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Echo characteristics of laser beams reflected from rough targets with a finite size in atmospheric turbulence
Optik - International Journal for Light and Electron Optics 225 (2021) 165861
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Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
Echo characteristics of laser beams reflected from rough targets with a finite size in atmospheric turbulence Ya-qing Li *, Li-guo Wang, Lei Gong, Qian Wang School of Optoelectronic Engineering, Xi’an Technological University, Xi’an, 710021, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Atmospheric turbulence Rough target Double-path propagation Covariance Scintillation index
Based on the extended Huygens-Fresnel principle, echo characteristics of the plane and spherical waves reflected from a rough target with a finite size in atmospheric turbulence were studied. The expressions of the intensity variance, covariance functions and scintillation index of the plane and spherical waves reflected from a rough target in double-path propagation through atmospheric turbulence were proposed considering the turbulence perturbations. The expressions were numerically calculated, and the results show that the intensity fluctuation caused by turbulence decreases with increasing target size. It is also found that under weak turbulence, it is mainly the log-amplitude fluctuations that affect the second-order statistical properties of the received beam intensity. Results obtained can potentially provide theoretical and technical guidance for the future applications of laser target detection and target recognition in space.
1. Introduction The term “speckle” generally refers to when a highly coherent laser is irradiated on a rough surface, and as a result the scattering field forms a speckle-like intensity fluctuation. In turbulence, there are two factors that form speckle, one of which is the speckle effect of the rough target in a vacuum, while the other is the speckle phenomenon caused by turbulence on the phase disturbance of the laser wave surface. The speckle carries a large amount of information regarding the target, and by measuring the speckle, the target features, such as the target geometry, target size and moving speed, can be obtained. In 1975, Goodman proposed a simplified scattering model of rough surface, and studied the influence of rough surface size, propagation distance and other factors on laser speckle characteristics [1]. In 1985, Ruffing investigated the relation between the correlation function of the intensities for partially or fully developed far-field speckle with two different wavelengths, along with the surface height distribution, wavelength difference and angle of incidence, and thus provided a simple way of performing surface-roughness measurements with a speckle [2]. In 1999, Cheng et al. studied the intensity covariance function of rough surface speckle, and proposed a method by which to calculate the target surface roughness and correlation length by using the intensity ratio of speckle images [3]. In the above studies, the influence of atmospheric turbulence on the speckle characteristics is usually ignored when the laser transmission distance is not too far. However, in many applications, such as the measurement of air missile target motion state by ground base station, or the measurement of ground target motion state by spaceborne radar, the laser transmission path is in a turbulent atmosphere for a long distance, and the turbulent effect cannot be ignored. Due to the complexity of the turbulence effect and target scattering interaction, the existing references on the target speckle field is generally limited to simple target situations, such as
* Corresponding author. E-mail address: [email protected] (Y.-q. Li). https://doi.org/10.1016/j.ijleo.2020.165861 Received 19 July 2020; Accepted 21 October 2020 Available online 25 October 2020 0030-4026/© 2020 Elsevier GmbH. All rights reserved.
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Fig. 1. Geometry for a laser propagation in double-path atmospheric turbulence.
diffuse plates or target containing multiple mirror glints [4–9]. Our research group analyzed the intensity correlation function of the reflected field from a rough surface with the extended Huygens-Fresnel principle, calculated the covariance and residual scintillation index of the received beam intensity for the cases of a collimated beam and a focused beam [10], proposed the statistical characteristics of a retro-reflector reflecting a Gaussian beam [11], presented the characteristics of the coherence length and intensity scintillation for the plane wave, spherical wave, and Gaussian beam reflected by the plane’s mirror and retro-reflector using the split-step Fourier method [12], and investigated the intensity fluctuation of the reflected field from a diffuse circular plate with a hard edge in turbulence by using the Rytov theory and the Extended Huygens-Fresnel principle [13]. In the present paper, the intensity variance and covariance functions of reflected waves from a rough target with a finite size propagation in atmospheric turbulence are derived considering the turbulence perturbations during the double-path propagation. The expressions were numerically calculated, and the results reveal the effects of the target size and strength of turbulence on the intensity covariance and scintillation index. 2. Correlation function of the intensity The geometry of laser propagation in double-path atmospheric turbulence is shown in Fig. 1. It is assumed that the laser (plane wave or spherical wave) propagates from the plane z = 0 of the laser device, which then propagates through the atmospheric tur bulence to the plane z = L, and illuminates the target with a diameter D. The laser reflected by the target passes through the atmo spheric turbulence a once gain and illuminates a detector in the plane z = 0. H is the height of the target above the ground, θ represents the zenith angle. For horizontal propagation, zenith angle θ = 90∘ . In this paper, the laser propagation under the condition of weak fluctuation is discussed. Therefore, under the Rytov approximation, the light field incident on the target can be written as [14] (1)
Ui (ρ) = U0 (ρ)exp[ψ i (ρ)],
where U0 (ρ) is the initial field at the plane z = 0. For plane waves, U0 (ρ) can be written as U0 (ρ) = U0 exp(ikL) and it can be written as U0 (ρ) = U0 exp[ikL + ikρ2 /(2L)] for a spherical wave. ψ i (ρ) denotes the field fluctuation caused by the atmospheric turbulence during the forward propagation, which can be expressed as ψ i (ρ) = χ i (ρ) + iSi (ρ). χ i (ρ) and Si (ρ) represent the fluctuation perturbations of the log amplitude and the phase caused by the atmospheric turbulence, respectively. The field at the target scattering from the target can be expressed as (2)
Us (ρ) = U0 (ρ)exp[χ i (ρ) + iSi (ρ) + iφ(ρ) ], where φ(ρ) is a random phase caused by the rough surface. Based on the extended Huygens-Fresnel principle, the field at the receiver can be expressed as [ ] ∫ k ik|p − ρ|2 Ut (p) = dρUs (ρ)exp exp(ikL) + ψ R (p, ρ) , 2πiL 2L ρ
(3)
where ψ R (p, ρ) is the field fluctuation of a laser beam propagating from point ρ on the target to the point p on the receiving plane in the atmospheric turbulence. The fourth-order statistical moment of the field, i.e. the correlation function of the beam intensity at the receiver, can be expressed as
2
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〈 BI (p1 , p2 ) = 〈I(p1 )I(p2 ) 〉 = Ut (p1 )Ut∗ (p1 )Ut (p2 )Ut∗ (p2 ) 〉 ( =
k
)4 ∫
2πL [ ik (
exp
2L
∫ ∫ ∫ dρ1 dρ2 dρ3 dρ4 〈Us (ρ1 )Us∗ (ρ2 )Us (ρ3 )Us∗ (ρ4 ) 〉
ρ1∼4
(4)
] ) |p1 − ρ1 |2 − |p1 − ρ2 |2 + |p2 − ρ3 |2 − |p2 − ρ4 |2 + H(ρ1 , ρ2 , ρ3 , ρ4 ; p1 , p2 ) ,
where 〈 H(ρ1 , ρ2 , ρ3 , ρ4 ; p1 , p2 ) = ψ R (p1 , ρ1 ) + ψ ∗R (p1 , ρ2 ) + ψ R (p2 , ρ3 ) + ψ ∗R (p2 , ρ4 ) 〉 [ ] 1 = exp − (D12 − D13 + D14 + D23 − D24 + D34 ) + 2Cχ 13 + 2Cχ 24 , 2
(5)
where the log-amplitude covariance function and the wave structure function are given by [10] ( ) Cχ ij = Cχ pij , ρij ∫
∫
1
= 0.132π2 k2 L 0
Cn2 (t)dt
∞
8
du · u− 3 sin2 0
) ( 2 ⃒) ( ⃒ u t(1 − t)L J0 u⃒pij t + ρij (1 − t)⃒ 2k
(∫ 1 { [ / / ] = 4.3374k2 LRe Cn2 (t)dt 0.5[iLt(1 − t)/k ]5/6 1 F 1 − 5 6; 1; ikρ2t (4Lt(1 − t) )
(6)
0
} − 0.1674ρ5/3 , t ( ) Dij = DϕR pij , ρij = 2.91Lk2
∫ 0
1
(7)
5
Cn2 (t)ρ3t dt,
⃒ ⃒ ⃒ ⃒ where p12 = p34 = 0, p13 = p14 = p23 = p24 = p1 − p2 , ρij = ρi − ρj , ρt = ⃒pij t + ρij (1 − t)⃒. C2n (t) is the model proposed by ITU-R for the
index-of-refraction structure constant with height used for the slant propagation [15]. 〈 The statistical moment Us (ρ1 )Us∗ (ρ2 )Us (ρ3 )U∗s (ρ4 ) 〉 is usually processed by assuming that the scattering fields are jointly Gaussian, however, this is not true for the case of double-path propagation. 〈 ( ) Us (ρ )U ∗ (ρ )Us (ρ )U ∗ (ρ ) 〉 = U0 (ρ1 )U0∗ (ρ2 )U0 (ρ3 )U0∗ (ρ4 ) × 〈 {1 s 2 ∗ 3 s 4 (8) exp ψ i (ρ1 ) + ψ i (ρ2 ) + ψ i (ρ3 ) + ψ ∗i (ρ4 ) + i[φ(ρ1 ) − φ(ρ2 ) + φ(ρ3 ) − φ(ρ4 ) ] } 〉 Assuming that the turbulence fluctuation and the surface fluctuation are statistically independent, and that the surface is perfectly rough, φ(ρ) averagely distributes over the range 0 ∼ 2π and is independent to that at other points. Consequently, Eq. (8) can be written as 〈 ( ) Us (ρ1 )Us∗ (ρ2 )Us (ρ3 )Us∗ (ρ4 ) 〉 = U0 (ρ1 )U0∗ (ρ2 )U0 (ρ3 )U0∗ (ρ4 ) × 〈 [ ] ∗ ∗ exp ψ i (ρ1 ) + ψ i (ρ2 ) + ψ i (ρ3 ) + ψ i (ρ4 ) 〉× 〈exp{i[φ(ρ1 ) − φ(ρ2 ) + φ(ρ3 ) − φ(ρ4 ) ] } 〉 ( / )2 ( ) = 4π k2 U0 (ρ1 )U0∗ (ρ2 )U0 (ρ3 )U0∗ (ρ4 ) × 〈 [ ] exp ψ i (ρ1 ) + ψ ∗i (ρ2 ) + ψ i (ρ3 ) + ψ ∗i (ρ4 ) 〉× [δ(ρ1 − ρ2 )δ(ρ3 − ρ4 ) + δ(ρ1 − ρ4 )δ(ρ3 − ρ2 ) ]
(9)
Using Eqs. (9) and (4), the correlation function of the beam intensity at the receiver can be written as ∫ ∫ 1 BI (p1 , p2 ) = 2 4 dρ dρ I0 (ρ1 )I0 (ρ3 )exp(4Cχi (ρ− ) )H1 (ρ1 , ρ3 , ; p1 , p2 ) π L ρ1
(10)
where ρ− = ρ1 − ρ3 , p− = p1 − p2 . Cχi (ρ− ) is the log-amplitude covariance function of the beam field on the target, which is written as
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(∫ 1 { [ / / ] Cχi (ρ− ) = 4.3374k2 LRe Cn2 (t)dt 0.5[iLt/k]5/6 1 F1 − 5 6; 1; ikρ2− (4Lt) 0 }) − 0.1674ρ5/3 , − For a plane wave incidence, and (∫ 1 { Cχi (ρ− ) = 4.3374k2 LRe Cn2 (t)dt 0.5[iLt(1 − t)/k ]5/6 × 0 } [ / ] 2/ − 5 6; 1; ik( (4Lt(1 − t) ) − 0.1674(ρ− t)5/3 , F ρ t) − 1 1
(11)
(12)
For a spherical wave incidence, and 〈 H1 = H(ρ1 , ρ1 , ρ3 , ρ3 ; p1 , p2 ) = ψ R (p1 , ρ1 ) + ψ ∗R (p1 , ρ1 ) + ψ R (p2 , ρ3 ) + ψ ∗R (p2 , ρ3 ) 〉 = exp[4Cχ (p− , ρ− ) ],
(13)
〈 H2 = H(ρ1 , ρ3 , ρ3 , ρ1 ; p1 , p2 ) = ψ R (p1 , ρ1 ) + ψ ∗R (p1 , ρ3 ) + ψ R (p2 , ρ3 ) + ψ ∗R (p2 , ρ1 ) 〉 = exp{2Cχ (p− , ρ− ) + 2Cχ (p− , ρ− ) − 1 [2DϕR (0, ρ− ) − DϕR (p− , − ρ− ) + 2DϕR (p− , 0) − DϕR (p− , ρ− ) ] 2
}
(14)
For a plane or a spherical wave incidence, it can be assumed that I0 (ρ1 ) = I0 (ρ3 ) = I0 . Using Eqs. (13) and (14), the correlation function of the intensity can be expressed as ∫ ∫ I2 BI (p− ) = 20 4 dρ dρ f (p , ρ ), (15) π L ρ1
(16)
(17)
where ⎧ { } (ρ )2 ]1/2 ( ) ( )[ 2 ⎪ ⎪ − ⎨ D cos− 1 ρ− − ρ− , ρ− ≤ D 1− 2 D D D K0 (ρ− , D) = ⎪ ⎪ ⎩ 0, ρ− > D
(18)
3. Normalized covariance function of the received intensity The normalized covariance function of the beam intensity is expressed by (19)
CI (p− ) = [BI (p− ) − 〈I(p1 ) 〉〈I(p2 ) 〉 ]/[〈I(p1 ) 〉〈I(p2 ) 〉 ], where ( 〈I(p) 〉 = Ut (p)Ut∗ (p) =
k 2πL
)2 ∫
∫
ρ1
〈
Us (ρ1 )Us∗ (ρ2 ) 〉× (20)
] [ ) 1 ik ( exp |p1 − ρ1 |2 − |p1 − ρ2 |2 − D(ρ1 − ρ2 ) 2L 2 As 〈
[ ] [ ] Us (ρ1 )Us∗ (ρ2 ) 〉 = U0 (ρ1 )U0∗ (ρ2 ) 〈exp ψ i (ρ1 ) + ψ ∗i (ρ2 ) 〉〈exp{i[φ(ρ1 ) − φ(ρ2 ) ] } 〉 ( / 2 )( ) [ ] = 4π k U0 (ρ1 )U0∗ (ρ2 ) 〈exp ψ i (ρ1 ) + ψ ∗i (ρ2 ) 〉δ(ρ1 − ρ2 ) 4
(21)
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Fig. 2. Normalized covariance of the received intensity under different turbulence conditions. (Horizontal path, L = 3000m, D = 5cm, λ = 1.38μm).
Fig. 3. Normalized covariance of the received intensity from targets with different diameters. (Horizontal path, L = 3000m, C2n = 1.23e − 15m-2/3 , λ = 1.38μm).
Then the averaged intensity becomes ∫ 1 D 2 I0 I dρ = . 〈I(p) 〉 = 2 πL ρ1
(22)
For a target with a finite diameter, the covariance of the received intensity can be numerically calculated using Eq. (19), CI (p− ) = [BI (p− ) − 〈I(p1 ) 〉〈I(p2 ) 〉 ]/[〈I(p1 ) 〉〈I(p2 ) 〉 ] /( ) ∫ / )2 ( I 2 π2 L4 dρ− K0 (ρ− , D)f (p− , ρ− ) − D2 I0 4L2 = ( 0 / )2 2 2 ρ−
(23)
where, Cχi (ρ− ), Cχ (p− , ρ− ) and DϕR (p− , ρ− ) contained in the integrand in Eq. (23) are all integrals, therefore Eq. (23) is a triple integral which can be numerically calculated with a computer program. As the results of the spherical wave incidence and the plane wave incidence are similar, only curves for the spherical wave incidence are presented in most cases. Fig. 2 presents the normalized covariance of the received intensity under different turbulence conditions. It can be found that in the absence of turbulence the normalized covariance is unity at p− = 0 and tends to be zero when p− → ∞, which is consistent with the results of the Gaussian field assumption. As turbulence becomes stronger, the normalized covariance will also increase. Thus in the presence of turbulence, the normalized covariance is larger than unity at p− = 0, which proves that the probability function of the field has changed, and tends to be a value greater than zero when p− → ∞, which is called the residual scintillation effect. However, the scale of the function does not change significantly as the turbulence strength increases. In fact, if the quadratic approximation is used in the wave structure function to make the exponent index 5/3 ≃ 2, Eq. (16) will become
5
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Fig. 4. Normalized covariance of the received intensity from targets with different diameters. (Slant path, θ = 60∘ , L = 100km, C0 = 2.21e − 13m-2/3 , λ = 1.38μm)
Fig. 5. Normalized covariance of the received intensity from targets with different zenith angles. (Slant path, H = 100km, D = 5m, C0 = 2.21e − 13m-2/3 , λ = 1.38μm).
f (p− , ρ− ) = exp{4Cχ i (ρ− ) + 4Cχ (p− , ρ− ) } [ ] ik +exp p− · ρ− exp{4Cχi (ρ− ) + 2Cχ (p− , − ρ− ) + 2Cχ (p− , ρ− ) }, L
(24)
2DϕR (0, ρ− ) − DϕR (p− , − ρ− ) + 2DϕR (p− , 0) − DϕR (p− , ρ− ) ≡ 0.
(25)
and Only terms reflecting the log-amplitude fluctuation remain in Eq. (24), which only increases the value of the normalized covari ance, and terms reflecting the phase fluctuation become negligible showing that the speckle size is not influenced by turbulence. Fig. 3 presents the normalized covariance of the received intensity from targets with different diameters D in the case of horizontal propagation, and Fig. 4 in the case of slant propagation. Both these figures show that as D increases, the fluctuations induced by turbulence decrease. It should be noted that Eq. (17) is similar to the expression of the aperture-averaged scintillation index [16], so there is also an effect called “target aperture” on the covariance similar to the aperture-average effect. It can be seen from Eq. (15) that the “aperture-average” factor is determined by f(p− , ρ− ) which is mainly dependent on the term Cχ (p− , ρ− ). In order to reflect the scale of Cχ (p− , ρ− ), the log-amplitude correlation length ρI is defined by Cχ (0, ρI )/Cχ (0) = e− 1 . For a target with D ≫ ρI , the normalized √̅̅̅̅̅̅̅̅ covariance tends to be the same value as in a vacuum. ρI is proportional to L/k, and the height distribution of the turbulence also make ρI larger in slant propagation, which is why the target in Fig. 4 (ρI = 3m) is much larger than that in Fig. 3 (ρI = 2.5cm), but the averaged covariance is still higher than that in Fig. 3. Fig. 5 shows the normalized covariance of the received intensity from a target with a fixed height but different zenith angles, which is often found in the situation where a target object in space orbits the earth. As the angle gets larger, the propagation length (and hence length of the path through turbulence) increase, thus the scale and the value of the covariance function both increase.
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Fig. 6. Normalized covariance of the received intensity from targets with different diameters using the small target approximation. For slant propagation, C0 = 2.21e − 13m-2/3 , θ = 60∘ L = 100km. For horizontal propagation C2n = 2.46e − 15m-2/3 , L = 3000m, λ = 1.38μm.
3.1. Small target If the target is so small (D ≪ ρI ) that the target “aperture-averaged” effect is not obvious, then Cχ (p− , ρ− ) ≃ Cχ (p− , 0), the corre lation function can be simplified as [ ( )] ∫ ∫ I2 ik BI (p− ) = 20 4 exp{4Cχ i (0) + 4Cχ (p− , 0) } dρ1 dρ3 1 + exp p− · ρ− L πL ρ2
(27)
where σ2Ii is the scintillation index of the field on the target, and CIR (p− ) is the normalized covariance of the intensity of a spherical wave propagating from the target to the receiving plane. Cs (p− ) is the normalized covariance of the intensity reflected by a target in a vacuum. As the speckle size of the reflected wave from a small target is large, which means that the scale of Cs (p− ) is larger than CIR (p− ), then ] [ the scale of σ2Ii + CIR (p− ) [1 + Cs (p− ) ] is mainly determined by CIR (p− ). Therefore the normalized intensity covariance function expressed by Eq. (27) is a modulation of fluctuations in two scales, one is the speckle scale of the reflected field by the target in vacuum, and the other is mainly influenced by turbulence. It is also found that as p− → ∞, CI (p− ) = σ 2Ii which means that the residual scin tillation index is that induced by turbulence in the forward propagation. Fig. 6 shows the normalized intensity covariance function of the reflected wave from targets with different diameters for the two cases of horizontal or slant propagation. These curves clearly show the two-scale structures with a break point nearly at ρ− = ρI ’, where ρI ’ is the intensity correlation length of a spherical wave propagating from the target to the receiver. 3.2. Scintillation index When p− = 0, the normalized covariance becomes the normalized variance, also called the scintillation index. And Eq. (17) becomes / ∫ ( 2 4) BI (0) = I02 πL dρ− K0 (ρ− , D)f (0, ρ− ) ρ−
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Fig. 7. Scintillation index versus the diameter of the target. Horizontal path, C2n = 2.46e − 15m-2/3 , L = 3000m.
Then scintillation index can be derived from Eq. (23) as, ]/ [ σ 2I = CI (0) = BI (0) − 〈I〉2 〈I〉2 ∫ /( / )2 = 2 πD2 4 dρ− K0 (ρ− , D)[1 + CIi (ρ− ) + CIR (ρ− ) ] − 1 ρ−
32 = 2 4 πD
∫ ρ−
dρ− K0 (ρ− , D) + 32
π 2 D4 =1+
32 π2 D4
32 π 2 D4
∫ dρ− K0 (ρ− , D)CIi (ρ− ) +
(29)
∫ ρ−
dρ− K0 (ρ− , D)CIR (ρ− ) − 1
∫ dρ− K0 (ρ− , D)CIi (ρ− ) + ρ−
ρ−
32 π2 D4
∫ dρ− K0 (ρ− , D)CIR (ρ− ) ρ−
It should be noticed that the last two terms in Eq. (29) are just expressions of the aperture-averaged scintillation index of a singlepass wave in turbulence. The last term corresponds to a spherical wave, while the first term corresponds to the incident wave (either plane or spherical). Then Eq. (27) can be simplified as { 1 + 2Ap (D)σ 2Ip + 2As (D)σ 2Is , plane incidence , (30) σ2I = 1 + 4As (D)σ2Is , spherical incedence where σ 2Ip is the scintillation index of a single-pass plane wave, and Ap (D) is the corresponding aperture averaging factor. σ2Is and As (D) are similarly those for a spherical wave [17]. ⎧ ( 2 )5/6 ( 2) ( 2 )2 ⎪ kD kD2 ⎪ ⎪ 1 + 1.711 kD − 0.040 kD − 2.257 , ≤1 ⎪ ⎨ 4L 4L 4L 4L Ap = , (31) ( 2 )− 7/6 ⎪ ⎪ kD kD2 ⎪ ⎪ ⎩ 0.932 , >1 4L 4L ⎧ ( 2 )5/6 ( 2 )2 ( 2) ⎪ kD kD kD kD2 ⎪ ⎪ − 2.093 , − 0.135 ≤1 ⎪ 1 + 1.711 ⎨ 4L 4L 4L 4L As = ( 2 )5/6 ⎪ ⎪ kD kD2 ⎪ ⎪ ⎩ 1 + 0.333 , >1 4L 4L
(32)
It can be found that the scintillation index consists of two parts; the first is always a value of unity due to the fluctuations caused by the rough surface, and the other is induced by the turbulence, which is averaged by the aperture of the target. It should be noted that the last part of the scintillation index is two times that introduced by the turbulence in the double-path propagation. Fig. 7 shows the scintillation index versus the diameter of the target. As the target becomes larger, the scintillation becomes smaller and eventually tends to unity. 4. Conclusions In this study, it was found that the turbulence along the forward propagation has a significant effect on the intensity fluctuation of 8
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the speckle field in the atmosphere. It contributes the same (for a spherical wave) or more (for a plane wave) intensity fluctuations than the turbulence along the backward propagation does. It was also found that the size of the target also has a great effect on the intensity fluctuations, which is similar as the aperture averaging effect. The fluctuations caused by turbulence can be averaged by a large target. If D ≫ ρI , then the covariance tends to the same value as in a vacuum. But for a small target or for a long propagation length, especially along a slant path, it probably meets the condition that D ≤ ρI , then turbulence effects must be considered as the covariance or variance of the received intensity are concerned. It should be noted that as the Rytov approximation was used, the results in the paper are only applicable in weak turbulence. The study results can provide theoretical and technical guidance for the application of laser target detection and recognition. Declaration of Competing Interest We declare that we have no financial and personal relationships with other people or organizations that can inappropriately in fluence our work, there is no professional or other personal interest of any nature or kind in any product, service and company that could be construed as influencing the position presented in, or the review of, the manuscript entitled. Acknowledgments This paper is supported by National Natural Science Foundation of China (Grant No. 61805190, 61704132), Young Talents Lifting Plan Project of Association for Science and Technology in Shaanxi Province (Grant No. 20190508), Scientific Research Plan Project of Shaanxi Education Department (Grant No. 19JK0406, 19JK0390) and Natural Science Foundation of Shaanxi Province (Grant No. 2019JQ-839, 2019JM-238). References [1] J.W. Goodman, Statistical properties of laser speckle patterns. Laser Speckle and Related Phenomena, Springer, Berlin, Heidelberg, 1975, pp. 9–75. [2] B. Ruffing, J. Fleischer, Spectral correlation of partially or fully developed speckle patterns generated by rough surfaces, J. Opt. Soc. Am. A 2 (10) (1985) 1637–1643. [3] C.F. Cheng, D.P. Qi, D.L. Liu, S.Y. Teng, The computational simulations of the Gaussian correlation random surface and its light-scattering speckle field and the analysis of the intensity probability density, Acta Phys. Sin. 48 (9) (1999) 1635–16439 (in chinese). [4] J.F. Holmes, M.H. Lee, J.R. Kerr, Effect of the log-amplitude covariance function on the statistics of speckle propagation through the turbulent atmosphere, J. Opt. Soc. Am. 70 (4) (1980) 335–340. [5] C.M. McIntyre, M.H. Lee, J.H. Churnside, Statistics of irradiance scattered from a diffuse target containing multiple glints, J. Opt. Soc. Am. A 70 (9) (1980) 1084–1091. [6] M.H. Lee, J.F. Holmes, Effect of the turbulent atmosphere on the autocovariance function for a speckle field generated by a laser beam with random pointing error, J. Opt. Soc. Am. A 71 (5) (1981) 559–565. [7] F. Amzajerdian, J.F. Holmes, Time-delayed statistics for a bistatic coherent lidar operating in atmospheric turbulence, Appl. Opt. 30 (21) (1991) 3029–3033. [8] V.S.R. Gudimetla, J.F. Holmes, M.E. Fossey, Covariance of the received intensity of a partially coherent laser speckle pattern in the turbulent atmosphere, Appl. Opt. 31 (9) (1992) 1286–1295. [9] H.Y. Wei, Z.S. Wu, H. Peng, Scattering from a diffuse target in the slant atmospheric turbulence, Acta Phys. Sin. 57 (10) (2008) 6666–6672 (in chinese). [10] H.L. Hou, L.G. Wang, Y.Q. Li, Intensity fluctuations of a laser beam reflected by a rough surface in atmospheric turbulence, Optik 157 (2018) 817–826. [11] Y.Q. Li, Z.S. Wu, Statistical characteristics of a Gaussian beam reflected by a retro-reflector in atmospheric turbulence, Optik 158 (2018) 1361–1369. [12] Y.Q. Li, L.G. Wang, Z.S. Wu, Numerical simulation for echo characteristics of laser beams reflected by retro-reflectors in atmospheric turbulence, Optik 179 (2019) 244–251. [13] L.G. Wang, M. Gao, Y.Q. Li, L. Gong, Intensity fluctuations of reflected wave from a diffuse target with a hard edge in atmospheric turbulence, J. Quant. Spectrosc. Radiat. Transf. 195 (2017) 141–146. [14] D.L. Fried, Propagation of a spherical wave in a turbulent medium, J. Opt. Soc. Am. 57 (1967) 175–180. [15] Y.Q. Li, L.G. Wang, Study on self-repairing and non-diffraction of Airy beams in slant atmospheric turbulence, Optica Applicata 48 (3) (2018) 435–447. [16] D.L. Fried, Aperture averaging of scintillation, J. Opt. Soc. Am. 57 (1967) 169–175. [17] L.C. Andrews, Aperture-averaging factor for optical scintillations of plane and spherical waves in the atmosphere, J. Opt. Soc. Am. A 9 (1992) 597–600.
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Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
Echo characteristics of laser beams reflected from rough targets with a finite size in atmospheric turbulence Ya-qing Li *, Li-guo Wang, Lei Gong, Qian Wang School of Optoelectronic Engineering, Xi’an Technological University, Xi’an, 710021, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Atmospheric turbulence Rough target Double-path propagation Covariance Scintillation index
Based on the extended Huygens-Fresnel principle, echo characteristics of the plane and spherical waves reflected from a rough target with a finite size in atmospheric turbulence were studied. The expressions of the intensity variance, covariance functions and scintillation index of the plane and spherical waves reflected from a rough target in double-path propagation through atmospheric turbulence were proposed considering the turbulence perturbations. The expressions were numerically calculated, and the results show that the intensity fluctuation caused by turbulence decreases with increasing target size. It is also found that under weak turbulence, it is mainly the log-amplitude fluctuations that affect the second-order statistical properties of the received beam intensity. Results obtained can potentially provide theoretical and technical guidance for the future applications of laser target detection and target recognition in space.
1. Introduction The term “speckle” generally refers to when a highly coherent laser is irradiated on a rough surface, and as a result the scattering field forms a speckle-like intensity fluctuation. In turbulence, there are two factors that form speckle, one of which is the speckle effect of the rough target in a vacuum, while the other is the speckle phenomenon caused by turbulence on the phase disturbance of the laser wave surface. The speckle carries a large amount of information regarding the target, and by measuring the speckle, the target features, such as the target geometry, target size and moving speed, can be obtained. In 1975, Goodman proposed a simplified scattering model of rough surface, and studied the influence of rough surface size, propagation distance and other factors on laser speckle characteristics [1]. In 1985, Ruffing investigated the relation between the correlation function of the intensities for partially or fully developed far-field speckle with two different wavelengths, along with the surface height distribution, wavelength difference and angle of incidence, and thus provided a simple way of performing surface-roughness measurements with a speckle [2]. In 1999, Cheng et al. studied the intensity covariance function of rough surface speckle, and proposed a method by which to calculate the target surface roughness and correlation length by using the intensity ratio of speckle images [3]. In the above studies, the influence of atmospheric turbulence on the speckle characteristics is usually ignored when the laser transmission distance is not too far. However, in many applications, such as the measurement of air missile target motion state by ground base station, or the measurement of ground target motion state by spaceborne radar, the laser transmission path is in a turbulent atmosphere for a long distance, and the turbulent effect cannot be ignored. Due to the complexity of the turbulence effect and target scattering interaction, the existing references on the target speckle field is generally limited to simple target situations, such as
* Corresponding author. E-mail address: [email protected] (Y.-q. Li). https://doi.org/10.1016/j.ijleo.2020.165861 Received 19 July 2020; Accepted 21 October 2020 Available online 25 October 2020 0030-4026/© 2020 Elsevier GmbH. All rights reserved.
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Fig. 1. Geometry for a laser propagation in double-path atmospheric turbulence.
diffuse plates or target containing multiple mirror glints [4–9]. Our research group analyzed the intensity correlation function of the reflected field from a rough surface with the extended Huygens-Fresnel principle, calculated the covariance and residual scintillation index of the received beam intensity for the cases of a collimated beam and a focused beam [10], proposed the statistical characteristics of a retro-reflector reflecting a Gaussian beam [11], presented the characteristics of the coherence length and intensity scintillation for the plane wave, spherical wave, and Gaussian beam reflected by the plane’s mirror and retro-reflector using the split-step Fourier method [12], and investigated the intensity fluctuation of the reflected field from a diffuse circular plate with a hard edge in turbulence by using the Rytov theory and the Extended Huygens-Fresnel principle [13]. In the present paper, the intensity variance and covariance functions of reflected waves from a rough target with a finite size propagation in atmospheric turbulence are derived considering the turbulence perturbations during the double-path propagation. The expressions were numerically calculated, and the results reveal the effects of the target size and strength of turbulence on the intensity covariance and scintillation index. 2. Correlation function of the intensity The geometry of laser propagation in double-path atmospheric turbulence is shown in Fig. 1. It is assumed that the laser (plane wave or spherical wave) propagates from the plane z = 0 of the laser device, which then propagates through the atmospheric tur bulence to the plane z = L, and illuminates the target with a diameter D. The laser reflected by the target passes through the atmo spheric turbulence a once gain and illuminates a detector in the plane z = 0. H is the height of the target above the ground, θ represents the zenith angle. For horizontal propagation, zenith angle θ = 90∘ . In this paper, the laser propagation under the condition of weak fluctuation is discussed. Therefore, under the Rytov approximation, the light field incident on the target can be written as [14] (1)
Ui (ρ) = U0 (ρ)exp[ψ i (ρ)],
where U0 (ρ) is the initial field at the plane z = 0. For plane waves, U0 (ρ) can be written as U0 (ρ) = U0 exp(ikL) and it can be written as U0 (ρ) = U0 exp[ikL + ikρ2 /(2L)] for a spherical wave. ψ i (ρ) denotes the field fluctuation caused by the atmospheric turbulence during the forward propagation, which can be expressed as ψ i (ρ) = χ i (ρ) + iSi (ρ). χ i (ρ) and Si (ρ) represent the fluctuation perturbations of the log amplitude and the phase caused by the atmospheric turbulence, respectively. The field at the target scattering from the target can be expressed as (2)
Us (ρ) = U0 (ρ)exp[χ i (ρ) + iSi (ρ) + iφ(ρ) ], where φ(ρ) is a random phase caused by the rough surface. Based on the extended Huygens-Fresnel principle, the field at the receiver can be expressed as [ ] ∫ k ik|p − ρ|2 Ut (p) = dρUs (ρ)exp exp(ikL) + ψ R (p, ρ) , 2πiL 2L ρ
(3)
where ψ R (p, ρ) is the field fluctuation of a laser beam propagating from point ρ on the target to the point p on the receiving plane in the atmospheric turbulence. The fourth-order statistical moment of the field, i.e. the correlation function of the beam intensity at the receiver, can be expressed as
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〈 BI (p1 , p2 ) = 〈I(p1 )I(p2 ) 〉 = Ut (p1 )Ut∗ (p1 )Ut (p2 )Ut∗ (p2 ) 〉 ( =
k
)4 ∫
2πL [ ik (
exp
2L
∫ ∫ ∫ dρ1 dρ2 dρ3 dρ4 〈Us (ρ1 )Us∗ (ρ2 )Us (ρ3 )Us∗ (ρ4 ) 〉
ρ1∼4
(4)
] ) |p1 − ρ1 |2 − |p1 − ρ2 |2 + |p2 − ρ3 |2 − |p2 − ρ4 |2 + H(ρ1 , ρ2 , ρ3 , ρ4 ; p1 , p2 ) ,
where 〈 H(ρ1 , ρ2 , ρ3 , ρ4 ; p1 , p2 ) = ψ R (p1 , ρ1 ) + ψ ∗R (p1 , ρ2 ) + ψ R (p2 , ρ3 ) + ψ ∗R (p2 , ρ4 ) 〉 [ ] 1 = exp − (D12 − D13 + D14 + D23 − D24 + D34 ) + 2Cχ 13 + 2Cχ 24 , 2
(5)
where the log-amplitude covariance function and the wave structure function are given by [10] ( ) Cχ ij = Cχ pij , ρij ∫
∫
1
= 0.132π2 k2 L 0
Cn2 (t)dt
∞
8
du · u− 3 sin2 0
) ( 2 ⃒) ( ⃒ u t(1 − t)L J0 u⃒pij t + ρij (1 − t)⃒ 2k
(∫ 1 { [ / / ] = 4.3374k2 LRe Cn2 (t)dt 0.5[iLt(1 − t)/k ]5/6 1 F 1 − 5 6; 1; ikρ2t (4Lt(1 − t) )
(6)
0
} − 0.1674ρ5/3 , t ( ) Dij = DϕR pij , ρij = 2.91Lk2
∫ 0
1
(7)
5
Cn2 (t)ρ3t dt,
⃒ ⃒ ⃒ ⃒ where p12 = p34 = 0, p13 = p14 = p23 = p24 = p1 − p2 , ρij = ρi − ρj , ρt = ⃒pij t + ρij (1 − t)⃒. C2n (t) is the model proposed by ITU-R for the
index-of-refraction structure constant with height used for the slant propagation [15]. 〈 The statistical moment Us (ρ1 )Us∗ (ρ2 )Us (ρ3 )U∗s (ρ4 ) 〉 is usually processed by assuming that the scattering fields are jointly Gaussian, however, this is not true for the case of double-path propagation. 〈 ( ) Us (ρ )U ∗ (ρ )Us (ρ )U ∗ (ρ ) 〉 = U0 (ρ1 )U0∗ (ρ2 )U0 (ρ3 )U0∗ (ρ4 ) × 〈 {1 s 2 ∗ 3 s 4 (8) exp ψ i (ρ1 ) + ψ i (ρ2 ) + ψ i (ρ3 ) + ψ ∗i (ρ4 ) + i[φ(ρ1 ) − φ(ρ2 ) + φ(ρ3 ) − φ(ρ4 ) ] } 〉 Assuming that the turbulence fluctuation and the surface fluctuation are statistically independent, and that the surface is perfectly rough, φ(ρ) averagely distributes over the range 0 ∼ 2π and is independent to that at other points. Consequently, Eq. (8) can be written as 〈 ( ) Us (ρ1 )Us∗ (ρ2 )Us (ρ3 )Us∗ (ρ4 ) 〉 = U0 (ρ1 )U0∗ (ρ2 )U0 (ρ3 )U0∗ (ρ4 ) × 〈 [ ] ∗ ∗ exp ψ i (ρ1 ) + ψ i (ρ2 ) + ψ i (ρ3 ) + ψ i (ρ4 ) 〉× 〈exp{i[φ(ρ1 ) − φ(ρ2 ) + φ(ρ3 ) − φ(ρ4 ) ] } 〉 ( / )2 ( ) = 4π k2 U0 (ρ1 )U0∗ (ρ2 )U0 (ρ3 )U0∗ (ρ4 ) × 〈 [ ] exp ψ i (ρ1 ) + ψ ∗i (ρ2 ) + ψ i (ρ3 ) + ψ ∗i (ρ4 ) 〉× [δ(ρ1 − ρ2 )δ(ρ3 − ρ4 ) + δ(ρ1 − ρ4 )δ(ρ3 − ρ2 ) ]
(9)
Using Eqs. (9) and (4), the correlation function of the beam intensity at the receiver can be written as ∫ ∫ 1 BI (p1 , p2 ) = 2 4 dρ dρ I0 (ρ1 )I0 (ρ3 )exp(4Cχi (ρ− ) )H1 (ρ1 , ρ3 , ; p1 , p2 ) π L ρ1
(10)
where ρ− = ρ1 − ρ3 , p− = p1 − p2 . Cχi (ρ− ) is the log-amplitude covariance function of the beam field on the target, which is written as
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(∫ 1 { [ / / ] Cχi (ρ− ) = 4.3374k2 LRe Cn2 (t)dt 0.5[iLt/k]5/6 1 F1 − 5 6; 1; ikρ2− (4Lt) 0 }) − 0.1674ρ5/3 , − For a plane wave incidence, and (∫ 1 { Cχi (ρ− ) = 4.3374k2 LRe Cn2 (t)dt 0.5[iLt(1 − t)/k ]5/6 × 0 } [ / ] 2/ − 5 6; 1; ik( (4Lt(1 − t) ) − 0.1674(ρ− t)5/3 , F ρ t) − 1 1
(11)
(12)
For a spherical wave incidence, and 〈 H1 = H(ρ1 , ρ1 , ρ3 , ρ3 ; p1 , p2 ) = ψ R (p1 , ρ1 ) + ψ ∗R (p1 , ρ1 ) + ψ R (p2 , ρ3 ) + ψ ∗R (p2 , ρ3 ) 〉 = exp[4Cχ (p− , ρ− ) ],
(13)
〈 H2 = H(ρ1 , ρ3 , ρ3 , ρ1 ; p1 , p2 ) = ψ R (p1 , ρ1 ) + ψ ∗R (p1 , ρ3 ) + ψ R (p2 , ρ3 ) + ψ ∗R (p2 , ρ1 ) 〉 = exp{2Cχ (p− , ρ− ) + 2Cχ (p− , ρ− ) − 1 [2DϕR (0, ρ− ) − DϕR (p− , − ρ− ) + 2DϕR (p− , 0) − DϕR (p− , ρ− ) ] 2
}
(14)
For a plane or a spherical wave incidence, it can be assumed that I0 (ρ1 ) = I0 (ρ3 ) = I0 . Using Eqs. (13) and (14), the correlation function of the intensity can be expressed as ∫ ∫ I2 BI (p− ) = 20 4 dρ dρ f (p , ρ ), (15) π L ρ1
(16)
(17)
where ⎧ { } (ρ )2 ]1/2 ( ) ( )[ 2 ⎪ ⎪ − ⎨ D cos− 1 ρ− − ρ− , ρ− ≤ D 1− 2 D D D K0 (ρ− , D) = ⎪ ⎪ ⎩ 0, ρ− > D
(18)
3. Normalized covariance function of the received intensity The normalized covariance function of the beam intensity is expressed by (19)
CI (p− ) = [BI (p− ) − 〈I(p1 ) 〉〈I(p2 ) 〉 ]/[〈I(p1 ) 〉〈I(p2 ) 〉 ], where ( 〈I(p) 〉 = Ut (p)Ut∗ (p) =
k 2πL
)2 ∫
∫
ρ1
〈
Us (ρ1 )Us∗ (ρ2 ) 〉× (20)
] [ ) 1 ik ( exp |p1 − ρ1 |2 − |p1 − ρ2 |2 − D(ρ1 − ρ2 ) 2L 2 As 〈
[ ] [ ] Us (ρ1 )Us∗ (ρ2 ) 〉 = U0 (ρ1 )U0∗ (ρ2 ) 〈exp ψ i (ρ1 ) + ψ ∗i (ρ2 ) 〉〈exp{i[φ(ρ1 ) − φ(ρ2 ) ] } 〉 ( / 2 )( ) [ ] = 4π k U0 (ρ1 )U0∗ (ρ2 ) 〈exp ψ i (ρ1 ) + ψ ∗i (ρ2 ) 〉δ(ρ1 − ρ2 ) 4
(21)
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Fig. 2. Normalized covariance of the received intensity under different turbulence conditions. (Horizontal path, L = 3000m, D = 5cm, λ = 1.38μm).
Fig. 3. Normalized covariance of the received intensity from targets with different diameters. (Horizontal path, L = 3000m, C2n = 1.23e − 15m-2/3 , λ = 1.38μm).
Then the averaged intensity becomes ∫ 1 D 2 I0 I dρ = . 〈I(p) 〉 = 2 πL ρ1
(22)
For a target with a finite diameter, the covariance of the received intensity can be numerically calculated using Eq. (19), CI (p− ) = [BI (p− ) − 〈I(p1 ) 〉〈I(p2 ) 〉 ]/[〈I(p1 ) 〉〈I(p2 ) 〉 ] /( ) ∫ / )2 ( I 2 π2 L4 dρ− K0 (ρ− , D)f (p− , ρ− ) − D2 I0 4L2 = ( 0 / )2 2 2 ρ−
(23)
where, Cχi (ρ− ), Cχ (p− , ρ− ) and DϕR (p− , ρ− ) contained in the integrand in Eq. (23) are all integrals, therefore Eq. (23) is a triple integral which can be numerically calculated with a computer program. As the results of the spherical wave incidence and the plane wave incidence are similar, only curves for the spherical wave incidence are presented in most cases. Fig. 2 presents the normalized covariance of the received intensity under different turbulence conditions. It can be found that in the absence of turbulence the normalized covariance is unity at p− = 0 and tends to be zero when p− → ∞, which is consistent with the results of the Gaussian field assumption. As turbulence becomes stronger, the normalized covariance will also increase. Thus in the presence of turbulence, the normalized covariance is larger than unity at p− = 0, which proves that the probability function of the field has changed, and tends to be a value greater than zero when p− → ∞, which is called the residual scintillation effect. However, the scale of the function does not change significantly as the turbulence strength increases. In fact, if the quadratic approximation is used in the wave structure function to make the exponent index 5/3 ≃ 2, Eq. (16) will become
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Fig. 4. Normalized covariance of the received intensity from targets with different diameters. (Slant path, θ = 60∘ , L = 100km, C0 = 2.21e − 13m-2/3 , λ = 1.38μm)
Fig. 5. Normalized covariance of the received intensity from targets with different zenith angles. (Slant path, H = 100km, D = 5m, C0 = 2.21e − 13m-2/3 , λ = 1.38μm).
f (p− , ρ− ) = exp{4Cχ i (ρ− ) + 4Cχ (p− , ρ− ) } [ ] ik +exp p− · ρ− exp{4Cχi (ρ− ) + 2Cχ (p− , − ρ− ) + 2Cχ (p− , ρ− ) }, L
(24)
2DϕR (0, ρ− ) − DϕR (p− , − ρ− ) + 2DϕR (p− , 0) − DϕR (p− , ρ− ) ≡ 0.
(25)
and Only terms reflecting the log-amplitude fluctuation remain in Eq. (24), which only increases the value of the normalized covari ance, and terms reflecting the phase fluctuation become negligible showing that the speckle size is not influenced by turbulence. Fig. 3 presents the normalized covariance of the received intensity from targets with different diameters D in the case of horizontal propagation, and Fig. 4 in the case of slant propagation. Both these figures show that as D increases, the fluctuations induced by turbulence decrease. It should be noted that Eq. (17) is similar to the expression of the aperture-averaged scintillation index [16], so there is also an effect called “target aperture” on the covariance similar to the aperture-average effect. It can be seen from Eq. (15) that the “aperture-average” factor is determined by f(p− , ρ− ) which is mainly dependent on the term Cχ (p− , ρ− ). In order to reflect the scale of Cχ (p− , ρ− ), the log-amplitude correlation length ρI is defined by Cχ (0, ρI )/Cχ (0) = e− 1 . For a target with D ≫ ρI , the normalized √̅̅̅̅̅̅̅̅ covariance tends to be the same value as in a vacuum. ρI is proportional to L/k, and the height distribution of the turbulence also make ρI larger in slant propagation, which is why the target in Fig. 4 (ρI = 3m) is much larger than that in Fig. 3 (ρI = 2.5cm), but the averaged covariance is still higher than that in Fig. 3. Fig. 5 shows the normalized covariance of the received intensity from a target with a fixed height but different zenith angles, which is often found in the situation where a target object in space orbits the earth. As the angle gets larger, the propagation length (and hence length of the path through turbulence) increase, thus the scale and the value of the covariance function both increase.
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Fig. 6. Normalized covariance of the received intensity from targets with different diameters using the small target approximation. For slant propagation, C0 = 2.21e − 13m-2/3 , θ = 60∘ L = 100km. For horizontal propagation C2n = 2.46e − 15m-2/3 , L = 3000m, λ = 1.38μm.
3.1. Small target If the target is so small (D ≪ ρI ) that the target “aperture-averaged” effect is not obvious, then Cχ (p− , ρ− ) ≃ Cχ (p− , 0), the corre lation function can be simplified as [ ( )] ∫ ∫ I2 ik BI (p− ) = 20 4 exp{4Cχ i (0) + 4Cχ (p− , 0) } dρ1 dρ3 1 + exp p− · ρ− L πL ρ2
(27)
where σ2Ii is the scintillation index of the field on the target, and CIR (p− ) is the normalized covariance of the intensity of a spherical wave propagating from the target to the receiving plane. Cs (p− ) is the normalized covariance of the intensity reflected by a target in a vacuum. As the speckle size of the reflected wave from a small target is large, which means that the scale of Cs (p− ) is larger than CIR (p− ), then ] [ the scale of σ2Ii + CIR (p− ) [1 + Cs (p− ) ] is mainly determined by CIR (p− ). Therefore the normalized intensity covariance function expressed by Eq. (27) is a modulation of fluctuations in two scales, one is the speckle scale of the reflected field by the target in vacuum, and the other is mainly influenced by turbulence. It is also found that as p− → ∞, CI (p− ) = σ 2Ii which means that the residual scin tillation index is that induced by turbulence in the forward propagation. Fig. 6 shows the normalized intensity covariance function of the reflected wave from targets with different diameters for the two cases of horizontal or slant propagation. These curves clearly show the two-scale structures with a break point nearly at ρ− = ρI ’, where ρI ’ is the intensity correlation length of a spherical wave propagating from the target to the receiver. 3.2. Scintillation index When p− = 0, the normalized covariance becomes the normalized variance, also called the scintillation index. And Eq. (17) becomes / ∫ ( 2 4) BI (0) = I02 πL dρ− K0 (ρ− , D)f (0, ρ− ) ρ−
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Fig. 7. Scintillation index versus the diameter of the target. Horizontal path, C2n = 2.46e − 15m-2/3 , L = 3000m.
Then scintillation index can be derived from Eq. (23) as, ]/ [ σ 2I = CI (0) = BI (0) − 〈I〉2 〈I〉2 ∫ /( / )2 = 2 πD2 4 dρ− K0 (ρ− , D)[1 + CIi (ρ− ) + CIR (ρ− ) ] − 1 ρ−
32 = 2 4 πD
∫ ρ−
dρ− K0 (ρ− , D) + 32
π 2 D4 =1+
32 π2 D4
32 π 2 D4
∫ dρ− K0 (ρ− , D)CIi (ρ− ) +
(29)
∫ ρ−
dρ− K0 (ρ− , D)CIR (ρ− ) − 1
∫ dρ− K0 (ρ− , D)CIi (ρ− ) + ρ−
ρ−
32 π2 D4
∫ dρ− K0 (ρ− , D)CIR (ρ− ) ρ−
It should be noticed that the last two terms in Eq. (29) are just expressions of the aperture-averaged scintillation index of a singlepass wave in turbulence. The last term corresponds to a spherical wave, while the first term corresponds to the incident wave (either plane or spherical). Then Eq. (27) can be simplified as { 1 + 2Ap (D)σ 2Ip + 2As (D)σ 2Is , plane incidence , (30) σ2I = 1 + 4As (D)σ2Is , spherical incedence where σ 2Ip is the scintillation index of a single-pass plane wave, and Ap (D) is the corresponding aperture averaging factor. σ2Is and As (D) are similarly those for a spherical wave [17]. ⎧ ( 2 )5/6 ( 2) ( 2 )2 ⎪ kD kD2 ⎪ ⎪ 1 + 1.711 kD − 0.040 kD − 2.257 , ≤1 ⎪ ⎨ 4L 4L 4L 4L Ap = , (31) ( 2 )− 7/6 ⎪ ⎪ kD kD2 ⎪ ⎪ ⎩ 0.932 , >1 4L 4L ⎧ ( 2 )5/6 ( 2 )2 ( 2) ⎪ kD kD kD kD2 ⎪ ⎪ − 2.093 , − 0.135 ≤1 ⎪ 1 + 1.711 ⎨ 4L 4L 4L 4L As = ( 2 )5/6 ⎪ ⎪ kD kD2 ⎪ ⎪ ⎩ 1 + 0.333 , >1 4L 4L
(32)
It can be found that the scintillation index consists of two parts; the first is always a value of unity due to the fluctuations caused by the rough surface, and the other is induced by the turbulence, which is averaged by the aperture of the target. It should be noted that the last part of the scintillation index is two times that introduced by the turbulence in the double-path propagation. Fig. 7 shows the scintillation index versus the diameter of the target. As the target becomes larger, the scintillation becomes smaller and eventually tends to unity. 4. Conclusions In this study, it was found that the turbulence along the forward propagation has a significant effect on the intensity fluctuation of 8
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the speckle field in the atmosphere. It contributes the same (for a spherical wave) or more (for a plane wave) intensity fluctuations than the turbulence along the backward propagation does. It was also found that the size of the target also has a great effect on the intensity fluctuations, which is similar as the aperture averaging effect. The fluctuations caused by turbulence can be averaged by a large target. If D ≫ ρI , then the covariance tends to the same value as in a vacuum. But for a small target or for a long propagation length, especially along a slant path, it probably meets the condition that D ≤ ρI , then turbulence effects must be considered as the covariance or variance of the received intensity are concerned. It should be noted that as the Rytov approximation was used, the results in the paper are only applicable in weak turbulence. The study results can provide theoretical and technical guidance for the application of laser target detection and recognition. Declaration of Competing Interest We declare that we have no financial and personal relationships with other people or organizations that can inappropriately in fluence our work, there is no professional or other personal interest of any nature or kind in any product, service and company that could be construed as influencing the position presented in, or the review of, the manuscript entitled. Acknowledgments This paper is supported by National Natural Science Foundation of China (Grant No. 61805190, 61704132), Young Talents Lifting Plan Project of Association for Science and Technology in Shaanxi Province (Grant No. 20190508), Scientific Research Plan Project of Shaanxi Education Department (Grant No. 19JK0406, 19JK0390) and Natural Science Foundation of Shaanxi Province (Grant No. 2019JQ-839, 2019JM-238). References [1] J.W. Goodman, Statistical properties of laser speckle patterns. Laser Speckle and Related Phenomena, Springer, Berlin, Heidelberg, 1975, pp. 9–75. [2] B. Ruffing, J. Fleischer, Spectral correlation of partially or fully developed speckle patterns generated by rough surfaces, J. Opt. Soc. Am. A 2 (10) (1985) 1637–1643. [3] C.F. Cheng, D.P. Qi, D.L. Liu, S.Y. Teng, The computational simulations of the Gaussian correlation random surface and its light-scattering speckle field and the analysis of the intensity probability density, Acta Phys. Sin. 48 (9) (1999) 1635–16439 (in chinese). [4] J.F. Holmes, M.H. Lee, J.R. Kerr, Effect of the log-amplitude covariance function on the statistics of speckle propagation through the turbulent atmosphere, J. Opt. Soc. Am. 70 (4) (1980) 335–340. [5] C.M. McIntyre, M.H. Lee, J.H. Churnside, Statistics of irradiance scattered from a diffuse target containing multiple glints, J. Opt. Soc. Am. A 70 (9) (1980) 1084–1091. [6] M.H. Lee, J.F. Holmes, Effect of the turbulent atmosphere on the autocovariance function for a speckle field generated by a laser beam with random pointing error, J. Opt. Soc. Am. A 71 (5) (1981) 559–565. [7] F. Amzajerdian, J.F. Holmes, Time-delayed statistics for a bistatic coherent lidar operating in atmospheric turbulence, Appl. Opt. 30 (21) (1991) 3029–3033. [8] V.S.R. Gudimetla, J.F. Holmes, M.E. Fossey, Covariance of the received intensity of a partially coherent laser speckle pattern in the turbulent atmosphere, Appl. Opt. 31 (9) (1992) 1286–1295. [9] H.Y. Wei, Z.S. Wu, H. Peng, Scattering from a diffuse target in the slant atmospheric turbulence, Acta Phys. Sin. 57 (10) (2008) 6666–6672 (in chinese). [10] H.L. Hou, L.G. Wang, Y.Q. Li, Intensity fluctuations of a laser beam reflected by a rough surface in atmospheric turbulence, Optik 157 (2018) 817–826. [11] Y.Q. Li, Z.S. Wu, Statistical characteristics of a Gaussian beam reflected by a retro-reflector in atmospheric turbulence, Optik 158 (2018) 1361–1369. [12] Y.Q. Li, L.G. Wang, Z.S. Wu, Numerical simulation for echo characteristics of laser beams reflected by retro-reflectors in atmospheric turbulence, Optik 179 (2019) 244–251. [13] L.G. Wang, M. Gao, Y.Q. Li, L. Gong, Intensity fluctuations of reflected wave from a diffuse target with a hard edge in atmospheric turbulence, J. Quant. Spectrosc. Radiat. Transf. 195 (2017) 141–146. [14] D.L. Fried, Propagation of a spherical wave in a turbulent medium, J. Opt. Soc. Am. 57 (1967) 175–180. [15] Y.Q. Li, L.G. Wang, Study on self-repairing and non-diffraction of Airy beams in slant atmospheric turbulence, Optica Applicata 48 (3) (2018) 435–447. [16] D.L. Fried, Aperture averaging of scintillation, J. Opt. Soc. Am. 57 (1967) 169–175. [17] L.C. Andrews, Aperture-averaging factor for optical scintillations of plane and spherical waves in the atmosphere, J. Opt. Soc. Am. A 9 (1992) 597–600.
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