A method for predicting surface characteristic parameters of an ultra-smooth optical element

Abstract

The application discloses a kind of ultra-smooth optical element surface characteristic parameter prediction method, comprising the following steps: step 1, based on MBK model, by comparing HS model and MBK model, establish GBK scalar scattering model;Step 2, based on GBK scalar scattering model, establish the relationship between the surface roughness and autocorrelation length and scattering rate of ultra-smooth and smooth optical element under the same incident angle;Step 3, based on two different incident angles, respectively measured corresponding incident angle under the scattering rate, by the relationship between surface roughness and autocorrelation length and scattering rate, obtain the set curve of surface roughness and autocorrelation length under different two incident angles, the intersection of two curves is the surface roughness and autocorrelation length of the optical element to be measured.The application is suitable for the rapid prediction of surface roughness and autocorrelation length of ultra-smooth and smooth optical element, with the characteristics of high prediction accuracy and high prediction efficiency.

Description

Technical Field

[0001] This invention belongs to the field of optical element measurement technology, and specifically relates to a method for predicting surface characteristic parameters of ultra-smooth optical elements. Background Technology

[0002] Commonly used models for describing the surface scattering characteristics of optical components include the Rayleigh-Rice (RR) model, the Beckmann-Kirchhoff (BK) model, the Harvey-Shack (HS) model, and the Modified Beckmann-Kirchhoff (MBK) model. Among these, the RR model is a vector scattering model, while the BK, HS, and MBK models are scalar scattering models. The RR model is suitable for scattering on smooth or ultra-smooth surfaces, but its vector model is relatively complex. The BK model is mainly suitable for Gaussian statistically distributed rough surfaces and requires modification to be applicable to ultra-smooth and smooth surfaces. The HS model cannot provide analytical solutions. The MBK model can only achieve the same effect as the HS model in calculating smooth surfaces and requires further modification.

[0003] Surface roughness is one of the most commonly used parameters for characterizing surface microstructure and has a significant impact on the scattering characteristics of optical element surfaces. Autocorrelation length is an important statistical parameter related to surface height undulations, providing a benchmark for evaluating the independence between two points on a surface. Therefore, surface roughness and autocorrelation length, as important parameters characterizing the surface properties of optical elements, directly affect the surface scattering characteristics of ultra-smooth and smooth optical elements. However, existing methods for measuring the surface characteristic parameters of optical elements cannot meet the measurement requirements of ultra-smooth and smooth surface optical elements, and the experimental measurement procedures are relatively complex. Using existing models describing the surface scattering characteristics of optical elements for measuring ultra-smooth and smooth surface optical elements also has limitations. It is necessary to modify and develop existing surface scattering models to establish scalar scattering models more suitable for ultra-smooth and smooth optical elements, thereby enabling rapid and accurate prediction of the surface characteristic parameters of ultra-smooth and smooth optical elements. Summary of the Invention

[0004] The technical problem to be solved by the present invention is to provide a method for predicting the surface characteristic parameters of ultra-smooth optical elements, which can quickly and accurately predict the surface characteristic parameters of ultra-smooth and smooth optical elements, in order to address the shortcomings of the prior art.

[0005] The technical solution adopted in this invention is: a method for predicting surface characteristic parameters of ultra-smooth optical elements, comprising the following steps:

[0006] Step 1: By comparing the HS model and the MBK model, extract the power spectral density function expression from the MBK model, and then modify the MBK model using the power spectral density function expression to establish the GBK scalar scattering model.

[0007]

[0008] Where g rel It can be represented as: In the formula, the dimensionless quantity Q is the polarization-dependent surface reflectivity; K is the normalization factor; θ i θ is the angle of incidence in spherical coordinates. s The scattering angle in spherical coordinates; φ s σ is the scattering azimuth angle in spherical coordinates; rel For relevant surface roughness; PSD m (f x ,f y ) is the relevant length l′ c =m –1 / 2 l c Fourier transform of the surface correlation function; l c is the half-width of the autocorrelation function at a height of 1 / e, representing the surface autocorrelation length; m is the convergence series; n1 is the refractive index of the incident medium; n2 is the refractive index of the exit medium;

[0009] Step 2: Based on the GBK scalar scattering model, establish the relationship between surface roughness and autocorrelation length and scattering rate for ultra-smooth and smooth optical elements at the same incident angle;

[0010] Step 3: Select two different incident angles and measure the scattering rate at each angle using angle-resolved scattering. Obtain the relationship between surface roughness and autocorrelation length and scattering rate at the two different incident angles:

[0011] {Sn1=f(σ s ,l c ,θ1),Sn2=f(σ s ,l c ,θ2)}

[0012] In the formula, θ1 and θ2 are two different incident angles; Sn1 and Sn2 are the scattering rates at two different incident angles; σ s For surface roughness; l c The autocorrelation length;

[0013] By substituting the scattering rate at the corresponding incident angle into the relationship between surface roughness and autocorrelation length and scattering rate at two different incident angles, we obtain the set curves of surface roughness and autocorrelation length at two different incident angles. The intersection of the two curves is the surface roughness and autocorrelation length of the optical element under test.

[0014] Preferably, the normalization factor K in the GBK scalar scattering model in step 1 is corrected to a fitting factor using a numerical fitting method, and then K is expressed as:

[0015]

[0016] As a preferred option, after establishing the GBK model in step 1, it can be verified using the RR model and the MBK model.

[0017] Preferably, step 2, establishing the relationship between the surface roughness and autocorrelation length of ultra-smooth and smooth optical elements and the scattering rate, includes the following steps:

[0018] S1, the GBK scalar scattering model represents the relationship between the angle-resolved scattering distribution and the initial parameters, which include surface roughness, autocorrelation length, and incident angle.

[0019] S2, the scattered light intensity is obtained by integrating the angle-resolved scattering distribution. The ratio of the scattered light intensity to the incident light intensity is the scattering rate, thus establishing the relationship between surface roughness and autocorrelation length and scattering rate under the same incident angle.

[0020] Preferably, after obtaining the predicted value in step 3, an error analysis can be performed between the predicted value and the calibration value.

[0021] Preferably, in step 3, by increasing the number of samples of surface roughness and autocorrelation length at the two incident angles, the prediction accuracy of the surface roughness and autocorrelation length of the optical element under test can be improved.

[0022] The HS model is the Harvey-Shack scalar scattering model, the MBK model is the Modified Beckmann-Kirchhoff scalar scattering model, the RR model is the Rayleigh-Rice vector scattering model, and the GBK model is the Generalized Beckmann-Kirchhoff scalar scattering model.

[0023] The beneficial effects of this invention are as follows:

[0024] (1) The GBK scalar scattering model is established by comparing it with the HS model based on the MBK model. It can provide analytical solutions and is used to quickly calculate the scattering characteristics of ultra-smooth and smooth optical elements. It is applicable to various roughness statistical distributions such as Gaussian, fractal and Cauchy-Lorentz.

[0025] (2) Establish the relationship between surface roughness, autocorrelation length and scattering rate under the same incident angle. Only the scattering rate of the optical element under two different incident angles needs to be measured to predict the surface roughness and autocorrelation length of ultra-smooth and smooth optical elements. It has the characteristics of high prediction accuracy and high prediction efficiency.

[0026] This invention is applicable to the rapid prediction of surface roughness and autocorrelation length of ultra-smooth and smooth optical elements, and features high prediction accuracy and high prediction efficiency. Attached Figure Description

[0027] Figure 1 This is a block diagram of the method of the present invention;

[0028] Figure 2 This is a comparison of the angle-resolved scattering distributions of the RR model, MBK model, and GBK scalar scattering model under different roughnesses in the embodiments of the present invention;

[0029] Figure 3 This is a distribution diagram of the scattering rate of the ultra-smooth and smooth optical elements corresponding to two incident angles in the embodiments of the present invention, as a function of surface roughness and autocorrelation length.

[0030] Figure 4 This is a schematic diagram of the set curves of surface roughness and autocorrelation length at two incident angles in an embodiment of the present invention. Detailed Implementation

[0031] The present invention will now be described in further detail with reference to the accompanying drawings and specific embodiments.

[0032] like Figure 1 As shown, the method for predicting surface characteristic parameters of ultra-smooth optical elements provided in this embodiment includes the following steps:

[0033] Step 1: According to the Beckmann-Kirchhoff model, assuming the microscopic profile of the ultra-smooth optical element surface follows a Gaussian distribution, the diffuse scattering intensity distribution of unit incident light radiation on a random surface can be expressed as:

[0034]

[0035] Harvey proposed the following modification to equation (1): (1) Remove the tilt factor F 2 (2) Introduce a normalization factor K to ensure energy conservation; (3) Let D{ρ} no longer represent the average scattering energy, but rather the diffraction radiance L, and then modify it to:

[0036]

[0037] In the formula, 1 / λ 2 Introduced by A. Krywonos, if the calculated volume of the scattering function is not equal to the total integrated scattering, then the scattering function needs to be renormalized according to the theory of non-paraxial scalar diffraction. The renormalization constant K of the scattering function is expressed by the following expression:

[0038]

[0039] According to the definition of diffraction radiance L, the bidirectional reflection distribution function of the scattering surface can be expressed as:

[0040]

[0041] This is the expression for the MBK model;

[0042] For ultrasmooth surfaces (g << 1), the MBK model is:

[0043]

[0044] Let l = l c / λ, then it simplifies to:

[0045]

[0046] For an ultrasmooth surface (g << 1), the expression for the HS model is:

[0047] f HS =QKBF{G}

[0048] In the formula, the Fourier transform of G and B can be expressed as:

[0049]

[0050]

[0051] In the formula, f x and f y It can be represented as:

[0052]

[0053] σ rel Defined as the relevant surface roughness, it can be obtained by taking the square root of the volume of the relevant portion of the surface:

[0054]

[0055] Therefore, f MBK and f HS The ratio is:

[0056]

[0057] For a Gaussian-distributed ultrasmooth surface, the power spectral density function can be expressed as:

[0058]

[0059] Substitute the power spectral density function into f MBK and f HS From the ratio formula, we can obtain:

[0060]

[0061] Based on the above derivation process, the GBK scalar scattering model is established, and its expression is:

[0062]

[0063] Where g rel It can be represented as: In the formula, the dimensionless quantity Q is the polarization-dependent surface reflectivity; K is the normalization factor; θ i θ is the angle of incidence in spherical coordinates. s The scattering angle in spherical coordinates; φ s σ is the scattering azimuth angle in spherical coordinates; rel For relevant surface roughness; PSD m (f x ,f y ) is the relevant length l′ c =m –1 / 2 l c Fourier transform of the surface correlation function; l c is the half-width of the autocorrelation function at a height of 1 / e, representing the surface autocorrelation length; m is the convergence series, the selection of which is related to the roughness, and as the roughness increases, more convergence series are selected; n1 is the refractive index of the incident medium; n2 is the refractive index of the exit medium;

[0064] Since the expression and calculation of the normalization factor K are relatively complex, which is not conducive to the rapid analysis of the scattering characteristics of ultra-smooth and smooth optical components, the numerical calculation results of the classical RR model are used as a reference. Through a large number of simulations, it was found that surface roughness and incident light wavelength are the main factors affecting the angle-resolved scattering distribution. The normalization factor K can be corrected into a fitting factor using a numerical fitting method, and K is expressed as:

[0065]

[0066] The GBK scalar scattering model was compared with the RR and MBK models under different surface roughness conditions. Figure 2 As shown, in this embodiment, a wavelength λ = 632.8 nm and a surface roughness σ are used. sThe values ​​are compared to 0.1λ, 0.01λ, 0.001λ, and 0.0001λ respectively.

[0067] Figure 2 Figure a shows a comparison of the angle-resolved scattering distributions of the GBK scalar scattering model, the RR model, and the MBK model when the surface roughness is 0.1λ.

[0068] Figure 2 Figure b shows a comparison of the angle-resolved scattering distributions of the GBK scalar scattering model, RR model, and MBK model when the surface roughness is 0.01λ.

[0069] Figure 2 Figure c shows a comparison of the angle-resolved scattering distributions of the GBK scalar scattering model, RR model, and MBK model when the surface roughness is 0.001λ.

[0070] Figure 2 The middle d figure is a comparison of the angle-resolved scattering distributions of the GBK scalar scattering model, RR model and MBK model when the surface roughness is 0.0001λ;

[0071] from Figure 2 As can be seen from the comparison of the four angle-resolved scattering distributions (a, b, c, d), the GBK scalar scattering model basically coincides with the curves of the RR model and the MBK model when the surface roughness corresponds to ultra-smooth and smooth surfaces. Therefore, the GBK scalar scattering model is suitable for optical components with ultra-smooth and smooth surface roughness.

[0072] Step 2: The GBK scalar scattering model represents the relationship between the angle-resolved scattering distribution and the initial parameters, including surface roughness, autocorrelation length, and incident angle. The scattered light intensity is obtained by integrating the angle-resolved scattering distribution, and the ratio of the scattered light intensity to the incident light intensity is the scattering rate. Thus, the relationship between surface roughness, autocorrelation length, and scattering rate is established for ultra-smooth and smooth optical elements at the same incident angle.

[0073] Step 3: Select two different incident angles and measure the scattering rate at each angle using angle-resolved scattering. Obtain the relationship between surface roughness and autocorrelation length and scattering rate at the two different incident angles:

[0074] {Sn1=f(σ s ,l c ,θ1),Sn2=f(σ s ,l c ,θ2)}

[0075] In the formula, θ1 and θ2 are two different incident angles; Sn1 and Sn2 are the scattering rates at two different incident angles; σ s For surface roughness; l cThe autocorrelation length;

[0076] By substituting the scattering rate at the corresponding incident angle into the relationship between surface roughness and autocorrelation length and scattering rate at two different incident angles, we obtain the set curves of surface roughness and autocorrelation length at two different incident angles. The intersection of the two curves is the surface roughness and autocorrelation length of the optical element under test.

[0077] This embodiment uses a graphical method, setting parameters n1 = 1, n2 = 1.5163, λ = 632.8 nm, and two incident angles θ1 = 5° and θ2 = 40°, σ s The range is 0.000947λ to 0.001013λ, l c The range is 3.9λ to 4.47λ. The angle-resolved scattering distribution of ultrasmooth and smooth optical elements can be obtained using the GBK scalar scattering model. Integrating this distribution yields the scattered light intensity. The ratio of this scattered light intensity to the incident light intensity is the corresponding scattering rate distribution map, as shown below. Figure 3 As shown;

[0078] Figure 3 Figure a shows the distribution of scattering rate of ultra-smooth and smooth optical elements as a function of surface roughness and correlation length when the incident angle θ1 = 5°.

[0079] Figure 3 Figure b shows the distribution of scattering rate of ultra-smooth and smooth optical elements as a function of surface roughness and correlation length when the incident angle θ1 = 40°.

[0080] The obtained surface roughness σ s and autocorrelation length l c By sampling at equal intervals within a given range and setting the sampling number to 200, the ensemble curves of surface roughness and autocorrelation length at two incident angles can be obtained using the sampling number. Figure 4 As shown; from Figure 4 As can be seen from the data, the coordinates of the intersection of the two set curves correspond to surface roughness and autocorrelation lengths of (0.0009918λ, 4.069λ) and (0.001008λ, 3.934λ), respectively. Therefore, the predicted surface roughness is between 0.0009918λ and 0.001008λ, and the autocorrelation length is between 3.934λ and 4.069λ.

[0081] Other measurement methods were used to measure the optical element under test, and the surface roughness of the ultra-smooth and smooth optical elements was found to be 0.001λ, with an autocorrelation length of 4λ. The predicted surface roughness endpoint value σ was selected. m =0.0009918λ and autocorrelation length endpoint value l mError analysis was performed using λ = 4.069. The relative error formula yielded the following result:

[0082]

[0083]

[0084] If the error does not meet the measurement requirements, the prediction accuracy of the surface roughness and autocorrelation length of the optical element under test can be improved by increasing the number of sampling points for surface roughness and autocorrelation length at the two incident angles.

[0085] The above description is only a preferred embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any modifications and substitutions based on the technical solutions and inventive concepts provided by the present invention should be covered within the scope of protection of the present invention.

Claims

1. A method for predicting surface characteristic parameters of ultra-smooth optical elements, characterized in that: Includes the following steps: Step 1: By comparing the HS model and the MBK model, extract the power spectral density function expression from the MBK model, and then modify the MBK model using the power spectral density function expression to establish the GBK scalar scattering model. Where g rel It can be represented as: In the formula, the dimensionless quantity Q is the polarization-dependent surface reflectivity; K is the normalization factor; θ i θ is the angle of incidence in spherical coordinates. s The scattering angle in spherical coordinates; φ s σ is the scattering azimuth angle in spherical coordinates; rel For relevant surface roughness; PSD m (f x ,f y ) is the relevant length l′ c =m –1 / 2 l c Fourier transform of the surface correlation function; l c is the half-width of the autocorrelation function at a height of 1 / e, representing the surface autocorrelation length; m is the convergence series; n1 is the refractive index of the incident medium; n2 is the refractive index of the exit medium; Step 2: Based on the GBK scalar scattering model, establish the relationship between surface roughness and autocorrelation length and scattering rate for ultra-smooth and smooth optical elements at the same incident angle; Step 3: Select two different incident angles and measure the scattering rate at each angle using angle-resolved scattering. Obtain the relationship between surface roughness and autocorrelation length and scattering rate at the two different incident angles: {Sn1=f(σ s ,l c ,θ1),Sn2=f(σ s ,l c ,θ2)} In the formula, θ1 and θ2 are two different incident angles; Sn1 and Sn2 are the scattering rates at two different incident angles; σ s For surface roughness; l c The autocorrelation length; By substituting the scattering rate at the corresponding incident angle into the relationship between surface roughness and autocorrelation length and scattering rate at two different incident angles, we obtain the set curves of surface roughness and autocorrelation length at two different incident angles. The intersection of the two curves is the surface roughness and autocorrelation length of the optical element under test.

2. The method for predicting surface characteristic parameters of an ultra-smooth optical element according to claim 1, characterized in that: In step 1, the normalization factor K in the GBK scalar scattering model is corrected to a fitting factor using a numerical fitting method. Therefore, K is expressed as:

3. The method for predicting surface characteristic parameters of an ultra-smooth optical element according to claim 1 or 2, characterized in that: Step 1: After establishing the GBK model, it can be verified using the RR model and the MBK model.

4. The method for predicting surface characteristic parameters of an ultra-smooth optical element according to claim 1, characterized in that: Step 2, establishing the relationship between the surface roughness and autocorrelation length of ultra-smooth and smooth optical elements and their scattering rate, includes the following steps: S1, the GBK scalar scattering model represents the relationship between the angle-resolved scattering distribution and the initial parameters, which include surface roughness, autocorrelation length, and incident angle. S2, the scattered light intensity is obtained by integrating the angle-resolved scattering distribution. The ratio of the scattered light intensity to the incident light intensity is the scattering rate, thus establishing the relationship between surface roughness and autocorrelation length and scattering rate under the same incident angle.

5. The method for predicting surface characteristic parameters of an ultra-smooth optical element according to claim 1, characterized in that: After obtaining the predicted value in step 3, an error analysis can be performed between the predicted value and the calibration value.

6. The method for predicting surface characteristic parameters of an ultra-smooth optical element according to claim 1 or 4, characterized in that: In step 3, by increasing the number of samples of surface roughness and autocorrelation length at the two incident angles, the prediction accuracy of surface roughness and autocorrelation length of the optical element under test can be improved.

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