Propagation of Bessel Gaussian beams through non-Kolmogorov turbulence based on Rytov theory

Wang Wanjun; Wu Zhensen; Shang Qingchao; Bai Lu

doi:10.1364/OE.26.021712

1. Introduction

Laser beams propagating in the turbulence have been widely studied and applied to the free-space optical communication [1–3], the laser radar systems [4, 5], and the imaging systems analysis [6, 7]. Among the various types of laser beams, Bessel beams are widely investigated due to their non-diffraction and self-healing properties [8, 9]. However, Bessel beams require an infinite amount of energy which is not feasible. To generate Bessel beams with finite energy, mostly a Gaussian exponential is chosen as a windowing function, thus Bessel Gaussian beams is introduced [10].

Bessel Gaussian beams propagating in the turbulence have been investigated by many scholars [11–14]. Based on the extended Huygens–Fresnel principle, the average intensity is simplified to a double integral [15–17]. Lots of theoretical and experimental studies show that Bessel beams are less affected than the Gaussian beams by the turbulence [18, 19]. And more information could be carried by Bessel beams than Gaussian beams for their orbital angular momentum [20, 21]. The study on the long-distance Bessel beam propagation through Kolmogorov turbulence shows that Bessel beams would be better than the Gaussian beams to overcome the loss of power through the turbulence [22]. The application of the adaptive compensation techniques in Bessel beams could reduce the inter-channel crosstalk, improve the bit-error rate performance, and recuperate the non-diffracting property of Bessel beams [23].

Based on the extended Huygens–Fresnel principle, the average intensity of Bessel Gaussian beams propagating through the turbulence is calculated under the condition that the wave structure function must be approximated to a quadratic function [24–26]. In addition, the Rytov theory could also be used to investigate the beams propagation through the turbulence [27–29]. However, no one has reported the applicable condition of the quadratic approximation for the Bessel Gaussian beam. Under this circumstance, this paper introduces another major method, the Rytov theory and the average intensity is derived by the Rytov theory without the quadratic approximation. On one hand, this study could be used to verify the accuracy of the quadratic approximation. Moreover, it provides the theoretical basis for other further research investigated by the Rytov theory, such as the slant path propagation, the channel crosstalk and the double-passage problems.

In this paper, the average intensity of the Bessel Gaussian beams propagating through the non-Kolmogorov turbulence is derived without the quadratic approximation. In order to obtain the average intensity, the second order statistical moments are also derived. The comparison between the intensity calculated by Rytov theory and that by extended Huygens–Fresnel principle is analyzed. The result indicates that the Rytov theory is better than the extended Huygens–Fresnel principle, especially when the beams propagate through the small inner scale turbulence. Except this case, there is a good agreement between the intensity calculated by the two methods. To verify the presumption, the intensities of Gaussian beams are calculated by two methods. After that the result is applied to investigate the Bessel Gaussian beam propagating through the different power spectrum atmosphere in slant path. There is an interesting finding which does exist in the Gaussian beams propagation. When the source width of the Bessel Gaussian beams is different, the beam intensity variation with the inner scale, can display different behaviors. The central intensities of some specific source width beams whose turbulence perturbation is zero do not vary with the inner sale after the inner scale increasing to a certain value. And the profile of this Bessel Gaussian beams has a flat top. This work provides the theoretical basis for the applications of the Bessel Gaussian beams propagation through the non-Kolmogorov turbulence.

2. Theoretical formulation

2.1 Average intensity based on Ratov theory

The field distribution of the high order Bessel Gaussian beams on the source plane with radial coordinates r and $φ_{r}$ can be written as Eq. (1) [10, 15].

U (r, φ_{r}) = J_{n} (β r) \exp (- k α r^{2}) \exp (- i n φ_{r})

where n denotes the order of the Bessel function,

β

is the width parameter, k is the wave number.

α = 1 / k w_{}^{2} + i / (2 F_{0})

,

w

is the Gaussian source width and

F_{0}

is the focusing parameter. In this paper, the beams are collimated beams (

F_{0} = \infty

).

The field distribution on a receiver plane of Bessel Gaussian beams propagating through the free space, which is used from the Eq. (4) of Ref [30], is reprinted here as Eq. (2) for the convenience.

U (r, L) = - \frac{\exp (i k L)}{1 + 2 i α L} \exp (- i n φ_{r}) \exp [- \frac{i β^{2} L + 2 α k^{2} r^{2}}{2 k (1 + 2 i α L)}] J_{n} (\frac{β r}{1 + 2 i α L})

According to the Rytov theory under the weak irradiance fluctuation conditions [27,31], the average intensity on a receiver plane of beams propagating through turbulence can be expressed as Eq. (2) [27].

〈 I (r, φ_{r}) 〉 = U (r, L) U^{*} (r, L) \exp [2 E_{1} (0,0) + E_{2} (r, r)]

where U is the field distribution of beams propagating through the free space shown as Eq. (2). E₁, E₂ are the second-order statistical moments.

2.2 Second order statistical moments

The second-order statistical moments have the relationship with the first and second order perturbations as Eqs. (4) and (5) [27].

E_{1} (r, r) = 〈 Φ_{2} (r, L) 〉

E_{2} (r_{1}, r_{2}) = 〈 Φ_{1} (r_{1}, L) Φ_{1}^{*} (r_{2}, L) 〉

The first and second order perturbations $Φ_{1} (r, L), Φ_{2} (r, L)$ can be expressed as Eqs. (6) and (7).

Φ_{1} (r, L) = \frac{k^{2}}{2 π} \int_{0}^{L} d z \int \int_{- \infty}^{\infty} d^{2} s \exp [i k (L - z) + \frac{i k {| s - r |}^{2}}{2 (L - z)}] \frac{U (s, z)}{U (r, L)} \frac{n_{1} (s, z)}{L - z}

Φ_{2} (r, L) = \frac{k^{2}}{2 π} \int_{0}^{L} d z \int \int_{- \infty}^{\infty} d^{2} s \exp [i k (L - z) + \frac{i k {| s - r |}^{2}}{2 (L - z)}] \frac{U (s, z)}{U (r, L)} \frac{Φ_{1} (s, z) n_{1} (s, z)}{L - z}

where

n_{1} (s, z) = \int \int_{- \infty}^{\infty} \exp (i K \cdot s) d v (K, z)

, K is the wave vector of scalar spatial frequency, and

κ = | K |

.

| |

is the vector magnitude.

In order to solve the integrals of Eqs. (6) and (7), there are Eqs. (8) and (9) benefiting from the 6.633(2) and 8.406(3) of Ref [32]. Equation (10) comes from the combination of 3.937(1) and (2) in Ref [32].

\int_{0}^{\infty} x \exp (- γ x^{2}) J_{n} (α x) J_{n} (β x) d x = \frac{1}{2 γ} \exp (- \frac{α^{2} + β^{2}}{4 γ}) I_{n} (\frac{α β}{2 γ})

I_{n} (z) = i^{- n} J_{n} (i z)

\int_{0}^{2 π} \exp (β \cos x) \exp (- i n x) d x = 2 π I_{n} (β)

The first order perturbation, Eq. (11) is derived by substituting Eqs. (2), (8), (9) and (10) into Eq. (6).

\begin{matrix} Φ_{1} (r, L) = i k J_{n} {(\frac{β r}{1 + 2 i α L})}^{- 1} \int_{0}^{L} d z \int \int_{- \infty}^{\infty} d v (K, z) \exp [i γ K \cdot r - \frac{i κ^{2} γ}{2 k} (L - z)] \\ \times J_{n} [- \frac{β (L - z)}{k (1 + 2 i α L)} | K - \frac{k r}{L - z} |] \exp [- i n (φ_{K r} - φ_{r})] \end{matrix}

where

γ = (1 + 2 j α z) / (1 + 2 j α L)

,

φ_{κ r}

is the angle of the vector

K - k (L - z) r

.

To ensure the statistical homogeneity of the refractive index, there is Eq. (12).

〈 d v (K, z) d v^{*} (K', z') 〉 = F_{n} (K, | z - z^{'} |) δ (K - K^{'}) d^{2} κ d^{2} κ^{'}

where

F_{n} (K, | μ |)

is the two dimensional spectral density and it have the relationship with the power spectrum as

2 π Φ_{n} (K) = \int_{- \infty}^{\infty} F_{n} (K, | μ |) d μ

. Based on the property of the

δ (K - K^{'})

, there is an equation

K = K^{'}

. Make the change of the variables,

μ = z - z'

,

η = (z + z') / 2

. According to the appreciable values of the function

F_{n} (K, | z - z^{'} |)

, there is an approximation

z ≅ z' ≅ η

. From the result above, the second order statistical moment E₂ is derived as Eq. (13) by substituting the Eqs. (11) and (12) into Eq. (5).

\begin{matrix} E_{2} (r, r) = 2 π k^{2} {| J_{n} (\frac{β r}{1 + 2 i α L}) |}^{- 2} \int_{0}^{L} d η \int \int_{- \infty}^{\infty} d^{2} κ Φ_{n} (κ) \\ \times \exp [i (γ - γ *) K \cdot r - \frac{i κ^{2}}{2 k} (γ - γ *) (L - η)] \\ \times J_{n} [\frac{β (L - η)}{k (1 + 2 i α L)} | K - \frac{k r}{L - η} |] J_{n}^{*} [\frac{β (L - η)}{k (1 + 2 i α L)} | K - \frac{k r}{L - η} |] \end{matrix}

The second order perturbation is expressed as Eq. (14) by substituting Eq. (11) into Eq. (7).

\begin{matrix} Φ_{2} (r, L) = \frac{i k^{3}}{2 π} J_{n} {(\frac{β r}{1 + 2 i α L})}^{- 1} \int_{0}^{L} d z \int_{0}^{z} d z^{'} \int \int_{- \infty}^{\infty} \int \int_{- \infty}^{\infty} \frac{d v (K, z) d v (K^{'}, z^{'})}{γ (L - z)} \\ \times \exp [\frac{i γ k r^{2}}{2 (L - z)} - \frac{i γ^{'} κ^{'}^{2}}{2 k} (z - z^{'})] \exp [\frac{i β^{2} (L - z)}{2 k (1 + 2 i α L) (1 + 2 i α z)}] \\ \times \int \int_{- \infty}^{\infty} d^{2} s \exp [\frac{i k s^{2}}{2 γ (L - z)}] \exp [i s \cdot (K + γ^{'} K^{'} - \frac{k r}{L - z})] \\ \times J_{n} [- \frac{β (z - z')}{k (1 + 2 i α z)} | K^{'} - \frac{k s}{z - z'} |] \exp [- i n (φ_{s} - φ_{r}) - i n (φ_{K' s} - φ_{s})] \end{matrix}

where

γ' = (1 + 2 j α z') / (1 + 2 j α z)

. In Eq. (14), the inequality

| K' (z - z') | < < | k s |

could be met at most of the integration interval. So assume that s is the main factor to the directions, and

φ_{K' s} \approx φ_{s}

,

| K' (z - z') - k s | \approx | k s |

. Using the same method as the first order perturbation, the second order perturbation can be simplified to the Eq. (15).

\begin{matrix} Φ_{2} (r, L) = - k^{2} J_{n} {(\frac{β r}{1 + 2 i α L})}^{- 1} \int_{0}^{L} d z \int_{0}^{z} d z^{'} \int \int_{- \infty}^{\infty} \int \int_{- \infty}^{\infty} d v (K, z) d v (K^{'}, z^{'}) \\ \times \exp [- i n (φ_{K K' r} - φ_{r})] J_{n} [\frac{β (L - z)}{k (1 + 2 i α L)} | K + γ^{'} K^{'} - \frac{k r}{L - z} |] \\ \times \exp [i γ (K + γ^{'} K^{'}) \cdot r - \frac{i {(K + γ^{'} K^{'})}^{2} γ}{2 k} (L - z) - \frac{i γ^{'} κ^{'}^{2}}{2 k} (z - z^{'})] \end{matrix}

Based on the fact $n_{1} (s, z)$ is a real function, the Eq. (16) can be found.

〈 d v (K, z) d v^{*} (K^{'}, z^{'}) 〉 = 〈 d v (K, z) d v^{*} (- K^{'}, z^{'}) 〉 = F_{n} (K, | z - z^{'} |) δ (K + K^{'}) d^{2} κ d^{2} κ^{'}

Based on $δ (K + K^{'})$ , there is $K + K^{'} = 0$ . With the approximation $z ≅ z' ≅ η$ , the equation $K + γ' K^{'} = 0$ can be obtained. Than the second order statistical moments E₁ is derived as Eq. (17).

E_{1} (r, r) = 〈 Φ_{2} (r, L) 〉 = E_{1} (0, 0) = - π k^{2} \int_{0}^{L} d η \int \int_{- \infty}^{\infty} d^{2} κ Φ_{n} (K)

The second order statistical moment E₂ of Bessel Gaussian beams is the same as the $E_{2}^{'}$ of Gaussian beams (19) under the condition (18) that K is much smaller around the axis.

| K - \frac{k r}{L - η} | ≅ \frac{k r}{L - η}

E_{2}^{'} (r, r) = 4 π^{2} k^{2} \int_{0}^{L} d η \int_{- \infty}^{\infty} d κ κ Φ_{n} (K) J_{0} [(γ - γ *) κ r] \exp [- \frac{i κ^{2}}{2 k} (γ - γ *) (L - η)]

When r = 0, the turbulence perturbation T of the 0 order Bessel Gaussian beams could be obtained by submitting the Eq. (13) and Eq. (17) into Eq. (3).

\begin{matrix} T = 2 E_{1} (0,0) + E_{2} (r, r) \\ = 4 π^{2} k^{2} \int_{0}^{L} d η \int_{0}^{\infty} d κ κ Φ_{n} (κ) {\exp [- \frac{i κ^{2}}{2 k} (γ - γ *) (L - η)] \\ \times J_{0} [- \frac{β (L - η) κ}{k (1 + 2 i α L)}] J_{0}^{*} [- \frac{β (L - η) κ}{k (1 + 2 i α L)}] - 1} \end{matrix}

When the $κ$ is small, there are two approximations $\exp (x) \approx 1 + x$ and $J_{0} (x) \approx 1 - 0.25 x^{2}$ . The Eq. (20) could be simplified to the Eq. (21) by submitting the approximations to the expression of the turbulence perturbation under the condition that the inner scale of the turbulence is not small and ignore the high order term of $κ$ .

T = 4 π^{2} k^{2} \int_{0}^{L} d η \int_{0}^{\infty} d κ κ Φ_{n} (κ) \frac{κ^{2} {(L - η)}^{2}}{2 k (1 + 4 α^{2} L^{2})} [\frac{β^{2} (4 α^{2} L^{2} - 1)}{k (1 + 4 α^{2} L^{2})} - 4 α]

Thus, the turbulence perturbation could equal to 0, and an interesting phenomenon could be observed that the 0 order Bessel Gaussian beams will not be affected by the turbulence under the condition that the Eq. (22) holds.

16 k α^{3} L^{2} - 4 α^{2} β^{2} L^{2} + 4 k α + β^{2} = 0

2.3 Average intensity based on extended Huygens–Fresnel principle

According to extended Huygens–Fresnel principle, the average receiver intensity $〈 I (r, φ_{r}) 〉$ of Bessel Gaussian beams, which is used from the Eq. (5) of Ref [15], is reprinted here as Eq. (23) for convenience.

\begin{matrix} 〈 I (r, φ_{r}) 〉 = \frac{b^{2}}{π (k α - i b + 1 / ρ_{0}^{2})} \exp [- \frac{a_{b}^{2} + 4 b r^{2}}{4 (k α - i b + 1 / ρ_{0}^{2})}] \\ \times \int_{0}^{\infty} \int_{0}^{2 π} d s_{2} d φ_{s_{2}} J_{n} (a_{b} s_{2}) \exp (i n φ_{s_{2}}) s_{2} \\ \times \frac{{[- i b r \exp (i φ_{r}) + s_{2} \exp (- i φ_{s_{2}}) / ρ_{0}^{2}]}^{n}}{{[- b^{2} r^{2} + s_{2}^{2} / ρ_{0}^{4} - 2 i b r s_{2} \cos (φ_{r} - φ_{s_{2}}) / ρ_{0}^{2}]}^{n / 2}} \\ \times J_{n} {\frac{a_{b} {[- b^{2} r^{2} + s_{2}^{2} / ρ_{0}^{4} - 2 i b r s_{2} \cos (φ_{r} - φ_{s_{2}}) / ρ_{0}^{2}]}^{1 / 2}}{k α - i b + 1 / ρ_{0}^{2}}} \\ \times \exp [- k α^{*} s_{2}^{2} - \frac{(i b k α + k α / ρ_{0}^{2} + b^{2}) s_{2}^{2} - 2 b r s_{2} (i k α + b) \cos (φ_{r} - φ_{s_{2}})}{k α - i b + 1 / ρ_{0}^{2}}] \end{matrix}

where b = 2k/L, ρ₀ is the spherical-wave spatial coherence radius. In order to get the integral (23), the wave structure function has been approximated into a quadratic function as the equation

D_{s p} (Q) = 2 ρ_{0}^{- 2} Q^{2}

, and the spherical-wave spatial coherence radius is expressed as the Eq. (24) [24, 25]. However, this approximation is bad for the turbulence with the small inner scale.

ρ_{0}^{- 2} = π^{2} k^{2} z / 3 \int_{0}^{\infty} d κ κ^{3} Φ_{n} (κ)

Based on the extended Huygens–Fresnel principle, the average intensity on a receiver plane of Gaussian beams propagating through the non-Kolmogorov turbulence can be obtained without the quadratic approximation. And it is expressed as Eq. (25) [27].

〈 I_{H G B} (r, L) 〉 = \frac{k^{2} w_{}^{2}}{4 L^{2}} \int_{0}^{\infty} Q d Q J_{0} (\frac{k r Q}{L}) \exp (- \frac{k Q^{2}}{4 Λ L}) \exp [- \frac{1}{2} D_{s p} (Q)]

where

Λ = \frac{2 L}{k w_{}^{2} [1 + {(2 L)}^{2} / {(k w_{}^{2})}^{2}]}

The wave structure function is shown as Eq. (27) under the condition $l_{0} \leq L_{0}$ [33].

\begin{array}{l} D_{s p} (Q) = 1.303 C_{n}^{2} k^{2} L κ_{x}^{- 5 / 3} {Γ (- 5 / 6) [1 -_{2} F_{2} (- 5 / 6, 1 / 2; 1, 3 / 2; - κ_{x}^{2} Q^{2} / 4)] \\ + a_{1} Γ (- 1 / 3) [1 -_{2} F_{2} (- 1 / 3, 1 / 2; 1, 3 / 2; - κ_{x}^{2} Q^{2} / 4)] \\ + a_{2} Γ (- 1 / 4) [1 -_{2} F_{2} (- 1 / 4, 1 / 2; 1, 3 / 2; - κ_{x}^{2} Q^{2} / 4)] - \frac{3}{5} κ_{x}^{5 / 3} κ_{0}^{1 / 3} Q^{2}} \end{array}

2.4 Power spectrum models and refractive index structure constant

The power spectrum models are shown as bellow.

Φ (κ, z) = 0.033 C_{n}^{2} (z) [1 + a_{1} (\frac{κ}{κ_{x}}) + a_{2} {(\frac{κ}{κ_{x}})}^{7 / 6}] \frac{\exp (- κ^{2} / κ_{x}^{2})}{{(κ^{2} + κ_{0}^{2})}^{11 / 6}}

where

κ_{0}^{} =2 π / L_{0}

, L₀ is the outer scale and l₀ is the inner scale. a₁ and a₂ are shown in Table 1 [33,34].

Table 1. Parameters of different atmospheric power spectrum

View Table

The non-Kolmogorov power spectrum is shown as bellow [35].

Φ (κ, z) = \frac{1}{4 π^{2}} Γ (α - 1) \cos (\frac{α π}{2}) {\tilde{C}}_{n}^{2} \frac{\exp (- κ^{2} / κ_{x}^{2})}{{(κ^{2} + κ_{0}^{2})}^{α / 2}}, 3 < α < 4

where

κ_{x}^{} = {[\frac{1}{6 π} Γ (α - 1) Γ (\frac{5 - α}{2}) \cos (\frac{α π}{2})]}^{1 / (α - 5)} / l_{0}

where the

{\tilde{C}}_{n}^{2}

is the generalized structure parameters with the units

m^{3 - α}

. When

α = 11 / 3

, the generalized power spectrum will reduced to the Karman spectrum.

One of the most widely used refractive index structure constant model is the Hufnagel-Valley (HV) model and it is described by [27].

C_{n}^{2} (h) = 8.148 \times 10^{- 56} v^{2} h^{10} e^{- h / 1000} + 2.7 \times 10^{- 16} e^{- h / 1500} + 1.7 * 10^{- 14} e^{- h / 100}

where h is the height in meters (m), v is the root mean square wind speed in meters per second (m/s).

3. Result and analysis

In this section, the average intensities calculated by Rytov theory through the non-Kolmogorov turbulence are derived, and compered with the intensities calculated by extended Huygens–Fresnel principle. The Gaussian beam intensities from making the Bessel function of the Bessel Gaussian beams equal to 1, are investigated to confirm the presumption. And the result is applied to study the intensity of the Bessel Gaussian beams in slant path and investigate the intensity varying with the inner scale. I_HF represents the average intensity on the receiver plane calculated by extended Huygens–Fresnel principle. I_Ry represents the intensity calculated by Rytov theory with the second order statistical moment of the Bessel Gaussian beam. I_GB represents the intensity of the Bessel Gaussian beams calculated by Rytov theory with the second order statistical moment of the Gaussian beam. I_HGB represents the average intensity of the Gaussian beams calculated by extended Huygens–Fresnel principle without the quadratic approximation shown in Eq. (22). $λ = 1.55 μm$ and $C_{n}^{2} = 1.727 \times 10^{- 14}$ .

Figures 1(a)-1(d) show the average intensity of the Gaussian beams on the axis calculated by different methods varying with inner scale. Without the quadratic approximation, the average intensity of Gaussian beams calculated by the Rytov theory coincides very well with that calculated by the extended Huygens Fresnel principle. Calculated by the extended Huygens–Fresnel principle with the quadratic approximation, the average intensity of the Gaussian beams, coming from the Bessel function of Bessel Gaussian beams equal to 1, has a large bias with other results, especially through the small inner sale turbulence. The average intensity of the Gaussian beams is monotonically incremental with the increase the inner scale.

Fig. 1 Average intensity on the axis calculated by different methods varying with inner scale in the Karman turbulence. (a) w = 0.02m, L = 500m, (b) w = 0.01m, L = 500m, (c) w = 0.02m, L = 1000m, (d) w = 0.02m, L = 1000m.

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In order to illustrate the difference between the average intensity calculated by different methods, the relative error of the intensities is shown in Figs. 2(a)-2(d). There is a big difference between the results calculated by two methods through the small inner scale turbulence. The relative error will enlarge with the propagating distance increasing in most case. Figure 2(c) shows when the difference caused by the inner scale is small as Fig. 2(a) shows, there will be good agreement of the intensity variation with the outer scale between the two methods, and the outer scale has little impact on the result. There are the some properties as described above shown in Fig. 2(d). When the $α$ is very small or large, the average intensity difference is small.

Fig. 2 Relative error between the two methods. (a) variation with inner scale L = 500m, (b) variation with inner scale L = 1000m, (c) variation with outer scale L = 500m, (d) variation in non-Kolmogorov turbulence.

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Figures 3(a) and 3(b) illustrate that a good agreement between the average intensity of zero order Bessel Gaussian beams of different methods. When w = 0.01 and β = 100, the beams will be more like Gaussian beams. And the intensity on the axis varying with radius of the beams is always maximum. When w = 0.02 and β = 200, the intensity on the axis will fail down lower than the outer value after beams propagating through a certain distance. This phenomenon is related to the beam properties. From the Eq. (13), with the distance increasing, the Bessel function will be more like the modified Bessel function, so the intensity will increase around the axis. When the r is small, two approximations are applied to study the relationship between the β and w, $J_{0} [β r / (1 + 2 i α L)] \approx 1 - 0.25 β^{2} r^{2} / {(1 + 2 i α L)}^{2}$ and $\exp (x) \approx 1 + x$ . Making the first order derivation of the intensity in free space larger then 0, the inequality $β^{2} (4 α^{2} L^{2} - 1) / (4 α^{2} L^{2} + 1) > 4 / w^{2}$ could be obtained. Therefore, to observe this phenomenon, the inequality $L > 1 / (2 α)$ must be hold and the condition $β > 2 / w$ is sufficient but not necessary.

Fig. 3 Average intensity of 0 order Bessel Gaussian beams calculated by different method at different propagating distance in the Karman turbulence. (a)L = 500m, (b) L = 1000m.

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The intensity calculated by Rytov theory with E₂ of the Bessel Gaussian beam coincides better with intensity of the extended Huygens–Fresnel principle than the intensity with E₂ of the Gaussian beam. Figure 3(b) shows that the average intensity of the Bessel Gaussian beams with the Gaussian beam statistical moments, diverges fast to infinite, thus the applicable conditions for the Gaussian beam statistical moments is only the narrow Gaussian source width or the small propagating distance which is much stricter than that of the Bessel Gaussian beams.

Figures 4(a) and 4(b) show similar properties of the first order Bessel Gaussian beams as those of zero order Bessel Gaussian beams in Fig. 3. The average intensity on the axis of high order Bessel Gaussian beams calculated by the Rytov theory is an infinite value. This is due to the field on the axis which equals to 0, does not meet the application conditions of the Rytov theory. Fortunately, the average intensity some distance away from the axis, coincides well with the intensity calculated by the extended Huygens–Fresnel principle, and the Rytov theory can be applied.

Fig. 4 Average intensity of the first order Bessel Gaussian beams calculated by different method at different propagating distance in the Karman turbulence. (a)L = 500m, (b) L = 1000m.

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Figures 5(a) and 5(b) illustrate the average intensity of Bessel Gaussian beams calculated by different methods coincide well with each other, no matter what atmospheric power spectrum turbulence the beams propagate through. The beams propagating through the Karman spectrum atmosphere diverges slowly and the peak intensity is highest. The maximum intensity in the marine spectrum atmosphere is smallest. And the maximum intensity propagating through the modified spectrum atmosphere is in the middle. Different orders Bessel Gaussian beams have similar properties. In this paper, assume that the intensity around the axis is incremental which could be proved with the same method below the Fig. 3, and the points which do not satisfy the assumption are removed.

Fig. 5 Average intensity of Bessel Gaussian beams in the different atmospheric power spectrum turbulence. (a) 0 order Bessel Gaussian beams 2000m away, (b) different order Bessel Gaussian beams 1000m away.

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Figure 6 illustrates the high order Bessel Gaussian beams propagating through the turbulence in slant path. For the refractive index structure constant decreasing when the height increase, the maximum intensity of the beams in downlink is larger than that in uplink.

Fig. 6 Average intensity of high order Bessel Gaussian beams in slant path through the Karman turbulence.

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Figures 7(a)-7(d) show that, when the source widths of the zero order Bessel Gaussian beams are different, their intensity variations on the axis with inner scale display different behaviors. In Figs. 7(a) and 7(b), the intensity of the beams with a small Gaussian source width is monotonously increasing. When the beams width increase to a certain value such as w = 0.0105, the intensity no longer varies with the inner scale when the inner scale increase to a certain value. With the Gaussian source width increasing, the intensity has a maximum value and the abscissa of the maximum intensity will move towards the axis. After this, the intensity decreases monotonously. For the beam with a larger width, the intensity changing with the inner scale appears the contrary property. When the width increase to w = 0.0196, the intensity also does not vary with the inner scale. With a larger source width, the intensity of the beam has a minimum value. With the beams width increasing continuously, the abscissa of the minimum intensity will move to the axis and then the intensity will become again monotonously increasing. Similar properties also can be obtained with the parameters in Figs. 7(c) and 7(d).

Fig. 7 Average intensity on the axis of 0 order Bessel Gaussian beams in the Karman turbulence. (a) large source width when L = 1000m. (b) small source width when L = 1000m. (c) large source width when L = 500m. (d) small source width when L = 500m.

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This phenomenon is due to the property of the Bessel Gaussian beams. The Eq. (22) could explain the phenomenon in theory. For some special width beams, the Eq. (22) holds when the inner scale is not small and turbulence perturbation on the axis equals to zero. When the b = 200 and L = 1000, the Eq. (22) have 6 roots. Among them, 2 roots are imaginary, 2 roots are negative and only 2 roots are available which respectively equal to 0.0194 and 0.0105. And those theoretical results are in a very good agreement with the results in Fig. 7(a). The same result could also be obtained with the parameters in Fig. 7(b) and the 2 roots respectively equal to 0.0140 and 0.0069. And they coincide very well with each other.

This phenomenon is not existed for all beams. And the condition is that, given any two of the three parameters of the Eq. (22), there will be at least one positive real root of this equation. Moreover, the Eq. (22) could also make the first order derivation of the central intensity in free space equal to 0, so the beam has a flat top or a flat valley around the axis shown in Fig. 8.

Fig. 8 Normalized average intensity of no turbulence affected Bessel Gaussian beams

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4. Conclusion

The average intensity of the Bessel Gaussian beams propagating through non-Kolmogorov turbulence is derived by the Rytov theory without the quadratic approximation. In order to obtain the average intensity, the second order statistical moments are also derived. And this method has an advantage in accuracy over the extended Huygens–Fresnel principle.

In this paper, the intensities calculated by the Rytov theory and the extended Huygens–Fresnel principle are compared. The comparison indicates that the Rytov theory is more accurate than the extended Huygens–Fresnel principle for the investigation of the Bessel Gaussian beams propagating through non-Kolmogorov turbulence, especially, when the beams have the large source width or the inner scale of the turbulence is small. Moreover, the comparison between the intensities of the Gaussian beams calculated by the two methods also implies that the quadratic approximation will create an appreciable deviation when the inner scale of the turbulence is small. Besides those cases, the intensity calculated by Rytov theory is in good agreement with extended Huygens–Fresnel principle.

This result is applied to investigate the intensity on the axis of the zero order Bessel Gaussian beam varying with inner scale. And an interesting phenomenon is found, when the beam source width is different, their intensity variations with the inner scale display different behaviors, which is different from the Gaussian beam. There are some Bessel Gaussian beams with special source widths which could satisfy the Eq. (22), and their intensities will become constant after the inner scale increasing to a certain value for their turbulence perturbation equaling to zero there. Moreover, the beams become to the flat top beams at this circumstance. The detailed analysis is illustrated in Figs. 7 and 8.

Funding

National Natural Science Foundation of China (61601355, 61701382, 61571355, 61401342, 61475123).

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atmospheric power spectrum	Karman spectrum	Modified atmospheric spectrum	Marine atmospheric spectrum
a₁	0	1.802	−0.061
a₂	0	−0.254	2.836
$κ_{x}$	$κ_{m} = 5.92 / l_{0}$	$κ_{l} = 3.30 / l_{0}$	$κ_{H} = 3.41 / l_{0}$

Propagation of Bessel Gaussian beams through non-Kolmogorov turbulence based on Rytov theory

Abstract

1. Introduction

2. Theoretical formulation

2.1 Average intensity based on Ratov theory

2.2 Second order statistical moments

2.3 Average intensity based on extended Huygens–Fresnel principle

2.4 Power spectrum models and refractive index structure constant

3. Result and analysis

4. Conclusion

Funding

References

Cited By

Figures (8)

Tables (1)

Equations (31)

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