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I'm trying to compile what I think is a pretty simple function that does matrix multilplications in a loop (see below). But Mathematica issues the message:

Could not complete external evaluation at instruction 1; proceeding with uncompiled evaluation

Could you please help.

Refl[λ_, d1_, d2_, n1_, n2_, Np_] :=
{
  M = {{0.5, 0.5}, {0.5, -0.5}};
  Do[
  {
    M = 
      M.{{Cos[2 Pi*n1*d1/λ], - ((I Sin[2 Pi*n1*d1/λ])/n1)}, 
         {-I n1*Sin[2 Pi*n1*d1/λ], Cos[2 Pi*n1*d1/λ]}};
    M = 
      M.{{Cos[2 Pi*n2*d2/λ], -((I Sin[2 Pi*n2*d2/λ])/n2)}, 
         {-I n2* Sin[2 Pi*n2*d2/λ], Cos[2 Pi*n2*d2/λ]}}
  }, {i, 1, Np}];
  M = M.{{1.0, 1.0}, {1.0, -1.0}};    
  Return[Abs[M[[2]][[1]]/M[[1]][[1]]]^2]
}

CompiledrRefl = 
  Compile[
    {{λ, _Real}, {d1, _Real}, {d2, _Real}, {n1, _Complex}, 
     {n2, _Complex}, {Np, _Integer}}, 
    Refl[λ, d1, d2, n1, n2, Np]]
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This will compile just fine, though as others have said, Compile is a tricky beast.

One key thing to remember is adding 0.0 I to your initialization of M, otherwise you will get a Compile::cset error message.

compiledrRefl = Compile[{{λ, _Real}, {d1, _Real}, {d2, _Real}, 
                       {n1, _Complex}, {n2, _Complex}, {Np, _Integer}},  
                  Module[{M},
                    M = {{0.5, 0.5}, {0.5, -0.5}} + 0.0 I;
                    Do[{M = M.{{Cos[2 Pi*n1*d1/λ], -((I Sin[2 Pi*n1*d1/λ])/n1)},
                               {-I n1*Sin[2 Pi*n1*d1/λ], Cos[2 Pi*n1*d1/λ]}};
                    M = M.{{Cos[2 Pi*n2*d2/λ], -((I Sin[2 Pi*n2*d2/λ])/ n2)},
                               {-I n2*Sin[2 Pi*n2*d2/λ], Cos[2 Pi*n2*d2/λ]}}}, {i, 1, Np}];
                    M = M.{{1.0, 1.0}, {1.0, -1.0}};
                    Abs[M[[2]][[1]]/M[[1]][[1]]]^2]]

Comparing it to your original function, which I've edited to remove the curly braces and wrapped in a module:

refl[λ_, d1_, d2_, n1_, n2_, Np_] := Module[{M},
                    M = {{0.5, 0.5}, {0.5, -0.5}} + 0.0 I;
                    Do[{M = M.{{Cos[2 Pi*n1*d1/λ], -((I Sin[2 Pi*n1*d1/λ])/n1)},
                               {-I n1*Sin[2 Pi*n1*d1/λ], Cos[2 Pi*n1*d1/λ]}};
                    M = M.{{Cos[2 Pi*n2*d2/λ], -((I Sin[2 Pi*n2*d2/λ])/ n2)},
                               {-I n2*Sin[2 Pi*n2*d2/λ], Cos[2 Pi*n2*d2/λ]}}}, {i, 1, Np}];
                    M = M.{{1.0, 1.0}, {1.0, -1.0}};
                    Abs[M[[2]][[1]]/M[[1]][[1]]]^2]]

Do[Refl[100, 2, 3, 5, 6, 1], {100}] // RepeatedTiming
(* 0.00710 seconds *)

Do[compiledrRefl[100, 2, 3, 5, 6, 1], {100}] // RepeatedTiming
(* 0.000216 seconds *)
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