Optical transfer function shaping and depth of focus by using a phase only filter Dina Elkind, Zeev Zalevsky, Uriel Levy, and David Mendlovic The design of a desired optical transfer function 共OTF兲 is a common problem that has many possible applications. A well-known application for OTF design is beam shaping for incoherent illumination. However, other applications such as optical signal processing can also be addressed with this system. We design and realize an optimal phase only filter that, when attached to the imaging lens, enables an optimization 共based on the minimal mean square error criterion兲 to a desired OTF. By combining several OTF design goal requirements, each represents a different plane along the beam propagation direction, an imaging system with an increased depth of focus is obtained. Because a phase only filter is used, high energetic efficiency is achieved. © 2003 Optical Society of America OCIS codes: 070.2580, 110.4850. 1. Introduction In modern optics diffractive elements play a major role. Diffractive optical element共s兲 共DOE兲共s兲 can be designed to utilize functions that would be difficult or impossible to achieve by conventional optical elements. Moreover, DOEs are characterized by lighter weight, smaller dimensions, and lower costs as compared with their refractive or reflective counterparts.1– 4 Unfortunately, DOEs are based on diffraction, and thus they are highly dispersive 共i.e., wavelength sensitive兲. For this reason DOEs are usually used in systems based on monochromatic illumination. Alternatively, one can use the highly depressive nature of DOEs to perform separations of wavelengths. In this paper we present what we believe is a novel approach for enlarging the depth of focus 共DOF兲 of an imaging system by use of a special DOE. A DOF defines the maximal acceptable deviation from the focal plane of an imaging system based on a resolution criterion. DOF is inversely proportional to the aperture size—large aperture results in a smaller DOF. There are several approaches that attempt to enlarge the DOF of imaging systems. One interesting approach is related to encoding the aperture plane with a cubic phase element.5– 8 This approach The authors are with the Department of Physical-Electronics, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978 Israel. Received 13 August 2002; revised manuscript received 16 December 2002. 0003-6935兾03兾111925-07$15.00兾0 © 2003 Optical Society of America produces an image that after proper digital postprocessing produces an in-focus image. The proposed technique is based on an iterative design of a phase only filter that is attached to the imaging lens. The iterative algorithm suggested in this paper is based on the Gerchberg–Saxton 共GS兲 algorithm.9 –11 The desired range for which the focus should be maintained is divided into N planes. The optical transfer function 共OTF兲 of each plane is now calculated, assuming that the aperture size is half of the actual aperture size to be used. This way, an improved DOF is obtained. Our goal is to design an OTF that enables the achieving of exactly such a DOF while keeping the original aperture dimensions, i.e., allowing much more light to be transferred by the imaging system. Note that the factor of half was chosen arbitrarily to demonstrate the increased DOF 共the DOF of half the aperture is approximately 4 times larger兲. Choosing a different factor could demonstrate an even larger DOF. In each iteration a phase only filter that generates the desired OTF of each transverse plane is computed. Then, N phase only filters that correspond to the N transverse planes are averaged and the result is converted into a phase only filter by setting its amplitude to be one, i.e., by neglecting the amplitude contribution. The obtained result is used as a starting point for the next iteration, and the process continues until the variation of the obtained results is smaller than the limit defined as convergence. The suggested algorithm allows setting a nonuniform weighting to the contributions from each plane to control the trade off between resolution and DOF. 10 April 2003 兾 Vol. 42, No. 11 兾 APPLIED OPTICS 1925 An analytic approach for the design of a desired OTF was already suggested.12 However, the iterative approach provides much more design freedom, and allows obtaining improved results. Note that iterative approaches are commonly used and previously discussed for filtering applications.13 Section 2 describes the effect of out-of-focus imaging on the OTF. Section 3 discusses theoretical aspects of the suggested approach. Computer simulations and experimental results are presented in Sections 4 and 5, respectively. Section 6 concludes the paper. plane to the perfect imaging plane. The path-length error W 共x, y兲 can then be determined by subtracting the ideal phase distribution from the actual phase distribution kW共 x, y兲 ⫽ 1 W共 x, y兲 ⫽ 2 OTF共 f x, f y兲 ⫽ exp jk W x ⫹ 冉 冊 1 1 w 2. ⫺ Za Zi W共 x, y兲 ⫽ W m x2 ⫹ y2 . w2 冊冉 冊 (5) , (6) P共 x, y兲dxdy where P共 x, y兲 is the pupil function and 共 fx, fy兲 are the coordinates of the OTF. Owing to the aberration, the effective pupil function is now multiplied by the phase delay caused by the aberration: P g共 x, y兲 ⫽ P共 x, y兲exp关 jkW共 x, y兲兴. (7) Now, it is possible to find the OTF of a system in the presence of the aberration, by substitution of Eq. 共7兲 into Eq. 共6兲: 冊 冉 Z i f x Z i f x Z i f y Z i f y ⫺W x⫺ , y⫹ , y⫺ 2 2 2 2 A共 fx,fy兲 APPLIED OPTICS 兾 Vol. 42, No. 11 兾 10 April 2003 (4) We can now estimate the effect of the focusing aberration on the OTF 共applicable for incoherent imaging兲. An ideal, aberration free OTF is given by 兰兰 A共00兲 1926 (3) The number Wm is a convenient indication of the severity of the focusing error. By use of the definition of Wm, the path-length error can be expressed by (1) where Za is the distance between the aperture plane to the observation plane. Generally speaking, Za ⫽ Zi , where Zi is the distance between the aperture OTF共 f x, f y兲 ⫽ W m ⫽ 12 兰兰 共 x 2 ⫹ y 2兲, Z a 兰兰 再 冋 冉 冊 1 1 ⫺ 共 x 2 ⫹ y 2兲. Za Zi Z i f x Z i f x Z i f y Z i f y P x⫺ dxdy , y⫹ , y⫺ 2 2 2 2 ray as it passes from the reference sphere to the actual wave front. The error can be positive or negative, depending on whether the actual wave front lies to the left-hand side or to the right-hand side 共respectively兲 of the reference sphere. Thus, the phase distribution across the exit pupil is of the form of 共 x, y兲 ⫽ 冉 By assuming a square aperture of width 2w, the maximal path-length error at the edge of the aperture along the x or y axes is given by The influence of out-of-focus imaging on the OTF is well known and can be found in many textbooks.1 Nevertheless, we feel that a short description is still helpful. When the imaging system is diffraction limited, the amplitude point-spread function consists of the Fraunhofer diffraction pattern of the exit pupil, centered on the ideal image plane. However, when the observation plane is out of focus, wave-front errors exist. This case can be described by an exit pupil, which is illuminated by a perfect spherical wave. By tracing a ray backwards from the ideal image point to the coordinates 共x, y兲 in the exit pupil, the aberration function W共x, y兲 is the path-length error accumulated by that P x⫹ (2) The path-length error is thus given by 2. Optical Transfer for Out-of-Focus Function Imaging 兰兰 冉 共 x 2 ⫹ y 2兲 ⫺ 共 x 2 ⫹ y 2兲. Z a Z i 冊册冎 dxdy , dxdy (8) Fig. 1. 共a兲 Schematic drawing of phase of the Gerchberg–Saxton algorithm, 共b兲 the specific algorithm applied for the presented iterative approach. where A共 fx, fy兲 is the area of integration, i.e., the overlapping area between two shifted apertures 关the shift is a function of 共 fx, fy兲兴. A共0, 0兲 is the area of the lens aperture 共the shift is zero兲. By substituting Eq. 共5兲 into Eq. 共8兲 and performing several straightforward manipulations one obtains OTF共 f x, f y兲 ⫽ ⌳ 冉 冊冉 冊 冋 冉 冊冉 冋 冉 冊冉 fx fy ⌳ 2f 0 2f 0 8W m f x ⫻ sinc 2f 0 ⫻ sinc 8W m f y 2f 0 兩 f x兩 1⫺ 2f 0 1⫺ 兩 f y兩 2f 0 冊册 冊册 where ⌳ denotes a triangular function. It can be easily seen that a diffraction-limited OTF is indeed obtained for the case of Wm ⫽ 0. However, for values of Wm ⱖ 兾2, which represent a significant defocusing error, sign reversal of the OTF occurs. As can be seen, a gradual attenuation of contrast and a number of contrast reversals are obtained for high spatial frequencies. 3. Description of the Algorithm and Filter Design , (9) This GS algorithm is used for phase retrieval of a wave function whose intensity in the lens and the imaging planes is known. The basic algorithm is an iterative procedure that is shown schematically on Fig. 1共a兲. 10 April 2003 兾 Vol. 42, No. 11 兾 APPLIED OPTICS 1927 Fig. 2. 共a兲 Three OTF for comparison at the left-hand side edge of the DOF range, 共b兲 three OTF for comparison at the right-hand side edge of the DOF range, 共c兲 three OTF for comparison at the focal plane. The procedure starts by the choosing of a random phase function that is multiplied by a rectangular amplitude function representing the shape of the lens aperture. Inverse fast Fourier transform 共IFFT兲 of this synthesized complex discrete function is then performed. The phase of the obtained result is kept while the amplitude is set to be the square of the inverse Fourier transform of the desired OTF distribution that corresponds to twice the smaller aperture. The distributions 共computed for each of the N out-of-focus planes兲 are Fourier transformed. The result is averaged, and the amplitude is set to be a rectangular function as in the previous iteration. A weighted average can be obtained by multiplying the 1928 APPLIED OPTICS 兾 Vol. 42, No. 11 兾 10 April 2003 contribution of each plane by a different weighting coefficient 关see Fig. 1共b兲兴. The process is repeated until the variations from one iteration to the other are bounded. By varying the weights of the different planes and by changing the desired OTF distribution constrained per each plane, one may determine the resolution of the designed system obtained within different positions in the DOF range. During the iterative algorithm, Eq. 共9兲 is used to generate the real OTF as well as the desired OTF. The desired OTF has an aperture width of half the width of the real aperture of the imaging lens 共smaller apertures result in a larger depth of focus range兲. By applying such a constraint one obtains an in- Fig. 3. The experimental setup. creased depth of focus 共that corresponds to half the width aperture兲, while maintaining high energetic efficiency 共corresponding to a full width aperture兲 because a phase-only filter is used. 4. Computer Simulations As a test case, we picked up 11 uniformly spaced out-of-focus planes. The focal length used for the simulation and the phase-mask design was 20 cm. The diameter of the lens was 2 cm. The DOF range was ⫾2 mm around the in-focus plane. The wavelength was 630 nm. Simulation results are given in Fig. 2. As can be seen, three OTFs are plotted, one on top of the other. The three curves of Fig. 2共a兲 correspond to the desired OTF 共with aperture of half the diameter兲, the OTF obtained when the filter is not used, and the OTF obtained by using the suggested filter. The plot of the results is obtained at the left-hand side edge of the desired extended DOF range. As a criterion for resolving this information we chose a contrast threshold of 0.1. Thus a line representing this value was added to the figures. By observing the results one may notice that the real OTF cutoff frequency is smaller without the filter than with the use of the filter. This result is expected because resolution and DOF are inversely proportional. Indeed, the DOF of the system with the filter is even larger than twice the DOF of the system with a twice smaller aperture. In the other edge of the extended DOF range one may see that no improvement is achieved. In Fig. 2共b兲 it can be easily seen that the dashed curve 共OTF with the use of the filter兲 and the dashed-dotted curve 共OTF without the filter兲 are nearly the same, and their spatial cutoff frequencies are identical. Figure 2共c兲 shows all three OTF at the focus plane. As ex- pected, the real OTF is twice as wide as the desired OTF because its aperture is twice as wide. Note that those two OTFs are ideal triangles, as expected. However, the OTF that is received by use of the filter is much thinner and distorted. This is expected as well, because the reference in the algorithm was the desired OTF, and that is thinner than the physical aperture of the real lens. In addition the distortions are also caused by the restrictions to obtain the increased DOF. 5. Experimental Results To further verify the validity of the proposed approach, an optical experiment was carried out. The parameters used for the experiment are similar to the one specified in the simulation in Section 4. A phase filter with 8 gray phase levels was generated by using a lithographic recording. The 8-level DOE was obtained with 3 binary masks. The masks were created by use of a Dolev plotter with 3600 dpi and then reduced by a factor of 10 to a milimask by use of a high-resolution imaging setup. Moving the imaged object generated the introduced defocusing. The movement range was of a few centimeters. The edge of the range at which a focused image was obtained was allocated. Fig. 3 depicts the setup of the experiment. The setup contained a light source, an imaging lens with the phase filter attached to it, and a CCD camera. The image of the object is obtained by the CCD and displayed on the monitor. The object is a transparent film with regions of different spatial frequencies. Because a He–Ne light source, which generates monochromatic illumination, is a coherent light source, a diffuser was added to create incoherent illumination. The results are shown in Figs. 4共a兲– 4共c兲. 10 April 2003 兾 Vol. 42, No. 11 兾 APPLIED OPTICS 1929 Fig. 4. 共a兲 Perfectly focused image 共without the filter兲, 共b兲 misfocused image without the filter, 共c兲 with the filter. Figure 4共a兲 presents a perfect imaging that was obtained by focusing the CCD on the perfect imaging plane 共Z ⫽ Zi 兲. In this case, no filter was used. Fig. 4共b兲 was taken by focusing the CCD on an out-of-focus plane. Again, no filter was used. As can be seen, it is very difficult to identify the high frequencies. Figure 4共c兲 is identical to Fig. 4共b兲. The only difference is the use of the suggested filter. One can clearly see that using the filter improves the image quality. The high frequency, that cannot be observed without the filter is clearly resolved after attaching the filter to the imaging lens. The two stripes with the lower spatial frequencies that are hardly seen without the filter due to low contrast can be clearly identified, because better contrast is now achieved. The experiment demonstrates that employing the filter increases the depth-of-focus range. In summary, the experimental results have demonstrated a DOF im1930 APPLIED OPTICS 兾 Vol. 42, No. 11 兾 10 April 2003 provement of approximately one order of magnitude 共The DOF range was more than 3.3 cm兲. Let us now add several insightful remarks regarding the suggested technique. The number and the concentration of the planes used for the computation of the phase-only filter is determined such that the DOF obtained for a single plane will be larger than the separation distance between adjacent planes. Obviously, having too many constraint planes increases the number of constraints, and eventually the averaging of the various OTFs cause the degradation of the anticipated performance obtained due to a constraint of a single plane. Because no digital processing, such as inverse filtering, is applied, the sensitivity to noise is not significant. Spectrally, wideband illumination will smear the image, however the smearing effect is similar to the sensitivity of a regular DOE dealing with a single plane. 6. Conclusions In this paper we have demonstrated what is to our knowledge a novel technique for an iterative OTF design to obtain an imaging system with an improved depth of focus. Computer simulations as well as experimental results have demonstrated the improved depth of focus obtained by use of the suggested approach. References 1. J. W. Goodman, Introduction to Fourier Optics, 2nd. edition 共McGraw-Hill, San Francisco, 1996兲. 2. R. S. 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Saxton, “Phase determination for image and diffraction plane pictures in the electron microscope,” Optik 共Stuttgart兲 34, 275–284 共1971兲. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 共Stuttgart兲 35, 237–246 共1972兲. Z. Zalevsky, D. Mendlovic, and R. G. Dorsch, “Gerchberg– Saxton algorithm applied in the fractional Fourier or the Fresnel domain,” Opt. Lett. 21, 842– 844 共1996兲. Z. Zalevsky, D. Mendlovic, and G. Shabtay, “Optical transfer function design by use of a phase-only coherent transfer function,” Appl. Opt. 36, 1027–1032 共1997兲. J. N. Mait and W. T. Rhodes, “A pupil function design algorithm for bipolar incoherent spatial filtering,” Appl. Opt. 28, 1474 –1488 共1989兲. 10 April 2003 兾 Vol. 42, No. 11 兾 APPLIED OPTICS 1931

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