The dove has long been a symbol of the Holy Spirit due to the accounts of Jesus' Baptism when the Spirit descended upon Him like a dove.  However, the dove also has strong symbolism in the Old Testament as well, which originates back to the time of Noah.  In this window we have a dove carrying an olive branch with the backdrop of a rainbow.  Just as the dove brought Noah hope and peace during times of turbulence and destruction by bearing an olive branch in its mouth, (a sign that the waters were receding and land was to be found), so the Holy Spirit likewise brings the message of hope in peace of a new beginning with God in Christ.  And just as the rainbow becames God's promise to humankind that the world would never be destroyed by flood again the Holy Spirit serves to remind us that Christ promises us the way to avoid destruction by our sins, and that He is the source of absolute peace and absolute hope.