some extensions of the gaussian beam expansion: radiation fields of the rectangular and the...

Some extensions of the Gaussian beam expansion: Radiationfields of the rectangular and the elliptical transducer

Desheng Ding,a) Yu Zhang, and Jinqiu LiuDepartment of Electronic Engineering, Southeast University, Nanjing 210096, People’s Republic of China

~Received 22 November 2002; revised 5 March 2003; accepted 17 March 2003!

A straightforward extension of Gaussian beam expansion is presented for calculation of the Fresnelfield integral @J. J. Wen and M. A. Breazeale, J. Acoust. Soc. Am.83, 1752–1756~1988!#. Thesource distribution function is expanded into the superposition of a series of two-dimensionalGaussian functions. The corresponding radiation field is expressed as the superposition of thesetwo-dimensional Gaussian beams and is then reduced to the computation of these simple functions.This treatment overcomes the limit that the shape of source is of circular axial-symmetry. Thenumerical examples are presented for the field of the~uniform! elliptical and the rectangular pistontransducers and agree well with the results given by complicated computation. ©2003 AcousticalSociety of America.@DOI: 10.1121/1.1572144#

PACS numbers: 43.20.Rz, 43.20.Bi, 43.20.El@MO#

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I. INTRODUCTION

The Fresnel field integral~or Fresnel transform! appearsin many relevant problems of diffraction, for instance, ttheory of the sound field radiated from an ultrasonic traducer and of the laser beam field diffracted by an apertThe exact solution of these problems is described bywell-known Rayleigh surface integral in general, orKing’s integral in the case of circular axial-symmetry.practical situations, the Fresnel field integral is often usinstead of these two solutions. It is a good approximatwhen the size of source is much larger than the wavelen~about ten times! and the observation point is located in thparaxial region. For most of the practical ultrasonic transders, this integral is possibly invalid only in the extremenear region of the transducers. In the most general case,ever, the Fresnel field integral, two-dimensional and stronoscillatory, has to be numerically performed in the evaluatof the field of an arbitrary distributed source. This is alwaa time-consuming thing. To overcome this difficulty, Coet al. first presented the beam~or function! expansionmethod that the field integral is expressed as the superption of Gaussian–Laguerre functions~or Gaussian–Hermiteones in the rectangular coordinate! and obtained a satisfactory result, compared with the numerical solution of the Raleigh integral, of the sound field from a circular uniforpiston transducer.1,2 Wen and Breazeale made some extesions of the method in which the expansion functions uinvolve only the Gaussian ones.3 Since then, many other extensions have been made in various ways.4–15 An obviousadvantage of this approach is that the analytical descripfor the ~linear! sound field distribution, and even for thsecond-order field nonlinearly generated, is obtainablethus the computation is greatly reduced. In previous papnearly all examples are analyzed for the sound fields wcircular axial symmetry, but few are analyzed for the lackthis symmetry.4

a!Electronic mail: [email protected]

J. Acoust. Soc. Am. 113 (6), June 2003 0001-4966/2003/113(6)/3

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This paper presents an alternative approach, a furextension of Breazeale’s, to evaluate the ultrasonic fieldtransducers. The source function is generally expressethe superposition of a series of two-dimensional Gaussfunctions. The corresponding sound field is then reducedthe computation of these simple functions. It avoids the limthat the shape of the source is of the circular axial symmeNumerical examples are given for the uniform elliptical arectangular piston transducers and compared with the rein references, demonstrating the high efficiency of omethod in computer run time.

II. THEORY

A. Field integral

Here we write only the Fresnel field integral andalternative form in nondimensional variables. The detaiderivation of this formula and the discussion of its appliction limits can be found in much of the literature and texbooks. Assume that the source or the aperture is placed inplanez50 in a rectangular coordinate, and here the quanu(x8,y8) is the pressure or normal velocity at the transduer’s surface and may be defined as zero in other regionmay also represent the others, such as the light beam dbution or the components of electromagnetic field intensIn the Fresnel approximation, the radiated sound field canexpressed by

u~x,y,z!51

ilz E2`

` E2`

`

expS ip

l

~x2x8!21~y2y8!2

z D3u~x8,y8! dx8 dy8, ~1!

i.e., the Fresnel field integral. Herel is wavelength and thepropagating factor exp@2i(vt2kz)# has been omitted. Withthe dimensionless variables

j5x/AS, z5y/AS, and h5lz/pS, ~2a!

Sbeing simply related to the area of source, we write Eq.~1!as

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u~j,z,h!51

iph E2`

` E2`

`

expS i~j2j8!21~z2z8!2

h D3u~j8,z8!dj8 dz8. ~2b!

B. Two-dimensional Gaussian beams

A two-dimensional Gaussian source has the form

u~j8,z8!5exp@2~Bxj821Byz82!#, ~3!

whereBx and By are generally complex numbers with threal parts greater than zero. The two-dimensional Gausbeam field, with a special notationG2 , is

G2~j,z,h;Bx ,By!

51

iph E2`

` E2`

`

expS i~j2j8!21~z2z8!2

h D3exp@2~Bxj821Byz82!#dj8 dz8. ~4!

Using

E2`

`

expS 2t2

4b2gt Ddt52Apb exp~bg2!, ~5!

and making some arrangements, we have

G2~j,z,h;Bx ,By!5F 1

A11 iBxhexpS 2

Bxj2

11 iBxhD G

•F 1

A11 iByhexpS 2

Byz2

11 iByhD G

5G1~j,h;Bx!•G1~z,h;By!. ~6!

In Eq. ~6!, the G1’s denote the terms in the square brackand are indeed the expression of a one-dimensional Gaubeam. WhenBx5By5B, Eq. ~6! is just a usual~circular!Gaussian beam.

In one paper by Wen and Breazeale, they expresseddistribution function of an axial-symmetric~circular! sourceas a linear superposition of complex Gaussian functionsreduced the field solution to the calculation of simple Gauian functions. Mathematically, this expansion means thaclass of functions can be expressed approximately by aof a set of Gaussian functions, namely,

f ~x!5 (k51

N

Ak exp~2Bkx2!, ~7!

where the expansion and Gaussian coefficients,Ak andBk ,can be found out by computer optimization. Specially, Wand Breazeale obtained one of the coefficient sets, contaionly ten terms of coefficients (N510), to match the circfunction. These coefficients on the right-side of Eq.~7!, anapproximation of this function, are listed in Table I of Ref.

3044 J. Acoust. Soc. Am., Vol. 113, No. 6, June 2003


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moreover, they calculated the sound field distribution ouniform circular piston source. It is found that an excelleagreement between a ten-term Gaussian beam solutionthe results of numerical integration is obtained throughthe beam field, and discrepancies exist only in the extrenear field~,0.12 times the Fresnel distance!. In this paper,we take Eq.~7! as a known result,an approximation to thecirc function, and we express an arbitrary source functionthe superposition of a series of two-dimensional Gaussfunctions and show how to apply this approach in the ficalculation for the elliptical and rectangular transducers.

C. Rectangular piston

Assume that the center of a rectangular piston traducer, with a uniform distribution excitation, is located at torigin of the coordinates. The source distribution functionthen defined as

u~x8,y8!5H 1 if ux8u<a, uy8u<b,

0 elsewhere.~8!

The quantitiesa andb are the half-major and half-minor axeof the transducer. Using the nondimensional variables,~2a!, with S5ab, we get

FIG. 1. Comparison of the sound intensity distributions of the rectangpiston along the acoustic axis~z!, computed by use of the Gaussian expasion and the exact expression.~a! Square transducer (b/a51); ~b! rectan-

gular (b/a523). ~a! and~b! correspond to Figs. 3 and 5 of Ref. 17. Note th

the axial distance is normalized toZ054a2/l.

Ding et al.: Transducer radiation field


FIG. 2. Comparison of the off-axis pressure field distributions~in thex-zplane! at the different axial distances. Hereh is as defined in the text@Eq. ~20!#. Noother results may be compared.

thian

d
u~j8,z8!5H 1 if uj8u<Aa/b, uz8u<Ab/a,
0 elsewhere.~9!

The above function can be written

u~j8,z8!5circ~Ab/aj8!•circ~Aa/bz8!

5H (k51

N

Ak expF2BkS b

aD j82G J•H (

k51

N

Ak expF2BkS a

bD z82G J . ~10!

In the next treatment it is not necessary to further expandinto the formal summation of two-dimensional Gauss

J. Acoust. Soc. Am., Vol. 113, No. 6, June 2003


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functions like Eq.~3!. Using Eq.~6!, we express the sounfield of the rectangular piston transducer as

u~j,z,h!5F (k51

N

AkG1S j,h;b

aBkD G

•F (k51

N

AkG1S z,h;a

bBkD G . ~11!

The field distribution, directly from Eq.~2!, may be ex-pressed by

3045Ding et al.: Transducer radiation field


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iph Ej852Aa/b

Aa/b Ez852Ab/a

Ab/a

3expF i~j2j8!21~z2z8!2

h Gdj8 dz8

51

iph Ej852Aa/b

Aa/bexpF i

~j2j8!2

h Gdj8

3Ez852Ab/a

Ab/aexpF i

~z2z8!2

h Gdz8. ~12!

One further obtains

u~j,z;h!51

2i FFSA 2

ph S j1Aa

bD D 2FSA 2

ph

3S j2Aa

bD D GFFSA 2

ph S z1Ab

aD D2FSA 2

ph S z2Ab

aD D G . ~13!

Here F(z)5C(z)1 iS(z) is the Fresnel function, a kind othe special functions widely used in the diffraction proble

We now give a comparison of the results for the rectgular piston, calculated respectively from Eqs.~11! and~13!.

FIG. 3. The on-axis field distributions with different values ofd5b/a. Herethe normalized distanceSdefined by Ref. 4 has the relation with the variabh of Eq. ~2a! in the text: S5(p/2)(d11/d)h. And DS/S5(12d2)/(11d2). ~a! d50.7738,DS/S50.3; ~b! d50.25, DS/S50.882. ~a! and ~b!correspond to Figs. 6~c! and 7~c! of Ref. 4.~a! has some discrepancies in thregion ofS'0.3– 0.6 from Fig. 6~c! of Ref. 4; the others presented in Fig.of Ref. 15 agree fairly well with those of Ref. 4.



.-

Figure 1 shows the intensity distribution of the on-axis fiewith the different values ofb/a, the ratio of the minor tomajor side. One can see that the results with the Gausexpansion technique agree well with those of the analytsolution. Discrepancies exist only in the extreme near-fieTo compare with the earlier results16,17 ~see Figs. 3 and 5 oRef. 17!, the calculation parameters in this example are tathe same as in Ref. 17. Figure 2 is typically for the pressdistribution of the off-axis field. In all the graphs, the resuare compared with numerical computation of the Fresfunction, Eq.~13!. A very good agreement is again obtainewith the exception of little loss of the fine structures in textreme near-field region.

D. Elliptical piston

For a uniform elliptical piston field, the source functiois

u~x8,y8!5H 1 if ~x8/a!21~y8/b!2<1,

0 elsewhere.~14!

Without loss of generality, herea andb are the semi-majorand semi-minor axes of an elliptical piston. LettingS5aband using dimensionless variables, we write

u~j8,z8!5H 1 ~b/a!j821~a/b!z82<1,

0 elsewhere.~15!

FIG. 4. The axial fields for a cylindrically focused circular piston (d5b/a51) obtained from the Gaussian expansion.~a! Fxl/4a250.372,Fy

5`; ~b! Fxl/4a250.744,Fy5`. Fx andFy are the focal lengths defined inRef. 4.



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The source function of the uniform elliptical piston canwritten

u~j8,z8!5circS b

aj821

a

bz82D

5 (k51

N

Ak expF2S b

aBkj821

a

bBkz82D G . ~16!

From Eqs.~6! and ~7!, the radiated field distribution is thuexpressed as

u~j,z,h!5 (k51

N

AkG2S j,z,h;b

aBk ,

a

bBkD . ~17!

As in the case of a uniform circular piston transducthere is no analytical expression for the uniform ellipticpiston field, except in the far field~Fraunhofer region! wherethe field distribution may be simply described in terms of tfirst-order Bessel function of the first kind. An integral reresentation of the on-axis radiation field from the elliptictransducer is given by Thompsonet al.4 This result is repro-duced here with our notation,

u~0,0,h!

51

2p E0

2pF12expF i2d

h@~11d2!1~12d2! cos 2u#G G du,

~18!

whered5b/a is the ratio of the semi-minor to semi-majoaxis. For the off-axis field, they expanded it into the supposition of Gaussian–Hermite functions up to about 20~for instance, 47347) terms, using the method of Cooket al.

We now present the calculated results for the elliptipiston field. For comparison, all parameters are the samin Ref. 4. Figure 3 demonstrates the on-axis field ofuniform elliptical pistons with the different values ofd. Theexact results from Eq.~18! are also shown in this figure. It iobvious that our results with the only ten-term Gaussianpansion technique are better than those calculated usin347- or 41341-term Gaussian-Hermite functions.4 ~SeeFigs. 6 and 7 of Ref. 4.! Unfortunately, there is no body oliterature for the off-axis near-field of elliptical transducethat permits comparison.

The present method is easy to apply to the case ofsound field of the elliptical piston focused by a bicylindriclens.4 The effect of focusing is~approximately! equivalent toa spatial modulation of the plane piston distribution. A prcedure fully applicable here has been given in Refs. 8 analthough there the field of the circular focused piston traducer was analyzed. We do not repeat here and merelysome results for the focused elliptical pistons, as shownFig. 4. In all calculations we apply the ten pairs of coefcients in Table I of Ref. 3.

Finally, it is worth pointing out that the present methomay be applied to the other circumstances. For examplesound field by a strip piston may be included as a specase of the rectangular transducer.18,19 A more general dis-cussion will be given in the Appendix. Besides, the integrepresentation of field, Eqs.~1! and~2!, appear in many othe

J. Acoust. Soc. Am., Vol. 113, No. 6, June 2003


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branches of physics and engineering application involvthe problem of diffraction, such as optics, the propagationelectromagnetic fields, etc. It is expected that the presenan improved method is still applicable in these fields.

III. CONCLUSION

In conclusion, we have presented a new approachtreats efficiently the radiation problem for the ultrasontransducers. It reduces the Fresnel field integral to the suposition of a set of two-dimensional Gaussian beams. Tradiated fields of the elliptical and the rectangular pistonexemplified and compared with the results in referencdemonstrating the high efficiency of this method. The posbility of the application to the other cases is also discuss

ACKNOWLEDGMENTS

This work is supported by the National Natural ScienFoundation of China under Grant No. C-A040505-199040and 10274010. DD would like to express his thanks to Pfessor M. A. Breazeale. Much work is inspired by his letures presented at the Institute of Acoustics of Nanjing Uversity many years ago.

APPENDIX: FURTHER EXTENSION

Notice that those several examples given in the textstill very special. The source function has the propeu(x,y)5u(6x,6y), that is, the source~or beam! distribu-tion is always symmetrical with respect ofx and y axes.Naturally, the question rises: Is there an expansion similaEq. ~10! or ~16!, or does the present method remain appcable, for an arbitrary, real source distribution without thsymmetry? To answer this question, we give the followistatement.

Let the functionf (x,y) be square integrable and~piece-wise continuous! in the entire region~x-y plane!, i.e.,*2`

` *2`` f 2(x,y)dx dy,1`. Then this function is always

expressed as the summation form of the two-dimensioGaussian functions, in a sense of an average convergencother words, if

Q5E2`

` E2`

` F f ~x,y!2 (k51

N

AkG2~x2ak ,

y2bk ;Bxk ,Byk!G2

dx dy, ~A1!

thenQ→0 whenN→`. The functionG2 here is the sameform as that of Eq.~3!, defined by

G2~x2ak ,y2bk ;Bxk ,Byk!5exp$2@Bxk~x2ak!2

1Byk~y2bk!2#%, ~A2!

andAk , Bxk , Byk , ak andbk are a set of coefficients to bdetermined for the functionf (x,y) given. Further,ak andbk

are always assumed to be real-valued. There is a geomexplanation for real-valuedak andbk . For a Gaussian beamdescribed by Eq.~A2!, its center shifts from the point~0,0! to(ak ,bk). In principle, these coefficients of Eq.~A1! are ob-tained by computer optimization.3

3047Ding et al.: Transducer radiation field


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We have some difficulty to mathematically prove thstatement. However, the statement seems to hold goodintuition. Under this assumption, the approach presentethe text is extended to a more general case. An arbitsource function is decomposed to

u~j,z!5 (k51

N

AkG2~j2ak ,z2bk ;Bxk ,Byk!. ~A3!

The radiation field is calculated from

u~j,z,h!5 (k51

N

AkG2~j2ak ,z2bk ,h;Bxk ,Byk! ~A4!

of G2 with the same form as Eq.~3!.An example at hand strongly supports the above ar

ment. We assume that the quarter of a uniform, elliptical~orcircular! piston is located in the first region of the rectangucoordinate. Its source function, no longer symmetric, ispressed as

u~j8,z8!5H 1 ~b/a!j821~a/b!z82<1, j8>0, z8>0,

0 elsewhere.~A5!

Note that Eq.~A5! may be written as

u~j8,z8!5 f 1~j8,z8! f 2~j8! f 3~z8!, ~A6a!

where

f 1~j8,z8!5circS b

aj821

a

bz82D

5 (k51

N

Ak expF2S b

aBkj821

a

bBkz82D G , ~A6b!

f 2~j8!5circS 2Ab

a Uj821

2U D5 (

k51

N

Ak expF24BkS b

aD S j821

2D 2G , ~A6c!

and

f 3~z8!5circS 2Aa

b Uz821

2U D5 (

k51

N

Ak expF24BkS a

bD S z821

2D 2G . ~A6d!

The above coefficients are known from Table I of Ref.Then Eq.~A5! is expressed in the form of Eq.~A3!, and itscoefficients, although not optimized, are the combinationthe known Gaussian and expansion coefficientsBk andAk ,and the parameterd5b/a.



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This is of course an imagined example. Fortunately,geometry of transducers or sources is nearly~considered!regular in many practical uses and the theory of diffractproblems. We do not say much.

1B. D. Cook and W. J. Arnoult III, ‘‘Gaussian-Laguerre/Hermite formultion for the nearfield of an ultrasonic transducer,’’ J. Acoust. Soc. Am.59,9–11 ~1976!.

2E. Cavanagh and B. D. Cook, ‘‘Gaussian-Laguerre description of ulsonic fields-Numerical Example: Circular piston,’’ J. Acoust. Soc. Am.67,1136–1140~1980!.

3J. J. Wen and M. A. Breazeale, ‘‘A diffraction beam field expressed assuperposition of Gaussian beams,’’ J. Acoust. Soc. Am.83, 1752–1756~1988!.

4R. B. Thompson, T. A. Gray, J. H. Rose, V. G. Kogan, and E. F. Lop‘‘The radiation of elliptical and bicylindrically focused piston transduers,’’ J. Acoust. Soc. Am.82, 1818–1828~1987!.

5D. Ding, J. Lin, Y. Shui, G. Du, and D. Zhang, ‘‘An analytical descriptioof ultrasonic field produced by circular piston transducer,’’ Acta Acu~Beijing! 18, 249–255~1993!.

6D. Ding, Y. Shui, J. Lin, and D. Zhang, ‘‘A simple calculation approacfor the second harmonic sound field generated by an arbitrary axsymmetric source,’’ J. Acoust. Soc. Am.100, 727–733~1996!.

7D. Ding and Z. Lu, ‘‘A simplified method to calculate the sound fieldpistonlike source,’’ Chin. J. Acoust.15, 213–222~1996!, also appearing inActa Acust.~Beijing! 21 ~Suppl. 4!, 421–428~1996!.

8D. Ding and Z. Lu, ‘‘A simplified calculation for the second-order fieldgenerated by axial-symmetric sources at bifrequency,’’ inNonlinearAcoustics in Perspective: Proceedings of the 14th International Symsium on Nonlinear Acoustics, edited by R. J. Wei~Nanjing U. P., Nanjing,1996!, pp. 183–188.

9D. Ding, ‘‘A simplified algorithm for the second-order sound fields,’’Acoust. Soc. Am.108, 2759–2764~2000!.

10D. Ding and X. Liu, ‘‘Approximate description for Bessel, Bessel-Gauand Gaussian beams with finite aperture,’’ J. Opt. Soc. Am. A16, 1286–1293 ~1999!.

11M. D. Prange and R. G. Shenoy, ‘‘A fast Gaussian beam descriptionultrasonic fields based on Prony’s method,’’ Ultrasonics34, 117–119~1996!.

12O. B. Matar, J. P. Rernenieras, C. Bruneel, A. Roncin, and F. Patat, ‘trasonic sensing of vibrations,’’ Ultrasonics36, 391–396~1998!.

13D. Huang and M. A. Breazeale, ‘‘A Gaussian finite-element methoddescription of sound diffraction,’’ J. Acoust. Soc. Am.106, 1771–1781~1999!.

14M. Spies, ‘‘Transducer field modeling in anisotropic media by superpotion of Gaussian base function,’’ J. Acoust. Soc. Am.105, 633–638~1999!.

15Y. Zhang, J. Liu, and D. Ding, ‘‘Sound field calculations of ellipticapistons by the superposition of two-dimensional Gaussian beams,’’ CPhys. Lett.19, 1825–1827~2002!.

16A. Freedman, ‘‘Sound field of a rectangular piston,’’ J. Acoust. Soc. A32, 197–201~1960!.

17J. Marini and J. Rivenez, ‘‘Acoustical fields from rectangular ultrasotransducers for nondestructive testing and medical diagnosis,’’ Ultraso12, 251–256~1974!.

18F. D. Martin and M. A. Breazeale, ‘‘A simple way to eliminate diffractiolobes emitted by ultrasonic transducers,’’ J. Acoust. Soc. Am.49, 1668–1669 ~1971!.

19M. A. Breazeale, F. D. Martin, and B. Blackburn, ‘‘Reply to ‘Radiatiopattern of partially electroded piezoelectric transducers,’’’ J. Acoust. SAm. 70, 1791–1793~1981!.



some extensions of the gaussian beam expansion: radiation fields of the rectangular and the...

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