Abstract

A fluctuation-characterization method using an interferometer is ordinarily classically formulated by considering frequency fluctuations. However, it is preferable to treat it quantum mechanically because the method measures q-number quadrature fluctuations at the same time. We formulated an interferometric fluctuation-characterization method by treating quadrature fluctuations as q-number variables and frequency fluctuations as a c-number stochastic variable, and we gave the power spectral density of the fluctuations for pulse light. This method measures fluctuations that are calibrated with shot noise level, and it makes it possible to estimate the fluctuation angle in phase space, which is the quantity that determines transmission characteristics in optical communications. Fluctuations in a laser diode were characterized as an example of measurement.

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References

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  1. T. Tomaru, “Secret key distribution protocol for practical optical channels using a preshared key and phase fluctuations,” Jpn. J. Appl. Phys. 49, 074401 (2010).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  6. U. Leonhardt and H. Paul, “Measuring the quantum state of light,” Prog. Quant. Electr. 19, 89–130 (1995).
    [CrossRef]
  7. U. Leonhardt, Measuring the Quantum State of Light(Cambridge University Press, 1997).
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    [CrossRef]
  9. K. Kikuchi and T. Okoshi, “Measurement of FM noise, AM noise, and field spectra of 1.3 μm InGaAsP DFB lasers and determination of the linewidth enhancement factor,” IEEE J. Quantum Electron. 21, 1814–1818 (1985) and references within.
    [CrossRef]
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    [CrossRef]
  11. C. H. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. 18, 259–264 (1982).
    [CrossRef]
  12. Y. Yamamoto and H. A. Haus, “Commutation relations and laser linewidth,” Phys. Rev. A 41, 5164–5170 (1990).
    [CrossRef] [PubMed]
  13. P. W. Anderson, “A mathematical model for the narrowing of spectral lines by exchange or motion,” J. Phys. Soc. Jpn. 9, 316–339 (1954).
    [CrossRef]
  14. R. Kubo, “Note on the stochastic theory of resonance absorption,” J. Phys. Soc. Jpn. 9, 935–944 (1954).
    [CrossRef]
  15. T. Tomaru, “Femtosecond pulse squeezing limited by stimulated-Raman process in optical fibers,” Opt. Commun. 273, 263–271 (2007).
    [CrossRef]

2010 (1)

T. Tomaru, “Secret key distribution protocol for practical optical channels using a preshared key and phase fluctuations,” Jpn. J. Appl. Phys. 49, 074401 (2010).
[CrossRef]

2007 (1)

T. Tomaru, “Femtosecond pulse squeezing limited by stimulated-Raman process in optical fibers,” Opt. Commun. 273, 263–271 (2007).
[CrossRef]

1995 (1)

U. Leonhardt and H. Paul, “Measuring the quantum state of light,” Prog. Quant. Electr. 19, 89–130 (1995).
[CrossRef]

1990 (1)

Y. Yamamoto and H. A. Haus, “Commutation relations and laser linewidth,” Phys. Rev. A 41, 5164–5170 (1990).
[CrossRef] [PubMed]

1986 (1)

R. W. Tkach, “Phase noise and linewidth in an InGaAsP DFB laser,” IEEE J. Lightwave Technol. LT-4, 1711–1716 (1986) and references within.
[CrossRef]

1985 (2)

J. Shapiro, “Quantum noise and excess noise in optical homodyne and heterodyne receivers,” IEEE J. Quantum Electron. 21, 237–250 (1985).
[CrossRef]

K. Kikuchi and T. Okoshi, “Measurement of FM noise, AM noise, and field spectra of 1.3 μm InGaAsP DFB lasers and determination of the linewidth enhancement factor,” IEEE J. Quantum Electron. 21, 1814–1818 (1985) and references within.
[CrossRef]

1984 (1)

1983 (2)

1982 (1)

C. H. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. 18, 259–264 (1982).
[CrossRef]

1966 (1)

1954 (2)

P. W. Anderson, “A mathematical model for the narrowing of spectral lines by exchange or motion,” J. Phys. Soc. Jpn. 9, 316–339 (1954).
[CrossRef]

R. Kubo, “Note on the stochastic theory of resonance absorption,” J. Phys. Soc. Jpn. 9, 935–944 (1954).
[CrossRef]

Anderson, P. W.

P. W. Anderson, “A mathematical model for the narrowing of spectral lines by exchange or motion,” J. Phys. Soc. Jpn. 9, 316–339 (1954).
[CrossRef]

Armstrong, J. A.

Chan, V. W. S.

Haus, H. A.

Y. Yamamoto and H. A. Haus, “Commutation relations and laser linewidth,” Phys. Rev. A 41, 5164–5170 (1990).
[CrossRef] [PubMed]

Henry, C. H.

C. H. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. 18, 259–264 (1982).
[CrossRef]

Kikuchi, K.

K. Kikuchi and T. Okoshi, “Measurement of FM noise, AM noise, and field spectra of 1.3 μm InGaAsP DFB lasers and determination of the linewidth enhancement factor,” IEEE J. Quantum Electron. 21, 1814–1818 (1985) and references within.
[CrossRef]

Kubo, R.

R. Kubo, “Note on the stochastic theory of resonance absorption,” J. Phys. Soc. Jpn. 9, 935–944 (1954).
[CrossRef]

Leonhardt, U.

U. Leonhardt and H. Paul, “Measuring the quantum state of light,” Prog. Quant. Electr. 19, 89–130 (1995).
[CrossRef]

U. Leonhardt, Measuring the Quantum State of Light(Cambridge University Press, 1997).

Okoshi, T.

K. Kikuchi and T. Okoshi, “Measurement of FM noise, AM noise, and field spectra of 1.3 μm InGaAsP DFB lasers and determination of the linewidth enhancement factor,” IEEE J. Quantum Electron. 21, 1814–1818 (1985) and references within.
[CrossRef]

Paul, H.

U. Leonhardt and H. Paul, “Measuring the quantum state of light,” Prog. Quant. Electr. 19, 89–130 (1995).
[CrossRef]

Schumaker, B. L.

Shapiro, J.

J. Shapiro, “Quantum noise and excess noise in optical homodyne and heterodyne receivers,” IEEE J. Quantum Electron. 21, 237–250 (1985).
[CrossRef]

Tkach, R. W.

R. W. Tkach, “Phase noise and linewidth in an InGaAsP DFB laser,” IEEE J. Lightwave Technol. LT-4, 1711–1716 (1986) and references within.
[CrossRef]

Tomaru, T.

T. Tomaru, “Secret key distribution protocol for practical optical channels using a preshared key and phase fluctuations,” Jpn. J. Appl. Phys. 49, 074401 (2010).
[CrossRef]

T. Tomaru, “Femtosecond pulse squeezing limited by stimulated-Raman process in optical fibers,” Opt. Commun. 273, 263–271 (2007).
[CrossRef]

Yamamoto, Y.

Y. Yamamoto and H. A. Haus, “Commutation relations and laser linewidth,” Phys. Rev. A 41, 5164–5170 (1990).
[CrossRef] [PubMed]

Yuen, H. P.

IEEE J. Lightwave Technol. (1)

R. W. Tkach, “Phase noise and linewidth in an InGaAsP DFB laser,” IEEE J. Lightwave Technol. LT-4, 1711–1716 (1986) and references within.
[CrossRef]

IEEE J. Quantum Electron. (3)

C. H. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. 18, 259–264 (1982).
[CrossRef]

J. Shapiro, “Quantum noise and excess noise in optical homodyne and heterodyne receivers,” IEEE J. Quantum Electron. 21, 237–250 (1985).
[CrossRef]

K. Kikuchi and T. Okoshi, “Measurement of FM noise, AM noise, and field spectra of 1.3 μm InGaAsP DFB lasers and determination of the linewidth enhancement factor,” IEEE J. Quantum Electron. 21, 1814–1818 (1985) and references within.
[CrossRef]

J. Opt. Soc. Am. (1)

J. Phys. Soc. Jpn. (2)

P. W. Anderson, “A mathematical model for the narrowing of spectral lines by exchange or motion,” J. Phys. Soc. Jpn. 9, 316–339 (1954).
[CrossRef]

R. Kubo, “Note on the stochastic theory of resonance absorption,” J. Phys. Soc. Jpn. 9, 935–944 (1954).
[CrossRef]

Jpn. J. Appl. Phys. (1)

T. Tomaru, “Secret key distribution protocol for practical optical channels using a preshared key and phase fluctuations,” Jpn. J. Appl. Phys. 49, 074401 (2010).
[CrossRef]

Opt. Commun. (1)

T. Tomaru, “Femtosecond pulse squeezing limited by stimulated-Raman process in optical fibers,” Opt. Commun. 273, 263–271 (2007).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. A (1)

Y. Yamamoto and H. A. Haus, “Commutation relations and laser linewidth,” Phys. Rev. A 41, 5164–5170 (1990).
[CrossRef] [PubMed]

Prog. Quant. Electr. (1)

U. Leonhardt and H. Paul, “Measuring the quantum state of light,” Prog. Quant. Electr. 19, 89–130 (1995).
[CrossRef]

Other (1)

U. Leonhardt, Measuring the Quantum State of Light(Cambridge University Press, 1997).

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Figures (9)

Fig. 1
Fig. 1

Model of the interferometer.

Fig. 2
Fig. 2

Schematic fluctuations in phase space.

Fig. 3
Fig. 3

Block diagram of the experimental setup.

Fig. 4
Fig. 4

Noise was measured at a pulsed operation in a case of LD current of 70 mA . (a), (b) Measured with a free-space interferometer. The resolution bandwidth was 300 kHz . (c) Measured with an asymmetric interferometer module. The resolution bandwidth was 1 MHz . (a1), (b1)  3 dB below SNL. (a2) SNL. (b2), (c2) Phase fluctuations. (a3) Amplitude fluctuations. (b3), (c3) Phase fluctuation-mixed amplitude fluctuations. The meanings of trace numbers are common in Figs. 4, 5, 7, 8.

Fig. 5
Fig. 5

Noise was measured in a cw operation. (c2) Phase fluctuations show a dip at 2.5 GHz , which is different behavior from a pulsed-operation case. (c3) Phase fluctuation-mixed amplitude fluctuations.

Fig. 6
Fig. 6

Schematic figure of a projection measurement. When the projection axis tilts by ϕ, δ x is measured as fluctuations on a minor axis instead of δ q .

Fig. 7
Fig. 7

Noise was measured at a near-threshold cw operation. Amplitude fluctuations (c3) as well as phase fluctuations (c2) oscillate according to Eqs. (32, 34), respectively.

Fig. 8
Fig. 8

Noise was measured in a near-threshold pulsed operation. Trace numbers have the same meanings as those in Fig. 4. Because amplitude fluctuations are enlarged, trace (a2) does not indicate SNL due to the effect shown in Fig. 6. SNL is 3 dB above trace (a1).

Fig. 9
Fig. 9

Fluctuation angle δ φ is schematically shown, where the second term of Eq. (34) is neglected.

Tables (2)

Tables Icon

Table 1 Relations between Spectrum Analyzer Outputs and Fluctuations of Light

Tables Icon

Table 2 Measured Noise and Fluctuation Angle

Equations (69)

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H ^ = k ω k a ^ k ( t ) a ^ k ( t ) .
H ^ = k [ ω k + ω r ( t ) ] a ^ k ( t ) a ^ k ( t ) .
d d t a ^ k ( t ) = i [ H ( t ) , a ^ k ( t ) ] .
a ^ k ( t ) = exp ( i ω k t i φ ( t ) ) a ^ k ( 0 ) ,
φ ( t ) = 0 t ω r ( t ) d t .
a ^ ( t ) = k t k a ^ k ( 0 ) exp ( i ω k t ) exp ( i φ ( t ) ) .
a ^ c ( t ) = k t k a ^ k ( 0 ) exp ( i ω k t )
a ^ ( t ) = a ^ c ( t ) exp ( i φ ( t ) ) .
( c ^ ( t ) d ^ ( t ) ) = ( cos θ 1 sin θ 1 sin θ 1 cos θ 1 ) ( a ^ ( t ) b ( t ) ) ,
( e ^ ( t ) f ^ ( t ) ) = ( cos θ 2 sin θ 2 sin θ 2 cos θ 2 ) ( c ^ ( t + τ ) d ( t ) ) .
I ^ ( t ) ξ e ^ ( t ) e ^ ( t ) η f ^ ( t ) f ^ ( t ) = k 1 2 ( ξ + η ) [ a ^ ( t + τ ) a ^ ( t ) + a ^ ( t ) a ^ ( t + τ ) ] + k 1 2 ( ξ + η ) [ b ^ ( t + τ ) b ^ ( t ) + b ^ ( t ) b ^ ( t + τ ) ] + k 2 2 ( ξ + η ) [ cos 2 θ 1 ( a ^ ( t + τ ) b ^ ( t ) + b ^ ( t ) a ^ ( t + τ ) ) sin 2 θ 1 ( a ^ ( t ) b ^ ( t + τ ) + b ^ ( t + τ ) a ^ ( t ) ) ] .
I ^ ( t ) = k 1 2 ( ξ + η ) [ a ^ c ( t + τ ) a ^ c ( t ) exp ( i δ φ ( t , τ ) ) + a ^ c ( t ) a ^ c ( t + τ ) exp ( i δ φ ( t , τ ) ) ] + k 1 2 ( ξ + η ) [ b ^ ( t + τ ) b ^ ( t ) + b ^ ( t ) b ^ ( t + τ ) ] + k 2 2 ( ξ + η ) [ cos 2 θ 1 ( a ^ c ( t + τ ) b ^ ( t ) exp ( i φ ( t + τ ) ) + b ^ ( t ) a ^ c ( t + τ ) exp ( i φ ( t + τ ) ) ) sin 2 θ 1 ( a ^ c ( t ) b ^ ( t + τ ) exp ( i φ ( t ) ) + b ^ ( t + τ ) a ^ c ( t ) exp ( i φ ( t ) ) ) ] .
a ^ c ( t ) = ( a ^ 1 ( t ) + i a ^ 2 ( t ) ) exp ( i ω 0 t ) ,
b ^ ( t ) = ( b ^ 1 ( t ) + i b ^ 2 ( t ) ) exp ( i ω 0 t ) .
a ^ 1 ( t ) = a ¯ 1 ( t ) + δ a ^ 1 ( t ) ,
I ^ ( t ) k 1 2 ( ξ + η ) { a ¯ 1 ( t + τ ) a ¯ 1 ( t ) [ exp ( i ω 0 τ ) ( 1 + i δ φ ( t , τ ) ) + exp ( i ω 0 τ ) ( 1 i δ φ ( t , τ ) ) ] + [ a ¯ 1 ( t + τ ) ( δ a ^ 1 ( t ) + i a ^ 2 ( t ) ) + a ¯ 1 ( t ) ( δ a ^ 1 ( t + τ ) i a ^ 2 ( t + τ ) ) ] exp ( i ω 0 τ ) + [ a ¯ 1 ( t + τ ) ( δ a ^ 1 ( t ) i a ^ 2 ( t ) ) + a ¯ 1 ( t ) ( δ a ^ 1 ( t + τ ) + i a ^ 2 ( t + τ ) ) ] exp ( i ω 0 τ ) } + k 2 2 ( ξ + η ) { cos 2 θ 1 [ a ¯ 1 ( t + τ ) ( b ^ 1 ( t ) + i b ^ 2 ( t ) ) exp ( i ω 0 τ ) exp ( i φ ( t ) ) + a ¯ 1 ( t + τ ) ( b ^ 1 ( t ) i b ^ 2 ( t ) ) exp ( i ω 0 τ ) exp ( i φ ( t ) ) ] sin 2 θ 1 [ a ¯ 1 ( t ) ( b ^ 1 ( t + τ ) + i b ^ 2 ( t + τ ) ) exp ( i ω 0 τ ) exp ( i φ ( t ) ) + a ¯ 1 ( t ) ( b ^ 1 ( t + τ ) i b ^ 2 ( t + τ ) ) exp ( i ω 0 τ ) exp ( i φ ( t ) ) ] } .
I ^ ( t ) = k 2 2 ( ξ + η ) [ cos θ 1 ( a ^ ( t ) d ^ ( t ) + a ^ ( t ) d ^ ( t ) ) + sin θ 1 ( b ^ ( t ) d ^ ( t ) + b ^ ( t ) d ^ ( t ) ) ] .
I ^ ( t ) cos θ 1 k 2 ( ξ + η ) a ¯ 1 ( t ) d ^ 1 ( t ) cos φ ( t ) d ^ 2 ( t ) sin φ ( t ) ,
I ^ ˜ ( f ) = T / 2 T / 2 I ^ ( t ) exp ( i 2 π f t ) d t .
| I ^ ˜ ( f ) | 2 / T cos 2 θ 1 k 2 2 ( ξ + η ) 2 E S ^ b 1 0 ,
I ^ ( t ) k 1 ( ξ + η ) [ a ¯ 1 ( t + τ ) a ¯ 1 ( t ) + a ¯ 1 ( t + τ ) δ a ^ 1 ( t ) + a ¯ 1 ( t ) δ a ^ 1 ( t + τ ) ] ± k 2 ( ξ + η ) [ cos 2 θ 1 a ¯ 1 ( t + τ ) ( b ^ 1 ( t ) cos φ ( t ) b ^ 2 ( t ) sin φ ( t ) ) sin 2 θ 1 a ¯ 1 ( t ) ( b ^ 1 ( t + τ ) cos φ ( t ) b ^ 2 ( t + τ ) sin φ ( t ) ) ] .
| I ^ ˜ ( f ) | 2 / T k 1 2 ( ξ + η ) 2 | R ˜ ( f ) | 2 / T + 4 k 1 2 ( ξ + η ) 2 cos 2 ( π f τ ) E n = S a 1 ( n f r ) E S ^ δ a 1 ( f n f r ) δ f + k 2 2 ( ξ + η ) 2 E S ^ b 1 0 k φ s ( f , τ ) ,
R ˜ ( f ) n = a ¯ ˜ 1 ( n f r ) a ¯ ˜ 1 ( f n f r ) δ f .
| I ^ ˜ ( f ) | 2 / T k 1 2 ( ξ + η ) 2 | R ˜ ( f ) | 2 / T + 4 k 1 2 ( ξ + η ) 2 E n = S a 1 ( n f r ) E S ^ δ a 1 ( f n f r ) δ f .
I ^ ( t ) ± k 1 ( ξ + η ) [ a ¯ 1 ( t + τ ) a ¯ 1 ( t ) δ φ ( t , τ ) + a ¯ 1 ( t + τ ) a ^ 2 ( t ) a ¯ 1 ( t ) a ^ 2 ( t + τ ) ] k 2 ( ξ + η ) [ cos 2 θ 1 a ¯ 1 ( t + τ ) ( b ^ 1 ( t ) sin φ ( t ) + b ^ 2 ( t ) cos φ ( t ) ) + sin 2 θ 1 a ¯ 1 ( t ) ( b ^ 1 ( t + τ ) sin φ ( t ) + b ^ 2 ( t + τ ) cos φ ( t ) ) ] .
| I ^ ˜ ( f ) | 2 / T k 1 2 ( ξ + η ) 2 E n = | R ˜ ( n f r ) | 2 E T S δ φ τ ( f n f r ) δ f + 4 k 1 2 ( ξ + η ) 2 sin 2 ( π f τ ) E n = S a 1 ( n f r ) E S ^ a 2 ( f n f r ) δ f + k 2 2 ( ξ + η ) 2 E S ^ b 1 0 k φ c ( f , τ ) .
| I ^ ˜ ( f ) | 2 / T k 2 2 ( ξ + η ) 2 E S ^ b 1 0 ,
S δ φ τ ( f ) = 4 π 2 τ 2 ( sin ( π f τ ) / π f τ ) 2 S F ( f )
δ x 2 = δ q 2 cos 2 ϕ + δ p 2 sin 2 ϕ .
δ φ ( 1 / 2 ) S ^ δ φ τ / S ^ b 2 0 / ( n ¯ η ) 4 P ( f ) / P 0 ( f ) / ( 2 n ¯ η ) ,
a ¯ 1 ( t ) = n = a ¯ ˜ 1 ( f ) exp ( i 2 π n f r t ) δ f , b ^ 1 ( t ) = j = b ^ ˜ 1 ( f j ) exp ( i 2 π f j t ) δ f , b ^ 2 ( t ) = j = b ^ ˜ 2 ( f j ) exp ( i 2 π f j t ) δ f ,
cos φ ( t ) = j = Φ ˜ c ( f j ) exp ( i 2 π f j t ) δ f , sin φ ( t ) = j = Φ ˜ s ( f j ) exp ( i 2 π f j t ) δ f .
I ^ ˜ ( f ) = cos θ 1 k 2 ( ξ + η ) T / 2 T / 2 a ¯ 1 ( t ) [ b ^ 1 ( t ) cos φ ( t ) b ^ 2 ( t ) sin φ ( t ) ] exp ( i 2 π f t ) d t = cos θ 1 k 2 ( ξ + η ) n = j = a ¯ ˜ 1 ( n f r ) [ Φ ˜ c ( f j ) b ^ ˜ 1 ( f n f r f j ) Φ ˜ s ( f j ) b ^ ˜ 2 ( f n f r f j ) ] ( δ f ) 2 .
| I ^ ˜ ( f ) | 2 / T cos 2 θ 1 k 2 2 ( ξ + η ) 2 × n = j = S a 1 ( n f r ) ( | Φ ˜ c ( f j ) | 2 T S ^ b 1 ( f n f r f j ) + | Φ ˜ s ( f j ) | 2 T S ^ b 2 ( f n f r f j ) ) ( δ f ) 2 ,
S a 1 ( f ) = | a ¯ ˜ 1 ( f ) | 2 / T , S ^ b 1 ( f ) = | b ^ ˜ 1 ( f ) | 2 / T , S ^ b 2 ( f ) = | b ^ ˜ 2 ( f ) | 2 / T .
| I ^ ˜ ( f ) | 2 / T cos 2 θ 1 k 2 2 ( ξ + η ) 2 × n = S a 1 ( n f r ) δ f S ^ b 1 0 j = ( | Φ ˜ c ( f j ) | 2 + | Φ ˜ s ( f j ) | 2 ) δ f / T .
1 = T / 2 T / 2 cos 2 φ ( t ) + sin 2 φ ( t ) d t / T = j = ( | Φ ˜ c ( f j ) | 2 + | Φ ˜ s ( f j ) | 2 ) δ f / T .
| I ^ ˜ ( f ) | 2 / T cos 2 θ 1 k 2 2 ( ξ + η ) 2 E S ^ b 1 0 ,
E = n = S a 1 ( n f r ) δ f .
δ a ^ 1 ( t ) = j = δ a ^ ˜ 1 ( f j ) exp ( i 2 π f j t ) δ f .
T / 2 T / 2 a ¯ 1 ( t + τ ) a ¯ 1 ( t ) exp ( i 2 π f t ) d t = n = a ¯ ˜ 1 ( n f r ) a ¯ ˜ 1 ( f n f r ) δ f = R ˜ ( f ) ,
T / 2 T / 2 ( a ¯ 1 ( t + τ ) δ a ^ 1 ( t ) + a ¯ 1 ( t ) δ a ^ 1 ( t + τ ) ) exp ( i 2 π f t ) d t 2 exp ( i π f τ ) cos ( π f τ ) n = a ¯ ˜ 1 ( n f r ) δ a ^ ˜ 1 ( f n f r ) δ f ,
T / 2 T / 2 ( a ¯ 1 ( t + τ ) b ^ 1 ( t ) a ¯ 1 ( t ) b ^ 1 ( t + τ ) ) cos φ ( t ) exp ( i 2 π f t ) d t 2 i n = j = a ¯ ˜ 1 ( n f r ) Φ ˜ c ( f j ) b ^ ˜ 1 ( f n f r f j ) exp [ i π ( f f j ) τ ] sin [ π ( f f j ) τ ] ( δ f ) 2 ,
T / 2 T / 2 ( a ¯ 1 ( t + τ ) b ^ 2 ( t ) a ¯ 1 ( t ) b ^ 2 ( t + τ ) ) sin φ ( t ) exp ( i 2 π f t ) d t 2 i n = j = a ¯ ˜ 1 ( n f r ) Φ ˜ s ( f j ) b ^ ˜ 2 ( f n f r f j ) exp [ i π ( f f j ) τ ] sin [ π ( f f j ) τ ] ( δ f ) 2 ,
| I ^ ˜ ( f ) | 2 / T k 1 2 ( ξ + η ) 2 | R ˜ ( f ) | 2 / T + 4 k 1 2 ( ξ + η ) 2 cos 2 ( π f τ ) E n = S a 1 ( n f r ) E S ^ δ a 1 ( f n f r ) δ f + k 2 2 ( ξ + η ) 2 E S ^ b 1 0 k φ s ,
S ^ δ a 1 ( f ) = | δ a ^ ˜ 1 ( f ) | 2 / T ,
k φ s ( f , τ ) = j = ( | Φ ˜ c ( f j ) | 2 + | Φ ˜ s ( f j ) | 2 ) sin 2 [ π ( f f j ) τ ] δ f / T .
δ φ ( t , τ ) = j = δ φ ˜ τ ( f j ) exp ( i 2 π f j t ) δ f , a ^ 2 ( t ) = j = a ^ ˜ 2 ( f j ) exp ( i 2 π f j t ) δ f .
T / 2 T / 2 a ¯ 1 ( t + τ ) a ¯ 1 ( t ) δ φ ( t , τ ) exp ( i 2 π f t ) d t = n = m = a ¯ ˜ 1 ( n f r ) a ¯ ˜ 1 ( m f r ) δ φ ˜ τ ( f n f r m f r ) ( δ f ) 2 = n = R ˜ ( n f r ) δ φ ˜ τ ( f n f r ) δ f ,
T / 2 T / 2 ( a ¯ 1 ( t + τ ) a ^ 2 ( t ) a ¯ 1 ( t ) a ^ 2 ( t + τ ) ) exp ( i 2 π f t ) d t = 2 i exp ( i π f τ ) sin ( π f τ ) n = a ¯ ˜ 1 ( n f r ) δ a ^ ˜ 2 ( f n f r ) δ f ,
T / 2 T / 2 ( a ¯ 1 ( t + τ ) b ^ 1 ( t ) + a ¯ 1 ( t ) b ^ 1 ( t + τ ) ) sin φ ( t ) exp ( i 2 π f t ) d t 2 n = j = a ¯ ˜ 1 ( n f r ) Φ ˜ s ( f j ) b ^ ˜ 1 ( f n f r f j ) exp [ i π ( f f j ) τ ] cos [ π ( f f j ) τ ] ( δ f ) 2 ,
T / 2 T / 2 ( a ¯ 1 ( t + τ ) b ^ 2 ( t ) + a ¯ 1 ( t ) b ^ 2 ( t + τ ) ) cos φ ( t ) exp ( i 2 π f t ) d t 2 n = j = a ¯ ˜ 1 ( n f r ) Φ ˜ c ( f j ) b ^ ˜ 2 ( f n f r f j ) exp [ i π ( f f j ) τ ] cos [ π ( f f j ) τ ] ( δ f ) 2 ,
| I ^ ˜ ( f ) | 2 / T k 1 2 ( ξ + η ) 2 E n = | R ˜ ( n f r ) | 2 E T S δ φ τ ( f n f r ) δ f + 4 k 1 2 ( ξ + η ) 2 sin 2 ( π f τ ) E n = S a 1 ( n f r ) E S ^ a 2 ( f n f r ) δ f + k 2 2 ( ξ + η ) 2 E S ^ b 1 0 k φ c ,
S δ φ τ ( f ) = | δ φ ˜ τ ( f ) | 2 / T ,
S ^ a 2 ( f ) = | a ^ ˜ 2 ( f ) | 2 / T ,
k φ c ( f , τ ) = j = ( | Φ ˜ s ( f j ) | 2 + | Φ ˜ c ( f j ) | 2 ) cos 2 [ π ( f f j ) τ ] δ f / T .
f r ( t ) = j = F ˜ ( f j ) exp ( i 2 π f j t ) δ f ,
i j = δ φ ˜ τ ( f j ) f j exp ( i 2 π f j t ) δ f = j = F ˜ ( f j ) ( exp ( i 2 π f j τ ) 1 ) exp ( i 2 π f j t ) δ f .
δ φ ˜ τ ( f ) = 2 π τ exp ( i π f τ ) sin ( π f τ ) π f τ F ˜ ( f ) .
S δ φ τ ( f ) = 4 π 2 τ 2 ( sin ( π f τ ) / π f τ ) 2 S F ( f ) .
P ( f ) P 0 ( f ) = S ^ δ a 1 ( f ) S ^ b 1 0
P ( f ) P 0 ( f ) = cos 2 ( π f τ ) S ^ δ a 1 ( f ) S ^ b 1 0
P ( f ) P 0 ( f ) = 1
P ( f ) P 0 ( f ) = 1 4 S ^ δ φ τ ( f ) S ^ b 1 0 + sin 2 ( π f τ ) S ^ a 2 ( f ) S ^ b 1 0
P ( f ) P 0 ( f ) = 1 2
S ^ δ a 1 ( f ) n = S a 1 ( n f r ) E S ^ δ a 1 ( f n f r ) δ f ,
S ^ a 2 ( f ) n = S a 1 ( n f r ) E S ^ a 2 ( f n f r ) δ f ,
S ^ δ φ τ ( f ) n = | R ˜ ( n f r ) | 2 E T S δ φ τ ( f n f r ) δ f ,
S δ φ τ ( f ) = 4 π 2 τ 2 sinc 2 ( π f τ ) S F ( f ) .

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