我可以'；t在wordpress中保存Facebook用户的电子邮件ID

我使用以下代码允许访问者通过Facebook加入我的博客。用户可以成功登录，但问题是WordPress没有存储电子邮件id。可能有一个愚蠢的错误，但我没有得到它。

function.php

// Facebook Login Button
require_once("inc/facebookoauth.php");
class Facebook_Login_Widget extends WP_Widget
{
    public function __construct()
    {
        parent::__construct("facebook_login_widget", "Facebook Login", array("description" => __("Display a Facebook Login Button")));
    }
    public function form( $instance )
    {
        // Check values
        if($instance)
        {
            $title = esc_attr($instance['title']);
            $app_key = $instance['app_key'];
            $app_secret = $instance['app_secret'];
        }
        else
        {
            $title = '';
            $app_key = '';
            $app_secret = '';
        }
        ?>
        <p>
            <label for="<?php echo $this->get_field_id('title'); ?>"><?php _e('Title', 'facebook_login_widget'); ?></label>
            <input class="widefat" id="<?php echo $this->get_field_id('title'); ?>" name="<?php echo $this->get_field_name('title'); ?>" type="text" value="<?php echo $title; ?>" />
        </p>
        <p>
            <label for="<?php echo $this->get_field_id('app_key'); ?>"><?php _e('App ID:', 'facebook_login_widget'); ?></label>
            <input type="text" class="widefat" id="<?php echo $this->get_field_id('app_key'); ?>" name="<?php echo $this->get_field_name('app_key'); ?>" value="<?php echo $app_key; ?>" />
        </p>
        <p>
            <label for="<?php echo $this->get_field_id('app_secret'); ?>"><?php _e('App Secret:', 'facebook_login_widget'); ?></label>
            <input type="text" class="widefat" id="<?php echo $this->get_field_id('app_secret'); ?>" name="<?php echo $this->get_field_name('app_secret'); ?>" value="<?php echo $app_secret; ?>" />
        </p>
        <?php
    }
    public function update( $new_instance, $old_instance )
    {
        $instance = $old_instance;
        $instance['title'] = strip_tags($new_instance['title']);
        $instance['app_key'] = strip_tags($new_instance['app_key']);
        $instance['app_secret'] = strip_tags($new_instance['app_secret']);
        update_option("facebook_app_id", $new_instance['app_key']);
        update_option("facebook_app_secret", $new_instance['app_secret']);
        return $instance;
    }
    public function widget( $args, $instance )
    {
        extract($args);
        $title = apply_filters('widget_title', $instance['title']);
        echo $before_widget;
        if($title)
        {
            echo $before_title . $title . $after_title ;
        }
        if(is_user_logged_in())
        {
            ?>
                <a href="<?php echo wp_logout_url( get_permalink() ); ?>" title="Logout"><input type="button" value="Logout" /></a>
            <?php
        }
        else
        {
            ?>
                <a href="<?php echo site_url() . '/wp-admin/admin-ajax.php?action=facebook_oauth_redirect'; ?>"><input type="button" value="Login Using Facebook" /></a>
            <?php
        }
        echo $after_widget;
    }
}
register_widget("Facebook_Login_Widget");

inc/facebookauth.php

<?php
session_start();
function facebook_oauth_redirect()
{
    global $wp, $wp_query, $wp_the_query, $wp_rewrite, $wp_did_header;
    require_once("../wp-load.php");
    //construct URL and redirect
    $app_id = get_option("facebook_app_id");
    $redirect_url = get_site_url() . "/wp-admin/admin-ajax.php?action=facebook_oauth_callback";
    $permission = "email,name";
    $final_url = "https://www.facebook.com/dialog/oauth?client_id=" . urlencode($app_id) . "&redirect_uri=" . urlencode($redirect_url) . "&permission=" . $permission;
    header("Location: " . $final_url); 
    die();
}
add_action("wp_ajax_facebook_oauth_redirect", "facebook_oauth_redirect");
add_action("wp_ajax_nopriv_facebook_oauth_redirect", "facebook_oauth_redirect");
function generateRandomString($length = 10) {
    $characters = '0123456789abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ';
    $randomString = '';
    for ($i = 0; $i < $length; $i++) {
        $randomString .= $characters[rand(0, strlen($characters) - 1)];
    }
    return $randomString;
}
function facebook_oauth_callback()
{
    global $wp, $wp_query, $wp_the_query, $wp_rewrite, $wp_did_header;
    require_once("../wp-load.php");
    if(isset($_GET["code"]))
    {   
        $token_and_expire = file_get_contents("https://graph.facebook.com/oauth/access_token?client_id=" . get_option("facebook_app_id") . "&redirect_uri=". get_site_url() . "/wp-admin/admin-ajax.php?action=facebook_oauth_callback" . "&client_secret=" . get_option("facebook_app_secret") . "&code=" . $_GET["code"]);
        parse_str($token_and_expire, $_token_and_expire_array);

        if(isset($_token_and_expire_array["access_token"]))
        {   
            $access_token = $_token_and_expire_array["access_token"];
            $user_information = file_get_contents("https://graph.facebook.com/me?access_token=" . $access_token . "&fields=email,name");
            $user_information_array = json_decode($user_information, true);
            $email = $user_information_array["email"];
            $name = $user_information_array["name"];
            if(username_exists($name))
            {
                $user_id = username_exists($name);
                wp_set_auth_cookie($user_id);
                update_user_meta($user_id, "facebook_access_token", $access_token);
                header('Location: ' . get_site_url());
            }
            else
            {
                //create a new account and then login
                wp_create_user($name, generateRandomString(), $email);
                $user_id = username_exists($name);
                wp_set_auth_cookie($user_id);
                update_user_meta($user_id, "facebook_access_token", $access_token);
                header('Location: ' . get_site_url());
            }
        }
        else
        {   
            header("Location: " . get_site_url());
        }
    }
    else
    {
        header("Location: " . get_site_url());
    }
    die();
}
add_action("wp_ajax_facebook_oauth_callback", "facebook_oauth_callback");
add_action("wp_ajax_nopriv_facebook_oauth_callback", "facebook_oauth_callback");

我尝试在update_user_meta($user_id, "facebook_access_token", $access_token);中添加$email，但也不起作用。