F.

Stern, A.

A. Stern and B. Javidi, “Analysis of practical sampling and reconstruction from Fresnel fields,” Opt. Eng. 43, 239–250 (2004).
[CrossRef]

Umeton, C.

Verdeyen, J. T.

J. T. Verdeyen, Laser Electronics (Prentice-Hall, 2000).

Vicari, L.

Voelz, D. G.

Wimpenny, J. W.

J. W. Wimpenny, “Growth and form of bacterial colonies,” J. Gen. Microbiol. 114, 483–486 (1979).
[PubMed]

Wimpenny, J. W. T.

M. W. A. Lewis and J. W. T. Wimpenny, “The influence of nutrition and temperature on the growth of colonies of Escherichia coli K12,” Can. J. Microbiol. 27, 679–684 (1981).
[CrossRef] [PubMed]

Wu, J. J.

S. H. Chen, J. Y. Fan, and J. J. Wu, “Novel diffraction phenomenon with the first order Fredericksz transition in a nematic liquid crystal film,” J. Appl. Phys. 83, 1337–1340 (1998).
[CrossRef]

Yan, P. Y.

Appl. Opt. (4)

Can. J. Microbiol. (1)

M. W. A. Lewis and J. W. T. Wimpenny, “The influence of nutrition and temperature on the growth of colonies of Escherichia coli K12,” Can. J. Microbiol. 27, 679–684 (1981).
[CrossRef] [PubMed]

Chin. J. Phys. (1)

R. P. Pan, S. M. Chen, and C. L. Pan, “A quantitative study of the far-field laser-induced ring pattern from nematic liquid-crystal films,” Chin. J. Phys. 30, 457–466 (1992).

IEEE Signal Process. Lett. (1)

A. Brodzik, “On the Fourier transform of finite chirps,” IEEE Signal Process. Lett. 13, 541–544 (2006).
[CrossRef]

J. Appl. Phys. (1)

S. H. Chen, J. Y. Fan, and J. J. Wu, “Novel diffraction phenomenon with the first order Fredericksz transition in a nematic liquid crystal film,” J. Appl. Phys. 83, 1337–1340 (1998).
[CrossRef]

J. Biomed. Opt. (2)

E. Bae, P. P. Banada, K. Huff, A. K. Bhunia, J. P. Robinson, and E. D. Hirleman, “Analysis of time-resolved scattering from macroscale bacterial colonies,” J. Biomed. Opt. 13, 014010(2008).
[CrossRef] [PubMed]

E. Bae, N. Bai, A. Aroonnual, J. P. Robinson, A. K. Bhunia, and E. D. Hirleman, “Modeling light propagation through bacterial colonies and its correlation with forward scattering patterns,” J. Biomed. Opt. 15, 045001 (2010).
[CrossRef] [PubMed]

J. Gen. Microbiol. (2)

J. W. Wimpenny, “Growth and form of bacterial colonies,” J. Gen. Microbiol. 114, 483–486 (1979).
[PubMed]

R. S. Kamath and H. R. Bungay, “Growth of yeast colonies on solid media,” J. Gen. Microbiol. 134, 3061–3069 (1988).
[PubMed]

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (2)

Opt. Eng. (2)

L. Onural, “Some mathematical properties of the uniformly sampled quadratic phase function and associated issues in digital Fresnel diffraction simulations,” Opt. Eng. 43, 2557–2563 (2004).
[CrossRef]

A. Stern and B. Javidi, “Analysis of practical sampling and reconstruction from Fresnel fields,” Opt. Eng. 43, 239–250 (2004).
[CrossRef]

Opt. Lett. (2)

Other (4)

N. Bai, E. Bae, A. Aroonnual, A. K. Bhunia, J. P. Robinson, and E. D. Hirleman, “Development of a real-time system of monitoring bacterial colony growth and registering the forward-scattering pattern,” in Sensing for Agriculture and Food Quality and Safety (SPIE, 2009), pp. 73150Z–73158.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

A. E. Siegman, Lasers (University Science, 1986).

J. T. Verdeyen, Laser Electronics (Prentice-Hall, 2000).

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Figures (8)

Fig. 1
Fig. 1

Coordinate definition for modeling diffraction from a bacterial colony. Colony is located at the aperture plane ( X a , Y a ) with a half-domain size of L a . The incident beam is a Gaussian beam with beam waist radius of w b , and the colony is assumed to have a Gaussian shape with a diameter of D, a center height of H 0 , and refractive index of n. The imaging plane ( X i , Y i ) is located at a distance of z with a maximum half-diffraction angle of θ / 2 and a half-domain size of L i .

Fig. 2
Fig. 2

Effect of colony morphology on the minimum value of the one-sided length L i of the imaging plane. (a) Aliasing case where the imaging plane is not large enough to capture the high-frequency terms. This can be avoided either via (b) decreasing L a (with constant N) or (c) decreasing d x .

Fig. 3
Fig. 3

Unwrapped phase function plots of the Gaussian phase profile and their DFT phase plots. (a) Effect of undersampling in space coordinates; (b) phase plot of the DFT of (a). (c) Oversampling in space coordinates; (d) phase plot of the DFT of (c).

Fig. 4
Fig. 4

(a), (b) Magnitude and phase plots of the chirp function and (c), (d) the Gaussian phase function with undersampling (1.25 d x 0 ), optimal ( d x 0 ), and oversampling (0.75 d x 0 ) cases.

Fig. 5
Fig. 5

(a), (b) Magnitude and phase plots of the chirp function and (c), (d) the Gaussian phase function with nonideal ( P 1 , P 2 ) and ideal ( P optimal ) values.

Fig. 6
Fig. 6

Comparison of the predicted diffraction patterns via the FA with and without a Gaussian window. (a)  N = 1024 , (b)  N = 1344 , (c)  N = 1800 without the Gaussian window before DFT. (d) 1D plot across Y i = 0 for (a)–(c). (f)  N = 1024 , (g)  N = 1344 , (h)  N = 1800 with the Gaussian window before DFT. (e) 1D plot cross Y i = 0 for (f)–(h). Common variables are L a = 1.6 mm , H 0 = 0.025 mm , and z = 12 mm . Optimal value [ λ z / ( 2 L a ) ] for the sampling interval d x 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).

Fig. 7
Fig. 7

Predicted diffraction patterns via the AS method: (a)  N = 1024 , (b)  N = 1344 , (c)  N = 1800 . Common variables are L a = 1.6 mm , H 0 = 0.025 mm , and z = 12 mm . Optimal value [ λ z / ( 2 L a ) ] for the sampling interval d x 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 propagation term for three cases, which shows oscillations ( N = 1024 ) or cutoff ( N = 800 ) for higher-frequency components. (e) Resulting 1D diffraction pattern that shows peak locations at the imaging plane.

Fig. 8
Fig. 8

Comparison of FA and AS for different wavelengths. (a)  P diff versus the imaging distance z. (b) Imaging domain size versus distance z. For AS, the aperture domain and imaging domain size remain the same, while FA can be increased.

Tables (1)

Tables Icon

Table 1 Simulation Parameters for Aliased and Normal Cases

Equations (21)

Equations on this page are rendered with MathJax. Learn more.

E a ( x a , y a , z 1 ) = E 0 exp ( ( X a 2 + X a 2 ) w 2 ( z 1 ) ) exp [ j k z 1 ] exp [ j k ( X a 2 + X a 2 ) 2 R ( z 1 ) ] ,
w 2 ( z ) = w 0 2 [ 1 + ( z z 0 ) 2 ] , R ( z ) = z [ 1 + ( z 0 z ) 2 ] ,
E i ( X i , Y i ) = 1 j λ Σ E a ( X a , Y a ) exp [ j k ( ϕ ( X a , Y a ) ) ] exp [ j k r a i r a i ] cos θ d X a d Y a ,
E i ( X i , Y i ) C 1 Σ exp [ ( X a 2 + Y a 2 ) w 2 ( z 1 ) ] exp [ j k ( X a 2 + Y a 2 ) 2 R ( z 1 ) ] exp [ j k ( X a 2 + Y a 2 ) 2 z 2 ] × exp [ j k ϕ ( X a , Y a ) ] exp [ i 2 π ( f x X a + f y Y a ) ] d X a d Y a ,
E i ( X i , Y i ) Σ t exp [ j ϕ r ] exp [ j ϕ c ] exp [ j ϕ g ] exp [ i 2 π ( f x X a + f y Y a ) ] d X a d Y a ,
t ( X a , Y a ) = exp [ ( X a 2 + Y a 2 ) w 2 ( z 1 ) ] ,
ϕ r ( X a , Y a ) = k ( ( X a 2 + Y a 2 ) 2 R ( z 1 ) ) ,
ϕ c ( X a , Y a ) = k ( ( X a 2 + Y a 2 ) 2 z 2 ) ,
ϕ g ( X a , Y a ) = k ( n 1 1 ) H 0 exp ( ( X a 2 + Y a 2 ) w b 2 ) .
θ / 2 max 1 k ( d Δ Φ d r ) max ,
L i z tan 1 ( θ / 2 ) max ,
F HS L i λ z ,
L a N F HS ,
2 w z 2 L a 2 N F HS ,
d x c = λ z 2 L a ,
d x g | d Δ Φ d r | π ,
d x g π | k ( n 1 1 ) H 0 2 x w b 2 exp ( x 2 w b 2 ) | max ,
d x g = λ z 2 L a ,
h ( x ) = exp ( i k L a 2 2 z ) ,
( 2 π λ L a 2 2 z ) = 2 π P ,
P diff = | P FA - P AS | P AS ,

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