Acousto-optic method for ultrasonic pulse characterization

Acousto-optic method for ultrasonic pulse characterization W.G. Mayer and T.H. Neighbors* Physics Department, Georgetown University, Washington, DC 2 0 0 5 7 , USA

Received 15 September 1986 The light distribution in a diffraction pattern produced by a sequence of ultrasonic pulses depends on a number of parameters characterizing the pulse shape, modulation and sequencing as well as changes in the spectral composition due to non-linearities in the propagation medium. Based on calculations of such light diffraction patterns it is possible to consider the inverse problem, i.e. the determination of parameters mentioned above by evaluating certain features of the diffraction pattern produced by the pulses. A number of examples are given to illustrate these possibilities. K e y w o r d s : pulse c h a r a c t e r i z a t i o n ;

mathematical

Shortly after Debye and Sears 1 and Lucas and Biquard 2 published their experimental findings concerning the interaction of light and low M H z ultrasonic waves travelling in a transparent liquid, R a m a n and Nath 3 provided a theoretical explanation of this acousto-optic interaction phenomenon. This theory made possible the calculation of the shape of light diffraction patterns for continuous, low amplitude, and relatively low frequency M H z ultrasonic waves. Extensions and refinements of this basic theory 4 to include higher frequency continuous ultrasonic waves and finite amplitude distorted waves widened the scope of applicability. The R a m a n - N a t h theory was reconsidered by Hargrove s to include the interaction of Gaussian light beams with arbitrary ultrasonic waves which in turn was used by ZitteP to investigate light diffraction by symmetric short ultrasonic pulses. Subsequent experiments 7 showed qualitative agreement between experiment and the predictions by Zitter. Since short ultrasonic pulses produced by a standard transducer are rarely symmetric, with inertia and ringing introducing non-negligible rise and fall times, the idealcase considerations of Hargrove and Zitter were extended 8 so that more c o m m o n variations in a pulse sequence are included in the theory. These variations include pulse modulation frequency, repetition rate, rise and fall times, absorption and the possible influence of medium non-linearities on these quantities. With these theories the shape and form of a light diffraction pattern produced by any given sequence of ultrasonic pulses can be calculated. While it is relatively simple to calculate the light distribution in a diffraction pattern produced by a sequence of analytically definable pulses, it is presently not quite possible to deduce all of the pertinent pulse characteristics from an evaluation of a light diffraction pattern produced by a sequence of ultrasonic pulses. *Permanent address: Falcon Associates, 6862 Elm Street, McLean,

VA 22101, USA 0041-624X/87/020083-04 $03.00 © 1987 Butterworth f:t Co (Publishers) Ltd

models

This Paper describes which pulse parameters can be determined directly from a light diffraction pattern evaluation and which parameters can be deduced to first order. Basic theory

In this section we present the features of the general theory needed for the evaluation of a given diffraction pattern. Derivational details and comments about higher-order approximations required for numerically evaluating products of infinite series of Bessel functions are presented elsewhere 8. Possible pulse shapes are indicated in Figure 1. The modulation frequency of the pulse is F which, for the example used here, is 3 MHz. The pulse repetition rate is determined by the frequency,f, which for this example is 150 kHz. With these values, F ----nfwith n = 20. Since the pulse shapes shown in Figure I are periodic in time, they can be expanded in terms of their respective Fourier series as

K

1/~

)1

Figure 1 (a) Idealized pulse sequence; (b) typical pulse sequence in medium

Ultrasonics 1 9 8 7 Vol 25 March

83

Acousto-optic pulse characterization." W.G. Mayer and T.H. Neighbors 04

rL\. t) = ~ "

(I)

a , sin[n(gSt- awl + © , -

H =0

where a,, a n d ¢),, are the a m p l i t u d e a n d p h a s e of the nth c o m p o n e n t . T h e quantity ~ is equal to 2rrf, and K = ~k', with c being the ultrasonic velocity in the liquid medium. W h e n a b e a m of width 2L is incident n o r m a l l y on the ultrasonic wave which p r o p a g a t e s in the x direction and has a time histo U d e t e r m i n e d by E q u a t i o n (1). a light diffraction pattern is formed in the far-field region. The light a m p l i t u d e distribution as a function of the diffraction angle. 0, is given by s

0?"

/1

..t{0./) = Cexp(ico/)

]

..... ,,i'll

exp(ikx sinO) exp( iwvCv, t )) d.v

2O

Figure 2

Pulse frequency spectrum after 0 cm travel m water

(2) wherc ~o a n d k are the light frequency a n d wave vector, respectively; C is a n o r m a l i z a t i o n constant: a n d rt is the R a m a n - N a t h p a r a m e t e r , a quantity p r o p o r t i o n a l to the m a x i m u m c h a n g e in the m e d i u m index of refraction, i.e. directly p r o p o r t i o n a l to the a m p l i t u d e o f the ultrasonic

,1 .......................................................................

WaVe.

T h e ultrasonic wave u n d e r c o n s i d e r a t i o n here has m a n y c o m p o n e n t s , given by the a p p r o p r i a t e form o f E q u a t i o n (1). Substitution o f E q u a t i o n (1) into the diffraction integral, E q u a t i o n (2) yields a solution for the light a m p l i t u d e d i s t r i b u t i o n in the far-field given by l(O ) =

J

.01."

iPl

irl

(3)

(sin &'2m/&"~m) 2 ],2n

i -40

where

=

II]Jrrt

12

1,n = a n t I and

¢,m=

~,,~ r:z=-

~

• ..

.Irl (m)(Vl) " .. Jrn

( Fn ) " . .

r n

• "" exp [i r l ( m ) +

"'"

+r n "'" + ""

]

(4)

with r~ (m) = m - 2r 2 - . . . - n r n - . . . a n d Jr the r t h - o r d e r Bessel function of the first kind. T h e diffraction pattern has a light intensity distribution which shows distinct m a x i m a at diffraction angles 0 which satisfy 8 sin0 = ±mK/'k Sample

(5

evaluation

Repetition rate determination C o n s i d e r a pulse sequence, as i n d i c a t e d in F i g u r e l b , with F = 3 M H z , f = 150 kHz, a 1/3 #s rise time. a n d a decay

84

0 ORDER NUMBER

20

40

Figure 3 Diffraction pattern produced by pulse sequence with components as shown in Figure 2

£Zrn = (ksin0 -mK ) Jill

-20

Ultrasonics 1987

Vol 25 March

such that the a m p l i t u d e decreases to l/e in 2.33 #s. F o r this pulse a p e a k a m p l i t u d e of 0.7 atm c o r r e s p o n d s to a R a m a n - N a t h p a r a m e t e r of ~ 2. T h e F o u r i e r series for the pulse sequence, given by E q u a t i o n (1), is a s u m m a t i o n of sines whose a r g u m e n t s increase in steps of 150 kHz, the repetition frequency. A frequency c o m p o n e n t of 3 M H z c o r r e s p o n d s to n = 20, the strongest c o m p o n e n t of the series. The relative a m p l i tudes, a, are shown in F i g u r e 2. T h e a m p l i t u d e a n d p h a s e terms in the F o u r i e r series, w h e n substituted into the diffraction integral, E q u a t i o n (2), p r o d u c e a light intensity distribution as a function o f the diffraction angle, O, E q u a t i o n (3), as shown in F i g u r e 3. It is i m p o r t a n t to note that the abscissa of this graph can be labelled in m a n y ways. F r o m E q u a t i o n (5) one would expect that the light intensity m a x i m a would occur at discrete angles 0. Since these angles are v e ~ small. sin0 ~ O, a n d the light intensity m a x i m a occur in regular intervals where the s e p a r a t i o n is governed exclusively by the repetition frequency. T h e light intensity.' m a x i m a located at o r d e r n u m b e r s - 2 0 and +20 are those associated with a frequency 20 times the repetition rate, i.e. the 3 M H z pulse m o d u l a t i o n frequency. I f a 3 M H z c o n t i n u o u s ultrasonic wave had been used instead of the pulse the diffraction pattern would have consisted o f positive a n d negative orders which a p p e a r at the two locations labelled - 2 0 a n d +20. For this p a r t i c u l a r

Acousto-optic pulse characterization: W.G. Mayer and f a n d F, the location of order number 20 can be labelled '3 MHz'. In turn, order number 19 is equivalent to 2.85 MHz, order number 18 is equivalent to 2.70 MHz, etc. The spacing of the orders surrounding the strongest light intensity orders is a direct reading of the repetition rate. The higher the repetition frequency, the bigger the spacing, with the reference frequency being the modulation frequency of the pulse. The location of the central diffraction order does not change and the angular location of the order representing 3 M H z does not change its location. Any change in the repetition rate changes the spacing of the satellite orders around the fixed location of the main order. Therefore, one can read the pulse repetition rate directly from the spacing of the diffraction orders.

Estimation of spectral composition due to nonlinear/ties It is well known that medium non-linear/ties change the wave shape of an ultrasonic wave as it propagates. The amount of change depends on a number of parameters and conditions, i.e. the initial amplitude of the signal, its frequency, and the distance through which the signal has travelled 9. Applied to high amplitude ultrasonic pulses, this implies that the frequency components indicated by Equation (1) are subject to changes as the pulse travels. Such changes should also change the light intensity distribution in a diffraction pattern. It is possible to obtain a good estimate of the non-linear changes in the pulse modulation if one interprets the asymmetries in the diffraction pattern. The causes for non-linear changes in the frequency spectrum are contained in the modified version of Burgers' equation used by Haran and Cook ~° to describe the change in the nth component of the particle velocity as the pulse propagates incrementally along the x direction. The differential change in particle velocity, U, is given by aU

_

Ox

_ /36 U _aU + a -~)2- U

c2

OT

T.H. Neighbors

harmonics and subharmonics of the pulse modulation frequency, F. The frequency spectrum is shown in Figure4 when the pulse described above has travelled 50 cm. Comparing this graph with Figure 2 clearly shows the creation of second harmonics (6 MHz components), and the resulting diffraction pattern, Figure 5, shows distinct differences when compared to Figure3. When one follows the development in the derivation of the expressions for the light intensities in the various orders 8, one notes that in Equations (3) and (4) the most important contribution to the light intensity in the 'main' orders (numbers - 4 0 and +40) occurs because there is a significant increase in the frequency components which, at 0 cm, were essentially non-existent. Provided the overall ultrasonic amplitudes are not so high that the R a m a n - N a t h parameter exceeds a value of ~ 2 , one can see from Equation (4) that the main contributing factor to the light intensity in the order ~ 40 is proportional to the difference between the square root of the intensity in the positive and negative nth order, all normalized to the central order. This then results in a formula for the estimate of the magnitude of the 2nd harmonic created in the pulse due to non-linear processes. With due regard for the meaning of the subscript n, this :amplitude is a.

I(In) ~ - (z_.) ~ i/n(10) ~

=

(8)

0.4

0.2

(6)

~T2

where/3 = 1 + B/2A, the non-linear parameter, and a is the attenuation coefficient. Taking a Taylor series expansion and keeping only first-order terms, Cook and Haran obtain

,,,,,,,,,rlF Ip,,,,, ,,,,r111,,

. . . . .

20

40

,,, ....

60

80

n

Figure

4

Pulse frequency

spectrum

water

a f t e r 5 0 c m t r a v e l in

Un(x +dx) = Un(x) + I(i{36/c~) (/~= 1 /UjUn-/

/=n+/ giving the incremental change of the pulse spectral components. Here dx is the incremental distance and * indicates the complex conjugate. If we now recognize that the change in the amplitude of the Fourier components is proportional to the change of the particle velocities associated with these components, we can use the fact that Equation (1) is proportional to Equation (7). Based on this, one can calculate the components of the frequency spectrum of the pulse sequence and the resulting diffraction pattern for any distance of propagation. The changes in the light intensity distribution are related to the growth and decay of the

.1

.....................................................................

.01 . . . . . . . . . . . . . . . . . . . . .

--¢

i -40

Figure 5

." ...........

r

! "

-20

Diffraction

- ~" . . . . . . . . . . . .

pattern

I

0

URGERNUMBER

• .....................

.

.

.

.

.

20

40

a f t e r 5 0 c m t r a v e l in w a t e r

U l t r a s o n i c s 1 9 8 7 Vol 2 5 M a r c h

85

Acousto-optic pulse characterization: W.G. Mayer and T.H. Neighbors

Conclusions

r

t

.~ i

"

t

~

i

J

i

i

I

I

L± 5

50 CM

Acknowledgements

.4

I

I

<=. ,

0ii

i

4

i i I iii i

m ' I

i 35

]'his work was supported in part by the Office of Nava~ Research, US Navy. W G M gratefully acknowledges financial assistance by NATO Scientific Affairs Division.

i

4O COMPONENI n

45

,

35

4fi COMPONENT°

'

J

45

Figure 6 Calculated (graphs on the left) and estimated (graphs on the right) frequency spectrum amplitudes in the vicinity of second harmonic (n = 40) for 25 cm (top graphs) and 50 cm (bottom graphs) travel in water

if the differential magnitude in the diffraction pattern is attributed to the generated harmonics and absolute phasing is ignored. As a check on the validity of Equation (8), a comparison of the results of applying this equation and the calculated results using Burgers" equation is shown in Figure 6.

86

It is shown that the evaluation of a light diffraction pattern produced by a series of ultrasonic pulses can be used to describe certain parameters describing the pulse sequence and some changes in the pulses caused by non-linear effects. The example given for the latter case shows acceptable agreement between the evaluation ot Burgers" equation and a comparison of the light intensities in the pulse diffraction pattern.

Ultrasonics 1 9 8 7 Vol 25 March

References ~3

2 3 4 5 6 7 8 9 10

Lucas, R. and Biquard, P. C RAcadSci(Part~) il932! 195 121 Raman, C.V. and Nath, N.S.N. Proc Indian Acad 5ci (1935) 2 Klein, W.R., Cook, B.D. and Mayer, W.G. Acustica (1965) 15 67 Hargrove, L.E. JAcoust Soc Am (1968) 43 847 Zitter, R.N. J Acoust Soc Am (1968) 43 864 Hiiusler, E., Mayer, W.G. and Schwartz, M. Acoust LeH ( 1981 ) 4 180 Neighbors, Ill, T.H. and Mayer, W.G. JAcoust Soc Am (1983) 74 146 Beyer, R.T. and Letcher, S.V. Physical Ultrasonic~ Academic Press. New York, USA (1969) Ch. 7 Haran, M.E. and Cook. B.D. J Acoust Soc Am (1983) 73 774