Temporal focusing by use of composite X waves - OSA Publishing

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J. Opt. Soc. Am. A / Vol. 20, No. 8 / August 2003

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Temporal focusing by use of composite X waves Amr M. Shaarawi* The Physics Department, The American University in Cairo, P.O. Box 2511, Cairo 11511, Egypt

Ioannis M. Besieris The Bradley Department of Electrical and Computer Engineering, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061

Tarek M. Said Cairo Higher Institute, Helioplis, Golf Region, Cairo, Egypt Received October 17, 2002; revised manuscript received March 10, 2003; accepted March 13, 2003 It is shown that highly focused pulses can be shaped by exciting a finite aperture with a spread-out pulse train of X waves. The basis of the proposed scheme is that the peaks of X waves, characterized by different apex angles, travel at different velocities. This property allows one to vary the temporal starting points of the initial excitations of a sequence of X waves so that all their peaks meet at a chosen focusing point. It is demonstrated that this simple criterion can be effective in producing a highly focused, composite X-wave pulse that exhibits a slower decay behavior than the individual X-wave components used in synthesizing it. © 2003 Optical Society of America OCIS codes: 320.5550, 320.5540.

1. INTRODUCTION Over the past decade, the introduction of X wave solutions by Lu and Greenleaf has spurred notable activity directed to the understanding of this class of ultrawideband, slowly dispersing field.1–8 Such wave solutions have an extended focus depth that renders them very effective in numerous applications ranging from high-resolution imaging, identification of buried objects, remote sensing, and secure communications. The original X-wave solution is an ultrawideband superposition of spectral plane wave components traveling along wave vectors restricted to a conic surface of apex halfangle ␰.9–11 There have been several investigations of the generation of X waves from finite apertures, as well as studies of their scattering from objects and transmission through dispersive media and multilayered structures.1,12–22 In the present work, we demonstrate that a simple superposition of several X waves having different axicon angles ␰ can be used to tailor pulses so that they converge at a specific focusing point. For the analysis presented in this work, we should emphasize two essential features exhibited by an X wave generated by a circular source of diameter D. First, the launched X wave exhibits an extended focused depth in the near field up to a distance z d ⫽ D/(2 tan ␰) independent of the spectral bandwidth of the elements of the source.1,12 The peak of the generated X wave stays unchanged until it reaches z d . Beyond z d , the peak of the X wave starts decaying at a very fast rate (much faster than 1/z). At farther distances, the decay slows down to the usual 1/z rate. The second feature exhibited by an X wave is that its peak travels at the superluminal speed 1084-7529/2003/081658-08$15.00

c/cos ␰ in the near-field range.1,23 As it approaches the z d limit, the peak of the X wave slows down and asymptotically approaches the speed of light at distances larger than z d . 23 To vary the localization range of the original X wave, one is restricted either to using a larger aperture or changing the parameter ␰. This limited control on the decay range of the original X-wave solution has sparked several attempts to introduce new synthesis methods for tailoring ultrawideband pulses of the X-wave type. These methods use superpositions over higher-order X waves, or over X waves characterized by different ␰ parameters.24–26 The second type has been discussed briefly in an earlier work by Lu et al. who introduced the term ‘‘composite X waves’’ to describe them.24 This type of superposition is crucial to the focusing technique advocated here. Our main objective is to use the fact that the peak of an X wave travels at a speed c/cos ␰ that depends on the spectral apex half-angle ␰. Thus X waves characterized by distinct spectral apex angles will travel at different speeds over a distance of order z d from the source. A train of X waves with varying ␰ angles launched at designated times can produce a highly focused, composite Xwave (CXW) pulse formed at a chosen distance z ⫽ z f and time t ⫽ t f . This happens when all the individual Xwave components reach z ⫽ z f simultaneously at t ⫽ t f . The plan of this work is as follows: In Section 2, our focusing scheme is illustrated in connection with a sourcefree X-wave train. The proposed concept is then extended to the generation of CXW pulses from finite apertures. Numerical examples illustrating the focusing capabilities of our approach are provided in Section 3. In © 2003 Optical Society of America

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addition, possible limitations on the focusing range are discussed. Conclusions and discussions of the prospects of the time-focusing scheme are included in Section 4.

2. FORMULATION Consider the following pulse train consisting of a spreadout, discrete sequence of N l first-order X waves:

冋兺 Nl

⌿ 共 ␳ , z, ct 兲 ⫽ Re

册

⌽ l 共 ␳ , z, ct; z l , ␰ l 兲 , (2.1a)

l⫽1

where ⌽ l 共 ␳ , z, ct; z l , ␰ l 兲 ⫽

⫺A 共 z l , ␰ l 兲兵 a 0 ⫹ i 关共 z ⫺ z l 兲 cos ␰ l ⫺ ct 兴其 (共 ␳ sin ␰ l 兲 2 ⫹ 兵 a 0 ⫹ i 关共 z ⫺ z l 兲 cos ␰ l ⫺ ct 兴其 2 )3/2

.

(2.1b) Here, ␰ l is the spectral apex half-angle of the lth X wave. The amplitude function A(z l , ␰ l ) can be used to emphasize contributions from certain X wave components. The peaks of the individual X waves exist at the distinct points z l at time t ⫽ 0 and move at different velocities ␯ l ⫽ c/cos ␰l . If the z l and ␰ l values are chosen so that all individual X waves arrive simultaneously at a predetermined point z f at a later time t ⫽ t f , then a highly focused CXW pulse will be obtained at that point. This concept is represented schematically in Fig. 1, and, accordingly, the following relationship should be satisfied by all individual X waves: 共 z f ⫺ z l 兲 cos ␰ l ⫽ ct f .

(2.2)

It should be noted that z f and t f are the same for all individual X waves. In designing the initial excitation of a finite source, one can start by choosing the focusing point z f . The simultaneous arrival time t f is determined by the specific choices of ␰ l and z l for l ⫽ 1. Subsequently, all other values of ␰ l and z l are determined by using Eq. (2.2). By combining Eqs. (2.1) and (2.2), one obtains

冋兺 Nl

⌿ 共 ␳ , z, ct 兲 ⫽ Re

l⫽1

⫺

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It should be emphasized that z f and t f appearing in Eq. (2.3) are related through Eq. (2.2). Therefore, once one of the two values is specified, the other can be determined for specific choices of z 1 and ␰ 1 . All other ␰ l values with l ⬎ 1 are subsequently deduced by use of Eq. (2.2).

3. SOURCE-FREE COMPOSITE X WAVES In this section, we provide examples of the behavior of the source-free CXW. Consider a pulse train consisting of N l ⫽ 1001 individual X waves characterized by uniformly distributed angular parameters 0.5° ⭐ ␰ l ⭐ 1.5°. Specifically, ␰ l ⫽ 0.001(l ⫺ 1) ⫹ 0.5°, where l acquires all integer values between 1 and 1001. The parameter a 0 in Eq. (2.3) determines the waist of the individual X waves as well as their spectral bandwidth.12,13 The latter two factors balance each other in such a way that a 0 does not influence the diffraction range z d . 1,12,13 For this reason, we choose a 0 ⫽ 10⫺6 m to be the same for all the individual X-wave components. The parameter z 1 is chosen arbitrarily to equal ⫺1 m. The focusing point z f ⫽ 3.4374 m and the corresponding value of t f is calculated by using Eq. (2.2). In Fig. 2, we show plots of the axial profile of the CXW pulse at ct ⫽ 1.0012, 2.3749, 4.4372, and 6.1559 m. In all numerical examples, we have chosen A(z l , ␰ l ) ⫽ 1. Therefore, the influence of this amplitude function is not considered in this work. One should note the initial spread of the CXW pulse at z ⫽ 0. At the focusing point z ⫽ z f ⫽ 3.4374 m, the focused pulse becomes very sharp and its peak amplitude equals 1000 times that of the initial CXW at z ⫽ 0. Finally, we stress that beyond z f the CXW pulse spread out again as the faster components overtake and move ahead of the slower ones. The focusing of the CXW pulse is further illustrated in the three-dimensional plots provided in Fig. 3 at (a) z ⫽ 0 and (b) z ⫽ 3.4374 m. For the former, we have plotted only the front half of the pulse because of difficulties with the resolution of an extended threedimensional plot. Behind the main X-shaped pulse extends an axial tail that ends with a smaller, inverted X-shaped pulse (compare the initial axial pulse in Fig. 2).

A 关 z f ⫺ 共 ct f /cos ␰ l 兲 , ␰ l 兴兵 a 0 ⫹ i 关共 z ⫺ z f 兲 cos ␰ l ⫺ c 共 t ⫺ t f 兲兴其 (共 ␳ sin ␰ l 兲 2 ⫹ 兵 a 0 ⫹ i 关共 z ⫺ z f 兲 cos ␰ l ⫺ c 共 t ⫺ t f 兲兴其 2 )3/2

册

.

(2.3)

The three-dimensional plot in Fig. 3(a) can be perceived as a representation of the excitation wave field on the source plane.

4. COMPOSITE X WAVES GENERATED BY A CIRCULAR APERTURE Fig. 1. Schematic representation of the temporal focusing scheme. X waves traveling at different speeds arrive simultaneously and add up at the focusing point z f .

The source-free CXW pulse illustrates in a straightforward manner the focusing scheme advocated by this work. However, it is well known that source-free X waves have infinite energy content, so finite-size sources

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Fig. 2. Behavior of a train of N l ⫽ 1001 source-free X waves designed to focus at z f ⫽ 3.4374 m. The parameters are a 0 ⫽ 10⫺6 m, z 1 ⫽ ⫺1 m and 0.5° ⭐ ␰ l ⭐ 1.5°. The plots show the axial profile of the CXW pulse at ct ⫽ 1.0012, 2.3749, 4.4372, and 6.1559 m.

cannot generate them. On the other hand, it has been shown that approximate X waves can be launched from finite sources.1,12,13 In what follows, we assume that CXW pulses are launched from a finite circular aperture of diameter D ⫽ 6 cm. The initial excitation of the aperture is a discretization of the waveform given in Eq. (2.3), and the radiated field is calculated by using the Rayleigh– Somerfeld I formula27 N

⌿ RSI共 0, z, t 兲 ⫽

兺

n⫽1

An 2␲Rn

关 ⫺⳵ z ⬘ ⌿ 共 ␳ ⬘ , z ⬘ ⫽ z 0 , t ⬘ 兲兴 .

(4.1)

Here, A n is the area of the annular section labeled n, N is the number of annular sections, ␳ n⬘ ⫽ n⌬ ␳ ⬘ is the radius of the nth annular section, and z is the point at which the field of the radiated pulse is observed. The distance from source points to an axial ( ␳ ⫽ 0) observation point is denoted by R n ⫽ 关 (n⌬ ␳ ⬘ ) 2 ⫹ (z ⫺ z 0 ) 2 兴 1/2. The square brackets in Eq. (4.1) indicate that the quantities enclosed are evaluated at the retarded time t ⬘ ⫽ t ⫺ R n /c. With the same parameters as in the case of the source-free X wave, we calculate the radiated field from a circular aperture situated at z 0 ⫽ 0. The axial profile of the radiated CXW pulse is plotted in Fig. 4 for

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z f ⫽ 3.4374 m at ct ⫽ 1.0012, 2.3749, 4.2654, and 6.1559 m. Notice that the pulse is focused at z ⫽ 0.95z f and that the peak amplitude of the CXW pulse is only 16 times greater than that of the initial excitation. This is because the focusing range z f is chosen to equal D/tan ␰0 ⬵ D/(2 tan ␰min), where ␰ 0 ⫽ 1° is the central value of the angular distribution and ␰ min ⫽ 0.5° is its minimum value. If we note that the individual diffraction ranges are given by z dl ⫽ D/(2 tan ␰l), it follows that all the individual X-wave components are propagating beyond their diffraction limit at the chosen value of z f . The fast roll-off in the amplitude of the individual X waves as they travel beyond their diffraction lengths z dl is, therefore, responsible for the decrease in the amplitude of the focused CXW pulse in comparison with that of the source-free CXW pulse. To clarify this issue, we provide in Fig. 5 plots of the axial profile of the CXW field for z f ⫽ 2.0624 m. Now, the designated focusing point lies in the middle between the minimum and maximum diffraction limits associated with the individual X waves. This is because the diffraction length changes between z dmin

Fig. 3. Surface plots of the time dependence of (a) the initial source-free CXW pulse at z ⫽ 0 and (b) the focused pulse at z f ⫽ 3.4374 m.


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⫽ 1.1456 m and z dmax ⫽ 3.4376 m as the apex angle ␰ acquires values between 1.5° and 0.5°, respectively. Fig. 5 shows axial profiles plotted for z f ⫽ 2.0624 m at ct ⫽ 1.0012, 2.0311, 3.0211, and 4.0935 m. For this case, the focusing occurs at a distance z ⫽ 0.98z f . This distance is closer to the designated focusing range z f than the z ⫽ 0.95z f value obtained for the CXW pulse considered in Fig. 4. The amplitude of the peak of the focused CXW pulse at z ⫽ 2.0212 m increases to about 80 times that of the initial excitation, but is still short of the 1000fold magnification obtained for the analogous source-free pulse. This maximum 1000-fold magnification is obtained when z f ⬍ z dmin ⫽ 1.1456, i.e., before any individual X wave component starts decaying. To clarify this point, consider the case when z f is chosen at a distance z ⫽ 2.0212 m. The plots in Fig. 6 illustrate that the focusing peak occurs at z ⫽ z f and the magnification increases to 220 times the initial excitation. Finally, we note that at z ⫽ 1.5z f , the CXW pulse in Fig. 4 does not spread out along the axial direction, as did the CXW pulse shown in Fig. 5. This is due to the fact that the peaks of the X waves propagate at superluminal speeds in the near-field range z ⬍ z d . Beyond this range, the velocity of the peaks asymptotically approaches the speed of light. Effectively, all individual X-wave components will be moving at the speed of light as they travel past the distance z dmax . In Fig. 5, individual X-wave components are still moving at different superluminal speeds after passing through the focusing point z f . This causes the spread observed in the CXW pulse at greater distances. To gain more insight into the decay behavior of CXW pulses, we provide in Fig. 7 logarithmic plots of the amplitudes of the peaks of such pulses versus the axial distance from the source. Here, we calculate the logarithms of the amplitudes of the peaks at various distances normalized with respect to the peak of the initial excitation. The plots illustrate the decay patterns of the peaks of the two CXW pulses considered in Figs. 4 and 5. As a reference for their performance, we have added a plot of the decay pattern of an individual X wave exhibiting the longest diffraction range z dmax ⫽ 3.4376 m (for ␰ ⫽ 0.5°). One should observe that the magnification in the focusing amplitude increases as we reduce z f . This is because more of the individual X-wave components contribute effectively to the focused peak of the CXW pulse. Furthermore, Fig. 7 illustrates that for a single X wave, the amplitude stays constant up to a distance z ⫽ z dmax ⫽ 3.4376 m. Beyond that distance, the amplitude of the peak of the pulse drops suddenly at a very fast rate before the decay slows down, and starts approaching asymptotically the anticipated 1/z rate at greater distances. In contradistinction, the roll-off in amplitude for z ⬎ z dmax is slower for CXW pulses when compared to a single X wave. The sudden drop in amplitude is softened significantly for z f ⫽ 2.0624 m, while it is not observed for z f ⫽ 3.4374 m. Finally, we would like to point out that a CXW pulse exhibiting higher magnification in the focusing region has a faster decay profile in the far-field region. Compared with the magnification of the amplitudes of the pulses in the focusing region, the two CXW pulses demonstrate a much slower decay rate in the far-field region than the single X wave. These advantages are achieved

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Fig. 4. Axial profile of the CXW pulse radiated from a circular aperture having diameter D ⫽ 6 cm. The pulse is designed to focus at z f ⫽ 3.4374 m. Plots show the axial profiles of the CXW pulse at ct ⫽ 1.0012, 2.3749, 4.2654, and 6.1559 m. The parameters characterizing the X waves are the same as in Fig. 2.

at the expense of an intricate excitation scheme of a source that consumes much higher energies.

5. CONCLUDING REMARKS It has been shown that highly focused CXW pulses can be achieved at predetermined distances from a source. The focusing technique makes use of the fact that the peaks of X waves characterized by different spectral angles ␰ travel at different speeds. A scheme has been described whereby one can arrange the initial times of excitation of the different individual X waves so that all of them arrive

simultaneously at a selected focusing point. For CXWs generated from a finite aperture, there are two factors limiting the focusing distance. First, individual X waves decay at a very fast rate once they pass their respective diffraction limits z dl ⫽ D/(2 tan ␰l). Second, the peak of each X-wave component travels initially at the superluminal velocity c/cos ␰l , which gradually decreases to c as the z dl limit is approached. These two factors prevent some of the X-wave components from arriving at the focusing point at the designated time, or from arriving there with appreciable amplitudes. The numerical examples examined in this work demon-

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strate that a high degree of focusing is attained at a point close to z f . In general, high focusing is achieved for distances smaller than z dmax . One should note that the magnification at the focusing point is reduced as z f is selected closer to z dmax . Another important behavior of CXWs is that they enter the usual 1/z decay region at closer distances than the slowest decaying individual Xwave component. This means that the sudden drop in the amplitude of an individual X wave can be avoided by using CXW pulses as an alternative. For applications requiring a pulse to arrive with a certain amplitude threshold at a certain distance from the source (e.g., the peak


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amplitude of the received pulse is required to be equal to that of the initial one), CXWs have major advantages over individual X waves. This can be seen by comparing the decay behavior of the X wave in Fig. 7 with that of the CXW pulse having z f ⫽ 3.4374 m. At the point where the peak of the CXW pulse has the same value as that of the initial excitation, the individual X wave has fallen to about 1/20 of the initial peak. At this stage, it is worthwhile pointing out that Song et al. have investigated the possibility of generating modified X waves for which the parameter ␰ ⫽ ␰ ( ␳ ) is varied as a function of the transverse radial distance of the source,28 and they have shown

Fig. 5. Axial profile of the CXW pulse radiated from a circular aperture having diameter D ⫽ 6 cm. The pulse is designed to focus at z f ⫽ 2.0624 m. Plots show the axial profiles of the CXW pulse at ct ⫽ 1.0012, 2.0311, 3.0211, and 4.0935 m. The parameters characterizing the X waves are the same as in Fig. 2.

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Fig. 6. Axial profile of the CXW pulse radiated from a circular aperture having diameter D ⫽ 6 cm. The pulse is designed to focus at z f ⫽ 2.0212 m. Plots show the axial profiles of the CXW pulse at ct ⫽ 1.0012 and 2.0311 m. The parameters characterizing the X waves are the same as in Fig. 2.

Fig. 7. Logarithmic plots of the normalized amplitudes of the peaks of the pulses shown in Figs. 4 and 5 versus the axial distance from the source. The decay patterns of the peaks of the two CXW pulses are compared to that of an individual X wave component exhibiting the longest diffraction range z dmax ⫽ 3.4376 m corresponding to ␰ 1 ⫽ 0.5°. The pulses are normalized with respect to their peak amplitude at the aperture.

that the resultant pulses have larger focusing depth and exhibit better lateral profiles than the original X waves. Their approach is different from that used in this paper. Whereas our method uses a superposition of many X waves having different ␰ values to excite a CXW pulse, their approach employs a single X-wave excitation characterized by a parameter ␰ that varies with the radius of the source. The method described in this work could be perceived as a pulse-shaping technique that facilitates the tailoring of the initial pulsed excitations to suit certain applications. Effectively, this focusing method is a pulse-

shaping approach in the time domain because it depends primarily on our understanding of the properties of the X wave solution in configuration space. The derivation of the spectral content of the excitation has been extensively discussed in the literature.12,13 Another advantage of using a superposition of X waves to tailor the excitation of a focused CXW pulse is that each individual X wave can be dealt with as an angular superposition over pulsed plane waves.5,17 This allows one to use available pulsed-planewave techniques for the analysis of diffraction and refraction problems involving X waves.13,20 This approach has proven to be extremely valuable in dealing with diffraction of X waves from objects, as well as their transmission and reflection from interfaces or multilayered media. For studies of diffraction, scattering, or refraction of CXWs, the aforementioned approach applies to individual X-wave components.17–20 Consequently, the diffracted, scattered, or refracted fields due to an incident CXW pulse can be calculated by summing up contributions from the constituent X waves. Corresponding author A. M. Shaarawi may be reached by e-mail at [email protected].

*On leave from the Department of Engineering Physics and Mathematics, Faculty of Engineering, Cairo University, Giza 12211, Egypt.

REFERENCES 1.

2. 3.

J-Y. Lu and J. F. Greenleaf, ‘‘Nondiffracting X waves—exact solutions to free-space scalar wave equation and their finite aperture realization,’’ IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19–31 (1992). J-Y. Lu and J. F. Greenleaf, ‘‘Experimental verification of nondiffracting X waves,’’ IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 441–446 (1992). I. M. Besieris, M. Abdel-Rahman, A. M. Shaarawi, and A. A.

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4. 5. 6. 7. 8.

9. 10. 11. 12.

13.

14.

15.

Chatzipetros, ‘‘Two fundamental representations of localized pulse solutions of the scalar wave equation,’’ Prog. Electromagn. Res. 19, 1–48 (1998). R. W. Ziolkowski, I. M. Besieris, and A. M. Shaarawi, ‘‘Aperture realizations of the exact solutions to homogeneouswave equations,’’ J. Opt. Soc. Am. A 10, 75–87 (1993). P. Saari and K. Reivelt, ‘‘Evidence of X-shaped propagationinvariant localized light waves,’’ Phys. Rev. Lett. 79, 4135– 4138 (1997). E. Recami, ‘‘On the localized ‘X-shaped’ superluminal solutions to Maxwell’s equations,’’ Physica A 252, 586–610 (1998). D. Mugnai, A. Ranfagni, and R. Ruggeri, ‘‘Observation of superluminal behaviors in wave propagation,’’ Phys. Rev. Lett. 84, 4830–4833 (2000). M. Zamboni-Rached, E. Recami, and H. E. HernandezFigueroa, ‘‘New localized superluminal solutions to the wave equations with finite total energies and arbitrary frequencies,’’ Eur. Phys. J. D 21, 217–228 (2002). J. Fagerholm, A. T. Friberg, J. Huttunen, D. P. Morgan, and M. M. Salomaa, ‘‘Angular-spectrum representation of nondiffracting X waves,’’ Phys. Rev. E 54, 4347–4352 (1996). J. Durnin, ‘‘Exact solutions for nondiffracting beams. I. The scalar theory,’’ J. Opt. Soc. Am. A 4, 651–654 (1987). J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, ‘‘Diffraction free beams,’’ Phys. Rev. Lett. 58, 1499–1501 (1987). A. A. Chatzipetros, A. M. Shaarawi, I. M. Besieris, and M. A. Abdel-Rahman, ‘‘Aperture synthesis of time-limited X-waves and analysis of their propagation characteristics,’’ J. Acoust. Soc. Am. 103, 2287–2295 (1998). A. M. Attiya, E. A. El-Diwany, and A. M. Shaarawi, ‘‘Generation of an Approximate TE Electromagnetic X-wave in free space using Huygens’s aperture,’’ Proceedings of the 16th National Radio Science Conference, NRSC’99 (Ain Shams University, Cairo, Egypt, 1999), pp. B3.1–11. A. M. Shaarawi, I. M. Besieris, A. M. Attiya, and E. A. ElDiwany, ‘‘Acoustic X-wave reflection and transmission at a planar interface: spectral analysis,’’ J. Acoust. Soc. Am. 107, 70–86 (2000). J-Y. Lu and J. F. Greenleaf, ‘‘Diffraction-limited beams and their applications for ultrasonic imaging and tissue characterization,’’ in New Developments in Ultrasonic Transducers and Transducer Systems, F. L. Lizzi, ed., Proc. SPIE 1733, 92–119 (1992).

Vol. 20, No. 8 / August 2003 / J. Opt. Soc. Am. A 16.

17.

18.

19.

20.

21. 22. 23. 24. 25. 26. 27. 28.

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J-Y. Lu, M. Fatemi, and J. F. Greenleaf, ‘‘Pulsed-echo imaging with X wave,’’ in Acoustical Imaging, P. Tortoli and L. Masotti, eds., (Plenum, New York, 1996), Vol. 22, pp. 191– 196. A. M. Attiya, E. A. El-Diwany, A. M. Shaarawi, and I. M. Besieris, ‘‘Reflection and transmission of X-waves in the presence of planarly layered media: the pulsed plane wave representation,’’ Prog. Electromagn. Res. 30, 191–211 (2000). A. M. Shaarawi, I. M. Besieris, A. M. Attiya, and E. A. ElDiwany, ‘‘Reflection and transmission of an electromagnetic X-wave incident on a planar air-dielectric interface: spectral analysis,’’ Prog. Electromagn. Res. 30, 213–250 (2000). A. M. Attiya, E. A. El-Diwany, and A. M. Shaarawi, ‘‘Transmission and reflection of TE electromagnetic X-wave normally incident on a lossy dispersive half-space,’’ Proceedings of the 17th National Radio Science Conference, NRSC’2000 (Minufiya University, Minufiya, Egypt, 2000), pp. B11.1–12. A. M. Attiya, ‘‘Transverse (TE) Electromagnetic X-waves: propagation, scattering, diffraction and generation problems,’’ Ph.D. thesis (Cairo University, Cairo, Egypt, 2001). ˜ najalg and P. Saari, ‘‘Suppression of temporal spread H. So of ultrashort pulses in dispersive media by Bessel beam generators,’’ Opt. Lett. 21, 1162–1164 (1996). M. A. Porras and I. Gonzalo, ‘‘Control of temporal characteristics of Bessel-X pulses in dispersive media,’’ Opt. Commun. 217, 257–64 (2002). A. M. Shaarawi and I. M. Besieris, ‘‘On the superluminal propagation of X-shaped localized waves,’’ J. Phys. A 33, 7227–7254 (2000). J. Y. Lu, ‘‘An X-wave transform,’’ IEEE Trans. Ultrason. Ferroelectr. Freq. Control 47, 1472–1481 (2000). J. Salo, A. T. Friberg, and M. Salomaa, ‘‘Orthogonal X-waves,’’ J. Phys. A 34, 9319–9327 (2001). J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, ‘‘Unified description of nondiffracting X and Y waves,’’ Phys. Rev. E 62, 4261–4275 (2000). J. W. Goodman, Introduction to Fourier Optics (McGrawHill, New York, 1968). T. K. Song, J-Y. Lu, and J. F. Greenleaf, ‘‘Modified X waves with improved field properties,’’ Ultrason. Imaging 15, 36–47 (1993).