Jipeng Sun

The ASM is the standard algorithm for simulating how light travels through space because it turns a massively complex 3D physics problem into a simple multiplication problem.

It works in three highly logical steps:

1. Decompose the Wave (The Fourier Transform)

Imagine trying to calculate how a chaotic, splashing wave moves across a pool. It is too complex to calculate all at once.

• Instead, the ASM takes the starting wave \(U(x,y,0)\) and applies a Fourier Transform (\(\mathcal{F}\)).
• This breaks the chaotic wave down into a set of perfectly flat, uniform plane waves traveling in different directions (the angular spectrum).

2. Fly the Planes (The Transfer Function)

Because a plane wave is perfectly flat and travels in a perfectly straight line, calculating its propagation is trivial. It doesn't change shape; it only changes its phase (the timing of its peaks and valleys) as it travels a distance \(z\).

• The algorithm multiplies the spectrum by a free-space transfer function \(H(f_x, f_y; z)\).
• Mathematically, this transfer function is just a complex exponential that applies a specific phase delay to each plane wave based on the angle it is traveling and the distance \(z\).

3. Reassemble the Wave (The Inverse Fourier Transform)

Once all the individual plane waves have been mathematically "flown" forward to distance \(z\) by shifting their phases, the algorithm uses an Inverse Fourier Transform (\(\mathcal{F}^{-1}\)) to add them all back together.

• The result is the exact, complex interference pattern \(U(x,y,z)\) at the new location.

The entire propagation is summarized in one elegant equation:

\[U(x,y,z) = \mathcal{F}^{-1}{ \mathcal{F}[U(x,y,0)] \cdot H(f_x, f_y; z) }\]