## 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

High power rare-earth doped fiber amplifiers offer several advantages compared to solid-state laser technologies including: high output powers, good power conversion efficiencies, broadband gain, good transverse mode stability and ease in thermal management. The maximum power output of the amplifier is primarily limited by fiber non-linearities. Narrow linewidth amplifiers are limited by stimulated Brillouin scattering (SBS). This is especially severe for signal linewidths less than the SBS gain bandwidth of ~75 MHz. Power levels exceeding the SBS threshold generate backscattered optical power that can limit the output power and damage optical components. Large mode area (LMA) optical fibers have been developed to increase the modal effective area Aeff and thereby increase the SBS threshold. A LMA fiber amplifier with A_{eff}=415 µm^{2} has been demonstrated [1] and a passive photonic crystal fiber with A_{eff}=1417 µm^{2} has recently been reported [2] offering the potential for further increases in the SBS threshold. In another approach, segmentation of the transverse acoustic properties of a LMA Yb-doped gain fiber resulted in a 6 dB increase in the SBS threshold [3].

Alternatively, higher-order-mode (HOM) optical fibers have demonstrated very large effective areas; up to 3200 µm^{2} at 1550 nm [4]. Light propagating in the HOM fiber exhibits low loss, high modal purity and is resistant to the bend-induced reduction in A_{eff} that may appear in other LMA fiber designs [5]. Presented here are SBS gain efficiency and threshold measurements for the LP_{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].

The report is structured in the following manner: Section 2 presents a brief description of the HOM fiber and the experimental arrangement. Brillouin spectra are taken with a high resolution (10 pm or 2.6 GHz) optical spectrum analyzer (OSA) and demonstrate the growth of the Stokes Brillouin peaks as a function of pump power. The Brillouin spectra are examined further with higher spectral resolution (3 MHz) in a heterodyne light scattering measurement that reveals a single phonon mode that contributes to the Stokes peak. Section 3 provides a modal analysis of the HOM fiber phonon spectra that identifies the vibrational mode that seeds the SBS process. The phenomenological model used to analyze the spectral data and extract the SBS gain efficiency C_{B} is developed in this section. This is followed by the conclusion is Section 4.

## 2. Experiment

The HOM fiber consists of a central core and an 86 µm diameter silica inner cladding that is surrounded by a lower index outer trench. Details of the fiber design, the LP_{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.

The experimental arrangement is shown in Fig. 2. A three-stage Yb-doped pulsed master-oscillator power-amplifier (MOPA) configuration generates the high power single frequency radiation. The seed source is a commercially available distributed feedback fiber laser exhibiting a linewidth <30 kHz and a center wavelength of ~1083.7 nm. The final amplifier stage utilizes a 20 µm mode-field-diameter Yb- doped LMA fiber. The amplifier was run at a 100 kHz repetition frequency and a pulse width of 1 µs. A mode-transformer injected the light from the LMA fiber into a SMF 1% tap where the length of the SMF was kept below 0.5 m to avoid SBS generation in the small core fiber. This arrangement provided a peak LP_{01} power of ~45 W at the tap output. The absence of SBS generation in the SMF was confirmed experimentally.

In the LP_{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] and g_{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]:

where *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.

In a second experiment, a 20 m length of HOM fiber was directly spliced onto the tap output to excite the LP_{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.

The Brillouin frequency shift of the Stokes radiation was determined in a heterodyne experiment where backscattered light was combined with the seed laser radiation with a 50% 2×2 coupler. Figure 5 shows the rf spectrum for the LP_{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}.

The full-width at half-maximum (FWHM) of the Stokes optical spectrum is determined by a non-linear least squares fit to the rf spectra utilizing a Gaussian trial function appropriate for the high gain G: exp[(*ν*-*ν*
_{B})^{2}/2σ^{2}] where ν is the frequency and σ is a fitting parameter [10]. The FWHM is given by: $\Delta \nu =2\sigma \sqrt{2\phantom{\rule{.2em}{0ex}}\mathrm{ln}\left(2\right)}$ and is found to be 33 MHz for the LP_{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

The SBS mechanism is illustrated in Fig. 6. Forward propagating light in either the LP_{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].

The thermal phonons or normal modes of vibration of the HOM fiber are given by the solutions to the Helmholtz equation [12, 13] with stress-free boundary conditions:

where ∇^{2}
_{⊥} is the transverse Laplacian operator, f is the acoustic frequency in Hz, Λ_{0} is the acoustic wavelength in silica, $f\xb7{\Lambda}_{0}={V}_{{\mathrm{SiO}}_{2}}$, ${V}_{{\mathrm{SiO}}_{2}}$ is the speed of sound in the silica inner cladding and N_{eff} is the effective acoustic index. The acoustic refractive index is given by $N\left(r\right)={V}_{{\mathrm{SiO}}_{2}}\u2044V\left(r\right)$ where V(r) is the speed of sound at a radius r. Brillouin scattering also requires that the Bragg condition be satisfied to ensure that the optical field resonantly excites the acoustic field, namely:

where *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}The local sound speed in the HOM optical fiber preform was measured with a scanning acoustic microscope [14]. Shown in Fig. 7 are an acoustic ‘time-of-flight’ image and the acoustic index profile of the central fiber region rescaled from the fiber preform diameter to the fiber cladding diameter. The preform measurement did not include the lower index outer trench region. The acoustic index of this region was estimated from the glass composition. The thermal Brillouin scattering will be dominated by the acoustic phonon that exhibits the maximum overlap with the optical mode intensity distribution [9]. This overlap may be expressed by the normalized overlap integral Γ_{m,n} for the m^{th} acoustic mode with the n^{th} optical mode:

The light intensity back-scattered from the forward propagating light in the optical mode by the density fluctuations *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.

Figures 9(a) and 9(b) show the normalized optical intensity modes and the normalized acoustic density fluctuation modes showing the greatest overlap for the LP_{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.

The SBS gain efficiency C_{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]:

$$\frac{d{P}_{S}}{\mathrm{dz}}=-{C}_{B}\xb7{P}_{P}\xb7{P}_{S}+\alpha \xb7{P}_{s}-\eta \xb7{\beta}_{S}\xb7{P}_{P}$$

where β is the total thermal Brillouin scattering coefficient per unit length, β_{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]. The quantities β_{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.

(6), ii) the thermal Brillouin scattering coefficient β is also small, so it can be neglected in Eq. (6a) and iii) at the onset of SBS (i.e. R_{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}This result is similar in form to that derived in reference [10] for a single mode fiber and reduces to the thermal limit at small gains G≪1, i.e. *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:

where Eq. (8) considers both possibilities for the effective area. The SBS gain coefficient g_{B} is given by [8]:

where p_{12} is the Pockel’s coefficient, ρ is the density, VS is the sound speed and Δ*ν*
* _{ph}* is the thermal phonon full-width at half-maximum (FWHM). The thermal phonon linewidths (FWHM) Δ

*ν*

*are determined from the rf spectral linewidth Δν by accounting for the gain-narrowing with the relation: $\Delta {\nu}_{\mathrm{ph}}=\Delta \nu \u2044\sqrt{\mathrm{ln}\left(2\right)\u2044G}$ [10]. They are found to be 123 and 105 MHz for the LP*

_{ph}_{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 |

The last two columns show the measured gain efficiencies (taken from Fig. 3 and 4) and the calculated gain efficiencies using measured thermal phonon linewidths and known parameters [15] with *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.

These measurements for the Brillouin gain coefficient were made in the very small reflectivity regime (i.e. R_{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):

where the gain coefficient CB is written in terms of the fundamental parameters and *e*
* ^{G}*≫1. Eq. (10) is a transcendental equation for the threshold powers P

_{th}and is of similar form to that first introduced by Smith [16]. It can be used to extrapolate the threshold powers to arbitrary reflectivities and fiber lengths, and yields the same results shown in Fig. 11. It is worth noting that the logarithmic term in the numerator of Eq. (10) evaluates to be 19.8 and 20.4 for the two optical modes at their respective threshold values. These values are close the value of 21 frequently quoted for unity reflectivity [8]. Note that the threshold power P

_{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

Low SBS reflectivity (<10^{-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

The authors thank Clifford Headley, David DiGiovanni and Alan McCurdy for helpful discussions.

## References

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

**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.

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