Scintillation characteristics of cosh-gaussian beams

Scintillation characteristics of cosh-gaussian beams Halil T. Eyyuboǧlu and Yahya Baykal By using the generalized beam formulation, the scintillation index is derived and evaluated for cosh- Gaussian beams in a turbulent atmosphere. Comparisons are made to cos-gaussian and Gaussian beam scintillations. The variations of scintillations against propagation length at different values of displacement and focusing parameters are examined. The dependence of scintillations on source size at different propagation lengths is also investigated. Two-dimensional scintillation index distributions covering the entire transverse receiver planes are given. From the graphic illustrations, it is found that in comparison to pure Gaussian beams cosh-gaussian beams have lower on-axis scintillations at smaller source sizes and longer propagation distances. The focusing effect appears to impose more reduction on the cosh-gaussian beam scintillations than those of the Gaussian beam. The distribution of the off-axis scintillation index values of the Gaussian beams appears to be uniform over the transverse receiver plane, whereas that of the cosh-gaussian beam is arranged according to the position of the slanted axis. 27 Optical Society of America OCIS codes: 1.133, 1.13, 1.331, 6.451. 1. Introduction The formulation and evaluation of the fluctuations of intensity in a turbulent atmosphere have been of interest, thus numerous theoretical and experimental results on weak and strong fluctuations are reported by many scientists. 1 13 These studies cover mainly the scintillations under the plane wave, the spherical wave, and the Gaussian beam wave (fundamental mode) excitations. Especially after the introduction of free-space optical atmospheric communication links into the telecommunications infrastructure, there appeared to be an increasing need for the calculation of the scintillation index in turbulence when different types of sources, other than plane, spherical, or Gaussian beam waves are used. The purpose is to search for the best excitation to minimize the degrading effects of turbulence in atmospheric optical links. In this respect, intensity fluctuations for annular 14 and flattopped beams 15,16 are evaluated. Currently, we are also working on the evaluation of the scintillation index under an arbitrary type of excitation. 17 In this The authors are with the Department of Electronic and Communication Engineering, Çankaya University, Öǧretmenler Caddesi No:14 Yüzüncüyıl 653 Balgat Ankara, Turkey. Y. Baykal s e-mail address is y.baykal@cankaya.edu.tr. Received 27 June 26; accepted 9 October 26; posted 12 October 26 (Doc. ID 72377); published 12 February 27. 3-6935/7/7199-8$15./ 27 Optical Society of America paper we examine the behavior of the scintillation in a turbulent atmosphere when cosh-gaussian beams are employed as the source in an atmospheric optical communications link. To form a basis for the current work in this paper, our earlier studies have already examined the log-amplitude correlations 18 and the average intensity of cosh-gaussian incidence 19 in atmospheric turbulence. 2. Formulation We contemplate a source plane held vertical to the axis of propagation z. On that source plane, the transverse position is designated by the vector s s x, s y.a general source beam field consisting of the lowestorder components and cocentric with the source point of s x s y, will be described by 2 u s s u s s x, s y l 1 N Al exp i l exp.5k xl s x 2 iv xl s x exp.5k yl s y 2 iv yl s y, (1) where A l and l refer to the amplitude and the phase, respectively, of the th beam in the summation, xl 1 k sxl2 i F xl, yl 1 k syl2 i F yl, (2) where sxl and syl are Gaussian source sizes, F xl and F yl are the source focusing parameters along the s x 1 March 27 Vol. 46, No. 7 APPLIED OPTICS 199

and s y directions, k 2 is the wavenumber with being the wavelength and i 1.5. V xl and V yl are used to create physical dislocations and phase rotations for the source field and will be hereafter named as displacement parameters. As explained in Ref. 2, a cosh-gaussian beam is acquired by setting the displacement parameters as purely imaginary quantities and implementing a summation over two terms, i.e., N 2 in Eq. (1). On a receiver plane located at a distance z L away from the source plane, the log-amplitude correlation function, B p, L of a general beam can be constructed as explained in Ref. 2. From there, for 1 k sl 2 i F l, the functions S 1 p, L,,, and S 2 p, L,,, can be stated in terms of the source and propagation parameters in the following manner: S 1 p, L,,, S N p, L,,, S N p, L,,,, D 2 p, L (4) S 2 p, L,,, S N p, L,,, S N * p, L,,, D p, L 2, where (5) N ik S N p, L,,, l 1 Al e i l 1 i l L exp k p 2 x p y2 l 2 1 i exp i V xlp x V ylp y l L 1 i l L exp i V 2 xl V yl2 L 2k 1 i l L exp i L V xl cos V yl sin k 1 i l L exp i 1 i l p x cos p y sin 1 i l L exp.5i L 1 i l 2 k 1 i l L, (6) N 1 D p, L l 1 Al e i l 1 i l L exp k p 2 x p y2 l 2 1 i exp i V xlp x V ylp y l L 1 i l L exp i V 2 xl V yl2 L 2k 1 i. (7) l L weak turbulence conditions, we obtain m 2 p, L, the scintillation index at a vectorial position p on the receiver plane as follows m 2 p, L 4B p, L L 4 Re d d 2 d S 1 p, L,,, S 2 p, L,,, n, (3) Here Re means the real part, is the distance variable along the propagation axis,, is the 2D spatial frequency in polar coordinates, and n is the spectral density for the index-of-refraction fluctuations. In Eq. (3), to make the weak turbulence approximation, which is m 2 p, L 4B p, L, requiring 4 2 1( 2 being the log-amplitude variance), it is necessary that the integrated contribution of the frequency, structure constant, and the path length should be relatively small. Using the definitions given in Ref. 2, and assuming xl yl l From the above notation, it is understood that x y symmetry is restricted to the source size sl and the focusing parameter F l, whereas the displacement parameters V xl and V yl are permitted to be asymmetric in x y. Upon substituting Eqs. (6) and (7) into Eqs. (4) and (5), and using the von Karman spectrum for the spectral density for the index-of-refraction fluctuations, i.e., n.33c n 2 exp 2 m2 2 2 11 6, (8) where C n 2 is the structure constant and m and are related to the inverse of the inner and outer scales of turbulence, respectively, the integration in Eq. (3) can be performed by using Eq. (3.937.2) of Ref. 21. The resulting expression contains two modified Bessel functions of the first kind. To solve the remaining integral over, we initially expand these Bessel functions into infinite series, then to each term of the expansion we apply Eq. (17) of Appendix II in Ref. 3. In the end the m 2 p, L expression of Eq. (3) becomes 11 APPLIED OPTICS Vol. 46, No. 7 1 March 27

m 2 p, L 1.328C n 2 k 2 Re D p, L 2 N N l1 1 l 2 1 r Al1 A * exp l2 i 1 1 l1 l2 1 i l1 L 1 i l2 *L 2r 5 3 exp k p 2 x p y2 l1 2 1 i l1 L k p 2 x p y2 l2 * 2 1 i exp i V xl1 p x V yl1 p y l2 *L 1 i l1 L exp i V 2 xl1 V yl12 L 2k 1 i l1 L i V xl 2 2 1 i l2 *L i L V xl2 * k 1 i l2 *L i 1 i l2 * p x i 1 i l2 * p 2 y 1 i l2 *L D 2 N p, L l1 1 r 2 * V yl22 * L 2k 1 i l2 *L L.25 r r! 1 i l2 *L i V xl2 *p x V yl2 *p y d i L V xl1 k 1 i l1 L i 1 i l1 p x 1 i l1 L i L V yl1 k 1 i l1 L i 1 i l1 p y i L V yl2 * 1 i l1 L k 1 i l2 *L U r 1, r 1 6,.5 L i 1 i l1 k 1 i l1 L i 1 i l2 * N Al1 l 2 1 r A exp l2 i 1 1 l1 l2 1 i l1 L 1 i l2 L 2 1 i l1 L k p 2 x p y2 l2 2 1 i exp i V xl1 p x V yl1 p y l2 L 1 i l1 L exp k p 2 x p y2 l1 exp i V 2 xl1 V yl12 L 2k 1 i l1 L i V 2 xl2 V yl22 L i L V xl2 k 1 i l2 L i 1 i l2 p x i 1 i l2 p 2 y 1 i l2 L r 2 1 i l2 L 2k 1 i l2 L L k 1 i l2 *L 1.25 r r! 2r 5 3 i V xl2 p x V yl2 p y 1 i l2 L d i L V xl1 k 1 i l1 L i 1 i l1 p x 1 i l1 L i L V yl1 k 1 i l1 L i 1 i l1 p y i L V yl2 1 i l1 L k 1 i l2 L U r 1, r 1 6,.5 L i 1 i l1 k 1 i l1 L i 1 i l2 k 1 i l2 L 1 m 2 m 2 2 2, 9 where! denotes the factorial and U is the confluent hypergeometric function of second kind. In the numerical calculations of the summation for r, eight terms are found to be sufficient for most cases. However, this number has to be increased as high as 25 if the computation involves excessively large values of V x and V y. Note that the denominator function, D p, L, as defined in Eq. (7), is free from integral variables and hence placed outside the integral sign. The consistency of Eq. (9) with previous results for a Gaussian beam can be verified by taking the limits of N 1, V x V y, m,, which means the plain Gaussian case with Kolmogorov spectrum. Then Eq. (9) after being divided by 4, will correctly reduce to Eq. (18 29) of Ref. 2. 3. Results and Discussions In this section, graphic illustrations are provided based on the numerical evaluation of Eq. (9). Although the scintillation index expression of Eq. (9) is able to generate results for any type of beam composed from the summation of different fundamental Gaussian beams, in the current study, we concentrate on the cosh-gaussian beam and its comparison to Gaussian and cos-gaussian cases. In Eq. (9), a cosh-gaussian beam is obtained by assigning N 2, V xl V yl iv x for the first beam, V xl V yl iv x for the second beam, and making the rest of the source parameters identical in both the first and second terms of the summations. A cos-gaussian beam is also constructed by taking two beams, i.e., N 2, but in this instance, the displacement parameters are transformed into purely real quantities, that is V xl V yl V x for the first beam and V xl V yl V x for the second beam. For all cases, the phase parameter, l is taken to be zero. The graphs are uniformly produced at a single structure constant and wavelength, which are C n 2 1 15 m 2 3, 1.55 m. Together with the propagation lengths employed, these choices make our results applicable to weak turbulence regimes. Unless otherwise, stated and displayed collimated beams, on-axis situations, zero inner turbulence scales, and infinity outer turbulence scales are considered. This means, F, p x p y, m,. As a general rule, our graphic illustrations will quote only the parameter values left unspecified here, with the indexing being arranged so as to remove the subscript l. Initially the effect of the displacement parameter on the scintillations of the cosh-gaussian beam is explored. To this end, Fig. 1 shows that, when compared to a Gaussian beam, raising the displacement parameter will cause the cosh-gaussian to attain lower scintillations at longer propagation distances. 1 March 27 Vol. 46, No. 7 APPLIED OPTICS 111

Fig. 1. Scintillation index of cosh-gaussian beam at selected values of displacement parameters and scintillation index of a single cos-gaussian beam versus propagation distance. However, higher-displacement parameters will simultaneously move the crossover point between cosh- Gaussian and Gaussian beams toward greater distances such that, for instance when the absolute value of the displacement parameter is much greater than 1 s, that is when V x 2i, this crossover point is pushed well beyond the propagation distances considered in the figure. Figure 1 also demonstrates that, for the chosen displacement parameter value, the cos-gaussian beam, when compared to the Gaussian beam is advantageous only at shorter propagation ranges, in this way it acts in a reciprocal manner to the cosh-gaussian beam. 22 Next, we examine the impact of the focusing parameter. Figure 2 is plotted by retaining all the source and propagation parameters of Fig. 1, but switching to a focusing parameter of F 1 m. Comparing Figs. 1 and 2, finite focusing seems to help reduce scintillations for all beams except the cos-gaussian one. On the other hand, the incremental scintillation reductions for the cosh-gaussian beam appear to be substantially larger than the incremental reduction for the Gaussian beam. This point is Fig. 2. Scintillation index of beams from Fig. 1 with focusing introduced. 112 APPLIED OPTICS Vol. 46, No. 7 1 March 27

Fig. 3. Scintillation index of beams from Fig. 1, off-axis case with p x p y.5 cm. particularly highlighted by the realization of a crossover point of V x 2i curve with the Gaussian curve in Fig. 2, while such a crossover point is not observed in Fig. 1. Figure 3 displays the variation of scintillations for the same group of beams belonging to Fig. 1, at an off-axis position, namely at p x p y.5 cm. By comparing Figs. 1 and 3, we notice that at an off-axis location, the scintillation characteristics may be quite different than those of the on-axis location. This observation is significant from a practical point of view as well, since a receiver aperture will attempt to capture the beam power over a finite area rather than a single spot. In Fig. 4, we find the graphs of the scintillation index plotted against the source size at two distinct propagation lengths of L 1 and 3 km. Figure 4 indicates that, compared to Gaussian beams, smaller source-sized cosh-gaussian beams will have less scintillations, while the reverse is true for cos- Gaussian beams. It is further revealed by the upper group of curves in Fig. 4 that, the useful region of cosh-gaussian beam, where it offers lower scintillations, is enlarged at extended propagation lengths. In all cases, however, the behavior of the scintillation Fig. 4. Scintillation index of cosh-gaussian, Gaussian, and cos-gaussian beams versus source size at selected values of propagation lengths. 1 March 27 Vol. 46, No. 7 APPLIED OPTICS 113

Fig. 5. Scintillation index distribution of collimated and focused cosh-gaussian and Gaussian beams over the receiver plane. index against the source size for both cosh-gaussian and cos-gaussian beams is similar to the behavior of the Gaussian beam. For the cosh-gaussian and cos- Gaussian beam plots of Fig. 4, the absolute value of the displacement parameter is set to the inverse of the respective Gaussian source size. In Fig. 5, we show the scintillation index distribution over the entire transverse receiver plane at L 2 km for Gaussian and cosh-gaussian beams where the source is commonly taken as s 1cm. Here the upper plots refer to collimated beams, while the lower ones are those of focused beams with F 1 m. Again, the previously mentioned positive effect of focusing is clearly visible for both Gaussian and cosh-gaussian beams. It is noted from Fig. 5 that for the Gaussian beam, the scintillation values at different p x, p y locations have rotational symmetry, whereas the scintillation values of cosh-gaussian beam are symmetric with respect to the two slanted axes. Furthermore the scintillation index distribution of Gaussian beam is relatively flat, that is, only slight increases are seen toward the edges. In comparison, the cosh-gaussian beam has much lower scintillation values especially around the on-axis region, with a tendency to rise toward the edges, differently along the two slanted axes. This is explained in more detail toward the end of this paragraph when discussing Fig. 6 in relation to Fig. 5. Figure 6 provides the intensity distribution of the same cosh- Gaussian beam belonging to Fig. 5 at the source and receiver planes arranged in the form of overlaid contour plots. Figure 6 indicates that after propagation the higher intensities of the beam become concentrated along the slanted axis, which is opposite to that of the source. In this manner, examining the second and fourth pictures (i.e., the two cosh- Gaussian beam pictures) of Fig. 5 together with Fig. 6, we witness that the alignment of scintillation index values is in perfect agreement with the intensity distributions of Fig. 6 and subsequently the orientation of the slanted axis. More specifically, the lower scintillations are encountered at the higher intensity points of Fig. 6, while the reverse happens for larger scintillations. To establish better cross-referencing, the slanted axis corresponding to lower scintillations, simultaneously to higher intensity points is also inserted as a broken line on the p x, p y plane in the second picture of Fig. 5. Finally, Fig. 7 shows how a cosh-gaussian beam is affected by the different levels of inner and outer scales of turbulence. We note that the inner and outer Fig. 6. Intensity distribution of collimated cosh-gaussian beam from to Fig. 5 at source and receiver planes. 114 APPLIED OPTICS Vol. 46, No. 7 1 March 27

Fig. 7. Scintillation index distribution of cosh-gaussian beam over the receiver plane at selected inner and outer scales of turbulence. scale parameters, l and L are related to m and of Eq. (8), respectively, by l 5.92 m and L 2. From the four successive pictures placed in Fig. 7, it is found as expected that the inner and outer scales of turbulence values other than l and L will produce smaller scintillations. 4. Conclusion The scintillation index of cosh-gaussian beams in turbulent atmosphere is formulated and evaluated at various source and turbulence parameters. Our evaluations at each step are compared with the known Gaussian beam scintillations and in some situations with cos-gaussian scintillations. For the collimated case, beyond a crossover point determined by the magnitude of the displacement parameter, it is observed that cosh-gaussian beams will have lower on-axis scintillations than the Gaussian beam on-axis scintillations at longer propagation distances. Below this crossover point, however, the reverse will take place. This crossover involving the propagation distance is advanced further for cosh beams with larger displacement parameters. When a finite focusing parameter is introduced, it is seen that on-axis scintillations of both cosh-gaussian and Gaussian beams are reduced as compared to the corresponding collimated beam on-axis scintillations. However, the incremental reduction in cosh-gaussian beam scintillation is higher as compared to the incremental reduction in Gaussian beam scintillation. When the scintillations are evaluated at the off-axis position in the receiver plane, it is observed that the scintillation characteristic is quite different than that of the on-axis scintillation. When the scintillation index of the cosh-gaussian beam is compared to the Gaussian beam scintillation index in terms of source sizes, it can be asserted that the small size cosh-gaussian sources will have less scintillations. The behavior of the scintillation index of the cosh-gaussian beam versus the source size follows a similar trend to that of the Gaussian beam scintillation. Examining the scintillations over the entire transverse receiver plane, we generally find lower scintillation values around the on axis region, with a tendency to rise toward the receiver plane edges. Again, beams with finite focusing parameters have smaller scintillations than the corresponding collimated beams, a property being equally valid for both cosh-gaussian and Gaussian beams. It is noted that for the Gaussian beam the scintillation values at different transverse receiver locations are in conformity with the associated intensity distributions. That is, lower scintillations will occur at points of higher intensity, whereas the opposite will be applicable for larger scintillations. Finally, the effect of the inner and outer scales of turbulence on the cosh-gaussian beam scintillations is investigated, and as expected, the scintillations evaluated at nonzero inner scale and finite outer scale are always lower compared with evaluations using the Kolmogorov spectrum. References 1. V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961). 2. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978), Vol. 2. 3. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, Bellingham, 25). 4. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE, 21). 5. V. A. Banakh and V. L. Mironov, Influence of the diffraction size of a transmitting aperture and the turbulence spectrum on 1 March 27 Vol. 46, No. 7 APPLIED OPTICS 115

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