A Modeling Method for a Ring-Shaped Composite Beam Laser Welding Heat Source

Abstract

This invention discloses a modeling method for a ring-shaped composite beam laser welding heat source model, relating to the field of laser welding technology. The modeling method includes: 1. Improving the existing Gaussian plane distribution heat source model by allowing arbitrary power proportions instead of the existing 95% power limit within the heat source radius; 2. Establishing a ring-shaped plane distribution heat source model; 3. Establishing a Gaussian volume distribution heat source model based on the improved Gaussian plane distribution heat source model; 4. Establishing a ring-shaped volume distribution heat source model based on the ring-shaped plane distribution heat source model. This solves the problem of accurately establishing the heat source in laser welding numerical simulation.

Description

Technical Field

[0001] This invention relates to the field of laser welding technology, specifically to a modeling method for a heat source model of annular composite beam laser welding. Background Technology

[0002] Laser welding technology, as a highly efficient and precise welding method, has experienced rapid development and widespread application globally. Its applications are gradually increasing in fields such as power batteries, automobiles, and consumer electronics, leading to a continuous expansion of the market. With the development of high-end precision manufacturing, the demand for laser welding in 3C products, automotive lightweighting, and power and energy storage batteries is constantly growing, demonstrating a broad market prospect. The heat source model is the foundation of numerical simulation for laser welding; it needs to accurately reflect the physical process of laser welding, including the distribution of laser energy on the workpiece. To improve the weld quality of laser welding, composite beam lasers are increasingly being applied in production. Figure 1 As shown, the optical fiber of the composite laser includes a central core and a ring core. The laser propagated in the central core forms a circular spot on the surface of the weldment, with a power density distribution conforming to a Gaussian distribution; the laser propagated in the ring core forms an annular spot on the surface of the weldment, with a power density distribution on the annular cross-section also conforming to a Gaussian distribution; the combination of the two forms a composite laser heat source for laser welding.

[0003] The power density distribution of a commonly used Gaussian plane heat source model is as follows:

[0004]

[0005] Among them: Q c r is the laser power. H is the radius of the circular light spot; r is the distance between any point inside the light spot and the center of the light spot.

[0006] There are three main types of existing laser beam annular spot heat source models, with the following power density distributions:

[0007] a. The power density is equal across the ring-shaped laser spot, meaning the laser energy is evenly distributed:

[0008]

[0009] Among them: Q r r is the total power of the ring-shaped light spot; o r is the outer diameter of the annular light spot. i denoted as the inner diameter of the annular light spot.

[0010] b. Modified from a circular light spot planar Gaussian heat source:

[0011]

[0012] in: This is the location of the maximum power density at the center of the ring-shaped light spot; It is half the width of the annular light spot.

[0013] c. The maximum power density q m Set as an undetermined value, the exponent part is modified from the circular spot formula:

[0014]

[0015] Where, q m This represents the maximum power density on the annular light spot.

[0016] Since the power density distribution of the laser follows a Gaussian distribution on the cross-section of the annular spot and on the circular spot, it is necessary to specify the proportion of power within a given range to the total laser power when defining the radius of the circular spot and the inner and outer diameters of the annular spot. For example... Figure 2 As shown, the commonly used Gaussian plane heat source model assumes that within a radius r... H The integral of the range power density accounts for 95% of the total power. However, in practice, different standards are used to determine the spot radius, and other ratios may be used, such as 86.5%. Taking a circular spot as an example, at different radii r... H Table 1 shows the relative error of the total power under the ratio of power within the region to the total power, using the existing Gaussian spot planar heat source model. The table shows that when the ratio is 0.95, the relative error is zero because it is the same as the assumed ratio in the existing model. The greater the deviation of the ratio from 0.95, the larger the relative error. Therefore, the existing Gaussian planar heat source model needs to be modified to broaden its applicability.

[0017]

[0018] For a planar heat source, the power density distribution must meet the following conditions:

[0019] (1) In the target area, i.e., in the circular light spot, the distance from the center is less than r H The circular region and the annular light spot are located at a distance greater than the inner diameter r from the center. i Smaller than outer diameter r o In the region where the integral of the power density accounts for a constant η of the total laser power,... c ;

[0020] Circular light spot:

[0021] Ring-shaped light spot:

[0022] (2) The integral of the power density over an infinite range equals the total laser power Q.

[0023] ∫0∞ q(r)2πrdr=Q

[0024] According to relevant experimental data, the power density distribution within the annular light spot is not uniform, so formula (1) is only suitable for rough calculations; the power density distribution formula for the annular light spot shown in formula (2) is obtained directly from the power density formula for a circular light spot without rigorous derivation. (The last part, "r," appears to be a typo and can be omitted.) p =0.5mm,r i For example, with a diameter of 0.3mm:

[0025]

[0026] Therefore, formula (2) does not satisfy condition (2). For the power density distribution formula of the annular spot shown in formula (3), according to condition (1), we have:

[0027]

[0028] The integral of power density over infinity is:

[0029]

[0030] Less than Q r This does not meet condition (2). Summary of the Invention

[0031] To address the shortcomings of existing technologies, this invention provides a modeling method for a ring-shaped composite beam laser welding heat source model, which solves the problems mentioned in the background art.

[0032] To achieve the above objectives, the present invention provides the following technical solution: a modeling method for a ring-shaped composite beam laser welding heat source model, specifically including the following steps:

[0033] S1. Formula for constructing a Gaussian plane distribution heat source for a circular laser spot:

[0034] A1. Assume the power density distribution of the Gaussian plane heat source is as follows:

[0035]

[0036] Where q m Where is the maximum power density, and K is the thermal energy concentration factor.

[0037] A2. The conditions (2) for the power density distribution of a planar heat source are:

[0038]

[0039] Solving for: A3. The conditions for the power density distribution of a planar heat source are (1):

[0040] Simplifying, we get: In conclusion: S2. Formula for constructing the Gaussian plane distribution heat source of the laser ring spot:

[0041] B1. Assume that the power density distribution on the cross-section of the annular light spot follows a Gaussian distribution, and the maximum power density is q. m The power density distribution of the annular light spot can be expressed as:

[0042]

[0043] Where σ is the standard deviation of the Gaussian distribution.

[0044] B2. According to the condition (1) for the power density distribution of a planar heat source, we have:

[0045]

[0046] Because the power density distribution on the annular light spot is about the center line of the ring, r = r p Since the distribution is symmetric and the integral of the Gaussian distribution over an infinite interval is 1, we can obtain: B3. Based on the condition (2) for the power density distribution of a planar heat source, let the ratio of the power in the region between the inner and outer diameters of the annular spot to the total power be η. r ,but:

[0047]

[0048] make Based on the properties of the Gaussian distribution, we can obtain η r The relationship curve with 1 / β, such as Figure 3 As shown. According to η r The value of β is determined from the curve.

[0049] In summary, the formula for the power density distribution of the annular light spot is:

[0050]

[0051] S3. Constructing the power density distribution of the planar distributed heat source model of the annular composite beam laser:

[0052] q f (r)=q c (r0+q r (r0; (3)

[0053] S4. Construct the Gaussian volume distribution formula for the heat source of a circular laser spot. The volume distribution model of the circular laser heat source is as follows: Figure 4 As shown.

[0054] C1. Basic assumptions: (1) The radius of the heat source's area decreases linearly with the increase of the molten pool depth; (2) The power density at the center of the plane at different molten pool depths z is equal.

[0055] C2. Assume the power density distribution formula for the volume distribution model of a circular laser heat source is:

[0056]

[0057] C3. According to the law of conservation of energy:

[0058]

[0059] C4. From the fact that the radius of the heat source's effective region decreases linearly with the increase of the molten pool depth, we can conclude that:

[0060]

[0061] C5. Formula for obtaining the power density volume distribution of a circular light spot:

[0062]

[0063] S5. Construct the Gaussian volume distribution heat source formula for the laser ring spot. The volume distribution model of the ring laser heat source is as follows: Figure 5 As shown.

[0064] D1. Basic assumptions: (1) The width of the annular region of the heat source area decreases linearly with the increase of the molten pool depth; (2) The position of the center of the annular region remains unchanged in the plane where the molten pool depth z is located; (3) The power density of the center of the annular region is the largest and equal in the plane where the molten pool depth z is located.

[0065] D2. Assume the power density distribution formula of the ring laser heat source volume distribution model is:

[0066]

[0067] It can be transformed into: D3. According to the law of conservation of energy:

[0068]

[0069] D4. Because on the annular cross section, the power density distribution is about r = r p Symmetry

[0070]

[0071] D5. Within the plane of the molten pool at depth z, there is

[0072]

[0073] We can obtain:

[0074]

[0075] make but:

[0076]

[0077] D6. According to condition (1), in the plane of the molten pool with depth z, there is

[0078]

[0079] Among them, z s and z b These are the values ​​of the molten pool surface and bottom surface in the z-axis direction, respectively; r so r bo r si and r bi These are the outer and inner diameters of the areas where the annular volumetric heat source acts on the surface and bottom of the molten pool, respectively.

[0080] w z =r zo -r zi

[0081] D7. Obtain the formula for the volume distribution of power density of the annular light spot:

[0082]

[0083] S6. Constructing the power density distribution of the volumetric heat source model for the annular composite beam laser:

[0084] q Vf (r)=q Vc (r)+q Vr (r). (6)

[0085] Preferably, the formula for constructing the Gaussian plane distribution heat source of the laser ring spot specifically includes the following steps:

[0086] B1. Assume that the power density distribution on the cross-section of the annular light spot follows a Gaussian distribution, and the maximum power density is q. m The power density distribution of the annular light spot can be expressed as:

[0087]

[0088] B2. According to the condition (1) for the power density distribution of a planar heat source, we have:

[0089]

[0090] Because the power density distribution on the annular light spot is about the center line of the ring, r = r p Since the distribution is symmetric and the integral of the Gaussian distribution over an infinite interval is 1, we can obtain:

[0091] B3. Based on the condition (2) for the power density distribution of a planar heat source, let the ratio of the power in the region between the inner and outer diameters of the annular spot to the total power be η. r ,but:

[0092]

[0093] make Based on the properties of the Gaussian distribution, we can obtain η r The curve relating to 1 / β;

[0094] According to η r The value of β is determined from the curve;

[0095] In summary, the formula for the power density distribution of the annular light spot is:

[0096]

[0097] Preferably, in A1, q m Where is the maximum power density, and K is the thermal energy concentration factor.

[0098] Preferably, in B1, σ is the standard deviation of a Gaussian distribution.

[0099] Preferably, in D6, z s and z b These are the values ​​of the molten pool surface and bottom surface in the z-axis direction, respectively; r so r bo r si and r bi These are the outer and inner diameters of the areas where the annular volumetric heat source acts on the surface and bottom of the molten pool, respectively.

[0100] Beneficial effects

[0101] This invention provides a modeling method for a ring-shaped composite beam laser welding heat source model. Compared with existing technologies, it has the following advantages:

[0102] The modeling method for the annular composite beam laser welding heat source model introduces a parameter η, which characterizes the proportion of energy within the radius of the circular spot to the total energy. c The derivation of any η c Below is a Gaussian plane heat source model. A parameter η is introduced to represent the proportion of energy within the bandwidth of the annular light spot to the total energy. rThe derivation of any η r Under these conditions, the power density distribution formula of the annular light spot strictly conforms to conditions 1 and 2, thus establishing a planar distribution heat source model for the annular light spot. Furthermore, based on the same inventive approach, a volumetric distribution heat source model was established, exhibiting high accuracy and providing valuable guidance. Attached Figure Description

[0103] Figure 1 This is a schematic diagram of the composite beam laser of the present invention;

[0104] Figure 2 This is a schematic diagram illustrating the definition of the heat source radius in this invention;

[0105] Figure 3 For the present invention η r The curve showing the relationship between 1 / β;

[0106] Figure 4 This is a schematic diagram of the volume distribution model of the circular laser heat source of the present invention;

[0107] Figure 5 This is a schematic diagram of the volume distribution model of the ring laser heat source of the present invention;

[0108] Figure 6 This is a planar distribution diagram of the laser power density of the central circular heat source in this invention;

[0109] Figure 7 This is a planar distribution diagram of the laser power density of the annular heat source of the present invention;

[0110] Figure 8 This is a planar distribution diagram of the laser power density of the composite heat source of the present invention;

[0111] Figure 9 This is a power density distribution diagram on the y=0 section under different power ratios of the central circular heat source of this invention;

[0112] Figure 10 This is a power density distribution diagram on the y=0 section under different power ratios of the annular heat source of the present invention;

[0113] Figure 11 This is a power density distribution diagram on the y=0 section under different power ratios of the composite heat source of the present invention;

[0114] Figure 12 This is a power density distribution diagram of the composite heat source volume distribution model on the surface of the weldment of the present invention;

[0115] Figure 13 This is a power density distribution diagram of the composite heat source distribution model inside the weldment of the present invention. Detailed Implementation

[0116] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0117] Please see Figure 1-5 This invention provides a technical solution: a modeling method for a ring-shaped composite beam laser welding heat source model, specifically including the following steps:

[0118] S1. Formula for constructing a Gaussian plane distribution heat source for a circular laser spot:

[0119] A1. Assume the power density distribution of the Gaussian plane heat source is as follows:

[0120]

[0121] Where q m Where is the maximum power density, and K is the thermal energy concentration factor.

[0122] A2. The conditions (2) for the power density distribution of a planar heat source are:

[0123]

[0124] Solving for: A3. The conditions for the power density distribution of a planar heat source are (1):

[0125]

[0126] Simplifying, we get: In conclusion: S2. Formula for constructing the Gaussian plane distribution heat source of the laser ring spot:

[0127] B1. Assume that the power density distribution on the cross-section of the annular light spot follows a Gaussian distribution, and the maximum power density is q. m The power density distribution of the annular light spot can be expressed as:

[0128]

[0129] Where σ is the standard deviation of the Gaussian distribution.

[0130] B2. According to the condition (1) for the power density distribution of a planar heat source, we have:

[0131]

[0132] Because the power density distribution on the annular light spot is about the center line of the ring, r = r p Since the distribution is symmetric and the integral of the Gaussian distribution over an infinite interval is 1, we can obtain: B3. Based on the condition (2) for the power density distribution of a planar heat source, let the ratio of the power in the region between the inner and outer diameters of the annular spot to the total power be η. r ,but:

[0133]

[0134] make Based on the properties of the Gaussian distribution, we can obtain η r The relationship curve with 1 / β, such as Figure 3 As shown. According to η r The value of β is determined from the curve.

[0135] In summary, the formula for the power density distribution of the annular light spot is:

[0136]

[0137] S3. Constructing the power density distribution of the planar distributed heat source model of the annular composite beam laser:

[0138] q f (r)=q c (r)+q r (r);

[0139] S4. Construct the Gaussian volume distribution formula for the heat source of a circular laser spot. The volume distribution model of the circular laser heat source is as follows: Figure 4 As shown.

[0140] C1. Basic assumptions: (1) The radius of the heat source's area decreases linearly with the increase of the molten pool depth; (2) The power density at the center of the plane at different molten pool depths z is equal.

[0141] C2. Assume the power density distribution formula for the volume distribution model of a circular laser heat source is:

[0142]

[0143] C3. According to the law of conservation of energy:

[0144]

[0145] C4. From the fact that the radius of the heat source's effective region decreases linearly with the increase of the molten pool depth, we can conclude that:

[0146]

[0147] C5. Formula for obtaining the power density volume distribution of a circular light spot:

[0148]

[0149] S5. Construct the Gaussian volume distribution heat source formula for the laser ring spot. The volume distribution model of the ring laser heat source is as follows: Figure 5 As shown:

[0150] D1. Basic assumptions: (1) The width of the annular region of the heat source area decreases linearly with the increase of the molten pool depth; (2) The position of the center of the annular region remains unchanged in the plane where the molten pool depth z is located; (3) The power density of the center of the annular region is the largest and equal in the plane where the molten pool depth z is located.

[0151] D2. Assume the power density distribution formula of the ring laser heat source volume distribution model is:

[0152]

[0153] It can be transformed into: D3. According to the law of conservation of energy:

[0154]

[0155] D4. Because on the annular cross section, the power density distribution is about r = r p Symmetry

[0156]

[0157] D5. Within the plane of the molten pool at depth z, there is

[0158]

[0159] We can obtain:

[0160]

[0161] make but:

[0162]

[0163] D6. According to assumption (1), in the plane of the molten pool with depth z, there is

[0164]

[0165] Among them, z s and z b These are the values ​​of the molten pool surface and bottom surface in the z-axis direction, respectively; r so r bo r si and r biThese are the outer and inner diameters of the areas where the annular volumetric heat source acts on the surface and bottom of the molten pool, respectively.

[0166] w z =r zo -r zi

[0167] D7. Obtain the formula for the volume distribution of power density of the annular light spot:

[0168]

[0169] S6. Constructing the power density distribution of the volumetric heat source model for the annular composite beam laser:

[0170] q Vf (r)=q Vc (r)+q Vr (r).

[0171] Preferably, the formula for constructing the Gaussian plane distribution heat source of the laser ring spot specifically includes the following steps:

[0172] B1. Assume that the power density distribution on the cross-section of the annular light spot follows a Gaussian distribution, and the maximum power density is q. m The power density distribution of the annular light spot can be expressed as:

[0173]

[0174] B2. According to the condition (1) for the power density distribution of a planar heat source, we have:

[0175]

[0176] Because the power density distribution on the annular light spot is about the center line of the ring, r = r p Since the distribution is symmetric and the integral of the Gaussian distribution over an infinite interval is 1, we can obtain:

[0177] B3. Based on the condition (2) for the power density distribution of a planar heat source, let the ratio of the power in the region between the inner and outer diameters of the annular spot to the total power be η. r ,but:

[0178]

[0179] make Based on the properties of the Gaussian distribution, we can obtain η r The curve relating to 1 / β;

[0180] According to η r The value of β is determined from the curve;

[0181] In summary, the formula for the power density distribution of the annular light spot is:

[0182]

[0183] In this invention, in A1, q m Where is the maximum power density, and K is the thermal energy concentration factor.

[0184] In this invention, σ in B1 is the standard deviation of the Gaussian distribution.

[0185] In this invention, in D6, z s and z b These are the values ​​of the molten pool surface and bottom surface in the z-axis direction, respectively; r so r bo r si and r bi These are the outer and inner diameters of the areas where the annular volumetric heat source acts on the surface and bottom of the molten pool, respectively.

[0186] In existing technologies, such as Figure 1 As shown, the ring-shaped composite laser beam is formed by irradiation through a double-layer optical fiber using lasers of different powers. Figure 2 As shown, the annular composite laser welding heat source consists of a central circular heat source and an outer annular heat source.

[0187] like Figure 3 As shown, the heat source model of the laser includes the planar power density distribution of the laser irradiating the surface of the weldment and the distribution of laser energy in the molten pool space, i.e., the volumetric power density distribution.

[0188] To address the shortcomings of the existing technology, this invention introduces a parameter η that characterizes the proportion of energy within the radius of a circular light spot to the total energy. c The derivation of any η c Below is a Gaussian plane heat source model. A parameter η is introduced to represent the proportion of energy within the bandwidth of the annular light spot to the total energy. r The derivation of any η r Under these conditions, the power density distribution formula of the annular light spot strictly conforms to conditions 1 and 2, thus establishing a planar distribution heat source model for the annular light spot. Based on the same inventive approach, a volumetric distribution heat source model was also established.

[0189] The process will be described in detail below with specific examples.

[0190] (1) Planar distribution model of laser heat source

[0191] According to formula (3), the variable parameters are set as follows, with the central laser power being Q. c =4000W; the power of the ring laser is power Q. r =2000W; the effective heating radius of the central laser is rH =1mm; the inner diameter of the effective heating area of ​​the ring laser heat source is r i =1mm; outer diameter is r o =3mm. Figure 6 , Figure 7 and Figure 8 The power density distributions in the composite planar laser heat source model are given for the central improved Gaussian circular planar heat source, the surrounding annular planar heat source, and the composite planar distributed heat source model of the sum of the two. Figure 9 , Figure 10 and Figure 11 The curves showing the power density as a function of the x-coordinate under different power density percentages within the y=0 cross section are presented.

[0192] (2) Laser heat source volume distribution model

[0193] According to formula (6), the variable parameters are set as follows: the central laser power is Qc = 2000W; the ring laser power is Qr = 10000W; and the effective heating radius of the central laser on the molten pool surface is r. sH =1mm; the inner diameter of the effective heating area of ​​the ring laser heat source is r si =1mm; outer diameter is r so =3mm; the effective heating radius of the laser at the center of the bottom of the molten pool is r. bH =0.5mm; the inner diameter of the effective heating area of ​​the ring laser heat source is r bi =1.4mm; outer diameter is r bo =2.6mm; the molten pool depth is H=3mm. Figure 12 and Figure 13 The spatial power density distribution of the composite heat source model is given.

[0194] It should be noted that, in this document, relational terms such as "first" and "second" are used only to distinguish one entity or operation from another, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Furthermore, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such process, method, article, or apparatus.

[0195] Although embodiments of the invention have been shown and described, it will be understood by those skilled in the art that various changes, modifications, substitutions and alterations can be made to these embodiments without departing from the principles and spirit of the invention, the scope of which is defined by the appended claims and their equivalents.

Claims

1. A modeling method for a ring-shaped composite beam laser welding heat source model, characterized in that: Specifically, the following steps are included: S1. Formula for constructing a Gaussian plane distribution heat source for a circular laser spot: A1. Assume the power density distribution of the Gaussian plane heat source is as follows: ； A2. The conditions (2) for the power density distribution of a planar heat source are: Solving for: ; A3. The conditions for the power density distribution of a planar heat source are (1): Simplifying, we get: ; In conclusion: ; S2. Construct the formula for the Gaussian plane distribution heat source of the laser ring spot; S3. Constructing the power density distribution of the planar distributed heat source model of the annular composite beam laser: ； S4. Formula for constructing the Gaussian volume distribution heat source of a circular laser spot: C1. Basic assumptions: (1) The radius of the heat source's area decreases linearly with the increase of the molten pool depth; (2) The power density at the center of the plane at different molten pool depths z is equal. C2. Assume the power density distribution formula for the volume distribution model of a circular laser heat source is: ； C3. According to the law of conservation of energy: ； C4. From the fact that the radius of the heat source's effective region decreases linearly with the increase of the molten pool depth, we can conclude that: ； C5. Formula for obtaining the power density volume distribution of a circular light spot: ； S5. Formula for constructing the Gaussian volume distribution heat source of the laser ring spot: D1. Basic assumptions: (1) The width of the annular region of the heat source area decreases linearly with the increase of the molten pool depth; (2) The position of the center of the annular region remains unchanged in the plane where the molten pool depth z is located; (3) The power density of the center of the annular region is the largest and equal in the plane where the molten pool depth z is located. D2. Assume the power density distribution formula of the ring laser heat source volume distribution model is: It can be transformed into: ; D3. According to the law of conservation of energy: ； D4. Because on the annular cross-section, the power density distribution is related to r= Symmetry ； D5. Within the plane of the molten pool at depth z: We can obtain: make but: ； D6. According to assumption (1), in the plane of the molten pool with depth z, we have: ； D7. Obtain the formula for the volume distribution of power density of the annular light spot: ； S6. Constructing the power density distribution of the volumetric heat source model for the annular composite beam laser: ； In A1, Where K is the maximum power density and K is the thermal energy concentration factor; In D6, and These are the values ​​of the molten pool surface and bottom surface in the z-axis direction, respectively; , , and These are the outer and inner diameters of the areas where the annular volumetric heat source acts on the surface and bottom of the molten pool, respectively.

2. The modeling method for a ring-shaped composite beam laser welding heat source model according to claim 1, characterized in that: The formula for constructing the Gaussian plane distribution heat source of the laser ring spot specifically includes the following steps: B1. Assume that the power density distribution on the cross-section of the annular light spot follows a Gaussian distribution, and the maximum power density is... The power density distribution of the annular light spot can be expressed as: ； B2. According to the condition (1) for the power density distribution of a planar heat source, we have: Because the power density distribution on the annular light spot is about the center line of the ring... Since the distribution is symmetric and the integral of the Gaussian distribution over an infinite interval is 1, we can obtain: ; B3. Based on the condition (2) for the power density distribution of a planar heat source, let the ratio of the power in the region between the inner and outer diameters of the annular spot to the total power be... ,but: ； make According to the properties of the Gaussian distribution, we can obtain and Relationship curve; according to The value is determined from the curve. The value; In summary, the formula for the power density distribution of the annular light spot is: ； In B1, denoted as the standard deviation of the Gaussian distribution.

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