1 Introduction

A renewed interest in diffractive optical elements (DOEs) has been experienced by the scientific community in the last years because these elements are essential as image forming setups that are used in THz applications [1, 2], soft X-ray microscopy [3, 4], astronomy [5, 6], lithography [7] and ophthalmology [8, 9], among others. A Zone Plate (ZP) [10], the simplest form of a diffractive lens, is characterized by a series of concentric transparent and opaque annular rings distributed periodically along the square of the radial coordinate, so the area of each annular zone is a constant. Under plane wave illumination, this distribution of zones produced by diffraction a set of foci located along the optical axis. To improve diffraction efficiency, binary phase ZPs [11] and kinoform lenses [12] (ZPs with a sawtooth profile) can be used. With the latter configuration, it can be achieved theoretically that all the energy concentrates on a single focus for the design wavelength. Photon Sieves [13] have been also implemented to improve diffraction efficiency. In this application, the transparent annular zones of amplitude ZPs are replaced by a disjoint set of holes.

Within this context, fractal zone plates (FZPs) were proposed by our group as a new promising structured diffractive lenses [14]. Under certain circumstances, a FZP can be considered as a conventional Fresnel zone plate with certain missing zones. The resulting structure is characterized by its fractal profile along the square of the radial coordinate. The axial irradiance provided by a FZP when illuminated with a parallel wavefront presents multiple foci creating a focal volume with an internal fractal structure, reproducing the self-similarity of the originating FZP. Imaging properties of FZP were also investigated. Interestingly, under polychromatic illumination a FZP produces a substantial increase in the depth of field and a noticeable reduction in the chromatic aberration compared with a Fresnel zone plate of the same focal distance [15]. Additionally, in order to design optimized optical systems, different mechanisms to control the diffraction efficiency of fractal lenses have been proposed in different publications. We have shown that the diffraction efficiency of a FZP can be controlled by amplitude modulation, resulting in new elements called fractal photon sieves [16] and also by fractal kinoform zone plates known as Devi’ls lenses [17]. These designs were the basis of new fractal intraocular lenses [18, 19] and fractal contact lenses [20, 21].

Fig. 1
figure 1

Generation of the silver mean sequence up to order \(n = 5\) and the transmittance function \(q(\zeta )\) of the corresponding SMZP

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In addition to fractal elements, there are numerous aperiodic sequences [22, 23] with which diffractive lenses can be structured. Diffractive lenses based on the Fibonacci [24] and m-Bonacci [25] sequences display focal splitting with respect to periodically structured ones. Other aperiodic sequences with which we have designed multifocal diffractive lenses are the Thue–Morse [26], the Rudin–Shapiro [27] and the Walsh functions [28]. Another interesting mathematical generator is the generalized precious mean sequence which includes the silver mean sequence [29, 30]. This sequence has been employed in the development of different aperiodic systems, such as photonic and phononic quasicrystals [31, 32].

In this paper, we show that silver mean zone plates (SMZPs), i.e., ZPs with a distribution of zones based on the silver mean sequence, are intrinsically quadrifocals. The corresponding foci are located at reduced axial positions given by de Pell numbers that can be correlated with the involved aperiodic sequence. This property is experimentally verified and compared with the theoretical prediction computed numerically. Both results are in excellent agreement.

2 Silver mean zone plates design

The Pell numbers, \(P_n=\{0,1,2,5,12,29,70,...\rbrace \) are obtained from the recurrence relation \(P_n=2P_{n-1}+P_{n-2}\) \((n\ge 2)\), starting from the seeds \(P_0=0\) and \(P_1=1\) [33, 34]. The silver ratio is defined as the limit of the ratio between two consecutive Pell numbers:

$$\begin{aligned} \phi =\lim _{n\rightarrow \infty } \frac{P_n}{P_{n-1}}=1+\sqrt{2} \end{aligned}$$
(1)

Based on Pell numbers, the so-called silver mean sequence can be generated from two seeds \(S_0=B\) and \(S_1=A\) as shown in Fig. 1. Each element of the series is obtained from the following concatenation rule: \(S_n=\{S_{n-1}S_{n-1}S_{n-2}\rbrace \) for \(n\ge 2\). This aperiodic sequence can also be generated iteratively by applying the substitution rules \(g(A)=AAB\) and \(g(B)=A\), that is, \(S_2= AAB\), \(S_3= AABAABA\), \(S_4= AABAABAAABAABAAAB\), etc. Note that the total number of elements of a given sequence of order n is \(P_n+P_{n-1}\), which results from the sum of \(P_n\) elements “A” plus \(P_{n-1}\) elements “B.”

When designing SMZPs, each of these sequences can be used to define the transmission generating function \(q(\zeta )\), with compact support on the interval \(\zeta \in [0,1]\). This interval is partitioned in \(P_n+P_{n-1}\) sub-intervals of length \(d_n = \frac{1}{P_n+P_{n-1}}\). The transmittance value, \(t_{n,j}\), that takes at the j-th sub-interval is associated with the element \(S_{n,j}\), being \(t_{n,j} = 1\), when \(S_{n,j}\), is “A” and \(t_{n,j} = 0\), when \(S_{n,j}\), is “B” (see Fig. 1). In mathematical terms, the transmittance function, \(q(\zeta )\), can be written as:

$$\begin{aligned} q(\zeta )=\sum _{j=1}^{P_n+P_{n-1}}t_{n,j} rect \left[ \frac{\zeta -(j-1/2) d_n}{d_n} \right] \end{aligned}$$
(2)

where \(\zeta =(\frac{r}{a})^2\) is the normalized quadratic radial coordinate, r is the radial coordinate, a is the radius of the lens and rect is the rectangular function. Figure 2 shows a ZP based on the silver mean sequence of order \(n=5\).

Note that the number of zones of a SMZP of order n coincides with \(P_n+P_{n-1}\), being \(P_n\) and \(P_{n-1}\) the number of transparent and opaque zones, respectively. We have found that not only the number of zones is related to the Pell numbers but also the foci distribution produced by the diffractive lens along the optical axis. For this, the focusing properties of a SMZP were studied numerically, using the Fresnel–Kirchhoff formula [35, 36]:

Fig. 2
figure 2

Silver mean zone plate of order \(n = 5\)

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Fig. 3
figure 3

Normalized irradiance versus reduced axial coordinate of silver mean zone plates of orders \(n=4,5,6\)

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$$\begin{aligned} U(x,y,z)&=-\frac{e^{-ikz} e^{-\frac{ik}{2z} (x^{2}+y^{2})}}{i\lambda z}\nonumber \\&\quad \iint _{-\infty }^{\infty }\ q(x_0,y_0)e^{\frac{-ik}{2z}(x_0^2+y_0^2)} e^{\frac{ik}{z}(xx_0+yy_0)} dx_0 dy_0 \end{aligned}$$
(3)

where U(x, y, z) is the amplitude of the diffracted field at position (x, y, z), \(k=\frac{2\pi }{\lambda }\) is the angular wave number, \(\lambda \) is the wavelength and the integration limits correspond to the opening dimensions. From Eq. (3), the irradiance can be obtained as the square of the amplitude for the diffracted field. Using normalized coordinates for the pupil plane as \(\bar{x}_0=\frac{x_0}{a}\), \(\bar{y}_0=\frac{y_0}{a}\), the irradiance is given by

$$\begin{aligned}&I(x,y,u)\nonumber \\&\quad =4u^2 \left| \iint _{-\infty }^{\infty }q(\bar{x}_0, \bar{y}_0)e^{i2\pi u(\bar{x}_0^2{+}\bar{y}_0^2)} e^{-i2\pi 2u(x\bar{x}_0{+}y\bar{y}_0)} \mathrm{{d}}\bar{x}_0 \mathrm{{d}}\bar{y}_0 \right| ^2 \end{aligned}$$
(4)

where \(u=\frac{a^{2}}{2\lambda z}\) is the reduced axial coordinate. If only the irradiance along the propagation axis is considered and the diffractive element has radial symmetry, Eq. (4) becomes

$$\begin{aligned} I(u)=4\pi ^2 u^2 \left| \int \limits _0^1 q(\zeta )e^{-i2\pi u\zeta } d\zeta \right| ^2 \end{aligned}$$
(5)

Figure 3 shows the axial normalized irradiance distributions, corresponding to the first-order diffraction foci, computed for SMZPs of orders \(n=4,5,6\). The axial irradiance was obtained from Eq. (5) using the 1D Fast Fourier Transform method, since this method requires low computation time compared to other numerical methods commonly used [36]. As can be seen, SMZPs produce the formation of four foci distributed along the axis. Note that the corresponding reduced focal lengths \(u_a,u_b,u_c,\) and \(u_d\) approximate to \(P_{n-1}\), \(P_{n-1}+P_{n-2}\), \(2P_{n-1}\) and \(P_n\), respectively. Therefore, the ratio between the focal distances is \(\frac{u_d}{u_a} \thickapprox \phi =1+\sqrt{2}\), \(\frac{u_c}{u_a} \thickapprox 2\), and \(\frac{u_b}{u_a}\thickapprox 1+\frac{1}{\phi }=\sqrt{2}\), respectively. The higher the order of the sequence, the better these approximations will be. For example, for \(n=6\), the discrepancy between these approximations and the corresponding ratio between the focal distances is less than 0.07\(\%\).

Fig. 4
figure 4

Scheme of the experimental setup used to study the focusing properties of SMZPs

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Fig. 5
figure 5

Comparison of numerical and experimental axial irradiance of a SMZP of order 5

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Fig. 6
figure 6

Evolution of the transverse intensity distribution produced by a SMZP of order 5

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3 Experimental setup and results

The focusing properties of SMZPs have been studied experimentally using the setup schemed in Fig. 4. The studied lens was implemented in the liquid crystal spatial light modulator (SLM) (Holoeye PLUTO, 1920 \(\times \) 1080 pixels, pixel size 8 \(\mu \)m, 8-bit gray-level) operating in phase-only modulation mode. A collimated and linearly polarized beam from an He-Ne Laser (\(\lambda = 633\) nm) illuminates the SLM. The SLM is programmed to show a linear phase grating in the zones with transmittance 1, while the rest of the display shows a constant phase. In this way, white zones are deflected to the first diffractive order of the grating, while black zones are reflected specularly (zero diffractive order). The SLM plane is imaged through a telescopic system (L2 and L3). The SLM is slightly tilted to correct the linear phase, and a pinhole (PH) is positioned at the Fourier plane to eliminate all diffraction orders of the linear phase grating except order +1. The PH also prevents noise from the specular reflection (zero diffractive order) and the pixelated structure of the SLM (higher diffraction orders). In this way, the studied lens transmittance is projected at the pupil plane and its focusing profile can be captured along the axis by a camera sensor mounted on a motorized stage.

We have experimentally characterized a SMZP of order \(n=5\) and a radius \(a = 2.01\) mm. The experimental axial irradiance provided by this ZP is shown in Fig. 5 along with the irradiance obtained numerically for comparison purposes. It can be seen that the SMZP provides four foci whose axial positions are \(z_a=26.47\) cm, \(z_b=18.91\) cm, \(z_c=13.12\) cm and \(z_d=10.91\) cm. The corresponding experimental reduced axial coordinates, \(u=\frac{a^2}{2\lambda z}\), are \(u_a=12.00\), \(u_b=16.80\), \(u_c=24.21\) and \(u_d=29.11\), respectively. As predicted from the theoretical analysis, the location of each of these foci approximates to \(u_a\thickapprox 12=P_4\), \(u_b\thickapprox 17=P_4+P_3\), \(u_c\thickapprox 24=2P_4\), and \(u_d\thickapprox 29=P_5\). The ratio between the positions of the fourth and first foci approximates to the silver ratio \(\phi \thickapprox \frac{u_d}{u_a}=2.426\), while \(\frac{u_b}{u_a}=1.44\thickapprox \sqrt{2}\) and \(\frac{u_b}{u_a}=2.02\thickapprox 2\).

To provide a more extensive study of the focusing characteristics of the SMZP, the transversal irradiance distribution in the xz plane was also computed numerically from Eq. (4) using the 2D Fast Fourier Transform method and captured experimentally (Fig. 6). As it can be seen, both results are in full agreement confirming the quadrifocal behavior of the lens as well as the consistent ratios between its focal positions.

4 Conclusions

A zone plate based on the silver mean sequence has been presented in this study. It is shown both theoretically and experimentally that a SMZP produces four foci along the optical axis. It has been demonstrated that these focal lengths are related to the Pell numbers and that the ratio between the fourth and first focus positions approximates the silver ratio. We believe that a diffractive lens based on this aperiodic sequence can have a wide range of applications, such as X-ray microscopy, THz imaging and ophthalmology. In fact, our next step is to design a kinoform-type diffractive structure based on this sequence to achieve an extended depth of focus (EDOF) intraocular lens.