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
The novel concepts of bacteria detection and identification are in focus of many international and national initiatives [1,2]. 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 [3–12]. 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 [13–19]. 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.
Experiments previously performed in our group have shown that analysis of bacteria colonies Fourier spectra, considered in general as diffraction patterns, can be used to estimate the bacteria colonies number and in consequence, to asses antimicrobial properties of different antimicrobial agents [20–22]. Moreover our previous works were focused also on novel optical system configuration with converging spherical wave illumination for characterization of light diffraction on bacterial colonies [23–27]. The proposed system exhibits some useful features, which can significantly improve the analysis of diffraction patterns of bacteria colonies. The main advantage of this optical system is enabling the registration of both Fresnel and Fraunhofer diffraction patterns of analyzed object. Although the Fresnel diffraction patterns of bacterial colonies exhibit unique features, which can be used for bacteria species identification with accuracy higher than 99% , the additional quadratic exponential phase modulation common for this kind of diffraction patterns is observed, which introduce the phase shift as a function of the distance. In consequence, it leads to the modulation of the Fresnel patterns for different locations of the observation plan, what requires tedious procedure of determination of proper localization of comparable Fresnel diffraction patterns for each bacteria species. The recording of Fraunhofer diffraction patterns of bacterial colonies allows to eliminate this problem, because in this case the patterns modulating quadratic exponential phase term is eliminated, if the diffraction patterns is recorded exactly in the Fourier plane. The use of the Fraunhofer diffraction patterns of bacterial colonies for their species identification enables the construction of optical processor with fixed location of the detector. However, the use of Fraunhofer patterns for this purpose is limited by the phase modulation of the incoming wave field caused by the profile of bacterial colony, which leads to the degeneration of Fraunhofer diffraction conditions in the proposed optical system. The profile of bacterial colonies affects the location of the Fourier plane, which is shifted along the optical axis. In consequence, for recording of Fraunhofer patterns of bacterial colonies the additional knowledge about colony profile, is necessary.
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 [15,16], 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 [28–33]. 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 . 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
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 , they are affected by additional modulating quadratic exponential phase term [23, 34, 35] 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), 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 and are describing the colony flat input surface (on a nutrient medium) and spherical output surface respectively. Therefore the total phase delay  can be expressed by
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:
Therefore as it was predicted in , 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, 35]. 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
The experiments were performed on bacteria colonies of Escherichia coli (0119) obtained from the microbiological laboratory of the Department of Environmental Engineering at Warsaw University of Technology. Bacteria suspensions were first incubated for 24 hours at the temperature of 37°C. Respective dilutions were seeded on the surface of the solid nutrient medium Columbia agar (Oxoid) in Petri dish, so as to obtain 12-20 colonies per plate, and were again incubated at 37 °C for next 18 hours. Next, the bacterial colonies located on Petri dish were placed in sample holder in optical system, where the measurements of bacterial colonies profile for 6 periods of time with 30 minutes step, were performed.
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 [29,31] (see Fig. 3). 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) . 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 .
The standard algorithm of phase conversion based on Thin Element Approximation (TEA)  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 . 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:21]). 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 . 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:34]:Fig. 3(b)).
Obtained results of the qualitative bacterial profile investigation performed by laser scanning confocal microscope (see Fig. 4) 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.
The main algorithms of the fringe pattern analysis and hologram reconstruction described in the previous section were applied for consecutive times of bacterial colonies incubation. In-focus conjugate optical field was captured after each period of time.
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 . The exemplary interferograms of bacterial colony recorded in the digital holographic system are presented in Fig. 5. 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.
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). 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.
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.
However, the qualitative measurements primary performed by scanning confocal microscope indicate that the Escherichia coli colonies have convex shape, which can be approximated by spherical surface, the results obtained by digital holographic microscope have shown that the bacterial colony surface exhibits more aspherical properties. In previous works  it was reported that Gaussian profile is correlated with the shape and the number of rings of scatter patterns. Therefore, it was decided to use the high order polynomial and Gaussian fitting to describe the experimental data of bacterial profile (see Fig. 7). Performed analysis have shown, that reconstructed topography of bacterial colony can be fitted with higher accuracy by high order polynomials, what is confirmed by low level deviation of the reconstructed profile of bacterial colony from the approximated profile. The RMS error obtained was equal 1.201 µm and 0.166 µm, respectively for Gaussian and 10-order polynomial profiles.
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). 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)
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)). 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.
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.
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.
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