Optical analogues of black-hole horizons

Yuval RosenbergDepartment of Physics of Complex Systems,

Weizmann Institute of Science, Rehovot 761001, Israel.

April 14, 2020

Abstract

Hawking radiation is unlikely to be measured from a real black-hole, but can be tested inlaboratory analogues. It was predicted as a consequence of quantum mechanics and generalrelativity, but turned out to be more universal. A refractive index perturbation produce anoptical analogue of the black-hole horizon and Hawking radiation that is made of light. Wediscuss the central and recent experiments of the optical analogue, using hands-on physics.We stress the roles of classical fields, negative frequencies, ’regular optics’ and dispersion.Opportunities and challenges ahead are shortly discussed.

1 Introduction

Analogue systems are used to test theories of predicted phenomena that are hard to observedirectly [1]. Hawking [2, 3] predicted black-holes to thermally radiate from the quantum vac-uum. Its measurement is unlikely, having a temperature inversely related to the black-holemass. Stellar and heavier black-holes’ radiation is much weaker than the cosmic microwavebackground (CMB) fluctuations [4, 5]. Hypothetical microscopic primordial black-holes mightemit significant Hawking radiation [6], but it is highly model-dependent, with rates seriouslybounded by the lack of its observation [7–9].

Volovik and Unruh [10,11] have shown that a transonic fluid flow is analogous to the space-time geometry surrounding a black-hole, and should emit sound waves analogous to Hawkingradiation.

Wave kinematics control the Hawking process, which appears in the presence of an effectivehorizon [12, 13]. Black-hole dynamics and Einstein equations of gravity are not an essentialrequirement. The analogue of the event horizon is the surface separating the subsonic andsupersonic flows. The surface gravity determines the strength of Hawking radiation and isanalogous to the flow acceleration at the horizon. The horizon should also last long enoughsuch that the modes of Hawking radiation will have well defined frequencies [13].

Many analogues have been proposed [14, 15] with realisations using water waves [16–21],Bose-Einstein condensates (BEC) [22–25], and optics [26–33]. The analogues gave new per-spectives in both gravity and the analogue systems [14], extending Hawking’s theory to wavedynamics in moving media. The universality of the Hawking effect shows it persists in real-world scenarios, where complicated dynamics replace simplified and possibly fine-tuned theories.Hawking’s original derivation seems questionable, since the radiation originates from infinitelyhigh frequencies, neglecting unknown high energy physics (known as the ’trans-Planckian prob-lem’). The microscopic physics of the analogue systems is well known, allowing to test the roleof high frequencies in Hawking predictions [34,35].

This paper discusses the optical analogues that use a refractive index perturbation to es-tablish an artificial black-hole horizon. Extensive reviews of analogue gravity exist [14,15], butthey pay little attention to optical analogues and substantial progress was made after their pub-lication. Additional useful resources include Refs. [36–39]. This paper aims to present the main

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experiments that transformed the optical analogues from simple ideas to established reality. Itaims to do so in the light of the recent developments, but in terms of hands-on physics. Webelieve that experiments reach the state where their data may lead the way for new questionsand discoveries in the physics of Hawking radiation.

Section 2. briefly describes early proposals for optical analogues using slow light and whythey could not materialise. Transformation optics is also mentioned, where stationary dielectricsare viewed to change the spatial parts of the metric. Section 3. explains the standard theoryof optical analogues discussed in this paper and the first demonstration of optical horizons.Further measurements of the frequency shift at group velocity horizons are briefly discussed,and additional related work is mentioned. Section 4. focuses on attempts in bulk optics ratherthan optical fibres, stressing the roles of group-velocity and phase-velocity [40] horizons. Section5. discusses the first measurements of negative frequencies in optics, a phenomenon closelyrelated to the Hawking effect. Its theory is also presented and shortly explained. Section 6.considers additional interpretations of the optical horizon. Theory and experiments are shownto directly relate the effect to cascaded four-wave mixing, but analysis in the time domain seemsto be unavoidable. The use of temporal analogues of reflection and refraction, and numericalsolutions are also mentioned. Section 7. discusses the first demonstration of stimulated Hawkingradiation in an optical analogue, and lessons learnt from it. Section 8. concludes the paper witha brief outlook to the future.

2 Early attempts

Fresnel’s drag [41] was an ether-based theory of light propagation in transparent media, ’con-firmed’ by Fizeau [42]. The drag effect is just a relativistic velocity addition [43], but Fresnel’swrong theory was based on correct intuition of velocity addition. The ether was replaced by thespace-time geometry and the quantum vacuum, which continued to have an intimate connec-tion to moving media and produce puzzling phenomena [2,3,44–46]. Analogue gravity took thisconnection a step further, but despite theoretical progress in the 1990’s, no practical realizationof analogue black-holes was suggested at that time [14].

Technology drove ideas in the right directions (optics and Bose-Einstein condensates) aroundthe year 2000 [47–51]. In optics, Leonhardt and Piwnicki [47–49] suggested to slow down lightsuch that its medium could be moved in super-luminal velocities and form a horizon. These ideasused the technology of ’slow light’ [52, 53], where incredibly low group velocities are producedusing electromagnetically induced transparency (EIT, see Fig. 3a). However, the phase velocityof ’slow light’ is fast, preventing the crucial formation of a phase velocity horizon (where themedia moves at the phase velocity of light. See below) [54]. Another problem with realisingthese ideas is the narrow bandwidth of light that can be slowed [54], and severe absorptionaround it [40]. The inevitable conclusion was to move the medium in relativistic velocities.

Despite missing key concepts of Hawking radiation, these ideas have pushed analogue grav-ity forward and beyond the scope of relativity (or relativitists). They showed that an analogueblack-hole metric can be made in optics. Similarly, stationary dielectrics mimic spatial geome-tries. This interpretation inspired transformation optics, and the development of meta-materialstechnology extended the range of practical geometries [55–60].

3 Changing a reference frame

The groups of Leonhardt and Konig [26] started a new approach that follows a simple idea [61]:light itself travels at the speed of light. A light pulse (also denoted pump) travels in a dielectricmedium with group velocity u . In the reference frame co-moving with the pulse (’the co-moving frame’), the medium seems to flow in the opposite direction with velocity magnitude |u|(Fig. 1b). Probe light of a different frequency differ in velocity due to dispersion. The nonlinear

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Black-Hole

! < 0

! > 0

n = n0 + n2I

(a) (b)

u

Figure 1: Main optical analogue concepts. (a) A black-hole emit Hawking radiation from itsevent horizon: positive norm waves escape outside, while negative norm waves drift towardsits singularity. The surface gravity κ represented by a falling weight determines the radiation’sstrength. (b) The medium seems to flow in the pulse co-moving frame with its group velocity u.A pair of black-hole and white-hole horizons are formed due to nonlinear refraction index beingproportional to the pulse local intensity. Positive frequency probe light is shifted, and negativefrequency waves are formed at both horizons according to the dispersion relation (see Fig. 2).The magnitude of the Hawking effect is determined by the pulse steepness, being analogous tothe surface gravity at the horizon (see text).

Kerr effect [62] slows down the probe upon interaction with the pulse1 – the refractive indexchange as n = n0 +n2I, where n0 is the linear refractive index, n2 is a parameter being typically10−16cm2/W [62], and I is the light intensity. A group velocity horizon forms where the probegroup velocity obey vg = |u|, blocking the probe from further entering the pump. A black-holehorizon is formed in the leading end of the pump, where the flow is directed into the pulse. Itstime reversal, a white-hole horizon with outward flow, is formed in the pump trailing end [26,63].

If the pump is slowly varying [13], the probe co-moving (and Doppler shifted) frequency,

ω′ = γ(

1− nuc

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is conserved. Here γ =(1− u2/c2

)−1/2is the Lorentz factor, c is the vacuum speed of light, n

is the refraction index, and ω is the probe frequency in the laboratory frame. Photon pairs ofHawking radiation are produced at each horizon: one with positive ω′, and its negative partnerwith −ω′ [38, 64]. For positive ω, ω′ is positive only if the probe phase velocity, vφ, is greaterthan |u|. A phase velocity horizon forms where vφ = |u| so ω′ = 0. The Hawking radiationoutgoing from the horizon mix incoming radiation of positive and negative frequencies [38].Quantum mechanically, it is described by a Bogoliubov transformation

b± = αa± + βa†∓ ; |α|2 − |β|2 = 1, (2)

where the sign of the operators equals the sign of ω′. Time dependent annihilation and creationoperators for incoming modes, a±, a

†∓ mix to form annihilation operators for outgoing modes,

b±. This extracts metric energy (pump energy) to amplify the radiation, thus spontaneouslycreating outgoing radiation from incoming vacuum [33, 38]. The transformation parameters, αand β, give the flux of Hawking radiation. When neglecting dispersion, the radiation effective

1This has a similar effect to the flow acceleration in the fluid analogues, changing the probe velocity relativeto the flow.

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Figure 2: Schematic Doppler curve of the fibre optic analogue with phase matching conditions.Frequency ω′ at the reference frame co-moving with the pump pulse is plotted against thelaboratory frequency ω (Eq. 1). The pump pulse is in a local minima, where the dispersion isanomalous. The phase matching condition of dispersive waves (DW) corresponds to conservationof ω′pump (upper dotted line). Negative frequency resonant radiation (NRR) conserve −ω′pump

(lower dotted line. See also Fig. 3d,3f). Probe light (black-diamond) is red-shifted (whitediamond) at black-hole horizon, conserving its co-moving frequency ω′probe (upper solid line).At the white-hole horizon the probe (now white diamond) is blue-shifted (black diamond), asseen also in Fig. 3b. At the local maxima the group-velocity matches that of the pump, u.Negative Hawking radiation (NHR) conserve −ω′probe (lower solid line. See also Fig. 3f). Thephase horizon (PH), where the phase velocity equals u, corresponds to ω′ = 0.

temperature is proportional to the analogue of the surface gravity – the steepness of the pulse(giving the probe velocity gradient in the co-moving frame [13, 38], which is related to therefractive index (and pulse intensity) gradient or rate of change [26,38,63]).

In [26], an optical fibre called a photonic crystal fibre (PCF) provided both the desireddispersion (Fig. 2) and nonlinearity. Its unique structure guided light in a very small region– the fibre’s core [65, 66]. This increased the light intensity and the fibre’s nonlinear response.Its nonlinear parameter was γ (ω0) = ω0n2/cAeff = 0.1W−1m−1 at ω0 corresponding to 780 nmwavelength, where Aeff is the fibre effective mode area and ε0 is the vacuum permittivity [66].The fibre’s structure was engineered to change its dispersion relation, n(w), to have two pointswith matching group velocities [66]: one in a normal dispersion region, where n(w) is increasing;and another in an anomalous dispersion region, where n(w) is decreasing. The anomalousdispersion included the pump spectra, generated by a Ti:Sapphire mode-locked laser. Self-phase modulation (SPM) due to the nonlinear refraction index counteracted the anomalousdispersion and formed stable solitons [66].

The solitons in [26] created horizons. Continuous wave (CW) probe was added to the fibre,being red-detuned from the frequency of matching group velocity (white diamond in Fig. 2).The fast probe mainly interacted with the pump trailing end, gradually slowed down, and blue-shifted (frequency up-conversion, see Fig. 3b). This pump-probe interaction is known as cross

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phase modulation (XPM), and induces chirp that depends on the pump pulse shape [66]. Ifδn = n2I is large enough to form a horizon, the shifting probe become slower than the pulseand they separate. In the co-moving frame the probe is reflected (Fig. 1b) while conserving itsco-moving frequency ω′probe, but not ω (Fig. 2). This reflection demonstrated the existence ofoptical horizons [26].

Philbin et.al. [26] used 70 fs duration FWHM (full width at half maximum), 800 nm carrierwavelength pulses with peak power of about 50 W. The PCF was 1.5 m long with core diametersmaller than 2 µm. The CW probe was tunable around 1500 nm wavelength with 100− 600 µWpower. Pump power was kept low to maintain a stable single soliton and reduce high-ordernonlinearities, such as the Raman effect [66]. Self-steepening was hoped to realize high Hawkingtemperatures.

This set-up achieved a mere 10−3 percent efficiency for probe shifting. Only 10−4 of the CWprobe power was expected to interact with the pump over the 1.5 m fibre, and another orderof magnitude reduction was attributed to tunnelling of the probe through the narrow pumpbarrier [26]. While clearly showing probe blue-shifting at the white-hole horizon, this set-upcould not produce detectable negative frequency partners. It was unclear whether they shouldform around the phase velocity horizon, where ω′ = 0, or around the supported frequency −ω′in(Fig. 2).

Choudhary and konig [67] reported probe red-shifting and studied its tunnelling. Theyused similar experimental parameters to [26], with tuneable 50 fs pump pulses and visible CWprobe at 532 nm (to avoid the difficulty of synchronisation). Tunnelling was minimal for probedetuning up to twice the soliton bandwidth (relative to the matching frequency), but limitedpump-probe interaction kept the conversion efficiency small. Tartara [68] derived both pumpand probe from the same 105 fs source and propagated them in a 1.1 m fibre, demonstratingconversion efficiencies of tens of percents. An optical parametric amplifier produced the tuneableprobe.

The Raman effect cause the pump to decelerate, and was shown to only slightly changethe shifted probe spectra [69]. Shifting of dispersive waves [70, 71] and trapping the probelight [72–76] were also related to the Raman effect. An optical ’black-hole laser’ that use boththe white and black-hole horizons was predicted and analysed [77–81]. The horizon dynamicswas also related to the formation of optical rogue waves and champion solitons [82–85], and theability to form all-optical transistors [86]. Similar ’front-induced transitions’ were studied inother areas of photonics [87], and the quest for measuring analogue Hawking radiation continued.

4 Horizonless emissions

Faccio’s group realized optical phase velocity horizons, without a group velocity horizon, inbulk optics [27, 88]. They constructed pulses of super-luminal group velocities, by using theirthree dimensional nature [89]. A pulsed Bessel beam is one such example. It is composed ofinfinitely many plane waves on a cone with a constant angle θ with the propagation axis. It canbe made with an axicon lens [89]. At the apex of the cone, the plane waves interfere to forma bright spot that moves with super-luminal velocity that depends on θ. Belgiorno et al. [27]used super-luminal pulses that travelled as narrow and powerful filaments inside fused silica.Looking perpendicularly to their propagation direction, they detected radiation around thephase velocity horizon – where the radiation phase velocity matched the pulses group velocity(Fig. 3c). They could not measure along the propagation direction, because vast radiation wasproduced by the powerful pulses there (peak intensity was as high as 1013W/cm2 centred at1055 nm wavelength).

Spurious radiation was a key issue even for observation at 90 degrees. Special care wastaken to tune the radiation into spectral windows of minimal noise. Correctly identifying allthe signals in such a set-up is another major challenge. Concerns were raised [90–92] and in-

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depth analysis related the radiation to a horizonless super-luminal perturbation, possibly linkedto Hawking radiation [93, 94]. This work stressed the importance of a blocking group velocityhorizon to the Hawking effect. Thermal Hawking radiation was predicted for super-luminalpulses in materials with linear dispersion at low energies (like diamond) [93, 94], but is yet tobe measured.

Leonhardt and Rosenberg [95] related the emission in [27] to the physics of Cherenkovradiation2 in a surprising way. The pulse was modelled as a super-luminal light bullet [97], andfound to behave like a moving magnetic dipole that is predicted to emit Cherenkov radiationwith a discontinuous spectrum [98,99]. This discontinuity is exactly at the phase horizon – wherethe effective dipole velocity equals the phase velocity of light and ω′ = 0 (Fig. 2). Interferencesturn the discontinuity into a peak if the dipole is an extended object, like the light bullet.

5 The role of negative frequencies

Negative frequencies are an integral part of Hawking radiation and can also appear as dispersivewaves. Solitons emit dispersive waves (also known as resonant radiation) at a shifted frequencywhen disturbed by higher order dispersion. Non-dispersive three dimensional light bullets emitsimilar radiation [89]. Both emissions can be viewed to originate from self-scattering of thepulse by its nonlinear refractive index barrier. Conservation of momentum dictates the emittedfrequency through a phase matching condition [66, 100]. In the reference frame co-movingwith the stable pulse, this condition becomes a conservation of energy, given by the pulseco-moving frequency, ω′pulse (Fig. 2). The frequency shifting at optical horizons extends thispicture to the scattering of probe light. The prediction of negative Hawking modes suggestedthat negative dispersive waves should similarly appear. Rubino et al. [28] had measured thisnegative-frequency resonant radiation (NRR) after it was neglected and disregarded for a longtime. This required very rapid (non-adiabatic) temporal changes of the pulse envelope.

Intense pulses created NRR through steep shocks in two ways [28]: 7 fs high order solitons[66] of about 300 pJ energy and 800 nm carrier wavelength were placed in 5 mm PCFs; andpulsed Bessel beams of 60 fs duration, about 20 µJ energy and 800 nm carrier wavelength weresent through a 2 cm long bulk calcium fluoride (CaF2), as seen in Fig. 3d. Despite NRR was seenboth in optical fibres and in bulk, further work was needed to exclude other possible mechanismsfor the effect – contributions due to high order spatial modes, and possible interactions betweenthe complex pulse and additional radiation.

Rubino et al. [101] later used a relativistic scattering potential to directly explain NRRformation, and Petev et al. [93] further related it to Hawking radiation. Conforti et al. [102]extended the theory to pulses in materials of normal dispersion (where the pulse experiencedispersive broadening) and dominating second order nonlinearity, which effectively creates anonlinear refractive index. The same conditions for phase matching and non-adiabatic tempo-ral evolution were found. Analytic derivation of the NRR [103] directly related it to interactionsbetween fields and their conjugates (or the mixing of positive and negative frequency modes).It further strengthened the connection between NRR and Hawking radiation, which originatefrom a Bogoliubov transformation that mixes creation and annihilation operators [38,64]. McLe-naghan and Konig [104] studied NRR for different PCF lengths and input chirps, and inferredUV propagation loss of ∼ 2dB/mm. We stress that the negative frequency (and negative norm)of the Hawking modes beyond the horizon is key to the Hawking amplification process [38],extracting energy from the background metric in accordance to a Bogoliubov transformation.

2Not to be confused with dispersive waves [96]

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6 ’It’s just optics’

Hawking radiation is a universal geometric effect that emerge due to conversion of modes at ahorizon, regardless of the microscopic physics that create the background space-time geometry[13]. The same (generalized) derivation applies for both astrophysical black-holes and analoguesystems [13, 26, 105–110]. This established universality proved insightful for both gravity andoptics (arguably solving the trans-Planckian problem and discovering negative frequencies inoptics are two examples [14,15]). However, some researchers from both the optics and gravity (orquantum field theory on curved space-times) communities are still not comfortable with analogueHawking radiation, claiming that the effect is ’just optics’ 3. Such claims do not undermine(analogue) Hawking radiation, but complete our understanding of it. It’s also possible to directlyexplain the effect using the underlying microscopic physics – unknown quantum gravity for realblack-holes, and nonlinear optics for the optical analogues.

The classical dynamics at optical horizons can be captured using numerical solutions thattake into account the entire electric field (including negative frequencies), the dispersion relationand all nonlinearities [66, 111–113]. These are extensions and alternatives for solving the usualgeneralised nonlinear Schrodinger equation (GNLSE), which is used extensively in standarddescriptions of ultrafast nonlinear fibre optics [66,100,114].

Another approach directly relates the phenomena to discrete photonic interactions. It usescascaded four-wave mixing of discrete spectra, and takes the continuum limit for compari-son [115, 116]. A CW probe and a pair of beating quasi-CWs (of 1 ns duration) positionedsymmetrically around the pump central frequency were being mixed continuously (Fig. 3e). Ateach step of the cascaded process, the probe shifted by the pair’s detuning, and all three gener-ated an equidistant frequency comb. A resonant amplification was produced at the frequencycorresponding to conservation of ω′probe. Similar amplification appeared at the dispersive waveresonance, at ω′pump. The experiments used low cascade orders (n = 5 − 7 for the probe shiftand about n = 15 for the dispersive waves) and were supported by numerical analysis. Thispicture shows how energy is being transferred from the pump to the shifting probe, by effectivelyabsorbing pump photons of one frequency and emitting at another. The cascaded four wavemixing also details the back reaction mechanism, where the pump undergoes spectral recoil.

The studies [115, 116] addressed the relevant phase matching conditions, but the efficiencyof the cascaded process was found only in [117], for dispersive waves. Remarkably, it showedthat the known concepts from horizon physics are crucial ingredients even when the pumpis made of beating quasi-CWs that form a frequency comb: the mixing was analysed using(temporal) soliton fission dynamics, being efficient only when the frequency spacing betweenthe CWs effectively generated compressing (non-adiabatic) high order solitons. The frequency-domain analysis was intractable for high cascade orders. The maximally compressed beat cyclecorresponded to about 102 fs duration FWHM in the efficient regime, of high cascading orders.Conversion efficiencies of 10−4 over a 100 m fibre were reported.

The generation of NRR in media with quadratic nonlinearity was also related to a cascadingprocess [102], which must also be the case for cubic materials. However, negative frequencieswere not demonstrated yet using a beating CW pump. It would be interesting to test how thecascading mixing behaves as it reaches negative frequencies, past the phase velocity horizon.

Temporal analogues of reflection and refraction [118, 119] are also used to explain the fre-quency shifts at optical horizons. It compares the frequency change due to XPM to the wavenumber change during refraction. The reflection at the horizon is analogous to total internalreflection. The tunnelling through the refractive index barrier is compared to frustrated totalinternal reflection, where an evanescent wave extends beyond the pulse and tunnels through.

3The author have personally encountered such allegations from both communities, and believe that theypartially originate from not separating ’gravity’, predicted to create the black-hole event horizon, and ’kinematics’,which are responsible for the Hawking effect once a horizon is formed [13].

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The optical horizon can be seen as a temporal beam splitter for probe light (acting also as anamplifier when considering negative frequencies as well).

Comparing the mode mixing and squeezing of parametric amplification [120] to the Hawkingeffect is also useful [38]. The origin of the two phenomena is completely different, and even in theoptical analogue, simple down conversion obey different phase matching conditions that conservetotal lab frequency [62, 66] and cannot explain Hawking radiation. However, the two shareintuition and the Bogoliubov transformation between incoming and outgoing modes. In bothcases a pump creates ’signal’ and ’idler’ waves: the vacuum spontaneously generates quantumemissions, while stimulating the effect with classical light amplifies the effect for specific modes.

7 Observing stimulated Hawking radiation

Spontaneous Hawking radiation originate from the quantum vacuum. Classical laser fields canreplace the vacuum and stimulate the effect. Drori et al. [33] observed stimulated negativeHawking radiation in an optical analogue (Fig. 3f). The idea of [26] was used, but with moresuitable pump pulses and a pulsed probe. Analysis of the fibre dispersion (schematically seen inFig. 2) and simulations of the experiment allowed to optimise the experimental parameters. Thepump was a high order few-cycle soliton that compressed and collapsed due to soliton fission [66].Its peak power reached a few hundred kilo-watts, with 800 nm central wavelength. Very non-adiabatic dynamics generated NRR in the mid UV (Fig. 3f), at the negative of the pumpco-moving frequency, −ω′pump. This ensured the formation of steep refractive index variations,increasing the analogous surface gravity and reducing the probability of probe tunnelling. Theprobe was derived from the pump’s laser, and was generated through cascaded Raman scatteringin a meter long PCF. Negative prechirp was used to enhance the Raman induced frequencyshift [121], whose wavelength was continuously tuned up to 1650 nm by varying the input power.Peak powers were around 1 kW.

Efficient and broad-band probe frequency shifts at the horizon accompanied negative Hawk-ing radiation in the mid-UV, conserving the probe co-moving frequency ω′probe and its conjugate−ω′probe, respectively. Since the fibre had strong dispersion, with different notions for phase andgroup velocity horizons, the spectrum was not Planckian [33, 64, 107]. Varying ω′probe, bothsignals shifted according to theory, supporting the correct interpretation of the negative Hawk-ing radiation. By this, also the interpretation of the NRR was supported, and its analytictheory [103] was generalised to include the Hawking process [113]. Linear relation was verifiedbetween the probe power and that of the negative Hawking signals up to a point where thesignal saturated. This saturation was related to a back-reaction of the probe on the pulse,which is a prerequisite for Hawking radiation (since its energy is drawn from the pump, or theblack-hole that curves the metric). It also slightly reduced the magnitude of the NRR (Fig. 3f).

The rate of spontaneous Hawking emission was estimated based on the magnitude of thestimulated effect [33]. It was found to be too low for measurement in the system used, calling todesign an improved setup. The predicted spontaneous effect is minuscule and is overwhelmed bythe noise in the system, which for the negative Hawking radiation is dominated by fluctuationsof the overlapping NRR. The multi-mode nature of the PCF in the UV [66] was found to reducethe observed signal power by up to four orders of magnitude.

The experiment [33] showed how robust the Hawking effect is: it appeared despite theextreme nonlinear dynamics of the collapsing soliton. Since time in the co-moving frame isrelated to the propagation distance along the fibre, rapid variations in the pump profile do notviolate the required slow evolution of the metric as long as they appear over a length scalemuch larger than the Hawking radiation wavelength [13]. As such, pump deceleration due tothe Raman effect could be accounted by simply correcting its central frequency.

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Figure 3: Central measurements. (a) Schematic absorption (dotted line) and refractive index(solid line) for generating ’slow light’ using electromagnetically induced transparency (EIT), asthe inset depicts. The refractive index has small but rapid variation at the narrow resonance.The slow group velocity is inversely related to this steep change [52, 53]. (b) Frequency blue-shift of probe light at fibre optical white-hole event horizon. 70 fs pump solitons were used in1.5 m long nonlinear PCF. A small fraction of CW probe at ω1 is converted to ω2, conservingits co-moving frequency, ω′probe. From [26]; reprinted with permission from c©2008 AAAS. (c)Horizonless emissions from a Bessel filament creating a super-luminal refractive index pertur-bation in bulk silica [27, 93]. The radiation was seen around the phase velocity horizon (whereω′ = 0) and increased with pump energy (in µJ). The wavelength shift (black line) and broad-ening was in agreement with the increased nonlinear refractive index. From [27]; reprinted withpermission from c©2010 APS (d) Negative-frequency resonant radiation (NRR) in bulk CaF2.16 µJ Bessel pump beam at 800 nm created dispersive-waves (around 600 nm) and NRR (around350 nm) under highly non-adiabatic temporal evolution. From [28]; reprinted with permissionfrom c©2012 APS. (e) Comparison of frequency translation induced by a pump soliton (orangecurve) and a pair of quasi-CW fields (purple curve). Probe light is being blue-shifted (’Idler’) ina similar way in both cases, relating the effect to cascaded four-wave mixing. Similar dispersive-waves (DW) are also seen. From [115]; reprinted with permission from c©2014 Springer Nature.(f) Negative frequency stimulated Hawking radiation. A compressing high-order few-cycle soli-ton [33] generated horizons and NRR (green dotted-dashed curve). When a pulsed probe at1450 nm interacted with the black-hole horizon, it red shifted (not shown) and produced neg-ative Hawking radiation that overlapped the NRR (black solid curve). Subtracting the NRRrevealed the UV Hawking signal (blue curve with 2σ error bars). From [33]; reprinted withpermission from c©2019 APS.

8 Beyond the horizon?

Analogue gravity in optics has come a long way: from erroneous visionary ideas, to carefulanalysis of experimental demonstrations. It drove new research directions and understandingsin optics. The reality of optical horizons and negative frequencies became clear, and multipleoptical interpretations made their physics more tangible. The crucial role of dispersion wasstressed, determining the Hawking spectrum (Fig. 2). A group velocity horizon is blockingthe radiation modes and allow their efficient conversion. A phase velocity horizon marks thesupport of non-trivial negative frequency modes, that allow for the Hawking amplification. Theoptical and other analogues, especially in water waves and BECs, provide extensive insights forgravity through concrete real-world examples. The universality of the Hawking effect separatesit from gravity and high-energy physics, and emphasises the central role of classical fields to theprocess. This hints to more possibilities in gravity [15] and in optics [122], even before goingfully quantum.

The main challenge for observing the spontaneous (quantum) effect in optics is its low power.Noise from spurious radiation makes matters worse. Using knowledge from the stimulatedeffect [33] could help overcome these challenges.

The demonstrated robustness of the Hawking effect is calling for new experiments to lead theway. Our increased confidence in the interpretation of the results allows to further separate fromthe traditional scheme of Hawking radiation, developing bolder questions and ideas. Opticaltechnologies allow to realise wild ideas and to study them with high precision. We call for morecross-fertilisation between research of the different black-hole analogues, which might open newhorizons in these fields.

9 Funding

Fellowship of the Sustainability and Energy Research Initiative (SAERI) program, the Weiz-mann Institute of Science; European Research Council; and the Israel Science Foundation.

10 Acknowledgements

I am grateful for discussions and comments from David Bermudez, Jonathan Drori and UlfLeonhardt. I acknowledge valuable discussions with the participants of the scientific meeting’The next generation of analogue gravity experiments’ at the Royal Society (London, December2019), and thank its organizers: Maxime Jacquet, Silke Weinfurtner and Friedrich Konig.

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