Well done for identifying this important motif in this play. You might like to consider how the whole play in a sense is built around various characters and groups of characters that oppose each other or act as doubles for each other. What is interesting to note is that nearly every character trait that is presented in the play finds its opposite in another character. For example, Puck likes to play pranks and Bottom is the victim of pranks. Helena is tall, Hermis is short. Titania is beautiful, Bottom is ugly.
This sense of doubling though is not just restricted to character traits. You might like to think about the way in which the three main groups of characters, the fairies, the mechanicals and the lovers, are depicted to clash strongly with each other. Just consider the other-worldly beauty and grace of the fairies compared with the clumsy nature of the mechanicals, for example. Perhaps the most poignant scene that demonstrates the importance of doubles and the conflict that such instances include is the picture of the radiantly beautiful Titania weaving flowers into Bottom's hair when he is disguised with the head of an ass. The profound clash created by this double serves to underline the importance of doubling as a motif, but also reinforces the play's central message of how love makes us act stupidly and do things we would never normally do as humans. Let us remember Hermia's wise words in Act I scene 1:
Things base and vile, holding no quantity,
Love can transpose to form and dignity.
The picture of Titania and Bottom captures the truth of this speech perfectly, and acts as an extreme example of how love can overwhelm our reason.