Related Works and Basic Considerations

Abstract

The characterization and measurement of surface profiles is one of the most basic metrology tasks in industrial manufacturing. What started with mechanical stylus profilometers has developed with ultrasonic transducers towards optical instruments. These are capable of appropriate resolutions to enable nanometrology, [14].

2.1 Profilometry

The characterization and measurement of surface profiles is one of the most basic metrology tasks in industrial manufacturing. What started with mechanical stylus profilometers has developed with ultrasonic transducers towards optical instruments. These are capable of appropriate resolutions to enable nanometrology, [14]. As surface metrology is well established, basic terms and parameters are defined in corresponding norms such as ISO 4288 (assessment of surface texture) and ISO 4287 (terms, definitions and surface texture parameters), [17, 18]. Established optical instruments are classified as areal integrating, line profiling and areal topography instruments in ISO 25187, [19]. According to Leach et al. [14], sensors with the capability to record areal information about surface shape, waviness and roughness simultaneously are the most desirable in nanometrology. These sensors are usually defined as 3D-sensors where the topography information is gathered as the local distance z at a specific coordinate (x, y). Apart from the classification mentioned in [19], numerous variants exist, [20]. On the most basic level, the common denominator for optical approaches is that all of them follow a simple interaction model where light from a light source interacts with an object of interest. The result of this interaction can be observed separately on an observation plane. The properties of the light source such as intensity, polarization state, temporal behavior, coherence and spectral behavior can be utilized to encode information about the object surface topography. Depending on the surface structure of the object of interest and the design of the observation method, different limitations or artifacts occur. One example is the occurrence of so-called speckles on optical rough surfaces under coherent illumination. While most metrology approaches suffer from this effect in terms of a decreased signal-to-noise ratio (SNR), some techniques such as coherence scanning interferometry or speckle interferometry make use of this effect. Production integrated metrology becomes increasingly important in areas where the critical dimensions decrease from the µm- to the nm-range. Industrial sectors such as MEMS, power semiconductors or photovoltaics require a continuous, precise monitoring of production to maintain quality, [14]. A variety of measurement technologies to characterize e.g. surface roughness, topography or film thickness have been reported, [21]. Apart from specific measurement machines with pm resolution for research purposes which require high efforts in building, calibration and maintenance, a broad range of technologies is known which is more applicable as production-accompanying tool, [22]. The following section is intended to give an overview of these approaches.

Häussler and Ettl classified optical 3D sensors in [21] mainly according to their limitations and their dominant noise sources. Following that, four main classes of sensors can be associated:

• Type I sensors (e.g. laser triangulation)

• Type II sensors (e.g. coherence scanning interferometry)

• Type III sensors (e.g. phase-shifting interferometry)

• Type IV sensors (e.g. deflectometry)

As Type I sensors mostly incorporate technologies which measure a lateral perspective shift on local details, the uncertainties are determined by the uncertainty of this shift measurement. This means that the uncertainty scales with the inverse square of the distance to the detail, [21]. Techniques include laser triangulation, fringe projection, confocal microscopy and others. For Type II sensors such as coherence scanning interferometry, the measurement relies not on the detection of the phase of the signal but on the correlation of single speckles reflected from a rough surface. Therefore the statistical noise, mainly composed of the standard deviation of the object surface, is the primary source of noise, [23, 24]. In contrast, Type III sensors involve classical interferometric technologies such as phase-shifting interferometry [25] and digital holographic microscopy [26]. These are phase measuring technologies which enable sub-nm axial resolution and are only limited to photon noise. Furthermore, [21] separates deflectometry as a Type IV sensor. With limitations of photon noise and developments towards nanoscale accuracy, this technology is promising for industrial applications, [14].

This section analyses important technologies from every sensor class as well as atomic force microscopy (AFM) as one important, high-resolution, stylus-based technology in order to evaluate open questions in nano-profilometry. All of them have been widely qualified for the measurement of nm-scale surface parameters in a lab environment, [27, 28].

2.1.1 Atomic Force Microscopy

AFM, although not an optical metrology approach, has become an important and highly precise tool in nano-engineering and science, [29–31]. It is a technological development which goes beyond the capabilities of classical stylus instruments in terms of its resolution. It provides an axial resolution down to 0.01 nm while the lateral resolution can be in the range of 0.1 – 10 nm, [32]. These features made it exceptionally important in areas such as cell biology and molecular sciences where it enables the precise study of molecular processes and interactions, [33, 34]. Industrial and technical applications make use of it for the characterization of nano-structured materials and features, [35–37]. AFM can deliver interesting additional features like force sensing and surface modification, [38]. Several developments such as multi-cantilever arrangements and resolution-enhancing methods also have enabled the technology to investigate cm²-large structures with high aspect ratios. In this context, aspect ratios of a axial measurement range of 15 µm with a deviation of ± 60 nm could be achieved while the lateral measurement range was 15 x 100 µm², [39]. Using an array of cantilevers in a 32x1 chip design, Minne et al. [40] were able to scan large areas of 2000 x 2000 µm² within 30 minutes while the axial measurement range was 2.5 µm at a resolution of 3.5 nm. Most of these developments have the need for complex setups and measurements last several minutes. Additionally, AFM is sensitive to systematic errors and contamination of samples. These circumstances prevent using AFM-based approaches for production accompanying tasks.

2.1.2 Confocal Laser Scanning Microscopy

Another established method for the characterization of surface profiles is confocal laser scanning microscopy (CLSM), [41, 42]. By aligning the focal plane of the illumination of a single point on a sample with the imaging plane of that same point on a pinhole, information is solely captured from this point, [43]. By scanning a depth range, the height profile of a sample can be acquired with a high SNR even if the surface is strongly scattering. As scanning has to be performed in the lateral dimension as well, the acquisition of a full areal height map requires a long time, meaning several minutes. In order to achieve a high axial resolution, the so-called auto-focus method is applied. In this method, the focal plane is adjusted for the highest intensity for a given point. The method is repeated in a given axial range of the sample while the position of the focal points is recorded correspondingly, [44]. A faster, yet less precise, option is the so-called intensity method. This method correlates a measured intensity to a calibrated height-intensity curve. According to [44], this results in measurement times in the order of compared to the auto-focus method. Also, due to the limited linearity of the calibration curve, the measurement range and precision is also significantly reduced. Furthermore, the calibration curve has to be recaptured for every new sample material in combination with the microscope objective used for the investigation.

Depending on the translation stages and magnification, a lateral resolution of (100 – 500) nm is typically achieved to capture surface parameters such as roughness, waviness and form error in an areal fashion, [45]. Larger lateral measurement ranges such as 1.3 x 1.3 mm are usually captured with a lower resolution of e.g. 10 µm in order to speed up the measurement process. Buajarern et al., [42], have shown that the axial resolution is not only limited by the resolution of the translation stages used to perform the auto-focus determination, but even more by the depth of focus / resolution of the microscope objective used for imaging. Using high numerical aperture (NA), large magnification objectives, the axial resolution is optimal but restricted to about 150 nm.

Different works have shown the applicability of CLSM to characterizations in biology, for dental as well as engineering materials [46–48]. Besides the need for mechanical scanning and the interdependence of measurement range and resolution, a major disadvantage of CLSM is the high effort necessary for adjustment of parameters in order to achieve optimal results. Tomovich et al., [49], have demonstrated that the adjustment of parameters is highly dependent on the reflective and structural properties of the sample. From this perspective, CLSM is a versatile tool in research labs but is limited in the usage for production accompanying tasks.

In addition to classical confocal microscopy, several developments have been made in order to circumvent some of the drawbacks such as the need for scanning. Chromatic confocal microscopy is one approach. It particularly targets scanning along the z-dimension. While dispersion in the optical system leads to different foci for each wavelength, height information can be decoded in spectral variations, [41, 50]. In terms of measurement range and resolution, both — the spectral range and spectral intensity stability—are important. Typical light sources include halogen and xenon lamps as well as LED sources. Other works have demonstrated the usability of supercontinuum sources as well, [51]. Using the linearized wavelength calibration, the calculation of height can be performed in different ways. The fastest and computational most efficient technique, yet the one with the lowest resolution, is to determine the maximum intensity of every spectral line. The highest computational effort is necessary when fitting the intensity distribution with e.g. a Gaussian approach, though these approaches enable the highest possible resolutions. A possible compromise between speed and accuracy is the so-called bary-center calculation to estimate the height, [21]. The limitations regarding the lateral resolution and the resolution of slopes are very similar to other microscopy techniques. Depending on the light sources, the NA of the objective and the size of the pinhole, typical spot sizes are 5 – 10 µm for vertical ranges of < 1 mm and 10 – 30 µm for vertical ranges > 1 mm, [21] can be achieved. The resolution of slopes is usually defined by the half aperture angle where the range is between $\pm \,{18}^{\circ }$ to $\pm \,{44}^{\circ }$ for NAs of 0.3 to 0.7 respectively.

The extension of this method in order to capture areal information normally relies on scanning approaches. Some problems occur due to dynamic stitching errors and spherical aberrations from scanning, [52, 53].

The method can also be used to measure the profile and thickness of multiple, semi-transparent layers. In this case, the analysis has to take into account the refractive index of the transparent material at the different wavelengths. Faster approaches utilize multi-probe setups in order to parallelize data acquisition. Nonetheless, they are strongly limited regarding the lateral resolution, [54, 55].

2.1.3 Digital Holographic Microscopy

Digital holographic microscopy (DHM) is a fast, robust and full-frame-capturing technology to evaluate the surface topography of samples. While surface data from an interferometric approach is captured by a camera as full-field information (spatially, intensity, phase) within the integration time of the device, all aberration correction, exact focusing and surface reconstruction can be done digitally within post-production. Stroboscopic illumination can be used additionally to speed up acquisition from conventional frame rates such as 20 – 60 fps to 25 MHz, [21]. The axial accuracy is dependent on the calibration of the light sources wavelength/frequency which leads to achievable axial resolutions of 0.1 nm with stabilized laser sources, [21]. One of the limiting factors is the axial measurement range which can be extended by the use of multiple wavelengths to several µms while keeping a sub-nm resolution, [56]. Furthermore, the combination with other techniques, such as reflectometry, enables the analysis of parameters such as layer thickness, refractive index profiles and topography of multilayered structures, [57]. The signal formation and analysis is generally described in two steps where one is the data acquisition and the other is the so-called reconstruction, [26]. A typical holographic setup contains e.g. a Mach-Zehnder interferometer as its main component. In this setup, the object wave, which is either reflected or transmitted from the object, is imaged close to the camera plane. The reference wave interferes with the object wave and its intensity can be detected and analyzed. The subsequent reconstruction is performed in two steps. First, the complex wave is reconstructed in order to separate the real and virtual hologram from the zeroth-order signal. This can be performed in hardware by introducing defined phase jumps (in inline-holography) or by introducing an angle between the object and the reference wave (in off-axis holography). Furthermore, the separation of the zeroth-order term [58] or the twin-image term [59] can be done by FFT approaches in software. Afterwards, the filtered hologram is digitally illuminated by a reference wave. During a second process step, numerical propagation is performed by approaches like the single Fourier approach [60], the angular spectrum approach [61] and the convolution approach [62]. In typical implementations, two approaches to define numerical lenses for the correction of higher order aberrations:

• recording of a physical reference hologram on a plane sample and by transmission through air, [60, 62]

• fitting of Zernike polynomials in an assumed flat area of a measured sample, [63–65].

It is quite common to use combinations of both solutions in order to achieve optimal results. The reference wave is used to correct all phase and tilt aberrations as well as for curvature of waves and de-focus correction. In particular, this technique has the advantage that only a single hologram needs to be acquired while the correct focal point can be tuned by numerical propagation. This can also be used to increase the depth of field, [66]. Furthermore, it indicates that DHM is a full-field metrology where all relevant information is captured in one image acquisition.

A common problem in DHM is $2\pi$/phase ambiguity as it limits the usable measurement range. An efficient method to overcome this problem is the so-called multiple-wavelength holography, [67, 68]. In this approach, data is captured with slightly different wavelengths which allows the numerical construction of a new, artificial wavelength which can be used to ‘construct’ unambiguous measurement ranges. DHM is also capable of recording dynamic events which occur in a repetitive fashion. For this purpose, the events of interest are observed using stroboscopic techniques. It could be shown that an acquisition of events with a repetition frequency of up to 25 MHz is possible, [69]. Extensions to DHM in order to evaluate functional parameters such as reflectometry have been reported, [21]. However, complex wavefront reconstructions or fitting procedures are necessary to separate the information of different layers. Currently, this prevents the use of this approach in industry.

2.1.4 Phase-shifting Interferometry

Phase-shifting interferometry (PSI), is an more elaborate variant of traditional (high coherence) interferometry for the determination of surface information, [25]. Depending on the light source, the detector and the optical setup, lateral resolution of 2 µm and axial resolutions up to 1 nm are reachable, [21]. PSI enhances classical interferometry by introducing multiple controlled phase shifts in order to determine the phase-dependent surface height of an object, [70] . Furthermore, it overcomes the directional ambiguity of interferometric signals by generating quadrature signals, since back-reflected light from a surface in combination with light from the reference arm generates an areal interferometric fringe pattern, [21]. Due to some simplifications data analysis of the generated quadrature signals can be accelerated by fitting sine and cosine signal components to determine the phase with high accuracy. A common approach for the data analysis of these signals is the so-called auto-correlation or synchronous detection, [21]. Both, sine and cosine components are described as integrals of a full cycle of the phase shift. Depending on the number of introduced phase shifts, various algorithms can be used, [25]. Furthermore, the use of non-linear e.g. sinusoidal manipulation of the phase shifts has been reported to be accurate and computational more efficient than linear manipulation, [71]. In addition to temporal PSI, other works have described alternative methods to encode and detect phase-shifts such as polarization encoding, [72, 73].

A common problem in interferometry occurring in PSI is the determination of phase using an arctan function which is only possible in the range of 0–$2\pi$. Surface heights corresponding to multiples of $2\pi$ will therefore be calculated within the given range resulting in phase jumps, known as a wrapped phase. Several algorithms are currently being researched, [74–77]. More recent approaches on phase-shift interferometry extend the standard setup with multiple sensors, [73, 74, 78]. By capturing interference signals at different polarizations, phase angles and wavelengths, an axial resolution of 2 nm on a 3 µm height step in an area of 0.5 x 0.5 µm could be achieved. Due to the need for multiple channels, high efforts have to be made to synchronize and handle the data of up to 6 cameras. The evaluation of the phase shift of the total interference contrast (TIC) is an alternative phase-shifting technique which is capable of providing sub-nm axial resolution in a full-field manner, [79]. As it is not based on coherence properties of the light source it is reliant on the knowledge of an material model of the tested sample. Therefore, the method is limited to mainly laboratory usage, [80].

2.1.5 Coherence Scanning Interferometry

Belonging to so-called type II sensors, coherence scanning interferometry (CSI) distinctively deviates from classical interferometry. This approach utilizes light of low temporal coherence not to measure the phase of the signal, but to estimate the correlogram of individual speckles on a surface, [21]. The technique traditionally scans one interferometer arm in order to evaluate the signal contrast which is equivalent to the envelope of the correlogram, [70]. The coherence length of the light source is the main limitation as it restricts the possible measurement range. Thus, only height variations within the coherence length can be detected continuously. This can be an issue on very rough surfaces or steep slopes, [81]. Usually setups are designed to capture the full lateral information using a camera which detects the correlogram envelope at every point P(x, y), while a scanner ensures the axial movement in the height range, [82, 83]. While the typical scanning range is (10 – 200) µm with piezo-based scanners and > 1 mm for mechanical scanners, the interferometer part of the system is most commonly designed as an microscope objective of the Michelson, Mirau or Linnik type. The approach is also known by the names of coherence radar, coherence scanning, white-light interferometry, vertical scanning interferometry and others, [70].

According to [84], the simplest model for the description of the signal in CSI is an incoherent superposition. The signal shape in this case is determined by the spectral and spatial distribution of light in the pupil plane. Using this model, several boundary cases can be found. For example the properties of a system with low NA (e.g. 0.2), narrow bandwidth (e.g. $\Delta \lambda$ = 100 nm) will basically be dominated by the spectral distribution of the light source with high fringe visibility at the zero scan position. If the system has a high NA (e.g. 0.6) and narrow bandwidth (e.g. $\Delta \lambda$ = 20 nm), in contrast, the coherence properties are dominated by the spatial distribution in the pupil plane, hence the focusing. In real systems both effects will be present, whereas it can be stated that in low NA systems the spectral contribution and in high NA systems the spatial contribution of the light source primarily influences the signal.

One of the distinguishing features of CSI is the lack of a $2\pi$ ambiguity. This makes it well suited for measurements of rough surfaces and steep slopes. In general, it can be said that the slowly modulated envelope of the interference fringes is an indicator of the signal strength as well as a measure for the surface profile. But it has to be clear that the peak of the envelope is not equal to the maximum height of the sample. It represents only the point where the optical path difference (OPD) equals zero. More precisely, it is the point where the dispersion affected path length difference in the setup equals zero. Therefore, the approximation of the envelope signal is only an idealization as in reality dispersion effects and scan dependent distortions have to be taken into account. Signal processing in CSI is usually performed in multiple steps, where the start of the scan position is determined by the maximum signal strength or fringe visibility. The scan range is determined by sectioning the images according to the expected height of the sample. Several methods to perform the envelope detection including peak finding of the maximum have evolved over the years. While [85] have shown the application of digital filters and an demodulation technique, other groups such as [86] and [87] have shown how to remove the carrier signal to separate the envelope. According to approaches of [88] and [89], who performed a centroid determination of the square of the signal derivative, a higher resistance to noise on the signal is achievable.

Noise, e.g. from optical aberrations, the scanning process, diffraction or vibration can significantly influence the sensitivity of the technique. Therefore, more advanced signal processing algorithms use the result of the envelope detection only as a rough estimate of the fringe order and call the result a first topography map. Based on this, a fine estimation using a phase analysis of the fringes results in a phase map. This method increases the accuracy significantly although other problems such as $2\pi$ ambiguities on thin film structures, surface roughness or sharp edges occur during phase analysis. Some algorithms take this into account, [90, 91].

More sophisticated approaches developed methods for the simultaneous envelope detection and phase analysis by the correlation of the intensity data with a complex kernel. These kernels have been derived fromPSI algorithms and use different approaches such as wavelet techniques or least-squares fitting, [88, 92–94]. Furthermore, recent works have also demonstrated that sub-nm axial resolution is possible through advanced signal modeling techniques, although there remain limitations due to scanning and the relatively small axial measurement range, [95, 96]. Coherence Scanning Interferometry is well suited for the determination of the surface topography on rough surfaces, [97]. One major disadvantage is the necessity for scanning along one physical dimension of a sample as this causes errors and slows down the measurement procedure.

2.1.6 Low-coherence Interferometry

One of the most advanced technologies for high resolution surface profilometry and tomography is low-coherence interferometry (LCI). Optical-coherence tomography (OCT) is a commonly known implementation, [98, 99]. Although OCT was initially used in medicine, developing it as a tool for industrial purposes has become increasingly important, [100–103]

The range of different OCT approaches reaches from time-domain optical-coherence tomography (TD-OCT) and spectral-domain optical-coherence tomography (SD-OCT) to frequency-domain optical-coherence tomography (FD-OCT), [104]. While most approaches are implemented as point sensors with the possibility to mechanically scan a surface, they vary in speed, measurement range and accuracy. More fundamentally, the accuracy and measurement range in the axial dimension are defined by the coherence length $l_c$ of the light source used, [105],

$$\begin{aligned} l_c= \frac{2 \cdot \ln {2}}{\pi } \frac{\lambda _c^2}{\Delta \lambda }, \end{aligned}$$

(2.1)

where $\lambda _c$ denotes the center wavelength of the light source and $\Delta \lambda$ its spectral range, assuming that the light source has a Gaussian-shaped spectrum. The typical axial measurement range, using broadband light sources, is a few hundred µm with a resolution of about 1 µm, [106]. These values are usually fixed within the design of the specific setup and can not be adjusted during operation. In order to achieve higher axial measurement ranges as well as a higher resolution some hardware problems must be solved. The most notable influences on the resolution are environmental disturbances on the interferometer arms as well as the repeatability of the scanning system used to gather areal information, [107]. An extension of the measurement range in TD-OCT is achieved by the introduction of multiple reflecting surfaces in the reference arm, [108]. This modification enables the measurement of a large axial range while keeping actual mechanical scanning to a minimum. Another approach merges both reference and sample arm in a FD-OCT configuration, [107]. In this approach, a reflective surface in the vicinity of the sample is used as a reference. Environmental changes have the same influence on both optical paths. This hardware adaption in combination with algorithmic frequency and phase evaluation leads to an accuracy of 0.1 nm in a measurement range of about 3 mm in the axial dimension.

Other approaches aiming to extend the measurement range and to increase the resolution rely on dispersion compensation and are known by the term dispersion-encoded full-range OCT (DEFR-OCT), [109, 110]. The authors applied a numerical dispersion compensation to remove complex conjugates and therefore extend the measurement range. Another advantage is that the point of highest sensitivity of the setup is shifted to the center of the measurement range. As the initial algorithm relied on multiple iterations of Fourier transforms, new approaches have been developed to speed up processing times. Some of them introduce artificial dispersion into the setup which can be compensated by the algorithm with only one Fourier transform and a convolution, [111]. Current works applying these ideas show high accuracy in the tomographic analysis of nano-structured domain walls in ferro-electric media, [103].

In contrast to medical applications, industrial measurement tasks often demand the characterization of relatively large areas (mm² instead of µm²). Scanning of samples is therefore necessary, but introduces issues regarding the accuracy and repeatability of results, [112]. Approaches to overcome these limitations are based on a setup incorporating a camera and a high resolution translation stage; summarized by the term full-field OCT (FF-OCT), [113–116]. In this particular approach scanning is still required in the reference arm in order to observe changes in the interference data due to different path lengths.

More recently, full-field approaches that avoid scanning altogether have been developed. Based on a hyperspectral imager and an etalon, Zhu et al., [117] have shown high precision surface measurements. While one dimension of the sample is encoded in the bandwidth of the etalon-generated spectral slices, the second dimension is directly imaged on the camera. The surface profile is calculated from the phase information of the interference fringes. The resolution and measurement range of this setup is predominantly defined by the wavelength spacing of the etalon as well as by the resolution of the spectrometer. Both tuning options are opposing each other.

A more advanced method substitutes the etalon for a microlens array in order to decode one areal dimension on the hyperspectral imager which increases the measurement range and light efficiency, [118]. The setup was capable of measuring an axial range of up to 880 µm with an resolution of 0.49 µm while having a lateral measurement area of 3.5 x 3.5 mm. Further developments by the same group have demonstrated the possibility to acquire 2500 independent probing points which increased the measurement range, light efficiency as well as the tilt angle acceptance, [119]. One drawback is that the method is only able to make use of about 50% of the detector size to image µm²-sized samples which decreases its lateral resolution. The axial measurement range was about 825 µm where a resolution of 6 nm was achieved. According to these works, the dynamic range (DR) is defined as the inverse ratio of the resolution to the measurement range. The analyzed works of recent, high-dynamic range approaches to LCI and OCT demonstrate a progression in technology and dynamic range, Tab. 2.1.

Table 2.1 Comparison of current LCI approaches regarding measurement range, resolution and dynamic range

In direct comparison, it can be deduced that one aim is to develop a full-field, areal approach to surface profilometry which is capable of measuring large distances with sub-nm resolution as Koch et al. have shown in a point sensor. Recent works such as of Reichold et al. have already demonstrated the potential of spectrally-encoded interferometry but still lack sub-nm resolution.

The current work aims to develop a system which is able to detect surface profiles with sub-nm resolution and a high-dynamic range in the axial as well as lateral dimension incorporating sub-nm resolution.

2.2 Polymer Cross-linking Characterization

The control of mechanical, electrical and optical properties of polymers during fabrication is necessary to ensure their performance, [121]. As cross-linking is a crucial process step in order to optimize properties and fabrication parameters, it is necessary to monitor its degree, [122]. Long-term mechanical resistance, temperature stability as well as functional parameters such as refractive index are adjusted with the cross-linking process, [123–125].

Interesting industrial applications of polymers are centered around lithographic processing such as resist coatings and optical waveguides, [126, 127]. Polymer-based optical waveguides are usually processed by patterning (photoresist-based or direct lithography), soft lithography or printing techniques [128, 129] in order to achieve defined cross-linking and refractive index differences. From the analysis of reaction kinetics of polymers it is known that the density of a material as well as its refractive index changes during cross-linking, [123]. This relationship can be described by the Lorentz-Lorenz equation, [130, 131],

$$\begin{aligned} R_m = \frac{(n^2 - 1)M_w}{(n^2 + 2)\rho } \end{aligned}$$

(2.2)

where $R_m$ represents the molar refractivity, n the refractive index, $M_w$ the mass-averaged molar mass and $\rho$ the density. Although it is known that the Lorentz-Lorenz relation is only an approximation, it has been proven applicable to a variety of polymers and was correlated with other methods such as hardness measurements, [131, 132]. For their fabrication, optically-cured polymers have to fulfill several requirements such as optical transparency and chemical as well as thermal stability [133, 134]. Advancing from conventional thermoplastics such as polymethyl methacrylate, polystyrene, polycarbonate and polyurethane, research has been geared towards the development of new polymers which exhibit lower absorption losses and higher stability, [128]. Promising classes of polymers are halogenated polyacrylates [135], fluorinated polyimides [136] or polysiloxanes [137]. In particular, applications such as polymeric waveguides or direct laser writing on wafers make use of this effect to generate functional properties with refractive index changes of about $10^{-2}$, [138]. Žukauskas et al. applied this effect to generate gradient-index lens elements with a size of 50 x 50 x 10 µm³, [139].

In order to characterize these functional properties alongside with the degree of cross-linking and their spatial distribution, different metrology approaches are known from literature.

2.2.1 Soxhlet-type Extraction

A very common method to determine the degree of cross-linking is the Soxhlet-type extraction, [140]. For that purpose a sample is exposed to a solvent, typically a xylene isomer, in which non-cross-linked material will dissolve after a few hours. After drying, the degree of cross-linking can be calculated from the respective weights of cross-linked and non-cross-linked material. The result is very precise whereas the process is time-consuming, destructive and not spatially resolving. Oreski et al. found that the time for the extraction is at least 18 hrs while drying takes another 24 hrs, [141]. Furthermore, Hirschl et al. found that the extraction time and other process parameters can have a huge influence on the repeatability of the measured degree of cross-linking, especially in weakly cross-linked samples. They determined that the repeatability ranges from 2–4 $\%$, [142].

2.2.2 Differential Scanning Caliometry

A method to determine cross-linking in polymers as a measure of thermal features such as heat-flow, melting point and reaction enthalpy is differential scanning caliometry (DSC), [121, 143, 144]. Similar to chemical-based methods, this method does not allow spatially resolved measurements, works destructively and is time-consuming. A typical measurement cycle in the so called dual-run mode takes 2 x 45 min during which a defined heating profile is applied, [141]. A comparative study has shown that different approaches for referencing the measurements to other methods might apply and also that errors in the repeatability can be $10 \%$ and larger, [145]. Especially weakly cross-linked samples require slower heating profiles, hence longer measurement times, and produce inherently larger errors, [143, 146].

2.2.3 Dynamic Mechanical Analysis

Dynamic mechanical analysis (DMA) is a well established laboratory approach to characterize thermal and mechanical properties of materials. It is especially well suited for visco-elastic materials such as polymers, [121]. During analysis a small dynamic force (e.g. in tension, compression, bending or shear mode) is applied to a well defined sample, typically in a sinusoidal fashion, [147]. The measurement time, the temperature as well as the oscillation frequency are typically variable parameters dependent on the sample types to be measured, [148]. In response to the introduced stress $\sigma$, the samples strain $\varepsilon$ is measured dynamically. Here, these values are measured as the amplitudes of the sinusoidal signals. Furthermore, the phase difference between both signals is measured as $\Phi$. From these, the complex modulus of the sample $E^*$ $= \frac{\sigma }{\varepsilon }$ can be calculated. It holds information on the real and imaginary components which are used to characterize the elastic properties of a material. The real component $E^\prime$ is known as the storage modulus which is proportional to the elastic deformation work stored by the sample during deformation. The imaginary component, known as the loss modulus $E^{\prime \prime }$, is a measure for the thermo-mechanical loss of deformation work due to internal friction among other effects. Both parameters can be referred to the phase difference $\Phi$ by

$$\begin{aligned} E^\prime = |E^{*}| \cdot \cos \Phi \end{aligned}$$

(2.3)

$$\begin{aligned} E^{\prime \prime } = |E^*| \cdot \sin \Phi \end{aligned}$$

(2.4)

From this relation, it becomes clear that the phase difference between excitation and response is crucial for the understanding of certain material properties. While fully elastic materials, such as steel, typically show a phase difference of $\Phi$ = 0, fully viscous materials show a phase difference close to $\Phi \,=\,{90}^{\circ }$. In the case of visco-elastic polymers any value in between may occur. Another important measure in DMA is the determination of the glass-transition temperature $T_G$. It describes the transition from a state where molecular networks are stiff and allow only elastic deformations to a state where non-elastic deformations are also possible. The $T_G$ can be calculated with a variety of approaches, some of which are comparable to DSC measurements, [121].

In order to calculate the applied stress to a specific sample, the knowledge of its geometry is important. Usually standard sample sizes are fabricated for the purpose of DMA measurements, [148]. The investigation of fabricated products or components is not possible. As the dimensional parameters depend on the kind of stress applied, their input can be significant. Any error in the determination of the samples dimension will have a strong influence on the resulting stress on the sample as well as on comparability between samples.

The time for a measurement is very significantly dependent on the chosen experimental parameters. Usually the temperature range as well as the heating rate are subject to changes while the excitation frequency is kept fixed. Typical investigations of polymeric materials work in ranges from -75 - ${{150}^{\circ }}\mathrm{C}$ with a heating rate of , [149] to ranges of -150 - ${{200}^{\circ }}\mathrm{C}$ at a heating rate of , [143].

One of the main advantages of DMA is the ability to directly measure the degree of cross-linking, [145]. Typically the degree of cross-linking is determined as a weighted ratio of the storage modulus, the sample density and the monomer molecular weight, [150]. Other works propose the calculation as a quotient of the logarithmic storage modulus of cross-linked and non-cross-linked material, [151].

DMA is, amongst others, a standard testing procedure in material development for novel polymers, fabrication technologies or the understanding of cross-linking mechanisms, [152–154]. With its high accuracy, the capability to determine the degree of cross-linking directly and the long measurement times, DMA is a predominantly laboratory-based method. Applications in production accompanying tasks are not established.

2.2.4 Spectroscopy-based Methods

Non-destructive measurements of cross-linking can be obtained by using optical metrology such as Raman spectroscopy, [155] or luminescence spectroscopy, [156]. The analysis of spectral features of reactional groups and bonds or the luminescence intensity of characteristic peaks can be utilized to calculate the degree of cross-linking. Recent works have shown that these technologies are able to characterize cross-linking of coatings on solar cells. In a comparative study, Hirschl and co-workers, [157], have demonstrated that Raman spectroscopy gives comparable results to classical methods like Soxhlet-extraction. Although it has to be noted that the measured errors of the degree of cross-linking were up to 15%, especially for samples with weak cross-linking. Furthermore, acquisition times for Raman spectra depend very much on the SNR of relevant spectral intensity peaks and hence require a large amount of averaging. Recent studies report acquisition times for single-point measurements between 50 – 100 s, [139, 157]. Peike et al. [155] point out that Raman analysis is very material-specific and can be complex with different peaks overlaying each other. Additionally, they found that the SNR decreases with peaks at higher wavelengths as $I_{peak}$  . This can be critical for weakly cross-linked material or materials with a low number of reactional groups.

A recent work by Schlothauer et al. has qualified luminescence spectroscopy as a tool for cross-linking characterization with an accuracy of 4 – 6%, [156]. However, the method requires a large amount of averaged spectra in a point-by-point scanning fashion. Acquisition times for a 16 x 16 cm² were about 80 minutes.

2.2.5 Low-coherence Interferometry and Other Optical Methods

As refractometry is a well established method to measure refractive indices, it is also suited to evaluate cross-linking in polymers, [158]. It has been used to directly determine cross-linking progress during curing within thermoset polymers used as a matrix material for composites, [159]. For the purpose of these examinations, optical fibers are declad over some probing area and integrated in a to-be cured polymeric matrix in order to measure the change in refractive index in terms of a change in transmission intensity, [158]. Another common approach is the integration of cleaved optical fibers in a polymeric matrix where the Fresnel reflection is measured in relation to the change in refractive index, [160, 161]. Both approaches are well suited for the characterization of thermoset polymers as the change in refractive index is rather high during cross-linking with $\Delta n$ = 2 - $6 \times {10^{-2}}$.

Classical optical-coherence tomography has been used to examine structural defects such as bubbles or phase separation during cross-linking by scanning a sample within a few seconds, [162]. Other interferometric techniques such as spectrally-resolved white-light interferometry, frequency domain interferometry or digital holographic interferometry have been utilized to measure refractive indices with accuracies in the range of ${10^{-5}}$- ${10^{-6}}$, [163, 164] as well as mechanical deformations on the nm-scale in material and biomedical engineering, [165–167]. Methods based on phase-sensitive OCT have been used to characterize photo-elasticity on polymeric composite materials and might also be suitable for cross-linking characterization, [168]. More recent works made use of the combination of LCI and confocal approaches in order to measure refractive index of transparent media from a distance, [169].

2.2.6 Spatially-resolved Approaches

None of the technologies for cross-linking characterization presented so far are inherently spatially resolving. Spatial resolution for chemical methods such as DSC or DMA is usually realized by cutting samples into defined sub-samples, [170, 171]. Especially spectroscopic technologies such as Raman spectroscopy gain spatial resolution by scanning over a sample. Due to the need for integration over multiple spectra at each point, the measurement times of a sample having 330 x 150 probing points can be as long as 82 minutes, [170]. Although this technology can be used to cover an area of a few cm², the typical lateral resolution limit is about 2 µm, [172]. Consequently, other groups have shown a significant increase in lateral resolution down to 35 nm by combining Fourier-transform infrared spectroscopy (FTIR) with AFM. This approach increases the measurement time significantly so that an area of 1 x 1.5 µm² was measured in 7.4 hrs, [173, 174]. None of these technologies have been used for cross-linking characterization so far. Among the LCI techniques, Guerrero et al., [175] published an approach which was able to reach a spatial resolution of about 17 µm. It was based on a phase estimation of intensity extremes and shows a theoretical refractive index resolution of ${10^{-4}}$ which did not take any thermal or noise influences into account. Shortcomings of the method are the restriction to measurements of the differential refractive index as well as its dependence on intensity measurements which are influenced by noise.

Beside these technologies, some other imaging approaches might be interesting for cross-linking analysis. Singh et al. have shown that corneal cross-linking can be indirectly imaged by using an OCT-based approach, [176]. An evaluation of the mechanical stiffness was performed by analyzing the damping vibrational response of the cornea by an OCT system. Other works have shown the applicability of nuclear magnetic resonance (NMR) imaging for polymer characterization, [177]. The resolution of NMR is strongly dependent on the natural line width of the molecules that are measured, and the magnetic field gradient. The method is primarily suited to characterize samples in a laboratory environment. A different approach would be time-of-flight secondary ion mass spectroscopy (ToF-SIMS) where spatial resolution in the axial domain is achieved. By structuring the sample under test with an ion beam, the cross-linking depth profile is acquired in depths up to a few nanometers, [178].

None of these technologies is suited for the in-line characterization of polymers in production or for products.

The main ambition in the characterization of polymeric cross-linking is to establish a measurement technique which is capable to measure non-destructively on product level, is capable of a high refractive index resolution (better than ${10^{-3}}$), to measure fast enough to be process-integrated and offers spatial resolution at the same time. None of the technologies known from literature combine these characteristics so far.

2.3 Film Thickness Measurement

Appropriate metrological tools for in-line characterization have to fulfill different requirements in distinction to classical lab-based technologies. In particular, tools are often required to measure a certain area ($\approx$ some µm²) with high axial resolution in accordance with the speed of the respective processing step and perform measurements autonomously for a large variety of materials. As the amount of data gathered is usually large, appropriate algorithms have to monitor key values constantly and only report on deviations. This applies especially to the production of high volume, complex multi-material systems in roll-to-roll processes as in the production of thin-film systems, [179].

State-of-the-art technologies for thin-film characterization most often incorporate variants of ellipsometry, [180] and reflectometry,[181]. While both methods are capable of being production accompanying tools [182], they usually cannot provide high lateral resolution, [183, 184]. As experimental works showed, higher resolutions and the ability to measure multi-layer systems, one of the most challenging problems in both technologies, is the knowledge of material constants and models in order to find or fit correct start parameters as well as converging criteria for fits, [185–187].

Alternatively, optical and mechanical properties of thin-films have been characterized by interferometric approaches, [188]. While reaching high lateral resolutions, these technologies tend to be slow, as they rely on z-scanning of the OPD between the sample and reference arm, [189]. With the development of k-scanning approaches this disadvantage was overcome as mechanical scanning in the axial dimension was substituted by a tuneable light source to scan through k-space, [190, 191]. However, these approaches are limited at both ends of the thickness measurement range. On the one end, the typical spectral scanning range of 20 nm limits the maximum measurable thickness to about 15 µm. On the other end, the minimal resolvable thickness is limited by the distinguishability of Fourier peaks which are unique for every reflection of a material interface. The thickness as well as the amount of dispersion of every material contributes to the width and position of each Fourier peak. Ghim et al. demonstrated that 500 nm is a typical minimal film thickness which can be resolved with this technique, [192].

2.3.1 Spectral Reflectometry

A common method for the determination of film thickness and material parameters of thin-films in the semiconductor industry is spectrally-resolved reflectometry, [181]. This technique captures the spectrum of a broadband light source back-reflected from a sample’s surface and tries to fit it to a model spectrum that was calculated beforehand. During measurement, a reference spectrum, from e.g. a pure silicon surface, is captured first in order to reference absolute intensity values later. Afterwards, the reflection spectrum of a substrate with a thin-film is captured by illuminating the sample with a broad spectrum under a defined angle $\theta$ (typical $\theta \,=\,{0}^{\circ }$). The back-reflected spectrum is typically collected with a fiber and spectrally decomposed using a grating spectrometer. For the determination of the film thickness, the real and imaginary part of the substrate’s refractive index as well as that of the film material one needs to know. By applying e.g. a brute-force fitting routine, the error between measured and calculated spectrum is minimized to obtain the film thickness. In order to calculate all possible reflection spectra for different thicknesses of a material system, the transfer-matrix approach is typically applied, [193]. Using this approach, the E-field is described as a matrix for every layer of material as well as for every transition between different materials. Each matrix M accounts for the reflectivity and transmission characteristics of every material and transition, [181].

In practice, a variety of theoretical spectra can be calculated with an assumed set of different film thicknesses in order to evaluate the measured film thickness. Usually, the root-mean square error (RMS) is determined between measured and calculated values. Additionally, it is possible to perform calculations for other parameters like the refractive index (real and imaginary parts) in relation to the wavelength on the basis of knowledge of the film thickness. The technique is advantageous because of the low instrumental effort, the lack of sample preparation and the fast data acquisition times. However, the method is tied to the knowledge of the material model of a sample and caution is necessary when handling measurements from samples with changing material parameters which can occur in a production environment. Some research has shown, that the parallel acquisition of data under a range of polarization angles in combination with a camera might enable the usage of spectral reflectometry in process-accompanying for complex multi-layer samples, [194]. Currently, research has proven the possibility to combine technologies such as spectral reflectometry with other techniques such as low-coherence interferometry, spectroscopic ellipsometry and Raman spectroscopy, [195–199]

2.3.2 Spectroscopic Ellipsometry

A widely used method to determine the dispersion properties of thin-films is spectroscopic ellipsometry (SE). This technique evaluates the change of polarization of light reflected from one or more layers of a thin-film of material. In particular, the technique is based on the determination of the complex quotient of the reflection coefficient of both polarization components. Typically, the amplitude quotient $\Psi$ and the phase difference $\Delta$ of s- and p-polarized light are captured as a measure, [200]. The data of these measures can be gathered either at a discrete wavelength or at a broad spectral range using a spectrometer as detector. In order to determine the dispersion of a thin-film, the geometrical parameters such as film’s thickness and angle of incidence as well as the refractive index of the substrate material must be known. By using an appropriate mathematical model of the material system, the real and complex part of the films’ refractive index can be calculated. Under the assumption of a measurement in a broad spectral range, this approach can be used to derive the dispersion of the film material. As [180] proved, appropriate models can be used to determine the dispersion of various materials in multi-layer material systems as well. Park, [201] has shown that the determination of the real and imaginary part of the dielectric function, hence the refractive index, the extinction coefficient and the dispersion in the range of (190 – 1000) nm is possible. The measurements were performed in an angular range from ${45}^{\circ }$ – ${75}^{\circ }$ in ${5}^{\circ }$ intervals. In the model, the authors took into account the material properties of the surroundings, a rough surface, a thin-film of cadmium sulfide (CdS) as well as for a substrate material. The film thickness of the rough surface and the CdS layer were determined by minimizing the error between calculated and measured values for the amplitude quotient $\Psi$ and the phase difference $\Delta$. Additionally, the gathered data in relation to the measured spectral range could be used to determine other material characteristics such as characteristic peaks which are dependent on the crystal structure of the material. Furthermore, the determination of the dispersion (according to the Wemple-DiDomenico model [202]) could be used to evaluate the bandgap of the material. Other works, such as [203], demonstrated the determination of optical properties on a variety of thin-film materials under the usage of different dispersion models. Obviously, the knowledge of material properties and the application of the correct model are important in order to obtain precise results using SE. Some developments proved that an application of the technology as a process monitoring tool could be possible. Fried et al. [182] demonstrated that by a modification of the setup and the simplification of the fit function, significantly faster data acquisition and analysis are possible. For example, in order to evaluate copper indium gallium selenide (CIGS) solar cells, a correlation of the fit values with the amount of gallium (Ga) in the material was performed. This reduces the number of adjustable fit parameters to one and speeds up the calculation. The authors show that a modification of the fit model can be done for other materials which are relevant for solar cells as well. It was also shown that a modification of the mechanical setup from using a collimated beam to the usage of a so-called expanded beam can decrease the time needed for data acquisition. This setup allows for the illumination and data acquisition on a large area and over various angles of incidence at the same time. Also, the simultaneous acquisition of data under several wavelengths and angles was demonstrated. When the sample is translated under the measurement spot, the fast measurement of areas with several cm² becomes possible. The authors of [183, 184] showed maps of film thickness for various material systems on a wafer. Additionally it was shown that a line projection of the expanded beam can be utilized to perform film thickness measurement as process monitoring in a roll-to-roll production environment. Although the results are promising, it has to be clarified that this approach only works in settings were the material is well known and the production focuses on a small number of material systems (in [182] the manufacturing of thin-film solar cells).

According to [187], the typical usable spectral range of SE is (130 – 2000) nm whereas most practical setups use a reduced range. While the lateral resolution of production accompanying is rather low, experimental works have shown that a lateral resolution of 4 µm can be reached using a microscope objective, [186]. However, this makes scanning of the sample a necessity in order to measure reasonably large samples.

Modifications of spectroscopic ellipsometry have been combined with scatterometry in order to perform the control of optically critical dimensions in the semiconductor industry, [204]. The measurement of the smallest lithographic structures like the 7-nm-node, introduces new challenges to metrology as e.g. dielectric functions for layers thinner < 10 nm. According to [187], future challenges for SE lie primarily in the increase of accuracy of the optical models as well as in the generation of new models. Furthermore, approaches for the combination with other techniques, [205], as well as for parallel acquisition of different parameters over a large spectral and angular range exist, [206]. In this area, it will be necessary to develop new and adapted components such as light sources with shorter emitting wavelengths,[207].

Some current research has tried to minimize the disadvantages in terms of the need for scanning by developing one-shot techniques using broadband light sources, a modulated carrier-frequency from an interferometer and a common spectrometer, [208]. Other works have tried to extend known analysis models by new mathematical approaches or by the combination of data from SE and other technologies such as reflectometry, [209–211]. Some works have shown imaging SE techniques which suffer from poor lateral resolution ($\approx$ 60 µm) and the need for temporal effort to perform the measurement and analyze the data ($\approx$ 8 s per wavelength in $\Delta \lambda$ = (400 – 700) nm), [212, 213]. Additionally, some work has been done to combine spectral ellipsometry with low-coherence or phase-shifting interferometry in order to get information on the surface profile of the sample alongside with the thin-film thickness, [214, 215]. While the resolution for film thickness and surface profile were in the nm-range, the combination of multiple technologies imposed new obstacles in terms of data fusion. Spectral ellipsometry is especially powerful in laboratory situations where the material model of a sample is well understood. In this case, thickness resolutions in the sub-nm can be achieved. In situations like an production environment, SE suffers from its low-lateral resolution and time-consuming data acquisition/analysis.

2.4 Material dispersion

The refractive index, $n_{abs}$, of a material is a measure of the refraction of light traveling through a material. Since it is defined as the relation of the vacuum light velocity $c_0$ to the light velocity in the material c and can be described as the optical resistance of the material, [216] pp. 94,

$$\begin{aligned} n_{abs} = \frac{c_0}{c}. \end{aligned}$$

(2.5)

As most practical applications are operated in air rather than in vacuum, the term relative refractive index, $n_{rel}$, is more common. This relative index is defined as the refractive index in relation to the refractive index of air, $n_{air}$,

$$\begin{aligned} n_{rel} = \frac{n_{abs}}{n_{air}}. \end{aligned}$$

(2.6)

The dispersion of a material describes the relation of the refractive index, and therefore the phase velocity of the light, with respect to the wavelength. This implies that the refractive index is wavelength dependent $n = n(\lambda )$. Normal and anomalous dispersion can occur. In the case of glass or most polymers, normal dispersion is present in the visible wavelength range. In this case the material shows comparatively high refractive indices ranging from (300 – 500) nm and significantly lower refractive index values in higher spectral regions (e.g. (500 – 900) nm). Anomalous dispersion consequently shows an opposing behavior. In materials with relatively low optical density such as glasses or polymers, the dispersion characteristic is closely related to the absorption behavior. In general, the material behavior can be described using a quantum mechanical model. The application of a simplified description utilizing an electro-magnetic model is feasible for materials with low optical density, [193]. This model describes the influence of an electro-magnetic wave on the moleculary structure of a material. Each bond charge of a material’s molecular structure has a specific resonant frequency $\omega _j$ which defines its absorption behavior. Dependent on the material properties, an incoming electro-magnetic wave excites the molecular structure which than leads to a specific refraction behavior. The resulting behavior of the real and imaginary part of the refractive index, n and $k^\prime$, can be described mathematically using a resonator model of the following form dependent on the angular frequency $\omega$, [217],

$$\begin{aligned} (n- \text {i}k^\prime )^2 = \frac{N_me^2}{\varepsilon _0m}\sum _{j}^{}\frac{F_j}{\left( \omega _j^2-\omega ^2 \right) +\text {i}\gamma _j\omega }, \end{aligned}$$

(2.7)

where $N_m$ is the number of molecules per unit volume, e and m are the charge and mass of the electron, $\varepsilon _0$ is the vacuum permittivity, $F_j$ is the strength of the absorption and $\gamma _j$ is a measure for the frictional force at the resonance frequency. The Kramers-Kronig model takes this relation into account, [218]. It describes the relation of absorption and dispersion of light in a material by combining the real and the imaginary components into one complex model, Fig. 2.1.

Figure 2.1

Depiction of the slope of the real part $n(\omega )$ and the imaginary part $k^\prime (\omega )$ of the refractive index for a simplified resonator model with a resonant frequency $\omega _j$

This relation has a general validity for all materials. For spectral ranges with negligible absorption of the specific material ($k^\prime \rightarrow 0$), the resonance frequency tends to differ significantly from the frequency of the incident light. For this case the relation can be simplified and expressed in terms of the wavelength $\lambda$/$\lambda _j$

$$\begin{aligned} n^2 - 1 = \frac{N_me^2}{\varepsilon _0m}\sum _{j}^{}\frac{F_j}{\omega _j^2-\omega ^2 } = \frac{N_me^2}{4\pi ^2 c^2\varepsilon _0m}\sum _{j}^{}\frac{F_j\lambda _j^2\lambda ^2}{\lambda ^2-\lambda _j^2 }, \end{aligned}$$

(2.8)

where one can abbreviate the material specific terms with $A_j$

$$\begin{aligned} A_j = \frac{N_me^2F_j\lambda _j^2}{4\pi ^2 c^2\varepsilon _0m}, \end{aligned}$$

(2.9)

which leads to the simplified equation

$$\begin{aligned} n^2 - 1 = \sum _{j}^{}\frac{A_j\lambda ^2}{\lambda ^2-\lambda _j^2 }. \end{aligned}$$

(2.10)

When analyzing real materials like glasses or polymers, the absorption characteristics differ very much from the simplified model. Most notably, multiple absorption peaks exist in contrast to only one single absorption peak at one resonant frequency. Furthermore, these peaks inherit a distinct structure which leads to a complex dispersion characteristic. For this reason, a number of approximation relations exist. These relations are always valid in a well defined spectral range which is most likely far away from a strong absorption band. An extrapolation outside the valid spectral range is not considered valid. Over the years a broad range of approximation relations have emerged for different materials, spectral ranges and achievable accuracies, Tab. 2.2.

The measurement of the refractive index of a medium is usually defined in relation to the refractive index of air, Eq. (2.6). For this reason, the conditions of the surrounding air should be well defined in terms of temperature, pressure and humidity. Calculations are based on typical values for laboratory conditions, Tab. 2.3.

In order to perform these measurements for refractive indices at defined single wavelengths with high accuracy, methods like refractometry (accuracy ± ${10^{-6}}$) and interferometry (accuracy ± ${10^{-7}}$) are preferred. Measurements are typically performed at one or few defined wavelengths of a gas discharge lamp such as the sodium line at $\lambda _D$ = 589.592 nm. The influence of the temperature T, pressure p and humidity w on the refractive index of air and therefore on the relative refractive index measurements of materials can be described as follows

$$\begin{aligned} \frac{\mathrm{d}n_{air}}{\mathrm{d}p}= & {} \frac{n_{air}-1}{p} = +0.268\times 10^{-6}\text { hPa}^{-1}\end{aligned}$$

(2.11)

$$\begin{aligned} \frac{\mathrm{d}n_{air}}{\mathrm{d}T}= & {} -\alpha \frac{n_{air}-1}{1+\alpha T} = -1.071\times 10^{-6}\text { K}^{-1}\end{aligned}$$

(2.12)

$$\begin{aligned} \frac{\mathrm{d}n_{air}}{\mathrm{d}w}= & {} \frac{41 \times 10^{-9}hPa^{-1}}{1+\alpha T} = -0.039 \times 10^{-6}hPa^{-1}. \end{aligned}$$

(2.13)

where the thermal expansion coefficient is introduced as $\alpha$. From these equations, it can be calculated that minor changes in the environmental conditions such as a temperature change of $\text {d}T$ = ± 5 K or a pressure change of $\text {d}p$ = ± 20 hPa can have a significant influence on $n_{air}$. For comparability, the joint commission for spectroscopy has developed an engineering equation for the determination of the refractive index of dry air (with 0.03 % $CO_2$ volume content) $n_{air}$ at T = ${{15}^{\circ }}\mathrm{C}$ and the reference pressure $p_0$ = 760 Torr in the spectral range between (200 – 1350) nm, as well as for other temperatures of air under the same pressure in the visible spectral range, [217].

Table 2.2 Models for the description of material dispersion according to [217]

Table 2.3 Typical environmental conditions for the calculation of influences regarding the refractive index according to [217]

2.4.1 Thermo-optic coefficient

Analogous to the behavior of air, the refractive index of a material is dependent on its temperature where the discrete form can be noted

$$\begin{aligned} \frac{n\left( \lambda _k,T_{i+1}\right) -n\left( \lambda _k,T_{i}\right) }{T_{i+1}-T_i} = \frac{\Delta n\left( \lambda _k,T_{i,i+1}\right) }{\Delta T_{i,i+1}}. \end{aligned}$$

(2.14)

For practical reasons this dependency is often evaluated by measuring the refractive index at different Fraunhofer lines $\lambda _k$ at different temperatures of interest $T_i$, [219–221]. An appropriate fit of this equation has to determine six parameters when calculating a single temperature. When incorporating the correct thermal dependency, that number of necessary fit parameters increases to eighteen. In order to simplify the calculation, some boundary conditions have to be implemented. As the influence of temperatures on the refractive index is relatively low, it can be described by using a simplified Sellmeier model. Hoffmann et al., [222], have shown, that a reduction to a one-term model ($i=1$) is sufficient where measurements are performed at only one wavelength $\lambda$ in reference to a base wavelength $\lambda _0$

$$\begin{aligned}&\phantom { \left( D_0 + 2D_1(T-T_0) + 3D_2(T-T)\right. } \frac{\text {d}n(\lambda ,T)}{\text {d}T} = \frac{n^2(\lambda ,T_0)-1}{2n(\lambda ,T_0)}\times \\ \nonumber&\left( D_0 + 2D_1(T-T_0) + 3D_2(T-T_0)^2 + \frac{E_0+2E_1(T-T_0)}{\lambda ^2-\lambda _{0}^2}\right) , \end{aligned}$$

(2.15)

which results in

$$\begin{aligned} {\begin{matrix} \Delta n(\lambda ,T-T_0) = \frac{n^2(\lambda ,T_0)-1}{2n(\lambda ,T_0)} \times \\ \left( D_0(T-T_0) + D_1(T-T_0)^2 + D_2(T-T_0)^3 + \frac{E_0(T-T_0)+E_1(T-T_0)^2}{\lambda ^2-\lambda _{0}^2}\right) , \end{matrix}} \end{aligned}$$

(2.16)

after integration. Usually, calculations are performed with a reference Temperature $T_0$ and a single measurement temperature T in order to fit the values for the thermo-optic coefficients $D_0,D_1,D_2,E_0,E_1$. For commonly used glasses these coefficients can be found in tables, e.g. in [220].

2.4.2 Photo-elastic influences

A second important influence on the refractive index and therefore on the dispersion is (mechanical) stress acting on a material. While materials like glasses or polymers are usually showing isotropic behavior, the application of mechanical stress $\sigma$ can lead to anisotropic behavior of the refractive index. The reason for this is that the refractive index is dependent on the electric field vector in relation to the stress plane in a sample. For a calculation, the relation of the refractive index to the stress plane can be described in parallel, $n_\parallel$ and perpendicular, $n_\perp$ orientation, [217],

$$\begin{aligned} n_{\parallel } = n + \frac{\text {d}n_\parallel }{\text {d}\sigma }\sigma = n + K_{\parallel } \end{aligned}$$

(2.17)

and

$$\begin{aligned} n_{\perp } = n + \frac{\text {d}n_\perp }{\text {d}\sigma }\sigma = n + K_{\perp }. \end{aligned}$$

(2.18)

If the deformation as a result of stress is elastic, Hookes law can be applied. In this case $\text {d}n/\text {d}\sigma$ can be written as the stress optical coefficients $K_\parallel$ and $K_\perp$. These coefficients are known for most common materials and are to be determined for example in a 4-point-bending test. In materials like glasses, the refractive index typically changes equally for both directions in relation to the stress plane. In the case of hydrostatic pressure p, not only the refractive index of the material itself but also of its surrounding medium will change. In this case the change is determined as

$$\begin{aligned} \frac{\text {d}n}{\text {d}p} = K_{\parallel } + 2K_{\perp }. \end{aligned}$$

(2.19)

Acousto-optical modulators are applications that make use of this effect. The stress-optical constant K can be found for a variety of materials in tables such as in [223]. In some cases, it might be necessary to determine it experimentally for a specific material. An experimental setup would consist of a sample under uni-axial load such as pressure or tension where a polarized beam of light illuminates the sample. The angle of the polarization will be ${{45}^{\circ }}$ towards the main stress axis. The polarized beam of light can be denoted as a superposition of one component perpendicular to the main stress axis and one component parallel to this axis. By applying stress on the sample, the two components will experience an optical path difference $\Delta s$ dependent on the sample length l which can be described with

$$\begin{aligned} \Delta s = \left( n + \frac{\text {d}n_\parallel }{\text {d}\sigma }\sigma - n - \frac{\text {d}n_\perp }{\text {d}\sigma }\sigma \right) l = (K_{\parallel } - K_{\perp })\sigma l. \end{aligned}$$

(2.20)

This can also be written as a phase difference $\Delta \Phi$ between the two components with

$$\begin{aligned} \Delta \Phi = \frac{2\pi }{\lambda }(K_{\parallel } - K_{\perp })\sigma l = \frac{2\pi }{\lambda }K\sigma l \end{aligned}$$

(2.21)

$$\begin{aligned} \text {with }K = K_{\parallel } - K_{\perp }. \end{aligned}$$

(2.22)

Clearly, a material’s stress optical constant is also dependent on the dispersion of the material, [224]. Furthermore, the stress optical coefficient K shows a temperature sensitivity. Hoffmann et al. [225] showed that usually only the stress-thermo-optical coefficients $A_0,A_1,A_2,B$ of the equation:

$$\begin{aligned} K(\lambda ,\Delta T) = A_{0} + A_{1}\cdot \Delta T + A_{2}\cdot \Delta T^2 + \frac{B}{\lambda ^2 - \lambda _{0}^2}\end{aligned}$$

(2.23)

$$\begin{aligned} \text {with } \Delta T = T- T_0 \end{aligned}$$

(2.24)

are relevant for glasses. For most materials, the temperature tends to have a minor influence until the glass transition temperature is reached, [226, 227].

2.4.3 Characterization of dispersion

The knowledge of the dispersion of materials and optical components plays a significant role in several areas of photonics. For example, the slope of dispersion in optical fibers in communications, due to different mechanisms like waveguide and material dispersion, determines the bandwidth and range of a transmission system, [228]. During the construction of laser sources with ultra short pulses the knowledge and control of the dispersion of components is crucial in order to ensure the generation of short pulses with high energy in a determined spectral range, [229]. Components such as fibers and mirrors have high demands regarding the determination of their dispersion behavior, [230]. According to the specific requirements of each application, different metrology approaches are common.

Time-of-flight measurement and phase-shift method

In the field of optical communications, methods which utilize the direct measurement of the propagation time of an optical pulse as a measure for the dispersion characteristics are common and known as time-of-flight measurements. As communication lengths are typically very long (tens to hundreds of kilometers), dispersion has a heavy impact on the propagation characteristics. Dispersion can lead to time delays of pulses and to spectral broadening of the optical pulses, [231]. In order to increase the transmission speed and data rates, modern systems implement approaches such as wavelength division multiplexing (WDM) where information is sent as spectrally fine separated pulses. The occurrence of dispersion influences the ability to clearly separate the pulses and therefore the data sent. Additional compensation mechanism have to be implemented, [231]. As a method of characterization, the direct measurement of time delays between sent pulses is typically used in optical communications to characterize the dispersion behavior. A major advantage of such methods is that it can be used also in already existing fiber transmission installments, [232]. In this way, not only the dispersion of the fiber, but also of the system as a whole can be determined. In the case that fiber sections are 4 km or longer, a significant amount of dispersion is present, so conventional metrology can be used to determine the time delay between pulses. The measured delay is a superposition of different dispersion mechanisms such as material and waveguide dispersion.

Typical light sources used in this context are Raman lasers, superluminescent diode (SLDs), erbium-doped fiber lasers and semiconductor lasers. According to [232], the main requirements for these light sources are a broad spectral range, a good tunability of a certain center wavelength and its full width half maximum as well as a high spectral power density.

One major disadvantage of this method is the requirement of appropriate hardware in order to achieve the necessary temporal resolution. In the case of samples with relatively low dispersion (circa ) it is necessary to have samples with a substantial length in order to characterize them with standard equipment. Fast photo detectors are able to work at rise times of a few picoseconds (circa 15 ps according to [233]). Although [233] could implement an autocorrelator in order to realize the measurement of very small phase differences, the shortest measurable fiber length was not shorter than 100 m. An application of the time-of-flight method to characterize samples with very low dispersion or very short lengths in the range of a few mm or µm is not possible.

A further development of the time-of-flight method is a phase-sensitive detection. In this approach, light sent through a sample is modulated with a sine signal which can be detected by an appropriately designed phase-selected amplifier circuit, [234]. This technique, in contrast, allows to delay measurements, the detection of much smaller differences in the dispersion related delays (with a resolution of about 10 ps). Problems arise from low light intensities and a bad SNR, [235]. Despite these disadvantages, this method is widely used as a resolution enhancement of time-of-flight measurements with rather long fiber lengths in the telecommunications industry.

Time-domain interferometric measurements

A method to cope with the requirements regarding the temporal resolution in time-of-flight measurements is interferometry in the time domain. For this approach, an interferometer (e.g. of the Michelson type) with one fixed and one movable arm is set up, [236]. Usually broadband light sources with low coherence lengths, such as SLDs and gas-discharge lamps filtered using a monochromator, are used in these setups. In a typical experiment, the movable reference arm is translated in a continuous fashion while the corresponding combined intensity signal of both arms is recorded with a photo diode. The shape of the recorded interferogram depends on the coherence properties of the light source. This method is similar to coherence scanning interferometry used for profilometry, see subsection 2.1.5. The execution of similar experiments using different center wavelengths is the basis for the dispersion measurements. After these experiments in a dispersion-free interferometer, a sample of interest can be introduced in one of the interferometer arms. After the recording of several new interferograms at interesting wavelengths, a temporal delay of the maxima of the interferograms in relation to the dispersion-free measurements can be done, [237–239]. A fit, using an appropriate dispersion model such as the Sellmeier model, can be performed using the maxima of the recorded interferograms. The temporal resolution in this approach is strongly dependent on the mechanical resolution of the translation stage. The step width and its repeatability define the possible measurable temporal differences between different interferograms.

The main disadvantage of this approach is the effort necessary to perform the measurements. Although broadband light sources can be used, the dispersion characterization over a large spectral range with a high spectral resolution takes time. Also, the appropriate filtering for each measurement has to be carried out. Furthermore, the accuracy of common translation stages limits the temporal resolution of the approach. A characterization of small dispersion, i.e. of thin structures (µm to mm) is not possible due to instrumental constraints. Therefore, the approach is mostly used for the characterization of optical fibers with at least a few centimeters of length, [240–242].

Frequency-domain interferometric measurements

Analogous to time-domain measurements, interferometry in the frequency domain uses broadband light sources as well. In contrast, these sources are not spectrally constricted. Furthermore, the interferometer is usually built using two fixed arms. The detection of the signal is performed on the complete spectral bandwidth of the light sources using a spectrometer, [243]. A typical signal of the recombined intensity of both arms leads to a modulated signal over a certain spectral bandwidth with a characteristic frequency. The frequency is dependent on the difference in the arm lengths of the interferometer as well as on the dispersion of the system, [244]. The insertion of an dispersive element leads to a characteristic change in the modulation and its frequency. The determination of dispersion can be done by different methods in this approach. On the one hand, the modulation can be analyzed using a Fast Fourier-transform (FFT). On the other hand, the analysis can be based on the determination of the so-called stationary phase point.

The usage of an FFT-based approach enables the fast determination of very small dispersion which can occur due to material characteristics or small amounts of media. The analysis is similar to FD-OCT, although the broadening of the Fourier-peaks due to dispersion can be neglected if dispersion is very low. Otherwise, complex dispersion compensation methods have to be implemented. Liebermann et al. [245] showed an experiment to determine the dispersion characteristics of distilled water using a Michelson interferometer in a free-space configuration, utilizing a supercontinuum light source as well as two combined spectrometers —(350 – 1100) nm and (900 – 1780) nm—to cover a large measurement range. For the analysis, two FFT-based approaches were compared. The so-called indirect approach transforms the signal initially from the frequency into the time-domain using zero-padding in order to achieve an even distribution of the signal after back transformation. The transformed signal is filtered afterwards and Fourier transformed for a second time in order to extract the spectral phase. A Taylor expansion enables the calculation of the dispersion coefficients. The second analysis method discussed in [245] is the so-called direct approach. In this approach, the spectral data is Fourier transformed into the non-even distributed time-domain and filtered using a Heaviside-Filter. After the back transformation and a Taylor expansion the dispersion can be determined. According to the authors, the indirect approach is computationally more time-consuming but also more suitable to characterize samples with large dispersion. Opposing to this, the direct approach is relatively fast and suitable for samples with small dispersion in process-accompanying problems.

Another approach to characterize refractive indices, is to fit dispersive interferometer data from experiments, [246]. The authors assumed that the frequency chirp of the spectral intensity was a measure for the dispersion. By fitting this data the fringe periodicity was analyzed for different sample configurations. Under the utilization of a mathematical model (e.g. Sellmeier) the wavelength-dependent refractive index was estimated. In order to increase the fitting quality, more complex fit methods or correction algorithms could be used, [247, 248].