Diffraction by a Circular Aperture — Question 8

Question 8

Detector sampling must accommodate the optical spatial-frequency cutoff

For an ideal incoherent circular-aperture microscope, take object-space cutoff frequency $f_c=2\mathrm{NA}/\lambda_0$ in cycles per meter. The image magnification is M. Nyquist sampling requires at least two pixels per shortest transmitted sinusoidal period. Let $\mathrm{NA}=0.80$, $\lambda_0=500\,\mathrm{nm}$, and $M=40$.

1. Find the object cutoff and largest permitted image-plane pixel pitch.

2. For a $6.5\,\mu\mathrm m$ pixel, determine the minimum magnification meeting the criterion.

Question 8 - Solution

Detector sampling must accommodate the optical spatial-frequency cutoff

(a) Method and calculation

$$\boxed{f_c=2(0.80)/(500\times10^{-9})=3.20\times10^6\,\mathrm{m^{-1}}}.$$$$f_{c,\mathrm{image}}=f_c/M\Longrightarrow\boxed{p\leq M/(2f_c)=M\lambda_0/(4\mathrm{NA})=6.25\,\mu\mathrm m}.$$

(b) Interpretation and further calculation

$$M\geq2pf_c=4p\mathrm{NA}/\lambda_0.$$$$\boxed{M_{\min}=41.6}.$$

Physical check

The cutoff criterion concerns sampling a continuous image, while Rayleigh resolution concerns separating point sources; the numerical constants need not coincide.