[Clydeuk]

All the recent TV's I have seen have one designated HDMI port that offers Digital Output usually to connect a Home Cinema system or soundbar. The one on my set is labelled ARC (audio return channel?). When connected this way it disables the TV's speakers and routes through the Home Cinema. It also allows operation of the home cinema unit from the TV remote control, powering the TV on or off will do likewise to the home cinema. The optical output is more a legacy connector nowadays.

Using the headphone socket to the audio input of the amplifier would work but will disable the internal speakers, which might not be a problem in this case, but I think the optical to analogue converter would be the best option quality wise.