Abstract
Analytic expression is derived for the cross-spectral density matrix of a stochastic electromagnetic beam truncated by a slit aperture and passing through the turbulent atmosphere. The new formula can be used in study of the modulation in the spectral degree of polarization of the electromagnetic Gaussian Schell-model beam on propagation. We find that the spectral degree of polarization in the output plane can be directly controlled by the width of the slit aperture. The effect of polarization shaping is also illustrated by numerical examples.
©2009 Optical Society of America
1. Introduction
The propagation of light beams through the turbulent atmosphere is important in many applications such as tracking, remote sensing, and optical communication. After the unified theory of coherence and polarization presented by Wolf [1], the changes in the spectral density, the spectral degree of coherence, and the spectral degree of polarization of stochastic electromagnetic beams propagating in deterministic or random media can be determined by the 2×2 cross-spectral density matrix [2]. Since then the statistical properties of stochastic electromagnetic beams passing through the turbulent atmosphere have been extensively investigated [3–6]. We have also studied the propagation of stochastic anisotropic electromagnetic beams [7] and stochastic electromagnetic array beams [8] through the turbulent atmosphere.
In practical instance as well as optical communication through the turbulent atmosphere, apertures of different size or shape are usually present. The cosh-Gaussian beam diffracted by a circular aperture and passing through the turbulent atmosphere has been studied [9] and the effect of turbulence on the quality of the apertured beam has also been discussed [10]. The apertures may directly influence the propagation properties of light beams but the polarization modulation effect of truncated stochastic electromagnetic beams on propagation through the turbulent atmosphere has not, to the best of our knowledge, been previously adequately addressed.
In this paper, we introduce a slit aperture function with the help of the tensor method and study the propagation of an electromagnetic Gaussian Schell-model beam truncated by a slit aperture and passing through the turbulent atmosphere. The slit aperture is considered as a modulator and the changes in spectral degree of polarization are illustrated.
2. Theoretical analysis
Suppose the beam in the source plane is truncated by a hard-aperture of a slit shape as shown in Fig. 1. It is known that the aperture function can be approximated by a multi-Gaussian series [11], so the slit aperture function in x and y dimensions can be expressed as
where N is the order of the multi-Gaussian series, βx is the spot size of the Gaussian function component. In two-dimensional case, the slit aperture function can be found by using tensor method as
where
k = 2π/λ, is the wave number, λ is the wavelength. The width of the slit aperture along x direction is 2σx with αx = Nβx · ρ′T 1 = (x′1, x′2) and n T 1 = (n 1, 0), T denotes matrix transposition. Figure 2 shows that Eq. (2) is an appropriate approximate expression for the slit aperture function with N = 10 adopted in the numerical calculations.
We have studied the propagation of stochastic electromagnetic beam through the turbulent atmosphere [7], if the beam in the source plane truncated by a slit aperture the cross-spectral density matrix in the output plane can be given by
where ρ′T 12 = (ρ′T 1, ρ′T 2) = (x′1, y′1, x′2, y′2) and ρ T 12 = ρ′T 12 = (ρ T 1, ρ T 2) = (x 1, y 1, x 2, y 2) are four-dimensional vectors, Det means the determinant,
where I is a 2×2 unit matrix, ρ 0 = (0.545C 2 n/k 2 z)-3/5 is the coherence length of a spherical wave propagating in the turbulent medium and C 2 n is the structure parameter of the refractive index. Āp (ρ′12) = A * p (ρ′1)Ap(ρ′2) is a four-dimensional aperture function of the form
where
and n T 12 = (n T 1, n T 1) = (n 1, 0, n 2, 0)
Assume that a beam is generated by an electromagnetic Gaussian Schell-model source. The cross-spectral density matrix in the source plane is given in tensor form as [7]
where M′-1 ij is a 4×4 complex matrix as
The coefficients Ai, Aj, Aj, Bij and the variances σi, σj, δij are independent of position but may depend on frequency.
On substituting from Eqs. (6) and (8) into Eq. (4), and after a vector integration, we obtain the analytic expression for the cross-spectral density matrix of a stochastic electromagnetic beam truncated by a slit aperture and passing through the turbulent atmosphere as
where I is a 4×4 unit matrix. If αx = ∞, Eq. (10) represents the cross-spectral density function of the beam passing through the turbulent atmosphere without aperture and reduces to the form
(11) It is in agreement with Eq. (11) shown in Ref. [7].
3. Numerical calculations and discussions
In this section we are going to study the changes in the spectral degree of polarization of a stochastic electromagnetic beam truncated by a slit aperture and passing through the turbulent atmosphere. The spectral degree of polarization at the point (ρ, z) with ρ T 12 = (ρ T, ρ T) is defined by the formula [1, 2]
In Fig. 3, the stochastic electromagnetic beam is truncated by slit apertures of the same width, but the structure parameters are different. In the condition of the beam propagating through the turbulent atmosphere, the spectral degrees of polarization in the far field tend to an identical value, which is different from the condition of propagating in free space with C 2 n =0 and is independent of the local strength of atmospheric turbulence. The similar phenomenon has been shown in Fig. 4(b) of Ref. [7] for the stochastic anisotropic electromagnetic beam propagating in non-truncated case. Now the effect of slit aperture on the changes in polarization of the beam on propagation is additionally illustrated.
In Fig. 4, the stochastic electromagnetic beam is truncated by slit apertures of different widths and passing through the turbulent atmosphere with the same structure parameter. The dashed curve, dash-dot curve and dotted curve are calculated from Eqs. (10) and (12), and the solid curve is calculated from Eqs. (11) and (12). When the width is sufficiently large, it is similar to that in the limiting non-truncated case. The figure also shows that we can modulate the spectral degree of polarization in the output plane especially in the far-field by simply controlling the width of the slit aperture.
In Fig. 5, we choose the same structure parameter of the turbulent atmosphere and the same output plane of z = 1 km, but the beam is truncated by different slit apertures. When the width is large the distribution of the spectral degree of polarization is axially symmetrical shown in Fig. 5(a). As the width decreasing, the distribution of the spectral degree of polarization becomes more changeless in x direction shown in Fig. 5(b) and it can be regarded as uniformity in this direction shown in Fig. 5(c) when the width is sufficiently small. This phenomenon is induced by the diffraction effect of slit aperture and we consider it as a convenient method for shaping the spectral degree of polarization of the stochastic electromagnetic beam on propagation.
4. Conclusions
We conclude by saying that we have found a way of modulating the spectral degree of polarization of the stochastic electromagnetic beam on propagation through the turbulent atmosphere. The two-dimensional slit aperture function is given and the analytic expression for the cross-spectral density matrix is obtained. By virtue of numerical calculations, we have shown that the spectral degree of polarization of the electromagnetic Gaussian Schell-model beam in the output plane can be modulated by controlling the width of the slit aperture and the effect of polarization shaping is also illustrated.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (10874150 and 60478041) and the Program for New Century Excellent Talents in University (NCET-07-0760).
References and links
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