Observation of the inverse Doppler effect innegative-index materials at optical frequenciesJiabi Chen1, Yan Wang1,2, Baohua Jia3, Tao Geng1*, Xiangping Li3, Lie Feng1, Wei Qian1,

Bingming Liang1, Xuanxiong Zhang1, Min Gu3* and Songlin Zhuang1

The Doppler effect is a fundamental frequency shift phenom-enon that occurs whenever a wave source and an observer aremoving with respect to one another. It has well-establishedapplications in astrophotonics, biological diagnostics, weatherand aircraft radar systems, velocimetry and vibrometry. Thecounterintuitive inverse Doppler effect was theoretically pre-dicted in 1968 by Veselago1 in negative-index materials2.However, because of the tremendous challenges of frequencyshift measurements inside such materials, most investigationsof the inverse Doppler effect have been limited to theoreticalpredictions and numerical simulations3–7. Indirect experi-mental measurements have been conducted only in non-linear transmission lines at ∼1–2 GHz (ref. 8) and in acousticmedia at 1–3 kHz (ref. 9). Here, we report the first experi-mental observation of the inverse Doppler shift at an opticalfrequency (l 5 10.6 mm) by refracting a laser beam in aphotonic-crystal prism that has the properties of a negative-index material.

The most commonly experienced Doppler shift (DfD) ariseswhen a vehicle passes an observer. When the vehicle approachesthe observer, the detected frequency ( f ′ ) is higher than thesource frequency ( f0) (Fig. 1a). In contrast, a signal with lower fre-quency is detected when the wave source is moving away. Thisoccurs because the wave is propagating in a naturally existing posi-tive refractive index media, and the group velocity vg and the wavevector k are parallel; that is, vg

. k . 0. A fascinating inverseDoppler effect has been predicted in tailored artificial materials2,such as a photonic bandgap material exhibiting negative index(vg

. k , 0). In negative index materials (NIM), the electric fieldE, magnetic field H and wave vector k are composed of a set ofleft-handed coordinates instead of the conventional right-handedcoordinates. The phase velocity of the light wave propagates in theopposite direction of the energy flow, leading to subversion of thefundamental physical rules and, consequently, intriguing phenom-ena2. These include backward wave propagation, reversed Cerenkovradiation and the inverse Doppler effect, as depicted in Fig. 1b, inwhich a redshifted signal is detected for a wave source approachingthe detector.

Experimental observation of such an inverse Doppler effect inthe optical region represents a great challenge, because the fre-quency is too high to be measured directly. The Doppler shift isusually detected using heterodyne interferometry. However, withnormal heterodyne interferometry, even if the inverse Dopplereffect has occurred, it cannot be identified by recording theDoppler shifts simply because the recorded Doppler shifts DfD¼|f ′ – f0| are absolute values, and are identical for waves propagating

in a positive medium or in a negative medium. Therefore, it is criti-cal to find a way to distinguish the normal and anomalous Dopplereffects explicitly.

For this purpose, we designed a highly sensitive two-channelheterodyne interferometric experimental setup (shown in Fig. 2)to measure the anomalous Doppler effect in a two-dimensionalphotonic-crystal prism with NIM property at 10.6 mm. The pro-posed photonic-crystal prism consisted of a triangular array ofsilicon rods in air (Fig. 3a). The incident light was along the GMdirection, as shown in the inset of Fig. 3a. The photonic crystalhad a rhombus geometry with sides of 5 mm and a vertex angleof 608. The silicon rods had a radius of r¼ 0.2a (a¼ 5 mm is thelattice constant) and height of d¼ 50 mm.

The band diagram was calculated using a plane wave method10

for an ideal two-dimensional photonic crystal (symmetric dielectricpillars with an infinite height) with the same geometry, and revealed

Detector

Source

Inverse Doppler effect

b

Normal Doppler effect

f ’ = f0 − ΔfD

f ’ = f0 + ΔfD f0

f0

a

n > 0

Sourcen < 0

Figure 1 | Schematic showing the Doppler effect. a, Normal Doppler effect

in normal materials (n . 0). b, Inverse Doppler effect in NIMs (n , 0).

1Shanghai Key Lab of Contemporary Optical System, Optical Electronic Information and Computer Engineering College, University of Shanghai for Scienceand Technology, Shanghai 200093 China, 2College of Physics and Communication Electronics, Jiangxi Normal University, Nanchang 330022 China,3Center for Micro-Photonics and CUDOS, Swinburne University of Technology, John Street Hawthorn, Victoria 3122, Australia.

*e-mail: max_geng@yahoo.com.cn; mgu@swin.edu.au

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that complete bandgaps exist between the first and second bandsand the third and fourth bands for the transverse magnetic (TM)-polarized (with the E field parallel to the silicon rods) incidentbeam (Fig. 3b). Equifrequency surface (EFS) contours in the kspace of the photonic crystal at several relevant frequencies in thesecond band are shown in Fig. 3c. The EFS contour shrinks whenthe frequency increases, clearly indicating that the region in thesecond band, extending from f¼ 0.436 (l¼ 11.47 mm) to f¼0.542 (l¼ 9.23 mm), with vg

. k , 0, is of particular interest11,12.The fact that the EFS in the first Brillouin zone is almost sphericalsuggests that the photonic crystal is a strongly modulated photoniccrystal11, so the light propagation angle can be derived simply fromSnell’s law (see Supplementary Information). The effective phaseindex np (ref. 13) for the second band along the GM direction is pre-sented in Fig. 3d. At the investigated frequency ( f¼ 0.4717; l¼10.6 mm), np ≈ –0.7.

Transmission measurements were performed to verify the nega-tive refraction of the photonic-crystal prism. As shown in Fig. 2, aTM polarized beam from a CO2 laser at the desired wavelength(l¼ 10.6 mm) was incident along the GM direction to thephotonic-crystal interface. To ensure that most of the energy ofthe beam (�3–4 mm) propagated through the photonic-crystalprism (height, 50 mm), we used a ZnSe lens with a focal distanceof 50 mm and a convergence angle of 48 to focus the incidentbeam into the sample. Both the photonic-crystal prism and thedetector were placed on computer-controlled rotation and trans-lation stages to allow individual positioning and angle adjustment.The detector was rotated around the rotation centre O to recordthe intensity of the refracted beam at all rotation angles u, asshown in the inset of Fig. 4a, so that the desired negative refractiveangle could be found.

The experimental result is presented in Fig. 4a. At a refraction angleof u≈ –268 (incident angle, 608), a high-intensity signal could bemeasured, clearly indicating that the photonic-crystal prism was oper-ating in the negative refraction region, with a measured np of –0.5062.The result was reproducible for different samples of the same geome-try, with a+48 error range for the refraction angle. The measured npwas slightly smaller than the theoretical estimation (–0.7), mainlydue to the finite height of the two-dimensional photonic crystal inthe experiment (infinite height was assumed for the rods in the calcu-lation14–16; see Supplementary Information).

To measure the Doppler effect, the set-up of Fig. 2 was used. Thephotonic-crystal prism was fixed at the measured negative refraction

angle, and the translation stage was moved uniformly along thex-axis to generate a relative velocity with respect to the detector(set to be perpendicular to the beam propagation direction).During the experiment, the CO2 laser and the detector were bothstationary. To distinguish between the normal and anomalousDoppler effects, a reference beam was used in the system.Beamsplitter 2, which was placed on the same translation stage asthe photonic-crystal prism (depicted in the inset of Fig. 2), wasused to combine, in a parallel manner, the reference beam andthe signal beam refracted by the photonic-crystal prism.

Considering the velocity of the translation stage to be along thepositive x-axis (see Supplementary Information for negative x-axisdisplacement), a schematic diagram of the light propagating inthe photonic-crystal prism is shown in Fig. 2. Because the directionof propagation of the CO2 laser beam is perpendicular to the inci-dent interface of the prism, the interface velocity in the directionof light propagation is zero. Therefore, no Doppler shift occursat this incident interface. For the exit interface of the prism, thevelocity in the direction of light propagation can be written asv1¼ v . cot(p/6), where v is the velocity of the translation stage.Therefore, the Doppler frequency shift at the second interface canbe calculated simply as

f1 = f0 1 − v1

cnp

( )= f0 1 − v cot(p/6)

cnp

( )(1)

where f0 is the original frequency of the CO2 laser, and c is the vel-ocity of light in vacuum. When np is negative, f1 is larger than f0,and so the inverse Doppler effect occurs.

By taking the relative movement between the photonic-crystalprism and the detector into consideration, the exit interface of theprism, which can be considered as an effective source of the out-going light, moves towards the detector. Because the transmittingmedium is air with a positive index of unity, the second Dopplereffect is normal. The recorded Doppler frequency shift at the detec-tor surface can therefore be calculated as

f2 = f1c

c − v2

( )= f1

cc − v cot(p/6) · cos(p/3 + u)

( )= f1k (2)

where a new parameter k is defined as k = c/(c − v cot(p/6) ·cos(p/3 + u)), which is always positive and close to 1 in thenon-relativistic regime (v ≪ c).

For the reference beam, because beamsplitter 2 is mounted onthe same translation stage as the photonic-crystal prism, it has thesame displacement along the x-axis. The velocity of beamsplitter 2in the beam incident direction can be written asv′1 = sin(2p/3 − b/2 − u)/sin(p/2 − b/2)v, where b is the anglebetween the incident beam and the reflected beam (Fig. 2, inset).Therefore, the first Doppler frequency shift at the beamsplittersurface can be calculated as

f ′1 = f0 1 + v′1c

( )(3)

For the relative movement between beamsplitter 2 and the detector,beamsplitter 2 can also be considered an effective source of reflectedlight, and its velocity in the direction of light reflection can bewritten as v′2¼ v′1 cos b. Therefore, the reference Doppler frequencyshift at the detector surface can be calculated as

f ′2 = f ′1c

c − v′1 cosb)

( )= f0 1 + v′1

c

( )c

c − v′1 cosb

( )(4)

CO2 laser BS 1

BS 2

PC prism

Lens

Translation stage

ND

Reference beam

Reflector

Detector

x

y

3 θ

60°ΓM

x

y

β3

Figure 2 | Experimental set-up for detection of the inverse Doppler effect.

BS, beamsplitter; ND, neutral density filter; PC, photonic crystal. The purple

outline highlights how the reference beam and the signal beam combine at

the beamsplitter.

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The frequency difference between the signal frequency f2 from thephotonic-crystal prism and the reference frequency f2

′ from beam-splitter 2 can be obtained from equations (2) and (4) as

Df = f ′2 − f2

∣∣ ∣∣ = f ′2 − f0k( )

− f1 − f0

( )k

∣∣ ∣∣ (5)

In equation (5), the first part on the right-hand side, f ′2 2 f0k, isindependent of the effective phase index np, and can be calculatedeasily according to the particular experimental conditions.With u¼ –268 and b¼ 458, the calculated values of f ′2 2 f0kat four experimental velocities (0.0123, 0.0245, 0.0488 and0.0732 mm s21) are all positive, and equal to 1.90, 3.77, 7.51 and

9.5 10.0 10.5 11.0 11.5λ (μm)

0.0

Γ K M Γ0.0

0.2

0.4

0.6

f (a/

λ)

0.8

1.0ba

c d

–0.4

–0.8

6

4

2

642

0

0

k z n p–2

–2

–4

–4

–6

–6

–1.2

kx

0.4457

0.4669

0.4881

0.5093

0.5305

θ

60°

ΓK

ΓMΓK

20.0 mm

Figure 3 | Negative refractive index photonic crystal. a, Scanning electron microscopic image of the two-dimensional photonic crystals (top view). Right

inset: side view. Left inset: geometry of the entire photonic crystal. b, Calculated band diagram of the photonic crystal. c, EFS contours of the photonic crystal

at several relevant frequencies in the second band. d, Effective phase index np versus wavelength l for the second band along the GM direction.

−40 −20 00

20

20

40

40

a b c

Pow

er (μ

W)

60Refraction angle (deg) Translation speed (mm s−1)

0.00

Freq

uenc

y sh

ift, Δ

f (H

z)

–12

–6

6

12

0

Freq

uenc

y sh

ift, Δ

f (H

z)

–12

–6

6

12

0

0.03 0.06Time (s)

0

0 2 4 6 8 10 0 2 4 60.0

0.4

0.8

1.2

Inte

nsity

(nor

mal

ized

)

0.0 0.6 1.2 1.81 2

Inverse Normal

Translation speed (mm s−1)0.00 0.06 0.12

f2’ − f0kΔf

f2’ − f0kΔf

Δf=1.83 HzΔf=0.89 Hz

Δf=5.14 HzΔf=3.65 Hz

0.0

0.4

0.8

1.2

Detectorθ < 0

θ > 0

θ = 0

60°ΓM

ΓK

ΓK

Figure 4 | Observation of inverse Doppler effect in negative refractive index photonic crystal. a, Measured transmission power as a function of the

refraction angle u for a normally incident beam with respect to the first interface of the photonic crystal (the incident angle at the exit interface is 608). Inset:

schematic of the experimental setup. b, Measured frequency shifts Df in the NIM photonic-crystal prism compared with the value of f ′2 2 f0k. Inset: recorded

signals with the photonic-crystal prism at v¼0.0123 mm s21 (Df¼0.89 Hz), v¼0.0245 mm s21 (Df¼ 1.83 Hz), v¼0.0488 mm s21 (Df¼ 3.65 Hz) and

v¼0.0732 mm s21 (Df¼ 5.14 Hz), respectively. c, Measured frequency shifts Df in the positive-index ZnSe prism compared with the value of f ′2 2 f0k.

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11.27 Hz, respectively, as shown in Fig. 4b (see SupplementaryInformation). Hence, if the measured beat frequency Df , f ′2 2 f0k(note that k . 0), the Doppler shift DfD¼ f1 2 f0 can only be posi-tive (that is, f1 . f0), which indicates that the Doppler frequency isblueshifted and larger than the original frequency of the CO2laser when the optical path becomes larger in the NIM. The anom-alous Doppler effect occurring in the NIM photonic-crystal prismcan therefore be demonstrated explicitly.

Figure 4b (inset) presents the recorded signals from the detectorat the abovementioned four velocities. The frequency differences Dfcan be obtained from fast Fourier transforms (FFTs) of the recordedsignals, and are equal to 0.89, 1.83, 3.65 and 5.14 Hz, respectively;these values match well with the theoretically calculated values,with a relative error of ,5% if a measured negative index of np¼–0.5062 is considered (see Supplementary Information). It can beclearly seen that all the measured frequency differences Df are lessthan f ′2 2 f0k at the corresponding velocities. This result clearlyindicates that the observed Doppler effect is anomalous, and theinverse Doppler effect has been explicitly observed at optical fre-quencies for the first time. The experimental results are reproduciblefor different photonic-crystal prisms with the same crystal structure.

To further verify the results and the reliability of our experimen-tal setup, we conducted comparison experiments with a positive-index ZnSe prism (np¼ 2.403) using a method similar to the onementioned above. As expected, the measured frequency differencesDf for the four velocities were all less than | f ′2 2 f0k| (as shown inFig. 4c; note that f ′2 2 f0k , 0, here), which clearly demonstratesthat the Doppler effect measured in the ZnSe prism is normal(see Supplementary Information). In addition, the relative errorsbetween the theoretical expectation and the experimental measure-ment were less than 3% (see Supplementary Information), indicat-ing that the transmission method is as suitable as the reflectionmethod for Doppler shift measurement. The experimental resultshave therefore demonstrated that our setup can explicitly andreliably distinguish normal and anomalous Doppler effects.

Received 16 August 2010; accepted 10 January 2011;published online 6 March 2011

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AcknowledgementsThis work is supported by the National Basic Research Program of China (2007CB935303,2011CB707504), the National Natural Science Foundation of China (60778031, 61008044)and the Shanghai Leading Academic Discipline Project (S30502). Min Gu acknowledgessupport from the Australian Research Council (ARC) under the Centres of Excellenceprogramme and the Laureate Fellowship scheme (FL100100099). Baohua Jia thanks theARC for support through APD grant DP0987006.

Author contributionsJ.C., Y.W., T.G. and S.Z. designed and performed experiments, analysed data and wrote thepaper. B.J., X.L. and M.G. designed the supplemental experiment and the physical strategyof the numerical calculations, analysed data and wrote the paper. B.L. designed andperformed experiments. L.F. and W.Q. performed experiments. X.Z. prepared the photoniccrystal prisms.

Additional informationThe authors declare no competing financial interests. Supplementary informationaccompanies this paper at www.nature.com/naturephotonics. Reprints and permissioninformation is available online at http://npg.nature.com/reprintsandpermissions/.Correspondence and requests for materials should be addressed to T.G. and M.G.

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