May 2015

NASA/TM–2015-218762

Temperature and Pressure Dependence of Signal 

Amplitudes for Electrostriction Laser-Induced Thermal 

Acoustics 

Gregory C. Herring 

Langley Research Center, Hampton, Virginia 
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May 2015

NASA/TM–2015-218762

Temperature and Pressure Dependence of Signal 

Amplitudes for Electrostriction Laser-Induced Thermal 

Acoustics 

Gregory C. Herring 

Langley Research Center, Hampton, Virginia 
Available from: 

NASA STI Program / Mail Stop 148 

NASA Langley Research Center 

Hampton, VA 23681-2199 

Fax: 757-864-6500 

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Space Administration. 
Contents 

Abstract .............................................................................................. 1

Introduction .......................................................................................... 1

Method and Apparatus.............................................................................. 2

Results ................................................................................................ 3

Discussion and Summary........................................................................... 4

Acknowledgment.................................................................................... 5

References ........................................................................................... 5

Tables................................................................................................. 8

Figures ............................................................................................... 9

List of Figures 

Figure 1 Schematic of the laser beam crossing geometry.

Figure 2 Relative decrease in heterodyned LITA signal Sh as a function of 

 Increasing Mach number. 

Figure 3 Fractional change in theoretical df/f versus the fractional change in 

observed heterodyned signal.

iii
Abstract 

The relative signal strength of electrostriction-only (no thermal grating) laserinduced thermal acoustics (LITA) in gas-phase air is reported as a function of 

temperature T and pressure P. Measurements were made in the free stream of a variable 

Mach number supersonic wind tunnel, where T and P are varied simultaneously as Mach 

number is varied. Using optical heterodyning, the measured signal amplitude (related to 

the optical reflectivity of the acoustic grating) was averaged for each of 11 flow 

conditions and compared to the expected theoretical dependence of a pureelectrostriction LITA process, where the signal is proportional to the square root of 

[P*P /( T*T*T)].

Introduction 

Laser-induced thermal acoustics (LITA), elsewhere called laser-induced thermal 

grating spectroscopy (LITGS) or transient grating spectroscopy (TGS), is actively being 

developed [1-8] for flow measurement [9-13] including combustion applications [5, 14, 

15]. Because the signals are strong, a majority of LITA efforts [5-10, 14, 15] utilize 

thermal gratings that are dominated by temperature gradients. Other studies [1-4, 11-13]

use pure electrostriction gratings (i.e., pure acoustic waves without a thermal grating) in 

spite of the weaker signals. The electrostrictive approach makes the LITA technique 

more universal since it is a nonresonant process and does not require an absorption 

transition within the gas species of interest. The global collection of wind tunnels uses a 

variety of different working fluids, so the non-thermal-grating approach makes the use of 

a single instrument possible for many different flow facilities. As part of the LITA 

development effort at NASA Langley Research Center (LaRC), off-body flow velocity, 

temperature and pressure measurements were recently demonstrated [13] in a variable 

Mach number supersonic wind tunnel. In addition to the primary goal of proof-ofconcept of LITA as a supersonic flow diagnostic, the work of [13] also makes possible 

the present study of the LITA signal amplitude dependence on gas density. 

 The signal level S of many light scattering techniques typically has either a linear 

or a quadratic density dependence (i.e., S  P/T or P2

/T2 respectively) for pressure P and 

temperature T. In contrast electrostriction-based LITA is expected [1, 16-18] to have a 

dependence of S  P

2

/T3. Furthermore an optically-heterodyned variation of the 

technique [4, 7], that mixes a local oscillator with the raw LITA signal, produces a final 

signal Sh that varies as Sh  (P2/T3)

1/2 = P/T3/2

. In LITA, T is readily derived from the 

frequency of the signal oscillations, while P is difficult to derive simultaneously from the 

signal decay rate at typical pressures of ~ 1 atm or smaller. Note that at larger pressures 

(i.e., P >> 1 atm), pressure can be derived from the decay rate as shown as shown in [15]. 

One potential application for amplitude measurements is to use them for an indirect 

pressure measurement for lower-pressure conditions. Confident knowledge of the P2/T3

1
relation is first necessary to infer pressure from measurements of T and S when P is not 

easily measured from the decay rate. 

Grating time decay [8, 15, 19] and pressure dependence of LITA signal 

amplitudes [1, 15 20] have been studied in some detail, but there are few studies of the 

pure temperature dependence. One study [16] that reports a temperature dependence of 

T

3

uses raw data that has roughly a factor-of-three noise superimposed on the temperature 

trend. This naturally leads to a desire for better data to confirm this temperature 

dependence with better accuracy. The lack of other studies investigating the temperature 

dependence motivated the present effort in this report to compare the previouslymeasured LITA signal variation with theory. The work of Reference 13 was originally 

intended to measure only Mach number and temperature of the wind tunnel flow, but 

after publication of Reference 16, it was natural to ask how the signal amplitudes of [13]

vary with P and T, since the noise of [13] was less than the noise of [16].

In this report, the theoretical dependence on pressure P and temperature T is 

compared to data previously [13] obtained in the NASA Langley Unitary Plan Wind 

Tunnel (UPWT). Ideally, P and T would be varied independently as in [1, 16, 20] to

check the P and T dependencies. However in the UPWT, P and T cannot be varied 

independently because they were forced to vary simultaneously as the wind tunnel Mach 

number M was changed by varying the nozzle expansion ratio. Thus instead of looking 

for a pure T3dependence at constant P in the current work, the dependence of the LITA 

signal is studied when changing P and T simultaneously. Fortunately the flow conditions 

P and T are both known with confidence [21].

Method and Apparatus 

Two common-path versions of optical-heterodyne LITA, from LaRC, were 

described for subsonic (VF < VS) [4] and supersonic (VF > VS) [13] fluid motion at 

velocity VF and sound speed VS = VS. Two crossed beams from a single 10-nsec 

pulsed laser (pump) electrostrictively create two counter propagating 1-sec duration 

acoustic plane wave packets in the medium volume defined by the beam crossing (30 Hz 

repetition rate). The volume of the ellipsoidal overlap region is about 1 cm by 200 m by 

200 m. For stationary or subsonic gas, illumination of these traveling wave packets 

(i.e., density gratings) with a second laser beam (probe) at frequency L that crosses the 

initial density-grating ellipsoid and generates a Bragg-diffracted signal laser beam that 

consists of two copropagating components that are spectrally distinct. These two 

components are distinguished by their different Doppler shifts   determined by the 

counter propagating geometry of the two acoustic wave packets. The beating of the two 

separate components, at frequencies L  , produces a modulation of this Braggdiffracted signal beam at frequency 2 . If the sound wave reciprocal wavelength is k 

(i.e., wave vector difference of the two pump beams), the two Doppler shifts are  = k

 (VS), where VS are the velocities of the two counter propagating wave packets and 

VS = the speed of sound (bold-faced quantities denote vectors). 

2
Measurement of the beat frequency 2 , with the known grating wavelength 

1/k, yields the sound speed. If the mass composition of the fluid is known, translational 

temperature is determined since T  VS

2

. Any non-zero bulk fluid motion at velocity VF

is also readily obtained from LITA with a heterodyne [10-12] approach. A local 

oscillator beam at the probe frequency L is introduced onto the detector. The local 

oscillator mixes nonlinearly with the original signal, and the detected signal then shows 

modulation at three frequencies: 2  and   (k  VF). VF is derived from the 

measured frequency difference, 2 , of the two new components, where  = k 

VF. The signal is digitized, and the frequencies can be extracted by a variety of methods 

to determine the velocity and temperature of the fluid. The fluid pressure can also be 

determined with LITA from decay rates of either thermal [15] or electrostriction [13] 

grating signals. 

For supersonic gas flow, the subsonic setup was modified by offsetting the probe 

beam Xo  0.25 mm downstream of the crossed pumps to optimize collection of the 

reflected light signals from the two acoustic Bragg gratings as they are swept 

downstream. (Note that this present estimate of Xo  0.25 mm is more accurate than the 

previous estimate of Xo  0.5 mm given in Reference 13, which was specified only to 

describe the experimental setup. The value 0.5 mm was not used in Reference 13 to 

derive the relative pressures reported in that work, but the value 0.25 mm is used in the 

present work to calculate the absorption of the acoustic wave packets.) A schematic of 

the supersonic version of LITA is shown in Figure 1. Measurements were made in the 

UPWT, a large-scale (1.3 by 1.3-meter test section) closed-circuit supersonic facility that 

has been widely used for 50 years [21]. Mach number was varied over 11 values from 

1.47 to 2.16 to give 11 different combinations of P and T. 

 Offsetting the probe laser beam downstream relative to the crossed pumps slightly 

changes the temporal profile of the LITA signal. A fast acoustic wave packet travels at

Mach number M+1 and a slower packet travels at M-1 in the detector reference frame. 

For each laser pulse the diffracted signal consists of two components (one from each 

grating) that are separated in time by about 500 ns. Each of the two components exhibits 

a different characteristic frequency due to distinct Doppler shifts. In the analysis 

presented here, the total LITA signal S is taken as the sum of these two sequential 

signals. Each single-pulse waveform was first Fourier transformed to frequency domain. 

Then about 500 spectra were averaged. In this analysis, the raw amplitudes are corrected 

for transit time and absorption effects. Finally the amplitudes of the two separate signals 

(from the fast and slow acoustic wave packets) were added together to generate the final 

initial signal amplitude Sh for each different P and T combination. 

Results

 Writing the non-heterodyned LITA signal as S  g(P, T) where g is a function of 

P and T, four choices of g are considered: g  P/T, g  P

2

/T2, g  P4/T4, and g  P

2

/T3, 

which correspond to linear, quadratic, and quartic density dependences, and the expected 

LITA dependence respectively. For g = f2 and heterodyned LITA signals Sh, these

dependences become Sh  f  (P/T)1/2

, f  P/T, and f  (P/T)2and f  P/T

3/2

3
respectively. Figure 2 shows calculated results for the reduction of the heterodyned 

signal as a function of Mach number (i.e., P and T are both decreasing with increasing M)

for the four different choices of function f. The present measured fractional changes in 

signal amplitudes dSh/Sh are also shown by individual filled-circle symbols. These 15 

measurements covering 11 Mach numbers are from the same 15 runs presented in 

Reference 13, where M, P and T were reported (but the signal amplitudes were not 

reported). Two flow conditions were repeated and the precision of the amplitude Sh of 

each point is estimated to be about + 30% from these limited repetitions. 

The same data as shown in Figure 2 are plotted in a different manner in Figure 3,

where the fractional theoretical signal is plotted versus the fractional change in observed 

signal. One expects an equality df/f = dSh/Sh (i.e., linear relation with slope = 1), for the 

correct theoretical dependence on P and T. The expected equality is shown with the solid 

line in each plot. 

Results of linear fitting to the four comparisons of Figure 3 are shown in Table 1, 

where all uncertainties are 68% confidence (i.e. + 1). Note that the fitted lines through 

the data sets are not shown in Figure 3. The results of the linear fitting are shown only in 

Table 1. The solid lines of Figure 3 are not fits; they simply show the equality df/f = 

dSh/Sh . At face value, the slope for f  P/T (i.e. S  P

2

/T2 which is quadratic with 

density) is closest to unity and the only slope consistent with unity at 68% confidence. 

The other three options are also ruled out at 95% confidence (+ 2).

Discussion and Summary 

 First the spread in the data of the present work is about +30%, much less than the 

factor-of-three spread in the data of Reference 16. Unfortunately, the changes in P and T 

for the present work are less than the factor-of-three change in temperature used in 

Reference 16. Thus the sensitivity to temperature of the current study in is probably not 

better than the sensitivity of Reference 16. 

Since optical heterodyne detection was used in the work of [13], f  P/T

3/2 is the 

expectation for a physical process like LITA with a straightforward, non-heterodyned 

dependence S  P

2

/T3. However, in Figure 2, the function f  P/T best matches the data 

compared to the other options. Also, in Figure 3 and Table 1, the same function is also 

the best match to the equality between df/f and dSh/Sh, with a unity slope. Because of the 

significant spread in the data and the uncertainties in the fitted slopes, the function f 

P/T3/2

 is also plausibly consistent with the data, but on face value, comes in second place 

to the best match, f  P/T. The two functions f  (P/T)1/2 and f  P

2

/T2are not consistent 

with the data to 95% confidence. 

 The bottom line is that, in this work, f  P/T provides a better match than f 

P/T3/2 for the UPWT data set. Theoretically [1, 18], S  P

2

/T3 (i.e., Sh  f  P/T3/2) is 

expected. The first possible conclusion is that LITA signals vary as S  P

2

/T2, i.e. 

density squared, and not as P2/T3

as reported in the literature. This conclusion is difficult 

to accept based on the literature, including the 1/T3 measurement result of Reference 16. 

4
Note that the heat capacity ratio and the Gladstone-Dale coefficient are both required to 

be temperature independent to get the P2/T3result. But these assumptions should be 

relatively good approximations for the pressures and temperatures of this work. A 

second possible conclusion is that there is an error in the present analysis. Note that 

Reference 17 reports a decay of electrostriction gratings faster than 1/T3(i.e., 1/T4.25) and 

discusses possible effects from turbulent convection, beam steering, and averaging with 

non-linear effects. In the present work, the observed decay is slower than 1/T3

. Other 

possibilities for errors in the present work are related to the pump-probe laser beam 

separation Xo and imperfect beam alignments. Finally an incorrect result may be due to 

systematic errors in the absorption estimate (between the time of formation and the time 

of measurement of the density gratings) in the data analysis. 

On the other hand, the absorption calculation is the same one that allowed the 

correct determinations of the wind tunnel free-stream relative pressures to within +4% in 

Reference 13. This argues that the absorption calculation is likely to be reasonable and 

that there may be other unknown systematic errors in the present analysis. Because of the 

potential errors in the present experiment and the 1/T3

result of Reference 16, it is 

reasonable to continue to assume that S  P

2

/T3is correct. With respect to the pressure 

measurement method proposed above in the introduction, significantly better data than 

those used in [13] or [16] must be obtained before the amplitude dependence on 

temperature can be confirmed accurately enough to provide confidence in future 

measurements of S to be used as an indirect pressure probe in gas samples, with pressures 

near atmospheric or below. 

Acknowledgement 

I gratefully thank R. J. Balla, M. T. Fletcher, and R. C. Hart for work on the 

experimental portion of the present study (i.e. the work of Reference 13). 

References 

1. D. E. Govoni, J. A. Booze, A. Sinha, and F. F. Crim, “The nonresonant signal in laserinduced grating spectroscopy of gases,” Chem. Phys. Lett. 216, 525-529 (1993). 

2. W. Hubschmid, B. Hemmerling, and A. Stampanoni-Panariello, “Rayleigh and 

Brillouin modes in electrostrictive gratings,” J. Opt. Soc. Am. B 12, 1850–1854 (1995). 

3. A. Stampanoni-Panariello, B. Hemmerling, and W. Hubschmid, “Electrostrictive 

generation of non-resonant gratings in the gas phase by multimode lasers,” Phys. Rev. A 

51, 655–662 (1995). 

4. R. C. Hart, G. C. Herring, and R. J. Balla, “Common-path heterodyne laser-induced 

thermal acoustics for seedless laser velocimetry,” Opt. Lett. 27, 710-712 (2002). 

5. S. Williams, L. A. Rahn, P. H. Paul, J. W. Forsman, and R. N. Zare, “Laser-induced 

thermal grating effects in flames,” Opt. Lett. 19, 1681–1683 (1994). 

5
6. M. A. Buntine, D. W. Chandler, and C. C. Hayden, “Detection of vibrational overtone 

excitation in water via laser-induced grating spectroscopy,” J. Chem. Phys. 102, 2718–

2726 (1995). 

7. A. A. Maznev, K. A. Nelson, and J. A. Rogers, “Optical heterodyne detection of laserinduced gratings,” Opt. Lett. 23, 1319-1321 (1998). 

8. E. B. Cummings, H. G. Hornung, M. S. Brown, and P. A. DeBarber, “Measurement of 

gas-phase sound speed and thermal diffusivity over a broad pressure range using laserinduced thermal acoustics,” Opt. Lett. 20, 1577–1579 (1995). 

9. D. J. W. Walker, R. B. Williams, and P. Ewart, “Thermal grating velocimetry,” Opt. 

Lett. 23, 1316–1318 (1998). 

10. S. Schlamp, E. B. Cummings, and T. H. Sobota, “Laser-induced thermal-acoustic 

velocimetry with heterodyne detection,” Opt. Lett. 25, 224-226 (2000). 

11. B. Hemmerling, D. N. Kozlov, and A. Stampanoni-Panariello, “Temperature and 

flow velocity measurements by use of laser-induced electrostrictive gratings,” Opt. Lett. 

25, 1340-1342 (2000). 

12. R. C. Hart, R. J. Balla R J, and G. C. Herring, “Simultaneous velocimetry and 

thermometry of air by use on nonresonant heterodyned laser-induced thermal acoustics,” 

Appl. Opt. 40, 965-968 (2001). 

13. R. C. Hart, G. C. Herring, and R. J. Balla, “Pressure measurement in supersonic 

airflow by differential absorptive laser-induced thermal acoustics,” Opt. Lett. 32, 1689-

1691 (2007). 

14. M. S. Brown and W. L. Roberts, “Single-point thermometry in high-pressure, 

sooting, premixed combustion environments,” J. Propulsion Power 15, 119-127 (1999).

15. R. Stevens and P. Ewart, “Simultaneous single shot measurement of temperature and 

pressure along a one dimensional line by use of laser-induced thermal grating 

spectroscopy,” Opt. Lett. 31, 1055-1058 (2006). 

16. A. Meier, and S. Schlamp, “Improved signal intensity for transient grating 

spectroscopy using a PIV laser,” AIAA 36th Fluid Dynamics Conference, San Francisco, 

CA (June 5-8, 2006) paper 2006-2973. 

17. S. Schlamp, R. T. Rosgen, D. N. Kozlov, C. Rakut, P. Kasal, and J. von Wolferdorf, 

“Transient grating spectroscopy in a hot turbulent compressible free jet,” J. Propulsion 

Power 21, 1008-1018 (2005).

6
18. R. J. D. Miller, R. Casalegno, K. A. Nelson, and M. A. Fayer, “Laser-induced 

ultrasonics: a dynamic holographic approach to the measurement of weak absorptions, 

optoelastic constants and acoustic attenuation,” Chem. Phys. 72, 371-379 (1982). 

19. Y. Li, W. L. Roberts, and M. S. Brown, “Investigation of acoustic gaseous damping 

rates by transient grating spectroscopy,” AIAA Journal 40, 1071-1077 (2002). 

20. F. Bake and F. Lehmann, “Preliminary investigations on thermometry in thermal 

flows via transient grating spectroscopy,” 12th International Symposium on Applications 

of Laser Techniques to Fluid Mechanics, Lisbon, Portugal (July 12-15, 2004) paper 12.4 

21. C. M. Jackson Jr., W. A. Corlett, and W. J. Monta “Description and calibration of the 

Langley unitary plan wind tunnel,” NASA TP-1905 (November 1981). 

7
 Table 1 

Function g Function f Slope Intercept R2 

P

4

/T4 P2/T2 2.7 + 0.35 0.015 + 0.13 0.82 

P

2

/T2 P/T

 1.01 + 0.12 0.028 + 0.05 0.83 

P

2

/T3 P/T

3/2 0.65 + 0.08 0.02 + 0.03 0.83 

P/T (P/T)1/2 0.44 + 0.05 0.017 + 0.02 0.84 

8
Figure 1. Schematic of (A) pump beam crossing geometry at the grating formation time 

to and (B) a short time later t-, after the two gratings (thick vertical lines) have propagated 

downstream and the slower grating is reflecting a fraction (i.e. signal A-) of the probe 

beam (dashed lines) into the detector [probe beam and detector are omitted from part (A) 

for clarity]. 

9
Figure 2. Relative decrease in heterodyned LITA signal Sh as a function of increasing 

Mach number for four different functional dependences f of pressure P and temperature T 

and the measurements from the UPWT [13]. 

10
Figure 3. Fractional change in theoretical df/f versus the fractional change in observed 

heterodyned signal dSh /Sh for the four different P and T dependences denoted on each 

panel. 

11
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Temperature and Pressure Dependence of Signal Amplitudes for 

Electrostriction Laser-Induced Thermal Acoustics 

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Herring, Gregory C.

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14. ABSTRACT

The relative signal strength of electrostriction-only (no thermal grating) laser-induced thermal acoustics (LITA) in gas-phase 

air is reported as a function of temperature T and pressure P. Measurements were made in the free stream of a variable Mach 

number supersonic wind tunnel, where T and P are varied simultaneously as Mach number is varied. Using optical 

heterodyning, the measured signal amplitude (related to the optical reflectivity of the acoustic grating) was averaged for each

of 11 flow conditions and compared to the expected theoretical dependence of a pure-electrostriction LITA process of 

heterodyned signal is proportional to the square root of [P*P /( T*T*T)]. 

15. SUBJECT TERMS

Electrostriction; Grating reflectivity; Laser-induced thermal acoustics; Pressure; Temperature dependence of laser-induced; 

Transient grating supectroscopy

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