There
are objects whose physical parameters of the structures are described by
periodic functions. Such objects include diffractive elements of integrated and
fiber optics, photonic crystal structures, transparent metamaterials, etc.
[1-5]. The physical parameters of such objects change both in space and in
time. Examples of such objects are photoinduced domain structures, electrically
controlled liquid crystal elements, dynamic magnon crystals, etc. [6-8]. In
addition to creating new technologies for manufacturing such structured
objects, the task is to measure their parameters. One of the urgent tasks of
quality control of structured objects is the identification, visualization and
assessment of the macrodefects of a periodic structure.

The
macroscopic defect of a periodic structure is understood to be the areas of the
object in which the period of the structure deviates from a certain average
value, as well as the deviation of the surface profile of the structure from a
certain reference one. The dimensions of these defects significantly exceed the
structure period. Marked macrodefects, in contrast to random formations, such
as pollution or rupture of structural elements, can be determined by moiré,
interference, and also shadow methods [9-12]. The implementation of these
techniques is carried out with sources of coherent light for moiré and
interference methods and non-coherent sources for moiré and shadow
methods. However, it should be noted that the patterns that visualize the
macrodefects of periodic structures obtained in incoherent light are of higher
quality than the patterns obtained in coherent light. The impairment in the
second case is due to the presence of speckle noise in images formed by
coherent light. In this case, when using photosensitive arrays [13] for
recording moiré patterns and subsequent digital processing, it is
desirable to use incoherent light sources [14].

This article proposes an easy-to-technical method for visualizing macrodefects
of a transmissive dynamic periodic structure using non-coherent lighting. This
method consists of two stages. At the first stage, a series of snapshots of a
dynamic periodic structure is recorded at various times during deformation
surface. At the second stage of the method implementation, a picture of moiré
fringes is observed on a matte screen background, when a selected pair of shots
are combined, and the moiré patterns are displayed on a personal
computer screen.

Previously,
the use of an incoherent light source in the visualization of macrodefects of
periodic structures made it possible to obtain moiré and interference
patterns of increased sensitivity with a sufficiently high quality [15-17].
However, in these works, the implementation of the method required the use of a
reference snapshot of the periodic structure under study, and incoherent
illumination was applied only at the final stage of the optical processing.

Let
us consider the simplest case of a dynamic periodic structure, which is an
object in the form of a thin amplitude transparency, the spatial transmission
of which is described by a periodic function. In this case, we assume that the
parameters describing the periodic structure of such an object change not only
in space, but also in time.

In Fig.
1a shows the optical scheme explaining the registration of the *S*
snapshot of the studied dynamic periodic structure *O*. A series of shots
is recorded in optically conjugated with the periodic structure *O* under
study, through a lens *L* plane, in which the photographic material is set
on a transparent basis. A diffuse scatterer *D* was used to ensure uniform
illumination of the periodic structure *O*, which was illuminated by a
diverging incoherent light source *Ls*. A film camera can be used as *L*
and *S* elements.

We
choose the directions of the axes plane coordinate system so that the y-axis
was parallel to the strokes of the pattern of the periodic structure in the
plane of the snapshot. In this case, the intensity distribution of light in the
plane of the snapshot of the periodic structure under study can be represented
as a Fourier series:

,

(1)

where
*a*_{n} are coefficients, T is a period of the structure, *φ(x,
y, t)* is a function that determines the distortions of the periodic structure
and describes the macrodefect. The function can be represented as the sum of
the individual components:

,

(2)

where
*φ*_{1}(x, y, t) is responsible for the violation of the periodicity
of the elementary structure, due to the deviation of the period of strokes from
a certain average value, and also because of the bending of strokes and
contains all the information about the position of the stroke [18], *φ*_{2}
(x, y, t) characterizes the deviation of the surface shape of the structure
from the plane and determines the change in the surface topography in space and
time [19].

Fig. 1. Optical schemes explaining: a -
registration of the snapshot *S* of the studied dynamic periodic structure,
b - registration of the moiré pattern when combining a pair of snapshot *S*_{1}
and *S*_{2}: *Ls* is the source of white light, *D* is
the diffuse scatterer, *O* is the periodic structure under study, *L*
is the lens, *S* - recorded snapshot, *S*_{1} and *S*_{2}
- registered snapshots, *CCD* - digital camera, *PC* - personal
computer.

Thus,
against the background of the diffuse scatterer *D*, a series of snapshots
*S* of the periodic structure *O* examined was recorded at different
points in time. After chemical processing of the photographic material, the
amplitude transmittances of the series of snapshots are *T*_{1} (x, y,
t_{1}), T_{2} (x, y, t_{2}), ..., T_{N} (x, y,
t_{N}). Amplitude transmission coefficient of *k*-th snapshot is
calculated using the formula:

,

(3)

where
*b*_{n} are coefficients.

To
determine the behavior of the dynamics of the function *φ (x, y, t)*,
one can use optical methods based on the use of a reference periodic structure
with a period close to the period T of the structure under study. Moiré fringes
that visualize the behavior of the function *φ (x, y, t)* in space
and in time are formed when patterns of the studied and the reference periodic
structures are superimposed [10]. In the absence of a reference periodic
structure for the formation of a moiré pattern that visualizes
macrodefects, one can use the overlay of two patterns of a periodic structure
that are shifted relative to each other. Previously, this approach was used to
determine the spatial position of surfaces with a periodic texture and also to
identify macro and microdefects of amplitude masks [19,20].

To
determine changes in the function

,

(4)

that
occurred between the registration of two snapshots at times *t*_{l}
and *t*_{k}, the approach used in differential holographic
interferometry can be used [21]. In differential holographic interferometry,
the interference pattern is reconstructed using a hologram recorded on one
medium by the method of two exposures at times *t*_{l} and *t*_{k},
or two separate holograms recorded at corresponding times. However, in contrast
to the differential interferometry method, the registration of snapshots of a
dynamic periodic structure does not require optical recording schemes that are
complex in the technical implementation, as well as optical processing. Fig. 1b
shows an optical scheme that allows forming a moiré pattern in
incoherent light that reflects the behavior of the function *∆φ*_{lk}
(x, y) during using two snapshots of a dynamic periodic structure with
amplitude transmittances *T*_{l} (x, y, t_{l}) and *T*_{k}
(x, y, t_{k}) recorded at times *t*_{l} and *t*_{k}.
The moiré pattern is observed when combining a pair of *S*_{1}
and *S*_{2} snapshots against the background of the diffuser *D*
when the latter is illuminated with the light source *Ls*. When combining
a pair of snapshots, the resulting amplitude transmittance *T*_{lk}
(x, y) is determined as *T*_{l} (x, y, t_{l}) and *T*_{k}
(x, y, t_{k}) [22] or

.

(5)

The
distribution of the light intensity *I*_{lk} (x, y) at the output
of the combined snapshots *S*_{1} and *S*_{2} will be
proportional to the amplitude transmittance *T*_{lk} (x, y), which
is defined by the expression (5). To find the equation describing the middle of
the moiré fringes corresponding to the regions with minimal visibility
of the periodic structure, we restrict ourselves to the harmonics in expression
(5), not higher than the main one. In this case

(6)

where
*c*_{0} and *c*_{1} are coefficients. The last term
in expression (6) describes the moiré fringes corresponding to the areas
of the combined snapshots in which the visibility of the periodic structure is
minimal. Equating the last term in expression (6) to zero, we get the
expression:

,

(7)

taking
into account the expression (5), we obtain the equations describing the middle
of the moiré fringes:

,

(8)

where *m*=0,1,2,...

Thus,
moiré fringes correspond to areas of the pattern in which the visibility
of a periodic structure with a period *T* takes on a minimum value.

To
confirm the operability of the visualization method of macrodefects of dynamic
periodic structures, a periodic metal mask was chosen as the object under
study, the surface of which was subjected to mechanical deformation. When the
surface of the metal mask was deformed, its surface deviated from the plane. A
series of snapshots of this object was recorded with Zenit TTL film mirror
camera on Micrat 300 film. The developer was Kodak D-19 contrast developer
which provides a high-contrast image of the periodic structure under study.

The
restoration of the moiré patterns which reflect the dynamics of the
development of the deformation of the surface of the metal mask under
investigation using a pair of snapshots recorded at different points in time
was carried out in the scheme shown in Fig. 1b. Moiré patterns were
recorded with a *CCD* digital camera while focusing on the plane of the
combined *S*_{1} and *S*_{2} snapshots set against
the diffuse scatterer *D*. It should be noted that during creating the
device layouts shown in Fig. 1a and Fig. 1b, a common lighting system was used.
A LED A60 lamp with a 7 W matt bulb was used as the *Ls* light source. The
combination of snapshots *S*_{1} and *S*_{2} was
carried out in a special optical device designed to combine holograms. This
device provided an absolute error of relative shift of images at the level of 5
microns [21].

In Fig.
2 shows photographs of moiré patterns obtained in the scheme (Fig. 1b)
when using pairs of snapshots recorded at different points in time *t*_{1}
= 1s, *t*_{2} = 3s, *t*_{3} = 5s, *t*_{4}
= 8s and *t*_{5} = 12s after the onset of deformation of the metal
surface masks.

a)
b) c)

Fig. 2. Photos of moiré patterns
of pairs of snapshots recorded
at different points in time: a) *t*_{1}
and *t*_{3},
b) *t*_{3}and *t*_{5},
c) *t*_{2}and *t*_{4}

Moiré
patterns show changes in the surface profile of a metal mask that occurred due
to the deformation of its surface over time intervals, corresponding to the
registration of pairs of snapshots with
amplitude transmittances: *T*_{1} (x, y, t_{1}) and *T*_{3}
(x, y, t_{3}) (Fig. 2a ); *T*_{3} (x, y, t_{3})
and *T*_{5}* (x,
y, t*_{5}*)*
(Fig. 2b); *T*_{2}*
(x, y, t*_{2}*)*
and *T*_{4}* (x,
y, t*_{4}*)*
(Fig. 2c).

A
relatively simple from the point of view of technical implementation method of
visualizing macrodefects of a transient dynamic periodic structure
is theoretically substantiated and experimentally confirmed.
The method is based on the effect of the formation of moiré fringes
when snapshots of
the studied structure recorded at different points in time
are combined. Affordable
equipment is used
for prototyping of devices.
To obtain quantitative information moiré patterns can be processed using
known algorithms for the digital processing of interferograms [23].

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