Abstract

A solution to the problem of plane electromagnetic waves focused by an ellipsoidal or a hyperboloidal lens is derived from the Stratton–Chu integral by solving a boundary-value problem. The current method is more rigorous than those hitherto published in the literature. Results show that for linearly polarized incident illumination and in the vicinity of the focus, the distribution of the time-averaged electric energy density is almost fully transverse electric.

© 2002 Optical Society of America

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References

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  1. E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. London Ser. A 253, 349–357 (1959).
    [CrossRef]
  2. B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
    [CrossRef]
  3. J. A. Stratton, L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56, 99–107 (1939).
    [CrossRef]
  4. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
  5. P. Varga, P. Török, “Electromagnetic focusing by a paraboloid mirror. I. Theory,” J. Opt. Soc. Am. A 17, 2081–2089 (2000).
    [CrossRef]
  6. P. Varga, P. Török, “Electromagnetic focusing by a paraboloid mirror. II. Numerical results,” J. Opt. Soc. Am. A 17, 2090–2095 (2000).
    [CrossRef]
  7. A. Hardy, D. Treves, “Structure of the electromagnetic field near the focus of a stigmatic lens,” J. Opt. Soc. Am. 63, 85–90 (1973).
    [CrossRef]
  8. P. Török, P. D. Higdon, T. Wilson, “On the general properties of polarising conventional and confocal microscopes,” Opt. Commun. 148, 300–315 (1998).
    [CrossRef]
  9. N. Sergienko, V. Dhayalan, J. J. Stamnes, “Comparison of focusing properties of conventional and diffractive lenses,” Opt. Commun. 194, 225–234 (2001).
    [CrossRef]
  10. A. Sommerfeld, Optics, Vol. IV in Lectures on Theoretical Physics (Academic, New York, 1954).

2001

N. Sergienko, V. Dhayalan, J. J. Stamnes, “Comparison of focusing properties of conventional and diffractive lenses,” Opt. Commun. 194, 225–234 (2001).
[CrossRef]

2000

1998

P. Török, P. D. Higdon, T. Wilson, “On the general properties of polarising conventional and confocal microscopes,” Opt. Commun. 148, 300–315 (1998).
[CrossRef]

1973

1959

E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. London Ser. A 253, 349–357 (1959).
[CrossRef]

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

1939

J. A. Stratton, L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56, 99–107 (1939).
[CrossRef]

Chu, L. J.

J. A. Stratton, L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56, 99–107 (1939).
[CrossRef]

Dhayalan, V.

N. Sergienko, V. Dhayalan, J. J. Stamnes, “Comparison of focusing properties of conventional and diffractive lenses,” Opt. Commun. 194, 225–234 (2001).
[CrossRef]

Hardy, A.

Higdon, P. D.

P. Török, P. D. Higdon, T. Wilson, “On the general properties of polarising conventional and confocal microscopes,” Opt. Commun. 148, 300–315 (1998).
[CrossRef]

Richards, B.

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

Sergienko, N.

N. Sergienko, V. Dhayalan, J. J. Stamnes, “Comparison of focusing properties of conventional and diffractive lenses,” Opt. Commun. 194, 225–234 (2001).
[CrossRef]

Sommerfeld, A.

A. Sommerfeld, Optics, Vol. IV in Lectures on Theoretical Physics (Academic, New York, 1954).

Stamnes, J. J.

N. Sergienko, V. Dhayalan, J. J. Stamnes, “Comparison of focusing properties of conventional and diffractive lenses,” Opt. Commun. 194, 225–234 (2001).
[CrossRef]

Stratton, J. A.

J. A. Stratton, L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56, 99–107 (1939).
[CrossRef]

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

Török, P.

Treves, D.

Varga, P.

Wilson, T.

P. Török, P. D. Higdon, T. Wilson, “On the general properties of polarising conventional and confocal microscopes,” Opt. Commun. 148, 300–315 (1998).
[CrossRef]

Wolf, E.

E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. London Ser. A 253, 349–357 (1959).
[CrossRef]

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

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

P. Török, P. D. Higdon, T. Wilson, “On the general properties of polarising conventional and confocal microscopes,” Opt. Commun. 148, 300–315 (1998).
[CrossRef]

N. Sergienko, V. Dhayalan, J. J. Stamnes, “Comparison of focusing properties of conventional and diffractive lenses,” Opt. Commun. 194, 225–234 (2001).
[CrossRef]

Phys. Rev.

J. A. Stratton, L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56, 99–107 (1939).
[CrossRef]

Proc. R. Soc. London Ser. A

E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. London Ser. A 253, 349–357 (1959).
[CrossRef]

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

Other

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

A. Sommerfeld, Optics, Vol. IV in Lectures on Theoretical Physics (Academic, New York, 1954).

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Figures (4)

Fig. 1
Fig. 1

Schematic diagram showing notation for (a) ellipsoidal and (b) hyperboloidal lenses.

Fig. 2
Fig. 2

Curves showing the distribution of the x component Ex of the electric field along the x axis for the (a) PV, (a) RW, and (a) HT theories. Panels (b), (b), and (b) show the corresponding values of the z component Ez. Panels (c), (c), and (c) show the modulus of Ex along the z optic axis. Panels (d), (d), and (d) show the phase along the optic axis. All values were computed for a convergence angle of α=30°.

Fig. 3
Fig. 3

Curves showing the distribution of the x component Ex of the electric field along the x axis for the (a) PV, (a) RW, and (a) HT theories. Panels (b), (b), and (b) show the corresponding values of the z component Ez. Panels (c), (c), and (c) show the modulus of Ex along the z optic axis. Panels (d), (d), and (d) show the phase along the optic axis. All values were computed for a convergence angle of α=48.8°.

Fig. 4
Fig. 4

Curves showing the distribution of the intensity |Ex|2+|Ez|2 along the x axis for the (a), (b) PV; (a) (b) RW; (a) (b) HT theories. The set of curves (a) and (b) were computed for a convergence angle of α=30° and α=48.8°, respectively.

Equations (76)

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n2ρ+n1(f-|zs|)=n2f
foraconcavesurfaceobservedfromfocus,
n2ρ=n1(|zs|-f)+n2f
foraconvexsurfaceobservedfromfocus.
ρ=1-n21n21 cos θ+1f,
n21=n1/n2.
sin[π-(π/2-θ)]=n2/n1,
m=1ν(s+n21ez),
s=(sin θ cos ϕ, sin θ sin ϕ, cos θ),
ν=(n212+1+2n21 cos θ)1/2,
Ai(S)exp-iωt-n1csi(S)r(S),
Ar(S)exp-iωt-n1csr(S)r(S),
At(S)exp-iωt-n2cst(S)r(S).
H=-ns×E,
Ei=ex exp[-i(ωt-k1z)],
Hi=n1ey exp[-i(ωt-k1z)],
q=ex sin ϕ-ey cos ϕ.
Ei,s=q(ex·q)exp(ik1zs)=(ex sin2 ϕ-ey sin ϕ cos ϕ)exp(ik1zs),
Ei,p=Ei-Ei,s=[ex(1-sin2 ϕ)+ey sin ϕ cos ϕ]exp(ik1zs).
Ei,s=sin ϕ exp(ik1zs),
Ei,p=cos ϕ exp(ik1zs).
cos βi=m·ez=1ν(n21+cos θ).
cos βt=-1ν(n21 cos θ+1).
Et,s=sin ϕTs exp(ik2zs),
Et,p=cos ϕTp exp(ik2zs),
Tp=2n1 cos βin2 cos βi+n1 cos βt,
Ts=2n1 cos βin1 cos βi+n2 cos βt,
Tp=2n21(cos θ+n21)(1-n212)cos θ,Ts=2n21(cos θ+n21)1-n212.
Tp=νmz(1-n212)cos θ,Ts=-νmz1-n212.
Et,s=Et,s·q.
Et,p=Et,pq×s.
Et(S)=Et,p+Et,s=2n211-n212νmz×exp(ik2zs)cos ϕcos θq×s-sin ϕq.
Et(S)=2n211-n212νmzszexp(ik2zs)(-szex+sxez).
Ht(S)=n22n211-n212νmzszexp(ik2zs)×[-sxsyex+(sx2+sz2)ey-syszez].
E(P)=-14πS[ik0(m×H)G+(m×E)×G+(m·E)G]dS-14πik0n2G(H·dqˆ),
H(P)=-14πS[ik0n2(m×E)G+(m×H)×G+(m·H)G]dS-14πik0G(E·dqˆ),
G(u)=exp(iku)u,
G(u)=ik1-1ikuG(u)g(x, y, z; xs, ys, ys),
g(x, y, z; xs, ys, zs)=Δxuex+Δyuey+Δzuez,,
u=[(xs-x)2+(ys-y)2+(zs-z)2]1/2.
g=s.
G(u)=ikG(u)g(x, y, z; xs, ys, zs).
dS=ρ2mzsin θdθdϕ.
Esurf(P)=-ik24πexp[i(k2-k1)f]απ02πexp[ik2(u-ρ)]u×VE(θ, ϕ)ρ2sin θmzdϕdθ,
VE=2n211-n212mzsz{[sz(1+n21sz)+sz(gxsx+gysy+gzsz+n21gz)+n21sx(sx-gx)+sx(gzsx-gxsz)]ex+[n21sx(sy-gy)+sy(gzsx-gxsz)]ey-[sx(gxsx+gysy+gzsz+1)+sz(gxsz-gzsx)+n21(gxsz+sxgz)]ez}.
H·dqˆ=2n211-n212n2(cos α+n21)sin αcos α×exp[ik12z(α)]ρ(α)cos ϕdϕ,
Econt(P)=12πn211-n212n2(cos α+n21)sin αcos αexp[ik2z(α)]×ρ(α)02πG(u)Δxuex+Δyuey+Δzuezdϕ.
VE=4n211-n212mzsz[sz(n21sx+1)ex-sx(n21sz+1)ez].
Esurf(F)=i2n211+n21k2f(1+cos α)exp[ik0(n2-n1)]ex.
Econt(F)=-n211-n212n2(cos α+n21)×sin α exp[iksz(α)]ez,
VH=2sxsy(1+n21sz)ex-(1+sx2)(1+n21sz)ey+2sysz(1+n21sz)ez.
exp[ik(u-ρ)]exp-ik22ρρ_·r,
r=(x, y, z)=(r sin Θ cos Φ, r sin Θ sin Φ, r cos Θ).
g(sx-x/ρ)ex+(sy-y/ρ)ey+(sz-z/ρ)ez,
V¯E=2n211-n212mzsz2(1+n21sz)(szex-sxez)+1ρ{[n21sxx-syszy-(sx2+n21sz+sz2)]ex+(syszx+n21sxy-sxsyz)ey-[(sx2+n21sz+sz2)x+sxsyy+n21sxz]ez}.
Esurf=-ik24πexp[i(k2-k1)f]απ02πexp-ik2ρ_·rρ×ρV¯E(θ, ϕ)sin θνmzdϕdθ.
Esurf,d=-2ik2fn211+n21×exp[i(k2-k1)f]απexp(ikrLC)tan θ×[cos θ J0(krLs)ex-i cos Φ sin θ J1(krLs)ez]dθ,
Esurf,m=ik2n211-n212exp[i(k2-k1)f]rαπexp(ikrLC)tan θ×(Fxex+Fyey+Fzez)dθ,
Fx=-cos Θ(1+2n21 cos θ+cos2 θ)J0(krLS)-i sin Θ(n21 cos2 Φ+sin2 Φ)×sin θ(1-cos θ)J1(krLS)+cos Θ cos 2Φ sin2 θ J2(krLS),
Fy=sin 2Φ[i sin Θ sin θ(n21+cos θ)J1(krLS)-cos Θ sin2 θ J2(krLS)],
Fz=[sin Θ cos Φ(1+2n21+cos2 θ)]J0(krLS)+in21 cos Φ sin θ cos θ J1(krLS)-12sin Θ cos Φ sin2 θ J2(krLS),
S=-4πn2c(R{Et})2s.
P=SS·dA=SS·mdA.
P=-4πn2c2n211+n212f2απ02π1n21 cos θ+1νmzcos2 θ×cos2(ωt-k1zs)(sx2+sz2)sin θ dϕdθ.
Ex=-i(I0+I2 cos 2ϕ),
Ey=-i(I2 sin 2Φ),
Ez=-2I1 cos Φ,
I0=0αcos θ sin θ(1+cos θ)J0(krLS)exp(ikrLC)dθ,
I1=0αcos θ sin2 θ J1(krLS)exp(ikrLC)dθ,
I2=0αcos θ sin θ(1-cos θ)J2(krLS)exp(ikrLC)dθ.
I0=0α2 sin θn12 cos θ-1n12-cos θJ0(krLS)exp(ikrLc)dθ,
I1=0αsin θ tan θn12 cos θ-1n12-cos θJ1(krLS)×exp(ikrLC)dθ,
I2=0,
I0=απsin θ J0(krLS)exp(ikrLC)dθ,
I1=απsin θ tan θJ1(krLS)exp(ikrLC)dθ,
I2=0.

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