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
Self-splitting of beams is a subject of constant intense investigation due to a fascinating phenomena encountered and their potential applications in optical processing, such as atoms guidance , particle trap , high power superluminescent diodes designation [3,4], etc. A number of theoretical and experimental techniques have been reported to predict and observe this phenomena, such as self-spitting of beams in planar waveguides [5,6], in a non-Kerr medium , in a saturable nonlinear medium , by a plane-parallel absorptive slab , etc. These methods are realized by special medium or beam splitter. It is well know the coherence properties of a field in the source plane are closely related to the propagation characteristics of optical fields and the transverse intensity distribution of the far field . Therefore, it is possible to generate prescribed far field intensity distributions by choosing specified spatial correlation functions of the source field, such as the multi-Gaussian Schell-model sources generating far fields with tunable flat profiles [11,12], the non-uniformly correlated partially coherent sources leading to self-focusing and laterally shifted intensity maxima [13–16]; the modulated Gaussian Schell-model sources generating a dark-hollow profile [17,18], the non-uniformly cosine-Gaussian correlated source generating self-focusing beams of variable focal length , the rectangular Gaussian Schell-model source producing far fields with rectangular flat profiles , the rectangular cosine-Gaussian Schell-model generating four-beamlets array profiles , and so on. On the other hand, the interactions between random light fields and turbulent atmosphere certainly affect the evolving beam’s characteristics. The propagation theory based on either the Rytov method or the extended Huygens-Fresnel principle plays an important role in predicting statistically averaged beam characteristics. Based on this theory, some interesting researches were performed to analyze the main set of the statistical properties of the light field propagating in turbulent atmosphere [22–32]. In this article, we introduce a class of random sources producing far fields self-splitting intensity profiles with variable and unequal spacing between the and directions by modeling the source degree of coherence with the help of two asymmetric one-dimensional cosine-Gaussian Schell-model, derive the beam conditions for ensuring the sources satisfaction to generate a beam, and explore and comparatively analyze the behavior of the beams on free-space and non-Kolmogorov’s atmospheric turbulence propagation.
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 and and at the oscillation angular frequency (for brevity that is not explicitly shown in the following equation) can be described by means of the cross-spectral density (CSD) function . For a Schell-model source, the CSD function has the form 33] that for a CSD to be genuine, i.e., physically realizable, it suffices to have a superposition integral of the formEqs. (1) and (4), we can find that the nonnegative function and the spectral degree of coherent are a Fourier transform pair.
Let us now consider a simple variation to the degree of coherence in Ref . and set the spectral degree of coherence in the source plane to beFigure 1 shows the absolute value of the degree of coherence (5) for several values of parameters and : (a) , which represent the conventional Gaussian Schell-model; (b) and , only the degree of coherence in the direction is modulated; (c) and , only the degree of coherence in the direction is modulated; (d) and , the modulation of the degree of coherence in the direction is severer than in the direction; (e) and , the modulation of the degree of coherence in the direction is severer than in the direction; (f) and , 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 in Eq. (2), we calculate the Fourier transform of Eq. (5) and arrive at
Let us also set the Gaussian profile for function :Eq. (7) together with the weight function Eq. (6), we obtain on substituting them into Eq. (4) the CSD function of the form
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 and specified by position vectors and () can be written as the following expression 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 sourceEq. (12) must be negligible except for directions within a narrow solid angle about the z axis . Since the value of hyperbolic cosine function is always greater than 1, this is so if
3. Propagation laws in non-Kolmogorov turbulence and free space
The paraxial propagation of the fields defined by Eq. (8) can be studied by using the Huygens-Fresnel principle, the CSD function propagating to the positive z direction at two positions and in any transverse z plane of the half-space , filled with turbulent atmosphere is related to those in the source plane as Eq. (15) is shown to be approximated by the expression26]Eq. (17) is a generalized refractive-index structure parameter with units . With the power spectrum in Eq. (17) the integral in Eq. (16) becomesEq. (22), thus, for the free space propagation, Eq. (22) can be express as
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.
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) . Next, we analyze the impact of the different modulation coefficient to spectral density distribution in far field, as is shown in Fig. 3.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 , the light field split into four equally spaced beams in far-field, as shown in Fig. 3(f).
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 and structure constant of turbulent medium. The inner and outer scales of the turbulent atmosphere are chosen to be and 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 and . 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. 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 and . 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 and . The value of is greater, the effects of turbulence is more obvious and the four split beams are merged more quickly. The dependence on is non-monotonic, 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 and the slope 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 and for in the vicinity of value 3.1.
The research is supported by the National Natural Science Foundation of China (NSFC) (11247004).
References and links
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]
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]
10. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
12. Z. Mei, O. Korotkova, and E. Shchepakina, “Electromagnetic multi-Gaussian Schell-model beams,” J. Opt. 15(2), 025705 (2013). [CrossRef]
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]
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]
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]