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Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 11 — Jun. 2, 2014
  • pp: 13029–13040
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Light sources generating self-splitting beams and their propagation in non-Kolmogorov turbulence

Zhangrong Mei  »View Author Affiliations


Optics Express, Vol. 22, Issue 11, pp. 13029-13040 (2014)
http://dx.doi.org/10.1364/OE.22.013029


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Abstract

A class of random sources producing far fields self-splitting intensity profiles with variable spacing between the x and y directions is introduced. The beam conditions for ensuring the sources to generate a beam are derived. Based on the derived analytical expression, the evolution behavior of the beams produced by these families of sources in free space and turbulence atmospheric are explored and comparatively analyzed. By changing the modulation parameters n and m, the degree of coherence of Gaussian Schell-model source in the x and y directions are modulated respectively, and then the number of splitting beams and the spacing between splitting beams can be adjusted. It is illustrated that the self-splitting intensity profile is stable when beams propagate in free space, but they eventually transformed into a Gaussian profiles when it passes at sufficiently large distances from its source through the turbulent atmosphere.

© 2014 Optical Society of America

1. Introduction

2. Light source model and beam conditions

The spatial coherence properties of an optical field at a pair of points in the source plane with position coordinates ρ1=(x1,y1) and ρ2=(x2,y2) and at the oscillation angular frequency ω (for brevity that is not explicitly shown in the following equation) can be described by means of the cross-spectral density (CSD) function W(0)(x1,y1,x2,y22;ω). For a Schell-model source, the CSD function has the form [10

10. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

]
W(0)(x1,y1,x2,y2)=[S(0)(x1,y1)S(0)(x2,y2)]1/2μ(0)(x1x2,y1y2),
(1)
S(0) is the spectral density and μ(0) is the spectral degree of coherent. It was proven in [33

33. F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007). [CrossRef] [PubMed]

] that for a CSD to be genuine, i.e., physically realizable, it suffices to have a superposition integral of the form
W(0)(x1,y1,x2,y2)=p(vx,vy)H0(x1,y1,vx,vy)H0(x2,y2,vx,vy)dvxdvy,
(2)
where H0 is an arbitrary kernel and p is a nonnegative weight function. For Schell-model corrections, the kernel H0 is chosen as a Fourier-like structure, i.e.,
H0(x,y,vx,vy)=τ(x,y)exp[i2π(vxxvyy)],
(3)
and hence W(0) becomes
W(0)(x1,y1,x2,y2)=τ(x1,y1)τ(x2,y2)p˜x(x1x2)p˜y(y1y2),
(4)
where τ is an amplitude profile function which can be chosen at will, the tilde denotes the Fourier transform. Comparison of Eqs. (1) and (4), we can find that the nonnegative function p and the spectral degree of coherent μ(0) are a Fourier transform pair.

Let us now consider a simple variation to the degree of coherence in Ref [21

21. C. Liang, F. Wang, X. Liu, Y. Cai, and O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014). [CrossRef] [PubMed]

]. and set the spectral degree of coherence in the source plane to be
μ(0)(x1x2,y1y2)=cos[n2π(x2x1)δx]exp[(x2x1)22δx2]×cos[m2π(y2y1)δy]exp[(y2y1)22δy2],
(5)
where δx and δy are rms correlation widths along the x and y directions, n and m are arbitrary real constants, which determine the modulation mode of the degree of coherence of Gaussian Schell-model source in the x and y directions, respectively.
Fig. 1 Absolute value of the degree of coherence calculated from Eq. (5) for several values of n and m.
Figure 1 shows the absolute value of the degree of coherence (5) for several values of parameters n and m: (a) n=m=0, which represent the conventional Gaussian Schell-model; (b) n=0 and m=5, only the degree of coherence in the y direction is modulated; (c) n=5 and m=0, only the degree of coherence in the x direction is modulated; (d) n=3 and m=5, the modulation of the degree of coherence in the y direction is severer than in the x direction; (e) n=5 and m=3, the modulation of the degree of coherence in the x direction is severer than in the y direction; (f) n=5 and m=5, the modulation of the degree of coherence in the two directions is symmetrical.

The choice of the mathematical form of spatial correlation function for optical fields is restricted by the constraint of nonnegative definiteness. So not any degree of coherence defines a physically meaningful random source, the sufficient condition for a genuine CSD is that it must be expressed by the integral form (2). In order to determine the nonnegative function p(vx,vy) in Eq. (2), we calculate the Fourier transform of Eq. (5) and arrive at
p(vx,vy)=2πδxδycosh[n(2π)3/2δxvx]cosh[m(2π)3/2δyvy]×exp[2π2(δxvx+δyvy)(n2+m2)π].
(6)
Since the hyperbolic cosine function and the exponential function is nonnegative for all the values of their arguments, the function p(vx,vy) is manifestly nonnegative. Thus, the source with correlation (5) is physically realizable.

Let us also set the Gaussian profile for function τ:
τ(x,y)=exp(x2+y24σ2),
(7)
σ is its rms source width. The amplitude profile function Eq. (7) together with the weight function Eq. (6), we obtain on substituting them into Eq. (4) the CSD function of the form
W(0)(x1,y1,x2,y2)=exp(x12+x224σx2y12+y224σy2)cos[n2π(x2x1)δx]×exp[(x2x1)22δx2]cos[m2π(y2y1)δy]exp[(y2y1)22δy2].
(8)
We will term such a source the orthogonal cosine-Gaussian Schell-model (OCGSM) source.

Let us now impose some restrictions on the parameters in Eq. (8) to ensure that it generates a beam. The CSD functions of radiated field in the far zone at points P(r1) and P(r2) specified by position vectors r1=r1s1 and r2=r2s2 (s12=s22=1) can be written as the following expression [6

6. A. Suryanto and E. Van Groesen, “Self-splitting of multisoliton bound states in planar Kerr waveguides,” Opt. Commun. 258(2), 264–274 (2006). [CrossRef]

]
W()(r1s1,r2s2)=(2πk)2cosθ1cosθ2W˜(0)(ks1,ks2)exp[ik(r2r1)]r1r2,
(9)
where k=2π/λ is the wave number, and
W˜(0)(f1,f2)=(12π)4W(0)(ρ1,ρ2)exp[i(f1ρ1+f2ρ2)d2ρ1d2ρ2,
(10)
is the four-dimensional Fourier transform of CSD function in the source field. The quantities s1(s1x,s1y,0) and s2(s2x,s2y,0) are projections, considered as two-dimensional vectors, of the unit vectors s1 and s2 on source plane z=0, W˜(0)(ks1,ks2)=W˜(0)(ks1x,ks1y,ks2x,ks2y), and θ1 and θ2 are angles which the unit vectors s1 and s2 make with the positive z-axis. On substituting from Eq. (8) into Eq. (10) and then into (9) we finally obtain the following expression for the CSD function in the far zone generated by an OCGSM source
W()(r1s1,r2s2)=k2σxσy2axaycosθ1cosθ2exp[ik(r2r1)]r1r2×cosh[n2π4axδxk(s1x+s2x)]cosh[m2π4ayδyk(s1y+s2y)]×exp[k2σx22(s1xs2x)2k216ax(s1x+s2x)2n2π2axδx2]×exp[k2σy22(s1ys2y)2k216ay(s1y+s2y)2m2π2ayδy2],
(11)
where aα=1/(8σα2)+1/(2δα2),α=x,y. The spectral density at a point P with position vector r (r1=r2=r) in the far zone take the form
S()(rs)=k2σxσy2r2axaycos2θexp[n2π2axδx2m2π2ayδy2]×cosh(n2π2axδxksx)cosh(m2π2ayδyksy)exp(k2sx24ax)exp(k2sy24ay).
(12)
In order to the source (8) to generate a beam-like field, its spectral density in Eq. (12) must be negligible except for directions within a narrow solid angle about the z axis [10

10. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

]. Since the value of hyperbolic cosine function is always greater than 1, this is so if
exp(k2sx24ax)0andexp(k2sy24ay)0.
(13)
Hence, the beam conditions for the OCGSM sources are

14σx2+1δx2<<2π2λ2and14σy2+1δy2<<2π2λ2.
(14)

3. Propagation laws in non-Kolmogorov turbulence and free space

On substituting from Eqs. (8) and (16) to Eq. (15) we obtain the expression
W(x1,x2,y1,y2,z)=k2σxσy4z2Δx(z)Δy(z)exp[ik2z(x12+y12x22y22)]×exp[(x1x2)2Rx(z)(y1y2)2Ry(z)][exp(γx+2Δx(z))+exp(γx2Δx(z))]×[exp(γy+2Δy(z))+exp(γy2Δy(z))],
(21)
where
1Rα(z)=k2σα22z2+k2π2z30κ3Φn(κ)dκ,α=x,y
(22)
Δα(z)=18σα2+12δα2+1Rα(z),α=x,y
(23)
γα±=(3k2σα24z212Rα(z))(α1α2)+ik4z(α1+α2)±in2π2δα,α=x,y.
(24)
It follows from above derivation that the CSD on propagation due to random medium is only included in the second term in the right side of Eq. (22), thus, for the free space propagation, Eq. (22) can be express as
Rα(z)=2z2/(k2σα2).
(25)
The behavior of the spectral density at the point (x,y,z) of the OCGSM beams on free-space and turbulent atmosphere propagation can be illustrated by the following formula
S(x,y,z)=W(x,x,y,y,z)=k2σxσy4z2Δx(z)Δy(z)[exp(γx+2Δx(z))+exp(γx2Δx(z))]×[exp(γy+2Δy(z))+exp(γy2Δy(z))],
(26)
with

γα±=ikα4z±in2π2δα,α=x,y.
(27)

4. Numerical results

We will now numerically determine the propagation-induced intensity changes of the OCGSM beams in free-space and turbulent atmosphere calculated from Eq. (26) by MARLAB software programming. In order to facilitate the reader to read and understand, Table 1 lists the values of all calculated parameters in this paper.

Table 1. Values of all calculated parameters in this paper

table-icon
View This Table

Fig. 2 Evolution of the spectral density of an OCGSM beam with n = 5 and m = 5 propagating in free space. (a) z = 0; (b) z = 40m; (c) z = 70m; (d) z = 200m.
Let us first consider the case of free space propagation. Figure 2 illustrate typical evolution of the spectral density of an OCGSM beam with n = 5 and m = 5 at several distances z from the source plane on propagation free space. One clearly sees that the transverse distribution of the beam’s spectral density from a Gaussian distribution of source plane gradually split into four beams with the increase of transmission distance. So we can term this light beam generated by the novel family of source with Gaussian spectral density and orthogonal cosine-Gaussian Schell-model correlation as self-splitting beams. The reason for this feature is that the Gaussian Schell correlation model is modified by the cosine function in x and y directions, respectively. The experimental realization of a random light source with the orthogonal cosine-Gaussian Schell-model correlation can be made with the help of the spatial light modulator (SLM) [21

21. C. Liang, F. Wang, X. Liu, Y. Cai, and O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014). [CrossRef] [PubMed]

]. Next, we analyze the impact of the different modulation coefficient to spectral density distribution in far field, as is shown in Fig. 3.
Fig. 3 Spectral density of the OCGSM beams with parameters as in Fig. 1 at the plane z = 5km.
Figure 3(a) shows Gaussian distribution of the spectral density which corresponds to the conventional Gaussian Schell-model source. Figures 3(b) and 3(c) indicate the Gaussian distribution of source plane are split into two beams due to the degree of coherence of the source field only is modulated in x or y one direction. Figures 3(d) and 3(e) show the spacing between the x and y directions are not equal due to the modulation factor in two directions are not equal. Therefore, the modulation factors n and m to the degree of coherence of source determine the spacing between the split beams. When n=m, the light field split into four equally spaced beams in far-field, as shown in Fig. 3(f).

Fig. 4 Comparison of the spectral density of an OCGSM beam with n = 5 and m = 5 propagating in free space and atmosphere turbulence with α=3.667 and C˜n2=1012m3α.
The interactions of a partially coherent beam with turbulent atmosphere are affected by the correlation-induced of the source and the turbulence-induced of the medium. We will now examine difference of the spectral density distribution of this new beam in non-Kolmogorov turbulence and free space, and trackle their dependence on slope parameter α and structure constant C˜n2 of turbulent medium. The inner and outer scales of the turbulent atmosphere are chosen to be l0=103m and L0=1m,respectively, and other parameters are specified in figure captions. Figure 4 shows the comparison of the spectral density of an OCGSM beam with n = 5 and m = 5 propagating in free space and atmosphere turbulence with α=3.667 and C˜n2=1012m3α. The left parts of Fig. 4 indicate the beam in the free space remains the stable the splitting beam in far field. For the case the right parts of Fig. 4 of the atmosphere turbulence the spectral density is gradually merged with an increasing transmission distance and eventually formed to resemble rectangular Gaussian distribution.
Fig. 5 Changes in spectral density of an OCGSM beam with n = 5 and m = 5 for different parameters α and C˜n2 at the plane z = 5km in the non-Kolmogorov turbulence.
Figure 5 shows the transverse distribution of the spectral density of an OCGSM beam with n = 5 and m = 5 at propagation distance z = 5km in the non-Kolmogorov turbulence for different values of parameters α and C˜n2 . It can be seen from Fig. 5 that the atmosphere turbulence modifies the intensity distribution of beam, the strength of the effect being dependent on α and C˜n2. The value of C˜n2 is greater, the effects of turbulence is more obvious and the four split beams are merged more quickly. The dependence on α is non-monotonic, α=3.1 is a singular point and the beam’s spectral density is destroyed the most at this point.

5. Concluding remarks

In this article we have introduced a class of random sources with properly chosen degree of coherence which can produce foursquare or rectangular or two self-splitting intensity profile in the far field. The suggested form of the degree of coherence (5) is a product of two separable function, which are two modulated Gaussian Schell-model by cosine functions with different modulation parameters n and m in x and y directions. The parameters n and m play key role to the formation of splitting beam and provides a convenient tool for adjusting the number of splitting beams and the spacing between splitting beams. The beam conditions for such source a beamlike are derived and discussed. The analytical formula for the cross-spectral density function of beams on propagation in free and in turbulent atmosphere is derived and used to explore and comparatively analyzed the evolution behavior of the spectral density. We have found that the novel source can produce a self-splitting intensity distribution in the far field in free place as well as at short distance in the turbulence atmosphere, depending on the values of the refractive-index structure parameter C˜n2 and the slope α of the turbulence power spectrum. The results also illustrated that the self-splitting intensity profile is stable when beams propagate in free space, but it is destroy by the atmosphere turbulent, they eventually transformed into a Gaussian profiles when it passes at sufficiently large distances from its source through the turbulent atmosphere. The beams are influenced the most for sufficiently large C˜n2 and for α in the vicinity of value 3.1.

Acknowledgment

The research is supported by the National Natural Science Foundation of China (NSFC) (11247004).

References and links

1.

D. Cassettari, B. Hessmo, R. Folman, T. Maier, and J. Schmiedmayer, “Beam splitter for guided atoms,” Phys. Rev. Lett. 85(26), 5483–5487 (2000). [CrossRef] [PubMed]

2.

A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11(5), 288–290 (1986). [CrossRef] [PubMed]

3.

Z. Zang, T. Minato, P. Navaretti, Y. Hinokuma, M. Duelk, C. Velez, and K. Hamamoto, “High-power (>110mW) superluminescent diodes by using active multimode interferometer,” IEEE Photon. Technol. Lett. 22(10), 721–723 (2010). [CrossRef]

4.

Z. Zang, K. Mukai, P. Navaretti, M. Duelk, C. Velez, and K. Hamamoto, “Thermal resistance reduction in high power superluminescent diodes by using active multi-mode interferometer,” Appl. Phys. Lett. 100(3), 031108 (2012). [CrossRef]

5.

J. P. Torres and L. Torner, “Self-splitting of beams into spatial solitons in planar waveguides made of quadratic nonlinear media,” Opt. Quantum Electron. 29(7), 757–776 (1997). [CrossRef]

6.

A. Suryanto and E. Van Groesen, “Self-splitting of multisoliton bound states in planar Kerr waveguides,” Opt. Commun. 258(2), 264–274 (2006). [CrossRef]

7.

A. W. Snyder, A. V. Buryak, and D. J. Mitchell, “Beam splitting on weak illumination,” Opt. Lett. 23(1), 4–6 (1998). [CrossRef] [PubMed]

8.

V. Tikhonenko, J. Christou, and B. Luther-Davies, “Three dimensional bright spatial soliton collision and fusion in a saturable nonlinear medium,” Phys. Rev. Lett. 76(15), 2698–2701 (1996). [CrossRef] [PubMed]

9.

P. Halevi, “Beam splitting by a plane-parallel absorptive slab,” Opt. Lett. 7(10), 469–470 (1982). [CrossRef] [PubMed]

10.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

11.

S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012). [CrossRef] [PubMed]

12.

Z. Mei, O. Korotkova, and E. Shchepakina, “Electromagnetic multi-Gaussian Schell-model beams,” J. Opt. 15(2), 025705 (2013). [CrossRef]

13.

H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011). [CrossRef] [PubMed]

14.

Z. Tong and O. Korotkova, “Nonuniformly correlated light beams in uniformly correlated media,” Opt. Lett. 37(15), 3240–3242 (2012). [CrossRef] [PubMed]

15.

Z. Tong and O. Korotkova, “Electromagnetic nonuniformly correlated beams,” J. Opt. Soc. Am. A 29(10), 2154–2158 (2012). [CrossRef] [PubMed]

16.

Z. Mei, Z. Tong, and O. Korotkova, “Electromagnetic non-uniformly correlated beams in turbulent atmosphere,” Opt. Express 20(24), 26458–26463 (2012). [CrossRef] [PubMed]

17.

Z. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013). [CrossRef] [PubMed]

18.

Z. Mei and O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. 38(14), 2578–2580 (2013). [CrossRef] [PubMed]

19.

Z. Mei, “Light sources generating self-focusing beams of variable focal length,” Opt. Lett. 39(2), 347–350 (2014). [CrossRef] [PubMed]

20.

O. Korotkova, “Random sources for rectangular far fields,” Opt. Lett. 39(1), 64–67 (2014). [CrossRef] [PubMed]

21.

C. Liang, F. Wang, X. Liu, Y. Cai, and O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014). [CrossRef] [PubMed]

22.

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). [CrossRef]

23.

X. Du, D. Zhao, and O. Korotkova, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere,” Opt. Express 15(25), 16909–16915 (2007). [CrossRef] [PubMed]

24.

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). [CrossRef]

25.

G. Zhou and X. Chu, “Propagation of a partially coherent cosine-Gaussian beam through an ABCD optical system in turbulent atmosphere,” Opt. Express 17(13), 10529–10534 (2009). [CrossRef] [PubMed]

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). [CrossRef] [PubMed]

27.

F. Wang and Y. Cai, “Second-order statistics of a twisted Gaussian Schell-model beam in turbulent atmosphere,” Opt. Express 18(24), 24661–24672 (2010). [CrossRef] [PubMed]

28.

A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Some limitations on optical communication reliability through Kolmogorov and non-Kolmogorov turbulence,” Opt. Commun. 283(7), 1229–1235 (2010). [CrossRef]

29.

O. Korotkova and E. Shchepakina, “Color changes in stochastic light fields propagating in non-Kolmogorov turbulence,” Opt. Lett. 35(22), 3772–3774 (2010). [CrossRef] [PubMed]

30.

X. Ji and X. Li, “M2-factor of truncated partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 28(6), 970–975 (2011). [CrossRef] [PubMed]

31.

Z. Mei, E. Shchepakina, and O. Korotkova, “Propagation of cosine-Gaussian-correlated Schell-model beams in atmospheric turbulence,” Opt. Express 21(15), 17512–17519 (2013). [CrossRef] [PubMed]

32.

Z. Mei and O. Korotkova, “Electromagnetic cosine-Gaussian Schell-model beams in free space and atmospheric turbulence,” Opt. Express 21(22), 27246–27259 (2013). [CrossRef] [PubMed]

33.

F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007). [CrossRef] [PubMed]

OCIS Codes
(010.1330) Atmospheric and oceanic optics : Atmospheric turbulence
(030.1640) Coherence and statistical optics : Coherence
(030.6600) Coherence and statistical optics : Statistical optics

ToC Category:
Coherence and Statistical Optics

History
Original Manuscript: April 18, 2014
Revised Manuscript: May 12, 2014
Manuscript Accepted: May 14, 2014
Published: May 21, 2014

Citation
Zhangrong Mei, "Light sources generating self-splitting beams and their propagation in non-Kolmogorov turbulence," Opt. Express 22, 13029-13040 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-11-13029


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References

  1. D. Cassettari, B. Hessmo, R. Folman, T. Maier, J. Schmiedmayer, “Beam splitter for guided atoms,” Phys. Rev. Lett. 85(26), 5483–5487 (2000). [CrossRef] [PubMed]
  2. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11(5), 288–290 (1986). [CrossRef] [PubMed]
  3. Z. Zang, T. Minato, P. Navaretti, Y. Hinokuma, M. Duelk, C. Velez, K. Hamamoto, “High-power (>110mW) superluminescent diodes by using active multimode interferometer,” IEEE Photon. Technol. Lett. 22(10), 721–723 (2010). [CrossRef]
  4. Z. Zang, K. Mukai, P. Navaretti, M. Duelk, C. Velez, K. Hamamoto, “Thermal resistance reduction in high power superluminescent diodes by using active multi-mode interferometer,” Appl. Phys. Lett. 100(3), 031108 (2012). [CrossRef]
  5. J. P. Torres, L. Torner, “Self-splitting of beams into spatial solitons in planar waveguides made of quadratic nonlinear media,” Opt. Quantum Electron. 29(7), 757–776 (1997). [CrossRef]
  6. A. Suryanto, E. Van Groesen, “Self-splitting of multisoliton bound states in planar Kerr waveguides,” Opt. Commun. 258(2), 264–274 (2006). [CrossRef]
  7. A. W. Snyder, A. V. Buryak, D. J. Mitchell, “Beam splitting on weak illumination,” Opt. Lett. 23(1), 4–6 (1998). [CrossRef] [PubMed]
  8. V. Tikhonenko, J. Christou, B. Luther-Davies, “Three dimensional bright spatial soliton collision and fusion in a saturable nonlinear medium,” Phys. Rev. Lett. 76(15), 2698–2701 (1996). [CrossRef] [PubMed]
  9. P. Halevi, “Beam splitting by a plane-parallel absorptive slab,” Opt. Lett. 7(10), 469–470 (1982). [CrossRef] [PubMed]
  10. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  11. S. Sahin, O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012). [CrossRef] [PubMed]
  12. Z. Mei, O. Korotkova, E. Shchepakina, “Electromagnetic multi-Gaussian Schell-model beams,” J. Opt. 15(2), 025705 (2013). [CrossRef]
  13. H. Lajunen, T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011). [CrossRef] [PubMed]
  14. Z. Tong, O. Korotkova, “Nonuniformly correlated light beams in uniformly correlated media,” Opt. Lett. 37(15), 3240–3242 (2012). [CrossRef] [PubMed]
  15. Z. Tong, O. Korotkova, “Electromagnetic nonuniformly correlated beams,” J. Opt. Soc. Am. A 29(10), 2154–2158 (2012). [CrossRef] [PubMed]
  16. Z. Mei, Z. Tong, O. Korotkova, “Electromagnetic non-uniformly correlated beams in turbulent atmosphere,” Opt. Express 20(24), 26458–26463 (2012). [CrossRef] [PubMed]
  17. Z. Mei, O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013). [CrossRef] [PubMed]
  18. Z. Mei, O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. 38(14), 2578–2580 (2013). [CrossRef] [PubMed]
  19. Z. Mei, “Light sources generating self-focusing beams of variable focal length,” Opt. Lett. 39(2), 347–350 (2014). [CrossRef] [PubMed]
  20. O. Korotkova, “Random sources for rectangular far fields,” Opt. Lett. 39(1), 64–67 (2014). [CrossRef] [PubMed]
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