Calculation method for the scattering characteristics of blue-green laser after passing through a ship's bubble wake

Abstract

The present invention discloses a method for calculating the scattering characteristics of a blue-green laser after passing through a ship bubble wake. The method comprises the following steps: Step 1: Based on the light field function expression of a Bessel-Gaussian beam in the source plane, the electric field components of the Bessel-Gaussian beam in the xyz directions in a spatial rectangular coordinate system are obtained; Step 2: The radial components of the electric and magnetic fields of the Bessel-Gaussian beam in a spherical coordinate system are solved; Step 3: The beam factor in the transverse magnetic and transverse electric modes is solved based on the radial components of the electric and magnetic fields obtained in Step 2; Step 4: The expanded field of the Bessel-Gaussian beam scattered by a single spherical particle is solved using the generalized Lorentz Mie theory; Step 5: The electric field of the Bessel-Gaussian beam scattered by a ship bubble wake is solved; Step 6: The scattering characteristics are solved using the obtained scattered electric field. This method can be used to analyze the scattering characteristics of a blue-green laser after being scattered by a ship bubble wake and has important reference value for rationally selecting beam parameters to improve transmission quality and ship wake identification.

Description

Technical Field

[0001] The present invention belongs to the technical field of wireless laser communications, and in particular relates to a method for calculating the scattering characteristics of blue-green lasers after passing through a ship's bubble wake. Background Art

[0002] In modern maritime military and civilian navigation, the identification and tracking of ship wakes has long been an important research area. Wakes contain a wealth of physical and chemical information that can be used to identify and track ships. Traditional wake identification methods primarily rely on technologies such as sonar and radar. While these technologies perform well in many respects, they still have limitations, such as significant susceptibility to environmental noise and limited identification range. Compared to sound waves, light waves have shorter wavelengths and higher propagation speeds. They exhibit good directionality when propagating in seawater, are particularly sensitive to bubbles in wakes, and are less susceptible to external interference. Wireless optical communication, as an emerging communication technology, offers advantages such as high bandwidth, low latency, and resistance to electromagnetic interference. Applying wireless optical communication technology to ship wake identification not only overcomes the shortcomings of traditional methods but also provides a more efficient and accurate solution.

[0003] Although the relevant theories of ship wakes were proposed early, research on the scattering characteristics of light beams in ship bubble wakes is still relatively scarce. Most existing studies use plane waves as input and obtain the scattered light intensity through Mie theory, or use Monte Carlo simulation methods to study optical scattering characteristics by photon simulation scattering. However, these methods still have certain shortcomings in terms of accuracy, applicability and operability. In addition, although there have been basic studies on the scattering characteristics of lasers on single particles, research on the scattering characteristics of vortex light beams in ship bubble wakes is still relatively scarce. Based on the generalized Lorentz Mie theory, this study analyzes the scattering characteristics of vortex light beams in ship bubble wakes, providing certain theoretical support for the identification of ship bubble wakes. Summary of the Invention

[0004] The purpose of the present invention is to provide a calculation method for the scattering characteristics of blue-green laser after passing through the bubble wake of a ship. This method can be used to analyze the scattering characteristics of blue-green laser after being scattered by the bubble wake of a ship, and has important reference value for rationally selecting beam parameters to improve transmission quality and ship wake identification.

[0005] The technical solution adopted by the present invention is a method for calculating the scattering characteristics of a blue-green laser after passing through a ship bubble wake, which specifically includes the following steps: Step 1, based on the light field function expression of a Bessel-Gaussian beam in the source plane, expand it using an angular spectrum to obtain the electric field components of the Bessel-Gaussian beam in the xyz directions in a spatial rectangular coordinate system; Step 2, solve the radial components of the electric field and magnetic field of the Bessel-Gaussian beam in a spherical coordinate system; Step 3, based on the radial components of the electric field and magnetic field obtained in Step 2, solve the beam factor in the transverse magnetic and transverse electric modes; Step 4, use the generalized Lorentz Mie theory to solve the expanded field of the Bessel-Gaussian beam scattered by a single spherical particle; Step 5, solve the electric field of the Bessel-Gaussian beam after passing through a ship bubble wake; Step 6, use the generalized Lorentz Mie theory, combined with the beam factor obtained in Step 3 and the scattered electric field obtained in Step 5, to solve the scattering characteristics.

[0006] The present invention is also characterized in that:

[0007] Step 1 is implemented according to the following specific steps:

[0008] Step 1.1. First, determine the light field function E(x, y, 0) of the Bessel-Gaussian beam at the source plane (emitter). E(x, y, 0) is expressed in the spatial rectangular coordinate system O-xyz as:

[0009]

[0010] Where E0 is the electric field amplitude, J l (·) is the lth-order Bessel function, k = 2π / λ is the wave number, λ is the wavelength in the medium, ρ 2 =x 2 +y 2 , α b is the cone angle, i is the imaginary unit, and the integer l represents the quantum number of orbital angular momentum. is the azimuth angle, ω0 represents the beam waist radius, and x, y, and z are the coordinates of the rectangular coordinate system;

[0011] When z≥0 in the free half-space, the electric field amplitude of the Bessel-Gaussian beam is expressed according to the angular spectrum as:

[0012]

[0013] Among them, E x (x,y,z) is the electric field component of the Bessel-Gaussian beam in the x direction, E y (x,y,z) is the electric field component of the Bessel-Gaussian beam in the y direction, E z (x,y,z) is the electric field component of the Bessel-Gaussian beam in the z direction, U x (k x ,k y) represents the complex factor in the x direction, U y (k x ,k y ) represents the complex factor in the y direction, k x 、k y 、k z represent the direction cosines in the x, y, and z directions respectively, and:

[0014]

[0015] Complex factor U x (k x ,k y ) and U y (k x ,k y ) are determined by Fourier transform and are given by:

[0016]

[0017] in

[0018]

[0019] where γ = ksinα b , θ is the azimuth angle, sinθ is the ratio of the electric field component in the x direction to the total electric field; cosθ is the ratio of the electric field component in the y direction to the total electric field;

[0020] Step 1.2: Calculate the expression of the complex factors in the angular spectrum representation of the Bessel-Gaussian beam to obtain the electric field components in the x, y, and z directions in the spatial rectangular coordinate system after free space transmission. The calculation process is as follows:

[0021] First, replace E in formula (5) x Substitute (x,y,0) into U in formula (4) x (k x ,k y )get:

[0022]

[0023] Among them, the Bessel function has the following properties:

[0024]

[0025] So, by substituting formula (7) into formula (6), we get:

[0026]

[0027] Where ρ 2 =x 2 +y 2is a plane z = 0 and The radial coordinate at time ;

[0028] Then, according to the integral formula of Bessel function:

[0029]

[0030] Among them, I l (·) represents the l-th order modified Bessel function. Substituting formula (9) into formula (8), we get:

[0031]

[0032] E in formula (5) y Substitute (x,y,0) into U in formula (4) y (k x ,k y ), repeat the above calculation process and get:

[0033]

[0034] Substituting formula (10) and formula (11) into formula (2), we can obtain the electric field expression after the Bessel-Gaussian beam propagates, which is as follows:

[0035]

[0036] There are still integrals in formula (12) and formula (13) that have not been solved. The steady-phase method is used to solve the integrals. According to the steady-phase method, the following formula is obtained:

[0037]

[0038] A(x,y) on the left side of formula (14) refers to f(x,y) refers to k in formula (12) x x+k y y+k z z; in A(x0,y0) and f(x0,y0) on the right side of the formula, x0 is the stable phase point, and its value is y0 is the stable phase point, and its value is At the steady-state point,

[0039] in,

[0040] Substituting formula (14) into formula (12) and formula (13) yields the electric field expression:

[0041]

[0042] Step 2 is implemented according to the following specific steps:

[0043] Step 2.1, transform the electromagnetic field expression in the rectangular coordinate system into the spherical coordinate system (r, θ, is the spherical coordinate system), and calculate the radial component E of the electric field of the Bessel-Gaussian beam r (x,y,z) and the radial component of the magnetic field H r (x,y,z):

[0044]

[0045] The integral local approximation method is used to solve the problem, where kr→n+1 / 2, n is an integer, and θ→π / 2. The magnetic field distribution is as follows:

[0046]

[0047]

[0048] Where μ is the magnetic permeability and ε is the dielectric constant. The radial components of the electric and magnetic fields of the Bessel-Gaussian beam are obtained as follows:

[0049]

[0050] Where H0 is the magnetic field amplitude.

[0051] Step 3 is implemented according to the following specific steps:

[0052] Substitute the radial component of the electric field (Equation (20)) and the radial component of the magnetic field (Equation (21)) in the spherical coordinate system obtained in step 2 into the integral expression of the beam factor in the transverse magnetic and transverse electric modes, and replace E r (x,y,z) and H r (x,y,z) is abbreviated as E r and H r , as shown below:

[0053]

[0054] Where, is the beam factor in the transverse magnetic mode, is the beam factor in the transverse electric mode, is the composite normalization factor, m is an integer, and n is the maximum number of iterations for Mie scattering calculation:

[0055]

[0056] And using the orthogonality of the exponential function and the trigonometric function, the beam factor of the beam incident on the axis is calculated, and the expression is as follows:

[0057] When m=l±1:

[0058]

[0059] When m≠l±1:

[0060] In step 4, the expanded field includes the electric field E of the incident field i and the magnetic field H of the incident field i , the electric field E of the scattered field s and the magnetic field H of the scattered field s , the electric field E of the internal field t and the magnetic field H of the internal field t , the specific expression is as follows:

[0061]

[0062] Where η is the characteristic impedance of seawater, η' is the characteristic impedance of bubble particles relative to seawater, and a mn 、b mn 、c mn d mn is the expansion coefficient, and is the spherical vector wave function, the superscripts 1 and 3 represent the first and third types of spherical vector wave functions, respectively. and Use M and N instead;

[0063]

[0064] The spherical vector wave function is defined as:

[0065]

[0066] in, is a scalar function, a is a constant vector, and the spherical vector wave function is specifically expanded as follows:

[0067]

[0068] Wherein the subscripts o and e represent The parity of is the first kind of associated Legendre function of order n and degree m, zn(kr) represents the 4-type spherical Bessel function j n (kr), y n (kr), one of the.

[0069] Step 5 is implemented according to the following specific steps:

[0070] Step 5.1. Calculate the expansion coefficient. Consider a single bubble in the ship bubble wake as a dielectric sphere. The boundary conditions of the dielectric sphere are:

[0071] When r=r1, Where r1 is the radius of the dielectric sphere, is the θ component of the incident electric field and scattered electric field in the spherical coordinate system; are the incident electric field and scattered electric field in spherical coordinate system Quantity; is the θ component of the incident magnetic field and the scattered magnetic field in the spherical coordinate system; are the incident magnetic field and the scattered magnetic field in the spherical coordinate system Components, expand the boundary conditions to obtain the expansion coefficients:

[0072]

[0073]

[0074] Where k'=kn, n is the refractive index of the spherical bubble particle relative to seawater; η'=k' / ω0μ;

[0075] Step 5.2: Calculate the electric field of far-field scattering. When calculating far-field scattering, the scattered electric field only has θ and Quantity and Therefore, the total scattered electric field is:

[0076]

[0077] The θ and The component is expressed as

[0078]

[0079] Among them, S1 and S2 are the amplitudes of the scattered electric field, and their expressions are:

[0080]

[0081] where a n and b n For the sake of simplicity, we take it as Mie coefficient.

[0082]

[0083] is the first kind of associated Legendre function.

[0084] Step 6 is implemented according to the following specific steps:

[0085] Step 6.1. Calculate the scattering efficiency factor Q of the scattered electric field according to the following formula: sca , extinction efficiency factor Q ext, absorption efficiency factor Q abs And the differential scattering cross section σ:

[0086]

[0087] in, is the θ component of the scattered energy density in spherical coordinates and Component, the superscript * of the magnetic field indicates conjugation;

[0088] Step 6.2: Obtain the volume scattering coefficient based on the scattering efficiency factor and the distribution function of the scattering medium. The calculation formula for the volume scattering coefficient is as follows:

[0089]

[0090] The formula for the differential scattering cross section is as follows:

[0091]

[0092] Considering that the bubbles in the ship bubble wake exist in the form of particle groups, the average differential scattering cross section is calculated, and its expression is:

[0093]

[0094] Among them, n(r) is the fractional function of the bubble group, which is as follows

[0095]

[0096] Among them, the relationship between the bubble spectral density n0 and the bubble number density N0 is:

[0097]

[0098] In step 6, select the minimum radius r min =10μm, maximum radius r max =80μm, most probable radius r peak =20μm.

[0099] The beneficial effects of the present invention are as follows: the method for calculating the afterscattering characteristics of a blue-green laser after passing through a ship's bubble wake utilizes the advantages of blue-green lasers in ship wake transmission, and based on the generalized Lorentz Mie theory, derives and calculates the electric field distribution of a Bessel-Gaussian beam scattered by a ship's wake, and analyzes the characteristics of the Bessel-Gaussian beam scattered by a ship's wake. The method of the present invention is easy to understand, is not limited by experimental conditions, and is easy to implement. The purpose of the present invention is to provide a method for calculating the afterscattering characteristics of a blue-green laser after passing through a ship's bubble wake. The method can be used to analyze the characteristics of blue-green lasers after being scattered by a ship's wake under different beam property conditions and different bubble radii in the ship's bubble wake, and has important reference value for ship wake identification. BRIEF DESCRIPTION OF THE DRAWINGS

[0100] Figure 1 This is a graph showing how the scattering efficiency factor of a Bessel-Gaussian beam scattered by a single spherical bubble particle changes with the scale parameter at different cone angles (0 rad and 3 rad);

[0101] Figure 2 The volume scattering coefficient of the Bessel-Gaussian beam after scattering by the ship's bubble wake varies with the particle radius at different transmission distances (50 μm and 100 μm);

[0102] Figure 3 This is a graph showing the change in the differential scattering cross section of a Bessel-Gaussian beam under different topological charge numbers (l=1; l=2; l=3) after being scattered by the ship bubble wake. DETAILED DESCRIPTION

[0103] The present invention will be described in detail below with reference to the accompanying drawings and specific embodiments.

[0104] The method for calculating the scattering characteristics of blue-green laser after passing through a ship bubble wake of the present invention specifically comprises the following steps:

[0105] Step 1: Based on the light field function expression of the Bessel-Gaussian beam in the source plane, expand it using the angular spectrum to obtain the electric field components of the Bessel-Gaussian beam in the xyz direction in the spatial rectangular coordinate system;

[0106] Step 1 is implemented according to the following specific steps:

[0107] Step 1.1. First, determine the light field function E(x, y, 0) of the Bessel-Gaussian beam at the source plane (emitter). E(x, y, 0) is expressed in the spatial rectangular coordinate system O-xyz as:

[0108]

[0109] Where E0 is the electric field amplitude, J l (·) is the lth-order Bessel function, k = 2π / λ is the wave number, λ is the wavelength in the medium, ρ 2 =x 2 +y 2 , α b is the cone angle, i is the imaginary unit, and the integer l represents the quantum number of orbital angular momentum. is the azimuth angle, ω0 represents the beam waist radius, and x, y, and z are the coordinates of the rectangular coordinate system;

[0110] According to the angular spectrum representation, any electric field amplitude can be expanded into the angular spectrum of a plane wave entering the free half-space z ≥ 0. Therefore, when the free half-space z ≥ 0, the electric field amplitude of the Bessel-Gaussian beam is expressed in terms of the angular spectrum as:

[0111]

[0112] Among them, E x (x,y,z) is the electric field component of the Bessel-Gaussian beam in the x direction, E y (x,y,z) is the electric field component of the Bessel-Gaussian beam in the y direction, E z (x,y,z) is the electric field component of the Bessel-Gaussian beam in the z direction, U x (k x ,k y ) represents the complex factor in the x direction, U y (k x ,k y ) represents the complex factor in the y direction, k x 、k y 、k z represent the direction cosines in the x, y, and z directions respectively, and:

[0113]

[0114] exist In the region, every plane wave is homogeneous and every plane wave is evanescent; and In the region, each plane wave decays exponentially along the z axis. Due to the long transmission distance, this part can be ignored. The complex factor U x (k x ,k y ) and U y (k x ,k y ) are determined by Fourier transform and are given by:

[0115]

[0116] in

[0117]

[0118] where γ = ksinα b , θ is the azimuth angle, sinθ is the ratio of the electric field component in the x direction to the total electric field; cosθ is the ratio of the electric field component in the y direction to the total electric field;

[0119] Step 1.2: Calculate the expression of the complex factors in the angular spectrum representation of the Bessel-Gaussian beam to obtain the electric field components in the x, y, and z directions in the spatial rectangular coordinate system after free space transmission. The calculation process is as follows:

[0120] First, replace E in formula (5) x Substitute (x,y,0) into U in formula (4) x (k x,k y )get:

[0121]

[0122] Among them, the Bessel function has the following properties:

[0123]

[0124] So, by substituting formula (7) into formula (6), we get:

[0125]

[0126] Where ρ 2 =x 2 +y 2 is a plane z = 0 and The radial coordinate at time ;

[0127] Then, according to the integral formula of Bessel function:

[0128]

[0129] Among them, I l (·) represents the l-th order modified Bessel function. Substituting formula (9) into formula (8), we get:

[0130]

[0131] E in formula (5) y Substitute (x,y,0) into U in formula (4) y (k x ,k y ), repeat the above calculation process and get:

[0132]

[0133] Substituting formula (10) and formula (11) into formula (2), we can obtain the electric field expression after the Bessel-Gaussian beam propagates, which is as follows:

[0134]

[0135] There are still integrals in formula (12) and formula (13) that have not been solved. The steady-phase method is used to solve the integrals. According to the steady-phase method, the following formula is obtained:

[0136]

[0137] In formula (14), A(x,y) on the left side refers to f(x,y) refers to k in formula (12) x x+ky y+k z z; in A(x0,y0) and f(x0,y0) on the right side of the formula, x0 is the stable phase point, and its value is y0 is the stable phase point, and its value is At the steady-state point,

[0138] in

[0139] Substituting formula (14) into formula (12) and formula (13) yields the electric field expression:

[0140]

[0141] Step 2: Solve the radial components of the electric field and magnetic field of the Bessel-Gaussian beam in the spherical coordinate system based on the complex factors of step 1 (Formula (10) and Formula (11));

[0142] Step 2.1, transform the electromagnetic field expression in the rectangular coordinate system into the spherical coordinate system (r, θ, is the spherical coordinate system), and calculate the radial component E of the electric field of the Bessel-Gaussian beam r (x,y,z) and the radial component of the magnetic field H r (x,y,z):

[0143]

[0144] The integral local approximation method is used to solve the problem, where kr→n+1 / 2, n is an integer, and θ→π / 2. The magnetic field distribution is as follows:

[0145]

[0146] Where μ is the magnetic permeability and ε is the dielectric constant. The radial components of the electric and magnetic fields of the Bessel-Gaussian beam are obtained as follows:

[0147]

[0148] Where H0 is the magnetic field amplitude.

[0149] Step 3: Based on the radial components of the electric and magnetic fields obtained in step 2, the beam factor in the transverse magnetic and transverse electric modes is solved. The specific process is as follows:

[0150] Substitute the radial component of the electric field (Equation (20)) and the radial component of the magnetic field (Equation (21)) in the spherical coordinate system obtained in step 2 into the integral expression of the beam factor in the transverse magnetic and transverse electric modes, and replace E r (x,y,z) and H r (x,y,z) is abbreviated as Er and H r , as shown below:

[0151]

[0152] Where, is the beam factor in the transverse magnetic mode, is the beam factor in the transverse electric mode, is the composite normalization factor, m is an integer, and n is the maximum number of iterations for Mie scattering calculation:

[0153]

[0154] And using the orthogonality of the exponential function and the trigonometric function, the beam factor of the beam incident on the axis is calculated, and the expression is as follows:

[0155] When m=l±1, we have:

[0156]

[0157] When m≠l±1:

[0158] Step 4: Use the generalized Lorentz Mie theory to solve the expanded field of the Bessel-Gaussian beam scattered by a single spherical particle. The expanded field includes the electric field E of the incident field. i and the magnetic field H of the incident field i , the electric field E of the scattered field s and the magnetic field H of the scattered field s , the electric field E of the internal field t and the magnetic field H of the internal field t , the specific expression is as follows:

[0159]

[0160]

[0161] Where η is the characteristic impedance of seawater, η' is the characteristic impedance of bubble particles relative to seawater, and a mn 、b mn 、c mn d mn is the expansion coefficient, and is the spherical vector wave function, the superscripts 1 and 3 represent the first and third types of spherical vector wave functions, respectively. and Use M and N instead;

[0162]

[0163] The spherical vector wave function is defined as:

[0164]

[0165] in, is a scalar function, a is a constant vector, and the spherical vector wave function is specifically expanded as follows:

[0166]

[0167] Wherein the subscripts o and e represent The parity of is the first kind of associated Legendre function of order n and degree m, zn(kr) represents the 4-type spherical Bessel function j n (kr), y n (kr), one of the.

[0168] Step 5: Based on the beam factor formula (24) obtained in step 3 and the expanded formulas (25) to (27) obtained in step 4, the expansion coefficients of formulas (26) and (27) are solved by the boundary conditions, and then the electric field of the scattered field of the Bessel-Gaussian beam after passing through the ship bubble wake is solved;

[0169] Step 5 is implemented according to the following specific steps:

[0170] Step 5.1. Calculate the expansion coefficient. Consider a single bubble in the ship bubble wake as a dielectric sphere. The boundary conditions of the dielectric sphere are:

[0171] When r=r1, Where r1 is the radius of the dielectric sphere, is the θ component of the incident electric field and scattered electric field in the spherical coordinate system; are the incident electric field and scattered electric field in spherical coordinate system Quantity; is the θ component of the incident magnetic field and the scattered magnetic field in the spherical coordinate system; are the incident magnetic field and the scattered magnetic field in the spherical coordinate system Components, expand the boundary conditions to obtain the expansion coefficients:

[0172]

[0173]

[0174] Where k'=kn, n is the refractive index of the spherical bubble particle relative to seawater; η'=k' / ω0μ;

[0175] Step 5.2: Calculate the electric field of far-field scattering. When calculating far-field scattering, the scattered electric field only has θ and Quantity and Therefore, the total scattered electric field is:

[0176]

[0177] The θ and The component is expressed as

[0178]

[0179] Among them, S1 and S2 are the amplitudes of the scattered electric field, and their expressions are:

[0180]

[0181] where a n and b n For the sake of simplicity, we take it as Mie coefficient.

[0182]

[0183] is the first kind of associated Legendre function.

[0184] Step 6: Use the generalized Lorentz Mie theory to solve the scattering characteristics by combining the beam factor obtained in step 3 (Equation (24)) and the scattered electric field obtained in step 5 (Equation (35)).

[0185] Step 6 is implemented according to the following specific steps:

[0186] Step 6.1. Calculate the scattering efficiency factor Q of the scattered electric field according to the following formula: sca , extinction efficiency factor Q ext , absorption efficiency factor Q abs And the differential scattering cross section σ:

[0187]

[0188] in, is the θ component of the scattered energy density in spherical coordinates and Component, the superscript * of the magnetic field indicates conjugation;

[0189] Step 6.2: Obtain the volume scattering coefficient based on the scattering efficiency factor and the distribution function of the scattering medium. The calculation formula for the volume scattering coefficient is as follows:

[0190]

[0191] The formula for the differential scattering cross section is as follows:

[0192]

[0193] Considering that the bubbles in the ship bubble wake exist in the form of particle groups, the average differential scattering cross section can also be calculated, and its expression is:

[0194]

[0195] Among them, n(r) is the fractional function of the bubble group, which is as follows

[0196]

[0197] Among them, the relationship between the bubble spectral density n0 and the bubble number density N0 is:

[0198]

[0199] The present invention selects the minimum radius r min =10μm, maximum radius r max =80μm, most probable radius r pea k=20μm.

[0200] According to equations (39), (40), and (42) in step 6, MATLAB simulation was used to analyze the scattering characteristics of the Bessel-Gaussian beam after passing through the ship's wake. For the scattering efficiency factor, according to the simulation results, the scattering efficiency factor of a single spherical bubble particle shows a trend of first rapidly rising and then slowly decreasing as the scale parameter increases. The volume scattering coefficient, to a certain extent, indicates the attenuation effect of the scattering particle on the beam. Therefore, the simulation results of the beam efficiency factor provide a theoretical reference for how to select beam parameters so that the beam can decay more slowly in the transmission medium and transmit farther. According to the simulation results, as the transmission distance increases, the change trend of the volume scattering coefficient slows down and the peak value decreases. This is because the increase in transmission distance increases the propagation loss. For the differential scattering cross section, according to the simulation results, as the topological charge increases, the differential scattering cross section decreases. This is because when the topological charge increases, the dark spot in the center of the beam increases, the energy that can be scattered by the particles decreases, and thus the differential scattering cross section decreases.

[0201] Compared with the existing Monte Carlo method for simulating the scattering characteristics of ship wakes, the present invention uses a numerical analysis method to construct a calculation method for the scattering characteristics of blue-green lasers after passing through ship bubble wakes. This method can support the analysis of the scattering characteristics of blue-green lasers after passing through ship bubble wakes. The method is simple and easy to understand.

[0202] In order to verify the correctness of the calculation method of the blue-green laser scattering characteristics after passing through the ship bubble wake, the changes in the scattering characteristics of the Bessel-Gaussian beam under different parameters after passing through the ship wake are analyzed.

[0203] Example 1

[0204] This embodiment mainly analyzes how the scattering efficiency factor of a Bessel-Gaussian beam after being scattered by a single spherical particle changes with the cone angle. The specific steps are as follows: Step 1: According to formula (39), the scattering efficiency factor of the Bessel-Gaussian beam after passing through a single spherical bubble can be calculated; Step 2: Without further explanation, the MATLAB simulation parameters are set as follows: the wavelength of the Bessel-Gaussian beam is 550 μm; the beam waist radius is 1 μm; the transmission distance is 10 μm; the topological charge is 0; the cone angle is 0°, and the relative refractive index is 1.75+0.44i; Step 3: According to formula (39) and the parameter settings of step 2, the scattering efficiency factor of the Bessel-Gaussian beam after being scattered by a single spherical particle under different parameters is simulated and analyzed, as shown in FIG. Figure 1 As shown:

[0205] Figure 1 This is a graph showing the variation of the scattering efficiency factor of a Bessel-Gaussian beam scattered by a single spherical particle at different cone angles; Figure 1 It can be seen that the scattering efficiency factor of the Bessel Gaussian beam after being scattered by a single spherical particle will show a curve change that rises rapidly and falls slowly as the particle scale parameter changes. At this time, l = 0, the cone angle is 0, the beam degenerates into a Gaussian beam, and the peak light intensity is at the center of the beam. As the scale parameter increases, the scattering efficiency factor begins to increase rapidly. When the scale parameter is about 2, the scattering efficiency factor reaches a peak, and then the scattering efficiency factor begins to gradually decrease. This is consistent with the change of the scattering efficiency factor of the Gaussian beam after being scattered by a single spherical particle, which verifies the correctness of the method of the present invention.

[0206] Example 2

[0207] This embodiment mainly analyzes the variation of the volume scattering coefficient of the Bessel-Gaussian beam after being scattered by the ship's bubble wake with the transmission distance. The specific steps are as follows: Step 1 Without explanation, set the MATLAB simulation parameters as follows: the wavelength of the Bessel-Gaussian beam is 532 μm; the beam waist radius is 1 μm; the transmission distance is 50 μm; the topological charge is 0; the relative refractive index is 0.75+0i, and the cone angle is 15°; Step 2 According to formula (40) and the parameter settings of step 1, simulate and analyze the variation of the volume scattering coefficient of the Bessel-Gaussian beam after being scattered by the ship's wake at different transmission distances, as shown in FIG. Figure 2 As shown: Figure 2 is the variation of the volume scattering coefficient of the Bessel-Gaussian beam after scattering by the ship wake at different transmission distances; Figure 2It can be seen that when the transmission distance is 50μm, the volume scattering coefficient of the Bessel-Gaussian beam after being scattered by the ship wake shows a curve that first rises and then rapidly decreases with the change of the particle radius. As the particle radius increases to around 50μm, the two curves begin to overlap. As the transmission distance increases (50μm→100μm), the upward and downward trends of the volume scattering coefficient slow down, and the peak value of the volume scattering coefficient decreases. This is because the increase in transmission distance increases the propagation loss. This scattering characteristic can provide some guidance for identifying the distance of the ship wake.

[0208] Example 3

[0209] This embodiment mainly analyzes the changes in the differential scattering cross-section of Bessel-Gaussian beams with different topological charges after being scattered by the ship bubble wake. The specific steps are as follows: Step 1 is not explained. The MATLAB simulation parameters are set as follows: the wavelength λ of the Bessel-Gaussian beam is 532 μm; the beam waist radius is 5λ; the transmission distance is 10 μm; and the relative refractive index is 0.75+0i. Step 2 According to formula (42) and the parameter settings of step 1, the changes in the differential scattering cross-section of the Bessel-Gaussian beam after being scattered by the ship bubble wake when the topological charges are 1, 2, and 3, respectively, are simulated and analyzed. Figure 3 As shown:

[0210] Figure 3 This is a graph showing the changes in the differential scattering cross section of Bessel-Gaussian beams with different topological charges after being scattered by the ship's bubble wake. Figure 3 It can be seen that when the topological charge remains constant, the differential scattering cross section first decreases and then increases with increasing scattering angle, which is consistent with the change in the differential scattering cross section when a beam is scattered by a particle. As the topological charge increases, the differential scattering cross section decreases. This is because when the topological charge increases, the dark spot at the center of the beam (the area near the phase singularity) becomes larger, reducing the amount of energy available for scattering, and thus reducing the differential scattering cross section. This scattering characteristic can provide guidance for selecting beam parameters to improve transmission quality in ship wakes.

Claims

1. The calculation method of the scattering characteristics of blue-green laser after passing through the bubble wake of a ship is characterized by: The specific steps include: Step 1: Based on the light field function expression of the Bessel-Gaussian beam in the source plane, expand it using the angular spectrum to obtain the electric field components of the Bessel-Gaussian beam in the xyz direction in the spatial rectangular coordinate system; Step 2: Calculate the radial components of the electric and magnetic fields of the Bessel-Gaussian beam in the spherical coordinate system. Step 3: Solve the beam factor in the transverse magnetic and transverse electric modes based on the radial components of the electric and magnetic fields obtained in step 2; Step 4: Use the generalized Lorentz Mie theory to solve the expanded field of the Bessel-Gaussian beam scattered by a single spherical particle; Step 5, solve the electric field of the Bessel-Gaussian beam scattered after passing through the ship bubble wake; Step 6: Use the generalized Lorentz Mie theory to solve the scattering characteristics by combining the beam factor obtained in step 3 and the scattered electric field obtained in step 5.

2. The method for calculating the scattering characteristics of blue-green laser after passing through a ship bubble wake according to claim 1 is characterized in that: Step 1 is implemented according to the following specific steps: Step 1.

1. First, determine the light field function E(x, y, 0) of the Bessel-Gaussian beam at the source plane. E(x, y, 0) is expressed in the spatial rectangular coordinate system O-xyz as: Where E0 is the electric field amplitude, J l (·) is the lth-order Bessel function, k = 2π / λ is the wave number, λ is the wavelength in the medium, ρ 2 =x 2 +y 2 , α b is the cone angle, i is the imaginary unit, and the integer l represents the quantum number of orbital angular momentum. is the azimuth angle, ω0 represents the beam waist radius, and x, y, and z are the coordinates of the rectangular coordinate system; When z≥0 in the free half-space, the electric field amplitude of the Bessel-Gaussian beam is expressed according to the angular spectrum as: Among them, E x (x,y,z) is the electric field component of the Bessel-Gaussian beam in the x direction, E y (x,y,z) is the electric field component of the Bessel-Gaussian beam in the y direction, E z (x,y,z) is the electric field component of the Bessel-Gaussian beam in the z direction, U x (k x ,k y ) represents the complex factor in the x direction, U y (k x ,k y ) represents the complex factor in the y direction, k x 、k y 、k z represent the direction cosines in the x, y, and z directions respectively, and: Complex factor U x (k x ,k y ) and U y (k x ,k y ) are determined by Fourier transform and are given by: in where γ = ksinα b , θ is the azimuth angle, sinθ is the ratio of the electric field component in the x direction to the total electric field; cosθ is the ratio of the electric field component in the y direction to the total electric field; Step 1.2: Calculate the expression of the complex factors in the angular spectrum representation of the Bessel-Gaussian beam to obtain the electric field components in the x, y, and z directions in the spatial rectangular coordinate system after free space transmission. The calculation process is as follows: First, replace E in formula (5) x Substitute (x,y,0) into U in formula (4) x (k x ,k y )get: Among them, the Bessel function has the following properties: So, by substituting formula (7) into formula (6), we get: Where ρ 2 =x 2 +y 2 is a plane z = 0 and The radial coordinate at time ; Then, according to the integral formula of Bessel function: Among them, I l (·) represents the l-th order modified Bessel function. Substituting formula (9) into formula (8), we get: E in formula (5) y Substitute (x,y,0) into U in formula (4) y (k x ,k y ), repeat the above calculation process and get: Substituting formula (10) and formula (11) into formula (2), we can obtain the electric field expression after the Bessel-Gaussian beam propagates, which is as follows: There are still integrals in formula (12) and formula (13) that have not been solved. The steady-phase method is used to solve the integrals. According to the steady-phase method, the following formula is obtained: In formula (14), A(x,y) on the left side refers to f(x,y) refers to k in formula (12) x x+k y y+k z z; in A(x0,y0) and f(x0,y0) on the right side of the formula, x0 is the stable phase point, and its value is y0 is the stable phase point, and its value is At the steady-state point, in, Substituting formula (14) into formula (12) and formula (13) yields the electric field expression:

3. The method for calculating the scattering characteristics of blue-green laser after passing through a ship bubble wake according to claim 2 is characterized in that: Step 2 is implemented according to the following specific steps: Step 2.1, transform the electromagnetic field expression in the rectangular coordinate system into the spherical coordinate system. is the spherical coordinate system, and calculate the radial component E of the electric field of the Bessel-Gaussian beam r (x,y,z) and the radial component of the magnetic field H r (x,y,z): The integral local approximation method is used to solve the problem, where kr→n+1 / 2, n is an integer, and θ→π / 2. The magnetic field distribution is as follows: Where μ is the magnetic permeability and ε is the dielectric constant. The radial components of the electric and magnetic fields of the Bessel-Gaussian beam are obtained as follows: Where H0 is the magnetic field amplitude.

4. The method for calculating the scattering characteristics of blue-green laser after passing through a ship bubble wake according to claim 3 is characterized in that: Step 3 is implemented according to the following specific steps: Substitute the radial component of the electric field and the radial component of the magnetic field in the spherical coordinate system obtained in step 2 into the integral expression of the beam factor in the transverse magnetic and transverse electric modes, and replace E r (x,y,z) and H r (x,y,z) is abbreviated as E r and H r , as shown below: Where, is the beam factor in the transverse magnetic mode, is the beam factor in the transverse electric mode, is the composite normalization factor, m is an integer, and n is the maximum number of iterations for Mie scattering calculation: And using the orthogonality of the exponential function and the trigonometric function, the beam factor of the beam incident on the axis is calculated, and the expression is as follows: When m=l±1: When m≠l±1:

5. The method for calculating the scattering characteristics of blue-green laser after passing through a ship bubble wake according to claim 4 is characterized in that: In step 4, the expanded field includes the electric field E of the incident field i and the magnetic field H of the incident field i , the electric field E of the scattered field s and the magnetic field H of the scattered field s , the electric field E of the internal field t and the magnetic field H of the internal field t , the specific expression is as follows: Where η is the characteristic impedance of seawater, η' is the characteristic impedance of bubble particles relative to seawater, and a mn 、b mn 、c mn d mn is the expansion coefficient, and is the spherical vector wave function, the superscripts 1 and 3 represent the first and third types of spherical vector wave functions, respectively. and Use M and N instead; The spherical vector wave function is defined as: in, is a scalar function, a is a constant vector, and the spherical vector wave function is specifically expanded as follows: Wherein the subscripts o and e represent The parity of is the first kind of associated Legendre function of order n and degree m, zn(kr) represents the 4-type spherical Bessel function j n (kr), y n (kr), one of the.

6. The method for calculating the scattering characteristics of blue-green laser after passing through a ship bubble wake according to claim 5 is characterized in that: Step 5 is implemented according to the following specific steps: Step 5.

1. Calculate the expansion coefficient. Consider a single bubble in the ship bubble wake as a dielectric sphere. The boundary conditions of the dielectric sphere are: When r=r1, Where r1 is the radius of the dielectric sphere, is the θ component of the incident electric field and scattered electric field in the spherical coordinate system; are the incident electric field and scattered electric field in spherical coordinate system Quantity; is the θ component of the incident magnetic field and the scattered magnetic field in the spherical coordinate system; are the incident magnetic field and the scattered magnetic field in the spherical coordinate system Components, expand the boundary conditions to obtain the expansion coefficients: Where k'=kn, n is the refractive index of the spherical bubble particle relative to seawater; η'=k' / ω0μ; Step 5.2: Calculate the electric field of far-field scattering. When calculating far-field scattering, the scattered electric field only has θ and Quantity and Therefore, the total scattered electric field is: The θ and The component is expressed as Among them, S1 and S2 are the amplitudes of the scattered electric field, and their expressions are: where a n and b n For the sake of simplicity, we take it as Mie coefficient. is the first kind of associated Legendre function.

7. The method for calculating the scattering characteristics of blue-green laser after passing through a ship bubble wake according to claim 5, characterized in that: Step 6 is implemented according to the following specific steps: Step 6.

1. Calculate the scattering efficiency factor Q of the scattered electric field according to the following formula: sca , extinction efficiency factor Q ext , absorption efficiency factor Q abs And the differential scattering cross section σ: in, is the θ component of the scattered energy density in spherical coordinates and Component, the superscript * of the magnetic field indicates conjugation; Step 6.2: Obtain the volume scattering coefficient based on the scattering efficiency factor and the distribution function of the scattering medium. The calculation formula for the volume scattering coefficient is as follows: The formula for the differential scattering cross section is as follows: Considering that the bubbles in the ship bubble wake exist in the form of particle groups, the average differential scattering cross section is calculated, and its expression is: Among them, n(r) is the fractional function of the bubble group, which is as follows Among them, the relationship between the bubble spectral density n0 and the bubble number density N0 is:

8. The method for calculating the scattering characteristics of blue-green laser after passing through a ship bubble wake according to claim 7 is characterized in that: In step 6, select the minimum radius r min =10μm, maximum radius r max =80μm, most probable radius r peak =20μm.

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