Login Logout in JSF 2

Question

I tried to do login and logout method refer this article http://www.itcuties.com/j2ee/jsf-2-login-filter-example/ Login method is fine but logout doesn't work!

LoginBean.java

@ManagedBean
@SessionScoped
public class LoginBean implements Serializable {

    @EJB
    private UsersFacadeLocal usersFacade;

    public LoginBean() {
    }

    //
    public String doLogin() {
        String redirect = "";
        if (usersFacade.loginUser(email, password)) {
            loggedIn = true;
            redirect = navigationBean.redirectToMain();
        } else {
            FacesContext facesContext = FacesContext.getCurrentInstance();
            facesContext.addMessage(null, new FacesMessage(FacesMessage.SEVERITY_WARN, "Username or password is incorrect", null));
            redirect = navigationBean.redirectToLogin();
        }
        return redirect;
    }

    //
    public String doLogout() {
       ExternalContext ectx = FacesContext.getCurrentInstance().getExternalContext();
        HttpServletResponse response = (HttpServletResponse)ectx.getResponse();
        HttpSession session = (HttpSession)ectx.getSession(false);
        session.invalidate();
        loggedIn = false;
        return navigationBean.redirectToLogin();
    }
    //
    private String email;
    private String password;
    private boolean loggedIn;
    @ManagedProperty(value = "#{navigationBean}")
    private NavigationBean navigationBean;

    public String getEmail() {
        return email;
    }

    public void setEmail(String email) {
        this.email = email;
    }

    public String getPassword() {
        return password;
    }

    public void setPassword(String password) {
        this.password = password;
    }

    public boolean isLoggedIn() {
        return loggedIn;
    }

    public void setLoggedIn(boolean loggedIn) {
        this.loggedIn = loggedIn;
    }

    public NavigationBean getNavigationBean() {
        return navigationBean;
    }

    public void setNavigationBean(NavigationBean navigationBean) {
        this.navigationBean = navigationBean;
    }
}

LoginFilter.java

public class LoginFilter implements Filter{

    @Override
    public void init(FilterConfig filterConfig) throws ServletException {
    }

    @Override
    public void doFilter(ServletRequest request, ServletResponse response, FilterChain chain) throws IOException, ServletException {
        HttpServletRequest req = (HttpServletRequest)request;
        HttpServletResponse res = (HttpServletResponse)response;
        LoginBean session = (LoginBean)req.getSession().getAttribute("loginBean");
        String url = req.getRequestURI();
        if(session == null || session.isLoggedIn() == false)
        {
            if(!url.equals("/login.xhtml"))
            {
                res.sendRedirect(req.getContextPath() + "/login.xhtml");
            }
            res.sendRedirect(req.getContextPath() + "/login.xhtml");
        }
        else
        {
            chain.doFilter(request, response);
        }
    }

    @Override
    public void destroy() {
    }

Web.xml

<?xml version="1.0" encoding="UTF-8"?>
<web-app version="3.1" xmlns="http://xmlns.jcp.org/xml/ns/javaee" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:schemaLocation="http://xmlns.jcp.org/xml/ns/javaee http://xmlns.jcp.org/xml/ns/javaee/web-app_3_1.xsd">
    <filter>
        <filter-name>LoginFilter</filter-name>
        <filter-class>ejb.DAO.LoginFilter</filter-class>
    </filter>

    <filter-mapping>
        <filter-name>LoginFilter</filter-name>
        <url-pattern>/faces/*</url-pattern>
    </filter-mapping>
    <servlet>
        <servlet-name>Faces Servlet</servlet-name>
        <servlet-class>javax.faces.webapp.FacesServlet</servlet-class>
        <load-on-startup>1</load-on-startup>
    </servlet>
    <servlet-mapping>
        <servlet-name>Faces Servlet</servlet-name>
        <url-pattern>*.xhtml</url-pattern>
    </servlet-mapping>

    <welcome-file-list>
        <welcome-file>faces/main.xhtml</welcome-file>
    </welcome-file-list>
</web-app>

After logged in, when I click logout button it's redirect to login page. but when I copy and paste the main page url to a new tab in browser the session is still there.

Any suggestion. Thanks in advance