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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. REFERENCES 1. 2. ⋅ ∆S ≡ ∆ω o (20) 3. L.V. Hau, S. Harries, Z. Dutton, and C. H. Behroozi, Nature 397 594 (1999). A. V. Turukhin, V.S. Sudarshanam, M.S. Shahriar, J.A. Musser, B.S. Ham, and P.R. Hemmer, Phys. Rev. Lett. 88 023602 (2002). G. E. Stedman, Rep. Prog. Phys. 60 615 (1997). 4. 5. E. J. Post Rev. Mod. Phys. 39 475 (1967). F. Zimmer and M. Fleischhauer, Phys. Rev. Letts. 92 253201 (2004). 6. A. Einstein, Astron. Nachr. 199 9 and 47 (1914) 7. U. Leonhardt and P. Piwnicki, Phys. Rev. A 62, 055801 (2000). 8. L.J. Wang, A. Kuzmich, and A. Dogariu, Nature, 406, 277 (2000). 9. G.B. Malykin, Physics-Uspekhi 43 1229 (2000). 10. E. 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