1. Introduction

Recently, the propagation characteristics of pulsed beams have attracted much attention due to many applications, such as remote sensing, lidar operation and high-speed free-space optical communication (FSOC) links. Berman et al. [1] proposed a design of FSOC links with Gbps data rate based on spectral amplitude encoding of radiation from a Ti:Sapphire pulsed laser. As is well known, the performance of FSOC links is significantly reduced by atmospheric turbulence; controlling the spatial coherence of transmitted beams has been suggested as a means of improving the statistics of the received optical signal [2,3]. Hence, the propagation characteristics of spatially partially coherent pulsed beams through atmospheric turbulence are of great interest in practical FSOC applications transmitting pulsed beams. Traditionally, the Gaussian Schell-model (GSM) beam [4] is used as a mathematical tool for the description of spatially partially coherent beam-like optical fields.

For pulsed beams, which belong to non-stationary optical fields, it is commonly assumed that the exact temporal shape of the pulses in a pulse train is fixed [5], i.e., the spectral components are considered as being completely coherent. However, in practice, this assumption does not generally hold true, and pulsed beams exhibit spectrally partial coherence. Wang et al. [6] investigated the Gaussian Schell-model pulsed (GSMP) beam, which can be used to characterize pulsed optical fields that are both spatially and spectrally partially coherent, and its propagation formula was derived by using the partially coherent light theory. Lajunen et al. [5] derived an analytical formula describing the free-space propagation of the cross-spectral density of a GSMP beam and also presented its coherent-mode representation. An analytical expression for the far-field spectrum of a diffracted GSMP beam was derived by Ding et al. [7], and the spectral shifts and the spectral switches in the far field were numerically calculated. It is noted that none of these studies have taken atmospheric turbulence into account.

The mutual coherence function and the cross-spectral density function [4], which form a Fourier-transform pair, are two important second-order statistical quantities in coherence theory of stationary optical fields, which were also extended to characterize the coherence of a pulsed beam by Ref. [5], and from them, other statistics, such as the average spectral density and the spectral degree of coherence, can be deduced. The need for understanding the propagation behaviors of a GSMP beam in atmospheric turbulence is important for the successful engineering of related applications. Considering a narrow-band, planar source with an arbitrary coherence state radiating into atmospheric turbulence, Fante [8] derived a general expression in the form of a multiple integral for the two-frequency, two-position mutual coherence function of the optical field. The effects of atmospheric turbulence on the spectral switches of a diffracted GSMP beam and on the polarization properties of a stochastic electromagnetic GSMP beam were formulated by Ding et al. [9,10]. Due to the difficulties to obtain a general analytical expression for the cross-spectral density function of spatially and spectrally partially coherent pulsed beams propagating through atmospheric turbulence, Mao et al. [11] assumed the different spectral components of the pulsed beams to be completely uncorrelated and obtained an analytical formula for the cross-spectral density function of a broadband partially coherent flat-topped beam with a certain angular frequency; however, the light beams used in many practical applications are usually not spectrally completely incoherent. To the best of our knowledge, no results have been reported up until now on the analytical solution for the mutual coherence function or the cross-spectral density function of a GSMP beam propagating through atmospheric turbulence; thus, a deeper investigation on this issue is desirable.

The purpose of this paper is to study the second-order propagation characteristics of a GSMP beam in atmospheric turbulence. The propagation of the GSMP beam is considered in the space-frequency domain. Based on the extended Huygens-Fresnel principle and the quadratic approximation of the random phase structure function, we obtain novel analytical expressions for the cross-spectral density function of a GSMP beam propagating through atmospheric turbulence. With the help of the expressions, the average spectral density and the spectral degree of coherence are in turn investigated.

2. Analytical expressions for the cross-spectral density function

Suppose that a GSMP beam is generated in the source plane z = 0 and propagates into the half-space z > 0, nearly parallel to the positive z direction. The mutual coherence function of a GSMP beam in the source plane is given by [5]

Assuming that any source fluctuations are statistically independent of the fluctuations of atmospheric turbulence and using the extended Huygens-Fresnel principle [12] together with the paraxial approximation [12], the cross-spectral density function of a GSMP beam in the half-space z > 0 can be written as

In most practical applications, we are interested in the statistics of a GSMP beam in an observation plane transverse to the direction of propagation. Hence, we let z ₁ = z ₂ = L, where L is the distance from the source plane to the observation plane. In the case of homogeneous turbulence, considering the quadratic approximation of the random phase structure function which gives a good approximation of the second-order statistical properties of optical fields in turbulence under many circumstances [13], <exp[ψ*(r ₁,ρ ₁,L,ω ₁) + ψ(r ₂,ρ ₂,L,ω ₂)]>_m can be written by [11,13–15]

Inserting Eq. (7) into Eq. (6), making use of the identity [16]

Equation (9) shows the important theoretical results obtained by this paper. Note that C_n ² = 0 corresponds to the absence of turbulence in which Γ₂ = 1 and ρ ₀ = ∞; under this condition, Eq. (9) readily proves to be consistent with the formulae obtained by Lajunen et al. (see Eqs. (50)−(60) of [5]) except that our expression for η(g ₁,g ₂) has a negative sign before arctan(∙). Indeed, Eq. (60) of [5] contains a mathematical error of lacking this negative sign.

3. Average spectral density

Setting ρ ₁ = ρ ₂ and ω ₁ = ω ₂ in Eq. (9), we obtain the average spectral density of a GSMP beam propagating through atmospheric turbulence as follows:

Note that Eqs. (10)–(11) are identical with Eqs. (61)−(62) of Ref. [5] except for a different definition of w_c. In fact, S(ρ,L,ω) is a product of the average spectral density of a GSM beam and the spectrum of a Gaussian pulse. The average spectral density of a GSMP beam decreases as the propagation distance L increases. We can see from Eq. (10) that the reduction in the average spectral density caused by the propagation of a GSMP beam is independent of the spectrally partial coherence of the source. It can be readily found from Eq. (11) that the effect of atmospheric turbulence, associated with ρ ₀, on the spectral expansion coefficient resembles the effect of spatially partial coherence of the source, associated with σ ₀. In addition, the contribution of ρ ₀ and σ ₀ toward the spectral expansion coefficient does not couple together as a GSMP beam propagates through atmospheric turbulence; this manifests that atmospheric turbulence is equivalent to spatially partial coherence of the source in enlarging the spectral expansion coefficient.

4. Spectral degree of coherence

The spectral degree of coherence of a GSMP beam, used to quantitatively characterize the strength of the field correlation at a pair of positions with two arbitrary angular frequencies, is defined as the normalized cross-spectral density function and given by

The substitution of Eqs. (9) and (10) into Eq. (12) leads to

Equation (13) shows another important theoretical contribution of this paper. It can be readily found that the terms Γ₂ and exp(−|ρ ₁ − ρ ₂|²/ρ ₀ ²) in Eq. (13) are two turbulence-induced attenuation factors associated with the angular frequency separation |ω ₁ − ω ₂| and the position separation |ρ ₁ − ρ ₂|, respectively, which have an influence on the modulus of the spectral degree of coherence. Furthermore, the argument of the spectral degree of coherence is the same as the cross-spectral density function.

Now, we numerically determine the spectral degree of coherence of a GSMP beam propagating through atmospheric turbulence and analyze its dependence on the turbulence strength measured by the atmospheric spatial coherence length ρ ₀. The greater the turbulence strength becomes, the smaller ρ ₀ is. We consider a GSMP beam of which the central wavelength λ ₀, corresponding to the central angular frequency ω ₀, is 1.55μm, and the expectation value of the initial pulse width T ₀ is 200fs. Note that the angular frequency ω is related to the wavelength λ by ω = 2πc/λ. If T_c = ∞, the spectral width Ω₀ = 5 × 10³GHz which corresponds to 40nm, and for the case of T_c = 100fs, the spectral width Ω₀ = 1.5 × 10⁴GHz which corresponds to 120.3nm.

Figure 1 illustrates the dependence relationships between |μ(ρ ₁,ρ ₂,L,ω,ω)| and the wavelength λ with different combinations of ρ ₀ and σ ₀. Note that the quantity |μ(ρ ₁,ρ ₂,L,ω,ω)| actually describes the frequency-varying degree of transverse spatial coherence of the beam, and indeed, |μ(ρ ₁,ρ ₂,L,ω,ω)| does not depend on T ₀ and T_c. It can be seen that |μ(ρ ₁,ρ ₂,L,ω,ω)| ≡ 1 if a spatially completely coherent pulsed beam propagates through free space without atmospheric turbulence, i.e., ρ ₀ = ∞ and σ ₀ = ∞; this result agrees with what one might expect. For other combinations of ρ ₀ and σ ₀, |μ(ρ ₁,ρ ₂,L,ω,ω)| shows a very slow monotonic growth with the increasing wavelength; the maximum values of |μ(ρ ₁,ρ ₂,L,ω,ω)|_{λ = 1.61μm} − |μ(ρ ₁,ρ ₂,L,ω,ω)|_{λ = 1.49μm} are 5.9 × 10⁻³ and 1.1 × 10⁻² for Figs. 1(a) and 1(b), respectively. This variation behavior of |μ(ρ ₁,ρ ₂,L,ω,ω)| with the wavelength is also consistent with the known theory for a GSM beam (see Eqs. (5.6-84) and (5.6-107) of [4]) when ρ ₀ = ∞. Based on these numerical results, one can say that in the presence of spatially partial coherence of the source and (or) atmospheric turbulence, the degree of transverse spatial coherence in an observation plane z > 0 is slightly smaller for a spectral component of a GSMP beam with a higher frequency than one with a lower frequency, that is, as opposed to the source plane in which the transverse correlation length for all spectral components is the same, the degree of transverse spatial coherence in the observation plane for a given spectral component of a GSMP beam is dependent on the frequency. Moreover, it can be observed from Fig. 1 that |μ(ρ ₁,ρ ₂,L,ω,ω)| reduces as the values of ρ ₀ and (or) σ ₀ decrease with the same wavelength λ.

 

Fig. 1 Dependence relationships between |μ(ρ ₁,ρ ₂,L,ω,ω)| and the wavelength λ with different combinations of ρ ₀ and σ ₀.

Download Full Size | PDF

Figure 2 shows |μ(ρ,ρ,L,ω ₀,ω ₁)| as a function of the wavelength separation δ_λ associated with the two spectral components of a GSMP beam, where the central angular frequency ω ₀ = 2πc/λ ₀ is fixed and the angular frequency ω ₁ = 2πc/(λ ₀ + δ_λ) varies with δ_λ. It should be noted that the quantity |μ(ρ,ρ,L,ω ₀,ω ₁)| is a measure of spectral coherence at the position ρ for the frequencies of ω ₀ and ω ₁. Figure 2(a) corresponds to a spatially and spectrally completely coherent pulsed beam, and Fig. 2(b) represents a spatially and spectrally partially coherent pulsed beam. We can see from Figs. 2(a) and 2(b) that |μ(ρ,ρ,L,ω ₀,ω ₁)| reduces monotonically as ρ ₀ decreases with the same δ_λ; this variation behavior of |μ(ρ,ρ,L,ω ₀,ω ₁)| with ρ ₀ manifests the fact that atmospheric turbulence induces the decrease of spectral coherence of a pulsed beam, which is just the reason for the temporal pulse broadening. It needs to be pointed out that the curves of |μ(ρ,ρ,L,ω ₀,ω ₁)| are actually not rigorously symmetric with respect to δ_λ = 0; this fact proves that |μ(ρ,ρ,L,ω ₀,ω ₁)| is determined exactly by a given combination of ω ₀ and ω ₁ instead of the frequency separation |ω ₀ − ω ₁|.

 

Fig. 2 |μ(ρ,ρ,L,ω ₀,ω ₁)| as a function of the wavelength separation δ_λ with ρ ₀ = ∞, 3cm, 2cm and 1cm.

Download Full Size | PDF

Figure 3 illustrates |μ(ρ ₁,ρ ₂,L,ω ₀,ω ₁)| in terms of the relative atmospheric spatial coherence length ρ ₀/[w ₀ ²∙Δ(L,ω ₀)]^1/2 which is defined as the ratio of the atmospheric spatial coherence length to the beam width. In Fig. 3(a), the frequency separation |ω ₀ − ω ₁| is fixed and the position separation d = |ρ ₁ − ρ ₂| is specified as 0cm, 2cm, 4cm and 6cm, respectively. In Fig. 3 (b), the position separation |ρ ₁ − ρ ₂| = 2cm is fixed and the frequency separation |ω ₀ − ω ₁| is specified by various values of the wavelength separation δ_λ. We can see from Figs. 3(a) and 3 (b) that |μ(ρ ₁,ρ ₂,L,ω ₀,ω ₁)| rises quickly as the relative atmospheric spatial coherence length increases. In Fig. 3(a), for the case of |ρ ₁ − ρ ₂| = 0, |μ(ρ ₁,ρ ₂,L,ω ₀,ω ₁)| approaches to saturation as the relative atmospheric spatial coherence length increases past 1, and for other cases, with the increase of |ρ ₁ − ρ ₂|, the saturation point moves gradually toward the right side. However, in Fig. 3(b), we can find that almost for all values of the wavelength separation δ_λ, |μ(ρ ₁,ρ ₂,L,ω ₀,ω ₁)| approaches to saturation as the relative atmospheric spatial coherence length increases beyond 1.

 

Fig. 3 |μ(ρ ₁,ρ ₂,L,ω ₀,ω ₁)| in terms of the relative atmospheric spatial coherence length ρ ₀/[w ₀ ²∙Δ(L,ω ₀)]^1/2.

Download Full Size | PDF

5. Conclusions

In this paper, novel analytical expressions for the cross-spectral density function of a GSMP beam propagating through atmospheric turbulence have been derived by using the extended Huygens-Fresnel principle together with the quadratic approximation of the random phase structure function, which coincide with the expressions obtained previously for a GSMP beam in free space if the absence of atmospheric turbulence is assumed. From the cross-spectral density function, the expressions for the average spectral density and the spectral degree of coherence of a GSMP beam were obtained. It has been shown that atmospheric turbulence is equivalent to spatially partial coherence of the source in enlarging the spectral expansion coefficient; in the presence of spatially partial coherence of the source and (or) atmospheric turbulence, the degree of transverse spatial coherence in an observation plane z > 0 for a given spectral component of a GSMP beam depends on the frequency, and the higher the frequency is, the slightly smaller the degree of transverse spatial coherence becomes; the spectral degree of coherence is determined exactly by a given combination of the two frequencies instead of the corresponding frequency separation; the modulus of the spectral degree of coherence rises quickly with the increasing relative atmospheric spatial coherence length and then approaches to saturation as the relative atmospheric spatial coherence length increases beyond a given value determined by the parameters of the positions and the frequencies.

The research work in this paper provides a better understanding of the second-order statistics of a GSMP beam propagating though atmospheric turbulence in the space-frequency domain and our results are useful for applications involving spatially and spectrally partially coherent pulsed beams. It should be mentioned that our analytical expressions for the cross-spectral density function also provide a basis for numerically studying the propagation characteristics of a GSMP beam through atmospheric turbulence in the space-time domain, for instance, by using the fast-Fourier-transform algorithm.