## SBS gain efficiency measurements and modeling in a 1714 µm^{2} effective area LP_{08} higher-order mode optical fiber

Optics Express, Vol. 15, Issue 24, pp. 15952-15963 (2007)

http://dx.doi.org/10.1364/OE.15.015952

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### Abstract

The stimulated Brillouin scattering (SBS) gain efficiencies were measured in the LP_{08} and LP_{01} modes of a higher-order-mode optical fiber. Gain efficiencies C_{B} of 0.0085 and 0.20 (m-W)^{-1} were measured for the LP_{08} and LP_{01} modes at 1083 nm, respectively. C_{B} is inversely proportional to the optical effective area A_{eff} and the same core-localized acoustic phonon seeds the SBS process in each case. An acoustic modal analysis and a distributed phenomenological model are presented to facilitate the data analysis and interpretation. The LP_{08} mode exhibits a threshold power-length product of 2.5 kW-m.

© 2007 Optical Society of America

## 1. Introduction

_{eff}=415 µm

^{2}has been demonstrated [1

1. A. Liem et al., “100-W single-frequency master-oscillator fiber power amplifier,” Opt. Lett. **28**, 1537 (2003). [CrossRef] [PubMed]

_{eff}=1417 µm

^{2}has recently been reported [2

2. W. S. Wong et al., “Breaking the limit of maximum effective area for robust single-mode propagation in optical fibers,” Opt. Lett. **30**. 2855 (2005). [CrossRef] [PubMed]

3. Ming-Jun Li et al., “Al/Ge co-doped large mode area fiber with high SBS threshold,” Opt. Express **15.**8290 (2007). [CrossRef] [PubMed]

^{2}at 1550 nm [4

4. S. Ramachandran et al., “Light propagation with ultralarge modal areas in optical fibers,” Opt. Lett. **31**, 1797 (2006). [CrossRef] [PubMed]

_{eff}that may appear in other LMA fiber designs [5

5. J. M. Fini and S. Ramachandran, “Natural bend-distortion immunity of higher-order-mode large-mode-area fibers,” Opt. Lett. **32**, 748 (2007). [CrossRef] [PubMed]

_{08}mode in a 50 m length of a passive HOM fiber with A

_{eff}=1714 µm

^{2}and for the LP

_{01}mode in a 20 m length of the same HOM fiber with A

_{eff}=61.5 µm

^{2}. It is found that the SBS threshold is determined by the ratio of the modal power P to the modal effective area, P/A

_{eff}, and not by the intensity maximum of the LP

_{08}mode whose peak intensity, at the same modal power, can exceed P/A

_{eff}by 12 dB. Furthermore, heterodyne measurements demonstrate that the same core-localized acoustic phonon generates the thermal Brillouin scattering event that seeds the SBS process in each case, thereby elucidating the SBS mechanism. The identification of the underlying SBS mechanism may be useful in the design of HOM fibers that exhibit further reductions in the SBS efficiency C

_{B}and higher SBS thresholds, thereby facilitating the development of higher power, narrow linewidth fiber amplifiers. This report expands upon recent initial measurements of the SBS gain efficiency in HOM and single-mode-fiber (SMF) fibers [6

6. M. D. Mermelstein, S. Ramachandran, and S. Ghalmi, “SBS Gain Efficiency Measurements in a 1714 µm^{2} Effective Area LP_{08} Higher Order Optical Fiber,” in *Conference on Lasers and Electro- Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies*, OSA Technical Digest Series (CD) (Optical Society of America, 2007), paper CTuS1.

_{B}is developed in this section. This is followed by the conclusion is Section 4.

## 2. Experiment

_{08}and LP

_{01}intensity profiles and a near-field image of the LP

_{08}mode after 50 m of propagation in the HOM fiber are shown in Fig. 1.

_{01}power of ~45 W at the tap output. The absence of SBS generation in the SMF was confirmed experimentally.

_{08}experiment, light is delivered to the 50 m HOM module in the LP

_{01}mode and coupled to the LP

_{08}mode with a long period grating (LPG), as shown in Fig. 2. The coupling efficiency is >99% and the output modal image was monitored to confirm that the signal remains in the desired LP

_{08}mode after 50 m of propagation. Shown in Fig. 3 are backscatter optical spectra as a function of the injected peak pump power P

_{P}. These spectra were taken with a 10 pm resolution optical spectrum analyzer (OSA). The reciprocity and high conversion efficiency of the LPG insures that the optical radiation appearing at the backscatter port of the tap originated from light propagating in the LP

_{08}mode in the HOM fiber. The growth of the Stokes peak is evident at the expected Brillouin shift frequency of ~15 GHz. The Stokes powers P

_{S}were extracted from the optical spectra after corrections for the tap splitting ratio, splice losses and other experimental details. The SBS reflectivity is defined as: R

_{SBS}=P

_{S}/P

_{P}. The data points in the inset to Fig. 3 show the SBS reflectivity as a function of the single pass gain G=C

_{B}P

_{p}(0)L (i.e. a normalized pump power) where C

_{B}=γ g

_{B}/A

_{eff}is the SBS gain efficiency,

*γ*is a polarization factor approximately equal to 1 at high gain G in the low-birefringent fiber [7

7. M. O. van Deventer and A. J. Boot, “Polarization properties of stimulated Brillouin scattering in single-mode fibers,” J. Lightwave Technol. **12**, 585 (1994) [CrossRef]

_{B}is the SBS gain coefficient. The fitting function R

_{SBS}(G) is developed in Section 3 and is given by Eq. (7). The modal effective area A

_{eff}is defined as [8]:

*E*

_{n}*r*is the electric field distribution of the n

^{th}fiber mode (either LP

_{08}or LP

_{01}) and the angular brackets <…> represent an integration over the cross section of the fiber. A SBS reflectivity equal to 10

^{-5}is achieved at a peak power of ~25.5 W. The maximum reflectivity was determined by the available peak pump power and the need to avoid an excessive amount of counter propagating light from entering the 3

^{rd}stage amplifier and generating parasitic lasing. No pump depletion at the fiber output was observed.

_{01}mode. Figure 4 shows the growth of the backscattered LP

_{01}Stokes spectral component as a function of the LP

_{01}pump power and the inset shows the R

_{SBS}vs. G. In this case the SBS reflectivity equals 10

^{-5}at a peak power of ~2.5 W.

_{08}and LP

_{01}modes exhibiting Brillouin shifts ν

_{B}of 15.19 and 15.20 GHz, respectively. These measurements were taken at an SBS reflectivity approximately equal to 10

^{-5}which is 30 to 40 dB above the thermal Brillouin reflectivity of 10

^{-9}to 10

^{-8}.

*ν*-

*ν*

_{B})

^{2}/2σ

^{2}] where ν is the frequency and σ is a fitting parameter [10

10. R. W. Boyd et al., “Noise initiation of stimulated Brillouin scattering,” Phys. Rev. A **42**, 5514 (1990). [CrossRef] [PubMed]

_{08}mode at G=7.0 and 32 MHz for the LP

_{01}mode at G=10.4. The absence of additional spectral peaks corresponding to other fiber acoustic modes in the rf spectrum and the ~30 MHz FWHM of these spectral features suggest that these measurements are taken sufficiently within the SBS regime where the scattering process is dominated by a single acoustic phonon. This indicates that the Stokes powers extracted from the low resolution OSA measurements correspond to scattering from a single acoustic mode. This is examined more closely in the following section which calculates the phonon spectra.

## 3. Modeling and Analysis

_{08}or LP

_{01}modes is scattered by density fluctuations exhibiting a transverse distribution

*ρ*(

*r*), associated with the acoustic phonons, in a thermal Brillouin scattering event and the frequency of the scattered radiation is shifted by the Doppler frequency. A fraction of the backscattered radiation is captured by the fiber modes and propagates in the backward direction The counter propagating electric fields generate a forward traveling intensity interference pattern which generates a forward propagating pressure wave by means of the electrostrictive effect. This pressure wave can induce stimulated scattering in two manners. The signal laser light is Brillouin scattered by the electrostrictively-generated pressure wave. The mixing of this scattered light with the signal radiation further enhances the pressure wave, leading to stimulated scattering. Also, the pressure wave can mechanically drive the thermal phonon that caused the original thermal Brillouin scattering event, thereby increasing the stimulated scattering [11].

12. E. Peral and A. Yariv, “Degradation of modulation and noise characteristics of semiconductor lasers after propagation in optical fiber due to phase shift induced by stimulated Brillouin scattering” IEEE J. Quantum Elect. **35**, 1185 (1999). [CrossRef]

^{2}

_{⊥}is the transverse Laplacian operator, f is the acoustic frequency in Hz, Λ

_{0}is the acoustic wavelength in silica,

_{eff}is the effective acoustic index. The acoustic refractive index is given by

*λ*

_{0}is the optical wavelength in vacuum and n

_{eff}is the effective optical index. The normal modes of vibration

*ρ*

*(*

_{m}*r*)are labeled by the index m and determined by Eq. (2). Those acoustic modes that also satisfy the Bragg condition given by Eq. (3) will participate in the thermal Brillouin scattering event. The acoustic eigenfrequencies are expressed in terms of the acoustic effective index

*N*

^{m}*:*

_{eff}9. Y. Koyamada, et al., “Simulating and designing Brillouin Gain Spectrum in single-mode fibers,” J. Lightwave Technol. **22**, 631 (2004). [CrossRef]

_{m,n}for the m

^{th}acoustic mode with the n

^{th}optical mode:

*ρ*

*(*

_{m}*r*) and captured by the same optical mode is proportional to the overlap integral Γ

_{m,n}. Shown in Figs. 8(a) and 8(b) are stem plots of the acousto-optic overlap integrals Γ

_{m,n}as a function of the modal acoustic frequency for the LP

_{08}and LP

_{01}optical modes, respectively. The acoustic modes with the greatest overlap with the LP

_{08}and LP

_{01}optical modes appear at frequencies of 15.30 GHz and 15.35 GHz and differ from the measured frequencies presented in Fig. 5 by 0.11 GHz and 0.15 GHz, respectively. This corresponds to a discrepancy of <1% and is within experimental error of the sound speed measurements.

_{08}and LP

_{01}optical modes, respectively. Hence, the rf spectra and phonon mode structure indicate that the acoustic phonon that seeds the SBS process is localized in the optical core region.

_{B}may be extracted from the SBS reflectivity data presented in Figs. 3 and 4 with the aid of a modified Brillouin amplifier model. This is accomplished by including a distributed thermal Brillouin scattering source term in the coupled differential equations for the pump and Stokes radiation [8]:

_{S}=β/2 (equal Stokes and anti-Stokes components) is the total Stokes scattering coefficient per unit length, and η is a capture fraction equal to the ratio of backscatter power captured by the fiber mode to the total thermal Stokes Brillouin scattered power. Note that the thermal Brillouin scattering coefficient is proportional to the overlap integral Γ

_{m,n}[9

9. Y. Koyamada, et al., “Simulating and designing Brillouin Gain Spectrum in single-mode fibers,” J. Lightwave Technol. **22**, 631 (2004). [CrossRef]

_{S}and η are left as phenomenological coefficients. The Brillouin amplifier model is illustrated in Fig. 10. Equations (6) may be solved analytically under the following conditions: i) the intrinsic fiber loss is negligible for short fiber amplifier lengths so α may be set equal to zero in Eqs.

_{SBS}≪1) there is negligible pump depletion. In this case the pump power is nearly constant and equal to the injected power, i.e.

*P*

*(*

_{P}*L*)=

*P*

*(0). Note that βS is retained in Eq. (6b) since it provides the distributed source. Eq. (6b) may be solved analytically with the additional boundary condition that no Stokes power is provided at the far end of the amplifier, i.e.*

_{P}*P*

*(*

_{S}*L*)=0. The SBS reflectivity

*R*

*=*

_{SBS}*P*

*(0)/*

_{S}*P*

*(0) can then be written as:*

_{P}10. R. W. Boyd et al., “Noise initiation of stimulated Brillouin scattering,” Phys. Rev. A **42**, 5514 (1990). [CrossRef] [PubMed]

*R*

_{0}=

*η*·

*β*

*·*

_{S}*L*. It is used to analyze the SBS reflectivity data presented in Figs. 3 and 4. The curves appearing in these figures are two-parameter non-linear least squares fit of R

_{SBS}to the data. The two parameters are

*η*·

*β*

*and C*

_{S}_{B}. The results of the fitting routine are shown in Figs. 3 and 4. Recently, an acousto-optic effective area [Ref.13, Eq. (8)]

*A*

^{ao}_{m,n}=

*A*

*/Γ*

_{eff}_{m,n}has been introduced to replace the modal effective area A

_{eff}appearing in the gain efficiency C

_{B}and the propagation differential equations Eqs. (6). The products of the SBS gain efficiency and either the optical effective area or acousto-optic effective area is solely a function of the fiber material parameters, optical parameters and the thermal phonon lifetime, and is independent of the acoustic and optical mode structure:

_{B}is given by [8]:

_{12}is the Pockel’s coefficient, ρ is the density, VS is the sound speed and Δ

*ν*

*is the thermal phonon full-width at half-maximum (FWHM). The thermal phonon linewidths (FWHM) Δ*

_{ph}*ν*

*are determined from the rf spectral linewidth Δν by accounting for the gain-narrowing with the relation:*

_{ph}10. R. W. Boyd et al., “Noise initiation of stimulated Brillouin scattering,” Phys. Rev. A **42**, 5514 (1990). [CrossRef] [PubMed]

_{08}and LP

_{01}modes, respectively. The independently measured optical effective areas and the smallest calculated acousto-optic effective areas are shown in columns 2 and 3 of Table I for the LP

_{08}and LP

_{01}modes. Equation (8) suggests that the gain efficiency-area product is nearly a constant. This is consistent with the experimentally determined gain efficiency-optical effective area products for the two modes shown in column 4, which are nearly equal. However, when the acousto-optic effective area is used, the product for LP

_{08}mode is 5.4 times greater than that for the LP

_{01}mode, conflicting with Eq. (8). Therefore, the measured SBS gain efficiency C

_{B}for the HOM fiber scales with the optical effective area A

_{eff}.

Table I | C_{B} [m-W]^{-1}
| |||||

A_{eff} [µm^{2}] | A_{m,n}
^{ao} [µm^{2}] | C_{B} A_{eff}
| C_{B}A^{ao}
_{m,n}
| meas. | calc. | |

LP_{08}
| 1714 | 7166 | 14.6 | 87 | 0.0085 | 0.0077 |

LP_{01}
| 61.5 | 80 | 13.1 | 16 | 0.20 | 0.21 |

15.
See ref. [8], the refractive index n=1.48, the Pockel’s coefficient p_{12}=0.27, c is the speed of light in vacuum, λ=1083 nm is the wavelength, ρ=2221 kg/m^{3} is the density, V_{S}=5661 m/s is the sound speed and the phonon FWHM is 123 MHz for the LP_{08} mode and 105 MHz for the LP_{01} mode.

*γ*=1. The calculated results for C

_{B}presented in Table I are in good agreement with the measured values. Calculated results for ηβ

_{S}are not presented since the capture fractions η and scattering coefficients β

_{S}are not available for Brillouin scattering in an HOM fiber.

_{SBS}<10

^{-4}). Generally, the onset of SBS is quantified by a threshold power P

_{th}corresponding to a chosen SBS reflectivity for a given fiber length. The SBS threshold powers at higher reflectivities may be determined by extrapolating Eq. (7) to higher pump powers. Shown in Fig. 11 is a plot of the SBS reflectivity as a function of pump power for the LP

_{08}and LP

_{01}modes for a 20 m length of HOM fiber using Eq. (7) and the measured parameters given in Figs. 3 and 4.. It is seen that the SBS threshold powers at 1% reflectivity are 117 W for the LP

_{08}mode and 4.3 W for the LP

_{01}mode. The ratio of threshold powers P

^{(08)}

_{th}:

*P*

^{(01)}

_{th}: is 27 and is nearly equal to the ratio of the independently measured modal effective areas

*A*

^{(08)}

*:*

_{eff}*A*

^{(01)}

*of 28, further confirming the scaling of the threshold power with A*

_{eff}_{eff}. Alternatively, an expression for the threshold power at an arbitrary SBS reflectivity R

_{SBS}≪1 and fiber length L may be found by manipulating Eq. (7):

*e*

*≫1. Eq. (10) is a transcendental equation for the threshold powers P*

^{G}_{th}and is of similar form to that first introduced by Smith [16

16. R.G. Smith, “Optical power handling capacity of low loss optical fibers as determined by stimulated Raman and Brillouin scattering,” Appl. Opt. **11**, 2489–2494 (1972). [CrossRef] [PubMed]

_{th}exhibits a weak logarithmic dependence on the overlap integral Γ

_{m,n}through the scattering coefficient β

_{S}. Eq. (10) may also be written in a more compact form in terms of the threshold gain Gth and thermal Brillouin reflectivity:

*G*

_{th}=In[(

*R*

*/*

_{SBS}*R*

_{0})·

*G*

*] where*

_{th}*G*

*=*

_{th}*C*

*·*

_{B}*P*

*·*

_{th}*L*. Ref. [13] introduces a correction factor to the value 21 accounting for the possible contribution to SBS my several acoustic phonons. However, this correction is small for the HOM fiber acoustic mode spectra presented in Fig. 8.

## 4. Conclusion

^{-4}) measurements of the SBS gain efficiency for the LP

_{08}and LP

_{01}modes in an HOM fiber have been presented. These results indicate that the SBS generation in these fibers originates from a core-localized acoustic phonon and confirm that the gain efficiency is governed primarily by the modal effective area Aeff and not the peak intensity of the highly structured optical mode. This understanding of the SBS mechanism in the HOM fiber may be useful in designing HOM fibers with enhanced SBS suppression [13]. A useful figure-of-merit for amplifier applications is the threshold power-length product:

*P*

*·*

_{th}*L*≅/

*C*

*[8] which characterizes the trade-off between peak power and amplifier length. Hence, it is desirable to minimize C*

_{B}_{B}and therefore maximize A

_{eff}. The very large modal effective area of 1714 µm

^{2}of the LP

_{08}mode yields a threshold power-length product of 2.5 kW-m, making this waveguide an excellent candidate for high-power single-frequency lasers and amplifiers.

## Acknowledgments

## References

1. | A. Liem et al., “100-W single-frequency master-oscillator fiber power amplifier,” Opt. Lett. |

2. | W. S. Wong et al., “Breaking the limit of maximum effective area for robust single-mode propagation in optical fibers,” Opt. Lett. |

3. | Ming-Jun Li et al., “Al/Ge co-doped large mode area fiber with high SBS threshold,” Opt. Express |

4. | S. Ramachandran et al., “Light propagation with ultralarge modal areas in optical fibers,” Opt. Lett. |

5. | J. M. Fini and S. Ramachandran, “Natural bend-distortion immunity of higher-order-mode large-mode-area fibers,” Opt. Lett. |

6. | M. D. Mermelstein, S. Ramachandran, and S. Ghalmi, “SBS Gain Efficiency Measurements in a 1714 µm |

7. | M. O. van Deventer and A. J. Boot, “Polarization properties of stimulated Brillouin scattering in single-mode fibers,” J. Lightwave Technol. |

8. | G. P. Agrawal, |

9. | Y. Koyamada, et al., “Simulating and designing Brillouin Gain Spectrum in single-mode fibers,” J. Lightwave Technol. |

10. | R. W. Boyd et al., “Noise initiation of stimulated Brillouin scattering,” Phys. Rev. A |

11. | I. L. Fabelinskii, |

12. | E. Peral and A. Yariv, “Degradation of modulation and noise characteristics of semiconductor lasers after propagation in optical fiber due to phase shift induced by stimulated Brillouin scattering” IEEE J. Quantum Elect. |

13. | A. Kobyakov et al., “Design concept for optical fibers with enhanced SBS threshold,” Opt. Express |

14. | Courtesy of James Hou, Sonix Inc., Springfield, VA. |

15. |
See ref. [8], the refractive index n=1.48, the Pockel’s coefficient p |

16. | R.G. Smith, “Optical power handling capacity of low loss optical fibers as determined by stimulated Raman and Brillouin scattering,” Appl. Opt. |

**OCIS Codes**

(060.2320) Fiber optics and optical communications : Fiber optics amplifiers and oscillators

(060.4370) Fiber optics and optical communications : Nonlinear optics, fibers

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: August 22, 2007

Revised Manuscript: October 2, 2007

Manuscript Accepted: October 12, 2007

Published: November 16, 2007

**Citation**

M. D. Mermelstein, S. Ramachandran, J. M. Fini, and S. Ghalmi, "SBS gain efficiency measurements and modeling in a 1714 μm^{2} effective area LP_{08} higher-order mode optical fiber," Opt. Express **15**, 15952-15963 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-24-15952

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### References

- A. Liem, et al., "100-W single-frequency master-oscillator fiber power amplifier," Opt. Lett. 28, 1537 (2003). [CrossRef] [PubMed]
- W. S. Wong, et al., "Breaking the limit of maximum effective area for robust single-mode propagation in optical fibers," Opt. Lett. 30. 2855 (2005). [CrossRef] [PubMed]
- Ming-Jun Li, et al., "Al/Ge co-doped large mode area fiber with high SBS threshold," Opt. Express 15. 8290 (2007). [CrossRef] [PubMed]
- S. Ramachandran et al., "Light propagation with ultralarge modal areas in optical fibers," Opt. Lett. 31, 1797 (2006). [CrossRef] [PubMed]
- J. M. Fini and S. Ramachandran, "Natural bend-distortion immunity of higher-order-mode large-mode-area fibers," Opt. Lett. 32, 748 (2007). [CrossRef] [PubMed]
- M. D. Mermelstein, S. Ramachandran, and S. Ghalmi, " SBS Gain Efficiency Measurements in a 1714 µm2 Effective Area LP08 Higher Order Optical Fiber," in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies, OSA Technical Digest Series (CD) (Optical Society of America, 2007), paper CTuS1.
- M. O. van Deventer and A. J. Boot, "Polarization properties of stimulated Brillouin scattering in single-mode fibers," J. Lightwave Technol. 12, 585 (1994) [CrossRef]
- G. P. Agrawal, Non-Linear Optics (Academic Press, San Diego, 1995).
- Y. Koyamada, et al., "Simulating and designing Brillouin Gain Spectrum in single-mode fibers," J. Lightwave Technol. 22, 631 (2004). [CrossRef]
- R. W. Boyd et al., "Noise initiation of stimulated Brillouin scattering," Phys. Rev. A 42, 5514 (1990). [CrossRef] [PubMed]
- I. L. Fabelinskii, Molecular Scattering of Light, (Plenum Press, 1968).
- E. Peral and A. Yariv, "Degradation of modulation and noise characteristics of semiconductor lasers after propagation in optical fiber due to phase shift induced by stimulated Brillouin scattering" IEEE J. Quantum Elect. 35, 1185 (1999). [CrossRef]
- A. Kobyakov, et al., "Design concept for optical fibers with enhanced SBS threshold," Opt. Express 14, 5388 (2005).
- Courtesy of James Hou, Sonix Inc., Springfield, VA.
- See Ref. [8], the refractive index n=1.48, the Pockel’s coefficient p12=0.27, c is the speed of light in vacuum, λ=1083 nm is the wavelength, ρ=2221 kg/m3 is the density, VS=5661 m/s is the sound speed and the phonon FWHM is 123 MHz for the LP08 mode and 105 MHz for the LP01 mode.
- R. G. Smith, "Optical power handling capacity of low loss optical fibers as determined by stimulated Raman and Brillouin scattering," Appl. Opt. 11, 2489-2494 (1972). [CrossRef] [PubMed]

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