## Abstract

In this paper we present a new approach of all-optical extended depth of focus providing two (or more) discrete ranges of focused imaging for close as well as far ranges. The fact that the extended depth of focus is not continuous allows obtaining improved contrast in the two (or more) axial regions of extended depth of focus. The design is aimed for the cell phone camera applications where dual range extended depth of focus can allow simultaneous reading of business cards at very short distance as well as very high contrasted imaging at far range.

© 2007 Optical Society of America

## 1. Introduction

Extending the depth of focus for imaging systems was widely explored during the recent years and a large variety of technologies were developed. Some approaches involve digital post processing [1–5], aperture apodization by absorptive mask [6–10] or diffraction optical phase elements such as multi focal lenses or spatially dense distributions that suffer from significant divergence of energy into regions that are not the regions of interest [11–13]. Other interesting approaches included tailoring the modulation transfer functions with high focal depth [14] and usage of logarithmic asphere lenses [15].

In previous papers [16,17] we have presented an all-optical way for extending the depth of focus (EDOF) by attaching a phase-affecting element. The attached element is constructed from a binary phase pattern with spatially low frequency transitions that codes the entrance pupil of the lens. The presented approach had several important features: since this optical element contains low spatial frequencies, it is not sensitive to wavelengths and dispersion (as other diffractive optical elements do) and it does not scatter energy outside the relevant region of the field of view. In addition its fabrication is simple and cheap. The optical element is a phase-only element and thus it does not cause apodization resulting with high-energetic efficiency. In addition, the phase element has no spatial high frequencies and thus there is no energy loss due to diffraction orders, resulting with an energetic efficiency that is close to 100%. Since the optical element does not require digital post processing it is adequate for ophthalmic applications as well.

The approach presented in this paper is based on the mathematical as well as logic concepts discussed in Ref. [17] and thus it exhibits all the previously specified advantages. However, the performance of the method that is discussed in this continuation paper is significantly improved. The reason for that stems from new trade-off definitions that can be made in specific applications (such as the cellular cameras market). The payment in the trade-offs significantly improves the contrast and the performance in the close and far range imaging.

Current technology in cell phone cameras provide imaging at ranges starting from a typical minimum object distance of 50cm up to infinity. Yet for the purpose of reading business cards and barcodes, it is required to allow imaging at close ranges of few cm as well. Thus, we will design a dual range system rather than a continuous range extended depth of focus. This is the main difference in comparison to Refs. [16,17]. The requirement is imaging in close range (e.g. 12.5cm) in order to allow reading of business cards and perfect quality of imaging in far field, while there is no need for focused imaging within all distances in between (30cm–50cm). We will use this apriori information in our mathematical constraints. The mathematical requirement forced in the formulation of the element is such that the axial super resolution has two (dual) ranges. The converged solution is similar to the one that was obtained in Refs. [16.17]. It is still a phase mask constructed of transparent areas and binary phase lines (e.g. grid) and/or one or more binary phase circles, modulating the entrance pupil of the imaging lens [the Coherent Transfer Function (CTF) plane].

The principle of operation is also the same i.e. under spatially incoherent illumination the out of focus effect is expressed as a quadratic phase distortion that is added to the Optical Transfer Function (OTF). The positions of the binary phase transitions in the element are appropriately selected to generate invariance to quadratic phase distortions that contribute to defocusing.

The position of binary phase transitions is computed using an iterative algorithm. During these iterations, various combinations of positioning the phase lines and/or circles are examined. The algorithm converges eventually to those spatial locations that provide maximal contrast of the OTF under a set of chosen out of focus axial regions. The meaning of OTF’s contrast optimization is actually having the out of focused OTF bounded as much as possible away from zero, however this time this constraint is applied in two (or more) separated longitudinal (axial) ranges.

The optical performance of the element is similar to a multi-focal lens but the element itself has no optical power and it contains only low spatial frequencies.

The element presented here was fabricated and tested experimentally. Indeed it provides significant improvement of the closest possible in-focus near field as well as very high image quality in the far field (infinity) imaging.

Section 2 presents the theoretical background and the mathematical derivation. In section 3 one may see the recently obtained experimental results. The paper is concluded in section 4.

## 2. Theoretical derivation

The mathematical derivation that we present here is based on the presentation we provided in Ref. [17]. In that paper we obtained that the OTF, which is the auto-correlation of the CTF equals to:

where *a _{n}* are binary coefficients equal either to zero or to a certain phase modulation depth:

*a*=(

_{n}*0, Δϕ*) of the phase-only element that we design.

*Δϕ*is the phase depth of modulation.

*Δx*represents the spatial segments of the element.

*λ*is the wavelength and

*µ*is the coordinate of the OTF plane.

*P*is the aperture of the lens having coordinates of

*x*(the plane of the CTF) and

*Z*is the distance between the imaging lens and the sensor. Since we do not want to create a diffractive optical element, i.e. spatial high frequencies and periodicity (since we wish to have no wavelength dependence) we force

_{i}*Δx≫λ*. Figure 1(a) shows an imaging system with a phase-element that is attached to the imaging lens, an object at distance

*Z*from the lens and a detection array that is positioned at distance

_{o}*Z*from the imaging lens. The OTF function that is given in Eq. (1) can be used to characterize the optical system that is depicted in Fig. 1(a).

_{i}Let us start with a general explanation and then show the relevant mathematical formulation. In our formulation, as well as the experimental analysis, we aimed to design a dual range element. However the same formulation can be easily extended to any number of discrete focused axial regions.

Generally speaking the formulation for the optimization criteria should be as follows: compute a phase-only element that will provide maximum for the minimal value of the OTF within the desired spectral (spatial spectrum) region of interest. The OTF that is being computed is composed from two terms. The first term is where the OTF exhibits strong defocusing deformation with large parameter *W _{m}*. The second term has a small defocusing deformation (denoted as

*βW*in Fig. 1(b) where

_{m}*β≪1*). The coefficient

*W*determines the severity of the defocusing error and is defined as:

_{m}where *ψ* is a phase factor represents the severity of out of focus:

Parameter *2b* denotes the diameter of the lens, *λ* is the wavelength, *Z _{o}* is the distance between the imaging lens and the object,

*Z*the distance between the imaging lens and the sensor and

_{i}*F*is the focal length. When imaging condition is fulfilled,

*ψ*is zeroed, since:

The optical alignment will be such that regular in-focus plane that satisfied imaging condition without an element will not be in-focus anymore. The requirement for improved resolution for small and large *W _{m}* will now determine two new in-focus regions that are positioned before and after the previous plane (see Fig. 1(b)).

Note that the requirement in this newly presented approach is different from the derivation that we performed in Ref. [17] where the two terms of optimization included the defocused as well as the in-focus MTF. Here both terms are of defocused MTF. The fact that here we replaced the second term with slightly defocused MTF is due to the fact that we wish to have focusing in two regions and a defocused image in between those regions.

Obviously if instead of dual focused range, the aim is to converge into multiple set of focused regions, then the optimization expression should contain larger number of terms (a summation of terms).

The result obtained in Ref. [17] was of an annular-like shape with a phase *Δϕ=π/2*, which stems from the constraint of a continuous focused region. That element was able to cancel the sign inversions of the quadratic phase that was generated in the aperture plane due to defocusing. Correction was obtained by the addition of proper phase to the spatial frequencies where inversion appeared. Although it seems like in order to cancel sign inversions one has to add a phase of *π*, the best solution in this case was obtained with a phase of *π/2*. The reason for a phase of *π/2* was due to the requirement for a continuous focused region. In a continuous focused region solution, one has to cancel the inversions of quadratic phase due to defocusing while not creating new sign inversion to the in-focus plane. Thus, a phase of *Δϕ=π/2* contributes equally both to the defocused planes as well as the in-focus plane.

When we wish to have dual focusing system, the optimal phase will be close to *π*, since now we wish to cancel only the sign inversions of the quadratic defocusing phase in the two regions that are located before and after the original in-focus plane. In this approach, the previously in-focus plane will now be distorted. The fact that two non-continuous regions are required, improves the contrast performance in each of the two regions in comparison to the case of Ref. [17] where a continuous requirement was applied. A schematic simplification of the above explanation is brought in Fig. 1(c).

Note that in the case of two focused regions, the phase-only element will have one annular shape, dealing with the sign inversions of certain quadratic phase corresponding to a certain defocusing plane. When multiple, non continuous focused regions are required the phase-only element will have a plurality of annular shapes, corresponding to spectral (spatial spectrum) position of sign inversions (i.e. positions in the coordinates set of the aperture plane) that are due to different quadratic phase factors. Each annular will correct a distinct sign inversion.

Mathematically, the above dual-range and multi-range criteria corresponds to an approach where the goal of the designer is to maximize the minimal value of the MTF within a desired dual- or multi- axial region, for a certain range of spatial frequencies. Maximization is performed by varying parameters of the phase mask and of the imaging system, i.e. varying the number of phase transitions, their modulation depths, shape size and coordinates of phase segments (i.e. varying the phase mask layout), the aperture and the focal distance of the lens and the distance to the photo detection array (PDA). Maximization may only be partial, in accordance with one or more of the above criteria. Such an approach can be formalized for example with the help of an indicator:

In Eq. (5), *H(µx,Z0)* is the OTF that primarily depends on object’s distance *Z _{o}* and spatial frequency

*µ*(without a loss of generality, a one dimensional case is considered). In this example, the considered spatial frequencies are those being smaller than maximal spatial frequency of the photo detecting surface

_{x}*µ*; the region of interest is dual and constructed of separate axial regions of interest

_{d}*R*and

_{1}*R*. The OTF is also dependent on the parameters of the optical system and phase mask; varying these parameters allows optimizing the indicator of Eq. (5).

_{2}The indicator of Eq. (5) may be generalized, so as to take into account the eventual decline in OTF due to the growth of spatial frequency. The indicator can take a form:

where *K(µ _{x})* is a weight function corresponding to utility of each spatial frequency. Higher spatial frequencies may be assigned lower utility.

Indicators of Eq. (5) may be generalized differently. Considering, for example, an imaging system application, in which a contrast in the near range is tailored for recognition of documents, while a contrast in the far range is tailored for landscape photographing, it may be noted, that targets from regions of *R _{1}* and

*R*may be decomposed into different spatial frequencies ranges. Then indicator of Eq. (5) may take a form:

_{2}where *K(µ _{x},Z_{o})* is a utility function dependent both on region of interest and spatial frequency.

Optimization may be aimed at maximization of a minimum average contrast for various spatial frequencies:

The latter form can be reduced to a form:

Here *E _{1}* and

*E*are boundaries (edges) of the near region and

_{2}*K*and

_{1}*K*are weights assigned to these boundaries. The corresponding far region in the first order of approximation is symmetrical to the near region, relatively to the in-focus plane of the lens (the plane with zero defocusing). Here, symmetry is based on defocusing and not the length of the regions. Optimization of Eq. (9) is similar to Eq. (8), but requires less computation. The value maximized in Eq. (9) is composed of two terms with different degrees of defocusing and phase factors.

_{2}This approximate symmetry is illustrated in Fig. 1(b). A plane * P_{i}* is an in-focus plane of the lens, for the selected PDA distance (i.e. distance between the lensing section and the PDA). From imaged space, the magnitude of geometrical defocusing is the largest either at the left edge of near region or at the right edge of far region. At the right edge of the near region and at the left edge of the far region the geometrical defocusing is the smallest. It should be noted that this smallest defocusing is not zero, as it would be in an EDOF application, but

*βW*, where

_{m}*β*is a number between zero and one. This is due to the effect of the phase mask. The effect of phase mask thus relates to overextending depth of focus.

Note that mathematically, in case that multiple set of *N* focused axial regions is to be generated the optimization expression takes the form of:

In the performed simulations *Δx* was close to 1/8 of the lens aperture. The numerical optimization of the mathematical condition of Eq. (9) was done using Zemax. The parameter *β* was chosen such that the obtained focusing ranges will be 12.5cm–25cm and 50cm to infinity for camera with F-number of 3 and focal length of 4.8mm. Numerical simulations of the new design reveal the dual region behavior of the designed element.

Figures 2(a) and 2(b) show the through focus MTF plots where in Fig. 2(a) the object is placed at a distance of 15cm from the camera and in Fig. 2(b) the object is simulated at infinity. The obtained contrast is maximized on the desired working regions (12.5–25cm and 50cm-infinity) whereas there is some contrast degradation in the non constrained axial region of 30–50cm. The new approach allowed us to obtain larger working range in comparison to the previously presented approach of Ref. [17]. The simulation of the through focus MTF was performed for high spatial frequency of 60 cycles/mm.

Additionally, it is possible to use non-equal weighing functions, *K _{1}* and

*K*of Eq. (9), in order to obtain an unbalanced response. Examples of two different asymmetric systems are shown below. Figure 3(a) shows the through focus MTF of a system having better near field response compared to the far field response. Figure 3(b) shows the through focus MTF of a system having better far field response compared to the near field response. The simulation of the through focus MTF was performed for spatial frequency of 60 cycles/mm.

_{2}As previously mentioned the same mathematical approach can be used for plurality of discrete focusing regions. In Fig. 4 we show simulations of an optical system with three focal regions around 13cm, 25cm and infinity as presented in the through focus OTF plots. The simulation of the through focus MTF was performed for spatial frequency of 150 cycles/mm.

## 3. Experimental results

We have fabricated the dual-region element that was designed and simulated in Fig. 2. In the experiment we verified the dual-region EDOF element. The dual-range system was capable of providing simultaneous imaging for far range objects (infinity) with performance that is comparable to a regular lens focused at infinity and at the same time providing a close range imaging around 12.5cm. The overall range of focused images was obtained for objects at 12.5cm–25cm and 50cm to infinity simultaneously. The element had two phase-discontinuities in its cross section and the design as well as the experiment was performed for cell phone camera with focal length of 4.8mm and F-number of 3.

A schematic sketch of the experimental setup is given in Fig. 5(a). The binary phase element was placed in the entrance plane of the imaging lens. Figure 5(b) presents the image captured when an object was placed at 15cm and yet there is scenery at far field about 7km away. One may see that the near object is in high contrast and the label of “made in France” can be read easily although the letters are very small (font size of 1mm). The far field objects are imaged with good contrast as well. Figure 5(c) shows an experiment where a business card is placed at range of 20cm from the camera. Once again the image contains both close and far range objects. The left image is the result obtained with the dual-region phase mask and the right image is from a reference camera with no element attached. One may see that the image from the dual-range system displays simultaneously the business card as well as the far field objects with good contrast whereas the regular system is capable of providing a good contrast only for the far field objects.

Figure 5(d) shows the contrast behavior along the Z-axis (obtained with the EDOF element). The image contains targets of line-widths of 0.5,1,2,4 and 8pt that are positioned at distances of 10,20,40,80 and 160cm respectively. The line-widths are adjusted according to the varying optical magnification so that they are imaged with equal size on the sensor. One can see that although the lines are imaged on sensor with same width, their contrast is not equal. The line positioned at 20cm exhibits high contrast whereas the line at 40cm has some contrast degradation, which is increased again when moving along the Z-axis towards infinity. This dual region contrast behavior is with full agreement to the through focus plots given in Figs 2(a) and 2(b). The experiment of Fig. 5(d) verified the theoretical prediction for the dual-region design.

## 4. Conclusions

In this paper we have demonstrated a novel approach that is based upon the technique of Refs. [16,17], that optimizes the extended depth of focus performance in two (or more) ranges.

Numerically as well as experimentally we have demonstrated improved performance in two ranges of 12.5cm–25cm and 50cm to infinity. The fact that in the optimization process we allowed contrast degradation in the range of 30cm–50cm, resulted with significant enhancement of the performance at close and far ranges. The extended depths of focus images are obtained in an all-optical manner. The phase-element is binary and contains only low spatial frequencies and thus it is simple to fabricate, it does not exhibit chromatic aberrations and its energetic efficiency is high.

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