Analysis of laser induced acoustic pulse probing of charge distributions in dielectrics

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Analysis of laser induced acoustic pulse probing of

charge distributions in dielectrics

C. Alquié, J. Lewiner, G. Dreyfus

To cite this version:

C. Alquié, J. Lewiner, G. Dreyfus.

Analysis of laser induced acoustic pulse probing of charge

distributions in dielectrics. Journal de Physique Lettres, Edp sciences, 1983, 44 (4), pp.171-178.

�10.1051/jphyslet:01983004404017100�. �jpa-00232176�

Analysis

of laser

induced acoustic

_pulse

probing

of

_charge

distributions

in

dielectrics

C.

_Alquié,

J. Lewiner and G.

_Dreyfus

Laboratoire d’Electricité _Générale,

Ecole _Supérieure de

_Physique

et de Chimie Industrielles de la Ville de Paris, 10, rue _Vauquelin, 75005 Paris, France

(Re~u le 17 novembre 1982, accepte le 23 decembre 1982)

Résumé. - Trois

aspects de la méthode de détermination des distributions de

_champs

ou de _charges

électriques

dans les

_{diélectriques,}

utilisant une onde de _pression créée par un laser

_pulsé

sont

_analysés.

1) Une méthode

_{d’analyse numérique}

_peut être utilisée

_qui

conduit à une résolution

spatiale

ne

dépendant

que du temps de montée de l’onde de _pression et non de sa durée. Des mesures sont

pré-sentées,

qui

illustrent cette

_analyse.

2) Les mécanismes de création de l’onde de _pression sont étudiés ; un

_dispositif

est décrit

_qui

_permet de

_produire

des centaines _{d’impulsions} de _pression sans variation notable de la forme de ces

_impulsions,

avec un bon rendement de conversion

_{d’énergie.}

₃₎ L’influence des

_{propriétés élastiques}

du matériau sur l’atténuation et la

_dispersion

de l’onde est considérée. Des mesures effectuées sur des échantillons de téflon montrent _que ces deux effets ne _{peuvent pas être}

négligés

si le _temps de montée de la

_pression

est

_{beaucoup plus}

court que le temps de transit de l’onde dans l’échantillon.

Abstract 2014 Some features of the laser induced acoustic

pulse method for the determination of electric field or

_charge

distributions in dielectrics are

analysed.

Three _aspects of this method are described.

1) The electric field can be derived

by

a numerical method whose resolution _depends

essentially

on

the rise time of _{the pressure} wave. Measurements are described which _support this _{analysis. 2)} The

mechanisms of _{generation of the pressure} wave are _studied; a method is described which allows the

production

of hundreds of

_pulses

without noticeable alteration, with a _{good energy conversion}

effi-ciency.

3) The influence of the elastic

_properties

of the material on the attenuation and

_dispersion

of the wave is considered. Measurements

performed

on

_{Polyfluoroethylenepropylene}

_(FEP)

_samples

show that one has to take these parameters into account _{if the rise time of the pressure is much shorter}

than the transit time. Classification Physics Abstracts 06. 30L - 72.20 - 77. 50 - 42.60 - 77. 30 1. Introduction. -

During

the last few _years, new

_approaches

to the

_study

of

_charge

_storage and

_transport

_phenomena

in dielectrics have been

_proposed

with a

_particular

interest in the determination of the

_spatial

distribution of electric

_charges

or fields inside the material.

Three methods

[1-3]

are

being currently

developed. They

all use the diffusion or

_propagation

of an

inhomogeneous

perturbation

of the

charges.

First Collins

_{[1] proposed}

to use the

_{inhomogeneous}

strain field associated with the diffusion of heat

_pulses

in the

_sample;

this

method,

known as the « thermal

_pulse

method » can

give

the

L-172 JOURNAL DE _{PHYSIQUE -} LETTRES

localization of the

_charge

centroid and a few coefficients of the distribution

[4].

The « electron-beam » method

_{proposed by}

Sessler et al.

_[3]

_permits

a

_high

resolution determination of the distribution but is destructive and is not

_applicable

to all materials since it

_{implies assumptions}

as to the conduction mechanisms involved.

In

1977,

we

_{proposed [2]}

to use the

propagation

of a _step function _pressure wave to

_produce

an

inhomogeneous

deformation of a

_sample

and showed that the induced

_voltage

or

_charge

variation are

_directly

related to the

_spatial

distribution of the

_potential

inside the

_sample.

Several

_attempts

were made to _generate such a _pressure

_{discontinuity using}

shock tubes

_[5],

_spark-gap

_[6]

and quartz

crystal [7] techniques.

The first two methods were limited

by

the

_difficulty

of

creating

reproducible

short-rise time

_perturbation

and the

_spatial

resolution obtained was

_typically

100 _gm. The electrical excitation of a

_quartz

_crystal

_{proposed by Eisenmenger [7]}

is more

success-ful,

it leads to a

_good

resolution

(a

few

microns).

Two

_important

_steps were made _very

_{recently, stimulating}

interest in the acoustic

_pulse

method.

First

it was shown

_{[6, 8]}

that an ideal

_{step-function}

_pressure

_profile

is not

_compulsory

since if the

shape

of the _pressure wave is

_known,

the electric field distribution can be derived from the observed

signal

through

a mathematical derivation which leads to a

unique

solution.

Second,

the

impact

of a short-duration laser

pulse

incident on an

_absorbing

_target

_produces

short-rise time _pressure

pulses

which can be transmitted in various _ways to the

_sample

under

_{investigation}

_[8-11].

In the case of references

_[9]

and

_[10],

since the duration of the laser

_pulse

is

_respectively

25 ns

and 15 ns, the

spatial

resolution is not better than 50 ~m and the

analysis

is restricted to a few mm

thick

_samples.

_By

_using

much shorter laser

_pulses

_(1.5

ns in Ref.

_[8],

75 ps to 1 ns in Ref.

_[11]),

a

_higher

resolution is obtained and the

_technique

is

_applicable

to thin film

_samples.

In this _paper, we discuss several

_aspects

of this laser induced pressure

pulse

method for the determination of field and

_charge

distribution in dielectrics.

First we

_analyse

the various methods which have been

_proposed

to derive electric field or

charge

distributions from the observed

signals, depending

on the

_spatial

extent of the _pressure wave and the thiekness of the

_sample

and show that if the

shape

of the pressure

pulse

is

_accurately

_measured,

the

_spatial

resolution of the method is related to the rise time of this

_pulse

but does not

_depend

critically

on its duration. Then we describe various

experimental

configurations

which can be

used to

_produce

short-rise time _pressure waves

_using

a

_pulsed

laser and characterize the

corres-ponding

pressure wave.

Finally,

we show that in _many

situations,

the attenuation and

dispersion

of the pressure wave must be taken into account. This is illustrated

_by

data obtained with FEP

samples.

2.

_Analysis

of the data. -

It has been shown

[2, 8]

that the electric field distribution inside a

dielectric

_sample

can be derived from the time

_dependence

of the

_{open-circuit voltage V(t)}

which appears

during

the

propagation

of a _pressure wave inside this

_sample.

If the pressure

_profile

is

described

_{by p(z, t)}

and if the

_dependence

of the dielectric constant with _pressure is

_neglected,

V(t)

is

_{given by :}

where _X is the

_{compressibility}

of the

_{sample, do}

its

_thickness,

_zf

_{the position}

of the wave front and

E(z,

0)

the electric field distribution before

penetration

of the wave.

A _very similar

_expression

was obtained

_{by Migliori [6] taking}

into account a _pressure

_dependent

dielectric constant. This

_expression

was written down

_explicitly

in reference

_[10]

in the case of a dielectric

_obeying

a Clausius Mossotti relation.

The electric field distribution can be

_numerically

derived from the measurements of

_V(t)

and of the _pressure

_profile

_p(z,

_t).

If the attenuation in the

_sample

can be

_neglected,

it is

only

_necessary

This was done in references

_[8]

and

_[10].

The electric field was

derived,

in reference

_{[8], by}

a

numerical

analysis

of the pressure, and in reference

[10] by

an

analytical

_{representation}

of this

pressure. In this later case _{the pressure} was

_supposed

to have a

_negligible

rise time followed

_by

an

exponential decay

e-~~t°,

where to

= 0.388

~s.

We will now show

_how,

_{using equation (1)}

and a limited number of

points

to describe the pressure rise either

numerically

or

_{analytically,}

it is

_possible

to

_improve

the

_spatial

resolution. As was done in reference

[8],

we transform the measured pressure

_p(do,

_t)

and the measured

signal V(t)

into a set of 2 N dimensionless values

_{p’ ,}

_Y,,

with 1 n N. These values are filtered

by

a

piecewise smoothing algorithm

which we have

_developed

for this _purpose.

_Finally,

the

inversion process is carried out with an iterative

_improvement,

which

_yields

the

_required

values of the electric field. Since the initial values of the pressure and of the measured

voltage

are

critical,

the influence of the

_step

_T/N

on the resolution of the process was

_{investigated systematically.}

The best results were obtained with a time increment of the order of 0.2 _fr. It can be noticed that

by improving

either the deconvolution method or the

_precision

of the measurement

_{of p}

and V

at short times the resolution can

_clearly

be

_improved.

For

instance,

in the case of a _pressure

pro-file such as described

by

curve 4 of

figure

1,

the

_optimum

time increment was found to be of the order of 1.3 ns, which

corresponds

to a

_spatial

resolution of 2 jmi in 25 _~,m thick FEP

_samples.

From this

_analysis,

it is clear that if

_{equation (1)}

is used in order to derive the electric field distri-bution in the

_sample,

the

_spatial

resolution is limited

_by

_{the rise time of the pressure} wave and the

response time and

dynamic

range of the electronic circuits. It does not

_{depend critically}

on the

duration of the pressure wave. The

charge

distribution can be obtained

by differentiating

the

electric field distribution with

respect

to _{the space coordinates.}

3.

_Analysis

of the

_generation

_{of laser induced pressure}

_pulses.

- The

generation

of

_high

ampli-tude pressure waves

_by

the

_impact

of a laser

_pulse

on an

absorbing

_target

originates

from a

_rapid

heating

of the surface of this

target.

The mechanisms involved in this

_{generation depend}

on the energy of the

pulse,

on the

_{absorbing properties}

of the _target and on the mechanical conditions

imposed

to the

_absorbing

_surface,

and will be discussed in the

_following.

The

_absorbing

surface must be made of a material

_having

a

_{high absorption}

coefficient at the

wavelength

emitted

by

the

laser,

a low thermal

conductivity

and a low heat of

_{vaporization.}

It can

be for instance a metal as used in references

[8-9],

or

_carbon,

as in references

[10-11].

Aluminum

is one of the most

_commonly

used metals. Zinc is also well suited for

_operation

with a Nd-YAIG laser

_emitting

1.06 _~m radiation.

Fig. 1. - Normalized pressure

profile p(t)

obtained with various _{targets :} 1)

evaporated

aluminum; 2) and ₃₎

_evaporated

aluminum covered with an aerolized _{graphite layer; 4)} 15 um thick aluminum foil.

L-174 JOURNAL DE _{PHYSIQUE -} LETTRES

The choice of a laser suitable for

studying

charge

distributions in dielectrics is

_governed

_by

two

conditions.

First,

the

_homogeneity

of the laser beam must

_be

as

_good

as

_possible

to

_give

a

good

uniformity

of the pressure

pulse

over the whole irradiated area. For this reason, solid-state lasers

are

_preferable

to _gas lasers.

Second,

the duration of the laser

_pulse

must be less than a few

nano-seconds,

in order to

_permit

a resolution of a few microns since if the sound

velocity

in the

_sample

is 2 000

_m/s,

2 ns

_corresponds

to a transit time of 4 _~m. The

development

of

_easily

available,

short-duration

_(typically

30 _ps to a few

_ns)

Nd-YAIG

_pulsed

lasers makes them convenient for

the

_experiments

described in this paper. The best

spatial homogeneity

can be obtained if the last stage of

amplification

is made of a

_Nd-glass

rod.

We have used a Nd-YAIG laser from

QUANTEL,

emitting

7 mJ to 350 mJ in 3 ns, at 1.06 ~m,

with a beam diameter of 8 mm. The same kind of

_laser,

_emitting

shorter

_pulses

₍₇₅

_ps to 1

_ns)

is used

by

Sessler et al.

_[11].

The time

_dependence

of the pressure

pulse,

which can be characterized

by

its rise time _tr and its full width at half

_{magnitude (FWHM),}

may differ

significantly

from that of the

laser,

and is

influenced

by

the thickness of the

_target

and the

_coupling

between the

_target

and the

_sample.

Moreover the duration of the

_{pulse depends strongly}

on the _energy of the laser

_pulse

and on the environment of the

_absorbing

surface.

In the

_{simplest configuration,}

the laser beam

_impinges

on a bare surface in air. In this case, the

width of the

_pulse

is related to the

_properties

of the ionized

_plasma

created near the

_absorbing

surface

_{by evaporation}

of a small amount of material. A _power

_density

of the order of

108

_{W Icm2}

is

_generally

_necessary to

_produce

this

_evaporation.

This _type of mechanism has been used

by

various authors

[8, 9,11],

either with a _target bounded to the

_sample

or on

_targets

_directly

depo-sited on it. This last method has the

_advantage

of

avoiding

a

coupling layer

between the

_sample

and the

_target

and leads to _very short duration pressures

pulses [11];

however it _may induce

some new

_problems

as is shown in the

_following.

We have

measured,

using

a

piezoelectric

_quartz transducer as

_already

mentioned in

refe-rence

_[8],

the pressure

profile produced by

a 3 ns laser

_pulse

incident on an aluminized quartz

covered with an aerolized

graphite layer.

The normalized _pressure obtained for a _power

_density

of 0.14

_GW/cm2

is shown Qn

figure

1. For a bare aluminum film

(curve

1),

the width of the

pulse

is

the same as that of the

light

_{pulse. Decreasing}

the duration of the

_{light pulse}

decreases

corres-pondingly

the width of the pressure

pulse, thereby increasing

the resolution of the method.

Unfortunately

the aluminum film is

_{destroyed by}

a

single pulse,

so that it cannot be used as a

convenient

_target.

When the aluminum film is

_{protected by}

a

_{graphite layer}

_(curves

2 and

3),

the

_{irregularities}

of the

thickness

of the

film,

which are difficult to

_control,

_produce

a

_broadening

of the

_pulse

which

_depends

on the

_quality

of the

_coating

and varies when

_repetitive

shots are

applied

due to the

_evaporation

of the

_layer.

To be

_applicable

to the

_study

of the evolution

_{of charge}

distributions in

_dielectrics,

_{it is necessary}

to control the

_quality

and

_{repeatability}

of the

_{absorbing layer.}

For

_comparison,

we show on curve 4 of

figure

1 the pressure obtained at the rear surface of

a 15 11m thick aluminum foil irradiated in

air,

and bounded to the transducer with

Nonacq

grease. The thickness of this

target

is sufficient to allow a

_large

number of shots without noticeable

variations of its thickness. Nevertheless its surface must be

_slightly

abraded in order to increase the

_absorption

of the laser beam. This leads to a measured

_peak

_pressure

_amplitude

of 80 bars for a

power

density

of 0.14

GW/cm2,

instead of

typically

500 bars for an aerolized

_graphite

_target.

In order to

_improve

the conversion from

_{electromagnetic}

to _{mechanical energy, it is}

_possible

to cover the

_target

with a

_transparent

overlay,

thus

constraining

the

absorbing

surface

[12].

The local and

_rapid

thermal

_expansion

of this

_layer

creates a

_{large gradient}

of

_displacement

and

consequently

generates

a stress wave. In this case the energy of the laser

pulse

can be

_considerably

reduced,

and it is not _necessary to

_evaporate

a

_part

of the

_target

in order to obtain

_peak

_pressures of a few tens of bars. Such an effect is used

_by

_{Migliori [10],}

who describes an

_experimental

_set-up

For small dimensions

_samples,

we have obtained _very

_reproducible

results with the _set-up

described on

figure

2a. An

absorbing layer

of colloidal

graphite (Aquadag)

is covered with a 3 mm

thick fused

_quartz

_plate

which is

_transparent

at 1.06 _J.1m. The thickness of this

_layer,

which pro-vides a

good

_energy transmission to the

_sample,

is controlled

_using

a 10 _pm thick

_{spacing ring}

against

which the

_sample

or the transducer are

_pressed.

The

corresponding

_pressure

profile,

obtained with

_only

5 mJ

_pulses,

and a _power

density

of 3 x

106

W/cm2,

is shown on

_{figure 2b ;}

the rise time is 2.8 ns and the FWHM is 5 ns. In these conditions no

_evaporation

occurs so that

the

_shape

and the

_amplitude

of the _pressure

_pulse

are

_{perfectly reproducible,}

even after hundreds of

_shots,

_{making signal}

_averaging

_{techniques possible.}

Moreover these results remain constant

from one

_mounting

to another.

Fig.

2. - Pressure _wave obtained with _a confined

layer of

_colloidal

_graphite.

_a) Schematic

_drawing

of the

target assembly ; b) Time

_dependence

of the

_{corresponding}

_pressure.

4. Influence of the attenuation of the _pressure wave. -

During

the

_propagation

in the dielectric

material,

the _pressure wave

_profile

is

_altered,

_depending

on the elastic

_properties

of the material.

The time

_dependence

of the pressure wave before

_entering

the

_sample

can be

_expressed

in terms

of its Fourier

_components

_p(do,

w)

as :

If the attenuation and

_dispersion

of the electric waves in the

_sample

are

neglected,

all these

components

propagate

at the same

_{velocity v,}

so that in a

plane

located at z, the Fourier

compo-nent _{of the pressure}

distribution,

of

_pulsation

c~, becomes :

L-176 JOURNAL DE PHYSIQUE - LETTRES

In this case, the pressure wave

_propagates

in the

_sample

without _any deformation of its

_profile.

When attenuation and

_dispersion

occur, the

phase velocity

v becomes

_frequency

_dependent

and is related to the attenuation

_x(~)

_by

_{Kramers-Kronig}

_{relationship [ 13].}

A

_{good approximation}

of this

relation,

which is valid in _many cases, is

given

in reference

[13] by :

where vo is the sound

velocity

at a reference

_pulsation

_{c~o. In these}

_conditions,

the _pressure

_profile

p(z, t)

becomes :

Knowing

its elastic

_properties,

this

_{expression completely}

describes the evolution _{of the pressure} distribution inside a material. In order to be

_used,

it

_requires

that ultrasonic attenuation and sound

velocity

data are

_available,

in the

_frequency

_range for which the Fourier

_components

of the pres-sure

profile

are

_large.

If this is not the case, it is however

possible

to evaluate

_simply

the influence of attenuation and

_dispersion

on the pressure,

_{by measuring}

the pressure transmitted

through

samples

of

_increasing

thicknesses.

We have

_performed

such an

_analysis

on F.E.P.

_samples.

The pressure wave was

produced using

a

_CO2

_laser,

as

already

described in reference

[8],

and was transmitted to the

_samples’by

a

_bonding

layer (conductive

adhesive E solder

_3021).

The influence of this

_layer

has been taken into account

by comparing

the pressure obtained

directly

at the rear surface of the

_target

(curve

a of

Fig. 3)

to that transmitted

_through

12.5 _~m thick

_samples

_(curve

_b).

In this case we have assumed that the attenuation and

broadening

of the _pressure wave are

caused

_by

the

_bonding;

the measurements show that the

_amplitude

of the pressure wave decreases

typically

in a 4 : 1 ratio and the maximum of the pressure occurs at about 15 ns instead of 9 ns.

Curves c and d of

figure

3 are the pressure waves obtained after

_{propagation through}

125 ~m and

250 _~m thick

_samples.

The

_dispersion

and attenuation of the

_{high frequency}

_components of the

Fig.

3. - Time

dependence

of _{the pressure transmitted}

_through

F.E.P.

_samples

of different thicknesses.

a)

Directly

at the rear surface of the _target. In order to be

_comparable

to the other curves, the

amplitude

was

divided

_by

4 to take into account the influence of the

_bonding

_{layer ; b)} after 12.5 Jlm ; c) after 125 urn;

pressure wave

_propagating

in teflon

produce

an increase of the rise time and a

_broadening

of the

pressure

pulse

which cannot be

_neglected

for 125 _J.1m thick

_samples.

We have used these results to

_analyse

the distribution of the electric field which was created in a

250 gm thick F.E.P. film. The

sample

was metallized on one

_side,

_{charged by}

a

negative

corona

process, heated for a few hours to broaden and stabilize the distribution. It was then metallized on the other side and bonded on both sides to 1.5 mm thick aluminum

plates

so that the whole

assembly

was

_{completely symmetrical}

and the pressure wave could be created either near the

charged

surface or near the

_opposite

one. _{The open} circuit

_voltage

was measured for a

_propagation

of the pressure wave in both directions and the electric field distribution was calculated from

equation (1).

In a first derivation the attenuation was

neglected

and the _pressure

profile p(z,

t)

was derived from curve b

of figure

_3,

which was obtained in a similar

_experimental

_arrangement.

The

_{corresponding}

electric fields are shown on

_figure

4a. In two

_{regions extending}

over 70 _J.lm near the

_surfaces,

a

_discrepancy

_{appears between the distributions obtained for the} two directions of

_propagation

of the wave.

This can be

explained by

the fact that the attenuation of the _pressure wave at the end of its transit

_through

the

_sample

is not

_negligible.

In order to check this

_assumption,

we have constructed

the matrix

_~(z~,

_~)

_using

curves

b,

c and d of

figure

3. When the wave front _propagates from 0 to

125 _{~,m, the pressure}

_profile

is

_{linearly interpolated}

from curve b to curve c, while in the second

part

of the

_sample,

it varies

linearly

from curve c to curve d. In these

conditions,

the electric field distributions derived from

_{equation (1)}

are shown on

_figure

4b. A

good

_agreement

is

obtained,

for the two directions of

_propagation

of the wave, even near the faces of the

_sample.

This

_clearly

shows that in F.E.P.

_samples,

the attenuation of the

_{high frequency}

_components of the pressure wave must be taken into account for thicknesses

larger

than 125 _{~m. If very short} pressure

pulses

are

_used,

_{higher frequency}

_components

are involved and are even more

_quickly

attenuated,

so that attenuation and

_dispersion

will have to be considered for thinner

_samples.

In reference

[11],

Sessler et al.

_proposed

a method for

_{measuring directly}

the

_charge

_{distribution,}

using

a _very short duration pressure

_{pulse. Assuming}

that the

_spatial

extent of the

_pulse

is _very short

_compared

to _{any Fourier}

_component

of the

_charge

_{distribution,}

_they

obtain this distribution from the short-circuit current

_flowing

in the external circuit

_during

the

_propagation

of the _pressure

wave in the

sample.

This _very

simplified

model is

only

valid if the duration of the _pressure

pulse T

Fig.

4. - Electric field distributions obtained from the

measurement of the _{open-circuit voltage,} for two

opposite directions of

_propagation

_{of the pressure} wave. _a)

_Neglecting

the attenuation and

_dispersion

in the

L-178 JOURNAL DE _{PHYSIQUE -} LETTRES

before

_entering

the

_sample

is such that n

_corresponds

to a sufficient resolution.

Moreover,

from what we have

just

shown,

it is restricted to rather thin

_samples,

in which

_pulse

attenuation

_and.

broadening during propagation

in the dielectric can be

_neglected.

By

measuring

the

_{pulse broadening}

of _pressure waves such as shown on curve 2 of

_figure

1,

after a transit

_through

25 _Ilm and 125 _~m F.E.P.

_samples,

we have evaluated the maximum thickness of F.E.P. films associated with the above limitation to _{be in the range of 50 ~m.}

5. Conclusion. - In _{this paper,} we have

_analysed

some

_important

features of the laser-induced pressure

pulse

method for the

_{investigation}

of electric field or

_charge

distributions in dielectrics. We have shown

that,

when the electric field is derived from the

_signal

measured

_during

_the

pro-pagation

of the _pressure wave _{and from the pressure}

_{profile, using}

a numerical

method,

it is

possible

to obtain

_by

an

_appropriate

choice of the

_sampling

_interval,

a resolution better than VTr9

where

is the rise time of the _pressure wave and v the sound

_velocity.

We have then

_analysed

the

_generation

of _pressure waves

_{using pulsed}

lasers and have shown that the energy of the laser can be

_considerably

reduced if the

_absorbing

surface is constrained. Because in this case the

absorbing

material is not

_{evaporated by}

the

_impact

of the

laser,

the

reproducibility

of the _pressure wave is much better than in the case of a free surface.

Finally

we have studied the influence of the elastic

_properties

of the material on the pressure

distribution in rather thick

_samples.

In order to obtain the field distribution with a

_good

_resolution,

it is necessary to take into account the attenuation and

_dispersion

of the

_{high frequency}

compo-nents _{of the pressure} wave.

We have shown that in F.E.P.

_samples,

for _pressure waves

_having

a rise time of 8 ns, these

effects must be taken into account for thicknesses of the order of or

_larger

than 125 _Ilm. More

generally,

it _appears that in F.E.P. the

_broadening

of the _pressure

_pulse

is

_important

if the transit time

_through

the

_sample

is one order of

magnitude larger

than the rise time of the

_pulse.

Should that be a

_limitation,

it would be

_possible

to _{reduce the attenuation of the pressure} wave

by operating

at low

_{temperatures.}

References

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[2]

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