laser-excited broadside array generated from a spherically spreading laser beam

Laser-excited broadside array generated from a spherically spreading laser beam

Richard S. Larson*

Applied Research Laboratories, The University of Texas at Austin, Austin, Texas 78712 and Department of Physics, Brown University, Providence, Rhode Island 02912 (Received 18 February 1975; revised 12 May 1975)

The problem of the generation of sound from an intensity-modulated laser is generalized to include the effect of spherical spreading of the propagating laser beam. It is shown that divergence of the laser beam reduces the sound generated perpendicular to the axis of the laser beam and increases the beamwidth of the sound wave.

Subject Classification: 35.65.

INTRODUCTION

It has been demonstrated both theoretically t and ex- perimentally •' that a modulated, attenuated laser beam may act as a directional array generating sound in the form of a narrow fan perpendicular to the axis of the laser beam. If the laser beam has a constant cross sec-

tion and is propagating in the z direction of a Cartesian coordinate system, the sound generated is, $ in the farfield,

iw/3P 0 exp (ikr - iw t) (1) P = - 4rrrc s 1 + i(k/•) cos0' where r is the radial coordinate in spherical coordinates, 0 is the polar angle, w is the angular modulation frequency of the laser, P0 is the power output of the laser, c s is the specific heat per unit mass, a is the attenuation coefficient for the laser radiation, •3 is the logarithmic coefficient of thermal expansion, k = w/c, and c is the speed of sound. The angle at which the acoustic intensity is reduced to one-half the maximum at 0 = •r/2 is 4

.

Equation 2 predicts that very narrow fans of sound may be formed in the farfield. For example in water, with • = 0.1 m 't, corresponding to absorption of blue-green light, s and a modulation frequency of 106 Hz,

0t/•. = 2.4 x 10 's tad. (3) In the atmosphere, for • = 10 '$ m 't, corresponding to absorption of 3.39 /•m HeNe laser radiation by methane, 6 and a modulation frequency of 104 Hz,

Ot/2= 3.2X10 '6 rad. (4) Implicit in the derivation of Eqs. 1 and 2 was the assumption that the laser beam was perfectly collimated and, in view of the narrow acoustic beamwidth predicted in Eqs. 3 and 4, this assumption must be reexamined. The problem of the generation of sound from a spherically spreading laser beam is formulated in Sec. I; and solutions are obtained in Sec. H and HI in the farfield.

The results are discussed in Sec. IV.

I. FORMULATION

The problem to be considered is illustrated in Fig. 1; the angle of spherical spreading of the laser beam is de-

fined by the polar angie 00. If Ir is the energy per unit time crossing a surface of unit area at a distance r from the origin, conservation of energy for an unattenuated beam requires that

2•rr•(1 - cos00) It= 2•r•(1 - cos00) I 0 --- P0 , (5)

where P0 is the power output of the laser. If the intensity is exponentially decreasing, Eq. 5 is replaced by

P0 exp(-ar) I t = 2•r•(1 _ cos00) , (6)

and the energy absorbed per unit time per unit volume is then

aPaexp(-ar-iwt) H=, 2•r•.(1 _ cos00) ' (7)

for a laser modulated at frequency w. The sound generated from laser heating is obtained via solution of the equation •

( 100• )p,= /3 OH (8) V2- •- - c s Ot ' The solution to Eq. 8 with source term given by Eq.

7 is

iaf•wPo Inexp(ikr - iwt) , (9) pt = _ 8 •rec, (1 - cos00) where

In = exp(- ikr) f exp(- ar t + ik [ • - r w I ) dSr ' IF- 7'1 r, = . (10) If I•- •tl is expanded in powers of rt/r and only first- order terms retained, one may write Eq. 10 as

In =- exp[- ikr • sin0 sin0 • cos(q - •o •)

dSr • - ikr' cos0 cos0 • - ar' ] rt•. , (11)

where 0 and •o are defined in Fig. 1. For Eq. 11 to ac- curately predict phase cancellations from different por- tions of the thermal volume, the first neglected term in the expansion of I•- r•l must be much smaller that 2•r/k. In terms of the included angle ¾ between • and i•, this restriction may be written

•rr sin• • << 1,

1009 J. Acoust. Soc. Am., Vol. 58, No. 5, November 1975 Copyright (D 1975 by the Acoustical Society of America 1009

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1010 R.S. Larson: Laser•xcited broadside acoustic array 1010

x

FIG. 1. Coordinate frame for a spherically spreading broadside array.

or since sin(7/2) -<- 1 ,

--<< 1 . (12)

In Eq. 12 the range of the variable r' has been approxi- mated by 1/c•. The integration over r ' may be immedi- ately performed yielding

=lff sinO'dO'dq' r a + ik (sinO' cosO cosq ' -cosO • sinO) ' (13)

where for convenience the substitution O-0 + •r/2 was made, and the angle qo was set equal to zero since the problem has azimuthal symmetry.

II. PEAK PRESSURE AND INTENSITY FOR A BROADSIDE ARRAY

For arbitrary 0 Eq. 13 may not be integrated in closed form; however, at 0 =0 (for the peak pressure), Eq. 13 becomes

•,o:lff sinO'dO'dq' r a +ik sinO' cos•' ' (14)

The integration over (•' gives

in o = 2 • f sinO' dO' (15) ar (1 +(k/a)2sin•O') •/a '

and the final integration over •' results in the expression

' i•13Pø D(O; a/k, 0o)ex•(ikr- iwt), (16) P = - 4;rrc} where D(O; a/k, 0o) is the direetivity function for an arbitrary angie 0 with D(O; a/k, 0o) given by

D(O; a/k, 0o)= -• resin k 2 1 + --C0S200) 1/2 0/2+ •

- • cos0 +(1 - cOSOo). (1•) .

In the limit 0o << 1, arcsin( ß ) may be expanded as (a/k)O•, and therefore

lim D(O; a/k, 0o)= 1 , (18) 8 0- o

and the expression for the maximum pressure in Eq. 16

becomes, with the use of Eqs. 17 and 18,

cø•3Pø exp(ikr - loot) . (19) P'I - e=0 4•rc•

If the angle of spherical spreading 0 o approaches zero, the peak pressure is the same as that obtained for a broadside array in Ref. 1.

For larger values of 00, ID(O; a/k, 00)l =, the directivity function for the intensity, may be plotted as a function of 00 for different values of a/k as shown in Fig. 2. From Fig. 2, if a/k, which is the beamwidth obtained from a broadside array formed from a laser beam of constant cross section, is held constant, as the angle of spherical spreading 00 increases, the peak intensity (at 0 = 0) is reduced. Qualitatively, this feature is expected since, for the pressure generated from each thermal volume along the array to add up coherently, the width of the laser beam must be much less than an acoustic wave-

length. As the laser beam divergence increases, the array is less efficient in the generation of sound due to the violation of this restriction.

III. PRESSURE AND INTENSITY FOR A BROADSIDE ARRAY

To study the effect of spherical spreading of the laser beam on the acoustic beamwidth, one must calculate the integral in Eq. 13 for arbitrary 0. The integrations over 0' and •o' may not both be performed exactly al- though by a suitable series of transformations the integration over •o' may be exactly calculated leaving a sim- ple numerical integration over •' in the range 0_-< •'--< •0. The evaluation of the integral in Eq. 13 is carried out in the Appendix. The result, from Eqs. A10-A12 is that the pressure at arbitrary angle • is given in terms of D(O; a/k, 0o) by

pt= _ iwl•Po D(O; a/k, Oo)exp(ikr-iwt ) , 4•r%

(20)

0.6

oo•

0.2

o o 0.02 o. oa 0.06 0.08

O 0 - RADIANS

0.10

FIG. 2. Directivity factor [D(O;ot/b, 00)] 2 plotted as a-function of 0 0 for various values of a/k.

J. Acoust. Soc. Am., Vol. 58, No. 5, November 1975


1011 R.S. Larson: Laser-excited broadside acoustic array 1011

8.0

7.0

6.0

5.0

4.0

3.0

2.0 0.06

1.0 0 0.02 0.0,4 0.06 0.08 0.10

0 o - RADIANS

FIG. 3. Ratio k01/2/c• for a broadside array plotted as a function of 0 0 for various values of c•/k.

where

I j' F(O'; (•/k, 0o, O)dO • D(•; a/k, 0o)- 2• I -cosOo , (21)

and

F(O'; (•/k, 0o, 0)= •r sinO'

5m a sin • [O +i--k cosO' sinO) (l + rn•)

- i(3) sinO ' cosO(1- m•)l . (22) The expressions for •, •, and • are found following Eq. A4, and the expressions for m and • are found following Eq. A5.

For small 0o, with O=O, the function F(O'; a/k, Oo, O) is almost a straight line given by

F • 2•0' (23)

and consequently D(O; a/k, 0o) , calculated from Eq. 21, is

lira D(O; a/k, 00)= 1 . (24) 80'0

This is precisely the same result obtained in Sec. H.

The real and imaginary parts of the function F(O'; •/ k, 00, 0) are smoothly varying functions of 0 ', and the final integration over O' in Eq. 21 was performed numeri- cally using Simpson' s Ruleß 9 The acoustic beamwidth in the Introduction was defined as that angie 0t/z which re- dueed the intensity to one half the peak value (at 0 = 0), and in Fig. 3 the ratio kOt/z/a , were a/k is the beamwidth calculated in Ref. 1 for a laser beam of constant

cross section, is plotted as a function of 00 and (•/k. The angie 0t/z was defined implicitly by the equation

[D(Ol/z,r•/k, 0o)] z • • ß = il D(0; o•/k, 00) [ . (25)

For fixed a/k, as 0o increases the beamwidth of the acoustic wave increasesß When a/k is small, any spher- spherical spreading sharply increases the real beamwidth 0t/• (see, for example, the curve (•/k=O. 01 in Fig. 3), while for relatively nondirectional beams (see the curve a/k=O. 1 in Fig. 3), a small amount of spherical spreading is relatively unimportant. It is interest- ing to note that over the range of Fig. 3 that when the spherical spreading 00 is the same as the beamwidth calculated in Ref. 1, (•/k, the ratio kOt/z/a is invariant; that is,

Ot/•= 1 4 •-- (26) ß k ß

Iv. DISCUSSION

A theory was presented in Ref. 1 for the generation of highly directional beams of sound from a collimated intensity-modulated laser. Applications of this device include use in acoustical holography, underwater commu- nications, and remote sensing of atmospheric pollu- tants. x0

In the present analysis the results of Ref. 1 have been extended to include the effect of spherical spreading of the propagating laser beam on the beamwidth of the generated sound. The results illustrated in Fig. 3 show that a moderate amount of spherical divergence in the laser beam may significantly affect the acoustic beamwidth. In fact, over the range of variables in Fig. 3, if the amount of spherical divergence of the laser beam is as large as the acoustic beamwidth calculated in Ref. 1, the real beamwidth calculated from Eq. 25 is 40% larger.

ACKNOWLEDGMENT

This work formed part of the author's Ph.D. dissertation at Brown University; it was supported in part by the U.S. Office of Naval Research.

APPENDIX A.

The denominator in Eq. 13 is first rationalized yielding

In = sinO' dO' d•o

[ 1 - i(k/a)(sinO' cos•o' cosO- cosO' sinO)] x L1 + (k/a)•'(sinO 'cos•o' cosO- cosO 'sinO)ZJ '

This may be rewritten as

In 1 / sinO ' dO ' [(l + i k ' ) = --cosO sinO I t

-i k sinW cosOIz] , where

' 2drp' I 1 --- , , COSa(p ' , a + btcosqo'+c

(A1)

(A2)

f0' 2 cos•o ' dq ' a +b cosrp +c cos•q '



1012 R.S. Larson' Laser-excited broadside acoustic array 1012

and

a = 1 + cos•'0' sin20,

b'= - 2 sin0 ' sin0 cos0 ' cos0,

c = sin28' cos20.

Under the transformation z = tan(qo '/2), I• and 12 become

fo '• (1 + z2)dz (A3) I• = 4 • q. •g2 q. •Z4 and

fo ' (1 - z 2) dz b.=4 a + •z•. + tSz • , where

•= 1 + (sin0'cos0 - cos0' sin0) 2 ,

•= 211+(•)2 (cos28' sin20 - sin20' cos20)] , and

(A4)

• = 1 + (cos0' sin0 + sin0' cos0)•.

The integration of I with respect to go • requires one to first obtain expressions for f(dz/R) and f (ffdz/R} where R = • + [z2+ •z 4' the first integral may be found in tables ø in the form

• dz 1 Is • z•'+2mz cos(•/2)+m 2 'R - 4am a sin• in • In zz _ 2rnz eos(•/2) + m z

where

m=

and

• z 2 _ m 2 + 2 cos • aretan 2rnz sin(•/2) ' (A5)

cos• = - z(aa)•/" ' Clearly for the expression to be valid •/• > 0 and I •/2(•5)x/21 < 1; examination of the expressions for }, •, and'• shows that these inequalities are satisfied. The second integral fz 2 dz/R for •2-4}• <0 may be derived from Eq. A5. Since

d z •' - m 2 2m•(z 2 + rn2),sin(•/2) d-• arctan 2zrn sin(•/2) = R

then

arctan z•"-m 2 c(fzdz 2zm sin(•/2) = 2m• sin • R +m2 y -•). In terms of Eq. A5, this may be rewritten as

f z2 dz 1 [ • z2 - m 2 R - 42m sinf 2 cos •aretan 2mz sin(•/2)

• z 2 + 2mz cos(•/2) + m2 ß (A7) - sin • h zZ _ 2mz cos(•/2) + The solutions for the integrals Ix and 12 may now be obtained by evaluating Eqs. A3, A4, AS, and A7 at the end- points 0 and oo. the result is

• (1 + m 2) (A8) It = 8rna sin(•/2) and

12= 'Srn a sin(•/2) (1 -m 2) . (A9) After the substitution of Eqs. A8 and A9 for Ia and 12 into Eq. A2, one obtains

In: 2•r D(O; a/k, 00) (1 -cos00) , (A10) where

O(O; or/k, 0o)- I IF(O '; or/k, 00, O)dO' (Al!) 2• 1 - cos0 o

and

F(O•;øt/k, 0o, 0)= •masin(•/2) +i•cos0•sin (l+m 2)

-i(•) sin0 'coso(1- m2)]. (A12) The expressions for a, b, and c are found following Eq. A4, and the expressions for m and • are found following Eq. A5.

*Present address: Pratt-Whitney Aircraft, East Hartford, CT 06108.

1p. j. Westervelt and R. S. Larson, J. Acoust. Soc. Am. 54, 121 (1973).

2T. G. Muir, J. Acoust. Soc. Am. 54, 298(A) (1973). aRef. 1, Eq. 5. 4Ref. 1, Eq. 7. 5j. p. Mutschlecner, D. K. Burge, and E. Regelson, Appl.

Opt. 2, 1202 (1963). 6B. Edwards and D. E. Barch, J. Opt. Soc. Am. 55, 174 (1965). ?P.M. Morse and K. Uno Ingard, Theoretical Acoustics

(McGraw--Hill, New York, 1968), p. 324. SW. Laska, Sammlung yon Formelen der reinen und augewand-

ten Mathematik (Viewig, Braunschweig, Germany, 1888), p. 146.

9K. S. Kurtz, Numerical Analysis (McGraw--Hill, New York, 1957), p. 146.

1øR. S. Larson, "Optoacoustic Interactions in Fluids," Ph.D. dissertation, Brown U. (1974).



laser-excited broadside array generated from a spherically spreading laser beam

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