In the experiment, fused silica samples were 50 × 50 × 5 mm^{3}in size, with the surface roughness (RMS) of less than10 nm. The samples were immersed in HF solution first to remove the redeposition on the surface. Then, irradiated by the 355 nm wavelength and 300 μm diameter laser pulse, a mass of damage sites occur. The range of laser fluence used in the experiment is 5-27 J/cm^{2} with the fluctuation of fluence ±3%. The damaged fused silica samples were immersed in ultra-high-purity HF solution repeatedly to remove the fragments, and irradiated by 97 W CO_{2} laser with 7 mmin diameter and the irradiation time for 4 seconds. Subsequently, damage sites became smooth again. The mitigated sites has a Gaussian spatial profile, and their lateral sizes range from 50 μm to 550*μ*m. Lastly, the deflection of the probe beam (He-Ne laser) passing the mitigated sites is detected by the location detector, which is shown in Figure 4.

Figure 4 Deflection of the probe beam passing the mitigated sites

The characteristics of mitigated sites *H*(*r*) is denoted as [9]

$$\begin{array}{}{\displaystyle H(r)={H}_{0}\mathrm{exp}\left(-\frac{{r}^{2}}{{R}_{0}^{2}}\right)}\end{array}$$(1)

where, *H*_{0} is the maximum repair depth, and *R*_{0} is the effective half width (at 1/e).

When the probe beam is vertically incident to a mitigated site as shown in Figure 4, the tangent value of incidence angle *α* is denoted as

$$\begin{array}{}{\displaystyle \mathrm{tan}(\alpha )=\frac{dH(r)}{dr}={H}_{0}\frac{2r}{{R}_{0}^{2}}\mathrm{exp}\left(-\frac{{r}^{2}}{{R}_{0}^{2}}\right)}\end{array}$$(2)

where

$$\begin{array}{}{\displaystyle \left\{\begin{array}{l}\mathrm{sin}(\alpha )=\mathrm{tan}(\alpha )/\sqrt{1+{\mathrm{tan}}^{2}(\alpha )}\\ \mathrm{cos}(\alpha )=1/\sqrt{1+{\mathrm{tan}}^{2}(\alpha )}\end{array}\right.}\end{array}$$(3)

From the refraction theorem:

$$\begin{array}{}{\displaystyle \mathrm{sin}(\beta )=\mathrm{sin}(\alpha )/n=\mathrm{tan}(\alpha )/n\sqrt{1+{\mathrm{tan}}^{2}(\alpha )},}\end{array}$$(4)

$$\begin{array}{}{\displaystyle \mathrm{cos}(\beta )=\sqrt{1-{\mathrm{tan}}^{2}(\alpha )/{n}^{2}[1+{\mathrm{tan}}^{2}(\alpha )]},}\end{array}$$(5)

Where *β* is the refraction angle and *n* is the refractive index of the medium. According to angle conversion, the sine value of the emergent angle *γ* can be denoted as

$$\begin{array}{}{\displaystyle \mathrm{sin}(\gamma )=n\mathrm{sin}(\alpha -\beta )}\\ {\displaystyle \phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}=n\left[\mathrm{sin}(\alpha )\mathrm{cos}(\beta )-\mathrm{sin}(\beta )\mathrm{cos}(\alpha )\right].}\end{array}$$(6)

The distance from the samples to the position detector is *d*, and the deflection displacement can be denoted as

$$\begin{array}{}{\displaystyle S=d\mathrm{tan}(\gamma )=d\mathrm{sin}(\gamma )/\sqrt{1-{\mathrm{sin}}^{2}(\gamma )}.}\end{array}$$(7)

When *d*[tan(*α*)]/*dr* = 0, which means that
*r* = (1/2)^{1/2} *R*_{0}, the deflection displacement takes its maximum value. Then, equation (2) can be rewritten as

$$\begin{array}{}{\displaystyle \mathrm{tan}(\alpha )={H}_{0}\frac{\sqrt{2}}{{R}_{0}}\mathrm{exp}\left(-\frac{1}{2}\right).}\end{array}$$(8)

The data measured by the stylus profilometer indicates that the depth of mitigated sites is much smaller than their width (*H*_{0} << *R*_{0}, that is, tan(*α*) <<), and the results of equations (3), (4) and (5) can be simplified as

$$\begin{array}{}{\displaystyle \left\{\begin{array}{l}\mathrm{sin}(\alpha )\approx {H}_{0}\frac{\sqrt{2}}{{R}_{0}}\mathrm{exp}\left(-\frac{1}{2}\right)\\ \mathrm{sin}(\beta )\approx {H}_{0}\frac{\sqrt{2}}{n{R}_{0}}\mathrm{exp}\left(-\frac{1}{2}\right)\\ \mathrm{cos}(\alpha )\approx 1\\ \mathrm{cos}(\beta )\approx 1\end{array}\right.}\end{array}$$(9)

Applying equations (6) and (9), the maximum deflection displacement according to equation (7) can be expressed as

$$\begin{array}{}{\displaystyle {S}_{m}\approx d(n-1){H}_{0}\frac{\sqrt{2}}{{R}_{0}}\mathrm{exp}\left(-\frac{1}{2}\right).}\end{array}$$(10)

From equation (10), the maximum value of deflection displacements grows with the growing repair points pitch depth. However, the maximum value of deflection displacements decreases with the enlarging repair points width.

Figure 5 indicates the deflection displacements of two different types of mitigated sites at different points. With the probe beam moving toward repairing pitches, the deflection displacements increase first and then decrease. With the beam moving to the bottoms of repairing pitches, the deflection displacements reversely increase first and then decrease. With the beam shifting out of mitigated site, deflection signals return to the equilibrium position.

Figure 5 Deflection displacements of two different types of mitigated sites at different positions (Site A *R*_{0} = 130 μm, *H*_{0} = 3.2 μm; Site B *R*_{0} = 280 μm, *H*_{0} = 4.5 μm)

Figure 6 reveals the changes of maximum deflection with depth-to-width ratios. The experiment values have good agreement with the results from equation (10), which indicates the maximum deflection displacement increases linearly with the increase of depth-to-width ratio. According to the equation (10), the depth-to-width ratios of mitigated sites can be obtained based on the measured maximum deflection displacement. The mitigated sites left from the mitigation process by CO_{2} laser which will result in light modulation to the downstream optics and the modulation intensity is affected by the morphology of mitigated sites. Thus, the morphology characteristics of mitigated sites obtained by beam deflection method could assess instantaneously damage resistance of the downstream optics in situ.

Figure 6 Maximum deflection varies with depth-to-width ratio

## Comments (0)

General note:By using the comment function on degruyter.com you agree to our Privacy Statement. A respectful treatment of one another is important to us. Therefore we would like to draw your attention to our House Rules.