## Chromatic compensation of programmable Fresnel lenses

Optics Express, Vol. 14, Issue 13, pp. 6226-6242 (2006)

http://dx.doi.org/10.1364/OE.14.006226

Acrobat PDF (711 KB)

### Abstract

Two proposals to compensate chromatic aberration of a programmable phase Fresnel lens displayed on a liquid crystal device and working under polychromatic illumination are presented. They are based on multiplexing a set of lenses, designed with a common focal length for different wavelengths, and a multicolor filter that makes each sublens work almost monochromatically. One proposal uses spatial multiplexing with mosaic aperture. The other uses a rotating scheme, a color filter against an array of lens sectors, and hybrid spatial-time integration. The central order focalization has a unique location at the focal plane. We have drastically reduced the transversal chromatic aberration of the polychromatic point spread function by properly adjusting the pupil size of each sublens. Depth of focus curves have been made coincident too for the selected wavelengths.

© 2006 Optical Society of America

## 1. Introduction

1. K. Miyamoto, “The phase Fresnel lens,” J. Opt. Soc. Am. **51**, 17–20 (1961). [CrossRef]

2. J.A. Jordan, P.M. Hirsch, L.B. Lesem, and D.L. Van Rooy, “Kinoform lenses,” Appl. Opt. **9**, 1883–1887 (1970). [PubMed]

*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

1. K. Miyamoto, “The phase Fresnel lens,” J. Opt. Soc. Am. **51**, 17–20 (1961). [CrossRef]

*λ*/

*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

4. R.A. Hyde, “Eyeglass.1. Very large aperture diffractive telescopes,” Appl. Opt. **38**, 4198–4212 (1999). [CrossRef]

5. P. Andrés, V. Climent, J. Lancis, G. Mínguez-Vega, E. Tajahuerce, and A.W. Lohmann, “All-incoherent dispersion- compensated optical correlator,” Opt. Lett. **24**, 1331–1333 (1999). [CrossRef]

6. E. C. Tam, S. Zhou, and M. R. Feldman, “Spatial-light-modulator-based electro-optical imaging system,” Appl. Opt. **31**, 578–580 (1992). [CrossRef] [PubMed]

7. J. A. Davis, D. M. Cottrell, R. A. Lilly, and S. W. Connely, “Multiplexed phase-encoded lenses written on spatial light modulators,” Opt. Lett. **14**, 420–422 (1989). [CrossRef] [PubMed]

8. J. A. Davis, D. M. Cottrell, J. E. Davis, and R. A. Lilly, “Fresnel lens-encoded binary phase-only filters for optical pattern recognition,” Opt. Lett. **14**, 659–661 (1989). [CrossRef] [PubMed]

9. D. M. Cottrell, J. A. Davis, T. R. Hedman, and R. A. Lilly, “Multiple imaging phase-encoded optical elements, written as programmable spatial light modulators,” Appl. Opt. **29**, 2505–2509 (1990). [CrossRef] [PubMed]

11. M. Hain, W. von Spiegel, M. Schmiedchen, T. Tschudi, and B. Javidi, “3D integral imaging using diffractive Fresnel lens arrays,” Opt. Express **13**, 315–326 (2005). [CrossRef] [PubMed]

12. E. Carcolé, J. Campos, and S. Bosch, “Diffraction theory of Fresnel lenses encoded in low-resolution devices,” Appl. Opt. **33**, 162–174 (1994). [CrossRef] [PubMed]

14. I. Moreno, C. Iemmi, A. Márquez, J. Campos, and M. J. Yzuel, “Modulation light efficiency of diffractive lenses displayed in a restricted phase-mostly modulation display,” Appl. Opt. **43**, 6278–6284 (2004). [CrossRef] [PubMed]

15. V. Arrizón, E. Carreón, and L. A. González, “Self-apodization of low-resolution pixelated lenses,” Appl. Opt. **38**, 5073–5077 (1999). [CrossRef]

16. A. Márquez, C. Iemmi, J.C. Escalera, J. Campos, S. Ledesma, J. A. Davis, and M.J. Yzuel, “Amplitude apodizers encoded onto Fresnel lenses implemented on a phase-only spatial light modulator,” Appl. Opt. **40**, 2316–2322 (2001). [CrossRef]

6. E. C. Tam, S. Zhou, and M. R. Feldman, “Spatial-light-modulator-based electro-optical imaging system,” Appl. Opt. **31**, 578–580 (1992). [CrossRef] [PubMed]

11. M. Hain, W. von Spiegel, M. Schmiedchen, T. Tschudi, and B. Javidi, “3D integral imaging using diffractive Fresnel lens arrays,” Opt. Express **13**, 315–326 (2005). [CrossRef] [PubMed]

13. V. Laude, “Twisted-nematic liquid-crystal pixilated active lens,” Opt. Commun. **153**, 134–152 (1998). [CrossRef]

13. V. Laude, “Twisted-nematic liquid-crystal pixilated active lens,” Opt. Commun. **153**, 134–152 (1998). [CrossRef]

*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

11. M. Hain, W. von Spiegel, M. Schmiedchen, T. Tschudi, and B. Javidi, “3D integral imaging using diffractive Fresnel lens arrays,” Opt. Express **13**, 315–326 (2005). [CrossRef] [PubMed]

17. A. Márquez, C. Iemmi, J. Campos, and M.J. Yzuel, “Achromatic diffractive lens griten onto a liquid cristal display,” Opt. Lett. **31**, 392–394 (2006). [CrossRef] [PubMed]

17. A. Márquez, C. Iemmi, J. Campos, and M.J. Yzuel, “Achromatic diffractive lens griten onto a liquid cristal display,” Opt. Lett. **31**, 392–394 (2006). [CrossRef] [PubMed]

18. J. Bescós, J. H. Altamirano, J. Santamaria, and A. Plaza, “Apodizing filters in colour imaging,” J. Optics (Paris) **17**, 91–96 (1986). [CrossRef]

*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

18. J. Bescós, J. H. Altamirano, J. Santamaria, and A. Plaza, “Apodizing filters in colour imaging,” J. Optics (Paris) **17**, 91–96 (1986). [CrossRef]

*f*

_{0}>0), it is also valid for a diverging lens (

*f*

_{0}<0). Simulation results are provided and discussed.

## 2. Mosaic aperture

*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

*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},

*R*=

*M*Δ/2 is against the SLM screen. For the sake of simplicity, the active area of a pixel is represented by a rectangle of dimensions Δ

*x’*,Δ

*y’*, that is, by rect

*x’*,Δ

*y’*<Δ(Fig. 1).

*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

*τ*

_{i}(Δ

*λ*) is the amplitude transmittance of the quasimonochromatic filter that selects a narrow bandwidth centred in

*λ*

_{i}(consequently,

*τ*

_{i}(Δ

*λ*)≈0 except for Δ

*λ*=|

*λ*-

*λ*

_{i}|≈0); the circ function corresponds to a circular pupil of radius

*R*

_{i}(with

*R*

_{i}≤

*R*=

*M*Δ/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.

*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 Δ

*l*×Δ

*s*with Δ

*l*Δ

*s*=

*N*Δ

^{2}. 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.

*λ*

_{i}impinges the aperture, the amplitude distribution behind the lens is

*l*andΔ

*s*, it is worth to point out that the reference focal length is no longer given by Eq. (2), but by the quantities (

*M*Δ

*l*Δ/

*λ*

_{i}) or (

*M*Δ

*s*Δ/

*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}≥

*λ*

_{i})=max{(

*M*Δ

*l*Δ/

*λ*

_{i}), (

*M*Δ

*s*Δ/

*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

*T*

_{i}is convolved with a diverging wave

*Z*

_{i}(

*x*,

*y*)=exp{

*jπ*(

*x*

^{2}+

*y*

^{2})/

*λ*

_{i}

*f*

_{0}}.

*L*

_{i}and

*Z*

_{i}(

*L*

_{i}(

*x*,

*y*)=

*Z**

_{i}(

*x*,

*y*)), we obtain

*M̃*

_{i}is the Fourier transform of the mosaic sampling function, that is,

*λ*

_{i}

*f*

_{0}/Δ

*l*) in the

*u*axis, and (

*λ*

_{i}

*f*

_{0}/Δ

*s*) 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 [Δ

*x’*+(1.22

*λ*

_{i}

*f*

_{0}/

*R*

_{i})], [Δ

*y’*+(1.22

*λ*

_{i}

*f*

_{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}, Δ

*l*, Δ

*s*, Δ, Δ

*x’*, Δ

*y’*involved.

12. E. Carcolé, J. Campos, and S. Bosch, “Diffraction theory of Fresnel lenses encoded in low-resolution devices,” Appl. Opt. **33**, 162–174 (1994). [CrossRef] [PubMed]

*Z*

_{i}varies less than either (

*π*Δ

*x’*/Δ

*l*)or(

*π*Δ

*y’*/Δ

*s*), 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}≈Δ

*x’*, Δ

*y’*≈10

^{-5}m;

*λ*

_{i}≈10

^{-6}m, and

*f*

_{0}≈10

^{-1}m and it yields a variation of ≈10

^{-3}×2

*π*. 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

*d*

_{i}=

*R*

_{i}/

*λ*

_{i}

*f*

_{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*

_{i}

*f*

_{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

18. J. Bescós, J. H. Altamirano, J. Santamaria, and A. Plaza, “Apodizing filters in colour imaging,” J. Optics (Paris) **17**, 91–96 (1986). [CrossRef]

*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}(Δ

*λ*)

*d*

_{i}

*R*

_{i}=constant or, equivalently,

*τ*

_{i}(Δ

*λ*)

*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}(Δ

*λ*) 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.

*ML*(

*x*,

*y*) can be described as the multiplexing (or addition) of the different single

*λ*

_{i}-channel lenses

*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 Δ

*l*=Δ

*s*=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}=

*λ*

_{4})=2

*M*Δ

^{2}/

*λ*

_{4}=2

*f*

_{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.

*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}(Δ

*λ*)=

*τ*(Δ

*λ*). 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)).

*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}/

*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

*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.

*L*

_{i}is described by the circular sector function

*r*,

*θ*)represents the polar coordinates

*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}=2

*π*). 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*=

*M*Δ/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

*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,

*Q*

_{i}(

*x*,

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

*T*

_{i}(

*x*,

*y*) given by Eq. (13). The development of the resulting expression gives two added terms,

*u*,

*v*)+

*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*,

*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

*x*,

*y*) of Eq. (12),

*u*,

*v*) is

*FT*indicates Fourier transform. Neglecting the slow varying phase terms inside function rect[(

*u*/Δ

*x’*),(

*v*/Δ

*y’*)], as we did in Section 2 (see also Ref. [12

12. E. Carcolé, J. Campos, and S. Bosch, “Diffraction theory of Fresnel lenses encoded in low-resolution devices,” Appl. Opt. **33**, 162–174 (1994). [CrossRef] [PubMed]

*λ*

_{i}

*f*

_{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

*u*,

*v*), defined by

*u*,

*v*). Let us analyze then the variation of

*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}(Δ

*λ*)

*λ*

_{i}=onstant, which coincides with the PSF

_{I}-condition established in Section 2. Thirdly,

*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

*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 〈|

^{2}〉, that is, the time average of |

^{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

*ML*in polar coordinates

*T*

_{i}is given by Eq. (13),

*ω*is the angular speed, t is time,

*x*=

*r*cos

*θ*,

*y*=

*r*sin

*θ*, with

*θ*=

*ωt*. We assume that the rotation period is shorter than the integration time

*T*

_{0}of the system sensor, that is, (2

*π*/

*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.

*f*

_{0}of Section 2 but with rotating aperture. Terms

*u*,

*v*), which contain the contributions of the blank subsectors, have been estimated in our example and they would alter terms

*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}(Δ

*λ*)

*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}(Δ

*λ*)=

*τ*(Δ

*λ*). 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.

*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.

*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.

## 4. Conclusions

*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}(Δ

*λ*)

*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.

17. A. Márquez, C. Iemmi, J. Campos, and M.J. Yzuel, “Achromatic diffractive lens griten onto a liquid cristal display,” Opt. Lett. **31**, 392–394 (2006). [CrossRef] [PubMed]

## Acknowledgments

## References and Links

1. | K. Miyamoto, “The phase Fresnel lens,” J. Opt. Soc. Am. |

2. | J.A. Jordan, P.M. Hirsch, L.B. Lesem, and D.L. Van Rooy, “Kinoform lenses,” Appl. Opt. |

3. | D. Faklis and G.M. Morris, “Broadband imaging with holographic lenses,” Opt. Eng. |

4. | R.A. Hyde, “Eyeglass.1. Very large aperture diffractive telescopes,” Appl. Opt. |

5. | P. Andrés, V. Climent, J. Lancis, G. Mínguez-Vega, E. Tajahuerce, and A.W. Lohmann, “All-incoherent dispersion- compensated optical correlator,” Opt. Lett. |

6. | E. C. Tam, S. Zhou, and M. R. Feldman, “Spatial-light-modulator-based electro-optical imaging system,” Appl. Opt. |

7. | J. A. Davis, D. M. Cottrell, R. A. Lilly, and S. W. Connely, “Multiplexed phase-encoded lenses written on spatial light modulators,” Opt. Lett. |

8. | J. A. Davis, D. M. Cottrell, J. E. Davis, and R. A. Lilly, “Fresnel lens-encoded binary phase-only filters for optical pattern recognition,” Opt. Lett. |

9. | D. M. Cottrell, J. A. Davis, T. R. Hedman, and R. A. Lilly, “Multiple imaging phase-encoded optical elements, written as programmable spatial light modulators,” Appl. Opt. |

10. | E. Carcolé, M.S. Millán, and J. Campos, “Derivation of weighting coefficients for multiplexed phase-diffractive elements,” Opt. Lett. |

11. | M. Hain, W. von Spiegel, M. Schmiedchen, T. Tschudi, and B. Javidi, “3D integral imaging using diffractive Fresnel lens arrays,” Opt. Express |

12. | E. Carcolé, J. Campos, and S. Bosch, “Diffraction theory of Fresnel lenses encoded in low-resolution devices,” Appl. Opt. |

13. | V. Laude, “Twisted-nematic liquid-crystal pixilated active lens,” Opt. Commun. |

14. | I. Moreno, C. Iemmi, A. Márquez, J. Campos, and M. J. Yzuel, “Modulation light efficiency of diffractive lenses displayed in a restricted phase-mostly modulation display,” Appl. Opt. |

15. | V. Arrizón, E. Carreón, and L. A. González, “Self-apodization of low-resolution pixelated lenses,” Appl. Opt. |

16. | A. Márquez, C. Iemmi, J.C. Escalera, J. Campos, S. Ledesma, J. A. Davis, and M.J. Yzuel, “Amplitude apodizers encoded onto Fresnel lenses implemented on a phase-only spatial light modulator,” Appl. Opt. |

17. | A. Márquez, C. Iemmi, J. Campos, and M.J. Yzuel, “Achromatic diffractive lens griten onto a liquid cristal display,” Opt. Lett. |

18. | J. Bescós, J. H. Altamirano, J. Santamaria, and A. Plaza, “Apodizing filters in colour imaging,” J. Optics (Paris) |

19. | J. W. Goodmann, |

**OCIS Codes**

(050.1970) Diffraction and gratings : Diffractive optics

(220.1000) Optical design and fabrication : Aberration compensation

(230.3720) Optical devices : Liquid-crystal devices

(230.6120) Optical devices : Spatial light modulators

**ToC Category:**

Optical Design and Fabrication

**History**

Original Manuscript: March 21, 2006

Revised Manuscript: June 16, 2006

Manuscript Accepted: June 18, 2006

Published: June 26, 2006

**Citation**

María S. Millán, Joaquín Otón, and Elisabet Pérez-Cabré, "Chromatic compensation of programmable Fresnel lenses," Opt. Express **14**, 6226-6242 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-13-6226

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### References

- K. Miyamoto, "The phase Fresnel lens," J. Opt. Soc. Am. 51, 17-20 (1961). [CrossRef]
- J. A. Jordan, P. M. Hirsch, L. B. Lesem, and D. L. Van Rooy, "Kinoform lenses," Appl. Opt. 9, 1883-1887 (1970). [PubMed]
- D. Faklis, G. M. Morris, "Broadband imaging with holographic lenses," Opt. Eng. 28, 592-598 (1989).
- R. A. Hyde, "Eyeglass. 1. Very large aperture diffractive telescopes," Appl. Opt. 38, 4198-4212 (1999). [CrossRef]
- P. Andrés, V. Climent, J. Lancis, G. Mínguez-Vega, E. Tajahuerce, and A. W. Lohmann, "All-incoherent dispersion- compensated optical correlator," Opt. Lett. 24, 1331-1333 (1999). [CrossRef]
- E. C. Tam, S. Zhou, and M. R. Feldman, "Spatial-light-modulator-based electro-optical imaging system," Appl. Opt. 31, 578-580 (1992). [CrossRef] [PubMed]
- J. A. Davis, D. M. Cottrell, R. A. Lilly, and S. W. Connely, "Multiplexed phase-encoded lenses written on spatial light modulators," Opt. Lett. 14, 420-422 (1989). [CrossRef] [PubMed]
- J. A. Davis, D. M. Cottrell, J. E. Davis, and R. A. Lilly, "Fresnel lens-encoded binary phase-only filters for optical pattern recognition," Opt. Lett. 14, 659-661 (1989). [CrossRef] [PubMed]
- D. M. Cottrell, J. A. Davis, T. R. Hedman, and R. A. Lilly, "Multiple imaging phase-encoded optical elements, written as programmable spatial light modulators," Appl. Opt. 29, 2505-2509 (1990). [CrossRef] [PubMed]
- E. Carcolé, M. S. Millán, and J. Campos, "Derivation of weighting coefficients for multiplexed phase-diffractive elements," Opt. Lett. 20, 2360-2362 (1995). [CrossRef] [PubMed]
- M. Hain, W. von Spiegel, M. Schmiedchen, T. Tschudi, and B. Javidi, "3D integral imaging using diffractive Fresnel lens arrays," Opt. Express 13, 315-326 (2005). [CrossRef] [PubMed]
- E. Carcolé, J. Campos, and S. Bosch, "Diffraction theory of Fresnel lenses encoded in low-resolution devices," Appl. Opt. 33, 162-174 (1994). [CrossRef] [PubMed]
- V. Laude, "Twisted-nematic liquid-crystal pixilated active lens," Opt. Commun. 153, 134-152 (1998). [CrossRef]
- I. Moreno, C. Iemmi, A. Márquez, J. Campos, and M. J. Yzuel, "Modulation light efficiency of diffractive lenses displayed in a restricted phase-mostly modulation display," Appl. Opt. 43, 6278-6284 (2004). [CrossRef] [PubMed]
- V. Arrizón, E. Carreón, and L. A. González, "Self-apodization of low-resolution pixelated lenses," Appl. Opt. 38, 5073-5077 (1999). [CrossRef]
- A. Márquez, C. Iemmi, J. C. Escalera, J. Campos, S. Ledesma, J. A. Davis, and M. J. Yzuel, "Amplitude apodizers encoded onto Fresnel lenses implemented on a phase-only spatial light modulator," Appl. Opt. 40, 2316-2322 (2001). [CrossRef]
- A. Márquez, C. Iemmi, J. Campos, and M. J. Yzuel, "Achromatic diffractive lens griten onto a liquid cristal display," Opt. Lett. 31, 392-394 (2006). [CrossRef] [PubMed]
- J. Bescós, J. H. Altamirano, J. Santamaria, and A. Plaza, "Apodizing filters in colour imaging," J. Opt. (Paris) 17, 91-96 (1986). [CrossRef]
- J. W. Goodmann, Introduction to Fourier Optics, 2nd edition (McGraw-Hill, New York, 1996).

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