Propagation Factor of a Stochastic Electromagnetic Gaussian Schell-Model Beam

Analytical formula is derived for the propagation factor (known as 2 M -factor) of a stochastic electromagnetic Gaussian Schell-model (EGSM) beam in free space and in turbulent atmosphere. In free space, the 2 M -factor of an EGSM beam is mainly determined by its initial degree of polarization, r.m.s. widths of the spectral densities and correlation coefficients, and its value remains invariant on propagation. In turbulent atmosphere, the 2 M -factor of an EGSM beam is also determined by the parameters of the turbulent atmosphere, and its value increases on propagation. The relative 2 M -factor of an EGSM beam with lower correlation factors, larger r.m.s. widths of the spectral densities and longer wavelength is less affected by the atmospheric turbulence. Under suitable conditions, an EGSM beam is less affected by the atmospheric turbulence than a scalar GSM beam (i.e. fully polarized GSM beam). 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Over the past several decades, many works have been carried out concerning the propagation of various laser beams through the turbulent atmosphere due to their wide applications, e.g. in free-space optical communication, laser radar, atmospheric imaging systems and remote sensing, and it has been found that the behavior of a laser beam in a turbulent atmosphere is closely related to its initial beam profile, coherence and polarization properties .Behavior of the statistical properties including the averaging intensity, coherence, degree of polarization, state of polarization and scintillation index of an EGSM beam, propagating in turbulent atmosphere has been studied in details [40][41][42][43][44][45][46][47][48].It was found that the EGSM beams may have reduced levels of scintillations compared to the scalar GSM beams [47], which is useful for free-space optical communications and laser radar systems (LIDARs) [48].To our knowledge no results have been reported up until now on the propagation factor of EGSM beams passing through the turbulent atmosphere.
The propagation factor (also known as the 2 M -factor) proposed by Siegman is a particularly useful property of an optical laser beam, and plays an important role in the characterization of beam behavior on propagation.Several methods have been developed to obtain the propagation factors of the laser beams in free space [49][50][51].The definition of propagation factor in terms of second-order moments of the Wigner distribution function has been adopted widely for characterizing laser beams [52][53][54].Recently, this method was extended for the analysis of the propagation factor of a laser beam traveling in the turbulent atmosphere [55][56][57].The purpose of this paper is to investigate, based on this recently extended method, the M 2 -factor of an EGSM beam propagating in the turbulent atmosphere, by deriving the explicit expression for the propagation factor of an EGSM beam for both free space propagation and atmospheric propagation, and exploring them comparatively.
(2) Within the validity of the paraxial approximation, the propagation of the trace of the crossspectral density matrix of an EGSM beam in the turbulent atmosphere can be studied with the help of the following generalized Huygens-Fresnel integral [26,27] ( , ; ) , 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 and H ȡ ȡ z in Eq. ( 3) 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 [27].
Equation ( 3) can be expressed in the following alternative form [55-57] where is the position vector in spatial-frequency domain.We can express tr ( '', ;0) The Wigner distribution function of an EGSM beam in turbulent atmosphere can be expressed in terms of the trace of the cross-spectral density matrix by the formula [55] 2 2 tr tr ( , ; ) ( ) ( , ; )exp( ) , 2 where ( , ) 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. ( 6) and ( 7) into Eq.( 8) and applying following integral formula, we obtain (after tedious integration) the expression Based on the second-order moments of the Wigner distribution function, the 2  M -factor of an EGSM beam can be defined as follows [49][50][51][52][53][54][55][56][57] where After substituting from Eqs. ( 14)-( 17) into Eq.( 11) we obtain the following expression for the 2  M -factor of an EGSM beam travelling in turbulent atmosphere Under the condition of ( ) 0 n N ) , Eq. ( 19) reduces to following expression for the free space 2 M -factor of an EGSM beam One finds from Eq. ( 20) that the 2  M -factor of an EGSM beam in free space is independent of z, so its value remains invariant on propagation, while being closely determined by parameters , and .20) reduces to the following expression for the 2 M -factor of a scalar GSM beam (i.e., fully polarized GSM beam) Equation ( 21) agrees well with the result reported in [51].Under the condition of DD G f , the right side of Eq. ( 21) reduces to unity, coinciding with the 2 M -factor of a coherent scalar Gaussian beam propagating in free space [49].

Numerical examples
In this section we study the 2 M -factor of an EGSM beam in free space and in turbulent atmosphere numerically.In the following examples, we choose the Tatarskii spectrum for the spectral density of the index-of-refraction fluctuations, which is expressed as [27] 2 2 1 1/3 2 ( ) 0.033 exp , where 2 n C is the structure constant of the turbulent atmosphere, 0 5.92 / m l N with 0 l being the inner scale of the turbulence.Substituting from Eq. ( 22) into Eq.( 18), we obtain the formula Substituting Eq. ( 23) into Eq.( 19), we now can calculate the 2 M -factor of an EGSM beam numerically.
For the convenience of analysis, we only consider the EGSM beam that is generated by an EGSM source whose cross-spectral density matrix is diagonal, i.e. of the form ( , ;0) 0 ( , ;0) .0 ( , ; 0 ) The degree of polarization of the initial source beam at point ' ȡ can be expressed as follows [40][41][42][43][44][45][46][47][48]     or decreases with increase of the degree of polarization for the case of , the EGSM beam reduces to a scalar GSM beam with 0 ( ;0) 1 P ' ȡ .

Fig. 1 .
Fig. 1.Dependence of the degree of polarization at source plane on y A with 1 x A

Figure 1
shows the dependence of the degree of polarization in the source plane on y A .It is clear from Fig.1that the degree of polarization in the source plane varies as the value of y A changes, any nonzero 0 P can be achieved either for y

Fig. 2 . 2 MFig. 3 .of the 2 M 2 M 2 M
Fig. 2. Dependence of the 2 M -factor of an EGSM beam in free space on its degree of polarization at source plane for two different cases (a) y

Figure 3 shows the dependence of the 2 M. 2 M 2 M
Figure3shows the dependence of the2  M -factor of an EGSM beam in free space on the r.m.s width ( x V ) of the spectral density along x direction with y

Fig. 4 . 2 M 2 M
Fig. 4. Dependence of the 2 M -factor of an EGSM beam in free space on the r.m.s width is the r.m.s width of the spectral density along D direction; xx G , yy G and xy G are the r.m.s.widths of auto-correlation functions of the x component of the field, of the y component of the field and of the mutual correlation function of x and y field components, respectively,