Radiation Modes in FRB 20220912A Microshots and a Crab PSR nanoshot

A microshot from FRB 20220912A \citep{H23} satisfies the uncertainty relation $\Delta \omega \Delta t \ge 1$ by a factor of only $\lessapprox 3$. A Crab pulsar nanoshot \citep{HE07} exceeds this bound by a similar factor. The number of orthogonal plasma modes contributing to the coherent radiation is also $\approx \Delta \omega \Delta t$, placing constraints on their excitation and growth.


INTRODUCTION
The recent discovery in Westerbork observations of FRB 20220912A (Hewitt et al. 2023) of microshots with temporal width ≤ 31.25 ns in a spectral channel of width 16 MHz leads to the question of how close microshots can approach the uncertainty bound ∆ω∆t = 2π∆ν∆t ≳ 1.The instrumental temporal and spectral resolutions are only upper bounds on the actual pulse duration and bandwidth, so only an upper bound ∆ω∆t ≲ 3 can be set.
This uncertainty bound is inescapable mathematics, although its quantitative value depends on the pulse shape and the definitions of ∆ω and ∆t.For a Gaussian pulse, if ∆ω and ∆t are defined as full widths at 1/e of maximum, then ∆ω∆t = 4 π . (1) The condition is satisfied for a Gaussian pulse if ∆ω and ∆t are defined as the full widths at exp − π/4 ≈ 0.412 of maximum.
If ∆ω∆t > 1 we consider the natural generalizations of the Gaussian to larger widths, the Gauss-Hermite functions, the eigenstates of the one-dimensional harmonic oscillator with Hamiltonian H = (kx 2 /2 + p 2 /2m): where Hn(x) is the n-th Hermite polynomial.Like the Gaussian (G0(x)), the Gauss-Hermite functions are their own Fourier transforms (Cincotti, Gori & Santasiero 1992;Horikis & McCallum 2006), as they must be because the harmonic ⋆ E-mail katz@wuphys.wustl.edu oscillator Hamiltonian is symmetric under the interchange kx 2 ←→ p 2 /m.Here we replace √ kx by t and p/ √ m by ω.It is readily seen, either from the properties of the Hermite polynomials or from the fact that the n-th excited state of the harmonic oscillator has energy En = (n + 1/2)ℏω and classical width 2En/k, that for these more complex pulse frequency and temporal profiles ∆ω∆t ∼ n. (4) Eq. 4 gives the approximate number of orthogonal eigenmodes whose superposition makes a pulse with ∆ω∆t > 1.

THE OBSERVATIONS
The lower panel of Fig. 5 of Hewitt et al. (2023) shows microshots in the 1304 MHz band at approximately 11 µs (with respect to the arbitrary zero time of the plot) whose temporal widths are no more than a single 31.25 ns resolution element and that are almost completely confined to a single 16 MHz wide spectral band.Their intensity in the 1288 MHz band is at least an order of magnitude less than in the 1304 MHz band and their intensity in the 1320 MHz band is a few times less than at 1304 MHz, implying an intrinsic ∆ν less than the band spacing of 16 MHz (the quantitative value depending on the assumed spectral shape).Conservatively taking ∆ν = 16 MHz implies ∆ω∆t ⪅ 3.By Eq. 4, no more than about three modes of the electromagnetic field, and of the plasma waves that coherently radiated it, contributed significantly to the observed microshot.A similar result holds for a ≤ 0.2 ns nanoshot of the Crab pulsar (Hankins & Eilek 2007).

PLASMA PHYSICS
Coherent emission, necessary to explain the extraordinary brightness temperatures of FRB (Katz 2014) and of pulsars, requires "bunching" of the radiating charges that must result from the exponential growth of plasma waves.The many efoldings of exponential growth raise the the amplitudes of the few fastest growing modes far above those of other modes.An observed brightness temperature of 10 36 K requires N ⪆ 50 efoldings if the initial brightness temperature was mec 2 /kB (an arbitrary but plausible initial thermal value).

Linear Growth
Assume a plasma instability grows exponentially with growth rate where ζ is a parameter of the plasma wave (perhaps its wavevector) and ∆ζ the width of γ(ζ).This is a general form that assumes nothing about the specifics of the plasma instability, but that does require that the plasma modes interact only weakly; the governing equations can be linearized so that eigenmodes grow exponentially and essentially independently of each other.From the fact that after N e-folds the width of the microshots of FRB 20220912A the fact that ∆ν/ν ≈ 0.01, and the plausible assumption that dζ/dν ∼ ζ/ν, we estimate This is an approximate constraint that can be placed on any linearized theory of the plasma instability.

Coupled Waves
In an alternative model, the plasma waves are strongly coupled as they approach saturation, resembling a soliton (Zabusky & Kruskal 1965), rather than being described as a superposition of several weakly interacting eigenmodes.This would suggest that ∆ω∆t ≈ 1 because the radiation is produced by a single nonlinear wave, rather than by the sum of multiple weakly interacting waves; n ≈ 1 would be required.
The observations of FRB microshots (Hewitt et al. 2023) and Crab pulsar nanoshots (Hankins & Eilek 2007) cited here are consistent with this hypothesis, which is supported by the fact that both of these extreme phenomena are described by similar bounds on ∆ω∆t.

BASE-BAND SIGNALS
If base-band voltages are measured in a microshot or nanoshot, they will show ∼ ω∆t ∼ ∆ω∆t(ω/∆ω) ∼ n(ω/∆ω) cycles of oscillation.This is a mathematical consequence of ∆ω∆t ∼ n, and does not depend on a physical model.However, such a direct measurement might provide other illuminating information about the radiation process and the plasma physics that drives it.The dependence of electric field on time may permit distinguishing the weakly and strongly coupled models.
As an example of what might be seen in base-band signals, Fig. 1 shows the time dependence of the electric field Even if the signal-to-noise ratio is insufficient to resolve the base-band oscillations, their envelope may reveal the number of contributing modes n = ∆ω∆t.The signal-to-noise ratio of the envelope is greater than that of the base-band signal by a factor O( ω0/n∆ω), which is O(16/ √ n) for the parameters shown; use of a matched filter would eliminate the factor of 1/ √ n.

DISCUSSION
In a weakly coupled wave model, because N is large a narrowly peaked distribution of amplitudes of modes with ∆ν ≪ ν does not require a narrowly peaked growth rate.Eq. 6 may be inverted to find N from the spectral width of a FRB microshot; spectral narrowness demands many e-folds of exponential growth of the underlying plasma waves.The secondary intensity peaks (in the envelope of the ≈ 1308 MHz oscillation in Fig. 1) are required by values of n = Radiation Modes in FRB 20220912A Microshots 3 ∆ω∆t ≳ 2. Hence the number n of contributing modes may be inferred from the time dependence of the intensity if it is sufficiently resolved, as well as from direct measurement of ∆ω and∆t of the central peak.The temporal resolutions of the microshots of Hewitt et al. (2023) and the nanoshot of Hankins & Eilek (2007) were insufficient to determine n, but future observations with better temporal resolution should be able to show this structure and determine the number of contributing modes.

Figure 1 .
Figure 1.Baseband signals for ∆ω∆t = n = 1, 2, 3 and ∆t = 31.25 ns.The wider bandwidth pulses have narrower main peaks (a mathematical necessity).The secondary peaks may be observable as frequency-averaged functions of intensity vs. time even if noise prevents measurement of the baseband signal.Fine structure is the result of beats between the signal and the discrete resolution of the graphics screen, and is not physical.