how frequency become orthogonal to each other
Imagine, each OFDM symbol time is T seconds. It is composed of two time intervals one is called Guard time or Guard interval which is Tg and the other one is Orthogonality period demonstrated by Tort.
The relationship between these three parameters is as follow:
T = Tg + Tort
Tg is usually less than 25 % of the symbol time and is defined as amost the transient time of the signal for each symbol in the channel. The inverse of the orthogonality Period or 1/Tg is called the frequency spacing of the OFDM Modulation the carriers placed 1/Tg in frequency apart are orthogonal to each other. The mathematical description of the orthogonality is as follow:
∫cos(2* pi*fc*t + n* Δf)*cos(2*pi*fc*t + k *Δf)dt the integral is from Tg to T
the cosine can be replaced by sin function as well.
if Δf = 1/Tg then the above integral shall equal to 0 if n ≠ k which describes the orthogonality of the carriers in the OFDM.
Refer to books on Modern Digital Communications and see OFDM. you get a more detailed view on the subject.