CN101241042A - Strong laser system optical manufacture error and light beam quality relationship quantitative analysis method - Google Patents

Abstract

The invention discloses a quantitative analysis method of optical manufacturing error and beam quality relation of powerful laser system, comprising following steps: 1) adopting the optical detecting device to obtain manufacturing error of the detected optical element; 2) calculating error grads distribution according to the manufacturing error of the optical element; 3) calculating transfer function of the error grads to beam phase modulation; 4) calculating transmission influence of the optical system to the laser beam according to structural parameter of the optical system; 5) calculating beam quality index of target according to quantitative relation mode between the manufacturing error and beam quality of the optical element; 6) judging qualification of the optical element according to request of the powerful laser system to the beam quality for providing data analytical result to instruct modified process, which is significant to improve working performance of the powerful laser system, to design and control the manufacturing error of the optical element under confirmative processing condition.

Description

The quantitative analysis method of strong laser system optical foozle and beam quality relation

Technical field

The invention belongs to the optical assessment method, relate to the analytical approach of quantitative relationship between the optical element foozle that is used for strong laser system and the beam quality.

Background technology

In strong laser system, by optical transmitting system laser beam is outputed on the focal plane of far field, thereby reduce the angle of divergence, and laser beam is focused on, to realize high cumulative requirement by Focused Optical system.There is face shape error in optical element in manufacture process, according to the Small-scale Self-focusing theory, the optical element foozle will form the small sample perturbations to light field amplitude and phase place, after transmission, because non-linear growth, the disturbance of amplitude and phase place all will form the modulation to light intensity; Optical element intermediate frequency, high frequency error be as the noise source of beam intensity and phase perturbation, also is to cause one of main root that focal spot secondary lobe and Nonlinear Self-Focusing destroy.The parameter that is usually used in estimating beam quality also has M except the Si Telieer ratio ²The factor, far-field divergence angle and encircled power compare etc.; Compare with other index, encircled power is than more being applicable to energy transport, coupled mode application scenario.Quantitative relationship at optical element foozle and beam quality is analyzed, for the serviceability that improves strong laser system, the foozle of optical element is significant under the determinacy processing conditionss such as appropriate design and the digital control processing of control small abrasive nose, magnetorheological polishing, ion beam polishing.

The U.S. is in the process of development laser ICF engineering " national portfire (NIF) ", the LLNL laboratory has proposed to estimate the new method of optical element foozle: adopt error gradient GRMS to characterize the low-frequency range of face shape error, but it does not provide the quantitative formula with the beam quality index.Photoelectric Technology Inst., Chinese Academy of Sciences starts with from the frequency domain distribution of optical element foozle, utilize power spectral density function PSD, some diffraction function to analyze encircled power, with the energy loss situation in definite corresponding band error range, and then the qualification of evaluation optical element; But this method is based on frequency-domain analysis, and what provide is a kind of globality evaluation conclusion, can't determine the distribution character of optical element error gradient and far field light intensity, thereby can not provide guidance for next step carries out optics correction processing.

In strong laser system, what beam quality indexs such as encircled power ratio, light distribution were played main influence is the face shape error gradient of optical element.At present than between also do not have clear and definite, general quantitative analysis method about error gradient and encircled power both at home and abroad, this area research still belongs to blank.

We once proposed the influence relation (referring to be published in " optical precision engineering " paper of 2007 year 9th phase " optics face shape error to the influence of encircled power ratio ") of optical component surface shape error to far field encircled power ratio, its emphasis is to utilize Gaussian phase error simulated optical element mismachining tolerance, analyze its influence, and compare to determine the scope of application of encircled power than computing formula by theoretical analysis and simulation result to far field encircled power ratio.

The weak point of this evaluation method is:

One: only analyze at optical transmitting system, analysis result not necessarily is applicable to other optical system;

Its two: only Theoretical Calculation and simulation result are compared, and do not have concrete experimental result to verify;

Its three: be not applied to during concrete example estimates.

Summary of the invention

The technical problem to be solved in the present invention: overcome the deficiency of existing assessment technique and method, propose the quantitative analysis method of a kind of strong laser system optical foozle and beam quality relation.This method is for the serviceability that improves strong laser system, and the foozle of optical element is significant under appropriate design, the control determinacy processing conditions.

Realization the object of the invention technical scheme is finished by following steps:

1) adopt optical detection apparatus to obtain the foozle data of the measured optical unit;

2), calculate its error gradient and distribute according to the foozle data of optical element;

3) according to the optical element foozle laser beam is carried out the characteristics of phase modulation (PM), by formula:

$H_s(u,v)=\exp [-{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="S200810030820XE00011.png"\; state="\{\{state\}\}">k</figure-callout>}}^2{\mathrm{\&sigma;}}_{\mathrm{\&Delta;}}^2({\mathrm{<figure-callout\; id="2"\; label="u"\; filenames="S200810030820XE00011.png"\; state="\{\{state\}\}">u</figure-callout>}}^2+v^2)]$

Error of calculation gradient is to the transport function of light beam phase modulation (PM), and in the formula, k=2 π/λ is a wave number, and λ is an optical maser wavelength, σ _ΔBe the root-mean-square value of optical element foozle gradient, (u v) is the volume coordinate of optical element;

4) according to the structural parameters of optical system, by formula:

$H_0(u,v)=T(u,v)\&CenterDot;\frac{\mathrm{\&pi;}{\mathrm{<figure-callout\; id="0"\; label="A"\; filenames="S200810030820XE00012.png,S200810030820XE00031.png"\; state="\{\{state\}\}">A</figure-callout>}}_0^2{\mathrm{\&omega;}}_0^2}{2}\&CenterDot;\exp [-\frac{L_{12}({\mathrm{<figure-callout\; id="2"\; label="u"\; filenames="S200810030820XE00011.png"\; state="\{\{state\}\}">u</figure-callout>}}^2+v^2)}{8{\mathrm{<figure-callout\; id="0"\; label="L"\; filenames="S200810030820XE00012.png,S200810030820XE00031.png"\; state="\{\{state\}\}">L</figure-callout>}}_0{\mathrm{\&omega;}}_0^2{\mathrm{<figure-callout\; id="2"\; label="L"\; filenames="S200810030820XE00011.png"\; state="\{\{state\}\}">L</figure-callout>}}_2^2}]$

The calculating optical system is to the transmission influence of laser beam, and in the formula, (u v) is a transmission factor to T, A ₀, ω ₀Be respectively the amplitude and the waist width of incident Gaussian beam, L ₁, L ₂Be transmission range, L ₀=k ₂ω ₀ ⁴+ 4L ₁ ², L ₁₂=4L ₀L ₂ ²+ 8L ₀L ₁L ₂+ L ₀ ²For different optical systems, and corresponding transmission factor T (u, v) different.

5) by the quantitative relationship model between optical element foozle and the beam quality:

$\mathrm{PIB}\left(r\right)=\frac{{\&Integral;}_0^r\mathrm{\&rho;}\&CenterDot;I\left(\mathrm{\&rho;}\right)\&CenterDot;\mathrm{d\&rho;}}{{\&Integral;}_0^{+\&infin;}\mathrm{\&rho;}\&CenterDot;I\left(\mathrm{\&rho;}\right)\&CenterDot;\mathrm{d\&rho;}}$

Calculate the beam quality index on the target surface, described quantitative relationship model is according to the light distribution of laser beam on optical element surface:

Make ρ ²=x ²+ y ²=λ ²L ₂ ²(f _x ²+ f _y ²) after obtain;

6) according to the requirement of strong laser system to beam quality, judge the qualification of optical element, provide the data analysis result for instructing to revise to process.

The encircled power of correspondence is more qualified than being considered as greater than 84% when focal spot radius r=w (w is the laser beam beamwidth).When not meeting the demands, determine the bigger zone of optical element surface error gradient distribution amplitude, provide the data analysis result for instructing to revise to process.

The inventive method compared with prior art, advantage is:

1, the inventive method is based on the Gradient distribution of optical element foozle, directly set up the relational model between itself and the beam quality index, realize the numerical solution of beam quality indexs such as far-field intensity distribution, encircled power under the determinacy optics processing conditions compare, and unqualified in the specific light channel structure of this class of optical transmitting system;

2, method is verified analysis result by experiment;

3, can analyze the optical element foozle respectively to the transmission influence of the modulation effects of laser beam, optical system structure parameter by step 3), step 4) to laser beam.

The inventive method is for the serviceability that improves strong laser system, and the foozle of optical element is significant under appropriate design, the control determinacy processing conditions.

Description of drawings

Fig. 1 is the strong laser system light channel structure synoptic diagram of measuring optical element foozle to laser beam impact;

Fig. 2 is the optical element foozle distribution plan that utilizes the Wyko wavefront interferometer to measure;

Fig. 3 is the analogous diagram of optical element foozle to the far-field intensity distribution influence;

Fig. 4 is after incoming laser beam is influenced by the optical element foozle, the far-field intensity distribution figure that adopts the light intensity analyser to measure;

Fig. 5 is the influence curve comparison diagram of optical element foozle gradient to the encircled power ratio.

Embodiment

Embodiment:

By the inventive method, the face shape error of analyzing Φ 100mm devitrified glass level crossing after the I.B.M. under the laser beam effect to the quantitative effect of beam quality.

The applied strong laser system light channel structure of the inventive method comprises laser instrument 1, diaphragm 2, optical element 3, long focus lens 4 and light intensity analyser 5 to be tested as shown in Figure 1.Analyze the quantitative relationship of optics foozle and beam quality by the inventive method.

1. adopt the Wyko wavefront interferometer to detect the face shape error data of Φ 100mm devitrified glass level crossing after I.B.M., and utilize Wyko wavefront interferometer Survey Software to carry the influence that analytic function is removed constant term and inclination item, measurement result as shown in Figure 2.As can be seen from the figure the face shape error at irradiation area center (promptly being caused by temperature distortion) maximum illustrates and is heated at most, and is consistent with the Gaussian incident beam.

Also can adopt optical detection apparatuss such as Hartmann's array, shearing interferometer, also can be equipped with corresponding catoptron in case of necessity, remove the face shape error constant term in measurement or the analytic process and the influence of the item that tilts from standard apparatus or LASER Light Source etc.

2. according to the measurement data of optical element foozle, calculate its Gradient distribution.

By the Wyko wavefront interferometer obtain optical element the foozle data W (x, y) after, obtain the error gradient W of each measurement point X, Y two directions respectively _X(x _i, y _j) and W _Y(x _i, y _j).Because error gradient is very responsive to noise,, need to adopt high-precision gradient calculation formula in order to guarantee computational accuracy.We adopt Richard's extrapolation formula to find the solution:

$W_X(x_i,y_j)=\frac{4}{3}\frac{W(x_{i+1},y_j)-W(x_{i-1},y_j)}{2h_x}-\frac{1}{6}\frac{W(x_{i+2},y_j)-W(x_{i-2},y_j)}{2h_x}$

$W_Y(x_i,y_j)=\frac{4}{3}\frac{W(x_i,y_{j+1})-W(x_i,y_{j-1})}{2h_y}-\frac{1}{6}\frac{W(x_i,y_{j+2})-W(x_i,y_{j-2})}{2h_y}$

In the formula, (x y) is the foozle of optical element, h to W _x, h _yRepresent the sampling interval of margin of error strong point respectively in X, Y direction.Sampling interval is more little, and the error of calculation of following formula is more little.H in an embodiment _x=h _y=109.37 μ m, following formula can obtain sufficiently high computational accuracy.

Based on the aforementioned calculation result, the Gradient distribution of optical element foozle is:

$\mathrm{\&Delta;}(x,y)=\sqrt{{W_X}^2(x,y)+{W_Y}^2(x,y)}$

3. according to the optical element foozle laser beam is carried out the characteristics of phase modulation (PM), calculate its modulation transfer function.

Under determinacy optics processing conditionss such as the digital control processing of computer control small abrasive nose, magnetorheological polishing, ion beam polishing, optical element foozle gradient to the transport function that light beam carries out phase modulation (PM) is:

H _s(u，v)＝E{exp[i2kW(x-u，y-v)-i2kW(x，y)]}

The employing first approximation is expressed as:

$H_s(u,v)\&cong;E\left\{\exp \right[\mathrm{<figure-callout\; id="2"\; label="i"\; filenames="S200810030820XE00011.png"\; state="\{\{state\}\}">i</figure-callout>}2\mathrm{ku}\frac{\&PartialD;W(x,y)}{\&PartialD;x}+\mathrm{<figure-callout\; id="2"\; label="i"\; filenames="S200810030820XE00011.png"\; state="\{\{state\}\}">i</figure-callout>}2\mathrm{kv}\frac{\&PartialD;W(x,y)}{\&PartialD;y}\left]\right\}$

According to Probability Statistics Theory, can get:

$H_s(u,v)=\exp [-2k^2(u^2{\mathrm{\&sigma;}}_{\mathrm{\&Delta;x}}^2+{\mathrm{<figure-callout\; id="2"\; label="v"\; filenames="S200810030820XE00011.png"\; state="\{\{state\}\}">v</figure-callout>}}^2{\mathrm{\&sigma;}}_{\mathrm{\&Delta;<figure-callout\; id="2"\; label="y"\; filenames="S200810030820XE00011.png"\; state="\{\{state\}\}">y</figure-callout>}}^2+2\mathrm{ruv}{\mathrm{\&sigma;}}_{\mathrm{\&Delta;x}}{\mathrm{\&sigma;}}_{\mathrm{\&Delta;y}})]$

Generally speaking, the root-mean-square value approximately equal and the correlativity of X, Y deflection error Gradient distribution are very little, and σ is arranged this moment _{Δ x}≈ σ _{Δ y}, r ≈ 0, i.e. σ _Δ ²≈ 2 σ _{Δ x} ²≈ 2 σ _{Δ y} ²Thereby, have:

$H_s(u,v)=\exp [-{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="S200810030820XE00011.png"\; state="\{\{state\}\}">k</figure-callout>}}^2{\mathrm{\&sigma;}}_{\mathrm{\&Delta;}}^2({\mathrm{<figure-callout\; id="2"\; label="u"\; filenames="S200810030820XE00011.png"\; state="\{\{state\}\}">u</figure-callout>}}^2+v^2)]$

In the formula, k=2 π/λ is a wave number, and λ is an optical maser wavelength, σ _ΔBe the gradient root-mean-square value of optical element foozle, (u v) is the volume coordinate of optical element.

4. according to the structural parameters of optical system, the calculating optical system is to the transmission influence of laser beam.

In strong laser system, incident beam commonly used is a Gaussian beam.At strong laser system shown in Figure 1, owing to before face shape error is modulated laser beam, do not exist other optical element that laser beam is produced the transmission influence, so transmission factor T (u, v)=1, thereby optical system to the transmission influence of Gaussian beam is:

$H_0(u,v)=\underset{-\&infin;}{\overset{+\&infin;}{\&Integral;}}\underset{-\&infin;}{\overset{+\&infin;}{\&Integral;}}{P}_D(x,y){P_D}^{*}(x-u,y-v)\exp \left[\frac{\mathrm{ik}}{2L_2}(2\mathrm{xu}+2\mathrm{yv}-u^2-v^2)\right]\mathrm{dxdy}$

In the formula, P _D(x y) is subjected to light intensity distribution of amplitudes behind the design of Optical System error effect for laser beam.

$P_D(x,y)=\frac{A_{0}k{{\mathrm{\&omega;}}_0}^2}{k{{\mathrm{\&omega;}}_0}^2+i2{\mathrm{<figure-callout\; id="1"\; label="L"\; filenames="S200810030820XE00011.png,S200810030820XE00012.png"\; state="\{\{state\}\}">L</figure-callout>}}_1}\exp (\mathrm{ik}{L}_1-\frac{k(x^2+y^2)}{k{{\mathrm{\&omega;}}_0}^2+i2L_1})$

In the formula, A ₀, ω ₀Be respectively the amplitude and the waist width of incident Gaussian beam, L ₁, L ₂Be the Laser Transmission distance.

With P _D(x, y) expression formula substitution H ₀(u, v) in and carry out Fourier integral and obtain:

$H_0(u,v)=\frac{\mathrm{\&pi;}{A}_0^2{\mathrm{\&omega;}}_0^2}{2}\&CenterDot;\exp [-\frac{L_{12}({\mathrm{<figure-callout\; id="2"\; label="u"\; filenames="S200810030820XE00011.png"\; state="\{\{state\}\}">u</figure-callout>}}^2+v^2)}{8{\mathrm{<figure-callout\; id="0"\; label="L"\; filenames="S200810030820XE00012.png,S200810030820XE00031.png"\; state="\{\{state\}\}">L</figure-callout>}}_0{\mathrm{\&omega;}}_0^2{\mathrm{<figure-callout\; id="2"\; label="L"\; filenames="S200810030820XE00011.png"\; state="\{\{state\}\}">L</figure-callout>}}_2^2}]$

In the formula, L ₀=k ²ω ₀ ⁴+ 4L ₁ ², L ₁₂=4L ₀L ₂ ²+ 8L ₀L ₁L ₂+ L ₀ ²

5. calculate the beam quality index on the target surface.

After the transmission influence of laser beam process optical system and the modulation effects of optics foozle, the light distribution on target surface is:

Light channel structure at Fig. 1 calculates:

$I(f_x,f_y)=\frac{4{\mathrm{\&pi;}}^2{{\mathrm{<figure-callout\; id="0"\; label="A"\; filenames="S200810030820XE00012.png,S200810030820XE00031.png"\; state="\{\{state\}\}">A</figure-callout>}}_0}^2{\mathrm{<figure-callout\; id="0"\; label="L"\; filenames="S200810030820XE00012.png,S200810030820XE00031.png"\; state="\{\{state\}\}">L</figure-callout>}}_0{{\mathrm{\&omega;}}_0}^4}{{\mathrm{\&lambda;}}^2(L_{12}+8k^2{\mathrm{<figure-callout\; id="0"\; label="L"\; filenames="S200810030820XE00012.png,S200810030820XE00031.png"\; state="\{\{state\}\}">L</figure-callout>}}_0{{\mathrm{\&omega;}}_0}^2{{\mathrm{<figure-callout\; id="2"\; label="L"\; filenames="S200810030820XE00011.png"\; state="\{\{state\}\}">L</figure-callout>}}_2}^2{{\mathrm{\&sigma;}}_{\mathrm{\&Delta;}}}^2)}\&CenterDot;\exp \{-\frac{8{{\mathrm{<figure-callout\; id="2"\; label="L"\; filenames="S200810030820XE00011.png"\; state="\{\{state\}\}">L</figure-callout>}}_2}^2{{\mathrm{\&omega;}}_0}^2{\mathrm{<figure-callout\; id="0"\; label="L"\; filenames="S200810030820XE00012.png,S200810030820XE00031.png"\; state="\{\{state\}\}">L</figure-callout>}}_0{\mathrm{\&pi;}}^2({f_x}^2+{f_y}^2)}{L_{12}+8k^2{\mathrm{<figure-callout\; id="0"\; label="L"\; filenames="S200810030820XE00012.png,S200810030820XE00031.png"\; state="\{\{state\}\}">L</figure-callout>}}_0{{\mathrm{\&omega;}}_0}^2{{\mathrm{<figure-callout\; id="2"\; label="L"\; filenames="S200810030820XE00011.png"\; state="\{\{state\}\}">L</figure-callout>}}_2}^2{{\mathrm{\&sigma;}}_{\mathrm{\&Delta;}}}^2}\}$

In the formula, f _x=x/ λ L ₂, f _y=y/ λ L ₂

For circular bore optical element, make ρ ²=x ²+ y ²=λ ²L ₂ ²(f _x ²+ f _y ²), then the encircled power in the radius r than PIB is:

$\mathrm{PIB}\left(r\right)=\frac{{\&Integral;}_0^r\mathrm{\&rho;}\&CenterDot;I\left(\mathrm{\&rho;}\right)\&CenterDot;\mathrm{d\&rho;}}{{\&Integral;}_0^{+\&infin;}\mathrm{\&rho;}\&CenterDot;I\left(\mathrm{\&rho;}\right)\&CenterDot;\mathrm{d\&rho;}}$

The substitution of I (ρ) expression formula found the solution:

$\mathrm{PIB}\left(r\right)=1-\exp \{-\frac{2{{{\mathrm{\&omega;}}_0}^{2}L}_0{k}^2{r}^2}{L_{12}+8k^2{\mathrm{<figure-callout\; id="0"\; label="L"\; filenames="S200810030820XE00012.png,S200810030820XE00031.png"\; state="\{\{state\}\}">L</figure-callout>}}_0{{\mathrm{\&omega;}}_0}^2{{\mathrm{<figure-callout\; id="2"\; label="L"\; filenames="S200810030820XE00011.png"\; state="\{\{state\}\}">L</figure-callout>}}_2}^2{{\mathrm{\&sigma;}}_{\mathrm{\&Delta;}}}^2}\}$

According to the above-mentioned relation model, can draw the quantitative relationship between the optical element foozle and beam quality index under the determinacy processing conditions easily.The far-field intensity distribution figure of Fig. 3 for coming out at the optical element foozle emulation that Fig. 2 surveyed finds that its distribution is approximated to Gaussian, and be closely similar with incident beam.

6. according to the requirement of strong laser system, judge the qualification of optical element, provide the data analysis result for instructing to revise to process to beam quality.

In strong laser system shown in Figure 1, the encircled power of correspondence is more qualified than being considered as greater than 84% when focal spot radius r=w (w is the laser beam beamwidth).When not meeting the demands, determine the bigger zone of error gradient distribution amplitude, provide the data analysis result for instructing to revise to process.

In the present embodiment, the encircled power of calculating ratio is 86.47%, greater than 84%, so meet the demands.

By the target practice experiment of indoor laser system, analyze the influence relation of optical element foozle gradient, experimental result such as Fig. 4, shown in Figure 5 to light distribution on the target surface, encircled power ratio.The incoming laser beam of measuring with PRIMES light intensity analyser is subjected to the far-field intensity distribution situation after the optical element foozle modulation effects as seen from Figure 4, and Fig. 5 represents to adopt resulting three encircled powers of model formation Theoretical Calculation, numerical simulation and experimental technique than curve comparison diagram respectively.We find three curve basically identicals, and this has verified the correctness of the inventive method.

According to the evaluation procedure that the inventive method is set up, can not practice shooting to test and just learn accurately that the optical element foozle waits the quantitative effect of beam quality index to encircled power ratio by the outfield; And then, judge the qualification of optical element according to the requirement of strong laser system to beam quality, provide the data analysis result for instructing to revise to process.

Claims (1)

1, the quantitative analysis method of a kind of strong laser system optical foozle and beam quality relation is characterized in that finishing by following steps:

1) adopt optical detection apparatus to obtain the foozle data of the measured optical unit;

2), calculate its error gradient and distribute according to the foozle data of optical element;

3) according to the optical element foozle laser beam is carried out the characteristics of phase modulation (PM), by formula

$H_s(u,v)=\exp [-k^2{\mathrm{\&sigma;}}_{\mathrm{\&Delta;}}^2(u^2+v^2)]$

Error of calculation gradient is to the transport function of light beam phase modulation (PM), and in the formula, k=2 π/λ is a wave number, and λ is an optical maser wavelength, σ _ΔBe the root-mean-square value of optical element foozle gradient, (u v) is the volume coordinate of optical element;

4) according to the structural parameters of optical system, by formula:

$H_0(u,v)=T(u,v)\&CenterDot;\frac{\mathrm{\&pi;}{A}_0^2{\mathrm{\&omega;}}_0^2}{2}\&CenterDot;\exp [-\frac{L_{12}(u^2+v^2)}{8L_0{\mathrm{\&omega;}}_0^2{L}_2^2}]$

The calculating optical system is to the transmission influence of laser beam, and in the formula, (u v) is a transmission factor to T, A ₀, ω ₀Be respectively the amplitude and the waist width of incident Gaussian beam, L ₁, L ₂Be transmission range, L ₀=k ²ω ₀ ⁴+ 4L ₁ ², L ₁₂=4L ₀L ₂ ²+ 8L ₀L ₁L ₂+ L ₀ ²For different optical systems, and corresponding transmission factor T (u, v) different;

5) by the quantitative relationship model between optical element foozle and the beam quality:

$\mathrm{PIB}\left(r\right)=\frac{{\&Integral;}_0^r\mathrm{\&rho;}\&CenterDot;I\left(\mathrm{\&rho;}\right)\&CenterDot;\mathrm{d\&rho;}}{{\&Integral;}_0^{+\&infin;}\mathrm{\&rho;}\&CenterDot;I\left(\mathrm{\&rho;}\right)\&CenterDot;\mathrm{d\&rho;}}$

Calculate the beam quality index on the target surface, described quantitative relationship model is according to the light distribution of laser beam on optical element surface:

Make ρ ²=x ²+ y ²After obtain;

6) according to the requirement of strong laser system to beam quality, judge the qualification of optical element, provide the data analysis result for instructing to revise to process.

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