I'm trying to understand the encouragement design but I have some doubts. For example, if I want to introduce a voucher for teachers that can be used for professional development(the treatment) and I want to estimate its impact on student performance(dependent variable) obviously I have some problems, since not all the individuals in the treatment group will actually use the voucher. How can I introduce the encouragement design in this case? for example is it right if I send (randomly in the treatment group) some invitations to some professional developments program and I use their participation to these invitations as instrumental variable? And what's the difference with intention to treat?
The fact that you can only encourage but not force the teachers to participate puts you squarely in the world of Local Average Treatment Effect (LATE) and Intention to Treat analysis.
Essentially if you use the voucher as a instrumental variable, you will get an estimate of the effect of the voucher on "the sort of person who you can encourage to take a class by giving them a voucher".
Read the Instrumental Variables section in Mostly Harmless Economics, to get a good easy to understand explanation of exactly the situation you are interested in.