INTRODUCTION

Devices for controlling the shape of femtosecond pulses have a high demand in many fields of science. Modulated pulses are used to control chemical reactions [1], in quantum optics applications [2], coherent photon generation [3], studies of the molecular electron density [4] and molecular fragmentation [5], generation of ultrashort pulses [6], and other applications.

Due to the high demand, the task of forming light pulses with an arbitrary envelope has been relevant for more than 35 years [7, 8]. There are several techniques for pulse envelope formation [9]. These techniques are typically based on the spectral decomposition of the initial pulse, the amplitude and/or phase modulation of individual spectral components, and combination of the spectral components into a new pulse of a desired shape [10]. Pairs of prisms, diffraction gratings, or other dispersive elements combined into a so-called 4f systems are used experimentally for the spectral decomposition of a pulse and its recomposition [11]. Different spectral components of the pulse propagate between the dispersion elements along different trajectories. The components pass through a multichannel modulator such as a liquid crystal matrix [12, 13], an acousto-optic filters [14], a deformable mirror [15], or a digital micromirror [16], currently used in movie projectors.

The formation of the femtosecond pulse envelope is theoretically described [9, 17] by the Fourier transform of the desired pulse envelope function in order to obtain the coefficients of its spectral decomposition. However, spatial modulators and other components of 4f systems can have a complex spectral response, the relation between the amplitude and phase, and other technical features that need to be taken into account in calculations. Therefore, more sophisticated techniques involving phase and amplitude selection using evolutionary algorithms [18, 19], neural networks [20, 21], and other machine learning algorithms are applied.

4f systems are large, are assembled on an optical table from expensive components, and are sensitive to tuning; for this reason, alternative ways are sought to generate femtosecond pulses. For example, arrays of chirped Bragg fibers [22], waveguides made by direct laser writing [23], metamaterials and metasurfaces [25], and nanoantennas [26] are used. Thin photonic crystals are utilized to chirp laser pulses [27].

In this work, for shaping of a femtosecond pulse during its reflection from a photonic crystal with refractive index modulation in the form of a sum of harmonic functions, it is proposed is to selectively control the amplitude and phase of the spectral components of the pulse using the amplitude and phase of the corresponding harmonic functions. The proposed scheme operates on reflection from a single optical element, does not require focusing of light, and therefore is suitable to form high-power pulses.

PROBLEM FORMULATION

We consider a transparent medium with the optical thickness L, refractive index \({{n}_{0}} = 1.5\), and flat interfaces. Let light propagate only along the x axis normal to the boundaries; i.e., the problem is one-dimensional. For the convenience of calculations, the coordinates inside the medium are measured in units of the optical path. The refractive index of the medium can deviate from the mean value and have some dependence \(n(x)\).

Let the function \(F(t)\) be real and continuous and have a finite spectrum. It is necessary to find a function \(n(x)\) such that a Gaussian optical reflected from the medium has an envelope corresponding to the function \(F(t)\).

SOLUTION

Let the refractive index of this medium be spatially modulated by a single harmonic function

$${{n}_{1}}(x) = {{n}_{0}} + {{A}_{1}}\cos ({{k}_{1}}x + {{\phi }_{1}}),$$
(1)

where \({{A}_{1}}\), \({{\phi }_{1}}\), and \({{k}_{1}}\) are the amplitude, phase, and wave vector (in the case of a one-dimensional problem, the wavenumber) of the grating of the refractive index, respectively. It is known (see, for example, [28]) that an incident wave with a wave vector of \({{k}_{1}}\)/2 undergoes diffraction reflection from this grating because the wave vector of the grating is added to the wave vector of the wave, resulting in the reversal of the direction of motion of the wave. The amplitude and phase of the reflected wave are proportional to the amplitude and phase of the refractive index modulation of the grating, respectively.

To illustrate the choice of the function \(F(t)\), we consider a rectangular optical pulse with a duration of 800 fs and a central wavelength of λ0 = 800 nm (Fig. 1a). According to the properties of the Fourier transform, the spectrum of this pulse has the form of the Fourier image \(F(\omega )\) shifted by the frequency of optical oscillations of the light wave \({{\omega }_{0}} = 2\pi c{\text{/}}{{\lambda }_{0}}\), where \(c\) is the speed of light. The absolute value of the complex function \(F(\omega - {{\omega }_{0}})\) is plotted in Fig. 1b in the form of the dependence on the wavelength, i.e., \(F(2\pi c{\text{/}}\lambda - 2\pi c{\text{/}}{{\lambda }_{0}})\) for clarity instead of the frequency dependence.

Fig. 1.
figure 1

(Color online) Step-by-step illustration of the method: (a) the desired shape of the optical pulse envelope, (b) the pulse spectrum with this envelope, and (c) the photonic crystal structure constructed according to the proposed method in the first and last 4 µm.

Full size image

Each Fourier component of the function \(F(\omega - {{\omega }_{0}})\) can be associated with the spatial harmonic

$${{n}_{\omega }}(x) = {{n}_{0}} + \delta n{{A}_{\omega }}\cos \left( {\frac{{2\omega x}}{c} + {{\phi }_{\omega }}} \right),$$
(2)

where the amplitude and phase coincide with the respective parameters of the Fourier component:

$${{A}_{\omega }} = {\text{|}}F(\omega - {{\omega }_{0}}){\text{|}},\quad {{\phi }_{\omega }} = \arg F(\omega - {{\omega }_{0}}).$$
(3)

A factor of 2 in the cosine argument in Eq. (2) means that the wave vector of the grating is twice the wave vector of the reflected wave. In Eq. (2), \(\delta n\) is the modulation coefficient, taken as 0.0005 for the example.

The combination of all the spatial harmonics gives

$$n(x) = {{n}_{0}} + \delta n\int\limits_{ - \infty }^\infty {{A}_{\omega }}\cos \left( {\frac{{2\omega x}}{c} + {{\phi }_{\omega }}} \right)d\omega .$$
(4)

The dependence \(n(x)\) on the optical thickness x allows us to take into account the dispersion of the medium; in this case, the dependence of the refractive index in the optical path \({{x}_{\omega }}\) on the frequency should be taken into account in the integrand.

An example of the dependence \(n(x)\) for the rectangular pulse is shown in Fig. 1c; the dependence has the form of a quasiharmonic function that is the sum of harmonic functions with close frequencies. The problem is solved.

The Fourier transform was calculated using the fast Fourier transform algorithm, and 8192 time points in a time step of 1 fs were used to calculate temporal characteristics.

VERIFICATION OF THE SOLUTION

Using a combination of known methods, the time optical response of this medium to the incident Gaussian pulse can be determined from the known dependence \(n(x)\). We first find the reflection coefficient spectrum of this structure. For an arbitrary function \(n(x)\), it is most convenient to use the transfer matrix method [29] or the recurrence method [30]. To discretize \(n(x)\), we approximate it by a piecewise constant function, which is physically equivalent to the separation of the optical medium into layers. The correctness of this approximation was discussed in [31]. We choose a coordinate step \(\delta x\) = 20 nm much smaller than the optical wavelength. The coordinates can take the values \({{x}_{j}} = j\delta x\); thus, the refractive indices of the layers are equal to \({{n}_{j}} = n({{x}_{j}})\), and their thicknesses are \({{d}_{j}} = \delta x{\text{/}}{{n}_{j}}\). Using the set of \({{n}_{j}}\) and \({{d}_{j}}\), the amplitude spectrum of the complex reflection coefficient \(r(\lambda )\) was calculated by the transfer matrix method. The absolute value of this spectrum is shown in Fig. 2a. One can clearly see that the plot \({\text{|}}r(\lambda ){\text{|}}\) is very similar to the spectrum of the function \(F\).

Fig. 2.
figure 2

(Color online) (a) Spectrum of the reflection coefficient of the modulated medium in comparison with the spectrum of the input signal (amplitude values). (b) Input and output signal envelopes.

Full size image

The reflection coefficient spectrum of the modulated medium has a relatively small maximum value of 0.15; in addition, most (in area under the plot) of the spectrum of the original pulse falls in the region of even lower reflection coefficient. Thus, most of the initial signal will pass through the medium. This is due to the small modulation amplitude \(\delta n = 0.0005\). To increase the conversion efficiency, this value can be increased, but for the purpose of illustrating the method, this value is also sufficient.

We now calculate the response of this multilayer structure to the pulse with the Gaussian envelope \(G(t)\) (Fig. 2b, the pulse duration chosen for the example is 30 fs). The calculation is performed by the well-known spectral decomposition method [9]. Namely, we find the Fourier image \(G(\omega - {{\omega }_{0}})\) of the pulse (Fig. 2a), then multiply each Fourier component by the reflection coefficient \(r(2\pi c{\text{/}}\omega )\), and perform the inverse Fourier transform to obtain the time form of the envelope of the pulse reflected from the medium:

$$S(t) = \int\limits_{ - \infty }^\infty r(2\pi c{\text{/}}\omega )G(\omega - {{\omega }_{0}}){{e}^{{i\omega t}}}d\omega .$$
(5)

The calculation result is shown in Fig. 2b. The time dependence of the envelope has the form of a rectangular function with smoothed fronts and a FWHM of 780 fs, which repeats the form of the function \(F(t)\).

RESULTS AND DISCUSSION

It can be seen in Fig. 2b that the fronts of the resulting pulse are smoothed; more specifically, the rise time from the 20 to 80% level is 30 fs. The reason is that the original function \(F(t)\) has an infinite spectrum: its fronts are absolutely sharp, while the spectrum of the input pulse has a Gaussian form and drops rapidly as it moves away from the central frequency. Therefore, harmonics necessary for the formation of sufficiently sharp fronts are absent in the input pulse, and according to Eq. (5), these harmonics will be absent in the output pulse as well.

Figure 3 shows how the leading edge of the output pulse changes as the duration of the input pulse increases. The front edge becomes smoother, and the rise time increases: Fig. 3 shows the rise time from 20 to 80%. This time coincides with the duration of the original pulse with a high accuracy, and a small difference is explained by rounding to the nearest discrete time value. Figure 3 additionally shows the reflection coefficient spectrum of the medium compared to the power spectrum of the original pulse for the 100 fs pulse. It is seen that almost all features of the medium spectrum are cut off due to the narrow spectrum of the input pulse. In the general case, features (peaks, fronts, drops) shorter in time than the duration of the initial pulse cannot be created in the output pulse.

Fig. 3.
figure 3

(Color online) Input and output signal envelopes (the leading edge is shown, the output pulse is shifted in time for visual comparison with the input pulse) at input pulse durations of 20, 30, 50, and 100 fs (indicated in the panels). Inset: the reflection spectrum of the optical medium in comparison with the spectrum of the initial pulse.

Full size image

DEPENDENCE ON THE THICKNESS OF THE OPTICAL MEDIUM

We calculate the distribution \(n(x)\) that ensures a sequence of four reflected Gaussian pulses with a FWHM of 500 fs. The corresponding function \(F(t)\) is shown in Fig. 4a. A 30-fs Gaussian pulse was used as the input pulse; the variation of the input duration in the range of 30–200 fs insignificantly changes the results. The resulting output pulse envelope (Fig. 4a) expectedly looks like four consecutive Gaussian pulses of the same amplitude, and the time delay between them corresponds to the desired one and is equal to τ = (1630 ± 1) fs.

Fig. 4.
figure 4

(Color online) (a) Preset time response function and (b) the corresponding spatial distribution of the refractive index of the optical medium. (c) Output time response function for a 30-fs Gaussian pulse reflected from the modulated medium.

Full size image

The profile of the refractive index \(n(x)\) has four Gaussian maxima each formed by harmonic functions with close frequencies. These maxima have a Gaussian shape because the Fourier transform of a Gaussian function is the Gaussian function. The spatial distance between the maxima is Δx = (242 ± 2) nm.

Each of the maxima of the function \(n(x)\) is responsible for a particular pulse in the sequence, with the deeper maxima being responsible for the later arriving pulses. To demonstrate this, we artificially cut the optical medium, retaining only the first one, two, and three maxima (Fig. 5). Removal of the corresponding maxima of the refractive index modulation function leads to the disappearance of pulses in the time response.

Fig. 5.
figure 5

(Color online) Time response functions for reflection from an optical medium artificially clipped at points of (from top to bottom) 860, 620, and 370 µm.

Full size image

We note that Δx and τ are related as 2Δx = cτ. Thus, the output pulses are time delayed because they are reflected from different depths of the optical medium. Consequently, the maximum possible time length of the received pulse sequence is \({{T}_{{\max }}} = 2L{\text{/}}c\).

EFFECTS OF OPTICAL ABSORPTION

Absorption or scattering inevitably occurring in real optical media worsens the considered optical effects. To estimate the effect of optical absorption, we repeated the calculation with the complex refractive index \({{n}_{0}} = 1.5 + 0.0001i\). Figure 6a shows the response function of the absorbing medium. It has the form of four Gaussian maxima decreasing exponentially with time, and more delayed pulses are more attenuated because they travel longer optical paths.

Fig. 6.
figure 6

(Color online) (a) Time response function for reflection from an absorbing optical medium. (b) Refractive index profile with absorption compensation. (c) Output function for absorption-compensated medium.

Full size image

To compensate for absorption, we propose to modify the given function \(F(t)\) by artificially increasing the amplitude of the later components. Figure 6b shows the profile of \(n(x)\), constructed for a given function in which the amplitude of each successive Gaussian pulse is 1.31 times larger than the preceding one. The pulse reflected from such a medium is converted into a sequence (Fig. 6c), in which the optical absorption is compensated by increasing the reflection coefficient of the medium, so that the optical response again corresponds to the desired one.

POSSIBILITIES OF EXPERIMENTAL REALIZATION

The described method of modulation of the optical medium to form femtosecond pulses can be implemented in the experiment using the following methods of manufacturing photonic structures, in which an arbitrary continuous or quasi-continuous modulation of the refractive index can be created.

• Electrochemical etching of porous silicon [32], aluminum [33, 34], and titanium [35]. The methods allow the fabrication of optical media up to 1 mm thick, the refractive index contrast reaches 0.2, the media do not absorb light, but light scattering on pores is essential.

• Two-photon laser lithography [36, 37]. It can be used to create media with a gradient refractive index. A promising method of grating creation is two-photon lithography in oxidized porous silicon [38]. The refractive index contrast does not exceed 0.05, but calculations show that this value is acceptable. The optical thickness in this case is limited by about 1 cm, which is determined mainly by the time cost of the setup.

• Fiber Bragg gratings created by direct laser writing [39]. Longitudinal modulation of the refractive index is achieved by intense laser irradiation of the fiber core [40]. The refractive index contrast does not exceed 10–4–10–3, but this can be compensated for by the length of the fiber. The linear size of optical fibers in the case of the present problem can be considered as unlimited.

CONCLUSIONS

A method of constructing an optical medium capable to form a reflected femtosecond pulse or a reflected sequence of pulses with a given envelope has been demonstrated theoretically. The following limitations of the method have been established. First, the spectrum of the desired sequence should not be wider than the spectrum of the pulse incident on the medium and, correspondingly, the temporal features (maxima, minima, fronts) of the desired sequence should not be shorter than the duration of the input pulse. Second, the total length of the sequence cannot exceed twice the optical thickness of the medium divided by the speed of light.

In addition, the influence of optical absorption on the operation of the method has been considered, and a technique has been shown to compensate for absorption by increasing the reflection coefficient.