1. Introduction

The limited range of focus is a significant disadvantage of incoherent imaging systems. The depth of focus is especially important when three-dimensional scenes or three-dimensional objects are imaged. Extending depth of field of optical imaging systems has been a subject of intensive investigations. Application of optical power-absorbing apodizers can increase the depth of focus [1-4]. This method leads to some disadvantages. Apodization limits an effective aperture of the imaging set-up what causes a substantial loss of an incident light energy and reduces significantly a resolution of imaging. The elements of this kind can be only used in a case of sufficiently strong illumination what limits their practical applications. Another possible method of extending depth of field is based on the two-step process [5-8]. The optical imaging system is designed in such a way that the point-spread function (PSF) is insensitive to misfocus while the optical transfer function (OTF) has no regions of zero values within its passband. Then the electronic processing of inverse filtration is used to restore the image formed by the optical system. The same electronic processing restores the image for all values of misfocus because the OTF is insensitive to misfocus. The electronic processing of an optical image seriously limits application of this method. The electronic stage makes impossible imaging in a real time and complicates a construction of an imaging set-up.

The most promising optical elements for imaging with extended depth of focus in real-time seem to be optical elements focusing an incident plane wave into a focal line segment. These elements can be regarded as modified lenses with controlled aberrations. The modification should lead to output images characterized by the possible highest contrast, brightness and sharpness. A fixed point of the focal segment is connected to a proper input plane or an output plane in an imaging process. The optical elements of this kind were intensively studied lately in many papers. The authors of them attempted to solve the problem of extended focal depth by different methods leading to different optical structures as for example axicons [9-13], elements defined by a numerical iterative approach [14-16] or optical diffusers [17]. Some works published in recent years demonstrated usefulness of the axilens in optical systems with a long focal depth [18-21]. The analyzed axilens was based on design proposed by Davidson at al. [22].

All the above mentioned optical elements exhibit the radial symmetry. According to the results presented for imaging set-ups with such elements, it is very difficult to maintain high quality of output images when defocus becomes large [13, 14, 16, 17]. This difficulty lies in a nature of the image formation process. According to the geometrical optics light focused around a point of an optical axis diverges quickly substantially spoiling neighboring focal points of the segment. Therefore a quality of the focal depth is limited, especially when a numerical aperture of an imaging system is large. Some disadvantages of elements with the symmetry of revolution can be overcome by an application of the angular modulation. The angular modulation offers an additional degree of freedom during element’s design and modifies harmful diffraction effects corresponding to focusing.

The aim of this paper is to illustrate usefulness and advantages of optical elements with angular modulation for imaging with extended depth of focus. For this purpose we study and compare imaging properties of three following optical elements: the forward logarithmic axicon [9] (FLA), the axilens [22] (AXL) and the light sword optical element [23] (LSOE). The LSOE is a modified convergent lens with an angular modulation while two others elements represent lenses with radial modulation. The terms lens modulation or lens modification used in this paper means modulation or modification of a phase transmittance of a lens. Particularly, we analyze in detail quality of output images by means of calculated modulation transfer functions (MTFs) corresponding to all elements and different object planes in the imaging set-up. We have intentionally chosen the parameters of an imaging arrangement similar to those described in works dealing with a presbyopia correction [13, 14], since it enables to evaluate usefulness of investigated elements for ophthalmologic applications.

2. Analyzed elements and an imaging set-up.

An assumed imaging set-up includes a thin optical imaging element. An imaging plane is placed in a fixed distance q=20 mm behind the element. The assumed distances between the input objects and the element vary and belong to the range p∊〈25 cm,∞). According to geometrical optics, the thin optical element focusing incident plane wave into a proper light segment makes possible to realize imaging with extended depth of field in the above set-up. Points of the focal segment should be situated from a distance f ₁=18.5 mm up to a distance f ₂=f ₁+Δf=20 mm behind the element in order to cover the assumed range of the object distances. Hence Δf=1.5 mm denotes a length of the focal segment. We designed the FLA, the AXL and the LSOE fulfilling the above conditions. In all cases we have assumed circular apertures of the elements with a radius R=2 mm and the wavelength λ=632.8 nm of a monochromatic illumination corresponding to a He-Ne laser. Because of moderate numerical apertures corresponding to focusing we have used the Fresnel paraxial approximation during our design and numerical calculations.

2.1 Forward logarithmic axicon (FLA)

The element is designed by means of geometrical optics and the principle of energy conservation [9]. According to the ray tracing the FLA exactly focuses light in an assumed focal segment with a uniform intensity distribution. The phase transmittance of the element has the following form:

where a=Δf/R ², k=2π/λ; λ is a wavelength of light and r denotes a radial coordinate in an element plane. Potential abilities of the FLA or its simplified version for imaging with extended depth of focus were confirmed numerically and experimentally [11-13].

2.2 Axilens (AXL)

The AXL was proposed by Davidson at al. [22]. The phase transmittance of the element is defined by the following phase:

Design leading to the above phase function violates the law of the energy conservation [24] therefore the AXL does not focus exactly incident light in an assumed fragment of an optical axis. Nevertheless, latest intensive investigations demonstrated that the AXL can be successfully used as a lens with a large focal depth [18-21]. According to the published results of these investigations the AXL seem to be especially suitable for imaging with extended depth of focus. Therefore we have decided to analyze imaging properties of the AXL and to compare them with those corresponding to the optical element with the angular modulation.

2.3 Light sword optical element (LSOE)

The LSOE is a counterpart of the AXL where the radial modulation of the lens was substituted by an angular one. Preliminary results illustrating abilities of the LSOE with a small numerical aperture for imaging with extended depth of focus were reported elsewhere [25]. The phase defining transmittance of the LSOE is given as follows:

where θ is an azimuthal coordinate in an element’s plane. The LSOE corresponds to a limiting case of the element focusing light into a curve lying on a lateral surface of a cylinder with its radius going to zero [26]. The element described by Eq. (3) forms approximately an assumed focal segment even within the geometrical optics. According to the paraxial ray tracing implemented to polar coordinates points (r, θ) of the infinitesimal angular sector θ=const of the element are connected to the following points (ρ, φ) of the output plane z=f ₁ +Δfθ/2π:

 

Fig. 1. Geometry of focusing by the LSOE. The infinitesimal angular sector of the element focuses an incident plane wave into a segment PP₁ oriented perpendicularly to the sector.

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The geometry of light focusing by the LSOE according to the ray tracing method is shown in Fig. 1. The infinitesimal angular sector of the element corresponding to the angular coordinate θ focuses light into a small line segment PP₁ instead into an assumed point P with coordinates (0,0, f ₁+Δfθ/2π) in the Cartesian coordinate system OXYZ. The length L of the segment is defined by the following relation:

where R denotes a radius of the LSOE’s aperture. The segment PP₁ is oriented perpendicularly to the angular sector, i.e. the segment’s direction is defined by a semi-line corresponding to an angular coordinate θ+π/2. Taking into account this geometrical approach, the assumed parameters R=2 mm, f ₁=18.5 mm, Δf=1.5 mm, λ=632.8 nm, θ∊[0,2π) and Eq. (5) the lengths of the segments PP₁ belong to the range [11.94 µm, 12.90 µm]. For comparison, diameters of the central Airy spots formed by lenses with the same aperture and focal distances correspond to the range [7.14 µm, 7.72 µm].

3. Numerical and experimental results

The assumed optical set-up resembles that used in an model of the human eye [27]. Hence we have analyzed the images of optotypes of Snellen with an angular dimension 5 minutes of arc and the smallest details equal to 1 minute of arc. The Snellen optotypes are commonly used in the ophthalmology for vision acuity examinations. The satisfying recognition of the Snellen optotypes with an angular dimension 5’ corresponds to the standard 20/20 vision acuity. We have selected for our analysis the input objects in a form shown in Fig. 2. The singular object was consisted of four optotypes in a shape of a capital E oriented in different directions. Three parallel strips of the letter E form a fragment of the Ronchi grating. According to the parameters of the assumed optical set-up an ideal image of grating in the output plane corresponds to the fundamental spatial frequency ν=86 lines/mm.

 

Fig. 2. The form of input object used in numerical simulations and experiments. Each letter E has the same angular dimension 5 minutes of arc. Singular strips of the letter have an angular width 1 minute of arc.

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In order to analyze in detail imaging abilities of the studied optical elements we have prepared numerical simulations corresponding to the following 12 different object distances p given in millimeters: 250, 300, 350, 400, 450, 500, 600, 700, 800, 1000, 1500, 2000. Numerical simulations were conducted using a diffractive modeling package working according to the modified convolution approach [28] on a matrix of 4096x4096 points. Making calculations we have assumed that the input object has been illuminated by a monochromatic spatially incoherent light of a He-Ne laser with a wavelength λ=632.8 nm. The columns AXL-s, FLA-s and LSOE-s of the Fig. 3 show the intensity distributions of output images of the Snellen optotypes created by the AXL, the FLA and the LSOE and obtained in numerical simulations. The intensities are presented in a gray scale with 256 different levels. All these images have the same maximal intensity defined by the highest level.

The transfer functions are useful tools for an analysis of imaging properties. Therefore we computed MTFs corresponding to the numerical results presented in Fig. 3. The MTFs shown in Figs. 4-5 were calculated as moduli of Fourier transforms of incoherent point spread functions. The LSOE does not exhibit a symmetry of revolution then in a case of this element we present two cross-sections of MTFs corresponding to perpendicular directions ν_x and ν_y in a frequency domain (ν_x, ν_y). The cross-sections of MTFs showed in Fig. 5 are representative.

 

Fig. 3. Intensity distributions of the output images formed by the AXL, FLA and LSOE for different object distances p given in milimeters. The columns AXL-s, FLA-s and LSOE-s includes results of numerical simulations. The remaining distributions correspond to experimental verifications.

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Fig. 4. MTFs calculated for the AXL and the FLA for different object distances p given in millimeters.

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Fig. 5. MTFs calculated for the LSOE and different object distances p given in millimeters. The column LSOE H corresponds to a direction ν_x in the spatial frequencies domain and the column LSOE V to a direction ν_y.

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The plots for other cross-sections have a similar character.

Then we have verified experimentally results of our numerical simulations. The elements assumed in the simulations were fabricated as binary-phase diffractive structures by electron beam lithography at the Institute of Electronic Materials in Warsaw by the technique described below in more detail.

The glass plate with a conductive layer (ITO) was covered with electron resist by its pulverization on the rotating substrate. The thickness of the resist was controlled by the velocity of rotation and should result in a phase shift equal to π (in our case, compared with the light propagating in air). A substrate prepared in this way was exposed with the help of an electron beam lithography device with variable shaped e-beam system (ZBA-20 by Jenoptik GmbH). After developing, the exposed areas were removed to achieve the binary phase element. The described process enables us to fabricate 0.5-µm-spot-sized structures with accuracy of 0.1 µm in both the x and y directions.

The produced binary-phase diffractive structures have limited diffraction efficiency theoretically equal to 40.5%. The output images are formed only by the (+1) orders of structures. Nevertheless in our imaging system the other orders have negligible influence for output intensity distributions. The (-1) order produces divergent wave-fronts and higher orders correspond to very small diffraction efficiencies. The columns AXL-e, FLA-e and LSOE-e of the Fig. 3 present output images formed by the fabricated elements in an optical set-up and captured by a CCD camera. The object was illuminated by He-Ne laser. In order to obtain spatially incoherent illumination, a rotated ground glass was inserted between the object and a He-Ne laser.

4. Discussion of the obtained results

According to the results shown in Fig. 3, the experiment confirmed numerical simulations. Generally the elements form images with different brightness and contrast. We registered the output images in the experiment using the same illumination intensities making possible to compare a brightness and a contrast of the output images in an optical set-up. Therefore otherwise to simulations the experimental results are counterparts of non-normalized intensity images. Numerical and experimental imaging results shown in Fig. 3 present satisfied coincidence. The structures of images are almost the same. The slight difference is probably caused by imperfections in the fabrication of diffractive structures and their limited diffractive efficiencies. Good agreement between numerical simulations and experiments justifies our assumption about the used paraxial approach.

Characteristic features of output images can be explained using the MTFs shown in Fig. 4-5. Only the LSOE forms recognizable images of Snellen optotypes for all object distances. MTFs corresponding to the LSOE have not zeros for spatial frequencies smaller than 100 lines/mm. The nonzero ranges of the MTFs are substantially wider than these corresponding to the FLA and the AXL. Some MTFs of the FLA and the AXL have zeros around the characteristic spatial frequency of the optotypes ν=86 lines/mm. Therefore the relative output images are completely blurred for object distances 400 mm, 600mm in a case of the FLA and 500mm, 1000mm in a case of the AXL. Additionally MTFs corresponding to the FLA and the AXL exhibit narrow maxima around the zero spatial frequency. Moreover these MTFs have smaller values for higher frequencies than in a case of the LSOE. Hence generally the images created by the FLA and the AXL demonstrate lower contrasts. This effect is especially recognizable in a case of the AXL where the MTFs decrease the most rapidly from the central maximum. The images created by the FLA and the AXL are considerably blurred when object distances become longer than 500 mm. The blurs correspond to rapid oscillations appearing in MTFs plots. These oscillations may cause substantial contrast inversions of output images.

The obtained numerical and experimental results prove superiority of the LSOE for imaging with extended depth of focus in the analyzed set-up. Generally the LSOE forms images with better resolution, contrast and higher brightness than those created by the FLA and the AXL. Surprisingly good abilities of the LSOE for imaging with extended depth of field are probably connected with the flow of energy during focusing. According to ray tracing shown in Fig. 1 the focusing process has an off-axis character. The maximum intensity of the focal spot lies outside the optical axis OZ and waves around it [25]. This phenomenon causes a displacement of the output images. Taking into account the assumed parameters corresponding to our simulations and experiments, the angular displacement is equal approximately to 2 minutes of arc. The flow of energy changes its main direction during focusing and positions of output images rotate around the optical axis. Then the effect of mutual disturbance between neighboring images corresponding to different focal lengths or different object distances is less harmful than in cases of the FLA and the AXL where the flow of energy during focusing has the same main direction along the optical axis.

5. Conclusion

We have analyzed in detail abilities of lenses with the radial and angular modulation for imaging with extended depth of focus. We have chosen for our analysis the imaging set-up similar to that representing a model of the human eye and three following optical elements: the forward logarithmic axicon and the axilens exhibiting radial modulation of the lens transmittance as well as the light sword optical element representing a lens with angular modulation. The structures of output images in numerical simulations and experiments were discussed by means of the calculated MTFs. The LSOE contradictory to the FLA and the AXL forms well recognizable output images for a wide range of object distances. The angular modulation of the lens transmittance used in a case of the LSOE modifies the flow of light energy during focusing what improves quality of imaging.

Nevertheless the LSOE exhibits some disadvantages. It forms slightly stretched focal spots without radial symmetry [25] what generates characteristic blur of output images. These images waves around the optical axis. The above disadvantages do not seriously limit usefulness of the LSOE for many imaging applications. Moreover the LSOE is only one relatively simple example of the lens with an angular modulation of a transmittance. Imaging abilities of the LSOE can be probably substantially improved by optimization of its design. Analytical modification can be realized by a substitution of the linear function of θ in a denominator of the phase transmittance given in Eq. (3) by an another, properly selected angular function. According to the lately published works especially promising seem to be iterative optimization methods [14-16].

The presented results give evidence that modified lenses with angular modulation of phase transmittances can be very useful tools for imaging with extended focal depth. Generally the angular modulation offers the additional degree of freedom in a design process. This modulation changes the flow of energy during focusing what can improve an imaging quality. According to the presented results angularly modified lenses of moderate numerical apertures can be at least used successfully in machine vision and ophthalmologic applications.