1104:
4318:
1878:
4330:
4342:
2573:
63:) of the unperturbed cavity slightly change. Analytically predicting how the perturbation changes the optical response is a classical problem in electromagnetics, with important implications spanning from the radio-frequency domain to present-day nano-optics. The underlying assumption of cavity perturbation theory is that electromagnetic fields inside the cavity after the change differ by a very small amount from the fields before the change. Then
611:) relies on stored energy considerations. The latter are intuitive since common sense dictates that the maximum change in resonant frequency occurs when the perturbation is placed at the intensity maximum of the cavity mode. However energy consideration in electromagnetism is only valid for Hermitian systems for which energy is conserved. For cavities, energy is conserved only in the limit of very small leakage (infinite Q's), so that Expression (
4354:
4306:
468:
1771:
978:
1362:
2122:
2615:
mode is well known. Ideally, the material to be measured is introduced into the cavity at the position of maximum electric or magnetic field. When the material is introduced at the position of maximum electric field, then the contribution of magnetic field to perturbed frequency shift is very small
42:
These performance changes are assumed to be caused by either introduction of a small foreign object into the cavity, or a small deformation of its boundary. Various mathematical methods can be used to study the characteristics of cavities, which are important in the field of microwave systems, and
4287:
Ogunlade, O.; Yifan Chen; Kosmas, P.; "Measurement of the complex permittivity of microbubbles using a cavity perturbation technique for contrast enhanced ultra-wideband breast cancer detection," Engineering in
Medicine and Biology Society (EMBC), 2010 Annual International Conference of the IEEE,
46:
There are many industrial applications for cavity resonators, including microwave ovens, microwave communication systems, and remote imaging systems using electro magnetic waves. How a resonant cavity performs can affect the amount of energy that is required to make it resonate, or the relative
3379:
Similarly, if the material is introduced into the cavity at the position of maximum magnetic field, then the contribution of electric field to perturbed frequency shift is very small and can be ignored. In this case, we can use perturbation theory to derive expressions for complex material
3815:
241:
2539:
is necessary in order to extrapolate material properties, these measurement techniques generally make use of standard resonant cavities where resonant frequencies and electromagnetic fields are well known. Two examples of such standard resonant cavities are rectangular and circular
4267:
A. Sklyuyev; M. Ciureanu; C. Akyel; P. Ciureanu; D. Menard; A. Yelon; "Measurement of
Complex Permeability of Ferromagnetic Nanowires using Cavity Perturbation Techniques," Electrical and Computer Engineering, 2006. CCECE '06. Canadian Conference on, vol., no., pp.1486–1489, May
1611:
787:
3618:
2495:
1127:
4257:
Qing Wang; Xiaoguang Deng; Min Yang; Yun Fan; Weilian Wang; "Measuring glucose concentration by microwave cavity perturbation and DSP technology," Biomedical
Engineering and Informatics (BMEI), 2010 3rd International Conference on, vol.3, no., pp.943–946, 16–18 Oct.
2262:
1893:
4277:
Wang, Z.H.; Javadi, H.H.S.; Epstein, A.J.; "Application of microwave cavity perturbation techniques in conducting polymers," Instrumentation and
Measurement Technology Conference, 1991. IMTC-91. Conference Record., 8th IEEE, vol., no., pp.79–82, 14–16 May
774:
products that are complex quantities, the imaginary part being related to the leakage. To emphasize the difference between the normal modes of
Hermitian systems and the resonance modes of leaky systems, the resonance modes are often referred to as
4247:
Wenquan Che; Zhanxian Wang; Yumei Chang; Russer, P.; "Permittivity
Measurement of Biological Materials with Improved Microwave Cavity Perturbation Technique," Microwave Conference, 2008. EuMC 2008. 38th European, vol., no., pp.905–908, 27–31 Oct.
3250:
2921:
232:
is the inverse of the mode lifetime. Cavity perturbation theory has been initially proposed by Bethe-Schwinger in optics , and
Waldron in the radio frequency domain. These initial approaches rely on formulae that consider stored energy
2789:
3331:
2676:
463:{\displaystyle {\frac {{\tilde {\omega }}-{\tilde {\omega }}_{0}}{{\tilde {\omega }}_{0}}}\thickapprox -{\frac {\iiint _{V}\Delta \epsilon |E_{0}|^{2}dv}{\iiint _{V}(\mu _{0}|H_{0}|^{2}+\epsilon |E_{0}|^{2})dv}}\,}
1766:{\displaystyle {\frac {\omega -\omega _{0}}{\omega _{0}}}\thickapprox -{\frac {1}{W}}\iiint _{V}({\frac {\Delta \epsilon }{\epsilon }}\cdot {\bar {w_{e}}}+{\frac {\Delta \mu }{\mu }}\cdot {\bar {w_{m}}})dv\,}
973:{\displaystyle {\frac {{\tilde {\omega }}-{\tilde {\omega }}_{0}}{{\tilde {\omega }}_{0}}}\thickapprox -{\frac {\iiint _{V}\Delta \epsilon E_{0}^{2}dv}{\iiint _{V}(\mu _{0}H_{0}^{2}+\epsilon E_{0}^{2})dv}}\,}
3440:
684:
3641:
1357:{\displaystyle {\frac {\omega -\omega _{0}}{\omega _{0}}}\thickapprox -{\frac {\iiint _{V}(\Delta \mu |H_{0}|^{2}+\Delta \epsilon |E_{0}|^{2})dv}{\iiint _{V}(\mu |H_{0}|^{2}+\epsilon |E_{0}|^{2})dv}}\,}
2382:
3450:
129:
2117:{\displaystyle {\frac {\omega -\omega _{0}}{\omega _{0}}}\thickapprox {\frac {\iiint _{\Delta V}(\mu |H_{0}|^{2}-\epsilon |E_{0}|^{2})dv}{\iiint _{V}(\mu |H_{0}|^{2}+\epsilon |E_{0}|^{2})dv}}\,}
2531:
Microwave measurement techniques based on cavity perturbation theory are generally used to determine the dielectric and magnetic parameters of materials and various circuit components such as
2154:
230:
176:
3903:
712:
The problems stems from the fact that a cavity is an open non-Hermitian system with leakage and absorption. The theory of non-Hermitian electromagnetic systems abandons energy, i.e.
549:
709:
is complex, i.e. only if the perturber is absorbent. Clearly this is not the case and it is well known that a dielectric perturbation may either increase or decrease the Q factor.
2548:
resonators . Cavity perturbation measurement techniques for material characterization are used in many fields ranging from physics and material science to medicine and biology.
513:
55:
When a resonant cavity is perturbed, e.g. by introducing a foreign object with distinct material properties into the cavity or when the shape of the cavity is changed slightly,
1588:
707:
1859:
1823:
3374:
3858:
3168:
2338:
2308:
179:
3136:
1565:
1432:
1534:
2612:
2369:
1398:
3099:
3072:
3045:
3018:
2991:
2964:
1486:
1459:
1067:
746:
603:
576:
1514:
1093:
772:
2812:
2686:
2616:
and can be ignored. In this case, we can use perturbation theory to derive expressions for real and imaginary components of complex material permittivity
3178:
67:
for original and perturbed cavities can be used to derive analytical expressions for the resulting resonant frequency shift and linewidth change (or
4066:
Yang, J.; Giessen, H.; Lalanne, P. (2015-04-06). "Simple
Analytical Expression for the Peak-Frequency Shifts of Plasmonic Resonances for Sensing".
1002:
has been verified in a variety of complicated geometries. For low-Q cavities, such as plasmonic nanoresonators that are used for sensing, equation
2520:
3273:
4010:
Lalanne, P.; Yan, W.; Vynck, K.; Sauvan, C.; Hugonin, J.-P. (2018-04-17). "Light interaction with photonic and plasmonic resonances".
3810:{\displaystyle \mu _{r}''=({\frac {\lambda _{g}^{2}+4a^{2}}{16a^{2}}})({\frac {V_{c}}{V_{s}}})({\frac {Q_{c}-Q_{s}}{Q_{c}Q_{s}}})\,}
2619:
2490:{\displaystyle {\frac {\omega -\omega _{0}}{\omega _{0}}}\thickapprox {\frac {({\bar {w_{m}}}-{\bar {w_{e}}})\cdot \Delta V}{W}}\,}
626:
3109:
3613:{\displaystyle \mu _{r}'-1=({\frac {\lambda _{g}^{2}+4a^{2}}{8a^{2}}})({\frac {f_{c}-f_{s}}{f_{s}}})({\frac {V_{c}}{V_{s}}})\,}
1866:
4232:
1885:
When a general shape of a resonant cavity is changed, a corresponding change in resonant frequency can be approximated as:
605:
are the electromagnetic fields of the unperturbed mode (permeability change is not considered for simplicity). Expression (
82:
3976:
2257:{\displaystyle {\frac {\omega -\omega _{0}}{\omega _{0}}}\thickapprox {\frac {\Delta W_{m}-\Delta W_{e}}{W_{m}+W_{e}}}\,}
4217:
Cavity perturbation technique for complex permittivity measurement of dielectric materials at X-band microwave frequency
185:
3383:
2581:
1008:
has been shown to predict both the shift and the broadening of the resonance with a high accuracy, whereas equation
134:
3863:
4379:
1537:
1116:
4384:
4219:. International Conference on Recent Advances in Microwave Theory and Applications. IEEE. pp. 836–838.
2146:) for change in resonant frequency can additionally be written in terms of time-average stored energies as:
518:
489:
4389:
4296:
1570:
689:
4374:
1862:
1828:
1792:
3352:
3836:
3147:
2523:
that are different for the displacement-field and electric-field vectors at the shape boundaries.
2313:
2283:
3114:
1547:
1410:
1519:
64:
56:
3108:
Once the complex permittivity of the material is known, we can easily calculate its effective
2584:
2351:
2536:
1383:
4146:
4127:
Cognée, K.C. (2019-03-20). "Mapping complex mode volumes with cavity perturbation theory".
4085:
4029:
3941:
3077:
3050:
3023:
2996:
2969:
2942:
2532:
1464:
1437:
1036:
715:
581:
554:
2916:{\displaystyle \epsilon _{r}''={\frac {V_{c}}{4V_{s}}}{\frac {Q_{c}-Q_{s}}{Q_{c}Q_{s}}}\,}
1499:
8:
1072:
751:
29:
4150:
4089:
4033:
3945:
2519:) can be gained by incorporating local field corrections, which simply results from the
4334:
4322:
4310:
4162:
4136:
4109:
4075:
4045:
4019:
3957:
3931:
1404:
59:
inside the cavity change accordingly. This means that all the resonant modes (i.e. the
2784:{\displaystyle \epsilon _{r}'-1={\frac {f_{c}-f_{s}}{2f_{s}}}{\frac {V_{c}}{V_{s}}}\,}
1103:
4358:
4317:
4228:
4205:
Mathew, K. T. 2005. Perturbation Theory. Encyclopedia of RF and
Microwave Engineering
4166:
4101:
2993:
represent resonant frequencies of original cavity and perturbed cavity respectively,
1877:
1590:
are changes in original permeability and permittivity introduced by material change.
1401:
4113:
4049:
3961:
71:
change) by referring only to the original unperturbed mode (not the perturbed one).
4220:
4154:
4093:
4037:
3988:
3949:
3245:{\displaystyle \sigma _{e}=\omega \epsilon ''=2\pi f\epsilon _{0}\epsilon _{r}''\,}
1015:
776:
60:
2535:. Since ex-ante knowledge of the resonant frequency, resonant frequency shift and
4097:
4346:
4184:
3996:
1600:
1493:
1489:
551:
are the complex frequencies of the perturbed and unperturbed cavity modes, and
4224:
4368:
2545:
617:) is only valid in this limit. For instance, it is apparent that Expression (
4158:
4105:
4041:
3992:
3953:
3142:
1541:
1112:
779:. In this framework, the frequency shift and the Q change are predicted by
21:
17:
4329:
3922:
Bethe, H. A.; Schwinger, J. (2018). "Perturbation theory for cavities".
2513:
Considerable accuracy improvements of the predictive force of
Equation (
1119:), a corresponding change in resonant frequency can be approximated as:
3139:
3047:
represent volumes of original cavity and material sample respectively,
1014:
is inaccurately predicting both. For high-Q photonic cavities, such as
4341:
3326:{\displaystyle tan\delta ={\frac {\epsilon _{r}''}{\epsilon _{r}'}}\,}
2374:
This expression can also be written in terms of energy densities as:
1024:
accurately predicts both the shift and the Q change, whereas equation
2541:
36:
79:
It is convenient to denote cavity frequencies with a complex number
4305:
4141:
4080:
4024:
3936:
3102:
2345:
2341:
68:
4353:
2572:
2580:
For rectangular waveguide cavity, field distribution of dominant
1018:
cavities or microrings, experiments have evidenced that equation
32:
4187:, Microwave Engineering, 2nd edition, Wiley, New York, NY, 1998.
4215:
Vyas, A.D.; Rana, V.A.; Gadani, D.H.; Prajapati, A.N. (2008).
1789:
where W is the total energy stored in the original cavity and
2671:{\displaystyle \epsilon _{r}=\epsilon _{r}'+j\epsilon _{r}''}
2576:
Material sample introduced into rectangular waveguide cavity.
679:{\displaystyle Im({\tilde {\omega }}-{\tilde {\omega }}_{0})}
4214:
1069:
products; however, they should be understood with the
4294:
4009:
3866:
3839:
3644:
3453:
3386:
3355:
3276:
3181:
3150:
3117:
3080:
3053:
3026:
2999:
2972:
2945:
2815:
2689:
2622:
2587:
2556:
2385:
2354:
2316:
2286:
2157:
1896:
1831:
1795:
1614:
1573:
1550:
1522:
1502:
1467:
1440:
1413:
1386:
1130:
1075:
1039:
790:
754:
718:
692:
629:
584:
557:
521:
492:
244:
188:
137:
124:{\displaystyle {\tilde {\omega }}=\omega -i\gamma /2}
85:
4288:vol., no., pp.6733–6736, Aug. 31 2010-Sept. 4 2010
4065:
3897:
3852:
3809:
3612:
3434:
3368:
3325:
3244:
3162:
3130:
3093:
3066:
3039:
3012:
2985:
2958:
2915:
2783:
2670:
2606:
2489:
2363:
2332:
2302:
2256:
2116:
1853:
1817:
1765:
1582:
1559:
1528:
1508:
1480:
1453:
1434:is the resonant frequency of the original cavity,
1426:
1392:
1356:
1087:
1061:
972:
766:
740:
701:
678:
597:
570:
543:
507:
462:
224:
170:
123:
43:more generally in the field of electro magnetism.
3105:of original and perturbed cavities respectively.
4366:
4061:
4059:
2521:interface conditions for electromagnetic fields
225:{\displaystyle \gamma =2Im({\tilde {\omega }})}
3921:
3435:{\displaystyle \mu _{r}=\mu _{r}'+j\mu _{r}''}
171:{\displaystyle \omega =Re({\tilde {\omega }})}
4056:
4201:
4199:
4197:
4195:
4193:
3898:{\displaystyle \lambda _{g}={\frac {2l}{n}}}
1111:When a material within a cavity is changed (
4180:
4178:
4176:
3977:"Perturbation theory of resonant cavities"
4190:
4140:
4079:
4023:
3935:
3806:
3609:
3349:where f is the frequency of interest and
3322:
3241:
2912:
2780:
2486:
2253:
2113:
1762:
1353:
969:
459:
4173:
2571:
1876:
1102:
1098:
1033:The following sections are written with
47:stability or instability of the system.
3974:
3924:N.D.R.C. RPT. D1-117 Cornell University
3860:is the guide wavelength (calculated as
4367:
4126:
544:{\displaystyle {\tilde {\omega }}_{0}}
1872:
1095:products of quasinormal mode theory.
996:The accuracy of the seminal equation
623:) predicts a change of the Q factor (
3635:
3444:
3267:
3172:
2806:
2680:
2376:
2148:
1887:
1605:
1121:
1030:accurately predicts the shift only.
781:
235:
28:describes methods for derivation of
13:
2474:
2355:
2317:
2287:
2212:
2196:
1940:
1719:
1679:
1574:
1551:
1221:
1185:
869:
693:
508:{\displaystyle {\tilde {\omega }}}
323:
14:
4401:
3975:Waldron, R. A. (September 1960).
2418:
2190:
1929:
1647:
1163:
850:
304:
74:
4352:
4340:
4328:
4316:
4304:
3376:is the free space permittivity.
1583:{\displaystyle \Delta \epsilon }
748:products, and rather focuses on
702:{\displaystyle \Delta \epsilon }
4281:
4271:
4261:
4251:
2526:
1599:) can be rewritten in terms of
50:
4241:
4208:
4120:
4003:
3968:
3915:
3803:
3749:
3746:
3719:
3716:
3661:
3606:
3579:
3576:
3534:
3531:
3476:
2468:
2462:
2440:
2424:
2101:
2091:
2075:
2058:
2042:
2035:
2014:
2004:
1988:
1971:
1955:
1948:
1854:{\displaystyle {\bar {w_{m}}}}
1845:
1818:{\displaystyle {\bar {w_{e}}}}
1809:
1753:
1747:
1707:
1673:
1341:
1331:
1315:
1298:
1282:
1275:
1254:
1244:
1228:
1208:
1192:
1182:
1055:
1041:
957:
908:
836:
816:
800:
734:
720:
673:
661:
645:
636:
529:
499:
447:
437:
421:
404:
388:
374:
346:
330:
290:
270:
254:
219:
213:
204:
165:
159:
150:
92:
1:
4012:Laser & Photonics Reviews
3908:
3369:{\displaystyle \epsilon _{0}}
35:for performance changes of a
4098:10.1021/acs.nanolett.5b00771
3853:{\displaystyle \lambda _{g}}
3823:
3626:
3339:
3258:
3163:{\displaystyle \tan \delta }
2568:rectangular waveguide cavity
2333:{\displaystyle \Delta W_{e}}
2303:{\displaystyle \Delta W_{m}}
1107:Cavity material perturbation
7:
3131:{\displaystyle \sigma _{e}}
2929:
2797:
2551:
2515:
2503:
2270:
2142:
2130:
1779:
1595:
1560:{\displaystyle \Delta \mu }
1427:{\displaystyle \omega _{0}}
1370:
1026:
1020:
1010:
1004:
998:
986:
619:
613:
607:
476:
10:
4406:
180:angular resonant frequency
26:cavity perturbation theory
4225:10.1109/amta.2008.4763128
1881:Cavity shape perturbation
1529:{\displaystyle \epsilon }
1407:of the perturbed cavity,
2607:{\displaystyle TE_{10n}}
2364:{\displaystyle \Delta V}
1869:densities respectively.
4159:10.1364/OPTICA.6.000269
3981:Proc. Inst. Electr. Eng
2340:represent time-average
1393:{\displaystyle \omega }
4380:Electrical engineering
4042:10.1002/lpor.201700113
3993:10.1049/pi-c.1960.0041
3954:10.1002/lpor.201700113
3899:
3854:
3811:
3614:
3436:
3370:
3327:
3246:
3164:
3132:
3095:
3068:
3041:
3014:
2987:
2960:
2917:
2785:
2672:
2608:
2577:
2537:electromagnetic fields
2491:
2365:
2348:energies contained in
2334:
2304:
2258:
2118:
1882:
1855:
1819:
1767:
1584:
1561:
1530:
1510:
1482:
1455:
1428:
1394:
1358:
1108:
1089:
1063:
974:
768:
742:
703:
680:
599:
572:
545:
509:
464:
226:
172:
125:
57:electromagnetic fields
3999:on February 18, 2020.
3900:
3855:
3812:
3615:
3437:
3371:
3328:
3247:
3165:
3133:
3096:
3094:{\displaystyle Q_{s}}
3069:
3067:{\displaystyle Q_{c}}
3042:
3040:{\displaystyle V_{s}}
3015:
3013:{\displaystyle V_{c}}
2988:
2986:{\displaystyle f_{s}}
2961:
2959:{\displaystyle f_{c}}
2918:
2786:
2673:
2609:
2575:
2533:dielectric resonators
2492:
2366:
2335:
2305:
2259:
2119:
1880:
1856:
1820:
1768:
1585:
1562:
1531:
1511:
1483:
1481:{\displaystyle H_{0}}
1456:
1454:{\displaystyle E_{0}}
1429:
1395:
1359:
1106:
1099:Material perturbation
1090:
1064:
1062:{\displaystyle |E.E|}
975:
769:
743:
741:{\displaystyle |E.E|}
704:
681:
600:
598:{\displaystyle E_{0}}
573:
571:{\displaystyle H_{0}}
546:
510:
465:
227:
173:
126:
4385:Microwave technology
3864:
3837:
3642:
3451:
3384:
3353:
3274:
3179:
3148:
3115:
3078:
3051:
3024:
2997:
2970:
2943:
2813:
2687:
2620:
2585:
2383:
2352:
2314:
2284:
2155:
1894:
1829:
1793:
1612:
1571:
1548:
1544:respectively, while
1520:
1509:{\displaystyle \mu }
1500:
1465:
1438:
1411:
1384:
1128:
1073:
1037:
788:
752:
716:
690:
627:
582:
555:
519:
490:
242:
186:
135:
83:
4151:2019Optic...6..269C
4090:2015NanoL..15.3439Y
4034:2018LPRv...1200113L
3946:2018LPRv...1200113L
3681:
3657:
3496:
3466:
3431:
3412:
3319:
3306:
3240:
2828:
2702:
2667:
2648:
1488:represent original
1088:{\displaystyle E.E}
956:
935:
889:
767:{\displaystyle E.E}
65:Maxwell's equations
3895:
3850:
3807:
3667:
3645:
3610:
3482:
3454:
3432:
3419:
3400:
3366:
3323:
3307:
3294:
3242:
3228:
3160:
3128:
3091:
3064:
3037:
3010:
2983:
2956:
2913:
2816:
2781:
2690:
2668:
2655:
2636:
2604:
2578:
2487:
2361:
2330:
2300:
2254:
2114:
1883:
1873:Shape perturbation
1851:
1815:
1763:
1580:
1557:
1526:
1506:
1478:
1451:
1424:
1405:resonant frequency
1390:
1354:
1109:
1085:
1059:
970:
942:
921:
875:
764:
738:
699:
676:
595:
568:
541:
505:
460:
222:
168:
121:
4390:Radio electronics
4234:978-1-4244-2690-4
3893:
3831:
3830:
3801:
3744:
3714:
3634:
3633:
3604:
3574:
3529:
3347:
3346:
3320:
3266:
3265:
2937:
2936:
2910:
2859:
2805:
2804:
2778:
2754:
2511:
2510:
2484:
2465:
2443:
2416:
2278:
2277:
2251:
2188:
2138:
2137:
2111:
1927:
1848:
1812:
1787:
1786:
1750:
1729:
1710:
1689:
1661:
1645:
1378:
1377:
1351:
1161:
994:
993:
967:
848:
839:
819:
803:
664:
648:
532:
502:
484:
483:
457:
302:
293:
273:
257:
216:
162:
95:
4397:
4375:Electromagnetism
4357:
4356:
4345:
4344:
4333:
4332:
4321:
4320:
4309:
4308:
4300:
4289:
4285:
4279:
4275:
4269:
4265:
4259:
4255:
4249:
4245:
4239:
4238:
4212:
4206:
4203:
4188:
4182:
4171:
4170:
4144:
4124:
4118:
4117:
4083:
4074:(5): 3439–3444.
4063:
4054:
4053:
4027:
4007:
4001:
4000:
3995:. Archived from
3972:
3966:
3965:
3939:
3919:
3904:
3902:
3901:
3896:
3894:
3889:
3881:
3876:
3875:
3859:
3857:
3856:
3851:
3849:
3848:
3825:
3816:
3814:
3813:
3808:
3802:
3800:
3799:
3798:
3789:
3788:
3778:
3777:
3776:
3764:
3763:
3753:
3745:
3743:
3742:
3733:
3732:
3723:
3715:
3713:
3712:
3711:
3698:
3697:
3696:
3680:
3675:
3665:
3653:
3636:
3628:
3619:
3617:
3616:
3611:
3605:
3603:
3602:
3593:
3592:
3583:
3575:
3573:
3572:
3563:
3562:
3561:
3549:
3548:
3538:
3530:
3528:
3527:
3526:
3513:
3512:
3511:
3495:
3490:
3480:
3462:
3445:
3441:
3439:
3438:
3433:
3427:
3408:
3396:
3395:
3375:
3373:
3372:
3367:
3365:
3364:
3341:
3332:
3330:
3329:
3324:
3321:
3315:
3302:
3293:
3268:
3260:
3251:
3249:
3248:
3243:
3236:
3227:
3226:
3205:
3191:
3190:
3173:
3169:
3167:
3166:
3161:
3137:
3135:
3134:
3129:
3127:
3126:
3100:
3098:
3097:
3092:
3090:
3089:
3073:
3071:
3070:
3065:
3063:
3062:
3046:
3044:
3043:
3038:
3036:
3035:
3019:
3017:
3016:
3011:
3009:
3008:
2992:
2990:
2989:
2984:
2982:
2981:
2965:
2963:
2962:
2957:
2955:
2954:
2931:
2922:
2920:
2919:
2914:
2911:
2909:
2908:
2907:
2898:
2897:
2887:
2886:
2885:
2873:
2872:
2862:
2860:
2858:
2857:
2856:
2843:
2842:
2833:
2824:
2807:
2799:
2790:
2788:
2787:
2782:
2779:
2777:
2776:
2767:
2766:
2757:
2755:
2753:
2752:
2751:
2738:
2737:
2736:
2724:
2723:
2713:
2698:
2681:
2677:
2675:
2674:
2669:
2663:
2644:
2632:
2631:
2613:
2611:
2610:
2605:
2603:
2602:
2567:
2505:
2496:
2494:
2493:
2488:
2485:
2480:
2467:
2466:
2461:
2460:
2451:
2445:
2444:
2439:
2438:
2429:
2422:
2417:
2415:
2414:
2405:
2404:
2403:
2387:
2377:
2370:
2368:
2367:
2362:
2339:
2337:
2336:
2331:
2329:
2328:
2309:
2307:
2306:
2301:
2299:
2298:
2272:
2263:
2261:
2260:
2255:
2252:
2250:
2249:
2248:
2236:
2235:
2225:
2224:
2223:
2208:
2207:
2194:
2189:
2187:
2186:
2177:
2176:
2175:
2159:
2149:
2132:
2123:
2121:
2120:
2115:
2112:
2110:
2100:
2099:
2094:
2088:
2087:
2078:
2067:
2066:
2061:
2055:
2054:
2045:
2034:
2033:
2023:
2013:
2012:
2007:
2001:
2000:
1991:
1980:
1979:
1974:
1968:
1967:
1958:
1947:
1946:
1933:
1928:
1926:
1925:
1916:
1915:
1914:
1898:
1888:
1860:
1858:
1857:
1852:
1850:
1849:
1844:
1843:
1834:
1824:
1822:
1821:
1816:
1814:
1813:
1808:
1807:
1798:
1781:
1772:
1770:
1769:
1764:
1752:
1751:
1746:
1745:
1736:
1730:
1725:
1717:
1712:
1711:
1706:
1705:
1696:
1690:
1685:
1677:
1672:
1671:
1662:
1654:
1646:
1644:
1643:
1634:
1633:
1632:
1616:
1606:
1589:
1587:
1586:
1581:
1566:
1564:
1563:
1558:
1535:
1533:
1532:
1527:
1515:
1513:
1512:
1507:
1487:
1485:
1484:
1479:
1477:
1476:
1460:
1458:
1457:
1452:
1450:
1449:
1433:
1431:
1430:
1425:
1423:
1422:
1399:
1397:
1396:
1391:
1372:
1363:
1361:
1360:
1355:
1352:
1350:
1340:
1339:
1334:
1328:
1327:
1318:
1307:
1306:
1301:
1295:
1294:
1285:
1274:
1273:
1263:
1253:
1252:
1247:
1241:
1240:
1231:
1217:
1216:
1211:
1205:
1204:
1195:
1181:
1180:
1170:
1162:
1160:
1159:
1150:
1149:
1148:
1132:
1122:
1094:
1092:
1091:
1086:
1068:
1066:
1065:
1060:
1058:
1044:
1016:photonic crystal
988:
979:
977:
976:
971:
968:
966:
955:
950:
934:
929:
920:
919:
907:
906:
896:
888:
883:
868:
867:
857:
849:
847:
846:
841:
840:
832:
828:
827:
826:
821:
820:
812:
805:
804:
796:
792:
782:
777:quasinormal mode
773:
771:
770:
765:
747:
745:
744:
739:
737:
723:
708:
706:
705:
700:
685:
683:
682:
677:
672:
671:
666:
665:
657:
650:
649:
641:
604:
602:
601:
596:
594:
593:
577:
575:
574:
569:
567:
566:
550:
548:
547:
542:
540:
539:
534:
533:
525:
514:
512:
511:
506:
504:
503:
495:
478:
469:
467:
466:
461:
458:
456:
446:
445:
440:
434:
433:
424:
413:
412:
407:
401:
400:
391:
386:
385:
373:
372:
362:
355:
354:
349:
343:
342:
333:
322:
321:
311:
303:
301:
300:
295:
294:
286:
282:
281:
280:
275:
274:
266:
259:
258:
250:
246:
236:
231:
229:
228:
223:
218:
217:
209:
177:
175:
174:
169:
164:
163:
155:
130:
128:
127:
122:
117:
97:
96:
88:
61:quasinormal mode
37:cavity resonator
4405:
4404:
4400:
4399:
4398:
4396:
4395:
4394:
4365:
4364:
4363:
4351:
4339:
4327:
4315:
4303:
4295:
4293:
4292:
4286:
4282:
4276:
4272:
4266:
4262:
4256:
4252:
4246:
4242:
4235:
4213:
4209:
4204:
4191:
4183:
4174:
4125:
4121:
4064:
4057:
4008:
4004:
3973:
3969:
3920:
3916:
3911:
3882:
3880:
3871:
3867:
3865:
3862:
3861:
3844:
3840:
3838:
3835:
3834:
3794:
3790:
3784:
3780:
3779:
3772:
3768:
3759:
3755:
3754:
3752:
3738:
3734:
3728:
3724:
3722:
3707:
3703:
3699:
3692:
3688:
3676:
3671:
3666:
3664:
3649:
3643:
3640:
3639:
3598:
3594:
3588:
3584:
3582:
3568:
3564:
3557:
3553:
3544:
3540:
3539:
3537:
3522:
3518:
3514:
3507:
3503:
3491:
3486:
3481:
3479:
3458:
3452:
3449:
3448:
3423:
3404:
3391:
3387:
3385:
3382:
3381:
3360:
3356:
3354:
3351:
3350:
3311:
3298:
3292:
3275:
3272:
3271:
3232:
3222:
3218:
3198:
3186:
3182:
3180:
3177:
3176:
3149:
3146:
3145:
3122:
3118:
3116:
3113:
3112:
3103:quality factors
3085:
3081:
3079:
3076:
3075:
3058:
3054:
3052:
3049:
3048:
3031:
3027:
3025:
3022:
3021:
3004:
3000:
2998:
2995:
2994:
2977:
2973:
2971:
2968:
2967:
2950:
2946:
2944:
2941:
2940:
2903:
2899:
2893:
2889:
2888:
2881:
2877:
2868:
2864:
2863:
2861:
2852:
2848:
2844:
2838:
2834:
2832:
2820:
2814:
2811:
2810:
2772:
2768:
2762:
2758:
2756:
2747:
2743:
2739:
2732:
2728:
2719:
2715:
2714:
2712:
2694:
2688:
2685:
2684:
2659:
2640:
2627:
2623:
2621:
2618:
2617:
2595:
2591:
2586:
2583:
2582:
2570:
2566:
2557:
2554:
2529:
2456:
2452:
2450:
2449:
2434:
2430:
2428:
2427:
2423:
2421:
2410:
2406:
2399:
2395:
2388:
2386:
2384:
2381:
2380:
2353:
2350:
2349:
2324:
2320:
2315:
2312:
2311:
2294:
2290:
2285:
2282:
2281:
2244:
2240:
2231:
2227:
2226:
2219:
2215:
2203:
2199:
2195:
2193:
2182:
2178:
2171:
2167:
2160:
2158:
2156:
2153:
2152:
2095:
2090:
2089:
2083:
2079:
2074:
2062:
2057:
2056:
2050:
2046:
2041:
2029:
2025:
2024:
2008:
2003:
2002:
1996:
1992:
1987:
1975:
1970:
1969:
1963:
1959:
1954:
1939:
1935:
1934:
1932:
1921:
1917:
1910:
1906:
1899:
1897:
1895:
1892:
1891:
1875:
1867:magnetic energy
1839:
1835:
1833:
1832:
1830:
1827:
1826:
1803:
1799:
1797:
1796:
1794:
1791:
1790:
1741:
1737:
1735:
1734:
1718:
1716:
1701:
1697:
1695:
1694:
1678:
1676:
1667:
1663:
1653:
1639:
1635:
1628:
1624:
1617:
1615:
1613:
1610:
1609:
1601:stored energies
1572:
1569:
1568:
1549:
1546:
1545:
1521:
1518:
1517:
1501:
1498:
1497:
1472:
1468:
1466:
1463:
1462:
1445:
1441:
1439:
1436:
1435:
1418:
1414:
1412:
1409:
1408:
1385:
1382:
1381:
1335:
1330:
1329:
1323:
1319:
1314:
1302:
1297:
1296:
1290:
1286:
1281:
1269:
1265:
1264:
1248:
1243:
1242:
1236:
1232:
1227:
1212:
1207:
1206:
1200:
1196:
1191:
1176:
1172:
1171:
1169:
1155:
1151:
1144:
1140:
1133:
1131:
1129:
1126:
1125:
1101:
1074:
1071:
1070:
1054:
1040:
1038:
1035:
1034:
951:
946:
930:
925:
915:
911:
902:
898:
897:
884:
879:
863:
859:
858:
856:
842:
831:
830:
829:
822:
811:
810:
809:
795:
794:
793:
791:
789:
786:
785:
753:
750:
749:
733:
719:
717:
714:
713:
691:
688:
687:
667:
656:
655:
654:
640:
639:
628:
625:
624:
589:
585:
583:
580:
579:
562:
558:
556:
553:
552:
535:
524:
523:
522:
520:
517:
516:
494:
493:
491:
488:
487:
441:
436:
435:
429:
425:
420:
408:
403:
402:
396:
392:
387:
381:
377:
368:
364:
363:
350:
345:
344:
338:
334:
329:
317:
313:
312:
310:
296:
285:
284:
283:
276:
265:
264:
263:
249:
248:
247:
245:
243:
240:
239:
208:
207:
187:
184:
183:
154:
153:
136:
133:
132:
113:
87:
86:
84:
81:
80:
77:
53:
12:
11:
5:
4403:
4393:
4392:
4387:
4382:
4377:
4362:
4361:
4349:
4337:
4325:
4313:
4291:
4290:
4280:
4270:
4260:
4250:
4240:
4233:
4207:
4189:
4172:
4135:(3): 269–273.
4119:
4055:
4018:(5): 1700113.
4002:
3987:(C): 272–274.
3967:
3930:(5): 1700113.
3913:
3912:
3910:
3907:
3892:
3888:
3885:
3879:
3874:
3870:
3847:
3843:
3829:
3828:
3819:
3817:
3805:
3797:
3793:
3787:
3783:
3775:
3771:
3767:
3762:
3758:
3751:
3748:
3741:
3737:
3731:
3727:
3721:
3718:
3710:
3706:
3702:
3695:
3691:
3687:
3684:
3679:
3674:
3670:
3663:
3660:
3656:
3652:
3648:
3632:
3631:
3622:
3620:
3608:
3601:
3597:
3591:
3587:
3581:
3578:
3571:
3567:
3560:
3556:
3552:
3547:
3543:
3536:
3533:
3525:
3521:
3517:
3510:
3506:
3502:
3499:
3494:
3489:
3485:
3478:
3475:
3472:
3469:
3465:
3461:
3457:
3430:
3426:
3422:
3418:
3415:
3411:
3407:
3403:
3399:
3394:
3390:
3363:
3359:
3345:
3344:
3335:
3333:
3318:
3314:
3310:
3305:
3301:
3297:
3291:
3288:
3285:
3282:
3279:
3264:
3263:
3254:
3252:
3239:
3235:
3231:
3225:
3221:
3217:
3214:
3211:
3208:
3204:
3201:
3197:
3194:
3189:
3185:
3159:
3156:
3153:
3125:
3121:
3088:
3084:
3061:
3057:
3034:
3030:
3007:
3003:
2980:
2976:
2953:
2949:
2935:
2934:
2925:
2923:
2906:
2902:
2896:
2892:
2884:
2880:
2876:
2871:
2867:
2855:
2851:
2847:
2841:
2837:
2831:
2827:
2823:
2819:
2803:
2802:
2793:
2791:
2775:
2771:
2765:
2761:
2750:
2746:
2742:
2735:
2731:
2727:
2722:
2718:
2711:
2708:
2705:
2701:
2697:
2693:
2666:
2662:
2658:
2654:
2651:
2647:
2643:
2639:
2635:
2630:
2626:
2601:
2598:
2594:
2590:
2569:
2561:
2555:
2553:
2550:
2546:coaxial cables
2528:
2525:
2509:
2508:
2499:
2497:
2483:
2479:
2476:
2473:
2470:
2464:
2459:
2455:
2448:
2442:
2437:
2433:
2426:
2420:
2413:
2409:
2402:
2398:
2394:
2391:
2360:
2357:
2327:
2323:
2319:
2297:
2293:
2289:
2276:
2275:
2266:
2264:
2247:
2243:
2239:
2234:
2230:
2222:
2218:
2214:
2211:
2206:
2202:
2198:
2192:
2185:
2181:
2174:
2170:
2166:
2163:
2136:
2135:
2126:
2124:
2109:
2106:
2103:
2098:
2093:
2086:
2082:
2077:
2073:
2070:
2065:
2060:
2053:
2049:
2044:
2040:
2037:
2032:
2028:
2022:
2019:
2016:
2011:
2006:
1999:
1995:
1990:
1986:
1983:
1978:
1973:
1966:
1962:
1957:
1953:
1950:
1945:
1942:
1938:
1931:
1924:
1920:
1913:
1909:
1905:
1902:
1874:
1871:
1847:
1842:
1838:
1811:
1806:
1802:
1785:
1784:
1775:
1773:
1761:
1758:
1755:
1749:
1744:
1740:
1733:
1728:
1724:
1721:
1715:
1709:
1704:
1700:
1693:
1688:
1684:
1681:
1675:
1670:
1666:
1660:
1657:
1652:
1649:
1642:
1638:
1631:
1627:
1623:
1620:
1579:
1576:
1556:
1553:
1525:
1505:
1496:respectively,
1494:magnetic field
1475:
1471:
1448:
1444:
1421:
1417:
1389:
1376:
1375:
1366:
1364:
1349:
1346:
1343:
1338:
1333:
1326:
1322:
1317:
1313:
1310:
1305:
1300:
1293:
1289:
1284:
1280:
1277:
1272:
1268:
1262:
1259:
1256:
1251:
1246:
1239:
1235:
1230:
1226:
1223:
1220:
1215:
1210:
1203:
1199:
1194:
1190:
1187:
1184:
1179:
1175:
1168:
1165:
1158:
1154:
1147:
1143:
1139:
1136:
1100:
1097:
1084:
1081:
1078:
1057:
1053:
1050:
1047:
1043:
992:
991:
982:
980:
965:
962:
959:
954:
949:
945:
941:
938:
933:
928:
924:
918:
914:
910:
905:
901:
895:
892:
887:
882:
878:
874:
871:
866:
862:
855:
852:
845:
838:
835:
825:
818:
815:
808:
802:
799:
763:
760:
757:
736:
732:
729:
726:
722:
698:
695:
675:
670:
663:
660:
653:
647:
644:
638:
635:
632:
592:
588:
565:
561:
538:
531:
528:
501:
498:
482:
481:
472:
470:
455:
452:
449:
444:
439:
432:
428:
423:
419:
416:
411:
406:
399:
395:
390:
384:
380:
376:
371:
367:
361:
358:
353:
348:
341:
337:
332:
328:
325:
320:
316:
309:
306:
299:
292:
289:
279:
272:
269:
262:
256:
253:
221:
215:
212:
206:
203:
200:
197:
194:
191:
167:
161:
158:
152:
149:
146:
143:
140:
120:
116:
112:
109:
106:
103:
100:
94:
91:
76:
75:General theory
73:
52:
49:
9:
6:
4:
3:
2:
4402:
4391:
4388:
4386:
4383:
4381:
4378:
4376:
4373:
4372:
4370:
4360:
4355:
4350:
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4331:
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4319:
4314:
4312:
4307:
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4301:
4298:
4284:
4274:
4264:
4254:
4244:
4236:
4230:
4226:
4222:
4218:
4211:
4202:
4200:
4198:
4196:
4194:
4186:
4181:
4179:
4177:
4168:
4164:
4160:
4156:
4152:
4148:
4143:
4138:
4134:
4130:
4123:
4115:
4111:
4107:
4103:
4099:
4095:
4091:
4087:
4082:
4077:
4073:
4069:
4062:
4060:
4051:
4047:
4043:
4039:
4035:
4031:
4026:
4021:
4017:
4013:
4006:
3998:
3994:
3990:
3986:
3982:
3978:
3971:
3963:
3959:
3955:
3951:
3947:
3943:
3938:
3933:
3929:
3925:
3918:
3914:
3906:
3890:
3886:
3883:
3877:
3872:
3868:
3845:
3841:
3827:
3820:
3818:
3795:
3791:
3785:
3781:
3773:
3769:
3765:
3760:
3756:
3739:
3735:
3729:
3725:
3708:
3704:
3700:
3693:
3689:
3685:
3682:
3677:
3672:
3668:
3658:
3654:
3650:
3646:
3638:
3637:
3630:
3623:
3621:
3599:
3595:
3589:
3585:
3569:
3565:
3558:
3554:
3550:
3545:
3541:
3523:
3519:
3515:
3508:
3504:
3500:
3497:
3492:
3487:
3483:
3473:
3470:
3467:
3463:
3459:
3455:
3447:
3446:
3443:
3428:
3424:
3420:
3416:
3413:
3409:
3405:
3401:
3397:
3392:
3388:
3380:permeability
3377:
3361:
3357:
3343:
3336:
3334:
3316:
3312:
3308:
3303:
3299:
3295:
3289:
3286:
3283:
3280:
3277:
3270:
3269:
3262:
3255:
3253:
3237:
3233:
3229:
3223:
3219:
3215:
3212:
3209:
3206:
3202:
3199:
3195:
3192:
3187:
3183:
3175:
3174:
3171:
3157:
3154:
3151:
3144:
3141:
3123:
3119:
3111:
3106:
3104:
3086:
3082:
3059:
3055:
3032:
3028:
3005:
3001:
2978:
2974:
2951:
2947:
2933:
2926:
2924:
2904:
2900:
2894:
2890:
2882:
2878:
2874:
2869:
2865:
2853:
2849:
2845:
2839:
2835:
2829:
2825:
2821:
2817:
2809:
2808:
2801:
2794:
2792:
2773:
2769:
2763:
2759:
2748:
2744:
2740:
2733:
2729:
2725:
2720:
2716:
2709:
2706:
2703:
2699:
2695:
2691:
2683:
2682:
2679:
2664:
2660:
2656:
2652:
2649:
2645:
2641:
2637:
2633:
2628:
2624:
2614:
2599:
2596:
2592:
2588:
2574:
2565:
2560:
2549:
2547:
2544:cavities and
2543:
2538:
2534:
2524:
2522:
2518:
2517:
2507:
2500:
2498:
2481:
2477:
2471:
2457:
2453:
2446:
2435:
2431:
2411:
2407:
2400:
2396:
2392:
2389:
2379:
2378:
2375:
2372:
2358:
2347:
2343:
2325:
2321:
2295:
2291:
2274:
2267:
2265:
2245:
2241:
2237:
2232:
2228:
2220:
2216:
2209:
2204:
2200:
2183:
2179:
2172:
2168:
2164:
2161:
2151:
2150:
2147:
2145:
2144:
2134:
2127:
2125:
2107:
2104:
2096:
2084:
2080:
2071:
2068:
2063:
2051:
2047:
2038:
2030:
2026:
2020:
2017:
2009:
1997:
1993:
1984:
1981:
1976:
1964:
1960:
1951:
1943:
1936:
1922:
1918:
1911:
1907:
1903:
1900:
1890:
1889:
1886:
1879:
1870:
1868:
1864:
1840:
1836:
1804:
1800:
1783:
1776:
1774:
1759:
1756:
1742:
1738:
1731:
1726:
1722:
1713:
1702:
1698:
1691:
1686:
1682:
1668:
1664:
1658:
1655:
1650:
1640:
1636:
1629:
1625:
1621:
1618:
1608:
1607:
1604:
1602:
1598:
1597:
1591:
1577:
1554:
1543:
1539:
1536:are original
1523:
1503:
1495:
1491:
1473:
1469:
1446:
1442:
1419:
1415:
1406:
1403:
1387:
1374:
1367:
1365:
1347:
1344:
1336:
1324:
1320:
1311:
1308:
1303:
1291:
1287:
1278:
1270:
1266:
1260:
1257:
1249:
1237:
1233:
1224:
1218:
1213:
1201:
1197:
1188:
1177:
1173:
1166:
1156:
1152:
1145:
1141:
1137:
1134:
1124:
1123:
1120:
1118:
1114:
1105:
1096:
1082:
1079:
1076:
1051:
1048:
1045:
1031:
1029:
1028:
1023:
1022:
1017:
1013:
1012:
1007:
1006:
1001:
1000:
990:
983:
981:
963:
960:
952:
947:
943:
939:
936:
931:
926:
922:
916:
912:
903:
899:
893:
890:
885:
880:
876:
872:
864:
860:
853:
843:
833:
823:
813:
806:
797:
784:
783:
780:
778:
761:
758:
755:
730:
727:
724:
710:
696:
668:
658:
651:
642:
633:
630:
622:
621:
616:
615:
610:
609:
590:
586:
563:
559:
536:
526:
496:
480:
473:
471:
453:
450:
442:
430:
426:
417:
414:
409:
397:
393:
382:
378:
369:
365:
359:
356:
351:
339:
335:
326:
318:
314:
307:
297:
287:
277:
267:
260:
251:
238:
237:
234:
210:
201:
198:
195:
192:
189:
181:
156:
147:
144:
141:
138:
118:
114:
110:
107:
104:
101:
98:
89:
72:
70:
66:
62:
58:
48:
44:
40:
38:
34:
31:
27:
23:
19:
4283:
4273:
4263:
4253:
4243:
4216:
4210:
4132:
4128:
4122:
4071:
4068:Nano Letters
4067:
4015:
4011:
4005:
3997:the original
3984:
3980:
3970:
3927:
3923:
3917:
3832:
3821:
3624:
3378:
3348:
3337:
3256:
3143:loss tangent
3110:conductivity
3107:
2938:
2927:
2795:
2579:
2563:
2558:
2530:
2527:Applications
2514:
2512:
2501:
2373:
2279:
2268:
2141:
2140:Expression (
2139:
2128:
1884:
1788:
1777:
1594:
1593:Expression (
1592:
1542:permittivity
1538:permeability
1379:
1368:
1117:permeability
1113:permittivity
1110:
1032:
1025:
1019:
1009:
1003:
997:
995:
984:
711:
618:
612:
606:
485:
474:
78:
54:
51:Introduction
45:
41:
30:perturbation
25:
15:
4335:Engineering
4323:Mathematics
4311:Electronics
4185:David Pozar
22:electronics
18:mathematics
4369:Categories
4359:Technology
4142:1811.11726
4081:1505.04877
4025:1705.02433
3937:1705.02433
3909:References
3140:dielectric
3101:represent
686:) only if
4167:119439374
3869:λ
3842:λ
3766:−
3669:λ
3647:μ
3551:−
3484:λ
3468:−
3456:μ
3421:μ
3402:μ
3389:μ
3358:ϵ
3309:ϵ
3296:ϵ
3287:δ
3230:ϵ
3220:ϵ
3213:π
3200:ϵ
3196:ω
3184:σ
3158:δ
3155:
3120:σ
2875:−
2818:ϵ
2726:−
2704:−
2692:ϵ
2657:ϵ
2638:ϵ
2625:ϵ
2542:waveguide
2475:Δ
2472:⋅
2463:¯
2447:−
2441:¯
2419:≈
2408:ω
2397:ω
2393:−
2390:ω
2356:Δ
2318:Δ
2288:Δ
2213:Δ
2210:−
2197:Δ
2191:≈
2180:ω
2169:ω
2165:−
2162:ω
2072:ϵ
2039:μ
2027:∭
1985:ϵ
1982:−
1952:μ
1941:Δ
1937:∭
1930:≈
1919:ω
1908:ω
1904:−
1901:ω
1846:¯
1810:¯
1748:¯
1732:⋅
1727:μ
1723:μ
1720:Δ
1708:¯
1692:⋅
1687:ϵ
1683:ϵ
1680:Δ
1665:∭
1651:−
1648:≈
1637:ω
1626:ω
1622:−
1619:ω
1578:ϵ
1575:Δ
1555:μ
1552:Δ
1524:ϵ
1504:μ
1416:ω
1388:ω
1312:ϵ
1279:μ
1267:∭
1225:ϵ
1222:Δ
1189:μ
1186:Δ
1174:∭
1167:−
1164:≈
1153:ω
1142:ω
1138:−
1135:ω
940:ϵ
913:μ
900:∭
873:ϵ
870:Δ
861:∭
854:−
851:≈
837:~
834:ω
817:~
814:ω
807:−
801:~
798:ω
697:ϵ
694:Δ
662:~
659:ω
652:−
646:~
643:ω
530:~
527:ω
500:~
497:ω
418:ϵ
379:μ
366:∭
327:ϵ
324:Δ
315:∭
308:−
305:≈
291:~
288:ω
271:~
268:ω
261:−
255:~
252:ω
214:~
211:ω
190:γ
160:~
157:ω
139:ω
131:, where
111:γ
105:−
102:ω
93:~
90:ω
4114:11999899
4106:25844813
4050:51695476
3962:51695476
3655:″
3464:′
3429:″
3410:′
3317:′
3304:″
3238:″
3203:″
2826:″
2700:′
2665:″
2646:′
2552:Examples
2346:magnetic
2342:electric
1863:electric
1490:electric
69:Q factor
33:formulae
4347:Science
4297:Portals
4147:Bibcode
4086:Bibcode
4030:Bibcode
3942:Bibcode
1402:angular
1400:is the
1115:and/or
178:is the
4231:
4165:
4129:Optica
4112:
4104:
4048:
3960:
3833:where
2939:where
2280:where
1380:where
486:where
4163:S2CID
4137:arXiv
4110:S2CID
4076:arXiv
4046:S2CID
4020:arXiv
3958:S2CID
3932:arXiv
4278:1991
4268:2006
4258:2010
4248:2008
4229:ISBN
4102:PMID
3442:as:
3170:as:
3138:and
3074:and
3020:and
2966:and
2678:as:
2344:and
2310:and
1865:and
1861:are
1825:and
1603:as:
1567:and
1540:and
1516:and
1492:and
1461:and
578:and
515:and
182:and
20:and
4221:doi
4155:doi
4094:doi
4038:doi
3989:doi
3985:107
3950:doi
3905:).
3152:tan
16:In
4371::
4227:.
4192:^
4175:^
4161:.
4153:.
4145:.
4131:.
4108:.
4100:.
4092:.
4084:.
4072:15
4070:.
4058:^
4044:.
4036:.
4028:.
4016:12
4014:.
3983:.
3979:.
3956:.
3948:.
3940:.
3928:12
3926:.
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3701:16
3627:12
3340:11
3259:10
2597:10
2562:10
2559:TE
2371:.
39:.
24:,
4299::
4237:.
4223::
4169:.
4157::
4149::
4139::
4133:6
4116:.
4096::
4088::
4078::
4052:.
4040::
4032::
4022::
3991::
3964:.
3952::
3944::
3934::
3891:n
3887:l
3884:2
3878:=
3873:g
3846:g
3826:)
3822:(
3804:)
3796:s
3792:Q
3786:c
3782:Q
3774:s
3770:Q
3761:c
3757:Q
3750:(
3747:)
3740:s
3736:V
3730:c
3726:V
3720:(
3717:)
3709:2
3705:a
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3683:+
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3625:(
3607:)
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3580:(
3577:)
3570:s
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3535:(
3532:)
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3498:+
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3488:g
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3290:=
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3257:(
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3207:=
3193:=
3188:e
3124:e
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3002:V
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2932:)
2930:9
2928:(
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2901:Q
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2836:V
2830:=
2822:r
2800:)
2798:8
2796:(
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2764:c
2760:V
2749:s
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2741:2
2734:s
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2717:f
2710:=
2707:1
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2642:r
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2504:7
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2131:5
2129:(
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2105:d
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2064:2
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2048:H
2043:|
2036:(
2031:V
2021:v
2018:d
2015:)
2010:2
2005:|
1998:0
1994:E
1989:|
1977:2
1972:|
1965:0
1961:H
1956:|
1949:(
1944:V
1923:0
1912:0
1841:m
1837:w
1805:e
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1780:4
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1757:d
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1641:0
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1474:0
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1420:0
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1371:3
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1193:|
1183:(
1178:V
1157:0
1146:0
1083:E
1080:.
1077:E
1056:|
1052:E
1049:.
1046:E
1042:|
1027:1
1021:2
1011:1
1005:2
999:2
989:)
987:2
985:(
964:v
961:d
958:)
953:2
948:0
944:E
937:+
932:2
927:0
923:H
917:0
909:(
904:V
894:v
891:d
886:2
881:0
877:E
865:V
844:0
824:0
762:E
759:.
756:E
735:|
731:E
728:.
725:E
721:|
674:)
669:0
637:(
634:m
631:I
620:1
614:1
608:1
591:0
587:E
564:0
560:H
537:0
479:)
477:1
475:(
454:v
451:d
448:)
443:2
438:|
431:0
427:E
422:|
415:+
410:2
405:|
398:0
394:H
389:|
383:0
375:(
370:V
360:v
357:d
352:2
347:|
340:0
336:E
331:|
319:V
298:0
278:0
220:)
205:(
202:m
199:I
196:2
193:=
166:)
151:(
148:e
145:R
142:=
119:2
115:/
108:i
99:=
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