Propagation Effects in the FRB 20121102A Spectra

Fast radio bursts (FRB) still mystify researchers due to unknown nature of their progenitors and anticipation for new propagation effects that may enrich their signals with information on the traversed medium; see the reviews by Popov et al. (2018), Petroff et al. (2019), and Cordes & Chatterjee (2019). Sky distribution of the registered bursts is isotropic (Thornton et al. 2013; Shannon et al. 2018), implying that they travel cosmological (Gpc) distances, and localization of several FRB sources confirms that; see Chatterjee et al. (2017), Marcote et al. (2017),Tendulkar et al. (2017), Bannister et al. (2019), Ravi et al. (2019), Prochaska et al. (2019), Marcote et al. (2020), and Bhandari et al. (2020). As a consequence, the FRB propagation effects include multiscale scintillations (Rickett 1990; Narayan 1992; Lorimer & Kramer 2004; Woan 2011), i.e., diffractive scattering of radio waves on the turbulent plasma clouds in the FRB host galaxy, Milky Way, and in the intervening galactic halos. In addition, the FRB waves may be lensed by refractive plasma clouds with smooth profiles (Clegg et al. 1998; Cordes et al. 2017), or lensed gravitationally by exotic massive compact objects like primordial black holes or dense minihalos (Zheng et al. 2014; Muñoz et al. 2016; Eichler 2017; Katz et al. 2020). Thus, studying the FRB signals, one may hope to learn something about the cosmological parameters (Deng & Zhang 2014; Yu & Wang 2017; Walters et al. 2018; Macquart et al. 2020), intergalactic medium (Zheng et al. 2014; Zhou et al. 2014; Akahori et al. 2016; Fujita et al. 2017), exotic inhabitants of the intergalactic space (Zheng et al. 2014; Muñoz et al. 2016; Eichler 2017; Katz et al. 2020), and plasma in the far-away galaxies.

The scales of the FRB events point at extreme conditions in their progenitors (Platts et al. 2019) that are hard to achieve in realistic astrophysical settings; see Ghisellini & Locatelli (2018), Lu & Kumar (2018), Katz (2018), Yang & Zhang (2018), and Wang et al. (2019). Millisecond durations of the bursts (Cho et al. 2020) constrain the progenitor sizes to be 100 km or less in the absence of special relativistic effects. Besides, the FRB microsecond substructure observed by Nimmo et al. (2021a) limits the instantaneous emission regions down to 1 km. High spectral fluxes (∼Jy) at GHz frequencies give record brightness temperatures T ∼ 10³⁵–10⁴¹ K (Cordes & Chatterjee 2019; Nimmo et al. 2021b) and therefore support nonthermal, presumably coherent emission mechanisms. The fluxes imply strong (Yang & Zhang 2020) electromagnetic fields 10¹³ V m⁻¹ near the sources that, nevertheless, should not halt the emission (Ghisellini & Locatelli 2018). In addition, the periodic activity of the two repeating FRB sources (CHIME/FRB Collaboration et al. 2020a; Cruces et al. 2020) points at the rotational motions of compact objects and further restricts the progenitor models. Recently an unusually intense radio burst was observed from a Galactic magnetar SGR 1935+2154 (CHIME/FRB Collaboration et al. 2020b; Bochenek et al. 2020; Kirsten et al. 2021a) thus marking these objects as main candidates for the FRB sources. The urge to explain the FRB properties and the absence of generally accepted theoretical models means new data is required. And this field progresses rapidly; see e.g., Tendulkar et al. (2021), Pleunis et al. (2021), Kirsten et al. (2021b), and Rafiei-Ravandi et al. (2021).

Critical information on the FRB central engines can be delivered by their spectra, although a careful data analysis is needed to separate the propagation effects. So far, spectral properties of FRB received undeservedly little attention in the literature, where promising results started to appear only recently; see Gajjar et al. (2018), Hessels et al. (2019), Chawla et al. (2020), Majid et al. (2020), Pearlman et al. (2020), and Pleunis et al. (2021).

With this paper, we develop methods for studying FRB spectral structures and spectral propagation effects; see also Katz et al. (2020). We apply these tools to investigate the frequency spectra of the repeating FRB 20121102A commonly known as FRB 121102 (Spitler et al. 2016; Chatterjee et al. 2017). The bursts were registered at 4–8 GHz by the Breakthrough Listen digital backend at the Green Bank Telescope by Gajjar et al. (2018).

The paper is organized as follows. We introduce the FRB spectra in Section 2 and consider their dominating 7.1 GHz peak in Section 3. The wideband pattern and the progenitor spectrum are considered in Section 4. In Section 5, we study narrowband scintillations. A new periodic spectral structure is analyzed in Section 6. In Section 7, we compare our narrowband spectral analysis with the other results. We summarize in Section 8. Appendices describe theoretical models used to fit the experimental data.

Public data of the Breakthrough Listen Science Team (2018) give a dedispersed spectral flux density ⁵ f(t, ν) of FRB 20121102A as a function of time t and frequency ν. The provided time intervals include 18 bursts ⁶ within the first 60 minutes of the 6 hr observations on 2017 August 26. The bursts are assigned identifiers ⁷ 11A through 11R and 12A through 12C, in order of their arrival. In the wideband analysis, we suppress the instrumental noise using the Gaussian average over the moving frequency window σ = 10 MHz,

$$\begin{eqnarray}&&{\bar{f}}_{\sigma }(t,\nu )=\int d\nu ^{\prime} \,f(t,\nu +\nu ^{\prime} )\,\displaystyle \frac{{{\rm{e}}}^{-{\nu }^{{\prime} 2}/2{\sigma }^{2}}}{\sigma \sqrt{2\pi }}.\end{eqnarray} \tag{ 1 }$$

By construction, ${\bar{f}}_{10}$ fairly represents f on scales exceeding ${\rm{\Delta }}\nu \geqslant \pi \sigma \sqrt{2}\approx 45\,{\rm{MHz}}$. The modulations at smaller scales Δν are suppressed by a factor $\exp (-2{\pi }^{2}{\sigma }^{2}/{\rm{\Delta }}{\nu }^{2})$. We stress that this smoothing is not used in the analysis of narrowband scintillations. Note that, unlike the simplest binning, Equation (1) does not rely on a preselected grid of frequencies and therefore does not create bias in the discussion of the spectral periodicity. ⁸ The color–coded smooth flux densities ${\bar{f}}_{10}(t,\nu )$ of the bursts 11A and 11Q are shown ⁹ in Figure 1.

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To further visualize the FRB spectra, we integrate f over the burst duration t₁ < t < t₂ and obtain its spectral fluence F(ν), i.e., the burst total energy per unit frequency,

$$\begin{eqnarray}&&{\bar{F}}_{\sigma }(\nu )=\underset{{t}_{1}(\nu )}{\overset{{t}_{2}(\nu )}{\int }}{dt}\,{\bar{f}}_{\sigma }(t,\nu ),\end{eqnarray} \tag{ 2 }$$

where the bar again denotes the smoothing Equation (1). The signal region t₁(ν) < t < t₂(ν) (tilted lines in Figure 1) is chosen in Appendix A to minimize the background noise. The size of this region is about a millisecond, varying from burst to burst. Outside of the signal region, ${\bar{f}}_{10}$ fluctuates around zero. The spectral fluences ${\bar{F}}_{10}$ of the bursts 11A, 11D, and 11Q are demonstrated in Figure 2, where the shaded areas near the graphs represent instrumental errors. Apparently, the latter are small, and we will ignore them in what follows.

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The spectra in Figure 2 expose a number of unusual features. First, the graphs 11A and 11D include a high and narrow main peak at ν ≈ 7.1 GHz. In fact, 10 out of the 18 spectra have this feature precisely at the same position. In some events, e.g., 11D or 11F, this peak carries most of the burst energy. On the other hand, the remaining 8 bursts have no 7.1 GHz peak at all; see the graph 11Q in Figure 2. Second, almost all spectra display forests of smaller peaks of width ≲100 MHz. The forests have physical origin, since their presence is not sensitive to the smoothing window σ in Equation (1). Third, the envelopes of the forests have distinctive near-parabolic forms with cutoffs at low and high frequencies. Below we study and explain these three properties.

The dominating feature of most spectra is a high and narrow main peak at 7.1 GHz; see the top two panels in Figure 2. The same peak was observed previously by Gajjar et al. (2018), but it was never explained. Let us argue that it may result from a propagation of the FRB signal through a plasma lens (Clegg et al. 1998; Cordes et al. 2017). Indeed, the latter usually splits the radio wave into multiple rays. Even if the interference of the rays is not relevant, their coalescence at certain frequencies—the lens caustics—may produce high spectral spikes of a specific form.

We consider the lens of Clegg et al. (1998) and Cordes et al. (2017) with a dispersion measure depending on one transverse coordinate x: $\mathrm{DM}(x)={\mathrm{DM}}_{l}\,{{\rm{e}}}^{-{x}^{2}/{a}^{2}}$; see Figure 3. It has two parameters: the size a and central dispersion DM_l . Such one-dimensional lenses are often used for modeling plasma overdensities; see Cordes et al. (2017). Note that occulting AU-sized structures are expected to be present in turbulent galactic plasmas, and one of them may be encountered by the FRB 20121102A signal. In fact, lensing on such structures is consistent with extreme scattering events (ESEs) observed in the light curves of some active galactic nuclei (Fiedler et al. 1987; Bannister et al. 2016) and perturbations in pulsar timings (Coles et al. 2015). Alternatively, the lens may represent long ionized filament in the supernova remnant from the host galaxy; see Graham Smith et al. (2011) and Michilli et al. (2018). Generically, the lens is located in the FRB host galaxy or in the Milky Way, at distances d_pl from the source and d_lo from us; d_po = d_pl + d_lo .

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The radio wave receives a dispersive time delay in the lens and, as a consequence, propagates along the bended path p x o in Figure 3. These two effects give the phase shift (see Clegg et al. 1998; Cordes et al. 2017; and Appendix B for details),

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{l}({\boldsymbol{x}})=\displaystyle \frac{1}{2{r}_{F,l}^{2}}\left[-{\alpha }_{l}{a}^{2}\,{{\rm{e}}}^{-{x}^{2}/{a}^{2}}+{\left({\boldsymbol{x}}-\tilde{{\boldsymbol{x}}}\right)}^{2}\right],\end{eqnarray} \tag{ 3 }$$

where the second term comes from the ray geometry; x = (x, y) is the transverse coordinate in the lens plane, and its value

$$\begin{eqnarray}&&\tilde{{\boldsymbol{x}}}=({{\boldsymbol{x}}}_{p}{d}_{{lo}}+{{\boldsymbol{x}}}_{o}{d}_{{pl}})/{d}_{{po}}.\end{eqnarray} \tag{ 4 }$$

corresponds to a straight propagation between the transverse positions x _p and x _o of the progenitor and the observer. Note that $\tilde{{\boldsymbol{x}}}$ coincides with x _p when the lens resides in the FRB host galaxy, and $\tilde{{\boldsymbol{x}}}\approx {{\boldsymbol{x}}}_{o}$ if it is close to us. We also introduced the lens Fresnel scale ${r}_{F,l}={({d}_{{pl}}{d}_{{lo}}/2\pi \nu {d}_{{po}})}^{1/2}$ and a dimensionless parameter

$$\begin{eqnarray}&&{\alpha }_{l}=\displaystyle \frac{2{e}^{2}{r}_{F,l}^{2}}{{a}^{2}{m}_{e}\nu }\,{\mathrm{DM}}_{l}\end{eqnarray} \tag{ 5 }$$

characterizing the lens dispersion. Note that ${\alpha }_{l}^{-1/2}\propto \nu$ is a dimensionless analog of frequency.

In Clegg et al. (1998) and Cordes et al. (2017), the lens Equation (3) was solved in the limit of geometric optics a ≫ r_F,l . We shortly describe this solution below and give a detailed review in Appendix B. In the geometric limit, the radio waves go along the definite paths x _j = (x_j , y_j ) extremizing the phase in Equation (3). Since the one-dimensional lens bends the rays only in the x direction, ${y}_{j}=\tilde{y}$ corresponds to a straight propagation, and x_j satisfies the nonlinear equation ∂_x Φ_l = 0. One may solve the latter graphically, by plotting Φ_l (x) and identifying the extrema.

Within this approach, one can explicitly see that, as long as the shift $\tilde{x}$ is small, there exists only one extremal radio path x = x₁ at any α_l . But above the critical shift $\tilde{x}\gt {\tilde{x}}_{\mathrm{cr}}=a{(3/2)}^{3/2}$, another two solutions x₂ and x₃ appear at ${\alpha }_{-}(\tilde{x})\lt {\alpha }_{l}\lt {\alpha }_{+}(\tilde{x})$, i.e., inside a certain frequency interval. Thus, the radio waves with these frequencies propagate along three different paths. The two additional paths coincide, x₂ = x₃, at the interval boundaries α_±. Besides, the three-path frequency interval is vanishingly small (α₊ = α₋) at $\tilde{x}={\tilde{x}}_{\mathrm{cr}}$ but becomes larger in size at larger shifts $\tilde{x}$.

From the observational viewpoint, the lens focuses or disperses the FRB signal along each path multiplying the intrinsic progenitor fluence F_p with the gain factor: F = G(ν)F_p . In the refractive one-dimensional case the theoretical gain factor

$$\begin{eqnarray}&&G(\nu )={r}_{F,l}^{-2}\sum _{j}| {\partial }_{x}^{2}{{\rm{\Phi }}}_{l}({x}_{j}){| }^{-1}\end{eqnarray} \tag{ 6 }$$

involves second derivatives of the phase at x = x_j , where we ignore the interference. Thus, the function G(ν) is infinite at the lens caustics ${\partial }_{x}^{2}{{\rm{\Phi }}}_{l}={\partial }_{x}{{\rm{\Phi }}}_{l}=0$ where two radio paths—the extrema x_j of the phase—coalesce. This regime takes place at $\tilde{x}\gt {\tilde{x}}_{\mathrm{cr}}$ when G becomes infinite at the frequencies ${\alpha }_{\pm }^{-1/2}$ due to coalescence of the paths x₂ and x₃. The respective graph of G has a particular two-spike form shown in the inset of Figure 4(b). In reality, the singularities of G(ν) are regulated by the instrumental resolution/smoothing in Equation (1) (dashed line in the figure) and the wave effects. The regime $\tilde{x}\lt {\tilde{x}}_{\mathrm{cr}}$ is entirely different, however. In this case, the function G(ν) is smooth; see the inset in Figure 4(a).

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The shape of the main peak in the experimental data looks similar to G(ν). It is particularly tempting to identify the side spikes ν₀ ± Δν/2 of this peak with the positions of the lens caustics in the regime $\tilde{x}\gt {\tilde{x}}_{\mathrm{cr}}$. The maxima of graph 11D in Figure 4(b) give ν₀ = 7.095 GHz and Δν/ν₀ = 0.0137. In Appendix B.3, we derive analytic expressions for the caustic positions at Δν ≪ ν₀: Equations (B18) and (B19). With the above experimental numbers, they give the source (observer) shift $\tilde{x}/a\approx 1.885$ and a combination of the lens parameters

$$\begin{eqnarray}&&\beta \equiv \left(\displaystyle \frac{{{\rm{DM}}}_{l}}{{\rm{pc}}\cdot {\mathrm{cm}}^{-3}}\right){\left(\displaystyle \frac{a}{{\rm{au}}}\right)}^{-2}\left(\displaystyle \frac{\min \{{d}_{{pl}},{d}_{{lo}}\}}{{\rm{kpc}}}\right)\end{eqnarray} \tag{ 7 }$$

entering α_l in Equation (5): β ≈ 0.0355. Notably, the latter value is consistent with the parameters of the AU-sized structures explaining the ESEs (Fiedler et al. 1987; Coles et al. 2015; Bannister et al. 2016) and parameters of the supernova filaments; see Graham Smith et al. (2011).

There is another, qualitatively different fit of the main peak with the lens spectrum. Namely, if $\tilde{x}$ is slightly below critical, the function G(ν) has a narrow maximum ν = ν₀ with half-height width ${\rm{\Delta }}\nu ^{\prime} \ll \nu$ near the point where the caustics are about to appear; see the inset in Figure 4(a). One can, therefore, interpret the major part of the 7.1 GHz peak as the effect of the lens with $\tilde{x}\lt {\tilde{x}}_{\mathrm{cr}}$, ignoring the side spikes. In Section 6, we will see that the latter spikes correlate with the short-scale periodic structure of the spectra, so they may be unrelated to a refractive lensing, indeed. We read off ν₀ ≈ 7.066 GHz and ${\rm{\Delta }}\nu ^{\prime} /{\nu }_{0}\approx 0.014$ from the spectrum 11D in Figure 4(a) and use Equations (B20) and (B21) of Appendix B to compute the lens parameters in this regime: β ≈ 0.031 and $\tilde{x}/a\approx 1.81$.

It is worth recalling that the experimental spectra in Figure 4 involve smoothing over the frequency intervals σ = 10 MHz. To perform the precise comparison, we smooth the theoretical lens spectra G(ν)F_p in the same way and then fit them to graph 11D; see the dashed lines in Figure 4. This procedure hardly affects the one-peak fit in Figure 4(a) but essentially modifies the caustics in the double-peaked lens spectrum in Figure 4(b). We obtain an improved estimate of the lens parameters in the latter case: $\tilde{x}/a\approx 1.890$ and β ≈ 0.0359. In what follows, we determine $\tilde{x}/a$ and β using the smoothed double-peaked lens spectrum.

So far we completely disregarded the interference of the lensed radio rays that may lead to oscillations of the gain factor with frequency. Note, however, that multiple rays exist only at $\tilde{x}\gt {\tilde{x}}_{\mathrm{cr}}$ in the narrow frequency interval between the caustics, and we do not see any oscillatory behavior there. To no surprise, the smoothing Equation (1) destroys any oscillations on scales below 45 MHz. Requiring the interference period to be smaller, we obtain a constraint a/r_F,l ≳ (ν/σ)^1/2 ∼ 25 or ${(a/{\rm{au}})}^{2}\gtrsim 0.007\cdot \min ({d}_{{pl}},{d}_{{lo}})\,{\mathrm{kpc}}^{-1}$.

To test the lens hypothesis, we compare the main peaks in different FRB spectra. Generically, one expects to find almost time-independent β and linearly evolving $\tilde{x}/a$ due to the transverse motion of the source/observer with respect to the lens. In Figure 5 we plot these parameters extracted from the double-peaked fits (Figure 4(b)) of different spectra. All values of β and $\tilde{x}/a$ are almost identical, except for the burst 12B; see the rightmost points in Figure 5(a)–(b). The main peak in the latter burst is slightly different from the others; see Figure 4(b). It may or may not represent the same spectral structure. If it does, the shift of its parameter $\tilde{x}$ represents motion of the source relative to the lens. In that case, we obtain the relation between the lens relative velocity $v=d\tilde{x}/{dt}$ and its size: a ∼ 0.3 au · v/(200 km s⁻¹).

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It is worth noting that the narrow bandwidth of the registered FRB spectra and large cosmological ν⁻² dispersion precludes analysis of another important lens characteristic: the dispersive time delay of the transmitted FRB signal. The latter depends on frequency in a nontrivial way, distorting the FRB image in the t–ν plane into a peculiar recognizable form; see Clegg et al. (1998), Cordes et al. (2017), and Figure 1.

Overall, the plasma lens hypothesis is very appealing. However, it has visible inconsistencies. First, the spikes in Figure 4 do not exactly match the main peak slopes and therefore the theoretical fit. Second, the height of this peak relative to the nearby spectrum strongly varies from burst to burst; see the bursts 11A and 11D in Figure 4. Third and final, some bursts do not have the main peak at all (see the graph 11Q in Figure 2) as if the lens voluntarily disappears and then appears again with precisely the same parameters.

Two of the above properties will be explained in the forthcoming sections. First, the spectra in Figure 2 include oscillations, mostly chaotic, at scales below 100 MHz. They certainly deform the main peak. Second, we will observe that the FRB spectra have narrow bandwidth, and their central frequency changes from burst to burst. This makes the 7.1 GHz peak vanish if it is outside of the signal band.

It is remarkable that the FRB spectra of Gajjar et al. (2018) are localized in the relatively narrow bands ν_bw ∼ GHz, but their central frequencies differ significantly. In fact, the same properties were observed before in the measured spectra of FRB 20121102A (Law et al. 2017; Gajjar et al. 2018; Gourdji et al. 2019; Hessels et al. 2019; Majid et al. 2020) and another repeater FRB 20180916B (Chawla et al. 2020; Pearlman et al. 2020). In this section, we are going to show that the wideband envelopes of our FRB 20121102A spectra are essentially distorted by a spectacular propagation phenomenon similar to wideband scintillations. Separating this effect, we will give a strong argument that the progenitor spectra themselves are narrowband and have strongly variable central frequencies.

To remove the effect of the 7.1 GHz lens, we divide all registered fluences F(ν) by the lens gain factor G(ν) determined from the double-peaked fit of the spectrum 11D in Figure 4(b). After that we compute the central (center-of-mass) frequency of the burst, ¹⁰

$$\begin{eqnarray}&&{\nu }_{c}\equiv \left({\int }_{{\nu }_{1}}^{{\nu }_{2}}\nu \,d\nu \,\displaystyle \frac{F(\nu )}{G(\nu )}\right)/\left({\int }_{{\nu }_{1}}^{{\nu }_{2}}d\nu \,\displaystyle \frac{F(\nu )}{G(\nu )}\right),\end{eqnarray} \tag{ 8 }$$

where the integrations are performed over the entire signal region ν₁ < ν < ν₂ with positive fluence. It is worth remarking that Equation (8) uses the original unsmoothed fluence F(ν).

The central frequency Equation (8) of the bursts is plotted in Figure 6 as a function of their arrival time t. Notably, the dependence of ν_c (t) is not chaotic! Rather, the central frequencies are attracted to one of the three parallel inclined bands ν_i (t) (dashed lines in Figure 6) with seemingly random choice of the band.

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The bands at ν ≈ 6 and 7 GHz have already been noticed by Gajjar et al. (2018) in the summed 4–8 GHz spectrum of FRB 20121102. However, their linear evolution with time has never been observed. Although both observations are made on the basis of limited statistics, they can be tested in the future using larger data samples.

Now, Figure 6 strongly suggests that the three bands represent a propagation effect, e.g., strong scintillations of the FRB signal in the turbulent interstellar plasma. In this case, linear time evolution appears due to the relative motion of the observer and progenitor with respect to the scintillating medium. From the physical viewpoint, the scintillations are caused by a refraction of the radio waves on the plasma fluctuations that makes them propagate via multiple paths. Interference between the paths then distorts the registered FRB spectra into a pattern of alternating peaks and dips. The bands ν_i (t) in Figure 6 may represent the scintillation maxima. Then the typical distance between them ${\nu }_{d}^{\prime} \sim {\nu }_{i+1}-{\nu }_{i}\sim 0.95\,{\rm{GHz}}$ estimates the decorrelation bandwidth.

One traditionally characterizes the scintillating plasma with the diffractive length—the typical transverse distance r_diff at which the correlations between the radio rays die away. This quantity is related to the decorrelation bandwidth as ${\nu }_{d}^{\prime} /\nu ={({r}_{\mathrm{diff}}^{{\prime} }/{r}_{F,S}^{{\prime} })}^{2}$, where ${r}_{F,S}^{{\prime} }$ is the respective Fresnel scale; see Narayan (1992) and Appendix C. We obtain ${r}_{\mathrm{diff}}^{{\prime} }\sim 0.4\,{r}_{F,S}^{{\prime} }$. A benchmark property of strong diffractive scintillations is an order-one modulation of the spectra that is observed here, indeed: the regions with strong signal form isolated islands of GHz width, and fluence between them is negligibly small; see Figure 2.

Importantly, the scintillation pattern is expected to evolve smoothly with time if the source (observer) ¹¹ has a nonzero relative velocity v with respect to the scintillating plasma. This is precisely what we see in Figure 6: the maxima ν_i (t) drift linearly with the characteristic timescale ${t}_{\mathrm{diff}}^{{\prime} }\simeq 1350\,{\rm{s}}$. Equating ${t}_{\mathrm{diff}}^{{\prime} }\sim {r}_{\mathrm{diff}}^{{\prime} }/v$, we relate the velocity v to ${r}_{F,S}^{{\prime} }={(d^{\prime} /2\pi \nu )}^{1/2}$ and hence to the typical distance $d^{\prime} \sim {v}_{200}^{2}\cdot 2\,{\rm{kpc}}$ between the scintillating plasma and the source (observer), where we introduced v₂₀₀ = v/(200 km s⁻¹).

There remains a question, why the registered spectra have the form of a single relatively narrow signal region despite the fact that the two or three scintillation maxima are usually present in the observation band 4 GHz < ν < 8 GHz; see Figures 2 and 6. This can happen only if the FRB progenitor has a comparably narrow spectrum with bandwidth ν_bw ≈ GHz that is capable of illuminating only one maximum. The same conjecture explains another feature of Figure 6: all bursts with the main peak (empty circles) have central frequencies within the GHz band around 7.1 GHz (the shaded region in Figure 6). Note that it would be impossible to explain the disappearance of the main peak in some bursts by destruction of the lens: the shape of this peak remains stable prior to disappearance and recovers later with precisely the same parameters; see Figures 5 and 6.

To sum up, we have argued that the spectrum of the FRB progenitor has ν_bw ≈ GHz bandwidth, and its central frequency is changing from burst to burst—chaotically or on a short timescale. Note that our argument is based on the separation of the progenitor properties from the wideband propagation phenomena that strongly distort this spectrum with the unknown frequency shifts of order GHz.

At 1.4 GHz, the registered spectra of FRB 20121102A also occupy a narrow band ν_bw ∼ 200 MHz or ν_bw/ν ∼ 20%, as was observed by Hessels et al. (2019). Note, however, that the entire bandwidth of their instrument is comparable to ν_bw. In that case, the spectral minima at the band boundaries may be provided by the wideband scintillations similar ¹² to ours. This means that the true bandwidth of the progenitor spectrum may be larger at 1.4 GHz: ν_bw ≳ 200 MHz. A different, interesting possibility is that the bandwidth of the progenitor spectrum always constitutes 20% of its central frequency. But this latter assumption still has to be tested with the wideband measurements.

Despite distortions, one can search for the periodic evolution of the progenitor central frequency ν_c (t). We performed this search using the periodogram method described in Zechmeister & Kurster (2009) and Ivanov et al. (2019) at timescales 60–1000 s. The best-fit value for a period is 112 s, but the effect is not statistically significant.

At shorter scales Δν ≲ 100 MHz, the spectra in Figure 2 display a seemingly chaotic pattern of alternating peaks and dips. It would be natural to explain this random behavior with another, narrowband kind of strong interstellar scintillations. Let us show that the latter are indeed present in the FRB 20121102A spectra.

It is natural to treat the scintillations statistically, i.e., average the spectra over a large ensemble of turbulent plasma clouds and then compare the mean observables to the theory. We recall (Rickett 1990; Narayan 1992; Lorimer & Kramer 2004; Woan 2011) that different-frequency waves refract differently and therefore go along different paths though statistically independent volumes of the turbulent medium. This makes the scintillating radio spectra uncorrelated at frequencies ν and ν + Δν if Δν exceeds the decorrelation bandwidth ν_d . As a consequence, the statistical average can be performed by integrating over many ν_d intervals. Below we regard the fluence ${\bar{F}}_{50}(\nu )$ smoothed with large window σ = 50 MHz in Equation (1) as a statistical mean. This quantity indeed delineates a wideband envelope of the spectrum in Figure 7 (dashed line) with no trace of the erratic short-scale structure. Note, however, that ${\bar{F}}_{50}$ should be interpreted with care, since smoothing in Equation (1) destroys any oscillatory behavior, erratic or not, at frequency periods below $\pi \sigma \sqrt{2}\sim 220\,{\rm{MHz}}$.

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We introduce the autocorrelation function (ACF) characterizing the statistical dependence of the spectral fluctuations $\delta F=F-{\bar{F}}_{50}$ at frequencies ν and ν + Δν,

$$\begin{eqnarray}&&{\rm{ACF}}({\rm{\Delta }}\nu )={ \mathcal N }\underset{{\nu }_{1}}{\overset{{\nu }_{2}-{\rm{\Delta }}\nu }{\int }}d\nu \,\,\displaystyle \frac{\delta F(\nu )\delta F(\nu +{\rm{\Delta }}\nu )}{{\nu }_{2}-{\nu }_{1}-{\rm{\Delta }}\nu },\end{eqnarray} \tag{ 9 }$$

where the data are averaged over the signal bandwidth ν₁ < ν < ν₂ by integrating and dividing by the integration interval, whereas the normalization factor ${ \mathcal N }$ makes ACF(0) = 1. We will see that the observable in Equation (9) is sensitive both to scintillations and to periodic structures in the spectra.

In Figure 8, we plot ACF(Δν) for the burst 11A. It decreases at first indicating that δ F(ν) and δ F(ν + Δν) are less correlated at larger Δν. But surprisingly, at Δν ≳ 50 MHz, the ACF develops a set of wide almost equidistant maxima suggesting that the coherence partially returns! We will consider this effect in the next section.

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To interpret the data, we theoretically computed the ACF for the radio waves strongly refracted in the turbulent plasma with the standard Kolmogorov-type distribution of free electrons; see Appendix C. The result ¹³ is,

$$\begin{eqnarray}&&{\rm{ACF}}={ \mathcal N }\underset{{\nu }_{1}}{\overset{{\nu }_{2}-{\rm{\Delta }}\nu }{\int }}\displaystyle \frac{{\bar{F}}_{50}^{2}(\nu )d\nu }{{\nu }_{2}-{\nu }_{1}-{\rm{\Delta }}\nu }{\left|h\left(\displaystyle \frac{2{\rm{\Delta }}\nu }{{\nu }_{d}(\nu )}\right)\right|}^{2},\end{eqnarray} \tag{ 10 }$$

where ${\bar{F}}_{50}\approx \langle F\rangle$ again substitutes the statistical average, while ∣h(w)∣² is a universal hat-like function depicted with the solid line in Figure 9. Strictly speaking, h is given by the integral (C30), but in practice one can use a very good approximation

$$\begin{eqnarray}&&h(w)\approx {\left[1+a{\left({iw}\right)}^{5/6}+{\left({iw}/b\right)}^{7/4}\right]}^{-4/7}\end{eqnarray} \tag{ 11 }$$

capturing the small- and large-w asymptotics of this function and therefore correctly representing it at the intermediate values as well; see the dashed line in Figure 9. In Equation (11), we used the numerical coefficients $a=\tfrac{7}{8}{\rm{\Gamma }}(11/6)$, b = Γ(11/5)2^6/5, and Euler gamma–function Γ(z).

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The only fitting parameter of the theoretical model Equation (10) is the value of the decorrelation bandwidth ν_d (ν) at a given frequency, say, ν_d,6 ≡ ν_d (6 GHz). At other frequencies the bandwidth is determined from the Kolmogorov scaling

$$\begin{eqnarray}&&{\nu }_{d}\,(\nu )={\nu }_{d,6}{\left(\displaystyle \frac{\nu }{6\,{\rm{GHz}}}\right)}^{4.4}=\nu \,{\left({r}_{\mathrm{diff}}/{r}_{F,S}\right)}^{2}.\end{eqnarray} \tag{ 12 }$$

The first of these equations is convenient in practice, while the second relates ν_d to the parameters of the scintillating medium: the Fresnel scale r_F,S characterizing its distancing from the source or the observer and diffractive length r_diff—the transverse separation at which the radio paths decohere inside the medium; see Equations (C7) and (C3).

We stress that the theoretical expressions (10) and (11) are new. Previous studies of Hessels et al. (2019) and Majid et al. (2020) traditionally assumed a Lorentzian ACF profile, ¹⁴

$$\begin{eqnarray}&&{h}_{L}^{2}={ \mathcal N }\left[c+{\left({\rm{\Delta }}{\nu }^{2}+{\nu }_{d,L}^{2}\right)}^{-1}\right],\end{eqnarray} \tag{ 13 }$$

which has two parameters: the bandwidth ν_d,L and an additive constant c. At c = 0, the fit of our theoretical prediction ∣h(w)∣² with this function gives the dotted line in Figure 9 and ν_d,L ≈ 1.2 ν_d . Notably, the Lorentzian profile works very well at intermediate frequency lags but, being regular at Δν → 0, deviates from the theory at small w. As a consequence, even at c = 0, it underestimates the height of the ACF, overestimates its half-height width, and gives 20% larger value of ν_d . We will see below that the two-parametric Lorentzian fits with arbitrary c are much worse.

Importantly, our theoretical ∣h∣² decreases with Δν from h(0) = 1 to zero reaching 1/2 at Δν ≈ 0.96 ν_d . The full ACFs in Equation (10) have similar profiles and, in particular, monotonically fall off at large Δν. This is the only possible behavior because random fluctuations in the turbulent plasma suppress correlations between the different-frequency waves, and the suppression is stronger at larger Δν. As a consequence, Equation (10) (dashed line) fits well the initial falloff of the experimental ACF in Figure 7(b) giving ν_d,6 = 3.5 MHz. But the same theory fails to explain the peaks at larger Δν that will be considered in the next section.

Overall, we find that the ACFs of the 12 most powerful bursts match Equation (10) at small Δν whereas the weaker bursts 11B, C, G, J, K, and M are dominated by the instrumental noise and do not produce discernible correlation patterns at all. From these fits, we obtain the 12 values of ν_d,6 in Figure 10(a). The data points group around a constant

$$\begin{eqnarray}&&{\nu }_{d}\,(6\,{\rm{GHz}})=4.3\pm 0.9\,{\rm{MHz}}\end{eqnarray} \tag{ 14 }$$

despite the fact that their spectra are localized in essentially different-frequency regions. Rescaling the spectral bandwidths ν_d to the burst central frequencies ν = ν_c via Equation (12), we obtain the function ν_d (ν_c ) in Figure 10(b) that closely follows the Kolmogorov scaling (dashed line).

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The mean value of ν_d fixes the parameter r_diff/r_F,S ≈ 0.027 ± 0.003 of the scintillating plasma. Assuming a galactic distance d to it, we obtain a reasonable diffractive length ${r}_{\mathrm{diff}}\sim {(1.3\pm 0.1)\times {10}^{9}\,{\rm{cm}}\times (d/{\rm{kpc}})}^{1/2}$; see Equation (C3).

It is worth noting that the frequency integral is an important part of Equation (10) because the decoherence bandwidth ν_d strongly depends on ν. The theoretical result simplifies, however, in the case ν₂ − ν₁ ≪ ν when

$$\begin{eqnarray}&&{\rm{ACF}}={\left|h(2{\rm{\Delta }}\nu /{\nu }_{d})\right|}^{2}\end{eqnarray} \tag{ 15 }$$

is completely determined by the hat-like function in Equation (11), while ν_d is computed either at (ν₂ + ν₁)/2 or at ν_c if the spectrum itself is narrowband. The price to pay, however, is larger statistical fluctuations in the smaller data set. Since our data are relatively narrow band, we fitted the ACFs of the 12 strongest bursts by Equation (15). The respective values of ν_d (ν_c ) were consistent with Figure 10(b). Using them in Equation (12), we arrived to ν_d (6 GHz) = 4.0 ± 0.8 GHz—almost the same result as before.

We finish this Section with an extra argument, why erratic narrowband structure stems from a propagation effect, and it is not just a stochastic variation of the progenitor spectrum. First, we compute the correlation functions (CFs) between the pairs of the burst spectra by substituting their fluences $\delta {F}_{i}(\nu )\equiv {F}_{i}-{\bar{F}}_{50,i}$ and δ F_j (ν) into Equation (9) instead of the two identical δ F's. In particular, in Figure 11 we plot CF(Δν) between the bursts 11A and 11I. The highest peak of this function occurs at nonzero Δν = Δν_AI suggesting that the erratic structures in the spectra are shifted with respect to each other. Moreover, the frequency shift Δν_ij between the different burst pairs linearly depends on the time elapsed between them; see Figure 12: ∂_t Δν_ij ≈ −2.52 · 10⁻² MHz s⁻¹ (dashed line). Like in the previous section, we explain this effect by a relative motion of the observer (source) with respect to the scintillating medium (Rickett 1990; Narayan 1992; Lorimer & Kramer 2004; Woan 2011). The velocity is then roughly estimated as v ∼ r_diff ∂_t Δν_ij /ν_d ∼ 76 km s⁻¹ (d/kpc)^1/2, where the experimental values for r_diff and ν_d are substituted. We obtained a reasonable galactic velocity.

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So far we have argued that the peaks of the ACF in Figure 8 cannot originate from the scintillations because the latter introduce a stronger suppression at larger Δν. The same peaks, however, are naturally explained by the wave diffraction. Indeed, suppose that before or after hitting the scintillating medium, the FRB signal passes through the lens—a plasma cloud or a vicinity of a compact gravitating body—that splits it into two radio rays,

$$\begin{eqnarray}&&f={f}_{p}\left({G}_{1}^{1/2}\,{{\rm{e}}}^{i{{\rm{\Phi }}}_{1}}+{G}_{2}^{1/2}\,{{\rm{e}}}^{i{{\rm{\Phi }}}_{2}}\right),\end{eqnarray} \tag{ 16 }$$

(see one of these rays in Figure 13). We introduced the intrinsic progenitor signal f_p (ν), gain factors G_1,2 of the rays, and their phases Φ_1,2. As a consequence of Equation (16), the net FRB fluence F = ∣f∣² includes an interference term proportional to $\cos ({{\rm{\Phi }}}_{1}-{{\rm{\Phi }}}_{2})$ that oscillates as a function of frequency with the period T_ν = 2π∣∂_ν (Φ₂ − Φ₁)∣⁻¹. This enhances correlations between F(ν) and F(ν + Δν) at frequency lags equal to the multiples of the frequency period, Δν = nT_ν , and therefore produces equidistant peaks in the ACF in Equation (9).

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Scintillations obscure the above picture by adding a random component to the wave Equation (16). In Appendix C, we develop an analytic model for the radio waves propagating through the lens and the scintillating plasma. The spectral fluence in this case equals (see Equation (C16))

$$\begin{eqnarray}\begin{array}{rcl}F & = & {\bar{F}}_{50}\left[1+{A}_{\mathrm{osc}}\,{{\rm{e}}}^{-\tfrac{1}{2}{\left(| {\tilde{{\boldsymbol{\xi }}}}_{1}-{\tilde{{\boldsymbol{\xi }}}}_{2}| /{r}_{\mathrm{diff}}\right)}^{5/3}}\cos ({{\rm{\Phi }}}_{1}-{{\rm{\Phi }}}_{2})\right]\\ & & +\,\delta ^{\prime} F,\end{array}\end{eqnarray} \tag{ 17 }$$

where we extracted a smooth envelope ${\bar{F}}_{50}(\nu )$ of the spectrum, denoted the relative oscillation amplitude by ${A}_{\mathrm{osc}}\equiv 2\sqrt{{G}_{1}{G}_{2}}/{({G}_{1}+{G}_{2})}^{-1}\lt 1$, and introduced the transverse distance $| {\tilde{{\boldsymbol{\xi }}}}_{2}-{\tilde{{\boldsymbol{\xi }}}}_{1}|$ between the radio rays inside the scintillating medium. The first line in Equation (17) represents the statistically averaged contributions of the radio rays and their interference, whereas $\delta ^{\prime} F$ denotes fluctuations of the fluence. Scintillations suppress the interference term if the rays go too far apart. We are interested in the unsuppressed regime. ¹⁵

$$\begin{eqnarray}&&| {\tilde{{\boldsymbol{\xi }}}}_{2}-{\tilde{{\boldsymbol{\xi }}}}_{1}| \lt {r}_{\mathrm{diff}}.\end{eqnarray} \tag{ 18 }$$

Notably, in Appendix C, we find out that this inequality easily holds for scintillations occurring in our galaxy if the lens is outside of it; see Equation (C31). Or vice versa: one may imagine that the scintillations happen in the FRB host galaxy, and the lens is either in the intergalactic space or in the Milky Way.

But even if Equation (18) holds, the interference peaks of F(ν) are hidden in the sea of erratic fluctuations $\delta ^{\prime} F$. The latter have order-one amplitude if the scintillations are strong: $\delta ^{\prime} F\sim F$; see Figure 7. To separate the two effects, one uses the ACF in Equation (9) where the frequency integral averages the fluctuations away. In Appendix C, we derive ACF for the theoretical model with scintillations+lensing,

$$\begin{eqnarray}\begin{array}{rcl}{\rm{ACF}} & = & { \mathcal N }{\displaystyle \int }_{{\nu }_{1}}^{{\nu }_{2}-{\rm{\Delta }}\nu }d\nu \,\displaystyle \frac{{\bar{F}}_{50}(\nu ){\bar{F}}_{50}(\nu +{\rm{\Delta }}\nu )}{{\nu }_{2}-{\nu }_{1}-{\rm{\Delta }}\nu }\\ & & \times \,\left[\,| h{| }^{2}+\displaystyle \frac{1}{2}{A}_{\mathrm{osc}}^{2}\left(1+| h{| }^{2}\right)\,\cos (2\pi {\rm{\Delta }}\nu /{T}_{\nu })\right],\end{array}\end{eqnarray} \tag{ 19 }$$

where h ≡ h(2Δν/ν_d (ν)) is the same scintillation function (11) as before, ν_d (ν) is given by Equation (12), and the frequency period T_ν is already extracted from Φ₂ − Φ₁. Note that Equation (19) is valid at ν_d ≪ ν with the corrections of order ${({\nu }_{d}/\nu )}^{1/3}\sim 8 \%$.

The theoretical expression (19) describes both the initial falloff of the ACF due to scintillations and the periodic peaks at Δν = nT_ν caused by the two-ray interference. It fits well the experimental data in Figure 8 (solid line) giving ν_d ≈ 2.4 MHz, T_ν ≈ 110.3 MHz, and A_osc ≈ 0.5 for the burst 11A.

In fact, the ACFs of the strongest bursts, e.g., 11A or 11D, display easily recognizable sets of periodic peaks; see Figure 14, and many other bursts include hints of those. Fitting ACFs of the 12 most powerful spectra ¹⁶ with Equation (19), we obtain the respective decorrelation bandwidths ν_d , frequency periods T_ν , and amplitudes A_osc in ¹⁷ Figures 15 and 16.

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The frequency periods in Figure 16(a) group within the 15% interval around the mean value T_ν ≈ 95 ± 16 MHz and do not indicate any dependence on frequency. At the same time, the jumps of A_osc ² in Figure 16(b) are much larger. This sensitivity can be explained by the fact that some weak ACFs include only hints of the periodic patterns and give very small A_osc, while the other have barely discernible initial scintillation falloffs, hence an overestimate of A_osc. In what follows ,we use A_osc ≈ 0.5 obtained by averaging the 12 points in Figure 16(b) (dashed line).

In Figure 17, we plot the frequency periods of different bursts versus their arrival times. The data points are consistent with the time-independent T_ν (dashed line), although a slow evolution with dT_ν /dt ≈ −1.1 · 10⁻² MHz s⁻¹ is also possible (solid line).

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The fit Equation (19) gives an improved estimate of the burst scintillation bandwidths ν_d,6 and ν_d (ν_c ); see Figures 15(a) and (b), and recall Figure 10. Now, the data group a bit closer, and they still agree with the prediction of the Kolmogorov turbulence Equation (12) (dashed lines). The averaging gives ν_d (6 GHz) ≈ (3.3 ± 0.6)MHz.

Now, we can explicitly visualize the periodic pattern in the burst spectra. In Figure 7, we plotted the smoothed fluence ${\bar{F}}_{10}(\nu )$ of the burst 11A together with the lattice of the vertical dotted lines separated by T_ν = 110.3MHz—a frequency period of the respective ACF in Figure 8. By construction, smoothing with σ = 10 MHz kills almost all narrowband scintillations because ${\nu }_{d}\ll \pi \sigma \sqrt{2}$. As a consequence, the highest maxima in Figure 7 should belong to the periodic structure. Indeed, too many of them are close to the dotted lines. Moreover, the side spikes of the main 7.1 GHz peak seem to be a part of the same structure.

We demonstrated that the two-ray interference correctly describes the leading periodic behavior of the spectra. Nevertheless, there are visible inconsistencies. First, the experimental ACFs in Figures 8(a) and 14 deviate from the fits in some places. Second, there are burst-to-burst variations in ACFs leading to 15% spread of T_ν values in Figure 16(a). Third, some maxima in Figure 7 are off the periodic grid.

All these effects are expected. On the one hand, strong erratic GHz- and MHz-scale scintillations exist in the spectra; without any doubt, the ones with ν_d ∼ 100 MHz are present as well. They slightly shift the maxima in Figure 7 and add smaller peaks. Moreover, unlike the strongest narrowband fluctuations that are averaged via the frequency integral in Equation (9), the ones with larger ν_d remain almost random. They stochastically distort the expected cos-like behavior of the ACFs in Figures 8(a) and 14 and penetrate into the fit results for T_ν . On the other hand, we use the simplest two-wave interference model and disregard the subdominant rays altogether. But generically, the latter are present in the data along with their subdominant interference contributions distorting the graphs. One can take these features into account at the cost of adding new parameters to the fits.

In fact, the structure similar to what we see in Figure 7 was observed in the High-Frequency Interpulses (HFI) of the Crab pulsar; see Hankins et al. (2016). Namely, the spectra of HFI consist of many isolated bands with an inter-band distance ν_band ≃ 0.06 ν. In turn, every band includes ∼3 sub-bands. At 6 GHz, this gives ν_band ≃ 344 and ∼100 MHz of sub-band distance. Intriguingly, the wideband envelope ${\bar{F}}_{50}$ of the FRB spectrum in Figure 7 has several maxima separated by ν_band ≃ 330 MHz that consist, at a higher spectral resolution, of the equidistant peaks with period T_ν ≈ 110 MHz. This resemblance may suggest similar mechanisms operating in Crab and in FRB 20121102A. Note that T_ν ∝ ν does not contradict to the points in Figure 16(a) that have a spread.

Let us argue that the entire 100 MHz spectral pattern, including the periodic structure and subleading peaks, originates from the propagation phenomena. We divide the smoothed spectra ${\bar{F}}_{10}(\nu )$ by their total energy releases making the areas under their graphs equal to one. After that some of the normalized spectra look almost identical to each other, like the twin brothers; see graphs 11A and 11E in Figure 18(a), or 11C and 11F in Figure 18(b). Notably, these coinciding bursts are not sequential, e.g., 11A is followed by the bursts B to D, and only then by the burst E. Such similarity would be very hard to explain by the intrinsic properties of the emission mechanism. In the model with diffractive lensing and scintillations, the effect is provided by small velocities of the lenses and the scintillating media and by the randomized central frequency of the FRB progenitor. The FRB signals acquire the same narrowband spectral structure if they are localized in the same frequency band and occur shortly after one another, so that the lens and the medium do not have enough time to evolve.

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We perform another important test by summing up the normalized spectra of the 12 most powerful bursts. The result is more noisy ¹⁸ than the strongest 11D and 11A spectra because the weaker bursts give the same-order contributions into the normalized sum. The summed spectrum is visualized in Figure 8, where smoothing with σ = 10 MHz is used. One finds that many of its maxima appear near the periodic lattice of dashed vertical lines. Besides, the ACF of this summed spectrum (Figure 19(b)) includes many almost equidistant peaks at large Δν. Fitting this function with Equation (19), we obtain the frequency period T_ν ≈ 97 MHz and decorrelation bandwidth ν_d ≈ 4.45 MHz that are close to our previous results. Notably, the fit is pretty good: note that the positions of the ACF peaks in Figure 19(b) correlate with the periodic maxima of the fitting function over many periods. We conclude that the periodic structure is stable in time and not peculiar to the strongest bursts.

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It is worth discussing possible theoretical models for the lens that splits the FRB wave into two rays and creates the periodic spectral structure. First, it may be formed by a plasma residing, say, in the FRB host galaxy. Consider, e.g., the one-dimensional Gaussian lens of Section 3 with the phase shift,

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{l}=\displaystyle \frac{1}{2{r}_{F,l}^{{\prime} 2}}\left[{\left(x-\tilde{x}^{\prime} \right)}^{2}+{\alpha }_{l}^{{\prime} }{a}^{{\prime} 2}\,{{\rm{e}}}^{-{x}^{2}/{a}^{{\prime} 2}}\right].\end{eqnarray} \tag{ 20 }$$

Here we equipped all the lens parameters with the primes and ignored the trivial dependence on y leaving only one transverse coordinate x. We also changed the sign in front of the second term, so now the lens with ${\alpha }_{l}^{{\prime} }\propto -\delta {n}_{e}\gt 0$ describes an underdensity of free electrons. For simplicity, below we assume $\mathrm{ln}{\alpha }_{l}^{{\prime} }\gg 1$ and $\tilde{x}^{\prime} \sim a^{\prime} \sqrt{\mathrm{ln}{\alpha }_{l}^{{\prime} }}$—a strong lens relatively far away from the line of sight.

Note that the underdensities are expected to appear in the interstellar medium due to heating, e.g., by the magnetic reconnections. They were often used to explain the pulsar data. For example, Pen & King (2012) interpreted the pulsar ESEs as lensing on the Gaussian underdensities Equation (20). Another type of underdensity lenses in the form of corrugated plasma sheets was suggested to cause pulsar scintillation arcs in Simard & Pen (2018). We will demonstrate that the lens Equation (20) can explain the diffractive peaks in our data.

The phase delay Equation (20) is plotted in Figure 20. It has three extrema corresponding to three radio rays. In Appendix B.4, we argue that the ray with x ≈ 0 has a small gain factor and can be ignored, while the other two give Equation (16). Computing the parameters of these rays, we obtain

$$\begin{eqnarray}&&{A}_{\mathrm{osc}}\approx \sqrt{1-\displaystyle \frac{{\tilde{x}}^{{\prime} 2}}{{a}^{{\prime} 2}\mathrm{ln}{\alpha }_{l}^{{\prime} }}},\qquad {T}_{\nu }\approx \displaystyle \frac{\pi \nu {r}_{F,l}^{{\prime} 2}}{a^{\prime} \tilde{x}^{\prime} \sqrt{\mathrm{ln}{\alpha }_{l}^{{\prime} }}}.\end{eqnarray} \tag{ 21 }$$

The experimental values A_osc ≈ 0.5 and T_ν ≈ 95 MHz then give $\tilde{x}\approx a^{\prime} \sqrt{\mathrm{ln}{\alpha }_{l}^{{\prime} }}$ and ${r}_{F,l}^{{\prime} }\approx 0.07\,a^{\prime} \sqrt{\mathrm{ln}{\alpha }_{l}^{{\prime} }}$, where realistically, $\mathrm{ln}{\alpha }_{l}^{{\prime} }\sim 3\mbox{--}10$.

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The second option includes lensing of the FRB signal on the gravitating compact (pointlike) object of mass M, e.g., a primordial black hole or a dense minihalo, like in Katz et al. (2020). The total phase delay in this case has the form similar to Equation (20) (see Peterson & Falk 1991; Matsunaga & Yamamoto 2006; Bartelmann 2010):

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{l}=\displaystyle \frac{1}{2{r}_{F,l}^{{\prime} 2}}\left\{{\left({\boldsymbol{x}}-{\tilde{{\boldsymbol{x}}}}^{{\prime} }\right)}^{2}-2{\left({d}_{{lo}}^{\prime} {\theta }_{E}\right)}^{2}\mathrm{ln}\left|\displaystyle \frac{{\boldsymbol{x}}-{{\boldsymbol{x}}}_{o}}{{d}_{{lo}}^{\prime} }\right|\right\},\end{eqnarray} \tag{ 22 }$$

where ${d}_{{lo}}^{\prime}$ is the distance to the object, ${\tilde{{\boldsymbol{x}}}}^{{\prime} }$ is given by Equation (4) with the parameters of the new lens, ${r}_{F,l}^{{\prime} }$ is the respective Fresnel scale, and ${\theta }_{E}={(4{{GMd}}_{{pl}}^{\prime} /{d}_{{lo}}^{\prime} {d}_{{po}})}^{1/2}$ is the Einstein angle. Now, the second term in the phase shift is caused by gravity rather than refraction. That is why it is proportional to the frequency of the radio wave and mass of the lens: ${({\theta }_{E}/{r}_{F,l}^{{\prime} })}^{2}\propto \nu M$.

Generically, the pointlike gravitational lens splits the radio wave into two rays in Equation (16). Computing the respective eikonal solutions, one finds,

$$\begin{eqnarray}\begin{array}{rcl}{A}_{\mathrm{osc}} & = & \displaystyle \frac{2}{{\zeta }^{2}+2},\qquad \\ {T}_{\nu } & = & \displaystyle \frac{1}{4{GM}}{\left[\zeta \sqrt{{\zeta }^{2}+4}+2\mathrm{ln}(\zeta /2+\sqrt{{\zeta }^{2}/4+1})\right]}^{-1},\end{array}\end{eqnarray} \tag{ 23 }$$

(for details, see Appendix B.5; Peterson & Falk 1991; Matsunaga & Yamamoto 2006; Bartelmann 2010; Katz et al. 2020). Here we introduced the angular separation of the source from the lens in units of the Einstein angle $\zeta =| {{\boldsymbol{x}}}_{p}-{{\boldsymbol{x}}}_{o}| /({\theta }_{E}{d}_{{lo}}^{\prime} )$. Substituting the mean experimental values of A_osc and T_ν , we obtain ζ ≈ 1.4 and M ≈ 1.1 · 10⁻⁴ M_⊙.

Let us guess where the gravitational lens lives. It is natural to assume that such exotic objects constitute a part γ of dark matter. Then the probability of meeting one of them in the intergalactic space at distance $| {{\boldsymbol{x}}}_{p}-{{\boldsymbol{x}}}_{o}| \lesssim \zeta {\theta }_{E}{d}_{{lo}}^{\prime}$ from the line of sight is of order $\gamma {\zeta }^{2}G{\rho }_{m}{d}_{{po}}^{2}\sim {10}^{-2}\,\gamma$, where we substituted ζ ≈ 1.4, the distance to the source d_po ∼ Gpc, and the mean dark matter density ρ_m ∼ 3 · 10⁻⁶ GeV cm⁻³. Thus, the gravitational lensing of one FRB signal is relatively improbable even if all dark matter consists of lenses. However, the probability of the respective event inside the galactic halo of Mpc size is ∼10 times smaller despite the larger density. Thus, the gravitational lensing is generically expected to occur on the way between the galaxies.

It would be great to discriminate between the above two lenses on the basis of the spectral data alone. For example, one may assume that the dependence of the frequency period T_ν on frequency ν is different in the two cases. Indeed, universality of the gravitational lensing gives ${T}_{\nu }(\nu )={\rm{const}}$, whereas refraction of radio waves in the plasma is essentially ν–dependent. Note, however, that the first (geometric) term of the plasma lens phase shift Equation (20) is proportional to the frequency, just like the gravitational shift Equation (22). If it is important, the dependence of the frequency period on ν may be extremely weak. An example is provided above by the strong underdensity lens. In this case, the values of A_osc and T_ν in Equation (21) logarithmically depend on ν via the lens strength ${\alpha }_{l}^{{\prime} }\propto {\nu }^{-2}$ and become indistinguishable from constants at ${\alpha }_{l}^{{\prime} }\gtrsim 5$.

Nevertheless, it is worth stressing that the experimental data do not indicate any dependence of the spectral oscillation parameters on frequency. Indeed, the values of T_ν in Figure 16(a) are almost the same for the bursts with essentially different central frequencies ν_c .

Our analysis of the narrowband spectral structure essentially differs from the previous ones. Let us explain the distinction and place our results in the context of the other FRB 20121102A studies.

The unusual features were observed in the burst ACFs before, but a conclusive evidence for their diffractive origin has never appeared. For example, Majid et al. (2020) published ¹⁹ two ACFs of the FRB 20121102A bursts B1 and B6 measured by the DSS–43 telescope of the Deep Space Network at frequency 2.24 GHz; see Figure 21. At large Δν, these functions display several recognizable maxima; see Figure 21, panels (b) and (d). Interpreting the latter as a manifestation of the two-wave interference, one can formally fit the B1 and B6 ACFs with the cos-like function in the integrand of Equation (19) plus a constant. The results of this fit are shown by the dash–dotted lines in Figure 21, panels (b) and (d). The respective best-fit frequency periods are T_ν ≈ 3.5 and 1.4 MHz for the bursts B1 and B6, respectively.

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Note, however, that unlike the Breakthrough Listen digital backend of the Green Bank Telescope, the instrument of Majid et al. (2020) has narrow 8 MHz noncontiguous sub-bands. As a consequence, the graphs B1 and B6 in Figure 21 include only 2 and 4 peaks in the available frequency interval, and one cannot judge whether they are periodic or not. Compare this to the strongest bursts, 11A and 11D, in Figures 8(a) and 14, every one of which displays ∼10 approximately equidistant ACF maxima. Besides, the apparent frequency periods T_ν of the bursts B1 and B6 mismatch by a factor of 2. The data of the previous section were consistent: the values T_ν of the 12 strongest bursts were grouping within the 15% interval near the central value. Finally, the widths of the maxima in Figure 21 are comparable to the entire frequency interval ν₂ − ν₁ = 8 MHz. In fact, the presence of the unsuppressed random fluctuations is expected ²⁰ at these scales, since the frequency integral in the ACF effectively kills only the noise with ν_d ≪ ν₂ − ν₁; see Equation (9).

We conclude that the periodic structures cannot be distinguished from the random spectral behavior in the Majid et al. (2020) data. To do that, one needs wideband measurements of many spectra, like the ones performed by the Green Bank Telescope.

Now, let us compare our new method of studying the scintillations with the previous ones. Gajjar et al. (2018) performed the original analysis of the 4–8 GHz Green Bank Telescope data by fitting the sub-band ACFs with the Gaussian profiles at Δν < 200 MHz. The resulting values of ν_d —the half width at half maximum of the fitting function—were found to be consistent with ν_d ≈ 24 MHz at 6 GHz, which is 6–7 times larger than our result. This huge discrepancy is related to the fact that the fitting interval Δν < 200 MHz in Gajjar et al. (2018) includes the first equidistant ACF maximum; see Figure 8. This new feature is enveloped by the fitting function making the latter wider. On the other hand, we fitted only the initial part of the ACF to the left of its first minimum. ²¹

Majid et al. (2020) obtained the decorrelation bandwidth of the two FRB 20121102A bursts, B1 and B6, using the 2.4 GHz DSS–43 telescope data. To this end, the initial falloffs of the respective ACFs were fitted with the Lorentzian profile Equation (13) at Δν < 0.84 MHz, where ${ \mathcal N }$, ν_d , and c served as the fit parameters; see Cordes et al. (1985). The result was ν_d ≈ 180 kHz and 280 kHz for the bursts B1 and B6, respectively. We replot the Majid et al. (2020) data and their Lorentzian fits in Figure 21 (steps and solid lines). Their procedure is different from ours in two important respects. First, we fix c—the constant part of the ACF—with the subtraction procedure that effectively means that c tracks the mean value of this function at large Δν; see Equation (9). Indeed, if one leaves this parameter free in the fit, its value would essentially depend on the Δν interval and, crudely speaking, would pick up the minimal value of the ACF at the interval boundary. This sensitivity is an artifact of the unexpected ACF oscillations at large Δν, and it is not correct. One can fix the arbitrariness in our manner by fitting the ACFs at large Δν with the constant c (dotted horizontal lines in Figure 21) and then performing the stable Lorentzian fit at small Δν. The fit result is ν_d (2.24 GHz) ≈ 155 ± 14 kHz and 162 ± 20 kHz for for the bursts B1 and B6, respectively. Notably, the two values of the decorrelation bandwidth now agree within the error bars obtained from the fits.

Second, we use new theoretical profile in Equations (11) and (15) for the ACF that has sharper behavior as Δν → 0; see Figure 9. As discussed in the previous section, this method generically gives 20% smaller value of ν_d . Indeed, for the bursts B1 and B6, we obtain ν_d ≈ 128 kHz and 133 kHz at a reference frequency 2.24 GHz; see the dashed lines in Figure 21, panels (a) and (c).

Now, we rescale the decorrelation bandwidth ν_d ≈ 130 kHz at 2.24 GHz to our frequencies. Using the Kolmogorov formula (12) with ν_d ∝ ν^4.4, we obtain ν_d ≈ 9.9 MHz at 6 GHz, which is 2–3 times larger than our values: recall that the scintillations and scintillations+lensing fits of the previous sections give ν_d (6 GHz) ≈ 4.3 and 3.3 MHz, respectively. This means that non-Kolmogorov frequency dependence ν_d ∝ ν^α with α < 4.4 is favored by the data. In particular, rescaling with α = 4 gives ν_d ≈ 6.7 MHz at 6 GHz that differs from our values by the reasonable factor of 2. At even smaller α = 3.5, one finds ν_d (6 GHz) ≈ 4.1 MHz in agreement with our result.

Note that non-Kolmogorov scaling ν_d ∝ ν^α with α < 4.4 was observed in the Milky Way pulsar signals; see, e.g., Bhat et al. (2004). Moreover the pulsar data coming from certain directions suggest α = 3 − 4 is not theoretically possible for weakly coupled turbulent plasma. Our result is of this kind.

Another value ν_d ≈ 58 kHz at 1.65 GHz was obtained by Hessels et al. (2019) using the European VLBI Network data for one FRB 20121102A burst. This result corresponds to ν_d ≈ 17, 10, and 5.3 MHz at 6 GHz in the cases α = 4.4, 4, and 3.5, respectively. The result at α = 4 is again twice larger than ours while the one with α = 3.5 is a good match.

Finally, let us compare the value of ν_d with the prediction of the NE2001 model for the Milky Way distribution of free electrons (Cordes & Lazio 2002). Extracting the scattering measure in the direction of FRB 20121102A from the provided software and using ²² Equation (10) of Cordes & Lazio (2002), we obtain the scintillation bandwidth in our galaxy: ν_{d, gal} ≈ 21 MHz at 6 GHz. This result again assumes Kolmogorov scaling, and it is 5–6 times larger than our experimental values. To feel how the prediction of the model would change in the non-Kolmogorov case, recall that Cordes & Lazio (2002) mostly use the pulsar data with typical frequencies in the GHz range; see Majid et al. (2020). Thus, scaling with powers α = 4 and 3.5 would give 6^4.4−α times smaller bandwidths ν_{d, gal} ≈ 10 and 4.2 MHz. Once again, we obtain a difference by a factor of 2 and a perfect match at α = 4 and 3.5, respectively.

To sum up, the bandwidth ν_d of our narrowband scintillations differs by a factor from the two measurements at other frequencies and from the prediction of the NE2001 model if the Kolmogorov scaling is assumed. In the case of non-Kolmogorov frequency dependence ν_d ∝ ν⁴, the four results agree up to a factor of two. If ν_d ∝ ν^3.5, all results match perfectly. Note that non-Kolmogorov scaling with α = 3.5 or 4 does not contradict to our data in Figures 10(b) and 15(b) (dotted lines) and in fact is observed in the pulsar measurements (Bhat et al. 2004; Geyer et al. 2017).

In this paper, we reanalyzed the spectra of FRB 20121102A measured by Gajjar et al. (2018). We developed practical theoretical tools to study the random spectral components, regular peaks, and periodic spectral structures that either may be caused by interstellar scintillations, refractive lensing, and diffractive lensing or, alternatively, can be intrinsic to the FRB progenitor.

We saw that the caustics of the refractive lens produce a spectral peak of a distinctive recognizable form (Clegg et al. 1998; Cordes et al. 2017) that can be directly fitted to the spectra. On the other hand, separation of diffractive lensing from scintillations requires calculation of an integral observable: the spectral ACF in Equation (9). The scintillations are responsible for the monotonic falloff of this function with frequency lag Δν, while the two-ray diffraction introduces a distinctive oscillatory behavior, i.e., the series of pronounced equidistant maxima. We derived explicit theoretical expressions for the ACFs that include the effects of Kolmogorov-type scintillations and scintillations on top of diffractive lensing, Equations (10), (11), and (19), respectively. The latter expressions can be used to interpret the experimental data and, in fact, fit them quite nicely. An alternative data analysis may involve a Fourier transformation as in Katz et al. (2020), the periodogram method in Zechmeister & Kurster (2009), Ivanov et al. (2019), or the Kolmogorov–Smirnov–Kuiper test in Press et al. (2007).

Using the above tools, we identify and explain several remarkable features in the FRB 20121102A spectra. First and most importantly, we discover a set of almost equidistant spectral peaks separated by T_ν = 95 ± 16 MHz. This periodicity is a benchmark property of wave diffraction, and we show that it may be relevant, indeed. On the one hand, the peaks may be caused by diffractive gravitational (femto)lensing of the FRB signals on a compact object of mass 10⁻⁴ M_⊙, e.g., a primordial black hole or a dense minihalo; see Katz et al. (2020). Theoretically, such events are expected to occur with relatively small probability ∼ 10⁻² in the intergalactic space if all dark matter consists of lenses. On the other hand, the periodic peaks may originate from the diffractive lensing on a plasma underdensity in the host galaxy. The respective lenses—the holes in the electron density—may be expected to appear due to plasma heating and, in fact, are discussed in the literature; see, e.g., Pen & King (2012) and Simard & Pen (2018).

Yet another suggestion would be to attribute the periodic structure to the progenitor spectrum. Notably, the banded pattern resembling our periodic structure has been observed in the spectra of the Crab pulsar; see, e.g., Hankins et al. (2016). This may point at the same physical origin of the two effects or similar propagation effects near the sources. Over the years, propagation and direct emission models were proposed for explanation of the Crab bands, but none of them has become universally accepted by now; see the discussion in Hankins et al. (2016).

The second spectral feature is a strong, almost monochromatic peak at 7.1 GHz dominating the spectra of most bursts. This peak was also spotted by Gajjar et al. (2018). We demonstrated that it can be produced by the refractive lensing of the FRB wave on a one-dimensional Gaussian plasma cloud. The latter may represent long ionized filament from the supernova remnant in the host galaxy (Michilli et al. 2018) or an AU-sized elongated turbulent overdensity that are expected to be abundant in galactic plasmas; see Fiedler et al. (1987), Bannister et al. (2016), and Coles et al. (2015). Note also that the origin of the lens may be essentially different. For example, FRB 20180916B—a repeating source very similar to FRB 20121102A—is possibly a high-mass X-ray binary system that includes a neutron star interacting with the ionized wind of the companion (Pleunis et al. 2021; Tendulkar et al. 2021). In this case, extreme plasma lensing may occur on the wind; see Main et al. (2018).

One can still imagine that the agreement of the 7.1 GHz peak profile with the expected spectrum of the lens is a coincidence, and this feature belongs to the intrinsic spectrum of the FRB progenitor. Going to the extreme, one can even assume that the progenitor produces a single line of powerful monochromatic emission at 7.1 GHz, and all other frequencies add up afterwards in the course of nonlinear wave propagation through the surrounding plasma. The generation mechanisms for the monochromatic signals use cosmic masers (Lu & Kumar 2018) or Bose stars made of dark matter axions (Tkachev 1986) that decay into photons. The latter process may occur explosively in strong magnetic fields (Iwazaki 2015; Tkachev 2015; Pshirkov 2017) or in the situation of parametric resonance (Tkachev 1986, 2015; Hertzberg & Schiappacasse 2018; Amin & Mou 2021; Hertzberg et al. 2020; Levkov et al. 2020). However, the axion-related mechanisms still belong to the speculative part of the FRB theory, whereas the cosmic masers with realistic parameters fail to provide the required FRB luminosity; see Lu & Kumar (2018).

Third, the FRB signals illuminate parts of a global GHz-scale spectral structure; see Sobacchi et al. (2021). The imprint of this structure was previously noticed by Gajjar et al. (2018) in the summed 4–8 GHz spectrum of FRB 20121102A. We demonstrate that the structure drifts linearly with time and, therefore, presumably represents a propagation effect e.g., GHz-scale scintillations. Of course, even this last feature may belong to the FRB source if the emission region itself evolves linearly.

Fourth, all pieces of the propagation scenario fit together if the spectrum of the FRB 20121102A progenitor has a relatively narrow bandwidth ν_bw ∼ GHz, and its central frequency changes rapidly and significantly from burst to burst. The same properties were observed before in the registered spectra of FRB 20121102A (Law et al. 2017; Gajjar et al. 2018; Gourdji et al. 2019; Hessels et al. 2019; Majid et al. 2020) and FRB 20180916B (Chawla et al. 2020; Pearlman et al. 2020), but in these studies, the intrinsic progenitor properties were not separated from the propagation phenomena. We perform the separation and, in fact, use the propagation effects as landmarks for studying the progenitor spectrum.

In particular, the main peak at 7.1 GHz, which we attribute to the strong lens, disappears in some spectra reappearing in the later-coming bursts at the same position and with the same form. We explain that this happens precisely because the progenitor spectrum has a GHz bandwidth and variable central frequency. Indeed, all the spectra with the main peak are located within the GHz band around 7.1 GHz, and the spectra without it have the major power outside of this band. Further, the bursts happening at different times illuminate different parts of the linearly evolving wideband spectral pattern introduced above. Reconstructing the pattern, we estimate the bandwidth of the progenitor.

Finally, we develop new generalized framework for the analysis of strong interstellar scintillations. We obtain their decorrelation bandwidth ν_d by fitting the experimental ACFs with the new theoretically derived profile. Notably, the ACF data agree with the theory, the value of ν_d crudely respects the predicted power-law scaling, and the overall scintillation pattern slowly drifts in frequency due to motion of the observer relative to the scintillating clouds. Our result for ν_d slightly depends on the assumptions on the above-mentioned periodic spectral structure. If we ignore it and use the scintillations-only model, the best-fit value is ν_d = 4.3 ± 0.9 MHz at the reference frequency 6 GHz. Adding the periodic structure to the fit, we obtain a consistent value ν_d (6 GHz) = 3.3 ± 0.6 MHz. One can conservatively consider the difference between the two results as a systematic error, although we do suggest that the last result is more consistent. We believe that our method for extracting ν_d is more reliable than the previous ones because it uses the theoretically predicted ACF profile.

Note that the narrowband scintillations were observed in the FRB 20121102A spectra before by Hessels et al. (2019) at 1.65 GHz and Majid et al. (2020) at 2.24 GHz. We perform comparison by scaling their two values of ν_d to 6 GHz with the power law ν_d (ν) ∝ ν^α . In the Kolmogorov case, α = 4.4; the scaling gives 5 and 3 times larger bandwidths than our result, respectively. Thus, the data strongly favor non-Kolmogorov frequency dependence. The smallest theoretically motivated power for the weak turbulence is α = 4. In this case, the values of Hessels et al. (2019) and Majid et al. (2020) at 6 GHz differ from ours by the factors of 3 and 2, respectively, which is already tolerable given large experimental uncertainties and different instruments. If α = 3.5, the three experimental results agree.

Do the scintillations appear in the Milky Way or in the FRB host galaxy? The model NE2001 (Cordes & Lazio 2002) predicts Milky Way scintillations with ν_{d, gal}(6 GHz) ≈ 21 MHz in the direction of FRB 20121102A, and that is 6 times larger than our result. But the same model assumes Kolmogorov scaling of ν_d with α = 4.4. Thus, the discrepancy can be again attributed to deviations from this law. Indeed, following Majid et al. (2020), we crudely account for arbitrary α in the model and arrive at estimates ν_d (6 GHz) ∼ 10 and 4 MHz at α = 4 and 3.5, which are closer to our result. Thus, the scintillations presumably originate in our galaxy.

To conclude, the studies of the FRB spectra are still in their infancy, but they evolve fast. With clever data analysis and separation of propagation effects, they soon may be able to purify the pristine chaos of the present–day theory for the FRB engines down to a single graceful picture.

We thank V. Gajjar for help, S. Sibiryakov for discussions, and the Referee for criticism. Scintillations in the FRB 20121102A spectra were studied within the framework of the RSF grant 16-12-10494. Investigation of the FRB lensing was supported by the Ministry of Science and Higher Education of the Russian Federation under the contract 075-15-2020-778 (State project Science). The rest of this paper was funded by the Foundation for the Advancement of Theoretical Physics and Mathematics, BASIS. Numerical calculations were performed on the Computational cluster of Theory Division of the Institute for Nuclear Research of the Russian Academy of Sciences.

Let us explain the computation of the spectral fluence Equation (2) in detail. The main idea is to choose the signal region t₁(ν) < t < t₂(ν) that minimizes the instrumental noise.

Most of the bursts are tilted in the t − ν plane, even after the dedispersion described in Gajjar et al. (2018); see burst 11A in Figure 22. We, therefore, use the linear boundaries ${t}_{1}(\nu )={\tilde{t}}_{1}+c\nu$ and ${t}_{2}(\nu )={\tilde{t}}_{2}+c\nu$ of the signal region with the frequency-independent signal duration t₂ − t₁.

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To determine the parameters ${\tilde{t}}_{1}$, ${\tilde{t}}_{2}$, and c, we employ two auxiliary technical steps. First, we Gauss-average the signal f(t, ν) over the moving time and frequency windows σ_t = 0.02 ms and σ = 10 MHz; see Equation (1). Second, for every burst, we preselect the frequency interval ν₁ < ν < ν₂ including the signal (dashed lines in Figure 22).

Once this is done, we integrate the smoothed density ${\bar{f}}_{10}$ along the inclined line,

$$\begin{eqnarray*}&&{J}_{c}(\tilde{t})=\underset{{\nu }_{1}}{\overset{{\nu }_{2}}{\int }}d\nu \,{\bar{f}}_{10}(\tilde{t}+c\nu ,\nu ).\end{eqnarray*}$$

This function is positive if the line $t=\tilde{t}+c\nu$ crosses the signal. Outside of the signal region, ${J}_{c}(\tilde{t})$ oscillates near zero due to noise. We, therefore, select ${\tilde{t}}_{1}$ and ${\tilde{t}}_{2}$ to be the first zeros of this function surrounding its global maximum; see Figure 22(b).

To choose the optimal value of the tilt c, we note that the integral ${E}_{c}={\int }_{{\tilde{t}}_{1}}^{{\tilde{t}}_{2}}{J}_{c}(\tilde{t})\,d\tilde{t}$ estimates the energy within the signal region. We maximize the average signal power ${E}_{c}/({\tilde{t}}_{2}-{\tilde{t}}_{1})$ with respect to c, thus choosing the minimal region t₁ < t < t₂ with a major part of the total energy; see Figure 22(c).

Once the signal region is identified, we perform integrations in Equations (1) and (2) obtaining ²³ the spectral fluence ${\bar{F}}_{10}(\nu )$. Note that this quantity is computed at all frequencies, even outside of the auxiliary interval ν₁ < ν < ν₂. At ν < ν₁ and ν > ν₂, the spectral fluence fluctuates near zero due to noise.

The experimental errors are estimated assuming that the statistical properties of the measuring device are time-independent. For every frequency ν, we select a sufficient number of random time points outside of the signal region, combine the data at these points into an artificial interval t₁(ν) < t < t₂(ν), then use Equations (1) and (2). This gives the random fluence ${\rm{\Delta }}{\bar{F}}_{10}(\nu )$ of the noise. We finally subtract the statistical mean of ${\rm{\Delta }}{\bar{F}}_{10}$ from Equation (2) and use its standard deviation to estimate the errors (shaded areas in Figure 2).

B.1. Eikonal Approximation

In this appendix, we review the effects of plasma and gravitational lenses on the propagating FRB signals. A plasma cloud equips the radio wave with a dispersive phase shift φ_l ,

$$\begin{eqnarray}&&{\varphi }_{l}=-\displaystyle \frac{{r}_{e}}{\nu }\,{{\rm{DM}}}_{l}({\boldsymbol{x}})=-\displaystyle \frac{{r}_{e}}{\nu }\int {dz}\,\delta {n}_{e}({\boldsymbol{x}},z),\end{eqnarray} \tag{ B1 }$$

where δ n_e is the overdensity of free electrons, r_e ≡ e²/m_e is the classical electron radius, and the integral runs along the line of sight. Notably, the dispersive phase Equation (B1) is inversely proportional to frequency.

We will use the standard (Rickett 1990; Katz et al. 2020) assumption that the entire dispersive shift Equation (B1) is acquired on the relatively thin lens screen halfway through the plasma cloud; see the dashed line in Figure 3. This approach explicitly separates the geometric and dispersive effects. It is justified if the lens is spatially separated from the other propagation phenomena.

Now, the dispersive phase Equation (B1) is a function of the two-coordinate x = (x, y) on the lens screen. Besides, the radio ray p x o in Figure 3 consists of two straight parts giving the extra geometric phase shift

$$\begin{eqnarray}&&{\phi }_{l}=2\pi \nu ({r}_{p{\boldsymbol{x}}}+{r}_{{\boldsymbol{x}}o}-{r}_{{po}})\approx {\left({\boldsymbol{x}}-\tilde{{\boldsymbol{x}}}\right)}^{2}/2{r}_{F,l}^{2}.\end{eqnarray} \tag{ B2 }$$

Here r_{p x} , r _{x o} , and r_po are the distances between the respective points in Figure 3; in the last equality, we used the small-angle approximation ${r}_{{ij}}\approx {d}_{{ij}}+{({{\boldsymbol{x}}}_{i}-{{\boldsymbol{x}}}_{j})}^{2}/2{d}_{{ij}}$ and collected the total square. Recall that x _p and x _o are the progenitor and observer shifts, $\tilde{{\boldsymbol{x}}}$ is defined in Equation (4), and r_F,l is the lens Fresnel scale; see Section 3. The total phase shift

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{l}({\boldsymbol{x}})={\phi }_{l}({\boldsymbol{x}})+{\varphi }_{l}({\boldsymbol{x}})\end{eqnarray} \tag{ B3 }$$

is the sum of Equations (B2) and (B1).

The gravitational lens modifies the phases of radio waves in a different way: by gravitationally attracting the radio rays and changing their length. For convenience we will also divide its phase shift Φ_l into the naive geometric part in Equation (B2) and a correction φ_l ( x ) at the lens screen. Both ϕ_l and φ_l , in this case, are proportional to the frequency, in contrast to ν⁻¹ behavior of the dispersive shift in Equation (B1).

The complex amplitude of the observed FRB signal is given by the Fresnel integral

$$\begin{eqnarray}&&{f}_{\nu }={f}_{p}\int \displaystyle \frac{{d}^{2}{\boldsymbol{x}}}{2\pi {{ir}}_{F,l}^{2}}\,\,{{\rm{e}}}^{i{{\rm{\Phi }}}_{l}({\boldsymbol{x}})},\end{eqnarray} \tag{ B4 }$$

where f_p (ν) is the signal of the FRB progenitor.

In this paper, we consider lensing in the limit of geometric optics ∣ x ∣ ≫ r_F,l when the integral (B4) receives main contributions near the stationary points x = x _j of the total phase. These points satisfy the equation,

$$\begin{eqnarray}&&{{\boldsymbol{\nabla }}}_{{\boldsymbol{x}}}{{\rm{\Phi }}}_{l}({\boldsymbol{x}})\equiv {r}_{F,l}^{-2}({\boldsymbol{x}}-\tilde{{\boldsymbol{x}}})+{{\boldsymbol{\nabla }}}_{{\boldsymbol{x}}}{\varphi }_{l}({\boldsymbol{x}})=0.\end{eqnarray} \tag{ B5 }$$

The signal (B4) is then given by the saddle–point formula,

$$\begin{eqnarray}&&{f}_{\nu }={f}_{p}\sum _{j}\,{G}_{j}^{1/2}\,{{\rm{e}}}^{i{{\rm{\Phi }}}_{l}({{\boldsymbol{x}}}_{j})},\end{eqnarray} \tag{ B6 }$$

where we introduced the gain factors of the radio paths,

$$\begin{eqnarray}&&{G}_{j}(\nu )=| \det ({\delta }_{\alpha \beta }+{r}_{F,l}^{2}{\partial }_{\alpha }{\partial }_{\beta }{\varphi }_{l}){| }_{{\boldsymbol{x}}={{\boldsymbol{x}}}_{j}}^{-1}\end{eqnarray} \tag{ B7 }$$

with ∂_α ≡ ∂/∂x_α . Note that the determinant in Equation (B7) is not necessarily positive. For convenience we keep G_j > 0 and include the phase of the determinant into Φ_l ( x _j ) → Φ_l ( x _j ) − π n_j /2, where n_j = 0, 1, 2 if x = x _j is a minimum, a saddle point, and a maximum of Φ_l , respectively. Once f_ν is computed, one obtains the fluence F(ν) ≡ ∣f_ν ∣².

B.2. Refractive and Diffractive Lenses

We see that every radio path x _j adds the term ΔF = G_j ∣f_p ∣² to the fluence, while its interference with other paths produces oscillating terms proportional to $\cos ({{\rm{\Phi }}}_{j}-{{\rm{\Phi }}}_{k})$, where Φ_j ≡ Φ_l ( x _j ). For example, in the case of the two trajectories Equation (B6) gives,

$$\begin{eqnarray}&&F={F}_{p}\left[{G}_{1}+{G}_{2}+2{\left({G}_{1}{G}_{2}\right)}^{1/2}\,\cos ({{\rm{\Phi }}}_{1}-{{\rm{\Phi }}}_{2})\right],\end{eqnarray} \tag{ B8 }$$

with F ≡ ∣f_ν ∣² and F_p ≡ ∣f_p ∣² denoting the registered and source fluences, respectively. The last term in Equation (B8) represents diffraction. As a function of frequency, it oscillates with the period

$$\begin{eqnarray}&&{T}_{\nu }=2\pi | {\partial }_{\nu }({{\rm{\Phi }}}_{1}-{{\rm{\Phi }}}_{2}){| }^{-1}\sim O(2\pi \nu \,{r}_{F,l}^{2}/{a}^{2}),\end{eqnarray} \tag{ B9 }$$

where a ∼ ∣ x ₂ − x ₁∣ is the typical lens size.

In Section 3, we consider a refractive lens with extremely large a/r_F,l ≫ (ν/σ)^1/2. In this case, the oscillatory term in Equation (B8) is exponentially dumped by the instrumental smoothing Equation (1) with window σ. As a consequence, the main effect of this lens is to multiply the source fluence with the sum of the gain factors in Equation (6).

In Section 6, we interpret the periodic spectral structures with T_ν ∼ 100 MHz as the oscillating term in Equation (B8). Of course, all these structures may be killed by smoothing with a sufficiently large window, say, σ = 50 MHz. In this case, ${\bar{F}}_{50}\approx {F}_{p}({G}_{1}+{G}_{2})$. The theoretical lens signal Equation (B8) then can be rewritten as

$$\begin{eqnarray}&&F\approx {\bar{F}}_{50}\left[1+{A}_{\mathrm{osc}}\cos ({{\rm{\Phi }}}_{1}-{{\rm{\Phi }}}_{2})\right],\end{eqnarray} \tag{ B10 }$$

where

$$\begin{eqnarray}&&{A}_{\mathrm{osc}}=2{\left({G}_{1}{G}_{2}\right)}^{1/2}\,{\left({G}_{1}+{G}_{2}\right)}^{-1}\leqslant 1\end{eqnarray} \tag{ B11 }$$

is the relative oscillation amplitude. In the main text, we also compute the correlation function Equation (9) by multiplying $\delta F(\nu )=F-{\bar{F}}_{50}$ at close by frequencies ν and ν + Δν and integrating over ν. This procedure exponentially suppresses the terms oscillating with the integration variable ν leaving

$$\begin{eqnarray}&&\begin{array}{l}{\rm{ACF}}({\rm{\Delta }}\nu )=\displaystyle \frac{{ \mathcal N }}{2}\underset{{\nu }_{1}}{\overset{{\nu }_{2}-{\rm{\Delta }}\nu }{\displaystyle \int }}d\nu \,\,\displaystyle \frac{{\bar{F}}_{50}(\nu ){\bar{F}}_{50}(\nu +{\rm{\Delta }}\nu )}{{\nu }_{2}-{\nu }_{1}-{\rm{\Delta }}\nu }\\ \ \times {A}_{\mathrm{osc}}(\nu ){A}_{\mathrm{osc}}(\nu +{\rm{\Delta }}\nu )\,\cos (2\pi {\rm{\Delta }}\nu /{T}_{\nu }).\end{array}\end{eqnarray} \tag{ B12 }$$

In the case of narrow bandwidth spectra with ν₂ − ν₁ ≪ ν and Δν ≪ ν, one can ignore the frequency dependence of the period and find

$$\begin{eqnarray}&&{\rm{ACF}}\approx \cos (2\pi {\rm{\Delta }}\nu /{T}_{\nu }),\end{eqnarray} \tag{ B13 }$$

where the normalization was performed, ACF(0) = 1. In practice, Equation (B12) accounts for the wideband envelope ${\bar{F}}_{50}$ of the spectrum and, therefore, better fits the experimental data, though Equation (B13) is simpler and may be used on preparatory stages.

B.3. Gaussian Overdensity Lens

In Section 3, we consider a Gaussian lens with the profile ${\varphi }_{l}=-({r}_{e}/\nu )\,{{\rm{DM}}}_{l}\,{{\rm{e}}}^{-{x}^{2}/{a}^{2}}$ depending only on one transverse coordinate x. The total phase shift Equations (B2) and (B3), in this case, reduces to Equation (3). The y component of the eikonal Equation (B5) gives ${y}_{j}=\tilde{y}$ implying that the lens bends the radio waves only in the x direction. The other, the x component, has the form

$$\begin{eqnarray}&&f(u)\equiv u-\tilde{u}+\alpha u{{\rm{e}}}^{-{u}^{2}}=0,\end{eqnarray} \tag{ B14 }$$

where u = x/a and $\tilde{u}=\tilde{x}/a$. We denote the solutions of this equation by u_j . The net gain factor of Equation (6) equals G(ν) = ∑_j ∣∂_u f(u_j )∣⁻¹.

Let us show that the lens caustics—solutions u_* of Equation (B14) with infinite gain factor G—exist only if $\tilde{x}/a=\tilde{u}$ exceeds the critical value ${\tilde{u}}_{\mathrm{cr}}={(3/2)}^{3/2}$. Indeed, by definition u_* satisfy equations f(u_*) = ∂_u f(u_*) = 0 that can be written in the form ${\alpha }_{l}={{\rm{e}}}^{{u}_{* }^{2}}/(2{u}_{* }^{2}-1)$ and $\tilde{u}=2{u}_{* }^{3}/(2{u}_{* }^{2}-1)$. The right-hand side of the last equation is bounded from below by the global minimum $\tilde{u}\geqslant {\tilde{u}}_{\mathrm{cr}}$ that occurs at ${u}_{* }={u}_{\mathrm{cr}}\equiv \sqrt{3/2}$ and ${\alpha }_{l}={\alpha }_{\mathrm{cr}}\equiv \tfrac{1}{2}{{\rm{e}}}^{3/2}$. We conclude that the lens caustics exist only for overcritical lens shifts, $\tilde{u}\geqslant {\tilde{u}}_{\mathrm{cr}}$. At $\tilde{u}={\tilde{u}}_{\mathrm{cr}}$, they appear at the critical frequency ${\alpha }_{l}^{-1/2}={\alpha }_{\mathrm{cr}}^{-1/2}$ and move apart as $\tilde{u}$ grows.

Consider the near-critical situation when $\tilde{u}$ slightly exceeds ${\tilde{u}}_{\mathrm{cr}}$. This corresponds to a nearby pair of caustics in the lens spectrum with α_*± and u_*± close to α_cr and u_cr. Performing the Taylor series expansion in u − u_cr and α_l − α_cr, we rewrite the lens equation as

$$\begin{eqnarray}\begin{array}{rcl}f & \approx & {\tilde{u}}_{\mathrm{cr}}-\tilde{u}-({\alpha }_{l}/{\alpha }_{\mathrm{cr}}-1)(u-3{u}_{\mathrm{cr}}/2)\\ & & +\,{\left(u-{u}_{\mathrm{cr}}\right)}^{3}\,=\,0.\end{array}\end{eqnarray} \tag{ B15 }$$

Now, we can explicitly solve the caustic equations f(u_*) = ∂_u f(u_*) = 0 with respect to u_* and frequency α_l finding

$$\begin{eqnarray}&&{u}_{* }\,\pm \approx {u}_{\mathrm{cr}}\pm \displaystyle \frac{1}{\sqrt{3}}{\left({\alpha }_{* }\,\pm /{\alpha }_{\mathrm{cr}}-1\right)}^{1/2},\end{eqnarray} \tag{ B16 }$$

$$\begin{eqnarray}&&{\alpha }_{* \,\pm }\approx {\alpha }_{\mathrm{cr}}\left[3\tilde{u}/{\tilde{u}}_{\mathrm{cr}}-2\pm \displaystyle \frac{4}{{u}_{\mathrm{cr}}}{\left(\tilde{u}/{\tilde{u}}_{\mathrm{cr}}-1\right)}^{3/2}\right],\end{eqnarray} \tag{ B17 }$$

where expansion in $\tilde{u}-{\tilde{u}}_{\mathrm{cr}}$ was performed, again. Since α_l ∝ ν⁻², the last expression fixes the positions of caustics ν₀ ± Δν of the lens spectrum: α_l (ν₀) = (α_{* +} + α_{* −})/2 and Δν/ν₀ = ∣α_{* +} − α_{* −}∣/2α_cr. We obtain

$$\begin{eqnarray}&&\tilde{u}\approx {\tilde{u}}_{\mathrm{cr}}+{\tilde{u}}_{\mathrm{cr}}{\left(\displaystyle \frac{{\rm{\Delta }}\nu \sqrt{3}}{4{\nu }_{0}\sqrt{2}}\right)}^{2/3},\end{eqnarray} \tag{ B18 }$$

$$\begin{eqnarray}&&{\alpha }_{l}({\nu }_{0})\approx {\alpha }_{\mathrm{cr}}\left(3\tilde{u}/{\tilde{u}}_{\mathrm{cr}}-2\right).\end{eqnarray} \tag{ B19 }$$

In the main text, these expressions are used to compute $\tilde{u}=\tilde{x}/a$ and β.

Now, suppose the lens shift $\tilde{u}$ is slightly below ${\tilde{u}}_{\mathrm{cr}}$. In this case, Equation (B15) has only one solution, u = u₁, and the gain factor $G={[3{({u}_{1}-{u}_{\mathrm{cr}})}^{2}+1-{\alpha }_{l}/{\alpha }_{\mathrm{cr}}]}^{-1}$ is smooth. Nevertheless, G(u₁) has a sharp maximum at u₁ = u_cr. Half-height of the maximum is reached at ${u}_{1}-{u}_{\mathrm{cr}}=\pm \tfrac{1}{\sqrt{3}}{(1-{\alpha }_{l}/{\alpha }_{\mathrm{cr}})}^{1/2}$. Substituting these points into Equation (B15), we find the frequency of the maximum α_l = α_l (ν₀) and its half-height width ${\rm{\Delta }}\nu ^{\prime} /{\nu }_{0}={\rm{\Delta }}{\alpha }_{l}/2{\alpha }_{\mathrm{cr}}$,

$$\begin{eqnarray}&&\tilde{u}\approx {\tilde{u}}_{\mathrm{cr}}-{\tilde{u}}_{\mathrm{cr}}{\left(\displaystyle \frac{{\rm{\Delta }}\nu ^{\prime} \sqrt{3}}{8{\nu }_{0}\sqrt{2}}\right)}^{2/3},\end{eqnarray} \tag{ B20 }$$

$$\begin{eqnarray}&&{\alpha }_{l}({\nu }_{0})\approx {\alpha }_{\mathrm{cr}}\left(3\tilde{u}/{\tilde{u}}_{\mathrm{cr}}-2\right).\end{eqnarray} \tag{ B21 }$$

These equations relate the main spectral peak to the parameters of the lens with $\tilde{x}\lt {\tilde{x}}_{\mathrm{cr}}$.

It is worth reminding that the analytic treatment of this appendix is applicable for narrow lens spectra, Δν/ν₀ ≪ 1. Notably, in this case, the two- or one-peaked lens contributions are easily recognizable on the experimental graphs.

B.4. Diffractive Underdensity Lens

In Section 6, we study diffraction of radio rays split by the plasma lens. We use the same Gaussian profile of the dispersive phase shift φ_l as before, but with a different sign in front of it; see Equation (20). This lens describes a hole in the interstellar plasma, i.e., an underdensity of free electrons: ${\alpha }_{l}^{{\prime} }\propto -\delta n\gt 0$.

The lens equation, Equation (20), has too many parameters and easily fits the experimental data, so in the main text, we voluntarily choose the simplest and most illustrative regime: a strong lens relatively far away from the line of sight, $\mathrm{ln}{\alpha }_{l}^{{\prime} }\gg 1$ and $\tilde{x}^{\prime} /a^{\prime} \sim \sqrt{\mathrm{ln}{\alpha }_{l}^{{\prime} }}$. Then the eikonal Equations (B5) and (20) give three rays (see Figure 20),

$$\begin{eqnarray}&&{x}_{\mathrm{1,2}}\approx \pm a^{\prime} \sqrt{\mathrm{ln}{\alpha }_{l}^{{\prime} }},\qquad {x}_{3}\approx -\tilde{x}/{\alpha }_{l}^{{\prime} },\end{eqnarray} \tag{ B22 }$$

where corrections to x_1,2 are suppressed by ${(\mathrm{ln}{\alpha }_{l}^{{\prime} })}^{-1}$. Recall that the one-dimensional lens does not bend the rays in the y direction: ${y}_{j}=\tilde{y}^{\prime}$. Plugging the eikonal equation into Equation (B7), we simplify the expression for the lens gain factor: $G=| 2{x}^{2}/{a}^{{\prime} 2}-2x\tilde{x}^{\prime} /{a}^{{\prime} 2}+\tilde{x}^{\prime} /x{| }^{-1}$. Three solutions (B22) then give,

$$\begin{eqnarray}&&{G}_{\mathrm{1,2}}\approx \displaystyle \frac{1}{2\mathrm{ln}{\alpha }_{l}^{{\prime} }}{\left(1\mp \displaystyle \frac{\tilde{x}^{\prime} }{a^{\prime} \sqrt{\mathrm{ln}{\alpha }_{l}^{{\prime} }}}\right)}^{-1},\,\,{G}_{3}\approx \displaystyle \frac{1}{{\alpha }_{l}^{{\prime} }}.\end{eqnarray} \tag{ B23 }$$

Notably, in our regime $\mathrm{ln}{\alpha }_{l}^{{\prime} }\gg 1$, the contribution of the third radio path can be ignored, and we obtain the two-wave interference in Equation (16).

Computing the phases in Equation (20) of the two remaining solutions, we obtain

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{2}-{{\rm{\Phi }}}_{1}\approx 2a^{\prime} \tilde{x}^{\prime} \sqrt{\mathrm{ln}{\alpha }_{l}^{{\prime} }}/{r}_{F,l}^{{\prime} 2}.\end{eqnarray} \tag{ B24 }$$

Now, the expressions (B9) and (B11) give the frequency period T_ν and relative amplitude A_osc of the interference oscillations. We derived Equation (21) from the main text.

B.5. Gravitational Lens

In the main text, we speculate that the periodic spectral structure may be explained by gravitational lensing of the FRB signals on a compact object hiding at distance ${d}_{{lo}}^{\prime}$ from us. A phase shift of the radio waves in the gravitational field of a pointlike lens is given by Equation (22); see also Peterson & Falk (1991), Matsunaga & Yamamoto (2006), Bartelmann (2010), and Katz et al. (2020). The latter expression still has the form (B3) and (B2), like in the case of refraction, but with a specific φ_l term. We treat the gravitational lens in the same eikonal approximation as before.

The lens equation, Equation (B5), has two solutions,

$$\begin{eqnarray}&&{{\boldsymbol{x}}}_{\mathrm{1,2}}={{\boldsymbol{x}}}_{o}+\displaystyle \frac{1}{2}\,{\theta }_{E}\,{d}_{{lo}}^{\prime} \,{\boldsymbol{\zeta }}\left(1\pm \sqrt{1+4{{\boldsymbol{\zeta }}}^{-2}}\right),\end{eqnarray} \tag{ B25 }$$

where we introduced the vector ζ = ( x _p − x _o )/(d_po θ_E ) characterizing the angular shift of the source from the lens in units of θ_E . Equations (B7) and (22) give

$$\begin{eqnarray*}&&\begin{array}{l}{G}_{\mathrm{1,2}}=\displaystyle \frac{{\zeta }^{2}+2}{2\zeta \sqrt{{\zeta }^{2}+4}}\pm \displaystyle \frac{1}{2},\\ {{\rm{\Phi }}}_{2}-{{\rm{\Phi }}}_{1}={\rm{\Omega }}\left[\zeta \sqrt{{\zeta }^{2}+4}+2\mathrm{ln}(\zeta /2+\sqrt{{\zeta }^{2}/4+1})\right]-\displaystyle \frac{\pi }{2},\end{array}\end{eqnarray*}$$

where ζ ≡ ∣ ζ ∣, and we denoted Ω = 8π ν GM. Using, finally, Equations (B11) and (B9), we obtain the parameters of spectral oscillations in Equation (23) of the main text.

C.1. Adding the Scintillation Screen

In practice regular structures coexist in the FRB spectra with random scintillations caused by refraction of radio waves in the turbulent interstellar clouds. To describe the latter effect theoretically, we add a thin transverse scintillation screen that equips any propagating wave with a random phase φ_S ( ξ ), where ξ = (ξ_x , ξ_y ) is a two-coordinate on the screen; see Figure 13. For definiteness, we assume that the scintillations occur between the lens and the observer at distances d_lS and d_So ; we will comment on the other choice below. Physically, the scintillations may happen in the FRB host galaxy and / or the Milky Way.

Since the scintillation phase φ_S is caused by refraction, it is given by Equation (B1). But now δ n_e is a random component of the electron overdensity. It is customary to assume that this component has a homogeneous and isotropic Kolmogorov turbulent spectrum (see Cordes et al. 1985; Rickett 1990; Narayan 1992; Lorimer & Kramer 2004; Woan 2011; Katz et al. 2020):

$$\begin{eqnarray}&&\langle \delta {n}_{e}({\boldsymbol{X}})\delta {n}_{e}({\boldsymbol{X}}^{\prime} )\rangle =\,{C}_{n}^{2}{\displaystyle \int }_{{\kappa }_{\mathrm{out}}}^{{\kappa }_{{in}}}{d}^{3}{\boldsymbol{\kappa }}\,| {\boldsymbol{\kappa }}{| }^{-11/3}\,{{\rm{e}}}^{i{\boldsymbol{\kappa }}({\boldsymbol{X}}^{\prime} -{\boldsymbol{X}})},\end{eqnarray} \tag{ C1 }$$

where X and ${\boldsymbol{X}}^{\prime}$ represent the three-dimensional space coordinates and κ_in ≫ κ_out—the cutoff scales for turbulence. Angular brackets in Equation (C1) average over realizations of the turbulent ensemble, e.g., volumes within the galaxy. Using Equation (B1), one can turn Equation (C1) into a correlator of two φ_S 's (Rickett 1990; Katz et al. 2020),

$$\begin{eqnarray}&&{S}_{\nu }({\boldsymbol{\xi }}-{\boldsymbol{\xi }}^{\prime} )\equiv \langle {\left[{\varphi }_{S}({\boldsymbol{\xi }})-{\varphi }_{S}({\boldsymbol{\xi }}^{\prime} )\right]}^{2}\rangle =\displaystyle \frac{| {\boldsymbol{\xi }}-{\boldsymbol{\xi }}^{\prime} {| }^{5/3}}{{r}_{\mathrm{diff}}^{5/3}},\end{eqnarray} \tag{ C2 }$$

where we introduced the diffractive length scale

$$\begin{eqnarray}\begin{array}{rcl}{r}_{\mathrm{diff}} & = & 3.63\cdot {10}^{10}\,{\rm{cm}}{\left(\displaystyle \frac{{C}_{n}^{2}\cdot L}{{10}^{-4}\,{{\rm{m}}}^{-20/3}\cdot {\rm{kpc}}}\right)}^{-3/5}\\ & & \times \,{\left(\displaystyle \frac{\nu }{6\,{\rm{GHz}}}\right)}^{6/5},\end{array}\end{eqnarray} \tag{ C3 }$$

and thickness of the scintillation screen L ∼ kpc. Equation (C2) implies that the rays crossing the screen at distance r_diff receive relative random phases of an order of 1. This makes them incoherent at $| {\boldsymbol{\xi }}-{\boldsymbol{\xi }}^{\prime} | \gtrsim {r}_{\mathrm{diff}}$. Technically, it will be important for us that the correlator equation, Equation (C2), is translationally invariant, i.e., depends on $| {\boldsymbol{\xi }}-{\boldsymbol{\xi }}^{\prime} |$, and proportional to ν⁻²; see Equation (B1).

In this appendix, we describe the scintillations statistically, i.e., compute the mean FRB spectra and their ACFs. Recall that, in the main text, we average the experimental data over the frequency relying on the fact that they become statistically independent if ν is shifted by ν_d —the decorrelation bandwidth. This approach is applicable for narrowband scintillations of Section 5 that have small ν_d compared to the total FRB bandwidth ν_bw ∼ GHz. However, the same description is at best qualitative, such as in the case of the wideband scintillations considered in Section 4. Below we heavily rely on the expansion in ν_d /ν ≪ 1.

In the thin-screen approach of Figure 13, the radio ray p x ξ o consists of three straight parts. Its geometric phase shift can be written as

$$\begin{eqnarray}&&\phi ={\phi }_{l}({\boldsymbol{x}})+{\phi }_{S}({\boldsymbol{x}},{\boldsymbol{\xi }}),\end{eqnarray} \tag{ C4 }$$

where ϕ_l is the same lens shift equation, Equation (B2), as before, and

$$\begin{eqnarray}&&{\phi }_{S}=2\pi \nu ({r}_{{\boldsymbol{x}}{\boldsymbol{\xi }}}+{r}_{{\boldsymbol{\xi }}o}-{r}_{{\boldsymbol{x}}o})\approx {\left({\boldsymbol{\xi }}-\tilde{{\boldsymbol{\xi }}}\right)}^{2}/2{r}_{F,S}^{2}\end{eqnarray} \tag{ C5 }$$

accounts for the additional turn at the point ξ of the scintillation screen. We introduced the coordinate

$$\begin{eqnarray}&&\tilde{{\boldsymbol{\xi }}}({\boldsymbol{x}})=({d}_{{So}}{\boldsymbol{x}}+{d}_{{lS}}{{\boldsymbol{x}}}_{o})/{d}_{{lo}}\end{eqnarray} \tag{ C6 }$$

corresponding to straight propagation between x and o. Besides,

$$\begin{eqnarray}&&{r}_{F,S}={\left({d}_{{lS}}{d}_{{So}}/2\pi \nu {d}_{{lo}}\right)}^{1/2}\end{eqnarray} \tag{ C7 }$$

is the Fresnel scale for scintillations.

To sum up, we consider the radio wave that sequentially crosses the lens and the scintillating medium acquiring the total phase Φ = ϕ_l + φ_l + ϕ_S + φ_S . The Fresnel integral for this wave has the form (see Equation (B4))

$$\begin{eqnarray}&&{f}_{\nu }={f}_{p}\int \displaystyle \frac{{d}^{2}{\boldsymbol{x}}}{2\pi {{ir}}_{F,l}^{2}}\,\,{{\rm{e}}}^{i{\phi }_{l}({\boldsymbol{x}})+i{\varphi }_{l}({\boldsymbol{x}})}\,{{ \mathcal A }}_{\nu }({\boldsymbol{x}}),\end{eqnarray} \tag{ C8 }$$

where the new factor ${{ \mathcal A }}_{\nu }$ in the integrand accounts for scintillations,

$$\begin{eqnarray}&&{{ \mathcal A }}_{\nu }=\int \displaystyle \frac{{d}^{2}{\boldsymbol{\xi }}}{2\pi {{ir}}_{F,S}^{2}}\,\,{{\rm{e}}}^{i{\phi }_{S}({\boldsymbol{x}},{\boldsymbol{\xi }})+i{\varphi }_{S}({\boldsymbol{\xi }})}.\end{eqnarray} \tag{ C9 }$$

Recall that φ_S and hence f_ν are the random quantities.

C.2. Mean Fluence

The Fresnel integral for the averaged fluence $\langle F\rangle =\langle {f}_{\nu }{f}_{\nu }^{* }\rangle$ runs over x , ξ , and ${\boldsymbol{x}}^{\prime} ,\,{\boldsymbol{\xi }}^{\prime}$ that come from the integrals (C8) and (C9) for f_ν and ${f}_{\nu }^{* }$, respectively. In this expression, the statistical mean 〈 · 〉 acts on the random phase $\exp [i{\varphi }_{S}({\boldsymbol{\xi }})-i{\varphi }_{S}({\boldsymbol{\xi }}^{\prime} )]$ in the integrand. Recall that we consider weakly interacting turbulent plasma with φ_S ∝ δ n_e behaving as a Gaussian random quantity. Any correlator of such quantity can be computed in terms of the two-point function (C2). In particular,

$$\begin{eqnarray}&&\langle {{\rm{e}}}^{i{\varphi }_{S}({\boldsymbol{\xi }})-i{\varphi }_{S}({\boldsymbol{\xi }}^{\prime} )}\rangle ={{\rm{e}}}^{-{S}_{\nu }({\boldsymbol{\xi }}-{\boldsymbol{\xi }}^{\prime} )/2}.\end{eqnarray} \tag{ C10 }$$

Thus, the scintillation factor in the integrand of 〈F〉 equals

$$\begin{eqnarray}&&\langle {{ \mathcal A }}_{\nu }{{ \mathcal A }}_{\nu }^{{\prime} * }\rangle =\int \displaystyle \frac{{d}^{2}{\boldsymbol{\xi }}\,{d}^{2}{\boldsymbol{\xi }}^{\prime} }{4{\pi }^{2}{r}_{F,S}^{4}}\,\,{{\rm{e}}}^{i{\phi }_{S}({\boldsymbol{x}},{\boldsymbol{\xi }})-i{\phi }_{S}({\boldsymbol{x}}^{\prime} ,{\boldsymbol{\xi }}^{\prime} )-{S}_{\nu }({\boldsymbol{\xi }}-{\boldsymbol{\xi }}^{\prime} )/2},\end{eqnarray} \tag{ C11 }$$

where ${{ \mathcal A }}_{\nu }^{\prime} \equiv {{ \mathcal A }}_{\nu }({\boldsymbol{x}}^{\prime} )$. From Equation (C5), one learns that ${\boldsymbol{\xi }}+{\boldsymbol{\xi }}^{\prime}$ enters linearly the exponent. As a consequence, the integral over this combination produces ${\delta }^{(2)}[{\boldsymbol{\xi }}-{\boldsymbol{\xi }}^{\prime} -\tilde{{\boldsymbol{\xi }}}({\boldsymbol{x}})+\tilde{{\boldsymbol{\xi }}}({\boldsymbol{x}}^{\prime} )]$, and we obtain

$$\begin{eqnarray}&&\langle {{ \mathcal A }}_{\nu }{{ \mathcal A }}_{\nu }^{{\prime} * }\rangle =\exp \left(-\displaystyle \frac{| {\boldsymbol{x}}-{\boldsymbol{x}}^{\prime} {| }^{5/3}}{2{\tilde{r}}_{\mathrm{diff}}^{5/3}}\right),\end{eqnarray} \tag{ C12 }$$

where a projection

$$\begin{eqnarray}&&{\tilde{r}}_{\mathrm{diff}}={d}_{{lo}}{r}_{\mathrm{diff}}/{d}_{{So}}\end{eqnarray} \tag{ C13 }$$

of the diffractive scale onto the lens screen was introduced.

We conclude that the net effect of scintillations is to ruin coherence in 〈F〉, i.e., suppress contributions of radio paths x and ${\boldsymbol{x}}^{\prime}$ if the distance between them exceeds ${\tilde{r}}_{\mathrm{diff}}$. Using Equations (C8) and (C12), we write the mean fluence as

$$\begin{eqnarray}&&\langle F\rangle ={F}_{p}\int \displaystyle \frac{d{\boldsymbol{x}}d{\boldsymbol{x}}^{\prime} }{4{\pi }^{2}{r}_{F,l}^{4}}\,{{\rm{e}}}^{i{{\rm{\Phi }}}_{l}-i{{\rm{\Phi }}}_{l}^{{\prime} }-\tfrac{1}{2}{\left(| {\boldsymbol{x}}-{\boldsymbol{x}}^{\prime} | /{\tilde{r}}_{\mathrm{diff}}\right)}^{5/3}},\end{eqnarray} \tag{ C14 }$$

where F_p ≡ ∣f_p ∣², Φ_l ≡ ϕ_l ( x ) + φ_l ( x ), and ${{\rm{\Phi }}}_{l}^{{\prime} }\equiv {{\rm{\Phi }}}_{l}({\boldsymbol{x}}^{\prime} )$. Note that the lens lurking in the FRB host galaxy is almost insensitive to the scintillations in the Milky Way: ${\tilde{r}}_{\mathrm{diff}}/{r}_{\mathrm{diff}}={d}_{{lo}}/{d}_{{So}}\sim {10}^{6}$.

If the scintillations occur between the source and the lens, the integral (C14) is still valid, but the projected diffractive scale equals ${\tilde{r}}_{\mathrm{diff}}={d}_{{pl}}{r}_{\mathrm{diff}}/{d}_{{pS}}$. Then the scintillations in the FRB host galaxy are geometrically suppressed if the lens is near us.

Now, recall that we consider large-size lenses in the limit of geometric optics Φ_l ≫ 1; see Appendix B. At the same time, we are interested in detectably large contributions, i.e., in the situation when the scintillation factor is not too small, i.e., $| {\boldsymbol{x}}-{\boldsymbol{x}}^{\prime} | /{\tilde{r}}_{\mathrm{diff}}\lesssim {\rm{few}}$. In this case, the integral is dominated by the same stationary points ²⁴ x = x _j and ${\boldsymbol{x}}^{\prime} ={{\boldsymbol{x}}}_{j^{\prime} }$ —the paths of radio rays— as in the case without the scintillations. We obtain

$$\begin{eqnarray}\begin{array}{rcl}\langle F\rangle & = & {F}_{p}\sum _{{jj}^{\prime} }{\left({G}_{j}{G}_{j^{\prime} }\right)}^{1/2}\,{{\rm{e}}}^{i{{\rm{\Phi }}}_{l}({{\boldsymbol{x}}}_{j})-i{{\rm{\Phi }}}_{l}({{\boldsymbol{x}}}_{j^{\prime} })}\\ & & \times \,{{\rm{e}}}^{-\tfrac{1}{2}{\left(| {{\boldsymbol{x}}}_{j}-{{\boldsymbol{x}}}_{j^{\prime} }| /{\tilde{r}}_{\mathrm{diff}}\right)}^{5/3}}\end{array}\end{eqnarray} \tag{ C15 }$$

where x _j solve the lens equation, Equation (B5), and the gain factors G_j are given by the same determinant as before. The effect of the scintillations is represented by the exponent in the second line of Equation (C15).

In the simplified case of two radio paths, one obtains the analog of Equation (B10),

$$\begin{eqnarray}&&\langle F\rangle ={\bar{F}}_{50}\left[1+{A}_{\mathrm{osc}}\,{{\rm{e}}}^{-\tfrac{1}{2}{\left(| {{\boldsymbol{x}}}_{1}-{{\boldsymbol{x}}}_{2}| /{\tilde{r}}_{\mathrm{diff}}\right)}^{5/3}}\cos ({{\rm{\Phi }}}_{1}-{{\rm{\Phi }}}_{2})\right],\end{eqnarray} \tag{ C16 }$$

where the only new factor is a scintillating exponent suppressing interference between the trajectories with $| {{\boldsymbol{x}}}_{1}-{{\boldsymbol{x}}}_{2}| \gtrsim {\tilde{r}}_{\mathrm{diff}}$. Note that it is hard to compare Equation (C16) with the experiment. We already explained that the only ²⁵ practical way to perform the statistical average is to smooth over the frequency. But—alas—the same procedure kills the oscillating interference terms. Thus, one either has to consider sophisticated statistical methods like Kolmogorov–Smirnov–Kuiper test (Press et al. 2007) or analyze more involved spectral correlators.

C.3. Autocorrelation Function

Now, consider the mean product of fluences at nearby frequencies ν and ν₁ = ν + Δν,

$$\begin{eqnarray}&&\langle F(\nu )F({\nu }_{1})\rangle =\langle {f}_{\nu }{f}_{\nu }^{* }{f}_{{\nu }_{1}}{f}_{{\nu }_{1}}^{* }\rangle .\end{eqnarray} \tag{ C17 }$$

Using Equations (C8) and (C9) for the amplitudes, we write it in the form of a Fresnel integral over ( x , ξ ), $({\boldsymbol{x}}^{\prime} ,\,{\boldsymbol{\xi }}^{\prime} )$, ( x ₁, ξ ₁), and $({{\boldsymbol{x}}}_{1}^{{\prime} },\,{{\boldsymbol{\xi }}}_{1}^{{\prime} })$. Here and below, we mark all quantities related to the last three amplitudes ${f}_{\nu }^{* }$, ${f}_{{\nu }_{1}}$, and ${f}_{{\nu }_{1}}^{* }$ by prime, 1, and 1-prime, respectively.

The integrand of the above four-wise Fresnel integral involves a statistical average of four random phases

$$\begin{eqnarray}&&{{\rm{e}}}^{-{S}_{4}/2}\equiv \langle {{\rm{e}}}^{i{\varphi }_{S}-i{\varphi }_{S}^{\prime} +i{\varphi }_{S1}-i{\varphi }_{S1}^{{\prime} }}\rangle ,\end{eqnarray} \tag{ C18 }$$

where ${\varphi }_{S1}^{{\prime} }\equiv {\varphi }_{S}({{\boldsymbol{\xi }}}_{1}^{{\prime} })$, etc. Like before, this correlator can be computed by recalling that φ_S is a Gaussian random variable and exploiting Equation (C2),

$$\begin{eqnarray}\begin{array}{rcl}{S}_{4} & = & \langle {\left({\varphi }_{S}-{\varphi }_{S}^{\prime} +{\varphi }_{S1}-{\varphi }_{S1}^{{\prime} }\right)}^{2}\rangle \\ & = & {S}_{\nu }({\boldsymbol{\xi }}-{\boldsymbol{\xi }}^{\prime} )+\displaystyle \frac{{\nu }^{2}}{{\nu }_{1}^{2}}\,{S}_{\nu }({{\boldsymbol{\xi }}}_{1}-{{\boldsymbol{\xi }}}_{1}^{{\prime} })+\displaystyle \frac{\nu }{{\nu }_{1}}{S}_{\nu }({{\boldsymbol{\xi }}}_{1}^{{\prime} }-{\boldsymbol{\xi }})\\ & & +\,\displaystyle \frac{\nu }{{\nu }_{1}}\left[{S}_{\nu }({{\boldsymbol{\xi }}}_{1}-{\boldsymbol{\xi }}^{\prime} )-{S}_{\nu }({{\boldsymbol{\xi }}}_{1}-{\boldsymbol{\xi }})-{S}_{\nu }({{\boldsymbol{\xi }}}_{1}^{{\prime} }-{\boldsymbol{\xi }}^{\prime} )\right],\end{array}\end{eqnarray} \tag{ C19 }$$

where we explicitly used the fact that φ_S ∝ ν⁻¹; see Equation (B1). Expression (C19) looks complicated. That is why we visualize it in Figure 23 by drawing every ${S}_{\nu }({\boldsymbol{\xi }}-{\boldsymbol{\xi }}^{\prime} )$ with an attractive spring between ξ and ${\boldsymbol{\xi }}^{\prime}$, and ( − S_ν )—with a repulsive solid line.

Zoom In Zoom Out Reset image size

The exponent (C18) cuts off the ξ integrations in regions with large S₄. Generically, this happens if the distances between ξ 's exceed r_diff. However, there are two valleys—large integration regions with small S₄. First, one can keep $({\boldsymbol{\xi }},\,{\boldsymbol{\xi }}^{\prime} )$ and $({{\boldsymbol{\xi }}}_{1},{{\boldsymbol{\xi }}}_{1}^{{\prime} })$ in tight pairs increasing the distance R = ξ ₁ − ξ between them; see Valley I in Figure 23(a). In this case,

$$\begin{eqnarray}&&{S}_{4}{| }_{\mathrm{valley}\,{\rm{I}}}\approx {S}_{\nu }({\boldsymbol{\xi }}-{\boldsymbol{\xi }}^{\prime} )+{S}_{{\nu }_{1}}({{\boldsymbol{\xi }}}_{1}-{{\boldsymbol{\xi }}}_{1}^{{\prime} }),\end{eqnarray} \tag{ C20 }$$

with corrections of order ${({r}_{\mathrm{diff}}/| {\boldsymbol{R}}| )}^{1/3}$. Second, we can move apart the pairs $({\boldsymbol{\xi }},{{\boldsymbol{\xi }}}_{1}^{{\prime} })$ and $({{\boldsymbol{\xi }}}_{1},{\boldsymbol{\xi }}^{\prime} )$ in Figure 23(b). This gives

$$\begin{eqnarray}\begin{array}{rcl}{S}_{4}{| }_{\mathrm{valley}\,\mathrm{II}} & \approx & {\left({\rm{\Delta }}\nu /\nu \right)}^{2}\,{S}_{\nu }({{\boldsymbol{\xi }}}_{1}-{\boldsymbol{\xi }})+{S}_{\nu }({\boldsymbol{\xi }}-{{\boldsymbol{\xi }}}_{1}^{{\prime} })\\ & & +\,{S}_{\nu }({{\boldsymbol{\xi }}}_{1}-{\boldsymbol{\xi }}^{\prime} ),\end{array}\end{eqnarray} \tag{ C21 }$$

with Δν ≡ ν₁ − ν and similar corrections as before. Soon we will see that ²⁶ ${({r}_{\mathrm{diff}}/R)}^{1/3}\sim {({\nu }_{d}/\nu )}^{1/3}\sim 8 \%$, where the numerical estimate is performed for the narrowband scintillations of Section 5.

Now, consider the scintillation factor ${{ \mathcal A }}_{4}\equiv \langle {{ \mathcal A }}_{\nu }{{ \mathcal A }}_{\nu }^{{\prime} * }{{ \mathcal A }}_{{\nu }_{1}}{{ \mathcal A }}_{{\nu }_{1}}^{{\prime} * }\rangle$ in the Fresnel integral for Equation (C17). It involves four ξ –integrals,

$$\begin{eqnarray}&&{{ \mathcal A }}_{4}=\int \displaystyle \frac{{d}^{2}\{{\boldsymbol{\xi }}{\boldsymbol{\xi }}^{\prime} {{\boldsymbol{\xi }}}_{1}{{\boldsymbol{\xi }}}_{1}^{{\prime} }\}}{{\left(2\pi {r}_{F,S}^{2}\right)}^{4}}\,\,{{\rm{e}}}^{i{\phi }_{S}-i{\phi }_{S}^{\prime} +i{\phi }_{S1}-i{\phi }_{S1}^{{\prime} }-{S}_{4}/2};\end{eqnarray} \tag{ C22 }$$

see Equation (C9). Valleys I and II with paired ξ 's give major contributions into ${{ \mathcal A }}_{4}$. From the technical viewpoint, these valleys appear because the four scintillation factors correlate in pairs, and their average product almost equals the sum of $\langle {{ \mathcal A }}_{\nu }{{ \mathcal A }}_{\nu }^{{\prime} * }\rangle \langle {{ \mathcal A }}_{{\nu }_{1}}{{ \mathcal A }}_{{\nu }_{1}}^{{\prime} * }\rangle$ and $\langle {{ \mathcal A }}_{\nu }{{ \mathcal A }}_{{\nu }_{1}}^{{\prime} * }\rangle \langle {{ \mathcal A }}_{{\nu }_{1}}{{ \mathcal A }}_{\nu }^{{\prime} * }\rangle$, with two-point correlators decaying exponentially at far-away ξ 's.

Let us start with the valley I, Equation (C20). In this case, the averaged scintillation phase $\exp (-{S}_{4}/2)$ depends separately on $({\boldsymbol{\xi }},{\boldsymbol{\xi }}^{\prime} )$ and $({{\boldsymbol{\xi }}}_{1},{{\boldsymbol{\xi }}}_{1}^{{\prime} })$. Integrating over these pairs in the same way as in Section C.2, we find

$$\begin{eqnarray}&&{{ \mathcal A }}_{4}{| }_{\mathrm{valley}\,{\rm{I}}}\approx {{\rm{e}}}^{-\tfrac{1}{2}{\left(| {\boldsymbol{x}}-{\boldsymbol{x}}^{\prime} | /{\tilde{r}}_{\mathrm{diff}}\right)}^{5/3}-\displaystyle \frac{1}{2}{\left(| {{\boldsymbol{x}}}_{1}-{{\boldsymbol{x}}}_{1}^{{\prime} }| /{\tilde{r}}_{\mathrm{diff}}\right)}^{5/3}},\end{eqnarray} \tag{ C23 }$$

where ${\tilde{r}}_{\mathrm{diff}}$ is the projected diffraction scale equation, Equation (C13). Due to Equation (C23), the entire correlator in Equation (C17), factorizes along valley I into

$$\begin{eqnarray}&&\langle F(\nu )F({\nu }_{1})\rangle {| }_{\mathrm{valley}\,{\rm{I}}}\approx \langle F(\nu )\rangle \langle F({\nu }_{1})\rangle ;\end{eqnarray} \tag{ C24 }$$

see Equation (C14).

The valley II is a bit different. The exponent S₄ in Equation (C21) weakly depends on the variable R ≡ ξ ₁ − ξ along this valley because the random phases do not completely compensate in the products ${{ \mathcal A }}_{\nu }{{ \mathcal A }}_{{\nu }_{1}}^{{\prime} * }$ and ${{ \mathcal A }}_{{\nu }_{1}}{{ \mathcal A }}_{\nu }^{{\prime} * }$ at ν₁ ≠ ν. Nevertheless, we will see that the R –dependence factorizes: the factor (Δ ν/ν)² ≪ 1 in front of the first term in Equation (C21) makes it insensitive to the small displacements $\delta {\boldsymbol{\xi }}={\boldsymbol{\xi }}-{{\boldsymbol{\xi }}}_{1}^{{\prime} }$ and $\delta {{\boldsymbol{\xi }}}_{1}={{\boldsymbol{\xi }}}_{1}-{\boldsymbol{\xi }}^{\prime}$ transverse to the valley. Indeed, notice that due to the shift symmetry, the center-of-mass variable ${\boldsymbol{\xi }}+{\boldsymbol{\xi }}^{\prime} +{{\boldsymbol{\xi }}}_{1}+{{\boldsymbol{\xi }}}_{1}^{{\prime} }$ enters linearly in the exponent of Equation (C22) and, therefore, gives

$$\begin{eqnarray*}&&{\delta }^{(2)}\left[{\boldsymbol{R}}-\tilde{{\boldsymbol{R}}}+\displaystyle \frac{\nu }{{\rm{\Delta }}\nu }(\delta {\boldsymbol{\xi }}+\delta {{\boldsymbol{\xi }}}_{1}-\delta \tilde{{\boldsymbol{\xi }}}-\delta {\tilde{{\boldsymbol{\xi }}}}_{1})\right],\end{eqnarray*}$$

where we introduced $\tilde{{\boldsymbol{R}}}={\tilde{{\boldsymbol{\xi }}}}_{1}-\tilde{{\boldsymbol{\xi }}}$, $\delta \tilde{{\boldsymbol{\xi }}}=\tilde{{\boldsymbol{\xi }}}-{\tilde{{\boldsymbol{\xi }}}}_{1}^{{\prime} }$, etc., like in the case without the tildes. This fixes the value of R leaving only the integrals over δ ξ and δ ξ ₁ in Equation (C22). Notably, δ ξ , δ ξ ₁ ∼ r_diff, and thus ∣ R ∣/r_diff ∼ ν/Δν ∼ ν/ν_d , like was announced previously. Moreover, we indeed can use ${\boldsymbol{R}}\approx \tilde{{\boldsymbol{R}}}\equiv ({{\boldsymbol{x}}}_{1}-{\boldsymbol{x}}){d}_{{So}}/{d}_{{lo}}$ in the first term of Equation (C22) ignoring corrections because this term is already suppressed by the factor (Δν/ν)².

Once this is done, dependences on δ ξ and δ ξ ₁ factorize, and we arrive to

$$\begin{eqnarray}\begin{array}{rcl}{{ \mathcal A }}_{4}{| }_{\mathrm{valley}\,\mathrm{II}} & = & \exp \left\{-\displaystyle \frac{{\rm{\Delta }}{\nu }^{2}}{2{\nu }^{2}}\,\displaystyle \frac{| {\boldsymbol{x}}-{{\boldsymbol{x}}}_{1}{| }^{5/3}}{{\tilde{r}}_{\mathrm{diff}}^{5/3}}\right\}\\ & & \times \,{{ \mathcal J }}_{{\rm{\Delta }}\nu }\left(\displaystyle \frac{| {\boldsymbol{x}}-{{\boldsymbol{x}}}_{1}^{{\prime} }| }{{\tilde{r}}_{\mathrm{diff}}}\right){{ \mathcal J }}_{{\rm{\Delta }}\nu }^{* }\left(\displaystyle \frac{| {{\boldsymbol{x}}}_{1}-{\boldsymbol{x}}^{\prime} | }{{\tilde{r}}_{\mathrm{diff}}}\right),\end{array}\end{eqnarray} \tag{ C25 }$$

where the functions ${{ \mathcal J }}_{{\rm{\Delta }}\nu }(\rho )$ and ${{ \mathcal J }}_{{\rm{\Delta }}\nu }^{* }({\rho }_{1})$ include integrals over δ ξ and δ ξ ₁. The latter can be written in the simplified form ²⁷

$$\begin{eqnarray}\begin{array}{rcl}{{ \mathcal J }}_{{\rm{\Delta }}\nu }(\rho ) & = & -i\,{{\rm{e}}}^{i{\rho }^{2}/w}\\ & & \times \,{\displaystyle \int }_{0}^{\infty }d\zeta \,{{\rm{e}}}^{i\zeta -\tfrac{1}{2}{\left(w\zeta \right)}^{5/6}}{J}_{0}\left(2\rho \sqrt{\zeta /w}\right).\end{array}\end{eqnarray} \tag{ C26 }$$

We introduced the rescaled frequency lag w ≡ 2Δν/ν_d , decorrelation bandwidth ${\nu }_{d}\equiv \nu {({r}_{\mathrm{diff}}/{r}_{F,S})}^{2}$, the argument ρ of ${{ \mathcal J }}_{{\rm{\Delta }}\nu }$ measuring distances between x 's in units of ${\tilde{r}}_{\mathrm{diff}}$, and the Bessel function J₀. Note that the ${ \mathcal J }$–factors in Equation (C25) do the same job as the exponent in Equation (C23): they force $| {\boldsymbol{x}}-{{\boldsymbol{x}}}_{1}^{{\prime} }|$ and $| {{\boldsymbol{x}}}_{1}-{\boldsymbol{x}}^{\prime} |$ to be smaller than ${\tilde{r}}_{\mathrm{diff}}$. Indeed, ${{ \mathcal J }}_{{\rm{\Delta }}\nu }(\rho )\to \exp \{-\tfrac{1}{2}{\rho }^{5/3}\}$ as ρ → + ∞ . At ρ ≲ w this function remains unsuppressed and essentially depends on the frequency lag w ∝ Δν.

We finally consider the integrals over the lensed radio paths x , ${\boldsymbol{x}}^{\prime}$, x ₁, and ${{\boldsymbol{x}}}_{1}^{{\prime} }$ in the Fresnel representation of the correlator equation, Equation (C17); see Equation (C8). The scintillations supply the factor in the integrand,

$$\begin{eqnarray}&&{{ \mathcal A }}_{4}({\boldsymbol{x}},{\boldsymbol{x}}^{\prime} ,{{\boldsymbol{x}}}_{1},{{\boldsymbol{x}}}_{1}^{{\prime} })={{ \mathcal A }}_{4}{| }_{\mathrm{valley}\,{\rm{I}}}+{{ \mathcal A }}_{4}{| }_{\mathrm{valley}\,\mathrm{II}},\end{eqnarray} \tag{ C27 }$$

given by the sum of Equations (C23) and (C25).

Like in the previous sections, we consider only large-size lenses in the limit of geometric optics. In this case, the phase shift is large, ${{\rm{\Phi }}}_{l}\sim | {\boldsymbol{x}}-\tilde{{\boldsymbol{x}}}{| }^{2}/{r}_{F,l}^{2}\gg 1$, and the Fresnel integral is dominated by the set of distinguished rays { x _j } satisfying the lens equation (B5). Notably, one can roughly estimate the typical lens phase via Equation (B9) obtaining Φ_l ∼ 2π ν/T_ν , where T_ν is a period of spectral oscillations. Thus, the geometric optics is, indeed, valid in the case T_ν ≪ ν considered in the main text. The saddle-point integration gives

$$\begin{eqnarray}&&\begin{array}{l}\langle F(\nu )F({\nu }_{1})\rangle ={F}_{p}^{2}\sum _{{jj}^{\prime} {j}_{1}{j}_{1}^{{\prime} }}{\left({G}_{j}{G}_{j^{\prime} }{G}_{{j}_{1}}{G}_{{j}_{1}^{{\prime} }}\right)}^{1/2}\\ \ \ \times {{\rm{e}}}^{i{{\rm{\Phi }}}_{l}({{\boldsymbol{x}}}_{j})-i{{\rm{\Phi }}}_{l}({{\boldsymbol{x}}}_{j^{\prime} })+i{{\rm{\Phi }}}_{l}({{\boldsymbol{x}}}_{{j}_{1}})-i{{\rm{\Phi }}}_{l}({{\boldsymbol{x}}}_{{j}_{1}^{{\prime} }})}\cdot {{ \mathcal A }}_{4},\end{array}\end{eqnarray} \tag{ C28 }$$

where the sum runs four-wise over the radio paths x _j , the scintillation factor ${{ \mathcal A }}_{4}$ depends on the four of them, and we introduced the lens phases Φ_l ( x _j ) and the gain factors G_j ; see Section B. Note that ${{ \mathcal A }}_{4}$ selectively suppresses the interference terms with far-away x 's.

Despite the complex form, Equation (C28) is easy to use. Indeed, it involves the same radio rays as in the previous section, whereas the suppression factor ${{ \mathcal A }}_{4}$ is explicitly given by Equations (C27), (C23), and (C25). Now, we continue with examples.

C.4. ACF for Scintillations Only

Suppose first that the lens is absent, φ_l = 0. In this case, there exists only one radio ray ${\boldsymbol{x}}=\tilde{{\boldsymbol{x}}}$ corresponding to straight propagation between the source and the scintillation screen. We obtain Φ_l = 0 and G = 1. Expression (C28) then reduces to the scintillation factors (C23) and (C25),

$$\begin{eqnarray}&&\langle F(\nu )F(\nu +{\rm{\Delta }}\nu )\rangle ={F}_{p}^{2}+{F}_{p}^{2}{\left|h\left(2{\rm{\Delta }}\nu /{\nu }_{d}\right)\right|}^{2},\end{eqnarray} \tag{ C29 }$$

where the decorrelation bandwidth ν_d (ν) is given by Equation (12), and we introduced the hat-like function

$$\begin{eqnarray}&&h(w)\equiv {{ \mathcal J }}_{{\rm{\Delta }}\nu }(0)=-i{\int }_{0}^{+\infty }d\zeta \,{{\rm{e}}}^{i\zeta -\tfrac{1}{2}{\left(w\zeta \right)}^{5/6}}\end{eqnarray} \tag{ C30 }$$

shown in Figure 9 (solid line). One can explicitly check that in the asymptotic regions w → 0, + ∞ this function coincides with the approximation (11), used in Section 5. The latter correctly represents h(w) even at finite w; see the dashed line in Figure 9.

Importantly, ∣h(w)∣ falls off from 1 at w = 0 to zero at large frequency lags. As a consequence, the correlator equation, Equation (C29), monotonically decreases from $2{F}_{p}^{2}$ at Δν = 0 to F_p ² at large Δν reflecting the fact that F(ν) and F(ν₁) are not correlated at far-away frequencies. Besides, ∣h∣² = 1/2 at Δν ≈ 0.96 ν_d . Thus, up to a factor 0.96, our definition of ν_d coincides with the standard one.

Using ${F}_{p}\approx {\bar{F}}_{50}(\nu )$ and Equation (9), we obtain the ACF Equation (10) from the main text. Note that the argument w = 2Δν/ν_d of h sharply depends on the frequency: ν_d ∝ ν^22/5; see Equation (C3). When comparing with the experiment, one has to take this dependence into account, like we do in Section 5.

C.5. ACF for Scintillations and Lensing

Now, suppose the lens produces two coherent images of the source x ₁ and x ₂ with phases Φ₁, Φ₂ and gain factors G₁, G₂. After passing the lens, these rays go through the scintillating plasma. One can imagine two opposite situations.

First, the rays may cross the plasma at close distances $| {\tilde{{\boldsymbol{\xi }}}}_{2}-{\tilde{{\boldsymbol{\xi }}}}_{1}| \ll {r}_{\mathrm{diff}}$ i.e., $| {{\boldsymbol{x}}}_{2}-{{\boldsymbol{x}}}_{1}| \ll {\tilde{r}}_{\mathrm{diff}}$. The scintillations do not decohere these paths, and ${{ \mathcal A }}_{4}$ can be evaluated at zero transverse shifts in Equations (C23) and (C25). Factorizing ${{ \mathcal A }}_{4}=1+| h{| }^{2}$, one obtains Equation (19) from the main text—a generalization of Equation (B12) to the model with scintillations. Note that all terms oscillating with frequency disappear from Equation (19) due to the overall integral over ν.

It is worth recalling that the interpath distance is related to the frequency period T_ν of interference oscillations by Equation (B9): $| {{\boldsymbol{x}}}_{2}-{{\boldsymbol{x}}}_{1}{| }^{2}\sim 2\pi \nu {r}_{F,l}^{2}/{T}_{\nu }$. With this formula, one can rewrite the coherent-paths condition $| {{\boldsymbol{x}}}_{2}-{{\boldsymbol{x}}}_{1}| \ll {\tilde{r}}_{\mathrm{diff}}$ as

$$\begin{eqnarray}&&\displaystyle \frac{{r}_{F,l}^{2}}{{r}_{F,S}^{2}}\ll \displaystyle \frac{{\nu }_{d}\,{T}_{\nu }}{2\pi {\nu }^{2}}\,\displaystyle \frac{{d}_{{lo}}^{2}}{{d}_{{So}}^{2}},\end{eqnarray} \tag{ C31 }$$

where we used Equations (C13) and (12) and assumed that scintillations occur between the lens and the observer. The experimental values of Sections 5 and 6 give ν_d /ν ∼ 10⁻³ and T_ν /ν ∼ 10⁻². The inequality (C31) is then satisfied, and the scintillations are irrelevant, say, if they occur in our galaxy, and the lens hides in the FRB host galaxy: r_F,l ∼ r_F,S and d_lo /d_So ∼ 10⁶. The other possibilities include Milky Way scintillations and the lens in the intergalactic space, or scintillations in the FRB host galaxy and the lens outside of it. ²⁸

Second, at $| {{\boldsymbol{x}}}_{2}-{{\boldsymbol{x}}}_{1}| \gg {\tilde{r}}_{\mathrm{diff}}$ the decoherence of radio rays is relevant and scintillations kill the majority of the interference terms. Expression (C28) takes the form

$$\begin{eqnarray}&&\begin{array}{c} \langle F(\nu )F({\nu }_{1}) \rangle ={\bar{F}}_{50}^{2}[1+| h{| }^{2}+\frac{{A}_{{\rm{o}}{\rm{s}}{\rm{c}}}^{2}}{2}\,| h{| }^{2}\\ \,\times \,(-1+\cos (2\pi {\rm{\Delta }}\nu /{T}_{\nu })\\ \,\,\,\times \,\exp \{-\frac{{\rm{\Delta }}{\nu }^{2}}{2{\nu }^{2}}\frac{| {{\boldsymbol{x}}}_{2}-{{\boldsymbol{x}}}_{1}{| }^{5/3}}{{\tilde{r}}_{{\rm{d}}{\rm{i}}{\rm{f}}{\rm{f}}}^{5/3}}\})],\end{array}\end{eqnarray} \tag{ C32 }$$

where we introduced the same ${\bar{F}}_{50}$ and A_osc as before, and h ≡ h(2Δν/ν_d ). Notably, the oscillating term persists in this case due to imperfect cancellation between the phases of f_ν at different frequencies. However, the amplitude of oscillations decreases with Δν becoming invisibly small at Δν ≫ ν_d or ${\rm{\Delta }}\nu \gg \nu \,{\tilde{r}}_{\mathrm{diff}}^{5/6}/| {{\boldsymbol{x}}}_{2}-{{\boldsymbol{x}}}_{1}{| }^{5/6}$. This means that Equation (C32) is relevant only at T_ν ≪ ν_d , not in the case of Section 5.