Design of a superluminal ring laser gyroscope using multilayer optical coatings with huge group delay

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 sensitivity-enhancement factor of ~103; while, the broadband FWHM of the enhancement factor can reach 20 MHz. This superluminal RLG is based on a traditional RLG with minimal re-engineering, and beneficial for miniaturization according to theoretical calculation. The idea of using GD coatings as a fast-light medium will shed lights on the design and application of fast-light sensors.

C ontrolling 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 fast-light 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 fast-light enhanced gyroscope might be able to detect the gravitational frame-dragging effect terrestrially via measuring the Lense-Thirring 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 (dual-pumped Brillouin gain in a fiber or fiber-coupled 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 White-Light-Cavity that operates at 1064 nm in the application of a LIGO-like gravitational wave detector 10 . Coupled optical resonators 9,14 can be conveniently employed to provide critical anomalous dispersion within a narrow-band 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 fast-light enhanced RLG, novel materials or systems for fast-light are needed which are compact and request minimal re-engineering; 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, low-loss, 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 Gires-Tournois 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 high-reflectivity 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 jH(v)j is the light amplitude, Q(v) is the phase shift when light propagates through the multilayer coatings. Q(v) can be expanded using Taylor series around v 0 22 : where dv 2 is the group delay dispersion, f and v are the frequency and angular frequency. Let us consider a square RLG with four high-reflectivity 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 re-engineering for practical updating from old designs. Due to the huge GD induced by M 4 , Sagnac effects of the system need to be re-calculated as follows: For an active RLG shown in Figure 1, the clockwise (CW) and counter-clockwise (CCW) ring laser modes have different frequen-cies (Df 5 f 2 2 f 1 ) because of the difference in effective round-trip optical path lengths caused by the rotation of the cavity. Here, we denote f 6 , l 6 and AEL 6 ae as the frequencies, wavelengths and effective optical cavity lengths seen by the CW and CCW propagating beams, respectively. AEL 6 ae can be expressed as follows: where AELae is the round-trip optical path length of the RLG without rotation, r 0 is the radius of the beam path for ring cavities, V 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 Df will induce additional phase difference. Thus, the resonance conditions for the CW and CCW propagating beams can be rewritten as: where a is a factor valued between 0 and 1, q 1 and q 2 are integers, l 1 5 c/f 1 , l 2 5 c/f 2 . Combining equation (3) and (4), we can obtain equation (5): where m 5 q 1 2 q 2 . Assuming m 5 0 and r 0 V = c, we can express equation (5) using Taylor series and the first order terms can be expressed by: Under such approximation, we have f 1 1 f 2 5 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 A~1 2 Lr 0 is the area enclosed by the beam path, l is the wavelength of the CW and CCW beams at zero rotation. For (7) becomes the formula for a usual RLG. For GD~{ dw dv =0 (the case shown in our design), a sensitivity enhancement factor S enh for the superluminal RLG can be calculated as Therefore, when {1v 1z( where H and L indicate quarter-wave optical layers with high and low refractive-index at 45u angle of incidence at a center wavelength of 632.8 nm. Figure 2(a) shows the theoretical refractive-index profile of a high-reflectivity 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 high-reflectivity mirrors, Ta 2 O 5 is usually selected as high refractive-index material (n 5 2.125), and SiO 2 as low refractive-index material (n 5 1.46). In fact, the multilayer structure of G/ (HL) 25 H2L(HL) 14 /A is composed of a 23-layer high-reflectivity mirror ((HL) 11 H) and a 57-layer 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 high-reflectivity multilayer-coating 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 high-reflectivity 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 Df using parameters provided above. The peak value of S enh < 1029 occurs at Df 5 0 (v 5 v 0 5 2pc/l 0 , where l 0 5 632.8 nm), and the FWHM bandwidth of enhancement factor S enh is ,20 MHz. For a regular round trip loss s , 300 ppm of the RLG, the linewidth of the ring laser cavity is Dn~c d pvLw <0:1 MHz. 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 fast-light enhanced RLG with respect to angular rotation rate V r . The relationship between the beat frequency of a standard RLG and the rotation rate is f b 5 4AV r /(L r l) according to equation (7), neglecting the effects of frequency lock-in. For a fast-light enhanced RLG, this expression needs to be modified to include the increased sensitivity term S enh : f b 5 4AS enh V r /(L r l). Figure 3(b) shows the beat frequency of both fast-light 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 V r plot for fast-light enhanced RLG with a rotation rate ranged from 230 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 fast-light 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  bandwidth of GD(l) 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(l) (in black), the FWHM bandwidth of S enh (Df) (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(l) 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(l) 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 fast-light 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 multilayer-coating system with GD is based on the traditional RLG with updating of only one mirror, which requires minimal re-engineering and will not introduce additional backscattering. Through optimizing the design of multilayer optical-coating 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 fast-light enhanced RLG is beneficial for miniaturization. With the development of micro-cavity technology such as vertical cavity emission lasers (VCSEL) [24][25][26] and integrated micro RLGs [27][28][29][30] , micro fast-light 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 re-engineering, and advantageous for miniaturization. The idea of using GD coatings as fast-light media will shed lights on the design and application of fast-light 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.