Abstract

A study is presented giving the response of three types of fiber-optic interferometers by which a standing wave through an object is investigated. The three types are a Sagnac, Mach–Zehnder and Michelson–Morley interferometer. The response of the Mach–Zehnder interferometer is similar to the Sagnac interferometer. However, the Sagnac interferometer is much harder to study because of the fact that one input port and output port coincide. Further, the Mach–Zehnder interferometer has the advantage that the output ports are symmetric, reducing the systematic effects. Examples of standing wave light absorption in several simple objects are given. Attention is drawn to the influence of standing waves in fiber-optic interferometers with weak-absorbing layers incorporated. A method is described for how these can be theoretically analyzed and experimentally measured. Further experiments are needed for a thorough comparison between theory and experiment.

© 2011 Optical Society of America

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References

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  1. F. A. Jenkins and H. E. White, Fundamentals of Optics (McGraw-Hill, 1981).
  2. E. Hecht and A. Zajac, Optics (Addison–Wesley, 1979).
  3. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  4. J. E. M. Lekner, Theory of Reflection (Martinus Nijhoff, 1987).
  5. O. Wiener, “Stehende Lichtwellen und die Schwingungsrichtung polarisirten Lichtes,” Ann. Phys. (Berlin) 276, 203(1890).
    [CrossRef]
  6. H. E. Ives and T. C. Fry, “Standing light waves; repetition of an experiment by Wiener, using a photoelectric probe surface,” J. Opt. Soc. Am. 23, 73–83 (1933).
    [CrossRef]
  7. E. W. Silvertooth and S. F. Jacobs, “Standing wave sensor,” Appl. Opt. 22, 1274–1275 (1983).
    [CrossRef] [PubMed]
  8. L. Carraresi, E. A. De Souza, D. A. B. Miller, W. Y. Jan, and J. E. Cunningham, “Wavelength-selective detector based on a quantum well in a standing wave,” Appl. Phys. Lett. 64, 134–136 (1994).
    [CrossRef]
  9. M. Sasaki, X. Mi, and K. Hane, “Standing wave detection and interferometer application using a photodiode thinner than optical wavelength,” Appl. Phys. Lett. 75, 2008 (1999).
    [CrossRef]
  10. H.-J. Büchner, H. Stiebig, V. Mandryka, E. Bunte, and G. Jäger, “An optical standing-wave interferometer for displacement measurements,” Meas. Sci. Technol. 14, 311(2003).
    [CrossRef]
  11. Y. Li, X. Mi, M. Sasaki, and K. Hane, “Precision optical displacement sensor based on ultra-thin film photodiode type optical interferometers,” Meas. Sci. Technol. 14, 479 (2003).
    [CrossRef]
  12. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media in Vol. 8 of Course of Theoretical Physics, 2nd ed. (Elsevier, 1984).
  13. C. L. Chen, Foundations for Guided-Wave Optics (Wiley, 2007).

2003 (2)

H.-J. Büchner, H. Stiebig, V. Mandryka, E. Bunte, and G. Jäger, “An optical standing-wave interferometer for displacement measurements,” Meas. Sci. Technol. 14, 311(2003).
[CrossRef]

Y. Li, X. Mi, M. Sasaki, and K. Hane, “Precision optical displacement sensor based on ultra-thin film photodiode type optical interferometers,” Meas. Sci. Technol. 14, 479 (2003).
[CrossRef]

1999 (1)

M. Sasaki, X. Mi, and K. Hane, “Standing wave detection and interferometer application using a photodiode thinner than optical wavelength,” Appl. Phys. Lett. 75, 2008 (1999).
[CrossRef]

1994 (1)

L. Carraresi, E. A. De Souza, D. A. B. Miller, W. Y. Jan, and J. E. Cunningham, “Wavelength-selective detector based on a quantum well in a standing wave,” Appl. Phys. Lett. 64, 134–136 (1994).
[CrossRef]

1983 (1)

1933 (1)

1890 (1)

O. Wiener, “Stehende Lichtwellen und die Schwingungsrichtung polarisirten Lichtes,” Ann. Phys. (Berlin) 276, 203(1890).
[CrossRef]

Büchner, H.-J.

H.-J. Büchner, H. Stiebig, V. Mandryka, E. Bunte, and G. Jäger, “An optical standing-wave interferometer for displacement measurements,” Meas. Sci. Technol. 14, 311(2003).
[CrossRef]

Bunte, E.

H.-J. Büchner, H. Stiebig, V. Mandryka, E. Bunte, and G. Jäger, “An optical standing-wave interferometer for displacement measurements,” Meas. Sci. Technol. 14, 311(2003).
[CrossRef]

Carraresi, L.

L. Carraresi, E. A. De Souza, D. A. B. Miller, W. Y. Jan, and J. E. Cunningham, “Wavelength-selective detector based on a quantum well in a standing wave,” Appl. Phys. Lett. 64, 134–136 (1994).
[CrossRef]

Chen, C. L.

C. L. Chen, Foundations for Guided-Wave Optics (Wiley, 2007).

Cunningham, J. E.

L. Carraresi, E. A. De Souza, D. A. B. Miller, W. Y. Jan, and J. E. Cunningham, “Wavelength-selective detector based on a quantum well in a standing wave,” Appl. Phys. Lett. 64, 134–136 (1994).
[CrossRef]

De Souza, E. A.

L. Carraresi, E. A. De Souza, D. A. B. Miller, W. Y. Jan, and J. E. Cunningham, “Wavelength-selective detector based on a quantum well in a standing wave,” Appl. Phys. Lett. 64, 134–136 (1994).
[CrossRef]

Fry, T. C.

Hane, K.

Y. Li, X. Mi, M. Sasaki, and K. Hane, “Precision optical displacement sensor based on ultra-thin film photodiode type optical interferometers,” Meas. Sci. Technol. 14, 479 (2003).
[CrossRef]

M. Sasaki, X. Mi, and K. Hane, “Standing wave detection and interferometer application using a photodiode thinner than optical wavelength,” Appl. Phys. Lett. 75, 2008 (1999).
[CrossRef]

Hecht, E.

E. Hecht and A. Zajac, Optics (Addison–Wesley, 1979).

Ives, H. E.

Jacobs, S. F.

Jäger, G.

H.-J. Büchner, H. Stiebig, V. Mandryka, E. Bunte, and G. Jäger, “An optical standing-wave interferometer for displacement measurements,” Meas. Sci. Technol. 14, 311(2003).
[CrossRef]

Jan, W. Y.

L. Carraresi, E. A. De Souza, D. A. B. Miller, W. Y. Jan, and J. E. Cunningham, “Wavelength-selective detector based on a quantum well in a standing wave,” Appl. Phys. Lett. 64, 134–136 (1994).
[CrossRef]

Jenkins, F. A.

F. A. Jenkins and H. E. White, Fundamentals of Optics (McGraw-Hill, 1981).

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media in Vol. 8 of Course of Theoretical Physics, 2nd ed. (Elsevier, 1984).

Lekner, J. E. M.

J. E. M. Lekner, Theory of Reflection (Martinus Nijhoff, 1987).

Li, Y.

Y. Li, X. Mi, M. Sasaki, and K. Hane, “Precision optical displacement sensor based on ultra-thin film photodiode type optical interferometers,” Meas. Sci. Technol. 14, 479 (2003).
[CrossRef]

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media in Vol. 8 of Course of Theoretical Physics, 2nd ed. (Elsevier, 1984).

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Mandryka, V.

H.-J. Büchner, H. Stiebig, V. Mandryka, E. Bunte, and G. Jäger, “An optical standing-wave interferometer for displacement measurements,” Meas. Sci. Technol. 14, 311(2003).
[CrossRef]

Mi, X.

Y. Li, X. Mi, M. Sasaki, and K. Hane, “Precision optical displacement sensor based on ultra-thin film photodiode type optical interferometers,” Meas. Sci. Technol. 14, 479 (2003).
[CrossRef]

M. Sasaki, X. Mi, and K. Hane, “Standing wave detection and interferometer application using a photodiode thinner than optical wavelength,” Appl. Phys. Lett. 75, 2008 (1999).
[CrossRef]

Miller, D. A. B.

L. Carraresi, E. A. De Souza, D. A. B. Miller, W. Y. Jan, and J. E. Cunningham, “Wavelength-selective detector based on a quantum well in a standing wave,” Appl. Phys. Lett. 64, 134–136 (1994).
[CrossRef]

Sasaki, M.

Y. Li, X. Mi, M. Sasaki, and K. Hane, “Precision optical displacement sensor based on ultra-thin film photodiode type optical interferometers,” Meas. Sci. Technol. 14, 479 (2003).
[CrossRef]

M. Sasaki, X. Mi, and K. Hane, “Standing wave detection and interferometer application using a photodiode thinner than optical wavelength,” Appl. Phys. Lett. 75, 2008 (1999).
[CrossRef]

Silvertooth, E. W.

Stiebig, H.

H.-J. Büchner, H. Stiebig, V. Mandryka, E. Bunte, and G. Jäger, “An optical standing-wave interferometer for displacement measurements,” Meas. Sci. Technol. 14, 311(2003).
[CrossRef]

White, H. E.

F. A. Jenkins and H. E. White, Fundamentals of Optics (McGraw-Hill, 1981).

Wiener, O.

O. Wiener, “Stehende Lichtwellen und die Schwingungsrichtung polarisirten Lichtes,” Ann. Phys. (Berlin) 276, 203(1890).
[CrossRef]

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Zajac, A.

E. Hecht and A. Zajac, Optics (Addison–Wesley, 1979).

Ann. Phys. (Berlin) (1)

O. Wiener, “Stehende Lichtwellen und die Schwingungsrichtung polarisirten Lichtes,” Ann. Phys. (Berlin) 276, 203(1890).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (2)

L. Carraresi, E. A. De Souza, D. A. B. Miller, W. Y. Jan, and J. E. Cunningham, “Wavelength-selective detector based on a quantum well in a standing wave,” Appl. Phys. Lett. 64, 134–136 (1994).
[CrossRef]

M. Sasaki, X. Mi, and K. Hane, “Standing wave detection and interferometer application using a photodiode thinner than optical wavelength,” Appl. Phys. Lett. 75, 2008 (1999).
[CrossRef]

J. Opt. Soc. Am. (1)

Meas. Sci. Technol. (2)

H.-J. Büchner, H. Stiebig, V. Mandryka, E. Bunte, and G. Jäger, “An optical standing-wave interferometer for displacement measurements,” Meas. Sci. Technol. 14, 311(2003).
[CrossRef]

Y. Li, X. Mi, M. Sasaki, and K. Hane, “Precision optical displacement sensor based on ultra-thin film photodiode type optical interferometers,” Meas. Sci. Technol. 14, 479 (2003).
[CrossRef]

Other (6)

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media in Vol. 8 of Course of Theoretical Physics, 2nd ed. (Elsevier, 1984).

C. L. Chen, Foundations for Guided-Wave Optics (Wiley, 2007).

F. A. Jenkins and H. E. White, Fundamentals of Optics (McGraw-Hill, 1981).

E. Hecht and A. Zajac, Optics (Addison–Wesley, 1979).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

J. E. M. Lekner, Theory of Reflection (Martinus Nijhoff, 1987).

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Figures (18)

Fig. 1
Fig. 1

Reflection and transmission of a plane wave with incident angle θ i on a substrate boundary at y = 0 .

Fig. 2
Fig. 2

Reflection and transmission geometry for a standing wave at a multilayered object.

Fig. 3
Fig. 3

Sketch of fiber-optic bidirectional coupler.

Fig. 4
Fig. 4

Sketch of fiber-optic Sagnac interferometer. The different directions of the light waves on top and at the bottom of the object are only for display purposes.

Fig. 5
Fig. 5

Sketch of fiber-optic Mach–Zehnder inter ferometer including a standing wave at the object. The different directions of the light waves on top and at the bottom of the object are only for display purposes.

Fig. 6
Fig. 6

Sketch of fiber-optic Michelson–Morley interferometer including a standing wave at the object. The different directions of the light waves on top and at the bottom of the object are only for display purposes.

Fig. 7
Fig. 7

Visibilities of Mach–Zehnder (left) and Michelson–Morley (right) interferometers including a single layer (refractive index n = 1.45 ) with a thickness corresponding to an optical phase of π / 2 (top) and π (bottom) as function of the two optical phases ϕ s , ϕ m and ϕ E , ϕ m , respectively.

Fig. 8
Fig. 8

Visibilities of Mach-–Zehnder (left) and Michelson–Morley (right) interferometers including a single titanium (top) and silver (bottom) layer of 15 nm thickness as function of the two optical phases ϕ s , ϕ m and ϕ E , ϕ m , respectively.

Fig. 9
Fig. 9

Absorbed fraction in a Mach–Zehnder (left) and Michelson–Morley (right) interferometers including a single titanium (top) and silver (bottom) layer of varying thicknesses as function of the optical phase between the propagating waves, ϕ s and ϕ E , respectively.

Fig. 10
Fig. 10

Absorbed fraction in a Mach–Zehnder (left) and Michelson–Morley (right) interferometers including a single layer of varying thicknesses as function of the optical phase between the propagating waves, ϕ s and ϕ E , respectively. The refractive index of the layer is n = 1.8 + 0.02 i .

Fig. 11
Fig. 11

Schematic of Fabry–Pérot cavity with length, L. Each semitransparent mirror consists of a thin metal layer on top of a glass substrate. The mirrors can be positioned in four different ways to form the cavity: (a) glass substrates facing each other, (b) and (c) glass substrate facing metal layer, and (d) metal layers facing each other.

Fig. 12
Fig. 12

Absorbed fraction in a Mach–Zehnder interferometer as function of the optical phase between the propagating waves, ϕ s and the varying length of a Fabry–Pérot cavity for geometries (a) to (d) (see Fig. 11). The semitransparent mirrors consist of a single layer of 15 nm titanium of top of a 3 mm glass substrate.

Fig. 13
Fig. 13

Absorbed fraction in a Michelson–Morley interferometer as function of the optical phase between the propagating waves, ϕ E and the varying length of a Fabry–Pérot cavity for geometries (a) to (d) (see Fig. 11). The semitransparent mirrors consist of a single layer of 15 nm titanium of top of a 3 mm glass substrate.

Fig. 14
Fig. 14

Absorbed fraction in a Mach–Zehnder interferometer as function of the optical phase between the propagating waves, ϕ s and the varying length of a Fabry–Pérot cavity for geometries (a) to (d) (see Fig. 11). The semitransparent mir rors consist of a single layer of 15 nm silver of top of a 3 mm glass substrate.

Fig. 15
Fig. 15

Absorbed fraction in a Michelson–Morley interferometer as function of the optical phase between the propagating waves, ϕ E and the varying length of a Fabry–Pérot cavity for geometries (a) to (d) (see Fig. 11). The semitransparent mirrors consist of a single layer of 15 nm silver of top of a 3 mm glass substrate.

Fig. 16
Fig. 16

Measured (top) and theoretical (bottom) renormalized visibility of a Mach–Zehnder interferometer as function of the voltage applied to fiber stretchers B and C. (a) without object; (b) object consisting of a single layer of 14.7 nm silver on top of a 3 mm glass substrate.

Fig. 17
Fig. 17

Measured renormalized sum of intensities of outputs of a Mach–Zehnder interferometer as function of the voltage applied to fiber stretchers B and C. (a) without object; (b) object consisting of a single layer of 14.7 nm silver on top of a 3 mm glass substrate.

Fig. 18
Fig. 18

Measured (top) and theoretical (bottom) renormalized intensity of output of a Michelson–Morley inter ferometer as function of the voltages applied to mirror piezo stacks. (a) without object; (b) object consisting of a single layer of 14.7 nm silver on top of a 3 mm glass substrate.

Equations (69)

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Ψ ( r , t ) = Ψ ^ e i ( ϕ + k · r ω t ) ,
ω = k v p ,
v p = c n .
Ψ ( r , t ) = Ψ ^ 1 e i ( ϕ 1 + k 1 · r ω 1 t ) + Ψ ^ 2 e i ( ϕ 2 + k 2 · r ω 2 t ) .
| Ψ ( r , t ) | 2 = Ψ ^ 1 2 + Ψ ^ 2 2 + 2 Ψ ^ 1 Ψ ^ 2 cos ( ϕ 1 ϕ 2 + ( k 1 k 2 ) · r ( ω 1 ω 2 ) t ) .
Ψ ( y ) = Ψ 0 + e i q 0 y + Ψ 0 e i q 0 y
y j + 1 y y j :     Ψ ( y ) = Ψ j + e i q j ( y y j ) + Ψ j e i q j ( y y j ) ,
Ψ ( y ) = Ψ m + 1 + e i q m + 1 ( y y m + 1 ) + Ψ m + 1 e i q m + 1 ( y y m + 1 ) ,
q j 2 = k 0 2 ( n j 2 sin 2 θ i ) ,
q j = k 0 n j .
( Ψ m + 1 + Ψ m + 1 ) = ( m 11 m 12 m 21 m 22 ) ( Ψ 0 + Ψ 0 ) ,
( m 11 m 12 m 21 m 22 ) = M m M m 1 M 1 M 0 ,
M j = 1 2 ( 1 + q j q j + 1 1 q j q j + 1 1 q j q j + 1 1 + q j q j + 1 ) ( e i q j d j 0 0 e i q j d j ) ,
( 0 τ t ) = ( m 11 m 12 m 21 m 22 ) ( ρ t 1 )
( 1 ρ b ) = ( m 11 m 12 m 21 m 22 ) ( τ b 0 )
( m 11 m 12 m 21 m 22 ) = 1 τ ( 1 ρ t ρ b τ 2 ρ b ρ t ) .
Q ( y ) = ω ϵ 0 | Ψ ( y ) | 2 ( n ( y ) ) ( n ( y ) ) ,
A j = y j + 1 y j Q ( y ) d y ,
S ( y ) = c ϵ 0 2 | n ( y ) | | Ψ ( y ) | 2 .
S ref = c ϵ 0 2 ( | Ψ 0 | 2 + | Ψ m + 1 + | 2 )
α j = 2 k 0 d j ( n j ) ( n j ) | Ψ j + | 2 f ( ( q j ) d j ) + | Ψ j | 2 f ( ( q j ) d j ) + 2 ( ( Ψ j + ) * Ψ j f ( i ( q j ) d j ) ) | Ψ 0 | 2 + | Ψ m + 1 + | 2 ,
f ( x ) = 1 e 2 x 2 x = 1 x + O ( x 2 ) ,
α j = 2 k 0 d j ( n j ) ( n j ) | Ψ j + + ( Ψ j ) * | 2 | Ψ 0 | 2 + | Ψ m + 1 + | 2 .
( o 1 o 2 ) = 1 1 + η 2 ( η e i ϕ η i i η e i ϕ η ) ( i 1 i 2 ) ,
η = sin 2 δ sin 2 ζ + cos 2 ζ cos 2 δ sin 2 ζ = cot ζ + δ 2 sin 2 ζ + O ( δ 4 ) ,
tan ϕ η = tan ζ sin δ ,
( o 1 o 2 ) = 1 1 + η 2 ( η e i ϕ η i ) 1 2 ( 1 i ) .
( o 1 o 2 ) = 1 2 ( 1 e i Δ ϕ i e i Δ ϕ i ) ,
( a 0 b m + 1 ) = ( e i ϕ A 0 0 e i ϕ B ) ( o 1 o 2 ) .
( b 0 a m + 1 ) = ( ρ t τ τ ρ b ) ( a 0 b m + 1 ) .
( i 3 i 4 ) = ( e i ϕ A 0 0 e i ϕ B ) ( b 0 a m + 1 ) .
( o 3 o 4 ) = 1 1 + η 2 ( η e i ϕ η i i η e i ϕ η ) ( i 3 i 4 ) .
( o 3 o 4 ) = e i ( ϕ A + ϕ B ) 1 + η 2 × ( e i ϕ η ( ρ t η 2 e i ϕ s ρ b e i ϕ s 2 i τ η ) ( η 2 1 ) τ η i ( ρ t e i ϕ s + ρ b e i ϕ s ) ) ,
| o 3 | 2 + | o 4 | 2 = T | 1 + η χ t | 2 + | η + χ b | 2 1 + η 2 ,
χ t = ρ t τ e i ( ϕ s + π / 2 ) and χ b = ρ b τ e i ( ϕ s + π / 2 ) .
| o 3 | 2 + | o 4 | 2 = T + R = 1.
| o 3 | 2 + | o 4 | 2 = T + R 4 η T R cos ( arg ρ ) sin ϕ s 1 + η 2 .
( i 3 i 4 ) = ( e i ϕ A o 1 0 ) ,
( i 5 i 6 ) = ( e i ϕ B o 2 0 ) .
( a 0 b m + 1 ) = ( e i ϕ E 0 0 e i ϕ F ) ( o 3 o 5 ) .
( i 7 i 8 ) = ( e i ϕ E b 0 0 ) ,
( i 9 i 10 ) = ( e i ϕ F a m + 1 0 ) .
( i 13 i 14 ) = ( e i ϕ A 0 0 e i ϕ B ) ( o 7 o 9 ) .
( o 4 o 6 ) = 1 1 + η 2 ( i η e i ( ϕ η + ϕ A ) e i ϕ B ) .
( o 13 o 14 ) = η 2 e i ( 2 ϕ η + ϕ A + ϕ B + ϕ E + ϕ F ) ( 1 + η 2 ) 2 × ( e i ϕ η ( ρ t η 2 e i ϕ s ρ b e i ϕ s 2 i τ η ) ( η 2 1 ) τ η i ( ρ t e i ϕ s + ρ b e i ϕ s ) ) ,
( i 11 i 12 ) = ( e i ϕ C 0 0 e i ϕ D ) ( o 8 o 10 ) .
( o 11 o 12 ) = i η e i ( ϕ η + ϕ E + ϕ F + ( ϕ A + ϕ B + ϕ C + ϕ D ) / 2 ) ( 1 + η 2 ) 2 × ( e i ϕ η ( ρ t η 2 e i ϕ s ρ b e i ϕ s 2 i τ η cos ϕ m ) ( η 2 e i ϕ m e i ϕ m ) τ η i ( ρ t e i ϕ s + ρ b e i ϕ s ) ) ,
| o 11 | 2 + | o 12 | 2 = η 2 T ( 1 + η 2 ) 3 ( | 1 + η χ t | 2 + | η + χ b | 2 ) ,
χ t = ρ t τ e i ( ϕ s + π / 2 ) and χ b = ρ b τ e i ( ϕ s + π / 2 )
| o 11 | 2 + | o 12 | 2 = η 2 ( 1 + η 2 ) 2 1 4
V = | o 11 | 2 | o 12 | 2 | o 11 | 2 + | o 12 | 2 = η 2 1 η 2 + 1 V 1 + V 2 .
V 1 = | 1 + η χ t | 2 | η + χ b | 2 | 1 + η χ t | 2 + | η + χ b | 2 ,
V 2 = υ cos ( 2 ϕ m arg ( 1 + η χ t ) + arg ( η + χ b ) ) ,
υ = 4 η η 2 + 1 | ( 1 + η χ t ) ( η + χ b ) | | 1 + η χ t | 2 + | η + χ b | 2 .
V 1 = ( 1 η 2 ) ( 2 T 1 ) + 4 η T { χ t } 1 + η 2 ,
υ = 4 η T ( η 2 + 1 ) 2 | ( 1 + η χ t ) ( η + χ t ) | .
( o 8 o 10 ) = i η e i ( ϕ η + ϕ E + ϕ F + ϕ B ) ( 1 + η 2 ) ( 3 / 2 ) × ( η ρ t e i ϕ s i τ e i ( ϕ F ϕ E ) ( η e i ϕ s ρ b i ) ) ,
| 2 T 1 | = X Y .
ξ = m 12 + m 22 e 2 i ϕ E m 11 + m 21 e 2 i ϕ E .
( o 3 o 4 ) = e 2 i ϕ B 1 + η 2 ( 1 ξ η 2 e i ϕ m η i e i ϕ η ( 1 + ξ e i ϕ m ) ) ,
| o 3 | 2 + | o 4 | 2 = 1 + η 2 | ξ | 2 1 + η 2 ,
| o 3 | 2 | o 4 | 2 | o 3 | 2 + | o 4 | 2 = ( 1 η 2 ) ( 1 η 2 | ξ | 2 ) 4 η 2 { ξ e i ϕ m } ( 1 + η 2 ) ( 1 + η 2 | ξ | 2 ) .
| o 3 | 2 + | o 4 | 2 = 1 ,
| o 3 | 2 | o 4 | 2 | o 3 | 2 + | o 4 | 2 = ( 1 η 2 ) 2 4 η 2 cos ( ϕ m + arg ξ ) ( 1 + η 2 ) 2 .
ρ = e 2 i n q 0 t 1 1 ρ F 2 e 2 i n q 0 t ρ F ,
ρ F = n 1 n + 1 .
τ = e i n q 0 t 1 ρ F 2 e 2 i n q 0 t τ F ,
τ F = 4 n ( n + 1 ) 2 .
ρ τ = i sin ( n q 0 t ) ρ F τ F .

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