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Analyzing the phenomena associated with the usage of ultrashort laser pulses with a duration about single cycle of electrical field, one cannot ignore the influence of the carrier-envelope phase (CEP), that is critically important for the generation of extremely short pulses [1,2], high order harmonics generation [3] and attosecond physics [4]. In high-intensity regimes beyond perturbative approximation, not only such physical effects as harmonics generation and ultrafast tunneling ionization depend on CEP, but also self-phase modulation and soliton self-compression leading the spectral and temporal CEP dependence on the forming ultrashort pulses. In our work, we have demonstrated the influence of the carrier-envelope phase on the supercontinuum spectrum and the characteristics of extremely short pulses formed as a result of nonlinear optical conversion of pump pulses in an argon-filled hollow anti-resonant waveguide. In our experiments, we used Ti:Sapphire laser system and optical parametric amplifier to produce tunable pulses with an energy about 180 μJ, central wavelength 2000 nm and duration 60 fs. These pulses are coupled into anti-resonant hollow-core fiber (AR HCF) filled with argon. The sequence of nonlinear transformations of the femtosecond pump pulse in an AR HCF leads to spectral broadening (supercontinuum generation (SC)) and near single-cycle waveform generation. The spectrum broadening follows the soliton self-compression (SSC) scenario, with additional enhancement from the self-steepening effect and parametric generation of four-wave components in the blue wing of the soliton spectrum [5,6]. In such condition it is possible to form very short pulses with the duration less than on cycle of the field, and for such pulses the influence of CEP could play noticeable role. Experimental and theoretical analysis shows that as a result of soliton self-compression of pump pulses radiation with an initial central wavelength of about 2 μm, a pulse with a duration of about one optical period is formed. The spectrum of such pulse is broadened to the region of 400–800 nm, where interference with the broadband third harmonic generated by the same pulse is observed. The interference pattern turns out to be sensitive to CEP of the laser pulse. An analysis of the interference pattern provides information on the difference between the spectral phases of the soliton and the third harmonic in the spectral range greater than an octave, and also makes it possible to control the duration of pulses formed in the process of soliton self-compression. The work was supported by the Russian Science Foundation with grants No.22-12-00149.