1. Introduction

Phase Fresnel [1] or kinoform [2] lenses are highly appreciated because of their well known properties of large collecting aperture, light weight, ease of replication, and additional degrees of freedom in correcting aberrations in comparison with equivalent conventional refractive lenses. A serious drawback, however, is their severe chromatic aberration caused by the large dispersion or wavelength dependence of the optical power. Phase Fresnel lenses have this characteristic in common with other diffractive elements and, in consequence, these optical components are used for narrow spectral bandwidth applications, power transmission, illumination, and optical communications, but they have been considered of little usefulness in broadband imaging applications. The focal length of the phase Fresnel lens is given by f(λ)=(λ ₀/λ) f ₀, where λ is the illumination wavelength and f ₀ is the focal length for the design wavelength λ ₀. Phase Fresnel lenses have been used to compensate for the secondary spectrum of a doublet lens [1]. As stated in Ref. 2, it is possible to bring two colors to a common focus using two properly spaced kinoform or phase Fresnel lenses, but only collection efficiencies of 5% are expected for bandwidths Δλ/λ ₀≃0.01. An interesting solution for a broadband imaging system consists of a combination of two phase Fresnel lenses and refractive achromatic doublets properly spaced [3]. The resulting imaging system is well-corrected for paraxial chromatic aberration over an illumination bandwidth of approximately 50% in the visible. This solution provided the keys to design the Eyeglass [4], a very large aperture (25–100 meters) diffractive telescope. Other solutions, based on hybrid diffractive-refractive configurations consisting of a number of properly separated lenses, have been proposed to obtain a spatially incoherent, dispersion-compensated imaging systems and correlators [5].

For broadband optical processors with high demands of compactness or for applications where there is no possibility to use a combination of spaced elements, diffractive and refractive, it is interesting to investigate how to obtain an achromatic phase Fresnel lens.

Liquid-crystal displays, working as electronically addressed spatial light modulators (SLM), have been widely used to generate programmable diffractive Fresnel lenses. These diffractive optical components are capable of dynamically changing their focal length by addressing the proper phase function onto a well characterized display. From a very large number of papers, we mention a few examples that report on the principles of using SLM to implement phase Fresnel lenses in imaging systems [6,7], optical correlators [8], and multiple imaging systems [9–11]. The mathematical models to encode a phase Fresnel lens function in a device constrained by its pixelated structure, phase quantization, amplitude and phase modulation coupling, and phase modulation depth, often below 2π, can be found in Refs. [12–14]. A self-apodization effect of the Fresnel lens displayed on the SLM appears as an intrinsic consequence of its pixelated pupil [15]. The advantages of being programmable permit to compensate for this effect as well as to encode a variety of non-uniform amplitude transmission filters jointly with a lens onto a single SLM [16]. For polychromatic illumination, as obtained from a white-light source, the performance of a diffractive lens encoded on SLM shows the expected severe chromatic aberration [6, 11, 13]. Following the analysis outlined in Ref. [13], three different chromatism effects can be identified, the refractive index chromatism, the diffraction chromatism, and the quantization chromatism. The first is the usual dispersion caused by the wavelength dependence of the ordinary and extraordinary refractive indices of the liquid-crystal. The second accounts for the dependence of the point spread function (PSF) with wavelength. The third is due to the modulo-2π definition of phase in the function displayed on the SLM that entails a reset to 0 when the function reaches 2π for the design wavelength λ ₀. In such a case, another wavelength λ<λ ₀ (λ>λ ₀) reads a phase higher (lower) than the phase read by λ ₀ when resetting the function. This problem also appears when the phase modulation depth does not reach 2π for the design wavelength. Under limited circumstances, the chromatic aberration of phase Fresnel lenses encoded on a SLM can be tolerated [11] but, in general, for broadband illumination, it deteriorates quickly the image. To the best of our knowledge, no filters or other optical components have been proposed yet to combine with the SLM encoded lens for chromatic compensation when the system works under broadband or white light illumination. It must be mentioned, however, a very recent paper by Márquez et al. [17] that can be considered the closest precedent to this work. They propose a spatial multiplexing scheme to design a diffractive lens with the same focal length at three discrete wavelengths in the R, G, and B regions of the visible spectrum. The resulting lens is programmed to be displayed on a liquid crystal device whose performance, in a phase-only regime, has been optimized for the three wavelengths. Under the illumination of the three wavelengths, the multiplexed lenses produce multiple monochromatic focalizations in the central order: each subaperture focuses each wavelength in a different location and with different PSF profile, although three of these focalizations are more efficient than others and coincide in the same plane, which is the focus plane of design. We think that some filtering of the spectral band that illuminates each subaperture becomes necessary to avoid such multiple focalizations that would worsen the situation in case of broadband or white light illumination. Another nondesired artifact is the potential presence of colored sidelobes in the focal plane that Márquez et al. reduce by using a random distribution of pixels [17]. According to the idea proposed by Bescós et al. [18], we will show a way to have fine control of the transversal chromaticity of the PSF and depth of focus by acting on the pupil size of each particular lens.

In this work, we present two different ways to compensate the chromatic aberration of a programmable phase Fresnel lens working under broadband or white light illumination. More specifically, the chromatic aberration can be compensated for a set of N discrete wavelengths that are properly selected by a set of filters within the broadband spectrum of visible light in such a way that the color content of images can be suitably conveyed. To this end we take advantage of the properties of SLM as programmable devices to display space and time variant images that can be refreshed at frame rate. Our method is based on designing a multichannel phase Fresnel lens that works nearly monochromatically in each channel and has a common focal point where the different focusing wavefronts add with temporally incoherent superposition. We design a set of N different phase Fresnel lenses L_i , with i=1..N, centered on a common optical axis, with their apertures placed at the same plane, and with the same focal length f ₀ for the respective design wavelengths λ ₁…λ _N . The lenses are then combined, or multiplexed, by carrying out some spatial integration or a hybrid spatial and time integration simultaneously as we describe in the following sections. Under broadband illumination, it must be ensured that only a narrow band of light centered on the wavelength λ_i will exclusively impinge the aperture of the lens L_i , which was designed for such wavelength. This can be accomplished by placing a color filter with the proper spectral transmittance against the aperture of each lens. Such filters select the wavelengths that focalize in the focal plane and no additional monochromatic focalizations of other wavelengths of the spectrum are obtained spatially separated in the central order. Moreover, the pupil size of each lens needs to be calculated to produce an individual PSF with desired size in order to compensate for transversal dispersion [18]. By sampling appropriately the spectrum of visible light, and by obtaining the temporally incoherent superposition of all the wavefronts emerging from the multiplexed lenses on their common focal point, it is possible to obtain a novel phase Fresnel lens with its polychromatic PSF compensated for chromatic aberration both longitudinal (axial) and transversal. The fast development of SLM in the last decades to produce devices with smarter features and higher efficiency lead us to propose how to design a programmable diffractive lens compensated from chromatic aberration for high quality polychromatic imaging purposes. We explore two possibilities to design the multichannel phase Fresnel lens:

- Mosaic aperture (static in time, spatial integration only): it uses a mosaic color filter against multilens with mosaic aperture

- Rotating multisector aperture (spatial and time integration): it uses a color filter and a multilens aperture, both divided into multiple circular sectors. Color filter and multilens rotate synchronized.

In the following sections we analyze and compare these two possibilities. Although our method will be applied to design a focusing lens (f ₀>0), it is also valid for a diverging lens (f ₀<0). Simulation results are provided and discussed.

2. Mosaic aperture

Firstly we study the case of a single channel defined by the wavelength λ_i and later on, we will extend our study to all the channels λ ₁…λ_N . Let us consider a converging lens L_i , placed at the plane of rectangular coordinates (x, y), with a quadratic phase function given by

where f ₀ is the focal length and λ_i the design wavelength. Let us consider that this lens function is sampled with a sampling period given by the pixel space (or pixel pitch). The lens is displayed on a M×M pixel array SLM, with square pixel pitch Δ and fill factor less than unity. We assume that the lens pattern reaches, at most, the Nyquist frequency at the circular contour of the aperture and, consequently, no secondary lenses appear. This implies that the focal length f ₀ has to be longer than or equal to the reference focal f_r (called critical distance in Ref. 9) that depends on the sampling period (Δ), the number of samples (M) and the wavelength λ_i ,

This condition sets the strongest constraint for the shortest wavelength. A circular aperture of maximum radius R=MΔ/2 is against the SLM screen. For the sake of simplicity, the active area of a pixel is represented by a rectangle of dimensions Δx’,Δy’, that is, by rect $(\frac{x}{\Delta x\prime },\frac{y}{\Delta y\prime })$ although other pixel shape function could be alternatively taken into consideration. Since the fill factor is less than unity, it follows that both Δx’,Δy’<Δ(Fig. 1).

Let us define a set of N phase Fresnel lenses, L_i,i=1…N that have the same focal length f ₀ for the set of wavelengths λ ₁…λ_N . In this section we design a mosaic sampling function M(λ,x,y) to combine these L_i lenses in the same aperture and analyze the polychromatic PSF of the resulting diffractive multichannel lens in the common focal plane. The mosaic sampling function for the wavelength λ_i is defined by

where τ_i (Δλ) is the amplitude transmittance of the quasimonochromatic filter that selects a narrow bandwidth centred in λ_i (consequently, τ_i (Δλ)≈0 except for Δλ=|λ-λ_i |≈0); the circ function corresponds to a circular pupil of radius R_i (with R_i ≤R=MΔ/2) and the summation corresponds to a 2D-comb function that establishes the positions of the sampling points. The mosaic sampling function of Eq. (3) represents a quite common mosaic color filter like, for instance, a Bayer patterned filter stuck on the camera sensor in digital photography.

 

Fig. 1. (a) Mosaic color filter placed against the SLM. (b) Mosaic basic pattern consisting of N=6 cells with distances defined in the text. The i-cell, centred at the point (a_i ,b_i ) is characterized by its amplitude transmittance τ_i (Δλ) with a very narrow bandwidth around λ_i .Behind the filter, the pixelated structure of the SLM with a fill factor less than unity is shown. The pixel size is Δ×Δ, but its active area is a smaller rectangle of size Δx’×Δy’.

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The mosaic filter of Eq. (3) and Fig. 1 originates from a basic pattern that replicates throughout the filter aperture. We assume that this basic pattern consists of N similar elements of size Δ×Δ that coincides with the SLM pixel size. Moreover, the grid of the mosaic color filter is assumed to perfectly match the SLM grid (Fig. 1). The basic pattern has rectangular dimensions Δl×Δs with ΔlΔs=NΔ². The point of coordinates (a_i , b_i ) gives the position of the i-cell containing the λ_i - quasimonochromatic filter inside the basic pattern.

If the phase Fresnel lens function of Eq. (1) is displayed on the SLM (phase quantization effects are not considered in this work) and a uniform plane wave of λ_i impinges the aperture, the amplitude distribution behind the lens is

where symbol ⊗ indicates convolution. Because the sampling period is now given by ΔlandΔs, it is worth to point out that the reference focal length is no longer given by Eq. (2), but by the quantities (MΔlΔ/λ_i ) or (MΔsΔ/λ_i ). Thus, for any symmetrical spherical phase Fresnel lens encoded in the SLM with the mosaic aperture (Eq. 4), its focal length has to meet the modified condition f ₀≥$f_r^{\mathit{\text{mosaic}}}$ (λ_i )=max{(MΔlΔ/λ_i ), (MΔsΔ/λ_i )}. Clearly, this condition limits the range of programmable focal lengths more than Eq. (2) does. Calculating the Fresnel propagation of the amplitude distribution of Eq. (4) in the focal plane [19], it gives

where function T_i is convolved with a diverging wave Z_i (x, y)=exp{jπ(x ²+y ²)/λ_if ₀}.

Introducing Eqs. (1), (3) in Eq. (4) and the result in Eq. (5), taking into account the associative and commutative properties of convolution, and the complex conjugate relationship between L_i and Z_i (L_i (x, y)=Z* _i (x, y)), we obtain

where M̃_i is the Fourier transform of the mosaic sampling function, that is,

The normalized version of the Bessel function, with value unity at the origin, has been used in Eq. (7). The last two equations represent a distribution of maxima separated a distance (λ_if ₀/Δl) in the u axis, and (λ_if ₀/Δs) in the v axis. The shape of each maximum is given by the convolution of Bessel J ₁ function of Eq. (7) by the rectangle function of Eq. (6), and it can be approximated by the dimensions [Δx’+(1.22λ_if ₀/R_i )], [Δy’+(1.22λ_if ₀/R_i )] in the u,v axis of the focal plane. It can be assumed that these maxima will appear spatially separated because of the relative dimensions of R_i , Δl, Δs, Δ, Δx’, Δy’ involved.

We are clearly interested in the maximum of the central order. Following an analysis analogous to that carried out in Ref. [12], it is possible to simplify Eqs.(6) and (7) by neglecting the phase factors within the central order. The quadratic phase factor Z_i varies less than either (πΔx’/Δl)or(πΔy’/Δs), which is less than a single half oscillation of the complex exponential inside the rectangle function and central lobe of the Bessel J ₁ function. We have estimated the variation of the linear phase factor of Eq. (7) inside the rectangle function, for a_i , b_i ≈Δx’, Δy’≈10^-5m; λ_i ≈10^-6m, and f ₀≈10^-1 m and it yields a variation of ≈10^-3×2π. Consequently, it can be considered constant too. Neglecting then the slow varying phase terms, the amplitude of the central order in the focal plane U _00i is

where d_i =R_i /λ_if ₀. Eq. (8) is the convolution of a wavelength dependent term by a wavelength independent rectangle function. The variation of the central lobe of U _00i with λ can therefore be analyzed through the variation of the first term of Eq. (8) with λ. The width of the central lobe of the Bessel J ₁ function in Eq. (8) is 0 1.22 λ_if ₀/R_i and its height is weighted by the precedent factor. Although the focal plane is the same for all λ_i , the central lobe of U _00i shows different sizes and, as a result, transversal chromatism is produced unless the condition d_i =constant or, equivalently, R_i /λ_i =constant, is fulfilled. Since this condition leads to having a PSF of the same width for all the wavelengths λ_i of the set, hereafter we refer to it as the same size of the PSF condition or just the PSFS-condition. It implies that the pupil sizes of the lens apertures for different wavelengths have to be different. This result agrees with the result obtained by Bescós et al. [18]. Since we use a mosaic filter and a pixelated display of a programmable SLM, this condition will not be particularly difficult to meet in our case. Note that if the PSFS-condition is met, then PSF profiles of the same width are obtained for all the wavelengths λ_i even when the focal length of design is changed from the programmed f ₀ to another value. This is a good property because it remains invariant under variations of the programmed focal length. On the other hand, the central order focalization of Eq.(8) shows different maximum intensity value with wavelength unless the multiplicative factor meets the condition τ_i (Δλ)d_iR_i =constant or, equivalently, τ_i (Δλ)$R_i^2$ /λ_i =constant, which hereafter we refer to as the same maximum intensity of the PSF condition or just the PSF_I-condition. Note that, again, if the PSF_I-condition is met, then PSF profiles of the same maximum height are obtained for all the wavelengths λ_i even if the focal length of design varies from the programmed f ₀ to another value. This is a second good property of invariance against variations of the programmed focal length. Both the PSF_S and the PSF_I conditions can be simultaneously fulfilled if the amplitude transmittance of the quasimonochromatic filter τ_i (Δλ) is taken as an additional degree of freedom for each focal length f ₀. This is not the most general case and, in fact, we will see that if the system fulfils the PSF_I condition, then PSF profiles of equal height and reasonably similar width can be obtained for the wavelengths of the set.

Under polychromatic illumination, the mosaic multichannel phase Fresnel lens ML(x, y) can be described as the multiplexing (or addition) of the different single λ_i -channel lenses

where L_i , M_i functions are defined in Eq. (1) and Eq. (3), respectively. We have computed a lens for N=4, Δ=26µm, M=256, sampling period of the mosaic pattern Δl=Δs=2Δ, and wavelength λ_i range comprising red λ ₁=632nm, green λ ₂=543nm, blue λ ₃=488nm, and violet λ ₄=458nm. The focal length of design for all the wavelengths is f ₀=$f_r^{\mathit{\text{mosaic}}}$ (λ ₄)=2MΔ²/λ ₄=2f_r (λ ₄)≈75cm. Figure 2 shows a simplified version of the lens for 64×64 pixels. Note that in all lenses L_i , the centre has been set to the same phase value, corresponding to an extreme of the phase range of modulo-2π assumed in this example, in order to have the first discontinuity as far from the optical axis as possible. The multichannel lens of Fig. 2 meets the PSF_S-condition (R_i /λ_i =constant). Pixels whose position in the mosaic pattern corresponds to the lens L_i , but whose distance from the center is longer than the R_i value given by the PSF_S-condition are assigned a constant phase value (CPV in Fig. 2(b)) which is equivalent to leave them blank. These pixels with constant phase value allow us to control the PSF diameter, but they also contribute with a background noise to the focal plane. We have estimated this noise for the lens of Fig. 2 and, in the worst case, represented by the sublens for the violet channel of λ ₄=458nm, the background noise in the focal plane is less than 3%. Therefore, we consider it negligible in our approach.

 

Fig. 2. Schematic diagram of the mosaic multichannel phase Fresnel lens building. (a) Phase Fresnel lenses L_i (partial) with i=1..4 (λ ₁=632nm, λ ₂=543nm, λ ₃=488nm, λ ₄=458nm). The radius R_i marked in each lens fulfils the PSF_S-condition R_i /λ_i =constant. (b) λ_i -channels lenses, obtained from L_i of (a), after a double discretization of pixelation and mosaic filtering. Pixels whose distance from the center is longer than R_i , are assigned a constant phase value (CPV). (c) Integration of λ_i -channels lenses by spatial multiplexing according to the basic pattern (magnified). The result is the mosaic multichannel Fresnel lens.

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Figure 3 shows the intensity cross-section profiles of the central order in the focal plane. Profiles are plotted for the four wavelengths in three geometrical conditions of the pupil size: when all single λ_i -channel lenses have the same aperture (R_i =R, constant) (Fig. 3(a)), when the PSF_S-condition (R_i /λ_i =constant) is fulfilled (Fig. 3(b)), and when the PSF_I-condition is fulfilled (Fig. 3(c)). For simplicity, it has been assumed that all color filters have equally shaped spectral transmittance curves centered in their respective λ_i , that is, τ_i (Δλ)=τ(Δλ). It can be seen that the transversal chromatic aberration of the PSF when all lenses L_i have the same pupil size (Fig. 3(a)) is compensated when the PSFS-condition is fulfilled (Fig. 3(b)), and almost compensated when PSFI-condition is fulfilled (Fig. 3(c)).

On the other hand, these two kinds of compensation imply a noticeable decrease of the maximum intensity in comparison with that obtained in Fig. 3(a). Comparing Figs. 3(b) and 3(c), it becomes clear that fulfilling the PSFI-condition would be preferable in practice although a slight residual chromatism would also appear in such a case.

Figure 4 shows the intensity cross-section profiles of the central order along the optical axis. Profiles are plotted for the four wavelengths in the same three geometrical conditions of the pupil size as before. The graphs are approximately symmetrical around the best image plane (given by the focal length f ₀), and their width along the axis represents the depth of focus in each case. Analogous comments to those of Fig. 3 can be made again for the three cases considered. It can be seen that Fig. 4(c), for which the PSFI-condition is fulfilled, represents a fully compensation of longitudinal chromatism, at least under paraxial approximation, because the axial scaling factor is proportional to λ_i /$R_i^2$ . Consequently, if the PSFI-condition is met, then the depth of focus is invariant for the wavelengths considered.

Figure 5 shows the total intensity of the polychromatic PSF computed from the superposition of the intensity distributions obtained for the set of wavelengths in all the cases considered in Figs. 3 and 4. They are represented in the focal plane defined by f ₀ (Fig. 5(a)) and along the optical axis (Fig. 5(b)). In both representations, the option R_i =R yields the most intense maximum, the option represented by fulfilling the PSF_I-condition obtains the second intense maximum values and, finally, the option represented by fulfilling the PSF_S-condition obtains the third intense maximum values. Regarding transversal resolution (Fig. 5(a)), the plots corresponding to R_i =R and PSF_I-condition obtain total intensities of the PSF with very similar width, thus providing similar resolution. When PSF_S-condition is fulfilled, however, the PSF slightly broadens transversally, consequently with a slight loss of transversal resolution. Regarding axial resolution (Fig. 5(b)), the smallest depth of focus is obtained for the R_i =R option, but it is very close to the depth of focus obtained when the PSF_I-condition is fulfilled. The option represented by fulfilling the PSF_S-condition obtains the longest depth of focus. As a conclusion from the analysis of Figs. 3–5, it appears that the aperture configuration of lenses L_i whose pupils fulfill PSFI-condition would be the most advantageous for both transversal and axial chromatic compensation in most practical cases.

3. Rotating multisector aperture

In this section we use a rotating multisector color filter against the SLM and multiplexe the N phase Fresnel lenses L_i by considering an array of circular sectors in the aperture plane. The focal length of design f ₀ is common for all the wavelengths. As before, let us firstly consider a single lens L_i , defined by the lens function of Eq. (1), in the channel defined by λ_i and then, we will extend the result to the rest of channels. All the SLM pixels belonging to a circular sector display a single lens function L_i , which is now sampled with period Δ (pixel pitch) inside the sector. Consequently, the focal length f ₀ has to meet the general condition of Eq. (2), which is less restrictive than the modified condition obtained in Section 2 for the mosaic aperture.

 

Fig. 3. Intensity distribution of the central order in the focal plane of the lens with mosaic aperture (Fig. 2) for λ_i , i=1..4 and (a) constant radius R_i =R (b) PSF_S-condition, and (c) PSF_I-condition.

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Fig. 4. Intensity distribution of the central order along the optical axis of the lens with mosaic aperture (Fig. 2) for λi, i=1.. 4. (a) Constant radius Ri=R, (b) PSFS-condition, and (c) PSFI-condition. The four plots coincide in (c).

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Fig. 5. Total intensity of the polychromatic PSF of the lens with mosaic aperture (Fig. 2), computed from the superposition of the intensities obtained in Figs. 3 and 4: (a) in the focal plane and (b), along the optical axis.

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The pupil of lens L_i is described by the circular sector function

where (r,θ)represents the polar coordinates $r={(x^2+y^2)}^{\frac{1}{2}},\theta ={\tan }^{-1}(y⁄x)$. Eq. (10) is a circular sector shaped pupil with vertex on the optical axis, in the coordinate origin, and angular extension A_i around the angle θ_i . It is assumed that sectors do not overlap between them and that they all complete the circle (∑ A_i =2π). Let us consider N=4 as in the example of Section 2. It represents a multicolor filter consisting of four sectors with equal angular amplitude A_i =π/2 and radius of at least R=MΔ/2 to cover the SLM size (Fig. 6(a)). The lens L_i is centered in the optical axis, limited by a quadrant shaped pupil, and has a radial extension of R_i ≤R. The sector defined by R_i ≤r≤R is left blank, or equivalently, it introduces a constant phase ϕ. The whole element could be mathematically described by

where H(x,y) is the 2D Heaviside step function and I(·) is the integer part of the argument. Eq. (11) can be rewritten as the addition of two terms corresponding to the lens and the blank subsectors,

When the element Q_i (x, y) of Eq. (11) is sampled and displayed on the pixelated liquid crystal device (Fig. 6), it turns out

Fresnel propagation leads us again to Eq. (5), but now with T_i (x, y) given by Eq. (13). The development of the resulting expression gives two added terms, $U_i^L$ (u, v)+$U_i^B$ (u, v), corresponding to the Fresnel propagation of the amplitudes transmitted by the lens and the blank subsectors, respectively. But only the first term $U_i^L$ (u, v) becomes interesting in our analysis. The second term contributes with a slow varying noise in the area of focalization and, consequently, it will be neglected. Taking into account $Q_i^L$ (x, y) of Eq. (12), $U_i^L$ (u, v) is

where FT indicates Fourier transform. Neglecting the slow varying phase terms inside function rect[(u/Δx’),(v/Δy’)], as we did in Section 2 (see also Ref. [12]), we obtain

Equation (15) represents a distribution of maxima separated a distance (λ_if ₀/Δ) in both the u and v axis. Note that this distance is longer (a factor two in our example) than the distance between the diffraction orders originated by the mosaic pattern of Section 2. This is a clear advantage with respect to the mosaic multichannel lens because it allows an enlargement of the image field without overlapping with higher diffraction orders. The amplitude of the central order in the focal plane is given by

which can be rewritten as

where the term $W_{00i}^L$(u, v), defined by

concentrates the wavelength dependence of the central lobe $U_{00i}^L$(u, v). Let us analyze then the variation of $W_{00i}^L$(u, v) with wavelength. Firstly, the condition to fix the PSF width cannot be established just by taking d_i =constant, or equivalently R_i /λ_i =constant, as we did in Eq. (8) of Section 2, because now, in Eq. (18), the term with square brackets is additionally convolved by the Fourier transform of the Heaviside function whose variable is not scaled by d_i . We will see in the simulation results, however, that the effects of this additional convolution in broadening the term in brackets are insignificant and therefore, we will keep the PSF_S-condition as R_i /λ_i =constant. Secondly, from Eq. (18), the condition to fix the same PSF maximum intensity for all the wavelengths of the set is τ_i (Δλ)$R_i^2$ /λ_i =onstant, which coincides with the PSF_I-condition established in Section 2. Thirdly, $W_{00i}^L$(u, v) has not circular symmetry because of the Fourier transform of the Heaviside function, H̃, which is not symmetric with circular symmetry. Figure 7(a) displays the intensity of the PSF corresponding to the sublens T ₁(x,y) of the circular sector in the top right quadrant. This lack of circular symmetry can be compensated if $W_{00i}^L$(u, v) rotates around the optical axis (Fig. 7(b)). It can be accomplished by rotating the multisector color filter and, synchronized with it, the multilens array displayed on the SLM also rotates. Note that the programmable facilities of SLM allow us to rotate the addressed multilens array but there is no need to actually rotate the SLM screen (Fig. 6(b)). If the angular speed is high enough to have a time period shorter than the integration time of the sensor, then the intensity of the detected signal will be 〈|$U_{00i}^L$|²〉, that is, the time average of |$U_{00i}^L$|². When polychromatic illumination is used, we build the multichannel lens ML(x, y) as the addition of the different single λ_i -channel diffractive lenses with rotating circular sector shaped pupils. At each instant of time the function is sampled according to the pixelization of the liquid crystal display, which does not rotate. The resulting multichannel phase Fresnel lens can be described by

for a given instant of time. To include rotation, we write ML in polar coordinates

where T_i is given by Eq. (13), ω is the angular speed, t is time, x=rcosθ, y=rsinθ, with θ=ωt. We assume that the rotation period is shorter than the integration time T ₀of the system sensor, that is, (2π/ω)<T ₀. Since rotation affects the color filter and the multilens, but it does not affect the sampling function nor the SLM pixel positions, the distribution of the different order focalizations is stationary in the focal plane.

 

Fig. 6. Scheme of the multichannel phase Fresnel lens with rotating aperture. (a) Color filter consisting of N=4 circular sectors with narrow band transmittance centered at the wavelengths λ ₁=632nm, λ ₂=543nm, λ ₃=488nm, and λ ₄=458nm. Each color filter is against the SLM that displays a part of the sublens L_i with i=1..4. The radius of each lens fulfils the PSF_S-condition (R_i /λ_i =constant). A constant phase value is assigned to pixels beyond R_i . (b) The λ_i -channel lenses are multiplexed using a hybrid spatial and time integration. The result is the multichannel Fresnel lens with rotating aperture. (604 KB).

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For an easy comparison with the results obtained for the multichannel lens with mosaic aperture (Figs. 3–5 of Section 2) we show now the time-average cross-section intensity profiles of the central order focalization for the lens with the same focal length f ₀ of Section 2 but with rotating aperture. Terms $U_{00i}^B$(u, v), which contain the contributions of the blank subsectors, have been estimated in our example and they would alter terms $U_{00i}^L$(u, v)less than 3% in the focalization area. This fact justifies we neglect them in our calculations. In Figs. 8 and 9 the profiles are plotted for the four wavelengths in three geometrical conditions of the pupil size: when all single λ_i -channel lenses have the same maximum aperture (R_i =R), when the PSF_S-condition R_i /λ_i =constant is fulfilled, and when the PSF_I-condition τ_i (Δλ)$R_i^2$ /λ_i =constant is fulfilled. As before, it has been also assumed that all color filters have a spectral transmittance curve with a common shape centered in their respective λ_i , that is, τ_i (Δλ)=τ(Δλ). The results obtained in Fig. 8 for the transversal compensation of chromatism are similar to those shown in Fig. 3 except for the expected broadening of the PSF width caused by the circular quadrant shaped aperture of the rotating scheme. This decrease of resolution can be considered as the price to pay if we want to have the first diffraction orders more distant from the central order. Although the PSFs-condition does not make the central lobe width exactly equal for all the wavelengths of the set, the aforementioned effects introduced by the convolution with the Fourier transform of the Heaviside function are hardly appreciated (Fig. 8(b)). Consequently, it can be said that the transversal chromatic aberration of the PSF when all lenses L_i have the same pupil (Fig. 8(a)) is compensated when the PSF_S-condition is fulfilled (Fig. 8(b)), and almost compensated when PSF_I-condition is fulfilled (Fig. 8(c)). Again, these two kinds of compensation imply a noticeable decrease of the maximum intensity in comparison with that obtained in Fig. 8(a). Comparing Figs. 8(b) and 8(c), it becomes clear that fulfilling the PSF_I-condition would be preferable in practice although a slight residual chromatism would also appear in such a case.

 

Fig. 7. (a) Intensity of the PSF of the T ₁(x,y) sublens, in the top right quadrant circular sector (Fig. 6). The lack of circular symmetry is compensated when $W_{00i}^L$(u, v) rotates around the optical axis (b).

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Figure 9 shows the time-average cross-section intensity profiles of the central order focalization along the optical axis. They all are very similar to those profiles of Fig. 4 and then, the comments made there are still valid for Fig. 9. The graphs are approximately symmetrical around the best image plane (given by f ₀), and their width along the axis represents the depth of focus in each case. When the PSF_I-condition is fulfilled (Fig. 9(c)), a fully compensation of longitudinal chromatism is achieved.

Figures 10(a) and 10(b) show the total intensity of the polychromatic PSF computed from the superposition of the intensities in the cases considered in Figs. 8 and 9. As in the results obtained in Section 2 for the lens with mosaic aperture (Fig. 5), the option R_i =R yields the most intense maximum in both (a) transversal and (b) axial representations, the option represented by fulfilling the PSF_I-condition is second intense, and the option represented by fulfilling the PSF_S-condition is third intense. Regarding axial resolution (Fig. 10(b)), the smallest depth of focus is obtained for the R_i =R option, very close to the depth of focus obtained when the PSF_I-condition is met. From the analysis of Figs. 8–10, lens pupils that fulfill the PSF_I-condition would be the most advantageous for both transversal and axial chromatic compensations. This conclusion is the same as that we reached in Section 2 for the mosaic aperture. A joint representation of Figs. 5(a) and 10(a) allows us to compare the polychromatic PSF obtained for the mosaic and the rotating aperture schemes in the focal plane. Concerning efficiency, we appreciate that the maximum values reached by both schemes are very similar, but the energy concentrated in the central order focalization is greater for the rotating aperture scheme. It is also clear the transversal enlargement of the PSF for the rotating aperture with respect to that obtained for mosaic aperture.

 

Fig. 8. Intensity distribution of the central order in the focal plane of the lens with rotating aperture (Fig. 6) for λ_i , i=1..4 and (a) constant radius R_i =R(b) PSF_S-condition, and (c) PSF_I-condition.

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Fig. 9. Intensity distribution of the central order of the lens with rotating aperture (Fig. 6) along the optical axis for λ_i , i=1.. 4 and (a) constant R_i =R(b) PSF_S-condition, and (c) PSF_I-condition. The four plots coincide in (c).

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Fig. 10. Total intensity of the polychromatic PSF computed from the superposition of the intensity distributions of Figs. 8 and 9. (a) In the focal plane, (b) along the optical axis. (c) Joint representation of Figs. 5(a) and 10(a).

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4. Conclusions

We have proposed two different ways to compensate the notorious chromatic aberration of a programmable diffractive phase Fresnel lens working under polychromatic illumination. Both proposals are based on a multiplexing of lenses, designed with the same focal length for a set of wavelengths, and a spectral filtering of the light that impinges each sublens. Since each sublens operates as a channel for a given wavelength, the combination of sublenses results in an achromatic multichannel phase Fresnel lens that would allow one to use such a diffractive lens for high quality polychromatic imaging purposes.

One of the proposals uses a spatial multiplexing with mosaic aperture. The other uses a rotating scheme with a filter of circular color sectors against an array of lens sectors displayed on the SLM. The second proposal applies a hybrid spatial and time integration. Both schemes achieve a common focal plane for all the wavelengths of the set. The focalization of the central order has a unique location at that plane. In addition to this, we have drastically reduced the transversal chromatic aberration of the PSF in the central order by properly adjusting the pupil size of each sublens. The axial distribution of energy in this order has been improved in the same way. Three conditions for the sublens pupil size have been considered: one, a constant radius for all sublenses (R_i =R to cover the addressable size of the display); two, a radius that fixes the same PSF size for all sublenses (PSF_S-condition: R_i /λ_i =)constant, and three, a radius that fixes equal maximum intensities of the PSF for all sublenses (PSF_I-condition: τ_i (Δλ)$R_i^2$ /λ_i =constant). For both the mosaic and the rotating aperture schemes, the PSF reaches its maximum intensity when all sublenses have the radius R_i =R, but a transversal dispersion and a variation in the depth of focus also appear for the wavelengths of the set. A very significant improvement is achieved when the PSF_I-condition is fulfilled. In such a case, the dispersion is compensated along the optical axis, with depths of focus coincident for all the wavelengths, and the transversal dispersion is almost compensated in the focal plane. The PSF_I-condition leads then to obtain the best results for both the mosaic and rotating aperture schemes and, consequently, it must be preferably met.

The multichannel phase Fresnel lens based on a mosaic aperture scheme allows one to have smaller PSF sizes than the rotating aperture scheme, thus contributing to a better resolution in an imaging system. On the other hand, the first diffraction orders are closer to the central order, thus limiting the extension of the image field more than the rotating scheme. Although the PSF maximum intensity value is similar for both schemes, the total energy in the central order focalization is highest for the rotating scheme. Concerning the range of programmable focal lengths, the rotating aperture has higher range. But, from the point of view of optomechanical requirements, the rotating aperture scheme could be more complex to implement than the static mosaic aperture.

Finally, although a significant improvement is expected with the two proposals, we point out several aspects that should be treated in future research if an experimental realization is pursued. It must be taken into account that SLMs are still devices with low efficiency in general, although their technical characteristics are improving, particularly for reflection-based liquid-crystal devices. Another important aspect is the spectral bandwidth of the color filters. It plays a key role to achieve good results. Their bandwidth must be narrow enough to make the secondary spectrum acceptable for a particular application. But the narrower the bandwidth transmittances, the lesser the efficiency of the resulting multichannel phase Fresnel lens. Another aspect is the choice of the wavelengths and the number of channels. We have done this choice arbitrarily in our examples, but it should be done carefully in order to optimize the white balance and the color content to convey in a given application. And, as Márquez et al. do in their work [17], the configuration of the SLM (phase-only modulation) has to be optimized for the set of wavelengths. In terms of experimental implementation, it is not trivial to achieve a good synchronism between the rotating multisector filter and the time-variant distribution of phase displayed by the SLM. This difficulty also depends on the time integration of the detector and the capability of the SLM for a reliable display. All these aspects must be taken into consideration when implementing the proposals experimentally.