php-google captcha not working

i have php login file and i want to put the google captcha in that login form and i use php and html together(not sending the form data to other php page.) the problem is when users don't verify captcha , users still can log in to the website.(even captcha not verify) here is my code :

                    <form action="" method="post" id="form">                        
                    <div class="form-group">
                        <div class="input-group">
                            <span class="input-group-addon"><i class="fa fa-user"></i></span>
                            <input type="text" class="form-control" name="form-username" id="form-username" placeholder="Username" required="">
                        </div>
                    </div>                        
                    <div class="form-group">
                        <div class="input-group">
                            <span class="input-group-addon"><i class="fa fa-lock"></i></span>
                            <input type="password" class="form-control" name="form-password" placeholder="Password" id="form-password" required="">
                        </div>
                    </div> 

  <div class="form-group">

                        <div class="g-recaptcha" data-theme="dark" data-sitekey="my site key"></div>

                    <div class="form-group no-border margin-top-20">
                     <input type="submit" name="submit" value="Sign In !" class="submit_button4 btn btn-primary btn-block" >

   </form>

<?php


    if($_SERVER["REQUEST_METHOD"] === "POST")
    {
        //form submitted

        //check if other form details are correct

        //verify captcha
        $recaptcha_secret = "my secret key";
        $response = file_get_contents("https://www.google.com/recaptcha/api/siteverify?secret=".$recaptcha_secret."&response=".$_POST['g-recaptcha-response']);
        $response = json_decode($response, true);
        if($response["success"] === true)
        {
            echo "Logged In Successfully";


        }
        else
        {
            echo "You are a robot";


        }
    }
?>

Note : also,this code <script src='https://www.google.com/recaptcha/api.js'></script> is in my head tag (and all code is in one page). sorry for my bad english.