1. Introduction

The propagation of laser beams in free space and in atmospheric turbulence is a topic that has been of considerable theoretical and practical interest for a long time [1, 2]. In 1978 Collett and Wolf predicted that Gaussian Schell-model (GSM) beams may have the same directionality as a fully coherent laser beam in free space [3, 4], which was confirmed experimentally later [5, 6]. It means that full spatial coherence is not a necessary condition for highly directional light beams. Recently, much work has been carried out concerning the spreading of laser beams in atmospheric turbulence [7–10]. In 2003 Wolf and his collaborators extended the result in Refs.[3, 4] from free space to atmospheric turbulence [11]. Very recently, we have found that, besides the equivalent GSM beams, there also exist other equivalent partially and fully coherent beams which may have the same directionality as a fully coherent laser beam in free space and also in atmospheric turbulence, such as the equivalent partially and fully coherent Hermite-Gaussian (H-G) beams [12], and the equivalent partially and fully coherent Hermite-cosh-Gaussian (H-ChG) beams and cosh-Gaussian (ChG) [13]. Thus, an interesting question arises: Does the result hold true for general beams (i.e., general partially and fully coherent beams)? The aim of this paper is to study the directionality of general beams in free space and in atmospheric turbulence based on the partial-coherence theory. Some general results are obtained, which are illustrated by examples.

2. Theoretical model

Within the framework of partial-coherence theory and under the quasi-monochromatic assumptions, the cross-spectral density of general beams is expressed as [14, 15]

where

with λ _i1,_j1 is the corresponding mode coherence coefficients (MCC’s), i1 and j1 are the mode indices. φ(x) represents the field distribution, which can be expressed as a series of orthogonal basis modes, for example, the H-G modes at the plane z=0 in the rectangular coordinate system, that is,

with w _0h being the waist width of the basis Gaussian mode, and H _l denoting the lth order Hermite polynomial. W ⁽⁰⁾(y ^′ ₁, y ^′ ₂) can be obtained if x, λ _i1,_j1, i1 and j1 are replaced by y, λ _i2,_j2, i2 and j2 in Eq. (2).

By using the orthogonality of H-G series, the MCC’s is obtained from Eq. (2), i.e.,

λ _i2,_j2 can be obtained if x, i1 and j1 are replaced by y, i2 and j2 in Eq. (4).

Based on the extended Huygens-Fresnel principle, the averaged intensity of the general beam represented by Eq.(1) propagating in atmospheric turbulence at the plane z is given by the expression (Sec.12.2 of Ref. 2)

where k is the wave number related to the wave length λ by k=2π/λ, ψ (r,r ^′, z) is the phase function that depends on the properties of the medium, and < >_m denotes average over the ensemble of the turbulent medium, which is given by (Sec.12.2.3 of Ref. 2)

with J ₀ being the Bessel function of the first kind and order zero, and Φ _n(κ) being the spatial power spectrum of the refractive-index fluctuations of the turbulent medium.

To obtain the analytical results, the new variables of integration are introduced as

Eq. (5) then becomes

The mean-squared beam width in the x direction is defined as [16]

where

On substituting from Eq. (8) into Eq. (10), and recalling the integral formulae

where δ denotes the Dirac delta function, and δ ^″ is its second derivative, we obtain

Recalling the integral formulae

where L ^α _i denotes the generalized Laguerre polynomial, f is an arbitrary function and f ^″ is its second derivative, and performing the integration with respect to ν _y, u _y, u _x and ν _x in turn in Eq.(13) yields

where

On substituting from Eq. (19) into Eq. (17), using the formula λ _i1,_j1=λ _j1,_i1, and considering f ^″(0)=0 for |j1-i1| ≥ 3, after some lengthy mathematical manipulations, we obtain

The right-hand side of Eq.(11) is the total power of the beam at distance z, because of energy conservation, it is equal to the total power of the source. Considering the orthonormality of H-G series, we obtain

Consequently, in view of Eq. (9), we obtain the mean-squared beam width in the x direction of general beams in atmospheric turbulence, i.e.,

where β _i,_i=λ _i,_i/∑^+∞ _i=0 λ _i,_i and β _{i,i +2}=λ _i,_{i +2}=λ _i,_{i +2}/∑^+∞ _i=0 λ _i,_i are the weighting factors of all basis modes and the weighting factors of the MCC’s of the ith and the (i+2)th, respectively. Similarly, we can obtain the mean-squared beam width in the y direction, and omitted here.

The first two terms on the right-hand side of Eq. (22) denote the spread of the beam width in free space due to diffraction, which indicate that the mean-squared beam width of general beams in free space is a parabola profile. The third (cubic) term in Eq. (22) presents the spreading of general beams due to atmospheric turbulence, which is independent of the MCC’s.

From Eq. (22), the angular spread in the x direction of general beams propagating through atmospheric turbulence turns to be

The first term on the right-hand side in Eq. (23) represents the angular spread of general beams in free space, which is dependent on not only β _i,i but also β _i,i+2 if the H-G modes are adopted; while the second term describes the effect of turbulence on the angular spread, which does not depend on the MCC’s, i.e., is independent of the beam parameters.

For i=0, Eq. (22) reduces to the mean-squared width of a fully coherent Gaussian beam specified by subscripts ‘Gs’, which is given by

and Eq.(23) reduces to the angular spread of the fully coherent Gaussian beam, i.e.,

From a comparison of Eqs. (23) and (25), we conclude that a general beam will generate the same angular spread as a fully coherent Gaussian beam both in free space and in turbulence if the condition

is satisfied. The result is independent of the spatial power spectrum Φ _n(κ). Such general beams are called the equivalent general beams [3, 4].

3. Examples

The examples are given to illustrate the result obtained above, in which fully coherent beams (e.g., Sinh-Gaussian beams) and partially coherent beams (e.g., GSM beams) are considered.

If the modified Von Karman spectrum is adopted as a model of the atmospheric turbulence, i.e., [2]

where κ _m=5.92/I ₀, I ₀ and L ₀ are the turbulence inner scale and outer scale, respectively. C ² _n is the refraction index structure constant, which describes how strong the turbulence is.

Letting l ₀=0.01m and L ₀=10m yields

(I) Sinh-Gaussian beam

The cross-spectral density function of Sinh-Gaussian (ShG) beams at the plane z=0 is expressed as [17]

where w ₀ is the waist width of the Gaussian part, Ω₀ is the parameter associated with the sinh part.

On substituting from Eqs. (3) and (29) into Eq. (4), after very tedious integral calculations, the analytical expression for the MCC’s can be derived as

where i, j=odd numbers, δ=w ² ₀Ω² ₀ and γ=2(w ₀/w _0h)².

Figure 1 gives the mean squared beam width w _x(z) of an equivalent ShG beam and of the corresponding fully coherent Gaussian beam propagating both in free space and in atmospheric turbulence. As can be expected, the equivalent ShG beam exhibits the same directionality as the corresponding fully coherent Gaussian beam in free space and also in atmospheric turbulence.

(‖) GSM beam

The cross-spectral density function of GSM beams at the plane z=0 is characterized by

where σ ₀ the spatial correlation length of GSM beams at the plane z=0.

Substituting Eqs. (3) and (31) into Eq. (4), after very tedious integral calculations, we obtain the MCC’s

where i, j and k have the same parity, or λ _{i, j}=0 when i and j have different parity; and

The mean-squared width w _x(z) of an equivalent GSM beam and of the corresponding fully coherent Gaussian beam versus propagation distance z is shown in Fig. 2. As can be seen, the equivalent GSM beam generate the same directionality as the corresponding fully coherent Gaussian beam not only in free space but also in turbulence as well.

 

Fig. 1. w _x(z) versus z. a: the corresponding fully coherent Gaussian beam with w _{0 Gs}=9.7mm, λ _Gs=1.54µm; b: an equivalent ShG beam with w ₀=4mm, γ=0.5, δ=6, λ=0.6328µm.

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Fig. 2. w _x(z) versus z. a: the corresponding fully coherent Gaussian beam with w _{0 Gs}=7.5mm, λ _Gs=1.54µm; b: an equivalent GSM beam with w ₀=4mm, γ=1.28, α=3.06, λ=0.6328µm.

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4. Concluding remarks

The directionality of general beams propagating in free space and in atmospheric turbulence has been studied. Based on the partial-coherence theory, the analytical expressions for the mean-squared width and the angular spread of general beams propagating in atmospheric turbulence have been derived by using the integral transform technique. The propagation of general beams in free space can be treated as special cases. It has been shown that the mean-squared width and the angular spread are dependent not only on the weighting factors of all basis modes but also on the weighting factors of the MCC’s of the ith and the (i+2)th if the H-G modes are adopted, i.e., β _i,_i and β _i,i+2. It has been found that under condition (26) there exist equivalent general beams which may have the same directionality as a fully coherent Gaussian beam both in free space and in turbulence, which is valid, irrespective of the turbulence model characterized by Φ_n(κ). As compared with the previous publications, the results obtained in this paper are more general, which hold true both for general fully coherent beams and for general partially coherent beams.