A scattering microscope was developed to investigate single cells and biological microstructures by light scattering measurements. The spectrally resolved part of the setup and its validation are shown in detail. The analysis of light scattered by homogenous polystyrene spheres allows the determination of their diameters using Mie theory. The diameters of 150 single polystyrene spheres were determined by the spectrally resolved scattering microscope. In comparison, the same polystyrene suspension stock was investigated by a collimated transmission setup. Mean diameters and standard deviations of the size distribution were evaluated by both methods with a statistical error of less than 1nm. The systematic errors of both devices are in agreement within the measurement accuracy.
© 2011 Optical Society of America
Scattered light is often an annoying phenomenon in nature, whether it is fog outdoors or unwanted stray light inside the optics laboratory. Nevertheless, many conclusions on the structure, the size or the optical properties of a medium can be drawn by the analysis of scattered light. The scattering patterns can be observed spectrally resolved [1, 2], angular resolved [3, 4], polarization dependent , time resolved [6, 7] or spatially resolved [8, 9]. Moreover, various publications of different combinations of these methods do exist [10, 11]. Light scattering measurements are non-invasive, therefore they are adequate for the investigation and diagnosis of biological tissue . The contrast is given only by the scattering, so the technique is marker-free and no further enhancement is necessary. Microscopic setups are essential for the study of single cells  and chromosomes . Experiments have been made with confocal microscopes [15, 16], brightfield microscopes , darkfield microscopes  or evanescent illumination , just to name a few different methods.
In this contribution a scattering microscope is presented that combines spectroscopic and angular resolved measurements, similar to the setup presented by Cottrell et al. . But in contrast to this, the here shown setup includes several relevant differences. First of all, the presented microscope works on the basis of a reflected darkfield illumination. This is advantageous for the observation of thicker or strongly absorbing samples. Moreover, in case of Mie scattering, the spectral and angular patterns of the backward scattered light contain more information. In addition, Cottrell et al., as well as e.g. Smith and Berger , are using an illumination that is rotationally symmetric to the optical axis. On the contrary, the here shown scattering microscope is using an unidirectional illumination beam. Therefore the geometrical orientation of a non-spherical sample and the direction of illumination can be rotated against each other, which increases the versatility of the measurements. Further, in comparison to Cottrell’s setup  or the 4D-ELF setup published by Roy et al. , the range of detected scattering angles is enlarged in our setup (from 93° to 157°).
This paper focuses on the spectrally resolved analysis of elastically scattered light. Further information about the angular resolved measurements performed by the here shown scattering microscope can be found in Rothe et al. .
Before starting studies on biological cells, a new setup has to be evaluated by well-known samples. Biological tissue is a complex medium, as it often contains multiple layers with miscellaneous structures having different scattering and absorption coefficients. Inner structures as cell cores and filaments can be approximated by spheres [22, 23], cylinders  or mixtures of these . Light scattered by a homogenous sphere can be described analytically by Mie theory, which is a solution of Maxwell’s equations . Additionally, analytical solutions exist for an infinite cylinder .
Therefore, spheres and cylinders are an ideal reference sample for single scatterers. For the experiment, spherical microparticles are available in various sizes and materials. In many contributions, e.g. [2, 17], the determined sphere diameters are compared with the manufacturer values, which are commonly given in the form of a Gaussian size distribution. Thus, for the precise determination of systematic errors, the measurement of one or a few single spheres is insufficient. Here, polystyrene beads in suspension with a nominal mean diameter νn = 4.21 μm and standard deviation σn = 0.07 μm were first analyzed by a well-approved collimated transmission setup . Then, 150 single beads from the same stock were evaluated separately by the spectrally resolved scattering microscope. The agreement of both methods was verified by comparing the mean diameter ν and the standard deviation σ of the evaluated particle size distributions. By this approach, the statistical errors are reduced which enables a very precise measurement of systematic errors between both setups. In this case, systematic errors of less than 1nm can be detected without the need of any complex or expensive setup as e.g. an electron microscope.
The here shown spectroscopic experiments are based on elastic light scattering by single, homogenous, spherical particles. Therefore, Mie theory provides an exact analytical solution. It is valid for any ratio of particle size to wavelength. In contrast to this, Rayleigh scattering or Fraunhofer diffraction are only reasonable approximations if particles are small or large compared to the wavelength λ, respectively. Input parameters for the Mie calculations are the diameter of the sphere D, the wavelength of the electromagnetic wave λ and the refractive indices of the sphere ns [29, 30] and the medium surrounding it nm . Moreover, the imaginary part of the refractive indices has to be taken into account. However, the absorption of polystyrene is very low in the visible regime , thus it is neglected. Output parameters are the phase function p and the scattering cross section Cs. The phase function p is proportional to the amount of scattered light in a unit solid angle of a specific direction. Whereas the scattering cross section Cs is proportional to the likelihood of interaction between particle and plane electromagnetic wave. The theory of both experimental methods is further described in the following two subsections.
2.1. Particles in suspension measured by the collimated transmission setup
The extinction coefficient μext (λ) can be measured by the collimated transmission setup. In the case of polystyrene bead suspensions the absorption coefficient μa(λ) can be neglected. Thus, the extinction coefficient μext (λ) is equal to the scattering coefficient μs(λ). For a monodisperse suspension of spheres, the scattering coefficient is given by
2.2. Single particles measured by the spectrally resolved scattering microscope
The following steps describe the theory of the spectrally resolved scattering microscope. The calculation of the theoretical spectra is performed by Mueller matrices. The scattering matrix M is calculated by Mie theory27]. They have to be calculated for varying scattering angles Θ, sphere diameters D and wavelengths λ. The scattering angle Θ is given by the normalized wave vectors of the incident light k⃗i and of the scattered light k⃗s, Fig. 1). The vectors k⃗i and k⃗s span a plane. For φ = 0° or φ = 180°, this plane is equal to the plane spanned by the x- and the z-axis. Otherwise it is rotated by an angle ξ about the vector k⃗i. Thus, their normal vectors n⃗φ=0 and n⃗(φ, ϑ) are rotated in the same way, 27] 33]. For a single interface, it is
3. Materials and methods
The results are based on two different methods, particles in suspension measured by the collimated transmission and single particles measured by the scattering microscope. Hence, the following issues are presented separately for both methods: a detailed explanation of the setup, a short paragraph concerning the sample preparation, an instruction of the measurement procedure and, finally, a description of the raw data analysis.
3.1. Particles in suspension measured by the collimated transmission setup
3.1.1. Collimated transmission setup
As described before, with the collimated transmission setup, it is possible to measure the extinction coefficient μext of semi-transparent fluids and solids. A scheme of the collimated transmission setup is shown in Fig. 2. A collimated light beam passes through the sample, in this case a filled cuvette, having a path length d = 10mm placed in an appropriate holder. The beam has a width of 3mm, provided by a fiber based halogen lamp (HL-2000, OceanOptics, Dunedin, FL, USA) and a collimating lens. Parts of the light are scattered according to the scattering coefficient of the suspension inside the cuvette which is related to the scattering cross sections of its included particles. In a relatively long distance behind the cuvette, here 45cm, a lens focuses the unscattered light onto the small aperture of an integrating sphere. A spectrometer (USB2000, OceanOptics, Dunedin, FL, USA) is linked to the inner sphere surface via fiber optics. The complete setup is boxed, only the sample chamber is accessible for the operator. Therefore, it is very resistant and a suitable device for the comparison with other methods, e.g. the spectrally resolved scattering microscopy.
3.1.2. Sample preparation
Mie oscillations in the extinction spectrum are quenched, if the diameter distribution of the sphere suspension is too broad. The pattern of these oscillations ensures a high accuracy in the determination of this distribution. Therefore, monodisperse polystyrene particles having a relatively small size distribution are taken as samples (PS/Q-F-L1086, microparticles GmbH, Berlin, Germany). The particle size distribution is assumed by a Gaussian distribution, having a nominal mean diameter νn = 4.21 μm and a nominal standard deviation σn = 0.07 μm. This stock suspension is given in an ultrasonic bath for 30 minutes and afterwards a diluted intermediate stock is prepared. Its volume concentration should be high enough to measure a significant extinction, but low enough to avoid any side effects by multiple or dependent scattering. For this experiment a volume concentration fV ≈ 10−4 is suitable.
3.1.3. Measurement procedure
First, a reference signal I0(λ) is taken by using a carefully cleaned cuvette filled with pure water to consider any reflections at the surface of the cuvette. Moreover, a dark spectrum ID(λ) is measured by closing the shutter of the lamp. Each measurement is performed with an integration time of 400ms and averaged 10 times. The cuvette is filled with 1ml of the intermediate stock. The transmitted intensity I(λ) is measured multiple times to check for any temporal errors due to sinking particles or intensity fluctuations of the halogen bulb.
3.1.4. Data analysis
The light transmission T(λ) is given byEq. 2. Therefore a set of Gaussian distributions g(D) is created having different mean values ν = 4...4.5 μm (Δν = 0.1nm) and standard deviations σ = 0...100nm (Δσ = 0.1nm). With this, a set of theoretical extinction curves μext,T (λ, ν, σ) is created and divided by the experimental extinction curve μext,E (λ)
3.2. Single particles measured by the spectrally resolved scattering microscope
3.2.1. Scattering microscope setup
The scattering microscope enables the measurement of scattered light by single particles. Its setup was developed in a way that both, spectrally and angular resolved measurements, are possible. However, only the part of the spectrally resolved setup is explained herein. The setup is based on an inverted microscope (see Fig. 3). A reflected darkfield illumination is realized by a collimated beam that is provided by a supercontinuum laser source (SuperK Blue, NKT Photonics A/S, Birkerød, Denmark). Therefore, integration times below 100ms are possible. As shown in earlier works , a common broadband source can also be used, with the main drawback of much longer integration times. The angle of illumination ϑi = 124° is not in the detectable range of the objective ϑmax = 9.2° which is given by the numerical aperture NA = 0.16 (EC Plan-Neofluar 5x/0.16, Carl Zeiss AG, Oberkochen, Germany). Thus, no reflected light by the coverslip but only scattered light by the sample can be detected by the objective. The front focal plane of the tube lens L1 (f1 = 160mm) is situated in the back focal plane F of the objective. An iris is placed in the first intermediate image O′ which is equal to the back focal plane of the tube lens L1. Another lens L2 (f2 = 100mm) is placed 300mm behind the tube lens L1. Therefore an intermediate plane F′ of the Fourier plane can be found in its back focal plane and an image plane O″ can be found 490mm behind the first intermediate image O′. In this plane one end of a glass fiber with a core diameter of 1000 μm is positioned. The other end is connected to a CCD spectrometer (MCS-CCD-Lab, Carl Zeiss AG, Oberkochen, Germany). Alternatively, a mirror can be slid into the optical path, so instead of the fiber a camera with RGB sensor (NS1300CU, NET GmbH, Finning, Germany) acquires the object in the image plane O″. The overall magnification is given by the objective, the focal length f1 of the tube lens L1, the position and the focal length f2 of lens L2. Hence, the calculated overall magnification is 12.5. The imaging resolution of the setup is limited due to the low numerical aperture of the objective (working distance 18.5mm) but is still good enough to align samples as single polystyrene spheres or cells and cell cores. In case of spectrally resolved scattering microscopy, the small NA is an advantage as the integration over a small range of scattering angles does not cancel spectral oscillations and thus information content is preserved.
3.2.2. Sample preparation
Exactly the same stock suspension which is measured by the collimated transmission setup is reused for the single particle samples. Therefore, the suspension is diluted again by a factor of 10 with pure water and homogenized in an ultrasonic bath. Afterwards a drop of this suspension having a volume of 20 μl is placed on a coverslip. This sample is air-dried in a clean box to protect it from disturbing dust particles.
3.2.3. Measurement procedure
The coverslips are placed – with the polystyrenes on top – onto the microscope stage. Thus, forward scattered light by a particle – which is in general much stronger than backscattered light – is not reflected at the coverslip and therefore not detected by the objective. The selection of suitable single particles is done manually and randomly by the operator with help of the motorized stage and the camera (see Fig. 4). The only restriction is the minimum distance of 80 μm to the nearest particle which is dependent on the core diameter of the fiber and the overall magnification of the system. Sufficient particles are measured separately to obtain a significant statistic. Moreover, for each particle, the spectrum of a spot nearby is measured to subtract the background signal of scattered light caused by the coverslip. A reference signal, obtained by the measurement of a reflection standard, was not taken because it is not implicitly needed for the following analysis.
3.2.4. Data analysis
The analysis of the spectra is automated by a self-written MATLAB code to achieve fast, reproducible and objective results . A set of theoretical spectra IT (λ, D) with varying particle diameters D = 3...5.5 μm (ΔD = 1nm) is calculated in advance as shown in section 2.2. Each experimental spectrum IE,n(λ) is compared to this set (n is the number of sphere).
The experimental IE,n(λ) and the theoretical spectra IT (λ, D), are differentiated. The correlation Cn(D) of these derivatives is calculated for wavelengths in the range between λs = 450nm and λe = 800nm
4. Results and discussion
4.1. Particles in suspension measured by the collimated transmission setup
Suspensions from the intermediate stock were measured three times by the collimated transmission setup as explained in section 3.1.3. The mean value νCT = 4.1468±0.0007 μm and the standard deviation σCT = 0.0208 ± 0.0004 μm of an assumed Gaussian size distribution were obtained with the self-written algorithm explained in section 3.1.4. Figure 5 presents the experimentally measured extinction spectra averaged over all three measurements. Below 450nm and above 800nm, the signal to noise ratio of the spectrum is decreasing due to lack of light intensity and detector sensitivity. Moreover, the corresponding theoretical curve is plotted in dashed lines. Both curves are in very good agreement to each other. The largest deviations can be found between 500nm and 550nm, with relative differences smaller than 3%.
4.2. Single particles measured by the spectrally resolved scattering microscope
In total, 150 single polystyrene beads were measured by the spectrally resolved scattering microscope. All spectra were analyzed by the self-written correlation algorithm. The solid line in Fig. 6 represents a typical experimental spectrum of a single polystyrene bead (number n = 121 of 150). In addition, the corresponding theoretical curve obtained by the correlation algorithm is plotted. In this case the experimentally identified diameter is D121 = 4.145 μm. The experimental curve was referenced by a tenth-order polynomial and normalized onto the theoretical curve. The intensity values of the characteristic Mie oscillations show some discrepancies. However, very good agreement can be found for the spectral positions of the Mie oscillations which is important for a correct size determination of the sphere. Figure 7 gives a section of the corresponding correlation functions C(D) of the experimental and the theoretical curve shown in Fig. 6. The global maximum of both curves is at D121 = 4.145 μm. The full width at half maximum of the main lobe is 27nm for the experimental and 24nm for the theoretical spectrum, respectively. Therefore, the experimentally determined size resolution is very close to the best possible value that is given by the theory for this setup.
The diameters Dn of all 150 spheres are plotted in a histogram (see Fig. 8). Their mean diameter is νSM = 4.1442 μm and the standard deviation is σSM = 0.0269 μm. In comparison, the Gaussian distribution obtained by the collimated transmission setup is plotted as well (νCT = 4.1468 μm and σCT = 0.0208 μm). The difference of both methods in mean diameter and standard deviation are 2.6nm and 6.1nm, respectively.
In test measurements, the statistical error ΔD = 2.3nm was determined by measuring the diameter D of an identical particle several times. Deviations are caused by imperfect centering of a polystyrene sphere in the x-y-direction and varying focal planes of the objective. It is assumed that every diameter Dn is measured with the same statistical error ΔDn = ΔD. Hence, by applying the law of error propagation, the statistical error of the calculated mean diameter νSM can be derived as
Therefore, the results of both measurements show deviations in mean diameter and standard deviation that are larger than the statistical errors. Moreover, the size distribution of the single spheres differs from a Gaussian shape. Especially, the relatively large amount of spheres having a diameter Dn smaller than νSM – 3 ·σSM = 4.0635 μm is noticeable. Their corresponding experimentally measured spectra IE,n(λ) show relatively weak correlations Cn(Dn) with the theoretically calculated spectra IT (λ, Dn). A threshold Cmin guarantees that only spectra with a sufficiently good correlation are taken into account and ensures an analysis on well-approved data. This threshold Cmin was calculated by the mean value νC and the standard deviation σC of all 150 correlation values Cn(Dn)
Considering the threshold Cmin, 137 out of 150 spectra remained, the diameters Dn of these are plotted in a second histogram (see Fig. 9). For these, the mean diameter is ν′SM = 4.1471 ± 0.0002 μm and the standard deviation is σ′SM = 0.0206 ± 0.0002 μm. It still does not have a perfect Gaussian shape, but it is similar enough. Therefore it is proper to analyze the collimated transmission measurements in Fig. 5 with the assumption of a Gaussian size distribution. Mean value and standard deviation of both methods differ by 0.3nm and 0.2nm, respectively, which is in agreement within the measurement accuracy.
A novel setup of a spectrally resolved scattering microscope was presented in detail and evaluated by comparing results with a well-approved second setup, the collimated transmission. Both methods are based on spectrally resolved elastic light scattering. Diameters of polystyrene beads were determined by the analysis of the spectrally resolved scattering pattern using Mie theory. Hence, the size distribution was determined for both methods. The results are in excellent agreement, the systematic error of the mean diameters and standard deviations is 0.3nm and 0.2nm, respectively, which is within the measurement accuracy (outliers are neglected).
In comparison, other validations of spectroscopic light scattering experiments which can be found in literature show deviations in the range of several nanometers [10, 17], although similar sphere suspensions were used. Further, the nominal values given by the manufacturer (νn = 4.21 μm and σn = 0.07 μm) differ by more than 50nm. Besides, the resulting histogram from Fig. 8 includes additional information about the mismatch between the assumed and the actual size distribution, which is rarely published by the manufacturers.
It should be kept in mind that for a single measurement the statistical error of both methods depends on the size of the particles and can be in the range of a few nanometer. Nevertheless, this is still a remarkable resolution with errors in the per mil range. Thus, scattering microscopy should not only be able to detect diameters within a few nanometers of different spheres  but also temporal changes of one and the same particle or sample . This feature might be used e.g. to observe the growth of neoplastic cells  or to monitor the apoptosis of living cells . The non-spherical shape of cells and the lower differences in the refractive indices are going to be new challenges in future work.
As mentioned before the scattering microscope is suitable for angular resolved measurements, too. Details on that part are going to be published in a separate contribution.
This work was financed by the Baden-Württemberg Stiftung gGmbH and the Deutsche Forschungsgemeinschaft (DFG).
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