Abstract

Optical parametric oscillators (OPO’s) have evolved to operate and be pumped by increasingly shorter pulses, so modeling them with a quasi-stationary approximation has become inappropriate. We derive a nonstationary model of an OPO and an optical parametric amplifier that takes into account the first-derivative terms of second- and third-order nonlinearity. We have rewritten this model entirely in the spectral domain. We applied the model to a KTP OPO and found good agreement with a well-characterized experiment. Further calculations were performed with a large diversity of parameters such as reflectivity of the output coupler, length and nonlinear index of refraction of the nonlinear crystal, length of the resonator, and duration of the pump pulse.

© 2000 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. C. Edelstein, E. S. Wachman, and C. L. Tang, “Broadly tunable high repetition rate femtosecond optical parametric oscillator,” Appl. Phys. Lett. 54, 1728–1730 (1989).
    [CrossRef]
  2. G. Mark, Q. Fu, and H. M. van Driel, “Externally pumped high repetition rate femtosecond infrared optical parametric oscillator,” Appl. Phys. Lett. 60, 542–544 (1992).
    [CrossRef]
  3. Q. Fu, G. Mark, and H. M. van Driel, “High-power, 62-fs infrared optical parametric oscillator synchronously pumped by a 76-MHz Ti:sapphire laser,” Opt. Lett. 17, 1006–1008 (1992).
    [CrossRef] [PubMed]
  4. W. S. Pelouch, P. E. Powers, and C. L. Tang, “Ti:sapphire-pumped, high-repetition-rate femtosecond optical parametric oscillator,” Opt. Lett. 17, 1070–1072 (1992).
    [CrossRef] [PubMed]
  5. J. Hebling, H. Giessen, S. Linden, and J. Kuhl, “Mirror-dispersion-compensated femtosecond optical parametric oscillator,” Opt. Commun. 141, 229–236 (1997).
    [CrossRef]
  6. D. T. Reid, M. Ebrahimzadeh, and W. Sibbett, “Design criteria and comparison of femtosecond optical parametric oscillators based on KTiOPO4 and RbTiOAsO4,” J. Opt. Soc. Am. B 12, 2168–2179 (1995).
    [CrossRef]
  7. G. M. Gale, M. Cavallari, and F. Hache, “Femtosecond visible optical parametric oscillator,” J. Opt. Soc. Am. B 15, 702–714 (1998).
    [CrossRef]
  8. S. T. Yang, R. C. Eckardt, and R. L. Byer, “Power and spectral characteristics of continuous-wave parametric oscillators: the doubly to singly resonant transition,” J. Opt. Soc. Am. B 10, 1684–1695 (1993).
    [CrossRef]
  9. A. Agnesi, G. C. Reali, V. Kubecek, S. Kumazaki, Y. Takagi, and K. Yoshihara, “β-Barium borate and lithium triborate picosecond parametric oscillators pumped by a frequency-tripled passive negative-feedback mode-locked Nd:YAG laser,” J. Opt. Soc. Am. B 10, 2211–2217 (1993).
    [CrossRef]
  10. J. D. V. Khaydarov, J. H. Andrews, and K. D. Singer, “Pulse-compression mechanism in a synchronously pumped optical parametric oscillator,” J. Opt. Soc. Am. B 12, 2199–2208 (1995).
    [CrossRef]
  11. R. Laenen and T. Roth, “Numerical study and experimental results for a synchronously pumped optical parametric os-cillator in the saturation regime,” J. Opt. Soc. Am. B 14, 454–459 (1997).
    [CrossRef]
  12. E. C. Cheung and J. M. Liu, “Theory of a synchronously pumped optical parametric oscillator in steady-state operation,” J. Opt. Soc. Am. B 7, 1385–1401 (1990).
    [CrossRef]
  13. G. Görer and R. Laenen, “Femtosecond optical parametric oscillators: numerical study on phase-dependent pulse formation and experimental results,” Opt. Commun. 152, 429–435 (1998).
    [CrossRef]
  14. R. Laenen, K. Wolfrum, A. Seilmeier, and A. Lauberau, “Parametric generation of femtosecond and picosecond pulses for spectroscopic applications,” J. Opt. Soc. Am. B 10, 2151–2161 (1993).
    [CrossRef]
  15. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley-Interscience, New York, 1991), Eqs. (19.1–6) and (19.1–7).
  16. I. Shoji, T. Kondo, A. Kitamoto, M. Shirane, and R. Ito, “Absolute scale of second-order nonlinear-optical coefficients,” J. Opt. Soc. Am. B 14, 2268–2294 (1997).
    [CrossRef]
  17. K. Kato, “Parametric oscillation at 3.2 μm in KTP pumped at 1.064 μm,” IEEE J. Quantum Electron. 27, 1137–1140 (1991).
    [CrossRef]
  18. F. Hache, A. Zéboulon, G. Gallot, and G. M. Gale, “Cascaded second-order effects in the femtosecond regime in β-barium borate: self-compression in a visible femtosecond optical parametric oscillator,” Opt. Lett. 20, 1556–1558 (1995).
    [CrossRef] [PubMed]
  19. L. P. Chen, Y. Wang, and J. M. Liu, “Singly resonant optical parametric oscillator synchronously pumped by frequency-doubled additive-pulse mode-locked Nd:YLF laser pulses,” J. Opt. Soc. Am. B 12, 2192–2198 (1995).
    [CrossRef]
  20. D. S. Butterworth, S. Girard, and D. C. Hanna, “A simple technique to achieve active cavity-length stabilization in a synchronously pumped optical parametric oscillator,” Opt. Commun. 123, 577–582 (1996).
    [CrossRef]
  21. J. M. Dudley, D. T. Reid, M. Ebrahimzadeh, and W. Sibbett, “Characteristics of a noncritically phasematched Ti:sapphire pumped femtosecond optical parametric oscillator,” Opt. Commun. 104, 419–430 (1994).
    [CrossRef]
  22. S. E. Harris, “Tunable optical parametric oscillators,” Proc. IEEE 57, 2096–2113 (1969).
    [CrossRef]

1998

G. M. Gale, M. Cavallari, and F. Hache, “Femtosecond visible optical parametric oscillator,” J. Opt. Soc. Am. B 15, 702–714 (1998).
[CrossRef]

G. Görer and R. Laenen, “Femtosecond optical parametric oscillators: numerical study on phase-dependent pulse formation and experimental results,” Opt. Commun. 152, 429–435 (1998).
[CrossRef]

1997

1996

D. S. Butterworth, S. Girard, and D. C. Hanna, “A simple technique to achieve active cavity-length stabilization in a synchronously pumped optical parametric oscillator,” Opt. Commun. 123, 577–582 (1996).
[CrossRef]

1995

1994

J. M. Dudley, D. T. Reid, M. Ebrahimzadeh, and W. Sibbett, “Characteristics of a noncritically phasematched Ti:sapphire pumped femtosecond optical parametric oscillator,” Opt. Commun. 104, 419–430 (1994).
[CrossRef]

1993

1992

1991

K. Kato, “Parametric oscillation at 3.2 μm in KTP pumped at 1.064 μm,” IEEE J. Quantum Electron. 27, 1137–1140 (1991).
[CrossRef]

1990

1989

D. C. Edelstein, E. S. Wachman, and C. L. Tang, “Broadly tunable high repetition rate femtosecond optical parametric oscillator,” Appl. Phys. Lett. 54, 1728–1730 (1989).
[CrossRef]

1969

S. E. Harris, “Tunable optical parametric oscillators,” Proc. IEEE 57, 2096–2113 (1969).
[CrossRef]

Agnesi, A.

Andrews, J. H.

Butterworth, D. S.

D. S. Butterworth, S. Girard, and D. C. Hanna, “A simple technique to achieve active cavity-length stabilization in a synchronously pumped optical parametric oscillator,” Opt. Commun. 123, 577–582 (1996).
[CrossRef]

Byer, R. L.

Cavallari, M.

Chen, L. P.

Cheung, E. C.

Dudley, J. M.

J. M. Dudley, D. T. Reid, M. Ebrahimzadeh, and W. Sibbett, “Characteristics of a noncritically phasematched Ti:sapphire pumped femtosecond optical parametric oscillator,” Opt. Commun. 104, 419–430 (1994).
[CrossRef]

Ebrahimzadeh, M.

D. T. Reid, M. Ebrahimzadeh, and W. Sibbett, “Design criteria and comparison of femtosecond optical parametric oscillators based on KTiOPO4 and RbTiOAsO4,” J. Opt. Soc. Am. B 12, 2168–2179 (1995).
[CrossRef]

J. M. Dudley, D. T. Reid, M. Ebrahimzadeh, and W. Sibbett, “Characteristics of a noncritically phasematched Ti:sapphire pumped femtosecond optical parametric oscillator,” Opt. Commun. 104, 419–430 (1994).
[CrossRef]

Eckardt, R. C.

Edelstein, D. C.

D. C. Edelstein, E. S. Wachman, and C. L. Tang, “Broadly tunable high repetition rate femtosecond optical parametric oscillator,” Appl. Phys. Lett. 54, 1728–1730 (1989).
[CrossRef]

Fu, Q.

G. Mark, Q. Fu, and H. M. van Driel, “Externally pumped high repetition rate femtosecond infrared optical parametric oscillator,” Appl. Phys. Lett. 60, 542–544 (1992).
[CrossRef]

Q. Fu, G. Mark, and H. M. van Driel, “High-power, 62-fs infrared optical parametric oscillator synchronously pumped by a 76-MHz Ti:sapphire laser,” Opt. Lett. 17, 1006–1008 (1992).
[CrossRef] [PubMed]

Gale, G. M.

Gallot, G.

Giessen, H.

J. Hebling, H. Giessen, S. Linden, and J. Kuhl, “Mirror-dispersion-compensated femtosecond optical parametric oscillator,” Opt. Commun. 141, 229–236 (1997).
[CrossRef]

Girard, S.

D. S. Butterworth, S. Girard, and D. C. Hanna, “A simple technique to achieve active cavity-length stabilization in a synchronously pumped optical parametric oscillator,” Opt. Commun. 123, 577–582 (1996).
[CrossRef]

Görer, G.

G. Görer and R. Laenen, “Femtosecond optical parametric oscillators: numerical study on phase-dependent pulse formation and experimental results,” Opt. Commun. 152, 429–435 (1998).
[CrossRef]

Hache, F.

Hanna, D. C.

D. S. Butterworth, S. Girard, and D. C. Hanna, “A simple technique to achieve active cavity-length stabilization in a synchronously pumped optical parametric oscillator,” Opt. Commun. 123, 577–582 (1996).
[CrossRef]

Harris, S. E.

S. E. Harris, “Tunable optical parametric oscillators,” Proc. IEEE 57, 2096–2113 (1969).
[CrossRef]

Hebling, J.

J. Hebling, H. Giessen, S. Linden, and J. Kuhl, “Mirror-dispersion-compensated femtosecond optical parametric oscillator,” Opt. Commun. 141, 229–236 (1997).
[CrossRef]

Ito, R.

Kato, K.

K. Kato, “Parametric oscillation at 3.2 μm in KTP pumped at 1.064 μm,” IEEE J. Quantum Electron. 27, 1137–1140 (1991).
[CrossRef]

Khaydarov, J. D. V.

Kitamoto, A.

Kondo, T.

Kubecek, V.

Kuhl, J.

J. Hebling, H. Giessen, S. Linden, and J. Kuhl, “Mirror-dispersion-compensated femtosecond optical parametric oscillator,” Opt. Commun. 141, 229–236 (1997).
[CrossRef]

Kumazaki, S.

Laenen, R.

Lauberau, A.

Linden, S.

J. Hebling, H. Giessen, S. Linden, and J. Kuhl, “Mirror-dispersion-compensated femtosecond optical parametric oscillator,” Opt. Commun. 141, 229–236 (1997).
[CrossRef]

Liu, J. M.

Mark, G.

Q. Fu, G. Mark, and H. M. van Driel, “High-power, 62-fs infrared optical parametric oscillator synchronously pumped by a 76-MHz Ti:sapphire laser,” Opt. Lett. 17, 1006–1008 (1992).
[CrossRef] [PubMed]

G. Mark, Q. Fu, and H. M. van Driel, “Externally pumped high repetition rate femtosecond infrared optical parametric oscillator,” Appl. Phys. Lett. 60, 542–544 (1992).
[CrossRef]

Pelouch, W. S.

Powers, P. E.

Reali, G. C.

Reid, D. T.

D. T. Reid, M. Ebrahimzadeh, and W. Sibbett, “Design criteria and comparison of femtosecond optical parametric oscillators based on KTiOPO4 and RbTiOAsO4,” J. Opt. Soc. Am. B 12, 2168–2179 (1995).
[CrossRef]

J. M. Dudley, D. T. Reid, M. Ebrahimzadeh, and W. Sibbett, “Characteristics of a noncritically phasematched Ti:sapphire pumped femtosecond optical parametric oscillator,” Opt. Commun. 104, 419–430 (1994).
[CrossRef]

Roth, T.

Seilmeier, A.

Shirane, M.

Shoji, I.

Sibbett, W.

D. T. Reid, M. Ebrahimzadeh, and W. Sibbett, “Design criteria and comparison of femtosecond optical parametric oscillators based on KTiOPO4 and RbTiOAsO4,” J. Opt. Soc. Am. B 12, 2168–2179 (1995).
[CrossRef]

J. M. Dudley, D. T. Reid, M. Ebrahimzadeh, and W. Sibbett, “Characteristics of a noncritically phasematched Ti:sapphire pumped femtosecond optical parametric oscillator,” Opt. Commun. 104, 419–430 (1994).
[CrossRef]

Singer, K. D.

Takagi, Y.

Tang, C. L.

W. S. Pelouch, P. E. Powers, and C. L. Tang, “Ti:sapphire-pumped, high-repetition-rate femtosecond optical parametric oscillator,” Opt. Lett. 17, 1070–1072 (1992).
[CrossRef] [PubMed]

D. C. Edelstein, E. S. Wachman, and C. L. Tang, “Broadly tunable high repetition rate femtosecond optical parametric oscillator,” Appl. Phys. Lett. 54, 1728–1730 (1989).
[CrossRef]

van Driel, H. M.

Q. Fu, G. Mark, and H. M. van Driel, “High-power, 62-fs infrared optical parametric oscillator synchronously pumped by a 76-MHz Ti:sapphire laser,” Opt. Lett. 17, 1006–1008 (1992).
[CrossRef] [PubMed]

G. Mark, Q. Fu, and H. M. van Driel, “Externally pumped high repetition rate femtosecond infrared optical parametric oscillator,” Appl. Phys. Lett. 60, 542–544 (1992).
[CrossRef]

Wachman, E. S.

D. C. Edelstein, E. S. Wachman, and C. L. Tang, “Broadly tunable high repetition rate femtosecond optical parametric oscillator,” Appl. Phys. Lett. 54, 1728–1730 (1989).
[CrossRef]

Wang, Y.

Wolfrum, K.

Yang, S. T.

Yoshihara, K.

Zéboulon, A.

Appl. Phys. Lett.

D. C. Edelstein, E. S. Wachman, and C. L. Tang, “Broadly tunable high repetition rate femtosecond optical parametric oscillator,” Appl. Phys. Lett. 54, 1728–1730 (1989).
[CrossRef]

G. Mark, Q. Fu, and H. M. van Driel, “Externally pumped high repetition rate femtosecond infrared optical parametric oscillator,” Appl. Phys. Lett. 60, 542–544 (1992).
[CrossRef]

IEEE J. Quantum Electron.

K. Kato, “Parametric oscillation at 3.2 μm in KTP pumped at 1.064 μm,” IEEE J. Quantum Electron. 27, 1137–1140 (1991).
[CrossRef]

J. Opt. Soc. Am. B

L. P. Chen, Y. Wang, and J. M. Liu, “Singly resonant optical parametric oscillator synchronously pumped by frequency-doubled additive-pulse mode-locked Nd:YLF laser pulses,” J. Opt. Soc. Am. B 12, 2192–2198 (1995).
[CrossRef]

R. Laenen, K. Wolfrum, A. Seilmeier, and A. Lauberau, “Parametric generation of femtosecond and picosecond pulses for spectroscopic applications,” J. Opt. Soc. Am. B 10, 2151–2161 (1993).
[CrossRef]

I. Shoji, T. Kondo, A. Kitamoto, M. Shirane, and R. Ito, “Absolute scale of second-order nonlinear-optical coefficients,” J. Opt. Soc. Am. B 14, 2268–2294 (1997).
[CrossRef]

D. T. Reid, M. Ebrahimzadeh, and W. Sibbett, “Design criteria and comparison of femtosecond optical parametric oscillators based on KTiOPO4 and RbTiOAsO4,” J. Opt. Soc. Am. B 12, 2168–2179 (1995).
[CrossRef]

G. M. Gale, M. Cavallari, and F. Hache, “Femtosecond visible optical parametric oscillator,” J. Opt. Soc. Am. B 15, 702–714 (1998).
[CrossRef]

S. T. Yang, R. C. Eckardt, and R. L. Byer, “Power and spectral characteristics of continuous-wave parametric oscillators: the doubly to singly resonant transition,” J. Opt. Soc. Am. B 10, 1684–1695 (1993).
[CrossRef]

A. Agnesi, G. C. Reali, V. Kubecek, S. Kumazaki, Y. Takagi, and K. Yoshihara, “β-Barium borate and lithium triborate picosecond parametric oscillators pumped by a frequency-tripled passive negative-feedback mode-locked Nd:YAG laser,” J. Opt. Soc. Am. B 10, 2211–2217 (1993).
[CrossRef]

J. D. V. Khaydarov, J. H. Andrews, and K. D. Singer, “Pulse-compression mechanism in a synchronously pumped optical parametric oscillator,” J. Opt. Soc. Am. B 12, 2199–2208 (1995).
[CrossRef]

R. Laenen and T. Roth, “Numerical study and experimental results for a synchronously pumped optical parametric os-cillator in the saturation regime,” J. Opt. Soc. Am. B 14, 454–459 (1997).
[CrossRef]

E. C. Cheung and J. M. Liu, “Theory of a synchronously pumped optical parametric oscillator in steady-state operation,” J. Opt. Soc. Am. B 7, 1385–1401 (1990).
[CrossRef]

Opt. Commun.

G. Görer and R. Laenen, “Femtosecond optical parametric oscillators: numerical study on phase-dependent pulse formation and experimental results,” Opt. Commun. 152, 429–435 (1998).
[CrossRef]

J. Hebling, H. Giessen, S. Linden, and J. Kuhl, “Mirror-dispersion-compensated femtosecond optical parametric oscillator,” Opt. Commun. 141, 229–236 (1997).
[CrossRef]

D. S. Butterworth, S. Girard, and D. C. Hanna, “A simple technique to achieve active cavity-length stabilization in a synchronously pumped optical parametric oscillator,” Opt. Commun. 123, 577–582 (1996).
[CrossRef]

J. M. Dudley, D. T. Reid, M. Ebrahimzadeh, and W. Sibbett, “Characteristics of a noncritically phasematched Ti:sapphire pumped femtosecond optical parametric oscillator,” Opt. Commun. 104, 419–430 (1994).
[CrossRef]

Opt. Lett.

Proc. IEEE

S. E. Harris, “Tunable optical parametric oscillators,” Proc. IEEE 57, 2096–2113 (1969).
[CrossRef]

Other

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley-Interscience, New York, 1991), Eqs. (19.1–6) and (19.1–7).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1
Fig. 1

Calculated (a) output spectra of the OPO and (b) temporal pulse shapes. For the parameters used, see text.

Fig. 2
Fig. 2

Dependence of OPO pulse parameters on round-trip time mismatch for three values of intracavity GDD compensation: (a) signal output efficiency, (b) signal pulse duration, (c) signal wavelength deviation from the phase-matching condition for 4× compensation. The insets show (b) unstable operation at the local minimum of efficiency characteristic (4×) and (c) the spectral shapes at several OPO cavity lengths.

Fig. 3
Fig. 3

Solid curve, effect of the nonlinear index of refraction on the spectral bandwidth. Dotted curve, pulse duration and inset, time–bandwidth product of the OPO signal pulse.

Fig. 4
Fig. 4

Calculated pump energies corresponding to OPO threshold at four durations of the pump pulse as a function of normalized crystal length. The equation indicates the relationship between the geometrical and the normalized crystal lengths.

Fig. 5
Fig. 5

Dependence of (a) OPO signal output efficiency and (b) signal pulse duration on the round-trip time mismatch for several durations of the pump pulse and several crystal lengths.

Fig. 6
Fig. 6

Dependence of (a) OPO signal output efficiency and (b) signal pulse duration on round-trip time mismatch for several pump pulse energies. The labels indicate by how many times the pump energy exceeds the operational threshold (Eth).

Fig. 7
Fig. 7

Dependence of (a) OPO signal output pulse duration, (b) time–bandwidth product of the signal pulse, and (c) signal output efficiency on GDD compensation for four pump pulse durations. The inset in (a) compares the dependence of pulse duration on GDD compensation for negligible and significant SPM. The inset in (b) compares the stability at two values of GDD compensation.

Fig. 8
Fig. 8

Solid curve, signal output pulse efficiency and dashed curve, pulse duration as a function of OPO cavity loss.

Fig. 9
Fig. 9

Dependence of solid curves, OPO signal output pulse efficiency and dashed curves, pulse duration on round-trip time mismatch for two cavity losses.

Equations (51)

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

2E(t, z)z2-μ2E(t, z)t2=μ2PNL(t, z)t2,
2e(ω, z)z2+k2e(ω, z)=-ω20c2pNL(t, z).
κ(Ω)=k(ω)-k0,
e(ω, z)=a(Ω, z)exp(-ik0z).
2az2+κ2a-i2k0az+iκa
=-ω20c2pNL exp(ik0z).
az+iκa=-iω20ncpNL exp(ik0z).
asz+iκsas=-iωsds4πcnsapai-iωsnNL2πcjsas,
aiz+iκiai=-iωidi4πcniapas-iωinNL2πcjiai,
apz+iκpap=-iωpdp4πcnpasai-iωpnNL2πcjpap.
jl=12πm=s,i,p(2-δlm)γmamam
ωp0=ωs0+ωi0.
as(Ω, 0)=as(Ω, L)R exp-iΩΔTM-Lvs×exp-iGDD2Ω2.
Δν1=L-1(vi-1-vs-1)-1,
Δν2=vi-1-vp-1vi-1-vs-1Δνp
Δνs[(Δν1)2+(Δν2)2]1/2.
Fouriertransformation(FT-),
f(ω)=-F(t)exp(-iωt)dt,
InverseFouriertransformation(FT+),
F(t)=12π-f(ω)exp(iωt)dω,
Convolutionintergral,
fg=-f(w)g(ω-w)dw,
Cross-correlationintegral,
fg=-f(w)g*(w-ω)dw,
dnFdtn(iω)nf,
FG12πfg,
FG*12πfg.
e(ω, z)=s(Ω, z)exp(-ikz)=a(Ω, z)exp(-ik0z).
s(Ω, z)=a(Ω, z)exp(iκz).
2sz2=2az2+κ2a+i2κaz+iκaexp(iκz).
2az2+κ2a+i2κaz+iκa0.
E(t)=E0(t)exp[i(ωt-kz)],
Pj(2)=½0χjklEkEl=0djklEkEl.
Ps(2)=0dsEpEi*ps(2)=0ds2πepei,
Pi(2)=0diEpEs*pi(2)=0di2πepes,
Pp(2)=0dpEsEipp(2)=0dp2πesei.
ps(2)=0ds2πapai exp[-i(kp0-ki0)z]=0ds2πapai exp(-iks0z),
pi(2)=0di2πapas exp[-i(kp0-ks0)z]=0di2πapas exp(-iki0z),
pp(2)=0dp2πasai exp[-i(ks0+ki0)z]=0dp2πasai exp(-ikp0z).
ks0+ki0=kp0.
ωm=ωj-ωj+ωm,
Pm(3)=180χ(3)j=s,i,p(2-δjm)EjEj*Em.
pm(3)=0nmnNLπj=s,i,p(2-δjm)γjejejem.
pm(3)=0nmnNLπj=s,i,p(2-δjm)γjajaiam exp(-ikm0z).
νp=νs+νi,
npνp=nsνs+niνi.
Δνp=Δνs+Δνi,
Δnpνp+npΔνp=Δnsνs+nsΔνs+Δniνi+niΔνi.
Δνpnp+νpΔnpΔνp=Δνsns+νsΔnsΔνs+Δνini+νiΔniΔνi.
Δνpvp=Δνsvs+Δνp-Δνsvi.
Δνs=vp-1-vi-1vs-1-vi-1Δνp.

Metrics