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A Generalized Sagnac-Wang-Fizeau formula
A. Ori, J.E. Avron Dept. of Physics, Technion, Israel
March 3, 2016

arXiv:1601.01448v2 [physics.optics] 2 Mar 2016

Abstract
We present a special-relativistic analysis of deformable interferometers where counter propagating beams share a common optical path. The optical path is allowed to change rather arbitrarily and need not be stationary. We show that, in the absence of dispersion the phase shift has two contributions. To leading order in v/c one contribution is given by Wang empirical formula for deformable Sagnac interferometers. The second contribution is due to the stretching of the optical path and we give an explicit formula for this stretch term valid to ﬁrst order in v/c. The analysis provides a unifying framework incorporating the Sagnac, Wang and Fizeau eﬀects in a single scheme and gives a rigorous proof of Wang empirical formula.

1 Introduction

In this work we present a general framework for treating interferometers where two counter-propagating beams share a common path which starts and ends at the same point. Three interferometers in this class are Sagnac [8], Fizeau [3] and Wang [10]. Fizeau’s work from 1851 played a role in the early experimental validation of the theory of relativity. The Sagnac eﬀect lies at the heart of the modern Fiber Optics Gyro and the Ring Laser Gyro [7, 2, 6]. The Sagnac eﬀect was originally born in the context of optics. However, similar ideas apply to other wave phenomena and in particular to matter waves [6, 4, 5]. The 290 references in the review [6] and over 100 references in [2] attest the vitality of the ﬁeld.
The Sagnac interferometer is shown schematically in Fig. 1. It rotates in the lab as a rigid body with an angular velocity Ω. The phase diﬀerence of the two beams at the detector, ∆Φ, is proportional to Ω. In the approximation γ ∼= 1, Sagnac area law says:

∆Φ

∼=

4ω c2

Ω

·

A.

(1)

Here A is the area enclosed by the path of two beams, ω is the frequency of the light source and c the velocity of light in vacuum. In applications, the Sagnac eﬀect is used to measure Ω. It is remarkable that the Sagnac eﬀect is independent of the refractive index n [7, 1], suggesting a universal geometric origin for the eﬀect; Hence the area law

1

Ω

Figure 1: The Sagnac interferometer rotates as a rigid body in the lab with angular velocity Ω. The beam splitter is at the bottom left corner. The two counter-propagating beams are marked by red arrows; One beam propagates with and the other against the rotation. The optical path may contain co-moving dielectric medium shown as a blue rectangle. The incoming source and outgoing signal are marked by dashed arrows.

(1) holds also when a co-moving dielectric medium is put in the path of the light beams

[7, 6]. In particular it holds for interferometers made with optical ﬁbers.

It is instructive to contrast Sagnac with Fizeau’s interferometer [3], shown schemat-

ically in Fig. 2. Here the source, mirrors and detector are at rest at the lab while a

section of the path goes through a ﬂowing liquid at (essentially uniform) velocity v0, so that one beam propagates with the ﬂow and a counter-propagating beam against it.

The phase shift ∆Φ is given by von Laue formula which, in the absence of dispersion,

reads [9]

∆Φ

=

−

2ω c2

v0Lw

(n2

−

1),

(2)

where Lw is the length of the optical path in water. There are some superﬁcial similarities between Eq. (1) and Eq. (2), but more inter-

esting are the diﬀerences: Eq. (2) has a term that depends on the refractive index and

a term that does not while Sagnac has one term independent of n. One of our aims is

to put both Sagnac and Fizeau in a common framework that would give both formulas

from a uniﬁed point of view and a single formula.

For a rigid rotation

v = Ω×r,

(3)

Sagnac area law can be written as a line integral [10]

4ω c2

˛

Ω

·

dA

=

2ω c2

˛

∇

×

v

·

dA

=

2ω c2

˛

v · dℓ

(4)

along the optical path. The ﬁrst equality uses Eq. (3) and the last Stokes formula. For rigid body motion writing the area law as a line integral seems like a futile exercise. The

2

ℓ2 Lw
ℓ1

Figure 2: Schematic Fizeau interferometer. The three mirrors and beams splitter are at rest in the lab. The two counter-propagating beams are denoted by the red arrows. Fluid, say water, is ﬂowing in one arm of the interferometer so one beam is moving against the ﬂow and one with the ﬂow.

advantage of the line integral and the key observation of Wang et. al. [10] is that while Sagnac area law does not even make formal sense for deformable interferometers, the line integral does. We therefore deﬁne

∆Φwang

=

2ω c2

˛

v · dℓ

(5)

In the case of non-stationary motions the integral should be understood as an integral

at the time of detection in the lab.

This bring us to the third type of interferometer that we shall consider: The Wang

interferometer, an example of which is shown schematically in Fig. 3. The Wang inter-

ferometer has a ﬂexible, non-stretching ﬁber, moving rather arbitrarily. (The detector

and light-source move with the ﬁber.) Wang et. al. [10] veriﬁed empirically that Eq. (5)

correctly describes the interference for a variety of ﬂexible interferometers. However, no

derivation of the formula has been given there, and its domain of validity has therefore

not been clariﬁed. One of our aims is to prove Wang formula and establish its domain

of validity. As we shall show Eq. (5) holds for non-stretching ﬁbers to ﬁrst order in v/c.

Here we present a unifying framework which covers the Sagnac, Fizeau and Wang

eﬀects. For ﬁber optics interferometers this means that we allow for ﬂexible and stretch-

ing ﬁbers and also allow the optical path to go through ﬂowing ﬂuids. The analysis is

in principle elementary.

To leading order in v/c and in the absence of dispersion we ﬁnd two contributions to

∆Φ:

∆Φ ∼= ∆Φwang + ∆Φstretch.

(6)

∆Φwang is Wang line integral of Eq. (5). In essence, it expresses the geometric properties of Minkowski space-time under Lorentz transformation and is independent of n.

3

Figure 3: An example of a Wang interferometer: The optical ﬁber shown as the red curve moves in the direction marked by the cyan arrows. The black dot represents the comoving beam-splitter and the two dashed red arrows the light source and detector. The counter-propagating light-beams share a common path and are marked by red arrows.

∆Φstretch represent the contributions of stretching. It is given by

ˆL ∆Φstretch ∼= ω
0

dℓ

ˆℓ
0

dℓ˜

s(ℓ) − s(ℓ˜) u′(ℓ˜)u′(ℓ)

,

(7)

where the integration variables ℓ and ℓ˜ are length parameters along the loop, L is the total length, u′(ℓ) = c/n(ℓ) is the velocity of light in the optical medium, and s(ℓ) is the local stretching rate, deﬁned in Eq. (11) below. In the case that the motion of the optical path is non-stationary, the integral should be understood as the integral at the time of detection in the lab. It expresses the eﬀect of change of length of given interval of the ﬁber between visits of the two beams. When n is constant, ∆Φstretch is proportional to n2.
In Wang experiments the stretching is quite small and the ﬁrst term dominates. In the Fizeau experiment stretching occurs at the ends of the pipe where the velocity of the water changes. Here both terms make signiﬁcant contribution and the two terms in Eq. (6) correspond to the two terms in Eq. (2) (in diﬀerent order).

2 Setup of the generalized Sagnac eﬀect
For the sake of concreteness consider a set-up of the generalized Sagnac eﬀect involving a closed ﬂexible and possibly stretching optical ﬁber moving in space. The ﬁber may have an arbitrary but non-dispersive refractive index n ≥ 1.
When the ﬁber’s radius is much smaller than the size of the loop we can think of it as a one-dimensional curve. The ﬁber’s conﬁguration at any given moment t is thus described, relative to the lab frame S, by the function r(θ, t). Here 0 ≤ θ ≤ θmax is a co-moving parametrization of the ﬁber, namely ﬁxed θ corresponds to a ﬁxed (material) point of the ﬁber. Since the ﬁber is a closed loop for all times t we have r(0, t) = r(θmax, t). Note that θ is deﬁned up to re-parametrization. Although some of the quantities deﬁned below depend on the choice of parameter θ, the ﬁnal result is of course independent of parametrization. We denote the overall ﬁber’s length by L.
4

+

−

θ=0

θ

e
Figure 4: The parametrized curve r(θ, t) is, for each ﬁxed time t, a closed loop in space representing a ﬁber loop. θ is a comoving coordinate, designating ﬁxed material points on the ﬁber. Shown are the tangent e at a point θ of the loop. The two counter-propagating beams are marked with the red arrows and the ± signs. The black dot represents the co-moving beam splitter located at θ = 0 and the two dashed red arrows the co-moving source and detector.

We choose a parametrization where the beam splitter is located at θ = 0. It is useful to imagine that the black triangle in the ﬁgures represents a pair of (putative) source and detector co-located at θ = 0 with world-line (t, r(0, t)) and rest frame S0. The source emits monochromatic waves with frequency ω0 (as measured in S0).
Two light beams, emanating from the beam splitter, propagate in opposite directions and make one round each along the loop. The phase diﬀerence ∆Φ = Φ+ − Φ− between the beams as they return to the beam splitter depends, among other things, on the ﬁber’s shape and motion. Being measurable, ∆Φ must be a Lorentz-invariant quantity.
Consider an inﬁnitesimal ﬁber’s segment dθ, and denote its momentary rest frame by S′. The full diﬀerential of r(θ, t) takes the form

dr = (∂θr) dθ + (∂tr) dt ≡ e dθ + v dt

(8)

where e is the tangent vector and eˆ the corresponding unit vector. We denote the inﬁnitesimal spatial displacement vector dℓ = e dθ. The segment’s velocity with respect to S (and hence the velocity of S′ with respect to S) is v (|v| < c). Note that e depends on parametrization, but v and eˆ are parametrization invariant.
We denote the length of the inﬁnitesimal segment in the lab frame by

dℓ = |dℓ| = |e|dθ

(9)

The segment’s proper length (namely its length in its own rest frame S′) is denoted dℓ′. Owing to Lorentz contraction, these two quantities are related by

dℓ′ =

dℓ2⊥ + γ2dℓ2

1/2 =

γ γ⊥

dℓ

,

(10)

5

where dℓ⊥, dℓ denote the projections of dℓ perpendicular and parallel to v. We use here γ = 1 − v2/c2 −1/2 and γ⊥ = 1 − v⊥2/c2 −1/2, where v⊥ is the component of v perpendicular to e. By its very deﬁnition dℓ (like dℓ′) is parametrization invariant.
If the ﬁber is non-stretching, dℓ′ of any segment is independent of time. In the case
of stretching the local stretching rate — or simply the stretch — is naturally deﬁned in
the Newtonian approximation by

s ≡ (dℓ)−1∂t(dℓ).

(11)

We can re-write this as

s = ∂t log dℓ = ∂t log |e| = ˆe · ∂ℓv .

(12)

The last equality follows from ∂te = ∂t∂θr = ∂θv. The partial derivative ∂ℓ is taken at ﬁxed t (namely ∂ℓ = |e|−1∂θ).
In the relativistic framework, the stretch is naturally deﬁned in the same manner as
above, as a scalar, but in the local rest frame, namely

s ≡ (dℓ′)−1∂t′ (dℓ′) = γ∂t log(dℓ′)= γ∂t log

γ γ⊥

|e|

which diﬀers from the Newtonian expressions (11,12) in second order of v/c. The phase velocity of the wave with respect to the segment’s rest frame S′, shall be
denoted by u′. It is determined by the ﬁber’s refractive index n, through 1

u′ = c/n.

We allow n to vary along the ﬁber, but we assume that it is time-independent (at any given segment), polarization independent and dispersion free. 2 Under these assumptions, the wave’s velocity at a given segment is the same for the two directions of light propagation, namely u′+ = u′− ≡ u′.
Our main concern is the phase diﬀerence ∆Φ = Φ+ − Φ− between waves in the two directions of propagation. This phase diﬀerence results from the diﬀerence in travel times in the two directions along the loop. We denote these travel times by T±, and their diﬀerence by ∆T :

∆T ≡ T+ − T− . With ω the lab-frame frequency of the source,

∆Φ = ω ∆T.

(13)

1 For single-mode ﬁbers whose width is comparable to λ, u′ is the actual phase velocity along the ﬁber, and n = c/u′.
2Allowing for chromatic dispersion would complicate the analysis and obscure the simple roots of the physics involved. Chromatic dispersion is a rather small correction in Fizeau eﬀect and is negligible in
Sagnac.

6

In a relativistic treatment, the lab frequency 3 ω and the rest-frame frequency ω0

diﬀer by time dilation: ω = ω0/γ0, where γ0 ≡ γ(θ = 0). The relativistic expression for

the phase diﬀerence is therefore

∆Φ

=

ω0 γ0

∆T

.

(14)

All that is left is to calculate the travel-time diﬀerence ∆T . We point out that Eq. (14) is accurate to order (v/c)3. This is because γ0 is
generally time dependent, and may undergo tiny changes during the short time interval

∆T . In experiments this ambiguity in the value of γ0 is totally unimportant — even its very deviation from 1 is too small to be detected. For the sake of completeness we

present the precise expression for ∆Φ, taking into account the time variation of γ0, in the Appendix. One can write this precise expression in the form (14) but with γ0 replaced by an appropriately averaged quantity γ¯0. The precise expression for ∆Φ is given in Eq. (44).

3 Calculation of travel times and phase diﬀerence
In this section we ﬁrst analyze the travel times of the two counter-propagating signals along an inﬁnitesimal ﬁber’s segment. Then we use this result to calculate the integrated time diﬀerence ∆T and phase diﬀerence ∆Φ along the closed loop.

3.1 Contribution of inﬁnitesimal segment

Let us consider an inﬁnitesimal ﬁber’s segment dθ. The segment has a momentary velocity v with respect to S, its lab-frame length is dℓ, Eq. (9), and its proper length is dℓ′, Eq. (10). The two counter-propagating waves visit the segment dθ at two diﬀerent moments t±, and have two diﬀerent (lab-frame) traversal times which we denote dt±. We shall now explicitly calculate dt+, the calculation of dt− will then follow analogously. For notational simplicity, we shall now omit the “+” index in the various quantities, and recover it later at the end of this subsection, adding the “±” suﬃx to the relevant quantities.
It is easiest to calculate the crossing time in the rest frame S′, because it is in that frame that the phase velocity takes its canonical form u′ = c/n. We obtain right away

dt′

=

dℓ′ u′

.

(15)

However, to compute T± we need to calculate the corresponding lab-frame travel time dt. This quantity is easily obtained from dt′ via Lorentz transformation, as we now
describe.

3By lab frequency we mean the lab-frame frequency of the oscillations in the laser cavity, not that of the emitted light beam. More explicitly, the laser’s lab frequency is deﬁned to be the ﬁeld’s oscillation frequency at a ﬁxed co-moving point in the laser (e.g. the beam’s exit mirror), as measured by the lab-frame clocks.

7

For concreteness let us consider a nodal point of the propagating wave, and follow its

travel across the segment dθ. Let A denote the event that the node enters the segment

and B the event that it leaves it, as shown in Fig. 5. In the lab frame, the time diﬀerence

between these two events is dt (≡ dtAB = tB − tA), and their spatial diﬀerence will be

denoted

drAB

≡

r B

−

r A

.

The

standard

Lorentz-transformation

formula

for

dt′,

applied

to the spacetime interval A-B, is

dt′ = γ

dt

−

v

·

drAB c2

.

(16)

θ

θ + dθ

t

B

drAB A

dℓ

C v dt

x

Figure 5: Space-time diagram describing the travel of a nodal point across an inﬁnitesimal ﬁber’s segment dθ. The diagram shows the projections of all 3D vectors in one direction. The two parallel black lines are the world-lines of the two segment’s edges. The red line is the world-line of the node. The lines intersect at the events A and B when the node enters and leaves the segment. Also shown are the vectors edθ = dℓ, drAB and vdt which participate in Eq. (17).

We denote by dℓ the spatial displacement vector between the two edges of the segment
(at an S-frame moment of simultaneity); namely dℓ ≡ drAC = e dθ. From Fig. 5 it is clear that drCB = vdt, and hence 4

drAB = drAC + drCB = dℓ + vdt.

(17)

Extracting dt from Eq. (16) and substituting Eq. (17) we obtain

dt =

dt′ γ

+

v · drAB c2

=

dt′ γ

+

v · dℓ c2

+

v2dt c2

.

4Note that the vectors drAB ,drAC ,drCB need not be co-linear.

(18)

8

We can now extract dt once again:

dt

=

γdt′

+

γ

2

v

· dℓ c2

=

γ

dℓ′ u′

+

γ

2

v

· dℓ c2

.

So far we considered the travel time for the “+” node. When considering a “−” node the only change is that now dℓ is to be replaced by −dℓ. We therefore obtain the exact, fully relativistic, expression for the contribution of an inﬁnitesimal segment:

dt±

=

γ±

dℓ′± u′

±

γ±2

v±

· dℓ± c2

.

(19)

Recall that the ± nodes visit the segment at two diﬀerent times t±(θ). The “±” suﬃx in

the various time-dependent quantities in Eq. (19) reﬂects this diﬀerence between values

at t+ and t−. The phase velocity u′ in a given segment is independent of time.

We denote by δt the contribution of the inﬁnitesimal segment to the overall time

diﬀerence ∆T :

δt ≡ dt+ − dt−.

(20)

The overall travel-time diﬀerence is thus

˛

∆T = δt .

(21)

3.2 The schedules t±(θ)
The expression (19) for dt± can be interpreted as a ﬁrst-order ODE which determines the exact phase schedule, namely the unknowns t±(θ). In the Appendix we describe in more details this ODE and its boundary condition and construct an exact formal expression for ∆Φ. However, for the rest of this paper we shall not really need this formulation. Instead, we shall now restrict the analysis to ﬁrst order in the small parameter v/c (which provides an excellent approximation for all lab experiments so far).

3.3 δt to ﬁrst order in v/c

To ﬁrst order in v/c we may set γ± ∼= γ0 ∼= 1 and dℓ′ ∼= dℓ in Eq. (19) which then

reduces to

dt±

=

dℓ± u′

±

v±

· dℓ± c2

.

(22)

The contribution of the inﬁnitesimal segment to the overall time diﬀerence ∆T is

δt

=dt+

−

dt−

=

dℓ+

− u′

dℓ−

+

v+

·

dℓ+

+ c2

v−

·

dℓ−

.

(23)

In general, the quantities dℓ±, dℓ±, v± on the R.H.S. may be time dependent, and are diﬀerent for the ± beams hence their “±” suﬃx. However, typically the light cycle

9

time is so short that the velocity and length of a given segment hardly change. 5 We can then safely approximate the schedule by the detection time, t±(θ) ∼= tdet. This gives

δt

∼=

dℓ+

− u′

dℓ−

+

2

v

· dℓ c2

.

(24)

We shall refer to the two terms in the R.H.S. as the stretch and Wang’s terms, which we shall respectively denote δtst and δtw. In the next two sections we address these two contributions.

4 Non-stretching ﬁbers: Wang formula

In the non-stretching case dℓ of a given segment is ﬁxed and hence dℓ+ = dℓ−. The ﬁrst term in the R.H.S. of Eq. (24) then cancels out, yielding

δt

=

2

v

· dℓ c2

≡

δtw

.

(25)

Accumulating the δt from the inﬁnitesimal segments of the loop we obtain

∆T

=

2˛ c2

v · dℓ ,

(26)

therefore we proved Wang formula, Eq. (5):

∆Φ

=

2ω c2

˛

v · dℓ ≡ ∆Φwang .

(27)

The integration is carried out with lab time set to the detection time tdet. Note that to leading order in v/c we do not need to distinguish between ω and ω0.

5 Stretching media

In the case of stretching media dℓ changes with time, and dℓ+ − dℓ− although small, no longer vanishes. However, the smallness of v compared to c (or more precisely, compared to u′ = c/n) implies that these quantities hardly change during the short time interval [t+(θ), t−(θ)]. We can then approximate

dℓ+ − dℓ− ∼= (t+ − t−) ∂t(dℓ).

(28)

From Eqs. (11) and (12)

∂t(dℓ) = s dℓ = dℓ ˆe · ∂ℓv .

(29)

This shows that s dℓ is ﬁrst order in v and hence to ﬁrst order in v/c

δtst

=

dℓ+ − dℓ− u′

∼=

(t+

− t−)

s u′

dℓ .

(30)

5 One can verify that dℓ+ − dℓ− = O(n dℓ v/c). Since we assume v/c ≪ 1 and compute ∆Φ to ﬁrst order in v/c, the change in the segment lengths can be neglected.

10

We still need to evaluate t+ − t−. Since s is proportional to v, we only need to
evaluate t± at zeroth order in v/c, that is, we may pretend that the ﬁber is static. We thus simply use dt± = ±dℓ/u′ = ±(|e|/u′)dθ, yielding

t±(θ)

∼=

ˆ ±

θ

|e(θ′ u′

)|

dθ′

,

(31)

with boundary conditions

t+(θmax) = t−(0) = tdet .

This integral may be viewed as the zeroth order of the scheduling equation (42). We therefore obtain

t+(θ)

−

t−(θ)

=

ˆθ
0

|e(θ′)| u′(θ′)

dθ′

−

ˆ θmax
θ

|e(θ′)| u′(θ′)

dθ′

=

ˆ θmax
0

sgn(θ

−

θ′)

|e(θ′)| u′(θ′)

dθ′

.

which we may conveniently re-express as

t+(ℓ)

−

t−(ℓ)

=

ˆL
0

dℓ˜

sgn(ℓ − u′(ℓ˜)

ℓ˜)

(32)

using the length parameter ℓ along the ﬁber, along with the corresponding integration variable ℓ˜.

The quantity δtst in Eq. (30) is the contribution from a given inﬁnitesimal segment dℓ. Integrating along the ﬁber and multiplying by ω we obtain the overall stretch

contribution

∆Φstretch

∼=

ω

ˆL
0

[t+(ℓ)

−

t−(ℓ)]

s(ℓ) u′(ℓ)

dℓ

.

(33)

Recall that to leading order in v/c we need not distinguish between ω and ω0. Substituting Eq. (32) in the integrand, we bring this expression to the more explicit form

ˆL ∆Φstretch ∼= ω
0

dℓ

ˆL
0

dℓ˜

s(ℓ) sgn(ℓ − u′(ℓ˜)u′(ℓ)

ℓ˜)

=

ωˆ L 20

dℓ

ˆL
0

dℓ˜

s(ℓ) − s(ℓ˜) u′(ℓ˜)u′(ℓ)

sgn(ℓ

−

ℓ˜)

(34)

which is equivalent to Eq. (7). We make the following observations:

• ∆Φstretch = 0 if s and u′ are symmetric, i.e. s(ℓ) = s(L − ℓ), and u′(ℓ) = u′(L − ℓ).

• In the special case where n is constant along the ﬁber the R.H.S. of Eq. (32) becomes (2ℓ − L)/u′ and Eq. (33) reduces to

∆Φstretch

∼=

ωn2 c2

ˆL s(ℓ)(2ℓ
0

− L) dℓ.

(35)

11

It is proportional to the ﬁrst moment of s(l) relative to the middle of the ﬁber. One can further verify that for the validity of the last equation it is suﬃcient that n is constant throughout the support of s.

• If n is constant and the stretch is localized to a single point, s(ℓ) = v0δ(ℓ − ℓ1),

then

∆Φstretch

=

ωn2v0 c2

(2ℓ1

− L) .

(36)

It thus gives information on the distance of the stretching point from the mid-point

L/2.

6 The Fizeau experiment

The framework we have described is suﬃciently general to accommodate Fizeau’s experiment. We consider a straight pipe section in which the water ﬂows. The laser beam enters the pipe through a glass window located at ℓ = ℓ1, and leaves it at another glass window at ℓ = ℓ2 with ℓ the length coordinate along the light beam. The optical path then has a section with ﬂowing water and a complementary static section where the beam-splitter and mirrors are. The entire system is stationary.
We shall only care here about the tangential velocity component v = ˆe · v of the water (the transversal velocity has no contribution at linear order). Note that v (ℓ) vanishes at the two windows at ℓ = ℓ1,2.
From Eq. (27) we obtain for Wang’s term:

∆Φwang

=

2ω c2

ˆ ℓ2
ℓ1

v

(ℓ)dℓ.

(37)

Turning next to analyze ∆Φstretch, we ﬁrst observe that the stretch occurs entirely inside the water so we can use Eq. (35). Since the laser beam is straight line inside the water,
ˆe is constant there and Eq. (12) yields s = ∂ℓv . Equation (35) therefore reads

∆Φstretch ∼=

ωn2 c2

ˆ ℓ2 (2ℓ
ℓ1

−

L)

∂ℓv

dℓ .

(38)

In a typical Fizeau experiment the pipe has a uniform cross section (as in Fig. 2), hence v has an approximately constant value v0 along the pipe—except near the windows where it abruptly drops to zero. In such a case the entire stretch contribution comes from the near-window region, where ∂ℓv has a δ-function shape (with amplitude v0 at ℓ ≈ ℓ1 and −v0 at ℓ ≈ ℓ2), which allows a simple evaluation of the last integral. It is simpler, however, to integrate Eq. (38) by parts:

∆Φstretch

=

−

2ωn2 c2

ˆ ℓ2
ℓ1

v

(ℓ)dℓ

.

(39)

(recalling that v always vanishes at the two windows).

12

Summing the two contributions (37,39) we obtain the overall Fizeau phase diﬀerence

∆Φ

=

∆Φwang

+

∆Φstretch

=

2ω c2

(1

−

ˆ ℓ2 n2)
ℓ1

v

(ℓ)dℓ .

(40)

This holds for an arbitrary v (ℓ). Hence the last expression reduces to

∆Φ

=

2ωv0Lw c2

1 − n2

,

(41)

where Lw ≡ ℓ2 − ℓ1 is the length of the light’s orbit in water, and v0 is v (ℓ) averaged over length (between ℓ1 and ℓ2). This reproduces Laue’s [9] classic result, Eq. (2).

7 Conclusion
We have described a uniﬁed framework for the phase shift in interferometers where counter-propagating beams share a common optical path, and gave a rigorous proof of Wang formula. We have shown that, neglecting dispersion, the phase shift to leading order in v/c has, in general, two contributions:
• The Wang term ∆Φwang, reﬂecting the geometry of Minkowski space and Lorentz transformations. It is accurate to ﬁrst order in v/c, is independent of n and holds if the ﬁber does not stretch.
• The stretch term ∆Φstretch which, to order v/c is given by Eq. (7). For constant n the stretch term is proportional to n2.
Our framework thus encompasses the experimental setting of Sagnac, Fizeau and Wang and sheds light on all three.

Appendix: Exact expression for ∆Φ

We shall present here the exact relativistic expression for the phase diﬀerence ∆Φ, which accounts for the time variation of the various quantities during the wave propagation along the ﬁber.
Suppose that we want to predict the value of ∆Φ at a given detection moment t = tdet. Then we need to follow the two constant-phase curves t±(θ) backward in time, along the entire closed loop, from t = tdet to the moments of emission te± at which the two counter-propagating constant-phase curves have left the source. ( Both the “source” and the “detector” are realized by the beam-splitter, located at θ = 0 or equivalently θmax.)
The schedule equation is a ﬁrst-order ODE for the unknowns t±(θ) in the range 0 ≤ θ ≤ θmax, which follows from Eqs. (19,10):

dt± dθ

=

±γ±2

|e|± (γ⊥)± u′

±

v± · e± c2

.

(42)

13

The diﬀerential equation is non-linear since the quantities on the R.H.S may be rather arbitrary functions of t± and θ (u′ only depends on θ.)
Recalling that the “+” and “−” directions respectively correspond to increasing and decreasing θ, the boundary conditions are

t+(θmax) = t−(0) = tdet.

(43)

The curves t±(θ) are determined, throughout the interval 0 ≤ θ ≤ θmax, by the

ﬁrst-order ODE (42) along with the boundary condition (43). The emission moments

te± are then given by

te+ = t+(0) , te− = t−(θmax) .

Note that the two travel times are T± = tdet − te±, and hence ∆T = te− − te+. Knowledge of te± will allow the precise calculation of ∆Φ. We assume that the

source emits a wave with ﬁxed rest-frame frequency ω0. The desired quantity ∆Φ is the

diﬀerence in the phase of the emitted wave between the moments t = te+ and t = te−. This

phase diﬀerence is ω0 multiplied by the proper time ∆τ of the source between these two

moments. Since the source speed v0 is in general time-dependent, so is it Lorentz factor

γ0, and ∆τ is thus given by an integral of 1/γ0(t) between the two relevant moments.

Therefore,

∆Φ

=

ˆ te− ω0
te+

dt γ0(t)

.

(44)

This result may also be re-expressed, similar to Eq. (14), as

∆Φ = ω0 ∆T,

(45)

γ¯0

where 1/γ¯0 is deﬁned to be the time average of 1/γ0(t) in the range between te+ and te−
(that is, the integral in Eq. (44) divided by ∆T ). Note that a fully-precise determination of ∆Φ requires knowledge of the two emission times te±, not just their diﬀerence ∆T .

Acknowledgment
The research is supported by ISF. We thank R. Wang, S. Lipson and O. Kenneth for useful discussions.

References
[1] HJ Arditty and Herbert C Lefevre. Sagnac eﬀect in ﬁber gyroscopes. Optics letters, 6(8):401–403, 1981.
[2] WW Chow, J Gea-Banacloche, LM Pedrotti, VE Sanders, Wo Schleich, and MO Scully. The ring laser gyro. Reviews of Modern Physics, 57(1):61, 1985.

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[3] MH Fizeau. Sur les hypoth`eses relatives `a l’´ether lumineux, et sur une exp´erience qui parait d´emontrer que le mouvement des corps change la vitesse avec laquelle la lumi`ere se propage dans leur int´erieur. SPIE milestone series, 28:445–449, 1991.
[4] TL Gustavson, A Landragin, and MA Kasevich. Rotation sensing with a dual atominterferometer Sagnac gyroscope. Classical and Quantum Gravity, 17(12):2385, 2000.
[5] Franz Hasselbach and Marc Nicklaus. Sagnac experiment with electrons: Observation of the rotational phase shift of electron waves in vacuum. Physical Review A, 48(1):143, 1993.
[6] Grigorii B Malykin. The Sagnac eﬀect: correct and incorrect explanations. Physics-Uspekhi, 43(12):1229, 2000.
[7] Evert Jan Post. Sagnac eﬀect. Reviews of Modern Physics, 39(2):475, 1967. [8] Georges Sagnac. L’´ether lumineux d´emontr´e par l’eﬀet du vent relatif d’´ether dans
un interf´erom`etre en rotation uniforme. CR Acad. Sci., 157:708–710, 1913. [9] Max von Laue. Zum versuch von f. Harress. Annalen der Physik, 367:448–463,
1920. [10] Ruyong Wang, Yi Zheng, and Aiping Yao. Generalized Sagnac eﬀect. Physical
review letters, 93(14):143901, 2004.
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