1. Introduction

In 1978 Collett and Wolf predicted that partially coherent beams, like Gaussian Schell-model (GSM) beams, may have the same directionality as a fully coherent laser beam [1, 2]. It means that full spatial coherence in free space is not a necessary condition for highly directional light beams. The theoretical prediction was confirmed by the experiments [3, 4]. In 2003 Wolf and his collaborators showed that under a certain condition there exist the equivalent GSM beams in turbulence which may generate the same angular spread as a fully coherent laser beam [5]. Very recently, we have found that, besides the equivalent GSM beams, there also exist the equivalent partially coherent Hermite-Gaussian (H-G) beams which may have the same directionality as a fully coherent laser beam in free space and also in atmospheric turbulence [6].

On the other hand, Hermite-sinusoidal-Gaussian (H-SG) beams are solutions of the paraxial wave equation [7]. H-SG beams cover a broad range of beams such as Hermite sinh-(or cosh-) Gaussian (H-ShG or H-ChG), and Hermite sin- (or cos-) Gaussian (H-SiG or H-CoG) beams. The production and propagation of H-SG beams in free space were studied in Ref. [8]. The propagation of laser beams in atmospheric turbulence is a topic that has been of considerable theoretical and practical interest for a long time. The propagation of H-G and Laguerre-Gaussian (L-G) beams, CoG and ChG beams, H-ChG beams, and H-SiG and H-ShG beams etc. through the turbulent atmosphere were investigated by Youngs, Eyyuboğlu, Cai and their coworkers in Refs. [9–15]. However, the studies are limited to the spatially fully coherent case. The complex degree of coherence and degree of polarization for partially coherent general beams in atmospheric turbulence, and the influence of turbulence on the propagation of partially coherent ChG, CoG and twisted anisotropic GSM beams was recently studied in Refs. [16–19]. It was shown both theoretically and experimentally that partially coherent beams are less affected by turbulence than the fully coherent laser beams [18, 20, 21].

The aim of this paper is to study the angular spread of partially coherent H-ChG beams propagating through atmospheric turbulence. In comparison with previous work our study is more general. The main results obtained in this paper are illustrated by numerical examples and interpreted physically.

2. Angular spread of partially coherent Hermite-cosh-Gaussian beams in turbulence

The field distribution of H-ChG beams at the source plane z=0 in the Cartesian coordinate system reads as [7, 8]

where ρ≡(ρ_x, ρ_y) is the two-dimensional position vector at the source plane z=0, H_m(·) and H_n(·) denote the mth and nth order Hermite polynomials, w ₀ is the waist width of the Gaussian part, Ω₀ is the beam parameter associated with the cosh part.

The fully coherent beam can be extended to the partially coherent one by introducing a Gaussian term of the spectral degree of coherence [22]. The cross spectral density function of partially coherent H-ChG beams at the plane z=0 is expressed as [7, 8, 22]

where σ₀ is the spatial correlation length of the source at the plane z=0.

Using the extended Huygens-Fresnel principle, the cross spectral density function of partially coherent H-ChG beams propagating through atmospheric turbulence is expressed as [23]

where ρ′≡(ρ′_x, ρ′_y) is the two-dimensional position vector at the plane z, k is the wave number related to the wave length λ by k=2π/λ, ψ(ρ, ρ′) represents the random part of the complex phase of a spherical wave due to the turbulence, <·> _m denotes the average over the ensemble of the turbulent medium statistics, and [24–27]

with

where ρ ₀ is the spatial coherence radius of a spherical wave propagating in turbulence, C ² _n is the refraction index structure constant which describes how strong the turbulence is [23]. It is noted that a quadratic approximation of Rytov’s phase structure function was used in Eq. (4) to obtain an analytical result. This approximation has been shown to be a good approximation in practice [24, 28].

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

By letting ρ′ ₁=ρ′ ₂=ρ′ in Eq. (3), and making use of Eqs. (2), (4) and (6), the intensity of partially coherent H-ChG beams in turbulence at the z plane is given by

where

The normalized rms beam width is defined as [5]

By using some integral transform techniques (see Appendix A), we obtain the normalized rms beam width of partially coherent H-ChG beams in turbulence. The final result is arranged as

where

where L^α_j(●) (j=0, 1, 2…) denotes the generalized Laguerre polynomial with indices α, j, P and Q are independent of propagation distance z. The first two terms on the right-hand side of Eq. (10) denote the spread of the beam width in free space due to diffraction, where the first term P is independent of the propagation distance z, whereas the second term $\frac{Q}{k^2}{z}^2$ increases with z ², and the third term $4{\left(0.545C_n^2{k}^{\frac{1}{3}}\right)}^{\frac{6}{5}}{z}^{\frac{16}{5}}$ indicates how the turbulence affects the beam spread, and increases with z ^16/5.

From Eq. (10) the angular spread of partially coherent H-ChG beams propagating through atmospheric turbulence turns out to be

Equations (10) and (16) are the main analytical results obtained in this paper. The first term on the right-hand side in Eq. (16) represents the angular spread of partially coherent H-ChG beams in free space which is dependent on the spatial correlation length σ₀ and beam parameters w ₀, Ω₀, m, n, but independent of the propagation distance. The second term describes the effect of turbulence on the angular spread, which increases with C ² _n and propagation distance z, but does not depend on σ₀, w ₀, Ω₀ and m, n. It means that the turbulence plays a dominant role in the angular spread as the propagation distance becomes large enough.

 

Fig. 1. Angular spread θ _sp of a partially coherent H-ChG beam versus the spatial correlation length σ₀. The calculation parameters are z=10km, m=n=1, w ₀=3cm, Ω₀=100m^-1.

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Fig. 2. Angular spread θ _sp of a partially coherent H-ChG beam versus the waist width w ₀. The calculation parameters are z=10km, m=n=1, σ₀=1.732cm, Ω₀=100m^-1, C ² _n=10^-14m^-2/3.

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Fig. 3. Angular spread θ _sp of a partially coherent H-ChG beam versus the beam parameter Ω₀. The calculation parameters are z=10km, m=n=1, w ₀=3cm, σ₀=4cm, C ² _n=10^-15m^-2/3.

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Fig. 4. Angular spread θ _sp of a partially coherent H-ChG beam versus the beam order m, n. The calculation parameters are z=10km, w ₀=1cm, σ₀=2 cm, Ω₀=300m^-1, C ² _n=10^-14m^-2/3.

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Numerical examples are given by using Eq. (16) to show the influence of turbulence on the angular spread of partially coherent H-ChG beams, where λ=1.06µm is kept fixed. The angular spread θ _sp versus the spatial correlation length σ₀ for different values of C ² _n is shown in Fig. 1. From Fig. 1 it can be seen that θ _sp increases with decreasing σ₀ and increasing C ² _n. In addition, the difference between θ _sp in turbulence and θ _sp in free space is smaller for partially coherent H-ChG beams with smaller value of σ₀ than for those with larger value of σ₀. For example, for C ² _n=10^-14m^-2/3, σ₀=1 and 6cm, we have θ _sp|_turb/θ _sp|_free=1.59 and 3.43, where θ _sp|_turb and θ _sp|_free are the angular spread in turbulence and in free space respectively. Figure 2 gives the angular spread θ _sp versus the waist width w ₀. Figure 2 indicates that θ _sp decreases with increasing w ₀. Furthermore, the larger the waist width w ₀ is, the larger the difference between θ _sp in turbulence and θ _sp in free space exhibits. For instance, for w ₀=1 and 4cm we have θ _sp|_turb/θ _sp|_free=1.32 and 2.30, respectively. The angular spread θ _sp versus the parameter Ω₀ is depicted in Fig. 3, which shows that θ _sp decreases with increasing Ω₀, and the difference between θ _sp in turbulence and θ _sp in free space increases with increasing Ω₀, e.g., for Ω₀=20 and 120m^-1 we have θ _sp|_turb/θ _sp|_free=1.15 and 1.26 respectively. Figure 4 gives the angular spread θ _sp versus the beam order m, n. Figure 4 indicates that θ _sp increases with increasing m, n, and the difference between θ _sp in turbulence and θ _sp in free space decreases with increasing m, n, e.g., for m=n=1 and 15 we have θ _sp|_turb/θ _sp|_free=1.48 and 1.26, respectively. Therefore, the smaller the spatial correlation length σ₀, waist width w ₀, beam parameter Ω₀, and the larger the beam orders m, n are, the less partially coherent H-ChG beams are affected by turbulence.

Generally, the angular spread of beams in turbulence is affected by two mechanisms. One is the free-space diffraction, and the other is the atmospheric turbulence [20]. Physically, the existence of any substantial original angular spread reduces the effect of atmospheric turbulence. Figures 1–4 indicate that in free space θ _sp decreases with increasing σ₀, w ₀, Ω₀, and decreasing m, n. Namely, the smaller σ₀, w ₀, Ω₀, and larger m, n mean the larger substantial original angular spread. This is the physical reason why the angular spread of partially coherent H-ChG beams with smaller σ₀, w ₀, Ω₀, and larger m, n is less affected by turbulence than that of partially coherent H-ChG beams with larger σ₀, w ₀, Ω₀, and smaller m, n.

3. Equivalent beams

Partially coherent H-ChG beams represent a more general type of beams. There are some partially coherent beams and fully coherent beams which can be treated as their special cases.

(I) For Ω₀=0, Eq. (16) reduces to

where

Equation (17) is the angular spread of partially coherent H-G beams in turbulence, which is in agreement with Eq. (29) in Ref. [6].

(II) For m=n=0, Eq. (16) reduces to the angular spread of partially coherent ChG beams in turbulence, i.e.,

where

(III) For Ω₀=0 and m=n=0, Eq. (16) reduces to

where

Equation (21) is the angular spread of GSM beams propagating through atmospheric turbulence. It is noted that Z ^6/5 occurs in the second term of Eq. (21) because a quadratic approximation of the Rytov’s phase structure function is used in this paper, otherwise z appears as Eq. (14) of Ref. [5].

(IV) For σ₀→∞, Eq. (17) reduces to

where

Equation (23) is the the angular spread of fully coherent H-ChG beams in turbulence.

(V) For Ω₀=0 and σ₀→∞, Eq. (16) reduces to the angular spread of fully coherent H-G beams in turbulence, which is given by

where

(VI) For m=n=0 and σ₀→∞, Eq. (16) reduces to

where

Equation (27) is the angular spread of fully coherent ChG beams.

(VII) For Ω₀=0, m=n=0 and σ₀→∞, Eq. (16) reduces to the angular spread of fully coherent Gaussian beams, i.e.,

where

In comparison of Eqs. (16), (17), (19), (21) and (29), we conclude that four partially coherent beams, i.e., partially coherent H-ChG, H-G, ChG and GSM beams will have the same angular spread as a fully coherent Gaussian beam if the condition

is fulfilled, provided that the wavelength λ is kept fixed. Such four partially coherent beams are called the equivalent partially coherent H-ChG, H-G, ChG and GSM beams, respectively [1, 2].

 

Fig. 5. Normalized rms width w(z) of the four equivalent partially coherent beams and the corresponding fully coherent Gaussian beam versus propagation distance z in free space and in turbulence. a: the corresponding fully coherent Gaussian beam; b: the equivalent GSM beam; c: the equivalent partially coherent H-G beam: d: the equivalent partially coherent H-ChG beam; e: the equivalent partially coherent ChG beam. The calculation parameters are listed in Table 1, and the other parameters are λ=1.06µm, C ² _n=10^-14m^-2/3.

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Table 1. Beam parameters relating to Fig. 5.

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Figure 5 gives the normalized rms width w(z) of the four equivalent partially coherent beams mentioned above and the corresponding fully coherent Gaussian beam propagating both in free space and in atmospheric turbulence. The calculation parameters are listed in Tab. 1. As can be expected, under the condition (31) the four equivalent partially coherent beams exhibit the same directionality as the corresponding fully coherent Gaussian beam in free space and also in turbulence.

Except for the equivalent partially coherent beams, there also exist equivalent fully coherent beams. From a comparison of Eqs. (23), (25), (27) and (29), we see that, three fully coherent beams, i.e., fully coherent H-ChG, H-G and ChG beams will have the same angular spread as a fully coherent Gaussian beam if the condition

is satisfied, provided that the wavelength λ is kept fixed. Such three fully coherent beams are referred to as the equivalent fully coherent H-ChG, H-G and ChG beams, respectively [1, 2].

 

Fig. 6. Normalized rms width w(z) of the three fully coherent beams and the corresponding fully coherent Gaussian beam versus propagation distance z. a: the corresponding fully coherent Gaussian beam; b: the equivalent fully coherent ChG beam; c: the equivalent fully coherent H-ChG beam; d: the equivalent fully coherent H-G beams. The calculation parameters are listed in Table 2, and the other parameters are λ=1.06µm, C ² _n=10^-14m^-2/3.

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Table 2. Beam parameters relating to Fig. 6.

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Figure 6 gives the normalized rms width w(z) of the three equivalent fully coherent beams mentioned above and the corresponding fully coherent Gaussian beam propagating both in free space and in atmospheric turbulence. The calculation parameters are compiled in Tab. 2. As can be seen, equivalent fully coherent H-ChG, H-G and ChG beams have also the same directionality as the corresponding fully coherent Gaussian beam both in free space and in turbulence, if the condition (32) is satisfied.

The results can be physically explained as follows. In fact, partially coherent H-ChG beams are characterized by four parameters (i.e., σ₀, w ₀, Ω₀, m, n), and partially coherent H-G, ChG and GSM beams are characterized by (σ₀, w ₀, m, n), (σ₀, w ₀, Ω₀) and (σ₀, w ₀), respectively. From Figs. 1–4 it follows that for partially coherent H-ChG beams the smaller σ₀ means the worse spatial coherence, which result in a larger angular spread. The smaller the waist width w ₀ and the beam parameter Ω₀ are, the larger the free-space diffraction spread is; while the smaller the beam orders m, n are, the smaller the free-space diffraction spread is. In addition, Eq. (16) indicates that the angular spread due to turbulence is independent of σ₀, w ₀, Ω₀ and m, n. Therefore, both in free space and in turbulence there are two competing mechanisms for partially coherent beams regarding to the directionality, i.e., the spatial coherence and diffraction, which can be balanced by a suitable choice of beam parameters σ₀, w ₀, Ω₀ and m, n, whereas for fully coherent beams discussed above, the angular spread resulting from diffraction can be compensated by appropriately adjusting beam parameters w ₀, Ω₀ and m, n to achieve the same directionality as a fully coherent Gaussian beam in free space and also in turbulence.

4. Concluding remarks

In this paper, the propagation of partially coherent H-ChG beams in turbulence has been studied in detail. The analytical expression for the angular spread of partially coherent H-ChG beams propagating through atmospheric turbulence has been derived by using the extended Huygens-Fresnel principle, quadratic approximation of Rytov’s phase structure function and integral transform techniques. In comparison with previous work our result is more general, because the angular spread of partially coherent H-G, ChG and GSM and fully coherent H-ChG, H-G, ChG, Gaussian beams can be treated as special cases of Eq. (16). It has been shown that the angular spread of partially coherent H-ChG beams with smaller σ₀, w ₀, Ω₀, and larger m, n is less affected by turbulence than that of partially coherent H-ChG beams with larger σ₀, w ₀, Ω₀, and smaller m, n. Partially coherent H-ChG, H-G, ChG and GSM beams may have the same angular spread as a fully coherent Gaussian beam in free space and also in atmospheric turbulence if the condition (31) is satisfied. Under the condition (32) there exist the equivalent fully coherent H-ChG, H-G and ChG beams which may have the same directionality as a fully coherent Gaussian beam both in free space and in atmospheric turbulence. The physical interpretation has been given to show the validity of our results.