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We report optical fiber experiments allowing to investigate integrable turbulence in the focusing regime of the one dimensional nonlinear Schrödinger equation (1D-NLSE). Our experiments are very similar in their principle to water tank experiments with random initial conditions (see M. Onorato et al. Phys. Rev. E 70 067302 (2004)). Using an original optical sampling setup, we measure precisely the probability density function (PDF) of optical power of partially coherent waves rapidly fluctuating with time. The PDF is found to evolve from the normal law to a strong heavy-tailed distribution, thus revealing the formation of rogue waves in integrable turbulence. Numerical simulations of 1D-NLSE with stochastic initial conditions reproduce quantitatively the experiments. Our investigations suggest that the statistical features experimentally observed rely on the stochastic generation of coherent analytic solutions of 1D-NLSE such as Peregrine solitons.The field of nonlinear optics has recently grown as a favorable laboratory to investigate both statistical properties of nonlinear random waves and hydrodynamic-like phenomena [1][2][3]. In particular, several recent works point out analogies between hydrodynamics and nonlinear fiber optics in the observation of supercontinuum generation [4], undular bores [2], optical turbulence, laminarturbulent transition [1] or oceanographic rogue waves (RW) [5,6].Rogue waves (RW), also called freak waves, are extremely large amplitude waves occurring more frequently than expected from the normal law [7,8] As stressed out in the recent review [6], there is no obvious analogy between most of the optical experiments on extreme events and oceanography. However a direct correspondence between nonlinear optics and hydrodynamics is provided by the one-dimensional nonlinear Schrödinger equation (1D-NLSE) (see Eq. (1)) that describes various wave systems [6,18]. In particular, the focusing 1D-NLSE describes at leading order the physics of deep-water wave trains and it plays a central role in the study of RW [6-8, 18, 19].Modulational instability (MI) is believed to be a fundamental mechanism for the formation of RW [8,20]. Moreover, analytical solutions of the integrable 1D-NLSE such a Akhmediev breathers (AB), Peregrine solitons or Kuznetsov-Ma solitons (KMs) are now considered as possible prototypes of RW [19,[21][22][23]. These coherent structures have been generated from very specific, carefullydesigned coherent initial conditions in optical fiber experiments [21,23,24].On the contrary, oceanic RW emerge from the interplay of incoherent waves in turbulent systems. The occurrence of RW in wave turbulence has been theoretically studied in optics [25,26] and in hydrodynamics [27]. In hydrodynamical experiments made with one-dimensional water tanks, non gaussian statistics of the wave height has been found to emerge from random initial conditions [20,28].The appropriate theoretical framework combining a statistical approach of random waves together with the 1D-NLSE is integrable turbulence...