You do not need to use AJAX to download a file. If the
content-disposition header is set, it will be downloaded and the current page will not be reloaded.
Just add this to your php script that creates the PDF:
header('Content-Disposition: attachment; filename=' . $MyFileName);
And instead of AJAX, use a regular anchor tag link:
As someone noted in the comments, if you need to post you can still use a simple form with the same effect:
<form method="POST" ACTION="tcpdf/examples/example_061.php"> <input type="hidden" name="myPostItem" value="My POSt VALUE" /> <input type="submit" value="download"> </form>