A paraxial wave is a wave which makes a small angle

A paraxial wave is a wave which makes a small angle (?) to the optical axis and lies near to the axis throughout the system. A wave is called to be paraxial if the wave front normal of this wave are paraxial rays. The way of constructing a paraxial wave is to begin with a plane wave A.exp (-jkz), like a “carrier” wave, and modulate its complex envelope A, and making it a slowly varying function of position, A(r), so the complex amplitude of the modulated wave becomes
U (r)= A( r ).exp(-jkz) (1.6)
The variation of the envelope A(r) with position and its derivative with position z must be slow within the distance of a wavelength ?=2???/k? so that the wave approximately maintains its plane-wave nature.