Coherence Properties of Radition in Non-Linear Optics and Lasers

Abstract

During last three years our group has carried out research work in the field of non-linear optics and lasers. Our group has published journals of high repute. Three research officers were supported by the PSF project. All of these research officers have submitted their Ph.D. dissertations. These dissertations have been evaluated by international referees and they have strongly recommended to accept these dissertations. The advanced studies and research board have approved these reports and these research officers are waiting for their final oral defense. The name of these research officers and their dissertations are as follows: 1. Mr. Khalid Zaheer: Noise quenching in optical amplifier via coherent superposition. 2. Mr. Nadeem Akram Ansari: A study in the theory of four-wave mixing. 3. Miss Imrana Ashraf: Quantum theory of two-photon micro maser and laser. In his dissertation Mr. Khalid Zaheer has studied the phase sensitive dynamics of laser amplifier. The calculations show that for a certain choice of the relative phase of the atomic dipole and the cavity field, the one of the side mode is completely quenched. A nonlinear theory of a correlated emission laser is also developed and in coherent superposition of states the spontaneous emission noise is quenched in the relative phase of the laser under certain conditions. Mr. Nadeem A. Ansari has studied the four-wave mixing process and the violation of the Cauchy-schwarz and Bell’s inequalities and higher-order squeezing are predicted in non-degenerate four-wave mixing under suitable parameters. A linear theory of two-level quantum-beat laser is also developed. It is shown that the strong pump field is responsible for coherence in such a way that the mean phase of the two side modes is locked and spontaneous emission noise is quenched. A non-linear theory of a three-mode laser is developed and condition are derived under which the quantum fluctuations of the pump field produces quantum-beat effect. Miss. Imrana Ashraf, in her dissertation, studied the quantum theory of two-photon micro maser and laser. The calculations show that the interaction of a quantized single mode field with an effective two-level atom through intermediate states produces a periodic collapse and revival type behavior in mean photon number and photon number fluctuations. The photon statistics of two-photon laser is also discussed and it is shown that the photon distribution function is wider than the Poisson distribution function. In addition the effect of cooperative atomic interaction in laser is studied. The dynamics of a single two-level atom in the presence of another atom and the behavior of population inversion of two atoms, photon statistics and the natural linewidth in a single mode laser is studied. Our experimental group has also started research work in the field of optical computing. Summary of Work: During the report period, 49 papers were submitted and published, all in the international journals of repute. These publications were based on the following studies and investigations. I. Effect of Cooperative Atomic Interaction In Lasers Most studies in atom-field interactions and lasers involve the independent interaction of atoms with the field. However, certain important features appear when the interactions among the atoms or the cooperative effects are also included. During the report period, a detailed investigation of the effects of cooperative atomic interactions in various systems was made. These include the dynamics of a single atom in the presence of another atom and the behavior of population inversion of two atoms interacting mutually and with single-mode quantized electromagnetic field, the spontaneous emission from two atoms in a non-deal cavity under different coupling conditions and the effect of cooperativity on photon statistics and the natural linewidth in a single-mode laser. We have studies the spontaneous emission from two-level atoms inside a demand cavity. It is found that the spontaneous emission from one atom is intimately dependent upon the coupling strength of the second atom and is reduced considerably for a weaker coupling of the second atom. Certain interesting features appear in the bad cavity (low Q) limit. Recently, the quantum theory of linear amplifiers has drawn renewed interest because of its applications in noise-free amplification and in photon correlation experiments. Since amplification is always associated by noise, the scheme followed in such amplifiers is to feed all the addictive noise in one of the phase sensitive quadrature of the field and the quadrature of interest remains noise free. We have shown that a two-photon linear amplifier adds a phase-sensitive noise to the input signal. It is therefore possible to amplify a squeezed signal quadrature with reduced added noise compared to standard phase insensitive amplifiers. We have also proposed a two-level phase sensitive amplifier in which atoms are injected in pairs. We have shown that under certain conditions on gain coefficient, coupling constants and relative phases, the additive noise is zero. Some further aspects of these noise-free linear amplifiers are being studied.

II. Atom- Field Interactions using Path-Integral Methods We extended the path-integral formalism to spin-Bose problems in quantum optics and studied the dynamics of a two-level atom interacting with a single-mode quantized radiation field without the rotating-wave approximation. Using this formalism, we obtained fully quantum-mechanical, analytic expressions for the dynamics of the atom as well as the field and have shown that even under the conditions in which this approximation is considered to be justified, there is significant contribution due to the energy non-conserving terms to the level population inversion. It was also shown that the energy of the atom-field system is not conserved in time but exhibits periodic collapse and revivals along with rapid oscillations. III. Non Linear Optics In the general area of nonlinear optics we investigated the following problems in detail. • The effect of laser linewidth associated with the phase fluctuations of the pump field on certain correlation functions of the signal mode in degenerate parametric amplification process. It was found that as a result of phase fluctuations of the pump field, the photon ant bunching effect decreases and for sufficient increase of these fluctuations, photon anti bunching disappears totally. We have also shown that the conversion efficiency increases due to phase fluctuations. • We have studied certain non-classical effects in the context of four-wave mixing and have predicted that Cauchy-Schwarx and Bell’s inequalities are violated for large detuning of pump frequency from the side-mode frequency10,11. We have also calculated the higher-order squeezing in non-degenerate four-wave mixing. We have predicted that the amount of squeezing can be increased with the degree of squeezing in non-degenerate four-wave mixing. • We have made an extensive study of the natural linewidth of various lasers and investigated the effect of different parameters in these lasers. These include a laser with saturable absorber, dye laser and a two-level, two-photon laser. We have shown that the absorber in the laser with saturable absorber contributes to increase the natural linewidth, whereas, the built-in nature of the absorber in dye molecules tend to reduce the natural linewidth in a dye laser. In the case of two-photon laser, it was found that the linewidth is independent of mean number of photons. In addition, two-photon laser and maser have been studied with the inclusion of dynamical stark shifts. We have also derived an exact expression for the photon distribution function of the three-level two-photon micro maser and study the behavior of mean photon number fluctuation. • Efforts have been made recently to somehow, step around the fundamental quantum limit imposed by the uncertainty relations in ultra-small measurements. We have developed a general, fully non-linear theory of a CEL and have shown that for certain values of the laser parameters, the spontaneous emission events of the two modes are correlated in such a way that their relative phase noise is quenched. • A linear theory of a three-level quantum beat laser is developed, we have shown that under suitable condition the relative phase between the two modes is locked and spontaneous emission noise is quenched from the relative phase angle. The other mean phase angle varies with some definite amount. We have also shown that squeezing in quantum-beat laser is also possible under suitable conditions. A linear theory of a two-level quantum beat laser is also developed. Our calculations show that under suitable values of detunings, dimensions less pump intensity, detuning of the pump frequency from the atomic transition frequency and the mode spacing between the adjacent modes the mean angle of the two modes is locked to a particular value. The relative phase angle of the two modes however diffuses with some definite value. The quantum fluctuations of the pump mode is also considered in order to develop a fully quantum theory of a three-mode laser. Our calculations show that the pump frequency is not only responsible for the mode locking between the adjacent modes but it also produces coherence. The relative phase angle between the three modes is locked to π for suitable values of dimensions less intensity, detunings and mode spacing. The other mean phase angle of the three modes varies with some definite amount. Thus we obtain a two level quantum-beat laser. • It has been generally believed that coherent trapping does not occur in two-level atoms. Considering a system of a two-level atom, we have shown that coherent trapping does occur in such a system for suitable initial conditions and that the emission spectrum under these conditions changes considerably and one of the side modes is completely quenched.

IV. Coherence Properties of Schell- Model Sources We studied some important coherence properties of the beams generated by partially coherent sources. In particular, we considered a Gaussian Schell-model source. We predicted that under suitable conditions, such a partially coherent source can increase the conversion efficiency in second harmonic generation as compared to coherent laser source. In addition, the diffraction properties of a Bessel-Gaussian beam and different conditions under which such a beam becomes diffraction less were investigated. We also considered a generalized Schell-model source and the coherence function, the intensity distribution and the degree of coherence of the wave field generated by such a source were studied. We also considered the second harmonic generation by a partially coherent beam. It is shown that the ratio between the mean square beam radius and mean square coherence radius of the second-harmonic beam depends on simply, the ratio of the beam radius and the phase correlation length of the pump beam in the entrance plane. V. Optical Computing Aside from working on the above mentioned problems, which were proposed in our project, we have also started experimental research activity in the field of optical computing. Using spatial filtering techniques, particularly, the polarization encoded shadow-casting scheme, we have designed adders and sub tractors, decoders and code converters. The designed logic elements are capable of parallel operation at maximum speed, i.e., the speed of light with separate and simultaneous generation of the outputs. Research papers from 30 to 35 are related to this work. In addition research papers from 36 to 49 are also published or submitted from our group during this period.