Propagation of multiple Bessel Gaussian beams through weak turbulence

Wanjun Wang; Zhensen Wu; Qingchao Shang; Bai Lu

doi:10.1364/OE.27.012780

1. Introduction

Bessel beams are widely investigated for their non-diffraction and self-healing properties [1–3]. To generate Bessel beams with finite energy, the Gaussian exponential is chosen as a windowing function and the Bessel Gaussian beam is proposed [4]. The property of the Bessel Gaussian beam propagating through the turbulence has been investigated by many scholars [5–8]. The Bessel beams have the advantage over the Gaussian beams in overcome the power loss after the long distance propagation [9] and Bessel beams are less influence by the turbulence [10,11]. And more information could be carried by Bessel beams with the usage of the multiple orbital angular momentum (OAM) states [12]. Recently, one property of the Bessel Gaussian beams is proposed and studied [13], at a certain propagation distance, it is the specific width Bessel Gaussian beams whose central average phase perturbation could be equal to 0 that will not be influenced by the turbulence there.

In recent years, the modified Bessel beams [14,15] and the Lommel beams [16–18] are studied for their similar properties as the Bessel beam. The modified Bessel Gaussian beams is proposed and expressed as the product of a Gaussian function and the summation of modified Bessel functions [15,19]. There are side lobes of the average intensity distribution of the modified Bessel Gaussian beams. With the propagating distance increasing, the modified Bessel Gaussian beam will transform into the Bessel Gaussian beam [15]. The asymmetric Bessel beam including a summation of nth-order Bessel beams is proposed for its special propagation characteristic [20]. Its transverse intensity distribution presents crescent shape and the beams can have both integer and fractional orbital angular momentum. After that, the asymmetric Bessel-Gauss beams with finite energy are proposed and they have similar properties to the asymmetric Bessel beams [21]. Based on the same theory as the asymmetric Bessel beam, the Lommel beam is proposed as a non-diffraction beam and it shows an advantage at the controllability of its field distribution [16]. To generate Lommel beam with finite energy, the Lommel Gaussian beam is proposed with the same reason as the Bessel Gaussian beam [22]. To study this type of beams and the interaction of the different order beams for the beam multiplexing, the multiple Bessel Gaussian beams (mBGBs) including more than two Bessel Gaussian beams are investigated in this paper. The Lommel Gaussian beam could be regarded as one special type of the mBGBs. The modified Bessel Gaussian beams are also similar to the mBGBs, and only by a few derivations, the average intensity of Bessel Gaussian beams could transform to that of the modified Bessel beams.

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 [23–25], therefore the result is less accurate than the that calculated by the Rytov theory especially when the inner scale of the turbulence is small. However, there is still a disadvantage for the Rytov theory that the central average intensity of the high order Bessel Gaussian beams is an infinite value for the field intensity on the axis does not meet the application conditions of the Rytov theory [13,26], and these values need to be removed. In this case, based on the Rytov theory, it is the approximation that the phase perturbation of the high order Bessel Gaussian beams is approximated by that of the 0 order Bessel Gaussian beams, is proposed. Then the central average intensity calculated by this method could meet the theoretical prediction that the paraxial value is finite and incremental along the radius.

In this paper, under the weak turbulence circumstance which is the applicable condition of the Rytov theory, based on the Rytov theory and the extended Huygens-Fresnel principle, the average intensity of the mBGBs propagating through the turbulence is derived and an approximation of the statistical moment of the phase perturbation is proposed. The comparison among the average intensity of the approximated method, the Rytov theory and the extended Huygens-Fresnel principle is analyzed. There are side lobes of the average intensity distribution of the mBGBs. The factors which have influence on the average intensity distribution, such as the beam order, the beam amplitude, the initial phase and the beam number are investigated. Specially, the average intensity distribution of the mBGBs with two sub beams appears the rotational symmetry with the controllable rotational direction. Moreover, the angular frequency is the order difference of the sub beams and it could be used detect the sub beam order. This paper provides a theoretical support for the beam multiplexing and the mBGBs propagation through the turbulence.

2. Theory and formulation

2.1 Average intensity based on the Rytov theory

The field distribution of the multiple Bessel Gaussian beams (mBGBs) on the source plane consist of the summation of the Bessel function and a Gaussian exponential function with radial coordinates s and $φ$ can is proposed as Eq. (1) [4,15].

U (r, φ_{r}) = \sum_{n = N_{1}}^{N_{2}} a_{n} \exp (- k α r^{2}) \exp [- i n (φ_{r} - φ_{0 n})] J_{n} (β r)

where n denotes the order of the Bessel function,

φ_{0 n}

is the initial phase.

β_{n}

is the width parameter, k is the wave number,

α = 1 / k w_{0}^{2}

and

w_{0}

is the Gaussian source width. a_n is the sub beam amplitude, and

A_{n} = a_{n} \exp (i n φ_{0 n})

for convenient.

According to the extended Huygens-Fresnel principle, the field distribution on a receiver plane of Bessel Gaussian beams propagating through the free space is derived as below [27].

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

The intensity in free space is defined as the product of the field intensity in the free space and its conjugate value expressed as Eq. (3).

\begin{matrix} I (r, L) = U (r, L) U^{*} (r, L) \\ = \frac{1}{1 + 4 α^{2} L^{2}} \exp [- \frac{2 α k^{2} r^{2} + 2 α β_{}^{2} L^{2}}{k (1 + 4 α^{2} L^{2})}] \\ \times \sum_{n = N_{1}}^{N_{2}} \sum_{n' = N_{1}}^{N_{2}} A_{n} A_{n'}^{*} J_{n} (\frac{β r}{1 + 2 i α L}) J_{n'}^{*} (\frac{β r}{1 + 2 i α L}) \exp [- i (n - n') φ_{r}] \end{matrix}

Based on the Rytov theory within the weak turbulence condition, the average received intensity on a receiver plane of beams propagating through turbulence can be expressed as Eq. (4) [26].

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

where

I (r, L)

is the intensity of the mBGBs propagating through the free space shown as Eq. (3). E₁, E₂ are the second order statistical moments. And there is the relationship between the second order statistical moments and the phase perturbations as Eqs. (5) and (6) [26].

E_{1} (r, r) = 〈 Φ_{2} (r, L) 〉

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

Based on the theory, the first and second order perturbations $Φ_{1} (r, L), Φ_{2} (r, L)$ expressed as Eqs. (36) and (40) in [13] can be derived as Eqs. (7) and (8) with the usage of Eqs. (6).633(2), 8.406(3), 3.937(1) and 3.937 (2) of [28]. And the derivation process is the same as that of the single beam [13].

\begin{matrix} Φ_{1} (r, L) = i k \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 \frac{\sum_{n = N_{1}}^{N_{2}} A_{n} J_{n} [- \frac{β (L - z)}{k (1 + 2 i α L)} | K - \frac{k r}{L - z} |] \exp (- i n φ_{K r})}{\sum_{n = N_{1}}^{N_{2}} A_{n} J_{n} (\frac{β r}{1 + 2 i α L}) \exp (- i n φ_{r})} \end{matrix}

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

where

d v (K, z)

is the random amplitude of the refractive index fluctuations,

γ = (1 + 2 i α z) / (1 + 2 i α L)

and

γ' = (1 + 2 i α z') / (1 + 2 i α z)

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

κ = | K |

.

| |

is the vector magnitude.

φ_{κ}

is the angle of the vector K,

φ_{r}

is the angle of the vector r.

φ_{K r}

is the angle of the vector

K - k {(L - z)}^{- 1} r

and it can be expressed as Eq. (9).

φ_{K K' r}

is the angle of the vector

K + γ^{'} K^{'} - k {(L - z)}^{- 1} r

.

\begin{matrix} \exp (- i φ_{κ r}) = \frac{[κ (L - z) \exp (- i φ_{κ}) - r k \exp (- i φ_{r})]}{| κ (L - z) \exp (- i φ_{κ}) - r k \exp (- i φ_{r}) |} \\ = \frac{{κ (L - z) \exp [- i (φ_{κ} - φ_{r})] - r k}}{{[κ^{2} {(L - z)}^{2} - 2 κ r k (L - z) \cos (φ_{κ} - φ_{r}) + r^{2} k^{2}]}^{0.5}} \cdot \exp (- i φ_{r}) \end{matrix}

The derivation of the statistical moment is also the same as the single Bessel Gaussian beam in [13]. After the variable change $η = (z + z') / 2$ and $μ = z - z'$ , the second order statistical moment E₂ is derived as Eq. (10) by substituting Eq. (7) to Eq. (6) with the equation $K - K' = 0$ .

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

where

Φ_{n} (κ)

is the spatial power spectral density of the refractive index fluctuation. In this paper, the Karman power spectrum model is used [29].

E₁ is derived as Eq. (11) by submitting Eq. (8) to Eq. (5) with the equation $K + γ^{'} K^{'} = 0$ .

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 expression $\exp [E_{1} (0, 0)]$ is the turbulence influence on the mean field [26] and it is independent of the $φ_{r}$ . Therefore, the average turbulence fluctuation can’t cause a destructive influence on the orbital angular momentum of the mBGBs.

The average intensity of the mBGBs could also be simplified to that of the single beam.

2.2 Approximation of the second order statistical moment

Based on the Rytov theory, the central average intensity of the high order Bessel Gaussian beams is an infinite value for the field intensity on the axis does not meet the application conditions of the Rytov theory. Therefore, the central value needs to modify. Considering the similarity of the Bessel function, an assumption is proposed that the statistical moments of the high order Bessel Gaussian beams are approximated to those of the 0 order Bessel Gaussian beam as Eq. (12). In addition, under the applicable condition of the Rytov theory, the approximation $| K (L - η) - k r | \approx | k r |$ could be satisfied in most case [13], and the value of Eq. (12) is close to 1. Then the second order statistical moments E₂ is simplified as Eq. (13).

\frac{J_{n} [\frac{β (L - η)}{k (1 + 2 i α L)} | K - \frac{k r}{L - η} |] \exp (- i n φ_{K r})}{J_{n} (\frac{β r}{1 + 2 i α L}) \exp (- i n φ_{r})} \approx \frac{J_{0} [\frac{β (L - η)}{k (1 + 2 i α L)} | K - \frac{k r}{L - η} |]}{J_{0} (\frac{β r}{1 + 2 i α L})}

\begin{matrix} E_{2} (r, r) = 2 π k^{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_{0} [\frac{β (L - η)}{k (1 + 2 i α L)} | K - \frac{k r}{L - η} |] J_{0}^{*} [\frac{β (L - η)}{k (1 + 2 i α L)} | K - \frac{k r}{L - η} |] {| J_{0} (\frac{β r}{1 + 2 i α L}) |}^{- 2} \end{matrix}

Equation (13) and Eq. (11) indicate that the statistical moment just varies with the beam radius and presents the cylindrical symmetry. Therefore, it could reduce much time to obtain the average intensity distribution of the mBGBs.

2.3 Average intensity based on the extended Huygens-Fresnel principle

According to extended Huygens-Fresnel principle, the average receiver intensity of mBGBs is derived as Eq. (13) based on Eq. (5) of [23], which includes an approximation that the wave structure function is approximated to a quadratic function [23,24].

\begin{matrix} 〈 I (r, φ_{r}) 〉 = \frac{b^{2}}{π (k α - i b + 1 / ρ_{0}^{2})} \exp [- \frac{β^{2} + 4 b^{2} r^{2}}{4 (k α - i b + 1 / ρ_{0}^{2})}] \\ \times \sum_{n_{1} = N_{1}}^{N_{2}} \sum_{n_{2} = N_{1}}^{N_{2}} \int_{0}^{\infty} \int_{0}^{2 π} d s_{2} d φ_{s_{2}} A_{n_{1}} A_{n_{2}}^{*} s_{2} J_{n_{2}} (β s_{2}) \exp (i n_{2} φ_{s_{2}}) \\ \times \frac{{[- i b r \exp (i φ_{r}) + s_{2} \exp (- i φ_{s_{2}}) / ρ_{0}^{2}]}^{n_{1}}}{{[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_{1} / 2}} \\ \times J_{n_{1}} {\frac{β {[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 = k/(2L), ρ₀ is the spherical-wave spatial coherence radius and it is expressed as the Eq. (15).

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

3. Result and analysis

In this paper, the Karman power spectrum model is used [29]. RyEJ₀ represents the result calculated by the Rytov theory with the statistical moment approximation as Eq. (12). Rytov represents the result calculated by the Rytov theory without the approximation. HF represents the result calculated by the extended Huygens-Fresnel principle. J_n represents n_th order Bessel Gaussian beam. A_n = 1 means all the elements of the matrix A_n equal to 1, and φ₀ = 0 means all the elements of the matrix φ₀ equal to 0. w₀ = 0.02m, $λ = 1.55 \times 10^{- 6} m$ , and $C_{n}^{2} = 1.727 \times 10^{- 14} m^{- 2 / 3}$ . Expect Figs. 1 and 2, the outer scale of the turbulence is L₀ = 2m and the inner scale is l₀ = 0.02m.

Fig. 1 Average intensity of the Bessel Gaussian beam. (a) l₀ = 0.02m, (b) l₀ = 0.002m.

Download Full Size | PDF

Fig. 2 RMSPE of the average intensity of the Bessel Gaussian beam calculated by different methods. (a) variation with L, (b) variation with l₀.

Download Full Size | PDF

Figure 1(a) shows when the inner scale of the turbulence is small, there is a large bias between the maximum average intensity calculated by the extended Huygens-Fresnel principle and those of the other two Rytov methods. In the other case, there is good agreement between the three methods shown in Fig. 1(b). The Rytov theory with the statistical moment approximation also maintains the advantage that there is a better accuracy of the result when the inner scale of the turbulence is small. The theoretical average intensity around the axis is incremental and close to 0, but the central average intensity of the high order Bessel Gaussian beams calculated by the Rytov theory is an infinite value, for the field intensity on the axis equals to 0 which does not meet the application conditions of the Rytov theory. Therefore, the points which do not satisfy the theoretical prediction are removed. Fortunately, the approximated statistical moment method could overcome this disadvantage and provide a result which meets the theoretical prediction without removing the bad points.

To compare the accuracy of the different methods, the root mean square percentage error (RMSPE, ${(\bar{σ_{I}^{2}})}^{1/2}$ ) is used and expressed as Eq. (16). The RMSPE could measure the bias of the bright lobe value and the lobe location calculated by the different methods.

{(\bar{σ_{I}^{2}})}^{1/2} = {(\frac{\sum s_{m} {(I_{m}^{'} / I_{m} -1)}^{2}}{\sum s_{m}})}^{1/2}

where

I_{m}

is the average intensity calculated by the Rtyov theory, and

I_{m}^{'}

is the average intensity of the extended Huygens-Fresnel principle or the approximated statistical moment method.

s_{m}

is the area of the m-th element corresponding to the

I_{m}

. In order to study the accuracy of the bright zone, the interval at where the average intensity calculated by the Rtyov theory is larger than the half of the maximum value, is chosen.

Figure 2 shows the root mean square percentage error (RMSPE) of the Bessel Gaussian beam average intensity calculated by the different methods. Within the weak turbulence circumstance which is also the limitation of the Rytov theory, the application condition of the statistical moment approximation is small beam order. The bias between the results of the two Rytov methods will enlarge with the beam order increasing. This result also confirms that the approximated method maintains an advantage of the Rytov theory that its result is more accurate when the beam propagates through the small inner scale turbulence.

Figure 3 shows that there is good agreement among the average intensity distribution calculated by the three methods. The average intensity distribution of the mBGBs with two sub beams presents the rotational symmetry which does not appear in the distribution of the mBGBs with three sub beams unless Fig. 5. Limited by the application conditions of the Rytov theory as the description below Fig. 1, the central average intensity of the mBGBs calculated by the Rytov theory are approximated to 0 for visualization in the figures of the average intensity distribution.

Fig. 3 Average intensity of the mBGBs (A_n = 1, L = 1000m, β = 200m⁻¹) (a) RyEJ₀, n = [2,5], (b) Rytov, n = [2,5], (c) HF, n = [2,5], (d) RyEJ₀, n = [3,6], (e) Rytov, n = [3,6], (f) HF, n = [3,6], (g) RyEJ₀, n = [0, 2, 5], (h) Rytov, n = [0, 2, 5], (i) HF, n = [0, 2, 5].

Download Full Size | PDF

Figure 4 illustrates that there is good agreement of the average intensity calculated by the three methods. Moreover, the abscissa of the maximum value and minimum value of the average intensity is also the same which indicates that there is the same symmetry of the average intensity distribution calculated by the three methods. The average intensity selected in Fig. 4 distributes in the bright lobe as Fig. 3 shows.

Fig. 4 Average intensity of the mBGBs. (a) n = [2,5], (b) n = [3,6], (c) n = [0, 2, 5].

Download Full Size | PDF

Table 1 shows the RMSPE of the mBGBs with the parameters L = 1000m and β = 200m⁻¹. The RMSPE is less than 5% which indicates that the three methods coincide very well with one another. For the complex expression of the extended Huygens-Fresnel principle, it is hard to summarize the turbulence influence on the average intensity distribution. The Rytov theory can’t compute the central average intensity and it costs too much time to calculate the distribution. Therefore, the approximated moment Rytov theory is chosen to calculate the later result and the RMSPE provides a theoretical support for this method.

Table 1. RMSPE of the mBGBs average intensity calculated by different methods

View Table

Figures 5(a)-5(d) also confirm that the mBGBs with two sub beams could present the rotational symmetry and the angular frequency is the order difference between the sub beams. Particularly, for the mBGBs with 3 or more sub beams, its average intensity distribution will present the rotational symmetry only when the order difference of any two sub beams is the integer multiples of the minimum order difference. And the angular frequency is the minimum order difference as Figs. 5(e) and 5(f) show. Given the lowest order sub beam of the mBGBs, the higher order component could be detected by the angular frequency of the average intensity. This phenomenon in Fig. 5 could be explained for that the rotational symmetry of the distribution depends on the intensity in the free space and the turbulence has no influence on the angular frequency of the mBGBs. From the expression of the free space intensity as Eq. (3), when the order difference of any two sub beams is the integer multiples of the minimum order difference, there is the rotational symmetry and the angular frequency equals to the minimum order difference. For the perturbation caused by the turbulence shown as Eq. (13) which is cylindrical symmetry has also no influence on the distribution symmetry. The same conclusion could also be obtained from Eq. (10) and it shows that the turbulence moment distribution also presents the same rotational symmetry as the intensity distribution in the free space. For the integrand function of the integral in Eq. (10) is a periodic function of the $φ_{κ}$ , and a $φ_{r}$ translation of the integral interval does not change the result of this integral in Eq. (9). And the period of the $\exp (- i φ_{K r})$ is the same as that of the $\exp (- i φ_{r})$ .

Fig. 5 Average intensity of mBGBs with different order sub beams (RyEJ₀, A_n = 1, L = 1000m, β = 200m⁻¹). (a) n = [2,5], (b) n = [0, 2], (c) n = [0, 5], (d) n = [3,6], (e) n = [0, 3, 6], (f) n = [1,3,5].

Download Full Size | PDF

Figure 6 shows that the average intensity distribution of the mBGBs will change with the amplitude of the sub beams, but the angular frequency still maintains. From Eq. (3), the period of the free space intensity does not vary with the beam amplitude except that one component of the mBGBs is zero. And the turbulence doesn’t influence the angular distribution. Therefore, when the amplitude of the one of sub beams is much larger than the other sub beams, the average intensity distribution will appear the cylindrical symmetry for the average intensity distribution of the Bessel Gaussian beam presents the cylindrical symmetry. The low order sub beam makes larger contribution to the average intensity distribution of the mBGBs. The symmetry of the average intensity distribution could also be used to measure the quality of the single Bessel Gaussian beam. And the average intensity distribution is more similar to the cylindrical symmetry distribution, the beams is more close to single Bessel Gaussian beam with less other order component.

Fig. 6 Average intensity of the mBGBs with two different amplitude sub beams. (RyEJ₀, n = [2,5], L = 1000m, β = 200m⁻¹). (a) An = [0.01, 1], (b) An = [0.1, 1], (c) An = [1, 0.5], (d) An = [1, 0.1].

Download Full Size | PDF

Figure 7 shows the distribution variation with the amplitude is complex. For the mBGBs with two more sub beams, the bright lobe number of the average intensity not only depends on the sub beam order, but also is related to the sub beam amplitude. The amplitude variation of the sub beams is equivalent to the sub beam recombination of the initial mBGBs. The bright lobe number could be the order difference of any two sub beams. Sometimes the value of some bright lobes may be too small to distinguish as Fig. 7(c). This phenomenon could also be confirmed by the zero-point number of the first derivative of the free space intensity and the zero-point number is usually twice of the bright lobe number.

Fig. 7 Average intensity of the mBGBs with three different amplitude sub beams. (RyEJ₀, n = [0, 2, 5], L = 1000m, β = 200m⁻¹). (a) A_n = [0.5, 1, 1], (b) A_n = [0.2, 1, 1], (c) A_n = [1, 0.6, 1], (d) A_n = [1, 0.2, 1].

Download Full Size | PDF

Figure 8 shows that for the mBGBs with two sub beams, the initial phase increase could make the average intensity distribution of the beam rotate, and the average intensity remains the same. The initial phase increase of the low order sub beam will cause the distribution rotate left, but the initial phase increase of the high order sub beam will lead the distribution rotate right. This distribution variation is the same as that of the free space intensity and the sign of the initial phase of the different order sub beams of the mBGBs in the free space is opposite. For the mBGBs with three or more sub beams, this phenomenon does not appear. Therefore, there is an advantage that when the receiver is less than the size of the beam, the average intensity of the mBGBs with two sub beams could still be obtained by rotating the source laser.

Fig. 8 Average intensity of the mBGBs with two different initial phase sub beams. (RyEJ₀, n = [2,5], a_n = 1, L = 1000m, β = 200m⁻¹) (a) φ_0n = [0, 0], (b) φ_0n = [π/4, 0], (c) φ_0n = [π/2, 0], (d) φ_0n = [3π/4, 0], (e) φ_0n = [0, π/10], (f) φ_0n = [0, π/5], (g) φ_0n = [0, 3π/10].

Download Full Size | PDF

Figure 9. shows when the beam propagates through the turbulence, its side lobes at the radial direction will combine to a larger bright lobe after the beams propagate to a certain distance, and the angular frequency of the average intensity distribution remains the same. With the distance increasing, the Bessel function in Eq. (3) will transform to the modified Bessel function. The modified Bessel function is monotonically increasing along the radius and the Gaussian function is monotonically decreasing. Therefore, there is only one peak at a long radius.

Fig. 9 Average intensity of the mBGBs. (RyEJ₀, n = [2,5], A_n = 1, β = 500m⁻¹) (a) L = 50m, (b) L = 100m, (c) L = 200m, (d) L = 500m.

Download Full Size | PDF

4. Conclusion

In this paper, the average intensity of the mBGBs propagating through the turbulence is studied based on Rytov theory and the extended Huygens-Fresnel principle. The Rytov theory is better than the extended Huygens-Fresnel principle at the small inner scale circumstance and easy to summarize the turbulence influence. But it could not calculate the central average intensity for its application conditions. To improve this disadvantage of the Rytov theory, an approximation that the statistical moment of the high order Bessel Gaussian beams is approximated to that of the 0 order Bessel Gaussian beams, is proposed. It takes much less time to calculate the average intensity distribution than the Rytov theory and the result could meet the theoretical prediction with the error less than 5%. The applicable condition of the approximated statistical moment method of the Rytov theory is that the turbulence fluctuation is weak and the sub beam order of the mBGBs is small.

With a mathematical way, it is proved that the weak turbulence just leads the mBGBs diverge and it has no influence on the angular distribution of both the mean field and the average intensity. For the mBGBs with two sub beams, its angular symmetry maintains whatever the sub beam amplitude is, and the initial phase change of different sub beam could make the average intensity distribution rotate in opposite direction. For the mBGBs with two more sub beams, there are not these phenomena. The rotational symmetry of the average intensity distribution of the mBGBs will appear only when the order difference between each of two sub beams is the integer multiples of the minimum order difference and the angular frequency is the minimum order difference. This may be useful to the sub beam detect and the receiver size selection. With the propagation distance increasing, the side lobes at the radial direction will combine to a large lobe with the same symmetry as its source. This paper provides the theoretical basis for the mBGBs propagation and the application for the Bessel Gaussian beam multiplexing.

Funding

National Natural Science Foundation of China (No. 61601355, 61875156, 61701382 and 61571355).

References

1. J. Durnin, J. Miceli Jr., and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987). [CrossRef] [PubMed]

2. J. Durnin, J. H. Eberly, and J. J. Miceli, “Comparison of Bessel and Gaussian beams,” Opt. Lett. 13(2), 79–80 (1988). [CrossRef] [PubMed]

3. J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. 197(4-6), 239–245 (2001). [CrossRef]

4. F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64(6), 491–495 (1987). [CrossRef]

5. C. Bao-Suan and P. Ji-Xiong, “Propagation of Gauss-Bessel beams in turbulent atmosphere,” Chin. Phys. B 18(3), 1033–1039 (2009). [CrossRef]

6. Y. Cai and X. Lü, “Propagation of Bessel and Bessel-Gaussian beams through an unapertured or apertured misaligned paraxial optical systems,” Opt. Commun. 274(1), 1–7 (2007). [CrossRef]

7. M. Cheng, L. Guo, J. Li, and Q. Huang, “Propagation properties of an optical vortex carried by a Bessel-Gaussian beam in anisotropic turbulence,” J. Opt. Soc. Am. A 33(8), 1442–1450 (2016). [CrossRef] [PubMed]

8. X. Ji and B. Lü, “Focal shift and focal switch of Bessel–Gaussian beams passing through a lens system with or without aperture,” Opt. Laser Technol. 39(3), 562–568 (2007). [CrossRef]

9. P. Birch, I. Ituen, R. Young, and C. Chatwin, “Long-distance Bessel beam propagation through Kolmogorov turbulence,” J. Opt. Soc. Am. A 32(11), 2066–2073 (2015). [CrossRef] [PubMed]

10. R. E. Meyers, K. S. Deacon, A. D. Tunick, and Y. Shih, “Virtual ghost imaging through turbulence and obscurants using Bessel beam illumination,” Appl. Phys. Lett. 100(6), 061126 (2012). [CrossRef]

11. V. D. Salakhutdinov, E. R. Eliel, and W. Löffler, “Full-field quantum correlations of spatially entangled photons,” Phys. Rev. Lett. 108(17), 173604 (2012). [CrossRef] [PubMed]

12. A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015). [CrossRef]

13. W. Wanjun, W. Zhensen, S. Qingchao, and B. Lu, “Propagation of Bessel Gaussian beams through non-Kolmogorov turbulence based on Rytov theory,” Opt. Express 26(17), 21712–21724 (2018). [CrossRef] [PubMed]

14. S. Ruschin, “Modified Bessel nondiffracting beams,” J. Opt. Soc. Am. A 11(12), 3224–3228 (1994). [CrossRef]

15. H. T. Eyyuboğlu and F. Hardalaç, “Propagation of modified Bessel-Gaussian beams in turbulence,” Opt. Laser Technol. 40(2), 343–351 (2008). [CrossRef]

16. A. A. Kovalev and V. V. Kotlyar, “Lommel modes,” Opt. Commun. 338, 117–122 (2015). [CrossRef]

17. L. Yu, B. Hu, and Y. Zhang, “Intensity of vortex modes carried by Lommel beam in weak-to-strong non-Kolmogorov turbulence,” Opt. Express 25(16), 19538–19547 (2017). [CrossRef] [PubMed]

18. L. Tang, H. Wang, X. Zhang, and S. Zhu, “Propagation properties of partially coherent Lommel beams in non-Kolmogorov turbulence,” Opt. Commun. 427, 79–84 (2018). [CrossRef]

19. L. Wang, X. Wang, and B. Lü, “Propagation properties of partially coherent modified Bessel–Gauss beams,” Optik (Stuttg.) 116(2), 65–70 (2005). [CrossRef]

20. V. V. Kotlyar, A. A. Kovalev, and V. A. Soifer, “Asymmetric Bessel modes,” Opt. Lett. 39(8), 2395–2398 (2014). [CrossRef] [PubMed]

21. V. V. Kotlyar, A. A. Kovalev, R. V. Skidanov, and V. A. Soifer, “Asymmetric Bessel-Gauss beams,” J. Opt. Soc. Am. A 31(9), 1977–1983 (2014). [CrossRef] [PubMed]

22. L. Ez-zariy, F. Boufalah, L. Dalil-Essakali, and A. Belafhal, “Effects of a turbulent atmosphere on an aperture Lommel-Gaussian beam,” Optik (Stuttg.) 127(23), 11534–11543 (2016). [CrossRef]

23. H. T. Eyyuboğlu, “Propagation of higher order Bessel-Gaussian beams in turbulence,” Appl. Phys. B 88(2), 259–265 (2007). [CrossRef]

24. T. Shirai, A. Dogariu, and E. Wolf, “Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 20(6), 1094–1102 (2003). [CrossRef] [PubMed]

25. Y. Zhu, M. Chen, Y. Zhang, and Y. Li, “Propagation of the OAM mode carried by partially coherent modified Bessel-Gaussian beams in an anisotropic non-Kolmogorov marine atmosphere,” J. Opt. Soc. Am. A 33(12), 2277–2283 (2016). [CrossRef] [PubMed]

26. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Medium, 2nd ed. (SPIE, Washington, 2005).

27. H. T. Eyyuboğlu, E. Sermutlu, Y. Baykal, Y. Cai, and O. Korotkova, “Intensity fluctuations in J-Bessel-Gaussian beams of all orders propagating in turbulent atmosphere,” Appl. Phys. B 93(2–3), 605–611 (2008). [CrossRef]

28. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 7th Ed. (Elsevier/Academic, 2007).

29. L. C. Andrews, “An analytical model for the refractive index power spectrum and its application to optical scintillations in the atmosphere,” J. Mod. Opt. 39(9), 1849–1853 (1992). [CrossRef]

RMSPE	J₂ + J₅	J₃ + J₆	J₀ + J₂ + J₅	0.5J₀ + J₂ + J₅	J₀ + 0.5J₂ + J₅
RyEJ₀/Rytov	3.52%	4.47%	2.90%	3.39%	3.40%
HF/Rytov	1.15%	1.15%	1.44%	1.35%	1.30%

Propagation of multiple Bessel Gaussian beams through weak turbulence

Abstract

1. Introduction

2. Theory and formulation

2.1 Average intensity based on the Rytov theory

2.2 Approximation of the second order statistical moment

2.3 Average intensity based on the extended Huygens-Fresnel principle

3. Result and analysis

4. Conclusion

Funding

References

Cited By

Figures (9)

Tables (1)

Equations (16)

Optics Express