Appl. Phys. 14, 99--115 (1977)
Applied
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-7405Eng-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 stagnationzone 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 zs. 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 zS; 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 improbable 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 appropriate 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
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)~.
Method of solving propagation equation
Symmetrized split operator, finite Fourier series, fast Fourier transform (FFT)
algorithm
e x p ( - idz
2\
/ iAz-\
V?expt1
d~
e x p ( - iAz vi)#
z\ ,
Z = k2(n2 - 1).
Hydrodynamics for steady-state cw problems
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
.
Transonic slewing
Steady-state calculation valid for all Mach numbers except M = 1. Code can be
used arbitrarily close to M = 1.
Treatment of time-dependent stagnation-zone phenomena
for cw beams
Time-dependent isobaric approximation. Transient succession of steady-state
density changes; i.e., solves
~?t
Nonsteady treatment of multipulse density changes
c~2
3x
Changes in density from previous pulses in train are calculated with isobaric
approximation using
&
~x
y-1
~- ~ ~
Cs
rL(x, y)a(t- t~
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,
Method of calculating density change for individual
pulse in train
Takes two-dimensional Fourier transform of
cdtp f
" 2 [:c,(k~+ky)
I
2 2 112tp]~
sin
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.
Treatment of steady-state multipulse blooming
Previous pulses in train are assumed to be periodic replications of current pulse.
Solves
~Oi
@i
@i
Ot F Vx ~xx + V~~ y
-
c~
S~ a ( t .
t.).
Pulse self-blooming is treated as in the nonsteady-state case.
102
J. A. Fleck Jr. et al.
Table 1 (continued)
Treatment of turbulence
Uses phase-screen method of Bradley and Brown with Von Karman spectrum [1 ].
Phase screen determined by
dkyexp(iky).a(k~, ky)~,1/2(kx, k,),
F ( x , y ) = 5 dkxexp(ikx)
~0o
-oo
where a is a complex random variable and ~, is spectral density of index
fluctuations.
Lens transformation and treatment of lens optics
Compensates for a portion of lens phase front with cylindrical Ta]anov lens
transformation [1]. Uses in spherical case
1
1
1
Zf
zT
ZL
where z f is focal length of lens, z r is focal length compensated for by Talanov
transformation, and zL is focal length of initial phase front.
Treatment of nondiffraction limited beams
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 lo is correlation length and cr2 phase variance.
Adaptive lens transformation
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.
Selection of z-step
Adaptive z-step selection based on limiting gradients in nonlinear contribution
to phase. Constant z-step over any portion of range also possible.
Scenario capability
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.
Treatment of multiline effects
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.
Treatment of beam jitter
Takes convolution of intensity in target plane with Gaussian distribution
{
Ij,~,o~= S I e x p / - x'2~+y'21
"I ( x ~ X t 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
-~-Oi~j[~f~Vli"~ ~Xi
2
~31~_V;v~)] ,
(lb)
d
~-. ( e l -- C2~01) -----('~ -- 1 ) ~ I ,
(lc)
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 tl v2de,t
(3)
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'
-= [v~],
vx
(8b)
.,_ v~
(8c)
the difference equation satis fled by ~ij = ~1(iA x,jA y) can
be written
1
We are interested in the steady state or the case in which
(3) becomes
~ij = fl~i-i',j-j '+
-'[-
Z]y .
(i-
3 Go
(2
+V,~y ~ = ( ? - I ) a V 2 I ,
~Oi-i',j-j"
=
for
fi>l;
(9a)
(4)
e~ = flO~-.,j-y + (1 - fi)q~-,.i
ZJy
where
~t
1
-~)~i,j-j'
+(1-~)~ij_y}
4 ~ V 2 v,~
103
+~
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
where vx2 +vy2 <c~.
In that case (4) is elliptic and its
solution can be expressed in terms of a finite Fourier
series representation
.
.
(e*J+ fie~-",J-;
+(1-fl)~_v.j}
for
fl<l.
(9b)
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
~I(X, y) = E ~l(kx, ky) e x p
kx,ky
[i(k~x +
k,y)].
(6)
_
7-1
c~ ~ ~. ~fl.(x, y)~(t- t.),
(to)
The coefficients ~(kx, ky) satisfy
q
4.
~
2 ,
c2
;
(7)
104
J.A. Fleck Jr. et al.
10
I
I
I
i
i
Satgnaoitn/
5
point
8
0
ot-
I--
I
-100
1
t
I
0.5
1.0
1.5 2.0
Axial distance - km
2.5
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
2 cd(kx,k,)
Cs
OG~p
Ot + i(Gk~ + v'k')~v
-
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 target9
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 avoided9
The summation in (12) may be evaluated directly, and
~ can be expressed in the form
Y~w~lc~j(kx, k , ) Z f ( t - t , ) .
Cs
9e x p[[ - iNp+I
~A t(k~G + k,G)J1
(11)
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]~
N, = MIN N,.p,t, [[GIA-~]'[Iv,lAt--~--]]'
Np
sin -~- A t(kxG + krvy)
n
At
(14)
sin ~- (kxvx + kyv~)
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 pulses9 This procedure requires bilinear interpolation and is less efficient and less accurate than
the corresponding Fourier method9 The Fourier method described in the previous section, on the other
hand, becomes a necessity when wind speeds approach
the speed of sound.
(13)
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 them9 An input value of Np is useful if
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 motion9 The motion of the
stagnation zone' turns out to be unimportant, but the
Propagation of High-Energy Laser Beams Through the Atmosphere: II
C~ Vacuum, corrected for linear absorption
Absorbing atmosphere
%
~
105
200
I
I
I
I
i
g
150
o
-~ 100
50
g
(a)
I
I
0.2
0.4
(b)
0.6 0
Time- s
I
I
0.2
0.4
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
??b-~
i):, (I I
0.1 s
0.2s
((
I,, I)
)lJ <i//
0.3s
I l, ig
;:2ii
'i>';--~',//
;~N~r "/
0.4s
I{,.
r,
/ ,
"I)':,b,,: cJ (() ) )
<;5~2~< )1
0.5 s
0.6s
Fig. 3. Isointensity contours as a function of pulse time for v = 10 s-1
results are nonetheless indicative of what a stagnation
zone can do to a high power multipulse beam. The
pertinent physical data are the following
These data can be expressed in terms of the following
dimensionless numbers
N F = k a 2 / f = 2.96,
Power, P
53 kW
Range, R
2.5 km
Focal length, f
4.5 km
Absorption coefficient, c~
0.25 k m - 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-
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~2a 3(
) af
'
106
J . A . Fleck Jr. et al.
80
Table 2. Comparison of multipulse intensities with and without
stagnation-zone motion
Time
Time-averaged intensity [W/cm 2]
E
Is]
No motion
Motion
Average
Peak
Average
Peak
168.5
78.9
60.7
53.7
49.6
47.3
234.0
117.3
100.7
92.5
88.7
84.1
168.5
78.7
60.8
55.3
52.4
50.2
234.0
117.3
103..~
96.8
92.7
87.5
I
g
0. t
0.2
0.3
0.4
0.5
0.6
60
40
,
2"<--
20 ~ l ~_ s .' 1~ 50s
I
i
I
0.2
0.4
Time- s
I
0.6
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
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 nomotion 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
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
30
I
I
B
10
,
g
"-
I
0
/@ \\tit
?.:,',_Ji!!/;!
,,'t,V
o
-10
'//!/*
i l/;;:V-',- ,,' ! !
_.o-
-30 -
-40
-20
I
=10
0
10
x coordlnate- cm
20
Fig. 4. Fluence contours for case of v = 1 0 s - t , motion included
Propagation of High-Energy Laser Beams Through the Atmosphere: II
Iii / .
107
fli/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 pulserepetition 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.
80,
t
I
'
I
m -
E
O
-~
60
o
o
40
Q;
20
E
O
e~
0
0.2
0.4
0.6
Time- s
Fig. 7. Width of beam in direction perpendicular to the wind at
various pulse-repetition frequencies
108
J. A. Fleck Jr. et al.
I
I
E~ 5I /
_1o r
x-Stagnati~
point
/
I
I
9
10 s-1
25 s-1
50 s - I
100 s -1
I
E
I
-0.5
g
0
o -I.0
-1.5
Fig. 9. Changing shapes of isointensity contours as a function of
pulse-repetition rate for noncoplanar scenario; laser at 10-m elevation
I
l
I
I
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 velocityas 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. Effect of Noncoplanarity of Propagation
of Multipulse B e a m s Through Stagnation 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. Single-Pulse Thermal B l o o m i n g
in the Triangular Pulse Approximation
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
repetition
frequency, v
[ s - 1]
Minimum
half-power area
(stagnation
point,
non-coplanar
screnario)
[cm- z]
Time to steady
state
(non-coplanar
scenario)
[s]
Overlap
number at
stagnation point
(non-coplanar
scenario)
Peak intensity
at target
(coplanar
scenario)
[W/cm 2]
Peak intensity
at target
(non-coplanar
scenario)
[W/cm 2]
Intensity
averaged over
minimum
half-power area
(coplanar
scenario)
[W/cm s]
Intensity
averagedover
minimum
half-power
area (noncoplanar
scenario)
[W/cm 2]
10
25
50
100
131
116
104
140
0.19
0.18
0.17
0.19
1.9
4.49
8.49
19.0
85.Y
59.5:
28.7 d
30.4 a
287"
116
70.9
49.0
52.0 c
32,5 r
17.8 d
13.2 ~
181 u
65.6
42.3
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 rate--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 c m - 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 . 0 k m ,
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
I
I
I
~E 3.0
U
~'2.5
2.0
x
"~ 1.5
1.0
0.5
0
25
50
Time - p s
75
100
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 o f on-axis fiuence due to strong pulse thermal
blooming
J. A. Fleck Jr. et al.
110
r
I
6.7
I1
-~
,2
,,P
0
10
20
30
Radlus - - cm
Fig. 12. Three-dimensionalplot of intensity as a function of time and
radius corresponding to Figure 10 and 11
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 analytically in terms of a finite Fourier series representation of
the laser intensity. The Fourier transform of the
nonisobarically induced density change is
T+p
I
"
9 1
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
2
1
2
2 1/2
[gcs~p(k~+k,) 3[
~
[ cs p(kx+k,)
] J,
sm
~ f ~ v - ,
(15)
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 transform expression
N
~]P(jAx, kA y) = (2N)- 2
Z
sp/7.crrt ~ )
0t ~-L-'
re, n = = N + 1
/
250
1
I
I
200
I
m
150
100
._
•
?
(~ 50
o
25
50
[
75
mj + nk\
9e x p / 2 ~ i ~ )
1O0
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= 1.5tr Erratic behavior is
due to development of spikes in the intensity pattern as a function
of transverse position
,
(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
Propagation of High-Energy Laser Beams Through the Atmosphere: II
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 onaxis 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/zll
I(t) =l
[0
(17)
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)
"
2,
=_[2N(7-1)~z2Epe-~] -1/3
[- ~ ~
-j
,
I
E~
100
E
"E ~..
.-=
a
75 I
~ 50
~
25
0
0
I
I
I
25
50
75
Pulse length
100
- gs
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 ts is
calculated from
= (_2N(7_ 1)az2i,,i-1/3
t~ [
3a4D(z )
]
.
(23)
If one is dealing with a multipulse laser with pulserepetition frequency v, (22) can be used to define a
critical input power with
(19)
Pcrlt = •Ecrit
where f is the focal distance and a is the radius of the
original Gaussian beam. The saturation time t~ at onaxial position z is given by
t~
I
(18)
Here c~is the absorption coefficient and
D(z):(l-f)z+(~a2)
125
111
(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
= 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.
1.00
I
I
1
I
~u 0 . 7 5 -
0,0
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.
00.25
Appendix A
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
\
Fluence
I
1
I
I
0, t
0 0.5 1.0 1.5 2.0 2.5
Normalized input - pulse energy
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
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,
~zz((xl- ( x l ) ) z ) / = 0,
(A.la)
i = 1, 2,
(A. lb)
X1 ~X, X2~y~
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: t h e triangular pulse approximation
should provide reasonably accurate fluence results for
pulse energies up to the values where strong thermal
where
(u)~
dxdyl(x, y)u
~dxdyI(x,y)
(A.lc)
(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~)exp [! - -iAz
~ e x p ( - i A z V~)g,,
- Zr
= exp l- 4k-
z=k2(n2-1),
\
2k
~-
(A.2)
Propagation of High-Energy Laser Beams Through th e Atmosphere: II
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
113
From (A.9) and (A.6) one obtains
O
a
0
i
2
2
(A.lOa)
0
(A.lOb)
Thus the reference phase front must satisfy
=0
(A.11a)
Equation (A.1) can be written
1
( xi> = ~ [.dxl dx2x~lg'(xl, x2, z)l 2 ,
1
@25 = p J dxldx2x'~lg(xl,
x2, z)l2 ,
(x,- <~,51~ , ~
(A.4a)
i = 1, 2
(A.4b)
(A.11b)
= 0.
Let us define a new phase variable
4;(xi, x2, z,+ 1/2)= G(xl, x2, z,+ 1/2)
where P is the beam power given by
2
P = J dxldx2l~(x 1, X 2 ) l 2 -
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),
~z(X,> = - ~Idxldx2l,~(xz,x2,z)12~x r
where the phase
+ ~ [oci(xi-(x,>)2+[31(xi-(xi>)],
(A.5)
(A.6a)
(A.6b)
where ~o(Xl, x2, z. + 1/2) represents the phase qS(x1, 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,
OXi
I
(~(xz,x2,z) is defined by
"~~@i ~)o(XDX2,Zn+l/2)>
~b(xz, x2, z) = Im {lng(xl, x2, z)}.
(A.6c)
1
=
~
Fp I S
(0
<~)
> <xl)
(A.7a)
k '
From (A.11b) we obtain
c?
2'
~zz (x2 > = - ~ Im {f I dwa&e2x,l~(x,, ~2, z)l 2
a
" ~ ~P(~q, x2, z)},
(A.13)
or
In Gspace one can similarly derive
3
(A.12)
i=l
<(x~-(xl))~x r
x2,z,+ ,/2)I =2oq<(x,-(xl)) 2>
+fldx~- <x~>>=0
(A.7b)
or
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)
O~ i - -
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.
2((Xi- (X15)25 '
i=1,2.
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
(A.9b)
q~o- ~b'= - ~ [cq(x i - (xi>) z +fli(xl- (x/>)]
(A.15)
i=1
Equation (A.9b) is equivalent to
C
~ <x,~>- 2<x,> ~ <x,> =0.
(A.14)
(A.9c)
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
(a~=(2AxE)-lIml~,,(~jk--~j-lk)(~yk
,~x/
g(x, y, z)
(1
(
I..~
Az '
Az '
1---1---
Zx
exp [i- ~
2 z~-- Az
+
z y - Az
Zy
~(~, xy, z) = ~(K~, toy,z - Az) exp i
2k
Z
zl - 1-(Az/z,,~)'
+
Z21~y
2k
,
(A. 16b)
(A.16c)
z
1 -(Az/zr) "
= 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 mesh9 In order to avoid this, fl~/A~c x
and/32~Artyare 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 steps9
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
j,k
<x> =
lej~l~,
1
~-~ x,slgj,,<l2 ,
1
<(x- <x>?> = ~ Z x~l~j~12- <~>~ ,
j,.k
]~X- j,k
Z
j,k
I~ z
*
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 average9
Appendix B:
An Adaptive
(A.16d)
The generalized focal lengths z~ and zy are determined by combining
the reciprocals of the current focal lengths,
E= ~
*
(A. 16a)
where
z2
J
((~-<~>)~-> =(2A/e)-'
Imf~j xj(~ jk--
9~
+ ~ i - 1,~)/ '
"
(A.19)
the Axial
Algorithm
Space
for Selecting
Increment
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 equation9 The numerical
accuracy of the vacuum propagators in the symmetrically split
solution operator,
o~"+ 1 = exp ( -iAz~V~)2k e x p ( _ '~_~j
iAz
[ iAz 2 \ g .
explZZVq ,
is independent of Az if the solution is based on a discrete Fourier
transform. The imposition of the phase front,
dz2
2k '
Ar =
(B.2)
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-tozone phase differences violate
Ig~,(Ar < 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 V2 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 ,
/=1
(B.43
Propagation of High-Energy Laser Beams Through the Atmosphere: II
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
max {lOj,k + 1 -
0:~1,10j+
1,k- 0jkl}(i~~ f'Imax"
(B.5)
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 , < A z , + 1 < 1.2Az",
(B.6)
AZmin < A z n+ l ~ A z . . . .
(B.7)
A : +~<
~ ~
2~/~D +
,
zr
(B.8)
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 z x = m i n ( 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.
115
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. UCRL51826 (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