A Brewster route to Cherenkov detectors

Cherenkov detectors enable a valuable tool to identify high-energy particles. However, their sensitivity and momentum coverage are limited by the refractive index of host materials. Especially, identifying particles with energy above multiple gigaelectronvolts requires host materials with a near-unity refractive index, which are limited to bulky gas chambers. Overcoming this fundamental material limit is important for future particle detectors yet remains a long-standing challenge. Here, we propose a different paradigm for Cherenkov detectors that utilizes the broadband angular filter made from stacks of variable one-dimensional photonic crystals. Owing to the Brewster effect, the angular filter is transparent only to Cherenkov photons from a precise incident angle. Particle identification is achieved by mapping each Cherenkov angle to the peak-intensity position of transmitted photons in the detection plane. Such angular filtering effect, although decreases the photon number collected in the detection plane, enables the realization of a non-dispersive pseudo refractive index over the entire visible spectrum. Moreover, the pseudo refractive index can be flexibly designed to different values close to unity. Our angular-selective Brewster paradigm offers a feasible solution to implement compact and highly sensitive Cherenkov detectors especially in beam lines with a small angular divergence using regular dielectrics.

including those close to unity.Our angular-selective Brewster paradigm offers a feasible solution to implement compact and highly sensitive Cherenkov detectors especially in beam lines and it can cover a wide momentum range using readily available dielectric materials.
A charged particle travelling in a transparent host medium emits photons when it travels faster than the phase velocity of the photons in that medium.This phenomenon is known as Cherenkov radiation, which is first observed experimentally by P. A. Cherenkov (under the guidance of S. Vavilov) [1,2] and later interpreted theoretically by I. M. Frank and I. Tamm [3,4].Remarkably, Cherenkov radiation [5][6][7][8] has enabled the invention of Cherenkov detectors [9][10][11][12][13][14] for identifying particles over a large momentum range in high-energy physics and astrophysics.The Cherenkov detector has played an essential role during the discovery of many elementary particles, including anti-protons [15], J/ particles [16], neutrino oscillations [17], etc.
According to Frank and Tamm's theory of Cherenkov radiation [18], the particle velocity  can be determined by measuring the light emission angle  CR (also known as the Cherenkov angle) where  is the refractive index of the host medium, and  is the speed of light in vacuum.Although larger refractive indices give rise to lower Cherenkov thresholds and higher photon yield, they are not always desirable in Cherenkov detectors.The reason is that large refractive index decreases the sensitivity of the Cherenkov angle to small changes in the velocity.In general, the highest detection sensitivity is obtained at  CR → 0 or  → / (The detection sensitivity is defined as , which reaches its maximum at  CR → 0 ).In other words, to identify high-energy particles (i.e. →  ), transparent dielectrics with a near-unity refractive index are often required.Such a constraint limits the host materials used in many Cherenkov radiators to low-index.This is for example the case in ring imaging Cherenkov (RICH) detectors that use gases for the identification of particles with momenta larger than 10 GeV/ [14,19,20].
Another limitation in state-of-the-art Cherenkov detectors is related to the photon emission efficiency: the use of low-index materials inevitably leads to low efficiency (i.e. the photon yield is 2  2 according to Frank and Tamm's theory).Especially when operating nearthreshold (i.e. → / ), the photon yield approaches zero.Consequently, traditional gas radiators generally require bulky gas chambers to produce sufficient photons for detection [14].Regular dielectrics offer a possible route to increase the photon emission efficiency, but their large refractive indices (generally far above unity) effectuate different types of high-energy particles to generate the same Cherenkov angle, namely  CR → cos −1 (1/), which thus makes particle identification impossible.As an example, quartz has a refractive index around 1.4, and its corresponding momentum coverage is limited below 3.5 GeV/ [21].All high-energy particles with momenta above this value emit light at  CR ∈ [44.30°, 44.42°], independent of the particle velocity and particle type.Under this scenario, measuring  CR cannot lead to the identification of the corresponding particle type.
Recently, several theoretical attempts have been made to relax the material limitations in Cherenkov detectors by using modern concepts from nanophotonics and metamaterials [22,23].One attempt proposes metal-based anisotropic metamaterials with one component of the effective refractive index close to unity [22].Another study makes use of all-dielectric one-dimensional (1D) photonic crystals, where the constructive interference of resonance transition radiation is adopted to control the effective Cherenkov angle [23].These nanophotonic Cherenkov detectors can achieve an enhanced sensitivity for any desired momenta range, however only at a specific working frequency.In fact, the working frequency is the major drawback of all nanophotonics-based Cherenkov detectors, i.e. they all have a narrow working bandwidth, resulting from the inherent chromatic dispersion of the constitutive materials (e.g., metal) and the resonant nature of periodic structures (e.g., photonic crystals).There are many more recent advances and ongoing efforts on the Cherenkov effect in nanophotonic settings and in novel material platforms [24][25][26][27][28][29][30][31][32][33][34][35].Nevertheless, the design of a broadband Cherenkov detector using regular transparent dielectrics and at the same time with enhanced performance has remained a longstanding scientific challenge.
Here we propose a new paradigm for Cherenkov detectors by exploiting a broadband angular filter.This broadband angular filter is comprised of stacks of many 1D photonic crystals of different periodicities but identical constituent materials [Fig.S1a].As a result, the Brewster effect makes the angular filter transparent only to p-polarized (i.e.transverse magnetic, TM) light incident at the Brewster angle, while the light with other polarizations or incident at other angles is totally reflected.Moreover, we can readily tailor the band gaps of these 1D photonic crystals to cover a broad spectral range, thus also making the angular selectivity broadband, spanning the entire visible spectrum [36][37][38][39].After transmitting through the broadband angular filter, the Cherenkov radiation in the detection plane features a pseudo Brewster-Cherenkov angle  BCR , namely the angle between the particle velocity and the tangential wavevector parallel to the detection plane [Fig.1].Remarkably, we find that the measurement of  BCR can provide an approach for particle identification at any momentum coverage with wide bandwidth and high sensitivity.This approach thus can tackle the key drawback listed above for nanophotonic Cherenkov detectors.Our approach is especially useful for the identification of a beam of charged particles with different momenta, even when the flux of charged particles is high.
We now proceed to analyze the essential role of the Brewster effect in our proposed particle detectors.As schematically shown in Fig. 1, we consider the charged particle travelling at a constant velocity  in a host material with the relative permittivity of  h and along a trajectory parallel to the surface of the broadband angular filter.Without loss of generality, we set the broadband angular filter composed of two regular transparent dielectrics with relative permittivities  r1 and  r2 [Fig.S1a].The detection plane is located at  =  0 beneath the broadband angular filter, and it is parallel to but far away from the particle trajectory [Fig.1].The Cherenkov radiation transmitting through the broadband angular filter has a tangential wavevector  ̅ BCR =  ̂ + k  parallel to the detection plane.On the one hand,   = / is fixed by the kinematic feature of the charged particle [23,40].On the other hand, the magnitude of  ̅ BCR is locked by the intrinsic electromagnetic property of the broadband angular filter.To be specific, only the p-polarized light incident at the Brewster angle can transmit through our angular filter for the entire spectral range of 400 to 700 nm [Figs.S1-S3], and our particle detector will operate over this broadband wavelength range.
According to the Brewster effect, | ̅ BCR | = √  r1  r2  r1 + r2   [18], which enables us to define a pseudo refractive index ( We highlight that the broadband angular filter is treated as a realistic multilayered structure, instead of an effectively homogeneous material, and hence  BCR [see Methods for its derivations] is completely different from the effective refractive index obtained using the standard homogenization theory.
Interestingly,  BCR is related to the pseudo Brewster-Cherenkov angle  BCR [see inset of In what follows, we shall explore the potential applications of this generalized Frank-Tamm formula for particle identification.As our approach exploits the Brewster effect, we refer to the particle detector depicted in Fig. 1 as the Brewster-Cherenkov detector.
To facilitate the conceptual demonstration, Fig. 2  We highlight that such sensitivity improvement occurs over a broadband frequency range because the pseudo refractive index  BCR in the generalized Frank-Tamm formula can be made approximately non-dispersive using regular transparent dielectrics that have a negligible dispersion in the visible range [41].To illustrate this point, we plot  BCR as a function of the Brewster angle.As shown in Fig. 3,  BCR can be flexibly engineered to arbitrary values (including those close to unity) by a suitable choice of  r1 and  r2 for the two constituent materials of the broadband angular filter.For example, according to equation (2),  r1 = 2.18 (e.g., SiO2) and  r2 = 3.07 (Al2O3) [41] give rise to a pseudo refractive index  BCR = 1.13, which is much smaller (and hence more close to unity) than the lowest refractive index found in natural solid materials (i.e.1.37 for MgF2) [19,20,[41][42][43][44].A pseudo refractive index even closer to unity can be achieved using other material combinations.For example,  BCR = 1.0026, if taking polymers with  r1 = 1.8578 (tetrafluoroethylene-co-hexafluoropropylene-co-vinylidene fluoride (THV, 3M Dyneon 221AZ)) and  r2 = 2.1904 (poly(methyl methacrylate), namely PMMA) [45][46].
A key feature of Brewster-Cherenkov detectors is that the pseudo refractive index  BCR determines their sensitivity and momentum range.Figure 4a shows the relation between the pseudo Brewster-Cherenkov angle  BCR and particle momenta for four elementary particles, namely electron, pion, kaon, and proton.In this exemplary case, we take  BCR = 1.As a final remark, we highlight that our Brewster approach has several unique advantages over the traditional methods for particle identification.First, the Brewster approach eliminates the strict requirement of near-unity-index host materials for the design of Cherenkov radiators.In other words, distinct from traditional Cherenkov detectors such as the RICH detector [13][14][15][16][17], our Brewster approach does not have any special requirements on the refractive index of the host material where the particle is travelling, since the sensitivity of Brewster-Cherenkov detectors is directly determined by  BCR of the broadband angular filter.Consequently, high-index transparent solids or gases with low atomic numbers can now be used as the host material.Such a high-index host material can significantly enhance the number of photons penetrating the broadband angular filter and reaching the detection plane, enabling a higher-efficiency Cherenkov detector.
Another important advantage of our Brewster approach in comparison with all previous nanophotonic approaches [22,23] is that the broadband angular filter does not need to be placed in the path of the high-energy particle beam.The charged particles can travel at a large distance away from the surface of the broadband angular filter so that all the Cherenkov photons are produced in the host material in which the particle is travelling.This way, the generation of secondary particles from the broadband angular filter can be effectively reduced.Last but not least, the performance of the proposed particle detector is robust to fabrication imperfections and geometric fluctuations of the broadband angular filter (see analysis in Refs.[36][37][38][39]).Therefore, this Brewster approach provides promising options to facilitate the design of advanced Cherenkov detectors with enhanced sensitivity, large bandwidth, miniaturized size, ultralight weight and wide momentum coverage, all using readily available regular dielectrics.These enhanced capabilities are especially attractive in the identification of high-energy particles in beam lines.
has tan Brewster = √ r2 / r1 [18].At this Brewster angle, the wavevector component of light parallel to the interface has  BCR =  .Due to the momentum matching at each interface, the value of  BCR is the same for different regions in the broadband angular filter, when the light transmits through the angular filter.In addition, we highlight that all the calculations in this treat the broadband angular filter as a realistic layered structure, instead of an effectively homogenized material by using the effective medium theory.These discussions are shown in section S2 of the supporting information.
Design of the broadband angular filter.The broadband angular filter [Fig.S1a] is comprised of many stacks (i.e. stacks) of 1D photonic crystals.All 1D photonic crystals are made of two regular transparent dielectrics, which have a relative permittivity of  r1 and  r2 , respectively.The  th 1D photonic crystal has a pitch of   =  1 +  2 , where  1 and  2 are the thickness of two dielectric slabs in each pitch.All these 1D photonic crystals have a pitch number of N and a thickness ratio of  1 / 2 = 3/2.This way, the band gap of each 1D photonic crystal can be flexibly tunable by changing   .The light transmission through the broadband angular filter is both angle-dependent and polarization-dependent.For the p-polarized light with arbitrary incident angle (except the one equal to the Brewster angle), the light is almost fully reflected by judiciously overlapping the band gaps of these 1D photonic crystals [Figs.S1 & S2].For the p-polarized light incident at the Brewster angle, it can safely pass through the broadband angular filter with no reflection [Figs.S1 & S2].For the s-polarized light, the transmission through the designed broadband angular filter is negligible for arbitrary incident angle [Fig.S3].The detailed design strategy is shown in sections S2-S4 of the supporting information.
Cherenkov radiation in the detection plane.For Cherenkov radiation passing through the broadband angular filter, while the direction of their in-plane wavevector  ̅ BCR =  ̂ + k  has a pseudo Brewster-Cherenkov angle with respect to the particle trajectory [Fig.S4], their motion in the detection plane is parallel to the particle trajectory (i.e.along the  axis); see the discussion of Cherenkov radiation in the detection plane in section S5 of the supporting information and their dynamics in the detection plane in Movie S1.Robustness of the performance of Brewster-Cherenkov detectors to particle's trajectory.The Brewster-Cherenkov detector has the potential to infer the projection of particle trajectory in the  plane (or the detection plane at  =  0 ).This is because the intensity distribution of the transmitted Cherenkov radiation in the detection plane is symmetric with respect to the projection of the particle trajectory in the  plane [Fig.2a-e].Due to this unique feature, the sensitivity of Brewster-Cherenkov detectors is in principle insensitive to the direction of particle velocity, if the particle velocity is parallel to the surface of the broadband angular filter.On the other hand, for the Brewster-Cherenkov detector, we can always set the particle trajectory very far away from the top surface of the broadband angular filter.Then if the particle velocity has a very small angle with respect to the surface of the broadband angular filter (but the particle would not penetrate through the filter), the performance of Brewster-Cherenkov detectors would not be degraded, since the feature of the transmitted Cherenkov radiation in the detection plane is mostly preserved.

Peak-intensity position of
Influence of the finite thickness of the broadband angular filter on the performance of Brewster-Cherenkov detectors.More discussions on the performance of Brewster-Cherenkov detectors are provided in section S7 and Figs.S8-S9 of the supporting information.When the stack number  of 1D photonic crystals and the periodicity number  of each 1D photonic crystal are finite, the p-polarized light incident at the angles very close to the Brewster angle can also safely pass through the broadband angular filter.This way, there is a small angular (and thus spatial) spread of the transmitted Cherenkov radiation in the detection plane, such as those shown in Fig. S1b-e.This phenomenon would degrade the sensitivity of Brewster-Cherenkov detectors.However, the sensitivity of Brewster-Cherenkov detector can still be guaranteed by effectively avoiding this phenomenon, through increasing both the values of  and  in the practical implementation [Fig.S9].

Fig. 1
generalizes the regular Frank-Tamm formula (i.e.equation (1)), providing a general route to engineer the Cherenkov radiation through the Brewster effect.

||
13 and a particle momentum fixed at 2 GeV/.The resulting values of  BCR are 27.8 o for electron, 27.5 o for pion, 24.3 o for kaon, and 12.2 o for proton [Fig.4a].Such a variation in  BCR indicates that  BCR = 1.13 is suitable for the identification of particles with a momentum less than 10 GeV/c.In comparison,  BCR = 1.0026 gives rise to a Brewster-Cherenkov detector capable of identifying particles with a momentum larger than 10 GeV/ .More interestingly, when  BCR = 1.000001, the corresponding Brewster-Cherenkov detector can even identify particles with ultra-high momenta in the TeV/ range.These results clearly demonstrate that the proposed Brewster-Cherenkov detector can achieve arbitrary momentum coverage with high sensitivity through a proper design of the pseudo refractive index.The pseudo Brewster-Cherenkov angle and the particle velocity can be determined by directly measuring the peak-intensity position  BCR of Cherenkov radiation in the detection plane [Figs.4b & S6-S7].Mathematically, we have |  0 ) , where ∆( 0 ) is the normalized displacement resulting from the refraction of light through the broadband angular filter.For simplicity but without loss of generality, we choose  BCR = 1.13 and  0 = 2.3 mm, and plot in Fig. 4b the ratio |  BCR  0 as a function of momentum for the four elementary particles.At the fixed momentum of 2 GeV/, gradually varies from 0.49, 0.48, 0.43 to 0.22 for electrons, pions, kaons and protons, respectively [e.g., see the corresponding intensity distributions of Cherenkov radiation in the detection plane in Fig. 2a-e].

|
Cherenkov radiation in the detection plane at  =   .The normalized peak-intensity position |  0 ) is analytically calculated according to the ray tracing theory.Here, ∆( 0 ) = − ( displacement induced by the refraction of light through the broadband angular filter, where   is the total thickness of dielectric regions with  r2 in the designed broadband angular filter.Since ∆( 0 ) ∝    0 , we have ∆( 0 ) → 0 if   ≪  0 .In other words, if the detection plane is far away from the particle trajectory and if their distance is much larger than the finite thickness of the broadband angular filter, we have | for a fixed broadband angular filter under different values of  0 is shown in section S6 and Figs.S5-S7 in the supporting information.

Figure 1 |
Figure 1 | Schematic of the proposed Brewster-Cherenkov detector based on a broadband angular

Figure 2 |
Figure 2 | Cherenkov radiation in the detection plane of Brewster-Cherenkov detectors.Here we

Figure 3 |
Figure 3 | Engineering the pseudo refractive index   for Brewster-Cerenkov detectors, achieving parameter regimes that do not exist in natural materials.For the transmitted Cherenkov radiation in the detection plane, its wavevector component parallel to the detection plane has  BCR =  BCR /,

Figure 4 |
Figure 4 | Performance of Brewster-Cherenkov detectors in the identification of high-energy particles.a, Pseudo Brewster-Cherenkov angle  BCR versus particle momentum.This figure is plotted according to the generalized Frank-Tamm formula cos BCR =   BCR  , by transforming the particle velocity to the momentum.The values of pseudo Brewster-Cherenkov angles for four elementary particles, 1 sin Brewster , where  1 = √  r1 / is the wavevector of light in region 1 and  Brewster = √  r2 / √  r1 +  r2 .In other words, we have  BCR =