Not exactly. Let's switch to continuous domain for simplicity.Din wrote: I understand the total disturbance on the hologram is the sum of individual disturbances caused by individual points. Basically, it's a sum of sinusoidal components for each point. Isn't this just a Fourier sum?
The Fourier transform G(fx, fy) of a function g(x, y) is given by
G(fx, fy) = \int \int_{-∞}^{∞} g(x, y) exp(-i 2π [x fx + y fy]) dx dy
and the inverse Fourier transform is given by
g(x, y) = \int \int_{-∞}^{∞} G(fx, fy) exp(i 2π [x fx + y fy]) dfx dfy
This sum can be interpreted as a sum of plane waves in the plane xy. Each plane wave has complex amplitude given by G(fx, fy) and direction given by a unit vector
[λ fx, λ fy, sqrt(1 - {λ fx}^2 - {λ fy}^2)]
Let us also assume that G(fx, fy) = 0 for fx^2 + fy^2 > 1 / λ^2. That is, if the square root in the z component of the direction vector gets imaginary, G(fx, fy) is zero anyway (i.e. no evanescent waves).
On the other hand, if we have point light sources located at z = 0, light in z = z0 can be expressed by
U(x, y, z0) = \int \int_{-∞}^{∞} U(ξ, η, 0) exp(i k r) / r dξ dη
where
r = sqrt[ (x - ξ)^2 + (y - η)^2 + z0^2]
The difference is clear:
- The inverse Fourier transform represents a sum of "sinusoidal" patterns. Each pattern is a "sinusoid" looks like corrugated metal sheet, each component has certain frequency, amplitude, phase and angle of rotation.
- Summation of point light sources is a sum of radially symmetric "sinusoidal" profiles, each profile looks like a wave on a water surface whan you throw a pebble inside. Each profile is shifted in xy, has certain amplitude and phase.
(To be continued...)