A POSSIBLE METHOD FOR ATTAINING PERIODIC TRAINS OF SHORT HEAT PULSES

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A POSSIBLE METHOD FOR ATTAINING PERIODIC TRAINS OF SHORT HEAT PULSES

N. Shiren

To cite this version:

N. Shiren. A POSSIBLE METHOD FOR ATTAINING PERIODIC TRAINS OF SHORT HEAT PULSES. Journal de Physique Colloques, 1972, 33 (C4), pp.C4-45-C4-47. �10.1051/jphyscol:1972410�.

�jpa-00215087�

JOURNAL DE PHYSIQUE Colloque C4, supplkment a u no 10, Octobre 1972, page C4-45

A POSSIBLE METHOD FOR ATTAINING PERIODIC TRAINS OF SHORT HEAT PULSES

N. S. SHTREN

[BM Thomas J. Watson Research Center Yorktown Heights, New Y o r k 10598

R&umB. - Rkcemment, nous avons obtenu une solution analytique presque exacte de I'equa- tion d'onde du second ordre qui d&rit la diffusion des ondes de dkplacement, de haute frequence et faible amplitude, par une onde de pompage acoustique de basse frCquence et haute amplitude.

La solution donne la dependance, en fonction du temps et de I'espace, du champ haute frequence.

Ainsi, elle n'est pas limitee aux premikres bandes laterales de la pompe, comme c'est le cas dans le calcul habitue1 de I'interaction entre trois phonons. La precision de la solution augmente comme le rapport de la haute frequence et de la frkquence du pompage, ce qui la rend trks approprih pour un calcul de I'effet de I'onde acoustique sur les ondes thermiques. La solution met en evidence une forte compression de I'amplitude et de la phase des phonons en fonction du temps et de I'es- pace. En integrant par rapport a toutes les ondes thermiques, nous montrons qu'il est possible d'engendrer des trains periodiques de pulsations balistiques de chaleur a partir des fonds ther- miques, ou d'accentuer la compression des pulsations de chaleur injectees.

Abstract. - In a recent calculation we obtained a nearly exact analytic solution of the second order wave equation describing the scattering of high frequency, low amplitude, displacement waves by a low frequency, high amplitude, acoustic pump wave. The solution gives the total space and time depcndent high frequency strain field. It is thus not limited to first side-bands of the pump, as is the usual three phonon interaction calculation. The accuracy of the solution increases with increasing ratio of the high frequency to the pump frequency. It is therefore very suitable to cal- culating the effect of an acoustic wave o n thermal waves. The solution is characterized by strong phonon bunching ; i. e., both amplitude and phase are compressed in time and space. By inte- grating over all thermal waves, we show that it should be possible to generate periodic trains of ballistic heat pulses from the thermal background, or to further compress injected heat pulses.

In a recent publication [1] we havc found a solution prescribed high frequency input strain o n t h c boundary, to thc one dimensional wave equation for the displace- x = 0 ; the function O(x, rl), must be evaluated from ment, with parametric source term ;

1 d

U x x - - U l l =

D

^- (u, E,,,) ,

u2 ax (1)

(subscripts x a n d t indicate differentiation).

This is the appropriatc equation describing the effect of a constant large amplitude strain wave, E,,(x

-

ct), o n a small amplitude displacement field, u(x, t ) . c is the velocity o f t h e modulating wave, a n d v is the velocity of the high frequency waves in the absence of the intcr- action.

fi

is a ratio of third order t o second order elastic constants. O u r solution is more nearly exact the highcr the frequencies in u compared with those in E,,,. Thus, it should be valid for a description of the interaction of a large amplitude acoustic wave o n thermal waves such a s occur i n heat pulses. Similarly, i t makes possible a nearly exact treatment of acoustic attenuation.

The solution of eq. (1) for the strain e(x, t) = u, is,

F o r a monochromatic modulating wave, Em = E cos (Kx

-

Qt), with y = PE

4

1 (the usual physically realizable case) g sz v. (1 - y/2 cos Kq).

Then eq. (2) may be approximated as,

and

a is the velocity mismatch a n d b is the effective non- linearity.

F o r a monochromatic input, e(0, t )

--

ei'"' (0 % Q), 0, is the normalized instantaneous frequency. T h u s both the amplitude a n d frequency i n e(x, t ) are periodi-

{d:" [it'

^{[ g ( -} c t f f 1 2 e ( ~ ,

.

( 2 ) cally bunched a s seen i n figures 1 a n d 2, where 0, is

plotted us x for two (alb). 0, is periodic in q, b u t grows q = x

-

c t ; g(q) = o.(l - P E , ( ~ ) ) " ~ ; e(0, t) is the exponentially as exp[Kx Jb2 - a 2 ] for b2 > a2. F o r

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972410

C4-46 N. S. SHIREN Then :

FIG. 1. -Normalized instantaneous frequency and/or strain amplitude, $, vsKx/n for a = 0 (matched velocities).

FIG. 2.

-

Normalized instantaneous frequency and/or strain amplitude, Q,, vsKx/n for

I

^alb

I <

1.

larger

I

a/b

I

the bunching begins to fall off as the waves get out of phase, but increases again as they come back into phase, thus causing a slow periodicity to be super- posed on the bunches, figure 3.

FIG. 3.

-

Normalized instantaneous frequency and/or strain amplitude, Qq, vsKx/n for

I

a/b

I >

^1.

For the thermal input, we take e(0, t ) = A, ei""'

n

The A, are complex amplitudes, which in equilibrium are assumed random. Their ensemble averages

<

A,

> ,

as well as cross correlations

<

A, A,

>

^{n # m,}

are zero. However, the autocorrelations are determined by the equilibrium energy densities,

En

( x = 0) = Pv2

<

A, A,*

>.

Each A, corresponds to a thermal mode with propagation vector, k,, in the absence of the modulating wave. We consider an isotropic material and longitudinal thermal waves only, since the transverse waves cannot be velocity matched t o any acoustic wave (the collinear transverse interaction has /? r 0).

q, is the polar angle of k, with respect to the propaga- tion direction of the acoustic wave.

In order to handle the apparently three dimensional problem in our one dimensional formalism we proceed as follows. Neglect changes in polarization or propaga- tion direction of the thermal waves due to the inter- action. In the wave equation for each direction, x,, write the modulating function as cos (Kx, cos q, - o t

+

^+,),

with $, = Ky, sin q,. This is equivalent to considering solutions of eq. (1) with

in eq. (3) for each direction, q, ;

and

We assume ballistic heat propagation. Then it is not necessary to average over a scattering probability density along x. The energy density per unit solid angle at angle rp is given by the integral of eq. ( 5 ) over do,.

Since 19 and 7 are independent of on we have,

.n

.

( k ~ ) ~ G(x, cp) dQ = --

4 v3 ti3 I9:(x, q ) [g(e(xy g(fl)

'))I

^dl2

^.

⁽⁶⁾

Thus a steady state heat input is formed into a train of periodic heat pulses at the modulation period, 2 n/Q.

In fact, because of the exponential gain factor in 8,, it should be possible to generate pulses by bunching the ambient thermal background. An input heat pulse, e(0, t ) = s ( t )

C

A, ei""', will be compressed if it is

n

injected at the proper phase of the modulation.

For the modulation, cos K(x

-

ct), and for the phase matched case, a ⁼ 0, the bunching maxima occur at K(x - ct) ⁼ (2 n -

4)

n ; the minima occur at K(x - ct) = (2 n

+ ^f)

n. The width of these bunches at the

5

amplitude points is

Maximum bunching will occur at the angle for velocity matching, cos q = clv. At very low tempera- tures the dominant longitudinal phonons will have u ^% c,, the longitudinal acoustic velocity. Then, if the modulation is also longitudinal, the matching angle

A POSSIBLE METHOD FOR ATTAINING PERIODIC TRAINS C4-47 is q~ = 0. This corresponds to collinear acoustic atte-

nuation of longitudinal waves. Corresponding to Landau-Rumer attenuation of transverse waves is the case of a transverse modulating wave for which the matching angle is given by cos q~ = c,/c,, where c, is the transverse acoustic velocity.

Bunching will occur to a lesser degree over a wide range of angles, as indicated by the curves, where the values of a may now be interpreted as corresponding to different cp. The position or timing of the bunches relative to the modulating wave varies with the degree of mismatch. Thus there will be an apparent spreading if the detector encompasses too large a solid angle ; therefore, narrow angle detectors are desirable.

Our solution is not valid if the pump acoustic wave is allowed to generate harmonics through the nonlinear interaction. Since the coherence length for harmonic generation is of the same order as the exponential

bunching gain constant we need to suppress harmonic generation. This is not a problem for transverse modu- lating waves since collinear harmonic generation on a pure mode axis is then strictly forbidden by symmetry.

For longitudinal pumps, harmonic generation may be suppressed by introducing dispersion ; e. g., with a n impurity paramagnetic resonance [2]. It may also be suppressed by resonating the pump wave in the host crystal with free surfaces at both ends [3].

Finally, we consider a numerical example. A typical value of

/3

may be

--

10. At 100 MHz a pump strain of

-

lod4 is attainable. Thus, y

--

Taking c z 3 x 10' cm/s, + y K = bk x 1. In a length of 2 cm the peak amplitude will be a factor 7.5 larger than the equilibrium value and the width of the bunch will be At x 4 x 10-'" s.

I wish to acknowledge several important discussions of these topics with my colleague, Dr. J. A. Armstrong.

References

[ l ] SHIREN (N. S.) and ARMSTRONG (J. A.), Proceedings of the International Conference on Phonon Scattering in Solids (July 1972), to be published ;

ARMSTRONG (J. A.) and SHIREN (N. S.), J. Quan- tum EIectronics, 1972, to be published.

[2] SHIREN (N. S.), Proc. IEEE, 1965, 10, 1540.

[3] B R E A ~ E A L E (M. A.), 5 e Congres International d'Acous- tique, Likge, 1965, D 18.

DISCUSSION C. J. ADKINS. ^- The phase matching condition

worries me. If you use a single type of wave for the pump and the wave to be bunched the simplest perio- dic lattice gives a decreasing phase velocity with frequency so that matching will only occur with an imaginary angle. Presumably one has to use different wave types or a material with a more complicated dispersion relation ?

N. SHIREN. - This is correct. To achieve perfect phase matching in most solids will, in general, require a transverse polarized pump wave modulating the longitudinal thermal phonons at an angle, cos 8 = c/v, as in Landau-Rumer attenuation. However, exponen- tial bunching occurs so long as a < 6, though reduced from the case a = 0. This is just the same as in colinear attenuation of longitudinal waves : viz. as long as the phonon momentum deficit is not too large, energy is transfered monotonically, from the pump acoustic waves to the phonons.

R. J. VON GUTFELD. - What is the longest time constant tolerable for a detector to observe the bunching effects you describe ?

10 MHz pump. Then a s response time would be sufficient. For the example analyzed, 100 MHz and 1 cm, a lo-' s response would be required. On general I think one would need a response time

x

1/10 the period of the pump wave.

A. ZYLBERSZTUN. - HOW sensitive is the process to differences in the low frequency wave and high frequency wave velocities, and could this be used a s a tool for measuring small acoustic dispersion effects ?

N. SHIREN.

-

Perhaps if narrow-angle and fast detectors were used. The amount by which the peak of the bunch lags or leads the zero of the pump wave is a sensitive measure of v - c.

K. WIGMORE.

-

Can phonon bunching also explain the non-exponential decay of X-band ultrasonic pulses observed with a bolometer ?

N. SHIREN. - Possibly. The time average energy absorbed from the pump by conversion to thermaI phonons is proportional to e x p ( ~ J n ) , i. e. it varies exponentially with the pump strain. If the bolometer detects these converted phonons, its output N. SIIIREN.

-

This depends on the phonon mean would not be expected to be exponential because the free path, which limits the interaction length. Thus if pump strain appears in the exponent above. However ballistic propagation could be observed over a length further analysis is required to see just what the expected of 10 cm appreciable bunching could be had with a decay is like.