Abstract
We propose and analyze a superluminal ring laser gyroscope (RLG) using multilayer optical coatings with huge group delay (GD). This GD assisted superluminal RLG can measure the absolute rotation with a giant sensitivityenhancement factor of ~10^{3}; while, the broadband FWHM of the enhancement factor can reach 20 MHz. This superluminal RLG is based on a traditional RLG with minimal reengineering, and beneficial for miniaturization according to theoretical calculation. The idea of using GD coatings as a fastlight medium will shed lights on the design and application of fastlight sensors.
Introduction
Controlling the speed at which light propagates has been the focus of numerous studies during the past two decades^{1,2,3}. In addition to its fundamental importance, the tunability of light speed also opens up new avenues for diverse applications ranging from optical data buffering to enhanced precision in interferometry. Recently, it has been shown that a fastlight medium can be used to realize an absolute rotation sensor whose sensitivity is enhanced with an enhancement factor as high as 10^{6} ^{4}. In this case, the fastlight enhanced gyroscope might be able to detect the gravitational framedragging effect terrestrially via measuring the LenseThirring rotation^{4,5}. This enhancement is induced by a frequency dependent phase shift within a Ring Laser Gyroscope (RLG), which has been shown both theoretically^{5,6} and experimentally^{7,8} by means of increased “mode pushing” or “mode pulling” effects. Various systems have been investigated in order to achieve optimal performances for superluminal gyroscope applications, including alkali metal vapor cells^{7,8}, coupled optical resonators^{9}, photorefractive crystals^{10}, optical fibers (dualpumped Brillouin gain in a fiber or fibercoupled whispering gallery resonators)^{11}, spectral hole burning^{12}, and rare atomic gases^{13}.
Alkali metal vapor cells have been widely investigated as anomalous dispersion system. However, additional components are requested facing significant engineering challenges, which is disadvantageous for precise measurement. Moreover, since the operating wavelength is confined by the alkali metal atoms, the applications for alkali metal vapor cells are limited. For example, it is unable to realize a WhiteLightCavity that operates at 1064 nm in the application of a LIGOlike gravitational wave detector^{10}. Coupled optical resonators^{9,14} can be conveniently employed to provide critical anomalous dispersion within a narrowband spectral range with the assistance of some passive elements. However, the FWHM bandwidth for the sensitivity enhancement factor is narrow, and the gyro is often too complex with a coupled cavity to accurately control its cavity length^{14}. The photonic structures such as photorefractive crystals and optical fibers are more suitable to integrate with practical systems, but they all suffer from fundamental limitations to provide long delays of short pulses^{15,16,17}. The nonlinear effect via spectral hole burning has been employed to induce dispersion which results in a scale factor enhancement of only 0.33~2, and an additional laser is needed^{12}. In general, existing anomalous dispersion systems are still far from practical applications. Additional components such as a vapor cell, a pumping laser or a coupled cavity are needed, which makes the gyroscope complex and unstable, leading to additional errors such as backscattering. Therefore, to reach the full potential of fastlight enhanced RLG, novel materials or systems for fastlight are needed which are compact and request minimal reengineering; the FWHM bandwidth of the enhancement factor should be large (broadband); and the novel materials or systems for fast light should not induce additional backscattering.
Multilayer optical coatings are most effective systems to modulate both the amplitude and phase of light^{18} by means of inducing group delay (GD) and group delay dispersion (GDD). For phase modulation, compared with other dispersive medium, multilayer coatings are compact, lowloss, and convenient to be integrated with other systems. Thus, they have been widely used in the field of dispersion enhancement and dispersion compensation, such as Chirped mirrors and GiresTournois mirrors^{19,20,21,22,23}. In this paper, we demonstrate a new design of a broadband superluminal ring laser gyro by employing multilayer optical coatings with huge GD on one of the mirrors while keeping its highreflectivity property. Meanwhile, we have computed the enhancement factor (S_{enh}) of the superluminal gyro and examined a number of parameters including the FWHM bandwidth for S_{enh}, the scale factor linearity, and the cavity length dependence for miniaturization issues. This GD involved superluminal RLG can greatly enhance the sensitivity of rotation measurement by a factor of 10^{3}; while, the broadband FWHM can reach 20 MHz. On contrast to all its merits, this superluminal RLG is compact and beneficial for miniaturization.
Results
The transformation function for multilayer optical coatings can be written as: where H(ω) is the light amplitude, ϕ(ω) is the phase shift when light propagates through the multilayer coatings. ϕ(ω) can be expanded using Taylor series around ω_{0}^{22}: where is defined as the group delay, is the group delay dispersion, f and ω are the frequency and angular frequency. Let us consider a square RLG with four highreflectivity mirrors (M_{1}, M_{2}, M_{3} and M_{4}), in which M_{4} is the one with a huge GD induced by multilayer coatings and M_{3} is the output mirror, as illustrated in Figure 1. The lead zirconium titanate (PZT) plate on M_{1} is used to control the laser cavity length. Compared to traditional RLGs, the only change is the multilayer coatings on M_{4}, which requires minimal reengineering for practical updating from old designs. Due to the huge GD induced by M_{4}, Sagnac effects of the system need to be recalculated as follows:
For an active RLG shown in Figure 1, the clockwise (CW) and counterclockwise (CCW) ring laser modes have different frequencies (Δf = f_{−} − f_{+}) because of the difference in effective roundtrip optical path lengths caused by the rotation of the cavity. Here, we denote f_{±}, λ_{±} and 〈L_{±}〉 as the frequencies, wavelengths and effective optical cavity lengths seen by the CW and CCW propagating beams, respectively. 〈L_{±}〉 can be expressed as follows: where 〈L〉 is the roundtrip optical path length of the RLG without rotation, r_{0} is the radius of the beam path for ring cavities, Ω is the angular velocity rotating about the normal axis through the center of the interferometer, c is the speed of light. When considering the huge GD effects of M_{4}, the frequency difference Δf will induce additional phase difference. Thus, the resonance conditions for the CW and CCW propagating beams can be rewritten as: where α is a factor valued between 0 and 1, q_{1} and q_{2} are integers, λ_{+} = c/f_{+}, λ_{−} = c/f_{−}. Combining equation (3) and (4), we can obtain equation (5): where m = q_{1} − q_{2}. Assuming m = 0 and r_{0}Ω ≪ c, we can express equation (5) using Taylor series and the first order terms can be expressed by:
Under such approximation, we have f_{+} + f_{−} = 2 f_{0}, where f_{0} is the frequency of the CW and CCW beams in RLG without rotation. Then we can obtain the following result: where is the area enclosed by the beam path, λ is the wavelength of the CW and CCW beams at zero rotation. For , , equation (7) becomes the formula for a usual RLG. For (the case shown in our design), a sensitivity enhancement factor S_{enh} for the superluminal RLG can be calculated as
Therefore, when , S_{enh} > 1, the scale factor of RLG is enlarged. For , the enhancement factor reaches its maximum. Therefore, it is critical to design multilayer optical coatings with GD around , while maintaining the highreflectivity property of the coatings for RLG. For multilayer optical coatings, there are many designs meeting above requirements. Considering practical applications, we choose the quarterwave multilayer structure as follows: where H and L indicate quarterwave optical layers with high and low refractiveindex at 45° angle of incidence at a center wavelength of 632.8 nm. Figure 2(a) shows the theoretical refractiveindex profile of a highreflectivity Ta_{2}O_{5}SiO_{2} multilayer coatings (G/(HL)^{25}H2L(HL)^{14}/A). In the case of ion beam sputtering (IBS) for the highreflectivity mirrors, Ta_{2}O_{5} is usually selected as high refractiveindex material (n = 2.125), and SiO_{2} as low refractiveindex material (n = 1.46). In fact, the multilayer structure of G/(HL)^{25}H2L(HL)^{14}/A is composed of a 23layer highreflectivity mirror ((HL)^{11}H) and a 57layer narrow bandpass filter ((LH)^{14}2L(HL)^{14}). Figure 2(b) and (c) show the computed reflectivity and GD curves as functions of wavelength for the RLG design illustrated in Figure 2(a). The calculated results suggest that this is a broadband highreflectivity multilayercoating system with huge GD (964228.61851 fs at maximum) at the central wavelength of 632.8 nm. At the maximum GD, for the cavity optical length of 289.55 mm, the enhancement factor S_{enh} can reach as high as 1029 according to equation (8). Both the highreflectivity mirrors in RLG and the bandpass filters in dense wavelength division multiplexing (DWDM) are widely used and the technology is mature, therefore this kind of superluminal RLG should be achievable.
Considering practical applications of the superluminal RLG, the FWHM bandwidths of GD and S_{enh} are very important, which need to be larger than the linewidth of the laser cavity, otherwise extremely accurate control of cavity length is required^{14}. Figure 3(a) shows the computed GD (in green) and S_{enh} (in black) versus detuning of laser frequency Δf using parameters provided above. The peak value of S_{enh} ≈ 1029 occurs at Δf = 0 (ω = ω_{0} = 2πc/λ_{0}, where λ_{0} = 632.8 nm), and the FWHM bandwidth of enhancement factor S_{enh} is ~20 MHz. For a regular round trip loss σ ~ 300 ppm of the RLG, the linewidth of the ring laser cavity is . Thus, the FWHM bandwidth of S_{enh} of our design is two orders of magnitude higher than the linewidth of ring laser cavity. We have also calculated the beat frequency f_{b} of fastlight enhanced RLG with respect to angular rotation rate Ω_{r}. The relationship between the beat frequency of a standard RLG and the rotation rate is f_{b} = 4AΩ_{r}/(L_{r}λ) according to equation (7), neglecting the effects of frequency lockin. For a fastlight enhanced RLG, this expression needs to be modified to include the increased sensitivity term S_{enh}: f_{b} = 4AS_{enh}Ω_{r}/(L_{r}λ). Figure 3(b) shows the beat frequency of both fastlight enhanced (black curve) and standard HeNe (red curve) RLGs with respect to the angular rotation rate. The inset gives an expanded view of the f_{b} vs Ω_{r} plot for fastlight enhanced RLG with a rotation rate ranged from −30 to 30 rad/s. It shows a perfect linear relation within this rotation speed range with its linear fitting shown in dashed blue line. This linear dynamic range of fastlight enhanced gyroscope is more than enough for usual navigation applications.
More importantly, equation (8) indicates that, in order to reach the same S_{enh}, for shorter optical cavity length, the GD value required is smaller, which reduces the number of layers in the mirror coatings and greatly lowers the design and fabrication difficulty in applications. Figure 4(a) shows the calculated GD curves with respect to the wavelength for ring lasers at various optical cavity lengths of 289.55 mm (black line), 30.47 mm (red line), 3.206 mm (blue line), and 0.3371 mm (green line), respectively. The corresponding multilayer coatings are 80 (G/(HL)^{25}H2L(HL)^{14}/A), 70 (G/(HL)^{23}2H(LH)^{11}L/A), 60 (G/(HL)^{20}H2L(HL)^{9}/A) and 50 layers (G/(HL)^{18}2H(LH)^{6}L/A), respectively. This suggests that, when the optical cavity length <L> decreases, the required number of layers in the mirror coatings and their GD value decease. Meanwhile, the bandwidth of GD(λ) increases (Figure 4(a)), so that this type of fastlight enhanced RLG is beneficial for miniaturization. Figure 4(b) shows the FWHM bandwidth of GD(λ) (in black), the FWHM bandwidth of S_{enh}(Δf) (in red) and the layers of multilayer coatings (in blue) with respect to the laser optical cavity length <L>, respectively, where the maximum S_{enh} is kept as constant at ~1000. It is obvious to see that, when the <L> decreases from 289.55 mm to 0.3371 mm, the number of layers required for the coatings deceases from 80 to 50; while, the GD(λ) FWHM bandwidth increases from ~0.01 Å to 7.64 Å. Therefore, the multilayer coatings with GD used for superluminal gyroscope can be realized more easily for RLGs with shorter optical cavity lengths. This multilayer optical coatings with a GD(λ) FWHM bandwidth of 7.64 Å can be easily realized by modern IBS method with optical monitoring of the central wavelength.
Discussion
Compared with other fastlight media or systems for superluminal gyroscopes such as alkali metal vapor cells^{7,8}, coupled optical resonators^{9}, photorefractive crystals^{10}, optical fibers^{11}, spectral hole burning^{12} and rare atomic gasses^{13}, the GD induced superluminal gyroscope has significant advantages as follows. Firstly, the multilayercoating system with GD is based on the traditional RLG with updating of only one mirror, which requires minimal reengineering and will not introduce additional backscattering. Through optimizing the design of multilayer opticalcoating systems, the operating wavelength of this GD induced RLG can be tunable to meet different applications. Secondly, the FWHM bandwidth of the enhancement factor S_{enh} is much larger than the linewidth of the ring laser cavity, which increases the tolerance of the system for deviations in cavity length. Thirdly, this type of fastlight enhanced RLG is beneficial for miniaturization. With the development of microcavity technology such as vertical cavity emission lasers (VCSEL)^{24,25,26} and integrated micro RLGs^{27,28,29,30}, micro fastlight enhanced RLG is promising to be realized in future with the idea of using multilayer coatings with huge GD.
In summary, we have proposed and analyzed a superluminal ring laser gyro using multilayer optical coatings with huge GD. This type of superluminal RLG has a strong sensitivity enhancement and a broadband enhancement factor that requires minimal reengineering, and advantageous for miniaturization. The idea of using GD coatings as fastlight media will shed lights on the design and application of fastlight sensors.
Methods
Considering the GD of multilayer coatings, Sagnac effects of the superluminal RLG are calculated according to the principle of laser physics directly as shown in the text, which is simple, clear and accurate. The design and computation of multilayer coatings are done by OptiLayer Thin Film Software, two targets (reflectivity and GD) are set for evaluation.
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Acknowledgements
This work was supported by the National Science Foundation of China (Grant No. 11304384), Research Project of National University of Defense Technology (Grant No. JC130702), and Open Research Fund Program of the State Key Laboratory of LowDimensional Quantum Physics.
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College of Optoelectric Science and Engineering, National University of Defense Technology, Changsha 410073, China
 Tianliang Qu
 , Kaiyong Yang
 , Xiang Han
 , Suyong Wu
 , Yun Huang
 & Hui Luo
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Contributions
T.L.Q. designed the superluminal gyroscope and performed the calculation. K.Y.Y. directed the research. X.H. carried out the calculation of Sagnac effects of the superluminal RLG when considering the GD of multilayer coatings. S.Y.W. assisted in the design and computation of multilayer coatings by Optilayer Software. T.L.Q. and K.Y.Y. prepared the manuscript and refined the paper. Y.H. and H.L. provided advices and helpful theoretical discussion. All authors discussed the results and contributed to the refinement of the paper.
Competing interests
The authors declare no competing financial interests.
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Correspondence to Tianliang Qu or Kaiyong Yang.
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