time-dependent propagation of high-energy laser … · time-dependent propagation of high-energy...

Appl. Phys. 14, 99--115 (1977) Appl ied Physics

@ by Springer-Verlag 1977

Time-Dependent Propagation of High-Energy Laser Beams Through the Atmosphere" II*

J. A. Fleck Jr., J. R. Morris, and M. D. Feit

University of California, Lawrence Livermore Laboratory, Livermore, CA 94550, USA

Received 14 September 1976/Accepted 20 March 1977

Abstract. Various factors that can effect thermal blooming in stagnation zone situations are examined, including stagnation-zone motion, longitudinal air motion in the neighborhood of the stagnation zone, and the effects o f scenario noncoplanarity. O f these effects, only the last offers reasonable hope o f reducing the strong thermal blooming that normally accompanies stagnation zones; in particular, noncoplanarity should benefit multipulse more than cw beams. The methods of treating nonhorizontal winds hydrodynamically for cw and multipulse steady-state sources are discussed. Aspects of pulse "self-blooming" are also considered.

PACS Code: 42.10

This is the second paper in a series dealing with the general problem of time-dependent thermal blooming o f multipulse and cw laser beams [ 1]. Time dependence is essential for describing the propagation of laser beams through stagnation zones, which are created whenever the motion of the laser platform and the slewing of the laser beam combine to create a null effective transverse wind velocity at some location along the propagation path. The location of vanishing transverse wind we shall call the stagnation point, and the term stagnation zone will refer to the portion of the propagation path, extending in both directions from the stagnation point, where the transverse wind has not yet had time to blow completely across the beam. The lack of wind at the stagnation point creates a steadily decreasing density and a thermal lens whose strength grows with time. This paper will continue the study of stagnation zones begun in [1]. Both pivoted-absorption-cell measurements [2, 3] and detailed numerical calculations of the experimental

* Work was performed under the auspices of the U. S. Energy Research and Development Administration under contract W-7405- Eng-48, U. S. Navy Contract N00014-76-F-0017 and U. S. Army Contract W31U31-73-8543.

arrangements [1, 4] give evidence that the blooming effects of stagnation zones tend to saturate with time. Thus the beam characteristics seem to approach a kind of quasi-steady state, which is possibly a result of the steady reduction in length of the stagnation zone with time [3]. Despite the existence of these quasi-steady states, calculations for high-power beams show that stagnation zones can lead to severe beam degradation. The notion of a stagnation zone requires that the transverse wind velocity vanish at at least one position along the propagation path. There are always present, however, a number of additional effects that can prevent a completely stagnant wind condition from occurring at any position. These effects are:

1) Natural convection. 2) Motion of the stagnation point with time. 3) Longitudinal air motion at the stagnation point. 4) Vertical air motion due to noncoplanar scenario geometry.

A realistic appraisal o f the in fluence o f stagnation zones on beam propagation requires that each o f these effects be assessed and possibly incorporated into the computational model.

100 J .A. Fleck Jr. et al.

Natural convection flow at the stagnation point is negligible for practical beam sizes and power levels, and therefore will not be considered further. Under most conditions the stagnation point is not stationary but moves in the same general direction as the target with a velocity that is little different from the target's. The parcel of air that sees a null wind speed changes with time and thus does not heat up in the manner of a stationary parcel. The influence of this stagnation-point motion on beam propagation has been found to be minimal for a cw wave-form [-1]. Stagnation-point motion turns out to be unimportant for a multipulse case examined in this paper as well. It is concluded that stagnation-point motion is unlikely to have any noticeable effect in alleviating stagnation- zone blooming, since, despite the motion, a substantial propagation path exists over which wind velocities are negligible. The existence of a null transverse wind-velocity component at a stagnation point in no way guarantees a vanishing magnitude of the wind vector because a nonvanishing longitudinal component almost always exists there. Any air parcel found within the beam at the stagnation point will, as a result, exit from the beam in a finite length of time. Indeed, in coplanar geometries all wind-flow trajectories should cross the beam in two locations: one for values of z (longitudinal position) below the stagnation point zs, and the other for values above z s. The wind flow may be in either the positive- or negative-z direction. In the neighborhood of the stagnation point, the wind-flow trajectories will enter one side o f the beam, reverse direction within the beam, and exit on the same side [5]. The residence time in the beam for fluid parcels passing through the beam center at the stagnation point will necessarily depend on scenario parameters, but for some typical beam sizes and scenarios this time can be of the order of 0.5-1 s [6]. Longitudinal flow should thus be about as effective as natural convection in controlling density changes at the stagnation point. One consequence of these re-entrant wind-flow trajectories is that air densities for z values greater than zs could be influenced hydrodynamically by densities for z values less than z S; but, in practical cases, the re-entrant times--except perhaps in the immediate neighborhood of the stagnation point-- would be considerably longer than times of interest. Consequently, hydrodynamic coupling between points above and below zs can be safely neglected. The existence of a position where the transverse wind velocity vanishes presupposes the extremely improb- able coplanarity of the laser beam and the trajectories

of its platform and the target--a situation that is clearly a limiting case of real-world situations, which are invariably noncoplanar. In the more general case of noncoplanar geometry, only the wind component along a certain transverse axis can be expected to vanish. The wind vector in the transverse plane will rotate and attain its minimum magnitude at the stagnationopoint. Since this minimum magnitude can never vanish, except in a space of measure zero, a steady state can always be defined for the governing hydrodynamic equations. Small amounts of noncoplanarity should not be expected to greatly improve cw laser performance in stagnation-zone situations but should contribute to ease in understanding and predicting it. The case of multipulse beams is another matter. As pulse-repetition frequencies are lowered, a small vertical wind component at the stagnation point becomes more and more effective in sweeping out the air between pulses. The benefits of noncoplanarity in stagnation-zone situations should thus be greater for multipulse beams than for cw beams. Methods for obtaining steady state solutions of the hydrodynamic equations for arbitrary transverse wind directions, based on Fourier techniques, are discussed in Secs. 1 and 2. The propagation of multi-pulse beams through stagnation zones is treated in Secs. 3 and 4. Scenario coplanarity is assumed in Sec. 3, but full noncoplanarity is assumed in Sec. 4. Section 5 deals with aspects of short pulse thermal blooming. Appendix A describes in detail an adaptive coordinate or lens transformation that allows a second order phase to be removed from the beam by means of an approp- riate coordinate transformation. Such adaptive transformations greatly improve computational accuracy, flexibility, and economy. Appendix B describes the adaptive selection of the axial integration step size. The ingredients of a versatile atmospheric propagation computer code are summarized in Table 1 for the convenience of the interested reader. Table 1 is an updated version of a similar one that appears in [1].

1. Steady-State Solutions of Hydrodynamic Equations for Arbitrary Transverse Wind Velocities: cw Steady State

Noncoplanar scenarios create effective winds whose orientation in the transverse plane varies with propagation distance z. All symmetry in the transverse plane is lost, and the x axis can no longer serve as the wind axis. The linearized hydrodynamic equations must be solved for a wind having an arbitrary direction.

Propagation of High-Energy Laser Beams Through the Atmosphere : II 101

Table 1. Basic outline of atmospheric propagation computer code

Variables

Form of propagation equation

Method of solving propagation equation

Hydrodynamics for steady-state cw problems

Transonic slewing

Treatment of time-dependent stagnation-zone phenomena for cw beams

Nonsteady treatment of multipulse density changes

Method of calculating density change for individual pulse in train

Treatment of steady-state multipulse blooming

x, y, z, t, where x, y are transverse coordinates and z is axial displacement

Scalar wave equation in parabolic approximation

2ik ~z = V~ ~ + k2(n 2 - 1)~.

Symmetrized split operator, finite Fourier series, fast Fourier transform (FFT) algorithm

d~ e x p ( - idz 2 \ / iAz - \ V?expt- 1 e x p ( - iAz z\ , vi)#

Z = k2( n2 - 1).

Uses exact solution to linear hydrodynamic equations. Fourier method for M < 1. Characteristic method for M > 1. Solves

@1 301 v_,~x + % ~ y + o o g z ' h =0 ,

/ &l ~vA

9 0 2 . ( , ~x + V, v ) ( P l - C s O , ) = ( ) ' - l) cd

Steady-state calculation valid for all Mach numbers except M = 1. Code can be used arbitrarily close to M = 1.

Time-dependent isobaric approximation. Transient succession of steady-state density changes; i.e., solves

2 ~?t 3x c~

Changes in density from previous pulses in train are calculated with isobaric approximation using

& ~x

y - 1 ~ - ~ ~ rL(x, y)a(t- t~ Cs n

where "cI.(x, y) is nth pulse fluence. Density changes resulting from the same pulse are calculated using acoustic equations and triangular pulse shape,

Takes two-dimensional Fourier transform of

cdtp f " 2 I 2 2 112 sin [:c,(k~+ky) tp]~

where I is Fourier transform of intensity, and tp is the time duration of each pulse. Source aperture should be softened when using this code provision.

Previous pulses in train are assumed to be periodic replications of current pulse. Solves

~Oi @i @i F Vx ~xx Ot + V~ ~ y

- S ~ a ( t - t .) . c~ .

Pulse self-blooming is treated as in the nonsteady-state case.

102

Table 1 (continued)

Treatment of turbulence

Lens transformation and treatment of lens optics

Treatment of nondiffraction limited beams

Adaptive lens transformation

Selection of z-step

Scenario capability

Treatment of multiline effects

Treatment of beam jitter

J. A. Fleck Jr. et al.

Uses phase-screen method of Bradley and Brown with Von Karman spectrum [1 ]. Phase screen determined by

F ( x , y ) = 5 dkxexp(ikx) dkyexp(iky).a(k~, 1/2 ky)~, (kx, k,), ~0o -oo

where a is a complex random variable and ~, is spectral density of index fluctuations.

Compensates for a portion of lens phase front with cylindrical Ta]anov lens transformation [1]. Uses in spherical case

1 1 1

Z f z T Z L

where z f is focal length of lens, z r is focal length compensated for by Talanov transformation, and z L is focal length of initial phase front.

Spherical-aberration phase determined by

CsA= 2~_ (x 2 +y2)2,

or phase-screen method of Hogge et al. [6]. Phase determined as in turbulence, only

-212

where l o is correlation length and cr 2 phase variance.

Removes phase

2

F~ [~i(xi- (xi)) 2 +fli(xi- (xi))] i = 1

through lens transformation and deflection of beam. Here x a = x, x2 = y, averages are intensity weighted, el and fi~ are calculated to keep the intensity centroid at mesh center, and intensity weighted rms values of x and y are constant with z.

Adaptive z-step selection based on limiting gradients in nonlinear contribution to phase. Constant z-step over any portion of range also possible.

General noncoplanar scenario geometry capability involving moving laser platform, moving target, and arbitrary wind direction. In coplanar case, wind can be function of t and z.

Calculates average absorption coefficient based on assumption of identical field distributions for all lines [6]

Z ~ exp( -~ iz )

~f~ exp(-~,z) ' i

where fi is fraction of energy in line i at z =0.

Takes convolution of intensity in target plane with Gaussian distribution

{ x'2 +y'21 Ij,~,o~= S I e x p / - ~ " I ( x ~ Xt y ~ y l ) d x l d y l ~

where ~2 is variance introduced by jitter,

Propagation of High-Energy Laser Beams Through the Atmosphere: II

The linearized hydrodynamic equations to be solved are

d~x d~- + 0~ -V'-v~ = 0, (la)

d

2 -~-Oi~j[~f~Vli"~ ~X i ~31~_V;v~) ] , ( l b )

d ~-. ( e l - - C2~01) ----- ('~ - - 1 ) ~ I , ( l c )

where ~,_v~, and pt represent the density, velocity, and pressure perturbations induced by laser heating, q is the

d viscosity, and the total derivative ~ is defined by

d c~ 8 0 dt - #t + V ~ x + V ~ y . (2)

Elimination of Pl and v~ yields the following equation for ~1

d / d ~ 4 t l v2de, t (3)

We are interested in the steady state or the case in which (3) becomes

( 2 4 ~ V 2 v,~ +V,~y ~ = ( ? - I ) a V 2 I , (4) 3 Go

where

~t v~ g01 0~1 = ~ - + , ~ ~ . (5)

The solution for ~x is carried out in two steps : first (4) is solved with ~1 as dependent variable, and then (5) is solved for Q1. We shall restrict our attention to the subsonic case,

2 2 2 where v x +vy <c~. In that case (4) is elliptic and its solution can be expressed in terms of a finite Fourier series representation

103

where [(kx, ky) is the Fourier transform of I(x, y), and where all Fourier transform operations are evaluated by means of the FFT algorithm. The function ~l(x,y) thus obtained then becomes the source term for (5), which can be solved using the Carlson method [1] for integration along characteristics. The solution of (5) is obtained by a difference method in configuration space in preference to a Fourier transform method because the transform of Ql(x,y) will have poles whenever vxk ~ + vyky = 0. The numerical evaluation of the inverse transform will be troublesome, since these poles must be avoided. If we define

t)xA ~ '

i' -= vx [v~], (8b)

. , _ v~ (8c)

the difference equation satis fled by ~ij = ~1( iA x,jA y) can be written

1 ( i - 1 ~ij = fl~i-i',j-j '+ -~)~i,j-j'

Z]y . -'[- ~Oi-i',j-j"

+ ( 1 - ~ ) ~ i j _ y } for f i > l ; (9a)

e~ = flO~-.,j-y + (1 - fi)q~-,.i

ZJy . . + ~ (e*J + fie~-",J-;

+(1-f l )~_v . j} for f l < l . (9b)

~I(X, y) = E ~l(kx, ky) e x p [ i (k~x + k,y)]. (6) kx,ky

The coefficients ~(kx, ky) satisfy

q 4. ~ 2 ,

c2 ;

2. Steady-State Solutions of Hydrodynamic Equations for Arbitrary Transverse Wind Velocities: Multipulse Steady State

Isobaric density changes induced by multipulse heating are governed by the equation

Dr n~ ~ ~x -t-Vy ~y

_ 7 - 1 c~ ~ ~. ~fl.(x, y)~(t- t.), (to)

(7)


I

8

t- o I - -

10 I I i i

Stagnation / point

5

0

I 1 t I

-100 0.5 1.0 1.5 2.0 2.5 Axial distance - km

Fig. 1. Transverse wind velocity as a function of axial distance

where zfl,(x, y) represents the fluence of the nth pulse, and zp represents the pulse width. If "steady-state" conditions prevail, it can be assumed that I, does not vary from pulse to pulse. Hence I,(x, y)= l(x, y). Taking the Fourier transform of (10) with respect to x and y yields

OG~p Ot + i(Gk~ + v'k')~v

- Y~w~lc~j(kx, k , ) Z f ( t - t , ) . (11) C s n

Solving (11) for ~'P at a time t=mAt, where m is any integer larger than Np (below) and At is the time interval between successive pulses, gives

~1 (kx, kr, mA t) - Y - 21 o~(kx ' ky ) Cs

Np �9 ~ exp [ - inA t(k~G + kyvy)]. (12)

n = l

The exponentials in (12) correspond to translations of the individual-pulse fluence distributions in configuration space by wind motion. The summation begins with n = 1 because the isobaric density changes created by a given pulse do not have time to develop during the pulse width, zp. The upper limit Ne is based on numerical considerations and is determined by

( [ N ~ ] [NAy]~ (13) N, = MIN N,.p,t, [[GIA-~]' [Iv, lAt--~--]]'

where N is the numerical length of the mesh used for solving the wave equation. In (13) MIN signifies the minimum of the arguments, and the square brackets represent the integer part of the arguments inside them�9 An input value of Np is useful if

a true stagnation point is encountered along the propagation path. In such cases the total density change at the stagnation point can be kept bounded. For example, Ninv, t might be set equal to the actual number of pulses in a given train, in which case a true lower bound could be assigned to the intensities at the target�9 The remaining arguments in (12) prevent any pulse fluence distribution from affecting the density calculation if it has been translated by more than the minimum (physical) dimension of the computational mesh for the wave equation, i.e. MIN(NAx, NAy). The density calculation itself is carried out on a 2N x 2N mesh, which has a buffer of length N in both the x and y directions. Thus if Np satisfies condition (13), periodic "wrap-around" or positional aliasing of the density contributions by past pulses in the train is avoided�9 The summation in (12) may be evaluated directly, and ~ can be expressed in the form

2 cd(kx, k,) Cs

[ iNp+I 1 �9 exp[- ~ - A t(k~G + k,G)J

Np sin -~- A t(kxG + krvy )

At sin ~- (kxv x + kyv~)

(14)

The density ~)I(X, y) is obtainable from (14) by an inverse Fourier transform. This solution method is accurate, easy to implement, and efficient when used with a set of one dimensional tables of trigonometric functions. A satisfactory solution of(10) is also obtainable directly in configuration space by summing along wind characteristics the contributions to the density change from previous pulses�9 This procedure requires bilinear in- terpolation and is less efficient and less accurate than the corresponding Fourier method�9 The Fourier method described in the previous section, on the other hand, becomes a necessity when wind speeds approach the speed of sound.

3. Propagation of Multipulse Laser Beams Through Stagnation Zones in Coplanar Geometry

The following coplanar scenario was chosen to determine the sensitivity of stagnation zone thermal blooming to stagnation zone motion�9 The motion of the stagnation zone' turns out to be unimportant, but the

Propagation of High-Energy Laser Beams Through the Atmosphere: II 105

% ~ 200

i g

150

o

-~ 100

50

g

C~ Vacuum, corrected for linear absorption Absorbing atmosphere

I I I I

(a) I I (b) I I 0.2 0.4 0.6 0 0.2 0.4

T ime- s 0.6

Fig. 2a and b. Area-averaged target intensity as a function of pulse time : six pulses at v = 10 s- < (a) No stagnation-zone motion included. (b) Stagnation-zone motion included

0.1 s 0.2s

i):, (I I (( I,, I) )lJ <i//

0.3s

??b-~ I l, I { , . r , / , i g )':, (( ;:2ii " 'i >';--~',// I b, ,: cJ ) ) )

;~N~r "/ <;5~2~< )1

0.4s 0.5 s

Fig. 3. Isointensity contours as a function of pulse time for v = 10 s-1

0.6s

results are nonetheless indicative of what a stagnation zone can do to a high power multipulse beam. The pertinent physical data are the following

Power, P 53 kW Range, R 2.5 km Focal length, f 4.5 km Absorption coefficient, c~ 0.25 km- 1 Wavelength, 2 10.6 gm Aperture diameter, 2e (Gaussian at 1/e 2) 30 cm Slewing rate, f2 7.44 mrad/s Pulse-repetition rate, v 10, 25, 50, and

100s-

These data can be expressed in terms of the following dimensionless numbers

N F = k a 2 / f = 2.96,

N o = 2 a / v o A t = 3.2 x 10-av,

N s = f 2 f / v o = 3.6,

N A = ~f = 1.125,

1 05 N o N A N p ( ? - 1)Ep = 100

N ~ 2 0~~ c~2 3( a ) a f '


Table 2. Comparison of multipulse intensities with and without stagnation-zone motion

Time Time-averaged intensity [W/cm 2] Is]

No motion Motion

Average Peak Average Peak

0. t 168.5 234.0 168.5 234.0 0.2 78.9 117.3 78.7 117.3 0.3 60.7 100.7 60.8 103..~ 0.4 53.7 92.5 55.3 96.8 0.5 49.6 88.7 52.4 92.7 0.6 47.3 84.1 50.2 87.5

80

60 E

I 40 g

20

,

2"<-- ~ l ~ s . 1~ 50 s _ ' ~

I i I I 0.2 0.4 0.6 Time - s

Maximum increase resulting from stagnation-zone motion: av, 6.1%; peak, 4.0 %.

Fig. 5. Space-aberaged intensity as a function of pulse time for various pulse-repetition rates

Table 3. Comparison of fluences with and without stagnation-zone motion

Fluence [J/cm 2]

No motion Motion

Average Peak Average Peak

30.61 42.1 30.1 40.7

Decrease resulting from stagnationzone motion: av, 1.7%; peak, 3.3%.

40 I I I

3 0 B

10 , /@ \\tit g 0 ! "- ?.:,',_Ji!!/; o ,,'t, V '//!/*

-10 i l/;;:V-',- ,,' ! !

_ . o -

-30 -

-40 I -20 =10 0 10 20

x coordlnate- cm

Fig. 4. Fluence contours for case of v= 10s- t , motion included

where N e , N o, N s, N A, and N D represent, respectively, the Fresnel, overlap, slewing, absorption, and distortion numbers [7]. The value of the distortion number N e is quite high, and the resulting thermal blooming is about the maximum that the code can accommodate. In the present scenario, the stagnation zone and target move with a speed of 300 m/s. For a multipulse beam with v--100 s - i , the stagnation zone moves 3 m between pulses, whereas for v= 10s -1, the stagnation zone moves 30 m between pulses. It would be hoped that, in the case of lower pulse-repetition frequency, the greater movement of the stagnation zone would lead to a reduced buildup of stagnant-air density changes. This effect turns out to be minimal. The time dependence of the average intensity on target (averaged over the minimum half-power area) is shown for the case of no stagnation-zone motion in Fig. 2a and for the case with stagnation-zone motion in Fig. 2b. The calculation is carried out for six pulses. The isointensity contours for the six pulses are shown in Fig. 3 for the moving stagnation zone. The contours in the nonmoving case are so similar that they are not shown. The performance in the two cases is summarized pulse by pulse in Table 2, where the intensity values have been averaged over the interpulse separation time, and the percent improvements in intensity indicated are for the last pulse in the train. Surprisingly, the improvements in peak and average fluence go in the opposite direction. The peak and average fluences are actually slightly higher in the no- motion case, as shown in Table 3. This behavior is due to the large contribution that the first pulse in the train makes to the total fluence. The isofluence contours for the case with motion are


Iii / . f l i / I ,')! I/tl it,'% l))/] l

0.01 s 0 .05 s 0 .10 s

0 .15 s 0 .20 s 0 .25 s

Fig. 6. Isointensity contours for pulses sampled from v = 100 s- ~ train

displayed in Fig. 4. The central fluence peak contains the maximum value and makes the largest contribution to the fluence averaged over the minimum half-power area. Apparently in the no-motion case the subsequent pulses in the train make a greater contribution in the central region than do the corresponding pulses in the case with motion. This small difference in peak and average fluences is of little or no practical importance, and is indicative of the fact that stagnation-zone motion plays no vital role in determining thermal blooming in stagnation zones. Figure 5 shows the dependence of average intensity on time at the target range for different values of pulse- repetition rate, v. Each curve begins with the time of arrival of the second pulse. (The first pulse would create a time-averaged intensity of 189 W/cm2.) It is clear that reducing v diminishes the effect of the stagnation zone. The reason obviously is that for smaller values of v the air can be swept out by wind between pulses over a greater proportion of the propagation path. Sample pulse-isointensity contours for v = 100 s- 1 are displayed in Fig. 6; these should be compared with those for v = 10 s-1 in Fig. 3. At the lower repetition rate, the beam has divided into two distinct spots. At the higher rate, lateral peaks are also formed but they are

much less distinct. The lateral spreading of the contours as a function of time is shown in Fig. 7. The width perpendicular to the wind is determined by measuring the maximum distance perpendicular to the wind direction between 30 % contours.

8 0 , t I ' I m -

E O

-~ 60

o

o 40 Q;

20

E O

e~

0 0.2 0 .4 0.6 T i m e - s

Fig. 7. Width of beam in direction perpendicular to the wind at various pulse-repetition frequencies

E

I - 0 . 5

g

0 o -I.0


I

I / x-Stagnati~ E~ 5 I / point

_1o r I I I

-1 .5 I l I I

108

9 10 s -1 25 s -1

50 s - I 100 s -1

Fig. 9. Changing shapes of isointensity contours as a function of pulse-repetition rate for noncoplanar scenario; laser at 10-m elevation

I I [ i

8

E

I 6

*~ 4

2

I I I I 0.5 1.0 1.5 2.0 2.5 Axial distance - km

Fig. 8a-c. Transverse wind velocity as a function of axial distance for mu~tipulse beam. (a) x component ; (b) y component; (c) magnitude

In conclusion, the performance of a multipulse laser under stagnation-zone conditions can be improved by lowering the pulse-repetition frequency, but, with or without motion of the stagnation point, the thermal blooming is likely to be substantial.

4. Ef fect o f N o n c o p l a n a r i t y o f P r o p a g a t i o n o f Mult ipulse B e a m s Through S t a g n a t i o n Z o n e s

We consider again the scenario of Sec. 3. With all problem parameters the same, except that the laser is now assumed to be elevated 10m above the scenario plane. Figure 8 shows the vertical and horizontal components and magnitude of the transverse wind velocity as functions of propagation distance. Figure 9

represents, the isointensity contours in the target plane for the various repetition rates. Table 4 compares laser performance as a function of pulse-repetition frequency for the coplanar scenario and the noncoplanar scenario with a laser elevation of 10 m. In the absence of complete steady-state data for the coplanar case, we have used in Table 4 intensity values corresponding to the final times exhibited in Fig. 5 for a given value of v. Thus the improvements due to noncoplanarity shown in Table 4 are conservative estimates. It is seen from Table 4 that improvements of at least a factor of 2, conservatively estimated, are possible for all values of v. In the case of v = 10 s-1 the laser performance is even better than it would be in a vacuum. The reason is that for this pulse-repetition frequency the overlap numbers at the stagnation point is only 2, and for overlap numbers in the range 1-2 such enhancement effects for multipulse beams are well known [-6]. To summarize: stagnation-zone blooming for multipulse beams can be minimized by a combination of elevating the laser aperture above the scenario plane and lowering the pulse-repetition frequency.

5. S ing le -Pul se T h e r m a l B l o o m i n g in the Tr iangular Pu l se A p p r o x i m a t i o n

The isobaric approximation for changes in air density is invalid for a single laser pulse whose duration is comparable to or less than the transit time of sound

Propagation of High-Energy Laser Beams Through the Atmosphere : II 109

Table 4. Comparison of multipulse beam properties for coplanar and non-coplanar scenarios. Power = 53 kW, range = 2.5 kin, 2 = 10.6 gin, elevation h = 10 m, and vertical wind speed at stagnation point = 0.62 m/s

Pulse Minimum Time to steady Overlap Peak intensity Peak intensity repetition half-power area state number at at target at target frequency, v (stagnation (non-coplanar stagnation point (coplanar (non-coplanar [ s - 1] point, scenario) (non-coplanar scenario) scenario)

non-coplanar [s] scenario) [W/cm 2] [W/cm 2] screnario) [cm- z]

Intensity Intensity averaged over averagedover minimum minimum half-power area half-power (coplanar area (non- scenario) coplanar [W/cm s] scenario)

[W/cm 2]

10 131 0.19 1.9 85.Y 287" 52.0 c 181 u 25 116 0.18 4.49 59.5: 116 32,5 r 65.6 50 104 0.17 8.49 28.7 d 70.9 17.8 d 42.3

100 140 0.19 19.0 30.4 a 49.0 13.2 ~ 30.3

" Vacuum beam has value 238. b Vacuum beam has value 170.

t = 0.6 s, steady state has not been reached. a t = 0.32 s, steady state has not been reached.

t = 0.2 s, steady state has not been reached.

across thebeam. In this time regime--referred to as the ta-regime because of the time dependence of density changes arising from an applied constant laser-energy absorption ra te-- the air-density changes must be determined from the complete set of time-dependent hydrodynamic equations (1) [1, 9]. At late times in the pulse, t 3 thermal blooming tends to reduce the on-axis intensity relative to what it would be if the beam were propagating in vacuum. This reduction increases with time, and for sufficiently late times a depression appears in the center of the beam. Energy added to the pulse at later times will contribute only marginally to the on-axis fluence. Thus, for a specific peak pulse intensity, the on-axis t]uence appears to saturate as the pulse duration is stretched out more and more. These properties are best illustrated by a numerical example. Let us consider a beam that is Gaussian at z = 0 with 1/e2-intensity radius 25 cm. The beam, which is focused at 2.5kin, is assumed to be 2 x diffraction limited (2-sCaled) with )~ = 10.59 lam and

= 0.3 x 10- s cm- t. The pulse is square-shaped in time and lasts 100 gs. The choice of a square-shaped pulse is convenient because a single calculation contains the complete information for all square pulses of duration shorter than the one chosen. Figure 10 shows the on-axis intensity at z=2 .0km, obtained by detailed numerical solution [1] of (1). The on-axis intensity clearly drops to a negligible value before the end of the pulse, and, as a consequence, the on-axis fluence saturates as the pulse width increases, as

3.5

~E 3.0 U

~ '2 .5

2.0 x "~ 1.5

1.0

I I I

0.5

0 25 50 75 100 Time - p s

Fig. 10. On-axis intensity as a function of time. The pulse is taken to be square-shaped in time. Thermal blooming reduces on-axis intensity to a negligible value after a sufficiently long time

150~ , ,

" oS0 �9 ~ 25

0 25 50 75 Time-gs

100

Fig. 11. Saturation of on-axis fiuence due to strong pulse thermal blooming

1 1 0

r

I

6.7

I1 -~ ,,P

,2 0 10 20 30

Radlus - - cm

Fig. 12. Three-dimensional plot of intensity as a function of time and radius corresponding to Figure 10 and 11

0 0 10 20 30

Radlus -- em

Fig. 13. Intensity as a function of radius for increasing time in pulse corresponding to Figures 10 and 11

250


I

200

I 150

100 . _

•

? (~ 50

1 I I

m

o [ 25 50 75 1 O0

Pulse length -/ss

Fig. 14. On-axis fluence as a function of pulse length, as calculated with triangular pulse approximation (x's) and by detailed numerical solution of hydrodynamic equations for a square pulse in time (solid curve). The triangular pulse approximation breaks down as saturated-fluenee condition sets in at zp = 1.5t r Erratic behavior is due to development of spikes in the intensity pattern as a function of transverse position

can be seen in Fig. 11. The detailed temporal evolution of the spatial shape of the beam is shown in Figs. 12 and 13. Figure 12 is a three-dimensional plot of the laser intensity as a function of time and radius. Figure 13 shows the radial intensity profiles for increasing values of time. The opening up of a hole in the back of the pulse is clear from both Figs. 12 and 13. Calculations of the type represented in Figs. 11-13 become impractical if one is treating a multipulse beam. The determination of nonisobaric contributions to the density is greatly simplified by the triangular pulse approximation [1], in which the dependence o f the laser intensity on time is represented as an isosceles triangle with base equal to 2%. The density is required only at time t ='cp, since the laser intensity is assumed to vanish for t = 0 and t > 2~;. The density change at t = zp can be evaluated analyti- cally in terms of a finite Fourier series representation of the laser intensity. The Fourier transform of the nonisobarically induced density change is

T+p

" 2 1 2 2 1 / 2 sm [gcs~p(k~+k ,) 3[ (15)

�9 1 ~ f ~ v - , ~ , [ cs p(kx+k,) ] J

where [ is the spatial Fourier trans form o f the intensity. The corresponding density changes at the grid points are given by the discrete Fourier t ransform expression

N sp/7.crrt ~ ) ~]P(jA x, kA y) = (2N)- 2 Z 0t ~-L-'

re, n = = N + 1

/ mj + nk\ �9 e x p / 2 ~ i ~ ) , (16)

where the basis functions are periodic on a square of side 2L. This allows for a buffer region that extends an additional distance L in both the x and y directions from the region of interest. Comparison of the triangular pulse approximation and detailed pulse thermal-blooming calculations for Gaussian-shaped pulses in time have shown good agreement between the calculated fluences for weak or moderate thermal blooming [-1]. Figure 14 shows the on-axis fluence calculated for the previous example with the triangular pulse approximation (x's) and the detailed solution of (1) for square pulses in time (solid line). Despite the difference in assumed pulse shapes, the agreement between the two types of calculation is very good up until time t~50~ts, which is well above the saturation time ts = 38 gs predicted by the perturbation


theory of Ulrich and Hayes [10] based on the work of Aitken et al. [11]. Above 55 I~S, or approximately 1.5t~, the beam abruptly develops spikes in its transverse spatial dependence; this indicates the breakdown o f the triangular pulse approximation, which is no longer valid When strong saturation behavior sets in. Figure 15 shows the fluence averaged over the minimum half-energy area (the area within the one-half peak energy contour) calculated with the triangular pulse approximation and with the detailed solution of (1) for square pulses. Both calculations increase initially, reach a maximum, and then turn over with increasing time. This is in part due to the increase of the area within the one-half peak energy contour with time. There is, however, no point in believing the triangular pulse approximation beyond the time when the average fluence curve has reached a maximum, which also coincides with the onset of erratic behavior in the on- axis fluence (Fig. 14). The perturbation theory alluded to earlier [10, 11] describes the on-axis fluence saturation for a beam that is initially Gaussian in shape and for a pulse shape that is square in time. In this theory, the expression for the on-axis intensity is

/,o/z ll I(t) =l (17)

[0 t > t , ,

where Io(z ) is the on-axis intensity for a Gaussian beam propagating in vacuum, or

lo(z ) - Io(0) e - ~ D ( z ) " (18)

Here c~ is the absorption coefficient and

D ( z ) : ( l - f ) z + ( ~ a 2 ) 2, (19)

where f is the focal distance and a is the radius of the original Gaussian beam. The saturation time t~ at on- axial position z is given by

=_[2N(7-1)~z2Epe-~] -1/3 t~ [- ~ ~ -j , (20)

where N is the refractivity, Ep is the pulse energy, and zp is the pulse duration. Since the fluence cannot be increased for pulses longer than t~, it can be argued that nothing is accomplished by making the pulse longer than t~. The fluence must be maximized instead by maximizing the product lo(z)t ~ or, equivalently, by

125

E ~ 100 E

"E ~..

.-= a 75 I

~ 50

~ 25

0 0

I I I 25 50 75

P u l s e l e n g t h - gs

I I

100

Fig. 15. Fluence averaged over minimum area containing one half of total beam energy, as a function o fpulse length. Solid curve is detailed calculation for square pulse, x's represent triangular pulse approximation

maximizing Io(z ). The maximum allowable value of Io(z ) at point z is normally determined by the condition that it not exceed the breakdown intensity, or

max lo(z ) = I~D. (21)

This maximum allowable intensity in turn determines a critical input pulse energy at z---0 given by

Eerit = ~ca 2 tflBDD(Z)e =~ , (22)

where (37) has been made use of, and where t s is calculated from

= (_2N(7_ 1)az2i , , i -1/3 t~ [ 3a4D(z ) ] . (23)

If one is dealing with a multipulse laser with pulse- repetition frequency v, (22) can be used to define a critical input power with

Pcrlt = •Ecrit

= zca2vtflBoD(z)e ~ �9 (24)

The self-consistency of the triangular pulse approximation, on the other hand, prevents the on-axis intensity from ever becoming negative, but, as previously remarked, the triangular pulse approximation breaks down for pulse energies greater than the value that maximizes the space-averaged target fluence. For this pulse energy, the average and on-axis fluences should be saturated, and further increases in pulse energy would give no return. Figures 16 and 17 have been calculated with the data on which Figures 10-15 are based, but with the following differences : the ranges for Figures 16 and 17 are 1.5 km and 2.0 kin, respectively ; the values

112 J.A. Fleck Jr. et al.

1 . 0 0 I I 1 I

~u 0 . 7 5 -

0,0 00.25

0(~' 0.5 1.0 1.5 2.0 2.5 Normalized input-pulse energy

Fig. I6. On-target fluence from triangular pulse approximation averaged over area containing (1 - i/e) fraction of total beam energy. Range = 1.5 kin, IBD = 1.6 x 106 W/cm 2

1.00~

"2~ 0.75 og 8a- ! ! 0.50

E~ i 0.25

O 0,

\

Fluence

t I 1 I I 0 0.5 1.0 1.5 2.0 2.5 Normalized input - pulse e n e r g y

Fig. 17. On-target space-averaged fiuence and intensity as functions of input pulse energy for triangular pulse approximation. Range = 2km, IBD=3 x 106 W/cm z

blooming saturates the on-axis fluence. The breakdown of the approximation will be indicated by the development of spikes in the transverse spatial dependence of the beam intensity as well as by a sharp falloff in the fluence averaged over some area as a function of pulse energy.

Appendix A

Adaptive Lens Transformation

One key to the successful implementation of a laser-propagation code is finding a coordinate transformation that keeps the laser beam away from the calculational mesh boundary and at the same time prevents the beam from contracting to an unreasonably small fraction of the total mesh area at the focus. If one is solving the Fresnel equation by

, the finite Fourier transform method, one may alternatively view the problem in terms of complementarity : one wishes to find a transformation that simultaneously keeps the beam intensity small on the mesh boundaries in configuration space and keeps the Fourier spectrum small on the mesh boundaries in k-space. If these two conditions are met, one knows from sampling theory that the numerical solution is highly accurate. The following adaptive procedure is designed to keep the intensity centroid at the center of the mesh and the intensity-weighted rms values of x and y constant with propagation distance z. These conditions can be written

~z <xl)i=O, (A.la)

~zz ((xl- (x l ) )z ) /= 0, i= 1, 2, (A. lb)

assigned (somewhat arbitrarily) to IBD at these ranges are 3 x 106 W/cm 2 and 1.5 x 106 W/cm 2. Both the on-target space-averaged fluence and intensity (Fig. 17) are plotted as functions of the input pulse energy normalized to Ecrit given in (23). The space averaging is over the area contained within the 1/e energy contour. The indicated maxima of the average fluences in both Figures 16 and 17 occur at an input pulse energy equal to 1.7 Eerit. The space-averaged fluence curves in Figures 16 and 17 are smoother than those displayed in Figure 14 because the former are averaged over larger areas. The scaling implications of the perturbation theory described in (16)-(22) are apparently valid for the triangular pulse approximation, although the maximum useful pulse energy predicted by the latter is about 50 % greater than that predicted by the perturbation theory. In summary: the triangular pulse approximation should provide reasonably accurate fluence results for pulse energies up to the values where strong thermal

X1 ~X, X2~y~

where

dxdyl(x, y)u (u)~ (A.lc)

~dxdyI(x,y) (Hereafter, all averages will be assumed to be intensity-weighted, and the subscript I will be dropped.) Conditions (A.1) also apply to the adaptive coordinate transformation of Bradley and Hermann [-12] which differs from the one we have implemented only in that it is preceded by a transformation to the coordinates of an arbitrary Gaussian beam propagating in vacuum. It should be evident, in any case, that such adaptive transformations are restricted to steady-state problems, since for time-dependent problems no single transformation will apply to all time values. To solve time-dependent problems one must employ a Talanov transformation that is optimized to all time values. This optimization is accomplished by a combination of trial and error and intuition. The splitting algorithm for solving the Fresnel wave equation can be written formally as

~,+1 [ iAz V~) [ iAz ~ e x p ( - i A z V~)g,, exp ! - - - Zr (A.2)

= exp l - 4 k - \ 2k ~ -

z=k2(n2-1) ,

Propagation of High-Energy Laser Beams Through th e Atmosphere: II 113

where the middle exponential on the right-hand side o f (A.2) contains the changes in phase resulting from hydrodynamic changes in density, turbulence, etc. Immediately after this step in the calculation, a quadratic reference phase front is determined and is removed from by means of a Talanov transformation and a deflection of the beam coordinates. These operations are carried out as part of the vacuum propagation step. During vacuum propagation the solution is advanced by solving

2ik - - = ~ $ . (A,3) Oz

Equation (A.1) can be written

1 ( xi> = ~ [. dxl dx2x~lg'(xl, x2, z)l 2 , (A.4a)

1 @25 = p J dxldx2x'~lg(xl, x2, z)l 2 , i= 1, 2 (A.4b)

where P is the beam power given by

P = J dxldx2l~(x 1, X 2 ) l 2 - (A.5)

By differentiating (A.4a) and (A.4b) with respect to z and making use of the Fresnel equation (A.3), one obtains the following relations

2 <x25 = - ~ I dxzdx2'e(xl, x2, z)'2x, ~ x ~ ~b(xl, x2, z), (A.6a)

~z(X,> = - ~Idxldx2l,~(xz,x2,z)12~x r (A.6b)

where the phase (~(xz,x2,z) is defined by

~b(xz, x2, z) = Im {lng(xl, x2, z)}. (A.6c)

In Gspace one can similarly derive

3 1 ~ < ~ ) (A.7a) = Fp I S k '

c? 2' ~zz (x2 > = - ~ Im {f I dwa&e2x,l~(x,, ~2, z)l 2

a " ~ ~P(~q, x2, z)}, (A.7b)

where ~(xl, ~c2, z) is the Fourier transform of g(xl, xl, z), and

~P(~I, ~2, z)= Im { ln~0q , tc2, z)}. (A.8)

We now wish to determine a phase front that will preserve the following conditions

~z <xi> = O, (A.9a)

3 ~z<(X-<Xi>)2>=O, i = 1 , 2 . (A.9b)

Equation (A.9b) is equivalent to

C ~ <x,~> - 2<x,> ~ <x,> =0. (A.9c)

From (A.9) and (A.6) one obtains

O i 0

a 2 2 0

(A.lOa)

(A.lOb)

Thus the reference phase front must satisfy

= 0

(x , - <~,51 ~ , ~ = 0.

Let us define a new phase variable

(A.11a)

(A.11b)

4;(xi, x2, z,+ 1/2) = G(xl , x2, z,+ 1/2)

2

+ ~ [oci(xi-(x,>)2+[31(xi-(xi>)], (A.12) i = l

where ~o(Xl, x2, z. + 1/2) represents the phase qS(x 1, x2, z,+ ~/2) at z,+ 1/2 before the vacuum propagation operator has been applied, and where cq and fll are determined so as to make conditions (A. 11 a) and (A.1 l b) hold for the phase front r 1, x2, z. + l/z). From (A. 11a) we obtain

< ~-~'(Xl,X2, gn+ l/2)~ =i'i<Xi--<Xi>)+fl, OX i I

"~ ~@i ~)o(XD X2, Zn+ l/2)> (A.13)

o r

( 0 > <xl)

From (A.11b) we obtain

<(x~-(xl))~x r x2,z,+ ,/2) I =2oq<(x,-(xl)) 2>

+fldx~- <x~>> =0 o r

O~ i - - 2((X i- (X15)25 ' i = 1 , 2 . (A.14)

Equations (A.13) and (A.14), which determine the desired reference phase-front parameters (A.12), can be shown to be completely equivalent to the relations used by Bradley and Hermann 1-12]. If the optimal phase front r is now substituted for the original phase front q~0 at z,+1/2, the phase increment

q~o- ~b'= - ~ [cq(x i - (xi>) z +fli(xl- (x/>)] (A.15) i = 1

must be compensated for in some way in order to preserve the original field. The quadratic contribution in (A.15) is compensated for by a generalized Talanov transformation, which involves a rescaling of •,

114 J.A. Fleck Jr. et al.

the mesh, and dz, according to

g(x, y, z)

(1 ( �9 ~ exp [ i - ~ +

Az ' Az ' 2 z~-- Az z y - Az 1 - - - - 1 - - -

Z x Zy

(A. 16a)

where

Z21~y ~(~, xy, z) = ~(K~, toy, z - Az) exp i + , (A. 16b)

2k 2k Z

zl - 1-(Az/z, ,~)' (A.16c)

z z2 1 - ( A z / z r ) " (A.16d)

The generalized focal lengths z~ and zy are determined by combining the reciprocals of the current focal lengths,

= 2~l /k , (A.17a)

z~, = 2~z/k , (A. 17b)

with those remaining from previous propagation steps [see argument of exponential in (A.16a)]. The linear term in (A. 15) corresponds to solving (A.3) in a coordinate system that has been rotated in x - y - z space. If this rotation is assumed to be small, it can be represented by a net deflection in the x and y coordinates given by

6x = - (fll/k) Az , (A.18a)

6y = - (fl2/k)Az . (A.18b)

The contribution Xi[31(xl- <xi>) must also be added to r before the vacuum propagation calculation, but this operation may correspond to a translation of the Fourier transform g(~c~, ~cy) by a nonintegral number of steps on the k-space mesh�9 In order to avoid this, fl~/A~c x and/32~Arty are both rounded off to the nearest integer, and g(K~, tcr) is then translated on its mesh in the x and y directions by the corresponding number of steps�9 The numerical implementation of (A.13) and (A.14) requires the following computations, where j and k represent the numerical coordinates of the mesh points

E = ~ lej~l~, j,k

1 <x> = ~ - ~ x,slgj,,<l 2 ,

1 <(x- <x>?> = ~ Z x~l~j~12- <~> ~ ,

j,.k

]~X- j,k

Z I~ z j ,k

(A.19)

( a ~ = ( 2 A x E ) - l I m l ~ , , ( ~ j k - - ~ j - l k ) ( ~ y k + ~ i - 1,~)/ ' ,~x / I..~ " J

((~-<~>)~-> =(2A/e)-'

Imf~j x j (~ jk-- * * I /c~r \

The computations involving the variable y are carried out in an analogous manner. In the calculation o f the average phase derivative, the phase derivative is monitored at each point and limited in magnitude to a fraction of ~. This prevents rapid phase fluctuations near the mesh boundary, where intensities may be weak, from contributing disproportionately to the average�9

Appendix B:

A n A d a p t i v e A l g o r i t h m f o r S e l e c t i n g

t h e A x i a l S p a c e I n c r e m e n t

It is desirable to select the subsequent axial space increment Az at a given axial position on the basis of requirements for numerical accuracy in the solution of the wave equation�9 The numerical accuracy of the vacuum propagators in the symmetrically split solution operator,

o~" + 1 = exp ( -iAz~V~)2k exp(_ iAz '~_~j

[ iAz 2 \ g . expl- ZZV q ,

is independent of Az if the solution is based on a discrete Fourier transform. The imposition of the phase front,

dz2 (B.2) Ar = 2k '

at z = z,+ ~/2, which is equivalent to passing the beam through a lens, will make the solution meaningless if any of the transverse zone-to- zone phase differences violate

Ig~, (A r < f~z, (B.3)

lay (Ach)l <~ f ~z,

0 < f < l .

It will always be necessary then to restrict the value of Az so that conditions (B.3) are met. While violating conditions (B.3) destroys the numerical integrity of the solution, satisfying them does not completely guarantee accuracy, since errors can also result from the noncommutation of V 2 and 3, and from upgrading ~ too infrequently. These errors must be controlled externally by inputting a maximum allowable value of Az. In practice, part of the effect of the phase front (B.2) is removed by the adaptive lens transformation. It would therefore be too restrictive to limit Az on the basis of conditions (B.3). As an alternative one can restrict the value of Az so as to control transverse gradients in the phase variable

2 k 0r 0 = ~ ei(x,- <xi>,) 2 - ~ ~ A z " 0 1 , (B.43

/=1


which is that part of the nonlinear phase front at z, + 1/2 that cannot be removed by the adaptive lens transformation. The next spatial increment A z "+ 1 is then chosen in terms of the current value A : by means of the relation

Az" + J = f Az"rt (B.5) max {lOj, k + 1 - 0:~1,10j+ 1,k - 0jkl }(i~ ~ f'Imax"

The arguments of the maximum function in the denominator of expression (B.5) are restricted to those mesh points where the intensity is greater than a certain fraction f ' of the maximum intensity. The final value of Az" + 1, however, must satisfy the additional constraints

0 . 8 A z , < Az ,+ 1 < 1.2Az", (B.6)

AZmin < A z n+ l ~ A z . . . . (B.7)

A : + ~ < ~ ~ , (B.8) 2~/~D + zr

k[<(x- <x>)-'> + <(y- <y>)~>]

with (B.6) taking precedence over (B.5), (B.7) over (B.6), and (B.8) over (B.7). In (B.8), fD~0.005 is an input fraction and zx=min ( I z x l , z y l ) .

Condition (B.8) is designed to reduce A z near a focus, where the geometric-optics scaling of the mesh by the Talanov transformation may result in an excessive shrinkage of the mesh. By updating the Talanov transformation sufficiently often, one can usually avoid a geometric-optics catastrophe. The adaptive z-step algorithm just described adds greatly to the convenience o frunning problems; it often improves problem running i time, and avoids large nonlinear phase changes that can invalidate the calculation. It should not, however, be regarded as a panacea. For sufficiently high beam power and strong enough thermal blooming, the criteria (B.5)-(B.8) can be satisfied and yet the problem still goes bad. In such cases, large non-quadratic transverse zone-to-zone phase differences can accumulate over many z-steps.

References

1. J.A.Fleck Jr., J.R.Morris, M.D.Feit: Appl. Phys. 10, 129 (1976); see also J. A. Fleck Jr., J. R. Morris, M. D. Feit : "Time-Dependent Propagation of High Energy Laser Beams Through the Atmosphere," Lawrence Livermore Laboratory, Rept. UCRL- 51826 (1975)

2. R.T.Brown, P.J.Berger, F.G.Gebhardt, H.C.Smith: "Influence of Dead Zones and Transonic Slewing on Thermal Blooming", United Aircraft Research Laboratory, East Hartford, Conn., Rept. N921724-7 (1974)

3. P.J.Berger, F.G.Gebhardt, D.C.Smith :"Thermal Blooming Due to a Stagnation Zone in a Slewed Beam", United Aircraft Research Laboratory, East Hartford, Conn., Rept. N921724-12 (1974)

4. P.J.Berger, P.B.Ulrich, J.T.Ulrich, F.G.Gebhardt: "Transient Thermal Bloomimg of a Slewed Laser Beam Containing a Regime of Stagnant Absorber", Appl. Opt. 16, 345 (1977)

5. The possibility of such curved flow trajectories in the neighborhood of the stagnation point was pointed out to one of the authors by B. Hogge (private communication)

6. J.A.Fleck Jr., J.R.Morris, M.D.Feit: "Time Dependent Propagation of High-Energy Laser Beams Through the Atmosphere: II ' , Lawrence Livermore Laboratory, Rept. UCRL-52071 (1976)

7. LeeC.Bradley, Jan Herrmann: Appl. Opt. 13, 331 (1974) 8. J.Wallace, J.R.Lilly: "Thermal Blooming of Repetitively Pulsed

Laser Beams", J. Opt. Soc. Am. 64, 1651 (1974) 9. P.B.Ulrich, J.Wallace: J. Opt. Soc. Am. 63, 8 (1973)

10. P.B.Ulrich, J.N.Hayes: U. S. Naval Research Laboratory, Washington, D. C., unpublished internal report (1974)

11. A.H.Aitken, LN.Hayes, P.B.Ulrich: Appl. Opt. 12, 193 (1973) 12. L.C.Bradley, J.Hermann: "Change of Reference Wavefront",

Massachusetts Institute of Technology, Lincoln Laboratory, Lexington, Mass., unpublished internal report

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