I can't understand why we need to use a ground plane at 1/4 wavelength and 1/2 wavelength not needed.
A full dipole has voltage and current waves like this: -
Picture from Wikipedia.
Now, if you focus your eye at dead centre of the picture you will see that the voltage is always zero volts. This is because a dipole is optimally driven with a balanced voltage source (\$V_O\$). A balanced voltage source is preferred for a dipole antenna. In fact, the voltage and electric field is zero all along the length of the green line below: -
This means you can optionally regard that green line as earth (providing the line is driven in a balanced way.
Now if you were to cut the above picture in half you'd have a quarter wave monopole driven with an unbalanced voltage source. An unbalanced voltage source is one that has typically 0 volts on one leg while the other leg does the voltage driving: -
And, not surprisingly, it has one half of the impedance presented by the half wave dipole.