1 Introduction

Optical systems for the generation of ultrashort pulse sequences can be used to control chemical reactions [1], or optical manipulation of molecular motion in solids [2], also for the generation of tunable narrowband terahertz electromagnetic radiation [3] and for transferring information [4].

There are a number of methods for obtaining the sequence of frequencies above 1 THz [4, 5, 7–10]. Formation of a sequence of pulses with a terahertz repetition rate, for example, is possible as a result of interaction of the two spectral supercontinuum in a nonlinear medium [4]. In [5], femtosecond sequence is generated using a femtosecond pulse shaping. For this was used 4f shaper representing a double monochromator with dispersion subtraction. Spatial light modulator was located in the spectral plane of the shaper. Spectrally decomposed femtosecond pulse was filtered by modulator, and the second part of the spectral shaper carried out the spectral composition and the synthesis of the sequence with the form, which is the Fourier transform of the pulse that has passed through the modulator [6]. Another method of generating a sequence is based on the diffraction of spectral decomposition wave of the reference pulse on a photorefractive crystal, located in the plane of the spectral shaper, on which a dynamic hologram spatial signal was recorded with a continuous laser [7]. The same team has proposed a new method for generating a sequence based on the diffraction of spectral decomposition wave of the reference pulse on the nonlinear crystal located in the plane of the spectral shaper that is based on four-wave mixing between the optical waves of spectrally decomposed ultrashort pulses and spatially Fourier-transformed quasi-monochromatic images [8].

The alternative experimental possibility of forming an entire single sequence of ultrashort light subpulses with a terahertz repetition rate as a result of the interference of two femtosecond pulses with phase modulation by the time delay between the interfering pulses, lower than their duration, is shown in [9, 10]. Subpulse repetition rate in these works can reach 8 THz and depends on the time delay between the pulses and the coefficient of the phase modulation. The use of such sequences is possible for time division multiplexing (TDM) systems. However, in these studies, there was nothing said about the possibility of controlling the substructure of each pulse separately within the sequence. For demultiplexing temporal signal in TDM systems and sequence management control, it is necessary to converse it to the spatial. Ultrafast time-to-spatial conversion based on spectral nonlinear optics has been proposed in [11] and demonstrated experimentally on the basis of four- [12, 13] and three-wave interactions [14–17].

In this paper, we investigated the possibility of individual subpulses in sequence with a terahertz repetition rate formed by the interference of two chirped pulses by spectral-temporal multiplexing controlling. Demultiplexing of the sequence was carried out by temporal-space conversion with the method of spectral nonlinear optics. The coherent image obtained as a result of conversion corresponds to temporal form of formed and controlled sequence. The advantage of proposed coding technique is that we send in the shaper preformed sequence of femtosecond pulses with a repetition rate of terahertz and thus we are not limited by the time window of the shaper [2], which in turn is determined by the spectral resolution of the shaper. The time window for the standard shaper is tens of ps as can be seen in the article [6]. The proposed method will fill the interpulse period for standard laser, due to the fact that you can use chirped nanosecond pulses. Time-to-space conversion for nanosecond pulses was realized earlier in [7]. This technique allowed us to encode the time signal of such sequence for demultiplexing TDM systems with terahertz repetition rate.

2 Experimental setup

Experimental setup for formation, control, encoding and time-to-space conversion of subpulse sequence is shown in Fig. 1. As a source of pulses, we use femtosecond laser system on the base of titan-sapphire regenerative amplifier (Regulus 35f1k, Avesta-Project). The pulse duration is up to 30 fs, repetition rate is 1 kHz, and energy of single pulse is up to 2 mJ.

Fig. 1
figure 1

Experivental setup. M1, M2 mirrors interferometr Michelson (IM); BS1, BS2 beam splitters; DL1, DL2 delay lines; G1, G2, G3, G4 gratings; L1, L2, L3, L4 lens; SLM spatial light modulator; N nonlinear crystal; CCD camera

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The pulse sequence is formed by Michelson interferometer Fig. 1 by interference of two linearly chirped pulses, where in propagation difference between pulses was less than pulse duration, and represented quasi-discrete sequence, both in time and in spectrum [9]. To introduce chirp in the pulse, we disadjust compressor. Michelson interferometer consists of mirrors M1, M2 and the beam splitter BS2. M2 mirror was fixed on the linear translator DL2 providing a time delay between the interfering pulses. Control and encoding sequence was accomplished by 4f shaper [5] Fig. 1. Shaper consists of gratings G1, G2 (eshelet 300 g/mm), L1, L2 lens with a focal length of 10 cm and a spatial light modulator SLM. The obtained sequence was incident on the grating G1 where the angular decomposition of comb spectrum took place. The spectrum of a sequence represented by the quasi-discrete structure [9, 10] formed in the spectral plane of shaper where modulator SLM was located. With the help of the spatial modulator, filtering of individual spectral bands in the quasi-discrete spectrum happened, which in the output of the shaper has led to changes in the temporal structure of the formed sequence. The second part of the shaper consisting of a lens L2 and grating G2 carried out the spectral composition of the filtered sequence.

To decode the controlled sequence, we used time-to-space conversion device. As a reference pulse, we used pulse from the femtosecond system. Formed by interferometer and encoded by shaper, subpulses sequence was used as a signal pulse. The right part of time-to-space conversion device is a two spectral instrument consisting of a diffraction gratings G3, G4 (600 g/mm), arranged one above the other, and a common lens L3 with 20 cm focus length. The reference and signal pulses each fall to their grating G3, G4. The pulses are diffracted from the grating and formed near the spectral plane spectral decomposition waves [18, 19]. The incidence angles of the reference and signal pulses on their gratings are selected so that the linear dispersions of the spectral decomposition waves in the spectral plane are equal in magnitude but opposite in sign [14, 15]. This means that the “blue” edge of the spectrum of the reference pulse corresponds to the “red” edge of the signal pulse, and vice versa. The interaction of the spectral decomposed reference and signal pulse in a nonlinear crystal N of beta barium borate (BBO), placed in spectral plane of spectral devices, resulted in generating noncollinear second harmonic. Adjustable time delay DL1 has been established in the beam of the reference pulse to ensure the temporal coincidence of the reference and signal pulses on a nonlinear crystal.

Since the dispersions of the spectral decomposition of the reference and signal pulse are equal in magnitude but opposite in sign, a subtraction of dispersion took place, and the generated noncollinear second harmonic became quasi-monochromatic up to a resolution of the spectral device. The spatial distribution of the amplitude and phase of the generated second harmonic in the crystal plane is equal to the multiplication of the amplitudes and the sum of the phases of the reference and signal pulse considering the dispersion in the opposite direction. Since the reference signal has a homogeneous spectrum, spatial distribution of amplitude and phase for the newly generated monochromatic radiation at spectral plane corresponds to the amplitude and phase distribution in the spectrum of the signal pulse. In other words, the newly generated monochromatic wave is an analogue of the spectrum of the signal pulse.

The left part of time-to-space conversion device is a Fourier processor, and a lens L4 with a 20 cm focus length performs inverse Fourier transformation of quasi-chromatic spatial frequency spectrum from front focal plane, where the nonlinear crystal N was placed, into a coherent image in the back focal plane of the lens, near which the CCD is located. This conversion is analogical to the transformation of the spectrum of the temporal signal into a spatial signal. So the amplitude-phase distribution of monochromatic radiation formed by the lens L4 on the back focal plane reproduces the temporal dependence of the amplitude and phase of the signal pulse and is determined by the convolution of the temporal form of the signal and reference pulses.

3 Experimental results

The duration of interfering chirped pulses, forming a sequence of subpulses in the Michelson interferometer, was 1 ps. Figure 2 presents the received spatial distribution of the subpulse sequences with a terahertz repetition rate registered by a CCD in a plane F. Temporal calibration is conducted by means of microscrew of movable mirror of the Michelson interferometer.

Fig. 2
figure 2

The spatial distribution of temporal shape of a subpulse sequence with a delay time a 350 fs, b 590 fs, c 1100 fs

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As can be seen from Fig. 2, when changing the delay time between two chirped pulses in the Michelson interferometer, the repetition rate changes [9]. The result is the formation of a sequence in the spectrum, and in time, with it there is a direct quantitative correspondence between the spectral [20] and a temporary form of subpulses sequence [9]. By increasing the time delay, the number of bands is changing in the spectrum without changing its width, and the total duration of the temporal structure and duration of subpulses change in time. For a time delay of 350 fs, a pulse repetition frequency is 4.4 THz, the duration of the subpulses is 225 fs, and the duration of a sequence is 1350 fs. For a time delay of 590 fs, a pulse repetition frequency is 6.4 THz, the duration of the subpulses is 160 fs, and the duration of a sequence is 1590 fs. With the further increase in the time delay between the interfering pulses, the number of subpulses in the sequence increases, but the contrast of subpulses decreases, and they are formed in the region of their intersection, and on the edges unmodulated “wings” of temporary structure are formed. Figure 2c shows an image and a graph of the spatial distribution of temporal shape of subpulse sequence for the repetition frequency of 12 THz at the intersection of interfering pulses, the duration of the subpulses is 85 fs, and the duration of a sequence of 2100 fs. This sequence is obtained by the temporary delay of 1100 fs. Next, using a spatial light modulator in the 4f shaper, we “cut” individual spectral bands in the quasi-discrete spectrum, thereby producing the coding sequence of time. Figure 3 presents the encoding of time sequence.

Fig. 3
figure 3

The spatial temporal distribution for uncoded (left) and coded (right) subpulses sequence with time delays: a ,b 350 fs, c 590 fs, d 1100 fs

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As can be seen from the figures for the first two cases (Fig. 3a–d), control subpulses sequence occurs, when removing spectral bands in the quasi-discrete spectrum of corresponding subpulses. In Fig. 3a, spectrally filtered bitwise sequence is coded 01111, in Fig. 3b—11011, in Fig. 3c—111110011. It should be noticed that, as a result of a spectral filtering, the amplitude of the signal in the one whole sequence (where subpulses previously were) does not drop to zero. The bands in the spectrum are formed by two phase-modulated pulses with a time delay between them, which leads to the fact that interference occurs between the different spectral components of these pulses. It gives rise to crosstalk in the region of zero bits. It should be noted that the bitmap encoding (1’s and 0’s) is possible with respect to the encoded signal to a maximum of the temporal signal at least 15%. Also, it leads to a shift of the maxima of the subpulse sequence as a result of the encoding. With more time shift, not interfering “wings” in a temporary structure are formed. That leads to the impossibility of controlling the time sequence. As shown in Fig. 3d, when cutting multiple spectral lines from one of the edges of the quasi-discrete spectrum, there is a drop in the contrast, the blur and the destruction of the whole temporal structure.

4 Simulation

According to the results of the experiment, a numerical simulation of the sequence control method was carried out. Consider the interference of two pulses with linear chirp:

$$\begin{aligned} E_\mathrm{int}(t)=E_{1}(t)+E_{2}(t+\Delta \tau ) \end{aligned}$$
(1)

where the phase-modulated pulse is given by the equation

$$\begin{aligned} E_{1,2}(t)=E_{0}*\exp (-2*t^{2}/\tau _{0}^{2})*\sin (\omega _{0}*(1+\alpha _{0}*t/\tau _{0})*t) \end{aligned}$$
(2)

\(\alpha _{0}\)—coefficient of phase modulation, \(\omega _{0}\)—the central pulse frequency, \(\tau _{0}\)—the pulse width.

Figure 4 shows the results of modeling the formation and spectral-time coding information in formed subpulses sequence. The duration of the phase-modulated interfering pulses was 1000 fs, phase modulation factor-1, 350-fs time shift (Fig. 4a) and 750 fs (Fig. 4b).

Fig. 4
figure 4

The results of modeling the formation and spectral-time coding information in a formed subpulses sequence; input spectrum interferece (left-solid) and coding spectrum interferece (left-dotted); input subpulses sequence (rightblack) and coding subpulses sequence (right-red); a 350 fs time shift; b 750 fs

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From Fig. 3d, it is clear that when at appearing unmodulated “wings” in the time structure Fig. 4b (right-red) the sequence control becomes impossible.

For a linear chirp, the frequency shift of the interfering spectral components from the time delay is determined by the following expression:

$$\begin{aligned} \delta \omega =\alpha _{0}*\Delta \tau *\omega _{0}/\tau _{0} \end{aligned}$$
(3)

In turn, the modulation spectrum is determined [20]:

$$\begin{aligned} I(\Delta \tau )=I_{0}*(1+\cos (\omega *\Delta \tau )) \end{aligned}$$
(4)

where I(\(\Delta \tau\))—intesity interference spectrum. Hence, the width of the spectral band subpulse in quasi-discrete spectrum is determined by the formula:

$$\begin{aligned} \Delta \omega =2*\pi /\Delta \tau \end{aligned}$$
(5)

The limitation of the control method formed by the sequence when the spectral-temporal coding is a condition where \(\delta \omega\) < \(\Delta \omega\).

It is easy to obtain an expression for the time delay:

$$\begin{aligned} \Delta \tau <\sqrt{2*\pi *\tau _{0}/(\alpha _{0}*\omega _{0})} \end{aligned}$$
(6)

This expression limits the time delay in the formation of subpulses sequence. For the case of the experiment described above, the time delay should not exceed 650 fs. It should be noted that by using the supergaussian pulses the visibility pattern of the modulated sequence is significantly improved for the same parameters of the chirp and duration for a Gaussian pulses. Bitmap coding region is significantly increased in the case of supergaussian pulses.

5 Conclusion

In this paper, experimental and numerical modeling techniques demonstrated the possibilities of the spectral-time encoding and decoding for time division multiplexing formed as a result of the interference of two phase-modulated pulses of sequence of femtosecond subpulses with THz repetition rate. It is shown that when not interfering “wings” are formed at the temporal interference, a sequence control method by the removal of the corresponding spectral bands in the quasi-discrete spectrum does not work. The conditions of work limitations of the demonstrated method are determined. In comparison with the method demonstrated in [5], where the duration of the formed sequence is limited by reverse spectral resolution of the instrument [6, 19], our method is limited to the duration of chirped pulse, which can be a period of dual bypass resonator of a femtosecond laser, as well as its coefficient of phase modulation. For example, in this study we demonstrated the formation, encoding and decoding subpulses sequence with a repetition rate of up to 6.4 THz. Thus, obtained controlled spatiotemporal signal sequence may be used for ultrafast optical signal processing.