Average intensity and spreading of cosh-Gaussian laser beams in the turbulent atmosphere

Average intensity and spreading of cosh-Gaussianlaser beams in the turbulent atmosphere

Halil T. Eyyuboglu and Yahya Baykal

The average intensity and spreading of cosh-Gaussian laser beams in the turbulent atmosphere areexamined. Our research is based principally on formulating the average-intensity profile at the receiverplane for cosh-Gaussian excitation. The limiting cases of our formulation for the average intensity arefound to reduce correctly to the existing Gaussian beam wave result in turbulence and the cosh-Gaussianbeam result in free space (in the absence of turbulence). The average intensity and the broadening of thecosh-Gaussian beam wave after it propagates in the turbulent atmosphere are numerically evaluatedversus source size, beam displacement, link length, structure constant, and two wavelengths of 0.85 and1.55 �m, which are most widely used in currently employed free-space-optical links. Results indicate thatin turbulence the beam is widened beyond its free-space diffraction values. At the receiver plane,analogous to the case of free space, this diffraction eventually leads to transformation of the cosh-Gaussian beam into an oscillatory average-intensity profile with a Gaussian envelope. © 2005 OpticalSociety of America

OCIS codes: 010.1330, 010.1300, 010.3310, 060.4510.

1. Introduction

The spreading of laser beams in the turbulent atmo-sphere has been studied by many researchers. Feizu-lin and Kravtsov1 did a diffraction study of thebroadening of a spatially bound laser beam. Wangand Plonus2 investigated the beam spread for thelowest-order �TEM00� Gaussian beam wave source.The spreading of the optical waves of the higher-order mode was reported by Young et al.3 Usingpartially coherent beams of the Gaussian Schellmodel, Shirai et al.4 examined the spreading of eachcoherent mode. Angular spread and the rms width ofpartially coherent beams were analyzed by Ponomar-enko et al.,5 revealing the conditions in which par-tially coherent beams become less sensitive toturbulence compared with coherent beams. Dogariuand Amarande6 experimentally evaluated the mean-squared width of the spatially partially coherentbeam in turbulence normalized (at the same receiverplane) with the mean-squared width of the beam in

the absence of turbulence and reported that a par-tially coherent beam is less affected than a spatiallycoherent beam.

On the other hand, the adverse effects of atmo-spheric turbulence on free-space-optical (FSO) com-munication systems in broadband access networksand methods for reducing the degradations imposedby the turbulent atmosphere have been subjects forinvestigation.7–9 We recently introduced10 theHermite-sinusoidal-Gaussian (HSG) laser beams inthe turbulent atmosphere, formulating the log–amplitude, phase correlation, and structure func-tions. A special case of the HSG laser beams is cosh-Gaussian excitation that confines a high portion of itsenergy to the outer lobes of the beam. The propaga-tion in free space (i.e., in the absence of turbulence) ofthe general HSG beams11 and the cosh-Gaussianbeam12–14 have been reported.

In this paper we are interested in finding theaverage-intensity profile and the spreading charac-teristics of the cosh-Gaussian beam for practical FSOlinks. To this end, the receiver plane average-intensity profile and beam width in the turbulentatmosphere are derived, starting with cosh-Gaussianlaser-beam excitation. In limiting cases our formula-tion correctly reduces to the known forms of theGaussian beam wave in the turbulent atmosphereand the cosh-Gaussian beam solution obtained in freespace (i.e., in the absence of turbulence).

H. Eyyuboglu and Y. Baykal ([email protected]) are withthe Department of Electronic and Communication Engineering,Çankaya University, Ögretmenler Cad., 14 Yüzüncüyil, 06530Balgat Ankara, Turkey.

Received 26 March 2004; revised manuscript received 13 Octo-ber 2004; accepted 22 October 2004.

0003-6935/05/060976-08$15.00/0© 2005 Optical Society of America

976 APPLIED OPTICS � Vol. 44, No. 6 � 20 February 2005

2. Parameters of the cosh-Gaussian Beam Source

The associated geometry is depicted in Fig. 1. Thesource and receiver planes are defined by the coordi-nates �s, z � 0� and �p, z � L�, respectively, wheres � �sx, sy� and p � �px, py� are the transverse sourceand receiver coordinates, respectively, and z is thepropagation axis. Thus z � 0 (called the source plane)denotes the exit plane of the laser source, while z� L is the receiver plane, where L is the link length.

At the source plane �z � 0� the field of a cosh-Gaussian beam us can be obtained as the special caseof a general HSG beam solution11 and is written as

us(sx, sy, z � 0)

� 0.5A exp(�i�)exp{�0.5[(sx��sx)2

� (sy��sy)2]}[exp(Vxisx � Vyisy)

� exp(�Vxisx � Vyisy)], (1)

where A is the amplitude of the field at the origin ofthe source plane (i.e., at sx � sy � z � 0); � is theconstant phase factor; i � ��1�1�2; �sx and �sy are thesource sizes of the Gaussian beam in the sx and sy

directions, respectively; and Vxi, Vyi are the imaginarycomponents of the complex displacement parametersassociated with the Gaussian part of the beam in thesx, sy directions, respectively. In the writing of Eq. (1)the focal lengths along both sx and sy are taken asinfinity.

Using Eq. (1) and setting A to unity, we obtain theintensity distribution at the exit plane of the laser as

Is(sx, sy, z � 0)

� 0.5A exp[�(sx2��sx

2 � sy2��sy

2)]� {1 � cosh[2(Vxisx � Vyisy)]}

� exp[�(sx2��sx

2 � sy2��sy

2)]cosh2[(Vxisx

� Vyisy)]. (2)

Using Eq. (2), we define the following basic parame-ters of the cosh-Gaussian beam. In our research weare interested in cosh-Gaussian beams exhibiting adistinct separation of energy in the outer lobes of thebeam; so, in defining the following parameters, weresort to the necessary approximations needed forsuch beams.

A. Coordinates of the Peak Intensity of thecosh-Gaussian Beam

The coordinates of the peak intensity of the cosh-Gaussian beam are found by taking ��sx and ��sx ofEq. (2) and setting both of them to zero. Discardingthe common terms of exp��sx

2��sx2 � sy

2��sy2�� and

cosh�Vxisx � Vyisy�, we are left with the following pairof equations coupled by sxp and syp:

�(2sxp��sx2)cosh(Vxisxp � Vyisyp) � 2Vxi

� sinh(Vxisxp � Vyisyp) � 0, (3)

�(2syp��sy2)cosh(Vxisxp � Vyisyp) � 2Vyi

� sinh(Vxisxp � Vyisyp) � 0, (4)

where sxp and syp correspond to the x and y compo-nents of the transverse source coordinates at whichthe peak value of the cosh-Gaussian beam is located.Keeping in mind that for cosh-Gaussian beams ex-hibiting a distinct separation of energy in the outerlobes of the beam, the hyperbolic cosine term in Eqs.(3) and (4) is approximated around the peak location,sxp and syp:

cosh(arg) � |sinh(arg)| � 0.5 exp(arg), (5)

where arg refers to the collection of terms in theargument of the hyperbolic cosine function. At thesource plane, arg � Vxisxp � Vyisyp. Note that approx-imation (5) is no longer accurate for positive arg val-ues of less than 2. However, for cosh-Gaussian beamsexhibiting a distinct separation of energy in the outerlobes of the beam, arg 2 is satisfied around thepeak.

When approximation (5) is applied to Eqs. (3) and(4), the Eq. (6) and (7) are produced:

(2sxp��sx2) 2Vxi � 0, (6)

(2syp��sy2) 2Vyi � 0. (7)

From Eqs. (6) and (7) sxp and syp are found to be sxp

� Vxi�sx2 and syp � Vyi�sy

2.

B. Adjustment of the cosh-Gaussian Beam Profile

Here we find the combinations of Vxi, Vyi, �sx, and �sy

that cause the cosh-Gaussian beam to exhibit a dis-tinct separation of energy in the outer lobes of thebeam. For this purpose we define a normalized inten-sity Io at the origin of the exit plane of the laser as

Io � Is(sx � 0, sy � 0, z � 0)�Is(sx � sxp, sy � syp, z � 0)(8)

and take Io �� 1. Using Eq. (2), and sxp � Vxi�sx2,

syp � Vyi�sy2 (obtained in Subsection 2.A) in Eq. (8)

and also utilizing approximation (5), we find thatVxi � �ln�2�Io

1�2��1�2��sx and Vyi � �ln�2�Io1�2��1�2��sy,

which are valid for the case of symmetrical beam

Fig. 1. Propagation geometry.

20 February 2005 � Vol. 44, No. 6 � APPLIED OPTICS 977

sizes along sx and sy. Otherwise the ln argument hasto be adjusted accordingly.

C. Beam Size

The beam size at the source plane is defined as thepoint where the intensity has fallen to exp��2� of thepeak intensity. Hence

Is[sx � �xbs(z � 0), sy � �ybs(z � 0), z � 0]�Is(sx � sxp, sy � syp, z � 0) � exp(�2), (9)

where in general �abs�z� and �ybs�z� are the beam sizesat the transverse plane located at z along the x and ydirections, respectively. Thus Eq. (9) evaluated at z� 0 defines the beam size at the source plane. Bysubstituting Eq. (2) into Eq. (9) and utilizing the ap-proximation given by Eq. (5), we arrive at

�xbs(z � 0) � �sx(Vxi�sx � 1), (10)

�ybs(z � 0) � �sy(Vyi�sy � 1). (11)

Comparing Eqs. (10) and (11) with Eq. (2), we resolvethat the hyperbolic cosine beam has retained the orig-inal beam size of the Gaussian beam with a simpleshifting of the peaks.

To apprehend better the parameters defined above,we give the normalized intensity distribution of atypical cosh-Gaussian beam at the source plane inFig. 2 as a three-dimensional view and in Fig. 3 as thecontour plot of the same beam. The plots in Figs. 2and 3 demonstrate the variation of the normalizedintensity, IsN�sx, sy, z � 0�, at the source plane, z� 0, defined as

IsN(sx, sy, z � 0) � Is(sx, sy, z � 0)�Is(sx � sxp, sy � syp, z � 0). (12)

3. cosh-Gaussian Beam in a Turbulent Atmosphere

When the Huygens–Fresnel principle is applied, thefield at the receiver plane �z � L� is given by

�u(p, L, t) � k exp(ikL)�(2�iL)

��

��

d2sus(s)exp[ik(p � s)2

�(2L) � �(s, p) � i2�ft], (13)

where k is the wave number, us�s� is the field of acosh-Gaussian beam at the source plane �z � 0� asprovided by Eq. (1), and ��s, p� is the solution to theRytov method representing the random part of thecomplex phase of a spherical wave propagating fromthe source point �s, z � 0� to the receiver point�p, z � L�, f is the frequency, and t denotes time.

The average intensity at the receiver plane is�I�p, L� � �u�p, L, t�u*�p, L, t�, where * and� � refer to the complex conjugate and the ensembleaverage over the medium statistics, respectively.When Eqs. (13) and (1) are combined, �I�p, L� isformulated as

�I(p, L) � k2�(2�L)2��

��

��

��

d2s1d2s2

� us(s1)us*(s2)exp{ik[(p � s1)2

� (p � s2)2]�(2L)} � exp[�(s1, p)

� �*(s2, p)] . (14)

The ensemble average term in Eq. (14) is2

�exp[�(s1, p) � �*(s2, p)] � exp[�0.5D�(s1 � s2)]

� exp[��0�2(s1

� s2)2], (15)

where D��s1 � s2� is the wave structure function and�0 � �0.545Cn

2k2L��3�5 is the coherence length of aspherical wave propagating in the turbulent medium,Cn

2 being the structure constant.When Eqs. (1) and (15) are replaced in Eq. (14) and

the transverse source and transverse receiver coordi-

Fig. 2. Normalized intensity of a cosh-Gaussian beam at thesource plane.

Fig. 3. Contour plot at the source plane of the same cosh-Gaussian beam as in Fig. 2.


nates are subsequently expanded into their corre-sponding x and y components, the average intensityat the receiver plane becomes

�I(p, L) � 0.25k2�(2�L)2

��

��

��

��

ds1xds1yds2xds2y

� exp[�(s1x2 � s2x

2)�(2�sx2) � (s1y

2

� s2y2)�(2�sy

2)]{exp[Vxi(s1x � s2x)� Vyi(s1y � s2y)] � exp[�Vxi(s1x � s2x)� Vyi(s1y � s2y)] � exp[Vxi(s1x � s2x)� Vyi(s1y � s2y)] � exp[�Vxi(s1x � s2x)

� Vyi(s1y � s2y)]}exp[0.5(ik�L)(s1x2

� 2pxs1x � s2x2 � 2pxs2x � s1y

2 � 2pys1y

� s2y2 � 2pys2y)]exp[��0

�2(s1x2 � 2s1xs2x

� s2x2 � s1y

2 � 2s1ys2y � s2y2)]. (16)

Repeatedly applying Eq. (3.323.2) of Ref. 15 in Eq.(16) and pursuing some lengthy algebra, we developthe average intensity at the receiver plane into

�I(p, L) � 0.5(k�L)2�04(DsxDsy)

�1�2 exp{�(�04k2�L2)

�[px2�(�sx

2Dsx) � py2�(�sy

2Dsy)]}

�(exp{2�02[Vxi

2(�02asx � 1)�Dsx)

� Vyi2(�0

2asy � 1)�Dsy]}

�cosh[(2�04k2�L2)(Vxipx�Dsx

� Vyipy�Dsy)] � exp{�04[Vxi

2�(�sx2Dsx)

� Vyi2�(�sy

2Dsy)]}cos{(2�04k�L)

�[Vxipx�(�sx2Dsx) � Vyipy�(�sy

2Dsy)]})(17)

where asx � ��0.5��sx2� � �1��0

2��, asy � ��0.5��sy2�

� �1��02��, Dsx � 4��0

4�asx2 � �k�2L�2� � 1, and Dsy

� 4��04�asy

2 � �k�2L�2� � 1.Using Eq. (17), we compute the coordinates of the

peak intensity of the cosh-Gaussian beam and thesize of the cosh-Gaussian beam after it has propa-gated through the turbulent media. These are ob-tained by following similar procedures outlined inSection 2 with the inclusion of atmospheric turbu-lence.

Within the range of approximation given by Eq. (5)the coordinates of the peak intensity of the cosh-Gaussian beam after propagating through atmo-spheric turbulence is found to be pxp � Vxi�sx

2 andpyp � Vyi�sy

2, hence independent of atmospheric tur-bulence. Here pxp and pyp denote the x and the ycomponents of the transverse receiver coordinates atwhich the peak value of the cosh-Gaussian beam islocated.

Similarly the size of the cosh-Gaussian beam afterit propagates through the turbulent media is found as

�xbs(z � L) � Vxi�sx2 � �sxLDsx

0.5�(�02k)

� Vxi�sx2 � (k�sx)

�1[L2 � (4�sx2L2��0

2)

� k2�sx4]0.5, (18)

�ybs(z � L) � Vyi�sy2 � �syLDsy

0.5�(�02k)

� Vyi�sy2 � (k�sy)

�1[L2 � (4�sy2L2��0

2)

� k2�sy4]0.5, (19)

where �xbs�z � L� and �ybs�z � L� denote the size of thecosh-Gaussian beam at the receiver plane �z � L� inthe px and py directions, respectively. Note that theparameters (�sx, �sy, Vxi, Vyi, L, Cn

2 and k) in Eqs. (18)and (19) should be chosen so that approximation (5) isnot violated where the term arg in the presence ofturbulence is expressed as

arg � 2{[Vxi2�sx

2�(k�2L2�sx�4 � 4k�2L2�sx

�2�0�2 � 1)]

� [Vyi2�sy

2�(k�2L2�sy�4 � 4k�2L2�sy

�2�0�2 � 1)]}.

(20)

After the beam propagates through the turbulentatmosphere, the beam size in the px direction growsfrom �xbs�z � 0� to a new size, �xbs�z � L�, as suppliedby Eq. (18), and similarly the beam size in the py

direction increases from �ybs�z � 0� to a new size,�ybs�z � L�, as given by Eq. (19). Thus, by deducting�sx and �sy from the second terms in Eqs. (18) and (19),respectively, we are able to express the amount ofspreading for a cosh-Gaussian laser beam at the re-ceiver plane owing to propagation in turbulence.

4. Limiting Cases

In this section we verify Eq. (17) for the followinglimiting cases:

(1) The average-intensity distribution at the re-ceiver plane given by Eq. (17) is evaluated for the zerolink length, i.e., for L � 0. In this limit the averageintensity at the receiver plane expressed by Eq. (17)reduces to the intensity at the source plane given byEq. (2) as expected. Note that px and py should, re-spectively, be changed to sx and sy when approachingthe source plane.

(2) To find the limit of the Gaussian beam in freespace, i.e., in the absence of turbulence, Cn

2 � 0, oralternatively �0→ , and Vxi � Vyi � 0 are insertedinto Eq. (17), and also the symmetrical beam is takenby choosing �sx � �sy � �s. In these conditions theaverage intensity in Eq. (17) reduces to

�I(p, L) � [k2�s4�(L2 � k2�s

4)]exp{�[k2�s2�(L2

� k2�s4)](px

2 � py2)}. (21)

Equation (21) matches the free-space propagationlimit examined in Ref. 3. The exact correspondence ofEq. (17) to Eq. (5) of Ref. 3 is established by notingthat k2�s

2��L2 � k2�s4� � 2�W2, 20.5�s � W0, px

� x, and py � y.


(3) Here we check our formula in Eq. (17) with theexisting result of the Gaussian beam in turbulence.For this case we set Vxi � Vyi � 0 and constitute thesymmetry in the x and y directions so that �sx � �sy

� �s. In these conditions Eq. (17) becomes

�I(p, L) � [k2�s4�0

2�(�02L2 � 4�s

2L2

� k2�s4�0

2)]exp[�k2�s2�0

2(px2

� py2)�(�0

2L2 � 4�s2L2 � k2�s

4�02)].

(22)

Equation (22) agrees with Eq. (12) of Ref. 2. Toachieve this agreement, the parameters in Eq. (12) ofRef. 2 are adopted as pd � 0, pc

2 � px2 � py

2, F � ,� � 0, A0 � 1, and �o � �s.

(4) As our last verification we compare our resultwith the cosh-Gaussian beam formulation in freespace (in the absence of turbulence). For this purposewe set Cn

2 � 0 thus the average intensity in Eq. (17)becomes

�I(p, L) � 0.5k2�sx2�sy

2gx0.5gy

0.5 exp[�k2(�sx2gxpx

2

� �sy2gypy

2)]exp[L2(�sx2Vxi

2gx

� �sy2Vyi

2gy)]{cosh[2k2(�sx4Vxigxpx

� �sy4Vyigypy)] � cos[2kL(�sx

2Vxigxpx

� �sy2Vyigypy)]}, (23)

where gx � �L2 � k2�sx4��1 and gy � �L2 � k2�sy

4��1.

When we revert to a single coordinate system, Eq.(23) becomes identical with an intensity equivalent ofthe field expression in Eq. (10) of Ref. 12 after thecorrespondence of k�sx

2��L� � Nw, �sx � wo�20.5,px � 20.5�sxx�, Vxi � a��20.5�sx� � �o is established.

5. Results

Our numerical evaluations are performed for thepractically used FSO links where two operatingwavelengths, � of 0.85 and 1.55 �m, and link lengthsL as long as 5 km, are mainly employed. However, inone particular case, in order to investigate the limit-ing behavior of the average-intensity profile for ex-treme link lengths, L � 20 km is also utilized. Ourformulation is valid for cosh-Gaussian beams that areasymmetrical along the x and y directions. However,the results in this paper are generated for symmet-rical beams (�sx � �sy and Vxi � Vyi).

The normalized average intensity at the receiverplane, IrN � �px, py, z � L�, is defined as

IrN(px, py, z � L) � � I(px, py, z � L)

�Is(sx � sxp, sy � syp, z � 0). (24)

The normalized intensity at the source plane,IsN�sx, sy, z � 0�, and the normalized average inten-sity at the receiver plane, IrN�px, py, z � L�, versusthe slanted axis are shown in Fig. 4 for a single

cosh-Gaussian beam. Here the slanted axis is the axispassing through the peak locations of the cosh-Gaussian beam as shown in the lower left inset inFig. 4. The broadening of the cosh-Gaussian beam inthe turbulent atmosphere can be observed in Fig. 4for the chosen parameters of source size �sx � �sy

� 3 cm, link length L � 5 km, wavelength �� 1.55 �m, the imaginary components of the complexdisplacement parameters associated with the Gauss-ian part of the beam in the sx, sy directions, Vxi

� Vyi � 55 cm�1, and the structure constant Cn2

� 1 � 10�15 m�2�3. Another view of Fig. 4 is shown inFig. 5, this time displaying IsN�sx, sy, z0� andIrN�px, py, z � L� as contour plots with the values onthe contour lines indicating the normalized intensi-ties.

The difference in the beam sizes at the receiver andsource planes in the x direction ��xbs is defined as

��xbs � �xbs(z � L) � �xbs(z � 0)

� (k�sx)�1[L2 � (4�sx

2L2��02) � k2�sx

4]0.5 � �sx.(25)

Fig. 4. Normalized intensity at the source plane and the normal-ized average intensity at the receiver plane for a cosh-Gaussianbeam.

Fig. 5. Contour plot of the normalized intensity at the sourceplane and the normalized average intensity at the receiver planefor the same cosh-Gaussian beam of Fig. 4.


Similarly ��ybs denotes the difference in the beamsizes at the receiver and source planes in the y direc-tion obtained by replacing x by y in Eq. (25). ��xbs

and ��ybs can be regarded as the measures of broad-ening in beam sizes along the x and the y directions,respectively, from the source plane to the receiverplane.

In Fig. 6 ��xbs is plotted versus the link length L fortwo wavelengths of 0.85 and 1.55 �m and for thesame cosh-Gaussian source with �xs � 3 cm. Twocurves, one in the absence of turbulence and the otherin the presence of turbulence, with Cn

2

� 1 � 10�15 m�2�3 are drawn for each wavelength.Note that the curves in Figs. 6 and 7 give the differ-ence in the beam sizes at the receiver and sourceplanes in the x direction only. Hence a similar per-formance should be envisaged for the y direction. Ateach link length the broadening of the cosh-Gaussianbeam beyond free space (in the absence of turbulence)diffraction is clearly seen for both wavelengths. Asexpected, for a fixed link length, the broadening both

in free space and in turbulence is larger at �� 1.55 �m than the broadening at � � 0.85 �m.

In Fig. 7, ��xbs is examined versus the source size�xs for two wavelengths of 0.85 and 1.55 �m and forthe link length, L � 5 km. Two curves, one in theabsence of turbulence and the other in the presence ofturbulence, with Cn

2 � 1 � 10�15 m�2�3 are drawn foreach wavelength. The broadening of the cosh-Gaussian beam above free-space (in the absence ofturbulence) diffraction is observed at each source sizefor both wavelengths. For a fixed source size thebroadening both in free space and in turbulence islarger at � � 1.55 �m compared with the broadeningat � � 0.85 �m, as expected. Large broadening forsmall source sizes (spherical wave limit) and smallbroadening for large source sizes (plane wave limit)are clearly visible in Fig. 7. The variation of ��xbs

versus Cn2 is provided in Fig. 8 for three FSO wave-

lengths of 0.85, 1.3, and 1.55 �m where it is observedthat broadening is more effective in relatively stron-ger levels of turbulence. As expected, broadening alsoincreases by a scale of magnitude for larger wave-lengths.

In Fig. 9 the normalized average intensity at the

Fig. 6. Difference in beam sizes at the receiver and the sourceplanes in the x direction ��xbs versus link length in turbulence andin free space for a fixed source size and at FSO wavelengths of 0.85and 1.55 �m.

Fig. 7. Difference in beam sizes at the receiver and the sourceplanes in the x direction ��xbs versus the source size in turbulenceand in free space for a fixed link length and at FSO wavelengths of0.85 and 1.55 �m.

Fig. 8. Difference in beam sizes at the receiver and the sourceplanes in the x direction ��xbs versus the structure constant Cn

2

evaluated at source size �sx � 3 cm, Vxi � 55 cm�1 and link lengthL � 2 km and for three FSO wavelengths of 0.85, 1.3, and1.55 �m.

Fig. 9. Normalized average intensity at the receiver transverseplane �z � L� at various link lengths.


receiver plane, IrN�px, py, z � L�, is plotted at variouslink lengths L for a given cosh-Gaussian beam with�sx � �sy � 3 cm, Vxi � Vyi � 55 cm�1, and at �� 1.55 �m and Cn

2 � 1 � 10�15m�2�3. Again theslanted axis is the axis passing through the peaklocations of the cosh-Gaussian beam as shown in thelower left inset in Fig. 9. The change in the average-intensity profile as the cosh-Gaussian beam propa-gates through turbulence can be observed in Fig. 9,starting as a distinct cosh-Gaussian beam at thesource plane �L � 0 km�, deviating gradually fromthis original cosh-Gaussian shape, and eventually ap-proaching a Gaussian average-intensity profile forsufficiently long link lengths, which is L � 20 km inthis specific case. This deformation of the cosh-Gaussian behavior occurs because each lobe of thecosh-Gaussian beam broadens along the propagationdistance, a nonzero average intensity is built aroundthe beam origin, eventually leading to formation of apeak average intensity around the origin of the beam,thus changing the average-intensity profile to aGaussian shape. To introduce variation in the aver-age intensity at various structure constants Cn

2, theaverage intensity at the receiver transverse plane�z � L� is normalized with respect to the maximumvalue of the free-space received intensity as

IrC(px, py, z � L) � � I(px, py, z � L)

�max[�I(px, py, z � L) ]Cn2�0. (26)

In Fig. 10 IrC�px, py, z � L� as defined in Eq. (26) isplotted at several structure constants Cn

2. It can beseen that the cosh-Gaussian beam profile becomesmore deformed as Cn

2 increases. Note that Gaussianbehavior is observable within the vicinity of the ori-gin of the receiver plane. However, when the obser-vation view along the slanted axis is extended,oscillatory behavior is revealed, a subject exploredfurther in Fig. 11.

To assess this oscillatory case properly, an averageintensity at the receiver plane normalized with re-spect to the average intensity at the origin of the

receiver plane, Ir0�px, py, z � L� is defined as

Ir0(px, py, z � L) � � I(px, py, z � L)

�� I(px � 0, py � 0, z � L) . (27)

In Fig. 11, Ir0�px, py, z � L� is plotted versus theslanted axis for different Vxi values for the samesource and link parameters of �sx � 0.1 cm, L� 1 km, � � 1.55 �m, Cn

2 � 1 � 10�15m�2�3. Here theoscillatory behavior of the average-intensity profile inthe turbulent atmosphere is observed whose varia-tions follow the cosine term in Eq. (17) with a Gauss-ian envelope. A similar result is reported in Fig. 2 ofRef. 12 in free space (in the absence of turbulence).Moreover larger complex displacement parameters(Vxi and Vyi) lead to a smaller spread in turbulence, ifonly the central lobe of the beam at the receiver planeis considered.

6. Conclusion

We have formulated and numerically evaluated theaverage-intensity profile of a cosh-Gaussian beam ina turbulent atmosphere and its associated beambroadening. The limiting cases of our formulation forthe average intensity are found to correctly reduce tothe existing Gaussian beam wave result in turbu-lence and the cosh-Gaussian beam result in freespace (in the absence of turbulence).

Our derivations indicate that the cosh-Gaussianbeams with the major part of their energy concen-trated in the outer lobes have similar attributes tothat of two Gaussian beams separated by the complexdisplacement parameter. In this sense the resultingparameters of cosh-Gaussian beams are definable interms of those belonging to pure Gaussian beams.Such an attribute is evident when the intensity pro-files are plotted along the slanted axis, passingthrough the peak locations.

The average intensity and the broadening of thecosh-Gaussian beam wave after it propagates in theturbulent atmosphere have been numerically evalu-ated versus beam parameters (source size, beam dis-

Fig. 10. Average intensity at the receiver transverse plane �z� L� normalized with respect to the maximum value of the free-space-received intensity evaluated at various structure constantsCn

2.

Fig. 11. Ir0�px, py, z � L� for different Vxi values at the fixedsource and link parameters of �sx � 0.1 cm, L � 1 km, �

� 1.55 �m, Cn2 � 1 � 10�15 m�2�3.


placement) and link parameters (link length,structure constant, wavelength). Here the mostwidely used FSO wavelengths of 0.85 and 1.55 �mhave been employed. Our results show that thespreading of the cosh-Gaussian beam increases whenthe individual link parameters comprising turbu-lence strength (link length, structure constant, andwavelength of operation) increase.

Our numerical evaluations also demonstrate thatthe spreading of the cosh-Gaussian beam decreaseswhen the beam parameters (source size and complexdisplacement) increase. Note particularly that, if onlythe central lobe of the beam at the receiver plane isconsidered, larger complex displacement parameters(Vxi and Vyi) lead to a smaller spread in turbulence.

Our results also reveal that akin to the behaviorobserved in free space, the cosh-Gaussian beam inturbulent atmosphere eventually transforms into acosine beam with a Gaussian envelope. Again thiseffect may be retarded somewhat by suitable adjust-ment of the complex displacement parameter.

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Average intensity and spreading of cosh-Gaussian laser beams in the turbulent atmosphere

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