LASER SPECTROSCOPY Direct Observation of the Shift of the 4 P

Transcription

1 ISSN X, Laser Physics, 2006, Vol. 16, No. 8, pp MAIK Nauka /Interperiodica (Russia), Original Text Astro, Ltd., LASER SPECTROSCOPY Direct Observation of the Shift of the 4P(1/2) Electronic Level in Atomic Potassium Vapor upon the Saturation (Redistribution of the Population of Levels) of the 4S(1/2) 4P(3/2) Transition under the Conditions for Nearly Resonant Laser Pumping V. E. Ogluzdin Prokhorov General Physics Institute, Russian Academy of Sciences, ul. Vavilova 38, Moscow, Russia Received December 5, 2005 Abstract The saturation of the population of transitions can cause the shift of electronic levels of an atomic shell in addition to the Stark effect. Experimental results obtained for atomic potassium vapor and demonstrating a shift of the 4P(1/2) electronic level upon the saturation of the 4S(1/2) 4P(3/2) transition in the presence of the field of a nearly resonant laser pumping are discussed. The processes under study are clearly interpreted based on the classical model of the Lorentz harmonic oscillator, supplemented with data regarding the multiphoton bleaching of the medium in the spectral range under study. PACS numbers: 32.80, Wr, k DOI: /S X INTRODUCTION The propagation of high-intensity laser radiation through a cell containing the atomic vapor of an alkali metal in the vicinity of resonant transitions is accompanied by the filamentation of the laser beam, the generation of various conic structures, and a variation in the spectrum of the transmitted radiation. The spectrum of the radiation having passed through such a nearly resonant medium exhibits both spectral broadening and several new scattering components, whose frequencies νj are shifted relative to the laser pump frequency ν. Variations are observed in both the frequency angular structure of the spectrum of the quasi-monochromatic radiation used in such experiments and the characteristics of the electronic shells of irradiated atoms. For example, the laser-induced equalization of populations (the saturation of one of the dipole-allowed electronic transitions) may cause a deformation of the remaining part of the electronic shell and, hence, an energy shift of other levels. The purpose of this work is a new interpretation of the line of radiation scattered in a cell with potassium vapor (λ' line), whose spectral position is related to the shift of the 4P(1/2) level relative to the original position by 3 10 cm 1. Note that the visualization of this shift and the new position of the line corresponding to the 4S(1/2) 4P(1/2) transition becomes possible due to the multiphoton bleaching of the medium and the four-photon parametric scattering (FPPS) of light. The 4S(1/2) 4P(1/2) transition frequency is ν 02 = cm 1 (D1 line), and the 4S(1/2) 4P(3/2) transition frequency is ν 01 = cm 1 (D2 line), so that ν 01 ν 02 = = 57 cm 1. The results from [1] illustrate the deformation of the electronic shell of the alkali-metal atom in the absence (due to saturation) of the 4S(1/2) 4P(3/2) transition. A new interpretation is proposed, since the nature of the λ' line was not comprehended in [1]. The transition is eliminated owing to the saturation and the multiphoton bleaching of the medium (atomicpotassium vapor) that take place when a high-power coherent light pulse with a nonuniform (nearly Gaussian) intensity distribution over the beam cross section propagates through the vapor. The multiphoton bleaching results from the six-photon parametric scattering (SPPS) in a nearly resonant two-level medium [2, 3] (see Section 4). A new modified state of the electronic shell can be observed after the elimination of the transition due to the simultaneous FPPS process [1]. This process is inherent in a three-level medium and makes it possible to detect and to measure the shift of the 4P(1/2) level on the frequency scale. In this case, the FPPS Stokes component serves as the probe radiation that detects and measures the shift of the 4P(1/2) level. A combination of the two-level and three-level models is valid owing to the features of the doublet structure of the 4S 4P transition [4]. Attempts at using the FPPS model in the interpretation of the experimental results obtained for alkalimetal vapor in the frequency range of the fundamental doublet date back to 1970 [5, 6]. However, the symme- 1178

2 DIRECT OBSERVATION OF THE SHIFT 1179 try of the output frequency spectrum expected in accordance with the energy conservation in the FPPS process 2ν = ν S + (1) is not observed in the illustrations from [5, 6]. Thus, the experimentally observed asymmetry contradicts the equidistant character of the Stokes ν S and anti-stokes components relative to the pump frequency predicted by the model. It is demonstrated in [1] that the model of the FPPS process is sufficient for the interpretation of some of the experimental results in a few frequency intervals in the vicinity of the 4S(1/2) 4P(1/2,3/2) transition under study. This is due to the application of a tunable highpower coherent source (optical parametric oscillator OPO). First, consider the narrow frequency range between the D1 and D2 lines, which corresponds to the 4S(1/2) 4P(1/2, 3/2) transition under study, where the refractive index of the vapor n(ν) is close to unity owing to the compensation of the two lines. In this case, the Stokes and anti-stokes frequency detunings of the generated frequency components of light beams relative to the pump frequency ν appear equal to the distance between the 4P(1/2) and 4P(3/2) levels [1]: ν S = ν and = ν +. Second, in the case when the pump frequency coincides reasonably well with the absorption line of any of the 4S(1/2) 4P(1/2,3/2) transitions (a doublet with the same detuning as in the previous case), the output radiation contains both Stokes and anti-stokes components. Note that a cascade process can be realized, when a series of several (up to seven [1]) lines equidistant by (high-order Stokes and anti-stokes components) are observed. In this case, the transition under study is not split into sublevels and remains three-level. Third, the aforementioned feature of the refractive index (n(ν) is close to unity) which facilitates the FPPS observation is realized when the pump frequency is high-frequency-detuned relative to the D2 line (Fig. 1a). In this case, in the FPPS observation, the frequency detuning ' ( ' ) of the Stokes and anti- Stokes components is determined by the phase-matching conditions and seems not to be directly related to any of the interlevel distances of the system under study. The estimates of the FPPS phase mismatch from [1] (Fig. 1b) δk = K K S K AS (K, K S, and K AS are the wave numbers of the pump, Stokes, and anti-stokes components, respectively) for the λ' line, which emerges in the vicinity of the D1 line and exhibits an anti-stokes (Stokes) shift relative to the D1 (pumping) line (Fig. 1c), prove the validity of the FPPS model and correspond to the proposed scheme. In [1], the λ' line is detected and interpreted as the FPPS Stokes component. This line is tuned in the range cm 1 with a variation in the excitation radiation frequency from to cm 1. Note n 1 δk B, arb. units (d) (e) SPPS FPPS ν 02 D1 λ' D2 ν 01 (a) (b) (c) ν'' S ν, cm 1 ν, cm ν'' AS ν, cm 1 ν S ν Fig. 1. (a) Plot of the refractive index n(ν) of potassium vapor vs. frequency ν in the range of the principal doublet (D1 and D2). Dashed line shows the dispersion curve resulting from a deformation of the electronic shell. (b) Plot of the FPPS phase mismatch δk = 2K K S K AS vs. pump frequency for potassium vapor in the case ' = ν = ν ν S = 94 cm 1 (T = 230 C) and ν > ν 01. (c) Superimposed microphotograms that were obtained by processing spectrograms. D1 and D2 reference lines are measured with a potassium spectral lamp. The distance between the doublet lines is 57.7 cm 1. Spectral contour 1 corresponds to the pumping radiation incident on the cell with potassium vapor (OPO spectrum consisting of several narrow lines). Contour 2 corresponds to the broadened frequency spectrum of the radiation having passed through the cell with potassium vapor. λ' line corresponds to the radiation having passed through the cell in the second transparency window. Blackening of the photographic layer B is plotted on the vertical axis in arbitrary units. (d) Asymmetry of the output radiation spectrum related to the SPPS process ν + ν + ν = 3ν = ''. (e) Diagram of the FPPS process ν + ν = + ν S.

3 1180 OGLUZDIN that the λ' line exhibits a blueshift with an increase in the potassium-vapor temperature (and, hence, the saturated vapor density) at a fixed pump frequency. A method for the calculation of the refractive index n(ν) of the alkali-metal atomic vapor needed for the estimation of the phase mismatch δk under nearly resonant conditions can be found in [7]. Such an estimate using the Selmeier formula with neglect of the saturation effect due to the nonuniform intensity distribution across the beam allows a linear approximation at least for a part of the beam. Later (1984) observations of similar processes in sodium vapor were discussed in [8] using the model of nondegenerate four-wave mixing (FWM) in a two-level medium. However, the authors of the FWM model and the authors of [9, 10] failed to interpret the transparency window and the nature of the λ' line. A new interpretation is needed for the problem of radiation corresponding to the λ' line, which was seemingly solved in [1]. A new study became possible due to the simultaneous analysis of the experimental results published by different authors in [1, 8 10]. In this work, we interpret the λ' line using the model of multiphoton bleaching in addition to the ideas from [1]. In my opinion, the λ' line emerges due to the FPPS process typical of a three-level medium and the additional processes of the three-photon electronic stimulated Raman scattering (SRS) [1, 11, 12] and SPPS [2, 3], which provide for the saturation of the transition between the atomic levels and, hence, the bleaching of the medium and the transparency windows upon nearly resonant pumping. The saturation involves an equilibrium between the radiation and the population of the two levels, as well as a variation in the refractive index (n 1). Note that the dispersion of the refractive index should be dynamically compensated for. In this work, we will demonstrate that the FPPS Stokes components (λ' line) indicates the position and shift of the 4P(1/2) level relative to its unperturbed (original) state in the case of the high-frequency detuning of nearly resonant high-intensity pumping that saturates the 4S(1/2) 4P(3/2) transition relative to the D2 line. We propose a qualitative interpretation. The calculations are beyond the scope of this work and will be performed later. Preliminary, let us note a few facts. The Gaussian intensity distribution over the beam cross section leads to a variation in the nearly resonant interaction of the radiation with the medium with increasing distance from the beam axis. Nonlinear optical processes are effectively realized on the beam axis (the refractive index depends on radiation field strength E [13]). However, the interaction of the radiation with the atomic medium can remain linear at a relatively large distance from the beam axis. For example, the conditions for the FPPS phase matching are calculated in the linear approximation. For the analysis, we employ the simplest model of the interaction of radiation with a resonant medium and consider each atom as a classical Lorentz harmonic oscillator [14]. To clearly illustrate the multiphoton bleaching and saturation, we employ the commonly accepted notation for the levels that reflect the doublet structure of the 4S 4P atomic transition. Such a combination of the classical and quantum approaches helps to describe the experimental results. A nearly resonant saturation of the transition in a medium containing two-level atoms is reported in [15, 16]. It is demonstrated that the results obtained using the model of a Lorentz harmonic oscillator are substantially modified in the case when the pump field saturates the transition. In particular, the line shape and, hence, the dispersion and reflection spectra are varied. 2. EXPERIMENT The experimental setup contains a cell with potassium vapor, a coherent source OPO tunable in the range of the potassium D1 and D2 lines (λ1 = 7699 Å and λ2 = 7665 Å), and a detection system. The laser pulse duration and energy are 10 ns and mj, respectively. The spectral width of the OPO line is comparable with the Doppler width of the atomic-potassium line under study. The OPO spectrum contains from one to three lines. The spectra of the incident and transmitted radiation are measured using a DFS-8 spectrograph with a linear dispersion of 6 A/mm. The potassium saturated vapor density is estimated based on the temperature measurements and the tabulated data (see [1 3, 12] for details of the experimental setup and the spectral detection methods). Figure 1c demonstrates a microphotogram of the spectral fragment of the radiation scattered in potassium vapor. Black contour 1 corresponds to the pumping radiation. Reference D1 and D2 lines are obtained with the radiation of a potassium lamp additionally incident on the entrance slit of the spectrograph. Spectrum 2 of the radiation having passed through the cell is asymmetrically broadened: a stronger broadening is observed for the anti-stokes wing. The spectral broadening in the vicinity of the resonance was extensively discussed, and a commonly accepted interpretation can be found in [11, 13]. In the transmitted radiation, the spectral component (λ') under study exhibits a Stokes shift relative to the pump frequency. This line lies in the vicinity of the D1 line and is blue-shifted relative to this line to the range where, in the linear case, n < 1 (Fig. 1a). Figure 2 shows the dependence of the λ' line detuning relative to the D1 line on the excitation frequency. A similar behavior of the λ' line in sodium vapor was observed in [9, 10]. It is also demonstrated in the experiments from [1] (as well as in the later experiments from [9]) that the position of the λ' line depends on the vapor temperature (density) (Fig. 2).

4 DIRECT OBSERVATION OF THE SHIFT DISCUSSION The radiation, having passed through the cell with potassium vapor, exhibits the λ' line in the vicinity of the reference D1 line (Fig. 1c) in the frequency range where, in the linear approximation for the refractive index, we have n(ν) < 1 (Fig. 1a) [14]. Hence, either reflection [14] or destructive interference [17] impedes the propagation of the monochromatic radiation. Thus, the above facts indicate the existence of a transparency window with n 1. Apparently, we may assume a shift of the D1 line. In the presence of a nearly resonant field of laser radiation, a transformation of the electronic shell of the atoms under study takes place. Consider two mechanisms that may cause a frequency shift of the 4P(1/2) level related to the D1 line. One of them involves the saturation (a variation in the populations of the upper and lower levels) of the 4S(1/2) 4P(3/2) transition due to nonlinear optical processes in the course of the propagation of the pumping radiation with frequency ν (lying on the high-frequency side relative to the D2 line (Fig. 1c)) through the atomic medium. The saturation is accompanied by the multiphoton bleaching of the medium and the appearance of the first transparency window in the vicinity of the pump frequency ν. The three-photon SRS (see expression (2)) and the related SPPS process (see expressions (3) and (4)) serve as mechanisms providing for the equalization of the populations of the 4S(1/2) and 4P(3/2) levels [2, 3, 12] (Fig. 1d). The saturation leads to the deformation of the electronic shells and the corresponding shift of the upper 4P(1/2) level. In this case, the λ' line marks a new position of the 4P(1/2) level that coincides with the second transparency window. The dashed dispersion curve in Fig. 1a corresponds to this scenario. In this case, the Stokes component ν S (Fig. 1d) of the FPPS process is downshifted with respect to the pump frequency by cm 1. For the three frequencies involved in the FPPS process (ν, ν S, and ), the refractive index n 1 tends to unity upon the multiphoton bleaching. This may facilitate the phase matching for the FPPS process in the nonlinear case as well. The second mechanism also involves a cascade process. For the pump frequency ν in the range cm 1, the radiation energy density that saturates the 4S(1/2) 4P(3/2) transition in accordance with [18] is about several millijoules per square centimeter, whereas the radiation energy density needed for the saturation of the 4S(1/2) 4P(1/2) transition excited with the λ' line ( cm 1 ) is two to three orders of magnitude lower due to a smaller detuning. Therefore, in the case of an effective FPPS transformation of the fundamental radiation into the Stokes component ν S = ν ' (Fig. 1e), this component of the output radiation may provide (as in the previous case) for the saturation of the second line of the D1 doublet and open up a new ν S ν 02, Òm ν ν 01, Òm 1 Fig. 2. Plots of the experimentally observed shift of the 4P(1/2) level (position of the λ' line relative to the reference D1 line) vs. the detuning of the frequency ν of the highpower laser pumping from the reference D2 line for T = (1) 300 and (2) 260 C. A decrease in the vapor temperature leads to a decrease in the vapor density. In this case, the shift of the λ' line relative to the D1 line decreases. transparency window that corresponds to its new position. Irrespective of the relative contributions of the above mechanisms, the presence of the radiation at frequency ν S (λ' line) in the beam having passed through the cell indicates the appearance of the second transparency window (n 1) and the multiphoton bleaching of the medium in the range of the D1 line (in Fig. 1c, the reference D1 line and the λ' line correspond to the original and new positions of the 4P level, respectively). Figure 1b demonstrates the phase mismatch δk = 2K K S K AS calculated for the FPPS process in the case when the pump frequency ν is high-frequencyshifted relative to the D2 line and the detunings ' of the Stokes and anti-stokes components relative to the pump frequency are 94 cm 1. Here, K, K S, and K AS are the wave numbers of the pump ν, Stokes ν S (see expression (4)), and anti-stokes (see expression (2)) components. Equation δk = 0 determines a narrow spectral fragment on the high-frequency side of the D2 line where the above FPPS process is allowed for the given detuning '. Thus, we must conclude that the position of the transparency window (which corresponds to the λ' line) in the vicinity of the reference marker of the D1 line 1 2

5 1182 OGLUZDIN determines a new position of the 4P(1/2) term. At this frequency, part of the radiation is transmitted through the cell. It is seen from Fig. 2 that, in the spectral fragment under consideration, a decrease in the detuning of the pump frequency relative to the D2 line corresponds to a decrease in the detuning of the λ' line relative to the reference D1 line. A similar dependence was observed in [10] for sodium vapor at small high-frequency detunings of the pumping radiation relative to the D2 line. In each of the above interpretations, the refractive index of the medium at the frequency of the λ' line is close to unity in the presence of the pumping radiation. Thus, the radiation at this frequency can be transmitted by the cell in spite of the fact that, for a weak field, n(ν) < 1 at this spectral fragment in the linear case. 4. SIMPLEST NONLINEAR OPTICAL EFFECTS IN A MEDIUM CONSISTING OF TWO-LEVEL ATOMS [2, 3, 12] Consider the nonlinear optical processes that facilitate the propagation of nanosecond laser beams with Gaussian intensity distributions over the cross section of a nearly resonant two-level medium at the spectral fragments ν > ν 0i, where the refractive index is n(ν) < 1 in the linear case [7, 14]. In this case, in the framework of wave optics, the pumping radiation must rapidly decay due to destructive interference [17]. From the corpuscular point of view, there exists a barrier for the pump photons in this spectral range for high-frequency detunings from the resonant transition ν 0i of up to 50 cm 1 (in potassium vapor). The monochromatic pump photons cannot overcome this barrier: in the opposite case, we arrive at a contradiction lying in the fact that these photons move forward at a velocity V = c/n(ν) that is greater than the velocity of light c. At the interface of the atomic medium, this barrier can be overcome in two stages. At the first stage, high-energy (blue) photons emerge and play the role of precursor photons. The first stage involves the generation of a precursor that can propagate in the vicinity of the resonance at the allowed velocity c (the refractive index n(ν) approaches unity when the detuning increases towards higher frequencies relative to the D2 line (Fig. 1a)). The radiation at the frequency of the three-photon SRS ( '' ) is [1, 4, 11] '' = 2ν ν 01, (2) where ν > ν 01 may serve as such a precursor. In the case under study, ν 01 is the frequency of the 4S(1/2) 4P(3/2) transition. This radiation is high-frequency-shifted relative to the pump frequency to the range where n(ν) 1 (Fig. 1a). The presence of this radiation indicates that, at the beam axis where the beam intensity reaches maximum, the population ratio of the 4S(1/2) and 4P(3/2) levels is varied. Finally, process (2) leads to the leveling of the populations and a narrow spatial channel (slit) is formed along the beam axis. The properties of the medium are varied along this channel, so that n(ν) = 1. At the second stage, the pump photons with frequency ν can propagate inside the narrow channel following the photons with frequency '' (the scenario resembles a drag race) [12]. In general, the generation of photons with frequency '' (2) in a medium with cubic nonlinearity in the presence of high-power pumping radiation corresponds to the nonlinear optical SPPS process [2, 3, 12] in accordance with energy conservation: ν+ ν+ ν = 3ν = ''. (3) Here, ν > ν 01, n(ν) < 1, and ν S '' is given by expression (4). Note that, in the presence of a strong laser field (at the beam axis), the refractive index of vapor tends to unity owing to the saturation effect. Thus, we consider a two-component two-mode model of the medium. The first component, whose refractive index is n(ν) < 1, is the original state of the medium in the presence of a weak light field. The second component at the beam axis corresponds to the modified state of the medium: in the case of saturation, the populations of the levels become equal and n(ν) = 1. Finally, the SRS process (2) is transformed into the parametric process (3), where ν S '' = ( ν 01 + ν)/2. (4) Parametric process (3) leads to the equalization of the populations of the levels and provides for the saturation of the transition (n = 1) and, hence, the free transmission of some of the pump photons through the medium in the spectral range under study. 5. CONCLUSIONS The model of the multiphoton bleaching of the medium (atomic potassium vapor) makes it possible to interpret the behavior of high-power quasi-monochromatic radiation in the spectral fragments where n(ν) < 1. In potassium vapor, transparency windows are formed in this spectral range in the vicinity of the 4S 4P transitions due to the three-photon SRS and SPPS, which provide for the leveling of the populations and maintenance of the population equilibrium of the ground and excited states while the radiation passes through the medium. The FPPS contribution is observed in several spectral fragments. In particular, the Stokes component of this process ν S represents the anomalous λ' line reported in [1]. This line emerges in the spectral range where the propagation of weak monochromatic radiation is inhibited owing to n(ν) < 1 in the presence of a weak field. However, a strong pump field saturates the 4S(1/2) 4P(3/2) transition due to process (3) and causes a deformation of the electronic shell and a shift of the 4P(1/2) term. The electronic shell of an atom is transformed

6 DIRECT OBSERVATION OF THE SHIFT 1183 over a time interval corresponding to the saturation of the transitions in the presence of the nearly resonant radiation. One of the reasons for such a transformation of the shell is a decrease in the population of the ground level of the 4S(1/2) atom. The FPPS Stokes component (Fig. 1e) serves as a probe radiation source that makes it possible to determine the transparency window in the range of the 4S(1/2) 4P(1/2) transition (Fig. 1c). The saturation of the dipole-allowed transition under the conditions for nearly resonant pumping provides for different shifts of the atomic shell terms depending on the original detuning of the excitation frequency from the 4S(1/2) 4P(3/2) transition frequency (Fig. 2). Therefore, let us note the following fact. Photographic measurements of spectrograms do not allow analysis of the dynamics of the effect under study. Hence, the experimentally observed [1, 8 10] shift of the doublet levels is primarily related to the nearly resonant saturation of populations, whereas the role and saturation of the Stark effect [19], which is proportional to the light field strength in the laser beam, was not studied in detail in the above experiments. ACKNOWLEDGMENTS I am grateful to L.L. Chaikov for valuable remarks. REFERENCES 1. V. I. Anikin, S. V. Kryuchkov, and V. E. Ogluzdin, Kvantovaya Elektron. 1, 1923 (1974) [Sov. J. Quantum Electron. 4, 1065 (1975)]. 2. V. E. Ogluzdin, Zh. Eksp. Teor. Fiz. 79, 361 (1980) [Sov. Phys. JETP 52, 181 (1980)]. 3. V. E. Ogluzdin, Pis ma Zh. Tekh. Fiz. 1, 563 (1975) [Sov. Tech. Phys. Lett. 1, 255 (1975)]. 4. P. P. Sorokin, V. S. Shiren, J. H. Lancard, et al., Appl. Phys. Lett. 10, 44 (1967). 5. Yu. M. Kirin, S. G. Rautian, A. E. Semenov, and B. M. Chernobrod, Pis ma Zh. Eksp. Teor. Fiz. 11, 340 (1970) [JETP Lett. 11, 226 (1970)]. 6. A. N. Bonch-Bruevich, V. A. Khodovoi, and V. V. Khromov, Pis ma Zh. Eksp. Teor. Fiz. 11, 431 (1970) [JETP. Lett. 11, 290 (1970)]. 7. R. W. Ditchburn, Light, 2nd ed. (Blackie, London, 1963; Nauka, Moscow, 1965). 8. D. J. Harter and R. W. Boyd, Phys. Rev. A 29, 739 (1984). 9. I. Golub, R. Shuker, and G. Erez, J. Phys. B 20, L63 (1987). 10. A. G. Leonov, A. A. Panteleev, A. N. Starostin, and D. I. Chekhov, Kvantovaya Elektron. 21, 165 (1994) [Quantum Electron. 24, 154 (1994)]. 11. A. M. Badalyan, A. A. Dabagyan, and M. E. Movsesyan, Zh. Eksp. Teor. Fiz. 70, 1178 (1976) [Sov. Phys. JETP 43, 612 (1976)]. 12. V. E. Ogluzdin, Bull. Lebedev Phys. Inst. 9, 1 (2002). 13. S. A. Akhmanov, A. I. Kovrigin, M. E. Maksimov, and V. E. Ogluzdin, Pis ma Zh. Eksp. Teor. Fiz. 15, 186 (1972) [JETP Lett. 15, 129 (1972)]. 14. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983; Mir, Moscow, 1986). 15. E. Pavelec, W. Gavlik, B. Samson, and K. Musiol, Phys. Rev. A 54, 913 (1996). 16. K.-D. Zhu and W.-S. Li, Appl. Phys. B 70, 65 (2000). 17. F. S. Growford, Waves. Berkeley Physics Course (McGraw-Hill; Nauka, Moscow, 1974), Vol R. B. Miles and S. E. Harris, IEEE J. Quantum Electron (1973). 19. J. Reintjes, Nonlinear Optical Parametric Processes in Liquids and Gases (Academic, New York, 1984; Mir, Moscow, 1987).