<?php
...
...

if ( $username && $password )
{
  $data['response'] = 'valid';
else
{
  $data['response'] = 'invalid';
}

echo json_encode($response);

$.ajax({
    type: 'POST',
    url: 'url',
    data: {username : username, password: password},
    success: function(response){
             if(response.response === valid){
                window.location.replace('admin-page');
             }else{
                //error
             }
      }
});