Tham khảo tài liệu 'laser pulse phenomena and applications part 10', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Intensity Effects and Absolute Phase Effects in Nonlinear Laser-Matter Interactions. 261 We summarize some basic properties of the displacement operator D a and the coherent states denoted by a which are generated from vacuum under the action of the classical current. In order to simplify these formulae we do not display the mode index k s where k is the propagation vector and s is the polarization index of the scattered radiation D a exp aat-a a D a O a skak k 1 2 k exp -1 a 2 a a a a 23 Finally we note that the the displacement property Dt a aD a a a means a shift of the quantized amplitudes. By putting explicit form of the electron current in the integral on the right hand side of Eq. 22 and using the Jacobi-Anger formula for the generation of the ordinary Bessel functions we obtain aks t i fi cepo 2nfi rokL3 1 2 so-E k s S J poXo Eo-k poXo Eo-k Jdtexp i rok n o t . 24 The time integral for large interaction times give a disctrete sequence of resonant frequencies rok nroo which are just the high-harmonic frequencies. In general if the quantized field is initially on the vacuum state thus we do not consider induced processes . O to 10 then the quantum state O t developing due to the interaction with the oscillating electron will be a multimode coherent state. Because of the frequency condition coming from the resonant time integral in Eq. 24 each harmonic components will be in a coherent state regardless of their average occupation which is governed by the size of the Bessel functions . The expectation value of the energy of a particular component is given by the expression ntoo at mosa mos ro2 s Eo 2 n4 2J z z 2 s k k z npo s-Eo . 25 The formula Eq. 25 is equivalent with the the classical formula for the high-harmonic production in nonlinear Thomson scattering however it contains only the first moment of the photon distribution so it should be considered as a mean value. Of course each harmonics of the scattered quantum field is loaded by inherent .
