The results of theoretical research and numerical simulation of the difference frequency radiation (DFR) pulse generation in 5–25 μm wavelength range ...

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Pulse Compression of Difference Frequency Radiation by Liquid Crystal Phase Transparent D. L. Hovhannisyana*, A. H. Vardanyanb, and G. D. Hovhannisyanb a

National Institute of Metrology, Yerevan, Armenia b Yerevan State University, Yerevan, Armenia * [email protected] Received April 27, 2017

Abstract⎯The results of theoretical research and numerical simulation of the difference frequency radiation (DFR) pulse generation in 5–25 µm wavelength range in the field of the pumping femtosecond laser pulse are presented. The pulse has the following characteristics: the central wavelength is 2 µm, duration is 34 fs and electric field amplitude is 70.71 MV/m. The pulse propagates in the ZnTe/air periodic structure with the number of periods along the normal to the (110) plane of the ZnTe crystal equal to 13; and the efficiency of the DFR generation is 1.11 × 10–4. It is shown that the use of the liquid crystal phase transparent, placed in the focal plane of the frequencyspatial shaping system, allows one to realize the DFR pulse compression at which the maximum intensity is increased by a factor of six. DOI: 10.3103/S1068337217040053 Keywords: liquid crystal, periodic structure, difference frequency radiation

1. INTRODUCTION The nonlinear optical methods for generation a frequency-modulated IR pulse are of practical importance for the nonstationary IR spectroscopy of polyatomic molecules, investigation of the processes of excitation and relaxation of polyatomic molecules, development of methods for obtaining the nonequilibrium intramolecular excitations, and study the physics of narrow-band semiconductors, and etc. [1−3]. The compression and control of the time profile of the frequency-modulated IR pulse with the duration of several oscillations is a matter of principle in the nonstationary IR spectroscopy, where the absorption spectrum of radiation by a substance is determined by the transition of the energy of the radiation quanta to the energy of the rotational and vibrational levels of the molecule [4]. At the generation of the difference frequency radiation (DFR) during the three-photon interaction of a femtosecond laser pulse with a nonlinear optical crystal with an irregular domain structure, as shown in [5, 6], it is possible to control the law of frequency modulation of the broadband IR pulse of the DFR in the wavelength range from 5 up to 15 μm. The use of an irregular domain structure with a linearly varying (increasing or decreasing) value of the domain period enables to control the sign of the DFR frequency modulation. At the same time, the value of the domain thickness is varied in such a way as to have the quasi-synchronous generation of the frequency-modulated DFR over a wide spectral range as the femtosecond laser pulse propagates. The quasi-synchronous generation of the DFR is also realizable by way of use a traditional dispersion mechanism of compensation of the phase detuning due to a change (increase or decrease) of the effective refractive index of the medium at the edges of the forbidden band of the one-dimensional periodic structure, of a photonic crystal (PC) [7−9]. 335

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In the present work, we consider the generation process of the broadband frequency-modulated DFR in the field of a femtosecond laser pulse propagating in a one-dimensional periodic structure of ZnTe/air. Theoretical studies and numerical simulation of the process of compression and control of the temporal profile of the pulsed DFR with the use of the liquid crystal phase transparent placed in the focal plane of the frequency-spatial transform system consisting of diffraction gratings and the lenses located between them, representing the Fourier image of the pulse into the spatial pattern in the focal plane are carried out. 2. GENERATION OF THE DFR PULSE IN THE FIELD OF FEMTOSECOND PULSE PROPAGATING IN PERIODIC ZnTe/AIR STRUCTURE The numerical integration of the system of nonlinear Maxwell equations has been carried out by the finite time difference method describing the process of the DFR generation by the linearly polarized femtosecond laser pulse at a central wavelength λ0 = 2 µm propagating in the periodic ZnTe/air structure. The periodic structure ZnTe/air, а one-dimensional PC, considered in the work, consists of parallel layers of ZnTe and air with the thicknesses lz and lo and the refractive indices nz = nZnTe and no = 1, respectively. The faces of the isotropic ZnTe crystal with the cubic crystal structure of the symmetry group 43m are parallel to the (110) planes.

Fig. 1. (a) Periodic structure of ZnTe/air and (b) the mutual orientation of the crystallographic (XYZ) and laboratory (xyz) coordinate systems.

Figure 1a shows the structure of PC, and Figure 1b shows the mutual orientation of the crystallographic (XYZ) and laboratory (xyz) coordinate systems. According to Fig. 1a, the period of the change of the refractive index is Λ = lo + lz. Let us consider a linearly polarized femtosecond laser pulse with the plane wave front and the components of the electromagnetic field (0, Ey, 0) and (0, 0, Hz), propagating along the x-axis coinciding with the normal to the (110) plane in the periodic structure ZnTe/air. In the quasistatic approximation, the nonlinear polarization of the ZnTe/air periodic structure in the transparency band of the ZnTe crystal is defined as [9] PzNL ( t ) = ε 0 d14 E y2 ( t ) , PyNL ( t ) = 2ε0 d14 E y ( t ) Ez ( t ) ,

(1)

where at m ( lo + lz ) < x < m ( lo + lz ) + lo , the nonlinear susceptibility tensor d14 = d14(air) = 0, and at m ( lо + lz ) + lо < x < m ( lо + lz ) + lо + lz , according to [10], d14 = d14(ZnTe) = 92.15 pm/V (m∈Z is the natural number). In the process of nonlinear interaction of the y-polarized laser pulse with the PC, JOURNAL OF CONTEMPORARY PHYSICS (ARMENIAN Ac. Sci.)

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according to (1), the z-polarized radiation is generated with the components of the electromagnetic field (0, 0, Ez) and (0, Hy, 0), which contains the spectral components at the sum and difference frequencies. The algorithm for the numerical solution of the system of the Maxwell equations describing this process is given in our work [9]. As shown in [7], in the PC, in which one of the subsystems is a set of plates without the nonlinearity with the negligible dispersion, and the second, is a quadratically nonlinear medium with the frequency-dependent refractive index, phase and group synchronism for the process of second-harmonic generation are achieved owing to the compensation of the dispersion of the ZnTe material with the dispersion of the periodic structure. Taking into account the dispersion of the ZnTe crystal in the wavelength range 5–30 μm for the PC under consideration, the layer thickness was chosen from the conditions lo = λ IR 4 = 2.5 µm and lz = λ IR 2nz ( λ IR ) = 1.8628 µm, where λIR = 5λ0 = 10 µm, and nz ( λ IR ) = 2.6839.

Fig. 2. Dispersive characteristics of the ZnTe/air periodic structure with the number of periods equal to 13.

Figure 2 shows the main dispersion characteristics of the structure under consideration with the number of periods equal to 13. Figure 2a shows the dependence of the real part of the following expression

⎛ 2πlo cos ( K ( λ ) Λ ) = cos ⎜ ⎝ λ

⎛ 2πnz ( λ ) lz ⎞ ⎞ ⎟ ⎟ cos ⎜ λ ⎠ ⎝ ⎠

⎞ ⎛ 2πlo 1⎛ 1 − ⎜⎜ + nz ( λ ) ⎟⎟ sin ⎜ 2 ⎝ nz ( λ ) ⎠ ⎝ λ

JOURNAL OF CONTEMPORARY PHYSICS (ARMENIAN Ac. Sci.)

(2)

⎞ ⎛ 2πnz ( λ ) lz ⎞ ⎟ ⎟ sin ⎜ λ ⎠ ⎝ ⎠

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on the wavelength λ, where Λ = lo + lz. According to the results of computation and, as can be seen from Fig. 2, for the PC ZnTe/air under consideration with the period d = lo + lz = 4.3628 μm and the number of periods 13 for the DFR with λ > 30 μm, there are no forbidden bands, that is, the PC behaves itself as a homogeneous medium with an average refractive index. Figure 2b shows the dependence of the real part of K(λ) on the wavelength, Fig. 2c represents the dependence of the real part of the refractive index on the −1 wavelength np ( λ ) = c ν p , where vp ( 2πc λ ) = ( K ( 2πc λ ) 2 πc λ ) is the phase velocity, c is the speed of light in vacuum; Fig. 2d shows the dependence of the ratio of the speed of light in vacuum to the real −1 part of the group velocity in the wavelength interval up to 30 μm vg ( 2πc λ ) = ( dK ( 2πc λ ) d ( 2πc λ ) ) . According to the results of computation obtained in [11] by the transfer matrix method and, as can be seen from Fig. 2d, the forbidden bands of the PC under consideration correspond to the wavelength bands: 12.18−20.56, 5.033−5.065, 2.870−3.196, 2.108−2.261, 1.685−1.704, 1.355−1.428, 1.169−1.204 and 1.021−1.040 µm. In the spectral ranges corresponding to forbidden bands, the group velocity is equal to zero. In the spectral ranges where the group velocity is negative, that is, the effective refractive index c/vg is less than zero, the radiation pulse leaves the medium before it completely enters in it [12]. This phenomenon, which contradicts our usual notions, was first recorded in a linearly absorbing medium [13]. Based on the calculation results 2 K ( ωIR ) = K ( 2ωIR ) and ωIR = 2πc/λIR, therefore, for the structure under consideration, there are the equality of the phase velocities for the spectral components at the wavelengths 10 и 5 µm. According to the computation results and, as can be seen from Fig. 2d, the group velocity of the femtosecond laser pulse at the central wavelength λ0 = 2 µm is equal to the value of the group velocity of the DFR pulses at wavelengths 10.79, 9.49, 5.02, 4.22, 2.80 and 2.55 µm. Consequently, for the structure under consideration, the group synchronism condition is fulfilled simultaneously for all indicated wavelengths. During the propagation of the femtosecond pulse in the PC, the nonlinear generation of the spectral continuum in the IR wavelength range occurs at which the group velocities of the femtosecond pulse and the DFR pulses at the above wavelengths. In addition, the generation of the remaining frequency components of the spectral IR supercontinuum occurs in the absence of both phase and group synchronism. The values of the DFR pulses wavelengths for which the group velocity is equal to the group velocity of the femtosecond pulse can be varied by changing the thickness of the layers of the PC. According to [9], the nonlinear addition to the refractive index of the ZnTe layer for z-polarized radiation with the change of the amplitude of the electric field from 70.71 to 100 MV/m varies from 1.218 × 10−3 to 1.723 × 10−3, and for the y-polarized radiation from 1.810 × 10−3 to 2.559 × 10−3, respectively. According to the calculation results, at the indicated values of the nonlinear additions to the refractive index, the displacements of the dispersion curves along the frequency axis (Fig. 2) can be neglected. In the field of the femtosecond laser pulse at the central wavelength λ0, the generation of the sum frequency radiation (SFR) at the wavelengths λSFR = λSλ L ( λ L + λS ) and the generation of the DFR at the wavelengths λ DFR = λSλ L ( λ L − λS ) , where λS, and λL are the short- and long-wavelength spectral components within the width of the spectrum of a femtosecond laser pulse. In order to separate the DFR in the wavelength range from 5 to 30 μm, we filter the z-polarized pulse in the course of the numerical experiment at the crystal output by a filter that can be experimentally realized in the form of a dielectric bandpass filter [14].

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3. COMPRESSION AND CONTROL OF THE TIME PROFILE OF THE DFR PULSE. RESULTS OF NUMERICAL COMPUTATIONS AND DISCUSSION The initial conditions for the numerical solution of the system of nonlinear Maxwell equations are chosen in the form ⎛ t2 E y ( t , x = 0 ) = E0 exp ⎜ − 2 ⎝ τ0

⎞ ⎛ 2πc ⎞ ⎟ cos ⎜ λ t ⎟ , ⎝ 0 ⎠ ⎠

(3)

where E0 = 70.71 MV/m is the amplitude of the IR pulse, 2τ0 = 34 fs is the pulse duration and λ0 = 2 μm is the central wavelength. The step of the spatial cell Δx is chosen equal to λ0/400 = 5 nm, and the time step Δt is determined by the Courant condition Δt = Δx/2c = 0.0083 fs.

Fig. 3. Scheme of generation, control and measurement of the DFR pulse: FLS is the fiber femtosecond laser, BS is the beam splitter, M is the totally reflecting mirror, 1D PC is the PC, DG is the diffraction grating, BaF2 is the barium fluoride lens, LCPT is the liquid crystal phase transparent, BPF is the band-pass filter, Ge is the germanium plate, SiO2 is the fused quartz plate, HPF is the optical highpass filter, PMT is the photodetector, Sp is the spectrum analyzer.

Figure 3 shows the DFR pulse generation scheme. The pulses of the fiber femtosecond laser (FLS) with a duration of 34 fs at a wavelength of 2 μm are directed into a beam splitter (BS). A part of the pulse passes through the splitter and is directed to the one-dimensional PC described above, where the SFR and JOURNAL OF CONTEMPORARY PHYSICS (ARMENIAN Ac. Sci.)

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the DFR are generated. The DFR is separated from the SFR by a band-pass filter (BPF) providing transmission in the wavelength range from 5 to 25 μm. In the course of numerical simulation, the spectral filtration of the electric field of the z-polarized pulse at the PC output was carried out in series: first with a high-pass filter, and then using a low-pass filter with the cutoff frequencies fHP = 60 THz (λHP = 5 µm) and fLP = 12 THz (λLP = 25 µm), respectively [9].

Fig. 4. Spectral-temporal characteristics of the z-polarized DFR pulse at the output of the BPF filter: (a) the dashed line shows the dependence of the pulse wavelength on time, and the solid line is the result of the interpolation of the given dependence by the cubic polynomial, (b) the time dependence of the field, (c) is the spectrum, and (d) is the SWVPD.

Figure 4 shows the spectral-temporal characteristics of the z-polarized pulse at the output from the filter. The dotted line in Fig. 4a shows the dependence of the wavelength of the DFR pulse on time, and the solid line represents the interpolation result of this dependence by the cubic polynomial. In accordance with the results of the computations and, as can be seen from given dependence, the short-wave spectral components outpace the temporal long-wave components by ∼1 ps. Figure 4b shows the time dependence of the field of the z-polarized pulse at the output from the filter, in Fig. 4c the dependence of the spectral density on the wavelength, and in Fig. 4d is the smoothened Wigner–Ville pseudo-distribution (SWVPD), which, with the aim of the temporal smoothing regardless of the frequency smoothing, contains a window for both frequency and temporal smoothings [15]. The DFR pulse length is about 1 ps, and the maximum field amplitude is Ezmax = 7.5 × 105 V/m. Thus, the efficiency of the DFR generation, defined as (Ezmax/E0)2 V/m, is equal to 1.11 × 10−4 (10lg(Ezmax/E0)2 = −39.57 dB). JOURNAL OF CONTEMPORARY PHYSICS (ARMENIAN Ac. Sci.)

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After the filter, according to Fig. 3, the z-polarized pulse with the aid of the mirror (M) is directed at a certain angle to the normal to the diffraction grating (DG) of the transform system, which consists of diffraction gratings and the lens located between them, representing the Fourier transform of the pulse into the spatial picture in the focal (Fourier) plane. As lenses, the lenses from the barium fluoride (BaF2) were used, which are transparent in the indicated IR range. In the focal plane of the system, there is an electrically controlled liquid crystal phase transparency (LCPT) to control the time profile of the DFR pulse, which has been implemented by the phase modulation of the pulse using a phase transparent placed in the transform system. The spectrally decomposed beam, after reflecting from the diffraction grating (DG) located in the front focal plane of the first lens of the formation system, is projected onto a phase transparent located in the back focal plane of the first lens. In addition, there is a one-to-one correspondence between the value of the frequency ω of the spectrally decomposed DFR and the transverse coordinate of the transparent x. Let us consider an electrically controllable phase transparent consisting of structured layers of a nematic liquid crystal (NLC) with the different refractive index values (Fig. 5). The transparent is a liquid-crystal cell of thickness d with two glass substrates coated with the transparent conductive films ITO (indium–tin oxide) [16]. On one of the substrates a continuous layer of ITO is deposited, and on the second layer (this layer is structured in the form of parallel electrically connected electrodes) with a width L. When voltages V1, V2, V3, ... VN are applied between the substrates, the reorientation of the NLC molecules is taking place, which is determined by their electro-optically induced birefringence. The phase difference of the spectral components of the DFR λ1 and λ2 that have passed through the layers with different refractive indices neff (V1 ) and neff (V2 ) can be represented in the form neff (V1 ) d λ1 − neff (V2 ) d λ 2 . With an appropriate choice of the values of the voltages V1, V2, V3, ... VN applied to the N electrodes, one can obtain the desired phase-frequency characteristic of the phase transparent under consideration. To compress the DFR pulse, we considered a set of values of control voltages at which the phase frequency characteristic of the transparent corresponds to the phase-frequency characteristic of the

Fig. 5. Schematic representation of the cross section of the liquid crystal phase transparent.

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DFR pulse approximated by the cubic polynomial and taken with a minus sign. The group delay of the transparent is defined as τ PT ( ω) = −

∂φappr ( ω) ∂ω

,

(4)

where ϕappr(ω) is the phase-frequency characteristic of the z-polarized DFR pulse, determined as a result of numerical integration of the system of the Maxwell nonlinear equations with the subsequent filtration and approximation by the cubic polynomial. In the numerical experiment, the diffraction gratings with a constant of 30 lines/mm and lenses of BaF2 with a focal length f = 5 cm are considered. The angle of incidence of the DFR pulse on the input diffraction grating is chosen to be 7° relative to the normal. At the width of the DFR pulse spectrum from 5 to 25 μm, the width of the DFR beam in the focal plane is 33 mm, which corresponds to the electrode width L = 110 μm for the number of electrodes N = 300 (Fig. 5). In the course of the computations, the NLC 6CHBT with the thickness d = 200 μm is considered. According to [16], when the value of the control voltage is changed by ΔV = 3 V, the refractive index can vary from 1.52 to 1.67 owing to the electro-optically induced birefringence. Thus, if the short-wave component of the DFR at the wavelength of 5 μm passes through the electrode of the phase transparent, to which a voltage equal to the threshold value is applied (the voltage 10.83 V corresponding to the Fredericks transition for the NLC 6CHBT with a thickness of 200 μm [16]), and the long-wavelength component at the wavelength of 25 μm passes through an electrode to which a voltage greater by ΔV = 3 V of the threshold voltage is applied, then in accordance with [16], the group delay between these components is equal to 1 ps. Therefore, with the number of electrodes N = 300, the step of changing the control voltage from the electrode to electrode is 10 mV. For a finite-diameter σ of the DFR beam in the plane of the input diffraction grating, the spectral component of frequency ω deviates by an angle β(ω) ≈ ∂θ(ω)/∂ω × ω, where ∂θ(ω)/∂ω is the angular dispersion coefficient of the lattice, and has an angular divergence proportional to λ/σ, which corresponds to a spatial spreading fλ/σ along the x-axis directed along the electrodes of the transparent [17]. For spectral components of the DFR at wavelengths of 5 and 25 μm, the spatial spreadings are 8.3 and 42 μm, respectively, which is less than the width of the electrode L = 110 µm. After leaving the phase transparent, according to Fig. 3, the DFR pulse is focused by the second BaF2 lens onto the output diffraction grating, at the output of which the filtered and collimated DFR pulse is formed. Figure 6 shows the spectral-temporal characteristics of the z-polarized DFR pulse at the output of the transform system. The time dependence of the wavelength (Fig. 6a), the temporal profile of the electric field (Fig. 6b), the spectrum (Fig. 6c) and the SWVPD (Fig. 6d) are given. It can be seen that the length of the DFR pulse at the output of the transform system is smaller than the pulse width DFR at the input of the system and is about 0.44 ps, and the maximum of the field amplitude Ezmax is 1.84 × 106 V/m. Thus, the application of the phase liquid crystal transparent placed in the focal plane of the frequency-spatial transform system makes it possible to realize such compression of the DFR pulse length at which the relative peak intensity of the DFR is increased by 10lg(Ezmax/E0)2 − 10lg(Ezmax0/E0)2 = 7.8 dB. According to Fig. 6a, at a time instant corresponding to the maximum of the field, a sharp increase in the wavelength from 6 to 16.5 μm occurs, and at the time intervals corresponding to the leading and the trailing edges of the pulse, the wavelength λDFR is about 12.5 µm. JOURNAL OF CONTEMPORARY PHYSICS (ARMENIAN Ac. Sci.)

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Fig. 6. Spectral-temporal characteristics of the z-polarized DFR pulse at the output of the frequency-spatial display system: (a) the dependence of the pulse wavelength on time, (b) the temporal dependence of the field, (c) the spectrum, and (d) the SWVPD.

It is proposed to measure the temporal profile of the DFR pulse with the use the cross-correlation technique based on the four-wave interaction (FWI) of the type 2 × 2πc/λ0 − 2πc/λDFR = 2πc/λ of highpower pulses of the fiber-optic femtosecond laser with the frequency of 2πc/λ0 and the DFR with the frequency 2πc/λDFR [14]. Such the FWI process reduces the problem of characterization of the DFR pulses to the metering of cross-correlation and spectral characteristics of pulses in the wavelength range concentrated in the vicinity of the wavelength λ0/2. Really, at λ0 = 2 μm the wavelength of the pulse generated as a result of this FWI process is in the region 1.16–1.21 μm, which allows one to record it by the standard semiconductor photodetectors and to perform the signal spectrum analysis using the common spectral analyzers designed to work in the wavelength range from 0.9 to 1.3 μm. To perform the cross-correlation measurements, some part of the radiation from the output of the fiber-optic femtosecond laser reflected from a beam splitter and transmitted through an optical delay line consisting of two mirrors is matching to the DFR on the germanium plate, after which both beams are focused by the BaF2 lens onto the nonlinear medium. As the nonlinear medium, whose center is coincided with the front focal plane of the first lens, the fused quartz (SiO2) or a gaseous medium can be used. Further, the radiation from the output of the nonlinear crystal is collimated with the aid of second BaF2 lens located at a distance equal to the focal length from the center of the crystal. For a fixed time delay τ between the y-polarized femtosecond pulse Ey(t) and the z-polarized pulse Ez(t), the nonlinear polarization JOURNAL OF CONTEMPORARY PHYSICS (ARMENIAN Ac. Sci.)

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of the medium is defined as 3) 3 PzNL ( t , τ ) ∞χ(zyyz ⎣⎡ E y ( t − τ ) + E z ( t ) ⎤⎦ .

(5)

The separation of the spectral components of the DFR in the vicinity of the frequency 2πc/λ = 2 × 2πc/λ0 − 2πc/λDFR can be realized by means of an optical bandpass filter installed after the second lens. 2

The sum of products E y ( t − τ ) Ez ( t ) dt taken for each fixed value of the delay τ corresponds to the 2

function of cross-correlation 2

I I FWM ( τ ) ∞ ∫ E y ( t − τ ) Ez ( t ) dt 2

∞

∫I

FWM

( τ, ω) d ω

(6)

and is registered by a photodetector. The two-dimensional function IFWM(τ,ω) in expression (6) corresponds to the dynamic spectrogram [18] 2

I FWM ( τ, ω) ∞ ∫ E y ( t − τ ) Ez ( t ) exp ( jωt ) dt . 2

(7)

The sum of products S y ( ω1 ) S z ( ω − ω1 ) d ω1 taken for each fixed value of frequency ω, corresponds to the convolution of the Fourier transforms of the interacting pulses 2

I II FWM ( ω ) ∞ ∫ S y ( ω1 ) S z ( ω − ω1 ) d ω1 2

∞

∫I

FWM

( τ, ω) d τ

(8)

and is registered by a spectral analyzer. The Fourier-images of interacting impulses in expression (8) are ∞

∞

−∞

−∞

defined as S y ( ω) = ∫ E y2 ( t ) exp ( jωt ) dt and S z ( ω) = ∫ Ez ( t ) exp ( jωt ) dt , respectively. It should be noted that the temporal resolution is defined by the step of changing the coordinates of the mirrors entering in the optical delay line, which, in turn, is defined by the accuracy of positioning the coordinates of the mechanical mirror holders. For holders with an accuracy of positioning equal to Δl = 1 μm, for example, the temporal resolution is equal to δτ = Δl/c ≈ 3.3 fs, which is half the period of the pulse oscillations E y ( t ) , it is 5.15 times shorter than the pulse duration and 151.5 times shorter than the pulse length Ez ( t ) . The accuracy of registering the convolution of Fourier transforms (8) is determined by the spectral resolution of the analyzer, which, in particular, can reach 0.1 nm for spectraoscope with a diffraction grating [19]. According to the foregoing, unlike the measurement method based on the frequency-resolved optical gating (FROG) used to register ultrashort laser pulses, in the case under consideration the time resolution is independent on the spectral resolution, and the performed measurements are multiple, realized separately for each delay value m × τ between pulses, where m∈Z. Figure 7 shows the spectral-temporal characteristics of the z-polarized DFR pulse at the output from the transform system without a liquid-crystalline phase transparent, obtained as a result of numerical simulation of the cross-correlation measurement process. The dependence of the wavelength on time is shown in (Fig. 7a), the temporal profile of the electric field normalized to its maximum value (Fig. 7b), the spectrum of the signal generated as a result of the FWI process (Fig. 7c) and the dynamic spectrogram (Fig. 7d). In Fig. 7b, the dashed line shows the envelope of the DFR pulse obtained from the crosscorrelation function (6). In the course of the numerical experiment, the dynamic spectrogram is formed as a matrix in which the intensity distribution along each column corresponds to the convolution of the Fourier images of the interacting pulses (8), and the column number m corresponds to the time delay between the interacting pulses. JOURNAL OF CONTEMPORARY PHYSICS (ARMENIAN Ac. Sci.)

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Fig. 7. Spectral-temporal characteristics of the z-polarized DFR pulse at the output of the transform system without the liquid crystal phase transparent, obtained as a result of cross-correlation: (a) the time dependence of wavelength, (b) the temporal profile of the electric field normalized to its maximum value (solid line), and the envelope (dashed line), (c) the spectrum of the signal generated as a result of the FWI process, and (d) the dynamic spectrogram.

Figure 8 shows spectral-temporal characteristics of a time-compressed z-polarized DFR pulse at the output from the transform system with the liquid crystal phase transparent, obtained as a result of numerical simulation of the cross-correlation measurement process. The temporal profile of the intensity of the compressed pulse is shown by the solid line, and the dotted line is the temporal profile of the DFR pulse before compression (Fig. 8a); the temporal profile of the electric field (Fig. 8b), the spectrum of the signal generated as a result of the FWI process (Fig. 8c), and the dynamic spectrogram (Fig. 8d) are also represented. In Fig. 8b, the dashed line shows the envelope of the compressed DFR pulse obtained from the cross-correlation function (6). As can be seen from Fig. 8a, the peak intensity of the compressed pulse is 6 times that of the DFR pulse before compression. 4. CONCLUSION The results are represented of the theoretical study and numerical simulation of generation process of the DFR pulse in the wavelength range from 5 to 30 μm in the field of the femtosecond laser pumping pulse at the wavelength of 2.0 μm, the duration of 34 fs, and the amplitude of 70.71 MW/m, propagating in the periodic ZnTe/air along the normal to the (110) plane of the ZnTe crystal. The results of

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Fig. 8. Spectral-temporal characteristics of the z-polarized DFR pulse at the output from the liquid crystal phase transparent of the transform system obtained as a result of cross-correlation: (a) the temporal profile of the intensity of the compressed pulse (solid line) and the time profile of the DFR pulse before compression (dashed line), (b) the time profile of the electric field normalized to its maximum value (solid line) and the envelope (dashed line), (c) the spectrum of the signal generated as a result of the FWI process, and (d) the dynamic spectrogram.

computations of the main dispersion characteristics of the periodic structure under consideration with the thicknesses of the air and ZnTe layers equal to 2.5 and 1.8628 μm, respectively, and with the number of periods equal to 13 are represented. It is shown that the efficiency of the DFR generation is equal to 1.11 × 10−4. The results of calculating the DFR pulse compression process using the transform system with a liquid-crystal phase transparency is controlled electrooptically. It is shown that the use of the phase liquid crystal transparent placed in the focal plane of a frequency-spatial transform system allows one to realize such compression of the DFR pulse length at which the relative peak intensity of the DFR increases by 7.8 dB. The results of the computation of the frequency-time characteristics of the DFR pulse with the help of the smoothened Wigner–Ville pseudo-distribution are also presented, as well as the results of a numerical experiment describing the process of cross-correlation measurement of the temporal profile of the DFR pulse obtained as a result of the FWI. REFERENCES 1. Petrov, V., Rotermund, F., and Noack, F., J. Opt. Pure Appl. Opt., 2001, vol. 3, p. R1. 2. Rotermund, F., Petrov, V., and Noack, F., Opt. Commun., 2000, vol. 185, p. 177. JOURNAL OF CONTEMPORARY PHYSICS (ARMENIAN Ac. Sci.)

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JOURNAL OF CONTEMPORARY PHYSICS (ARMENIAN Ac. Sci.)

Vol. 52

No. 4

2017