Abstract

We investigated surface phonon polariton in cesium iodide with terahertz time-domain attenuated total reflection method in Otto configuration, which gives us both information on amplitude and phase of surface electromagnetic mode directly. Systematic experiments with precise control of the distance between a prism and an active material show that the abrupt change of π-phase jump appears sensitively under polariton picture satisfied when the local electric field at the interface becomes a maximum. This demonstration will open the novel phase-detection terahertz sensor using the active medium causing the strong enhancement of terahertz electric field.

©2008 Optical Society of America

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References

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  1. E. Burstein, A. Hartstein, J. Schoenwald, A. A. Maradudin, D. L. Mills, and R. F. Wallis, “Surface plariton-Electromagnetic waves at interfaces,” in Polaritons - Prceedings of the First Taormina Research Conference on the Structure of Matter, E. Burstein and F. D. Martina, eds. (Pergamon, New York, 1974), pp. 89–108.
  2. S. Y. Wu, H. P. Ho, W. C. Law, C. Lin, and S. K. Kong, “Highly sensitive differential phase-sensitive surface Plasmon resonance biosensor based on the Mach-Zehnder configuration,” Opt. Lett. 29, 2378–2380 (2004).
    [Crossref] [PubMed]
  3. F. J. García-Vidal and J. B. Pendry, “Collective Theory for Surface Enhanced Raman Scattering,” Phys. Rev. Lett. 77, 1163–1166 (1997).
  4. S. Nie and S. R. Emory, “Probing Single Molecules and Single Nanoparticles by Surface-Enhanced Raman Scattering,” Science 275, 1102–1106 (1997).
    [Crossref] [PubMed]
  5. A. N. Grigorenko, P. I. Nikitin, and A. V. Kabashin, “Phase jumps and interferometric surface plasmon resonance imaging,” Appl. Phys. Lett. 75, 3917–3919 (1999).
    [Crossref]
  6. D. Mittleman, Sensing with Terahertz Radiation (Springer-Verlag, Berlin Heidelberg, 2003).
  7. B. Fischer, M. Hoffmann, H. Helm, R. Wilk, F. Rutz, T. Kleine-Ostmann, M. Koch, and P. Jepsen “Terahertz time-domain spectroscopy and imaging of artificial RNA,” Opt. Express 13, 5205–5215 (2005).
    [Crossref] [PubMed]
  8. P. C. Upadhya, Y. C. Shen, A. G. Davies, and E. H. Linfield, “Terahertz Time-Domain Spectroscopy of Glucose and Uric Acid,” J. Biol. Phys. 29, 117–121 (2003).
    [Crossref]
  9. K. Sakai, Terahertz Optoelectronics (Springer-Verlag, Berlin, 1998).
  10. H. Hirori, K. Yamashita, M. Nagai, and K. Tanaka, “Attenuated total reflection spectroscopy in time domain using terahertz coherent pulses,” Jpn. J. Appl. Phys. 43, L1287–L1289 (2004).
  11. H. Hirori, M. Nagai, and K. Tanaka, “Attenuated Total Reflection Spectroscopy in Time Domain Using Terahertz Coherent Pulses,” Opt. Express 13, 10801–10814 (2005).
    [Crossref] [PubMed]
  12. A. Otto, “Excitation of nonradiative surface plasma waves in silver by the method of frustrated total reflection,” Z. Phys. 216, 398–410 (1968).
    [Crossref]
  13. P. U. Jepsen and B. M. Fischer, “Dynamic range in terahertz time-domain transmission and reflection spectroscopy,” Opt. Lett. 30, 29–31 (2005).
    [Crossref] [PubMed]
  14. A. Rice, Y. Jin, X. Ma, X. Zhang, D. Bliss, J. Larkin, and M. Alexander, “Terahertz optical rectification from<110>zinc-blende crystals,” Appl. Phys. Lett. 64, 1324–1326 (1994).
    [Crossref]
  15. A. Nahata, A. Weling, and T. Heinz, “A wideband coherent terahertz spectroscopy system using optical rectification and electro-optic sampling,” Appl. Phys. Lett. 69, 2321–2323 (1996).
    [Crossref]
  16. J. F. Vetelino, S. S. Mitra, and K. V. Namjoshi, “Lattice Dynamics, Mode Grüneisen Parameters, and Coefficient of Thermal Expansion of CsCl, CsBr, and CsI,” Phys. Rev. B2, 2167–2175 (1970).
  17. V. M. Agranovich and D.L. Mills, Surface polaritons (North-Holland Publishing Company, Amsterdam, NewYork, Oxford, 1982).
  18. P. G. Johannsen, “Refractive index of the alkali halides. I. Constant joint density of states model,” Phys. Rev. B 55, 6856–6864 (1997).
    [Crossref]
  19. A. Otto , “The surface polariton response in attenuated total reflection,” in Polaritons: Prceedings of the First Taormina Research Conference on the Structure of Matter, E. Burstein and F. D. Martina , eds. (Pergamon, New York, 1974), pp. 117–121.
  20. R. F. Wallis and A. A. Maradudin, “Lattice anharmonicity and optical absorption in polar crystals. III. quantum mechanical treatment in the linear approximation,” Phys. Rev. 125, 1277–1282 (1962).
    [Crossref]

2005 (3)

2004 (1)

2003 (1)

P. C. Upadhya, Y. C. Shen, A. G. Davies, and E. H. Linfield, “Terahertz Time-Domain Spectroscopy of Glucose and Uric Acid,” J. Biol. Phys. 29, 117–121 (2003).
[Crossref]

1999 (1)

A. N. Grigorenko, P. I. Nikitin, and A. V. Kabashin, “Phase jumps and interferometric surface plasmon resonance imaging,” Appl. Phys. Lett. 75, 3917–3919 (1999).
[Crossref]

1997 (2)

S. Nie and S. R. Emory, “Probing Single Molecules and Single Nanoparticles by Surface-Enhanced Raman Scattering,” Science 275, 1102–1106 (1997).
[Crossref] [PubMed]

P. G. Johannsen, “Refractive index of the alkali halides. I. Constant joint density of states model,” Phys. Rev. B 55, 6856–6864 (1997).
[Crossref]

1996 (1)

A. Nahata, A. Weling, and T. Heinz, “A wideband coherent terahertz spectroscopy system using optical rectification and electro-optic sampling,” Appl. Phys. Lett. 69, 2321–2323 (1996).
[Crossref]

1994 (1)

A. Rice, Y. Jin, X. Ma, X. Zhang, D. Bliss, J. Larkin, and M. Alexander, “Terahertz optical rectification from<110>zinc-blende crystals,” Appl. Phys. Lett. 64, 1324–1326 (1994).
[Crossref]

1970 (1)

J. F. Vetelino, S. S. Mitra, and K. V. Namjoshi, “Lattice Dynamics, Mode Grüneisen Parameters, and Coefficient of Thermal Expansion of CsCl, CsBr, and CsI,” Phys. Rev. B2, 2167–2175 (1970).

1968 (1)

A. Otto, “Excitation of nonradiative surface plasma waves in silver by the method of frustrated total reflection,” Z. Phys. 216, 398–410 (1968).
[Crossref]

1962 (1)

R. F. Wallis and A. A. Maradudin, “Lattice anharmonicity and optical absorption in polar crystals. III. quantum mechanical treatment in the linear approximation,” Phys. Rev. 125, 1277–1282 (1962).
[Crossref]

Agranovich, V. M.

V. M. Agranovich and D.L. Mills, Surface polaritons (North-Holland Publishing Company, Amsterdam, NewYork, Oxford, 1982).

Alexander, M.

A. Rice, Y. Jin, X. Ma, X. Zhang, D. Bliss, J. Larkin, and M. Alexander, “Terahertz optical rectification from<110>zinc-blende crystals,” Appl. Phys. Lett. 64, 1324–1326 (1994).
[Crossref]

Bliss, D.

A. Rice, Y. Jin, X. Ma, X. Zhang, D. Bliss, J. Larkin, and M. Alexander, “Terahertz optical rectification from<110>zinc-blende crystals,” Appl. Phys. Lett. 64, 1324–1326 (1994).
[Crossref]

Burstein, E.

E. Burstein, A. Hartstein, J. Schoenwald, A. A. Maradudin, D. L. Mills, and R. F. Wallis, “Surface plariton-Electromagnetic waves at interfaces,” in Polaritons - Prceedings of the First Taormina Research Conference on the Structure of Matter, E. Burstein and F. D. Martina, eds. (Pergamon, New York, 1974), pp. 89–108.

Davies, A. G.

P. C. Upadhya, Y. C. Shen, A. G. Davies, and E. H. Linfield, “Terahertz Time-Domain Spectroscopy of Glucose and Uric Acid,” J. Biol. Phys. 29, 117–121 (2003).
[Crossref]

Emory, S. R.

S. Nie and S. R. Emory, “Probing Single Molecules and Single Nanoparticles by Surface-Enhanced Raman Scattering,” Science 275, 1102–1106 (1997).
[Crossref] [PubMed]

Fischer, B.

Fischer, B. M.

García-Vidal, F. J.

F. J. García-Vidal and J. B. Pendry, “Collective Theory for Surface Enhanced Raman Scattering,” Phys. Rev. Lett. 77, 1163–1166 (1997).

Grigorenko, A. N.

A. N. Grigorenko, P. I. Nikitin, and A. V. Kabashin, “Phase jumps and interferometric surface plasmon resonance imaging,” Appl. Phys. Lett. 75, 3917–3919 (1999).
[Crossref]

Hartstein, A.

E. Burstein, A. Hartstein, J. Schoenwald, A. A. Maradudin, D. L. Mills, and R. F. Wallis, “Surface plariton-Electromagnetic waves at interfaces,” in Polaritons - Prceedings of the First Taormina Research Conference on the Structure of Matter, E. Burstein and F. D. Martina, eds. (Pergamon, New York, 1974), pp. 89–108.

Heinz, T.

A. Nahata, A. Weling, and T. Heinz, “A wideband coherent terahertz spectroscopy system using optical rectification and electro-optic sampling,” Appl. Phys. Lett. 69, 2321–2323 (1996).
[Crossref]

Helm, H.

Hirori, H.

H. Hirori, M. Nagai, and K. Tanaka, “Attenuated Total Reflection Spectroscopy in Time Domain Using Terahertz Coherent Pulses,” Opt. Express 13, 10801–10814 (2005).
[Crossref] [PubMed]

H. Hirori, K. Yamashita, M. Nagai, and K. Tanaka, “Attenuated total reflection spectroscopy in time domain using terahertz coherent pulses,” Jpn. J. Appl. Phys. 43, L1287–L1289 (2004).

Ho, H. P.

Hoffmann, M.

Jepsen, P.

Jepsen, P. U.

Jin, Y.

A. Rice, Y. Jin, X. Ma, X. Zhang, D. Bliss, J. Larkin, and M. Alexander, “Terahertz optical rectification from<110>zinc-blende crystals,” Appl. Phys. Lett. 64, 1324–1326 (1994).
[Crossref]

Johannsen, P. G.

P. G. Johannsen, “Refractive index of the alkali halides. I. Constant joint density of states model,” Phys. Rev. B 55, 6856–6864 (1997).
[Crossref]

Kabashin, A. V.

A. N. Grigorenko, P. I. Nikitin, and A. V. Kabashin, “Phase jumps and interferometric surface plasmon resonance imaging,” Appl. Phys. Lett. 75, 3917–3919 (1999).
[Crossref]

Kleine-Ostmann, T.

Koch, M.

Kong, S. K.

Larkin, J.

A. Rice, Y. Jin, X. Ma, X. Zhang, D. Bliss, J. Larkin, and M. Alexander, “Terahertz optical rectification from<110>zinc-blende crystals,” Appl. Phys. Lett. 64, 1324–1326 (1994).
[Crossref]

Law, W. C.

Lin, C.

Linfield, E. H.

P. C. Upadhya, Y. C. Shen, A. G. Davies, and E. H. Linfield, “Terahertz Time-Domain Spectroscopy of Glucose and Uric Acid,” J. Biol. Phys. 29, 117–121 (2003).
[Crossref]

Ma, X.

A. Rice, Y. Jin, X. Ma, X. Zhang, D. Bliss, J. Larkin, and M. Alexander, “Terahertz optical rectification from<110>zinc-blende crystals,” Appl. Phys. Lett. 64, 1324–1326 (1994).
[Crossref]

Maradudin, A. A.

R. F. Wallis and A. A. Maradudin, “Lattice anharmonicity and optical absorption in polar crystals. III. quantum mechanical treatment in the linear approximation,” Phys. Rev. 125, 1277–1282 (1962).
[Crossref]

E. Burstein, A. Hartstein, J. Schoenwald, A. A. Maradudin, D. L. Mills, and R. F. Wallis, “Surface plariton-Electromagnetic waves at interfaces,” in Polaritons - Prceedings of the First Taormina Research Conference on the Structure of Matter, E. Burstein and F. D. Martina, eds. (Pergamon, New York, 1974), pp. 89–108.

Mills, D. L.

E. Burstein, A. Hartstein, J. Schoenwald, A. A. Maradudin, D. L. Mills, and R. F. Wallis, “Surface plariton-Electromagnetic waves at interfaces,” in Polaritons - Prceedings of the First Taormina Research Conference on the Structure of Matter, E. Burstein and F. D. Martina, eds. (Pergamon, New York, 1974), pp. 89–108.

Mills, D.L.

V. M. Agranovich and D.L. Mills, Surface polaritons (North-Holland Publishing Company, Amsterdam, NewYork, Oxford, 1982).

Mitra, S. S.

J. F. Vetelino, S. S. Mitra, and K. V. Namjoshi, “Lattice Dynamics, Mode Grüneisen Parameters, and Coefficient of Thermal Expansion of CsCl, CsBr, and CsI,” Phys. Rev. B2, 2167–2175 (1970).

Mittleman, D.

D. Mittleman, Sensing with Terahertz Radiation (Springer-Verlag, Berlin Heidelberg, 2003).

Nagai, M.

H. Hirori, M. Nagai, and K. Tanaka, “Attenuated Total Reflection Spectroscopy in Time Domain Using Terahertz Coherent Pulses,” Opt. Express 13, 10801–10814 (2005).
[Crossref] [PubMed]

H. Hirori, K. Yamashita, M. Nagai, and K. Tanaka, “Attenuated total reflection spectroscopy in time domain using terahertz coherent pulses,” Jpn. J. Appl. Phys. 43, L1287–L1289 (2004).

Nahata, A.

A. Nahata, A. Weling, and T. Heinz, “A wideband coherent terahertz spectroscopy system using optical rectification and electro-optic sampling,” Appl. Phys. Lett. 69, 2321–2323 (1996).
[Crossref]

Namjoshi, K. V.

J. F. Vetelino, S. S. Mitra, and K. V. Namjoshi, “Lattice Dynamics, Mode Grüneisen Parameters, and Coefficient of Thermal Expansion of CsCl, CsBr, and CsI,” Phys. Rev. B2, 2167–2175 (1970).

Nie, S.

S. Nie and S. R. Emory, “Probing Single Molecules and Single Nanoparticles by Surface-Enhanced Raman Scattering,” Science 275, 1102–1106 (1997).
[Crossref] [PubMed]

Nikitin, P. I.

A. N. Grigorenko, P. I. Nikitin, and A. V. Kabashin, “Phase jumps and interferometric surface plasmon resonance imaging,” Appl. Phys. Lett. 75, 3917–3919 (1999).
[Crossref]

Otto, A.

A. Otto, “Excitation of nonradiative surface plasma waves in silver by the method of frustrated total reflection,” Z. Phys. 216, 398–410 (1968).
[Crossref]

A. Otto , “The surface polariton response in attenuated total reflection,” in Polaritons: Prceedings of the First Taormina Research Conference on the Structure of Matter, E. Burstein and F. D. Martina , eds. (Pergamon, New York, 1974), pp. 117–121.

Pendry, J. B.

F. J. García-Vidal and J. B. Pendry, “Collective Theory for Surface Enhanced Raman Scattering,” Phys. Rev. Lett. 77, 1163–1166 (1997).

Rice, A.

A. Rice, Y. Jin, X. Ma, X. Zhang, D. Bliss, J. Larkin, and M. Alexander, “Terahertz optical rectification from<110>zinc-blende crystals,” Appl. Phys. Lett. 64, 1324–1326 (1994).
[Crossref]

Rutz, F.

Sakai, K.

K. Sakai, Terahertz Optoelectronics (Springer-Verlag, Berlin, 1998).

Schoenwald, J.

E. Burstein, A. Hartstein, J. Schoenwald, A. A. Maradudin, D. L. Mills, and R. F. Wallis, “Surface plariton-Electromagnetic waves at interfaces,” in Polaritons - Prceedings of the First Taormina Research Conference on the Structure of Matter, E. Burstein and F. D. Martina, eds. (Pergamon, New York, 1974), pp. 89–108.

Shen, Y. C.

P. C. Upadhya, Y. C. Shen, A. G. Davies, and E. H. Linfield, “Terahertz Time-Domain Spectroscopy of Glucose and Uric Acid,” J. Biol. Phys. 29, 117–121 (2003).
[Crossref]

Tanaka, K.

H. Hirori, M. Nagai, and K. Tanaka, “Attenuated Total Reflection Spectroscopy in Time Domain Using Terahertz Coherent Pulses,” Opt. Express 13, 10801–10814 (2005).
[Crossref] [PubMed]

H. Hirori, K. Yamashita, M. Nagai, and K. Tanaka, “Attenuated total reflection spectroscopy in time domain using terahertz coherent pulses,” Jpn. J. Appl. Phys. 43, L1287–L1289 (2004).

Upadhya, P. C.

P. C. Upadhya, Y. C. Shen, A. G. Davies, and E. H. Linfield, “Terahertz Time-Domain Spectroscopy of Glucose and Uric Acid,” J. Biol. Phys. 29, 117–121 (2003).
[Crossref]

Vetelino, J. F.

J. F. Vetelino, S. S. Mitra, and K. V. Namjoshi, “Lattice Dynamics, Mode Grüneisen Parameters, and Coefficient of Thermal Expansion of CsCl, CsBr, and CsI,” Phys. Rev. B2, 2167–2175 (1970).

Wallis, R. F.

R. F. Wallis and A. A. Maradudin, “Lattice anharmonicity and optical absorption in polar crystals. III. quantum mechanical treatment in the linear approximation,” Phys. Rev. 125, 1277–1282 (1962).
[Crossref]

E. Burstein, A. Hartstein, J. Schoenwald, A. A. Maradudin, D. L. Mills, and R. F. Wallis, “Surface plariton-Electromagnetic waves at interfaces,” in Polaritons - Prceedings of the First Taormina Research Conference on the Structure of Matter, E. Burstein and F. D. Martina, eds. (Pergamon, New York, 1974), pp. 89–108.

Weling, A.

A. Nahata, A. Weling, and T. Heinz, “A wideband coherent terahertz spectroscopy system using optical rectification and electro-optic sampling,” Appl. Phys. Lett. 69, 2321–2323 (1996).
[Crossref]

Wilk, R.

Wu, S. Y.

Yamashita, K.

H. Hirori, K. Yamashita, M. Nagai, and K. Tanaka, “Attenuated total reflection spectroscopy in time domain using terahertz coherent pulses,” Jpn. J. Appl. Phys. 43, L1287–L1289 (2004).

Zhang, X.

A. Rice, Y. Jin, X. Ma, X. Zhang, D. Bliss, J. Larkin, and M. Alexander, “Terahertz optical rectification from<110>zinc-blende crystals,” Appl. Phys. Lett. 64, 1324–1326 (1994).
[Crossref]

Appl. Phys. Lett. (3)

A. N. Grigorenko, P. I. Nikitin, and A. V. Kabashin, “Phase jumps and interferometric surface plasmon resonance imaging,” Appl. Phys. Lett. 75, 3917–3919 (1999).
[Crossref]

A. Rice, Y. Jin, X. Ma, X. Zhang, D. Bliss, J. Larkin, and M. Alexander, “Terahertz optical rectification from<110>zinc-blende crystals,” Appl. Phys. Lett. 64, 1324–1326 (1994).
[Crossref]

A. Nahata, A. Weling, and T. Heinz, “A wideband coherent terahertz spectroscopy system using optical rectification and electro-optic sampling,” Appl. Phys. Lett. 69, 2321–2323 (1996).
[Crossref]

J. Biol. Phys. (1)

P. C. Upadhya, Y. C. Shen, A. G. Davies, and E. H. Linfield, “Terahertz Time-Domain Spectroscopy of Glucose and Uric Acid,” J. Biol. Phys. 29, 117–121 (2003).
[Crossref]

Opt. Express (2)

Opt. Lett. (2)

Phys. Rev. (2)

J. F. Vetelino, S. S. Mitra, and K. V. Namjoshi, “Lattice Dynamics, Mode Grüneisen Parameters, and Coefficient of Thermal Expansion of CsCl, CsBr, and CsI,” Phys. Rev. B2, 2167–2175 (1970).

R. F. Wallis and A. A. Maradudin, “Lattice anharmonicity and optical absorption in polar crystals. III. quantum mechanical treatment in the linear approximation,” Phys. Rev. 125, 1277–1282 (1962).
[Crossref]

Phys. Rev. B (1)

P. G. Johannsen, “Refractive index of the alkali halides. I. Constant joint density of states model,” Phys. Rev. B 55, 6856–6864 (1997).
[Crossref]

Science (1)

S. Nie and S. R. Emory, “Probing Single Molecules and Single Nanoparticles by Surface-Enhanced Raman Scattering,” Science 275, 1102–1106 (1997).
[Crossref] [PubMed]

Z. Phys. (1)

A. Otto, “Excitation of nonradiative surface plasma waves in silver by the method of frustrated total reflection,” Z. Phys. 216, 398–410 (1968).
[Crossref]

Other (7)

A. Otto , “The surface polariton response in attenuated total reflection,” in Polaritons: Prceedings of the First Taormina Research Conference on the Structure of Matter, E. Burstein and F. D. Martina , eds. (Pergamon, New York, 1974), pp. 117–121.

V. M. Agranovich and D.L. Mills, Surface polaritons (North-Holland Publishing Company, Amsterdam, NewYork, Oxford, 1982).

F. J. García-Vidal and J. B. Pendry, “Collective Theory for Surface Enhanced Raman Scattering,” Phys. Rev. Lett. 77, 1163–1166 (1997).

D. Mittleman, Sensing with Terahertz Radiation (Springer-Verlag, Berlin Heidelberg, 2003).

K. Sakai, Terahertz Optoelectronics (Springer-Verlag, Berlin, 1998).

H. Hirori, K. Yamashita, M. Nagai, and K. Tanaka, “Attenuated total reflection spectroscopy in time domain using terahertz coherent pulses,” Jpn. J. Appl. Phys. 43, L1287–L1289 (2004).

E. Burstein, A. Hartstein, J. Schoenwald, A. A. Maradudin, D. L. Mills, and R. F. Wallis, “Surface plariton-Electromagnetic waves at interfaces,” in Polaritons - Prceedings of the First Taormina Research Conference on the Structure of Matter, E. Burstein and F. D. Martina, eds. (Pergamon, New York, 1974), pp. 89–108.

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Figures (8)

Fig. 1.
Fig. 1. Schematic diagram of the coupling prism to produce an evanescent wave on the internal total reflection surface. The distance d is determined by interference measurement with visible light.

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Fig. 2.
Fig. 2. ATR reflectivity (top) and phase shift Δφ (bottom) using the prism in (a) p-polarized incidence and (b) s-polarized incidence.

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Fig. 3.
Fig. 3. The sample-prism distance d dependence of the ATR reflectivity. Solid curves and broken curves show experimental and theoretical data, respectively.

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Fig. 4.
Fig. 4. The sample-prism distance d dependence of the phase shift Δφ. Solid curves and broken curves show experimental and theoretical data, respectively.

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Fig. 5.
Fig. 5. Real (bottom) and imaginary part (top) of the dielectric constant ε 3 of CsI measured by the THz TD-ATR technique (solid lines) and calculated by the Lorentz model (broken lines).

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Fig. 6.
Fig. 6. Calculation of absorption 1 - R (black), dip frequency shift f S (red), and halfwidth of reflectivity dip Γ (blue) as a function of gap distance d. Triangles, circles, and squares are experimental data of the absorption, frequency shift, and halfwidth experimentally, respectively.

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Fig. 7.
Fig. 7. The experimental result of B(d, f) (solid line). The calculated curve (broken lines) is also shown. An inserted figure shows schematic drawing of the distribution of the electric field on the surface of CsI.

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Fig. 8.
Fig. 8. The enhancement of electric field B(d, f) as a function of the damping constant. The square indicates the experimental data of CsI crystal. The triangle is theoretically estimated value in the case of temperature at 4 K (γ=10 GHz).

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Equations (10)

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

r 123 ( d , f ) = r 12 + r 23 exp ⁡ ( − 2 η 2 d ) 1 + r 12 r 23 exp ⁡ ( − 2 η 2 d ) ,
ε 3 ( f ) = ε ∞ + ε ∞ f LO 2 − f TO 2 f TO 2 − f 2 − i γ f ,
R ( d , f ) = ∣ r 123 ( d , f ) ∣ 2 = 1 − 4 φ 33 φ 13 exp ⁡ ( − 2 η 20 d ) ( f − f 0 − φ 11 exp ⁡ ( − 2 η 20 d ) ) 2 + ( φ 33 + φ 13 exp ⁡ ( − 2 η 20 d ) ) 2 ,
φ 11 = 4 ∣ ε 3 R 0 ∣ 2 Re [ r 12 ( f 0 , k 0 ) ] ( ∣ ε 3 R 0 ∣ + ε 2 ) ε ′ 3 R 0 > 0 ; φ 13 = 4 ∣ ε 3 R 0 ∣ 2 Im [ r 12 ( f 0 , k 0 ) ] ( ∣ ε 3 R 0 ∣ + ε 2 ) ε ′ 3 R 0 < 0 ; φ 33 = − ε 3 I 0 ε ′ 3 R 0 < 0 ;
                    ε 3 R 0 = Re [ ε 3 ( f 0 ) ] ; ε 3 I 0 = Im [ ε 3 ( f 0 ) ] ; ε ′ 3 R 0 = 1 2 π d Re [ ε 3 ( f ) ] df | f = f 0 ; η 20 = η 2 ( f 0 ) ,
r 123 ( d , f ) = r 12 + B ( d , f ) exp ⁡ ( − η 2 ( f ) d ) t 21 ,
r 12 + r 12 exp ⁡ ( − 2 η 2 d ) = 0 .
tan ⁡ − 1 ( r 23 ( f ) ) − tan − 1 ( r 12 ) = π ,
∣ r 23 ( f ) ∣ = exp ⁡ ( 2 η 2 ( f ) d ) .
B ( d , f ) = r 23 · − r 12 t 21 .

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