Method for realizing coherent detection by using Laguerre-Gaussian beam

Abstract

The invention discloses a method for realizing coherent detection by using a Laguerre-Gaussian beam, which comprises the following steps of: 1, deducing a light field expression of an LG beam transmitted by atmospheric turbulence according to a transmission theory of the LG beam in the atmospheric turbulence, Rytov approximation and an atmospheric turbulence refractive index power spectrum model; and step 2, deducing a mixing efficiency expression of coherent detection when the LG light beam is used as a signal light beam in combination with a light field expression of the LG light beam transmitted by the atmospheric turbulence in the step 1. The method can be used for analyzing the change condition of the mixing efficiency of the vortex light beam after being transmitted by the atmospheric turbulence under different light source parameters and turbulence intensities, and has important reference value for reasonably regulating and controlling laser parameters, improving the anti-interference capability of the light beam after being transmitted by the atmospheric turbulence and keeping better transmission characteristics for actual communication.

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 the atmospheric turbulence transmission medium, a series of atmospheric turbulence effects such as weakened laser coherence, beam expansion, and spot jitter will occur, which greatly reduces the communication performance of the FSO system in order to achieve long-distance space communication. In addition, coherent detection (heterodyne detection) technology is a very important research direction in the field of free space optical communication. In actual communication, coherent detection is suitable for small signal detection, but it will be affected by atmospheric turbulence effects during channel transmission. In the atmospheric turbulence channel, the random fluctuations of the amplitude and phase of the optical signal caused by the random fluctuations of the atmospheric refractive index will have a significant impact on the system performance, thereby restricting the stability of the FSO system.

[0003] Vortex beams carrying orbital angular momentum (OAM) have a relative advantage in transmission due to their phase singularity, 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 law of vortex beams in complex environments and improving the anti-interference ability and stability of beams. Summary of the invention

[0004] The purpose of the present invention is to provide a method for realizing coherent detection using a Laguerre-Gaussian beam, which can be used to analyze the change in the mixing efficiency of a vortex beam after being transmitted through atmospheric turbulence under different light source parameters and turbulence intensities, and has important reference value for reasonably regulating laser parameters, improving the anti-interference ability of the beam after being transmitted 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 transmission theory of LG beam in atmospheric turbulence, Rytov approximation and atmospheric turbulence refractive index power spectrum model, derive the light field expression of the LG beam after transmission through atmospheric turbulence;

[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: According to the Rytov method, the light field of the LG beam after it is transmitted through atmospheric turbulence is solved. Its expression is for:

[0014]

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

[0016] ψ(r,L)=ψ1(r,L)+ψ2(r,L)(6);

[0017] Among them, 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 can be 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 It is 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, A L ,ω L and They 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 It is 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 DC components, 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 the sum frequency component, the intermediate frequency signal current output by the photodetector is:

[0042]

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

[0044]

[0045] Where s is the photosensitive surface area 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 It is expressed as:

[0047]

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

[0049] By introducing the signal-to-noise ratio and combining equation (16) with 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 output by the photodetector 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 by 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:

[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 not affected by 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 It is 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 model of the photodetector in step 2.1 is: BPD230.

[0065] In formula (17) of step 2.2: P S is the signal beam power, 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 realizing coherent detection using a vortex beam. The transmission advantage of the vortex beam in atmospheric turbulence is utilized. Based on the Rytov approximation principle and the atmospheric turbulence power spectrum, the expression of the mixing efficiency of the LG beam after atmospheric turbulence transmission is derived. The method of the present invention can analyze the influence of the light source parameters, transmission distance and the photosensitive surface radius of the photodetector on the mixing efficiency during the atmospheric turbulence transmission of the LG beam, and can support the quantitative analysis of the mixing efficiency when the LG beam is transmitted in 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 research on the reasonable regulation of laser parameters, the improvement of the anti-interference ability of the LG beam after atmospheric turbulence transmission, and the subsequent coherent detection system. BRIEF DESCRIPTION OF THE DRAWINGS

[0071] Figure 1 The schematic diagram of the method for realizing coherent detection using Laguerre-Gaussian beam of the present invention;

[0072] Figure 2 is a 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 is a schematic diagram of the variation of the mixing efficiency with the transmission distance z when the topological charge number is l=1, 2, and 3 respectively after the LG light beam is transmitted through atmospheric turbulence in Example 2 of the present invention;

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

[0075] Figure 5 It is a schematic diagram of the variation of the mixing efficiency 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 is 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 transmission theory of LG beams in atmospheric turbulence, Rytov approximation and atmospheric turbulence refractive index power spectrum model, derive the light field expression of the LG beam after transmission 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 can be 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 It is expressed as:

[0087]

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

[0089] Step 1.2: When the LG beam is transmitted in the atmospheric turbulent 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 atmospheric turbulence transmission is solved. Its expression is for:

[0090]

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

[0092] ψ(r,L)=ψ1(r,L)+ψ2(r,L)(6);

[0093] Among them, 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 the LG beam is used as the 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, A L ,ω L and They 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 a 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 of the present invention is the BPD230 series photodetector) is p It is 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 DC components, 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 the sum frequency component, the intermediate frequency signal current output by the photodetector is:

[0112]

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

[0114]

[0115] Where s is the photosensitive surface area 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, since the shot noise caused by the local oscillator beam is much larger than other noises, it can be generally assumed that the noise after coherent detection comes from the local oscillator beam. Therefore, the power of the noise is It 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, expressed as:

[0122]

[0123] For an ideal coherent detection system, the signal-to-noise ratio SNR0 output by the photodetector 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 by 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 not affected by 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 It is 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 beam after transmission through atmospheric turbulence, the mixing efficiency of LG beam and Gaussian beam was compared and analyzed, and the variation of mixing efficiency with transmission distance under different light source parameters and photodetector radius after LG beam was transmitted through atmospheric turbulence was studied.

[0137] Example 1

[0138] This embodiment mainly analyzes the variation of the mixing efficiency of the LG beam and the Gaussian beam with the 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 is R = 0.02m.

[0140] 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 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 beam and Gaussian beam with topological charge number 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 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 embodiment mainly analyzes the change of mixing efficiency of LG beams after being transmitted through atmospheric turbulence under different topological charges, radial indices, and photosensitive surface radii of photodetectors. 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 is transmitted through atmospheric turbulence under different topological charges, radial indices and photodetector photosensitive surface radii can be simulated and analyzed, as shown in FIG. Figure 3 to Figure 5 As shown:

[0146] Figure 3 is the variation of mixing efficiency with transmission distance when the topological charge number l = 1, 2, and 3 after the LG beam is transmitted through atmospheric turbulence; Figure 3 It can be seen that the larger the topological charge, the higher the mixing efficiency of the LG beam after being transmitted through atmospheric turbulence, and as the transmission distance of the beam increases, the mixing efficiency gradually decreases. When the transmission distance is within one kilometer, the mixing efficiency is relatively high. This is because the beam is affected by atmospheric turbulence during transmission. The longer the transmission distance, the worse the signal quality. Figure 4 is the variation of mixing efficiency with topological charge when radial index p = 1, 2, 3 after LG beam is transmitted through atmospheric turbulence; 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 serious the beam expansion, the worse the beam turbulence suppression effect, the worse the beam quality, and the lower the mixing efficiency. Therefore, the mixing efficiency can be improved by changing the topological charge of the signal light. Figure 5 It is the variation of mixing efficiency with transmission distance when the radius of the photosensitive surface of the photodetector R = 0.02m, 0.03m, and 0.05m after the LG beam is transmitted through atmospheric turbulence; Figure 5 It can be seen that as the radius of the photosensitive surface of the photodetector 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 being transmitted through atmospheric turbulence under different light source parameters and turbulence intensities, and has important reference value for rationally regulating laser parameters, improving the anti-interference ability of the light beam after being transmitted through atmospheric turbulence, and maintaining better transmission characteristics for actual communication.

[0148] Example 3

[0149] The method for realizing coherent detection using Laguerre-Gaussian beam comprises the following steps:

[0150] Step 1, based on the transmission theory of LG beam in atmospheric turbulence, Rytov approximation and atmospheric turbulence refractive index power spectrum model, derive the light field expression of the LG beam after transmission through atmospheric turbulence;

[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] The method for realizing coherent detection using Laguerre-Gaussian beam comprises the following steps:

[0154] Step 1, based on the transmission theory of LG beam in atmospheric turbulence, Rytov approximation and atmospheric turbulence refractive index power spectrum model, derive the light field expression of the LG beam after transmission through atmospheric turbulence;

[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: According to the Rytov method, the light field of the LG beam after it is transmitted through atmospheric turbulence is solved. Its expression is for:

[0160]

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

[0162] ψ(r,L)=ψ1(r,L)+ψ2(r,L)(6);

[0163] Among them, 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] The method for realizing coherent detection using Laguerre-Gaussian beam comprises the following steps:

[0177] Step 1, based on the transmission theory of LG beam in atmospheric turbulence, Rytov approximation and atmospheric turbulence refractive index power spectrum model, derive the light field expression of the LG beam after transmission through atmospheric turbulence;

[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 can be 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 It is expressed as:

[0187]

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

[0189] Step 1.2: According to the Rytov method, the light field of the LG beam after it is transmitted through atmospheric turbulence is solved. Its expression is for:

[0190]

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

[0192] ψ(r,L)=ψ1(r,L)+ψ2(r,L)(6);

[0193] Among them, 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] The method for realizing coherent detection using Laguerre-Gaussian beam comprises the following steps:

[0206] Step 1, based on the transmission theory of LG beam in atmospheric turbulence, Rytov approximation and atmospheric turbulence refractive index power spectrum model, derive the light field expression of the LG beam after transmission through atmospheric turbulence;

[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, A L ,ω L and They 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 It is 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 DC components, 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 the sum frequency component, the intermediate frequency signal current output by the photodetector is:

[0216]

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

[0218]

[0219] Where s is the photosensitive surface area 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 It is expressed as:

[0221]

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

[0223] By introducing the signal-to-noise ratio and combining equation (16) with 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 output by the photodetector 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 by 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:​

[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 not affected by 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 It is 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 realizing coherent detection using a Laguerre-Gaussian beam, characterized in that: The following steps are involved: Step 1, based on the transmission theory of LG beam in atmospheric turbulence, Rytov approximation and atmospheric turbulence refractive index power spectrum model, derive the light field expression of the LG beam after transmission through atmospheric turbulence; 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.

2. The method for realizing coherent detection using a Laguerre-Gaussian beam according to claim 1, characterized in that: 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; Step 1.2: According to the Rytov method, the light field of the LG beam after it is transmitted through atmospheric turbulence is solved. Its expression is for: Where L represents the beam transmission path, ψ(r,L) represents the complex phase disturbance caused by atmospheric turbulence. Usually, only the first-order and second-order complex phase disturbances need to be considered, which can be expressed as: ψ(r,L)=ψ1(r,L)+ψ2(r,L) (6); Among them, 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 realizing coherent detection using Laguerre-Gaussian beam according to claim 2, characterized in that: In formula (1) of step 1.1: is the generalized Laguerre polynomial, which can be 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 It is expressed as: Where λ is the wavelength of the LG beam.

4. The method for realizing coherent detection using a Laguerre-Gaussian beam according to claim 2, characterized in that: 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, A L ,ω L and They 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 It is 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 DC components, 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 the sum frequency component, the intermediate frequency signal current output by the photodetector is: According to the basic principle of coherent detection and equation (14), the power of the intermediate frequency current is IF 2 > is represented as:​ Where s is the photosensitive surface area 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 It is expressed as: Where B is the bandwidth of the photodetector; By introducing the signal-to-noise ratio and combining equation (16) with equation (15) in step 2.1, the signal-to-noise ratio (SNR) of the photodetector output is expressed as: For an ideal coherent detection system, the signal-to-noise ratio SNR0 output by the photodetector 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 by 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: 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 not affected by 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; Substituting equations (21) and (22) into equation (20) in step 2.2, the mixing efficiency η is het It is expressed as: 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.

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

6. The method for realizing coherent detection using Laguerre-Gaussian beam according to claim 4, characterized in that: The model of the photodetector in step 2.1 is: BPD230.

7. The method for realizing coherent detection using Laguerre-Gaussian beam according to claim 4, characterized in that: In formula (17) of step 2.2: P S is the signal beam power, expressed as:

8. The method for realizing coherent detection using Laguerre-Gaussian beam according to claim 4, characterized in that: In formula (23) of step 2.3, is the Laguerre polynomial, expressed as:

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