Downloading file from PHP not working when using AJAX

I am trying to download a file, say an image, to a client's computer in PHP.

download.php

<?php
   $file = "path_to_file";
   header('Content-Disposition: attachment; filename="'.basename($file).'"');
   readfile($file);
   exit;
?>

Using the code above works fine, as long as I open the PHP file "manually" in the browser. When using AJAX to execute the PHP code from a JS file (see below), however, it simply does nothing.

var request = $.ajax({
   url: "download.php",
   type: "post"
});

How could I fix this?
Thank you for your answers.