Electromagnetic sinc Schell-model beams and their statistical properties Zhangrong Mei,1,* and Yonghua Mao2 1
2
Department of Physics, Huzhou University, Huzhou 313000, China Department of Physics, Zhejiang University of Technology, Hangzhou 310014, China *[email protected]
Abstract: A class of electromagnetic sources with sinc Schell-model correlations is introduced. The conditions on source parameters guaranteeing that the source generates a physical beam are derived. The evolution behaviors of statistical properties for the electromagnetic stochastic beams generated by this new source on propagating in free space and in atmosphere turbulence are investigated with the help of the weighted superposition method and by numerical simulations. It is demonstrated that the intensity distributions of such beams exhibit unique features on propagating in free space and produce a double-layer flat-top profile of being shape-invariant in the far field. This feature makes this new beam particularly suitable for some special laser processing applications. The influences of the atmosphere turbulence with a non-Kolmogorov power spectrum on statistical properties of the new beams are analyzed in detail. ©2014 Optical Society of America OCIS codes: (030.1640) Coherence; (010.1300) Atmospheric propagation; (010.1330) Atmospheric turbulence; (260.5430) Polarization.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.
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#216580 - $15.00 USD Received 8 Jul 2014; revised 29 Aug 2014; accepted 29 Aug 2014; published 10 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.022534 | OPTICS EXPRESS 22534
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1. Introduction Schell-model sources have played a role of utmost importance in optical coherence theory since it was introduced in 1967 [1]. Among the most classic models is the Gaussian Schellmodel source which comes with the degree of coherence of Gaussian distribution [2]. Some other augmented models for scalar random sources have been recently introduced, in which the classic Gaussian degree of coherence is modulated by a well-behaved function, such as Multi-Gaussian Schell-model sources [3], Bessel-Gaussian and Laguerre-Gaussian Schellmodel sources [4], cosine-Gaussian Schell-models sources [5], etc. Several random scalar sources producing far field with rectangular intensity profiles have also been introduced by modeling the source degree of coherence with the help of two one-dimensional Schell-model sources [6–8]. These scalar sources and beams are specified by a single correlation function, while the electromagnetic random beams are characterized by the 2 × 2 cross-spectral density (CSD) matrices. It has been shown that the changes in the propagation of the spectral density, the spectral degree of coherence, and the spectral degree of polarization of a random electromagnetic beam can be determined from knowledge of a CSD matrix [9]. This matrix has already been used in studies of the behaviors of random electromagnetic beams propagating through turbulent atmosphere [10,11], an optical resonator [12] and optical focusing systems [13,14], etc. For an electromagnetic stochastic beam, the spatial correlation functions must satisfy additional restrictions, leading to restrictions for source parameters known as the realizability conditions [15]. Roychowdhury and Korotkova had derived realizability conditions for electromagnetic Gaussian Schell-model (EM GSM) sources [16]. Recently, a basic theory on devising genuine CSD matrices for an electromagnetic stochastic beam was exploited and extended [17,18]. Based on this method, several models for scalar random sources have been extended to electromagnetic domain, such as the electromagnetic Multi-Gaussian Schell-model sources [19], electromagnetic non-uniformly correlated sources [20,21], electromagnetic cosine-Gaussian Schell-models sources [22], and so on. The degree of coherence of all the known Schell-model sources is Gaussian or based on the Gaussian degree of coherence. Recently, we introduced a new random source class with sinc Schell-model (SSM) correlated function [23], whose degree of coherence in the source plane presents sinc function profile rather than the traditional Gaussian distribution. The
#216580 - $15.00 USD Received 8 Jul 2014; revised 29 Aug 2014; accepted 29 Aug 2014; published 10 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.022534 | OPTICS EXPRESS 22535
beams generated by SSM sources have been found to form a stable single layer flat profile in the far filed. In this paper, we use a scalar model proposed in [23] as a building for developing electromagnetic sources with SSM correlations, terming the novel class of beams the electromagnetic sinc Schell-model (EM SSM) beams, in which all the correlations are prescribed with the help of the scalar SSM distributions. As we illustrate by numerical examples, the EM SSM source can produce a double-layer flat-top intensity distribution and remain shape-invariant in the far field. It is not like the known partially coherent flat-top beams produce a single layer flat-top intensity profile and disappear gradually in propagation and becomes Gaussian in the far field. The double-layer flat-top intensity profile may be of use in applications dealing with laser processing. After extending the scalar SSM sources to the full electromagnetic domain and deriving the realizability conditions, our major task in this study is to explore the behavior of the main statistical properties for EM SSM beams propagating in free space and atmosphere turbulence. Usually, it is difficult to obtain a general analytical formula for the propagated CSD matrix of EM SSM beams. However, the integral structure of the elements of the CSD matrix constructed through the mode superposition retains its form on propagation. The elementary modes change on propagation, while the weight functions remain unchanged [18,20]. Therefore, we can consider separately the propagation of the kernel functions and apply the weighted superposition method to study the statistical properties of EM SSM beams by a set of numerical examples. 2. Electromagnetic Sinc Schell-model source The second-order correlation properties of a statistically stationary electromagnetic source at two points ρ1′ and ρ 2′ will be described by the 2 × 2 CSD matrix Wˆ (0) ( ρ1′, ρ 2′ ; ω ) , whose element Wαβ(0) ( ρ1′, ρ 2′ ; ω ) are given by [9] Wαβ(0) ( ρ1′, ρ 2′ ; ω ) = Eα* ( ρ1′; ω ) Eβ ( ρ 2′ ; ω ); (α = x, y; β = x, y ),
(1)
where Eα ( ρ ′; ω ) is the fluctuating electric field component along the α axis at point ρ ′ and the angular brackets denote an ensemble average. From now on, the angular frequency dependence of all the quantities of interest will be omitted but implied, and each of the symbols α and β as indexes can be either x and y . Recall the non-negative definiteness condition for CSD matrices to be physically realizable, it suffices to have an integral of the form [18] Wαβ(0) ( ρ1′, ρ 2′ ) = pαβ (v) H α∗ ( ρ1′, v) H β ( ρ 2′ , v)dv,
(2)
where pαβ (v) are the elements of the weigh matrix pˆ (v) . Hα ( ρ1′, v) and H β ( ρ 2′ , v) are two arbitrary kernels, they have plenty of choices and each choice is likely to lead to distinct classes of CSD matrices. A simple and significant class of the CSD matrices, leading to the classic Schell-model correlation sources, can be obtain by assigning to the kernels Hα ( ρ1′, v) and H β ( ρ 2′ , v) Fourier-like structure, viz., H α ( ρ1′, v) = Aατ ( ρ1′) exp(−2π iv ρ1′),
(3)
H β ( ρ 2′ , v) = Aβ τ ( ρ 2′ ) exp(−2π iv ρ 2′ ),
(4)
where Aα and Aβ are the amplitude of the field component, τ ( ρ ′) is a profile function. Then, Eq. (2) lends to Wαβ(0) ( ρ1′, ρ 2′ ) = Aα Aβ τ ∗ ( ρ1′)τ ( ρ 2′ ) p αβ ( ρ1′ − ρ 2′ ).
(5)
#216580 - $15.00 USD Received 8 Jul 2014; revised 29 Aug 2014; accepted 29 Aug 2014; published 10 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.022534 | OPTICS EXPRESS 22536
where tilde symbol denotes the Fourier transform of pαβ . The choice of the element pαβ (v) of the matrix pˆ (v) defines a family of sources with different correlation functions. Let us suppose pαβ (v) to be of the form pαβ (v) = Bαβ δαβ rect(δαβ v),
(6)
where δ αβ are the characteristic source correlations which are the positive real constants. Bαβ =| Bαβ | e
iφαβ
is the single-point correlation coefficient. rect( x) is the rectangular function,
which equals 1 for x ≤ 1/ 2 and 0 otherwise. On substituting Eq. (6) into Eq. (5) and setting the Gaussian profile exp[− | ρ ′ |2 /(2σ 02 )] with the r.m.s. source width σ 0 for the function τ ( ρ ′) , one finds the explicit form of the CSD matrix elements: ρ ′2 + ρ ′2 Wαβ(0) ( ρ1′, ρ 2′ ) = Aα Aβ Bαβ exp − 1 2 2 2σ 0
ρ1′ − ρ 2′ . sin c δαβ
(7)
Equation (7) represents a new family of sources that may be Electromagnetic Sinc Schellmodel (EM SSM) sources. The CSD matrix of a physically realizable field must be quasi-Hermitian, i.e. that (0) (0) Wαβ ( ρ1′, ρ 2′ ) = Wβα ( ρ 2′ , ρ1′) . It is sufficient that the condition holds if Bxx = Byy = 1, | Bxy |=| Byx |, δ xy = δ yx .
(8)
Furthermore, the non-negative definiteness constraint for the genuine CSD matrices means that, the following inequalities pαα (v) ≥ 0,
(9)
pxx (v) p yy (v) − pxy (v) p yx (v) ≥ 0,
(10)
have to be satisfied for any v . We readily find from Eq. (6) that inequality (9) is always satisfied, and on substituting from Eq. (6) into inequalities (10) implies that it is satisfied if
δ xxδ yy rect(δ xx v)rect(δ yy v) ≥| Bxy |2 δ xy2 [rect(δ xy v)]2 .
(11)
Since function rect(ax) equals 1 for x ≤ 1 / (2a) and 0 otherwise, it is not difficult to find that the realizability condition is expressed by the following fork inequality: max{δ xx , δ yy } ≤ δ xy ≤
δ xxδ yy | Bxy |
.
(12)
3. Propagation laws for the beams generated by the new class of sources in linear random medium
Suppose that source (7) generates a beam-like field propagating into half-space z > 0 filled with turbulent atmosphere. According to the extended Huygens-Fresnel integral principle adjusted for propagation in linear random medium, the elements of the CSD matrix at two points ( ρ1 , z ) and ( ρ 2 , z ) in the same transverse plane are related to those in the source plane as [24–26] Wαβ ( ρ1 , ρ 2 , z ) = Wαβ(0) ( ρ1′, ρ 2′ )K ( ρ1′, ρ 2′ , ρ1 , ρ 2 , z )d 2 ρ1′d 2 ρ 2′ ,
(13)
#216580 - $15.00 USD Received 8 Jul 2014; revised 29 Aug 2014; accepted 29 Aug 2014; published 10 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.022534 | OPTICS EXPRESS 22537
where the propagation kernel K ( ρ1′, ρ 2′ , ρ1 , ρ 2 , z ) , depended on the Green’s function of the random medium, is given by 2
( ρ1 − ρ1′) 2 − ( ρ 2 − ρ 2′ ) 2 k K ( ρ1′, ρ 2′ , ρ1 , ρ 2 , z ) = exp −ik 2z 2π z (14) 2 2 ∞ π k z 2 2 3 ( ρ1 − ρ 2 ) + ( ρ1 − ρ 2 )( ρ1′ − ρ 2′ ) + ( ρ1′ − ρ 2′ ) κ Φ n (κ )dκ . × exp − 0 3
Here, k = 2π / λ , with λ being the wavelength. Φ n (κ ) is the one-dimensional power spectrum of fluctuations in the refractive index of the turbulent medium. On inserting Eq. (2) into Eq. (13) we obtain, after interchanging the orders of integrals, the expression Wαβ ( ρ1 , ρ 2 , z ) = pαβ (v) Hα* ( ρ1 , v, z ) H β ( ρ 2 , v, z )dv,
(15)
where H α* ( ρ1 , v, z ) H β ( ρ 2 , v, z ) = H α* ( ρ1′, v)H β ( ρ 2′ , v) K ( ρ1′, ρ 2′ , ρ1 , ρ 2 , z )d 2 ρ1′d 2 ρ 2′ . (16) It is seen from Eq. (15) that the structure of the elements of the CSD matrix constructed through the superposition rule specified by Eq. (2) retains its form on propagation through turbulent medium, the Wαβ can be expressed as an incoherent superposition of elementary modes H α weighted by the function pαβ . H α change on propagation and the weight function pαβ remain unchanged. On substituting from Eqs. (3), (4) and (14) into Eq. (15), after long integral calculations, we obtain, for any mode, the formula Hα* ( ρ1 , v, z ) H β ( ρ 2 , v, z ) =
Aα Aβ σ 02 w2 ( z )
k 2σ 02 k 2π 2 z ∞ 3 2 ik × exp − ( ρ12 − ρ 22 ) exp − + κ Φ n (κ )dκ ( ρ1 − ρ 2 ) 2 0 3 2z 4 z 2 ρ + ρ 2 2π zv kσ 0 kπ 2 z 2 × exp − 1 + −i − 3 k 2z 2
∞
0
κ 3 Φ n (κ )dκ ( ρ1 − ρ 2 )
2
(17) w2 ( z ) ,
where 4π 2 z 3 ∞ 3 (18) κ Φ n (κ )dκ . 3 0 k 2σ 02 To model the atmosphere turbulence, a non-Kolmogorov spectrum is used [26–28] w2 ( z ) = σ 02 +
z2
+
Φ n (κ ) = A(α )C n2 exp[−(κ 2 / κ m2 )] / (κ 2 + κ 02 )α / 2 ,
0 ≤ κ < ∞, 3 < α < 4, (19)
where C n2 is a generalized refractive-index structure parameter with units m3−α , κ 0 = 2π / L0 and κ m = c(α ) / l0 , L0 and l0 being the outer and the inner scale of turbulence, and c(α ) = [Γ(5 −
α 2
) A(α )
2π 1/ (α − 5) ] , 3
(20)
#216580 - $15.00 USD Received 8 Jul 2014; revised 29 Aug 2014; accepted 29 Aug 2014; published 10 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.022534 | OPTICS EXPRESS 22538
A(α ) = Γ(α − 1) ⋅
cos(απ / 2) , 4π 2
(21)
with Γ( x) being the Gamma function, the integral in Eqs. (17) and (18) becomes
∞
0
κ 3 Φ n (κ )dκ =
κ2 A(α ) 2 2 −α α κ2 Cn κ m β exp( 02 )Γ(2 − , 02 ) − 2κ 04 −α , 2(α − 2) 2 κm κm
(22)
where β = 2κ 02 − 2κ m2 + ακ m2 and Γ denotes the incomplete Gamma function. It follows from above derivation that the propagated kernel of the elements of CSD matrix on propagation due to random medium is only included in the integral term in the right side of Eqs. (17) and (18), thus, for the free space propagation, Eqs. (17) and (18) can be express as Hα* ( ρ1 , v, z ) H β ( ρ 2 , v, z ) = Aα Aβ σ 02
k 2σ 02 2 ik 2 2 ρ ρ ρ − ρ2 ) − − exp exp ( ) − 1 2 2 2 ( 1 w ( z) 2z 4z
(23)
2 2 ρ1 + ρ 2 2π zv ikσ 0 × exp − + − ( ρ1 − ρ2 ) w2 ( z ) , k 2 2z
where w2 ( z ) = σ 02 + z 2 (k 2σ 02 ). (24) Based on the weighted superposition (15) and the propagating formulae of the modes (17) and (23), the spectral density S , the spectral degree of coherence μ and the spectral degree of polarization P of the electromagnetic stochastic beam in the turbulent atmosphere and free space can be calculated by the expressions [9] S ( ρ , z ) = TrWˆ ( ρ , ρ , z ), 4DetWˆ ( ρ , ρ , z ) , [TrWˆ ( ρ , ρ , z )]2
(26)
TrWˆ ( ρ1 , ρ 2 , z ) , TrWˆ ( ρ1 , ρ1 , z )TrWˆ ( ρ 2 , ρ 2 , z )
(27)
P( ρ , z ) = 1 −
μ ( ρ1 , ρ 2 , z ) =
(25)
where Det and Tr stand for the determinant and the trace of the matrix. 4. The statistical properties of EM SSM beams in free space and non-Kolmogorov atmosphere turbulence
We will now illustrate the behavior of the statistical properties, including spectral density, the degree of coherence and the degree of polarization, of typical beams generated by source (7) on propagating in free space and atmosphere turbulence by a set of numerical examples. Without loss of generality, the values of source and medium parameters are chosen to be Ax = Ay = 1 , Bxy = 0.2 , σ 0 = 10mm , k = 107 m −1 , δ xx = 1mm , L0 = 1m , l0 = 1mm , otherwise specified in the text and captions.
#216580 - $15.00 USD Received 8 Jul 2014; revised 29 Aug 2014; accepted 29 Aug 2014; published 10 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.022534 | OPTICS EXPRESS 22539
A. Spectral density S
Figure 1 shows typical evolution of the spectral density S of an EM SSM beam with δ yy = 2.5mm in the transverse beam cross-sections at several distances z from the source plane on propagating in free space. One clearly sees that the Gaussian profile of source field gradually transforms into a double-layer flat-top distribution with increasing propagation distance. This is because the coherence properties of light field in the source plane are closely related to the propagation characteristics and the transverse intensity distribution of the far field. Two flat-top profiles correspond to the spectral densities’ distributions of x and y components of electric field with different r.m.s. correlation width. In order to demonstrate the dependence of the spectral density behavior on the degree of coherence, we plot in Fig. 2 its evolution in the transverse beam cross-section at the plane z = 1km in free space for different δ yy . Note that for the case δ yy = δ xx = 1mm the transverse beam cross-section is a single-layer flat-top profile, this is due to the degree of coherences of x and y components are the same, which results in two directions with the same far-field spectral density distributions. However, for the cases δ yy ≠ δ xx shown as Figs. 2(b) and 2(c), the far-field spectral density distributions of x and y directions are different, resulting in the appearance of double-layer profile, the lower layer corresponds to the x component and the higher layer corresponds to the y component. This is the same reason that the electromagnetic cosineGaussian Schell-model beams possess double-ring intensity profiles.
Fig. 1. Transverse distribution of the spectral density S of an EM SSM beam with δ yy = 2.5mm at several different propagation distances in free space. (a) z = 0m; (b) z =
250m; (c) z = 350m; (d) z = 500m.
#216580 - $15.00 USD Received 8 Jul 2014; revised 29 Aug 2014; accepted 29 Aug 2014; published 10 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.022534 | OPTICS EXPRESS 22540
Fig. 2. Transverse distribution of the spectral density S of EM SSM beams with different δ yy propagating in free space at the plane z=1km. (a) δ yy = 1mm ; (b) δ yy = 2.5mm ; (c)
δ yy = 5mm .
Fig. 3. Transverse distribution of the spectral density S of an EM SSM beam with δ yy = 2.5mm propagating in the atmosphere at the distance z=5km for different values of atmosphere parameters.
Figure 3 shows the transverse distribution of the spectral density S of an EM SSM beam with δ yy = 2.5mm propagating in the non-Kolmogorov turbulence atmosphere at the distance z = 5km for different values of turbulence parameters C n2 and α . The longitudinal orientations indicate the changes in the Kolmogorov turbulence as the values of α increase and the transverse orientations indicate the changes in the Kolmogorov turbulence as the
#216580 - $15.00 USD Received 8 Jul 2014; revised 29 Aug 2014; accepted 29 Aug 2014; published 10 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.022534 | OPTICS EXPRESS 22541
values of C n2 increase. One can see from Fig. 3 that, for the weakest turbulence the spectral density keeps the double-layer distribution as shown in Fig. 3(c), while for Figs. 3(a) and 3(f) with decreasing α and increasing C n2 the double-layer profile of spectral density is destroyed gradually. Finally, in Fig. 3(d) for substantially strong turbulence the upper layer and lower layer are blended completely and the beams’ intensity resembles Gaussian profile. Figures 3(b) and 3(e) show the spectral density distributions is destroyed the most when α = 3.1 , which just like other beams [22, 26]. B. Spectral degree of polarization P
We now turn our attention to the changes in the spectral degree of polarization of EM SSM beams in free space and non-Kolmogorov atmosphere turbulence. Figure 4 illustrates the typical distributions of the polarization properties for the most general case of partially correlated electric field components, as beam propagates at 1km in free space. Note that for the case δ xx = δ yy = δ xy the spectral degree of polarization does not change upon propagation. It is the same as the source plane and uniform for all the points at the any propagation plane, which was also shown to be critical for other beams. However, for other cases, unlike the degree of polarization of EM GSM beams to form an inverted Gaussian distribution in the far field, Figs. 3(b) and 3(c) clearly show the formation of an inverted flat distribution with prominent and uniform polarization distribution in the central region. The height of the prominent part increases with the increase of the coherence length.
Fig. 4. Transverse distribution of the degree of polarization of the beams with different δ yy and δ xy propagating in free space at the plane z=1km. (a) δ yy = δ xy = 1mm ; (b)
δ yy = δ xy = 2.5mm ; (c) δ yy = δ xy = 5mm .
Figure 5 shows the transverse distribution of the degree of polarization P of an EM SSM beam with δ yy = δ xy = 2.5mm at the propagation distance z = 5km in the turbulence atmosphere for different values of parameters C n2 and α . For the case Fig. 5(c) of the weakest turbulence, the distribution of the degree of polarization is similar to the case of free space. With the strengthening of turbulence, the steepness of edges gradually become decline and the flat central prominent region change into a Gaussian profile, as shown in Figs. 5(a), 5(f), 5(b) and 5(d). For the case Fig. 5(e) of the strongest turbulence, the central projection disappears and the distribution of the degree of polarization present the inverted Gaussian profile.
#216580 - $15.00 USD Received 8 Jul 2014; revised 29 Aug 2014; accepted 29 Aug 2014; published 10 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.022534 | OPTICS EXPRESS 22542
Fig. 5. Transverse distribution of the degree of polarization P of an EM SSM beam with δ yy = δ xy = 2.5mm propagating in the atmosphere at the distance z=5km for different values of turbulent parameters.
Fig. 6. Changes in the degree of polarization P along the z-axis of the same beam as in Fig. 5 propagating in the atmosphere turbulence for different values of turbulent parameters. (a) α = 3.1 ; (b) C 2 = 10−12 m3−α . n
In Fig. 6 we illustrate the changes in the spectral degree of polarization along the z-axis of the same beam as in Fig. 5 propagating in the atmosphere turbulence for different values of turbulent parameters. For the case of free space, i.e., C n2 = 0 , the spectral degree of polarization increases to a certain value with increasing propagation distance and remains unchanged. However, in atmosphere turbulence, this quantity returns to its value in the source plane after propagating a certain distance, the shorter distance required for the stronger turbulence. This result is similar to the case of EM GSM beams [29].
#216580 - $15.00 USD Received 8 Jul 2014; revised 29 Aug 2014; accepted 29 Aug 2014; published 10 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.022534 | OPTICS EXPRESS 22543
C. Spectral degree of coherence μ
We will now turn to the analysis of the spectral degree of coherence of the EM SSM beam as it travels in the atmosphere turbulence. Figure 7 illustrates the behavior of the degree of coherence μ of an EM SSM beam with δ yy = 2.5mm as a function of ρ d =| ρ1 − ρ 2 | between two points symmetric with respect to the optical axis, i.e. ρ1 = − ρ 2 = ρ d / 2 , for different values of atmosphere parameters and propagation distances. We can see from Fig. 7 that there are two effects occurring with the degree of coherence, as either the propagation distance or the strength of turbulence increase. With the propagation distance increases, the fluctuations of the degree of coherence profiles gradually weaken and eventually degenerate into a Gaussian shape, and the width of profile is reduced in the further field. And the width of Gaussian profile also decreases with the strengthening of turbulence effects. In Fig. 8 we illustrate the changes in the spectral degree of coherence μ along the z-axis of the same beam as in Fig. 7 propagating in the atmosphere turbulent for different values of atmosphere parameters when ρ d = 0.5mm . For the case of C n2 = 0 , corresponding to the free space, the spectral degree of coherence tends to 1 in the far-zone. On the contrary, this quantity tends to zero after propagating a certain distance in atmosphere turbulence. This result implies the atmosphere turbulence also destroys spectral coherence of EM SSM beams and it is so at smaller distances from the source for the stronger turbulence.
Fig. 7. Evolutions of the degree of coherence μ as a function of ρ d , for an EM SSM beam with δ yy = 2.5mm propagating in the atmosphere turbulent with different values of atmosphere parameters.
#216580 - $15.00 USD Received 8 Jul 2014; revised 29 Aug 2014; accepted 29 Aug 2014; published 10 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.022534 | OPTICS EXPRESS 22544
Fig. 8. Changes in the degree of coherence μ along the z-axis of the same beam as in Fig. 7 propagating in the atmosphere turbulent for different values of atmosphere parameters when ρ d = 0.5mm . (a) α = 3.1 ; (b) C n2 = 10−12 m3−α .
5. Concluding remarks
In this article, we have introduced a new class of stochastic electromagnetic sources with Sinc Schell-model correlations properties, and derived the realizability conditions for such sources. And then, we have explored the variation of the three main statistical properties of the EM SSM beams: the spectral density, and the states of coherence and polarization, on propagating in free space and in atmosphere turbulence based on the weighted superposition method and by numerical simulations. It is shown that, for such novel source which can initially have any intensity distribution, as Gaussian profile in our examples, can produce a double-layer flat-top intensity distribution in the far field in free space. The difference of height and diameter of two layer intensity distribution can be controlled by adjusting the coherent properties of the source field. The results also illustrated that the double-layer flat-top profile is preserved for any propagation distances in free space and weak atmosphere turbulence but it is destroyed by the strong atmosphere turbulence. The double-layer flat-top profile gradually disappears and the Gaussian-like distribution is produced when it propagates in the strong turbulent atmosphere over sufficiently long distances. The spectral degree of polarization and the degree of coherence are also destroyed by the turbulence. The distributions in far field eventually transformed into a divergent inverted Gaussian profile for the spectral degree of polarization and a shrink Gaussian profile for the degree of coherence relative to free space. Just like for other beam classes, the influence of the turbulence to statistical properties of EM SSM beams is the most for sufficiently large C n2 and for α in a region about 3.1. The propagation of light beams in non-Kolmogorov weak turbulence depends on the choice of the length unit [30,31]. We state that the turbulence structure parameter in this paper uses meter as a unit. If a different unit is selected, the plots would be different. The electromagnetic random beams can be produced with help of the interferometric technique involving two spatial light modulators described in Ref [32]. For the EM SSM the phase correlation function of the modulators should take forms of Sinc functions instead of Gaussian function as suggested in Ref [32]. for generation of EM GSM model beams. Unlike conventional flat light field with a single layer flat-top profile, the EM SSM source can produce a double-layer flat-top intensity distribution and remain shape-invariant in the far field in free space and weak atmosphere turbulence, and can be controlled the structure of two layer by adjusting the coherent properties of the source field. This feature makes this new beam particularly suitable for applications involving some laser processing, such as laser drilling for some stepped complex hole with different aperture and depth.
#216580 - $15.00 USD Received 8 Jul 2014; revised 29 Aug 2014; accepted 29 Aug 2014; published 10 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.022534 | OPTICS EXPRESS 22545
Acknowledgments
The research is supported by the National Natural Science Foundation of China (NSFC) (11247004).
#216580 - $15.00 USD Received 8 Jul 2014; revised 29 Aug 2014; accepted 29 Aug 2014; published 10 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.022534 | OPTICS EXPRESS 22546
2
Department of Physics, Huzhou University, Huzhou 313000, China Department of Physics, Zhejiang University of Technology, Hangzhou 310014, China *[email protected]
Abstract: A class of electromagnetic sources with sinc Schell-model correlations is introduced. The conditions on source parameters guaranteeing that the source generates a physical beam are derived. The evolution behaviors of statistical properties for the electromagnetic stochastic beams generated by this new source on propagating in free space and in atmosphere turbulence are investigated with the help of the weighted superposition method and by numerical simulations. It is demonstrated that the intensity distributions of such beams exhibit unique features on propagating in free space and produce a double-layer flat-top profile of being shape-invariant in the far field. This feature makes this new beam particularly suitable for some special laser processing applications. The influences of the atmosphere turbulence with a non-Kolmogorov power spectrum on statistical properties of the new beams are analyzed in detail. ©2014 Optical Society of America OCIS codes: (030.1640) Coherence; (010.1300) Atmospheric propagation; (010.1330) Atmospheric turbulence; (260.5430) Polarization.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.
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#216580 - $15.00 USD Received 8 Jul 2014; revised 29 Aug 2014; accepted 29 Aug 2014; published 10 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.022534 | OPTICS EXPRESS 22534
19. Z. Mei, O. Korotkova, and E. Shchepakina, “Electromagnetic multi-Gaussian Schell-model beams,” J. Opt. 15(2), 025705 (2013). 20. Z. Tong and O. Korotkova, “Electromagnetic nonuniformly correlated beams,” J. Opt. Soc. Am. A 29(10), 2154–2158 (2012). 21. Z. Mei, Z. Tong, and O. Korotkova, “Electromagnetic non-uniformly correlated beams in turbulent atmosphere,” Opt. Express 20(24), 26458–26463 (2012). 22. Z. Mei and O. Korotkova, “Electromagnetic cosine-Gaussian Schell-model beams in free space and atmospheric turbulence,” Opt. Express 21(22), 27246–27259 (2013). 23. Z. Mei, “Two types of sinc Schell-model beams and their propagation characteristics,” Opt. Lett. 39(14), 4188– 4191 (2014). 24. X. Ji, E. Zhang, and B. Lü, “Changes in the spectrum and polarization of polychromatic partially coherent electromagnetic beams in the turbulent atomosphere,” Opt. Commun. 275(2), 292–300 (2007). 25. J. Pu and O. Korotkova, “Propagation of the degree of cross-polarization of a stochastic electromagnetic beam through the turbulent atmosphere,” Opt. Commun. 282(9), 1691–1698 (2009). 26. E. Shchepakina and O. Korotkova, “Second-order statistics of stochastic electromagnetic beams propagating through non-Kolmogorov turbulence,” Opt. Express 18(10), 10650–10658 (2010). 27. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 026003 (2008). 28. A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Propagation of electromagnetic waves in Kolmogorov and nonKolmogorov atmospheric turbulence: three-layer altitude model,” Appl. Opt. 47(34), 6385–6391 (2008). 29. X. Du and D. Zhao, “Polarization modulation of stochastic electromagnetic beams on propagation through the turbulent atmosphere,” Opt. Express 17(6), 4257–4262 (2009). 30. M. Charnotskii, “Intensity fluctuations of flat-topped beam in non-Kolmogorov weak turbulence: comment,” J. Opt. Soc. Am. A 29(9), 1838–1840, discussion 1841–1842 (2012). 31. I. Toselli, “Introducing the concept of anisotropy at different scales for modeling optical turbulence,” J. Opt. Soc. Am. A 31(8), 1868–1875 (2014). 32. T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 7(5), 232–237 (2005).
1. Introduction Schell-model sources have played a role of utmost importance in optical coherence theory since it was introduced in 1967 [1]. Among the most classic models is the Gaussian Schellmodel source which comes with the degree of coherence of Gaussian distribution [2]. Some other augmented models for scalar random sources have been recently introduced, in which the classic Gaussian degree of coherence is modulated by a well-behaved function, such as Multi-Gaussian Schell-model sources [3], Bessel-Gaussian and Laguerre-Gaussian Schellmodel sources [4], cosine-Gaussian Schell-models sources [5], etc. Several random scalar sources producing far field with rectangular intensity profiles have also been introduced by modeling the source degree of coherence with the help of two one-dimensional Schell-model sources [6–8]. These scalar sources and beams are specified by a single correlation function, while the electromagnetic random beams are characterized by the 2 × 2 cross-spectral density (CSD) matrices. It has been shown that the changes in the propagation of the spectral density, the spectral degree of coherence, and the spectral degree of polarization of a random electromagnetic beam can be determined from knowledge of a CSD matrix [9]. This matrix has already been used in studies of the behaviors of random electromagnetic beams propagating through turbulent atmosphere [10,11], an optical resonator [12] and optical focusing systems [13,14], etc. For an electromagnetic stochastic beam, the spatial correlation functions must satisfy additional restrictions, leading to restrictions for source parameters known as the realizability conditions [15]. Roychowdhury and Korotkova had derived realizability conditions for electromagnetic Gaussian Schell-model (EM GSM) sources [16]. Recently, a basic theory on devising genuine CSD matrices for an electromagnetic stochastic beam was exploited and extended [17,18]. Based on this method, several models for scalar random sources have been extended to electromagnetic domain, such as the electromagnetic Multi-Gaussian Schell-model sources [19], electromagnetic non-uniformly correlated sources [20,21], electromagnetic cosine-Gaussian Schell-models sources [22], and so on. The degree of coherence of all the known Schell-model sources is Gaussian or based on the Gaussian degree of coherence. Recently, we introduced a new random source class with sinc Schell-model (SSM) correlated function [23], whose degree of coherence in the source plane presents sinc function profile rather than the traditional Gaussian distribution. The
#216580 - $15.00 USD Received 8 Jul 2014; revised 29 Aug 2014; accepted 29 Aug 2014; published 10 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.022534 | OPTICS EXPRESS 22535
beams generated by SSM sources have been found to form a stable single layer flat profile in the far filed. In this paper, we use a scalar model proposed in [23] as a building for developing electromagnetic sources with SSM correlations, terming the novel class of beams the electromagnetic sinc Schell-model (EM SSM) beams, in which all the correlations are prescribed with the help of the scalar SSM distributions. As we illustrate by numerical examples, the EM SSM source can produce a double-layer flat-top intensity distribution and remain shape-invariant in the far field. It is not like the known partially coherent flat-top beams produce a single layer flat-top intensity profile and disappear gradually in propagation and becomes Gaussian in the far field. The double-layer flat-top intensity profile may be of use in applications dealing with laser processing. After extending the scalar SSM sources to the full electromagnetic domain and deriving the realizability conditions, our major task in this study is to explore the behavior of the main statistical properties for EM SSM beams propagating in free space and atmosphere turbulence. Usually, it is difficult to obtain a general analytical formula for the propagated CSD matrix of EM SSM beams. However, the integral structure of the elements of the CSD matrix constructed through the mode superposition retains its form on propagation. The elementary modes change on propagation, while the weight functions remain unchanged [18,20]. Therefore, we can consider separately the propagation of the kernel functions and apply the weighted superposition method to study the statistical properties of EM SSM beams by a set of numerical examples. 2. Electromagnetic Sinc Schell-model source The second-order correlation properties of a statistically stationary electromagnetic source at two points ρ1′ and ρ 2′ will be described by the 2 × 2 CSD matrix Wˆ (0) ( ρ1′, ρ 2′ ; ω ) , whose element Wαβ(0) ( ρ1′, ρ 2′ ; ω ) are given by [9] Wαβ(0) ( ρ1′, ρ 2′ ; ω ) = Eα* ( ρ1′; ω ) Eβ ( ρ 2′ ; ω ); (α = x, y; β = x, y ),
(1)
where Eα ( ρ ′; ω ) is the fluctuating electric field component along the α axis at point ρ ′ and the angular brackets denote an ensemble average. From now on, the angular frequency dependence of all the quantities of interest will be omitted but implied, and each of the symbols α and β as indexes can be either x and y . Recall the non-negative definiteness condition for CSD matrices to be physically realizable, it suffices to have an integral of the form [18] Wαβ(0) ( ρ1′, ρ 2′ ) = pαβ (v) H α∗ ( ρ1′, v) H β ( ρ 2′ , v)dv,
(2)
where pαβ (v) are the elements of the weigh matrix pˆ (v) . Hα ( ρ1′, v) and H β ( ρ 2′ , v) are two arbitrary kernels, they have plenty of choices and each choice is likely to lead to distinct classes of CSD matrices. A simple and significant class of the CSD matrices, leading to the classic Schell-model correlation sources, can be obtain by assigning to the kernels Hα ( ρ1′, v) and H β ( ρ 2′ , v) Fourier-like structure, viz., H α ( ρ1′, v) = Aατ ( ρ1′) exp(−2π iv ρ1′),
(3)
H β ( ρ 2′ , v) = Aβ τ ( ρ 2′ ) exp(−2π iv ρ 2′ ),
(4)
where Aα and Aβ are the amplitude of the field component, τ ( ρ ′) is a profile function. Then, Eq. (2) lends to Wαβ(0) ( ρ1′, ρ 2′ ) = Aα Aβ τ ∗ ( ρ1′)τ ( ρ 2′ ) p αβ ( ρ1′ − ρ 2′ ).
(5)
#216580 - $15.00 USD Received 8 Jul 2014; revised 29 Aug 2014; accepted 29 Aug 2014; published 10 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.022534 | OPTICS EXPRESS 22536
where tilde symbol denotes the Fourier transform of pαβ . The choice of the element pαβ (v) of the matrix pˆ (v) defines a family of sources with different correlation functions. Let us suppose pαβ (v) to be of the form pαβ (v) = Bαβ δαβ rect(δαβ v),
(6)
where δ αβ are the characteristic source correlations which are the positive real constants. Bαβ =| Bαβ | e
iφαβ
is the single-point correlation coefficient. rect( x) is the rectangular function,
which equals 1 for x ≤ 1/ 2 and 0 otherwise. On substituting Eq. (6) into Eq. (5) and setting the Gaussian profile exp[− | ρ ′ |2 /(2σ 02 )] with the r.m.s. source width σ 0 for the function τ ( ρ ′) , one finds the explicit form of the CSD matrix elements: ρ ′2 + ρ ′2 Wαβ(0) ( ρ1′, ρ 2′ ) = Aα Aβ Bαβ exp − 1 2 2 2σ 0
ρ1′ − ρ 2′ . sin c δαβ
(7)
Equation (7) represents a new family of sources that may be Electromagnetic Sinc Schellmodel (EM SSM) sources. The CSD matrix of a physically realizable field must be quasi-Hermitian, i.e. that (0) (0) Wαβ ( ρ1′, ρ 2′ ) = Wβα ( ρ 2′ , ρ1′) . It is sufficient that the condition holds if Bxx = Byy = 1, | Bxy |=| Byx |, δ xy = δ yx .
(8)
Furthermore, the non-negative definiteness constraint for the genuine CSD matrices means that, the following inequalities pαα (v) ≥ 0,
(9)
pxx (v) p yy (v) − pxy (v) p yx (v) ≥ 0,
(10)
have to be satisfied for any v . We readily find from Eq. (6) that inequality (9) is always satisfied, and on substituting from Eq. (6) into inequalities (10) implies that it is satisfied if
δ xxδ yy rect(δ xx v)rect(δ yy v) ≥| Bxy |2 δ xy2 [rect(δ xy v)]2 .
(11)
Since function rect(ax) equals 1 for x ≤ 1 / (2a) and 0 otherwise, it is not difficult to find that the realizability condition is expressed by the following fork inequality: max{δ xx , δ yy } ≤ δ xy ≤
δ xxδ yy | Bxy |
.
(12)
3. Propagation laws for the beams generated by the new class of sources in linear random medium
Suppose that source (7) generates a beam-like field propagating into half-space z > 0 filled with turbulent atmosphere. According to the extended Huygens-Fresnel integral principle adjusted for propagation in linear random medium, the elements of the CSD matrix at two points ( ρ1 , z ) and ( ρ 2 , z ) in the same transverse plane are related to those in the source plane as [24–26] Wαβ ( ρ1 , ρ 2 , z ) = Wαβ(0) ( ρ1′, ρ 2′ )K ( ρ1′, ρ 2′ , ρ1 , ρ 2 , z )d 2 ρ1′d 2 ρ 2′ ,
(13)
#216580 - $15.00 USD Received 8 Jul 2014; revised 29 Aug 2014; accepted 29 Aug 2014; published 10 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.022534 | OPTICS EXPRESS 22537
where the propagation kernel K ( ρ1′, ρ 2′ , ρ1 , ρ 2 , z ) , depended on the Green’s function of the random medium, is given by 2
( ρ1 − ρ1′) 2 − ( ρ 2 − ρ 2′ ) 2 k K ( ρ1′, ρ 2′ , ρ1 , ρ 2 , z ) = exp −ik 2z 2π z (14) 2 2 ∞ π k z 2 2 3 ( ρ1 − ρ 2 ) + ( ρ1 − ρ 2 )( ρ1′ − ρ 2′ ) + ( ρ1′ − ρ 2′ ) κ Φ n (κ )dκ . × exp − 0 3
Here, k = 2π / λ , with λ being the wavelength. Φ n (κ ) is the one-dimensional power spectrum of fluctuations in the refractive index of the turbulent medium. On inserting Eq. (2) into Eq. (13) we obtain, after interchanging the orders of integrals, the expression Wαβ ( ρ1 , ρ 2 , z ) = pαβ (v) Hα* ( ρ1 , v, z ) H β ( ρ 2 , v, z )dv,
(15)
where H α* ( ρ1 , v, z ) H β ( ρ 2 , v, z ) = H α* ( ρ1′, v)H β ( ρ 2′ , v) K ( ρ1′, ρ 2′ , ρ1 , ρ 2 , z )d 2 ρ1′d 2 ρ 2′ . (16) It is seen from Eq. (15) that the structure of the elements of the CSD matrix constructed through the superposition rule specified by Eq. (2) retains its form on propagation through turbulent medium, the Wαβ can be expressed as an incoherent superposition of elementary modes H α weighted by the function pαβ . H α change on propagation and the weight function pαβ remain unchanged. On substituting from Eqs. (3), (4) and (14) into Eq. (15), after long integral calculations, we obtain, for any mode, the formula Hα* ( ρ1 , v, z ) H β ( ρ 2 , v, z ) =
Aα Aβ σ 02 w2 ( z )
k 2σ 02 k 2π 2 z ∞ 3 2 ik × exp − ( ρ12 − ρ 22 ) exp − + κ Φ n (κ )dκ ( ρ1 − ρ 2 ) 2 0 3 2z 4 z 2 ρ + ρ 2 2π zv kσ 0 kπ 2 z 2 × exp − 1 + −i − 3 k 2z 2
∞
0
κ 3 Φ n (κ )dκ ( ρ1 − ρ 2 )
2
(17) w2 ( z ) ,
where 4π 2 z 3 ∞ 3 (18) κ Φ n (κ )dκ . 3 0 k 2σ 02 To model the atmosphere turbulence, a non-Kolmogorov spectrum is used [26–28] w2 ( z ) = σ 02 +
z2
+
Φ n (κ ) = A(α )C n2 exp[−(κ 2 / κ m2 )] / (κ 2 + κ 02 )α / 2 ,
0 ≤ κ < ∞, 3 < α < 4, (19)
where C n2 is a generalized refractive-index structure parameter with units m3−α , κ 0 = 2π / L0 and κ m = c(α ) / l0 , L0 and l0 being the outer and the inner scale of turbulence, and c(α ) = [Γ(5 −
α 2
) A(α )
2π 1/ (α − 5) ] , 3
(20)
#216580 - $15.00 USD Received 8 Jul 2014; revised 29 Aug 2014; accepted 29 Aug 2014; published 10 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.022534 | OPTICS EXPRESS 22538
A(α ) = Γ(α − 1) ⋅
cos(απ / 2) , 4π 2
(21)
with Γ( x) being the Gamma function, the integral in Eqs. (17) and (18) becomes
∞
0
κ 3 Φ n (κ )dκ =
κ2 A(α ) 2 2 −α α κ2 Cn κ m β exp( 02 )Γ(2 − , 02 ) − 2κ 04 −α , 2(α − 2) 2 κm κm
(22)
where β = 2κ 02 − 2κ m2 + ακ m2 and Γ denotes the incomplete Gamma function. It follows from above derivation that the propagated kernel of the elements of CSD matrix on propagation due to random medium is only included in the integral term in the right side of Eqs. (17) and (18), thus, for the free space propagation, Eqs. (17) and (18) can be express as Hα* ( ρ1 , v, z ) H β ( ρ 2 , v, z ) = Aα Aβ σ 02
k 2σ 02 2 ik 2 2 ρ ρ ρ − ρ2 ) − − exp exp ( ) − 1 2 2 2 ( 1 w ( z) 2z 4z
(23)
2 2 ρ1 + ρ 2 2π zv ikσ 0 × exp − + − ( ρ1 − ρ2 ) w2 ( z ) , k 2 2z
where w2 ( z ) = σ 02 + z 2 (k 2σ 02 ). (24) Based on the weighted superposition (15) and the propagating formulae of the modes (17) and (23), the spectral density S , the spectral degree of coherence μ and the spectral degree of polarization P of the electromagnetic stochastic beam in the turbulent atmosphere and free space can be calculated by the expressions [9] S ( ρ , z ) = TrWˆ ( ρ , ρ , z ), 4DetWˆ ( ρ , ρ , z ) , [TrWˆ ( ρ , ρ , z )]2
(26)
TrWˆ ( ρ1 , ρ 2 , z ) , TrWˆ ( ρ1 , ρ1 , z )TrWˆ ( ρ 2 , ρ 2 , z )
(27)
P( ρ , z ) = 1 −
μ ( ρ1 , ρ 2 , z ) =
(25)
where Det and Tr stand for the determinant and the trace of the matrix. 4. The statistical properties of EM SSM beams in free space and non-Kolmogorov atmosphere turbulence
We will now illustrate the behavior of the statistical properties, including spectral density, the degree of coherence and the degree of polarization, of typical beams generated by source (7) on propagating in free space and atmosphere turbulence by a set of numerical examples. Without loss of generality, the values of source and medium parameters are chosen to be Ax = Ay = 1 , Bxy = 0.2 , σ 0 = 10mm , k = 107 m −1 , δ xx = 1mm , L0 = 1m , l0 = 1mm , otherwise specified in the text and captions.
#216580 - $15.00 USD Received 8 Jul 2014; revised 29 Aug 2014; accepted 29 Aug 2014; published 10 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.022534 | OPTICS EXPRESS 22539
A. Spectral density S
Figure 1 shows typical evolution of the spectral density S of an EM SSM beam with δ yy = 2.5mm in the transverse beam cross-sections at several distances z from the source plane on propagating in free space. One clearly sees that the Gaussian profile of source field gradually transforms into a double-layer flat-top distribution with increasing propagation distance. This is because the coherence properties of light field in the source plane are closely related to the propagation characteristics and the transverse intensity distribution of the far field. Two flat-top profiles correspond to the spectral densities’ distributions of x and y components of electric field with different r.m.s. correlation width. In order to demonstrate the dependence of the spectral density behavior on the degree of coherence, we plot in Fig. 2 its evolution in the transverse beam cross-section at the plane z = 1km in free space for different δ yy . Note that for the case δ yy = δ xx = 1mm the transverse beam cross-section is a single-layer flat-top profile, this is due to the degree of coherences of x and y components are the same, which results in two directions with the same far-field spectral density distributions. However, for the cases δ yy ≠ δ xx shown as Figs. 2(b) and 2(c), the far-field spectral density distributions of x and y directions are different, resulting in the appearance of double-layer profile, the lower layer corresponds to the x component and the higher layer corresponds to the y component. This is the same reason that the electromagnetic cosineGaussian Schell-model beams possess double-ring intensity profiles.
Fig. 1. Transverse distribution of the spectral density S of an EM SSM beam with δ yy = 2.5mm at several different propagation distances in free space. (a) z = 0m; (b) z =
250m; (c) z = 350m; (d) z = 500m.
#216580 - $15.00 USD Received 8 Jul 2014; revised 29 Aug 2014; accepted 29 Aug 2014; published 10 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.022534 | OPTICS EXPRESS 22540
Fig. 2. Transverse distribution of the spectral density S of EM SSM beams with different δ yy propagating in free space at the plane z=1km. (a) δ yy = 1mm ; (b) δ yy = 2.5mm ; (c)
δ yy = 5mm .
Fig. 3. Transverse distribution of the spectral density S of an EM SSM beam with δ yy = 2.5mm propagating in the atmosphere at the distance z=5km for different values of atmosphere parameters.
Figure 3 shows the transverse distribution of the spectral density S of an EM SSM beam with δ yy = 2.5mm propagating in the non-Kolmogorov turbulence atmosphere at the distance z = 5km for different values of turbulence parameters C n2 and α . The longitudinal orientations indicate the changes in the Kolmogorov turbulence as the values of α increase and the transverse orientations indicate the changes in the Kolmogorov turbulence as the
#216580 - $15.00 USD Received 8 Jul 2014; revised 29 Aug 2014; accepted 29 Aug 2014; published 10 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.022534 | OPTICS EXPRESS 22541
values of C n2 increase. One can see from Fig. 3 that, for the weakest turbulence the spectral density keeps the double-layer distribution as shown in Fig. 3(c), while for Figs. 3(a) and 3(f) with decreasing α and increasing C n2 the double-layer profile of spectral density is destroyed gradually. Finally, in Fig. 3(d) for substantially strong turbulence the upper layer and lower layer are blended completely and the beams’ intensity resembles Gaussian profile. Figures 3(b) and 3(e) show the spectral density distributions is destroyed the most when α = 3.1 , which just like other beams [22, 26]. B. Spectral degree of polarization P
We now turn our attention to the changes in the spectral degree of polarization of EM SSM beams in free space and non-Kolmogorov atmosphere turbulence. Figure 4 illustrates the typical distributions of the polarization properties for the most general case of partially correlated electric field components, as beam propagates at 1km in free space. Note that for the case δ xx = δ yy = δ xy the spectral degree of polarization does not change upon propagation. It is the same as the source plane and uniform for all the points at the any propagation plane, which was also shown to be critical for other beams. However, for other cases, unlike the degree of polarization of EM GSM beams to form an inverted Gaussian distribution in the far field, Figs. 3(b) and 3(c) clearly show the formation of an inverted flat distribution with prominent and uniform polarization distribution in the central region. The height of the prominent part increases with the increase of the coherence length.
Fig. 4. Transverse distribution of the degree of polarization of the beams with different δ yy and δ xy propagating in free space at the plane z=1km. (a) δ yy = δ xy = 1mm ; (b)
δ yy = δ xy = 2.5mm ; (c) δ yy = δ xy = 5mm .
Figure 5 shows the transverse distribution of the degree of polarization P of an EM SSM beam with δ yy = δ xy = 2.5mm at the propagation distance z = 5km in the turbulence atmosphere for different values of parameters C n2 and α . For the case Fig. 5(c) of the weakest turbulence, the distribution of the degree of polarization is similar to the case of free space. With the strengthening of turbulence, the steepness of edges gradually become decline and the flat central prominent region change into a Gaussian profile, as shown in Figs. 5(a), 5(f), 5(b) and 5(d). For the case Fig. 5(e) of the strongest turbulence, the central projection disappears and the distribution of the degree of polarization present the inverted Gaussian profile.
#216580 - $15.00 USD Received 8 Jul 2014; revised 29 Aug 2014; accepted 29 Aug 2014; published 10 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.022534 | OPTICS EXPRESS 22542
Fig. 5. Transverse distribution of the degree of polarization P of an EM SSM beam with δ yy = δ xy = 2.5mm propagating in the atmosphere at the distance z=5km for different values of turbulent parameters.
Fig. 6. Changes in the degree of polarization P along the z-axis of the same beam as in Fig. 5 propagating in the atmosphere turbulence for different values of turbulent parameters. (a) α = 3.1 ; (b) C 2 = 10−12 m3−α . n
In Fig. 6 we illustrate the changes in the spectral degree of polarization along the z-axis of the same beam as in Fig. 5 propagating in the atmosphere turbulence for different values of turbulent parameters. For the case of free space, i.e., C n2 = 0 , the spectral degree of polarization increases to a certain value with increasing propagation distance and remains unchanged. However, in atmosphere turbulence, this quantity returns to its value in the source plane after propagating a certain distance, the shorter distance required for the stronger turbulence. This result is similar to the case of EM GSM beams [29].
#216580 - $15.00 USD Received 8 Jul 2014; revised 29 Aug 2014; accepted 29 Aug 2014; published 10 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.022534 | OPTICS EXPRESS 22543
C. Spectral degree of coherence μ
We will now turn to the analysis of the spectral degree of coherence of the EM SSM beam as it travels in the atmosphere turbulence. Figure 7 illustrates the behavior of the degree of coherence μ of an EM SSM beam with δ yy = 2.5mm as a function of ρ d =| ρ1 − ρ 2 | between two points symmetric with respect to the optical axis, i.e. ρ1 = − ρ 2 = ρ d / 2 , for different values of atmosphere parameters and propagation distances. We can see from Fig. 7 that there are two effects occurring with the degree of coherence, as either the propagation distance or the strength of turbulence increase. With the propagation distance increases, the fluctuations of the degree of coherence profiles gradually weaken and eventually degenerate into a Gaussian shape, and the width of profile is reduced in the further field. And the width of Gaussian profile also decreases with the strengthening of turbulence effects. In Fig. 8 we illustrate the changes in the spectral degree of coherence μ along the z-axis of the same beam as in Fig. 7 propagating in the atmosphere turbulent for different values of atmosphere parameters when ρ d = 0.5mm . For the case of C n2 = 0 , corresponding to the free space, the spectral degree of coherence tends to 1 in the far-zone. On the contrary, this quantity tends to zero after propagating a certain distance in atmosphere turbulence. This result implies the atmosphere turbulence also destroys spectral coherence of EM SSM beams and it is so at smaller distances from the source for the stronger turbulence.
Fig. 7. Evolutions of the degree of coherence μ as a function of ρ d , for an EM SSM beam with δ yy = 2.5mm propagating in the atmosphere turbulent with different values of atmosphere parameters.
#216580 - $15.00 USD Received 8 Jul 2014; revised 29 Aug 2014; accepted 29 Aug 2014; published 10 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.022534 | OPTICS EXPRESS 22544
Fig. 8. Changes in the degree of coherence μ along the z-axis of the same beam as in Fig. 7 propagating in the atmosphere turbulent for different values of atmosphere parameters when ρ d = 0.5mm . (a) α = 3.1 ; (b) C n2 = 10−12 m3−α .
5. Concluding remarks
In this article, we have introduced a new class of stochastic electromagnetic sources with Sinc Schell-model correlations properties, and derived the realizability conditions for such sources. And then, we have explored the variation of the three main statistical properties of the EM SSM beams: the spectral density, and the states of coherence and polarization, on propagating in free space and in atmosphere turbulence based on the weighted superposition method and by numerical simulations. It is shown that, for such novel source which can initially have any intensity distribution, as Gaussian profile in our examples, can produce a double-layer flat-top intensity distribution in the far field in free space. The difference of height and diameter of two layer intensity distribution can be controlled by adjusting the coherent properties of the source field. The results also illustrated that the double-layer flat-top profile is preserved for any propagation distances in free space and weak atmosphere turbulence but it is destroyed by the strong atmosphere turbulence. The double-layer flat-top profile gradually disappears and the Gaussian-like distribution is produced when it propagates in the strong turbulent atmosphere over sufficiently long distances. The spectral degree of polarization and the degree of coherence are also destroyed by the turbulence. The distributions in far field eventually transformed into a divergent inverted Gaussian profile for the spectral degree of polarization and a shrink Gaussian profile for the degree of coherence relative to free space. Just like for other beam classes, the influence of the turbulence to statistical properties of EM SSM beams is the most for sufficiently large C n2 and for α in a region about 3.1. The propagation of light beams in non-Kolmogorov weak turbulence depends on the choice of the length unit [30,31]. We state that the turbulence structure parameter in this paper uses meter as a unit. If a different unit is selected, the plots would be different. The electromagnetic random beams can be produced with help of the interferometric technique involving two spatial light modulators described in Ref [32]. For the EM SSM the phase correlation function of the modulators should take forms of Sinc functions instead of Gaussian function as suggested in Ref [32]. for generation of EM GSM model beams. Unlike conventional flat light field with a single layer flat-top profile, the EM SSM source can produce a double-layer flat-top intensity distribution and remain shape-invariant in the far field in free space and weak atmosphere turbulence, and can be controlled the structure of two layer by adjusting the coherent properties of the source field. This feature makes this new beam particularly suitable for applications involving some laser processing, such as laser drilling for some stepped complex hole with different aperture and depth.
#216580 - $15.00 USD Received 8 Jul 2014; revised 29 Aug 2014; accepted 29 Aug 2014; published 10 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.022534 | OPTICS EXPRESS 22545
Acknowledgments
The research is supported by the National Natural Science Foundation of China (NSFC) (11247004).
#216580 - $15.00 USD Received 8 Jul 2014; revised 29 Aug 2014; accepted 29 Aug 2014; published 10 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.022534 | OPTICS EXPRESS 22546
