1. Introduction

Since 60s, when the world has seen the first laser [1,2], the synthesis and production of a multilayer dielectric coatings becomes to play an important role in the laser physics. A dielectric coating could provide much higher reflectivity in comparison to a metal mirror. Basing on successes in the multilayer coatings [3], laser physicists obtain shorter durations and higher intensities of pulses. One can notice that significant breakthroughs [4–16] in the ultrafast physics in the past 20 years are associated with constitutive steps in the thin-films optics [17–34]. One of constitutive steps, namely, the invention of chirped mirrors [17] makes possible accurate phase control. Another constitutive step, the invention [35,36] and further development [37–39] of the needle optimization technique makes possible design of very complicated multilayer coatings with superior spectral characteristics. As a result, more general multilayer structures, so-called dispersive dielectric mirrors were designed [40,41] and successfully used for precise pulse control [15, 17–34]. Nowadays dispersive dielectric multilayer mirrors are the key elements able to control a phase of high-intensity femtosecond laser pulses. This achievement resulted in generation of sub-5 fs pulses directly from a Ti:sapphire cavity [42,43].

The ultrafast physics becomes one of the most rapidly developing fields in optics in last two decades. The first chirped mirror was fabricated with an electron-beam evaporation technology in spite of its low accuracy of the layer thickness control. An improvement in the control of layer thicknesses becomes possible due to use of magnetron-sputtering and ion-beam-sputtering processes [29–34] for the layer deposition. Both processes provide extremely high density of coating layers and very stable deposition rates. Sputtered layers have excellent homogeneity and demonstrate high stability to environment conditions that is not the case for layers produced by evaporation processes. All mentioned features of the sputtering process make possible the employment of a time control during mirror deposition. With proper calibration, the time control provides an ultimate accuracy of layer thicknesses [44]. A simultaneous improvement of designing algorithms [36–39] and manufacturing capabilities [44,45] allows one to compress a pulse down to 3-fs [46].

Nevertheless quite often designs for precise phase control are impossible to be manufactured due to their extreme sensitivity to the deposition errors. Therefore, a further improvement of designs with respect to the sensitivity to deposition errors is required. One of possible approaches consists in attempts to decrease the sensitivity of existing designs by including sensitivity of the design into formulation of the design problem [47–49]. It can be performed by adding special penalty function [47] describing sensitivity of the coating to manufacturing errors. Adjusting the weight of this penalty function, it is possible to decrease the sensitivity of the design to a certain level. Alternatively, the design problem can be reformulated as a minimax problem and solved by a semi-heuristic method [48,49]. The main disadvantage of the mentioned approaches consists in necessity to have a good starting design. Therefore, these methods have all limitations inherent to classical refinement methods, in particular, convergence to some local minimum, not necessary providing a solution with satisfactory spectral characteristics.

In this paper, we apply a new robust synthesis method [50,51] to the design of dispersive dielectric mirrors. The advantage of this method consists in the ability to design a dispersive mirror with a low sensitivity to manufacturing errors without using any specific starting design. As a result, we designed and manufactured dispersive mirrors that were impossible to produce with acceptable performance by using conventional needle optimization technique [36, 38,39]. Efficiency of the proposed robust synthesis method is demonstrated practically. For this purpose we fabricated two different dispersive mirrors, one designed with conventional approached and another designed with the new robust synthesis method. Comparison of obtained spectral characteristics demonstrates significantly lower sensitivity of the robust dispersive mirror to the manufacturing errors and, as the result, much better reproducibility of target spectral characteristics.

2. Robust synthesis (theory)

The proposed method of the robust synthesis can be considered as a generalization of very efficient needle optimization and gradual evolution techniques [35–39]. It is based on a simultaneous optimization of spectral characteristics of multiple designs located in a small neighborhood of a main or pivotal design. We call these additional designs a design cloud. The method does not use results of the perturbation theory for taking into account thickness variations and therefore is more suitable for the most complicated design problems including design of dispersive mirrors.

We designate layer thicknesses of a pivotal design as $d_j$, $j=1,\dots ,N$, where N is a number of design layers. Layer thicknesses of the j-th design in the cloud of designs is represented as

A concept of the merit function characterizing a proximity of theoretical spectral characteristics to target spectral characteristics can be generalized for the design cloud specified by Eq. (1). For the design $X_j$ with layer thicknesses $d_i^{(j)}$ we consider the corresponding partial merit function ${\Phi }_j$:

Here partial merit function ${\Phi }_0$ corresponds to the pivotal design that is also included to averaging operation.

For the merit function Φ introduced by Eq. (4) it is not difficult to derive formulas for analytical gradient, Hesse matrix [37], and perturbation function [36] necessary for efficient implementation of refinement, needle optimization and gradual evolution procedures. Therefore, for any fixed set of thickness perturbations ${\delta }_{ij}$efficient and well-proven synthesis method [38,39] can be applied.

There are several complications connected with the described approach. First, the sum $d_i+{\delta }_{ij}$in Eq. (1) may become negative for a thin layer of a pivotal design. In this case, corresponding thickness of the perturbed design is assumed to be zero. Second, the concept of a needle layer should be reconsidered, since now we are working with a cloud of designs. In our approach we consider a needle layer for each design in the cloud, and introduce the perturbation function as a sum of perturbation functions of each design. This approach corresponds to the definition of the merit function Φ, expressed as a sum of squares of particular merit functions. Third, a special strategy for regenerating of the cloud of designs should be applied. If the cloud is regenerated at every iteration of the refinement algorithm, then this degrade the convergence of the refinement procedure. Computational experiments show that the cloud should be regenerated from time to time for preventing adaptation of the synthesis procedure to a specific shape of the cloud.

The proposed robust synthesis method is not a deterministic one. Therefore it is difficult to detect the condition when the design is not improved further, since a regeneration of the cloud changes its shape and as a consequence changes the value of the merit function and a trajectory of the optimization procedure. In the current version, termination of the computational procedure is performed manually, when the design performance is not improved in the course of gradual evolution iterations for a long time.

Computational time of the described robust synthesis technique is proportional to the number of designs in the cloud M. For moderate-level design problems (100–300 target points L and 40–60 design layers) typical computational time is from 10 to 40 minutes even with large enough number of designs in a cloud (M = 200) (3GHz dual-core Intel CPU). The described technique is well suited for parallelization at multi-core and multi-processor high-performance computing systems. Therefore, it has a high potential for solving even the most complicated design problems.

3. Design of the robust high dispersive mirrors

High-dispersive mirrors (HDM) are able to compensate tens of thousands femtoseconds squared with just a few percents of total losses [34] in pulse compressors with 50 and more bounces. In this paper, we present HDM designs, that are able to compensate GDD depicted in Fig. 1 with magenta crosses. The level of GDD at 800 nm corresponds to −300 fs². HDM with these GDD specifications is able to compensate a dispersion of micro objective lenses consisting of different glass types in the wavelength region 690−890 nm.

 

Fig. 1 Reflectance (black curve), target reflectances (red crosses), GDD (blue curve), target GDD (magenta crosses) and a corridor of GDD errors (grey area) for the conventional design.

Download Full Size | PDF

We consider B260 glass as a substrate, tantalum pentoxide (Ta₂O₅) and silicon dioxide (SiO₂) as layer materials. Refractive indices of the glass substrate and both layer materials are specified by Cauchy formula $n(\lambda )=n_{\infty }+A/{\lambda }^2+B/{\lambda }^4$ with coefficients presented in Table 1 .

Table 1. Cauchy formula coefficients for the substrate and layer materials (wavelength in the Cauchy formula should be expressed in microns), and refractive index values at 0.8µm.

View Table

Using gradual evolution technique [39] and OptiLayer software [52], we obtained a design with GDD and reflectance depicted in Fig. 1. Of course, this design is not optimized with respect to sensitivity to layer thickness errors and is referred below as a conventional design.

The same target values ${\widehat{R}}_l$ and $G\widehat{D}{D}_l$ (Eq. (3)) were used with the robust design synthesis approach. The number of designs in the cloud was 50. In this case we considered only absolute errors (relative errors were excluded, ${\sigma }_{rel}=0$) and selected the size of the cloud ${\sigma }_{abs}=0.5$nm. According to our experience, this number is sufficient to reduce sensitivity of the design to layer thickness errors. Increasing of the number of designs in the cloud by factor of two does not bring any significant improvement in this certain application. With the robust synthesis approach, we obtained the design with reflectance and GDD depicted in Fig. 2 . Both designs have comparable level of GDD oscillations and comparable level of reflectances in the wavelength region 690-890 nm.

 

Fig. 2 Reflectance (black curve), target reflectances (red crosses), GDD (blue curve), target GDD (magenta crosses) and a corridor of GDD errors (grey area) for the robust design.

Download Full Size | PDF

To estimate the sensitivity of obtained designs to the layer thickness errors we scrutinized our HDM designs by means of computational error analysis. Layer thicknesses of both designs were randomly varied according to Eqs. (1)-(2) with ${\sigma }_{abs}=0.5$nm and ${\sigma }_{rel}=0$. For each perturbed design we computed the GDD and performed a statistical analysis of the obtained GDD dependencies. The grey area confined by green curves represents a corridor of errors that encloses GDD values with a probability of 68.3%. Comparing both designs, we observed significant reduction of the sensitivity to layer thickness errors for the robust design (Fig. 2) with respect to the conventional one (Fig. 1).

Why significant improving of the design stability is possible? To answer this question we analyzed the design structures of the conventional (Fig. 3a ) and of the robust (Fig. 3b) designs. Delays in a multilayer stack are mainly controlled by two effects: i) penetration depth effect utilized in chirp mirrors [17] and ii) resonance effect also known as Gires-Tournois interferometer [21,22]. Modern HDM designs uses a combination of the both effects [31, 34].

 

Fig. 3 Layer thicknesses of the 92 layer conventional HDM design (a) and layer thicknesses of the 74 layer robust HDM design (b).

Download Full Size | PDF

HDM designs that mostly use resonance effect are much more sensitive to layer thickness errors due to high sensitivity of resonance conditions to thickness variations of layers forming and surrounding resonance cavities. One can notice that the number of cavities in the case of the robust design (Fig. 3b) is less than for conventional design (Fig. 3a). This fact can also be confirmed by analyzing the distribution of an intensity of the electric field inside the multilayer structure as shown in Fig. 4 and Fig. 5 . In these figures zero coating coordinate corresponds to the boundary between the substrate and the coating and maximum coating coordinate corresponds to the boundary between the coating and the incident medium. Resonances are accompanied by significant enhancement of the electric field intensity. In the case of conventional design, one can see much higher intensities of the electric field (Fig. 4) in more locations at different wavelengths comparing to the robust design (Fig. 5). Therefore, for the robust design, resonances are less pronounced and the number of cavities is reduced. As the result, the design becomes more stable with respect to layer thickness errors. Additional advantage of this fact is significantly better stability of robust designs to high intensities of laser pulses [51].

 

Fig. 4 The penetration of electric field intensity versus coating coordinate of the conventional HDM.

Download Full Size | PDF

 

Fig. 5 The penetration of electric field intensity versus coating coordinate of the robust HDM.

Download Full Size | PDF

4. Realization and comparison

In order to demonstrate experimentally better manufacturability of the robust design, we deposited both designs (Fig. 3), obtained in the previous section. The accuracy of the layer thicknesses better than 0.5 nm can be reached with state-of-the-art magnetron sputtering machine (Helios, Leybold Optics GmbH, Alzenau, Germany), which is currently one of the most precise devices available for the production of dispersive optics [29–34].

Figure 6 represents theoretical GDD of the conventional design with corridor of errors (see also Fig. 1). GDD values of manufactured coatings were measured with white-light interferometer (WLI) [53]. Since WLI measurements are very sensitive to mechanical noises of the environment that are highly unpredictable, we performed averaging of 3-5 measurements carried out at different days. Obtained GDD values are indicated by the red crosses. One can notice big deviations of measured GDD values from theoretical ones. This difference between theoretical and measured GDD values can be explained by high sensitivity of the conventional design to layer thickness errors. Even small layer thickness errors about 0.5 nm are increasing GDD oscillations in the order of magnitude comparing to the level of oscillations of the theoretical design. Many GDD values are outside 68.3% corridor of errors because not all disturbing factors presenting in a real deposition process are included into error analysis procedure.

 

Fig. 6 The theoretical GDD (blue curve), measured GDD (red crosses) and a corridor of GDD errors (grey area) for the conventional design.

Download Full Size | PDF

The robust design Fig. 3b demonstrated much better manufacturability. The measured GDD of the robust design is shown in Fig. 7 in comparison with theoretical GDD and the corridor of GDD errors. For the robust design, GDD measurements are in a good agreement with the theoretical GDD. Figure 7 reveals that the actual layer thickness accuracy during this deposition may be estimated as better than 0.5 nm, since most of GDD measurements belongs to 68.3% corridor of errors. High thickness accuracy during deposition allows us to fabricate robust HDM designs in a highly reproducible and reliable way. We confirmed this with an independent second coating run (see orange curve in Fig. 7), which also provided acceptable GDD.

 

Fig. 7 The theoretical GDD (blue curve), measured GDD (first run - red crosses, second run – orange crosses) and a corridor of GDD errors (grey area) for the robust design.

Download Full Size | PDF

5. Conclusions

The novel robust synthesis technique was applied to the design of a high dispersive mirror. The robust design is compared with a conventional design obtained with needle optimization technique for the same problem. The sensitivities of conventional and novel robust designs to layer thickness errors are compared. The robust design demonstrates much lower perturbations of GDD caused by variations of layer thicknesses. We also demonstrate differences in multilayer structures of both designs, in particular, less pronounced resonance effects in the robust design.

We performed experimental depositions of both designs with magnetron-sputtering in order to show the efficiency of the robust design technique in practice. GDD measurements of fabricated mirrors show that the robust design has superior performance in comparison to conventional design. In particular, the level of GDD oscillations is an order of magnitude lower for the robust design comparing to the conventional design. Also the robust design demonstrates a good manufacturability, providing very similar GDD in repeated deposition process. As an additional benefit, the robust design provides lower intensities of electric field inside the coating and therefore higher laser damage threshold values can be expected.