Introduction

Thin-film-filter technology provides high performance and low cost dense-wavelength-division-multiplexing (DWDM) devices. It is not surprising that the narrow-band-pass-filter (NBPF) applications have come to dominate the 200- and 100-GHz DWDM markets. However, there is difficulty in achieving the close manufacturing tolerances necessary to ensure a high level of 50-GHz thin-film device performance [1–4]. Design of NBPFs consisting of more than 100 dielectric films is based on several Fabry-Perot cavities [5–8]. The most important problems are thickness monitoring and prediction of turning points during filter deposition, i.e., points in which deposition of a particular film terminates. Correct predictions of turning points during deposition of narrow band pass filters are crucial for overall filter performance. Sensitivity to thickness errors and thickness error compensation have been discussed by Bousquet et al. [9], Willey [10], and Sullivan and Dobrowolski [11, 12].

There are two different approaches for monitoring of turning-points using optical transmittance data. The first one commonly used is based on a local searching of transmittance extremes using, for example, a fitting of the transmittance to a polynomial, monitoring of turning-points overflowing, or zero searching of the transmittance derivatives [6]. The second approach is based on a fitting of all transmittance data during the film deposition to a model based on light interference in the filter structure. The later approach suggested by Schroedter [13] and Willey [14] is applied in this work. The technology of NBPF is known as the most advanced in the area of thin-film optical coatings. Consequently, complex optical phenomena have to be included in the model and the filter simulation requires sophisticated theory.

In this paper, we propose a method to model the most important optical phenomena affecting fitted transmittance signal and simultaneously keep computation time sufficiently short. The proposed approach uses: (i) recurrence giving fast computation, (ii) fitting of relative quantity which eliminates inaccuracy of the absolute transmittance measurement, (iii) numerical correction of the effects of partial coherence originating from monochromator bandwidth, (iv) correction of divergence of the testing incident beam, and (v) fitting of the measured data to the model using robust non-linear Levenberg-Marquardt algorithm.

2. Theory

The structure of the 100-GHz NBPF (central wavelength of 1550 nm) used for DWDM consisting of four-cavity and 128 layers is used for theoretical modeling. Silicon dioxide and tantalum pentoxide have been considered for the low- and high- refractive-index materials, respectively. Figure 1 shows calculated transmittance data during filter deposition and schematic description of the filter structure.

 

Fig. 1. Calculated transmittance during filter deposition and filter structure are shown (H, L, and G represent high-, low-refractive index layers, and substrate, respectively). The red and blue lines correspond to the Ta₂O₅ and SiO₂ layers, respectively. Four minima correspond to the Fabry-Perot cavities.

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During thickness monitoring of a particular layer, structure of thin films already prepared remains unchanged and only thickness of the layer actually deposited increases. Consequently, the recurrence approach is suitable for fitting of the transmittance data during deposition and saves computation time. Interference phenomena in the complicated deposited multilayer are included in the reflection and transmission coefficients [15]. Substrate can be considered as a thick layer in which the interference phenomena are averaged by variation of thickness. Back reflection from the substrate can be neglected because of efficient back-side antireflection coating, or included into recurrent calculation. The recurrence method is more simplified for nonabsorbing layer. In this case, only the reflection coefficient r can be calculated, and the transmittance is obtained in the form: T=1-|r|².

Obtained transmittance is a function of the layer thickness d related with a deposition rate and time. The deposition rate R ₁ can be considered either as a constant during deposition of one layer, or its fluctuation can be corrected from crystal monitor data [13, 14]. We also include an additional parameter - the initial layer thickness R ₀ describing thickness errors of previous layer and inhomogeneity near interface:

where m and τ denotes the sampling index and the sampling time. The parameter R ₀ enables automatic thickness error compensation. Consequently, the main fitting parameters are the deposition rate R ₁, the refractive index of the layer n and the thickness error compensation R ₀.

Important results of this paper are related to the partial coherence phenomena [16] originating from a finite monochromator bandpass, which is usually omitted in modeling of thin-film optical coatings. The resolution of the monochromator is characterized by the instrumental line profile (instrumental bandpass of monochromator), which can be obtained by measurement of the response from a monochromatic light source. The monochromator bandpass depends on many factors including the width of the grating, system aberrations, spatial resolution of the detector, and entrance and exit slit widths. The monochromator bandpass of 0.15 nm is estimated in our case. The effects of a finite monochromator bandpass correspond to the situation of filter illumination by partially coherent polychromatic light. Consequently, the monochromator bandpass is equivalent to the spectral linewidth of the incident polychromatic light. Corresponding coherence length l_c=16 mm (for λ=1550 nm) much exceeds the total layer thickness. However, we show that the spectral width of the monochromator function significantly affects the transmittance measurement. This originates from strong resonance character of NBPF.

 

Fig. 2. Influence of the partial coherence phenomena originating from the finite monochromator bandpass to the spectral response of the filter (a) and to the transmittance measured during deposition of the layer number 31. Lorentzian profile was used in the calculation and Δλ denotes the FWHM (full width at half maximum) spectral linewidth.

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Figure 2(a) shows total spectral transmittance modeled for the NBPF for the monochromator bandpass ranging from 0 to 0.2 nm. Lorentzian profile was used in the calculation. Figure 2b shows how the monochromator bandpass affects the amplitude and shape of the transmittance during deposition of the 31st layer. This coupling (matching) layer shows strong sensitivity to the partial coherence effects originating from the finite monochromator bandpass. Figure 2(b) shows that finite monochromator bandwidth could results in an additional extreme which does not correspond to the termination point. Consequently, the monochromator bandpass has been considered as an additional fitted parameter and reasonable average value of 0.18 nm was obtained. Our experience shows that the Lorentzian line profile is more suitable that the Gaussian one. We note that transmittance during deposition of cavity and near-cavity layers is almost insensitive to the partial coherence phenomena. The partial coherence phenomena may compensate also effects of thickness non-uniformity.

Other fitting problem may originate from divergence of a testing beam and consequent angle-of-incidence variation. Numerical transmittance averaging is expressed in the form:

where f(λ) and g(φ) denote the weight functions for the wavelength λ and the angle of incidence φ, respectively. We note that both phenomena exhibit different symmetry.

 

Fig. 3. Measured relative transmittance data (red lines) are compared with the best fit model (blue dashed lines) for the layer number 15. Plot (b) shows a detail near the turning point.

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There are also another optical phenomena and experimental errors affecting absolute transmittance. It is sometimes difficult to determine corresponding parameters precisely and use appropriate model. Consequently we propose to fit the relative transmittance, which is commonly used to obtain optical function spectra [17]. It is useful to relate transmittance to the initial, or final point. If we relate the transmittance to the initial point, the fit is less sensitive to data noise and shows good stability and convergence. However, the result may be affected by some deposition instability at the beginning. If we relate the transmittance to the final point, the results of the fit are more precise, but more sensitive to noise of the measured signal.

Proposed approach of the turning-point prediction represents the least squares fitting to the non-linear function. The standard Levenberg-Marquardt algorithm is used to minimize the error function and obtain the fitted parameters.

3. Results and discussion

Measured transmittance data were obtained during preparation of the NBPF filter (see Fig. 1) using ion-assisted deposition. The 10 mm thick BK7-glass substrate rotated by speed of 1200 rpm. The transmittance was measured using a system consisting of the polychromatic light source, optical fiber, collimating optics, monochromator, and lock-in detector.

Figure 3(a) shows an example of the measured data and obtained fit during deposition of 15th layer. This is the tantalum pentoxide near-cavity layer. It is known that the cavity and near-cavity layers are very sensitive to thickness errors and even a small uncompensated error may easily reduce total filter performance. Figure 3(b) shows detail of the measured data and fit of relative transmittance near turning point. We can see that the global fitting of the measured data to the physically based realistic model gives good agreement with the data.

Turning points have to be predicted in advance in order to terminate the layer deposition exactly corresponding to the required thickness. Therefore we have studied the convergence of the predicted turning point, i.e. ability to predict the turning point with sufficient precision. Good convergence, as we believe, is the strongest argument for the approach proposed. Figure 4(a) shows the convergence test for the layer number 15, which represents the layer near the first cavity. Obtained turning-point position is plotted as a function of time (or sampling) during layer deposition (compare x-scale with Fig. 3(a)). The turning point can be estimated in advance with high accuracy and the estimation shows good convergence. From Fig. 4(a) we can deduce that the turning point can be predicted with precision of 2 s (0.5 nm) even 150 s in advance. The good convergence of the presented approach gives possibility to deposit precisely layers of arbitrary thickness.

 

Fig. 4. Convergence tests describing predictability of the turning points are shown. Plot (a) corresponds to the layer number 15, which represents a layer near cavity. Plot (b) shows the convergence tests for layer number 30. Blue solid and red dashed lines correspond to the model with and without inclusion of partial coherence phenomena.

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Convergence tests showing some trend and fluctuations give sign that the model or data precision has to be improved. Figure 4b shows the convergence test for the SiO₂ layer number 30, which represent a layer near the coupling layer. Response during deposition of such layers is sensitive to the partial coherence effects discussed in previous section. Red dashed line was obtained using the thin film model neglecting partial coherence effects. Slow convergence giving inaccurate prediction is evident from Fig. 4(b). However, the model including the partial coherence effects originating from a finite monochromator bandpass (blue solid line) gives precise and stable prediction (70 s in advance in this case). Equation (2) was used for calculation of the total transmittance, where f(λ) represents the power spectral density. Lorentzian line profile was assumed in the model. Reasonable value of additional fitting parameter Δλ=0.23 nm was obtained from the final data fit. The model including partial coherence effects gives better convergence and precision than the simple thin-film model.

4. Conclusion

We conclude by summarizing the advantages of the proposed approach. (1) The method gives good convergence proposing precise prediction of the termination point in advance. (2) The model includes partial coherence phenomena (effects of the monochromator bandpass) and the influences of incidence angle non-uniformity. (3) Fitting of relative transmittance reduces sensitivity to absolute signal errors and some model imperfections. (4) The layer thicknesses are not restricted to an integer number of quarter-wave thickness. (5) Crystal sensor data can be included, which compensates deposition rate inhomogeneity. (6) The approach includes automatic error compensation. (7) The calculation can be performed in real time which is enabled by the fast recurrence approach.