Diffraction by a Circular Aperture — Question 10

Question 10

Diffraction and aperture-dependent aberration can favor a finite opening

In a supplied engineering model, total angular blur is $\theta_{\mathrm{tot}}^2=(A/D)^2+(BD^2)^2$, with $A=1.22\lambda$, $\lambda=500\,\mathrm{nm}$, and $B=0.0100\,\mathrm{m^{-2}}$. The two blur contributions add in quadrature; this is an approximate model.

1. Derive the aperture diameter minimizing the total blur.

2. Evaluate the optimum diameter and minimum blur, and find the ratio of diffraction blur to aberration blur there.

Question 10 - Solution

Diffraction and aperture-dependent aberration can favor a finite opening

(a) Method and calculation

$$\frac{d\theta_{\mathrm{tot}}^2}{dD}=-\frac{2A^2}{D^3}+4B^2D^3=0\Longrightarrow D^6=A^2/(2B^2).$$$$\boxed{D_{\mathrm{opt}}=\left(\frac A{\sqrt2B}\right)^{1/3}=35.0702\,\mathrm{mm}}.$$

(b) Interpretation and further calculation

$$\boxed{\theta_d=17.3937\,\mu\mathrm{rad},\quad\theta_a=12.2992\,\mu\mathrm{rad},\quad\theta_{\min}=21.3028\,\mu\mathrm{rad}}.$$$$A^2/D^2=2B^2D^4\Longrightarrow\boxed{\theta_d/\theta_a=\sqrt2}.$$

Physical check

The optimum does not occur at equal blur contributions because the two terms have different powers of aperture diameter.