Angular spectrum method

The Angular Spectrum Method (ASM) is a technique used in optics, particularly in the analysis of wave propagation and diffraction of light. It is based on the principle of representing a wavefront as a superposition of plane waves. This method is widely used in computer science and engineering fields, especially in image processing, optical system design, and simulation of wave propagation in various media.

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Angular spectrum method by

Codex  0 2026-11-05

The angular spectrum representation expresses a monochromatic wave as a superposition of plane waves. In a homogeneous, lossless, source-free half-space with real wavenumber

k = 2 π / λ > 0

, transverse Fourier transform components propagate independently, providing a useful solution of the Helmholtz equation.^[ref]

Use angular frequency

ω

and time dependence

e^{- iω t}

U (k_{x}, k_{y}; 0) = \int\int U (x, y, 0) e^{- i (k_{x} x + k_{y} y)} d x d y

. Then

U (x, y, z) = \frac{1}{( 2 π ) ^{2}} \int\int U (k_{x}, k_{y}; 0) e^{i (k_{x} x + k_{y} y + k_{z} z)} d k_{x} d k_{y}, k_{z} = k^{2} - k_{x}^{2} - k_{y}^{2} .

The outgoing branch of the angular spectrum has nonnegative real and imaginary parts of

k_{z}

for propagation toward

z > 0

. A propagating angular-spectrum component has real

k_{z}

; an evanescent angular-spectrum component has

k_{z} = iκ

and decays as

e^{- κ z}

. The source-plane amplitudes and their complex arguments matter, not just their squared magnitudes.^[ref]

For an original example, a transverse period

λ /2

gives

k_{x} = 2 k

,

k_{y} = 0

and

κ = 3 k

. After distance

z = λ /4

, its amplitude is multiplied by $e^{- π 3 /2} ≃ 0.0658$ . Such fine spatial detail does not propagate with unchanged amplitude to the far field.

The paraxial angular-spectrum propagator uses

k_{z} ≃ k - (k_{x}^{2} + k_{y}^{2}) / (2 k)

. For the source-plane Gaussian beam

U_{0} = e^{- (x^{2} + y^{2}) / w_{0}^{2}}

, its transverse spectrum is proportional to

e^{- w_{0}^{2} (k_{x}^{2} + k_{y}^{2}) /4}

. The on-axis Gaussian integral therefore gives

U (0, 0, z) = \frac{e ^{ik z}}{1 + i z / z _{R}}, z_{R} = k w_{0}^{2} /2.

The denominator produces the axial Gouy phase

- arctan (z / z_{R})

. This connects the phase shift with interference of transverse spectral components, rather than an abrupt phase jump at the focus.^[ref]

The scalar propagation formula does not authorize arbitrary independent vector components. A physical electromagnetic field must satisfy the transverse polarization constraint in an angular spectrum; in a uniform source-free medium,

k \cdot E = 0

. Boundary conditions and source consistency accompany a vector reconstruction.

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