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Recently published episodic spectral (Levesque and Massey, 2020) and high angular resolution (Montarges et al, 2020) observations of Betelgeuse suggest that the deep minimum of 2019-2020 was caused by an enhanced dust abundance in the stellar atmosphere. Detailed monitoring of such events may prove useful for constructing consistent physical models of mass loss by evolved stars. For such observations it is fundamentally important to employ methods resolving an inhomogeneous stellar atmosphere. We present the differential speckle polarimetric observations of Betelgeuse at 2.5-m telescope of Caucasian Mountain Observatory of SAI MSU covering the period of 2019-2020 minimum. The observations were secured on 17 dates at wavelengths 465, 550, 625 and 880 nm. The circumstellar reflection nebula with the angular size of $\approx0.1^{\prime\prime}$ was detected for all the dates and at all wavelengths. The morphology of the nebula changed significantly over the observational period. Net polarized brightness of the envelope remained constant until February 2020, while the stellar $V$ band flux decreased 2.5 times. Starting from mid-February 2020, polarized flux of the envelope rose 2.1 times, at the same time the star returned to the pre-minimum state of October 2019. Basing on these data and our low resolution spectrum obtained on 2020-04-06 we confirm a conclusion that the minimum is caused by the formation of a dust cloud located on the line of sight. A quantitative characterisation of this cloud will be possible when the data on its thermal radiation are employed.