## Abstract

We present a single-shot line sensor based on spectral interferometry. Light of a broadband laser source is chromatically dispersed by a grating and focused onto a line on the surface such that each focal point on this line is formed by another wavelength. The entire height profile is obtained by applying a phase evaluation algorithm to the registered interference signal, followed by a model-based approach. The sensor concept is finally verified by experimental results.

© 2011 Optical Society of America

## 1. Introduction

In the last decade, confocal and interferometric measurement techniques became established in order to realize noncontact and high-precision topographical surveys of technical surfaces. By using confocal microscopy or white-light interferometry, a mechanical scan must be executed to measure the distance of one point on the surface. However, chromatic confocal microscopy [1] and spectral interferometry (SI) [2] overcome this disadvantage such that a single-shot point measurement is feasible.

In order to provide high-speed sensors for surface measurements, it is desirable to parallelize the sampling of single points. Therefore, different approaches propose to illuminate one line on the object’s surface with chromatically dispersed light from a broadband source. Then, it is possible to create a line sensor based on confocal microscopy [3, 4] as well as on optical coherence tomography [5, 6, 7]. However, in order to get the requested height information, schemes [4, 5, 6, 7] require a high-resolution mechanical scan in the reference arm, respectively, a scan of the focal plane [3], while the sensor presented in [7] needs a two-dimensional spectrometer and suffers from low lateral resolution. In this paper, we propose a line sensor without a full-range mechanical scan that uses the same illumination technique combined with Fourier domain SI and covers a measurement range of approximately $1\text{\hspace{0.17em}}\mathrm{mm}\times 0.25\text{\hspace{0.17em}}\mathrm{\mu m}$.

This paper is organized as follows: The measurement principle together with an explanation of the experimental setup is given in Section 2. A physical modeling of the sensor is introduced in Section 3. Based on these results, Section 4 is dedicated to the presentation of the signal evaluation. First, the phase information has to be extracted from the interference signal; afterwards, different model-based approaches are proposed in order to obtain the line’s height profile from the phase. Section 5 deals with measurement results in order to verify the theoretical thoughts and gives an overview about the sensor’s capabilities. Our conclusions are presented in Section 6.

## 2. Measurement Principle

The laterally chromatically dispersed, spectrally encoded interferometer (LCSI) is a single-shot line sensor concept based on Fourier domain SI where the height information is encoded in a single spectrometer line. The schematic of the sensor principle is shown in Fig. 1.

A broadband, fiber-coupled light source (super luminescence diode, 800–$870\text{\hspace{0.17em}}\mathrm{nm}$ single-mode fiber, and $16\text{\hspace{0.17em}}\mathrm{mW}$ fiber output) is collimated (two achromatic near-infrared (NIR) doublets, ${f}_{\text{combined}}=40\text{\hspace{0.17em}}\mathrm{mm}$). The collimated light then passes a beam splitter ($50/50$) where it is split into a reference and an object wave. The object beam hits a grating ($600\text{\hspace{0.17em}}\mathrm{grooves}/\mathrm{mm}$, tilt angle $\alpha =2.7\xb0$). Different wavelengths are reflected into its respective diffraction orders and focused onto the measurement object by an achromatic objective ($f=25\text{\hspace{0.17em}}\mathrm{mm}$, clear aperture $12.5\text{\hspace{0.17em}}\mathrm{mm}$). Because of the wavelength-dependent angle of incidence, the objective forms a focus line in the object space with each wavelength *λ* encoding a different lateral position $x(\lambda )$ on the line. An aperture stop in the back focal plane of the objective is used to achieve a telecentric imaging to avoid variations of the lateral position of the individual wavelengths within the measurement volume. The back-reflected light from the object passes the grating again and is coupled back into the fiber. A $1\times 2$ single-mode fiber coupler and a fiber-coupled spectrometer (Avantes AvaSpec-3648, resolution $0.08\text{\hspace{0.17em}}\mathrm{nm}$) is used to acquire spectrally resolved interference signals. The relation between the pixel *j* on the spectrometer’s CCD, its respective wavelength *λ*, and lateral position $x(\lambda )$ on the illuminated line is determined by calibration.

The reference arm consists of a neutral density filter for adjusting the intensity and a reference mirror. This arm has a different optical path length in comparison to the object arm to achieve a constant carrier frequency in the spectrally resolved signal. The lateral measurement range of the sensor setup is $0.9\text{\hspace{0.17em}}\mathrm{mm}$. The optical resolution according to the Rayleigh criteria is about $16\text{\hspace{0.17em}}\mathrm{\mu m}$ diameter and a single pixel represents between $0.9\text{\hspace{0.17em}}\mathrm{\mu m}$ and $1.1\text{\hspace{0.17em}}\mathrm{\mu m}$ in object space. With the given setup, the axial measurement range is limited by the confocal filtering of the object wave, which reduces the interference contrast. The experimental FWHM of the interference signal is $120\text{\hspace{0.17em}}\mathrm{\mu m}$.

## 3. Physical Modeling

The physical modeling of the LCSI line sensor is based on the models introduced by Davidson *et al.* [8], Kino and Chim [9], and Sheppard and Larkin [10]. Papastathopoulos *et al.* [11] extended the formalism for chromatic confocal SI (CCSI), where the light is chromatically dispersed in the axial direction in contrast to the lateral dispersion in the current setup. In both cases, light, being propagated to the object and reference surface, is supposed to be randomly polarized and low coherent. The total interference signal is the incoherent sum of all interfering ray bundles of different wavelengths, which are reflected from the object surface at an angle of incidence *ψ* after having passed through the pupil plane of the objective.

The interference contribution of rays having the same wave number $k=2\pi /\lambda $ is supposed to be reflected from one single point of the illuminated line with a distance $z(k)$ between this point on the object surface and the conjugated plane with respect to the reference. Following the two-beam interference analysis, this contribution as a function of the angle of incidence *ψ* is expressed by

The height profile $z(k)$ of one line on the sample’s surface is obtained by measuring the wavenumber-dependent optical path difference (OPD) $d(k)$ between the optical paths in the reference and object arm. The relationship between the OPD $d(k)$ and the heights $z(k)$ is assumed to be

since the gap between the objective and the sample’s surface is filled with air. With respect to Eq. (1), the OPD information is contained in the phase of the cosine term. Because of the limited numerical aperture of the objective and the additional confocal filtering while coupling light beams into the single-mode fiber, the maximal incident angle*ψ*is assumed to be small. Hence, the term $\mathrm{cos}(\psi )$ in Eq. (1) is neglected in the course of this paper. Consequently, the phase term $\mathrm{cos}(\phi )$ with becomes independent of the angle of incidence and can be taken out of the integral in the interference signal in Eq. (2). In contrast to a point sensor based on the CCSI method or ordinary white light interferometry (WLI) where the whole spectrum addresses the same object point, the OPD $d(k)$ in Eq. (4) of the presented LCSI line sensor is dependent on the wavenumber

*k*. Therefore, an adapted signal evaluation strategy is necessary to obtain the desired information.

## 4. Signal Evaluation

The signal evaluation of an acquired intensity signal involves a number of processing steps. The basic scheme of this evaluation is depicted in Fig. 2 and explained in the course of this section.

First, the measurement data is processed in a signal preprocessing step to meet the requirements of the subsequent phase evaluation, where the phase term of Eq. (4) is determined. Afterwards, the reference phase of a planar mirror measurement is subtracted from the phase *φ* such that the nonlinear term $\delta (k)$ is eliminated. Finally, the course of the OPD and, ultimately, the height profile is calculated.

Figure 3a shows a measurement signal of a resolution standard (solid line) and the reference signal of the superluminescent diode (SLD) source, recorded by disabling the object arm of the interferometer. In the preprocessing step, the reference is subtracted from the measurement signal, such that the dc part and low-frequency parts of the signal are removed, which is depicted in Fig. 3b.

The normalized signal is now passed to a phase evaluation algorithm. A global overview of different phase detection principles is given in Debnath *et al.* [12]. One possible method to determine the spectral phase *φ* is based on temporal phase shifting, which is executed by axially moving the reference mirror with a specific constant offset. However, this axial offset does not lead to the same phase shift at each point of the line due to the spectral dispersion. Therefore, a pointwise compensation factor must be determined first [13]. Using temporal phase shifting, the measured phase ${\phi}_{\text{meas}}$ as a function of the wavenumber *k* is given by

*m*.

On the one hand, temporal phase shifting is a precise phase detection technique; on the other hand, it requires at least three different measurement steps. By using one-shot frequency analysis methods that are able to handle chirped signals, e.g., wavelet transformation [14], windowed Fourier transformation [15], or chirplet transformation [16], it is possible to determine the phase from only one interference signal of a single frame. However, using these methods, it is not possible to get information about the sign of the real spectral phase because the interference signal of a phase with either positive or negative sign leads to the same result due to the symmetry of the cosine function. Hence, the measured phase ${\phi}_{\text{meas}}$ is described by

In the case of CCSI and WLI, the ambiguity of Eq. (6) does not exist because of different facts. First, the OPD *d* is assumed to be constant for the whole spectral range in the spectrometer. In order to guarantee a good phase detection quality, Schnell *et al.* [17] pointed out that a certain OPD offset ${d}_{0}$ has to be established between the reference and object arm such that the measured OPD *d* meets the condition

*k*to the spectrometer pixel is monotonic, the derivative of the real phase with respect to

*k*does not change the sign within the measurement range under the assumption that the nonlinear term $\delta (k)$ is only slightly varying with respect to the wavenumber. However, in the context of the LCSI sensor, the OPD $d(k)$ is dependent on

*k*. Therefore, the sign of the real phase’s gradient depends on the topography of the object and may change within one interference signal, hence, the single-shot phase evaluation techniques have to be extended. On the one hand, this can be done by recording a second measurement with a light source having a different light spectrum; one the other hand, a small axial offset $\mathrm{\Delta}z$ can be applied to the reference mirror before acquiring a second signal with the same light source, which has been done in the course of this work to simplify matters. This offset should be smaller than one-half of the wavelength, such that the additional phase shift is lower than $2\pi $ in order to avoid problems with the cosine periodicity. The decision about the sign of the real phase can be made by comparing both measured phases with all possible combinations of real phases. This extended version of frequency analysis algorithms allows finding the unique phase representation of Eq. (5) with at most two phase-shifted measurements.

The purpose of the penultimate step in the signal evaluation chain is to eliminate the nonlinear term $\delta (k)$, which is independent from surfaces lying within the objective’s depth of focus. Therefore, the phase of a planar mirror measurement is subtracted from the phase in Eq. (5). This also has the effect that the OPD $d(k)$ and the corresponding height profile is now representing the (optical path) distance between the measured line profile and the subtracted planar mirror. By this step, Eq. (5) yields

Unfortunately, the OPD $d(k)$ cannot be determined directly from Eq. (8) because of the $2\pi $ ambiguity. Supposing the object’s height profile along the line is smooth, *m* is incrementing whenever a $2\pi $ phase wrap occurs. By using an ordinary unwrapping algorithm, the measured phase ${\phi}_{\text{meas}}$ will be adapted such that *m* is constant within the measurement range. Now it is possible to eliminate the term $2\pi m$ by differentiating Eq. (5) with respect to the wavenumber *k*:

In WLI and CCSI, the phase derivative of Eq. (9) is equal to the desired constant OPD *d*. However, in the scope of the present setup, Eq. (9) becomes an ordinary differential equation of first-order, which yields an initial value problem. Different approaches to solve this differential equation in order to calculate the entire height profile are given in the following subsections.

#### 4A. Model-Based Approach

For the presentation of a model-based approach, it is supposed that the measured line on the surface is divided into a set of *n* discrete points, each illuminated by different wavenumbers. Monochromatic light being reflected from one of these points hits a specific pixel *j* on the spectrometer’s CCD. The precise mapping of wavenumber to detector pixel is obtained from calibration. Figure 4 shows a section of the whole line profile with two exemplary sampling points. Each point is hit by rays of wavenumber ${k}_{j}$ and is described with the parameters ${\gamma}_{j}$, ${d}_{j}$, and ${b}_{j}$ according to Eq. (9).

In order to derive a model-based representation of the line profile, the relation between adjacent points has to be determined. By using Taylor’s theorem, it is possible to describe ${d}_{j+1}$ with respect to the fixed point (${k}_{j}$, ${d}_{j}$) and analogously ${d}_{j}$ with respect to (${k}_{j+1}$, ${d}_{j+1}$). Hence, the OPD difference between two adjacent sampling points is the mean value of the first-order approximations to the OPD:

Now, Eqs. (9, 10) are evaluated at each pixel $j\in [1,n]$, which yields the following system of linear equations:

Since *A* is a $(n-1)\times (n)$ matrix, the system of linear equations has one degree of freedom and, hence, the solution vector $X=[B,D{]}^{T}$, obtained by a common linear solver, lies within a one-dimensional vector plane:

In order to determine *τ* such that the appropriate solution vector *X* is found, at least one further linear equation has to be added to the system of Eq. (11). This additional condition needs to be created from some *a priori* information about the macroscopic structure of the surface. The simplest form for this condition is the knowledge of the OPD *d* of one sampling point. Experiments showed that it is usually difficult to find the right OPD value for one point. However, *a priori* information can also be found by classifying objects. For instance, if it is known that a surface is symmetrical with respect to the perpendicular of the optical axis, then the mean value of the surface’s gradients is supposed to be zero. Hence, the mean value of ${b}_{1}\dots {b}_{n}$ can be set to zero too.

With this model-based approach, the measurement value of each sampling point contributes to the overall solution set in equal parts. This property bears the risk that an error in one point influences the whole measurement. On the other hand, if a continuous interference signal is recorded, measurement noise is somehow averaged out and the height profile can be determined with high accuracy.

#### 4B. Combination of Multiple Laterally Shifted Measurements

If no *a priori* information is available for solving the initial value problem, the line profile can also be determined if every point on the sample’s surface is scanned by at least two measurements with different wavenumbers. This can be achieved by synchronously using two different light sources or, to simplify matters, by shifting the object parallel to the sampled line and acquiring separate measurements, which has been applied in the course of this paper. By considering the calibration, the recorded intensity signals and the corresponding phase data can be interpolated, such that for a regular grid of points ${x}_{j}(j\in [1,n])$ along the line *m* tuples $(m>1)$,

Consequently, the OPD *d* and its slope at a certain lateral position ${x}_{j}$ is obtained by solving the following, clearly determined system of linear equations:

If different wavenumbers for each position are obtained by shifting the sample and not by illuminating it with a completely different light spectrum, the parameters *η* in Eq. (15), obtained with the aid of the calibration data, are only varying by about 3.5% with respect to the experimental setup. This leads to a very bad condition number for matrix *A*. Consequently, *d* can only be determined with a very low accuracy, especially if the phase evaluation is affected by a noisy input signal. The condition number will highly improve if the difference between the wavenumbers illuminating the same position in different measurements is bigger than in the current demonstrator setup.

#### 4C. Model-Based Approach with Multiple Shifted Measurements

By combining both methods that have been proposed in the previous subsections, it is possible to obtain accurate measurement results without the requirement of *a priori* information about the object’s surface. With respect to the tuple set in Eq. (13), a model-based solution set ${X}_{i}$ can be determined for each measurement $i\in [1,m]$:

*i*is supposed to be equal because of the interpolation mentioned in the Section 4B, the vectors ${D}_{1}\dots {D}_{m}$ can be set equal. This yields a system of linear equations whose solution vector is $[{\tau}_{1}\dots {\tau}_{m}]$. Finally, the OPD ${d}_{j}$ for each sampling point is calculable, for instance by taking the mean value of ${D}_{1}\dots {D}_{m}$.

## 5. Exemplary Measurements

In this section, measurement results of two exemplary specimens are given in order to verify the theoretical idea of the sensor concept and show the performance of the demonstrator setup. First, a resolution standard [18] (RS-M, SiMETRICS GmbH) has been measured. This standard consists of a set of gratings. The pitch of the measured area of the grating is $200\text{\hspace{0.17em}}\mathrm{\mu m}$ and the grating depth is stated with approximately $90\text{\hspace{0.17em}}\mathrm{nm}$. No information about the height confidence interval is given since this standard is a lateral and not an axial resolution standard; furthermore the shape of the sidewalls is denoted “rounded”.

Figure 5 shows the result obtained by applying the model-based approach without lateral shift. Since this section of the specimen is symmetrical with respect to the horizontal and vertical axes, it could be assumed as an additional condition that the mean gradient value *b* of all sampling points should be equal to zero. To obtain the depicted result, a best-fit line has been subtracted from the calculated OPD solution set. By fitting a line through both the upper and lower segments, a mean OPD of $87.9\text{\hspace{0.17em}}\mathrm{nm}$ has been determined, which is 2.4% smaller than the stated nominal value. With an ordinary edge detection algorithm, an average pitch of $200.3\text{\hspace{0.17em}}\mathrm{\mu m}$ has been calculated, which differs from the nominal value by only 0.15%.

In a further experiment, the same specimen has been measured by combining five laterally shifted measurements with an offset of $100\text{\hspace{0.17em}}\mathrm{\mu m}$. Using the combined model-based approach yields the result in Fig. 6. Here, an average OPD of $87.3\text{\hspace{0.17em}}\mathrm{nm}$ with a standard deviation of $0.69\text{\hspace{0.17em}}\mathrm{nm}$ was found. The mean grating pitch is $199.5\text{\hspace{0.17em}}\mathrm{\mu m}$ (standard deviation of $0.87\text{\hspace{0.17em}}\mathrm{\mu m}$). In both presented measurement results, the noiseless axial resolution is only obtained because every sampling point participates in the calculation of the overall OPD profile, which can be compared to the effect of averaging over many measurement points.

As a second object, a chirped calibration specimen, described in Krüger-Sehm *et al.* [19], has been measured using the model-based approach without lateral shift. The height profile of one line at *y* position $2.45\text{\hspace{0.17em}}\mathrm{mm}$ is depicted in Fig. 7a while a three-dimensional measurement result is shown in Fig. 7b, obtained by a scan in the *y* direction.

The comparison of the measured and the target periods is given in Table 1. It is obvious that the “fringes” on the calibration specimen could be measured with high accuracy. However, both the single line measurement in Fig. 7a and the three- dimensional representation in Fig. 7b show reconstruction errors starting at the eighth or ninth fringe. The reason for this is the violation of the Shannon sampling theorem with increasing gradients. Assuming that at least three independent samples per period have to be resolved, the following condition for the phase difference $\mathrm{\Delta}\phi $ between adjacent sampling points must hold:

With respect to the current demonstration and the basic properties of the chirped calibration specimen, the condition in Eq. (17) will be violated for the first time at the highest gradient in the ninth fringe. The calculated height profile in Fig. 7a confirms that the reconstruction starts to fail at the indicated position.

## 6. Conclusions

In this paper, it was shown that the LCSI line sensor allows determining the height profile of one line on a sample’s surface without continuous mechanical scan. Since both the lateral position and its height are encoded in one interference signal, an enhanced signal evaluation with a phase evaluation algorithm and a model-based approach has been developed. On the one hand, this approach showed good reconstruction results; on the other hand, *a priori* information about the object’s surface is necessary and partial registration errors will badly influence the determination of the entire height profile.

To overcome these constraints, each point of the line on the surface has to be sampled by at least two different wavelengths, which was done in this paper by laterally shifting the object and recording multiple interference signals. Then the height profile can be determined by means of a pointwise analysis or a combined method. However, it was shown that the lateral shift leads to high numerical uncertainties and noise if using the pointwise evaluation due to the limited bandwidth of the light source. In an outlook, we propose to simultaneously use a second light source with a spectrum of 400–$435\text{\hspace{0.17em}}\mathrm{nm}$ in order to avoid the numerical problems and realize a line measurement without overall dependencies and need of *a priori* information. In the current setup, the vertical measurement range is limited by the small depth of focus of the objective. This range can be increased using Bessel beams, which will be topic of a future publication.

We thank the Deutsche Forschungsgesell schaft (DFG), OS 111/21-2 “Chromatisch-konfokale Spektral-Interferometrie zur dynamischen Profilerfassung,” for providing financial support to this work.

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