## Abstract

Based on the intensity moments and Wigner distribution function, the second-order moments for broadband partially coherent flat-topped (BPCFT) beams in atmospheric turbulence are studied. The beam width of BPCFT beams in atmospheric turbulence is larger than that in free space. The beam width of BPCFT beams in atmospheric turbulence is larger than that of broadband fully coherent flat-topped (BFCFT) beams in atmospheric turbulence. The broader the bandwidth is, the larger the beam width of BPCFT beams becomes. Similar conclusion can be obtained by analyzing the divergence angle and beam propagation factor of BPCFT beams. The beam width of BPCFT beams in atmospheric turbulence is less affected by the broad spectral bandwidth than that in free space. The beam width of BFCFT beams in atmospheric turbulence is less affected by the broad spectral bandwidth than that of BPCFT beams in atmospheric turbulence.

©2010 Optical Society of America

## 1. Introduction

As for many practical applications, such as remote sensing, tracking, and long-distance optical communications, it is necessary to study the propagation and parametric characteristics for a laser beam in atmospheric turbulence. Earlier in the 90th of last century, based on the research work by Ishimaru [1,2], Wu and Boardman [3,4] pointed out that Gaussian Schell-mode beams (GSM) with a lower degree of global coherence are less affected by the turbulence than those with a higher degree of global coherence; GSM beams are less sensitive to the effects of turbulence than fully coherent Gaussian beams are. From then on, many theoretical and experimental studies on the partially coherent beams in atmospheric turbulence have been extensively done. Wolf *et al*. have studied the spreading of partially coherent beams in random media by using the method of equivalent source [5,6]. Lu *et al*. have studied the mean-squared beam width and angular spread of partially coherent beams in atmospheric turbulence by using the Wigner distribution function (WDF) and some mathematical technique [7,8]. Zhou and Chu have studied the propagation of a partially coherent cosine-Gaussian beam through a paraxial ABCD optical system in turbulent atmosphere [9]. From the viewpoint of physics, Shirai *et al*. [10] explained the above results well by use of the coherent-mode representation for GSM beams, i.e., a GSM beam of a lower degree of global coherence consists of more Hermite-Gaussian modes, whereas in atmospheric turbulence the relative spreading of the higher order modes is smaller than that of the lower-order modes. Experiments supporting these results have also been performed recently [11]. Very recently, the parametric characteristics for partially coherent beams in atmospheric turbulence have also been studied. The beam propagation factor (*M*
^{2} factor) of coherent Gaussian beams and of partially coherent flat-topped (PCFT) beams in turbulence has been studied by a numerical method [12] and by an analytical method [13], respectively.

Above research work focuses on the propagation problem for monochromatic spatially partially coherent beams in atmospheric turbulence. With the development of laser technology, laser beams can achieve a more broadband spectrum [14,15]. Studies [16–20] show that the two frequency correlation functions are the key to solve the propagation characteristics of a pulse in a random medium. In 2002, Pääkkönen *et al*. [21] established a novel class of partially coherent non-stationary fields, which may be called Gaussian-Schell model pulses, which feature Gaussian distributions of both the spectrum and the complex degree of spectral coherence. They facilitate a smooth transition from spectrally fully coherent pulses to statistically stationary polychromatic fields with the same spectrum. In 2005, Lajunen *et al*. [22] derived a coherent-mode representation for spatially and spectrally partially coherent pulses both in the space-frequency domain and in the space-time domain. To the best of our knowledge, it is difficult to obtain an analytical expression for the two frequency correlation functions for a pulsed beam propagating in atmospheric turbulence. So it is difficult to do some research on the second-order intensity moments for broadband spatially and spectrally partially coherent pulsed beams in atmospheric turbulence. To simplify the problem, the broadband light beams considered in this paper is an extreme case. The optical field is assumed to be a temporally stationary polychromatic light beam. So different frequencies of the beam are completely uncorrelated as shown by Ref [23].

Researches show that there are many advantages of broadband laser beams in an optical system over monochromatic ones [24,25]. To the best of our knowledge, there is little research on this topic [26,27]. So it is interesting to find out the propagation characteristics for broadband spatially partially coherent laser beams in atmospheric turbulence. In this paper, the broadband partially coherent flat-topped (BPCFT) beams is taken as an example. This paper is organized as follows. In section 2, the analytical expression for the cross-spectral density of a partially coherent flat-topped (PCFT) beam with a certain angular frequency in atmospheric turbulence is derived. In section 3, the second-order intensity moments for a broadband partially coherent flat-topped (BPCFT) beam in atmospheric turbulence are obtained by using the Wigner distribution function and the definition of second-order intensity moments. In section 4, numerical calculation and physical explanation are given. Finally, a simple conclusion is outlined in section 5.

## 2. Cross-spectral density for a PCFT beam with a certain angular frequency in atmospheric turbulence

On the basis of the beam model proposed by Li [28,29], the modified flat-topped beam of any order containing equal power can be written as [30]

where *L* denotes the beam order of a flat-topped beam, *w*
_{0} is the beam width, the binomial coefficient

the scaling factor

The spatial distribution of broadband light beams considered in this paper is flat-topped as shown by Eq. (1). The spatial correlation function is assumed to be a Gaussian function. Let *s*(*ω*) be the spectral distribution function of the light beam and it is a real function. For simplicity, the broadband light beam is assumed to be a temporally stationary light beam, so different frequencies of the light beam are completely uncorrelated [23]

where *E*(ŕ_{i}, *v*) denotes the field distribution of one component with its frequency *v* at the point ŕ_{i}. So the cross-spectral density function of PCFT beams at the plane of *z* = 0 can be expressed as

where ŕ_{i} ≡ (*x*́_{i}, *y*́_{i}) denotes the two-dimensional position vector at the input plane *z* = 0. *δ*(*ω*
_{1} - *ω*
_{2}) denotes the Dirac delta function. Inserting Eq. (1) into Eq. (5), the cross-spectral density function of a PCFT beam with a certain angular frequency in the space-frequency domain at the plane *z* = 0 is expressed as

where *σ*
_{0} denotes the spatial correlation length of the source and may be related to the angular frequency *ω*. For simplicity *σ*
_{0} is always assumed to be a constant in numerical calculation.

Assume that the atmosphere is statistically homogeneous and isotropic. Using the paraxial form of the extended Huygens-Fresnel principle, the cross-spectral density function of a PCFT beam with its frequency *ω* propagating in atmospheric turbulence reads as [31]

where *k* = *ω*/*c*, r_{i} ≡ (*x _{i}*,

*y*) denotes two vectors defining points at the

_{i}*z*plane.

*ψ*is a random phase factor representing the effect of turbulence on the propagation of a spherical wave, the sharp brackets with subscript

*m*denotes the average over an ensemble of statistical realizations of the turbulent atmosphere. The asterisk means the complex conjugation. The last term in Eq. (7) can be written as [32–36]

where *ρ*
_{0} is the spherical-wave lateral coherence radius due to the turbulence of the entire optical system and defined as

where *C _{n}*

^{2}denotes the refraction index structure constant of the atmospheric turbulence.

On substituting from Eqs. (6) and (8) into Eq. (7), and using the integral formula [37]

the analytical expression of the cross-spectral density function for a PCFT beam with its frequency *ω* in atmospheric turbulence is obtained

where

So the spectral density at the output plane *z* is obtained by supposing r_{1} = r_{2} = R

The intensity distribution at the output plane *z* for a BPCFT beam is obtained

When *L* = *L*’ = 1 and *ρ*
_{0} → +∞ in mathematics, Eq. (11) reduces to the cross-spectral density for a GSM beam in free space and is expressed as

## 3. Second-order intensity moments for a BPCFT beam in atmospheric turbulence

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*(r, r_{d}, *z*, *ω*) as [38]

where

so Eq. (11) can also be written as

where

On substituting from Eq. (21) into Eq. (18), the analytical expression for the WDF of a PCFT beam in atmospheric turbulence is obtained

where vector θ = (*θ _{x}*,

*θ*),

_{y}*kθ*and

_{x}*kθ*are the wave vector components along the

_{y}*x*-axis and

*y*-axis, respectively. Hence

*θ*= (

*θ*

_{x}^{2}+

*θ*

_{y}^{2})

^{1/2}represents an angle of propagation (without taking the evanescent waves into account). Integration of function

*h*over the angular variables gives the beam intensity, and its integral over the spatial variables

*x*and

*y*is proportional to the radiant intensity of the field.

The moments of the order [*n*
_{1} + *n*
_{2} + *m*
_{1} + *m*
_{2}) of WDF *h* for three-dimensional beams with its angular frequency *ω* are given by the expression [38,39]

where *P* = ∫^{+∞}
_{-∞} ∫^{+∞}
_{-∞} ∫^{+∞}
_{-∞} ∫^{+∞}
_{-∞}
*h*(r,θ,*z*,*ω*)d^{2} rd^{2} θ is the total irradiance of the beams. According to the properties of WDF, Eq. (25) can be obviously generalized to the case of a broadband light beam, the moments of the order (*n*
_{1} + *n*
_{2} + *m*
_{1} + *m*
_{2}) of WDF *h* for a three-dimensional broadband light beam are given by

where

The second-order moments for a three-dimensional beam are given by

Obviously, all the first-order moments turn out to be zero.

By using the integral formulae [37]

and inserting Eq. (24) into Eq. (26), the second-order moments are obtained

where

According to the definition of root-mean-square (rms) spatial width of light beams [39], the rms spatial width of BPCFT beams in turbulence can be expressed as

the divergence angle can be defined as

the beam propagation factor for a three-dimensional beam is defined as

where *k*
_{0} = *ω*
_{0}/*c* denotes the wave number corresponding to the central angular frequency *ω*
_{0}. On substituting corresponding equations into above equations, the beam width, the divergence angle and the beam propagation factor for a BPCFT beam in atmospheric turbulence are obtained.

The relative beam width *w*/*w*
_{0} is defined as the ratio of the beam width for a BPCFT beam or broadband fully coherent flat-topped (BFCFT) beam in atmospheric turbulence or in free space to the beam width of Gaussian part in the input plane. The relative beam width *w*
_{turb}/*w*
_{free} is defined as the ratio of the beam width for a BPCFT beam in atmospheric turbulence to that in free space. The relative beam width *w*
_{pc}/*w*
_{tc} is defined as the ratio of the beam width for a BPCFT beam to that for a BFCFT beam. The relative divergence angle and relative beam propagation factor can also be defined in this way.

To simplify the numerical calculation, the spectrum density function in the input plane is assumed to be a rectangular function

## 4. Numerical calculation and physical explanation

Parameters are chosen as follows: the central wavelength of BPCFT or BFCFT beams *λ*
_{0} =1054.3nm, the beam order of BPCFT or BFCFT beams *L* = 10, the beam width of Gaussian part *w*
_{0} = 2 cm.

Figure 1 shows the relative beam width of BPCFT beams with different spectral bandwidth in atmospheric turbulence and in free space. The spatial correlation length *σ*
_{0} = 2*w*
_{0}. The lines marked by the solid signs in Fig. 1(a) and 1(b) denote the cases of *C _{n}*

^{2}= 1.0 × 10

^{-14}m

^{-2/3}and

*C*

_{n}^{2}= 1.0 × 10

^{-15}m

^{-2/3}, respectively. The lines marked by the hollow signs in Fig. 1(a) and 1(b) denote the case of free space. It can be seen from Fig. 1(a) and 1(b) that the relative beam width

*w*/

*w*

_{0}of BPCFT beams in atmospheric turbulence or in free space increases with the increasing of propagation distance. Moreover, the stronger the atmospheric turbulence is, the larger the beam width becomes. The BPCFT beams propagating in atmospheric turbulence become more spreading than those in free space. The stronger the atmospheric turbulence is, the more spreading of BPCFT beams becomes. In the case of

*C*

_{n}^{2}= 1.0 × 10

^{-14}m

^{-2/3}as shown by the solid signs in Fig. 1(a), the relative beam width

*w*/

*w*

_{0}of BPCFT beams increases quite slowly with the broadening of spectral bandwidth. In the case of

*C*

_{n}^{2}= 1.0×10

^{-15}m

^{-2/3}as shown by the solid signs in Fig. 1(b), the relative beam width

*w*/

*w*

_{0}of BPCFT beam with its spectral bandwidth Δ

*ω*/

*ω*

_{0}= 0.4 in atmospheric turbulence is almost the same as that in free space. The relative beam width

*w*/

*w*

_{0}of BPCFT beam with its spectral bandwidth Δ

*ω*/

*ω*

_{0}= 0.01 in atmospheric turbulence is almost the same as that in free space. Numerical calculation shows that the relative beam width

*w*/

*w*

_{0}of BPCFT beams in atmospheric turbulence or in free space increases with the increasing of spectral bandwidth. Figure 1(c) and 1(d) show that the relative beam width

*w*

_{turb}/

*w*

_{free}of BPCFT beams decreases with the increasing of spectral bandwidth. What’s more, the relative beam width

*w*

_{turb}/

*w*

_{free}decreases more obviously with the increasing of spectral bandwidth as the propagation distance increases. Thus the beam width of BPCFT beams in free space increases more sharply with the increasing of spectral bandwidth than that in atmospheric turbulence. The spreading of BPCFT beams propagating in free space is mainly caused by the diffraction in free space, the broad spectral bandwidth and the spatial correlation length of the beams; the spreading of BPCFT beams propagating in atmospheric turbulence is mainly caused by the atmospheric turbulence, the broad spectral bandwidth and the spatial correlation length of the beams. With the increasing of propagation distance, the spreading of BPCFT beams due to atmospheric turbulence is dominant [5] and the influence of atmospheric turbulence on the spreading of beams is none of spectral bandwidth [27], thus the relative beam width

*w*

_{turb}/

*w*

_{free}of BPCFT beams decreases with the increasing of spectral bandwidth. Thus the atmospheric turbulence can hold back the relative spreading induced by the broad spectral bandwidth. The beam width of BPCFT beams in atmospheric turbulence is less affected by the broad spectral bandwidth than that in free space. The beam with of BPCFT beams in atmospheric turbulence increases more slowly with the increasing of spectral bandwidth than that in free space.

Figure 2 shows the divergence angle and relative divergence angle of BPCFT beams with different spectral bandwidth in atmospheric turbulence and in free space. The captions for lines are the same as those for Fig. 1. As shown by Fig. 2(a) and 2(b), the divergence angle maintains constant when BPCFT beams propagate in free space. In the cases of spectral bandwidth Δ*ω*/*ω*
_{0} = 0.01, Δ*ω*/*ω*
_{0} = 0.4, and Δ*ω*/*ω*
_{0} = 0.7 , the divergence angles of BPCFT beams with corresponding spectral bandwidth in free space are *θ* = 0.1449 × 10^{-4} rad *θ* = 0.1479 × 10^{-4} rad , and *θ* = 0.1547 × 10^{-4} rad, respectively. While the divergence angle of BPCFT beams in atmospheric turbulence increases with the increasing of propagation distance. What’s more, the divergence angle of BPCFT beams in atmospheric turbulence is larger than that in free space and it increases with the increasing of atmospheric structure constant *C _{n}* as shown by Fig. 2(a) and 2(b). The divergence angle of BPCFT beams in atmospheric turbulence or in free space increases with the increasing of spectral bandwidth. In the case of strong atmospheric turbulence such as

*C*

_{n}^{2}= 1.0×10

^{-14}m

^{-2/3}as shown by Fig. 2(a), the divergence angles of BPCFT beams with different spectral bandwidth in strong atmospheric turbulence are almost the same However the relative divergence angle

*θ*

_{turb}/

*θ*

_{free}of BPCFT beams decreases with the increasing of spectral bandwidth.

Figure 3 shows the relative beam width of BPCFT and BFCFT beams with different spectral bandwidth in atmospheric turbulence. The atmospheric structure constant is assumed to be *C _{n}*

^{2}= 1.0×10

^{-15}m

^{-2/3}. The lines marked by the solid signs in Fig. 3(a) and 3(b) denote the case of

*σ*

_{0}= 0.2

*w*

_{0}and

*σ*

_{0}= 2

*w*

_{0}, respectively. The lines marked by the hollow signs in Fig. 3(a) and 3(b) denote the case of BFCFT beams. The relative beam width

*w*/

*w*

_{0}of BPCFT or BFCFT beams in atmospheric turbulence increases with the propagation distance. The relative beam width

*w*/

_{pc}*w*is larger than 1 when BPCFT beams propagate in atmospheric turbulence as shown by Fig. 3(c) and 3(d). The smaller the spatial correlation length is, the more spreading BPCFT beams becomes. So the BPCFT beams in atmospheric turbulence become more spreading than BFCFT beams do. Figure 3(b) shows that the beam width of a BPCFT beam with narrow spectral bandwidth may be equal to that of a BFCFT beam with broad spectral bandwidth. The BPCFT beams with smaller spatial correlation length and broader spectral bandwidth become more spreading. The relative beam width

_{fc}*w*/

*w*

_{0}of BPCFT or BFCFT beams increases with the increasing of spectral bandwidth. The relative beam width

*w*/

_{pc}*w*of BPCFT beams increases with the increasing of spectral bandwidth. So the beam width of BPCFT beams in atmospheric turbulence increases more sharply with the spectral bandwidth than that of BFCFT beams. The beam width of BFCFT beams in atmospheric turbulence is less affected by broad spectral bandwidth than that of BPCFT beams.

_{fc}Figure 4 shows the divergence angle and relative divergence angle of BPCFT and BFCFT beams with different spectral bandwidth in atmospheric turbulence. The atmospheric structure constant is assumed to be *C _{n}*

^{2}= 1.0 × 10

^{-15}m

^{-2/3}. The caption for the lines is the same as that for Fig. 3. The divergence angle increases when BPCFT or BFCFT beams propagate in atmospheric turbulence. The smaller the spatial correlation length is, the more divergent BPCFT or BFCFT beams become. The relative divergence angle as shown by Fig. 4(c) and 4(d) is larger than 1 and decreases with the increasing of propagation distance. So the divergence of BPCFT beams is less affected by the atmospheric turbulence than that of BFCFT beams is. Figure 4(b) shows that the divergence angle of a BPCFT beam with narrow spectral bandwidth can be equal to that of a BFCFT beam with broad spectral bandwidth. The divergence angle of BPCFT or BFCFT beams in atmospheric turbulence increases with the increasing of spectral bandwidth. The relative divergence angle

*θ*/

_{pc}*θ*also increases with the increasing of spectral bandwidth. So the divergence of BFCFT beams in atmospheric turbulence is less affected by the spectral bandwidth than that of BPCFT beams.

_{fc}Figure 5 shows the change of beam propagation factor and relative beam propagation factor with the propagation distance for BPCFT beams with different spectral bandwidth in atmospheric turbulence and in free space. The spatial correlation length is assumed to be *σ*
_{0} = 2*w*
_{0}. The caption for lines is the same as that for Fig. 1. The beam propagation factor of BPCFT beams in free space maintains constant. The beam propagation factor of BPCFT beams with spectral bandwidth Δ*ω*/*ω*
_{0} = 0.01, Δ*ω*/*ω*
_{0} = 0.4, Δ*ω*/*ω*
_{0} = 0.7 are *M _{free}*

^{2}= 1.2923,

*M*

_{free}^{2}= 1.3190,

*M*

_{free}^{2}=1.3796, respectively. The beam propagation factor of BPCFT beams in atmospheric turbulence increases with the increasing of propagation distance. The beam propagation factor of BPCFT beams in strong atmospheric turbulence is larger than that in weak atmospheric turbulence. The beam propagation factor of BPCFT beams in atmospheric turbulence is larger than that in free space. The relative beam propagation factor

*M*

_{turb}^{2},/

*M*

_{free}^{2}of BPCFT beams increases with the increasing of propagation distance. The effect of broad spectral bandwidth of BPCFT beams on the beam propagation factor is weak in the case of strong atmospheric turbulence. The beam propagation factor of BPCFT beams with different spectral bandwidth in atmospheric turbulence or in free space increases with the broadening of spectral bandwidth. The relative beam propagation factor

*M*

_{turb}^{2}/

*M*

_{free}^{2}of BPCFT beams decreases with the increasing of spectral bandwidth. Thus the beam propagation factor of BPCFT beams in atmospheric turbulence increases more slowly with the increasing of spectral bandwidth than that in free space. The beam propagation factor of BPCFT beams in atmospheric turbulence is less affected by the broad spectral bandwidth than that in free space.

Figure 6 shows the change of beam propagation factor and relative beam propagation factor of BPCFT and BFCFT beams with different spectral bandwidth in atmospheric turbulence. The caption for the lines is the same as that for Fig. 3. The beam propagation factor of BPCFT beams or BFCFT beams in atmospheric turbulence increases with the increasing of propagation distance. The beam propagation factor of BPCFT beams is larger than that of BFCFT beams. The smaller the spatial correlation length is, the larger the beam propagation factor is. The relative beam propagation factor decreases with the increasing of propagation distance. Thus the beam propagation factor of BPCFT beams is less affected by atmospheric turbulence than that of BFCFT beams is. The beam propagation factor of BPCFT or BFCFT beams increases with the increasing of spectral bandwidth and the relative beam propagation factor also increases with the increasing of spectral bandwidth. The beam propagation factor of BFCFT beams is less affected by the broad spectral bandwidth than that of BPCFT beams. The beam propagation factor of a BPCFT beam with narrow spectral bandwidth can be equal to that of a BFCFT beam with broad spectral bandwidth.

## 5. Conclusion

In this paper, taking the temporally stationary broadband light beams as an example, the second-order intensity-moment characteristics for BPCFT and BFCFT beams in atmospheric turbulence and in free space are studied theoretically and numerically based on the WDF and the second-order intensity moments. The beam width, the divergence angle and the beam propagation factor of BPCFT beams in atmospheric turbulence all increase with the increasing of propagation distance. The divergence angle and the beam propagation factor of BPCFT beams in free space both maintain constant. The beam width of BPCFT beams in atmospheric turbulence is larger than that in free space. The divergence angle of BPCFT beams in atmospheric turbulence is larger than that in free space. The beam propagation factor of BPCFT beams in atmospheric turbulence is larger than that in free space. The second-order moments of BPCFT beams in atmospheric turbulence are less affected by the broad spectral bandwidth than those in free space. The second-order moments of BTCFT beams are less affected by the broad spectral bandwidth than those of BPCFT beams. The second-order moments of BPCFT beams are less affected by the atmospheric turbulence than those of BTCFT beams. The second-order moment characteristics for stationary broadband light beams in atmospheric turbulence have been studied in this paper, the influence of spectral correlation can be studied in further research work. The research work in this paper may provide us with a theoretical basis for applications.

## Acknowledgment

This work was supported by the National Natural Science Foundation of China (10874150), Program for New Century Excellent Talents in University (NCET-07-0760), and Zhejiang Provincial Natural Science Foundation of China (R1090168).

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