In this paper, interaction of an ultrashort single-cycle pulse (USCP) with a bound electron without ionization is reported for the first time. For a more realistic mathematical description of USCPs, Hermitian polynomials and combination of Laguerre functions are used for two different single cycle excitation cases. These single cycle pulse models are used as driving functions for the classical approach to model the interaction of a bound electron with an applied electric field. A new novel time-domain technique was developed for modifying the classical Lorentz damped oscillator model in order to make it compatible with USCP excitation. This modification turned the Lorentz oscillator model equation into a Hill-like function with non-periodic time varying damping and spring coefficients. Numerical results are presented for two different excitation models and for varying spring and damping constants. Our two driving model excitations provide quite different time response of the bound electron. Different polarization response will subsequently result in relative differences in the time dependent index of refraction.
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Ultrashort light pulse research has led to the creation of laser systems generating pulses only a few cycles in duration . Now that these ultrashort few-cycle EM pulses exist experimentally, the need for mathematical models to describe these short pulse interactions with matter becomes very important . There is a growing need to model and to understand the interaction of single ultrashort pulses or a train of ultrashort pulses with matter below the point where strong field effects dominate. This need is driven by the advances made in femtosecond (fs) and attosecond (as) laser technologies. Applications of these ultra short pulses range from free space communications, 3D depth profiling in biological samples, optical communication, high resolution/precision atomic and molecular scale imaging, high speed electronics and optoelectronics in terahertz (THz) regime, behavior of electrons in quantum structures, relativistic physics, high-energy physics, astrophysics to medical applications. Furthermore, ultrafast few cycle lasers are expected to be a promising solution to probe the fastest events in atomic, molecular, biochemical, and solid state systems due to their unique property of being the shortest controlled bursts of energy ever developed [3–11].
Basic physics of the pulse-matter interaction depends strongly on the ratio of the pulse duration and the characteristic response time of the medium (as well as on the pulse intensity and energy). This ratio is the key term in the polarization response of the medium. The goal of this study is to provide insight in the linear polarization response of dispersive materials to ultrashort single cycle pulses. This paper is concerned with the case where the electric field strength is low enough to not produce ionization. Since the energy is below the ionization threshold of the medium, there is not any plasma effect during the interaction of the applied field with the matter. Understanding the linear polarization response is extremely crucial in order to formulate a realistic field integral. This realistic field integral will provide a more realistic propagation model of optical pulses through dispersive media [12–24]. The interaction of an ultra short pulse with matter involves the interaction of the incident electric field with the electrons of the material. In this study, classical approaches to this problem are modified for solving the interaction of a single-cycle ultrashort laser pulse with a bound electron without ionization.
2. Mathematical model
In order to make an original contribution for the analysis of the interaction of an ultrashort single-cycle pulse (USCP) with a bound electron without ionization, first it is necessary to find a realistic model for a USCP. Such pulses have a rather different structure from conventional modulated quasi-monochromatic signals with a rectangular or Gaussian envelope [25–28]. Due to the following main reasons associated with USCPs, combination of Laguerre functions and Hermitian polynomials (Mexican Hat) are used in this study for modeling applied EM field:
- i) Arbitrary transient steepness: The rising and the falling times of the signal can be essentially unequal.
- ii) Varying zero spacing: The distances between zero-crossing points may be essentially unequal.
- iii) Both the waveform envelope and its first spatial and temporal derivatives are continuous.
- iv) Arbitrary envelope asymmetry: USCP waveforms can be classified conventionally for two groups.
The combination of Laguerre functions for defining the spatiotemporal profile of a USCP is defined as where is a single Laguerre function with order m and . Here, c is the velocity of light in vacuum, z is the propagation direction and is the time scale of the pulse. In this study, the combination of 2nd and 4th order Laguerre functions are used to define a single USCP:Fig. 1(a) ].
Figure 1(b) shows the first derivative of the applied field and it is clearly seen that the analytical expression in Eq. (1) satisfies the conditions of arbitrary transient steepness and arbitrary envelope asymmetry. From Fig. 1(a), it is also clearly seen that it satisfies the condition of varying zero spacing for a USCP. In addition to these, time profile of the Laguerre USCP is almost fulfilling the integral property:Fig. 2(a) ], the following definition is used:Figure 2(b) illustrates that the Hermitian pulse satisfies the above concerns.
In addition to the question how to formulate ultrashort single cycle transients, it is also natural to ask how these pulses propagate in optical medium. In this study, USCP means the smallest possible single cycle piece (unity source) of a wave packet. It is the part of an actual carrier field and does not contain any other carrier fields in itself. For a USCP, it is difficult to introduce the concept of an envelope and it is not possible to define a group velocity. For such short pulses the distinction between carrier oscillations and slowly varying envelope (SVE), which have two different temporal scales that are peculiar to quasi-monochromatic pulses, becomes diffuse or meaningless [22,29–31]. Jumping from many cycle optical waves to single cycle optical pulses in dealing with light-matter interaction, the mathematical treatments should be revised. The traditional analysis of pulsed EM phenomena is questionable [25–28]. If the applied field is a USCP, the shortest possible field as explained above, then it is impossible to separate the applied source into pieces to find the effect of each part (or piece) by superposing as being suggested in the models explained in many fundamental textbooks .
In order to understand the USCP-medium interaction phenomenon, we must acquire certain special features such as operating directly with Maxwell equations beyond the scope of Fourier representations [25–28]. Since the situations occur where the time scale of the pulse is equal or shorter than the relaxation time of the medium, material has no time to establish its response parameters during the essential part of the pulse continuance [23,33–36]. These parameters, which govern the polarization response of the media, change their values during the pulse continuance [23,33]. Thus, solutions of Maxwell equations with time-dependent coefficients are required for the analysis of the wave dynamics [35,36].
In our study, we consider an approach such that under a single USCP excitation, the change in the relative position of a bound electron to its parent atom without ionization will change the amplitude of the dipole in the atom and so forth the instantaneous polarization. As a result of this fluctuation in the polarization, the index of refraction will change in the duration of the single USCP excitation during which the propagation dynamics of the same applied USCP and the other USCPs coming after the first one will be evaluated. So physically, we consider a case where the medium is including the source. This is a common situation especially in optical communication. In addition to this, we can associate this approach to some diagnostic techniques in ultrafast optics such as pump-probe experiments where both pump and probe pulses propagate and evaluate the time varying physical parameters of the medium. But before diving into Maxwell equations, we have to figure out how the polarization response of the medium must be handled for the interaction of a USCP EM field with a bound electron. Understanding the polarization response of the material under the excitation of a USCP EM field is one of the most important, not clearly answered yet, core question of today and near future ultrafast laser engineering.
Polarization is a very crucial physical term, especially for optical communication, since it defines the change in the index of refraction in the material due to the applied field [23,33,34,37,38]. In terms of permittivity, we can write index of refraction (for a nonmagnetic material) as:8,10,39–41].
The starting point of all these dynamics is the inhomogeneous wave equation:
For USCP excitation, unlike the long pulse excitation fields, the response (oscillation) of the electron must be handled in a different manner. Since, both due to the mass of inertia of the electron and the shortness of the USCP compared to the relaxation time of the medium, the electron will not sense the applied field exactly at the leading edge point of the pulse. The response of the electron to the applied field will increase gradually. During this sense, the electron will not follow the oscillation profile of the applied electric field. So, the oscillation field of the electron will not only have a difference in the phase but also will have a different time profile (time-dependency) with the applied field. In regular cases, if the applied field is in the form of time-dependency, then we assume that the oscillation of the electron will be in the same time-dependency form. In the literature, Lorentz oscillator model is directly used in time-dependency . But for a USCP excitation, not only the time-dependency is not valid, but also the oscillation field will have a different waveform than the applied field waveform (time-dependency). This means that, the term in Eq. (6), that is the oscillation field of the electron, will have a modified form of time-dependency with respect to the applied USCP. In order to define the modified function , we developed a new time domain technique that we call “Modifier Function Approach”. In this approach, we define the oscillation field of the electron as the multiplication of the applied USCP with the modifier function:Eq. (7) into Eq. (6), we obtainEq. (8), we obtain Eq. (10):Eq. (10) as:Eq. (11) that it has a similar form with a Hill type equation where for a regular Hill equation, and terms are periodic and the right side is zero. A linear equation of this type occurs often when a system exhibiting periodic motion is perturbed in some way . This type of equation was first derived by G.W. Hill to describe the effect of perturbations on the orbit of the Moon, and it occurs in many other places in physics, including the quantum motion of electrons in a periodic potential of a crystal . The band theory of solids is based on a similar equation, as is the theory of propagating electromagnetic waves in a periodic structure . Other applications include parametric amplifiers. Although and terms are periodic in a Hill equation, in our case they are not. So, in our model, Eq. (11) is a Hill-like equation which has a dc source on its right side and a time-dependent damping coefficient (12) and a time-dependent spring coefficient (13) in terms of a damped forced oscillator model. The objective of Eq. (11) is to find the modifier function which can be then used to define the oscillation field (polarization response) of the material. Due to the time-dependent damping and spring coefficients, the modifier function is totally coupled with the time dependency or time profile of the applied field.
Equation (6) could also have been solved directly in the temporal domain, in which case we would have lost the analogy with the Hill-like equation. But the appropriateness of using the more complicated approach with the modifier function has solid physical reasons. In the case of a USCP excitation, the polarization response of the material is not unique all through the pulse continuance. Due to the shortness of the duration of the applied USCP comparing to the relaxation time of the bound electron, the interaction dynamics and the ability of the material to sense and follow the applied USCP field during its continuance will be completely different than the conventional matter-field interaction approach. In Eq. (6), physical parameters (damping and spring coefficients) are constant. However, the interaction dynamics will not be constant during the USCP excitation. So, in order to penetrate the effect of the applied field into the oscillator model via these physical parameters to have a better understanding of the oscillation response of the material under USCP excitation, we must find the definition of these physical parameters in terms of the applied field and the physical constants of the system (material). Equation (12) and Eq. (13) are these definitons. They are being used in Eq. (11) to find the modifier function which has been embedded into Eq. (6). The physical dimension of the modifier function is a dipole moment per unit force. It frames the time dependency and the phase delay of the oscillation field of the bound electron under USCP excitation.
3. Numerical results and discussions
In Fig. 4 , different interaction characteristics of Laguerre and Hermitian pulses are shown for a fixed, relatively low value of damping constant (). Due to the definition: , ( is the mass of electron, is the spring constant for bound electron), t he free oscillation frequency of material is in UV range for spring constant values of 4 N/m, 9 N/m, 325 N/m, 525 N/m [Figs. 4(a), 4(b), 4(c), 4(d), 4(e), 4(f), 4(g), 4(h)], 650 N/m [Fig. 8(b) ] and 750 N/m [Figs. 5(b) , 8(c)]. For spring constant values of 1500 N/m [Fig. 5(c)], 2500 N/m [Figs. 4(i), 4(j)] and 7500 N/m [Fig. 8(d)], the free oscillation frequency is in X-ray range. As it is clearly seen in Fig. 4, the Hermitian interaction has a more tendency to oscillation than the Laguerre interaction for relatively low values of spring constant [see Figs. 4(a), 4(b), 4(c), 4(d)]. As the spring constant is increased, Laguerre interaction gains a more oscillatory profile [see Figs. 4(e), 4(g)] while the oscillation due to the Hermitian pulse interaction stabilizes and its time profile settles down into the inverted phase time profile of the excitation pulse (inverted Mexican Hat) [see Figs. 4(f), 4(h), 4(j)]. Here, the amplitude of oscillation or the amplitude of trembling-like motion of the electron is in the range of 10−20 m – 10−21 m which is in the scale of electron radius length. Finally, as the spring constant is increased to relatively higher values, the Laguerre interaction settles down into the inverted phase time profile of the excitation pulse, too (inverted Laguerre pulse) [see Fig. 4(i)]. Figure 4 shows a very clear distinction between the interaction characteristics of Laguerre and Hermitian USCPs until the spring constant is 2500 N/m (after this value, we obtain only the inverted phase time profile of the excitation source for the oscillation). The oscillation characteristics of bound electron under different single USCP sources originates from modifier function approach. The Hill-like equation, which is the result of the modification on the classic Lorentz damped oscillator model with the modifier function approach, causes the time varying physical parameters to come into play during the interaction process. Since these physical parameters (time varying damping and spring coefficients) are absolutely source dependent, they behave differently in the pulse duration of each different USCP source. As a result of this, we see different oscillation profiles for a bound electron under a single Laguerre and Hermitian USCP excitations.
In Fig. 5, response of a bound electron is shown for a Laguerre pulse excitation for varying values of spring constant with a fixed, relatively higher damping constant value (1x1016) than the previous case (Fig. 4). An interesting feature here in Fig. 5(a) and Fig. 4(g) is that although they are at the same spring constant value, they show different oscillation characteristics. Due to a higher dampimg coefficient in Fig. 5(a), while the oscillation attenuates quicker at the second half cycle of the Laguerre USCP than in Fig. 4(g), it hits to a higher peak at the first half cycle of the excitation pulse than in Fig. 4(g). So, for a reasonable value of spring constant, while relatively higher damping coefficient makes the first half cycle of the Laguerre USCP more efficient in the means of interaction, it makes the second half cycle less efficient. In order to compare oscillation results more detailly between Figs. 5(a) and 4(g), it is necessary to look at their physical parameter solutions such as time varying damping and time varying spring coefficients. As it is explained above, these time varying parameters come into play due to the nature of “Modifier Function Approach”. In Fig. 6 , time varying damping coefficient, time varying spring coefficient and the modifier function solutions ofFigures 4(g) and 5(a) are shown respectively for two different damping constant values with a fixed spring constant at 525 N/m. In Figs. 6(a) and 6(c), a sudden jump is seen in the time varying damping coefficient profiles at the time point where the excitation pulse changes itspolarization direction. Although they look identical, the magnified views [see Figs. 7(a) , 7(b), 7(c), 7(d)] of the left and right wings of the damping coefficient show the difference between two different damping constant cases. Here, the left wing corresponds to the first half cycle, right wing corresponds to the second half cycle of the Laguerre excitation pulse. Comparing the amount of the change on the y-axis with the time duration on the x-axis between Figs. 7(a) - 7(b), and 7(c) - 7(d), it is easy to see the reasonable amount of difference to affect the solution of modifier function [see Figs. 6(i), 6(j)]. For time varying spring coefficients [see Figs. 6(e), 6(g)], a significant difference is seen in the time profile although the spring constant values are the same for both cases. The jump in Fig. 6(g) hits a higher peak than the jump in Fig. 6(e). This can be a reasonable explanation for a relatively low oscillation tendency in the second half cycle of Fig. 5(a) than the Fig. 4(g). It can be said that, due to the dissipation of higher energy, this jump causes a lower oscillation profile for the bound electron during its interaction with the second half cycle of the Laguerre pulse in Fig. 5(a) than in Fig. 4(g). In Fig. 5(c), as the spring constant is increased to a relatively higher values, same as in Fig. 4(i), the oscillation profile settles down into the inverted time phase profile of the excitation pulse. Different from Fig. 4(i), the oscillation settles down at a relativley lower spring constant value. So, it can be said that, for a higher damping constant, a lower spring constant is enough to stabilize the oscillation profile in time domain.
For a damping constant value of 1x1017 (Fig. 8), very different oscillation behaviors are seen than the previuos cases (Fig. 4) of Hermitian pulse excitation. The most prominent feature in Figs. 8(a), 8(b) and 8(c) is the high frequency oscillation profile with a phase delay wrt. excitation pulse. In Fig. 8, spring constant is increased gradually from 8(a) to 8(c) while keeping the damping value constant. For a relatively low value of spring constant in Fig. 8(a), the main lobe and the trailing tail of the excitation pulse have almost no effect on the oscillation of the electron. The bound electron starts sensing the leading tail of the Hermitian excitation after a phase delay of 5 fs. In Fig. 9 , the modifier function solutions for the Hermitian pulse excitation for Fig. 8 is shown. As it is clearly seen in Fig. 9(a), modifier function suppresses the interaction effect of main lobe and the trailing tail of Hermitian function. As a result of this, the bound electron starts sensing the excitation pulse with a phase delay [Fig. 8(a)] associated with the modifier function. Same behaviour of the modifier function is seen in Figs. 9(b) and 9(c), too. As a result of this, approximately 2fs phase delay occurs in Figs. 8(b) and 8(c). In Fig. 9(d), the type of modifier function is seen that gives a completely phase inverted time profile of the excitation pulse for the oscillation of the bound electron. In Fig. 8(d), the stabilized oscillation profile is seen as a result of this modifier function.
The results of this work indicate that if the applied field is a USCP, then it is not possible to separate the field into pieces to find the polarization effect of each part of the applied field on a bound electron since the USCP cannot be further broken down into separate pieces of the applied field. The traditional Fourier method of multiplying the Delta function response with the applied field and integrating (superposing) this product in time can only be used for SVE approximation which is not realistic for single cycle pulses of unity femtosecond and attosecond applied fields. In a USCP case, the Lorentz oscillator model must be modified in order to find the polarization effect of a single USCP. Since a USCP is extremely broadband, it is not realistic to use a center frequency in the calculations as is done in the Fourier series expansion approach. Results in this work are presented on the transient response of the system during the USCP duration without switching to frequency domain. In order to accomplish this mathematically, we developed a new technique we label as the “Modifier Function Approach”. The modifier function is embedded in the classic Lorentz damped oscillator model and by this way, we upgrade the oscillator model so that it is compatible with the USCP on its right side as the driving force. Results of this work also provide a new modified version of the Lorentz oscillator model for ultrafast optics. The results also indicate that the time response of the two models used to represent the USCP can alter the time dependent polarization of the material as it interacts with a single cycle pulse.
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