A　scheme　is　proposed　for　generating　NOON　states　for　two　atomic　samples　trapped　in　two　distant　cavities　connected　by　a　third　cavity　and　optical　fibers.　In　the　scheme,　all　the　atoms　are　always　populated　in　the　two　degenerate　ground　states,　so　the　atoms＇　spontaneous　emission　can　be　omitted　approximately.　During　the　operation　neither　the　cavity　modes　nor　fiber　modes　are　excited,　which　is　important　in　view　of　decoherence.　The　scheme　does　not　include　projective　measurement　and　the　NOON　state　is　generated　deterministically.