Information transfer via temporal convolution in nonlinear optics


The general layout of the exploited FSHG system is illustrated in Fig. 1a. It consists of a 4f.-setup with a nonlinear crystal in the frequency plane13,14,15. In this plane, the spectrum is dispersed spatially along a focal line as in a spectrometer. The frequencies are separated within focussed narrow spectral slices.

Figure 1

FSHG optical system. (a) The field is projected into the Fourier plane using a grating and a focusing lens separated by the focal distance. Through the χ(2) interaction with the nonlinear crystal located in the Fourier plane (green line), a field at the second harmonic is generated. The total field is recombined after the second lens and grating. (b) The fundamental field entering the 4f-setup consists of a pair of pulses having a relative delay (Delta t=2tau ). (c) From the temporal convolution of the fundamental pulse pair, three second harmonic pulses are generated at the output, separated by (Delta t=tau ). (de) In frequency, the corresponding spectra are characterized by interference modulations with twice the period at the central frequency (2{omega }_{0}.)

As a consequence, there is no cross-talk between the spectral components, and the second harmonic field is given by12:

$$stackrel{sim }{mathbf{E}}left(2{varvec{upomega}}right)propto stackrel{sim }{mathbf{E}}left({varvec{upomega}}right)cdot stackrel{sim }{mathbf{E}}left({varvec{upomega}}right)$$

(1)

This equation is equivalent to a temporal convolution:

$$mathbf{E}left(frac{mathbf{t}}{2}right)propto mathbf{E}left(mathbf{t}right)otimes mathbf{E}left(mathbf{t}right)$$

(2)

In practice, this means that while spectral cross-talk is switched off, temporal cross-talk is enabled. This is in strong contrast to conventional time-domain SHG where the second harmonic field results from the product ({varvec{E}}({varvec{t}})cdot {varvec{E}}({varvec{t}})), so that temporal convolution operations cannot be accessed. To our knowledge, the operations of Eqs. (1) and (2) are unique to a spectral device employing optical nonlinearities in the Fourier domain.

Here, employing frequency domain nonlinear conversion, such temporal convolution of an optical field is demonstrated experimentally. To illustrate our concept, we consider two femtosecond pulses delayed by a time 2τ much longer than their respective durations, as shown in Fig. 1b. Adapting the concept of a temporal convolution, one would expect at the output of the FSHG setup the sequence of three distinct second harmonic pulses depicted in Fig. 1c, as we will discuss in detail later.

Experimentally, we initially consider a sequence of two identical fundamental pulses of 35-fs duration and separated by ~ 4 ps generated with a Michelson interferometer. They are frequency doubled by propagating through a FSHG setup after which all pulses (fundamental and second harmonic) are characterized through XFROG measurements obtained with a weak reference pulse at 790 nm10. This reference pulse has the effect of subtracting the average group delay of the overall system for all measurements. The XFROG spectrograms of all output pulses are shown in Fig. 2a for the fundamental (XFROG signal at 395 nm) and Fig. 2b for the second harmonic pulses (XFROG signal at 263 nm) coming out of the FSHG system. The central wavelength of the XFROG signal is determined from the sum-frequency generation process with the reference pulse at the fundamental central wavelength λ0 = 790 nm. Therefore, we have signals at λXFROG = λ0/2 = 395 nm and λXFROG = λ0/3 = 263 nm, for the fundamental and second-harmonic pulses, respectively. At 395 nm, the pair of fundamental pulses delayed by 4 ps is observed and each individual pulse is referred to as F1 and F2. At 263 nm, the spectrogram exhibits three pulses: An early pulse, referred to as H1 and leaving the system at the same time as F1, a second pulse H12, leaving the system 2 ps later, thus right in between F1 and F2, and finally, a third pulse H2 coincident with F2.

Figure. 2
figure2

X-FROG spectrograms of the pulse sequence after the FSHG stage. (a) Pair of identical fundamental pulses with 35-fs duration, separated by 4 ps representing the input pulses; (b) Triplet of identical second-harmonic pulses measured with the same reference pulse. The FSHG pulses are spaced by 2 ps.

With a simple test, we validate the respective contributions from the fundamental pulses to the generated second-harmonic pulses. For instance, one arm of the Michelson interferometer is blocked to let only one fundamental pulse going through the system, either the pulse F1 or F2. It is observed that if F1 is blocked, the harmonic pulses H1 and H12 are suppressed, with H2 remaining. While if F2 is blocked, pulses H12 and H2 disappear and H1 remains. This test confirms that the harmonic pulse H12 is a cross-term originating from both F1 and F2. Inversely, H1 and H2 are non-cross terms respectively and exclusively originating from F1 and F2. Those experimental observations are presented in the Supplementary Information document.

In a further step, the control of the cross-term pulse is demonstrated both in amplitude and phase by shaping the most delayed fundamental pulse F2. For instance, by inserting an optical element in the corresponding arm of the Michelson interferometer, F2 is modified either in spectral amplitude (Fig. 3) or in spectral phase (Fig. 4). In both cases, the relative delay between the two fundamental pulses is re-adjusted to ~ 4 ps after the insertion of the optical element.

Figure 3
figure3

Coherent transfer of information within the laser field through interferometric FSHG. An amplitude filter is inserted in the longer Michelson arm before the nonlinear stage; XFROG spectrograms featuring the amplitude distribution among both pulse sequences: (a) fundamental pulses and (c) second-harmonic pulses. (b) Spectrum (solid line) and phase (dashed line) of individual pulses (black F1, red F2) retrieved from XFROG reconstruction. The spectrum of the delayed pulse at ω0 (F2) has a sharp cut-off. (d) The sharp cut-off is transferred from F2 at 2ω0 to two pulses at two different delays, H12 and H2. Notably, H12 is leaving the system 2 ps before F2.

Figure 4
figure4

Coherent transfer of the phase through interferometric FSHG within the time window available for temporal convolution: A spectral phase filter (a 50 mm fused silica rod introducing positive group-delay dispersion) is inserted in the delayed Michelson arm before the nonlinear stage. Left column: XFROG spectrograms of the fundamental (a) and FSHG pulses (c) revealing the distribution of both pulse sequences at ω0 and 2ω0. (b) Spectrum (solid line) and phase (dashed line) of individual pulses (black F1, red F2) retrieved from XFROG reconstruction. The delayed pulse at ω0 (F2) has a positive chirp. (d) Its quadratic phase is transferred from F2 to the second harmonic pulses H12 and H2, where the phase stroke is doubled. H12 is leaving the system 2 ps before F2.

The effect of those added optical components on the fundamental pulses is visible from the XFROG spectrograms, as F2 significantly differs from F1: in the case of the amplitude filter in Fig. 3a, a spectral filter (Semrock Inc.) is blocking any wavelength above 785 nm. In consequence, the corresponding XFROG signal of F2 is truncated compared to F1. In the situation of phase variation with a dispersive component consisting of a 50-mm long fused silica rod (Fig. 4a), a positive linear chirp is observed for F2. To quantify those variations, each individual pulse is retrieved with a XFROG reconstruction algorithm. In Fig. 3b, the retrieved spectrum of pulse F2 exhibits the sharp edge introduced by the amplitude filter (red solid line). In Fig. 4b, the retrieved spectral phase of F2 shows the positive quadratic phase introduced by the fused silica rod (red dashed line).

The second-harmonic pulse sequence after the amplitude and phase modifications of F2 is also characterized by XFROG. In each situation, the pulse H1 remains identical. For the amplitude filter, the sharp edge of the spectrum of F2 is transferred to both H12, the central pulse and to H2, the latest second-harmonic pulse, as observed in the spectrogram (Fig. 3c) and also in the retrieved pulses (Fig. 3d, blue and red solid lines). In the case of the dispersive component, the linear chirp of F2 is transferred in the same manner to both pulses H12 and H2, see Fig. 4c. Considering the retrieved phase in Fig. 4d, we observe a flat spectral phase for H1 (black dashed line) as expected, since F1 is transform-limited. In contrast, for H12 and H2, a quadratic phase is observed with the cross-term H12 having the phase of F2, and H2 having twice this phase. This transfer of linear chirp is visible directly from the XFROG spectrogram of Fig. 4c, with the slope of H2 being identical to that of F2 and twice that of H12. Combining these results admits the linear transfer of amplitude and phase information along the temporal axis of the second harmonic laser field.

In summary, in our experiments, we process two identical pulses delayed by several picoseconds and we obtain a sequence of three distinct second harmonic pulses. Then, by modifying the most delayed fundamental pulse, we observe the coherent transfer of information from the fundamental to the second harmonic pulses at different delays. As such, it is confirmed that one of the generated pulses is effectively the product of a temporal convolution of the initial field, carrying the amplitude and phase of both fundamental pulses.

In a last experiment, we demonstrate the ability to isolate the convolved term by controlling the polarizations and adapting the phase-matching conditions of the nonlinear interaction. For this, we consider frequency doubling two delayed pulses having orthogonal polarizations. For this specific experiment, the amplitude and phase of the harmonic pulses were not retrieved because the crystals involved were 700 µm thick and the phase-matching bandwidth was limited compared to the full bandwidth 35 fs pulses. We were interested here in the polarization transfer and the generated pulse sequence through the convolution operation. In this case, employing a type-II crystal, it is assumed that only the pulse H12 fulfills the phase-matching conditions. This situation is illustrated in Fig. 5c and the experimental result is confirmed in Fig. 5d, where only the pulse H12 is generated. In comparison, the situation of F1 and F2 with identical polarizations and type-I crystal is illustrated in Fig. 5a. The result with three pulses is shown in Fig. 5b with corresponding pulse delays (F1 and F2 coincide with H1 and H2). The shape of the pulse H12 is different in each case due to limitation of bandwidth associated with the 700 µm thick crystals. As such, it is observed that the pulse H12 in type-I phase-matching has more bandwidth and is shorter (see Fig. 5b), while the pulse H12 in type-II phase-matching has less bandwidth and is longer (see Fig. 5d). Nevertheless, this result demonstrates the possibility not only to generate but also to isolate the product of time convolution between two optical pulses and, thus, the capability to perform linear filtering operations with optical signals.

Figure 5
figure5

XFROG spectrograms of the pulse sequence obtained from FSHG of a pair of delayed pulses with parallel and orthogonal polarization states in type-I and type-II nonlinear crystals. (a) Both fundamental pulses have parallel polarization (arrows indicating directions of polarization). Three second harmonic pulses are generated in type I phase-matching, H1, H12 and H2. (b) Experimental FSHG pulses corresponding to (a). (c) One fundamental pulse has parallel polarization and the other, delayed, has orthogonal polarization. Using a type-II BBO crystal, phase-matching is possible only for orthogonal polarization states. This condition is fulfilled only for the cross-term H12 at the intermediate delay between the two fundamental pulses. (d) Experimental isolated FSHG pulse corresponding to (c).



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