1. Introduction

The propagation of light beams in atmospheric turbulence has been extensively studied for a long time [1-3]. In recent years, because of many practical applications such as free-space optical communications, it is especially interesting that what types of light beams and under what conditions the effects of the turbulent atmosphere on the beams are the smallest [4-8]; i.e., variations of the parameters (such as beam width etc.) characterizing beams due to turbulence are the smallest. Preliminary works [4, 5] have shown that the Gaussian-Schell model (GSM) beam with the worse coherence is less affected by turbulence. Consequently, lots of theoretical studies on the spatial and angular spreading of partially coherent beams in atmospheric turbulence have been carried out, indicating that partially coherent beams are less influenced by turbulence than fully coherent beams [6-8]. Subsequently, experiments supporting these findings have been performed [9]. It is well known that the Wigner distribution function (WDF) can characterize partially coherent beams in space and in spatial frequency domain simultaneously [10-13]. Consequently, the beam propagation factor (M ²-factor) for characterizing laser beams can be studied. The M ²-factor of partially coherent beams propagating in free space and in gain or absorbing media has been investigated [14-18]. Very recently, the M ²-factor of fully coherent Gaussian beams in atmospheric turbulence has been studied by numerical approach [19]. However, to our knowledge, there have been no reports on the beam propagation factor of partially coherent beams propagating in atmospheric turbulence by analytical approach.

The main purpose of this paper is to generalize the definition of M ²-factor in terms of second-order moments of the WDF to the case in atmospheric turbulence and to study the propagation law of M ²-factor for partially coherent flat-topped (PCFT) beams with circular symmetry in a turbulent atmosphere. First, by use of the extended Huygens–Fresnel principle and the definition of the WDF, an expression for the WDF of PCFT beams in a turbulent atmosphere has been given. Then, according to method of second-order moments of the WDF, the analytical formulas for the rms spatial width, the rms angular width and the M ²-factor of PCFT beams in turbulence have been derived, which can be applied to cases of different types of spatial power spectra of the refractive index fluctuations. Finally, the rms angular width and the M ²-factor of PCFT beams in turbulence have been discussed with numerical examples.

2. WDF of PCFT beams in a turbulent atmosphere

According to the beam model proposed by Li for a flat-topped (FT) beam [20, 21] and assuming that FT beams of any order contain equal power and under the assumption of the quasi-monochromatic scalar field, the field of an FT beam with circular symmetry can be expressed as [22]

where σ is the radial coordinate, w ₀ is the waist width of the Gaussian beam, N is the order of the FT beam,

and

For the sake of convenience, a constant is omitted in Eq. (1). For the case of N=1, Eqs. (1) and (3) are reduced to the field of a Gaussian beam and p ₁=1, respectively.

The fully coherent beam can be extended to the partially coherent beam by introducing a Gaussian coherence function, and this type of partially coherent beam can be produced by the fully coherent beam passing through a random phase plate or a liquid crystal [23, 24]. Consequently, based on the beam model described by Eqs. (1)-(3) for FT beams and the Schell model used for describing the partially coherent light [25, 26], the cross-spectral density function of PCFT beams at the plane of z=0 can be written as [22, 23]

where the angle brackets with subscript m denote averaging over the field ensemble, ρ ^′ ₁ and ρ ^′ ₂ are two vectors defining points on the transverse plane of z=0, w ₀ represents the waist width of the Gaussian beam, σ ₀ is the correlation length of the source. For the case of N=1, Eq. (4) is reduced to the cross-spectral density function of GSM beams.

Let us consider a PCFT beam propagating in a turbulent atmosphere from the plane z=0 into the half-space z>0. By using the paraxial form of the extended Huygens–Fresnel principle [3], the cross-spectral density function of PCFT beams through the turbulence can be expressed as [2, 3]

where z is the propagation distance, k=2π/λ is the wave number, λ is the wavelength, and

where the central abscissa coordinate systems are chosen, that is,

In Eq. (5) the term exp[-H(ρ _d, ρ ^′ _d, z)] represents the effect of the turbulence, and H can be written as

where J₀ is the Bessel function of the first kind and zero order, Φ _n is the spatial power spectrum of the refractive index fluctuations of the turbulent atmosphere.

It is well known that the WDF is especially suitable for the treatment of partially coherent beams. The WDF can be expressed in terms of the cross-spectral density W(ρ, ρ _d,z) as [11, 12]

where vector θ=(θ _x,θ _y), k θ _x and kθ _y is the wave vector component along the x-axis and y-axis, respectively. Hence θ=(θ ² _x+θ ² _y)^1/2 represents an angle of propagation (without taking the evanescent waves into account). Integration of function h over the angular variables θ _x and θ _y gives the beam intensity, and its integral over the spatial variables x and y is proportional to the radiant intensity of the field.

On the basis of inverse Fourier transform of the Dirac delta function and its property of even function (Chap. 5 of Ref. [27]), we obtain

Then, the cross-spectral density of the beams in the source plane (z=0) can be rewritten as

On substituting from Eq. (12) into Eq. (5) and using Eq. (11) and the property of the Dirac delta function (p. 75 of Ref. [27])

we obtain

where the term $W(\mathit{\rho }″,{\mathit{\rho }}_d+\frac{z}{k}{\mathit{\kappa }}_d,0)$ is given by the expression from Eqs. (4) and (6), i.e.,

On substituting from Eq. (14) into Eq. (10) and calculating the integral with respect to ρ ^″ by using the formula (3.3232) in Ref. [31]

the WDF of PCFT beams through a turbulent atmosphere can be expressed as

where κ _d=|κ _d|=(κ ² _dx+κ ² _dy)^1/2, and

where

denotes the degree of global coherence of the source.

3. The rms spatial and angular spreads of PCFT beams in a turbulent atmosphere

The moments of the order n ₁+n ₂+m ₁+m ₂ of WDF h for three-dimensional beams are given by the expression [12, 13]

where P=∬h(ρ,θ,z)d² ρd² θ is the total irradiance of the beams, and P=πw ² ₀/2 for the case of PCFT beams.

As far as the second-order field correlations are concerned, a partially coherent beam can be characterized by the WDF. It can be seen from Eq. (23) that the moments at the plane of z=const are determined by the WDF at the same plane. Consequently, using Eqs. (17) and (23), we can obtain the moments of PCFT beams at the plane of z=const in turbulence.

On substituting from Eq. (17) into Eq. (23), and recalling the formulas (p. 82, p. 95, and p. 141 of Ref. [27])

(where δ ⁽ⁿ⁾ is nth derivative of the Dirac delta function, f(x) is an arbitrary function and f ⁽ⁿ⁾(x) is its nth derivative), some second-order moments of PCFT beams in turbulence turn out to be

whereas the other second-order moments (i.e., <xy>, <θ _x θ _y>, <xθ _y> and <yθ _x>) and all the first-order moments turn out to be zero. In Eqs. (27)-(29), the quantity T is given by the expression, i.e.,

which denotes the intensity of the turbulent atmosphere.

It should be noted that the quantity T, which can only be determined by the spatial power spectrum of the refractive index fluctuations, is independent of the parameter of the beams.

According to the definition of rms spatial and angular width of light beams [12], the rms spatial and angular width of PCFT beams in turbulence can be expressed as

Equation (31), which is same as Eq. (44) of Ref. [22], represents the spatial spreading of PCFT beams in turbulence; and it is also consistent with the formula (30) of Ref. [6] derived by method of the “equivalent source”. Equation (32) represents the angular spreading of PCFT beams in turbulence, and is consistent with the formula (25a) of Ref. [7] derived by method of the “phase-space dynamics of beam”. The angular width given by Eq. (32) can be regarded as the width of the beams in spatial frequency domain. In other words, the angular width θ _N(z ^′) given by Eq. (32) is equal to the divergence of the “source” at the plane of z=z′ in the free space, but the “source” is described by the WDF h(ρ,θ,z ^′) in turbulence, i.e., Eq. (17).

It can be readily seen from Eq. (32) that the rms angular spread of PCFT beams propagating in turbulence increases with the propagation distance z, whereas it maintains invariance in free space (i.e., for the case of T=0).

For the sake of comparison, the normalized rms angular width, which represents the relative angular spreading of the beams, is defined as

From Eq. (33), it is clear that the normalized rms angular spreading of PCFT beams in turbulence decreases with decreasing α; i.e., the beams with the worse coherence are less affected by turbulence. For a given propagation distance z, Eq. (33) indicates that the relative angular spreading increases with waist width w ₀ and decreasing wavelength λ, whereas based on Eq. (31) the relative spatial spreading w _N(z)/w _N increases with decreasing w ₀ and increasing λ. That is, the influence of atmospheric turbulence on the relative angular spreading is larger for PCFT beams with the larger w ₀ and the smaller λ, but the influence on the relative spatial spreading is smaller. Consequently, in these cases neither the normalized rms spatial nor angular spreading can represents comprehensively the effect of turbulence on PCFT beams.

For the case of N=1, Eqs. (31)-(33) are easily simplified to the results of GSM beams, i.e.,

Taking into account the modified von Karman spectrum [3], Eq. (34) is equal to Eq. (13) in [28]. For the case of α→∞, Eqs. (34)-(36) are further reduced to the expressions of fully coherent Gaussian beams in turbulence.

4. Beam propagation M2-factor of PCFT beams in a turbulent atmosphere

For three-dimensional beams, the beam propagation M ²-factor can be defined as [12]

It should be noted that in Eq. (37) the M ²-factor of a beam can be defined in terms of second-order moments of the WDF in the same plane, and it can also be calculated from the WDF h at any plane of z=const. Consequently, the definition of M ²-factor, i.e. Eq. (37), is valid for the case of the presence of the turbulence. In general, the value of M ²(z) in turbulence is equal to the one that is measured at the distance z.

On substituting from Eqs. (27)-(29) into Eq. (37), the beam propagation M ²-factor of PCFT beams through a turbulent atmosphere can be expressed as

where T is given by Eq. (30), and

and

Note that the value of M ²(z ^′) given by Eq. (38) is equal to the M ²-factor of the “source” at the plane of z=z′ in the free space, but the “source” is described by the WDF h(ρ,θ,z ^′) described by Eq. (17) in turbulence. Equation (38), which is the main result of this paper, indicates that the M ²-factor of PCFT beams in turbulence increases with the propagation distance z. For the case of T=0 (in free space), Eq. (38) shows that the M ²-factor, which is only determined by the beam order and coherent parameter of the source, maintains constant (cf. Refs. [29, 30]).

For the sake of convenience, the normalized M ²-factor, which represents the relative M ²-factor of the beams, is introduced by the expression

It can be shown from Eqs. (40) and (41) that in Eq. (42), the coefficient A only depends on N, the coefficient B only depends on N and α, and the value of B increases with decreasing α. Further, it can be readily seen from Eq. (42) that the normalized M ²-factor of PCFT beams decreases with decreasing α and increasing λ. Consequently, the effect of atmospheric turbulence on the normalized M ²-factor of PCFT beams for the worse coherence and the longer wavelength is smaller. It can be also seen from Eq. (42) that for a given propagation distance z, there exists a waist width w ₀=w _0min corresponding to the minimal value of the normalized M ²-factor for the beams with different waist widths.

For the cases mentioned in section 3, neither the spatial nor angular spreading can denote comprehensively the effect of atmospheric turbulence on PCFT beams at the plane of z=const for different values of w ₀ and of λ, but now we can use the normalized M ²-factor for these cases. Namely, in these cases the smaller value of the normalized M ²-factor of a beam indicates that the beam is less affected by turbulence. Therefore, the normalized M2-factor can show more comprehensively the effect of the turbulence on the optical beams than the normalized spatial and angular width.

For the case of N=1, Eqs. (38) and (42) are simplified to the M ²-factor and the normalized M ²-factor of GSM beams, respectively, namely

The formula (43) is a generalization of known formula for the M ²-factor of GSM beams in free space [16]. Further, for the case of α→∞, Eqs. (43) and (44) are reduced to the M ²-factor and the normalized M ²-factor of fully coherent Gaussian beams in turbulence respectively.

5. Numerical calculation results and analysis

In this section, the Tatarskii spectrum is used as a model for atmospheric turbulence [3], i.e.,

where C ² _n is the structure constant of the refractive index fluctuations of the turbulence and κ _m=5.92/l ₀ with l ₀ being the inner scale turbulence.

On substituting from Eq. (45) into Eq. (30) and by using the formula 3.4621 in Ref. [31], the quantity T can be easily simplified to

Numerical calculations for the rms angular width and normalized rms angular width of PCFT beams were performed on the basis of Eqs. (32), (33) and (46). Some illustrative examples are compiled in Figs. 1 and 2. The calculation parameters are α=2, λ=850 nm, w ₀=0.05 m, C ² _n=10^-15 m^-2/3.

Figure 1 gives the rms angular and normalized rms angular widths for different values of beam order N versus the propagating distance in atmospheric turbulence, respectively. It can be seen from Fig. 1(a) that differing from the case of free space (absence of the turbulence), the rms angular width in turbulence obviously increases with the propagation distance, and it is larger for the beams with a higher beam order. Figure 1(b) implies that the normalized rms angular width in turbulence is smaller for the beams with a higher beam order; i.e., the relative angular spreading is less affected by atmospheric turbulence for the beams with a higher beam order.

 

Fig. 1. (a). rms angular width and (b) normalized rms angular width versus propagation distance for different values of beam order N, respectively. The calculation parameters are α=2, λ=850 nm, w ₀=0.05 m, C ² _n=10^-15 m^-2/3.

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Figure 2 shows the variation of the normalized rms angular width for different values of coherent parameter α on the propagating distance in turbulence. It follows from Fig. 2 that the relative angular spreading of PCFT beams in turbulence decreases with decreasing α, meaning that the relative angular spreading for the beams with the worse spatial coherence is less affected by turbulence.

 

Fig. 2. Normalized rms angular width versus propagation distance for different values of α. The calculation parameters are N=10, λ=850 nm, w ₀=0.05 m, C ² _n=10^-15 m^-2/3.

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Moreover, based on Eqs. (38), (42) and (46), numerical calculations for the M ²-factor and normalized M ²-factor of PCFT beams in turbulence were carried out. Some typical examples are compiled in Figs. 3-6.

Figure 3 shows the M ²-factor and normalized M ²-factor for different values of beam order N versus the propagating distance in atmospheric turbulence, respectively. The calculation parameters are α=2, λ=850 nm, w ₀=0.05 m, C ² _n=10^-15 m^-2/3. It can be shown in Fig. 3(a) that the M ²-factor in turbulence apparently increases with the propagation distance, indicating that the beam quality obviously becomes worse with increasing propagation distance in turbulence. Figure 3(b) implies that the normalized M ²-factor in turbulence is smaller for PCFT beams with a higher beam order, showing the less influence of the turbulence on the relative M ²-factor of the beams with a higher beam order.

 

Fig. 3. (a). M ²-factor and (b) normalized M ²-factor versus propagation distance for different values of beam order N, respectively. The calculation parameters are α=2, λ=850 nm, w ₀=0.05 m, C ² _n=10^-15 m^-2/3.

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Figure 4 gives the normalized M ²-factor for different values of coherent parameter α versus the propagating distance in atmospheric turbulence. The calculation parameters are N=10, λ=850 nm, w ₀=0.05 m, C ² _n=10^-15 m^-2/3. It can be easily seen that the normalized M ²-factor of PCFT beams in turbulence decreases with decreasing α, showing the less influence of the turbulence on the relative M ²-factor of the beams with the worse spatial coherence.

 

Fig. 4. Normalized M ²-factor versus propagation distance for different values of α. The calculation parameters are N=10, λ=850 nm, w ₀=0.05 m, C ² _n=10^-15 m^-2/3.

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Figure 5 gives the normalized M ²-factor for different values of beam order N at the plane of z=1 km versus the waist width w ₀ of the GSM beam. The calculation parameters are α=2, z=1 km, C ² _n=10^-15 m^-2/3, λ=850 nm. It can also be seen from Fig. 5 that there exists a minimum of the normalized M ²-factor as the waist width w ₀ increases. For example, for the case of N=1 (i.e. GSM beams), the minimal value of the normalized M ²-factor is of about 1.11, whereas for the case of N=10, the minimal value of the normalized M ²-factor is of about 1.10.

 

Fig. 5. Normalized M ²-factor versus w ₀ for different values of beam order N. The calculation parameters are α=2, z=1 km, C ² _n=10^-15 m^-2/3, λ=850 nm.

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Figure 6 represents the normalized M ²-factor for different values of beam order N at the plane of z=1 km versus the wavelength of PCFT beams. The calculation parameters are α=2, z=1 km, C ² _n=10^-15 m^-2/3, w ₀=0.05 m. It can be clearly seen from Fig. 6 that the normalized M ²-factor decreases with increasing wavelength λ, indicating that the relative M ²-factor of the beams with the longer wavelength is less affected by turbulence.

 

Fig. 6. Normalized M ²-factor versus wave length λ for different values of beam order N. The calculation parameters are α=2, z=1 km, C ² _n=10^-15 m_-2/3, w ₀=0.05 m.

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6. Conclusions

In the present paper, by use of the extended Huygens–Fresnel principle and definition of the WDF, an expression for the WDF of PCFT beams with circular symmetry in a turbulent atmosphere has been given. The analytical formulas for the rms spatial width, the rms angular width, and the beam propagation M ²-factor of PCFT beams in turbulence have been derived based on the method of second-order moments of the WDF. For the case of beam order N=1, all results obtained in this paper can be simplified to those of GSM beams. Furthermore, the rms angular width and M ²-factor in turbulence have been discussed with numerical examples.

It can be shown that the beam propagation M ²-factor of PCFT beams depends on the beam order, the degree of global coherence of the source, the waist width, the wavelength, and the spatial power spectrum of the refractive index fluctuations of atmospheric turbulence. Moreover, the beam propagation M ²-factor of PCFT beams increases obviously with the propagation distance in turbulence. From the viewpoint of relative variation of the M ²-factor due to the turbulent atmosphere, the effect of the turbulence on PCFT beams is smaller for beams with a higher order, a lower degree of global coherence of the source, and a larger wavelength.