Suppose the RF frequency of the driver of an AOM is 40 MHz. I understand that it will Doppler shift the diffracted first order beam by 40 MHz but how does it modulate the amplitude of that first order beam by 40 MHz. I am just trying to understand that whether the Doppler shifting and the amplitude modulation both take place simultaneously or are in some way related phenomenon.
Doppler shift and amplitude modulation are two separate characteristics of the modulator.
For the first order, laser frequency is shifted by 40 MHz as you stated.
Laser amplitude is modulated by the RF envelope, not RF frequency.
Changes in RF envelope are slower than 40 MHz. They may be up to 30% of the RF frequency and usually much lower. This rate of envelope changes depends on the RF generator bandwidth, electro-acoustic transducer bandwidth, laser spot size and more.