Propagation of various dark hollow beams in a turbulent atmosphere

Yangjian Cai; Sailing He

doi:10.1364/OE.14.001353

Optics Express
Vol. 14,
Issue 4,
pp. 1353-1367
(2006)
•https://doi.org/10.1364/OE.14.001353

Propagation of various dark hollow beams in a turbulent atmosphere

Yangjian Cai and Sailing He

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About this Article
History
- Original Manuscript: December 22, 2005
- Revised Manuscript: January 22, 2006
- Manuscript Accepted: February 5, 2006
- Published: February 20, 2006

Abstract

Propagation of a dark hollow beam (DHB) of circular, elliptical or rectangular symmetry in a turbulent atmosphere is investigated. Analytical formulas for the average intensity of various DHBs propagating in a turbulent atmosphere are derived in a tensor form based on the extended Huygens-Fresnel integral. The intensity and spreading properties of the DHBs in a turbulent atmosphere are studied numerically. It is found that after a long propagation distance a dark hollow beam of circular or non-circular eventually becomes a circular Gaussian beam (without dark hollow) in a turbulent atmosphere, which is much different from its propagation properties in free space. The conversion from a DHB to a circular Gaussian beam becomes quicker and the beam spot in the far field spreads more rapidly for a larger structure constant, a shorter wavelength, a lower beam order and a smaller waist size of the initial beam.

1. Introduction

In recent years, dark-hollow beams (DHBs) have attracted more and more attentions because of their wide applications in modern optics and atomic optics.¹ DHBs have been widely studied in both experimental and theoretical aspects. Several theoretical models have been proposed to describe DHBs, such as the ${TEM}_{01}^{*}$ beam, some higher-order Bessel beams, and the hollow Gaussian beam, etc. [2–9]. A DHB can be expressed as a finite sum of Laguerre-Gaussian beams or Gaussian beams. Propagation properties of a DHB through free space or a paraxial optical system have been widely studied.

Investigations of the propagation properties of laser beams in a turbulent atmosphere become more and more important because of their wide applications in e.g. free-space optical communications.[10, 11]. Much work has been carried out concerning the spreading of a laser beam in a turbulent atmosphere. Feizulin et al studied the broadening of a spatially bounded laser beam. [12]. Young et al. studied the spreading of a higher-order mode laser beam. [13]. The propagation and spreading of partially coherent beams in a turbulent atmosphere have also been studied. [14–16]. Eyyuboglu and Baykal investigated the properties of cos-Gaussian, cosh-Gaussian, Hermite-sinusoidal-Gaussian, Hermite -cosine-Gaussian and Hermite–cosh-Gaussian laser beams in a turbulent atmosphere [17–21].

Dark hollow beam is a special case of higher-order annular beams whose log-amplitude and phase fluctuations in a turbulent medium have been studied recently. [22] In this paper, we study the propagation properties of a DHB of circular or non-circular symmetry in a turbulent atmosphere by using a tensor method, which is a convenient method for the propagation of coherent and partially coherent astigmatic laser beam [23–27]. Analytical formulas for the average intensity of a DHB propagating in a turbulent atmosphere are derived, and numerical examples are given.

2. Dark hollow beams of circular and non-circular symmetries

In this section, we first outline briefly the definition of DHBs of circular and non-circular symmetries. The electric field of a DHB of circular symmetry at z=0 can be expressed as the following finite sum of Gaussian beams [8]

E_{N} (x, y, 0) = \sum_{n = 1}^{N} \frac{{(- 1)}^{n - 1}}{N} (\binom{N}{n}) [\exp (- \frac{n x^{2} + n y^{2}}{w_{0}^{2}}) - \exp (- \frac{n x^{2} + n y^{2}}{p w_{0}^{2}})],

where $(\binom{N}{n})$ denotes a binomial coefficient, w ₀ determines mainly the beam waist width, N is the order of a circular dark hollow beam, and p < 1. The area of the dark region across a DHB increases as N or p increases. Figure 1 shows the contour plots of the normalized irradiance distribution of a circular DHB for two different N values with p=0.9 and w ₀ =2cm.

Fig. 1. Contour plots of the normalized intensity distribution of a circular DHB for two different N values with p=0.9 and w ₀ =2cm. (a) N=3; (b) N=15.

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Fig. 2. Contour plots of the normalized intensity distribution of a rectangular DHB for two different sets of (H, N). (a) H=N=5 (b) H=N =15.

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Similarly, we can express the field of a DHB of rectangular symmetry at z=0 as the following finite sum of elliptical Gaussian beams

E_{HN} (x, y, 0) = \sum_{h = 1}^{H} \sum_{n = 1}^{N} \frac{{(- 1)}^{h + n}}{HN} (\binom{H}{h}) (\binom{N}{n}) [\exp (- \frac{h x^{2}}{w_{0 x}^{2}} - \frac{n y^{2}}{w_{0 y}^{2}}) - \exp (- \frac{h x^{2}}{p w_{0 x}^{2}} - \frac{n y^{2}}{p w_{0 y}^{2}})],

where w ₀ and w ₀ are related to the waist sizes of an elliptical Gaussian beam in the x and y directions, respectively. H and N indicate the orders of a rectangular DHB. The area of the dark region increases as H, N or p increases. Figure 2 shows the contour plots of the normalized irradiance distribution of a rectangular dark hollow beam for two different sets of (H, N) with w _0x = 1cm, w _0y = 2cm and p=0.9

For the more general case, we can express a DHB of elliptical symmetry at z=0 as the following finite sum of astigmatic elliptical Gaussian beam

E_{N} (x, y, 0) = \sum_{n = 1}^{N} \frac{{(- 1)}^{n - 1}}{N} (\binom{N}{n}) [\exp (- \frac{n x^{2}}{w_{0 x}^{2}} - \frac{n y^{2}}{w_{0 y}^{2}} - \frac{2 nxy}{w_{0 xy}^{2}}) - \exp (- \frac{n x^{2}}{p w_{0 x}^{2}} - \frac{n y^{2}}{p w_{0 y}^{2}} - \frac{2 nxy}{p w_{0 xy}^{2}})]

where w _0x, w _0y and w _0xy are related to the beam waist sizes of an astigmatic elliptical Gaussian beam in x direction, y direction and xy-coupled direction, respectively. The orientation of the elliptical beam spot is controlled by w _0x, w _0y and w _0xy. Figure 3 shows the contour plots of the normalized intensity distributions of an elliptical DHB for two different sets of w _0x, w _0y and w _0xy with N=10 and p=0.9.

Fig. 3. Contour plots of the normalized intensity distributions of an elliptical DHB for two different sets of (w _0x, w _0y, w _0xy) with N=10 and p=0.9. (a) w _0x = 1cm , w _0y = 2cm, w _0xy = 2cm; (b)w _0x = 2cm, w _0y = 1cm, w _0xy =2cm.

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After some manipulation, we can express Eqs. (1) and (3) in the following tensor form

E_{N} (r, 0) = \sum_{n = 1}^{N} \frac{{(- 1)}^{n - 1}}{N} (\binom{N}{n}) [\exp (- \frac{ik}{2} r^{T} Q_{1 n}^{- 1} r) - \exp (\frac{ik}{2} r^{T} Q_{1 np}^{- 1} r)],

where k = 2π/λ is the wave number, λ is the wavelength of the beam, r is the position vector given by r^T = (x y), $Q_{1 n}^{- 1}$ is a 2×2 matrix called the complex curvature tensor for an astigmatic Gaussian beam, [23–26] which is an extension of a complex curvature 1/q (q is the radius of the wavefront) for a stigmatic Gaussian beam. (Note that Q ^-1 _1n is the inverse of complex radius matrix Q _1n.) The introduction of the complex curvature tensor allows us to treat the propagation of a DHB through a paraxial astigmatic optical system [5] and a turbulent atmosphere (as shown later) conveniently through some vector integration and tensor operation. More information about the complex curvature tensor can be found in Refs. [23–26]. For a circular DHB, Q ^-1_1n and Q ^-1 _1np are given by

Q_{1 n}^{- 1} = \frac{2 n}{ik w_{0}^{2}} I, Q_{1 np}^{- 1} = \frac{2 n}{ipk w_{0}^{2}} I,

where I is a 2×2 unit matrix. For an elliptical DHB, Q ^-1 _1n and Q ^-1 _1np are given by

Q_{1 n}^{- 1} = (\begin{matrix} \frac{2 n}{ik w_{0 x}^{2}} & \frac{2 n}{ik w_{0 xy}^{2}} \\ \frac{2 n}{ik w_{0 xy}^{2}} & \frac{2 n}{ik w_{0 y}^{2}} \end{matrix}), Q_{1 np}^{- 1} = (\begin{matrix} \frac{2 n}{ipk w_{0 x}^{2}} & \frac{2 n}{ikp w_{0 xy}^{2}} \\ \frac{2 n}{ikp w_{0 xy}^{2}} & \frac{2 n}{ik {pw}_{0 y}^{2}} \end{matrix}) .

Similarly, Eq. (2) can be expressed in the following tensor form

E_{HN} (x, y, 0) = \sum_{h = 1}^{H} \sum_{n = 1}^{N} \frac{{(- 1)}^{h + n}}{HN} (\binom{H}{h}) (\binom{N}{n}) [\exp (- \frac{ik}{2} r^{T} Q_{1 hn}^{- 1} r) - \exp (- \frac{ik}{2} r^{T} Q_{1 hnp}^{- 1} r)],

where

Q_{1 hn}^{- 1} = (\begin{matrix} \frac{2 h}{ik w_{0 x}^{2}} & 0 \\ 0 & \frac{2 n}{ik w_{0 y}^{2}} \end{matrix}), Q_{1 hnp}^{- 1} = (\begin{matrix} \frac{2 h}{ikp w_{0 x}^{2}} & 0 \\ 0 & \frac{2 n}{ik {pw}_{0 y}^{2}} \end{matrix}) .

3. Analytical formulas for the average intensity of a dark hollow beam in a turbulent atmosphere

The propagation of a laser beam in a turbulent atmosphere can be studied with the following extended Huygens-Fresnel integral [17–21]

E (ρ, z, t) = - \frac{ik}{2 πz} \exp (- ikz) \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} E (r_{1}, 0) \exp [- \frac{ik}{2 z} {(r_{1} - ρ)}^{2} + Ψ (r_{1}, ρ) - i 2 πft] d r_{1},

where E(r ₁,0) and E(ρ,z,t) are the electric fields of the laser beam in the source plane (z=0) and the output plane, respectively, z is the propagation distance, Ψ(r ₁,ρ) represents (in the Rytov method) the random part (due to the turbulence of the atmosphere) of the complex phase of a spherical wave propagating from the source plane to the output plane. f is the frequency, and t denotes the time.

The average intensity at the output plane is given by 〈I(ρ,z)〉 = 〈E(ρ,z,t)E ^*(ρ,z,t)〉, where^* and 〈〉 denote the complex conjugate and the ensemble average over the medium statistics, respectively. From Eq. (9), we obtain

〈 I (ρ, z) 〉 = \frac{k^{2}}{4 π^{2} z^{2}} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} E (r_{1}, 0) E^{*} (r_{2}, 0) \exp [- \frac{ik}{2 z} {(r_{1} - ρ)}^{2} + \frac{ik}{2 z} {(r_{2} - ρ)}^{2}]

The ensemble average term in Eq. (10) can be expressed as [13, 14, 15, 17,]

〈 \exp [Ψ (r_{1}, ρ) + Ψ^{*} (r_{2}, ρ)] 〉 = \exp [- 0.5 D_{Ψ} (r_{1} - r_{2})] = \exp [- \frac{1}{ρ_{0}^{2}} {(r_{1} - r_{2})}^{2}]

where D _Ψ(r ₁-r ₂) is the phase structure function in Rytov’s representation ρ = (0.545 $C_{n}^{2}$ k ² z)^-3/5 is the coherence length (inversely proportional to the magnitude of the turbulence) of a spherical wave propagating in the turbulent medium ( $C_{n}^{2}$ is the structure constant). Note that in the derivation of Eq. (11), we have employed a quadratic approximation (see e.g. Eqs. (14) and (15) in Ref [13]) of the actual 5/3 power law for Rytov’s phase structure function in order to obtain an analytical result through the tensor analysis (as shown below). If the 5/3 power law is used directly (instead of the quadratic approximation), it will be difficult to make an analytical tensor analysis for the propagation of DHBs in a turbulent atmosphere. Here we note that this quadratic approximation has been shown to be reliable in e.g. Refs. [14] and [15], and has been used widely (see e.g. Refs. [13]–[15], [17]–[21]). Our numerical results (not presented here to save space) have also shown that the difference between the normalized irradiance distribution obtained with this quadratic approximation and the one calculated numerically with the 5/3 power law is quite small, and they give the same trends.

After some manipulation, we can express Eq. (10) in the following tensor form

〈 I (ρ, z) 〉 = \frac{k^{2}}{4 π^{2} {[\det (\tilde{B})]}^{\frac{1}{2}}} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} E (r_{1}, 0) E^{*} (r_{2}, 0)

where r˜^T = (r^T ₂ r^T ₂), ρ˜^T =(ρ^T ρ^T), and

\tilde{B} = (\begin{matrix} z I & 0 \\ 0 & - z I \end{matrix}), \tilde{P} = \frac{2}{ik ρ_{0}^{2}} (\begin{matrix} I & - I \\ - I & I \end{matrix}) .

Using Eq. (4), we can express E (r ₁, 0) E ^*(r ₂,0) in Eq. (12) as follows

E_{N} (r_{1}, 0) E_{N}^{*} (r_{2}, 0) = \sum_{n = 1}^{N} \sum_{m = 1}^{N} \frac{{(- 1)}^{n + m}}{N^{2}} (\binom{N}{n}) (\binom{N}{m}) [\exp (- \frac{ik}{2} {\tilde{r}}^{T} {\tilde{Q}}_{1 nm}^{- 1} \tilde{r}) - \exp (- \frac{ik}{2} {\tilde{r}}^{T} {\tilde{Q}}_{2 nm}^{- 1} \tilde{r})

where

{\tilde{Q}}_{1 nm}^{- 1} = (\begin{matrix} Q_{1 n}^{- 1} & 0 \\ 0 & - {(Q_{1 m}^{- 1})}^{*} \end{matrix}), {\tilde{Q}}_{2 nm}^{- 1} = (\begin{matrix} Q_{1 n}^{- 1} & 0 \\ 0 & - {(Q_{1 mp}^{- 1})}^{*} \end{matrix}),

Then substituting Eq. (14) into Eq. (12), we obtain (after some tedious vector integration) the following average intensity of the DHB of circular or elliptical symmetry in the output plane

〈 I (ρ, z) 〉 = \sum_{n = 1}^{N} \sum_{m = 1}^{N} \frac{{(- 1)}^{n + m}}{N^{2}} (\binom{N}{n}) (\binom{N}{m}) [\det {[S_{1}]}^{- \frac{1}{2}} \exp (- \frac{ik}{2} {\tilde{ρ}}^{T} {\tilde{Q}}_{o 1 nm}^{- 1 - 1} \tilde{ρ}

where

S_{i} = \tilde{I} + \tilde{B} ({\tilde{Q}}_{inm}^{- 1} + \tilde{P}), {\tilde{Q}}_{oinm}^{- 1} = {[({\tilde{Q}}_{inm}^{- 1} + \tilde{P}) + \tilde{B}]}^{- 1} .

Here I˜ is a 4×4 unit matrix.

Similarly, we can express E(r ₁,0) is E ^*(r ₂,0) in Eq. (12) (for a DHB of rectangular symmetry) as follows [using Eq. (7)]

E_{HN} (r_{1}, 0) E_{HN}^{*} (r_{2}, 0) = \sum_{h = 1}^{H} \sum_{l = 1}^{H} \sum_{n = 1}^{N} \sum_{m = 1}^{N} \frac{{(- 1)}^{h + l + n + m}}{N^{2} H^{2}} (\binom{H}{h}) (\binom{H}{l}) (\binom{N}{n}) (\binom{N}{m}) [\exp (- \frac{ik}{2} {\tilde{r}}^{T} Q_{1 hnlm}^{- 1} \tilde{r})

where

{\tilde{Q}}_{1 hnlm}^{- 1} = (\begin{matrix} Q_{1 hn}^{- 1} & 0 \\ 0 & - {(Q_{1 lm}^{- 1})}^{*} \end{matrix}), {\tilde{Q}}_{2 hnlm}^{- 1} = (\begin{matrix} Q_{1 hn}^{- 1} & 0 \\ 0 & - {(Q_{1 lmp}^{- 1})}^{*} \end{matrix}),

Substituting Eq. (18) into Eq. (12), we obtain the following average intensity of the rectangular DHB in the output plane after some tedious but straightforward vector integration and manipulation

〈 I (ρ, z) 〉 = \sum_{h = 1}^{H} \sum_{l = 1}^{H} \sum_{n = 1}^{N} \sum_{m = 1}^{N} \frac{{(- 1)}^{h + l + n + m}}{N^{2}} (\binom{H}{h}) (\binom{H}{l}) (\binom{N}{n}) (\binom{N}{m}) [\det {[S_{1}]}^{- \frac{1}{2}} \exp (- \frac{ik}{2} {\tilde{ρ}}^{T} {\tilde{Q}}_{o 1 hnlm}^{- 1}^{- 1} \tilde{ρ})

where

S_{i} = \tilde{I} + \tilde{B} ({\tilde{Q}}_{ihnlm}^{- 1} + \tilde{P}), {\tilde{Q}}_{o ihnlm}^{- 1} = {[{({\tilde{Q}}_{ihnlm}^{- 1} + \tilde{P})}^{- 1} + \tilde{B}]}^{- 1}, (i = 1,2,3,4) .

Eqs. (16) and (20) are convenient for use in analyzing the spreading of a DHB in a turbulent atmosphere. In the absence of turbulence (ρ ₀ → ∞ i.e., $C_{n}^{2}$ = 0), P˜ =0 and consequently Eqs. (16) and (20) are reduced to the propagation formulas of a DHB in free space. If we set N=1, p=0, w _0xy =∞, w _0x =w _0y = w ₀ in Eqs. (16) and (20), then Eqs. (16) and (20) are reduced to the following propagation formula for a Gaussian beam of fundamental mode in a turbulent atmosphere [17–21]

〈 I (ρ, z) 〉 = \frac{k^{2} ρ_{0}^{2} w_{0}^{4}}{k^{2} ρ_{0}^{2} w_{0}^{4} + 4 (ρ_{0}^{2} + {2 w}_{0}^{2}) z^{2}} \exp [- \frac{2 k^{2} ρ_{0}^{2} w_{0}^{2} (ρ_{x}^{2} + ρ_{y}^{2})}{k^{2} ρ_{0}^{2} w_{0}^{4} + 4 (ρ_{0}^{2} + 2 w_{0}^{2}) z^{2}}] .

Note that formulas (16) and (20) give concise forms of the average intensity for a DHB of circular or non-circular symmetry. Expansion of the present matrix form will end up with more complicated formulas for the anverage intensity (especially for an elliptical DHB whose variables x and y are coupled) and won’t provide any further useful or physical information about the features of the resultant beams. Since many computational softwares, such as Matlab and Mathematics, can be used convenienlty to carry out directly matrix calcuations and numerical simlations, we keep the concise matrix forms of formulas (16) and (20). The numerical calculation in the present paper is carried out with the software of Matlab. Useful (physical) information, such as the spreading properties of the DHBs in a turbulent atmosphere, can be found through some numerical evaluation of formulas (16) and (20). The tensor method is convenient for treating the propagation of coherent and partially coherent complex beams (see e.g. Refs.[23–27]).

4. Propagation properties of dark hollow beams in a turbulent atmosphere

Now we study the propagation properties of various DHBs in a turbulent atmosphere by using the analytical formulas derived in the previous section.

Figure 4 shows the cross line (y=0) of the normalized average intensity I(x,0,z)/I(x,0,z)_max [calculated with Eq. (16)] of a circular DHB at several propagation distances in a turbulent atmosphere for different structure constants $C_{n}^{2}$ with w ₀ = 2cm, N=3, p=0.9, λ = 632.8nm (the wavelength of a commonly used He-Ne laser). The normalization in Fig. 4 and the rest figures is carried out with respect to the peak of each beam individually, i.e. they are not normalized with respect to the peak value of the source beam. For comparison, the corresponding intensity distributions in free space are also given in Fig. 4. One can see from Fig. 4 that the evolution properties of the intensity distribution of the circular DHB in the turbulent atmosphere are similar to that in free space in the near field, namely, the hollow beam profile disappears gradually (and the central intensity increases) as propagation distance z increases. While in the far field, the circular DHB becomes a circular Gaussian beam (without a central hollow) in a turbulent atmosphere [see the dashed or dotted lines in Fig. 4(e) and 4(f)], which is different from its propagation properties in free space [there is a small bright ring around the brightest center in free space; see the solid lines in Fig. 4(e) and 4(f)]. Furthermore, from Fig. 4, one can also see that the conversion from a circular DHB to a circular Gaussian beam becomes quicker and the beam spot in the far field spreads more rapidly for a larger structure constant. The evolution speed of the intensity of the DHB in a turbulent atmosphere is also closely related with beam parameters. Figure 5 shows the cross line (y=0) of the normalized average intensity of a circular DHB for two different λ values (1.55 um and 632.8nm) at several propagation distances in a turbulent atmosphere with w ₀ = 2cm, N=3,p=0.9 and $C_{n}^{2}$ = 10^-14 m ^-2/3. Figure 6 shows the cross line (y=0) of the normalized average intensity of a circular DHB for two different w ₀ values (1 cm and 2 cm) with λ = 632.8nm , N=3, p=0.9 and $C_{n}^{2}$ = 10^-14 m ^-2/3. Figure 7 shows the cross line (y=0) of the normalized average intensity of a circular DHB for two different N values (N=2 and 15) with λ = 632.8nm , p=0.9, w ₀ = 2cm and $C_{n}^{2}$ = 10^-14 m ^-2/3. From Figs. 5–7, one can see that the conversion from a circular DHB to a circular Gaussian beam becomes quicker in a relatively short distance range (see the second distances in these figures; e.g., in Fig. 5 the beam profile has become Gaussian at the second distance for λ = 632.8nm but not for λ = 1.55μm) and the beam spot in the far field range spreads more rapidly for a shorter wavelength, a lower beam order and a smaller waist width of the initial beam.

Fig. 4. Cross line (y=0) of the normalized average intensity of a circular DHB at several different propagation distances in a turbulent atmosphere (for two different values of structure constant $C_{n}^{2}$ ) (a) z=0; (b) z=0.5km; (c) z=1.1km; (d) z=1.5km; (e) z=5km; (f) z=10km.

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Fig. 5. Cross line (y=0) of the normalized average intensity of a circular DHB for two different λ values at several propagation distances in a turbulent atmosphere.

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Fig. 6. Cross line (y=0) of the normalized average intensity of a circular DHB for two different w ₀ values at several propagation distances in a turbulent atmosphere.

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Fig. 7. Cross line (y=0) of the normalized average intensity of a circular DHB for two different N values at several propagation distances in a turbulent atmosphere.

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To learn more about the spreading of a DHB in a turbulent atmosphere, we study the broadening of the effective beam size of a DHB. By use of twice the variance of x or y, the effective beam size of a DHB at plane z can be defined as [13, 28]

W_{s} (z) = \sqrt{\frac{2 \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} s^{2} I (x, y, z) dxdy}{\int_{- \infty}^{\infty} \int_{- \infty}^{\infty} I (x, y, z) dxdy}}, .

Here W_x (z), W_y (z) are the effective beam sizes of a DHB in the x and y directions, respectively. For a circular DHB, we have W_x (z) = W_y (z) = W(z). The difference between the effective beam sizes in the source plane (z=0) and output plane is defined as

Δ W_{s} (z) = W_{s} (z) - W_{s} (0) .

Applying Eqs. (16), (23), and (24), we calculate the difference between the effective beam sizes of a circular DHB at the source plane and output plane (z=10km) in a turbulent atmosphere with various values of w ₀,λ,N, $C_{n}^{2}$ , and the results are shown in Table 1. For comparison, the corresponding results in free space are also shown in Table 1. One can see that broadening of the effective beam size of a circular DHB in a turbulent atmosphere is larger for a larger $C_{n}^{2}$ , lower N, smaller w ₀ and shorter λ. These calculation results are consistent with Figs. 4–7. Broadening of the effective beam size of a circular DHB in a turbulent is larger than that in free space. From Table 1, one can also find an interesting result that broadening of effective beams size of a circular DHB in a turbulent atmosphere is larger for a shorter λ, which is quite different from that in free space.

Table 1. Difference between the effective beam sizes of a circular DHB at the source plane and output plane (z=10km) for different values of w ₀, λ, N, $C_{n}^{2}$

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Fig. 8. Normalized average intensity of an elliptical DHB at several different propagation distances in a turbulent atmosphere. (a) z=0; (b) z=1.1km; (c) z=10km. (d) z=10km (free space).

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Figure 8. shows the normalized average intensity [calculated by Eq. (16)] of an elliptical DHB (λ = 632.8nm , p=0.9, w _0x = 2cm, w _0y = 3cm and w _0xy = 4cm) at several different propagation distances in a turbulent atmosphere with N=3 and $C_{n}^{2}$ =10^-14 m ^-2/3. For comparison, we also calculated the far field distribution of the same elliptical DHB in free space [see Fig. 8(d)]. From Fig. 8(a) and 8(b) one can see that the elliptical hollow profile disappears gradually as z increases in the near field, which is similar to the case of free space. However, in the far field, the beam spot eventually becomes a circular Gaussian beam spot [see Fig. 8(c)]; and then spreads as z increases further), which is quite different from that in free space (an elliptical DHB will retain its elliptical symmetry in free space and there is a small bright elliptical ring around the brightest center in free space; see Fig. 8(d).

Fig. 9. Normalized average intensity of a rectangular DHB at z=10km (a) in a turbulent atmosphere with $C_{n}^{2}$ = 10^-14 m ^-2/3; (b) in free space

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Figure 9 shows the normalized average intensity [calculated with Eq. (20)] of rectangular DHB at z=10km in a turbulent atmosphere and in free space with w _0x = 2cm, w _0y = 3cm, N=H=3, p=0.9, λ = 632.8nm. One can see from Fig. 9 that the rectangular DHB also becomes a circular Gaussian beam [see Fig. 9(a)], which is mu different from it properties in free space [a rectangular DHB will become a beam with som small peaks (rectangular symmetry) around the brightest center (elliptical symmetry) aft propagating over a long distance in free space; see Fig. 9(b)].

Table 2. Difference between the effective beam sizes of an elliptical DHB at the source plane and output plane (z=10km) in a turbulent atmosphere with different values of w _0x, w _0y, λ, N, $C_{n}^{2}$

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Table 3. Difference between the effective beam sizes of a rectangular DHB at the source plane and output plane (z=10km) in a turbulent atmosphere with different values of w _0x, w _0y, λ, N, M, $C_{n}^{2}$

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Similar to the case of a circular DHB, conversion from a DHB of non-circular symmetry to a circular Gaussian beam becomes quicker in a relatively short distance range and the beam spot in the far field spreads more rapidly for a smaller beam width, a lower beam order of the initial beam, a larger structure constant and a shorter wavelength. Table 2 shows the difference between the effective beam sizes of an elliptical DHB at the source plane and output plane (z=10km) in a turbulent atmosphere for different values of w _0x, w _0y, w _0xy, λ, N, $C_{n}^{2}$ . Table 3 shows the difference between the effective beam sizes of a rectangular DHB at the source plane and output plane (z=10km) in a turbulent atmosphere for different values of w _0x, w _0y, w _0xy, λ, N, M, $C_{n}^{2}$ . One can see quantitatively from Table 2 that broadening of the effective beam size of a DHB of non-circular symmetry in a turbulent is larger for a larger $C_{n}^{2}$ , lower N, smaller w ₀ and shorter λ.

5. Conclusion

In conclusion, we have derived some analytical formulas for the average intensity for a DHB of circular or non-circular (elliptic or rectangular) symmetry propagating in a turbulent atmosphere by using a tensor method. The intensity and spreading of a DHB of circular or non-circular symmetry in the turbulent atmosphere have been investigated numerically. We have found that the evolution properties of the intensity distribution of a DHB of circular or non-circular symmetry are similar to those in free space in the near field. However, in the far field, a DHB of circular or non-circular symmetry becomes a circular Gaussian beam spot (without the central dark hollow) under the influence of the atmosphere turbulence. This phenomenon is very interesting, and is quite different from the propagation properties of a DHB in free space. The evolution properties are closely related with the parameters of the beam and the structure constant of the turbulent atmosphere. Our analytical formulas provide an effective and convenient way for analyzing the propagation of a DHB in a turbulent atmosphere. The derivation is based on Rytov’s representation of the phase structure function (i.e., the first equation of Eq. (11)). Such an representation is valid for the cases of weak turbulence (0.307 $C_{n}^{2}$ k ^7/6 z ^11/6≪1) and strong turbulence (0.307 $C_{n}^{2}$ k ^7/6 z ^11/6 >0.5)[17]

Appendix A. Derivation of propagation formulas (16) and (21)

Substituting Eq. (14) into Eq. (12) and after some rearrangement, we obtain

〈 I (ρ, z) 〉 = \frac{k^{2}}{4 π^{2} {[\det (\tilde{B})]}^{\frac{1}{2}}} \sum_{n = 1}^{N} \sum_{m = 1}^{N} \frac{{(- 1)}^{n + m}}{N^{2}} (\begin{matrix} N \\ n \end{matrix}) (\begin{matrix} N \\ m \end{matrix}) \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} [\exp (- \frac{ik}{2} {\tilde{r}}^{T} {\tilde{Q}}_{1 nm}^{- 1} \tilde{r}) - \exp (- \frac{ik}{2} {\tilde{r}}^{T} {\tilde{Q}}_{2 nm}^{- 1} \tilde{r})

Then applying the following integral formula

\int_{- \infty}^{\infty} \exp ({- ax}^{2}) d x = \sqrt{\frac{π}{a}},

Eq. (A1) reduces to (after some integration)

〈 I (ρ, z) 〉 = {[\det (\tilde{B})]}^{- \frac{1}{2}} \sum_{n = 1}^{N} \sum_{m = 1}^{N} \frac{{(- 1)}^{n + m}}{N^{2}} (\begin{matrix} N \\ n \end{matrix}) (\begin{matrix} N \\ m \end{matrix})

We can reduce Eq. (A3) to Eq. (16) by applying the following operations

{[\det (\tilde{B})]}^{- \frac{1}{2}} {[\det ({\tilde{Q}}_{1 nm}^{- 1} + {\tilde{B}}^{- 1} + \tilde{P})]}^{- \frac{1}{2}} = {[\det (\tilde{I} + \tilde{B} ({\tilde{Q}}_{1 nm}^{- 1} + \tilde{P}))]}^{- \frac{1}{2}}, (i = 1,2,3,4)

{\tilde{B}}^{- 1} - {\tilde{B}}^{- 1 T} {({\tilde{Q}}_{inm}^{- 1} + {\tilde{B}}^{- 1} + \tilde{P})}^{- 1} {\tilde{B}}^{- 1} = {\tilde{B}}^{- 1} - {\tilde{B}}^{- 1 T} {(\tilde{B} {\tilde{Q}}_{inm}^{- 1} + \tilde{I} + \tilde{B} \tilde{P})}^{- 1}

and by setting

S_{i} = \tilde{I} + \tilde{B} ({\tilde{Q}}_{inm}^{- 1} + \tilde{P}), {\tilde{Q}}_{oinm}^{- 1} = {[{({\tilde{Q}}_{inm}^{- 1} + \tilde{P})}^{- 1} + \tilde{B}]}^{- 1} .

Here I˜ is a 4×4 unit matrix. In the above operations, we have used the relation B˜ =B˜ ^-1. Substituting Eq. (18) into Eq. (12), we can obtain the propagation formula (21) in a similar way. More information about vectorial integrations and tensor operations can be found in Refs. [23–27].

Acknowledgments

This research is partially supported by the National Basic Research Program (973) of China (2004CB719800).

References and links

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Fig. 1. Contour plots of the normalized intensity distribution of a circular DHB for two different N values with p=0.9 and w ₀ =2cm. (a) N=3; (b) N=15.

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Fig. 2. Contour plots of the normalized intensity distribution of a rectangular DHB for two different sets of (H, N). (a) H=N=5 (b) H=N =15.

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Fig. 4. Cross line (y=0) of the normalized average intensity of a circular DHB at several different propagation distances in a turbulent atmosphere (for two different values of structure constant

C_{n}^{2}

) (a) z=0; (b) z=0.5km; (c) z=1.1km; (d) z=1.5km; (e) z=5km; (f) z=10km.

View in Article | Download Full Size | PDF

Fig. 5. Cross line (y=0) of the normalized average intensity of a circular DHB for two different λ values at several propagation distances in a turbulent atmosphere.

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Fig. 6. Cross line (y=0) of the normalized average intensity of a circular DHB for two different w ₀ values at several propagation distances in a turbulent atmosphere.

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Fig. 7. Cross line (y=0) of the normalized average intensity of a circular DHB for two different N values at several propagation distances in a turbulent atmosphere.

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Fig. 8. Normalized average intensity of an elliptical DHB at several different propagation distances in a turbulent atmosphere. (a) z=0; (b) z=1.1km; (c) z=10km. (d) z=10km (free space).

View in Article | Download Full Size | PDF

Fig. 9. Normalized average intensity of a rectangular DHB at z=10km (a) in a turbulent atmosphere with

C_{n}^{2}

= 10^-14 m ^-2/3; (b) in free space

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Tables (3)

Table 1. Difference between the effective beam sizes of a circular DHB at the source plane and output plane (z=10km) for different values of w ₀, λ, N, $C_{n}^{2}$

View Table | View all tables in this article

Equations (52)

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E_{N} (x, y, 0) = \sum_{n = 1}^{N} \frac{{(- 1)}^{n - 1}}{N} (\binom{N}{n}) [\exp (- \frac{n x^{2} + n y^{2}}{w_{0}^{2}}) - \exp (- \frac{n x^{2} + n y^{2}}{p w_{0}^{2}})],

E_{HN} (x, y, 0) = \sum_{h = 1}^{H} \sum_{n = 1}^{N} \frac{{(- 1)}^{h + n}}{HN} (\binom{H}{h}) (\binom{N}{n}) [\exp (- \frac{h x^{2}}{w_{0 x}^{2}} - \frac{n y^{2}}{w_{0 y}^{2}}) - \exp (- \frac{h x^{2}}{p w_{0 x}^{2}} - \frac{n y^{2}}{p w_{0 y}^{2}})],

E_{N} (x, y, 0) = \sum_{n = 1}^{N} \frac{{(- 1)}^{n - 1}}{N} (\binom{N}{n}) [\exp (- \frac{n x^{2}}{w_{0 x}^{2}} - \frac{n y^{2}}{w_{0 y}^{2}} - \frac{2 nxy}{w_{0 xy}^{2}}) - \exp (- \frac{n x^{2}}{p w_{0 x}^{2}} - \frac{n y^{2}}{p w_{0 y}^{2}} - \frac{2 nxy}{p w_{0 xy}^{2}})]

E_{N} (r, 0) = \sum_{n = 1}^{N} \frac{{(- 1)}^{n - 1}}{N} (\binom{N}{n}) [\exp (- \frac{ik}{2} r^{T} Q_{1 n}^{- 1} r) - \exp (\frac{ik}{2} r^{T} Q_{1 np}^{- 1} r)],

Q_{1 n}^{- 1} = \frac{2 n}{ik w_{0}^{2}} I, Q_{1 np}^{- 1} = \frac{2 n}{ipk w_{0}^{2}} I,

Q_{1 n}^{- 1} = (\begin{matrix} \frac{2 n}{ik w_{0 x}^{2}} & \frac{2 n}{ik w_{0 xy}^{2}} \\ \frac{2 n}{ik w_{0 xy}^{2}} & \frac{2 n}{ik w_{0 y}^{2}} \end{matrix}), Q_{1 np}^{- 1} = (\begin{matrix} \frac{2 n}{ipk w_{0 x}^{2}} & \frac{2 n}{ikp w_{0 xy}^{2}} \\ \frac{2 n}{ikp w_{0 xy}^{2}} & \frac{2 n}{ik {pw}_{0 y}^{2}} \end{matrix}) .

E_{HN} (x, y, 0) = \sum_{h = 1}^{H} \sum_{n = 1}^{N} \frac{{(- 1)}^{h + n}}{HN} (\binom{H}{h}) (\binom{N}{n}) [\exp (- \frac{ik}{2} r^{T} Q_{1 hn}^{- 1} r) - \exp (- \frac{ik}{2} r^{T} Q_{1 hnp}^{- 1} r)],

Q_{1 hn}^{- 1} = (\begin{matrix} \frac{2 h}{ik w_{0 x}^{2}} & 0 \\ 0 & \frac{2 n}{ik w_{0 y}^{2}} \end{matrix}), Q_{1 hnp}^{- 1} = (\begin{matrix} \frac{2 h}{ikp w_{0 x}^{2}} & 0 \\ 0 & \frac{2 n}{ik {pw}_{0 y}^{2}} \end{matrix}) .

E (ρ, z, t) = - \frac{ik}{2 πz} \exp (- ikz) \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} E (r_{1}, 0) \exp [- \frac{ik}{2 z} {(r_{1} - ρ)}^{2} + Ψ (r_{1}, ρ) - i 2 πft] d r_{1},

〈 I (ρ, z) 〉 = \frac{k^{2}}{4 π^{2} z^{2}} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} E (r_{1}, 0) E^{*} (r_{2}, 0) \exp [- \frac{ik}{2 z} {(r_{1} - ρ)}^{2} + \frac{ik}{2 z} {(r_{2} - ρ)}^{2}]

\times 〈 \exp [Ψ (r_{1}, ρ) + Ψ^{*} (r_{2}, ρ)] 〉 d r_{1} d r_{2} .

〈 \exp [Ψ (r_{1}, ρ) + Ψ^{*} (r_{2}, ρ)] 〉 = \exp [- 0.5 D_{Ψ} (r_{1} - r_{2})] = \exp [- \frac{1}{ρ_{0}^{2}} {(r_{1} - r_{2})}^{2}]

〈 I (ρ, z) 〉 = \frac{k^{2}}{4 π^{2} {[\det (\tilde{B})]}^{\frac{1}{2}}} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} E (r_{1}, 0) E^{*} (r_{2}, 0)

\times \exp [- \frac{ik}{2} ({\tilde{r}}^{T} {\tilde{B}}^{- 1} \tilde{r} - 2 \tilde{r} {\tilde{B}}^{- 1} \tilde{ρ} + {\tilde{ρ}}^{T} {\tilde{B}}^{- 1} \tilde{ρ})] \exp [- \frac{ik}{2} {\tilde{r}}^{T} \tilde{P} \tilde{r}] d \tilde{r},

\tilde{B} = (\begin{matrix} z I & 0 \\ 0 & - z I \end{matrix}), \tilde{P} = \frac{2}{ik ρ_{0}^{2}} (\begin{matrix} I & - I \\ - I & I \end{matrix}) .

E_{N} (r_{1}, 0) E_{N}^{*} (r_{2}, 0) = \sum_{n = 1}^{N} \sum_{m = 1}^{N} \frac{{(- 1)}^{n + m}}{N^{2}} (\binom{N}{n}) (\binom{N}{m}) [\exp (- \frac{ik}{2} {\tilde{r}}^{T} {\tilde{Q}}_{1 nm}^{- 1} \tilde{r}) - \exp (- \frac{ik}{2} {\tilde{r}}^{T} {\tilde{Q}}_{2 nm}^{- 1} \tilde{r})

- \exp (- \frac{ik}{2} {\tilde{r}}^{T} {\tilde{Q}}_{3 nm}^{- 1} \tilde{r}) + \exp (- \frac{ik}{2} {\tilde{r}}^{T} {\tilde{Q}}_{4 nm}^{- 1} \tilde{r})],

{\tilde{Q}}_{1 nm}^{- 1} = (\begin{matrix} Q_{1 n}^{- 1} & 0 \\ 0 & - {(Q_{1 m}^{- 1})}^{*} \end{matrix}), {\tilde{Q}}_{2 nm}^{- 1} = (\begin{matrix} Q_{1 n}^{- 1} & 0 \\ 0 & - {(Q_{1 mp}^{- 1})}^{*} \end{matrix}),

{\tilde{Q}}_{3 nm}^{- 1} = (\begin{matrix} Q_{1 np}^{- 1} & 0 \\ 0 & - {(Q_{1 m}^{- 1})}^{*} \end{matrix}), {\tilde{Q}}_{4 nm}^{- 1} = (\begin{matrix} Q_{1 np}^{- 1} & 0 \\ 0 & {- (Q_{1 mp}^{- 1})}^{*} \end{matrix}) .

〈 I (ρ, z) 〉 = \sum_{n = 1}^{N} \sum_{m = 1}^{N} \frac{{(- 1)}^{n + m}}{N^{2}} (\binom{N}{n}) (\binom{N}{m}) [\det {[S_{1}]}^{- \frac{1}{2}} \exp (- \frac{ik}{2} {\tilde{ρ}}^{T} {\tilde{Q}}_{o 1 nm}^{- 1 - 1} \tilde{ρ}

- \det {[S_{2}]}^{- \frac{1}{2}} \exp (- \frac{ik}{2} {\tilde{ρ}}^{T} {\tilde{Q}}_{o 2 nm}^{- 1 - 1} \tilde{ρ}) - \det {[S_{3}]}^{- \frac{1}{2}} \exp (- \frac{ik}{2} {\tilde{ρ}}^{T} {\tilde{Q}}_{o 3 nm}^{- 1 - 1} \tilde{ρ})

+ \det {[S_{4}]}^{- \frac{1}{2}} \exp (- \frac{ik}{2} {\tilde{ρ}}^{T} {\tilde{Q}}_{o 4 nm}^{- 1 - 1} \tilde{ρ})],

S_{i} = \tilde{I} + \tilde{B} ({\tilde{Q}}_{inm}^{- 1} + \tilde{P}), {\tilde{Q}}_{oinm}^{- 1} = {[({\tilde{Q}}_{inm}^{- 1} + \tilde{P}) + \tilde{B}]}^{- 1} .

E_{HN} (r_{1}, 0) E_{HN}^{*} (r_{2}, 0) = \sum_{h = 1}^{H} \sum_{l = 1}^{H} \sum_{n = 1}^{N} \sum_{m = 1}^{N} \frac{{(- 1)}^{h + l + n + m}}{N^{2} H^{2}} (\binom{H}{h}) (\binom{H}{l}) (\binom{N}{n}) (\binom{N}{m}) [\exp (- \frac{ik}{2} {\tilde{r}}^{T} Q_{1 hnlm}^{- 1} \tilde{r})

- \exp (- \frac{ik}{2} r_{1}^{T} Q_{2 hnlm}^{- 1} r_{1}) - \exp (- \frac{ik}{2} {\tilde{r}}^{T} Q_{3 hnlm}^{- 1} \tilde{r}) + \exp (- \frac{ik}{2} {\tilde{r}}^{T} {\tilde{Q}}_{4 hnlm}^{- 1} \tilde{r})],

{\tilde{Q}}_{1 hnlm}^{- 1} = (\begin{matrix} Q_{1 hn}^{- 1} & 0 \\ 0 & - {(Q_{1 lm}^{- 1})}^{*} \end{matrix}), {\tilde{Q}}_{2 hnlm}^{- 1} = (\begin{matrix} Q_{1 hn}^{- 1} & 0 \\ 0 & - {(Q_{1 lmp}^{- 1})}^{*} \end{matrix}),

{\tilde{Q}}_{3 hnlm}^{- 1} = (\begin{matrix} Q_{1 hnp}^{- 1} & 0 \\ 0 & - {(Q_{1 lm}^{- 1})}^{*} \end{matrix}), {\tilde{Q}}_{4 hnlm}^{- 1} = (\begin{matrix} Q_{1 hnp}^{- 1} & 0 \\ 0 & {- (Q_{1 lmp}^{- 1})}^{*} \end{matrix}) .

〈 I (ρ, z) 〉 = \sum_{h = 1}^{H} \sum_{l = 1}^{H} \sum_{n = 1}^{N} \sum_{m = 1}^{N} \frac{{(- 1)}^{h + l + n + m}}{N^{2}} (\binom{H}{h}) (\binom{H}{l}) (\binom{N}{n}) (\binom{N}{m}) [\det {[S_{1}]}^{- \frac{1}{2}} \exp (- \frac{ik}{2} {\tilde{ρ}}^{T} {\tilde{Q}}_{o 1 hnlm}^{- 1}^{- 1} \tilde{ρ})

- \det {[S_{2}]}^{- \frac{1}{2}} \exp (- \frac{ik}{2} {\tilde{ρ}}^{T} {Q_{o 2 hnlm}^{- 1}}^{- 1} \tilde{ρ}) - \det {[S_{3}]}^{\frac{1}{2}} \exp (- \frac{ik}{2} {\tilde{ρ}}^{T} {Q_{o 3 hnlm}^{- 1}}^{- 1} \tilde{ρ})

+ \det {[S_{4}]}^{- \frac{1}{2}} \exp (- \frac{ik}{2} {\tilde{ρ}}^{T} {\tilde{Q}}_{o 4 hnlm}^{- 1}^{- 1} \tilde{ρ})],

S_{i} = \tilde{I} + \tilde{B} ({\tilde{Q}}_{ihnlm}^{- 1} + \tilde{P}), {\tilde{Q}}_{o ihnlm}^{- 1} = {[{({\tilde{Q}}_{ihnlm}^{- 1} + \tilde{P})}^{- 1} + \tilde{B}]}^{- 1}, (i = 1,2,3,4) .

〈 I (ρ, z) 〉 = \frac{k^{2} ρ_{0}^{2} w_{0}^{4}}{k^{2} ρ_{0}^{2} w_{0}^{4} + 4 (ρ_{0}^{2} + {2 w}_{0}^{2}) z^{2}} \exp [- \frac{2 k^{2} ρ_{0}^{2} w_{0}^{2} (ρ_{x}^{2} + ρ_{y}^{2})}{k^{2} ρ_{0}^{2} w_{0}^{4} + 4 (ρ_{0}^{2} + 2 w_{0}^{2}) z^{2}}] .

W_{s} (z) = \sqrt{\frac{2 \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} s^{2} I (x, y, z) dxdy}{\int_{- \infty}^{\infty} \int_{- \infty}^{\infty} I (x, y, z) dxdy}}, .

Δ W_{s} (z) = W_{s} (z) - W_{s} (0) .

〈 I (ρ, z) 〉 = \frac{k^{2}}{4 π^{2} {[\det (\tilde{B})]}^{\frac{1}{2}}} \sum_{n = 1}^{N} \sum_{m = 1}^{N} \frac{{(- 1)}^{n + m}}{N^{2}} (\begin{matrix} N \\ n \end{matrix}) (\begin{matrix} N \\ m \end{matrix}) \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} [\exp (- \frac{ik}{2} {\tilde{r}}^{T} {\tilde{Q}}_{1 nm}^{- 1} \tilde{r}) - \exp (- \frac{ik}{2} {\tilde{r}}^{T} {\tilde{Q}}_{2 nm}^{- 1} \tilde{r})

- \exp (- \frac{ik}{2} {\tilde{r}}^{T} {\tilde{Q}}_{3 nm}^{- 1} \tilde{r}) + \exp (- \frac{ik}{2} {\tilde{r}}^{T} {\tilde{Q}}_{4 nm}^{- 1} \tilde{r})] \exp [- \frac{ik}{2} ({\tilde{r}}^{T} {\tilde{B}}^{- 1} \tilde{r} - 2 \tilde{r} {\tilde{B}}^{- 1} \tilde{ρ} + {\tilde{ρ}}^{T} {\tilde{B}}^{- 1} \tilde{ρ})] \exp [- \frac{ik}{2} {\tilde{r}}^{T} \tilde{P} \tilde{r}] d \tilde{r}

= \frac{k^{2}}{4 π^{2} {[\det (\tilde{B})]}^{\frac{1}{2}}} \sum_{n = 1}^{N} \sum_{m = 1}^{N} \frac{{(- 1)}^{n + m}}{N^{2}} (\begin{matrix} N \\ n \end{matrix}) (\begin{matrix} N \\ m \end{matrix}) {\exp [- \frac{ik}{2} {\tilde{ρ}}^{T} {\tilde{B}}^{- 1} \tilde{ρ} + \frac{ik}{2} {\tilde{ρ}}^{T} {\tilde{B}}^{- 1 T} ({\tilde{Q}}_{1 nm}^{- 1} + {\tilde{B}}^{- 1} + \tilde{P}) {\tilde{B}}^{- 1} \tilde{ρ}]

\times \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \exp [- \frac{ik}{2} {∣ {({\tilde{Q}}_{1 nm}^{- 1} + {\tilde{B}}^{- 1} + \tilde{P})}^{\frac{1}{2}} \tilde{r} - {({\tilde{Q}}_{1 nm}^{- 1} + {\tilde{B}}^{- 1} + \tilde{P})}^{- \frac{1}{2}} {\tilde{B}}^{- 1} \tilde{ρ} ∣}^{2}] d \tilde{r} - \exp [- \frac{ik}{2} {\tilde{ρ}}^{T} {\tilde{B}}^{- 1} \tilde{ρ} + \frac{ik}{2} {\tilde{ρ}}^{T} {\tilde{B}}^{- 1 T} {({\tilde{Q}}_{2 nm}^{- 1} + {\tilde{B}}^{- 1} + \tilde{P})}^{- 1} {\tilde{B}}^{- 1} \tilde{ρ}]

\times \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \exp [- \frac{ik}{2} {∣ {({\tilde{Q}}_{2 nm}^{- 1} + {\tilde{B}}^{- 1} + \tilde{P})}^{\frac{1}{2}} \tilde{r} - {({\tilde{Q}}_{2 nm}^{- 1} + {\tilde{B}}^{- 1} + \tilde{P})}^{- \frac{1}{2}} {\tilde{B}}^{- 1} \tilde{ρ} ∣}^{2}] d \tilde{r} - \exp [- \frac{ik}{2} {\tilde{ρ}}^{T} {\tilde{B}}^{- 1} \tilde{ρ} + \frac{ik}{2} {\tilde{ρ}}^{T} {\tilde{B}}^{- 1 T} {({\tilde{Q}}_{3 nm}^{- 1} + {\tilde{B}}^{- 1} + \tilde{P})}^{- 1} {\tilde{B}}^{- 1} \tilde{ρ}]

\times \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \exp [- \frac{ik}{2} {∣ {({\tilde{Q}}_{3 nm}^{- 1} + {\tilde{B}}^{- 1} + \tilde{P})}^{\frac{1}{2}} \tilde{r} - {({\tilde{Q}}_{3 nm}^{- 1} + {\tilde{B}}^{- 1} + \tilde{P})}^{- \frac{1}{2}} {\tilde{B}}^{- 1} \tilde{ρ} ∣}^{2}] d \tilde{r} + \exp [- \frac{ik}{2} {\tilde{ρ}}^{T} {\tilde{B}}^{- 1} \tilde{ρ} + \frac{ik}{2} {\tilde{ρ}}^{T} {\tilde{B}}^{- 1 T} {({\tilde{Q}}_{4 nm}^{- 1} + {\tilde{B}}^{- 1} + \tilde{P})}^{- 1} {\tilde{B}}^{- 1} \tilde{ρ}]

\times \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \exp [- \frac{ik}{2} {∣ {({\tilde{Q}}_{4 nm}^{- 1} + {\tilde{B}}^{- 1} + \tilde{P})}^{\frac{1}{2}} \tilde{r} - {({\tilde{Q}}_{4 nm}^{- 1} + {\tilde{B}}^{- 1} + \tilde{P})}^{- \frac{1}{2}} {\tilde{B}}^{- 1} \tilde{ρ} ∣}^{2}] d \tilde{r}} .

\int_{- \infty}^{\infty} \exp ({- ax}^{2}) d x = \sqrt{\frac{π}{a}},

〈 I (ρ, z) 〉 = {[\det (\tilde{B})]}^{- \frac{1}{2}} \sum_{n = 1}^{N} \sum_{m = 1}^{N} \frac{{(- 1)}^{n + m}}{N^{2}} (\begin{matrix} N \\ n \end{matrix}) (\begin{matrix} N \\ m \end{matrix})

{{[\det ({\tilde{Q}}_{1 nm}^{- 1} + {\tilde{B}}^{- 1} + \tilde{P})]}^{- \frac{1}{2}} \exp [- \frac{ik}{2} {\tilde{ρ}}^{T} {\tilde{B}}^{- 1} \tilde{ρ} + \frac{ik}{2} {\tilde{ρ}}^{T} {\tilde{B}}^{- 1 T} {({\tilde{Q}}_{1 nm}^{- 1} + {\tilde{B}}^{- 1} + \tilde{P})}^{- 1} {\tilde{B}}^{- 1} \tilde{ρ}]

{- [\det ({\tilde{Q}}_{2 nm}^{- 1} + {\tilde{B}}^{- 1} + \tilde{P})]}^{- \frac{1}{2}} \exp [- \frac{ik}{2} {\tilde{ρ}}^{T} {\tilde{B}}^{- 1} \tilde{ρ} + \frac{ik}{2} {\tilde{ρ}}^{T} {\tilde{B}}^{- 1 T} {({\tilde{Q}}_{2 nm}^{- 1} + {\tilde{B}}^{- 1} + \tilde{P})}^{- 1} {\tilde{B}}^{- 1} \tilde{ρ}]

{- [\det ({\tilde{Q}}_{3 nm}^{- 1} + {\tilde{B}}^{- 1} + \tilde{P})]}^{- \frac{1}{2}} \exp [- \frac{ik}{2} {\tilde{ρ}}^{T} {\tilde{B}}^{- 1} \tilde{ρ} + \frac{ik}{2} {\tilde{ρ}}^{T} {\tilde{B}}^{- 1 T} {({\tilde{Q}}_{3 nm}^{- 1} + {\tilde{B}}^{- 1} + \tilde{P})}^{- 1} {\tilde{B}}^{- 1} \tilde{ρ}]

{+ [\det ({\tilde{Q}}_{4 nm}^{- 1} + {\tilde{B}}^{- 1} + \tilde{P})]}^{- \frac{1}{2}} \exp [- \frac{ik}{2} {\tilde{ρ}}^{T} {\tilde{B}}^{- 1} \tilde{ρ} + \frac{ik}{2} {\tilde{ρ}}^{T} {\tilde{B}}^{- 1 T} {({\tilde{Q}}_{4 nm}^{- 1} + {\tilde{B}}^{- 1} + \tilde{P})}^{- 1} {\tilde{B}}^{- 1} \tilde{ρ}]} .

{[\det (\tilde{B})]}^{- \frac{1}{2}} {[\det ({\tilde{Q}}_{1 nm}^{- 1} + {\tilde{B}}^{- 1} + \tilde{P})]}^{- \frac{1}{2}} = {[\det (\tilde{I} + \tilde{B} ({\tilde{Q}}_{1 nm}^{- 1} + \tilde{P}))]}^{- \frac{1}{2}}, (i = 1,2,3,4)

{\tilde{B}}^{- 1} - {\tilde{B}}^{- 1 T} {({\tilde{Q}}_{inm}^{- 1} + {\tilde{B}}^{- 1} + \tilde{P})}^{- 1} {\tilde{B}}^{- 1} = {\tilde{B}}^{- 1} - {\tilde{B}}^{- 1 T} {(\tilde{B} {\tilde{Q}}_{inm}^{- 1} + \tilde{I} + \tilde{B} \tilde{P})}^{- 1}

= [{\tilde{B}}^{- 1} (\tilde{B} {\tilde{Q}}_{inm}^{- 1} + \tilde{I} + \tilde{B} \tilde{P}) - {\tilde{B}}^{- 1 T}] {(\tilde{B} {\tilde{Q}}_{inm}^{- 1} + \tilde{I} + \tilde{B} \tilde{P})}^{- 1}

= {[{({\tilde{Q}}_{inm}^{- 1} + \tilde{P})}^{- 1}]}^{- 1} {[\tilde{I} + \tilde{B} ({\tilde{Q}}_{inm}^{- 1} + \tilde{P})]}^{- 1} = {[{({\tilde{Q}}_{inm}^{- 1} + \tilde{P})}^{- 1} + \tilde{B}]}^{- 1}, (i = 1,2,3,4)

S_{i} = \tilde{I} + \tilde{B} ({\tilde{Q}}_{inm}^{- 1} + \tilde{P}), {\tilde{Q}}_{oinm}^{- 1} = {[{({\tilde{Q}}_{inm}^{- 1} + \tilde{P})}^{- 1} + \tilde{B}]}^{- 1} .

w ₀	20mm	20mm	20mm	10mm	20mm
λ	632.8nm	632.8nm	1.55μm	632.8nm	632.8nm
N	2	5	2	2	2
$C_{n}^{2}$	10^-14 m ^-2/3	10^-14 m ^-2/3	10^-14 m ^-2/3	10^-14 m ^-2/3	10^-15 m ^-2/3
ΔW(z)(atmosphere)	323.002mm	319.725mm	311.048mm	364.577mm	81.067mm
ΔW(free space)	29.185mm	29.061mm	116.496mm	105.723mm	29.1856

w _0x	20mm	20mm	20mm	10mm	20mm
w _0y	30mm	30mm	30mm	15mm	30mm
w _{0_xy}	40mm	40mm	40mm	20mm	40mm
λ	632.8nm	632.8nm	1.55μm	632.8nm	632.8nm
N	2	5	2	2	2
$C_{n}^{2}$	10^-14 m ^-2/3	10^-14 m ^-2/3	10^-14 m ^-2/3	10^-14 m ^-2/3	10^-15 m ^-2/3
Δ_x W(z)	323.106mm	314.688mm	314.405mm	365.754mm	83.58mm
Δ_y W(z)	305.274mm	293.507mm	296.573mm	356.837mm	65.747mm

w _0x	20mm	20mm	20mm	10mm	20mm
w _0y	30mm	30mm	30mm	15mm	30mm
λ	632.8nm	632.8nm	1.66μm	632.8nm	632.8nm
N	3	5	3	3	3
M	3	5	3	3	3
$C_{n}^{2}$	10^-14 m ^-2/3	10^-14 m ^-2/3	10^-14 m ^-2/3	10^-14 m ^-2/3	10^-15 m ^-2/3
Δ_x W(z)	320.146mm	306.281mm	311.445mm	364.273mm	80.619mm
Δ_y W(z)	300.833mm	286.142mm	292.132mm	354.617mm	61.306mm

w ₀	20mm	20mm	20mm	10mm	20mm
λ	632.8nm	632.8nm	1.55μm	632.8nm	632.8nm
N	2	5	2	2	2
$C_{n}^{2}$	10^-14 m ^-2/3	10^-14 m ^-2/3	10^-14 m ^-2/3	10^-14 m ^-2/3	10^-15 m ^-2/3
ΔW(z)(atmosphere)	323.002mm	319.725mm	311.048mm	364.577mm	81.067mm
ΔW(free space)	29.185mm	29.061mm	116.496mm	105.723mm	29.1856

w _0x	20mm	20mm	20mm	10mm	20mm
w _0y	30mm	30mm	30mm	15mm	30mm
w _{0_xy}	40mm	40mm	40mm	20mm	40mm
λ	632.8nm	632.8nm	1.55μm	632.8nm	632.8nm
N	2	5	2	2	2
$C_{n}^{2}$	10^-14 m ^-2/3	10^-14 m ^-2/3	10^-14 m ^-2/3	10^-14 m ^-2/3	10^-15 m ^-2/3
Δ_x W(z)	323.106mm	314.688mm	314.405mm	365.754mm	83.58mm
Δ_y W(z)	305.274mm	293.507mm	296.573mm	356.837mm	65.747mm

Abstract

1. Introduction

2. Dark hollow beams of circular and non-circular symmetries

3. Analytical formulas for the average intensity of a dark hollow beam in a turbulent atmosphere

4. Propagation properties of dark hollow beams in a turbulent atmosphere

5. Conclusion

Appendix A. Derivation of propagation formulas (16) and (21)

Acknowledgments

References and links

Cited By

Figures (9)

Tables (3)

Equations (52)

Optics Express