Download Observation of the inverse Doppler effect

An experimental method for demonstrating the inverse
Doppler effect in a photonic crystal
Jiabi Chen and Songlin Zhuang
University of Shanghai for Science and Technology
College of Opto-electronic information and computer engineering
Shanghai, China, 200093
Abstract
The inverse Doppler effect was theoretically predicated in 1968 by Veselago
in the so-called the negative-index material (NIM) and there are lots of
difficulties of Doppler frequency shift measurements inside the NIM. A 2-D
photonic crystal (PC) 60O prism which has effective negative refraction index for
the wave length λ=10.6 µm was made from a silicon wafer. Then a laser beam is
refracting by the prism moving perpendicular to one of its sides, the light
refracted throughout the prism should have suffered Doppler effect. By
heterodyne interferometry, the Doppler frequency shift was measured and the
really abnormal was observed.
Key Words: inverse Doppler effect, effective negative refraction index,
photonic crystal prism, heterodyne interferometry
1. Introduction
In 1968, Veselago [1] theoretically predicted the existence of the left-handed material (LHM), in
which the electric field E , the magnetic field H and the wave vector k are not composed of a set
of right-handed coordinates but a set of left-handed coordinates. Within LHM, the phase velocity of
light wave propagates in the opposite direction of the energy flow and it is also called negative-index
material (NIM). Thus, some intriguing or even counterintuitive phenomena in NIM have attracted
much attention, for example, the inverse Doppler effect. However, because of difficulties of frequency
shift measurements inside of the NIM, most researchers used numerical simulations to study
anomalous Doppler shifts [2-5]. To our best knowledge, the only indirect experimental result which has
been reported by Seddon and Bearpark [6] in nonlinear transmission lines is at 1~2 GHz and
nonrelativistic reversed Doppler shifts have never been observed at optical frequencies. Here, we report
an explicit observation of inverse Doppler shift in a photonic crystal (PC) prism, which has NIM
property at 10.6 m wavelength.
The proposed structure of the PC prism consists of a triangular array of Si rods in air, as shown in
fig.1. Its shape is a rhombus with side of 5mm and vertex angle of 60  . The radius of the Si rods is
r  0.2a , where a  5m is the lattice constant. The height of the Si rods is 50m . According to our
previous work [7], the sample exhibited negative refraction property at 10.6 m wavelength.
(a) Vertical View
Fig.1
(b) Side View
SEM Photo of the etched sample
Fig. 2 shows the diagram of experimental setup. Since the light of the CO2 laser is invisible, a
He-Ne Laser with the same optical axis is used for alignment of the sample. The laser beam is incident
upon the side of the prism perpendicularly. The PC prism is fixed on a translating stage, which can
move uniformly along x direction. During the experiment, the CO2 laser and the detector are both static
and the surface of the detector is normal to the emergent light. When the object stage moves to the
detector along x direction, the optical paths in the PC prism will be changed and produce the Doppler
frequency shift.
Reflector
He-Ne laser
Translating stage
PC prism
y
CO2 laser
Beam splitter1

v
Ge lens
x
Ge reflector
Beam splitter2
θ
Reference
beam
Reflector
β
Attenuator
Detector
Fig. 2Diagram of experimental setup
Firstly, longer optical paths inside the PC prism makes the incident beam upon the side of prism
from inside of PC prism suffered the Doppler shift. The Doppler shifted frequencies can be calculated
simply:
f1  f 0 (1 
, where
v tan  3
n)
c
(1)
f 0 is the output frequency of CO2 laser, v is the translational velocity ratios of the
translating stage, n is the effective index of the PC prism and c is the light velocity in the vacuum.
When n is negative, f1 will larger than f0, and the inverse Doppler effect will occur.
Considering the relative movement between the PC prism and the detector, the output surface of
the PC prism, which can be considered as a effect source of the output light, moves towards the
detector. Since the transmitting medium is air with positive index unity, the second Doppler effect is
normal. . The observed frequency f2, which is received by detector, is:
f 2  f1 (
c
)  f1 k
c  v tan  3 sin(  6   )
(2)
, where  is refraction angle of light passed through PC (as shown in fig. 2) and a new parameter k is
defined as k 
c
, which is always positive and very close to 1. Because
c  v tan  3 sin(  6   )
f2 is too high to be measured directly, the Doppler shift was detected using the heterodyne
interferometry. Since the beam splitter on the translating stage moves together with PC prism, the
observed frequency
f 2 of the reference light can be obtained using a similar analysis and the Doppler
effect is also normal:
f 2  f1(
c
v
c
)  f 0 (1  )(
)
c  v cos  )
c c  v cos 
(3)
, where  is the angle between the incident light and the reflected light from the beam splitter 2 and
v  is a relative velocity ratios, which is defined as v  v
The frequency difference between
sin(2 3   2   )
.
sin( 2   2)
f 2 and f 2 can be obtained from Eq. 2 and Eq. 3:
f  f 2  f 2   f 2  f 0 k    f1  f 0 k
(4)
In fact, during the whole experiment,  and  have no change, which were kept at 19  180 and
5 18 respectively. Therefore, in Eq.4, the first part in the absolute value f 2  f 0 k should be a
constant for a given velocity ratio and can be calculated easily. The calculation values of
f 2  f 0 k at
two different translational velocity ratios are shown in Table 1, and equal to 1.66Hz and 3.30Hz,
respectively. It can be seen that both values are positive. Hence, if the measured beat frequency
f  f 2  f 0 k , note that k  0 , f1  f 0 can and can only be positive, i.e. f1  f 0 , which
indicates that the Doppler shifted frequency is larger than original frequency of CO 2 laser when the
optical path becomes larger in NIM. Then, it can be concluded that the Doppler effect, which occurred
in the PC prism, is anomalous.
Fig. 3 (a) and (b) show the recorded signal from the detector at two different translational velocity
ratios. The velocities of translating stage were measured by a double frequency laser interferometer.
The beat frequencies can be calculated using the fast Fourier transform (FFT) from the experiment data,
as shown in Table 1.The values are quite small in the order of Hz, however, the most important is that
both measured frequency difference  f at two different translational velocity ratios are less than
f 2  f 0 k . The experiment was repeated more than 20 times using 8 different PC prisms with same
crystal structure. All results are similar.
Fig. 3 Recorded signal with different translational velocity: (a) v=0.012275 mm  s 1 , (b) v=0.024465
mm  s 1
It is obvious that the observed Doppler effect is anomalous. In the other words, it is proved that the
inverse Doppler effects were explicitly observed in our work at optical frequencies first time. In
addition, if the Snell's law is suitable here, the theoretical effective index n of the PC prism is –0.376.
Therefore, using Eq. (1)-(4), the theoretical frequency difference can be calculated and the results are
0.8984Hz and 1.7852Hz with v=0.012275 mm  s 1 and v=0.024465 mm  s 1 respectively. As can be
seen, both results are very close to the experimental results, and the relative error between the
theoretical values and the experimental values are less than 4%. In conclusion, our works shows that
the inverse Doppler effect is really exist within NIM, and we hope that the result can provide a
common foundation on the further research of LHM.
Table 1. The values of
f 2  f 0 k and f with two translational velocities.
v / mm  s 1
f 2  f 0 k /Hz
f / Hz (experimental)
0.0123
1.66
0.93
0.0245
3.30
1.86
This work is supported by the National Basic Research Program of China (2007CB935303,
2005CB724304), National Natural Science Foundation of China (60778031), and Shanghai Leading
Academic Discipline Project (S30502).
Reference:
[1] V.G. Veselago, The electrodynamics of substances with simultaneously negative values of
permittivity and permeability. Sov. Phys. Usp. 10, 509-514(1968)
[2] Kevin M. K. H. Leong, Anthony Lai, Tatsuo Itoh, Demonstration of reverse Doppler effect using
a left-handed transmission line. Micro. Optic. Tech. Lett. 48, 545-547 (2006)
[3] Alexander B. Kozyrev, Daniel W. van der Weide, Explanation of the inverse Doppler effect
observed in nonlinear transmission lines. Phys. Rev. Lett. 94, 203902 (2005)
[4] Chiyan Luo, Mihai Ibanescu, Evan J. Reed, Steven G. Johnson, J. D. Joannopoulos, Doppler
radiation emitted by an oscillating dipole moving inside a photonic band-Gap crystal. Phys. Rev.
Lett. 96, 043903 (2006)
[5] A.V. Kats, Sergey Savel’ev, V. A. Yampol’skii, Franco Nori, Left-handed interfaces for
electromagnetic surface waves. Phys. Rev. Lett. 98, 073901 (2007)
[6] N. Seddon, T. Bearpark, Observation of the inverse Doppler effect. Science 302, 1537-1540
(2003)
[7] Jiabi Chen, Binming Liang, Dawei Zhang, Songlin Zhuang, Experiments of negative-index
refraction in optical frequency region. Proc. Of ISMTII-2009 Vol.4 PP323-327, Saint-Petersburg,
Russia (2009)