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Annalen der Physik

Amplituden‐ und Intensitätsschwankungen nach Verstärkung mit einem Laserverstärker

A. Bandilla, Henryk Paul

Published Jan 1, 1970• 6 citations
Publisher

Abstract

Abstract In a preceding paper [1] the phase uncertainty of an electromagnetic wave amplified by an (ideal) laser amplifier has been evaluated. Similar calculations are now performed for the amplitude and the intensity of the radiation. A special result is the conclusion that a field with minimum intensity fluctuations after amplification corresponds uniquely to a signal with a sharp photon number. The same holds approximately for the amplitude, the approximation being exactly the same as in [1] which allowed us to identify the phase uncertainty after amplification with the expression following formally from the phase operator applied to the incident field.

DOI
10.1002/andp.19704790304

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Extending recent calculations of GLASSGOLD and HOLLIDAY [3] the amplification process by means of a laser amplifier is treated theoretically in case of an arbitrary quantum mechanical state of the incident electromagnetic field. An explicit expression for GLAUBER's P‐function representing the state of the field after amplification, is found. Interpreting this function as a classical distribution function for the complex amplitude of the field, a distribution function W(φ; t) for the phase is obtained by averaging over the real amplitude. With the help of W(φ; t) the phase uncertainty (δφ)2 for the amplified field is calculated. The result approximately agrees with the expression for(δφ)2 following from the phase operator originally introduced by HEITLER if applying it to the incident field. Thus a connection between the phase operator and an experimental arrangement is established.

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Physical Review A•Feb 1, 1978•30 citations
Abstract In a preceding paper [1] the phase uncertainty of an electromagnetic wave amplified by an (ideal) laser amplifier has been evaluated. Similar calculations are now performed for the amplitude and the intensity of the radiation. A special result is the conclusion that a field with minimum intensity fluctuations after amplification corresponds uniquely to a signal with a sharp photon number. The same holds approximately for the amplitude, the approximation being exactly the same as in [1] which allowed us to identify the phase uncertainty after amplification with the expression following formally from the phase operator applied to the incident field.
Extending recent calculations of GLASSGOLD and HOLLIDAY [3] the amplification process by means of a laser amplifier is treated theoretically in case of an arbitrary quantum mechanical state of the incident electromagnetic field. An explicit expression for GLAUBER's P‐function representing the state of the field after amplification, is found. Interpreting this function as a classical distribution function for the complex amplitude of the field, a distribution function W(φ; t) for the phase is obtained by averaging over the real amplitude. With the help of W(φ; t) the phase uncertainty (δφ)2 for the amplified field is calculated. The result approximately agrees with the expression for(δφ)2 following from the phase operator originally introduced by HEITLER if applying it to the incident field. Thus a connection between the phase operator and an experimental arrangement is established.
Abstract A novel approach to the quantum mechanical description of phase properties of a one‐mode low‐intensity radiation field is presented. The procedure is based upon the correspondence between Glauber states and classical waves of fixed phase ϕ and amplitude a, and, in contrast to the conventional quantum mechanical phase concepts, allows to make contact between certain operators ℰm corresponding to the classical quantities exp {imφ} (m = ± 1, ± 2, …) and realistic measurements. In the observation, a large amplification of the microscopic field by means of a linear quantum amplifier plays a decisive role. Since the amplified field is a macroscopic one, the measurement of phase is not problematic, and hence, in particular, average values «exp {imϕ}» can be determined experimentally. The main result of our analysis is that the mean values «exp {imϕ}» are equal to the quantum mechanical expectation values for the operators ℰm with respect to the field before amplification. To obtain this result it became necessary to calculate Glauber's P function for the amplified field in case of an arbitrary state for the initial field.
Abstract Going ahead with investigations started by ideas of H. Paul it is examined how different quantum properties in the initial field state show up in the strongly amplified output field. As examples of extreme quantum fields the number state and the recently introduced phase optimized state are used. For the linear laser amplifier the inevitable noise is fully taken into account. These results are compared with those for a coherent input field. The characteristic differences in the macroscopic output fields allow unique conclusions onto the initial field. For the calculations the positive definite Glauber's P‐representation after strong amplification is applied.
Abstract Extending the method developed in a previous paper [1] the correlation functions both for electric field strength and intensity are calculated. In the latter case the result is obtained quite easily by solving a differential equation. In the first case, on the contrary, a detailed knowledge of the photon creation operator b+ is necessary, especially of the different temporal behaviour with respect to phase and amplitude. In the main, the results are in agreement with those published by other authors. Of special interest is the result that the damping constant for intensity correlations tends to the value corresponding to the case of thermal radiation, in the region far from threshold. In addition, the ansatz b+ = (ro + ϱ) exp{i χ(t)} (ro c‐number) introduced by HAKEN is discussed from a quantum mechanical point of view.
The phase of an optical field inside a linear amplifier is widely known to diffuse with a diffusion coefficient that is inversely proportional to the photon number. The same process occurs in lasers which limits its intrinsic linewidth and makes the phase uncertainty difficult to calculate. The most commonly used simplification is to assume a narrow photon-number distribution for the optical field (which we call the small-noise approximation). For coherent light, this condition is determined by the average photon number. The small-noise approximation relies on (i) the input to have a good signal-to-noise ratio, and (ii) that such a signal-to-noise ratio can be maintained throughout the amplification process. Here we ask: For a coherent input, how many photons must be present in the input to a quantum linear amplifier for the phase noise at the output to be amenable to a small-noise analysis? We address these questions by showing how the phase uncertainty can be obtained without recourse to the small-noise approximation. It is shown that for an ideal linear amplifier (i.e. an amplifier most favourable to the small-noise approximation), the small-noise approximation breaks down with only a few photons on average. Interestingly, when the input strength is increased to tens of photons, the small-noise approximation can be seen to perform much better and the process of phase diffusion permits a small-noise analysis. This demarcates the limit of the small-noise assumption in linear amplifiers as such an assumption is less true for a nonideal amplifier.
The amplification of noise in a laser amplifier is treated theoretically. The model for the active medium and its description using density-matrix techniques, are taken from the theory of laser operation. The fundamental concern of this investigation is the spectral behavior of the radiation in the nonlinear regime; hence, the formalism is written from the onset in the frequency domain. The statistics of the light are gradually modified by the nonlinear amplification process, and expressions are derived for the rate of change of fluctuations in intensity as a measure of statistical changes. These expressions should be easily susceptible to detailed experimental observations. In addition, the range of validity of Litvak's Gaussian-statistics approximation is discussed with some detail. In the homogeneous-broadening case, the evolution of initially broadband Gaussian radiation toward quasimonochromatic oscillations with laserlike statistics is extensively explored in several numerical examples. The connections of this study with the time-domain work of Risken and Nummedal, on self-pulsing in a ring-laser configuration, are clearly established. Finally, spectral-narrowing and -rebroadening effects in Doppler-broadened media are discussed both analytically and with numerical examples. These examples show the distinct contribution of pulsations in the population ("Raman-type terms"), and saturation phenomena. The predicted narrowing-rebroadening rate is compared with expressions found in previous literature.