Multifocal diffractive lens generating several fixed foci at different design wavelengths

We propose a method for designing multifocal diffractive lenses generating prescribed sets of foci with fixed positions at several different wavelengths. The method is based on minimizing the difference between the complex amplitudes of the beams generated by the lens microrelief at the design wavelengths, and the functions of the complex transmission of multifocal lenses calculated for these wavelengths. As an example, a zone plate generating three fixed foci at three different wavelengths was designed, fabricated, and experimentally investigated. The proof-of-concept experimental results confirm the formation of foci with fixed positions at the design wavelengths. The obtained results may find applications in the design and fabrication of novel multifocal contact and intraocular lenses with reduced chromatic effects. © 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement OCIS codes: (050.1965) Diffractive lenses; (050.1970) Diffractive optics; (220.3630) Lenses. References and links 1. V. A. Soifer, V. V. Kotlyar, and L. L. Doskolovich, Iterative Methods for Diffractive Optical Elements Computation (Taylor & Francis, 1997). 2. M. A. Golub, L. L. Doskolovich, N. L. Kazanskiy, S. I. Kharitonov, and V. A. Soifer, “Computer generated diffractive multi-focal lens,” J. Mod. Opt. 39(6), 1245–1251 (1992). 3. M. A. Golub, “Laser beam splitting by diffractive optics,” Opt. Photonics News 15(2), 36–41 (2004). 4. S. Ravikumar, A. Bradley, and L. N. Thibos, “Chromatic aberration and polychromatic image quality with diffractive multifocal intraocular lenses,” J. Cataract Refract. Surg. 40(7), 1192–1204 (2014). 5. D. Gatinel, C. Pagnoulle, Y. Houbrechts, and L. Gobin, “Design and qualification of a diffractive trifocal optical profile for intraocular lenses,” J. Cataract Refract. Surg. 37(11), 2060–2067 (2011). 6. V. Akondi, C. Dorronsoro, E. Gambra, and S. Marcos, “Temporal multiplexing to simulate multifocal intraocular lenses: theoretical considerations,” Biomed. Opt. Express 8(7), 3410–3425 (2017). 7. S. S. Lane, M. Morris, L. Nordan, M. Packer, N. Tarantino, and R. B. Wallace 3rd, “Multifocal intraocular lenses,” Ophthalmol. Clin. North Am. 19(1), 89–105 (2006). 8. P. Valle, J. Oti, V. Canales, and M. Cagigal, “Visual axial PSF of diffractive trifocal lenses,” Opt. Express 13(7), 2782–2792 (2005). 9. V. Osipov, L. L. Doskolovich, E. A. Bezus, T. Drew, K. Zhou, K. Sawalha, G. Swadener, and J. S. W. Wolffsohn, “Application of nanoimprinting technique for fabrication of trifocal diffractive lens with sine-like radial profile,” J. Biomed. Opt. 20(2), 25008 (2015). 10. U. Hinze, A. El-Tamer, L. L. Doskolovich, E. A. Bezus, S. Reiß, H. Stolz, R. F. Guthoff, O. Stachs, and B. Chichkov, “Additive manufacturing of a trifocal diffractive-refractive lens,” Opt. Commun. 372, 235–240 (2016). 11. T. Kohnen, “First implantation of a diffractive quadrafocal (trifocal) intraocular lens,” J. Cataract Refract. Surg. 41(10), 2330–2332 (2015). 12. H. Dammann, “Color separation gratings,” Appl. Opt. 17(15), 2273–2279 (1978). 13. B.-Z. Dong, G.-Z. Yang, B.-Y. Gu, and G.-Q. Zhang, “Diffractive phase elements that implement wavelength demultiplexing and spatial annular focusing simultaneously,” J. Opt. Soc. Am. A 14(1), 44–48 (1997). Vol. 26, No. 4 | 19 Feb 2018 | OPTICS EXPRESS 4698 #307788 https://doi.org/10.1364/OE.26.004698 Journal © 2018 Received 22 Sep 2017; revised 31 Oct 2017; accepted 11 Nov 2017; published 14 Feb 2018 14. B.-Z. Dong, G.-Q. Zhang, G.-Z. Yang, B.-Y. Gu, S.-H. Zheng, D.-H. Li, Y.-S. Chen, X.-M. Cui, M.-L. Chen, and H.-D. Liu, “Design and fabrication of a diffractive phase element for wavelength demultiplexing and spatial focusing simultaneously,” Appl. Opt. 35(35), 6859–6864 (1996). 15. J. Bengtsson, “Kinoforms designed to produce different fan-out patterns for two wavelengths,” Appl. Opt. 37(11), 2011–2020 (1998). 16. L. L. Doskolovich and P. M. Repetto, “Design of DOEs for wavelength demultiplexing and spatial focusing,” J. Opt. A, Pure Appl. Opt. 2, 488–493 (2000). 17. V. A. Soifer, L. L. Doskolovich, D. L. Golovashkin, N. L. Kazanskiy, S. I. Kharitonov, S. N. Khonina, V. V. Kotlyar, V. S. Pavelyev, R. V. Skidanov, V. S. Solovyev, G. V. Uspleniev, and A. V. Volkov, Methods for Computer Design of Diffractive Optical Elements (Wiley, 2002). 18. D. W. Sweeney and G. E. Sommargren, “Harmonic diffractive lenses,” Appl. Opt. 34(14), 2469–2475 (1995). 19. S. I. Kharitonov, S. G. Volotovsky, and S. N. Khonina, “Geometic-optical calculation of the focal spot of a harmonic diffractive lens,” Comput. Opt. 40(3), 331–337 (2016). 20. V. M. Gurenko, L. B. Kastorsky, V. P. Kiryanov, A. V. Kiryanov, S. A. Kokarev, V. M. Vedernikov, and A. G. Verkhogliad, “Laser writing system CLWS-300/C-M for microstructure synthesis on axisymmetric 3D surfaces,” Proc. SPIE 4900, 320–325 (2002).


Introduction
A diffractive multifocal lens enables focusing an incident beam to several focal points along the optical axis, providing a required energy distribution between the foci [1,2]. Such multifocal lenses are important in ophthalmic applications, optical sensors and microscopy systems [3]. In particular, contact and intraocular lenses (IOLs) that provide clear vision at near, intermediate and far distances are usually based on multifocal diffractive lenses, which create a prescribed set of foci corresponding to different vision distances [2,[4][5][6][7][8][9]. The multifocal properties are achieved either by dividing the IOL into different zones that provide refraction into different foci [4][5][6][7] or by combining a refractive lens of fixed focal length with a diffractive zone plate, which generates additional foci in several diffraction orders [2,3,[6][7][8][9][10]. Until recently, the majority of the commercially available multifocal IOLs were designed to generate two foci at different distances. Trifocal lenses have now been introduced to the marketplace, which, in comparison with bifocal IOLs, provide a significant improvement in the intermediate vision [5]. Trifocal lenses contain a "segmented" diffractive relief consisting of alternating diffractive steps of different heights generating two foci in addition to the main focus. Implantation of a lens of this type was discussed in [11]. In general, the use of a segmented relief leads to the increase in the diffractive spot size.
In our previous works [9,10], we designed trifocal lenses corresponding to the superposition of a conventional lens and a trifocal zone plate. In this case, the whole aperture of the element contributes to all the three generated foci thus providing the minimum achievable diffractive spot size.
The diffraction relief used in the existing multifocal IOLs provides the formation of additional foci (one or two) with a given position for only one given wavelength. When the wavelength is changed, the positions of the additional foci (as well as the energy distribution between the foci) are also changed. This may lead to a significant deterioration in the quality of the images formed at different wavelengths.
In the present work, we propose a method for designing 'spectral multifocal diffractive lenses', which generate prescribed sets of foci retaining their positions at several different wavelengths. As an example, we designed and fabricated a spectral trifocal zone plate having fixed optical powers at three design wavelengths. As a fabrication technique, direct laser writing in photoresist was used. Simulations and a proof-of-concept experiment confirm the feasibility of the proposed design and demonstrate the formation of three foci with fixed positions at three wavelengths.

Design method
The multifocal lens discussed here consists of two optical elements: a conventional refractive lens and a multifocal diffractive zone plate. However, these elements can be combined into a single multifocal lens [1,10].
Let us first consider the calculation of a multifocal diffractive zone plate for a single given wavelength λ . A general method for designing multifocal zone plates was described in [1,2].
In this case, the phase function of the multifocal zone plate takes the form where R is the zone plate aperture radius, the function For a binary-relief trifocal zone plate, the nonlinear transformation function can be defined in the following form [1,2]: where ( ) can be defined as a sine function [8,9]: The multiplicative constant in Eq. (5) provides a uniform energy distribution between the three foci:  provide any required energy distribution between a preset number of foci [1,2]. Diffractive phase elements (DPEs) are generally designed for a single given wavelength. At the same time, DPEs generating different or identical patterns for different wavelengths (usually, for two or three) are also widely known [12][13][14][15][16][17][18][19]. The design of such DPEs is in most cases based on sophisticated iterative optimization procedures.
To the best of our knowledge, the first diffractive optical elements proposed for working with several wavelengths are the so-called color separation gratings (CSGs) [12,16]. These CSGs enable to separate three wavelengths 1 λ − , 0 λ , and 1 λ + related by the expression where N is an integer, between the -1st, 0th, and + 1st diffraction orders: the incident radiation with the wavelength 0 λ is directed to the 0th diffraction order, while the radiation with the wavelengths 1 λ ± is directed to the ± 1st orders, respectively. In order to explain the method for designing the spectral multifocal zone plate intended for generation of several foci at several different wavelengths, let us first consider the CSG working principle. For the specified wavelengths 0 1 , λ λ ± related by Eq. (6), the CSG has N equal-width steps per period, and their heights can be calculated analytically as [12] where n is the grating refractive index at the wavelength 0 λ . Disregarding the material dispersion, the relation between the grating microrelief height and the phase delay ϕ generated at the wavelength λ will be defined as follows [12, 16]: According to Eq. (8), the grating of Eq. (7) generates the following phase delays for the incident plane waves with the wavelengths 0 Taking into account the phase 2π -periodicity, the phase delays of Eq. (9) may be rewritten as According to Eq. (10), the phase delay for the wavelength 0 λ is equal to zero, which means that this wavelength is directed to the 0th diffraction order. For the wavelengths 1 λ ± , the phase delays in Eq. (10) correspond to the linear phase functions quantized into N levels. These phase functions coincide with the phase delays introduced by quantized diffractive prisms designed for each of these wavelengths and providing the incident beam deflection by the angles corresponding to the -1-st (for the wavelength 1 λ − ) or + 1-st (for the wavelength 1 λ + ) diffraction order of the grating. Thus, the presented analysis confirms that the grating of Eq. (7) separates the three wavelengths related by Eq. (6) between the -1st, 0th, and + 1st diffraction orders. It is important to note that the CSG of Eq. (7) minimizes the function ( ) ( ) where ( )  (11) and the CSG of Eq. (7) suggests that a function similar to Eq. (11) can be used as a merit function for the design of diffractive phase elements generating required patterns for several different wavelengths. In a general case, this merit function will represent the difference between the complex amplitudes of the beams generated by the element at the design wavelengths and the complex transmission functions providing the formation of the required patterns for these wavelengths. Following this approach, let us consider the calculation of the profile of a spectral multifocal zone plate (SMZP), which generates several prescribed foci at several arbitrarily chosen wavelengths , 1,..., l l L λ = [not necessarily related by any analytical formulas similar to Eq. (6)]. In this case, the SMZP profile is found by minimizing the difference between the complex amplitudes of the beams generated at the design wavelengths l λ , and the complex transmission functions of the multifocal lenses  calculated for these wavelengths. Assume that the radial profile of the SMZP has N steps with the same width R N Δ = and the heights , 0, 1 where l w are the weight coefficients ( )  .
Thus, the calculation of the SMZP is performed using Eq. (14). In order to obtain a spectral multifocal zone plate with three fixed foci

Trifocal zone plate for the three wavelengths 450, 540, and 580 nm
In order to assess the efficiency of the proposed approach, we designed a spectral trifocal zone plate defined by Eqs. (1) . The values of max h and M were chosen according to the capabilities of the technological equipment (direct laser writing system CLWS-300) used for the fabrication of the considered spectral trifocal zone plate.
As the design wavelengths, we chose the following three wavelengths approximately corresponding to the peak sensitivities of the three types of cones in the retina:  14) is shown in Fig. 1(a). It is worth mentioning that this profile contains of about 90 "irregular" height levels, which constitute a subset of the initially chosen 256 M = uniformly spaced levels. Let us also note that the maximum profile height of the spectral zone plate is about 5 times the height of the conventional diffractive microrelief working at a single wavelength. Typically, an increase in the max h value enables improving the performance of the zone plate, but makes its fabrication more complicated (mainly due to an increase in the aspect ratio).
Let the spectral multifocal lens consist of a conventional refractive lens with the focus  The intensity distribution generated by the spectral multifocal lens can be calculated using the Fresnel-Kirchhoff diffraction integral [ where z denotes the coordinate along the optical axis, r is the radial coordinate in the plane perpendicular to the optical axis,

( )
A ρ is the amplitude of the incident beam,  Fig. 2(a) and demonstrate high-quality focusing into three spots. The foci positions stay nearly the same for the design wavelengths. Let us note that the fabrication and experimental investigation of a multifocal lens containing the designed trifocal zone plate for three wavelengths is discussed below in Section 4.
According to Eqs. (12)-(14), the spectral trifocal zone plate approximates the complex transmission functions of the trifocal zone plates defined by Eqs. (1) and (5) for the given wavelengths. Let us note that the trifocal zone plate of Eqs. (1) and (5)   For comparison, Fig. 2(b) shows the normalized intensity distributions along the optical axis generated by a trifocal zone plate of Eqs. (1) and (5) Fig. 3(a)]. The normalized intensity distributions along the optical axis generated by this spectral trifocal lens are shown in Fig.  3(b) and demonstrate focusing into three spots with the same positions for all the four design wavelengths. Note that as the number of the design wavelengths increases, the focal peak quality decreases, namely, the side-lobes increase and the energy distribution between the foci changes. These effects can be compensated by increasing the maximum height of the microrelief. However, as mentioned above, this will make the fabrication of such microrelief more complicated.

Experimental results
The trifocal three-wavelength spectral zone plate described above in Subsection 3.1 [ Fig.  1(a)] was fabricated by direct laser writing in photoresist using the laser writing system CLWS-300 [20]. First, a thick 6 μm layer of a positive photoresist AZ6632 was deposited on a quartz substrate. The thick layer was obtained by spin-coating two 3 μm layers, one on top of another. For the fabrication, a grayscale image corresponding to the designed zone plate was prepared [ Fig. 1(b)]. The radial discretization step used in the fabrication was set to 1 μm. White areas in the image receive maximum exposure, and thus correspond to the deepest (lowest) parts of the relief. The microrelief profile of the fabricated spectral zone plate measured using a KLA-Tencor P-16 + profilometer is shown in Fig. 4(a). Figure 4(b) shows the comparison of fragments of the theoretical (blue) and measured (red) profiles. The profile of the fabricated SMZP is smoother and can be well described by the convolution of the theoretical profile with a 10 μm-wide rectangular window (green). In a future work, this model may be utilized for the compensation of the relief fabrication errors similarly to the optical proximity correction approach used in lithography. In this case, a profile has to be found, which, after the convolution with a rectangular window, approximates the theoretically calculated profile in the best way in accordance with a certain criterion.  Figure 5 shows the optical setup used for the investigation of the focusing properties of the fabricated trifocal spectral lens. As a light source 1, tunable laser Ekspla NT242 was used. The output beam was expanded up to 5 mm diameter corresponding to the SMZP diameter and collimated using two lenses 2. The obtained beam impinged on the fabricated trifocal spectral zone plate 3 placed immediately before a collecting lens 4 with focal length 50 mm. The generated intensity distributions were registered with a CCD camera 5 mounted on a motorized stage 6. The images were taken at different distances from the focusing lens in 50 μm increments. The exposure at each wavelength was adjusted so that the registered signal did not exceed the dynamic range of the used CCD sensor. The longitudinal intensity distributions generated by the investigated lens along the optical axis for the design wavelengths are presented in Fig. 6 (blue solid lines) and demonstrate formation of three sharp foci for each of the three wavelengths. Similarly to Fig.  2(a), the intensity distributions in Fig. 6 are normalized by maximum values for each of the wavelengths l λ , 1, 2, 3 l = . Note that the lens 4 and the collimating lenses 2 possess chromatic aberrations affecting the position of the central focus. Thus, to compare the positions of the additional foci with respect to the central focus, the longitudinal intensity distributions in Fig. 6 were shifted so that the central focus in all the three cases corresponded to 50 mm z = . Taking into account this correction of the central focus positions, the positions of the additional focal peaks are almost the same for the design wavelengths and are in good agreement with the peaks of the theoretical distributions (shown in Fig. 6 with red dashed lines). At the same time, there are some discrepancies between the calculated and registered intensity distributions, namely, the registered peaks are broader and have different relative intensities than in the theoretically calculated distributions. The observed discrepancies result from the spherical aberrations of the refractive spherical lenses used in the experimental setup [9], as well as from the defects in the profile of the fabricated spectral multifocal zone plate.  . For illustrative purposes, these circles are shown not in the zero-intensity plane, but in the plane corresponding to the normalized intensity value of 0.6. The sizes of the measured peaks are in good agreement with the theoretical values. Again, a certain increase in the size of the measured peaks can be attributed to the spherical aberrations of the used refractive spherical lenses and to the fabrication defects.

Conclusion
A method for designing diffractive spectral multifocal lenses generating a set of fixedposition foci at several different wavelengths has been proposed. The presented method is based on minimizing the difference between the complex amplitudes of the beams generated by the microrelief of the spectral lens at the design wavelengths , 1,..., l l L λ = , and the complex transmission functions of "monochromatic" multifocal diffractive lenses calculated for these wavelengths.
As an example, a trifocal zone plate with 5 mm diameter and optical powers 1 3D P − = , has been designed and fabricated using direct laser writing in photoresist. The results of proof-of-concept experimental investigations of the trifocal spectral lens consisting of a conventional refractive lens and the fabricated spectral zone plate have confirmed the formation of three fixed-position foci for the design wavelengths. The obtained results may find applications in the design and fabrication of novel multifocal intraocular lenses with reduced chromatic effects.

Funding
Ministry of Education and Science of Russian Federation; Russian Foundation for Basic Research (RFBR) (16-29-11683).