Abstract

Owing to the limited spectral response of the fiber directional coupler used in a balanced optical coherence tomography configuration, the spectra are different in the two outputs. This affects unfavorably operation of the balanced photodetector unit. Excess photon noise makes a larger contribution than a directional coupler with a flat spectral response. A theoretical model is developed that shows that an optimum set of parameters may be defined to maximize the achievable signal-to-noise ratio. The model leads to a redefinition of the effective noise bandwidth, which takes into account the nonflat response of the directional coupler used. The model also predicts a limitation on the signal-to-noise ratio even when the stray reflectances in the interferometer are brought to zero.

© 2004 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
  3. P. R. Morkel, R. I. Laming, D. N. Payne, “Noise characteristics of high-power doped-fiber superluminescent sources,” Electron. Lett. 26, 96–97 (1990).
    [CrossRef]
  4. V. Sorin, D. M. Baney, “A simple intensity noise reduction technique for optical low-coherence reflectometry,” IEEE Photonics Technol. Lett. 4, 1404–1406 (1992).
    [CrossRef]
  5. J. A. Izatt, M. D. Kulkarni, H.-W. Wang, K. Kobayashi, M. V. Sivak, “Optical coherence tomography and microscopy in gastrointestinal tissues,” IEEE J. Sel. Top. Quantum Electron. 2, 1017–1028 (1996).
    [CrossRef]
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    [CrossRef] [PubMed]
  7. J. A. Izaat, M. R. Hee, G. M. Owen, E. A. Swanson, J. G. Fujimoto, “Optical coherence microscopy in scattering media,” Opt. Lett. 19, 590–592 (1994).
    [CrossRef]
  8. W. Drexler, U. Morgner, R. K. Ghanta, F. X. Kartner, J. S. Schuman, J. G. Fujimoto, “Ultrahigh-resolution ophthalmic optical coherence tomography,” Nature Med. 7, 502–507 (2001).
    [CrossRef] [PubMed]
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    [CrossRef]
  16. J. A. Rogers, A. Podoleanu, G. Dobre, D. A. Jackson, F. W. Fitzke, “Topography and volume measurements of the optic nerve using en-face optical coherence tomography,” Opt. Express 9, 533–545 (2001), http://www.opticsexpress.org .
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2004 (2)

2002 (2)

G. Genty, M. Lehtonen, H. Ludvigsen, J. Broeng, M. Kaivola, “Spectral broadening of femtosecond pulses into continuum radiation in microstructured fibers,” Opt. Express 10, 1083–1098 (2002), http://www.opticsexpress.org .
[CrossRef]

A. F. Fercher, C. K. Hitzenberger, M. Sticker, R. Zawadzki, B. Karamata, T. Lasser, “Dispersion compensation for optical coherence tomography depth-scan signals by a numerical technique,” Opt. Commun. 204, 67–74 (2002).
[CrossRef]

2001 (2)

W. Drexler, U. Morgner, R. K. Ghanta, F. X. Kartner, J. S. Schuman, J. G. Fujimoto, “Ultrahigh-resolution ophthalmic optical coherence tomography,” Nature Med. 7, 502–507 (2001).
[CrossRef] [PubMed]

J. A. Rogers, A. Podoleanu, G. Dobre, D. A. Jackson, F. W. Fitzke, “Topography and volume measurements of the optic nerve using en-face optical coherence tomography,” Opt. Express 9, 533–545 (2001), http://www.opticsexpress.org .
[CrossRef]

2000 (1)

1999 (1)

1998 (1)

K. Takada, “Noise in optical low-coherence reflectometry,” IEEE J. Quantum Electron. 34, 1098–1108 (1998).
[CrossRef]

1996 (1)

J. A. Izatt, M. D. Kulkarni, H.-W. Wang, K. Kobayashi, M. V. Sivak, “Optical coherence tomography and microscopy in gastrointestinal tissues,” IEEE J. Sel. Top. Quantum Electron. 2, 1017–1028 (1996).
[CrossRef]

1994 (1)

1992 (2)

E. A. Swanson, D. Huang, M. R. Lee, J. G. Fujimoto, C. P. Lin, C. A. Puliafito, “High-speed optical coherence domain reflectometry,” Opt. Lett. 17, 151–153 (1992).
[CrossRef] [PubMed]

V. Sorin, D. M. Baney, “A simple intensity noise reduction technique for optical low-coherence reflectometry,” IEEE Photonics Technol. Lett. 4, 1404–1406 (1992).
[CrossRef]

1991 (1)

K. Takada, A. Himeno, K. Yukimatsu, “Phase-noise and shot-noise operations of low coherence optical time domain reflectometry,” Appl. Phys. Lett. 59, 2483–2485 (1991).
[CrossRef]

1990 (1)

P. R. Morkel, R. I. Laming, D. N. Payne, “Noise characteristics of high-power doped-fiber superluminescent sources,” Electron. Lett. 26, 96–97 (1990).
[CrossRef]

1980 (1)

Baney, D. M.

V. Sorin, D. M. Baney, “A simple intensity noise reduction technique for optical low-coherence reflectometry,” IEEE Photonics Technol. Lett. 4, 1404–1406 (1992).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1989), pp. 503–505.

Broeng, J.

Cucu, R. G.

Dobre, G.

Dobre, G. M.

Dogariu, A.

Drexler, W.

W. Drexler, U. Morgner, R. K. Ghanta, F. X. Kartner, J. S. Schuman, J. G. Fujimoto, “Ultrahigh-resolution ophthalmic optical coherence tomography,” Nature Med. 7, 502–507 (2001).
[CrossRef] [PubMed]

Fercher, A. F.

A. F. Fercher, C. K. Hitzenberger, M. Sticker, R. Zawadzki, B. Karamata, T. Lasser, “Dispersion compensation for optical coherence tomography depth-scan signals by a numerical technique,” Opt. Commun. 204, 67–74 (2002).
[CrossRef]

Fitzke, F. W.

Fujimoto, J. G.

Genty, G.

Ghanta, R. K.

W. Drexler, U. Morgner, R. K. Ghanta, F. X. Kartner, J. S. Schuman, J. G. Fujimoto, “Ultrahigh-resolution ophthalmic optical coherence tomography,” Nature Med. 7, 502–507 (2001).
[CrossRef] [PubMed]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), pp. 40–44.

Hee, M. R.

J. A. Izaat, M. R. Hee, G. M. Owen, E. A. Swanson, J. G. Fujimoto, “Optical coherence microscopy in scattering media,” Opt. Lett. 19, 590–592 (1994).
[CrossRef]

M. R. Hee, “Optical coherence tomography: theory,” in Handbook of Optical Coherence Tomography, B. Bouma, G. Tearney, eds. (Marcel Dekker, New York, 2001), pp. 41–66.
[CrossRef]

Himeno, A.

K. Takada, A. Himeno, K. Yukimatsu, “Phase-noise and shot-noise operations of low coherence optical time domain reflectometry,” Appl. Phys. Lett. 59, 2483–2485 (1991).
[CrossRef]

Hitzenberger, C. K.

A. F. Fercher, C. K. Hitzenberger, M. Sticker, R. Zawadzki, B. Karamata, T. Lasser, “Dispersion compensation for optical coherence tomography depth-scan signals by a numerical technique,” Opt. Commun. 204, 67–74 (2002).
[CrossRef]

Huang, D.

Izaat, J. A.

Izatt, J. A.

J. A. Izatt, M. D. Kulkarni, H.-W. Wang, K. Kobayashi, M. V. Sivak, “Optical coherence tomography and microscopy in gastrointestinal tissues,” IEEE J. Sel. Top. Quantum Electron. 2, 1017–1028 (1996).
[CrossRef]

M. D. Kulkarni, S. Yazdanfar, J. A. Izatt, “Coherent signal analysis in optical coherence tomography,” in Coherence Domain Optical Methods in Biomedical Science and Clinical Applications II, V. V. Tuchin, J. A. Izatt, eds., Proc. SPIE3251, 22–26 (1998).
[CrossRef]

Jackson, D. A.

Kaivola, M.

Karamata, B.

A. F. Fercher, C. K. Hitzenberger, M. Sticker, R. Zawadzki, B. Karamata, T. Lasser, “Dispersion compensation for optical coherence tomography depth-scan signals by a numerical technique,” Opt. Commun. 204, 67–74 (2002).
[CrossRef]

Kartner, F. X.

W. Drexler, U. Morgner, R. K. Ghanta, F. X. Kartner, J. S. Schuman, J. G. Fujimoto, “Ultrahigh-resolution ophthalmic optical coherence tomography,” Nature Med. 7, 502–507 (2001).
[CrossRef] [PubMed]

Kobayashi, K.

J. A. Izatt, M. D. Kulkarni, H.-W. Wang, K. Kobayashi, M. V. Sivak, “Optical coherence tomography and microscopy in gastrointestinal tissues,” IEEE J. Sel. Top. Quantum Electron. 2, 1017–1028 (1996).
[CrossRef]

Kulkarni, M. D.

J. A. Izatt, M. D. Kulkarni, H.-W. Wang, K. Kobayashi, M. V. Sivak, “Optical coherence tomography and microscopy in gastrointestinal tissues,” IEEE J. Sel. Top. Quantum Electron. 2, 1017–1028 (1996).
[CrossRef]

M. D. Kulkarni, S. Yazdanfar, J. A. Izatt, “Coherent signal analysis in optical coherence tomography,” in Coherence Domain Optical Methods in Biomedical Science and Clinical Applications II, V. V. Tuchin, J. A. Izatt, eds., Proc. SPIE3251, 22–26 (1998).
[CrossRef]

Laming, R. I.

P. R. Morkel, R. I. Laming, D. N. Payne, “Noise characteristics of high-power doped-fiber superluminescent sources,” Electron. Lett. 26, 96–97 (1990).
[CrossRef]

Lasser, T.

A. F. Fercher, C. K. Hitzenberger, M. Sticker, R. Zawadzki, B. Karamata, T. Lasser, “Dispersion compensation for optical coherence tomography depth-scan signals by a numerical technique,” Opt. Commun. 204, 67–74 (2002).
[CrossRef]

Lee, M. R.

Lehtonen, M.

Lin, C. P.

Ludvigsen, H.

Morgner, U.

W. Drexler, U. Morgner, R. K. Ghanta, F. X. Kartner, J. S. Schuman, J. G. Fujimoto, “Ultrahigh-resolution ophthalmic optical coherence tomography,” Nature Med. 7, 502–507 (2001).
[CrossRef] [PubMed]

Morkel, P. R.

P. R. Morkel, R. I. Laming, D. N. Payne, “Noise characteristics of high-power doped-fiber superluminescent sources,” Electron. Lett. 26, 96–97 (1990).
[CrossRef]

Mujat, C.

Owen, G. M.

Payne, D. N.

P. R. Morkel, R. I. Laming, D. N. Payne, “Noise characteristics of high-power doped-fiber superluminescent sources,” Electron. Lett. 26, 96–97 (1990).
[CrossRef]

Podoleanu, A.

Podoleanu, A. G.

Puliafito, C. A.

Rashleigh, S. C.

Rogers, J. A.

Rosen, R. B.

Schuman, J. S.

W. Drexler, U. Morgner, R. K. Ghanta, F. X. Kartner, J. S. Schuman, J. G. Fujimoto, “Ultrahigh-resolution ophthalmic optical coherence tomography,” Nature Med. 7, 502–507 (2001).
[CrossRef] [PubMed]

Sivak, M. V.

J. A. Izatt, M. D. Kulkarni, H.-W. Wang, K. Kobayashi, M. V. Sivak, “Optical coherence tomography and microscopy in gastrointestinal tissues,” IEEE J. Sel. Top. Quantum Electron. 2, 1017–1028 (1996).
[CrossRef]

Sorin, V.

V. Sorin, D. M. Baney, “A simple intensity noise reduction technique for optical low-coherence reflectometry,” IEEE Photonics Technol. Lett. 4, 1404–1406 (1992).
[CrossRef]

Sticker, M.

A. F. Fercher, C. K. Hitzenberger, M. Sticker, R. Zawadzki, B. Karamata, T. Lasser, “Dispersion compensation for optical coherence tomography depth-scan signals by a numerical technique,” Opt. Commun. 204, 67–74 (2002).
[CrossRef]

Swanson, E. A.

Takada, K.

K. Takada, “Noise in optical low-coherence reflectometry,” IEEE J. Quantum Electron. 34, 1098–1108 (1998).
[CrossRef]

K. Takada, A. Himeno, K. Yukimatsu, “Phase-noise and shot-noise operations of low coherence optical time domain reflectometry,” Appl. Phys. Lett. 59, 2483–2485 (1991).
[CrossRef]

Ulrich, R.

Wang, H.-W.

J. A. Izatt, M. D. Kulkarni, H.-W. Wang, K. Kobayashi, M. V. Sivak, “Optical coherence tomography and microscopy in gastrointestinal tissues,” IEEE J. Sel. Top. Quantum Electron. 2, 1017–1028 (1996).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1989), pp. 503–505.

Yariv, A.

A. Yariv, Optical Electronics, 4th ed. (Saunders, Philadelphia, Pa., 1991), pp. 519–529.

Yazdanfar, S.

M. D. Kulkarni, S. Yazdanfar, J. A. Izatt, “Coherent signal analysis in optical coherence tomography,” in Coherence Domain Optical Methods in Biomedical Science and Clinical Applications II, V. V. Tuchin, J. A. Izatt, eds., Proc. SPIE3251, 22–26 (1998).
[CrossRef]

Yukimatsu, K.

K. Takada, A. Himeno, K. Yukimatsu, “Phase-noise and shot-noise operations of low coherence optical time domain reflectometry,” Appl. Phys. Lett. 59, 2483–2485 (1991).
[CrossRef]

Zawadzki, R.

A. F. Fercher, C. K. Hitzenberger, M. Sticker, R. Zawadzki, B. Karamata, T. Lasser, “Dispersion compensation for optical coherence tomography depth-scan signals by a numerical technique,” Opt. Commun. 204, 67–74 (2002).
[CrossRef]

Appl. Opt. (3)

Appl. Phys. Lett. (1)

K. Takada, A. Himeno, K. Yukimatsu, “Phase-noise and shot-noise operations of low coherence optical time domain reflectometry,” Appl. Phys. Lett. 59, 2483–2485 (1991).
[CrossRef]

Electron. Lett. (1)

P. R. Morkel, R. I. Laming, D. N. Payne, “Noise characteristics of high-power doped-fiber superluminescent sources,” Electron. Lett. 26, 96–97 (1990).
[CrossRef]

IEEE J. Quantum Electron. (1)

K. Takada, “Noise in optical low-coherence reflectometry,” IEEE J. Quantum Electron. 34, 1098–1108 (1998).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

J. A. Izatt, M. D. Kulkarni, H.-W. Wang, K. Kobayashi, M. V. Sivak, “Optical coherence tomography and microscopy in gastrointestinal tissues,” IEEE J. Sel. Top. Quantum Electron. 2, 1017–1028 (1996).
[CrossRef]

IEEE Photonics Technol. Lett. (1)

V. Sorin, D. M. Baney, “A simple intensity noise reduction technique for optical low-coherence reflectometry,” IEEE Photonics Technol. Lett. 4, 1404–1406 (1992).
[CrossRef]

Nature Med. (1)

W. Drexler, U. Morgner, R. K. Ghanta, F. X. Kartner, J. S. Schuman, J. G. Fujimoto, “Ultrahigh-resolution ophthalmic optical coherence tomography,” Nature Med. 7, 502–507 (2001).
[CrossRef] [PubMed]

Opt. Commun. (1)

A. F. Fercher, C. K. Hitzenberger, M. Sticker, R. Zawadzki, B. Karamata, T. Lasser, “Dispersion compensation for optical coherence tomography depth-scan signals by a numerical technique,” Opt. Commun. 204, 67–74 (2002).
[CrossRef]

Opt. Express (3)

Opt. Lett. (3)

Other (5)

A. Yariv, Optical Electronics, 4th ed. (Saunders, Philadelphia, Pa., 1991), pp. 519–529.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1989), pp. 503–505.

M. R. Hee, “Optical coherence tomography: theory,” in Handbook of Optical Coherence Tomography, B. Bouma, G. Tearney, eds. (Marcel Dekker, New York, 2001), pp. 41–66.
[CrossRef]

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), pp. 40–44.

M. D. Kulkarni, S. Yazdanfar, J. A. Izatt, “Coherent signal analysis in optical coherence tomography,” in Coherence Domain Optical Methods in Biomedical Science and Clinical Applications II, V. V. Tuchin, J. A. Izatt, eds., Proc. SPIE3251, 22–26 (1998).
[CrossRef]

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Figures (12)

Fig. 1
Fig. 1

Balanced OCT configuration.

Fig. 2
Fig. 2

Directional coupler with a nonconstant cross-coupling coefficient.

Fig. 3
Fig. 3

Experimental characterization of the couplers: N1, N2, standard narrowband 50/50 couplers centered at 820 and 850 nm, respectively, and bandwidth ±10 nm; B1, B2, broadband couplers operating above 800 nm. In the top, left graph the spectrum of the laser, L, is shown. The spectrum of both outputs, o1 and o2, is shown for each coupler as well as the ratio o1/o2 (right axis), denoted N1, N2, B1, and B2.

Fig. 4
Fig. 4

Factor ξ [Eq. (14)] for couplers N1, N2, B1, and B2 as a function of the optical source central wavelength for a source of Δλ = 30-nm bandwidth and a Gaussian profile.

Fig. 5
Fig. 5

Maximum SNR versus the attenuation factor σ in the reference arm. The source spectrum is assumed to be Gaussian with mean wavelengths of 800 and 40 nm FWHM. Fiber-end reflectivity is set at R = 4% or γ1 = 0.5, and the power to the object is PP th [Eq. (29b)].

Fig. 6
Fig. 6

Optimum reference launching coefficient for two values of the fiber-end reflectivity as a function of γ1.

Fig. 7
Fig. 7

Maximum achievable SNR with varying total fiber-end reflectivity (σ = 0.5).

Fig. 8
Fig. 8

Comparison of the excess photon noise [parameter A in (28e)] as a function of fiber-end reflectivity R for narrowband coupler N1 and wideband coupler B2 (σ = 0.5 and γ1 = 0.5).

Fig. 9
Fig. 9

Comparison of the curves for excess photon noise [parameter A′ in Eq. (28e)] versus reference attenuation σ in the reference path for narrowband coupler N1 and wideband coupler B2 (γ1 = 0.5 and R = 4%).

Fig. 10
Fig. 10

Comparison of the SNR for different types of couplers as a function of optical power.

Fig. 11
Fig. 11

Signal-to-noise ratio for different values of the cross-coupling coefficient of the first coupler γ1, considering B2 as the coupler in the balanced stage, R = 4%, R L = 10 kΩ, and σ = 1.

Fig. 12
Fig. 12

Signal-to-noise ratio versus power to the object for two different bandwidth couplers and different values of the Johnson noise with σ = 1, γ1 = 0.5, and R = 4%.

Tables (2)

Tables Icon

Table 2 Fitted Profiles for the Coupler Cross-Coupling Efficiency γ(ν) as Determined from the Experimental Data in Fig. 3

Equations (75)

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

Γτ=Et+τE*t=4 0+ Gνexp-i2πντdν.
Γ0=EtE*t=4 0+ Gνdν,
P  I=4 0+ Gνdν.
E+/OUT=12E1l-iE2x, E-/OUT=12E2l-iE1x.
it=2αPRefPobj1/2 cos2πf0t-φ0,
2 E+=E1l-iE2x, 2 E-=E2l-iE1x.
Pi=κ×4 0+ Giνdν.
Pix=κ×4 0+ γνGiνdν,Pil=κ×4 0+1-γνGiνdν.
G1ν=g1Gν, G2ν=g2Gν,
it=i+t-i-t,
i+t=αE1lt-iE2xtE1l*t+iE2x*t=α|E1lt|2+|E2xt|2-iE2xtE1l*t+c.c.,i-t=αE2lt-iE1xtE2l*t+iE1x*t=α|E2lt|2+|E1xt|2-iE1xtE2l*t+c.c..
idc=i+-i-dc=αP1l-P1x+P2x-P2l=αΔP1-ΔP2,
idc=4αg2-g10+1-2γνGνdν.
iact=i+-i-ac=+α-iE2xtE1l*t+c.c.+iE1xtE2l*t+c.c.,
iE2xtE1l*t+c.c.=2×4g2g10+1-γν×Gνdν0+ γνGνdν1/2 cos φt,iE1xtE2l*t+c.c.=2×4g1g20+1-γν×Gνdν0+ γνGνdν1/2 cos φt.
iact=16αg2g10+1-γνGνdν×0+ γνGνdν1/2 cos φt.
it=4αg1-g201-2γνGνdν+16αg1g21/20γνGνdν×01-γνGνdν1/2 cos φt.
i2=α28g1g21-ξξP2
ξ=0 γνGνdν0 Gνdν.
ΔiJohnson2=4kBTRL B.
ΔiSN2=2eBαP.
i+=αP1l+P2x,i-=αP1x+P2l.
ΔiSN2=2αeBg1+g2P0.
ΔiEPN2=it+τit=idct+τ+iact+τidct+iact=idct+τidct+iact+τiact+idct+τiact+iact+τidct.
ΔiEPN2=2α2BP1P2Δν0+P12+P22Δν1,
Δν0-1=8 0 γν1-γνG2νdν0 Gνdν2,Δν1-1=01-2γν2G2νdν0 Gνdν2
Δi2=4kBTRL B+2αeBP1+P2+2α2BP1P2Δν0+P12+P22Δν1.
Δi2=2αeBg1+g2P0+2α2g1g2P02Δν0+g12+g22P02Δν1+4kBTRL B.
Δi2=ΔiJohnson2+Δishot2+ΔiEPN2=4kBTRL+2αeP1+P2+2α2I1I2ΔνB,
Δν-1=2 0 G1νG2νdν0 G1νdν0 G2νdν=2 0 G2νdν0 Gνdν2.
SNideal=2αγ12OP024kBTRL1-γ1+2αγ12RΔν P02+2eαγ1P0B.
SN=GP2AP2+DP+CB,
G=2αγ12O,A=2αγ12RΔν0,D=2eαγ1,C=4kBTRL1-γ1σ.
BOSNMAX/ideal=ΔνR.
SNR=8α2g1g2P021-ξξ4kBTR B1-γ1+2αeBg1+g2_noiseP0+2α2Bg1g2_noiseΔν0+g12+g2_noise2Δν1P02,
g1=γ11-γ1 σ,g2=γ1R+O,g2_noise=γ1R.
BOSN=8αγ121-ξξP024kBTRL1-γ1σ+2αγ12RΔν0+σ1-γ1+R2σ1Δν1P02+2eαγ1P0.
SN=GP2AP2+DP+CB,
G=4G1-ξξ,A=A+2αγ12Δν1σ1-γ1+R2σ,D=D,C=C.
BOSNMAX=GA=41-ξξRΔν0+σ1-γ1+R2σ1Δν1.
APth2=DPth+C.
BOSNR0MAX41-ξξ1-γ1Δν1σ.
σopt=R1-γ11/2.
u1u2u3*u4*=u1u3*u2u4*+u1u4*u2u3*
iact+τiact=-α2×-E2xt+τE1l*t+τ+E1xt+τE2l*t+τ+c.c.×-E2xtE1l*t+E1xtE2l*t+c.c.,
iact+τiact=α2×-iE2xt+τE1l*t+τ+iE1xt+τE2l*t+τ×-iE2xtE1l*t+iE1xtE2l*t+c.c.+-iE2xt+τE1l*t+τ+iE1xt+τE2l*t+τ×+iE2x*tE1lt-iE1x*tE2lt+c.c.,
iact+τiact=α2×-E2xt+τE1l*t+τE2xtE1l*t+E2xt+τE1l*t+τE1xtE2l*t+E1xt+τE2l*t+τE2xtE1l*t-E1xt+τE2l*t+τE1xtE2l*t+α2{E2xt+τE1l*t+τE2x*tE1lt-E2xt+τE1l*t+τE1x*tE2lt-E1xt+τE2l*t+τE2x*tE1lt+E1xt+τE2l*t+τE1x*tE2lt+c.c.
iact+τiact=α20+E2xt+τE2l*tE1xtE1l*t+τ+E1xt+τE1l*tE2xtE2l*t+τ + 0+ E2xt + τE2x*tE1ltE1l*t + τ + 0+ 0 + E1xt + τE1x*tE2ltE2l*t + τ + c.c.
Emst+τEmt*tEnsttEnt*t+τ+c.c.=16 0 Gmstνexp-i2πντdν×0 Gnstνexp+i2πντdν.
Emst+τEmt*tEnstEnt*t+τ+c.c.=64 0 GmstνGnstνdν.
E2xt+τE2l*tE1xtE1l*t+τ =64g1g20 γν1-γνG2νdν,
E1xt+τE1l*tE2xtE2l*t+τ =64g1g20 γν1-γνG2νdν,
E1xt+τE1x*tE2ltE2l*t+τ=64g1g20 γν1-γνG2νdν,
E2xt+τE2x*tE1ltE1l*t+τ=64g1g20 γν1-γνG2νdν.
iact+τiact=4α2×64g1g20 γν1 -γνG2νdν.
idct+τidct =ΔP1t+τΔP1t+ΔP2t+τΔP2t-ΔP1t+τΔP2t-ΔP2t+τΔP1t.
ΔPit+τΔPit =Pilt+τ-Pixt+τPilt-Pixt =Pilt+τPilt+Pixt+τPixt-Pilt+τPixt-Pixt+τPilt.
Pilt+τPilt =Eilt+τEil*t+τEiltEil*t =Eilt+τEil*t+τEiltEil*t+Eilt+τEil*t+EiltEil*t+τ, Pilt+τPilt=Pil2+32 0 Gil2νdν.
Pilt+τPilt=Pil2+32 0 Gil2νdν, Pixt+τPixt=Pix2+32 0 Gix2νdν, Pilt+τPixt=PixPil+32 0 GilνGixνdν, Pixt+τPilt=PixPil+32 0 GilνGixνdν.
ΔP1t+τΔP1t+ΔP2t+τΔP2t=P1l-P1x2+32 0G1lν-G1xν2dν+P2l-P2x2+32 0G2lν-G2xν2dν
ΔP1t+τΔP1t+ΔP2t+τΔP2t=ΔP12+ΔP22+32g12+g2201-2γνGν2dν.
ΔPit+τΔPjt=Pilt+τ-Pixt+τPjlt-Pjxt=Pilt+τPjlt+Pixt+τPjxt-Pilt+τPjxt-Pixt+τPjlt.
Pist+τPjtt=Eist+τEis*t+τEjttEjt*t+Eist+τEjt*t+EjttEis*t+τ,Pist+τPjtt=PisPjt,
+Pilt+τPjlt=+PilPjl, +Pixt+τPjxt=+PixPjx, -Pilt+τPjxt=-PilPjx, -Pixt+τPjlt=-PixPjl,
ΔP1t+τΔP2t+ΔP2t+τΔP1t=2ΔP1ΔP2.
idct+τidct=α2B|ΔP1|-|ΔP2|2+32g12+g22 ×01-2γνGν2dν.
idct+τiact =iΔP1t+τ-iE2xtE1l*t+c.c. +iE1xtE2l*t+c.c.-iΔP2t+τ×-iE2xtE1l*t+c.c.+iE1xt×E2l*t+c.c..
ΔP1t+τ-E2xtE1l*t =-P1lE2xtE1l*t-E1lt+τE1l*t×E2xtE1l*t+τ+P1xE2xtE1l*t+E1xt+τE1l*tE2xtE1x*t+τ.
ΔiEPN2=it=τit-i2=α2×4×64g1g20 γν1-γνG2νdν+32g12+g2201-2γν2G2νdν.
ΔiEPN2=α24×64 0 γν1-γνG1νG2νdν+32 01-2γν2G12ν+G22νdν
ΔiEPN2=2BP1P2Δν0+P12+P22Δν1,
Δν0-1=4×64 0 γν1-γνG1νG2νdν2×420 G1νdν0 G2νdν=8 0 γν1-γνG2νdν0 Gνdν2,
Δν1-1=32 01-2γν2G12νdν2×4 0 G1νdν2=32 01-2γν2G22νdν2×4 0 G2νdν2=01-2γν2G2νdν0 Gνdν2.
γλ=0.354+0.062×exp-λ-775.72488.1+0.146×exp-λ-827.421579.6
γλ=0.21+0.15×exp-λ-77321352+0.245×exp-λ-83724463

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