Https two-way authentication with server using a public signed cert, but client using a private CA

Question

I'm a node-js guy but I think this is about Certificate/CA only.

I want to set up an https server using a certificate which is signed by a public CA, so that all the browsers can visit my website without certificate error. At the same time, I want my server to provide two-way https authentication, so that the server can recognize my clients if my clients is using a certificate. Client certificate is signed by CA created by myself.

When I let the client connect to the server, it gets an error called Error: CERT_UNTRUSTED. But I have set up the "ca" & "agent" option for both the server and the client, so I can't figure out my mistake.

I have installed my self-signed CA in my windows 8 Root Certificates, altough I don't think it's really needed.

My Code:

Server

var options = {
    key:keyForCertificate,
    cert:certFromPublicCA,
    ca:[PublicCA, self-signedCA],
    requestCert: true,
    rejectUnauthorized: false
};
var server = require('https').Server(options, require('express')());
server.listen(443);

Client

require('https').request({ host: "www.publicWebsite.com"
    , method: "GET"
    , port: 443
    , headers: { host: "www.publicWebsite.com" }
    , ca:[PublicCA, self-signedCA],
    , path: "/" }, function (res) {
    if (res.client.authorized) {
        console.log("node test: OK")
    } else {
        throw new Error(res.client.authorizationError)
    }
}).end()