Effects of the Atmosphere on the Propagation of10.6-/, Laser Beams

J. N. Hayes, P. B. Ulrich, and A. H. Aitken

This paper gives an overview of the use of a wave optics computer code to model the propagation ofhigh power C0 2 laser beams in the atmosphere. The nonlinear effects of atmospheric heating and ki-netic cooling phenomena are included in the analysis. Only steady-state, nonturbulent cases are studied.Thermal conduction and free convection are assumed negligible compared to other effects included inthe calculation. Results showing the important effect of water vapor concentration on beam qualityare given. Beam slewing is also studied. Comparison is made with geometrical optics results, and goodagreement is found with laboratory experiments that simulate atmospheric propagation.

The subject of the present investigation is a theoreti-cal study of the nonlinear propagation of a high powerCO2 laser beam through a nonturbulent atmosphereusing wave optics. Ray optics treatments of thisproblem have been reported by Wallace and CamacI andby Gebhardt and Smith.2 Extensions of these eikonalmethods and treatments similar to the one presentedhere are presently under investigation at severallaboratories. 3

The nonlinear effect that is considered here is theheating of the atmosphere by absorption of energyfrom the beam. The resultant density changes alterthe index of refraction and thereby modify the beampath. Absorption in a motionless atmosphere pro-duces defocusing, which is time-dependent and is re-ferred to as thermal blooming. If a steady transversewind is present, however, the heated air is swept out ofthe beam and a steady state evolves. Beam slewingleads also to a steady state for a similar reason.

A detailed discussion of the equations appropriate tothis problem has been published elsewhere by theauthors.4 A brief rsum6 of the salient features of theproblem and some typical results are presented here.

The atmosphere is taken to be a nonviscous, perfectgas. Heat conduction and free convection are not in-cluded. Heat conduction can play a significant role inbeams whose dimensions become small enough that thePeclet number' (locally) is small. Such instances ariseparticularly in laboratory simulation experiments.In such cases a concentration of beam energy can evolve

The authors are with the U.S. Naval Research Laboratory,Washington, D.C. 20390.

Received 2 July 1971.CLEA paper 2.6.

downrange for which it is no longer valid to neglectconduction effects. These particular situations areexcluded from this report.

The atmosphere (or laboratory-simulated atmo-sphere) can therefore be described by the fluid equationsfor conservation of mass, momentum, and energy to-gether with the perfect gas law. If the absorbed energyis taken to be converted to thermal energy in timesshort compared with the times describing fluid motion,the internal energy per unit mass is assumed propor-tional to the temperature. Due to the presence ofnitrogen in the atmosphere, however, it can take anappreciable time for the energy stored in excited vibra-tional states to be distributed among all the degrees offreedom. This effect leads to the kinetic coolingphenomenon described by Wood et al.6 Thus, thetotal internal energy per unit mass is written as the sumof the energy that is in thermal equilibrium and thevibrational energy. A further equation describingthe approach to equilibrium of the vibrational energyis needed in this case.

For the computer runs modeling the atmosphere thedensity changes due to beam heating are always lessthan 1% of the ambient density, and so the fluid equa-tions are linearized. The steady-state solution isfound by ignoring all time dependence. Taking thewind along the y axis, the density change is given by

p1 -(a/cpvoTo)

X| {i -. exp[-(y - y')/vor]}I(x,y,z)dy', (1)

where a is the linear absorption coefficient, c is thespecific heat at constant pressure, vo is the wind speed,To is the ambient temperature, 8 is a constant propor-tional to the ratio of absorption coefficient for CO2 tothe total absorption coefficient, is the relaxation

February 1972 / Vol. 11, No. 2 APPLIED OPTICS 257

time of the excited vibrational levels, and I is the beamintensity.

For the computer runs modeling the simulated-at-mosphere laboratory experiments the density changescan be large (typically 30-100%). Accordingly, linear-ization of the equations is replaced by an isobarictreatment together with an assumed linear dependenceof the absorption coefficient on the density. For theseexperiments there is no kinetic cooling. The densitychanges in these cases are given by

Pi = [(acpVoTo) I I(xy'z)dy'

[1 + (a/cpVoTopo) fl I(xy',z)dy']. (2)

The case of a slewing beam is treated in the rotatingframe of reference. This gives rise to an apparentwind which increases linearly with distance downrange.

The light beam itself is described by the wave equa-tion for the scalar amplitude s, neglecting the term b2 y/az 2 ,

2ik(o/az) + [(a2/aX2 ) + (a2/y 2)]b + k2(n2 - 1)(P = 0, (3)

where k is the wavenumber and n is the index of re-fraction. It follows from Eq. (3) that

ff dxdy *,po = constant in z.

0.0 km

0.6 km

0.2 km

0.8 km

Fig. 1. Isoirradiance contours for six downrange distancesbetween laser aperture and focal point. Beam parameters:power, 100 kW; radius, 10 cm; focal length, 1 km; no beam

rotation. Atmospheric parameters: wind speed, 200 cm/sec,blowing from left to right; partial pressure H20, 8 Torr.

(4)

The beam intensity is given by I = e-z~so*,p.The interaction of the beam and the fluid medium is

described by the Lorentz-Lorenz law, which, writtenin terms of deviation from ambient values, is

n2- I = 3Rp1, (5)

where R is the refractivity. Equation (3), using Eq.(5) and either Eq. (1) or Eq. (2), was programmed forcomputer analysis. The resulting propagation equa-tion was solved by an explicit marching algorithm, de-tails of which appear in Ref. 4.

Some typical results of the computer studies willnow be described. All beams are taken to be Gaussianat the laser aperture. The beam behavior is displayedby a set of isoirradiance contours for selected down-range distances. The intensities at each range arenormalized to the peak intensity at that range and eachcontour represents a change of 10% relative to the peakintensity. All contours in a given figure are to thesame scale. The left-hand side of Eq. (4) is calculatedfor each iteration and monitored for constancy as anaccuracy check on the code. Acceptable runs arethose which violate this condition by no more than afew parts in 104 and which, in addition, have no lessthan six sample points per oscillation of the beam ampli-tude.

Figure 1 is an example of a situation in which theheating causes severe distortion of the beam. Thewind is blowing from left to right. The beam and at-mospheric parameters are given in the caption. Thekinetic cooling has little compensating effect, since theproduct of relaxation time r and wind speed vo for this

0.28 km 0.55 km

0.0 km

0.82 km 1 .09 km 1.36 km

Fig. 2. Isoirradiance contours for six downrange distances asindicated. All parameters are as in Fig. 1 except that for thiscase the beam is slewing at the rate 0.15 rad/sec into the wind,

which is blowing from left to right.

run is about 0.4 cm. This means that the region inwhich most of the beam intensity is confined is alwaysmuch greater than this vor. Furthermore, 8 [cf. Eq.(1) ] is less than 1, so that the beam is always heating

the air. The blooming transverse to the wind gives abeam about eight times the diffraction-limited spotsize at the focal point. Such severe blooming requiredsolution of this particular case in an expanding co-ordinate system.

The addition of beam slewing increases the relativewind velocity and leads to a marked change in beambehavior as shown in Fig. 2. Here the slewing rate is0.15 rad/sec. Thus, in addition to the 200 cm/secsteady wind there is a wind of 150 m/sec at the focalpoint. This reduces the magnitude of the nonlinearheating term and also increases the product of relaxa-tion time and wind speed. Hence, at the focal pointthe beam looks more like a diffraction-limited beam,which in turn means that it is reduced in size relative to

258 APPLIED OPTICS / Vol. 11, No. 2 / February 1972

the product vr. Further distortion, except for diffrac-tion, occurs only on the downwind side of the beam.

The contours shown in Fig. 3 are for propagation ofidentical beams through atmospheres containing threedifferent partial pressures of water vapor. The con-tours are displayed at the focal point, and the diffrac-tion-limited spot size and scale are shown on the left-hand contour. The increased water vapor leads toenhanced collision rates with the excited nitrogenmolecules, which shortens the effective relaxationtime and reduces the kinetic cooling and simultane-ously increases the heating of the air.

The relaxation times used in all of these runs weresubject to some uncertainty, and a test was made of thesensitivity of the results to choice of relaxation time,all else being equal. Results are shown in Fig. 4.The central contours are at the focal point calculatedwith the best theoretical estimate for T available,8while the contours on the left represent a run with thisrelaxation time arbitrarily halved and the contourson the right show the effect of arbitrarily doubling r.It is apparent that these times must be known to withinless than a factor of 2 for practical applications of thiscode.

The kinetic cooling effects can be maximized bychoosing zero water vapor pressure. This determinesa and in Eq. (1). The wind velocity is then chosento make the product V0T greater than the initial beamsize. Other parameters were chosen so that compari-son could be made with the geometrical optics resultsreported by Wallace and Camac.' They presentedtheir data at z = 0.6 km. Figure 5 shows the resultsof the present calculation of this case. The contoursat 0.6 km agree in detail with the results of Ref. 1.The geometrical optics solution is not valid beyond thispoint but the wave optics solution proceeds and appearsto self-focus, in spite of the fact that the beam wasinitially collimated.

Finally, some results of a comparison with a labora-tory simulation of atmospheric propagation are pre-sented. The work of Gebhardt and Smith9 is par-ticularly appropriate, since they took pains to reduceto a minimum the effects of free convection and thermalconduction. A detailed report of this comparison hasbeen made by the authors.' The experiments weredone in a 50-cm CO2 gas cell which was doped withpropylene to enhance the absorption. Twelve separatecomputer runs were done with differing wind velocities.The data are plotted in Fig. 6. The quantity IREL isthe ratio of the peak on-axis intensity of the beam atthe detector to the peak intensity expected there forabsorbed, but unaberrated, vacuum propagation. Theparameter N is a quantity that naturally occurs in ageometrical optics treatment 2" 0 and was used byGebhardt and Smith to display their results. It isproportional to the power and inversely proportionalto the wind speed. The products at are shown, where tis the cell length and a is the absorption coefficient.

In addition to the above data Gebhardt and Smith2have measured beam deflections in a different experi-

DiffractionLimited Spot

5 cmi~~~~~

2.25 Torr 4.00 Torr 8.00 Torr

Fig. 3. Isoirradiance contours for beams at the focal point forthree different partial pressures of water vapor as indicated in thefigure. The beam parameters for all three runs are as in Fig. 1.

X...

T = 0.0034 see 0.0068 seac 0.0136 seac

Fig. 4. Isoirradiance contours for three identical beams at thefocal point for which the theoretically determined relaxationtime has been altered arbitrarily. The central contours arefor the beam of Fig. 1 except that the water vapor pressure is 3Torr. For the run producing the contours on the left the re-laxation time for the 3-Torr run was halved and on the right it

was doubled, as indicated.

0.2 km

0. - ein

0.4 km

1.0 kmFig. 5. Isoirradiance contours for a situation where the kineticcooling effect has been maximized. Beam parameters: power,390 kW; radius, 10 cm; unfocused initial beam; no beamslewing. Atmospheric parameters: wind speed, 600 cm/see,

blowing from left to right; partial pressure H 20, 0.0.

A i-

February 1972 Vol. 11, No. 2 / APPLIED OPTICS 259

I.I

GEOMETRICAL OPTICSTHEORY

DATA OF GEBHARDT a SMITH{

/ WAVE OPTICS THEORY

IRE 1.2

"08/O.8H xx ACAo`%X X

0.6 K- ;a Lo hA 0 AA A 0 *x

Fig. 6. Plot of IEL vs N (see textfor definitions) for experimental dataand for twelve computer runs. In boththe experiment and for the computerruns, the value of N was changed by

changing the wind speed vo.A

I I 1 _ . . . 1

2 3 4

N

z 1.60

>1.4-

a 1.4-JLI.

a1.2

C 1.0

<0.80a 0.6N

§ 0.4M0

0.2

Sz

.

0 0

* 0

o

0o

a

S

o DATA OF GEBHARDTa SMITH FOR CS,, at= 2.4

* WAVE OPTICS THEORYFOR CO,, at = 2.6

.

06

0

S

I I! I I I I I

Fig. 7. Plot of normalized beam deflection vs N for a liquid CS2

absorbing cell and for the computer model of twelve CO2 ex-

periments. In the laboratory the value of N was changed byvarying the input power, while in the computer runs the value of

N was changed by varying the wind speed vo.

ment in which the absorber was liquid CS2. Since thesesame data were not available for the CO2 experiment,the numerically computed deflections for the lattercase are compared with those for the CS2 experiment,which had approximately the same value of at. Thecomparison is shown in Fig. 7. The agreement be-tween the computed points and the observed data isstriking. According to geometrical optics, N is auniversal dimensionless parameter which should de-scribe a wide variety of experimental situations. How-ever, the disagreement between the experimental andcomputed points of Fig. 6 and the geometrical opticsprediction shows that diffraction is playing an impor-tant role for N > 1. Note, however, that the de-flections computed for the CO2 experiment coincideremarkably well with those of the CS2 experiment whenplotted with N as independent variable, despite themarked differences in experimental arrangements.In addition all points in Fig. 7 lie on the curve com-

puted according to geometrical optics. It wouldappear that deflections are not sensitive to diffractioneffects.

This paper has given an overview of the wide rangeof problems in atmospheric propagation which thewave optics code will handle. The agreement with thelaboratory experiments is confirmation of its validityfor situations where thermal conduction and free con-vection are not deemed important effects.

The authors have had useful conversations withL. C. Bradley, F. C. Gebhardt, J. Herrmann, P. M.Livingston, D. C. Smith, and J. Wallace, Jr.

References1. J. Wallace, Jr., and E. M. Camac, J. Opt. Soc. Am. 60, 1587

(1970).2. F. G. Gebhardt and D. C. Smith, IEEE J. Quantum Electron.

QE-7, 63 (1971).3. J. Wallace, Jr., at AVCO-Everett, Everett, Mass.; L. C.

Bradley and J. Herrmann, at Lincoln Laboratories, Lincoln,Mass.; P. M. Livingston and H. Breaux, at Ballistic Re-search Lab., Aberdeen, Md.; C. B. Hogge et al., at AFWL,Kirtland AFB, Albuquerque, N.M.; F. G. Gebhardt andD. C. Smith, at United Aircraft Research, Hartford, Conn.;see also F. G. Gebhardt and D. C. Smith, Appl. Opt. 11,244 (1972).

4. A. H. Aitken, J. N. Hayes, and P. B. Ulrich, NRL Report7293, 18 May 1971.

5. The Peclet number is defined as the ratio vo/(x/pocpa),where vo is the wind speed, x is the thermal conductivity, po is

the ambient density, c, is the specific heat at constantpressure, and a is the beam radius. It is a measure of therelative importance of convective heat transfer as opposed tothermal conduction. The cases considered in this workhave Peclet numbers > 10.

6. A. Wood, M. Camac, and E. Gerry, AVCO-Everett ResearchLaboratory Report 350, June 1970.

7. P. B. Ulrich, J. N. Hayes, and A. H. Aitken (to be published).8. See, e.g., R. L. Taylor and S. Bitterman, Rev. Mod. Phys.

41, 26 (1969) and references therein.9. F. G. Gebhardt and D. C. Smith, Quarterly Report to Army

Contract DAAB07-70-C-0204; United Aircraft ResearchLaboratories Report UAR-J921004-2, 16 November 1970.

10. P. M. Livingston, Appl. Opt. 10, 426 (1971).

260 APPLIED OPTICS / Vol. 11, No. 2 / February 1972

2.4K

2.0-

1.6 e

o at-o.9,2.6x at .8A at= 0.24* at= 2.6

0.2

0 I Il

I

0.4 r

2