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Enhancement of interferometric precision using fast light
M.S. Shahriar, G.S. Pati, R. Tripathi, V. Gopal, and K. Salit
Department of Electrical and Computer Engineering, Northwestern University
2145 N. Sheridan Rd, Evanston IL 60208
It is generally accepted that the use of slow light does not enhance the sensitivity of a conventional
interferometer. Here, we show that the use of fast light, which makes use of anomalous dispersion,
can be employed to enhance the sensitivity of optical interferometry under certain conditions. In
particular, we show that a dual-chamber Fabry-Perot Interferometer with a shared mirror-pair can be
operated in a way so that its sensitivity is increased by a factor of 1/ng, where ng is the so called group
index. By operating the device near the critically anomalous dispersion where ng can be much less
than unity, it is possible to achieve an enhancement factor that can be as high as 109 under realistic
conditions.
PACS Codes: 0.37.-a, 0.07.-a, 45.40.Cc
It is well known that slow light does not affect the
sensitivity of an interferometer such as an MachZehnder Interferometer (MZI). Generally this
conclusion also holds for the Sagnac interferometer
except in a special case where the medium is rotating
with respect to the interferometer. This special case
does not have an extension to MZI for applications to
general purpose interferometry. However, we have
shown (GD) that when fast light is used in a passive
ring resonator, the rotation sensitivity is enhanced by a
factor ~ ng-1 where ng is the group index, which can be
very large for fast light. Here, we show that the fast
light induced enhancement can also be implemented
for general purpose sensing by using a dual-chamber
Fabry-Perot (FP) resonator
If the arms of the interferometer are filled with a
slowing medium on one sid then under continuous
wave condition, the index dispersion simply shifts the
null-point. It does not enhance the sensitivity by ∆n.
There is no enhancement in the interferometer
sensitivity under pulse operation as a pulse can be
represented by a sum of continuous wave beams. When
the MZI is analyzed in the Fourier transform basis, it is
easy to show that there is no enhancement of
sensitivity. This, again, is due to the fact that the phase
velocities of the component waves remain very close to
the free space value.
It has generally been perceived that group
velocity does not play any role in an optical
interferometer. The basic argument behind this stems
from the fact that while the group velocity represents
the speed of the envelope of a pulse, the carrier wave
underneath propagates with the phase velocity, and the
interference process is insensitive to the behavior of the
envelope. It has been determined that while this
conclusion is valid in most cases, interferometers
designed under special configurations can display an
increase in sensitivity that depends explicitly on the
ratio of the phase velocity to the group velocity, called
the group index (ng). Since ng can be tuned anywhere
from 1010 to large, negative values, this opens up the
possibility of drastic enhancements in the performance
of optical interferometric sensing and Sagnac effect
based rotation sensing.
Let us review the reason as to why an MZI
does not show enhanced sensitivity when slow light is
used. The basic argument behind this result stems from
the fact that the group velocity represents the speed of
the envelope of a pulse, while the carrier wave
underneath propagates with the phase velocity, and the
interference process is insensitive to the behavior of the
envelope.
BS2
Det
Source
BS
Figure 1: The basic setup for a
Mach Zehnder Interometer
Optical,
acoustic,
or
matter-wave
interferometry can be employed to measure the
absolute rate of rotation around an axis perpendicular
to the plane of an interferometer, via the Sagnac effect.
For illustrative purposes, the simplest configuration for
such a sensor is a Mach-Zehnder interferometer (MZI),
illustrated schematically in figure 1. The Sagnac effect
is very controversial [3,4,9], and many scientists over
the years have offered various physical explanations
that are all often (but not always) correct to lowest
order, but are sometimes wrong fundamentally. The
review by Malykin [9] summarizes these controversies.
As such, before we proceed, it is important to recall the
correct mechanism behind the Sagnac effect.
Assume, for simplicity of analysis, the MZI to
be circular in shape, with a radius of R. Consider a
general situation where a wave is propagating in both
directions around the interferometer, which is assumed
to be rotating at a rate Ω around the axis of the loop
(the result is independent of the axis of rotation, but
this choice makes the analysis simpler). The wave
could be of any kind: optical, acoustic, or matter wave.
Let us assume that the velocity of a phase front (PF) is
VP in the absence of rotation. In an inertial frame, for
the CW(+) and CCW(-) directions, the relativistic
±
velocities of the PF are denoted by VR , the time for
the PF to travel from the first beamsplitter (BS) to the
±
second one are denoted by T , and the effective
distance from the first BS to the second are denoted by
L± . These quantities are related as follows:
VP ± v
VR± =
,
L± = πR ± vT ± ,
2
1 ± VP v / Co
T ± = L± / VR±
(1)
where v=ΩR is the tangential velocity of the second
BS, Co is the velocity of light in vacuum, and the law
of addition of velocities in special relativity (SR) has
VR± . These
±
expressions can be solved easily to determine T , and
been used in deriving the expression for
the difference between them is found to be:
∆t ≡ T + − T − =
2AΩ
≈ 2AΩ / C o2 ≡ ∆t o
2
C (1 − β )
2
o
(for β ≡ v / C o << 1)
(2)
where A=πR2 is the area enclosed, and β is the boost
parameter, assumed to be much less than unity for
typical rotations. Note that this expression does not
depend at all on the velocity of the wave. Instead, it
involves the free space velocity of light only, even if
acoustic waves or matter waves are used. In the case
of optical waves, this results is independent of the
refractive index of the medium of propagation.
The form of the time delay in eqn. 2 attests to
the fact that this time delay is simply a geometric
effect, attributable to relativistic time dilations. One
way to measure this time delay is through a phase shift
in the interference observed at the location of the
second BS, given by:
∆φ = ω∆t = 4πfAΩ / C o2
(generic Sagnac phase shift )
(3)
For the case of an optical wave, this can be reexpressed
as
∆φ = 4πAΩ /(λ o C o ) ≡ ∆φ o
(optical Sagnac phase shift )
(4)
This result was derived under the assumption that the
axis of rotation coincides with the center of the loop.
However, the result remains unchanged if the axis of
rotation is displaced. This is illustrated graphically in
figure 2. Briefly, the off-axis rotation is equivalent to
an on-axis rotation, plus a movement of the center of
the loop around the true axis of rotation. The
movement of the center of the loop affects the CW and
CCW paths in the same way; therefore, it does not
produce any additional time delay or phase shift.
Therefore, the Sagnac effect is independent of the axis
of rotation.
Finally, note that the Sagnac effect is a
manifestation of SR, and as such consistent with
General Relativity (GR), since GR incorporates SR.
The application of GR is manifestly necessary only
when a significant gradient in the gravitational
potential is present, as discussed by Malykin [9]. This
is not the case for the Sagnac effect employing small
loops. As such, all the relevant analysis of the Sagnac
effect can be performed without invoking GR.
In the case of optical wave, the phase velocity
in the absence of rotation (VP) is given by Co/n, where
n is the index of refraction of the medium of
±
propagation. The relativistic velocities, V R , of the
CW and CCW phase fronts, as given by the first term
in eqn. 1, can then be expressed as (again in the limit of
β<<1):
VR± =
Co
m vα F ;
n
α F = (1 −
1
) (5)
n2
where the term αF is called the Fresnel drag coefficient,
and v=ΩR is the tangential velocity. The time delay
and the phase shift are then given by
(
)
∆t ≈ n 2 (1 − α F ) ∆to ;
(
)
∆φ = n 2 (1 − α F ) ∆φo
(6)
Note that when the expression αF from eqn. 5 is
inserted in eqn. 6, we recover the same result as in
eqns. 2 and 4 (as it must be, since we have simply
reexpressed results in a different way.) Thus, for this
particular form of the drag coefficient, the phase shift,
∆φ, is independent of n and αF. Thus, the Sagnac effect
incorporates the Fresnel drag effect, but its magnitude
is independent of the Fresnel drag coefficient, as well
as the refractive index. This issue was first clarified by
Einstein[6].
Consider the MZI depicted in figure 1 where
several modifications are done on the basicsetup. First,
the input to the MZI comes through an optical fiber,
thus making it possible to rotate the MZI without
rotating the source. Second, we assume that the
medium inside the MZI paths are allowed to move with
respect to the MZI frame (i.e., the mirrors and
beamsplitters).
Consider first the conventional case where the
laser, the frame, and the medium are all co-rotating at
the same rate of Ω. In this case, there is no Doppler
shift of any kind, as seen by the output BS. As such,
Doppler shift plays no role, and the phase-shift
expression of eqn. 4 holds. Consider next the situation
where the medium and the frame are co-rotating, but
the laser is stationary. In this case, the first BS will see
a Doppler shift. Hoewever, since this BS now
effectively serves as the source of radiation for both
arms, this shift does not have a first order effect on the
phase shift, and eqn. 4 still holds.
Finally, consider the case where the laser and the frame
are co-rotating at the rate of Ω, but the medium is is
moving at a velocity of VM with respect to the MZI
frame. As seen by the medium, the CW(+) and CCW() beams are Doppler shifted by equal and opposite
amounts, given by
∆ω ± = ±ωVM / Co . In this case,
the the relativistic velocities can be rewritten by
expanding the index n around ω, to get:
Co
∆ω ± ∂n
V ≈
(1 −
) m vα F
no
n o ∂ω
±
R
=
=
Co
ω ∂n
m VM 2
m vα F
no
n o ∂ω
(7)
(n g −n o )
Co
m VM
m vα F ;
no
n o2
where v=ΩR is the radial velocity, no is the index at the
input frequency of ω,
α F = (1 − 1 / no2 ) is the Fresnel
In
the
∂n / ∂ω
ng>>no [which results when
>> (no / ωo ) ], a condition characteristic of
limit
systems that produce slow-light, the expressions for
the time delay and the phase shift simplify to:
∆t ≈ ng ∆to ;
∆φ = ng ∆φo (10)
Thus, under this condition, the rotation sensitivity
scales directly with the slowing factor, which can be
very large (more than 1010 in a recent experiment using
photorefractive crystals[10]).
As discussed earliar, the fundamental feature
of the Sagnac effect is the difference between the
effective travel time of two phase fronts propagating in
opposite directions. The MZI architecture allows the
detection of this time difference in the form of a phase
shift. Another way this difference can be detected is by
making use of a ring cavity. The ring cavity can be
operated in either active (e.g. ring laser) or passive
mode[14]. In what follows, we consider primarily the
passive ring cavity (PRC), which is illustrated
schematically in figure 2. Just as in the case of the
MZI in figure 1, we simplify our analysis by assuming
that the interferometer is round with a radius of R.
Consider first a situation where the rate of
rotation is vanishing. In that case, a given frequency
that will resonate in the cavity in the CW direction will
also resonate in the cavity in the CCW direction. Let
us call this frequency ωo, given by:
ωo =
ω ± = VE± ⋅
(
)
∆t ≈ n 2 (1 − α L ) ∆to ;
∆φ = (n 2 (1 − α L ) )∆φo
(9)
Note that in the absence of dispersion (i.e, ng=no) αL
reduces to αF, and we recover the same results as in
eqns. 2 and 4.
∆ω o
2πN
≡ ωo ±
;
P
2
VE± = VR± ± v;
∆ω o =
±
R
(8)
where αL is the so-called Laub drag coefficient,
verified experimentally with great precision by the
group of S. Ezekiel [13]. The expressions for the time
delay and the phase shift are given by:
Co 2πN
no P
(11)
where N is an integer, and P = 2πR is the perimeter.
In the presence of rotation, the resonance frequencies
(adjacent to ωo) are different for the CW(+) and
CCW(-), and are given by:
drag coefficient, and ng is the group index, defined as
the ratio of the free space velocity of light and the
group velocity at ω. If the medium is stationary, then
VM ≅ (-v), The relativistic velocities in this case are
thus given by:

C
1 (n g − n o )
V ≈ o m vα L ; α L = 1 − 2 −

no
n o2
 no

of
(12)
ω o ΩA
2ΩRω o
=
⋅
Con o
Co n o P
±
where VR are the relativistic velocities, v=ΩR is the
±
tangential velocity, VE are the effective velocities, and
A=πR2 is the area. In a passive cavity, the beat
frequency, ∆ωo can be determined as follows. The
frequency of AOM1(AOM2) is adjusted to maximize
the cavity transmission in the CCW(CW) direction.
Parts of the transmitted signals are mixed on a detector
to produce the beat note.
Note that even though the basic mechanisms
for the MZI and the PRC are the same, there is a key
difference in the result: the signal in the case of the
PRC depends explicitly on the index of the medium.
The expression for the beat note derived in
eqn. 13 above is strictly true only when there is no
dispersion, i.e., the index is independent of frequency.
When the effect of dispersion is taken into account, the
result changes significantly. In what follows, we
derive this result, and discuss the strong implications
that follow.
Without loss of generality, we can write
ω ± = ωo ±
∆ω
2
= VE± ⋅
2πN
P
(13)
where ∆ω is considered a parameter whose amplitude
±
E ,
is to be determined. The effective velocities, V
be written as:
VE± = VR± ± v =
(14)
In the same manner as in section 1.3, we can now
expand the value of n around no, to get:
C 
v
∆ω 
n
m~
VE± = o ⋅ 1 ±
;
2  (15)
n o  Co n o
~
n ≡ [∂n / ∂ω] / n
o
diff.
Σ
Laser
Ω
VCO1
1 << ξ << C o n o / v;
∆f
(17)
v = ΩR ; ∂n / ∂ω = −(n o / ωo )[1 − ξ −1 ]
As an example, consider a ring cavity with R=1 meter,
a rotation rate of ~73 micro-radian per second (earth
rate), and no=1.5, the enhancement factor can be as
high as 1012 while still satisfying the constraints.
Note that this enhancement happens nears the vicinity
of the critical anomalous condition where
∂n / ∂ω = −(no / ωo ) . As such, we designate this
phenomenon as enhancement of rotational sensitivity
based on critically anomalous dispersion (CAD).
We have showed in detail how, by using a
medium near the Critically Anomalous Dispersion
(CAD), the rotational sensitivity of a Passive Cavity
Ring Resonator can be enhanced by a factor that can
easily exceed 1012. Here, we propose a novel type of
interferometer that can make use of the CAD in
enhancing the sensitivity of a general purpose sensor.
diff.
AOM2
AOM1
∆ω = ∆ω o ⋅ ξ;
diff.
V1
factor to be ξ so that ng=no/ ξ, assuming ξ>>1, and
applying the condition that ∆n<<1, we get the
following result:
can

Co 
v
⋅ 1 ±
±
± 
n(ω )  Co n(ω ) 
beat
det
case, we see that this result implies a potentially very
significant enhancement of sensitivity. In order to
quantify the bounds of this enhancement, we note first
that the analysis is subject to the condition that
∆n = no n~∆ω << 1 . Defining the enhancement
beat
det
Σ
V1
VCO2
Σ
V2
Laser
diff.
Test
Chamber
~ is defined as [∂n / ∂ω ] / n . Inserting eqn. 16
where n
o
into eqn. 14, we get a set of self-consistent relations
involving ∆ω, which yield our key result:
∆ωo
n
= ∆ωo ⋅ o
~
1 + ωo n
ng
Reference
Chamber
VCO1
Figure 2: Schematic illustration of the passive
ring cavity gyroscope. See text for details.
∆ω =
∆f
(16)
For systems that yield slow-light, ng>>no [i.e.,
∂n / ∂ω >> (no / ωo ) ], so that this result implies a
substantial reduction in rotational sensitivity. On the
other hand, it is just as easily possible to have a
condition where 0 < ng << 1 (characteristic of a
medium that produces the so-called fast light). In that
AOM1
AOM2
VCO2
Σ
V2
Figure 3: Schematic illustration of a general
pupose sensing interferometer with a
sensitivity enhanced by the CAD process.
See text for details.
The basic configuration of this CADenhanced Interferometer (CADI) is illustrated in figure
3. Briefly, the interferometer consists primarily of a
high-Q Fabry-Perot resonator, filled with a dispersive
medium. We assume that the operating optical
frequency of the interferometer is in the vicinity of the
CAD, as defined in section 1.4. Furthermore, we
assume that the index of this medium changes linearly,
independent of frequency (over a small bandwidth), as
a function of the physical parameter S (for example,
magnetic field, electric field, density of a non-reactive
chemical agent, temperature, pressure, etc.) to be
sensed. The volume inside is separated in two parts.
The part on the right is the refernce volume, shielded in
a way so that it does not see the effect of S. The
volume on the left is exposed to the effect of S. Two
distinct frequencies are transmitted through these
regions, and each tuned independently to the peak of
the Fabry-Perot resonance. The observable quantity
is the beat note between these two frequencies. This
model can be represented quantitatively by expressing
the indices in these regions as follows:
ref region : n (ω) = n o + ∆ω ⋅
∂n
;
∂ω
{∂n / ∂ω = −(n o / ωo )[1 − ξ −1 ]; ξ =
no
>> 1}
ng
(18)
∂n
∂n
+ ∆S ⋅ ;
test region : n (ω) = n o + ∆ω ⋅
∂ω
∂S
{∂n / ∂S ≡ σ, independent of ω}
(19)
|2>
probe
Bi-frequency
pump
|3>
|1>
Figure 4: Schematic illustratin of the BPRGD process
used for generating critically anomalous dispersion
necessary for enhanced rotation sensing in a passive
Obviously, when S=0, the two chambers are identical
so that the beat frequency is zero, as it should be. The
frequency for each zone will be , ω o = C o /( 2no L) ,
where L is the distance between the two mirrors.
When S is non-zero, consider first the case where there
is no dispersion. The beat frequency is then given by
∆ω = ω o ⋅
σ
no
For the general case, the remaining analysis is
essentially similar to the steps shown in section 1.4,
and the beat frequency is given by:
n
{ξ =
∆ω = ∆ωo ⋅ ξ;
>> 1; the CAD condition}
ng
(21)
Thus, the sensitivity of the sensor can be enhanced by a
very large factor as long as the value of ng is near the
vicinity of the CAD condition.
Finally, note that the particular arrangement shown in
figure 3 is one of many possible configurations that can
be employed to achieve this result. The choice of the
configuration will be dictated strongly by the effect one
wants to measure.
Before we proceed, it is instructive to recall briefly the
process of bi-frequency pumped Raman gain doublet
(BPRGD), which leads to the desired anomalous
dispersion. Figure 4illustrates schematically the basic
mechanism of BPRGD [8]. Briefly, the diagram on the
left shows a typical Λ system, which consists of two
metastable states (|1> and |3>) coupled to an excited
state (|2>) through electric dipole interactions. In the
presence of a steady-state Raman-type population
inversion (more atoms in 1 than in 3), which occurs
naturally for proper choice of parameters [11], and a
single-freuency pump, the probe is amplified when the
two-photon resonance condition is satisfied. If there
are two frequencies present in the pump, there are two
gain peaks (top-right figure), each corresponding to the
two photon resonance condition for one of the pumps.
From the Kramer-Koenig relations, it then follows that
at the center of these two gain peaks, the index profile
displays anomalous dispersion, (i.e., ∂n / ∂ω < 0 ), as
illustrated in the bottom-right figure. The slope of the
dispersion can be tuned by controlling the strength of
the pumps.
The so-called Critically Anomalous
Dispersion
(CAD)
occurs
when
∂n / ∂ω = −(no / ωo ) .
Thus, we have demonstrated that by using critically
anomalous dispersion (0<ng<<1), i.e., fast light in a
partitioned FP resonator, one can enhance the
interferometric sensitivity to perturbing effects by a
factor of (1/ng). This work was supported by the ARO
grant # DAAD19-001-0177 under the MURI program,
and by the AFOSR grant # FA9550-04-1-0189.
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