Method for measuring coherence length of correlation structure light beams of Shell model

Abstract

The invention relates to a method for measuring the coherence length of a Shell model correlation structure light beam, which belongs to the field of laser space coherence measurement, and comprises the following steps: S1, calculating the coherence lengths # imgabs0 # and # imgabs1 # in x and y directions by using the analytic relationship between the coherence length of the Shell model correlation structure light beam and the spatial coherence modulus; s2, a double-aperture mask is arranged on a light beam shaping transmission path of the laser processing system, the aperture positions are symmetrical, the distance d is adjustable, and after light beams are focused, intensity signals # imgabs2 are collected at a focal plane detector; and S3, extracting an intensity signal # imgabs3 # to obtain a double-aperture interferogram, performing Fourier transform on the interferogram, establishing a corresponding relation between a low-frequency component in a frequency spectrum and a characteristic frequency of an aperture spacing d, and calculating to obtain a spatial coherence modulus # imgabs4 #. According to the invention, based on partially coherent light dual-aperture interference and Fourier spectrum analysis, a direct analytical relationship between the spatial coherence degree and the coherence length is established, and a new way is provided for beam quality control in laser processing.

Description

Technical Field

[0001] The invention belongs to the field of spatial coherence length, and in particular relates to a method for measuring the coherence length of a Scherrer model correlation structure light beam. Background Art

[0002] The concept of optical coherence is at the root of many scientific and applied fields. In scalar descriptions, the cornerstone of optical coherence theory is the mutual coherence function. Zernike first measured the coherence of light using a Young interferometer and introduced the term "coherence" in 1938, defining the maximum contrast of interference obtained from two points in the wave field as their coherence. He used a simple statistical method to derive a general formula for coherence from illumination data. However, Zernike only considered the temporal coherence of light, whereas in practice, the characterization of spatial coherence is crucial. Wolf introduced a complete description of classical optical coherence theory, including time delays, and the corresponding quantum coherence theory was formulated by Glauber. He showed that the coherence is given by the visibility of the interference fringes multiplied by a factor related to the intensity of each aperture. In 2007, Mejía and González proposed measuring spatial coherence using a multi-aperture mask. In recent years, this method has been simplified to measurements using a dual-aperture mask. During this period of development, the most widely used light source model was the Schell-model. The radiation field of this type of model closely resembles that of commonly observed light sources in nature. In 1978, Collett and Wolf proposed the Gaussian Schell-model (GSM) beam. The spatial intensity and coherence distributions of the source exhibit Gaussian distributions, and the far-field analytical solution is easily calculated. The Gaussian Schell model is a classic model for describing partially coherent light fields. In this model, spatial coherence is characterized by the coherence length, which represents the characteristic scale at which the coherence between two points in the light field decays with distance.

[0003] In the field of laser processing, the spatial coherence of a beam directly determines its focusing performance and energy distribution. Traditional methods for measuring spatial coherence length (such as Young's double-slit interferometry and Michelson interferometer) have the following drawbacks:

[0004] 1. Reliance on highly stable interference fringes: Vibration and temperature fluctuations in industrial environments can easily cause fringes to drift, affecting measurement reliability. 2. High system complexity: Precision optical adjustments and complex photoelectric detection equipment are required, making integration into production lines difficult for real-time monitoring. 3. Limited applicability: The ability to analyze partially coherent light fields (such as Gauss-Scherr model beams) is insufficient, making it impossible to directly derive the coherence length from a formula. In recent years, laser processing has evolved towards high precision and intelligent processing, creating an urgent need for a fast, analytical coherence length measurement method applicable to partially coherent light fields. Summary of the Invention

[0005] This paper addresses the challenges of existing technologies and proposes a method for measuring the coherence length of a beam with a Scherrer model correlation structure. Based on aperture mask theory, this method uses Fourier transforms to obtain corresponding spectral characteristics. By utilizing the relationship between the spatial coherence modulus and the coherence length, this method provides a method for measuring the coherence length of a beam. By combining double-aperture interferometry with Fourier transforms, the coherence length can be rapidly determined without iterative fitting, meeting the real-time and precision requirements of industrial scenarios.

[0006] In order to achieve the above object, the present invention is implemented by the following technical solution: A method for measuring the coherence length of a Scherrer model correlation structure beam is performed according to the following steps:

[0007] Step S1: Calculate the coherence length in the x and y directions using the analytical relationship between the coherence length of the Schell model beam and the spatial coherence modulus. and ;

[0008] ;

[0009] ;

[0010] in, is the distance between two points in the light field in the x direction, is the distance between two points in the light field in the y direction, is the spatial coherence modulus;

[0011] Step S2: Set a double aperture mask on the beam transmission path of the laser processing system, with the aperture positions symmetrical and the spacing d adjustable, and collect the far-field intensity of the far-field interference pattern through the focal plane detector. ;

[0012] Step S3: Extract the intensity signal to obtain the double-aperture interferogram, perform Fourier transform on the interferogram, establish a corresponding relationship between the low-frequency component in the spectrum and the characteristic frequency of the double-aperture spacing d, and calculate the spatial coherence modulus. , and substituting it into the analytical formula for solving the coherence length in step S1, the coherence length under the corresponding coherence modulus can be obtained.

[0013] Furthermore, the calculation process of the analytical relationship between the coherence length and the spatial coherence modulus in step S1 is as follows:

[0014] In the Gauss-Scherr model beam, the spatial coherence can be characterized by the coherence degree, i.e.

[0015] (1)

[0016] in, are two points in the light field and The coherence between the two values ​​ranges from [0,1]. When , it means that there is complete coherence between the two points, that is, the phase difference between the two points is constant and the visibility of the interference fringes is maximum; when When , it means that the two points are completely unrelated and cannot interfere with each other; when When , it indicates partial coherence, and the degree of coherence increases with the value. and is the coherence length in the x and y directions;

[0017] , , when the light field is isotropic, that is , total spacing , formula (1) can be transformed into:

[0018] (2)

[0019] From formula (2), by taking the logarithm and solving the equation, the coherence length can be obtained: and Relationship:

[0020] (3)

[0021] Arranged:

[0022] (4)

[0023] because is a positive real number, and , and finally get the following formula:

[0024] (5)

[0025] The limiting case is: when , When , it means that the coherence has not decayed; when , When , it means that the coherence decays instantly; similarly, when the light field is anisotropic , calculate the coherence length in the x and y directions respectively:

[0026] (6)

[0027] (7)

[0028] in, is the spatial coherence modulus, is the distance between the two apertures in the x direction, is the distance between the two apertures in the y direction.

[0029] Furthermore, the specific calculation process of the intensity of the light beam at the focal plane after focusing in step S2 is as follows:

[0030] First, the cross spectral density CSD function of the twisted Gaussian Scherrer mode beam TGSM is:

[0031] (8)

[0032] in, is the beam waist width, is the coherence length, is the distortion factor, and represent two arbitrary position vectors of the source plane, k is the wave number, is an imaginary unit used to describe the phase characteristics of the wave;

[0033] The beam passes through a mask with a symmetrical aperture. The instantaneous field behind the mask is written as:

[0034] (9)

[0035] in, is a function describing the geometry of each aperture, a is the aperture radius, represents the light field within the aperture, represents the Dirac function, r m (m=1,2) represents the center position of the aperture, The symbol represents the convolution operation;

[0036] The far-field interference pattern generated by formula (9) can be obtained by placing a lens behind the mask, and the CSD function at the focal plane is obtained by using the generalized Collins formula:

[0037] (10)

[0038] in, is the wavelength, is the focal length, and are two lateral position vectors on the focal plane, is the ensemble average, is an imaginary unit;

[0039] Substituting formula (9) into formula (10) and integrating, we can obtain:

[0040] (11)

[0041] in, is the Bessel function, yes The complex conjugate of is the Bessel function of the first kind, is the CSD function of TGSM;

[0042] when When , the intensity distribution produced by the aperture pair is:

[0043] (12)

[0044] Among them, Re represents the real part, Indicates and The strength at two points, express and The coherence between two points, , the second term in formula (12) represents the interference caused by the aperture pair. According to the δ function approximation of formula (9), the aperture diameter D (i.e., D = 2a) has been eliminated in the far-field intensity formula (12) and subsequent spectrum analysis. Therefore, this method approximately ignores the change of the diameter D when the aperture spacing d is much larger than the aperture diameter D.

[0045] Furthermore, the specific process of step S3 is as follows:

[0046] Performing Fourier transform on formula (12) yields the Fourier spectrum of the interference pattern:

[0047] (13)

[0048] in, yes The Fourier transform is also the autocorrelation function of h(r), v represents the coordinates in the spatial frequency domain, and represents the light intensity at two points, represents the coherence between two points, yes The complex conjugate of represents the Dirac function;

[0049] According to the properties of convolution and Dirac function and because the double aperture is symmetrically distributed, formula (13) is simplified to obtain the following formula:

[0050] (14)

[0051] in, yes The Fourier transform of , v represents the coordinates in the spatial frequency domain, and Represents the light intensity at two points, d is the distance between the apertures, and when d>D, The values ​​of are very close to zero, so the complex amplitude components of the Fourier spectrum of the central peak and the secondary peak are obtained from formula (14):

[0052] (15)

[0053] (16)

[0054] in, is the phase of the central peak, Represents the Fourier transform value at spatial frequency (0,0), that is, the frequency domain of the total intensity, is the autocorrelation function of a single aperture function and its value at the origin is related to the aperture geometry, and represents the light intensity at two points, is the phase of the non-central peak, which directly corresponds to the coherent phase difference of the double-aperture light field. is the interference pattern at spatial frequency The Fourier transform value at , d is the double aperture spacing, is the wavelength, is the focal length, the frequency corresponds to the spatial separation information of the aperture pair, is the complex coherence modulus, is the coherence phase difference, so the spatial coherence modulus is given by Equation (17):

[0055] (17)

[0056] in, and are the complex amplitude components of the Fourier spectrum of the central peak and the secondary peak, and Indicates the light intensity at two points.

[0057] Compared with the prior art, the beneficial effects of the present invention are as follows: the present invention uses a method of double-aperture interference combined with Fourier spectrum analysis, and the corresponding relationship between the coherence modulus and the coherence length is given by the Gauss-Scherr model beam. The two are combined to establish a corresponding analytical relationship, without the need for iteration or numerical fitting, and direct calculation simplifies the process; this formula is applicable to light fields that satisfy any Gauss-Scherr model, and can cover a wider range of scenarios through anisotropy. BRIEF DESCRIPTION OF THE DRAWINGS

[0058] Figure 1 Interference pattern and amplitude spectrum with varying aperture distance. DETAILED DESCRIPTION

[0059] The following will describe the embodiments of the present invention in detail with reference to the accompanying drawings and examples, so as to fully understand and implement the process of how the present invention applies technical means to solve technical problems and achieve technical effects.

[0060] Based on dual-aperture interferometry and Fourier spectrum analysis, this paper establishes a direct analytical relationship between the spatial coherence modulus and the coherence length, providing a new approach for beam quality control in laser processing. The following describes a method for measuring the coherence length of a beam with a Scherrer model correlation structure. The method is carried out in the following steps:

[0061] Step S1: Calculate the coherence length in the x and y directions using the analytical relationship between the coherence length of the structured beam and the spatial coherence modulus using the Scherrer model. and , the specific calculation process is as follows:

[0062] In the Gauss-Scherr model beam, the spatial coherence is characterized by the coherence degree, i.e.

[0063] (1)

[0064] in Describing two points in a light field and The coherence between the two is in the range of [0,1]. When , it means that there is complete coherence between the two points, that is, the phase difference between the two points is constant and the visibility of the interference fringes is maximum; when When , it means that the two points are completely unrelated and cannot interfere with each other; when When , it indicates partial coherence, and the degree of coherence increases with the value. and is the coherence length in the x and y directions;

[0065] make , , if the light field is isotropic, that is , and define the total spacing , formula (1) can be transformed into:

[0066] (2)

[0067] From formula (2), by taking the logarithm and solving the equation, the coherence length can be obtained: and Relationship:

[0068] (3)

[0069] Arranged:

[0070] (4)

[0071] because is a positive real number, and , and finally get the following formula:

[0072] (5)

[0073] The limiting case is: when , When , it means that the coherence has not decayed; when , When , it means that the coherence decays instantly; similarly, when the light field is anisotropic , calculate the coherence length in the x and y directions respectively:

[0074] (6)

[0075] (7)

[0076] in, is the spatial coherence modulus, is the distance between the two apertures in the x direction, is the distance between the two apertures in the y direction.

[0077] Step S2: The laser processing system uses a Scherrer model beam. A double-aperture mask is set on the beam shaping transmission path of the laser processing system. The aperture positions are symmetrical and the spacing d is adjustable. After the beam is focused, the intensity signal is collected at the focal plane detector. , the specific calculation process is as follows:

[0078] First, the cross-spectral density (CSD) function of the twisted Gaussian Schell-model (TGSM) beam is:

[0079] (8)

[0080] in, is the beam waist width, is the coherence length, is the distortion factor, and represent two arbitrary position vectors of the source plane, k is the wave number, is an imaginary unit used to describe the phase characteristics of the wave;

[0081] The beam passes through a mask with a symmetrical aperture. The instantaneous field behind the mask is written as:

[0082] (9)

[0083] in, is a function that describes the geometry of each aperture, a is the aperture radius, represents the light field within the aperture, represents the Dirac function, r m (m=1,2) represents the center position of the aperture, The symbol represents the convolution operation;

[0084] The far-field interference pattern generated by formula (9) can be obtained by placing a lens behind the mask, and the CSD function at the focal plane is obtained by using the generalized Collins formula:

[0085] (10)

[0086] in, is the wavelength, is the focal length, and are two lateral position vectors on the focal plane, is the ensemble average, Is an imaginary unit.

[0087] Substituting formula (9) into formula (10) and integrating, we can obtain:

[0088] (11)

[0089] in, is the Bessel function, yes The complex conjugate of is the Bessel function of the first kind, is the CSD function of TGSM;

[0090] when When , the far-field intensity generated by the aperture is:

[0091] (12)

[0092] Among them, Re represents the real part, Indicates and The strength at two points, express and The coherence between two points, , the second term in formula (12) represents the interference generated by the aperture pair. According to the δ function approximation of formula (9), the aperture diameter D (i.e., D = 2a) has been eliminated in the far-field intensity formula (12) and subsequent spectrum analysis. Therefore, this method approximately ignores the change in diameter D when the aperture spacing d is much larger than the aperture diameter D.

[0093] Step S3: Extract the intensity signal to obtain a dual-aperture interferogram, perform Fourier transform on the interferogram, establish a corresponding relationship between the low-frequency component in the spectrum and the characteristic frequency of the corresponding dual-aperture spacing d, and calculate the spatial coherence modulus. The specific process is as follows:

[0094] Perform Fourier transform on formula (12) in step S2 to obtain the Fourier spectrum of the interference pattern

[0095] (13)

[0096] in, yes The Fourier transform is also the autocorrelation function of h(r), v represents the coordinates in the spatial frequency domain, and represents the light intensity at two points, represents the coherence between two points, yes The complex conjugate of represents the Dirac function.

[0097] According to the properties of convolution and Dirac function and because the double aperture is symmetrically distributed, formula (13) can be simplified to the following formula:

[0098] (14)

[0099] in, yes The Fourier transform of , v represents the coordinates in the spatial frequency domain, and Represents the light intensity at two points, d is the distance between the apertures, and when d>D, The values ​​of are very close to zero. Therefore, from formula (14), the complex amplitude components of the Fourier spectrum of the central peak and the secondary peak are:

[0100] (15)

[0101] (16)

[0102] in, is the phase of the central peak, Represents the Fourier transform value at spatial frequency (0,0), that is, the frequency domain of the total intensity, is the value of the autocorrelation function of a single aperture function at the origin, which is related to the aperture geometry, and represents the light intensity at two points, is the phase of the non-central peak, which directly corresponds to the coherent phase difference of the double-aperture light field. is the interference pattern at spatial frequency The Fourier transform value at , d is the aperture spacing, is the wavelength, is the focal length, the frequency corresponds to the spatial separation information of the aperture pair, is the complex coherence modulus of the double-aperture light field, is the coherent phase difference of the double-aperture light field, which is determined by the spatial distribution of the light field and affects the position of the interference fringes. Therefore, the spatial coherence modulus is given by Equation (17):

[0103] (17)

[0104] in, and are the complex amplitude components of the Fourier spectrum of the central peak and the secondary peak, and Indicates the light intensity at two points.

[0105] The effectiveness of the method of the present invention is verified by using specific cases below.

[0106] like Figure 1 As shown, the coordinates of the center point of the double aperture are taken as , , changing the distance d between the apertures. Graphs are plotted for d = 0.002m, 0.003m, 0.005m, and 0.01m, respectively. As shown in Figures (a1) to (d1), x represents the horizontal coordinate and y represents the vertical coordinate, both in meters. As the distance d between the apertures increases, the spatial period of the fringes decreases, meaning the distance between the fringes shortens, the number of observable fringes increases, and the visibility of the fringes decreases, consistent with theoretical results that spatial coherence decreases with increasing distance. Figures (a2) to (d2) are the amplitude plots corresponding to the intensity interferograms. As d increases, the secondary peak signal, which determines the complex coherence modulus, weakens. Table 1 also shows that the modulus decreases with increasing distance. The data in Table 1 conform to the exponential decay law of the Gauss-Scherr model. Furthermore, the coherence length measurement error rate is less than or equal to 0.01%, indicating that this measurement method is insensitive to changes in aperture spacing and is suitable for industrial scenarios.

[0107]

[0108] The above description is merely a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any changes or substitutions that can be easily conceived by a person skilled in the art within the technical scope disclosed in the present invention should be included in the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be based on the scope of protection of the claims.

Claims

1. A method for measuring the coherence length of a Scherrer model correlation structure beam, characterized by: Follow these steps: Step S1: Calculate the coherence length in the x and y directions using the analytical relationship between the coherence length of the Schell model beam and the spatial coherence modulus. and ; ； ； in, is the distance between two points in the light field in the x direction, is the distance between two points in the light field in the y direction, is the spatial coherence modulus; Step S2: Set a double aperture mask on the beam transmission path of the laser processing system, with the aperture positions symmetrical and the spacing d adjustable, and collect the far-field intensity of the far-field interference pattern through the focal plane detector. ; Step S3: Extract the intensity signal to obtain the double-aperture interferogram, perform Fourier transform on the interferogram, establish a corresponding relationship between the low-frequency component in the spectrum and the characteristic frequency of the double-aperture spacing d, and calculate the spatial coherence modulus. , and substituting it into the analytical formula for solving the coherence length in step S1, the coherence length under the corresponding coherence modulus can be obtained.

2. The method for measuring the coherence length of a Scherrer model correlation structure light beam according to claim 1, characterized in that: The calculation process of the analytical relationship between the coherence length and the spatial coherence modulus in step S1 is as follows: In the Gauss-Scherr model beam, the spatial coherence can be characterized by the coherence degree, i.e. （1） in, are two points in the light field and The coherence between the two values ​​ranges from [0,1]. When , it means that there is complete coherence between the two points, that is, the phase difference between the two points is constant and the visibility of the interference fringes is maximum; when When , it means that the two points are completely unrelated and cannot interfere with each other; when When , it indicates partial coherence, and the degree of coherence increases with the value. and is the coherence length in the x and y directions; , , when the light field is isotropic, that is , total spacing , formula (1) can be transformed into: （2） From formula (2), by taking the logarithm and solving the equation, the coherence length can be obtained: and Relationship: （3） Arranged: （4） because is a positive real number, and , and finally get the following formula: （5） The limiting case is: when , When , it means that the coherence has not decayed; when , When , it means that the coherence decays instantly; similarly, when the light field is anisotropic , calculate the coherence length in the x and y directions respectively: （6） （7） in, is the spatial coherence modulus, is the distance between the two apertures in the x direction, is the distance between the two apertures in the y direction.

3. The method for measuring the coherence length of a Scherrer model correlation structure light beam according to claim 2, characterized in that: The specific calculation process of the intensity of the light beam at the focal plane after focusing in step S2 is as follows: First, the cross-spectral density (CSD) function of the twisted Gaussian Schell-model (TGSM) beam is: （8） in, is the beam waist width, is the coherence length, is the distortion factor, and represent two arbitrary position vectors of the source plane, k is the wave number, is an imaginary unit used to describe the phase characteristics of the wave; The beam passes through a mask with a symmetrical aperture. The instantaneous field behind the mask is written as: （9） in, is a function describing the geometry of each aperture, a is the aperture radius, represents the light field within the aperture, represents the Dirac function, r m (m=1,2) represents the center position of the aperture, The symbol represents the convolution operation; The far-field interference pattern generated by formula (9) can be obtained by placing a lens behind the mask, and the CSD function at the focal plane is obtained by using the generalized Collins formula: （10） in, is the wavelength, is the focal length, and are two lateral position vectors on the focal plane, is the ensemble average, is an imaginary unit; Substituting formula (9) into formula (10) and integrating, we can obtain: （11） in, is the Bessel function, yes The complex conjugate of is the Bessel function of the first kind, is the CSD function of TGSM; when When , the intensity distribution produced by the aperture pair is: （12） Among them, Re represents the real part, Indicates and The strength at two points, represents the coherence between points r1 and r2, , the second term in formula (12) Represents the interference produced by the aperture pair.

4. The method for measuring the coherence length of a Scherrer model correlation structure light beam according to claim 3, characterized in that: The specific process of step S3 is as follows: Performing Fourier transform on formula (12) yields the Fourier spectrum of the interference pattern: （13） in, yes The Fourier transform is also the autocorrelation function of h(r), v represents the coordinates in the spatial frequency domain, and represents the light intensity at two points, represents the coherence between two points, yes The complex conjugate of represents the Dirac function; According to the properties of convolution and Dirac function and because the double aperture is symmetrically distributed, formula (13) is simplified to obtain the following formula: （14） in, yes The Fourier transform of , v represents the coordinates in the spatial frequency domain, and Represents the light intensity at two points, d is the distance between the apertures, and when d>D, The values ​​of are very close to zero, so the complex amplitude components of the Fourier spectrum of the central peak and the secondary peak are obtained from formula (14): （15） （16） in, is the phase of the central peak, Represents the Fourier transform value at spatial frequency (0,0), that is, the frequency domain of the total intensity, is the autocorrelation function of a single aperture function and its value at the origin is related to the aperture geometry, and represents the light intensity at two points, is the phase of the non-central peak, which directly corresponds to the coherent phase difference of the double-aperture light field. is the interference pattern at spatial frequency The Fourier transform value at , d is the double aperture spacing, is the wavelength, is the focal length, the frequency corresponds to the spatial separation information of the aperture pair, is the complex coherence modulus, is the coherence phase difference, so the spatial coherence modulus is given by Equation (17): （17） in, and are the complex amplitude components of the Fourier spectrum of the central peak and the secondary peak, and Indicates the light intensity at two points.

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