Three-dimensional array diffraction-limited foci from Greek ladders to generalized Fibonacci sequences

Junyong Zhang

doi:10.1364/OE.23.030308

1. Introduction

Three-dimensional array foci, as the name implies, mean that two-dimensional array diffraction-limited foci are distributed as required at multi-focal planes. Focusing and imaging of light has many applications in ophthalmology, biological cell, virtual display technique, high-resolution microscopy, spectroscopy, and lithography. Unfortunately, a conventional refractive lens cannot focus soft x-rays and EUV as a consequence of the strong absorption of solid materials in the x-ray and EUV spectral regions. Although a traditional Fresnel zone plate (FZP) can be used for this kind of focusing, it has inherent limitations [1–5]. Some aperiodic zone plates, generated by fractal Cantor set, have been proposed to overcome these limitations [6–8]. In 2001, a new concept – photon sieve (PS) was proposed [9], which is a FZP with the transparent zones replaced by a great number of completely separated pinholes to overcome the disadvantages of traditional zone plate. Then many kinds of PS were designed and studied in detail [10–19].

Actually there is only one focus at a single focal plane no matter for FZP or PS. In 2013, an aperiodic Fibonacci sequence was introduced into the planar diffractive optical elements (DOEs) and generated two foci with a definite relationship along the optical axis [20–24]. The other interesting array illuminator is the Dammann grating [25–28]. We find that the former functionality is achieved by quasi-periodic Fibonacci structure, but the latter Dammann grating is a periodic structure. In our previous work, we proposed the generalized Fibonacci PS, and produced two adjustable foci on axis according to the mathematical character of generalized Fibonacci sequences [24]. This technology can produce multiple foci on the optical axis.

In this paper, we propose a new family of mathematical sequence named by Greek ladder [29–31], and introduce the multi-dimensional quasi-periodic Greek ladder to nano-photonic device, furthermore design the corresponding device to generate two, three and four foci on axis, respectively. By analogy, more than four-focal device can be designed. Meanwhile the focal distances can be freely adjusted. Then under some special conditions, the two-dimensional quasi-periodic classical Greek ladder can be reduced to one-dimensional quasi-periodic generalized Fibonacci sequences. By means of proper encoding and binary, ternary or quaternary phase modulation, the Greek-ladder devices do generate the diffraction-limited array foci at different multi-focal planes. Essentially, three-dimensional array focusing can be achieved because the periodic and symmetric device is replaced by the aperiodic and asymmetric structure which leads to the advent of quasi-steady intermediate field between the near field and the far field.

2. Mathematical description of Greek ladders

For traditional FZP, the optical path difference (OPD) between two adjacent zones is equal to λ/2, where λ is the incident wavelength. If the OPD λ/2 is replaced with Kλ/2 where K is a positive real number and denotes the optical path difference scaling factor (OPDSF), we can get the following OPD in the case of a monochromatic plane wave incidence

\sqrt{r_{m}^{2} + f_{0}^{2}} - f_{0} = m K λ, (m \in N_{+}, K > 0)

Where f₀ is the expected focal length, wavelength λ; r_m is the radius of m^th zone; K denotes the OPDSF.

When a light source is incident on the optical device, the diffracted field can be given by Rayleigh-Sommerfeld diffraction integral

E_{1} (x, y, z) = \frac{- 1}{2 π} \iint E_{0} (ξ, η, 0) t (ξ, η) \frac{\partial}{\partial z} [\frac{\exp (i k R)}{R}] d ξ d η

Where R = [(x-ζ)² + (y-η)² + z²]^1/2, E₀(ζ,η,0) denotes the incident light, t(ζ,η) is the corresponding transmission function of each transparent region.

Now let us introduce the Greek ladder which has the initial seeds

r_{1, j} = α_{j}, j = 1, 2, ... N

Where α_j is a real number.

And has the following linear recursion relations

{\begin{cases} r_{n + 1, 1} = r_{n, 1} + r_{n, 2} + ... + r_{n, N} \\ r_{n + 1, 2} = r_{n + 1, 1} + (C - 1) r_{n, 1} \\ r_{n + 1, 3} = r_{n + 1, 2} + (C - 1) r_{n, 2} \\ ... \\ r_{n + 1, N} = r_{n + 1, N - 1} + (C - 1) r_{n, N - 1,} C \in R^{+} \end{cases}

Where n is a positive integer. r_n,j(j = 1,2,…N) denotes the nth Greek-ladder vector.

We can easily get the math limit of Eq. (3) by the following definition

γ = \lim_{n \to \infty} r_{n, k + 1} / r_{n, k} = \sqrt[N]{C}, k = 1, 2, ..., N - 1

In order to determine the initial seeds of Greek ladder, we introduce the best asymptotical n^th convergent by use of the finite continued fraction. In number theory, any one of irrational number can be approximated by the continued fraction $\sqrt{C} = [a_{0}, a_{1}, a_{2}, ...]$ . So for the classical Greek ladder, we can get the best asymptotic fraction by the method proposed by Luogeng Hua [32].

{\begin{cases} a_{1} = [x], R_{1} = x - a_{1} \\ a_{n} = [1 / R_{n - 1}], R_{n} = 1 / R_{n - 1} - a_{n}, n \geq 2 \\ P_{0} = a_{0}, P_{1} = a_{1} a_{0} + 1, P_{i} = a_{i} P_{i - 1} + P_{i - 2} \\ Q_{0} = 1, Q_{1} = a_{1}, Q_{i} = a_{i} Q_{i - 1} + Q_{i - 2} \end{cases}

Obviously, the n^th convergent

\lim_{i \to \infty} P_{i} / Q_{i}

approximates the value of

\sqrt{C}

. In a particular case r_1,j = 1, j = 1, 2, the Greek ladder can approximate the simple continued fraction

\sqrt{C} = [1; C - 1, 2; C - 1, 2; C - 1, 2; ...]

From the above analysis, we can come to a conclusion that a continued fraction of this type is eventually periodic of period 2. There is a possibility that the Greek-ladder device can be divided into two incommensurable periods in the case of proper encoding.

3. Longitudinal focusing of Greek-ladder devices

Figure 1 shows the schematic of a Greek-ladder nano-photonic device, and Table 1 gives the first 5th Greek-ladder terms. The diameter of each pinhole is equal to the width of the zone where the pinhole is located. The 1st and the 3rd quadrants have 94 opaque and 103 transparent zones respectively, while the 2nd and the 4th quadrants have 101 opaque and 111 transparent zones respectively. The total transmission function is the linear superposition of each transparent area. The calculation parameters are as follow: the incident wavelength equals to 632.8nm, K = 0.5, the expected focal length f₀ = 3.5cm, C = 1.25, the math limit $\sqrt[3]{1.25} \approx 1.0772$ , the initial seeds (α₁,α₂,α₃) = (1,2,3) and the encoded seeds (α¹,α²,α³) = (1,01,010). The two focal distances are f₁ = 3.567cm and f₂ = 3.439cm, respectively. Obviously, the ratio of the two focal distances equals to f₁/f₂≈1.0372 approximating $\sqrt[3]{1.25}$ .

Fig. 1 Schematic of Greek-ladder device in polar or Cartesian coordinates and their normalized intensity against the axial distance, f₁ = 3.567cm, f₂ = 3.439cm.

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Table 1. 3-order Greek ladder with the third root of 1.25

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When r_1,j = α_j, j = 1,2, the above expression is reduced to the traditional Greek ladder or Theon’s ladder which (circa 140 A.D.) had described a remarkably simple way to calculate rational approximations to $\sqrt{2}$ . Actually due to the concatenation operation of the encoded seeds, the former term is the subset of the latter one, so we can choose any column as the encoded switching. The simulation parameters are similar to the calculation in Fig. 1. For this condition, Greek-ladder zone plate (ZP) with square root of 1.25 has 85 transparent and 95 opaque zones. The initial seeds are (α₁,α₂) = (1,1). The result is illustrated in Table 2 and Fig. 2. The two focal distances are f₁ = 3.707cm and f₂ = 3.315cm, respectively. The ratio of the two focal distances equals to f₁/f₂≈1.1182 approximating $\sqrt{1.25}$ . The other results of Greek-ladder devices with square root of 2, 3 and 4 are also shown in Table 2 and Fig. 2. To clarify, the maximum intensities of axial foci for four cases are nearly equal.

Table 2. Greek-ladder ZP with square root of 1.25, 2, 3 and 4

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Fig. 2 Axial normalized intensity of Greek-ladder ZP with square root of 1.25, 2, 3 and 4.

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The number of axial foci will increase with the increment of the OPDSF, and the emerging foci satisfy the following Eq. (7). Taking the Greek-ladder device with square root of 2 into account, the results are indicated in Table 3 and Fig. 3. Although the values of K are different, the maximum intensity on axis keeps the same value under the condition of the plane wave incidence. The similar conclusions are also applied to other cases mentioned above with respect to the different OPDSF.

f_{m} = \frac{f_{0} \times K}{[\frac{m - 1}{2}] + \frac{γ \times \mod (m - 1, 2) + 1 \times \mod (m, 2)}{γ + 1}}, f_{m} < f_{m - 1} < ... < f_{1}

Where [x] denotes the nearest integers less than or equal to x, mod(X,2) represents the modulus after division 2, and m denotes the m^th axial focus.

Table 3. Greek-ladder devices with square root of 2 against K = 0.50, 1.24 and 2.25

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Fig. 3 Axial intensity of Greek-ladder devices with square root of 2 against K = 0.50, 1.24 and 2.25.

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Equation (7) represents that there are another two techniques to generate multiple equidistant foci with equal intensity on axis. One is to scale proportionally focal length as required by OPDSF. The other is to alter the expected focal length while keeping the same OPDSF. Taking trifocal distances into account, the Greek-ladder device with square root of 2 has two focal distances which equals to f₄ = 2.987cm and f₂ = 4.225cm, respectively. The mean of the two focal distances is 3.606cm. Compared with Greek-ladder device with K = 0.5 in the 1st and the 3rd quadrants, the value of K equals to 3.606/3.5*0.5 = 0.5151 in the 2nd and the 4th quadrants. As for four-focal distances, the Greek-ladder device with square root of 2 is also arranged in the 1st and the 3rd quadrants. If we expect that the third focal length f₃ is 3.606cm, then the fourth focal length f₁ satisfies the relation f₁-f₂ = f₃-f₄. The value of f₁ is 4.844cm. On the basis of Eq. (7), the expected focal length is 4.1343cm. In the 2nd and the 4th quadrants, we can obtain the Greek-ladder device with the square root of C = (f₁/f₃)² = 1.8045. If so, the four-focal nano-photonic device has been successfully designed. The results of the two examples are all given in Table 4 and Fig. 4, and the axial foci are located at 2.984cm, 3.635cm, 4.221cm and 4.783cm, respectively.

Table 4. Greek-ladder device with trifocal or four-focal distances

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Fig. 4 Axial normalized intensity of Greek-ladder device with three or four foci.

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4. Transverse focusing at multi-focal planes of Greek-ladder devices

Based on the concept of Fourier optics, the different phase stands for the different angular spectrum in the spatial domain. In this case, when the amplitude modulation is replaced by the phase modulation in the right way, the single focal spot may be split into multiple spots. Figure 5 shows the schematic of the Greek-ladder device with three-dimensional array foci. The value of zero still denotes the opaque pinhole, but the transparent pinhole is completely replaced with binary phase. The red pinhole denotes the zeros phase, while the blue pinhole denotes the π phase. For Cartesian Greek-ladder device, the transparent pinholes is set to zeros phase if the absolute value of the center of pinhole is less than the quarter full width of device; otherwise, the transparent pinholes is set to π phase. For polar Greek-ladder device, the transparent pinholes is set to zeros phase if the polar radius of pinhole is less than the 0.574 times the radius of device; otherwise, the transparent pinholes is set to π phase. The intensity distributions corresponding to the model are illuminated in Figs. 6(a)-6(d). The Greek-ladder device does generate 2*2 array foci at two focal planes. The locations of the two focal planes are at 2.987cm and 4.225cm, respectively. For polar Greek-ladder device, the two focal planes are at the same location.

Fig. 5 Schematic of Greek-ladder device (the first 12 zone) with (a) (2*2)*2 array foci, (b) (2*1)*2 annular foci.

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Fig. 6 (a) (2*2)*2 array foci, (c) (2*1)*2 annular foci, (b) and (d) intensity contour at their own two focal planes.

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In order to get the anisotropic focal spots, the area of zeros phase along the x-axis must be different from that along the y-axis. Taking Cartesian Greek-ladder device into account, the transparent pinholes is set to zeros phase when the absolute value of the center of pinhole is less than the quarter full width of device along the x-axis and 0.35 or 0.40 times full width of device along the y-axis. The array foci are shown in Fig. 7. As the increment of the area of zeros phase along the y-axis, the two focal spots gradually blend together along the vertical direction.

Fig. 7 Intensity contour at two focal planes. (a, b) 0.35 times width on y-axis. (c, d) 0.40 times width on y-axis.

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For direct comparison to (2*2)*2 array foci under the same model in Cartesian coordinates, the transparent pinholes is set to zeros phase if the absolute value of the center of pinhole is less than the 0.325 times full width of device; otherwise, the transparent pinholes is set to π phase. In addition, the packed density is set to the value of 0.9 in the area of zeros phases. Figures 8(a) and 8(b) shows the intensity contour at two focal planes. The two focal planes are located at 2.987cm and 4.225cm, respectively. Again for the model of Fig. 5(b), the transparent pinholes is set to zeros phase if the polar radius of pinhole is less than 0.26 times and larger than 0.66 times the radius of device; otherwise, the transparent pinholes is set to π phase. Three-ring spots are obtained at two focal planes located at 2.987cm and 4.225cm, respectively [see Figs. 8(c) and 8(d)].

Fig. 8 Intensity contour at two focal planes. (a, b) stair-stepping array foci. (c, d) (3*1)*2 annular foci.

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5. Three-dimensional array foci from Greek ladder to generalized Fibonacci

Finally let us consider a special form of classical Greek ladder,

{\begin{cases} F_{1} = α, G_{1} = β \\ F_{n + 1} = G_{n} \\ G_{n + 1} = m F_{n + 1} + k F_{n}, (m, k \in R) \end{cases}

The characteristic equation of Eq. (8) is a quadratic equation. That is

γ^{2} - m γ - k = 0

From Eqs. (8) and (9), we know

\lim_{n \to \infty} G_{n} / F_{n} = \lim_{n \to \infty} G_{n} / G_{n - 1} = \lim_{n \to \infty} F_{n} / F_{n - 1} = γ

.

Based on the algebraic equation, we can easily get another equivalent recursion relation named generalized Fibonacci sequence

{\begin{cases} F_{1} = α, F_{2} = β \\ F_{n + 1} = m F_{n} + k F_{n - 1} \end{cases}

In particular, all the methods in the analysis process of Greek-ladder devices are all applied to the generalized Fibonacci structures.

In Fig. 7, the anisotropic focal spots can be achieved by adjusting the area of zeros phases in Cartesian coordinates. Here we propose another method to get the anisotropic focal spots by multiple Fibonacci sequences distributed in different regions in polar coordinates. One Fibonacci device with (m,k) = (1,1) has 92 transparent and 149 opaque zones in the 1st and the 3rd quadrants, the other Fibonacci device with (m,k) = (−2,0.3) has 77 transparent and 164 opaque zones in the 2nd and the 4th quadrants. The initial seeds are all (2,3). In the 1st and 3rd quadrants, the transparent pinholes is set to zeros phase if the absolute value of the center of pinhole is less than the 0.307 times full width of device; otherwise, the transparent pinholes is set to π phase. The same principle also applies to the quadrants (2,4) except that (0, π) phases are replaced with the corresponding (0.5π,1.5π). The four focal planes are located at 2.567cm, 2.830cm, 4.584cm and 5.494cm, respectively. Figures 9(a)-9(d) shows the intensity contour at four focal planes, and there are four foci at each focal plane. When the transparent pinholes in quadrants (2,4) are all set to the value of 0.5π phase and the quadrants (1,3) remain the same, at this time we successfully obtain different number of foci at each focal planes as shown Figs. 9(e)-9(h). It should be pointed out that when the 0.5π phase in quadrants (2,4) is replaced with 1.5π phase, the spots at each focal plane will be integrally rotated by 90 degree, but the diffraction pattern remain the same.

Fig. 9 Intensity contour at four focal planes (a-d and e-h) with f = 2.567cm, 2.830cm, 4.584cm, 5.494cm.

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Taking the diffraction efficiency into account, the amplitude-only DOE is worse than the phase-only DOE. For binary optical element (BOE) with phase modulation, if the opaque areas in the same partition are replaced with the opposite phase, the diffraction efficiency will be improved greatly. In addition, it should be noted that the phase modulation in visible wavelengths cannot be fit for EUV and soft x-rays in that their wavelengths are too short. Fortunately, the optical theory tells us that the phase equals to the product of the wave number and its own OPD. Obviously, the OPD of two pinholes, which are located at the two adjacent Fresnel zone, is equal to half wavelength. So we can move the pinholes in odd Fresnel zone to that in even Fresnel zone in the partition as required, and vice versa. In this way, phase modulation for EUV and soft x-rays can be realized by amplitude modulation.

Furthermore, the characteristic equation of Fibonacci sequence is a quadratic equation; the math limit is proportional to the square roots. While for Greek ladder, the math limit is proportional to the nth roots. It means that the Greek ladder converges at a faster rate than Fibonacci sequences under the same condition. At this point the device with a very short distance between two focal planes can be easily constructed by Greek ladder. The submicro-to-micro sized distance between two focal planes is more suitable for biological cell, microscopy, lithography, etc.

6. Conclusion

We have proposed a kind of new technology to generate array foci at the multi-focal planes. Different from the mono-focal Fibonacci diffractive lenses with the ratio of two focal distances approaching the golden mean, the Greek ladder is introduced into the diffractive optics for the first time, and successfully produces the multiple foci on axis with the ratio of them approaching the nth root of C, where C is a positive real number. Meanwhile we can be free to adjust the focal locations as required by OPDSF. A special form of the Greek ladders can be reduced to the generalized Fibonacci sequences. With these technologies and the multi-nary phase coding, we can be free to design various kinds of multi-focal nano-photonic devices with any number and any location. In visible wavelengths, this technology has many applications in ophthalmology, virtual display technique, array imaging, laser micromachining and spectroscopy. In EUV and soft x-ray region, it can be applied for biological cell, microscopy, lithography, etc.

Acknowledgments

This work is supported by National Natural Science Foundation of China (NSFC) (No. 61205212).

Reference and links

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r₁	r₂	r₃	r₂/r₁	r₃/r₂
1.0000	2.0000	3.0000	2.0000	1.5000
6.0000	6.2500	6.7500	1.0417	1.0800
19.0000	20.5000	22.0625	1.0789	1.0762
61.5625	66.3125	71.4375	1.0772	1.0773
199.3125	214.7031	231.2813	1.0772	1.0772

C	sqrt(C)	T/O	(α₁,α₂)	f₂/cm	f₁/cm	f₁/f₂
1.25	1.1180	85/95	(1,1)	3.315	3.707	1.1182
2.00	1.4142	74/106	(1,1)	2.987	4.225	1.4145
3.00	1.7321	76/208	(2,1)	2.759	4.780	1.7325
4.00	2.0000	61/182	(2,1)	2.623	5.253	2.0027

	K	f₆/cm	f₅/cm	f₄/cm	f₃/cm	F₂/cm	f₁/cm	f₁/f₂
simulation	0.50	-	-	-	-	2.987	4.225	1.4145
	1.42	-	-	3.132	3.514	8.506	12.030	1.4143
	2.25	3.041	3.260	4.978	5.585	13.500	19.100	1.4148
based on Eq. (7)	0.50	-	-	-	-	2.987	4.225	1.4145
	1.42	-	-	3.134	3.514	8.484	11.999	1.4143
	2.25	3.045	3.262	4.966	5.569	13.444	19.012	1.4142

C	sqrt(C)	T/O	(α₁,α₂)	f₂/cm	f₁/cm	f₁/f₂	f₀
2.0000	1.4142	70/99	(1,1)	2.987	4.225	1.4145	3.5000
-	-	85/85	(1,1)	-	-	-	3.5000
1.8045	1.3433	66/87	(1,1)	3.606	4.844	1.3326	4.1343

r₁	r₂	r₃	r₂/r₁	r₃/r₂
1.0000	2.0000	3.0000	2.0000	1.5000
6.0000	6.2500	6.7500	1.0417	1.0800
19.0000	20.5000	22.0625	1.0789	1.0762
61.5625	66.3125	71.4375	1.0772	1.0773
199.3125	214.7031	231.2813	1.0772	1.0772

Three-dimensional array diffraction-limited foci from Greek ladders to generalized Fibonacci sequences

Abstract

1. Introduction

2. Mathematical description of Greek ladders

3. Longitudinal focusing of Greek-ladder devices

4. Transverse focusing at multi-focal planes of Greek-ladder devices

5. Three-dimensional array foci from Greek ladder to generalized Fibonacci

6. Conclusion

Acknowledgments

Reference and links

Cited By

Figures (9)

Tables (4)

Equations (11)

Optics Express