Real time modulable multifocality through annular optical elements

We present and analyze new multifocal optical elements based on an annular distribution of the transmittance. These elements provide selectable number of foci and can be designed to work between two fixed positions or even to provide extended focal depth. The energy of the foci can be modulated through a single parameter that controls the area of each ring. In our study we analyze the quality of the peaks and also the limit number of foci that can be obtained. The properties shown by these elements make them usable in instrumental optics or in ophthalmic optics, as new intraocular implants, where multifocal elements are required. The implementation has been done on a twisted nematic spatial light modulator, thus allowing real time reconfiguration of the element. ©2008 Optical Society of America OCIS codes: (220.3620) Lens system design; (110.2990) Image formation theory; (110.0110) Imaging systems. References and links 1. A. Kolodziejczyk, S. Bara, Z. Jaroszewicz, and M. Sypek, “The light sword optical element -a new diffraction structure with extended depth of focus,” J. Mod. Opt. 37, 1283-1286 (1990). 2. H. Luo and C. Zhou, “Comparison of superresolution effects with annular phase and amplitude filters,” Appl. Opt. 43, 6242-6247 (2004). 3. J. Monsoriu, W. D. Furlan, P. Andrés, and J. Lancis, “Fractal conical lenses,” Opt. Express 14, 9077–9082 (2006). 4. I. Golub, “Fresnel axicon,” Opt. Lett. 31, 1890-1892 (2006). 5. V. F. Canales, and M. P. Cagigal, “Pupil filter design by using a Bessel functions basis at the image plane,” Opt. Express 14, 10393-10402 (2006). 6. T. R. M. Sales and G. M. Morris, “Diffractive superresolution elements,” J. Opt. Soc. Am. A 14, 1637-1646 (1997). 7. A. Burvall, K. Kolacz, Z. Jaroszewicz, and A. Friberg , “Simple lens axicon,” Appl. Opt. 43, 4838-4844 (2004). 8. A. Flores, M. Wang, and J. J. Yang, “Achromatic hybrid refractive-diffractive lens with extended focal length”, Appl. Opt. 43, 5618-5630 (2004). 9. J. A. Davis, C. S. Tuvey, O. López-Coronado, J. Campos, M. J. Yzuel, and C. Iemmi, “Tailoring the depth of focus for optical imaging systems using a Fourier transform approach,” Opt. Lett. 32, 844–846 (2007). 10. D. Mas, J. Espinosa, J. Perez, and C. Illueca, “Three dimensional analysis of chromatic aberration in diffractive elements with extended depth of focus,” Opt. Express 15, 17842-17854 (2007). 11. 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). 12. J. Leach, G. M. Gibson, M. Padgett, E. Exposito, G. McConell, A. J. Wright, and J. M. Girkin, “Generation of achromatic Bessel beams using a compensated spatial light modulator,” Opt. Express 14, 5581-5587 (2006). 13. C. Iemmi, J. Campos, J. C. Escalera, O. Lopez-Coronado, R. Gimeno and M. J. Yzuel, “Depth of focus increase by multiplexing programmable diffractive lenses,” Opt. Express 14, 10207-10217 (2006). 14. V. F. Canales, J. E. Oti, and M. P. Cagigal, “Three-dimensional control of the focal light intensity distribution by analytically-designed phase masks,” Opt. Commun. 247, 11-18 (2005). 15. P. J. Valle, J. E. Oti, V. F. Canales, and M. P. Cagigal, “Visual axial PSF of diffractive trifocal lenses,” Opt. Express 13, 2782–2792 (2005). 16. D. Mas, J. Pérez, C. Hernández, C. Vázquez, J. J. Miret, and C. Illueca, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. 227, 245-258 (2003). (C) 2008 OSA 31 March 2008 / Vol. 16, No. 7 / OPTICS EXPRESS 5095 #92801 $15.00 USD Received 14 Feb 2008; revised 13 Mar 2008; accepted 13 Mar 2008; published 28 Mar 2008 17. G. Mikula, Z. Jaroszewicz, A. Kolodziejczyk, K. Petelczyc, and M. Sypek, “Imaging with extended focal depth by means of lenses with radial and angular modulation,” Opt. Express 15, 9184-9193 (2007). 18. J. A. Davis, I. Moreno, and P. Tsai, “Polarization Eigenstates for Twisted-Nematic Liquid-Crystal Displays,” Appl. Opt. 37, 937-945 (1998).


Introduction
In the last recent years proposal of amplitude and phase only filters for designing the three dimensional response of an optical system has been a recurrent topic.Among all proposals, radial-symmetric filters have been usually preferred for the ease of its fabrication and analysis [1][2][3][4][5].These filters are use in order to achieve depth of focus, achromatic system, diffractive superresolution elements, etc. [6][7][8][9][10].The use of SLMs to implement diffractive optical elements (DOEs) has been reported to be a useful tool in real-time optical processing [11][12][13].
In some applications, e.g. in Ophthalmic Optics, a number of studies of visual performance for diffractive lenses has been developed [14].Diffractive lenses focus light into several points simultaneously.A nearly periodic modulation of the surface of these lenses is used in order to diffract light into several directions.The two principal applications of diffractive multifocal lenses in Ophthalmology are the contact lenses and the intraocular lenses implanted during cataract surgery to replace the crystalline lens.Recently, Valle, et al., proposed and theoretically analyzed a diffractive trifocal lens design with adjustable add powers and light distribution in the foci [15].
The aim of this paper is to present and analyze new real time multifocal optical elements based on an annular distribution of the transmittance.In Section 2 we theoretically determine the different annular areas performance.We compute the axial intensity as well as the theoretical limit to the number of isolated foci that can be obtained.In Section 3 we give technical details of the experimental setup and results.Special attention is paid to the comparison between the experimental results and the theoretical simulations, finding a good correlation.Finally, in Section 4, we outline the main results of this paper and future works are proposed.

Theoretical considerations
The basic structure of the proposed elements are composed by n concentric rings with n > 2, each ring being characterized by its optical power P m its boundary radial limits a m-1 and a m and an amplitude function A m .Thus, transmittance for the m-ring can be written as: If we illuminate such element with a collimated monochromatic beam of wavelength λ, the obtained on-axis amplitude distribution can be written as sinc(x) being the normalized sinc function and λ 0 the diffractive optical element design wavelength.This expression is similar to the one obtained in [5].From here, a convenient choice of the parameters will provide different focal structures on the optical axis.In general, such structures are used to produce focalization on a determined region of the space [13].In our case, we pretend to cover this range of the space with a series of well-defined equidistant peaks or even with a constant focusing segment.To this end, we find that a convenient choice consists of taking a recurrence relation which, in general, takes the form: with g(m) being a modulation transmittance function of the m-ring.In principle, there is no restriction about the g(p) function, whenever it is continuous in the [1,m-1] interval.Different g(p) functions which determine the power steps between each mask zones can be used.The simplest relation is that resulting from taking g(p)=1, ∀p, which is maintained through the remaining communication.
We may consider that the shape of the whole MFM element is circular with a pupil aperture of 2R p .The area of each ring is directly related to the relative energy between peaks.Since the energy transmitted by the element is proportional to the area of the element, equal area ring would be the trivial solution.Nevertheless, it would be convenient to redistribute energy between the peaks.The simplest form to control the energy ratio between peaks is by redistributing the zone areas.To this end we have introduced an area modulation factor of the form: ( ) By imposing that the outer ring is limited by the size of the element, a n =R p , we deduce that (see appendix): We should notice that h=0 implies that all the rings have the same area.Modifying the size of each ring will have different consequences on the energy distribution around each focus.If we consider the recurrence relation in Eq. ( 3), previous expression is reduced to: The condition 2 0 , m a m ≥ ∀ , provides an upper limit to the parameter h given by (see appendix): And, finally, by substituting Eq. ( 6) in Eq. ( 2) we obtain the axial amplitude distribution of a MFM: Mathematical properties of the sinc function state that as b m tends to zero, the function reaches a delta peak at z m =λ 0 /P m λ.Thus, for small values of b m the axial distribution presents well defined energy peaks, each one corresponding to a different ring of the MFM.The value of b m increases with the number of rings, being m=1 the most unfavourable case in all configurations.
In order to obtain as isolated as possible punctual peaks we must impose that side lobes of two consecutive peaks must not overlap.To this end, it is sufficient condition to impose that the first zero at right side of one peak, coincides with the first zero at left side of the following peak.As sinc(ξ)=0 if ξ ∈ Z , the first zero on the right side of the peak number m will be taken into: Similarly, the first zero to the left of the peak m+1, can be found at: ( ) Therefore, the following condition must be met: Having ( 3) and ( 9) in mind, and after some basic mathematical manipulation, the following condition is obtained: ) Expression ( 13) fulfilled for any m assuming fulfillment for m=1.That is, if it is wanted to have separate foci axis, the maximum number of foci is limited by the relation: This restriction imposes a limitation on the number of zones that a MFM can admit.In Fig. 1 we depict left and right sides of the inequality (14).The limits in the number of foci for two different diameters in the particular case of h=0 are indicated there.Figure 2 represents the theoretical value of the axial intensity for the case of a phase only MFM (A m =1) with the following parameters: P 1 =1/20 mm -1 , P n =1/25 mm -1 , Rp=3.5 mm, h=0 and with a number of rings from n =2 to n=9.According to what we explained above, we can observe there that as we approach to the limit number of foci (n=8), the quality of the peaks is getting worse.For the case beyond the limit, although we obtain a series of peaks, axial aliasing does not allow multifocal implementations.We have also analyzed the effect of the amplitude factor A m in the transmittance function.In Fig. 3, we compare the same phase mask with and without amplitude modulation.In Fig. 3(a), we show the axial irradiance for the only phase case.There, we can see that for the case h=0 the intensity of the peaks diminishes for larger distances from the element.For the last peak, it is observable an amplitude decay in 40%.The equivalent case in Fig. 3(b), with a factor A m =1/P m shows a clear stabilization of the energy of the peaks along the optical axis.Thus, this factor also acts as energy modulator.Different choices in A m have different effects on the axial distribution energy.Nevertheless, optical implementation of amplitude and phase mask is difficult, and it results easier to implement phase only filter.That is why we have introduced the area modulation factor, as we anticipated in Eq. ( 4).Therefore, it is feasible to find a factor h that compensates the intensity of each focus.Taking the same parameters of the previous figures, the upper limit for h results h C =0.00680556 mm.With A m =1, we have represented in Fig. 4 the axial amplitude for a 4 zones MFM for different h values.As it can be observed, changes in the value of this parameter modify the width and relative height of the peaks.One can also see that as the value of h increases, the further peaks gets higher and narrower, and the contrary happens for the closer peaks.In the limit case h=h C -see Eq. ( 7)-first maximum is completely lost.Once we have defined all the parameters in the MFM determination, we have also evaluated the optical quality of the proposed element through the calculation of the MTF at each peak location.To do that, we have evaluated Fresnel patterns following the criteria explained in [16].In Fig. 5 we depict the radial MTF of the patterns obtained with a MFM with n=4 zones, an optical power interval of P 1 =50 D, P 4 =40 D, and a pupil diameter of 7.00 mm, at distances where maxima are expected (z=20.00,21.43, 23.08 and 25.00 mm) and h=h C /9.The presence of multiple foci degrades the MTF with respect to the one of a single lens.Fast decay in very low frequencies is due to the presence of off-axis light corresponding to defocused patterns.Thus, we expect background noise superposed to the image object.This noise can be removed by subtracting a constant value to the whole image.Comparing with other works where multifocality is proposed, we find that MFM presents slightly better response than elements presented in Iemmi, et al., [13] and Mikula, et al. [17].
It is interesting to note that, theoretically, for a large number of rings, all peaks overlap and axial distribution tends to form a continuous focal segment between two well defined distances.In Fig. 6 we present the axial intensity distribution for n=1000, h=h C /3, P 1 =1/20 mm -1 and P n =1/25 mm -1 .provided that the number of rings is high enough.This application is of major importance in ophthalmologic implants, where such elements are currently being designed.

Experimental results
Experimental implementation of MPMs has been done through a twisted nematic spatial light modulator (TN-SLM).Although the Nyquist limit of these modulators limits the optical power of the rings, these systems provide real time manipulation possibilities.Thus, the area of the different rings can be changed at video rate and thus increasing the applications of the MPM masks.The scheme of the experimental setup can be seen in Fig. 7.The core of the experimental setup is a twisted nematic liquid crystal display (TN-LCD) from CRL-Opto, model XGA3.This device has a resolution of 1024 x 768 squared pixels, being the pixel pitch 18x18 μm 2 , the mask is addressed through a standard VGA output.In order to use these transmission modulators as pure phase elements we used the arrangement described by J. A. Davis, et al., [18], who used two quarter wave-plates D1 and D2 and two polarizers P1 and P2.Previously, the angles of these four elements were properly fitted so that the phase modulation provided by the modulator was the most efficient one for used wavelength.With the aim of achieving a relatively large phase modulation, it was used an Ar laser, F, of a wavelength of 488 nm together with a spatial filter, SF, used to expand the beam.The lens L with 25 cm of focal length allow to move the object point in order to obtain the images provided by the different ring on a fixed semitransparent screen S, located at 1.25 m from the modulator.The first polarizer was placed against an iris diaphragm, ID, which limits the diameter of the beam to 13.9 mm.Finally, an 8-bit camera, C, captured the image on S. In order to not saturating the image detected by the camera, a grey filter, GF, was located to the left of the screen.
A four zones (n=4) phase mask was implemented in the modulator.In order to conduct the experimental verification, the chosen zone powers (P 1 =1.82D, P n =0.40 D) followed Eq. ( 3).Those values were limited by the number and size of the pixels of the used modulator.In Fig. 8 we represent the phase mask implemented in the modulator.The radial limits of the four areas were in this case: 2.32, 3.90, 5.43, 6.95 mm and the area parameter used for modulator was h=h C /1.8=0.00211425mm.Repeating the process successively, we obtain the following recurrence relation: By imposing that the outer ring is limited by the size of the element, a n =R p , we deduce that:  expression from which we can finally deduce the equation that provides the zone radius values (Eq.( 5)): In second place, to obtain the Eq. ( 7), if we consider the recurrence relation in Eq. ( 3) with g(m)=1, ∀m, it results: ( )  The condition 2 0 , m a m ≥ ∀ , provides an upper limit to the parameter h given by: ( ) ( )( ) that is the Eq. ( 7),

Fig. 1 .
Fig.1.Limit number of zones in a MFM according to Eq. (14).Bold lines represent the left side of the inequality for two different pupil sizes and the dot line corresponds to the right side of the inequality.

8 )
Making the summation, the above expression is reduced to: