Atmospheric propagation of high-power laser pulses by self-channelling

Atmospheric propagation of high-power laser pulses by self-channelling G.P. AG RAWAL The atmospheric propagation of a sequence of short, intense laser pulses is considered. The pulses propagate in a self-induced low-density channel, the effectiveness of which depends on a set of three variables. Using a simple model for the laser-atmosphere interaction, the range of successive pulses is determined by numerical solutions to the paraxial wave equation. The dependence of the length, width, and depth of the created channel on the important variables is explored. Laser propagation in the atmosphere is a subject of continuing interest owing to its technological applications in optical communication, range-finding, and target destruction. The beam intensity and range are in general affected by atmospheric turbulence, scattering by aerosol particles (rain, fog or dust), molecular absorption, and diffraction. Some of the previous work has been reviewed in references l-3. Recently, the possiblity of self-chamrelling of high-power laser pulses has been getting attention.4-6 In one of the proposed schemes4 density reduction is achieved by optical breakdown. Alternatively thermal heating of the atmosphere, resulting from the absorbed electromagnetic energy, may be used to create a low-density channel. Such a channel might be useful for minimizing the propagation losses of other types of beams, eg charged particle beams. It may also be used to extend the range of a laser pulse by sacrificing the leading portion of the pulse to create a channel in which the trailing portion travels with less attenuation.’ An interesting possibility is to use a sequence of intense, short pulses. The first few pulses are absorbed within a few absorption lengths but the later pulses have increasing range due to the low-density channel created by the preceding pulses. In this case, the pulse train is so configured that the pulse duration is short compared to the hydrodynamic response time (of the order of time taken by the air to cross the beam radius at the speed of sound) and pulse spacing is longer than the response time. Under these conditions, the density remains constant and the pressure and temperature increase during the time that the pulse deposits its energy. After the pulse has passed, the atmosphere expands adiabatically to establish pressure equilibrium lowering the density near beam centre. The density profile encountered by a given pulse is determined by the interaction of all preceding pulses with the atmosphere. From a practical point mining the best values which the self-induced lar one wants to know

of view, the interest lies in deterof the various variables involved for channel is most effective. In particuthe minimum number of pulses

OPTICS AND LASER TECHNOLOGY

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Laser-induced density reduction An understanding of the channelling process requires a knowledge of the pressure rise induced by a laser pulse. Part of the laser pulse energy is absorbed by molecules in the air and a fraction f of the absorbed energy is converted to translational kinetic energy by various inter- and intramolecular processes. When the pulse duration is short compared to the hydrodynamic response time, the heat energy density et is found to be6 : Et = *”

EP

kBT

0

,&

d(r) Z(r) -do -3I,,

(1)

where pa is the normal atmospheric pressure, EP is the pulse energy, kBT is the thermal energy, u is the absorption crosssection area per atom and aw2 is the laser beam crosssectional area. Further, d andl represent, respectively, the density and the (time-integrated) intensity at the point r where the interaction is taking place; do is the intial density at the pressure pa and I,, is the intensity at the beam centre. In deriving equation (1) the incident laser pulse was assumed to be Gaussian in time as well as in the radial direction. To calculate the change in the atmospheric density, the first law of thermodynamics dQ = uC,,dT + pdV

pressure and

(2)

is used together with the perfect gas law pdV + Vdp = uRodT

The author is with Quantel Lasers et Electro-Optique, 17 Avenue de I’Atlantique ZI,91400 Orsay, France. The paper was presented at the annual meeting of the Optical Society of America, Chicago, 15 October 1980. Received 22 December 1980.

0030-3992/81/030141-04

required before the laser starts depositing its energy at a predetermined distance. This paper summarizes the results of a numerical investigation carried out to answer some of these questions. It should be pointed out that the model and techniques developed for the atmospheric propagation problem may also find applications in laser medicine, although the time scales involved are quite different.

(3)

Here Q, p, V and T stand, respectively, for the heat, pressure, volume and temperature. Further, Y is the number of moles per unit volume, Cv is the specific heat at constant volume and R. is the universal gas constant. On eliminating dT in

$02.00 0 1981 IPC Business Press Ltd 141

equations

where w is the initial spot size and QL= k(lmX) is the attenuation constant. Equation (10) becomes

(2) and (3) we obtain

(Y - 1)

dQ = -HV+

vdp,

(4) 2i(a$/az)

where y E C&v = (1 t R,/C,) is the specific heat ratio. For short laser pulses the electromagnetic energy is absorbed at constant volume because the air molecules do not move significantly in this time scale. This condition is easily satisfied for nanosecond pulses. Putting dI’=O in equation (4) we obtain the instantaneous pressure increase

dp = b-1) (dQ/V = b-1) et

(5)

that takes place after the passage of a laser pulse. The atmosphere then expands adiabatically (dQ=O). Using equation (4) we find that the quantity p P’ remains constant during the expansion. The final density d’ is therefore related to the initial density d by the relation d’P

= KP, + dp)hl

-“+Y

(6)

If a normalized density D=d/d,, is defined, equations (5) and (6) give D’/D = [l t BD Z(r)&] -“y,

(l),

(7)

where

B = CT-l)f(E,lk~J? (u/~M.‘~)

(8)

is the dimensionless variable that governs the strength of the laser-atmosphere interaction. It will be seen later that it plays an important role in determining the range of the laser pulse train. It is important to note from equation (7) that the density is related to the pulse intensity in a nonlinear fashion. Laser pulse propagation Atmospheric propagation of each individual pulse requires solution of Maxwell’s equations. It is assumed that the pulses propagate along the z axis and are linearly polarized along the x-axis. The electric field is given by E = i Re]$ exp [i&z-at)]]

where k=u/c, o is the laser frequency of light. In the paraxial approximation equation 2ik(&$@z)

(9) and c is the velocity J/ satisfies the

+ V 2 $ = -k2xD$

(10)

whereVt2 = @“/ax’ t a2/ay2) is the transverse Iaplacian, x is the susceptibility at the normal atmospheric density d,, and D(r) is the dimensionless density which the pulse encounters during its propagation at any point r. For the first pulse D=l and equation (10) is solved numerically to obtain the intensity I(r) = I$I12.This intensity is used in equation (7) to obtain the density which will be seen by the second pulse. Its value is then used in (10) to obtain the intensity I(r) of the second pulse. The process is repeated for each pulse. For numerical purposes it is convenient sionless variables, X = x/w, Y=y/wandZ=oz,

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to introduce

dimen-

t EVA $ =D(R-i)$

where we have introduced e = (&w2)-’

two dimensionless

and R = -(Rex/lmx)

(12)

quantities: (13)

In physical terms they govern the role of diffraction and refraction, respectively, during pulse propagation. The positive and negative values of R produce focusing and defocusing respectively, of the laser beam. It may be remarked that for some laser systems, such as a CO, laser, R is large and negative. The defocusing effects then give rise to the wellknown thermal blooming of the laser beam. Numerical

procedure

Equation (12) was solved numerically using the Fourier transform method. To take advantage of the cylindrical symmetry a recently developed quasi-fast Hankel transform algorithm7’s was used. A radial mesh of 128 points was used to represent the field $ on any transverse x - y plane. A pulse is followed until its energy becomes less than a predetermined value (taken to be 0.1% of the incident energy); the pulse is then assumed to be completely absorbed. The corresponding distance gives the pulse range 2,. During the propagation of a given pulse its intensity I(r) is used in equation (7) to obtain the density D(r) which will be encountered by the next pulse. In general the density is reduced near the beam centre. Each successive pulse therefore suffers less attenuation and has an increased range 2,. Since the density profiles at many transverse planes are required, it was necessary to store the density values on the disc. To start the computation the initial and boundary conditions must be supplied as well as appropriate values of the three input variables. It is assumed that initially the atmosphere has a uniform density and the transverse intensity profile of each pulse at the input plane z=O is Gaussian. Choice of variables As already mentioned, the atmospheric propagation of laser pulses is governed by three dimensionless variables B, E, and R and is given by equations (8) and (13). The values of these can vary over a considerable range depending upon the particular values for the pulse energy E,, the spot size w and the laser wavelength. In order to give an idea of plausible values of three quantities two extreme cases, of a CO2 laser and that of an hypothetical x-ray laser, are taken For CO2 laser we take E, = 50 kJ and w = 0.1 m. At 10.6 pm wavelength, the susceptibility of the atmosphere is Rex = 6 x 10s4 and lmx = 4.2 x lo-“. The attenuation constant is found to be (Y= 2.5 x 10q4 m-l, reflecting the presence of the familiar atmospheric window near 10 pm. The three parameters are E = 0.68, R = -1.4 x 106, and B = 1.5 x 10m3. The small values of B imply that the laser-atmosphere interaction is weak and little charmelling is expected in this case. Further, a large and negative value of R indicates strong defocusing giving rise to the thermal blooming effect. At the other extreme of the electromagnetic spectrum a narrow pulse of X-rays with E, = 10s4 J, w = lo-’ m and k = 5.1 x lo7 cm-’ is considered. At this wavelength of

OPTICS AND LASER TECHNOLOGY

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1.2 nm, Re,x = -4.9 x 10m7 and lmx = 8 x lo-‘. The values of the three parameters are E = 4.8 x 10m3,R = 6.2 and B = 490. The large value of the B indicates that significant channelling is possible in this case.

2.5

2.0

For our numerical calculations we have chosen E = 0.4 while the R and B were varied between 1 and 100. 1.5

Results and discussion Figure 1 shows the radial distribution D(r) of the density at the propagated distance Z = I,5 and 10 (in units of the absorption length) for several pulses. The effectiveness of the laser-induced channel is determined by its length, width and depth. For instance, for the tenth pulse the channel is wide and deep at Z = 1 but becomes shallow at Z = 10. The reason is that the preceding pulses either did not reach, or were not intense enough to produce any significant density ‘reduction at, Z = 10. An increase in the pulse energy increases B. Numerical results show that for larger values of B the channel is wider, deeper and longer for the same number of pulses. Variation of the effective beam radius (spot size) with the distance Z is also of interest. In Fig. 2 this quantity is plotted normalized to w, the spot size at Z = 0. We note that the first pulse diffracts without focusing as no channel is yet created. The value of Z at which each curve terminates corresponds to the range of that pulse - the pulse energy is 0.1% of the incident energy at that point. The beam radius oscillates back and forth as the pulse propagates inside the channel: a manifestation of the competing effects of diffraction and the focusing arising from the non-uniform density profile. The oscillations persist until, near the end of the channel, the density gradient is insufficient to refocus the pulse. The spot size then increases continuously and the pulse loses its energy. Very large values of R lead to oscillations in the density profile of Fig. 1. In this case the focusing is not uniform; the central portion of the beam may be focusing while the outer portion defocuses,

.-9 :: a

I.0

1

0.5

0 0

I

I

I

I

I

2

4

6

6

IO

12

z Fig. 2 Variation of the effective beam radius (spot size) with the propagated distance 2 (measured in units of the absorption length) for odd-numbered pulses. The parameters are those of Fig. 1

From a practical point of view one wants to know how the range Z, - distance Z at which a pulse is completely absorbed - of successive pulses depends on the choice of input variables. This information is displayed in Fig. 3 where the range Z, is plotted for 10 pulses for four sets of parameters. It is evident from Fig. 3 that large values of B and relatively small values of R are desirable for the atmospheric propagation. Conclusions This paper considers the possibility of atmospheric propagation of intense short laser pulses by creating a self-induced lowdensity channel in the vicinity of propagation path. It 30

r

I .2 25

I .o 20 0.8

h

2 ‘t 0.6 0”

15

IO

-

5

d

I .o

2.0

3.0

Radius Radial distribution of the atmospheric density during Fig. 1 propagation of several successive short laser pulses. The curve labeled as (2, n) stands for nth pulse at a distance Z measured in units of the absorption length. The parameters used are e = 0.4, R=lOandB=lO

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0

0

I

I

I

I

2

4

6

6

1

IO

Pulse number, N Fig. 3 The range, maximum propagated distance in units of the absorption length, for a set of ten successive pulses for four choices oftheparametersBandR:a-lOO,lO;b-lOO,l00;c-10.10; d - 10, 100; respectively. E = 0.4 in each case

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was shown that the propagation characteristics are determined by a set of three variables, B, E and R, given by equations (8) and (13). Numerical results show that large values of B are helpful in creating a deep and wide channel. This dictates the need for high energy pulses, as one should expect. On the other hand very large values of R are not helpful in increasing the range of successive pulses. Further, R should be positive and it requires that Rex is negative. This suggests that channelling in the atmosphere might be achieved, for example, from a high-power tunable laser which operates at a frequency which is just above a resonant molecular absorption frequency for nitrogen. Alternatively, the laser frequency should be higher than all of the bound resonant frequencies of the atmosphere species. This will be the case if one operates in the far ultraviolet region or above. In conclusion, it should be stressed that the present work is based on a simple model for the laser-atmospheric interaction. It is assumed that the absorbed electromagnetic energy is converted into heat almost instantaneously. The filling of the laser-induced channel by mechanisms such as convective cooling is ignored. Electrostrictive forces, arising from transverse intensity gradients, also play a role in modifying density distribution. Finally, the wind effects ignored

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here may be important considered.

when more realistic situations

are

Acknowledgements The author acknowledges contributions of M. Lax and J.H. Batteh to this work. The research was supported in part by a grant from the US Army Research Office during the author’s stay at the City College of the City University of New York. References Zuev, V.E. ‘Propagation of visible and infrared radiation in the atmosphere’ (J. Wiley and Sons, New York, 1974) Strohbehn, J.W., Ed ‘Laser beam propagation in the atmosphere’ (Springer-Verlag, Berlin, 1978) Fleck, J.R., Morris, J.R., Feit, hi.D. Appl Phys 10 (1976) 129 Askar’yan, G.A., Tarasova, N.M. JETP Lerr 20 (1974) 123 Agrawal, G.P., Lax, M., Batteh, J.H. Proc tenth Pittsburg conf Edited by Vogt W.G., Mickle, M.H. Instrum Sot Am 10 part 3 (1979) 1241 I+, M., Batteh, J.H., Agrawal, G.P. J Appl Phys to be published Siegman, A.E. Opt Letr 1 (1977) 13 Agrawal, G.P., Lax, M. Opt Left to be published

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