How do I find the results of varphi?

Questions
,
1

messageImage_1686547771630.jpg
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I want to find relationship between azimuthal angle and wavelength. How do I revise it?
[1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=messageImage_1686547771630.jpg&userId=2934717

2

It seems that Solve needs a little help: the equation becomes easy by expressing Cos[2 \[CurlyPhi]] in terms of Cos[ \[CurlyPhi]]:

eq = \[Lambda] == (-d Cos[\[CurlyPhi]] + 
     Sqrt[(d^2 Cos[2 \[CurlyPhi]] + 2 r^2 - d^2)/2])/
    m (Sqrt[(\[Epsilon] nd^2)/(\[Epsilon] - nd^2)] - ni Sin[\[Alpha]]);
expandedEq = 
  eq /. Cos[2 \[CurlyPhi]] :> TrigExpand[Cos[2 \[CurlyPhi]]];
FullSimplify[Solve[expandedEq, \[CurlyPhi]]]

It is disappointing that TrigExpand does not expand the Cos in simple cases like this:

TrigExpand[Sqrt[Cos[2 x]]]