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We derive an analytical expression for the propagation factor (known as M 2-factor) of electromagnetic concentric rings Schell-model (EM CRSM) beams in non-Kolmogorov turbulence by utilizing the extended Huygens–Fresnel diffraction integral formula and the second-order moments of the Wigner distribution function (WDF). Our results show that the EM CRSM beam has advantage over the scalar CRSM beam for reducing the turbulence-induced degradation under suitable conditions. The EM CRSM beam with multi-rings far-fields in free space is less affected by the turbulence than the one with dark-hollow far-fields or the electromagnetic Gaussian Schell-model (EGSM) beam. The dependence of the M 2-factor on the beam parameters and the turbulence are investigated in detail.
Since the important application in free-space and atmospheric optical communication, the propagation properties of laser beams through atmospheric turbulence have been widely investigated.[1–11] Studies show that partially coherent beams on propagation in atmospheric turbulence are less affected than spatially coherent beams. Superposition rule for random sources presented in Ref. [12] has provided an effective approach to generate new genuine sources. Those sources can radiate beam-like fields with special intensity profiles, such as self-focusing far-fields,[13–15] flat-top far-fields,[16] ring-shaped far-fields,[17, 18] four-petal far-fields,[19, 20] lattice-like far-fields,[21, 22] etc. The behavior of these scalar beams (regardless of the polarization properties) in free space and atmospheric turbulence are closely related to their spatial correlation functions.[13–22] Based on the unified theory of coherence and polarization, the electromagnetic partially coherent beams can be described by the cross-spectral density (CSD) matrix in space-frequency domain,[1] and can be obtained as an extension of the corresponding scalar partially coherent beams. However, due to the constraint of non-negative definiteness of the CSD matrix,[23] only several kinds of electromagnetic partially coherent Schell-model beams have been proposed, such as electromagnetic Gaussian Schell-model (EGSM) beam,[24] electromagnetic multi-Gaussian Schell-model (EM MGSM) beam,[25, 26] electromagnetic cosine-Gaussian Schell-model (EM CGSM) beam,[27] electromagnetic sinc Schell-model (EM SSM) beam,[28] which all demonstrate special distributions of intensity and polarization properties.
The propagation factor (known as M 2-factor) of laser beams is an important parameter for characterizing the beam quality in many practical applications.[29] Studies of the M 2-factor of several partially coherent Schell-model beams in atmospheric turbulence[30–35] indicate that the EGSM beam and partially coherent Schell-model beams with nonconventional correlation functions are all less affected by the turbulence than the classical GSM beams, and the beam quality is closely related to the spatial correlation function and the intensity profile.
Recently, an electromagnetic concentric rings Schell-model (EM CRSM) beam has been proposed.[36] The intensity profile of this beam demonstrates multi-rings or dark-hollow (single ring) far-fields by the modulating of the special correlation function, which is a sum of series of cosine function. Since the properties of coherence and polarization have been studied, this paper tends to investigate the beam quality of EM CRSM beams on propagation in non-Kolmogorov turbulence. By employing the extended Huygens-Fresnel diffraction integral formula and second-order moments of the WDF, we derive the analytical expression for the M 2-factor of EM CRSM beams propagating in non-Kolmogorov turbulence. By which we numerically analyzed the influence of beam parameters and turbulence exert on the M 2-factor of EM CRSM beams, and the classical GSM beam as well for a reference.
The second-order statistical properties of an EM CRSM beam can be characterized by the
(1) |
(2) |
Within the validity of the paraxial approximation, the trace of the CSD of an EM CRSM beam propagating in non-Kolmogorov turbulence can be calculated by the following generalized Huygens–Fresnel integral[37]
(3) |
(4) |
(5) |
(6) |
Substituting Eqs. (
(7) |
(8) |
Utilizing Eqs. (
(9) |
Because the spectral degree of polarization is a constant, the polarization properties across the initial plane are uniform. Under the condition of
Figure
The WDF of an EM CRSM beam in non-Kolmogorov turbulence can be expressed in terms of the trace of the CSD as follows:[30]
(10) |
The M 2-factor is regarded as a beam quality factor in many practical applications, which is defined by the second-order moments of the WDF as follows:[39, 40]
(11) |
(12) |
(13) |
For partially coherent Schell beams, the propagation of the second-order moments of the WDF in the turbulence obey the general formulae as follows:[41]
(14) |
(15) |
(16) |
Substituting Eqs. (
(17) |
(18) |
(19) |
For the case of
(20) |
When
(21) |
In this section, we will investigate the M
2-factor of an EM CRSM beam in non-Kolmogorov turbulence by applying the formulae derived in above section. Without loss of generality, the parameters of the EM CRSM beam are the same as those given in Fig.
Firstly, we study the influence of the initial degree of polarization on the normalized M
2-factor of the EM CRSM beam through the non-Kolmogorov turbulence. Figure
Figure
The effect of initial spatial correlation factors δ
xx
, δ
yy
, beam width σ, and wavelength λ of EM CRSM beams on the normalized M
2-factor is considered in Figs.
The influence of wavelength λ on the normalized M
2-factor of EM CRSM beams in the non-Kolmogorov turbulence is shown in Fig.
In the following section, we analyze the influence of the turbulent parameters on the normalized M
2-factor of an EM CRSM beam in the non-Kolmogorov turbulence. Figure
Figure
The effect of outer scale L
0 and inner scale l
0 of the non-Kolmogorov turbulence on the normalized M
2-factor of the EM CRSM beam is shown in Fig.
The analytical formula for the M 2-factor of EM CRSM beams on propagation in the non-Kolmogorov turbulence is derived. By which, the dependence of the normalized M 2-factor on beam parameters and non-Kolmogorov turbulence have been investigated numerically. We found that EM CRSM beams are less affected by the turbulence than the scalar CRSM beam under suitable conditions. The beam quality of the EM CRSM beam with multi-rings far-fields in free space is better than the one with dark-hollow far-fields or the EGSM beam, especially for the EM CRSM beam with larger radius of each concentric ring. Furthermore, the effect of the turbulence on the EM CRSM beam can be reduced by choosing lower initial degree of polarization P, longer wavelength λ, larger beam width σ, or weaker turbulence. Our results will be useful for the practical applications in free-space and atmospheric optical communications.
[1] | |
[2] | |
[3] | |
[4] | |
[5] | |
[6] | |
[7] | |
[8] | |
[9] | |
[10] | |
[11] | |
[12] | |
[13] | |
[14] | |
[15] | |
[16] | |
[17] | |
[18] | |
[19] | |
[20] | |
[21] | |
[22] | |
[23] | |
[24] | |
[25] | |
[26] | |
[27] | |
[28] | |
[29] | |
[30] | |
[31] | |
[32] | |
[33] | |
[34] | |
[35] | |
[36] | |
[37] | |
[38] | |
[39] | |
[40] | |
[41] |