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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 10 — May. 14, 2007
  • pp: 5919–5924
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Intracavity pulse dynamics and stability for passively mode-locked lasers

Cristian Antonelli, Jeff Chen, and Franz X. Kärtner  »View Author Affiliations


Optics Express, Vol. 15, Issue 10, pp. 5919-5924 (2007)
http://dx.doi.org/10.1364/OE.15.005919


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Abstract

We derive a general characterization of the intracavity pulse dynamics for passively mode-locked fiber lasers based on the use of the variational principle. As a first application this method is used for an efficient simulation of the laser dynamics of stretched pulse and similariton lasers and evaluation of its stability.

© 2007 Optical Society of America

1. Introduction

The dynamics of many lasers can no longer be described satisfactorily by the master equation of modelocking [1

1. H. A. Haus, “Mode-Locking of Lasers,” IEEE J. Sel. Top. Quantum Electron. 6, 1173–1185 (2000). [CrossRef]

], which is based on reasonably small changes in the lasers pulse shape within one roundtrip. This is especially so for fiber lasers operating in the stretched pulse mode [2

2. H. A. Haus, K. Tamura, L. E. Nelson, and E. P. Ippen “Stretched-Pulse Additive Pulse Mode-Locking in Fiber Abstruct- Stretched-pulse Ring Lasers: Theory and Experiment,” IEEE J. Quantum Electron. 31, 591–598 (1995). [CrossRef]

] or similariton mode [3

3. V. I. Kruglov, A. C. Peacock, J. M. Dudley, and J. D. Harvey, “Self-similar propagation of high-power parabolic pulses in optical fiber amplifiers,” Opt. Lett. 25, 1753–1755 (2000). [CrossRef]

, 4

4. F. O. Ilday, J. R. Buckley, W. G. Clark, and F.W. Wise, “Self-Similar Evolution of Parabolic Pulses in a Laser,” Phys. Rev. Lett. 92, 213902 (2004). [CrossRef] [PubMed]

]. The variational method is largely used for the analysis of dispersion managed transmission systems [5

5. J. N. Kutz, P. Holmes, S. G. Evangelides, and J. P. Gordon, “Hamiltonian dynamics of dispersion-managed breathers,” J. Opt. Soc. Am. B 15, 87–96 (1998). [CrossRef]

]. It was recently applied to the analysis of fiber and solid state mode-locked lasers [6

6. C. Jirauscheck, U. Morgner, and F. X. Kärtner, “Variational analysis of spatio-temporal pulse dynamics in dispersive Kerr media,” J. Opt. Soc. Am. B 19, 1716–1721 (2002). [CrossRef]

, 7

7. C. Jirauscheck, U. Morgner, and F. X. Kärtner, “Spatiotemporal Gaussian pulse dynamics in Kerr-lens mode-locked lasers,” J. Opt. Soc. Am. B 20, 1356–1368 (2003). [CrossRef]

, 8

8. C. Jirauscheck and F. X. Kärtner, “Gaussian pulse dynamics in gain media with Kerr nonlinearity,” J. Opt. Soc. Am. B 23, 1776–1784 (2006). [CrossRef]

, 9

9. J- G. Caputo, N. Flytzanis, and M. P. Sorensen, “Ring laser configuration studied by collective coordinates,” J. Opt. Soc. Am. B 12, 139–145 (1995). [CrossRef]

, 10

10. S. Waiyapot and M. Matsumoto, “Jitter and time stability of an actively mode-locked dispersion-managed fiber laser,” Opt. Commun. 188, 167–180 (2001). [CrossRef]

]. To gain insight into dynamics and pulse stability it seems a natural alternative to time consuming full numerical simulations. In this paper we show that this method provides a general framework for a semi-analytical characterization of the intra-cavity pulse dynamics of passively mode-locked lasers with large pulse shaping per roundtrip and their linear stability, and apply it to the two relevant regimes of stretched pulse [2

2. H. A. Haus, K. Tamura, L. E. Nelson, and E. P. Ippen “Stretched-Pulse Additive Pulse Mode-Locking in Fiber Abstruct- Stretched-pulse Ring Lasers: Theory and Experiment,” IEEE J. Quantum Electron. 31, 591–598 (1995). [CrossRef]

] and similariton laser operation [4

4. F. O. Ilday, J. R. Buckley, W. G. Clark, and F.W. Wise, “Self-Similar Evolution of Parabolic Pulses in a Laser,” Phys. Rev. Lett. 92, 213902 (2004). [CrossRef] [PubMed]

].

2. Theory and system modeling

Pulse propagation is described by the extended nonlinear Schrodinger equation (NLSE):

uz=gs(z)u+gs(z)Ωg22ut2+iβ(z)22ut2u2u.
(1)

where u = u(z,t) is the complex pulse envelope and t is the time in the co-moving frame. We denote by β (z) the group velocity dispersion, γ(z) is the self phase modulation coefficient and gs(z) is the saturated gain coefficient given by gs(z) = g 0(z)/[1+E(z)/E sat], where g 0 (z) is the small signal gain coefficient, E(z) is the pulse energy and E sat is the gain saturation energy. We assume a parabolic spectral gain profile, (z, ω) ≃ gs(z) [1 - (ωg)2]1, although a similar analysis could be performed for a generic gain profile [8

8. C. Jirauscheck and F. X. Kärtner, “Gaussian pulse dynamics in gain media with Kerr nonlinearity,” J. Opt. Soc. Am. B 23, 1776–1784 (2006). [CrossRef]

]. Due to the heterogeneity of the resonator, each parameter of the NLSE is periodically dependent on z in a step-wise fashion. The conservative part of the NLSE can be derived from the Lagrangian [5

5. J. N. Kutz, P. Holmes, S. G. Evangelides, and J. P. Gordon, “Hamiltonian dynamics of dispersion-managed breathers,” J. Opt. Soc. Am. B 15, 87–96 (1998). [CrossRef]

] ∫+∞ -∞ = ℒ(u,uz;u *,u * z)dt, with the lagrangian density ℒ(u,uz;u *,u * z) defined as follows:

(u,uz;u*,uz*)=i2(uu*zu*uz)+β2ut2+γ2u4,
(2)

where subscript z denotes differentiation with respect to z. The experimental evidence of a linear chirp [2

2. H. A. Haus, K. Tamura, L. E. Nelson, and E. P. Ippen “Stretched-Pulse Additive Pulse Mode-Locking in Fiber Abstruct- Stretched-pulse Ring Lasers: Theory and Experiment,” IEEE J. Quantum Electron. 31, 591–598 (1995). [CrossRef]

, 4

4. F. O. Ilday, J. R. Buckley, W. G. Clark, and F.W. Wise, “Self-Similar Evolution of Parabolic Pulses in a Laser,” Phys. Rev. Lett. 92, 213902 (2004). [CrossRef] [PubMed]

] in both stretched and similariton lasers suggests adopting the following general ansatz in order to evaluate the evolution of its parameters in the fiber ring:

u=Eτf(tt0τ)exp[iρ2(tt0)2+iΩt+],
(3)

Ik=skf2(s)ds
(4)
Jk=λkf˜(λ)2dλ2π.
(5)

Defining also P 4 = ∫+∞ -∞ f 4 (s)ds, it is straightforward algebra to calculate the Lagrangian for the ansatz (3):

L=E[I22τ2dρdz+dϕdz+t0dΩdz]+γP42E2τ+β2[J2τ2+I2τ2ρ2+Ω2]E,
(6)

which expectedly depends only on the moments I 2, J 2 and P 4. The equations of motion for the pulse parameters can be derived by applying the Euler-Lagrange equations, appropriately modified to include the non-conservative terms [9

9. J- G. Caputo, N. Flytzanis, and M. P. Sorensen, “Ring laser configuration studied by collective coordinates,” J. Opt. Soc. Am. B 12, 139–145 (1995). [CrossRef]

]:

LxddzLxz=2Im{+gs(u+1Ωg22ut2)u*xdt},
(7)

dEdz=2gs(z)E2gs(z)Ω2+2Ωg2E,
(8)
dτdz=β(z)ρτ+(1S+I2J2I2I4I22I2ρ2τ4)gs(z)τΩg2,
(9)
dρdz=β(z)(ρ2J2I2τ4)gs(z)Ωg2ρτ24S1I2γP42I2Eτ3,
(10)
dΩdz=4gs(z)2Ωg2Ω
(11)
dt0dz=2[β(z)2+2I2ρτ2gs(z)Ωg2]Ω
(12)

where we have made use of the additional definition S=+s2f2ds=+λ2f˜2dλ2π..

The equations of motion have useful insights. From Eq. (8) one can see that the energy is increased by the saturated gain and decreased by the gain filtering in proportion to the mean square bandwidth and to the square frequency deviation. Note that this equation could be directly derived from the NLSE multiplying both sides by u * and integrating the real part [11

11. M. Horowitz and C. R. Menyuk, “Analysis of pulse dropout in harmonically mode-locked fiber lasers by use of the Lyapunov method,” Opt. Lett. 25, 40–42 (2000). [CrossRef]

]. Equation (9) shows that the gain filtering contributes to the change in pulse width depending on the sign of the quantity 1 - S + I 2 J 2. In fact, for 1 - S + I 2 J 2 < 0 the pulse width is reduced by gain filtering, whereas for 1 - S + I 2 J 2 > 0 there is a critical value of the chirp, βC = √(1 - S + I 2 J 2) / (I 4 -I 2 2) such that for ∣β∣ < βC the gain filtering broadens the pulse. This value discriminates the two different ways the gain filtering influence the pulse width, namely increasing the pulse width when filtering for β < βC and reducing the pulse width by reducing the chirp for β > βC (for the Gaussian ansatz, for instance, βC = 1). From Eq. (11) one can see immediately that the frequency offset is forced to vanish by a permanent restoring force; in consequence of this the derivative of the time shift Eq. (12) also vanishes and t 0 tends to a constant value that can be set to zero without loss of generality. In what follows we can safely set Ω = 0 and t 0 = 0.2

Fig. 1. (a)Schematic diagram for the stretched pulse laser. (b) Intracavity pulse dynamics: (i) Energy, (ii) rms pulsewitdth and (iii) bandwidth. Solid lines are the solution to Eqs. (8)-(10); dashed lines refer to full numerical simulations.
A+A=(1l01+f02A2Psat)
(13)
τ2τ+2=1+2l0f02A2Psat(1+f02A2Psat)(1+f02A2Psatl0),
(14)

where f 0 = f(0) and we have denoted by the subscript “+” the parameters of the transmitted pulse u +(t). The energy of the transmitted pulse is obtained by using E + = A 2 +τ+ along with Eqs. (13), (14), whereas the parameter ρ is not affected by the saturable absorber.

3. Stretched pulse laser and similariton laser: intracavity pulse dynamics

In order to apply our analysis to the stretched pulse laser and to the similariton laser we need to choose an appropriate ansatz. While the choice seems to be mandatorily gaussian for the first case, self-similar pulses are of slightly different shape [3

3. V. I. Kruglov, A. C. Peacock, J. M. Dudley, and J. D. Harvey, “Self-similar propagation of high-power parabolic pulses in optical fiber amplifiers,” Opt. Lett. 25, 1753–1755 (2000). [CrossRef]

, 12

12. C. Jirauschek and F. ö. Ilday, “Theory of the Self-Similar Laser Oscillator,” CLEO 2005, Paper JWB65 (2005).

]. Nevertheless we use also for the similariton laser a gaussian pulse shape, having found no significant changes occurring by different choices. The design of the cavity for the stretched pulse laser is based on ref. [13

13. K. Tamura, E. P. Ippen, H. A. Haus, and L. Nelson, “77-fs pulse generation from a stretched-pulse mode-locked all-fiber ring laser,” Opt. Lett. 18, 1080–1082 (1993). [CrossRef] [PubMed]

], whereas for the similariton laser is based on ref. [4

4. F. O. Ilday, J. R. Buckley, W. G. Clark, and F.W. Wise, “Self-Similar Evolution of Parabolic Pulses in a Laser,” Phys. Rev. Lett. 92, 213902 (2004). [CrossRef] [PubMed]

]. The steady state values of the pulse parameters at any location in the resonator are found by using the shooting method [14

14. Introduction to numerical analysis, ser. Texts in Applied Mathematics, vol. 12, Springer (2002).

] and then are assumed as the initial condition to integrate the equations for one roundtrip. The results are shown in Figs. 1 and 2, where we have plotted the evolution of the energy, the root mean square pulse width and bandwidth versus the normalized position in the ring. We have compared the results of the variational analysis to the results of full numerical simulations based on the split step method, which are reported by thick dashed lines. The evident agreement with the numerical simulations confirms the validity of this approach. The small deviation in terms of pulse width and bandwidth in Fig. 2 seems to be due to the fact that the shape of the similariton is also evolving in the cavity, but this feature is not included in the present analysis . An estimate of the maximum energy in the cavity, E max, can also be obtained by neglecting the term accounting for gain filtering in the integration of Eq. (8). This is allowed by the fact that the energy dynamics is dominated by gain saturation [11

11. M. Horowitz and C. R. Menyuk, “Analysis of pulse dropout in harmonically mode-locked fiber lasers by use of the Lyapunov method,” Opt. Lett. 25, 40–42 (2000). [CrossRef]

]. Neglecting also the change in pulse width at the saturable absorber, we obtain E maxEL [2g 0 Lg + log(ΓSA(1 -l 0)2)] /[1 - ΓSA(1 - l 0)2]. For the systems of Figs. 1 and 2 this gives E max ≃ 0.36nJ and E max ≃ 34nJ respectively, which are very close to the real values.

Fig. 2. Same as fig. 1 for the similariton laser oscillator.

4. Linear stability analysis

As an application of the theory, we study the stability of the two systems under scrutiny against perturbations and against the growth of cw radiation. In the first regard, solving the equations of motion over many map periods and for different initial conditions (which requires a few seconds on a personal computer) insures this kind of stability. We may also quantify the stability by computing the Floquet coefficients [15

15. Advanced Synergetics, ser. Springer Series, vol. 20, Springer-Verlag (1983).

] of the system. To do so we linearize the equations of motion (8-10) around the steady state solutions and obtain a set of linear equations with periodically varying coefficients:

dΔxdz=[A(z)+BLnδ(zzLnLt)]Δx
(15)

where Δx is the perturbation column vector, Δx = (ΔE, Δτ, Δρ)T, A(z) is a 3 × 3 periodic matrix with period L t and the matrix B L accounts for the lumped contribution of the saturable absorber and also the grating pair in the similariton laser. The matrix Q(z) of the independent solutions of (15) can be written as Q(z) = U(z)exp(λz), being U(z) a periodic matrix with period L t. The diagonal matrix λ contains the Floquet coefficients λi, i= 1,2,3: their real parts determine the decay rate of the perturbations, whereas their imaginary parts may account for relaxation oscillation type of behavior. They can be evaluated by using that Q(Lt) = exp(λLt) for Q(0) = 1. The Floquet coefficient with the smallest absolute real part determines the stability of the perturbations. The Floquet coefficient for the frequency shift, which plays a key role in the analysis of the timing jitter, can be evaluated explicitely linearizing Eq. (11): λΩ = - 4 ∫0 Lt gs(z)ℬ2(z)/Ωg 2dz/L t. With regard to the stability against cw radiation, we assume as a stability criterion that the net (power) gain for the cw radiation must be less than unit for the laser to be stable (this criterion is equivalent to the one assumed in ref. [6]):

Fig. 3. Solid lines are the boundary of the stable region, whereas dashed lines correspond to the stable cw gain G cw = 0.9 for (a) stretched pulse laser and (b) similariton laser. Corresponding, insets show the negative real part of the Floquet coefficients λi (solid lines) and λΩ (dot-dashed lines) normalized to the total fiber length L t.
Gcw=exp[20Lggs(z)dz](1l0)2ΓSA<1,
(16)

where gs(z) is determined by the steady state pulse and Lg is the length of the gain fiber. The gain for cw radiation is sensitive to the modulation depth of the saturable absorber and to the gain bandwidth. The results of the analysis of such dependence are reported in Fig. 3, where we have plotted the boundary of the stable region in the plane (l 0g/2π) for (a) the stretched pulse laser and for (b) the similariton laser. In the insets, for the values of l 0 (and Ωg) corresponding to G cw = 0.9, we have plotted the negative real part of the Floquet coefficients λi (solid lines, only two because two coefficients are complex conjugate) and λ Ω (dot-dashed lines), normalized to the ring length L t. The plots show that, for the range of parameters under scrutiny, the two systems manifest the same robustness against perturbations (which is set by the negative real part of the Floquet coefficient closest to zero), despite the significantly higher energy of the similariton laser, which, interestingly, appears to be even more robust against frequency fluctuations.

5. Conclusion

In conclusion, we have developed a variational theory to study the dynamics and stability of passively mode-locked lasers that show large temporal and spectral changes in the pulse within one roudtrip of the cavity. The theory is used to study the dynamics of the perturbations and the stability of these systems against the growth of cw radiation.

Acknowledgement

This research has been supported by ONR N00014-02-1-0717 and AFOSR FA9550-04-1-0011. C. Antonelli acknowledges fellowship support from “Fondazione Ferdinando Filauro.” The authors thank A. Mecozzi and F. Ö. Ilday for invaluable discussions.

Footnotes

1 It is worth reminding that the quadratic term in the frequency domain is equivalent to second derivative in the time domain, ω 2 2/∂t 2.
2 The equation for the phase, inessential to this paper, is: dϕdz=gsΩg2[(4S1)ρ+42t0Ω]β(J2τ2+Ω22)5γP44Eτ

References and links

1.

H. A. Haus, “Mode-Locking of Lasers,” IEEE J. Sel. Top. Quantum Electron. 6, 1173–1185 (2000). [CrossRef]

2.

H. A. Haus, K. Tamura, L. E. Nelson, and E. P. Ippen “Stretched-Pulse Additive Pulse Mode-Locking in Fiber Abstruct- Stretched-pulse Ring Lasers: Theory and Experiment,” IEEE J. Quantum Electron. 31, 591–598 (1995). [CrossRef]

3.

V. I. Kruglov, A. C. Peacock, J. M. Dudley, and J. D. Harvey, “Self-similar propagation of high-power parabolic pulses in optical fiber amplifiers,” Opt. Lett. 25, 1753–1755 (2000). [CrossRef]

4.

F. O. Ilday, J. R. Buckley, W. G. Clark, and F.W. Wise, “Self-Similar Evolution of Parabolic Pulses in a Laser,” Phys. Rev. Lett. 92, 213902 (2004). [CrossRef] [PubMed]

5.

J. N. Kutz, P. Holmes, S. G. Evangelides, and J. P. Gordon, “Hamiltonian dynamics of dispersion-managed breathers,” J. Opt. Soc. Am. B 15, 87–96 (1998). [CrossRef]

6.

C. Jirauscheck, U. Morgner, and F. X. Kärtner, “Variational analysis of spatio-temporal pulse dynamics in dispersive Kerr media,” J. Opt. Soc. Am. B 19, 1716–1721 (2002). [CrossRef]

7.

C. Jirauscheck, U. Morgner, and F. X. Kärtner, “Spatiotemporal Gaussian pulse dynamics in Kerr-lens mode-locked lasers,” J. Opt. Soc. Am. B 20, 1356–1368 (2003). [CrossRef]

8.

C. Jirauscheck and F. X. Kärtner, “Gaussian pulse dynamics in gain media with Kerr nonlinearity,” J. Opt. Soc. Am. B 23, 1776–1784 (2006). [CrossRef]

9.

J- G. Caputo, N. Flytzanis, and M. P. Sorensen, “Ring laser configuration studied by collective coordinates,” J. Opt. Soc. Am. B 12, 139–145 (1995). [CrossRef]

10.

S. Waiyapot and M. Matsumoto, “Jitter and time stability of an actively mode-locked dispersion-managed fiber laser,” Opt. Commun. 188, 167–180 (2001). [CrossRef]

11.

M. Horowitz and C. R. Menyuk, “Analysis of pulse dropout in harmonically mode-locked fiber lasers by use of the Lyapunov method,” Opt. Lett. 25, 40–42 (2000). [CrossRef]

12.

C. Jirauschek and F. ö. Ilday, “Theory of the Self-Similar Laser Oscillator,” CLEO 2005, Paper JWB65 (2005).

13.

K. Tamura, E. P. Ippen, H. A. Haus, and L. Nelson, “77-fs pulse generation from a stretched-pulse mode-locked all-fiber ring laser,” Opt. Lett. 18, 1080–1082 (1993). [CrossRef] [PubMed]

14.

Introduction to numerical analysis, ser. Texts in Applied Mathematics, vol. 12, Springer (2002).

15.

Advanced Synergetics, ser. Springer Series, vol. 20, Springer-Verlag (1983).

OCIS Codes
(140.3510) Lasers and laser optics : Lasers, fiber
(140.4050) Lasers and laser optics : Mode-locked lasers

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: February 23, 2007
Revised Manuscript: April 23, 2007
Manuscript Accepted: April 25, 2007
Published: April 30, 2007

Citation
Cristian Antonelli, Jeff Chen, and Franz X. Kartner, "Intracavity pulse dynamics and stability for passively mode-locked lasers," Opt. Express 15, 5919-5924 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-10-5919


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References

  1. H. A. Haus, "Mode-Locking of Lasers," IEEE J. Sel. Top. Quantum Electron. 6, 1173-1185 (2000). [CrossRef]
  2. H. A. Haus, K. Tamura, L. E. Nelson and E. P. Ippen "Stretched-Pulse Additive Pulse Mode-Locking in Fiber Abstruct- Stretched-pulse Ring Lasers: Theory and Experiment," IEEE J. QuantumElectron. 31, 591-598 (1995). [CrossRef]
  3. V. I. Kruglov, A. C. Peacock, J. M. Dudley and J. D. Harvey, "Self-similar propagation of high-power parabolic pulses in optical fiber amplifiers, "Opt. Lett. 25, 1753-1755 (2000). [CrossRef]
  4. F. Ö. Ilday, J. R. Buckley, W. G. Clark and F.W. Wise, "Self-Similar Evolution of Parabolic Pulses in a Laser," Phys. Rev. Lett. 92, 213902 (2004). [CrossRef] [PubMed]
  5. J. N. Kutz, P. Holmes, S. G. Evangelides, J. P. Gordon, " Hamiltonian dynamics of dispersion-managed breathers," J. Opt. Soc. Am. B 15, 87-96 (1998). [CrossRef]
  6. C. Jirauscheck, U. Morgner and F. X. Kärtner, "Variational analysis of spatio-temporal pulse dynamics in dispersive Kerr media," J. Opt. Soc. Am. B 19, 1716-1721 (2002). [CrossRef]
  7. C. Jirauscheck, U. Morgner and F. X. Kärtner, "Spatiotemporal Gaussian pulse dynamics in Kerr-lens modelocked lasers," J. Opt. Soc. Am. B 20, 1356-1368 (2003). [CrossRef]
  8. C. Jirauscheck and F. X. Kärtner, "Gaussian pulse dynamics in gain media with Kerr nonlinearity," J. Opt. Soc. Am. B 23, 1776-1784 (2006). [CrossRef]
  9. J- G. Caputo, N. Flytzanis, M. P. Sorensen, "Ring laser configuration studied by collective coordinates," J. Opt. Soc. Am. B 12, 139-145 (1995). [CrossRef]
  10. S. Waiyapot and M. Matsumoto, "Jitter and time stability of an actively mode-locked dispersion-managed fiber laser," Opt. Commun. 188, 167-180 (2001). [CrossRef]
  11. M. Horowitz and C. R. Menyuk, "Analysis of pulse dropout in harmonically mode-locked fiber lasers by use of the Lyapunov method," Opt. Lett. 25, 40-42 (2000). [CrossRef]
  12. C. Jirauschek and F. Ö. Ilday, "Theory of the Self-Similar Laser Oscillator," CLEO 2005, Paper JWB65 (2005).
  13. K. Tamura, E. P. Ippen, H. A. Haus and L. Nelson, "77-fs pulse generation from a stretched-pulse mode-locked all-fiber ring laser," Opt. Lett. 18, 1080-1082 (1993). [CrossRef] [PubMed]
  14. Introduction to numerical analysis, ser. Texts in Applied Mathematics, vol. 12, Springer (2002).
  15. Advanced Synergetics, ser. Springer Series, vol. 20, Springer-Verlag (1983).

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