Answer
Based on official Jordanian legal texts
A non-custodial father in Jordan has significant rights combining his bond with his children and his guardianship over them. Article 181 of the Personal Status Law establishes his right to see, gather, host, and accompany the child, and to keep in touch through modern means of communication when direct visitation is not feasible.
Article 184 provides that the guardian (walī) has the right to oversee the child's affairs, upbringing, and guidance, so custody is not separated from the father's role as guardian. Article 177 states that the custodian's travel with the child outside the Kingdom requires the guardian's consent or the court's permission if he refuses, so the father has a say in such pivotal decisions.
A non-custodial father's rights thus include regular visitation, oversight of education and guidance, participation in major decisions such as travel, and being informed about his children's affairs. These rights lapse only by a reasoned judicial ruling where harm to the child is established.
Organizing and detailing these rights remains within the Sharia Court's competence according to the child's interest and the circumstances of each case.
This is a general explanation based on Jordanian Personal Status Law and does not replace advice from a qualified lawyer in a specific dispute.