Absorption measurements of transparent materials for a visible power laser

Hervé Piombini; Amira Guediche; Gilles Dammame

doi:10.1364/OL.45.000519

The optical absorption $A$ in material sciences is responsible for light losses in several applications, including mirrors and optical fibers. The absorption comprises two terms: one for the scattering $A_S$ which matches the spray of the light in free space, and the other for the thermal absorption $A_T$, responsible for heating when components are submitted to high-power lasers. This temperature rise induces distortions of wavefronts [1], depolarizations [2], self-focusing [3], and laser damage [4]. The high-power multi-wavelength laser bundles need achromatic lenses made of highly transparent materials such as silica or fluorite and excellent coatings in order to minimize the effects of thermal absorption. The good transparency of silica is well known by its applications in the field of optical fibers [5]. The fluorite is known for its great transparency over a large spectral range [0.13–10 µm] [6]. However, the difficulty to measure the absorption of silica or fluorite still remains because of the use of spectrophotometers [7] to make these measurements: these devices have about 1000 to 2000 ppm of accuracy [8]. To detect a weak absorption, it is possible to use the guided waves [7] or the cavity ring down [9]. Another method uses a pressure change induced by laser heating [7,10]. Interferometric calorimetry (IC) can be used to measure the absorption of several optical materials: it is the method described in this Letter. As for high-power lasers, the thermal absorption coefficient must be known. In this Letter, the wavelength is taken equal to 510 nm; one of the two lines of a cupper vapor laser (CVL), the measuring methods, and the experimental setup are described. The results obtained on some materials are given.

Table 1. Samples

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The absorption of a power laser by a material given by the Beer Lambert’s law induces a local temperature gradient which produces a refractive index variation, a mechanical expansion, and an increase of stresses [11]. All these variations engender a difference in the optical path (DOP) [12] which may be measured by interferometery [13], Shack–Hartmann [14], or beam deflection [15]. These variations along the propagation direction $z$ according to the temperature $T$ for the materials studied have been previously detailed [11]. Different materials given in Table 1 have been used for IC to determine their DOP and, thus, their optical absorption. The advantage of our IC is the perfect collinearity between the pump and probe beam [16], the use of thick samples, and a high average power laser. The pulse duration of the CVL, equal to 60 ns, is shorter than the heat diffusion time [16] given for a beam radius of 1 mm (Table 1).

A facility has been implemented to measure the absorption at 510 nm. The setup used is described in Fig. 1. Here the pump and probe beams are perfectly collinear, which is not generally the case with other photothermal methods. The laser beam of power P0 was a polarized beam coming from a CVL channel (510 and 578 nm) of 500 W. The power variation was made by the rotation of the first polarizer P1. The second polarizer P2 set the laser beam polarization. The initial beam which was 40 mm in diameter was reduced by two afocal magnifiers (L1-L2 and L3-L4) to obtain in the sample a beam around ${ r1} = {1}$ or 1.5 mm in radius. Only the green line of the CVL was kept: the yellow line was eliminated at each reflection using dichroic mirrors Di. They reflected the green (510 nm) and transmitted the yellow (578 nm) and the red (633 nm) lines. The s-polarization was chosen to minimize the losses in transmission of the main beam on the dichroic mirrors. The parasite reflections due to the surfaces of the samples were rejected by the dichroic mirror ${{\rm D}_5}$ in order to avoid the damage of the video tube caused by the reflections entering into the Zygo interferometer. ${{\rm D}_4}$ and ${{\rm D}_6}$ are used to adjust the collinearity between the probe beam and the pump beam. The mirror ${{\rm M}_2}$ sent the laser beam into a power meter. The mirror ${{\rm M}_1}$ was the cavity mirror of the Zygo interferometer. The diameter of the analysis beam was given by the diaphragm filter (D.F.).

Fig. 1. Setup to measure the absorption at 510 nm using a CVL with continuous heating and a Zygo interferometer at 633 nm.

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Each sample was measured with and without the laser power. The first measurements with the power laser were carried out after more than 1 h of heating. Several wavefront acquisitions were achieved with the Zygo Mark 3 interferometer. The recorded wavefronts without a laser beam were called “Background”; the ones with a laser beam were called “Image.” A homemade program was computed for each material to get the average of Background and Image files to perform the subtraction “Image - Background” of these two previous files. This last file was analyzed in columns and lines to find the beam impact location (column and line numbers). These two lines were processed to evaluate the physical sizes to obtain four half-cuttings that were averaged. This final average was saved in a file. This one was used with a spreadsheet to determine ${A_T}$, the thermal absorption coefficient of the tested material. To evaluate ${A_T}$, a fit between the measurements of $\Delta({ r})$ representing the variation of the DOP [11] and the analytical model is made but, before doing that, we must know the temperature profile.

As said, a first numerical approach has been previously detailed to determine numerically the temperature profile along the radius of the sample [11]. An analytical approach can also be used to determine the temperature profile from the heat equation supposing that the problem is in a plane, axisymmetric and stationary. Moreover, the problem is supposed to be independent of the material thickness, $ t $. The heat energy losses are supposed to be essentially radial convection and radiative losses. Thus, from the heat equation, we obtain [Eq. (1)]:

(1)$$\begin{split}&{r^2}\,\frac{{{\partial ^2}T(r)}}{{\partial {r^2}}} + r\frac{{\partial T(r)}}{{\partial r}} - \frac{{2 h {r^2}}}{{t {\lambda _c}}}\\&\times\left( {T(r) - \frac{{{P_{\rm deposited}} t}}{{2 h}} - {T_{\rm ambient}}}\! \right) = 0,\end{split}$$

which is the standard form of the modified Bessel equation of the order zero. Here ${ h}$ is the transfer coefficient, ${\lambda _c}$ is the heat conductivity, $t$ is the thickness of the material, ${ r}$ is the radius, ${ T}$ is the temperature, and ${ P}_{\rm{deposited}}$ is defined as follows:

(2)$${\rm If}\,\,\space r< r_1 \space {P_{\rm deposited}} = \frac{{P_0} {A_T}}{{\pi {r_1}^2}}, {\rm else} \space {{P_{\rm deposited}} = 0},$$

with ${{ r}_1}$ being the radius of the beam supposed to have a top hat profile. Thus, the solution to the heat transfer is given by

(3)$$\begin{split}&T( r) - {T_{\rm ambient}} - \frac{{{P_{\rm deposited}} t}}{{2 h}} \\&\qquad= \left( {{U_i} \frac{{{I_0}(M r)}}{{{I_0}(M {r_1})}} + {V_i} \frac{{{K_0}(M r)}}{{{K_0}(M {r_1})}}} \right) \left( {{T_1} - {T_{\rm ambient}}} \right)\!,\end{split}$$

where ${I_0}$, ${K_0}$ are the modified Bessel functions of the first and second kind, respectively; ${M^2} = \frac{{2h}}{{{\lambda _c} t}}$, ${{ U}_i}$, and ${{ V}_i}$ are two factors with ${ i}={1}$ if ${ r} \lt {{r}_1}$; otherwise, ${i} = {2}$. For ${r} \lt {{r}_1}$, the solution given by Eq. (3) becomes Eq. (4):

(4)$$\begin{split}T(r) - {T_{\rm ambient}} &= \frac{{{I_0} ( {M r} )}}{{{I_0} ( {M {r_1}} )}} ( {{T_1} - {T_{\rm ambient}}} ) \\&\quad+ \left( {1 - \frac{{{I_0} ( {M r} )}}{{{I_0} ( {M {r_1}} )}}} \right)\frac{{{P_0} t }}{{2 \pi h {r_1}^2}}{A_T}.\end{split}$$

Here the second constant ${{V}_1}$ from Eq. (3) is equal to 0 to prevent the divergence of ${{ K}_0}$ at ${ r}={0}$. ${{T}_1}$ is the temperature at ${r}={{r}_1}$. For ${ r} \gt {{ r}_{1}}\,{{ P}_{\rm deposited}}={0}$, the solution (3) becomes [Eq. (5)]:

(5)$$\begin{split}T(r) {-} {T_{\rm ambient}} = \left( {{U_2}\frac{{{I_0} ( {M r} )}}{{{I_0}( {M {r_1}} )}} {+} {V_2}\frac{{{K_0}( {M r} )}}{{{K_0}( {M {r_1}} )}}} \right)\! ( {{T_1} {-} {T_{\rm ambient}}} ).\end{split}$$

${{U}_2}$ and ${{V}_2}$ are two constants such that ${{U}_2}+{{ V}_2}={1}$, determined by the continuity of the flux at ${ r}={{ r}_1}$ [Eq. (6)]:

(6)$$\begin{split}\frac{{\partial T({r_1})}}{{\partial r}} & = M \left( {{U_2}\frac{{{I_1} ( {M {r_1}} )}}{{{I_0} ( {M {r_1}} )}} - {V_2}\frac{{{K_1} ( {M {r_1}} )}}{{{K_0} ( {M {r_1}} )}}} \right) ( {{T_1} - {T_{\rm ambient}}} )\\ & = M\frac{{{I_1} ( {M {r_1}} )}}{{{I_0} ( {M {r_1}} )}}\left( { ( {{T_1} - {T_{\rm ambient}}} ) - \frac{{{P_0} t }}{{2 \pi h {r_1}^2}}{A_T}} \right)\!.\end{split}$$

In addition, the Wronskian relation $W(M r)$ associated to Eq. (3) is given by Eq. (7):

(7)$$W(M r) = {I_0} ( {M r} ){K_1} ( {M r} ) + {K_0} ( {M r} ).{I_1} ( {M r} ) = \frac{1}{{M r}}.$$

Thus, from Eqs. (6) and (7), we find out [Eq. (8)]:

(8)$$\!\!\!\!{V_2} = M \frac{{{I_1} ( {M {r_1}} ){K_0} ( {M {r_1}} ){P_0} \cdot t}}{{2 \pi h {r_1} ( {{T_1} - {T_{\rm ambient}}} )}}{A_T}\quad {\rm with }\,\,{V_2} = 1 - {U_2}.$$

If the medium is infinite (sample radius ${R} \gg {{r}_1}$), ${{U}_2}={0}$ to prevent the divergence of ${{ I}_0}$ at ${ r} \rightarrow \infty $, and ${{V}_2}={1}$: the temperature ${{T}_1}$ is fully determined [Eq. (9)]:

(9)$${T_1} - {T_{\rm ambient}} = M {I_1}( {M {r_1}} ){K_0}( {M {r_1}} )\frac{{{P_0}t}}{{2 \pi h {r_1}}}{A_T}.$$

Figure 2 shows a comparison of the radial profile of the temperature between the numerical and the analytical calculations of 8 mm in thickness and 76 mm in diameter substrate made of BK7 having an absorption coefficient ${{A}_T}={27.{10}^{ - 4}}\,\,{{\rm cm}^{ - 1}}$ heated by a laser of 100, 40, and 10 W of power with a uniform intensity profile on a 1 mm radius and ${ h}={10.71}\;{{\rm W} \cdot {\rm m}^{ - 2}}$. The results from the two approaches are almost similar.

Fig. 2. Comparison of the radial profile of the temperature between numerical (N) [11] and analytical (A) calculations of an 8 mm in thickness and 76 mm in diameter substrate made of BK7 having an absorption coefficient ${{A}_T}={27.{10}^{ - 4}}\,\, {\rm cm^{-1}}$ heated by a laser of 100, 40, and 10 W of power with a uniform intensity profile (1 mm radius).

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Figure 3 illustrates the different steps followed with a diaphragm diameter fixed at 40 mm. First, the background has been measured three times and averaged; it corresponds to the different distortions given by the three dichroic mirrors ${{\rm D}_5}$, ${{\rm D}_6}$, and ${{\rm D}_7}$; the sample without laser power; the mirror ${{\rm M}_1}$; the standard flat from Zygo; and the disturbance of the air.

Fig. 3. Different wavefronts corresponding to the measurements of the Background, the Image (the distortion of the wavefront with a power CVL of 59.5 W), and the subtraction of these two wavefronts.

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Then two measurements have been carried out with a laser power of 59.5 W after thermal stabilization (2 h) and averaged to form the “Image.” The heating by the laser increases the refractive index and the length of the sample at the center of the sample, the wave at the center is delayed compared to the wave at the edge which explains the hole at the center of the “Image” interferogram. The subtraction of this Image with the Background gives the distortion $\Delta(r)$ produced by the power.

This wavefront was recorded in a data table (${100} \times {100}$ elements). The line ${{ i}_{\max}}$ [Fig. 4(a)] and the column ${{ j}_{\max}}$ [Fig. 4(b)] corresponding to the beam maximum (distortion maximum) are extracted from this table.

Fig. 4. Distortion expressed in a fraction of $\lambda $ (633 nm) of (a) the line 47 and (b) the column 46 corresponding to the center of the laser beam expressed in a fraction of $\lambda $ (633 nm). (c) Comparison between the experimental measurements and fit with the analytical model for a BK7 sample.

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This line and this column were averaged from their top to give a single curve, which has been fitted with our analytical model [Eq. (10)]. The parameters are the beam radius ${{r}_1}$ and the absorption coefficient ${{A}_T}$; by using a spreadsheet (see Fig. 4(c)), we get the value of the absorption:

\begin{split}&{A_T} = \frac{{2 \pi h {r_1}^2\int_0^R {\Delta ( r )\delta r} }}{{{\xi _{{ w \,{\rm or}\, r}}} {P_0} t ( {\int_0^{{r_1}} {f(r) \delta r + \int_{{r_1}}^R {g(r) \delta r} } } )}}\\&f(r) = \frac{{{I_0} ( {M r } )}}{{{I_0} ( {M {r_1} } )}}M {r_1} {I_1} ( {M {r_1} } ){K_0} ( {M {r_1} } ) + \left( {1 - \frac{{{I_0}\left( {M r } \right)}}{{{I_0} ( {M {r_1} } )}}} \right)\\& g(r) = \frac{{{K_0}( {M r } )}}{{{K_0} ( {M {r_1} } )}}M {r_1} {I_1} ( {M {r_1} } ){K_0} ( {M {r_1} } )\end{split}

and ${\rm if}\,t \lt 2R$:

(10)$$\begin{split}& {\xi _{{w }}} \,\,=\,\, \left[ t \left( \beta \,\,+ \,\, \left( {n - 1} \right)\alpha \left( {1 + \upsilon } \right) + \frac{{{n^3} \alpha E}}{4} \left( {{q_{//}} + {q_ \bot }} \right) \right) \right]\\&{{\rm or}}\,\, {\rm else}\, t \gt 2R:\,\, {\xi _{{\rm r}}} = \left[ {t \left( {\beta + \frac{{{n^3} \alpha E}}{{4\left( {1 + \upsilon } \right)}}\left( {{q_{//}} + 3 {q_ \bot }} \right)} \right)} \right]\!,\end{split}$$

with α being the mean linear expansion coefficient in the range [$-30{-}70^\circ {\rm C}$], $\beta $ being the relative temperature coefficient in the range [20–40°C], E being the Young’s modulus, $\upsilon $ being the Poisson’s ratio, ${ R}$ being the sample radius, ${ n}$ being the initial optical index, and ${{ q}_{//}}$ and ${{ q}_ {- }}$ being the parallel and perpendicular elasto-optical constants, respectively. The values can be found in previous papers with the full equations [6,11].

Table 2. Summary and Results of Our Measurements

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The first absorption value is fed back in our numerical model to evaluate the numerical DOP, which is compared to the measurement to get the final absorption of the BK7 (see Table 2).

Figure 5(a) shows the different steps followed with a diaphragm diameter fixed at 40 mm with a 7940 silica sample. The Background has been measured six times and then averaged. Three measurements have been carried out with a laser power of 62.2 W after thermal stabilization (2 h) and averaged to form the “Image.” The hole exists, but it is less perceptible.

Fig. 5. (a) Wavefronts corresponding to the background, the image (the distortion of the wavefront with a power CVL of 62.2 W), and the subtraction of these two wavefronts. (b) Distortion of the line 46 and the column 46 corresponding to the center of the laser expressed in a fraction of $\lambda $ (633 nm).

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The line ${{ i}_{\max}} = {46}$ and the column ${{ j}_{\max}} = {46}$ corresponding to the beam maximum (distortion maximum) for the 7940 silica sample are shown in Fig. 5(b) and permit us to give a single curve (Fig. 6) that has been fitted with our analytical model.

Fig. 6. Comparison: measurement and fit for silica.

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Although the laser power was higher for silica than for BK7, the distortion of the wavefront was lower [11].

The first absorption value is fed back in our numerical model to evaluate the numerical DOP, which is compared to the measurements to get the final absorption of the silica (see Table 2).

The overall results for the other samples are summarized in Table 2; all the results were obtained with the same method.

The measured values for both samples in silica give 2.4 (5.5) and 11 dB/km (24 ppm/cm) in attenuation. These attenuations can be compared to 10 dB/km (thermal absorption) given at 514.5 nm for optical fiber [10]. These values are close to values of silica absorption measured by Virgo’s team (15 or 18 ppm/cm at 1064 nm [14,15]). For fluorite, the measurements of the Merk sample confirm that $\beta$ is negative [17]. For the Anapole sample, the distortion induced by the power laser beam is imperceptible. To obtain the extinction coefficient for this sample, the distortion due to dichroic mirrors under high laser flux (Background line in Table 2) is taken away, and a third-order smoothing is performed. The laser powers are indicated in Table 2; they correspond to measured powers after the sample compensated for losses coming from the dichroic mirror ${{\rm D}_7}$, the mirror ${{\rm M}_2}$, and the reflections of both faces of the sample. It is the power inside the sample; it is also the incidence power (transmission of a Fabry–Perrot without absorption).

In conclusion, interferometric calorimetry is employed to measure the absorption of bulk material. The setup has been presented; it uses a laser with a high repetition rate (CVL). The accuracy of our method depends on the interferometer ($\lambda /{20}$), the power meter (5%), and the parameter values with an accuracy of 30% of our results. To assist the interpretation of measurements, a thermal modelling has been developed with an analytical part presented here and a digital part [11] taking into account thermal losses linked to radiative and convective fluxes leading to a lower temperature. The temperature rises of the material linked to the laser also induce thermal stresses which affect the DOP. These mechanical stresses are also taken into account by our computer code. Thanks to theoretical and experimental aspects, the absorptions of weakly absorbent materials such as silica, BK7, and fluorite have been measured at 510 nm. The obtained values remain coherent with the literature, depending on the type of silica [10]. These results can be very useful for achromatic lenses used in different systems such as dye lasers, optical benches using optical parametric oscillators, or image transportation.

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Material	Supplier	Diameter	Thickness	Heat Diffusion Time
BK7 glass	Schott	76 mm	8 mm	0.60 s
7940 Silica	Corning	100 mm	100 mm	0.59 s
Suprasil Silica	Heraeus	50 mm	153 mm	0.59 s
Anapole Fluorite	Sorem	50 mm	140 mm	6.59 s
Fluorite	Merck	50 mm	150 mm	6.59 s

Material	Number of Backgrounds/ Images taken	$P . V . > 0$ in $λ$	$P_{0}$ in W	$A_{T}$ in ${c m}^{- 1}$
BK7 glass	3/2	0.5278	59.49 W	${30.10}^{- 4}$
7940 Silica	6/3	0.0276	62.16 W	${5.5.10}^{- 6}$
Suprasil Silica	3/4	0.1347	61.62 W	${24.10}^{- 6}$
Anapole Fluorite	8/7	0.0140	56.47 W	${1.5.10}^{- 4}$
Merck Fluorite	4/3	0.0254	64.46 W	${3.3.10}^{- 4}$
Background	5/3	0.0064	60.5 W	—

Material	Supplier	Diameter	Thickness	Heat Diffusion Time
BK7 glass	Schott	76 mm	8 mm	0.60 s
7940 Silica	Corning	100 mm	100 mm	0.59 s
Suprasil Silica	Heraeus	50 mm	153 mm	0.59 s
Anapole Fluorite	Sorem	50 mm	140 mm	6.59 s
Fluorite	Merck	50 mm	150 mm	6.59 s

Material	Number of Backgrounds/ Images taken	$P . V . > 0$ in $λ$	$P_{0}$ in W	$A_{T}$ in ${c m}^{- 1}$
BK7 glass	3/2	0.5278	59.49 W	${30.10}^{- 4}$
7940 Silica	6/3	0.0276	62.16 W	${5.5.10}^{- 6}$
Suprasil Silica	3/4	0.1347	61.62 W	${24.10}^{- 6}$
Anapole Fluorite	8/7	0.0140	56.47 W	${1.5.10}^{- 4}$
Merck Fluorite	4/3	0.0254	64.46 W	${3.3.10}^{- 4}$
Background	5/3	0.0064	60.5 W	—

Absorption measurements of transparent materials for a visible power laser

Abstract

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Equations (11)

Optics Letters