Abstract

We present a complete electromagnetic study, which includes electric, magnetic, and Poynting vector fields of diffracted convergent spherical waves under all possible polarization states compatible with Maxwell’s equations. Exit pupil boundary conditions for these polarizations were obtained by means of Hertz potentials. Using these boundary conditions, two-dimensional Luneburg diffraction integrals for the three components of electric and magnetic fields were formulated, and after some approximations, we showed that the complete electromagnetic description of the inhomogeneous polarization states of spherical waves is reduced to the knowledge of seven one-dimensional integrals. The consistency of the method was tested by comparison with other previously reported methods for linearly polarized (LP), TE, and TM polarizations, while the versatility of the method was showed with the study of nonstandard polarization states, for example, that resulting from the superposition of TE and TM dephased spherical waves, which shows a helicoidal behavior of the Poynting vector at the focalization region, or the inhomogeneous LP state that exhibits a ring structure for the Poynting vector at the focal plane.

© 2013 Optical Society of America

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References

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  1. B. Richards and E. Wolf, “Electromagnetic diffraction of optical system II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 253, 358–379 (1959).
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  11. C. Sheppard, “Focal distribution and Hertz potentials,” Opt. Commun. 160, 191–194 (1999).
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  13. C. Sheppard and S. Saghafi, “Transverse-electric and transverse magnetic beam modes beyond the paraxial approximation,” Opt. Lett. 24, 1543–1545 (1999).
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  15. A. April, Ultrashort, Strongly Focused Laser Pulses in Free Space, Coherence and Ultrashort Pulse Laser Emission(InTech, 2010).
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  17. J. Guo, X. Zhao, and Y. Min, “The general integral expressions for on-axis nonparaxial vectorial spherical waves diffracted at a circular aperture,” Opt. Commun. 282, 1511–1515 (2009).
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  18. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1994).
  19. A. V. Novitsky and D. V. Novitsky, “Negative propagation of vector Bessel beam,” J. Opt. Soc. Am. A 24, 2844–2849 (2007).
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2012

2010

2009

J. Guo, X. Zhao, and Y. Min, “The general integral expressions for on-axis nonparaxial vectorial spherical waves diffracted at a circular aperture,” Opt. Commun. 282, 1511–1515 (2009).
[CrossRef]

2008

2007

2006

2004

2002

A. L. Sokolov, “Polarization of spherical waves,” Opt. Spectrosc. 92, 936–942 (2002).
[CrossRef]

2000

1999

1959

B. Richards and E. Wolf, “Electromagnetic diffraction of optical system II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 253, 358–379 (1959).
[CrossRef]

April, A.

A. April, Ultrashort, Strongly Focused Laser Pulses in Free Space, Coherence and Ultrashort Pulse Laser Emission(InTech, 2010).

Bokor, N.

Brown, T.

Chen, Z.

Climent, V.

Davidson, N.

Dorn, R.

S. Qusabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[CrossRef]

Eberler, M.

S. Qusabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[CrossRef]

Glockl, O.

S. Qusabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1994).

Guo, J.

J. Guo, X. Zhao, and Y. Min, “The general integral expressions for on-axis nonparaxial vectorial spherical waves diffracted at a circular aperture,” Opt. Commun. 282, 1511–1515 (2009).
[CrossRef]

Khonina, S. N.

Kitamura, K.

Lancis, J.

Leuchs, G.

S. Qusabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[CrossRef]

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (University of California, 1964).

Mendoza-Yero, O.

Min, Y.

J. Guo, X. Zhao, and Y. Min, “The general integral expressions for on-axis nonparaxial vectorial spherical waves diffracted at a circular aperture,” Opt. Commun. 282, 1511–1515 (2009).
[CrossRef]

Nishimoto, M.

Noda, S.

Novitsky, A. V.

Novitsky, D. V.

Qusabis, S.

S. Qusabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[CrossRef]

Richards, B.

B. Richards and E. Wolf, “Electromagnetic diffraction of optical system II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 253, 358–379 (1959).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1994).

Saghafi, S.

Sakai, K.

Sheppard, C.

Sheppard, C. J. R.

Sokolov, A. L.

A. L. Sokolov, “Polarization of spherical waves,” Opt. Spectrosc. 92, 936–942 (2002).
[CrossRef]

Tahjahuerce, E.

Takayama, N.

Vega, G. M.

Volotovsky, S. G.

Wolf, E.

B. Richards and E. Wolf, “Electromagnetic diffraction of optical system II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 253, 358–379 (1959).
[CrossRef]

Youngworth, K. S.

Zhao, D.

Zhao, X.

J. Guo, X. Zhao, and Y. Min, “The general integral expressions for on-axis nonparaxial vectorial spherical waves diffracted at a circular aperture,” Opt. Commun. 282, 1511–1515 (2009).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

J. Guo, X. Zhao, and Y. Min, “The general integral expressions for on-axis nonparaxial vectorial spherical waves diffracted at a circular aperture,” Opt. Commun. 282, 1511–1515 (2009).
[CrossRef]

S. Qusabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[CrossRef]

C. Sheppard, “Focal distribution and Hertz potentials,” Opt. Commun. 160, 191–194 (1999).
[CrossRef]

Opt. Express

Opt. Lett.

Opt. Spectrosc.

A. L. Sokolov, “Polarization of spherical waves,” Opt. Spectrosc. 92, 936–942 (2002).
[CrossRef]

Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.

B. Richards and E. Wolf, “Electromagnetic diffraction of optical system II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 253, 358–379 (1959).
[CrossRef]

Other

A. April, Ultrashort, Strongly Focused Laser Pulses in Free Space, Coherence and Ultrashort Pulse Laser Emission(InTech, 2010).

R. K. Luneburg, Mathematical Theory of Optics (University of California, 1964).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1994).

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Figures (7)

Fig. 1.
Fig. 1.

Intensity and behavior of the real and imaginary parts of polarization (E⃗ep) at the exit pupil plane for cases (a) r⃗e=(1,0,0), r⃗m=(0,1,0) (LP), (b) r⃗e=(1,0,0), r⃗m=(0,1,0) (ILP), (c) r⃗e=(0,0,0), r⃗m=(0,0,1) (TE), (d) r⃗e=(0,0,1), r⃗m=(0,0,0)(TM), and (e) r⃗e=(0,i,0), r⃗m=(0,0,1) TE–TM.

Fig. 2.
Fig. 2.

Electromagnetic normalized field distribution versus the radial coordinate at the focal plane for angles ξ=0, ξ=π/3, ξ=2π/3, and ξ=π. |E| and |H| are obtained using Eqs. (25) and (26) in the blue and red continuous lines, respectively, and |E| and |H| are obtained using [1] in the dashed blue and red lines, respectively.

Fig. 3.
Fig. 3.

Normalized Poynting vector at the focal plane for the LP (a) and ILP (b) polarization states.

Fig. 4.
Fig. 4.

Electromagnetic normalized fields at the focal plane. |Eρ| and |Ez| obtained using Eqs. (29) and (30) in continuous and dashed lines, respectively, using [2].

Fig. 5.
Fig. 5.

Electromagnetic normalized fields at the focal plane. |Eϕ| obtained using Eqs. (35) and (36) in continuous and dashed lines, respectively, using [2].

Fig. 6.
Fig. 6.

Poynting vector at the focal region for TM and TE polarization.

Fig. 7.
Fig. 7.

Poynting vector at the focal region for a π/2 dephased TM–TE polarization.

Equations (43)

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ψ(x,y,z)=exp(iωcx2+y2+z2)x2+y2+z2=exp(iωcR)R,
Π⃗e=ψ(x,y,z)(Xex^+Yey^+Zez^)=ψ(x,y,z)r⃗e,
Π⃗m=η01ψ(x,y,z)(Xmx^+Ymy^+Zmz^)=η01ψ(x,y,z)r⃗m,
2Π⃗e+ω2c2Π⃗e=0⃗,2Π⃗m+ω2c2Π⃗m=0⃗,
E⃗=××Π⃗e+iωμ0×Π⃗m,H⃗=××Π⃗miωϵ0×Π⃗e,
E⃗ep=Ex(xo,yo)x^+Ey(xo,yo)y^,
H⃗ep=Hx(xo,yo)x^+Hy(xo,yo)y^,
Ex(x,y,z)=(z+f)2πΣEx(xo,yo)ikRf1Rf3exp(ikRf)dxodyo,Ey(x,y,z)=(z+f)2πΣEy(xo,yo)ikRf1Rf3exp(ikRf)dxodyo,Ez(x,y,z)=12πΣ((xxo)Ex(xo,yo)+(yyo)Ey(xo,yo))ikRf1Rf3exp(ikRf)dxodyo,
Hx(x,y,z)=(z+f)2πΣHx(xo,yo)ikRf1Rf3exp(ikRf)dxodyo,Hy(x,y,z)=(z+f)2πΣHy(xo,yo)ikRf1Rf3exp(ikRf)dxodyo,Hz(x,y,z)=12πΣ((xxo)Hx(xo,yo)+(yyo)Hy(xo,yo))ikRf1Rf3exp(ikRf)dxodyo,
E⃗ep=(××Π⃗e+iωμ0×Π⃗m)(xo,yo,f),H⃗ep=(××Π⃗miωϵ0×Π⃗e)(xo,yo,f).
E⃗ep=Me·r⃗e+Mm·r⃗m,
H⃗ep=1η0(Mm·r⃗e+Me·r⃗m).
Me=exp(ikG(0,ρ))2G(0,ρ)5(P1P2ρ2Cos(2θ)P2ρ2Sin(2θ)2fP2ρCos(θ)P2ρ2Sin(2θ)P1+P2ρ2Cos(2θ)2fP2ρSin(θ)),Mm=exp(ikG(0,ρ))2G(0,ρ)5(02fQ2QρSin(θ)2fQ02QρCos(θ)),
P1(ρ)=2f4k2+ρ2(1+iG(0,ρ)k+k2ρ2)+f2(22iG(0,ρ)k+3k2ρ2),P2(ρ)=3+G(0,ρ)k(3i+G(0,ρ)k),Q(ρ)=G(0,ρ)2k(i+G(0,ρ)k).
RfG(z,ρ)rρCos(θξ)G(z,ρ).
ikRf1Rf3exp(ikRf)(ikG(z,ρ)1)exp(ikG(z,ρ))G(z,ρ)3exp(ikrρCos(θξ)G(z,ρ)).
exp(ikrρCos(θξ)G(z,ρ))=J0(H(z,r,ρ))+2p=1ipJp(H(z,r,ρ))Cos(p(θξ)),
E⃗=Ie·r⃗e+Im·r⃗m,
H⃗=1η0(Im·r⃗e+Ie·r⃗e),
Ie=((I00+I2Cos(2ξ))I2Sin(2ξ)2ifI10Cos(ξ)I2Sin(2ξ)I00+I2Cos(2ξ)2fI10Sin(ξ)r(I00+I2)+I11f+zCos(ξ)r(I00+I2)+I11f+zSin(ξ)(2ifrI10I01)f+z),
Im=(02fI022iI12Sin(ξ)2fI0202iI12Cos(ξ)2f(iI12rI02)Sin(ξ)f+z2f(iI12rI02)Cos(ξ)f+z0),
I00(r,z,a)=0aKc(ρ,z)P1(ρ)J0(H(z,r,ρ))dρI01(r,z,a)=0aKc(ρ,z)2fP2(ρ)ρ2J0(H(z,r,ρ))dρI02(r,z,a)=0aKc(ρ,z)Q(ρ)J0(H(z,r,ρ))dρI10(r,z,a)=0aKc(ρ,z)P2(ρ)J1(H(z,r,ρ))ρdρI11(r,z,a)=0aKc(ρ,z)iρ(P1(ρ)+P2(ρ)ρ2)J1(H(z,r,ρ))dρI12(r,z,a)=0aKc(ρ,z)Q(ρ)J1(H(z,r,ρ))ρdρI2(r,z,a)=0aKc(ρ,z)P2(ρ)J2(H(z,r,ρ))ρ2dρ
Kc(ρ,z)=(f+z)(ikG(z,ρ)1)exp(ik(G(0,ρ)G(z,ρ)))2G(0,ρ)5G(z,ρ)3ρ.
S⃗=Re(E⃗×H⃗*2),
E⃗=((I00+2fI02+I2Cos(2ξ))I2Sin(2ξ)(I112ifI12+(I00+2fI02+I2)r)Cos(ξ)f+z),
H⃗=1η0(I2Sin(2ξ)I002fI02+I2Cos(2ξ)(I112ifI12+(I00+2fI02+I2)r)Sin(ξ)f+z).
E⃗=((I002fI02+I2Cos(2ξ))I2Sin(2ξ)(I11+2ifI12+(I002fI02+I2)r)Cos(ξ)f+z),
H⃗=1η0(I2Sin(2ξ)I00+2fI02+I2Cos(2ξ)(I11+2ifI12+(I002fI02+I2)r)Sin(ξ)f+z).
E⃗=(2ifI10Cos(ξ)2ifI10Sin(ξ)(I01+2ifrI10)f+z),
H⃗=1η0(2iI12Sin(ξ)2iI12Cos(ξ)0).
Tρ=TxCos(ξ)+TySin(ξ),
Tϕ=TyCos(ξ)TxSin(ξ),
E⃗=(EρEϕEz)=(2ifI100(I01+2ifrI10)f+z),
H⃗=(HρHϕHz)=1η0(02iI120)
E⃗=(2iI12Sin(ξ)2iI12Cos(ξ)0),
H⃗=1η0(2ifI10Cos(ξ)2ifI10Sin(ξ)(I01+2ifrI10)f+z).
E⃗=(EρEϕEz)=(02iI120),
H⃗=(HρHϕHz)=1η0(2ifI100(I01+2ifrI10)f+z),
E⃗=(2i(fI10Cos(ξ)peexp(iνe)+I12Sin(ξ)pmexp(iνm))2i(fI10Sin(ξ)peexp(iνe)+I12Cos(ξ)pmexp(iνm))(I01+2ifrI10)f+zpeexp(iνe)),
H⃗=1η0(2i(fI10Cos(ξ)pmexp(iνm)I12Sin(ξ)peexp(iνe))2i(fI10Sin(ξ)pmexp(iνm)+I12Cos(ξ)peexp(iνe))(I01+2ifrI10)f+zpmexp(iνm)).
E⃗=(EρEϕEz)=(2ifI10peexp(iνe)2iI12pmexp(iνm)(I01+2ifrI10)f+zpeexp(iνe)),
H⃗=(HρHϕHz)=1η0(2ifI10pmexp(iνm)2iI12peexp(iνe)(I01+2ifrI10)f+zpmexp(iνm)).
S⃗=(SρSϕSz)=12η0(f+z)((pe2+pm2)(Im[I12I01*]+2frRe[I12I10*])2fpepmRe[I01I10*]Sin(νeνm)2f(f+z)(pe2+pm2)Re[I10I12*]).

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