INTRODUCTION

Today, the demand on flat optical components with specification on final surface flatness that must be smaller than λ/20 or λ/30 peak to valley, for various applications including linear accelerators [1,2], space applications [3]…, needs an accurate knowledge on the mechanical stress induced in thin films. In order to design correctly stacks of thin films on both side of a substrate and achieve perfect stress compensation, it is thus obvious that the first step is to precisely characterize the stress in single layers. Then from the characterization of single layer, we can model stress induced in multlilayers and develop strategies to optimize the stack design on both sides of the substrate to control the flatness of the final component.

I.

EXPERIMENTAL SET UP

All the optical components were manufactured with the help of a HELIOS 4” machine, developed by Bühler/Leybold Optics, where low and high refractive index materials were both deposited through Plasma Assisted Reactive Magnetron Sputtering (PARMS) [4,5]. As seen in Fig. 1., the main chamber of the machine is subdivided in 4 deposition/treatment zones, two dedicated for dielectric materials (mid frequency (MF) magnetron sputtering), one for metallic deposition (DC magnetron sputtering, not used here) and the last one for oxygen plasma assistance (PBS).

Fig. 1.

HELIOS deposition process chamber.

The substrates are set on a 12-position rotating sample holder (rotating at 240 rpm). According to the deposited material, either low or high refractive index, the corresponding MF magnetron sputtering cathode is switched on; the oxygen plasma assistance is used for densification of the coating and also to respect the stoichiometry of the layers. Typical deposition rates of low and high index materials are respectively around 0.40-0.45 nm.s^-1 and 0.50-0.60 nm.s^-1, which means that less than an atomic layer of material is deposited at each turn of the sample holder. Thicknesses of the different deposited layers are optically controlled, in a transmission mode, at a precisely selected wavelength through the Optical Monitoring System OMS5000 developed by Bühler/Leybold Optics [6].

Mechanical stress was then extracted from the substrate radius of curvature evolution before and after films deposition. These radiuses of curvature were measured with white light interferometry (Zygo NewView 7300).

II.

STRESS IN SINGLE LAYERS

Single layers made of SiO₂ and Nb₂O₅ with different thicknesses have been deposited. For each deposition, substrate thicknesses t_S were measured with a digital micrometer and radii of curvature on both sides of the substrate, before (R_S) and after (R_S+f) deposition, were measured with white light interferometry. Stoney equation [7] (equation 1) was redefined to simply extract the information on the stress value (equation 2). In this study, the substrates were made of fused silica (Φ = 25 mm), where the Young modulus E_s = 73 GPa, the Poisson coefficient v_s = 0.16 and the thickness t_s = 1.00±0.05 mm.

In equation (2), is a constant term that depends on the mechanical properties of the substrate,

is the substrate thickness normalized radius of curvature directly induced by the deposited layer.

As we plotted, in Fig.2, the term K_S/R_Norm as a function of film thickness for both SiO₂ and Nb₂O₅ layers, we could perform two different fits on the experimental data. In a first time, we could consider a linear evolution of the data that means a constant value of the mechanical stress as a function of the film thickness: σ = K_i [8] (i stands for either SiO₂ or Nb₂O₅). In a second time we used a second order polynomial to fit the experimental data, we obtained a slightly better fit with a thickness dependence (expressed in nanometers) of the stress: σ_i = A_i + B_i t_i. The results of both fits are detailed in Tab. 1.

Fig. 2.

Normalized radius of curvature evolution as a function of the film thickness for Nb₂O₅ (left) and SiO₂ (right). Linear fit (blue line) and second order polynomial fit (red line).

The merit function to determine the quality of the fit is described hereafter in equation 3, in which N is the number of points and X stands here for K_S/R_Norm:

Tab. 1.

Stress parameters obtain from a linear fit and from a second order polynomial fit.

III.

STRESS IN MULTILAYERS

Based on the results obtained on single layers of high and low refractive index materials, we analyzed if those one can be applied to multilayer structures. We studied the evolution of stress while fabricating a mirror centered at 515 nm with R>99.95% for both s and p polarization with an angle of incidence of 22°. The designed mirror is a 21-layer quarter-wave mirror centered at 532 nm ((HL)¹⁰ H), with Nb₂O₅ as high index material and SiO₂ as low index material.

In order to test the different models of the stress in single layers, we inserted several substrates in the deposition machine and removed progressively samples to obtain mirrors with an increasing number of layers (M5, M9, M11, M15, M17, M19 and M21) and therefore characterized the mirror deformation during its construction. This approach is possible with the HELIOS machine thanks to the load-lock that allows removing some of the samples being coated, without opening the whole deposition chamber and therefore without affecting the other samples being coated. With this procedure, all the samples were processed within the same batch.

In order to model the experimental data, we assumed that the K_S/R_Norm ratio evolves as expressed in equation 4 NH and NL are respectively the number of high (H) and low (L) refractive index material layers in the structure.

In order to have a better idea of this ratio, we focused our study on the sag of the sample after deposition. In that case, we could write again equation 4 in a different way to extract the experimental value of R_Norm (equation 5).

Then from this experimental normalized radius of curvature we could calculate the sag as shown in equation 6 with the sign norm described in figure 3.

Fig. 3.

Definition of the sample sag δ induced by the stress due to the deposited layers.

In equation 5, + sign will be used if R_Norm is positive and – sign if R_Norm is negative.

In Fig. 4, we plotted the experimental data measured at different steps of the mirror under fabrication on both sides of the substrate and we compared them with both models described in section 2 of this paper. The model with a thickness dependency of the stress is validated (MF = 0.017) as it allows accurately predicting the stress-induced sag, while the model with constant stress values overestimates the deformation as it supposes a larger value of the stress induced that would be expected in thicker layers (MF = 0.024). From this accurate model, it is now possible to precisely model the stress induced deformation by any single or multilayer structures.

Fig. 4.

Evolution of K_S/R_Norm as a function of the number of deposited layers for a M21 mirror centered at 532 nm. Comparison between experimental data and models.

IV

STRESS COMPENSATION

For many advanced applications requiring complex optical filters, one of the requirements is the flatness of the component after fabrication. In order to obtain stress compensation in multilayer optical elements, one of the common approaches consists of depositing two optical functions with similar stress-induced deformation, on each faces of the substrate [9]. To achieve, the stress compensation should respect the equation 7.

Supposing an already A coated face, to fulfill the requirements of equation 7, there are a wide range of possible solutions for the B-face coating.

We considered in this paper, the first two different approaches:

A.

Stress compensation with a single layer

Let us suppose that the face A has already been coated with the previously defined mirror (M21 @ 532 nm), equation 7 can be matched with the deposition of one single layer of SiO₂ or Nb₂O₅ that compensates the face A. In that case, equation 7 can be written as equation 8 in which x can stand either for SiO₂ or Nb₂O₅.

If we assume a second order polynomial model for the stress induced by the layers, we have to solve equation 9.1 (if we consider theoretical data) or 9.2 (if we consider experimental stress measurements on the sample after deposition on face A) to find the thickness t_x of material that will compensate the deformation on the opposite face.

As an example, we used this method to compensate the native deformation of a bare substrate with typical deformations in the range of λ/4 peak to valley at 532 nm. In Fig.5, we show the result of the minimization of the substrate deformation on face A with an optimized thickness of SiO₂ (t_SiO2 = 42 nm) deposited on face B. The optimized thickness of SiO₂ was calculated from equation 9.2 considering one layer of SiO₂ and an infinite value of R_S+f since it is a bare substrate. The radius of curvature before deposition were respectively of -986 m and -732 m for face A and B and after deposition we measured a radius of curvature of -34000 m for face A and -419 m for face B. If we calculate the residual radius of curvature on face B according to the initial radius of curvature and the thickness of the SiO₂ layer deposited on face A, we obtained -419.9 m which is in a very good agreement with experimental data.

Fig. 5.

Deformation, on orthogonal direction, of a substrate before and after deposition of a SiO₂ compensating layer.

However, this solution works for a restricted range of temperature and does not allow controlling the spectral properties of the face B. Another solution is to deposit the same mirror on the second face.

CONCLUSIONS

We have validated a new thickness-dependent model for the stress induced in SiO₂ and Nb₂O₅ layers deposited by plasma assisted reactive magnetron sputtering (PARMS). This model was used to predict the evolution of the deformation of an optical component under manufacturing (e.g. a dielectric quarterwave mirror). From this accurate modeling, and the initial substrate deformation, it is thus possible to design a thin film stack that minimizes the deformation of the final component.