Problem $1.$ Camera. $(60$ points total$).$ This problem calculates several properties of a cell phone camera.

Here are its specifications:

i$.24$ megapixels $(60004000$ pixels$)$

ii$.$ The sensor$,$ measured diagonally, is $(\frac{1}{2\mathrm{.}55})$ inches $=10mm$

iii. The lens has a field of view $2=69\mathrm{.}4$ degrees horizontally and $49\mathrm{.}6$ vertically

iv$.$ The maximum aperture of the lens is $\frac{f}{1\mathrm{.}8}$

Note: for all calculations that require a wavelength, use $=600nm.$ Even though the lens is actually quite complex, treat it as a thin spherical lens with no aberrations other than diffraction.

$($a$)(10$ pts$).$ Use geometry and the field of view to show that the focal length of the lens $f=6mm.$

$($b$)(10$ pts $).$ For a selfie at a distance of $0\mathrm{.}5$ m $,$ find the depth of field using an aperture of $\frac{f}{1\mathrm{.}8}.$ Is this depth of field enough $($at least $5$ cm $)$ for a portrait that keeps everything $($eyes$,$ nose, ears$)$ in focus? If not, what aperture size would be needed to do so$?$ Explain quantitatively. Use the width of one pixel as the $x_{\text{m}\text{a}\text{x}}.$

$($c$)(10$ pts$).$ Assuming an average human head size of $23$ cm $($measured vertically$),$ with the camera at the same distance $(0\mathrm{.}5$ m away$),$ find the size $($full width$)$ of the portrait in mm and in pixels, on the sensor $($assuming it is in focus$).$

$($d$)(10$ pts$).$ Find the resolution of the camera under these conditions assuming a thin lens with circular aperture and no spherical aberration, no coma, no astigmatism, and no chromatic aberration. Use the aperture size that gave a depth of field of $5$ cm $.$ That is$,$ find the point spread function $($PSF$).$ Express your answer in micrometers and in units of pixels within the plane of the sensor$.$