Interferometric approach

Let us accede now two cellophane patterns with a adverse I that varies with a sinusoidal law:

I_1(x) = I_0 \cdot \sin (2\pi \cdot k_1 \cdot x)

I_2(x) = I_0 \cdot \sin (2\pi \cdot k_2 \cdot x)

(the accomplish are appropriately p1 = 1/k1 and p2 = 1/k2), if the patterns are superimposed, the consistent acuteness (interference) is

I(x) = I_0 \cdot ( \sin (2\pi \cdot k_1 \cdot x) + \sin (2\pi \cdot k_2 \cdot x) )

with the Euler's formula:

I(x) = I_0 \cdot 2 \cos \left ( 2\pi \frac{(k_1-k_2)}{2} \cdot x \right ) \cdot \sin \left ( 2\pi \frac{(k_1+k_2)}{2} \cdot x \right )

We can see that the consistent acuteness is fabricated of a atrium law with a top "spatial frequency" (wave number) which is the boilerplate of the spatial frequencies of the two patterns, and of a atrium law with a low spatial abundance which is the bisected of the aberration amid the spatial frequencies of the two patterns. This additional basic is an "envelope" for the aboriginal atrium law. The amicableness λ of this basic is the changed of the spatial frequency

\frac{1}{\lambda} = \frac{k_1 - k_2}{2} = \frac{1}{2} \cdot \left ( \frac{1}{p_1} - \frac{1}{p_2} \right )

if we accede that's p1 = p and p2 = p+δp:

\lambda = 2\frac{p_1 p_2}{p_2 - p_1} \approx 2\frac{p^2}{\delta p} .

The ambit amid the zeros of this envelope is λ/2, and the maxima of amplitude are aswell spaced by λ/2; we appropriately access the aforementioned after-effects as the geometrical approach, with a alterity of p/2 which is the ambiguity affiliated to the advertence that is considered: arrangement 1 or arrangement 2. This alterity is negligible if δp << p.

This abnormality is agnate to the stroboscopy.

edit Rotated patterns

Let us accede two patterns with the aforementioned footfall p, but the additional arrangement is angry by an bend α. Seen from far, we can aswell see aphotic and anemic lines: the anemic curve accord to the curve of nodes, that is, curve casual through the intersections of the two patterns.

If we accede a corpuscle of the "net", we can see that the corpuscle is a rhombus: it is a parallelogram with the four abandon according to d = p/sin α; (we accept a appropriate triangle which hypothenuse is d and the ancillary against to the α bend is p).

Unit corpuscle of the "net"; "ligne claire" agency "pale line"

Effect of alteration angle.

The anemic curve accord to the baby askew of the rhombus. As the diagonals are the bisectors of the neighbouring sides, we can see that the anemic band makes an bend according to α/2 with the erect of the curve of anniversary pattern.

Additionally, the agreement amid two anemic curve is D, the bisected of the big diagonal. The big askew 2D is the hypothenuse of a appropriate triangle and the abandon of the appropriate bend are d(1+cos α) and p. The Pythagorean assumption gives:

(2D)2 : d2(1+cos α)2 + p2

id est

(2D)^2 = \frac{p^2}{\sin^2 \alpha}(1+ \cos \alpha)^2 + p^2 = p^2 \cdot \left ( \frac{(1 + \cos \alpha)^2}{\sin^2 \alpha} + 1\right )

thus

(2D)^2 = 2 p^2 \cdot \frac{1+\cos \alpha}{\sin^2 \alpha}

Effect on arced lines.

When α is actual baby (α < π/6), the afterward approximations can be done:

sin α ≈ α

cos α ≈ 1

thus

D ≈ p / α.

We can see that the abate the α, the extreme the anemic lines; if the both patterns are alongside (α = 0), the agreement amid the anemic curve is "infinite" (there is no anemic line).

There are appropriately two means to actuate α: by the acclimatization of the anemic curve and by their spacing

α ≈ p / D

If we accept to admeasurement the angle, the final absurdity is proportional to the altitude error. If we accept to admeasurement the spacing, the final absurdity is proportional to the changed of the spacing. Thus, for the baby angles, it is best to admeasurement the spacing.