zotero/storage/9A8V7C4E/.zotero-ft-cache

The physical origin of the Fresnel drag of light by a moving dielectric medium
A. Drezet∗ Institut fu¨r Experimentalphysik, Karl Franzens Universita¨t Graz, Universita¨tsplatz 5 A-8010 Graz, Austria
We present a new derivation of the Fresnel-Fizeau formula for the drag of light by a moving medium using a simple perturbation approach. We focus particulary on the physical origin of the phenomenon and we show that it is very similar to the Doppler-Fizeau effect. We prove that this effect is, in its essential part, independent of the theory of relativity. The possibility of applications in other domains of physics is considered.
PACS numbers: 03. 30. +p; 32. 80. -t;11. 80. -m;42. 25. -p

arXiv:physics/0506004v1 [physics.optics] 1 Jun 2005

I. INTRODUCTION

It is usual to consider the famous experiment of Fizeau (1851)1 on the drag of light by a uniformly moving
medium as one of the crucial experiments which, just
as the Michelson-Morley experiment, cannot be correctly
understood without profound modification of Newtonian
space-time concepts (for a review of Einstein’s relativity
as well as a discussion of several experiments the reader is invited to consult2,3,4). The result of this experiment which was predicted by Fresnel5, in the context of elas-
tic theory, is indeed completely justified by well known arguments due to von Laue (1907)6. He deduced the
Fresnel-Fizeau result for the light velocity v in a medium,
corresponding to a relativistic first order expansion of the
Einstein velocity transformation formula:

v

=

c n0

+

ve

1

+

ve cn0

≃

c n0

+ ve

1

−

1 n20

+O

v2 c2

.

(1)

Here, n0 represents the optical index of refraction of the dielectric medium in its proper frame, and we suppose
that the uniform medium motion with velocity ve is parallel to the path of the light and oriented in the same
direction of propagation. In the context of electromagnetic theory7,8 all derivations of this effect are finally
based on the invariance property of the wave operator ∂µ∂µ[...] in a Lorentz transformation. It is easy to write the wave equation [∂x2′ − n20/c2 ∂t2′ ]ψ = 0 in the comoving frame R’(x′, t′) of the medium in covariant form11 [∂µ∂µ + n20 − 1 (vν ∂ν)2]ψ = 0 which is valid in all inertial frames and which for a plane waves, implies the
result of Eq. 1. In this calculation we obtain the result
v = c/n0 +ve if we use the Galilean transformation which proves the insufficiency of Newtonian dynamics.
However, the question of the physical meaning of this
phenomenon is not completely clear. This fact is in part
due to the existence of a derivation made by Lorentz (1895)9 based on the mixing between the macroscopic
Maxwell’s equations and a microscopic electronic oscil-
lator model which is classical in the sense of the New-
tonian dynamics. In his derivation Lorentz did not use
the relativistic transformation between the two coordi-
nate frames: laboratory and moving medium. Conse-
quently, the relativistic nature of the reasoning does not

appear explicitly. Following the point of view of Einstein (1915)10 the Lorentz demonstration must contain an implicit hypothesis of relativistic nature, however, this point has not been studied in the literature. Recent developments in optics of moving media11,12,13,14 allows us to consider this question as an important one to understand the relation between optics, relativity and newtonian dynamics. This constitutes the subject of the present paper. Here, we want to analyze the physical origin of the Fresnel-Fizeau effect. In particular we want to show that this phenomenon is, in its major part, independent of relativistic dynamics. The paper is organized as follows. In section II we present the generalized Lorentz “microscopicmacroscopic” derivation of the Fresnel formula and the principal defect of this treatment. In section III we show how to derive the Fresnel result in a perturbation approach based on the Lorentz oscillator model and finally in IV we justify this effect independently from all physical assumptions concerning the electronic structure of matter.

II. THE LORENTZ ELECTRONIC MODEL AND ITS GENERALIZATION

In this part, we are going to describe the essential con-

tents of the Lorentz model and of its relativistic exten-

sion. Let ξ (x, t) be the displacement of an electron from

its equilibrium position at rest, written as an explicit

function of the atomic position x and of the time t. In

the continuum approximation we can write the equation

of motion for the oscillator as ∂t2ξ (x, t) + ω02ξ (x, t) ≃

−

e m

E0e−i(ωt−k·x)

where

the

supposed

harmonic

electric

incident field appears and where the assumption of small

velocity allows us to neglect the magnetic force term. In

the case of a non relativistic uniformly moving medium

we have

(∂t

+

ve

·

∇)2

ξ+ω02ξ

≃

−

e m

E0

+

ve c

×

B0

e−i(ωt−k·x)

(2)

which includes the magnetic field B = ck × E/ω of the

plane wave and the associated force due to the uniform

motion with velocity ve. The equation of propagation

of the electromagnetic wave in the moving medium has

2

an elementary solution when the velocity of the light

and of the medium are parallel. If we refer to a carte-

sian frame k = keˆx, ve = veeˆx we have in this case

E

=

E0 e−i(ωt−kx) eˆy ,

B

=

c

k ω

E0

e−i(ωt−kx)

eˆz

for

the

elec-

tromagnetic field and

ξ

=

−

e m

E0 ω02 −

1

−

kve ω

(ω − kve)2

e−i(ωt−kx)

(3)

for the displacement vector parallel to the y axis. The relativistic extension of this model can be obtained directly putting ve = 0 in Eq. 2 or 3 and using a Lorentz transformation between the moving frame and the laboratory one. We deduce the displacement

ξ

=

−

e m

γe

ω02

E0

1

−

kve ω

− γe2 (ω − kve)2

e−i(ωt−kx)

(4)

where γe = 1/ (1 − ve2/c2). We could alternatively obtain the same result considering the generalization of

the Newton dynamics i. e. by doing the substitutions

m → mγe and ω0 → ω0γe−1 in Eq. 2. The dispersion relation is then completely fixed by the Maxwell equation

∂2 ∂x2

E

−

1 c2

∂2 ∂t2

E

=

4π c2

∂ ∂t

J,

where

the

current

density

J

is

given by the formula J = −eN (∂t + ve∂x) ξ depending

on the local number of atoms per unit volume N sup-

posed to be constant. Using J and Eq. 3 or 4 we obtain a

dispersion relation k2 = n2 (ω) ω2/c2 where the effective

refractive index n (ω) depends on the angular frequency

ω and on the velocity ve. The more general index ob-

tained using Eq. 4 is defined by the implicit relation

n2

(ω)

=

1

+

γe2[n20

(ω′)

−

1][1

−

n

(ω) c

ve

]2.

(5)

Here ω′

=

ω

1

−

nve c

γe,

and

n20 (ω)

=

1+

4πN0e2/ ω02 − ω2 /m is the classical Lorentz index (also

called Drude index) which contains the local proper den-

sity which is defined in the frame where the medium is

immobile by N0 = N γe−1. These relativistic equations imply directly the correct relativistic formula for the ve-

locity of light in the medium: Writing n20 − 1 = (n2 − 1)(1 − ve2/c2)/(1 − nve/c)2 = (n − ve/c)2/(1 − nve/c)2 − 1 we deduce

c n0

=

c/n − ve

1

−

ve cn

.

(6)

which can be easily transformed into

v

=

c n

=

c/n0 (ω′) + ve

1

+

ve cn0 (ω ′ )

.

(7)

It can be added that by combining these expressions we deduce the explicit formula

n2

(ω)

=

1

+

γe−2

[n20 [1 +

(ω′) − 1]

n0

(ω′ c

)ve

]2

.

(8)

The non relativistic case can be obtained directly from Eq. 3 or by writting γe = 1 in Eqs. 5,7. This limit

v

=

c n

≃

c n0

+ ve[1 −

1 n20

+

ω

d

ln n0 dω

]

+O

ve2 c2

(9)

is the Fresnel-Fizeau formula corrected by a “frequencydispersion” term due to Lorentz9. For our purpose, it
is important to note that in the non-relativistic limit of
Eq. 5 we can always write the equality

c n

=

c

− ve n′

+

ve

(10)

where n′ = n (1 − ve/c) / (1 − nve/c) is the index of refraction defined relatively to the moving medium. We
then can see directly that the association of Maxwell’s
equation with Newtonian dynamics implies a modifica-
tion of the intuitive assumption “c/n0 + ve” used in the old theory of emission. In fact, the problem can be un-
derstood in the Newtonian mechanics using the absolute time t = t′ and the transformation x = x′ + vet′. In the laboratory frame the speed of light, which in vacuum
is c, becomes c/n0 in a medium at rest. In the moving frame the speed of light in vacuum is now c − ve2. However, due to invariance of acceleration and resultant force
in a galilean transformation we can interpret the pres-
ence of the magnetic term in Eq. 2 as a correction to the
electric field in the moving frame. This effective electric
field affecting the oscillator in the moving frame is then
transformed into E (1 − nve/c). It is this term which essentially implies the existence of the effective optical index n′ = n0 and the light speed (c − ve) /n′ in the moving frame. It can be observed that naturally Maxwell’s
equations are not invariant in a Galilean transformation.
The interpretation of E (1 − nve/c) as an effective electric field is in the context of Newtonian dynamics only
formal: This field is introduced as an analogy with the case ve = 0 only in order to show that n′ must be different from n0.

III. PERTURBATION APPROACH AND OPTICAL THEOREM
The difficulty of the preceding model is that the Lorentz derivation does not clarify the meaning of the Fresnel-Fizeau phenomenon. Indeed we justify Eq. 1 using a microscopical model which is in perfect agreement with the principle of relativity. However we observe that at the limit ve ≪ c the use of the non relativistic dynamics of Newton (see Eq. 2) gives the same result. More precisely one can see from Eq. 2 that the introduction of the magnetic force −eve × B/c in addition to the electric force is already sufficient to account for the Fresnel-Fizeau effect and this even if the classical force formula F = mx¨(t) is conserved. Since the electromagnetic force contains the ratio ve/c and originates from Maxwell’s equations this is already a term of relativistic

3

FIG. 1: Representation of a linearly polarized electromagnetic plane wave travelling in a moving slab perpendicular to the x axis. The velocity of the slab is cβe, and the three spatial regions in front, in and after the slab are denoted by 1, 2, and 3, respectively. We have plotted in addition a typical observation point P(x,0,0).

nature (Einstein used indeed this fact to modify the dynamical laws of Newton15). The derivation of Lorentz is then based on Newton as well as on Einstein dynamics.
It is well know in counterpart that the Doppler-Fizeau effect, which includes the same factor 1 − ve/c, can be understood without introducing Einstein’s relativity. Indeed this effect is just a consequence of the invariance of the phase associated with a plane wave when we apply a Galilean transformation (see2, Chap. 11) as well as a Lorentz transformation. We must then analyze further
in detail the interaction of a plane wave with a moving dipole in order to see if the Fresnel phenomenon can be
understood independently of the specific Lorentz dynamics.
We consider in this part a different calculation based on a perturbation method and inspired by a derivation of the optical theorem by Feynman16,17. Consider a thin slab of thickness L perpendicular to the x axis. Let this slab
move along the positive x direction with the constant velocity veeˆx. Let in addition E0e−iω(t−x/c) be the incident electric field of a plane wave which pursues the moving slab (see Fig. 1). Therefore, the electric field after the
slab can be formally written as

Eafter = E0e−iω(t−δtve −x/c),

(11)

where δtve appear as a retardation time produced by the interaction of light with the slab and where all reflections
are neglected ( Eafter = Ebefore ). For a “motionless” slab (i. e., the case considered by Feynman) we can write
the travel time of the light through the slab as ∆τ0 = L0/c + δt0 = n0 · L0/c and therefore δt0 = (n0 − 1) · L0/c where L0 defines the proper length of the slab in the frame where it is at rest. For the general case of a moving slab of reduced length L = L0γe−1 we find for the travel

time:

∆τe

=

(L

+

ve∆τe) c

+

δtve

=

n·

(L

+

ve∆τe) c

(12)

and therefore the perturbation time is

δtve

=

(n (c

− −

1) L nve)

.

(13)

We can obtain this result more rigorously by using Maxwell’s boundary conditions at the two moving interfaces separating the matter of the slab and the air (see Appendix A).
In order to evaluate the diffracted field which is Eafter− Ebefore we can limit our calculation to a first order approximation. Thereby, each dipole of the Lorentz model as discussed above can be considered as being excited directly by the incident electromagnetic wave and where we can neglect all phenomena implying multiple interactions between light and matter. In this limit Eq. 11 reduces to

Eafter

=

E0

e−iω(t−x/c)

e+iω(n−1)

L c−nve

≃ E0e−iω(t−x/c)

1

+

iω

(n

−

1)

c

L − nve

.

(14)

If the distance between the slab and an observation point
is much larger than L we can consider the slab as a
2D continuous distribution of radiating point dipoles.
The vector potential Arad radiated by a relativistically moving point charge e is in according with the LienardWiechert’s formula given by2:

Arad (x, t) = e

v/c 1 − Rˆ · β

|ret . R

(15)

Here R = x − x0 (t) is the distance separating the observation point x (denoted by P) and the point charge

position located at x0 (t) at the time t; additionally v (t) = x˙ 0 (t) is the velocity of the point charge and Rˆ (t) is the unit vector (x − x0 (t)) /R (t). In this formula, in agreement with causality, all point charge variables are
evaluated at the retarded time tret = t − R (tret) /c. In the present case the motion of the point charge can

be decomposed into a uniform longitudinal component

vet oriented along the positive x direction and into a

transversal

oscillating

part

ξ

(t)

=

ξ0

e−iω(1−

ve c

)t

obeying

the condition ξ˙ (t) /c ≪ 1. Owing to this condition we

can identify 1 − Rˆ · v/c with 1 − Rˆ · ve/c . Conse-

quently, in the far-field the contribution of the electron uniform velocity is cancelled by the similar but opposite

contribution associated with the nucleus of the atomic

dipole: Only the vibrating contribution of the electron

survives at a long distance from the diffraction source. If

we add the contribution of each dipole of the slab act-

ing on the observation point P at the time t we obtain

then the total diffracted vector potential Adiff produced

4

by the moving medium:

Adiff

(x,

t)

≃

−2πγeN0Liω

e c

(1

−

βe)

ξ0

+∞
ρdρ

e−iω(1−βe )(t−R(tret )/c)

,

0

1 − Rˆ (tret) · ve/c R (tret)

(16)

Here ρ is the radial coordinate in a cylindrical coordinate system using the direction x as a revolution axis, and the quantity γeN0L2πρdρ is the number of dipoles contained in the cylindrical volume of length L and of radius varying between ρ and ρ + dρ if we consider a local dipole density given by γeN0. In this formula the retarded distance R (tret) is a function of ρ and we have (see the textbook of Jackson2)

R (tret) = γe−1 1 − Rˆ (tret) · ve/c −1 ρ2 + γe2 (x − vet)2.
(17) This expression shows that the minimum Rmin is obtained for a point charge on the x axis, and that:

Rmin = (x − vet) / (1 − βe) .

(18)

In order to evaluate the integral in Eq. 16 we must use in addition the following relation (see Appendix C):

R (tret) = γe2βe (x − vet) + γe ρ2 + γe2 (x − vet)2 (19)

Hence, we obtain the following integral :

Adiff

(x,

t)

≃

−2πiωγe2N0L

e c

(1

−

βe)

ξ0

·e−iω(1−β√e)(t−γe2βe(x−vet))

·

+∞

ρdρ

ei

ω c

(1−βe )γe

ρ2 +γe2 (x−ve t)2 }
,

0

ρ2 + γe2 (x − vet)2

(20)

where we have used the relations Eq. 17, Eq. 19 in the denominator and in the exponential argument of the right hand side of Eq. 16, respectively. The diffracted field is therefore directly calculable by using the variable u =

ρ2 + γe2 (x − vet)2. We obtain the result

Adiff

≃

2πγe

LN0

e c

ξ0e−iω(t−x/c).

(21)

The total diffracted electric field Ediff is obtained using Maxwell’s formula E = − (1/c) ∂tA, which gives:

Ediff

≃

2πiγeLN0ω

e c

ξ0

e−iω(t−x/c).

(22)

The final result is given substituting Eq. 4 in Eq. 22 and implies by comparison with Eq. 14

n

≃

1

+

2πN0γe2

e2 m

ω02

−

1− γe2ω2

ve c
1

2
−

ve c

2.

(23)

This equation constitutes the explicit limit N0 → 0 of
Eq. 5 and implies the correct velocity formula Eq. 7 when we neglect terms of O[N02]. It can again be observed that the present calculation can be reproduced in the non relativistic case by neglecting all terms of order (ve/c)2.

IV. PHYSICAL MEANING AND DISCUSSION

The central fact in this reasoning is “the travel condition” given by Eqs. 12,13. Indeed, of the same order in power of N0 we can deduce the relation

δtve = γeδt0 (1 − ve/c)

(24)

and consequently the condition Eq. 12 reads

∆τe

=

L

+

ve

(∆τe c

−

δt0γe)

+

δt0γe

=

n

·

(L

+

ve c

∆τe)

.

(25)

If we call δt0 the time during which the energy contained

in a plane of light moving in the positive x direction is

absorbed by the slab at rest in the laboratory, δt0γe is evidently the enlarged time for the moving case. During

the period where this plane of light is absorbed by the

slab its energy moves at the velocity ve. This fact can be directly deduced of the energy and momentum conserva-

tion laws. Indeed, let M γeve be the momentum of the

slab of mass M before the collision and ǫ the energy of

the plane of light, then during the interaction the slab is

in a excited state and its energy is now E∗ = ǫ + M γec2 and its momentum P ∗ = ǫ/c + M γeve. The velocity of the excited slab is defined by w = c2P ∗/E∗ and we can

see that in the approximation M → ∞ used here w ≃ ve (we neglect the recoil of the slab). During δt0γe the slab

moves along a path length equal to veδt0γe and thus the travel condition of the plane of energy in the moving slab

can be written

c (∆τe − γeδt0) = L + ve (∆τe − γeδt0) , (26)

which is an other form for Eq. 25. Now eliminating directly ∆τe in Eq. 26 give us the velocity v of the wave:

v

=

ve

+

1

+

c
cδt0 L

− ve (1 −

βe) γe

,

(27)

i. e.

v

=

ve

+

1

+

(n0

c −

− 1)

ve (1

−

βe) γe2

,

(28)

which depends on the optical index n0 = 1+cδt0/L0. After straightforward manipulations this formula becomes

v

=

c/n0 1+

+ ve
ve cn0

(29)

which is the Einstein formula containing the Fresnel re-
sult as the limit behavior for small ve. It can be observed that this reasoning is even more nat-
ural if we think in term of particles. A photon moving
along the axis x and pursuing an atom moving at the
velocity ve constitutes a good analogy to understand the Fresnel phenomenon. This analogy is evidently not limited to the special case of the plane wave eiω(t−x/c). If

5

for example we consider a small wave packet which before the interaction with the slab has the form

Ebefore (x, t) =

dω aω ei(kx−ωt) ,

∆ω

(30)

where ∆ω is a small interval centered on ωm, then after the interaction we must have:

Eafter (x, t) =

dω aω ei(kx−ω[t−δt(ω)) ,

∆ω

(31)

where δt (ω) is given by Eq. 13. After some manipulation we can write these two wave packets in the usual approximative form:

Ebefore ≃ ei(kmx−ωmt)

dωaω e−i(ω−ωm)[t−∂k/∂ωmx]
∆ω
= ei(kmx−ωmt)F (t − x/vg)

Eafter ≃ ei(kmx−ωm[t−δt(ωm)])F (t − x/vg − δtg) . (32)

Here, vg = ∂ωm/∂km = c is the group velocity of the pulse in vacuum and δtg = ∂ (ωmδt (ωm)) /∂ωm is the perturbation time associated with this group motion.
This equation for F possesses the same form as Eq. 11 and then the same analogy which implies Eq. 25 is pos-
sible. This can be seen from the fact that we have

δt (ω) = γeδt0(ω′) (1 − ve/c)

(33)

with ω′ = γeω(1 − ve/c). We deduce indeed

δtg = γeδt0g(ωm′ ) (1 − ve/c) ,

(34)

where we have δt0g(ωm′ ) = ∂ (ωmδt0 (ωm′ )) /∂ωm i. e. δt0g(ωm′ ) = ∂ (ωm′ δt0 (ωm′ )) /∂ωm′ . Since Eq. 33 and Eq. 34 have the same form the Fresnel law must be true for the group velocity. It is important to remark that all this reasoning conserves its validity if we put γe = 1 and if we think only in the context of Newtonian dynamics. Since the reasoning with the travel time does not explicitly use the structure of the medium involved (and no more the magnetic force −eve × B/c) it must be very general and applicable in other topics of physics concerning for example elasticity or sound. Consider as an illustration the case of a cylindrical wave guide with revolution axis x and of constant length L pursued by a wave packet of sound. We suppose that the scalar wave ψ obeys the equation [c2∂2/∂r2 −∂2/∂t2]ψ = 0 where c is the constant sound velocity. The propagative modes in the cylinder considered at rest in the laboratory are characterized by the classical dispersion relation

ω2/c2 = γn2,m + kx2

(35)

where the cut off wave vector γn,m depend only of the
two “quantum” numbers n, m and of the cross section area A of the guide (γ2 ∼ 1/A). The group velocity

∂ω/∂kx of the wave in the guide is defined by vg =

c2/ω

ω2/c2 − γ2

≃

c[1 −

1 2

c2γ

2

/ω2

]

and

the

travel

time

∆τ

by

L/vg

≃

L[1 +

1 2

c2γ

2

/ω2]/c

which

implies

δt0

=

1 2

Lcγ

2/ω2.

In

the

moving

case

where

the

cylinder

possesses the velocity ve we can directly obtain the con-

dition given by Eq. 25 (with γe = 1) and then we can

deduce the group velocity of the sound in the guide with

the formula

v

=

ve

+

1

+

c − ve

cδt0 L

(1

−

. βe)

(36)

This last equation give us the Fresnel result if we put the effective sound index n0 = 1+cδt0/L. We can control the self consistency of this calculation by observing that the dispersion relation Eq. 35 allows the definition of a phase index nphase = ck/ω ≃ 1 − c2γn2,m/(2ω2) which is equivalent to Eq. 25 when ω0 = 0 and 2πN0e2/m = c2γn2,m/2. This reveals a perfect analogy between the sound wave propagating in a moving cylinder and the light wave propagating in a moving slab. It is then not surprising that the Fresnel result is correct in the two cases. The principal limitation of our deduction is contained in the assumption expressed above for the slab example:
Eafter ≃ Ebefore i. e. the condition of no reflection supposing the perturbation on the motion of the wave to be small. Nevertheless, the principal origin of the Fresnel effect is justified in our scheme without the use of the Einstein relativity principle. We can naturally ask if the simple analogy proposed can not be extended to a dense medium i. e. without the approximation of a weak density N0 or of a low reflectivity. In order to see that it is indeed true we return to the electromagnetic theory and we suppose an infinite moving medium like the one considered in the second section. In the rest frame of the medium we can define a slab of length L0. The unique difference with the section 3 is that now this slab is not bounded by two interfaces separating the atoms from the vacuum but is surrounded by a continuous medium having the same properties and moving at the same velocity ve. In the laboratory frame the length of the moving slab is L = L0γe−1. We can write the time ∆τe taken by a signal like a wave packet, a wave front or a plane of constant phase to travel through the moving slab:

c∆τe = n · (L + ve∆τe) .

(37)

The optical index n can be the one defined in section 2 for the case of the Drude model but the result is very general. We can now introduce a time δt0 such that Eq. 25, and consequently Eq. 27, are true by definition. We conclude that this last equation Eq. 27 is equivalent to the relativistic Eq. 29 if, and only if, we define the time δt0 by the formula

δt0 = (n0 − 1) · L0/c.

(38)

In other terms we can always use the analogy with a photon pursuing an atom since the general formula Eq. 29

6

is true whatever the microscopic and Electrodynamics model considered. In this model - based on a retardation effect- the absorbtion time δt0 is always given by Eq. 38. This opens new perspectives when we consider the problem of a sound wave propagating in an effective moving medium. Indeed there are several situations where we can develop a deep analogy between the propagation of sound and the propagation of light. This implies that the conclusions obtained for the Fresnel effect for light must to a large part be valid for sound as well. This is in particular true if we consider an effective meta material like the one that is going to be described now: We consider a system of mirrors as represented in Fig. 2A, at rest in the laboratory. A beam of light propagates along the zigzag trajectory A0, B0, A1, ..., An, Bn, .... The length AnBn is given by (L20 + D2) where the distance L0 and D are represented on the figure. The time ∆τ0 spent by a particle of light to move along AnBn is then
(L20 + D2)/c. We can equivalently define an effective optical index n0 such that we have

(L20 + D2) c2

=

∆τ02

=

L20n20 c2

.

(39)

This implies

n20

=

1

+

D02 L20

.

(40)

We consider now the same problem for a system of mir-
rors moving with the velocity ve. In order to be consistent with relativity we introduce the reduced length L = L0γe−1. The beam propagating along the path A0, B0, A1, ..., An, Bn, ... must pursue the set of mirrors. We then define the travel time ∆τe along an elementary path AnBn by

((L

+ ve∆τe)2 c2

+ D2)

=

∆τe2

=

((L

+

ve∆τe c2

)2n2

,

(41)

where n is the effective optical index for the moving
medium. From this equation we deduce first ∆τe = (L/c)n/(1 − ven/c) and then

n2

−

1

=

(n20

−

1)

(1 (1

− −

ve n c

)2

(

ve c

)2

)

(42)

which finally give us the formula

c n

=

c/n0 1+

+ ve
ve cn0

(43)

We can again justify the Fresnel formula at the limit ve/c ≪ 1. The simplicity of this model is such that it does not depend on the physical properties of atoms, electrons and photons but only on geometrical parameters. Clearly we can make the same reasoning for a sound wave by putting γe = 1. This still gives us the Fresnel formula when we neglect terms equal or smaller than O ve2/c2 . In addition this model allows us to conclude that the essential element justifying the Fresnel-Fizeau result is the

FIG. 2: An ideal meta material equivalent to a medium with an effective index. A wave represented by an arrow propagates between the mirrors A0, B0, A1, ..., An, Bn, .... A) when the mirrors are at rest in the laboratory the travel time δτ0 = AnBn/c is dependent only on the distances L0 and D. B) When the mirrors move at the velocity ve relatively to the laboratory the travel time δτe = AnBn/c is affected by the motion and depends on ve as well as on D and L = L0/γe.
emergence of a delay time – a retardation effect– when we consider the propagation of the signal at a microscopic or internal level. The index n which characterizes the macroscopic or external approach is then just a way to define an effective velocity without looking for a causal explanation of the retardation. The essential message of our analysis is that by taking explicitly into account the physical origin of the delay we can justify the essence of the Fresnel-Fizeau effect in a non relativistic way. The Fresnel-Fizeau effect is then a very general phenomenon. It is a consequence of the conservation of energy and momentum and of the constant value of the wave velocity in vacuum or in the considered medium. The so called travel condition (Eq. 26) which is a combination of these two points can be compared to the usual demonstration for the Doppler effect. In these two cases of light pulses pursuing a moving particle the perturbation time δtve ≃ δt0 (1 − ve/c) is a manifestation of the Doppler phenomenon. It should be emphasized that the analogy between sound and electromagnetic waves discussed in this article could be compared to the similarities between sound wave and gravitational waves discussed in particular by Unruh. On this subject and some connected discussions concerning the acoustic Aharonov-Bohm effect (that is related to the optical Aharonov-Bohm effect that follows from the Fizeau effect) the reader should consult18,19.
V. SUMMARY
We have obtain the Fresnel-Fizeau formula using a perturbation method based on the optical theorem and in a more general way by considering the physical origin of

7

the refractive index. The modification of the speed of light in the medium appears then as a result of a retardation effect due to the duration of the interaction or absorbtion of light by the medium, and the Fresnel-Fizeau effect, as a direct consequence of the medium’s flight in front of the light. These facts rely on the same origin as the Doppler-Fizeau effect. We finally have shown that it is not correct to assume, as frequently done in the past, that a coherent and “Newtonian interpretation” of these phenomena would be impossible. On the contrary, the results do not invalidate the derivation of the Fresnel-Fizeau effect based on the principle of relativity but clarify it. We observe indeed than all reasoning is in perfect agreement with the principle of relativity. We must emphasize that even if the Fizeau/Fresnel effect is conceptually divorced from relativity it strongly motivated Einstein’s work (more even than the Michelson and Morley result). The fact that the Fizeau as well as the Michelson-Morley experiment can be justified so easily with special relativity clearly show the advantages of Einstein’s principle to obtain quickly the correct results. Nevertheless, if we look from a dynamical point of view, as it is the case here, this principle plays a role only for effects of order ve2/c2 which however are not necessary to justify the Fresnel formula.

Considering the second interface (II-III) in a similar way we obtain the following conditions

ω3

=

ω2

1

−

n 1

(ω, βe) − βe

βe

=

ω

δ3

=

Lc

n (ω, βe) − 1 − n (ω, βe) cβe

(A3)

where the 2nd equality is Eq. 13.

APPENDIX B

Acknowledgments

The author acknowledges S. Huant, M. Arndt, J. Krenn, D. Jankowska as well as the two anonymous referees for interesting and fruitful discussions during the redaction process.

APPENDIX A

Maxwell’s equations impose the continuity of the elec-
tric field on each interface of the slab. More pre-
cisely these boundary relations impose: Emedium A|S = Emedium B|S where S is one of the two moving interfaces separating vacuum and matter. Hence we obtain an
equality condition between the two phases φmedium A and φmedium B valid for all times at the interface. Let Φ1 = −iω (t − x/c) be the phase of the plane wave before the
slab. In a similar way let Φ2 = −iω2 (t − nω2,βe x/c − δ2) and Φ3 = −iω3 (t − x/c − δ3) be the phases in the slab and in vacuum after traversing the slab, respectively. In
these expressions there appear two retardation constants,
δ2,3 and the optical index of the slab. On the first interface denoted by (I-II) we have x = cβet and consequently

ω (1 − βe) t = ω2 (1 − n (ω, βe) βe)

·

t

−

1

−

n

δ2 (ω2, βe)

βe

,

(A1)

which is valid for each time and possesses the unique

solution:

ω2

=

ω

1

−

1 − βe n (ω, βe)

βe

, δ2 = 0.

(A2)

FIG. 3: In this figure Q is the position of the particle at the retarded time tret and P is the observation point at the time t. The particle moves uniformly on the the x line QR following the trajectory vet and Q’ is the position of the particle at the time t separated of P by the distance r. In addition, if we call R the projection of P on QR, then x and ρ are the coordinates of the observation point in the plane of the figure.

Using geometrical considerations (see Fig. 3) we can deduce the relation

R (tret)2 = ρ2 + (βeRret + x − vet)2 ,

(B1)

which is equivalent after manipulations to the other:

ρ2 + γe2 (x − vet)2 = 1 − βe2 Rret − βeγe2 (x − vet) 2 . (B2)
We can in a second step rewrite this equality as follows:

Rret = γe2βe (x − vet) + γe ρ2 + γe2 (x − vet)2 (B3) which is Eq. 19.

8

∗ Electronic address: aurelien.drezet@uni-graz.at 1 H. Fizeau, C. R. Acad. Sci. (Paris) 33, 349 (1851). 2 J. D. Jackson, Classical Electrodynamics, third edition
(J. Wiley and Sons, New York, 1999). 3 J. van Bladel, Relativity and engineering (Springer-Verlag,
Berlin, 1984). 4 J. H. Smith, Introduction to special relativity (Benjamin,
New York, 1965). 5 A. Fresnel, Ann. Chim. Phys. 9, 57 (1818). 6 H. von Laue, Ann. Phys. (Leipzig) 23, 989 (1907). 7 H. Minkowsky, Nachr. d. K. Ges. d. Wiss. Zu Gott., Math. -
Phys. Kl. 53 (1908). 8 W. Pauli, Theory of relativity (Pergamon press, London,
1958). 9 H. A. Lorentz, The theory of electrons and its application to
the phenomena of light and radiant heat, 2nd edn. , chapter
5 (B. G. Teubner, Stutters, 1916). 10 A. Einstein, U¨ ber die spezielle und die allgemeine rel-
ativita¨tstheorie gemeinverst¨anlich (Vieweg, Brunswick,
1917). 11 U. Leonhardt and P. Piwinicki, Phys. Rev. A60, 4301
(1999); J. Fiurazek, U. Leonhardt and R. Parentani,
Phys. Rev. A65, 011802 (2002); U. Leonhardt and P. Pi-
winicki, Phys. Rev. Lett. 84, 822 (2000); U. Leonhardt,

Phys. Rev. A62, 012111 (2000); U. Leonhardt and
P. Ohberg, Phys. Rev. A67, 053616 (2003). 12 M. Wilkens, Phys. Rev. Lett. 72, 5 (1994); H. Wei, R. Han
and X. Wei, ibid. 75 2071, (1995); G. Spavieri, ibid. 82,
3932 (1999). 13 I. Carusotto, M. Artoni, G. C. La Rocca, and F. Bassani,
Phys. Rev. Lett. 86, 2549 (2001); M. Artoni and I. Caru-
sotto, Phys. Rev. A68, 011602 (2003); I. Carusotto, M. Ar-
toni, G. C. La Rocca, and F. Bassani, Phys. Rev. A68,
063819 (2003). 14 D. Strekalov, A. B. Matsko. N. Yu. and L. Maleki,
Phys. Rev. Lett. 93, 023601 (2004). 15 A. Einstein, Ann. Phys. (Leipzig) 17, 891 (1905). 16 R. P. Feynman, R. B. Leighton and M. L. Sands, The Feyn-
man Lectures on Physics, chapters 30-31 (Addison-Wesley,
Massachusetts, 1963). 17 H. C. van de Hulst, Light scattering by small particles,
chapter 4 (John Wiley and Sons Inc. , New York, 1957). 18 W. G. Unruh, Phys. Rev. Lett. 46, 1351 (1981); M. Visser,
Class. Quant. Grav. 15, 1767 (1998). 19 S. E. P. Bergliaffa, K. Hibberd, M. Stone, and M. Visser,
Physica D 191, 121 (2004); P. Roux, J. de Rosny, M. Tan-
ter, and M. Fink, Phys. Rev. Lett. 79, 3170 (1997).