Abstract

A method of measuring double resonant two-photon signal and background from a single cavity ring-down decay is introduced. This is achieved by modulating the double resonance loss via one of the light sources exciting the transition. The noise performance of the method is characterized theoretically and experimentally. The addition of a new parameter to the fitting function introduces a minor noise increase due to parameter correlation. However, the concurrent recording of the background can extend the stable measurement time. Alternatively, the method allows a faster measurement speed, while still recording the background, which is often advantageous in double resonance measurements. Finally, the method is insensitive to changes in the cavity decay rate at short timescales and can lead to improved performance if they have significant contribution to the final noise level compared to the detector noise.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. D. Romanini, A. A. Kachanov, N. Sadeghi, and F. Stoeckel, “CW cavity ring down spectroscopy,” Chem. Phys. Lett. 264, 316–322 (1997).
    [Crossref]
  2. F. V. Englich, Y. He, and B. J. Orr, “Continuous-wave cavity-ringdown detection of stimulated Raman gain spectra,” Appl. Phys. B 94, 1–27 (2008).
    [Crossref]
  3. J. Karhu, M. Vainio, M. Metsälä, and L. Halonen, “Frequency comb assisted two-photon vibrational spectroscopy,” Opt. Express 25, 4688–4699 (2017).
    [Crossref] [PubMed]
  4. W. K. Bischel, P. J. Kelly, and C. K. Rhodes, “Observation of Doppler-free two-photon absorption in the ν3 bands of CH3F,” Phys. Rev. Lett. 34, 300–303 (1975).
    [Crossref]
  5. M. Siltanen, M. Metsälä, M. Vainio, and L. Halonen, “Experimental observation and analysis of the 3ν1(Σg) stretching vibrational state of acetylene using continuous-wave infrared stimulated emission,” J. Chem. Phys. 139, 054201 (2013).
    [Crossref]
  6. A. Callegari, H. K. Srivastava, U. Merker, K. K. Lehmann, G. Scoles, and M. J. Davis, “Eigenstate resolved infrared-infrared double-resonance study of intramolecular vibrational relaxation in benzene: First overtone of the CH stretch,” J. Chem. Phys. 106, 432–435 (1997).
    [Crossref]
  7. H. Huang and K. K. Lehmann, “Long-term stability in continuous wave cavity ringdown spectroscopy experiments,” Appl. Opt. 49, 1378–1387 (2010).
    [Crossref] [PubMed]
  8. G. Giusfredi, S. Bartalini, S. Borri, P. Cancio, I. Galli, D. Mazzotti, and P. De Natale, “Saturated-absorption cavity ring-down spectroscopy,” Phys. Rev. Lett. 104, 110801 (2010).
    [Crossref] [PubMed]
  9. K. K. Lehmann, “Theoretical detection limit of saturated absorption cavity ring-down spectroscopy (SCAR) and two-photon absorption cavity ring-down spectroscopy,” Appl. Phys. B 116, 147–155 (2013).
    [Crossref]
  10. T. Hausmaninger, G. Zhao, W. G. Ma, and O. Axner, “Depletion of the vibrational ground state of CH4 in absorption spectroscopy at 3.4 μm in N2 and air in the 1–100 Torr range,” J. Quant. Spectros. Radiat. Transfer 205, 59–70 (2018).
    [Crossref]
  11. J. E. Bjorkholm and P. F. Liao, “Line shape and strength of two-photon absorption in an atomic vapor with a resonant or nearly resonant intermediate state,” Phys. Rev. A 14, 751–760 (1976).
    [Crossref]
  12. K. K. Lehmann and H. Huang, “Optimal signal processing in cavity ring-down spectroscopy,” in Frontiers of Molecular Spectroscopy (Elsevier, 2009), pp. 623–658.
  13. M. Siltanen, M. Vainio, and L. Halonen, “Pump-tunable continuous-wave singly resonant optical parametric oscillator from 2.5 to 4.4 μm,” Opt. Express 18, 14087–14092 (2010)
    [Crossref] [PubMed]

2018 (1)

T. Hausmaninger, G. Zhao, W. G. Ma, and O. Axner, “Depletion of the vibrational ground state of CH4 in absorption spectroscopy at 3.4 μm in N2 and air in the 1–100 Torr range,” J. Quant. Spectros. Radiat. Transfer 205, 59–70 (2018).
[Crossref]

2017 (1)

2013 (2)

K. K. Lehmann, “Theoretical detection limit of saturated absorption cavity ring-down spectroscopy (SCAR) and two-photon absorption cavity ring-down spectroscopy,” Appl. Phys. B 116, 147–155 (2013).
[Crossref]

M. Siltanen, M. Metsälä, M. Vainio, and L. Halonen, “Experimental observation and analysis of the 3ν1(Σg) stretching vibrational state of acetylene using continuous-wave infrared stimulated emission,” J. Chem. Phys. 139, 054201 (2013).
[Crossref]

2010 (3)

2008 (1)

F. V. Englich, Y. He, and B. J. Orr, “Continuous-wave cavity-ringdown detection of stimulated Raman gain spectra,” Appl. Phys. B 94, 1–27 (2008).
[Crossref]

1997 (2)

A. Callegari, H. K. Srivastava, U. Merker, K. K. Lehmann, G. Scoles, and M. J. Davis, “Eigenstate resolved infrared-infrared double-resonance study of intramolecular vibrational relaxation in benzene: First overtone of the CH stretch,” J. Chem. Phys. 106, 432–435 (1997).
[Crossref]

D. Romanini, A. A. Kachanov, N. Sadeghi, and F. Stoeckel, “CW cavity ring down spectroscopy,” Chem. Phys. Lett. 264, 316–322 (1997).
[Crossref]

1976 (1)

J. E. Bjorkholm and P. F. Liao, “Line shape and strength of two-photon absorption in an atomic vapor with a resonant or nearly resonant intermediate state,” Phys. Rev. A 14, 751–760 (1976).
[Crossref]

1975 (1)

W. K. Bischel, P. J. Kelly, and C. K. Rhodes, “Observation of Doppler-free two-photon absorption in the ν3 bands of CH3F,” Phys. Rev. Lett. 34, 300–303 (1975).
[Crossref]

Axner, O.

T. Hausmaninger, G. Zhao, W. G. Ma, and O. Axner, “Depletion of the vibrational ground state of CH4 in absorption spectroscopy at 3.4 μm in N2 and air in the 1–100 Torr range,” J. Quant. Spectros. Radiat. Transfer 205, 59–70 (2018).
[Crossref]

Bartalini, S.

G. Giusfredi, S. Bartalini, S. Borri, P. Cancio, I. Galli, D. Mazzotti, and P. De Natale, “Saturated-absorption cavity ring-down spectroscopy,” Phys. Rev. Lett. 104, 110801 (2010).
[Crossref] [PubMed]

Bischel, W. K.

W. K. Bischel, P. J. Kelly, and C. K. Rhodes, “Observation of Doppler-free two-photon absorption in the ν3 bands of CH3F,” Phys. Rev. Lett. 34, 300–303 (1975).
[Crossref]

Bjorkholm, J. E.

J. E. Bjorkholm and P. F. Liao, “Line shape and strength of two-photon absorption in an atomic vapor with a resonant or nearly resonant intermediate state,” Phys. Rev. A 14, 751–760 (1976).
[Crossref]

Borri, S.

G. Giusfredi, S. Bartalini, S. Borri, P. Cancio, I. Galli, D. Mazzotti, and P. De Natale, “Saturated-absorption cavity ring-down spectroscopy,” Phys. Rev. Lett. 104, 110801 (2010).
[Crossref] [PubMed]

Callegari, A.

A. Callegari, H. K. Srivastava, U. Merker, K. K. Lehmann, G. Scoles, and M. J. Davis, “Eigenstate resolved infrared-infrared double-resonance study of intramolecular vibrational relaxation in benzene: First overtone of the CH stretch,” J. Chem. Phys. 106, 432–435 (1997).
[Crossref]

Cancio, P.

G. Giusfredi, S. Bartalini, S. Borri, P. Cancio, I. Galli, D. Mazzotti, and P. De Natale, “Saturated-absorption cavity ring-down spectroscopy,” Phys. Rev. Lett. 104, 110801 (2010).
[Crossref] [PubMed]

Davis, M. J.

A. Callegari, H. K. Srivastava, U. Merker, K. K. Lehmann, G. Scoles, and M. J. Davis, “Eigenstate resolved infrared-infrared double-resonance study of intramolecular vibrational relaxation in benzene: First overtone of the CH stretch,” J. Chem. Phys. 106, 432–435 (1997).
[Crossref]

De Natale, P.

G. Giusfredi, S. Bartalini, S. Borri, P. Cancio, I. Galli, D. Mazzotti, and P. De Natale, “Saturated-absorption cavity ring-down spectroscopy,” Phys. Rev. Lett. 104, 110801 (2010).
[Crossref] [PubMed]

Englich, F. V.

F. V. Englich, Y. He, and B. J. Orr, “Continuous-wave cavity-ringdown detection of stimulated Raman gain spectra,” Appl. Phys. B 94, 1–27 (2008).
[Crossref]

Galli, I.

G. Giusfredi, S. Bartalini, S. Borri, P. Cancio, I. Galli, D. Mazzotti, and P. De Natale, “Saturated-absorption cavity ring-down spectroscopy,” Phys. Rev. Lett. 104, 110801 (2010).
[Crossref] [PubMed]

Giusfredi, G.

G. Giusfredi, S. Bartalini, S. Borri, P. Cancio, I. Galli, D. Mazzotti, and P. De Natale, “Saturated-absorption cavity ring-down spectroscopy,” Phys. Rev. Lett. 104, 110801 (2010).
[Crossref] [PubMed]

Halonen, L.

Hausmaninger, T.

T. Hausmaninger, G. Zhao, W. G. Ma, and O. Axner, “Depletion of the vibrational ground state of CH4 in absorption spectroscopy at 3.4 μm in N2 and air in the 1–100 Torr range,” J. Quant. Spectros. Radiat. Transfer 205, 59–70 (2018).
[Crossref]

He, Y.

F. V. Englich, Y. He, and B. J. Orr, “Continuous-wave cavity-ringdown detection of stimulated Raman gain spectra,” Appl. Phys. B 94, 1–27 (2008).
[Crossref]

Huang, H.

H. Huang and K. K. Lehmann, “Long-term stability in continuous wave cavity ringdown spectroscopy experiments,” Appl. Opt. 49, 1378–1387 (2010).
[Crossref] [PubMed]

K. K. Lehmann and H. Huang, “Optimal signal processing in cavity ring-down spectroscopy,” in Frontiers of Molecular Spectroscopy (Elsevier, 2009), pp. 623–658.

Kachanov, A. A.

D. Romanini, A. A. Kachanov, N. Sadeghi, and F. Stoeckel, “CW cavity ring down spectroscopy,” Chem. Phys. Lett. 264, 316–322 (1997).
[Crossref]

Karhu, J.

Kelly, P. J.

W. K. Bischel, P. J. Kelly, and C. K. Rhodes, “Observation of Doppler-free two-photon absorption in the ν3 bands of CH3F,” Phys. Rev. Lett. 34, 300–303 (1975).
[Crossref]

Lehmann, K. K.

K. K. Lehmann, “Theoretical detection limit of saturated absorption cavity ring-down spectroscopy (SCAR) and two-photon absorption cavity ring-down spectroscopy,” Appl. Phys. B 116, 147–155 (2013).
[Crossref]

H. Huang and K. K. Lehmann, “Long-term stability in continuous wave cavity ringdown spectroscopy experiments,” Appl. Opt. 49, 1378–1387 (2010).
[Crossref] [PubMed]

A. Callegari, H. K. Srivastava, U. Merker, K. K. Lehmann, G. Scoles, and M. J. Davis, “Eigenstate resolved infrared-infrared double-resonance study of intramolecular vibrational relaxation in benzene: First overtone of the CH stretch,” J. Chem. Phys. 106, 432–435 (1997).
[Crossref]

K. K. Lehmann and H. Huang, “Optimal signal processing in cavity ring-down spectroscopy,” in Frontiers of Molecular Spectroscopy (Elsevier, 2009), pp. 623–658.

Liao, P. F.

J. E. Bjorkholm and P. F. Liao, “Line shape and strength of two-photon absorption in an atomic vapor with a resonant or nearly resonant intermediate state,” Phys. Rev. A 14, 751–760 (1976).
[Crossref]

Ma, W. G.

T. Hausmaninger, G. Zhao, W. G. Ma, and O. Axner, “Depletion of the vibrational ground state of CH4 in absorption spectroscopy at 3.4 μm in N2 and air in the 1–100 Torr range,” J. Quant. Spectros. Radiat. Transfer 205, 59–70 (2018).
[Crossref]

Mazzotti, D.

G. Giusfredi, S. Bartalini, S. Borri, P. Cancio, I. Galli, D. Mazzotti, and P. De Natale, “Saturated-absorption cavity ring-down spectroscopy,” Phys. Rev. Lett. 104, 110801 (2010).
[Crossref] [PubMed]

Merker, U.

A. Callegari, H. K. Srivastava, U. Merker, K. K. Lehmann, G. Scoles, and M. J. Davis, “Eigenstate resolved infrared-infrared double-resonance study of intramolecular vibrational relaxation in benzene: First overtone of the CH stretch,” J. Chem. Phys. 106, 432–435 (1997).
[Crossref]

Metsälä, M.

J. Karhu, M. Vainio, M. Metsälä, and L. Halonen, “Frequency comb assisted two-photon vibrational spectroscopy,” Opt. Express 25, 4688–4699 (2017).
[Crossref] [PubMed]

M. Siltanen, M. Metsälä, M. Vainio, and L. Halonen, “Experimental observation and analysis of the 3ν1(Σg) stretching vibrational state of acetylene using continuous-wave infrared stimulated emission,” J. Chem. Phys. 139, 054201 (2013).
[Crossref]

Orr, B. J.

F. V. Englich, Y. He, and B. J. Orr, “Continuous-wave cavity-ringdown detection of stimulated Raman gain spectra,” Appl. Phys. B 94, 1–27 (2008).
[Crossref]

Rhodes, C. K.

W. K. Bischel, P. J. Kelly, and C. K. Rhodes, “Observation of Doppler-free two-photon absorption in the ν3 bands of CH3F,” Phys. Rev. Lett. 34, 300–303 (1975).
[Crossref]

Romanini, D.

D. Romanini, A. A. Kachanov, N. Sadeghi, and F. Stoeckel, “CW cavity ring down spectroscopy,” Chem. Phys. Lett. 264, 316–322 (1997).
[Crossref]

Sadeghi, N.

D. Romanini, A. A. Kachanov, N. Sadeghi, and F. Stoeckel, “CW cavity ring down spectroscopy,” Chem. Phys. Lett. 264, 316–322 (1997).
[Crossref]

Scoles, G.

A. Callegari, H. K. Srivastava, U. Merker, K. K. Lehmann, G. Scoles, and M. J. Davis, “Eigenstate resolved infrared-infrared double-resonance study of intramolecular vibrational relaxation in benzene: First overtone of the CH stretch,” J. Chem. Phys. 106, 432–435 (1997).
[Crossref]

Siltanen, M.

M. Siltanen, M. Metsälä, M. Vainio, and L. Halonen, “Experimental observation and analysis of the 3ν1(Σg) stretching vibrational state of acetylene using continuous-wave infrared stimulated emission,” J. Chem. Phys. 139, 054201 (2013).
[Crossref]

M. Siltanen, M. Vainio, and L. Halonen, “Pump-tunable continuous-wave singly resonant optical parametric oscillator from 2.5 to 4.4 μm,” Opt. Express 18, 14087–14092 (2010)
[Crossref] [PubMed]

Srivastava, H. K.

A. Callegari, H. K. Srivastava, U. Merker, K. K. Lehmann, G. Scoles, and M. J. Davis, “Eigenstate resolved infrared-infrared double-resonance study of intramolecular vibrational relaxation in benzene: First overtone of the CH stretch,” J. Chem. Phys. 106, 432–435 (1997).
[Crossref]

Stoeckel, F.

D. Romanini, A. A. Kachanov, N. Sadeghi, and F. Stoeckel, “CW cavity ring down spectroscopy,” Chem. Phys. Lett. 264, 316–322 (1997).
[Crossref]

Vainio, M.

Zhao, G.

T. Hausmaninger, G. Zhao, W. G. Ma, and O. Axner, “Depletion of the vibrational ground state of CH4 in absorption spectroscopy at 3.4 μm in N2 and air in the 1–100 Torr range,” J. Quant. Spectros. Radiat. Transfer 205, 59–70 (2018).
[Crossref]

Appl. Opt. (1)

Appl. Phys. B (2)

F. V. Englich, Y. He, and B. J. Orr, “Continuous-wave cavity-ringdown detection of stimulated Raman gain spectra,” Appl. Phys. B 94, 1–27 (2008).
[Crossref]

K. K. Lehmann, “Theoretical detection limit of saturated absorption cavity ring-down spectroscopy (SCAR) and two-photon absorption cavity ring-down spectroscopy,” Appl. Phys. B 116, 147–155 (2013).
[Crossref]

Chem. Phys. Lett. (1)

D. Romanini, A. A. Kachanov, N. Sadeghi, and F. Stoeckel, “CW cavity ring down spectroscopy,” Chem. Phys. Lett. 264, 316–322 (1997).
[Crossref]

J. Chem. Phys. (2)

M. Siltanen, M. Metsälä, M. Vainio, and L. Halonen, “Experimental observation and analysis of the 3ν1(Σg) stretching vibrational state of acetylene using continuous-wave infrared stimulated emission,” J. Chem. Phys. 139, 054201 (2013).
[Crossref]

A. Callegari, H. K. Srivastava, U. Merker, K. K. Lehmann, G. Scoles, and M. J. Davis, “Eigenstate resolved infrared-infrared double-resonance study of intramolecular vibrational relaxation in benzene: First overtone of the CH stretch,” J. Chem. Phys. 106, 432–435 (1997).
[Crossref]

J. Quant. Spectros. Radiat. Transfer (1)

T. Hausmaninger, G. Zhao, W. G. Ma, and O. Axner, “Depletion of the vibrational ground state of CH4 in absorption spectroscopy at 3.4 μm in N2 and air in the 1–100 Torr range,” J. Quant. Spectros. Radiat. Transfer 205, 59–70 (2018).
[Crossref]

Opt. Express (2)

Phys. Rev. A (1)

J. E. Bjorkholm and P. F. Liao, “Line shape and strength of two-photon absorption in an atomic vapor with a resonant or nearly resonant intermediate state,” Phys. Rev. A 14, 751–760 (1976).
[Crossref]

Phys. Rev. Lett. (2)

W. K. Bischel, P. J. Kelly, and C. K. Rhodes, “Observation of Doppler-free two-photon absorption in the ν3 bands of CH3F,” Phys. Rev. Lett. 34, 300–303 (1975).
[Crossref]

G. Giusfredi, S. Bartalini, S. Borri, P. Cancio, I. Galli, D. Mazzotti, and P. De Natale, “Saturated-absorption cavity ring-down spectroscopy,” Phys. Rev. Lett. 104, 110801 (2010).
[Crossref] [PubMed]

Other (1)

K. K. Lehmann and H. Huang, “Optimal signal processing in cavity ring-down spectroscopy,” in Frontiers of Molecular Spectroscopy (Elsevier, 2009), pp. 623–658.

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Figures (7)

Fig. 1
Fig. 1 Energy diagrams for (a) ladder-type double resonant absorption, (b) V-type double resonance, and (c) Λ-type double resonance.
Fig. 2
Fig. 2 Principle of the SMRS. In the beginning of the ring-down decay, one light source, referred to as the pump, is exciting one step of the double resonance, and the second step is excited by the light source of the CRDS measurement. During the initial phase of the decay, the ring-down time is affected by the cavity loss, linear absorption of the sample gas and possible double-resonant two-photon loss. In the middle of the decay, the pump laser is turned off. After this point, the ring-down time is determined by the cavity loss and the linear absorption only. The two ring-down times can be used to calculate the double resonance loss.
Fig. 3
Fig. 3 Standard deviation of the difference between kon and koff, compared to the standard deviation of decay rate k from a single exponential decay, as a function of the switching time. Switching time ts is the point at which the modulated decay changes from rate kon to koff. The time axis has been normalized to the decay rate.
Fig. 4
Fig. 4 Schematic of the measurement setup. BS: beam sampler, WM: wavemeter, DM: dichroic mirror, PD: photodiode, TP: thermopile detector.
Fig. 5
Fig. 5 a) An example ring-down spectrum recorded with the SMRS without averaging. The OPO was on at the beginning of the ring-down decay, corresponding to the kon trace. The end of the decay is affected only by the empty cavity losses, as well as any linear absorption of the ECDL that coincides with the double resonance line (koff trace). As shown by the shape of the latter trace, here the double resonance line resides at the side of a linear absorption line. b) The double resonance attenuation coefficient calculated from the spectrum on the left.
Fig. 6
Fig. 6 Allan-Werle deviation of attenuation coefficient measured with traditional CRDS and SMRS. The CRDS-trace is the deviation for the variable k/c, where k is the ring-down rate, measured with normal CRDS. The SMRS-trace is the deviation for double resonance absorption coefficients from modulated decay fits. The data for both models were the same set of ring-down decays, recorded over about 5 hours. The ECDL was kept free running at around a set wavelength during the whole measurement and OPO was off the whole time.
Fig. 7
Fig. 7 Allan-Werle deviations for double resonance absorption measured either using two separate single exponential decays (CRDS-trace) or SMRS method, to measure the total losses and only the linear losses. The ECDL was again kept at constant wavelength and OPO was not used. The ECDL wavelength was chosen so that it coincided with a side of an absorption line. a) the ECDL is free-running, with its typical narrow linewidth. b) excess noise has been added to the ECDL wavelength, resulting in about 10 MHz linewidth. The wavelength noise couples into variation of the ring-down rate between decays.

Equations (7)

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α dr = n c ( k on k off ) k off = ( c / n ) [ ( 1 R ) / L + α linear ]
S i = A exp ( k on Δ t i ) + B 0 i < N 1
S i = A exp ( k on Δ t N 1 k off Δ t j ) + B 0 i < N 2
σ 2 ( Δ k ) = 4 k 3 Δ t e 2 k t s e 4 k t s ( 2 k 2 t s 2 + 2 k t s + 1 ) ( σ d A ) 2
S i B = 1 S j B = 1 S i A = a 1 i S j A = a 1 N 1 a 2 j S i k 1 = A Δ t i a 1 i S j k 1 = A Δ t N 1 a 1 N 1 a 2 j S i k 2 = 0 S j k 2 = A Δ t j a 1 N 1 a 2 j
α B , B = N 1 + N 2 α A , B = 1 a i N 1 1 a i + a 1 N 1 1 a 2 N 2 1 a 2 α k 1 , B = A Δ t [ a 1 ( 1 a 1 N 1 ) N 1 a 1 N 1 ( 1 a 1 ) ( 1 a 1 ) 2 + N 1 a 1 N 1 1 a 2 N 2 1 a 2 ] α k 2 , B = A Δ t a 1 N 1 [ a 2 ( 1 a 2 N 2 ) N 2 a 2 N 2 ( 1 a 2 ) ( 1 a 2 ) 2 ] α A , A = 1 a 1 2 N 1 1 a 1 2 + a 1 2 N 1 1 a 2 2 N 2 1 a 2 2 α k 1 , A = A Δ t [ a 1 2 ( 1 a 1 2 N 1 ) N 1 a 1 2 N 1 ( 1 a 1 2 ) ( 1 a 1 2 ) 2 + N 1 a 1 2 N 1 1 a 2 2 N 2 1 a 2 2 ] α k 2 , A = A Δ t a 1 2 N 1 [ a 2 2 ( 1 a 2 2 N 2 ) N 2 a 2 2 N 2 ( 1 a 2 2 ) ( 1 a 2 2 ) 2 ] α k 1 , k 1 = ( A Δ t ) 2 [ a 1 2 ( 1 + a 1 2 ) a 1 2 N 1 [ N 1 2 ( 1 a 1 2 ) 2 + 2 N 1 a 1 2 ( 1 a 2 2 ) + a 1 2 ( 1 + a 1 2 ) ] ( 1 a 1 2 ) 3 + N 1 2 a 1 2 N 1 1 a 2 2 N 2 1 a 2 2 ] α k 1 , k 2 = ( A Δ t ) 2 N 1 a 1 2 N 1 [ a 2 2 ( 1 a 2 2 N 2 ) N 2 a 2 2 N 2 ( 1 a 2 2 ) ( 1 a 2 2 ) 2 ] α k 2 , k 2 = ( A Δ t ) 2 a 1 2 N 1 [ a 2 2 ( 1 + a 2 2 ) a 2 2 N 2 [ N 2 2 ( 1 a 2 2 ) 2 + 2 N 2 a 2 2 ( 1 a 2 2 ) + a 2 2 ( 1 + a 2 2 ) ] ( 1 a 2 2 ) 3 ]
β B = i y i + j y j A [ 1 a 1 N 1 1 a 1 + a 1 N 1 1 a 2 N 2 1 a 2 ] B [ N 1 + N 2 ] β A = i y i a 1 i + j y j a 2 j A [ 1 a 1 2 N 1 1 a 1 2 + a 1 2 N 1 1 a 2 2 N 2 1 a 2 2 ] B [ 1 a 1 N 1 1 a 1 + a 1 N 1 1 a 2 N 2 1 a 2 ] β k 1 = A Δ t [ i y i i a 1 1 + N 1 a 1 N 1 j y j a 2 j ] + A 2 Δ t [ a 1 2 ( 1 a 1 2 N 1 ) N 1 a 1 2 N 1 ( 1 a 1 2 ) ( 1 a 1 2 ) 2 + N 1 a 1 2 N 1 1 a 2 2 N 2 1 a 2 2 ] + B A Δ t [ a 1 ( 1 a 1 N 1 ) N 1 a 1 N 1 ( 1 a 1 ) ( 1 a 1 ) 2 + N 1 a 1 N 1 1 a 2 N 2 1 a 2 ] β k 2 = A Δ t a 1 N 1 j y j j a 2 j + A 2 Δ t a 1 2 N 1 [ a 2 2 ( 1 a 2 2 N 2 ) N 2 a 2 2 N 2 ( 1 a 2 2 ) ( 1 a 2 2 ) 2 ] + A B Δ t a 1 N 1 [ a 2 ( 1 a 2 N 2 ) N 2 a 2 N 2 ( 1 a 2 ) ( 1 a 2 ) 2 ]

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