Judge SACK files a separate opinion concurring in part.
McLAUGHLIN, Circuit Judge.
Bryant Smith filed a petition for a writ of habeas corpus pursuant to 28 U.S.C. § 2254. In it he claims that the state trial court’s exclusion of his brother and sister during the testimony of two undercover officers, unless they consented to sitting behind a screen, violated his Sixth Amendment right to a public trial. The United States District Court for the Eastern District of New York (Korman, C.J.) denied Smith’s petition in an oral opinion. We vacate the district court’s decision because we find that the state court failed to make the requisite particularized findings necessary to justify the exclusion, but remand for a supplemental evidentiary hearing [535]*535pursuant to Nieblas v. Smith, 204 F.3d 29 (2d Cir.1999).
BACKGROUND
In September 1997, in the Bushwick section of Brooklyn, Smith allegedly sold two glassine envelopes of heroin to an undercover police officer (the “Undercover?) for $20 of prerecorded buy money. The police back-up team, which included a second undercover (the “Ghost”), immediately arrested Smith.
In 1998, Smith stood trial in New York State Supreme Court, Kings County, on one count of criminal sale of a controlled substance in the third degree, two counts of criminal possession of a controlled substance in the third degree, and two counts of criminal possession of a controlled substance in the seventh degree.
The prosecution made an application to seal the courtroom during the testimony of the Undercover and the Ghost (together the “undercovers”). In response, the court suggested a “semi-seal,” excluding the general public, but allowing Smith’s family to attend the proceedings. Smith’s counsel consented to such a closure. The prosecution, however, objected, arguing that other measures needed to be taken because “these undercovers still operated in this vicinity, and not only their safety, but their ability to operate effectively as undercovers [would] be jeopardized if anyone were to see them.” Specifically, the prosecution proposed placing a chalkboard between Smith’s relatives and the witness stand. In response, Smith’s counsel emphasized that his client’s family “do[es] not live in the vicinity. They live some distance away,” and thus the prosecutor’s assertion that they “may jeopardize [the undercovers’] safety and effectiveness ... is not really sufficient.” The prosecution responded that “they still work in [an undercover] capacity in the [Bushwick] neighborhood, and that their- ability to act effectively .as undercovers will be jeopardized, and their physical safety will be jeopardized unless the courtroom- is sealed.” The prosecutor added that he would not allow the undercovers to testify if Smith’s relatives were -not behind a “chalkboard at the very least, if not.exclude[d] altogether.”
The following colloquy then ensued:
THE COURT: So what’s the point? Do we have to have a hearing? ...
PROSECUTOR: Say we have the hearing and that’s what they testify to? What then?
COUNSEL: I believe I am given an opportunity to question th[ese] undercover[s], and I believe, then, that I am given the opportunity under the law to make an argument,that this is a public courtroom and,, that there, is no threat [that this] is credible.
THE COURT: So, you are saying-
PROSECUTOR: What the undercover says is not going to affect who these characters are who are sitting in the courtroom.
COUNSEL: I’m going to say that if [the prosecutor] refers to my client’s family as characters or drug dealers again ... there will be a contempt problem for pie because ... that’s totally unacceptable and boyish [and] improper.
Following this exchange, the court ordered a hearing pursuant to People v. Hinton, 31 N.Y.2d 71, 334 N.Y.S.2d 885, 286 N.E.2d 265 (1972), which requires' courts to conduct a hearing to determine the propriety and scope of a closure. Both under-covers testified.
The Ghost testified that he had been an undercover for two years, and he had worked only in Brooklyn, specifically in Bushwick and Bedford Stuyvestant. Additionally, the Ghost intended to return to [536]*536work in those neighborhoods in the future. He stated that he had been involved in “over a hundred” narcotics purchases during his two years as an undercover, and had approximately fifteen “lost subjects,” i.e., people who sold him drugs but were not arrested by his back-up team. The Ghost further stated that he had testified before a grand jury over a hundred times and during those appearances he had never once used his real name to identify himself, giving only his badge number.
The Ghost went on to admit that the grand jurors, some of whom came from communities where he had performed undercover work, had an unobstructed view of him during his grand jury testimony. During his court appearances he wore civilian clothes, never appeared in public in the company of uniformed officers, traveled in an unmarked police car, and on the day of the hearing, he entered the courtroom via a backdoor entrance. He had approximately twenty “open cases” pending in Kings County Supreme Court.
As to the threat he faced by testifying publicly, the Ghost said that testifying in an open courtroom would jeopardize his safety because it would “give the general public an idea of what he looked like ... because [he was] still an undercover ... [and he] would maybe be purchasing narcotics from someone who’s sitting in the courtroom.” The Ghost believed that he y/ould be physically harmed if it was discovered that he was an undercover. He added that testifying in open court could jeopardize his effectiveness because “psychologically [he] didn’t know if [he] would be prepared to go back to the street and purchase drugs as an undercover.”
On cross-examination, the Ghost made three notable concessions: he knew neither Smith nor any of his family members; he had not been threatened in this case; and he could not recall ever being threatened by the family members of a defendant against whom he had testified.
After the Ghost’s testimony, the court ruled that Smith’s elderly mother, common-law wife, and child would be able to view the testimony of the Ghost without a screen.1 Any remaining family members could remain in the courtroom, but only if Smith consented to their sitting behind a screen that would obscure them view of the Ghost.
Smith’s counsel immediately objected. He argued that the court was applying an improper standard and erred by failing to hear evidence regarding where Smith’s family members lived. Smith’s counsel then pointed out that both Smith’s brother and sister lived in Queens, while his mother, common-law wife, and child all lived in Flatbush. It was his contention that by residing in Queens, the siblings posed no threat, to the Brooklyn undercovers. The court rejected this argument. Smith’s counsel then rejected the offer to use the screen because “it suggests danger coming from the audience ... and that will prejudice my client.”
The Undercover testified next at the Hinton hearing. His testimony closely tracked that of the Ghost. He was assigned to Brooklyn North Narcotics and worked in several Brooklyn neighborhoods, including Bushwiek. He too planned to return to undercover work in Bushwiek. The Undercover entered the courtroom through a rear entrance, had approximately ten cases pending in the [537]*537same courthouse, and approximately seven lost subjects. He too believed that testifying in an open courtroom would jeopardize both his safety and efficacy.
The Undercover did' not claim to fear Smith’s relatives and admitted that they had not threatened him. In fact, the Undercover admitted that he had never been threatened in more than fifty undercover drug purchases. Nor had he seen Smith since the day of the incident. Much like the Ghost, the Undercover had testified before the grand jury on countless occasions without a screen and, quite often, some of those grand jurors came from communities in which he worked.
Following the hearing, Smith’s counsel renewed his argument that the courtroom should be left open to Smith’s entire family, asserting that “there is no rational basis to distinguish between family members.” The court, however, disagreed and issued the same ruling that it made following the Ghost’s testimony: limited closure' permitting only Smith’s 'mother, common-law wife, and child to view the testimony without a screen. Other relatives or friends would have to remain behind an “unobtrusively placed screen in the back” during the testimony of the undercovers.2
Both undercovers testified at trial. The record is silent on whether anyone chose to remain in the courtroom behind a screen. Smith was eventually convicted' and -sentenced to nine to eighteen years’ imprisonment.
Smith appealed his conviction to the Appellate Division, Second Department, arguing that the trial court violated his Sixth-Amendment right to a public trial when it conditioned the attendance of his brother and sister at the undercovers’ testimony upon their sitting behind a screen. The Appellate Division affirmed Smith’s conviction, holding that “the trial court’s alternative to closure ... did not deprive him of his right to a public trial:” People v. Smith, 282 A.D.2d 626, 723 N.Y.S.2d 396 (2d Dep’t 2001). Leave to appeal-to the. New York Court of Appeals was denied. People v. Smith, 96 N.Y.2d 942, 733 N.Y.S.2d 382, 759 N.E.2d 381 (2001) (Wesley, J.).
In July 2002, Smith sought a writ of habeas corpus in the United States District Court for the Eastern District of New York. He renewed his Sixth Amendment argument.
In April 2003, in an oral decision, the district court denied the petition. The district court concluded that the siblings’ remaining outside the courtroom during the undercovers’ testimony was not .the result of the court’s order but was instead simply “self-exclusion,” designed to create a Sixth Amendment argument on appeal. The district, court also observed that the trial court had permitted several members of the public to remain in the courtroom, without obstruction, and thus, Smith “had a public trial [and] the exclusion of two of five family members is sufficiently dimini-mus [sic] for a brief period of time ... that it doesn’t even amount to a violation of the public trial clause.” Additionally, the district court' opined that the Second Circuit “spends too much time on these frivolous arguments. But this one is particularly absurd” in light of the fact that' three members of the family remained in attendance. Finally, the district court held that, even if the trial court’s ruling was erroneous, the error was harmless. The district court did, however, grant a Certificate of Appealability “on the issues whether the Public T]rial Clause was violated, and, if so, whether the error was harmless.”
[538]*538Smith, who is currently incarcerated, now appeals.
DISCUSSION
Smith contends that the district court erred by denying his petition for a writ of habeas corpus. We agree that the state court failed to make the requisite particularized findings necessary to justify the exclusion of Smith’s brother and sister. We remand, however, to the district court for a supplemental evidentiary hearing pursuant to Nieblas v. Smith, 204 F.3d 29 (2d Cir.1999).
We review state court determinations in federal habeas corpus proceedings under the standard delineated by 28 U.S.C. § 2254(d)(l)(2005), which provides that the writ cannot be granted “unless the adjudication of the claim ... resulted in a decision that was contrary to, or involved an unreasonable application of, clearly established Federal law, as determined by the Supreme Court of the United States.”3 28 U.S.C. § 2254(d)(1); see also Williams v. Taylor, 529 U.S. 362, 402-03, 120 S.Ct. 1495,146 L.Ed.2d 389 (2000).
A state court adjudication is “contrary to” clearly established federal law only if “the state court arrives at a conclusion opposite to that reached by [the Supreme] Court on a question of law or if the state court decides a case differently than [the Supreme] Court has on a set of materially indistinguishable facts.”4 Williams, 529 U.S. at 413, 120 S.Ct. 1495.
“Under the ‘unreasonable application’ clause, a federal habeas court may grant the writ if the state court identifies the correct governing legal principle from [the Supreme] Court’s decisions but unreasonably applies that principle to the facts of the prisoner’s case.” Id. Under this standard, “a federal court may not issue the writ simply because that court concludes on its independent judgment that the relevant state-court decision applied clearly established federal law erroneously or incorrectly.” Id. at 411, 120 S.Ct. 1495. “Some increment of incorrectness beyond error is required.” Francis S. v. Stone, 221 F.3d 100, 111 (2d Cir.2000). “[H]ow-ever, ... the increment need not be great; otherwise, habeas relief would be limited to state court decisions so far off the mark as to suggest judicial incompetence.” Id. (internal quotations omitted).
Smith concedes that the trial court properly looked to Waller v. Georgia, 467 U.S. 39, 104 S.Ct. 2210, 81 L.Ed.2d 31 (1984), for the legal standard governing courtroom closures. He argues, however, that the court misapplied that standard. Thus, our initial question is whether the trial court “unreasonably appli[ed]” the Waller standard. Additionally, Smith argues that the district court erred when it held, in the alternative, that if the trial court did in fact err by excluding Smith’s siblings, that error was harmless. We address both issues in turn.
A. Exclusion of Smith’s Siblings
Under Waller, in order to justify a courtroom closure “(1) the party seeking to close the courtroom must advance an overriding interest that is likely to be prejudiced, (2) the closure must be no broader [539]*539than necessary to protect that interest, (8) the trial court must consider reasonable alternatives to closing the proceedings, and (4) [the trial court] must make adequate findings to support the closure.” Ayala v. Speckard, 131 F.3d 62, 69 (2d Cir.1997) (en banc) (citing Waller, 467 U.S. at 48, 104 S.Ct. 2210).
Smith concedes, as he must, that the state court’s order excluding the general public was consistent with Waller. Thus, we need only consider whether the trial court’s order, requiring Smith’s siblings to remain outside the courtroom unless they agreed to sit behind a screen, was an unreasonable application of Waller.
Recently, in Rodriguez v. Miller, this Court scrutinized the tension between the Sixth Amendment and the exclusion of family members. 439 F.3d 68 (2d Cir.2006). In granting the writ to a defendant whose family’s attendance was conditioned upon their willingness to sit behind a screen during the testimony of an. undercover, this Court expressed its strong devotion to the preservation of an individual’s right to have family and friends present at his trial. Id. at 73 (“Intrinsic to the public trial right is an individual’s right to have family members and friends present at his trial, a right this Court takes very seriously.”); see also Carson v. Fischer, 421 F.3d 83, 91 (2d Cir.2005)(“[T]his -Court takes very seriously a defendant’s right to have family members present at his- trial.”); Guzman v. Scully, 80 F.3d 772, 776 (2d Cir.1996) (“The exclusion of courtroom observers, especially a defendant’s family members and friends, even from part of a criminal trial, is not a step to be taken lightly.”); Vidal v. Williams, 31 F.3d 67, 69 (2d Cir.1994) (noting “a special' concern for assuring the attendance of family members of the accused”). The Rodriguez Court also reminded us, however, that our “devotion” is not unbridled, as it “may give way in certain cases to other rights or interests.” Waller, 467 U.S. at 45, 104 S.Ct. 2210.
Looking to the first prong of Waller, prdsecutory powers often have 'an overriding interest in protecting the safety and efficacy of undercover officers. See Brown v. Artuz, 283 F.3d 492, 501-02 (2d Cir.2002) (the “safety of a police officer working undercover surely constitutes an overriding interest”); Nieblas v. Smith, 204 F.3d 29, 33 (2d Cir.1999) (“important interest” necessary for closure of a courtroom includes the need to protect an undercover officer’s anonymity and safety); Ayala, 131 F.3d at 72 (“The state interest in maintaining the continued effectiveness of an undercover officer is an extremely substantial interest .... ”). Under Waller and its progeny, courts must undertake a more exacting inquiry when excluding family members, as distinguished from the general public:
Waller prevents a court from denying a family member’s request to be exempted from a courtroom closure order unless the court is convinced that the exclusion of that particular relative is necessary to protect the overriding interest at stake. Indeed, it would be an unreasonable interpretation of Waller for a court to deny such a request if the exclusion of that particular relative, under the specific circumstances at issue, is not necessary to promote the overriding interest.
Yung v. Walker, 341 F.3d 104, 111 (2d Cir.2003). Thus, courts are constrained to make a “particularized inquiry into whether the exclusion [of a family member] was necessary to advance” the overriding interest. Carson, 421 F.3d at 91.
Here the trial court failed to ensure that the siblings’ exclusion was necessary to advance the overriding interest. The record demonstrates that the court, in excluding the siblings, relied primarily on [540]*540the undercovers’ fear that the family members would point out the officers to unspecified “people” Smith “knows,” in the vicinity of the courthouse or their homes. Such “naked assertion[s], without more do[] not justify excluding a defendant’s family members.” Rodriguez, 439 F.3d at 75. Moreover, the trial court failed to make even the most rudimentary inquiry-let alone a particularized one-into whether the exclusion of Smith’s siblings was even necessary. Without such a particularized inquiry, the Appellate Division’s affir-mance of the trial court’s exclusion order is an unreasonable application of Waller. See Yung, 341 F.3d at 111.
Furthermore, we conclude that the trial court’s exclusion order was essentially arbitrary. The court permitted the family members who lived in the same borough in which the undercovers worked to observe the testimony, but required a screen for the siblings who lived in a different borough. It is hard to discern how allowing the Brooklyn residents to see the under-covers while putting a screen in front of the Queens residents serves the prosecution’s interest in protecting the safety and efficacy of the undercovers.
B. Trivial Closure
Ruling that the petition should be denied, the district court found, alternatively, that any potential violation of the Sixth Amendment was harmless. We do not agree that this is the appropriate standard.
The denial of a public trial is a “structural” error and accordingly is not subject to a harmless error analysis. Peterson v. Williams, 85 F.3d 39, 42 (2d Cir.1996). Instead, when addressing whether an unjustified closure is a Sixth Amendment violation, a “triviality standard” is the proper benchmark. Id.; see also Carson, 421 F.3d at 95 (addressing the distinction between the harmless error and triviality standards). This differs greatly from a harmless error analysis.
A triviality standard does not “dismiss a defendant’s claim on the grounds that the defendant was guilty anyway or that he did not suffer ‘prejudice’ or ‘specific injury.’ ” Peterson, 85 F.3d at 42. Rather, a triviality inquiry looks to “whether the actions of the court and the effect that they had on the conduct of the trial deprived the defendant-whether otherwise innocent or guilty-of the protections conferred by the Sixth Amendment.” Id. This analysis turns on whether the closure subverts the values the drafters of the Sixth Amendment sought to protect: “1) to ensure a fair trial; 2) to remind the prosecutor and judge of their responsibility to the accused and the importance of their functions; 3) to encourage witnesses to come forward; and 4) to discourage perjury.” Id. at 43.
Recently, in Carson, this Court held that the values underlying the Sixth Amendment were not implicated when a defendant’s former mother-in-law was excluded during an informant’s testimony. We excused the trial court’s failure to make particularized findings on whether the exclusion was truly necessary. 421 F.3d at 91-93. We concluded that the presence of, inter alios, four of the defendant’s family members and three members of a co-defendant’s family sufficiently safeguarded the public trial right. Id. In reaching this result, the Court emphasized that the informant’s testimony was discussed at length during the prosecution’s summation and overlapped that of another witness, whose testimony was open to the public. Accordingly, the Carson Court found it to be one of the “rare circumstances” where “the exclusion of a family member or friend, may ... not implicate the Sixth Amendment public trial guarantee.” Id. at 94.
[541]*541This case is not one of the “rare circumstances” contemplated by Carson. The undercovers here provided the great bulk, of the trial testimony against Smith. The Undercover was the principal prosecution witness. He was the only witness to identify Smith as having been involved in the drug sale. His testimony, combined With the Ghost’s, encompassed approximately 114 pages of the 312 total pages of testimony. The trial court order thus effectively excluded the siblings from the crux of the case against Smith.
On these facts, the exclusion of Smith’s brother and sister during the testimony of the undercovers was not trivial, apd, in short, “the values underlying, the Sixth Amendment” were implicated. Id.
C. Supplemental Evidentiary Hearing
It is clear that the state court failed to make the requisite particularized findings necessary to justify the exclusion of Smith’s brother and sister. We think it appropriate, nevertheless, to remand-the case to the district court for an additional evidentiary hearing with the possibility of avoiding “the windfall of a new trial where the alleged constitutional violation does not affect the fairness of the outcome at trial.”, Nieblas v. Smith, 204 F.3d 29, 32 (2d Cir.1999). A district court considering ag habeas petition “retains broad discretion tq, hear additional evidence on behalf of the' state.” Yung, 341 F.3d at 112. The district court should exercise that discretion here on remand and hold an evidentiary hearing to determine whether (here is additional evidence supporting the closure.
Although the mandate shall issue forthwith, we retain jurisdiction pursuant to United States v. Jacobson, 15 F.3d 19 (2d Cir.1994). Accordingly, either party may seek subsequent -appellate review after the district court holds a. supplemental eviden-tiary hearing by notifying the Clerk of the Court within thirty days .of entry of the district court’s judgment. See, e.g., Galviz Zapata v. United States, 431 F.3d 395, 400 (2d Cir.2005). Upon such notification, the case will be heard by this, panel upon letter briefs to be filed according to a schedule set by the-Clerk.
CONCLUSION
Because the state court failed to make the requisite particularized findings necessary to justify the exclusion of Smith’s brother and sister-, we VACATE the judgment of the district court and REMAND with directions to hold a supplemental evi-dentiary hearing pursuant to Nieblas v. Smith.