Is it considered unethical/bad etiquette to contact another potential PhD supervisor while waiting for a response from my first choice?

Question

I have contacted a potential PhD supervisor and currently he is reviewing my work, however this evaluation process seems to be taking too long.

Is it ethical to give my research proposal to another professor (another university) while he is reviewing my work? Will this reflect negatively on me if he finds out about it?
There is a good chance that these two professors are in contact with each other ( though I can't be sure!)

He is the foremost expert in my field and I would prefer to have him as supervisor. I just don't want to lose my other opportunities.

Note : I have already secured funding for my PhD. I am only looking for a reputable school (and supervisor). I am considering two institutions in Europe, in Netherlands and Germany.

Answer 1

Finding a PhD advisor, like finding a job, is not a serial process. You can—and should—have multiple applications out simultaneously. If you get an offer you find acceptable, then you should take it.

If you are concerned about missing out on an opportunity with your first-choice advisor if someone else makes you an offer, then you can send a polite email informing the first-choice advisor something like:

I would really like to work with you, but I am starting to receive offers of admission from other advisors. They would like to have an answer by <date>. While I understand you need time to make a decision, would it be possible for you to give me an answer by then?

If they're serious about having you as a PhD student, they'll respond.

Answer 2

I'm assuming you are a normal student, i.e. you don't have single-author papers in science or something similar. If you have, contact your only choice again, if he has money he should hire you.

Assuming you are a normal student, then yes, contact other Professors and programs as well. This is normal (at least in the natural sciences) and increases your chances of actually having a position in the future. Don't worry about this reflecting negatively. If someone does, you maybe wouldn't want to work with this person anyway. Honestly, however, there are many applicants that individual withdrawels don't bother us too much.

Also, consider being less picky. If your choice of Prof. is indeed the foremost expect the (prior) probability of him actually hiring you is rather low. To increase the (posterior) probability of being hired, you need to stand out.