Fourier Optics & the Abbe Theory
Fourier Optics & the Abbe Theory
A thin lens performs an optical Fourier transform: a coherent field $U(x,y)$ in the front focal plane appears as $\tilde{U}(f_x,f_y)$ in the back focal plane, where spatial frequency $f_x=x/(\lambda f)$. This underpins spatial filtering, holography, and modern microscopy.
Ernst Abbe (1873) showed a microscope objective captures only a finite spatial-frequency band; the highest captured frequency sets the resolution $d_{\min}=\lambda/(2\,\text{NA})$. Higher NA or shorter $\lambda$ improves resolution.
Optical Fourier Transform
Back focal plane of lens (focal length $f$): $$U_\text{out}(x\prime,y\prime)=\frac{e^{ikf}}{i\lambda f}\,\hat{U}\!\!\left(\frac{x\prime}{\lambda f},\frac{y\prime}{\lambda f}\right).$$
Abbe Resolution Limit
$$d_{\min}=\frac{\lambda}{2\,\text{NA}},\quad \text{NA}=n\sin\theta_{\max}.$$ Collecting more diffraction orders recovers finer spatial detail.
Example 1
Objective NA$=1.4$ in oil ($n=1.515$), $\lambda=488\,\text{nm}$. Find $d_{\min}$.
Solution: $d_{\min}=488/(2\times1.4)\approx174\,\text{nm}$.
Example 2
A 4-$f$ system blocks all frequencies above $f_c$. What does the output look like?
Solution: Output is the input convolved with the point-spread function of the aperture — a low-pass blurred version of the original coherent field.
Practice
- What physical operation does a converging lens perform on a coherent wavefield?
- How does a 4-$f$ system implement a spatial filter?
- Why does higher NA improve microscope resolution?
- Sketch the Fourier-plane pattern for a sinusoidal grating.
Show Answer Key
1. A converging lens performs a 2D Fourier transform of the input field at its back focal plane (for coherent illumination at the front focal plane). Each point in the focal plane corresponds to a spatial frequency component of the input.
2. Input at plane $P_1$ → lens $L_1$ (FT at $P_2$, the Fourier/filter plane) → apply a mask/filter → lens $L_2$ (inverse FT at $P_3$, the output plane). By blocking or modifying spatial frequencies at $P_2$, one performs operations like low-pass filtering (blur), high-pass (edge detection), or matched filtering.
3. Resolution limit (Rayleigh): $\delta = 0.61\lambda/\text{NA}$ where $\text{NA} = n\sin\theta$ is the numerical aperture. Higher NA collects higher spatial frequencies (larger angles), improving resolution. Oil immersion ($n>1$) increases NA beyond the air limit of ~1, achieving sub-micron resolution.
4. A sinusoidal grating $t(x) = \frac{1}{2}(1+\cos(2\pi f_0 x))$ produces three spots in the Fourier plane: one at the origin (DC, zeroth order) and two symmetric spots at $\pm f_0$ (first orders). The spacing of the spots is proportional to $f_0$ (grating spatial frequency).