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An improved method to experimentally determine temperature and pressure behind laser-

induced shock waves at low Mach numbers

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IOP PUBLISHING JOURNAL OF PHYSICS D: APPLIED PHYSICS

J. Phys. D: Appl. Phys. 44 (2011) 145501 (6pp) doi:10.1088/0022-3727/44/14/145501

An improved method to experimentallydetermine temperature and pressurebehind laser-induced shock waves at lowMach numbers

Mohammad Hendijanifard and David A Willis

Department of Mechanical Engineering, Bobby B. Lyle School of Engineering, Southern Methodist University,

Dallas, TX, USA

E-mail: dwillis@lyle.smu.edu

Received 19 October 2010, in final form 6 January 2011

Published 22 March 2011

Online at stacks.iop.org/JPhysD/44/145501

Abstract

Laser–matter interactions are frequently studied by measuring the propagation of shock waves caused

by the rapid laser-induced material removal. An improved method for calculating the thermo-fluid

parameters behind shock waves is introduced in this work. Shock waves in ambient air, induced by

pulsed Nd : YAG laser ablation of aluminium films, are measured using a shadowgraph apparatus.

Normal shock solutions are applied to experimental data for shock wave positions and used to

calculate pressure, temperature, and velocity behind the shock wave. Non-dimensionalizing the

pressure and temperature with respect to the ambient values, the dimensionless pressure and

temperature are estimated to be as high as 90 and 16, respectively, at a time of 10 ns after the ablationpulse for a laser fluence of F = 14.5 J c m−2. The results of the normal shock solution and the

Taylor–Sedov similarity solution are compared to show that the Taylor–Sedov solution under-predicts

pressure when the Mach number of the shock wave is small. At a fluence of 3.1 J cm−2, the shock 

wave Mach number is less than 3, and the Taylor–Sedov solution under-predicts the non-dimensional

pressure by as much as 45%.

(Some figures in this article are in colour only in the electronic version)

1. Introduction

Many applications require knowledge of the conditions in

a laser ablation plume, including laser micromachining[1, 2], laser cleaning [3, 4], laser shock processing [5],

pulsed laser deposition [6], surface nanostructuring [7, 8] and

nanofabrication[9, 10]. Differentmethods have been in useformeasuring pressure during laser–matter interaction, including

hydrophones, piezoelectric transducers, fast photography,interferometry, probe beamdeflection and photoacoustic probe

beam deflection method using a broadband piezoelectric

transducer [11–20]. Methods for measuring the temperatureinclude optical thermometry, spectroscopy and spectrally

resolved pyrometry [21–24]. The earliest studies of laser-

induced shock waveswere published in the late 1960s [25–29].A common method of studying laser-induced shock waves

is to measure the shock wave velocity as a function of time, for example using shadowgraph imaging [29]. Using

measured shock wave propagation data, theories such as theTaylor–Sedov similarity solution [30–33] are used to fit thespherical shock wave propagation data and calculate pressure

and temperature behind the shock wave [34–37]. This work uses time-resolved shadowgraph imaging for capturing thelaser-induced shock waves during laser ablation of aluminiumfilms on glass substrates. The normal shock wave solutionsare applied to the transient shock wave positions to determinepressure, temperature, and velocity behind the shock wave.Due to the low shock Mach numbers, it is shown that thecurrent solution is more accurate than theresults obtained usingthe Taylor–Sedov blast wave solution since the assumption of negligible background pressure was removed.

2. Experimental method and calculations

In this work the transient positions of laser-induced shock waves are measured using a shadowgraph technique. The

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J. Phys. D: Appl. Phys. 44 (2011) 145501 M Hendijanifard and D A Willis

Figure 1. Schematic diagram of the time-resolved shadowgraphapparatus.

shock wave positions are used with normal shock solutions to

extract pressure, temperature and velocity. Figure 1 displays

the schematic diagram of the shadowgraph imaging apparatus.

A Gaussian Nd : YAG laser (M 2 = 6) at 1064 nm and 7 ns

at full width half maximum (FWHM) is focused to a spot size

diameter of 63µm on a 200 nm aluminium film. The Nd : YAGlaser is synchronized with an illumination laser through a pulse

delay generator and a computer program. The illumination

laser is a nitrogen laser-pumped dye laser with a 3 ns (FWHM)

pulseduration, tuned to a wavelengthof 500 nm. TheNd : YAG

laser has a Gaussian temporal distribution, while the temporal

distribution of the dye laser is closer to a top-hat distribution.

The time delay between the two lasers is defined as the time

between the beginning of the ablation pulse and the beginning

of the illumination pulse, thus all time delays are measured

with respect to the beginning of the laser heating process by

the Nd : YAG laser. Further details are given in a previous

publication by one of the authors [38]. The setup allowsresolution of the order of magnitude of several nanoseconds

for capturing snapshots at a known time delay, a sequence of 

which comprisesthe laser-induced shock position as a function

of time for a given laser fluence. Due to the finite duration of 

the laser pulses, the rise time of the photodiode, oscilloscope

and uncertainty from measurements of the distances between

photodiodes and the sample, there is a maximum of 10%

error induced into the time delay measurements. The laser

fluences are determined from the measured pulse energies and

the measured spot size diameters, hence a maximum of 12%

error is expected for the fluence values.

Figure 2 displays a sample sequence of snapshots for alaser fluence of F  = 14.5 J c m−2. Analysing the snapshots,

the shock wave position is obtained as a function of the delay

time. The measured shock radii in the normal and lateral

directions show that the shock waves are semi-hemispherical.

The shock wave radii in the normal direction are shown in

figure 3. The shock wave radii were fit to power law functions

and differentiated to obtain shock velocity as a function of 

time, as displayed in figure 4. The reason for fitting power

law functions to the measured shock radii-time graphs is that

the Taylor–Sedov similarity solution predicts a power law for

the shock wave expansions. By knowing the shock speed as a

function of time, the normal shock solutions can be employed

to calculate the thermo-fluid parameters immediately behindthe shock waves. From now on, we simply refer to the location

Figure 2. Time-resolved images of shock wave propagation data atF  = 14.5 J c m−2 for Al with 200 nm thickness. The pictures are

taken at the delay times of (a) 17.6, (b) 43.2, (c) 130.4 and (d )331.2 ns after the ablation laser pulse, respectively.

Figure 3. Shock wave radius as a function of delay time for 200 nmaluminium films.

Figure 4. Shock wave speed, Rsh, as a function of delay time for200 nm aluminium films.

immediately behind and ahead of the shock wave as behind and

ahead of the shock wave, respectively.

Since the shock wave behaviour is unsteady, the negative

of the temporal speed of the shock is superimposed to all flow

speeds(i.e. shock speed andflow speed behind andaheadof theshock), making the shock wave equations similar to a standing

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J. Phys. D: Appl. Phys. 44 (2011) 145501 M Hendijanifard and D A Willis

Figure 5. Schematic diagram of the fixed and relative axes.

normal shock. In other words, in the shock wave relative axisof figure 5 (the prime coordinates), the solution to the standing

normal shock equations of momentum, continuity, energy andstate would simplify to [39]

M 

22 = M 

2

1 + 2/(γ −

1)[2γ/(γ − 1)]M 

21 − 1

, (1)

p2 = p11 + γM 

21

1 + γM 22

, (2)

T 2 = T 11 + (γ − 1)M 

21 /2

1 + (γ − 1)M 22 /2

. (3)

Subscripts 1 and 2 in these equations and the rest of this paperrefer to ahead of and behind the shock wave, respectively.

Prime notation shows that the solutions are in the relativecoordinate systems. In equations (1)–(3), γ  refers to the

specific heat capacity ratio, M  to the Mach number, p to

the pressure and T  to the temperature. Knowing the ambientconditions,p1 andT 1 ahead of theshockwave, andsubstitutingthe measured value of the shock Mach number, M 1, into

equations (1)–(3), the thermo-fluid parameters behind theshock can be attained. Note that since the background air

speed is zero, in the prime coordinates, the speed of air aheadof the shock wave is equal and in the opposite direction of the

shock speed, hence M 1 is equal to shock Mach number. Thevelocity behind the shock can also be obtained by switching

back to the stationary axis as

V 2 = −M 2a

2 + Rsh, (4)

in which a

2, the speed of sound behind the shock wave, iscalculated from

a2 =

γRT 2, (5)

where Rsh is the spherical shock speed, R is the gas constant,

V  is the flow velocity and a is the speed of sound.

Note that in equation (4), for finding the absolute velocitybehind the shock wave, we first find the relative velocity,

the first term, and then subtract it from the negative of thesuperimposed velocity, the second term.

3. Results and discussion

Non-dimensionalizing the pressure with the ambient pressure,pamb, figure 6 displays the dimensionless pressure behind the

Figure 6. Non-dimensional pressure, p2/p1, behind shock wave as afunction of time for 200 nm aluminium films, p1 = pamb = 100kPa.

Figure 7. Non-dimensional temperature, T 2/T 1, behind shock wave

as a function of delay time for 200 nm aluminium films,T 1 = T amb = 300 K.

shock wavedetermined by equation (2). As seenin figure 6, the

pressurebehind theshockwave reaches 9.2 MPa at a time delay

of 10 ns for a fluence of 14.5 J cm−2. Further from the sample,

the shock wave reaches the acoustic limits (acoustic limit is

where the shock pressure reaches the background pressure;

here 100 kPa). Non-dimensionalizing the temperature with the

ambient temperature, T amb, figure 7 displays the dimensionless

temperature behind the shock wave as a function of laser

fluence. As seen in figure 7, the temperature behind the

shock reaches more than 5000 K in less than 10 ns at F =

14.5 J c m−2. Although thetemperatureabove thesurface of the

sample and earlier delay times might be higher than 5000 K,

the earliest measured shock data point is captured at 10 ns due

to the experimental limitations. Non-dimensionalizing the

velocity with the measured shock speeds, the dimensionless

velocity behind the shock wave is displayed in figure 8. As

seen in figure 8, the dimensionless vapour velocities behind the

shock are as high as 0.8, or 2600 m s−1, for F  = 14.5 J c m−2.

The shock speed is 1.2 times larger than the velocity behind

the shock wave for these cases.

We now compare our results with the Taylor–Sedov

solutions for spherical shock waves. The Taylor–Sedov

solution for spherical shock is chosen since the experiments

showed a lateral to normal shock radii ratio of around 0.9. For

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J. Phys. D: Appl. Phys. 44 (2011) 145501 M Hendijanifard and D A Willis

Figure 8. Non-dimensional velocity behind shock wave, V 2/V shock ,as a function of delay time for 200 nm aluminium films.

Figure 9. Comparing results for non-dimensional pressure, p2/p1,

as a function of delay time for F  = 3.1 J c m−2.

acquiring the Taylor–Sedov solutions, we have fitted power

law functions of the form Rsh = C1t 0.4, where C1 is a

constant, to the radii-time values in figure 3 such that thestandard deviation is minimized. Differentiating the radii-time

function, the shock speed is found as a function of time. The

pressure behind the shock wave is found through employing

[40] p2 = (2/(γ  + 1))R2shρ1, where Rsh is the shock radius, t 

is the time and ρ1 is the density ahead of the shock wave.

The dimensionless pressure values calculated using the

present method, equation (2), and the Taylor–Sedov solution

are plotted in figures 9 and 10 for laser fluences of F  = 3.1and 11.7 J cm−2. The plots are divided on the graph into three

separate regions; the close-range, C, the mid-range, M, and

the far-range, F. Taylor–Sedov’s similarity solution is not validwhen the shock wave is still close to the blast origin, since

the mass encompassed by the shock wave is the same order

of magnitude as of the exploded mass, the C range [41]. As

the shock wave expands past the close-range the Taylor–Sedov

solution is commonly accepted as being valid. However, as it

expands far from the target its strength decreases and it finally

reaches the limit for which its Mach number becomes close

to 1. This is called the acoustic limit and is defined as the F

range, for which the solution is no longer valid.

As shown in figures 9 and 10, the Taylor–Sedov solutionunder-predicts the pressure evolution in the M range where

Figure 10. Comparing results for non-dimensional pressure, p2/p1,as a function of delay time for F  = 11.7 J c m−2.

the solution is considered valid. For the higher laser fluence,figure 10 compared with figure 9, the error in Taylor–Sedov’s

spherical blast solution tends to damp out in the M range. Wehave plotted the shock Mach number on the right-hand side of figures 9 and 10. Comparing the Mach numbers in figures 9and 10, it is observed that the Taylor–Sedov relations have lessunder-prediction for higher Mach numbers. The percentage bywhich the Taylor–Sedov solution under-predicts the pressureis listed in figures 9 and 10 at several time delays.

Now the question arises why the current solutions areconsidered more accurate than the Taylor–Sedov solutions.The answer lies in some major but subtle assumptions of the Taylor–Sedov method. As mentioned in the works of Sedov [33], the similarity solutions are targeted towards shock 

waves of a blast origin, meaning a point explosion expandingintoa background withnegligible pressure. Sinceour approachmakes no assumptions regarding the level of the backgroundpressure, or if the shock originates from a point source, ourmodel is considered more general. It is also noted that as thefluence increases, our model approaches the results based onthe Taylor–Sedov theory. This is a result of the increased ratioof the shock pressure to the background pressure, for whichthe Taylor–Sedov theory is valid.

Fornon-zerobackground pressures, as a primary scale, thestrength of the shock wave can be envisioned as the speed of the shock wave. In other words, we would expect strongershocks to have higher Mach numbers. As a result, it is

reasonable to expect a lower limit for the shock Mach numberfor which the Taylor–Sedov solution is not valid anymore.Our solution is valid even for shocks expanding in non-zerobackground pressures, as well as for shocks which are notperfectly spherical. Hence, it is considered as a more generalsolution for estimating the thermo-fluid parameters.

4. Conclusion

The results of this work have demonstrated that all of the thermo-fluid parameters behind a laser-induced shock wave can be determined using the combined experimentaland analytical solutions described in the text. The time-dependent pressure, temperature and velocity behind the laser-induced shock waves are calculated for different laser fluences.

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J. Phys. D: Appl. Phys. 44 (2011) 145501 M Hendijanifard and D A Willis

The non-dimensional pressure and temperature behind shock 

waves (F  = 14.5 J c m−2) reach as high as 90 and 16,

respectively, in less than 10 ns after the laser pulse. For the

same fluence, the non-dimensional velocity behind the shock 

wave reaches as high as 0.8 of the shock speed in less than

10 ns after the laser pulse. The results show that for low

Mach numbers, the use of the Taylor–Sedov theory resultsin considerable error when compared with the generalized

method used in this work.

Acknowledgments

The authors would like thank to Ms Sandra Setnick Zucker, the

research librarian at the central university libraries of SMU, for

her help with the science index books. This work was partially

funded through NSF Grant # 0500401.

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