Statistics of pulse enrgy fluctuations in a solid-state Raman laser

In this paper, we present the results of the study of the statistics of pulse energy fluctuations in a Raman laser under optical pump by the multimode nanosecond pulses. A system of coupled differential equations for slowly varying envelopes of the pump field and first three Stokes lines was integrated numerically with taking into account spatial inhomogeneity of the pump beam, spontaneous noise, and optical feedback. Data of the numerical simulation revealed a sharp increase in the fluctuation amplitude in the nonlinear regime of Raman frequency conversion when the optical length of the Raman cavity was matched with the cavity length of the multimode pump laser. At a mean 1st Stokes conversion efficiency of 3.5–3.8 %, the calculations showed an increase in the coefficient of variation (CV) of a random value from 9 % to 118 %. In the linear regime of Raman frequency conversion, when the conversion efficiency was 0.2–0.03 %, a further increase in the CV value up to 270–500 % was predicted. It is also numerically shown that the fluctuation statistics under the conditions of the cavity length matching is essentially non-Gaussian and described by the L-type probability density distributions (PDDs) with long tails and maxima located near zero. The numerical data were quantitatively confirmed by an experiment for a Raman laser on a barium nitrate crystal operated near the Raman threshold, when the 1st Stokes conversion efficiency did not exceed 0.3 %. A Raman cavity was formed by two flat mirrors providing a double-pass pump configuration. The Raman laser was excited by the linearly polarized frequency-doubled radiation of a Q-switched Nd:YAG laser generating multimode pulses with a duration of 7–8 ns. A Raman laser operating regime characterized by the hyperexponential PDDs with CVs reaching 480 %, which is 2–2.5 times higher than those observed earlier for the single-pass conditions of stimulated Raman scattering, was realized.

stimulated Raman scattering (SRS), Raman laser, statistics of fluctuations, coefficient of variation, probability density distribution

1. Walker D. A. G., Taylor P. H., Taylor R. E. The shape of large surface waves on the open sea and the Draupner New Year wave. Applied Ocean Research, 2004, vol. 26, no. 3–4, pp. 73–83. https://doi.org/10.1016/j.apor2005.02.001

2. Abarbanel H., Koonin S., Levine H., MacDonald G., Rothaus O. Statistics of Extreme Events with Application to Climate. JASON, 1992, JSR-90-30S. https://doi.org/10.1016/j.apor.2005.02.001

3. Alvarado E., Sandberg D. V., Pickford S. G. Modeling Large Forest Fires as Extreme Events. Northwest Science, 1998, vol. 72, pp. 66–75.

4. Embrechts P., Klüppelberg C., Mikosch T. Statistical Methods for Extremal Events. Modelling extremal events for insurance and finance. Berlin, Spring Verlag, 1997, pp. 283–370. https://doi.org/10.1007/978-3-642-33483-2

5. Orsini F., Gecchele G., Gastaldi M., Rossi R. Collision prediction in roundabouts: a comparative study of extreme value theory approaches. Transportmetrica A: Transport Science, 2019, vol. 15, no. 2, pp. 556–572. https://doi.org/10.1080/23249935.2018.1515271

6. Carreras B. A., Newman D. E., Dobson I. North American Blackout Time Series Statistics and Implications for Blackout Risk. IEEE Transactions on Power Systems, 2016, vol. 31, no. 6, pp. 4406–4414. https://doi.org/10.1109/TPWRS.2015.2510627

7. Zebrev G. I, Galimov A. M., Useinov R. G., Fateev I. A. Extreme Value Based Estimation of Critical Single Event Failure Probability. arXiv, 2019. Available at: https://arxiv.org/abs/1909.07804v1

8. Solli D. R., Ropers C., Koonath P., Jalali B. Optical rogue waves, Nature, 2007, vol. 450, no. 7172, pp. 1054–1058. https://doi.org/10.1038/nature06402

9. Birkholz S., Nibbering E. T. J., Brée C., Skupin S., Demircan A., Genty G., Steinmeyer G. Spatiotemporal Rogue Events in Optical Multiple Filamentation. Physical Review Letters, 2013, vol. 111, no. 24, pp. 243903. https://doi.org/10.1103/PhysRevLett.111.243903

10. Kasparian J., Béjot P., Wolf J-P., Dudley J.M. Optical rogue wave statistics in laser filamentation. Optics Express, 2009, vol. 17, no. 14, pp. 1270–1275. https://doi.org/10.1364/OE.17.012070

11. Montina A., Bortolozzo U., Residori S., Arecchi F. T. Non-Gaussian statistics and extreme waves in a nonlinear optical cavity. Physical Review Letters, 2009, vol. 103, no. 17, pp. 173901. https://doi.org/10.1103/PhysRevLett.103.173901

12. Hammani K., Finot C., Millot G. Emergence of extreme events in fiber-based parametric processes driven by a partially incoherent pump wave. Optics Letters, 2009, vol. 34, no. 8, pp. 1138–1140. https://doi.org/10.1364/OL.34.001138

13. Soto-Crespo J. M., Grelu Ph., Akhmediev N. Dissipative rogue waves: extreme pulses generated by passively mode-locked lasers. Physical Review E, 2011, vol. 84, no. 1, pp. 016604. https://doi.org/10.1103/PhysRevE.84.016604

14. MacPherson D. C., Swanson R. C., Carlsten J. L. Quantum Fluctuations in the Stimulated-Raman-Scattering Linewidth. Physical Review Letters, 1988, vol. 61, no. 1, pp. 66–69. https://doi.org/10.1103/PhysRevLett.61.66

15. Raymer M. G., Li Z. W., Walmsley I. A. Temporal quantum fluctuations in stimulated Raman scattering: Coherent-modes description. Physical Review Letters, 1989, vol. 63, no, 15, pp. 1586–1589. https://doi.org/10.1103/PhysRevLett.63.1586

16. Duncan M. D., Mahon R., Tankersley L. L., Reintjes J. Control of transverse spatial modes in transient stimulated Raman amplification. Journal of the Optical Society of America B, 1990, vol. 7, no. 7, pp. 1336–1345. https://doi.org/10.1364/JOSAB.7.001336

17. Hammani K., Picozzi A., Finot C., Extreme statistics in Raman fiber amplifiers: From analytical description to experiments. Optics Communications, 2011, vol. 284, no. 10–11, pp. 2594–2603. https://doi.org/10.1016/j.optcom.2011.01.057

18. Aalto A., Genty G., Toivonen J. Extreme-value statistics in supercontinuum generation by cascaded stimulated Raman scattering. Optics Express, 2010, vol. 18, no. 2, pp. 1234–1239. https://doi.org/10.1364/OE.18.001234

19. Monfared Y. E., Ponomarenko S. A. Non-Gaussian statistics and optical rogue waves in stimulated Raman scattering. Optics Express, 2017, vol. 25, no. 6, pp. 5941–5950. https://doi.org/10.1364/OE.25.005941

20. Fabricius N., Nattermann K., D. von der Linde. Macroscopic Manifestation of Quantum Fluctuations in Transient Stimulated Raman Scattering. Physical Review Letters, 1984, vol. 52, no. 2, pp. 113–116. https://doi.org/10.1103/PhysRevLett.52.113

21. Walmsley I.A., Raymer M. G. Observation of Macroscopic Quantum Fluctuations in Stimulated Raman Scattering. Physical Review Letters, 1983, vol. 50, no. 13, pp. 962–965. https://doi.org/10.1103/PhysRevLett.50.962

22. Raymer M. G., Walmsley I. A. III The quantum coherence properties of stimulated Raman scattering, Progress in Optics, 1990, vol. 28, pp. 247-255. https://doi.org/10.1016/S0079-6638(08)70290-7

23. Apanasevich P. A., Gakhovich D E, Grabchikov A. S., Kilin S. Y., Kozich V P, Kontsevoĭ B. L., Orlovich V A. Statistical characteristics of the energies of pulses of forward and backward stimulated Raman scattering under linear, intermediate, and nonlinear scattering conditions. Soviet Journal of Quantum Electronics, 1992, vol. 22, no. 9, pp. 822–827. https://doi.org/10.1070/qe1992v022n09abeh003607

24. Grabtchikov A. S., Vodtchits A. I., Orlovich V. A., Pulse-energy statistics in the linear regime of stimulated Raman scattering with a broad-band pump. Physical Review A, 1997, vol. 56, no. 2, pp. 1666–1669. https://doi.org/10.1103/PhysRevA.56.1666

25. Borlaug D., Fathpour S., Jalali B. Extreme Value Statistics in Silicon Photonics. IEEE Photonics Journal, 2009, vol. 1, no. 1, pp. 33–39. https://doi.org/10.1109/JPHOT.2009.2025517

26. Betlej A., Schmitt P., Sidereas P., Tracy R., Goedde C. G., Thompson J. R. Increased Stokes pulse energy variation from amplified classical noise in a fiber Raman generator. Optics Express, 2005, vol. 13, no. 8, pp. 2948–2960. https://doi.org/10.1364/OPEX.13.002948

27. Chang J., Baiocchi D., Vas J., Thompson J. R. First Stokes pulse energy statistics for cascade Raman generation in optical fiber. Optics Communications, 1997, vol. 139, no. 4–6, pp. 227-231. https://doi.org/10.1016/S0030-4018(97)00060-6

28. Headley C., Agrawal G. P. Noise Characteristics and Statistics of Picosecond Stokes Pulses Generated in Optical Fibers Through Stimulated Raman Scattering. IEEE Journal of Quantum Electronics, 1995, vol. 31, no. 11, pp. 2058–2067. https://doi.org/10.1109/3.469288

29. Zverev P. G., Basiev T. T., Osiko V. V., Kulkov A. M., Voitsekhovskii V. N. Physical, chemical, and optical properties of barium nitrate Raman crystal. Optical Materials, 1999, vol. 11, no. 4, pp. 315–334. https://doi.org/10.1016/S0925-3467(98)00031-7

30. Lisinetskii V. A., Busko D. N., Chulkov R. V., Grabchikov A. S., Apanasevich P. A., Orlovich V. A. Self-mode locking at multiple Stokes generation in the Raman laser. Optics Communications, 2010, vol. 283, no. 7, pp. 1454–1458. https://doi.org/10.1016/j.optcom.2009.11.047

31. Battle P. R., Swanson R. C., Carlsten J. L. Quantum limit on noise in a Raman amplifier. Physical Review A, 1991, vol. 44, no. 3, pp. 1992–1930. https://doi.org/10.1103/PhysRevA.44.1922

32. Karamzin Y. N., Sukhorukov A. P., Trophimov V. A. Mathematical Modeling in Nonlinear Optics. Moscow, Publishing House of Moscow State University, 1989. 154 p. (in Russian).

33. Chulkov R. V., Markevich V. Y., Orlovich V. A., El-Desouki M.M. Steady-state Raman gain coefficients of potassium-gadolinium tungstate at the wavelength of 532 nm. Optical Materials, 2015, vol. 50, pp. 92–98. https://doi.org/10.1016/j.optmat.2015.10.004

34. Lisisnetskii V. A., Mish-kel’ I. I., Chulkov R. V., Grabtchikov A. S., Apanasevich P. A., Eichler H. J., Orlovich V. A. Raman gain coefficient of barium nitrate measured for the spectral region of Ti:Sapphire laser. Journal of Nonlinear Optical Physics & Materials, 2005, vol. 14, no. 1, pp. 107–114. https://doi.org/10.1142/s0218863505002530

35. Chulkov R. V., Markevich V. Y., Alyamani A. Y., Cheshev E. A., Orlovich V. A. Cavity length matching and optical resonances in a Raman laser with the multimode pump source. Optics Letters, 2017, vol. 42, no. 23, pp. 4824–4827. https://doi.org/10.1364/OL.42.004824

36. Apanasevich P. A., Gakhovich D. E., Killin S. Y., Kozich V. P., Kontsevoi B. L., Orlovich V. A. Statistical characteristics of the pulse energies for forward and backward SRS in linear, intermediate, and nonlinear scattering modes. Quantum Electronics, 1992, vol. 19, pp. 884–890.