We theoretically analyze the relationship between the electric field envelope shape of an optical pulse launched into an optical fiber and the power spectrum of the spontaneous Brillouin backscattered light it produces. The electric field envelope is characterized by the pulse width, leading-trailing time, and steepness. The peak power of the launched, pulsed-light power spectrum is proportional to the square of the pulse width regardless of the pulse leading-trailing time and steepness, and the power spectrum broadens in inverse proportion to the pulse width. The peak power of the spontaneous Brillouin backscattered light produced by the launched, pulsed light is proportional to the pulse width when it is above approximately 100 ns and is proportional to the square of the pulse width when it is below approximately 1 ns. The power spectrum of the spontaneous Brillouin backscattered light also broadens rapidly corresponding to the pulse width, when the pulse width falls below approximately 30 ns. As the pulse leading-trailing time is shortened or the pulse leading-trailing part becomes steep, the Brillouin backscattered-light power spectrum broadens greatly, even if the launched pulse width remains constant. Our analysis showed that an optical pulse with a triangular-shaped electric field envelope forms the Brillouin backscattered-light power spectrum with the narrowest profile and consequently gives the minimum error in measuring the peak-power frequency, when the pulse width is below approximately 50 ns. The measurement error with the triangular-shaped pulsed light is 1/sq root times smaller than that for a rectangular-shaped pulsed light, when the pulse width falls below several nanoseconds. By contrast, the rectangular-shaped envelope gives the minimum error when the pulse width exceeds ～50 ns.
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