1. Introduction
Acoustic focusing is a hot topic in a wide range of industrial [
1] and medical applications [
2]. One of the newest alternatives devised to focus acoustic waves are acoustic metasurfaces, which can be implemented using different types of structures, such as coiling-up space channels [
3,
4,
5], Helmholtz resonators [
6], or subwavelength slits [
7]. In recent years, researchers developed a new kind of holographic device based on 3D printed structures capable of focusing complex pressure waves with arbitrary shapes [
8,
9,
10,
11]. One of the main drawbacks of these types of structures is that they require a complete 3D design of each one of their unit cells in the case of metasurfaces and a complex 3D propagation and back-propagation optimization algorithm to obtain the desired holographic plate in the case of acoustic holograms. Other interesting alternatives to focus ultrasound waves include spheres [
12] or cylinders [
13] filled with different liquids, which are able to produce acoustic jets with lateral resolutions beyond the diffraction limit. However, these kind of structures can only focus in very close range applications, which limits their potential.
In this sense, Fresnel Zone Plates (FZPs) are a much simpler yet flexible and powerful alternative to focus acoustic waves. FZPs are monofocal planar lenses employed in a wide range of fields, ranging from optical [
14], to microwave [
15], and acoustic [
16,
17,
18] applications. This kind of lens consist of a series of concentric rings with decreasing widths, known as Fresnel regions, which can be implemented by either alternating pressure blocking regions with transparent regions (Soret FZPs) [
16,
18], or phase-reversal regions with transparent regions [
19]. In addition, in recent years, a novel kind of lense based on applying a binary sequence to the different Fresnel regions of a conventional FZP [
20] has been presented, increasing the versatility of this kind of devices. If Fibonacci or M-bonacci binary sequences are employed, the resulting lens achieves a bifocal focusing profile [
21,
22,
23], while if fractal Cantor binary sequences are used, multifocal focusing profiles with interesting self-similarity properties are achieved [
24,
25,
26,
27]. Analogously, binary Thue–Morse sequences provide focusing profiles with bifocal fractal properties [
28,
29].
It is well known that FZP focusing profiles are sensitive to fabrication errors during the manufacturing stage. In this work, we investigate the effect of introducing controlled errors in the design of the Fresnel rings. We refer to this novel technique as predistortion, due to the fact that the FZP layout is willfully distorted in advance to achieve a certain effect on its focusing profile. These intentional errors, when produced in certain patterns, can be used to modify the shape of the FZP focusing profile in a desirable way. The main parameters of the predistortion technique are the number and location of the Fresnel regions where the predistortion is introduced, as well as the degree of the induced predistortion.
In this paper, we will show that the lens focusing profiles produced by our predistortion technique present certain similarities with the focusing profiles obtained when using Fractal zone plates [
24,
25,
26,
27]. The main advantage of our technique is its degree of flexibility, which allows the controling of the focusing profile by modifying either the number of predistortion regions where predistortion is introduced or the severity of the predistortion. Therefore, our technique can be used in all those applications where Fractal zone plates have been successfully applied [
25,
30,
31,
32].
Section 4 further develops on the different potential applications of the predistortion technique.
2. Predistortion Procedure
The governing equation used to design an FZP with point source excitation is given by
with
, being
N the number of Fresnel regions,
F the FZP desired focal length,
the operating wavelength,
d the separation between the point source and the FZP, and
the radius of each Fresnel region. This equation is easily derived from the schematic shown in
Figure 1. The Fresnel zone construction principle [
16] establishes that the path difference between the direct path (blue path) and the diffracted path through a Fresnel zone (green path) must be an integer multiple of
.
The Fresnel region radii can be directly obtained solving Equation (
1) as
If some of the Fresnel region radii obtained from Equation (
2) are slightly modified, either by increasing or decreasing their values, an intentional distortion is introduced during the design process of the FZP. This predistortion procedure affects the FZP focusing profile, resulting in some interesting effects. A
predistortion parameter, ranging from 0 to 1, is introduced in order to characterize the predistortion procedure. This
parameter is used to indicate the variation of the Fresnel radii, and thus the amount of introduced predistortion. The minimum (non-existent) predistortion corresponds to
, whereas
corresponds to the maximum (total) predistortion case.
The
parameter can be mathematically defined using the
parameter previously introduced in [
20,
24]. For the distortion-free conventional FZP, the
parameter takes the following values:
and Equation (
2) can then be expressed as
When a certain amount of controlled distortion is introduced in the design of the FZP lens, the
parameter is also distorted and denoted by
. Odd Fresnel regions (
, with
) increase the value of their
parameter, while those Fresnel regions corresponding to even numbers (
, with
) diminish their
parameter values. The corresponding equations are
Not every Fresnel radius needs to be predistorted. Predistortion can be limited to a certain area inside the FZP layout. In this paper, we analyze the effect of applying predistortion to both the inner Fresnel regions and the intermediate Fresnel regions of the FZP. Substituting the distorted values given by Equations (5) and (6) into Equation (4), the required predistorted Fresnel radii can be calculated.
In this work, Soret FZP lenses, where the Fresnel regions are either opaque or transparent, are employed. Moreover, the central Fresnel region is considered to be opaque. However, we have carried out additional simulations, and the results presented in this paper are also valid for both Soret FZP lenses with a central transparent region and phase-reversal FZPs with transparent and phase-reversal Fresnel regions.
When the predistortion
parameter is augmented, Fresnel opaque regions increase their sizes, while Fresnel transparent regions reduce them. This is due to the fact that odd Fresnel radii increase their values whereas even Fresnel radii diminish theirs. If the central Fresnel region is opaque as considered in this analysis, any opaque region in the FZP lens begins at an odd Fresnel radius and ends at the next even Fresnel radius, and the opposite is true for any transparent region. A curious phenomenon takes place when maximum distortion is considered (
). In this extreme case, Fresnel transparent regions disappear in the predistorted area, and therefore, Fresnel opaque regions cover the whole predistorted area, becoming a large whole opaque area. For instance, the fifth Fresnel radius (
) corresponds to the end of the third Fresnel opaque region and the beginning of the third transparent region, while the next radius (
) corresponds to the end of the third Fresnel transparent region and the beginning of the fourth Fresnel opaque region. When
, the distorted
parameters for
and
can be calculated as
and
, respectively. Both values are equal, which results in the disappearance of the third transparent Fresnel region, while the third opaque Fresnel region ends at the same radius where the fourth opaque Fresnel region begins without any discontinuity, becoming a larger opaque region. A very similar structure involving large inner opaque areas was previously proposed in [
33]. In this work, the FZP was modified by placing a pupil device in front of the FZP lens. As it will be discussed later in
Section 4, the effect that is achieved on the predistorted FZP focusing profile for the case of maximum predistortion (
) being very close to that experimentally demonstrated in this previous work.
Once an FZP is designed, the main parameters that must be considered to apply the predistortion technique are the number of Fresnel regions that are going to be predistorted (
L), their location, and the degree of the predistortion (
). In this paper, the location of the predistorted regions is going to be limited to two different situations: inner regions and intermediate regions. Inner regions refer to the Fresnel regions closer to the central opaque region, whereas intermediate regions refer to the Fresnel regions closer to the middle Fresnel region. The application of this predistortion technique is shown in
Figure 2, which depicts the Fresnel radii along the normalized
domain (top row), the Fresnel radii along the
r-axis (second row), and the corresponding FZP layouts (bottom row) for three cases: the conventional non-predistorted FZP (left column), the predistorted FZP whith inner regions (central column) and the predistorted FZP whith intermediate regions (right column). As can be observed from the top row of the figure, when the predistortion parameter
becomes higher, the predistorted transparent regions become smaller, their widths being
when depicted against the normalized
domain. The layouts shown in
Figure 2 correspond to
. When maximum predistortion is applied (
), this width becomes equal to 0 and the predistorted transparent regions totally disappear.
Thus, when considering predistorting inner regions, if for instance
(central column of
Figure 2), it results that the first seven regions, starting with the central opaque region (region 1), are going to be predistorted. Therefore, the first four opaque regions and the first three transparent regions become predistorted in this case. On the other hand,
Figure 2 (right column) shows the situation when intermediate regions are predistorted. For instance, if
, the middle Fresnel region is region 11, and if
, the predistorted regions would be in this case Fresnel regions
and 13.
4. Discussion
In this paper we have shown that the focusing profile of a conventional FZP lens can be modified by predistorting the Fresnel regions. If the inner Fresnel regions are predistorted, it is possible to control the focal depth of the focus by modifying the predistortion value
without changing the location of the main focus. When the predistorted Fresnel regions are shifted to intermediate positions, several secondary foci appear adjacent to the main focus. The level of these secondary foci can be adjusted using both the
L and
parameters. Thus, when predistortion is applied to intermediate regions, we can change the balance between the main and the secondary foci intensities. It is true that these types of focusing profile are not novel and can be achieved using alternative techniques such as Cantor [
24,
25,
26,
27] or Thue–Morse [
28] binary sequences, but we present our technique as an alternative method with a higher degree of flexibility. In our technique, the
L and
parameters allow one to finely adjust the degree of balance among the different foci, whereas this balance is fixed when using the binary sequence techniques. Therefore, all the applications in which the Cantor or Thue–Morse binary sequences have been successfully applied can also be approached with our predistortion technique, but as stated above, with a greater degree of flexibility. In this sense, multifocal Cantor fractal zone plates, which provide similar focusing profiles to those of a predistorted FZP at the intermediate regions, have been recently proposed as optical tweezers with multiple trapping planes [
30,
31]. Analogously, predistorted FZPs have potential applications as acoustical tweezers for particle trapping and manipulation in many biological and industrial fields [
34,
35,
36,
37,
38].
Although it is not shown in the paper, we have carried out additional simulations that show that these results can be totally translated from the acoustic domain into the optical domain, where FZPs are also a hot research topic. As stated above, the FZP focusing profile that is obtained when applying the predistortion technique for intermediate regions presents certain similarities with that obtained when Cantor or Thue–Morse binary sequences are used. It has been shown that this type of focusing profile has promising applications in the optical field in reducing the chromatic aberration and extending the depth of field [
25,
28]. Fractal zone plates based on Cantor binary sequences have also been successfully applied as multifocal intraocular lenses [
32]. Therefore, our predistortion technique is also suitable for this type of application, with the advantage of providing a higher degree of flexibility and control over the focusing profile.
In order to experimentally validate the analytical model and the simulations shown in this paper, we rely on some previous work from our research group ([
33]). This reference analyzes the use of a pupil in front of an FZP lens. The combination of the FZP lens and the pupil results in a very similar concept as the predistortion technique for the inner regions in the case of maximum predistortion (
). The experimental increase in the focal depth (FLHM) reported in [
33] is around
, while the FWHM slightly decreases, as predicted in our simulations. We have carried out the corresponding simulation with our predistortion technique using the same lens parameters (
,
,
mm) and operating wavelength (
kHz) as those used in the experiment. We have also used a piston transducer with the same active diameter (
mm) as the experimental ultrasonic source. A piston transducer emitter presents a certain radiation diagram that emphasizes the inner FZP regions over the outer regions and can cause certain deviations between experimental and simulation results, although we have verified that they are not very significant. The piston transducer is separated
mm from the FZP plane. The increase in the focal depth with the predistortion technique is of the order of 83.5% when
. The difference between our simulation result and that of the experiment can be due to several factors. In our opinion, the signal to noise ratio of the measurement presented in reference [
33] is not very high, and this could have led to some measurement errors. The lateral resolution is slightly enhanced as the FWHM becomes smaller when predistortion is applied. In this case, we have obtained much closer results between the simulation (
= 87.2%) and the experiment (
= 84.8%). We think that these results are much closer due to the fact that the measurement of the lateral resolution is of much better quality than that of the focusing profile. In any case, we consider that these experimental results demonstrate the feasibility of our predistortion technique and validate both the theoretical analysis and the simulations presented in this work.
5. Conclusions
In this work, we have shown that the introduction of controlled errors during the manufacturing stage of an FZP can be used to modified the FZP focusing profile with a certain degree of flexibility. Two different predistortion techniques have been proposed. The first technique deals with predistortion applied to the inner regions. In this case, a number of Fresnel regions, starting with the central region, are conveniently predistorted. It has been shown that, depending on the degree of predistortion applied (
) and the number of Fresnel regions that are predistorted (
L), the focal depth (FLHM) of the focusing profile can be significantly increased, while the lateral resolution is slightly improved. The second technique is used when the predistortion is applied to the intermediate Fresnel regions of the FZP. Here,
and
L are still the key parameters, but the effect on the FZP focusing profile is completely different, exciting a number of secondary foci adjacent to the main focus. In a way, this effect could still be considered as an enlargement of the main focus, although not as a continuous whole focus as in the case of predistorting the inner regions, but as a discretely distributed larger focus. It has been shown that the main secondary focus can achieve an intensity level as high as
of the main focus, although this intensity transfer from one focus to another results in the main focus diminishing its peak intensity value. The focusing profiles that are obtained when intermediate regions are employed present certain similarities with those from Fractal zone plates when Cantor binary sequences are used. Therefore, in all those applications where Fractal zone plates have already succeeded [
25,
30,
31,
32], the predistortion technique can be applied to achieve higher flexibility and control.
Summing up, we have introduced three new key parameters in the design of an FZP lens: the degree of predistortion (), the number of predistortion regions (L), and the location of the predistortion regions inside the FZP (inner regions or intermediate regions). These three parameters can be used together with the conventional design parameters of an FZP lens, such as the number of Fresnel regions (N), the location of the main focus (F) and the operating frequency (f) in order to have a higher degree of flexibility in the design of the FZP focusing profile. We can conclude that predistortion is a useful procedure that introduces additional flexibility in the FZP focusing profile design and can be appealing to research working on a wide range of applications.