Async operation in WPF

MainWindow.xaml

<Window x:Class="WpfApp1.MainWindow"
        xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
        xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"
        xmlns:d="http://schemas.microsoft.com/expression/blend/2008"
        xmlns:mc="http://schemas.openxmlformats.org/markup-compatibility/2006"
        mc:Ignorable="d"
        Title="MainWindow" Height="450" Width="800">
    <DockPanel>
        <Button DockPanel.Dock="Bottom" Click="Button_Click" Content="Append" />
        <TextBlock x:Name="input" />
    </DockPanel>
</Window>

MainWindow.xaml.cs

using Nito.AsyncEx;
using System.IO;
using System.Text;
using System.Threading;
using System.Threading.Tasks;
using System.Windows;

namespace WpfApp1
{
    public partial class MainWindow : Window
    {
        private readonly Timer timer;
        private readonly AsyncLock locker = new AsyncLock();

        public MainWindow()
        {
            InitializeComponent();

            this.timer = new Timer(timerCallback);
            this.Loaded += (s, e) => this.timer.Change(1000, 1000);
        }

        private async Task WriteTextAsync(string text)
        {
            using (await locker.LockAsync())
            {
                using (var fs = new FileStream("output.txt", FileMode.Append, FileAccess.Write, FileShare.None, 65536, true))
                {
                    var tw = new StreamWriter(fs, Encoding.UTF8);

                    await tw.WriteLineAsync(text);
                    await tw.FlushAsync();
                }
            }
        }

        private void timerCallback(object state)
        {
            var task = this.WriteTextAsync("Timer periodic.");
        }

        private void Button_Click(object sender, RoutedEventArgs e)
        {
            var task = this.WriteTextAsync(this.input.Text);
        }
    }
}