We summarize the basic problems in laser coherence and the concepts and techniques needed to solve them. Special techniques needed to handle partition noise in semiconductor lasers will be touched on. We address the problem of how a self-sustained oscillator works. Why is the laser line so narrow? Why does a small amount of noise produce a broadening of the line as opposed to additive side bands? What is the role of zero point oscillations and spontaneous emission in determining the line width? In what sense can the transition to a lasing state be described as a phase transition? The concepts used to deal with the response of a laser system involve a quasi-Markoffian description. This in turn validates a quantum regression theorem which relates the spectrum of the noise to the transport response even in a system far from equilibrium. A significant simplification is made by introducing non-commuting (quantum) noise sources. The techniques required in the solution of laser noise problems involve a multi-time correspondence between quantum variables and c-number random variables that reduces a quantum stochastic problem to a classical stochastic problem. Adiabatic elimination is used to eliminate the fast variables to obtain a reduced problem for the fields alone or the fields and some total atomic variables (total inversion and polarization). Reduction to a rotating wave version of the van der Pol oscillator and numerical solution of the Fokker-Planck equation yields (a) the line shape (b) the spectrum of intensity fluctuations and (c) the distribution of photocounts observed over some time interval. Comparison of theoretical results with later experiments will be presented.
A number of experimental investigations have been carried out over the past fifteen years which have been used to study the effect of the bandwidth of the laser on various nonlinear optical interactions. These are useful and interesting for a number of reasons. On the technical side laser bandwidth effects can limit the accuracy of measurements of the frequency of atomic transitions as well as the determination of the strength of the interaction. Of a more fundamental interest is the question of the behavior of the nonlinear atomic system when interacting with the non-monochromatic field. In the first experiments carried out to investigate these effects lasers were used which could be operated on a single longitudinal mode or on multiple modes corresponding to the presence or absence of an intra-cavity etalon. Following this a number of techniques were developed to modulate the amplitude phase and/or frequency of the output of a stabilized cw laser with a random (but well-characterized) signal. In this way laser fields could be generated in the laboratory which closely approximate models of laser fields on which theorists have based calculations of laser bandwidth effects. These works allow direct quantitative comparison of results with theory. A number of these field models which can now be generated in the laboratory will be discussed as well as measurements of the role they play in nonlinear interactions.
The dynamical behavior of an atom interacting with a coherent and a chaotic field is investigated. Langevin equations describing the time dependent behavior of the dipole moment and population inversion are numerically integrated using Monte-Carlo techniques. The model can account for a finite correlation time for the chaotic component. Results on fluctuations in the fluorescence intensity from an ensemble of two-level atoms and the power spectra of these intensity fluctuations are presented. The effect of detuning the pump field from the resonant frequency of the two-level atoms on the fluctuations in fluorescence is studied. The effect of field fluctuations on the macroscopic polarization produced by the atoms is also reported. The fluctuation behavior of the atomic fluorescence is found to be very sensitive to the correlation time of the stochastic component.
An extension of the non-Markovian jump model of laser noise to the case of correlated frequency jumps is considered. The model parameters are the mean jump time the rms jump size and a correlation parameter which quantifies the degree of correlation between two successive jumps. A technique to obtain exact expressions for correlation functions is described. While the basic jump processes are stationary both the phase and frequency fluctuations are not. The field autocorrelation function for the pure phase jump process is stationary for all values of the correlation parameter. For the frequency jump process the average over all initial times of the two time field autocorrelation function is non zero only for fully anticorrelated jumps.
. The results of a theoretical and experimental investigation of the steady state and dynamical unstable behaviour of a single-mode homogeneously broadened Raman laser is presented. Bistable features of the Raman emission and its operating conditions are determined. Parameterisation of the dynamical behaviour shows instabilities to exist over a broad range of operating conditions and for broad cavity detuning to the low frequency side of the gain-centre where the bistable emission also prevails. Notably period doubling routes to chaos and sustained oscillatory behaviour are found to occur. Single mode homogeneously broadened Raman lasers which are equivalent to far off-resonantly pumped three level lasers are mathematically reducible to a simple description similar to that for a detuned two level laser''. The Raman system is however physically quite distinct involving a two photon rather than a one photon interaction between levels in a non-inverted rather than an inverted state. Significantly these differences lead to considerably relaxed operating conditions for the emergence of instabilities compared to those for the two level laser. Periodic chaotic and bistable behaviour for the Raman laser has been identified occurring at or close to the first lasing threshold and asymmetrically with respect to the gain centre. Further instabilities were found to be most prevalent at or close to conditions for optimum lasing. As such these systems provide excellent opportunity for quantitative tests of nonlinear dynamics both through their simple
We discuss recent theoretical and experimental work concerning the phase jump instability in the He:Ne ring laser gyroscope. The exact solution to the deterministic third-order equations for the laser yields an expression for the phase difference between the two counterpropagating modes. This phase difference can be constant monotonically increase oscillate be unstable or exhibit discontinuities. Experiments and computer solutions of the stochastic laser equations show corresponding phase jumps which can be very rapid but not truly discontinuous. We discuss the behavior of the phase difference and of the timeaveraged beat frequency between the two modes with and without the effects of laser noise.
Recent experiments to investigate the noise properties of He:Ne lasers operating close to threshold at 633 nm are described. In the first set of experiments transient fluctuations during the turn on of the laser were studied by means of the first-passage-time technique. In another set of experiments intensity fluctuations of a laser with a saturable absorber were investigated. Measurements of intensity fluctuations in both the bistable and tricrical regimes of operation were made. Finally the effects of multiplicative white noise due to pump parameter fluctuations were investigated by means of photon counting and correlation techniques. Implications of these results for other laser systems are also discussed.
A recent reformulation of the laser field equations using the true modes of the laser cavity and including a noise polarization term to account for spontaneous emission shows that conventional understanding of amplifier noise figure oscillator linewidth and oscillator build-up time needs to be modified for laser amplifiers or oscillators with nonorthogonal transverse eigenmodes. Gain-guided lasers and unstable-resonator lasers in particular have nonorthogonal transverse modes and hence exhibit unconventional noise behavior. One important consequence of this theory is the appearance of an excess spontaneous emission factor (the " Petermann factor" ) which multiplies the well-known Schawlow-Townes formula for the laser linewidth. This factor can be substantially greater than unity for unstable resonators with large magnification or Fresnel number. We discuss our investigations of the consequences of the nonorthogonality of the transverse eigenmodes especially on laser linewidth in unstable-resonator lasers. We believe that the results of these investigations are important where the temporal coherence properties of the laser are of interest namely in coherent optical communications spectroscopy optical wavelength standards and also in the injection seeding of pulsed high-power unstable resonator lasers. Extensive calculations of the excess noise factor in real hard-edged unstable resonators have been carried out using a virtual-source approach to calculate the exact unstable-resonator eigenmodes. An experiment is also being carried out to demonstrate the linewidth enhancement in a small diode-pumped geometrically unstable laser oscillator.
The results of a quantum-electrodynamical theory of excess spontaneous emission noise in lossy resonators will be presented. The " Petermann K factor" does not enter into the spontaneous emission rate of a single atom in the cavity. The QED theory allows different interpretations of the K factor and we use this fact to justify semiclassical analyses and to provide in one example a simple derivation of K in terms of the amplification of the quantum vacuum field entering the resonator through its mirrors.
We describe the theory and practice of using coherent optical feedback from a simple external reflector or self-contained high-finesse resonator to reduce the time-averaged spectral width and FM noise level in a single-mode semiconductor injection laser.
We present the theory and implementation of various electro-optic feedback systems used to stabilize the intensity of lasers. These variably attenuate the beam and sample the result applying feedback to achieve a desired intensity despite laser fluctuations. In this way extremely stable beams (0. 01 variation) have been produced over the spectral range 257 nm - 10. 6 p. Applications include radiometric calibrations spectroscopy fluorescence-tagged detection laser heating controlled ablation and video disk mastering. Lasers stabilized this way include CW modelocked and Q-switched pulsed Nd:YAG Ar ion (both the fundamental and the second harmonic) C02 Ti:Sapphire He-Ne and He-Cd.
We demonstrate the use of a commercial mode-locked and Q-switched Nd:YAG laser in photon-noise reduction experiments. There are many frequency bands wherein the photodetection of such a laser is shot-noise limited. By appropriately filtering the photocurrent produced in direct detection of light from an optical parametric amplifier that is pumped with this laser we have successfully generated twin-beam states of light. We demonstrate that such twin beams can be used to make measurements below the shot-noise limit.
Using an extension of the Scully—Lamb laser theory we show that by insertion of nonlinear elements in the cavity, laser light with nonclassical sub—Poissonian photon statistics can be generated. For an N—photon absorber in the limit of small saturation the variance of the photon number distribution can be reduced by a factor of (+1/N) compared to Poissonian limit of the usual laser. When we generalize the type of such nonlinear elements to the case of a nonlinear feedback or Raman type nonlinear transition, the noise in the laser intensity can be reduced even further and approach the ideal limit, characterized by a Mandel Q parameter of Q = —1. Furthermore we show how this noise reduction via nonlinearities can be combined with a regular pump mechanism, which recently has been sugessted as a way to generate sub—Poissonian light. It turns out that in certain configurations the effect of both methods can be combined. Finally we show how this nonclassical behaviour is also reflected in a reduction of the output intensity power spectra below the shot noise level.
The fact that semiconductor lasers subject to external optical feedback are extremely prone to low-frequency instabilities has been known for more than two decades. However the rich variety of low-frequency intensity fluctuations ranging from purely stochastic phenomena to examples of deterministic chaos is not generally appreciated. Here we describe a few examples of this variety: undamped relaxation oscillations (self-pulsing) coherence collapse staircase fluctuations and the onset of mode beating noise self-pulsations subharmonic cascades and chaos. Apart from their significance to the fundamental science of nonlinear dynamics these phenomena are important because of the widespread application of diode lasers to optical communication and data storage.
In nearly every application of semiconductor lasers some externally reflected light is coupled back into the laser with a certain time delay. The noise and coherence properties are very sensitive to optical feedback. Coupling to external cavity modes induces frequency shifts linenarrowing and broadening competition among different external cavity modes or dynamical instabilities. Strongly dispersive gain makes instabilities efficiently generate phase noise leading to coherence collapse. We present an overview of these phenomena and their theoretical descriptions.
The semiconductor diode lasers that are used for optical data storage are known to show large intensity noise. One means of rejecting such noise is to use a differential split detector to read stored data as well as to follow servo tracks etc. Such a strategy is subject to beam position noise a hitherto obscure effect which cannot always be neglected. This paper will discuss the management of diode laser noise which -under ideal conditions - can actually be pushed below the shot noise limit.
Spontaneous emission is a major source of noise in semiconductor lasers. The noise phenomena such as relative intensity noise mode-partition noise and laser linewidth are discussed by using the Langevin rate equations. Particular attention is paid to the impact of intensity arid phase noise on the performance of optical communication systems.
The experimental and theoretical investigations of radiation dynamic in a system of two optically-coupled C02-lasers and in a periodically - pumped COlaser are carried out. In a optically-coupled lasers on the edge of theirs locke- band and in a periodically-pumped laser near threshold the transition of lasers to the chaotic generation through subsequent period-doubling bifurcations are experimentally revealed. A good agreement between experimental and theoretical results was obtained.
A theory is developed to describe the optical transient signals that arise when a sample of two-level atoms is irradiated by a sequence of two or three broad-bandwidth pulses is presented. The first two pulses have a relative delay time of order of the correlation time of the pulse fluctuations. These pulses whose temporal width is much greater than the delay time are fully correlated with one another and can be strong enough to saturate the two-level atomic transition. In the case of three-pulse transients the third pulse is weak non-correlated with the first two and is delayed in time so that it does not overlap them. Taking into account the effects ofinhomogeneous and homogeneous broadening we calculate the intensity of the transient signals as a function of delay time. The signals are found to depend dramatically on the intensities of the excitation pulses. It is shown that for strong excitation pulses a direct dependence of the signal on the cross-correlation time of the pulses r occurs that does not exist when the pulses are weak. In particular for saturating pulses the signals exhibit a peak of width of order r . In the case of the two-pulse transient the peak is found to disappear when the Doppler width of the atomic ensemble becomes sufficiently large. This peak can have a very narrow dip near its maximum whose width is much
The intensity fluctuations in the output of an intracavity doubled solid state laser have been shown to be deterministic in origin. We describe a simple technique to eliminate these fluctuations. Synchronized antiphase oscillations of the longitudinal mode intensities have been observed. Energy sharing between coupled modes is characterized by a statistical analysis of the chaotic fluctuations.
Calculations on the Hanle effect in a J - J:i atomic transition are performed for laser fields with a) Brownian motion phase diffusion b) Gaussian amplitude fluctuations and c) fluctuating real and imaginary parts of the laser electric field amplitude (chaotic field model). We use the Fokker-Planck operator approach to stochastic differential equations. The zero-field level-crossing dip broadens at high laser intensity significantly less so as the laser bandwidth increases or if the amp]itu(le fluctuates.
A general framework for the calculation of random switch-on times and transient intensity fluctuations is reviewed. Several applications including dye lasers and the detection of weak signals are considered. In particular single mode semiconductor laser gain switching is discussed: Results are given for the dispersion in the maximum value of the laser intensity at the first peak of the relaxation oscillations. Such dispersion is easily related to the jitter in the switch-on times.
Fluctuations in the laser light arise due to the quantum nature of the electromagnetic field in the laser cavity as well as due to environmental fluctuations such as pump noise. The mathematical tool for describing the noisy laser light is the laser Fokker-Planck equation, which we solve in terms of matrix continued fractions. Physical quantities such as moments and spectral densities are related to solutions of the Fokker-Planck
Methods of determining the frequency of a laser during its transient evolution such as during the switch-on of a laser are considered. This study is motivated by several illustrations of steady state laser systems which change their operating frequencies as their parameters are varied adiabatically. For evolving systems one would like an instantaneous measure of the frequency but for stochastic processes it is not possible. We discuss several different ways of characterizing the evolution of characteristic frequencies and phase fluctuations during transients that come as close as possible to instantaneous measures and which give evidence of the evolution that is taking place.
Our recent experiment shows that the oppositely directed spontaneous emission from a thin layer of a fluorescence source is highly coherent. It is proposed that a thin layer of gain medium be applied to a ring cavity and be located at the symmetric point from the output coupler so that the spontaneous emission in the laser outputs are correlated and can be removed from a laser beam by destructive interference technique. A theoretical analysis and a physical interpretation for such a spontaneous emission free laser beam are presented. The potential of using the device as an active laser gyro is discussed.
This paper describes a simple all-electronic noise cancellation scheme which allows wideband shot noise limited optical measurements at baseband with noisy lasers in many kinds of optical systems. With this system it is usually possible to achieve the performance of a complex heterodyne system with a much simpler homodyne approach. Although it is similar to earlier differential and ratiometric techniques its noise cancellation performance is much better and it is highly effective at modulation frequencies up to tens of megahertz. The basic idea is to subtract photocurrents directly under feedback control to cancel excess noise (i. e. noise above the shot noise level) and spurious modulation of the beam. A sample is split off from the beam at the laser and detected with a photodiode similar to the main detector at the system output. Most optical systems and detectors have very wide temporal bandwidths and excellent linearity thus at all frequencies of interest the sample photocurrent has exactly the same instantaneous fractional excess noise fluctuations as the laser beam itself with no differential gain or phase. If a fraction of the sample photocurrent is subtracted from the main detector output with feedback controlling the division ratio to keep the DC component of the result at zero the excess noise cancels identically. The actual noise cancellation bandwidth is very wide and does not depend on the feedback bandwidth only on that of