In a pitcher, combine the lime juice and sugar and stir until the sugar is dissolved. Place a small sieve on top of the opening of the pitcher, and in batches, press the raspberries through the sieve with a spoon. Discard the remaining raspberry pulp and seeds.

Add the sparkling water to the pitcher and stir everything together. Add the lime slices, raspberries and mint sprigs to the pitcher (optional). Chill until ready to serve.