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:

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

Fringe analysis and phase reconstruction from modulated intensity patterns

A novel method of determining phase from a modulated intensity pattern is described. A line integral of the gradient of the phase is used to reconstruct the phase, eliminating the necessity for complex methods of phase unwrapping. The new algorithm can be used with any technique that experimentally or theoretically yields the cosine and sine or the tangent of the phase. This phase-reconstruction process works effectively even in the regions of high-intensity gradients and is insensitive to the profile of the illuminating beams and to the shape of the domain boundaries.     

Influence of fluctuations of a laser pump on the intensity correlation of resonance fluorescence radiation

The influence of fluctuations of a relatively weak laser pump on the intensity correlation of resonance fluorescence radiation is studied for a laser acting both below and above the threshold. Below the threshold the radiation of the laser is characterized by Gaussian properties causing non-factorization of the intensity correlation function. Above the threshold the laser shows non-linear filter properties and, therefore, the deviation of the exact intensity correlation function from its factorized value is small.     

Diffraction from modulated structures IV. Working formulae for intensity and rectangular correlation function

The expression for the intensity diffracted on quasiperiodically modulated structures derived formerly is put in a form suitable for practical applications. This procedure consists in carrying out the summation of terms in the intensity expression and in expressing the functionsR(h) andS(h), determining the influence of quasiperiodic modulations on the intensity, in terms of the symmetrical rectangular correlation function.     

Simulation of a Kolmogorov phase screen

Abstract

Two new methods for modelling Kolmogorov phase fluctuations over a finite aperture are described. The first method relies on the incorporation of subharmonics in order to model accurately the low frequencies of the Kolmogorov spectrum. The second method provides a less accurate, but much faster method for simulating the Kolmogorov spectrum by using a midpoint displacement algorithm used in computer graphics.     

Simulation of a Kolmogorov phase screen

Two new methods for modelling Kolmogorov phase fluctuations over a finite aperture are described. The first method relies on the incorporation of subharmonics in order to model accurately the low frequencies of the Kolmogorov spectrum. The second method provides a less accurate, but much faster method for simulating the Kolmogorov spectrum by using a midpoint displacement algorithm used in computer graphics.     

Interference of speckle fields in the diffraction zone of a focused spatially modulated laser beam by a random phase screen

The processes of formation of average-intensity interference fringes upon diffraction by a random-phase object of a laser beam having interference fringes and focused on the surface of the object are considered. The dependences of the fringe contrast on the parameters of scattering inhomogeneities of the object and the parameters of the focused laser beam are established in analytical form for various diffraction regimes. Practical possibilities of a method of probing of scattering objects in problems of measuring the parameters of inhomogeneities and problems of interference-pattern formation in optical systems with scattering media are discussed.     

Focussing by a random phase screen

We present experimental measurements of the statistics of intensity scintillations in light scattered by a random phase screen. The phase screen was produced in the laboratory by turbulent mixing of hot and cold air; the parameters of this screen were such that geometrical focussing effects could be studied near the screen and speckle effects were clearly visible in the far field. Results are given for the propagation of both laser light and white light through the turbulent region. We compare experimental results with the theory for a phase screen with a gaussian phase correlation function. New theoretical work which allows comparison for mean-square phase fluctuation $\text{φ}{}^2$ å 1 is also outlined.