CN107462176A - White light reflection dynamic measurement film thickness method based on Mallat algorithms - Google Patents

Abstract

The invention discloses a method for dynamically measuring film thickness by white light reflection based on Mallat algorithm. This method decomposes the white light reflectance spectrum signal into three layers of wavelets, then extracts the approximation coefficients and detail coefficients of each layer, and uses the default threshold for denoising, and finally obtains the characteristic value of the white light reflectance spectrum, that is, it is at 190nm‑800nm Quickly judge the average value of the reflectance above to find the film thickness. The invention enhances the accuracy and speed of the thickness measurement of the film to be tested.

Description

White light reflection dynamic measurement method of thin film thickness based on Mallat algorithm

technical field

The invention belongs to the field of optical precision measurement and signal processing, and specifically designs a method for judging film thickness by processing white light reflectance spectrum signals by using wavelet transform based on Mallat algorithm.

Background technique

With the development of microelectronic devices, thin films have been widely used in optics and semiconductor fields. Effectively measuring film thickness is an important part of the manufacturing process. To protect the film from damage, non-contact measurements are required. White light reflectance dynamic measurement is one of the widely used non-contact methods.

The present invention proposes to use White Light Reflectance Spectroscopy (WLRS) to measure the film thickness. This method has the advantages of no damage, fast response, and high accuracy of measurement results. The specific principle of WLRS measurement of film thickness is shown in Figure 1: a beam of white light is nearly perpendicular to the film to be measured from point A, after several times of refraction and reflection on the surface of S1 and S2, its phase changes, and then forms its corresponding WLRS and displayed in the signal acquisition area. There is a one-to-one correspondence between the WLRS and the thickness of the film to be measured. Thus, the film thickness on the surface of the base layer can be measured in a non-contact manner according to the above principle. Since the precision can reach the nanometer level during the dynamic measurement of film thickness, a lot of noise will inevitably be introduced. These noises directly affect the accuracy of film thickness measurement. Therefore, how to quickly deal with the noise introduced by various reasons in the measurement process has become an important issue.

Traditional denoising methods, such as Fourier signal analysis, analyze the overall signal, but cannot deal with the subtle parts of the signal well. Wavelet transform (Wavelet Transform, WT) can achieve the purpose of effectively extracting information through local transformation of time and frequency. It does not need Fourier transform and is more convenient to use. WT processes the details of the signal, and can adapt to the time domain and frequency domain signals, so it is not difficult to achieve fine processing of the signal. The Mallat algorithm is a fast wavelet decomposition and reconstruction algorithm proposed by Mallat and Meyer in 1987 based on the multiresolution analysis (Multiresolution Analysis) theory. further development.

Wavelet transform can be defined as the inner product of function and wavelet basis, as shown in Equation 1:

(W _ψ f)(a,b)＝＜f(t),ψ _a,b (t)＞ (1)

Among them, (W _ψ f)(a,b) is the continuous wavelet transform, f(t) is the function to be transformed, and ψ _a,b (t) is the wavelet basis. Among them, a is the scaling coefficient of the wavelet transform. If the value of a is too large, the sampling will be too dense, resulting in new noise. Using the Mallat algorithm can effectively avoid this situation.

Contents of the invention

The invention proposes a method for quickly denoising the WLRS signal by using the wavelet transform based on the Mallat algorithm to measure the thickness of the film.

The present invention comprises the following steps:

1) Make the light source of white light incident on the film to be tested vertically to obtain the original signal of WLRS;

2) carrying out wavelet decomposition to the WLRS original signal gained in step 1;

3) Thresholding the wavelet coefficients in step 2;

4) Reconstruct the WLRS signal;

5) Extract eigenvalues from the denoised WLRS signal obtained in step 5, and obtain the film thickness of the film to be measured through the eigenvalues.

The beneficial effect of the present invention is that the denoising effect is good, the processing speed is fast, useful signals and noises can be better distinguished without introducing new noises, and the accuracy rate of judging film thickness by using characteristic values is better improved.

Description of drawings

Figure 1 schematic diagram;

Figure 2 overall block diagram;

Figure 3 WLRS dynamic measurement results of different film thicknesses before denoising (300-350nm);

Fig. 4 Distribution diagram of characteristic parameters before denoising (300-350nm);

Fig. 5 single original WLRS result map (film thickness: 300nm);

Figure 6 Original WLRS spectrum (film thickness: 300nm);

Figure 7 Low frequency and high frequency signal decomposition diagram (film thickness: 300nm);

Fig. 8 Decomposed spectrum diagram of low-frequency and high-frequency signals (film thickness: 300nm);

Figure 9 Reconstruction of low frequency and high frequency signals (film thickness: 300nm);

Figure 10 Reconstruction spectrum of low frequency and high frequency signals (film thickness: 300nm);

Figure 11 Comparison of reconstructed signal and original signal (film thickness: 300nm);

Figure 12 WLRS dynamic measurement results of different film thickness after denoising (300-350nm);

Fig. 13 Distribution diagram of characteristic parameters after denoising (300-350nm).

detailed description

The present invention will be further described below in conjunction with accompanying drawing.

1) Make the light source of white light incident on the film to be tested vertically to obtain the original signal of WLRS;

Specifically: the specific principle is shown in Figure 1. White light enters the film to be tested from point A near vertical (the maximum angle does not exceed ±5°). After several times of refraction and reflection _on the surfaces _of S1 and S2, its phase Variety. By collecting B ₁ , B ₂ , . . . , B _n , the original WLRS signal can be obtained. Let the model of the original signal be:

S(x)＝f(x)+n1(x)×n2(x) (2)

Among them, S(x) is a noisy signal, f(x) is a real signal, n1(x) is an additive noise, and n2(x) is a multiplicative noise. In this embodiment, the signal model can be simplified as:

S(x)=f(x)+n(x) (3)

Among them, S(x) is the noisy signal, f(x) is the real signal, and n(x) is the noise.

2) Decomposing the WLRS original signal obtained in step 1;

Specifically: After WLRS is decomposed by wavelet, its low frequency is a useful signal, while its high frequency is a noise signal. Therefore, WLRS can be decomposed into a certain number of layers, and then processed with different thresholds on each layer to achieve the purpose of denoising. The more the number of decomposition layers, the more the high-frequency part is removed, but too many layers will remove some useful signals together, that is, the decomposition is excessive. Therefore, the wavelet with 3 layers is selected to decompose the WLRS original signal f(x) by wavelet.

From the multiresolution theory, we can get:

where P _j f(t) is the smooth approximation of the function f(t) at resolution j, is the weight of the linear combination, φ _jn (t) is the discrete orthogonal wavelet basis.

Decompose a certain number of layers on the WLRS signal. Here we take the decomposition of the first layer as an example. The j in formula (4) is 0, and the approximate coefficient can be obtained from the decomposition coefficient<φ _0n (t), φ _1n (t)>=h _0(n-2k) for:

1 layer detail factor for:

Approximate coefficients and detail coefficients of the remaining layers can be obtained by using a recursive algorithm.

3) Thresholding the wavelet coefficients in step 2;

Specifically: use the calculated threshold δ _j to the detail part of the jth layer to process. The processing formula is as follows:

Instantly season when season to this Perform screening.

In general, the threshold is selected by the signal-to-noise ratio of the original signal. The present invention uses a fixed threshold. The threshold δ _j is determined by the following formula:

Where n is the length of the signal.

4) Reconstruct the WLRS signal;

Specifically: use the thresholded j-th layer approximation coefficients and detail coefficients to obtain the coefficients of the previous layer (that is, the j-1th layer) through formula (9), and restore them to obtain the real signal after removing the noise.

In the formula: is the approximate coefficient of the jth layer, is the detail coefficient of the jth layer, to pass with The upper approximation coefficients of the reconstruction, g ₀ and g ₁ are the reconstruction coefficients.

5) Extract eigenvalues from the denoised WLRS signal obtained in step 4, and obtain the film thickness of the film to be measured through the eigenvalues.

Specifically, the average reflectance of the denoised WLRS signal at 190nm-800nm is the characteristic value, and there is a certain linear relationship between the characteristic value and the film thickness, and the film thickness to be measured can be obtained through this linear relationship.

Further description will be given below through examples. In the range of film thickness 300nm-350nm, a dynamic measurement is performed every 1nm, and a total of 50 WLRSs are obtained. Select 4 WLRSs of 300nm, 315nm, 330nm and 345nm from the 50 WLRSs, as shown in Figure 3. During the high-precision dynamic measurement process, some noise will be introduced, which will cause some deviations when using the eigenvalues to calculate the actual film thickness. As shown in Figure 4, it is not difficult to see that there is a certain correlation between the eigenvalues and the film thickness. In other words, as the film thickness increases, its eigenvalues generally decrease. However, if we want to calculate the film thickness more accurately from the eigenvalues, we need to do some processing. Therefore, wavelet transform is used to denoise it. Taking the wavelet transform based on the Mallat algorithm for WLRS with a film thickness of 300nm as an example, Figure 5 is the WLRS result map when the film thickness is 300nm, and Figure 6 is its frequency spectrum. Firstly, the model of WLRS original signal is established according to formula (3), and then it is decomposed by three-layer wavelet, and the first layer is decomposed according to formula (5) and formula (6) to obtain approximate coefficients and detail factor The approximation coefficients and detail coefficients of other layers are calculated recursively by the coefficients of one layer. Accompanying drawing 7 is the low-frequency and high-frequency signal diagram after decomposition, and accompanying drawing 8 is its spectrum diagram. Then threshold value processing is performed on the wavelet coefficients of each layer. The total threshold value calculated by formula (8) is processed by formula (7) on the 1-3 layer details (when |d _j,k |>δ _j , let d _j,k =d _j,k ; when | When d _j,k |≤δ _j , set d _j,k =0). Finally, similar to the decomposition, the corresponding inverse operation is performed, that is, reconstruction, to restore the real signal and achieve the purpose of denoising. Figure 9 is a reconstructed low-frequency and high-frequency signal diagram, and Figure 10 is a corresponding frequency spectrum diagram. Formula (9) is used to restore the details of the upper layer during reconstruction, and the rest of the layers are obtained by recursive algorithm. Figure 11 is a comparison diagram between the original signal and the reconstructed signal after denoising processing. It is not difficult to see that the reconstructed WLRS has tended to be smooth, and the purpose of denoising is better achieved. The above-mentioned processing is performed on the 300-350nm signal, and the result is shown in Fig. 12 . Calculate the average value of the reflectance of the denoised WLRS at 190nm-800nm, and draw the characteristic value-film thickness curve, which is shown in Figure 13. It can be seen that the eigenvalues are basically linearly correlated with the film thickness. At this time, the film thickness can be calculated according to the characteristic value, so as to achieve the purpose of dynamic measurement.

Claims (6)

1. the white light reflection dynamic measurement film thickness method based on Mallat algorithms, it is characterised in that comprise the following steps：

1) the vertically incident film to be measured of light source of white light is made, to obtain white light reflectance spectrum WLRS primary signals；

2) the white light reflectance spectrum primary signal obtained by step 1) is subjected to wavelet decomposition；

3) threshold process is carried out to the wavelet coefficient in step 2)；

4) white light reflectance spectrum signal is rebuild；

5) the white light reflectance spectrum signal extraction characteristic value after the denoising obtained to step 5, film to be measured is tried to achieve by characteristic value Thickness.

2. the white light reflection dynamic measurement film thickness method according to claim 1 based on Mallat algorithms, its feature It is, the step 1 is specially：White light by point A closely vertically into film to be measured, in S₁And S₂Surface is by refraction for several times and instead After penetrating, its phase changes；By gathering the light B after reflecting reflection repeatedly₁、B₂、…、B_n, obtain WLRS primary signals； If the model of primary signal is：

S (x)=f (x)+n1 (x) × n2 (x)

Wherein, S (x) is signals and associated noises, and f (x) is actual signal, and n1 (x) is additive noise, and n2 (x) is multiplicative noise；Then signal Model simplification is：

S (x)=f (x)+n (x)

Wherein, S (x) is signals and associated noises, and f (x) is actual signal, and n (x) is noise.

3. the white light reflection dynamic measurement film thickness method according to claim 1 based on Mallat algorithms, its feature It is, the step 2 is specially：The decomposition of certain number of plies is carried out to WLRS primary signals, then using different on every layer Threshold value is handled, and reaches the purpose of denoising.

4. the white light reflection dynamic measurement film thickness method according to claim 1 based on Mallat algorithms, its feature It is, the step 3 is specially：Use threshold value δ_jTo the detail coefficients of jth layerHandled；It is as follows to handle formula：

Work asWhen, orderWhenWhen, orderIt is right with thisCarry out Screening Treatment；

Threshold value δ_jDetermined by following formula：

<mrow> <msub> <mi>&amp;delta;</mi> <mi>j</mi> </msub> <mo>=</mo> <msqrt> <mrow> <mn>2</mn> <mi>l</mi> <mi>o</mi> <mi>g</mi> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> </mrow> </msqrt> </mrow>

N is the length of signal in formula.

5. the white light reflection dynamic measurement film thickness method according to claim 1 based on Mallat algorithms, its feature It is, the step 4 is specially：Using the jth layer approximation coefficient and detail coefficients Jing Guo threshold process, the near of j-1 layers is obtained Like coefficient：

<mrow> <msubsup> <mi>x</mi> <mi>n</mi> <mrow> <mo>(</mo> <mi>j</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mo>=</mo> <munder> <mo>&amp;Sigma;</mo> <mi>k</mi> </munder> <msub> <mi>g</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mi>n</mi> <mo>-</mo> <mn>2</mn> <mi>k</mi> <mo>)</mo> </mrow> <msubsup> <mi>x</mi> <mi>n</mi> <mrow> <mo>(</mo> <mi>j</mi> <mo>)</mo> </mrow> </msubsup> <mo>+</mo> <munder> <mo>&amp;Sigma;</mo> <mi>k</mi> </munder> <msub> <mi>g</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>n</mi> <mo>-</mo> <mn>2</mn> <mi>k</mi> <mo>)</mo> </mrow> <msubsup> <mi>d</mi> <mi>n</mi> <mrow> <mo>(</mo> <mi>j</mi> <mo>)</mo> </mrow> </msubsup> </mrow>

In formula：For the approximation coefficient of jth layer,For the detail coefficients of jth layer,To pass throughWithThat rebuilds is upper Layer approximation coefficient, g₀And g₀It is reconstructed coefficients；

So as to recover to obtain the actual signal after removing noise.

6. the white light reflection dynamic measurement film thickness method according to claim 1 based on Mallat algorithms, its feature It is, the step 5 is specially：Reflectivity average value of the WLRS signals after denoising on 190nm-800nm is asked to be characterized value, Characteristic value, there is certain linear relationship, thickness to be measured is obtained by this linear relationship with thickness.