Abstract

A novel single-longitudinal-mode (SLM) fiber laser with a single cylindrical microresonator (CMR) as the filter is proposed. The CMR has a nanoscale variation in radius, in which high-Q localized modes interfere with continuum radiative modes, generating a peak-like transmission spectrum based on the Fano resonance effect. The peak linewidth is sufficiently small to select only one SLM oscillation in a fiber cavity. In an Er-doped fiber laser coupled with the CMR, we achieve a stable SLM laser operation by optimizing the coupling location of the tapered fiber. Stable lasing at 1568 nm with a 3-dB linewidth of ∼15 kHz is obtained. The laser slope efficiency reduction introduced by the CMR for SLM lasing is limited to 1.4%. This scheme can be a competitive approach for the development of compact, highly efficient, and cost-effective SLM fiber lasers.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2017 (5)

G. Lin, A. Coillet, and Y. K. Chembo, “Nonlinear photonics with high-Q whispering-gallery-mode resonators,” Adv. Opt. Photonics 9(4), 828–890 (2017).
[Crossref]

H. Wang, X. Fan, Z. Li, T. Tang, F. Wu, D. Shen, and X. Wu, “Stabilizing and tuning the laser frequencies in Er-doped fiber ring lasers based on microbubble resonators,” IEEE Photonics J. 9, 1502509 (2017).
[Crossref]

M. F. Limonov, M. V. Rybin, A. N. Poddubny, and Y. S. Kivshar, “Fano resonances in photonics,” Nat. Photonics 11(9), 543–554 (2017).
[Crossref]

S. Fu, W. Shi, Y. Feng, L. Zhang, Z. Yang, S. Xu, X. Zhu, R. A. Norwood, and N. Peyghambarian, “Review of recent progress on single-frequency fiber lasers,” J. Opt. Soc. Am. B 34(3), A49–A62 (2017).
[Crossref]

K. Zhang, Y. Wang, and Y. Wu, “Enhanced Fano resonance in a non-adiabatic tapered fiber coupled with a microresonator,” Opt. Lett. 42(15), 2956–2959 (2017).
[Crossref]

2016 (1)

Y. Miao, Y. Peng, Y. Xiang, M. Li, Y. Lu, and Y. Song, “Dynamic Fano resonance in thin fiber taper coupled cylindrical microcavity,” IEEE Photonics J. 8(6), 1–6 (2016).
[Crossref]

2015 (2)

2014 (1)

2013 (2)

2012 (3)

B. Li, Y. Xiao, C. Zou, X. Jiang, Y. Liu, F. Sun, Y. Li, and Q. Gong, “Experimental controlling of Fano resonance in indirectly coupled whispering-gallery microresonators,” Appl. Phys. Lett. 100(2), 021108 (2012).
[Crossref]

Z. Tian, P. Bianucci, and D. V. Plant, “Fiber ring laser using optical fiber microdisk as reflection mirror,” IEEE Photonics Technol. Lett. 24, 1396–1398 (2012).
[Crossref]

F. Monifi, J. Friedlein, Ş. Özdemir, and L. Yang, “A robust and tunable add–drop filter using whispering gallery mode microtoroid resonator,” J. Lightwave Technol. 30(21), 3306–3315 (2012).
[Crossref]

2011 (2)

B. Li, Y. Xiao, C. Zou, Y. Liu, X. Jiang, Y. Chen, Y. Li, and Q. Gong, “Experimental observation of Fano resonance in a single whispering-gallery microresonator,” Appl. Phys. Lett. 98(2), 021116 (2011).
[Crossref]

J. Ward and O. Benson, “WGM microresonators: sensing, lasing and fundamental optics with microspheres,” Laser Photonics Rev. 5(4), 553–570 (2011).
[Crossref]

2010 (1)

2009 (2)

B. Sprenger, H. G. L. Schwefel, and L. J. Wang, “Whispering-gallery-mode-resonator-stabilized narrow-linewidth fiber loop laser,” Opt. Lett. 34(21), 3370–3372 (2009).
[Crossref]

Y. Xiao, L. He, J. Zhu, and L. Yang, “Electromagnetically induced transparency-like effect in a single polydimethylsiloxane-coated silica microtoroid,” Appl. Phys. Lett. 94(23), 231115 (2009).
[Crossref]

2007 (1)

2005 (1)

Y. Louyer, D. Meschede, and A. Rauschenbeutel, “Tunable whispering-gallery-mode resonators for cavity quantum electrodynamics,” Phys. Rev. A 72(3), 031801 (2005).
[Crossref]

2003 (2)

K. J. Vahala, “Optical microcavities,” Nature 424(6950), 839–846 (2003).
[Crossref]

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber taper-coupled micro-resonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91(4), 043902 (2003).
[Crossref]

Andrés, M. V.

Benson, O.

J. Ward and O. Benson, “WGM microresonators: sensing, lasing and fundamental optics with microspheres,” Laser Photonics Rev. 5(4), 553–570 (2011).
[Crossref]

Bianucci, P.

Z. Tian, P. Bianucci, and D. V. Plant, “Fiber ring laser using optical fiber microdisk as reflection mirror,” IEEE Photonics Technol. Lett. 24, 1396–1398 (2012).
[Crossref]

Chembo, Y. K.

G. Lin, A. Coillet, and Y. K. Chembo, “Nonlinear photonics with high-Q whispering-gallery-mode resonators,” Adv. Opt. Photonics 9(4), 828–890 (2017).
[Crossref]

Chen, Y.

B. Li, Y. Xiao, C. Zou, Y. Liu, X. Jiang, Y. Chen, Y. Li, and Q. Gong, “Experimental observation of Fano resonance in a single whispering-gallery microresonator,” Appl. Phys. Lett. 98(2), 021116 (2011).
[Crossref]

Coillet, A.

G. Lin, A. Coillet, and Y. K. Chembo, “Nonlinear photonics with high-Q whispering-gallery-mode resonators,” Adv. Opt. Photonics 9(4), 828–890 (2017).
[Crossref]

Collodo, M. C.

Cruz, J. L.

Díez, A.

Dong, Y.

Fan, X.

H. Wang, X. Fan, Z. Li, T. Tang, F. Wu, D. Shen, and X. Wu, “Stabilizing and tuning the laser frequencies in Er-doped fiber ring lasers based on microbubble resonators,” IEEE Photonics J. 9, 1502509 (2017).
[Crossref]

Feng, Y.

Friedlein, J.

Fu, S.

Gong, Q.

B. Li, Y. Xiao, C. Zou, X. Jiang, Y. Liu, F. Sun, Y. Li, and Q. Gong, “Experimental controlling of Fano resonance in indirectly coupled whispering-gallery microresonators,” Appl. Phys. Lett. 100(2), 021108 (2012).
[Crossref]

B. Li, Y. Xiao, C. Zou, Y. Liu, X. Jiang, Y. Chen, Y. Li, and Q. Gong, “Experimental observation of Fano resonance in a single whispering-gallery microresonator,” Appl. Phys. Lett. 98(2), 021116 (2011).
[Crossref]

He, L.

L. He, S. K. Ozdemir, and L. Yan, “Whispering gallery microcavity lasers,” Laser Photonics Rev. 7(1), 60–82 (2013).
[Crossref]

Y. Xiao, L. He, J. Zhu, and L. Yang, “Electromagnetically induced transparency-like effect in a single polydimethylsiloxane-coated silica microtoroid,” Appl. Phys. Lett. 94(23), 231115 (2009).
[Crossref]

Jiang, X.

B. Li, Y. Xiao, C. Zou, X. Jiang, Y. Liu, F. Sun, Y. Li, and Q. Gong, “Experimental controlling of Fano resonance in indirectly coupled whispering-gallery microresonators,” Appl. Phys. Lett. 100(2), 021108 (2012).
[Crossref]

B. Li, Y. Xiao, C. Zou, Y. Liu, X. Jiang, Y. Chen, Y. Li, and Q. Gong, “Experimental observation of Fano resonance in a single whispering-gallery microresonator,” Appl. Phys. Lett. 98(2), 021116 (2011).
[Crossref]

Jin, X.

Kieu, K.

Kippenberg, T. J.

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber taper-coupled micro-resonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91(4), 043902 (2003).
[Crossref]

Kivshar, Y. S.

M. F. Limonov, M. V. Rybin, A. N. Poddubny, and Y. S. Kivshar, “Fano resonances in photonics,” Nat. Photonics 11(9), 543–554 (2017).
[Crossref]

Li, B.

B. Li, Y. Xiao, C. Zou, X. Jiang, Y. Liu, F. Sun, Y. Li, and Q. Gong, “Experimental controlling of Fano resonance in indirectly coupled whispering-gallery microresonators,” Appl. Phys. Lett. 100(2), 021108 (2012).
[Crossref]

B. Li, Y. Xiao, C. Zou, Y. Liu, X. Jiang, Y. Chen, Y. Li, and Q. Gong, “Experimental observation of Fano resonance in a single whispering-gallery microresonator,” Appl. Phys. Lett. 98(2), 021116 (2011).
[Crossref]

Li, M.

Y. Miao, Y. Peng, Y. Xiang, M. Li, Y. Lu, and Y. Song, “Dynamic Fano resonance in thin fiber taper coupled cylindrical microcavity,” IEEE Photonics J. 8(6), 1–6 (2016).
[Crossref]

Li, Y.

B. Li, Y. Xiao, C. Zou, X. Jiang, Y. Liu, F. Sun, Y. Li, and Q. Gong, “Experimental controlling of Fano resonance in indirectly coupled whispering-gallery microresonators,” Appl. Phys. Lett. 100(2), 021108 (2012).
[Crossref]

B. Li, Y. Xiao, C. Zou, Y. Liu, X. Jiang, Y. Chen, Y. Li, and Q. Gong, “Experimental observation of Fano resonance in a single whispering-gallery microresonator,” Appl. Phys. Lett. 98(2), 021116 (2011).
[Crossref]

Li, Z.

H. Wang, X. Fan, Z. Li, T. Tang, F. Wu, D. Shen, and X. Wu, “Stabilizing and tuning the laser frequencies in Er-doped fiber ring lasers based on microbubble resonators,” IEEE Photonics J. 9, 1502509 (2017).
[Crossref]

Limonov, M. F.

M. F. Limonov, M. V. Rybin, A. N. Poddubny, and Y. S. Kivshar, “Fano resonances in photonics,” Nat. Photonics 11(9), 543–554 (2017).
[Crossref]

Lin, G.

G. Lin, A. Coillet, and Y. K. Chembo, “Nonlinear photonics with high-Q whispering-gallery-mode resonators,” Adv. Opt. Photonics 9(4), 828–890 (2017).
[Crossref]

Liu, Y.

B. Li, Y. Xiao, C. Zou, X. Jiang, Y. Liu, F. Sun, Y. Li, and Q. Gong, “Experimental controlling of Fano resonance in indirectly coupled whispering-gallery microresonators,” Appl. Phys. Lett. 100(2), 021108 (2012).
[Crossref]

B. Li, Y. Xiao, C. Zou, Y. Liu, X. Jiang, Y. Chen, Y. Li, and Q. Gong, “Experimental observation of Fano resonance in a single whispering-gallery microresonator,” Appl. Phys. Lett. 98(2), 021116 (2011).
[Crossref]

Louyer, Y.

Y. Louyer, D. Meschede, and A. Rauschenbeutel, “Tunable whispering-gallery-mode resonators for cavity quantum electrodynamics,” Phys. Rev. A 72(3), 031801 (2005).
[Crossref]

Lu, Y.

Y. Miao, Y. Peng, Y. Xiang, M. Li, Y. Lu, and Y. Song, “Dynamic Fano resonance in thin fiber taper coupled cylindrical microcavity,” IEEE Photonics J. 8(6), 1–6 (2016).
[Crossref]

Mansuripur, M.

Meschede, D.

Y. Louyer, D. Meschede, and A. Rauschenbeutel, “Tunable whispering-gallery-mode resonators for cavity quantum electrodynamics,” Phys. Rev. A 72(3), 031801 (2005).
[Crossref]

Miao, Y.

Y. Miao, Y. Peng, Y. Xiang, M. Li, Y. Lu, and Y. Song, “Dynamic Fano resonance in thin fiber taper coupled cylindrical microcavity,” IEEE Photonics J. 8(6), 1–6 (2016).
[Crossref]

Monifi, F.

Norwood, R. A.

Ozdemir, S. K.

L. He, S. K. Ozdemir, and L. Yan, “Whispering gallery microcavity lasers,” Laser Photonics Rev. 7(1), 60–82 (2013).
[Crossref]

Özdemir, S.

Painter, O. J.

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber taper-coupled micro-resonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91(4), 043902 (2003).
[Crossref]

Peng, Y.

Y. Miao, Y. Peng, Y. Xiang, M. Li, Y. Lu, and Y. Song, “Dynamic Fano resonance in thin fiber taper coupled cylindrical microcavity,” IEEE Photonics J. 8(6), 1–6 (2016).
[Crossref]

Peyghambarian, N.

Plant, D. V.

Z. Tian, P. Bianucci, and D. V. Plant, “Fiber ring laser using optical fiber microdisk as reflection mirror,” IEEE Photonics Technol. Lett. 24, 1396–1398 (2012).
[Crossref]

Poddubny, A. N.

M. F. Limonov, M. V. Rybin, A. N. Poddubny, and Y. S. Kivshar, “Fano resonances in photonics,” Nat. Photonics 11(9), 543–554 (2017).
[Crossref]

Rauschenbeutel, A.

Y. Louyer, D. Meschede, and A. Rauschenbeutel, “Tunable whispering-gallery-mode resonators for cavity quantum electrodynamics,” Phys. Rev. A 72(3), 031801 (2005).
[Crossref]

Rivera-Pérez, E.

Rodríguez-Cobos, A.

Rybin, M. V.

M. F. Limonov, M. V. Rybin, A. N. Poddubny, and Y. S. Kivshar, “Fano resonances in photonics,” Nat. Photonics 11(9), 543–554 (2017).
[Crossref]

Schwefel, H. G. L.

Sedlmeir, F.

Shen, D.

H. Wang, X. Fan, Z. Li, T. Tang, F. Wu, D. Shen, and X. Wu, “Stabilizing and tuning the laser frequencies in Er-doped fiber ring lasers based on microbubble resonators,” IEEE Photonics J. 9, 1502509 (2017).
[Crossref]

Shi, W.

Song, Y.

Y. Miao, Y. Peng, Y. Xiang, M. Li, Y. Lu, and Y. Song, “Dynamic Fano resonance in thin fiber taper coupled cylindrical microcavity,” IEEE Photonics J. 8(6), 1–6 (2016).
[Crossref]

Spillane, S. M.

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber taper-coupled micro-resonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91(4), 043902 (2003).
[Crossref]

Sprenger, B.

Sumetsky, M.

Sun, F.

B. Li, Y. Xiao, C. Zou, X. Jiang, Y. Liu, F. Sun, Y. Li, and Q. Gong, “Experimental controlling of Fano resonance in indirectly coupled whispering-gallery microresonators,” Appl. Phys. Lett. 100(2), 021108 (2012).
[Crossref]

Svitlov, S.

Tang, T.

H. Wang, X. Fan, Z. Li, T. Tang, F. Wu, D. Shen, and X. Wu, “Stabilizing and tuning the laser frequencies in Er-doped fiber ring lasers based on microbubble resonators,” IEEE Photonics J. 9, 1502509 (2017).
[Crossref]

Tian, Z.

Z. Tian, P. Bianucci, and D. V. Plant, “Fiber ring laser using optical fiber microdisk as reflection mirror,” IEEE Photonics Technol. Lett. 24, 1396–1398 (2012).
[Crossref]

Vahala, K. J.

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber taper-coupled micro-resonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91(4), 043902 (2003).
[Crossref]

K. J. Vahala, “Optical microcavities,” Nature 424(6950), 839–846 (2003).
[Crossref]

Wang, H.

H. Wang, X. Fan, Z. Li, T. Tang, F. Wu, D. Shen, and X. Wu, “Stabilizing and tuning the laser frequencies in Er-doped fiber ring lasers based on microbubble resonators,” IEEE Photonics J. 9, 1502509 (2017).
[Crossref]

Wang, K.

Wang, L. J.

Wang, Y.

Ward, J.

J. Ward and O. Benson, “WGM microresonators: sensing, lasing and fundamental optics with microspheres,” Laser Photonics Rev. 5(4), 553–570 (2011).
[Crossref]

Wu, F.

H. Wang, X. Fan, Z. Li, T. Tang, F. Wu, D. Shen, and X. Wu, “Stabilizing and tuning the laser frequencies in Er-doped fiber ring lasers based on microbubble resonators,” IEEE Photonics J. 9, 1502509 (2017).
[Crossref]

Wu, X.

H. Wang, X. Fan, Z. Li, T. Tang, F. Wu, D. Shen, and X. Wu, “Stabilizing and tuning the laser frequencies in Er-doped fiber ring lasers based on microbubble resonators,” IEEE Photonics J. 9, 1502509 (2017).
[Crossref]

Wu, Y.

Xiang, Y.

Y. Miao, Y. Peng, Y. Xiang, M. Li, Y. Lu, and Y. Song, “Dynamic Fano resonance in thin fiber taper coupled cylindrical microcavity,” IEEE Photonics J. 8(6), 1–6 (2016).
[Crossref]

Xiao, Y.

B. Li, Y. Xiao, C. Zou, X. Jiang, Y. Liu, F. Sun, Y. Li, and Q. Gong, “Experimental controlling of Fano resonance in indirectly coupled whispering-gallery microresonators,” Appl. Phys. Lett. 100(2), 021108 (2012).
[Crossref]

B. Li, Y. Xiao, C. Zou, Y. Liu, X. Jiang, Y. Chen, Y. Li, and Q. Gong, “Experimental observation of Fano resonance in a single whispering-gallery microresonator,” Appl. Phys. Lett. 98(2), 021116 (2011).
[Crossref]

Y. Xiao, L. He, J. Zhu, and L. Yang, “Electromagnetically induced transparency-like effect in a single polydimethylsiloxane-coated silica microtoroid,” Appl. Phys. Lett. 94(23), 231115 (2009).
[Crossref]

Xu, S.

Yan, L.

L. He, S. K. Ozdemir, and L. Yan, “Whispering gallery microcavity lasers,” Laser Photonics Rev. 7(1), 60–82 (2013).
[Crossref]

Yang, L.

F. Monifi, J. Friedlein, Ş. Özdemir, and L. Yang, “A robust and tunable add–drop filter using whispering gallery mode microtoroid resonator,” J. Lightwave Technol. 30(21), 3306–3315 (2012).
[Crossref]

Y. Xiao, L. He, J. Zhu, and L. Yang, “Electromagnetically induced transparency-like effect in a single polydimethylsiloxane-coated silica microtoroid,” Appl. Phys. Lett. 94(23), 231115 (2009).
[Crossref]

Yang, Z.

Zhang, K.

Zhang, L.

Zhu, J.

Y. Xiao, L. He, J. Zhu, and L. Yang, “Electromagnetically induced transparency-like effect in a single polydimethylsiloxane-coated silica microtoroid,” Appl. Phys. Lett. 94(23), 231115 (2009).
[Crossref]

Zhu, X.

Zou, C.

B. Li, Y. Xiao, C. Zou, X. Jiang, Y. Liu, F. Sun, Y. Li, and Q. Gong, “Experimental controlling of Fano resonance in indirectly coupled whispering-gallery microresonators,” Appl. Phys. Lett. 100(2), 021108 (2012).
[Crossref]

B. Li, Y. Xiao, C. Zou, Y. Liu, X. Jiang, Y. Chen, Y. Li, and Q. Gong, “Experimental observation of Fano resonance in a single whispering-gallery microresonator,” Appl. Phys. Lett. 98(2), 021116 (2011).
[Crossref]

Adv. Opt. Photonics (1)

G. Lin, A. Coillet, and Y. K. Chembo, “Nonlinear photonics with high-Q whispering-gallery-mode resonators,” Adv. Opt. Photonics 9(4), 828–890 (2017).
[Crossref]

Appl. Opt. (2)

Appl. Phys. Lett. (3)

B. Li, Y. Xiao, C. Zou, X. Jiang, Y. Liu, F. Sun, Y. Li, and Q. Gong, “Experimental controlling of Fano resonance in indirectly coupled whispering-gallery microresonators,” Appl. Phys. Lett. 100(2), 021108 (2012).
[Crossref]

Y. Xiao, L. He, J. Zhu, and L. Yang, “Electromagnetically induced transparency-like effect in a single polydimethylsiloxane-coated silica microtoroid,” Appl. Phys. Lett. 94(23), 231115 (2009).
[Crossref]

B. Li, Y. Xiao, C. Zou, Y. Liu, X. Jiang, Y. Chen, Y. Li, and Q. Gong, “Experimental observation of Fano resonance in a single whispering-gallery microresonator,” Appl. Phys. Lett. 98(2), 021116 (2011).
[Crossref]

IEEE Photonics J. (2)

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Supplementary Material (3)

NameDescription
» Visualization 1       Dynamic signals in the F-P interferometer for the laser at free running
» Visualization 2       Dynamic signals in the F-P interferometer for the laser at mode hopping
» Visualization 3       Dynamic signals in the F-P interferometer for the laser at stable SLM

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Figures (6)

Fig. 1.
Fig. 1. (a) Excitation of only WGMs in the CMR. The transmission spectrum demonstrates a symmetric Lorentzian lineshape. (b) Excitation of both WGMs and radiation modes in the CMR. The transmission spectrum demonstrates an asymmetric Fano lineshape. (c)-(g) Field distributions of WGMs in the CMR with the quantum numbers [m, q, p] of [354, 1, 1], [343, 2, 5], [335, 3, 25], [354, 1, 15] and [335, 3, 1], respectively. (h) Resonance wavelength as a function of the axial quantum number.
Fig. 2.
Fig. 2. Evolution of the transmission spectrum of the CMR. The mode spectrum becomes less dense as the tapered fiber is moved away from the origin along the z-axis. However, the Q-factors of WGMs are reduced if the tapered fiber is far away from the origin (d), owing to that it needs a smaller coupling gap, which yields an increased loss for the WGMs.
Fig. 3.
Fig. 3. Zoom-in-image of one of WGMs at 1555.5 nm in Fig. 2(c). The mode evolve from the symmetric Lorentz lineshape (a)–(e) to the asymmetrical Fano-resonance lineshape (f)–(j) with the decrease in coupling gap. (k)-(o) Simulation results corresponding to the experimental results in (f)-(j), respectively.
Fig. 4.
Fig. 4. Experimental setup for the analysis of filter function of the CMR for the SLM Er-doped fiber laser. ISO: isolator, EDF: Er-doped fiber, WDM: wavelength division multiplexer. FPI: Fabry-Pérot interferometer, OSA: optical spectrum analyzer, ESA: electrical spectrum analyzer. The red and blue lines indicate the optical path used to tune on the fiber laser and test path for the transmission spectrum of the CMR when tuning off the fiber laser, respectively.
Fig. 5.
Fig. 5. (a) Fano resonance transmission spectrum when the tapered fiber is at the initial position and (b) cleaner spectrum after the optimization of the fiber location. (c), (d) are the signals in the ESA when the fiber laser is operated at the MLM and SLM, respectively. (e) Signals of the F–P interferometer for stable SLM lasing, Visualization 1, Visualization 2 and Visualization 3 show the dynamic signals in the F–P interferometer at free running, mode hopping and stable SLM lasing regimes, respectively. (f) Spectrum of the stable SLM lasing in the OSA. (g) Output power as a function of the pump power in three different cases: free running, SLM operation, and tapered fiber in touch with the microresonator.
Fig. 6.
Fig. 6. Heterodyne signal of the SLM fiber laser. The red line is the Lorentzian fitting of the measured data.

Equations (3)

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λ m , p , q = 2 π n e f f R 0 m 2 + m ( 2 p + 1 ) Δ k R 0 .
Δ λ m λ m + 1 , p , q λ m , p , q = λ m , p , q 2 2 π R 0 n e f f .
Δ λ p λ m , p , q λ m , p + 1 , q = Δ k λ m , p , q 2 2 π n e f f .

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