8. Macro- and microstructures in the superposition

Of course, no dot-screen equivalent ex ists for superpositions with an odd number m of gratings, like the case of m = 3 in Fig. 8.1(a).

Obviously, the period of the microstructure is always greater than or equal to the original screen periods: Since the impulses of the original screen frequencies are included in the compound nailbed, it is clear that the fundamental impulses of the compound nailbed can either coincide with the original screen frequencies (as in the singular (1,0,−1,0)-moiré or in the singular moiré shown in Fig. 5.7(a)), or fall even closer to the DC (as in the singular (1,2,−2,−1)-moiré; see Fig. 8.3(a)).

This follows directly from the fact that any irrational number, including tana, can be approximated by a series of closer and closer rational fractions \(\frac{n}{m},\) for example by taking more and more digits from its infinite decimal representation.

Singular states in which there is no clear visual distinction between “in-phase” and “counter-phase” microstructures do not produce off the singular state a visible macro-moiré in the superposition. This often happens in moirés of high orders, or in moirés involving many superposed layers.

We mention this case just for the sake of completeness, although we will not need it here.

Note that such periods are often themselves irrational numbers, just like the angle α itself (in degrees).

Note that the above discussion on rational vs. irrational screens with respect to an underlying device-pixel grid is, in fact, a particular case in which T ₁, T ₂ correspond to the device-pixel grid.

Note that this abuse of language may be quite misleading, since an angle α (in degrees) may be a rational number while tanα is irrational, and vice versa.

The explicit equation of this plane is most conveniently expressed in the parametric form by ζ₁= a ¹ x +b ₁ y, … ζ _m = a _m x+b _m y; by elimination of x and y it can be also expressed as a system of two linear equations \(c_{1,1} \xi _1 + \ldots + c_{m,1} \xi _m = 0,c_{1,2} \xi _1 + \ldots + c_{m,2} \xi _m = 0,\) i.e., as an intersection of two linear subspaces of dimension m−1.

It is interesting to note that if the superposition in the (x,y) plane consists of non-linearly curved layers (i.e., non-linear transformations of periodic functions; see Chapter 10), then the image of Ξis a curved 2D surface within R ^m .

More precisely: for any positive ε, be it as small as we may desire, we can find in the superposition rosettes (of either type) with a mismatch smaller than ε, provided that we go far enough from the origin.

It is interesting to note that the superposition of the third screen on top of the initial 2-screen superposition does not add new impulse locations in the spectrum support (compare the 2-screen spectrum support in Fig. 8.8(a) with the 3-screen spectrum support in Fig. 8.5(a)). The reason is that the new frequency vectors f ₅ and f ₆ are linear combinations of the original frequency vectors f ₁,f ₂,f ₃,f ₄, and therefore all the new convolution impulses which are generated in the spectrum owing to the superposition of the third screen are located on top of already existing impulses. Thus, each impulse in the spectrum of the 2-screen superposition turns into a compound impulse in the spectrum of the 3-screen superposition, and the non-singular 2-screen superposition turns into a singular 3-screen superposition.

Note that if one already observed from Eq. (8.8) that ζ₅ = ζ₄,−ζ₂, and ζ₆, = ζ₁,−ζ₃,, then Eq. (8.11) can be directly deduced from Eq. (8.9).

Note that during the previous discussion we considered the linear dependence (or independence) over Z of the scalars ζ _i However, this is equivalent to the linear dependence (or independence) over Z of the frequency vectors f _i , since: ∑k _i f _i = 0 ∀x (∑k _i f _i ) x = 0 ⇔ ∀x ∑k _i f _i x = 0 ∀x ∑k _i ζ _i = 0 (by Eq. (8.3)). An interesting result of this equivalence is that just as the spectral interpretation of a (k _i,…,k _m )-singular superposition is ∑k _i f _i = 0, its image-domain interpretation is that, for any point x in the x,y plane, ∑k _i ζ _i = 0 (provided that all the superposed layers are given in their initial phase). For example, in Fig. 8.12, which illustrates a (1,2,−2,−1)-singular superposition, any point x in the x,y plane satisfies: ζ₁+ 2ζ₂ − 2ζ₃ − ζ₄ = 0. (In the spectral domain we have, of course, f ₁ + 2f ₂ − 2f ₃ − f ₄ = 0.)

Note that the period-coordinate of any 1-fold periodic function (such as a single layer p _i (x) or the (k ₁,…,k _m )-moiré (x)) at point x) is given by a single number, while the period-coordinate of a superposition is given, by the vector (ζ₁,…,ζ _m ) of the period-coordinates of the individual layers at point x.