# Spectral composition of low-coherence interferograms at high numerical apertures

**Part of the following topical collections:**

## Abstract

Interference signals in coherence scanning interferometry at high numerical apertures and narrow bandwidth illumination are spectrally broadened. This enables phase analysis within a spectral range much wider than the spectral distribution of the light emitted by the light source. Consequently, different surface features can be resolved depending on the wavelength used for phase analysis of the interference signals.

In addition, the surface topography itself affects the spectral composition of interference signals in different ways. Signals related to tilted surfaces or step height structures show special spectral characteristics. Thus, spectral amplitude and phase analysis enables a better understanding of the underlying physical mechanisms and gives hints how to improve the measurement accuracy.

## Keywords

Coherence scanning interferometry Numerical aperture Interference microscopy Spectral analysis Lateral resolution## Introduction

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].

*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]:

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.

*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

*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 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.

## Methods

For our experimental investigations we used a pitch standard manufactured by the company Supracon AG in order to show the relationship between the evaluation wavelength and the corresponding spatial frequency of the almost rectangular surface structures [18]. The standard provides grating structures of 0.3 and 0.4 μm period and 25 nm depth, etched in a nanocrystalline silicon layer on a quartz substrate. These periods are close to the lateral resolution limit of 0.33 μm for the Mirau interferometer and 0.26 μm for the Linnik system. The resulting interference patterns are phase analyzed pixel by pixel using three different evaluation wavelengths. Results are discussed for the nominal wavelength of the light source (460 nm for the blue LED), the center wavelength of the interference signal (520 nm for the Mirau and 600 nm for the Linnik interferometer) occurring on a flat mirror, and longer wavelengths (650 nm for the Mirau and 800 nm for the Linnik system) corresponding to higher angles of incidence.

A piezo z-stage moves the pitch standard along the optical axis (z-axis) of the interferometer while a camera takes an image every 20 nm. The wavelength dependent spectral composition of CSI signals at steep edges will be demonstrated by use of the RS-N standard manufactured by Simetrics GmbH comprising a rectangular grating of 6 μm period with perpendicular edges and a nominal groove depth of 190 nm [19]. CSI measurements obtained with a Mirau system of NA = 0.55 under red and blue LED illumination will be compared.

In order to detect the phase of a CSI-signal for a certain evaluation wavelength we use a lock-in algorithm [20, 21].

## Results and discussion

_{eval}= 600 nm the surface profile looks quite noisy due to the rather low intensity value at this wavelength for a numerical aperture of 0.7 (see Fig. 2b).

Higher evaluation wavelengths providing better lateral resolution are accessible with the Linnik interferometer as displayed in Figs. 5 and 6.

Figure 5a and d show that an evaluation wavelength of 460 nm will not resolve the 0.3 μm grating structure, but the surrounding structures of lower spatial frequency are quite well resolved. At λ_{eval} = 600 nm (Fig. 5b and e) the amplitude of the low-frequency structure at the left and right hand side of the profile is reduced but the high frequency structure is still not resolved. Using an evaluation wavelength of 800 nm resolves the high frequency grating structure but the low-frequency contributions become blurred.

*θ*

_{e}resulting from the choice of the evaluation wavelength

*θ*

_{diff}corresponding to the first diffraction order of the reflection grating:

*Λ*is the grating period. For

*λ*= 460 nm (blue LED) the angle

*θ*

_{e}= 40° corresponds to λ

_{eval}= 600 nm. For λ

_{eval}= 800 nm an angle

*θ*

_{e}= 55° results. The first order diffraction angle is 35° for Λ = 0.4 μm and 50° for Λ = 0.3 μm. With respect to Fig. 6b and e the angle corresponding to the evaluation wavelength and the diffraction angle are closest to each other (Λ = 0.4 μm,

*θ*

_{diff}= 35°,

*θ*

_{e}= 40°). This explains the good resolution of the grating structure in this case.

Comparison of Figs. 4d and 6d) shows that the same structure in the surrounding of the grating looks somehow low-pass filtered in the result of the Mirau interferometer. We suppose that this is a consequence of the obscuration of the reference mirror in the Mirau system, which prevents light rays from striking the object or the reference mirror surface perpendicularly.

If the center wavelength of a CSI signal for a perfectly aligned surface, which is perpendicular to the optical axis, is chosen as evaluation wavelength a strong loss of signal intensity occurs at high surface tilts and, therefore, the maximum measurable tilt angle of a surface structure is typically much smaller than the angle *θ*_{max}.

## Conclusions

This contribution deals with the analysis of physical phenomena, which affect the spectral distribution of CSI signals and thus may reduce or enhance the accuracy of CSI measurements.

The low frequency or long wavelength contributions up to wavelengths of more than 900 nm occurring at a NA of 0.9 even in the case of blue LED illumination correspond to oblique angles of incidence. Thus, phase analysis at long evaluation wavelengths corresponding to long effective wavelengths enable high lateral resolution according to Abbe’s theory of microscopic imaging. As we demonstrate by example of a pitch standard providing line gratings with periods of 0.3 and 0.4 μm, phase analysis of interference signals at longer evaluation wavelength enable improved lateral resolution of the optical topography measurement. At first glance this seems to contradict the fact that better lateral resolution can be achieved by reducing the wavelength of light [12]. However, the wavelength we discuss here is the effective or evaluation wavelength, which results from higher NA and oblique incidence and corresponds to lower spatial frequency contributions occurring if the NA increases while the central wavelength of the used light source remains unchanged.

Moreover, our experimental results suggest that phase analysis of a CSI signal at a certain wavelength seems to comprise a kind of bandpass filtering, where only certain spatial frequency contributions of a measured surface structure will contribute to the measured surface profile.

It should be noted that the spatial frequency dependent analysis of measurement data is a unique opportunity due to the interferometric measurement, which in contrast to other optical profiling techniques such as confocal of focus scanning microscopy enables wavelength dependent signal processing.

With respect to height steps the spectral distribution of CSI signals occurring at an edge allows to adjust the center wavelength of the illuminating light in a way that artefacts such as batwings will be minimized.

Finally, steeper surface slopes lead to CSI signal spectra, which are shifted to higher wavelength, i. e. higher angles of incidence compared to signals obtained from perfectly adjusted mirror-like surfaces. For this reason care must be taken if steeper surface slopes are to be measured, since phase analysis will only lead to robust measurement results as long as there is enough signal power in the spectrum of a CSI signal at the chosen evaluation wavelength.

## Notes

### Acknowledgements

The authors gratefully acknowledge the funding of parts of this work by the German Research Foundation.

### Funding

Partially by the German Research Foundation, projects LE 992/6–2 and LE 992/15–1.

### Availability of data and materials

As given in the present paper.

### Authors’ contributions

All authors contributed to this work. PL analysed and simulated the occurring effects, interpreted the data and wrote the manuscript, ST wrote the measurement and data evaluation programs, BA performed the measurements according to Fig. 8, SH obtained the measurement results shown in Figs. 2b, 4 and 7, and LH obtained the measurement results shown in Figs. 5 and 6. All authors read and approved the final manuscript.

### Authors’ information

As given in the present paper.

### Competing interests

The authors declare that they have no competing interests.

### Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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