II.ii. Brevity - Olivier '48

The scene starts on the King and Queen looking into each other's eyes, hiding the figure of Polonius, showing how focused they are on each other compared to other things. As Polonius starts to speak, they separate and we see him between their heads, a shot reprised a number of times during this section. Olivier's staging is usually very deliberate, so we can infer that Polonius is creating an impediment to their couplehood. He separates them by bringing news of Hamlet, opening a can of worms that the royal couple were not necessarily willing to open. Remember, in this version, Rosencrantz and Guildenstern do not exist, so the royals are here triggered into addressing the Hamlet situation. They are not proactively recruiting agents. Polonius imposes this agenda on them, which will lead to the couple's undoing, as prefigured in the staging here.

Claudius, in fact, couldn't look less compassionate. He's stern and aloof, frequently looking at Gertrude's reactions for cues. While Gertrude does prod Polonius for more direct answers, Claudius isn't any more patient with his well-meaning, but tedious aide. When he says "Not that I know of", there is an underlying sense of mistrust. It's not that Polonius hasn't been a useful counselor, but that he's meddling in things the King might not want to be bothered with. He's only letting Polonius talk to keep the Queen happy. Gertrude, for her part, seems to have given up hope. There's trembling despair in her voice when she asks for "more matter", as if to tell Polonius that it is too serious a subject for him to play with words. Based on that reaction, the King agrees to some token spying. Olivier has Hamlet overhear the whole thing, which will have to be taken into account when discussing his meeting with Ophelia later. He will already know it's a set-up.
On matters related to the letter, in this version, we must accept that Ophelia went to her father and surrendered her correspondence on faith. No scene corroborates it, but Polonius' kindly attitude would not support the more sinister idea that he would have misappropriated them. Jean Simmons' Ophelia is too naive and child-like to keep anything from her parent.