J. Astrophys. Astr. (2018) 39:74 © Indian Academy of Sciences https://doi.org/10.1007/s12036-018-9565-y

Interstellar communication: Short pulse duration limits of optical SETI

MICHAEL HIPPKE

Sonneberg Observatory, Sternwartestr. 32, 96515 Sonneberg, Germany. E-mail: [email protected]

MS received 31 July 2018; accepted 19 October 2018; published online 30 November 2018

Abstract. Previous and ongoing searches for extraterrestrial optical and infrared nanosecond laser pulses and narrow line-width continuous emissions have so far returned null results. At the commonly used observation cadence of ∼ 10−9 s, sky-integrated starlight is a relevant noise source for large field-of-view surveys. This can be reduced with narrow bandwidth filters, multipixel detectors, or a shorter observation cadence. We examine the limits of short pulses set by the uncertainty principle, interstellar scattering, atmospheric scintillation, refraction, dispersion and receiver technology. We find that optimal laser pulses are time-bandwidth limited −12 Gaussians with a duration of t ≈ 10 s at a wavelength λ0 ≈ 1 µm, and a spectral width of λ ≈ 1.5nm. Shorter pulses are too strongly affected through Earth’s atmosphere. Given certain technological advances, survey speed can be increased by three orders of magnitude when moving from ns to ps pulses. Faster (and/or parallel) signal processing would allow for an all-sky-at-once survey of lasers targeted at Earth.

Keywords. Optical SETI—lasers—interstellar communication—Techniques: photometric—Instrumentation: photometers.

1. Introduction detectors). In order to reduce the noise levels, one could “know” (guess) the correct wavelength and use a narrow Optical SETI typically assumes laser signals to be either filter in addition to short time cadence. This scenario is pulsed or narrowband. Short pulses are usually assumed explored in paper 10 of this series. −9 to be tmin ≤ 10 s(Howard & Horowitz 2001)or Another option, which is explored in this paper, is to 10−14 ...10−12 s(Maire et al. 2016). Narrowband con- increase the time resolution. For one hemisphere, the tinuous emissions are generally taken as λ<10−2 nm total night sky radiance over a bandwidth of 1,000 nm (e.g., Tellis & Marcy 2017)orevenλ<10−12 nm is ≈ 1014 photons per second (section 7.1), or ≈105 (< Hz, Kingsley 1993b). photons per ns. This is the background flux which The justification for the feasibility of optical SETI competes with a laser signal in an all-sky survey, is that short pulses or narrowband emissions outshine and would overpower the typically assumed plausi- their blended host as the main noise source (e.g., ble OSETI signals. To make the background noise Howard et al. 2004). This is plausible, because a small per cadence, one could reduce the sky cov- laser focused and received through 10m telescopes erage, bandwidth, or a combination of both. At an would outshine a G2V host star for laser pulses observation cadence of order picosecond, the all-sky of kJ energy at ns cadence (or in a narrow spec- noise flux would reduce to ≈100 photons per cadence. tral channel), independently of distance, and there As typical wide-field telescopes have fields-of-view of exist MJ (PW) lasers on Earth (Holzrichter & Manes 5◦×5◦ (∼1/1,000 of the sky, section 6.2), noise lev- 2017). els reduce to 0.1 photons per ps cadence, which is Such examples offer a credible use-case for negligible. observations of individual one by one. However, a In this study, we determined the physical short end survey of the entire sky could be much faster by observ- time limit of laser pulses, set by the time-bandwidth ing a large field of view. Unfortunately, noise levels limit, barycentric corrections, interstellar scattering, increase with the number of stars observed and quickly atmospheric refraction, dispersion, and broadening, and require implausibly high signal power (or multipixel receiver technology. 74 Page 2 of 17 J. Astrophys. Astr. (2018) 39:74

2. Physical limits 103

2 2.1 Time-bandwidth limit 10 101 The Heisenberg uncertainty principle (time-bandwidth limit) prohibits an arbitrary combination of infinitely 100 short and narrow pulses: “the more precisely the posi- −1 tion is determined, the less precisely the momentum 10

Bandwidth (nm) Air Heisenberg is known in this instant, and vice versa” (Heisen- 10−2 berg 1927), so that E t ≥ h¯ /2whereE is −3 the standard deviation of the particle energy, t is 10 10−5 10−3 10−1 101 the time it takes the expectation value to change by Pulse width (ns) one standard deviation, and h¯ is the reduced Planck constant. A photon pulse with a temporal width t Figure 1. Parameter space of a Gaussian pulse with a min- can therefore not be monochromatic, but has a spec- imum time-bandwidth product for λ = µm (black line). trum. Both are related through a Fourier transform, Atmospheric effects (section 4) are shown in blue. Small and it can be shown that (Griffiths 2004; Rullière star symbols represent past and current OSETI experiments (section 6.1), which are ∼ 3 orders of magnitude from 2005) the time-bandwidth limit. The red star symbol indicates the OSETI sweet spot identified in this paper. λ2 t ≥ K 0 min λ (1) c 2.2 Minimum laser linewidth λ λ where 0 is the central wavelength, is the width The minimum laser linewidth is also a function of of the spectrum (FWHM), c is the speed of light power, which was known even before the first laser was ∼ . and K 0 441 for a Gaussian pulse shape. For experimentally demonstrated. The fundamental (quan- λ = example, a near-infrared (NIR) laser pulse ( 0 tum) limit for the linewidth of a laser is (Schawlow & µ λ = 1 m) with a 5% bandwidth ( 50 nm) has a Townes 1958) minimum width of t ∼29 × 10−15 s, or ∼29 fs min π ( )2 (Table 1). 2 hf fc flaser = , (2) As an example, Tellis and Marcy (2017) state that Pout “Laser lines as narrow as 1 Hz are already in use where flaser is the half width at half-maximum on Earth”. While this is a true statement, a Hz band- linewidth of the laser, fc is the half width of the reso- λ = µ width laser pulse at 1 m has a pulse duration nances of the laser resonator and Pout is the laser output  0.4 s, so that a choice between “broadband” short power. For OSETI, power levels will be large ( W) so −12 pulses and “continuous” wave narrowband emission is that flaser  Hz (i.e.,  10 nm), and Schawlow- required. The allowed parameter space for pulses is Townes produce no relevant limit. showninFig.1. There is no obvious distinction where one regimes begins and the other ends. To comply with 2.3 Laser pulse shape the literature, we will refer to “pulses” for signals with  < µ tmin s. Laser generation can produce pulses of various shapes in the time and frequency domains (Smith & Landon Table 1. Examples for time-bandwidth limited Gaussian 1970; Hao et al. 2013). Typically, the the temporal pulses at λ0 = 1 µm. intensity of short laser pulses is approximated with a Gaussian   λ (nm) λ/λ t Comment   0 min t 2 P(t) = PP exp −4ln2 (3) 50 0.05 29fs Shortest plausible pulse τ 1.5 2 × 10−3 1ps Atmospheric limit τ 0.01 1 × 10−5 0.15ns Normal spectroscopy where is the pulse duration and PP is the peak power. A 10−6 1 × 10−9 1.5 µs Extreme spectroscopy minor discrepancy with reality is that the tails of a Gaus- 3 × 10−12 3 × 10−15 0.4s Hz bandwidth sian function never actually reach zero (but the laser power does). Due to laser production technicalities, J. Astrophys. Astr. (2018) 39:74 Page 3 of 17 74

1.0 1.0

0.8 0.8

0.6 0.6

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0.2 0.2 Normalized pulse height Normalized power spectrum 0.0 0.0 −4 −2 0 2 4 −4 −2 0 2 4 Time from t0 (arbitrary units) Frequency from f0 (arbitrary units) Figure 2. Left: Comparison of pulse shapes with approximately the same energy integrals. Gaussian (black), sech2 (red), Lorentzian (blue). Right: Spectrum of a Gaussian (black) and a sech2 pulse for 0.5 <α<3 (red). pulses from mode-locked lasers often have a temporal 3. Interstellar effects shape better described with a squared hyperbolic secant (sech2) function (Lazaridis et al. 1995) Interstellar extinction is the absorption and scattering of photons by dust and gas. Absorption, mainly as a func- ( ) = PP . tion of wavelength and distance, is discussed in paper P t 2 t (4) cosh τ 10 of this series. Relevant for the short pulse case is interstellar plasma scattering, which produces a tempo- For pulses of the same amplitude and energy, a ral delay and pulse broadening in time and frequency. Gaussian has slightly weaker wings than the sech2- shaped pulse (Fig. 2, left). For same energy same 3.1 Dispersion full-width at half-mean (FWHM) pulses, the Gaussian’s peak power is higher, ∼ 0.94 the pulse energy divided Dispersive time delays from interstellar plasma are by the FWHM pulse duration compared to ≈ 0.88 well-known from pulsar studies and follow the relation for the sech2 pulse. The time-bandwidth product of a (Taylor & Cordes 1993) Gaussian is K ∼ 0.441 and K ∼ 0.315 for the sech2 pulse, which allows for slightly shorter (FWHM) time- −2 td ∼ 4.15 DM f (ms) (6) bandwidth limited sech2 pulses. The intensity spectrum of a Gaussian pulse is also a valid for pulses λ>µm(Shostak 2011), where f is the 2 Gaussian. For sech pulses, the spectrum depends on its frequency in GHz and DM is the dispersion measure in α − phase-amplitude-coupling factor , and the normalized units of pccm 3. The dispersion law corresponds to power spectrum is (Lazaridis et al. 1995) cold plasma and is valid where the photon frequency is     π π much larger than the plasma frequency. (ωτ + α) (ωτ − α) −3 sech 2 sech 2 A typical dispersion measure is DM=1pccm over E(ω) = . (5) − sech2(πα/2) 100pc in the solar neighborhood, DM = 30 pc cm 3 over kpc distances and DM = 100 pc cm−3 over kpc A variety of shapes for 0.5 <α<3isshownin towards the galactic center (Yao et al. 2017). For GHz −3 Fig. 2 (right), ranging from an approximately Gaus- frequencies and DM = 30 pc cm , the delay is td ≈ sian to a top-hat function. For larger values of α,the 0.1 s and decreases to ≈ 1psatλ = µm. spectrum approximates a rectangle, which might be Dispersive delays can approximately be removed by favorable given finite sized spectral windows of atmo- shifting the time-series of many narrow frequency chan- spheric transparency and low noise (see article titled nels with an estimated amount of DM for the source. “Interstellar communication: The colors of optical This correction is never perfect because of the finite SETI” by M. Hippke in this issue and Fig. 4). number of individual channels and time sampling, and A variety of other pulse shapes exists for other laser the imperfect knowledge of the true (time variable) DM technologies, such as the Lorentzian shape (Fig. 2,left). (Alder 2012). In a typical OSETI observation, where The observation of an alien laser pulse shape will tell the receiver is monochromatic (e.g., a photomultiplier, us a lot about the technology used for its production. PMT), de-dispersion is not possible. Even in case a fast The following calculations in this paper will assume spectrograph would be used, the correction can only be the most common Gaussian form. applied post-detection, as the DM is not known a 74 Page 4 of 17 J. Astrophys. Astr. (2018) 39:74 priori. Overall, the time delay is of order ps or less, turbulence model with a frequency scaling of f −4. and changes little (< 1%,i.e. 10 fs) over time. Pulsar scatter measurements closely follow this scaling over a large dispersion range, 2 < DM < 1,000 (Krish- 3.2 Scatter broadening nakumar et al. 2015). However, the majority of sources show a flatter index over a frequency range between − Dust and protons in the interstellar medium scatter and 0.1 ...1 GHz, with some down to f 3 (Krishnakumar absorb (then re-radiate) photons, reducing the pulse et al. 2017; Xu & Zhang 2017). amplitude and leaving an exponential scattering tail Using this scaling, the strongest scattering at GHz (Cordes & Lazio 2002). Such a tail could be modeled frequencies in our galaxy of ≈ 1s for DM = − − − using an exponentially modified normal distribution 1000 pc cm 3 scales to 10 25 satλ = µm (or 10 17 sat − (Grushka 1972) f 3). This indicates that scattering tails at optical and IR   wavelengths are irrelevant, because the time-bandwidth λ λ μ + λσ 2 − (2μ+λσ 2−2x) x f (x) = e 2 erfc √ (7) limit is stronger in all realistic cases. 2 2σ This is in agreement with radio observations of the where erfc is the complementary error function Crab pulsar which show pulses with high power (MJ) at f = 8.6 GHz which are unresolved at 2ns (Hankins ( ) = − ( ) erfc x 1 erf x et al. 2003) time resolution and 0.4ns time resolution.  ∞ 2 (8) = = √2 −t . Interstellar scattering models predict 0.05ns at f e dt . π x 8 6 GHz for the Crab Nebula’s DM of 56.7 (Hankins & Eilek 2007) at a distance of 1.9 ± 0.1 kpc (Trimble This is illustrated in Fig. 3. On the one hand, it 1973). has been argued that the effect “can be quite severe” At optical and IR wavelengths, practical measure- et al et al (Howard . 2000b, 2004; Horowitz . 2001). On ments only exist into the ms regime. The scattering tail the other hand, conflicting statements exist, “temporal is unknown in practice. Limits from the Crab pulsar dispersion (...) is entirely negligible at optical wave- show no detectable scattering tail at UV and optical et al lengths.” (Howard . 2000a). Both arguments are wavelengths for an optical millisecond pulse width and based on calculations in Cordes (2002, AppendixM.3) E(B − V ) = 0.52 (Sollerman et al. 2000; Hinton et al. e−τ which state that a fraction of the pulse, arrives 2006; Karpov et al. 2007; Lucarelli et al. 2008). − e−τ τ unscattered, while 1 is broadened in time, with These results indicate that the influence of the ISM is as the total optical depth. Therefore, scatter broadening irrelevant for temporal pulse broadening at optical and is small for optical wavelengths out to 100pc, and in IR wavelengths. the IR out to kpc. Frequency dependent scatter broadening is well known from pulsars, following a Kolmogorov (1941) 3.3 Spectral broadening

Spectral broadening by the ISM and interplanetary medium is along most lines of sight well approximated − / 1.0  ∼ . 6 5 as fbroad 0 1Hz fGHz (Cordes & Lazio 1991; Siemion et al. 2013, 2015). For λ = µm(f ∼ 1014 Hz) 0.8 −8 we get fbroad  10 Hz, which is negligible. 0.6

0.4 4. Atmospheric effects 0.2 Normalized pulse height Dispersion and turbulence absorb and distort pulses 0.0 traveling through Earth’s atmosphere. Pulses are −2 0 2 4 6 delayed compared to the speed of light in vacuum, Time (ps) broadened in length, and their temporal and spectral Figure 3. An original Gaussian pulse at FWHM=1ps shapes are distorted. In addition, their amplitudes vary (blue) is delayed and broadened (red). Pulse broadening is due to scintillation (“twinkling”), as discussed in paper < fs in practice and exaggerated by > 100× here for clarity. 10 of this series and shown in Fig. 4. J. Astrophys. Astr. (2018) 39:74 Page 5 of 17 74

1.0

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0.6

0.5

0.01 0.4

0.3 Fractional bandwidth (Δ 0.2

0.1

0.001 0.0 0 1 2 3 4 5 Wavelength (μm)

Figure 4. Atmospheric transparency as a function of wavelength and bandwidth, using a Gaussian convolution. Data from Noll et al. (2012) and Jones et al. (2013) for VLT Cerro Paranal at an altitude of 2640m at median observing conditions with a precipitable water vapor of 2.5 mm at zenith angle.

The effects mainly depend on wavelength, initial smaller variance with a factor of a few between optical pulse duration, and turbulence characteristics. The total and NIR wavelengths (Fig. 5). delay and pulse broadening effects are of order ps under The median coherence time is typically 10 ...50 ms most conditions. (Kenyon et al. 2006), much longer than the relevant time scales of expected OSETI pulses (ns). In longer expo- sures, the scintillation noise will be reduced by temporal 4.1 Atmospheric scintillation averaging (for details, see Osborn et al. 2015). A com- mon approximation for the time averaging is given by Scintillation is a variation of the optical refractive index, Young (1967): caused by anomalous refraction through small-scale fluctuations in air density due to temperature gradients. σ 2 = −5 −4/3 −1( )−3 (− / ) It enlarges the point spread function of the telescope Y 10 D t cos z exp 2h H (10) (Coulman et al. 1995) if not corrected for with adap- tive optics (Hardy 1998). The strength of scintillation is measured as the variance of the beam amplitude (the where D is the telescope aperture, z the zenith distance, Rytov variance, Andrews et al. 1988) t the observation cadence and h the altitude. For a 1s cadence of a meter-sized telescope under typical condi- 2 2 7/6 11/6 σ 2 ∼ −5 σ = 1.23C k H (9) tions, Y 10 . In practice, there is a cutoff point around the where k = 2π/λ is the wave number, H is the scale coherence time. The turning point is set by the amount height of the atmospheric turbulence (generally taken of spatial averaging of the scintillation (the telescope as H ∼ 8,000 m, Osborn et al. 2015), and C2 is the aperture) and the wind velocity. structure constant for refractive-index fluctuations as a For the 1m Jacobus Kapteyn Telescope on La Palma measure of the optical turbulence strength. Observed under typical conditions, the knee is observed around values are C2 = 1.7 × 10−14 m−2/3 at ground level and t ∼ 50 ms, below which scintillation changes to the C2 = 2 × 10−18 m−2/3 at a height of 14km (Coulman short-exposure regime with amplitude variations of ∼ et al. 1988; Kopeika et al. 2001; Zilberman et al. 2001). 20 % per 50ms cadence (Osborn et al. 2015). Another Common distinctions for turbulence levels are 10−13 station in Graz/Austria measured 200Hz (5ms) as the (strong), 10−15 (average), and 10−17 (weak) (Goodman maximum frequency of the atmospheric fluctuations 1985; Zhu & Kahn 2002). Turbulence is particularly low (Prochazka et al. 2004; Kral et al. 2006). Therefore, at Dome C in Antarctica, about 2 ...4× lower than at on ms and shorter timescales, scintillation will typi- other observatories such as Cerro Tololo, Cerro Pachón, cally cause abrupt signal variations of 10 ...50 % every La Palma, Mauna Kea, and Paranal (Kenyon et al. 2006; 1 ...50 ms (Fig. 5). Scintillation occurs as temporally Osborn et al. 2015). Longer wavelengths experience a autocorrelated (red) noise. 74 Page 6 of 17 J. Astrophys. Astr. (2018) 39:74

1.0 100

0.8 10−1 0.6 10−2 0.4

10−3 0.2 Scintillation (variance) Scintillation (variance)

0.0 10−4 0 500 1000 1500 2000 10−3 10−2 10−1 100 101 102 Wavelength (nm) Observation cadence (s) Figure 5. Left: Estimated scintillation as a function of wavelength, with curves for medium to low turbulence (10−15 < C2 < 10−17) from space to ground. The red symbol shows the measurements taken with the 1m Jacobus Kapteyn Telescope on La Palma under typical conditions (Osborn et al. 2015). Right: Scintillation as a function of observational cadence, scaled following Figure 10 in Osborn et al. (2015).

4.2 Wavelength change from atmospheric refraction from wavelength, temperature, pressure and humidity. During the course of an hour-long observing session, the While light travels at c=299,792,458 m s−1 in elevation angle changes, causing arrival time variations vacuum, its phase velocity v is lower in a medium. of typically 1 ...10 ns. Searches for periodic signals In air, the refractive index is n≈1.00027717, so that require a correction for these effects. Detailed models light travels at v = c/n ≈ 299,709,388 m s−1, caus- for these parameters exist with experimental validation ing a change in wavelength λ. The refractive index is at the < 1pslevel(Mendes 2004). Atmospheric correc- mainly a function of pressure (altitude), water vapor, tion models leave residuals of a few ps over an observing and carbon dioxide (Edlén 1966). At λ = 1 µm, the season (Wijaya & Brunner 2011). wavelength change amounts to λ ∼ 0.27 nm, which is of the same order as the bandwidth of a time-bandwidth 4.3.2 Variations of the refractive delay Fluctuations limited pulse. The transmitter or receiver may choose in the index of refraction of the air cause speed of light to adjust for this. Corrections are possible to better than variations that cause a different time delay for a laser 10−8, which makes the effect negligible (Edlén 1966). pulse (Currie & Prochazka 2014). In practice, the effect has been measured with laser ranging observations to 4.3 Arrival time variations from refraction the Envisat satellite with the Graz Satellite Laser Rang- ing station (Prochazka et al. 2004; Kral et al. 2006). The delay in the arrival time of a pulse can be critical or From an altitude of 500m to space and back, the atmo- . ... . irrelevant, depending on the magnitude of the effect and spheric timing jitter was 0 6 1 0 ps, consistent with the search paradigm. For single pulses strong enough to various models (Hulley & Pavlis 2007). For a pulse trav- ensure a significant detection, it is irrelevant whether eling from space to Earth once (not twice), the effect is . ... . these are recorded a little bit earlier or later - they are about half this value (0 3 0 5 ps). At higher altitudes , detected in any case. Single pulses, however, are of little (e.g., 5 000 m), the jitter is likely smaller by a factor of . value, because one can never be sure of their astrophys- a few, perhaps of order 0 1ps. ical origin. After such an initial detection, the search for a repeating pulse will be crucial. The most conve- 4.4 Pulse broadening through dispersion nient repetition scheme would be a constant repetition, where the variations in arrival time are smaller than the Since there are no perfectly monochromatic pulses instrumental measurement uncertainty. Therefore, it is (section 2.1), we have to consider the way in which interesting to determine the absolute delay, as well as a group of photons of different wavelengths travels the change in delay over time. through a medium such as Earth’s atmosphere. A laser pulse can be treated as an envelope of wave amplitudes which travels through a medium with a group velocity 4.3.1 Absolute refractive delay The approximate − π 2 c −24 −1 −1 delay for the travel time from space to ground for a Dλ = GVD × (10 ps nm km ) (11) refractive index of n ≈ 1.00027717 and an air scale λ2 height of H = 8,000 m is ≈ 7.5 ns at zenith (≈ 12 ns where GVD ≈0.030036 fs2 mm−1 is the estimated at 45◦ elevation angle), with slight (0.5ns) variations group velocity dispersion in air (Ciddor 1996). For λ = J. Astrophys. Astr. (2018) 39:74 Page 7 of 17 74

2 102 10 ) 0

1 /T 10 2 T

1 100 10

Dispersion (ps) 10−1 Pulse broadening ( 100 10−2 0 500 1000 1500 2000 10−13 10−12 10−11 10−10 Wavelength (nm) Initial pulse duration T0 (s) Figure 6. Left: Dispersion for a pulse with λ = 100 nm and air distance of 20 km as a function of wavelength. The dispersion is < ps for λ>300 nm. Right: Dispersive pulse broadening as a function of initial pulse duration. Colors show scenarios for travel distances from space to sea-level (red), mid (2,500 m, blue) and high (5,000 m, black) altitude. Initial pulses shorter than 10−12 s are very sensitive to dispersion, because the square of the pulse duration is smaller than the group delay dispersion. The effect becomes negligible for pulses longer than 10−12 s.

300 nm(λ = µm) and 8km of air, we get Dλ ≈ 0.5ps 103

(0.05 ps). ) 0

Dispersive pulse broadening also depends on the /T 2

< T initial pulse duration T0. Initial pulses ps are very 102 sensitive to dispersion, because the square of the pulse duration is smaller than the group delay dispersion. The > effect becomes negligible for pulses ps. For an orig- 101 inally unchirped Gaussian pulse with the duration T0, the pulse duration is increased to (Boyd 2013) Pulse broadening (   0 2 10 D 10−15 10−14 10−13 10−12 10−11 10−10 T = T 1 + 4ln2 2 (12) 2 0 2 Initial pulse duration T0 (s) T0 Figure 7. Pulse broadening (as a factor of the initial where the group delay dispersion D2 per unit length pulse duration) due to turbulence, as a function of initial 2 −1 (in units of s m ) is the group velocity dispersion. pulse length. Distance: Space to sea-level. Each series of For initial pulse durations T0 = 1 ps over 20 km of air, curves shows wavelengths of 300nm (top), 500nm (mid- T2 ≈ 1.3 ps. The effect becomes negligible (< 1%)for dle) and 1,000 nm (bottom). Red: Strong turbulence (C2 = pulses longer than a few ps (Fig. 6). 10−14 m, L = 10 m). Blue: medium turbulence (C2 = 10−16 m, L = 10 m). Black: Quiet air (C2 = 10−17 m, L = < > 4.5 Pulse broadening through turbulence 1 m). Pulse broadening becomes negligible ( 1%)forT0 ps in most cases and for T0>0.1 ns in all scenarios. Short optical signals passing through a turbulent ≈ 90 % of the air mass. Even in high turbulence, pulse medium such as Earth’s atmosphere are temporally dis- broadening becomes negligible (< 1%) for T > torted, so that the received pulse has a longer duration. 0 0.1 ns. In more quiet, photometric conditions, we expect A semi-analytical model by Young et al. (1998) valid pulse broadening to reduce to < 0.1ps(Fig.7). for the far-field (Tjin-Tham-Sjin et al. 1998)gives We note that Lu et al. (2012) calculate pulse 2 broadening to be more severe (about two orders of T2 = T + 8α (13) 0 magnitude, at 10ps). The authors use a theoretical where z is the distance so that framework by Marcuse (1981) for single-mode fibers − / which is likely not applicable to the atmosphere. Dis- σ 2 Q 5 6 z α = 0.322 , Q = . (14) agreements over which approximations describe reality ( )2 2 kc kL best can be resolved by real-world measurements. The model is valid for narrowband optical and IR These theoretical estimates are in agreement with −14 pulses (T0 > 2 × 10 s). For the stratosphere and practical results. The atmospheric scattering in free- up, pulse broadening is very low (< 1 %). Most rele- space laser communications (λ = 1.06 µm) is not vant is the lowest layer, the troposphere, which contains relevant in practice for 10ps (10−11 s) pulses unless 74 Page 8 of 17 J. Astrophys. Astr. (2018) 39:74 strong turbulence is present (Majumdar & Ricklin With spatial resolution, the source location still has has 2010). The dependence of turbulent pulse broadening uncertainties (δα, δβ) which give rise to periodic timing on wavelength is model-dependent and ranges between errors δtc resulting from incorrect barycentric correc- negligible Young et al. (1998) and a linear relation, with tion (Lyne et al. 2006) shorter wavelengths having stronger broadening (Kelly δ = δα (ω − α) β & Andrews 1999). In the latter model, temporal broad- tc A sin t cos ening is found to be very relevant for 10 ...30 fs pulses − A δβ cos(ωt − α)sinβ (15) and becomes irrelevant for > 100 fs under all condi- which can amount to an absolute delay delta of a few tions (far field, near field, horizontal, vertical, strong seconds per degree on the sky. A source which is turbulence) (Kelly & Andrews 1999). misidentified by one degree, and repeats periodically in the ETI reference frame, would change its period by 5. Barycentering  1 ms per hour in Earth’s reference frame. Repeating signals may be periodic. The period can be constant or changing. A constant period can be constant 5.2 Wavelength Doppler shift in the ETI reference frame, or can be corrected to be  −1  . −1 constant (by ETI) for the Sun’s or the Earth’s reference Earth’s orbit ( 30 km s ) and rotation ( 0 3kms ) frame. It is useful to understand the accuracy limits in result in Doppler shifts in the received radiation spec- order to perform the according periodicity tests for an trum. The corresponding shift over the course of half a λ = λ v −1 ∼ . assumed scenario. year is 0 c 0 1 nm. This is smaller than λ ∼ . It appears unlikely that a distant (e.g., kpc) periodic the atmospheric limit for pulses ( 1 5 nm for ps ∼ transmitter would be corrected for Earth’s barycentric pulses, Table 1). It is however larger by a factor of 10 motion due to the dynamics of the , such as than classical spectroscopy, and needs to be corrected nutation and precession of the Earth, over kyr of light in this case. time travel. As we do not know the ETI’s origin and thus reference frame (e.g., a transmitter in orbit around a star), we can 6. Instrumental effects only correct for the motion of known bodies. In our case, this is the Earth orbiting the sun. Before discussing telescope and detector choices, we The classical light travel time across the Earth’s orbit review the technology used in previous OSETI experi- (the R¿mer delay) has a magnitude of  498 cos β sec- ments. onds with β as the ecliptic latitude of the source, over − the course of half a year; a rate of < 3.2×10 5.Overthe 6.1 Previous searches course of a one hour observing session, this amounts to < 0.12 s, and could smear any periodicity (of periodic Searches for continuous and pulsed laser signals began signals) if uncorrected. Compared to this delay, other in the 1970s with efforts by Shvartsman (1977)and factors such as the Einstein and Shapiro delay are small Beskin et al. (1997). Using small telescopes, fast PMTs (< 1.6ms). entered the field in the 1990s (Kingsley 1993a, 1995) Simple barycentric correction codes offer corrections and were soon widely adopted (e.g., Wright et al. 2004; − down to 0.24 cm s 1, or 29ns over an hour (Wright & Stone et al. 2005). All modern searches for pulsed sig- Eastman 2014). More complex implementations such nals used PMTs with cadences of order ns (Table 2). as TEMPO2 (Hobbs et al. 2006) offer an accuracy of ≈ Searches with Cherenkov telescopes have been sug- 0.5 ns over 1 hr and ≈ 100 ns over years (Edwards et al. gested by Eichler and Beskin (2001)andHolder et al. 2006). These corrections have been validated against (2005), and performed by e.g., Abeysekara et al. (2016). many pulsars with timing residuals of ≈ 200 ns over Heliostats entered the field at about the same time (Ong 10years (Hobbs et al. 2010) et al. 1996; Hanna et al. 2009). Only recently, quantum efficiency became suffi- 5.1 Spatial resolution ciently high to allow for useful nanosecond IR detectors, in discrete avalanche photodiodes (DAPDs) (Wright In an all-sky at once observation (without high spatial et al. 2014; Maire et al. 2014, 2016). While this project resolution), barycentering corrections can not be made, searches mainly for individual strong pulses, it uses a because the ecliptic latitude of the source is unknown. (weak) MHz pulsed lasers for tests. J. Astrophys. Astr. (2018) 39:74 Page 9 of 17 74

Table 2. Selection of previous pulsed OSETI detectors.

Obervatory Cadence (ns) λ (nm) Sensitivity (γ m−2 ns−1) Reference

MANIA 6 m 100 300 ...800 n/a Shvartsman et al. (1993) Kingsley Columbus 0.25m 2 300 ...650 n/a Kingsley (1995) OZ Australia 0.4m 1 300 ...650 n/a Bhathal (2000, 2001) Lick 1 m Nickel 5 450 ...850 51 Wright et al. (2001) Harvard Oak Ridge 1.5m 5 450 ...650 100 Howard et al. (2004) Princeton Fitz Randolph 0.9m 5 450 ...850 80 Howard et al. (2004) STACEE heliostats 12 300 ...600 10 Hanna et al. (2009) Leuschner 0.8 m 5 300 ...700 41 Korpela et al. (2011) Harvard 1.8m 5 300 ...800 60 Mead (2013) Lick 1 m NIROSETI 1 950 ...1650 40 Maire et al. (2014) Veritas 12m 50 300 ...500 1 Abeysekara et al. (2016) Boquete 0.5m 5 350 ...600 67 Schuetz et al. (2016)

Table 3. Comparison of observation choices.

Bandwidth (nm) Cadence G2V star at d = 100 pc 5◦ × 5◦ All-sky

1,000 s 106 1011 1014 1,000 ns 10−3 100 105 1,000 ps 10−6 0.1 100 1ps10−9 10−4 0.1 Fluxes in photons per square meter

Observations have mostly focused on FGK stars otherwise temporal smearing occurs, reducing the and individual objects, such as the anomalous star signal amplitude, and thus sensitivity. We now examine KIC 8462852 (Schuetz et al. 2016; Abeysekara et al. three exemplary telescope designs relevant for OSETI. 2016), and exoplanet host stars Trappist-1, GJ 422 and Wolf 1061 (Welsh et al. 2018). 6.2.1 Classical telescopes In a parabolic telescope The Harvard all-sky survey used a 1.8m telescopes mirror, parallel rays are perfectly focused to a point (the with 16 PMTs, each with 64 pixels, scanning a field-of- mirror is free of spherical aberration), no matter where ◦ ◦ − view of 1.6 × 0.2 (4 × 10 6 of one sky hemisphere). they strike the mirror. However, this is only the case for Following Table 3, the expected broadband flux per rays that are parallel to the axis of the parabola, i.e. in the − pixel is 10 3 per 5ns cadence Mead (2013). This shows center of the field of view. Rays entering at an angle of that broadband large field-of-view observations are pos- a nonzero field of view will suffer from coma. This and sible even at “slow” ns cadence, using higher spatial higher order aberrations can be reduced with correctors, resolution. at the expense of additional throughput losses and costs. Spectroscopic (continuous wave) searches have been For typical parabolic reflectors, coma and astigmatism suggested (Betz 1993) and performed (Reines & Marcy are < 10 µm for a field of view of a few degrees. The 2002; Tellis & Marcy 2017)for∼ 5,600 FGKM stars corresponding light time travel difference is < 10 fs, i.e. with power thresholds between 3kW and 13MW in a negligible. For meter-sized telescopes, practical limits wavelength range 364<λ<789 nm. to the field of view are of order 5◦ (Roy 2005). The LSST will have a FOV of 5◦ (Claver et al. 2004; Neill et al. 6.2 Telescope 2016). These limits keep light time travel differences < 50 fs, which means that atmospheric effects are larger by Light travels very slowly at 30 cm ns−1.Thisis at least an order of magnitude. Similarly, mirror surface relevant at short cadence, where all light from a collec- roughness of typically <λ/4 results in negligible (< tor must arrive at the detector within the same cadence, fs) arrival time variations. 74 Page 10 of 17 J. Astrophys. Astr. (2018) 39:74

spherical structure towards the focal point, allowing for . 1 0 smaller aberrations off the optical axis compared to a parabolic design. Similarly, the H.E.S.S.-I telescope has 0.8 a t ∼ 5ns(rms ∼ 1.4ns)(Akhperjanian & Sahakian 0.6 2004; Schliesser & Mirzoyan 2005). Larger telescopes with the Davies-Cotton design would suffer from larger 0.4 time spread (Davies & Cotton 1957; Vassiliev et al. Transparency 2007). The effect could be reduced to  0.3ns by 0.2 mounting the tessellated parabolic mirror facets stag- geredindepth(Dravins et al. 2013). 0.0 0.5 1.0 1.5 2.0 Telescope designs such as the Schwarzschild-Couder Wavelength λ (μm) (Schwarzschild 1905) offer isochronous large unvi- gneted fields of view, up to 12◦ (Vassiliev et al. 2007; Figure 8. Transparency as a function of wavelength for Vassiliev & Fegan 2008). The isochronicity however acrylic Fresnel lenses. Source: Thorlabs. is only valid on axis. Rays from the large field of view have a delay of ∼ 0.3 ns per degree (Figure 10 from Vas- In case of multiple coincidence detectors, a precise siliev et al. 2007). In principle, this effect is correctable alignment is required which takes into account the light by delaying the pixels in the detector accordingly. If the travel time. For example, a ps cadence corresponds to a detector has many (104) pixels, the field of view per distance of 0.3mm. pixel is small (∼arcmin), so that the time smear effect per pixel is small (∼ 5 ps). There are plans to build 6.2.2 Fresnel lenses Fresnel lenses are thinner (typ- detectors with many (104)pixels(Actis et al. 2011; ically mm), lighter and cheaper than classical lenses, Acharya et al. 2013; Dickinson et al. 2018). as they divide the lens into a set of concentric annu- lar sections. The finite size of these sections (typically 6.2.4 Heliostats The problem of light time travel mm) makes them much cheaper (hundreds of USD variations is particularly severe for extended light col- for a meter-sized aperture), but also of inferior optical lectors such as heliostats, as used by STACEE, with 64 quality, compared to classical lenses or parabolic mir- individual heliostats (each 37 m2) and a total collecting rors. They can be made of plastics or acrylic, and their area of 2,300 m2 (Hanna et al. 2009). Synchroniza- transparency can be of order unity for optical and IR tion was increased with separate detectors for groups wavelengths (Fig. 8). Fresnel lenses are not diffraction of heliostats. Still, cadence was limited to 12ns. limited, but can focus light to a mm spot size, so that their isocronicity error is a few ps (Costa et al. 2007). 6.3 Detector They have a large field of view, up to 30◦ × 30◦.Ithas been suggested to build a dedicated OSETI instrument An ideal detector would have a 100% quantum in a dome, with an array of Fresnel lenses and detectors efficiency, zero noise and dead time after a signal, and to cover the entire sky (Covault 2013). There are also perfect time resolution. Commercial PMTs offer band- plans to use Fresnel lenses for Cherenkov telescopes widths of ≈ 100 %, quantum efficiencies of ≈ 50 %, (Cusumano et al. 2002; Maccarone et al. 2008; Arruda dark rates of a few hundred Hz when cooled, reset times & GAW Collaboration 2010). of ≈ 3 ns and timing jitter of ≈ 0.1ns(Abbasi et al. 2010). We show an overview of useful detector types 6.2.3 Cherenkov Telescopes Cherenkov Telescopes for OSETI in Table 4. for the detection of very-high-energy gamma-ray pho- Superconducting nanowire single-photon detectors tons offer large apertures for competitive prices. A (SNSPDs) require cooling to a few K (Zhang et al. Cherenkov telescope with a 12m aperture and 15◦ 2003) and provide ultrahigh counting rates exceeding field of view would exceed the étendue of LSST by 1 GHz (Tarkhov et al. 2008). Originally, their timing a factor of 10 (Vassiliev et al. 2007). However, these jitter was similar to PMTs, ∼ 0.15 ns (Gol’tsman et al. telescopes are not isochronous. For example, the com- 2001; Marsili et al. 2013). Latest improvements reduce mon Davies-Cotton telescope type with a total mirror timing jitter to very low values (< 18 ps FWHM, Shch- area of order 10 m2 located in 18 hexagonal facets of eslavskiy et al. 2016), at low intrinsic dark count rate 0.78m has an optical time spread of t  0.84 ns rms (kHz, Gemmell et al. 2017), and short recovery times (Moderski et al. 2013). Its primary reflector forms a (< 20 ns, Gemmell et al. 2017). Quantum efficiency is J. Astrophys. Astr. (2018) 39:74 Page 11 of 17 74

Table 4. Detector technologies available for OSETI.

Detector Wavelength Temp. (K) QE (%) Time jitter t (ps) Dark count rate Max count rate (MHz)

PMT Optical 300 40 300 100 Hz 10 PMT IR 200 2 300 200 kHz 10 Si SPAD Optical 250 65 400 25 Hz 10 InGaAs SPAD IR 240 10 55 16 kHz 100 Frequency up- IR 300 2 40 20 kHz 10 conversion SNSPD IR 1.5 57 30 1 Hz 1000 Data from Hadfield (2009) near unity in the optical and IR for single photons (93%, (21.9magarcsec−2 in V-band, Benn & Ellison 1998), Marsili et al. 2013). San Pedro Martir (Mexico) (21.84, Plauchu-Frayn et al. Single Photon Avalanche Photodiodes (SPADs) offer 2017), Calar Alto (22.01, Sánchez et al. 2007)and a time resolution of ∼ 50 ps, but a maximum count rate Dome A in Antarctica (23.4, Sims et al. 2012; Yang of 8MHz (125ns) (Billotta et al. 2009; Zampieri et al. et al. 2017). 2015). A detailed model of the night sky spectrum (Cerro Microwave Kinetic Inductance Detectors (MKIDs) Paranal Advanced Sky Model) is consistent with these provide large arrays at maximum count rates of ∼ 103 values1 (Noll et al. 2012; Jones et al. 2013). This is also counts/pixel/s with µs timing (McHugh et al. 2012; consistent with measurements using a large field-of- Mazin et al. 2012). view telescope with a PMT, resulting in 2×1012γ s−1 Timing resolutions of OSETI instruments have sr−1 m−2 for 300<λ<650 nm for dark sky regions such improved from 100ns (Shvartsman 1977)to20ns as Sculptor or Virgo, and about twice the value for (Shvartsman et al. 1997) and finally 1ns (Maire et al. bright regions such as Carina towards the galactic plane 2014), a decrease by two orders of magnitude over 50 (Hampf et al. 2011), in agreement with measurements years. from Namibia (Preu§ et al. 2002)andLaPalma(Mir- zoyan & Lorenz 1994). 6.4 Electronics Based on these counts, the total radiance for one sky hemisphere (2.7 × 1011 arcsec2)is≈ 1014 γ s−1 m−2. Detectors can only be as fast as the readout electronics. With a 1nm filter centered at λ = 1.064 µm, the radi- Typical commercial sampling equipment works at GHz ance is < 1011 γ s−1 m−2, or about half that value for frequencies, and devices up to 10GHz (0.1ns) are com- telescope and receiver efficiencies of 50%. This is still mon. Free-space optical communication is common at  100 γ ns−1 m−2. The all-sky background would be 0.56ns cadence (1.8GHz) (Brandl et al. 2014; Ferraro very small ( 0.1) at a picosecond observation cadence, et al. 2015). The fastest commercially available signal −11 or when observing a fraction of the sky. processing oscilloscopes sample at 100GHz, or 10 s In a typical wide-field telescopes with a fields-of- (Foster et al. 2008; Füser et al. 2012). In the labora- ◦ × ◦(∼ / , −15 view of 5 5 1 1 000 of the sky, section 6.2), tory, petahertz optical oscilloscopes (10 s) have been noise levels reduce to 0.1 photons per ps cadence for a demonstrated (Kim et al. 2013). bandwidth of 1,000 nm (Table 3). In practice, fast detectors such as SNSPDs can be ∼ sampled at 10 ps time resolution as long as the count 7.2 Short natural astrophysical pulses rate is  10 MHz (dead time 100ns), as demonstrated by Shcheslavskiy et al. (2016) using commercial equip- A detailed study of astrophysical phenomena found ment. that the shortest timescale of (known) natural signals in the optical appears to be of microsecond duration 7. Noise and longer (Howard & Horowitz 2001). If that is 7.1 Stochastic background

Measurements of the night sky brightness are 1For the conversion of light-intensity units, see Benn and Ellison available for many observatory sites, such as La Palma (1998, Appendix). 74 Page 12 of 17 J. Astrophys. Astr. (2018) 39:74 true, no optical “RFI” exists at ns or ps cadence. The 104 discovery of such a source would be of great interest in Observed 3 any case. 10 Expected Terrestrial interference comes from Air Cerenkov 102 flashes produced by cosmic rays and γ -rays. These have typical durations of ∼ 5 ns and can be distin- 101 guished by imageable tracks in multi-pixel detectors 0 (Eichler & Beskin 2001). Similarly, OSETI observa- 10 Numberofevents tions have been triggered by airplane positioning lights 10−1 with µs durations. These signals can be flagged with a dome camera which detects bright moving objects. 10−2 Such events are not periodic on the relevant time scales 0 1 2 3 4 5 6 7 8 Photons per bin (Mead 2013). Figure 9. Received photons per timeslot in the lunar laser ranging experiment (red, Murphy 2008) versus expected pho- 7.3 Poissonian photon pileup tons following a Poisson distribution (blue). To explain the 6, 7 and 8 photon events, the flux level has to be 6× higher, Using beamsplitters and multiple coincidence detectors, which can not be explained with better atmospheric trans- the false positive rate can be reduced. In the absence of parency which is of order 50%. The difference can only be any true pulsed signal, the expected number of coinci- due to laser power variations and pointing quality fluctua- dences R is (Wright et al. 2001; Coldwell 2002) tions.  r n − R = η T n 1 (16) n PMT buys more average laser power (at lower peak power). ≥ where n 1 is the number of detectors, r is the number This concept has been proposed for OSETI by Leeb η of photons per second, T is the pulse width, and is the et al. (2013). 6 detector efficiency. For example, a rate of 10 detected Lunar laser ranging observations have been found × 5 photons per second leads to 9 10 coincidences per to exhibit a strong shot-to-shot variability ascribed hour with two detectors, and 133 per hour with 3 detec- to speckle structure and other scintillation (“seeing”) η = η = . tors for 1. Using a more realistic 0 2, it reduces effects. At an average return rate of 0.23 photons per −1 to 27hr . The downside of this false positive signal pulse, statistics would expect < 20 % of the returns reduction by multiple coincidence detection is a loss in to be in multiple-photon bundles, and no events with < / sensitivity to 1 n. more than four photons in a 10,000-shot run. In practice For the search paradigm of repeating signals, false (Murphy 2008), 46 % of the returning photons were in positives (or a noise floor) are not critical, as these multiple-photon bundles, with up to 8 photons detected effects vanish with well-known search methods for in one cadence (Fig. 9). For OSETI, the pointing qual- periodicity, such as FFTs, periodograms and autocor- ity is irrelevant, as the ETI laser would illuminate all of relation. Earth equally. Atmospheric scintillation, however, may In practice, data might occur not to exhibit strict contribute 10 ...50 % of flux variation, and may lead Poisson noise, as is observed in lunar laser ranging, to higher than expected photon pile-up. which measures the distance between the Earth and the moon with the light travel time from Earth-based lasers bounced back by retro-reflectors placed on the lunar surface (Murphy 2008; Murphy et al. 2012). 8. Summary of results These observations originally used few high power pulses (Shelus 1985; Samain et al. 1998; Murphy 2013), All relevant effects are summarized in Table 5. but recently moved towards lower pulse energies at Interstellar effects are irrelevant in all cases. Pulses are much higher repetition frequency (80MHz) Nd:YAGs broadened by  0.3 ps by dispersion and  0.1psby at λ = 1064 nm and λ = 532 nm with typical widths turbulence in the atmosphere. Time of arrival variations λ ∼ 0.19 nm in combination with short (10ps) pulse are dominated by changes in refraction to  0.5ps. durations (Adelberger et al. 2017). Many weaker pulses Most atmospheric effects are also a function of wave- have the advantage that their arrival time errors can be length, and are factor of a few less severe for NIR averaged, and a finite amount of monetary investment compared to optical. J. Astrophys. Astr. (2018) 39:74 Page 13 of 17 74

Table 5. Summary of effects on short pulses (λ0 = 1 µm,λ = 1nm).

Component Effect Comment Section

Physical limits: 2 Ð Time-bandwidth limit ≈ ps Choice based on λ0 and λ 2.1 Ð Schawlow-Townes flaser  Hz Irrelevant 2.2 Ð Pulse shape ∼ 10 % of pulse duration Irrelevant 2.3 Interstellar: 3 Ð Dispersion ≈ ps Constant delay; variation < 1% 3.1 Ð Scatter broadening < fs Irrelevant 3.2 Ð Spectral broadening < 10−8 Hz Irrelevant 3.3 Atmospheric: 4 Ð Scintillation  20 % variance Typical frequency ≈ 200 Hz 4.1 Ð Wavelength change (refraction)  0.3 nm Correctable to < 10−4 nm 4.2 Ð Absolute refractive delay  10 ns Correctable to ∼ ps 4.3.1 Ð Refractive delay variations  0.5 ps Likely not correctable 4.3.2 Ð Pulse broadening (dispersion) T2  0.3psforT0 = 1 ps For medium turbulence 4.4 Ð Pulse broadening (turbulence) T2  0.1psforT0 = 1 ps For medium turbulence 4.5 Barycentering < 0.12 s per hour Correctable to  0.5 ns per hour 5 Ð Finite spatial resolution < ms per degree per hour Relevant for periodic signals 5.1 Ð Spectral Doppler shift < 0.1 nm per 6 months Correctable to < 10−4 nm 5.2 Telescope 6.2 Ð Parabolic reflector < 10 fs Instrument alignment relevant 6.2.1 Ð Fresnel lenses ∼ 3ps 6.2.2 Ð Cherenkov telescopes ∼ 0.3 ns per degree Correctable with multipixel detectors to ∼ 5ps 6.2.3 Ð Heliostats ∼ 10 ns 6.2.4 Detector > 30 ps Depending on type, see Table 46.3 Electronics > 10 ps See text 6.4

Given these atmospheric limits, pulses shorter than 9. Discussion ps are not realistic for ground-based detectors. Such ps pulses have time-bandwidth limits of λ ∼ nm at 9.1 Short signals through spectroscopy λ0 = 1 µm. This corresponds to a fractional bandwidth λ/λ0 ∼ 0.001. As can be seen in Fig. 4,therearewin- There is one additional method for OSETI, which is dows of atmospheric transparency (discussed in detail widely ignored in the literature. It is based on spectra in paper 10 of this series) of order unity even for large which are Fourier transformed to search for periodic bandwidths, λ/λ0  0.1. modulations. In this section, we explain the method and Periodic signals are additionally affected by barycen- discuss its advantages and issues. tering issues. While spectral Doppler shifts are cor- The spectral modulation of coherently separated laser rectable, current timing codes are limited to an accuracy pulses was first noted by Chin et al. (1992) and sub- of order ns per hour. This accuracy may be improved in sequently studied in Borra (2010a,b, 2012) with the the future with more detailed models and calibrations motivation to apply the method to astronomical data. by many pulsars. For repeating ps pulses which occur Astronomical spectra are sampled at equal wavelength e.g. at kHz repetition over a few seconds, current cor- intervals, λ = c/f where λ = const. To search for rections are sufficient. Over longer times (years), many signals with a constant temporal period, these spectra subtle adjustments would need to be made. For example, must be converted to equal frequency intervals f = c/λ continental drifts are a few cm per year, which translates with f = const. With constant frequency intervals, a to  100 ps year−1 (or  0.3psday−1). It appears sen- Spectral Fourier transform (SFFT) can me made which sible to restrict searches of periodic ps pulses to short produces power for signals with a constant temporal time durations of order minutes. period, such as repeating laser pulses. 74 Page 14 of 17 J. Astrophys. Astr. (2018) 39:74

Searches for such periodic temporal modulations cadence, in order to push noise to less than one photon were reported for SDSS spectra of individual stars per cadence. (Borra & Trottier 2016; Borra 2017) as well as galaxies Every cadence and channel could be sampled with (Borra 2013). Detections were claimed with repetition e.g., one byte of data to allow for 256 distinct values. frequencies (not pulse durations) of ≈ 10−13 s. The data stream is then 100 GB s−1. For comparison, the Following the standard values of atmospheric condi- Radio Data Recorder (MacMahon tions discussed in section 4.5, this appears problematic. et al. 2018)saves24GBs−1 of data to disk. Their com- For the data taken with the SDSS telescope at an alti- puting facilities also allow for real-time de-dispersion tude of 2,788 m at Apache Point Observatory in New and pulse search. It appears that an all-sky optical and Mexico, pulse broadening from atmospheric turbulence NIR survey at ps cadence would not require implausible might be significantly smaller. The exact value could be computing requirements. determined experimentally using laser pulses reflected off the moon or a satellite. If broadening occurs, it does not destroy the pulses, but weaken them, as the Fourier 10. Conclusion transform makes an average over the entire spectrum. In general, the method appears to be useful for signals We have examined the influence of interstellar and with longer repetition frequencies, but would require atmospheric effects on short pulses. We find that pulse independent confirmation with another method in case durations are limited to ps due to refraction and disper- a signal would be detected. sion. SFFTs are not sensitive to the pulse length, but With current technology, timing (t ∼ 10−9 s) is a ρ instead to the duration between pulses ( ), where better filter than frequency (∼ 10−6 λ/λ ). This might /ρ 0 1 is the pulse repetition rate. SFFTs’ sensitivity be countered by new spectroscopic technologies on the region is constrained by the spectral resolution, typi- receiver side, or power level advantages of continuous . × −15 ... × −12 cally 2 5 10 4 10 s for optical spectra with over pulsed lasers on the transmitter side. ≈ , R 10 000 and can be increased to longer spacings The optimal laser signals to maximize S/N appear −11 = , (10 s) with higher resolution (R 100 000) spec- to be time-bandwidth limited Gaussian t ≈ 10−12 s trographs. The spectrograph “ESPRESSO” is expected pulses at a wavelength λ ≈ 1 µm, and a spectral width = , < 0 to deliver a resolution of R 200 000 between 380 of λ ≈ 1.5 nm. An all-sky all the time survey at ps λ< 780 nm at 5% efficiency, a linewidth of 0.001nm cadence may be performed given certain technological (González Hernández et al. 2017). advances. The sensitivity of SFFTs is limited to about 10−5 of the stellar flux in SDSS spectra, if the star is blended with the hypothetical laser source. For a L = L Acknowledgements star and a competing laser with Dt = Dr = 10 m at λ = 1 µm, this SNR can be achieved with an average laser power of 3.5MW independently of distance. For MH is thankful to Marlin (Ben) Schuetz for useful comparison, military laser weapons are being developed discussions. with ≈ 0.2 MW power. For faster survey speeds, multi- ple spectra can be obtained in parallel, as was done by SDSS. References

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