Stimulated Brillouin Scattering: Mitigation Techniques and Applications

Written by Nicholas M. Luzod

Faculty Advisor Robert A. Norwood

Masters report submitted to the COLLEGE OF OPTICAL SCIENCES In partial fulfillment of the requirements for the degree of MASTER OF SCIENCE IN OPTICAL SCIENCES In the Graduate college of THE UNIVERSITY OF ARIZONA 2016

STATEMENT BY AUTHOR

The report titled Stimulated Brillouin Scattering: Mitigation Techniques and Applications prepared by Nicholas M. Luzod has been submitted in partial fulfillment of requirements for a master’s degree at the University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library.

Brief quotations from this report are allowable without special permission, provided that an accurate acknowledgement of the source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his or her judgment the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author.

SIGNED: Nicholas M. Luzod

Table of Contents 1. Introduction ...... 1 2. Background ...... 2 2.1. Optical Nonlinearities ...... 2 2.2. Intensity-dependent Refractive Index ...... 3 2.3. Electrostriction ...... 4 2.4. Stimulated Brillouin Scattering ...... 6 3. SBS Mitigation Techniques ...... 10 3.1. SBS Threshold ...... 10 3.2. Increased Mode Area ...... 12 3.3. Frequency Broadening ...... 16 3.3.1. Modulators ...... 16 3.3.2. Amplitude Modulation ...... 17 3.3.3. Frequency Modulation ...... 18 3.3.4. Phase Modulation...... 19 3.4. External Stimuli ...... 24 3.4.1. Thermal Gradient ...... 25 3.4.2. Strain Gradient ...... 26 3.5. Advanced Waveguides ...... 27 3.5.1. Acoustic Mode Tailoring ...... 27 3.5.2. Photoelastic Constant Reduction ...... 29 4. Practical Uses of SBS ...... 32 4.1. Brillouin Fiber Amplifiers ...... 32 4.1.1. Optical Filtering ...... 32 4.1.2. Distributed Fiber Sensors ...... 34 4.2. Brillouin Fiber Lasers ...... 35 4.2.1. Brillouin Laser Gyroscopes ...... 36 4.2.2. Frequency Comb Generation ...... 36 4.2.3. RF Frequency Generation ...... 37 4.3. Optical Phase Conjugation ...... 38 4.4. Beam Cleanup ...... 39 4.5. Brillouin Slow Light ...... 40 5. Conclusions ...... 43

6. References ...... 44

List of Figures and Tables Figures

Figure 1. Process by which SBS arises from an input optical wave and a scattered wave from spontaneously generated acoustic phonons [7]...... 7

Figure 2. Spatial and phase profiles of six Laguerre-Gaussian modes. The fundamental mode, 01 is shown at the upper left of the image [12]...... 12

Figure 3. Diagram showing splice between single mode fiber (SMF) and multi-mode fiber (MMF). The splice region has been adiabatically tapered to significantly reduce HOM content in the MMF [16]...... 13

Figure 4. Example of a cleaved hollow-core PCF fiber (a). The hollow core is surrounded by a cladding made of air holes, which is then surrounded by a solid glass structure. Detail showing 20.4um core diameter (b) [19]...... 14

Figure 5. CAD model of a 3C fiber with an octagonal core and eight side-cores (left). Cleaved endface showing cross-section of a triple-clad 3C fiber (right) [24]...... 15

Figure 6. Schematic showing an EAM integrated with a distributed feedback (DFB) laser diode [31]...... 16

Figure 7. Grey-coded 16-QAM constellation diagram. Each circle is a unique symbol and is labeled with the binary bit sequence it represents. The real and imaginary axes are labeled I and Q, respectively. Distance from the origin represents the signal amplitude, and the angle from the real axis is the phase shift...... 17

Figure 8. The first six orders of (), the Bessel function of the first kind [39]...... 20

Figure 9. Simulated (left) and experimental (right) results for multi-frequency sinusoidal modulation used to achieve eleven equal-amplitude sidebands. Odd harmonics are used with frequencies = 30,3 = 90, 5 = 150. [45]...... 21

Figure 10. Enhancement factor vs. normalized linewidth for Lorentzian (left) and 2 (right) lineshapes for white noise phase modulation at different fiber lengths [44]...... 22

Figure 11. Evolution of effective noise spectrum over a length of fiber. The white noise signal in the time domain, separated into sements of fiber length, (top) results in an optical spectrum (bottom) which is averaged along the length of the fiber. The effective spectrum is defined as the average of all spectra from segments leading up to and including the segment of interest. For the case that optical power changes along the fiber length, each window is weighted by the relative optical power contained in that fiber length and averaged to produce the effective spectrum [46]...... 23

iii

Figure 12. Power spectral density (PSD) of an optical signal modulated with a PRBS-3 pattern at a frequency of 2GHz (left). Zoomed portion of PSD clearly showing the discrete spectral features spaced at 2 −1 [44]...... 24

Figure 13. Calculated SBS threshold enhancement factor resulting from various applied temperature profiles applied to 100m of highly nonlinear fiber [50]...... 25

Figure 14. Measurement of backscattered power from SBS in a 20m core, ytterbium-doped amplifier with and without a strain gradient applied. 190W output power of pump-limited output power was achieved. The measured Brillouin gain spectrum is shown in the inset plot in the lower right, demonstrating discrete shifts in the Brillouin frequency shift from fiber segments with varying applied strain [54]...... 27

Figure 15. Brillouin gain spectrum from splicing two equal lengths of AAG. The two AAGs were designed to have complimentary individual Brillouin gain spectra, enabling the flat output observed here [59]...... 28

Figure 16. Design of segmented core for Yb-doped PCF (left). The center of the core and the outer ring are doped such that their Brillouin gain spectra are separated by 220MHz. The resulting combined Brillouin gain spectrum (right) contains two well-defined peaks. The Brillouin gain spectrum of a similar fiber without acoustic segmentation is superimposed for comparison [56]...... 29

Figure 17. Calculated value of the Brillouin gain coefficient in a sapphire-derived core fiber for varying average alumina concentration. Experimental results for fibers A-D, each with varying alumina concentration are superimposed on the calculated curve The model predicts a significant decrease in the Brillouin gain which peaks near 88 mol%, corresponding to 92.5 wt% [62]...... 30

Figure 18. Optical band pass filter (a) and notch filter (b) created in a Brillouin fiber amplifier. The bandwidth is tunable by changing power and modulation pattern of the pump source [65]...... 33

Figure 19. Experimental setup which utilizes a Brillouin-erbium laser to selectively attenuate a transmitted optical carrier signal. Modulated signal is amplified, passes through a carrier amplitude controller (CAC) block which includes a tunable erbium fiber amplifier, and is detected at the receiver [68]...... 33

Figure 20. Two Brillouin fiber laser configuations. A ring-cavity configuration (left) created using a beamsplitter and a directional coupler. A Fabry-Perot cavity configuration (right) which uses optical circulators as mirrors and to couple pump and output beams [8]...... 35

Figure 21. Cascaded SBS in a fiber ring cavity. Backward and forward direction spectra are superimposed, showing the frequency spacing of ~10GHz between each Stokes line [78]...... 36

Figure 22. Wide bandwidth frequency comb created using a short HLNF fiber in a Fabry-Perot configuration. Detail in the five inset plots show the individual SBS Stokes lines which make up the frequency comb [80]...... 37

Figure 23. Microwave output resulting from utilizing an optical heterodyne with the pump and Stokes waves in a Brillouin fiber laser. Top and bottom plots are generated from the same cavity at different temperatures [84]...... 37

Figure 24. Diagram illustrating the difference between a normal mirror and a phase conjugate mirror. An input beam is aberrated through a medium, is reflected, passes back through the medium, and is further aberrated (top). A beam follows the same sequence of events but is reflected by a phase conjugate mirror. After passing back through the medium the aberrations from the first pass are removed [86]...... 38

Figure 25. Diagram of SBS beam combination setup (top). Six off-axis pump beams are combined into a multimode fiber and the resultant Stokes beam exits the center of the system to the left. Contour plots showing experimental irradiance patterns (bottom). a) near field pump, b) far field pump, c) near-field Stokes, d) far-field Stokes [90]...... 40

Figure 26. Real (solid blue) and imaginary (dashed) parts of the Brillouin gain spectrum. The linear dependence of the imaginary component at the center of the spectrum is responsible for the added group delay [93]...... 41

Figure 27. Simulation of optical phase before and after 8GHz wide Brillouin slow light element is applied (left). Experimental result showing optical delay of 10.7GB/s DPSK signal adjusted by varying pump power. Distortion can be seen at the higher delays [98]...... 42

Tables

Table 1. Unique optical frequency terms of the second-order nonlinear polarization resulting from the application of an optical field containing two frequency components...... 3

Table 2. Typical values of parameters related to SBS ...... 9

Table 3. Common silica fiber dopants and their effects on optical and acoustic refractive index...... 28

1. Introduction

As is the case for many scientific phenomena, Brillouin scattering was predicted long before being observed. Léon Brillouin first published his work on this inelastic scattering effect in 1922, which predicted the interaction between acoustic phonons and optical fields now knows as Brillouin scattering [1]. Independently, Leonid Mandelstam simultaneously conducted a theoretical study of the same effect in Russia, finally publishing his work in 1926 [2]. Though some disagreement on which of these scientists first predicted the effect persists, Brillouin retains the namesake in the West. Stimulated Brillouin scattering (SBS) remained unobserved until 1964, when Chiao [3] successfully characterized SBS for the first time.

Optical fiber based on silicon dioxide (SiO) was first suggested by 1966 Kao and Hockham [4]. While initial attempts produced fibers with material inhomogeneities and geometric fluctuations, rapid advances in process control and glass processing lead to low-loss fiber with attenuation near 0.2dB/km by 1986 [5]. The increase in information bandwidth possible with optical fiber compared to heritage technologies enabled a revolution in communication. Driven in part by the rapid growth of the internet in the 1990’s, optical fiber networks now stretch across continents and through oceans. This proliferation of optical fiber further enabled many of the techniques discussed in this work, either out of necessity where SBS arose as a system limit which needed to be overcome, or out of opportunity where SBS was uniquely positioned to achieve a desired effect.

This work strives to outline the theoretical framework necessary to understanding SBS, present techniques to mitigate its negative effects, and review applications where it has proven useful. While SBS can occur in any number of materials, this work focuses on SBS in optical fibers. SBS is often the limiting effect for power scaling in both passive networks and fiber amplifiers, owing to the fact that the gain for this process is orders of magnitude larger than other well-known effects such as stimulated Raman scattering (SRS). The long lengths and low attenuations which describe modern fiber networks allow for long interaction lengths where SBS can be detrimental even for power levels near 1mW. High gain present in rare-earth doped fiber amplifiers similarly creates a situation where small SBS signals can be amplified, degrading signal quality or damaging upstream optical components.

2. Background

SBS is a nonlinear effect arising from thermally-excited acoustic phonons that provide a strong interaction between pump and scattered optical fields. The basis for this effect is the coupling between strong optical fields and material density fluctuations realized through electrostriction. The following section reviews the general concepts of optical nonlinearities including polarization, susceptibility, and the nonlinear refractive index. It then continues by connecting the spontaneous Brillouin scattering process to SBS through electrostriction and the optical and acoustic wave equations. The majority of the derivations presented are summarized from Boyd’s Nonlinear Optics [6], and supplemental information is included when deemed useful to the topic.

2.1. Optical Nonlinearities

In any optical system, the polarization of the optical medium, defined as the dipole moment per unit volume, depends on the amplitude of the applied electric field. In the general case, this induced polarization can be described by:

() () () () = () + () + () … (2.1)

() where is the permittivity of free space, is the nth-order susceptibility, and () is the () () amplitude of the optical field. The first-order term of the summation, () = (), is often referred to as the linear susceptibility, while the sum of the higher order terms is referred to as the nonlinear polarization and denoted (). The susceptibility is a tensor whose rank exceeds the order of the term by one, meaning () can be represented as a second rank tensor. Often the value of the nonlinear susceptibility is defined as a single constant. This is usually the element of the susceptibility tensor which is relevant to the fields and orientations used in a given application.

We can express any arbitrary input field as the sum of any number of discrete frequency components in the following form:

() = . . + (2.2)

If we examine the contribution from this combined optical field to just the second order nonlinear polarization we find that resultant equation contains six separate frequency components. These components each represent a unique combination of the input frequencies:

Table 1. Unique optical frequency terms of the second-order nonlinear polarization resulting from the application of an optical field containing two frequency components.

Term Physical Process

Second harmonic generation Second harmonic generation + Sum-frequency generation − Difference-frequency generation = − Optical rectification = − Optical rectification

Table 1 shows the four non-zero frequency components along with two static components, and the physical processes they represent. While each term represents the potential generation of a new field, in practice usually only one frequency component is selected through a carefully chosen phase-matching condition. The same exercise can be extended to determine the set of frequency components which can be generated in the third-order nonlinear polarization from the application of three optical fields. The additional degree of freedom significantly increases the complexity, resulting in 44 non-zero frequency components and their corresponding negatives.

It is clear from (2.1) that the term which relates the optical field to the total polarization is the electric susceptibility. This term reflects the ability of an electric field to polarize a given material. As previously mentioned, the susceptibility is described by a tensor whose rank is one larger than the order. This means that each unique frequency component contributing to the second-order polarization requires a 27 element tensor to describe the coupling between the fields and the polarization.

However, a number of mathematical and material symmetries exist which reduce the number of unique elements required to describe this nonlinear coupling, helping make the expressions more easily written and understood. One notable example of this is the distinction between centrosymmetric and non-centrosymmetric materials. Centrosymmetric materials possess the property of inversion symmetry, meaning they are symmetric in all directions from a central feature. It can be shown that for centrosymmetric materials all elements of the second-order susceptibility tensor are zero. This class of materials included fused silica, which is the basis for most optical fiber. Thus, effects such as second harmonic generation do not occur in optical fiber with the absence of special doping.

2.2. Intensity-dependent Refractive Index

The refractive index is a material property which defines the ratio between the speed of light in a vacuum and in a given material. While the refractive index is usually referred to as a frequency- dependent value, its value in many optical materials is also dependent on the intensity of interacting optical fields. The refractive index is related to the dielectric constant through the equation:

(ω) = ϵ() = 1 + () (2.3)

3 where, in general, () = + is a complex number. Under normal conditions, with weakly interacting fields, () may be replaced directly with only the linear susceptibility. However, when considering the contribution from the nonlinear susceptibility, the definition of the refractive index can be extended as

= + 2|()| (2.4)

where () is the applied optical field and is the nonlinear refractive index. The second term in (2.4) bears a relationship to the third-order susceptibility which can be understood after examining the total equation for the material polarization. When considering both first and third- order susceptibility, the total polarization can be expressed as a function of a combined effective susceptibility:

() () () = () + 3χ |()| () = () (2.5)

() () = + 3χ |()| (2.6)

By replacing () in (2.3) with and squaring both sides of the equation, we find that the nonlinear refractive index is described by

3() (2.7) = 4

Alternatively, the total refractive index is often described as

= + (2.8)

where is the time-averaged optical field intensity =2 |()|. Here the units of are which may sometimes be more easily understood than the units of . We can relate , , and () through the equations

3 () (2.9) = = 4

2.3. Electrostriction

Application of an electric field to a dielectric material results in compression of the material, and thus an increase in density, . Compression of a dielectric within the applied electric field changes the density by an amount Δ, which in turn changes the material dielectric constant by

Δ = Δ (2.10)

This change in the dielectric constant leads to a change in the field energy density, Δ, which is also equal to the amount of work performed to compress the material, Δ. We can express this relation as

1 1 Δ Δ = Δ = Δ = Δ = − (2.11) 2 2

where is the strictive pressure. From (2.11) we can see that =− , where the term

= (2.12) is referred to as the electrostrictive constant. This term plays an important role in the modulation of refractive index that leads to the generation of SBS. From the relation between strictive pressure and material density, we can relate the change in density back to the applied electric field strength through the equation

1 1 Δ = = 〈 ∙ 〉 (2.13) 2 2

where = is the material compressibility. The term with represents the case for a static applied field, while 〈 ∙ 〉 represents the time-averaged field from an optical input. We can combine the relation for Δ from (2.10) and (2.13), along with the knowledge that the susceptibility relates to the dielectric constant as Δ =Δ to find:

1 Δ = 〈 ∙ 〉 = ∙ ∗ (2.14) 2

Here, we see the explicit relation between material susceptibility, electrostrictive constant, and an applied optical field. From (2.1) we know that the change in the susceptibility relates to a change in the total polarization. Thus, the nonlinear polarization resulting from a change in susceptibility can be described by =Δ, leading to the equation:

P = ϵCγ|E| E (2.15)

(2.15) contains an interaction between three fields and is equivalent to the right side of the expression for total polarization found in (2.5). Therefore, we can rewrite it with a term for the third-order nonlinear susceptibility:

() P = 3ϵχ |E| E (2.16)

Comparison of (2.15) and (2.16) shows that the third-order nonlinear susceptibility is related to the electrostrictive constant by

1 χ() = (2.17) 3

This equation shows that the density modulation resulting from a strongly interacting optical field changes the third-order susceptibility. From (2.9) we know that a change in the susceptibility results in a modification of the refractive index. It will be shown in the following section these changes in the refractive index lay the basis for SBS.

2.4. Stimulated Brillouin Scattering

Stimulated Brillouin scattering arises from a spontaneous process in which an incident optical field with angular frequency and wavevector scatters and produces an acoustic wave with angular frequency Ω and wavevector . Spontaneous scattering can occur due to material inhomogeneity resulting from any combination of fluctuations in pressure, entropy, density, temperature, or concentration. However, Brillouin scattering primarily derives from changes in density.

In most descriptions of SBS in the literature, the scattering process is initiated from interactions between an incident optical wave and acoustic phonons generated from thermal noise. The thermal noise generates a pressure variation in the material which can be related to a density change through the equation

Δ = Δ (2.18) where Δ and Δ denote the incremental density and pressure changes, respectively. The light scattered by the pressure variations must obey the driven wave equation

1 (2.19) ∇ − = where the polarization is given by

(, ) = Δ(, )(, ) (2.20)

Combining (2.19) and (2.20) yields

∇ − = − ( − Ω) Δ∗()∙() (2.21) ()∙() + ( + Ω) Δ + . . where the two terms in the brackets on the right side of the equation represent the scattered Stokes and anti-Stokes waves. For the stokes wave, the scattered wave is described by a wavevector = − , and consequently = −Ω. Since Ω ≪ ,, ≅ and ≅ . Thus, the magnitude of the acoustic wavevector and frequency are described by

(2.22) |q | = | + | = 2|| sin 2

2nων (2.23) Ω = sin c 2

where is the angle between wavevectors and . For single-mode fibers, the scattered wave can only be guided in the forward (0°) or backwards (180°) directions.

From (2.23) we can see that the acoustic phonon frequency is a maximum for scattering in the backwards direction and approaches zero for forward scattered light, and thus does not occur. It should be noted that forward, co-propagating Brillouin scattering is possible as the input optical field can interact with transverse, rather than longitudinal, acoustic fiber modes in a phenomenon known as guided acoustic wave Brillouin scattering [7]. However, a full description of this effect is outside the scope of this work and will not herein be considered.

Since the incident and scattered optical fields differ by Ω = − , beating between the two fields also occurs at this frequency. For a strong incident optical field this interaction drives density modulations in the material through electrostriction which can be described physically as a traveling acoustic wave, also with frequency Figure 1. Process by which SBS arises from an input Ω. The density changes in the material function optical wave and a scattered wave from as a Bragg grating, scattering more of the input spontaneously generated acoustic phonons [7].

7 field and driving a feedback loop which constructively amplifies both the Stokes and acoustic fields.

The interaction between the input, Stokes, and acoustic fields can be described by a set of coupled amplitude equations derived from the wave equation in (2.19) and the corresponding acoustic wave equation:

− Γ∇ − ∇ = ∇ ∙ f (2.24) where f is the force per unit area and Γ′ is a damping parameter for the material. The relation between the Brillouin linewidth and the damping parameter is given by Γ = Γ′. The phonon lifetime is the reciprocal of the Brillouin linewidth, =Γ . The optical field inserted into the wave equation (2.19) is given by (, ) = (, ) + (, ), where pump and Stokes fields are

() (, ) = (, ) + . . (2.25)

() (, ) = (, ) + . . (2.26)

Similarly, the acoustic field is described by density fluctuations of the form

() (, ) = + (, ) + . . (2.27)

Under the approximations that the acoustic amplitude varies slowly and that hypersonic phonons are strongly damped, the acoustic amplitude relates to the optical amplitudes through the equation

∗ (2.28) ( ) , = Ω − Ω − ΩΓ

The acoustic amplitude can also be shown to be related to the polarization by

() (2.29) = + . .

∗ () (2.30) = + . .

Then, by substituting the optical fields and polarizations into the wave equation, we arrive at the coupled amplitude equations

|| (2.31) = 2 Ω − Ω − ΩΓ

− || (2.32) = 2 Ω − Ω + ΩΓ

∗ Finally, definition of the intensities as =2 allows for rearranging these equations into the form

(2.33) = − where we implicitly assume the pump is not depleted along the interaction length. The SBS gain factor in (2.33) is given by

(Γ⁄2) (2.34) (Ω) = (Ω − Ω) + (Γ⁄2) where the maximum gain is

(2.35) = Γ

The Brillouin gain spectrum in (2.34) is defined by a Lorentzian shape with a FWHM of Γ. Several typical values related to SBS are listed in the table below.

Table 2. Typical values of parameters related to SBS

Parameter Symbol Value Unit Brillouin Frequency Ω/2 11 @ 1550nm, 16 @ 1064nm GHz Shift Brillouin Gain 2.510 m/W Brillouin Linewidth Γ 20-100 MHz Acoustic velocity 5900-6000 m/s

3. SBS Mitigation Techniques

3.1. SBS Threshold

The SBS threshold is often referred to in the literature and is useful for generating an order of magnitude estimate of the power a fiber can carry before SBS causes the system to fail. In the analysis below, we assume a narrow linewidth pump wave propagates forward along a fiber of length L, generating SBS in the backwards direction. The coupled mode equations that describe the generation of the SBS are [8]:

= − − (3.1)

− = − (3.2)

Here (Ω) is the frequency-dependent Brillouin gain coefficient, is the pump intensity, and is the Stokes intensity. These equations are valid for a CW or quasi-CW input pump and assume that the fiber maintains polarization along the fiber for both pump and Stokes waves. Since the goal is to generate an approximate value for the SBS threshold, we can assume that the pump is not significantly depleted. Thus, the first term on the right of (3.1) goes to zero, allowing us to solve the equation and substitute it into (3.2), yielding:

− = − (3.3)

Solving (3.3) yields:

(3.4) (0) = ()

where = , with referring to the effective mode area of the waveguide, and is the effective length which is calculated as = in a passive fiber. The Stokes power can further be approximated by multiplying (3.4) by the photon energy ℏ and integrating over all frequencies. This integral determines the Stokes power by assuming an imaginary photon is incident at length L of the fiber, which has been shown to be equivalent to spontaneous Brillouin scattering events occurring as the pump propagates along the fiber length [9]. By recognizing that the integral is dominated by the region around the gain peak where = , the power can be expressed as:

( ) (3.5) (0) =

where Ω = − , =ℏ, and describes the effective bandwidth of , which can

be expressed formally as = / . Here, the relation for assumes that the fiber supports a single mode.

In this context, the SBS threshold is generally considered to be the power at which the Stokes and pump waves are equal at either end of the fiber. Since we are using the undepleted pump approximation, the pump and Stokes power are both equal to the pump power attenuated over the length, L, of the fiber:

(0) = () = (3.6)

We can then combine (3.5) and (3.6) to yield:

() (3.7) =

Under the assumptions that the above interaction takes place in a low loss fiber with = 20/, the Brillouin linewidth has a Lorentzian form, and the pump linewidth is less than the Brillouin linewidth such that the entire pump power can contribute to the generation of SBS; Smith [9] gives the following approximation for the SBS threshold:

(Ω) ≅ 21 (3.8)

Since (3.8) was initially derived in 1972, many researchers have created new models and conducted experiments to better understand the SBS threshold. Some researchers have found critical gain values in the 16-19 range [10] [7] rather than 21 as originally approximated by Smith. While the exact value is not unanimously agreed upon, the relation shown above still acts as a useful method to estimate the SBS threshold.

As previously stated, the above relations are valid only for a steady-state, CW system. For pulsed lasers, the pulse width and pulse repetition rate (PRF) can have a significant influence on the SBS dynamics [8]. For high PRF systems in the GHz regime, the SBS threshold is reduced, but only by a small factor. The short period for this type of system means that multiple pulses can interact with the same acoustic wave, contributing to SBS. However, if pulse widths are on the order of the acoustic phonon lifetime of approximately 10ns and the PRF is reduced to the MHz regime or lower, SBS is significantly reduced compared to the CW case. This is, in part, because the acoustic grating fully decays between pulses. Since the Stokes and pump waves propagate counter to each other, the effective interaction length, , reduces to approximately Δ/2 [9], where is the group velocity and Δ is the pulse width, increasing the SBS threshold significantly.

The SBS threshold can also be affected by factors such as the temporal shape of the pulses propagating in the fiber [11]. In many applications, pulse shape is critical to system performance. In high power fiber amplifiers, gain steepening causes the pulse to “lean” forward as it is amplified due to gain saturation. The pulse can be pre-shaped to “lean” backwards to mitigate this effect on the final output pulse. However, this pre-compensation of the pulse shape has been shown to reduce the SBS threshold.

3.2. Increased Mode Area

Rearranging (3.8) shows how the general parameters, , , and (Ω) affect the SBS threshold power :

21 ≅ (3.9) (Ω)

It is immediately apparent that there is a linear relationship with the effective mode area, . For long distance fiber networks that are used in communications applications, optical network designers are limited to small core fibers which only support the fundamental mode. A single mode fiber is defined as one which has a V parameter less than 2.405, where the V number is defined by = . Multimode fibers are generally not used in communication systems to avoid modal dispersion, which causes dramatic signal degradation over long distances.

However, in CW fiber amplifiers, high peak power fiber lasers, and other short length-fiber applications where modal dispersion may be a concern secondary to managing non-linear effects, the mode field area can be increased. These systems must manage the additional spatial modes supported by a larger fiber core so that these modes do not degrade that system’s set of requirements. The modal content within a fiber can be described as the superposition of one or more Laguerre- Gaussian modes, some of which are shown in Figure 2. These cylindrical modes are denoted as , where m denotes the number of azimuthal direction nodes and n denotes the number of nulls in the radial direction. The fundamental, or Gaussian, mode is thus referred to as . One system parameter which degrades with increased mode Figure 2. Spatial and phase profiles of six Laguerre-Gaussian modes. The content is beam quality, often fundamental mode, is shown at the upper left of the image [12].

12 characterized numerically by the parameter. The factor compares the measured beam waist and divergence to an ideal, single mode Gaussian beam at the same wavelength, and is described by the equation:

= 4 (3.10)

() Here is the beam waist radius, and is the variance of the spatial frequency given by = [12]. For an ideal Gaussian beam, the product = . Therefore, a perfect system is characterized by =1. Realistically, most applications which require the system output to propagate long distances in free space specify that the laser has ≤ 1.2. Since increasing the core size increases the number of modes a fiber supports at a given wavelength, a system with good beam quality must actively manage modal content to maintain beam quality. One way to control higher-order mode (HOM) content is through preferential mode filtering using controlled fiber bends. Generally, decreasing the allowable bend radius causes additional attenuation. However, the attenuation coefficient in a coiled fiber is larger for HOMs than for the fundamental mode [13], allowing for pseudo-single mode operation even while using fiber with a core size above the single-mode cutoff. In a fiber amplifier, controlling modal content at the input to the gain section is especially important, as the HOMs will decrease the effective amplifier gain for the fundamental mode.

In most fiber amplifiers a single type of fiber is used for a given amplification stage, meaning the mode area remains constant even as the laser intensity increases. The SBS Stokes shift is constant throughout the amplifier, so reflected SBS power from the end of the fiber seeds more SBS at the front of the fiber. A segmented amplifier, assembled by splicing two or more dissimilar fibers together, helps mitigate SBS by introducing a discontinuity in the physical fiber structure, disrupting buildup of the acoustic grating. The implementation can consist of multiple fibers with increasing mode field diameter, increasing the effective area as the signal is amplified. The splices between these dissimilar size fibers may require a taper so that excessive HOMs are not generated at the splice, causing beam quality degradation and potential thermal issues. A taper may also be introduced at or towards the end of an LMA fiber amplifier [14].

Figure 3. Diagram showing splice between single mode fiber (SMF) and multi-mode fiber (MMF). The splice region has been adiabatically tapered to significantly reduce HOM content in the MMF [15].

Photonic crystal fiber (PCF) uses a periodic structure of glass and air to create a waveguide tuned for a range of wavelengths. There are two general classes of PCF, separated distinguished by the mechanism which allows the fiber to act as a waveguide. The first class consists of a set of air holes embedded in a solid material, usually fused silica, arranged in a triangular pattern. One or more of the air gaps are removed at the center of the fiber, creating a core with an effective refractive index higher than the surrounding patterned air-hole structure [16]. This type of fiber is referred to as index-guided PCF. Compared to traditional step-index fibers, this type of PCF is capable of maintaining single-mode operation over a much wider range of wavelengths. Single mode operation has been shown from the visible to the infrared regimes in a single fiber [16]. The second class of PCF is referred to as photonic bandgap fiber (PBG) [17]. PBG fiber utilizes a symmetric hole pattern designed to create a photonic bandgap, as shown in Figure 4, where light is confined to the core. Unlike index-guided PCF, PBG fiber can be designed to guide light in a hollow core, leading to a number of unique applications.

The fiber preform for PCF fiber is assembled using an array of fused glass rods and glass capillaries arranged in a specific design. The fiber is then heated and pulled in a draw tower in a similar method to standard, solid fiber.

In the context of SBS suppression, PCF offers single- mode operation with core sizes much larger than in step-index fiber. For example, NKT photonics produces a commercially-available Yb3+- doped active fiber (DC-200/40- PZ-Yb) for applications near 1060nm which has a 29µm mode field diameter [18]. At this wavelength, step-index single Figure 4. Example of a cleaved hollow-core PCF fiber (a). The hollow core mode fiber like Corning PM980 is surrounded by a cladding made of air holes, which is then surrounded by a solid glass structure. Detail showing 20.4um core diameter (b) [19]. has a mode field diameter around 8µm; a simple calculation shows that the step-index equivalent fiber has approximately 7% the effective area of the commercial single-mode PCF. PCFs with even larger core diameters are available in the form of long, rigid glass rods. Fiber amplifiers created using PCF rods have been shown to support generation of pulses up to 1.35MW peak power in 100um diameter core Yb3+-doped PCF rods [19].

However, PCF does have some undesirable qualities. The air-glass structure makes splicing extremely difficult, as the low thermal mass makes it susceptible to overheating and collapse of the waveguide structure. While splicing PCF is achievable, it often requires expensive glass processing equipment and experienced operators. Coupled with high fiber cost, sometimes in the range of >$1k/m, PCF experimentation and development can be prohibitive to small companies and university groups that do not have the necessary experience or capital.

Chirally-coupled-core (CCC, or 3C) fiber represents another fiber innovation geared to increase core size while maintaining effective single mode operation. 3C fiber is designed as a solid structure, unlike the air-hole-filled PCF. The fiber contains a single large high index core with one or more smaller side-cores which wrap around the main core in a helical fashion longitudinally

14 down the fiber. This fiber structure is particularly interesting because it can be handled similarly to standard single-mode step-index fiber in that it can be easily spliced and routed without particular attention paid to maintaining a specific bend radius [20]. The distance between cores, core sizes, and the orbiting period are specifically tuned to couple higher order modes out of the main core, into the side core, and out into the cladding. The side core acts as a weakly coupled waveguide to the main core, where the interaction between the two is governed by their optical angular momentum. The geometry of the side core creates quasi-phase matching conditions which can be expressed as [21]:

(3.11) β − β ∙ 1 + − Δ ∙ = 0

Figure 5. CAD model of a 3C fiber with an octagonal core and eight side-cores (left). Cleaved endface showing cross-section of a triple-clad 3C fiber (right) [24].

Here is the propagation constant in the central core and is the propagation constant of the side core, which is accompanied by a correction factor for the helical structure. The constants and are the rotation rate of the side core and the offset from the central core, respectively.

The physical geometry of the side-core is accomplished by rapidly spinning the fiber at a precisely controlled rate during the fiber draw. 3C fiber has been demonstrated in both passive and Yb- doped active versions [22], meaning it can be used directly in fiber amplifiers. Implementations of various core sizes, including 33µm [23] and 55µm [22], have been demonstrated, proving scaling of the design larger mode areas. Pulse energies as high as 9.1mJ with 10ns pulse widths have been reported in 55µm core P-CCC fiber [24], which includes a hexagonal core and eight side cores, as shown in Figure 5.

Several other large mode area fibers have been developed and may be used in a similar way to reduce SBS. These fibers have been fabricated and demonstrated, but they have not yet entered into widespread use. This group of fibers includes modal sieve PCF, large pitch PCF, leaky channel fiber, gain guiding fiber, higher order mode (HOM) fiber, SHARC fiber, and all-solid photonic bandgap fiber.

3.3. Frequency Broadening

3.3.1. Modulators

Optical modulators are used in conjunction with laser diodes to achieve control of amplitude, frequency, and phase. Modulators can be integrated directly into laser sources or externally applied as a separate component. Several different modulation techniques have been developed in literature and in practice, but this section will cover a subset of well-known implementations utilized for many fiber applications.

External modulators can be utilized to avoid the frequency chirp resulting from directly modulating the laser electrical drive current [25]. In a fiber communications system, external amplitude modulation can be accomplished two main ways. The first method is by splicing a separate amplitude modulator component to the laser source. One popular implementation is an integrated Mach-Zehnder interferometer. This component equally splits the signal onto two paths, both of which are constructed using LiNbO3, and then recombines the signal. A voltage can be applied to one path, shifting the phase by 180 degrees and causing destructive interference when the two paths are recombined. Since the laser is maintained at a constant power output, this method does not cause significant chirping of the laser. One drawback of using a LiNbO3 modulator are the signal losses incurred from optical coupling material absorption. Commercial Mach-Zehnder-style modulators have insertion losses in the 3-5 dB range [26] [27].

Integrated Electro-Absorptive Modulators (EAM) avoid the optical losses caused by separated LiNbO3 modulators because they are integrated in the laser diode and are constructed with similar materials. An EAM functions by utilizing the Franz-Keldysh effect [28], in which an applied external electric field modifies the bandgap of the semiconductor material. If the laser wavelength lies near absorption edge of the material, modifying the bandgap energy can result in a significant change in the value of the attenuation coefficient, leading to laser intensity modulation based on the applied voltage. Since the EAM is integrated onto the semiconductor device, it has an effect on the laser gain which can lead to a small amount of chirp on the laser signal.

Phase modulators utilize the electro- optic effect to control optical phase. This is a non-linear effect in which an external electric field is applied across a suitable crystal, usually lithium niobate (LiNbO3) or potassium titanyl phosphate (KTP). The external field modifies the nonlinear refractive index, , of the material, which changes linearly with the electrical field strength. Thus, the phase of the output signal depends on the electric field Figure 6. Schematic showing an EAM integrated with a distributed strength applied to the non-linear feedback (DFB) laser diode [31]. crystal. Crystals like LiNO3 and KTP are typically used for these devices because of their strong nonlinear properties, which allows for a significant change in the refractive index at reasonable voltages, normally in the 2-20V range.

Modern telecommunications networks use a variety of modulation techniques to encode optical signals. Some encoding techniques combine both amplitude and phase modulation to increase the effective bitrate while maintaining a constant symbol rate. Increasing the number of bits transmitted per symbol has been critical to increasing data transmissions rates across both short and long-haul fiber networks. One widespread encoding method is Quadrature Amplitude Modulation (QAM) which itself has several implementations, each with a defined number of available amplitude and phase symbol combinations. Systems as high as 2048-QAM, named because it uses 2048 amplitude/phase combinations, have been successfully demonstrated [29].

3.3.2. Amplitude Modulation

In communication systems, the optical signal must be encoded in some manner to transmit the desired data. The most obvious approach is to encode the information on the intensity of the optical source by directly modulating the drive current. This method allows for signal generation using various return to zero (RZ) or non-return to zero (NRZ) configurations. The modulation frequency for communications using direct amplitude modulation is limited by the relaxation oscillation frequency of the laser diode source [25]. The relaxation oscillation frequency, determined by the physical design of the laser diode as well as the amount of current supplied, limits most laser diodes to modulation frequencies in the 1-20GHz Figure 7. Grey-coded 16-QAM constellation range, though some high-speed diodes have been diagram. Each circle is a unique symbol and is labeled with the binary bit sequence it represents. shown to produce signal frequencies greater than The real and imaginary axes are labeled I and Q, 24GHz [30]. respectively. Distance from the origin represents the signal amplitude, and the angle from the real axis is The amplitude modulation method works by the phase shift. applying a time-varying signal directly to the laser drive input. This produces the desired change in output intensity, but also induces a change in the laser gain by changing the carrier population [25]. This type of gain change in turn dynamically affects the refractive index, which finally results in modulation of the optical phase, which can be expressed as:

Φ 1 1 = Β ( − ) − (3.12) 2

Where = is the instantaneous angular frequency, Β is the enhancement factor, = is the gain derivative, is the carrier population, and is the photon lifetime. Thus, any direct amplitude modulation induces a change in the instantaneous frequency, the magnitude of which depends on both the amplitude modulation signal as well as the coupling between amplitude and phase, described by the enhancement factor.

The practical result from the amplitude-phase coupling in (3.12) is that amplitude modulation results in linewidth broadening. The SBS gain bandwidth, , has a finite width measured to be in the 10-40MHz range for single mode fiber operating near 1550nm [31] [32]. When the laser linewidth exceeds that of the SBS gain, the laser linewidth can be divided into individual bands of width , each effectively separately contributing to SBS generation. Thus when the laser is modulated, producing a broad linewidth signal of width Δ, the SBS threshold increases by a factor of [8]

Δ = 1 + (3.13) Δ

Since this linewidth broadening is achieved by modulating the laser power, it adds excess residual noise to systems which use amplitude to encode or decode information. In communications systems, this additional noise degrades the quality of the information link, often measured using the bit error rate (BER).

3.3.3. Frequency Modulation

Direct frequency modulation of the laser source accomplishes linewidth broadening while avoiding the potentially negative side effects from changing the signal amplitude. Instead of directly modulating the laser drive current, frequency modulation can be achieved by modulating the laser cavity optical path length. This can be accomplished in a number of ways, including utilizing a piezoelectric transducer (PZT) to drive a change in the physical cavity length [33], modulating laser gain, modulating semiconductor diode gratings, and by embedding an electro- optic phase modulator within a laser cavity [34]. Changing the optical path length drives a change in the output frequency which is described by [25]:

= + Δcos (2) (3.14)

where is the central frequency, Δ is the modulation amplitude, and is the modulation frequency. The modulation amplitude can be chosen to increase the SBS threshold by the factor indicated in (3.13).

Compared with amplitude modulation, frequency modulation with the laser held at a constant amplitude can improve system SNR [35]. Frequency modulators which are integrated on-chip can also provide a size decrease when compared to external modulation. While frequency modulation is not identical to phase modulation, in the context of linewidth broadening for SBS they are functionally similar: additional frequencies are created around the optical carrier frequency according to the input modulation signal. Both phase and frequency modulators are capable of operating in the 10-40GHz range and thus can produce similar effective linewidths. Due to their functional similarity, further discussion of will be deferred to the next section.

3.3.4. Phase Modulation

As discussed in Section 3.3.1, phase modulators are used to modify the optical phase of an input signal, resulting in a change in the instantaneous frequency of an optical signal. These devices are widely used in various laser applications requiring precision control of optical phase or frequency. Several unique phase modulation schemes for achieving SBS suppression have been demonstrated in the literature.

3.3.4.1. Sinusoidal

The time-varying electric field of a single-frequency optical carrier is described by the equation:

() = (2 + ) (3.15)

where A is the signal amplitude, is the carrier frequency, and is the optical phase. Application of a sinusoidal drive signal of voltage , frequency , and phase , to an in-line phase modulator changes the electric field expression to

() = A 2 + π cos(2 + ) (3.16)

where, is the voltage required by the modulator to achieve a phase shift. Driving the phase modulator with a sinusoidal voltage produces a signal with lineshape identical to the input at the carrier frequency that is accompanied by mirror images of itself, each separated by the modulation frequency, . The number of additional frequencies and the relative amplitudes of each are dependent on the total electrical power to the phase modulator as well as the component specification. The drive voltage and can be related by the more general modulation index parameter, = \ . A Fourier expansion of (3.16) yields an output field where the complex amplitudes of the fundamental and sideband frequencies can be expressed as

( ) = , (3.17) ∑

where is the Bessel function of the first kind of order n. The first six orders of () are shown in Figure 8. By adjusting the RF power input to the modulator to = , a spectrum consisting of a combination of all () is realized. In practice, the phase modulator drive amplitude is often adjusted in-situ to achieve the desired optical spectrum.

Since the SBS gain bandwidth is finite, the amount of signal within the gain bandwidth can be minimized to increase the SBS threshold. Increasing the drive voltage of the sinusoidal modulation signal will effectively broaden the signal by splitting power into more discrete sidebands. This method was shown to suppress the SBS response in a 50km optical transmission

line by 25dB when using a 20MHz modulation frequency to achieve a spectrum 120MHz wide with six discrete sidebands [36].

In communications networks, wavelength-division multiplexing (WDM) is often used to transmit multiple data streams on the same cable by splitting the data across multiple wavelengths. WDM exists in two general implementations: coarse WDM (CWDM) and dense WDM (DWDM). CWDM uses 20nm channel spacing [37] while Figure 8. The first six orders of (), the Bessel function of the first kind [39]. DWDM uses channel spacing as narrow as 12.5GHz [38], though 25 or 50GHz spacing is more common. Since DWDM channel spacing in most commercial systems is typically no less than 25GHz, modulation on the order of 100MHz does not have a significant effect on network performance [25].

For modulation frequencies much larger than the SBS gain bandwidth, the Stokes fields from each modulated sideband are independent, and the SBS threshold can be calculated from the highest power sideband [39]:

1 = (3.18) ,()

Here, ,() is the Bessel function of the first kind evaluated at the modulation index, , with order determined by the sideband with the highest power. Since only the highest power sideband matters, the ideal case for increasing the SBS threshold is having many sidebands each with equal power. For single frequency sinusoidal modulation, the maximum number of equal frequency sidebands is three, achieved with a modulation index = 1.435. This modulation index results in equal amplitude =−1,0,1 sidebands and ~3.5dB SBS threshold improvement [40].

There are a couple of practical limits to simply increasing the modulator input voltage with the aim of increasing SBS threshold. The modulator has a damage threshold for electrical input power which fixes the maximum phase offset at modulation index = . This defines the maximum number of discrete sidebands over which the optical signal can be split. Perhaps more importantly, the laser output may have a limited operational bandwidth for a given application.

Instead of splitting the laser power over a large number of sidebands, a more spectrally efficient modulation technique is to combine multiple sine waves with varying frequencies and phases. This technique is capable of producing more equal-amplitude sidebands to further increase SBS threshold while minimizing overall spectral width. To achieve this effect, the modulation frequencies must be integer multiples of a single fundamental frequency. For the case of utilizing

odd harmonics of the fundamental frequency, 2 +1 equal amplitude spectral sidebands can be created [41].

3.3.4.2. White Noise

While using single-frequency signals to drive a phase modulator produces an optical spectrum consisting of discrete sidebands, a white noise source (WNS) can be used to produce a broad spectrum which is characterized by a more continuous envelope. White noise has constant power spectral density across a wide frequency bandwidth. For all practical applications, white noise signals are defined within a spectral band of interest, deviating from the ideal infinite width description.

For optical phase modulation, an RF white noise source is filtered to achieve the desired drive spectrum. The filter may be tunable to vary the spectral width or the shape of the output spectrum. It is clear from (3.9) that the SBS threshold strongly depends on the signal spectral width. However, both experiments and simulations have shown that the lineshape of the filtered WNS waveform also affects the SBS threshold [40]. Two filtered WNS lineshapes, the and Lorentzian, have been studied in the literature.

An analytical approach was used to derive the effective Brillouin gain bandwidth for both lineshapes, and the results were compared those derived for a single-frequency input [40]. The effective Brillouin bandwidths were used to define an enhancement factor, defined as the increase in SBS threshold for a given lineshape. For a Lorentzian lineshape, the enhancement factor was found to be = 1+ , equivalent to the result obtained in (3.13). Here, Δ is the WNS linewidth and Γ is the Brillouin bandwidth. For a lineshape the enhancement factor was reduced, characterized by the equation:

= Δ (3.19) − 1 + Γ

For large modulation bandwidths, the first term in the denominator approaches 0.64Δ/Γ.

Figure 10. Enhancement factor vs. normalized linewidth for Lorentzian (left) and (right) lineshapes for white noise phase modulation at different fiber lengths [40].

From Figure 10 it is clear the enhancement factor for short fibers is significantly reduced compared to long fibers, where the enhancement factor approaches the analytical expressions above. This discrepancy can be attributed to the difference between the power spectral density of the white noise process itself and the actual spectrum of a WNS-modulated optical signal [42]. A random process like white noise produces a spectrum that when averaged over a long period of time, yields to the power spectral density. However, the spectrum in a finite window has an effective width narrower than the power spectral density. This explanation can be applied to fiber length, where long lengths of fiber act to smooth the overall modulated spectrum, approaching the white noise power spectral density. For short fibers, less averaging takes place yielding an effectively narrower spectrum which results in a decreased SBS threshold enhancement factor. Further analysis shows that the enhancement factor for a noise-broadened optical signal, regardless of lineshape, can be described as [43]:

= (3.20) +

3.3.4.3. Pseudo-Random Binary Sequence (PRBS)

A PRBS is a binary, deterministic pattern which has gained popularity in variety of fields because it exhibits properties similar to a random source. Even though the sequence is deterministic, the autocorrelation is zero everywhere except at the origin, meaning that the value of one bit cannot be predicted given the preceding bit. A PRBS pattern is referred to as PRBS- or PRBS, where is the word size and defines the total pattern length as 2 −1. PRBS- patterns comprise all non- zero binary words of length . The pattern can be generated using a simple linear feedback shift register (LFSR), where the input bit is generated by an exclusive-or (XOR) operation on two current bit values. The LFSR seed can be any non-zero binary sequence of length .

Unlike white noise, the power spectral density (PSD) of PRBS exhibits discrete spectral features where the spacing between individual sidebands is Δ = . Here, is the modulation frequency and is the PRBS word length. The envelope is defined by a function with nulls at multiples of the modulation frequency. Figure 12. Power spectral density (PSD) of an optical signal modulated with a PRBS-3 pattern at a frequency of 2GHz (left). Zoomed portion of PSD clearly showing the discrete spectral features spaced at Figure 12 shows that, unlike (2 −1) white noise, PRBS generates a well-defined PSD which is easily manipulated by changing the modulation frequency and pattern length.

In the context of optical phase modulation, a PRBS− signal is generated and used to drive a phase modulator such that the +1 pattern state results in + optical phase shift. This technique is generally known as Binary Phase Shift Keying (BPSK). Since the generated spectrum contains discrete sidebands, the SBS threshold enhancement factor rolls off when the separation between sidebands is appreciably greater than the Brillouin linewidth [40]. For a given modulation frequency, the enhancement factor achievable with PRBS patterns of increasing is not necessarily intuitive. One analysis showed that for a CW fiber amplifier consisting of 9m of 10m core fiber, the enhancement factor was greater for a PRBS-9 pattern than for patterns with = 3, 5 ,7, 17, and 31 [40]. This result was attributed to long, unchanging binary sequences present in the = 17, 31 patterns which significantly exceed the phonon lifetime. Thus, parts of these long sequence patterns create a situation in which the input laser is effectively unmodulated for a period of time during which significant SBS can be generated, lowering the enhancement factor.

In an experimental study, a four-stage fiber amplifier was constructed using a single frequency, phase modulated laser as the input source. Various PRBS patterns were tested over a range of modulation frequencies, and the results were compared to theoretical simulations. The researchers achieved a CW output power >1kW using PRBS-6 modulated at 6GHz, compared to an output of 37W achieved without modulation [44]. Subsequent experiments using the same optical setup where the final amplifier stage was modified to increase the fiber size from 20/400 to 25/400 were able to achieve 1.17kW output power using a PRBS-5 pattern modulated at 3GHz [45]. In this experiment, the authors note that the modulation frequency could be increased to enable additional power scaling were the system not power-limited by the final stage fused fiber combiner.

3.4. External Stimuli

The Brillouin frequency shift, , defines the frequency difference between an input field and the resulting Brillouin-shifted field. As stated in Section 2.4, the value of the Brillouin frequency shift depends on a number of factors, including waveguide material composition, temperature, and strain. Material composition for a given system is usually fixed, though it can be varied by combining different materials or structures, as mentioned with segmented fiber amplifiers in

Section 3.2. Since temperature and strain are not fundamental properties of system components, they can be varied much more easily, or even adjusted dynamically to achieve the desired system performance. Increasing the SBS threshold by controlling external factors such as temperature or strain may also be preferred compared to the popular methods of frequency or phase modulation as the spectral properties of the seed are not modified. Alternatively, the external factors may be controlled in tandem with other SBS suppression techniques to further reduce the SBS gain and achieve greater power output.

Control of the Brillouin frequency shift along the length of a fiber allows for reduction in overlap between Brillouin gain spectra in differently controlled segments. Since the Brillouin gain for the same frequency shift may vary along the fiber, the length over which the SBS acoustic grating can effectively build up is reduced, raising the SBS threshold. An alternate way of viewing this effect is that the shift in Brillouin frequency shift along the fiber leads to an effective broadening of the Brillouin gain spectrum and reducing its peak value.

The SBS threshold change resulting from a non-uniform Brillouin frequency shift distribution can be described by [46]

1 = −10 log − () (3.21) 1 + Δ⁄2

where is the SBS threshold enhancement factor in dB, () is the Brillouin frequency shift at normalized distance = / along the fiber, Δ is the Brillouin gain bandwidth, and is the frequency at the peak of the Brillouin gain spectrum. From this equation we can see the intuitive result that the enhancement factor increases for fibers with narrow Brillouin gain bandwidth and large changes in Brillouin frequency shift.

3.4.1. Thermal Gradient

In one study, measurements of the Brillouin gain spectrum in erbium-doped fiber were acquired at several fiber temperatures [47]. The results showed that, for the fiber under test, the Brillouin frequency shift increased linearly with temperature. Over the measured temperature range of 20°C to 100°C, the Brillouin frequency shift increased approximately 100MHz, representing a shift approximately equal to the Brillouin linewidth in this fiber. This compares well to other studies which have found the slope of the Brillouin frequency shift with Figure 13. Calculated SBS threshold enhancement factor temperature to be in the 1-2MHz/°C range. resulting from various applied temperature profiles applied to 100m of highly nonlinear fiber [50].

Another study directly measured the SBS threshold change resulting from applying different temperature gradients to 100m of highly nonlinear fiber. The researchers observed a maximum enhancement factor of 4.8dB which occurred at the maximum experimental temperature differential of 140°C [46]. These results closely agree with calculations based on (3.21). The temperature differential is not limited by the fiber itself, but by the polymer coating applied to most fibers to increase durability. This coating typically has a maximum temperature in the range of 80-120°C.

It should be noted that fiber amplifiers contain a natural thermal gradient in the active fiber resulting from both pump absorption and signal amplification. The thermal profile created is dependent on the both the pump configuration and the method of fiber heatsinking used, if any. The thermal gradient is maximized for an uncooled, counter-pumped amplifier design. This thermal gradient can increase the SBS threshold through the same mechanism as in actively- heated passive fiber. This effect was attributed to results of one experiment which successfully produced 264W of pump-limited output power from a counter-pumped ytterbium-doped fiber amplifier [48]. This result represented a significant departure from the predicted SBS threshold of approximately 70W, and the researchers attributed the increase to the thermal gradient created by absorbed pump along the active fiber. The addition of the Brillouin frequency shift induced by a 100°C thermal gradient in the model suggested that output powers >400W were achievable without modifying the amplifier design.

3.4.2. Strain Gradient

The change in SBS threshold given an applied strain gradient is fundamentally similar to that of a thermal gradient. A strain gradient can be achieved in practice by stretching the fiber by known values of Δ in segments short compared to the total fiber length, where the induced strain is = Δ/. One study applied two different strain distributions to 500m of passive fiber and measured the increase in SBS threshold [49]. For a periodic square-wave strain distribution with the fiber divided into 160 sections, the maximum increase in SBS threshold was 3dB. The maximum strain for this measurement was = 0.36%, but the SBS threshold was approximately constant for > 0.26%. For a strain distribution comprised of 40 steps with linearly increasing strain up to = 0.67% in the final segment, a threshold enhancement factor of 7.3dB was attained. Both distributions shows good correlation between model and experiment.

Unlike with thermal gradients, there is no natural strain produced by optical pumping in a fiber amplifier. However, a strain gradient can be separately applied to further enhance the suppression of SBS in a controlled manner. In one experiment, a strain gradient was applied in a ramp configuration with maximum strain value = 1% to a 12m long, 20m core ytterbium-doped fiber [50]. The strain gradient resulted in a 7x increase in achievable power output compared to the same amplifier without any applied strain, as shown in Figure 14. The maximum output power of 190W in the strained configuration was pump-limited, meaning the SBS threshold increase was likely larger than the measured value.

While tensile strain, achieved by stretching the fiber, is a straightforward way of creating a desired strain value, fiber reliability under high tensile load may be limited [50]. Alternatively, the fiber may be compressed to achieve the desired strain; this method has been successfully applied to the tuning of fiber Bragg gratings with strains as high as 9% achieved and as high as 2% shown to be stable over a one year time span [51]. Application of large compressive strains may have the potential for further increasing the SBS threshold in short fiber amplifiers by further broadening the effective Brillouin gain spectrum.

3.5. Advanced Waveguides

A number of methods have been proposed to suppress SBS by tailoring various fiber properties. Many of these techniques utilize unique waveguide designs or material compositions to effectively suppress SBS and allow for near single-frequency operation at high power. While many have been experimentally implemented and proven, they are generally not yet commercially available.

3.5.1. Acoustic Mode Tailoring

The Brillouin angular frequency has previously been shown to be Ω =2/. It is clear from this equation that both the index of refraction and acoustic velocity help define the Brillouin scattering response. However, while both the refractive index and acoustic velocity are dependent on the waveguide structure and material composition, the sensitivities of these parameters to

changes in the waveguide design vary [52]. In general, the acoustic velocity is more sensitive to material composition changes than the optical refractive index. Fine tailoring of these parameters can be accomplished by varying common fiber dopants such as germanium, phosphorus, titanium, boron, fluorine, and aluminum to tailor the fiber’s acoustic response. The optical and acoustic refractive index responses to these dopants are shown in [53].

Table 3. Common silica fiber dopants and their effects on optical and acoustic refractive index.

Optical Refractive Index ↑ ↑ ↑ ↓ ↓ ↑ Acoustic Refractive Index ↑ ↑ ↑ ↑ ↑ ↓

Here, the acoustic index is defined as = /. Since the transverse acoustic and optical indexes can be separately designed, the overlap between the acoustic and optical fields can be significantly reduced. For most standard step index fibers, the normalized overlap integral between the optical and acoustic fields, defined by [54]

∫ ∗∗ Θ I̅ = (3.22) ∗ ∗ ∫ () Θ∫ Θ

is close to unity, and is therefore usually ignored. However, the SBS threshold is inversely dependent on the value of the overlap integral; thus, a decrease in the overlap integral directly translates to an increase in the SBS threshold. In (3.22), is the fundamental optical mode and is the longitudinal mode field of order . In acoustically anti-guiding (AAG) fiber designs, the dopants in the core are adjusted such that the acoustic index is depressed relative to the cladding. This type of structure leads to increased waveguide loss for the acoustic field and broadens the Brillouin gain spectrum [55]. The shape of the Brillouin gain spectrum for AAG fibers is also characterized by two maxima, which represent separate core and cladding resonances. In one study, two AAG fibers with slightly different dopant concentrations were spliced together and the resulting combined Brillouin gain spectrum was measured. The fibers were designed such that the individual Brillouin gain spectra shapes were complimentary, resulting in a flat combined spectrum which varied <0.5dB over 200MHz [55].

Another method of SBS suppression is to separate a fiber core into regions which are characterized by a large differential of acoustic velocity but similar refractive index. This approach is commonly Figure 15. Brillouin gain spectrum from splicing two equal lengths of AAG. The two AAGs were designed to have referred to as tailored-core or segmented complimentary individual Brillouin gain spectra, enabling the core. Core segmentation leads to a flat output observed here [59]. Brillouin gain spectrum characterized by

28 two or more separate peaks whose relative maxima are governed by the nonlinear effective area in each segment. A segmented core design was realized in one study by utilizing the stack and draw technique to fabricate an acoustically segmented, Yb3+-doped, double clad PCF fiber. The core was comprised of seven rods stacked in a hexagonal pattern with the center rod having a larger acoustic index than the surrounding six [52], as shown in Figure 16. This core design resulted in two distinct peaks in the Brillouin gain spectrum, separated by >200MHz. Five meters of this fiber were used in the final stage of a high power, single-frequency fiber amplifier. The researchers were able to produce 494W output power with an <1.3 [56].

3.5.2. Photoelastic Constant Reduction

The Brillouin gain coefficient is defined as [8]

2 = (3.23) Δ

where is the photoelastic constant and is the mass density. It is clear from (3.23) that a reduction in the photoelastic constant reduces the Brillouin gain coefficient, thus raising the SBS threshold. Examination of the expressions for the Brillouin gain coefficient given in (2.35) and (3.23) shows the photoelastic constant relates to the electrostrictive constant through the equation

= (3.24)

Like the acoustic and optical refractive index, the value of the photoelastic constant can be modified by tailoring the fiber material composition. It has been shown that alumina has a large, negative photoelastic constant [57], and thus increased alumina doping can be used to decrease the overall fiber photoelastic constant. However, standard CVD techniques are limited to alumina dopant levels of approximately 12 wt%, which is too low to have a significant effect on the value of the Brillouin gain coefficient [58].

Recent research has shown the viability of fibers comprised of a doped crystal core embedded in a silica cladding [59]. These novel fibers can allow for the use of dopants which are unable to be integrated in traditional silica cores as well as an increase in dopant concentrations. Fiber cores derived from yttrium aluminum garnet (YAG) have been demonstrated in highly doped Er3+ and Yb3+-doped configurations. In the future these types of fibers may allow for decreased active fiber lengths and thus reduced nonlinearities. Sapphire may also be used as the crystal basis for a fiber core, with the added benefit that sapphire is capable of alumina concentrations much higher than both traditional silica fibers as well as those with YAG-derived cores.

In one study, a sapphire-core fiber with silica cladding was fabricated with various alumina concentrations up to 55 mol% [58]. Several fiber lengths were tested, including measurements of the Brillouin gain spectrum. The fibers showed a significant reduction in the Brillouin gain coefficient compared to SMF-28, with a maximum measured reduction of 19dB. The measured data fit well against a model where the average alumina content was varied, as shown in Figure 17. The deep minimum shown in the figure corresponds to the model prediction that the Brillouin coefficient reaches zero at approximately 88 mol%.

Several factors in this fiber contribute to the reduction in Brillouin gain coefficient, including broadening of the Brillouin gain spectrum, introduction of AAG waveguide properties, and an increase in mass density. However, the authors note that a significant portion of this SBS reduction comes directly from the decrease in the value of the photoelastic content driven by high alumina concentrations.

While these results are promising there still exist a number of practical issues associated with the Figure 17. Calculated value of the Brillouin gain coefficient in a production of these novel crystal sapphire-derived core fiber for varying average alumina fibers [59]. For example, large concentration. Experimental results for fibers A-D, each with varying differences between the coefficient of alumina concentration are superimposed on the calculated curve The thermal expansion (CTE) in the model predicts a significant decrease in the Brillouin gain which peaks near 88 mol%, corresponding to 92.5 wt% [62]. crystal core materials and the surrounding silica cladding can lead to cracks forming during the cooling process, leading to reduced fiber reliability. Additionally, the molten-core technique used to produce these fibers can lead to diffusion between core and

30 cladding during the draw process, resulting in undesired silica content in the core or a noticeable gradient between core and cladding refractive index. Overall, these and many other issues with the production of crystal-derived fibers may be attributed to the relatively recent spike in interest in this field and the corresponding lack of research available. With time, viable material compositions and production techniques will likely make these crystal-derived fibers more practical and accessible to the optical community.

4. Practical Uses of SBS

4.1. Brillouin Fiber Amplifiers

In the preceding discussion, the Stokes field was generated from scattering induced by thermally- excited acoustic phonons. Through electrostriction, an index grating was generated which then acted to reflect additional power from the input field into the counter-propagating Stokes field. However, the SBS process can instead by seeded with a counter-propagating signal at or near the Stokes wavelength. In this way, SBS can be used to efficiently amplify a narrowband signal.

There are a number of applications for SBS fiber amplifiers, including microwave photonics, radio over fiber (ROF), and fiber sensing [7]. The set of differential equations for spontaneous Brillouin scattering given in (3.1) and (3.2) remain applicable to Brillouin fiber amplifiers, though the boundary condition for the Stokes wave is now well defined as () = . However, a viable solution of these equations for Brillouin fiber amplifiers cannot be made by assuming loss is insignificant due to the long fiber lengths used in this application. The undepleted pump approximation also does not apply well here as the pump power is usually well above threshold where appreciable power can be coupled into the Stokes wave [7]. Despite these issues, a perturbative approach can be used to find an appropriate equation describing the Brillouin gain [60]

ln 1 − = 1−Λ+ (4.1)

This equation is valid for the depleted-pump case and agrees well with experiments for amplifiers <20km long. Here, = and Λ=−ln(), where is the peak SBS efficiency. The seed in this equation is assumed to be narrowband and at the center of the Brillouin gain spectrum.

4.1.1. Optical Filtering

The gain spectrum for Brillouin fiber amplifiers is narrow, but can be widened by modulating the pump field. This effect can be used to make tunable bandpass or notch filters with widths in the GHz regime [61]. The filter widths can be tuned by changing the modulation scheme or the power of the pump, and the required pump power can be in the W regime. In a densely-packed multi- channel system these Brillouin filters can be used to select individual channels provided that the channel spacing is wider than the Brillouin gain bandwidth.

Tunable optical bandpass filters have found strong demand in their application to microwave photonics [62]. The desirability for microwave signal processing in the optical domain arises from benefits such as low loss, elimination of electromagnetic interference, and flexibility. Single bandpass filters with strong contrast and stability are critical for further development in this space. In a recent study, rectangular bandpass filters width bandwidths between 1 and 3GHz were

32 demonstrated [62]. These filters were shown to provide contrast between pass and stop bands as high as 40dB.

While optical beating is well understood for measuring relative frequency differences between two signals, absolute frequency measurement can be much more difficult and requires a well-known reference signal. In the field of optical metrology, optical frequency combs have become a critical tool for absolute frequency measurements. Frequency combs generated from mode- locked lasers can be fully characterized by their frequency spacing and the optical carrier envelope offset. Since they can extend over bandwidths of 100’s of nm, frequency combs are an ideal reference source for frequency measurement. Brillouin fiber amplification can be used to selectively amplify one peak in the frequency comb to be used as a reference for a signal under test. Since the SBS process preserves the original frequency content of the peak, there is no degradation of the measurement. A beat signal between the selected peak and the signal under test can be accurately measured with resolution near 1Hz, allowing for very accurate frequency measurements using this technique [63].

Just as SBS can be used to selectively amplify a signal, the same signal can be selectively attenuated by tuning a counter-propagating SBS seed source to the signal’s Stokes frequency. This technique has been Figure 18. Optical band pass filter (a) and notch filter (b) created in a Brillouin fiber demonstrated for carrier filtering in optical amplifier. The bandwidth is tunable by communications networks [64]. While data is encoded changing power and modulation pattern of the via modulation of the optical carrier, the carrier itself pump source [65]. does not contain any information. In many systems, the modulated sidebands are significantly lower in power than the carrier in the range of 30dB. While an increase in optical power can be beneficial to signal quality, excesss power incident on the receiver photodiode can lead to unwanted thermal or nonlinear effects. Carrier filtering can reduce total power on optical receivers and can help increase the system dynamic range.

This concept was demonstrated by integrating a Brillouin-erbium fiber laser with a long length of single-mode fiber and a tunable optical filter (TOF), as shown in Figure 19. The optical filter Figure 19. Experimental setup which utilizes a Brillouin-erbium laser to selectively attenuate a transmitted optical carrier signal. is tuned such that the EDFA operates Modulated signal is amplified, passes through a carrier amplitude near the central wavelength of the controller (CAC) block which includes a tunable erbium fiber amplifier, and is detected at the receiver [68].

33 modulated signal. Since the carrier frequency contains much more power than the modulated sidebands, it generates SBS along the 11km of SMF which is then further amplified to further attenuate the carrier. The signal is coupled out through the 3dB coupler where it can be detected by the receiver and analyzed. This system allowed the researchers to achieve 55dB of carrier attenuation without degrading the data contained in the sidebands.

4.1.2. Distributed Fiber Sensors

As discussed in Section 3.4, characteristics of Brillouin scattering, including the Brillouin frequency shift, carry a dependence on external factors such a temperature and strain. This dependence has been used to develop several different sensors based on measuring Brillouin scattering along a length of fiber. Some of these sensors focus on measuring spontaneous Brillouin scattering, though in general their usefulness is limited by nonlinear effects for fibers longer than 50km. A number of established techniques fit into this passive regime, including Brillouin optical time domain reflectometers (BOTDR), Brillouin optical correlation domain reflectometers (BOCDR), and Landau-Placzek ratio sensors [65]. Active sensors based on SBS can be used to overcome some of these issues to provide sensing capability of >100km.

Brillouin optical time domain analysis (BOTDA) is based on a simple architecture consisting of counter propagating pump and probe beams which interact over a long length of fiber [66]. In most configurations, the pump source is pulsed while the probe source is operated in CW mode. The pump and probe lasers are tuned such that the difference in frequency is equal to the Brillouin frequency shift relevant to the fiber and wavelengths used. The pump pulses stimulate amplification of the CW probe beam, and the generated SBS signal can be analyzed using time domain techniques to determine information about temperature or strain along the fiber. The first demonstration of this type of sensor was able to measure temperature fluctuations along the fiber with a resolution of 3°C and 100m spatial resolution, however, both sensitivity and spatial resolution have improved significantly since then. The spatial resolution is limited to near 1m by the phonon lifetime, however, advanced versions of this technique which utilize different pulse patterns have been shown to have spatial resolutions on the order of 1cm [67].

In BOTDA systems, the pulse lengths are kept long so that the pulse bandwidth does not significantly exceed the Brillouin gain bandwidth, which would decrease the measurement fidelity. Alternatively, Brillouin optical correlation domain analysis (BOCDA) systems use wide- bandwidth pulses which interact coherently at a specific location along the fiber. This technique requires that the phases of the pump and probe beams are well controlled. While this technique limits the useful fiber length to near 1km, it can increase spatial resolution to near 1mm and increase the sampling rate to 1kHz. For reference, sampling times for many BOTDA systems are on the order of minutes. Another variation of the BOTDA technique is Brillouin optical frequency domain analysis (BOFDA). The relation between these two techniques is analogous to optical time domain reflectometry (OTDR) and optical frequency domain reflectometry (OFDR). For this technique, the probe beam is intensity modulated over a wide range of frequencies. A transfer function is determined from the collected data and a profile of temperature and strain is generated by taking its inverse Fourier transform.

Sensor systems based on the techniques outlined above have been used in commercially deployed structures. Their applications include monitoring of strain across structural members of bridges

34 and railways, temperature monitoring for buried pipelines, monitoring of stress in critical electrical components for power stations, and monitoring steel corrosion in buildings. As is obvious from the applications list, the long sensing lengths of active Brillouin techniques are well suited to continuous monitoring of large structures. These types of sensors can alert engineers to otherwise latent problems in critical infrastructure so that maintenance can be performed prior to failure.

4.2. Brillouin Fiber Lasers

Since the waveguide material acts as a gain medium for SBS, a new laser source can be created by enclosing it within an optical cavity with a pump source. In fiber, this can be accomplished in both ring and Fabry-Perot cavity configurations, as shown in Figure 20. For CW operation of a ring- cavity Brillouin laser, the SBS threshold is [8]

⁄ = 1 (4.2)

where is the fraction of the Stokes power which remains in the cavity after one round trip. In this configuration, the number of modes present in the cavity is determined by the fiber length. When the longitudinal mode spacing, giving by Δ = where is the effective mode index is greater than the Brillouin gain bandwidth, only a single longitudinal mode will exist. However, for the case that the Brillouin linewidth exceeds the longitudinal mode spacing, many modes can exist, and the output can become unstable unless actively managed [68].

In the Fabry-Perot configuration, the pump and any other generated waves propagate in both directions. This allows for cascaded generation of higher-order Stokes lines through SBS as well as the generation of anti-Stokes lines through the interaction between co-propagating pump and Stokes waves. Thus, the output spectrum can contain a large number of discrete sidebands which extend out on both sides of the pump.

4.2.1. Brillouin Laser Gyroscopes

Laser gyroscopes exploit the Sagnac effect to determine the angular rotation frequency of a system. This effect states that for two counter-propagating beams in a rotating loop, the beam propagating in the same direction as the rotation will take longer to reach the same point compared to its counterpart traveling in the opposite direction. The two lasers can be partially coupled from the loop and coherently recombined on a detector to determine the rotation rate from the beat frequency using the relation [69]:

4 = ∙ Ω (4.3)

where is the beat frequency, is the perimeter of the path along which the beams travel, is the area enclosed by the loop, and Ω is the angular rotation frequency.

The pump can be generated by a narrow linewidth source, such as an external cavity laser (ECL) or a DFB. As long as the pump exceeds the SBS threshold, Stokes signals will be generated in both directions. The pump can be stabilized such that it remains at the peak of a cavity resonance. The frequency of the generated Brillouin signal will change proportional to the rotation rate as the Sagnac effect results in a phase shift in the cavity. While Brillouin ring laser gyroscopes have the potential for high performance, they have suffered issues with drift and frequency lock-in at low rotation rates. A number of solutions have been proposed for these issues, at the cost of increased system complexity [70].

4.2.2. Frequency Comb Generation

Optical frequency combs have been used for a wide range of applications, including frequency metrology [71], precision spectroscopy [72], optical clocks, and DWDM communications systems [73]. In each of these applications, frequency combs are useful because they can offer known line spacing in the microwave regime, allowing them to be used as a reference to other laser signals. As previously mentioned, since SBS has a low power threshold it can be used to generate multiple Stokes sidebands, each spaced by the Brillouin frequency shift, Ω. This cascading can be combined with four wave mixing to generate additional anti-Stokes peaks, further increasing the bandwidth over which the Brillouin laser operates. Figure 21. Cascaded SBS in a fiber ring cavity. Backward and Cascaded SBS in a ring-cavity forward direction spectra are superimposed, showing the frequency spacing of ~10GHz between each Stokes line [78]. configuration results in Stokes lines

36 spaced by 2Ω as even and odd orders each propagate in a single direction. In one experiment [74], a modified ring configuration using a four port circulator and two distinct ring cavities was pumped to achieve four lines propagating in each direction. Each direction was separately characterized and then combined to achieve an eight frequency band output with spacing Ω ≅ 10, the expected Brillouin frequency shift for fused silica fiber operating near 1550nm.

SBS can be used to achieve significantly more than eight modes, especially when combined with other nonlinear effects such as four wave mixing [75]. For an experiment utilizing a CW 1550nm EDFA- amplified pump source in a short Fabry- Perot cavity with 5.2cm of highly non- linearly fiber, cascaded SBS generation was shown to fill >100nm bandwidth [76]. In this case, parametric generation due to Figure 22. Wide bandwidth frequency comb created using a short HLNF fiber in a Fabry-Perot configuration. Detail in the modulation instability in the fiber lead to five inset plots show the individual SBS Stokes lines which make initial 2THz mode spacing, which was up the frequency comb [80]. subsequently filled in by the cascaded SBS.

4.2.3. RF Frequency Generation

In all-optical fiber communications networks, a number of operations require the generation of signals in the microwave frequency domain. Optical generation of microwave signals offers advantages over traditional electrical generation in the areas of cost, reliability, and speed [77]. Since the Brillouin frequency shift for pump wavelengths near 1550nm is within the microwave band near 11GHz, mixing between pump and Stokes waves can be exploited for microwave generation.

The optical heterodyne technique [78] is well established and used in a multitude of commercial devices for both electrical and optical signals. The overall premise is that two signals with different frequencies are superimposed on a detector, and the difference frequency is detected at the output. The bandwidth of the detector only has to be wide enough to accurately detect the difference signal, making it possible to use the Figure 23. Microwave output resulting from utilizing technique with closely spaced optical signals. For an optical heterodyne with the pump and Stokes waves microwave frequency generation, the pump and in a Brillouin fiber laser. Top and bottom plots are generated from the same cavity at different Stokes signal generated from SBS are mixed onto temperatures [84].

37 a photodetector, creating a signal at − =Ω. RF generation can be accomplished using both ring and Fabry-Perot configurations, however, it has been shown that a Fabry-Perot configuration using fiber Bragg gratings (FBG) can achieve very high conversion efficiencies [79].

In one demonstration [80] of the use of SBS to achieve microwave generation, a Fabry-Perot cavity was built using 10m of SMF-28 spliced between two custom FBGs. A 1550nm DFB was injected through a circulator and used as the pump source, and reflected power from the Fabry- Perot cavity was coupled out using the third circulator port. The reflectivities of the FBGs at the pump and Stokes wavelengths were chosen such that the reflected powers in each wave were approximately equal. The light coupled out of the system using the circulator was incident on an 18GHz photodiode, and the electrical output was captured on a spectrum analyzer. A 4MHz wide output was achieved at 10.87GHz. The researchers also utilized temperature control of the SMF- 28 to tune the output microwave frequency over 100MHz by changing the temperature from 8°C to 70°. Models showed that by tuning the reflection band of the FBGs, the pump could be mixed with higher order Stokes waves, leading to the potential for RF frequencies in the millimeter-wave regime.

4.3. Optical Phase Conjugation

Optical phase conjugation is the process by which an optical beam which has accumulated some amount of distortion after traveling through a non-ideal medium can be interacted with through nonlinear optical processes and returned to its original, undistorted state after propagating backwards through the same medium. The optical element which interacts with the distorted beam reverses its direction and modifies the complex amplitude function to its conjugate. This element is usually referred to as a phase conjugate mirror. A wide variety of applications exist for SBS optical phase conjugation, including the correction of phase distortions in high power lasers, phase stabilization, beam combination, pulse compression, and arbitrary waveform generation [81].

Several different methods have been developed to achieve optical phase conjugation, including using four-wave mixing, three-wave mixing, and stimulated scattering [82]. SBS was actually the subject of the experiment which lead to the first recorded observation of optical phase conjugation. SBS is able to produce a phase conjugate Figure 24. Diagram illustrating the difference between a normal mirror and a phase conjugate mirror. An input beam is beam using only a strong input source aberrated through a medium, is reflected, passes back through above the SBS threshold without any the medium, and is further aberrated (top). A beam follows the special measures taken to ensure phase same sequence of events but is reflected by a phase conjugate mirror. After passing back through the medium the aberrations conjugation. The process by which from the first pass are removed [86]. stimulated scattering processes such as

SBS generate a phase-conjugated beam can be understood through the quasi-collinear four-wave mixing model. In this model, the aberrated pump beam can be represented as the superposition of an undistorted pump and a weaker, distorted pump. The two pump waves interact in the medium, creating a volume holographic grating formed with refractive index changes. The undistorted beam is more powerful and induces the stimulated scattering process, created an undistorted reflected beam. As for Brillouin scattering, this beam experiences a frequency downshift according to the Brillouin frequency shift of the medium. This reflected wave acts as a reading beam in the induced volume holographic grating, creating a distorted beam which is phase-conjugated to the original distorted beam. Both beams experience SBS gain since their frequencies are nearly the same.

SBS phase conjugation can be observed in both single-mode and multimode fibers. However, since the Stokes wave is frequency shifted, it cannot exhibit exact time reversal behavior along the full length of the fiber. Thus, efficient SBS phase conjugation in fibers can only occur with limited fiber lengths [83]. An equation was derived by Hellwarth [84] predicting the maximum waveguide length over which significant phase conjugation could be achieved using SBS:

6√1 − ≤ (4.4) Δ

where is the number of modes in the waveguide, is the fidelity of optical phase conjugation, defined as the fraction of the Stokes beam which is phase conjugate to the pump, and Δ is the difference in wavelength between pump and Stokes beams. Phase error due to the difference in wavelength accumulates over long fiber lengths, leading to a decrease in fidelity [83].

4.4. Beam Cleanup

Observed in many experiments with SBS optical phase conjugation, SBS beam cleanup is the process by which a highly aberrated pump beam injected into multimode fiber can produce a Stokes beam which is close to diffraction limited. It has been shown in the literature that while optical phase conjugation in fiber can be accomplished with high fidelity in short fiber lengths, longer lengths tend to reduce the effect. Likewise, beam cleanup is more effective in fibers that are long rather than short [85]. Even though the spatial characteristics of the Stokes was are not conjugate to the pump beam after a long length of fiber, one experiment showed that the polarization remains conjugated [85]. This was demonstrated by passing a polarized pump through a quarter wave plate prior to injection into the fiber and observing that the Stokes wave polarization was parallel to the input after passing through the same wave plate in the reverse direction.

The beam cleanup effect has also been shown to be applicable to both coherent and incoherent combination of multiple pump beams. Two pump beams ( = 807) with frequencies differing by ~100MHz were coupled into a long length of SMF-28, and a single, spatially coherent beam was observed in the backwards direction. The same effect was later demonstrated in 8.8km of 50m core multimode GRIN fiber, where four 1064nm pump beams generated using Yb3+-doped fiber amplifiers were coupled into the fiber in an off-axis configuration, as shown in Figure 25. The combined power of the beams was well above the SBS threshold, and a single Stokes wave with measured beam quality of =1.12 was generated [86]. Both modeling and experiments have shown that the SBS gain in GI fiber for lower-order modes was nearly twice the gain for higher order modes. This was determined to Figure 25. Diagram of SBS beam combination setup (top). Six contribute to beam clean up, as lower- off-axis pump beams are combined into a multimode fiber and order modes are effectively selected for the resultant Stokes beam exits the center of the system to the left. Contour plots showing experimental irradiance patterns and amplified preferentially [87]. (bottom). a) near field pump, b) far field pump, c) near-field Stokes, d) far-field Stokes [90]. 4.5. Brillouin Slow Light

Slow light is a relatively new area of research which focuses on utilizing dispersive media to slow down, or even stop, pulses of light. Applications for these techniques exist in optical computing, telecommunications, and quantum computing, among others [88].

Group velocity is often referred to as the velocity at which energy, or information, propagates. Group velocity can be related to group index by

() = (4.5) () where

() = () + () (4.6)

Here, the quantity / is the dispersion. In most cases, large increases in dispersion are associated with similarly large increases in attenuation. This is an issue for slow light applications, where attenuation must be kept to a minimum to preserve the signal fidelity. However, it is clear from (4.6) that an increase in the dispersion can be used to decrease group velocity. Several techniques have been developed to address the attenuation issue, including coherent population trapping, coherent population oscillation, polariton trapping, and stimulated scattering. While the stimulated scattering techniques have not achieved the decrease in group velocity achieved with the other techniques, they have generated interest because they do not require exotic materials or extreme environmental conditions for operation [89].

The principle of Brillouin slow light is derived from an added group delay acquired by SBS- amplified pulses. The complex Brillouin gain spectrum can be defined as [90]

1 2 (4.7) () = 1 − 2( − − Ω)/Γ

where is the pump amplitude and Γ is the SBS gain bandwidth. The real part of the spectrum determines the gain while the imaginary part determines the phase delay. The phase dependence at the center of the Brillouin gain spectrum is linear, which can be associated with a group delay given by [91]

= (4.8) Γ

It is clear from this equation that the group delay is strongly dependent on the pump strength, but decreases as the SBS bandwidth is increased. However, since the SBS bandwidth is only 20- 50MHz, the effect by itself cannot be used to control pulses in modern telecommunications networks, which have bandwidths in the multiple GHz regime. Thus, a large amount of research has been dedicated to broadening the spectrum to push SBS slow light towards the realm of useful applications.

Many researchers have attempted to broaden the Brillouin gain by using external modulators. Spectra created using random noise generation were shown to reach SBS gain bandwidths around

12GHz, which were able to delay 75ps pulses [92]. A combination of two pumps with careful spacing was used to further increase the SBS gain bandwidth to 25GHz [93]. Significant research is continuing to not only further increase bandwidth, but to reduce pulse distortion and increase signal to noise for the delayed signals.

5. Conclusions

For many fiber systems, stimulated Brillouin scattering is the lowest-threshold nonlinear effect. SBS arises from a spontaneous scattering process whereby thermally-excited acoustic phonons scatter a pump beam, creating a backwards-traveling Stokes wave. The Stokes and pump waves interfere, creating a traveling index grating which stimulates additional scattering and seeds a positive feedback process. SBS can be responsible for a reduction of signal quality in fiber optic communication networks, where error rates must remain extremely low for proper operation. SBS is also one of the primary limiting factors for power scaling in high power fiber amplifiers, where the presence of high gain can create an SBS response which leads to total system failure.

Several techniques have been developed to mitigate the effects of SBS, each of which affect the SBS threshold by manipulating one of several parameters. These techniques fit into several loose categories, including increasing effective mode area, broadening the pump spectrum beyond the Brillouin gain bandwidth, changing the Brillouin gain bandwidth along a length of fiber, designing waveguides which reduce optical and acoustic mode overlap, and utilizing materials which reduce coupling between optical fields and the nonlinear refractive index. The reduction of SBS can and has been accomplished through combining several of these techniques in a single system.

The unique properties of SBS have also been utilized in a positive way for various applications. For instance, SBS was used in the first demonstration of optical phase conjugation, exhibiting this time-reversal technique in optical systems that had not previously been realized. Some of these, like optical filtering, microwave generation, and Brillouin slow light, can be utilized for high performance, all-optical processing. The sensitivity of SBS to external stimuli has been shown to be useful for mitigating SBS and has also been combined with optical time and frequency-domain techniques to create distributed sensor systems operating over lengths which can exceed 100km.

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