Abstract

Near-field optical imaging mechanisms are investigated on an elementary level with the use of a coupled dipole formalism. Both optical probe and sample particles are considered as single dipolar cells. The sample particles are located on the surface of a layered substrate, and the optical probe is guided over the sample at constant height. The theory of the coupled dipole formalism with the use of Green's functions of a layered reference system is outlined, and asymptotic forms for far-field radiation are derived. Depending on the direction of observation, the recorded far-field radiation contains different information. It is shown that radiation emitted into the lower half-space at angles within the critical angle of total internal reflection (allowed light) provides reliable images that are only weakly correlated to the topography of the sample. Higher resolution is achieved with the radiation emitted at supercritical angles (forbidden light), but the high sensitivity to topographical variations makes image interpretation difficult. In order to image the optical properties of the sample, it is shown to be unfavorable to include the forbidden light in the optical signal.

© 1997 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. O. J. F. Martin, A. Dereux, and C. Girard, “Iterative scheme for computing exactly the total field propagating in dielectric structures of arbitrary shape,” J. Opt. Soc. Am. A 11, 1073–1080 (1994).
    [CrossRef]
  2. L. Novotny, D. W. Pohl, and P. Regli, “Light propagation through nanometer-sized structures: the two-dimensional-aperture scanning near-field optical microscope,” J. Opt. Soc. Am. A 11, 1768–1779 (1994).
    [CrossRef]
  3. R. Carminati and J. J. Greffet, “Influence of dielectric contrast and topography on the near field scattered by an inhomogeneous surface,” J. Opt. Soc. Am. A 12, 2716–2725 (1995).
    [CrossRef]
  4. D. A. Christensen, “Analysis of near field tip patterns including object interaction using finite-difference time-domain calculations,” Ultramicroscopy 57, 189–196 (1995).
    [CrossRef]
  5. J. L. Kann, T. D. Milster, F. Froehlich, R. W. Ziolkowski, and J. Judkins, “Numerical analysis of a two-dimensional near-field probe,” Ultramicroscopy 57, 251–256 (1995).
    [CrossRef]
  6. D. V. Labeke, D. Barchiesi, and F. Baida, “Optical characterization of nanosources used in scanning near-field optical microscopy,” J. Opt. Soc. Am. A 12, 695–703 (1995).
    [CrossRef]
  7. L. Novotny and D. W. Pohl, “Light propagation in scanning near-field optical microscopy,” in Photons and Local Probes, O. Marti and R. Möller, eds., Vol. 300 of NATO Advanced Study Institute, Series E (Kluwer, Dordrecht, The Netherlands, 1995), pp. 21–33.
  8. L. Novotny, “Allowed and forbidden light in near-field optics. I. A single dipolar light source,” J. Opt. Soc. Am. A 14, 91–104 (1997).
    [CrossRef]
  9. B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491–1499 (1994).
    [CrossRef]
  10. K. Lumme and J. Rahola, “Light scattering by porous dust particles in the discrete-dipole approximation,” Astrophys. J. 425, 653–667 (1994).
    [CrossRef]
  11. S. B. Singham and C. F. Bohren, “Light scattering by an arbitrary particle: the scattering-order formulation of the coupled-dipole method,” J. Opt. Soc. Am. A 5, 1867–1872 (1988).
    [CrossRef] [PubMed]
  12. M. A. Taubenblatt and T. K. Tran, “Calculation of light scattering from particles and structures on a surface by the coupled-dipole method,” J. Opt. Soc. Am. A 10, 912–919 (1993).
    [CrossRef]
  13. R. F. Harrington, Field Computation by Moment Methods (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1992).
  14. J. J. H. Wang, Generalized Moment Methods in Electromagnetics (Wiley, Chichester, UK, 1991).
  15. M. N. O. Sadiku, Numerical Techniques in Electromagnetics (CRC Press, Boca Raton, Fla., 1992), Chap. 5.7.
  16. D. E. Livesay and K. M. Chen, “Electromagnetic fields induced inside arbitrarily shaped biological bodies,” IEEE Trans. Microwave Theory Tech. MTT-22, 1273–1280 (1974).
    [CrossRef]
  17. M. F. Iskander, H. Y. Chen, and J. E. Penner, “Optical scattering and absorption by branched chains of aerosols,” Appl. Opt. 28, 3083–3091 (1989).
    [CrossRef] [PubMed]
  18. G. H. Goedecke and S. G. O’Brien, “Scattering by irregular inhomogeneous particles via the digitized Green’s function algorithm,” Appl. Opt. 27, 2431–2438 (1989).
    [CrossRef]
  19. A. Lakhtakia, “Strong and weak forms of the method method of moments and the coupled dipole method for scattering of time-harmonic electromagnetic fields,” J. Mod. Phys. C. 3, 583–603 (1992).
  20. W. C. Chew, Waves and Fields in Inhomogeneous Media, 2nd ed. (Institute of Electrical and Electronics Engineers, New York, 1995).
  21. A. D. Yaghjian, “Electric dyadic Green’s functions in the source region,” Proc. IEEE 68, 248–263 (1980).
    [CrossRef]
  22. J. V. Bladel, “Some remarks on Green’s dyadic for infinite space.” IRE Trans. Antennas Propagat. 9, 563–566 (1961).
  23. A. D. McLachlan, “Van der Waals forces between an atom and a surface,” Mol. Phys. 34, 381–388 (1964).
    [CrossRef]
  24. C. F. Bohren and D. R. Huffmann, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  25. B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
    [CrossRef]
  26. J. C. Ku, “Comparison of coupled-dipole solutions and dipole refractive indices for light scattering and absorption by arbitrary shaped or agglomerated particles,” J. Opt. Soc. Am. A 10, 336–342 (1993).
    [CrossRef]
  27. R. Loudon, The Quantum Theory of Light, 2nd ed. (Clarendon, Oxford, 1983).
  28. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1970).
  29. B. Hecht, D. W. Pohl, H. Heinzelmann, and L. Novotny, “Tunnel near-field optical microscopy: TNOM-2,” in Photons and Local Probes, O. Marti and R. Möller, eds., Vol. 300 of NATO Advanced Study Institute, Series E (Kluwer, Dordrecht, The Netherlands, 1995), pp. 93–107.
  30. G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, New York, 1985).
  31. N. Bleistein and R. A. Handelsman, Asymptotic Expansion of Integrals, 1st ed. (Rinehart & Winston, New York, 1975).
  32. L. Novotny, “Light propagation and light confinement in near-field optics,” Ph.D. dissertation 11420 (Swiss Federal Institute of Technology, Zurich, Switzerland, 1996).
  33. L. M. Brekhovskikh, Waves in Layered Media, 2nd ed. (Academic, New York, 1980).

1997 (1)

1995 (4)

R. Carminati and J. J. Greffet, “Influence of dielectric contrast and topography on the near field scattered by an inhomogeneous surface,” J. Opt. Soc. Am. A 12, 2716–2725 (1995).
[CrossRef]

D. A. Christensen, “Analysis of near field tip patterns including object interaction using finite-difference time-domain calculations,” Ultramicroscopy 57, 189–196 (1995).
[CrossRef]

J. L. Kann, T. D. Milster, F. Froehlich, R. W. Ziolkowski, and J. Judkins, “Numerical analysis of a two-dimensional near-field probe,” Ultramicroscopy 57, 251–256 (1995).
[CrossRef]

D. V. Labeke, D. Barchiesi, and F. Baida, “Optical characterization of nanosources used in scanning near-field optical microscopy,” J. Opt. Soc. Am. A 12, 695–703 (1995).
[CrossRef]

1994 (4)

1993 (2)

1989 (2)

1988 (2)

1980 (1)

A. D. Yaghjian, “Electric dyadic Green’s functions in the source region,” Proc. IEEE 68, 248–263 (1980).
[CrossRef]

1974 (1)

D. E. Livesay and K. M. Chen, “Electromagnetic fields induced inside arbitrarily shaped biological bodies,” IEEE Trans. Microwave Theory Tech. MTT-22, 1273–1280 (1974).
[CrossRef]

1964 (1)

A. D. McLachlan, “Van der Waals forces between an atom and a surface,” Mol. Phys. 34, 381–388 (1964).
[CrossRef]

Baida, F.

Barchiesi, D.

Bohren, C. F.

Carminati, R.

Chen, H. Y.

Chen, K. M.

D. E. Livesay and K. M. Chen, “Electromagnetic fields induced inside arbitrarily shaped biological bodies,” IEEE Trans. Microwave Theory Tech. MTT-22, 1273–1280 (1974).
[CrossRef]

Christensen, D. A.

D. A. Christensen, “Analysis of near field tip patterns including object interaction using finite-difference time-domain calculations,” Ultramicroscopy 57, 189–196 (1995).
[CrossRef]

Dereux, A.

Draine, B. T.

B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491–1499 (1994).
[CrossRef]

B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
[CrossRef]

Flatau, P. J.

Froehlich, F.

J. L. Kann, T. D. Milster, F. Froehlich, R. W. Ziolkowski, and J. Judkins, “Numerical analysis of a two-dimensional near-field probe,” Ultramicroscopy 57, 251–256 (1995).
[CrossRef]

Girard, C.

Goedecke, G. H.

Greffet, J. J.

Iskander, M. F.

Judkins, J.

J. L. Kann, T. D. Milster, F. Froehlich, R. W. Ziolkowski, and J. Judkins, “Numerical analysis of a two-dimensional near-field probe,” Ultramicroscopy 57, 251–256 (1995).
[CrossRef]

Kann, J. L.

J. L. Kann, T. D. Milster, F. Froehlich, R. W. Ziolkowski, and J. Judkins, “Numerical analysis of a two-dimensional near-field probe,” Ultramicroscopy 57, 251–256 (1995).
[CrossRef]

Ku, J. C.

Labeke, D. V.

Livesay, D. E.

D. E. Livesay and K. M. Chen, “Electromagnetic fields induced inside arbitrarily shaped biological bodies,” IEEE Trans. Microwave Theory Tech. MTT-22, 1273–1280 (1974).
[CrossRef]

Lumme, K.

K. Lumme and J. Rahola, “Light scattering by porous dust particles in the discrete-dipole approximation,” Astrophys. J. 425, 653–667 (1994).
[CrossRef]

Martin, O. J. F.

McLachlan, A. D.

A. D. McLachlan, “Van der Waals forces between an atom and a surface,” Mol. Phys. 34, 381–388 (1964).
[CrossRef]

Milster, T. D.

J. L. Kann, T. D. Milster, F. Froehlich, R. W. Ziolkowski, and J. Judkins, “Numerical analysis of a two-dimensional near-field probe,” Ultramicroscopy 57, 251–256 (1995).
[CrossRef]

Novotny, L.

O’Brien, S. G.

Penner, J. E.

Pohl, D. W.

Rahola, J.

K. Lumme and J. Rahola, “Light scattering by porous dust particles in the discrete-dipole approximation,” Astrophys. J. 425, 653–667 (1994).
[CrossRef]

Regli, P.

Singham, S. B.

Taubenblatt, M. A.

Tran, T. K.

Yaghjian, A. D.

A. D. Yaghjian, “Electric dyadic Green’s functions in the source region,” Proc. IEEE 68, 248–263 (1980).
[CrossRef]

Ziolkowski, R. W.

J. L. Kann, T. D. Milster, F. Froehlich, R. W. Ziolkowski, and J. Judkins, “Numerical analysis of a two-dimensional near-field probe,” Ultramicroscopy 57, 251–256 (1995).
[CrossRef]

Appl. Opt. (2)

Astrophys. J. (2)

K. Lumme and J. Rahola, “Light scattering by porous dust particles in the discrete-dipole approximation,” Astrophys. J. 425, 653–667 (1994).
[CrossRef]

B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

D. E. Livesay and K. M. Chen, “Electromagnetic fields induced inside arbitrarily shaped biological bodies,” IEEE Trans. Microwave Theory Tech. MTT-22, 1273–1280 (1974).
[CrossRef]

J. Opt. Soc. Am. A (9)

J. C. Ku, “Comparison of coupled-dipole solutions and dipole refractive indices for light scattering and absorption by arbitrary shaped or agglomerated particles,” J. Opt. Soc. Am. A 10, 336–342 (1993).
[CrossRef]

S. B. Singham and C. F. Bohren, “Light scattering by an arbitrary particle: the scattering-order formulation of the coupled-dipole method,” J. Opt. Soc. Am. A 5, 1867–1872 (1988).
[CrossRef] [PubMed]

M. A. Taubenblatt and T. K. Tran, “Calculation of light scattering from particles and structures on a surface by the coupled-dipole method,” J. Opt. Soc. Am. A 10, 912–919 (1993).
[CrossRef]

O. J. F. Martin, A. Dereux, and C. Girard, “Iterative scheme for computing exactly the total field propagating in dielectric structures of arbitrary shape,” J. Opt. Soc. Am. A 11, 1073–1080 (1994).
[CrossRef]

L. Novotny, D. W. Pohl, and P. Regli, “Light propagation through nanometer-sized structures: the two-dimensional-aperture scanning near-field optical microscope,” J. Opt. Soc. Am. A 11, 1768–1779 (1994).
[CrossRef]

R. Carminati and J. J. Greffet, “Influence of dielectric contrast and topography on the near field scattered by an inhomogeneous surface,” J. Opt. Soc. Am. A 12, 2716–2725 (1995).
[CrossRef]

D. V. Labeke, D. Barchiesi, and F. Baida, “Optical characterization of nanosources used in scanning near-field optical microscopy,” J. Opt. Soc. Am. A 12, 695–703 (1995).
[CrossRef]

L. Novotny, “Allowed and forbidden light in near-field optics. I. A single dipolar light source,” J. Opt. Soc. Am. A 14, 91–104 (1997).
[CrossRef]

B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491–1499 (1994).
[CrossRef]

Mol. Phys. (1)

A. D. McLachlan, “Van der Waals forces between an atom and a surface,” Mol. Phys. 34, 381–388 (1964).
[CrossRef]

Proc. IEEE (1)

A. D. Yaghjian, “Electric dyadic Green’s functions in the source region,” Proc. IEEE 68, 248–263 (1980).
[CrossRef]

Ultramicroscopy (2)

D. A. Christensen, “Analysis of near field tip patterns including object interaction using finite-difference time-domain calculations,” Ultramicroscopy 57, 189–196 (1995).
[CrossRef]

J. L. Kann, T. D. Milster, F. Froehlich, R. W. Ziolkowski, and J. Judkins, “Numerical analysis of a two-dimensional near-field probe,” Ultramicroscopy 57, 251–256 (1995).
[CrossRef]

Other (15)

L. Novotny and D. W. Pohl, “Light propagation in scanning near-field optical microscopy,” in Photons and Local Probes, O. Marti and R. Möller, eds., Vol. 300 of NATO Advanced Study Institute, Series E (Kluwer, Dordrecht, The Netherlands, 1995), pp. 21–33.

R. F. Harrington, Field Computation by Moment Methods (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1992).

J. J. H. Wang, Generalized Moment Methods in Electromagnetics (Wiley, Chichester, UK, 1991).

M. N. O. Sadiku, Numerical Techniques in Electromagnetics (CRC Press, Boca Raton, Fla., 1992), Chap. 5.7.

A. Lakhtakia, “Strong and weak forms of the method method of moments and the coupled dipole method for scattering of time-harmonic electromagnetic fields,” J. Mod. Phys. C. 3, 583–603 (1992).

W. C. Chew, Waves and Fields in Inhomogeneous Media, 2nd ed. (Institute of Electrical and Electronics Engineers, New York, 1995).

J. V. Bladel, “Some remarks on Green’s dyadic for infinite space.” IRE Trans. Antennas Propagat. 9, 563–566 (1961).

C. F. Bohren and D. R. Huffmann, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

R. Loudon, The Quantum Theory of Light, 2nd ed. (Clarendon, Oxford, 1983).

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1970).

B. Hecht, D. W. Pohl, H. Heinzelmann, and L. Novotny, “Tunnel near-field optical microscopy: TNOM-2,” in Photons and Local Probes, O. Marti and R. Möller, eds., Vol. 300 of NATO Advanced Study Institute, Series E (Kluwer, Dordrecht, The Netherlands, 1995), pp. 93–107.

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, New York, 1985).

N. Bleistein and R. A. Handelsman, Asymptotic Expansion of Integrals, 1st ed. (Rinehart & Winston, New York, 1975).

L. Novotny, “Light propagation and light confinement in near-field optics,” Ph.D. dissertation 11420 (Swiss Federal Institute of Technology, Zurich, Switzerland, 1996).

L. M. Brekhovskikh, Waves in Layered Media, 2nd ed. (Academic, New York, 1980).

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 (6)

Fig. 1
Fig. 1

Contribution of the induced dipole moment p k at r k to the field at r. The electric field is determined by the dyadic Green's function G(r, r k) and the amplitude and the orientation of the dipole.

Fig. 2
Fig. 2

Definition of the coordinates used in the text. r k is the location of the dipolar subunit Vk, and rk is its projection on the surface of the layered structure. The direction of observation is defined by the unit vector n r with the spherical components (ϑ, φ).

Fig. 3
Fig. 3

(a) Side view and (b) top view of the investigated system. The optical probe is represented by a dipolar particle locally illuminated by a horizontal electric field. The probe is guided at constant height z=h along the layered structure, carrying several sample particles. An optical scan image is obtained by recording of the far-field radiation for every position (x, y) of the probe.

Fig. 4
Fig. 4

Illustration of the different far-field signals that are used for separate scan images: light radiated into the upper half-space (P ), light coupled to the fundamental TE mode (Pm), allowed light ((Pa)), and forbidden light (Pf).

Fig. 5
Fig. 5

Forbidden-light scan image of an aluminum particle (=-34.5+8.5i) and a dielectric particle (=3). The particles have radii of a=10 nm and are located at x=-20 nm (aluminum particle) and x=20 nm (dielectric particle). The optical probe is a dielectric particle (=2.25, a=5 nm), which is locally illuminated by an x-polarized electric field (λ=488 nm) and scanned at a constant height of h=25 nm. The sample particles are sitting on a slab waveguide consisting of a 35-nm layer with =5 and a glass substrate with =2.25.

Fig. 6
Fig. 6

(a) Allowed-light scan image and (b) forbidden-light scan image for the same structure as that in Fig. 5 but with a modulation of the scan height (amplitude: 1 nm; periodicity: 10 nm; oblique direction of 45°).

Tables (1)

Tables Icon

Table 1 Spectral Composition of the Four Different Far-Field Signals P (Light Radiated into the Upper Half-Space), Pa (Allowed Light), Pf (Forbidden Light), and Pm (Light Coupled to the Fundamental TE Waveguide Mode)

Equations (58)

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

×E(r)=iωμoH(r),
×H(r)=-iωoref(r)E(r)+je(r),
je(r)=-iωo[(r)-ref(r)]E(r).
××E(r)-ko2ref(r)E(r)=iωμoje(r),
××G(r, r)-ko2ref(r)G(r, r)=Iδ(r-r).
E(r)=Eo(r)+iωμoVdV G(r, r)·je(r),rV,
H(r)=Ho(r)+VdV[×G(r, r)]·je(r),
rV.
je(r)=-iωpkδ(r-rk),
E(r)=ω2μoG(r, rk)·pk,
H(r)=-iω[×G(r, rk)]·pk.
G(r, rk)=Go(r, rk)+Gs(r, rk).
je(r)=-iωn=1Npnδ(r-rn).
pn=αnEexc(rn),
pk=αk·Eo(rk)+ω2μoαk·Gs(rk, rk)·pk
+ω2μon=1nkNαk·G(rk, rn)·pn,k=1, , N.
E(r)=Eo(r)+ω2μon=1NG(r, rn)·pn,rV,
H(r)=Ho(r)-iωn=1N[×G(r, rn)]·pn,rV.
αk=3Vkorefrkrk-refrkrk+2refrk×I-3kref2rkrk-refrkrk+2refrkMrk-1,
M(rk)=231kref2(rk){[1-ikref(rk)ak]×exp[ikref(rk)ak]-1}I.
P(ΔΩ)=Δφ dφ Δϑ dϑ(sin ϑ)p(ϑ, φ).
p(ϑ, φ)=|r|2S·nr,
S=12Re(E×H*)=120jμo |E|2nr.
lim|r|Go(r, rk)=exp(ikj|r|)|r|Go(rk, nr),
lim|r| Gs(r, rk)=exp(ikj|r|)|r|Gs(rk, nr).
p(ϑ, φ)=12ω4μo2ojμo n=1N [Go(rn, nr)+Gs(rn, nr)]·pn2.
·(E×H)+H·tB+E·tD+j·E=0.
12V da Re(E×H*)·n+12V dV Re(j*·E)=0.
Pk=ω2Im[pk*·E(rk)].
E(rk)=Eem(rk)+Eexc(rk),
Pok=pk212πωo1k13,
Pk=Pok+ω2Im[pk*·Eexc(rk)].
Eexc(rk)=Eo(rk)+ω2μoGs(rk, rk)·pk+ω2μo n=1nkN G(rk, rn)·pn.
Pm=Ptot-(Pa+Pf+P).
αc=arcsin(k1/k3),
1+r1,2(p)(k)r2,3(p)(k)exp(2ik2zd)=0,
1+r1,2(s)(k)r2,3(s)(k)exp(2ik2zd)=0,
kjz=kj2-k2withIm(kjz)>0.
E(r)=ω2μoG(r, rk)·pk.
G(r, rk)=Go(r, rk)+Gs(r, rk).
Go(r, rk)=14πexp(ik1|r-rk|)|r-rk|,
Gor, ro=I+1k12Gor, rk.
lim|r| |r-rk|=|r|-nr·rk=|r|-|rk|cos Θk,
lim|r| Go(r, rk)exp(ik1|r|)4π|r|exp(-ik1nr·rk).
lim|r| Gor, rk=expik1|r|4π|r|exp-ik1nr·rk×1-cos2 φ sin2 ϑ-sin φ cos φ sin2 ϑ-cos φ sin ϑ cos ϑ-sin φ cos φ sin2 ϑ1-sin2 φ sin2 ϑ-sin φ sin ϑ cos ϑ-cos φ sin ϑ cos ϑ-sin φ sin ϑ cos ϑsin2 ϑ,
Es=(kj2+·)s.
I(R)=limR Γ dz g(z)exp[Rf(z)]
f(z=zo)z=0.
I(R)=g(zo)exp[Rf(zo)]R-2πf(zo),
lim|r| Gsr, rk=expikj|r|4π|r|exp-ikjnr·rk×cos2 φ cos2 ϑΦ2+sin2 φΦ3 sin φ cos φ cos2 ϑΦ2-sin φ cos φΦ3 -cos φ sin ϑ cos ϑΦ2sin φ cos φ cos2 ϑΦ2-sin φ cos φΦ3 sin2 φ cos2 ϑΦ2+cos2 φΦ3 -sin φ sin ϑ cos ϑΦ2-cos φ sin ϑ cos ϑΦ1  -sin φ sin ϑ cos ϑΦ1  sin2 ϑΦ1,
Φ(1)=r(p)(k)exp(ik1h cos ϑ),
Φ(2)=-r(p)(k)exp(ik1h cos ϑ),
Φ(3)=r(s)(k)exp(ik1h cos ϑ),
Φ(1)=t(p)(k)1nkn cos ϑk12-k2exp(-iknδ cos ϑ)×exp(ihk12-k2),
Φ(2)=t(p)(k)1nexp(-iknδ cos ϑ)×exp(ihk12-k2),
Φ(3)=t(s)(k)1nkn cos ϑk12-k2exp(-iknδ cos ϑ)×exp(ihk12-k2),
r(p,s)=r1,2(p,s)+r2,3(p,s) exp(2ik2zd)1+r1,2(p,s)r2,3(p,s) exp(2ik2zd),
t(p,s)=t1,2(p,s)t2,3(p,s) exp(ik2zd)1+r1,2(p,s)r2,3(p,s) exp(2ik2zd).

Metrics