Imaging with extended focal depth by means of the refractive light sword optical element

The paper presents first experiments with a refractive light sword optical element (LSOE). A refractive version of the LSOE was prepared in photoresist by gray scale photolithography. Then we examined chromatic aberrations of the produced element and compared them with those corresponding to two different lenses. For this purpose we performed two experiments, the first one where white light illumination was used and the latter one by the help of monochromatic illumination with three different wavelengths. The obtained results lead to the conclusion that the refractive LSOE does not exhibit significant chromatic aberrations and can be successfully used for imaging with extended depth of focus in polychromatic illumination. ©2008 Optical Society of America OCIS codes: (080.2740) Geometric optics: Geometric optical design; (110.2990) Imaging systems: Image formation theory; (110.6890) Imaging systems: Three-dimensional image processing; (220.3620) Optical design and fabrication: Lens system design. References and links 1. M. Mino and Y. Okano, “Improvement in the optical transfer function of a defocused optical system through the use of shaded apertures,” Appl. Opt. 10, 2219-2225 (1971). 2. J. Ojeda-Castaneda, P. Andres, and A. Diaz, “Annular apodizers for low sensitivity to defocus and to spherical aberration,” Opt. Lett. 11, 487-489 (1986). 3. J. Ojeda-Castaneda, E. Tepichin, and A. Diaz, “Arbitrary high focal depth with a quasioptimum real and positive transmittance apodizer,” Appl. 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Introduction
The increased depth of focus can be a valuable advantage of incoherent imaging systems in many applications. The extended focal depth is especially important when three-dimensional scenes or three-dimensional objects should be properly imaged. Hence one can note in recent years a growing number of scientific publications devoted to this subject. The authors of these works attempted to solve the problem of the extended depth of focus in different ways and by help of different methods. Amplitude apodization can increase the focal depth [1][2][3][4] but this approach causes a substantial loss of the incident light energy and diminishes the effective aperture of the system limiting the resolution of imaging. The two step process based on the use of a proper phase filter in an optical set-up and a further electronic processing was applied successfully in microscopy [5][6][7][8][9]. Nevertheless the second electronic stage of this approach makes impossible to use the method for imaging in real time.
Especially well suited for real time imaging with extended focal depth seem to be optical elements focusing light into line focal segments. From a theoretical point of view they are equivalent to lenses with properly controlled aberrations. A focus stretched in the form of a line segment hereby makes possible to connect a certain range of input object planes with a certain range of output image planes according to the lens equation. The elements of this kind can be designed in different ways including geometrical approach [10][11][12][13][14][15], the ray tracing method completed by the energy conservation principle [16][17][18][19], iterative methods [20][21][22] or stochastic methods [23,24]. All these elements besides the structures obtained by stochastic methods exhibit symmetry of revolution, i.e., their transmittances depend only on the radial coordinate in the polar system. Lately we have presented numerical and experimental results concerning the optical properties of the light sword optical element (LSOE) where additionally an angular modulation of a transmittance was implemented [25]. Our work pointed out to the superiority of the diffractive LSOE over the other diffractive elements with radial modulation of the transmittance only, at least in an optical set-up simulating a human eye. Nevertheless diffractive elements including those focusing light in a line segment suffer from substantial chromatic aberrations. This effect reveals especially in a case of the diffractive LSOE forming at the same plane well separated output focal spots corresponding to different wavelengths [26]. Strong chromatic aberrations of the diffractive LSOE limit practically its imaging applications to monochromatic cases. Hence the only reasonable solution to this problem when polychromatic or white light illumination is used seems to be the substitution of the diffractive element by its refractive counterpart.
This paper presents first imaging experiments with the refractive LSOE. Our goal was an experimental study of chromatic aberrations of this element in an imaging set-up with white light illumination. For a better illustration we have compared the imaging properties of the refractive LSOE with those corresponding to two lenses with a similar numerical aperture. In the Section 2 we give the theoretical background describing the design of the LSOE and its focusing properties. Section 3 presents the experimental results. The last Section discusses the results of the experiments of the refractive LSOE.

Theoretical background
The LSOE is a thin optical element, whose phase transmittance has the following form in the polar coordinate system: where r, θ are respectively the radial and azimuthal coordinates; k=2π/λ and λ is the wavelength of light. For a fixed value of the angular argument θ the phase transmittance Φ (r) is equivalent to that of a thin lens with focal length f 1 +Δf θ /2π. Therefore in a first approximation the LSOE focuses an incident plane wave into a focal segment of width then the segment is stretched from a distance f 1 to a distance f 1 +Δf behind the LSOE plane. A more detailed analysis leads to a conclusion that the element described by Eq. (1) 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, 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 PP 1 instead into an assumed point P with coordinates in the Cartesian coordinate system OXYZ. A superiority of the diffractive LSOE for imaging with extended focal depth over lenses, axilenses and axicons was already proven numerically and experimentally [25,27,28]. A new concept of angular modulation of transmittance seems to be effective. 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 illustrated 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 [28]. This phenomenon causes a displacement of the output images. 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 elements with symmetry of revolution where the flow of energy during focusing has the same main direction along the optical axis. In consequence the output focal spots, point spread functions and modulated transfer functions corresponding to the LSOE almost conserve their shape and character for a wide range of defocusing parameters [25,28]. Therefore the LSOE with an angular modulation of its transmittance can exhibit its superiority for imaging with extended depth of focus over optical elements with symmetry of revolution where controlled spherical aberration is used. Nevertheless, the increased depth of focus comes at a cost. The LSOE exhibits some disadvantages. It forms slightly stretched focal spots without radial symmetry [28] what generates characteristic blur of output images. These images wave around the optical axis. The above disadvantages do not seriously limit usefulness of the LSOE for many imaging applications. According to the already presented results the diffractive LSOEs of moderate numerical apertures can be at least used successfully in machine vision and ophthalmologic applications [25]. Unfortunately, a diffractive LSOE can be used for imaging purposes only in monochromatic illumination what substantially limits possible applications especially in ophthalmology. As was shown lately the off-axis focusing corresponding to the LSOE generates strong chromatic aberrations [26]. The focal spots formed by different wavelengths are considerably mutually shifted in the same output plane. Then one can expect a poor quality of imaging when polychromatic or white light will be used. In order to avoid this serious drawback the LSOE can be fabricated in a thin refractive version. Chromatic aberrations of refractive elements are one order of magnitude smaller than diffractive ones or even more. Assuming that the refractive index is independent on the light frequency, the form of the phase transmittance given in Eq. (1) remains the same for the different wavelengths λ of visible light.

Experiment
The refractive elements were produced by gray scale photolithography of sol-gel photoresist deposited onto 1 mm thick circular glass substrates of 25.4 mm diameter. In this way we have fabricated two elements, the LSOE and a spherical lens for a comparison in an imaging set-up. The maximum height of the refractive profile of our elements was 20 μ m (peak-to-valley). Fig. 2 presents interferograms of the refractive LSOE and the refractive lens in a Mach-Zender interferometer. According to these interferograms the refractive structures exhibit a certain amount of spherical aberration caused by imperfections in the fabrication process, mainly due to the limited number of gray levels printable on gray scale mask, which was equal to 256. Both elements had a circular aperture of diameter 4.7 mm. Then we measured the focal parameters of the LSOE and the lens in an optical set-up. We obtained a focal length of the lens equal to f 2 =28.5 cm and the parameters f 1 =30 cm, Δ f=15 cm corresponding to the LSOE. Afterwards we performed two experiments illustrating the chromatic properties of the refractive LSOE. In all cases the input object consisted of four Snellen optotypes. Each optotype had the form of a properly oriented capital E with 2 mm height and a smallest detail of 0,4 mm. The input object was placed at a fixed distance 70 cm in front of the imaging element and the distance of the image plane behind the element was changed. According to the lens equation and the parameters given above the fabricated LSOE was designed to form output images lying in planes distant from 52.5 cm up to 126 cm what corresponds to defocusing 1.1 D measured in optical power.
During the first experiment we used as input objects a positive and a negative of the Snellen optotypes illuminated by the white light emitted by an ordinary filament lamp of power 35 W. Additionally we compared the imaging properties of the fabricated elements with those of a professional Carl Zeiss objective of focal length f 2 =30 cm. In order to have comparable numerical apertures we reduced the circular aperture of the objective to a diameter of 5 mm. The experimental results for different defocusing parameters are presented in Fig. 3 in form of color photographs. The defocusing, given in optical power, was calculated separately for each element according to the formula 1/f-1/p-1/q, where f, p, q denote the focal length of the element, the object distance and the image distance respectively. In case of the LSOE we assumed that f=f 1 . The experiment illustrates well the achromatic character of imaging realized by the refractive LSOE. The photographs make impression to be prepared in a gray-scale. No color is clearly visible except white, black and gray. What is most important, the refractive LSOE similarly to its diffractive version exhibits distinctly abilities for imaging with extended depth of focus. Images of optotypes formed by the LSOE are readable in a presented range of defocusing 0D -1D, otherwise to images created by lenses which are considerably blurred for substantial defocusing. One can observe that the focal depth of the spherical lens fabricated on photoresist is significantly bigger than that of the professional objective, because of the residual spherical aberration present in the former.
The main purpose of the paper is an illustration and an analysis of chromatic aberrations of a refractive LSOE and not a comparison of focal depths realized by the LSOE and the lenses. As we have already pointed out, a superiority of the LSOE for imaging with increased focal depth of focus over elements with symmetry of revolution was shown before [25,27,28]. Therefore we present in Fig. 3 output images created by lenses corresponding to only positive values of the defocusing. It is necessary to take into account that lenses can realize readable images also in a case of small negative values of the defocusing. On the other hand according to the paraxial theory of lenses the optical transfer function is an even function of the used defocusing parameter [29]. Hence in a case of negative defocusing values one can expect quality of images similar to these created by lenses and shown in Fig. 3. In order to illustrate the experimental results more completely we present in Fig. 4 images formed by the refractive LSOE in the full range of defocusing from 0D to 1D, corresponding to image distances from 52.5 cm to 110,5 cm.  For a more detailed analysis of the chromatic properties of the fabricated LSOE we performed the second experiment with a positive of the Snellen optotypes as input object. The