The concept of an effective apodization was introduced to describe the field pattern for the nonlinearly generated second harmonic within the focal zone using a linear propagation model. Our objective in this study was to investigate the validity of the concept of an effective apodization at as an approach to approximating the field of the second harmonic over a wide range of depths. Two experimental setups were employed: a vascular imaging array with a water path and an adult cardiac imaging array with an attenuating liver path. In both cases the spatial dependencies of the ultrasonic fields were mapped by scanning a point-like hydrophone within a series of planes orthogonal to the propagation direction. The sampling distances were located before, within, and beyond the focal zone. The signals were Fourier transformed and the complex values at were linearly backpropagated to the transmit plane in order to obtain an effective apodization. The measured results demonstrated a relatively constant effective apodization at as a function of propagation distance. Finite amplitude computer simulations were found to be in agreement with these measurements. Thus the measure of the effective apodization at provides an approximation to the second harmonic field outside the focal zone.
REFERENCES
1.
R. J.
Fedewa
, K. D.
Wallace
, M. R.
Holland
, J. R.
Jago
, G. C.
Ng
, M. R.
Rielly
, B. S.
Robinson
, and J. G.
Miller
, “Spatial coherence of the nonlinearly generated second harmonic portion of backscatter for a clinical imaging system
,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control
50
, 1010
–1022
(2003
).2.
R. J.
Fedewa
, K. D.
Wallace
, M. R.
Holland
, J. R.
Jago
, G. C.
Ng
, M. R.
Rielly
, B. S.
Robinson
, and J. G.
Miller
, “Spatial coherence of backscatter for the nonlinearly produced second harmonic for specific transmit apodizations
,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control
51
, 576
–588
(2004
).3.
R. J. Fedewa, “Nonlinear ultrasonic propagation through aberrating and nonaberrating media,” Ph.D. dissertation, Washington University, St. Louis, MO, 2002, available from UMI, 300 North Zeeb Road, PO Box 1346, Ann Arbor, MI 48106-1346 (http://www.umi.com/hp/Products/Dissertations.html).
4.
F. A.
Duck
, “Nonlinear acoustics in diagnostic ultrasound
,” Ultrasound Med. Biol.
28
, 1
–18
(2002
).5.
T.
Christopher
, “Finite amplitude distortion-based inhomogeneous pulse echo ultrasonic imaging
,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control
44
, 125
–139
(1997
).6.
B.
Ward
, A. C.
Baker
, and V. F.
Humphrey
, “Nonlinear propagation applied to the improvement of resolution in diagnostic medical ultrasound
,” J. Acoust. Soc. Am.
101
, 143
–154
(1997
).7.
V. F.
Humphrey
, “Nonlinear propagation in ultrasonic fields: measurements, modelling and harmonic imaging
,” Ultrasonics
38
, 267
–272
(2000
).8.
M. A. Averkiou, “Tissue harmonic imaging,” presented at the 2000 IEEE International Ultrasonics Symposium in San Juan, Puerto Rico, 2000, Vol. 00CH37121, pp. 1563–1572.
9.
K. D. Wallace, B. S. Robinson, M. R. Holland, M. R. Rielly, and J. G. Miller, “Experimental comparisons of the impact of abdominal wall aberrators on linear and nonlinear beam patterns,” 2004 IEEE International Ultrasonics Symposium, Montreal, Canada, 2004, Vol. 04CH37553C, pp. 866–869.
10.
B. D. Steinberg, Principles of Aperture and Array System Design (Wiley, New York, 1976), Chap. 3, pp. 40–57.
11.
D. Kourtiche, L. Ait Ali, A. Chitnalah, and M. Nadi, “Theoretical study of harmonic generation sound beams in case of uniform, exponential and cosinusoidal apertures,” presented at the 2001 Proceedings of the 23rd Annual EMBS International Conference, Istanbul, Turkey, 2001, pp. 2395–2398.
12.
K. D. Wallace, R. J. Fedewa, M. R. Holland, G. C. Ng, B. S. Robinson, J. R. Jago, M. R. Rielly, and J. G. Miller, “Measurements comparing the linearly propagated field using an effective apodization and the nonlinearly generated second harmonic field,” presented at the 2003 IEEE International Ultrasonics Symposium, Honolulu, HI, 2003, Vol. 03CH37476C, pp. 453–456.
13.
R.
Mallart
and M.
Fink
, “The Van Cittert–Zernike theorem in pulse echo measurements
,” J. Acoust. Soc. Am.
90
, 2718
–2727
(1991
).14.
D.-L. D. Liu, P. Von Behren, and J. Kim, “Single transmit imaging,” presented at the 1999 IEEE Ultrasonics Symposium in Caesars Tahoe, NV, 1999, Vol. 99CH37027, pp. 1275–1278.
15.
B. J. Geiman, “Transmit beam estimation in ultrasonic imaging,” Ph.D. dissertation, Duke University, Durham, NC, 2000, available from UMI, 300 North Zeeb Road, P.O. Box 1346, Ann Arbor, MI 48106-1346 (http://www.umi.com/hp/Products/Dissertations.html).
16.
E. W.
Cherin
, J. K.
Poulsen
, A. F. W.
van der Steen
, P.
Lum
, and F. S.
Foster
, “Experimental characterization of fundamental and second harmonic beams for a high-frequency ultrasound transducer
,” Ultrasound Med. Biol.
28
, 635
–646
(2002
).17.
P. C.
Li
and C. C.
Shen
, “Effects of transmit focusing on finite amplitude distortion based second harmonic generation
,” Ultrason. Imaging
21
, 243
–258
(1999
).18.
P. R.
Stepanishen
and K. C.
Benjamin
, “Forward and backward projection of acoustic fields using FFT methods
,” J. Acoust. Soc. Am.
71
, 803
–812
(1982
).19.
M. E.
Schafer
and P. A.
Lewin
, “Transducer characterization using the angular spectrum method
,” J. Acoust. Soc. Am.
85
, 2202
–2214
(1989
).20.
S.
Krishnan
, J. D.
Hamilton
, and M.
O’Donnell
, “Suppression of propagating second harmonic in ultrasound contrast imaging
,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control
45
, 704
–711
(1998
).21.
T.
Christopher
, “Source prebiasing for improved second harmonic bubble-response imaging
,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control
46
, 556
–563
(1999
).22.
K. D. Wallace, “Characterization of the nonlinear propagation of diffracting, finite amplitude ultrasonic fields,” Ph.D. dissertation, Washington University, St. Louis, MO, 2001, available from UMI, 300 North Zeeb Road, P.O. Box 1346, Ann Arbor, MI 48106-1346 (http://www.umi.com/hp/Products/Dissertations.html).
23.
P. T.
Christopher
and K. J.
Parker
, “New approaches to nonlinear diffractive field propagation
,” J. Acoust. Soc. Am.
90
, 488
–499
(1991
).24.
C. J. Vecchio, “Finite amplitude acoustic propagation modeling using the extended angular spectrum method,” Ph.D. dissertation, Drexel University, Philadelphia, PA, 1992, available from UMI, 300 North Zeeb Road, P.O. Box 1346, Ann Arbor, MI 48106-1346 (http://www.umi.com/hp/Products/Dissertations.html).
25.
R. T. Beyer, “The parameter B/A,” in Nonlinear Acoustics, edited by M. F. Hamilton and D. T. Blackstock (Academic, San Diego, CA, 1998), Chap. 2, pp. 25–40.
26.
B. J. Geiman, R. C. Gauss, and G. E. Trahey, “In vivo comparison of fundamental and harmonic lateral transmit beam shapes,” presented at the 2000 IEEE Ultrasonics Symposium in San Juan, Puerto Rico, 2000, Vol. 00CH37121, pp. 1669–1675.
27.
R. M.
Lerner
and R. C.
Waag
, “Wave space interpretation of scattered ultrasound
,” Ultrasound Med. Biol.
14
, 97
–102
(1988
).28.
W. F.
Walker
and G. E.
Trahey
, “The application of k-space in pulse echo ultrasound
,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control
45
, 541
–558
(1998
).29.
P. R.
Stepanishen
, “The time-dependent force and radiation impedance on a piston in a rigid infinite planar baffle
,” J. Acoust. Soc. Am.
49
, 841
–849
(1971
).30.
J. A.
Jensen
, “A model for the propagation and scattering of ultrasound in tissue
,” J. Acoust. Soc. Am.
89
, 182
–191
(1991
).31.
J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968), Chap. 4, pp. 73–75.
32.
S.
Saito
, “A property of the nonlinearly generated second-harmonic component of focused ultrasound detected by a concave receiver
,” Ultrasonics
34
, 537
–541
(1996
).
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