M2-Factor of Coherent and Partially Coherent Dark Hollow Beams Propagating in Turbulent Atmosphere

Analytical formula is derived for the 2 M -factor of coherent and partially coherent dark hollow beams (DHB) in turbulent atmosphere based on the extended Huygens-Fresnel integral and the second-order moments of the Wigner distribution function. Our numerical results show that the 2 M factor of a DHB in turbulent atmosphere increases on propagation, which is much different from its invariant properties in free-space, and is mainly determined by the parameters of the beam and the atmosphere. The relative 2 M -factor of a DHB increases slower than that of Gaussian and flat-topped beams on propagation, which means a DHB is less affected by the atmospheric turbulence than Gaussian and flat-topped beams. Furthermore, the relative 2 M -factor of a DHB with lower coherence, longer wavelength and larger dark size is less affected by the atmospheric turbulence. 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Introduction
The propagation factor (best known as 2M -factor) proposed by Siegman is a particularly important property of an optical laser beam [1] being regarded as a beam quality factor in many practical applications.Martinez-Herrero et al. developed the generalized second moments of hard-edge diffracted coherent laser beam to calculate its 2  M -factor [2,3].Gori et al. extended the definition of 2  M -factor from to the partially coherent beam-like fields, and studied the 2  M -factor of partially coherent beam in the absence of the aperture [4,5].Zhang et al. studied the 2  M -factor of hard-edge diffracted partially coherent beams [6,7].In other papers [8][9][10][11][12][13][14][15] various aspects have been related to the 2  M -factor of main classes of coherent and partially coherent beams.
Propagation characteristics of different types of laser beams in a turbulent atmosphere are being studied extensively due to their important applications in free-space optical communications, remote sensing of atmosphere and target tracking [40][41][42][43][44][45][46][47][48][49][50].It is necessary and important to find suitable ways to overcome or reduce the destructive effect of atmospheric turbulence in these applications.One possible way for reducing the effect of atmospheric turbulence is using partially coherent beam or electromagnetic partially coherent beam instead of coherent beam [40][41][42][43][44]. Another possible way is using laser beam with special beam profile, such as Helmholtz-Gauss beams [45], cosh-Gaussian beams [46], higher-order laser beams [47], flat-topped beams [48,49], DHBs [32,33,49] and so on.The average intensity and scintillation index of coherent DHBs in turbulent atmosphere have been studied in [32,33] (see also [41] where intensity, coherence and scintillation of an annular beam are discussed).It was shown in [32,33] that DHBs have advantage over a Gaussian beam and flat-topped beam for overcoming the destructive effect of atmospheric turbulence from the aspect of scintillation, and, hence, have important potential application in free-space optical communications.It was also shown in [49] that partially coherent DHB have advantage over coherent DHB.Up to now, to our knowledge, the 2  M -factor of coherent and partially coherent DHBs in turbulent atmosphere hasn't been reported.In fact, only few papers have been published on the 2  M -factor of laser beams in turbulent atmosphere [50][51][52].In this paper, our aim is to investigate the 2 M -factor of coherent and partially coherent DHBs in turbulent atmosphere.Analytical formula for the DHBs have advantage over a Gaussian beam and flat-topped beam for overcoming the destructive effect of atmospheric turbulence from the aspect of 2  M -factor

Formulation
The electric field of a DHB with circular symmetry at z = 0 can be expressed as the following finite sum of Gaussian modes [27] where denotes a binomial coefficient, N is the beam order of a circular DHB, ) is a scaling factor for controlling the dark size of the DHB.When N = 1 and p = 0, Eq. ( 1) reduces to the expression for the electric field of a fundamental Gaussian beam.When N>1 and p = 0, Eq. ( 1) reduces to the expression for the electric field of a flat-topped beam.Figure 1 shows the cross line (y = 0) of the normalized intensity distribution of a circular DHB for several different values of N and p with 0 1 w mm .One sees from Fig. 1 that the central dark size across a DHB increases as N or p increase., ȡ ȡ are two arbitrary points in the source plane, g V is the transverse coherence width.Under the condition of g V !f , a partially coherent DHB reduces to a coherent DHB.
Within the validity of the paraxial approximation, the propagation of the cross-spectral density of a partially coherent beam in the turbulent atmosphere can be studied with the help of the following generalized Huygens-Fresnel integral [40,41] where 2 / k S O is the wave number with O being the wavelength.In Eq. ( 3) we have used the following sum and difference vector notation where 1 2 , ȡ ȡ are two arbitrary points in the receiver plane, perpendicular to the direction of propagation of the beam.We can express the cross-spectral density in the source plane as follows In Eq. ( 3), the term ' ( , , ) is the contribution from the atmospheric turbulence expressed as )] ( ) , where 0 J is the Bessel function of zero order, n ) represents the one-dimensional power spectrum of the index-of-refraction fluctuations [41].
After some operations as shown in [50], Eq. ( 3) can be expressed in the following alternative form where is the position vector in spatial-frequency domain.For a partially coherent circular DHB, using Eq. ( 2), we can express the CSD ( '', ;0) where The Wigner distribution of a partially coherent beam on propagation in turbulent atmosphere can be expressed in terms of the cross-spectral density function by the formula [50] ( , ; ) ( ) ( , ; ) exp( ) , 2 where ( , ) x y T T { ș denotes an angle which the vector of interest makes with the z-direction, x kT and y kT are the wave vector components along the x-axis and y-axis, respectively.Substituting from Eqs. ( 7), ( 8) and (9) into Eq.( 10), we obtain (after tedious integration) where In above derivations, we have used the integral formula [54] Based on the second-order moments of the Wigner distribution function, the 2 M -factor of a partially coherent beam is defined as follows [2-7,50,51] where ( , , ) .
To check the validity of our formulae, we calculate in Fig. 2 the 2 M -factor of a coherent circular DHB in the source plane (z = 0) versus N and p using Eq. ( 25

O
. One finds that our results agree well with Fig. 3 of [27].
The 2 M -factor increases as p or N increases (i.e., the central dark size increases).For the convenience of comparison, we now study the normalized 2  M -factor of DHBs defined as 2    To learn about the dependence of 2 M -factor of a circular DHB in turbulent atmosphere on its initial beam parameters, we calculate in Fig. 5 the normalized 2  M -factor of a coherent circular DHB on propagation in turbulent atmosphere for different values of beam order N with 0 20 , 0.9    M -factor of flat-topped beam is larger than that of a Gaussian beam at short propagation distance (z<2.5km),but is smaller than that of a Gaussian beam at a long propagation distance, which means a flat-topped beam has advantage over a Gaussian beam for long-distance free-space optical communications.We also note that normalized 2  M -factor of a DHB is always smaller than that of Gaussian and flat-topped beams at any propagation distance except at z = 0, which means a DHB is less affected by the atmospheric turbulence than Gaussian and flat-topped beams.Our results agree well with those reported in [33], where we found that a DHB has advantage over Gaussian and flattopped beams for overcoming the destructive effect of atmospheric turbulence from the aspect of scintillation.The results presented in this paper will be useful in long-distance free-space optical communications We now turn to calculations relating to the 2 M -factor of a partially coherent circular DHB on propagation in turbulent atmosphere.Our numerical results (not shown here to save space) show that the dependence of the normalized 2 M -factor of a partially coherent circular DHB on the parameters ( M -factor of a partially coherent circular DHB also increases on propagation in turbulent atmosphere, but the increment are slower as its initial coherence width decreases, which means a DHB with lower coherence is less affected by the atmospheric turbulence.

Conclusion
We have derived the analytical formula for the 2  M -factor of coherent and partially coherent DHBs in turbulent atmosphere by means of the extended Huygens-Fresnel integral and the second-order moments of the Wigner distribution function.We have found that the 2  M factor of a DHB in turbulent atmosphere increases upon propagation, and these increases become accelerated as the structure constant of turbulence increases or as the inner scale decreases, which is very different from its properties in free space, where its value remains invariant on propagation.Our numerical results have shown that a DHB with lower coherence, longer wavelength and larger dark size is less affected by the atmosphere, and a DHB is less affected by the atmospheric turbulence than Gaussian and flat-topped beams, which might be very useful for free-space optical communications.

w
being the beam waist size of the fundamental Gaussian mode, p ( 0 1 p

Fig. 1 .
Fig. 1.Cross line (y = 0) of the normalized intensity distribution of a circular DHB for different values of N and p with 0 1 w mmA partially coherent beam which has a DHB intensity distribution and a Gaussian spatial correlation can be characterized by the cross-spectral density (CSD) [53] of the form[37,38]

2 M
Fig. 3 the normalized 2 M -factor of a coherent circular DHB on propagation using different values of the structure constant ( 2 n C ) of the turbulent atmosphere with

.
Figure 4 shows the normalized 2 M -factor of a coherent circular DHB on propagation in turbulent atmosphere for different values of inner scale of the turbulence ( 0 l ) with 0 20 , It is clear from Fig. 4 that the normalized 2 M -factor increases more rapidly as 0 l decreases.

Fig. 2 . 2 M
Fig. 2.2M -factor of a coherent circular DHB in the source plane (z = 0) versus N and p

Fig. 3 . 2 M 2 M
Fig. 3. Normalized 2 M -factor of a coherent circular DHB on propagation in turbulent atmosphere for different values of the structure constant ( 2 n C ) of the turbulent atmosphere

6 shows the normalized 2 M
-factor of a coherent circular DHB on propagation in turbulent atmosphere for different values of scaling factor p with .Figure7shows the normalized2  M -factor of a coherent circular DHB on propagation in turbulent atmosphere for different values of wavelength O with 0 20 , Figs. 5-7 that the normalized2  M -factor of a coherent circular DHB are closely related to its initial beam parameters, and its value increases slower on propagation as its beam order N, scaling factor and wavelength O increases, which means a DHB with larger beam order, larger scaling factor and longer wavelength O is less affected by the atmospheric turbulence.#114266 -$15.00USD Received 14 Jul 2009; revised 14 Aug 2009; accepted 8 Sep 2009; published 14 Sep 2009 (C) 2009 OSA 28 September 2009 / Vol.17, No. 20 / OPTICS EXPRESS 17353

Fig. 8 . 2 M
Fig. 8. Normalized 2 M -factors of coherent Gaussian beam, circular flat-topped beam and circular DHB on propagation in turbulent atmosphere.

Fig. 9 . 2 M
Fig. 9. Normalized 2 M -factor of a partially coherent circular DHB on propagation in turbulent atmosphere for different values of the initial transverse coherence width g V Equation (25) can also be used to calculate the 2 M -factor of coherent Gaussian (N = 1 and p = 0) or flat-topped beam (N>1 and p = 0).We calculate in Fig. 8 the normalized 2 M -factors of coherent Gaussian beam, circular flat-topped beam and circular DHB on propagation in turbulent atmosphere with 2 1 5 2 / 3 10 n C m , 0 20 w mm , 0 10 l mm and 632.8nmO .One finds from Fig. 8 that the normalized 2M -factor of flat-topped beam is larger than that of a Gaussian beam at short propagation distance (z<2.5km),but is smaller than that of a Gaussian beam at a long propagation distance, which means a flat-topped beam has advantage over a Gaussian beam for long-distance free-space optical communications.We also note that normalized2  M -factor of a DHB is always smaller than that of Gaussian and flat-topped beams at any propagation distance except at z = 0, which means a DHB is less affected by the atmospheric turbulence than Gaussian and flat-topped beams.Our results agree well with those reported in[33], where we found that a DHB has advantage over Gaussian and flattopped beams for overcoming the destructive effect of atmospheric turbulence from the aspect of scintillation.The results presented in this paper will be useful in long-distance free-space optical communicationsWe now turn to calculations relating to the 2 M -factor of a partially coherent circular DHB on propagation in turbulent atmosphere.Our numerical results (not shown here to save space) show that the dependence of the normalized2  M -factor of a partially coherent circular DHB on the parameters ( 2 19) O ) is similar to that of a coherent circular DHB.We calculate in Fig.9the normalized 2 M -factor of a partially coherent circular DHB on propagation in turbulent #114266 -$15.00USD Received 14 Jul 2009; revised 14 Aug 2009; accepted 8 Sep 2009; published 14 Sep 2009 (C) 2009 OSA 28 September 2009 / Vol.17, No. 20 / OPTICS EXPRESS 17355 atmosphere for different values of the initial transverse coherence width g