We introduce azimuthal Fourier expansion for functions with circular symmetry: we write \(f(r, \phi)\) as a sum of angular modes \(\hat{f}_n(r) e^{i n \phi}\), where \(\hat{f}_n(r)\) is obtained by integrating over \(\phi\) and measures the strength of mode \(n\) at radius \(r\). We then contrast this discrete angular Fourier series on \([0,2 \pi]\) with the usual continuous Fourier transform on \((-\infty, \infty)\), and we briefly note that this separation of radial and angular dependence is especially useful in optics, quantum mechanics, fluid dynamics, and image processing.
Basic Concept
When dealing with problems possessing circular or cylindrical symmetry, it's natural to work in polar coordinates \((r, \phi)\) rather than Cartesian coordinates \((x, y)\).
For a function \(f(r, \phi)\) defined on a disk, we can: - Fix the radius \(r\) - Treat it as a function of the angular coordinate \(\phi\) only: \(f_r(\phi)\) - Since \(\phi\) is periodic with period \(2\pi\), \(f_r(\phi)\) is a periodic function
Azimuthal Fourier Expansion
For each fixed radius \(r\), we expand \(f(r, \phi)\) as a Fourier series:
\[ f(r, \phi) = \sum_{n=-\infty}^{\infty} \hat{f}_n(r) e^{in\phi} \]
Where: - \(e^{in\phi}\) are the basis functions representing pure angular modes - \(n = 0\): axisymmetric mode (no angular dependence) - \(n = 1\): one complete sinusoidal variation around the circle - \(n = 2\): two complete variations, etc.
The n-th Azimuthal Fourier Coefficient
The coefficient \(\hat{f}_n(r)\) is called the "n-th azimuthal Fourier coefficient":
Forward Transform (Analysis):
\[ \hat{f}_n(r) = \frac{1}{2\pi} \int_0^{2\pi} f(r, \phi) e^{-in\phi} d\phi \]
Inverse Transform (Synthesis):
\[ f(r, \phi) = \sum_{n=-\infty}^{\infty} \hat{f}_n(r) e^{in\phi} \]
Physical Interpretation
- \(\hat{f}_n(r)\) represents the amplitude or strength of the angular mode \(n\) at radius \(r\)
- It's a function of \(r\) only, not \(\phi\)
- The decomposition: 2D function \(f(r, \phi)\) → 1D functions \(\hat{f}_n(r)\) for each angular mode \(n\)
Important Distinction: Two Different Fourier Frameworks
Case 1: Continuous Fourier Transform (Standard Definition)
For functions on the entire real line \((-\infty, +\infty)\):
Forward Transform: \[ F(\omega) = \int_{-\infty}^{+\infty} f(t) e^{-j\omega t} dt \]
Inverse Transform: \[ f(t) = \frac{1}{2\pi} \int_{-\infty}^{+\infty} F(\omega) e^{j\omega t} d\omega \]
Case 2: Azimuthal Fourier Analysis (Our Case)
For periodic functions on \([0, 2\pi]\):
Forward Transform: \[ \hat{f}_n(r) = \frac{1}{2\pi} \int_0^{2\pi} f(r, \phi) e^{-in\phi} d\phi \]
Inverse Transform: \[ f(r, \phi) = \sum_{n=-\infty}^{\infty} \hat{f}_n(r) e^{in\phi} \]
Comparison Table
| Aspect | Continuous Fourier Transform | Azimuthal Fourier Analysis |
|---|---|---|
| Domain | \((-\infty, +\infty)\) | \([0, 2\pi]\) (periodic) |
| Forward Transform | Integral | Integral |
| Inverse Transform | Integral | Series summation |
| Coefficient Location | \(\frac{1}{2\pi}\) in inverse transform | \(\frac{1}{2\pi}\) in forward transform |
| Basis | Continuous frequencies \(\omega\) | Discrete modes \(n\) |
Key Insights
- The integral formula for \(\hat{f}_n(r)\) is the forward transform, not the inverse transform
- The series formula is the inverse transform, not the forward transform
- The forward transform cannot be meaningfully expressed as a series - it's a fundamental definition
- This framework is particularly useful in optics, quantum mechanics, fluid dynamics, and any field dealing with circular symmetry
Applications
- Optics: Laser mode analysis, lens aberrations
- Quantum Mechanics: Hydrogen atom wavefunctions
- Fluid Dynamics: Cylindrical flows
- Image Processing: Circular harmonic analysis
This mathematical structure provides a powerful tool for decomposing and analyzing functions with circular symmetry by separating the radial and angular dependencies.