Abstract

The problem of vectorial diffraction of electromagnetic waves is addressed. An integral representation is obtained for a possibly high-aperture, finite-Fresnel-number lens and a homogeneous medium of propagation. The solution is given in terms of coherent superposition of plane electromagnetic waves with position coordinates scaled with the well-known Li–Wolf scaling factor [J. Opt. Soc. Am. A 1, 801 (1984)]. This integral representation is then used to obtain formulas for the case in which light is focused through a plane dielectric interface. The solution is given by the linear combination of three functions, each of which consists of only a single integral. The aberration function, representing spherical aberration, is shown to be analytical. Numerical examples are given to demonstrate the effectiveness of the solution.

© 1998 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. H. Ling, S. Lee, “Focusing of electromagnetic waves through a dielectric interface,” J. Opt. Soc. Am. A 1, 965–973 (1984).
    [CrossRef]
  2. P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (Pergamon, Oxford, UK, 1966).
  3. J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, 1986).
  4. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1975).
  5. S. F. Gibson, F. Lanni, “Experimental test of an analytical model of aberration in an oil-immersion objective lens used in three-dimensional light microscopy: errata,” J. Opt. Soc. Am. A 9, 154–166 (1992).
    [CrossRef] [PubMed]
  6. S. Hell, G. Reiner, C. Cremer, E. H. K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive-index,” J. Microsc. 169, 391–405 (1993).
    [CrossRef]
  7. 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]
  8. 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]
  9. P. Török, P. Varga, G. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indexes: structure of the electromagnetic field. I,” J. Opt. Soc. Am. A 12, 2136–2144 (1995).
    [CrossRef]
  10. P. Török, P. Varga, G. Németh, “Analytical solution of the diffraction integrals and interpretation of wave-front distortion when light is focused through a planar interface between materials of mismatched refractive indexes,” J. Opt. Soc. Am. A 12, 2660–2671 (1995).
    [CrossRef]
  11. P. Török, P. Varga, A. Konkol, G. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indexes: structure of the electromagnetic field. II,” J. Opt. Soc. Am. A 13, 2232–2238 (1996).
    [CrossRef]
  12. S. H. Wiersma, T. D. Visser, “Defocusing of a converging electromagnetic wave by a plane dielectric interface,” J. Opt. Soc. Am. A 13, 320–325 (1996).
    [CrossRef]
  13. S. Wiersma, P. Török, T. Visser, P. Varga, “Comparison of different theories for focusing through a plane interface,” J. Opt. Soc. Am. A 14, 1482–1490 (1997).
    [CrossRef]
  14. V. Dhayalan, J. J. Stamnes, “Focusing of electromagnetic waves into a dielectric slab: I. Exact and asymptotic results,” Pure Appl. Opt. 7, 33–52 (1998).
    [CrossRef]
  15. 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]
  16. R. G. Wenzel, “Effect of the aperture–lens separation on the focal shift in large-F-number systems,” J. Opt. Soc. Am. A 4, 340–345 (1987).
    [CrossRef]
  17. X. Gan, C. J. R. Sheppard, M. Gu, “Effects of Fresnel diffraction on confocal imaging with an annular lens,” Bioimaging 5, 153–158 (1997).
    [CrossRef]
  18. C. J. R. Sheppard, P. Török, “Dependence of Fresnel number on aperture stop position,” J. Opt. Soc. Am. A 15, 3016–3019 (1998).
    [CrossRef]
  19. P. Török, P. Higdon, T. Wilson, “On the general properties of polarising conventional and confocal microscopes,” Opt. Commun. 148, 300–315 (1998).
    [CrossRef]
  20. P. Török, T. Wilson, “Rigorous theory for axial resolution in confocal microscopes,” Opt. Commun. 137, 127–135 (1997).
    [CrossRef]
  21. J. Stratton, L. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56, 99–107 (1939).
    [CrossRef]
  22. J. Jackson, Classical Electrodynamics (Wiley, New York, 1962).
  23. These conditions are obtained as follows: The physical argument that Σ does not contribute to the field is given by Born and Wolf (Ref. 4, Sec. 8.3.2). When the Stratton–Chu integral, Eq. (1), is applied to the surface Σ, the resulting formula gives zero contribution to the field only if the conditions of Eq. (5) apply. From this surface integral it is also possible to prove that the conditions of Eq. (5) are identical to Sommerfeld’s radiation condition.
  24. W. Hsu, R. Barakat, “Stratton–Chu vectorial diffraction of electromagnetic fields by apertures with application to small-Fresnel-number systems,” J. Opt. Soc. Am. A 11, 623–629 (1994).
    [CrossRef]
  25. 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]
  26. 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, 349–357 (1959).
    [CrossRef]
  27. D. Y. Jiang, J. J. Stamnes, “Focusing at low Fresnel numbers in the presence of cylindrical and spherical aberration,” Pure Appl. Opt. 6, 85–96 (1997).
    [CrossRef]

1998 (3)

V. Dhayalan, J. J. Stamnes, “Focusing of electromagnetic waves into a dielectric slab: I. Exact and asymptotic results,” Pure Appl. Opt. 7, 33–52 (1998).
[CrossRef]

C. J. R. Sheppard, P. Török, “Dependence of Fresnel number on aperture stop position,” J. Opt. Soc. Am. A 15, 3016–3019 (1998).
[CrossRef]

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

1997 (4)

P. Török, T. Wilson, “Rigorous theory for axial resolution in confocal microscopes,” Opt. Commun. 137, 127–135 (1997).
[CrossRef]

S. Wiersma, P. Török, T. Visser, P. Varga, “Comparison of different theories for focusing through a plane interface,” J. Opt. Soc. Am. A 14, 1482–1490 (1997).
[CrossRef]

X. Gan, C. J. R. Sheppard, M. Gu, “Effects of Fresnel diffraction on confocal imaging with an annular lens,” Bioimaging 5, 153–158 (1997).
[CrossRef]

D. Y. Jiang, J. J. Stamnes, “Focusing at low Fresnel numbers in the presence of cylindrical and spherical aberration,” Pure Appl. Opt. 6, 85–96 (1997).
[CrossRef]

1996 (2)

1995 (3)

1994 (1)

1993 (1)

S. Hell, G. Reiner, C. Cremer, E. H. K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive-index,” J. Microsc. 169, 391–405 (1993).
[CrossRef]

1992 (1)

1987 (1)

1984 (2)

1959 (3)

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]

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, 349–357 (1959).
[CrossRef]

1939 (1)

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

Barakat, R.

Booker, G.

Booker, G. R.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1975).

Chu, L.

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

Clemmow, P. C.

P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (Pergamon, Oxford, UK, 1966).

Cremer, C.

S. Hell, G. Reiner, C. Cremer, E. H. K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive-index,” J. Microsc. 169, 391–405 (1993).
[CrossRef]

Dhayalan, V.

V. Dhayalan, J. J. Stamnes, “Focusing of electromagnetic waves into a dielectric slab: I. Exact and asymptotic results,” Pure Appl. Opt. 7, 33–52 (1998).
[CrossRef]

Gan, X.

X. Gan, C. J. R. Sheppard, M. Gu, “Effects of Fresnel diffraction on confocal imaging with an annular lens,” Bioimaging 5, 153–158 (1997).
[CrossRef]

Gibson, S. F.

Gu, M.

X. Gan, C. J. R. Sheppard, M. Gu, “Effects of Fresnel diffraction on confocal imaging with an annular lens,” Bioimaging 5, 153–158 (1997).
[CrossRef]

Hell, S.

S. Hell, G. Reiner, C. Cremer, E. H. K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive-index,” J. Microsc. 169, 391–405 (1993).
[CrossRef]

Higdon, P.

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

Hsu, W.

Jackson, J.

J. Jackson, Classical Electrodynamics (Wiley, New York, 1962).

Jiang, D. Y.

D. Y. Jiang, J. J. Stamnes, “Focusing at low Fresnel numbers in the presence of cylindrical and spherical aberration,” Pure Appl. Opt. 6, 85–96 (1997).
[CrossRef]

Konkol, A.

Laczik, Z.

Lanni, F.

Lee, S.

Li, Y.

Ling, H.

Németh, G.

Reiner, G.

S. Hell, G. Reiner, C. Cremer, E. H. K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive-index,” J. Microsc. 169, 391–405 (1993).
[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]

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, 349–357 (1959).
[CrossRef]

Sheppard, C. J. R.

C. J. R. Sheppard, P. Török, “Dependence of Fresnel number on aperture stop position,” J. Opt. Soc. Am. A 15, 3016–3019 (1998).
[CrossRef]

X. Gan, C. J. R. Sheppard, M. Gu, “Effects of Fresnel diffraction on confocal imaging with an annular lens,” Bioimaging 5, 153–158 (1997).
[CrossRef]

Stamnes, J. J.

V. Dhayalan, J. J. Stamnes, “Focusing of electromagnetic waves into a dielectric slab: I. Exact and asymptotic results,” Pure Appl. Opt. 7, 33–52 (1998).
[CrossRef]

D. Y. Jiang, J. J. Stamnes, “Focusing at low Fresnel numbers in the presence of cylindrical and spherical aberration,” Pure Appl. Opt. 6, 85–96 (1997).
[CrossRef]

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

Stelzer, E. H. K.

S. Hell, G. Reiner, C. Cremer, E. H. K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive-index,” J. Microsc. 169, 391–405 (1993).
[CrossRef]

Stratton, J.

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

Török, P.

C. J. R. Sheppard, P. Török, “Dependence of Fresnel number on aperture stop position,” J. Opt. Soc. Am. A 15, 3016–3019 (1998).
[CrossRef]

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

P. Török, T. Wilson, “Rigorous theory for axial resolution in confocal microscopes,” Opt. Commun. 137, 127–135 (1997).
[CrossRef]

S. Wiersma, P. Török, T. Visser, P. Varga, “Comparison of different theories for focusing through a plane interface,” J. Opt. Soc. Am. A 14, 1482–1490 (1997).
[CrossRef]

P. Török, P. Varga, A. Konkol, G. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indexes: structure of the electromagnetic field. II,” J. Opt. Soc. Am. A 13, 2232–2238 (1996).
[CrossRef]

P. Török, P. Varga, G. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indexes: structure of the electromagnetic field. I,” J. Opt. Soc. Am. A 12, 2136–2144 (1995).
[CrossRef]

P. Török, P. Varga, G. Németh, “Analytical solution of the diffraction integrals and interpretation of wave-front distortion when light is focused through a planar interface between materials of mismatched refractive indexes,” J. Opt. Soc. Am. A 12, 2660–2671 (1995).
[CrossRef]

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]

Varga, P.

Visser, T.

Visser, T. D.

Wenzel, R. G.

Wiersma, S.

Wiersma, S. H.

Wilson, T.

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

P. Török, T. Wilson, “Rigorous theory for axial resolution in confocal microscopes,” Opt. Commun. 137, 127–135 (1997).
[CrossRef]

Wolf, E.

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]

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, 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]

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1975).

Bioimaging (1)

X. Gan, C. J. R. Sheppard, M. Gu, “Effects of Fresnel diffraction on confocal imaging with an annular lens,” Bioimaging 5, 153–158 (1997).
[CrossRef]

J. Microsc. (1)

S. Hell, G. Reiner, C. Cremer, E. H. K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive-index,” J. Microsc. 169, 391–405 (1993).
[CrossRef]

J. Opt. Soc. Am. A (12)

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]

R. G. Wenzel, “Effect of the aperture–lens separation on the focal shift in large-F-number systems,” J. Opt. Soc. Am. A 4, 340–345 (1987).
[CrossRef]

C. J. R. Sheppard, P. Török, “Dependence of Fresnel number on aperture stop position,” J. Opt. Soc. Am. A 15, 3016–3019 (1998).
[CrossRef]

H. Ling, S. Lee, “Focusing of electromagnetic waves through a dielectric interface,” J. Opt. Soc. Am. A 1, 965–973 (1984).
[CrossRef]

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]

P. Török, P. Varga, G. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indexes: structure of the electromagnetic field. I,” J. Opt. Soc. Am. A 12, 2136–2144 (1995).
[CrossRef]

P. Török, P. Varga, G. Németh, “Analytical solution of the diffraction integrals and interpretation of wave-front distortion when light is focused through a planar interface between materials of mismatched refractive indexes,” J. Opt. Soc. Am. A 12, 2660–2671 (1995).
[CrossRef]

P. Török, P. Varga, A. Konkol, G. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indexes: structure of the electromagnetic field. II,” J. Opt. Soc. Am. A 13, 2232–2238 (1996).
[CrossRef]

S. H. Wiersma, T. D. Visser, “Defocusing of a converging electromagnetic wave by a plane dielectric interface,” J. Opt. Soc. Am. A 13, 320–325 (1996).
[CrossRef]

S. Wiersma, P. Török, T. Visser, P. Varga, “Comparison of different theories for focusing through a plane interface,” J. Opt. Soc. Am. A 14, 1482–1490 (1997).
[CrossRef]

W. Hsu, R. Barakat, “Stratton–Chu vectorial diffraction of electromagnetic fields by apertures with application to small-Fresnel-number systems,” J. Opt. Soc. Am. A 11, 623–629 (1994).
[CrossRef]

S. F. Gibson, F. Lanni, “Experimental test of an analytical model of aberration in an oil-immersion objective lens used in three-dimensional light microscopy: errata,” J. Opt. Soc. Am. A 9, 154–166 (1992).
[CrossRef] [PubMed]

Opt. Commun. (2)

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

P. Török, T. Wilson, “Rigorous theory for axial resolution in confocal microscopes,” Opt. Commun. 137, 127–135 (1997).
[CrossRef]

Phys. Rev. (1)

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

Proc. R. Soc. London, Ser. A (3)

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]

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, 349–357 (1959).
[CrossRef]

Pure Appl. Opt. (2)

D. Y. Jiang, J. J. Stamnes, “Focusing at low Fresnel numbers in the presence of cylindrical and spherical aberration,” Pure Appl. Opt. 6, 85–96 (1997).
[CrossRef]

V. Dhayalan, J. J. Stamnes, “Focusing of electromagnetic waves into a dielectric slab: I. Exact and asymptotic results,” Pure Appl. Opt. 7, 33–52 (1998).
[CrossRef]

Other (5)

P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (Pergamon, Oxford, UK, 1966).

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

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1975).

J. Jackson, Classical Electrodynamics (Wiley, New York, 1962).

These conditions are obtained as follows: The physical argument that Σ does not contribute to the field is given by Born and Wolf (Ref. 4, Sec. 8.3.2). When the Stratton–Chu integral, Eq. (1), is applied to the surface Σ, the resulting formula gives zero contribution to the field only if the conditions of Eq. (5) apply. From this surface integral it is also possible to prove that the conditions of Eq. (5) are identical to Sommerfeld’s radiation condition.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

Schematic diagram of the optical system for a homogeneous medium of propagation.

Fig. 2
Fig. 2

Schematic diagram of the optical system in the presence of a dielectric interface.

Fig. 3
Fig. 3

Axial line distributions of the time-averaged electric-energy density for focusing through a dielectric interface between media of mismatched refractive indices.

Fig. 4
Fig. 4

Contours of the time-averaged electric-energy density for focusing through a dielectric interface between media of mismatched refractive indices. The spacing between individual contour lines is 2.5 dB. (a) Case 1, (b) case 2.

Equations (60)

Equations on this page are rendered with MathJax. Learn more.

E(rP)=-14π S [ik0(m×H)G+(m×E)×G+(m·E)G]dS,
G=exp(ikr)r=exp(ik|rP-rQ|)|rP-rQ|,
limR G=exp(ik|rQ|)|rQ| exp(ikm·rP),
limR G=-ikmG
E(rP)=-14π S1[ik0(m×H)G+(m×E)×G+(m·E)G]dS1.
E,H=Ei, HionS10otherwise
QrQ=(xQ, yQ, zQ)=-f sin θ cos ϕi-f sin θ sin ϕj-f cos θk
PrP=(x, y, z)=(ρ cos γ, ρ sin γ, z),
Hi=Ei×m,
E(rP)=-ik4π S1[m×(Ei×m)+(m×Ei)×r+(m×Ei)·r]GdS1,
r=|rQ-rP|=[(xQ-x)2+(yQ-y)2+(zQ-z)2]1/2.
r=[(f+z)2+ρ2-2 fz+2 fz cos θ+2 fρ sin θ cos(γ-ϕ)]1/2f+z+12(f+z)[ρ2-2zf+2 fρ sin θ×cos(γ-ϕ)+2 fz cos θ+].
E(ρ, γ, z)=-ikE0f2×exp(iΦ)2π(z+f) 0α02πEig(θ)×expik fz+f[sin θ cos(γ-ϕ)]×expik fz+fz cos θsin θdθdϕ,
Φ=k(z+f)+ρ22(z+f)-fz+fz.
R=fz+fρ,Z=fz+fz,
s=sin θ cos ϕi+sin θ sin ϕj+cos θk=sxi+syj+szk,
P=R cos γi+R sin γj+Zk=Xi+Yj+Zk,
E=Asx2+sy21 g(sx, sy) Ei(sx, sy)sz exp(iks·P)dsxdsy,
A(x, y, z; k, f)=-ikE0f2 exp(iΦ)2π(z+f).
s1=s1xi+s1yj+s1zk=s1t+s1zk;
s1z=(1-s1x2-s1y2)1/2,
s2=s2xi+s2yj+s2zk=s2t+s2zk;
s2z=(1-s2x2-s2y2)1/2,
k1s1x=k2s2x;k1s1y=k2s2y,
s2z=1-n12n22(s1x2+s1y2)1/2.
P=Xi+Yj+Zk=rt+Zk
E1(x, y, z)=A(x, y, z; k1, f)Ω1W(s1t)×exp(ik1s1t·rt)exp(ik1s1zZ)ds1xds1y,
W(s1t)=g(s1t) Ei(s1t)s1z
E2(x, y,-d)=A(x, y,-d; k1, f)Ω1MW(s1t)×exp(ik1s1t·rt)×exp-ik1s1z fdf-dds1xds1y,
E2(x, y, z)=A(x, y, z; k2, f)Ω2F(s2t)×exp(ik2s2t·rt)exp(ik2s2zZ)ds2xds2y,
E2(x, y, z)=A(x, y, z; k2, f)Ω1F(s2t)×exp(ik1s1t·rt)×exp(ik2s2zZ)J(s1t, s2t)ds1xds1y,
J(s1t, s2t)=(k1/k2)2.
F(s2t)=A(x, y,-d; k1, f)A(x, y,-d; k2, f) MW(s1t)J(s1t, s2t)×exp-i fdf-d(k1s1z-k2s2z).
E2(x, y, z)=A(x, y, z; k2, f) A(x, y,-d; k1, f)A(x, y,-d; k2, f)×Ω1MW(s1t)exp(iΨ)exp(ik1s1t·rt)×exp(ik2s2zZ)ds1xds1y,
Ψ(k1, k2, d)=-dff-d(k1s1z-k2s2z).
c=MW=g(θ1)cos θ1×τp cos θ2 cos2 ϕ+τs sin2 ϕτp cos θ2 sin ϕ cos ϕ-τs sin ϕ cos ϕ-τp sin θ2 cos ϕ,
d=(2/μ2)1/2s2×c,
d=2μ21/2 g(θ1)cos θ1×-τp sin ϕ cos ϕ+τs cos θ2 sin ϕ cos ϕτp cos2 ϕ+τs sin2 ϕ cos θ2-τs cos θ2 sin ϕ.
E2x=-iK[I0(e)+I2(e) cos 2γ],
E2y=-iKI2(e) sin 2γ,
E2z=-2KI1(e) cos γ,
H2x=-iKn2I2(h) sin 2γ,
H2y=-iKn2[I0(h)-I2(h) cos 2γ],
H2z=-2Kn2I1(h) sin γ,
K=iE0k1 f2f+z exp{i[Φ1(k1, k2, f, d)+Φ2(k2, f, z)]},
Φ1(k1, k2, f, d)=(k1-k2)×(f-d)+ρ22(f-d)+fdf-d,
Φ2(k2, f, z)=k2(f+z)+ρ22(f+z)+fzf+z,
I0(e)=0αg(θ1)(τs+τp cos θ2)sin θ1J0k1 fρf+z sin θ1×exp(iΨ)expik2 fzf+z cos θ2dθ1,
I1(e)=0αg(θ1)τp sin θ2 sin θ1J1k1 fρf+z sin θ1×exp(iΨ)expik2 fzf+z cos θ2dθ1,
I2(e)=0αg(θ1)(τs-τp cos θ2)sin θ1J2k1 fρf+z sin θ1×exp(iΨ)expik2 fzf+z cos θ2dθ1,
I0(h)=0αg(θ1)(τp+τs cos θ2)sin θ1J0k1 fρf+z sin θ1×exp(iΨ)expik2 fzf+z cos θ2dθ1
I1(h)=0αg(θ1)τs sin θ2 sin θ1J1k1 fρf+z sin θ1×exp(iΨ)expik2 fzf+z cos θ2dθ1
I2(h)=0αg(θ1)(τp-τs cos θ2)sin θ1J2k1 fρf+z sin θ1×exp(iΨ)expik2 fzf+z cos θ2dθ1,
Ψ(k1, k2, d)=-dff-d(k1 cos θ1-k2 cos θ2)
I0=0αg(θ)(1+cos θ)sin θJ0k fρf+z sin θ×expik fzf+z cos θdθ,
I1=0αg(θ)sin2 θJ1k fρf+z sin θ×expik fzf+z cos θdθ,
I2>=0αg(θ)(1-cos θ)sin θJ2k fρf+z sin θ×expik fzf+z cos θdθ,
K=iE0k f2f+z exp[iΦ(k, f, z)],
Φ(k, f, z)=k(f+z)+ρ22(f+z)+fzf+z.
we(ρ, γ, z; d, k1, k2)(E·E*).

Metrics