Abstract

We derive a solution to the problem of a plane electromagnetic wave focused by a parabolic mirror. The solution is obtained from the Stratton–Chu integral by solving a boundary-value problem. Our solution can be considered self-consistent. We also derive the far-field, i.e., Debye, approximation of our formulas. The solution shows that when the paraboloid is infinite, its focusing properties exhibit a dispersive behavior; that is, the structure of the field distribution in the vicinity of the focus strongly depends on the wavelength of the illumination. We show that for an infinite paraboloid the confinement of the focused energy worsens, with the energy distribution spreading in the focal plane.

© 2000 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. A. Boivin, E. Wolf, “Electromagnetic field in the neighborhood of a focus of a coherent beam,” Phys. Rev. 138, B1561–B1565 (1965).
    [CrossRef]
  4. A. Boivin, J. Dow, E. Wolf, “Energy flow in the neighborhood of the focus of a coherent beam,” J. Opt. Soc. Am. 57, 1171–1175 (1967).
    [CrossRef]
  5. C. J. R. Sheppard, T. Wilson, “The image of a single point in microscopes of large numerical aperture,” Proc. R. Soc. London, Ser. A 379, 145–158 (1982).
    [CrossRef]
  6. P. Török, P. Varga, Z. Laczik, G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices—an integral representation,” J. Opt. Soc. Am. A 12, 325–332 (1995).
    [CrossRef]
  7. J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986), Sec. 16.1.
  8. W. Wang, A. T. Friberg, E. Wolf, “Structure of focused fields in systems with large Fresnel numbers,” J. Opt. Soc. Am. A 12, 1947–1953 (1995).
    [CrossRef]
  9. P. Török, “Focusing of electromagnetic waves through a dielectric interface by lenses of finite Fresnel number,” J. Opt. Soc. Am. A 15, 3009–3015 (1998).
    [CrossRef]
  10. V. S. Ignatowsky, “The relationship between geometrical and wave optics and diffraction of homocentrical beams,” Trans. Opt. Inst. 1(3), 1–30 (1920).
  11. V. S. Ignatowsky, “Diffraction by a paraboloid mirror with arbitrary aperture,” Trans. Opt. Inst. 1(5), 1–30 (1920).
  12. V. Galindo-Israel, R. Mittra, “A new series representation for the radiation integral with application to reflector antennas,” IEEE Trans. Antennas Propag. AP-25, 631–641 (1985).
  13. H. Ling, S.-W. Lee, P. T. C. Lam, W. V. T. Rusch, “Focal shift in parabolic reflectors,” IEEE Trans. Antennas Propag. AP-33, 744–748 (1977).
  14. J. Stratton, L. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56, 99–107 (1939).
    [CrossRef]
  15. F. Kottler, “Diffraction at a black screen, Part I,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1965), Vol. IV, Chap. VII, and F. Kottler, “Diffraction at a black screen, Part II,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1967), Vol. VI, Chap. I.
  16. P. Debye, “Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie,” Ann. Phys. (Leipzig) 30, 755–776 (1909).
    [CrossRef]
  17. R. K. Luneburg, Mathematical Theory of Optics (University of California, Berkeley, Berkeley, Calif., 1964), Sec. 47.
  18. D. J. Innes, A. L. Bloom, “Design of optical systems for use with laser beams,” Spectra-Phys. Laser Tech. Bull. 5, 1–10 (1966).
  19. C. J. R. Sheppard, A. Choudhury, J. Gannaway, “Electromagnetic field near the focus of wide-angular lens and mirror systems,” Microwave Opt. Acoust. 1, 129–132 (1977).
    [CrossRef]
  20. J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986), Sec. 16.1.2.
  21. P. Varga, P. Török, “Focusing of electromagnetic waves by paraboloid mirrors. II. Numerical results,” J. Opt. Soc. Am. A 17, 2090–2095 (2000).
    [CrossRef]
  22. Y. Li, E. Wolf, “Three-dimensional intensity distribution near the focus in systems of different Fresnel numbers,” J. Opt. Soc. Am. A 1, 801–808 (1984).
    [CrossRef]

2000

1998

1995

1985

V. Galindo-Israel, R. Mittra, “A new series representation for the radiation integral with application to reflector antennas,” IEEE Trans. Antennas Propag. AP-25, 631–641 (1985).

1984

1982

C. J. R. Sheppard, T. Wilson, “The image of a single point in microscopes of large numerical aperture,” Proc. R. Soc. London, Ser. A 379, 145–158 (1982).
[CrossRef]

1977

H. Ling, S.-W. Lee, P. T. C. Lam, W. V. T. Rusch, “Focal shift in parabolic reflectors,” IEEE Trans. Antennas Propag. AP-33, 744–748 (1977).

C. J. R. Sheppard, A. Choudhury, J. Gannaway, “Electromagnetic field near the focus of wide-angular lens and mirror systems,” Microwave Opt. Acoust. 1, 129–132 (1977).
[CrossRef]

1967

1966

D. J. Innes, A. L. Bloom, “Design of optical systems for use with laser beams,” Spectra-Phys. Laser Tech. Bull. 5, 1–10 (1966).

1965

A. Boivin, E. Wolf, “Electromagnetic field in the neighborhood of a focus of a coherent beam,” Phys. Rev. 138, B1561–B1565 (1965).
[CrossRef]

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

1920

V. S. Ignatowsky, “The relationship between geometrical and wave optics and diffraction of homocentrical beams,” Trans. Opt. Inst. 1(3), 1–30 (1920).

V. S. Ignatowsky, “Diffraction by a paraboloid mirror with arbitrary aperture,” Trans. Opt. Inst. 1(5), 1–30 (1920).

1909

P. Debye, “Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie,” Ann. Phys. (Leipzig) 30, 755–776 (1909).
[CrossRef]

Bloom, A. L.

D. J. Innes, A. L. Bloom, “Design of optical systems for use with laser beams,” Spectra-Phys. Laser Tech. Bull. 5, 1–10 (1966).

Boivin, A.

A. Boivin, J. Dow, E. Wolf, “Energy flow in the neighborhood of the focus of a coherent beam,” J. Opt. Soc. Am. 57, 1171–1175 (1967).
[CrossRef]

A. Boivin, E. Wolf, “Electromagnetic field in the neighborhood of a focus of a coherent beam,” Phys. Rev. 138, B1561–B1565 (1965).
[CrossRef]

Booker, G. R.

Choudhury, A.

C. J. R. Sheppard, A. Choudhury, J. Gannaway, “Electromagnetic field near the focus of wide-angular lens and mirror systems,” Microwave Opt. Acoust. 1, 129–132 (1977).
[CrossRef]

Chu, L.

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

Debye, P.

P. Debye, “Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie,” Ann. Phys. (Leipzig) 30, 755–776 (1909).
[CrossRef]

Dow, J.

Friberg, A. T.

Galindo-Israel, V.

V. Galindo-Israel, R. Mittra, “A new series representation for the radiation integral with application to reflector antennas,” IEEE Trans. Antennas Propag. AP-25, 631–641 (1985).

Gannaway, J.

C. J. R. Sheppard, A. Choudhury, J. Gannaway, “Electromagnetic field near the focus of wide-angular lens and mirror systems,” Microwave Opt. Acoust. 1, 129–132 (1977).
[CrossRef]

Ignatowsky, V. S.

V. S. Ignatowsky, “The relationship between geometrical and wave optics and diffraction of homocentrical beams,” Trans. Opt. Inst. 1(3), 1–30 (1920).

V. S. Ignatowsky, “Diffraction by a paraboloid mirror with arbitrary aperture,” Trans. Opt. Inst. 1(5), 1–30 (1920).

Innes, D. J.

D. J. Innes, A. L. Bloom, “Design of optical systems for use with laser beams,” Spectra-Phys. Laser Tech. Bull. 5, 1–10 (1966).

Kottler, F.

F. Kottler, “Diffraction at a black screen, Part I,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1965), Vol. IV, Chap. VII, and F. Kottler, “Diffraction at a black screen, Part II,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1967), Vol. VI, Chap. I.

Laczik, Z.

Lam, P. T. C.

H. Ling, S.-W. Lee, P. T. C. Lam, W. V. T. Rusch, “Focal shift in parabolic reflectors,” IEEE Trans. Antennas Propag. AP-33, 744–748 (1977).

Lee, S.-W.

H. Ling, S.-W. Lee, P. T. C. Lam, W. V. T. Rusch, “Focal shift in parabolic reflectors,” IEEE Trans. Antennas Propag. AP-33, 744–748 (1977).

Li, Y.

Ling, H.

H. Ling, S.-W. Lee, P. T. C. Lam, W. V. T. Rusch, “Focal shift in parabolic reflectors,” IEEE Trans. Antennas Propag. AP-33, 744–748 (1977).

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (University of California, Berkeley, Berkeley, Calif., 1964), Sec. 47.

Mittra, R.

V. Galindo-Israel, R. Mittra, “A new series representation for the radiation integral with application to reflector antennas,” IEEE Trans. Antennas Propag. AP-25, 631–641 (1985).

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]

Rusch, W. V. T.

H. Ling, S.-W. Lee, P. T. C. Lam, W. V. T. Rusch, “Focal shift in parabolic reflectors,” IEEE Trans. Antennas Propag. AP-33, 744–748 (1977).

Sheppard, C. J. R.

C. J. R. Sheppard, T. Wilson, “The image of a single point in microscopes of large numerical aperture,” Proc. R. Soc. London, Ser. A 379, 145–158 (1982).
[CrossRef]

C. J. R. Sheppard, A. Choudhury, J. Gannaway, “Electromagnetic field near the focus of wide-angular lens and mirror systems,” Microwave Opt. Acoust. 1, 129–132 (1977).
[CrossRef]

Stamnes, J. J.

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986), Sec. 16.1.

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986), Sec. 16.1.2.

Stratton, J.

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

Török, P.

Varga, P.

Wang, W.

Wilson, T.

C. J. R. Sheppard, T. Wilson, “The image of a single point in microscopes of large numerical aperture,” Proc. R. Soc. London, Ser. A 379, 145–158 (1982).
[CrossRef]

Wolf, E.

W. Wang, A. T. Friberg, E. Wolf, “Structure of focused fields in systems with large Fresnel numbers,” J. Opt. Soc. Am. A 12, 1947–1953 (1995).
[CrossRef]

Y. Li, E. Wolf, “Three-dimensional intensity distribution near the focus in systems of different Fresnel numbers,” J. Opt. Soc. Am. A 1, 801–808 (1984).
[CrossRef]

A. Boivin, J. Dow, E. Wolf, “Energy flow in the neighborhood of the focus of a coherent beam,” J. Opt. Soc. Am. 57, 1171–1175 (1967).
[CrossRef]

A. Boivin, E. Wolf, “Electromagnetic field in the neighborhood of a focus of a coherent beam,” Phys. Rev. 138, B1561–B1565 (1965).
[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]

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]

Ann. Phys. (Leipzig)

P. Debye, “Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie,” Ann. Phys. (Leipzig) 30, 755–776 (1909).
[CrossRef]

IEEE Trans. Antennas Propag.

V. Galindo-Israel, R. Mittra, “A new series representation for the radiation integral with application to reflector antennas,” IEEE Trans. Antennas Propag. AP-25, 631–641 (1985).

H. Ling, S.-W. Lee, P. T. C. Lam, W. V. T. Rusch, “Focal shift in parabolic reflectors,” IEEE Trans. Antennas Propag. AP-33, 744–748 (1977).

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Microwave Opt. Acoust.

C. J. R. Sheppard, A. Choudhury, J. Gannaway, “Electromagnetic field near the focus of wide-angular lens and mirror systems,” Microwave Opt. Acoust. 1, 129–132 (1977).
[CrossRef]

Phys. Rev.

A. Boivin, E. Wolf, “Electromagnetic field in the neighborhood of a focus of a coherent beam,” Phys. Rev. 138, B1561–B1565 (1965).
[CrossRef]

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

Proc. R. Soc. London, Ser. A

C. J. R. Sheppard, T. Wilson, “The image of a single point in microscopes of large numerical aperture,” Proc. R. Soc. London, Ser. A 379, 145–158 (1982).
[CrossRef]

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]

Spectra-Phys. Laser Tech. Bull.

D. J. Innes, A. L. Bloom, “Design of optical systems for use with laser beams,” Spectra-Phys. Laser Tech. Bull. 5, 1–10 (1966).

Trans. Opt. Inst.

V. S. Ignatowsky, “The relationship between geometrical and wave optics and diffraction of homocentrical beams,” Trans. Opt. Inst. 1(3), 1–30 (1920).

V. S. Ignatowsky, “Diffraction by a paraboloid mirror with arbitrary aperture,” Trans. Opt. Inst. 1(5), 1–30 (1920).

Other

F. Kottler, “Diffraction at a black screen, Part I,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1965), Vol. IV, Chap. VII, and F. Kottler, “Diffraction at a black screen, Part II,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1967), Vol. VI, Chap. I.

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986), Sec. 16.1.

R. K. Luneburg, Mathematical Theory of Optics (University of California, Berkeley, Berkeley, Calif., 1964), Sec. 47.

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986), Sec. 16.1.2.

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Figures (4)

Fig. 1
Fig. 1

Notation of the optical system.

Fig. 2
Fig. 2

Segment of the paraboloid and relevant notation.

Fig. 3
Fig. 3

Far-field projection on the surface of a sphere.

Fig. 4
Fig. 4

Various sections on the surface of the paraboloid for solving the boundary-value problem.

Equations (92)

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zs=xs2+ys24f-f,
zs+2 f=ρs.
nx=-xs2fρs,ny=-ys2fρs,nz=fρs.
ρs(θ)=2 f1-cos θs,
nx=-sin θscos ϕs[2(1-cos θs)]1/2,
ny=-sin θssin ϕs[2(1-cos θs)]1/2,
nz=1-cos θs21/2.
Er=2n(Ein)-Ei.
Hr=Hi-2n(Hin).
Ei=2n(Ern)-Er,Hi=Hr-2n(Hrn).
E(P)=14πA[ik(n×H)G+(n×E)G+(nE)G]dA.
E(P)=14πA[ik(n×H)G+(n×E)×G+(nE)G]dA+14πikCG(Hds),
G(u)=exp(iku)/u
u=[(xs-xp)2+(ys-yp)2+(zs-zp)2]1/2.
H(P)=14πA[ik(E×n)G+(n×H)×G+(nH)G]dA-14πikCG(Eds).
E(S)=Ei+Er=2n[nEi(S)],
H(S)=Hi+Hr=2{Hi(S)-n[nHi(S)]}.
Ei=(Ei,x, 0, 0),Ei,x=a exp(-ikz),
Hi=(0, Hi,y, 0),Hi,y=-a exp(-ikz),
a=exp[i(π/2-kf )].
n×H(S)=2n×Hi(S)=2(nzex-nxez)exp(-ikzs),
n×E(S)=0,
nE(S)=2nEi(S)=2nxexp(-ikzs),
G(u)=ik1-1iku×G(u)u (Δxex+Δyey+Δzez),
dA=1+zsxs2+zsys21/2dxsdys=ρsfdxsdys=ρs2nzsin θsdθsdϕs.
Esurf(P; δ1, δ2)=a exp(2ikf )2π ikδ1δ202πdθsdϕs×exp[ik(u-ρs)]u ρs2sin θs×1+1-1ikunxnzΔxuex+1-1ikunxnzΔyuey+-nxnz+1-1ikunxnzΔzuez,
nxnz=-sin θscos ϕs1-cos θs.
Econt(P)=1ik14πG(u)[H(S)  ds]
ds=2 f sin δ1-cos δ (-sin ϕsex+cos ϕsey)dϕs.
Hx=2anxnyexp[-ikz(δ)]=a(1+cos δ)sin ϕscos ϕsexp[-ikz(δ)],
Hy=-2a(1-ny2)exp[-ikz(δ)]=a[-2+(1+cos δ)sin2 ϕs]exp[-ikz(δ)],
Hz=2anynzexp[-ikz(δ)]=-a sin δ sin ϕsexp[-ikz(δ)],
Hs  ds=-4af sin δ1-cos δcos ϕsdϕs
Econt(P; δ)=-afexp(2ikf )πsin δ1-cos δ×02π1-1ikuexp{ik[u-ρs(δ)]}u×Δxuex+Δyuey+Δzuezcos ϕsdϕs.
S=c4πEr×Hr=c4π-xsρsesx-ysρsey-zsρsezexp(-2ikzs),
Er,x(S)=a(cos θscos2 ϕs-sin2 ϕs)exp(-ikzs),
Er,y(S)=a[(cos θs+1)sin ϕscos ϕs]exp(-ikzs),
Er,z(S)=-a sin θscos ϕsexp(-ikzs),
Hr,x(S)=a[(cos θs+1)sin ϕscos ϕs]exp(-ikzs),
Hr,y(S)=a(cos θssin2 ϕs-cos2 ϕs)exp(-ikzs),
Hr,s(S)=-a sin θssin ϕsexp(-ikzs).
|E^r(S)×H^r(S)|ρs2dΩ=|Eˆ(Q)×Hˆ(Q)|ρ02dΩ.
Eˆ(Q)=ρsρ0E^r(S),Hˆ(Q)=ρsρ0H^r(S).
exp[-ik(ρ0-ρs)]=exp[-ik(ρ0-zs-2 f )].
E(Q)=2 fρ0exp[-ik(ρ0-2 f )]1-cos θsEr(S),
H(Q)=2 f ρ0exp[-ik(ρ0-2 f )]1-cos θsHr(S).
ED(P)=-ika exp(2ikf )4π×απ02πE^D(Q)exp{ikrp[cos θscos θp+sin θssin θpcos(ϕs-ϕp)]}× ×sin θsdϕsdθs,
ED,x(P)=ika exp(2ikf )4 (I0+I2cos 2ϕp),
ED,y(P)=ika exp(2ikf )4 I2sin 2ϕp,
ED,z(P)=-ka exp(2ikf )2 I1cos ϕp,
I0=απsin θs J0(krpsin θs)exp(ikzpcos θs)dθs,
I1=απ(1+cos θs)sin θsJ1(krpsin θs)×exp(ikzpcos θs)dθs,
I2=απ1+cos θs1-cos θssin θsJ2(krpsin θs)×exp(ikzpcos θs)dθs.
Ei(in)(xs, ys, zs)=a exp[-i(ωt+kzs)]ex,
Hi(in)(xs, ys, zs)=-a exp[-i(ωt+kzs)]ey,
Er(out)(xs, ys, zs)=a*exp[-i(ωt-kzs)]ex,
Hr(out)(xs, ys, zs)=a*exp[-i(ωt-kzs)]ey.
E(in)(S)=Ei(in)(S)+Er(in)(S)=2n[n  Ei(in)(S)].
E(out)(S)=2n[n  Er(out)(S)].
E(S)=E(in)(S)+E(out)(S).
H(in)(S)=Hi(in)(S)+Hr(in)(S)=2{Hi(in)(S)-n[n  Hi(in)(S)]},
H(out)(S)=Hi(out)(S)+Hr(out)(S)=2{Hr(out)(S)-n[n  Hr(out)(S)]},
H(S)=H(in)(S)+H(out)(S).
E(i)=Esurf(P; π-α, π)
E(ii)=Econt*(P; π-α).
E(iii, in)=Esurf(P; α, π-α).
E(iii, out)=Esurf*(P; α, π-α).
E(iii)=Esurf(P; α, π-α)+Esurf*(P; α, π-α).
E(iv)=Econt(P; α)+Econt*(P; α).
E(P)=E(i)+E(ii)+E(iii)+E(iv).
Esurf(F; δ1; δ2)=-a exp(2ikf )π ikδ1δ202π(ρssin θs)×1+1+ikρsnxnzΔxρsex+1+ikρsnxnzΔyρsey-nxnz1-1+ikρsΔzρsezdϕsdθs.
Esurf(F; δ1, δ2)=-exp(ikf )(cos δ2-cos δ1)×kf-ikf (1+cos δ2+cos δ1).
Esurf(F; α, π)=exp(ikf )(1+cos α)kf-i2cos α.
Econt(F, α)=exp(ikf )(sin2 α)-14kf (1-cos α)+i2,
I=E  E*=(kf )2(1+cos α)2,
Iz=EE*z+E*Ez=2 ReEE*z.
Esurf(P; δ1, δ2)zP=F=k4exp(ikf )6g(δ)+2δ+i4kfg(δ)+2 cotδ2-364cos 2δ+164cos 6δδ1δ2ex,
g(δ)=52cos δ+58cos 2δ+16cos 3δ+132cos 4δ+4 logsinδ2.
Esurf(P;δ1, δ2)zP=F=1fexp(ikf )sin2 δ cos δ×-kf2-3(1-cos δ)28kf+3i4 (1-cos δ).
IzF=k2f1+cos α419316+2π-6g(α)-2α-2 sin2 α cos α-2(1+cos 2α)×19396-g(α)-2 cotα2.
Esurf(F)=(A cos kf+B sin kf )ex,
A=-2kf(cos δ2-cos δ1),
B=12 (cos δ2-cos δ1)(1+cos δ1+cos δ2).
Esurf(F)=(4kf cos kf+2 sin kf )ex;
u={ρs2+r2-2ρsr[cos θ cos θs+sin θ sin θscos(ϕ-ϕ2)]}1/2.
ukkρs-kr[cos θ cos θs+sin θ sin θscos(ϕ-ϕs)]+R,
r<1δ2fRk
2 Re[Esurf(P; δ1, δ2)]=-2kf(cos kf )[B0(r)ex+B2(r)×(cos 2ϕ ex+sin 2ϕey)]-4kf sin kf cos ϕ B1(r)ez,
B0(r)=δ1δ2sin θs J0(kr sin θs)dθs,
B1(r)=δ1δ2sin θs1-cos θs J1(kr sin θs)dθs,
B2(r)=δ1δ2sin θs1-cos θs (1+cos θs)×J2(kr sin θs)dθs.
2 Re[Esurf(P; δ1, δ2)]=2kf Re[exp(ikf )]×δ1δ2exp(ikr cos θs)sin θsdθs8kf cos[k(f-r)] sin krkr

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