t = {0.349, 0.698, 1.221};
a = 10^-3 * {0.935, 1.110, 1.260};
d = 1.96 * 10^-3 ;

n[t_, a_, d_] = Sqrt[a^2 * Sin[t]^2 + d^2 * Sin[2*t]^2] / a;

Sqrt[{D[n[t], {{t, a, d}}]*1.7*10^-3}^2 + {D[n[a], {{t, a ,d}}]*10^-3}^2 + {10^-3}]

n[1, 1, 1]
(* {1.38996, 1.85383, 1.37325} *)

n[5, 5, 5]
(* {1.38996, 1.85383, 1.37325} *)

n[t_, a_, d_] = Sqrt[a^2*Sin[t]^2 + d^2*Sin[2*t]^2]/a;

t = {0.349, 0.698, 1.221};
a = 10^-3*{0.935, 1.110, 1.260};
d = 1.96*10^-3;

n[1, 1, 1]

n[5, 5, 5]

n[x, y, z]

D[n[x, y, z], x]

grad = D[n[x, y, z], {{x, y, z}}] /. {x -> t, y -> a, z -> d}

t = {0.349, 0.698, 1.221};
a = 10^-3*{0.935, 1.110, 1.260};
d = 1.96*10^-3;

n[t_, a_, d_] := Sqrt[a^2*Sin[t]^2 + d^2*Sin[2*t]^2]/a;

error[t_, a_, d_] := Module[{refr, grad},
  refr = n[t, a, d];
  grad = Grad[n[v1, v2, v3], {v1, v2, v3}];
  grad = grad /. {v1 -> 1, v2 -> a, v3 -> d};
  Sqrt[grad^2. refr^2]
  ]

error[t, a, d]

(* {{1.91521, 1.4607, 1.1626}, {3457.36, 2375.13, 1788.53}, {1221.75,
  996.403, 851.707}} *)