1

Experimental Observation of Reversed Doppler

Effects in Acoustic Metamaterials

Shilong Zhai, Xiaopeng Zhao1*, Song Liu, Chunrong Luo

1Smart Materials Laboratory, Department of Applied Physics, Northwestern Polytechnical

University, Xi’an 710129 P. R. China,

*email: xpzhao@nwpu.edu.cn

This paper reports an experimental observation of broadband reversed Doppler effects using an

acoustic metamaterial with seven “flute-like” double-meta-molecule clusters. Simulations and

experiments verify that this locally resonant acoustic metamaterial with simultaneous negative

elastic modulus and mass density can realize negative refraction in a broad frequency range. The

constructed metamaterial exhibits broadband reversed Doppler effects. The frequency shift

increases continuously as the frequency increases. The assembling of double-meta-molecule

clusters introduces a new direction in designing double-negative acoustic metamaterials in an

arbitrarily broad frequency range and other various applications.

Keywords: acoustic metamaterials; double-meta-molecule; clusters; double-negative; negative

refraction; reversed Doppler effects; broad band.

2

The Doppler effect refers to the change in frequency from the wave source caused by the

relative motion between the wave source and the observer1, 2

. This phenomenon is applied in many

fields, including scientific research, space technology, traffic management, and medical

diagnosis3-6

. In 1968, Veselago7 theoretically predicted that metamaterials

8-10 with negative

refractions11, 12

can realize the reversed Doppler effect. Recent studies show that the reversed

Doppler shift in electromagnetic waves can be achieved using transmission line13-15

, backward

dipolar spin waves in a magnetic thin film16, 17

and photonic crystals18, 19

; in acoustics, phononic

crystal20

is a feasible material to obtain the reversed Doppler effect. However, this abnormal

phenomenon appears experimentally only in the frequency range corresponding to the energy gap,

and the band is narrow. Based on nonresonant elements, a quasi-1D double-negative acoustic

metamaterial can also realize the reversed Doppler effect in the low-frequency region21

.

In many applications, such as metamaterial absorber and cloak, broadband metamaterials are

required. Given that the mechanism of metamaterial is based on resonance in the periodic

structure22, 23

, the bandwidth of this resonance is narrow by nature. To achieve broadband

absorption, some projects employ nested elements, such as concentric square rings, and realize

double- and even triple-band absorption24, 25

; however, these frequency regions are not close

enough to combine into a broader band because of the limitation of element size. Simulated

studies have verified multiplexed configuration as a convenient approach to obtaining

multiresonance and broadband absorption26

. Xie et al. demonstrated that the resonating nature

resulting from space folding instead of local resonance within the unit cell creates an extremely

broad frequency field (i.e., more than 1000 Hz) of negative index27

. Nonresonant metamaterial

elements can also be used for broadband cloaks28, 29

. Gradient-index structures can be utilized for

3

broadband antireflection30, 31

employing phononic crystals. However, to some extent, the

multiple-scattering-based mechanism limits the working frequency region of acoustic waves in

phononic crystals32

. A general approach to realizing broadband effects is still required. However, a

general method of designing and fabricating a controllable broadband double negative acoustic

metamaterial is yet to be reported. Therefore, realizing arbitrarily broadband acoustic transmission

and obtaining the reversed Doppler effect using negative refraction materials are difficult.

Electromagnetic waves consist of many photonics with different energies and frequencies.

Visible white light (400 THz-800 THz) can be obtained via mixing disparate photons, with colors

corresponding to red, orange, yellow, green, cyan, blue, and purple. Analogous to photons, surface

plasmon polaritons exist in acoustic metamaterials, which can be generated by the resonance of a

sort of acoustic meta-molecule33

. The frequency of plasmon polaritons is related to the geometry

of structural units and responds to the region where the simultaneous negative mass density,

modulus, and the reversed Doppler effect of the meta-molecule material appear. Nevertheless, the

frequency bandwidth is relatively narrow33, 34

. Inspired by the fact that visible light is composed of

seven-color lights, we propose a “flute-like” model of acoustic meta-molecule cluster containing

seven double meta-molecules with different dimensions. A metamaterial sample is experimentally

constructed based on this model. Transmission and reflection results are obtained via experimental

measurements and numerical simulations, from which mass density and bulk modulus are derived

to be simultaneously negative in a broadband range. The calculated and measured refractions are

also negative in the broad resonant frequency region. Taking advantage of this broadband

double-negative sample, we experimentally realize reversed Doppler effects from 1.186 KHz to

6.534 KHz.

4

The analogy betwwen acoustic surface plasmon polaritons and photons is shown in Fig. 1.

The frequency ν of photon is determined by its energy ε, which is expressed as

ε=hν. (1)

No interactions occur among between photons. In the visible portion of the electromagnetic waves,

photons with different frequencies correspond to lights with seven different colors. Visible white

light is formed by mixing these seven colors. Acoustic metamaterials can generate oscillations of

surface plasmon polaritons near the resonant frequency (from ω0 - δω to ω0 + δω); such

oscillations directly lead to the abnormal effective parameters of materials, with mass density and

bulk modulus shown as follows33

:

2

t 11 0 2 2

1 t

1eff

F

i

， （2）

2

p 1

2 2

1 0 1 p

1 11

eff

F

E E i

. （3）

Experiments and theories have demonstrated no interactions among the surface plasmon

polaritons35-38

. Based on the phenomenon that acoustic artificial meta-molecules can

simultaneously produce negative mass density and bulk modulus33

and inspired by the fact that the

visible white light is formed by seven-color lights, we propose a “flute-like” acoustic

meta-molecule cluster model, we also design seven acoustic double meta-molecules with different

geometric dimensions to construct the cluster. The length ratios of the units are 1, 5/7, 4/7, 3.5/7,

3/7, 2.5/7, and 2/7. The aperture ratios are 1 and 2.

Our previous studies have concluded that intrinsic resonant frequencies, and the

double-negative frequency ranges of meta-molecules are determined by the tube length and

diameter of the side hole. That is to say, the working frequency of the metamaterial can be

5

modulated by changing the dimension of the meta-molecule. The locally resonant essence33, 39-41

of the present metamaterial guarantees an insignificant influence among adjacent units; therefore,

a broadband double-negative metamaterial can be realized by stacking meta-molecules with

different working frequencies. As discussed using simulations and experiments, the number and

structure size of the units are finally determined for this model. A meta-molecule cluster is

engineered by arranging 14 meta-molecules, which are divided into seven units, as shown in Fig.

1b. The intervals of the adjacent units in x- and y-axes are 1 and 2 mm, respectively.

Figure 1 Model and behavior of the acoustic meta-molecule cluster. (a) Relationship between rainbow and

visible white light. The frequency of a photon is determined by its energy, and lights with different frequencies

have different colors. When we mix seven-color lights, visible white light appears. (b) Relationship between the

meta-molecule units and the cluster. The units constructed with different dimensions generate acoustic surface

6

plasmon polaritons with different frequencies and anomalous acoustic properties. Seven meta-molecules are

combined to form a cluster, with every meta-molecule containing two fine structures to realize broadband

abnormal acoustic properties. Sound waves propagate along the positive direction of the y-axis. (c) Transmitted

and reflected ratios of the metamaterial sample as a function of frequency. (d) Transmitted and reflected phases of

sample. The black and red lines indicate the simulated and experimental transmittance curves, respectively. The

blue and purple lines refer to the simulated and experimental reflectances, respectively.

The transmission and reflection behavior of the fabricated metamaterial sample is

experimentally measured. The present cluster is also numerically simulated. The structural

dimension of the simulated cluster is identical with that of the actual sample. A good agreement

can be observed by comparing the experimental value and the simulated pattern, as shown in Fig.

1c, d, respectively. The transmittance curve shows a series of absorption peaks in a wide

frequency range, within which phase shifts appear. This phenomenon is due to the fact that

meta-molecule units with different dimensions correspond to different locally resonant frequencies.

Moreover, no interaction occurs among adjacent units; each of them can resonate independently.

The slight deviation of the transmission valleys within the experiments and simulations mainly

stems from the machining error in the preparation of practical sample. The magnitude of the

experimental transmission valley diverges from that of the simulated result because the

environmental parameter set in the simulation does not match real circumstances well.

7

Figure 2 Effective parameters of the metamaterial sample obtained by experiments and simulations. (a)

Effective refractive index. (b) Impedance. (c) Mass density. (d) Bulk modulus. The red and black lines represent

the real parts of the results derived from the experimental and simulated data, respectively. The mass density and

bulk modulus are negative in a very wide frequency band.

Figure 2 indicates the effective parameters (i.e., refractive index, impedance, mass density,

and bulk modulus) as a function of frequency; they are derived from the transmission and

reflection results based on the method in42

. The real parts of the mass density and bulk modulus of

the material are negative in a broad frequency range, which are 1.402 KHz to 6.56 KHz and 1.74

KHz to 6.635 KHz, respectively. The real part of refractive indices is negative from 1.472 KHz to

6.614 KHz.

A right triangle sample is fabricated based on the meta-molecule clusters. The refraction of

the sample is measured experimentally in the frequency range of 0.8 KHz to 7.5 KHz (with 18

8

discrete frequencies). Figure 3a shows the schematic for the experimental measurement and the

field distribution of sound pressure at 3.5 KHz. The transmitted and incident beams are on the

same side of the normal. Based on the propagation direction of transmitted waves, the refraction

of the sample is n = -0.577. Figure 3b exhibits the experimental and calculated results of the

refractive indices of the metamaterial as a function of frequency. The measured refractive indices

are negative within the frequencies between 1.238 and 6.214 KHz; this result matches the

calculated results well.

Figure 3 Results of refraction for the metamaterial sample. (a) Schematic for the measurement of refractive

indics. θ0 = 17.5°, and the length of right-angle sides is a × b = 950 mm × 300 mm. The field pattern plotted in this

figure is the experimental result at 3.5 KHz. (b) Relationship between refractive index and frequency. The black

solid line indicates the measured results, and the red dashed line refers to the real part of the refractive indices

derived from the transmission and reflection results.

One of the unique characteristics of double-negative metamaterials is the reversed Doppler

effect. The Doppler effect of the proposed metamaterial is investigated experimentally. The

experimental set-up is illustrated in Fig. 4a. The actual frequency of the moving source can be

obtained by calculating the wave number in unit time, from which the Doppler shift of the

constructed metamaterial can be derived. Experiments on the reversed Doppler effect are

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performed at frequencies ranging from 1.0 KHz to 6.7 KHz (16 discrete frequency points). Figure

4b displays the recorded signal by a stationary detector at 2.0 KHz. The moving source passes the

sample centre at the middle of time axis (red dashed line); that is, the left and right sides of the red

dashed line indicate the approaching and receding processes of the moving source, respectively.

The measured Doppler shifts of the broadband double-negative sample versus frequencies are

shown in Fig. 4c. Within a broad frequency band (1.5 KHz to 6.5 KHz), the Doppler shift of the

metamaterial is not normal.

Figure 4 Results of the reversed Doppler effect. (a) Sketch map of the experimental setup for the Doppler shifts.

(b) Oscillograms detected by a stationary microphone. The frequency of exciting sound wave is 2.0 KHz. When

the source approaches the microphone, the detected frequency is reduced by 1.6 Hz; as the source recedes, the

frequency increases by 1.2 Hz. (c) Doppler shifts of sample as a function of source frequency. The black boxes and

red circles refer to the results of the approaching and receding processes, respectively. The positive value means

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the detected frequency is larger than the source frequency, and the negative value implies the contrary situation.

The blue solid line and the purple dashed line indicate the approaching and receding results derived from the

refractive indices, respectively.

When sound waves propagate into the metamaterial from air, the relation between the sound

velocity and refractive indices of the metamaterial is n1v1 = n0v0, where n1 and n0 are the refractive

indices of the metamaterial and air, respectively; v1 and v0 represent the sound velocity of the

metamaterial and air, respectively. As the observer is motionless and the source moves, the

detected Doppler shift can be derived from the following equation1:

0 1 010 0 0

1 0 1 0 1

1 1 1s s s

v n vvf f f f

v v v n v v v n

, (4)

where ∆f is the frequency shift; vs and f0 are the speed and frequency of the sound source,

respectively. Equation (4) indicates that if the speed of the sound source vs is fixed, then the

Doppler shift is related only to the refractive index. The Doppler shifts calculated based on the

derived refraction results are also displayed in Fig. 4c. The calculated results match the

experimental results well.

Inspired by the fact that visible light is composed of multifrequency photons with seven

different colors and based on acoustic surface plasmon polaritons and the previously presented

meta-molecule fabricated by integrating a split hollow sphere meta-atom with negative bulk

modulus and a hollow tube with negative mass density, we propose a model of “flute-like”

acoustic double-meta-molecule cluster. We also provide a new approach to designing acoustic

metamaterials with an arbitrarily broad band. The meta-molecule cluster consists of seven acoustic

double meta-molecules with different dimensions. The length ratios of units are 1, 5/7, 4/7, 3.5/7,

3/7, 2.5/7, and 2/7. The aperture ratios are 1 and 2. Simulations and experiments verify that the

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metamaterial sample exhibits negative mass density in a frequency range of 1.402 KHz to 6.56

KHz, and a negative bulk modulus from 1.74 KHz to 6.635 KHz, realizing double negative in

broad band. The measured refractive index of the right triangle sample is negative at frequencies

ranging from 1.238 KHz to 6.214 KHz and changes continuously, which matches the derived

results from the effective parameters and simulated results. The experiments demonstrate that the

fabricated metamaterial sample exhibits a reversed Doppler effect from 1.5 KHz to 6.5 KHz, and

the frequency shift increases with frequency. Assembling broadband double-negative

metamaterials using double-meta-molecule clusters can be greatly tunable, which can be realized

only by changing the structure size of the unit. This method of preparing an acoustic metamaterial

paves a new way of designing and fabricating metamaterials with arbitrarily changing refractive

index and broadband double-negative parameters. The method also exhibits great potential in such

applications as broadband absorber and cloaking.

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Methods

Fabrication of Metamaterial samples. The lengths of the tubes in a cluster are 98, 67, 55,

48, 41, 32.5, and 29 mm. The side hole is 5 mm away from one end of the tube, with diameters of

1 and 2 mm. The external and internal diameters of the tube are 7 and 5 mm, respectively. The

material of tubes is plastic, which is hard enough for acoustic waves. The propagation medium of

the acoustic wave was air. The clusters are pasted periodically on the front and back surfaces of a

sponge substrate to form the metamaterial sample. The thickness of substrate is 20 mm. The

sponge is a non-dispersive sound medium suitable for use as an acoustic substrate22

. The

dimension of the constructed metamaterial sample is 415 mm × 415 mm × 34 mm, which is used

for the measurements of transmission, reflection, and Doppler shift. The dimension of the

right-triangle sample is 950 mm × 300 mm × 34 mm, with θ0 of 17.5°.

Experimental facilities. A plane wave driver (4510ND, BMS, Hannover, Germany) is

connected to a signal generator (MC3242, BSWA, Beijing, China) and a power amplifier (PA50,

BSWA, Beijing, China) to generate sine acoustic waves. A free field microphone (MPA416,

BSWA, Beijing, China) is connected to a lock-in amplifier (SR830, SRS, Sunnyvale, USA) to

record the amplitude and phase signals of sound wave. The samples are surrounded by sound

absorbing materials to eliminate the scattered waves.

Measurement methods of transmission and reflection of the sample. The detailed

measurement methods are presented in the reference 33.

Measurement method of refraction of the right-triangle sample. A planar wave driver is

placed next to the sample and generates sound beams perpendicular to the interface. The incident

beam forms an angle of 17.5° with the refraction surface. To map the sound field distribution of

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refracted waves on the X–Y plane, a microphone is fixed on a 3D translation stage with a scanning

area of 200 mm × 50 mm.

Measurement method of Doppler shifts of the metamaterial sample. The sound source,

launching sinusoidal acoustic signals, is mounted on a 1D motorized translation stage and moves

along X-axis at a speed of 500 mm/s. A microphone is located at the center of the sample to record

oscillograms as the source moves from one side of the sample to the other side (i.e., including the

approaching and receding of the sound source from the observer). The loudspeaker and

microphone are near the metamaterial surface, without contact.

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References

1. Doppler, C. J., Über das farbige Licht der Doppelsterne und einiger anderer Gestirne des

Himmels (About the coloured light of the binary stars and some other stars of the heavens).

Abh. KoniglichenBohmischenGes. Wiss. 2, 465–482 (1842).

2. Papas, C. H. Theory of Electromagnetic Wave Propagation. McGraw-Hill, New York, (1965).

3. Pätzold, M. & Bird, M. K. Velocity changes of the Giotto spacecraft during the comet flybys:

on the interpretation of perturbed Doppler data. Aerosol Sci. Technol. 5, 235–241 (2001).

4. Chen, V. C., Li, F., Ho, S.-S. & Wechsler, H. Micro-doppler effect in radar: Phenomenon,

model, and simulation study. IEEE T. Aero. Elec. Sys. 42, 2–21 (2006).

5. Davis, J. -Y., Jones, S. A. & Giddens, D. P. Modern spectral analysis techniques for blood

flow velocity and spectral measurements with pulsed Doppler ultrasound. IEEE T. Bio-med.

Eng. 38, 589–596 (1991).

6. Oh, J. K. et al. The noninvasive assessment of left ventricular diastolic function with

two-dimensional and doppler echocardiography. J. Am. Soc. Echocardiog. 10, 246–270

(1997).

7. Veselago, V. G. The electrodynamics of substances with simultaneously negative values of

permittivity and permeability. Sov. Phys. Usp. 10, 509–514 (1968).

8. Liu, H., Zhao, X. P., Yang, Y., Li, Q. W. & Lv, J. Fabrication of infrared left-handed

metamaterials via double template-assisted electrochemical deposition. Adv. Mater. 20, 2050

(2008).

15

9. Liu, B. Q., Zhao, X. P., Zhu, W. R., Luo, W. & Cheng, X. C. Multiple Pass-Band Optical

Left-Handed Metamaterials Based on Random Dendritic Cells. Adv. Funct. Mater. 18, 3523

(2008).

10. Zhao, X. P. Bottom-up fabrication methods of optical metamaterials. J. Mater. Chem. 22,

9439 (2012).

11. Pendry, J. B. Negative Refraction Makes a Perfect Lens. Phys. Rev. Lett. 85, 3966 (2000).

12. Veselago, V. G. & Narimanov, E. E. The left hand of brightness: past, present and future of

negative index materials. Nat. Mater. 5, 759–762 (2006).

13. Seddon, N. & Bearpark, T. Observation of the inverse Doppler effect. Science 302, 1537–

1540 (2003).

14. Reed, E. J., Soljacic, M., Ibanescu, M. & Joannopoulos, J. D. Comment on "Observation of

the inverse Doppler effect". Science 305, 778b (2004).

15. Leong, K., Lai, A. & Itoh, T. Demonstration of reverse Doppler effect using a left-handed

transmission line. Microwave Opt. Technol. Lett.48, 545 (2006).

16. Stancil, D. D., Henty, B. E., Cepni, A. G. & Van’tHof, J. P. Observation of an inverse Doppler

shift from left-handed dipolar spin waves. Phys. Rev. B 74, 060404 (2006).

17. Chumak, A. V., Dhagat, P., Jander, A., Serga, A. A. & Hillebrands, B. Reverse Doppler effect

of magnons with negative group velocity scattered from a moving Bragg grating. Phys. Rev. B

81, 140404 (2010).

18. Reed, E. J., Soljacic, M. & Joannopoulos, J. D. Reversed Doppler Effect in Photonic Crystals.

Phys. Rev. Lett. 91, 133901 (2003).

16

19. Chen, J. B. et al. Observation of the inverse Doppler effect in negative-index materials at

optical frequencies. Nat. Photonics 5, 239–245 (2011).

20. Hu, X. H., Hang, Z. H., Li, J., Zi, J. & Chan, C. T. Anomalous Doppler effects in phononic

band gaps. Phys. Rev. E 73, 015602 (2006).

21. Lee, S. H., Park, C. M., Seo, Y. M. & Kim, C. K. Reversed Doppler effect in double negative

metamaterials. Phys. Rev. B 81, 241102 (2010).

22. Ding, C. L., Hao, L. M. & Zhao, X. P. Two-dimensional acoustic metamaterial with negative

modulus. J. Appl. Phys.108, 074911 (2010).

23. Chen, H. J., Zeng, H. C., Ding, C. L., Luo, C. R. & Zhao, X. P. Double-negative acoustic

metamaterial based on hollow steel tube meta-atom. J. Appl. Phys. 113, 104902 (2013).

24. Ma, Y. et al. A terahertz polarization insensitive dual band metamaterial absorber. Opt. Lett.

36, 945–947 (2011).

25. Shen, X. et al. Polarization-independent wide-angle triple-band metamaterial absorber. Opt.

Express 19, 9401–9407 (2011).

26. Luo, H., Cheng, Y. & Gong, R. Numerical study of metamaterial absorber and extending

absorbance bandwidth based on multi-square patches. Eur. Phys. J. B 81, 387–392 (2011).

27. Xie, Y. B., Popa, B., Zigoneanu, L. & Cummer, S. A. Measurement of a broadband negative

index with space-coiling acoustic metamaterials. Phys. Rev. L 110, 175501 (2013).

28. Ma, H. F.& Cui, T. J. Three-dimensional broadband ground-plane cloak made of

metamaterials. Nature Commun. 1, 21 (2010).

17

29. Liu, R. et al. Broadband Ground-Plane Cloak. Science 323, 366–369 (2009).

30. Southwell, W. H. Gradient-index antireflection coatings. Opt. Lett. 8, 584 (1983).

31. Li, X., Xue, L. & Han, Y. Broadband antireflection of block copolymer/homopolymer blend

films with gradient refractive index structures. J. Mater. Chem. 21, 5817 (2011).

32. Qi, D. X. et al. Broadband enhanced transmission of acoustic waves through serrated metal

gratings. Appl. Phys. Lett. 106, 011906 (2015).

33. Zhai, S. L., Chen, H. J., Ding, C. L. & Zhao, X. P. Double-negative acoustic metamaterial

based on meta-molecule. J. Phys. D: Appl. Phys. 46, 475105 (2013).

34. Zeng, H. C. et al. Flute-model acoustic metamaterials with simultaneously negative bulk

modulus and mass density. Solid State Commun. 173, 14–18 (2013).

35. Zhao, X. P., Zhao, Q., Kang, L., Song, J. & Fu, Q. H. Defect effect of split ring resonators in

left-handed metamaterials. Phys. Lett. A 346, 87–91 (2005).

36. Zhao, X. P. & Song, K. Review Article: The weak interactive characteristic of resonance cells

and broadband effect of metamaterials. AIP Advances 4, 100701 (2014).

37. Zharov, A. A., Shadrivov, I. V. & Kivshar, Yu. S. Suppression of left-handed properties in

disordered metamaterials. J. Appl. Phys. 97, 113906 (2005).

38. Gorkunov, M. V., Gredeskul, S. A., Shadrivov, I. V. & Kivshar, Y. S. Effect of microscopic

disorder on magnetic properties of metamaterials. Phys. Rev. E 73, 056605 (2006).

18

39. Yang, Z., Mei, J., Yang, M., Chan, N. H. & Sheng, P. Membrane-Type Acoustic Metamaterial

with Negative Dynamic Mass. Phys. Rev. Lett. 101, 204301 (2008).

40. Yang, Z., Dai, H., Chan, N., Ma, G. & Sheng, P. Acoustic metamaterial panels for sound

attenuation in the 50-1000 Hz regime. Appl. Phys. Lett. 96, 041906 (2010).

41. Ma, G. C., Yang, M., Yang, Z. Y. & Sheng, P. Low-frequency narrow-band acoustic filter

with large orifice. Appl. Phys. Lett. 103, 011903 (2013).

42. Fokin, V., Ambati, M., Sun, C. & Zhang, X. Method for retrieving effective properties of

locally resonant acoustic metamaterials. Phys. Rev. B 76, 144302 (2007).

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Acknowledgements

This work was supported by the National Natural Science Foundation of China under

Grant Nos. 11174234 and 51272215 and the National Key Scientific Program of China

(under project No. 2012CB921503)

Author contributions

X.P.Z. and C.R.L. conceived the idea and designed the experiments; S.L.Z. and S.L.

performed the major experiments; S.L.Z. performed the simulation study; S.L.Z. and X.P.Z.

wrote the manuscript; S.L.Z. drafted the text and aggregated the figures; X.P.Z. and C.R.L.

discussed the results and revised the manuscript.

Additional information

Supplementary information accompanies this paper at http://

Competing financial interests

The authors declare no competing financial interests.