Scintillation characteristics of cosh-Gaussian beams

Scintillation characteristics of cosh-Gaussian beams

Halil T. Eyyuboglu 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 beamscintillations. The variations of scintillations against propagation length at different values of displace-ment and focusing parameters are examined. The dependence of scintillations on source size at differentpropagation lengths is also investigated. Two-dimensional scintillation index distributions covering theentire transverse receiver planes are given. From the graphic illustrations, it is found that in compar-ison to pure Gaussian beams cosh-Gaussian beams have lower on-axis scintillations at smaller sourcesizes and longer propagation distances. The focusing effect appears to impose more reduction on thecosh-Gaussian beam scintillations than those of the Gaussian beam. The distribution of the off-axisscintillation 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. © 2007Optical Society of America

OCIS codes: 010.1330, 010.1300, 010.3310, 060.4510.

1. Introduction

The formulation and evaluation of the fluctuations ofintensity in a turbulent atmosphere have been of in-terest, thus numerous theoretical and experimentalresults on weak and strong fluctuations are reportedby many scientists.1–13 These studies cover mainly thescintillations under the plane wave, the sphericalwave, and the Gaussian beam wave (fundamentalmode) excitations. Especially after the introduction offree-space optical atmospheric communication linksinto the telecommunications infrastructure, there ap-peared to be an increasing need for the calculation ofthe scintillation index in turbulence when differenttypes of sources, other than plane, spherical, or Gauss-ian beam waves are used. The purpose is to search forthe best excitation to minimize the degrading effectsof turbulence in atmospheric optical links. In thisrespect, intensity fluctuations for annular14 and flat-topped beams15,16 are evaluated. Currently, we arealso working on the evaluation of the scintillation in-dex under an arbitrary type of excitation.17 In this

paper we examine the behavior of the scintillation in aturbulent atmosphere when cosh-Gaussian beams areemployed as the source in an atmospheric optical com-munications link. To form a basis for the current workin this paper, our earlier studies have already exam-ined the log-amplitude correlations18 and the averageintensity of cosh-Gaussian incidence19 in atmosphericturbulence.

2. Formulation

We contemplate a source plane held vertical to theaxis of propagation z. On that source plane, the trans-verse position is designated by the vector s � �sx, sy�. Ageneral source beam field consisting of the lowest-order components and cocentric with the source pointof sx � sy � 0, will be described by20

us�s� � us�sx, sy� � �l�1

N

Al exp��i�l�

� exp��0.5k�xlsx2 � iVxlsx��

� exp��0.5k�ylsy2 � iVylsy��, (1)

where Al and �l refer to the amplitude and the phase,respectively, of the �th beam in the summation,

�xl � 1��k�sxl2� � i�Fxl, �yl � 1��k�syl

2� � i�Fyl, (2)

where �sxl and �syl are Gaussian source sizes, Fxl andFyl are the source focusing parameters along the sx

The authors are with the Department of Electronic and Com-munication Engineering, Çankaya University, Ögretmenler Cad-desi No:14 Yüzüncüyıl 06530 Balgat Ankara, Turkey. Y. Baykal’se-mail address is [email protected].

Received 27 June 2006; accepted 9 October 2006; posted 12October 2006 (Doc. ID 72377); published 12 February 2007.

0003-6935/07/071099-08$15.00/0© 2007 Optical Society of America

1 March 2007 � Vol. 46, No. 7 � APPLIED OPTICS 1099

and sy directions, k � 2�� is the wavenumber with� being the wavelength and i � ��1�0.5. Vxl and Vyl areused to create physical dislocations and phase rota-tions for the source field and will be hereafter namedas displacement parameters. As explained in Ref. 20,a cosh-Gaussian beam is acquired by setting the dis-placement parameters as purely imaginary quanti-ties and implementing a summation over two terms,i.e., N � 2 in Eq. (1).

On a receiver plane located at a distance z � Laway from the source plane, the log-amplitude corre-lation function, B�p, L� of a general beam can beconstructed as explained in Ref. 20. From there, for

weak turbulence conditions, we obtain m2�p, L�, thescintillation index at a vectorial position p on thereceiver plane as follows

m2�p, L� � 4B�p, L�

� 4� Re��0

L

d �0

�

�d� �0

2�

d��S1�p, L, , �, ��

� S2�p, L, , �, �� n��, (3)

Here Re means the real part, � is the distance vari-able along the propagation axis, ��, �� is the 2D spa-tial frequency in polar coordinates, and n�� is thespectral density for the index-of-refraction fluctua-tions. In Eq. (3), to make the weak turbulence ap-proximation, which is m2�p, L� � 4B�p, L�, requiring4�

2 � 1 (�2 being the log-amplitude variance), it is

necessary that the integrated contribution of thefrequency, structure constant, and the path lengthshould be relatively small. Using the definitionsgiven in Ref. 20, and assuming �xl � �yl � �l �

1��k�sl2� � i�Fl, the functions S1�p, L, , �, �� and

S2�p, L, , �, �� can be stated in terms of the sourceand propagation parameters in the following manner:

S1�p, L, , �, �� SN�p, L, , �, ��SN�p, L, , ��, ��

D2�p, L�,

(4)

S2�p, L, , �, �� SN�p, L, , �, ��SN*�p, L, , �, ��

�D�p, L��2 ,

(5)

where

From the above notation, it is understood that x–ysymmetry is restricted to the source size �sl and thefocusing parameter Fl, whereas the displacement pa-rameters Vxl and Vyl are permitted to be asymmetricin x–y.

Upon substituting Eqs. (6) and (7) into Eqs. (4)and (5), and using the von Karman spectrum for thespectral density for the index-of-refraction fluctua-tions, i.e.,

n�� 0.033Cn2 exp��2��m

2��2 � �02�11�6, (8)

where Cn2 is the structure constant and �m and �0 are

related to the inverse of the inner and outer scalesof turbulence, respectively, the � integration in Eq.(3) can be performed by using Eq. (3.937.2) of Ref.21. The resulting expression contains two modifiedBessel functions of the first kind. To solve the re-maining integral over �, we initially expand theseBessel functions into infinite series, then to eachterm of the expansion we apply Eq. (17) of AppendixII in Ref. 3. In the end the m2�p, L� expression of Eq.(3) becomes

SN�p, L, , �, �� l�1

N

Ale�i�l

ik1 � i�lL

exp�k�px

2 � py2��l

2�1 � i�lL� �exp�i�Vxlpx � Vylpy�

1 � i�lL �� exp�

i�Vxl2 � Vyl

2�L2k�1 � i�lL� �expi�L � ��Vxl cos � � Vyl sin ��

k�1 � i�lL� �� expi�1 � i�l��px cos � � py sin ��

1 � i�lL �exp�0.5i�L � ��1 � i�l��2

k�1 � i�lL� �, (6)

D�p, L� � �l�1

N

Ale�i�l

11 � i�lL

exp�k�px

2 � py2��l

2�1 � i�lL� �exp�i�Vxlpx � Vylpy�

1 � i�lL �� exp�

i�Vxl2 � Vyl

2�L2k�1 � i�lL� �. (7)

1100 APPLIED OPTICS � Vol. 46, No. 7 � 1 March 2007

where ! denotes the factorial and U is the confluenthypergeometric function of second kind. In the nu-merical calculations of the summation for r, eightterms are found to be sufficient for most cases. How-ever, this number has to be increased as high as 25if the computation involves excessively large valuesof Vx and Vy. Note that the denominator function,D�p, L�, as defined in Eq. (7), is free from integralvariables and hence placed outside the integral sign.

The consistency of Eq. (9) with previous results fora Gaussian beam can be verified by taking the limitsof N � 1, Vx � Vy � 0, �m → �, �0 � 0, which meansthe plain Gaussian case with Kolmogorov spectrum.Then Eq. (9) after being divided by 4, will correctlyreduce to Eq. (18–29) of Ref. 2.

3. Results and Discussions

In this section, graphic illustrations are providedbased on the numerical evaluation of Eq. (9). Al-though the scintillation index expression of Eq. (9) isable to generate results for any type of beam com-posed from the summation of different fundamentalGaussian beams, in the current study, we concen-trate on the cosh-Gaussian beam and its comparisonto Gaussian and cos-Gaussian cases. In Eq. (9), acosh-Gaussian beam is obtained by assigning N � 2,Vxl � Vyl � iVx for the first beam, Vxl � Vyl � �iVx for

the second beam, and making the rest of the sourceparameters identical in both the first and secondterms of the summations. A cos-Gaussian beam isalso constructed by taking two beams, i.e., N � 2, butin this instance, the displacement parameters aretransformed into purely real quantities, that is Vxl

� Vyl � Vx for the first beam and Vxl � Vyl � �Vx forthe second beam. For all cases, the phase parameter,�l is taken to be zero. The graphs are uniformly pro-duced at a single structure constant and wavelength,which are Cn

2 � 10�15 m�2�3, � � 1.55 �m. Togetherwith the propagation lengths employed, these choicesmake our results applicable to weak turbulence re-gimes. Unless otherwise, stated and displayed colli-mated beams, on-axis situations, zero inner turbulencescales, and infinity outer turbulence scales are consid-ered. This means, F → �, px � py � 0, �m → �,�0 � 0. As a general rule, our graphic illustrationswill quote only the parameter values left unspecifiedhere, with the indexing being arranged so as to re-move the subscript l.

Initially the effect of the displacement parameteron the scintillations of the cosh-Gaussian beam isexplored. To this end, Fig. 1 shows that, when com-pared to a Gaussian beam, raising the displacementparameter will cause the cosh-Gaussian to attainlower scintillations at longer propagation distances.

m2�p, L� � 1.3028Cn2k2 Re�D�p, L��2 �

l1�1

N

�l2�1

N

�r�0

�

Al1Al2* exp��i��l1 � �l2��1

�1 � i�l1L�1

�1 � i�l2*L��0.25�r

r!

� �02r��5�3� exp�

k�px2 � py

2��l1

2�1 � i�l1L��

k�px2 � py

2��l2*2�1 � i�l2*L� �exp�

i�Vxl1px � Vyl1py�1 � i�l1L

�i�Vxl2*px � Vyl2*py�

1 � i�l2*L �� exp��

i�Vxl12 � Vyl1

2�L2k�1 � i�l1L�

�i��Vxl2

2�* � �Vyl22�*�L

2k�1 � i�l2*L� ��0

L

d� i�L � �Vxl1

k�1 � i�l1L��

i�1 � i�l1�px

1 � i�l1L

�i�L � �Vxl2*k�1 � i�l2*L�

�i�1 � i�l2*�px

1 � i�l2*L �2

� i�L � �Vyl1

k�1 � i�l1L��

i�1 � i�l1�py

1 � i�l1L�

i�L � �Vyl2*k�1 � i�l2*L�

�i�1 � i�l2*�py

1 � i�l2*L �2r

U�r � 1, r � 1�6, 0.5�L � � i�1 � i�l1�k�1 � i�l1L�

�i�1 � i�l2*�k�1 � i�l2*L�

�1

�m2��0

2 � D�2�p, L� �

l1�1

N

�l2�1

N

�r�0

�

Al1Al2 exp��i��l1 � �l2��1

�1 � i�l1L�1

�1 � i�l2L��0.25�r

r! �02r��5�3�

� exp�k�px

2 � py2��l1

2�1 � i�l1L��

k�px2 � py

2��l2

2�1 � i�l2L� �exp�i�Vxl1px � Vyl1py�

�1 � i�l1L��

i�Vxl2px � Vyl2py��1 � i�l2L� �

� exp�i�Vxl1

2 � Vyl12�L

2k�1 � i�l1L��

i�Vxl22 � Vyl2

2�L2k�1 � i�l2L� ��

0

L

d� i�L � �Vxl1

k�1 � i�l1L��

i�1 � i�l1�px

1 � i�l1L

�i�L � �Vxl2

k�1 � i�l2L��

i�1 � i�l2�px

1 � i�l2L�2

� i�L � �Vyl1

k�1 � i�l1L��

i�1 � i�l1�py

1 � i�l1L�

i�L � �Vyl2

k�1 � i�l2L�

�i�1 � i�l2�py

1 � i�l2L�2r

U�r � 1, r � 1�6, 0.5�L � � i�1 � i�l1�k�1 � i�l1L�

�i�1 � i�l2�k�1 � i�l2L�

�1

�m2��0

2 �,

�9�


However, higher-displacement parameters will si-multaneously move the crossover point between cosh-Gaussian and Gaussian beams toward greaterdistances such that, for instance when the absolutevalue of the displacement parameter is much greaterthan 1��s, that is when Vx � 200i, this crossover pointis pushed well beyond the propagation distances con-sidered in the figure. Figure 1 also demonstratesthat, for the chosen displacement parameter value,the cos-Gaussian beam, when compared to the Gauss-ian beam is advantageous only at shorter propaga-

tion ranges, in this way it acts in a reciprocal mannerto the cosh-Gaussian beam.22 Next, we examine theimpact of the focusing parameter. Figure 2 is plottedby retaining all the source and propagation parame-ters of Fig. 1, but switching to a focusing parameter ofF � 1000 m. Comparing Figs. 1 and 2, finite focusingseems to help reduce scintillations for all beams exceptthe cos-Gaussian one. On the other hand, the incre-mental scintillation reductions for the cosh-Gaussianbeam appear to be substantially larger than the incre-mental reduction for the Gaussian beam. This point is

Fig. 1. Scintillation index of cosh-Gaussian beam at selected values of displacement parameters and scintillation index of a singlecos-Gaussian beam versus propagation distance.

Fig. 2. Scintillation index of beams from Fig. 1 with focusing introduced.


particularly highlighted by the realization of a cross-over point of Vx � 200i curve with the Gaussian curvein Fig. 2, while such a crossover point is not observedin Fig. 1.

Figure 3 displays the variation of scintillations forthe same group of beams belonging to Fig. 1, at anoff-axis position, namely at px � py � 0.5 cm. Bycomparing Figs. 1 and 3, we notice that at an off-axislocation, the scintillation characteristics may be quitedifferent than those of the on-axis location. This ob-servation is significant from a practical point of viewas well, since a receiver aperture will attempt to cap-

ture the beam power over a finite area rather than asingle spot. In Fig. 4, we find the graphs of the scin-tillation index plotted against the source size at twodistinct propagation lengths of L � 1 and 3 km. Fig-ure 4 indicates that, compared to Gaussian beams,smaller source-sized cosh-Gaussian beams will haveless scintillations, while the reverse is true for cos-Gaussian beams. It is further revealed by the uppergroup of curves in Fig. 4 that, the useful region ofcosh-Gaussian beam, where it offers lower scintilla-tions, is enlarged at extended propagation lengths. Inall cases, however, the behavior of the scintillation

Fig. 3. Scintillation index of beams from Fig. 1, off-axis case with px � py � 0.5 cm.

Fig. 4. Scintillation index of cosh-Gaussian, Gaussian, and cos-Gaussian beams versus source size at selected values of propagationlengths.


index against the source size for both cosh-Gaussianand cos-Gaussian beams is similar to the behavior ofthe Gaussian beam. For the cosh-Gaussian and cos-Gaussian beam plots of Fig. 4, the absolute value ofthe displacement parameter is set to the inverse ofthe respective Gaussian source size.

In Fig. 5, we show the scintillation index distri-bution over the entire transverse receiver plane atL � 2 km for Gaussian and cosh-Gaussian beamswhere the source is commonly taken as �s � 1 cm.Here the upper plots refer to collimated beams, whilethe lower ones are those of focused beams withF � 1000 m. Again, the previously mentioned posi-tive effect of focusing is clearly visible for both Gauss-ian and cosh-Gaussian beams. It is noted from Fig. 5that for the Gaussian beam, the scintillation valuesat different �px, py� locations have rotational symme-try, whereas the scintillation values of cosh-Gaussianbeam are symmetric with respect to the two slantedaxes. Furthermore the scintillation index distribu-tion of Gaussian beam is relatively flat, that is, onlyslight increases are seen toward the edges. In com-parison, the cosh-Gaussian beam has much lowerscintillation values especially around the on-axis re-gion, with a tendency to rise toward the edges, dif-ferently along the two slanted axes. This is explainedin more detail toward the end of this paragraph whendiscussing Fig. 6 in relation to Fig. 5. Figure 6 pro-vides the intensity distribution of the same cosh-Gaussian beam belonging to Fig. 5 at the source andreceiver planes arranged in the form of overlaid con-tour plots. Figure 6 indicates that after propagationthe higher intensities of the beam become con-centrated along the slanted axis, which is oppositeto that of the source. In this manner, examiningthe 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 dis-tributions of Fig. 6 and subsequently the orientationof the slanted axis. More specifically, the lower scin-tillations are encountered at the higher intensitypoints of Fig. 6, while the reverse happens for largerscintillations. To establish better cross-referencing,the slanted axis corresponding to lower scintillations,simultaneously to higher intensity points is also in-serted as a broken line on the �px, py� plane in thesecond picture of Fig. 5.

Finally, Fig. 7 shows how a cosh-Gaussian beam isaffected by the different levels of inner and outerscales of turbulence. We note that the inner and outer

Fig. 5. Scintillation index distribution of collimated and focused cosh-Gaussian and Gaussian beams over the receiver plane.

Fig. 6. Intensity distribution of collimated cosh-Gaussian beamfrom to Fig. 5 at source and receiver planes.


scale parameters, l0 and L0 are related to �m and �0of Eq. (8), respectively, by l0 � 5.92��m and L0 �2��0. From the four successive pictures placed inFig. 7, it is found as expected that the inner and outerscales of turbulence values other than l0 � 0 andL0 → � will produce smaller scintillations.

4. Conclusion

The scintillation index of cosh-Gaussian beams inturbulent atmosphere is formulated and evaluated atvarious source and turbulence parameters. Our eval-uations at each step are compared with the knownGaussian beam scintillations and in some situationswith cos-Gaussian scintillations.

For the collimated case, beyond a crossover pointdetermined by the magnitude of the displacementparameter, it is observed that cosh-Gaussian beamswill have lower on-axis scintillations than the Gauss-ian beam on-axis scintillations at longer propagationdistances. Below this crossover point, however, thereverse will take place. This crossover involving thepropagation distance is advanced further for coshbeams with larger displacement parameters. Whena finite focusing parameter is introduced, it is seenthat on-axis scintillations of both cosh-Gaussian andGaussian beams are reduced as compared to the cor-responding collimated beam on-axis scintillations.However, the incremental reduction in cosh-Gaussianbeam scintillation is higher as compared to the in-cremental reduction in Gaussian beam scintillation.When the scintillations are evaluated at the off-axisposition in the receiver plane, it is observed that thescintillation characteristic is quite different than thatof the on-axis scintillation.

When the scintillation index of the cosh-Gaussianbeam is compared to the Gaussian beam scintillationindex in terms of source sizes, it can be asserted that

the small size cosh-Gaussian sources will have lessscintillations. The behavior of the scintillation indexof the cosh-Gaussian beam versus the source sizefollows a similar trend to that of the Gaussian beamscintillation.

Examining the scintillations over the entire trans-verse receiver plane, we generally find lower scintil-lation values around the on–axis region, with atendency to rise toward the receiver plane edges.Again, beams with finite focusing parameters havesmaller scintillations than the corresponding colli-mated beams, a property being equally valid for bothcosh-Gaussian and Gaussian beams. It is noted thatfor the Gaussian beam the scintillation values at dif-ferent transverse receiver locations are in conformitywith the associated intensity distributions. That is,lower scintillations will occur at points of higher in-tensity, whereas the opposite will be applicable forlarger scintillations.

Finally, the effect of the inner and outer scales ofturbulence on the cosh-Gaussian beam scintillationsis investigated, and as expected, the scintillationsevaluated at nonzero inner scale and finite outerscale are always lower compared with evaluationsusing the Kolmogorov spectrum.

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Fig. 7. Scintillation index distribution of cosh-Gaussian beam over the receiver plane at selected inner and outer scales of turbulence.


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Scintillation characteristics of cosh-Gaussian beams

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