# Three-Dimensional Transfer Functions of Interference Microscopes

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Results

#### 2.1. Derivation of the 3D Transfer Function for Monochromatic Light

#### 2.2. Dependence of 3D Transfer Functions on Temporal Coherence

#### 2.3. 3D Transfer Functions and Surface Topography Reconstruction

- The 2D MTF can be derived from the 3D TF via integration with respect to the ${q}_{z}$ coordinate [19]. Expressing the 2D MTF by an autocorrelation of a uniformly-filled 2D circular pupil function assumes that both the illumination as well as the imaging pupil plane is uniformly-filled. As discussed before, this requires single point scatterers with certain scattering characteristics on the surface under investigation. In contrast, specularly reflecting surfaces such as typical surface standards lead to a non-uniformly filled image pupil function even if the illumination pupil is uniformly-filled (see Figure 5).
- The stack of interference images resulting from a CSI measurement can be analyzed at certain axial spatial frequencies. According to [5,10], this analysis is typically performed for the so-called equivalent wavelength ${\lambda}_{\mathrm{eq}}$, which corresponds to the axial spatial frequency value ${q}_{z,\mathrm{eq}}=4\pi /{\lambda}_{\mathrm{eq}}$. As a consequence, not the integration of $H\left(\mathbf{q}\right)$ with respect to ${q}_{z}$ plays a crucial role for surface reconstruction but the coarse of the function $H({q}_{x},{q}_{y},{q}_{z}=\mathrm{const}.)$ for a certain constant ${q}_{z}$-value. This value ${q}_{z,\mathrm{eval}}$ is related to what we call the ‘evaluation wavelength’ ${\lambda}_{\mathrm{eval}}$ by$${q}_{z,\mathrm{eval}}=4\pi /{\lambda}_{\mathrm{eval}}.$$Note that ${\lambda}_{\mathrm{eval}}$ may equal the equivalent wavelength ${\lambda}_{\mathrm{eq}}$ or not.

## 3. Discussion

## 4. Materials and Methods

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Malacara, D. (Ed.) Optical Shop Testing; John Wiley & Sons: Hoboken, NJ, USA, 2007. [Google Scholar]
- Coupland, J.; Lobera, J. Holography, tomography and 3D microscopy as linear filtering operations. Meas. Sci. Technol.
**2008**, 19, 074012. [Google Scholar] [CrossRef] [Green Version] - Coupland, J.; Mandal, R.; Palodhi, K.; Leach, R. Coherence scanning interferometry: Linear theory of surface measurement. Appl. Opt.
**2013**, 52, 3662–3670. [Google Scholar] [CrossRef] [Green Version] - Su, R.; Thomas, M.; Liu, M.; Drs, J.; Bellouard, Y.; Pruss, C.; Coupland, J.; Leach, R. Lens aberration compensation in interference microscopy. Opt. Lasers Eng.
**2020**, 128, 106015. [Google Scholar] [CrossRef] - de Groot, P.; Colonna de Lega, X. Fourier optics modeling of interference microscopes. J. Opt. Soc. Am. A
**2020**, 37, B1–B10. [Google Scholar] [CrossRef] [PubMed] - Lehmann, P.; Künne, M.; Pahl, T. Analysis of interference microscopy in the spatial frequency domain. IOP J. Phys. Photonics
**2021**, 3, 014006. [Google Scholar] [CrossRef] - Su, R.; Coupland, J.; Sheppard, C.; Leach, R. Scattering and three-dimensional imaging in surface topography measuring interference microscopy. J. Opt. Soc. Am. A
**2021**, 38, A27–A41. [Google Scholar] [CrossRef] - Pahl, T.; Hagemeier, S.; Künne, M.; Danzglock, C.; Reinhold, N.; Schulze, R.; Siebert, M.; Lehmann, P. Vectorial 3D modeling of coherence scanning interferometry. Proc. SPIE
**2021**, 11783, 117830G. [Google Scholar] - Künne, M.; Pahl, T.; Lehmann, P. Spatial-frequency domain representation of interferogram formation in coherence scanning interferometry. Proc. SPIE
**2021**, 11782, 117820T. [Google Scholar] - de Groot, P.; Colonna de Lega, X.; Su, R.; Coupland, J.; Leach, R. Fourier optics modelling of coherence scanning interferometers. Proc. SPIE
**2021**, 11817, 118170M. [Google Scholar] - Beckmann, P.; Spizzichino, A. The Scattering of Electromagnetic Waves from Rough Surfaces; Artech House, Inc.: Norwood, MA, USA, 1987. [Google Scholar]
- Born, M.; Wolf, E. Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light; Cambridge University Press: Cambridge, UK, 2013. [Google Scholar]
- McCutchen, C.W. Generalized aperture and the three-dimensional diffraction image. J. Opt. Soc. Am.
**1964**, 54, 240–244. [Google Scholar] [CrossRef] - Wombell, R.J.; DeSanto, J.A. Reconstruction of rough-surface profiles with the Kirchhoff approximation. J. Opt. Soc. Am. A
**1991**, 8, 1892–1897. [Google Scholar] [CrossRef] - Sheppard, C.; Connolly, T.; Gu, M. Imaging and reconstruction for rough surface scattering in the Kirchhoff approximation by confocal microscopy. J. Mod. Opt.
**1993**, 40, 2407–2421. [Google Scholar] [CrossRef] - Quartel, J.C.; Sheppard, C.J.R. Surface reconstruction using an algorithm based on confocal imaging. J. Mod. Opt.
**1996**, 43, 469–486. [Google Scholar] [CrossRef] - Sheppard, C. Imaging of random surfaces and inverse scattering in the Kirchoff approximation. Waves Random Media
**1998**, 8, 53–66. [Google Scholar] [CrossRef] - Xie, W.; Lehmann, P.; Niehues, J. Lateral resolution and transfer characteristics of vertical scanning white-light interferometers. Appl. Opt.
**2012**, 51, 1795–1803. [Google Scholar] [CrossRef] [PubMed] - Lehmann, P.; Pahl, T. Three-dimensional transfer function of optical microscopes in reflection mode. J. Microsc.
**2021**, 284, 45–55. [Google Scholar] [CrossRef] - Goodman, J.W. Introduction to Fourier Optics; Roberts and Company Publishers: Greenwood Village, CO, USA, 2005. [Google Scholar]
- Wilson, T. (Ed.) Confocal Microscopy; Academic Press, Inc.: London, UK, 1990. [Google Scholar]
- Corle, T.R.; Kino, G.S. Confocal Scanning Optical Microscopy and Related Imaging Systems; Academic Press, Inc.: San Diego, CA, USA, 1996. [Google Scholar]
- Krüger-Sehm, R.; Bakucz, P.; Jung, L.; Wilhelms, H. Chirp Calibration Standards for Surface Measuring Instruments. tm-Tech. Mess.
**2007**, 74, 572–576. [Google Scholar] [CrossRef] - Singer, W.; Totzeck, M.; Gross, H. Physical Image Formation. In Handbook of Optical Systems; Wiley-VCH: Weinheim, Germany, 2006; Volume 2. [Google Scholar]
- Garces, D.H.; Rhodes, W.T.; Pena, N.M. Projection-slice theorem: A compact notation. J. Opt. Soc. Am. A
**2011**, 28, 766–769. [Google Scholar] [CrossRef] - Sheppard, C.; Larkin, K. Effect of numerical aperture on interference fringe spacing. Appl. Opt.
**1995**, 34, 4731–4734. [Google Scholar] [CrossRef] [PubMed] - de Groot, P.; Colonna de Lega, X. Signal modeling for low-coherence height-scanning interference microscopy. Appl. Opt.
**2004**, 43, 4821–4830. [Google Scholar] [CrossRef] [PubMed] - de Groot, P.; Colonna de Lega, X.C. Interpreting interferometric height measurements using the instrument transfer function. In Fringe 2005; Springer: Berlin/Heidelberg, Germany, 2006; pp. 30–37. [Google Scholar]
- Lehmann, P.; Tereschenko, S.; Allendorf, B.; Hagemeier, S.; Hüser, L. Spectral composition of low-coherence interferograms at high numerical apertures. J. Eur. Opt. Soc.-Rapid Publ.
**2019**, 15, 5. [Google Scholar] [CrossRef] [Green Version] - de Groot, P.; Deck, L. Surface profiling by analysis of white-light interferograms in the spatial frequency domain. J. Mod. Opt.
**1995**, 42, 389–401. [Google Scholar] [CrossRef] - Fleischer, M.; Windecker, R.; Tiziani, H. Fast algorithms for data reduction in modern optical three-dimensional profile measurement systems with MMX technology. Appl. Opt.
**2000**, 39, 1290–1297. [Google Scholar] [CrossRef] [PubMed] - Tereschenko, S. Digitale Analyse Periodischer und Transienter Messsignale Anhand von Beispielen aus der Optischen Präzisionsmesstechnik. Ph.D. Thesis, University of Kassel, Kassel, Germany, 2018. [Google Scholar]
- Lehmann, P.; Xie, W.; Allendorf, B.; Tereschenko, S. Coherence scanning and phase imaging optical interference microscopy at the lateral resolution limit. Opt. Express
**2018**, 26, 7376–7389. [Google Scholar] [CrossRef] [PubMed] - Gao, F.; Leach, R.K.; Petzing, J.; Coupland, J.M. Surface measurement errors using commercial scanning white light interferometers. Meas. Sci. Technol.
**2008**, 19, 015303. [Google Scholar] [CrossRef] [Green Version] - Hagemeier, S.; Schake, M.; Lehmann, P. Sensor characterization by comparative measurements using a multi-sensor measuring system. J. Sens. Sens. Syst.
**2019**, 8, 111–121. [Google Scholar] [CrossRef] [Green Version] - Künne, M.; Hagemeier, S.; Käkel, E.; Hillmer, H.; Lehmann, P. Investigation of measurement data of low-coherence interferometry at tilted surfaces in the 3D spatial frequency domain. tm-Tech. Mess.
**2021**, 88, 65–70. [Google Scholar] [CrossRef]

**Figure 1.**Scattering geometry [6] (

**a**) in the $xz$-plane; (

**b**) in the $xy$-plane; (

**c**) Ewald sphere construction for a reflection-type microscope assuming plane wave illumination incident under an angle ${\theta}_{\mathrm{in}}$ taking all angles of incidence and all scattering angles that are covered by the objective’s NA into account; (

**d**) Ewald limiting sphere construction showing the vertical line corresponding to specular reflection, an outer sphere of radius $2{k}_{0}$ and an axial low-frequency limit given by ${q}_{z}=2{k}_{0}\sqrt{1-{\mathrm{NA}}^{2}}$. The blue line represents the maximum transverse spatial frequency bandwidth along the ${q}_{x}$ axis at a certain ${q}_{z}$ value.

**Figure 2.**Geometry for the derivation of the 3D TF according to [19]: (

**a**) two spherical caps corresponding to wave vectors of incident waves ${\mathbf{k}}_{\mathrm{in}}$ and scattered waves ${\mathbf{k}}_{\mathrm{s}}$ are correlated. The point P is defined by vector $\mathbf{q}$ with coordinates ${q}_{x}$ and ${q}_{z}$ in the spatial frequency domain representing the shift of the centres of the two spheres of radius ${k}_{0}$; (

**b**) the circle of intersection of the two spheres is characterized by the radius $\rho $ and the tilt angle $\alpha $. The vectors $-{\mathbf{k}}_{\mathrm{in},1}$, $-{\mathbf{k}}_{\mathrm{in},2}$, ${\mathbf{k}}_{\mathrm{s},1}$, and ${\mathbf{k}}_{\mathrm{s},2}$ corresponding to point P are located in the plane of incidence (${q}_{x}{q}_{z}$-plane); (

**c**) top view representing the area ${A}_{1}^{\prime}$ for height shifts between ${q}_{z,0}$ and ${q}_{z,\mathrm{max}}$ as well as area ${A}_{2}^{\prime}$ for height shifts between ${q}_{z,\mathrm{min}}$ and ${q}_{z,0}$.

**Figure 3.**3D representations of cross sections of the transfer functions $H({q}_{x},{q}_{z},{k}_{0})$ for $\mathrm{NA}=0.55$ and $\lambda =500$ nm: (

**a**) for scattering objects; (

**b**) for specularly reflecting objects; and (

**c**) cross sectional view of the difference $\Delta H({q}_{x},{q}_{z},{k}_{0})$ between (

**b**) and (

**a**).

**Figure 4.**3D representations of cross sections of the transfer functions $H({q}_{x},{q}_{z},{k}_{0})$ for $\mathrm{NA}=0.9$ and $\lambda =500$ nm: (

**a**) for scattering objects; (

**b**) for specularly reflecting objects; and (

**c**) cross sectional view of the difference $\Delta H({q}_{x},{q}_{z},{k}_{0})$ between (

**b**) and (

**a**).

**Figure 6.**Cross sectional views of the transfer functions $H({q}_{x},{q}_{z})$: (

**a**–

**c**) for $\mathrm{NA}=0.55$, $\lambda =550$ nm and spectral FWHM of 2.5 nm (

**a**); 25 nm (

**b**); and 100 nm (

**c**); (

**d**–

**f**) for $\mathrm{NA}=0.9$, $\lambda =550$ nm and spectral FWHM of 2.5 nm (

**d**); 25 nm (

**e**); and 100 nm (

**f**).

**Figure 7.**3D representations of cross sections of the transfer functions $H({q}_{x},{q}_{z})$ according to Figure 6: (

**a**–

**c**) for $\mathrm{NA}=0.55$, $\lambda =550$ nm and spectral FWHM of 2.5 nm (

**a**); 25 nm (

**b**); and 100 nm (

**c**); (

**d**–

**f**) for $\mathrm{NA}=0.9$, $\lambda =550$ nm and spectral FWHM of 2.5 nm (

**d**); 25 nm (

**e**); and 100 nm (

**f**).

**Figure 8.**Numerical aperture factors: ${f}_{1}$ is related to the equivalent wavelength, ${f}_{2}$ corresponds to the maximum bandwidth wavelength, and ${f}_{3}$ to the maximum effective wavelength.

**Figure 9.**Cross sectional views of the transfer functions $H({q}_{x},{q}_{z}=\mathrm{const}.)$ for different spectral bandwidth given by the FWHM, (

**a**) for $\mathrm{NA}=0.55$, $\lambda =550$ nm, and ${q}_{z}=21.2\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}{\mathrm{m}}^{-1}$, (

**b**) for $\mathrm{NA}=0.9$, $\lambda =550$ nm, and ${q}_{z}=16.6\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}{\mathrm{m}}^{-1}.$.

**Figure 10.**Surface profile reconstruction using the transfer function $H({q}_{x},{q}_{z}=\mathrm{const}.)$ for $\mathrm{NA}=0.55$ and $\lambda =550$ nm: (

**a**,

**c**) cosinusoidal input profile $s\left(x\right)$ of 0.5 µm period length and 55 nm PV-amplitude and reconstructed profiles ${s}_{\mathrm{rec}}\left(x\right)$; (

**b**,

**d**) corresponding partial transfer functions $H({q}_{x},{q}_{z}=\mathrm{const}.)$ for ${q}_{z,\mathrm{eval}}=21.2$ µ${\mathrm{m}}^{-1}$, absolute value of the electric field $|{U}_{0}\left({q}_{x}\right)|$ and product $H({q}_{x},{k}_{0})\left|{U}_{0}\left({q}_{x}\right)\right|$, (

**a**,

**b**) for FWHM = 2.5 nm, (

**c**,

**d**) for FWHM = 100 nm.

**Figure 11.**Surface profile reconstruction using the transfer function $H({q}_{x},{q}_{z}=\mathrm{const}.)$ for $\mathrm{NA}=0.55$ and $\lambda =550$ nm: (

**a**,

**c**) cosinusoidal input profile $s\left(x\right)$ of 5 µm period length and 550 nm PV-amplitude and reconstructed profiles ${s}_{\mathrm{rec}}\left(x\right)$; (

**b**,

**d**) corresponding partial transfer functions $H({q}_{x},{q}_{z}=\mathrm{const}.)$ for ${q}_{z,\mathrm{eval}}=21.2$ µ${\mathrm{m}}^{-1}$, absolute value of the electric field $|{U}_{0}\left({q}_{x}\right)|$ and product $H({q}_{x},{k}_{0})\left|{U}_{0}\left({q}_{x}\right)\right|$, (

**a**,

**b**) for FWHM = 2.5 nm, (

**c**,

**d**) for FWHM = 100 nm.

**Figure 12.**Surface profile reconstruction using the transfer function $H({q}_{x},{q}_{z}=\mathrm{const}.)$ for $\mathrm{NA}=0.9$ and $\lambda =500$ nm: (

**a**,

**c**) cosinusoidal input profile $s\left(x\right)$ of 0.3 µm period length and 50 nm PV-amplitude and reconstructed profiles ${s}_{\mathrm{rec}}\left(x\right);$ (

**b**,

**d**) partial transfer functions $H({q}_{x},{q}_{z}=\mathrm{const}.)$ for FWHM = 25 nm, absolute value of the electric field $|{U}_{0}\left({q}_{x}\right)|$ and product $H({q}_{x},{k}_{0})\left|{U}_{0}\left({q}_{x}\right)\right|$, (

**a**,

**b**) for ${q}_{z}=18.9\phantom{\rule{0.166667em}{0ex}}$ µ${\mathrm{m}}^{-1}$, (

**c**,

**d**) for ${q}_{z}=13.0\phantom{\rule{0.166667em}{0ex}}$ µ${\mathrm{m}}^{-1}$.

**Figure 13.**(

**a**) Cross sectional view of the transfer function $H({q}_{x},{q}_{z})$ for $\mathrm{NA}=0.9$, $\lambda =500$ nm and FWHM of 25 nm, (

**b**) $H\left({q}_{z}\right)$ for ${q}_{x}=0\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}{\mathrm{m}}^{-1}$ (blue line) and ${q}_{x}$ = 15.7 $\mathsf{\mu}{\mathrm{m}}^{-1}$ (red line).

**Figure 14.**Surface profile reconstruction by coherence peak detection using the transfer function ${H}_{\mathrm{env}}({q}_{x},{q}_{z,\mathrm{eval}})$ for $\mathrm{NA}=0.9$ and $\lambda =500$ nm: (

**a**) cosinusoidal input profile $s\left(x\right)$ of 0.8 µm period length and 50 nm PV-amplitude and reconstructed coherence profile ${s}_{\mathrm{rec}}\left(x\right)$; (

**b**) partial transfer function ${H}_{\mathrm{env}}({q}_{x},{q}_{z,\mathrm{eval}})$ for ${q}_{z,\mathrm{eval}}=19.0\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}{\mathrm{m}}^{-1}$ and FWHM = 25 nm, absolute value of the electric field $|{U}_{0}\left({q}_{x}\right)|$ and ${H}_{\mathrm{env}}({q}_{x},{q}_{z,\mathrm{eval}})\left|{U}_{0}({q}_{x},{q}_{z,\mathrm{eval}})\right|$; (

**c**) cosinusoidal input profile $s\left(x\right)$ of 0.4 µm period length and 50 nm PV-amplitude and reconstructed coherence profile ${s}_{\mathrm{rec}}\left(x\right)$; (

**d**) same corresponding partial transfer function ${H}_{\mathrm{env}}({q}_{x},{q}_{z,\mathrm{eval}})$, absolute value of the electric field $|{U}_{0}\left({q}_{x}\right)|$ and ${H}_{\mathrm{env}}({q}_{x},{q}_{z,\mathrm{eval}})\left|{U}_{0}({q}_{x},{q}_{z,\mathrm{eval}})\right|$.

**Figure 15.**Measured 3D topographies of a diamond milled aluminum mirror using a Mirau CSI with NA = 0.55, a red LED with a central wavelength of 630 nm for illumination and an evaluation wavelength ${\lambda}_{\mathrm{eval}}$ of 670 nm, (

**a**) result of envelope evaluation; (

**b**) result of phase evaluation.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Lehmann, P.; Hagemeier, S.; Pahl, T.
Three-Dimensional Transfer Functions of Interference Microscopes. *Metrology* **2021**, *1*, 122-141.
https://doi.org/10.3390/metrology1020009

**AMA Style**

Lehmann P, Hagemeier S, Pahl T.
Three-Dimensional Transfer Functions of Interference Microscopes. *Metrology*. 2021; 1(2):122-141.
https://doi.org/10.3390/metrology1020009

**Chicago/Turabian Style**

Lehmann, Peter, Sebastian Hagemeier, and Tobias Pahl.
2021. "Three-Dimensional Transfer Functions of Interference Microscopes" *Metrology* 1, no. 2: 122-141.
https://doi.org/10.3390/metrology1020009