52 Polarization describes the relationship of the *x* and *y*-component of the complex wave vector: Linearly polarized light is defined by equal phases in the two orthogonal components, which have a phase difference of 2 0 . Circular polarization is observed, if the phases are different by 2 ( 4 ) and the amplitudes are same. If the relative vertical amplitude *y J* is zero and the relative horizontal amplitude *x J* is non-zero, the wave is horizontally polarized – vertical polarization is defined analogously with 0, 0 *x y J J* . Any intermediate form with non-equal amplitudes and phases is called elliptical polarization.

 According to these measurements the Stokes vector t *S* results in

$$
\bar{S}\_1 = \begin{pmatrix} s\_1 \\ s\_2 \\ s\_3 \\ s\_4 \end{pmatrix} = \begin{pmatrix} I\_{\text{tot}} \\ I\_{\text{hor}} - I\_{\text{ver}} \\ I\_{45} - I\_{135} \\ I\_{\text{RZ}} - I\_{1Z} \end{pmatrix}. \tag{2.75}
$$

Both the Stokes and the Jones formalisms allow to describe changes in the state of polarization by using matrices that transform the input polarization vector to the output vector. In case of multiple optical elements passed, the matrices have to be concatenated by matrix multiplication. In the Stokes calculus, the transformation matrices are named Mueller matrices, in the Jones calculus, Jones matrices. Accordingly, light of polarization 1*J* passing optical elements with Jones matrices 1...n **J** is transformed to light with Jones vector

$$
\bar{J}\_2 = \mathbf{J}\_n \cdots \mathbf{J}\_1 \bar{J}\_1. \tag{2.76}
$$

Expressed in Stokes vectors, light of polarization 1 *S* passes elements with Mueller matrices **M**1...n and is transformed to

$$
\bar{S}\_2 = \mathbf{M}\_\mathbf{n} \cdots \mathbf{M}\_1 \bar{S}\_1. \tag{2.77}
$$

Both calculi are linked and Mueller matrices can be derived from Jones matrices [52]. The reverse calculation of Jones matrices from Mueller matrices only succeeds in the case without depolarization.

The purpose of polarization considerations within this work is the description of polarization effects in optical media including the system itself and the sample under test. As OCT measurements record only the coherent part of the backscattered light, depolarized light will not be detected. Hence, the Jones calculus is sufficient for the description of PS-OCT and it allows direct access to the horizontal and vertical field components.

### **2.2.2 PS-OCT concepts**

The very general aim of a PS-OCT measurement is the depth-dependent determination of polarization properties of a sample [53]. Depending on the application, it suffices to only determine birefringence [54], dichroism, the optical axis orientation [55] or a set of Stokes parameters [56] for a single incident polarization. This leads to simplifications in the optical setup and reduced measurement time, as compared to more comprehensive PS-OCT systems with capability of extracting the full Jones [57] or Mueller [58] matrix of the material. Early systems were based on conventional TD-OCT approaches [53], whereas later polarization-sensitive systems also adopt the SD-OCT or SS-OCT concept [59].

### **2.2.3 Applications of PS-OCT**

PS-OCT systems can serve for various applications. In dental OCT, buried caries lesions can be detected by PS-OCT, as demineralization accompanying carious defects leads to anisotropically ordered structures, which cause crosspolarized backscatter signals [60]. Dermatologic applications comprise analysis of burn depths that can be determined by characterization of the

birefringence difference between unaffected skin layers and layers with former heat impact [61]. This is possible, since collagen proteins in the skin are originally birefringent, but denaturate from temperatures of around 60°C onwards, while losing their polarization properties. PS-OCT is further applied to the detection of basal cell carcinoma, which lead to a local alteration of skin structure and thus to local changes in dermal birefringence [62]. Recent studies combined this analysis with a machine-learning algorithm and achieved a detection sensitivity and specificity of 93 % and 87 %, respectively [63]. An important application field of PS-OCT is ophthalmology, aiming at, e.g., earlystage detection of glaucoma. This degeneration of the retinal nerve fibre layer is in some cases preceded by a necrosis of ganglion cells inducing a replacement of nerve axons by tissue of a lower degree of organization. Hence, the birefringence of the retinal nerve fibre layer changes. A PS-OCT measurement can detect this birefringence modification and thus allows for early diagnosis of glaucoma [64]. As a use case in material sciences, PS-OCT is applied to determine mechanical stress in glass/epoxy composites, where the birefringence of the material is stress dependent in both orientation and magnitude [65]. The PS-OCT analysis offers the prospect of early failure detection.

### **2.2.4 Measurement principle**

A free-space optical setup of a PS-OCT system is depicted in Figure 2.12. With a single measurement of both received polarizations, this setup is capable of determining the birefringent properties of the sample [53].

In contrast to the conventional free-space TD-OCT, see Figure 1.2, the incident light is horizontally polarized by a polarizer (Pol) before being split among sample and reference arm. The part towards the sample passes a quarter-wave plate (QWP) at 45°, which leads to circularly polarized light incident on the sample. This can be seen from the following analysis:

A QWP with its fast axis in the horizontal direction is described by the Jones matrix [66]

$$\mathbf{J}\_{\rm QWP} = \mathbf{e}^{\frac{j\pi}{4}} \begin{pmatrix} 1 & 0 \\ 0 & \mathbf{j} \end{pmatrix}. \tag{2.78}$$

Using the rotation matrix [67]

$$\mathbf{R}\begin{pmatrix}\mathcal{G}\end{pmatrix} = \begin{pmatrix}\cos\mathcal{G} & \sin\mathcal{G} \\ -\sin\mathcal{G} & \cos\mathcal{G}\end{pmatrix},\tag{2.79}$$

the optical element, here the rotated QWP with fast axis at an angle to the horizontal direction, can be expressed by the Jones matrix

$$\begin{aligned} \text{Section, can be expressed by the Jones matrix} \\ \begin{aligned} \mathbf{J}\_{\text{QWP}}(\mathcal{G}) &= \mathbf{R} \left( -\mathcal{G} \right) \mathbf{J}\_{\text{QWP}} \mathbf{R} \left( \mathcal{G} \right) \\ &= \mathbf{e}^{\frac{1}{4}\pi} \begin{pmatrix} \cos^{2}\mathcal{G} + \text{j}\sin^{2}\mathcal{G} & \text{(1-j)}\sin\mathcal{G}\cos\mathcal{G} \\ \text{(1-j)}\sin\mathcal{G}\cos\mathcal{G} & \sin^{2}\mathcal{G} + \text{j}\cos^{2}\mathcal{G} \end{pmatrix}. \end{aligned} \tag{2.80}$$

55 j **Figure 2.12:** Free-space polarization-sensitive OCT setup. Pol: Polarizer, BS: (non-polarizing) beam splitter, QWP: quarter-wave plate at given orientations of the fast axis with respect to the horizontal direction, PBS: polarization beam splitter. The setup is a modification of the conventional TD-OCT setup shown in Figure 1.2. The sample comprises various reflections at depths *z*s,1, *z*s,2 and *z*s,3. The polarizer lets horizontally polarized light pass, and the sample is illuminated by circular polarization. The light passing the 22.5°-QWP in the reference arm forth and back returns to the BS linearly polarized at −45°. The interference signal is split at the PBS in horizontal and vertical polarization and is detected.

With 45 this results in

$$\mathbf{J}\_{\rm QWP} \left( 45^{\circ} \right) = \frac{1}{2} \mathbf{e}^{\frac{1}{4} \frac{\pi}{4}} \begin{pmatrix} 1+\mathbf{j} & 1-\mathbf{j} \\ 1-\mathbf{j} & 1+\mathbf{j} \end{pmatrix} . \tag{2.81}$$

Applied to horizontally polarized input light, as is the case here, the output polarization travelling towards the sample is

$$\begin{aligned} \text{avelling towards the sample is} \\ \bar{J}\_s = \frac{1}{2} \mathbf{e}^{\frac{j\pi}{4}} \begin{pmatrix} 1+\mathbf{j} & 1-\mathbf{j} \\ 1-\mathbf{j} & 1+\mathbf{j} \end{pmatrix} \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \frac{1}{2} \mathbf{e}^{\frac{j\pi}{4}} \begin{pmatrix} 1+\mathbf{j} \\ 1-\mathbf{j} \end{pmatrix} = \frac{\sqrt{2}}{2} \begin{pmatrix} \mathbf{j} \\ 1 \end{pmatrix}, \end{aligned} \tag{2.82}$$

which represents a circular polarization state, characterized by equal amplitudes in each of the orthogonal axes and 90° optical phase delay. Circular polarization is of advantage over linear polarization, as an arbitrary linear polarization could coincide with the optical axis of the sample. In this unwanted case, no birefringence information could be extracted, as the difference in propagation constants of slow and fast axes would not be measurable.

The light returning from the reference arm has passed a QWP at 22.5° in forward and backward direction, which results in −45° linearly polarized light, as is explained in the following calculation:

The reference arm represents a concatenation of QWP, plane mirror and QWP in reverse direction of propagation, leading to the total Jones matrix of the reference arm

$$\mathbf{J}\_{\rm ref} \left( \begin{array}{c} \mathfrak{G} \end{array} \right) = \mathbf{J}\_{\rm QWP, rev} \left( \begin{array}{c} \mathfrak{G} \end{array} \right) \mathbf{J}\_{\rm M} \mathbf{J}\_{\rm QWP} \left( \begin{array}{c} \mathfrak{G} \end{array} \right) \begin{array}{c} \begin{array}{c} \left( \begin{array}{c} \end{array} \right) \end{array} \end{array} \tag{2.83}$$

The Jones matrix **J**QWP of a rotated QWP has been evaluated in Eq. (2.80) and is unitary, as a QWP represents a loss- and gainless element [68] leading to an expression of the type

$$\mathbf{J}\_{\rm QWP}\left(\mathcal{G}\right) = \begin{pmatrix} \boldsymbol{\mu}\_1 & \boldsymbol{\mu}\_2 \\ -\boldsymbol{\mu}\_2^\* & \boldsymbol{\mu}\_1^\* \end{pmatrix} \tag{2.84}$$

with the matrix elements

$$\begin{aligned} \mu\_1 &= \mathbf{e}^{\frac{\mathbf{j}\cdot\mathbf{\pi}}{4}} \left( \cos^2 \mathcal{G} + \mathbf{j} \sin^2 \mathcal{G} \right), \\ \mu\_2 &= \mathbf{e}^{\frac{\mathbf{j}\cdot\mathbf{\pi}}{4}} \left( (1-\mathbf{j}) \sin \mathcal{G} \cos \mathcal{G} \right). \end{aligned} \tag{2.85}$$

We choose a right-handed Cartesian coordinate system, with the +z-direction being always parallel to the direction of propagation of light. Hence, when light is incident on a plane mirror, the reflected light is described by a coordinate system rotated about the *x*-axis, and the Jones matrix of a plane mirror at perpendicular incidence becomes [68]

$$\mathbf{J}\_{\mathbf{M}} = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}. \tag{2.86}$$

The Jones matrix rev **J** of reverse propagation of any optical reciprocal material with Jones matrix **J** , when taking into account a coordinate system as described above, can be written as [69]

$$\begin{aligned} \text{be written as [69]}\\ \mathbf{J}\_{\text{rev}} = \begin{pmatrix} m\_1 & -m\_3 \\ -m\_2 & m\_4 \end{pmatrix}, \quad \text{with } \mathbf{J} = \begin{pmatrix} m\_1 & m\_2 \\ m\_3 & m\_4 \end{pmatrix}. \end{aligned} \tag{2.87}$$

In case of a unitary reciprocal material like the QWP, compare Eq. (2.84), this leads to

$$\mathbf{J}\_{\text{QWP}, \text{rev}} = \begin{pmatrix} u\_1 & u\_2^\* \\ -u\_2 & u\_1^\* \end{pmatrix}. \tag{2.88}$$

Now, the total Jones matrix of the reference path can be calculated from Eq. (2.83),

57 ref QWP,rev M QWP 2 \*2 \* 1 2 1 2 1 2 \* \*2 2 1 2 1 2 1 2 cos 2 sin 2 j . sin 2 cos 2 *u u u u u u u u u u u u* **J J J J** (2.89)

This combination of forward-pass of the QWP, the mirror and the reverse pass of the QWP acts similarly to a half-wave plate (HWP) with

$$\mathbf{J}\_{\rm HWP}(\mathcal{G}) = \mathbf{j} \begin{pmatrix} \cos(2\mathcal{G}) & \sin(2\mathcal{G}) \\ \sin(2\mathcal{G}) & -\cos(2\mathcal{G}) \end{pmatrix},\tag{2.90}$$

where the second row of the Jones matrix is negated as compared to **J**ref . With the fast axis of the QWP at 22.5 with respect to the horizontal axis, the overall reference path matrix is

$$\mathbf{J}\_{\rm ref} \left( 22. \mathcal{S}^{\diamond} \right) = \mathbf{j} \frac{\sqrt{2}}{2} \begin{pmatrix} 1 & 1 \\ -1 & 1 \end{pmatrix}. \tag{2.91}$$

Applied to horizontally polarized input light, this results in

$$\vec{J}\_{\mathbf{r}} = \mathbf{j} \frac{\sqrt{2}}{2} \begin{pmatrix} 1 & 1 \\ -1 & 1 \end{pmatrix} \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \mathbf{j} \frac{\sqrt{2}}{2} \begin{pmatrix} 1 \\ -1 \end{pmatrix} \tag{2.92}$$

referring to as linear polarization at −45°. Hence, with equal amplitudes in horizontal and vertical direction, the reference field could interfere with both horizontally and vertically polarized components of the field backscattered from the sample. The polarizing beam splitter in the detection path splits both polarizations and guides them to separate detectors. Birefringence of the sample would emerge as oscillation of signal strength between the two detection polarizations versus measured sample depth. With rotation of the sample or of the optic elements in the system, the optic axis of the sample could be determined.

In free-space optical systems, the polarization can be controlled exactly at any point of the system but is sensitive with respect to mechanical misalignment. In contrast to that, fibre-based systems are more robust, but propagation of light in single-mode fibres (SMF) influences the optical phase in both orthogonal polarizations in an undetermined way and thus modifies the polarization state. In case of commonly used standard SMF, this impact is highly dependent on the exact positioning of the fibres, fibre bend stress and temperature. Thus, without calibration, the Jones matrix of a fibre is unknown. The use of polarization-maintaining fibres could circumvent this calibration step, as the incident polarization state can be maintained. This polarization-maintaining behaviour is possible by minimizing cross-talk between signals in the slow and the fast axis of the fibre, which is achieved by stress-induced differences in the individual propagation constants.

In OCT practice, polarization-maintaining fibres are difficult to deploy, since the strong group delay differences between the slow and the fast axis leads to ghost images at a relative amplitude, which corresponds to the polarization cross talk, which may, e.g., occur at each fibre-to-fibre connection. From the moment of polarization mode coupling, each portion of the respective polarization travels with a distinct speed and arrives at the detector with a respective time delay. This is equivalent to different imaging depths in the sample. With these ghost images, OCT signal analysis would become severely disturbed. For high-end PM connectors, cross-talk can be kept smaller than -30 dB. Still, with dynamic ranges of more than 60 dB, this is insufficient for many OCT applications. As a consequence, the use of polarization-maintaining fibres is avoided and we use standard single-mode fibres instead. A polarization calibration process is then mandatory.

A fibre-based system faces two challenges: First providing a method for characterization and compensation of the SMF-induced polarization change, and second, supplying the sample with incident light, with polarization different to the principal state of the sample medium. The latter property is important in order to measure material birefringence by observing a difference in propagation constants of the material. For a more comprehensive analysis of the material, e.g., when Mueller or Jones matrices are to be measured, the system needs to be able to vary the state of polarization incident on the sample.

In the following we present a system concept, which satisfies the abovementioned conditions for a fibre-based Jones matrix PS-OCT, see Figure 2.13. Light emitted by the frequency-swept light source (SS) is sent through a first polarizer, thus guaranteeing a constant input polarization during one measurement. This light propagates either towards the sample measurement head or to the calibration setup and leaves the fibre section in the sample path (SP) with polarization in *J* . The portion of the input light travelling through the reference path (RP) is adjusted in polarization, such that it later splits in equal parts on the two axes of the PBS in the detection path. This is controlled by monitoring the received power at each of the single photodiodes. This setting ensures that a received sample field can interfere with the reference field, independent of its polarization. When a sample is characterized, the sample is illuminated with a set of different polarizations, arising from different rotational positions Q of the fast axis of the QWP (QWP 2) with respect to the horizontal axis (0° to 165° with a step size of 15) with corresponding Jones matrices **J**QWP2 Q . In the sample, the light undergoes a depth-dependent polarization change s **J** *z* . This Jones matrix incorporates forward propagation to the backscattering plane at depth *z*, the scattering process itself, and backpropagation. Dependent on the material to be investigated, either material propagation, e.g., in birefringent material like skin, or the scattering processes, e.g., at elongated nanoparticles, dominate the Jones matrix. Backscattered light exiting the sample towards the fibre-optic system reversely passes the QWP 2. The reverse propagation leads to the Jones matrix **J**QWP2,rev Q , see Eqs. (2.88) and (2.85). The following fibre-optic propagation back through the SP and then towards the PBS in the receiver arm is taken into account by the matrix sys **J** . Finally, the Jones vector as incident on the PBS, and thus measured by the detector is

$$
\bar{\mathbf{J}}\_{\rm m} = \mathbf{J}\_{\rm sys} \mathbf{J}\_{\rm QWP2, rev} \left( \mathcal{G}\_{\rm Q} \right) \mathbf{J}\_{\rm s} \mathbf{J}\_{\rm QWP2} \left( \mathcal{G}\_{\rm Q} \right) \bar{J}\_{\rm in} \,. \tag{2.93}
$$

The matrices **J**QWP2 Q and **J**QWP2,rev Q are defined by the rotation angle Q of the QWP 2. The matrix of interest s **J** is unknown, as well as the system matrix sys **J** and the Jones vector in *J* of the input light. Assuming sys **J** and in *J* would be known, s **J** could be determined from a set of measurements of *J*m with different QWP-settings. However, both sys **J** and in *J* depend on the unknown polarization transformation in the fibre optic system, which makes a preceding calibration step necessary. Once both are determined, the calibration holds until fibres are moved or the temperature changes.

**Figure 2.13:** Fibre-based swept-source polarization-sensitive OCT setup. SS: Swept source, Pol 0,1: Polarizer, CPL: 3-dB fibre coupler (polarization-insensitive), SP: sample path, RP: reference path, FC: fibre collimator, QWP 1,2: quarter-wave plates with Jones matrix **J**QWP1,2 Q , PBS: polarization beam splitter, PD: photodetector, PolC Polarization controller. Jones vectors and matrices are denoted in blue colour. in *J* : Jones vector of the light exiting the fibre section towards the sample, **J**Pol1 P : Jones matrix of the polarizer in the calibration setup, **J**s: Jones matrix of the sample, **J**sys: Jones matrix of the fibre optic system connecting the FC in the sample path to the PBS, *J*m : measured Jones vector at the PBS. For polarization calibration a calibration setup consisting of a rotatable QWP (QWP 2), a polarizer , and a mirror (M2) is used in the sample arm.

In order to perform this calibration, the sample is replaced by a calibration setup, comprising a rotatable polarizer and a plane mirror, see Figure 2.13. By rotation of the polarizer POL 1 to the angular positions P 0 to 150° in steps of 30°, and by rotation of the QWP 1 to the angular positions Q 0 to 165° in steps of 15°, several polarization states can be generated and the fibreoptic system can be probed with these. During this calibration, the Jones vector received by the PBS is given by

$$\begin{array}{l} \text{undamentals} \\\\ \bar{\mathbf{J}}\_{\text{m,cal}} = \mathbf{J}\_{\text{sys}} \mathbf{J}\_{\text{QWP1,rev}} \left( \boldsymbol{\upbeta}\_{\text{Q}} \right) \mathbf{J}\_{\text{Poll},rev} \left( \boldsymbol{\upbeta}\_{\text{P}} \right) \mathbf{J}\_{\text{M}} \mathbf{J}\_{\text{Poll}} \left( \boldsymbol{\upbeta}\_{\text{P}} \right) \mathbf{J}\_{\text{QWP1}} \left( \boldsymbol{\upbeta}\_{\text{Q}} \right) \bar{\mathbf{J}}\_{\text{in}}, \end{array} \tag{2.94}$$

62 *J J* m,cal sys QWP1,rev Q Pol1,rev P M Pol1 P QWP1 Q in **J J J J J J** with the Jones matrices **J**Pol1 P of the polarizer, **J**Pol1,rev P of the polarizer in reverse direction of propagation, and the Jones matrix **J**M of the plane mirror. The mirror can be replaced by any strongly reflecting surface, e.g. by the surface of a glass cuvette containing the sample. The determination of sys **J** and in *J* results from a least-squares fit of the system model in Eq. (2.94) for several angular settings of the QWP 1 and of the polarizer.

# **3 Multi-scale dispersion-state characterization of nanocomposites using OCT**

The following chapter corresponds to publication [P2] with slight adaptations to match the format and the structure of this thesis.

*[Begin of Publication P2]*

*This article is licensed under a Creative Commons Attribution 4.0 International License*

#### **Multi-scale dispersion-state characterization of nanocomposites using optical coherence tomography**

*Scientific Reports; Volume 6; 31733; August 2016* DOI: 10.1038/srep31733

Nanocomposite materials represent a success story of nanotechnology. However, development of nanomaterial fabrication still suffers from the lack of adequate analysis tools. In particular, achieving and maintaining welldispersed particle distributions is a key challenge, both in material development and industrial production. Conventional methods like optical or electron microscopy need laborious, costly sample preparation and do not permit fast extraction of nanoscale structural information from statistically relevant sample volumes. Here we show that optical coherence tomography (OCT) represents a versatile tool for nanomaterial characterization, both in a laboratory and in a production environment. The technique does not require sample preparation and is applicable to a wide range of solid and liquid material systems. Large particle agglomerates can be directly found by OCT imaging, whereas dispersed nanoparticles are detected by model-based analysis of depthdependent backscattering. Using a model system of polystyrene nanoparticles, we demonstrate nanoparticle sizing with high accuracy. We further prove the viability of the approach by characterizing highly relevant material systems based on nanoclays or carbon nanotubes. The technique is perfectly suited for in-line metrology in a production environment, which is demonstrated using a state-of-the-art compounding extruder. These experiments represent the first demonstration of multi-scale nanomaterial characterization using OCT.

## **3.1 Introduction**

Nanomaterials represent an emerging multi-billion dollar market driven by a vast variety of applications that range from mechanical and civil engineering to energy storage and life sciences. Examples comprise nanocomposite polymers with enhanced mechanical or electronic properties [11,12], functional coatings [70], flame-retardant materials [71], advanced drug-delivery systems [13], and anode materials for Li-ion batteries [72]. These applications mostly rely on the unique properties of nanosize particles, namely huge surface-to-volume ratios, enhanced tensile strengths and superior electrical conductivities as shown by carbon nanotubes (CNT) or other nanofibers [73], or outstanding barrier properties of nanoplatelets [74]. Properties of

nanocomposites depend not only on the size and shape of the particles but also on their dispersion state, characterized by the degree of agglomeration when immersed into a liquid or solid host material. The dispersion state is governed by the nanoparticle properties, by the composition and the physical parameters of the host material, as well as by the processing route adopted for dispersing the nanoparticles in the host material. In order to ensure consistent product quality, the dispersion state must be continuously monitored during fabrication, which has been identified a key challenge both in nanomaterial development and industrial production [18]. In solid media, dispersion-state characterization mainly relies on microscopic imaging. For small nanoparticles, this requires high-resolution techniques such as scanning-electron microscopy (SEM) of specially prepared sample surfaces [19], or transmission electron microscopy (TEM) of microtome slices [20]. In both cases, sample preparation is laborious and costly and not well suited for quality control or process development, where sample processing and analysis have to be continuously iterated. Moreover, SEM and TEM are limited to small sample volumes, which are not necessarily representative of the entire batch. Light microscopy, on the other hand, can reduce the experimental effort [21], but is limited to the identification of large agglomerates in sufficiently transparent samples and areas close to the surface. In contrast to image-based methods, light scattering techniques have proven to be viable tools for measuring particle size distributions from large sample volumes. Static light scattering (SLS) relies on angleresolved and/or spectrally resolved detection of scattered light [22,23] and is based on a rather complex optical setup, especially when large scattering angles have to be taken into account. Moreover, for solid samples, it is challenging to separate scattering inside the sample volume from contributions of the rough sample surface. Dynamic light scattering (DLS) methods exploit temporal fluctuations of interference patterns of scattered light to calculate the Brownian motion and the hydrodynamic diameter of particles within the respective solvent [75]. This technique is limited to low concentrations, and can only be applied to liquid media. In addition, both SLS and DLS suffer from limitations in dealing with mixtures of particles having vastly different diameters that may range, e.g., from a few nanometres to hundreds of micrometres,

as is often the case for nanocomposites with poorly dispersed nanoparticles. Partial wave spectroscopic microscopy (PWS) [76] has been used for investigation of nanoscale refractive index fluctuations, which can be an early indicator of carcinogenesis, yet without providing a link to the size of the scatterers. X-ray diffraction (XRD) [77] or small-angle X-ray scattering (SAXS) [78] finally rely on diffraction or scattering of X-rays in the sample and are able to reveal the atomic or molecular arrangement inside the material. However, high instrumental effort and a small probing region limit the application range of these methods to the laboratory environment. Hence, none of the aforementioned techniques can meet the stringent requirements associated with industrial process development and quality control, which comprise robustness of the measurement method, fast analysis, the capability to characterize representative sample volumes without laborious sample preparation, good mechanical and thermal robustness of the measurement system, and the possibility to integrate the measurement system into the processing line for enabling in-line process control. The lack of adequate analysis methods for dispersionstate characterization is considered one of the major obstacles towards largescale industrial processing and exploitation of nanomaterials.

As an alternative, optical coherence tomography (OCT) [1,28,79] was proposed as a tool for nanocomposite characterization [80,81]. OCT provides three-dimensional imaging data from the bulk of the sample and avoids expensive sample preparation. However, previous demonstrations have been limited to image-based analysis of composites containing rather large microparticles combined with wavelet-based processing of the image data [81]. Due to the limited resolution of OCT imaging, this method cannot provide information about the nanoscopic structure of the nanocomposite such as the particle size. A further method uses a spectral-domain OCT setup and measures the size of particles suspended in a fluid by the temporal variation of the optical phase due to particle diffusion [82,83]. Particle sizes range from 15 nm to 625 nm. The main disadvantage is that only particles suspended in a fluid can be investigated, and that parameters like viscosity and temperature need to be tightly controlled. Another technique is angle-resolved OCT [84], where the angular dependence of the scattered intensity is evaluated according to Mie's theory.

However, the angular range is limited to about 0.5 rad due to practical reasons, and the particle size which can be detected is not smaller than 5 µm in diameter. A further approach is low-coherence spectroscopy, which allows the extraction of wavelength-dependent scattering coefficients of the investigated samples and compares well with Mie scattering calculations [85]. The aforementioned techniques serve well for the determination of scattering parameters from nanoparticle samples, but miss the multi-scale approach, including imaging of large agglomerates. A technique using a setup similar to spectraldomain OCT is super-resolution imaging relying on spectral encoding of spatial frequency (SESF) [86]. With that approach, sub-micrometre imaging has been demonstrated, but exact nanoparticle sizes cannot be extracted due to the still limited resolution. In summary, OCT-based quantitative and qualitative characterization of composites at the nanoscale still remains to be shown.

In this paper we demonstrate that OCT represents an attractive tool for fast and robust dispersion-state characterization of composite materials over a wide range of particle and agglomerate sizes both on the micrometre and on the nanometre scale. The technique pursues a multi-scale approach: Using a theoretical model of light scattering in the sample, we accurately measure particle sizes down to 140 nm. Particle agglomerates with sizes of up to hundreds of micrometres can be easily detected by applying dedicated image processing techniques to the OCT data. Both methods can be performed *in situ*, without prior sample preparation, in both liquid and solid materials, and are applicable to laboratory investigations as well as to in-line process control in industrial production. Using a model system of polystyrene nanoparticles dispersed in water, we prove the reliability and accuracy of our sizing technique. We further apply the technique to an epoxy resin filled with multi-wall carbon nanotubes (MWCNT). The results of OCT-based scattering analysis show good correlation with independently measured material properties, thereby outperforming conventional characterization techniques based on light microscopy. Finally, we show that our technique is also perfectly suited for in-line metrology in a production environment. To this end, we integrate our system with a state-of-the-art industrial compounding extruder using a dedicated optical probe. The OCT system operates reliably during the compounding process and

allows to immediately examine the impact of extruding parameters on the dispersion state of the material. We believe that OCT will pave the path towards industrially viable nanomaterial characterization and process control.

## **3.2 Materials and Methods**

## **3.2.1 Swept-source optical coherence tomography system**

Optical coherence tomography (OCT) evolved greatly in the past decades. The technique provides microscopic resolution in volumetric imaging and highly sensitive detection of backscattered optical power. OCT opened a wide field of applications reaching from ophthalmology in medical diagnostics [87] to particle and defect characterization in material sciences [80,88]. Among various implementations, swept-source OCT (SS-OCT) offers a particularly attractive combination of highest sensitivity and high imaging speed [28].

The SS-OCT setup used in this work is depicted in Figure 3.1 (a). In general, an OCT system measures the position and the strength of a multitude of scatterers along a light path in a sample. To this end, the electric field that is backreflected from a sample is compared in amplitude and phase to a reference field. Both the sample and the reference field are derived from the same optical swept-wavelength source (SS). In our experiments, we use an SS laser with central wavelength of 1315 nm and a wavelength scanning range of 1260−1370 nm, a scan rate of 1 kHz, and 10 dBm average output power (model s3, Micron Optics Inc., Atlanta, GA, USA). The scan range of the laser corresponds to a theoretical depth resolution of 7 µm, which compares well to the experimentally observed resolution of 11 µm. A first fibre-based directional coupler (CPL1, splitting ratio 50:50) is used to split the power among the sample path (SP) and the reference path (RP). The reference path contains a free-space section allowing precise matching of the RP and SP length. Backscattered light from the sample and light travelling along the RP is superimposed in a second fibre-based coupler (CPL2, splitting ratio 50:50) and coupled to a balanced photodetector (BD, model PDB430C, Thorlabs, Munich, Germany). The output current of the BD contains patterns resulting from

interference of the backscattered field with the reference field. The BD suppresses intensity noise from the strong RP signal and enables a large dynamic range and a high sensitivity, defined by the lowest detectable backscatter from the sample, of −110 dB. The electrical signal is digitized by a 16 bit analogueto-digital converter (ADC, model ATS660, Alazar Technologies Inc., Pointe-Claire, Canada) and processed in a personal computer (PC). The amplitude and the position of the backscatter along the light path can be obtained by Fourier analysis of the photocurrent as a function of optical frequency [28]. One depth-scan (A-scan) consists of 768 measurement points with 8 µm step size. We extract the backscatter strength, which denotes the ratio of backscattered optical power to optical power incident on the sample.

In the course of this work, the SP of the SS-OCT system is connected to two different scan heads: First, to a conventional OCT scan head for off-line characterization of laboratory samples, Figure 3.1 (b), and second, to a specially developed probe head for in-line dispersion characterization during the compounding process in an industrial extruder, Figure 3.1 (c). This probe is designed for the harsh environmental conditions at a nanocomposite production line and must tolerate vibrations, high temperatures of 250°C, and high pressures of 200 bar. The probe features a titanium shaft and a sapphire window towards the sample, and is designed to allow the adjustment of focal length and the axial position of the focus within the sample.

## **3.2.2 Model-based dispersion-state analysis and sizing of nanoparticles**

Big agglomerates of nanoparticles with dimensions larger than the resolution limit of the OCT system can be detected directly by imaging. However, size information on nanoscale agglomerates and single particles is relevant as well. In this section we show that model-based analysis of OCT backscatter measurements allows to extract scattering parameters that are correlated with the dispersion state of the material so that even small particle sizes can be determined.

**Figure 3.1:** Fibre-based swept-source optical coherence tomography (SS-OCT) setup for laboratory measurements and in-line process control. **(a)** Schematic setup of the SS-OCT. SS: Swept-wavelength source, scanning range 1260-1370 nm; CPL1,2: fibre-based directional coupler; SP: sample path; RP: reference path; FC: fibre collimator; PolC: polarization controller; BD: balanced photodetector; ADC: analogue-to-digital converter; PC: personal computer. **(b)** Laboratory scan head with galvo-based scanners and a scan lens for 3D-imaging. The red line indicates the light beam towards the sample. **(c)** Extruder probe for in-line measurements. The probe consists of a titanium shaft for a high temperature environment and comprises a sapphire window towards the sample.

The analysis relies on a single-scattering model assuming that incident light is scattered at maximum once within the medium, similar to the approach used by Kodach *et al.* [89]. This model is found to be appropriate for weakly scattering samples and for an analysis of moderate scattering depths [90]. As depicted in Figure 3.2 (a), a light beam with input power *P*in enters the sample. At each particle (grey), a first portion (blue) of the incident light in the respective depth *z* is scattered back into the numerical aperture of the optical system, a second portion (red) is scattered into all other directions or is absorbed, and the remaining third portion (black) is propagating deeper into the medium. The total scattering s and absorption cross section a can be described by the extinction cross section t s a of a single particle, or by the extinction coefficient t t *N* for an ensemble of particles with volume number density *N*. In analogy, the backscatter is described by the backscatter cross section b, or by the backscatter coefficient b b *N* of the material.

The depth-dependent decay t d d *P z z P z* of the forward-travelling power *P z* is dictated by the extinction coefficient t , leading to an exponential decay of optical power t ine *z P z P* inside the sample. For a given depth *z*, the power in a depth element *z* as scattered back into the numerical aperture of the optical system amounts to t b b e *z P z z P z* , where t e *z* accounts for the extinction of the backscattered light during backpropagation. The signal measured in an OCT scan corresponds to the depthdependent backscatter *P z P* b in . In the presence of a depth-independent noise floor *R*n, the depth-dependent backscatter signal measured by the OCT system is

$$\mathcal{R}\_{\mathbf{s}}\left(z\right) = \mu\_{\mathbf{b}} \,\,\delta z \,\,\mathbf{e}^{-2\mu\_{\mathbf{t}}z} + \mathcal{R}\_{\mathbf{n}}\,. \tag{3.1}$$

A semi-logarithmic plot *R z R z* dB s 10 lg of this backscatter signal is depicted in Figure 3.2 (b). The background noise term *R*n defines the sensitivity limit of the OCT system. For real OCT systems, this background noise is sometimes dominated by relative intensity noise (RIN) of the swept source, which may lead to a depth-dependent noise floor. In contrast to that, the noise floor in our system originates from thermal noise of the receiver electronics, which exhibits a white power spectrum and is hence constant over the depth range of interest. Note that all measurements for particle sizing are taken under oblique incidence of the OCT beam on the sample surface. This avoids occurrence of isolated reflection peaks at the sample surface such that the signal model according to Eq. (3.1) can be directly used to fit the measurement data.

In real OCT measurements, the measured backscatter depends on further parameters, which need to be determined in a calibration step. This comprises the divergence of the measurement beam, the decay of the coherence function of the swept laser, and the absorption of the matrix material which surrounds the scatterers. These influences are approximated by including two calibration factors [80,91] *q* and *Q* in the single-scattering model according to Eq. (3.1),

$$R\_{\mathbf{s}}\left(z\right) = \mathcal{Q}\,\mu\_{\mathbf{b}}\,\,\delta z \,\,\mathbf{e} \,\, \mathbf{e}^{-2z\left(\mu\_{\mathbf{t}} + q\right)} + R\_{\mathbf{n}}\,.\tag{3.2}$$

Both calibration factors are determined by comparing measured backscatter of a NIST-traceable polystyrene particle standard (246 nm diameter polystyrene nanospheres in 0.5 wt.% aqueous dispersion, BS-Partikel GmbH, Wiesbaden, Germany) with the model calculations according to Eq. (3.1).

In the special case of spherical scatterers with a size in the order of the wavelength, the scattering cross section s can be modelled by means of Mie's theory [50]. As an example, Figure 3.2 (c) shows direction-dependent scattering lobes of water-dispersed polystyrene nanospheres with diameters 143 nm and 506 nm. The larger sphere (506 nm diameter) shows stronger total scattering (red), but less pronounced fractional backscatter as compared to the small sphere (143 nm diameter). For a small aperture of the scan head (1° half-angle, corresponding to a theoretical and experimental lateral resolution of 28 µm and 36 µm, respectively), the Mie scattering lobes can be parameterized using the total scattering cross section s and the backscattering cross section b. Figure 3.2 (d) shows both cross sections s and b as a function of the sphere diameter *d* for polystyrene (PS) spheres (refractive index *n*PS = 1.57) dispersed in water (refractive index H O2 *n* 1.33 ) at a wavelength of 1315 nm. In the limit of small diameters *d*, Mie scattering can be approximated by Rayleigh scattering. In this regime, both scattering cross sections increase in proportion to *d* 6 . For larger diameters, the spheres show resonances, which lead to dips in the backscattering cross section b, whereas the total scattering cross section remains unaffected. The relationship between backscattering cross section b and particle size is unambiguous only if the particle diameter is smaller than 460 nm, which corresponds to roughly half the material wavelength in the polystyrene spheres.

In real measurements, the total scattering cross section s and the backscattering cross section b cannot be assessed directly. Instead, only the extinction coefficient t t *N* and the backscatter coefficient b b *N* are extracted from the backscatter signal *R*s(*z*). For non-absorbing particles in the Rayleigh scattering regime, particle size and concentration cannot be separately evaluated, since both s and b increase with *d* 6 , Figure 3.2 (d). As an example, for particle sizes of less than 150 nm (/ 9), we can safely assume Rayleigh

scattering, and the backscattering probability *p*b = b / s stays constant within 10 %. As a consequence, a low concentration of bigger particles cannot be distinguished from a high concentration of smaller particles. However, for the case of nanomaterial characterization, the total volume of nanoparticles added to the sample is usually known.

**Figure 3.2:** Concept of model-based dispersion state analysis and sizing of nanoparticles. **(a)** Single scattering model: Incident light is scattered at maximum once inside the medium. A light beam with power *P*in enters the sample. At each particle (grey), a first portion (b, blue) of the incident light in the respective depth *z* is scattered back into the numerical aperture of the optical system, a second portion (t, red) is scattered into all other directions or is absorbed, and the remaining third portion (black) propagates deeper into the medium. **(b)** Schematic profile of the logarithmic backscattering factor *R z R z* dB s 10 lg . **(c)** Direction-dependent scattering lobes according to Mie's theory, plotted here for the example of polystyrene (PS) nanospheres (diameters 506 nm and 143 nm, refractive index PS *n* 1.57 at 1315 nm) in aqueous dispersion ( H O2 *n* 1.33 ). s: total scattering (red); b: backward scattering (blue). **(d)** Total scattering s and backscattering b cross sections for non-absorbing polystyrene spheres in aqueous dispersion.

For an increasing degree of dispersion, the average size *d* of the agglomerates decreases and their volume number density *N* increases in proportion to *d* −3 . Together with the *d* 6 -dependence of the scattering cross sections in the Rayleigh regime, this leads to an overall decrease of the scattering coefficients in proportion to *d* 3 , which allows robust separation of particle size *d* and volume number density *N*. Note that this applies to non-absorbing particles only. If the extinction coefficient t is dominated by the contribution of absorption rather than scattering, an increased degree of dispersion could lead to an increase of t. This is due to the fact that the breaking-up of agglomerates exposes more particles from the inner region to the incident light. These particles from the inner region were formerly shielded from light by the absorbing outer shell, and did therefore not contribute to attenuation. If absorption dominates, the effect of increasing extinction with increasing number of separated absorbing particles can be exploited for the analysis of the dispersion state. We use this approach in the section on dispersion analysis of epoxy-CNT composites.

### **3.2.3 Image-based dispersion-state analysis**

Although nanocomposites ideally feature a homogeneous distribution of nanosized fillers, in practice, microscale agglomerates cannot be completely avoided. The size of the agglomerates could reach several hundreds of micrometres, especially at the beginning of the so-called compounding process, which usually exploits shear forces to break up particle agglomerates into their nanoscale constituents. Therefore, a dispersion-state analysis suitable for process monitoring should be able to cope with nanometre to micrometre sized objects. This section is dedicated to imaging and analysis of agglomerates in the micrometre range.

Optical coherence tomography can be used to visualize agglomerates, if their dimensions exceed the spatial resolution *x*, *y*, *z* of the imaging system in *x*, *y* and *z*-direction (about 10 µm), and if the backscattering is stronger than the background noise. Below these limits, an analysis based on a scattering-model has to be applied. To identify agglomerate regions with stronger backscatter within the OCT image, we use an image segmentation algorithm based on

seeded region growing [92]. For quantitative information related to agglomerate size and number, two independent parameters are extracted from the segmented images, namely the area fraction (AF) and the perimeter-to-area ratio (PAR) of the agglomerates, see Figure 3.3. The area fraction relates the image area covered by all identified agglomerates with individual area *Ai* to the entire imaging cross section *A*tot,

$$\text{AF} = \frac{\sum\_{i} A\_{i}}{A\_{\text{tot}}}.\tag{3.3}$$

For a given volume fraction of the nanosize filler, and assuming spherical agglomerate shapes which are accurately detected by an ideal measurement system, the AF would be independent of the agglomerate diameter. This can be understood by the following consideration: Assume that each spherical agglomerate splits and decreases in radius by a factor of . The total amount of material remains constant, therefore the number of (smaller) agglomerates in the volume increases by 3 , whereas the number of agglomerates in the measurement plane increases by 2 . The average area *Ai* of each individual cross section in the measurement plane decreases in proportion to 2 . Therefore the radius change of the agglomerates has no effect on AF. This would render AF as non-indicative for agglomerate analysis. For real measurement systems, however, a decreasing average agglomerate size will increase the number of agglomerates that are smaller than the detection threshold of the image-based analysis technique. In this case, a decrease of AF is observed which is correlated with a decreasing agglomerate size. In our experiments, the detection threshold is set to three times the standard deviation of the background noise floor in the image. While the threshold influences the measured percentage of the area fraction, our choice of the threshold level suffices to judge the dispersion of a certain sample type.

As a further parameter, the perimeter-to-area ratio (PAR) relates the sum of all agglomerate perimeters *si* to the sum of all agglomerate areas *Ai*,

$$\text{PAR} = \frac{\sum\_{i} s\_i}{\sum\_{i} A\_i} \tag{3.4}$$

Assuming agglomerates of equally distributed shapes, the PAR depends only on the average size of the agglomerates and is otherwise independent of the number density. A change in the average shape of all agglomerates during the dispersion process would influence the PAR, but this is not to be expected for typical dispersion processes that rely on milling or shear-strain-induced breaking of agglomerates. If the dimensions of each agglomerate decrease by a factor of , the perimeter decreases by , while the cross sectional area decreases by 2 . Accordingly, the perimeter-to-area ratio of each agglomerate increases by a factor of , and the same applies to the overall PAR, independently of the filler content. Image-based analysis is applied to OCT measurements of nanocomposites both offline and during production.

**Figure 3.3:** Illustration of agglomerate areas and perimeters in a cross-sectional OCT image. The total image cross section amounts to *A*tot; the quantities *A*1…*An* denote the areas (grey) and the quantities *s*1…*sn* (red) denote the perimeters of the agglomerates. The area fraction (AF) is a measure for total agglomerate content and relates the agglomerate areas *A*1…*An* to the total image cross section *A*tot. The perimeter-to-area ratio (PAR) is a measure for the size of the found agglomerates and relates perimeters *s*1…*sn* to the areas *A*1…*An*.

## **3.3 Results and discussion**

To prove the viability of OCT as a tool for characterization of nanoparticles and nanocomposite materials, we performed a series of experiments comprising accurate nanoparticle sizing as well as nanocomposite dispersion state analysis in liquid and solid materials. Our characterization employs image evaluation and a simplified model-based scattering analysis. We show that OCT methods are useful for measuring nanoscopic particle sizes as well as for

analyzing the dispersion of agglomerates in the micro- and millimetre regime. The technique can even cope with highly absorbing CNT-loaded materials and is well suited for in-line process control.

#### **3.3.1 Model-based sizing of nanoscale particles**

Polystyrene (PS) nanospheres (refractive index *n*PS = 1.57) in aqueous dispersion (refractive index H O2 *n* 1.33 ) are used as a model system to prove the viability of OCT as a tool for model-based nanoparticle sizing. This model system offers higher refractive index contrast than the sample systems investigated in following sections (polyamide/clay: *n*PA = 1.59, *n*Clay = 1.54; polypropylene/clay: *n*PP = 1.49, *n*Clay = 1.54). All these samples offer sufficient backscattering levels. In this experiment, we characterize dispersed NISTtraceable polystyrene particles (BS-Partikel GmbH, Wiesbaden, Germany) with diameters of 143 nm and 246 nm. The results are depicted in Figure 3.4. The particle size is determined by fitting the calibrated single scattering model according to Eq. (3.2) to measured OCT depth scans as described in the previous section. We use three different samples S1, S2, and S3 with different diameters and particle concentrations expressed in weight-% (wt. %) of PS particles in the dispersion, see Figure 3.4 (d) and (e) for scanning electron images of the dried particles. The nominal size and concentration of the investigated dispersed particles amount to 143 nm at 2.0 wt. % for S1, 143 nm at 0.5 wt. % for S2, and 246 nm at 0.2 wt. % for S3, respectively. Figure 3.4 (a) depicts averaged OCT depth scans (light coloured circles) along with the curves of the fitted single-scattering model (bright colours). For the fit, measurement data have been taken into account only up to a depth of 1.8 mm (225 measurement points) to avoid inaccuracies by multiple scattering [93]. The extinction coefficient t and the backscattering coefficient b can be extracted from the fit: Sample S1 (143 nm, 0.5 wt.%): t1 = (85 ± 23) m −1 , b1 = (97.0 ± 7.4)×10−4 m −1 ; Sample S2 (143 nm, 2.0 wt.%): t2 = (352 ± 26) m −1 , b2 = (53.6 ± 5.8)×10−3 m −1 ; Sample S3 (246 nm, 0.2 wt.%): t3 = (125 ± 7.5) m −1 , b3 = (18.0 ± 2.0)×10−3 m −1 . The error bounds refer to the standard deviation of the averages. Since the particle concentrations and hence the number densities *N* are known, we can translate

these coefficients directly to the scattering cross sections s and b, assuming that absorption of the PS particles can be neglected at the measurement wavelength around 1315 nm. The following total scattering s and backscattering cross sections b have been extracted: Sample S1 (143 nm, 0.5 wt.%): s1 = (25.9 ± 7.0) nm 2 , b1 = (29.7 ± 2.3)×10−4 nm 2 ; Sample S2 (143 nm, 2.0 wt.%): s2 = (26.9 ± 2.0)nm2 , b2 = (41.0 ± 4.4)×10−4 nm 2 ; Sample S3 (246 nm, 0.2 wt.%): s3 = (488 ± 29) nm 2 , b3 = (70.1 ± 7.7)×10−3 nm 2 . The values given include the standard deviation of the averages. The cross sections s and b can be related to the diameters of the respective particles using Mie's theory [50], see Figure 3.4 (b). The total scattering cross section s (left axis, red) and the backscattering cross section b (right axis, blue) of polystyrene nanospheres dispersed in water are calculated and plotted as a function of the sphere diameter, assuming incident light of 1315 nm wavelength and a system aperture of 0.018 (1° half-angle).

**Figure 3.4:** Size determination of sub-wavelength nanoparticles with OCT. **(a)** Measured OCT depth scans of aqueous dispersions of polystyrene (PS) nanospheres (S1: 143 nm diameter, 0.5 wt.%, grey circles: measurement, black line: fit; S2: 143 nm diameter, 2.0 wt.%, light green circles: measurement, green line: fit; S3: 246 nm diameter, 0.2 wt.%, light orange circles: measurement, orange line: fit), fitted with the calibrated single-scattering model according to Eq. (3.2). **(b)** Total scattering (red) and backscattering (blue) cross sections, according to Mie's theory for polystyrene spheres in water. Both scattering cross sections are compared with the measurements, depicted only for the samples S1 (s1: total scattering cross section, b1 backscattering cross section) and S3 (s3, b3, respectively). Both parameters can be attributed to sphere diameters, horizontal axis. **(c)** Comparison of nominal particle sizes with measured sizes. Relative (absolute) deviations from the nominal values are maximal 4 % (5 nm). **(d,e)** Scanning electron micrograph (SEM) of the nanospheres having diameters of 143 nm and 246, respectively.

From the measurement fits in Figure 3.4 (a), the scattering parameters s1, b1 for sample S1 and s3, b3 for sample S3 are extracted. The sphere diameter can then be read from the Mie calculations displayed in Figure 3.4 (b), vertical arrows. For clarity, the analysis is only depicted for sample S1, but sample S2 yields very similar results.

The table in Figure 3.4 (c) summarizes the results as derived from scattering cross section s and backscattering cross section b, together with the associated standard deviations of the averages, and compares them with the nominal particle sizes and their standard deviations. The standard deviation measured by OCT can be lower than the standard deviation of the particle size distribution, since various particles contribute to one OCT measurement. Measurement errors may arise from refractive index uncertainties, where an uncertainty of 0.01 would lead to a size determination error of 2 nm to 4 nm. Relative deviations between nominally and measured particle sizes are below 4 %.

### **3.3.2 Model-based nanoscale dispersion-state analysis**

Owing to the high sensitivity of the OCT technique, even strongly absorbing nanomaterials such as carbon-nanotube (CNT) composites can be analysed. This is a key feature, as CNT-loaded composites represent an important market segment [94]. Standard light scattering techniques like DLS or SLS are not applicable because the scattered or transmitted optical powers are small. Conventional CNT dispersion analysis relies either on thin material layers that are investigated in a light microscope (LM), or on the determination of macroscopic material parameters like the dielectric permittivity that can also indicate the dispersion quality [95]. For our experiments, we use a multi-wall CNT (MWCNT) dispersion in an epoxy resin (Araldite LY 556, Huntsman Advanced Materials GmbH, Basel, Switzerland). The samples contain 0.12 wt. % of MWCNT (NC7000, Nanocyl S.A., Sambreville, Belgium), and were prepared by 1, 3 and 5 milling cycles in a three-roll mill. We compare the results of OCT-based backscattering and extinction parameter analysis with the results of standard LM analysis as well as with rheological and dielectric measurements. For OCT measurements, the CNT-filled resin samples were filled

into cuvettes and heated up to a temperature of 50°C for melting the resin, thereby avoiding scattering from resin crystals.

The results of the dispersion analysis are shown in Figure 3.5. Conventional dispersion analysis of thin composite layers by light microscopy, see Figure 3.5 (a), reveals significant agglomerates after the first milling cycle. After three milling cycles, the size of agglomerates reduces. After five milling cycles, the light microscope images could lead to the conclusion that dispersion quality seems to decrease again, but other measurement techniques, like OCT analysis and rheological characterization, contradict this finding. Light microscopy for dispersion analysis, while being a state-of-the-art technique, suffers from small imaging areas. This could lead to an erroneous interpretation, if the agglomerate concentration is not spatially homogeneous over the whole sample. The same samples were investigated using OCT, see Figure 3.5 (b), which shows extinction coefficients t and the backscattering coefficients b in a two-dimensional plot. For each of the samples, we take ten measurements, each consisting of 5000 depth-scans which were taken while laterally moving the sample over a distance of 2 mm. The depth scans are averaged, and the extinction coefficient t and the backscattering coefficient b are extracted by fitting Eq. (3.2) to the measurement data. This has been repeated at ten different regions of each sample, each region corresponding to a cross in Figure 3.5 (b). Note that the particle sizes cannot be extracted from these data: In contrast to the situation for pure nanosphere dispersions used as a reference, the CNT agglomerates exhibit a large variety of shapes, and a Mie scattering theory based on simple spheres cannot be applied. Yet, the analysis of the extinction and backscattering coefficients suffices for a qualitative dispersion analysis. The horizontal and vertical error bars in Figure 3.5 (b) represent the average (solid dots) and the standard deviation of the measurements obtained for each sample. Although it is not possible to exactly discriminate between scattering and absorption, we may assume that the high extinction values of 5 mm −1 up to 15 mm −1 arise mainly from absorption of the CNT and only marginally from scattering. This is supported by strong reported absorption [96] of 1000 mm −1 due to separated CNT in composites with comparable concentrations. Consequently, we did not take multi-scattering into account. A

clear tendency with respect to the number of milling cycles can be seen: The more agglomerates are broken, the more isolated CNT can contribute to absorption and hence to the extinction t, which rises significantly between one and three milling cycles, and increases slightly for five cycles. Simultaneously, the backscattering coefficient b increases with the number of milling cycles and shows the same tendency as the extinction t. The backscattering coefficient b = b *N*, however, does not necessarily increase if the dispersion quality increases so that the agglomerates become smaller and their number *N* becomes larger. It is impossible to state in general how smaller agglomerates contribute to the backscattering inside the light receiving aperture, because b strongly depends on the angle dependency of the backscattering for the respective agglomerate shape. In our specific case, b increases with an increasing degree of dispersion, and therefore allows a qualitative judgement of the state of the sample dispersion.

These results of the OCT analysis are confirmed by a rheological characterization of the relative permittivity r. For this purpose, the dispersions were investigated using a rheometer (MCR 501, Anton Paar GmbH, Graz, Austria) in combination with a programmable LCR bridge for measuring inductance, capacitance and resistance (HM 8118, HAMEG Instruments GmbH, Mainhausen, Germany). The rheometer consists of two parallel rotatable plates at a separation of *s* = 1 mm. The sample dispersion fills the gap between the plates, which are isolated electrically from each other and which are connected to the LCR bridge for measuring the capacitance. During a measurement, one of the plates performs an oscillating rotation. The amplitude *b*r of the oscillation at the outer radius of the plate, related to the plate separation *s* results in the strain amplitude r *b s* . Both the measurement setup and the measurement procedure are described in detail in Ref. [95]. The relative permittivity is measured at a frequency of 1 kHz and increases monotonically from the pure epoxy material over 1 to 3 to 5 milling cycles: This indicates a continuous improvement of dispersion quality, see Figure 3.5 (c), since an increasing number of CNT are separated from agglomerates and contribute to the electric polarization of the nanocomposite. Note that the rheological measurements themselves introduce shear strain into the samples, which could lead to further

exfoliation of CNT from the agglomerates and hence to an increase of the relative permittivity during characterization of the samples after three milling cycles. This additional exfoliation might be not possible for the mixture after one milling cycle (insufficient exfoliation) and five milling cycles (largely complete exfoliation). At higher strain amplitudes (not shown), the separated CNT align in parallel to the rheometer plates and thus perpendicularly to the electric field. In this case the CNT cease to influence the permittivity.

**Figure 3.5:** OCT characterization of highly absorbing epoxy/carbon nanotube (CNT) composite materials. Comparison to conventional characterization techniques and macroscopic material properties. The composite was milled over 1, 3 and 5 cycles in a three-roll-mill to improve dispersion. **(a)** Conventional dispersion state analysis by light microscopy of thin composite layers. Black areas are attributed to agglomerates. **(b)** OCT-determined backscattering b and extinction t coefficients, measured in 10 different spatial regions (crosses) of each sample (average: solid dots; standard deviations: horizontal and vertical error bars). **(c)** Rheologic dielectric characterization of the relative permittivity r as a function of applied shear strain after 1, 3, and 5 milling cycles. The relative permittivity increases monotonically from the pure epoxy material over 1 to 3 to 5 milling cycles, indicating an increase in the separation of the CNT.

In contrast to the microscopy analysis and the permittivity measurement, OCT characterizes the heated sample without any further preparation. The results correlate well with the permittivity measurement and with the number of dispersion cycles. In contrast to light microscopy, OCT also reveals the increase in dispersion quality between the third and the fifth dispersion cycle. This experiment demonstrates that OCT metrology is useful to characterize even highly absorbing nanomaterials, outperforming by far the elaborate and time consuming conventional light microscopy method.

### **3.3.3 Image-based dispersion-state analysis for microscale agglomerates**

For experimentally assessing the ability of the OCT technique to analyse nanocomposites with large agglomerates, composites of polyamide (PA; Badamid B70, Bada AG, Bühl, Germany) and nanoclay particles with 5 wt.% concentration were prepared in a compounding extruder [97]. For controlling the dispersion quality during fabrication, one batch of samples was prepared from a clay with modified surface (I.34, Nanocor Inc., Hoffman Estates, IL, USA), the other one was prepared from unmodified clay (PGN, Nanocor Inc.). The surface modification is expected to lead to an improved dispersion state and to decreased sizes of agglomerates as compared to the unmodified clay. After compounding, all samples are granulated without further sample processing. Cross-sectional images (B-scans) are taken from the granules with our laboratory OCT system. Representative B-scans from the PA/nanoclay composite without and with surface modification are shown in Figure 3.6 (a) and (b), where the pixel brightness indicates measured backscattering. The sample with no modification, Figure 3.6 (a) shows large lengthy bright areas, indicating large clay agglomerates that extend over several hundreds of micrometres, whereas the sample with modification, Figure 3.6 (b), features small bright spots only from which small agglomerates can be inferred. The image segmentation algorithm detects bright regions automatically; the corresponding borders are drawn as red lines.

For each sample type, three B-scans are taken, segmented, and the area fraction (AF) as well as the perimeter-to-area ratio (PAR) are calculated for each measurement. Figure 3.6 (c) and (d) display the average AF and the average PAR for both sample types along with the corresponding error bars showing the standard deviation of the averages. A significant decrease in the AF is obtained when the surface of the clay filler is modified (AFmod = 2.0 % ± 0.4 %) as compared to the filler without surface modifications (AFunmod = 5.3 % ± 0.8 %), Figure 3.6 (c). This indicates that a surface modification significantly decreases the agglomerate size such that a substantial fraction of agglomerates becomes smaller than the size detection threshold of the image analysis. This is in accordance with the observation that the cross-sectional image of the surface-modified clay composite exhibits a large number of bright spots, each of which contains only a few pixels, see Figure 3.6 (b). At the same time, the PAR increases significantly with modification of the filler surface (PARunmod = 132 mm −1 ±8.2 mm −1 , PARmod = 178 mm −1 ±4.1 mm −1 ) which confirms the decrease in average agglomerate size, Figure 3.6 (d).

**Figure 3.6:** Image-based dispersion-state analysis for microscale agglomerates, showing the impact of chemical surface modification on dispersibility. **(a)** Cross-sectional image (B-Scan) of a PA/nanoclay composite with 5 wt-% nanoclay content without surface modification. Bright spots (strong backscattering) with red borders indicate agglomerates. Right part: magnified section. **(b)** Cross-sectional image of a PA/nanoclay-composite with surface-modified clay particles having the same concentration as the sample in (a). Right part: magnified section. **(c)** Area fraction (AF) covered by detected agglomerates. The circles denote average values of the three samples, and the error bars indicate the standard deviation of the average. **(d)** Perimeter-to-area ratio (PAR) of the detected agglomerates. The circles denote average values of the three samples, and the error bars indicate the standard deviation of the average.

These experiments prove the viability of image-based OCT analysis to characterize the dispersion state of samples with relatively large agglomerates sized from a few micrometres to hundreds of micrometres. The quantitative evaluation of further dispersion-related parameters like agglomerate shape and number would further increase the reliability and robustness of our technique.

## **3.3.4 Demonstration of in-line dispersion-state analysis**

Nanocomposite development is hampered by rather long development cycles, which are dominated by time consuming off-line characterization. In this section, OCT is demonstrated to be well-suited for continuous dispersion-state monitoring in a production environment.

In the following experiment, a production-scale twin-screw compounding extruder (Leistritz GmbH, Nuremberg, Germany) is used for production of a polypropylene(PP)/nanoclay composite with a mass throughput of 6 kg/h. The applied host material is PP (R352-08R, Dow Chemical, Midland, MI, USA) and the nanosized filler is a nanoclay (Cloisite 15A, Southern Clay Products, Gonzales, TX, USA). The screws introduce shear into the polymer melt filled with nanoparticles and thus disperse the particles [97]. With this extruder, nanocomposites were dispersed with different revolution speeds. In general, increasing speed comes with higher energy input into the material and causes a better nanoparticle dispersion [21]. In order to characterize the dispersion state of the PP/clay melt during production, the extruder has been equipped with the OCT probe for in-line process monitoring, see Figure 3.1 (c). The probe is mounted to a sensor port of the extruder located at the end of the machine close to the die. The optical window of the probe is in direct contact with the main stream of the extruded nanoparticle loaded polymer melt. With the OCT probe, A-scans of the medium underneath the probe window are measured continuously. The flow of the melt causes a continuous movement of the material seen by the OCT measurement beam, thereby replacing a lateral movement of the measurement beam in conventional measurements. This results in a temporal change of the backscatter signal, which can be interpreted as 2D image data, where one dimension corresponds to time and the other

dimension is the usual imaging depth measured from the probe window, see Figure 3.7.

Figure 3.7 (a), (b) and (c) show the in-line OCT data for the screw revolution speeds 200 rpm, 500 rpm and 800 rpm, respectively. Note that, due to nonuniform mass flow as a function of depth, it not possible to directly map the measurement time (horizontal axis) to spatial coordinates of the sample. Bright spots indicate strong backscatter and are attributed to agglomerates. The straight horizontal line at *z* 0 originates from permanent reflections at the probe window. With higher screw speed and accordingly higher shear strain inside the material, fewer agglomerates are visible indicating an improvement in dispersion quality. At the highest screw speed of 800 rpm, almost no agglomerates are visible. This impression is also confirmed by quantitative dispersion-related parameters. Since the measurement is performed as a function of time rather than as a function of lateral position, a perimeter-toarea ratio in the strict sense as defined by Eq. (3.4) cannot be calculated.

**Figure 3.7:** Demonstration of in-line dispersion-state analysis during operation of a compounding extruder. The OCT measurement beam is fixed, the vertical axis of the OCT images indicates the imaging depth and the horizontal axis corresponds to the measurement time. Bright spots: strong backscatter of agglomerates. Straight horizontal line at *z* 0: permanent reflections at the probe window. Material system: polypropylene/nanoclay. Mass flow: 6 kg/h. Clay content: 1 %. Bottom: magnification after segmentation. **(a)** OCT scan at 200 rpm. **(b)** OCT scan at 500 rpm. **(c)** OCT scan at 800 rpm. **(d)** Quantitative analysis of the dispersion state by area fraction (AF) and modified perimeter-to-area ratio PAR from five OCT scans. The circles denote average values of the five scans, and the error bars indicate the standard deviation of the averages.

Instead, we use a modified quantity PAR , which essentially corresponds to the PAR except that both, the agglomerate perimeters *s*i and the agglomerate areas *A*i, are expressed by pixel numbers rather than by physical lengths and areas. Note that PAR is a dimensionless quantity in contrast to PAR, which has the unit m−1 . The results are depicted in Figure 3.7 (d), where the circles denote average values from five OCT scans, and the error bars denote the respective standard deviations of the averages. With increasing speed, the area fraction AF reduces, and the PAR rises. Both quantities indicate that the agglomerates inside the material flow become smaller corresponding to a better dispersion quality of the composite. These results reveal a clear relationship between operation parameters of the machine and OCT-measured dispersion parameters. This is the first demonstration of an in-line dispersion-state analysis in a nanocomposite production line. The results encourage the application of our technique to more material systems for controlling multiple production parameters.

## **3.4 Conclusions**

In this work, we present a novel and a highly attractive approach to nanomaterial analysis applied to nanoparticles and nanocomposites. The method uses optical coherence tomography (OCT) and model-based parameter extraction. Our approach enables detection and quantification of nanoparticles and agglomerates over a wide range of size scales: Image segmentation of OCT data sets is well suited for dispersion-state characterization of nanocomposites with agglomerates in the micrometre range, whereas model-based scattering analysis lends itself to size determination of nanoparticles below the resolution limit. We elaborate the measurement technique along with theoretical models and demonstrate the viability of the procedure in a series of proof-of-principle experiments. A wide variety of material systems is investigated by our experiments: A first demonstration shows accurate OCT-based nanoparticle sizing. An in-situ dispersion-state analysis characterizes strongly absorbing CNT composites in liquid media. Finally, we perform in-line monitoring of the compounding process in a state-of-the-art production line. Major challenges in

industrial applications are the stringent requirements with respect to mechanical stability and size of the OCT system. These requirements can be met by integration of interferometer and detection system on a silicon photonic chip [98]. We believe that OCT has the potential to fill a metrology gap in the emerging field of nanocomposite technology. We conclude that OCT metrology opens new directions in material analysis, both in laboratory and production environments.

*[End of Publication P2]*

# **4 Nanoparticle size and shape characterization using PS-OCT**

The following chapter is a manuscript intended for publication in a peerreviewed journal. Slight adaptions have been made to match the format and the structure of this thesis.

Nanoparticles of diverse shapes play an increasingly important role in medical and technical applications. Their tiny size comes along with unique physical properties, which are tightly coupled to their actual size and shape. In consequence, continuous quality control in nanoparticle fabrication is mandatory. Transmission or scanning electron-microscopy is a costly and time-consuming method, yielding results from small sample batches only. Here we show that polarization-sensitive optical coherence tomography allows to determine the size and the shape of nanoparticles in a quick process with small technical effort and with no need for laborious sample preparation. The technique relies on measuring the depth and polarization-dependent backscatter and on evaluating the results by a semi-analytical model. We found the backscatter measurement to be sufficiently sensitive on gold nanorod geometry to enable diameter determination with 6.2 nm accuracy and maximum deviation in aspect ratio measurement of 10 %. These experiments correspond to the first demonstration of quantitative nanoparticle shape determination with optical coherence tomography.

## **4.1 Introduction**

Nanoparticles are attractive for both scientific examinations and technical applications. Their size is in or below the range of the optical wavelength and is thus only one to four orders of magnitude bigger than an atomic diameter, thus leading to huge surface-to-volume ratios. As a consequence, in contrast to bulk materials, size and shape of the particles govern their physical properties

and become the essential parameter for targeted material design. As an example, the shape-dependent surface plasmon resonance frequency of metallic nanorods allows to use them as highly-sensitive refractive index sensors [99,100]. Moreover, gold and platinum nanoparticles can be used as chemical catalysts for which the reaction rate depends significantly on size and shape [14,101], and also the cellular uptake of gold nanorods into biological tissue, when used as imaging markers, turns out to be shape- and sizedependent [9]. The same is true for soft-matter nanorods as nanocarriers in tumour therapy [102]. Nanoclays form layered structures and thus feature a large lateral extension and small height. This special shape accounts for their outstanding barrier properties [74], when incorporated into a polymer host material. The aspect ratio of carbon nanotubes (CNT) determines their ability to build up electrically conductive paths inside isolating matrix materials. Transparent polymers can thus become conductive and might offer an alternative to expensive indium tin oxide (ITO) [12]. All these applications have in common that size and shape of the nanoparticles are crucial to the macroscopic material properties and thus to the final performance of the product. In this context, characterization of the nanostructure is of highest relevance, either indirectly by characterization of the final macroscopic properties or directly using imaging, scattering or filtration techniques.

State-of-the-art size and shape characterization of nanoparticles mainly relies on microscopic imaging. This requires high-resolution techniques such as scanning-electron microscopy (SEM) [16], transmission electron microscopy (TEM) [17] or atomic force microscopy (AFM) [103,104]. In all cases, sample preparation is laborious and costly and might create artefacts, e.g. agglomeration effects during the drying process in sample preparation. Moreover, SEM, TEM and AFM are limited to small sample volumes, which are not necessarily representative of the entire batch. Light scattering techniques, in contrast to imaging methods, are viable tools for determining particle size distributions from large sample volumes. Static light scattering (SLS) uses angle-resolved and/or spectrally resolved detection of scattered light [22,23] and needs a rather complex optical setup, when large scattering angles are to be taken into account. Dynamic light scattering (DLS) methods make use of temporal

fluctuations of interference patterns of scattered light to calculate the hydrodynamic diameter of particles within the respective solvent [75]. Relying on the observation of the Brownian motion of single particles, the technique is limited to rather low concentrations. Both SLS and DLS preferentially provide information on particle size, while particle shape cannot be determined [24]. X-ray diffraction (XRD) [77] or X-ray scattering (SAXS) [78] can be used to reveal the atomic or molecular structure inside the material. However, expensive instrumentation and a small probing region limit the application range of these methods to small batch sizes. Tunable resistive pulse shaping (TRPS) [25] measures the particle size by observing the ionic current through a small pore. Once a particle passes this pore the ionic flow is affected for a certain time. After calibration, this technique allows accurate determination of particle size distributions. Information on particle shape however cannot be extracted. Particle size distributions can also be measured using centrifugal sedimentation techniques, where particles with different dimension are separated by differences in their sedimentation speed in presence of centrifugal forces [25]. While this technique is applicable to a broad size range, high experimental effort is necessary and no information on particle shape is obtained.

Optical coherence tomography (OCT) represents a combination of light scattering techniques with optical imaging is [1,28,79]. OCT provides threedimensional imaging data from the bulk of a sample and avoids expensive sample preparation. Previous demonstrations of nanoparticle analysis using OCT have shown size determination with nanometre accuracy [10] and nanocomposite dispersion-state characterization [10]. Microparticles have been investigated using image-based analysis [81]. OCT has been applied to particle diffusion measurements and showed particle size-dependent temporal variations [82,83]. Angle-resolved OCT has proven to be able to measure particle sizes down to 5 µm [84], by evaluating the angular-dependent scattering lobes with Mie's theory. Gold nanorods have been differentiated from collagen cells, exploiting their strong polarization-cross talk in OCT measurements [105]. While micro- and nanometre particle size determination has been addressed by some OCT-based methods, the exact determination of

anisotropic particle shapes, e.g., particle sizes in long and short dimension, still needs to be shown.

In this section we demonstrate that polarization-sensitive OCT (PS-OCT) [59] allows accurate determination of both nanoparticle size and shape. By evaluation of backscatter signals in different polarizations and by comparison to electromagnetic field simulations of differently shaped particles, we demonstrate precise determination of length and diameter of gold nanorods in aqueous dispersion. Without the need for sample preparation, this technique allows to measure nanometre particle dimensions in statistically relevant volumes of several cubic millimetres. We determine size and shape of gold nanorods in a unimodal aqueous dispersion by comparing measured scattering parameters with simulation results. Our samples cover particles which range from 15 to 50 nm in particle diameter and from 1 to 16 in aspect ratio. In practice, this technique can be extended to the analysis of many nanoparticles with prolate or oblate shape as long as the refractive index contrast to the surrounding media is sufficient and absorption is moderate.

## **4.2 Measurement principle: Size- and shapedependent backscattering**

Light scattering methods allow size and shape measurements beyond the resolution of optical imaging systems. The analysis presented here links the geometry of the particles to their scattering properties and relates those to polarization-sensitive backscattering measurements with a PS-OCT system. First, we introduce the applied scattering model on the basis of nanospheres and their size-dependent scattering. Second, we expand the model to non-spherical scatterers using a polarization-dependent scattering model.

For the analysis, a single-scattering model [89] is used, under the assumption that incident light is scattered at maximum once inside the medium, which is appropriate for weakly scattering samples and for the analysis of moderate depths [90]. Figure 4.1 depicts the scattering model in two exemplary media: nanospheres, Figure 4.1 (a), and non-spherical particles Figure 4.1 (c), with

their respective polarization-dependent backscattering curves Figure 4.1 (b) and Figure 4.1 (d). As shown in the case of nanospheres Figure 4.1 (a), a light beam with incident power *P*in and polarization *E*in enters the sample. At each particle (grey), one part of the incident light at depth *z* is scattered back into the numerical aperture of the system, another part is scattered into other directions or is absorbed (black), and the remainder propagates deeper into the medium. Mie's theory [50] is applicable to the calculation of angle-dependent scattering amplitudes of spherical particles with sizes in the range of or smaller than the optical wavelength of the incident light. The scattering amplitudes are dependent on size of the spheres, refractive index of sphere and surrounding material, as well as on the optical wavelength. Thus, observation of the angle-dependent scattering can be used for size analysis of nanospheres. In the case of spherical scatterers, the polarization of the output field *E*out,m is the same as the incident polarization *E*in [50,106,107]. This is an indication of spherecity of the scatteres, if their size is far below the wavelength and multiple scattering can be neglected. The total scattering cross section s and the absorption cross section a of a particle sum up to the extinction cross section t. For an ensemble of particles with number density *N*, the extinction coefficient can be expressed as t = *N*t. Analogously, the backscattering cross section into the −180° direction is given by b, and the backscattering coefficient is b = *N*b. The depth-dependent decay d*P*(*z*)/d*z* = −t*P*(*z*) of the forward travelling power *P*(*z*) follows an exponential function t in e *z P z P* . For a given depth *z*, the power is scattered back from inside a depth element *z*, and is captured by the numerical aperture of the system, t b b e *z P z zP z* , where t e *z* accounts for the losses in the back propagation path. In an OCT measurement the depth-dependent backscattering factor *R*0(*z*) = *P*in/*P*b(*z*) + *R*n is measured, including a system dependent noise floor *R*n, which defines the sensitivity of the system expressed as minimum measurable sample reflectivity, see Section 2.1.4. In summary, the scalar single-scattering model, which omits polarization effects is given by

$$R\_0(z) = \mu\_\mathbf{b} \delta z \text{ e }^{-2\mu\_\mathbf{t} z} + R\_\mathbf{n} \text{ .} \tag{4.1}$$

In practice, the measured backscattering signal depends on further effects, which need to be determined in a calibration step. This comprises the decay of the coherence function of the frequency-swept laser, the divergence of the measurement beam, and the absorption of the material, which surrounds the scatterers. These influences are taken into account by including two calibration factors [10,91] *q* and *Q* in the single-scattering model according to Eq. (4.1),

$$R\_{\mathbf{s}}\left(z\right) = \mathcal{Q}\mu\_{\mathbf{b}}\delta z \text{ e } ^{-2\left(\mu\_{\mathbf{t}}+q\right)z} + R\_{\mathbf{n}}\tag{4.2}$$

Both calibration factors are determined by comparison of measured backscatter of a NIST-traceable polystyrene particle standard with 246 nm sphere diameter in 0.5 wt.% aqueous dispersion (BS-Partikel GmbH, Wiesbaden, Germany) to model calculations according to Eq. (4.1).

A sketch of a semi-logarithmic plot *R*dB(*z*) = 10 lg(*R*s(*z*)) of both the polarization-maintaining backscattering factor *R*m(z) (blue) and of the crosspolarization backscattering factor *R*c(z) (red) is shown in Figure 4.1 (b). With no polarization conversion, the cross-polarization backscatter is zero, and the measured signal is given by the associated noise level.

Light scattering of particles with non-spherical shape can lead to backscatter with altered polarization, see Figure 4.1 (c). In general, light scattering is caused by the excitation of dipole oscillations inside the particles and the associated radiation of light. The orientation of the dipoles and thus the polarization of the emitted light depend on both incident polarization and particle geometry [50]. For elongated particles, dipoles are preferentially excited in the direction of the long axis of the particle, thus showing a polarization change of backscattered with respect to the incoming light. With an ensemble of randomly oriented particles, all random backscatter polarizations sum up, and an average backscatter polarization is obtained, which differs the more from the incident polarization, the more the particles' shape deviate from that of a sphere.

**Figure 4.1:** Polarization-sensitive scattering-model-based size and shape analysis of nanoparticles. **(a)** Scattering of a linearly polarized laser beam with power *P*in inside a dispersion of spherical nanoparticles, with aspect ratio AR = 1. Concentration and refractive index contrast are low enough to assume single scattering inside the medium. The input polarization *E*in is considered vertical here (blue colour). Backscattering of spherical particles in the −180° direction maintains the polarization, so the backscattering cross section b = b,m equals the polarization-maintaining backscattering cross section. The total extinction cross section is given by t. Light that is neither scattered nor absorbed propagates deeper into the medium. *P*out,m denotes the output power of light in same polarization as of the input light, *E E* in out,m . **(b)** A singlescattering model leads to a polarization maintaining backscattering factor *R*m, describing the depth-dependent backscattering factor into the polarization of incidence. In case of spherical particles the cross-polarization backscattering factor *R*c is zero. Together with a systemdependent noise level *R*n, both factors are plotted in dB-scale along the depth axis *z*. **(c)** Scattering in a medium with non-spherical particles of aspect ratio AR > 1. The input polarization (vertical) is only partially maintained, and the backscattering cross section b = b,m + b,c is given by the sum of both the polarization-maintaining component b,m and the cross-polarized component b,c (red). Accordingly, output light is polarized partially in parallel to the input *E E* in out,m and orthogonal *E E* in out,c . *P*out,m and *P*out,c denote the corresponding backscattered powers. **(d)** In contrast to the case in (b), the cross-polarized backscattering factor *R*c is non-zero for non-spherical particles. The ratio *R*c/*R*m can be taken as a measure for nonsphericity of the investigated particles.

Analogously to Figure 4.1 (a), a beam with power *P*in and an electric field orientation *E*in is incident on the sample. Interaction of the non-spherical particles with the incident light is expressed by the total extinction cross section t, the polarization-maintaining backscattering cross sections b,m describing polarization maintaining backscatter, and the cross-polarization backscattering cross sectionb,c. The backscattered light is hence composed of light with maintained polarization with power *P*out,m and electric-field *E*out,m, which is polarized parallel to the incident field, and of light with power *P*out,c, and electricfield *E*out,c, which is polarized orthogonally to the incident field.

The semi-logarithmic depth-dependent backscatter signal of a sample with non-spherical particles is depicted in Figure 4.1 (d). In comparison to the scalar case, the backscattering coefficient b b *N* is vectorial,

$$
\bar{\mu}\_{\mathbf{b}} = \begin{pmatrix} \mu\_{\mathbf{b}, \mathbf{m}} \\ \mu\_{\mathbf{b}, \mathbf{c}} \end{pmatrix}, \tag{4.3}
$$

and the backscatter signal is given by

$$\begin{array}{l}\text{ter signal is given by} \\\\ \bar{R}\_{\text{s}}\left(z\right) = \begin{pmatrix} \mathcal{R}\_{\text{m}}\left(z\right) \\ \mathcal{R}\_{\text{c}}\left(z\right) \end{pmatrix} = \begin{pmatrix} \mu\_{\text{b,m}} \\ \mu\_{\text{b,c}} \end{pmatrix} \mathcal{Q}\delta z \text{ e } \text{e}^{-2\left(\mu\_{\text{l}} + q\right)z} + \mathcal{R}\_{\text{n}} \text{ .} \end{array} . \tag{4.4}$$

The difference of the two backscattering coefficients, b,m and b,c, leads to offsets between the logarithmic plots *R*m(*z*) and *R*c(*z*) in Figure 4.1 (d). This difference is denoted as polarization cross talk = 10 lg(b,c / b,m), which we define in dB. The decay of both curves is the same, since the particles in the sample are randomly oriented and the extinction is thus dependent on input or output polarization. While the backscatter curve in the scalar model contains information on particle size and number density, the vectorial polarizationdependent model additionally supplies information on particle shape. In the following sections, dispersions of particles with different diameters and aspect ratios are characterized by measuring their polarization-dependent backscattering signals.

## **4.3 Measurement results**

To determine the viability of our concept, we analyse aqueous dispersions of polystyrene nanospheres and aqueous dispersions of gold nanorods with

different aspect ratios. The analysis is performed using a polarization-sensitive OCT system, which is described in detail in Section 4.6. Measurements of the nanospheres show negligible polarization conversion, which is in accordance with theoretical considerations [50,106,107]. For nanorods, we find shapedependent polarization conversion. By comparing the measurement data to electromagnetic simulations, we find that diameter and aspect ratio (AR) of the rods can be determined with nanometre accuracy.

In our experiments, aqueous dispersions of gold nanorods with *d*G = 25 nm diameter and average lengths of *l*G1 = 102 nm (sample G1) and *l*G2 = 256 nm (sample G2; both types of nanorods from Nanopartz Inc., Loveland, Colorado, USA) have been characterized with the PS-OCT system. These rods feature respective aspect ratios of ARG1 = 4 and ARG2 = 10 and number densities of *N*G1 = 5.7×1018 m −3 and *N*G2 = 1.1×1017 m −3 , corresponding to weight concentrations of 0.54 wt. % and 0.026 wt. %, respectively. For comparison, additionally a sample (S4) with spherical particles (ARS4 = 1), containing an aqueous dispersion of polystyrene nanospheres of diameter of *d*S4 = 246 nm, number density of *N*S4 = 6.4×1017 m −3 and weight concentration of 0.5 wt. % is investigated. For preparation, all samples were filled into quartz glass cuvettes with broad optical transmittance. In Figure 4.2 (a-c), OCT measurement results are depicted along with the associated scanning electron microscope (SEM) images in Figure 4.2 (d-f) of the dried particles. The measurement results consist of two curves, polarization-maintaining backscatter *R*m(*z*) and cross-polarized backscatter *R*c(*z*), both as a function of depth inside the sample. Figure 4.2 (a) depicts the results of the microsphere sample (S4), with *R*m(*z*) showing an exponential decay inside the sample (z > 0). The data for z < 0 is defined by the noise level. By tilted incidence on the sample surface, facet reflections could be avoided. The solid curves represent a fit of the scattering model in Eq. (4.4) to the measured data (light coloured). Applying Mie's theory to *R*m(*z*), the spheres' sizes can be extracted with nanometre accuracy, as we have demonstrated previously [10]. The cross-polarized backscatter *R*c(*z*) is more than 20 dB below *R*m(*z*). Since the polarization extinction ratio of the system is around 20 dB, this data can be considered to be below the noise level and no scattering model fit is reasonable. The complete

polarization maintenance of the backscatter leads to the conclusion that the sample particles are completely spherical. The SEM picture, Figure 4.2 (d), of the dried particles of sample S4 confirms this finding. The gold nanorod sample G1 with AR of 4, see Figure 4.2 (b), shows an exponential decay both for the polarization maintaining backscatter *R*m(*z*) and the cross-polarized backscatter *R*c(*z*).

**Figure 4.2:** PS-OCT measurements and scanning-electron microscope (SEM) images of nanoparticles with different shapes. **(a)** Sample S4: PS-OCT depth scan of polystyrene nanospheres with *d*S4 = 246 nm diameter in 0.5 wt.-% aqueous dispersion. *R*m denotes polarization-maintaining backscatter (light red: measurement, bright red: single-scattering model fit), and *R*c denotes cross-polarized backscatter. The polarization maintaining curve is dominating the cross-polarized trace by more than 20 dB, hinting to an aspect ratio of the particles close to unity. Fluctuations of the cross-polarized part are due to the measurement noise around −100 dB. **(b)** Sample G1: PS-OCT measurement of gold nanorods with *d*G1 = 25 nm diameter and aspect ratio of ARG1 = 4 in 0.54 wt. % aqueous dispersion. The signal decay with depth corresponds to the total scattering and absorption cross sections of the gold nanorods. The cross-polarization curve (grey: measurement, black: single scattering fit) is below the polarization-maintaining backscatter, and the polarization conversion amounts to G1= −6.9 dB. The noticeable cross term is in accordance with the expected polarization conversion by scattering from non-spherical nanoscale objects. **(c)** Sample G2: PS-OCT measurement of gold nanorods with *d*G2 = 25 nm diameter and aspect ratio of ARG2 = 10 in 0.026 wt. % aqueous dispersion. In comparison to (b), the polarization conversion is stronger and amounts to G2 = −3.7 dB. **(d-f)** Scanning electron microscope images of the respective dried nanoparticles: **(d)** Sample S4: polystyrene nanospheres of *d*S4 = 246 nm diameter, **(e)** Sample G1: gold nanorods of *d*G1 = 25 nm diameter and ARG1 = 4, **(f)** Sample G2: gold nanorods of *d*G2 = 25 nm diameter and ARG2 = 10. As can be seen in (e) and (f), the length of the rods is not completely uniform and the nominal values can be regarded as average values.

The decay coefficient of both curves is same, as expected in Eq. (4.4), and the polarization cross talk is measured to be G1 = −6.9 dB. This noticeable cross talk is attributed to a clear deviation of the scatterers from a spherical shape, which is confirmed by the SEM image in Figure 4.2 (e). Compared to G1, the sample G2 shows even higher polarization cross talk G2 = −3.7 dB, see Figure 4.2 (c). This behaviour is expected as the AR is increased to 10. The SEM image in Figure 4.2 (f) confirms the elongated particle shape. Further, a variation of rod length around the nominal value of 256 nm can be noticed.

The following table summarizes the determined scattering coefficients and cross sections as well as the polarization cross talk.

Table 4.1: Scattering coefficients, cross sections and polarization cross talk, determined from the measurements in Figure 4.2 using the polarization-sensitive single scattering model in Eq. (4.4). Sample S4 is used for calibration of t and b,m, for this reason the values of and are nominal values.

## **4.4 Shape determination by comparison of simulation and measurements**

In the following, the measured scattering parameters are compared to simulation. For the determination of the particles' size and shape, a set of simulations with varying geometries is performed and used for comparison.

The selected electromagnetic simulation technique relies on a commercially available fully vectorial integral equation solver that is part of CST Microwave Studio [108]. The solver, which follows a Method of Moments (MoM) method, a rigorous full-wave technique for solving open-boundary electromagnetic problems, is used for calculating the surface integral of the electric and magnetic field at the scattering particle. This method is especially

recommended for the calculation of scattering cross sections and thus fits perfectly to our needs. An alternative numerical method, which allows scattering calculations of non-spherical particles is basing on the Discrete-Dipole Approximation (DDA) [109]. In this method, the specific particle geometry is adapted by discrete dipoles of which the mutual excitation is taken into regard along with the excitation by the incident field. The technique is used for nanorod investigations, but shows weaknesses in calculation of gold nanorods at infrared wavelengths [110], which is the regime of our analysis. A further calculation approach for non-spherical particles or small agglomerates thereof is to make use of the Rayleigh-Debye-Gans theory (RDG) [50]. In this approach, the arbitrarily shaped particle is subdivided into small Rayleigh scatterers, which are assumed to be unperturbed by each other. In a comparison [111] to other calculation techniques, RDG was found appropriate for limited size parameters *kl* < 0.3 and refractive index differences *n* < 2, with the angular optical wavenumber *k* = 2 / and the particles longest dimension *l*. Although the RDG approach is appropriate for some types of non-spherical particles, the aforementioned conditions do not hold for our sample types. A promising method for calculation of scattering and extinction of particles as used in our experiment is the T-Matrix Method [112,113], which relies on the expansion of scattered and initial field in suitable vector spherical wave functions, and on linking the respective expansion coefficients in a so-called T-matrix. This approach is applicable to many particle types and could be a further option in future calculations.

For the CST calculations, the selection of the proper material properties of gold turned out to be crucial to the simulation results. Muskens [114] identified the complex refractive indices measured by Johnson [115] to be fitting best to gold particles of nanometre size. These values were interpolated to a wavelength of 1.3 µm and used for a Drude material model [116] within the simulations. In the simulation environment, the nanorods were built up from a cylindrical body and two hemispherical tails. We investigate a variety of differently shaped nanorods with diameters of 15 to 50 nm and aspect ratios of 1 to 16. For these particles, the absorption cross section and the spatial and polarization-dependent distribution of the scattering power were simulated and

averaged over random rod orientations to represent the scattering properties of nanorods dispersed in water. Averaging of the scattered powers from the individual particles is possible, since the particle concentration is small and light can thus be assumed to be scattered at most once. From integrating these angular scattering cross sections over the complete solid angle of 4 the scattering cross section s is obtained. The extinction cross section t = s + a results from taking the absorption cross section a into account as well. The polarization-maintaining backscattering cross section b,m results from integration of the polarization-maintaining angular scattering cross section in the backward direction over 1° half-angle, which corresponds to the numerical aperture of the system. The polarization-resolving simulation allows to detect the cross-polarized backscatter separately from the polarization-maintaining backscatter, which is used for calculation of the polarization cross talk .

Figure 4.3 depicts the simulation results and compares them to the nanorod measurements. The subplots in Figure 4.3 (a-c) show simulated extinction and backscattering cross sections as well as polarization cross-talk as a function of diameter as well as AR. The term diameter is used here for the short axis of the particle, whereas AR×*d* gives the extent along the long axis, see Figure 4.3 (f). For particle diameters between 15 and 40 nm, simulations were performed with a step size of 5 nm, and for diameters between 40 and 50 nm, the step size was increased to 10 nm. The step size in AR was 1 for AR = 1 … 14, and 2 for AR = 14 … 16, leading to 105 data points in total. For display and comparison to measurements, all results have been interpolated using a 2D spline interpolation. Figure 4.3 (a) shows the extinction cross section t, Figure 4.3 (b) shows the polarization-maintaining backscattering cross section b,m and Figure 4.3 (c) shows the polarization cross talk . Within the simulation domain, t generally rises with both particle diameter and extinction ratio. In the vicinity of the plasmonic resonances around AR of 6 and 14 the extinction is highest. For b,m the trend is analogous, although not identical to the results in Figure 4.3 (a). The polarization cross-talk is almost independent from diameter but shows strong dependence on AR, especially if AR is small. The black lines in the three subfigures indicate the values measured for sample G1 and G2. Table 4.2 summarizes the simulation results for the

geometries given by samples G1 and G2. The measured values, see also Table 4.1, are denoted for comparison. Measurement and simulation of t and b,m show agreement with respect to the order of magnitude. The polarization cross-talk differs at most by 2.71 dB between simulation and measurement.

Table 4.2: Comparison of simulated and measured extinction and backscattering coefficients, t, b,m, respectively, as well as the polarization cross talk .

In principle, the determination of pairs of *d* and AR for a set of two measured parameters, e.g. t and b,m, is possible. However, in practice, each pair out of the three measured parameters (t, b,m, ) would lead to a different solution for *d* and AR, since measurement and simulation provide a limited accuracy. For this reason, all three parameters are taken into account by calculating the (*d*, AR)-pair which provides in sum the lowest relative difference of simulation and measurement. First, the single ratios of each measured parameter to the simulated parameters are calculated. Since the extinction and backscattering cross sections span a wide range of several magnitudes, comparison on a logarithmic scale seems to be favourable. The polarization cross talk is expressed in logarithmic dB-units already. Thus, the absolute logarithmic values of the ratios read

$$\begin{aligned} \Delta \sigma\_t &= \left| 10 \lg \left( \frac{\sigma\_t^{(\text{meas})}}{\sigma\_t^{(\text{sim})}} \right) \right| \\ \Delta \sigma\_{\text{b,m}} &= \left| 10 \lg \left( \frac{\sigma\_{\text{b,m}}^{(\text{meas})}}{\sigma\_{\text{b,m}}^{(\text{sim})}} \right) \right|, \\ \Delta \eta &= \left| \eta^{(\text{meas})} - \eta^{(\text{sim})} \right| \end{aligned} \tag{4.5}$$

where the superscript denotes either measurement (meas) or simulated (sim) data. The single ratios, Eq. (4.5), are divided by their individual maxima within the simulation domain, and after this normalization the total ratio is calculated

$$
\Delta\_{\rm tot} = \frac{1}{3} \left( \frac{\Delta \sigma\_{\rm t}}{\max \left( \Delta \sigma\_{\rm t} \right)} + \frac{\Delta \sigma\_{\rm b,m}}{\max \left( \Delta \sigma\_{\rm b,m} \right)} + \frac{\Delta \eta}{\max \left( \Delta \eta \right)} \right). \tag{4.6}
$$

The factor of 1/3 keeps the result in the [0,1] range but does not affect the calculation of the particle shape. Figure 4.3 (d-e) shows tot versus *d* and AR in colour-coded plots for both measurements of sample G1, Figure 4.3 (d), and of sample G2, Figure 4.3 (e). The white cross indicates the position of lowest total deviation. In the case of sample G1 with nominal diameter of 25 nm and nominal AR of 4, see Figure 4.3 (d), tot is low (blue coloured) in an area reaching from AR of 3 and diameter of 50 nm to AR of 6 and diameter of 15 nm. The rod geometry with lowest deviation from simulation is found at a diameter of 26.75 nm and at AR of 4.1. This deviates from the nominal values by only 1.75 nm in diameter and 2.5% in AR. In the case of sample G2 with nominal diameter of 25 nm and AR of 10, see Figure 4.3 (e), tot is low for AR > 6. In this case, the sensitivity in diameter is less pronounced. The smallest total ratio is found at a diameter of 18.8 nm and AR of 9. This amounts to a deviation of 6.2 nm in diameter and 10% in AR.

Table 4.3 summarizes the OCT measurement results, which are in good agreement with the particle parameters obtained from SEM imaging. Note that the OCT measurement captures backscatter from a multitude of particles at once, thus averaging out deviations of individual particles from the nominal shape values.

**Figure 4.3:** Particle shape determination by comparison of OCT-measured scattering parameters and electro-magnetic scattering simulations. Subfigures **(a-c)** show averaged scattering parameter simulations for an aqueous dispersion of randomly oriented gold nanorods. The horizontal and vertical axes denote aspect ratio AR and diameter *d*, respectively. **(a)** Total extinction cross section *t*, **(b)** polarization maintaining backscattering cross section b,m, and **(c)** polarization cross talk . Extinction cross section t and backscattering cross section b,m increase generally with particle diameter *d* and aspect ratio AR and show high values at the plasmonic resonances, e.g., around AR = 6. The crosstalk increases with AR. The black lines indicate the values extracted from the OCT measurements of sample G1 and sample G2, which contain nanorods with nominal diameter *d*nominal,1,2 = 25 nm and nominal aspect ratios ARnominal,1 = 4 and ARnominal,2 = 10. **(d,e)** Total ratio tot of measurement and simulation comprising all three parameters (t, b,m, ). **(d)** Total ratio tot of the simulations and the measured parameters extracted from sample G1, see Eq. (4.6). **(e)** Total ratio tot of the simulations and the measured parameters extracted from sample G2, see Eq. (4.6). The white crosses show minimum total ratios and thus indicate the estimated particle diameter *d*meas1,2 and aspect ratio ARmeas1,2. Table 4.3 lists the results for both samples. **(f)** Schematic drawing of both nanorod geometries G1 and G2, along with the nominal diameters *d*nominal,1,2 and the respective aspect ratios ARnominal,1 and ARnominal,2.

Table 4.3: Estimated particle diameters *d*meas and aspect ratios ARmeas compared to the nominal specifications *d*nominal, ARnominal. The absolute and relative deviations between measurement and nominal values are given by d and AR.

## **4.5 Discussion**

The ramp-up of nanotechnology-driven industry demands fast and reliable techniques for quality control. In this work, we demonstrated that polarizationsensitive OCT offers precise size and shape measurement of nanoparticles, along with no sample preparation. The large probing volume of several cubic millimetres and the prospect of integrability into production lines, as well as the instantaneous output of characterization results are unique among the nanoparticle shape characterization tools.

This work for the first time demonstrates quantitative characterization of size and shape of nanoparticles using PS-OCT. Nevertheless, the technology has some practical limitations, which we will discuss in the following. The technique is constrained to the analysis of light scattering media, which are neither fully transparent, nor strongly absorbing, however the range of possible materials is broad, as has been shown in the analysis of strongly absorbing CNTloaded materials [10]. A further restriction is given by the aspect ratio, which should not exceed AR = 10, when exact quantitative shape determination is aimed for. Aspect ratios below this limit cover most practically relevant nanoparticles from nanospheres to nanorods, but excludes, e.g., the differentiation of high-AR nanowhiskers from long nanorods. For larger aspect ratios, see Figure 4.3, the extinction cross section, the backscattering cross section and the polarization cross talk showed to be less sensitive to further aspect ratio changes. In our experiments, the particle size distributions have been narrow. Particles with broader size and shape variations could possibly be

distinguished by trend, but a quantitative shape determination down to nanometre accuracy, as has been shown here, would be hampered.

In conclusion, we have shown that PS-OCT allows to accurately determine size and shape of monodisperse gold nanorods having 25 nm diameter and lengths of 102 nm and 256 nm. We used a polarization-sensitive OCT setup, which we calibrated with respect to both power level and polarization. We measured three parameters: extinction cross section, backscattering cross section and polarization cross talk. Shape and size information were extracted by comparison to electro-magnetic scattering simulations of randomly dispersed particles, and the results are in fair agreement with the particle dimensions obtained from SEM-based imaging. We believe PS-OCT offers an attractive option for simple and rapid nanoparticle characterization with reasonable instrumentational effort as needed, e.g., in in-line or on-line quality control.

## **4.6 Methods**

The swept-source PS-OCT system used in this work is illustrated in Figure 4.4. In general, an OCT system measures the strength and the position of a multitude of scatterers along a light path in a sample. To this end, the electric field that is back reflected from the sample into the numerical aperture of the system is superimposed with a reference field originating from the same swept laser source. The superposition leads to optical frequency dependent interference patterns on a photodetector. The variation of signal amplitude with laser frequency is connected to the depth-dependent back reflection profile of the sample by a Fourier transform [28], see also Chapter 2.1.1. In our experiment we use a swept-source laser with central wavelength of 1315 nm and a wavelength scanning range of 1260−1370 nm, a scan rate of 1 kHz, and 10 dBm (10 mW) average output power (model s3, Micron Optics Inc., Atlanta, GA, USA). A first fibre-based directional coupler (CPL1) is used to split the power among the sample path (SP) and the reference path (RP). The reference path contains a free-space section allowing precise matching of the SP and RP lengths. Backscattered light from the sample and light travelling along the RP

is superimposed in a second fibre-based coupler (CPL2) and coupled to a polarization-sensitive balanced photodetector (PS-BD, model INT-POL-1300, Thorlabs Inc., Newton, NJ, USA) offering 15 MHz electrical bandwidth. Polarization controllers (PolC) are used for matching the polarization in both detector arms, and for setting the reference arm polarization to 45°, thus providing the same reference arm power in both orthogonal polarizations. The PS-BD consists of two polarizing beam splitters (PBS) and a balanced detector (BD) for each of the polarizations. The balanced detection principle suppresses relative intensity noise (RIN) from the strong RP light and enables a large dynamic range and a high sensitivity defined by the smallest measurable sample reflectivity of −110 dB. The electrical signal is digitized by a 16 bit analogue-to-digital converter (ADC, model ATS660, Alazar Technologies Inc., Pointe-Claire, Canada) and processed by a personal computer (PC). The sample path comprises fibre collimator (FC), rotatable quarter-wave plate (QWP) for polarization variation, a pair of galvo scan mirrors for lateral 2D beam scanning, a scan lens and the sample under test. The SP exists in two configurations: First the polarization calibration setup (top) with additional rotatable polarizer (Pol) and plane mirror instead of a sample, and second the measurement setup (bottom).

For the describing the polarization of light within the PS-OCT system, we use the Jones formalism [117], where the electric field *E t* , of polarized light is represented by the Jones vector *J* . The Jones vector consists of two complex elements *Jx* , *Jy* with the normalized field amplitudes cos , sin , of each of the orthogonal field components in *x* and *y* direction, as well as their common phase and individual phases and . Thus, the electric field is j j j j j , e e e e cos e sin *t kz t kz E t A J A e e x y* .

The transmission through a linear optical system is described by a Jones matrix **J**, transforming input Jones vectors to output Jones vectors. If multiple optical systems are concatenated, their individual Jones matrices are multiplied, and reverse transmission [69] of a unitary, optical reciprocal element with forward propagation matrix **J** results in

$$\mathbf{J}\_{\rm rev} = \begin{pmatrix} m\_1 & -m\_3 \\ -m\_2 & m\_4 \end{pmatrix}, \quad \text{with } \mathbf{J} = \begin{pmatrix} m\_1 & m\_2 \\ m\_3 & m\_4 \end{pmatrix}. \tag{4.7}$$

The aim of a PS-OCT measurement is the determination of the backscattering of the sample as a function of depth and polarization. This can be expressed by a Jones matrix

$$\mathbf{J}\_s = \frac{1}{R\_\mathbf{S}(z)} \begin{bmatrix} R\_{\mathbf{xx}}(z) & R\_{\mathbf{yx}}(z) \\ R\_{\mathbf{xy}}(z) & R\_{\mathbf{yy}}(z) \end{bmatrix} \tag{4.8}$$

with the backscattering coefficient *Rij*(*z*) denoting backscattering of incident polarization *i* into backscattered polarization *j*, where *i*, *j* can each be *x* (horizontal) or *y* (vertical orientation). The division by the total backscattering *R z* S is necessary to normalize the Jones matrix. According to Eq. (4.4), the sample backscatter of randomly oriented single-scattering particles only distinguishes polarization maintaining *R*m(*z*) from cross-polarized backscatter *R*c(*z*), thus in this case the sample Jones matrix can be simplified to

$$\mathbf{J}\_{\mathbf{s}} = \frac{1}{\mathcal{R}\_{\mathbf{S}}(z)} \begin{bmatrix} \mathcal{R}\_{\mathbf{m}}(z) & \mathcal{R}\_{\mathbf{c}}(z) \\ \mathcal{R}\_{\mathbf{c}}(z) & \mathcal{R}\_{\mathbf{m}}(z) \end{bmatrix}. \tag{4.9}$$

108 1 3 1 2 The measurement setup was built up with standard single-mode fibres (SMF) instead of polarization-maintaining fibres (PMF). In PMF, both principal polarization axes feature different group refractive indices, which imply different group velocities, which in turn cause spurious signal peaks in the OCT measurement. Using an SMF instead avoids spurious peaks but does not maintain the incident polarization. Hence, a polarization calibration step is necessary before performing a sample measurement. This is done using the calibration setup depicted in Figure 4.4 (top). In this step, the Jones matrix **J**sys describing the propagation of backscattered light through the fibre-optic system towards the PS-BD is determined as well as the input polarization in *J* of the light, which is emitted from the SP fibre towards the sample.

**Figure 4.4:** Setup schematic of the fibre-based polarization-sensitive swept-source optical coherence tomography setup used in this work. SS: swept-wavelength source, scanning range 1260-1370 nm; CPL1,2: fibre-based directional coupler; SP: sample path; RP: reference path; FC: fibre collimator; PolC: polarization controller; PS-BD: polarization-sensitive balanced detector, consisting of two polarization beam splitters (PBS) and two balanced detectors (BD); ADC: analogue-to-digital converter; PC: personal computer. For sample measurements, the sample arm comprises a free-space propagation section with a rotatable quarter-wave plate (QWP), scan mirrors, scan lens and sample. For calibration purposes, the sample is replaced by a plane mirror, and a rotatable polarizer (Pol) is added to the free-space section in the sample arm. Jones vectors and matrices are denoted in blue. in *J* : Jones vector of the input polarization leaving the FC towards sample or towards calibration setup; **J**QWP1,2: Jones matrix of the QWP1,2; **J**Pol: Jones matrix of the polarizer; **J**s: Jones matrix of the sample; **J**sys: Jones matrix of the fibre-optic system from FC at the sample or calibration measurement head to the PBS at the PS-BD; *m J* : Jones vector, measured by the PS-BD.

The polarization measured at the PS-BD results from the concatenation of all transmitted optical elements *J J* m,cal sys QWP1,rev Q Pol,rev P M Pol P QWP1 Q in **J J J J J J**

$$\bar{J}\_{\rm m,cal} = \mathbf{J}\_{\rm sys} \mathbf{J}\_{\rm QWP1, rev} \left(\mathcal{G}\_{\rm Q}\right) \mathbf{J}\_{\rm Pol, rev} \left(\mathcal{G}\_{\rm P}\right) \mathbf{J}\_{\rm M} \left(\mathcal{J}\_{\rm Pol}\right) \mathbf{J}\_{\rm QWP1} \left(\mathcal{G}\_{\rm Q}\right) \bar{J}\_{\rm in} \dots \bar{J}\_{\rm MW} \tag{4.10}$$

$$(4.10)$$

which describes propagation of in *J* through the quarter-wave plate with **J**QWP1 Q at an angle Q of the fast axis with respect to the horizontal direction, the polarizer with **J**Pol P at an angle P with respect to the horizontal direction, the back-reflection at the plane mirror described by **J**M, the propagation back through polarizer **J**Pol,rev P and QWP1 **J**QWP1,rev Q and finally through the fibre optic system with matrix **J**sys towards the detector.

To determine both **J**sys and in *J* , multiple measurements of m,cal *J* are necessary and performed at angular positions of the QWP1 of Q 0° to 165° in steps of 15° combined to angular positions of the polarizer of P 0° to 150° in steps of 30°. With 72 measurements in total, the system is over-defined, and a least-squares based fit allows robust determination of both **J**sys and in *J* . The calibration is valid until fibres are moved or until a temperature drift causes a change of the fiber birefringence. In practice, the system is calibrated right before a measurement session. Here, all sample measurements use the same calibration.

The configuration for the actual sample measurement is shown in Figure 4.4 (bottom). In contrast to the calibration setup, the polarizer is removed and the mirror is replaced by the sample. The measured Jones vector is given by

$$
\bar{J}\_{\rm m} = \mathbf{J}\_{\rm sys} \cdot \mathbf{J}\_{\rm QWP2, rev} \left( \mathcal{J}\_{\rm Q} \right) \mathbf{J}\_{\rm S} \, \mathbf{J}\_{\rm QWP2} \left( \mathcal{J}\_{\rm Q} \right) \bar{J}\_{\rm in} \,. \tag{4.11}
$$

Now, only **J**s is unknown, and for a robust determination thereof, *J*m is measured at various positions of the QWP2 from *Q* 0° to 165°, with a step size of 15°. A least-squares fitting procedure reveals **J**s, which we verified by using a linear polarizer as sample and determined its angular position with an accuracy of 3°.

# **5 Optical coherence tomography system on a silicon photonic chip**

This chapter corresponds to publication [P1] with slight adaptations to match the format and the structure of this thesis. With the preceding chapters OCT has proven to be a powerful tool for the characterization of nanoparticles and nanocomposite materials. However, current systems are not suitable for many industrial environments, where often robust, small-sized and cost-effective systems are required. These challenges can be addressed by integrated-optical systems, with silicon as the material platform of choice, see Chapter 1.3.

*[Begin of Publication P1]*

*Adapted with permission, © 2016 Optical Society of America.*

#### **Optical coherence tomography system mass-producible on a silicon photonic chip**

*Optics Express; Volume 24; Issue 2; pp. 1573-1586; January 2016* DOI: 10.1364/OE.24.001573

S. Schneider1 , M. Lauermann1 , P.-I. Dietrich1,2, C. Weimann1 ,

W. Freude1,2 and C. Koos1,2

Miniaturized integrated optical coherence tomography (OCT) systems have the potential to unlock a wide range of both medical and industrial applications. This applies in particular to multi-channel OCT schemes, where scalability and low cost per channel are important, to endoscopic implementations with stringent size demands, and to mechanically robust units for industrial applications. We demonstrate that fully integrated OCT systems can be realized using the state-of-the-art silicon photonic device portfolio. We present two different implementations integrated on a silicon-on-insulator (SOI) photonic chip, one with an integrated reference path (OCT*int*) for imaging objects in distances of 1 mm to 6 mm from the chip edge, and another one with an external reference path (OCT*ext*) for use with conventional scan heads. Both OCT systems use integrated photodiodes and an external swept-frequency source. In our proof-of-concept experiments, we achieve a sensitivity of −64 dB (−53 dB for OCT*int*) and a dynamic range of 60 dB (53 dB for OCT*int*). The viability of the concept is demonstrated by imaging of biological and technical objects.

## **5.1 Introduction**

Optical coherence tomography (OCT) [1,28] looks back on great advances in the past decades. Micrometer-range resolution in volumetric imaging opened a wide field of applications reaching from ophthalmology in medical diagnostics [87] to particle and defect characterization in materials science [80,88]. While current OCT systems are still assembled from discrete components, optical integration offers the prospect of compact low-cost mass-producible implementations. Especially in endoscopic applications as well as in industrial surface and distance metrology, integrated systems can comply with the demands in size, mechanical robustness, temperature stability and cost. An arrangement of multiple parallel OCT systems becomes feasible, allowing the simultaneous recording of depth scans at different positions (B-scan). Silicon photonics is a particularly attractive platform for integration of OCT systems, lending itself to large-scale photonic-electronic integration on the basis of mature high-yield CMOS processes that are offered by widely available foundry services [118].

Early in 2014, we introduced the first integrated silicon-photonic OCT system with surface imaging capabilities, comprising the interferometer and on-chip photodetectors [119]. In the same year, Yurtsever *et al.* demonstrated a silicon-photonic OCT system and showed a sensitivity of −62 dB, yet with offchip photodetectors [120]. Recently, a silicon photonic optical combiner with integrated photodiodes was used as a polarization-diverse receiver for a fiberbased swept-source OCT system [121]. However, for completing the OCT interferometer, external couplers and circulators are used. Other realizations of integrated OCT systems rely on silicon nitride and silicon oxynitride waveguides, where both swept-source [122] and spectral-domain implementations have been shown. These demonstrations comprise, e.g., spectral-domain OCT chips based on arrayed waveguide gratings (AWG) [123], which have been used for characterization of biological samples [124]. Yet, silicon nitride and silicon oxynitride integration platforms are limited to passive components only such as waveguides and power splitters, and hence must be always complemented by external photodetectors. This limits the potential for large-scale integration.