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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 22 — Nov. 4, 2013
  • pp: 26493–26505
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Degeneration of Fraunhofer diffraction on bacterial colonies due to their light focusing properties examined in the digital holographic microscope system

Igor Buzalewicz, Kamil Liżewski, Małgorzata Kujawińska, and Halina Podbielska  »View Author Affiliations


Optics Express, Vol. 21, Issue 22, pp. 26493-26505 (2013)
http://dx.doi.org/10.1364/OE.21.026493


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Abstract

The degeneration of Fraunhofer diffraction conditions in the optical system with converging spherical wave illumination for bacteria species identification based on diffraction patterns is analyzed by digital holographic methods. The obtained results have shown that the colonies of analyzed bacteria species act as biological lenses with the time-dependent light focusing properties, which are characterized and monitored by means of phase retrieval from sequentially captured digital holograms. This significantly affects the location of Fraunhofer patterns observation plane, which is continuously shifted across optical axis in time.

© 2013 Optical Society of America

1. Introduction

The novel concepts of bacteria detection and identification are in focus of many international and national initiatives [1

1. S. G. B. Amyes, “The rise in bacterial resistance,” BMJ 320(7229), 199–200 (2000). [CrossRef] [PubMed]

,2

2. S. B. Levy and B. Marshall, “Antibacterial resistance worldwide: causes, challenges and responses,” Nat. Med. 10(12Suppl), S122–S129 (2004). [CrossRef] [PubMed]

]. Among various methods commonly used in microbiological laboratories, the optical methods are intensively examined in order to reduce the cost of analysis and quickly detect bacterial pathogens in food, water or in clinical samples [3

3. D. Ivnitski, I. Abdel-Hamid, P. Atanasov, and E. Wilkins, “Biosensors for detection of pathogenic bacteria,” Biosens. Bioelectron. 14(7), 599–624 (1999). [CrossRef]

12

12. J. C. Auger, K. B. Aptowicz, R. G. Pinnick, Y.-L. Pan, and R. K. Chang, “Angularly resolved light scattering from aerosolized spores: Observations and calculations,” Opt. Lett. 32(22), 3358–3360 (2007). [CrossRef] [PubMed]

]. Over the past few years, it was demonstrated that the analysis of forward light scattering on bacterial colonies, mostly affected by diffraction effects, can be used for identification of different bacteria species [13

13. P. 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(17), 3639–3648 (2007).

19

19. E. Bae, A. Aroonnual, A. K. Bhunia, and E. D. Hirleman, “On the sensitivity of forward scattering patterns from bacterial colonies to media composition,” J Biophotonics 4(4), 236–243 (2011). [CrossRef] [PubMed]

]. Diffraction patterns of bacterial colonies exhibit some specific features, which are suitable for bacteria species characterization. Optical biosensors investigating the light diffraction on bacterial colonies offer the non-invasive and non-destructive detection, since in this case the amplitude and phase of light modulated by pathogens are analyzed, instead of pathogens themselves. Therefore, they can be used as first-step systems for bacteria species identification. If it is necessary the obtained results can be verified by an another method analyzing the same sample.

In presented paper we analyze the changes of bacterial colony profile determined by radius of curvature (ROC) and their influence on phase modulation of incoming optical waves. The use of ROC for defining the profile of bacterial colony was already reported in [15

15. P. P. Banada, S. Guo, B. Bayraktar, E. Bae, B. Rajwa, J. P. Robinson, E. D. Hirleman, and A. K. Bhunia, “Optical forward-scattering for detection of Listeria monocytogenes and other Listeria species,” Biosens. Bioelectron. 22(8), 1664–1671 (2007). [CrossRef] [PubMed]

,16

16. 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(1), 014010 (2008). [CrossRef] [PubMed]

], however its correlation with scattering pattern were there analyzed, while in our case the light focusing properties of bacterial colonies and their influence on the degeneration of Fraunhofer diffraction conditions in proposed optical system will be investigated. Moreover, since bacterial colonies are the biological objects and they are growing in time, the changes of their profile will be also analyzed to evaluate their light modulating properties. The profile of bacterial colony will be determined by digital holographic microscope (DHM), which is based on overlaying object coherent wave with reference beam and use the resulting interference fringes to deduce the relative phases of the two waves. The method is based on the analysis of the captured complex wavefront of optical field recorded in Mach-Zehnder interferometer, which was initially proposed for characterization of metrological micro-objects, especially microlenses [28

28. J. Kostencka, T. Kozacki, and K. Liżewski, “Autofocusing method for tilted image plane detection in digital holographic microscopy,” -,” Opt. Commun. 297, 20–26 (2013). [CrossRef]

33

33. T. Kozacki, M. Józwik, and R. Jóźwicki, ““Determination of optical field generated by a microlens using digital holographic method,” Opto-Electron. Rev. 17, 58–63 (2009).

]. Moreover, the performed experiments determining the profile of analyzed bacterial colonies reveal that these biological structures exhibit light focusing properties similar to classical optical lenses, what was initially suggested in [13

13. P. 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(17), 3639–3648 (2007).

]. However, according to our knowledge the time-dependent light focusing properties of bacterial colonies were not as so far explored in literature and it is the first attempt to determine their light transformation properties.

2. Theoretical consideration of the influence of bacterial colony profile on degeneration of Fraunhofer diffraction conditions

The proposed optical system for bacteria species identification based on converging spherical wave illumination (see Fig. 1
Fig. 1 The proposed optical system configuration for characterization of bacteria colonies diffraction patterns: L0 transforming lens in (x0,y0) plane, bacteria colonies on Petri dish in (x1,y1) plane, observation plane (x2,y2) [23].
) and its main properties were extensively analyzed in [23

23. I. Buzalewicz, A. Wieliczko, and H. Podbielska, “Influence of various growth conditions on Fresnel diffraction patterns of bacteria colonies examined in the optical system with converging spherical wave illumination,” Opt. Express 19(22), 21768–21785 (2011). [CrossRef] [PubMed]

].

Theoretical consideration has shown that illumination by spherical wave converging towards the plane z = f eliminates the need for large observation distances for recording the Fraunhofer pattern. If the location of the observation plane ranges from the object plane z = z1 to the Fourier transform plane z = f, then, it is possible to observe the Fresnel diffraction pattern of the bacterial colonies. Although, the Fresnel diffraction patterns of bacterial colony recorded in proposed optical system exhibit unique features, which can be used for bacteria species identification with high accuracy [27

27. A. Suchwałko, I. Buzalewicz, A. Wieliczko, and H. Podbielska, “Bacteria species identification by the statistical analysis of bacterial colonies Fresnel patterns,” Opt. Express 21(9), 11322–11337 (2013). [CrossRef] [PubMed]

], they are affected by additional modulating quadratic exponential phase term [23

23. I. Buzalewicz, A. Wieliczko, and H. Podbielska, “Influence of various growth conditions on Fresnel diffraction patterns of bacteria colonies examined in the optical system with converging spherical wave illumination,” Opt. Express 19(22), 21768–21785 (2011). [CrossRef] [PubMed]

, 34

34. J. W. Goodman, Introduction to Fourier Optics, Third edition, (Robert & Company Publishers, 2005).

, 35

35. J. D. Gaskill, Linear systems, Fourier transform and optics, (John Wiley & Sons, New York, 1978).

] depended on the observation distance, focal distance of transforming lens L0 and the wavelength of illuminating optical field.

Therefore, the Fresnel patterns are changing with the distance of observation (see Fig. 2
Fig. 2 Exemplary Fresnel diffraction patterns of Escherichia coli colony recorded in different location of the observation plane: (a) 2cm, (b) 3cm, (c) 4 cm (bacteria colony diameter: approx. 0.9 mm, beam diameter: approx. 1 mm).
), what is associated with the appropriate procedures of determination of proper localization of comparable Fresnel diffraction patterns for each bacteria species, which have to be applied in proposed method.

However, if the observation plane is near the Fourier transform plane, then the Fraunhofer diffraction pattern of bacterial colony can be observed. In this case the quadratic exponential phase term is eliminated and measurements of diffraction patterns can be performed in fixed observation distance, what can improve the application potential of proposed optical method of bacteria species identification. It should be pointed out, that this approach is appropriate in the case of the light diffraction on the flat transparent object only. In the case of biological structures such as bacterial colonies, many of them have spheroid shapes. Performed during this investigation preliminary measurements of bacterial colonies profile have shown, that it can be approximated by a convex shape, where the radiuses of curvatures r= and rb are describing the colony flat input surface (on a nutrient medium) and spherical output surface respectively. Therefore the total phase delay [34

34. J. W. Goodman, Introduction to Fourier Optics, Third edition, (Robert & Company Publishers, 2005).

] can be expressed by
ϕ(x1,y1,t)=knhMAX(t)k(n1)x12+y122(1r1rb(t))=knhMAX(t)kx12+y122Fb(t),
(1)
where Fb=1/fb=(n1)(r1rb1), fb is the focal distance of bacterial colony, n is the refractive index of the bacteria colony, h MAX is the thickness along optical axis and k is a wave vector.

Based on the wave optics we can see that the above expression is similar to the phase delay of conventional flat-convex light refracting optical elements. Therefore, the Eq. (1) indicates that bacterial colonies should exhibit the light focusing properties. The amplitude transmittance of bacterial colony can be described by the following expression:
tb(x,y,t)=tb0(x,y,t)exp{iϕ(x,y,t)}=tb0(x,y,t)exp{knhMAX(t)}exp{kx2+y22Fb(t)},
(2)
where tb0(x,y,t) expresses the two-dimensional transmission coefficient of the bacterial colony.

Therefore as it was predicted in [13

13. P. 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(17), 3639–3648 (2007).

], the bacterial colony acts as biological lens. However, contrary to the optical lens the bacterial colony is semi-transparent object and the light transmission trough colony is limited. Moreover, it should be pointed out that bacterial colony as a biological structure is evolving over the time, therefore it can be treated as an amplitude-phase objects variable in time, what will occurred by time-dependent light focusing. In consequence, in the considered situation the term Fb indicates that the phase modulation of bacterial colony due to its convex shape and the refractive index n, affects the conditions of Fraunhofer diffraction observation, what was extensively described in [23

23. I. Buzalewicz, A. Wieliczko, and H. Podbielska, “Influence of various growth conditions on Fresnel diffraction patterns of bacteria colonies examined in the optical system with converging spherical wave illumination,” Opt. Express 19(22), 21768–21785 (2011). [CrossRef] [PubMed]

, 35

35. J. D. Gaskill, Linear systems, Fourier transform and optics, (John Wiley & Sons, New York, 1978).

]. The location of the Fourier plane is shifted along the optical axis, respectively to the value of Fb. Therefore, it can be seen that for analysis of Fraunhofer patterns the additional information about bacteria colony profile is required, since the location of Fraunhofer patterns observation plane will be changing in time.

3. Materials and methods

3.1 Preparation of bacterial colonies samples

3.2. The investigation of bacterial colonies by scanning confocal microscope

The bacterial colony profile and surface roughness were investigated by scanning confocal microscope Olympus LEXT 3D Measuring Laser Microscope (laser 405 nm, objective: 5X, 2X) and were evaluated by LEXT image analysis software.

3.3. The experimental system for measuring the bacterial colony profile

The topography measurements of bacterial colonies were performed in the digital holographic system in a microscope configuration based on Mach-Zehnder interferometer [29

29. T. Kozacki, K. Liżewski, and J. Kostencka, “Holographic method for topography measurement of highly tilted and high numerical aperture micro structures,” Opt. Laser Technol. 49, 38–46 (2013). [CrossRef]

,31

31. K. Liżewski, T. Kozacki, M. Józwik, and J. Kostencka, “On topography characterization of micro-optical elements with large numerical aperture using digital holographic microscopy,” Proc. SPIE 8430, 8430 (2012).

] (see Fig. 3
Fig. 3 Experimental DHM setup based on Mach-Zehnder interferometer: (a) full system, (b) the configuration of an object arm. P1,P2- polarizer’s; HWPP1,HWPP2 - half wave plates; M1,M2,M3 - mirrors; C - colimator; BS1,BS2 - beam splitter cubes; MO - microscope objective; IL - imaging lens; PZT - piezotransducer, MS- translation stage.
). In the system, the spatially filtrated and collimated coherent light beam (λ = 632.8 nm) is divided into object and reference waves by the beam splitter BS1. The optical field scattered by the sample is imaged by an afocal imaging system consisting of a microscope objective MO (NA = 0.42, 20X) and an illumination lens IL (f = 200mm).This imaging setup has special features: firstly lateral magnification is constant and independent of the object plane position. Secondly field distributions defined on a plane perpendicular to the optical axis in the object space are imaged on a suitable plane perpendicular to the optical axis without any phase corrections.

The imaging system conjugates a chosen object plane with a CCD detector (resolution 2456 pixels x 2058 pixels, pixel size 3.45µm x 3.45µm) providing magnification of 22X. A reference beam is transmitted towards the reference mirror (M3) mounted on a piezoelectric transducer (PZT), which enables realization of the temporal phase shifting algorithm (TPS) [36

36. I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. 22(16), 1268–1270 (1997). [CrossRef] [PubMed]

]. The configuration enabled to reconstruct the phase from in-line holograms. The waves propagating in both channels are combined by a second beam splitter BS2. The result of the interference is registered, in the form of an in-line hologram, by the CCD camera. In our case a nutrient medium plane of the sample is optically conjugated with the image plane, which coincides with the CCD sensor. In this plane it is easy to detect sharp edges between the nutrient medium and convex part of bacterial colonies, which facilitates the determination of the sample position with regards to the microscope objective of imaging system. However, the measurements based on the use of digital holography are not only restricted to the nutrient plane but can characterize a complex object wave in any desired plane, as shown schematically in Fig. 3(b).

3.4 The algorithm of fringe pattern analysis and hologram reconstruction

In order to obtain quantitative information about bacteria colonies such as profile parameters: diameter, radius of curvature ROC and central bacterial colony thickness, the phase retrieved from holograms by applying the TPS algorithm, has to be converted to topography map of the sample taking into consideration refractive index of bacteria colonies. According to our knowledge, the characterization of refractive index distribution of bacterial colonies was not performed in literature, therefore in our approach it was assumed that the bacteria colony has average refractive index n = 1.35, which is an average value of the refractive indexes of the extracellular material and bacterial cell, which are forming the colony [18

18. 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(4), 045001 (2010). [CrossRef] [PubMed]

].

The standard algorithm of phase conversion based on Thin Element Approximation (TEA) [34

34. J. W. Goodman, Introduction to Fourier Optics, Third edition, (Robert & Company Publishers, 2005).

] is a source of significant errors of profile reconstruction in case of 3D sample and results in false parameters determination. To overcome this problem in topography reconstruction, the Local Ray Approximation (LRA) algorithm initially developed for shape reconstruction of metrological object (e.g. microlenses) taking into accounts the depth and high gradient of topography of a sample, have been exploited. In our case we aim into accurate characterization of bacteria colonies topography which features 3D geometry. These requirement are satisfied by LRA algorithm described in [32

32. T. Kozacki, M. Józwik, and K. Liżewski, “High-numerical-aperture microlens shape measurement with digital holographic microscopy,” Opt. Lett. 36(22), 4419–4421 (2011). [CrossRef] [PubMed]

]. The LRA method is based on the analysis of local ray transition through a sample and allows for computing the element topography (hLRA) by using following relations:
hLRA(x')=φ(x)(nk0k02kzn1)1,
(3)
xs=φ(x)φ(x)k01(nkznk0)1,
(4)
where vectors kn = [kxn,kyn,kzn] represents illumination wave and xs = x' -x, denotes transverse shifts of the imaging rays, respectively. Distribution φ(x) is an unwrapped phase of the measured object wave and n is the average refractive index of the sample (here n = 1.35 for bacterial colony [21

21. I. Buzalewicz, K. Wysocka-Król, and H. Podbielska, “Image processing guided analysis for estimation of bacteria colonies number by means of optical transforms,” Opt. Express 18(12), 12992–13005 (2010). [CrossRef] [PubMed]

]). Such reconstructed topography of bacteria colony at nutrient plane was the base for ROC calculation, determined for each state of bacteria growth. An algorithm that is used to find the best fitting sphere is based on a Gauss-Newton algorithm [37

37. A. B. Forbes, A.B. 1989, “Least-Squares Best-Fit Geometric Elements,” National Physical Laboratory Report DITC 140/89 (1989).

]. The Gauss-Newton algorithm is applied for solving a root finding problem F = 0, where F is a non-linear function, expresses distance between measurement and reference points as F = ri-r, where:
ri=(xx0)2+(yy0)2+(zz0)2,
(5)
denotes radius of sphere and (x0,y0,z0) are coordinates of the center of the sphere. In order to determine the position of focal point of bacterial colony, the complex object wave was numerically propagated to many successive planes perpendicular to optical axis in the range zrange = 500-800 µm within the volume of interest. In each plane the real amplitude distribution is reconstructed and analyzed in order to obtain the focal point. The maximal intensity determined in the volume of interest for each hologram captured was chosen as the criterion of finding the location of focal point. In particular z planes, the complex optical field U2 is calculated by Rayleigh–Sommerfeld integral, after applying the Fourier convolution theorem can be described by [34

34. J. W. Goodman, Introduction to Fourier Optics, Third edition, (Robert & Company Publishers, 2005).

]:
U2(ξ,η)=1{{U1(x,y)}{hLRA(x,y)}},
(6)
and
h(x,y)=12πexp(ikr)rzr(1ri2πλ),
(7)
where U1 is the source plane (in our case z = 0- substrate plane), h(x,y) is the general form of the Rayleigh–Sommerfeld impulse response, r = (x2 + y2 + z2)1/2 and {...},1{...}denote the 2D Fourier transform and the inverse Fourier transform respectively. Such optical fields in consecutive planes in the range zrange are assembled in 3D cube of optical field as a set of cross-sections of real amplitude of integrated optical field (Fig. 3(b)).

4. Results

Obtained results of the qualitative bacterial profile investigation performed by laser scanning confocal microscope (see Fig. 4
Fig. 4 Exemplary profile of Escherichia coli colony obtained by the confocal laser scanning microscope.
) have shown, that investigated Escherichia coli colonies exhibit the convex shape, what was initially predicted in Section 2. To obtain the quantitative information about bacterial colony profile the investigation by digital holographic microscope was performed.

Due to growing process of biological sample each registration of complex amplitude images was proceeded by the autofocus procedure allowing sharp imaging of the bacterial colony sample [28

28. J. Kostencka, T. Kozacki, and K. Liżewski, “Autofocusing method for tilted image plane detection in digital holographic microscopy,” -,” Opt. Commun. 297, 20–26 (2013). [CrossRef]

]. The exemplary interferograms of bacterial colony recorded in the digital holographic system are presented in Fig. 5
Fig. 5 The exemplary fringe pattern of Escherichia coli colony limited by rectangular aperture (a), phase modulo 2π (b) and topography reconstruction (c) obtained by LRA algorithm.
. The analysis of interferograms included the limitation of the recorded fringe patterns by rectangular aperture (see Fig. 5(a)) and further determination of phase modulo 2π pattern (see Fig. 5(b)) to reconstruct the topography of bacterial colonies by LRA algorithm (see Fig. 5(c)). The topography of bacterial colony and its calculated parameters such as ROC, focal distance f and diameter, were reconstructed. The registered fringe patterns and obtained bacterial colony topography suggest that the profile of colony has convex shape similar to aspherical surface. To investigate the influence of bacterial colony growth process on its profile and focusing properties, the temporal measurements for the same colonies were performed. The obtained results are presented in Table 1

Table 1. Changes of Bacterial Colony Properties in Time

table-icon
View This Table
.

The experiments indicate that the analyzed biological structures as Escherichia coli colonies exhibit the light focusing properties similar to classical optical microlenses, therefore for complete characterization, the captured integrated field at substrate plane of bacterial colonies was numerically propagated in order to evaluate the focus position. As a criterion of finding the focal point, an algorithm of determination of maximal intensity in marked region of interest was chosen. After the hologram reconstruction based on Rayleigh–Sommerfeld diffraction approximation, the 3D cube of optical field represented by real amplitude of integrated field in consecutive planes in the range zrange, were obtained. In order to illustrate the location of the focal point the cross-section of the 3D representation of optical field along z-axis was extracted (see Fig. 6
Fig. 6 The cross-sectional representations of intensity of integrated optical field transformed by bacterial colony in consecutive planes (zrange = 500-800µm) as obtained for six measurements separated by 30 min. time slot.
). The marked region of interest represents the location of the focal point.

The results confirmed, that the increase of bacterial colony diameter and central thickness hMAX is observed over the time. The geometrical structure of bacterial colony affects its interaction with incoming waves causing the light beam focusing. In another words, the bacterial colonies can be considered as light focusing elements. Moreover, the changes of bacterial colony profile and in consequence its focal distance f are increasing in time.

5. Discussion

The obtained results confirm that bacterial colonies exhibit the light focusing properties, which are changing over the time corresponding to the colony growth. This process is associated with the changes of bacterial colony geometrical structure, particularly the radius of curvature. Therefore, the bacterial colonies can be considered as variable in time light focusing biological objects.

Moreover, the bacterial colony surface exhibit the deviation from ideally spherical surface. The comparison of the reconstructed bacterial colony topography and ideal spherical surface is presented on Fig. 8(a)
Fig. 8 The exemplary comparison of the experimentally reconstructed Escherichia coli colony profile and approximated ideal spherical surface (a) and the quantitative analysis of the deviation between them (b) performed for the colony measured in t = 0 min.
. The obtained results have shown that significant deviation of reconstructed profile from approximated ideal spherical described by the residuals values (see Fig. 8(b)) is observed. The maximal deviation is equal 2.0 −2.4 µm and it is describing the deviation of the approximated radius of curvature (ROC) and the real profile of the bacterial colony. This indicates that analyzed Escherichia coli colony topography should be treated as aspherical surface. Therefore, according to the light focusing properties of bacterial colony, it can be considered as biological aspherical microlens.

To investigate the variations of approximated bacterial colony profile to the actual analysis results of bacterial colonies light focusing properties additional numerical simulations were performed. Firstly, the constant refractive index n = 1.35 of entire colony and different profiles of bacterial colony were assumed and the complex optical field was numerically propagated along optical axis to determine the location of the focal point. The following profiles of bacterial colony (t = 0 min.) were used: 10-th order polynomial (best fitted profile), spherical profiles (different radius of curvature) and Gaussian profiles. (Fig. 9
Fig. 9 Profiles of colony shape for t = 0 min. used for determination of the influence of bacterial colony profile variations on the simulated location of the focal point.
)

The focal point for simulation with the bacterial colony profile defined by 10-th order polynomial (see Fig. 7) is equal 673 µm, while for experimental data is 671 µm (Fig. 10(a)
Fig. 10 Location of the focal point in function of ROC (a) and refractive index n (b).
). Therefore it can be seen, that proposed fitting of bacterial colony profile by this polynomial provides numerical results most convergent with obtained experimental results. On the other hand, the obtained focal point of bacterial colony for profile defined by 2-nd order polynomial (spherical) is equal 683 µm for ROC = 230 µm and 701µm for ROC = 245 µm. These results suggest that the treatment of bacterial colony as spherical surface can lead to significant errors, which can affect the validity of performed numerical simulations. Moreover, it can be seen, that as expected the focal length grows as a function of increased ROC. Finally, the focal length obtained for the best fitted Gaussian profiles of bacterial colony defined as y = 27.1509·exp(-(x + 1.0568)2/(2·53.5851) is equal 198 µm and exhibits the lowest coincidence with experimental results.

To investigate the influence of the bacterial colony refractive index variation on the location of focal point, a complementary analysis as above was performed for the constant bacterial colony profile (fitted by 10-th order polynomial) and changed refractive index of colony in range 1.25-1.45 From the Fig. 10(b), the increase of bacterial colony refractive index causes the shortening of the focal length. For the refractive index change equals ∆n = 0.01, the 18 µm shift of focal point is observed while for ∆n = 0.1 it is more than 250 µm. From the obtained computational data it can be concluded that bacterial colony acts as microlenses and for increased optical power (change of refractive index) results in shorter focal length. It should be pointed out, that during this investigation the constant refractive index of bacterial colonies was assumed, however heterogeneous structure of such biological object as bacterial colony can suggest that the distribution of the refractive index overall the colony might not be uniform. This issue will be in focus of our future works.

The 2D cross-sectional representations of the intensity of the integrated optical field in different propagation distances from the colony (see Fig. 6), have shown that the focusing properties are changing in time. For analyzed period of time (150 min), the 39 μm shift of the focal point is observed. The obtained focal shift is consistent with calculated ROC values for consecutive stages of bacteria colonies growth. The ROC of Escherichia coli colony increases as a function of time and has changed value from 237 μm to 259 μm over the investigated time. The observed tendency of the increase of ROC values associated with the decrease of the focal length of the analyzed colonies, confirmed the thesis that bacteria colonies act as microlenses. It is worth emphasis that area of topography close to nutrient plane features tangent of topography relatively smaller for bacteria colonies with longer incubation time. Therefore, the changes of bacterial colony focal point locations in time indicate that during the colony growth process the location of Fourier plane, where the Fraunhofer diffraction pattern can be recorded, will be changing in time as well. To determine the location of the Fraunhofer patterns observation plane the continuous monitoring of bacterial profile, is necessary. This significantly affects the practical implementation of bacterial colonies Fraunhofer patterns analysis for their species identification based on the light diffraction on bacterial colonies.

Moreover, the numerical simulations have shown, that the intensity distribution in the focal point is more spread or extended in axial and lateral directions than in case of conventional optical lenses. This effect can be emerged by roughness of the bacterial colony surface as well as by the non-uniform structure (small local variations of refractive index) of these biological objects, which are causing the light scattering.

To perform the additional analysis of the bacterial colony output surface roughness the Escherichia coli colonies were investigated by laser scanning confocal microscope and analyzed by attached software. The surface roughness average Sa was changing in time from 8.710 to 19.879 μm, therefore it can be seen that the light scattering on bacterial colonies can be affected by the surface roughness and deviation from the optically smooth surfaces of conventional lenses. In our understanding, the bacterial colony surface roughness makes, that colony acts as classical optical diffusor, which adds additional high frequency noise (worse S/N ratio), but it will not influence Fraunhofer diffraction conditions, contrary to the change of bacterial colony ROC. The surface kurtosis Sku is a measure of surface flattening in comparison with the Gaussian distribution. For a Gaussian distribution Sku value is 3.0. Lower values represent the distribution of relatively flat (less concentrated), while higher values mean more convex (more concentrated) shape from a Gaussian distribution. In the case of Escherichia coli colony measured in 6 periods of time the surface kurtosis was changing in time from 3.56 to 4.7. This indicates that analyzed colony can be treated as convex surface, however not ideally spherical.

6. Conclusions

The performed experimental examination of the Escherichia coli colony profile has shown that analyzed biological object exhibits the light focusing properties similar to the conventional aspherical microlenses. It should be pointed out that the conventional optical lens does not change the intensity but induces a nonuniform phase shift to the wave, which are transformed into visible intensity variations during the propagation through a suitable distance. However, contrary to the classical lens, the bacterial colony is semi-transparent object and the light transmission through the colony will be limited, therefore besides the introduced phase shift, the additional amplitude modulation of incoming wave is observed and bacterial colony can be treated as amplitude and phase light modulator. Moreover, according to the roughness of analyzed bacterial colony surface, the bacterial colony can be considered also as optical diffusor, which are affecting the spread of focal point and spatial light intensity variations in focal plane. The obtained results have shown that the bacterial colony is evolving over the time, therefore its focusing properties are changing in time and it can be considered as an optical element with adaptive light focusing properties. This significantly affects the possibility of the use of bacterial colonies Fraunhofer diffraction patterns for their species identification, because the location of the Fourier plane due to the changing of the focal distance, is continuously shifting along the optical axis over the time. Therefore, in the proposed optical method for bacteria species identification based on diffraction patterns of bacterial colonies, the additional system of bacterial profile determination should be included. Moreover, also the use of Fresnel diffraction patterns of bacterial colonies for their species identification recorded in fixed collection distance is affected by light focusing properties of bacterial colonies as well, because the Fresnel patterns will be shifted along optical axis for bacterial colonies incubated in different times. To omit this problem it is necessary to additionally characterized the locations of Fresnel diffraction patterns observation plane for different times of bacterial colonies incubation. This factor should be considered as standardized parameter limiting the efficiency of bacteria species identification by the methods based on analysis of bacterial colony diffraction patterns. The most of the optical processors are working on the assumption of plane samples, what is not acceptable in case of bacterial colonies with specific convex profile and light focusing properties, what was shown in presented investigation. Therefore, our future works will be focused on the preparation of automatically corrected optical processor, which can be used to investigate the complex and variable in time (amplitude-phase) objects. The additional knowledge gained by measurement of phase map can be used for correcting it by spatial light modulator and bringing the solution in optical processor to the simple solution of recognition of amplitude object with constant phase as the phase will be compensating actively. This will allow use of the processor with constant location of the detector, which is not in both Fresnel and Fraunhofer cases, if we have variable in time complex objects.

Acknowledgments

This work was partially supported by of the European Union under the European Social Fund (No MK/SS/41/III/2012/U) and European Founds of Regional Development within the program TEAM of Foundation for Polish Science (TEAM/2011-7/7). The authors acknowledge Ms. Ewa Miaśkiewicz-Pęska from the Faculty of Environmental Engineering at Warsaw University of Technology for preparation of the bacterial colonies samples.

References and links

1.

S. G. B. Amyes, “The rise in bacterial resistance,” BMJ 320(7229), 199–200 (2000). [CrossRef] [PubMed]

2.

S. B. Levy and B. Marshall, “Antibacterial resistance worldwide: causes, challenges and responses,” Nat. Med. 10(12Suppl), S122–S129 (2004). [CrossRef] [PubMed]

3.

D. Ivnitski, I. Abdel-Hamid, P. Atanasov, and E. Wilkins, “Biosensors for detection of pathogenic bacteria,” Biosens. Bioelectron. 14(7), 599–624 (1999). [CrossRef]

4.

S. C. Hill, R. G. Pinnick, S. Niles, N. F. Fell Jr, Y. L. Pan, J. Bottiger, B. V. Bronk, S. Holler, and R. K. Chang, “Fluorescence from airborne microparticles: dependence on size, concentration of fluorophores, and illumination intensity,” Appl. Opt. 40(18), 3005–3013 (2001). [CrossRef] [PubMed]

5.

A. Maninen, M. Putkiranta, A. Rostedt, J. Saarela, T. Laurila, M. Marjamӓki, J. Keskinen, and R. Hernberg, “Instrumentation for measuring fluorescence cross-sections from airborne microsized particle,” Appl. Opt. 47(7), 110–115 (2008).

6.

A. Alimova, A. Katz, P. Gottlieb, and R. R. Alfano, “Proteins and dipicolinic acid released during heat shock activation of Bacillus subtilis spores probed by optical spectroscopy,” Appl. Opt. 45(3), 445–450 (2006). [CrossRef] [PubMed]

7.

J. Thomason, “Spectroscopy takes security into the field,” Photon. Spectra 38, 83–85 (2004).

8.

R. T. Noble and S. B. Weisberg, “A review of technologies for rapid detection of bacteria in recreational waters,” J. Water Health 3(4), 381–392 (2005). [PubMed]

9.

D. L. Rosen, “Airborne bacterial endospores detected by use of an impinger containing aqueous terbium chloride,” Appl. Opt. 45(13), 3152–3157 (2006). [CrossRef] [PubMed]

10.

S. J. Mechery, X. J. Zhao, L. Wang, L. R. Hilliard, A. Munteanu, and W. Tan, “Using bioconjugated nanoparticles to monitor E. coli in a flow channel,” Chem. Asian J. 1(3), 384–390 (2006). [CrossRef] [PubMed]

11.

J. Homola, J. Dostálek, S. Chen, A. Rasooly, S. Jiang, and S. S. Yee, “Spectral surface plasmon resonance biosensor for detection of staphylococcal enterotoxin B in milk,” Int. J. Food Microbiol. 75(1-2), 61–69 (2002). [CrossRef] [PubMed]

12.

J. C. Auger, K. B. Aptowicz, R. G. Pinnick, Y.-L. Pan, and R. K. Chang, “Angularly resolved light scattering from aerosolized spores: Observations and calculations,” Opt. Lett. 32(22), 3358–3360 (2007). [CrossRef] [PubMed]

13.

P. 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(17), 3639–3648 (2007).

14.

M. Venkatapathi, B. Rajwa, K. Ragheb, P. P. Banada, T. Lary, J. P. Robinson, and E. D. Hirleman, “High speed classification of individual bacterial cells using a model-based light scatter system and multivariate statistics,” Appl. Opt. 47(5), 678–686 (2008). [CrossRef] [PubMed]

15.

P. P. Banada, S. Guo, B. Bayraktar, E. Bae, B. Rajwa, J. P. Robinson, E. D. Hirleman, and A. K. Bhunia, “Optical forward-scattering for detection of Listeria monocytogenes and other Listeria species,” Biosens. Bioelectron. 22(8), 1664–1671 (2007). [CrossRef] [PubMed]

16.

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(1), 014010 (2008). [CrossRef] [PubMed]

17.

P. P. Banada, K. Huff, E. Bae, B. Rajwa, A. Aroonnual, B. Bayraktar, A. Adil, J. P. Robinson, E. D. Hirleman, and A. K. Bhunia, “Label-free detection of multiple bacterial pathogens using light-scattering sensor,” Biosens. Bioelectron. 24(6), 1685–1692 (2009). [CrossRef] [PubMed]

18.

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(4), 045001 (2010). [CrossRef] [PubMed]

19.

E. Bae, A. Aroonnual, A. K. Bhunia, and E. D. Hirleman, “On the sensitivity of forward scattering patterns from bacterial colonies to media composition,” J Biophotonics 4(4), 236–243 (2011). [CrossRef] [PubMed]

20.

I. Buzalewicz, K. Wysocka, and H. Podbielska, “„Exploiting of optical transforms for bacteria evaluation in vitro,” Proc. SPIE 7371, 73711H, 73711H-6 (2009). [CrossRef]

21.

I. Buzalewicz, K. Wysocka-Król, and H. Podbielska, “Image processing guided analysis for estimation of bacteria colonies number by means of optical transforms,” Opt. Express 18(12), 12992–13005 (2010). [CrossRef] [PubMed]

22.

I. Buzalewicz, K. Wysocka–Król, K. Kowal, and H. Podbielska, “Evaluation of antibacterial agents efficiency” in Information Technologies in Biomedicine 2, E. Pietka, J. Kawa ed. (Springer-Verlag, Berlin, 2010).

23.

I. Buzalewicz, A. Wieliczko, and H. Podbielska, “Influence of various growth conditions on Fresnel diffraction patterns of bacteria colonies examined in the optical system with converging spherical wave illumination,” Opt. Express 19(22), 21768–21785 (2011). [CrossRef] [PubMed]

24.

A. Suchwalko, I. Buzalewicz, and H. Podbielska, “Computer-based classification of bacteria species by analysis of their colonies Fresnel diffraction patterns,” Proc. SPIE 82120R, 82120R-13 (2012). [CrossRef]

25.

H. Podbielska, I. Buzalewicz, and A. Suchwalko, “Bacteria Classification by Means of the Statistical Analysis of Fresnel Diffraction Patterns of Bacteria Colonies,” Biomedical Optics (BIOMED) (2012).

26.

A. Suchwałko, I. Buzalewicz, and H. Podbielska, “Identification of bacteria species by using morphological and textural properties of bacterial colonies diffraction patterns,” Proc. SPIE 8791, 8791 (2013).

27.

A. Suchwałko, I. Buzalewicz, A. Wieliczko, and H. Podbielska, “Bacteria species identification by the statistical analysis of bacterial colonies Fresnel patterns,” Opt. Express 21(9), 11322–11337 (2013). [CrossRef] [PubMed]

28.

J. Kostencka, T. Kozacki, and K. Liżewski, “Autofocusing method for tilted image plane detection in digital holographic microscopy,” -,” Opt. Commun. 297, 20–26 (2013). [CrossRef]

29.

T. Kozacki, K. Liżewski, and J. Kostencka, “Holographic method for topography measurement of highly tilted and high numerical aperture micro structures,” Opt. Laser Technol. 49, 38–46 (2013). [CrossRef]

30.

K. Liżewski, T. Kozacki, and J. Kostencka, “Digital holographic microscope for measurement of high gradient deep topography object based on superresolution concept,” Opt. Lett. 38(11), 1878–1880 (2013). [CrossRef] [PubMed]

31.

K. Liżewski, T. Kozacki, M. Józwik, and J. Kostencka, “On topography characterization of micro-optical elements with large numerical aperture using digital holographic microscopy,” Proc. SPIE 8430, 8430 (2012).

32.

T. Kozacki, M. Józwik, and K. Liżewski, “High-numerical-aperture microlens shape measurement with digital holographic microscopy,” Opt. Lett. 36(22), 4419–4421 (2011). [CrossRef] [PubMed]

33.

T. Kozacki, M. Józwik, and R. Jóźwicki, ““Determination of optical field generated by a microlens using digital holographic method,” Opto-Electron. Rev. 17, 58–63 (2009).

34.

J. W. Goodman, Introduction to Fourier Optics, Third edition, (Robert & Company Publishers, 2005).

35.

J. D. Gaskill, Linear systems, Fourier transform and optics, (John Wiley & Sons, New York, 1978).

36.

I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. 22(16), 1268–1270 (1997). [CrossRef] [PubMed]

37.

A. B. Forbes, A.B. 1989, “Least-Squares Best-Fit Geometric Elements,” National Physical Laboratory Report DITC 140/89 (1989).

OCIS Codes
(050.1940) Diffraction and gratings : Diffraction
(120.4630) Instrumentation, measurement, and metrology : Optical inspection
(170.0170) Medical optics and biotechnology : Medical optics and biotechnology
(170.4580) Medical optics and biotechnology : Optical diagnostics for medicine
(090.1995) Holography : Digital holography

ToC Category:
Medical Optics and Biotechnology

History
Original Manuscript: August 2, 2013
Revised Manuscript: September 16, 2013
Manuscript Accepted: October 17, 2013
Published: October 28, 2013

Virtual Issues
Vol. 9, Iss. 1 Virtual Journal for Biomedical Optics

Citation
Igor Buzalewicz, Kamil Liżewski, Małgorzata Kujawińska, and Halina Podbielska, "Degeneration of Fraunhofer diffraction on bacterial colonies due to their light focusing properties examined in the digital holographic microscope system," Opt. Express 21, 26493-26505 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-22-26493


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References

  1. S. G. B. Amyes, “The rise in bacterial resistance,” BMJ320(7229), 199–200 (2000). [CrossRef] [PubMed]
  2. S. B. Levy and B. Marshall, “Antibacterial resistance worldwide: causes, challenges and responses,” Nat. Med.10(12Suppl), S122–S129 (2004). [CrossRef] [PubMed]
  3. D. Ivnitski, I. Abdel-Hamid, P. Atanasov, and E. Wilkins, “Biosensors for detection of pathogenic bacteria,” Biosens. Bioelectron.14(7), 599–624 (1999). [CrossRef]
  4. S. C. Hill, R. G. Pinnick, S. Niles, N. F. Fell, Y. L. Pan, J. Bottiger, B. V. Bronk, S. Holler, and R. K. Chang, “Fluorescence from airborne microparticles: dependence on size, concentration of fluorophores, and illumination intensity,” Appl. Opt.40(18), 3005–3013 (2001). [CrossRef] [PubMed]
  5. A. Maninen, M. Putkiranta, A. Rostedt, J. Saarela, T. Laurila, M. Marjamӓki, J. Keskinen, and R. Hernberg, “Instrumentation for measuring fluorescence cross-sections from airborne microsized particle,” Appl. Opt.47(7), 110–115 (2008).
  6. A. Alimova, A. Katz, P. Gottlieb, and R. R. Alfano, “Proteins and dipicolinic acid released during heat shock activation of Bacillus subtilis spores probed by optical spectroscopy,” Appl. Opt.45(3), 445–450 (2006). [CrossRef] [PubMed]
  7. J. Thomason, “Spectroscopy takes security into the field,” Photon. Spectra38, 83–85 (2004).
  8. R. T. Noble and S. B. Weisberg, “A review of technologies for rapid detection of bacteria in recreational waters,” J. Water Health3(4), 381–392 (2005). [PubMed]
  9. D. L. Rosen, “Airborne bacterial endospores detected by use of an impinger containing aqueous terbium chloride,” Appl. Opt.45(13), 3152–3157 (2006). [CrossRef] [PubMed]
  10. S. J. Mechery, X. J. Zhao, L. Wang, L. R. Hilliard, A. Munteanu, and W. Tan, “Using bioconjugated nanoparticles to monitor E. coli in a flow channel,” Chem. Asian J.1(3), 384–390 (2006). [CrossRef] [PubMed]
  11. J. Homola, J. Dostálek, S. Chen, A. Rasooly, S. Jiang, and S. S. Yee, “Spectral surface plasmon resonance biosensor for detection of staphylococcal enterotoxin B in milk,” Int. J. Food Microbiol.75(1-2), 61–69 (2002). [CrossRef] [PubMed]
  12. J. C. Auger, K. B. Aptowicz, R. G. Pinnick, Y.-L. Pan, and R. K. Chang, “Angularly resolved light scattering from aerosolized spores: Observations and calculations,” Opt. Lett.32(22), 3358–3360 (2007). [CrossRef] [PubMed]
  13. P. 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(17), 3639–3648 (2007).
  14. M. Venkatapathi, B. Rajwa, K. Ragheb, P. P. Banada, T. Lary, J. P. Robinson, and E. D. Hirleman, “High speed classification of individual bacterial cells using a model-based light scatter system and multivariate statistics,” Appl. Opt.47(5), 678–686 (2008). [CrossRef] [PubMed]
  15. P. P. Banada, S. Guo, B. Bayraktar, E. Bae, B. Rajwa, J. P. Robinson, E. D. Hirleman, and A. K. Bhunia, “Optical forward-scattering for detection of Listeria monocytogenes and other Listeria species,” Biosens. Bioelectron.22(8), 1664–1671 (2007). [CrossRef] [PubMed]
  16. 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(1), 014010 (2008). [CrossRef] [PubMed]
  17. P. P. Banada, K. Huff, E. Bae, B. Rajwa, A. Aroonnual, B. Bayraktar, A. Adil, J. P. Robinson, E. D. Hirleman, and A. K. Bhunia, “Label-free detection of multiple bacterial pathogens using light-scattering sensor,” Biosens. Bioelectron.24(6), 1685–1692 (2009). [CrossRef] [PubMed]
  18. 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(4), 045001 (2010). [CrossRef] [PubMed]
  19. E. Bae, A. Aroonnual, A. K. Bhunia, and E. D. Hirleman, “On the sensitivity of forward scattering patterns from bacterial colonies to media composition,” J Biophotonics4(4), 236–243 (2011). [CrossRef] [PubMed]
  20. I. Buzalewicz, K. Wysocka, and H. Podbielska, “„Exploiting of optical transforms for bacteria evaluation in vitro,” Proc. SPIE7371, 73711H, 73711H-6 (2009). [CrossRef]
  21. I. Buzalewicz, K. Wysocka-Król, and H. Podbielska, “Image processing guided analysis for estimation of bacteria colonies number by means of optical transforms,” Opt. Express18(12), 12992–13005 (2010). [CrossRef] [PubMed]
  22. I. Buzalewicz, K. Wysocka–Król, K. Kowal, and H. Podbielska, “Evaluation of antibacterial agents efficiency” in Information Technologies in Biomedicine 2, E. Pietka, J. Kawa ed. (Springer-Verlag, Berlin, 2010).
  23. I. Buzalewicz, A. Wieliczko, and H. Podbielska, “Influence of various growth conditions on Fresnel diffraction patterns of bacteria colonies examined in the optical system with converging spherical wave illumination,” Opt. Express19(22), 21768–21785 (2011). [CrossRef] [PubMed]
  24. A. Suchwalko, I. Buzalewicz, and H. Podbielska, “Computer-based classification of bacteria species by analysis of their colonies Fresnel diffraction patterns,” Proc. SPIE82120R, 82120R-13 (2012). [CrossRef]
  25. H. Podbielska, I. Buzalewicz, and A. Suchwalko, “Bacteria Classification by Means of the Statistical Analysis of Fresnel Diffraction Patterns of Bacteria Colonies,” Biomedical Optics (BIOMED) (2012).
  26. A. Suchwałko, I. Buzalewicz, and H. Podbielska, “Identification of bacteria species by using morphological and textural properties of bacterial colonies diffraction patterns,” Proc. SPIE8791, 8791 (2013).
  27. A. Suchwałko, I. Buzalewicz, A. Wieliczko, and H. Podbielska, “Bacteria species identification by the statistical analysis of bacterial colonies Fresnel patterns,” Opt. Express21(9), 11322–11337 (2013). [CrossRef] [PubMed]
  28. J. Kostencka, T. Kozacki, and K. Liżewski, “Autofocusing method for tilted image plane detection in digital holographic microscopy,” -,” Opt. Commun.297, 20–26 (2013). [CrossRef]
  29. T. Kozacki, K. Liżewski, and J. Kostencka, “Holographic method for topography measurement of highly tilted and high numerical aperture micro structures,” Opt. Laser Technol.49, 38–46 (2013). [CrossRef]
  30. K. Liżewski, T. Kozacki, and J. Kostencka, “Digital holographic microscope for measurement of high gradient deep topography object based on superresolution concept,” Opt. Lett.38(11), 1878–1880 (2013). [CrossRef] [PubMed]
  31. K. Liżewski, T. Kozacki, M. Józwik, and J. Kostencka, “On topography characterization of micro-optical elements with large numerical aperture using digital holographic microscopy,” Proc. SPIE8430, 8430 (2012).
  32. T. Kozacki, M. Józwik, and K. Liżewski, “High-numerical-aperture microlens shape measurement with digital holographic microscopy,” Opt. Lett.36(22), 4419–4421 (2011). [CrossRef] [PubMed]
  33. T. Kozacki, M. Józwik, and R. Jóźwicki, ““Determination of optical field generated by a microlens using digital holographic method,” Opto-Electron. Rev.17, 58–63 (2009).
  34. J. W. Goodman, Introduction to Fourier Optics, Third edition, (Robert & Company Publishers, 2005).
  35. J. D. Gaskill, Linear systems, Fourier transform and optics, (John Wiley & Sons, New York, 1978).
  36. I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett.22(16), 1268–1270 (1997). [CrossRef] [PubMed]
  37. A. B. Forbes, A.B. 1989, “Least-Squares Best-Fit Geometric Elements,” National Physical Laboratory Report DITC 140/89 (1989).

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