White-light or coherence scanning interferometry (CSI) is one of the most used optical profiling techniques [1,2,3,4,5]. Proper application of CSI instruments assumes that their transfer characteristics are well-known. However, this is true only for phase analysis of CSI signals and surface height differences, which are much smaller than a quarter of the wavelength of the illuminating light [6]. Nonetheless, due to the ongoing miniaturization in micro- and nano-technology the lateral resolution capabilities and transfer characteristics close to the lateral resolution limit of CSI instruments become increasingly relevant.

The general assumption that the spectrum of a CSI signal from a mirror-like measuring object equals the spectrum emitted by the light source weighted by the spectral sensitivity of the camera is no longer fulfilled for high NA (numerical aperture) objective lenses. In particular, a 100x Mirau interferometer with a NA of 0.7 (Nikon CF IC Epi Plan DI) and a Linnik interferometer equipped with objective lenses of NA = 0.9 (Olympus MPLFLN100XBDP) used throughout this study show the well-known NA effect. This effect leads to an increased fringe spacing, which results in a longer effective wavelength of the measured correlograms [7,8,9,10].

The phenomenon can be understood by the following consideration: The basic scheme of microscopic illumination known as Köhler illumination assumes an image of an extended incoherent light source in the pupil plane of the objective lens resulting in plane waves incident onto the surface of the measuring object under numerous angles [11]. The angular range spread out by these plane waves corresponds to the numerical aperture of the objective lens so that the maximum possible angle of incidence equals *θ*_{max} = arcsin(*NA*) if the surrounding medium is air. Consequently, the higher the maximum angle of incidence, the better is the lateral resolution of the microscopic imaging. If the object under investigation is a grating with a period close to the diffraction limit of the CSI system the corresponding situation is outlined in Fig. 1. A plane wave characterized by the wave vector \( {\overrightarrow{k}}_{\mathrm{obj}} \) hits the object under the angle *θ*_{e} with respect to the optical axis. The zero order diffracted wave characterized by \( {\overrightarrow{k}}_r \) and the minus first order diffracted light characterized by \( {\overrightarrow{k}}_{\mathrm{diff}} \) enter the objective lens and therefore contribute to the microscopic image. If *θ*_{e} = *θ*_{max} this leads to Abbe’s lateral resolution criterion of microscopic imaging [12]:

$$ \delta =\frac{\lambda }{2\sin {\theta}_{\mathrm{e}}}=\frac{\lambda }{2\; NA} $$

In the reference arm of the interferometer the wave characterized by the wave vector \( {\overrightarrow{k}}_{\mathrm{ref}} \) hits the reference mirror and the reflected wave propagates under an angle *θ*_{e} towards the objective lens (see Fig. 1b). During the depth scan the length of the measurement arm of the interferometer changes continuously and the resulting interference patterns are captured by the camera at certain positions of the so-called depth scanner, so that each camera pixel records an interference signal step by step.

However, with respect to the depth scan and the height structure of the measuring object only the z-component of the wave vectors contribute. The interference intensity for a certain point source in the pupil plane is thus given by:

$$ {\displaystyle \begin{array}{c}I={I}_{\mathrm{obj}}+{I}_{\mathrm{ref}}+2\sqrt{I_{\mathrm{obj}}{I}_{\mathrm{ref}}}\kern0.5em \left|\gamma \left(z-{z}_0\right)\right|\cos \left(2{\overrightarrow{k}}_{\mathrm{obj}}\cdot \widehat{z}\;z-2{\overrightarrow{k}}_{\mathrm{ref}}\cdot \widehat{z}\;{z}_0+{\varphi}_0\right)\\ {}={I}_{\mathrm{obj}}+{I}_{\mathrm{ref}}+2\sqrt{I_{\mathrm{obj}}{I}_{\mathrm{ref}}}\kern0.5em \left|\gamma \left(z-{z}_0\right)\right|\cos \left(\frac{4\pi }{\lambda_{\mathrm{eff}}}\left(z-{z}_0\right)+{\varphi}_0\right),\end{array}} $$

where \( \widehat{z} \) is the unit vector in *z*-direction, *z* − *z*_{0} is the optical path length difference between measurement and reference arm of the interferometer, and *γ*(*z* − *z*_{0}) is the temporal coherence function. For the second part of the above equation the relationships

$$ {\displaystyle \begin{array}{l}{\overrightarrow{k}}_{\mathrm{obj}}\cdot \widehat{z}=\frac{2\pi }{\lambda}\cos {\theta}_{\mathrm{e}}=\frac{2\pi }{\lambda_{\mathrm{e}\mathrm{ff}}}\\ {}{\overrightarrow{k}}_{\mathrm{ref}}\cdot \widehat{z}=\frac{2\pi }{\lambda}\cos {\theta}_{\mathrm{e}}=\frac{2\pi }{\lambda_{\mathrm{e}\mathrm{ff}}}\end{array}} $$

hold. Consequently, an effective wavelength *λ*_{eff} instead of the center wavelength *λ* of the illuminating light represents the height difference corresponding to two bright fringes in the fringe pattern. In practice, not only a single point source has to be considered but the complete spatial distribution of light in the pupil plane of the objective lens [10]. However, for the sake of simplicity we confine the theoretical description to a single point source generating plane wave illumination on the object’s surface here.

As mentioned above, throughout this study we use a Mirau and a Linnik interferometer with numerical apertures of 0.7 and 0.9, respectively. Assuming the light source is a blue LED with a center wavelength of 460 nm, this results in a maximum effective wavelength of 644 nm for the Mirau and 1055 nm for the Linnik interferometer. This is the maximum wavelength contributing to the CSI signals and thus the maximum wavelength contribution in the spectrum of a measured CSI signal.

Figure 2 confirms these considerations. Figure 2a shows spectra of simulated CSI signals assuming NA values of 0.14, and 0.9, respectively. For comparison the spectrum of the light source assumed for the simulation is plotted. The simulation model is based on a Kirchhoff approach and results in interference signals, which are analyzed with signal processing algorithms that are also used for experimental data [13, 14]. As demonstrated by Fig. 2 the absolute value of the Fourier transform of such signals shows that additional low frequency (long wavelength) contributions occur at high NA. These are attributed to oblique angles of incidence and appear even if a narrow band light source is used [15, 16]. For this reason the corresponding effect in the spatial domain is related to the longitudinal spatial coherence [16].

Figure 2b) depicts spectral distributions of CSI signals (i. e. the absolute values of the spectral coefficients calculated by discrete Fourier transformation) obtained with interferometers of different NA. The effect of spectral broadening clearly turns out. For NA = 0.9 wavelength contributions of more than 900 nm occur, for NA = 0.7 a maximum wavelength of more than 600 nm appears, and for NA = 0.55 the spectrum extends up to more than 500 nm.

In an earlier paper we documented the dependence of the lateral resolution of a phase measuring interference microscope on the so-called evaluation wavelength *λ*_{eval}, which is the wavelength that is used for phase analysis [15].

In addition, local surface features such as height steps, slopes, or curvatures of the surface of the measuring object affect the shape of the spectrum of interference signals [9, 13, 14]. This represents a potential source of measurement errors, which must be taken into account the more the measuring object differs from an optical flat.

In this context, it should be noted that the Fourier transform of the envelope of an interference signal is often interpreted as its spectral distribution centered at the central frequency of the interference signal [17]. Consequently, spectral changes primarily affect the shape of the envelope and measurement errors occur if the maximum position of the envelope is used for height determination. For this reason, phase analysis behaves more robust and thus we only discuss results of phase analysis of CSI signals in the following sections.