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Numerical Simulations of Chirped Excitation

Benjamin Iannitelli

Published Jan 1, 2012• 0 citations
OpenAlex

Abstract

In this project I developed a general method of finding the optimal laser excitation for an ensemble of two-level atoms with the primary goal of exciting as many atoms as possible, as quickly as possible, for as long as possible, in order of decreasing priority. Specifically, I simulated the laser excitation of a collection of Rubidium-87 atoms from (n=5, L=0, S=1/2, J=1/2) to (n=5, L=1, S=1/2, J=3/2), by finding numerical solutions to the optical Bloch equations. I optimized the parameters of a linear chirp paired with a Gaussian intensity pulse first neglecting and then including spontaneous emission, and then for a hyperbolic-tangent chirp paired with a squared-hyperbolic-secant intensity pulse including spontaneous emission. Comparing the optimal parameters for the linear chirp both with and without spontaneous emission demonstrated that neglecting spontaneous emission will lead to significant errors, at least when considering Rubidium-87. Comparing the linear chirp to the hyperbolic-tangent chirp (both with spontaneous emission) showed that both chirp shapes lead to excitation of greater than 96% of the population, and that based on my simulations, the linear chirp was slightly better at meeting all three goals of ceiling, speed and endurance. The best of all the excitations that I found using a linear chirp is to use a chirp rate between 0.2 and 0.65 GHz/ns and to use a pulse with a FWHM of 100 ns and a peak intensity of 10 W/cm^2.

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Pulse shaping is the technique which controls the ultra-short pulse shape, and it became of great technological interest because of its potential applications in laser pulse compression, digital communications, microscopy etc. We demonstrate the idea of pulse-shaping technique and pulse propagation with low energy losses in a resonant linear absorbing medium. This thesis presents the results of a study of the propagation of Gaussian and hyperbolic secant ultrashort chirped and chirp-free pulses in homogeneously and inhomogeneously broadened resonant linear absorbers. Changes to the pulse shape and energy loss factor are presented as the pulse propagates in the absorber. The Fast Fourier method is used to numerically determine both the normalized intensity profile and the pulse spectrum. Our results show that, for pulse durations shorter than the relaxation time, chirped pulses in absorbing media obey the area theorem, with their shape changing with the propagation distance. Simulation results of the spectra of chirped pulses clearly show the burning of a spectral ’hole’ as the pulse propagates, with the pulse energy pushed away towards the wings. When compared to chirp-free pulses, chirped pulses reshape faster and develop wings in their tail due to initial phase modulation. Simulation results of the energy loss factor show that chirped pulses propagating in resonant linear absorbers sustain less energy losses than do chirp-free pulses. A comparison of chirped secant and Gaussian pulses shows that secant pulses propagate with lower energy losses. Analytic solutions are presented for long-distance asymptotic expressions of initial rms spectral bandwidth as well as for the attenuation factor of chirped Gaussian pulses. These analytical results are in agreement with numerical simulations. The comparison of energy losses of short chirped Gaussian pulses and long pulses of any profile in linear absorbers is also discussed in the thesis.

•Dec 14, 2011•0 citations
In this project I developed a general method of finding the optimal laser excitation for an ensemble of two-level atoms with the primary goal of exciting as many atoms as possible, as quickly as possible, for as long as possible, in order of decreasing priority. Specifically, I simulated the laser excitation of a collection of Rubidium-87 atoms from (n=5, L=0, S=1/2, J=1/2) to (n=5, L=1, S=1/2, J=3/2), by finding numerical solutions to the optical Bloch equations. I optimized the parameters of a linear chirp paired with a Gaussian intensity pulse first neglecting and then including spontaneous emission, and then for a hyperbolic-tangent chirp paired with a squared-hyperbolic-secant intensity pulse including spontaneous emission. Comparing the optimal parameters for the linear chirp both with and without spontaneous emission demonstrated that neglecting spontaneous emission will lead to significant errors, at least when considering Rubidium-87. Comparing the linear chirp to the hyperbolic-tangent chirp (both with spontaneous emission) showed that both chirp shapes lead to excitation of greater than 96% of the population, and that based on my simulations, the linear chirp was slightly better at meeting all three goals of ceiling, speed and endurance. The best of all the excitations that I found using a linear chirp is to use a chirp rate between 0.2 and 0.65 GHz/ns and to use a pulse with a FWHM of 100 ns and a peak intensity of 10 W/cm^2.
The excitation of a two-level system by a chirped laser pulse is analyzed using an analytical approach (in the perturbation-theory limit) that involves the modified rotating-wave approximation and the numerical solution to the Bloch equation. The dependence of the population of the upper energy level on the chirp is studied for various radiation intensities and pulse durations. An exact solution is compared with the result obtained using the modified rotating-wave approximation and the perturbation theory. It is demonstrated that, for a certain range of parameters, the excitation probability of a two-level system can be effectively controlled via a variation in the chirp.
Via a non-perturbative method we study the population dynamics and photoelectron spectra of Cs atoms subject to intense chirped laser pulses, with Gaussian beams. We include above threshold ionization spectral peaks. The frequency of the laser is near resonance with the 6s-7p transition. Dominant couplings are included exactly, weaker ones accounted for perturbatively. We calculate the relevant transition matrix elements, including spin-orbit coupling. The pulse is taken to be a hyperbolic secant in time and the chirping a hyperbolic tangent. This choice allows the equations of motions for the probability amplitudes to be solved analytically as a series expansion in the variable , where is a measure of the pulse length. We find that the chirping changes the ionization dynamics and the photoelectron spectra noticeably, especially for longer pulses of the order of au. The peaks shift and change in height, and interference effects between the 7p levels are enhanced or diminished according to the amount of chirping and its sign. The integrated ionization probability is not strongly affected.
A study is presented for the time-dependent resonance-fluorescence spectrum of two-level atoms irradiated by hyperbolic-secant pulsed lasers, allowing for an arbitrary spontaneous-emission rate from the excited level to the ground level. Novel features of the temporally transient spectrum have been observed. The spectrum is dominated by a central peak which oscillates in time and is accompanied by multiple side peaks. Asymptotically (i.e., after the pulse has passed) the side peaks disappear, leaving a tail at the line center which displays a Lorentzian shape of width equal to the sum of the spontaneous-emission rate and the width of the spectrometer considered and of magnitude proportional to the instantaneous intensity of the pulse. The exact behavior of the spectrum is dictated by the area of the engaged pulse and by how the atom is prepared initially. In particular, it is found that the spectrum can be asymmetric even when the laser is tuned to exact resonance if the atoms are initially prepared in a certain class of proper coherent states, irrespective of how the field amplitude may vary in time. An analysis of the spectral symmetry without resorting to any explicit evaluation of the spectrum itself is also presented. Furthermore, the associated optical Bloch equations are solved analytically in terms of hypergeometric functions in a similar manner to that done previously for the undamped two-level Schr\"odinger equation. Specific numerical results are presented for pulses of area equal to \ensuremath{\pi}, 2\ensuremath{\pi}, 3\ensuremath{\pi}, 4\ensuremath{\pi}, and 5\ensuremath{\pi} for both symmetric and asymmetric cases.
We have found new analytic results for a wide variety of atomic excitation processes which involve the use of amplitude-and frequency-modulated laser pulses: (1) We have unified all the previously known solutions including those of Rosen and Zener,1 Landau2 and Zener,3 Rabi,4 Bambini and Berman,5 Demkov,6 McCall and Hahn,7 Allen and Eberly,8 and Hioe,9 among others, and have significantly extended it with expressions involving three free parameters and an arbitrary function to an infinite variety of amplitude-and frequency-modulated laser pulses.10 We have found a simple condition which the three essential parameters in the amplitude-and frequency-modulation functions must satisfy in order to achieve a complete population inversion, or to achieve a complete population restoration as in the case of self-induced transparency7 or soliton propagation.
Pulse shaping is the technique which controls the ultra-short pulse shape, and it became of great technological interest because of its potential applications in laser pulse compression, digital communications, microscopy etc. We demonstrate the idea of pulse-shaping technique and pulse propagation with low energy losses in a resonant linear absorbing medium. This thesis presents the results of a study of the propagation of Gaussian and hyperbolic secant ultrashort chirped and chirp-free pulses in homogeneously and inhomogeneously broadened resonant linear absorbers. Changes to the pulse shape and energy loss factor are presented as the pulse propagates in the absorber. The Fast Fourier method is used to numerically determine both the normalized intensity profile and the pulse spectrum. Our results show that, for pulse durations shorter than the relaxation time, chirped pulses in absorbing media obey the area theorem, with their shape changing with the propagation distance. Simulation results of the spectra of chirped pulses clearly show the burning of a spectral ’hole’ as the pulse propagates, with the pulse energy pushed away towards the wings. When compared to chirp-free pulses, chirped pulses reshape faster and develop wings in their tail due to initial phase modulation. Simulation results of the energy loss factor show that chirped pulses propagating in resonant linear absorbers sustain less energy losses than do chirp-free pulses. A comparison of chirped secant and Gaussian pulses shows that secant pulses propagate with lower energy losses. Analytic solutions are presented for long-distance asymptotic expressions of initial rms spectral bandwidth as well as for the attenuation factor of chirped Gaussian pulses. These analytical results are in agreement with numerical simulations. The comparison of energy losses of short chirped Gaussian pulses and long pulses of any profile in linear absorbers is also discussed in the thesis.