As the simplest for actual calculation, we will consider a little further the case where the glass is plane and parallel, of thickness t and index μ, and is supplemented by a lens at whose focus the source of light is placed. This lens acts both as collimator and as object-glass, so that the combination of lens and plane mirror replaces the concave mirror of Newton’s experiment. The retardation is calculated in the same way as for thin plates. In fig. 5 the diffracting particle is situated at B, and we have to find the relative retardation of the two rays which emerge finally at inclination θ, the one diffracted at emergence following the path ABDBIE, and the other diffracted at entrance and following the path ABFGH. The retardation of the former from B to I is 2μt + BI, and of the latter from B to the equivalent place G is 2μBF. Now FB = t sec θ′, θ′ being the angle of refraction; BI = 2t tan θ′sin θ; so that the relative retardation F is given by
R = 2μt {1 + μ−1 tan θ′ sin θ − sec θ′) = 2μt (1 − cos θ′).
If θ, θ′ be small, we may take
R = 2tθ2 / μ
(1).
as sufficiently approximate.
The condition of distinctness is here satisfied, since R is the same for every ray emergent parallel to a given one. The rays of one parallel system are collected by the lens to a focus at a definite point in the neighbourhood of the original source.
The formula (1) was discussed by Herschel, and shown to agree with Newton’s measures. The law of formation of the rings follows immediately from the expression for the retardation, the radius of the ring of nth order being proportional to n and to the square root of the wave-length.
§ 10. Interferometer.—In many cases it is necessary that the two rays ultimately brought to interference should be sufficiently separated over a part of their course to undergo a different treatment; for example, it may be desired to pass them through different gases.
| Fig. 6. |
| Fig. 7. |