. INTRODUCTIONwenty years ago, Wu1 demonstrated theoretically thatlectromagnetic missile solutions exhibiting slow energyecay exist. Such solutions have an energy decay ratelower than 1/R2 over an unlimited range as the distancerom a finite-size source approaches infinity; i.e., R→�.ubsequent experiments did not substantiate claims ofnlimited slow decay.2,3 In this paper, we study the char-cteristics of several known localized wave (LW) solutionshat seem to have a missilelike behavior. We consider LWolutions such as the modified power spectrum (MPS)ulse and the splash pulse4–6 (SP). It will be shown that,lthough the MPS pulse exhibits a slow missilelike decayn the source-free case, such behavior is not observedhen it is generated from a finite-size aperture. In con-
radistinction, an aperture-generated SP has a quasi-issile decay behavior over an extended intermediate
ange between the near- and the far-field regions; theeak of the SP decays at a slow rate before undergoing thesual 1/z roll-off along the z direction. The reason for this
s that the SP is not as tightly localized as the MPS pulse.o support this view, we introduce a new localized pulse,amely, the double-exponential (DEX) pulse, which hasissilelike decay properties comparable with those of the
P. The DEX pulse has the distinguishing feature that itonsists of a highly focused central pulse incorporated in
relatively large background field. Two distinct situa-ions will be considered and studied: specifically, the be-avior of the new LW in free space and its decay patternhen it is excited from a finite-size aperture.LW are solutions of the wave equation that are tempo-
ally localized along the direction of propagation and spa-ially confined within a finite waist. The introduction ofhe first LW, namely, the focus wave mode (FWM), byrittingham7 motivated several researchers to investi-ate some of the propagation properties of suchulses.4–6,8–12 The original FWM is a nonseparable elec-romagnetic wave solution to the homogeneous Maxwell’squations that has the form of a three-dimensional (3-D)ocalized pulse. Brittingham demonstrated that such anlectromagnetic transverse electric pulse is continuousnd nonsingular, propagates at the light velocity intraight lines, and does not disperse for all time. Later on,calar-valued FWM solutions were derived for the scalarave equation.2,6,8,9,12 Soon after its introduction, it was
ealized that the FWM pulse remains focused for all times it propagates, say, along the z axis, because it hasnfinite-energy content.9,13 To construct finite-energy LW,
006 Optical Society of America
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ov
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2040 J. Opt. Soc. Am. A/Vol. 23, No. 8 /August 2006 Shaarawi et al.
iolkowski and co-workers4,5 suggested using a superpo-ition of the FWM,
���r,t� =1
4��a1 + i��e−��2/�a1+i��ei��, �1�
ver a weighted function (spectrum) of the free variable �,iz.,
��r,t� =1
4��a1 + i���0
�
d�F���e−�s, �2�
here
s =�2
a1 + i�− i�; � = z − ct, � = z + ct. �3�
his superposition yields a large number of finite-energyW solutions under the condition that F����−1/2 is square
ntegrable.4,5 The MPS pulse is derived through thehoice of the spectrum
here Hs is the Heaviside unit-step function. The evalu-tion of the Laplace-type integration in Eq. (2) yields
��r,t� =1
4��a1 + i��
e−bs/p
�a2 + s/p�q . �5�
The free parameters appearing in the solution may bedjusted to obtain the desired pulsed structure. For prop-rly tuned parameters, the MPS pulse has a Gaussianransverse profile near the peak of the field ��=0,z=ct�.or q=1, the peak of the MPS pulse undergoes only localariations and does not decay up to the distance zpa2 /2. After that range, the peak decays at a very fast
ate before adjusting to the usual 1/z behavior. Theausal generation of the MPS pulse from a finite arrayas been investigated in several publications, and the lo-alization properties of this ultrawide-band LW pulse inomparison with more common cw radiation have beenssessed.5,10,14,15 The generation of LW solutions has beenemonstrated recently in the femtosecond optical domainnd is still attracting considerable attention.16–21 A vari-ty of FWM-like LW pulses can be derived using spectraimilar to the one given in Eq. (4). As an example, thehoice of the spectrum
F��� = �q−1 e−�a2 �6�
eads to Ziolkowski’s SP.4 The SP can also be considered apecial case of the MPS pulse under the parameter re-trictions b=0 and p=1.
In addition to the decay rates of the MPS pulse and theP, we shall study the behavior of the DEX pulse, a newW derived from a double-exponential spectrum. Numeri-al simulations will be used to demonstrate that the newulse has a peak decay rate slower than 1/z, thus exhib-ting a missilelike behavior over an extended intermedi-te region before the decay rate changes to the usual 1/zne. In Section 2, we introduce the DEX pulse. Its missile-ike decay behavior is investigated in Sections 3 and 4. Inhe former we consider the decay of the source-free solu-ions, and in the latter we study the decay of pulses gen-
rated from finite apertures. A comparison of the decayatterns of various LW missiles, including the MPS pulse,he SP, and the DEX pulse, is carried out in Section 5. Fi-ally, the decay patterns of LW missile solutions gener-ted from finite-size sources are quantified in Section 6.oncluding remarks and discussion are provided in Sec-
ion 7.
. DOUBLE-EXPONENTIAL PULSEnew LW pulse can be derived by substituting the spec-
rum
F��� = 4��q−1 sinh�a2��e−�a3, q − 1, �7�
n the Ziolkowski superposition given in Eq. (2). Thishoice is termed the DEX spectrum because it can be re-ritten as
F��� = 2��q−1�e−��a3−a2� − e−��a3+a2��. �8�
he substitution of F��� in Eq. (2), followed by the inte-ration over �, yields the DEX pulse
���,�,�� =��q�
2�a1 + i���1
� �2
a1 + i�− i� + a3 − a2q
−1
� �2
a1 + i�− i� + a3 + a2q , �9�
here the parameter a1 characterizes the bandwidth ofhe pulse, the parameters a2 and a3 characterize theiffraction-free limit, and q controls the decay rate of theulse in the intermediate range between the near- andhe far-field ranges. Essentially, this solution is the differ-nce of two SP.
It is known that all FWM-based LW solutions are com-osed of causal and acausal components.6,22 It has beenhown that, for �a11, the acausal components of theWM are negligible. Similar conditions have been identi-ed for other LW solutions.6,22 The DEX pulse has pre-ominantly causal components if a3�a2, while keeping a1ery small. In this case, the solution ��r , t� has negligibleackward-going spectral components, and it behaves as aocalized pulse moving in the positive z direction.
. DECAY PATTERN OF SOURCE-FREEOCALIZED WAVE MISSILE SOLUTIONShe decay behavior of several source-free LW solutions isonsidered in this section. It is shown that the MPS,plash, and DEX free-space solutions can all have missile-ike decay rates for specific choices of parameters. Fromq. (5), it is straightforward to realize that the peak of thePS pulse (�=0 and �=0) decays as z−q for z�pa2 /2. For
ractional values of q, the peak of the MPS decays at aate slower than 1/z. Furthermore, the SP is a specialase of the MPS pulse; specifically, it follows from the lat-er by substituting b=0 and p=1 in Eq. (5). Therefore, theP has the same missilelike decay as the MPS solution.
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elttdFust
Ff
Shaarawi et al. Vol. 23, No. 8 /August 2006 /J. Opt. Soc. Am. A 2041
At this stage, it should be noted that the DEX pulse issuperposition of two SPs having the following forms:
�sp�−���,�,�� =
1
2
��q�
�a1 + i��� �2
�a1 + i��− i� + a3 − a2−q
,
�10�
�sp�+���,�,�� =
1
2
��q�
�a1 + i��� �2
�a1 + i��− i� + a3 + a2−q
.
�11�
he parameters a2 and a3 characterize the diffraction-ree lengths, and q controls the decay rate of the pulses inhe intermediate region between the near- and the far-eld ranges. The SPs given in Eqs. (10) and (11) are ob-ained from the MPS pulse in Eq. (5) by substituting inhe latter b=0 and pa2→a3±a2. It follows then that, forractional values of q�1, the free-space SPs given aboveave missilelike decays. It can be easily shown that, for� �a3+a2� /2, the amplitude of the peak of a SP decayssymptotically as �sp� iq���q� /2q+1a1�z−q. To illustratehe missilelike decay behavior, the decay pattern of theeak of the SP given in Eq. (10) is plotted in Fig. 1 for dif-erent q�1 values. The pulse parameters are chosen sohat the decay patterns are comparable with later plotsrovided for the DEX pulse with a3−a2=990 cm and a12 10−5 cm. The free-space MPS pulse has exactly theame decay patterns as the ones shown in Fig. 1.
Fig. 1. (Color online) Missilelike decay of the SP.
ig. 2. (Color online) Decay rate of the peak of the DEX pulseor a −a =990 cm and different values of q.
3 2
In contrast to the MPS pulse and SP, the slow decay ofhe DEX pulse is not that obvious. To recognize the mis-ilelike decay rate of the DEX pulse, we change q for dif-erent values of a3−a2. The effect of choosing q=−0.9,0.5, −0.1, +0.1, +0.5, and +0.9 for a3−a2=990 cm on theecay rate is illustrated in Fig. 2. This figure shows thatlower decay rates are obtained for smaller values of q.here are two features that should be identified in thelots shown in Fig. 2. The first is that we obtain decayaster (slower) than 1/z for positive (negative) values of q.he second feature is that the difference a3−a2 deter-ines the distance after which the pulse starts decayingith slope z−�1+q�. The larger this difference, the larger is
he distance at which the decay starts increasing. Theseonclusions can be reached by examining the asymptoticehavior of the DEX pulse given in Eq. (9). Consider, inarticular, the axial ��=0� amplitude of the peak ��=0� ofhe DEX pulse, viz.,
��� = 0,� = 0,� = 2z� =��q�
2a1� 1
�− i2z + a3 − a2�q
−1
�− i2z + a3 + a2�q . �12�
he decay of the pulse starts when z �a3−a2� /2. Theausality condition, a3�a2, results in the relationship3−a2�a3+a2. Therefore, for z� �a3+a2� /2, the axial de-ay of the pulse peak has the leading term
��� = 0,� = 0,� = 2z� �iq��q�a3
2q+1a1
�− iq�
zq+1 . �13�
he above expression shows that the DEX pulse has aissilelike decay for −1�q�0.Three-dimensional plots of the DEX pulse for differentvalues at z=0 are shown in Figs. 3(a)–3(f). These plots
how the shape of the DEX pulse for a1=2 10−5 cm and=−0.9, −0.5, −0.1, +0.1, +0.5, and +0.9, with a3−a2 equalo 9 cm. The longitudinal �t−z /c� ranges are the same forll plots. In all figures, the ranges of the transverse vari-ble ��� are kept the same for all cases except for q=0.9 and +0.9. This is done because the transverse exten-ion of the DEX pulse increases for greater a3−a2 valuesnd larger absolute values of q. This transverse localiza-ion behavior of the DEX pulse can be inferred by compar-ng plots for the same value of the difference a3−a2 butifferent values of the parameter q. For example, whenhe plots in Figs. 3(a)–3(f) are compared, it can be seenhat the narrower pulses are the ones having smaller val-es of q . One can also see that negative values of the pa-ameter q result in pulses with larger transverse dimen-ions than those for positive q values.
Surface plots of the source-free DEX solution at differ-nt distances, z=1000, 100 000, and 10 000 000 cm, are il-ustrated in Figs. 4(a)–4(c). It is shown in Fig. 4(a) thathe field at distance z=1000 is rapidly distorted with posi-ive q values. For negative values of q, however, the fieldistribution of the DEX pulse remains unchanged. Inigs. 4(b) and 4(c), the field patterns with positive q val-es lose their localization around �=0, and their maximahift away from the axis of propagation. In contradistinc-ion, the peaks of the DEX pulses with negative q values
odF(n�fpcts
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Fq
2042 J. Opt. Soc. Am. A/Vol. 23, No. 8 /August 2006 Shaarawi et al.
ccur at �=0, and the pulses stay localized for very largeistances, even though their amplitudes are decreasing.inally, one should note that the expressions given in Eqs.
10) and (11) represent divergent (convergent) SPs foregative (positive) values of q. Surface plots of �sp
�−� and
sp�+� at z=1000 cm are shown in Fig. 5. The divergent pair
or q=−0.5 leading to the missilelike decay of the DEXulse is displayed in Fig. 5(a), while Fig. 5(b) shows theonvergent pair for q= +0.5. The differences betweenhese two pairs produce DEX pulses similar to the oneshown in Fig. 4(a).
ig. 3. (Color online) Field pattern of the DEX pulse for a3−a2=� �0.9.
. MISSILE LOCALIZED WAVE PULSESADIATED FROM FINITE APERTURES
he field generated by a finite-size source into the z0alf-space is usually calculated by applying the Huygensonstruction to the initial excitation of the source aper-ure. For a planar source placed at z�=0, the field at aistance R and time t inside the wavefront surfacebeing zero outside such surface) is given by theollowing integration over the area of the finiteperture:
Shaarawi et al. Vol. 23, No. 8 /August 2006 /J. Opt. Soc. Am. A 2043
���,z,t� =1
2��
0
2�
d���0
W0
d����
R�− �z����,z� = 0,t��
+z
R2����,z� = 0,t��
z
ig. 4. (Color online) DEX field pattern in free space for10,000,000 cm. For the figures on the left, q=−0.5, while the fi
+Rc
�t�����,z� = 0,t��t�=t−R/c
. �14a�
ere, R= ���2+�2−2��� cos ��+z2�1/2 and W0 is the aper-ure radius. The primed coordinates refer to source pointsn the aperture, while the unprimed ones refer to the ob-ervation points in the z0 half-space. Using the source-ree DEX pulse given in Eq. (9) as the excitation field inq. (14a), we can calculate the axial radiated field by us-
ng the discrete version of the Huygens formula, viz.,
=990 cm at distance (a) z=1000 cm, (b) z=100,000 cm, (c) zon the right have q= +0.5.
a3−a2
HataTpnct
==t1
wttefpl
tplctocs�poia
Fv
2044 J. Opt. Soc. Am. A/Vol. 23, No. 8 /August 2006 Shaarawi et al.
�H�0,z,t� = − ��=1
N�
�n=1
N An�
2�Rn�
���z��� − ��ct���z
Rn�
− ���z
Rn�2 � .
�14b�
ere, An� is the area of the �th angular section of the nthnnulus, N is the total number of annular sections, N� ishe total number of angular sections, and z is the distancet which the field of the radiated pulse is reconstructed.he distance from the source point to the observationoint is Rn�= ��n����2+�2−2��n����cos������+z2�1/2. Forumerical computations, the time history is calculated atertain instances expressed as the following discrete re-arded times: tn�,J� = tJ−Rn� /c.
The axial radiated field from an aperture of radius W010 cm is shown in Figs. 6(a)–6(c) for q=−0.5, a3−a2990 cm, and a1=2 10−5 cm. The plots show the shape of
he radiated pulse at distances 1000, 100 000, and
ig. 5. (Color online) SP with a3=105 cm, a1=2 10−5 cm, and aergent pairs.
0 000 000 cm. Comparing the aperture-generated fields
ith the corresponding free-space pulses, one can see thathe two pulses have the same shapes. It is only at the far-hest distance [cf. Fig. 6(c)] that one detects some differ-nce in shape between the aperture-generated and theree-space pulses. If one compares Figs. 6(c) and 4(c), theeak in the former appears to be smaller than that in theatter.
To gain a better understanding of the decay behavior ofhe DEX pulse, we compare in Fig. 7 the decay rate of theeak of the free-space DEX field with that of DEX pulsesaunched from apertures having different radii; specifi-ally, W0=1, 10, and 100 cm. The plots in Fig. 7 illustratehat, as the aperture increases in size, the decay patternf the generated DEX pulse resembles the source-free-ase pattern for farther distances from the source. Ithould also be noted that all fields start their decay at z�a3−a2�, similarly to the corresponding source-free DEX
ulses. However, unlike the source-free pulse, the peaksf the aperture-generated pulses exhibit a 1/z decay raten the far-field region. Before acquiring the 1/z decay, theperture-generated pulses go through an intermediate
04 cm at distance z=1000 cm. (a) The divergent and (b) the con-
2=9 1
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tprptpwdpiibac tt(
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Fw=
Shaarawi et al. Vol. 23, No. 8 /August 2006 /J. Opt. Soc. Am. A 2045
ange where they exhibit a slow, missilelike decay z−�1+q�.his intermediate missilelike decay range extends to
arger distances as the size of the aperture increases.
ig. 6. (Color online) Radiated DEX field from a finite apertureith radius=10 cm at (a) z=1000 cm, (b) z=100,000 cm, (c) z10,000,000 cm. The initial excitation has q=−0.5, a1=210−5 cm, and a3−a2=990 cm.
. COMPARISON OF THE DOUBLE-XPONENTIAL PULSE AND OTHEROCALIZED WAVESe have shown (cf. Fig. 1) that the free-space SP can ex-
ibit a missilelike slow decay. The same applies to thePS pulse given in Eq. (5), which reduces to the SP when
=0. The peaks of both LW pulses have missilelike decaysor q�1 values. In this section, we compare, first, the de-ay patterns of the DEX and the MPS pulses generatedrom finite-size apertures. Subsequently, we illustrate theossibility of having DEX pulses that behave differentlyrom SPs of comparable transverse waists and spectra. Torovide a quantitative comparison between the DEX andhe MPS pulses, both must have the same waist on theperture plane �z=0�, and their initial fields must excitehe same aperture size. For the MPS pulse, the waist ofts focused region can be determined from the followingelation: Rf=�pa1 /b. For p=648 105, b=104 cm−1, a1=210−5 cm, and a2=0.01 cm, the waist of the focused re-
ion of the MPS pulse equals 0.36 cm. This is comparableith the radius of the DEX pulse studied in Section 4, us-
ng a3−a2=990 cm and a1=2 10−5 cm. Since the param-ter a1 [cf. Eq. (20)] determines the maximum frequencyf both pulses, choosing the same value of this parameteror the two pulses ensures the same frequency content.
Using the pulses given in Eqs. (5) and (9) as the exci-ation fields of a 100 cm radius aperture, we calculate theeak amplitudes of the generated DEX and MPS pulses,espectively. The results are shown in Fig. 8, where it ap-ears that the DEX pulse starts decaying at a shorter dis-ance from the aperture ��1000 cm�. However, the MPSulse decays very rapidly around 106 cm after holding outithout any decay for this extended distance. This rapidrop in the amplitude of the MPS pulse reduces its am-litude to about 10−4 times that of the DEX pulse, whichs still undergoing a slow missilelike decay. We have alsoncluded the decay pattern of the SP derived by setting=0 in Eq. (5) and using the same parameters chosenbove. The decay pattern of the peak of the DEX pulse isalculated for q=−0.5, a3−a2=990 cm, and at a1=210−5 cm, which are the parameters used in earlier sec-
ions. For the SP to have a decay pattern comparable withhat of the DEX field, we have substituted q=0.5 in Eq.5). One should notice that the three MPS pulses evalu-
ig. 7. (Color online) Amplitude decay rate of the source-freeEX pulse and the aperture-generated ones. The DEX pulsesave q=−0.5, a =2 10−5 cm, and a −a =990 cm.
1 3 2
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2046 J. Opt. Soc. Am. A/Vol. 23, No. 8 /August 2006 Shaarawi et al.
ted for q=0.1, 0.5, and 1.0 have the same decay patterns.t is also clear that none of the three MPS pulses acquireshe slow missilelike decay of the SP and DEX pulse, de-pite the fact their free-space equivalents exhibit slow de-ays proportional to z−q. This behavior is due to the tightransverse localization of the MPS pulse. As can be in-erred from Eq. (5), the transverse roll-off of the MPSulse has an exponential dependence, whereas the DEXulse and SP have algebraic dependences. This points outpossible role being played by the wings of the DEX pulse
n producing the missilelike decay.Surface plots of the DEX and MPS pulses generated
rom a 100 cm aperture are provided in Fig. 9. The fasteterioration of the MPS pulse relative to the DEX pulses clear at the distance z=107 cm. At z=105 cm, bothulses are well localized even though the DEX pulse hasost some amplitude. The decay patterns shown in Fig. 8emonstrate that the DEX pulse and SP have similar de-ay behaviors. Recalling that the DEX pulse is a combi-ation of two SPs, one may wonder whether there coulde any differences in the decay patterns of the two solu-ions. We try to demonstrate in Fig. 10 that with a specialhoice of parameters we can achieve noticeable differ-nces between the decay patterns of the two pulses. Inree space, the DEX pattern resulting from q=−0.1 isimilar to that of a SP with q=0.9 as long as the values of1, a2, and a3 are the same. We choose, next, the values ofhe parameters a2 and a3 to take very large values of 9104 and 105 cm, respectively, together with a1=2 10−5.he differences between the decays of the two pulses arelearly shown in Figs. 10(a)–10(c), where surface plots ofoth pulses are provided at various distances from the ap-rture. The decay of the peaks of both pulses with dis-ance is provided in Fig. 11. It is seen from the plots thathe DEX pulse preserves its shape and amplitude at far-her distances better than the SP.
. DIFFRACTION LIMITS OF THEOCALIZED WAVE MISSILE PULSES
n this section, we shall try to explain the quasi-missileecay behavior and to quantify the three decay ranges,amely, the diffraction-free range, the missile-decayange, and the far-field range. The diffraction-free rangef LW is determined by a formula introduced by Hafizind Sprangle,23 which can be expressed as follows:
ig. 8. (Color online) Comparison of the decay of the aperture-enerated DEX and MPS pulses.
ZLW =1
2c�maxRfW0. �15�
ere, W0 is the aperture radius and Rf is the focused ra-ius of the pulse at z=0 and �max is the maximum effec-ive angular frequency. When the highly focused centralart of the excitation field of the LW pulse has a radiushat equals that of the aperture �W0=Rf�, Eq. (15) reduceso
ZRW =1
2c�maxW0
2, �16�
hich is the usual Rayleigh diffraction limit. In contrast,n ordinary pulse does not have the spatiotemporal cou-ling that characterizes the LW wave fields. Therefore,or a conventional pulse having transverse width Rf andxcited from a large aperture �W0�Rf�, the Rayleigh dis-ance becomes
ZRf =1
2c�maxRf
2. �17�
t can be easily seen that ZRf�ZLW�ZRW. Comparing theiffraction limits given in Eqs. (15) and (17), one can seehe advantage of the generation of highly focused LWulses compared with conventional pulses of the same fo-used waist. The spatiotemporal coupling of a LW excita-ion requires that different elements of the source be ex-ited at specific times by special waveforms in a coherentashion. The resulting diffraction length of a LW pulse isarger than the Rayleigh length associated with the fo-used portion of the field24 but smaller than the diffrac-ion length of the whole aperture.
To characterize the diffraction limit of the DEX pulse,e need to calculate the maximum frequency of the DEX
pectrum. Evaluating the temporal Fourier transform
t is clear that the value of the parameter a1 determineshe maximum frequency of the pulse because of the pres-nce of the exponential term exp�−�� /c�a1�. In the follow-ng, we choose the maximum frequency at which the spec-rum drops to e−4 of its maximum value. Applying thisriterion, we obtain
�max =4c
a1. �20�
or a1=2 10−5 cm, the parameter value used in previousxamples, it follows that �max=6 106 rad/ns.
As discussed in earlier sections, the decay of the DEXulse is divided into three regions. The first one is theiffraction-free region for z� �a3−a2� /2, where the pulseemains unaltered. In the second region, corresponding to �a −a � /2, the pulse decays as z−�1+q�. The farthest
3 2
ld1otTsttp
ppdc
fe1ma
Fz
Shaarawi et al. Vol. 23, No. 8 /August 2006 /J. Opt. Soc. Am. A 2047
imit for this missile-decay range is simply the Rayleighistance ZRW. For zZRW, the pulse undergoes the usual/z decay. To determine the influence of the aperture sizen the decay rate, we simulate the decay behavior forhree circular apertures having radii 1, 10, and 100 cm.he parameters a1, a2, a3, and q are chosen to yield theame DEX pulse considered in Figs. 6(a)–6(c). As men-ioned above, the maximum frequency of the DEX spec-rum is �max=6 106 rad/ns. The radius of the focusedart of the pulse is determined from the 3-D plot of the
ig. 9. (Color online) Comparison of the 3-D surface plots of the=1000 cm, (b) z=100,000 cm, (c) z=10,000,000 cm.
ower distribution of the DEX pulse � 2 at the aperturelane �z=0�, which is provided in Fig. 12. The focused ra-ius of the DEX excitation field is equal to Rf=0.36 cm,orresponding to the 1/e2 point.
The diffraction limits ZRW, ZLW, and ZRF are calculatedor apertures having radii 1, 10, and 100 cm. The initialxcitation of the aperture is the DEX pulse given in Fig.2 having �max=6 106 rad/ns and Rf=0.36 cm. The esti-ated values of the three diffraction patterns are evalu-
ted using Eqs. (15)–(17) and appear in Table 1. To be
and MPS pulses excited from a 100 cm aperture at distances (a)
DEX
acptsptwcs
dfaLhpLfigt
F
2048 J. Opt. Soc. Am. A/Vol. 23, No. 8 /August 2006 Shaarawi et al.
ble to judge the accuracy of the diffraction limits, weompare the decay patterns for the peaks of the DEXulses generated by the three apertures (cf. Fig. 7) withhe diffraction ranges in Table 1. Unlike most of the LWstudied in the literature, the decay of the peak of the DEXulse is not abrupt. This point was demonstrated in Sec-ion 5, where the decay of the DEX pulse was comparedith that of the MPS pulse. As the aperture radius in-
reases (cf. Fig. 7), the intermediate range exhibiting thelow missilelike decay becomes more pronounced, and the
ig. 10. (Color online) Comparison of the 3-D surface plots o104 cm at distances (a) z=1000 cm, (b) z=5000 cm, (c) z=100,0
ecay pattern becomes more similar to that of the source-ree case over distances farther away from the source. Inll cases, the 1/z decay starts at approximately z�ZRW. AW missile, such as the DEX pulse, seems to exhibit a be-avior intermediate between LW and conventional cwulses. The main advantage of the DEX pulse over otherW pulses is that it has a smooth transition into the far-eld range through the intermediate missile-decay re-ion. Finally, one should note at ZRf is the same for thehree pulses, and that the knee characterizing the transi-
DEX pulse and SP with a3=105 cm, a1=2 10−5 cm, and a2=9
f the00 cm.
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Ft=
Fpr
Shaarawi et al. Vol. 23, No. 8 /August 2006 /J. Opt. Soc. Am. A 2049
ion into the missile-decay region occurs at the same dis-ance for the three pulses. Even though one might beempted to make a connection between these two facts,here is a big difference in the numerical values of bothoints.
Table 1. Comparison of Diffraction Ranges forDifferent Aperture Sizes
iffraction LimitW0=1 cm
(cm)W0=10 cm
(cm)W0=100 cm
(cm)
RW= 12c�maxW0
2 105 107 109
LW= 12c�maxRfW0 3.6 104 3.6 105 3.6 106
Rf=12c�maxRf
2 1.3 104 1.3 104 1.3 104
ig. 11. (Color online) Decay rate of the DEX pulse �q−0.1� and SP �q= +0.9� with a3=105 cm, a1=2 10−5 cm, and2=9 104 cm.
ig. 12. (Color online) Power distribution of the source-freeEX solution at z=0.
ig. 13. (Color online) Effect of the parameter a1 on the decay ofhe peak of the DEX pulse for q=−0.5, a2=10 cm, and a31000 cm.
As explained above, the parameter a1 in the DEX pulseetermines the maximum frequency �max=4c /a1. Beforettempting to provide an explanation for the decay pat-ern of the DEX pulse, we demonstrate in Fig. 13 that therst knee occurring around 1000 cm does not depend on1. The plots show the decay of DEX pulses having q=0.5 and a3−a2=990 cm and generated from a 100 cm ap-rture. Decreasing a1 extends the distance at which theignal enters the far-field region.
In the following, we shall attempt to show that the mis-ilelike decay of the DEX pulse can be explained as a syn-hesis of decay patterns of two distinct parts of the pulse.ur approach is rooted in the observation that, unlikether LW pulses, a background field of relatively high am-litude surrounds the peak of the DEX pulse. We shallemonstrate that the slow missilelike decay can be ex-lained as a superposition of the decay of two distinctave fields. The first one represents the focused central
egion of the DEX pulse, and the second is the surround-ng background field. It is shown in Fig. 14 that the am-litude of the background wings becomes larger as thealue of q decreases. Furthermore, a longer missilelikeecay range results from the choice of a smaller q param-ter. This observation points to the possibility that theings of the DEX wave field affect the decay rate of theeak of the focused region.
ig. 14. (Color online) Effect of the parameter q on the DEXulse for a1=2 10−5 cm, a2=10 cm, and a3=1000 cm. (a) Decayate and (b) transverse dependence on the aperture plane.
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2050 J. Opt. Soc. Am. A/Vol. 23, No. 8 /August 2006 Shaarawi et al.
To show that the missilelike decay pattern of the DEXulse is due to a combination of the decays of the centralocused field and the background wings, we calculate theecay of the axial amplitude of the two parts of the initialxcitation. Specifically, we use the Huygens formula [Eq.14b)] to evaluate the decay due to the focused centraleld extending over the range [0, 0.4] cm and the back-round field that exists over the range [0.4, 100] cm. Thisecomposition is illustrated schematically in Fig. 15. Theenerated fields for the two sections have been calculatedor q=−0.5, a3−a2=990 cm, and a1=2 10−5 cm. The re-ults of this calculation are shown in Fig. 16 where, as ex-ected, the sum of the amplitudes of the two parts of thenitial excitation yields the same decay pattern as that ofhe total DEX wave field. However, we observe that therst knee ��1000 cm� is due primarily to the decay of theighly focused central portion of the DEX pulse. The slowissilelike decay, on the other hand, results solely from
he low-amplitude wings of the field.
ig. 15. (Color online) Annular section of the aperture showinghe central lobe and wings.
ig. 16. (Color online) Superposition of the decay rates of theain lobe and wings for q=−0.5, a1=2 10−5 cm, a2=10 cm, and
3=1000 cm.
ig. 17. (Color online) Dependence of the spectrum of the DEXulse on the radial distance from the center of the aperture.
It appears that the field outside the highly focused cen-ral part of the DEX pulse is distributed so that when con-ributions due to the inner sections are lost because of dif-raction they are partially compensated by contributionsrom the outer annular sections. To demonstrate that thisxplanation is valid, we divide the aperture into severalnnular sections, calculate the Rayleigh distance Z� forach annular section situated at radius �, and assume aecay rate for each of the annular sections according tohe following relation:
Amplitude =A�Z�
z + Z�
, where Z� =1
2c���2. �21�
ere, A� and �� are the amplitude of the field and itsaximum frequency of an annular section of the aperture
ituated a distance � from the center. To determine theaximum frequency of each annular section, the Fourier
ransform of the excitation of each section was evaluated.he dependence of the e−4 spectral cutoff, namely, ��, on
he radii of the annular sections is provided in Fig. 17.he Rayleigh distances Z� for the various sections werevaluated, and 1/z decays were assumed for the fieldsenerated by each section for zZ�. That was achievedsing the amplitude decay formula given in Eq. (21). Plotshowing the hypothetical decay rate [cf. Eq. (21)] for eachnnular section of the DEX pulse are shown in Fig. 18. Byumming up the contributions from the different sections,e derived the hypothetical decay pattern for the wholeperture shown in Fig. 19. A comparison of this hypotheti-al decay pattern with the one derived from the Huygensormula shows that the two are very close. This estab-ishes the fact that the wings of the excitation field of theEX pulse play a crucial role in producing the slow mis-
ilelike decay.
. CONCLUSION AND DISCUSSIONe have studied the salient properties underlying theissilelike behavior of the MPS pulse, the SP, and a new
ocalized wave, the DEX pulse. It has been shown that, al-hough a source-free MPS pulse exhibits a slow missile-ike decay, this is not the case when it is generated from anite-size aperture. In contradistinction, an aperture-enerated SP has a quasi-missile decay behavior over anxtended intermediate range between the near- and thear-field regions; specifically, the peak of the SP decayslong the z direction at a slow rate prior to undergoinghe usual 1/z roll-off. However, the new localized pulse,he DEX, has a double-exponential spectrum used in itsonstruction; this new localized wave pulse has missile-ike decay properties comparable with those of the SP andslower decay rate in the far field in comparison with theore conventional cw signals. Two distinct situationsave been considered in our studies of the slow, missile-
ike behavior of the DEX pulse: the source-free case andhe situation whereby the pulse has been generated by anite-size aperture. As the aperture increases in size, ourtudies have revealed that the decay pattern of the gen-rated DEX pulse resembles the source-free-case patternp to larger distances from the source and, also, that bothhe source-free and the aperture-generated fields start
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Shaarawi et al. Vol. 23, No. 8 /August 2006 /J. Opt. Soc. Am. A 2051
heir decay at the same distance z��a3−a2�, a2,3 beingarameters characterizing the DEX pulse. However, un-ike a source-free pulse, the peak of an aperture-enerated DEX pulse exhibits a 1/z decay rate in the far-eld range. Prior to acquiring the 1/z decay, the aperture-enerated pulse goes through an intermediate rangehere it exhibits a slow missilelike decay z−�1+q�, q beingnother parameter entering into the definition of theulse. This intermediate missilelike decay range extendso larger distances as the size of the aperture increases.his result establishes that aperture-generated DEXulses exhibit a quasi-missile decay rate over an extendedange intermediate between the near- and the far-field re-ions. The decay rate of the DEX pulse is divided intohree regions. The first one is the diffraction-free regionz� �a3−a2� /2� in which the pulse remains unaltered. Inhe second region �z �a3−a2� /2�, the pulse decays as−�1+q�. The upper limit for this missilelike decay range isimply the Rayleigh distance ZRW. For zZRW, the thirdegion, the pulse undergoes the usual 1/z decay in the far-eld range. Quantifying the three different decay rangeshrough a numerical simulation, we have found that theissilelike decay of the DEX pulse can be explained as a
ynthesis of decay patterns of two distinct parts of theulse. The first part represents the focused region of theEX pulse, and the second one is the surrounding back-round field. It appears that the field outside the highlyocused central part of the DEX pulse is distributed sohat when contributions due to the inner sections are lostecause of diffraction they are partially compensated fory contributions from the outer sections.
ig. 18. (Color online) Decay patterns for the annular sectionsf the DEX pulse for q=−0.5, a1=2 10−5 cm, a2=10 cm, and a31000 cm.
ig. 19. (Color online) Summation of the decay patterns contrib-ted by the annular sections of the DEX pulse for q=−0.5, a12 10−5 cm, a2=10 cm, and a3=1000 cm.
A comparison has been undertaken of the DEX pulse,PS pulse, and SP under the same conditions. This com-
arison has shown that the DEX pulse starts decaying atshorter distance from the aperture (�1000 cm for the
hosen parameters). However, the MPS pulse decays veryapidly around 106 cm after holding out without any de-ay for this extended distance. This rapid drop in the am-litude of the MPS pulse reduces its amplitude to about0−4 times that of the DEX pulse, which is still undergo-ng a slow missilelike decay. In general, the DEX pulsend SP have similar decay behaviors. Recalling that theEX pulse is a combination of two SP, one may wonderhether there can be any differences in the decay pat-
erns of the two solutions. It turns out that for differentarameters, the DEX pulse can preserves its shape andmplitude at farther distances better than the SP.Our work has made clear that a source-generated MPS
ulse does not acquire the slow missilelike decay of theP and DEX pulse, despite the fact that their source-freequivalents exhibit slow decays proportional to z−q. Thisehavior could be a result of the tight localization of thePS pulse. As can be inferred from the closed-form ex-
ressions of the three LW solutions, the transverse roll-offf the MPS pulse has an exponential dependence,hereas the DEX pulse and SP have algebraic dependen-
ies. We can conclude from the above discussion that theain advantage of the DEX pulse over other LW pulses is
hat it exhibits a smooth transition from an intermediateissilelike decay region into the far-field range and pre-
erves its shape and amplitude for distances farther fromhe source.
Corresponding author A. M. Shaarawi can be reachedy e-mail at [email protected].
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