Based on the dressed state formalism, we obtain the adiabatic criterion of the sum frequency conversion. We show that this constraint restricts the energy conversion between the two dressed fields, which are superpositions of the signal field and the sum frequency field. We also show that the evolution of the populations of the dressed fields, which in turn describes the conversion of light photons from the seed frequency to the sum frequency during propagation through the nonlinear crystal. Take the quasiphased matched (QPM) scheme as an example, we calculate the expected bandwidth of the frequency conversion process, and its dependence on the length of the crystal. We demonstrate that the evolutionary patterns of the sum frequency field’s energy are similar to the Fresnel diffraction of a light field. We finally show that the expected bandwidth can be also deduced from the evolution of the adiabaticity of the dressed fileds.
©2010 Optical Society of America
The nonlinear frequency conversion in crystals is one of the popular methods for generating the tunable optical radiation. This is a three-wave mixing process, during which the requirement of phase matching is critical to the efficiency of the frequency conversion. In the efficient frequency conversion of broadband optical signals, however, it is difficult to satisfy the restriction of phase matching due to its sensitivity to the inputting frequencies. Thus, exploring the efficient frequency conversion between the broadband optical signals is a meaningful thing .
In recent years, the crystal’s characteristics can be modulated by periodic or aperiodic electric field. Using this technique in QPM scheme one can solve the problem of the broadband frequencies conversion [2–5], but with small efficiency. However, Chirped QPM gratings as an efficient technique can achieve the robust efficiency conversion among broad frequencies range. This scheme can be used to manipulate short laser pulse in second harmonic generation [6–8], difference frequency generation , and in parametric amplification [7,10]. Comparing with the way of describing the interaction between a laser pulse and a two-level atomic system , Haim Suchowski et al. [12,13] recently proposed the concept of adiabatic frequency conversion, and successfully realized a robust, highly efficient broadband wavelength conversion in the laboratory. Although the proposed physical ideas are very novel and were successfully demonstrated in the lab, they did not illustrate which physical quantities are adiabatic and how the adiabatic restriction affects the sum frequency process.
In this letter, using the dressed state formalism [14–16], we obtain the adiabatic criterion of energy conversion during the sum frequency process through the propagation equation. This constraint is obeyed by the two dressed fields composed of the signal field and the sum frequency field . The relations among the sum frequency and the adiabatic- and diabatic-process are discussed. We show that the energy conversion points of the sum frequency correspond to the diabatic points. We calculate the expected bandwidth of the frequency conversion process, and its dependence on the length of the crystal. We demonstrate that the evolutionary patterns of the sum frequency field’s energy are similar to the Fresnel diffractions of a light field. We show that the expected bandwidth can be also deduced from the evolution of the adiabaticity of the dressed fileds. We also give a satisfying theoretical account for the previous experimental observation [12,13].
2. Propagation equation and its adiabatic solution during the sum frequecy
Consider the sum frequency generation process under the first kind phase-matching condition. The two input fields are called as pump field and signal field, respectively. It is assumed that the pump field is so strong that we regard it to be constant during its propagation. Under the undepleted pump approximation, the coupled equation can be simplified as [12,13]:Eq. (1) becomes with , and . is the derivative of to z, while is that of to z. The eigenvalues or the G are and , respectively. Defining, one may obtain the rotation matrix and its inversion . Do operation about as with R. Borrowing the name of the dressed-pulse fields , we call and as dressed fields. The dressed fields obey the new propagation equation14–16], the adiabatic criterion about the dressed fields and isEquation (3) is a general expression for the sum frequency conversion. When the adiabatic condition (3) holds, Eq. (2) becomesEquation (4) means that there is no energy conversion between the dressed fields and , and the dressed fields are entirely determintated by the boundary condition. Evidently, , . Then Eq. (3) can be replaced by , which is identical to that presented by the literature . It shows that the dressed fields and satisfies the adiabatic restriction only when the variation rate of the phase mismatch with the position z is very slowly.
When the second order susceptibility is a constant, then the coupling coefficient q is real and independent of the position z, i.e., with . Given that the phase mismatch is also independent of z, one can easily show the adiabatic condition is satisfied well. Under the adiabatic criterion, the solution of Eq. (4) is , with. This solution manifestly shows that there is not any energy exchange between and , i.e., they are adiabatic. At the input side, the signal field is , then the dressed fields are and . In this way, the signal field and the sum frequency field are expressed asEquation (5) shows that energy conversion between the signal field and the sum frequency field occurs, i.e., they are diabatic. When, Eq. (5) is reduced to , . These two equations clearly show that the energy of the signal field can be completely transformed into that of the sum frequency field during the propagation. It also show that the energy oscillates with the propagation distance, which is similar to the oscillation of an electron on the two levels of an atomic system driven by an electronic field. Thus, the transformation efficiency between the signal field and the sum frequency field is maximal. In this case, corresponds to the Rabi frequency of the transition. Therefore, the coupling coefficient q completely governs the energy conversion between the signal field and the sum frequency field. For, shows that the energy transformation efficiency varies with , and cannot reach the optimal value. Therefore, one should amplify the value of q or decrease that of to improve the efficiency of frequency conversion.
3. Diabatic processes and sum frequency conversion
The quasi-phase matching technique allows us to design almost any desired function of the phase mismatched parameter . In particular, we choose just one specific case of adiabatic frequency conversion, namely, the case that implements the scheme using a linearly chirped grating. Here the second order susceptibility of crystal can be expressed as Eq. (6), is a constant, and with being the spatial chirp coefficient. If the frequency of the pump field is equal to the central frequency , then that of the signal field can be written as with a frequency variation . Thus, the frequency of the sum frequency field is . Given that the first dispersion approximation of the light field in the crystal, i.e., , , and , with being the group velocity of the pump field, signal field, and sum frequency field, respectively. Thus, the phase mismatch can be given as , with , , and the group velocity mismatch and . The first dispersion approximation of the light field in the crystal is called as group velocity mismatch effects. Using the quasi-phase-matching scheme, the coupling coefficient becomes with .
3.1. Energy conversion evolving with the propagation distance
Due to , the adiabatic criterion (3) becomes
The left side of the inequality (7) reaches the maximal value when , which corresponds to . From the dotted line of Fig. 1 one see that the adiabatic condition is not valid around , and thus there is energy conversion between the dressed fields and . Because is a function of the frequency variation , for the different frequencies of the signal field, the positions of the energy conversion are different. Thus, for the given , the adiabatic restraint is fully satisfied when , while it is not valid for the small value of . In order to well interprets the adiabaticity and the diabaticity, we divide the z axis into two regions, with one being called as diabatic region (I) and another one as adiabatic region (). In the diabatic region (I) the adiabatic restriction is not valid, while it holds in the adiabatic region (). The analytical solution of Eq. (1) in the adiabatic region () is equivalent to that of Eq. (4), which is readily reached by the same way as obtaining Eq. (5). However, it is difficult to solve the propagation Eq. (1) in the diabatic region (I). Numerically solving the Eq. (1) with the fourth-order Runge–Kutta method, the energy conversion evolution with the propagation distance can be obtained as shown in Fig. 2 . The dashed line represents the energy variation of the signal field, while the solid line corresponds to that of the sum frequency field. It is shown that the energy of the signal field can be completely transformed into that of the sum frequency field. The energy evolution of the sum frequency field is similar to the Fresnel diffraction of a light by a straight edge, while the diabatic point corresponds to the “straight edge”. The strong oscillation occurs at the two “sides” of the “edge”. From what discussed above, and are the two essential restriction for achieving efficient sum frequency conversion.
The effects of the diabatic process on the sum frequency field can be further illustrated by using the two dressed fields. Using , , and , then the dressed fields can be expressed asFigure 3(a) shows the energy transfer between the dressed fields and . The solid line shows the energy evolution of , while the dashed line corresponds to that of . Figure 3(b) gives the evolution of , and , which are denoted with the solid line attached with △, ○and □ respectively. In the adiabatic region (), the physical process is similar to the total population transfer principle. In this region, there is not energy conversion between the dressed fields. From Fig. 3(b) one see that and in the adiabatic region (), but plays a role only in the diabatic region (I). Consider the boundary condition and , then one may obtain and . After the complete transformation from the energy of the signal field into that of the sum frequency field, and becomes maximal, and thus , , with l being the length of the crystal. The intersection in Fig. 3(a) shows that the dressed fields are in the diabatic region (I), i.e., they are not adiabatic. At the diabatic poin , in terms of the definition of and , then with n being the integer, thus , . By properly choosing the parameters used in Fig. 1-3, the equation can be satisfied, where m is also a integer. Thus, Eq. (8) becomesEquation (9) indicates that the energy of the dressed fields meet the maximal coherence at the diabatic point, and under this condition the energy is transferred from the signal field to the smu frequency field completely. For different frequency variation , there will be different diabatic point . In this way, the restriction is critical to the total energy conversion between the signal field and the sum frequency field. The eigenvalues of the matrix in Eq. (2) varies nonlinearly, and meet . This shows that both conservation law of energy and that of momentum hold at the diabatic point during sum frequency. Furthermore, the momentum is minimum at this point.
How to interpret the concepts of the adiabatic and diabatic process? A thorogh understanding will be reached by the analogy of these processes with the atomic transition in a two-level system driven by a laser light field. If we consider that the signal field is similar to the groud state of the two-level system, while the sum frequency field resembles the excited state , then the pump field in the sum frequency process corresponds to the driving field. In this way, the frequency conversion between the signal field and the sum frequency field has strong similarity to the transition from the groud state to the excited state . Thus, the dressed states and constructed by and are similar to the dressed fields and in this paper. The adiabatic and diabatic process in the two-level atomic system are to the dressed states and , thus that of sum frequcny conversion ocuurs only between the dressed fields and . One can achieve the total population transfer with the theory of the rapid adiabatic passage in the two-level atomic system . Similarly, one also can achieve the complete frequency conversion during the sum frequency process, as shown in Fig. 2. The energy conversion is closely associated with the diabatic process, as shown in Fig. 2 and Fig. 3. Because the diabatic point is a function of the signal field’s frequency, thus one can achieve the broadband frequency conversion through adjusting the frequency of the signal field. This will be discussed in the following part.
3.2. Energy transfer evolving with the frequency of the signal field
Given that the length of the crystal is a fixed value and the frequency of the signal field is a variable, numerically solving Eq. (1) one can obtain the evolution of the dressed fields with the frequency. Figure 4 shows the intensity of the evolution of the sum frequency field with the frequency for different lengths of the crystal. When the length is very small, for example, as shown in Fig. 4(a), the evolving process resembles the Fresnel diffraction by a slit. With the increase of the crystal’s length, the shape of the intensity for the sum frequency field is similar to a superposition of two converse Fresnel diffractions with different straight edges, as shown in Figs. 4(b) or 4(c). When , the range of the bandwidth for the robust sum frequency transfer can be achieved, as shown in Fig. 4(c). This theoretical result can be demonstrated by the previous experimental result . The physical process may be well interpreted by using the non-adiabatic evolution of the dressed fields.
In terms of Eqs. (1) and (8) one may reach the calculated results described in Fig. 5 . Here, the central position of the crystal is set as the origin of coordinate, and the crystal’s length modulated by the external field is l. Figures 5(a) and 5(b) shows the energy evolution of the sum frequency and the left side of the inequality (7) with , respectively. The dashed- and solid- line denotes the energy evolution of and respectively. Figure 5(c) dictates the energy conversion between the two dressed fields with . It is shown from Figs. 5(a) and 5(c) respectively that the traces describing and are similar. The two intersections of and coincides with the two diabatic points, as shown in Figs. 5(b) and 5(c). Therefore, the energy transfer between and determines the energy conversion of (dashed line) and (solid line) as shown in Fig. 5(a).
Using the adiabatic condition (7) one may calculate the positions of the two diabatic points are and respectively. The distance between these two points is obtained as , which corresponds to the range of the frequency response for the generated field. This result is same as the outcome determined by the gating period [6–8,12]. However, our result may manifestly shows the energy conversion during the sum frequency. The range of the frequency response is proportional to the crystal’s length and thus confines the bandwidth of the frequency. Therefore, using the adiabatic and diabatic process may well explain the sum frequency.
In this paper we analyzed the adiabatic and diabatic processes of sum frequency conversion. We showed that the adiabatic and diabatic processes are to the dressed fields, which are superpositions of the signal field and the sum frequency field. A rigorous adiabatic criterion was obtained by using the dressed state formalism, rather than by the simply analogy with the two-level sysytem. We also demonstrated that the diabatic process occurs between the dressed fields when the adiabatic criterion is not satisfied, and that the positions of the energy conversion are entirely determined by the diabatic points. We demonstrated that the evolutionary patterns of the sum frequency field’s energy are similar to the Fresnel diffractions of a light field. Based on the intensity expression of the dressed fields, we showed that the dressed fields meet the maximal coherence at the diabatic points when the energy of the signal field is completely transfered to that of the sum frequency field. We calculated the expected bandwidth of the frequency conversion process, and its dependence on the length of the crystal. We finally showed that the expected bandwidth can be also deduced from the evolution of the adiabaticity of the dressed fileds.
In conclusion, we believe that understanding the sum frequency conversion (or other nonlinear optical processes) with the adiabatic and diabatic process is a new angle of view. From this profile, we can get more physical informations, and bring new physical insights into the process of the frequency conversion. This scheme is greatly propitious to the practical applications.
This research was supported by Doctoral Dissertations Foundation of Shaanxi Normal University (Grants No.X2009YB09).
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