Quantum theory of noise in gas and solid state lasers with an inhomogeneously broadened line. I

We present a quantum mechanical nonlinear treatment of the phase and amplitude flucutations of gas lasers, i.e. lasers with moving atoms, and of solid state lasers with an inhomogeneously broadened line. The atoms may possess an arbitrary number of levels. As in our preceding papers the noise due to the pump, incoherent decay, lattice vibrations or atomic collisions, as well as due to the thermal and zero point fluctuations of the cavity is completely taken into account. The linewidth (due to phase diffusion), and the intensity fluctuations (due to amplitude noise) are essentially expressed by the threshold inversion, the unsaturated inversion and the saturated population numbers of the two atomic levels, which support the laser modes. Our results apply to the whole threshold region and above up to essentially the same photon number, to which the previous semiclassical theories of inhomogeneously broadened lasers were applicable. For the example of a two-level system we also demonstrate the application of a new technique which allows us to eliminate rigorously the atomic variables (operators), yielding a set of nonlinear coupled equations for the lightfield operators alone. If the elimination procedure is carried out only partially and additional approximations are made, we find essentially the rate equations ofMcCumber, in a form derived byLax. When we neglect noise, the nonlinear equation may be solved exactly in the case of single mode operation. By a suitable expansion of the exact multimode equations we find a convenient set of equations, which reduce in the noiseless case to those derived and used previously byHaken andSauermann as well asLamb.     

Über die Temperaturabhängigkeit der Lichtabsorption durch Excitonen

The temperature dependence of the absorption of light by excitons is considered. The eigenfunctions of the excitons are those of the vibrating crystal, introduced byTjablikow, Lee, Low, andPines a. o. The model can only be used if the coupling between excitons and crystal vibrations is week. The results are expressions for the production of excitons and simultaneous absorption or emission of phonons.     

A criticism of the standard treatment of phonon drag

By the study of a simple example, namely the evolution in timet of an electron-phonon system with fixed, total momentum, it is shown that the “standard” treatment of “phonon drag”, which involves solving the (linearized and spatially homogeneous) coupled electron and phonon Boltzmann equations by an iteration procedure, is not always correct. In the asymptotic limit (t→∞), the iteration or “standard” procedure does not give the “correct” (i.e. the equilibrium statistical mechanical) result for the distribution of momentum between electrons and phonons. However, a proper treatment of the Boltzmann equations does lead to the “correct” sharing of momentum between electrons and phonons fort→∞. All the calculations in this paper are performed for metals at high temperatures (i.e.,T>Θ_D, the Debye temperature).     

Zur Deutung der Struktur der Bremsstrahl-Isochromate des Wolframs an der kurzwelligen Grenze

Ohlin's maximum of the tungsten isochromat may well be explained on the basis of the energy bands of tungsten calculated byManning andChodorow. The photons near the threshold voltage are then assumed to arise from transitions of the primary electrons to the unoccupied levels of the 5d and 6s bands. Using this explanation one has to drop, however, a selection rule formulated byNijboer, admitting only transitions to final s states. The validity of this ?s selection rule“ is critically examined. For a strict treatment one has to calculate the matrix elements connected with the transitions, using the exact wave functions of the tungsten crystal. These, however, are not known sufficiently. ThereforeNijboer appliedNedelsky's results for free-free transitions at the isolated atom to the transitions to bound states of the solid. Yet according to calculations made recently byGuggenberer the s selection rule is put into question even for free-free transitions. — A better approximation is obtained considering electron transitions from the continuum to the 5d and 6s states of the isolated atom. A calculation of the corresponding matrix elements is carried out. It shows that both kinds of transitions are of about the same probability at energies around 1 keV which are of special concern to us. Therefore it is allowed to use the 5d band for explaining the tungsten isochromat.     

On transport theory of Bose systems. I

The concept of thermodynamical correlation functions for non-equilibrium states used byKadanoff andBaym in transport theory is generalized by means ofBogoliubov's quasi-averages in order to describe transport phenomena in systems with Bose-Einstein condensation. The one-particle Green's function is decomposed in one part describing the condensed particles and a second one describing the elementary excitations. The coupled equations of motion for these functions are derived and some consequences for the case of slowly varying disturbances are discussed.     

Untersuchungen zur Mehrfach-Emission von Sekundärelektronen

CuBe- and NaCl-targets are bombarded by single electrons (100–600 eV). The secondary electrons accelerated by 40 kV strike the crystal of a scintillation counter, backed by a multichannel analyser. The probabilityP _n of emission ofn=0, 1, 2, 3, ... secondaries can be found from the pulse height distribution. The probability distributionP _n =f(n) shows a characteristic deviation from aPoisson's distribution. There was no evidence that there is a preference for even numbers ofn as found byBarrington andAnderson.     

Quantenmechanische Behandlung des optischen Masers

In the present paper we give a fully quantummechanical treatment of the self-sustained oscillation of one mode in solid-state lasers. The total laser system consists of various subsystems: The lasing mode is coupled to the atoms of the active material and to a loss mechanism. It is assumed to be in complete resonance with the homogeneously broadened atomic transition. The pump of the active atoms, which are assumed to have only two levels, is brought about by their interaction with a large system of negative temperature. The active atoms decay not only by induced and spontaneous emission into the lasing mode, but also by spontaneous emission into the continuum of nonlasing modes (and possibly by nonradiative transitions). This process is fully taken into account. The pumping process and the spontaneous emission into the continuum of nonlasing modes are treated as in a preceding paper. There we have shown that the coordinates of these fields can be eliminated in some sense and give rise to a mean dissipative motion of the atoms and to fluctuations. Using the Heisenberg picture we obtain a system of coupled nonlinear equations of motion for the atomic operators and for the creation operator of the oscillating mode. We then eliminate the atomic operators by the iteration procedure of the semiclassical laser theory. This leaves us with a nonlinear differential equation of the van-der-Pol type for the creation operator of the laser mode, which contains the fluctuations of the pumping process, the spontaneous emission into the continuum and the loss mechanism as inhomogeneities of operator character. Such an operator equation has previously been obtained byHaken, who has shown, that in the neighbourhood of the stationary saturated level of oscillation the amplitude is highly stabilized, whereas the phase undergoes an undamped diffusion process. This process takes the phase in the course of time arbitrarily far from any given initial value. We useHaken's method of solution and demonstrate that the correct commutation rules for the oscillating mode [b(t),b ⁺(t)]=1 are preserved for all times. Besides these quantum mechanical properties our solution contains all the well known results of the semiclassical theory. Our main result is the expression for the linewidth, which is caused by phase diffusion. The half width at half maximum power is in circular frequencies given by

$$\Delta \omega = \frac{{\hbar \omega }}{P}\kappa ^2 \left( {n_{TH} + \frac{1}{2} + \frac{1}{{2\sigma _k }}} \right) = \frac{{\hbar \omega }}{P}\kappa ^2 \left( {n_{TH} + \frac{1}{2} + n_{SP} } \right).$$     

Gruppentheoretische Behandlung der Aufspaltung von Kristalltermen vermöge der Wechselwirkung äquivalenter Gitterteilchen

First an analysis of the Hamiltonians related toBethe's crystal levels and to the energy levels of the whole crystal is given. Then the splitting ofBethe's levels by the interaction between the electrons of equivalent lattice particles is described in the group ring (factor group splitting,Davydov splitting). By thisWinston's rule derived for this splitting is deduced from a new group theoretical point of view. FinallyBethe's levels degenerated according to time reversal symmetry are considered and spin is regarded.     

Quantum mechanical solutions of the laser masterequation

We treat a laser consisting of one mode described by a running wave and a set of atoms with two optically active levels which are homogeneously broadened. We start from the laser density matrix equations ofWeidlich andHaake and define a distribution functionf for lightfield and atomic variables, where we use for the lightfield the coherent state representation and for the atomic system a modified version of the distribution function used bySchmid andRisken in a previous paper. We derive a partial differential equation forf which is completely exact and is of the type of a generalized Fokker-Planck equation, i.e. it contains higher derivatives. Using a recently stated theorem ofHaken andWeidlich we show that this distribution function allows to calculate single-time as well as multitime quantum mechanical correlation functions. If the leading terms of the generalized Fokker-Planck equation are retained we find the semiclassical Fokker-Planck equation ofRisken,Schmid andWeidlich. Our treatment can be extended to several modes connected with standing waves and multilevel atoms.     

A nonlinear theory of laser noise and coherence. I

We consider the interaction of a set of atoms at random lattice sites with a decaying resonator mode. The optical transition is supposed to possess a homogeneously broadened Lorentzian line. The pumping is taken into account explicitly as a stochastic process. After elimination of the atomic coordinates a second order nonlinear differential equation for the light amplitude is found. In between excitation collisions this equation can be solved exactly if the resonator width is large as compared to all other frequency differences. In contrast to linear theories there exists a marked threshold. Below it the amplitude decreases after each excitation exponentially and the linewidth turns out to be identical with those of previous authors (for instanceWagner andBirnbaum), if specialized to large cavity width. Above the threshold the light amplitude converges towards a stable value, whereas the phase undergoes some kind of undamped diffusion process. We then consider the general case with arbitrary cavity width. If the general equation of motion of the light amplitude is interpreted as that of a particle moving in two dimensions, it becomes clear that also in this case the amplitude oscillates above threshold around a stable value which is identical with that determined in previous papers byHaken andSauermann neglecting laser noise. This stable value may, however, undergo shifts, if there are slow systematic changes of the cavity width, inversion etc. On the other hand the phase still fluctuates in an undamped way. After splitting off the phase factor the equations can be linearized and solved explicitly. With these solutions simple examples of correlation functions are calculated in a semiclassical way, thus yielding expressions for the line width above threshold. The results can also be used to evaluate from first principles correlation functions for different laser beams. As an example the complex degree of mutual coherence of two laser beams is determined. It vanishes if one of the lasers is still below threshold and its value is close to unity well above threshold for observation times small compared to the inverse laser linewidth.     

Theory of noise in semiconductor laser emission

For the complex light field amplitude of a semiconductor laser an equation of a generalized van der Pol oscillator with a fluctuating driving term is derived from first principles. This equation is shown to be valid for optical band-to-band transitions with and withoutk-selection rule. Neglecting the nonlinearity in the saturation higher than of second order and also the intensity dependence of the dispersion this equation reduces to the standard van der Pol equation with a noisy driving term. From the general equation the linewidth and the noise of the intensity of the laser emission is calculated above and below threshold. The results are in agreement with the experimental data ofArmstrong andSmith.     

Theory of intensity and phase fluctuations of a homogeneously broadened laser

We extend our previous quantum mechanical nonlinear treatment of laser noise to the following problem: We consider a set of atoms each with three levels, which support laser action of one or several modes. The laser action can take place either between the upper or the lower two levels. The atomic line is assumed to be homogeneously broadened. The broadening can be caused by the decay into the nonlasing modes, by the pumping process, lattice vibrations and other, non specified sources. The fluctuations of the atomic variables (or operators) are taken into account in a quantum mechanically consistent way using results of previous papers byHaken andWeidlich as well asSchmid andRisken. The laser modes are coupled to the thermal resonator noise usingSenitzky's method. In the first part of the present paper, we treat quite generally multimode laser action. It is shown, that each light mode chooses a specificcollective atomic “mode” to interact with. We introduce a set of suitable collective atomic “modes”, which leads to a simplification of the equations of motion for theHeisenberg operators of the light field and the atomic operators. From the new equations we can eliminate all atomic operators. We are then left with a set of coupled nonlinear, integro-differential equations for the light field operators alone. These equations, which are completely exact and valid both for running and standing waves, represent a considerable simplification of the original problem. In the second part of this paper, these equations are specialized to single mode operation, which is studied above laser threshold. In the vicinity of the threshold the laser equation can be simplified to an operator-equation, whose classical analogue is vander-Pol's equation with a noisy driving force. With increasing inversion, the full equation must be treated, however. Using the method of our previous paper, we decompose the light amplitude into a phase-factor and a real amplitude, which is expanded around its stable value. We determine the Fourier-transform of the intensity correlation function and the total intensity of the fluctuating part of the amplitude. Somewhat above threshold this intensity drops down with the inverse of the photon output power,P, while the inherent relaxation frequency increases withP. The noise intensity stems in this region from the off-diagonal elements of the noise operators and not from the diagonal elements, which are responsible for the shot noise. This result is insofar remarkable, as a rate equation treatment would include only the latter ones. Under certain conditions the intensity fluctuations can show resonances with increasing output power,P. At high inversion the vacuum fluctuations of the light field are dominant, while the other noise sources give rise to contributions which vanish with the inverse of the output power. As a by-product our treatment yields the following formula for the linewidth (half width at half power) which is caused by phase fluctuations:

$$\Delta \nu = \frac{{\gamma _{3 2}^2 \kappa ^2 }}{{(\kappa + \gamma _{3 2} )^2 }}\frac{{\hbar \omega }}{P}\left( {\frac{1}{2}\frac{{(N_3 + N_2 )}}{{N_3 - N_2 }} + n_{Th} + \frac{1}{2}} \right)$$     

Deduction of generalised Kirchhoff's Laws from the basic principles of electromagnetism

The deduction of the so well-renowned and established Laws of Kirchhoff relating to the currents flowing in a network, which are considered to be almost axiomatic in electrical engineering sciences, seems to be preposterous and non-sensical at first sight. However, on a closer examination it will appear that these laws are based on two principles, viz, the steady-state condition under which accumulation of free electrical charge is precluded, and, the experimentalOhm's Law, which propounds a linear relation between the difference of potential across a conductor and the total current flowing through the same. But, there are two things, which are normally overlooked. The conductors inKirchhoff's Laws are “wires”, which are one-dimensional lines, and theOhm's Law is the macroscopicOhm's Law for total currents applied to these lumped resistors. Strictly speaking, the “Laws” are to be deduced from the more fundamental electromagnetic equations for continuous media and the microscopicOhm's Law. It is to be noted that theKirchhoff Nodal Law is but a consequence of the steady state condition derivable from these basic equations, viz, the current-density is divergence-free. For a continuous medium, what the form of theKirchhoff's Laws will be, is difficult to guess unless deduced from the basic equations. Once these are established from the basic equations of electromagnetism for a continuous medium, the usual form ofKirchhoff's Laws will follow as corollary of the general case, as has been shown here.     

Mobility of electrons in nondegenerate semiconductors considering electron-electron scattering

Spitzer andHärm [1] have investigated the velocity distribution of electrons in the presence of a weak electric field in an ionized gas. Introducing the concept of time of relaxation τ′, due to electron-ion and electron-electron scattering, and using the results ofSpitzer andHärm [1], the authors have obtained an expression for τ′ which is applicable to semiconductors with little modification. In this paper the authors have used this expression for τ′ to obtain the mobility of electrons in nondegenerate semiconductors, taking into account the scattering by lattice vibrations, electron-ion interactions and electron-electron interactions.     

Neuberechnung der Bindungsenergie des Wasserstoffmolekülions

The procedure of quantum mechanical treatment of a three-body problem described in the preceding paper byDiehl, Flügge, Schröder, Völkel andWeiguny has been specialized to the ground state in which the eigenfunction only depends upon the three distances between the particles. The solution of the Schrödinger equation has been approximated by variational methods, using the electronic functions ofFinkelstein andHorowitz, and ofGuillemin andZener, but including the nuclear vibration in the trial function.     

Elektronenterme von Kristallen bei nichttotalsymmetrischem Grundzustand der Gitterteilchen

The splitting ofBethe's crystal levels by the interaction between the electrons of equivalent lattice particles is discussed the ground-state ofBethe's levels being degenerated. A structured spectrum may be found for mixed crystals only regarding single foreign ions and pairs of foreign ions being next neighbours. A method is given to get the energy levels of pairs fromBethe's levels by means of group theory.     

Can one derive the Schwartzschild line-element without the field equations?

In 1944 W.Lenz has conceived an ingenious derivation of the Schwartzschild line Element for weak gravitational fields, without using the field equations ofEinstein. The derivation is based only on the Principle of Equivalence and the conservation of energy in the Newtonian approximation. Later M.Köhler raised some doubts how far this derivation is correct. Our aim is to show thatKohler's criticism can be met andLenz's derivation is correct.     

Broadening of the Mössbauer line for Gaussian spectra

A procedure described byBykov andHien is employed to obtain an approximation formula for the line broadening expected in a Mössbauer experiment for the case where the emission and absorption spectra have Gaussian shapes. For identical emission and absorption spectra, the broadening as given by this formula has an error of less than 4·5% for effective absorber thicknesses up toT _A =10. The case where the emission and absorption spectra have different half widths is also considered.     

Zur Elektronennachemission bei Ionenkristallen

The emission of electrons from NaCl-, LiF- and Zn-crystals after excitation by electrons and α-particles was investigated. The influence of external electrical fields on the emission was studied. The results of this investigations are compatible with the model developped byMatyás according to which an electrical dipole-layer is caused by the distribution of lattice defects in the surface.