Open Access
Add to BibSonomy
Abstract
Stimulated Brillouin scattering is one of the predominant limitations for fiber-based narrowband lasers, amplifiers, or passive transmission links. In addition to the output power limitation, stimulated Brillouin scattering imprints a broadband excess intensity noise on the signal for which several theoretical explanations have been reported. Some studies suggested the conversion from phase to intensity noise due to an asymmetric Brillouin gain spectrum. In this work, we present new and consistent measurements of such asymmetric gain profiles and corresponding intensity noise spectra. Based on our results, we were able to reconstruct the shape of the measured excess intensity noise.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
A variety of applications such as telecommunication data links, LIDAR, optical spectroscopy and atom cooling require high optical output power at a narrowband laser linewidth [1–4]. Therefore, the output power of single-frequency fiber-amplifiers in master oscillator power amplifier (MOPA) configuration has been continuously increased in the recent decade. In 2008 Mermelstein et al. presented 194 W output power [5], in 2011 Zhu et al. demonstrated up to 511 W output power [6] and Pulford et al. presented an output power level of 811 W [7]. In addition to that, single-frequency fiber amplifiers operating at 1064 nm are promising candidates to fulfill the challenging requirements regarding laser sources for the next generation of interferometric gravitational wave detectors [8]. To increase the sensitivity of these detectors, high optical output power up to 1 kW is required [9]. All of the above mentioned laser sources are limited for further power scaling by the nonlinear effect of stimulated Brillouin scattering (SBS).
SBS is an effect caused by the $\chi ^{(3)}$-nonlinearity and results from scattering of signal photons on thermally excited acoustic phonons in an optical medium. Due to the beating with the signal, counter propagating scattered photons create a periodic refractive index modulation along the fiber core through the electrostrictive effect, which constitutes a traveling Bragg grating. This grating leads to further light scattering and to the onset of SBS. Above a certain threshold power of the signal, SBS can reflect a significant part of the power of the incident light and limits the output power of fiber lasers and amplifiers. The transition from spontaneous to stimulated Brillouin scattering is connected with the Brillouin gain spectrum. The frequency shift due to the Doppler effect and the spectral shape of the Brillouin gain depends on the signal frequency and fiber characteristics such as core dimensions and glass composition [10]. Furthermore, the inelastical backscattered process shows stochastic dynamics [11].
A broadband excess intensity noise is also a limiting factor for many applications [8,12–14]. Several published theories provide models to explain the observed excess noise in the transmitted signal: Stokes and anti-Stokes Brillouin scattering [15] or a conversion from phase to intensity noise [16,17].
Here, we report new and consistent experimental studies, which suggest the conversion from phase to intensity noise of the transmitted signal as origin of excess noise due to an asymmetric Brillouin gain spectrum.
2. Experiment
As a test signal, a single-frequency non-planar ring oscillator (NPRO, Innolight Mephisto 2000NE, c.f. Fig. 1) with a narrow laser linewidth (< 1 kHz) was used. The NPRO emitted 2 W output power of continuous-wave at 1064 nm. The MOPA system consisted of a free space part and an amplification stage based on a rare-earth doped fiber. A 3 m polarization-maintaining large mode-area ytterbium doped fiber (Nufern, PLMA-YDF-10/125) allowed an amplification of the NPRO test signal up to 20 W preserving its narrow linewidth. The pump light for the amplification was supplied by two 30 W high-power laser diodes (II-VI, BMU30-975-01-R02) emitting at 976 nm and was injected into the cladding of the active fiber via a pump combiner (PC). Any residual pump light of the counter-propagating configuration was removed by an in-house fabricated cladding light stripper (CLS) [18].
Fig. 1. Experimental setup for intensity noise measurements of the transmitted light above the SBS threshold. The system consisted of an NPRO for the signal beam, a monolithic fiber amplifier and a single-mode fiber for the generation of SBS. The connection between the MOPA stage and the single-mode fiber was either a high power fiber isolator (at higher output power b)) or a circulator (at lower output power a)) to measure the Brillouin gain spectrum. CLS: cladding light stripper, PC: pump combiner, L$_{i}$: lens. Splices are represented by x.
The SBS signal was generated in a single-mode fiber (Raman-fiber from OFS Fitel) with a measured 4 ${\mathrm{\mu}}$ m core diameter and different fiber lengths (2.5 m, 5 m, 7.5 m and 20 m). Any potential back reflection into the fiber amplifier was suppressed by a circulator (c.f. Fig. 1(a)) or a high-power fiber isolator (c.f. Fig. 1(b)). The noise of the transmitted signal was measured with an InGaAs-photodiode (Thorlabs, PDA10CF-EC) with a bandwidth of 150 MHz and was analyzed with an electrical spectrum analyzer (Agilent, E4440).
The SBS signal and the backwards propagating Rayleigh scattering of the signal beat together and enable a monolithic heterodyne detection of the gain spectrum by a fast amplified photodiode with a bandwidth of 45 GHz (Newport Corp. 1014 and Newport Corp. 1421) at the output port of the circulator.
If the power level of the signal wave exceeds a certain threshold after the MOPA stage, the output power of the Stokes light increases significantly. In passive fibers the power threshold $P_{\textrm {B}}$ at which the Brillouin scattering becomes stimulated can be expressed by:
3. Results
3.1 Modulated light due to an asymmetric Brillouin gain spectrum
The modulation of light amplitude is visualized with a phasor diagram. The graphical interpretation of an amplitude and phase modulation is shown in Fig. 2. Here, the carrier (red arrow) as well as sidebands (blue arrows) are represented by rotating vectors on a complex plane. The amplitude of the carrier field is given by the length of the vector, which rotates clockwise with the rate $\omega$. The sideband vectors rotate in opposite directions among themselves with the rotation-frequency of the carrier field. The sum of these three vectors yields a complex vector, where the projection on the real axis represents the modulation of the amplitude of the light.
Fig. 2(a)) represents the amplitude modulation (AM). The vector sum of the modulated sidebands is always in phase with the carrier field and the time-dependent projection onto the real axis including the AM is depicted to the right of the phasor diagram. In case of a phase modulation (PM) in Fig. 2(b)), the vector sum of the sidebands oscillates at 90 $\circ$, i.e. $\pi /2$, phase shift with respect to the carrier. Thus, the resulting modulated oscillation vector has the same length as the carrier field vector. The subsequent pure-PM oscillation without any AM is depicted to the right of the diagram.
Fig. 2. Phasor diagram for amplitude (a) and phase (b) modulated light with the carrier field (red arrow) and the sideband vectors (blue arrows). The resulting modulated oscillation is shown in a green arrow. The time-dependent projection of the electric field is depicted to the right of the corresponding phasor diagram.
The idea of a conversion from phase to intensity noise based on an asymmetric Brillouin gain or an asymmetric loss spectrum for the transmitted signal was first reported by Zhang and Phillips [17]. Numerical simulations of the overlap between acoustic and optical modes have provided a potential explanation of such an asymmetric gain spectrum [16].
Fig. 3 presents the impact of an asymmetric Brillouin gain spectrum and the resulting asymmetric phase shift on the modulated oscillation vector of the light for a pure input PM. In the case of a symmetric Brillouin gain spectrum, the sidebands of the transmitted signal, inherently present due to the frequency noise of the laser source, experience an identical phase shift with opposite sign. Thus, their vector sum does not change its phase. If now an asymmetric Brillouin gain spectrum with an asymmetric phase distribution is assumed, the sidebands experience a differential loss and different phase shifts, so in the sum of the vectors the modulated oscillation does not have the same length as the carrier. Thus, the resulting electrical field now is also modulated in its amplitude. This induced amplitude modulation scales directly with the phase modulation it was generated from.
Fig. 3. Principle of the conversion from phase to intensity noise based on an asymmetric Brillouin loss and a corresponding asymmetric phase shift for the transmitted signal. A symmetric gain or loss spectrum has no influence on the amplitude and the sidebands experience a pure PM. Due to an asymmetric spectrum (right), the sidebands experience a differential loss and phase shift and the pure phase modulation is converted into a mixture of amplitude and phase modulation.
In general, assuming the Maxwell wave equations with material equations and nonlinear polarization, SBS is governed by coupled amplitude $A_{\textrm {j}}$ equations for the pump (j=p) and the Stokes wave (j=s), such that $|A_{\textrm {j}}|^2$ represents the power [22]. For example, the evolution of the pump wave is given by:
where $\chi (\nu )$ is the frequency-dependent susceptibility. Under steady-state conditions, applicable for a CW pump signal and a spectral pump-linewidth much smaller than the SBS bandwidth, the coupled mode equations can be simplified to the following equations for the power evolution along the fiber [22]:Using Eq. (2) the frequency-dependent phase shift can be derived as a Hilbert transformation of the real part of the susceptibility $\chi (\nu )$ with the Stokes power $P_{\textrm {Stokes}}$ and the mode field diameter of the fiber core:
Figure 4(a)) presents a measured gain profile at a frequency shift of around $\nu _{\textrm {B}}=$14.7 GHz after 20 m of single-mode fiber. The profile was recorded with the aforementioned heterodyne detection of the beating between the Stokes wave and the Rayleigh scattered part of the signal. Instead of a symmetric Lorentzian profile, the measured profile exhibits an asymmetric characteristic, which can be expressed by $n=210$ (c.f. Equation (7)) and a bandwidth of $\Omega _{\textrm {BS}}=21$ MHz based on a corresponding fit (Fig. 4(a), dashed red). In addition, further asymmetric broadening can be identified at different signal power (c.f. Fig. 4(b)).
Fig. 4. Measured stimulated Brillouin scattering gain spectrum: a) Spectrum of a 20 m single-mode fiber at a frequency shift of around 14.7 GHz and a fitted asymmetric Lorentzian profile. b) Gain profile at different signal power levels with a 20 m single-mode fiber.
The measured intensity noise of the amplified NPRO signal after the single-mode fiber for a fiber length of 20 m is shown in Fig. 5. The measurement below the SBS threshold $P_{\textrm {B}}$ is shown in black. An increase of the pump power above the SBS threshold resulted in a characteristic excess intensity noise up to 50 MHz. Additionally, above a certain signal power ($> 1.10 \cdot P_{\textrm {B}}$) corresponding resonances occured in the intensity noise spectrum. Due to small back reflections at the fiber end facets, the setup corresponds to a resonator with a low Q factor. One would assume that the resonator eigenfrequencies are given by the fiber length and the refractive index of the optical medium [23]. The resonances in the intensity noise spectrum correspond to the beating between the carrier and the eigenfrequencies or between the upper and lower eigenfrequencies among themselves. However, the measured resonances do not match with the corresponding fiber length, i.e. $\Delta \nu _0$ = 2.9 MHz would correspond to a fiber length of 35 m instead of 20 m. Previous explanations by modulation instabilities [24] can not explain this behavior for two reasons: On the one hand, modulation instabilities only occur in the intensity noise if they are also visible in the gain spectrum and on the other hand, the absolute frequency separation $\Delta \nu _i$ will be always larger than $2c/nL$ [24], but here it decreases with increasing output power.
Fig. 5. Intensity noise spectrum of the transmitted signal below and above the SBS threshold $P_{\textrm {B}}$ for a fiber length of 20 m. The difference of the eigenfrequencies are marked as $\Delta \nu _i$.
The total phase shift of one round trip $\Phi _{\textrm {RT}}$ with the refractive index $n_0$ of the fiber is given by:
Figure 6 illustrates the resulting effect of the phase shift of the round trip without (red dashed line) and with an additional phase shift due to SBS (blue line). A phase shift of $2\pi$ corresponds exactly to the distance between two longitudinal modes or an equivalent change of the frequency $\Delta \nu _{\textrm {i}}$, where the change is consistent with $\Delta \nu _i = \nu (\Phi _{\textrm {RT}}=(n+1)2 \pi ) - \nu (\Phi _{\textrm {RT}}=n 2\pi )$ with $n=0, 1, 2, 3,\ldots \,$. The insert in Fig. 6 visualizes the case for the shifted round trip of $2\pi$ and $4\pi$ with a corresponding eigenfrequency of 2.8 MHzand 5.7 MHz.
Fig. 6. Shift of the phase of the round trip without (red dashed line) and with an additional phase shift due to SBS (blue line). The insert visualizes the case for the shifted round trip of $2\pi$ and $4\pi$ with a corresponding eigenfrequency of 2.8 MHz and 5.7 MHz.
The measured eigenfrequencies in the intensity noise spectrum can be well predicted by Eq. 8. Table 1 gives an overview of the measured and calculated resonances at different fiber lengths: The results confirm the parameters of the Lorentzian profile fit (c.f. Fig. 4), where the uncertainties are determined especially by the accuracy of the length of the single-mode fiber.
Table 1. Measured and calculated resonator eigenfrequencies at different fiber lengths
The evolution of the resonator eigenfrequencies at 15 MHz is presented in Fig. 7 in detail for a fiber length of 20 m. Due to the asymmetric gain profile and the corresponding phase shift, the beat frequencies between the upper and lower eigenfrequencies with the carrier are not equal, which becomes noticeable in a splitting of the resonator peaks in the intensity noise spectrum. Consequently, this result is a first and new experimental evidence for an asymmetric gain profile.
Fig. 7. Intensity noise of the transmitted signal above the SBS threshold $P_{\textrm {B}}$ in a frequency range from 13 to 17 MHz. The splitting of the resonances expressed by the red arrows indicates an asymmetric gain profile.
3.2 Reconstruction of the low-frequency excess intensity noise
In the following it will be demonstrated that the results of the fitted asymmetric SBS gain profile can be used to reconstruct the characteristic shape of the excess intensity noise quite well. Behind the fiber the entire electric field $E_{\textrm {total}}$ can represented by:
In addition, several assumptions were made in the following. The frequency noise of the NPRO shows a $1/\nu$-characteristic [25] and only the contribution of the beating of the sidebands with the carrier are significant at MHz-frequencies, because the sideband-sideband interaction is too weak and will not be considered further. Then, it follows for the intensity:
The shape of the excess intensity noise is then given by the AC term of Eq. (10) with a scale factor $\kappa$ (to be fitted) and the $1/{\nu }$-frequency noise characteristic of the NPRO:
Figure 8 presents three intensity noise measurements at different signal power levels above the SBS threshold $P_{\textrm {B}}$ (blue) and one measurement below the threshold (black) on a logarithmic frequency scale. Similar to Fig. 5, the spectrum exhibits an increase of the broadband noise in the MHz-frequency range up to 50 MHz with increasing output power. At lower frequencies the NPRO relaxation peak at 450 kHz overlaps the broadband noise and at higher frequencies, the measurement is shot noise limited. For the three measurements and the corresponding fitted Lorentzian profile (Fig. 4(b)) the model $M(\nu )$ was calculated analytically with Eq. 10. The fit was only scaled according to the measured noise level with the parameter $\kappa$. For more asymmetric profiles, the influence of the differential phase shift distribution increases, so that in a regime of 10 MHz an excess noise can be observed. This regime correlates with the frequency range of the maximum of the phase shift. The frequency range from 1 to 10 MHz shows a discrepancy between the model and the experimental data. One possible explanation is the small phase difference around the inflection point near the carrier of the phase shift and the corresponding low intensity modulation (c.f. Fig. 6). Another more flexible model, like a Gaussian fit, could optimize the results. In summary, Fig. 8 shows for the first time the reconstruction of the characteristic shape of the excess intensity noise by SBS.
Fig. 8. Intensity noise spectrum of the transmitted light above and below the SBS threshold $P_{\textrm {B}}$. A model of the noise for three measurements based on an asymmetric Lorentzian gain spectrum and a conversion from phase to intensity noise is shown in red.
4. Conclusion
We have presented theoretical and experimental investigations regarding the broadband excess intensity noise observed above the SBS threshold in optical fibers. New and consistent results suggest the conversion from phase to intensity noise. Due to an asymmetry of the measured Brillouin gain spectrum, the shifted resonator eigenfrequencies of a single-mode fiber are perceptible in the intensity noise spectrum and show a splitting of the resonator peaks.
These results confirm the parameter of the fitted and modified Lorentzian profile. Based on the asymmetry of the Brillouin gain spectrum, we were able to reconstruct the shape of the excess intensity noise for the first time.
The knowledge of the excess intensity noise above the SBS threshold conduces to the comprehension of the noise properties for further laser stabilization for the next generation of gravitational wave detectors and a variety of other optical applications limited by SBS.
Disclosures
The authors declare no conflicts of interest.
References
1. G. Bonfrate, F. Vaninetti, and F. Negrisolo, “Single-frequency MOPA Er3+ DBR fiber Laser for WDM digital telecommunication systems,” IEEE Photonics Technol. Lett. 10(8), 1109–1111 (1998). [CrossRef]
2. L. G. Holmen, G. Rustad, and M. W. Haakestad, “Eye-safe fiber laser for long-range 3D imaging applications,” Appl. Opt. 57(23), 6760–6767 (2018). [CrossRef]
3. F. Ruschewitz, J. L. Peng, H. Hinderthr, N. Schaffrath, K. Sengstock, and W. Ertmer, “Sub-Kilohertz Optical Spectroscopy with a Time Domain Atom Interferometer,” Phys. Rev. Lett. 80(15), 3173–3176 (1998). [CrossRef]
4. C. W. Oates, K. R. Vogel, and J. L. Hall, “High Precision Linewidth Measurement of Laser-Cooled Atoms: Resolution of the Na 3p22P3/2 Lifetime Discrepancy,” Phys. Rev. Lett. 76(16), 2866–2869 (1996). [CrossRef]
5. M. D. Mermelstein, K. Brar, M. J. Andrejco, A. D. Yablon, M. Fishteyn, C. Headley III, and D. J. DiGiovanni, “All-fiber 194 W single-frequency single-mode Yb-doped master-oscillator power amplifier,” Proc. SPIE 6873, 68730L (2008). [CrossRef]
6. C. Zhu, I. Hu, X. Ma, and A. Galvanauskas, “Single-frequency and single-transverse mode Yb-doped CCC fiber MOPA with robust polarization SBS-free 511W output,” in Advances in Optical Materials, OSA Technical Digest (CD) (Optical Society of America, 2011).
7. B. Pulford, I. Dajani, and C. Robin, Near Diffraction-Limited, 811W, Single-Frequency Photonic Crystal Fiber Amplifier, in CLEO: 2014, OSA Technical Digest (online) (Optical Society of America, 2014).
8. T. Theeg, H. Sayinc, J. Neumann, and D. Kracht, “All-Fiber Counter-Propagation Pumped Single Frequency Amplifier Stage With 300-W Output Power,” IEEE Photonics Technol. Lett. 24(20), 1864–1867 (2012). [CrossRef]
9. ET Science Team, “Einstein Gravitational Wave Telescope Conceptual Design Study,” available at http://www.et-gw.eu/, Date of access: May. 06 2020, ET-0106A-10 0, (2011).
10. M. Niklas, L. Thavenaz, and P. A. Robert, “Brillouin Gain Spectrum Characterization in Single-Mode Optical Fibers,” J. Lightwave Technol. 15(10), 1842–1851 (1997). [CrossRef]
11. A. L. Gaeta and R. W. Boyd, “Stochastic dynamics of stimulated Brillouin scattering in an optical fiber,” Phys. Rev. A 44(5), 3205–3209 (1991). [CrossRef]
12. Y. Aoki, K. Tajima, and I. Mito, “Input power limits of single-mode optical fibers due to stimulated Brillouin scattering in optical communication systems,” J. Lightwave Technol. 6(5), 710–719 (1988). [CrossRef]
13. J. Zhao, F. Yang, F. Wei, X. Zhang, Z. Ding, R. Wu, and H. Cai, “Effect of linewidth on intensity noise induced by stimulated Brillouin scattering in single-mode fibers,” Opt. Express 28(10), 15025–15034 (2020). [CrossRef]
14. X. Guan, C. Yang, P. Ma, Q. Zhao, Z. Zhang, W. Lin, X. Wei, Z. Yang, and S. Xu, “Influence of stimulated Brillouin scattering on the noise evolution of high-power all-fiber single-frequency MOPA system,” Opt. Laser Technol. 128, 106212 (2020). [CrossRef]
15. M. Horowitz, A. R. Chraplyvy, R. W. Tkach, and J. L. Zyskind, “Broad-band transmitted intensity noise induced by Stokes and anti-Stokes Brillouin scattering in single-mode fibers,” IEEE Photonics Technol. Lett. 9(1), 124–126 (1997). [CrossRef]
16. E. Peral and A. Yariv, “Degradation of Modulation and Noise Characteristics of Semiconductor Lasers After Propagation in Optical Fiber Due to a Phase Shift Induced by Stimulated Brillouin Scattering,” IEEE J. Quantum Electron. 35(8), 1185–1195 (1999). [CrossRef]
17. J. Zhang and M. R. Phillips, “Modeling Intensity Noise Caused by Stimulated Brillouin Scattering in Optical Fibers,” Conference on Lasers and Electro-Optics (2005).
18. M. Wysmolek, C. Ottenhues, T. Pulzer, T. Theeg, H. Sayinc, M. Steinke, U. Morgner, J. Neumann, and D. Kracht, “Microstructured Fiber Cladding Light Stripper for Kilowatt-Class Laser Systems,” Appl. Opt. 57(23), 6640–6644 (2018). [CrossRef]
19. R. Smith, “Optical power handling capacity of low loss optical fibers as determined by stimulated Raman and Brillouin scattering,” Appl. Opt. 11(11), 2489–2494 (1972). [CrossRef]
20. R. Billington, NPL Report COEM 31, available at http://eprintspublications.npl.co.uk/, Date of access: May. 06 2020, (1999).
21. M. Hildebrandt, S. Basche, P. Wessels, M. Frede, and D. Kracht, “Brillouin scattering spectra in high-power single-frequency ytterbium doped fiber amplifiers,” Opt. Express 16(20), 15970 (2008). [CrossRef]
22. G. P. Agrawal, Nonlinear Fiber Optics4rd ed. (Academic Press, 2006).
23. L. Stapien, S. Randoux, and J. Zemmouri, “Origin of spectral hole burning in Brillouin fiber amplifiers and generators,” Phys. Rev. A 65(5), 053812 (2002). [CrossRef]
24. Y. Takushima and K. Kikuchi, “Spectral gain hole burning and modulation instability in a Brillouin fiber amplifier,” Opt. Lett. 20(1), 34–36 (1995). [CrossRef]
25. https://edge.coherent.com/assets/pdf/COHR_Mephisto_DS_0417_2.pdf, Datasheet of Innolight Mephisto 2000NE.
