The dramatic release of French archeologist Francoise Claustre after three years of captivity by Toubou rebels in northern Chad has exposed important elements of a new power struggle in the wastes of the Sahara.

Last Tuesday, while most television viewers focused on the fatigued, deeply suntanned faces of Claustre and her husband, Pierre, in a brief film clip showing them leaving the Toubou camp, military attaches and other professional North Africa watchers concentrated on the Libyan soldiers clearly shown by the film to have established themselves deep inside northern Chad.

The release of the Claustres was in itself important proof of a new alliance between Libya's unpredictable young leader, Col. Muammar Qaddafi, who was credited by Mrs. Claustre with arranging the release, and a newly pro-Libyan leader of the Toubou rebels, who control a major part of the Tibesti region.

African governments in the region are concerned that France has implicitly legitimized Qaddafi's deep instrusion into northern Chad in return for his help in freeing the couple, according to diplomatic sources.

The Claustres have gone into hiding since they were flown back to France from Tripoli, Libya, last Tuesday. After voicing some indirect criticisms of the handling of the case by the government of President Valery Giscard d'Estaing, they have preferred to fall silent than to face direct questions about their experiences.

Many details previously suppressed by worried officials are filtering out now that the couple has been freed, at a large price to France. The Claustre story, in its present form, has at least three levels.

The first has to do with the tangled politics of Chad, a poverty stricken former French colony at the geographical center of Africa with a government dominated by Christian or animist southerners since independence. In 1966, the Toubou tribesmen and other predominantly Moslem northerners began a revolt against the central government.

The movement, known as Frolinat, split in 1972 and a fierce Toubou nationlist, Hissen Habre, emerged as the leader of the strongest group in the Tibesti. He moved to oppose the first incursions by Qaddafi's troops in 1973, while continuing his war with the government in the Chad capital.

It was into this power struggle that Francoise Claustre stumbled, in circumstances that have led leftist French publications to suggest that either she or her husband work for French intelligence agencies.

The first demands for arms and money to ransom her were made shortly after Giscard was elected in May, 1974, and the new president chose to negotiate with the rebels and to involve the presidency deeply in her fate rather than to allow the Foreign Ministry or other government departments to handle it.

A French military officer trying to negotiate with the rebels was assassinated, the better part of a $2 million cash and weapons France sent to Chad as ransom disappeared and the Chad government ordered French military advisers out of the country as the negotiations dragged on, with no results except deepening cynicism about the case here.

The third phase began with the visit in March 1976, to Tripoli by then Prime Minister Jacques Chirac, who obtained a firm promise from Qaddafi to help in gaining the Claustres freedom.

Libyan efforts to court Habre failed, however. Last September Libyan troops began to move deeper into the Tibesti after establishing posts on the Chad side of the disputed frontier and a sharp clash between Habre's men and the Libyans erupted. It triggered a Libyan decision to move directly against the rebel chief.

It now appears that Qaddafi threw his support behind Habre's younger second in command, Goukouni Guddei, whose austere and strong commitment to Islam seems to have impressed the Libyan leader. Habre was soon forced out of the main Toubou camp and is now reportedly leading a splinter group in the Sahara.

It was Goukouni who agreed to liberate the Claustres. He joined them in a press conference in Tripoli last Tuesday, singing Qaddafi's praises. Qaddafi's careful staging of the even has raised concern here that the Claustre affair may be a sign of a significant shift in the uneasy balance between African and Arab rivalries in the southern band of the Sahara.