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.     

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.     

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:

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

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.     

Infrared active modes of cubic strontium titanate

It is shown that there are nine optically active modes of SrTiO₃ and not twelve as stated byNarayanan andVedam in a previous article in this Journal.     

Frequency shifts of laser modes in solid state and gaseous systems

The present paper calculates the frequency shifts of laser modes which are brought about by the nonlinear response of the active laser material. The active material is supposed to consist of a set of N atoms each with three or four levels either at random lattice sites or moving with a Gaussian velocity distribution. The formalism developed in our previous work in which a homogeneously broadened line was considered is now applied to a line which is in addition inhomogeneously broadened. The physically most important case of an inhomogeneous Gaussian is treated in detail up to orderγ/α, whereγ is the natural linewidth and α that of the superimposed inhomogeneous broadening. Starting from first principles our present microscopic theory represents especially a foundation of Bennetts concept of hole burning effects. Furthermore it contains the effects of time-dependent nonlinearities as well as that of polarization. Besides the well known mode pulling effect of an inhomogeneous Gaussian several new effects show up. If only one mode lasers an additional repulsive term occurs in both solid state and gaseous systems, which stems from the time independent response of the atomic system. It is small in the solid state case but can be pronounced in the gas laser if the mode frequency is close to the center of the broadened line. If two modes oscillate simultaneously, the mode interaction leads to a repulsive term which can be appreciable both for solid state and gaseous lasers. The results for a solid state laser are in qualitative agreement with line pushing as observed bySnitzer. For comparison with gas laser experiments the He-Ne-Laser is considered in detail. If wide angle scattering of Ne-atoms is slow, the frequency pushing is governed by the time independent response. Our theoretical results are in favorable agreement with various experimental findings byBennett. Finally, the stability of simultaneous laser modes is treated in detail both for the solid state and gaseous case.     

Quantummechanical solutions of the laser masterequation. II

A Fokker-Planck equation for a distribution function over the macroscopic observables of the laser essentially equivalent to that recently obtained byRisken,Schmid andWeidlich is derived from the fundamental quantummechanical laser masterequation. The general method used is the expansion of the statistical operator in a complete set of projection operators of the atoms and the lightfield. The assumptions leading from the microscopic equation of motion to the macroscopic semiclassical Fokker-Planck equation are explicitly introduced and discussed.     

Streuung mittelschneller Elektronen um kleinste Winkel — Vergleich von Absolutmessungen an Gasen mit theoretischen Ergebnissen nach Lenz

During the last years several experiments on small angle scattering have been made the results of which are not in agreement withLenz's theoretical results commonly used in electron microscopy. These measurements having been made on foils, a possible reason for the discrepancies seemed to be that scattering did not take place on isolated, spherical, neutral atoms as theory assumes. These conditions were fulfilled, however, at the experiments of the following paper. In spite of this there is no agreement with theoretical results basing onLenz's scattering cross sections except in the case of neon, whereLenz's cross sections give very reliable values. The discrepancies are shown not to be explainable by a breakdown ofBorn's approximation as seems to be obvious because of the systematic increase of the discrepancies with atomic number.     

Helicity antenna showers for hadron colliders

We present a complete set of helicity-dependent \(2\rightarrow 3\) antenna functions for QCD initial- and final-state radiation. The functions are implemented in the Vincia shower Monte Carlo framework and are used to generate showers for hadron-collider processes in which helicities are explicitly sampled (and conserved) at each step of the evolution. Although not capturing the full effects of spin correlations, the explicit helicity sampling does permit a significantly faster evaluation of fixed-order matrix-element corrections. A further speed increase is achieved via the implementation of a new fast library of analytical MHV amplitudes, while matrix elements from Madgraph are used for non-MHV configurations. A few examples of applications to QCD \(2\rightarrow 2\) processes are given, comparing the newly released Vincia 2.200 to Pythia 8.226.     

Calculation of laser modes in an active Perot-Fabry-Interferometer

The mode structure and the diffraction losses of a Perot-Fabry-Interferometer filled with an active material are calculated analytically with aGreen's function method. The active material is described by a complex susceptibility. The mode pattern and the diffraction losses derived here are in good agreement with the results of Fox andLi, who obtained the mode pattern and the diffraction losses of the decaying modes of a Perot-Fabry-Interferometer by a machine calculation. Furthermore the case of different dielectric constants between the active material and the outer space and the example of a Perot-Fabry-Interferometer with infinite mirrors are discussed. Throughout the paper only a single mode is treated and it is assumed that the inversion in the active material is constant.     

Concerning internal conversion coefficients in Hf180

The gamma-ray relative intensities from transitions in Hf¹⁸⁰ following the 5.5 hour Hf^180m decay have been measured using a bent-crystal gamma-ray diffraction monochromator and a least-squares fitting technique. From these measurements it was possible to deduce a value for the internal conversion coefficient for the 93.3-keV transition of α _T ⁹³ =4.91±0.23. From previous measurements of conversion electron intensities byEdwards andBoehm and our gamma-ray relative intensities, internal conversion coefficients for all transitions except the 57.5-keV transition were obtained. These coefficients agree well with the previous determinations byEdwards andBoehm, however, our measurements have improved precision, particularly in the case of 501.3-keV transition. The present measurements of α _K for the 215.3-, 332.5-and 443.8-keVE2 transitions are 11% lower than theoretical values while α_K for 93.3 keV E2 transition agrees closely with the theoretical value. These results are in close agreement with the previous measurements ofEdwards andBoehm. The present experimental α_K for the 501-keV transition agrees closely with the theoretical α_K for an E3 multipolarity.     

Theorie des stimulierten Raman-Effekts

We consider stimulated Raman emission in solids, placed in a plane laser beam external to the cavity. The Hamiltonian of the system of phonons, electrons and electromagnetic fields is derived within the framework of a generalized adiabatic approximation for electrons and nuclei. It contains terms due to nonlinear interactions between electrons and phonons. Because the usual time-dependent perturbation theory cannot describe coherence effects properly we turn toHeisenberg's equations of motion for the operators of photons, phonons and electron excitations. In order to solve these equations in the steady state we apply an iteration procedure. We start with the light waves which give rise to electron transitions. The electrons such excited create phonons which then react on the electrons. Finally the electrons are coupled again to the lightfield. This procedure yields besides the usual wellknown Raman process two main processes occurring in stimulated Raman emission: a coupled two step Raman process and a parametric process. In the first one two phonons are involved. If the linewidth of phonons is comparable to the phonon frequencies the non-resonant parts of the above processes also become important. In solving the set of coupled equations for the light amplitudes, obtained from the iteration procedure, we only consider terms due to the first Stokes, the first anti-Stokes and the laser line. We then find frequency shifts of these lines due to the stimulated emission which are of the order of the linewidth of photons if this linewidth is very much smaller than that of phonons as it is the case in solids. This means that the coupled two step Raman process is dominant, in good agreement with measurements ofChiao andStoicheff in calcite.     

Über die Entstehung von Plasmawellen in stark ionisierten Gasen

This paper concernes the problem of development of plaine waves in a gas plasma without external fields. The question has been recently treated byLandau, v.Kampen, Berz and o. m. The equation ofBoltzmann is discussed and solved for the case with initial conditions (Anfangswertproblem) explicitly. With the usual expression for the collision term exist no ?Eigenlösungen“. Nevertheless in a short time long waves develop withλ>30l _D (Debye-Length) from the initial distribution; the statistical damping cancels down the development of shorter waves. Changing the collision term, one finds true Eigenlösungen too. This different behaviour of the solutions seems to be less a physical problem than a question of the used mathematical formalism.     

Zum Einfluß der Schalenstruktur der Kerne auf die Spektren der μ-Mesonenatome

In two earlier investigations it has been shown that the discrepancy between some new measurements^{1, 2} of the spectrum of theμ-mesic Bi-atom and the calculations ofFord andWills as well as the discrepancy found by fitting the electron-scattering data from Bi to a density function of “Fermi”-type⁴, may be explained if one takes into account explicitly the charge density of the 83-rd proton in the shell-model^{5, 6}. The present paper deals with the Tl nucleus where a similar “shell structure effect” seems to occur.     

Helligkeit von ZnS-Phosphoren während kurzzeitiger Anregung

The theory ofRiehl andSchön for the luminescence of ZnS-phosphors is leading to a bimolecular mechanism. Thus in a starting range still far from reaching the saturation, the luminescence would increase in the second power with the stimulation intensity. The starting intensity of several phosphors was measured at stimulation time between 10^?5 to 1 sec. The theory ofRiehl andSchön was confirmed. Measuring results proving a higher increase than the second power could equally be explained by this theory.     

Polarization of atomic line radiation

In order to test the predictions of the general theory for the polarization of atomic radiation excited by electron impact, we measured the polarization of atomic lines in electron-alkali atom crossed-beam experiments. The results show a typical behaviour for the polarization of lithium and sodium resonance lines, which is contrary to the earlier measurements such as those ofSkinner andAppleyard: the polarization of the resonance lines increases monotonically from higher energies to the excitation threshold. The threshold polarization of the first resonance lines of Li⁶, Li⁷ and Na²³ is in good agreement with the calculations byFlower andSeaton. These results demonstrate that polarization is very sensitive to the natural level width, and to the fine and hyperfine structure separations of the excited states. Polarization of further lines has been investigated: an alkali line from anS to aP state is unpolarized as expected, but the polarization of a line from aD to aP state decreased in an unexpected manner near the threshold energy.     

Beugung elektromagnetischer Wellen in Braunbekscher Näherung. II

The extension, given in a previous paper, ofBraunbek's approximation method to the diffraction of electromagnetic waves, is applied to the near field of the circular aperture and the circular disk. The field distributions along the central axis and in the plane of the screen are calculated and compared with the results of both the rigorous solution andKirchhoff's approximation. The relation ofBraunbek's method to other approximation methods for the optical region is discussed.     

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

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

Ein neuer Variationsansatz für das Polaron

A new trial function for the self-energy of the polaron is presented which is a combination of the weak and strong coupling methods ofLee,Low andPines resp.Pekar. The result for the ground-state energy is a smooth interpolation between both methods which is better than either one but not as good asFeynman's result.     

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.