Calculating The Deviation Angle For A Prism

DEVIATION ANGLE FOR A PRISM

The total deviation ε of a light ray can be computed, given the incidence angle α. The same calculation allows for determination of the emergence angle δ.

We have:
ε=ζ+η
A=β+γ
=>A+ε=(ζ+β)+(γ+η)
=>A+ε=α+δ
=>ε=α+δ-A (1)
sin(δ)/sin(β)=n=>δ=sin^-1{n*sin(β)}
β=A-γ
=>δ=sin^-1{n*sin(A-γ)} (2)
Now β=sin^-1{sin(α)/n}, because sin(α)/sin(β)=n again,
so (2)=>δ=sin^-1{n*sin[A-sin^-1(sin(α)/n)]} (3)
Now sin[A-sin^-1(sin(α)/n)]=sin(A)*cos(sin^-1(sin(α)/n))-cos(A)*sin(α)/n
So δ=sin^-1{n*sin(A)*cos(sin^-1(sin(α)/n))-cos(A)*sin(α)}
=sin^-1{n*sin(A)*Ö[1-(sin(α)/n)²]-cos(A)*sin(α)}
=>δ=sin^-1{sin(A)*Ö[n²-sin²(α)]-cos(A)*sin(α)} (4)

And finally:
(1)(4)=>

(5) (5) gives the deviation angle ε, as a function of α, n_λ and A.

Here's a graph of the above equation, with A=60°, n_D=1.72803 (SF10 Crystal) and α ranging from 45° to 80°. Angles are measured in degrees on the graph. Note that the deviation has a minimum, close to 60°.

The above calculation gives the deviation for an arbitrary angle α. When the incidence angle and the emergence angle are the same, the prism is said to be in the position of minimum deviation, for a given wavelength. Because light gets dispersed into many wavelengths, this position can occur for a particular wavelength only. Below are the calculations that lead to the fundamental formula for prisms in the position of minimum deviation:

For the position of minimum deviation, the ray travels perpendicular to the bisect of angle A, and incidence and emergence angles are equal. Then:
α=δ and β=γ (6)
Also: A=β+γ, ε_min=(α-β)+(δ-γ) (7)
(7)(6)=>A=2*β, ε_min=2*α-2*β or
β=A/2 and ε_min/2=α-A/2 => α=ε_min/2+A/2 (8)
Now we know that n=sin(α)/sin(β) so from the last equation and (8) we get:

(9)

For n_D=1.72803 and A=60°, expression (9) gives ε_min=59.54084145°, which is exactly the minimum displayed on the graph above.

Because the light dispersion angle is very small, (on the order of 7°) practically all the rays suffer minimum deviation. When a prism spectroscope is designed, its prism is placed in that position for usually the yellow part of the spectrum. (Sodium D lines). The PHASMATRON's prisms are placed in minimum deviation position for the green line of λ=5615.97363281Angstroms, its resonating wavelength.

Back to Spectroscopy