Abstract

We have developed a blood velocimeter based on the principle of self-mixing in a semiconductor laser diode through an optical fiber. The intensity of the light is modulated by feedback from moving scattering particles that contain the Doppler-shift frequency. Upon feedback the characteristics of the laser diode change. The threshold current decreases, and an instable region may become present above the new threshold. The amplitude of the Doppler signal turns out to be related to the difference in intensity between situations with and without feedback. This amplitude is highest just above feedback. The suppression of reflection from the glass-fiber facets is of paramount importance in the obtaining of a higher signal-to-noise ratio. Using an optical stabilization of the feedback, we optimized the performance of the laser-fiber system and the Doppler modulation depth and clarified its behavior with a suitable physical model. We also investigated the effect of the finite coherence length of the laser. We tested the efficiency of the self-mixing velocimeter in vivo with the optical glass fiber inserted in the artery with endoscopic catheters, both in upstream and in downstream blood flow conditions. For the latter we used a special side-reflecting device solution for the fiber facet to allow downstream measurements.

© 2002 Optical Society of America

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References

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  1. M. Slot, M. H. Koelink, F. G. Scholten, F. F. M. de Mul, A. L. Weijers, J. Greve, R. Graaf, J. G. Aarnoudse, “Blood flow velocity measurements based on the self-mixing effect in a fibre-coupled semiconductor laser,” Med. Biol. Eng. Comput. 30, 441–446 (1992).
    [CrossRef] [PubMed]
  2. M. H. Koelink, M. Slot, F. F. M. de Mul, J. Greve, R. Graaf, A. C. M. Dassel, J. G. Aarnoudse, “Laser Doppler velocimeter based on the self-mixing effect in a fiber-coupled semiconductor laser: theory,” Appl. Opt. 31, 3401–3408 (1992).
    [CrossRef] [PubMed]
  3. M. H. Koelink, F. F. M. de Mul, A. L. Weijers, J. Greve, R. Graaf, A. C. M. Dassel, J. G. Aarnoudse, “Fiber-coupled self-mixing diode-laser Doppler velocimeter: technical aspects and flow velocity profile disturbances in water and blood flows,” Appl. Opt. 33, 5628–5641 (1995).
    [CrossRef]
  4. M. J. Rudd, “A laser Doppler velocimeter employing the laser as a mixer-oscillator,” J. Phys. E 1, 723–726 (1968).
    [CrossRef]
  5. T. Tanaka, G. B. Benedek, “Measurements of the velocity of blood flow (in vivo) using a fiber optic catheter and optical mixing spectroscopy,” Appl. Opt. 14, 189–196 (1975).
    [CrossRef] [PubMed]
  6. D. Kilpatrick, J. V. Tyberg, W. W. Parmleg, “Blood velocity measurements by fiber optic laser Doppler anemometry,” IEEE Trans. Biomed. Eng. 29, 142–145 (1982).
    [CrossRef] [PubMed]
  7. W. M. Wang, K. T. V. Grattan, A. W. Palmer, W. J. O. Boyle, “Self-mixing interference inside a single-mode diode laser for optical sensing applications,” J. Lightwave Technol. 12, 1577–1587 (1994).
    [CrossRef]
  8. L. Scalise, W. Steenbergen, F. F. M. de Mul, “Self-mixing feedback in a laser diode for intra-arterial optical blood flowmetry,” Appl. Opt. 40, 4608–4615 (2001).
    [CrossRef]
  9. N. Servagent, F. Gouaux, T. Bosh, “Measurements of displacement using the self-mixing interference in a laser diode,” J. Opt. (Paris) 29, 163–173 (1998).
    [CrossRef]
  10. K. Petermann, Laser Diode Modulation and Noise (Kluwer Academic, Dordrecht, The Netherlands, 1988).
    [CrossRef]
  11. G. H. B. Thompson, D. F. Lovelace, S. E. H. Turley, “Kinks in the light/current characteristic and near-field shifts in (GaAl)As-heterostructure stripe lasers and their explanation by the effect of self-focusing on a built in optical waveguide,” Solid State Electron. Dev. 2, 12–30 (1978).
  12. G. Arnold, K. Petermann, “Self-pulsing phenomena in (GaAl)As double-heterostructure injection lasers,” Opt. Quantum Electron. 10, 311–322 (1978).
    [CrossRef]
  13. H. Temkin, N. A. Olsson, J. H. Abeles, R. A. Logan, M. B. Panish, “Reflection noise in index-guided InGaAsP lasers,” IEEE J. Quantum Electron. QE-22, 286–293 (1986).
    [CrossRef]
  14. C. H. Henry, R. F. Kazarinov, “Instability of semiconductor lasers due to optical feedback from distant reflectors,” IEEE J. Quantum Electron. QE-22, 294–301 (1986).
    [CrossRef]
  15. R. Lang, K. Kobayashi, “External optical feedback effects on semiconductor lasers,” IEEE J. Quantum Electron. QE-16, 347–355 (1980).
    [CrossRef]
  16. M. Fujiwara, K. Kubota, R. Lang, “Low-frequency intensity fluctuation in laser diodes with external optical feedback,” Appl. Phys. Lett. 38, 217–220 (1981).
    [CrossRef]
  17. J. C. Dainty, Laser Speckle and Related Phenomena (Springer-Verlag, Berlin, 1975).
    [CrossRef]

2001 (1)

1998 (1)

N. Servagent, F. Gouaux, T. Bosh, “Measurements of displacement using the self-mixing interference in a laser diode,” J. Opt. (Paris) 29, 163–173 (1998).
[CrossRef]

1995 (1)

1994 (1)

W. M. Wang, K. T. V. Grattan, A. W. Palmer, W. J. O. Boyle, “Self-mixing interference inside a single-mode diode laser for optical sensing applications,” J. Lightwave Technol. 12, 1577–1587 (1994).
[CrossRef]

1992 (2)

M. Slot, M. H. Koelink, F. G. Scholten, F. F. M. de Mul, A. L. Weijers, J. Greve, R. Graaf, J. G. Aarnoudse, “Blood flow velocity measurements based on the self-mixing effect in a fibre-coupled semiconductor laser,” Med. Biol. Eng. Comput. 30, 441–446 (1992).
[CrossRef] [PubMed]

M. H. Koelink, M. Slot, F. F. M. de Mul, J. Greve, R. Graaf, A. C. M. Dassel, J. G. Aarnoudse, “Laser Doppler velocimeter based on the self-mixing effect in a fiber-coupled semiconductor laser: theory,” Appl. Opt. 31, 3401–3408 (1992).
[CrossRef] [PubMed]

1986 (2)

H. Temkin, N. A. Olsson, J. H. Abeles, R. A. Logan, M. B. Panish, “Reflection noise in index-guided InGaAsP lasers,” IEEE J. Quantum Electron. QE-22, 286–293 (1986).
[CrossRef]

C. H. Henry, R. F. Kazarinov, “Instability of semiconductor lasers due to optical feedback from distant reflectors,” IEEE J. Quantum Electron. QE-22, 294–301 (1986).
[CrossRef]

1982 (1)

D. Kilpatrick, J. V. Tyberg, W. W. Parmleg, “Blood velocity measurements by fiber optic laser Doppler anemometry,” IEEE Trans. Biomed. Eng. 29, 142–145 (1982).
[CrossRef] [PubMed]

1981 (1)

M. Fujiwara, K. Kubota, R. Lang, “Low-frequency intensity fluctuation in laser diodes with external optical feedback,” Appl. Phys. Lett. 38, 217–220 (1981).
[CrossRef]

1980 (1)

R. Lang, K. Kobayashi, “External optical feedback effects on semiconductor lasers,” IEEE J. Quantum Electron. QE-16, 347–355 (1980).
[CrossRef]

1978 (2)

G. H. B. Thompson, D. F. Lovelace, S. E. H. Turley, “Kinks in the light/current characteristic and near-field shifts in (GaAl)As-heterostructure stripe lasers and their explanation by the effect of self-focusing on a built in optical waveguide,” Solid State Electron. Dev. 2, 12–30 (1978).

G. Arnold, K. Petermann, “Self-pulsing phenomena in (GaAl)As double-heterostructure injection lasers,” Opt. Quantum Electron. 10, 311–322 (1978).
[CrossRef]

1975 (1)

1968 (1)

M. J. Rudd, “A laser Doppler velocimeter employing the laser as a mixer-oscillator,” J. Phys. E 1, 723–726 (1968).
[CrossRef]

Aarnoudse, J. G.

Abeles, J. H.

H. Temkin, N. A. Olsson, J. H. Abeles, R. A. Logan, M. B. Panish, “Reflection noise in index-guided InGaAsP lasers,” IEEE J. Quantum Electron. QE-22, 286–293 (1986).
[CrossRef]

Arnold, G.

G. Arnold, K. Petermann, “Self-pulsing phenomena in (GaAl)As double-heterostructure injection lasers,” Opt. Quantum Electron. 10, 311–322 (1978).
[CrossRef]

Benedek, G. B.

Bosh, T.

N. Servagent, F. Gouaux, T. Bosh, “Measurements of displacement using the self-mixing interference in a laser diode,” J. Opt. (Paris) 29, 163–173 (1998).
[CrossRef]

Boyle, W. J. O.

W. M. Wang, K. T. V. Grattan, A. W. Palmer, W. J. O. Boyle, “Self-mixing interference inside a single-mode diode laser for optical sensing applications,” J. Lightwave Technol. 12, 1577–1587 (1994).
[CrossRef]

Dainty, J. C.

J. C. Dainty, Laser Speckle and Related Phenomena (Springer-Verlag, Berlin, 1975).
[CrossRef]

Dassel, A. C. M.

de Mul, F. F. M.

Fujiwara, M.

M. Fujiwara, K. Kubota, R. Lang, “Low-frequency intensity fluctuation in laser diodes with external optical feedback,” Appl. Phys. Lett. 38, 217–220 (1981).
[CrossRef]

Gouaux, F.

N. Servagent, F. Gouaux, T. Bosh, “Measurements of displacement using the self-mixing interference in a laser diode,” J. Opt. (Paris) 29, 163–173 (1998).
[CrossRef]

Graaf, R.

Grattan, K. T. V.

W. M. Wang, K. T. V. Grattan, A. W. Palmer, W. J. O. Boyle, “Self-mixing interference inside a single-mode diode laser for optical sensing applications,” J. Lightwave Technol. 12, 1577–1587 (1994).
[CrossRef]

Greve, J.

Henry, C. H.

C. H. Henry, R. F. Kazarinov, “Instability of semiconductor lasers due to optical feedback from distant reflectors,” IEEE J. Quantum Electron. QE-22, 294–301 (1986).
[CrossRef]

Kazarinov, R. F.

C. H. Henry, R. F. Kazarinov, “Instability of semiconductor lasers due to optical feedback from distant reflectors,” IEEE J. Quantum Electron. QE-22, 294–301 (1986).
[CrossRef]

Kilpatrick, D.

D. Kilpatrick, J. V. Tyberg, W. W. Parmleg, “Blood velocity measurements by fiber optic laser Doppler anemometry,” IEEE Trans. Biomed. Eng. 29, 142–145 (1982).
[CrossRef] [PubMed]

Kobayashi, K.

R. Lang, K. Kobayashi, “External optical feedback effects on semiconductor lasers,” IEEE J. Quantum Electron. QE-16, 347–355 (1980).
[CrossRef]

Koelink, M. H.

Kubota, K.

M. Fujiwara, K. Kubota, R. Lang, “Low-frequency intensity fluctuation in laser diodes with external optical feedback,” Appl. Phys. Lett. 38, 217–220 (1981).
[CrossRef]

Lang, R.

M. Fujiwara, K. Kubota, R. Lang, “Low-frequency intensity fluctuation in laser diodes with external optical feedback,” Appl. Phys. Lett. 38, 217–220 (1981).
[CrossRef]

R. Lang, K. Kobayashi, “External optical feedback effects on semiconductor lasers,” IEEE J. Quantum Electron. QE-16, 347–355 (1980).
[CrossRef]

Logan, R. A.

H. Temkin, N. A. Olsson, J. H. Abeles, R. A. Logan, M. B. Panish, “Reflection noise in index-guided InGaAsP lasers,” IEEE J. Quantum Electron. QE-22, 286–293 (1986).
[CrossRef]

Lovelace, D. F.

G. H. B. Thompson, D. F. Lovelace, S. E. H. Turley, “Kinks in the light/current characteristic and near-field shifts in (GaAl)As-heterostructure stripe lasers and their explanation by the effect of self-focusing on a built in optical waveguide,” Solid State Electron. Dev. 2, 12–30 (1978).

Olsson, N. A.

H. Temkin, N. A. Olsson, J. H. Abeles, R. A. Logan, M. B. Panish, “Reflection noise in index-guided InGaAsP lasers,” IEEE J. Quantum Electron. QE-22, 286–293 (1986).
[CrossRef]

Palmer, A. W.

W. M. Wang, K. T. V. Grattan, A. W. Palmer, W. J. O. Boyle, “Self-mixing interference inside a single-mode diode laser for optical sensing applications,” J. Lightwave Technol. 12, 1577–1587 (1994).
[CrossRef]

Panish, M. B.

H. Temkin, N. A. Olsson, J. H. Abeles, R. A. Logan, M. B. Panish, “Reflection noise in index-guided InGaAsP lasers,” IEEE J. Quantum Electron. QE-22, 286–293 (1986).
[CrossRef]

Parmleg, W. W.

D. Kilpatrick, J. V. Tyberg, W. W. Parmleg, “Blood velocity measurements by fiber optic laser Doppler anemometry,” IEEE Trans. Biomed. Eng. 29, 142–145 (1982).
[CrossRef] [PubMed]

Petermann, K.

G. Arnold, K. Petermann, “Self-pulsing phenomena in (GaAl)As double-heterostructure injection lasers,” Opt. Quantum Electron. 10, 311–322 (1978).
[CrossRef]

K. Petermann, Laser Diode Modulation and Noise (Kluwer Academic, Dordrecht, The Netherlands, 1988).
[CrossRef]

Rudd, M. J.

M. J. Rudd, “A laser Doppler velocimeter employing the laser as a mixer-oscillator,” J. Phys. E 1, 723–726 (1968).
[CrossRef]

Scalise, L.

Scholten, F. G.

M. Slot, M. H. Koelink, F. G. Scholten, F. F. M. de Mul, A. L. Weijers, J. Greve, R. Graaf, J. G. Aarnoudse, “Blood flow velocity measurements based on the self-mixing effect in a fibre-coupled semiconductor laser,” Med. Biol. Eng. Comput. 30, 441–446 (1992).
[CrossRef] [PubMed]

Servagent, N.

N. Servagent, F. Gouaux, T. Bosh, “Measurements of displacement using the self-mixing interference in a laser diode,” J. Opt. (Paris) 29, 163–173 (1998).
[CrossRef]

Slot, M.

M. H. Koelink, M. Slot, F. F. M. de Mul, J. Greve, R. Graaf, A. C. M. Dassel, J. G. Aarnoudse, “Laser Doppler velocimeter based on the self-mixing effect in a fiber-coupled semiconductor laser: theory,” Appl. Opt. 31, 3401–3408 (1992).
[CrossRef] [PubMed]

M. Slot, M. H. Koelink, F. G. Scholten, F. F. M. de Mul, A. L. Weijers, J. Greve, R. Graaf, J. G. Aarnoudse, “Blood flow velocity measurements based on the self-mixing effect in a fibre-coupled semiconductor laser,” Med. Biol. Eng. Comput. 30, 441–446 (1992).
[CrossRef] [PubMed]

Steenbergen, W.

Tanaka, T.

Temkin, H.

H. Temkin, N. A. Olsson, J. H. Abeles, R. A. Logan, M. B. Panish, “Reflection noise in index-guided InGaAsP lasers,” IEEE J. Quantum Electron. QE-22, 286–293 (1986).
[CrossRef]

Thompson, G. H. B.

G. H. B. Thompson, D. F. Lovelace, S. E. H. Turley, “Kinks in the light/current characteristic and near-field shifts in (GaAl)As-heterostructure stripe lasers and their explanation by the effect of self-focusing on a built in optical waveguide,” Solid State Electron. Dev. 2, 12–30 (1978).

Turley, S. E. H.

G. H. B. Thompson, D. F. Lovelace, S. E. H. Turley, “Kinks in the light/current characteristic and near-field shifts in (GaAl)As-heterostructure stripe lasers and their explanation by the effect of self-focusing on a built in optical waveguide,” Solid State Electron. Dev. 2, 12–30 (1978).

Tyberg, J. V.

D. Kilpatrick, J. V. Tyberg, W. W. Parmleg, “Blood velocity measurements by fiber optic laser Doppler anemometry,” IEEE Trans. Biomed. Eng. 29, 142–145 (1982).
[CrossRef] [PubMed]

Wang, W. M.

W. M. Wang, K. T. V. Grattan, A. W. Palmer, W. J. O. Boyle, “Self-mixing interference inside a single-mode diode laser for optical sensing applications,” J. Lightwave Technol. 12, 1577–1587 (1994).
[CrossRef]

Weijers, A. L.

M. H. Koelink, F. F. M. de Mul, A. L. Weijers, J. Greve, R. Graaf, A. C. M. Dassel, J. G. Aarnoudse, “Fiber-coupled self-mixing diode-laser Doppler velocimeter: technical aspects and flow velocity profile disturbances in water and blood flows,” Appl. Opt. 33, 5628–5641 (1995).
[CrossRef]

M. Slot, M. H. Koelink, F. G. Scholten, F. F. M. de Mul, A. L. Weijers, J. Greve, R. Graaf, J. G. Aarnoudse, “Blood flow velocity measurements based on the self-mixing effect in a fibre-coupled semiconductor laser,” Med. Biol. Eng. Comput. 30, 441–446 (1992).
[CrossRef] [PubMed]

Appl. Opt. (4)

Appl. Phys. Lett. (1)

M. Fujiwara, K. Kubota, R. Lang, “Low-frequency intensity fluctuation in laser diodes with external optical feedback,” Appl. Phys. Lett. 38, 217–220 (1981).
[CrossRef]

IEEE J. Quantum Electron. (3)

H. Temkin, N. A. Olsson, J. H. Abeles, R. A. Logan, M. B. Panish, “Reflection noise in index-guided InGaAsP lasers,” IEEE J. Quantum Electron. QE-22, 286–293 (1986).
[CrossRef]

C. H. Henry, R. F. Kazarinov, “Instability of semiconductor lasers due to optical feedback from distant reflectors,” IEEE J. Quantum Electron. QE-22, 294–301 (1986).
[CrossRef]

R. Lang, K. Kobayashi, “External optical feedback effects on semiconductor lasers,” IEEE J. Quantum Electron. QE-16, 347–355 (1980).
[CrossRef]

IEEE Trans. Biomed. Eng. (1)

D. Kilpatrick, J. V. Tyberg, W. W. Parmleg, “Blood velocity measurements by fiber optic laser Doppler anemometry,” IEEE Trans. Biomed. Eng. 29, 142–145 (1982).
[CrossRef] [PubMed]

J. Lightwave Technol. (1)

W. M. Wang, K. T. V. Grattan, A. W. Palmer, W. J. O. Boyle, “Self-mixing interference inside a single-mode diode laser for optical sensing applications,” J. Lightwave Technol. 12, 1577–1587 (1994).
[CrossRef]

J. Opt. (Paris) (1)

N. Servagent, F. Gouaux, T. Bosh, “Measurements of displacement using the self-mixing interference in a laser diode,” J. Opt. (Paris) 29, 163–173 (1998).
[CrossRef]

J. Phys. E (1)

M. J. Rudd, “A laser Doppler velocimeter employing the laser as a mixer-oscillator,” J. Phys. E 1, 723–726 (1968).
[CrossRef]

Med. Biol. Eng. Comput. (1)

M. Slot, M. H. Koelink, F. G. Scholten, F. F. M. de Mul, A. L. Weijers, J. Greve, R. Graaf, J. G. Aarnoudse, “Blood flow velocity measurements based on the self-mixing effect in a fibre-coupled semiconductor laser,” Med. Biol. Eng. Comput. 30, 441–446 (1992).
[CrossRef] [PubMed]

Opt. Quantum Electron. (1)

G. Arnold, K. Petermann, “Self-pulsing phenomena in (GaAl)As double-heterostructure injection lasers,” Opt. Quantum Electron. 10, 311–322 (1978).
[CrossRef]

Solid State Electron. Dev. (1)

G. H. B. Thompson, D. F. Lovelace, S. E. H. Turley, “Kinks in the light/current characteristic and near-field shifts in (GaAl)As-heterostructure stripe lasers and their explanation by the effect of self-focusing on a built in optical waveguide,” Solid State Electron. Dev. 2, 12–30 (1978).

Other (2)

K. Petermann, Laser Diode Modulation and Noise (Kluwer Academic, Dordrecht, The Netherlands, 1988).
[CrossRef]

J. C. Dainty, Laser Speckle and Related Phenomena (Springer-Verlag, Berlin, 1975).
[CrossRef]

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Figures (17)

Fig. 1
Fig. 1

Self-mixing velocimeter with glass fiber, F. The moving object, MO, can be a rotating wheel covered with paper or scattering particles in a flow: DL, diode laser including photodiode; L1, L2, lenses. Mirrors: M1, M2, back and front facets of the laser crystal; M3, M4, fiber facets; M5, at the moving object. The lens surfaces have no mirror function because they are not in focus.

Fig. 2
Fig. 2

Effective-mirror approach to the fiber-coupled self-mixing setup: r, z2, amplitude reflection coefficients; L, lengths.

Fig. 3
Fig. 3

Light intensity versus current for moderate feedback for a distant reflector as measured with the setup used for the present experiments (see Section 3): Jth, threshold current at zero feedback; ΔJth = JC - Jth, ΔJth′ = JC′ - Jth, shifts in threshold current due to feedback; A, b, C, regions where instability time is short, intermediate, and long, respectively. In transition region b undulations may occur (kink).

Fig. 4
Fig. 4

Schematic drawing of the test setup. (a) The fiber, to be positioned between lens L2 and the wheel, W, is not shown. Side wings 1 and 2 are used to measure intensities. (b) Reduction of disturbing feedback from the first fiber facet into the laser cavity. D, diode laser in package with photodiode.

Fig. 5
Fig. 5

Intensity versus current relation for this setup with fiber and wheel without glass plate. The effect of feedback shifting the curve toward lower current values is clearly seen.

Fig. 6
Fig. 6

Doppler amplitude for various values of the driving current (see Fig. 5). Immediately above the new threshold the amplitude is maximum.

Fig. 7
Fig. 7

Comparison of measured and calculated Doppler amplitudes [see Fig. 5 and Eqs. (15)–(17)].

Fig. 8
Fig. 8

Comparison of measured and calculated modulation coefficient [see Fig. 5 and Eqs. (18) and (19)].

Fig. 9
Fig. 9

Intensity versus current relation for the setup with fiber and wheel and the glass plate.

Fig. 10
Fig. 10

Amplitude for various currents below (a) and above (b, c, d) the threshold.

Fig. 11
Fig. 11

Measured and calculated amplitude versus injection current for the setup in Fig. 9. Multiplication factor, 34.

Fig. 12
Fig. 12

Measured and calculated modulation depth for the setup in Fig. 9.

Fig. 13
Fig. 13

Idc2/〈Iac2〉 as a function of variable distance in the parallel beam.

Fig. 14
Fig. 14

Relationship between measured cutoff frequencies in the upstream and the downstream situation. The frequencies measured upstream correspond with actual velocities according to Eq. (2).

Fig. 15
Fig. 15

Mirror fiber. Oblique cleaving (35° ± 5°) of the fiber facet (diameters: core, 50 µm; cladding, 150 µm). Attachment layer: chrome, 2 nm. Reflecting layer: silver, 200 nm (reflectivity, ≈88%). Afterward the facet was covered with a silicon protecting layer (≈200 nm, not shown). beams A and b are at ≈95° and ≈20°, respectively, with the axis. beam A is by far the most intense. Overall fiber transmission (for beam A), ≈50%.

Fig. 16
Fig. 16

Mirror fiber in the catheter, downstream in the milk flow: (a) Doppler spectrum; (b) measured and calculated velocity; (c) Normal fiber, upstream; dots, flow profile in milk stream; curve, laminar flow profile.

Fig. 17
Fig. 17

Downstream measurements in the arteria pulmonalis of a healthy calf. Several curves (1–4) show measurements taken one after another. Cutoff frequency, 110 ± 10 kHz.

Equations (24)

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

Δνd=νs-νi=12πΔk·u=2νiucsin½φcos β,
Δνd=12π2kiu=2uμλ0cos θ,
Iz0, t  EF2+EB2+2EFEB cos2πΔνt+4πLext,0/λ+φB-φS,
gc-gth=-L-1j κj cos2πντj, j=f1, f2, ext,
z2k=r2+ηf11-r22rf1 exp-2ikμairLE1+ηf21-|r2|21-|rf1|2rf2×exp-2ikμairLE1+μcoreLF+ηext1-|r2|21-|rf1|21-|rf2|2rext×exp-2ikμairLE1+μcoreLF+μairLE2,
r1r21+κf1 exp-2ikμairLE1+κf2 exp-2ikμairLE1+μcoreLF+κext exp-2ikμairLE1+μcoreLF+μairLE2expg-αsLD-2iμldkLD=1,
Δφ=2πτLν-νth+1+α21/2j κj sin2πντj+arctan α=m2π; m1, 2,
μLnth+Δn=μth+χΔn=μth+dμ/dnthΔn, gnth+Δn=gth+ρΔn=gth+dg/dnthΔn,
2µLkth=2πM, gth=αL-LD-1 lnrL1rL2
Δk=-κext2µthLD1+α21/2 sin2θext+2ΔkLext,0-arctan α,
Δg=ρΔn=-κext cos2θext+2ΔkLext,0LD-1; θext=Lext,0+utk
JeV=nthτe+νggthQth, JeV=nτe+νggQ with JtheV=nthτe,
ΔQ=Q-Qth=-Δnνggthτe-1+νgρQ.
I=-VhννgQLDlnrL1rL2
ΔI=κextVhντeρLD-IlnrL1rL2cos2θextt+2ΔkLext,0.
κext=δβln1rL1rL2ΔJc1Jc1; δ=g/gn/nc1.
ΔI=δβΔJc1Jc1hνntheVδeτe+Icos2θextt+2ΔkLext,0=δβΔJc1Jc1hνJc1δe+Icos2θextt+2ΔkLext,0.
I=I01+m cos2θextt+2ΔkLext,0 with m=δβI0ΔJc1Jc1hνJc1δe+I
I0-Ic1=ηJ-Jc1,
m=δβΔJc1Jc1hνJc1δe+IIc1+ηJ-Jc1-1.
m=δβΔJc1Jc1hνJc1δe+IIc1+ηJ-Jc1-1×exp-δν2τextCspec.
ΔJJ=JI ΔIJ.
Iac2Idc2  11+Δf/Δf121/2.
Iac2Idc2  11+2L/lcoh21/2.

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