Conical diffraction: Hamilton's diabolical point at the ... - Physics home

46 Conical diffraction and Hamilton’s diabolical point [2, § 9
conical refraction, deflections are determined by the half-angle of the ray cone,
which from eq. (3.7) is
A = 1 2 arctan( n 2 2√
αβ
)
.
(A.1)
This is indeed small in practice, because of the near-equality of the three refractive
indices. For the Lloyd [1837] experiment on aragonite, the Berry, Jeffrey and
Lunney [2006] experiment on MDT, and the Raman, Rajagopalan and Nedungadi
[1941] experiment on naphthalene (whose cone angle is the largest yet reported),
the data in Table 1 give
1
24 A4 Lloyd = 3.3 × 10−9 ,
1
(A.2)
24 A4 Berry = 9.1 × 10−9 ,
1
24 A4 Raman = 8.5 × 10−6 .
As is often emphasized, internal and external conical refraction are associated
with different aspects of the geometry of the wave surface. But in the paraxial
regime the difference between the cone angles (A and A ext respectively) disappears.
To explore this, we first note that (Born and Wolf [1999])
A ext = 1 2 arctan( n 1 n 3
√
αβ
)
.
(A.3)
In terms of the refractive-index differences
μ 1 ≡ n 2 − n 1
, μ 3 ≡ n 3 − n 2
,
n 2 n 2
we have, to lowest order,
A ≈ A ext ≈ √ μ 1 μ 3 ,
(A.4)
(A.5)
which is proportional to the refractive-index differences. The difference between
the angles is
A − A ext ≈ √ [
μ 1 μ 3 μ 1 − μ 3 + 1 3μ
2
(A.6)
4(
1 − 2μ 1 μ 3 + 3μ 2 3) ] ,
which is proportional to the square of the index differences, except when the two
differences are equal, when it is proportional to the cube. For the aragonite, MDT
and naphthalene experiments,
A Lloyd − A ext,Lloyd = 8.9 × 10 −2 A Lloyd ,
A Berry − A ext,Berry =−4.4 × 10 −3 A Berry ,
A Raman − A ext,Raman =−2.7 × 10 −4 A Raman .
(A.7)