Fourier-transform spectral interferometry for in situ group delay dispersion monitoring of thin film coating processes

Sebastian Schlichting; Thomas Willemsen; Henrik Ehlers; Uwe Morgner; Detlev Ristau

doi:10.1364/OE.24.022516

1. Introduction

During the last years, short pulse lasers have gained of importance in many application fields as for example ultra-precise material processing, laser medicine, or fundamental research on nonlinear properties of optical materials. In the course of this development, the requirements on the output power and beam quality continuously increased and imposed critical demands on the optical components of fs-lasers. For instance, precisely coated mirrors with a flat broadband negative group delay dispersion (GDD) or very high negative GDD are needed to correct for the dispersion in the resonator. In respect to the high requirements on the quality, the process reproducibility, and the stability of optical constants, sputter processes are favored for production of such mirrors. For the deposition of complicated coating designs with more than hundred layers, different optical monitoring strategies including single wavelength techniques and especially broad band monitoring are state of the art. These techniques use wavelength resolved transmittance or reflection measurements during the coating process to derive data for a precise termination of every single layer with a defined optical thickness. In this context, error self-compensation effects can be considered as advantageous consequences of optical monitoring concepts. Typical beneficial mechanisms are based on the self-compensation of errors in refractive index by physical thickness variations or on thickness deviations in multilayer coatings caused by thickness errors of already deposited layers [1, 2]. With these effects the result will be normally slightly worse than the targeted performance but better than without any self-compensation effect. In case of the phase properties of chirped mirrors, self-compensation strategies based on transmittance or reflectance measurements can be counterproductive, because variations or inaccuracies in the dispersion properties have a large impact on the GDD and cannot be easily compensated by adjustments in layer thickness. Hence, the last layers of a chirped mirror, which represent an AR-Coating on the top of the mirror, have a particularly large influence on the GDD properties of the mirror and are often controlled by classical methods like quartz crystal monitoring or time. These methods are not very meaningful, and many coating iterations are needed to find a coating parameter set for an acceptable GDD result. Other methods are based on reengineering procedures. By evaluating recorded in situ transmittance and / or reflection data, the optical constants and thickness values of the actual layer structure can be recalculated. On the basis of these recalculated layer properties a forecast of the GDD result can be attained, and if necessary, corrections in thickness values of the following layers can be integrated to achieve the best GDD result [3]. However, this method is not appropriate for the last layers of the design, because they typically do not offer sufficient degrees of freedom to compensate the occurred deviations in GDD completely. With a broad band in situ phase measurement, additional phase information is attained as a first step towards a much better layer model and optimized refining procedures.

The motivation of this work is an online measurement system of the spectral phase information, the group delay (GD) and group delay dispersion (GDD) curve. In the first section the experimental setup is presented. Afterwards, the theoretical basis of the novel in situ measurement is established. Therefore, data preparation, evaluation, smoothing, and methods for reduction of transformation artifacts are introduced. Finally, the performance is demonstrated in comparison with a commercial ex-situ GDD measurement system.

2. Experimental setup and methods

2.1 Experimental setup

Figure 1 shows a schematic illustration of the measurement system. A fiber-based Michelson interferometer illuminates a Horiba iHR320 imaging spectrometer equipped with a fast and sensitive CMOS camera (pco.edge). It consists of a polarization maintaining fiber coupler, a super luminescence diode (SLED) as broadband light source, and a computer based control and calculation unit.

Fig. 1 Schematic of measurement equipment.

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The main advantage of the fiber coupled system is the enhanced flexibility in positioning of sample and reference channel of the interferometer. So, it is independent of the individual installation conditions related to flange positions or the specific construction of the substrate holder, and nearly every process chamber can be equipped with the measurement system. In our case, the interferometer supports a wavelength range from 770 to 1100 nm, the spectral response of the camera is restricted to 400 to 1000 nm, the SLED with a specified output power of 5 mW emits in a wavelength range between 790 and 900 nm, and the range covered by the spectrometer is 80 nm. In this combination, the operational static wavelength range of the present hardware results in an interval from 800 to 880 nm. The input light power to the spectrometer is approximately 50 µW in case of using silver mirrors as reference and sample. Nevertheless, the presented modular set-up is flexible, and all parts can be substituted independently by components for other wavelength ranges.

The fiber collimators in sample and reference arm are equipped with micrometer adjusters to impose a defined time delay $τ$ between the two arms to generate spectral interference fringes. As reference mirror a protected silver mirror was used with high reflectivity and zero dispersion. To compensate for the dispersion of the vacuum window of the sample collimator, an identical glass window is placed into the reference arm [4, 5]. Remaining deviations due to small differences between the fiber lengths in both arms are measured with an identical silver mirror at the substrate position and subtracted later.

Measuring an accurate GDD during the deposition process imposes essential requirements. First of all, the optical measurement system must be designed to minimize angle and vibration influences to get a stable reflection signal of the moving substrate. Second, a very fast measurement is necessary to catch the substrates in motion. The integration time of the detector must be lower than typical vibration time scales but high enough to ensure an adequate signal-to-noise ratio. In case of a too high integration time, the contrast of the interferometric measurement and accordingly the fringe pattern, which carries the phase information, decreases. Typically, a complete broad band measurement can be performed within 100 microseconds. Finally, dedicated algorithms have to be used that evaluate the interference contrast followed by a statistical analysis for selecting the meaningful results out of multiple measurements performed during substrate holder rotation.

2.2 Evaluation methods

The in situ GDD measurement approach we propose is based on Fourier-transform spectral interferometry (FTSI) [6]. In the following $E_{R} (T)$ and $E_{S} (T)$ symbolize the time dependence of two electrical fields reflected by the reference mirror and the measured sample in the two arms of the Michelson interferometer set-up. Their Fourier functions are denoted by $E_{R} (ω)$ and $E_{S} (ω)$ . The phase difference $Δ φ$ to be deduced from the measurement is given by

Δ φ = \arg [E_{S} (ω)] - \arg [E_{R} (ω)] .

A relative time delay

τ

between the two electric fields is introduced, and the fields are superimposed by a beam combiner to form the sum

E_{R} (T) + E_{S} (T - τ)

. This superposition is recorded by the spectrometer. The resulting spectral density is given by

\begin{matrix} S (ω) = {| E_{R} (ω) |}^{2} + {| E_{S} (ω) |}^{2} + 2 Re {E_{R}^{*} (ω) E_{S} (ω)} \\ = {| E_{R} (ω) |}^{2} + {| E_{S} (ω) |}^{2} + E_{R}^{*} (ω) E_{S} (ω) \exp [i (ω τ + Δ φ)] + c . c . . \end{matrix}

The last term contains the desired phase difference

Δ φ

coded in the spectral fringe pattern. In order to extract the phase difference

Δ φ (ω)

from the measurement given by Eq. (2), the inverse Fourier transform is applied

F T^{- 1} S (ω) = E_{R}^{*} (- T) \otimes E_{R} (T) + E_{S}^{*} (- T) \otimes E_{S} (T) + S (T - τ) + S {(- T - τ)}^{*} .

The first two terms are the crosscorrelation functions of the individual reflected signals from sample and reference mirror, centered at

T = 0

. In contrast, the term

S (T - τ)

is centered at and the last term at

T = - τ

, respectively. For values of

τ

large enough,

S (T)

does not overlap with self-crosscorrelation terms [7–9]. The separation of the function

S (T - τ)

in expression Eq. (3) can be achieved by multiplication with a super Gaussian window function centered around

w (T) = exp [- {(\frac{(T - τ)}{σ})}^{2 λ}] .

The parameter

σ

denotes the half-width of the function and

λ

the super Gaussian order [10, 11]. In the algorithm introduced by Takeda, the result of this transformation is then shifted by the time delay

- τ

to

T = 0

and is subject to a second Fourier transformation back to the spectral domain [12].

To reduce the effects of artefact oscillations caused by discontinuities at the beginning and end of the data series (so-called Gibbs phenomenon) in practice, different methods using Fourier approximation and Chebyshev methods can be found in literature [13–15].-Methods of reassignment like S-Transformation, Evolutionary Periodogram and Wavelet Transformation with various wavelets like Gabor or Wigner-Ville Distribution (WVD) [18–26] were considered. These methods lead to improved resolution in time and frequency or better transformation results by multistage Fourier transformations. However, this is achieved at the expense of computational time, generation of cross-components or high noise sensitivity. Therefore, these methods are not well adapted to rapid data reduction concepts for the intended in situ phase and group delay measurement, respectively. Finally, the Short Time Fourier Transformation (STFT) as time-frequency representation was analyzed in respect to the intended in situ group delay measurement concept. This signal analysis method is a linear time-frequency representation with an in time scale varying window function $h [t]$ . The STFT in the form

X (T, ω) = \int_{- \infty}^{\infty} h [t] S [t + T] \exp (- i ω t) d t

gives the spectral information of the signal within the window at its position

T

in time, and the window function

h [t]

with finite length is centered around zero and independent of time scale

T

. By sliding the window to different positions, the time-varying spectral characteristic of the signal is attained [18, 21, 23, 27–29]. The group delay is the negative derivative of phase

φ (T, ω)

with respect to the frequency

ω

. The GD is then given by

G D (T, ω) = - \frac{\partial}{\partial ω} \arg [X (T, ω)] .

For the present application, the time axis provides no additional information, because the signal to be analyzed is time-invariant. The time

T

can be eliminated, as Auger et al. [19] and Flandrin [20] describe by extracting the center of gravity of the spectrogram in direction of time. Thereby the GD is written as

G D (ω) = \frac{\int_{- \infty}^{\infty} T X (T, ω) d T}{\int_{- \infty}^{\infty} X (T, ω) d T} .

Finally, the group delay dispersion can be calculated by an additional derivative of the GD with respect to frequency

ω

[11, 4].

G D D (ω) = - \frac{\partial (G D)}{\partial ω}

Specific advantages of the STFT for GD-analysis include relatively small computational time and a very effective reduction of transformation artifacts and noise [23]. Therefore, the STFT method is often favored especially also considering the suppression of cross terms [27–32] and computational simplicity [33–35].

3. Results and discussion

One example of an interferometric measurement is shown in Fig. 2. Depending on rotation speed, size of the sample and size of the substrate holder, typically 20 to 60 of these measurements can be performed on one sample mirror during every rotation of the substrate holder. Based on the experimental experiences, around 80% of these measurements are reliable and are used to calculate GD and GDD curves. For the final result, an averaging of these curves follows.

Fig. 2 In situ interferometric measurement of a chirped mirror.

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Figure 3 shows the real part of the Fourier transformed interferometric measurement data depicted in Fig. 2 and the applied super Gaussian separation function [Eq. (4)]. The time delay $τ$ is adjusted to 500 fs. It has been found that the optimum delay $τ$ is the minimum value which is necessary to separate the interferometric term from the peak around zero. Too large values of $τ$ will result in a reduced signal-to-noise ratio leading to losses in information because of the limited wavelength resolution of the detector. For unreasonably low values of $τ$ the signal cannot be completely separated from the non-interferometric terms and the GDD result will be distorted. Additionally, variations of the shape and position of the interferometric peak have to be taken into account for the practical case of in situ measurements. Therefore, in case of drifts in the interferometer set-up the delay must be readjusted after several minutes. Furthermore, the half-width $σ$ of the super Gaussian function must be large enough to separate also broader peaks. In the next step of the data evaluation algorithm, the separated interferometric term has to be shifted to zero, to eliminate the time delay $τ$ [Fig. 4].

Fig. 3 Real part of Fourier transformed interferometric measurement data $S (ω)$ in time domain and super Gaussian separation function with $σ = 240 fs$ and $λ = 8$ .

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Fig. 4 Real part of the separated Fourier transformed signal shifted to zero to eliminate time delay $τ$ .

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Finally, a flat-top windowed STFT [Eq. (5)] is applied to the complex form of the signal from Fig. 4. The window function is a fifth order flat-top window

h [t] = a_{0} - a_{1} \cos (- \frac{2 π t}{n}) + a_{2} \cos (- \frac{4 π t}{n}) - a_{3} \cos (- \frac{6 π t}{n}) + a_{4} \cos (- \frac{8 π t}{n})

with coefficients

a_{0} = 0.21557895

,

a_{1} = 0.41663158

,

a_{2} = 0.277263158

,

a_{3} = 0.083578947

and

a_{4} = 0.006947368

which normalize the maximum of the window function to one [36]. The quotient n specifies the width of the window in number of points. The flat-top window function was preferred to the Gaussian window function, because the usage of Gaussian window produces more ripples in the final GDD result. In contrast, a flat-top window produces more artifacts in the edges of the wavelength range, but the result is much more reliable. This has been shown based on comparative evaluations in [17].

The output of the STFT [Eq. (5)] is a three dimensional time-frequency spectrogram [Fig. 5] of several Fourier transformations overlapping in time. This can be applied for smoothing and results in a much more stable signal in comparison to a single Fourier transformation [6]. Especially periodic artefacts of the Fourier transformation from frequency to time regime as well as the influence of noise, which results in large GDD oscillations, can be reduced by three to four orders of magnitude compared with a conventional FFT [17]. This approach is crucial for the presented in situ GD measurement, because permanent optimization of the shape of peak separation function and data windowing for transformation to time domain is not necessary anymore. An essential parameter governing the achievable resolution and degree of smoothing is an appropriate window size of the STFT window function $h [t]$ . The window width, 144 fs in the example Fig. 5, should be determined according to the balance between the linear phase approximation error and the noise level. An advantageous window shape can be found by adjusting the width n between 25 and 60 points which represents 120 fs and 300 fs to minimize ripples without changing the general characteristic of the measurement curve [37]. The slope of the spectrogram in frequency direction provides directly information about the group delay. It is calculated by Eq. (7). Figure 6 illustrates the GD of a chirped mirror made of 54 layers tantalum pentoxide (Ta₂O₅) and silicon dioxide (SiO₂) with a target GDD of −600 fs² between 830 and 855 nm wavelength. An in situ measured GD is shown in comparison to the design curve and an ex-situ measurement performed with a commercial time domain measurement system (Chromatis by KMLabs). As a first monitoring approach, the critical last 3 layers of this mirror were terminated manually based on a visual comparison of the measured GDD with the calculated GDD spectra of the theoretical design. In contrast, the layers 1 to 51 were controlled by an optical broad band monitoring system using transmittance measurements and automatic reengineering algorithms [3].

Fig. 5 STFT result of separated and shifted Fourier transformed signal, flat-top sampling window function with width 144 fs.

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Fig. 6 Comparison of GD measurements of a chirped mirror −600 fs² in the wavelength range of 830 to 855 nm wavelength, offset corrected.

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Figure 7 illustrates the GDD of the same mirror. The depicted spectra show averaged in situ GDD measurements with rotating and nonrotating substrate holder in comparison with a Chromatis ex-situ measurement and the calculated layer design with slightly modified thicknesses of the last layer to demonstrate the strong influence of thickness deviations. The in situ measured values are in very good agreement with the design and the ex-situ curve of the Chromatis system. The stability of the measurements is indicated by the error bars according to the standard deviation derived from approximately 150 successive GDD measurements with nonrotating substrate holder (gray curve) and approximately 150 GDD measurements during rotation of the substrate holder. The maximum standard deviation in the concerned wavelength range is 30 fs² for the non-rotating and also for the rotating measurement. This demonstrates the stability of the in situ measurement system, which is not negatively affected by the rotation of substrate holder.

Fig. 7 GDD measurement of a chirped mirror −600 fs² from 830 to 855 nm wavelength (design with 54 layers).

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In the presented experimental results, the small deviations between ex-situ and in situ measurement can be explained by different measurement spots and measurement positions on the sample. The spot size diameter of the in situ measurement system is in a range of 100 µm, and measurements are executed during rotation. In contrast, the spot diameter of the ex-situ measurement system is in centimeter range covering an extended area on the test sample. As a consequence, a slight lateral inhomogeneity of the coating on the test sample has a significant effect to both measurements [Fig. 7]. To illustrate the sensitivity of GDD to small variations in thickness caused for example by layer inhomogeneity or deformation of the substrate by excessive coating stress, the GDD spectra of the design with last layer thicknesses of 14.2 nm and 14.8 nm are shown in Fig. 7. The design with last layer 14.2 nm is very close to the in situ measurement. However, the design with a last layer thickness of 14.8 nm is closer to the ex-situ Chromatis measurement. In case of in situ measurement, there is also the effect of coating material on the vacuum window of the measuring head. Because of these unavoidable coating influences the measurement system has to be calibrated regularly, e.g. after five mirror coating runs. This can be achieved by monitoring a silver mirror at the sample position.

Finally, Fig. 8 illustrates as an example the in situ measured change in GDD during deposition of the last layer of a chirped mirror consisting of 58 layers. The decrease of the cavity around 830 nm over the deposition time of 9.5 minutes can be observed. Reaching the target thickness of 50 nm for this layer, a flat GDD of −600 fs² between 830 and 855 nm wavelength is achieved.

Fig. 8 In situ GDD measurements during deposition of last layer of a chirped mirror −600 fs² from 830 to 855 nm wavelength (design with 58 layers).

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4. Conclusions and outlook

This study demonstrates a new fiber based in situ Fourier transform group delay dispersion (GDD) measurement system for optical coating processes. With this system it is possible to measure the GDD directly on moving substrates in the coating plant during deposition with a precision comparable to conventional ex-situ time domain methods.

To achieve a balanced data reduction, several Fourier analysis methods were analyzed and finally a conventional Fourier transformation in the time domain and a STFT back in frequency domain have been proven as most appropriate alternative in terms of computational time and signal stability. This approach opens new horizons for the chirped mirror production and other optical coatings with challenging GDD specifications. Based on this additional information, new deposition control algorithms can be developed to improve the performance of complex coatings with predefined GDD and to optimize the yield. Furthermore, the GDD reacts very sensitive to optical thickness variations enabling an improvement in the precision of the layer termination techniques in future. Not least, because of the large effects of dispersion and refractive index variations on the GDD, in situ phase measurements could be considered as a versatile basis for an advanced differentiation algorithm for thickness versus refractive index in a coating.

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Fourier-transform spectral interferometry for in situ group delay dispersion monitoring of thin film coating processes

Abstract

1. Introduction

2. Experimental setup and methods

2.1 Experimental setup

2.2 Evaluation methods

3. Results and discussion

4. Conclusions and outlook

References and links

Cited By

Figures (8)

Equations (9)

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