Abstract

A resonant Raman line is shown to cause a redshift of the pulse spectrum within every pass through the resonator of a mode-locked laser. The filtering action of the mirror reflectivity or any other action that confines the spectrum counterbalances the shift. The net effect is that the center frequency of the pulse spectrum deviates from the center of the filter response. The net shift caused by a set of lines is the sum of the individual shifts. The Raman shift leads to a net gain reduction and thus to a limit on the achievable pulse width. The process is similar to yet different from the well-known soliton carrier frequency shift that is due to the delayed Kerr effect in fibers.

© 1998 Optical Society of America

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References

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  1. J. P. Gordon, “The soliton self-frequency shift,” Opt. Lett. 11, 664–666 (1986).
    [CrossRef]
  2. B. E. Lemoff and C. P. J. Barty, “Generation of high-peak-power 20-fs pulses from a regeneratively initiated, self-mode-locked Ti:sapphire laser,” Opt. Lett. 19, 1367–1369 (1992).
    [CrossRef]
  3. Ch. Spielmann, P. F. Curley, T. Brabec, E. Wintner, and F. Krausz, “Generation of sub-20 fs mode-locked pulses from Ti:sapphire laser,” Electron. Lett. 28, 1532–1533 (1992).
    [CrossRef]
  4. A. Kasper and K. T. Witte, “10-fs pulse generation from a unidirectional Kerr-lens mode-locked Ti:sapphire ring laser,” Opt. Lett. 21, 360–362 (1996).
    [CrossRef] [PubMed]
  5. L. Xu, Ch. Spielmann, A. Poppe, T. Brabec, F. Krausz, and T. W. Hänsch, “Route to phase control of ultrashort light pulses,” Opt. Lett. 21, 2008–2010 (1996).
    [CrossRef] [PubMed]
  6. M. J. P. Dymott and A. I. Ferguson, “Self-mode-locked diode-pumped Cr:LiSAF laser producing 34-fs pulses at 42-mW average power,” Opt. Lett. 20, 1157–1159 (1995).
    [CrossRef] [PubMed]
  7. S. Uemura and K. Miyazaki, “Femtosecond Cr:LiSAF laser pumped by a single diode laser,” Opt. Commun. 138, 330–332 (1997).
    [CrossRef]
  8. R. Mellisch, N. P. Barry, S. C. W. Hyde, R. Jones, P. M. W. French, and J. R. Taylor, “Diode-pumped Cr:LiSAF all-solid-state femtosecond oscillator and regenerative amplifier,” Opt. Lett. 20, 2312–2314 (1995).
    [CrossRef]
  9. I. T. Sorokina, E. Sorokin, E. Wintner, A. Cassanho, H. P. Jenssen, and R. Szipöcs, “Sub-20-fs pulse generation from the mirror dispersion controlled Cr:LiSGaF and Cr:LiSAF lasers,” Appl. Phys. B (1997).
    [CrossRef]
  10. I. T. Sorokina, E. Sorokin, E. Wintner, A. Cassanho, H. P. Jenssen, and R. Szipöcs, “14-fs pulse generation from prismless KLM Cr:LiSAF and Cr:LiSGaF lasers: observation of pulse self-frequency shift,” Opt. Lett. 22, 1716 (1997).
    [CrossRef]
  11. S. P. S. Porto and R. S. Krishnan, “Raman effect of corundum,” J. Chem. Phys. 47, 1009–1012 (1967).
    [CrossRef]
  12. K. J. Blow, N. J. Doran, and D. Wood, “Suppression of the soliton-self-frequency shift by bandwidth-limited amplification,” J. Opt. Soc. Am. B 5, 1301–1304 (1988).
    [CrossRef]
  13. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), Eq. (10.26), p. 151.
  14. H. A. Haus and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron. 29, 983 (1993).
    [CrossRef]
  15. H. A. Haus and Y. Lai, “Quantum theory of soliton squeezing: a linearized approach,” J. Opt. Soc. Am. B 7, 386–392 (1990).
    [CrossRef]
  16. A. K. McQuillan, W. R. L. Clements, and B. P. Stoicheff, “Stimulated Raman emission in diamond: spectrum, gain, angular distribution of intensity,” Phys. Rev. A 1, 628–635 (1970).
    [CrossRef]
  17. W. D. Johnston, Jr., I. P. Kaminov, and J. G. Bergman, Jr., “Stimulated Raman gain coefficients for Li6NbO3, Ba2NaNb5O15, and other materials,” Appl. Phys. Lett. 13, 190–193 (1968).
    [CrossRef]

1997 (2)

1996 (2)

1995 (2)

1993 (1)

H. A. Haus and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron. 29, 983 (1993).
[CrossRef]

1992 (2)

Ch. Spielmann, P. F. Curley, T. Brabec, E. Wintner, and F. Krausz, “Generation of sub-20 fs mode-locked pulses from Ti:sapphire laser,” Electron. Lett. 28, 1532–1533 (1992).
[CrossRef]

B. E. Lemoff and C. P. J. Barty, “Generation of high-peak-power 20-fs pulses from a regeneratively initiated, self-mode-locked Ti:sapphire laser,” Opt. Lett. 19, 1367–1369 (1992).
[CrossRef]

1990 (1)

1988 (1)

1986 (1)

J. P. Gordon, “The soliton self-frequency shift,” Opt. Lett. 11, 664–666 (1986).
[CrossRef]

1970 (1)

A. K. McQuillan, W. R. L. Clements, and B. P. Stoicheff, “Stimulated Raman emission in diamond: spectrum, gain, angular distribution of intensity,” Phys. Rev. A 1, 628–635 (1970).
[CrossRef]

1968 (1)

W. D. Johnston, Jr., I. P. Kaminov, and J. G. Bergman, Jr., “Stimulated Raman gain coefficients for Li6NbO3, Ba2NaNb5O15, and other materials,” Appl. Phys. Lett. 13, 190–193 (1968).
[CrossRef]

1967 (1)

S. P. S. Porto and R. S. Krishnan, “Raman effect of corundum,” J. Chem. Phys. 47, 1009–1012 (1967).
[CrossRef]

Barry, N. P.

Barty, C. P. J.

Bergman Jr., J. G.

W. D. Johnston, Jr., I. P. Kaminov, and J. G. Bergman, Jr., “Stimulated Raman gain coefficients for Li6NbO3, Ba2NaNb5O15, and other materials,” Appl. Phys. Lett. 13, 190–193 (1968).
[CrossRef]

Blow, K. J.

Brabec, T.

L. Xu, Ch. Spielmann, A. Poppe, T. Brabec, F. Krausz, and T. W. Hänsch, “Route to phase control of ultrashort light pulses,” Opt. Lett. 21, 2008–2010 (1996).
[CrossRef] [PubMed]

Ch. Spielmann, P. F. Curley, T. Brabec, E. Wintner, and F. Krausz, “Generation of sub-20 fs mode-locked pulses from Ti:sapphire laser,” Electron. Lett. 28, 1532–1533 (1992).
[CrossRef]

Cassanho, A.

Clements, W. R. L.

A. K. McQuillan, W. R. L. Clements, and B. P. Stoicheff, “Stimulated Raman emission in diamond: spectrum, gain, angular distribution of intensity,” Phys. Rev. A 1, 628–635 (1970).
[CrossRef]

Curley, P. F.

Ch. Spielmann, P. F. Curley, T. Brabec, E. Wintner, and F. Krausz, “Generation of sub-20 fs mode-locked pulses from Ti:sapphire laser,” Electron. Lett. 28, 1532–1533 (1992).
[CrossRef]

Doran, N. J.

Dymott, M. J. P.

Ferguson, A. I.

French, P. M. W.

Gordon, J. P.

J. P. Gordon, “The soliton self-frequency shift,” Opt. Lett. 11, 664–666 (1986).
[CrossRef]

Hänsch, T. W.

Haus, H. A.

H. A. Haus and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron. 29, 983 (1993).
[CrossRef]

H. A. Haus and Y. Lai, “Quantum theory of soliton squeezing: a linearized approach,” J. Opt. Soc. Am. B 7, 386–392 (1990).
[CrossRef]

Hyde, S. C. W.

Jenssen, H. P.

Johnston Jr., W. D.

W. D. Johnston, Jr., I. P. Kaminov, and J. G. Bergman, Jr., “Stimulated Raman gain coefficients for Li6NbO3, Ba2NaNb5O15, and other materials,” Appl. Phys. Lett. 13, 190–193 (1968).
[CrossRef]

Jones, R.

Kaminov, I. P.

W. D. Johnston, Jr., I. P. Kaminov, and J. G. Bergman, Jr., “Stimulated Raman gain coefficients for Li6NbO3, Ba2NaNb5O15, and other materials,” Appl. Phys. Lett. 13, 190–193 (1968).
[CrossRef]

Kasper, A.

Krausz, F.

L. Xu, Ch. Spielmann, A. Poppe, T. Brabec, F. Krausz, and T. W. Hänsch, “Route to phase control of ultrashort light pulses,” Opt. Lett. 21, 2008–2010 (1996).
[CrossRef] [PubMed]

Ch. Spielmann, P. F. Curley, T. Brabec, E. Wintner, and F. Krausz, “Generation of sub-20 fs mode-locked pulses from Ti:sapphire laser,” Electron. Lett. 28, 1532–1533 (1992).
[CrossRef]

Krishnan, R. S.

S. P. S. Porto and R. S. Krishnan, “Raman effect of corundum,” J. Chem. Phys. 47, 1009–1012 (1967).
[CrossRef]

Lai, Y.

Lemoff, B. E.

McQuillan, A. K.

A. K. McQuillan, W. R. L. Clements, and B. P. Stoicheff, “Stimulated Raman emission in diamond: spectrum, gain, angular distribution of intensity,” Phys. Rev. A 1, 628–635 (1970).
[CrossRef]

Mecozzi, A.

H. A. Haus and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron. 29, 983 (1993).
[CrossRef]

Mellisch, R.

Miyazaki, K.

S. Uemura and K. Miyazaki, “Femtosecond Cr:LiSAF laser pumped by a single diode laser,” Opt. Commun. 138, 330–332 (1997).
[CrossRef]

Poppe, A.

Porto, S. P. S.

S. P. S. Porto and R. S. Krishnan, “Raman effect of corundum,” J. Chem. Phys. 47, 1009–1012 (1967).
[CrossRef]

Sorokin, E.

Sorokina, I. T.

Spielmann, Ch.

L. Xu, Ch. Spielmann, A. Poppe, T. Brabec, F. Krausz, and T. W. Hänsch, “Route to phase control of ultrashort light pulses,” Opt. Lett. 21, 2008–2010 (1996).
[CrossRef] [PubMed]

Ch. Spielmann, P. F. Curley, T. Brabec, E. Wintner, and F. Krausz, “Generation of sub-20 fs mode-locked pulses from Ti:sapphire laser,” Electron. Lett. 28, 1532–1533 (1992).
[CrossRef]

Stoicheff, B. P.

A. K. McQuillan, W. R. L. Clements, and B. P. Stoicheff, “Stimulated Raman emission in diamond: spectrum, gain, angular distribution of intensity,” Phys. Rev. A 1, 628–635 (1970).
[CrossRef]

Szipöcs, R.

Taylor, J. R.

Uemura, S.

S. Uemura and K. Miyazaki, “Femtosecond Cr:LiSAF laser pumped by a single diode laser,” Opt. Commun. 138, 330–332 (1997).
[CrossRef]

Wintner, E.

I. T. Sorokina, E. Sorokin, E. Wintner, A. Cassanho, H. P. Jenssen, and R. Szipöcs, “14-fs pulse generation from prismless KLM Cr:LiSAF and Cr:LiSGaF lasers: observation of pulse self-frequency shift,” Opt. Lett. 22, 1716 (1997).
[CrossRef]

Ch. Spielmann, P. F. Curley, T. Brabec, E. Wintner, and F. Krausz, “Generation of sub-20 fs mode-locked pulses from Ti:sapphire laser,” Electron. Lett. 28, 1532–1533 (1992).
[CrossRef]

Witte, K. T.

Wood, D.

Xu, L.

Appl. Phys. Lett. (1)

W. D. Johnston, Jr., I. P. Kaminov, and J. G. Bergman, Jr., “Stimulated Raman gain coefficients for Li6NbO3, Ba2NaNb5O15, and other materials,” Appl. Phys. Lett. 13, 190–193 (1968).
[CrossRef]

Electron. Lett. (1)

Ch. Spielmann, P. F. Curley, T. Brabec, E. Wintner, and F. Krausz, “Generation of sub-20 fs mode-locked pulses from Ti:sapphire laser,” Electron. Lett. 28, 1532–1533 (1992).
[CrossRef]

IEEE J. Quantum Electron. (1)

H. A. Haus and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron. 29, 983 (1993).
[CrossRef]

J. Chem. Phys. (1)

S. P. S. Porto and R. S. Krishnan, “Raman effect of corundum,” J. Chem. Phys. 47, 1009–1012 (1967).
[CrossRef]

J. Opt. Soc. Am. B (2)

Opt. Commun. (1)

S. Uemura and K. Miyazaki, “Femtosecond Cr:LiSAF laser pumped by a single diode laser,” Opt. Commun. 138, 330–332 (1997).
[CrossRef]

Opt. Lett. (7)

Phys. Rev. A (1)

A. K. McQuillan, W. R. L. Clements, and B. P. Stoicheff, “Stimulated Raman emission in diamond: spectrum, gain, angular distribution of intensity,” Phys. Rev. A 1, 628–635 (1970).
[CrossRef]

Other (2)

I. T. Sorokina, E. Sorokin, E. Wintner, A. Cassanho, H. P. Jenssen, and R. Szipöcs, “Sub-20-fs pulse generation from the mirror dispersion controlled Cr:LiSGaF and Cr:LiSAF lasers,” Appl. Phys. B (1997).
[CrossRef]

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), Eq. (10.26), p. 151.

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Figures (3)

Fig. 1
Fig. 1

Raman spectrum of corundum.10

Fig. 2
Fig. 2

Normalized frequency shift G(ωRτ)/(ωRτ)2 as a function of normalized pulse bandwidth 1/ωRτR.

Fig. 3
Fig. 3

Normalized steady-state pulse carrier frequency shift G(ωRτ) as a function of normalized pulse bandwidth 1/ωRτR.

Tables (1)

Tables Icon

Table 1 Raman Shift per One Raman Line in Ti:sapphire and Cr:LiSGaF Lasers

Equations (71)

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z As=iQ*Ap,
z Ap=iQAs.
z (|Ap|2+|As|2)=0.
2t2 Q+2τr Q+ωR2Q=μApAs*,
Q=μApAs*ωR2-2i(ωp-ωs)/τR+(ωp-ωs)2.
z As=iμ|Ap|2/As,
Q=μApAs*(ωR+ωp-ωs)(ωR-ωp+ωs)-2i(ωp-ωs)/τRμApAs*2ωR(ωR-ωp+ωs)-2iωR/τr=i(μτR/2ωR)ApAs*1+i(ωR-ωp+ωs)τR.
z As=κτR|Ap|2As1-i(ωR-ωp+ωs)τR,
z Ap=-κτR|As|2Ap1+i(ωR-ωp+ωs)τR.
z As=κτR|Ap|21+(ωR-ωp+ωs)2τR2 As.
z Ap=-κτR|As|21+(ωR-ωp+ωs)2τR2 Ap.
z As=κτR|Ap|21+(ωR-ωp+ωs)2τR2-κτR|Ap|21+(-ωR-ωp+ωs)2τR2As.
z Ap=-κτR|As|21+(ωR-ωp+ωs)2τR2-κτR|As|21+(-ωR-ωp+ωs)2τR2Ap.
f(t)=nFn exp(-iωnt),
Fn=1To -T0/2To/2f(t)exp(iωnt)dt,
1To -To/2To/2|f(t)|2dt=1To -To/2To/2 m,nFn exp(-iωnt)Fm*×exp(iωmt)dt=n|Fn|2
Fn=AoTo πτ sechπωnτ2.
F.T.{|Ao|2 sech2(t/τ)}=|Ao|2To πτ ωnτsinhπωnτ2.
Asz=|Ao|2τTo m,n×πωnτsinhπωnτ2 κτR1+(ωR-ωn+ωs)2τR2-κτR1+(-ωR-ωn+ωs)2τR2As.
Asz=|Ao|2πκτ2 π(ωs+ωR)τsinhπ(ωs+ωR)τ2-π(ωs-ωR)τsinhπ(ωs-ωR)τ2As.
ΔAs=L|Ao|2πκτ2 π(ωs+ωR)τsinhπ(ωs+ωR)τ2-π(ωs-ωR)τsinhπ(ωs-ωR)τ2As.
ΔAs(ωn)=L|Ao|2πκτ2 π(ωn+ωR)τsinhπ(ωn+ωR)τ2-π(ωn-ωR)τsinhπ(ωn+ωR)τ2 AoTo πτ sechπωnτ2.
To T a(T, t)=iD 2t2 a(T, t)+iδ|a(T, t)|2a(T, t).
ao(T, t)=Ao sechtτexp(iδ|Ao|2T/2To),
1τ2=δ2D |Ao|2.
a(T, t)=ao(T, t)+Δa(T, t).
To T Δa(T, t)=iD 2t2 Δa(T, t)+2iδ|ao(T, t)|2Δa(T, t)+iδao2(T, t)Δa*(T, t)+Δp(T, t).
Δa(T, t)=[ΔA(T)fA(t)+Δθ(T)fθ(t)+Δt(T)ft(t)+Δω(T)fω(t)]exp(iδ|Ao|2T/2To)+Δac(T, t),
fω(t)=-iAot sechtτ.
f_ω(t)=-iAoτ2 tanhtτsechtτ.
Δω(T)=Redtf_ω*(t)Δp(T, t)exp(-iδ|Ao|2T/2To).
Δω=Redtf_ω*(t)Δa(t).
ΔωR=Redtf_ω*(t)Δa(t)=Re-To/2To/2dtnF_ω*(ωn)exp(iωnt)mΔAs(ωm)×exp(-iωmt)
=To RemF_ω*(ωm)ΔAs(ωm).
F_ω(ωn)=1To dt exp(iωmt)f-ω(t)=-1To iAoτ2 dt tanhtτsechtτexp(iωmt)=1To 1Aoτ2 πτωmτ sechπωmτ2.
ΔωR=To RemF_ω*(ωm)ΔAs(ωm)=Re m1Aoτ2 πτωmτ sechπωmτ2 L|Ao|2πκτ2×π(ωn+ωR)τsinhπ(ωn+ωR)τ2-π(ωn-ωR)τsinhπ(ω-ωR)τ2 AoTo πτ sechπωnτ2=L|Ao|2πκ4 dωτωτπ(ω+ωR)τsinhπ(ω+ωR)τ2-π(ω-ωR)τsinhπ(ω-ωR)τ2π sech2πωτ2,
ΔωR=L|Ao|2πκ2×dωτωτ π(ω+ωR)τsinhπ(ω+ωR)τ2 π sech2πωτ2=L|Ao|2πκ2 G(ωRτ),
G(ωRτ)=dxx π(x+ωRτ)sinhπ(x+ωRτ)2 π sech2πx2,
G(ωRτ)=2π sechπωRτ2π tanhπωRτ2-2ωRτ-πωRτ sechπωRτ2.
ΔωR=LDπκωR22δ 1(ωRτ)2 G(ωRτ).
a(ω)+Δa(ω)=11+Δω+ωΩf2 a(ω)1-Δω+ωΩf2a(ω).
Δa(t)=-ΔωΩf2a(t)+i2 ΔωΩf2 t a(t)+1Ωf2 2t2 a(t).
Δωf=Re dtf_ω*(t)Δa(t)=Re dtf_ω*(t)2i ΔωΩf2 t Ao sechtτ=-dt 2ΔωΩf2τ3 tanh2tτsech2tτ=-43 ΔωΩf2τ2.
To ddT Δω=-To Δωτf,
1τf=43Ωf2τ2 1To.
To ddT Δω=-To Δωτf+ΔωR.
Δω=τfTo ΔωR=3Ωf2τ24 ΔωR.
Δω=3Ωf2τ24 ΔωR=3Ωf2LDπκ8δ G(ωRτ).
ΔλR=-3λ2128c Ωf2gRδωRG(ωRτ)τWAL.
z As=iμ|Ap|2AsωR2-2i(ωp-ωs)/τR+(ωp-ωs)2.
z Asiμ|Ap|2AsωR2+2i(ωp-ωs)/τRiμωR2 [1-2i(ωp-ωs)/(ωR2τR)]|Ap|2As.
ΔAs=iδ|Ap|2As.
Δa(ω)=iδdωdωdω×a*(ω)a(ω)a(ω)δ(ω+ω-ω)=iδdωdωa*(ω)a(ω)a(ω-ω+ω).
dωΔa(ω)exp(-iωt)=iδdt exp[-i(ω-ω+ω)t]a(ω-ω+ω)×dωa*(ω)exp(iωt)dω exp(-iωt)a(ω)=iδ|a(t)|2a(t).
δδ[1+i(ω-ω)τG],
τG=2/(ωR2τR).
-δdωdω(ω-ω)τGa*(ω)a(ω)
×a(ω-ω+ω).
-δdt exp[-i(ω-ω+ω)t]a(ω-ω+ω)
×dωdω(ω-ω)τGa*(ω)a(ω)
×exp[i(ω-ω)t]
=iδτG ddt |a(t)|2a(t).
Δω(T=0)=Redtf-ω*(t)iδτG ddt |a(t)|2a(t)=δ|Ao|2τGdt 2τ3 tanh2tτsech4tτ=-1615 DτGτ4.
f(x)=dkf(k)exp(ikx),
f(k)=12π dxf(x)exp(-ikx).
12π dx exp(-ikx)dxf(x+x)g*(x)=12π dx exp(-ikx)dxdk×exp[ik(x+x)]f(k)dk exp(-ikx)g*(k)=dxdkdkδ(k-k)×exp[i(k-k)x]f(k)g*(k)=dxdk exp[i(k-k)x]f(k)g*(k)=2πdkδ(k-k)f(k)g*(k)=2πf(k)g*(k).
12π dx exp(ikx)G(x)
=12π dx exp(ikx)dx π(x+x)sinh π(x+x)2×πx sech2πx2
=-4i ddk ksinh(k)sech2(k).
G(ωrτ)=dk exp(ikωRτ)G(k)=-4idk exp(ikx) ddk ksinh(k)sech2(k)=4idk exp(ikx) ksinh(k)×[ix sech2(k)-2 sinh(k)sech3(k)]=-8idk exp(ikx)k sech3(k)-4xdk exp(ikx)k sech2(k)sinh(k)=-8idk exp(ikx)k sech3(k)-4xdk exp(ikx) ksinh(k)+4xdk exp(ikx)(k)sech(k).
G(ωRτ)=2π sechπx2π tanhπx2-2x-πx sechπx2.

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