Method for realizing coherent detection using Laguerre-Gaussian beam

Abstract

The present invention discloses a method for achieving coherent detection using a Laguerre-Gaussian beam, comprising the following steps: Step 1, based on the transmission theory of LG beams in atmospheric turbulence, the Rytov approximation, and the atmospheric turbulence refractive index power spectrum model, deriving a light field expression for the LG beam after atmospheric turbulence transmission; Step 2, combining the light field expression for the LG beam after atmospheric turbulence transmission in Step 1, deriving a mixing efficiency expression for coherent detection when the LG beam is used as the signal beam. This method can be used to analyze the variation in mixing efficiency of a vortex beam after atmospheric turbulence transmission under different light source parameters and turbulence intensities, and has important reference value for rationally controlling laser parameters, improving the anti-interference capability of the beam after atmospheric turbulence transmission, and maintaining better transmission characteristics for practical communications.

Description

Technical Field

[0001] The present invention belongs to the technical field of wireless laser communication, and in particular relates to a method for realizing coherent detection by using Laguerre-Gaussian beams (LG beams). Background Art

[0002] Free-space optical communication (FSO) is a technology that uses laser beams as information carriers to transmit data, voice, images, and other information. However, due to atmospheric turbulence, the transmission medium can cause a series of atmospheric turbulence effects, such as reduced laser coherence, beam expansion, and spot jitter, which greatly reduces the communication performance of FSO systems. In order to achieve long-distance space communication, coherent detection (heterodyne detection) technology is also a very important research direction in the field of free-space optical communication. In actual communications, coherent detection is suitable for detecting tiny signals, but it is affected by atmospheric turbulence effects during channel transmission. In atmospheric turbulence channels, random fluctuations in the atmospheric refractive index cause random fluctuations in the amplitude and phase of the optical signal. These fluctuations can have a significant impact on system performance, thereby restricting the stability of the FSO system.

[0003] Vortex beams carrying orbital angular momentum (OAM) have advantages during transmission due to their phase singularities, but they are still affected by atmospheric turbulence during long-distance transmission. Therefore, studying the mixing efficiency of vortex beams passing through atmospheric turbulence is of great significance for understanding the transmission laws of vortex beams in complex environments and improving the beam's anti-interference ability and stability. Summary of the Invention

[0004] The purpose of the present invention is to provide a method for achieving coherent detection using a Laguerre-Gaussian beam, which can be used to analyze the changes in the mixing efficiency of a vortex beam after transmission through atmospheric turbulence under different light source parameters and turbulence intensities. It has important reference value for rationally regulating laser parameters, improving the anti-interference ability of the beam after transmission through atmospheric turbulence, and maintaining better transmission characteristics for actual communication.

[0005] The technical solution adopted by the present invention is a method for realizing coherent detection using a Laguerre-Gaussian beam, comprising the following steps:

[0006] Step 1: Based on the propagation theory of LG beams in atmospheric turbulence, the Rytov approximation, and the atmospheric turbulence refractive index power spectrum model, the light field expression of the LG beam after propagation through atmospheric turbulence is derived;

[0007] Step 2: Combined with the light field expression of the LG beam after it is transmitted through atmospheric turbulence in step 1, derive the mixing efficiency expression of coherent detection when the LG beam is used as the signal beam.

[0008] The present invention is also characterized in that:

[0009] Step 1 is implemented according to the following specific steps:

[0010] Step 1.1: First determine the optical field of the LG beam when it propagates in free space The expression is:

[0011]

[0012] Where C is a constant, l is the topological charge, p is the radial index, r is the distance between the light and the transmission axis, k is the wave number, and k = 2π / λ, z is the transmission distance, i is the imaginary unit, represents the spiral phase factor of the LG beam, z R is the confocal parameter (i.e., Rayleigh length), λ is the wavelength of the LG beam;

[0013] Step 1.2: Use the Rytov method to solve the light field of the LG beam after it propagates through atmospheric turbulence. Its expression is for:

[0014]

[0015] Where L represents the beam propagation path, and ψ(r,L) represents the complex phase perturbation caused by atmospheric turbulence. Usually, only the first-order and second-order complex phase perturbations need to be considered, which can be expressed as:

[0016] ψ(r,L)=ψ1(r,L)+ψ2(r,L)(6);

[0017] Where ψ1(r,L) and ψ2(r,L) represent the first-order and second-order complex phase perturbations, respectively;

[0018] Substituting formula (6) into formula (5), formula (5) is expressed as:

[0019]

[0020] Among them, E1(0,0) represents the statistical moment of the first-order complex phase disturbance of the light field under the horizontal transmission path, namely:

[0021]

[0022] Among them, Φ n (κ) represents the atmospheric turbulence refractive index power spectrum, and its expression is:

[0023]

[0024] Among them, 0≤κ≤∞, κ m =5.92 / l0, κ0=2π / L0, l0 and L0 are the inner and outer scales of atmospheric turbulence respectively; C0 represents the atmospheric refractive index structure constant;

[0025] Substituting equations (7) to (9) and equation (1) in step 1.1 into equation (5), we can obtain the light field of the LG beam after it is transmitted through atmospheric turbulence: The expression is:

[0026]

[0027] In formula (1) of step 1.1: is the generalized Laguerre polynomial, which is expressed as:

[0028]

[0029] Where m is the summation variable, w(z) is the beam radius of the LG beam at the transmission distance z, and w(z) is expressed as:

[0030]

[0031] Where w0 is the waist radius of the LG beam at the transmission distance z = 0; z R is the confocal parameter, z R Expressed as:

[0032]

[0033] Where λ is the wavelength of the LG beam.

[0034] Step 2 is implemented according to the following specific steps:

[0035] Step 2.1: When the light field of the signal beam is E S (t), the light field of the local oscillator beam is E L (t), the expressions of the two are:

[0036]

[0037] Among them, A S 、ω S and Represent the amplitude, central angular frequency and phase of the signal beam respectively, A L 、ω L and represent the amplitude, central angular frequency and phase of the local oscillator beam respectively, and t is the time;

[0038] After the signal beam and the local oscillator beam pass through the 90° optical mixer, according to equations (11) and (12), the current signal i detected on the photodetector is p Expressed as:

[0039]

[0040] Where α is the photodetector responsivity, α = eη / hv, e is the electron charge, η is the photodetector quantum efficiency, h is the Planck constant, and v is the optical frequency. In formula (13), the first two terms are the DC component, the third term is the sum frequency component, and the fourth term is the difference frequency component. The frequency difference is ω S -ω L ;

[0041] For equation (13), after filtering out the DC component and sum frequency component, the intermediate frequency signal current output by the photodetector is:

[0042]

[0043] According to the basic principle of coherent detection and formula (14), the power of the intermediate frequency current is IF 2 > is represented as:

[0044]

[0045] Where s is the area of ​​the photosensitive surface of the photodetector, R is the radius of the photosensitive surface of the photodetector, r0 is the radial vector from any point on the photosensitive surface of the photodetector to the center, and θ is the angle between the wave vector and the polar coordinate axis in the polar coordinate system;

[0046] Step 2.2 assumes that the noise after coherent detection comes from the local oscillator beam, so the power of the noise is Expressed as:

[0047]

[0048] Where B is the bandwidth of the photodetector;

[0049] Introducing the signal-to-noise ratio, combined with equation (16) and equation (15) in step 2.1, the signal-to-noise ratio (SNR) of the photodetector output is expressed as:

[0050]

[0051] For an ideal coherent detection system, the signal-to-noise ratio (SNR0) of the photodetector output is:

[0052]

[0053] According to equation (17), the mixing efficiency of coherent detection is defined as: ​

[0054]

[0055] Step 2.3: The light field of the LG beam after being transmitted through atmospheric turbulence is used as the light field of the signal beam, and the light field of the LG beam without being affected by atmospheric turbulence is used as the light field of the local oscillator beam, that is:

[0056]

[0057] in, is the light field expression after atmospheric turbulence transmission, as shown in equation (10) in step 1; is the light field expression of the LG beam without the influence of atmospheric turbulence, that is:

[0058]

[0059] Where C is a constant, l is the topological charge, p is the radial index, and r is the distance between the light and the transmission axis. represents the spiral phase factor of the LG beam, w0 is the beam waist radius;

[0060] Substituting equations (21) and (22) into equation (20) in step 2.2, the mixing efficiency η is het Expressed as:

[0061]

[0062] Substituting equation (23) and equation (10) in step 1 into equation (25), the mixing efficiency of the LG beam after transmission through atmospheric turbulence can be simulated and calculated.

[0063] The model of the optical mixer in step 2.1 is COH24.

[0064] The photodetector model in step 2.1 is: BPD230.

[0065] In formula (17) of step 2.2: P S is the signal beam power, which is expressed as:

[0066]

[0067] In formula (23) of step 2.3, is the Laguerre polynomial, expressed as:

[0068]

[0069] The beneficial effects of the present invention are:

[0070] The present invention is a method for achieving coherent detection using a vortex beam. By utilizing the transmission advantages of a vortex beam in atmospheric turbulence, and based on the Rytov approximation principle and the atmospheric turbulence power spectrum, an expression for the mixing efficiency of an LG beam after transmission through atmospheric turbulence is derived. The method of the present invention can analyze the effects of light source parameters, transmission distance, and photodetector photosensitive surface radius on the mixing efficiency during the transmission of an LG beam through atmospheric turbulence, and can support quantitative analysis of the mixing efficiency of an LG beam during transmission through atmospheric turbulence. The method of the present invention is simple to construct, easy to understand, not limited by experimental conditions, and easy to implement. It has important reference value for the rational regulation of laser parameters, improving the anti-interference ability of an LG beam after transmission through atmospheric turbulence, and subsequent research on coherent detection systems. BRIEF DESCRIPTION OF THE DRAWINGS

[0071] Figure 1 Schematic diagram of the method for realizing coherent detection using Laguerre-Gaussian beams according to the present invention;

[0072] Figure 2 Schematic diagram of the variation of the mixing efficiency of the LG beam and the Gaussian beam with the transmission distance z after being transmitted through atmospheric turbulence in Example 1 of the present invention;

[0073] Figure 3 2 is a schematic diagram showing how the mixing efficiency of an LG beam after being transmitted through atmospheric turbulence in Example 2 of the present invention varies with the transmission distance z when the topological charge number is l=1, 2, and 3 respectively;

[0074] Figure 4 2 is a schematic diagram showing how the mixing efficiency of the LG beam after being transmitted through atmospheric turbulence in Example 2 of the present invention varies with the topological charge number l when the radial index is p=1, 2, and 3 respectively;

[0075] Figure 5 This is a schematic diagram of how the mixing efficiency changes with the transmission distance z when the radius of the photosensitive surface of the photodetector is R = 0.02m, 0.03m, and 0.05m respectively after the LG light beam is transmitted through atmospheric turbulence in Example 2 of the present invention. DETAILED DESCRIPTION

[0076] The present invention will be described in detail below with reference to the accompanying drawings and specific embodiments.

[0077] The present invention provides a method for realizing coherent detection using a Laguerre-Gaussian beam, which specifically comprises the following steps:

[0078] Step 1: Based on the propagation theory of LG beams in atmospheric turbulence, the Rytov approximation, and the atmospheric turbulence refractive index power spectrum model, derive the light field expression of the LG beam after propagation through atmospheric turbulence;

[0079] Step 1 is implemented according to the following specific steps:

[0080] Step 1.1: First determine the optical field of the LG beam when it propagates in free space The expression is:

[0081]

[0082] Where C is a constant, l is the topological charge, p is the radial index, r is the distance between the light and the transmission axis, k is the wave number, and k = 2π / λ, z is the transmission distance, i is the imaginary unit, represents the spiral phase factor of the LG beam, z R is the confocal parameter (i.e., Rayleigh length), λ is the wavelength of the LG beam; is the generalized Laguerre polynomial, which is expressed as:

[0083]

[0084] Where m is the summation variable, w(z) is the beam radius of the LG beam at the transmission distance z, and w(z) is expressed as:

[0085]

[0086] Where w0 is the waist radius of the LG beam at the transmission distance z = 0; z R is the confocal parameter (i.e. Rayleigh length), z R Expressed as:

[0087]

[0088] Where λ is the wavelength of the LG beam.

[0089] Step 1.2: When the LG beam propagates in the atmospheric turbulence random medium, it will cause atmospheric turbulence effects such as amplitude and phase disturbance of the beam. According to the Rytov method, the light field of the LG beam after propagation through the atmospheric turbulence is solved. Its expression is for:

[0090]

[0091] Where L represents the beam propagation path, and ψ(r,L) represents the complex phase perturbation caused by atmospheric turbulence. Usually, only the first-order and second-order complex phase perturbations need to be considered, which can be expressed as:

[0092] ψ(r,L)=ψ1(r,L)+ψ2(r,L)(6);

[0093] Where ψ1(r,L) and ψ2(r,L) represent the first-order and second-order complex phase perturbations, respectively;

[0094] Substituting formula (6) into formula (5), formula (5) is expressed as:

[0095]

[0096] Among them, E1(0,0) represents the statistical moment of the first-order complex phase disturbance of the light field under the horizontal transmission path, namely:

[0097]

[0098] Among them, Φ n (κ) represents the atmospheric turbulence refractive index power spectrum. The present invention mainly adopts the modified VonKarman spectrum, which is expressed as follows:

[0099]

[0100] Among them, 0≤κ≤∞, κ m =5.92 / l0, κ0=2π / L0, l0 and L0 are the inner and outer scales of atmospheric turbulence respectively; C0 represents the atmospheric refractive index structure constant;

[0101] Substituting equations (7) to (9) and equation (1) in step 1.1 into equation (5), we can obtain the light field of the LG beam after it is transmitted through atmospheric turbulence: The expression is:

[0102]

[0103] Step 2, such as Figure 1 As shown, combined with the light field expression of the LG beam after it is transmitted through atmospheric turbulence in step 1 The mixing efficiency expression of coherent detection when LG beam is used as signal beam is derived.

[0104] Step 2 is implemented according to the following specific steps:

[0105] Step 2.1: When the light field of the signal beam is E S (t), the light field of the local oscillator beam is E L (t), the expressions of the two are:

[0106]

[0107] Among them, A S 、ω S and Represent the amplitude, central angular frequency and phase of the signal beam respectively, A L 、ω L and represent the amplitude, central angular frequency and phase of the local oscillator beam respectively, and t is the time;

[0108] After the signal beam and the local oscillator beam pass through the 90° optical mixer (commonly used model is COH24), according to equations (11) and (12), the current signal i detected on the photodetector (commonly used model in the present invention is the BPD230 series photodetector) is p Expressed as:

[0109]

[0110] Where α is the photodetector responsivity, α = eη / hv, e is the electron charge, η is the photodetector quantum efficiency, h is the Planck constant, and v is the optical frequency. In formula (13), the first two terms are the DC component, the third term is the sum frequency component, and the fourth term is the difference frequency component. The frequency difference is ω S -ω L ;

[0111] For equation (13), after filtering out the DC component and sum frequency component, the intermediate frequency signal current output by the photodetector is:

[0112]

[0113] According to the basic principle of coherent detection and formula (14), the power of the intermediate frequency current is IF 2 > is represented as:

[0114]

[0115] Where s is the area of ​​the photosensitive surface of the photodetector, R is the radius of the photosensitive surface of the photodetector, r0 is the radial vector from any point on the photosensitive surface of the photodetector to the center, and θ is the angle between the wave vector and the polar coordinate axis in the polar coordinate system;

[0116] Step 2.2: When the power of the local oscillator beam is large, the shot noise caused by the local oscillator beam is much larger than other noises. Generally, the noise after coherent detection can be assumed to come from the local oscillator beam. Therefore, the power of the noise is Expressed as:

[0117]

[0118] Where B is the bandwidth of the photodetector;

[0119] In order to measure the quality of the signal output by the photodetector, the signal-to-noise ratio (SNR) is introduced. Combining equation (16) with equation (15) in step 2.1, the signal-to-noise ratio (SNR) of the photodetector output is expressed as:

[0120]

[0121] Among them, P S ​is the signal beam power, which is expressed as:

[0122]

[0123] For an ideal coherent detection system, the signal-to-noise ratio (SNR0) of the photodetector output is:

[0124]

[0125] According to equation (17), the mixing efficiency of coherent detection is defined as:

[0126]

[0127] Step 2.3: In the present invention, the light field of the LG beam after being transmitted through atmospheric turbulence is used as the light field of the signal beam, and the light field of the LG beam not affected by atmospheric turbulence is used as the light field of the local oscillator beam, that is:

[0128]

[0129] in, is the light field expression after atmospheric turbulence transmission, as shown in equation (10) in step 1; is the light field expression of the LG beam without the influence of atmospheric turbulence, that is:

[0130]

[0131] Where C is a constant, l is the topological charge, p is the radial index, and r is the distance between the light and the transmission axis. represents the spiral phase factor of the LG beam, w0 is the beam waist radius, is the Laguerre polynomial, expressed as:

[0132]

[0133] Substituting equations (21) and (22) into equation (20) in step 2.2, the mixing efficiency η is het Expressed as:

[0134]

[0135] Substituting equation (23) and equation (10) in step 1 into equation (25), the mixing efficiency of the LG beam after transmission through atmospheric turbulence can be simulated and calculated.

[0136] In order to study the mixing efficiency of LG beams after transmission through atmospheric turbulence, the mixing efficiency of LG beams and Gaussian beams was compared and analyzed. The variation of the mixing efficiency of LG beams with transmission distance under different light source parameters and photodetector radii after transmission through atmospheric turbulence was also studied.

[0137] Example 1

[0138] This example mainly analyzes how the mixing efficiency of an LG beam and a Gaussian beam varies with transmission distance after being transmitted through atmospheric turbulence. The specific steps are as follows:

[0139] Step 1: Unless otherwise specified, set the MATLAB simulation parameters as follows: LG beam wavelength λ = 632.8 nm, topological charge number l = 3, 5, radial index p = 1, beam waist radius w0 = 0.02 m; Gaussian beam wavelength λ0 = 632.8 nm, beam waist radius w0 = 0.02 m; atmospheric refractive index structure constant C0 = 1.7 × 10 -17 m -2 / 3 ;The radius of the photosensitive surface of the photodetector R = 0.02m.

[0140] Step 2: Based on equations (11) to (15), equation (25) and the parameter settings of step 1 of this embodiment, the variation of the mixing efficiency of the LG beam and the Gaussian beam with the transmission distance after being transmitted through atmospheric turbulence can be simulated and analyzed, as shown in FIG. Figure 2 As shown:

[0141] Figure 2 The mixing efficiency of LG beams and Gaussian beams with topological charge l = 3 and 5 after atmospheric turbulence transmission varies with the transmission distance. Figure 2 It can be seen that with the increase of transmission distance, the mixing efficiency of LG beam and Gaussian beam both decreases, but the mixing efficiency of LG beam is significantly higher than that of Gaussian beam. This is because the vortex beam has strong anti-interference ability to atmospheric turbulence and high signal quality, which is conducive to improving the mixing efficiency.

[0142] Example 2

[0143] This example mainly analyzes the changes in mixing efficiency of LG beams after they have been transmitted through atmospheric turbulence under different topological charges, radial indices, and photodetector photosensitive surface radii. The specific steps are as follows:

[0144] Step 1 Unless otherwise specified, set the MATLAB simulation parameters as follows: LG beam wavelength λ = 632 nm; topological charge l = 3; radial index p = 3; beam waist radius w0 = 0.02 m; transmission distance z = 5000 m; atmospheric refractive index structure constant C0 = 1.7 × 10 -17 m -2 / 3 ; The inner scale of atmospheric turbulence is l0 = 0.01m, the outer scale of atmospheric turbulence is L0 = 100m; the radius of the photosensitive surface of the photodetector is R = 0.02m;

[0145] Step 2: According to equations (11) to (15), equation (25) and the parameter settings of step 1 of this embodiment, the variation of the mixing efficiency of the LG beam after it passes through atmospheric turbulence under different topological charges, radial indices and photodetector photosensitive surface radii can be simulated and analyzed, as shown in FIG. Figures 3 to 5 As shown:

[0146] Figure 3 The variation of mixing efficiency with transmission distance when topological charge number l = 1, 2, 3 after LG beam is transmitted through atmospheric turbulence; Figure 3 It can be seen that the greater the topological charge, the higher the mixing efficiency of the LG beam after transmission through atmospheric turbulence. Furthermore, the mixing efficiency gradually decreases as the beam transmission distance increases, reaching a high efficiency for transmission distances under one kilometer. This is because the beam is affected by atmospheric turbulence during transmission, and the longer the transmission distance, the worse the signal quality. Figure 4 The mixing efficiency of the LG beam after it passes through atmospheric turbulence varies with the topological charge when the radial index p = 1, 2, and 3. Figure 4 It can be seen that the smaller the radial index, the higher the mixing efficiency of the LG beam, and with the increase of the topological charge, the mixing efficiency of the LG beam gradually increases. This conclusion is consistent with Figure 3 The conclusions are consistent. This is because the smaller the topological charge, the more severe the beam expansion, the worse the beam's turbulence suppression effect, the worse the beam quality, and thus the lower the mixing efficiency. Therefore, the mixing efficiency can be improved by changing the topological charge of the signal light. Figure 5 The LG beam is transmitted through atmospheric turbulence and the mixing efficiency changes with the transmission distance when the radius of the photosensitive surface of the photodetector is R = 0.02m, 0.03m, and 0.05m; Figure 5 It can be seen that as the radius of the photodetector's photosensitive surface increases, the mixing efficiency decreases. Therefore, in actual experiments, we can choose a suitable photodetector to improve the mixing efficiency.

[0147] The method of the present invention can be used to analyze the changes in the mixing efficiency of a vortex light beam after it is transmitted through atmospheric turbulence under different light source parameters and turbulence intensities. It has important reference value for rationally regulating laser parameters, improving the anti-interference ability of the light beam after it is transmitted through atmospheric turbulence, and maintaining better transmission characteristics for actual communication.

[0148] Example 3

[0149] A method for achieving coherent detection using a Laguerre-Gaussian beam comprises the following steps:

[0150] Step 1: Based on the propagation theory of LG beams in atmospheric turbulence, the Rytov approximation, and the atmospheric turbulence refractive index power spectrum model, the light field expression of the LG beam after propagation through atmospheric turbulence is derived;

[0151] Step 2: Combined with the light field expression of the LG beam after it is transmitted through atmospheric turbulence in step 1, derive the mixing efficiency expression of coherent detection when the LG beam is used as the signal beam.

[0152] Example 4

[0153] A method for achieving coherent detection using a Laguerre-Gaussian beam comprises the following steps:

[0154] Step 1: Based on the propagation theory of LG beams in atmospheric turbulence, the Rytov approximation, and the atmospheric turbulence refractive index power spectrum model, the light field expression of the LG beam after propagation through atmospheric turbulence is derived;

[0155] Step 1 is implemented according to the following specific steps:

[0156] Step 1.1: First determine the optical field of the LG beam when it propagates in free space The expression is:

[0157]

[0158] Where C is a constant, l is the topological charge, p is the radial index, r is the distance between the light and the transmission axis, k is the wave number, and k = 2π / λ, z is the transmission distance, i is the imaginary unit, represents the spiral phase factor of the LG beam, z R is the confocal parameter (i.e., Rayleigh length), λ is the wavelength of the LG beam;

[0159] Step 1.2: Use the Rytov method to solve the light field of the LG beam after it propagates through atmospheric turbulence. Its expression is for:

[0160]

[0161] Where L represents the beam propagation path, and ψ(r,L) represents the complex phase perturbation caused by atmospheric turbulence. Usually, only the first-order and second-order complex phase perturbations need to be considered, which can be expressed as:

[0162] ψ(r,L)=ψ1(r,L)+ψ2(r,L)(6);

[0163] Where ψ1(r,L) and ψ2(r,L) represent the first-order and second-order complex phase perturbations, respectively;

[0164] Substituting formula (6) into formula (5), formula (5) is expressed as:

[0165]

[0166] Among them, E1(0,0) represents the statistical moment of the first-order complex phase disturbance of the light field under the horizontal transmission path, namely:

[0167]

[0168] Among them, Φ n (κ) represents the atmospheric turbulence refractive index power spectrum, and its expression is:

[0169]

[0170] Among them, 0≤κ≤∞, κ m =5.92 / l0, κ0=2π / L0, l0 and L0 are the inner and outer scales of atmospheric turbulence respectively; C0 represents the atmospheric refractive index structure constant;

[0171] Substituting equations (7) to (9) and equation (1) in step 1.1 into equation (5), we can obtain the light field of the LG beam after it is transmitted through atmospheric turbulence: The expression is:

[0172]

[0173] Step 2: Combined with the light field expression of the LG beam after it is transmitted through atmospheric turbulence in step 1, derive the mixing efficiency expression of coherent detection when the LG beam is used as the signal beam.

[0174] Example 5

[0175] Example 5

[0176] A method for achieving coherent detection using a Laguerre-Gaussian beam comprises the following steps:

[0177] Step 1: Based on the propagation theory of LG beams in atmospheric turbulence, the Rytov approximation, and the atmospheric turbulence refractive index power spectrum model, the light field expression of the LG beam after propagation through atmospheric turbulence is derived;

[0178] Step 1 is implemented according to the following specific steps:

[0179] Step 1.1: First determine the optical field of the LG beam when it propagates in free space The expression is:

[0180]

[0181] Where C is a constant, l is the topological charge, p is the radial index, r is the distance between the light and the transmission axis, k is the wave number, and k = 2π / λ, z is the transmission distance, i is the imaginary unit, represents the spiral phase factor of the LG beam, z R is the confocal parameter (i.e., Rayleigh length), λ is the wavelength of the LG beam;

[0182] In formula (1) of step 1.1: is the generalized Laguerre polynomial, which is expressed as:

[0183]

[0184] Where m is the summation variable, w(z) is the beam radius of the LG beam at the transmission distance z, and w(z) is expressed as:

[0185]

[0186] Where w0 is the waist radius of the LG beam at the transmission distance z = 0; z R is the confocal parameter, z R Expressed as:

[0187]

[0188] Where λ is the wavelength of the LG beam.

[0189] Step 1.2: Use the Rytov method to solve the light field of the LG beam after it propagates through atmospheric turbulence. Its expression is for:

[0190]

[0191] Where L represents the beam propagation path, and ψ(r,L) represents the complex phase perturbation caused by atmospheric turbulence. Usually, only the first-order and second-order complex phase perturbations need to be considered, which can be expressed as:

[0192] ψ(r,L)=ψ1(r,L)+ψ2(r,L)(6);

[0193] Where ψ1(r,L) and ψ2(r,L) represent the first-order and second-order complex phase perturbations, respectively;

[0194] Substituting formula (6) into formula (5), formula (5) is expressed as:

[0195]

[0196] Among them, E1(0,0) represents the statistical moment of the first-order complex phase disturbance of the light field under the horizontal transmission path, namely:

[0197]

[0198] Among them, Φ n (κ) represents the atmospheric turbulence refractive index power spectrum, and its expression is:

[0199]

[0200] Among them, 0≤κ≤∞, κ m=5.92 / l0, κ0=2π / L0, l0 and L0 are the inner and outer scales of atmospheric turbulence respectively; C0 represents the atmospheric refractive index structure constant;

[0201] Substituting equations (7) to (9) and equation (1) in step 1.1 into equation (5), we can obtain the light field of the LG beam after it is transmitted through atmospheric turbulence: The expression is:

[0202]

[0203] Step 2: Combined with the light field expression of the LG beam after it is transmitted through atmospheric turbulence in step 1, derive the mixing efficiency expression of coherent detection when the LG beam is used as the signal beam.

[0204] Example 6

[0205] A method for achieving coherent detection using a Laguerre-Gaussian beam comprises the following steps:

[0206] Step 1: Based on the propagation theory of LG beams in atmospheric turbulence, the Rytov approximation, and the atmospheric turbulence refractive index power spectrum model, the light field expression of the LG beam after propagation through atmospheric turbulence is derived;

[0207] Step 2: Combined with the light field expression of the LG beam after it is transmitted through atmospheric turbulence in step 1, derive the mixing efficiency expression of coherent detection when the LG beam is used as the signal beam.

[0208] Step 2 is implemented according to the following specific steps:

[0209] Step 2.1: When the light field of the signal beam is E S (t), the light field of the local oscillator beam is E L (t), the expressions of the two are:

[0210]

[0211] Among them, A S 、ω S and Represent the amplitude, central angular frequency and phase of the signal beam respectively, A L 、ω L and represent the amplitude, central angular frequency and phase of the local oscillator beam respectively, and t is the time;

[0212] After the signal beam and the local oscillator beam pass through the 90° optical mixer, according to equations (11) and (12), the current signal i detected on the photodetector is p Expressed as:

[0213]

[0214] Where α is the photodetector responsivity, α = eη / hv, e is the electron charge, η is the photodetector quantum efficiency, h is the Planck constant, and v is the optical frequency. In formula (13), the first two terms are the DC component, the third term is the sum frequency component, and the fourth term is the difference frequency component. The frequency difference is ω S -ω L ;

[0215] For equation (13), after filtering out the DC component and sum frequency component, the intermediate frequency signal current output by the photodetector is:

[0216]

[0217] According to the basic principle of coherent detection and formula (14), the power of the intermediate frequency current is IF 2 > is represented as:

[0218]

[0219] Where s is the area of ​​the photosensitive surface of the photodetector, R is the radius of the photosensitive surface of the photodetector, r0 is the radial vector from any point on the photosensitive surface of the photodetector to the center, and θ is the angle between the wave vector and the polar coordinate axis in the polar coordinate system;

[0220] Step 2.2 assumes that the noise after coherent detection comes from the local oscillator beam, so the power of the noise is Expressed as:

[0221]

[0222] Where B is the bandwidth of the photodetector;

[0223] Introducing the signal-to-noise ratio, combined with equation (16) and equation (15) in step 2.1, the signal-to-noise ratio (SNR) of the photodetector output is expressed as:

[0224]

[0225] For an ideal coherent detection system, the signal-to-noise ratio (SNR0) of the photodetector output is:

[0226]

[0227] According to equation (17), the mixing efficiency of coherent detection is defined as:

[0228]

[0229] Step 2.3: The light field of the LG beam after being transmitted through atmospheric turbulence is used as the light field of the signal beam, and the light field of the LG beam without being affected by atmospheric turbulence is used as the light field of the local oscillator beam, that is:​

[0230]

[0231] in, is the light field expression after atmospheric turbulence transmission, as shown in equation (10) in step 1; is the light field expression of the LG beam without the influence of atmospheric turbulence, that is:

[0232]

[0233] Where C is a constant, l is the topological charge, p is the radial index, and r is the distance between the light and the transmission axis. represents the spiral phase factor of the LG beam, w0 is the beam waist radius;

[0234] Substituting equations (21) and (22) into equation (20) in step 2.2, the mixing efficiency η is het Expressed as:

[0235]

[0236] Substituting equation (23) and equation (10) in step 1 into equation (25), the mixing efficiency of the LG beam after transmission through atmospheric turbulence can be simulated and calculated.

Claims

1. A method for coherent detection using a Laguerre-Gaussian beam, characterized in that: The following steps are involved: Step 1: Based on the propagation theory of Laguerre-Gaussian LG beams in atmospheric turbulence, the Rytov approximation, and the atmospheric turbulence refractive index power spectrum model, the light field expression of the LG beam after propagation through atmospheric turbulence is derived; Step 2: Combined with the light field expression of the LG beam after atmospheric turbulence transmission in step 1, derive the mixing efficiency η of coherent detection when the LG beam is used as the signal beam. het The expression is used to analyze the variation of the mixing efficiency of the LG beam with the transmission distance under different light source parameters and photodetector radius after the beam is transmitted through atmospheric turbulence. Among them, the mixing efficiency η het Expressed as: Where, is the light field expression of the LG beam without the influence of atmospheric turbulence, is the light field expression after atmospheric turbulence transmission, r is the distance between the light and the transmission axis, represents the spiral phase factor of the LG beam, z is the transmission distance, r0 is the radial vector from any point on the photosensitive surface of the photodetector to the center, θ is the angle between the wave vector and the polar coordinate axis in the polar coordinate system, and R is the radius of the photosensitive surface of the photodetector.

2. The method for coherent detection using a Laguerre-Gaussian beam according to claim 1, wherein: Step 1 is implemented according to the following specific steps: Step 1.1: First determine the optical field of the LG beam when it propagates in free space The expression is: Where C is a constant, l is the topological charge, p is the radial index, r is the distance between the light and the transmission axis, k is the wave number, and k = 2π / λ, z is the transmission distance, i is the imaginary unit, represents the spiral phase factor of the LG beam, z R is the confocal parameter, λ is the wavelength of the LG beam; In formula (1) of step 1.1: is the generalized Laguerre polynomial, which is expressed as: Where m is the summation variable, w(z) is the beam radius of the LG beam at the transmission distance z, and w(z) is expressed as: Where w0 is the waist radius of the LG beam at the transmission distance z = 0; z R is the confocal parameter, z R Expressed as: Where, λ is the wavelength of the LG beam; Step 1.2: Use the Rytov method to solve the light field of the LG beam after it propagates through atmospheric turbulence. Its expression is for: Where L represents the beam propagation path, and ψ(r,L) represents the complex phase perturbation caused by atmospheric turbulence. Usually, only the first-order and second-order complex phase perturbations need to be considered, which can be expressed as: ψ(r,L)=ψ1(r,L)+ψ2(r,L) (6); Where ψ1(r,L) and ψ2(r,L) represent the first-order and second-order complex phase perturbations, respectively; Substituting formula (6) into formula (5), formula (5) is expressed as: Among them, E1(0,0) represents the statistical moment of the first-order complex phase disturbance of the light field under the horizontal transmission path, namely: Among them, Φ n (κ) represents the atmospheric turbulence refractive index power spectrum, and its expression is: Among them, 0≤κ≤∞, κ m =5.92 / l0, κ0=2π / L0, l0 and L0 are the inner and outer scales of atmospheric turbulence respectively; C0 represents the atmospheric refractive index structure constant; Substituting equations (7) to (9) and equation (1) in step 1.1 into equation (5), we can obtain the light field of the LG beam after it is transmitted through atmospheric turbulence: The expression is:

3. The method for coherent detection using a Laguerre-Gaussian beam according to claim 2, wherein: Step 2 is implemented according to the following specific steps: Step 2.1: When the light field of the signal beam is E S (t), the light field of the local oscillator beam is E L (t), the expressions of the two are: Among them, A S 、ω S and Represent the amplitude, central angular frequency and phase of the signal beam respectively, A L 、ω L and represent the amplitude, central angular frequency and phase of the local oscillator beam respectively, and t is the time; After the signal beam and the local oscillator beam pass through the 90° optical mixer, according to equations (11) and (12), the current signal i detected on the photodetector is p Expressed as: Where α is the photodetector responsivity, α = eη / hv, e is the electron charge, η is the photodetector quantum efficiency, h is the Planck constant, and v is the optical frequency. In formula (13), the first two terms are the DC component, the third term is the sum frequency component, and the fourth term is the difference frequency component. The frequency difference is ω S -ω L ; For equation (13), after filtering out the DC component and sum frequency component, the intermediate frequency signal current output by the photodetector is: According to the basic principle of coherent detection and formula (14), the power of the intermediate frequency current is IF 2 > is represented as:​ Where s is the area of ​​the photosensitive surface of the photodetector, R is the radius of the photosensitive surface of the photodetector, r0 is the radial vector from any point on the photosensitive surface of the photodetector to the center, and θ is the angle between the wave vector and the polar coordinate axis in the polar coordinate system; Step 2.2 assumes that the noise after coherent detection comes from the local oscillator beam, so the power of the noise is Expressed as: Where B is the bandwidth of the photodetector; Introducing the signal-to-noise ratio, combined with equation (16) and equation (15) in step 2.1, the signal-to-noise ratio (SNR) of the photodetector output is expressed as: In formula (17) of step 2.2: P S is the signal beam power, which is expressed as: For an ideal coherent detection system, the signal-to-noise ratio (SNR0) of the photodetector output is: According to equation (17), the mixing efficiency of coherent detection is defined as: Step 2.3: The light field of the LG beam after being transmitted through atmospheric turbulence is used as the light field of the signal beam, and the light field of the LG beam without being affected by atmospheric turbulence is used as the light field of the local oscillator beam, that is: in, is the light field expression after atmospheric turbulence transmission, as shown in equation (10) in step 1; is the light field expression of the LG beam without the influence of atmospheric turbulence, that is: Where C is a constant, l is the topological charge, p is the radial index, and r is the distance between the light and the transmission axis. represents the spiral phase factor of the LG beam, w0 is the beam waist radius; In formula (23) of step 2.3, is the Laguerre polynomial, expressed as: Substituting equations (21) and (22) into equation (20) in step 2.2, we can obtain the mixing efficiency η het ; Substituting equation (23) and equation (10) in step 1 into equation (25), the mixing efficiency of the LG beam after transmission through atmospheric turbulence can be simulated and calculated.

4. The method for coherent detection using a Laguerre-Gaussian beam according to claim 3, wherein: The model of the optical mixer in step 2.1 is COH24.

5. The method for coherent detection using a Laguerre-Gaussian beam according to claim 3, wherein: The photodetector model in step 2.1 is: BPD230.

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