Abstract
Rigorous expressions of resonant frequency shift (RFS) in anomalous dispersion enhanced resonant optical gyroscopes (ADEROGs) are deduced without making approximation, which provides a precise theoretical guidance to achieve ultra-sensitive ADEROGs. A refractive index related modification factor 
is introduced when considering special theory of relativity (STR). We demonstrate that the RFS will not be ”infinitely large” by using critical anomalous dispersion (CAD) and negative modification does not exist, which make the mechanism of anomalous dispersion enhancement clear and coherent. Although step change of RFS will happen when the anomalous dispersion condition varies, the amplification of RFS is limited by attainable variation of refractive index in practice. Moreover, it is shown that the properties of anomalous dispersion will influence not only the amplification of RFS, but also the detection range of ADEROGs.
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Introduction
Resonant optical gyroscopes (ROGs) operate on the principle of the Sagnac effect1, i.e. the rotation of the optical cavity will change its resonant frequency for the light propagating in it. The resonant frequency difference between clockwise (CW) and counterclockwise (CCW) directions can be measured accurately by optical heterodyne technique like in the cases of Ring Laser Gyroscopes (RLGs), which are active resonant optical gyroscopes. In recent years, there has been interest in the development of resonant optical gyroscopes with novel materials2,3,4, structures5,6,7,8,9,10,11,12 and physical effects13,14,15,16,17,18,19, among which the utilization of fast light for sensitivity enhancement is a potential technique15,16,17,18,19. Actually, it is the strongly anomalous dispersion that amplifies the Sagnac RFS as well as allows superluminal light propagation. There are two schemes to generate anomalous dispersion in ROGs, i.e. coupled-resonator structures15,16,17 and steep negative dispersion materials18,19.
In the superluminal helium–neon RLGs15,16,17, the amplification of RFS is achieved by replacing one mirror in the traditional ring laser cavity with auxiliary passive cavities, where a theoretical rotation sensitivity enhancement of about two orders of magnitude is demonstrated15. The performance of such gyros can be regulated by changing the mirrors’ properties in these coupled cavities16 or the round-trip length of the ring lasers and the coupled cavities17. However, the sensitivity of enhancement factor to rotation or equivalent frequency detuning requires real time control of complex reflection coefficient of the mirrors, which is hard to achieve in assembled devices. Moreover, the direct relation between RFS and coupled cavities induced anomalous dispersion is not revealed, because coupled condition is considered instead of dispersion condition in these works.
The optical resonator with intracavity dispersive cell is another scheme for ADEROGs18,19. Compared to structure induced dispersion, the dispersion in materials is well studied and a number of different media can carry out anomalous dispersion, such as semiconductors20, dye solutions21, room temperature solids22, optical fibres23 and atomic vapors24. The dispersive element can be treated as both an amplitude and phase nonlinear filter18. However, the time-dependent analysis of the cavity field is intuitionistic and hard to understand. A simpler approach can be used to analyze the influence of anomalous dispersion, where the effect of Fresnel drag is considered for preciseness19. According to their results, the dispersion-related modification factor of RFS is 1/ng, where ng = n(ω) − Kω is the group index. Obviously, the RFS will be amplified with the group velocity be faster than the speed of light in vacuum if ng is less than 1. This is the reason that such amplification is connected with fast light15,16,17,18,19. Nevertheless, the modification factor is not accurate as some approximations are made in their derivations, which lead to some inappropriate conclusions. For examples, it is demonstrated that RFS will be “infinitely large” by using CAD (corresponding to the group index approaching zero). Besides, confusing negative modification for RFS is possible when the group index is less than zero.
In this paper, we theoretical analyze the RFS in ADEROGs based on concise optical resonator theory. Results of non-dispersion case and anomalous dispersion case are presented and compared. To provide a strict derivation of the expression for the magnitude of the Sagnac effect in the framework of STR, Lorentz velocity transformation is used and no approximation is made in our derivations. The amplification factor of RFS is hence not the inverse of group index, which lead to some distinct conclusions. The influences of anomalous dispersion and relativistic velocities are presented by numerical calculation and detailed discussed.
Results
According to Sagnac effect1, any closed curve with “non-zero vector area” can compose the optical path for detecting rotation. However, path-integral approach is needed in the analysis of polygon-shaped cavities. To focus on the physical effect and avoid complicated mathematics, a ring cavity with radius R is considered. As is shown in Fig. 1, the cavity is filled with non-dispersive material (for example, vacuum) or anomalous dispersive one for comparison. We assume that the cavity clockwise rotates at angular velocity Ω, i.e. the rotation velocity is v = ΩR. Results of counterclockwise rotation can be calculated in the same way used below. In addition, the CW and CCW light are presented by the superscript “+” and “−” respectively.
Diagram of the single ring resonator with radius R.
The cavity is subject to rotation at an angular rate of Ω, and the medium inside the cavity is non-dispersive or anomalous dispersive material. Resonant light in CW and CCW directions counter-propagate in the cavity simultaneously.
The optical resonator is the sensitive element of ROGs. When the light propagating in the resonator satisfies the resonant condition, the transmission will reach minimum. The resonant condition means completely coherence of the circulating light, which requires the optical length to be an integral multiple of its wavelength:
where ω is the angular frequency of the light, vp is the equivalent phase velocity, m is arbitrary positive integer and LR is the effective optical length which will be modified by rotation in gyros. Illustrated below are the expressions for the changed path length LR in a laboratory (stationary) system of coordinates25:
Here, 
are the times spent by counterpropagating waves to complete one trip around the enclosed area of the ring, while 
is the corresponding equivalent phase velocity.
Apparently, the resonant frequency will be changed by rotation too. The increase of LR will lead to the decrease of ω, and vice versa. In other words, the resonant frequency ω0 at rest will become 
when the cavity rotates (Δω is the absolute value of RFS). The shift of the resonant frequency is the key factor that dispersion works, as it is proved that the Sagnac phase shift is independent of medium parameters when the frequencies of counter-propagating waves are same25. Things are different when frequency changes. Let us take the developing process of Δω+ as example. The rotation-induced increase of effective optical length requires the decrease of resonant frequency to make the equation (1) still true. Hence the phase velocity vp = c/n (c refers to the speed of light in vacuum and n is the refractive index) decreases owning to the negative correlation between refractive index and frequency in anomalous dispersion medium, which will further cause the decrease of frequency. This process is iterative until Δω+ reaching a steady-state value, which is greater than the original one. Accurate expressions are demonstrated below.
The equivalent phase velocity and effective optical length of the resonant light in stationary cavity are vp = c/n0 and LR = 2πR respectively, where n0 is the corresponding refractive index. So we can get the resonant frequencies of CW and CCW light:
For rotation case, owing to the dispersion, the phase velocities should change into:
where n± are the refractive index for 
. According to equations (1), (2) and (4), Δω± are given by (subscripts “1” corresponds to the non-relativistic RFS):
Actually, the STR works when two coordinate frames move at constant velocity relative to each other, which is just the case happening for light propagating in rotating cavity. The expressions for the Sagnac effect in the framework of STR can be derived by taking advantage of the invariance of the interval x2 + y2 + z2 = c2t2 (where x, y, z are the wave front coordinates, and t is the time)26. A simpler and physically more illustrative way is based on the relativistic law of velocity composition25:
The general expressions of RFS in consideration of STR are deduced by combining equations (1), (2) and (6) (subscripts “2” corresponds to the relativistic RFS):
where n± are nonanalytic above. We will introduce concrete expressions of n± in different kinds of media next to get computable RFSs. For cavity with non-dispersive medium, n(ω±) = n0, the RFSs become (subscripts “n” represents non-dispersive case):
Obviously, the RFS is modified when the effect of STR is considered. The influence is rotation related just as the relativistic phase velocity does. Because 
, we can also get 
, which is in accord with the previous research19.
For cavity with anomalous dispersive medium, n(ω±) = n0 ± Δn, where Δn± are the shifts of refractive index for frequency transfers from ω0 to 
for CW light or 
for CCW light. Normally, the shifts are regarded as linear variation27:
The width of such dispersion Δω and maximum dispersion gradient Kmax are main token parameters in experiments, which lead to limited variation of the refractive index Δn. For example, the Kmax varies from 10−14 to 10−9 with Δn up to 10−7~10−5 in typical anomalously dispersive atomic media24. The limited Δn will confine the width of RFS and is an important condition for our following discussion, as some results will go against it, i.e. is unpractical. The potential performance of ADEROGs is therefore influenced.
Substituting equation (10) into equation (5) while keeping in mind that Δω± are assumed to be positive, we can get the non-relativistic RFSs (subscripts “a” represents anomalous dispersive case):
Note that there is an implied condition for CCW light, i.e. Kn0vω0 − cKω0 + cn0 > 0, which can be written as K < cn0/(cω0 − ω0n0v). As the RFSs of CW and CCW light should be detected simultaneously, the restriction on the range of K for CCW light is of essential importance in the determination of properties of anomalous dispersion. In addition, the existence of two positive solutions for CW light requires careful discussion. We will demonstrate later that 
is unpractical and useless in detection.
Relativistic RFSs are obtained in similar way:
where −cKω0 + cn0 − v > 0 is the corresponding restriction.
The results above are somehow complicated, which involve all physical details. Obviously, the dispersion-related modification factor of RFS is no more the inverse of group refractive index, which eliminates the confusing infinite-enhancement and negative-modification of RFS. Detailed discussions will be presented in next section.
Discussion
In this section, we will analyze the performance of ADEROGs by numerical simulation. While a number of different media can carry out steep negative dispersion20,21,22,23,24, atomic media offer a flexible platform for such gyros through the ability to readily control and vary the atomic parameters that influence the dispersion. We will hence choose 85Rb vapor as the anomalous dispersion materials inside our resonator here. In addition, it is vacuum that acts as the comparative non-dispersion material. Thus, the refractive index n0 for resonant frequency at rest is 1. The perimeter of the resonator is set as 78 cm to make the original resonant wavelength center on 780 nm, i.e. ω0 ≈ 2.4166 × 1015 Hz, which is the typical wavelength that experiences strongly anomalous dispersion in related research24,28. In fact, any perimeter that satisfy equation (1) is proper.
Figure 2 presents the RFSs of non-dispersion cases 
and anomalous dispersion cases 
in different angular velocities, while the influence of STR is considered. According to equation (9), the RSF in non-dispersion condition is approximately equal to ω0ΩR/c, so 
is independent with the dispersion gradient K and is parallel to horizontal axis. Because radius R is fixed, 
is proportional to the angular velocity Ω, which agrees with the previous researches1,25,26. Similarly, 
will be proportional to the radius R when angular velocity Ω is fixed, which provides a practical method to increase the sensitivity of optical gyroscopes by increasing their size. For profiles of 
, there is a threshold value for K. When K is much smaller than the critical K0, i.e. the influence of anomalous dispersion is indistinctive, 
is almost equal to 
. Evident amplification of RFS is achieved when K approaches K0 and we call this scope as strong enhancement region (SER). If K > K0, step transition of 
will happen, and 
is amplified by great extent compared to 
.
Results of 
(dashed lines) and 
(solid curves) versus the dispersion gradient K in different angular velocities.
The SER refers to strong enhancement region and K0 is the threshold value where step change happens in 
. The range of K comes from the experiments24.
To explain this step change, we firstly rewrite the equation (14) as:
where 
, 
. Because 
, the value of T1 is hardly changed by rotation velocity v, while T2 is nearly proportional to v. As is shown in Fig. 3(a), there is a transition value for T1, which is exactly the critical K0 and its expression is 
. T1 is negative when K < K0 and positive when K > K0. And it can be seen from Fig. 3(b) that 
except on the scope near K0, which means that 
is slightly greater than |T1| in most cases. Thus, when K < K0, 
and the value of T2 plays an important role in the final result, which eventually leads to the shift of 
in different angular velocities. The influence of T2 increases as the ratio between 
and T2 decreases, resulting in the existence of SER. As for the cases of K > K0, 
, where the influence of tiny T2 is inconspicuous. It is important to emphasize that the great amplification of RFS when K > K0 is useless for detection as it is nearly same in different rotation velocities. In fact, it is unrealizable in practice, because the change of refractive index Δn is beyond the available value in experiments24.
(a) T1 versus the dispersion gradient K in different rotation velocities. The profiles of different rotation velocities overlap while K0 refers to the transition of T1 (b) Value of 
versus the dispersion gradient K in different rotation velocities. 
in most cases while 
decreases rapidly on the scope near K0.
The situation for 
is slightly different, as K is confined by additional condition: 
. As is shown in Fig. 4, the situations of 
and 
are similar to the those of 
and 
, where evident amplification of RFS is achieved when K approaches K0. The profiles of 
in different rotation velocities overlap, while the change of refractive index Δn is beyond the available value in experiments24 too by the time, so only light with frequency 
can exists in the CCW direction in practice.
Results of 
(dashed lines, bottom), 
(solid curves, bottom) and 
(dot dash curves, top) versus the dispersion gradient K in different angular velocities.
Profiles of 
of different angular velocities overlap while their corresponding Δn are beyond attainable value in experiments24. The SER refers to the strong enhancement region and K0 is the limitation for 
.
The physical meaning of K0 may be understood that it is also the critical value of group index. If K > K0, group index will be negative according to its definition and the pulse envelope appears to travel backward27, which may result in the dramatic change of RFS. Similar conclusion was early demonstrated19, but we need to point out that RFS will not be “infinitely large” when using CAD (ng = 0). Figure 5 clearly shows the difference between our results and the approximate amplification factor 1/ng. It can be seen that the approximation is inaccurate when K approaches the critical K0, which will result in overestimation of the amplification effect. Our rigorous expression of RFS avoids this defect and provides a precise theoretical guidance to achieve ultrasensitive ADEROGs. Moreover, there is no confusing “negative enhancement” in our results, which leads to clear and coherent explanations of the dispersion-enhanced mechanism.
Results of 1/ng (solid curves) and 
(dashed curves) versus the dispersion gradient K with angular velocity Ω = 10 rad/s.
As the signals of CW and CCW light should be detected simultaneously and the RFS is only effectively amplified in a tiny scope, where amplification factor rapidly changes as the variation of the dispersion gradient K, subtle control of the dispersion is hence of great importance when designing such ADEROGs. Furthermore, the properties of anomalous dispersion will influence not only the amplification of RFS, but also the detection range of such gyros. As we showed above, the critical K0 is approximately equal to n0/ω0 and thus can be manipulated by changing the original refractive index and resonant frequency. However, its magnitude is fixed at around 10−15 for traditional light sources and anomalous dispersion materials, which makes the SER be in the magnitude of 10−15 too. The attainable shift of refractive index Δn in practice will therefore confine the width of RFS according to 
. We can estimate the correspondingly maximum detectable angular velocity by combing the expression of K0 and Δω2n while assuming the enhancement factor as 1/ng:
We have discussed the results of relativistic RFSs above. The results and analyses for non-relativistic RFSs are similar to those for RFSs in relativistic cases except that a modification factor 
is introduced when considering STR, i.e. the RFS in non-relativistic cases is greater than those in non-relativistic cases. Figure 6 shows the results of 
, 
, 
and 
in different n0, while the perimeter of the resonator and original resonant wavelength are set as the prior cases. Obviously, the difference between 
and 
increases when the original refractive index n0 becomes larger. Although the influence of STR is negligible in the ADEROGs with atomic vapor as n0 ≈ 1, it matters in the gyroscopes’ design when other dispersive media are used.
RFSs of 
(blue dashed lines) and 
(blue solid curves), 
(red dashed lines), 
(red solid curves) versus the dispersion gradient K in different n0 with angular velocity Ω = 10 rad/s.
n0 in (a–c) is 1, 2, 4 respectively. 
overlaps 
while 
overlaps 
in (a) as modification factor is 
.
Methods
Theoretical analysis of the resonant frequency shift in anomalous dispersion enhanced resonant optical gyroscopes is carried out according to beam propagation theory. Numerical calculation is performed by MATLAB for showing the influence of parameters.
Additional Information
How to cite this article: Lin, J. et al. Theoretical analyses of resonant frequency shift in anomalous dispersion enhanced resonant optical gyroscopes. Sci. Rep. 6, 38759; doi: 10.1038/srep38759 (2016).
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Acknowledgements
This work was supported by the International S&T Cooperation Program of China (No. 2014DFA52000) and National Natural Science Foundation of China (Nos 11574021, 51372008, 11204154, 11574016) and Beijing Municipal Science and Technology Commission (Grant No. Z161100000216149).
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Jian Lin and Zhisong Xiao performed the calculations and wrote the manuscript. Jiaming Liu, Hao Zhang and Wenxiu Li refined the paper. Lu Zhao, Junjie Jin, Anping Huang and Xiaofu Zhang provided advices and helpful theoretical discussion. All authors discussed the results and contributed to review the manuscript.
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Lin, J., Liu, J., Zhang, H. et al. Theoretical analyses of resonant frequency shift in anomalous dispersion enhanced resonant optical gyroscopes. Sci Rep 6, 38759 (2016). https://doi.org/10.1038/srep38759
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DOI: https://doi.org/10.1038/srep38759
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