Synergetics pp 229-274 | Cite as

# Physical Systems

## Abstract

The laser is nowadays one of the best understood many-body problems. It is a system far from thermal equilibrium and it allows us to study cooperative effects in great detail. We take as an example the solid-state laser which consists of a set of laser-active atoms embedded in a solid state matrix (cf. Fig. 1.9). As usual, we assume that the laser end faces act as mirrors serving two purposes: They select modes in axial direction and with discrete cavity frequencies. In our model we shall treat atoms with two energy levels. In thermal equilibrium the levels are occupied according to the Boltzmann distribution function. By exciting the atoms, we create an inverted population which may be described by a negative temperature. The excited atoms now start to emit light which is eventually absorbed by the surroundings, whose temperature is much smaller than *ℏω/k* _{B} (where *ω* is the light frequency of the atomic transition and *k* _{B} is Boltzmann’s constant) so that we may put this temperature ≈ 0. From a thermodynamic point of view the laser is a system (composed of the atoms and the field) which is coupled to reservoirs at different temperatures. Thus the laser is a system far from thermal equilibrium.

## Keywords

Rayleigh Number Electric Field Strength Linear Stability Analysis Unstable Mode Saturable Absorber## Preview

Unable to display preview. Download preview PDF.

## References

## Physical Systems

- For related topics see H. Haken: Rev. Mod. Phys.
**47**, 67 (1975)MathSciNetADSCrossRefGoogle Scholar - and the articles by various authors in H. Haken, ed.: Synergetics (Teubner, Stuttgart 1973)zbMATHGoogle Scholar
- and the articles by various authors in H. Haken, M. Wagner, eds.: Cooperative Phenomena (Springer, Berlin-Heidelberg-New York 1973)zbMATHGoogle Scholar
- and the articles by various authors in H. Haken, ed.: Cooperative Effects (North Holland, Amsterdam 1974)Google Scholar
- and the articles by various authors in H. Haken (ed.): Springer Series in Synergetics Vols. 2–20 (Springer, Berlin-Heidelberg-New York)Google Scholar

## Cooperative Effects in the Laser. Self-Organization and Phase Transition

- The dramatic change of the statistical properties of laser light at laser threshold was first derived and predicted by H. Haken: Z. Phys.
**181**, 96 (1964)ADSCrossRefGoogle Scholar

## The Laser Equations in the Mode Picture

- For a detailed review on laser theory see H. Haken: In Encyclopedia of Physics, Vol. XXV/c: Laser Theory (Springer, Berlin-Heidelberg-New York 1970)Google Scholar

## The Order Parameter Concept

- Compare especially H. Haken: Rev. Mod. Phys.
**47**, 67 (1975)MathSciNetADSCrossRefGoogle Scholar

## The Single Mode Laser

- The dramatic change of the statistical properties of laser light at laser threshold was first derived and predicted by H. Haken: Z. Phys.
**181**, 96 (1964)ADSCrossRefGoogle Scholar - For a detailed review on laser theory see H. Haken: In Encyclopedia of Physics, Vol. XXV/c: Laser Theory (Springer, Berlin-Heidelberg-New York 1970)Google Scholar
- Compare especially H. Haken: Rev. Mod. Phys.
**47**, 67 (1975)MathSciNetADSCrossRefGoogle Scholar - The laser distribution function was derived by H. Risken: Z. Phys.
**186**, 85 (1965)ADSCrossRefGoogle Scholar - The laser distribution function was derived by R. D. Hempstead, M. Lax: J. Phys. Rev.
**161**, 350 (1967)ADSCrossRefGoogle Scholar - For a fully quantum mechanical distribution function cf. W. Weidlich, H. Risken, H. Haken: Z. Phys.
**201**, 396 (1967)ADSCrossRefGoogle Scholar - For a fully quantum mechanical distribution function cf. M. Scully, W. E. Lamb: Phys. Rev.
**159**, 208 (1967):ADSCrossRefGoogle Scholar - For a fully quantum mechanical distribution function cf. M. Scully, W. E. Lamb: Phys. Rev.
**166**, 246 (1968)ADSCrossRefGoogle Scholar

## The Multimode Laser

- H. Haken: Z. Phys.
**219**, 246 (1969)ADSCrossRefGoogle Scholar

## Laser with Continuously Many Modes. Analogy with Superconductivity

- For a somewhat different treatment see R. Graham, H. Haken: Z. Phys.
**237**, 31 (1970)MathSciNetADSCrossRefGoogle Scholar

## First-Order Phase Transitions of the Single Mode Laser

- J. F. Scott, M. Sargent III, C. D. Cantrell: Opt. Commun.
**15**, 13 (1975)ADSCrossRefGoogle Scholar - W. W. Chow, M. O. Scully, E. W. van Stryland: Opt. Commun.
**15**, 6 (1975)ADSCrossRefGoogle Scholar

## Hierarchy of Laser Instabilities and Ultrashort Laser Pulses

- We follow essentially H. Haken, H. Ohno: Opt. Commun.
**16**, 205 (1976)ADSCrossRefGoogle Scholar - We follow essentially H. Ohno, H. Haken: Phys. Lett.
**59A**, 261 (1976), and unpublished workADSGoogle Scholar - For a machine calculation see H. Risken, K. Nummedal: Phys. Lett.
**26A**, 275 (1968);ADSGoogle Scholar - For a machine calculation see H. Risken, K. Nummedal: J. appl. Phys.
**39**, 4662 (1968)ADSCrossRefGoogle Scholar - For a discussion of that instability see also R. Graham, H. Haken: Z. Phys.
**213**, 420 (1968)ADSCrossRefGoogle Scholar - For temporal oscillations of a single mode laser cf. K. Tomita, T. Todani, H. Kidachi: Phys. Lett.
**51A**, 483 (1975)ADSGoogle Scholar - For further synergetic effects see R. Bonifacio (ed.): Dissipative Systems in Quantum Optics, Topics Current Phys., Vol. 27 (Springer, Berlin-Heidelberg-New York 1982)Google Scholar

## Instabilities in Fluid Dynamics: The Bénard and Taylor Problems. 8.10 The Basic Equations. 8.11 Introduction of new variables. 8.12 Damped and Neutral Solutions (R ≤ Rc)

- Some monographs in hydrodynamics: L. D. Landau, E. M. Lifshitz: In Course of Theoretical Physics, Vol. 6: Fluid Mechanics (Pergamon Press, London-New York-Paris-Los Angeles 1959)Google Scholar
- Some monographs in hydrodynamics: Chia-Shun-Yih: Fluid Mechanics (McGraw Hill, New York 1969)Google Scholar
- Some monographs in hydrodynamics: G. K. Batchelor: An Introduction to Fluid Dynamics (University Press, Cambridge 1970)Google Scholar
- Some monographs in hydrodynamics: S. Chandrasekhar: Hydrodynamic and Hydromagnetic Stability (Clarendon Press, Oxford 1961)zbMATHGoogle Scholar
- Stability problems are treated particularly by Chandrasekhar l.c. and by C. C. Lin: Hydrodynamic Stability (University Press, Cambridge 1967)Google Scholar

## Solution Near R = Rc (Nonlinear Domain). Effective Langevin Equations. 8.14 The Fokker-Planck Equation and Its Stationary Solution

- We follow essentially H. Haken: Phys. Lett.
**46A**, 193 (1973)ADSGoogle Scholar - and in particular Rev. Mod. Phys. 47, 67 (1976)Google Scholar
- For related work see R. Graham: Phys. Rev. Lett.
**31**, 1479 (1973):ADSCrossRefGoogle Scholar - For related work see R. Graham: Phys. Rev.
**10**, 1762 (1974)ADSGoogle Scholar - A. Wunderlin: Thesis, Stuttgart University (1975)Google Scholar
- J. Swift, P. C. Hohenberg: Phys. Rev.
**A15**, 319 (1977)ADSGoogle Scholar - For the analysis of mode-configurations, but without fluctuations, cf. A. Schlüter, D. Lortz, F. Busse: J. Fluid Mech.
**23**, 129 (1965)MathSciNetADSzbMATHCrossRefGoogle Scholar - F. H. Busse: J. Fluid Mech.
**30**, 625 (1967)ADSzbMATHCrossRefGoogle Scholar - A. C. Newell, J. A. Whitehead: J. Fluid Mech.
**38**, 279 (1969)ADSzbMATHCrossRefGoogle Scholar - R. C. Diprima, H. Eckhaus, L. A. Segel: J. Fluid Mech.
**49**, 705 (1971)ADSzbMATHCrossRefGoogle Scholar - Higher instabilities are discussed by F. H. Busse: J. Fluid Mech.
**52**, 1, 97 (1972)ADSzbMATHCrossRefGoogle Scholar - Higher instabilities are discussed by D. Ruelle, F. Takens: Comm. Math. Phys.
**20**, 167 (1971)MathSciNetADSzbMATHCrossRefGoogle Scholar - Higher instabilities are discussed by J. B. McLaughlin, P. C. Martin: Phys. Rev.
**A12**, 186 (1975)ADSGoogle Scholar - Higher instabilities are discussed by J. Gollup, S. V. Benson: In Pattern Formation by Dynamic Systems and Pattern Recognition, (ed. by H. Haken), Springer Series in Synergetic Vol. 5 (Springer, Berlin-Heidelberg-New York 1979)Google Scholar
- where further references may be found. A review on the present status of experiments and theory give the books Fluctuations, Instabilities and Phase Transitions, ed. by T. Riste (Plenum Press, New York 1975)Google Scholar
- H. L. Swinney, J. P. Gollub (eds.): Hydrodynamic Instabilities and the Transitions to Turbulence, Topics Appl. Phys., Vol. 45 (Springer, Berlin-Heidelberg-New York 1981)Google Scholar
- For a detailed treatment of analogies between fluid and laser instabilities c.f. M. G. Velarde: In Evolution of Order and Chaos, ed. by H. Haken, Springer Series in Synergetics, Vol.17 (Springer, Berlin-Heidelberg-New York 1982) where further references may be found.Google Scholar

## A Model for the Statistical Dynamics of the Gunn Instability Near Threshold

- J. B. Gunn: Solid State Commun.
**1**, 88 (1963)ADSCrossRefGoogle Scholar - J. B. Gunn: IBM J. Res. Develop. 8, (1964)Google Scholar
- For a theoretical discussion of this and related effects see for instance H. Thomas: In Synergetics, ed. by H. Haken (Teubner, Stuttgart 1973)Google Scholar
- Here, we follow essentially K. Nakamura: J. Phys. Soc. Jap.
**38**, 46 (1975)ADSCrossRefGoogle Scholar

## Elastic Stability: Outline of Some Basic Ideas

- Introductions to this field give J. M. T. Thompson, G. W. Hunt: A General Theory of Elastic Stability (Wiley, London 1973)zbMATHGoogle Scholar
- K. Huseyin: Nonlinear Theory of Elastic Stability (Nordhoff, Leyden 1975)zbMATHGoogle Scholar