The generation of short pulses with a high repetition rate is of interest from the point of view of applications in high-speed optical communication [1]. Such pulses can be created directly by passive mode-locked lasers, but the parameters of these pulses are difficult to control. Among them, one can note the duration and shape of the pulses, as well as their position in time. Recently, sources of short pulses with a high repetition rate using microresonators have been proposed; see reviews [2, 3]. They make it possible to obtain highly stable short pulses with a repetition frequency, which, depending on the size of the microresonator, can vary from 10 GHz to 1 THz [4, 5]. Such pulses with a highly stable frequency comb spectrum can be used both in metrology and high resolution spectroscopy [6] and in optical communications, for example, for wave-division multiplexing [7], see also [8]. The generation of such pulses is based on the formation of a dissipative Kerr soliton in a microresonator. Due to the high quality factor and strong light confinement in microresonators, the threshold for nonlinear light conversion is significantly lowered, for example, through a nonlinear four-wave mixing in media with cubic nonlinearity. The generation of stable dissipative Kerr solitons by continuous single-frequency pumping in microresonators is achieved under a balance between gain and losses and a balance between dispersion and nonlinearity [2, 3]. To obtain a stable single-soliton state, fine tunings of the resonator and pump are necessary, as well as some manipulations to purify the radiation to a single-soliton one, followed by thermal stabilization of the resonator and control of the pump power [2]. Maintaining a delicate balance between the parameters of such a generator guarantees its stable operation. The amplitude, shape of the pulses and their duration are unique for each set of generator parameters, and their control is not trivial [9]. Meanwhile, the duration of pulses can be controlled, for example, by passing through a single-mode fiber of a certain length to compensate for the dispersion of group velocities of various components of the pulse spectrum and to change the ratio of their phases [10]. In microresonators with a quadratic nonlinearity, it is possible to generate broadband frequency combs with excellent stability and controllability using electro-optical modulators [11], or to create a soliton and a comb emission spectrum due to cascade generation of the second harmonic in the process of parametric generation [12].

There are alternative methods for generating pulses in the linear response mode, which allow one to control their parameters and implement, for example, time-division multiplexing using both phase and amplitude field modulation [13]. In the general case, such pulses are created from narrow-band radiation of a highly stabilized cw laser using modulators. There are amplitude and phase modulators of radiation. The latter have a number of advantages; see, for example, [14, 15]. Previously, many schemes for generating pulses using phase modulation were proposed and experimentally implemented [14–28]. This list of works is far from complete. The following were considered: harmonic [14–23], rectangular (i.e., binary, when the phase jumps between two values) [15, 24–26], and sawtooth modulation [27, 28] of the phase. Periodic modulation causes the narrow emission line to be transformed into a frequency comb with a distance between adjacent components equal to the modulation frequency. In most of the references listed, except for [14, 19, 22, 23], phase-modulated radiation is passed through a phase filter, for example, a chirped fiber Bragg grating or a sufficiently long single-mode fiber, which changes the phase of each spectral component of the comb according to a certain law, depending on the component position relative to the frequency of the laser radiation.

Meanwhile, there is an interesting method for generating short pulses and a series of pulses using resonant absorbers [29–33]. This method is universal and can be applied both to gamma radiation [29–31] and in the optical domain [32, 33]. As filters, optically thick resonant absorbers are proposed, which use Mössbauer nuclei [29–31] or a cloud of cold atoms [32, 33] with narrow homogeneously broadened absorption lines. The method is based on the following considerations. Monochromatic radiation in the process of propagation in a medium experiences coherent forward scattering on resonant particles (atoms, ions or molecules). Coherently scattered radiation is opposite in phase to the incident radiation. Even for resonant particles randomly located in a medium, a regular, and not random, destructive interference of the incident and coherently scattered radiation is always established. This leads to the field attenuation during propagation in a thick medium. As a result, at the exit from the medium, the amplitude of continuous radiation is attenuated according to the Bouguer–Lambert–Beer law as \(\exp ( - \alpha L{\text{/}}2)\), where α is the absorption coefficient of the medium, taking into account only resonant losses, and L is the length of the medium. The formation of the coherently scattered radiation takes time. Therefore, a fast, for example, stepwise switching on of the field leads to transient processes, which are called optical transient nutations. For optically thick samples, the rate of formation of the coherently scattered field can be roughly estimated as \({{\Gamma }_{c}} = D{\text{/}}{{T}_{2}}\), where \({{T}_{2}}\) is the phase relaxation time of resonant particles (inhomogeneous broadening is not considered here) and \(D = \alpha L\) is the effective thickness of the medium, see [29–32, 34–36]. If, after the formation of coherently scattered radiation in the medium, the phase of the incident radiation is switched fast to the opposite one, i.e. it is changed to π, then these fields will constructively interfere and a burst of radiation with a doubled amplitude will appear. The intensity of the radiation pulse is four times greater than the intensity of the incident radiation, the rise time of the pulse (the duration of its leading edge) depends on the rate of phase switching, and the trailing edge decreases with the rate of formation of a new coherently scattered radiation. Therefore, the pulse has an asymmetric shape. The pulse repetition rate during successive switching of the phase of the incident radiation is limited by the formation time of the coherently scattered radiation. For example, when generating a series of pulses after filtering through a cloud of cold atoms with an effective thickness \(D = 95\), the phase change repetition time \(0.24{{T}_{2}}\) was used [33]. This is about an order of magnitude longer than time \({{T}_{c}} = 1{\text{/}}{{\Gamma }_{c}}\), since the establishment of the stationary state of the coherently scattered field is determined by the slowly decaying zero-order Bessel function, which oscillates. In this scheme of pulse generation, it is impossible to make the interval between pulses shorter than the time of complete formation of the coherently scattered field after the next phase change. An analysis of the data given in [31–33] shows that the interval between successive phase switches should not be shorter than \({{T}_{2}}{\text{/}}\sqrt D \).

In this paper, it is proposed to overcome the aforementioned limitation and increase the pulse repetition rate by 3–5 orders of magnitude (depending on the value of \({{T}_{2}}\)), create pulses of a symmetrical shape, and significantly shorten their duration. In addition, in the new method of pulse generation, it is not necessary to use narrow-band resonant filters. Standard frequency filters used in optics can be applied.

If the phase of the radiation is rapidly changed with a period T by \(2\pi \), then most of the time the atoms will interact with the field that has the same phase. Only in short time intervals \(\tau \ll T\), during which phase changes take place, do transients occur. In the middle of these intervals, when the phase takes the value π, pulses will be formed, see Fig. 1. In the proposed scheme, atoms most of the time, \(1 - 2\tau {\text{/}}T\) in relative units, are in the field with the same phase, and only part of the time, \(2\tau {\text{/}}T\), experience transient processes. Therefore, a coherently scattered radiation field will always have time to be fully developed, regardless of the relation between the phase relaxation rate \({{T}_{2}}\) and the phase change period T. Pulses will appear at the moments of the passage of the phase of value π, their duration will depend on the time of passage of this value. For a linear law of phase change, the pulses will have a symmetrical shape. Below these qualitative arguments will be confirmed by mathematical calculations.

Fig. 1.
figure 1

(Color online) (a) Time evolution of the radiation phase \(\varphi (t)\) caused by a sequence of rectangular voltage pulses that are applied to the electro-optical modulator (dashed blue line). The durations of the phase rise and fall are equal to \(\tau = 0.05T\). The intensity \(I(t)\) of the sequence of radiation pulses that are formed after the removal of the central component of the spectrum of the phase-modulated field using the resonant absorber with effective thickness of \(D = 15\) is shown by red solid line. (b) An enlarged part of the graph (a) is shown to detail the pulse formation.

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For simplicity, let us consider the following periodic law of radiation phase change \(\varphi (t)\), which is governed by an electro-optical modulator: \({{E}_{M}}(t) = {{E}_{0}}\exp [ - i{{\omega }_{r}}t + ikr + i\varphi (t)]\), where \({{E}_{0}}\) is the field amplitude, \({{\omega }_{r}}\) and k are its frequency and wave number, r is the propagation distance,

$$\varphi (t) = \sum\limits_{n = - \infty }^{ + \infty } \phi (t - nT),$$
(1)
$$\phi (t) = \Delta \left\{ {\frac{{t - {{T}_{{ - - }}}}}{\tau }\theta (t - {{T}_{{ - - }}}) + \frac{{t - {{T}_{{ + + }}}}}{\tau }\theta (t - {{T}_{{ + + }}})} \right.$$
$$ + \;\left( {1 - \frac{{t - {{T}_{{ - - }}}}}{\tau }} \right)\theta (t - {{T}_{{ - + }}})$$
(2)
$$ - \;\left. {\left( {1 + \frac{{t - {{T}_{{ + + }}}}}{\tau }} \right)\theta (t - {{T}_{{ + - }}})} \right\},$$

n in an integer, Δ is the maximum phase shift, which is induced by periodic rectangular voltage pulses applied to the electro-optical modulator, \({{T}_{{ \pm \pm }}} = (T \pm {{T}_{p}} \pm \tau ){\text{/}}2\), T is the pulse repetition period, \({{T}_{p}}\) is the duration of the voltage pulses at half height, \(\tau \) is the duration of the time interval of linear increase/decrease of the phase on the leading and trailing edges of rectangular pulses, see Fig. 1b.

A periodic change of phase transforms a single narrow line of the cw laser to a frequency comb with a period equal to \(\Omega = 2\pi {\text{/}}T\). The amplitudes of the spectral components of the comb can be calculated by the expression

$${{A}_{n}} = \frac{{{{E}_{0}}}}{T}\int\limits_0^T {{{e}^{{i\varphi (t) + i2\pi nt/T}}}} dt,$$
(3)

where n is the number of the component with frequency \({{\omega }_{r}} + n\Omega \) (n is an integer). In the case under consideration, when \(\Delta = 2\pi \), the amplitude of the central component \(n = 0\) does not depend on the duration of rectangular voltage pulses, \({{T}_{p}}\), and is described by a simple expression, i.e.,

$$\frac{{{{A}_{0}}}}{{{{E}_{0}}}} = 1 - 2\frac{\tau }{T}.$$
(4)

The amplitudes of the even components \(n = 2k\), where k is an integer, but \(k \ne 0\), have the simplest form when \({{T}_{p}} = T{\text{/}}2\). i.e.,

$$\frac{{{{A}_{{2k}}}}}{{{{E}_{0}}}} = ( - {{1)}^{{k + 1}}}\frac{{2\tau }}{T}{\text{sinc}}\left( {\pi k\frac{{2\tau }}{T}} \right)\left[ {1 + \frac{{{{{(2k)}}^{2}}}}{{{{{\left( {\frac{T}{\tau }} \right)}}^{2}} - {{{(2k)}}^{2}}}}} \right],$$
(5)

and the amplitudes of the odd components \(n = 2k + 1\), where integer k can take the value 0, are described, respectively, by

$$\frac{{{{A}_{{2k + 1}}}}}{{{{E}_{0}}}} = 2i{{( - 1)}^{k}}{\text{sinc}}\left( {\pi k\frac{{2\tau }}{T}} \right)\frac{{2k + 1}}{{{{{\left( {\frac{T}{\tau }} \right)}}^{2}} - {{{(2k + 1)}}^{2}}}},$$
(6)

where \({\text{sinc}}(x) = \sin (x){\text{/}}x\).

It can be seen from the above expressions that the amplitude of the central component \({{\omega }_{r}}\) of the radiation slightly decreases compared to the amplitude of the radiation before modulation by the value \({{E}_{0}}2\tau {\text{/}}T\), which is proportional to the ratio of the duration of the phase change, \(2\tau \), to the modulation period, T. The amplitudes of the even components, \(n = 2k\), are proportional to this ratio and decrease with increasing number n as \({\text{sinc}}\left( {\frac{{\pi n\tau }}{T}} \right)\). Their values decrease to zero at \(n = \pm 2T{\text{/}}\tau \) due to the growing part in square brackets of Eq. (5), see Fig. 2. Therefore, the width of the radiation spectrum between the extreme zero values of the amplitudes at \(n = \pm 2T{\text{/}}\tau \) can be estimated as \(4\Omega T{\text{/}}\tau \). Thus, the field spectrum is significantly broadened. It should be noted that the sign of the even components changes sequentially from plus to minus as the number of the component increases. As regards the amplitudes of the odd components of the field spectrum, they are approximately \(T{\text{/}}\tau \) times smaller than the amplitudes of the even components and have an imaginary amplitude. It should be noted that in Eqs. (5) and (6) there are singularities at \(n = \pm T{\text{/}}\tau \), but they are artificial, since according to definition (3) the values of the spectral components are limited.

Fig. 2.
figure 2

(Color online) Envelopes of the dependences of the absolute value of the (solid red line) even, \(Ae(k) = {\text{|}}{{A}_{{2k}}}{\text{/}}{{E}_{0}}{\text{|}}\), and (dashed blue line) odd, \(Ao(k) = {\text{|}}{{A}_{{2k + 1}}}{\text{/}}{{E}_{0}}{\text{|}}\), components on k for \(\tau = 0.05T\). The central component is not shown because of large contrast. The spectrum is symmetrical with respect to the sign of the number of the spectral component, \({\text{|}}{{A}_{n}}{\text{|}} = {\text{|}}{{A}_{{ - n}}}{\text{|}}\). Therefore, its left side is also not shown.

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If radiation with such a spectrum is transmitted through an optically dense filter with a resonant frequency \({{\omega }_{r}}\) and a narrow absorption line with a half-width \(\gamma = 1{\text{/}}{{T}_{2}} \ll \Omega \), then the amplitude of the central component will decrease by a factor of \(\exp ( - D{\text{/}}2)\). This is true in the approximation of a linear response of each particle in the resonant filter [16, 17], when the intensity of continuous radiation is not sufficient for bleaching a dense medium. In this case, the amplitude of the radiation \({{E}_{F}}(t) = E(t)\exp ( - i{{\omega }_{r}}t + ikr)\) that has passed through the filter can be represented as follows

$$E(t) = {{E}_{0}}{{e}^{{i\varphi (t)}}} + {{A}_{0}}\left( {{{e}^{{ - D/2}}} - 1} \right),$$
(7)

where \({{A}_{0}}\exp ( - D{\text{/}}2)\) is the result of the filtering, and \( - {{A}_{0}}\) is introduced to remove the central component of the comb so as not to count it twice. As a result, we obtain a simple expression for the radiation intensity \(I(t) = {\text{|}}E(t){{{\text{|}}}^{2}}\) after filtering, i.e.,

$$I(t) = {{I}_{0}}\left[ {1 - 2a\cos \varphi (t) + {{a}^{2}}} \right],$$
(8)

where \({{I}_{0}} = {\text{|}}{{E}_{0}}{{{\text{|}}}^{2}}\) and \(a = (1 - {{e}^{{ - D/2}}}){{A}_{0}}{\text{/}}{{E}_{0}}\). It can be seen from Eq. (8) that when the phase \(\varphi (t)\) is equal to 0 or \(2\pi \), the radiation intensity due to interference with the effective field, which is the central component in antiphase, decreases to the value \({{(1 - a)}^{2}}{{I}_{0}}\). Since it is assumed that the value of \({{A}_{0}}\) is close to \({{E}_{0}}\), and the exponent \(\exp ( - D)\) is small, the resulting intensity drops significantly. If the effective thickness of the filter is very large and this exponent can be neglected, the intensity drops by a factor of \({{(T{\text{/}}2\tau )}^{2}}\). For example, for the values of the parameters \(\tau = 0.05T\) and \(D = 15\), the radiation intensity drops by a factor of 100.

In the time intervals when the phase increases from 0 to \(2\pi \) or decreases from \(2\pi \) to zero, the nature of the field interference changes. For example, when the phase \(\varphi (t)\) takes the value π, see Fig. 1b, the interference becomes constructive and a pulse appears, the maximum intensity of which is equal to \({{(1 + a)}^{2}}{{I}_{0}}\). For large values of the effective thickness of the resonant filter, this intensity is equal to \(4(1 - \tau {\text{/}}T{{)}^{2}}{{I}_{0}}\). For example, for the above values of the parameters, the pulse intensity is 3.6 times larger than the radiation intensity of the CW laser. The development of the pulse in the process of changing the phase is shown in Fig. 1b. It can be seen that the time interval during which the pulse develops is equal to τ.

If we now compare the obtained results with the case considered in [29–33], when the phase changes only by π; i.e., \(\Delta = \pi \) in Eq. (2), it can be seen that the central component of the spectrum A0/E0 = 1 – \(2({{T}_{p}}{\text{/}}T) + 4i(\tau {\text{/}}T)\) is very small. When \({{T}_{p}} = T{\text{/}}2\), it is proportional to the ratio \(\tau {\text{/}}T\). Therefore, for \(\tau \ll T\), the interference of the components of the field that has passed through the filter removing only the central component does not give a noticeable result. Only in the case when the period of voltage pulses is comparable with the long phase relaxation time, \({{T}_{2}}\), of resonant particles, large pulses appear on an almost zero background. This is due to the fact that under the given condition, \(T \sim {{T}_{2}}\), the comb spectrum has a frequency period \(\Omega \sim \gamma \) and many spectral components interact with the resonant particles of the filter in an extended medium. Therefore, the use of a standard optical filter that selectively suppresses a certain component will not give the desired result in this case.

In the case \(\Delta = 2\pi \), which is considered in the article, the amplitude of the central component of the spectrum \({{A}_{0}}\), see Eq. (4), does not depend on the duration of the rectangular voltage pulse \({{T}_{p}}\). Therefore, it can be changed without affecting the amplitude and duration of the pulses. A change in \({{T}_{p}}\) leads only to a change in the time interval between pulses. This opens up the possibility of creating pairs of closely spaced pulses separated by large time intervals. By changing the duration of the sequence of rectangular voltage pulses, one can create a sequence of radiation pulses, the distance between which contains information. For example, if a pair of pulses is separated by a time interval \(T{\text{/}}2\), then information bit 0 can be assigned to this pair. If the distance between pulses is T, then information bit 1 can be assigned to this pair. It is possible to divide time into N channels, i.e., a series of time slots following each other. Each slot can contain a pulse, which corresponds to bit 1, or not, which corresponds to bit 0. In turn, the series, i.e., “frames” also go sequentially one after another. This will enable time division multiplexing.

When the distance between pulses changes in a non-periodic way, the amplitude of the pulses can be estimated by calculating the average value of the period of the pulses, which is equal to the time interval of the pulse train divided by the average number of pulses in the sequence. Then, this average value of the period can be used to estimate the value of \({{A}_{0}}\), which directly determines the intensity of the pulses.

It is also possible to create single pulses by switching the phase once according to the expression

$${{\varphi }_{s}}(t) = \Delta \left[ {\frac{{{{t}_{ + }}}}{\tau }\theta \left( {{{t}_{ + }}} \right) + \left( {1 - \frac{{{{t}_{ + }}}}{\tau }} \right)\theta \left( {{{t}_{ - }}} \right)} \right],$$
(9)

where \({{t}_{ \pm }} = t - {{t}_{s}} \pm \tau {\text{/}}2\) and \({{t}_{s}}\) is the time when the phase acquires the value π. The remaining parameters have the same meaning as in Eq. (2). The time interval when a single phase changes is shown in Fig. 3 by a thin solid black line. Such a phase switch leads to the generation of a pulse, the intensity of which is described by Eq. (8). Let us assume that before a single phase switching there is enough time for the complete formation of coherently scattered radiation. Then we can assume that in Eq. (4)\(T \to \infty \) and \(\tau {\text{/}}T = 0\). Therefore, in (8) one can use the value \({{A}_{0}}\) equal to \({{E}_{0}}\). Time evolution of the pulse obtained in this way is shown by red solid line in Fig. 3.

Fig. 3.
figure 3

(Color online) Single phase jump from 0 to \(2\pi \) is shown by a thin black line, \({{t}_{s}} = 100\)ps is the time when the phase takes the value of π. Also, this time is taken as the time scale in the plot. The radiation field is turned on long time before the phase jump. On the graph, the phase \({{\varphi }_{s}}(t)\) is normalized to π. The dependence of the radiation intensity obtained using Eq. (10) for \({{T}_{2}} = 30\) ns, \(D = 15\), and \(\tau = 5\) ps is shown by dotted blue line. Solid red line shows the dependence obtained using Eq. (8).

Full size image

The same result can be obtained by solving the Maxwell–Bloch equations for continuous radiation with a time-varying phase in the linear response approximation for an individual resonant particle [29, 30, 32, 34–36], i.e.,

$${{E}_{s}}(t) = {{E}_{0}}\left[ {{{e}^{{i{{\varphi }_{s}}(t)}}} - b\int\limits_0^{ + \infty } {{e}^{{i{{\varphi }_{s}}(t - x) - \gamma x}}}\frac{{{{J}_{1}}\left( {2\sqrt {bx} } \right)}}{{\sqrt {bx} }}dx} \right],$$
(10)

where \(\gamma = 1{\text{/}}{{T}_{2}}\) is the phase relaxation rate of resonant particles in the absorber, \(b = \gamma D{\text{/}}2\), \({{J}_{1}}\left( {2\sqrt {bx} } \right)\) is the first-order Bessel function. Time evolution of the radiation intensity \({{I}_{s}} = {\text{|}}{{E}_{s}}(t){\text{|}}\) is shown in Fig. 3 by dotted blue line and completely coincides with that obtained using Eq. (8).

The generation of square wave voltage pulses with steep edges can be difficult, especially if the pulse repetition rate is high. For example, when choosing the frequency \(\Omega {\text{/}}2\pi = 10\) GHz, rather short voltage on/off times are required to create high-intensity pulses on an almost zero background. At the specified frequency, time \(\tau = 0.05T\) used in this work for illustration is 5 ps. Meanwhile, the Gunn diode or IMPATT diode allows switching in the electrical circuit at a rate of 1 THz [37, 38].

A periodic sequence of phase changes in the form of rectangular pulses with steep edges can be created using harmonic synthesis, i.e.,

$${{\varphi }_{h}}(t) = \pi - 4\sum\limits_{k = 0}^N \frac{{{{{( - 1)}}^{k}}\cos \left[ {(2k + 1)\frac{{2\pi t}}{T}} \right]}}{{2k + 1}},$$
(11)

where N is the number of harmonics from which a periodic sequence of a signal of the required shape is synthesized. The expansion in terms of harmonics (11) with \(N \to \infty \) was obtained for a sequence of rectangular pulses with stepwise phase change (1), where \(\tau = 0\). For a finite number N, the pulse fronts have a certain slope. In Fig. 4a, red solid line shows the approximation by five harmonics (\(N = 4\)) with frequencies \(\Omega \), \(3\Omega \), \(5\Omega \), \(7\Omega \), and \(9\Omega \). The synthesis of these harmonics gives a sequence of almost rectangular pulses, the trailing and leading edges of which ideally coincide with the fronts of \(\varphi (t)\) sequence in Eq. (1) with \(\tau = 0.05T\) (shown by blue dotted line in Figs. 1a and 4a).

Fig. 4.
figure 4

(Color online) (a) Radiation phase modulation \({{\varphi }_{h}}(t)\), see Eq. (11), synthesized from five harmonics, (solid red line), which is quite close to the rectangular phase modulation \(\varphi (t)\) and described by Eq. (1) with \(\tau = 0.05T\), is shown by blue dotted line. (b) Radiation pulses, \({{I}_{h}}(t)\), produced by the filtering the central component of the radiation spectrum generated by the phase modula-tion (11).

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The radiation intensity with phase \({{\varphi }_{h}}(t)\) after the filter can be obtained using formula (8), where the parameter \({{A}_{0}}\) is calculated using Eq. (3), where \(n = 0\). For \(N = 4\), numerical integration gives \({{A}_{0}}{\text{/}}{{E}_{0}} = 0.867\). The sequence of pulses after the resonant filter with \(D = 15\) is shown in Fig. 4b. The shape of the pulses almost coincides with that shown in Fig. 1a for radiation with phase modulation \(\varphi (t)\) under the condition \(\tau = 0.05T\) and the same effective thickness of the filter. The difference is revealed in the appearance of small oscillations on the pulse wings. They are caused by the oscillating nature of reaching the values 0 and \(2\pi \) in the evolution of phase \({{\varphi }_{h}}(t)\). The considered harmonic synthesis is technically possible for \(\Omega = 10\) GHz, since in this case a harmonic with a maximum frequency of \(9\Omega = 90\) GHz can be obtained using modern generators. They will make it possible to generate pulses with a FWHM of 2.5 ps and a repetition rate of 20 GHz.

The proposed method for generating stable short pulses with a controlled duty cycle differs fundamentally from that proposed earlier in [32] and experimentally demonstrated in [33]. In the previous method, the pulse repetition frequency could not be greater than the effective spectral width of the filter, since the filter must change the amplitudes and phases of all significant components of the spectrum of the frequency comb created by the phase modulation of the field. In the proposed method, the filter removes only the central component of the comb, and therefore its spectral width can be significantly smaller than the repetition frequency. The method makes it possible to generate pulses with a duration of several picoseconds with the period of ~50 ps using available electro-optical modulators. As frequency filters, one can use a cloud of cold atoms, for example, \({\text{S}}{{{\text{t}}}^{{88}}}\) at the \(^{1}{{S}_{0}} \to {{\;}^{3}}{\kern 1pt} {{P}_{1}}\) transition (689 nm) with the homogeneous line width \(2\gamma {\text{/}}2\pi = 7.5\) kHz [33] or Rb85, \({{D}_{1}}\) line (795 nm) with \(2\gamma {\text{/}}2\pi = 6\) MHz [39, 40]. Alkali metal vapors can also be used, for example, Rb85, \({{D}_{2}}\) line (589 nm) with a Doppler width of 1.69 GHz [16]. It is also possible to apply filters based on multipixel liquid crystal modulators [41]. In addition, the proposed method allows using one phase modulator to create and encode information, while in the previous methods of generating a periodic pulse sequence, information is encoded into this sequence using an additional amplitude modulator synchronized with the generator.