## 1 Introduction  
Polymer optical fibers (POFs) [1,2] have attracted considerable attention for the past several decades due to their extremely easy and cost-effective connection, high safety, and high flexibility [3] compared to conventional glass fibers. Therefore, in spite of their propagation loss being higher than that of silica glass fibers, POFs have been made use of in medium-range communication applications such as home networks and automobiles [4] as well as in high-strain monitoring applications [3,5].  
On the other hand, Brillouin scattering in optical fibers [6,7], one of the most significant nonlinear processes, has been extensively studied, and a great number of useful applications have been reported so far, such as optical amplification [7], lasing [7,8], optical comb generation [8], microwave signal processing [9], slow light generation [10], phase conjugation [11], tunable delay [12], optical storage [13], core alignment of butt-coupling [14], and strain/temperature sensing [15, 16, 17, 18]. For the purpose of improving the performance of these applications, Brillouin scattering has been investigated not only in silica single-mode fibers (SMFs) but also in some specialty fibers. They include, for example, silica multimode fibers (MMFs) [14], tellurite glass fibers [19,20], chalcogenide glass fibers [21,22], bismuth-oxide glass fibers [23,24], and photonic crystal fibers (PCFs) [25,26].  
Recently, we have succeeded in observing spontaneous Brillouin scattering (SpBS) in POFs, for the first time to the best of our knowledge, at the telecommunication wavelength of 1.55 \(\upmu\)m [27,28], which will offer many advantages of POFs to the conventional Brillouin application fields. The POFs used in the experiment were cyclic transparent optical polymer (CYTOP)-based perfluorinated graded-index (PFGI-) POFs, which are fabricated by replacing the hydrogen atoms of standard polymethyl methacrylate (PMMA)-based POFs with fluorine atoms. Their detailed structure and fabrication method are well summarized in [29]. Their Brillouin gain coefficient was estimated to be approximately 3.09 \(\times\)10\({}^{-11}\) m/W, which was almost the same as that of silica SMFs, indicating that Brillouin scattering in PFGI-POFs can be applied to a variety of practical devices and systems in the same way as Brillouin scattering in silica SMFs. We have also investigated the dependences of the Brillouin frequency shift (BFS) on strain and temperature in a PFGI-POF at 1.55 \(\upmu\)m, and found that SpBS in PFGI-POFs can be potentially utilized to develop high-accuracy temperature sensors with reduced strain sensitivity [30]. Small Brillouin Stokes signal was, however, a major problem to be solved for practical future applications as well as for detailed investigations of the Brillouin gain spectrum (BGS).  
In this review, we summarize two methods to enhance the Brillouin Stokes signal. First, stimulated Brillouin scattering (SBS) is induced using the so-called pump-probe technique [31], and its applicability to temperature sensors is discussed. Second, the influence of the core diameter and length of the PFGI-POFs is investigated [32], which will be a good guideline for designing PFGI-POF structures suitable for future applications.