Abstract
Analytical formula is derived for the propagation factor (known as-factor) of a stochastic electromagnetic Gaussian Schell-model (EGSM) beam in free space and in turbulent atmosphere. In free space, the -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 -factor of an EGSM beam is also determined by the parameters of the turbulent atmosphere, and its value increases on propagation. The relative -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). Our results will be useful in long-distance free-space optical communications.
©2010 Optical Society of America
1. Introduction
In 1994 it was found that the degree of polarization of a stochastic electromagnetic beam may change on propagation in vacuum, and such changes depend on the coherence properties of the source of the beam [1]. Electromagnetic Gaussian Schell-model (EGSM) beam [2–4] was introduced as an extension of scalar GSM beam [5–8]. Due to its importance in the theories of coherence and polarization of light, numerous theoretical and experimental papers relating to stochastic electromagnetic beams have been published in the past several years [9–24].
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 [25–46]. 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 [40–48]. 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 -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 [49–51]. The definition of propagation factor in terms of second-order moments of the Wigner distribution function has been adopted widely for characterizing laser beams [52–54]. Recently, this method was extended for the analysis of the propagation factor of a laser beam traveling in the turbulent atmosphere [55–57]. The purpose of this paper is to investigate, based on this recently extended method, the M2-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. Theory
The second-order statistical properties of an EGSM beam can be characterized by the cross-spectral density matrix specified at any two points with position vectors and in the source plane with elements [2–4]
where is the r.m.s width of the spectral density along α direction; , and 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, is the complex correlation coefficient between the x and y components of the electric field; parameters, , and are independent of position and, in our analysis, of frequency. The nine real parameters, , , , , , , and entering the general model are shown to satisfy several intrinsic constraints and obey some simplifying assumptions (e.g. the phase difference between the x- an y-components of the field is removable, i.e. ) [9,14,18]. The trace of the cross-spectral density matrix of an EGSM beam is expressed as [2–4,9,10]Within the validity of the paraxial approximation, the propagation of the trace of the cross-spectral 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 is the wave number withλ being the wavelength. In Eq. (3) we have used the following sum and difference vector notationwhere are two arbitrary points in the receiver plane, perpendicular to the direction of propagation of the beam and The term in Eq. (3) is the contribution from the atmospheric turbulence expressed aswhere is the Bessel function of zero order, 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 of an EGSM beam asThe 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]
where denotes an angle which the vector of interest makes with the z-direction, and 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 expressionwith , , ().Based on the second-order moments of the Wigner distribution function, the -factor of an EGSM beam can be defined as follows [49–57]
whereSubstituting Eq. (10) into Eqs. (12) and (13), we obtain (after tedious integration) the formulas for the moments
withAfter substituting from Eqs. (14)-(17) into Eq. (11) we obtain the following expression for the -factor of an EGSM beam travelling in turbulent atmosphere
Under the condition of , Eq. (19) reduces to following expression for the free space -factor of an EGSM beam
One finds from Eq. (20) that the -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 Under the condition of or , Eq. (20) reduces to the following expression for the -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 , the right side of Eq. (21) reduces to unity, coinciding with the -factor of a coherent scalar Gaussian beam propagating in free space [49].3. Numerical examples
In this section we study the -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]
where is the structure constant of the turbulent atmosphere, with being the inner scale of the turbulence. Substituting from Eq. (22) into Eq. (18), we obtain the formulaSubstituting Eq. (23) into Eq. (19), we now can calculate the-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
The degree of polarization of the initial source beam at pointcan be expressed as follows [40–48]Under the condition of or , the EGSM beam reduces to a scalar GSM beam with .In the following numerical examples, we set and unless stated otherwise. In this case, the polarization properties are uniform across the source plane with Figure 1 shows the dependence of the degree of polarization in the source plane on . It is clear from Fig. 1 that the degree of polarization in the source plane varies as the value ofchanges, any nonzero can be achieved either for or for .
First we study the properties of the -factor of an EGSM beam in free space. We calculate in Fig. 2 the dependence of the -factor of an EGSM beam on its degree of polarization in the source plane (z = 0) for two different cases with . One finds from Fig. 2 that the -factor of an EGSM beam is closely determined by its degree of polarization at z = 0. Its value increases as the degree of polarization increases for the case of or decreases with increase of the degree of polarization for the case of . This is caused by the fact that the contribution of the element to the -factor dominates that of the element for the case of , and the contribution of the element plays a dominant role otherwise.
Figure 3 shows the dependence of the -factor of an EGSM beam in free space on the r.m.s width () of the spectral density along x direction with , , , , . From Fig. 3 it is clearly seen that the value of the -factor of an EGSM beam in free space increases as its r.m.s. width of the spectral density increases. Figure 4 shows the dependence of the -factor of an EGSM beam in free space on the r.m.s width () of auto-correlation functions of the x component of the field with , , , . One finds from Fig. 4 that the value of the -factor of an EGSM beam decreases as the correlation factors and decrease, due to the fact that the spectral degree of coherence decreases with the decrease of the correlation factors.
In what follows we study the properties of the normalized-factor of an EGSM beam on propagation in turbulent atmosphere. It is clear from Eq. (19) that the -factor of an EGSM beam is now determined by both the parameters of the beam and of the turbulence together. We calculate in Fig. 5 the normalized -factor of an EGSM beam on propagation in turbulent atmosphere for different values of the initial degree of polarization with , , , , . As seen from Fig. 5, unlike its propagation-invariant properties in free-space, the normalized -factor increases on propagation in turbulent atmosphere, which means the beam quality of an EGSM beam degrades. It is clear from Fig. 5(a) that the value of the normalized -factor increases more rapidly as the initial degree of polarization decreases for the case of , which means that the beam quality of a scalar GSM beam is less affected by the atmospheric turbulence than that of an EGSM beam. From Fig. 5(b) we conclude that for the case of , the beam quality of an EGSM beam with low initial degree of polarization is less affected by the atmospheric turbulence than that of an EGSM beam with large degree of polarization or that of a scalar GSM beam.
Figure 6 shows the normalized -factor of an EGSM beam on propagation in turbulent atmosphere for different initial r.m.s. correlations and with, , , , . One finds from Fig. 6 that the value of the normalized -factor increases slower on propagation as the initial correlation factors decreases, implying that an EGSM beam with low initial spectral degree of coherence is less affected by the atmospheric turbulence. Figure 7 shows the normalized -factor of an EGSM beam on propagation in turbulent atmosphere for different initial r.m.s. widths and of the spectral densities with , ,, , , , . Figure 8 shows the normalized -factor of an EGSM beam on propagation in turbulent atmosphere for different values of wavelengthλ with , ,, , , , . The results in Fig. 7 and Fig. 8 lead to the conclusion that the EGSM beam with larger r.m.s. width of the spectral densities and longer wavelength λ is less affected by the atmospheric turbulence.
The parameters of turbulence also affect the evolution properties of the normalized -factor in turbulent atmosphere. We calculate in Fig. 9 the normalized -factor of an EGSM beam on propagation using different values of the structure constant () of the turbulent atmosphere with , , , , and . Figure 10 shows the normalized -factor of an EGSM beam on propagation in turbulent atmosphere for different values of inner scale of the turbulence () with , , , , and . As seen in Fig. 9 and Fig. 10, the normalized -factor increases more rapidly as increases or decreases.
4. Conclusion
We have derived the analytical formula for the -factor of an EGSM beam valid for both free space and atmospheric propagation by means of the extended Huygens-Fresnel integral and the second-order moments of the Wigner distribution function. Our numerical results show that the initial degree of polarization, the r.m.s. width of the spectral densities and the r.m.s. correlation coefficients together determine the -factor of an EGSM beam in free space, whose value is independent of the propagation distance. In turbulent atmosphere, the parameters of the turbulence also affect the -factor of this class of beam on propagation. Furthermore, the EGSM beam with lower degree of polarization, 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. Our results may find applications in long-distance free-space optical communications.
Acknowledgments
Yangjian Cai acknowledges the support by the National Natural Science Foundation of China under Grant No. 10904102, the Foundation for the Author of National Excellent Doctoral Dissertation of PR China under Grant No. 200928, the Natural Science of Jiangsu Province under Grant No. BK2009114 and the Huo Ying Dong Education Foundation of China under Grant No. 121009. O. Korotkova's research is funded by the US AFOSR (grant FA 95500810102) and US ONR (grant N0018909P1903).
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