Abstract

We report a novel cavity feedback mechanism for stabilizing single-longitudinal-mode (SLM) operation of diamond Raman lasers. Polarization-dependent Raman gain and in-grown stress birefringence in diamond were investigated as sources for Hänsch-Couillaud-type locking signals. The power range of SLM operation increased from 2.1 W to 7.2 W, compared to the free-running laser, for a simple standing-wave laser cavity without frequency-selective elements. Methods for further increasing power range and frequency stability are discussed.

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

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References

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    [Crossref]

2019 (1)

2017 (1)

2016 (2)

2015 (5)

2013 (1)

O. Lux, H. Fritsche, and H. J. Eichler, “Trace Gas Remote Sensing by Lasers,” Opt. Photonik 8(4), 48–51 (2013).
[Crossref]

2012 (1)

2011 (2)

P. Asenbaum and M. Arndt, “Cavity stabilization using the weak intrinsic birefringence of dielectric mirrors,” Opt. Lett. 36(19), 3720–3722 (2011).
[Crossref]

M. Vainio, J. E. Bernard, and L. Marmet, “Cavity-enhanced optical frequency doubler based on transmission-mode Hänsch–Couillaud locking,” Appl. Phys. B 104(4), 897–908 (2011).
[Crossref]

2010 (5)

2009 (1)

I. Friel, S. Clewes, H. Dhillon, N. Perkins, D. Twitchen, and G. Scarsbrook, “Control of surface and bulk crystalline quality in single crystal diamond grown by chemical vapour deposition,” Diamond Relat. Mater. 18(5-8), 808–815 (2009).
[Crossref]

2007 (1)

2005 (1)

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave Raman silicon laser,” Nature 433(7027), 725–728 (2005).
[Crossref]

1999 (1)

1998 (1)

1997 (1)

J. Boon-Engering, W. Van der Veer, E. Bente, and W. Hogervorst, “Stabilization of an optical cavity containing a birefringent element,” Opt. Commun. 140(4-6), 285–288 (1997).
[Crossref]

1993 (1)

N. Bobroff, “Recent advances in displacement measuring interferometry,” Meas. Sci. Technol. 4(9), 907–926 (1993).
[Crossref]

1988 (1)

1987 (1)

1980 (1)

T. Hansch and B. Couillaud, “Laser frequency stabilization by polarization spectroscopy of a reflecting reference cavity,” Opt. Commun. 35(3), 441–444 (1980).
[Crossref]

1975 (1)

T. W. Hänsch and A. L. Schawlow, “Cooling of gases by laser radiation,” Opt. Commun. 13(1), 68–69 (1975).
[Crossref]

Aasi, J.

J. Aasi, B. Abbott, R. Abbott, T. Abbott, M. Abernathy, K. Ackley, C. Adams, T. Adams, P. Addesso, and R. Adhikari, “Advanced ligo,” Class. Quantum Gravity 32(11), 115012 (2015).
[Crossref]

Abbott, B.

J. Aasi, B. Abbott, R. Abbott, T. Abbott, M. Abernathy, K. Ackley, C. Adams, T. Adams, P. Addesso, and R. Adhikari, “Advanced ligo,” Class. Quantum Gravity 32(11), 115012 (2015).
[Crossref]

Abbott, R.

J. Aasi, B. Abbott, R. Abbott, T. Abbott, M. Abernathy, K. Ackley, C. Adams, T. Adams, P. Addesso, and R. Adhikari, “Advanced ligo,” Class. Quantum Gravity 32(11), 115012 (2015).
[Crossref]

Abbott, T.

J. Aasi, B. Abbott, R. Abbott, T. Abbott, M. Abernathy, K. Ackley, C. Adams, T. Adams, P. Addesso, and R. Adhikari, “Advanced ligo,” Class. Quantum Gravity 32(11), 115012 (2015).
[Crossref]

Abernathy, M.

J. Aasi, B. Abbott, R. Abbott, T. Abbott, M. Abernathy, K. Ackley, C. Adams, T. Adams, P. Addesso, and R. Adhikari, “Advanced ligo,” Class. Quantum Gravity 32(11), 115012 (2015).
[Crossref]

Ackley, K.

J. Aasi, B. Abbott, R. Abbott, T. Abbott, M. Abernathy, K. Ackley, C. Adams, T. Adams, P. Addesso, and R. Adhikari, “Advanced ligo,” Class. Quantum Gravity 32(11), 115012 (2015).
[Crossref]

Adams, C.

J. Aasi, B. Abbott, R. Abbott, T. Abbott, M. Abernathy, K. Ackley, C. Adams, T. Adams, P. Addesso, and R. Adhikari, “Advanced ligo,” Class. Quantum Gravity 32(11), 115012 (2015).
[Crossref]

Adams, T.

J. Aasi, B. Abbott, R. Abbott, T. Abbott, M. Abernathy, K. Ackley, C. Adams, T. Adams, P. Addesso, and R. Adhikari, “Advanced ligo,” Class. Quantum Gravity 32(11), 115012 (2015).
[Crossref]

Addesso, P.

J. Aasi, B. Abbott, R. Abbott, T. Abbott, M. Abernathy, K. Ackley, C. Adams, T. Adams, P. Addesso, and R. Adhikari, “Advanced ligo,” Class. Quantum Gravity 32(11), 115012 (2015).
[Crossref]

Adhikari, R.

J. Aasi, B. Abbott, R. Abbott, T. Abbott, M. Abernathy, K. Ackley, C. Adams, T. Adams, P. Addesso, and R. Adhikari, “Advanced ligo,” Class. Quantum Gravity 32(11), 115012 (2015).
[Crossref]

Arndt, M.

Asenbaum, P.

Bai, Z.

Bente, E.

J. Boon-Engering, W. Van der Veer, E. Bente, and W. Hogervorst, “Stabilization of an optical cavity containing a birefringent element,” Opt. Commun. 140(4-6), 285–288 (1997).
[Crossref]

Bernard, J. E.

M. Vainio, J. E. Bernard, and L. Marmet, “Cavity-enhanced optical frequency doubler based on transmission-mode Hänsch–Couillaud locking,” Appl. Phys. B 104(4), 897–908 (2011).
[Crossref]

Bobroff, N.

N. Bobroff, “Recent advances in displacement measuring interferometry,” Meas. Sci. Technol. 4(9), 907–926 (1993).
[Crossref]

Bonnell, L. J.

Boon-Engering, J.

J. Boon-Engering, W. Van der Veer, E. Bente, and W. Hogervorst, “Stabilization of an optical cavity containing a birefringent element,” Opt. Commun. 140(4-6), 285–288 (1997).
[Crossref]

Brasseur, J.

Bretenaker, F.

Briles, T. C.

Brown, N.

Buikema, A.

Bulu, I.

Burek, M. J.

Calia, D. B.

D. B. Calia, Y. Feng, W. Hackenberg, R. Holzlöhner, L. Taylor, and S. Lewis, “Laser development for sodium laser guide stars at ESO,” The Messenger 139, 12–19 (2010).

Carlsten, J.

Cassidy, D. T.

Chang, C.

Chen, Y.

Cho, C.

Chow, J. H.

T. T.-Y. Lam, B. J. Slagmolen, J. H. Chow, I. C. Littler, D. E. McClelland, and D. A. Shaddock, “Digital laser frequency stabilization using an optical cavity,” IEEE J. Quantum Electron. 46(8), 1178–1183 (2010).
[Crossref]

Cingöz, A.

Clewes, S.

I. Friel, S. Clewes, H. Dhillon, N. Perkins, D. Twitchen, and G. Scarsbrook, “Control of surface and bulk crystalline quality in single crystal diamond grown by chemical vapour deposition,” Diamond Relat. Mater. 18(5-8), 808–815 (2009).
[Crossref]

Cohen, O.

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave Raman silicon laser,” Nature 433(7027), 725–728 (2005).
[Crossref]

Cong, Z.

Z. Liu, S. Men, Z. Cong, Y. Liu, H. Rao, S. Zhang, and X. Zhang, “Single-frequency Nd: GGG/BaWO4 Raman laser emitting at 1178.3 nm,” in CLEO: Sci. Innov., (Optical Society of America, 2016), SM3M. 3.

Couillaud, B.

T. Hansch and B. Couillaud, “Laser frequency stabilization by polarization spectroscopy of a reflecting reference cavity,” Opt. Commun. 35(3), 441–444 (1980).
[Crossref]

Dhillon, H.

I. Friel, S. Clewes, H. Dhillon, N. Perkins, D. Twitchen, and G. Scarsbrook, “Control of surface and bulk crystalline quality in single crystal diamond grown by chemical vapour deposition,” Diamond Relat. Mater. 18(5-8), 808–815 (2009).
[Crossref]

Dordevic, T.

Drag, C.

Eichler, H. J.

O. Lux, H. Fritsche, and H. J. Eichler, “Trace Gas Remote Sensing by Lasers,” Opt. Photonik 8(4), 48–51 (2013).
[Crossref]

Esherick, P.

Evans, M.

Fang, A.

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave Raman silicon laser,” Nature 433(7027), 725–728 (2005).
[Crossref]

Feng, Y.

Friel, I.

I. Friel, S. Clewes, H. Dhillon, N. Perkins, D. Twitchen, and G. Scarsbrook, “Control of surface and bulk crystalline quality in single crystal diamond grown by chemical vapour deposition,” Diamond Relat. Mater. 18(5-8), 808–815 (2009).
[Crossref]

Fritsche, H.

O. Lux, H. Fritsche, and H. J. Eichler, “Trace Gas Remote Sensing by Lasers,” Opt. Photonik 8(4), 48–51 (2013).
[Crossref]

Gray, M.

Hackenberg, W.

D. B. Calia, Y. Feng, W. Hackenberg, R. Holzlöhner, L. Taylor, and S. Lewis, “Laser development for sodium laser guide stars at ESO,” The Messenger 139, 12–19 (2010).

Hak, D.

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave Raman silicon laser,” Nature 433(7027), 725–728 (2005).
[Crossref]

Hansch, T.

T. Hansch and B. Couillaud, “Laser frequency stabilization by polarization spectroscopy of a reflecting reference cavity,” Opt. Commun. 35(3), 441–444 (1980).
[Crossref]

Hänsch, T. W.

T. W. Hänsch and A. L. Schawlow, “Cooling of gases by laser radiation,” Opt. Commun. 13(1), 68–69 (1975).
[Crossref]

Hausmann, B. J. M.

Heising, M.

Hempler, N.

Hogervorst, W.

J. Boon-Engering, W. Van der Veer, E. Bente, and W. Hogervorst, “Stabilization of an optical cavity containing a birefringent element,” Opt. Commun. 140(4-6), 285–288 (1997).
[Crossref]

Holzlöhner, R.

D. B. Calia, Y. Feng, W. Hackenberg, R. Holzlöhner, L. Taylor, and S. Lewis, “Laser development for sodium laser guide stars at ESO,” The Messenger 139, 12–19 (2010).

Huang, K.

Jasbeer, H.

Jones, R.

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave Raman silicon laser,” Nature 433(7027), 725–728 (2005).
[Crossref]

Kitzler, O.

Lam, T. T.-Y.

T. T.-Y. Lam, B. J. Slagmolen, J. H. Chow, I. C. Littler, D. E. McClelland, and D. A. Shaddock, “Digital laser frequency stabilization using an optical cavity,” IEEE J. Quantum Electron. 46(8), 1178–1183 (2010).
[Crossref]

Latawiec, P.

Lee, C.

Lewis, S.

D. B. Calia, Y. Feng, W. Hackenberg, R. Holzlöhner, L. Taylor, and S. Lewis, “Laser development for sodium laser guide stars at ESO,” The Messenger 139, 12–19 (2010).

Libson, A.

Lin, J.

Littler, I. C.

T. T.-Y. Lam, B. J. Slagmolen, J. H. Chow, I. C. Littler, D. E. McClelland, and D. A. Shaddock, “Digital laser frequency stabilization using an optical cavity,” IEEE J. Quantum Electron. 46(8), 1178–1183 (2010).
[Crossref]

Liu, A.

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave Raman silicon laser,” Nature 433(7027), 725–728 (2005).
[Crossref]

Liu, Y.

Z. Liu, S. Men, Z. Cong, Y. Liu, H. Rao, S. Zhang, and X. Zhang, “Single-frequency Nd: GGG/BaWO4 Raman laser emitting at 1178.3 nm,” in CLEO: Sci. Innov., (Optical Society of America, 2016), SM3M. 3.

Liu, Z.

Z. Liu, S. Men, Z. Cong, Y. Liu, H. Rao, S. Zhang, and X. Zhang, “Single-frequency Nd: GGG/BaWO4 Raman laser emitting at 1178.3 nm,” in CLEO: Sci. Innov., (Optical Society of America, 2016), SM3M. 3.

Loncar, M.

López, C. C.

Lux, O.

Malcolm, G. P.

Marmet, L.

M. Vainio, J. E. Bernard, and L. Marmet, “Cavity-enhanced optical frequency doubler based on transmission-mode Hänsch–Couillaud locking,” Appl. Phys. B 104(4), 897–908 (2011).
[Crossref]

Mayor, S. D.

McClelland, D.

McClelland, D. E.

T. T.-Y. Lam, B. J. Slagmolen, J. H. Chow, I. C. Littler, D. E. McClelland, and D. A. Shaddock, “Digital laser frequency stabilization using an optical cavity,” IEEE J. Quantum Electron. 46(8), 1178–1183 (2010).
[Crossref]

McKay, A.

Men, S.

Z. Liu, S. Men, Z. Cong, Y. Liu, H. Rao, S. Zhang, and X. Zhang, “Single-frequency Nd: GGG/BaWO4 Raman laser emitting at 1178.3 nm,” in CLEO: Sci. Innov., (Optical Society of America, 2016), SM3M. 3.

Mhibik, O.

Mildren, R. P.

X. Yang, O. Kitzler, D. J. Spence, R. J. Williams, Z. Bai, S. Sarang, L. Zhang, Y. Feng, and R. P. Mildren, “Single-frequency 620 nm diamond laser at high power, stabilized via harmonic self-suppression and spatial-hole-burning-free gain,” Opt. Lett. 44(4), 839–842 (2019).
[Crossref]

O. Kitzler, J. Lin, H. M. Pask, R. P. Mildren, S. C. Webster, N. Hempler, G. P. Malcolm, and D. J. Spence, “Single-longitudinal-mode ring diamond Raman laser,” Opt. Lett. 42(7), 1229–1232 (2017).
[Crossref]

O. Lux, S. Sarang, O. Kitzler, D. J. Spence, and R. P. Mildren, “Intrinsically stable high-power single longitudinal mode laser using spatial hole burning free gain,” Optica 3(8), 876–881 (2016).
[Crossref]

H. Jasbeer, R. J. Williams, O. Kitzler, A. McKay, S. Sarang, J. Lin, and R. P. Mildren, “Birefringence and piezo-Raman analysis of single crystal CVD diamond and effects on Raman laser performance,” J. Opt. Soc. Am. B 33(3), B56 (2016).
[Crossref]

O. Kitzler, A. McKay, D. J. Spence, and R. P. Mildren, “Modelling and optimization of continuous-wave external cavity Raman lasers,” Opt. Express 23(7), 8590–8602 (2015).
[Crossref]

O. Kitzler, A. McKay, and R. P. Mildren, “Continuous-wave wavelength conversion for high-power applications using an external cavity diamond Raman laser,” Opt. Lett. 37(14), 2790–2792 (2012).
[Crossref]

A. Sabella, J. A. Piper, and R. P. Mildren, “1240 nm diamond Raman laser operating near the quantum limit,” Opt. Lett. 35(23), 3874–3876 (2010).
[Crossref]

My, T. H.

Owyoung, A.

Pabœuf, D.

Paniccia, M.

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave Raman silicon laser,” Nature 433(7027), 725–728 (2005).
[Crossref]

Pask, H. M.

Perkins, N.

I. Friel, S. Clewes, H. Dhillon, N. Perkins, D. Twitchen, and G. Scarsbrook, “Control of surface and bulk crystalline quality in single crystal diamond grown by chemical vapour deposition,” Diamond Relat. Mater. 18(5-8), 808–815 (2009).
[Crossref]

Piper, J. A.

Rao, H.

Z. Liu, S. Men, Z. Cong, Y. Liu, H. Rao, S. Zhang, and X. Zhang, “Single-frequency Nd: GGG/BaWO4 Raman laser emitting at 1178.3 nm,” in CLEO: Sci. Innov., (Optical Society of America, 2016), SM3M. 3.

Repasky, K.

Rong, H.

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave Raman silicon laser,” Nature 433(7027), 725–728 (2005).
[Crossref]

Sabella, A.

Sarang, S.

Scarsbrook, G.

I. Friel, S. Clewes, H. Dhillon, N. Perkins, D. Twitchen, and G. Scarsbrook, “Control of surface and bulk crystalline quality in single crystal diamond grown by chemical vapour deposition,” Diamond Relat. Mater. 18(5-8), 808–815 (2009).
[Crossref]

Schawlow, A. L.

T. W. Hänsch and A. L. Schawlow, “Cooling of gases by laser radiation,” Opt. Commun. 13(1), 68–69 (1975).
[Crossref]

Schibli, T. R.

Shaddock, D.

Shaddock, D. A.

T. T.-Y. Lam, B. J. Slagmolen, J. H. Chow, I. C. Littler, D. E. McClelland, and D. A. Shaddock, “Digital laser frequency stabilization using an optical cavity,” IEEE J. Quantum Electron. 46(8), 1178–1183 (2010).
[Crossref]

Slagmolen, B. J.

T. T.-Y. Lam, B. J. Slagmolen, J. H. Chow, I. C. Littler, D. E. McClelland, and D. A. Shaddock, “Digital laser frequency stabilization using an optical cavity,” IEEE J. Quantum Electron. 46(8), 1178–1183 (2010).
[Crossref]

Spence, D. J.

Spuler, S. M.

Taylor, L.

D. B. Calia, Y. Feng, W. Hackenberg, R. Holzlöhner, L. Taylor, and S. Lewis, “Laser development for sodium laser guide stars at ESO,” The Messenger 139, 12–19 (2010).

Tuan, P.

Twitchen, D.

I. Friel, S. Clewes, H. Dhillon, N. Perkins, D. Twitchen, and G. Scarsbrook, “Control of surface and bulk crystalline quality in single crystal diamond grown by chemical vapour deposition,” Diamond Relat. Mater. 18(5-8), 808–815 (2009).
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M. Vainio, J. E. Bernard, and L. Marmet, “Cavity-enhanced optical frequency doubler based on transmission-mode Hänsch–Couillaud locking,” Appl. Phys. B 104(4), 897–908 (2011).
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J. Boon-Engering, W. Van der Veer, E. Bente, and W. Hogervorst, “Stabilization of an optical cavity containing a birefringent element,” Opt. Commun. 140(4-6), 285–288 (1997).
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Z. Liu, S. Men, Z. Cong, Y. Liu, H. Rao, S. Zhang, and X. Zhang, “Single-frequency Nd: GGG/BaWO4 Raman laser emitting at 1178.3 nm,” in CLEO: Sci. Innov., (Optical Society of America, 2016), SM3M. 3.

Zhang, X.

Z. Liu, S. Men, Z. Cong, Y. Liu, H. Rao, S. Zhang, and X. Zhang, “Single-frequency Nd: GGG/BaWO4 Raman laser emitting at 1178.3 nm,” in CLEO: Sci. Innov., (Optical Society of America, 2016), SM3M. 3.

Appl. Opt. (2)

Appl. Phys. B (1)

M. Vainio, J. E. Bernard, and L. Marmet, “Cavity-enhanced optical frequency doubler based on transmission-mode Hänsch–Couillaud locking,” Appl. Phys. B 104(4), 897–908 (2011).
[Crossref]

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J. Aasi, B. Abbott, R. Abbott, T. Abbott, M. Abernathy, K. Ackley, C. Adams, T. Adams, P. Addesso, and R. Adhikari, “Advanced ligo,” Class. Quantum Gravity 32(11), 115012 (2015).
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I. Friel, S. Clewes, H. Dhillon, N. Perkins, D. Twitchen, and G. Scarsbrook, “Control of surface and bulk crystalline quality in single crystal diamond grown by chemical vapour deposition,” Diamond Relat. Mater. 18(5-8), 808–815 (2009).
[Crossref]

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T. T.-Y. Lam, B. J. Slagmolen, J. H. Chow, I. C. Littler, D. E. McClelland, and D. A. Shaddock, “Digital laser frequency stabilization using an optical cavity,” IEEE J. Quantum Electron. 46(8), 1178–1183 (2010).
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Other (1)

Z. Liu, S. Men, Z. Cong, Y. Liu, H. Rao, S. Zhang, and X. Zhang, “Single-frequency Nd: GGG/BaWO4 Raman laser emitting at 1178.3 nm,” in CLEO: Sci. Innov., (Optical Society of America, 2016), SM3M. 3.

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Figures (10)

Fig. 1.
Fig. 1. Experimental arrangement of a cavity-locked DRL; BS - beam sampler, HWP - half-wave plate, FL - focusing lens, IC - input coupler, OC - output coupler, PZT - piezoelectric translation stage, DM - dichroic mirror, QWP - quarter-wave plate, PBS - polarizing beam splitter, PD1 and PD2- photodetectors.
Fig. 2.
Fig. 2. Calculated locking and reflected pump signals at (a) below, and (b) above Raman lasing threshold when the cavity length is scanned at the rate of 5 µm/s. The vertical and horizontal lines indicate lock-point and zero-crossing for signal, respectively.
Fig. 3.
Fig. 3. (a).Polarization angle of Stokes output (dashed lines) and depleted pump (solid lines) as a function of incident pump polarization angle. Red, green, blue, orange and black lines are modeled dependencies for 0.1%, 50%, 70%, 90% and 99% depletion, respectively. Blue triangles are data from [23] while red circles are data from this study. (b).Crystallographic directions in the plane normal to propagation.
Fig. 4.
Fig. 4. Depleted pump rotation as a function of pump depletion for fixed Stokes angle between −10° and 80°. For Stokes angles above 40°, the depletion rate of the pump decreases significantly after reaching a steady state angle, shown as dashed lines.
Fig. 5.
Fig. 5. Stokes power (triangles) and residual pump power (circles) characteristics without cavity stabilization.
Fig. 6.
Fig. 6. Time traces of pump resonances (black) and locking signal (red) (a) below and (b) above laser threshold when the cavity length is scanned at the rate of 10 µm/s. The vertical and horizontal lines indicate lock-point and zero-crossing for signal, respectively. The trace (blue) above each figure indicates the timing of the voltage ramp.
Fig. 7.
Fig. 7. FP scan (red) of the stabilized laser shows SLM operation (black) at 7.2 W Stokes power.
Fig. 8.
Fig. 8. (a) Stokes central wavelength fluctuations of the stabilized laser at 4.5 W power. The inset shows fluctuations in the output wavelength of about 125 MHz for about 2 mins. (b) FP scan of the laser output during the initial period. (c) FP scan of the laser output during de-stabilized operation, corresponding to the period after 120 seconds in (a). The red curves above the FP scans in (b) and (c) show the measured Stokes output power.
Fig. 9.
Fig. 9. Calculated locking signal for (a) below (b) above Raman threshold (for 50% depletion of the pump) for input mirror reflectivities R1 = 10%, 30%, 50%, 70% and 90%, respectively. The vertical and horizontal lines indicate lock-point and zero-crossing for signal, respectively.
Fig. 10.
Fig. 10. Calculated locking signals and pump resonance for round-trip dephasing values of 1.6 and 3.02 rad. The vertical and horizontal lines indicate lock-point and zero-crossing for signal, respectively.

Equations (10)

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E x , y r = E x , y i { R 1 T 1 F ( cos δ e , o F + isin δ e , o ) R 1 ( ( 1 F ) 2 + 4 F sin 2 ( δ e , o / δ e , o 2 2 ) ) }
( E a E b ) = ( cos θ sin θ sin θ cos θ ) ( 1 0   0 i ) ( cos ( θ ) sin ( θ )   sin ( θ ) cos ( θ ) ) ( E x r E y r )
d S k dz = g 2 ijl χ i j k l S i P j P l ,
d P l dz = g 2 η ijk χ i j k l S i P j S k ,
d S x dz = g 2 ( S x P x P x + S x P y P y + S y P x P y )
d S y dz = g 2 ( S y P x P x + S x P y P x )
d P x dz = g 2 η ( S x P x S x + S x P y S y + S y P x S y )
d P y dz = g 2 η ( S y P x S x + S x P y S x ) .
F x , y = R 1 R 2 ( 1 L ) 2 ( 1 F l o s s x , y ) 2
E x , y r = E x , y i { R 1 T 1 F x , y ( cos δ e , o F x , y + isin δ e , o ) R 1 ( ( 1 F x , y ) 2 + 4 F x , y sin 2 ( δ e , o / δ e , o 2 2 ) ) } .

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