Download Optical Doppler Effect in a Medium C. H. Chang, J. H. Hsiao, and

CHINESE JOURNAL OF PHYSICS
VOL. 47, NO. 4
AUGUST 2009
Optical Doppler Effect in a Medium
C. H. Chang, J. H. Hsiao, and T. M. Hong
Department of Physics, National Tsing Hua University,
Hsinchu 30043, Taiwan, Republic of China
(Received November 8, 2008)
The acoustic and optical Doppler effects are unified into one single formula, via which the
Doppler shift for EM waves in a medium can be obtained without having to address the issue
of complicated interactions with the dense matter. Since the medium provides an absolute
rest frame, this formula is expectedly different from that of the vacuum. But it also does not
trivially adopt the form for sound waves, because the latter fails to include the optical shift
in vacuum as a limiting case. A universal two-way Doppler shift is derived by solving the
wave equation in an infinitely hard well with a moving boundary. Since the mathematical
solution does not care about the detailed nature of the wave, its features are shared by sound
and light in vacuum or a medium.
PACS numbers: 03.30.+p
One common challenge we pose to students when lecturing on special relativity is
“Do we simply change c to c/n, where n denotes the refractive index of sea water, to obtain
the Lorentz transformation for Meno, the clownfish?” Instead of admitting our ignorance
of how to deal with the complicated interaction of EM waves with dense matter, we often
appeal to counterevidence such as “This cannot be correct because, otherwise, the factual
and Nobel-winning
(1958) phenomenum of Cherenkov radiation will render the omnipresent
p
factor 1 − [v/(c/n)]2 and the physical quantities imaginary”. When we go on to provide a
second evidence by proving that the time dilation factor inp
the tangible mirror experiment
if performed under water will remain the good old γ ≡ 1/ 1 − (v/c)2 without the n, the
students are sure to be left in awe.
FIG. 1: The schematic plot of the experimental setup to determine the time dilation under water.
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421
c 2009 THE PHYSICAL SOCIETY
OF THE REPUBLIC OF CHINA
OPTICAL DOPPLER EFFECT . . .
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VOL. 47
The schematic plot of the experimental setup is shown in Fig. 1, which includes a
light bulb at the origin and a mirror at y = h. Define the sequence of events as: begin by
turning on the bulb, and end upon receiving the reflected EM waves from the mirror. For
an observer sitting at the origin, the time it takes for the EM waves to travel a round-trip
distance is trivially ∆t = 2h/(c/n). To add to the drama, the lecturer may choose to do
it initially in an intuitive but wrong way for the second observer that travels with a fixed
velocity v relative to the first one. Namely, writing down [(c/n)(∆t0 /2)]2 = h2 + [v∆t0 /2]2
from the Pythagoras principles for the bent path of the EM waves. It is apparent that
∆t0 /∆t will depend on n in a complicated way and you can immediately detect the amused
look from the audience.
The time is ripe for you to point out the flaw in the above derivations: the refractive
index seen by the moving observer ought to be n0 instead of n, just as the transformed spacetime (x0 , y 0 , t0 ). In spite of the theoretical efforts of Fresnel and Lorentz, the famous Fizeau
effect [1] designed to measure the difference between n0 and n has not yet been accurately
measured experimentally so far. So let us forego this n0 and resort to the relativistic formula
for relative velocities:
ux − v
,
1 − ux v/c2
p
uy 1 − (v/c)2
0
,
uy =
1 − ux v/c2
u0x =
(1)
p
to determine the speed of EM waves in the moving frame as ~u0 = (−v, (c/n) 1 − (v/c)2 )
from ~u = (0, c/n) in the lab frame. Without bothering with the Pythagoras principles, the
time interval for the moving
observer can be directly determinedp
from the vertical motion
p
to be 2h/u0y = 2h/(c/n 1 − (v/c)2 ). In other words, ∆t0 = ∆t/ 1 − (v/c)2 without any
mentioning of n. Voila!
The indication seems to be that nothing needs to be changed in the Lorentz transformation. Why is that? Isn’t the speed of light changed in the medium? The reason is
that, when we postulated the speed of light in vacuum to be the same for different interial
frames, we did not say anything about our environment. So the existence of medium should
not affect the validity of the space-time transformation we derive from the postulates. Consequently, the time dilation, length contraction, mass enhancement, and Eq. (1) remain the
same as in the vacuum. Strictly speaking, the invariant speed specified in Einstein’s second
postulate of relativity is not the speed for visible light in the medium, but the maximum
speed of light, that is, the speed for light of infinite frequency. In a dispersive medium,
light of finite frequency has a frequency-dependent speed, and therefore this speed cannot
be regarded as universal. Light of infinite frequency is unaffected by the resonant frequencies of the atoms of the medium, its index of refraction is necessarily n = 1, and its speed
is universal. The Lorentz transformation permits us to make one, and only one speed invariant; and obviously, we must select the maximum speed of light (and of any other wave
or particle) as this invariant speed.
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C. H. CHANG, J. H. HSIAO, AND T. M. HONG
423
What about the optical Dopper effect? We know that this certainly will need modifications because the transformation for angular frequency:
ω − kx v
ω0 = q
2 ,
1 − vc
(2)
tells us to expect
v
1 − c/n
ω0 = ω q
2
1 − vc
(3)
for the longitudinal Doppler effect by substituting kx = ω/(c/n). But there is something
fishy about Eq. (3), because (1) it fails to distinguish whether it is the emitter or the
absorber that moves after an absolute frame has been established by the medium, (2) if
Eq. (3) is true, the accumulated shift when an EM waves bounces from an approaching wall
with velocity v will be
v 2
1 + c/n
0
ω =ω
2 ,
1 − vc
which violates the universal value of α to be derived later.
In any case, argument (1) is enough to rule out the possibility that the optical formula
for vacuum remains valid in a medium. Now one question is whether we can fix the problem
by distinguishing n and n0 in Eq. (3). Another question is that, with the presence of an
absolute frame, we would have expected the optical Doppler shift in a medium to resemble
sound more than EM waves in vacuum. But, in the meantime, it is easy to refute this
possibility, because the acoustic Doppler formula fails to include the optical shift in vacuum
as a limiting case when we substitute the sound velocity by c. How do we compromise these
two observations, and how can this study of EM waves in medium help us bridge the gap
between the different Doppler formulae for EM waves in vacuum and for sound?
It is misleading for many teachers to ascribe the difference between these two formulae
to the fact that the propagation of EM waves does not require a medium. As we shall point
FIG. 2: Schematic plot of the absorber and emitter who approach each other with velocity va and
ve , respectively.
OPTICAL DOPPLER EFFECT . . .
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VOL. 47
out shortly, their dissimilarity is only superficial. As soon as the time dilation is included
in the (classical) acoustic Doppler formulae, they become identical to the optical one. In
retrospect, it is not surprising that these seemingly different Doppler formulae can be unified
because, had we been solving a wave equation with moving boundaries, the mathematical
solution certainly would not care about the detail nature of the wave. Let us demonstrate
that, when it is the emitter that approaches with velocity ve (see Fig. 2), the relativistic
acoustic Doppler shift[4] becomes
r
1
ve 2
0
,
(4)
ν =ν
1−
1 − ve /v
c
where v denotes the speed of sound. Similarly, when it is the absorber that moves towards
the emitter with velocity va , the shift turns into
ν 0 = ν(1 + va /v) q
1
1−
va 2
c
.
(5)
Upon replacing the sound velocity v by c, both Eqs. (4) and (5) immediately reduce to the
optical Doppler formula
s
1 + vc
0
.
(6)
ν =ν
1 − vc
In general, when both ve and va are nonzero[5],
v
u
ve 2
va u
1
+
0
c t1 − c
ν =ν
,
1 − vce 1 − va 2
(7)
c
p
which can again be reduced to the optical formula ν (1 + u/c)/(1 − u/c) with u ≡ (ve +
va )/(1 + ve va /c2 ) being the correct relativistic relative velocity between the emitter and the
absorber.
It is illuminating to study the mathematical solution of the wave equation (which can
describe sound, EM waves in vacuum or in a medium, etc.)
2
∂ 2 ψ(x, t)
2 ∂ ψ(x, t)
=
u
∂t2
∂x2
(8)
in an infinitely hard well with one boundary that shrinks with velocity v. The boundary
conditions [2] (BC) require that
ψ(x = 0, t) = 0 = ψ(x = L − vt, t)
(9)
at all times. It is not hard to guess at the solution since (1) it has to be a function of
x + ut or x − ut, (2) BC, (3) it should remain unchanged when bounced from the moving
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C. H. CHANG, J. H. HSIAO, AND T. M. HONG
425
boundary one more time, namely, ψk (x, t) = ψkα (x, t) where α is the Doppler shift picked
up by the wave number. The solution to Eq. (8) is found to be:
h 2π` ln k|x + ut − uL/v| i
ψk,` (x, t) = exp i
ln α
h 2π` ln k|x − ut + uL/v| i
− exp i
,
(10)
ln α
where α = (u + v)/(u − v) and ` is an integer which will survive the continuous k to label
the discrete eigenmodes when u → 0.
Note that Eq. (10) and the universal value of α can be applied to all waves, including
the EM waves in a medium. The Doppler effect has been widely used in scientific, industrial,
and medical appliances. In normal circumstances, for instance, the velocity of blood flow
is certainly negligible in comparison to c/n when measured by the Doppler tomographic
imaging [3], and the approximate formula the medical doctors adopt to interpret their data
is enough. However, when higher accuracies are demanded or when the medium involves
whirling charged plasma, e.g., during the early stage of star formation, which may exhibit
very large n around certain frequencies, a rigorous understanding of this effect will be called
for. Since the factor α introduced by each collision with the moving boundary is universal,
we can check it first by our familiar acoustic Doppler formula:
q
2
1 − vc
1 + uv
q
(11)
α=
2 ,
1 − uv
1 − vc
where the first correction comes from Eq. (5) for va 6= 0 and the second one from Eq. (4)
for ve 6= 0. Their relativistic corrections will cancel out. This universal form indicates that
relativistic time-dilation effects play no role whatsoever in this Doppler shift. To reassure
ourselves, let us replace the wave velocity u by c to check the prediction of the optical
formula in vacuum:
s
s
1 + vc 1 + vc
,
(12)
α=
1 − vc 1 − vc
which is obviously identical to Eq. (11) and indeed universal.
Now we are confident to utilize this property for an EM wave in a medium. First,
Eq. (5) for va 6= 0 can be copied to obtain the Doppler shift for the first phase:
1+
ν = νq
1−
0
va
c/n
va 2
c
.
(13)
After the wave bounces and leaves the wall, we expect Eq. (4) for ve 6= 0 to work and the
second shift is
r
v 2
1
e
0
1
−
.
(14)
ν =ν
ve
1 − c/n
c
OPTICAL DOPPLER EFFECT . . .
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VOL. 47
Change both va and ve by v to fit the scenario of Eq. (8). It is obvious that the product of
Eqs. (13) and (14) satisfies the universal form of α, and so it is safe not to worry about n0
in either equation.
Note that Eq. (13) is exactly the form as predicted by Eq. (3) from the Lorentz
transformation. It is natural to ask if Eqs. (14) can also be reduced to the same form:
ve
1 + c/n
0
ν0 = ν q
,
ve 2
1− c
(15)
where n0 is the refractive index measured by the emitter. The emitter is now chasing the
EM waves, and so we expect the speed of light in his/her frame to be slower than c/n. The
different refractive indices can be defined through their respective speeds of light u0x = c/n0
and ux = c/n, and they are related by Eq. (1) as:
c
c
n −v
=
c .
v
n0
1 − nc2
(16)
We shall leave this as an exercise for the readers to check that Eq. (16) indeed establishes
the link between Eq. (15) and Eq. (14) as identical twins. The transverse optical Doppler
shift in vacuum can similarly be deduced.
We thank Professor Hsiuhau Lin and the late Professor Darwin Chang for useful
discussions. The support by National Science Council in Taiwan under grants 95-2120M007-008 and 96-2120-M-007-002 is gratefully acknowledged.
References
[1] I. Lerche, Am. J. Phys. 45, 1154 (1977); A. Gjurchinovski, ibid. 72, 934 (2004).
[2] D. J. Griffiths, Introduction to Quantum Mechanics, 2nd ed. (Benjamin Cummings, San Francisco, 2006), Problem 10.1.
[3] A. Chen, T. E. Milner, D. Dave, and J. S. Nelson, Opt. Lett. 22, 64 (1997).
[4] Richard A. Bachman, Am. J. Phys. 50(9), 816-818 (1982); ibid. 54(9), 848-849 (1986); ibid.
54(8), 717-719 (1986); ibid. 64(4), 493 (1996).
[5] Daniel R. Frankl, Am. J. Phys. 52(4), 374-375 (1984).