Why we analyse samples of optical rotation only at 589nm?

No, you can use any source of sufficiently monochromatic light, such as a mercury line or a laser.
Historically, the sodium line was used because it is very intense, and right in the middle of the visible spectrum, so it was easy to do the polarimetry “by eye,” which was the original practice in earlier days.
The degree of polarization rotation is wavelength dependent, and since most historical data uses the sodium line, it’s only natural for people to keep using it.