Abstract

The angular spectrum of a vectorial laser beam is expressed in terms of an intrinsic coordinate system instead of the usual Cartesian laboratory coordinates. This switch leads to simple, elegant, and new expressions, such as for the angular spectrum of the Hertz vectors corresponding to the electromagnetic fields. As an application of this approach, we consider axially symmetric vector beams, showing nondiffracting properties of these beams, without invoking the paraxial approximation.

© 2012 Optical Society of America

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References

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    [CrossRef]
  3. A. Nesterov and V. Niziev, “Vector solution of the diffraction task using the Hertz vector,” Phys. Rev. E 71, 046608 (2005).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  10. J. Durnin, “Exact solutions for nondiffracting beams.I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
    [CrossRef]
  11. A. Nesterov and V. Niziev, “Propagation features of beams with axially symmetric polarization,” J. Opt. B 3, S215–S219 (2001).
    [CrossRef]

2007

2006

2005

A. Nesterov and V. Niziev, “Vector solution of the diffraction task using the Hertz vector,” Phys. Rev. E 71, 046608 (2005).
[CrossRef]

2001

A. Nesterov and V. Niziev, “Propagation features of beams with axially symmetric polarization,” J. Opt. B 3, S215–S219 (2001).
[CrossRef]

1996

1987

1977

E. Essex, “Hertz vector potentials of electromagnetic theory,” Am. J. Phys. 45, 1099–1101 (1977).
[CrossRef]

1966

D. Rhodes, “On the stored energy of planar apertures,” IEEE Trans. Antennas Propag. 14, 676–683 (1966).
[CrossRef]

Chen, J.

Durnin, J.

Essex, E.

E. Essex, “Hertz vector potentials of electromagnetic theory,” Am. J. Phys. 45, 1099–1101 (1977).
[CrossRef]

Guo, H.

Li, C.-F.

Nesterov, A.

A. Nesterov and V. Niziev, “Vector solution of the diffraction task using the Hertz vector,” Phys. Rev. E 71, 046608 (2005).
[CrossRef]

A. Nesterov and V. Niziev, “Propagation features of beams with axially symmetric polarization,” J. Opt. B 3, S215–S219 (2001).
[CrossRef]

Niziev, V.

A. Nesterov and V. Niziev, “Vector solution of the diffraction task using the Hertz vector,” Phys. Rev. E 71, 046608 (2005).
[CrossRef]

A. Nesterov and V. Niziev, “Propagation features of beams with axially symmetric polarization,” J. Opt. B 3, S215–S219 (2001).
[CrossRef]

Rhodes, D.

D. Rhodes, “On the stored energy of planar apertures,” IEEE Trans. Antennas Propag. 14, 676–683 (1966).
[CrossRef]

Someda, C.

C. Someda, Electromagnetic Waves, 2nd ed. (CRC, 2006).

Török, P.

Van Bladel, J.

J. Van Bladel, Electromagnetic Fields (McGraw-Hill, 1964).

Varga, P.

Zhuang, S.

Am. J. Phys.

E. Essex, “Hertz vector potentials of electromagnetic theory,” Am. J. Phys. 45, 1099–1101 (1977).
[CrossRef]

IEEE Trans. Antennas Propag.

D. Rhodes, “On the stored energy of planar apertures,” IEEE Trans. Antennas Propag. 14, 676–683 (1966).
[CrossRef]

J. Opt. B

A. Nesterov and V. Niziev, “Propagation features of beams with axially symmetric polarization,” J. Opt. B 3, S215–S219 (2001).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Express

Opt. Lett.

Phys. Rev. A

C.-F. Li, “Unified theory for Goos-Hänchen and Imbert-Fedorov effects,” Phys. Rev. A 76, 013811 (2007).
[CrossRef]

Phys. Rev. E

A. Nesterov and V. Niziev, “Vector solution of the diffraction task using the Hertz vector,” Phys. Rev. E 71, 046608 (2005).
[CrossRef]

Other

J. Van Bladel, Electromagnetic Fields (McGraw-Hill, 1964).

C. Someda, Electromagnetic Waves, 2nd ed. (CRC, 2006).

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Equations (64)

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k=kxex+kyey+kzez=kter+kzez=ktcosφex+ktsinφey+kzez
k2=kx2+ky2+kz2=kt2+kz2.
R=xex+yey+zez=rer+zez=rcosαex+rsinαey+zez,
s=eφp=kzkerktkezm=k/k.
s=ez×k|ez×k|=ez×kktp=s×kk=s×m.
s=kyktex+kxkteyp=kzkxkktex+kzkykkteyktkezm=kxkex+kykey+kzkez.
ex=kykts+kxkzkktp+kxkmey=kxkts+kykzkktp+kykmez=ktkp+kzkm,
E=gradϕAtB=rotA,
divA+ε0μ0ϕt=0.
2P1c22Pt2=0;=e,m.
ϕ=divPeA=ε0μ0Pet
ϕ=0A=rotPm.
E=graddivPeε0μ02Pet2=rotrotPeB=μ0H=ε0μ0t(rotPe).
E=t(rotPm)B=μ0H=graddivPm1c22Pmt2=rotrotPm.
E=graddivPe1c22Pet2(rotPm)tB=μ0H=ε0μ0(rotPe)t+graddivPm1c22Pmt2.
Ex=2PezxzEy=2PezyzEz=2Pezz2ε0μ02Pezt2Hx=ε02PezytHy=ε02PezxtHz=0,
Pe=(0,0,Pez)
pez(kx,ky)=k2Pez(x,y)exp[j(kxx+kyy)]dxdy
Pez(x,y,z=0)=Pez(x,y)=14π2k2pez(kx,ky)exp[j(kxx+kyy)]dkxdky.
Pez(x,y,z,t)=exp(jωt)14π2k2pez(kx,ky)exp[j(kxx+kyy+kzz)]dkxdky,
E(R)=14π2k2e(kx,ky)exp[j(kxx+kyy+kzz)]dkxdky.
e(kx,ky)=k2E(x,y,0)exp[j(kxx+kyy)]dxdy
h(kx,ky)=k2H(x,y,0)exp[j(kxx+kyy)]dxdy.
e·k=h·k=0,
x[pe(x,y,0)]=pex|z=0,
e(kx,ky)=(k.pe)k+k2pekck×pm
h(kx,ky)=ε0kck×pe(k.pm)k/μ0+k2/μ0pm,
E,He,hPe,mpe,m.
p(kx,ky)=Π+Π=e,mk.Π=0k×Π=0.
e(kx,ky)=k2Πe(kx,ky)kck×Πm(kx,ky)h(kx,ky)=k2μ0Πm(kx,ky)+kcε0k×Πe(kx,ky).
{E,H}={ETE,HTE}+{ETM,HTM}ETE·ez=0;HTM·ez=0.
Pe(R)=Pez(R)ez;Pm(R)=Pmz(R)ez.
pe=pezez.
eTM(kx,ky)=pez[kz(kxex+kyey)(kx2+ky2)ez]=kktpez(kx,ky)p(kx,ky)aTMp
hTM(kx,ky)=ε0kck×ezpez=ε0kc(kyexkxey)pez=ε0μ0kktpez(kx,ky)s(kx,ky)bTMs.
pe=0pm=pmzez.
eTE(kx,ky)=kck×ezpmz=kc(kyexkxey)pmz=ckktpmz(kx,ky)s(kx,ky)aTEs
hTE(kx,ky)=1μ0pmz[kz(kxex+kyey)(kx2+ky2)ez]=1μ0kktpmz(kx,ky)p(kx,ky)bTEp.
eTM=k2Πe=kktpezp,
Πe(kx,ky)=ktkpez(kx,ky)p.
Πm(kx,ky)=ktkpmz(kx,ky)p.
eTM=k2ΠehTM=kcε0k×ΠeeTE=kck×ΠmhTE=k2μ0Πm.
E(R)=ETM(R)+ETE(R)
e(kx,ky)=eTM(kx,ky)+eTE(kx,ky).
ETM(R)=1(2π)2Πe(kx,ky)exp(jk.R)dkxdkyHTM(R)=cε0(2π)2kk×Πe(kx,ky)exp(jk.R)dkxdkyETE(R)=c(2π)2kk×Πm(kx,ky)exp(jk.R)dkxdkyHTE(R)=1(2π)2μ0Πm(kx,ky)exp(jk.R)dkxdky.
ETM(R)=1(2π)2kktpez(kx,ky)pexp(jk.R)dkxdkyHTM(R)=1(2π)2ε0μ0ktpez(kx,ky)sexp(jk.R)dkxdkyETE(R)=c(2π)2ktpmz(kx,ky)sexp(jk.R)dkxdkyHTE(R)=1(2π)2kμ0ktpmz(kx,ky)pexp(jk.R)dkxdky.
Ex(x,y,z)=14π2k2Ax(kx,ky)exp(jk.R)dkxdkyEy(x,y,z)=14π2k2Ay(kx,ky)exp(jk.R)dkxdkyEz(x,y,z)=14π2k2[kxkzAx(kx,ky)+kykzAy(kx,ky)]exp(jk.R)dkxdky.
E(R)=ETM(R)+ETE(R)=Ex(R)ex+Ey(R)ey+Ez(R)ezETM(R)=14π2k2[kzkkt(kxex+kyey)ktkez]ATM(kx,ky)exp(jk.R)dkxdkyETE(R)=14π2k2[kyktexkxktey]ATE(kx,ky)exp(jk.R)dkxdky.
ATM(kx,ky)=k2kktkz(kxEx+kyEy)exp[j(kxx+kyy)]dxdyATE(kx,ky)=k21kt(kyExkxEy)exp[j(kxx+kyy)]dxdy,
ETM(R)=14π2k2pATM(kx,ky)exp(jk.R)dkxdkyETE(R)=14π2k2sATE(kx,ky)exp(jk.R)dkxdky
A(kx,ky)=ATMp+ATEs,
B(kx,ky)=BTMs+BTEp.
E(R)=14π2k2A(kx,ky)exp[j(kxx+kyy+kzz)]dkxdkyH(R)=14π2k2B(kx,ky)exp[j(kxx+kyy+kzz)]dkxdky.
aTM=ATMbTM=BTMaTE=ATEbTE=BTE.
ATM=kktpezBTM=ε0μ0kktpez=ε0μ0ATM
ATE=ckktpmz=μ0ε0BTEBTE=1μ0kktpmz.
ATM=k2ΠeBTM=ε0μ0k2ΠeATE=μ0ε0k2ΠmBTE=k2Πm.
E(R)=Er(r,z)er+Eα(r,z)eα+Ez(r,z)ez=ETM(r,z)+ETE(r,z).
ETM(r,z)=12πk30k[jkzJ1(ktr)erktJ0(ktr)ez]ktATM(kt)exp[jkzz]dktETE(r,z)=jeα2πk20kATE(kt)J1(ktr)ktexp[jkzz]dkt,
ATM(kt)=2πk3jk2kt20Er(r,z=0)J1(ktr)rdrATE(kt)=2πjk20Eα(r,z=0)J1(ktr)rdr.
ETE=s(.)dkxdky=eφ(.)dkxdky=eα(.)dkt.
ETM=p(.)dkxdky=(kzkerktkez)(.)dkxdky=(J1kzkerJ0ktkez)(.)dkt.
u(x,y,z)=exp(jkzz)02πa(φ)exp[jkt(xcosφ+ysinφ)]dφ
ETE(r,z)=j2πJ1(ktr)exp(jkzz)eα.

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