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Revista brasileira de fisica

Dynamics of third harmonic generation

M. P. Sharma

Published Jan 1, 1983• 0 citations
OpenAlex

Abstract

Ttie build - up of the third harmonic and the depletion of the pump mode are investigated d escribing both the fields quantum mechani- cally. The Heisenberg equationsof motion describing the dynamics of the pump and oF the third harmonic mode are solved in the short-time appro- ximation. It is found that the pump mode shows antibunching effect due to the nonlinear interaction between the modes. l'hird harmonic generation has been studied both theoretically as well as experimental 1 y by severa1 workersl-l1 in the f ield of nonl i- near optics. The third harmonic is being used to produce ultraviolet ra - diation not available from existing lasers by nonlinear optical mixing . The initial build-up of the third harmonic has been studied theoretically, in general, using the parametric approximation i.e. considering the pump depletion to be negligible. It is a reasonable approximation so long as the conversion efficiency is quite low, but recent experiments have shom increased irnprovement in the conversion ef f iciency l l. Under these c i r - cumstances, it is necessary to include pump depletion while studying the harmonic generation. The present communication aims at investigating the dynamics of the third harmonic generation including pump depletion and treating both the pump and the harmonic fields quantum mechanically. The model Hamiltonian of the system is written along the line of shen12 and ~alls'~. The Heisenberg equations of motion describing the time evolution of the pump and the harmonic mode are solved in the short - time approxima- tion14. Our analysis shows thatasaresultof nonlinear interaction, the pump photons manifest antibunching, if the pump mode is in the coherent state and the harmonic mode in the vacum state initially. The photons generated in the harmonic mode are uncorrelated for the quoted i n i tia l conditions.

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Ttie build - up of the third harmonic and the depletion of the pump mode are investigated d escribing both the fields quantum mechani- cally. The Heisenberg equationsof motion describing the dynamics of the pump and oF the third harmonic mode are solved in the short-time appro- ximation. It is found that the pump mode shows antibunching effect due to the nonlinear interaction between the modes. l'hird harmonic generation has been studied both theoretically as well as experimental 1 y by severa1 workersl-l1 in the f ield of nonl i- near optics. The third harmonic is being used to produce ultraviolet ra - diation not available from existing lasers by nonlinear optical mixing . The initial build-up of the third harmonic has been studied theoretically, in general, using the parametric approximation i.e. considering the pump depletion to be negligible. It is a reasonable approximation so long as the conversion efficiency is quite low, but recent experiments have shom increased irnprovement in the conversion ef f iciency l l. Under these c i r - cumstances, it is necessary to include pump depletion while studying the harmonic generation. The present communication aims at investigating the dynamics of the third harmonic generation including pump depletion and treating both the pump and the harmonic fields quantum mechanically. The model Hamiltonian of the system is written along the line of shen12 and alls'. The Heisenberg equations of motion describing the time evolution of the pump and the harmonic mode are solved in the short - time approxima- tion14. Our analysis shows thatasaresultof nonlinear interaction, the pump photons manifest antibunching, if the pump mode is in the coherent state and the harmonic mode in the vacum state initially. The photons generated in the harmonic mode are uncorrelated for the quoted i n i tia l conditions.
We studied photon antibunching in the pump and the harmonic modes of the sixth harmonic generation process. The generalized interaction Hamiltonian is solved for several particular cases in the Heisenberg picture, and the possibility of observing photon antibunching is investigated using the short-time approximation technique. It is shown that the photon antibunching in the pump field depends on the number of pump photons, the interaction time, and the coupling of the field between the modes. With the same amount of pump photons, we deduced that third-order photon antibunching is more nonclassical than second- and first-order photon antibunching. In this process, the effect of photon antibunching is not seen in the harmonic mode over pump mode. It is shown that photon antibunching is more noticeable with shorter interaction times as the depth of nonclassicality increases and the second-order correlation function decreases. The first-order Hamiltonian interaction, which stimulates both pump and harmonic fields, is demonstrated to be more nonclassical than the second-order Hamiltonian interaction. It is clear that the coherent state, or vacuum state, of a pump field with a nonzero harmonic field creates photon clusters because the pump field causes interaction, which leads to bunching effects. It is interpreted that the degree of photon antibunching is maximum in the regions where the second correlation function is minimum. Photon antibunching has been shown to be one of the quantum properties of light.
With the help of second-order nonlinear interactions and hence Hamiltonian, we construct the equations of motion corresponding to pump and signal (second harmonic) fields in a second harmonic generation. The corresponding coupled differential equations involving the non-commuting field operators are not solvable in closed analytical forms. With the help of a perturbation method, we obtain analytical solutions of these field operators up to cubic orders in the interaction constant. In an appropriate limit (truncating the solution up to the second order in the interaction constant), these solutions lead to the existing solutions under the short-time approximation. The present analytical solutions are exploited to investigate the squeezing and the antibunching of photons of the input coherent light coupled to the said second-order nonlinear medium. We report the squeezing and antibunching effects of the pump field even for solutions up to the second order and hence the present results are consistent with earlier results. However, for the second harmonic mode, we report the squeezing involving the leading cubic term in the interaction constant.
In Part I [1], we presented a quantum theory of third-harmonic generation (THG) under two-photon absorption in a two-level atomic medium. We studied the time evolution of the third-harmonic field and its coherence properties by neglecting the depletion of the pump power. In this paper we extend the theory by taking the depletion of pump power into account. The density matrix for the third-harmonic field is derived. The study of the growth rate of the density matrix in time shows the quantum mechanical effects more clearly. The effects of detuning in the two-photon interactions and atomic motion are pointed out. The results are compared with the earlier results and differences are brought out.
The squeezing and sub-Poissonian eects of light in third harmonic generation are investigated based on the fully quantum mechanical approach up to the rst order Hamiltonian interaction in gt, where g is the coupling constant between the modes per second and t is the interaction time between the waves during the process in a nonlinear medium.The coupled Heisenberg equations of motion involving real and imaginary parts of the quadrature operators are established.The occurrence of amplitude squeezing eects in both the quadratures of the radiation eld in the fundamental mode is investigated and found to be dependent on the selective phase values of the eld amplitude.The photon statistics of the pump mode in this process have also been investigated and found to be sub-Poissonian in nature.It is shown that for particular phase values the amplitude squeezing and sub-Poissonian photon statistics of light occur simultaneously.It is observed that there is no possibility to produce squeezed light in the harmonic mode up to rst-order interaction in gt.Further, it is found that the normal squeezing in the harmonic mode directly depends upon the amplitude-cubed squeezing of the initial pump eld to the case of second-order interaction in gt.This gives a method of converting higher-order (amplitude-cubed) squeezing of the fundamental mode into normal squeezing of the harmonic mode and vice versa.
The quantum dynamics of a parametric interaction of the electromagnetic field with a nonlinear medium is considered. The three nonlinear coupled Heisenberg equations of motion are solved under the short-time approximation. The characteristic function for the normal ordering rule of association is evaluated. This function is then used to obtain the time dependence of the density operator. Assuming that the initial state of the system is a coherent state, explicit expressions for the diagonal coherent state representation of the reduced density operators for the pump as well as for the signal mode are obtained. For the pump mode, the initially coherent state remains coherent, whereas for the signal mode the diagonal coherent state representation is found to be a gaussian distribution whose variance is proportional to the average number of photons initially present in the pump mode.
The study investigates antibunching, sub-Poissonian statistics, and photon correlation functions in the pump and harmonic modes of the seventh harmonic generation process.We investigated various approaches to photon antibunching by solving the generalised interaction Hamiltonian in the Heisenberg picture using short-time approximation.It is inferred that the number of pump photons, coupling of the field strength, and interaction times significantly influence photon antibunching.It is observed that third-order antibunching in the pump mode has greater nonclassicality than second-and first-order antibunching.In the harmonic mode, the nonclassical effect is found to be less pronounced and becomes more evident as the interaction time decreases.The first-order Hamiltonian interaction induces a stronger nonclassical effect than the second-order interaction.The interaction of a coherent pump field with a nonzero harmonic field leads to photon clustering and bunching effects.Finally, photon antibunching appears most pronounced, indicating nonclassicality, as the second-order correlation function decreases.This confirms that antibunching is a fundamental quantum property of light.