Abstract

We introduce a new mathematical method, based on the inverse spectral theory of Gel’fand and Levitan, of designing dispersive coatings for use in femtosecond lasers. We fabricated an example in AlGaAs by metal-organic chemical-vapor deposition. The mirror has a high value of group delay dispersion over a bandwidth of 10 nm, reaching an extreme of -1200 fs2 at the center (805 nm) and falling to -800 fs2 at the edge of this range. In the same band the reflectivity remains over 95%. We created the design by identifying parameters in the spectral domain, numerically optimizing these parameters, and solving the resulting inverse problem to recover the refractive-index profile. Because we performed the optimization in the spectral domain, we needed only four parameters to obtain a good result. Numerical analysis shows that the excess optical delay at the red end of the stop band can be achieved by use of a mild optical resonance, with the optical field approximately twice the magnitude of that at the blue end.

© 1997 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef] [PubMed]
  3. M. Yamashita, K. Torizuka, T. Sato, “A chirp-compensation technique using incident angle changes of cavity mirrors in a femtosecond pulse laser,” IEEE J. Quantum Electron. QE-23, 2005–2007 (1987).
    [CrossRef]
  4. W. B. Jiang, R. Mirin, J. E. Bowers, “Mode-locked GaAs vertical cavity surface emitting lasers,” Appl. Phys. Lett. 60, 677–679 (1992).
    [CrossRef]
  5. Z. Zhang, K. Torizuka, T. Itatani, K. Kobayashi, T. Sugaya, T. Nakagawa, “Self-starting mode-locked femtosecond forsterite laser with a semiconductor saturable-absorber mirror,” Opt. Lett. 22, 1006–1008 (1997).
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    [CrossRef]
  18. R. Szipöcs, A. Köházi-Kis, “Design of dielectric high reflectors for dispersion control in femtosecond lasers,” in Optical Interference Coatings, F. Abeles, ed., Proc. SPIE2253, 140–149 (1994).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]

1997

1996

1994

1993

J. E. Roman, K. A. Winick, “Waveguide grating filters for dispersion compensation and pulse compression,” IEEE J. Quantum Electron. 29, 975–982 (1993).
[CrossRef]

1992

W. B. Jiang, R. Mirin, J. E. Bowers, “Mode-locked GaAs vertical cavity surface emitting lasers,” Appl. Phys. Lett. 60, 677–679 (1992).
[CrossRef]

F. Krausz, M. E. Fermann, T. Brabec, P. F. Curley, M. Hofer, M. H. Ober, C. Spielmann, E. Wintner, A. J. Schmidt, “Femtosecond solid state lasers,” IEEE J. Quantum Electron. 28, 2097–2122 (1992).
[CrossRef]

T. Brabec, Ch. Spielmann, F. Krausz, “Limits of pulse shortening in solitary lasers,” Opt. Lett. 17, 748–750 (1992).
[CrossRef] [PubMed]

1991

1990

1989

1988

G. M. L. Gladwell, S. R. A. Dods, S. K. Chaudhuri, “Non-uniform transmission line synthesis using inverse eigenvalue analysis,” IEEE Trans. Circuits Syst. 35, 659–666 (1988).
[CrossRef]

1987

G. M. L. Gladwell, S. R. A. Dods, “Examples of reconstruction of vibrating rods from spectral data,” J. Sound Vib. 119, 267–276 (1987).
[CrossRef]

M. Yamashita, K. Torizuka, T. Sato, “A chirp-compensation technique using incident angle changes of cavity mirrors in a femtosecond pulse laser,” IEEE J. Quantum Electron. QE-23, 2005–2007 (1987).
[CrossRef]

1985

1983

S. Adachi, K. Oe, “Internal strain and photoelastic effects in GaAlAs/GaAs and InGaAsP/InP crystals,” J. Appl. Phys. 54, 6620–6627 (1983).
[CrossRef]

1982

1980

1979

A. K. Jordan, S. Ahn, “Inverse scattering theory and profile reconstruction,” Proc. Inst. Electr. Eng. 126, 945–950 (1979).
[CrossRef]

1977

L. Sossi, “On the synthesis of interference coatings,” Eesti NSV Tead. Akad. Toim. Fuus. Mat. 26, 28–36 (1977).

1974

L. Sossi, “A method for the synthesis of multilayer interference coatings,” Eesti NSV Tead. Akad. Toim. Fuus. Mat. 23, 229–237 (1974).

1967

Adachi, S.

S. Adachi, K. Oe, “Internal strain and photoelastic effects in GaAlAs/GaAs and InGaAsP/InP crystals,” J. Appl. Phys. 54, 6620–6627 (1983).
[CrossRef]

Ahn, S.

A. K. Jordan, S. Ahn, “Inverse scattering theory and profile reconstruction,” Proc. Inst. Electr. Eng. 126, 945–950 (1979).
[CrossRef]

Bendow, B.

Bowers, J. E.

W. B. Jiang, R. Mirin, J. E. Bowers, “Mode-locked GaAs vertical cavity surface emitting lasers,” Appl. Phys. Lett. 60, 677–679 (1992).
[CrossRef]

Brabec, T.

F. Krausz, M. E. Fermann, T. Brabec, P. F. Curley, M. Hofer, M. H. Ober, C. Spielmann, E. Wintner, A. J. Schmidt, “Femtosecond solid state lasers,” IEEE J. Quantum Electron. 28, 2097–2122 (1992).
[CrossRef]

T. Brabec, Ch. Spielmann, F. Krausz, “Limits of pulse shortening in solitary lasers,” Opt. Lett. 17, 748–750 (1992).
[CrossRef] [PubMed]

Cassanho, A.

Chadan, K.

K. Chadan, P. C. Sabatier, Inverse Problems in Quantum Scattering Theory (Springer-Verlag, New York, 1989).
[CrossRef]

Chaudhuri, S. K.

G. M. L. Gladwell, S. R. A. Dods, S. K. Chaudhuri, “Non-uniform transmission line synthesis using inverse eigenvalue analysis,” IEEE Trans. Circuits Syst. 35, 659–666 (1988).
[CrossRef]

Coddington, E. A.

E. A. Coddington, N. Levinson, Theory of Ordinary Differential Equations (McGraw-Hill, New York, 1965).

Courant, R.

R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1962).

Curley, P. F.

F. Krausz, M. E. Fermann, T. Brabec, P. F. Curley, M. Hofer, M. H. Ober, C. Spielmann, E. Wintner, A. J. Schmidt, “Femtosecond solid state lasers,” IEEE J. Quantum Electron. 28, 2097–2122 (1992).
[CrossRef]

Delano, E.

Dobrowolski, J. A.

Dods, S. R. A.

S. R. A. Dods, “Bragg reflection waveguide,” J. Opt. Soc. Am. A 6, 1465–1476 (1989).
[CrossRef]

G. M. L. Gladwell, S. R. A. Dods, S. K. Chaudhuri, “Non-uniform transmission line synthesis using inverse eigenvalue analysis,” IEEE Trans. Circuits Syst. 35, 659–666 (1988).
[CrossRef]

G. M. L. Gladwell, S. R. A. Dods, “Examples of reconstruction of vibrating rods from spectral data,” J. Sound Vib. 119, 267–276 (1987).
[CrossRef]

Ferencz, K.

Fermann, M. E.

F. Krausz, M. E. Fermann, T. Brabec, P. F. Curley, M. Hofer, M. H. Ober, C. Spielmann, E. Wintner, A. J. Schmidt, “Femtosecond solid state lasers,” IEEE J. Quantum Electron. 28, 2097–2122 (1992).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in fortran (Cambridge U. Press, UK, 1992).

Fluck, R.

Gladwell, G. M. L.

G. M. L. Gladwell, S. R. A. Dods, S. K. Chaudhuri, “Non-uniform transmission line synthesis using inverse eigenvalue analysis,” IEEE Trans. Circuits Syst. 35, 659–666 (1988).
[CrossRef]

G. M. L. Gladwell, S. R. A. Dods, “Examples of reconstruction of vibrating rods from spectral data,” J. Sound Vib. 119, 267–276 (1987).
[CrossRef]

Hilbert, D.

R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1962).

Hille, E.

E. Hille, Lectures on Ordinary Differential Equations (Addison–Wesley, Reading, Mass., 1969).

Hofer, M.

F. Krausz, M. E. Fermann, T. Brabec, P. F. Curley, M. Hofer, M. H. Ober, C. Spielmann, E. Wintner, A. J. Schmidt, “Femtosecond solid state lasers,” IEEE J. Quantum Electron. 28, 2097–2122 (1992).
[CrossRef]

Itatani, T.

Jenssen, H. P.

Jiang, W. B.

W. B. Jiang, R. Mirin, J. E. Bowers, “Mode-locked GaAs vertical cavity surface emitting lasers,” Appl. Phys. Lett. 60, 677–679 (1992).
[CrossRef]

Jordan, A. K.

A. K. Jordan, L. S. Tamil, “Inverse scattering theory for optical waveguides and devices: synthesis from rational and nonrational reflection coefficients,” Radio Sci. 31, 1863–1876 (1996).
[CrossRef]

A. K. Jordan, S. Lakshmanasamy, “Inverse scattering theory applied to the design of single-mode planar optical waveguides,” J. Opt. Soc. Am. A 6, 1206–1212 (1989).
[CrossRef]

A. K. Jordan, S. Ahn, “Inverse scattering theory and profile reconstruction,” Proc. Inst. Electr. Eng. 126, 945–950 (1979).
[CrossRef]

Keller, U.

Kobayashi, K.

Köházi-Kis, A.

R. Szipöcs, A. Köházi-Kis, “Design of dielectric high reflectors for dispersion control in femtosecond lasers,” in Optical Interference Coatings, F. Abeles, ed., Proc. SPIE2253, 140–149 (1994).
[CrossRef]

Kopf, D.

Krausz, F.

Lakshmanasamy, S.

Levinson, N.

E. A. Coddington, N. Levinson, Theory of Ordinary Differential Equations (McGraw-Hill, New York, 1965).

Mirin, R.

W. B. Jiang, R. Mirin, J. E. Bowers, “Mode-locked GaAs vertical cavity surface emitting lasers,” Appl. Phys. Lett. 60, 677–679 (1992).
[CrossRef]

Mogi, K.

Moser, M.

Naganuma, K.

Nakagawa, T.

Ober, M. H.

F. Krausz, M. E. Fermann, T. Brabec, P. F. Curley, M. Hofer, M. H. Ober, C. Spielmann, E. Wintner, A. J. Schmidt, “Femtosecond solid state lasers,” IEEE J. Quantum Electron. 28, 2097–2122 (1992).
[CrossRef]

Oe, K.

S. Adachi, K. Oe, “Internal strain and photoelastic effects in GaAlAs/GaAs and InGaAsP/InP crystals,” J. Appl. Phys. 54, 6620–6627 (1983).
[CrossRef]

Piotrowski, S. H. C.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in fortran (Cambridge U. Press, UK, 1992).

Roman, J. E.

J. E. Roman, K. A. Winick, “Waveguide grating filters for dispersion compensation and pulse compression,” IEEE J. Quantum Electron. 29, 975–982 (1993).
[CrossRef]

Sabatier, P. C.

K. Chadan, P. C. Sabatier, Inverse Problems in Quantum Scattering Theory (Springer-Verlag, New York, 1989).
[CrossRef]

Sato, T.

M. Yamashita, K. Torizuka, T. Sato, “A chirp-compensation technique using incident angle changes of cavity mirrors in a femtosecond pulse laser,” IEEE J. Quantum Electron. QE-23, 2005–2007 (1987).
[CrossRef]

Schmidt, A. J.

F. Krausz, M. E. Fermann, T. Brabec, P. F. Curley, M. Hofer, M. H. Ober, C. Spielmann, E. Wintner, A. J. Schmidt, “Femtosecond solid state lasers,” IEEE J. Quantum Electron. 28, 2097–2122 (1992).
[CrossRef]

Shin, S. Y.

Song, G. H.

Sorokin, E.

Sorokina, I. T.

Sossi, L.

L. Sossi, “On the synthesis of interference coatings,” Eesti NSV Tead. Akad. Toim. Fuus. Mat. 26, 28–36 (1977).

L. Sossi, “A method for the synthesis of multilayer interference coatings,” Eesti NSV Tead. Akad. Toim. Fuus. Mat. 23, 229–237 (1974).

Spielmann, C.

R. Szipöcs, K. Ferencz, C. Spielmann, F. Krausz, “Chirped multilayer coatings for broadband dispersion control in femtosecond lasers,” Opt. Lett. 19, 201–203 (1994).
[CrossRef] [PubMed]

F. Krausz, M. E. Fermann, T. Brabec, P. F. Curley, M. Hofer, M. H. Ober, C. Spielmann, E. Wintner, A. J. Schmidt, “Femtosecond solid state lasers,” IEEE J. Quantum Electron. 28, 2097–2122 (1992).
[CrossRef]

Spielmann, Ch.

Sugaya, T.

Szipöcs, R.

Tamil, L. S.

A. K. Jordan, L. S. Tamil, “Inverse scattering theory for optical waveguides and devices: synthesis from rational and nonrational reflection coefficients,” Radio Sci. 31, 1863–1876 (1996).
[CrossRef]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in fortran (Cambridge U. Press, UK, 1992).

Torizuka, K.

Verly, P. G.

P. G. Verly, “Design of inhomogeneous and quasi-inhomogeneous optical coatings at the NRC,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. SPIE2046, 36–45 (1993).
[CrossRef]

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in fortran (Cambridge U. Press, UK, 1992).

Winick, K. A.

J. E. Roman, K. A. Winick, “Waveguide grating filters for dispersion compensation and pulse compression,” IEEE J. Quantum Electron. 29, 975–982 (1993).
[CrossRef]

Wintner, E.

I. T. Sorokina, E. Sorokin, E. Wintner, A. Cassanho, H. P. Jenssen, R. Szipöcs, “Prismless passively mode-locked femtosecond Cr:LiSGaF laser,” Opt. Lett. 21, 1165–1167 (1996).
[CrossRef] [PubMed]

F. Krausz, M. E. Fermann, T. Brabec, P. F. Curley, M. Hofer, M. H. Ober, C. Spielmann, E. Wintner, A. J. Schmidt, “Femtosecond solid state lasers,” IEEE J. Quantum Electron. 28, 2097–2122 (1992).
[CrossRef]

Yamada, H.

Yamashita, M.

K. Torizuka, M. Yamashita, “Third-order dispersion in a passively mode-locked continuous-wave dye laser,” J. Opt. Soc. Am. B 8, 2442–2448 (1991).
[CrossRef]

M. Yamashita, K. Torizuka, T. Sato, “A chirp-compensation technique using incident angle changes of cavity mirrors in a femtosecond pulse laser,” IEEE J. Quantum Electron. QE-23, 2005–2007 (1987).
[CrossRef]

Yeh, P.

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

Yukon, S.

Zhang, G.

Zhang, Z.

Appl. Opt.

Appl. Phys. Lett.

W. B. Jiang, R. Mirin, J. E. Bowers, “Mode-locked GaAs vertical cavity surface emitting lasers,” Appl. Phys. Lett. 60, 677–679 (1992).
[CrossRef]

Eesti NSV Tead. Akad. Toim. Fuus. Mat.

L. Sossi, “A method for the synthesis of multilayer interference coatings,” Eesti NSV Tead. Akad. Toim. Fuus. Mat. 23, 229–237 (1974).

L. Sossi, “On the synthesis of interference coatings,” Eesti NSV Tead. Akad. Toim. Fuus. Mat. 26, 28–36 (1977).

IEEE J. Quantum Electron.

M. Yamashita, K. Torizuka, T. Sato, “A chirp-compensation technique using incident angle changes of cavity mirrors in a femtosecond pulse laser,” IEEE J. Quantum Electron. QE-23, 2005–2007 (1987).
[CrossRef]

J. E. Roman, K. A. Winick, “Waveguide grating filters for dispersion compensation and pulse compression,” IEEE J. Quantum Electron. 29, 975–982 (1993).
[CrossRef]

F. Krausz, M. E. Fermann, T. Brabec, P. F. Curley, M. Hofer, M. H. Ober, C. Spielmann, E. Wintner, A. J. Schmidt, “Femtosecond solid state lasers,” IEEE J. Quantum Electron. 28, 2097–2122 (1992).
[CrossRef]

IEEE Trans. Circuits Syst.

G. M. L. Gladwell, S. R. A. Dods, S. K. Chaudhuri, “Non-uniform transmission line synthesis using inverse eigenvalue analysis,” IEEE Trans. Circuits Syst. 35, 659–666 (1988).
[CrossRef]

J. Appl. Phys.

S. Adachi, K. Oe, “Internal strain and photoelastic effects in GaAlAs/GaAs and InGaAsP/InP crystals,” J. Appl. Phys. 54, 6620–6627 (1983).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

J. Sound Vib.

G. M. L. Gladwell, S. R. A. Dods, “Examples of reconstruction of vibrating rods from spectral data,” J. Sound Vib. 119, 267–276 (1987).
[CrossRef]

Opt. Lett.

Proc. Inst. Electr. Eng.

A. K. Jordan, S. Ahn, “Inverse scattering theory and profile reconstruction,” Proc. Inst. Electr. Eng. 126, 945–950 (1979).
[CrossRef]

Radio Sci.

A. K. Jordan, L. S. Tamil, “Inverse scattering theory for optical waveguides and devices: synthesis from rational and nonrational reflection coefficients,” Radio Sci. 31, 1863–1876 (1996).
[CrossRef]

Other

E. A. Coddington, N. Levinson, Theory of Ordinary Differential Equations (McGraw-Hill, New York, 1965).

E. Hille, Lectures on Ordinary Differential Equations (Addison–Wesley, Reading, Mass., 1969).

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in fortran (Cambridge U. Press, UK, 1992).

R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1962).

K. Chadan, P. C. Sabatier, Inverse Problems in Quantum Scattering Theory (Springer-Verlag, New York, 1989).
[CrossRef]

P. G. Verly, “Design of inhomogeneous and quasi-inhomogeneous optical coatings at the NRC,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. SPIE2046, 36–45 (1993).
[CrossRef]

R. Szipöcs, A. Köházi-Kis, “Design of dielectric high reflectors for dispersion control in femtosecond lasers,” in Optical Interference Coatings, F. Abeles, ed., Proc. SPIE2253, 140–149 (1994).
[CrossRef]

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

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Figures (10)

Fig. 1
Fig. 1

Simple multilayer structure that makes some dispersion by using a mild resonance: Λ = 805.00 nm, Λr = 815.19 nm, nc = 3.35, and Lc = 0.6027 µm.

Fig. 2
Fig. 2

Delay and dispersion spectra of the multilayer of Fig. 1. The spectral parameters chosen for this system have the values λ1 = 813.52 nm, λ2 = 784.68 nm, ρ1 = 6.82 µm, and ρ2 = 36.83 µm.

Fig. 3
Fig. 3

Delay and dispersion spectra after optimization of the spectral parameters. The spectra of the auxiliary system (Fig. 1) and the target are also shown. After optimization the spectral parameters became λ1 = 814.10 nm, λ2 = 782.04 nm, ρ1 = 4.94 µm, and ρ2 = 178.00 µm.

Fig. 4
Fig. 4

Refractive-index profile of the optimized system of Fig. 3. The direction of the z axis is switched to have the substrate at zero, which is more convenient for use in the actual growing of the crystal.

Fig. 5
Fig. 5

Aluminum alloy fraction in the AlxGa1-xAs. The fraction was calculated from Fig. 4 by use of Ref. 27, setting the wavelength to 805 nm.

Fig. 6
Fig. 6

Measured delay and reflectivity spectra of the grown sample. The target delay of -1000 fs2 is also shown.

Fig. 7
Fig. 7

Measured reflectivity and GDD of the grown sample.

Fig. 8
Fig. 8

Normalized electric-field intensity inside the multilayer. For each wavelength the field strength is normalized such that the intensity of the incident light is unity.

Fig. 9
Fig. 9

Reflectivity and delay spectra given from Eq. (B1): k1 = 2π/(0.8 µm), k2 = 3k1/5, k3 = 7k1/5, ρ1 = 0.6 µm, ρ2 = 4.2 µm, ρ3 = 7.8 µm, and n = 1.8.

Fig. 10
Fig. 10

Refractive-index profile reconstructed from the spectral data of Fig. 9.

Equations (38)

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

Δτ=2tchirped-tqw/c,
d2Edz2+k2n2zEk, z=0 for z>0,
Ek, z=exp-jkz+Γk×expjkz for z<0.
s=0zntdt.
ddsa2dEds+k2a2Ek, s=0,
Δλ4Λπn1-n2n1+n246 nm.
ρn=0a2sE2kn, sds.
ρ˜n=0a˜2sE˜2k˜n, sds.
Z˜k=Zauxk+jkn0m=1M1/ρ˜mk˜m2-k2-1/ρmkm2-k2,
Ek, 0=1, Ek, 0=0.
asEk, s=φk, s,
ãsE˜k, s=asEk, s+0sKs, tatEk, tdt.
Ks, t+Fs, t+0sKs, uFu, tdu=0,
Fs, t=m=0MasEk˜m, satEk˜m, tρ˜m-asEkm, satEkm, tρm.
E0, s=E˜0, s=1,
ãs=as+0sKs, tatdt.
τλ=fλ/n,
dτdλ=fn1-λndndλ.
nx=0.3, λ=800 nm=3.454, nx=0.3, λ=850 nm=3.415,
Ek, s=exp-jks+Γkexpjks for s0.
Ek, 0-jkEk, 0=-2jk.
fs=- Fkϕk, sdρK,
Fk=0fsa2sϕk, sds.
Ek, s=-gk, kϕk, sdρK,
JE=0a2E2-k2a2E2ds+jka20×E2k, 0-4Ek, 0,
χk1, k2=0a2sϕk1, sϕk2, sds,
ηk1, k2=0a2sϕk1, sϕk2, sds.
Fk1=-χk1, k2Fk2dρK2,
k12Fk1=-ηk1, k2Fk2dρK2.
Jg=-K-Kg2k, kdρK+jka20×-gk, kdρK2-4-gk, kdρK.
gk, k=jka202-Ek, 0K-K.
Ek, 0=2-Ek, 0jka20-dρKK-K.
Ek, 0=1+Γk.
1+Γk1-Γk=jka20-dρKK-K.
Zk=jka20-dρKK-K.
Z1k-Z2k=jka20-dρ1K-ρ2KK-K.
Zk=1+jknm=131/ρmkm2-k2,
τm=limkkm1cddkargZ-1Z+1=4ρmnc,

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