Abstract

We study the quality factor of single-mode optical whispering gallery mode resonators using finite element method simulations, with a particular focus on the photonic belt resonator geometry. We experimentally observe a large difference between the quality factors of TM and TE modes in such resonators. Examining radiative losses, we conclude that the TM fundamental mode of single-mode resonators can have geometry related radiative losses caused by mode hybridization and coupling that limits their achievable quality factor. However, TE modes are free from mode hybridization radiative losses. This leads to much higher achievable Q factors for TE modes, only limited by fabrication and material quality. We experimentally observed photonic belt resonator quality factors on the order of one billion for TE modes, higher than in any other single mode optical resonator of similar dimensions.

© 2016 Optical Society of America

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References

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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]

2015 (3)

I. S. Grudinin and N. Yu, “Dispersion engineering of crystalline resonators via microstructuring,” Optica 2, 221–224 (2015).
[Crossref]

A. G. Griffith, R. K. W. Lau, J. Cardenas, Y. Okawachi, A. Mohanty, R. Fain, Y. Ho Daniel Lee, M. Yu, C. T. Phare, C. B. Poitras, Al. L. Gaeta, and M. Lipson, “Silicon-chip mid-infrared frequency comb generation,” Nat. Commun. 6, 6299 (2015).
[Crossref] [PubMed]

X. Yi, Q.-F. Yang, K. Y. Yang, M.-G. Suh, and K. Vahala, “Soliton frequency comb at microwave rates in a high-Q silica microresonator,” Optica 2, 1078–1085 (2015).
[Crossref]

2014 (3)

2013 (2)

M. I. Cheema and A. G. Kirk, “Accurate determination of the quality factor and tunneling distance of axisymmetric resonators for biosensing applications,” Opt. Express 21, 8724–8735 (2013).
[Crossref] [PubMed]

Q. Li, A. A. Eftekhar, Z. Xia, and A. Adibi, “Unified approach to mode splitting and scattering loss in high-Q whispering-gallery-mode microresonators,” Phys. Rev. A 88, 033816 (2013).
[Crossref]

2012 (1)

I. S. Grudinin and N. Yu, “Finite-element modeling of coupled optical microdisk resonators for displacement sensing,” J. Opt. Soc. Am. B 29, 221–224 (2012).
[Crossref]

2007 (2)

M. Oxborrow, “Traceable 2-D finite-element simulation of the whispering-gallery modes of axisymmetric electromagnetic resonators,” IEEE Trans. Microw. Theory Tech. 55, 1209–1218 (2007).
[Crossref]

J. E. Heebner, T. C. Bond, and J. S. Kalman, “Generalized formulation for performance degradations due to bending and edge scattering loss in microdisk resonators,” Opt. Express 15, 4452 (2007).
[Crossref] [PubMed]

2006 (3)

I. S. Grudinin, V. S. Ilchenko, and L. Maleki, “Ultrahigh optical Q factors of crystalline resonators in the linear regime,” Phys. Rev. A 74, 063806 (2006).
[Crossref]

A. B. Matsko and V. S. Ilchenko, “Optical resonators with whispering-gallery modes - Part I: Basics,” IEEE J. Sel. Top. Quantum Electron. 12, 3–14 (2006).
[Crossref]

A. A. Savchenkov, I. S. Grudinin, A. B. Matsko, D. Strekalov, M. Mohageg, V. S. Ilchenko, and L. Maleki, “Morphology-dependent photonic circuit elements,” Opt. Lett. 31, 1313–1315 (2006).
[Crossref] [PubMed]

2005 (2)

2003 (1)

K. J. Vahala, “Optical microcavities,” Nature 424, 839–846 (2003).
[Crossref] [PubMed]

2000 (1)

1997 (1)

F. L. Teixeira and W. C. Chew, “Systematic derivation of aniosotropic PML absorbing media in cylindrical and spherical coordinates,” IEEE Microw. Guided Wave Lett. 7, 371–373 (1997).
[Crossref]

1996 (1)

1939 (1)

R. D. Richtmyer, “Dielectric resonators,” J. Appl. Phys. 10, 391–398 (1939).
[Crossref]

Adibi, A.

Q. Li, A. A. Eftekhar, Z. Xia, and A. Adibi, “Unified approach to mode splitting and scattering loss in high-Q whispering-gallery-mode microresonators,” Phys. Rev. A 88, 033816 (2013).
[Crossref]

Bond, T. C.

Borselli, M.

Brasch, V.

V. Brasch, T. Herr, M. Geiselmann, G. Lihachev, M. H. P. Pfeiffer, M. L. Gorodetsky, and T. J. Kippenberg, “Photonic chip based optical frequency comb using soliton induced Cherenkov radiation,” arXiv:1410.8598.

Cardenas, J.

A. G. Griffith, R. K. W. Lau, J. Cardenas, Y. Okawachi, A. Mohanty, R. Fain, Y. Ho Daniel Lee, M. Yu, C. T. Phare, C. B. Poitras, Al. L. Gaeta, and M. Lipson, “Silicon-chip mid-infrared frequency comb generation,” Nat. Commun. 6, 6299 (2015).
[Crossref] [PubMed]

Cheema, M. I.

Chen, S.

Chew, W. C.

F. L. Teixeira and W. C. Chew, “Systematic derivation of aniosotropic PML absorbing media in cylindrical and spherical coordinates,” IEEE Microw. Guided Wave Lett. 7, 371–373 (1997).
[Crossref]

Chu, S. T.

Demchenko, A. A.

F. Ferdous, A. A. Demchenko, S. P. Vyatchanin, A. B. Matsko, and L. Maleki, “Microcavity morphology optimization,” Phys. Rev. A 90, 033826 (2014).
[Crossref]

Eftekhar, A. A.

Q. Li, A. A. Eftekhar, Z. Xia, and A. Adibi, “Unified approach to mode splitting and scattering loss in high-Q whispering-gallery-mode microresonators,” Phys. Rev. A 88, 033816 (2013).
[Crossref]

Eliyahu, D.

Fain, R.

A. G. Griffith, R. K. W. Lau, J. Cardenas, Y. Okawachi, A. Mohanty, R. Fain, Y. Ho Daniel Lee, M. Yu, C. T. Phare, C. B. Poitras, Al. L. Gaeta, and M. Lipson, “Silicon-chip mid-infrared frequency comb generation,” Nat. Commun. 6, 6299 (2015).
[Crossref] [PubMed]

Ferdous, F.

F. Ferdous, A. A. Demchenko, S. P. Vyatchanin, A. B. Matsko, and L. Maleki, “Microcavity morphology optimization,” Phys. Rev. A 90, 033826 (2014).
[Crossref]

Gaeta, Al. L.

A. G. Griffith, R. K. W. Lau, J. Cardenas, Y. Okawachi, A. Mohanty, R. Fain, Y. Ho Daniel Lee, M. Yu, C. T. Phare, C. B. Poitras, Al. L. Gaeta, and M. Lipson, “Silicon-chip mid-infrared frequency comb generation,” Nat. Commun. 6, 6299 (2015).
[Crossref] [PubMed]

Geiselmann, M.

V. Brasch, T. Herr, M. Geiselmann, G. Lihachev, M. H. P. Pfeiffer, M. L. Gorodetsky, and T. J. Kippenberg, “Photonic chip based optical frequency comb using soliton induced Cherenkov radiation,” arXiv:1410.8598.

Gorodetsky, M. L.

M. L. Gorodetsky, A. D. Pryamikov, and V. S. Ilchenko, “Rayleigh scattering in high-Q microspheres,” J. Opt. Soc. Am. B 17, 1051–1057 (2000).
[Crossref]

V. Brasch, T. Herr, M. Geiselmann, G. Lihachev, M. H. P. Pfeiffer, M. L. Gorodetsky, and T. J. Kippenberg, “Photonic chip based optical frequency comb using soliton induced Cherenkov radiation,” arXiv:1410.8598.

Griffith, A. G.

A. G. Griffith, R. K. W. Lau, J. Cardenas, Y. Okawachi, A. Mohanty, R. Fain, Y. Ho Daniel Lee, M. Yu, C. T. Phare, C. B. Poitras, Al. L. Gaeta, and M. Lipson, “Silicon-chip mid-infrared frequency comb generation,” Nat. Commun. 6, 6299 (2015).
[Crossref] [PubMed]

Grudinin, I. S.

I. S. Grudinin and N. Yu, “Dispersion engineering of crystalline resonators via microstructuring,” Optica 2, 221–224 (2015).
[Crossref]

I. S. Grudinin and N. Yu, “Finite-element modeling of coupled optical microdisk resonators for displacement sensing,” J. Opt. Soc. Am. B 29, 221–224 (2012).
[Crossref]

I. S. Grudinin, V. S. Ilchenko, and L. Maleki, “Ultrahigh optical Q factors of crystalline resonators in the linear regime,” Phys. Rev. A 74, 063806 (2006).
[Crossref]

A. A. Savchenkov, I. S. Grudinin, A. B. Matsko, D. Strekalov, M. Mohageg, V. S. Ilchenko, and L. Maleki, “Morphology-dependent photonic circuit elements,” Opt. Lett. 31, 1313–1315 (2006).
[Crossref] [PubMed]

Heebner, J. E.

Herr, T.

V. Brasch, T. Herr, M. Geiselmann, G. Lihachev, M. H. P. Pfeiffer, M. L. Gorodetsky, and T. J. Kippenberg, “Photonic chip based optical frequency comb using soliton induced Cherenkov radiation,” arXiv:1410.8598.

Ho Daniel Lee, Y.

A. G. Griffith, R. K. W. Lau, J. Cardenas, Y. Okawachi, A. Mohanty, R. Fain, Y. Ho Daniel Lee, M. Yu, C. T. Phare, C. B. Poitras, Al. L. Gaeta, and M. Lipson, “Silicon-chip mid-infrared frequency comb generation,” Nat. Commun. 6, 6299 (2015).
[Crossref] [PubMed]

Ilchenko, V. S.

Johnson, T. J.

Kalman, J. S.

Kippenberg, T. J.

V. Brasch, T. Herr, M. Geiselmann, G. Lihachev, M. H. P. Pfeiffer, M. L. Gorodetsky, and T. J. Kippenberg, “Photonic chip based optical frequency comb using soliton induced Cherenkov radiation,” arXiv:1410.8598.

Kirk, A. G.

Lau, R. K. W.

A. G. Griffith, R. K. W. Lau, J. Cardenas, Y. Okawachi, A. Mohanty, R. Fain, Y. Ho Daniel Lee, M. Yu, C. T. Phare, C. B. Poitras, Al. L. Gaeta, and M. Lipson, “Silicon-chip mid-infrared frequency comb generation,” Nat. Commun. 6, 6299 (2015).
[Crossref] [PubMed]

Leaird, D. E.

Li, Q.

Q. Li, A. A. Eftekhar, Z. Xia, and A. Adibi, “Unified approach to mode splitting and scattering loss in high-Q whispering-gallery-mode microresonators,” Phys. Rev. A 88, 033816 (2013).
[Crossref]

Liang, W.

Lihachev, G.

V. Brasch, T. Herr, M. Geiselmann, G. Lihachev, M. H. P. Pfeiffer, M. L. Gorodetsky, and T. J. Kippenberg, “Photonic chip based optical frequency comb using soliton induced Cherenkov radiation,” arXiv:1410.8598.

Lipson, M.

A. G. Griffith, R. K. W. Lau, J. Cardenas, Y. Okawachi, A. Mohanty, R. Fain, Y. Ho Daniel Lee, M. Yu, C. T. Phare, C. B. Poitras, Al. L. Gaeta, and M. Lipson, “Silicon-chip mid-infrared frequency comb generation,” Nat. Commun. 6, 6299 (2015).
[Crossref] [PubMed]

Little, B. E.

Liu, Y.

Maleki, L.

Matsko, A. B.

Metcalf, A. J.

Mohageg, M.

Mohanty, A.

A. G. Griffith, R. K. W. Lau, J. Cardenas, Y. Okawachi, A. Mohanty, R. Fain, Y. Ho Daniel Lee, M. Yu, C. T. Phare, C. B. Poitras, Al. L. Gaeta, and M. Lipson, “Silicon-chip mid-infrared frequency comb generation,” Nat. Commun. 6, 6299 (2015).
[Crossref] [PubMed]

Okawachi, Y.

A. G. Griffith, R. K. W. Lau, J. Cardenas, Y. Okawachi, A. Mohanty, R. Fain, Y. Ho Daniel Lee, M. Yu, C. T. Phare, C. B. Poitras, Al. L. Gaeta, and M. Lipson, “Silicon-chip mid-infrared frequency comb generation,” Nat. Commun. 6, 6299 (2015).
[Crossref] [PubMed]

Oxborrow, M.

M. Oxborrow, “Traceable 2-D finite-element simulation of the whispering-gallery modes of axisymmetric electromagnetic resonators,” IEEE Trans. Microw. Theory Tech. 55, 1209–1218 (2007).
[Crossref]

Painter, O.

Pfeiffer, M. H. P.

V. Brasch, T. Herr, M. Geiselmann, G. Lihachev, M. H. P. Pfeiffer, M. L. Gorodetsky, and T. J. Kippenberg, “Photonic chip based optical frequency comb using soliton induced Cherenkov radiation,” arXiv:1410.8598.

Phare, C. T.

A. G. Griffith, R. K. W. Lau, J. Cardenas, Y. Okawachi, A. Mohanty, R. Fain, Y. Ho Daniel Lee, M. Yu, C. T. Phare, C. B. Poitras, Al. L. Gaeta, and M. Lipson, “Silicon-chip mid-infrared frequency comb generation,” Nat. Commun. 6, 6299 (2015).
[Crossref] [PubMed]

Poitras, C. B.

A. G. Griffith, R. K. W. Lau, J. Cardenas, Y. Okawachi, A. Mohanty, R. Fain, Y. Ho Daniel Lee, M. Yu, C. T. Phare, C. B. Poitras, Al. L. Gaeta, and M. Lipson, “Silicon-chip mid-infrared frequency comb generation,” Nat. Commun. 6, 6299 (2015).
[Crossref] [PubMed]

Pryamikov, A. D.

Qi, M.

Rabiei, P.

Richtmyer, R. D.

R. D. Richtmyer, “Dielectric resonators,” J. Appl. Phys. 10, 391–398 (1939).
[Crossref]

Savchenkov, A. A.

Seidel, D.

Strekalov, D.

Suh, M.-G.

Teixeira, F. L.

F. L. Teixeira and W. C. Chew, “Systematic derivation of aniosotropic PML absorbing media in cylindrical and spherical coordinates,” IEEE Microw. Guided Wave Lett. 7, 371–373 (1997).
[Crossref]

Vahala, K.

Vahala, K. J.

K. J. Vahala, “Optical microcavities,” Nature 424, 839–846 (2003).
[Crossref] [PubMed]

Vyatchanin, S. P.

F. Ferdous, A. A. Demchenko, S. P. Vyatchanin, A. B. Matsko, and L. Maleki, “Microcavity morphology optimization,” Phys. Rev. A 90, 033826 (2014).
[Crossref]

Wang, J.

Wang, P.-H.

Weiner, A. M.

Xia, Z.

Q. Li, A. A. Eftekhar, Z. Xia, and A. Adibi, “Unified approach to mode splitting and scattering loss in high-Q whispering-gallery-mode microresonators,” Phys. Rev. A 88, 033816 (2013).
[Crossref]

Xuan, Y.

Xue, X.

Yang, K. Y.

Yang, Q.-F.

Yi, X.

Yu, M.

A. G. Griffith, R. K. W. Lau, J. Cardenas, Y. Okawachi, A. Mohanty, R. Fain, Y. Ho Daniel Lee, M. Yu, C. T. Phare, C. B. Poitras, Al. L. Gaeta, and M. Lipson, “Silicon-chip mid-infrared frequency comb generation,” Nat. Commun. 6, 6299 (2015).
[Crossref] [PubMed]

Yu, N.

I. S. Grudinin and N. Yu, “Dispersion engineering of crystalline resonators via microstructuring,” Optica 2, 221–224 (2015).
[Crossref]

I. S. Grudinin and N. Yu, “Finite-element modeling of coupled optical microdisk resonators for displacement sensing,” J. Opt. Soc. Am. B 29, 221–224 (2012).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

A. B. Matsko and V. S. Ilchenko, “Optical resonators with whispering-gallery modes - Part I: Basics,” IEEE J. Sel. Top. Quantum Electron. 12, 3–14 (2006).
[Crossref]

IEEE Microw. Guided Wave Lett. (1)

F. L. Teixeira and W. C. Chew, “Systematic derivation of aniosotropic PML absorbing media in cylindrical and spherical coordinates,” IEEE Microw. Guided Wave Lett. 7, 371–373 (1997).
[Crossref]

IEEE Trans. Microw. Theory Tech. (1)

M. Oxborrow, “Traceable 2-D finite-element simulation of the whispering-gallery modes of axisymmetric electromagnetic resonators,” IEEE Trans. Microw. Theory Tech. 55, 1209–1218 (2007).
[Crossref]

J. Appl. Phys. (1)

R. D. Richtmyer, “Dielectric resonators,” J. Appl. Phys. 10, 391–398 (1939).
[Crossref]

J. Lightwave Technol. (1)

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

M. L. Gorodetsky, A. D. Pryamikov, and V. S. Ilchenko, “Rayleigh scattering in high-Q microspheres,” J. Opt. Soc. Am. B 17, 1051–1057 (2000).
[Crossref]

I. S. Grudinin and N. Yu, “Finite-element modeling of coupled optical microdisk resonators for displacement sensing,” J. Opt. Soc. Am. B 29, 221–224 (2012).
[Crossref]

Nat. Commun. (1)

A. G. Griffith, R. K. W. Lau, J. Cardenas, Y. Okawachi, A. Mohanty, R. Fain, Y. Ho Daniel Lee, M. Yu, C. T. Phare, C. B. Poitras, Al. L. Gaeta, and M. Lipson, “Silicon-chip mid-infrared frequency comb generation,” Nat. Commun. 6, 6299 (2015).
[Crossref] [PubMed]

Nature (1)

K. J. Vahala, “Optical microcavities,” Nature 424, 839–846 (2003).
[Crossref] [PubMed]

Opt. Express (3)

Opt. Lett. (3)

Optica (3)

Phys. Rev. A (3)

Q. Li, A. A. Eftekhar, Z. Xia, and A. Adibi, “Unified approach to mode splitting and scattering loss in high-Q whispering-gallery-mode microresonators,” Phys. Rev. A 88, 033816 (2013).
[Crossref]

F. Ferdous, A. A. Demchenko, S. P. Vyatchanin, A. B. Matsko, and L. Maleki, “Microcavity morphology optimization,” Phys. Rev. A 90, 033826 (2014).
[Crossref]

I. S. Grudinin, V. S. Ilchenko, and L. Maleki, “Ultrahigh optical Q factors of crystalline resonators in the linear regime,” Phys. Rev. A 74, 063806 (2006).
[Crossref]

Other (2)

O. Pironneau, F. Hecht, A. Le Hyaric, and J. Morice, “FreeFem++,” http://www.freefem.org/

V. Brasch, T. Herr, M. Geiselmann, G. Lihachev, M. H. P. Pfeiffer, M. L. Gorodetsky, and T. J. Kippenberg, “Photonic chip based optical frequency comb using soliton induced Cherenkov radiation,” arXiv:1410.8598.

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Figures (9)

Fig. 1
Fig. 1

Simulations with 10 μm radius silica microspheres, examining the TE fundamental mode. PML layers are set along the top, bottom and right boundaries to allow determination of Q due to radiative losses. Window height is 32 μm. (a) and (b) show the simulated intensity distribution of the TE fundamental mode at a wavelength of 1.55 μm in a logarithmic scale (difference of 30 dB between contour lines with the highest value line located at half the maximum intensity). The location of the right border was adjusted to understand how PML affects simulation results, with the location varying from (a) the PML placed 1 μm away from the microsphere to (b) the PML being placed 20 μm away from the microsphere. The results of Q as determined for different PML locations are shown in (c).

Fig. 2
Fig. 2

Simulations with 10 μm radius MgF2 photonic belt resonators with 2 by 2 μm belts. (a), (b), and (d) show the simulated intensity distribution of the TE fundamental mode for various window sizes at a wavelength of 1.55 μm in a logarithmic scale (difference of 30 dB between contour lines with the highest value line located at half the maximum intensity). (a) and (b) correspond to simulations where the window height was set to 60 μm while the location of the right PML boundary was varied from (a) 13 μm to (b) 1 μm from the right edge of the belt. The simulated Q values for the window width sweep between (a) and (b) are shown in (c). We also performed a window height sweep from (a) 60 μm height to (d) 2.25 μm height while the left and right window boundaries were kept unchanged. The simulated Q results for the sweep from (a) to (d) are shown in (e).

Fig. 3
Fig. 3

Simulations of the TE and TM fundamental modes at a wavelength of 1.55 μm for 750 μm radius MgF2 photonic belt resonators. The belt is 7 μm wide and extends 5 μm above the cylinder. The window width is 48 μm. (a) and (b) show the simulated intensity distribution (difference of 30 dB between contour lines with the highest value line located at half the maximum intensity) for the TE fundamental mode, while (c) and (d) correspond to the logarithmic intensity distribution of the TM fundamental mode. The window height was varied from 120 μm for (a) and (c) to 10 μm for (b) and (d). The simulated Q values are shown in (e) and (f), corresponding to the TE and TM modes respectively. Note that, due to precision limitations in calculating Q with FreeFem++, (e) is limited to plotting simulation results with window heights no higher than 60 μm.

Fig. 4
Fig. 4

(a) Optical profilometer scan of the MgF2 cylinder with three photonic belt structures. Polishing traces and varying surface roughness are visible across different parts of the structures. (b) Profile of the middle structure from (a).

Fig. 5
Fig. 5

Simulation of the TE and TM fundamental mode at a wavelength of 1.55 μm for a 1337 μm radius MgF2 photonic belt resonator. The belt was set to be 10 by 10 μm. (a) shows the simulated TE mode intensity distribution (linear scale). (b) shows the intensity along the resonator edge for both the TE and TM modes. The labels A, B, and C on figures (a) and (b) correspond to the same points of the arc of the resonator edge, located at 0 μm, 5 μm, and 10 μm from the belt center along the arc. (c) shows the ratio of the TM fundamental mode intensity over the TE fundamental mode intensity along the resonator edge.

Fig. 6
Fig. 6

Simulations showing lines of equivalent magnitude for the electric field components of the TE and TM fundamental modes at a wavelength of 1.55 μm for a 750 μm radius MgF2 photonic belt resonator. The belt was set to be 7 by 5 μm. (a)–(c) correspond to components of the TE fundamental mode, while (e)–(g) correspond to the TM fundamental mode. (a) and (e) show the magnitude of the ER-components. (b) and (f) show the Eϕ -components. (c) and (g) show the EZ-components. (d) and (h) plot the magnitude of these components along the resonator edge, starting from the belt center and following the resonator arc upwards.

Fig. 7
Fig. 7

Simulation results where the resonator protrusion dimensions were varied. The optical field intensity is plotted along the arc of the resonator surface profile, starting from the belt center. We used a wavelength of 1.55 μm for a 750 μm radius MgF2 PBR. We denote the belt width by w, the belt height by h, and the belt area by A = w × h. (a) and (b) plot the TM mode intensities along the resonator edge for different protrusion dimensions. For (a), A was kept constant while w/h was varied. For (b), w/h was kept constant while A was varied.

Fig. 8
Fig. 8

Simulation of the TM fundamental mode with comparisons to the TE fundamental mode. The simulated wavelength was 1.55 μm with a 1337 μm radius MgF2 resonator. The resonator had a Gaussian protrusion with a full width at half maximum of 10 μm and a height of 5 μm. The simulated TM mode intensity distribution (linear scale) is given in (a) and its EZ-component is shown in (b). The intensity along the arc of the resonator edge is given in (c) for both the TE and TM modes. The plot in (d) shows the ratio of the intensity of the TM fundamental mode over the intensity of the TE fundamental mode along the resonator edge.

Fig. 9
Fig. 9

Simulation results where the resonator cylinder angle was varied. We used a wavelength of 1.55 μm for a 750 μm radius MgF2 resonator. The belt was 7 by 5 μm. (a)–(d) show the electric field Z-component of the TM fundamental modes, at edge angles relative to the Z-axis of (a) 0 degrees, (b) 5 degrees, (c) 10 degrees, and (d) 15 degrees. (e) and (f) respectively plot the TE and TM mode intensities along the resonator edge for different edge angles.

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