Abstract

The propagation characteristics of the fundamental Airy wave are obtained; the intensity distribution is the same as that for a point electric dipole situated at the origin and oriented normal to the propagation direction. The propagation characteristics of the modified fundamental Airy wave are determined. These characteristics are the same as those for the fundamental Gaussian wave provided that an equivalent waist is identified for the Airy wave. In general, the waves are localized spatially with the peak in the propagation direction.

© 2013 Optical Society of America

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References

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  1. M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47, 264–267 (1979).
    [CrossRef]
  2. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32, 979–981 (2007).
    [CrossRef]
  3. M. A. Bandres and J. C. Gutierrez-Vega, “Airy-Gauss beams and their transformation by paraxial optical systems,” Opt. Express 15, 16719–16728 (2007).
    [CrossRef]
  4. S. Yan, B. Yao, M. Lei, D. Dan, Y. Yang, and P. Gao, “Virtual source for an Airy beam,” Opt. Lett. 37, 4774–4776 (2012).
    [CrossRef]
  5. S. R. Seshadri, “Reactive power in the full Gaussian light wave,” J. Opt. Soc. Am. A 26, 2427–2433 (2009).
    [CrossRef]
  6. S. R. Seshadri, “Full-wave generalizations of the fundamental Gaussian beam,” J. Opt. Soc. Am. A 26, 2515–2520 (2009).
    [CrossRef]

2012 (1)

2009 (2)

2007 (2)

1979 (1)

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47, 264–267 (1979).
[CrossRef]

Balazs, N. L.

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47, 264–267 (1979).
[CrossRef]

Bandres, M. A.

Berry, M. V.

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47, 264–267 (1979).
[CrossRef]

Christodoulides, D. N.

Dan, D.

Gao, P.

Gutierrez-Vega, J. C.

Lei, M.

Seshadri, S. R.

Siviloglou, G. A.

Yan, S.

Yang, Y.

Yao, B.

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Figures (3)

Fig. 1.
Fig. 1.

Time-averaged power P± in watts transported by the modified fundamental Airy wave in the ±z direction as a function of kw0,eq for 1<kw0,eq<5. In the limit of the paraxial approximation, P± equals PFA±=1W.

Fig. 2.
Fig. 2.

Radiation intensity pattern Φ(θ+,ϕ) of the modified fundamental Airy wave for curves a, ϕ=0°; b, ϕ=30°; c, ϕ=60°; d, ϕ=90° as functions of θ+ for 0°<θ+<90°. Other parameters are kw0,eq=1.563. The total power in the modified fundamental Airy beam is 2 W. The total power in the modified fundamental Airy wave is 1.686 W.

Fig. 3.
Fig. 3.

Reactive power Pim of the modified fundamental Airy wave as a function of kw0,eq for 0.5<kw0,eq<3.5. The total power in the modified fundamental Airy beam is 2 W.

Equations (69)

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Ax0±(x,y,z)=exp(±ikz)ax0±(x,y,z),
Ex0±(x,y,z)=±Hy0±(x,y,z)=ikAx0±(x,y,z).
J0(x,y,z)=x^2ikAx0±(x,y,0)δ(z).
ax0±(x,y,0)=Ax0±(x,y,0)=NikAi(αx)Ai(αy)exp[2πa(x+y)].
a¯x0±(px,py,0)=Nik1α2exp{i3(2πα)3[(pxia)3+(pyia)3]}.
a¯x0±(px,py,z)=a¯x0±(px,py,0)exp[2π2k(px2+py2)i|z|].
Ax0±(x,y,z)=exp(±ikz)Nikf0(x,z)f0(y,z),
f0(u,z)=exp[2πu(a+α3i|z|4πk)+2π2a2i|z|kπaα3|z|2k2α6i|z|312k3]×Ai{α[u+i|z|k(2πa+iα3|z|4k)]}foru=x,y.
PC=Pre+iPim=c2dxdydzE0±(x,y,z)·J0*(x,y,z)=ck2dxdyax0±(x,y,0)ax0±*(x,y,0),
Pre=ck2dpxdpya¯x0±(px,py,0)a¯x0±*(px,py,0).
Pre=cN2α4dpxexp{i3(2πα)3[(pxia)3(px+ia)3]}×dpyexp{i3(2πα)3[(pyia)3(py+ia)3]}.
Pre=cN2α4exp[43(2πaα)3]dpxdpyexp[(2πα)32a(px2+py2)].
Pre=cN216π2αaexp[43(2πaα)3].
(2x2+2y2+2z2+k2)Ax±(x,y,z)=2NAi(αx)Ai(αy)exp[2πa(x+y)]δ(z).
(2z2+ζ2)A¯x±(px,py,z)=δ(z)2Nα2exp{i3(2πα)3[(pxia)3+(pyia)3]},
ζ=[k24π2(px2+py2)]1/2.
A¯x±(px,py,z)=Niα2exp{i3(2πα)3[(pxia)3+(pyia)3]}×ζ1exp(iζ|z|).
Ax±(x,y,z)=Niα2dpxdpyexp[i2π(pxx+pyy)]×exp{i3(2πα)3[(pxia)3+(pyia)3]}1ζexp[iζ|z|].
ζ=k2π2k(px2+py2).
Ex±(x,y,z)=ik(1+1k22x2)Ax±(x,y,z).
Ex±(x,y,z)=Nkα2dpxdpyexp[i2π(pxx+pyy)]×exp{i3(2πα)3[(pxia)3+(pyia)3]}×(14π2px2k2)1ζexp(iζ|z|).
PC=c2dxdydzEx±(x,y,z)Jx0*(x,y,z)
=c2dxdyNkα2dpxdpyexp[i2π(pxx+pyy)]×exp{i3(2πα)3[(pxia)3+(pyia)3]}×(14π2px2k2)1ζ(2)Nα2dp¯xdp¯yexp[i2π(p¯xx+p¯yy)]×exp{i3(2πα)3[(p¯x+ia)3+(p¯y+ia)3]}.
PC=cN2kα4dpxdpy(14π2px2k2)1ζ×exp{i3(2πα)3[(pxia)3(px+ia)3]}×exp{i3(2πα)3[(pyia)3(py+ia)3]}.
PC=cN2kα4exp[43(2πaα)3]dpxdpy(14π2px2k2)1ζ×exp[(2πα)32a(px2+py2)].
w0,eq=(8πaα3)1/2.
2πpx=pcosϕand2πpy=psinϕ.
PC=cN2k4π2α4exp[43(2πaα)3]0dpp02πdϕ(1p2cos2ϕk2)×1(k2p2)1/2exp[12w0,eq2p2].
PC=cN2k4π2α40dpp02πdϕ(1p2cos2ϕk2)(k2p2)1/2fora=0.
PC=Pre+iPim,
ζ=(k2p2)1/2fork2>p2=i(p2k2)1/2forp2>k2.
iPim=icN2k4π2α4kdpp02πdϕ(1p2cos2ϕk2)(p2k2)1/2fora=0.
iPim=icN2k24πα40dτ(1τ2)=fora=0.
Pre=cN2k4π2α40kdpp02πdϕ(1p2cos2ϕk2)(k2p2)1/2fora=0.
Pre=cN2k24π2α40π/2dθsinθ02πdϕ(1sin2θcos2ϕ)fora=0.
exp{i3(2πα)3[(pxia)3+(pyia)3]}1.
exp{i3(2πα)3[(p¯x+ia)3+(p¯y+ia)3]}1.
Φ(θ,ϕ)=1sin2θcos2ϕ
Pre=cN2k23πα4fora=0.
PC=w0,eq2kπ0dpp02πdϕ(1p2cos2ϕk2)×1(k2p2)1/2exp[12w0,eq2p2].
iPim=iw0,eq2kπkdpp02πdϕ(1p2cos2ϕk2)×1(p2k2)1/2exp[12w0,eq2p2].
iPim=ik2w0,eq2exp(12k2w0,eq2)×0dτ(1τ2)exp(12k2w0,eq2τ2).
iPim=i(π2)1/2kw0,eq(11k2w0,eq2)exp(12k2w0,eq2).
Pre=w0,eq2kπ0kdpp02πdϕ(1p2cos2ϕk2)×1(k2p2)1/2exp[12w0,eq2p2].
p=ksinθ+.
ζ=k|cosθ+|.
Pre=P++P,
P+=0π/2dθ+sinθ+02πdϕΦ(θ+,ϕ)
Φ(θ+,ϕ)=k2w0,eq2π(1sin2θ+cos2ϕ)exp(12k2w0,eq2sin2θ+).
P=ππ/2dθ+sinθ+02πdϕΦ(θ+,ϕ).
θ=πθ+.
Φ(πθ,ϕ)=Φ(θ,ϕ).
P=0π/2dθsinθ02πdϕΦ(θ,ϕ).
J0(x,y,z)=x^2Nexp(x2+y2w02)δ(z).
(2x2+2y2+2z2+k2)Ax±(x,y,z)=δ(z)2Nexp(x2+y2w02).
(2z2+ζ2)A¯x±(px,py,z)=δ(z)2Nπw02exp[π2w02(px2+py2)].
A¯x±(px,py,z)=iNπw02exp[π2w02(px2+py2)]ζ1exp(iζ|z|).
A¯x0±(px,py,z)=exp(±ikz)Nikπw02exp[π2w02(px2+py2)q±2],
q±2=1+i|z|b
b=12kw02
a¯x0±(px,py,0)=A¯x0±(px,py,0)=Nikπw02exp[π2w02(px2+py2)].
Pre=cN2π2w04dpxdpyexp[2π2w02(px2+py2)]=cN2πw022.
N=(4cπw02)1/2.
Ax±(x,y,z)=iNπw02dpxdpyexp[i2π(pxx+pyy)]×exp[π2w02(px2+py2)]ζ1exp(iζ|z|).
Ex±(x,y,z)=Nkπw02dpxdpyexp[i2π(pxx+pyy)]×exp[π2w02(px2+py2)]×(14π2px2k2)ζ1exp(iζ|z|).
PC=c2dxdydzEx±(x,y,z)Jx0*(x,y,z)=c2dxdyNkπw02dpxdpyexp[i2π(pxx+pyy)]×exp[π2w02(px2+py2)](14π2px2k2)ζ1×(2)Nπw02dp¯xdp¯yexp[i2π(p¯xx+p¯yy)]×exp[π2w02(p¯x2+p¯y2)].
PC=cN2kπ2w04dpxdpyexp[2π2w02(px2+py2)]×(14π2px2k2)1ζ.
PC=cN2kw0440dpp02πdϕ(1p2cos2ϕk2)×(k2p2)1/2exp(12w02p2).
PC=w02kπ0dpp02πdϕ(1p2cos2ϕk2)×(k2p2)1/2exp(12w02p2).

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