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On atmospheric optical communicationsEl-S. A. El-Badawy, F.Z. El-Halafawy

To cite this version:El-S. A. El-Badawy, F.Z. El-Halafawy. On atmospheric optical communications. Revue de PhysiqueAppliquee, 1984, 19 (10), pp.879-888. �10.1051/rphysap:019840019010087900�. �jpa-00245279�

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On atmospheric optical communications

El-S. A. El-Badawy (*,+) and F. Z. El-Halafawy (**,+)

(*) Dept. of Electronic Engineering, Faculty of Engineering, University of Alexandria, Alexandria, Egypt(**) Dept. of Wire Comm. Engineering, Faculty of Electronic Engineering, Menoufia University, Menouf, Egypt

(Reçu le 17 janvier 1984, révisé le 21 mai, accepté le 16 juillet 1984)

Résumé. 2014 La diffraction d’une porteuse puissante qui se propage dans une chaine de communication non guidée(atmosphère sèche) est plus rigoureusement étudiée. Quatre points pertinents sont examinés i) distribution radialede la température, ii) paramètre « largeur moyenne du faisceau », iii) facteur de mérite normalisé de distortionoptique, et iv) mise en forme d’un champ optique pour réduire la diffraction créée par les effets réfractifs.

Abstract 2014 This paper investigates, more rigorously, the diffraction of a powerful optical carrier propagating in anunguided communication channel (dry atmosphere). Four main relevant topics are considered i) the radial tempe-rature distribution, ii) the average beamwidth parameter, iii) a dimensionless optical distortion figure of merit,and iv) the irradiance tailoring technique as a method to reduce the diffraction caused by refractive effects.

Revue Phys. Appl. 19 (1984) 879-888 OCTOBRE 1984, PAGE 879

Classification

Physics Abstracts42.10 - 42.80 - 42.60

1. Introduction.

The diffraction caused by the thermal blooming phe-nomenon associated with the propagation of a CWlaser beam in an open medium constitutes one of the

important nonlinear propagation phenomena. Whena laser beam propagates in an absorbing médium,such as the atmosphere, heating of the medium takesplace. This heating results in refractive index gradientsacross the laser beam. These gradients act as a distri-buted lens which diffracts the beam and decreasesthe peak intensity along the propagation path [1].The problem of calculating the thermal blooming

effect was solved on the basis of geometrical optics andparaxial approximation [2]. Light propagating throughan inhomogeneous medium whose refractive indexhas weak changes with cylindrical symmetry aroundthe propagation path was studied by Majias [3].The cylindrical TEMoo and TEM 10 laser modeswere considered and explicit analytical expressionsfor the light ray and intensity profiles for short andlong time intervals were derived Jones andMcMordie [1] reported that the magnitude of theeffects of the thermal blooming could be characterizedby four dimensionless parameters or numbers : theFresnel number, an absorption number, a distortion

(+) Members of the Optical Society of America.

number, and a slewing number. Sodha et al. [4]deeply investigated the self-distortion of laser beamspropagation in dielectrics, plasmas, and semiconduc-tors. Based on the paraxial approximation, theyreduced the problem to a nonlinear differential

equation in a single characterizing parameter, namely,the « beamwidth parameter ». Wallace et aL [5]evaluated the impact of irradiance tailoring as amethod of reducing the influence of the refractiveeffects caused by the heating of the medium. Theprocess is to create, by irradiance shaping, onlycorrectable phase changes along the propagationpath. The practicality of designing lasers of a

specific irradiance distribution is, however, largelyunanswered Generally, irradiance tailoring may beachieved by special splitters and filters. El-Badawyand El-Halafawy [6, 7] investigated partially the ther-mal blooming phenomenon in dry atmosphere.

In the present work, the heating of clear air bypowerful Gaussian laser beams is studied The

impact of irradiance tailoring technique is applied todesign a suitable electric field distribution to reducethe diffraction caused by the refractive effects and toincrease the focal length. Thermal blooming pheno-menon is rigorously treated taking into account thecoupling between the laser and clear air only at theentrance. The affecting parameters are studied indetails over a wide range of variation. The correlations

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/rphysap:019840019010087900

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of a dimensionless optical distortion figure of meritand the other affecting parameters are also investi-gated

Section 2 describes the basic model. Section 3 dealswith mathematical formulation and analysis of theproblem. The results and discussions are exposed insection 4. Section 5 points out the main conclusions.

2. Basic modeL

The change in the intensity I(r, z) of an initially colli-mated laser beam propagating along the z-directionthrough a medium of variable refractive index n(r, z)and linear absorption coefficient Qa is derived as inreference [2] :

where Vt is the transverse gradient and z is the propagation distance.

The refractive index n and the pressure P are relatedto the medium (air) density p. and temperature T,respectively, through Gladestone’s law and the per-fect gas law as

and

where Ru is the universal gas constant and K is aconstant proportional to the medium polarizability.The temperature T is determined from the energybalance equation :

Based on data of Kumar [8], it could be assumed thatboth the thermal conductivity kt and the density p.are linearly related to the normalized temperatureTn ( Tn = TjT 0’ To being the initial temperature) inthe range of interest as :

and

with ko = 4.64 x 10- 3 W/mK, a = 4.263, po =

1.95 kg/m’, and fl = - 0.356.After Kroll and Watson [9], both the absorption

coefficient (fa and the polarizability K appear to beinversely proportional to the square of the laser fre-quency (J) through the relations :

and

where P is the pressure in standard atmosphericunits. For dry air, Qo = 3 x 10-4 m-1 [10],

Following our previous work [6], the spatial profileF(r, 0), defined as I(r, 0) = Io F(r, 0), is tailored underthe following forms.

2. 1 TEMoo MODE. - i) Gaussian distributions.

The negative sign is for the Gaussian distributionsand the positive sign for the inverse distributions.The index i equals 1, 2 and 3 for the normal, flat,and superflat distributions, respectively.ii) Parabolic distributions

where R is the beam radius, and oci is a parameterin the range ( - 1,10). The index i runs from 0 to 10.

iii) Biquadratic distributions

These spatial profiles have a maximum value fm ata fixed radial position rm where rm = R/2 andfm = 1 + a/4.

2.2 TEM10 MODE. - Following Majias [3], thedistributions are tailored as

The parameter m controls the radial position r2 atwhich F(r, 0) = 0. Thus r2 = R/m.

2.3 TE20 MODE.

where A is designed so that the maximum value ofF(r, 0) occurring at rm is always equal to unity whererm = R m/(1 + m).

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The above distributions have the saine maximum

intensity wherever it is.The dimensionless optical distortion figure of merit,

DODFM, introduced by the authors in reference [6]proved to be a good measuring tool to evaluate theimpact of irradiance tailoring as a method to reducethe diffraction caused by the refractive effects. Thisfigure is designed under the form

where Q is the power transferred per unit length fromthe hot propagation path to the surrounding coldblanket, AT" is the maximum radial temperaturedifference, on/oT ) is the average temperature rateof change of the refractive index, and H is the totalabsorbed power per unit length. From the opticalcommunication point of view, one must have high Qand small 0394T, ~ on/oT ), and H. The figure of meritFm designed in this manner makes a good propagationprocess (of less diffraction) be characterized by alarge value of Fm.

3. Analysis.3.1 TEMPERATURE DISTRIBUTION. - Assuming thatthe coupling between laser and air is only at theentrance of the medium, the heat equation is written as

Taking equation (5) into consideration, substitutingT. = T/To, p = r/R (for Gaussian distributions Ris the 1/e radius), and F(r, z) = F(r, 0) exp(- u. z),and assuming as a solution for equation (15)

equation (15) reduces to

where A is the dimensionless quantity (fa R2 Io/ko T o.The solution of equation (17) is assumed as a powerseries :

Such a solution satisfies the boundary condition

At the interface between the hot propagation pathand the cold surrounding air, the following boundarycondition is applied :

where h is the air-air heat transfer coefficient [12].Using equation (13) in (19) gives

where ô = 4/Nu 03C1w, with Nu the Nusselt No. (for airNu = 2 000 R [12]) and 03C1w the normalized radial

position at r = R. The solution of (20) yields bo asa function of both the medium properties and thelaser characteristics.

3.2 THERMAL BLOOMING ANALYSIS. - The solutionof (1) is assumed under a simple closed form forinitially Gaussian laser beam,

where f (r, z) is a single characterizing parameterknown as the « beamwidth parameter » ; it characte-rizes the phenomenon of thermal blooming bothquantitatively and qualitatively. At z = 0, f (r, 0) = 1.Using (21), (1) yields

Assuming that the transverse refractive index gradient,Otn, to be constant along the propagation path,substituting Fn = 1/g and taking the logarithmicdifferentiation w.r.t. z-coordinate, one obtains :

where

Using equation (3), where P is constant, with equa-tion (2) yields

Substituting equations (25) and (26) into (24), F,,is obtained using a computational technique andassuming decoupling between the laser beam and thepropagation path.

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4. Results and discussion.

The problem is numerically solved with high accuracyand low negligible truncation error in the series solu-tion. Monotonic relationships are found among theset of dependent variables { T, n, fa, DODFM } andthe set of independent variables { Eo, R, À, F(p, 0) }.The variations of the increase in the radial tempe-

rature, (T - T o), with the normalized radial position,p, at different values of the electric field, Eo, and theassumed sets of parameters are shown in figure 1.It is clear that the radial temperature, T, increaseswith the increase in Eo at any radial position. Thisis expected since the absorbed power is proportionalto the quantity Eg.The increase in Eo yields an increase in the radial

gradient of the refractive index, Orn, at any radialposition and results in an increase in the beam crosssection and consequently a decrease in the poweralong the propagation path. This is depicted in

figures 2 and 3, where the average beamwidth para-meter fa increases with the increase of Ea. fa is definedas :

where f (r, z) is the beamwidth parameter.

Fig. 1. - Variation of (T - T 0) with p for different valuesof E, and the assumed set of parameters.

Fig. 2. - Variation of V, n with p for different values of E,and the assumed set of parameters.

Fig. 3. - Variation of fa with z for différent values of E,and the assumed set of parameters.

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The variation of the dimensionless ôptical distor-tion figure of merit, DODFM, with E. for differentvalues of R and the assumed set of other parametersis displayed in figure 4, where the DODFM decreaseswith the increase in Eo.The axial normalized temperature (on-axis), bo,

may be approximately obtained by solving

or

taking into consideration that :

Equation (28) yields

It is clear that bo is nearly linearly related to thedimensionless quantity A = 03C3a R2 Iolko To. Thus itcould be concluded that bo and consequently T is

proportional to E02. The radial gradient of the tem-perature distribution is always negative (the heatflux is always outside the hot path). This in turnyields a positive radial gradient for the refractiveindex which can be derived from equations (2) and(3) as ;

The variations of the increase of the radial tempe-rature, ( T - To), with the normalized radial position,p, at different values of the beam radius, R, and theassumed set of other parameters are displayed infgure 5. This figure reveals that, at any radial position,both T and 1 V r T 1 increase with the increase of Rwhich is true on the basis of équation (30). As thebeam radius increases, the radial gradient of therefractive index Vrn increases also as it is clear in

figure 6. The average beamwidth parameter fa decrea-ses with the increase of R at any propagation distanceas shown in figure 7. From (24) one can say that V_,fis proportional to the inverse of R2. Thus, along thepropagation path, f increases as R decreases. In

spite of the increase of f, the beam radius at any pro-pagation distance (z > 0) decreases with the decreaseof the initial radius (at z = 0).

Fig. 4. - Variation of DODFM with E, for different valuesof R and the assumed set of parameters.

Fig. 5. -Variation of (T - To) with p for différent values ofR and the assumed set of parameters.

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Fig. 6. - Variation of V, n with p for different values of Rand the assumed set of parameters.

Fig. 7. - Variation of fa with z for different values of Rand the assumed set of parameters.

The variation of the DODFM with the beam

radius, R, for different values of Eo and the assumedset of other parameters is displayed in figure 8 ; fromwhich it is clear that the DODFM decreases with theincrease of RThe variations of the increase in the radial tempe-

rature, (T - T o), with p at different angular fre-

quencies Q) and the assumed set of other parametersare shown in figure 9.

Based on equation (7) and figure 9, it is revealedthat the interaction process between the opticalcarrier and the dry atmosphere is weakened as 03C9

increases. Thus, both the temperature T and itsradial gradient 1 V r T decrease, at any radial position,with the increase in the laser frequency. Consequently,the radial gradient of the refractive index Vrn decrea-ses as Q) increases as shown in figure 10.At any propagation distance, the average beam-

width parameter fa increases with the increase of thelaser frequency m as shown in figure 11. At anyfixed set of parameters { Eo, R, To, P }, the DODFMincreases with the increase in the laser frequency Q)as depicted in figure 12.The variations of the DODFM with different

spatial profiles for the TEMoo mode at a fixed assumedset of parameters { R, À, Eo, To, P } are illustrated infigures 13 to 16. For the Gaussian distribution,figure 13, the DODFM increases with the decrease

Fig. 8. - Variation of DODFM with R for different valuesof E, and the assumed set of parameters.

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Fig. 9. - Variation of (T - To) with p for différent valuesof úJ and the assumed set of parameters.

Fig. 11. - Variation of fa with z for different values of w,and the assumed set of parameters.

Fig. 10. - Variation of Or n with p for different values of coand the assumed set of parameters.

Fig. 12. - Variation of DODFM with Q) for different valuesof R and the assumed set of parameters.

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Fig. 13. - Variation of DODFM with m for the assumed

set of parameters.

in the radial gradient of F(p, 0). This is also valid forthe quadratic and the biquadratic distributions,respectively, shown in figures 14 and 15 and in figure 16.The effect of the TEM20 mode is shown in figure 17.As m increases the DODFM increases also, i.e., asthe peak of the distribution moves towards the coldregion (r > R), ~rF(03C1, 0) decreases and the DODFMincreases. The variation of the DODFM with diffe-rent spatial profiles of the TEM10 mode is shown infigure 18; from which it is clear that as m increases(i.e., the radial position of the zero value movestowards the centre of the beam), the DODFMincreases also. Moreover, as m increases V,F(p, 0)decreases. Figures 12 to 18 reveal that there is somecorrelation between the DODFM and ~rF(03C1, 0).

5. Conclusions.

In the present investigation the diffraction of anoptical carrier propagating in an unguided commu-nication channel (dry atmosphere) is rigorouslyanalysed taking into account the coupling betweenthe optical carrier and the medium only at the entranceof the medium. The density and thermal conductivityof the medium are both considered to be temperature-dependent quantities. Both the absorption coefficientand the polarizability of the medium are consideredinversely-proportional to the square of the angular

Fig. 14. - Variation of DODFM with a for different valuesof i and the assumed set of parameters.

Fig. 15. - Variation of DODFM with a for different valuesof i and the assumed set of parameters.

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Fig. 16. - Variation of DODFM with a for the assumed setof parameters.

Fig. 17. - Variation of DODFM with m for the assumed setof parameters.

REVUE DE PHYSIQUE APPLIQUÉE. - T. 19, N- 10, OCTOBRE 1984

Fig. 18. - Variation of DODFM with m for the assumed

set of parameters.

frequency. Four main relevant topics are deeplytreated, namely, 1) the radial temperature distribution,T(r, 0), 2) the average beamwidth parameter, fa,3) the dimensionless optical distortion figure of merit,DODFM, and 4) the irradiance tailoring techniqueas a method to reduce the diffraction caused by therefractive effects ; since one of the fundamental limitson the performance of the optical channel is set bydiffraction.

In general, T(r, 0), fa, and DODFM are all functionsof the optical electric field Eo, optical beam radius R,and optical frequency (JJ at constant pressure.The main conclusions of the present study are :

i) Both T(r, 0) and 1 VrT(r, 0) 1 increase with theincrease of Eo and/or R

ii) Both T(r, 0) and |~rT(r, 0)| decrease as (JJ

increases.

iii) ~rn increases with the increase of Eo and/or Rwhile it decreases with the increase of cv.

iv) The variation in the beam radius, or the diffrac-tion, along the propagation path due to the refractiveeffects increases with the increase of Eo and/or Rand decreases with the increase of 03C9.

v) The DODFM can be used as a measure for thediffraction caused by the refractive index variations ;as at any set of parameters, the DODFM is positivelycorrelated with ~rn.

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vi) The DODFM increases with the decrease ofEo and/or R and increases with the increase of 03C9.

vii) The DODFM increases with the decrease ofradial gradient of the spatial profile F(p, 0) whateverthe TEM mode is.

viii) From the optical communication point of

view, the optical carrier must possess a) low electricfield, b) small radius, c) high frequency, i.e., small

power and high frequency, and d) spatial profileF(p, 0) of small radial gradient (V F(p, 0) 0).

Two points shall be published elsewhere i) the cri-tical optical power i.e., the smallest optical electricfield and/or the smallest beam radius, and ii) thecoupling between the optical carrier and the mediumalong over the propagation path.

References

[1] JONES, A.T. and MCMORDIE, J.A., Thermal Bloomingof Continuous Wave Laser Radiation, J. Phys. D :Appl. Phys. 14 (1981) 163-172.

[2] SMITH, D.C., High Power Laser Propagation : ThermalBlooming, Proc. IEEE 65 n° 12 (1977) 1676-1714.

[3] MAJIAS, P.M., Light Propagation Through Inhomoge-neous Media with Radial Refractive Index : Appli-cation to Thermal Blooming, Appl. Opt. 20 n° 24(1981) 4287-95.

[4] SODHA, M.S., CHATAK, A.K. and TRIPATHI, V.K.,Self Focusing of Laser Beams in Dielectric, Plasma,and Semiconductors (Tata McGraw-Hill Pub. Co.Ltd., India) 1974.

[5] WALLACE, J., ITZKAN, I. and CAMM, J., Irradiance

Tailoring as a Method of Reducing Thermal Bloo-ming in an Absorbing Medium, J. Opt. Soc. A. 64n° 8 (1974) 1123-29.

[6] EL-S.A. EL-BADAWY and F.Z. EL-HALAFAWY, ThermalBlooming of an Optical Carrier in the Atmosphere,Proc. of MELECON, 83, IEEE-Region 8,pp. B4.08, May 24-26, 1983, Athena, Greece.

[7] EL-S.A. EL-BADAWY and F.Z. EL-HALAFAWY, On theReduction of Self Focusing of Laser Beams inIonized Communication Channels, Proc. of AFRI-CON’ 83, IEEE-Region 8, pp. D2-1, December 7-9,1983, Nairobi, Kenya.

[8] KUMAR, K. L., Eng. Tables and Charts (Khanna Publ.,India) 1981.

[9] KROLL, N. and WATSON, K.M., Theoretical Study ofIonization of Air by Intense Laser Pulses, Phys.Rev. A. 5 n° 4 (1972) 1883-1905.

[10] REICHART, J.D., WAGNER, W.G. and CHEN W.Y.,Instabilities of Intense Laser Beams in Air, J. Appl.Phys. 44 n° 8 (1973) 3641-46.

[11] LAWRANCE, R.S. and STROHBEHN, J.W., A Survey ofClear Air Propagation Effects Relevant to OpticalCommunications, Proc. IEEE 58 n° 10 (1970) 1523-1545.

[12] MIKHEYEV, M., Fundamentals of Heat Transfer (PeasePubl., Moscow, USSR) 1976.