Interferences by a blade of air

The interference by a blade of air is a phenomenon of Interférence obtained with a Interféromètre of Michelson. It is about a simplification of the phenomenon of Interférence by a thin layer of Index of refraction different from 1.

If one leaves the two mirrors perpendicular to the axes of the apparatus but that one draws aside from them one of a distance D , one causes a similar phenomenon called “interference by a blade of air”: all occurs as if there were a semi-transparent mirror in front of a mirror, spaced D .

The rays are considered in a symmetrical way.

Conditions of constructive interference

If one calls:

• θ the complementary one to the angle of incidence (i.e. the angle enters the ray and surface)
• L the way traversed by the ray between the two mirrors;
• L the way traversed by the ray reflected by the first mirror enters the reflection and the plan of the wave front of the emerging ray of the between-mirrors;
• p the distance separating the entrance point and the exit point from the ray traversing the inter-mirrors;

then one has two right-angled triangles, which make it possible to determine by the trigonometrical relations that:

$d\; =\; \backslash \; frac\; \{L\}\; \{2\}\; \backslash \; cdot\; \backslash \; sin\; \backslash \; theta$

$\backslash \; tan\; (\backslash \; theta)\; =\; \backslash \; frac\; \{2d\}\; \{p\}$

$l\; =\; p\; \backslash \; cdot\; \backslash \; cos\; \backslash \; theta$

that is to say

$L\; =\; \backslash \; frac\; \{2d\}\; \{\backslash \; sin\; \backslash \; theta\}$

$p\; =\; \backslash \; frac\; \{2d\}\; \{\backslash \; tan\; \backslash \; theta\}$

$l\; =\; \backslash \; frac\; \{2d\; \backslash \; cdot\; \backslash \; cos^2\; \backslash \; theta\}\; \{\backslash \; sin\; \backslash \; theta\}$

The difference of way optical δ between these two rays is thus

$\backslash \; delta\; =\; L\; -\; L\; =\; \backslash \; frac\; \{2d\}\; \{\backslash \; sin\; \backslash \; theta\}\; \backslash \; cdot\; (1\; -\; \backslash \; cos^2\; \backslash \; theta)\; =\; 2d\; \backslash \; cdot\; \backslash \; sin\; \backslash \; theta$

The rays being parallel, they meet “ad infinitum” (i.e. after a few meters on an optical scale). The interferences will be constructive if the Déphasage between the waves is a multiple of 2π Radian S, i.e. if the difference in walk is a multiple of the Wavelength λ. A checking maximum of intensity in the directions θ will thus be seen:

$2d\; \backslash \; cdot\; \backslash \; sin\; \backslash \; theta\; =\; N\; \backslash \; cdot\; \backslash \; lambda$

where N is a Integer called “order of interference”.

See too

• Law of Bragg

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