1. Introduction

The effects of photo-induced temperature are ubiquitous in many components and optical systems. These effects may be extremely detrimental, and they affect in particular the appearance of mechanical stress, planarity modification, drift or deterioration of optical properties, accelerated ageing, gradients of physical properties, etc. However, despite the ubiquity of these photo-induced processes, which take place well before the damage phenomena, their monitoring remains a difficult task and has not been routinely mastered to a sufficient extent; the spatiotemporal temperature distributions actually result from many parameters, including the formula of the filter, the pulse duration of the laser and its repetition frequency, the average wavelength, the polarization and incidence, the thermal parameters of thin-film materials, and lastly the nature of the substrate, and the materials and technologies used. Furthermore, these spatiotemporal temperature variations are not easy to measure with precision.

In this context, it is crucial to be able to rely on an accurate prediction of these photo-induced phenomena in a steady-state or transient regime. Several models have thus been developed for thin-film stacks, most of which are based on solving the heat equation with the aid of numerical methods such as the finite-element method [1–6]. There are also analytical models, insofar as they do not resort to numerical methods, although many of them assume a partial distribution of the heat source (volume density of absorption) [7,8] or a continuous or infinitely short laser regime [9,10].

The purpose of this work is to present an accurate model (within the scope of Fourier’s law) of the photo-induced heating in multilayer optical interference filters subjected to pulsed or continuous illumination. This modelling is essentially analytical insofar as it does not resort to numerical methods, but is primarily based on a double Fourier transform (in space and time). Furthermore, the solution method relies on the optical/thermal analogies which we have demonstrated in the second Fourier plane, relating to the equations of heat diffusion and optical propagation (in metallic media) [11]. These analogies allow the problem of heat diffusion in multilayer systems to be treated in exactly the same way as that of the bulk scattering of light by index heterogeneities in the same systems [12], while taking into account the boundary conditions at the interfaces. This means that several optical quantities (effective indices, standing fields, complex admittances, recurrence relations and transfer coefficients, etc.) are extended to the case of heat for direct application of the calculation codes already used for optics. More generally, this type of analytical modelling allows better tackling of a certain number of inverse problems such as that of extracting the thermal parameters of isolated thin films or ones inserted into an interference system. It also allows understanding of the effect of the various input data, namely the laser parameters (energy, pulse duration, illuminated surface) and the structural parameters (formula of the multilayer system, adsorption, diffusivity and thermal conductivity).

Note again that the scope of this work is the Fourier’s law since we consider pulse durations greater than the ps regime, while the thicknesses of the thin films are of the order of a visible wavelength. Therefore, we admit that the temperature is a fully defined quantity at each time and location in the structure, which is referred to as the local thermodynamic equilibrium condition, as is the case in the literature [1–6,8]. Note however that the layer thicknesses are not so far from the phonon mean free path in solid dielectrics (45 nm in Silicon at 300 K [13]). It is known that for materials with thicknesses less than the phonon mean free path (or for a Knudsen number above 1), the thermal transport becomes ballistic rather than diffusive and the notion of temperature can no longer be considered [13,14]. In other words, although the Fourier’s law is adapted to our problem, one must keep in mind that the border with another domain of Physics, where classical heat diffusion cannot be considered, is not far away.

This context of validity being specified, we have carried out our modelling and its formalism and numerical results are presented. It makes it possible to predict the width and the depth of the heat diffusion, quantities which are crucial to the concept of thermal resolution, in the pulse (single or repeating pulses) or steady-state regime. Results are presented for various types of components commonly used in the field of optical thin films (substrates, single layers, mirrors and filters, etc.) and for different types of pulse regime. The spatial geometry considered for these calculations is in 3 dimensions (x, y and z), which makes it possible to include the case of a heat source due to illumination at oblique incidence. The model makes it possible to describe the presence of “thermal fringes” in response to the optical fringes in a transverse plane (z = constant) which would result from the superposition of 2 incident beams. Lastly, it should be emphasized that these spatiotemporal temperature calculations are vital because they constitute a first step in addressing the study of thermal radiation in multilayer components ongoing within our laboratory.

This work is presented in the following way. Section 2 is devoted to the temperature formalism in a multilayer system, based on a double Fourier transform, which makes it possible to introduce the tools of thermal admittance and effective index. Accurate calculation of the heat source in the time-domain space and in the 2 Fourier planes is not trivial, and forms the subject of Section 3. In order to reduce the numerical calculation time, the particular solution is calculated analytically in the Supplement 1. The numerical results are then presented in Section 4 for metallic and dielectric monolayers subjected to pulsed illumination (from ms to ps). The temporal and spatial responses are analyzed in detail in order to quantify the response times, the thermal diffusion widths and depths, and the maximum temperature elevation as a function of the pulse duration and the width of the illumination spot. Section 5 is given over to 2 types of multilayers (mirror and narrow-band filter); the way in which the temperature distribution follows the standing electromagnetic field with short pulses is shown and is in accordance with the literature [2,5,6]. The temperature spectra are also given, i.e. the variations of the temperature with the illumination wavelength, for different pulse durations. All results are used to compare the thermal and electrical damage thresholds as a function of the illumination parameters and the imaginary indices of the thin-film materials. Lastly, the repeating regime is considered in Section 6, in which the effect of the repetition rate of the pulse source on the temperature is quantified. The concept of a thermal fringe is studied in Section 7, before the conclusion presented in Section 8.

2. Modelling of the temperature in multilayer systems in a spatiotemporal regime

2.1 Heat equation

The materials are assumed to be linear, homogeneous and isotropic from the thermal and optical viewpoints. The variations of the physical parameters with temperature are neglected. The effects of convection are not considered. The multilayer component is schematized in Fig. 1, and contains p thin films. The pair $({{e_i},{n_i}} )$ denotes the thickness and the complex optical index of layer i, while the pair $({{a_i},{b_i}} )$ denotes the diffusivity and conductivity of layer i. The medium with an index ${n_0}$, from which the light responsible for the photoinduced thermal effects comes, will be referred to as the superstrate. The substrate has an index ${n_s}$.

 

Fig. 1. Geometry of a Multilayer System (see text).

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One classical way to determine the laser-induced temperature is to solve the heat equation [1–6] which is written, in the sense of functions in the spatiotemporal regime, and in each medium (i) as:

This equation is usually solved thanks to numerical techniques and a full description of the model can be found in [1]. As mentioned earlier in the introduction, the choice here is to propose an analytical approach based on optical/thermal analogies, that can help to better understand the impact of the input data, and to provide an easier investigation of inverse problems.

The first step of the modelling is to consider Eq. (1) in the first Fourier Plane:

In what follows, the temporal angular frequency $\omega = 2\pi f$ may also be used. The same FT in time will be used for the optical calculations, so that the imaginary parts of the complex indices of the materials will generally be positive.

In the second Fourier plane, Eq. (2) is written as:

The same FT in space will be used in optics, and will lead to positive optical paths.

It will be noted at this stage that the formulation given by (4) is, from a mathematical viewpoint, the same whether governing diffusion of heat or optical propagation (apart from the coefficient $\alpha $, which will be different in optics) in the second Fourier plane. This will allow us in Section 2.5 to use the optical/thermal analogy [11] in order to solve this equation pragmatically, that is to say by using the models and calculation codes already employed for optics.

2.2 Homogeneous and particular solutions

The solution of (4) is given by the sum of the homogeneous solution $\hat{T}_i^h$ (with no second member) and a particular solution $\hat{T}_i^g$, i.e.:

The particular solution is given by:

Let us explain (9):

It will be noted here that each integral $\hat{T}_i^g\; $ relates to the thickness of a layer ($0 < z^{\prime} < {e_i}$). In the case of extreme media which are absorbent (i.e. also contain a heat source), it will also be necessary to integrate over the ranges ($- \infty < z^{\prime} < 0$) and ($\mathop \sum \limits_{i = 0}^p {e_i} < z^{\prime} < \infty $).

As regards the homogeneous solution, this is conventionally written as:

It is these latter constants $T_{i - 1}^ \pm $ which constitute the unknowns of the problem, and which are determined using the boundary conditions at the interfaces of the multilayer. The problem will be made easier by using the optical/thermal analogy [11], which will make it possible to introduce equally well the formalism of the optical or thermal admittances.

2.3 Introduction of the heat flux

The heat flux density vector is defined as:

It will be noted that, in comparison with Eq. (4) for the temperature, only the second member is different because of the spatial derivatives. Consequently, each component of the flux is expressed in the form of a homogeneous solution $\hat{q}_{u,i}^h$ and a particular solution $\hat{q}_{u,i}^g$, with expressions similar to those given by (11) and (12), i.e.:

As before, the unknowns of the problem are the constants $q_{u,i}^ \pm $, the particular solution being given by (16).

2.4 Boundary conditions

At this stage, we have analytical expressions of the temperature and the heat flux, which satisfy Eqs. (4) and (14). Since these equations are linear with respect to the source term ${\hat{S}_i}$, it is possible to solve the problem for each source term taken separately and to add the quantities obtained (temperature and flux) for all the source terms.

Let us consider the case in which there is only the volume source term ${\hat{S}_i}$. This is supported in z by the layer (i) bounded by the interfaces (i-1) and (i). Let us denote by ${\hat{T}_{ij}}$ the temperature created by this source ${\hat{S}_i}$ at the interface j. Let us write the quantities on either side of the volume source, from now on omitting variables $\vec{\nu }$ and f. At boundary (i-1), we have:

We now need to use the boundary conditions at each interface for the temperature ${\hat{T}_i}(z )\; $ and the normal flux ${\hat{q}_{z,i}}(z )$, two quantities which are known to be continuous here since the source does not have a singularity (in the sense of distributions). In what follows, we will denote by $X_i^{\prime}$ and ${X_i}$ the quantities at boundary (i) which are taken in media (i-1) and (i), respectively. The boundary conditions are thus written:

It should be noted that no thermal flux is imposed at the top (i = 0) or back (i = p) surfaces, since convection processes are not considered. Note also that interface absorption, which would modify these boundary conditions, is not considered. Indeed high precision coatings are today manufactured using energetic techniques (Double Ion Beam Sputtering, Magnetron Sputtering and others…) that result in dense layers with smooth interfaces that can be checked with SEM techniques on film cross-sections. Furthermore the polishing and cleaning techniques, which could be responsible for absorption at the substrate interface, are also well mastered. Another point is that the values of the optical field at the substrate surfaces is very low at the design wavelength of mirrors and Fabry-Perot filters, so that interface absorption would again be reduced at these surfaces. And finally it should be stressed that interface absorption can be directly taken into account with our model, since it is an approximation of a bulk absorption within a very thin layer.

2.5 Introduction of the thermal admittances with optical/thermal analogy

The solution of the problem will involve introducing the formalism of the complex admittances in multilayer systems, which is moreover widely used in optics for interference filters [15]. To this end, we will use the work developed in [11] relating to optical/thermal analogies, which will allow us to solve the problem by a method similar to the one we have developed for bulk scattering in optical filters [12].

In order to reduce the number of parameters, the fluxes are related to the temperatures with the aid of the thermal admittances [11]:

The admittances $Y_{k - 1}^\mathrm{^{\prime}}$ and ${Y_k}$ are defined in the regions free of sources, i.e. here outside layer (i). They will thus be calculated by recurrence relations which respectively involve only the half-spaces $z < {z_{i - 1}}$ (for $Y_{i - 1}^\mathrm{^{\prime}}$) and $z > {z_i}\textrm{}$(for ${Y_{i + 1}}$). These relations are written as:

To recall, relations (25) are initialized on the basis of the admittance vectors in the extreme media ($k\; = \; 0$) and ($k\; = \; s\; = \; p + 1$) where they are identified with the effective thermal indices ${m_k}$, which are equal to $- {m_0}$ for $Y_0^\mathrm{^{\prime}}$ and ${m_{p + 1}} = {m_s}$ for ${Y_p}$, with:

It will also be noted that the same Relation (25) may be used to obtain the admittance continuously (along z) in the half-stacks $z < {z_{i - 1}}$ and $z > {z_i}$.

At this stage, the admittance is therefore known and the system of equations (18-21) may be rewritten at boundary (i-1) as:

2.6 Resolution

The homogeneous components $\hat{T}_i^h$ and $\hat{q}_{z,i}^h\textrm{}$ are not continuous outside layer (i), but we can nevertheless relate their values at the interfaces (i-1) and (i) as:

Combining (28-31) with (32-33) leads to:

After resolution, the temperatures $\hat{T}_{i,i - 1}^{\prime}$ and ${\hat{T}_{i,i}}$ at the two ends of the layer (i) are written as:

It may also be noted that the source terms involved in (37-38) are related as:

This gives:

Thus, with knowledge of the source terms given by (41-42), the temperatures $\textrm{}\hat{T}_{i,i - 1}^{\prime}$ and ${\hat{T}_{i,i\textrm{}}}$ on each side of the layer (i) containing the source may be calculated with the aid of (36). It is then known to make this temperature diffuse in the half-spaces $\textrm{}z < {z_{i - 1}}$ et $z > {z_i}$ with the aid of the transfer coefficients as:

These same relations make it possible to have the temperature at each height z in the source-free half-spaces.

2.7 Calculation the temperature in the medium (i) with a source

In the previous section, the homogeneous components $\hat{T}_i^h$ and $\hat{q}_{z,i}^h\; $ were eliminated in order to calculate the temperature outside the layer (i) where the source is located. However, knowledge of them is also essential for calculating the temperature in this layer. To this end, Relations (28) and (30) may be reused as follows:

Since the homogeneous temperature at the interfaces is known, it may be expressed in the volume of layer (i) as:

Finally, the temperatures are entirely determined in the entire volume of the component, except for knowledge of the source terms which are involved in the quantities $\hat{T}_i^g(0 )$ and $\hat{T}_i^g({z = {e_i}} )$. To recall, these sources are the result of the convolution product of the heat source ${\hat{S}_i}$ with the Green’s function G, according to Relations (9-11).

3. Calculation of the heat source

It remains to calculate the heat source ${S_i}$, which is related to the absorption ${\mathrm{{\cal A}}_i}\textrm{}$ caused by the illumination beam in layer (i). In most practical cases, it is possible to make a few assumptions about the illumination beam that allow the source to be written in a simple way. These hypotheses, which are described in detail in Supplement 1, are quasi-mono-chromaticity and weak divergence. Furthermore, if the temporal and spatial envelopes of the beam are described by gaussian functions, the heat source in both Fourier plane can be written as (see Supplement 1):

It is this relation (50) which will be employed to calculate the source terms $\hat{T}_i^g(0 )$ and $\textrm{}\hat{T}_i^g({z = {e_i}} )$. The surface density (or fluence) of incident electromagnetic energy (close to $W/{L^2}$) or the average power $\textrm{W}/({\mathrm{\tau }{\textrm{L}^2}} )$ related to this density, are seen to appear in (50). At this stage, the modelling therefore makes it possible to describe the thermal diffusion volume ${v_{th}} \approx L_x^{th}L_y^{th}L_z^{th}$, where ${S_{th}} \approx L_x^{th}L_y^{th}$ is close to the diffusion surface and $L_z^{th}$ is the diffusion depth. These quantities are essential for analyzing the 3D thermal resolution as a function of the duration of the pulses.

4. Numerical results for a single layer subjected to different illumination regimes

In this section, the numerical simulations of the photo-induced temperature of a single thin film are presented. The temperature field $T({x,y,z,t} )$ has 4 variables, which are the time t and the three spatial variables (x,y,z). The transverse variable $\vec{r} = ({x,y} )$ lies in a plane parallel to the interfaces, while $\vec{z}$ represents the perpendicular direction. Beyond the specific temperature increases of the components and the illumination conditions, the numerical calculations make it possible to quantify the concepts of temporal, transverse and depth resolution of the temperature in these thin films.

4.1 Study of a metallic layer

We will start by analyzing the case of a metal layer, even though metals are seldom used in precision optical filters because of their high ohmic losses. This makes it possible on the one hand to establish the orders of magnitude of the temperature elevations $\mathrm{\Delta }T$, particularly in comparison with the results which will be presented for dielectric layers in the following sections. Furthermore, these calculations for metal layers allow comparison with the results obtained from the literature. The component is therefore in this case an aluminum thin film with a thickness of 100 nm deposited on a transparent BK7 substrate. The thermal and optical parameters of aluminum and BK7 are given in Table 1. The illumination conditions are indicated in Table 2. We are therefore working in TE polarization with normal incidence (${\theta _0} = 0^\circ $), at the central wavelength λ = 1064 nm. The incident energy is $W = 1\; mJ$, a value which will be used for all the numerical calculations in this article (except where otherwise indicated). The radius at 1/e² of the illumination spot is ${L_x} = {L_y} = L = 100\mu m$. It will be noted that from an optical viewpoint, the Al layer is opaque to the incident radiation. Its complex index is taken to be equal to $1.38 + 10.24j$ [16].

Table 1. Optical and Thermal Parameters Used for the Calculation of the Laser-Induced Temperature

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Table 2. Source Parameters for the 3 Different Regimes

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The temporal variations of $\mathrm{\Delta }T$ which were calculated on the upper interface (in contact with air) of the metal layer are plotted in Fig. 2 for a single-pulse mode and for three pulse regimes (ps, ns and ms). Taking into account the high absorption of the metal, the temperature elevation $\mathrm{\Delta }T$ for short pulse durations would be very large (70000 K and 11000 K in ps regime and ns regime, respectively) for this energy of 1 mJ, and the melting temperature of Al (933 K) would be very greatly exceeded. This will not be the case for dielectrics. For long pulse durations (ms), conversely, the variation $\mathrm{\Delta }T$ is much less (40 K), which reminds us of the importance of the pulse duration. It may also be noted that the decay time (normalized to the duration of the incident pulse) is shorter with long pulse durations. It will be gathered here that in a ps or ns regime, the melting temperature of the metal would be reached with an incident energy reduced by about 2 decades (ps regime) or one decade (ns regime), i.e. with respective energies of the order of 0.01 mJ (ps regime) and 0.1 mJ (ns regime). These values are characteristic of the thermal damage thresholds for Al.

 

Fig. 2. Temporal evolution of the temperature elevation ΔT calculated at the upper interface of an aluminum monolayer deposited on a transparent BK7 substrate and subjected to ps (left), ns (center) et ms (right) illumination. Dotted red: temporal evolution of the laser pulse (assumed to be Gaussian).

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4.2 Temporal response of a dielectric layer in ps, ns and ms regime

The component is now a single layer of SiO₂ deposited on a transparent glass substrate (BK7). The optical thickness of the layer is half the central wavelength of 1064 nm. The optical and thermal parameters of the materials used are given in Table 1. In general, it is assumed by default for the numerical calculation that the imaginary indices of the thin-film dielectric materials are of the order of $n^{\prime\prime} = {10^{ - 5}}$. This value does actually correspond to the order of magnitude measured for high-quality dielectric thin films [17]. It should, however, be indicated that it may be less (a few ${10^{ - 6}}$), as for example in the case of mirrors manufactured for the detection of gravitational waves [18]. This value may also be high (a few ${10^{ - 4}}$) in the case of components for MIR or near-UV, or for components with a large number of layers (of the order of one hundred), which require a significant deposition time.

As before, the temporal response of $\mathrm{\Delta }T$ is plotted as a function of the duration of the laser pulse (ps, ns and ms) in Fig. 3. This elevation of T° is calculated at the position where it is maximum, i.e. in the middle of the thin-film and at the center of the beam. The illumination conditions are still those of Table 2. In contrast to the previous case of the Al layer, the ΔT values are much lower and far below the melting T° of the material (2000K for SiO₂). It is close to 2.8 K in the ps and ns regimes, and 0.04 K in the ms regime. It may again be noted that ΔT follows overall the leading edge of the laser pulse, but with a much longer decay time. The latter is shorter with long pulse duration if it is normalized to the duration of the incident pulse. It will be noted here that, since the temperature is proportional to the incident energy and the imaginary index (at the first order) of the material (see section 4.5), these results make it possible to extrapolate the data of Fig. 3 to an arbitrary energy or absorption. Thus, for an imaginary index $n^{\prime\prime} \approx {10^{ - 4}}$ and for an energy W = 10 mJ in the ps or ns regime, there would be a ΔT of 280 K, which begins to be detrimental, particularly in terms of mechanical stresses.

 

Fig. 3. Temporal evolution of the temperature calculated in the middle of a weakly absorbent half-wave monolayer of SiO₂ deposited on a transparent BK7 substrate and subjected to ps (left), ns (center) et ms (right) illumination. Dotted red: temporal evolution of the laser pulse (assumed to be Gaussian).

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4.3 Response in the thickness of the component

The elevation ΔT of the temperature in the thickness of the monolayer is now studied. To this end, the temperature in the three regimes studied (ps, ns, ms) is plotted in Fig. 4 (left) as a function of the height z in the layer. In these three regimes, the temperature distributions are plotted at the instants t_max which the temperature reaches a maximum in Fig. 3.

 

Fig. 4. Distribution of the temperature ΔT (left) in the thickness of the layer, calculated for 3 different regimes (ps, ns, ms). Right: distribution of the square of the electric field normalized by the incident field.

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This distribution may be compared in Fig. 4 (right) with that of the normalized electric field inside the monolayer. As previously shown in [2,5,6], it may be noted that for short duration pulses (ps and ns regimes), the temperature profile within the layer is similar to that of the stationary electric field, which is not the case in the ms regime where it is quasi-constant in the thin film. This result will be easier to see with multilayers (see section 5.1). It applies to all the components and, except for the dispersion of the imaginary index, controls the variation of ΔT as a function of the central illumination wavelength. It is due overall to the thermal diffusion length, defined as [19]:

4.4 Transverse distribution of the temperature

It remains to study the spatial distribution of the photoinduced temperature in the (x,y) plane. We still assume a normal illumination incidence, which ensures symmetry of revolution of the temperature about the z axis. This makes it possible to observe jointly in Fig. 5 (top) the value ΔT as a function of time and the transverse abscissa x for the 2 regimes (ns and ms). The thermal radius ${R_{th}}$, defined at 1/e², of the maximum temperature may be deduced therefrom as a function of time and for the various regimes. In order to visualize the modification of this radius ${R_{th}}$ better, a narrower illumination spot ($L = 10\mu m$) was considered, which creates a higher elevation ΔT (280 K in the ns regime). For greater clarity, the maximum elevation ΔT is also plotted in Fig. 5 (bottom) as a function of the transverse axis x for the two pulse regimes studied. In the ns (and respectively ms) regime, this maximum elevation is reached 1.7 ns (and respectively 0.1 ms) after the maximum of the incident pulse. It may be noted that in the ns regime, the component is almost not heated outside the surface illuminated by the incident beam, which is represented by the dotted pink curve. The thermal surface is therefore identical to the illumination surface (${R_{th}} \approx L$). In the ms regime, conversely, the heat starts to propagate outside the illuminated surface (${R_{th}} > L$). More precisely, the thermal radius is ${R_{th}} = 10\; \mu m$ in the ns regime and ${R_{th}} = 19\; \mu m$ in the ms regime. It will, however, be noted that the thermal radius ${R_{th}}\; $ cannot be lower than the radius of the incident laser spot; for that reason the transverse diffusion is less marked than the depth diffusion.

 

Fig. 5. Top: temperature elevation of a monolayer component as a function of time and the transverse direction x in the ns (left) and ms (right) regimes. Bottom: transverse distribution of the maximum temperature elevation reached in the component with the ns regime in blue and the ms regime in red. The temperature values are normalized by their maximum (at x = 0). Dotted pink, the Gaussian spatial envelope of the incident laser beam.

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4.5 General evolution with the incident energy, the duration of the pulse, the illuminated surface and the imaginary index

From a practical viewpoint, information which is essential for the user relates to the maximum temperature elevation obtained in the layer as a function of the energy, the pulse duration, the illuminated surface and the imaginary index of the material. It is in fact these data which make it possible to grasp the importance of the photo-induced thermal phenomena which we are attempting to summarize and simplify here. The ΔT evolution as a function of the laser energy is clear. Specifically, according to Eq. (50), the heat source is proportional to the incident energy so that ΔT is proportional to this energy. The influence of the other parameters is more subtle, however, as illustrated in Fig. 6.

 

Fig. 6. Top: maximum elevation of the temperature in a monolayer as a function of the duration of the laser pulse and the imaginary part of the optical index of the thin film, $L = 100\; \mu m$. Bottom left: maximum elevation of the temperature plotted as a function of the imaginary index for 2 different pulse durations (ns regime in blue, ms regime in red), with $L = 100\; \mu m$. Bottom right: maximum elevation of the temperature plotted as a function of the illumination surface for 2 different pulse durations (ns regime in blue, ms regime in red) with constant absorption $({n^{\prime\prime} = {{10}^{ - 5}}} )$. Constant energy W = 1 mJ

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At the top left of this figure, ΔT has been plotted in the layer as a function of the imaginary index and the duration of the pulse for a radius $L = 100\mu m$ of the illumination spot, still with W = 1 mJ. For more clarity, $\mathrm{\Delta }T$ has also been plotted in Fig. 6 (top right) as a function of the duration of the pulse for two fixed values of the imaginary index ($n^{\prime\prime} = {10^{ - 5}}\textrm{and}{10^{ - 4}}$) corresponding to a dielectric material. Two zones can be seen. For short pulses ($\tau < \textrm{}10\textrm{}ns$), the temperature elevation is quasi-constant, so that photo-induced thermal effects are similar for these short pulses. Conversely, for longer pulses $\mathrm{\Delta }T$ decreases and gradually approaches a $1/\tau $ decay. This can be related to the similarity between long pulses and the continuous regime (see Eq. (53) of Sec.  6.1 ) where the maximum temperature is proportional to the illumination power.

Furthermore, it is known that for the low values of $\textrm{}n^{\prime\prime}$ the maximum temperature is proportional to the imaginary index of the material. This result is confirmed by Fig. 6 (bottom left), where the maximum temperature elevation ΔT is plotted as a function of the imaginary part $n^{\prime\prime}$ of the optical index for 2 different pulse regimes with a constant illumination surface ($L = 100\; \mu m$). These results are clearly calculated in the absence of thermal damage. The proportionality is quite valid over the range $n^{\prime\prime} < {10^{ - 3}}$. The linearity is then lost because of the metallic behavior of the material (strong reflection) and a maximum appears around $n^{\prime\prime} \approx 0.9$ for both ns and ms regimes.

Lastly, the effect of the illumination surface is illustrated at the bottom right of Fig. 6, where the quantity ΔT is plotted as a function of the spot radius L. The results show that the maximum temperature is inversely proportional to the illumination surface in the ns regime, but that this is no longer the case in the ms regime, at least below $L = 100\; \mu m$ where the slope of the curve is modified. It will be gathered overall from Fig. 6 that, with an energy of 1 mJ, we remain far from the degradation temperature of the component so long as the imaginary index does not exceed ${10^{ - 4}}$ (ps or ns regime) or a few ${10^{ - 2}}$ (ms regime). This result is clearly modified (with a quasi-proportionality) if the energy increases or if the illuminated surface decreases.

To conclude this subsection, we stress on the fact that it is important to verify certain properties by an analytical expression, because off the complexity of the numerical calculation. This is what is done in section 2 of Supplement 1, namely an analytical study of the behavior of the temperature as a function of the pulse duration and the illuminated surface. To do so, the photo-induced temperature is calculated in a homogenous medium, where the total absorption A is distributed over a depth ${L_z}$, which simplified greatly the equations. This allows us to determine a simple expression of the maximum temperature elevation induced by a short pulse (less than 10 ns) and to demonstrate the proportionality in $1/{L^2}$:

5. Numerical results for a multilayer under different illumination regimes

5.1 Case of a mirror and a Fabry-Perot

One major specificity of multilayers resides in the depth distribution of the standing electromagnetic field [15,21]. This field profile varies considerably with the formula of the component (mirror, filter, splitter, polarizer, high-pass, etc.) and depends closely on the illumination conditions (incidence, wavelength, polarization). The same is therefore true for the volume density absorption and for the heat source resulting therefrom. Consequently, depending on the pulse duration, the temperature will be constant in the component (ms regime) or will reproduce the shape of the standing wave in the component (ps and ns cases). In what follows, the temperature elevation ΔT is considered at the instant when it is maximum.

Two components were considered in Fig. 7. The first (top figure) is a quarter-wave mirror centered on 1064 nm, having a formula H(LH)⁴, deposited on a glass substrate (BK7) which is assumed to be non-absorbent. The high-index material is Nb₂O₅ and the low-index material is SiO₂. The optical and thermal properties of the different materials used are summarized in Table 1. The layers are assumed to be weakly absorbent ($n^{\prime\prime} = {10^{ - 5}}$). The standing wave plotted on the right is characteristic of the mirror, with a decrease of the maxima when moving away from the first interface. It may actually be seen in the left-hand figure that, for short pulse durations (ps and ns), ΔT follows this standing wave in the component with an accuracy which increases when the duration of the pulse decreases. For long pulse durations (ms), conversely, ΔT is constant in the volume of the component. This elevation is furthermore of the same order of magnitude as for the single dielectric layer.

 

Fig. 7. Profiles of temperature (left) and of the square of the electric field normalized by the incident field (right) in the thickness of the multi-dielectric component, calculated for 3 different laser regimes (ps, ns, ms). Top: case of the mirror H(LH)⁴; Bottom: case of the Fabry-Perot H(LH)²6LH(LH)².

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The results are not different (Fig. 7 bottom) for the second component, which is a Fabry-Perot composed of the same materials but with a formula H(LH)²6 L(HL)²H, deposited on the same substrate. Conventionally, the standing wave (on the right) is maximum in the cavity of the filter and decreases when approaching the end surfaces. As before, the left-hand figure shows that ΔT follows this standing wave for short pulse durations. This explains a greater elevation ΔT for the Fabry-Perot than for the mirror, since the ratio between the maximum values of these 2 standing fields is 11.32.

Lastly, this part is completed by plotting the temporal response of the 2 multilayers in Fig. 8. As when studying the single layer component, the temporal evolution of the temperature is plotted at the position where it reaches a maximum in the component. Since the materials forming these two filters are dielectric, the temporal response is similar to that observed for the single SiO₂ layer in Fig. 3. In the ms regime, the formula of the filter has almost no effect on the temperature elevation, so that a temporal response very similar to that obtained with a single layer is found.

 

Fig. 8. Temporal response of the 2 multilayers of Fig. 7, calculated at the temperature elevation maximum, for 3 pulse durations (ps, ns and ms). Top: case of the mirror H(LH)⁴; Bottom: case of the Fabry-Perot H(LH)²6 L(HL)²H.

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To conclude this sub-section, it is interesting to consider the calculation time with our model. The calculation is performed with the common following processor: Intel Core i5-8265U CPU @ 1.60GHz. Let us start with the ns regime. For Fig. 7, with a time vector of 501 values ranging from 0 ns to 50ns, the computation time is 4 minutes. In this configuration, the minimum possible number of values for the z-vector has been taken, which is 10 (one point per interface). To go further and plot the temperature within the layers, it is necessary to increase the number of values of the z-vector. With a doubled z-vector (one point per interface + one point per layer) the calculation time is 12 minutes. Thus, to obtain Fig. 8 with 4 points per layer, it is necessary to carry out 4 simulations of 12 minutes, which gives a total calculation time of 48 minutes. In the ms regime, the calculation time is greater because we need to increase the number of points in the spatial frequency vectors (to consider heat diffusion). Thus, it takes 12 mins to get Fig. 8 in the ms regime.

5.2 Multi-dielectric mirror on metal substrate

For reflective components, it may be advantageous to use a metal substrate in order to reduce the temperature (taking into account the high conductivity of the metal), even more so since the substrate may be cooled by convection in contact with a fluid. This advantage will arise for long pulse durations (ms or more) because, with short pulse durations, the temperature elevation remains concentrated in the first layers of the component (Fig. 7) and therefore does not depend on the thermal parameters of the substrate.

It is not, however, possible to consider a mirror of the M₉ type (i.e. with 9 layers) such as that studied in Section 6.1. This is because with a formula having 9 layers, the electric field still remains too large in the substrate and induces consequential absorption by the metal and therefore a large temperature elevation. Thus, with the M₉ mirror of Section 6.1 deposited on a metal substrate, the maximum temperature elevation induced by a ms laser is of the order of 0.5 K, which is much more than the elevation (0.04 K) observed for a dielectric substrate (Fig. 8).

In order for the temperature reduction to be effective, it is therefore necessary to consider dielectric mirrors having a larger number of layers; under these conditions, the electric field is sufficiently low in the substrate for the temperature elevation induced by the latter to be negligible. To this end, a mirror M₂₃ formed by 23 layers of the H(LH)¹¹ type is studied. As before, the layers are quarter-wave at the central wavelength of the incident laser. The properties of the materials are described in Table 1, while the properties of the laser are described in Table 2. In Fig. 9, the temporal temperature response of the mirror M₂₃ deposited on a dielectric substrate (BK7) and on a metal substrate (Al) is plotted for a pulse duration of 1 ms (left) and 10 ms (right). In the millisecond (and respectively 10 ms) regime, the maximum temperature obtained with a metal substrate (Al) is divided by a factor of 3.55 (and respectively 6.37) with respect to that obtained with a dielectric substrate (BK7). The larger factor in the case of a 10 ms pulse duration is explained by the fact that in the ms regime, for mirrors composed of a large number of layers, the temperature is not entirely uniform in the component and is slightly higher in the first layers, which reduces the effect of the substrate.

 

Fig. 9. Temporal response of two identical H(LH)¹¹ mirrors deposited on a dielectric substrate (blue curve) and on a metal substrate (red curve). Left: ms regime; Right: 10 ms regime.

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In conclusion, an (uncooled) metal substrate makes it possible to reduce the photoinduced temperature in reflective components for long pulse durations (> ms), so long as the multi-dielectric mirrors have a sufficiently large number of layers.

5.3 Variation of the temperature with the illumination wavelength

Interference filters may be used with a wide spectral range of optical sources. It is therefore useful to study the spectral dispersion of the temperature for these components, with a view to predicting the thermal behavior of interference filters over the entire wavelength range of optical operation.

It is not simple to analyze the spectral dispersion of the temperature, because it is a local quantity $T({\lambda ,\vec{r},z,t} )\; $ which varies in the depth z of the multilayer. Ideally, it would be necessary to address its maximum value (versus t and z) in the stack at each wavelength, but the numerical calculation here would be overburdened by this. This is why we will limit ourselves to calculating the temperature at the center of the beam on each interface (rather than in the layers), the maximum value of which with respect to time and the interfaces will be taken. The results are given in Fig. 10 for 2 components already used in Fig. 7, which are the mirror (Fig. 10, top) and the Fabry-Perot filter (Fig. 10, bottom) with identical illumination conditions (except for the wavelength, which is swept). For each component, two different pulse regimes are considered: ns regime in the left-hand figures and ms regime in the right-hand figures. The overall absorption of the components as a function of the wavelength (defined as $A = 1 - R - T$, with R and T being the reflection and transmission coefficients) is also plotted in dotted red. It may be seen that the maximum temperature elevation is in-phase with the absorption of the component, with slight differences for the ns regime.

 

Fig. 10. Variation of the temperature with the central illumination wavelength (see text). Case of a mirror in the ns regime (top left) and ms regime (top right); case of a narrowband filter in the ns regime (bottom left) and ms regime (bottom right). The 2 components are centered on 1064 nm and are those of Fig. 7. The right vertical scale is for the absorption.

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This spectral variation of the temperature is explained in the following way. The energy absorbed by the component is the source of the temperature elevation. Since this absorbed energy varies as a function of the wavelength, the same is true of the temperature. In the ns regime, the temperature is spatially confined (see Fig. 7), which explains the slight spectral variations of the temperature with respect to the overall absorption of the component. In the ms regime, conversely, the distribution of the temperature is homogeneous in the component because of the diffusion of heat; there is therefore a good agreement between the overall absorption and the temperature rise.

5.4 Melting temperature versus electric breakdown in single-pulse mode

Very many works have been devoted to the laser damage of components under flux [2,19,22–28]. Catastrophic optical damage is a complex process, mostly described by non-linear effects and run-away type phenomena. Hence the modelling presented until now is not suitable for the analysis of this phenomenon. However, the damage triggering processes are related to the high values of electromagnetic field and temperature, which compete with each other depending on materials (including impurities) and illumination parameters. In these conditions it is always interesting, even if only in a linear regime, to calculate the temperature rises of multilayers under high power. This allows to quantify a thermal damage threshold (at least its maximum value) and try to position it in regard to the electromagnetic threshold.

It is known that for short pulse durations (typically less than one ns), the damage process is not thermal but resembles the concept of dielectric breakdown caused by the high value of the electromagnetic field. Thus, it is common practice to define for these short pulses a laser damage threshold, or electromagnetic damage threshold, as a reminder that it is related to the value of the optical field (rather than the temperature). This threshold naturally depends on the materials and technologies used for manufacturing the thin films, as well as the formula of the component and the illumination conditions. It is also known that this threshold is related to the nature and the density of defects in the components. Experimental data at 1064nm in the ns regime have indicated an electric damage threshold ${F_{elm}}$ which has an order of magnitude of a few J/cm² for a multi-dielectric component, of 20 J/cm² or more than 100 J/cm² for a surface or volume of Suprasil substrate, respectively [25,26]. In recent work, a threshold of 60 J/cm² for an all silica mirror has been demonstrated experimentally in the ns regime [27].

This damage issue is similar for longer pulses, except that the damage process is thermal in nature. It should be noted that the thermal degradation of the component occurs far below the melting temperature ${T_f}$ of the thin-film materials, particularly in view of the thermo-induced mechanical stress phenomena or the presence of localized defects. In this context it is helpful to use our formalism in order to calculate the temperature value ${T_{elm}} = T({{F_{elm}}} )$ obtained at the electromagnetic damage thresholds of the materials, so that it can be compared to ${T_f}$. In other words, the question is to know whether, for increasing illumination fluences $F$, the temperature $T(F )\; $ exceeds the melting temperature ${T_f}$ before the electric threshold ${F_{elm}}$ occurs. If we denote by ${F_{th}}$ the thermal fluence threshold defined as ${T_f} = T({{F_{th}}} )$, this can be reduced to a comparison between ${F_{elm}}$ and ${F_{th}}.$

Thus, in Fig. 11 left, the maximum temperature elevation $\Delta T(F )$ of the M9 mirror of Section 5.1 is plotted as a function of the fluence F for different pulse durations (ns regime in blue and ms regime in red). For each pulse regime, this curve is plotted for different values of the imaginary index, a parameter of which it is known that to first order the temperature is proportional to it (see Section 4.5). The melting temperature of the dielectric material Nb2O5 is ${T_f} = 1512^\circ C$, which corresponds to a temperature elevation of $\mathrm{\Delta }{T_f} = $ 1765 K for an ambient temperature of 20°C. A horizontal bar in Fig. 11 left indicates this $\mathrm{\Delta }{T_f}\textrm{}$ temperature which would be reached at the fluences ${F_{th}}$ given by the intersection of this horizontal bar and the oblique curves. Similarly, a vertical bar indicates the electrical damage threshold ${F_{elm}}$, selected here at 10 J/cm² in the ns regime. This figure makes it possible to position the electric $({{F_{elm}}} )$ and thermal $({F_{th}})$ fluences for different values of the thin film imaginary indices.

 

Fig. 11. Left: maximum temperature elevation as a function of the laser fluence for a M₉ mirror in 2 different laser regimes and for different absorptions (see text). The blue and red curves are for the ns and ms regimes, respectively, the vertical and horizontal black bars are for the electrical damage thresholds (10J/cm² in ns regime) and for the melting temperature, respectively. Right: thermal damage thresholds (red curves) calculated versus the pulse duration (see text), for different imaginary indices of the thin film materials. Local defects are not considered. The dotted blue curve is an assumption of the dielectric threshold behavior at long pulses (see text).

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It is first clearly confirmed that, with low absorptions ($n^{{\prime}{\prime}}\; = \; 10^{-5},\; 10^{-4}$), the dielectric threshold is much less than the thermal one (${F_{elm}} \ll {F_{th}})$ in the ns regime (blue curves). Indeed, reaching a melting temperature would require a fluence beyond $5000\; J/c{m^2}$ (case $n^{{\prime}{\prime}}\; = \; 10^{-5})$ or $500\; J/c{m^2}$ ($n^{{\prime}{\prime}}\; = 10^{-4}$), which is much more than the electric threshold ($10\; J/c{m^2}$). However, this result is modified when the imaginary index increases. Thus, starting from $n^{\prime\prime} = {10^{ - 2}}$, in the ns regime we have ${F_{elm}} > {F_{th}}$, so that the thermal damage occurs before the dielectric one. Though this imaginary index value is high, this may happen in the presence of localized impurities. It should also be recalled that localized defects amplify the optical field (depending on their geometry) and create thermomechanical stress. Consequently, the conclusions which may be drawn from Fig. 11 need to be weighted with these comments, especially since the ns regime is a transitional regime between thermal and dielectric damage.

A similar discussion would be interesting in the ms regime, but this would require to introduce an electric threshold ${F_{elm}}\textrm{}$ in this regime. Experimental data [28] have revealed a variation in $\sqrt \tau $ of the dielectric threshold for pulse durations $\tau $ one or two decades below or above the ns regime, but no ${F_{elm}}\textrm{}$ data are available for longer pulse durations. What can be said here, as the nature of damage is known to be thermal [19,22] for long pulse durations, is that the thermal threshold is lower than the dielectric one $({F_{th}} < {F_{elm}})$. And since we know [19,22] that ${F_{th}}$ is proportional to the pulse duration $\tau $ in these long pulse regimes (see Fig. 6, top right), we conclude that the $\tau $ variation of the dielectric threshold is greater than ${\tau ^\alpha }$ with $\alpha \ge 1$ in these regimes.

Another way to summarize this information is to plot the thermal threshold ${F_{th}}$ versus pulse duration $\tau $. This is done in Fig. 11 right for different values of imaginary index. Recall that this thermal threshold is calculated following the condition ${\rm \; \Delta }T(F_{th},\tau ,n{\rm ^{{\prime}{\prime}}}) = {\rm \Delta }T_f$. It would be interesting to plot both thresholds versus $\tau $ but the variation ${F_{elm}}(\tau )$ is still missing above 10 ns. Despite this, to get an idea the oblique blue curve in Fig. 11 right is plotted for a ${F_{elm}}\textrm{}$ linear behavior above 10ns (${F_{elm}}\sim \tau $) and a square root variation below $\left( {{F_{elm}}\sim \sqrt \tau } \right)$.

To conclude this section we stress again on the fact that all values must be modified if the intrinsic values ${F_{el}}$ and ${F_{th}}$ of the materials are replaced by those of localized defects, responsible for lower thresholds. Furthermore, the damage process occurs well before the melting temperature of the materials, which also decreases the thermal threshold. Lastly, these curves are given in a single-pulse regime and may also be modified in the case of a repeating regime (see next section 6).

6. Study of the repeating pulse regime

So far, we have been dealing with a pulse regime of the single-pulse type. However, the repetition of these pulses cannot always be ignored, so that the repeating regime of the source must be considered. It may also be a continuous source modulated extrinsically by using modulators (mechanical, acousto-optical, electro-optical, etc.). If the optical repetition period is very long compared with the decay time of the temperature (which is generally the case), the results will not be very different from those of the previous sections. This means that an optical pulse creates a temperature pulse which is repeated identically. When the optical period decreases, however, the temperature pulses may overlap and this leads to overheating which tends towards a limit value, which it is useful to know accurately.

6.1 Continuous regime

Let us first recall that the temperature elevation induced by a continuous laser converges towards a maximum value [10,22,29]. The demonstration is given in section 2 of Supplement 1 where a homogenous medium with a total absorption A distributed over a depth ${L_z}$ is considered. In this configuration, the maximum temperature elevation is given by the following expression:

Note that (53) is approximated since it is given for a homogeneous medium. However, it will allow to check the magnitude order of the temperature limit of multilayers in the repeating regime (see next sub-section).

6.2 Repeating regime

We now come back to numerical calculation using our temperature model. The temperature elevation in a repeating pulse regime may now be found by adjusting the optical period. Figure 12 relates to the ns regime in a SiO₂ layer, under conditions identical to those of Fig. 3 ($W = 1mJ,\; L = 100\mu m,\; n^{\prime\prime} = {10^{ - 5}})$. To recall, the decay time of the temperature is much longer than the pulse duration of 1 ns. In order to ensure overlap of the thermal pulses, an optical repetition period of 1 µs is selected, i.e. a rate of 1 MHz, as illustrated in Fig. 12, left, where the maximum temperature elevation in the layer is plotted as a function of time for 10 pulses. We first observe that the temperature increases with each pulse. The maximum temperature elevation for each pulse is then plotted in Fig. 12, right. and tends towards a limit value of about 260 K. About 5 million pulses (i.e. 5 s) are needed to reach this limit value. It is interesting to notice that this maximum overheating is of the same magnitude order as the limit value of 160K given by (53) when assuming a power of 1 kW, corresponding to the average power calculated with a pulse duration given by the inverse repetition rate (that is, 1µs instead of 1 ns).

 

Fig. 12. Temperature in repeating pulse regime for a layer of SiO₂ (see text) - ns regime

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The same type of calculation is carried out in the ms regime for the SiO2 layer. In this case, the decay time of the temperature is similar to the pulse duration (see Fig. 3). This is why a continuous source modulated externally at a frequency of 1 kHz is considered. The optical repetition period is therefore 1 ms, which ensures overlap of the thermal pulses. The results are given in Fig. 13. It may be seen that the maximum temperature elevation tends towards a limit value of 0.15 K at the end of about 10,000 pulses (i.e. 10 s). This maximum overheating here again corresponds to the limit value of 0.16 K given by (53) when assuming a power of 1 W (calculated from the repetition rate).

 

Fig. 13. Temperature in modulated continuous regime, for a layer of SiO₂ (see text) - ms regime

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6.3 Melting temperature versus electric breakdown in repeating mode

In the previous section we showed that in the repeating mode, the maximum temperature elevation is much higher than in single pulse mode. We therefore made the same study than that of section 5.4 (Fig. 11) but in repeating mode. To reduce computational time, we chose to analyze a layer of SiO₂ rather than the mirror M₉ of Fig. 11. Recall that the maximum temperature elevation is in the same order of magnitude for these two components as can be seen in Fig. 3 and Fig. 8.

In Fig. 14, the maximum temperature elevation with respect to the fluence of the laser is plotted for different values of the imaginary index $n^{\prime\prime}$. The blue curves correspond to the ns regime with a repetition rate of 1 MHz (situation of Fig. 12) and the red curves correspond to the ms regime with a repetition rate of 1kHz (situation of Fig. 13). The dielectric breakdown is still taken at $10\; J/c{m^2}$, though this could be discussed. The main differences with the single pulse mode occurs in the ns regime. Indeed, while in the single pulse mode, a high imaginary index ($n^{\prime\prime} \ge {10^{ - 2}}$) was required to make the thermal damage dominant, in the repeating mode this condition must be written as $n^{\prime\prime} \ge {10^{ - 4}}.$

 

Fig. 14. Maximum temperature elevation as a function of the laser fluence for a single SiO₂ layer in 2 different laser regimes and for different absorptions (see text). The blue and red curves are for the laser configurations of Fig. 12 and Fig. 13, respectively. the vertical and horizontal black bars are for the electrical damage thresholds (10J/cm² in ns) and for the melting temperature, respectively.

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In conclusion, the repetition of ns laser pulses induces an increase of the maximum temperature elevation which may make the thermal damage dominant. Note that this analysis must be balanced by the phenomenon of incubation which has been demonstrated in numerous papers [22,30,31] and tends to decrease the electrical damage threshold.

7. Case of thermal fringes

The purpose of the final part is to illustrate and quantify the concept of a “thermal fringe”. For short pulse durations, the temperature elevation follows the standing wave both in depth and width. Consequently, this temperature should also follow the transverse modulation (in the xy plane) which would be created by an optical interference field, and this may have applications in a number of fields such as material growth in the form of microstructures, sensors, self-organization of nanoparticles [32,33] or 3D printing, etc. This leads us first to recalculate the heat source (transient and photo-induced) in the case of optical interference.

We here consider a basic interference consisting in the superposition of 2 coherent beams (obtained by splitting a single beam) illuminating the sample symmetrically, at angles of incidence ${\theta _0}$ and $- {\theta _0}$ with amplitudes $\beta $ and $1 - \beta $, respectively. The optical field resulting on the first interface of the component is written:

With the assumption of a quasi-monochromatic and weakly divergent source and by using the same tools as in Section 3 and in Supplement 1, we can write the heat source in medium (i) and in the various Fourier domains as:

As in the Supplement 1, we assume that the envelope of the incident beam may be written by two Gaussian functions (spatial and temporal). By returning to the same energy calculations, the heat source may be related to the parameters of the laser. We obtain:

The results are given in Fig. 15 for the dielectric SiO₂ layer of Fig. 3. In order to observe around 6 periods over the width of the laser beam (${L_x} = 100$ µm), a low illumination angle was selected, that is ${\theta _0} = \textrm{asin}\frac{1}{{60}} = 0.955^\circ $. The laser and layer parameters are those of Table 2 and Table 1 respectively. Lastly, a factor $\beta $ equal to 0.5 was selected, which corresponds to the situation in which the optical contrast is most pronounced.

Two pulse regimes are considered. Figure 15, left, shows in blue the maximum temperature elevation obtained in the SiO₂ layer in the ns regime. The results agree with those of Fig. 5: the temperature follows the spatial envelope of the heat source (dotted pink), which in this case is a diagram of optical interference due to the superposition of the two incident beams. In the ms regime (Fig. 15, right, blue curve), conversely, the thermal fringes are as expected less pronounced because of the diffusion of heat being greater (to recall, the thermal diffusion length is 27 µm calculated with (51), which is close to the modulation period $\mathrm{\Delta }x$). Note that larger illumination angles would reduce the optical inter-fringe; this reduced inter-fringe would be followed by the temperature in the ns regime but not in the ms regime. Also, all fringes exist within the whole coating depth.

 

Fig. 15. Optical interference (dotted pink) and thermal fringes (solid blue line). The curves are superimposed in the ns regime (left), in contrast to the ms regime (right).

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In conclusion, optical interference can create thermal fringes in a dielectric material if the period of the optical fringes is much greater than the thermal diffusion length. This situation is readily achievable in the ns regime, as demonstrated in this section.

8. Conclusion

We carried out an analytical method for calculating the photo-induced temperature $T({\vec{r},z,t} )$ in a multilayer interference filter subjected to laser illumination in a variety of regimes (continuous, pulse, repeating). This modelling is based on optical/thermal analogies [11] which allowed us to extend to thermal the concepts of effective index or of complex admittance which are widely used in thin-film optics [15]. Although this analogy had already been introduced in [11], it had not yet been used in the space-time regime, nor in the presence of a thermal source, which precluded its use to predict the photo-induced response of temperature. It should also be noted that this analogy allows numerous calculation codes developed for optics to be extended to heat.

In this context, it has been possible to employ the calculation methods developed for bulk scattering [12] in order to solve the equation of photo-induced heat in thin-film stacks. The modelling involves the 2 Fourier planes [15] and takes into account the volume density of absorption in the volume of the multilayer, which describes the depth variations of the heat source. The beam has been assumed to be weakly divergent with pulse durations of more than one hundred fs. The filter thin film materials have been assumed to be homogeneous.

This work has made it possible to analyze with accuracy the temperature elevations over time in various multilayer filters, while taking into account the thermal diffusion widths and depths in the component. The concept of temporal, transverse or depth resolution has also been considered. Various pulse regimes were analyzed, from ms to ps. As expected, the temperature elevations are less for longer pulses but last longer (in terms of pulse widths). Furthermore, for short pulses the spatial distribution of the temperature follows that of the stationary electromagnetic field in the stack, which leads to local overheating.

The effect of the repetition rate of the laser was also considered in detail, which is not so common. The effect is major if the repetition rate is too high in relation to the decay time of the temperature. In this case, the temperature increases rapidly to a much higher value (characteristic of the continuous regime), which may make the thermal thresholds dominant in the ns regime. All this work was carried out as a function of the imaginary index of the materials, and therefore applies to the dielectric or metallic case. The case of multi-dielectric stacks on a metal substrate was also studied. In general, a number of figures have been given so that the results may be extrapolated to an arbitrary illumination regime, which allows any user to ascertain the operating temperature of the component in a specific system. The same figures give an idea of the maximum thermal damage thresholds, in regard to the dielectric ones in the ns regime. Quantitative values were given to show that the ns regime may give rise to electric or thermal damage threshold, depending on the repetition rate and local impurities, imaginary indices…

Lastly, we also studied the variations of the temperature with the illumination surface for various pulse regimes. This led us to analyze the temperature which would be created by structured optical (interference) illumination. We showed the way in which the thermal fringes are superimposed on the optical fringes when the pulse duration decreases, which provides a technique to spatially microstructure the temperature.

We believe that this work may be very useful in the field of optical thin films, as well as for works relating to sensors and biosensors, nano-organization of particles under flux [32,33] or 3D printing.