**2.2 Large beam facilities**

62 Advanced Topics in Measurements

measurement and the damage density. Few results presented in section 6 illustrate the complementarity of procedures, the repeatability, the reproducibility and the

Measurements are currently performed on Q-switched Nd:YAG lasers supplying three

Two parameters are essential in damage tests: the beam profile and the energy of each shot. Equivalent surface *Seq* (i.e. defined as the surface given at 1/e for a Gaussian beam) of the beam is determined at the equivalent plane corresponding to distance between the focusing lens and the sample (see Fig. 1). At the focus point, beam is millimetric Gaussian shaped and diameter (given at 1/e) is around 0.5 to 1 mm. Shot-to-shot laser fluences fluctuations (about 15% at one standard deviation) are mainly due to fluctuations of the equivalent surface of



the beam. This is the reason why fluence has to be determined for each shot:

Pyroelectric detector

polarizer

Fig. 1. Typical experimental setup for laser damage testing (Nd:Yag facility) with small

Beam dump Beamblock Half wave plate +

3w

) harmonics. Laser injection seeding ensures a longitudinal monomode beam and a stable temporal profile. The lasers deliver approximately 1 J at a nominal repetition rate of 10Hz at 1. Beam is focused into the sample by a convex lens which focal length is few meters. It induces a depth of focus (DOF) bigger than the sample thickness, ensuring the beam shape

), the second (0.532µm, 2

) and third (0.355µm,

Sample photodiode

Specular reflexion

(Probe 488 nm)

Beam stop Scattered light

beam Ar beam

X-Y stages

Analysis beam *Sample table*

Pump

Towards sample

*Laser table*

Main beam

Probe beam (He-Ne)

representativeness.

**2. Tests facilities** 

3

**2.1 Small beam facilities** 

to be constant along the DOF.

each test.

longitudinal mode.

Lens CCD

High speed cell

Nd:YAG Laser

Mirror

beam.

wavelengths, the fundamental (1.064µm, 1

The advantage of tests with large beams is that they are representative of real shots on high power lasers. They are carried out on high power facilities where parametric studies are conducted: pulse duration or phase modulation (FM) effects. With energies about 100J at 1053 nm and 50J at 351nm and after size reduction of the beam on the sample, high fluences are delivered with beam diameters about 16 mm. Due to a contrast inside the beam itself (peak to average) of about few units, a shot at a given average fluence covers a large range of local fluences (Fig. 2). The laser front-end capability makes possible parametric studies like the effect of FM, temporal shapes and pulse durations on laser damage phenomenology. The characteristics of this kind of laser are quite similar to high-power lasers for fusion research: the front-end, the amplification stage, the spatial filters, the frequency converters crystals. Then laser damage measurements performed with this system should be representative of the damage phenomenon on high-power lasers.

Fig. 2. Damage test with a centimetre-sized beam. On the left, spatial profile of the 16mmdiametre beam at the sample plane as measured on CCD camera. On the right, the corresponding damage photography is reported. Matching the two maps allows to extract the fluence for each damage site.

#### **3. Procedures**

#### **3.1 1on1 procedure**

The 1/1 test is made on a limited number of sites (ISO standards, 2011). Results are generally given in terms of damage probability as a function of fluence. Since the beam sizes

Laser-Induced Damage Density Measurements of Optical Materials 65

In order to scan a large area, the sample to be tested is translated continuously along a first direction and stepped along a second direction (Fig.3), the laser working at predetermined control fluence. Repeating this test at several fluences on different areas permits to determine the number of damage sites versus fluence thus the damage density. But in the case of Nd:Yag tripled frequency, shot to shot fluence fluctuations have to be taken into account. Fluence can display a standard deviation of up to 15% for this kind of laser (Fig. 5 of reference (Lamaignère et al., 2007)). When thousands of shots are fired, modulation of a factor 2 in fluence is obtained. Thus, it is important to monitor the specific fluence of each shot in order to get a precise correlation between damage occurrence and laser fluence. During the test, energy, spatial and temporal profiles, beam position on the sample are recorded for each shot (at 10 Hz) in order to build up an accurate map of peak fluences

Much attention has to be paid to the position of the zero level in the beam image, because the value of fluence is very sensitive to small errors in that level. The determination of that position has to be checked very often, for example by verifying that the total energy integrated on the CCD is proportional to the pyroelectric cell measurement. The uncertainty on the zero level is responsible for the largest part of the total uncertainty on fluence

Depending on facilities, damage detection is realized in-situ after each shot during scans or after the irradiation with a "postmortem" observation. The "post-mortem" observation of irradiated areas is realized by means of a long working microscope (Fig. 6 of ref. (Lamaignère et al., 2007)). The minimum damage size detected is about 10 m whatever the morphology of damage. Below this value, it is difficult to discriminate between a damage site and a defect of the optic that did not evolve due to the shots. Damage sites might be of

different types: some are rather deep with fractures, others are shallower.

Fig. 3. On the left, the successive laser pulses overlap spatially to achieve an uniform

**Continous moving**

measurement. This point is treated on paragraph 5.1.

scanning. On the right, the beam overlap.

**3.3 Damage detection system** 

2° area, F2

1° area, F1

3° area, F3

**Fluence (J/cm²)**

**Silica sample under test**

1500 2000 2500 3000 3500 4000 4500 5000 5500 **Distance (m)**

Step shot to shot (300 m)

**overlap 66%**

1/e

**3.2 Rasterscan procedure** 

corresponding to the scan.

**Step by step moving**

of several benches are different, it is compulsory to convert probability into damage density. 1/1 tests are also done because with a small number of tested sites, a rapid result is achieved. Another advantage is that a relative comparison of several optical components is possible if the test parameters are unchanged. Generally, the 1/1 test is used instead of S/1 to avoid material ageing or conditioning effects.

#### **3.1.1 Small beam**

In the 1/1 procedure:


The next step is to convert the probability into damage density. This data treatment is presented on paragraph 4.1.

#### **3.1.2 Large beam**

The metrology is very close to small beam metrology: energy, temporal and spatial profiles are recorded. For the latest, a CCD camera is positioned at a plane optically equivalent to that of the sample. Beam profile and absolute energy measurement give access to energy density *F(x,y)* locally in the beam:

$$F(\mathbf{x}, \mathbf{y}) = \frac{E\_{\text{tot}}}{S\_{pix} \times \sum\_{i=\text{min}}^{i=\text{max}} \{n\_i \times i\}} \times i(\mathbf{x}, \mathbf{y}) \tag{1}$$

*Etot* : total energy *Spix* : CCD pixel area *i(x,y)* : pixel grey level

Figure 2 shows fluence spatial distribution of a large beam shot (beam diameter of about 16 mm). Due to a beam contrast (peak to average) of about 40%, a shot at a requested fluence (mean value) covers a large range of fluences. This make compulsory the exact correlation between local fluence and local event (damage or not). Data treatment is then completed by matching fluence and damage maps (map superposition is realized by means of reference points in the beam, hot spots). The two maps are compared pixel to pixel to connect a local fluence with each damage site detected. Then data are arranged to determine a damage density for each 1 J/cm² wide class of fluence. Damage density versus fluence is then given by the following relation:

$$\text{D(f)} = \text{ -ln (1 - P)} / \text{ S} \tag{2}$$

Where *P* is the damage probability: rate between the damage sites number (*Nd*) over the irradiated sites number (*Nt*) and *S* is the size of the pixel camera.

#### **3.2 Rasterscan procedure**

64 Advanced Topics in Measurements

of several benches are different, it is compulsory to convert probability into damage density. 1/1 tests are also done because with a small number of tested sites, a rapid result is achieved. Another advantage is that a relative comparison of several optical components is possible if the test parameters are unchanged. Generally, the 1/1 test is used instead of S/1




The next step is to convert the probability into damage density. This data treatment is

The metrology is very close to small beam metrology: energy, temporal and spatial profiles are recorded. For the latest, a CCD camera is positioned at a plane optically equivalent to that of the sample. Beam profile and absolute energy measurement give access to energy

max

( )

D(f) = -ln (1 – P) / S (2)

(1)

(,) (,)

 

*<sup>E</sup> Fxy ixy S ni* 

*tot i pix i i*

min

Figure 2 shows fluence spatial distribution of a large beam shot (beam diameter of about 16 mm). Due to a beam contrast (peak to average) of about 40%, a shot at a requested fluence (mean value) covers a large range of fluences. This make compulsory the exact correlation between local fluence and local event (damage or not). Data treatment is then completed by matching fluence and damage maps (map superposition is realized by means of reference points in the beam, hot spots). The two maps are compared pixel to pixel to connect a local fluence with each damage site detected. Then data are arranged to determine a damage density for each 1 J/cm² wide class of fluence. Damage density versus fluence is then given

Where *P* is the damage probability: rate between the damage sites number (*Nd*) over the

irradiated sites number (*Nt*) and *S* is the size of the pixel camera.

to avoid material ageing or conditioning effects.

versus fluence are determined.

lead to beam fluence and intensity for each shot.

**3.1.1 Small beam**  In the 1/1 procedure:

laser beam.

**3.1.2 Large beam** 

presented on paragraph 4.1.

density *F(x,y)* locally in the beam:

*Etot* : total energy *Spix* : CCD pixel area *i(x,y)* : pixel grey level

by the following relation:

In order to scan a large area, the sample to be tested is translated continuously along a first direction and stepped along a second direction (Fig.3), the laser working at predetermined control fluence. Repeating this test at several fluences on different areas permits to determine the number of damage sites versus fluence thus the damage density. But in the case of Nd:Yag tripled frequency, shot to shot fluence fluctuations have to be taken into account. Fluence can display a standard deviation of up to 15% for this kind of laser (Fig. 5 of reference (Lamaignère et al., 2007)). When thousands of shots are fired, modulation of a factor 2 in fluence is obtained. Thus, it is important to monitor the specific fluence of each shot in order to get a precise correlation between damage occurrence and laser fluence. During the test, energy, spatial and temporal profiles, beam position on the sample are recorded for each shot (at 10 Hz) in order to build up an accurate map of peak fluences corresponding to the scan.

Fig. 3. On the left, the successive laser pulses overlap spatially to achieve an uniform scanning. On the right, the beam overlap.

Much attention has to be paid to the position of the zero level in the beam image, because the value of fluence is very sensitive to small errors in that level. The determination of that position has to be checked very often, for example by verifying that the total energy integrated on the CCD is proportional to the pyroelectric cell measurement. The uncertainty on the zero level is responsible for the largest part of the total uncertainty on fluence measurement. This point is treated on paragraph 5.1.

### **3.3 Damage detection system**

Depending on facilities, damage detection is realized in-situ after each shot during scans or after the irradiation with a "postmortem" observation. The "post-mortem" observation of irradiated areas is realized by means of a long working microscope (Fig. 6 of ref. (Lamaignère et al., 2007)). The minimum damage size detected is about 10 m whatever the morphology of damage. Below this value, it is difficult to discriminate between a damage site and a defect of the optic that did not evolve due to the shots. Damage sites might be of different types: some are rather deep with fractures, others are shallower.

Laser-Induced Damage Density Measurements of Optical Materials 67

() . *d F <sup>m</sup> DF F*

*<sup>m</sup> F F* .

*DF F* .

*<sup>P</sup> D F <sup>S</sup>* 

For each energy density level *F*, the observed damage probability *P* is given by the equation:

where *(nD + nND)* is the total number of exposed sites and *nD* the number of damage sites.

*<sup>n</sup> <sup>P</sup> n n*

*m D ND*

*<sup>n</sup> <sup>F</sup>*

*<sup>n</sup> D F*

*D D ND*

. .

 . . . *D D ND*

from power law fitting of experimental data.

*nn S*

In order to determine the absolute damage density, the knowledge of the number of damage sites and the Gaussian beam equivalent area is sufficient. The final step consists in

For a top-hat beam, the absolute damage density is directly given by equation (13), the same

In this part, the calculations used to analyze the experimental data are described. When tests are done with small quasi-Gaussian beams, the overlap is not perfect (see the right insert

*nn S*

*D*

*eq g*

*eq g*

ln 1 .

 

 

From equations (7) and (8), the absolute damage density is obtained:

*D F*

are calculated from the best fit of the measurements.

The experimental curve of

Where and 

Or again:

Note that if P<<1, then relation (10) is equivalent to:

Then, if *nD*<< *nD + nND*, the measured damage density is:

Finally, the absolute damage density given in (11) is:

equations can be used than for Gaussian beams taking 1

determining the exponent

**4.2 Rasterscan procedure** 

*dF* 

.

*P*

*eq g*

. . *eq g*

*S*

*m(F)* can often be fitted by a power law of the energy density:

(7)

(8)

*<sup>m</sup>* (9)

(10)

(11)

(13)

(14)

(12)

In the case of the in-situ detection, a probe beam is focused at the same position as the pump laser on the sample. The specular reflexion is stopped by a pellet set at the centre of a collecting lens. The probe beam is scattered when damage occurs and the collecting lens redirects the stray light onto a photocell: that corresponds to a Schlieren diagnostic. Since the optic moves during the scan, the signal is recorded just before and just after the shot. The comparison of the two signals allows making a decision on damage occurrence or not. The smallest damage detected is again about 10 m.

Matching the maps of fluence and damage sites allows one to extract the peak fluence Fp for each damage site. A first data treatment is then realized by gathering damage sites in several fluence groups [(Fp-Fp/2) to (Fp+Fp/2)]. Knowing damage number and shot number, and attributing an area to each shot, the damage density is determined for each group (Fig. 4). Then with only one predetermined control fluence, damage density is then determined on a large fluence range, due to fluence fluctuations during scan. At this point, however, only the relation between damage sites and peak fluences is available. In next section, damage density is calculated as a function of local fluence.

#### **4. Data treatment**

#### **4.1 1/1 procedure**

The damage probability is converted into damage density. Note that interaction between materials defects is neglected. Thus, if defects damaging at a given energy density *F* are randomly distributed, the defect density *D(F)* follows a Poisson law. Then, on a given area *S*, damage probability *P(F,S)* and defect density *D(F)* are related by (Feit et al., 1999):

$$P(F, S) = 1 - \exp\left(-\mathcal{D}(\mathcal{F}).S\right) \tag{3}$$

The treatment is the same considering surface or volume damage densities. The total volume illuminated *V* is the product of the beam area by the optical component thickness (Lamaignère al al., 2009). *Veq.g* is defined as the Gaussian beam equivalent volume (*Seq.g* being the Gaussian beam equivalent area):

$$V\_{eq,g} = S\_{eq,g}.\theta\tag{4}$$

Defining *<sup>m</sup>* as the measured damage density which is then obtained from damage probability:

$$P = 1 - \exp\left(-\delta\_m S\_{eq,g}\right) \tag{5}$$

The measured damage density is then:

$$\mathcal{S}\_m = -\frac{\ln(1 - P)}{\mathcal{S}\_{eq,g}} \tag{6}$$

When a spatially Gaussian beam is used, the absolute damage density can be expressed as the logarithmic derivative of the measured density *m*:

$$D(F) = F.\frac{d\mathcal{S}\_m(F)}{dF} \tag{7}$$

The experimental curve of *m(F)* can often be fitted by a power law of the energy density:

$$
\delta\_m(F) = \alpha. (F)^\beta \tag{8}
$$

Where and are calculated from the best fit of the measurements. From equations (7) and (8), the absolute damage density is obtained:

$$D\left(F\right) = \beta \mathcal{L}\_m\left(F\right) \tag{9}$$

Or again:

66 Advanced Topics in Measurements

In the case of the in-situ detection, a probe beam is focused at the same position as the pump laser on the sample. The specular reflexion is stopped by a pellet set at the centre of a collecting lens. The probe beam is scattered when damage occurs and the collecting lens redirects the stray light onto a photocell: that corresponds to a Schlieren diagnostic. Since the optic moves during the scan, the signal is recorded just before and just after the shot. The comparison of the two signals allows making a decision on damage occurrence or not. The

Matching the maps of fluence and damage sites allows one to extract the peak fluence Fp for each damage site. A first data treatment is then realized by gathering damage sites in several fluence groups [(Fp-Fp/2) to (Fp+Fp/2)]. Knowing damage number and shot number, and attributing an area to each shot, the damage density is determined for each group (Fig. 4). Then with only one predetermined control fluence, damage density is then determined on a large fluence range, due to fluence fluctuations during scan. At this point, however, only the relation between damage sites and peak fluences is available. In next section, damage

The damage probability is converted into damage density. Note that interaction between materials defects is neglected. Thus, if defects damaging at a given energy density *F* are randomly distributed, the defect density *D(F)* follows a Poisson law. Then, on a given area *S*,

The treatment is the same considering surface or volume damage densities. The total volume illuminated *V* is the product of the beam area by the optical component thickness

(Lamaignère al al., 2009). *Veq.g* is defined as the Gaussian beam equivalent volume (*Seq.g*

*P S* 1 exp .

*m*

*<sup>m</sup>* as the measured damage density which is then obtained from damage

. ln(1 )

> *m*:

*eq g P*

*S*

When a spatially Gaussian beam is used, the absolute damage density can be expressed as

*PFS* ( , ) 1 exp D(F). *S* (3)

(4)

*m eq g*. (5)

(6)

damage probability *P(F,S)* and defect density *D(F)* are related by (Feit et al., 1999):

smallest damage detected is again about 10 m.

density is calculated as a function of local fluence.

being the Gaussian beam equivalent area):

The measured damage density is then:

. . . *V S eq g eq g*

the logarithmic derivative of the measured density

**4. Data treatment 4.1 1/1 procedure** 

Defining

probability:

$$D\left(F\right) = \beta.\left(-\frac{\ln\left(1 - P\right)}{S\_{eq,g}}\right) \tag{10}$$

Note that if P<<1, then relation (10) is equivalent to:

$$D\left(F\right) = \beta . \frac{P}{S\_{eq, \mathcal{g}}} \tag{11}$$

For each energy density level *F*, the observed damage probability *P* is given by the equation:

$$P = \frac{n^D}{n^D + n^{ND}} \tag{12}$$

where *(nD + nND)* is the total number of exposed sites and *nD* the number of damage sites. Then, if *nD*<< *nD + nND*, the measured damage density is:

$$\mathcal{S}\_m(F) = \frac{n^D}{\left(n^D + n^{ND}\right) \mathcal{S}\_{eq \cdot g}} \tag{13}$$

Finally, the absolute damage density given in (11) is:

$$D\left(F\right) = \beta . \frac{n^D}{\left(n^D + n^{ND}\right) S\_{eq, \mathcal{g}}} \tag{14}$$

In order to determine the absolute damage density, the knowledge of the number of damage sites and the Gaussian beam equivalent area is sufficient. The final step consists in determining the exponent from power law fitting of experimental data.

For a top-hat beam, the absolute damage density is directly given by equation (13), the same equations can be used than for Gaussian beams taking 1

#### **4.2 Rasterscan procedure**

In this part, the calculations used to analyze the experimental data are described. When tests are done with small quasi-Gaussian beams, the overlap is not perfect (see the right insert

Laser-Induced Damage Density Measurements of Optical Materials 69

Let D(F) be the damage density distribution that we are trying to measure, a function of local fluence F. We call m(Fp) the experimentally measured density at Fp. The number of damage sites created by a shot of maximum fluence Fp for a Gaussian beam is given by the

> . 0 0 2 . . () . . *Fp p eq g D F N F r dr D F r S dF*

The third member of eq. (17) is obtained by changing variables in the integral and using eq.

This permits to express easily the measured density as a function of the true damage density

 0

Thus D(F) can be easily expressed as the logarithmic derivative of the measured density m.

 *d f <sup>m</sup> Df f df* 

Since the experimental curve of m(Fp) is rather dispersed, it is better to use a functional approximation of m(Fp) to derive D(F). At high fluences (above Fcut), damage density is well

*mp p F F* .

and can be calculated from the best fit of the measurements. From eq. (20) and (21), one

Practically, above Fcut, to determine the absolute damage density for each fluence group, we have to know the number of damage sites and Gaussian effective area from the experiment, and then we have to determine the exponent from power law fitting of experimental data.

*N f Df f nf S* 

. . . *<sup>m</sup>*

 

 

*Fp*

. ( ). *p*

*N F*

*nF S*

*D F F dF F*

*p eq g*

.

.

*eq g*

*F*

(18)

(19)

(20)

(21)

(22)

(17)

Fcut: Transition fluence between "plateau" and power law behaviors

(16). As shown in figure 4, we defined the experimentally measured density by:

*m p*

*m p*

*F*

Seq.g = Gaussian beam equivalent area. Let us also define, for the following equations S(total): Total scanned surface area

N: Number of damage sites

n: Number of shots

fitted by a fluence power law:

obtains the absolute damage density:

relation:

D(F):

figure 3): a large area is irradiated at a fluence smaller than peak fluence. Then the precise beam shape and more precisely the Gaussian equivalent area have to be taken into account. Each shot is identified with its maximum fluence Fp. On the high fluence side, it is observed experimentally that the damage density varied rapidly with fluence, approximately as a power law (Fig. 4). The shape of the beam at the top is then the relevant function. Shots are thus characterized by the equivalent area of the Gaussian peak. When this is done, it appears that, schematically, the resulting curve could be divided in two parts (Fig. 4).

Fig. 4. Damage density versus fluence, after treatment taking care of beam shape and to derive experimental uncertainty. Diamonds are the raw data. Triangles represent treated data; in this case fluence is the local fluence. Data on plateau are issue from relation (23) and those in the high fluence range from relation (22). Error bars calculations are explained in §5.2.

#### **4.2.1 At high fluences, above Fcut, (see Fig. 4)**

Damage density increases quickly with fluence. Calculations in this section are made easy by considering spatially Gaussian shapes, for which fluence distribution can be expressed as a function of *r*, the distance to the peak. Each shot is identified with its peak fluence *Fp*.

$$F(r) = F\_p \exp(-\frac{\pi r^2}{\mathbf{S}\_{\text{eq,g}}}) \tag{15}$$

becomes, when derived

$$\frac{dF}{F} = \frac{2\pi r dr}{\mathbb{S}\_{\text{eq.g.}}}\tag{16}$$

where we defined:

Seq.g = Gaussian beam equivalent area.

Let us also define, for the following equations

S(total): Total scanned surface area

Fcut: Transition fluence between "plateau" and power law behaviors

N: Number of damage sites

n: Number of shots

68 Advanced Topics in Measurements

figure 3): a large area is irradiated at a fluence smaller than peak fluence. Then the precise beam shape and more precisely the Gaussian equivalent area have to be taken into account. Each shot is identified with its maximum fluence Fp. On the high fluence side, it is observed experimentally that the damage density varied rapidly with fluence, approximately as a power law (Fig. 4). The shape of the beam at the top is then the relevant function. Shots are thus characterized by the equivalent area of the Gaussian peak. When this is done, it appears that, schematically, the resulting curve could be divided in two parts (Fig. 4).

> 0 2 4 6 8 10 12 14 16 18 20 **Fluence (J/cm²)**

> > 2

(15)

(16)

eq.g

Fig. 4. Damage density versus fluence, after treatment taking care of beam shape and to derive experimental uncertainty. Diamonds are the raw data. Triangles represent treated data; in this case fluence is the local fluence. Data on plateau are issue from relation (23) and those in the

Damage density increases quickly with fluence. Calculations in this section are made easy by considering spatially Gaussian shapes, for which fluence distribution can be expressed as a function of *r*, the distance to the peak. Each shot is identified with its peak fluence *Fp*.

> ( ) exp( ) <sup>S</sup> *<sup>p</sup> <sup>r</sup> Fr F*

> > 2 S *dF rdr*

*F*

eq.g

high fluence range from relation (22). Error bars calculations are explained in §5.2.

0,01

becomes, when derived

where we defined:

**4.2.1 At high fluences, above Fcut, (see Fig. 4)** 

0,10

1,00

10,00

100,00

**Damage Density (/cm²)**

1000,00

10000,00

100000,00

raw data treated data plateau power law

Let D(F) be the damage density distribution that we are trying to measure, a function of local fluence F. We call m(Fp) the experimentally measured density at Fp. The number of damage sites created by a shot of maximum fluence Fp for a Gaussian beam is given by the relation:

$$N\left(F\_p\right) = \int\_0^{+\alpha} 2\pi r.d\boldsymbol{r}.D\left[F(r)\right] = \int\_0^{F\_p} S\_{eq\cdot g}\,dF. \frac{D\left[F\right]}{F} \tag{17}$$

The third member of eq. (17) is obtained by changing variables in the integral and using eq. (16). As shown in figure 4, we defined the experimentally measured density by:

$$\mathcal{S}\_m \left( F\_p \right) = \frac{N \left( F\_p \right)}{n \left( F\_p \right) \mathcal{S}\_{eq,g}} \tag{18}$$

This permits to express easily the measured density as a function of the true damage density D(F):

$$\mathcal{S}\_m \left( F\_p \right) = \int\_0^{r\_p} \frac{D \{ F \}}{F} dF \tag{19}$$

Thus D(F) can be easily expressed as the logarithmic derivative of the measured density m.

$$D\left(f\right) = f \cdot \frac{d\delta\_m\left(f\right)}{df} \tag{20}$$

Since the experimental curve of m(Fp) is rather dispersed, it is better to use a functional approximation of m(Fp) to derive D(F). At high fluences (above Fcut), damage density is well fitted by a fluence power law:

$$\mathcal{S}\_m \left( F\_p \right) = \alpha. \left( F\_p \right)^\beta \tag{21}$$

 and can be calculated from the best fit of the measurements. From eq. (20) and (21), one obtains the absolute damage density:

$$D(f) = \beta. \mathcal{S}\_m(f) = \beta. \frac{N(f)}{n(f). S\_{eq.\mathcal{g}}} \tag{22}$$

Practically, above Fcut, to determine the absolute damage density for each fluence group, we have to know the number of damage sites and Gaussian effective area from the experiment, and then we have to determine the exponent from power law fitting of experimental data.

Laser-Induced Damage Density Measurements of Optical Materials 71

correlation between the two sets of data (of 1000 shots each) is found with a relative

After reducing the total uncertainty on equivalent area measurement, a comparison between the total energy integrated on the CCD camera and the measurement from the pyroelectric cell has to be performed. The two signals should be proportional. The comparison gives a random error of about 5% (due to the determination of the zero level on the camera, determined at each shot and repeatability of the pyroelectric cell), contributor 5. Thus, considering the whole analysis of error measurements (the different contributors are summed up in Table I with the hypothesis that there are not correlated), it is appropriate to consider for the different facilities, that the absolute fluence values are known with an

390 400 410 420 430 440 450

Fig. 5. Optical paths comparison. Measurements of the beam equivalent areas (full symbols) at the sample and equivalent planes with the same camera, at several positions close to the focal point. At each position, the deviations between the two optical paths are indicated.

Contributor Error bar at 1

 Calorimeter 2 Deviation between cameras 4 Deviation between optical paths 4 Shot-to-shot correlation between optical paths 6 Deviation between camera and pyroelectric cell 5 Table 1. Synthesis of error margins for identified contributors [Error budget]. A quadratic

summation provides an accuracy around 10% for the determination of fluences.


**CCD positions (cm), from the lens**


0%

5%

10%

15%

**Standard deviation**

20%

Seq - equivalent plane Seq - sample plane Rel. Var. equiv. plane Rel. Var. sample plane


25%

variation of about 6% at 1, contributor 4 (see Fig. 4 of ref (Lamaignère et al., 2010 )).

accuracy around 10%.

0,0005

0.0005

0,0010

0.0010

0,0015

0.0015

0,0020

0.0020

**Equivalent area (cm²)**

0,0025

0.0025

0,0030

0.0030

#### **4.2.2 At low damage fluences, below Fcut**

The measured damage densitym(Fp) is nearly constant for fluences lower than Fcut. This plateau must be treated differently from the high fluence part. First of all, in this fluence range, and especially when damage sites are attributed to low fluence shots, Fp is not a local fluence maximum. Since m(Fp) shows a very weak variation with Fp, it is reasonable to assume that damage density D(F) is really a constant on this fluence range. Thus D(F) is simply obtained by taking the ratio of the total number of damage sites in this fluence range, by the total area covered by these shots (with Fp < Fcut). This area is proportional to the number of these shots.

$$D\left(F < F\_{cut}\right) = \frac{N\left(F < F\_{cut}\right)}{S\left(F < F\_{cut}\right)} = \frac{N\left(F < F\_{cut}\right)}{n\left(F < F\_{cut}\right)}\tag{23}$$

Damage density D is then once again almost m multiplied by a numerical factor . ( ). *n total Seq g <sup>S</sup>* ,

which is not far from 1. The result of this treatment can be visualized in figure 4. Since fluence is low, the number of damage sites is low too, and the result is dominated by the error bar, as we are going to see in paragraph 5.2.

#### **5. Error bars**

#### **5.1 Fluence error**

Synthesis of error margins is reported in Table 1. The first error (contributor 1) is due to the acquisition of the energy with the pyroelectric cell. This measurement is systematically compared with an absolute calorimeter calibrated yearly: the measurement uncertainty is given at 2% at 1 (contributor 1).

A special attention has to be paid to the "qualification" of the camera used on each facility. For example series of 1000 shots at different positions around the waist position are realized in order to measure the mean equivalent area. It appears that two identical cameras (same model) gave similar results within 4% (contributor 2). The same deviation is found between three different cameras of different providers (two being 8-bit analog cameras with rectangular pixels (13\*11.5 m²), the third one being a 12-bit digital camera with square pixels (9.9\*9.9 m²)). The fluctuations of cameras parameters (amplification, transmission factors) and the laser instability must be taken into account; they certainly play a part in these deviations. Thus, the different cameras give a rather similar result. These verifications were completed by checking that the effective area determined on the equivalent plane was the same as on the sample. It can be done by measuring around the waist position the areas on the two paths "camera" and "sample" (Fig. 5). For the latter, a camera is placed instead of the sample. Series of 1000 shots can then been realized for each position, each camera recording spatial profiles at the same time. It appears, in Fig. 5, a good correspondence between the two paths. From analysis, in the vicinity of the Rayleigh length, a deviation between the two paths of less than 4% was obtained (contributor 3). This includes the deviation coming from the cameras. It appears, in the same figure, that the deviation and fluctuations shot-to-shot are drastically reduced close to the waist position. Last, it is also important to check that the same fluctuations are recorded on the two paths. A good

The measured damage densitym(Fp) is nearly constant for fluences lower than Fcut. This plateau must be treated differently from the high fluence part. First of all, in this fluence range, and especially when damage sites are attributed to low fluence shots, Fp is not a local fluence maximum. Since m(Fp) shows a very weak variation with Fp, it is reasonable to assume that damage density D(F) is really a constant on this fluence range. Thus D(F) is simply obtained by taking the ratio of the total number of damage sites in this fluence range, by the total area covered by these shots (with Fp < Fcut). This area is proportional to the

Damage density D is then once again almost m multiplied by a numerical factor . ( ). *n total Seq g*

which is not far from 1. The result of this treatment can be visualized in figure 4. Since fluence is low, the number of damage sites is low too, and the result is dominated by the error bar, as

Synthesis of error margins is reported in Table 1. The first error (contributor 1) is due to the acquisition of the energy with the pyroelectric cell. This measurement is systematically compared with an absolute calorimeter calibrated yearly: the measurement uncertainty is

A special attention has to be paid to the "qualification" of the camera used on each facility. For example series of 1000 shots at different positions around the waist position are realized in order to measure the mean equivalent area. It appears that two identical cameras (same model) gave similar results within 4% (contributor 2). The same deviation is found between three different cameras of different providers (two being 8-bit analog cameras with rectangular pixels (13\*11.5 m²), the third one being a 12-bit digital camera with square pixels (9.9\*9.9 m²)). The fluctuations of cameras parameters (amplification, transmission factors) and the laser instability must be taken into account; they certainly play a part in these deviations. Thus, the different cameras give a rather similar result. These verifications were completed by checking that the effective area determined on the equivalent plane was the same as on the sample. It can be done by measuring around the waist position the areas on the two paths "camera" and "sample" (Fig. 5). For the latter, a camera is placed instead of the sample. Series of 1000 shots can then been realized for each position, each camera recording spatial profiles at the same time. It appears, in Fig. 5, a good correspondence between the two paths. From analysis, in the vicinity of the Rayleigh length, a deviation between the two paths of less than 4% was obtained (contributor 3). This includes the deviation coming from the cameras. It appears, in the same figure, that the deviation and fluctuations shot-to-shot are drastically reduced close to the waist position. Last, it is also important to check that the same fluctuations are recorded on the two paths. A good

*cut*

*DF F*

( )

*SF F nF F <sup>S</sup>*

*NF F NF F*

*cut cut*

*cut cut*

( )

(23)

*<sup>S</sup>* ,

*n total*

**4.2.2 At low damage fluences, below Fcut**

number of these shots.

**5. Error bars 5.1 Fluence error** 

we are going to see in paragraph 5.2.

given at 2% at 1 (contributor 1).

correlation between the two sets of data (of 1000 shots each) is found with a relative variation of about 6% at 1, contributor 4 (see Fig. 4 of ref (Lamaignère et al., 2010 )).

After reducing the total uncertainty on equivalent area measurement, a comparison between the total energy integrated on the CCD camera and the measurement from the pyroelectric cell has to be performed. The two signals should be proportional. The comparison gives a random error of about 5% (due to the determination of the zero level on the camera, determined at each shot and repeatability of the pyroelectric cell), contributor 5. Thus, considering the whole analysis of error measurements (the different contributors are summed up in Table I with the hypothesis that there are not correlated), it is appropriate to consider for the different facilities, that the absolute fluence values are known with an accuracy around 10%.

Fig. 5. Optical paths comparison. Measurements of the beam equivalent areas (full symbols) at the sample and equivalent planes with the same camera, at several positions close to the focal point. At each position, the deviations between the two optical paths are indicated.


Table 1. Synthesis of error margins for identified contributors [Error budget]. A quadratic summation provides an accuracy around 10% for the determination of fluences.

Laser-Induced Damage Density Measurements of Optical Materials 73

. when k 0 ; 0 when k 0

. ! 2

(25)

min

(26)

 min max ;

max

with an error rate of 2.3%. This number of sites must be translated into a density.

 

standard deviation (2) of a Gaussian variable = .0455, or /2 = .02275.

*k <sup>e</sup> <sup>d</sup> k* 

This means that we calculate a probability 1- for to lie in the interval between min and max. In this section, we know use specifically the confidence limits that corresponds with 2

The confidence limits are very far apart when the measured number of damage sites is low. Table 2 gives a numerical derivation of these limits for low k values. At k=0, when no damage site is detected, we can only say that the average number of sites is lower than 3.7

One should notice that these error bars are only given by the statistical variations due to the limited number of data (connected to the size of the sample). Potential errors due to

**k 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20**  min 0 0,2 0,6 1,1 1,6 2,2 2,8 3,4 4 4,7 5,4 6,1 6,8 7,6 8,3 9 9,8 11 11 12 13 max 3,7 5,6 7,2 8,8 10 12 13 15 16 17 19 20 21 22 24 25 26 27 29 30 31



Figure 6 shows results obtained on the same optical component tested with the 1/1 and rasterscan procedures, and on the same facility. Results are directly reported in terms of damage density with the presented formalism in §4.1 and 4.2. During rasterscan, about 6000 shots (this corresponds to a scanned area of about 6 cm²) are fired with fluences between 2.5 and 4.5 J/cm². During the 1/1 tests, only 200 shots have been fired with fluences between 4.5 and 6.5 J/cm². These results, which were presented, in the past, in terms of damage probabilities, are translated in terms of damage densities. The good complementarity of the two test results leads to validate the developed formalisms (for clarity, fluence error bars are


with

and

**6. Results** 


**6.1 Complementarity** 

min

*k <sup>e</sup> <sup>d</sup> k*

0

! 2

  

inaccurate damage detection are not taken into account.

Table 2. Interval of confidence of for a given measured value of k.

This chapter is dedicated to experimental results and illustrates:

depending on the information asked;

beams on high laser facilities.
