Spectro-angular light scattering measurements of individual microscopic objects

Abstract: The spectro-angular light scattering measurements of individual microscopic objects are presented. Using spectroscopic quantitative phase microscopy and Fourier transform light scattering, the 2D angle-resolved light scattering intensity and phase patterns are measured in a spectral range of 450-750 nm and an angular range of –70-70°. The spectro-angular light scattering measurements of individual polystyrene beads are demonstrated with high sensitivity and precision.


Introduction
Angle-resolved light scattering (ARLS) measurements enable access to information about the morphological and optical properties of scatters such as the refractive index (RI), size, and shape; it has been widely used in research fields including analytical chemistry, biotechnology, and material science [1-3].Traditionally, ARLS has been measured using goniometer-based instruments; thus, it is challenging to measure the ARLS using microscopic samples.Recently, Fourier transform light scattering (FTLS) was introduced to highly sensitively measure the ARLS signal from microscopic samples [4].In FTLS, the ARLS signal from individual scatters is obtained through numerically propagating the scalar electric field (E field) of the samples to the far fields, which is usually measured using quantitative phase imaging (QPI) techniques [5,6].The capability and sensitivity of FTLS has been demonstrated in several studies, including biological samples [7-10], colloidal clusters [11], and solar cells [12].However, the FTLS technique only measures the ARLS at a fixed wavelength and does not provide spectral information about the scattering signal.This constraint is unfortunate because spectral light scattering can provide abundant information about the scatters.Although several systems have measured angle-resolved or wavelengthdependent light scattering from individual samples [13][14][15][16][17][18][19][20], the simultaneous measurement of the spectro-angular light scattering from individual micrometer-sized samples has not been demonstrated, primarily due to the technical limitations including unavoidable azimuthal averaging, detector translation, and separate spectral channels for measuring each angular signal.
In this study, a novel light scattering measurement technique, referred to as swept source Fourier-transform light scattering (ssFTLS), is presented and it measures the spectro-angular light scattering.Using the principles of spectroscopic QPI [21] and FTLS [4], the ssFTLS measures the 2D ARLS from individual micrometer-sized samples with a spectral range of 450-750 nm and an angular range of -70-70°.The spectral and angular resolutions are less than 8 nm and 10 mrad, respectively.In addition, the optical phase information of the 2D ARLS is also obtained.The capability of ssFTLS is demonstrated through measuring the spectro-angular light scattering from individual polystyrene microspheres in a phosphate buffered saline (PBS) solution and a bovine serum albumin (BSA) solution.

Swept source Fourier-transform light scattering (ssFTLS)
ssFTLS employs the principle of spectroscopic QPI and FTLS.Spectroscopic QPI measures the E fields from a sample in the sample plane with various wavelengths, i.e.U(x,y; λ).Then, the far-field scattering signals (U s ) are calculated by numerically propagating the measured E fields to the far-field scattering plane.In FTLS, the scattered E field U s at the far-field and the E field in the sample plane are related using the Fourier transform, as follows: where λ is the wavelength of an illumination and k is the lateral spatial frequency corresponding to a scattering angle (θ) in a medium using the relationship k = 2π⋅n m ⋅sinθ/λ where n m is the refractive index of the medium.Several approaches have been reported for the spectroscopic QPI, including ones that use multiple coherent lasers [22], a broadband light source with band-pass filters [23], a wavelength-swept source [21], spectral filtering using a spatial light modulator [24], and spectral analysis of low-coherence interferogram [25].In principle, any spectroscopic QPI technique can be used for ssFTLS.In this study, the swept-source diffraction phase microscopy (ssDPM) [21] was used because it provides a wide spectral range with a high spectral resolution.The ssDPM, which is based on common-path full-field interferometric microscopy [5, 6], has a wavelength-swept source and it can measure multispectral E field imaging in the visible spectrum with a spectral range of 450-750 nm and an angular range of -70-70°.The lateral spatial resolution of the ssDPM is limited by diffraction, which is approximately 300 nm.The detailed experimental setup for ssDPM has been described elsewhere [21].
The procedure of ssFTLS is depicted in Fig. 1.First, using ssDPM, the holograms of a sample are recorded at various wavelengths [Fig.1(a)], from which the E fields that contain both the amplitude and phase information are retrieved using a field-retrieval algorithm [26] [Fig.1(b)].Then, the 2D ARLS is calculated for each wavelength using FTLS [Fig.1(c)].Then, the spectro-angular light scattering intensity signals (|U s (k x ,k y ;λ)| 2 ) can be obtained and plotted as a function of the scattering angle and wavelength after averaging in the azimuthal direction [Figs.1(d) and 1(e)].In general, ssFTLS can be used to measure 2D spectro-angular light scattering for samples with arbitrary shapes.However, only spherical samples were used in this study for simplicity without losing generality

Spectro-angular light scattering intensity of individual micro-sized beads
In order to demonstrate the feasibility and capability of ssFTLS, the spectro-angular light intensity scattering patterns of individual microspheres were measured.First, polystyrene microspheres (diameter = 3.0 μm; Invitrogen Inc., USA) that were submerged in a PBS solution were measured according the procedure described in Section 2.1.The measured spectro-angular light scattering patterns of the individual polystyrene beads are presented in Fig. 2(a).The ARLS signals at each wavelength demonstrate a characteristic oscillatory behavior.
In order to test the validity of the measurement, the measured scattering patterns were compared with the numerical simulation based on the Mie scattering theory [27].The theoretical result is presented in Fig. 2(b).The measured result was consistent with the theoretical scattering signal over the full range of wavelengths and scattering angles [Fig.2(c)].In particular, the scattering angles that corresponded to the local minima of the scattering signals were well matched for both the experiments and theory, as indicated by the arrowhead in Fig. 2(c).Furthermore, the extremely low values in the standard deviation imply excellent sensitivity and repeatability of the proposed system.Although the measured scattering intensities are slightly lower than the Mie calculation in the short wavelengths, the oscillatory features in the scattering signal remain in good agreement with the theory.In this system, the deviation may result from an uncorrected chromatic aberration.(Note that all optical lenses used in this system are achromatic.) In order to demonstrate the sensitivity of the proposed system, the spectro-angular light scattering of the polystyrene beads was measured after changing the surrounding medium from a PBS solution to 30% BSA solution (300 mg/ml).Then, the experiments were repeated for nine individual beads.As seen in Fig. 3, the measured and theoretical results are consistent over the full spectro-angular domain.Although the refractive index difference between the PBS and BSA solution was small (0.041 ± 0.001 for the spectral range of 450-750 nm), ssFTLS measured the spectro-angular light scattering with unprecedented precision and sensitivity.

Discussions and conclusions
A method has been proposed for measuring the spectro-angular light scattering from individual microscopic objects.Using the principle of spectroscopic QPI and FTLS, the quantitative and precise measurements of spectro-angular light scattering from polystyrene microbeads were demonstrated.It was also demonstrated that the optical phase of the spectroangular light scattering could be obtained.The measured spectro-angular light scattering intensity and phase patterns were well matched with the Mie theory.
In this study, the light scattering signal was obtained in the spectrum over 450-750 nm with a spectral resolution of less than 8 nm and in the scattering angle over -70-70° (-1.22 to 1.22 rad) with an angular resolution of less than 0.5° (10 mrad).The spectral range and resolution were determined using a light source and a wavelength-sweeping unit, respectively, which could be further enhanced using high-power wide-spectrum illumination (e.g. a supercontinuum laser).Furthermore, although the present ssFTLS has been demonstrated in visible wavelengths, it can be easily adapted for other spectral ranges.For example, through only modifying a part of the wavelength-swept source, the wavelength ranges can be adjusted from ultraviolet (UV) to visible and to infrared (IR).Hyper-spectral ssFTLS measurements are also possible.The scattering angle range is primarily limited through the numerical aperture of the objective lens, and the angular resolution is determined using the field-of-view in the QPI system.However, it can be further extended using an aperture-synthesis technique [30].
The temporal resolution of ssFTLS is primarily determined using the repetition rate of the wavelength-swept part.In the present system, the temporal resolution is approximately 5 s for one full spectro-angular measurement, which is limited due to the low spectral density of the light source used.However, the temporal resolution of the ssFTLS can be further enhanced using a high power source, and it may enable the investigation of dynamic spectro-angular light scattering measurements.
It is expected that ssFTLS can be used in various applications including surface inspections, resonance light-matter interactions in metallic nanostructures, and chemical compositions through the unprecedented spectro-angular resolving capability.Structural characteristics in nanoscale could be determined with high sensitivity and accuracy through a wealth of information from spectro-angular light scattering measurements [13,16].In particular, the analysis and study of spectro-angular light scattering from biological cells and tissues may enable investigations of the pathophysiology of several diseases such as malaria and sickle cell disease [31][32][33][34][35].In addition, the incorporation of ssFTLS into a commercial microscope is possible and straightforward using the principle of inline holography [36].Furthermore, when combined with a polarization-sensitive QPI technique [37], ssFTLS could be extended to investigate the spectro-angular light scattering of optically anisotropic samples.

Fig. 1 .
Fig. 1. ssFTLS procedure for measuring the spectro-angular light scattering.(a) The holograms of a sample are recorded at various wavelengths; (b) the corresponding optical fields are obtained using an appropriate retrieval algorithm; (c) the angle-resolved scattering signals at each wavelength are calculated using FTLS; and the scattering signals are presented as a function of the (d) scattering angle and (e) wavelength after averaging in the azimuthal direction.