Kins on "Add a widget zone as footer"

Hello.

I have some issues to add a widgetzone as my footer.
There are tons of tuto on the web but nothing helped:

I have a website with a theme child from thirteen theme.
I have modified the functions.php from parent theme (so the thirteen) and added some code:

register_sidebar(array(
        'name' => 'footer_widget',
        'before_widget' => '<div class="footer_widget">',
        'after_widget' => '</div>',
        'before_title' => '<h3>',
        'after_title' => '</h3>',
    ));

To function twentythirteen_widgets_init() {

Normaly, that would be enough to see a change in the backoffice... but nothing.

So i have added this in my footer.php:

<div id="footer_widget">
 <?php    if ( !function_exists( 'dynamic_sidebar' ) || !dynamic_sidebar('footer_widget') ) ?>
</div>

And nothing... what did i do wrong.
Thanks for your help.