Programmable apodizer to compensate chromatic aberration effects using a liquid crystal spatial light modulator

Programmable apodizers written on a liquid crystal spatial light modulator (LCSLM) offer the possibility of modifying the point spread function (PSF) of an optical system in monochromatic light with a high degree of flexibility. Extension to polychromatic light has to take into account the liquid crystal response dependence on the wavelength. Proper control of the chromatic properties of the LCSLM in combination with the design of the correct apodizer is necessary for this new range of applications. In this paper we report a successful application of a programmable amplitude apodizer illuminated with polychromatic light. We use an axial apodizing filter to compensate the longitudinal secondary axial color (LSAC) effects of a refractive optical system on the polychromatic PSF. The configuration of the LCSLM has been optimized to obtain a good amplitude transmission in polychromatic light. Agreement between experimental and simulated results shows the feasibility of our proposal. ©2005 Optical Society of America OCIS codes: (220.1000) Aberration compensation, (220.1230) Apodization, (230.6120) Spatial light modulators, (230.3720) Liquid-crystal devices. References and links 1. C.S Chung and H.H. Hopkins, “Influence of non-uniform amplitude on PSF,” J. Mod. Opt. 35, 1485-1511 (1988). 2. C. J. R. Sheppard and Z. S. Hegedus, “Axial behavior of pupil plane filters,” J. Opt. Soc. Am. A, 5, 643647 (1988). 3. M.J. Yzuel, J.C. Escalera and J. Campos, “Polychromatic axial behavior of axial apodizing and hyperresolving filters,” Appl. Opt. 29, 1631-1641 (1990). 4. D. M. de Juana, J. E. Oti, V. F. Canales and M. P. Cagigal, “Design of superresolving continuous phase filters,” Opt. Lett. 28, 607-609 (2003). 5. S. Ledesma, J. Campos, J. C. Escalera and M. J. 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Calvo, “A study of the possibility of image optimization by apodization filters with residual aberrations,” Optica Acta 26, 1397-1406 (1979). 11. J. Campos and M.J. Yzuel, “Axial and extra-axial response in aberrated optical systems with apodizers. Optimization of the Strehl ratio,” J. Mod. Opt. 36, 733-749 (1989). 12. J. C. Escalera, M. J. Yzuel and J. Campos, “Influence of amplitude-only filters in optical systems with longitudinal chromatic aberration,” J. Mod. Opt. 38, 1703-1720 (1991). 13. V. G. Chigrinov, Liquid crystal devices: physics and applications (Artech House, 1999). 14. J. Campos, A. Márquez, M. J. Yzuel, J. A. Davis, D. M. Cottrell and I. Moreno, “Fully complex synthetic discriminant functions written onto phase-only modulators,” Appl. Opt. 39, 5965-5970 (2000). 15. O. Gualdrón, J. A. Davis, J. Nicolás, J. Campos and M. J. Yzuel, “Complex encoding of rotation invariant filters onto a single phase-only SLM,” Appl. Opt. 42, 1973-1980 (2003) 16. V. Laude, “Twisted-nematic liquid-crystal pixelated active lens,” Opt. Commun. 153, 134-152 (1998). 17. M. J. Yzuel, J. Campos, A. Márquez, J. C. Escalera, J. A. Davis, C. Iemmi and S. Ledesma, “Inherent apodization of lenses encoded on liquid crystal spatial light modulators,” Appl. Opt. 39, 6034-6039 (2000). 18. J. McOrst, M.D. Sharma, C.J.R. Sheppard, E. West and K. Matsuda, “Hyperresolving phase-only filters with optically-addressable liquid crystal spatial ligth modulators,” Micron 34, 327-332 (2003). 19. R. Dou and M. K. Giles, “Closed-loop adaptive optics system with a liquid crystal television as a phase retarder,” Opt. Lett. 20, 1583-1585 (1995). 20. H.J. Coufal, D. Psaltis and B.T. Sincerbox, eds., Holographic Data Storage (Springer-Verlag, Berlin, 2000) 21. J. A. Davis, J. C. Escalera, J. Campos, A. Márquez, M. J. Yzuel and C. Iemmi, “Programmable axial apodizing and hyperresolving amplitude filters using a liquid crystal spatial light modulator,” Opt. Lett. 24, 628-630 (1999). 22. 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). 23. J. L. de Bougrenet de la Tocnaye and L. Dupont, “Complex amplitude modulation by use of liquid-crystal spatial light modulators,” Appl. Opt. 36, 1730-1741 (1997). 24. A. Márquez, J. Campos, M. J. Yzuel, I. Moreno, J. A. Davis, C. Iemmi, A. Moreno and A. Robert, “Characterization of edge effects in twisted nematic liquid crystal displays,” Opt. Eng. 39, 3301-3307 (2000). 25. A. Márquez, C. Iemmi, I. Moreno, J. A. Davis, J. Campos and M. J. Yzuel, “Quantitative prediction of the modulation behavior of twisted nematic liquid crystal displays based on a simple physical model,” Opt. Eng. 40, 2558-2564 (2001). 26. J. Nicolás, J. Campos and M. J. Yzuel, “Phase and amplitude modulation of elliptic polarization states by non-absorbing anisotropic elements: application to liquid crystal devices,” J. Opt. Soc. Am. A 19, 10131020 (2002). 27. A. Márquez, C. Cazorla, M. J. Yzuel and J. Campos, “Characterization of the retardance of a wave plate to increase the robustness of amplitude-only and phase-only modulations of a liquid crystal display,” J. Mod. Opt. 52, 633-650 (2005).


Introduction
Non-uniform transmission filters in the exit pupil of an optical system may produce different effects in the point-spread function (PSF), both in the transverse plane, generating apodization or hyperresolution [1], and along the axis, varying the depth of focus [2,3].Sheppard and Hegedus introduced the transverse and axial gains to study the influence of transmission filters in planes near the paraxial focus [2].These parameters have been generalized for phase filters when the focalization is produced near the paraxial focus by de Juana et al. [4] and by Ledesma et al. for phase filters in any defocused plane [5].We have also analysed some filters that give an equal axial response while providing a transverse hyperresolving or a transverse apodizing response [6].
In general, the response of optical systems varies with the wavelength.In various papers, we have analysed the behaviour of non-uniform transmission filters on the polychromatic response, focusing both on the irradiance distribution and on the chromaticity [3], and how they can be modified using amplitude-only [3,7] or annular colour filters [8].In general, an optical system can introduce a certain degree of longitudinal chromatic aberration, shifting the best image plane for the various wavelengths, and producing that the chromaticity in the point of maximum irradiance does not coincide with the chromaticity of the illuminant.In demanding broadband imaging systems these chromatic variations are taken into account in the design process and they are corrected [9].Even in this case the corrected optical system exhibits, in general, a residual chromatic aberration.We note that non-uniform transmission filters have demonstrated their capability to compensate different kinds of monochromatic and chromatic aberrations [10][11][12].
Advances in the technology of spatial light modulators (SLM), and particularly in liquidcrystal SLMs (LCSLM), make possible the implementation of programmable diffractive optical elements (DOE) whose characteristics can be changed dynamically.For these purposes, twisted-nematic LCSLMs are especially attractive.They are low-cost and easily available devices due to their extensive use as liquid crystal displays (LCD) [13].LCSLMs have been used in several applications such as image processing [14,15], diffractive optics [16][17][18], adaptive optics [19], and in holographic data storage [20].In previous works we proposed the use of SLMs to generate amplitude apodizers with monochromatic illumination.In particular we have used LCSLMs in two different complex amplitude regimes: amplitudeonly regime [21] and phase-only regime [22].In the latter case, we combined the non-uniform amplitude filter, with a quadratic phase to produce the focusing.Thus, the diffractive element modifies simultaneously the focus location, and the PSF of the optical system.We note that the performance of these programmable DOEs on the LCSLM was analysed for a single wavelength.
In this paper we report the first, to our knowledge, successful application of a programmable amplitude apodizer under polychromatic illumination.Specifically we compensate the longitudinal secondary axial color (LSAC) effects of a refractive optical system.To achieve this goal we design an axial apodizing filter that is displayed on a LCSLM acting as the exit pupil of the system.In Section 2 we consider the longitudinal chromatic aberration of a real refractive optical system to design an axial apodizing filter compensating the effects of this aberration.To sample the visible spectrum we consider three different wavelengths: 633, 514 and 458 nm.The axial apodizing filter allows for a homogeneization of the energy along the axial coordinate without increasing the transverse width of the PSF.In Section 3.1, we calculate the polarizers orientation enabling for an optimum amplitude-mostly regime with the LCSLM, valid for the whole visible range.In Section 3.2 and 3.3 we show respectively the simulated and the experimental results for the PSF obtained with a series of axial apodizing filters displayed onto the LCSLM.We demonstrate the compensation of the LSAC effects, being also able to control the PSF in the transverse plane (apodization or hyperresolution).Both simulated and experimental results agree, thus showing the feasibility of our proposal.The main conclusions of the paper are given in Section 4.

Point-spread function of an optical system with polychromatic illumination
To calculate the PSF for a rotationally symmetric system, when not very high numerical apertures are used, we square the complex amplitude by where λ is the wavelength of the incident beam.The complex amplitude F λ (ρ) is the Bessel transform of the exit pupil function f λ (r),

( ) ( ) ( )
where is the Bessel function of first class and first order.The exit pupil function is given by, ( ) where A is the pupil area, τ(r) is the amplitude pupil transmission, and W(r) is the wave aberration.Hereafter we consider that the system is aberration-free, i.e., W(r) = 0. Note that r and ρ are respectively the normalized radial coordinates at the exit pupil and at the image plane.In order to compute the PSF at defocused planes we introduce the defocus coefficient W 20 , thus we can compute the 3-D PSF, and . ( From these relations we see that non-uniform amplitude transmission filters can be used to modify the PSF of an optical system.To express the calculations with respect to the transverse, s, and to the axial, z, lab coordinates we have to scale the normalized PSF coordinates as follows, with NA the numerical aperture of the system.The analysis of Eqs.(4)(5)(6) shows that when polychromatic illumination is used the PSF can change with the wavelength due to three factors.First, the complex amplitude function in the exit pupil f λ (r) may vary for the different wavelengths because the support media of the applied filter may exhibit a wavelength dependent transmission.Second, the PSF given in Eq. ( 4) depends on the normalized coordinates ρ and W 20 , however if it is written in lab coordinates s and z the size of the PSF will depend on the wavelength according to Eq. (6).Finally, as given in Eq. ( 4), the magnitude of the intensity depends on the wavelength through the factor 1/λ 2 .These three factors must be taken into account when simulating the performance of an optical system under polychromatic illumination.To simulate the experiments we also have to consider the spectral distribution of the light source, and the sensitivity versus wavelength of the detector.

Apodizer design
In this subsection we analyze the chromatic aberration of a real optical system.Then, this information is used to design a non-uniform amplitude filter compensating the aberration effects.The refractive optical system that we consider in this paper is a Zoom Nikon ED, AF Nikkor 80-200 mm, with f number ranging from 2.8-22.We consider the Best Image Plane (BIP) as the plane where the value of the axial intensity is a maximum.We measure the BIP location for this system as a function of the wavelength along the whole visible spectrum.We use 4 different wavelengths obtained from an air-cooled Argon laser (green: 514 nm; blue: 488 nm; violet: 458 nm) and from a He-Ne laser (red: 633 nm).
To measure the BIP location in equal incident conditions the four different beams are aligned so that they can be filtered with the same spatial filter.The pinhole of the spatial filter serves as the point object for the imaging system.In Fig. 1(a) we show the scheme for the imaging system, that consists on the Zoom Nikon ED objective and a LCSLM (Liquid crystal display and polarizers) needed to display the apodizing filters.We note that the exit pupil of the system is located on the LCSLM.A more complete explanation of the optical system will be given in Section 3.3 when we deal with the details of the LCSLM in the experimental setup.At the moment we only need to know that the position of the BIP, measured with respect to the exit pupil of the system, varies with the wavelength as follows: 465 mm for the green and the blue, 466.21 mm for the violet, and 468.16 mm for the red.The system exhibits a longitudinal chromatic aberration plotted in Fig. 1(b), where we consider the origin located at the BIP corresponding to the green and the blue wavelengths.Longitudinal chromatic aberration for the imaging system expressed with respect to the BIP for the green (514 nm) and the blue (488 nm).
We use the previous data to simulate the 3-D PSF.Taking into account the location of the BIP for the various wavelengths with respect to the exit pupil of the system, and considering that the diameter of the exit pupil is 16.52 mm, we obtain that the NA of the system for the various wavelengths is approximately 0.017.We note that the system has a low NA, thus, to simulate the 3-D PSF of the optical system we can use Eq. ( 4) developed in the previous Section.In the simulations we consider an equienergetic source, i.e. equal irradiance along the visible spectrum.In the following, to sample the visible spectrum we use the following three wavelengths: red (633 nm), green (514 nm) and violet (458 nm).To quantify the response of the system in the polychromatic PSF we add the intensities for the three wavelengths and we call it total intensity.
In Fig. 2 we consider that the exit pupil is a clear aperture (no filter).This is the situation corresponding to the measurements given in Fig. 1(b).First, in Fig. 2(a) we show the uniform amplitude transmission in the exit pupil as a function of the normalized coordinate r for the three wavelengths.In Fig. 2(b) the intensity along the axial coordinate (in lab coordinates) is plotted for the three wavelengths.The values of intensity are normalized with respect to the maximum of intensity for the violet.We also plot the total intensity, normalized with respect to its maximum value.In the axial response we see that at the BIP for the green the intensity for the red wavelength is almost zero.The opposite happens at the BIP for the red, where the intensity is almost zero for the violet and for the green.We can see that the addition of the three intensities (total intensity) has decreased significantly at the BIP plane for the red wavelength.Its peak value is at z = 0.97 mm, thus, it is almost coincident with the peak value for the violet.Actually, the factor 1/λ 2 in Eq. ( 4), which increases for shorter wavelengths, makes that the violet has a larger weight in the addition to calculate the total intensity.
In Fig. 2(c) and 2(d) we plot respectively the PSF for the three wavelengths at the BIP for the green and at the BIP for the red, with respect to the transverse coordinate (in lab coordinates).The intensity values are normalized with respect to the axial peak intensity for the violet shown in Fig. 2(b).We also show the total intensity, normalized to its maximum value at each BIP.As given in Fig. 2(b), at the BIP for the green there is almost no red component, and at the BIP for the red there is almost no green component.Besides, the values of the diameters of the PSF up to the first minimum for each of the wavelengths at their focused BIPs are 43.5 µm for the red, 35.3 µm for the green, and 31.5 µm for the violet.This makes that the width of the total intensity is larger at the BIP for the red.
To compensate the effect of the longitudinal secondary axial color (LSAC) given in Fig. 1(b) we propose the use of an axial apodizing filter, i.e. a filter increasing the depth of focus (DOF).The family of the supergaussian filters is well suited for this task [6].The transmittance for the supergaussian filter is described by, where t = r 2 and r is the radial coordinate at the exit pupil.The shape of these filters can be varied by changing the value of the various parameters.The parameter t 0 , corresponds to the square of the radius, i.e. t 0 = r 0 2 , and sets the position for the maximum transmission.The parameter Ω is related with the width of the supergaussian, and α is the degree of the supergaussian and determines the slope of its edges.For α=1 the edges are smooth (similar to a gaussian function).As α increases the supergaussian becomes more similar to a step function.We have selected this family of filters because the parameters t 0 , Ω and α have a direct influence in different quality criteria [6].By changing t 0 , the width of each monochromatic PSF in a transversal plane is modified.When t 0 <0.5 the width is increased while for t 0 >0.5 is decreased.The parameter Ω directly influences the DOF.When Ω decreases, the DOF increases and the total transmitted energy decreases.When α=1 the axial distribution does not present secondary maxima: they appear for higher values of α, as the supergaussian filter becomes more similar to the annular filter.In the following we will see what is the effect of this kind of filters when applied to a system with longitudinal chromatic aberration as the refractive optical system described above.We have done an experimental search of the best values of Ω, α and t 0 to compensate the effects of this LSAC on the polychromatic PSF.This compensation is achieved by increasing the DOF of the different monochromatic responses.In the experimental search we have considered α=1 and we have varied the parameter Ω from 0.26 to 0.08 to improve the DOF (we have stopped at Ω=0.08 since the energy transmitted by the filter was too low).The parameter t 0 was varied from 0 to 1 in steps of 0.1.In the range [0-0.3]we loose transversal resolution.In the range [0.6-1] although the transversal resolution is increased, the secondary maxima were very high.The range [0.3-0.6]presents the best trade-off between resolution and the presence of side lobes.For numerical simulation the values we have selected are Ω=0.17,α=1, and t 0 =0.5.In Fig. 3 we show the simulated results for a supergaussian filter with these parameters.In Fig. 3(a) we plot the amplitude transmission of the filter at the exit pupil where we see that the peak transmission is located at r 0 =0.71 (t 0 =0.5).In these simulations we suppose that the amplitude transmission is the same for all the wavelengths.In Fig. 3(b) we give the axial intensity in lab coordinates.We see that the axial intensity varies slowly for all of the wavelengths, i.e. we have increased the depth of focus.Now, the intensity is not zero for any of the wavelengths at the BIPs for the red and for the green: thus, the chromaticity along the axis and specifically in the region of the maximum of the total intensity gets closer with respect to the original source.In comparison with Fig. 2(b) the maximum of the total intensity has broadened significantly.Its peak value happens at z = 1.07, very close to the peak value for the system with no filter.In Fig. 3(c) and 3(d) we give respectively the intensity at the axial planes z = 0 mm (BIP for the green) and z = 3.16 mm (BIP for the red).We note that, as opposed to Fig. 2(c) and 2(d), now the intensity for the three wavelengths is on the same order of magnitude.If we look at the transverse profile of the total intensity we see that it has the same shape and width at both planes.Thus, the transverse response along the axis is more uniform.The diameter of the PSF is about 30 µm.Taking into account the diameter of the spot for each of the wavelengths given by the system with no filter, this means that the increase in the DOF has not deteriorated the transverse resolution of the system.We have shown in Ref. [6] that the supergaussian filters centered at t 0 =0.5 do not change the transverse resolution in the BIP.This can be seen by comparing in Fig. 2 and 3 the green PSF in the BIP for the green (Fig. 2(c) and 3(c)) and the red PSF in the BIP for the red (Fig. 2(d) and 3(d)).We can observe that the width of these PSFs does not change when this filter is introduced.But as this filter increases the DOF, it has a strong influence on the width of the PSF in defocused planes.This can be seen, for instance, in the blue PSF in the BIP for the red.In the system without filter (Fig. 2(d)) there is not a minimum, but in the system with this filter (Fig. 3(d)) the minimum appears at about 15 µm.Thus, this filter improves the resolution of the defocused PSFs by concentrating the energy around the main maximum.As a consequence, the width of the central maximum of the total intensity is narrower with this filter (compare the black lines in Fig. 2(d) and 3(d)).Then the polychromatic PSF has been narrowed and the resolution has been increased.The simulations have shown that using axial apodizing filters may compensate the longitudinal chromatic aberration effects, thus obtaining a more balanced intensity in the 3-D PSF along the visible spectrum.We are interested in taking advantage of the programmable properties of spatial light modulators to generate these filters.Specifically we plan to use twisted-nematic LCSLMs for this purpose.They have been proven useful in the generation of apodizers with monochromatic light [17,21,22].However, the chromatic dispersion properties of the LCSLMs impose a new challenge, as we need to work with polychromatic light that we study in the following section.

Amplitude-mostly regime using the LCSLM with polychromatic illumination
In twisted nematic LCDs the amplitude and phase modulations are highly coupled [23], especially in newer LCDs where the thickness is decreased to enable the efficient addressing of the larger number of pixels.The reverse-engineering model that we proposed in [24] is able to account for the electro-optical properties of the twisted nematic LCSLMs, allowing the accurate calculation of its complex amplitude transmission as a function of the applied voltage.In this model we take into account that the LC molecules in the vicinity of the faces of the LCSLM (edge layers) have a constraint to reorient with the applied voltage V. Based on this model, together with a novel elliptically polarized light approach, we demonstrated an optimization procedure [25][26] leading to amplitude-only and phase-only modulations for thin LCDs.This optimization was performed for a single wavelength, as in general the chromatic dispersion of the liquid crystal produces a large variation of the complex amplitude transmission as a function of the wavelength.In a recent work [27] we also studied the robustness of the amplitude-only and the phase-only regimes to deviations in the orientation of the external polarization devices.
Our goal is to optimize the modulation of the LCSLM for the whole visible spectrum.In this work we use a Sony liquid crystal panel, model LCX012BL, extracted from a Sony videoprojector model VPL-V500, inserted between two polarizers.We are interested in using the LCSLM in the amplitude-only regime.We extend the optimization procedure proposed in Ref. [25] to broadband spectrum in order to obtain the orientation of the polarizing devices for the optimum amplitude-only configuration for all the wavelengths.For the optimization procedure we must define a figure of merit which takes into account the characteristic properties desired for the amplitude-only regime: the amplitude should have a large dynamic range and should be as linear as possible, the minimum of intensity should be ideally zero to avoid background light, and the phase dynamic range should be zero.With respect to previous works now we have to attain all these properties not at one wavelength but along the whole visible range.In particular, they should be attained simultaneously for the three wavelengths selected to sample the whole visible spectrum.The results obtained for the transmission axis of the input and output polarizers are respectively , ϕ 1 =90.3º and ϕ 2 =89.7º, expressed in each case with respect to the director axis at the input face and at the output face of the LCSLM.
In Fig. 4 we show the theoretical intensity and phase shift predicted from our model together with the experimental measurements for the three wavelengths 458 nm, 514 nm and 633 nm.In the experiments the voltage sent to the SLM is controlled by the gray level of the frame grabber used to display the images.We can see that there is a good agreement between theory and experiment for the three wavelengths.In the intensity plot, Fig. 4(a), we see that the minimum of intensity is very close to zero for the three wavelengths.The maximum value of transmission is attained at a different gray level for each of the three wavelengths.As we need a single-valued calibration curve to display the apodizers, we only use the gray level range from 0 to 210 (as indicated by the dashed line on the figure).
Looking at the phase graph, in Fig. 4(b), we see that the phase dynamic range is about 140º along the gray level range from 0 to 210.Nevertheless, we see that the larger phase variation takes place in the gray level range 0-40 where the intensity values are very low.So, the effective phase range can be considered much lower than 140º.In the gray level range 40-210 the phase variation is about 60º.We will see in the following section that this is not a limiting factor for the present application and we can consider this transmission as an amplitude mostly regime.

Simulated results
When the proposed ideal filter shown in Fig. 3(a) is displayed in the LCSLM, the real transmission will be affected by the modulation response of the LCSLM (Fig. 4), that is different for each wavelength.The amplitude and phase transmission of the filter for the three considered wavelengths is shown in Fig. 5(a) and 5(b) respectively.To connect the desired amplitude values for the filter with the available values from the LCSLM we apply a look-up table adapted for the violet (458 nm), which assigns a specific gray value to be addressed for each amplitude value in the filter.In Fig. 5(a) and 5(b) we can see that in the zones where the ideal filter has a low transmission, the real filter presents for all wavelengths a low amplitude transmission.It is important to obtain a transmission close to zero in all wavelengths because the results of the filter get better as more effectively the light is blocked.We may also note that the variation in the coupled phase transmission is very low, especially if we consider the zone were the amplitude transmission is significant.In Fig. 5(c) and 5(d), the intensity distribution along the axis for the different wavelengths is displayed.In the first case (Fig. 5(c)) we consider that there are no phase variations, while in the second case (Fig. 5(d)) the phase variations of Fig. 5(b) are taken into account.We can see that the results in both figures are almost identical showing the negligible effect of these little phase variations.
If we compare Fig. 5(c) (or Fig. 5(d)) with Fig. 2(b) we clearly validate the increase of the DOF given by the axial apodizing filter.We also see that the intensity values for the red wavelength in Fig. 5(d) are smaller than in the ideal case in Fig. 3(b): the LCSLM transmittance for this wavelength is lower than for the others.Finally, we note that in Fig. 5 the peak value for the total intensity happens at approximately the same plane as in Fig. 2(b): at z=0.99 mm in Fig. 5(c) and at z=0.98 mm in Fig. 5(d).Thus, the axial apodizing filter  Fig. 6.Simulated results for the 3-D PSF considering the exit pupil with the supergaussian filter Ω=0.17, α=1, t 0 =0.5 together with the amplitude transmission and coupled phase given by the LCSLM for each of the wavelengths.Intensity at the BIP (a) for the green (z=0 mm) and (b) for the red (z=3.16mm).Plots are in lab coordinates.
The intensity distributions at the BIPs corresponding to the green (z = 0 mm) and the red (z = 3.16 mm) are shown in Fig. 6(a) and 6(b) respectively.We note that the magnitudes of the transverse responses for the three wavelengths are now more equal than in the system without filter (Fig. 2), especially at the BIP for the red.This causes a shift in the dominant wavelength of the PSF towards the white zone corresponding to the equienergetic illuminant, which is our goal.If we look at the total intensity we also note that the width of the PSF is about 30 µm, as in Fig. 3(c) and 3(d).As we have commented at the end of Section 2, the energy for all the PSF is more concentrated when the filter is applied.Thus the transverse resolution of the polychromatic PSF has increased with respect to the system with no filter (Fig. 2(c) and 2(d)).

Experimental results
The scheme of the experimental setup is shown in Fig. 7.The two laser beams (Ar + laser and He-Ne laser) are aligned to pass through the spatial filter, whose pinhole acts as a point source for the optical system.Behind the optical system, a spatial light modulator system composed by the LCSLM inserted between two polarizers is placed.The optical system forms the image of the pinhole at about 50 cm from the LCSLM.The LCSLM determines the limiting aperture, so it is acting as the exit pupil.To study the PSF given by the optical system we capture it with a CCD camera coupled to a microscope objective to magnify it.To study all the monochromatic PSFs we use a black and white CCD camera, and the system is illuminated by monochromatic light by using the three wavelengths successively.By selecting the specific wavelength the different monochromatic PSFs can be obtained.This is the experimental system used to measure the location of the BIPs for the various wavelengths, represented in Fig. 1(b).To consider an equienergetic illuminant, before each measurement, we control with a radiometer that the intensity incident on the system from each of the wavelengths is equal.By shifting the CCD camera along the axis we register the PSF at different axial planes.The experimental images that we show in this Section correspond to the BIP for the green (z = 0 mm) and the BIP for the red (z = 3.16 mm).In the first three columns of Fig. 8 we show the images obtained with the system without filter for each of the wavelengths, in the BIP for the green (first row) and in the BIP for the red (second row).We see that the intensity for the various wavelengths is very different.We can observe that in the BIP for the green, the maximum intensity of the blue is higher than the maximum intensity of the red.It coincides with the numerical results of Fig. 2(c).In the BIP for the red (second row), the intensity for the green is very low as in Fig. 2(d).The fourth column of Fig. 8 corresponds to a pseudocolored image trying to give a direct impression of the unbalance of the energy between the PSF for the three wavelengths, given in the first three columns in gray level images.It is obtained by using these three first columns as the RGB channels of the image.In the pseudocolored image we clearly observe the unbalance of energy, obtaining a mostly green image at the BIP for the green, and a mostly red image at the BIP for the red.This is in accordance with the simulations shown in Fig. 2(c) and 2(d) for the system with no filter.polychromatic 458 nm 514 nm 633 nm Fig. 8. PSF obtained with the system without filter.The first and the second rows correspond to the BIP for the green (z=0 mm) and for the red (z=3.16mm) respectively.The first three columns are the monochromatic PSF for the 633 nm, 514 nm and 458 nm wavelengths.The last column is a pseudocolored image obtained from the three monochromatic ones.polychromatic 458 nm 514 nm 633 nm Fig. 9. PSF obtained with the supergaussian filter Ω=0.17, α=1, t 0 =0.5.The first and the second rows correspond to the BIP for the green (z=0 mm) and for the red (z=3.16mm) respectively.The first three columns are the monochromatic PSF for the 633 nm, 514 nm and 458 nm wavelengths.The last column is a pseudocolored image obtained from the three monochromatic ones.
In Fig. 9 we see the images obtained with the filter Ω=0.17, α=1, t 0 =0.5 (the corresponding simulations are given in Fig. (6)).We clearly see that the intensity at both BIPs is more homogeneous for the three wavelengths than in the case without filter (Fig. 8).When the monochromatic images are combined in a pseudocolored one (fourth column) we can see that the chromaticity is nearer to the white zone than in the system without filter, especially in the BIP for the red where we see that in shown in Fig. 6.By comparing the pseudocolored PSF in Fig. 8 and 9 we can see that the introduction of this filter improves the resolution in the polychromatic image because, due to the increase of the DOF, it narrows the PSF in defocused planes.With these results we demonstrate the possibility of compensating the effects produced by the LSAC by using a programmable apodizing filter which increases the DOF.Furthermore, comparing the pseudocolored images for the green and for the red BIPs in Fig. 9, we see that the width of both polychromatic PSFs is very similar.This also agrees with Fig. 6, where we could see that the width of the total intensity curve is practically the same in both BIPs.Thus, experimentally we have shown that we can both compensate the chromatic effects and obtain a very uniform transverse response along the axis.
We have studied in paper [6] the effects of supergaussian filters in monochromatic light.The parameter t 0 determines the position of the maximum transmission of the filter.By means of this parameter we can modify the resolution in the transverse plane.When the maximum transmission of the filter appears near the centre of the pupil (small values of t 0 ) the filter is more apodizing, and when the maximum transmittance is near the edge of the pupil (big values of t 0 ), the filter is more hyperresolving.The parameter Ω determines the width of the transmittance function of the filter.By means of this parameter we can control the DOF: the smaller the value of Ω, the higher the DOF.Now we are applying these filters with polychromatic light in order to equalize the chromaticity.We will see that we can control both the resolution in the transverse plane and the chromaticity.Fig. 10.Pseudocolored images of the PSF obtained in the BIP for the green, z=0 mm (first row), and for the red, z=3.16 mm (second row), for the system without filter (column 1), and with supergaussian filters (columns 2, 3 and 4) centered at the same position t 0 =0.4 and for different widths given by Ω.In all cases the degree of the supergaussian is α=1.
In Fig. 10 we show the pseudocolored images obtained at the BIPs for the green and for the red wavelengths for various filters.The parameter α of the supergaussian function (equation 7) is equal to 1 in all cases.The filters have equal position of the maximum, i.e. equal t 0 (t 0 = 0.4), but the width of the filter curve varies, i.e.Ω is different.In this way, the width of the central maximum remains constant in the transverse response, but the DOF varies, decreasing as Ω increases.Thus the filters with less DOF (Ω=0.26)do not homogenize enough the intensity for the various wavelengths, while the filters which produce high DOF (Ω=0.17,Ω=0.08) are able to equalize the intensity for the different wavelengths and they may produce a chromaticity near the white zone corresponding to the equienergetic illuminant.This effect has been shown in Fig. 6 Ω=0.17, t 0 =0.4 Ω=0.26, t 0 =0.4 wavelengths, in the BIP for the red, are more equal than in the system without filter (Fig.

2(d)).
Fig. 11.Pseudocolored images of the PSF obtained in the BIP for the green, z=0 mm (first row), and for the red, z=3.16 mm (second row), for the system without filter (column 1), and with supergaussian filters (columns 2, 3 and 4) of the same width Ω=0.17 and centered in different positions given by t 0 .In all cases the degree of the supergaussian is α=1.
Moreover we can also control, in some range, the resolution in the transverse plane.This can be done in this type of filters with the parameter t 0 .In Fig. 11 we show the pseudocolored images obtained with various filters.The PSFs are represented at the BIPs for the green and the red wavelengths.The parameter α is equal to 1 in all cases.The filters have equal width, i.e. equal Ω (Ω = 0.17), but the position of the maximum is different, i.e. t 0 varies.In all the cases the DOF should be similar, thus the pseudocolored image should have equal chromaticity at the maximum.This can be seen especially in the BIP for the red (second row of Fig. 11) where the chromaticity tends to the white zone.The difference when applying the different filters is the width of the PSF, which is more apodized for t 0 =0.3 than for t 0 =0.5.As we have commented at the end of Section 3, the increase in the DOF produced by these filters, causes that the energy of the PSF in defocused planes is more concentrated, being similar to the width in the BIP.Consequently the width of the polychromatic PSF can be controlled by changing the position of the filter maximum transmittance.With this experiment we demonstrate that we are able not only to compensate the chromatic variations along the axis by increasing the depth of focus, but we can also modify the resolution in the transverse plane.

Conclusions
In this work we have shown the successful application under polychromatic light of a programmable non-uniform amplitude apodizer written onto a LCSLM.The LCSLM is inserted between two polarizers and the orientation of their transmission axis is optimized to obtain amplitude-mostly modulation along the whole visible spectrum.Specifically, we have used an axial apodizing filter displayed on the LCSLM to compensate the effects of the longitudinal secondary axial color of a commercial refractive optical system on the polychromatic PSF.The axial apodizing filter increases the depth of focus of the system, thus equalizing the intensities for the different wavelengths.To control the depth of focus we have used a supergaussian filter.We have analyzed the improvements provided by the filter on the polychromatic PSF and we have found a very good agreement between theoretical and experimental results.Two characteristics of the polychromatic PSF have been improved.On no filter Ω=0.17, t 0 =0.3 Ω=0.17, t 0 =0.4 Ω=0.17, t 0 =0.5 z = 0 mm z = 3.16 mm one hand, we have compensated to a great extent the chromatic variations along the axis due to the LSAC of the refractive optical system.On the other hand, we have increased the transversal resolution of the polychromatic PSF, and this transverse resolution stays constant for a wide range of defocused planes.Furthermore, we have analyzed experimentally the performance under polychromatic light of a wide range of supergaussian filters written onto the LCSLM.The theory says that modifying the width of the supergaussian we control the depth of focus of the PSF, and modifying the position of its maximum transmission we control the resolution in the transverse plane of the PSF.These theoretical properties have been experimentally confirmed under polychromatic illumination using the LCSLM.
In conclusion, we have shown that a LCSLM working in the amplitude-mostly regime is a suitable device to display non-uniform amplitude filters under polychromatic illumination.

Fig. 1 .
Fig. 1.(a) Scheme of the imaging system.The pupil is located behind the optical system.(b)Longitudinal chromatic aberration for the imaging system expressed with respect to the BIP for the green (514 nm) and the blue (488 nm).

Fig. 2 .
Fig. 2. Simulated results for the 3-D PSF considering the exit pupil with no filter.Plots (b), (c) and (d) are in lab coordinates: (a) Amplitude transmission along the exit pupil with no filter.The transmission is equal to one for the three wavelengths; (b) Intensity along the axis.We note the defocus between the various wavelengths; Intensity at the BIP for (c) the green (z=0 mm) and (d) the red (z=3.16mm).

Fig. 3 .
Fig.3.Simulated results for the 3-D PSF considering the exit pupil with the supergaussian filter Ω=0.17, α=1, t 0 =0.5.Plots (b), (c) and (d) are in lab coordinates: (a) Amplitude transmission along the exit pupil; (b) Intensity along the axis.We note that the depth of focus has increased for the various wavelengths; Intensity at the BIP for (c) the green (z=0 mm) and (d) the red (z=3.16mm).

Fig. 4 .
Fig. 4. Modulation response of the LCSLM for the three wavelengths used in this work.Symbols correspond to the experimental measurements and continuous lines to the theoretical predictions.(a) Intensity transmission and (b) phase shift.

Fig. 5 .
Fig.5.Simulated results for the 3-D PSF considering the exit pupil with the supergaussian filter Ω=0.17, α=1, t 0 =0.5 together with the transmission curve of the LCSLM for each of the wavelengths.Plots (c) and (d) are in lab coordinates: (a) Amplitude transmission and (b) coupled phase profile along the exit pupil for the three wavelengths; Intensity along the axis considering that (c) there is no coupled phase, and (d) considering the coupled phase.

Fig. 7 .
Fig.7.Scheme of the imaging set-up with the optical system under analysis.The aperture of the LCSLM (16.52 mm of diameter) determines the exit pupil of the system.P1 and P2 are the polarizers; MO is the microscope objective.The distance d from the exit pupil to the image plane is about 50 cm; the distance d MO is fixed to capture magnified images with the same magnification.

Fig. 6 (
b) the intensities for the three wavelengths are more similar than in Fig.6(a).Then, the experimental results agree with the simulations z (b) where the intensities for the three z = 0 mm z = 3.16 mm no filter Ω=0.08, t 0 =0.4