Combine water, juice, and ginger in a blender; process until blended.
Line a strainer with cheesecloth. Strain mixture over a pitcher; discard solids. Add sugar to pitcher; stir until dissolved.
Add sparkling water just before serving. Serve over ice. Garnish with lime slices, if desired.
If you can't find superfine sugar, place granulated sugar in a blender, and process until fine. You can find bottled ground fresh ginger in the produce section of the supermarket.