In the format provided by the authors and unedited.

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Integrated Cherenkov Radiation Emitter Eliminating the Electron Velocity Threshold

Fang Liu*, Long Xiao, Yu Ye, Mengxuan Wang, Kaiyu Cui, Xue Feng, Wei Zhang, and

Yidong Huang**

*liu_fang@tsinghua.edu.cn; **yidonghuang@tsinghua.edu.cn Department of Electronic Engineering, Tsinghua National Laboratory for Information Science and Technology, Tsinghua University, Beijing 100084, China.

1. Effective Permittivity

The hyperbolic metamaterial (HMM) can be treated as a homogeneous medium effectively

with permittivity tensor components of x, y, and z, namely

z

0 0= 0 0

0 0

x

z

.

εx and εz are determined by Maxwell-Garnet theory27 as

, 11

m dx y z m d

m d

p pp p

, (S1)

where p is the filling ratio of metal, m and d are the permittivity of the metal and dielectric,

respectively.

Figure S1 The effective permittivity εx and εz of multilayers along x and z direction. The multilayers consisting of alternative Au and SiO2 film with metal filling ratio of p=0.5. When λ0>425nm, the εx>0, εz<0 indicates the multilayers is Type II HMM. When λ0<425nm, the εx>0, εz>0 indicates the multilayers is conventional anisotropic dielectrics.

300 600 900 1200 1500

-40

-30

-20

-10

0

10

Effe

ctiv

e pe

rmitt

ivity

Wavelength 0 (nm)

x

z

Integrated Cherenkov radiation emitter eliminating the electron velocity threshold

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SUPPLEMENTARY INFORMATIONDOI: 10.1038/NPHOTON.2017.45

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2

The multilayer HMM consists of alternative thin layers of Au and SiO2 with metal filling ratio

of p=0.5. According to Eq. (S1), the effective permittivity εx and εz are calculated and shown in

Fig. S1. Here the parameters of Au and SiO2 are based on Ref. (28).

2. Effective refractive index of Cherenkov radiation in HMM

As the mode in HMM is excited by the evanescent electromagnetic (EM) wave surrounding

free electrons, the effective refractive index of EM mode in the multilayers should be related to the

velocity of electrons u0. The analytical expression of effective refractive index as functions of u0

and permittivity of HMM is derived.

Similar to studying light propagation in HMM with a planar wave as the incidence, the wave

equation is written as2

2 2 2

0 0( ) ( ) 0k k E E k k E k E E , (S2)

where k is the wavevector, c is the speed of light in vacuum, 0 is the permeability in vacuum,

E is the electric field, and ω is the angular frequency.

By defining the effective refractive index as2

eff /n kc , (S3)

Eq. (S2) could be rewritten in matrix form 2 2 2 2 2 2 2 2

eff eff eff

2 2 2 2 2 2 2 2

eff eff eff

2 2 2 2 2 2 2 2

eff eff eff

( ) / / /

/ ( ) / / 0

/ / ( ) /

x y z x y x z x

x y z x z y z y

x z y z z x y z

n k k k n k k k n k k k E

n k k k n k k k n k k k E

n k k k n k k k n k k k E

. (S4)

After having developed the determinant, for eff xn , the wave equation can be further simplified

as 2 22 2

2 2 2

eff eff eff

+ =y zx

x z

k kk k

n n n

. (S5)

The evanescent EM wave surrounding electron bunch is assumed as the incidence with kz=ω/u0

and ky=0. Thus, the neff of CR in HMM,

2

eff

0

1 zz

x

cn

u

, (S6)

could be derived. As the evanescent wave of electron bunch is TM polarized, the solution of

eff xn corresponding to TE mode is discarded.

With evanescent EM field surrounding electron bunch as the incidence, the neff of CR in HMM

could be calculated according to Eq. (S6). Different from conventional medium, Fig. S2 reveals

3

that the neff of CR mode in HMM varies with incidence and smaller electron velocity u0 (or energy

E) corresponds to larger neff. For the case of electron energy E=0.1keV, the neff is even larger than

100 around the wavelength of λ0=1000nm. It is the characteristic that leads to the no-threshold CR

in HMM.

Figure S2 The effective refractive index of CR in HMM. The neff of CR when electron energy E=0.1keV

(red straight curve), 1.0keV (blue dashed curve) and 10keV (green dotted curve). Low-energy electrons

(E=0.1keV) could excite CR with super large effective refractive index (neff ~100 around λ0=1000nm). Here,

HMM has the effective permittivity in Fig. S1.

Table S1 Comparison of neff obtained by analytical and numerical methods

Electron energy E Analytical results of neff

(Shown in Fig. 2S)

Simulation results of λ and neff

10keV 8 98nm 8.2

1keV 28 31nm 26

0.1keV 85 14nm 57

The neff of CR in HMM obtained by Eq. (S6) is compared with that derived by simulated results.

For λ0=800nm, the analytical results of neff is 8 (E=10keV), 28 (E=1keV), and 85 (E=0.1keV) as

shown in Fig. S2. While according to the simulated field pattern, the wavelength of CR in HMM

and neff can be estimated and shown in the Table S1. It can be seen that the analytical and simulated

results are in good agreement when E=10keV and 1keV. The relative large deviation for E=0.1keV

might be due to the different models applied for calculation. In simulation, the multilayer structure

is considered and the thickness of Au/SiO2 would be comparable with the CR wavelength when E

500 750 1000 1250 15000

50

100

150neff

Wavelength l0 (nm)

0.1keV

1keV

10keV

4

is rather low. While, in analytical calculation, the effective permittivity is used for calculation with

Eq. (S6).

3. Condition of generating CR in HMM

As we know, the CR occurs only when the velocity of the electrons is larger than that of light

in the medium4. Namely, the condition of generating CR in HMM is that the wavevector of EM

mode in HMM (βHMM) should be larger than that of the evanescent EM wave around electron bunch

(βEB),

HMM EB . (S7)

Here, since the frequency of EM wave is far from intrinsic resonance of metal, the imaginary

part of neff for CR in HMM is much smaller than the real one25 and could be ignored. With

βHMM=neff∙k0, βEB=c/u0∙k0, and Eq. (S6), we have

2

0

0

1 1zz

x

uc

u c

(S8)

The situations are different for two types of HMM.

(1) For Type I HMM ( x <0, z >0), the above inequation is always satisfied, which means that

there is no threshold for velocity of electron bunch u0 to generate CR. Namely, at certain EM

frequency, once the medium could be treated as Type I HMM, CR could be obtained in such

medium no matter how slow the electron bunch moves.

(2) For Type II HMM ( x >0, z <0), the inequation can be simplified to

0 xu c （with 0z ）. (S9)

Compared with Eq. (1), the sign of inequality is reversed, which means that as long as the velocity

of electron bunch lower than a certain value, the CR could be generated in Type II HMM no matter

how small the u0 is.

For Type II HMM, although there is an upper limit of electron velocity, considering the blue

and gray area shown in Fig. 3a, the CR could always be generated in multilayers no matter how

slow or how fast the electrons fly and the electron velocity threshold is eliminated.

4. Angles of wavevector and power flow of CR in HMM

Here we use θk and θS to represent the angle of wavevector k and power flow S of CR in

HMM, respectively, with respect to z-direction. Based on the wavevector matching condition at

the interface between the vacuum and HMM, the angle θk could be calculated by

tan k x zk k . (S10)

5

While in HMM, the wavevector component kz and kx meet the relation24 22

2

0xz

x z

kkk

, (S11)

where kz is equal to the propagation constant of the evanescent EM wave around electron bunch26,

0/zk u . (S12)

Thus, according to Eq. (S10), (S11), and (S12), the angle θk could be derived as

2

0 1tan k z

x

u

c

. (S13)

Figure S3 Radiation angle in HMM. a, The angle of wavevector k (θk) and poynting vector S (θS) of

CR in HMM calculated analytically. The angle increases with wavelength and varies little when E<10keV

at a certain frequency. b, The angle difference between k and S (θS-θk), which increases with wavelength.

Here, HMM has the effective permittivity in Fig. S1.

6

Since the direction of k is given by D H and the direction of S is given by E H 2, the angle

θs could be obtained by

tan tanzS k

x

. (S14)

According to Eq. (S13) and (S14), Fig. S3 illustrates the angles of wavevector (θk) and power

flow (θs) of CR in HMM. The negative θk means k always points to the top of multilayers and

positive θS indicates the radiation energy transfers gradually from top to bottom of the multilayers.

The difference of angle between k and S , namely θS-θk, increases with λ0. This interesting

phenomenon is due to the abnormal dispersion relation of HMM. For a certain frequency, θk and

θS vary little with E when E<10keV. From simulation results in Fig. 3b and 3c, the θk could also

be obtained along the vertical direction of CR equiphase surface, which is in good agreement with

the analytical calculation in Fig. S3.

5. Calculation of local density of states

To analyze the factors related to the radiation efficiency of CR in HMM at different frequencies,

the local density of states (LDOS) of HMM was calculated analytically and compared with the

simulated radiation power spectrum. As free electron bunch could be treated as a diploe14, we

assume a dipole is placed above HMM with a distance of d. For perpendicular and parallel

orientation of the dipole to the interface, the corresponding LDOS isS1

0

0

3Re ( )

4

3Re ( )

2

dk D k

dk D k

. (S15)

The D and D are decided by

2

2 2

1 0 1 0

3

2

1 0

11 1

11

z z

z

ik d ik dzs p

z

ik d

p

z

k kD r e r e

k k k

kD r e

k k

, (S16)

where ε1 is the permittivity medium around electron bunch, kz is the component of the wavevector

along z axis; and ,p sr is the reflection coefficient of HMM for p- or s-polarized wave, respectively.

Applying Eq. (S16) in the case of multilayer substrates with incident light from air, the

reflection coefficient ,p sr is easy to be obtained via the transfer matrix methodS2. Thus,

7

1

23

LDOS (S17)

Figure S4 LDOS and spectrum of CR in HMM. a, Calculated LDOS as a function of wavelength

(frequency). b, The product of LDOS and single photon energy (blue curve), and power density spectrum

when E=10keV (red curve).

According to Eq. (S17), the LDOS of HMM at different wavelengths was calculated and

illustrated in Fig. S4a. Here we multiply LDOS by photon energy and find that the profile of

LDOS× curve is similar to the emission spectrum obtained by Frank-Tamm formula (Eq. S18).

Compared with the simulated radiation spectra when E is higher than 10keV as shown in Fig. 3d,

the analytical results in Fig. S4b have similar profile. The spectra difference between analytical

and simulation results might be because: (1) The analytical calculation is based on two-dimensional

model, while the simulation is based on three-dimensional model; (2) In the simulation model, the

gap d between electron bunch and HMM would greatly affect the radiation efficiency when E is

8

low. Longer wavelength has higher coupling efficiency of EM field from electron bunch to HMM.

Therefore, the LDOS of HMM is an important factor of deciding the CR generation efficiency in

multilayers at different frequency.

6. Calculation of CR energy in HMM

The radiation power of CR in HMM was simulated numerically considering the model in Fig.

1 and the parameters of practical fabricated device. Figure S5 illustrates that, for different electron

energy E, the radiation power does not change significantly. The power density of CR in HMM is

~1012W/cm2 with charge quantity of 3.2×10-15C in an electron bunch, which is several orders of

magnitude higher than the previous reported results16,S3. It is amazing that such a low-energy

electron (E=0.1-1keV) could radiate CR with super high power density.

Figure S5 Simulated power density of CR generated in HMM for different E. The low-energy electron

could radiate power density of ~1012W/cm2, which is several orders of magnitude higher than that of CR

generated by other nanostructures. Here, the charge quantity in electron bunch is assumed as 3.2×10-15C for

calculation.

The simulated radiation power spectrum corresponding to Fig. S5 is illustrated in Fig. 3d. To

understand how total radiation power changes with E and the profile of radiation spectrum, the

radiation energy is calculated by assuming gap d=0. The Frank-Tamm formula describes the total

energy W radiated by electron with velocity u0 asS4 2 2

2 2

0

( ) (1 )4 ( )

q l cdW d

u n

, (S18)

9

where q is the charge of electrons, μ(ω) is the medium permeability, n(ω) is the refractive index of

medium, and l is the length of electron interacting with medium.

With n(ω)=neff obtained by Eq. (S6) and shown in Fig. S2, the energy density of CR in HMM

could be calculated analytically. It is found that lower electron energy E could generate even higher

CR energy in HMM indicated by Fig. S6, which is in contrast to CR in conventional medium.

According to Eq. (S18), the power density of CR in water, for example, could be obtained and

higher electron energy results in higher radiation power, which is because n(ω) is unchanged at a

certain frequency and higher u0 would result in higher dW/dω. However, for CR in HMM, the

effective refractive index is changeable and lower energy E (or velocity u0) corresponds to higher

n(ω) (Fig. S2) and even larger u0×n(ω), which results in higher dW/dω (Fig. S6) as well as higher

radiation power. As discussed in Fig. S4, the radiation spectrum is related to the LDOS of HMM.

Figure S6 Energy density spectrum of CR generated in HMM according to F-T formula. Lower

electron energy corresponds to higher energy radiation.

For the simulation model, there is a gap between electrons and HMM. For electrons with low

E, the evanescent field is highly bounded surrounding the flying electrons, which results in the

relatively weak coupling of evanescent EM field into HMM to generated CR (i.e. the radiation

power for E=0.1keV in Fig. S5), even though dW/dω shown in Fig. S6 is rather high. For electrons

with high E, the influence of the gap could be neglected and lower dW/dω leads to lower radiation

power (i.e. the radiation power for E=40, 80keV in Fig. S5). Therefore, as shown in Fig. S5, the

radiation power increases first and then decreases with E. Similarly, the profile of radiation power

spectrum (Fig. 3d) could be easily understood.

500 750 1000 1250 15000

10

20

30

40

50

60

dW

/d

(

2/

c)

Wavelength 0 (nm)

0.1keV

1keV

10keV

20keV

40keV

10

The radiation efficiency could also be described by the number of photons (N) radiated by

electrons flying a certain distance. The radiated photon number in HMM is calculated and

compared with that in water as well as the simulation results. According to Eq. (S18), the N radiated

by electrons with charge q can be calculated, 2 2

2 2

0

( )(1 )4 ( )

q l cN d

u n

. (S19)

Assuming q=1.6×10-19C, l=1cm, and u0=c/50 (corresponding to electron energy E=~0.1keV), 620

photons/cm in the wavelength region of λ0=470nm~1500nm could be emitted in HMM by single

electron. While for traditional CR in water (assume n=1.33), this value is 475 photons/cm in the

same wavelength region with electron energy E=500keVS5. It is amazing that the CR emission

efficiency in HMM is higher than that in conventional medium with such low electron energy,

which is ascribed to the higher LDOS of photon in HMM. The simulated power density of

~1012W/cm2 (Fig. S5) corresponds to ~400 photons/cm emitted by single electron and is in good

agreement with the analytical results (620 photons/cm by single electron). The difference on photon

number is ascribed to the three-dimensional numerical model and two-dimensional analytical

model.

7. Light coupled from HMM to free space via plasmonic periodic nano-slits

To detect the CR, we have to couple CR in HMM to free space. Here a metal (Au) plasmonic

periodic nano-slitsS6-S8 located under multilayer HMM is applied. Different from conventional

metal grating, the width of the nano-slit is only 70nm which is only about 1/10 of the pitch of nano-

slits. This kind of nano-slits could support localized surface plasmon modeS6-S8, which would help

us couple the CR in HMM to light in free space for measurement. Figure S7a shows the simulation

result that CR in HMM is coupled to free space, and Fig. S7b illustrates the power density spectrum

of CR in HMM (black curve) and that of light coupled to free space with Pslit=530nm, 570nm, and

610nm (corresponding to red, green, and blue curve, respectively). Compared with CR in HMM,

only ~0.7% of the light energy could be coupled to free space. More energy could be coupled out

if an adiabatic structure is applied. Figure S7b shows that, for different Pslit, the light spectra

coupled to free space have similar profile and different central wavelength.

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Figure S7 Simulation results for CR in HMM with periodic metal nano-slits. a, Numerical simulation

(main component of electric field Ez at λ0=800nm) of CR in HMM and the light coupled to free space by

nano-slits. b, Simulation of power density spectra of CR in HMM (P1, black curve) and light spectra coupled

to free space when Pslit=530nm (P2, red curve), 570nm (P3, green curve) and 610nm (P4, blue curve). Here,

the electron energy E=1.0keV.

Compared with the simulation results, the peaks of spectrum shown in Fig. 2d are somewhat

different. The following reasons might result in the difference. (1) The pitch of nano-slits might

deviate from designed value by about 10nm since FIB milling could not control the pitch so

accurately. (2) The deviation of the thickness of PMMA polymer layer between HMM and nano-

slits might change the peak of spectrum. In experiment, if the thickness of PMMA layer is larger

than designed value and the light coupling between HMM and nano-slits becomes weaker, the

larger λ0 would correspond to higher coupling efficiency and the peak of the free-space light

spectrum would shift to longer wavelength. (3) The actual permittivity of Au in experiment should

12

be different from the value used in simulation, which would also lead to the deviation of spectrum

from the simulation.

8. Condition for the Au/SiO2 multilayer to be HMM

Figure S8 Iso-frequency contour (IFC) of Au/SiO2 multilayers for λ0=600nm. Black curve is hyperbola

IFC obtained by effective medium theory. Red, blue and green curves are the IFC obtained by transmission

matrix method when hm+hd=40nm, 20nm and 4nm (hm=hd), respectively.

According to the theoretical analysis based on effective medium theory (EMT), the CR has

no threshold in HMM. However, the effective medium theory will breakdown at some point. The

effective refractive index neff shown in Fig. S2 indicates that, when electron energy E is low, the

neff of CR in HMM is super high. If the effective wavelength of CR (λ0/neff) is comparable or

13

smaller than the thickness of Au/SiO2 layer, the EMT should not be available any more, which

means the Au/SiO2 multilayers could not be considered as HMM.

Thus, the transmission matrix methodS2 is applied to calculate the actual iso-frequency

contour (IFC) of Au/SiO2 multilayers. Figure S8 illustrates the IFC obtained by EMT (black

hyperbolic curve) and TMM (red, blue and green curve) when the thickness of Au/SiO2 are

different. It can be seen that, for multilayer Au/SiO2, the IFC deviates from hyperbolic curve when

kz/k0 is large and the finite kz/k0 would result in an electron energy E (velocity u0) threshold.

Considering the wavevector matching along z direction between evanescent field surrounding free

electrons and CR mode in multilayer, we could use the highest kz/k0 (defined as nmax) in Fig. S8 to

estimate the lowest umin (=c/nmax) allowing for CR generation. For the fabricated Au/SiO2 in this

paper (hm+hd=20nm), the blue curve in Fig. S8 indicates that nmax is ~50, and the umin is ~c/50

(corresponding to the lowest E ~0.1keV). Since 2nm-thick Au and SiO2 film could be realizedS9,

the Au/SiO2 multilayer with hm+hd=2nm might be realized and corresponding IFC is shown as the

green curve in Fig. S8. The nmax could reach ~200 and the corresponding umin is ~c/200

(corresponding to the lowest E ~0.0064keV). Such low electron energy could be considered that

the threshold to generate CR is eliminated.

9. Measurement setup

Figure S9 Schematic view of measurement system. Integrated CR emitter was placed in a high vacuum

chamber. The light originated from CR in HMM was coupled out by parabolic mirror, lens and fibers and

measured by power meter and spectrometer.

14

10. Influence of long-ranged Coulomb interaction

Besides CR effect, the long-ranged Coulomb interaction26 might also contribute to the radiation.

To indicate that the Coulomb interaction is much weaker, a test device illustrated in Fig. S10a was

fabricated and measured. The test device has the same structure as the CR-emitter, just except

replacing the multilayer HMM and nano-slits with an Au monolayer film. The thickness of the Au

monolayer is 30~50nm for exciting surface plasmon by free electrons and coupling excited surface

plasmon to the light in SiO2. Using the same measurement method described in the manuscript, the

radiation light power from the test device was measured and shown as the green circles in Fig.

S10b.

Figure S10 Experiment for evaluating the influence of long-ranged Coulomb interaction. a, Schematic

figure of test device by replacing multilayer HMM and nano-slits in Fig. 1a with an Au film. b, Measured

light output of test device without HMM (green circles, test device shown in a) and with HMM (black dots,

CR emitter shown in Fig. 1a). The black dots are same as those in Fig. 2a.

It can be seen that, under the same voltage Vca, the optical signal from the test device is much

lower than that from the CR-emitter (shown as black dots). When Vca is lower than 1keV (the

current Ica is less than 10μA), the optical signal is too weak to be observed by the power meter used

in our experiment (Thorlabs S130VC). When Vca is increased to 1.2~1.6keV (Ica is about 15~26μA),

15

the light output of only 3~4nW is obtained. It can be considered that, without HMM structure, CR

can not occur under such a low Vca and only the radiation power generated by long-ranged Coulomb

interaction was measured.

For the CR-emitter described in the manuscript, with the help of HMM structure, electron

velocity threshold can be eliminated and CR occurs at such a low Vca. That is why we observed

much high optical signal from CR-emitter. Considering that less than 1% radiation power could be

coupled out to free space in Fig. 1a (Supplementary Information, Section 7), the radiation power

generated by long-ranged Coulomb interaction should be about 3~4 orders of magnitude lower

than that generated by CR in HMM.

16

Table S2. Electron energy E and corresponding velocity. c is the light speed in vacuum.

Electron energy E Electron velocity u0

1000 keV 0.9411c

300 keV 0.7765c

100 keV 0.5482c

10 keV 0.1950c

1.4 keV 0.0739c

1.0 keV 0.0625c

0.25 keV 0.0313c

0.1 keV 0.0198c

0.01 keV 0.0063c

0.001 keV 0.0020c

17

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Express 15, 16222–16229 (2007).

S4. Frank, I. M. & Tamm, I. Coherent visible radiation of fast electrons passing through matter. C

R Acad Sci URSS 14, 109–114 (1937).

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2013).

S7. Sobhani, A. et al. Narrowband photodetection in the near-infrared with a plasmon-induced hot

electron device. Nat. Commun. 4, 1643 (2013).

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5683 (2014).

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