Abstract

We analyze the sensitivity to inertial rotations Ω of a micron scale integrated gyroscope consisting of a coupled resonator optical waveguide (CROW). We show here that by periodic modulation of the evanescent coupling between resonators, the sensitivity to rotations can be enhanced by a factor up to 109 in comparison to a conventional CROW with uniform coupling between resonators. Moreover, the overall shape of the transmission through this CROW superlattice is qualitatively changed resulting in a single sharp transmission resonance located at Ω = 0s−1 instead of a broad transmission band. The modulated coupling therefore allows the CROW gyroscope to operate without phase biasing and with sensitivities suitable for inertial navigation even with the inclusion of resonator losses.

© 2011 Optical Society of America

1. Introduction

Since the introduction of the first ring laser gyroscope (RLG) by Macek and Davis in 1963 and later the fiber optic gyroscope (FOG) by Vali and Shorthill in 1976, optical gyroscopes have become a mainstay of the global aerospace and defense industry being used in civilian and military aircraft, rockets, and missiles for inertial navigation and varying other applications such as vehicle and antenna stabilization. Both gyroscopes operate via the Sagnac effect by which light traveling around a closed path experiences in the presence of an inertial rotation an increased optical path length when co-propagating with the rotation and a decreased path length when counter-propagating relative to the rotation.

Despite their success, RLGs and FOGs are unsuitable for many portable device applications because of their relatively large size and weight. A typlical RLG weighs several kilograms with a volume exceeding 2000cm3 and uses around 10W of power [1] while FOGs are only slightly better weighing at least several hundred grams and utilizing a kilometer or more of optical fiber wrapped around a circular core with radius ∼ 10cm [2]. MEMS (microelectromechanical systems) gyroscopes are miniaturized mechanical gyros that can be integrated onto a standard semiconductor microchip and are currently used in smart phones, tablet computers, and digital cameras. However, the best MEMS gyros have sensitivities that are on the order of 10 deg per hour, which is far greater than the 0.01 deg per hour or better sensitivities needed for inertial navigation [1].

Here we propose a new type of on chip integrated optical gyroscope that offers the size advantages of MEMS along with sensitivities comparable to much larger commercial optical gyroscopes. Our gyroscope consists of a linear array of microring optical resonators coupled via evanescent waves with shot noise limited rotation sensitivities on the order of ∼ 0.001 deg per hour. Utilizing microresonators with radii 10–100μm and N = 10–100 resonators such as have been made with silicon on insulator waveguides [3, 4] or polymer rings [5, 6], the overall dimensions would be 1 – 0.1mm2, comparable to MEMS gyros.

In 2006, Scheuer and Yariv proposed an optical Sagnac gyroscope using a CROW [7]. A CROW is an array of high-Q microresonators coupled to each other via evanescent fields between nearest neighbors that create a wave guide mode through the entire structure, the so called CROW mode, which exhibits greatly reduced group velocities controllable by the evanescent coupling [8]. The authors argued that the slow optical group velocities in CROWs leads to an enhanced sensitivity to rotations. It was later shown however that the enhanced sensitivity was a result of an improper evaluation of the sensitivity and, in reality, the sensitivity of a CROW gyroscope is equal to a resonant FOG (RFOG) with the same enclosed area [9]. Since the sensitivity of a Sagnac gyroscope is proportional to the enclosed area and the area of a microresonator CROW gyroscope would be 105 – 106 smaller than commercial FOGs, it would seem that the utility of CROW gyroscopes would be quite limited.

We introduce here a method for enhancing the sensitivity of CROW gyroscopes by periodically modulating the evanescent coupling κ between resonators between weak and strong coupling. The addition of this new periodicity in the array forms an effective superlattice that narrows the transmission band leaving a single isolated transmission resonance centered at Ω = 0s−1. Rotation measurements in the vicinity of this resonance yield shot noise limited sensitivities 101 – 109 better than previously proposed CROW gyroscopes [7, 9] with 11–21 resonators. By increasing the difference between strong and weak coupling κ’s, one can achieve rotation sensitivities that would otherwise require a CROW with an orders of magnitude larger footprint.

κ, as used here, is the power coupling coefficient between resonators and represents the fraction of power coupled from one resonator to the adjacent resonator due to the penetration of the evanescent fields into the gap spacing between them [9, 10]. It can be evaluated in terms of the overlap integral between the electric fields in adjacent resonators and depends on the space between resonators, the distance along the resonators over which the fields overlap, and the difference of the indices of refraction in the resonators and in the gap between them [1113]. Increasing either the distance between resonators or decreasing the portion of the circumference over which the fields overlap will decrease κ and large variations of κ are possible due to the high sensitivity to the resonator spacing. Control of κ between microring resonators has already been explored experimentally using various techniques for the purpose of designing CROW optical filters [14]. Most commonly the gap spacing between resonators is varied [13] but other techniques that manipulate the indices of refraction have also been explored including laser induced photobleaching of the refractive index in polymer ring resonators [12], on-chip fluid cladding of resonators [15], and the thermo-optic effect whereby electrical heating elements locally control the index of refraction [16].

The outline of the remainder of this paper is as follows: In section 2, we present the transfer matrix approach for calculating the transmission through the resonator array in the presence of inertial rotations. In section 3, the modulation scheme for the evanescent couplings is presented and the its effect on the gyroscopic sensitivity is analyzed both for ideal resonators and for resonators with finite Q-factors. Finally, section 4 presents our conclusions and future outlook.

2. Model

The CROW gyro we will consider is illustrated in Fig. 1. It consists of a one dimensional array of microring resonators of radius R ∼ 10 – 100μm connected to a 3dB coupler into which an input light beam is split and then injected into the two ends of the CROW array. After propagating through the CROW structure in both directions the transmitted light is recombined again at the 3dB coupler producing two output signals. Inside of the resonators, a photon circulates around each resonator on average ∼ 1/2κ times before hopping to the adjacent resonator due to the overlap of the evanescent electromagnetic field between resonators. As a result of momentum conservation, the propagating right handed mode (RHM) of one resonator couples to the counter propagating left handed mode (LHM) of the adjacent resonator and vice versa. Besides phase matching of the electric field at the coupling junction, the resonators must share a common resonance frequency, ω0, equal to the injected light, ω for the electric field to propagate in the CROW mode without dispersion [9, 17]. Experiments have shown that size variations created during fabrication, which lead to resonance frequency variations, can be made significantly smaller than the intrinsic resonator line widths [4, 18] implying that the main obstacle limiting the maximum number of resonators in a CROW is the individual resonator losses. In microring resonators, the dominant loss mechanisms are propagation losses due to light scattering at rough sidewalls of the waveguide and bending losses in which the light leaks out at bends of the waveguide leading to intrinsic (unloaded) Q-factors that have been measured up to 500,000 [11, 19].

 figure: Fig. 1

Fig. 1 (a) Illustration of micro-resonator CROW gyroscope on planar substrate connected to a 3dB coupler with periodic modulation of evanescent coupling between resonators. (b) Schematic diagram of proposed modulation scheme for evanescent couplings κj. Note that the green and red discs represent the strength of the evanescent coupling between resonators, which alternates between weak (green) and strong (red) coupling or vice versa symmetrically about the center resonator.

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Rotating the CROW about an axis perpendicular to the plane of the device at the rate Ω introduces a phase shift between LHMs and RHMs due to the Sagnac effect. This phase shift is proportional to the enclosed area πR2 of the resonators and for a single resonator the round trip Sagnac phase shift is given by

ϕS=4πωΩR2c2
Since ϕSR2, larger rings will not only result in a larger phase shift but also lower bending losses and higher Q-factors due to the larger radius of curvature. Individually each microring behaves like an RFOG with the effect of κ being to multiply the overall phase shift per ring by a factor of 1/2κ, corresponding to the average number of round trips of a photon in each ring [9].

We utilize the transfer matrix approach for CROWs developed in Ref. [20] to obtain the optical transmission T through the device from which the sensitivity with respect to the rotation rate dT/dϕS can be evaluated. The transfer matrices for the input (Uin), output (Uout), and repetitive RHM (URHM) and repetitive LHM (ULHM) segments are shown in Eqs. (25) in terms of the coupling coefficient κj and optical phase shift in each ring ϕj,

Uin=iκ1(exp[iϕ1]1κ1exp[iϕ1]1κ1exp[iϕ1]exp[iϕ1])
Uout=iκN+1(exp[iϕN]1κN+1exp[iϕN]1κN+1exp[iϕN]exp[iϕN])(i1κNκNiκNiκNi1κNκN)
URHM(j)=(i1κjκjexp[iϕj]iκjexp[iϕj]iκjexp[iϕj]i1κjκjexp[iϕj]),
ULHM(j)=(i1κjκjexp[iϕj]iκjexp[iϕj]iκjexp[iϕj]i1κjκjexp[iϕj])

κj is the evanescent coupling between the j – 1st and jth resonator, which for j = 1 represents the coupling between the input bus waveguide and the first resonator and for j = N + 1 it represents the coupling between the Nth resonator and the output bus waveguide. Here we use the total phase shift in the jth ring ϕj = πβR +ϕS/2, which is a sum of the rotation induced Sagnac phase shift and the ordinary propagation phase. β = nω/c±ia/2 where a=(nω/c)Qint1 is the power attenuation coefficient per unit length in terms of the intrinsic quality factor Qint of the resonator. It is worth mentioning that the Sagnac phase shift is independent of the center of rotation so that the phase shift in each resonator will be the same regardless of its distance from the axis of rotation or geometry of the array [7].

We calculate the CROW transmission by multiplying the transfer matrices of the constituent segments to obtain the coupled cavity waveguide (CCW) matrix,

TCCW=Uoutl=1(N3)/2ULHM(2l+2)URHM(2l+1)ULHM(2)Uin
=(T11T12T21T22)
for a CROW with N resonators. The transmission function, which is proportional to the output intensity, is given by T(ϕS) = 1/|T22|2.

3. Results

The central feature of our proposal is a periodic modulation of κj that is symmetric about the central resonator of the array. The coupling alternates between κα and κβ starting on each side with the coupling between the two bus waveguides and edge resonators and moving inwards towards the central resonator. This symmetry about the central ring requires that the number of resonators N be odd. The coupling can be parameterized as

κj={καj=1,3,5,(N±1)/2κβj=2,4,6,(N1)/2καj=2+(N1)/2,4+(N1)/2,,N+1κβj=2+(N±)1/2,4+(N±1)/2,N
where the upper (lower) sign is used when (N + 1)/2 is odd (even). In the case that either κακβ or κακβ there is qualitative change in the transmission spectrum as a function of ϕS, which is shown in Fig. 2(a) for κα = 0.01 and κβ = 0.1 with Qint1=0. For comparison, the transmission of two uniform CROWs one with κα and the second with κβ for all of the couplings is also shown. One can see that the broad transmission band of the uniform CROW, consisting of Fabry-Perot resonances resulting from the finite size of the CROW [14], is replaced by a single transmission resonance centered about ϕS = 0 whose slope is noticeably larger than the uniform CROWs. The width of the central resonance decreases both with increasing N and with increasing |κακβ| as shown in Fig. 2(b). Consequently, by increasing |κακβ| alone one can achieve narrow resonances whose slopes are comparable to what would be obtained with an uniform CROW having a much larger number of resonators.

 figure: Fig. 2

Fig. 2 (a) Transmission spectrum as a function of Sagnac phase shift per resonator ϕS for the alternating symmetric coupling modulation with κα = 0.1 and κβ = 0.01 (red line). Notice the single transmission resonance at 0 phase. For comparison the transmission through CROWs with uniform coupling constants κ = 0.01 (blue line) and κ = 0.1 (green line) are also shown. In all plots N = 13. (b) Closeup of central transmission resonance for κα = 0.01 as a function of ϕS and κβ for N = 9 rings. One can see clearly the narrowing of the resonance for increasing |κακβ|.

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Additionally, the ordinary CROW gyro with uniform coupling achieves its maximum transmission slope not at ϕS but rather at the edges of the transmission band located symmetrically around zero phase at ϕS ≈ ±0.1 and ±0.3 for the specific cases in Fig 2(a). Thus, for the ordinary CROW gyro, phase biasing the location of the maximum slope towards ϕS = 0 would be necessary to achieve the maximum sensitivity for small rotations. Such phase biasing is not necessary with the coupling modulated CROW gyro since the maximum sensitivity is already obtained in the vicinity of ϕS = 0.

The slope dT/dϕS is proportional to the change in the measured output power per unit rotation and the maximum value of dT/dϕS is shown in Fig. 3 without resonator losses and compared to uniform CROWs. For N = 11 – 21 resonators, dT/dϕS is 10 to 109 larger than a comparable uniform coupling CROW. (It should be noted for the sake of comparison in Fig. 3 that for a uniform CROW, the slope decreases with increasing κ.) In Fig. 3(a), κακβ, which yields better maximal slopes than the opposite case κακβ shown in Fig. 3(b). The addition of four resonators to the array, causes a jump in the slope by nearly an order of magnitude, whereas the addition of only two resonators, results in only a small increase in slope. This is because for N = 5,9,13,... the coupling at the edges of the array is the same as the couplings to the center resonator whereas for N = 3,7,11,... the edge coupling and coupling to the center resonator are different.

 figure: Fig. 3

Fig. 3 (a) Maximum value of dT/dϕS in the vicinity ϕS = 0 as a function of N for various coupling modulation strengths with weak coupling at the edges, κακβ. Also shown is the maximum dT/dϕS for a uniform CROW with κ = 0.01 at the edge of the transmission band. In this figure Qint1=0. (b) The same as part (a) except that coupling is strong at the edges, κακβ.

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The shot noise limited minimum rotation rate can be calculated using the expression

Ωmin=λc8πAeff2hcΔfληPopt
where λ = 2πc/ω, Δf is the measurement bandwidth, Popt is the detected optical power, and η the quantum efficiency of the photodetector [1, 21]. Aeff = ξNπR2 is the effective area of the gyroscope where ξ is the enhancement factor of the CROW gyroscope compared to a standard FOG with the some total geometric area NπR2. It was shown in Ref. [9] that the effective area of CROW gyroscope with uniform κ’s is 1/2κ larger than the geometric area. By contrast, for periodic modulation of κj as shown in Fig. 3, the slope, which is proportional to ξN, is 101 to 109 larger than that of a uniform CROW. As an example, consider a CROW gyro with N = 21 resonators each with R = 50μm, which has a geometric area of 0.16mm2.Taking an intermediate estimate of 107 for the improvement over a uniform CROW gyroscope, one gets Ωmin = 0.002 deg per hour assuming λ = 1.55μm, Popt = 0.1mW, η = 1, and Δf = 1Hz. This is an order of magnitude better than the required sensitivities for inertial navigation but with a footprint that is 105 smaller than a typical FOG. It is also four orders of magnitude better than the best sensitivities of MEMS gyros but with a comparable footprint.

Next, we address the effect that cavity losses have on the central resonance and its slope, which is shown in Fig. 4. One can see directly from Fig. 4(a) that for Qint ≥ 106, there is virtually no change in the slope of the resonance and maximum transmission is still greater than 90% for the N = 11 rings shown in Fig. 4(a). However, the central resonance is visibly degraded when Qint ≤ 105 with transmission falling below 10% for Qint = 104. These conclusions are reflected in Fig. 4(b) where one can see the maximum value of dT/dϕS again. For micro-resonators with Qint > 106, there is no meaningful difference in the slope compared to Qint1=0 implying that our previous estimate of the minimum detectable rotation rate remains valid. For Qint = 105, by contrast, one can see that sensitivity is reduced by nearly a factor of 10 at N = 21 while the maximum transmission is reduced to about 20%. Using the same parameters as in the example from the previous paragraph, one would have Ωmin ≈ 0.05 deg per hour for Qint = 105 due both to the reduced slope dT/dϕS and reduced detected power, Popt.

 figure: Fig. 4

Fig. 4 (a) Effect of finite Qint on the transmission resonance of the alternating symmetric coupling modulated CROW for κα = 0.01 and κβ = 0.7 for N = 11 resonators. (b) Maximum value of dT/dϕS in the vicinity ϕS = 0 as a function of N for finite Qint and coupling constants modulations κα = 0.01 and κβ = 0.3 (corresponding to Fig. 3(a)) and κα = 0.3 and κβ = 0.01 (corresponding to Fig. 3(b)). Also shown is the maximum dT/dϕS for a uniform CROW with κ = 0.01 at the edge of the transmission band. In both figures the resonators have radii R = 25μm and index of refraction n = 1.6

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Finally, it is worth noting that these results are robust even if one includes random variations of the couplings κj up to 1% of their defined values. Random variations much larger than 1% tend to significantly broaden the central transmission resonance and thereby reduce the slope. Moreover, random variations of the couplings also lead to an irregular spacing of the transmission resonances and the appearance of new transmission resonances. The new resonances can in certain instances have transmission slopes comparable to what has been presented here but their random locations make them impractical for any types of measurements. Fortunately, variations of the coupling strengths could be actively compensated for using localized heating elements located at the junction between resonators that control the index of refraction using the thermo-optic effect [16].

4. Conclusions

In conclusion, we have examined the transmission through a CROW subject to inertial rotations. By periodic modulation of the evanescent coupling between microring resonators, shot noise limited sensitivities can be achieved that are orders of magnitude better than CROW gyros with uniform coupling and comparable to much larger FOG and RLG gyroscopes. This is opens up the tantalizing prospect of all optical on chip gyroscopes with sufficient sensitivity for inertial navigation. It remains to be seen the extent to which the dramatic enhancement provided by the coupling modulation is robust with respect to mechanical vibrations and thermal fluctuations of the device, which would effect the resonance frequencies of the individual resonators [9]. However, it should be pointed out that these effects are the dominant source of error in conventional FOGs [22]. The small size of the CROWs compared to FOGs should in this case be an advantage since for a given temperature or stress gradient, the variations across the dimensions of the CROW will be much smaller than across a FOG.

An immediate extension of this work would be to see if the modulation of the evanescent couplings proposed here combined with the previously proposed technique of chirping of the resonator sizes for improved gyroscopic sensitivities [23] can be combined for further improvement of they gyroscopic sensitivities of CROWs.

References and links

1. C. Ciminelli, F. Dell’Olio, C. E. Campanella, and M. N. Armenise, “Photonic technologies for angular velocity sensing,” Adv. Opt. Photon. 2, 370–404 (2010). [CrossRef]  

2. B. Culshaw, “The optical fibre Sagnac interferometer: an overview of its principles and applications,” Meas. Sci. Technol. 17, R1–R16 (2006). [CrossRef]  

3. F. Xia, L. Sekaric, M. O’Boyle, and Y. Vlasov, “Coupled resonator optical waveguides based on silicon-on-insulator photonic wires,” App. Phys. Lett. 89, 041122 (2006). [CrossRef]  

4. F. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photon. 1, 65–71 (2007). [CrossRef]  

5. J. K. S. Poon, L. Zhu, G. A. DeRose, and A. Yariv, “Polymer Microring Coupled-Resonator Optical Waveguides,” J. Lightwave Technol. 24, 1843–1849 (2006). [CrossRef]  

6. J. K. S. Poon, L. Zhu, G. A. DeRose, and A. Yariv, “Transmission and group delay of microring coupled-resonator optical waveguides,” Opt. Lett. 31, 456–458 (2006). [CrossRef]   [PubMed]  

7. J. Scheuer and A. Yariv, “Sagnac effect in coupled-resonator slow-light waveguide structures,” Phys. Rev. Lett. 90, 053901 (2006). [CrossRef]  

8. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24, 711–713 (1999). [CrossRef]  

9. M. A. Terrel, M. J. F. Digonnet, and S. Fan, “Performance limitation of a coupled resonant optical waveguide gyroscope,” J. Lightwave Technol. 27, 47–54 (2009). [CrossRef]  

10. J. Capmany, P. Munoz, J. D. Domenech, and M. A. Muriel, “Apodized coupled resonator waveguides,” Opt. Express 15, 10196–10206 (2007). [CrossRef]   [PubMed]  

11. J. K. S. Poon, J. Scheuer, Y. Xu, and A. Yariv, “Designing coupled-resonator optical waveguide delay lines,” J. Opt. Soc. Am. B 21, 1665–1673 (2004). [CrossRef]  

12. G. Gupta, Y.-H. Kuo, H. Tazawa, W. H. Steier, A. Stapleton, and J. D. O’Brien, “Analysis and demonstration of coupling control in polymer microring resonators using photbleaching,” App. Opt. 48, 5324–5335 (2009). [CrossRef]  

13. F. Xia, M. Rooks, L. Sekaric, and Y. Vlasov, “Ultra-compact high order ring resonator filters using submicron photonic wires for on chip optical interconnects,” Opt. Express 15, 11934–11941 (2007). [CrossRef]   [PubMed]  

14. H.-C. Liu and A. Yariv, “Synthesis of high-order bandpass filters based on coupled-resonator optical waveguides (CROWs),” Opt. Express 19, 17563–17668 (2011). [CrossRef]  

15. U. Levy, K. Campbell, A. Groisman, S. Mookherjea, and Y. Fainman, “On-chip microfluidic tuning of an optical microresonator,” App. Phys. Lett. 88, 111107 (2006). [CrossRef]  

16. A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. M. De La Rue, A. Samarelli, M. Sorel, and A. Melloni, “Silicon coupled-ring resonator structures for slow light applications: potential, impairments, and ultimate limits,” J. Opt. 12, 104008 (2010). [CrossRef]  

17. B. Z. Steinberg, J. Scheuer, and A. Boag, “Rotation-induced superstructure in slow-light waveguides with mode-degeneracy: optical gyroscopes with exponential sensitivity,” J. Opt. Soc. Am. B 24, 1216–1224 (2007). [CrossRef]  

18. M. L. Cooper, G. Gupta, M. A. Schneider, W. M. J. Green, S. Assefa, F. Xia, Y. A. Vlasov, and S. Mookherjea, “Statistics of light transport in 235-ring silicon coupled-resonator optical waveguides,” Opt. Exp. 18, 26505–26516 (2010). [CrossRef]  

19. L.-W. Luo, G. S. Wiederhecker, J. Cardenas, C. Poitras, and M. Lipson, “High quality factor etchless silicon photonic ring resonators,” Opt. Express 19, 6284–6288 (2011). [CrossRef]   [PubMed]  

20. J. K. S. Poon, J. Scheuer, S. Mookherjea, G. T. Paloczi, Y. Huang, and A. Yariv, “Matrix analysis of microring coupled-resonator optical waveguides,” Opt. Express 12, 90–103 (2004). [CrossRef]   [PubMed]  

21. S. Merlo, M. Norgia, and S. Donati, “Fiber gyroscope principles” in Handbook of optical fibre sensing technology ed. J. M. Lopez-Higuera (John Wiley and Sons, NY, 2002), 331–347.

22. P. B. Ruffin, “Fiber optics gyroscope sensors” in Fiber Optic Sensors, 2nd ed. edited by F. T. S. Yu, S. Yin, and P. B. Ruffin (CRC Press, 2008).

23. J. R. E. Toland, Z. A. Kaston, C. Sorrentino, and C. P. Search, “Chirped area coupled resonator optical waveguide gyroscope,” Opt. Lett. 36, 1221–1223 (2011). [CrossRef]   [PubMed]  

References

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  1. C. Ciminelli, F. Dell’Olio, C. E. Campanella, and M. N. Armenise, “Photonic technologies for angular velocity sensing,” Adv. Opt. Photon. 2, 370–404 (2010).
    [Crossref]
  2. B. Culshaw, “The optical fibre Sagnac interferometer: an overview of its principles and applications,” Meas. Sci. Technol. 17, R1–R16 (2006).
    [Crossref]
  3. F. Xia, L. Sekaric, M. O’Boyle, and Y. Vlasov, “Coupled resonator optical waveguides based on silicon-on-insulator photonic wires,” App. Phys. Lett. 89, 041122 (2006).
    [Crossref]
  4. F. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photon. 1, 65–71 (2007).
    [Crossref]
  5. J. K. S. Poon, L. Zhu, G. A. DeRose, and A. Yariv, “Polymer Microring Coupled-Resonator Optical Waveguides,” J. Lightwave Technol. 24, 1843–1849 (2006).
    [Crossref]
  6. J. K. S. Poon, L. Zhu, G. A. DeRose, and A. Yariv, “Transmission and group delay of microring coupled-resonator optical waveguides,” Opt. Lett. 31, 456–458 (2006).
    [Crossref] [PubMed]
  7. J. Scheuer and A. Yariv, “Sagnac effect in coupled-resonator slow-light waveguide structures,” Phys. Rev. Lett. 90, 053901 (2006).
    [Crossref]
  8. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24, 711–713 (1999).
    [Crossref]
  9. M. A. Terrel, M. J. F. Digonnet, and S. Fan, “Performance limitation of a coupled resonant optical waveguide gyroscope,” J. Lightwave Technol. 27, 47–54 (2009).
    [Crossref]
  10. J. Capmany, P. Munoz, J. D. Domenech, and M. A. Muriel, “Apodized coupled resonator waveguides,” Opt. Express 15, 10196–10206 (2007).
    [Crossref] [PubMed]
  11. J. K. S. Poon, J. Scheuer, Y. Xu, and A. Yariv, “Designing coupled-resonator optical waveguide delay lines,” J. Opt. Soc. Am. B 21, 1665–1673 (2004).
    [Crossref]
  12. G. Gupta, Y.-H. Kuo, H. Tazawa, W. H. Steier, A. Stapleton, and J. D. O’Brien, “Analysis and demonstration of coupling control in polymer microring resonators using photbleaching,” App. Opt. 48, 5324–5335 (2009).
    [Crossref]
  13. F. Xia, M. Rooks, L. Sekaric, and Y. Vlasov, “Ultra-compact high order ring resonator filters using submicron photonic wires for on chip optical interconnects,” Opt. Express 15, 11934–11941 (2007).
    [Crossref] [PubMed]
  14. H.-C. Liu and A. Yariv, “Synthesis of high-order bandpass filters based on coupled-resonator optical waveguides (CROWs),” Opt. Express 19, 17563–17668 (2011).
    [Crossref]
  15. U. Levy, K. Campbell, A. Groisman, S. Mookherjea, and Y. Fainman, “On-chip microfluidic tuning of an optical microresonator,” App. Phys. Lett. 88, 111107 (2006).
    [Crossref]
  16. A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. M. De La Rue, A. Samarelli, M. Sorel, and A. Melloni, “Silicon coupled-ring resonator structures for slow light applications: potential, impairments, and ultimate limits,” J. Opt. 12, 104008 (2010).
    [Crossref]
  17. B. Z. Steinberg, J. Scheuer, and A. Boag, “Rotation-induced superstructure in slow-light waveguides with mode-degeneracy: optical gyroscopes with exponential sensitivity,” J. Opt. Soc. Am. B 24, 1216–1224 (2007).
    [Crossref]
  18. M. L. Cooper, G. Gupta, M. A. Schneider, W. M. J. Green, S. Assefa, F. Xia, Y. A. Vlasov, and S. Mookherjea, “Statistics of light transport in 235-ring silicon coupled-resonator optical waveguides,” Opt. Exp. 18, 26505–26516 (2010).
    [Crossref]
  19. L.-W. Luo, G. S. Wiederhecker, J. Cardenas, C. Poitras, and M. Lipson, “High quality factor etchless silicon photonic ring resonators,” Opt. Express 19, 6284–6288 (2011).
    [Crossref] [PubMed]
  20. J. K. S. Poon, J. Scheuer, S. Mookherjea, G. T. Paloczi, Y. Huang, and A. Yariv, “Matrix analysis of microring coupled-resonator optical waveguides,” Opt. Express 12, 90–103 (2004).
    [Crossref] [PubMed]
  21. S. Merlo, M. Norgia, and S. Donati, “Fiber gyroscope principles” in Handbook of optical fibre sensing technology ed. J. M. Lopez-Higuera (John Wiley and Sons, NY, 2002), 331–347.
  22. P. B. Ruffin, “Fiber optics gyroscope sensors” in Fiber Optic Sensors, 2nd ed. edited by F. T. S. Yu, S. Yin, and P. B. Ruffin (CRC Press, 2008).
  23. J. R. E. Toland, Z. A. Kaston, C. Sorrentino, and C. P. Search, “Chirped area coupled resonator optical waveguide gyroscope,” Opt. Lett. 36, 1221–1223 (2011).
    [Crossref] [PubMed]

2011 (3)

2010 (3)

M. L. Cooper, G. Gupta, M. A. Schneider, W. M. J. Green, S. Assefa, F. Xia, Y. A. Vlasov, and S. Mookherjea, “Statistics of light transport in 235-ring silicon coupled-resonator optical waveguides,” Opt. Exp. 18, 26505–26516 (2010).
[Crossref]

A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. M. De La Rue, A. Samarelli, M. Sorel, and A. Melloni, “Silicon coupled-ring resonator structures for slow light applications: potential, impairments, and ultimate limits,” J. Opt. 12, 104008 (2010).
[Crossref]

C. Ciminelli, F. Dell’Olio, C. E. Campanella, and M. N. Armenise, “Photonic technologies for angular velocity sensing,” Adv. Opt. Photon. 2, 370–404 (2010).
[Crossref]

2009 (2)

M. A. Terrel, M. J. F. Digonnet, and S. Fan, “Performance limitation of a coupled resonant optical waveguide gyroscope,” J. Lightwave Technol. 27, 47–54 (2009).
[Crossref]

G. Gupta, Y.-H. Kuo, H. Tazawa, W. H. Steier, A. Stapleton, and J. D. O’Brien, “Analysis and demonstration of coupling control in polymer microring resonators using photbleaching,” App. Opt. 48, 5324–5335 (2009).
[Crossref]

2007 (4)

2006 (6)

J. K. S. Poon, L. Zhu, G. A. DeRose, and A. Yariv, “Polymer Microring Coupled-Resonator Optical Waveguides,” J. Lightwave Technol. 24, 1843–1849 (2006).
[Crossref]

J. K. S. Poon, L. Zhu, G. A. DeRose, and A. Yariv, “Transmission and group delay of microring coupled-resonator optical waveguides,” Opt. Lett. 31, 456–458 (2006).
[Crossref] [PubMed]

J. Scheuer and A. Yariv, “Sagnac effect in coupled-resonator slow-light waveguide structures,” Phys. Rev. Lett. 90, 053901 (2006).
[Crossref]

B. Culshaw, “The optical fibre Sagnac interferometer: an overview of its principles and applications,” Meas. Sci. Technol. 17, R1–R16 (2006).
[Crossref]

F. Xia, L. Sekaric, M. O’Boyle, and Y. Vlasov, “Coupled resonator optical waveguides based on silicon-on-insulator photonic wires,” App. Phys. Lett. 89, 041122 (2006).
[Crossref]

U. Levy, K. Campbell, A. Groisman, S. Mookherjea, and Y. Fainman, “On-chip microfluidic tuning of an optical microresonator,” App. Phys. Lett. 88, 111107 (2006).
[Crossref]

2004 (2)

1999 (1)

Armenise, M. N.

Assefa, S.

M. L. Cooper, G. Gupta, M. A. Schneider, W. M. J. Green, S. Assefa, F. Xia, Y. A. Vlasov, and S. Mookherjea, “Statistics of light transport in 235-ring silicon coupled-resonator optical waveguides,” Opt. Exp. 18, 26505–26516 (2010).
[Crossref]

Boag, A.

Campanella, C. E.

Campbell, K.

U. Levy, K. Campbell, A. Groisman, S. Mookherjea, and Y. Fainman, “On-chip microfluidic tuning of an optical microresonator,” App. Phys. Lett. 88, 111107 (2006).
[Crossref]

Canciamilla, A.

A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. M. De La Rue, A. Samarelli, M. Sorel, and A. Melloni, “Silicon coupled-ring resonator structures for slow light applications: potential, impairments, and ultimate limits,” J. Opt. 12, 104008 (2010).
[Crossref]

Capmany, J.

Cardenas, J.

Ciminelli, C.

Cooper, M. L.

M. L. Cooper, G. Gupta, M. A. Schneider, W. M. J. Green, S. Assefa, F. Xia, Y. A. Vlasov, and S. Mookherjea, “Statistics of light transport in 235-ring silicon coupled-resonator optical waveguides,” Opt. Exp. 18, 26505–26516 (2010).
[Crossref]

Culshaw, B.

B. Culshaw, “The optical fibre Sagnac interferometer: an overview of its principles and applications,” Meas. Sci. Technol. 17, R1–R16 (2006).
[Crossref]

De La Rue, R. M.

A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. M. De La Rue, A. Samarelli, M. Sorel, and A. Melloni, “Silicon coupled-ring resonator structures for slow light applications: potential, impairments, and ultimate limits,” J. Opt. 12, 104008 (2010).
[Crossref]

Dell’Olio, F.

DeRose, G. A.

Digonnet, M. J. F.

Domenech, J. D.

Donati, S.

S. Merlo, M. Norgia, and S. Donati, “Fiber gyroscope principles” in Handbook of optical fibre sensing technology ed. J. M. Lopez-Higuera (John Wiley and Sons, NY, 2002), 331–347.

Fainman, Y.

U. Levy, K. Campbell, A. Groisman, S. Mookherjea, and Y. Fainman, “On-chip microfluidic tuning of an optical microresonator,” App. Phys. Lett. 88, 111107 (2006).
[Crossref]

Fan, S.

Ferrari, C.

A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. M. De La Rue, A. Samarelli, M. Sorel, and A. Melloni, “Silicon coupled-ring resonator structures for slow light applications: potential, impairments, and ultimate limits,” J. Opt. 12, 104008 (2010).
[Crossref]

Green, W. M. J.

M. L. Cooper, G. Gupta, M. A. Schneider, W. M. J. Green, S. Assefa, F. Xia, Y. A. Vlasov, and S. Mookherjea, “Statistics of light transport in 235-ring silicon coupled-resonator optical waveguides,” Opt. Exp. 18, 26505–26516 (2010).
[Crossref]

Groisman, A.

U. Levy, K. Campbell, A. Groisman, S. Mookherjea, and Y. Fainman, “On-chip microfluidic tuning of an optical microresonator,” App. Phys. Lett. 88, 111107 (2006).
[Crossref]

Gupta, G.

M. L. Cooper, G. Gupta, M. A. Schneider, W. M. J. Green, S. Assefa, F. Xia, Y. A. Vlasov, and S. Mookherjea, “Statistics of light transport in 235-ring silicon coupled-resonator optical waveguides,” Opt. Exp. 18, 26505–26516 (2010).
[Crossref]

G. Gupta, Y.-H. Kuo, H. Tazawa, W. H. Steier, A. Stapleton, and J. D. O’Brien, “Analysis and demonstration of coupling control in polymer microring resonators using photbleaching,” App. Opt. 48, 5324–5335 (2009).
[Crossref]

Huang, Y.

Kaston, Z. A.

Kuo, Y.-H.

G. Gupta, Y.-H. Kuo, H. Tazawa, W. H. Steier, A. Stapleton, and J. D. O’Brien, “Analysis and demonstration of coupling control in polymer microring resonators using photbleaching,” App. Opt. 48, 5324–5335 (2009).
[Crossref]

Lee, R. K.

Levy, U.

U. Levy, K. Campbell, A. Groisman, S. Mookherjea, and Y. Fainman, “On-chip microfluidic tuning of an optical microresonator,” App. Phys. Lett. 88, 111107 (2006).
[Crossref]

Lipson, M.

Liu, H.-C.

Luo, L.-W.

Melloni, A.

A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. M. De La Rue, A. Samarelli, M. Sorel, and A. Melloni, “Silicon coupled-ring resonator structures for slow light applications: potential, impairments, and ultimate limits,” J. Opt. 12, 104008 (2010).
[Crossref]

Merlo, S.

S. Merlo, M. Norgia, and S. Donati, “Fiber gyroscope principles” in Handbook of optical fibre sensing technology ed. J. M. Lopez-Higuera (John Wiley and Sons, NY, 2002), 331–347.

Mookherjea, S.

M. L. Cooper, G. Gupta, M. A. Schneider, W. M. J. Green, S. Assefa, F. Xia, Y. A. Vlasov, and S. Mookherjea, “Statistics of light transport in 235-ring silicon coupled-resonator optical waveguides,” Opt. Exp. 18, 26505–26516 (2010).
[Crossref]

U. Levy, K. Campbell, A. Groisman, S. Mookherjea, and Y. Fainman, “On-chip microfluidic tuning of an optical microresonator,” App. Phys. Lett. 88, 111107 (2006).
[Crossref]

J. K. S. Poon, J. Scheuer, S. Mookherjea, G. T. Paloczi, Y. Huang, and A. Yariv, “Matrix analysis of microring coupled-resonator optical waveguides,” Opt. Express 12, 90–103 (2004).
[Crossref] [PubMed]

Morichetti, F.

A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. M. De La Rue, A. Samarelli, M. Sorel, and A. Melloni, “Silicon coupled-ring resonator structures for slow light applications: potential, impairments, and ultimate limits,” J. Opt. 12, 104008 (2010).
[Crossref]

Munoz, P.

Muriel, M. A.

Norgia, M.

S. Merlo, M. Norgia, and S. Donati, “Fiber gyroscope principles” in Handbook of optical fibre sensing technology ed. J. M. Lopez-Higuera (John Wiley and Sons, NY, 2002), 331–347.

O’Boyle, M.

F. Xia, L. Sekaric, M. O’Boyle, and Y. Vlasov, “Coupled resonator optical waveguides based on silicon-on-insulator photonic wires,” App. Phys. Lett. 89, 041122 (2006).
[Crossref]

O’Brien, J. D.

G. Gupta, Y.-H. Kuo, H. Tazawa, W. H. Steier, A. Stapleton, and J. D. O’Brien, “Analysis and demonstration of coupling control in polymer microring resonators using photbleaching,” App. Opt. 48, 5324–5335 (2009).
[Crossref]

Paloczi, G. T.

Poitras, C.

Poon, J. K. S.

Rooks, M.

Ruffin, P. B.

P. B. Ruffin, “Fiber optics gyroscope sensors” in Fiber Optic Sensors, 2nd ed. edited by F. T. S. Yu, S. Yin, and P. B. Ruffin (CRC Press, 2008).

Samarelli, A.

A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. M. De La Rue, A. Samarelli, M. Sorel, and A. Melloni, “Silicon coupled-ring resonator structures for slow light applications: potential, impairments, and ultimate limits,” J. Opt. 12, 104008 (2010).
[Crossref]

Scherer, A.

Scheuer, J.

Schneider, M. A.

M. L. Cooper, G. Gupta, M. A. Schneider, W. M. J. Green, S. Assefa, F. Xia, Y. A. Vlasov, and S. Mookherjea, “Statistics of light transport in 235-ring silicon coupled-resonator optical waveguides,” Opt. Exp. 18, 26505–26516 (2010).
[Crossref]

Search, C. P.

Sekaric, L.

F. Xia, M. Rooks, L. Sekaric, and Y. Vlasov, “Ultra-compact high order ring resonator filters using submicron photonic wires for on chip optical interconnects,” Opt. Express 15, 11934–11941 (2007).
[Crossref] [PubMed]

F. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photon. 1, 65–71 (2007).
[Crossref]

F. Xia, L. Sekaric, M. O’Boyle, and Y. Vlasov, “Coupled resonator optical waveguides based on silicon-on-insulator photonic wires,” App. Phys. Lett. 89, 041122 (2006).
[Crossref]

Sorel, M.

A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. M. De La Rue, A. Samarelli, M. Sorel, and A. Melloni, “Silicon coupled-ring resonator structures for slow light applications: potential, impairments, and ultimate limits,” J. Opt. 12, 104008 (2010).
[Crossref]

Sorrentino, C.

Stapleton, A.

G. Gupta, Y.-H. Kuo, H. Tazawa, W. H. Steier, A. Stapleton, and J. D. O’Brien, “Analysis and demonstration of coupling control in polymer microring resonators using photbleaching,” App. Opt. 48, 5324–5335 (2009).
[Crossref]

Steier, W. H.

G. Gupta, Y.-H. Kuo, H. Tazawa, W. H. Steier, A. Stapleton, and J. D. O’Brien, “Analysis and demonstration of coupling control in polymer microring resonators using photbleaching,” App. Opt. 48, 5324–5335 (2009).
[Crossref]

Steinberg, B. Z.

Tazawa, H.

G. Gupta, Y.-H. Kuo, H. Tazawa, W. H. Steier, A. Stapleton, and J. D. O’Brien, “Analysis and demonstration of coupling control in polymer microring resonators using photbleaching,” App. Opt. 48, 5324–5335 (2009).
[Crossref]

Terrel, M. A.

Toland, J. R. E.

Torregiani, M.

A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. M. De La Rue, A. Samarelli, M. Sorel, and A. Melloni, “Silicon coupled-ring resonator structures for slow light applications: potential, impairments, and ultimate limits,” J. Opt. 12, 104008 (2010).
[Crossref]

Vlasov, Y.

F. Xia, M. Rooks, L. Sekaric, and Y. Vlasov, “Ultra-compact high order ring resonator filters using submicron photonic wires for on chip optical interconnects,” Opt. Express 15, 11934–11941 (2007).
[Crossref] [PubMed]

F. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photon. 1, 65–71 (2007).
[Crossref]

F. Xia, L. Sekaric, M. O’Boyle, and Y. Vlasov, “Coupled resonator optical waveguides based on silicon-on-insulator photonic wires,” App. Phys. Lett. 89, 041122 (2006).
[Crossref]

Vlasov, Y. A.

M. L. Cooper, G. Gupta, M. A. Schneider, W. M. J. Green, S. Assefa, F. Xia, Y. A. Vlasov, and S. Mookherjea, “Statistics of light transport in 235-ring silicon coupled-resonator optical waveguides,” Opt. Exp. 18, 26505–26516 (2010).
[Crossref]

Wiederhecker, G. S.

Xia, F.

M. L. Cooper, G. Gupta, M. A. Schneider, W. M. J. Green, S. Assefa, F. Xia, Y. A. Vlasov, and S. Mookherjea, “Statistics of light transport in 235-ring silicon coupled-resonator optical waveguides,” Opt. Exp. 18, 26505–26516 (2010).
[Crossref]

F. Xia, M. Rooks, L. Sekaric, and Y. Vlasov, “Ultra-compact high order ring resonator filters using submicron photonic wires for on chip optical interconnects,” Opt. Express 15, 11934–11941 (2007).
[Crossref] [PubMed]

F. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photon. 1, 65–71 (2007).
[Crossref]

F. Xia, L. Sekaric, M. O’Boyle, and Y. Vlasov, “Coupled resonator optical waveguides based on silicon-on-insulator photonic wires,” App. Phys. Lett. 89, 041122 (2006).
[Crossref]

Xu, Y.

Yariv, A.

Zhu, L.

Adv. Opt. Photon. (1)

App. Opt. (1)

G. Gupta, Y.-H. Kuo, H. Tazawa, W. H. Steier, A. Stapleton, and J. D. O’Brien, “Analysis and demonstration of coupling control in polymer microring resonators using photbleaching,” App. Opt. 48, 5324–5335 (2009).
[Crossref]

App. Phys. Lett. (2)

F. Xia, L. Sekaric, M. O’Boyle, and Y. Vlasov, “Coupled resonator optical waveguides based on silicon-on-insulator photonic wires,” App. Phys. Lett. 89, 041122 (2006).
[Crossref]

U. Levy, K. Campbell, A. Groisman, S. Mookherjea, and Y. Fainman, “On-chip microfluidic tuning of an optical microresonator,” App. Phys. Lett. 88, 111107 (2006).
[Crossref]

J. Lightwave Technol. (2)

J. Opt. (1)

A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. M. De La Rue, A. Samarelli, M. Sorel, and A. Melloni, “Silicon coupled-ring resonator structures for slow light applications: potential, impairments, and ultimate limits,” J. Opt. 12, 104008 (2010).
[Crossref]

J. Opt. Soc. Am. B (2)

Meas. Sci. Technol. (1)

B. Culshaw, “The optical fibre Sagnac interferometer: an overview of its principles and applications,” Meas. Sci. Technol. 17, R1–R16 (2006).
[Crossref]

Nat. Photon. (1)

F. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photon. 1, 65–71 (2007).
[Crossref]

Opt. Exp. (1)

M. L. Cooper, G. Gupta, M. A. Schneider, W. M. J. Green, S. Assefa, F. Xia, Y. A. Vlasov, and S. Mookherjea, “Statistics of light transport in 235-ring silicon coupled-resonator optical waveguides,” Opt. Exp. 18, 26505–26516 (2010).
[Crossref]

Opt. Express (5)

Opt. Lett. (3)

Phys. Rev. Lett. (1)

J. Scheuer and A. Yariv, “Sagnac effect in coupled-resonator slow-light waveguide structures,” Phys. Rev. Lett. 90, 053901 (2006).
[Crossref]

Other (2)

S. Merlo, M. Norgia, and S. Donati, “Fiber gyroscope principles” in Handbook of optical fibre sensing technology ed. J. M. Lopez-Higuera (John Wiley and Sons, NY, 2002), 331–347.

P. B. Ruffin, “Fiber optics gyroscope sensors” in Fiber Optic Sensors, 2nd ed. edited by F. T. S. Yu, S. Yin, and P. B. Ruffin (CRC Press, 2008).

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Figures (4)

Fig. 1
Fig. 1 (a) Illustration of micro-resonator CROW gyroscope on planar substrate connected to a 3dB coupler with periodic modulation of evanescent coupling between resonators. (b) Schematic diagram of proposed modulation scheme for evanescent couplings κj. Note that the green and red discs represent the strength of the evanescent coupling between resonators, which alternates between weak (green) and strong (red) coupling or vice versa symmetrically about the center resonator.
Fig. 2
Fig. 2 (a) Transmission spectrum as a function of Sagnac phase shift per resonator ϕS for the alternating symmetric coupling modulation with κα = 0.1 and κβ = 0.01 (red line). Notice the single transmission resonance at 0 phase. For comparison the transmission through CROWs with uniform coupling constants κ = 0.01 (blue line) and κ = 0.1 (green line) are also shown. In all plots N = 13. (b) Closeup of central transmission resonance for κα = 0.01 as a function of ϕS and κβ for N = 9 rings. One can see clearly the narrowing of the resonance for increasing |κακβ|.
Fig. 3
Fig. 3 (a) Maximum value of dT/dϕS in the vicinity ϕS = 0 as a function of N for various coupling modulation strengths with weak coupling at the edges, κακβ. Also shown is the maximum dT/dϕS for a uniform CROW with κ = 0.01 at the edge of the transmission band. In this figure Q int 1 = 0. (b) The same as part (a) except that coupling is strong at the edges, κακβ.
Fig. 4
Fig. 4 (a) Effect of finite Qint on the transmission resonance of the alternating symmetric coupling modulated CROW for κα = 0.01 and κβ = 0.7 for N = 11 resonators. (b) Maximum value of dT/dϕS in the vicinity ϕS = 0 as a function of N for finite Qint and coupling constants modulations κα = 0.01 and κβ = 0.3 (corresponding to Fig. 3(a)) and κα = 0.3 and κβ = 0.01 (corresponding to Fig. 3(b)). Also shown is the maximum dT/dϕS for a uniform CROW with κ = 0.01 at the edge of the transmission band. In both figures the resonators have radii R = 25μm and index of refraction n = 1.6

Equations (9)

Equations on this page are rendered with MathJax. Learn more.

ϕ S = 4 π ω Ω R 2 c 2
U in = i κ 1 ( exp [ i ϕ 1 ] 1 κ 1 exp [ i ϕ 1 ] 1 κ 1 exp [ i ϕ 1 ] exp [ i ϕ 1 ] )
U out = i κ N + 1 ( exp [ i ϕ N ] 1 κ N + 1 exp [ i ϕ N ] 1 κ N + 1 exp [ i ϕ N ] exp [ i ϕ N ] ) ( i 1 κ N κ N i κ N i κ N i 1 κ N κ N )
U R H M ( j ) = ( i 1 κ j κ j exp [ i ϕ j ] i κ j exp [ i ϕ j ] i κ j exp [ i ϕ j ] i 1 κ j κ j exp [ i ϕ j ] ) ,
U L H M ( j ) = ( i 1 κ j κ j exp [ i ϕ j ] i κ j exp [ i ϕ j ] i κ j exp [ i ϕ j ] i 1 κ j κ j exp [ i ϕ j ] )
T C C W = U out l = 1 ( N 3 ) / 2 U L H M ( 2 l + 2 ) U R H M ( 2 l + 1 ) U L H M ( 2 ) U i n
= ( T 11 T 12 T 21 T 22 )
κ j = { κ α j = 1 , 3 , 5 , ( N ± 1 ) / 2 κ β j = 2 , 4 , 6 , ( N 1 ) / 2 κ α j = 2 + ( N 1 ) / 2 , 4 + ( N 1 ) / 2 , , N + 1 κ β j = 2 + ( N ± ) 1 / 2 , 4 + ( N ± 1 ) / 2 , N
Ω min = λ c 8 π A eff 2 h c Δ f λ η P opt

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