Abstract

A new effect of resonant tunneling of a fundamental mode of optical radiation through a subwavelength slit formed by smoothly shaped narrowing of a waveguide is examined. An effective energy transfer by evanescent waves is shown to be determined by the geometry of curvilinear narrowing. Spectra of reflectionless transmittance for waves 2.5–3 times longer than the width of the slit are found. Dependence of resonant tunneling on the geometrical parameters of narrowing and splitting of resonant maxima resulting in formation of two peaks of tunneling transmittance are demonstrated. An additional insight into the properties of narrowed waveguides has been obtained through the visualization of the variation of the field patterns associated with the modes for a specific configuration and the wavelength.

© 2010 Optical Society of America

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References

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  1. A. A. Eikhenwald, “Motion of energy in the effect of total internal reflection,” J. Russian Phys. Soc. 41, 131–157 (1909).
  2. L. I. Mandelstam, “Radiation of the source of light, located nearby the boundary of two transparent media,” Zeitshrift fur Physik 15, 220–225 (1914).
  3. G. A. Gamow, “Zur quantentheorie des atomkernes,” Zeitshrift fur Physik 51, pp. 204–212 (1928).
    [CrossRef]
  4. A. B. Mikhailovskii, Electromagnetic Instabilities in an Inhomogeneous Plasma (A. Hilger, 1992).
  5. A. Ranfagni, P. Fabeni, G. Pazzi, and D. Mugnai, “Anomalous pulse delay in microwave propagation: A plausible connection to the tunneling time,” Phys. Rev. E 48, 1453–1460 (1993).
    [CrossRef]
  6. F. de Fornel, Evanescent Waves: from Newtonian Optics to Atomic Optics, Springer Series in Opt. Sciences, Vol. 73 (Springer, 2001).
  7. A. Iwamoto, V. M. Aquino, and V. C. Aquilero–Nowarro, “Tunneling through rectangular plus linear barrier,” Int. J. Theor. Phys. 43, 483–495 (2004).
    [CrossRef]
  8. A. B. Shvartsburg, V. Kuzmiak, and G. Petite, “Optics of subwavelength gradient nanofilms,” Phys. Rep. 452, 33–88 (2007).
    [CrossRef]
  9. A. B. Shvartsburg, M. Marklund, G. Brodin, and L. Stenflo, “Superluminal tunneling of microwaves in smoothly varying transmission lines,” Phys. Rev. E 78, 016601 (2008).
    [CrossRef]
  10. M. Friedman and R. F. Fernsler, “Guiding radio frequency waves on metallic foils,” Appl. Phys. Lett. 74, 3468–3470 (1999).
    [CrossRef]
  11. H. I. Perez, C. I. Valencia, E. R. Mendez, and J. A. Sanchez-Gil, “On the transmission of diffuse light through thick slits,” J. Opt. Soc. Am. A 26, 909–918 (2009).
    [CrossRef]
  12. A. Shvartsburg, V. Kuzmiak, and G. Petite, “Polarization-dependent tunneling of light in gradient optics,” Phys. Rev. E 76, 016603 (2007).
    [CrossRef]
  13. O. V. Rudenko and A. B. Shvartsburg, “Nonlinear and linear wave phenomena in narrow pipes,” Acoustical J. 56, 429–434 (2010).
    [CrossRef]

2010 (1)

O. V. Rudenko and A. B. Shvartsburg, “Nonlinear and linear wave phenomena in narrow pipes,” Acoustical J. 56, 429–434 (2010).
[CrossRef]

2009 (1)

2008 (1)

A. B. Shvartsburg, M. Marklund, G. Brodin, and L. Stenflo, “Superluminal tunneling of microwaves in smoothly varying transmission lines,” Phys. Rev. E 78, 016601 (2008).
[CrossRef]

2007 (2)

A. Shvartsburg, V. Kuzmiak, and G. Petite, “Polarization-dependent tunneling of light in gradient optics,” Phys. Rev. E 76, 016603 (2007).
[CrossRef]

A. B. Shvartsburg, V. Kuzmiak, and G. Petite, “Optics of subwavelength gradient nanofilms,” Phys. Rep. 452, 33–88 (2007).
[CrossRef]

2004 (1)

A. Iwamoto, V. M. Aquino, and V. C. Aquilero–Nowarro, “Tunneling through rectangular plus linear barrier,” Int. J. Theor. Phys. 43, 483–495 (2004).
[CrossRef]

2001 (1)

F. de Fornel, Evanescent Waves: from Newtonian Optics to Atomic Optics, Springer Series in Opt. Sciences, Vol. 73 (Springer, 2001).

1999 (1)

M. Friedman and R. F. Fernsler, “Guiding radio frequency waves on metallic foils,” Appl. Phys. Lett. 74, 3468–3470 (1999).
[CrossRef]

1993 (1)

A. Ranfagni, P. Fabeni, G. Pazzi, and D. Mugnai, “Anomalous pulse delay in microwave propagation: A plausible connection to the tunneling time,” Phys. Rev. E 48, 1453–1460 (1993).
[CrossRef]

1992 (1)

A. B. Mikhailovskii, Electromagnetic Instabilities in an Inhomogeneous Plasma (A. Hilger, 1992).

1928 (1)

G. A. Gamow, “Zur quantentheorie des atomkernes,” Zeitshrift fur Physik 51, pp. 204–212 (1928).
[CrossRef]

1914 (1)

L. I. Mandelstam, “Radiation of the source of light, located nearby the boundary of two transparent media,” Zeitshrift fur Physik 15, 220–225 (1914).

1909 (1)

A. A. Eikhenwald, “Motion of energy in the effect of total internal reflection,” J. Russian Phys. Soc. 41, 131–157 (1909).

Aquilero–Nowarro, V. C.

A. Iwamoto, V. M. Aquino, and V. C. Aquilero–Nowarro, “Tunneling through rectangular plus linear barrier,” Int. J. Theor. Phys. 43, 483–495 (2004).
[CrossRef]

Aquino, V. M.

A. Iwamoto, V. M. Aquino, and V. C. Aquilero–Nowarro, “Tunneling through rectangular plus linear barrier,” Int. J. Theor. Phys. 43, 483–495 (2004).
[CrossRef]

Brodin, G.

A. B. Shvartsburg, M. Marklund, G. Brodin, and L. Stenflo, “Superluminal tunneling of microwaves in smoothly varying transmission lines,” Phys. Rev. E 78, 016601 (2008).
[CrossRef]

de Fornel, F.

F. de Fornel, Evanescent Waves: from Newtonian Optics to Atomic Optics, Springer Series in Opt. Sciences, Vol. 73 (Springer, 2001).

Eikhenwald, A. A.

A. A. Eikhenwald, “Motion of energy in the effect of total internal reflection,” J. Russian Phys. Soc. 41, 131–157 (1909).

Fabeni, P.

A. Ranfagni, P. Fabeni, G. Pazzi, and D. Mugnai, “Anomalous pulse delay in microwave propagation: A plausible connection to the tunneling time,” Phys. Rev. E 48, 1453–1460 (1993).
[CrossRef]

Fernsler, R. F.

M. Friedman and R. F. Fernsler, “Guiding radio frequency waves on metallic foils,” Appl. Phys. Lett. 74, 3468–3470 (1999).
[CrossRef]

Friedman, M.

M. Friedman and R. F. Fernsler, “Guiding radio frequency waves on metallic foils,” Appl. Phys. Lett. 74, 3468–3470 (1999).
[CrossRef]

Gamow, G. A.

G. A. Gamow, “Zur quantentheorie des atomkernes,” Zeitshrift fur Physik 51, pp. 204–212 (1928).
[CrossRef]

Iwamoto, A.

A. Iwamoto, V. M. Aquino, and V. C. Aquilero–Nowarro, “Tunneling through rectangular plus linear barrier,” Int. J. Theor. Phys. 43, 483–495 (2004).
[CrossRef]

Kuzmiak, V.

A. B. Shvartsburg, V. Kuzmiak, and G. Petite, “Optics of subwavelength gradient nanofilms,” Phys. Rep. 452, 33–88 (2007).
[CrossRef]

A. Shvartsburg, V. Kuzmiak, and G. Petite, “Polarization-dependent tunneling of light in gradient optics,” Phys. Rev. E 76, 016603 (2007).
[CrossRef]

Mandelstam, L. I.

L. I. Mandelstam, “Radiation of the source of light, located nearby the boundary of two transparent media,” Zeitshrift fur Physik 15, 220–225 (1914).

Marklund, M.

A. B. Shvartsburg, M. Marklund, G. Brodin, and L. Stenflo, “Superluminal tunneling of microwaves in smoothly varying transmission lines,” Phys. Rev. E 78, 016601 (2008).
[CrossRef]

Mendez, E. R.

Mikhailovskii, A. B.

A. B. Mikhailovskii, Electromagnetic Instabilities in an Inhomogeneous Plasma (A. Hilger, 1992).

Mugnai, D.

A. Ranfagni, P. Fabeni, G. Pazzi, and D. Mugnai, “Anomalous pulse delay in microwave propagation: A plausible connection to the tunneling time,” Phys. Rev. E 48, 1453–1460 (1993).
[CrossRef]

Pazzi, G.

A. Ranfagni, P. Fabeni, G. Pazzi, and D. Mugnai, “Anomalous pulse delay in microwave propagation: A plausible connection to the tunneling time,” Phys. Rev. E 48, 1453–1460 (1993).
[CrossRef]

Perez, H. I.

Petite, G.

A. Shvartsburg, V. Kuzmiak, and G. Petite, “Polarization-dependent tunneling of light in gradient optics,” Phys. Rev. E 76, 016603 (2007).
[CrossRef]

A. B. Shvartsburg, V. Kuzmiak, and G. Petite, “Optics of subwavelength gradient nanofilms,” Phys. Rep. 452, 33–88 (2007).
[CrossRef]

Ranfagni, A.

A. Ranfagni, P. Fabeni, G. Pazzi, and D. Mugnai, “Anomalous pulse delay in microwave propagation: A plausible connection to the tunneling time,” Phys. Rev. E 48, 1453–1460 (1993).
[CrossRef]

Rudenko, O. V.

O. V. Rudenko and A. B. Shvartsburg, “Nonlinear and linear wave phenomena in narrow pipes,” Acoustical J. 56, 429–434 (2010).
[CrossRef]

Sanchez-Gil, J. A.

Shvartsburg, A.

A. Shvartsburg, V. Kuzmiak, and G. Petite, “Polarization-dependent tunneling of light in gradient optics,” Phys. Rev. E 76, 016603 (2007).
[CrossRef]

Shvartsburg, A. B.

O. V. Rudenko and A. B. Shvartsburg, “Nonlinear and linear wave phenomena in narrow pipes,” Acoustical J. 56, 429–434 (2010).
[CrossRef]

A. B. Shvartsburg, M. Marklund, G. Brodin, and L. Stenflo, “Superluminal tunneling of microwaves in smoothly varying transmission lines,” Phys. Rev. E 78, 016601 (2008).
[CrossRef]

A. B. Shvartsburg, V. Kuzmiak, and G. Petite, “Optics of subwavelength gradient nanofilms,” Phys. Rep. 452, 33–88 (2007).
[CrossRef]

Stenflo, L.

A. B. Shvartsburg, M. Marklund, G. Brodin, and L. Stenflo, “Superluminal tunneling of microwaves in smoothly varying transmission lines,” Phys. Rev. E 78, 016601 (2008).
[CrossRef]

Valencia, C. I.

Acoustical J. (1)

O. V. Rudenko and A. B. Shvartsburg, “Nonlinear and linear wave phenomena in narrow pipes,” Acoustical J. 56, 429–434 (2010).
[CrossRef]

Appl. Phys. Lett. (1)

M. Friedman and R. F. Fernsler, “Guiding radio frequency waves on metallic foils,” Appl. Phys. Lett. 74, 3468–3470 (1999).
[CrossRef]

Int. J. Theor. Phys. (1)

A. Iwamoto, V. M. Aquino, and V. C. Aquilero–Nowarro, “Tunneling through rectangular plus linear barrier,” Int. J. Theor. Phys. 43, 483–495 (2004).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Russian Phys. Soc. (1)

A. A. Eikhenwald, “Motion of energy in the effect of total internal reflection,” J. Russian Phys. Soc. 41, 131–157 (1909).

Phys. Rep. (1)

A. B. Shvartsburg, V. Kuzmiak, and G. Petite, “Optics of subwavelength gradient nanofilms,” Phys. Rep. 452, 33–88 (2007).
[CrossRef]

Phys. Rev. E (3)

A. B. Shvartsburg, M. Marklund, G. Brodin, and L. Stenflo, “Superluminal tunneling of microwaves in smoothly varying transmission lines,” Phys. Rev. E 78, 016601 (2008).
[CrossRef]

A. Ranfagni, P. Fabeni, G. Pazzi, and D. Mugnai, “Anomalous pulse delay in microwave propagation: A plausible connection to the tunneling time,” Phys. Rev. E 48, 1453–1460 (1993).
[CrossRef]

A. Shvartsburg, V. Kuzmiak, and G. Petite, “Polarization-dependent tunneling of light in gradient optics,” Phys. Rev. E 76, 016603 (2007).
[CrossRef]

Zeitshrift fur Physik (2)

L. I. Mandelstam, “Radiation of the source of light, located nearby the boundary of two transparent media,” Zeitshrift fur Physik 15, 220–225 (1914).

G. A. Gamow, “Zur quantentheorie des atomkernes,” Zeitshrift fur Physik 51, pp. 204–212 (1928).
[CrossRef]

Other (2)

A. B. Mikhailovskii, Electromagnetic Instabilities in an Inhomogeneous Plasma (A. Hilger, 1992).

F. de Fornel, Evanescent Waves: from Newtonian Optics to Atomic Optics, Springer Series in Opt. Sciences, Vol. 73 (Springer, 2001).

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Figures (7)

Fig. 1
Fig. 1

Geometry of the narrowed waveguide. The following geometrical parameters are indicated: the narrowing length b, the distance between the waveguide walls d; the distance between the tops of the coordinate lines β 0 and β 0 determines minimum width s.

Fig. 2
Fig. 2

Dependence of the transmittance on the slit width s in the range 225 nm < s < 300 nm , while b = 1400 nm , d = 500 nm are kept fixed, and the wavelength range is determined according to Eq. (13): curve 1, indicated by dashed–dotted (magenta online), corresponds to s = 225 nm ; curve 2, dashed (blue on line), s = 250 nm ; curve 3, dotted (green online), s = 275 nm ; curve 4, solid (red online), s = 300 nm .

Fig. 3
Fig. 3

Dependence of the transmittance on the parameter d in the range 301 nm < d < 350 nm , while b = 1400 nm , s = 300 nm are kept fixed and the wavelength range varies with d according to Eq. (13): curve 1, solid (red online) corresponds to d = 301 nm ; curve 2, dashed (blue online) d = 325 nm ; curve 3, dashed–dotted (magenta online), d = 350 nm .

Fig. 4
Fig. 4

Dependence of the transmittance on the parameter b in the range 400 nm < d < 800 nm , while d = 500 nm , s = 300 nm are kept fixed: curve 1, dashed–dotted (black online), b = 400 nm ; curve 2, dashed (blue online),     b = 500 nm ; curve 3, dotted (green online), b =     600 nm ; curve 4, solid (red online), b =     700 nm ; curve 5, dashed–doted (magenta online),     b = 800 nm .

Fig. 5
Fig. 5

Dependence of transmittance on the permittivity ε in the range 1.5 < ε < 2.25 , while d = 500 nm , s = 300 nm , b = 1400 nm are kept fixed: curve 1, solid (red online) corresponds to ε = 2.25 ; curve 2, dotted (green online),     ε = 2 ; curve 3, dashed (blue online),     ε = 1.75 ; curve 4, dashed–doted (magenta online),     ε = 1.5 .

Fig. 6
Fig. 6

(a) Transmittance versus wavelength for the waveguide with the parameters b = 1400 nm , d = 400 nm , s = 300 nm . Wavelengths of the modes at λ = 975 nm , 935 nm , 910 nm , and 770 nm as shown in (b)–(e) are indicated by arrows. (b)–(d) distributions of the intensity of the electric field within the waveguide cross section associated with the modes at the wavelengths indicated in (a).

Fig. 7
Fig. 7

Intensity of the electric field | E y | 2 within the waveguide cross section associated with the mode (a) at λ = 955 nm for the waveguide characterized by the parameters b = 1600 nm , d = 500 nm , s = 300 nm ; (b) at λ = 950 nm for the waveguide characterized by the parameters b = 1700 nm , d = 500 nm , s = 300 nm .

Equations (64)

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x = a ch α sin β ; z = a sh α cos β .
β 0 = ± Arctg ( d 2 s 2 b ) ; a = s 2 | sin β 0 | .
E y = 1 c Ψ t ; H x = Ψ z ; H z = Ψ x .
d 2 F d α 2 + p 2 ( ch 2 α A ) F = 0 ; p 2 = ω 2 a 2 ε c 2 ;
d 2 f d β 2 + p 2 ( A sin 2 β ) f = 0 .
F 1 = n = 0 F 1 n α 2 n ; F 2 = n = 0 F 2 n α 2 n + 1 .
F 10 = 1 ; F 11 = p 2 ( 1 A ) 2 ; F 12 = p 2 12 [ p 2 ( 1 A ) 2 2 1 ] ;
F 20 = 1 ; F 21 = p 2 ( 1 A ) 6 ; F 22 = p 2 20 [ p 2 ( 1 A ) 2 6 1 ] .
f ( β 0 ) = f ( β 0 ) = 0 .
f = n = 0 f n β 2 n ; f 0 = 1 ; f 1 = p 2 A 2 ; f 2 = p 2 12 ( 1 + p 2 A 2 2 ) .
sin 2 β = β 2 β 4 3 + 2 β 6 45
A 0 = 2 ( p β 0 ) 2 .
d ε < λ < 2 d ε .
A 1 , 2 = 6 ( p β 0 ) 2 ( 1 ± 1 3 p 2 β 0 4 18 ) .
Ψ 1 = B 1 cos ( k x ) exp [ i ( γ z ω t ) ] ; k = π d ;
γ = ω 2 c 2 ε k 2 ,
E y = i ω B 1 c cos ( k x ) ; H x = i γ B 1 cos ( k x ) ;
H z = k B 1 sin ( k x ) .
Ψ 2 = B 2 [ F 1 ( α ) + Q F 2 ( α ) ] f ( β ) .
α ( x ) = ± Arcch ( q + g 2 ) ;
β ( x ) = ± Arc sin ( q g 2 ) ; x = u d 2 ;
q = 1 + ( b 2 a ) 2 + ( u d 2 a ) 2 ; g = q 2 ( u d a ) 2 .
B 1 ( 1 + R ) = B 2 d 2 ( χ 1 + Q χ 2 ) ;
χ 1 = 1 1 F 1 ( α ) f ( β ) cos ( π 2 u ) d u ;
χ 2 = 1 1 F 2 ( α ) f ( β ) cos ( π 2 u ) d u .
z = 1 a g ( ch α cos β α sh α sin β β ) .
i γ B 1 ( 1 R ) = B 2 d 2 a ( σ 1 + Q σ 2 ) ;
σ 1 = 1 1 d u g [ ch α cos β F 1 α f ( β ) sh α sin β F 1 ( α ) f β ] ;
σ 2 = 1 1 d u g [ ch α cos β F 2 α f ( β ) sh α sin β F 2 ( α ) f β ] .
R = i γ a χ 1 σ 1 + Q ( i γ a χ 2 σ 2 ) i γ a χ 1 + σ 1 + Q ( i γ a χ 2 + σ 2 ) .
| F 1 | z = b 2 = | F 1 | z = b 2 ; | F 1 α | z = b 2 = | F 1 α | z = b 2 ;
| F 2 | z = b 2 = | F 2 | z = b 2 ; | F 2 α | z = b 2 = | F 2 α | z = b 2 ;
| χ 1 | z = b 2 = | χ 1 | z = b 2 ; | χ 2 | z = b 2 = | χ 2 | z = b 2 ;
| σ 1 | z = b 2 = | σ 1 | z = b 2 ; | σ 2 | z = b 2 = | σ 2 | z = b 2 .
B 2 d 2 ( χ 1 Q χ 2 ) = B 3 ; B 2 d 2 a ( σ 1 + Q σ 2 ) = i γ B 3 .
Q = i γ a χ 1 + σ 1 i γ a χ 2 + σ 2 .
R = ( γ a ) 2 χ 1 χ 2 + σ 1 σ 2 Δ ;
Δ = ( γ a ) 2 χ 1 χ 2 σ 1 σ 2 i γ a ( χ 1 σ 2 + χ 2 σ 1 ) .
T = i γ a ( χ 1 σ 2 χ 2 σ 1 ) Δ .
| R | = | ( γ a ) 2 χ 1 χ 2 + σ 1 σ 2 | | Δ | ; | T | = 1 | R | 2 ;
φ r = Arctg [ γ a ( χ 1 σ 2 + χ 2 σ 1 ) ( γ a ) 2 χ 1 χ 2 σ 1 σ 2 ] ;
φ t = Arctg [ σ 1 σ 2 ( γ a ) 2 χ 1 χ 2 γ a ( χ 1 σ 2 + χ 2 σ 1 ) ] .
φ r φ t = π 2 .
F 1 F 1 ( A 1 ) + G 1 F 1 ( A 2 ) ; F 2 F 2 ( A 1 ) + G 2 F 2 ( A 2 ) .
Ψ 2 = B 2 { [ F 1 ( A 1 ) f ( A 1 ) + G 1 F 1 ( A 2 ) f ( A 2 ) ] + Q [ F 2 ( A 1 ) f ( A 1 ) + G 2 F 2 ( A 2 ) f ( A 2 ) ] } .
χ 1 χ 11 + G 1 χ 12 ; χ 2 χ 21 + G 2 χ 22 ;
σ 1 σ 11 + G 1 σ 12 ; σ 2 σ 21 + G 2 σ 22 .
χ 11 = 1 1 F 1 ( A 1 ) f ( A 1 ) cos ( π 2 u ) d u ;
χ 12 = 1 1 F 1 ( A 2 ) f ( A 2 ) cos ( π 2 u ) d u ;
χ 21 = 1 1 F 2 ( A 1 ) f ( A 1 ) cos ( π 2 u ) d u ;
χ 22 = 1 1 F 2 ( A 2 ) f ( A 2 ) cos ( π 2 u ) d u .
σ 11 = 1 1 d u g [ ch α cos β F 1 ( A 1 ) α f ( A 1 ) sh α sin β F 1 ( A 1 ) f ( A 1 ) β ] cos ( π 2 u ) .
σ 21 = 1 1 d u g [ ch α cos β F 2 ( A 1 ) α f ( A 1 ) sh α sin β F 2 ( A 1 ) f ( A 1 ) β ] cos ( π 2 u ) .
| Ψ | z 0.5 b = B exp [ γ 2 ( z + b 2 ) ] cos ( k 2 x ) ; | Ψ | z 0.5 b = P exp [ γ 2 ( z b 2 ) ] cos ( k 2 x ) ;
γ 2 = ( 2 π d ) 2 ( ω c ) 2 ε .
1 γ 2 = a [ ς 11 + G 1 ς 12 + Q ( ς 21 + G 2 ς 22 ) ] δ 11 + G 1 δ 12 + Q ( δ 21 + G 2 δ 22 ) .
ς 11 = 1 1 F 1 ( A 1 ) f ( A 1 ) cos ( π u ) d u .
δ 11 = 1 1 d u g [ ch α cos β F 1 ( A 1 ) α f ( A 1 ) sh α sin β F 1 ( A 1 ) f ( A 1 ) β ] cos ( π u ) .
δ 21 = 1 1 d u g [ ch α cos β F 2 ( A 1 ) α f ( A 1 ) sh α sin β F 2 ( A 1 ) f ( A 1 ) β ] cos ( π u ) .
1 γ 2 = a [ ς 11 + G 1 ς 12 Q ( ς 21 + G 2 ς 22 ) ] δ 11 G 1 δ 12 + Q ( δ 21 + G 2 δ 22 ) .
G 1 = δ 11 τ 2 ς 11 τ 2 ς 12 δ 12 ; G 2 = δ 21 τ 2 ς 21 τ 2 ς 22 δ 22 ; τ 2 = γ 2 a .
Q = i τ M 1 + K 1 i τ M 2 + K 2 ; τ = γ a ;
M 1 = χ 11 + G 1 χ 12 ; M 2 = χ 21 + G 2 χ 22 ; K 1 = σ 11 + G 1 σ 12 ; K 2 = σ 21 + G 2 σ 22 .
R = τ 2 M 1 M 2 + K 1 K 2 τ 2 M 1 M 2 K 1 K 2 i τ ( M 1 K 2 + M 2 K 1 ) .

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