A

Step 1: To achieve weak transmission through the thin film via interference, we need to consider the condition for minimal transmission due to destructive interference. In a thin film, the condition for destructive interference (for the first minimum) can be given by the formula:
$$ 2nt = (m + \frac{1}{2})\lambda $$
where:
- $n$ is the refractive index of the film,
- $t$ is the thickness of the film,
- $m$ is the order of interference (0, 1, 2,...),
- $\lambda$ is the wavelength of light in vacuum.

Step 2: Since we want the minimum thickness that causes weak transmission, we can set $m = 0$ (first order minimum):
$$ 2nt = \frac{1}{2}\lambda $$
or,
$$ t = \frac{\lambda}{4n} $$

Step 3: Substituting the values:
- $\lambda = 560 \text{ nm} = 560 \times 10^{-9} \text{ m}$
- $n = 1.4$

$$ t = \frac{560 \times 10^{-9}}{4 \times 1.4} $$
$$ t = \frac{560 \times 10^{-9}}{5.6} $$
$$ t = 100 \times 10^{-9} \text{ m} $$

Thus, the minimum thickness of the film so that it weakly transmits the light is 100 nm.