April 15, 2003 / Vol. 28, No. 8 / OPTICS LETTERS
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Design of superresolving continuous phase filters Daniel M. de Juana, José E. Oti, Vidal F. Canales, and Manuel P. Cagigal Departamento de Física Aplicada, Universidad de Cantabria, Los Castros S/N, 39005. Santander, Spain Received October 10, 2002 We present a procedure for designing rotationally symmetric pupil-plane masks to control the threedimensional light-intensity distribution near focus. Our method is based on the use of a series of figures of merit that are properly defined to describe the effect of general complex pupil functions. As a practical implementation, we have applied our method to obtain superresolving continuous smoothly varying phase-only filters. The advantages of these kinds of filters are that they do not produce energy absorption and they are easy to build with a phase-controlling device such as a deformable mirror. Results of comparisons between the performance of our method and that of other phase-filter designs are provided. © 2003 Optical Society of America OCIS codes: 100.6640, 110.1220, 220.1230, 350.5730.
A number of methods for the design of superresolving pupil filters have been proposed. At first, these filters were based on variable-transmittance pupils.1 – 3 However, in recent years, attention has centered on the design of phase-only prof iles. This is due to the fact that, in general, phase-only f ilters yield better performance than transmittance f ilters.4 Many phase prof iles that achieve transverse superresolution, such as the diffractive superresolution elements (DSEs) proposed by Sales and Morris5 and, more recently, the three-zone binary phase f ilters reported by Wang et al.6,7 (designed to increase the data storage density in the next-generation DVDs), are based on annular designs. In contrast, to our knowledge, only one method for the design of continuous superresolving phase-only prof iles has been proposed in the literature. This method is the global/local united search algorithm8 (GLUSA), which combines genetic/annealing algorithms with the hill-climbing method. This algorithm provides high-performance superresolution phase masks with small spot sizes while maintaining acceptable Strehl ratio values and low sidelobe intensities. However, this performance is obtained only with extremely complex phase masks. The goal of the research reported in this Letter is to f ind a procedure for designing general pupil masks for controlling focal volume. As an example of the power of our method, we have applied it to find simple slowly varying phase-only f ilters. The advantage of this kind of filter is that it can be implemented dynamically in optical instruments by use of a phase-controlling device such as a deformable mirror or a LCD. This technique could be useful in many different fields, for example, in astronomical telescopes to obtain superresolution or in optical tweezers to control the width and size of the spotlight. Our design method consists of the def ining a series of f igures of merit that when properly used in a f itting procedure will provide the parameters that describe the phase-mask shape. The figures of merit that we have selected are the Strehl ratio, S, the sidelobe intensity, Ir , and the axial and transverse superresolution gain factors,9 GA and GT , respectively. S is a relevant parameter for analyzing image quality and is def ined as the ratio of the intensity at the focal 0146-9592/03/080607-03$15.00/0
point to that corresponding to an unobstructed pupil. Ir is def ined as the maximum intensity of the first ring in the superresolution intensity pattern relative to the peak intensity. It is desirable to keep this value as low as possible. Usually, values greater than 10% of the peak intensity are not acceptable for most applications. Finally, GA and GT give a measure of the superresolution performance in the transverse and axial directions, respectively. These factors are normalized so that for the unobstructed pupil the two gains are equal to unity. In a particular direction the f ilter is superresolving when the corresponding gain is greater than unity and is an apodizer when the corresponding gain is lower than unity. The f ilters that we propose achieve performance that is comparable with performance provided by the GLUSA method; the proposed filters have the advantage of potential practical applications because of their simple implementation. The theoretical development of our design method is a generalization for complex pupil functions of that stated by Sheppard and Hegedus,9 who analyzed the diffracted intensity distribution near the geometrical focus by use of rotationally symmetric amplitude pupil functions within the framework of scalar diffraction theory. Let us consider a general complex pupil function P 共r兲 苷 T 共r兲 exp关 jf共r兲兴, where r is the normalized radial coordinate over the circular pupil, T 共r兲 is the transmittance function, and f共r兲 is the phase function. For a converging monochromatic spherical wave front passing through the center of the pupil, the amplitude U in the focal region may be written as Z 1 P 共r兲J0 共vr兲exp共 jur 2兾2兲rdr , (1) U 共v, u兲 苷 2 0
where v and u are radial and axial dimensionless optical coordinates, given by v 苷 k NA r ,
(2)
u 苷 k NA z ,
(3)
2
where k 苷 2p兾l, NA is the numerical aperture of the pupil, and r and z are the radial and axial distances, respectively, shown in Fig. 1. The intensity distribution provided by this f ilter is expanded in © 2003 Optical Society of America
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OPTICS LETTERS / Vol. 28, No. 8 / April 15, 2003
respect to this point: GT 苷 2
Re共I0 I1 ⴱ 兲 2 uF Im共I0 ⴱ I2 兲 , jI0 j2 2 uF Im共I0 ⴱ I1 兲
GA 苷 12
Re共I2 ⴱ I0 兲 2 jI1 j2 . jI0 j2 2 uF Im共I0 ⴱ I1 兲
(8) (9)
Furthermore, an expression for the Strehl ratio can also be obtained: S 苷 jI0 j2 2 uF Im共I0 ⴱ I1 兲 .
Fig. 1. Illustration of the notation used in determining the structure of the focal region. Point Q is located on a converging spherical wave front centered on F and passing through the center, O, of the pupil; P is a point near the focus, F; f is the focal length; and a is the pupil radius. The coordinates r and z of P are taken from the geometrical focus of the pupil.
series near the geometrical focus. The transverse and axial intensity distributions can be expressed, to the second order, as I 共v, 0兲 苷 jI0 j2 2
1 Re共I0 I1 ⴱ 兲v2 , 2
(4)
1 I 共0, u兲 苷 jI0 j2 2 Im共I0 ⴱ I1 兲u 2 关Re共I2 ⴱ I0 兲 2 jI1 j2 兴u2 , 4 (5) where ⴱ denotes complex conjugate and In is the nth moment of the pupil function, defined as In 苷
Z
1 0
P 共t兲tn dt .
(6)
In Eq. (6) the variable t 苷 r 2 is introduced. From Eqs. (4) and (5), it can be seen that the transverse intensity is symmetrical with respect to the geometrical focus (v 苷 0, u 苷 0). However, for the axial intensity this is not true in general. Furthermore, the intensity maximum in the axial direction is displaced from the geometrical focus by an amount equal to uF 苷 22
Im共I0 ⴱ I1 兲 . Re共I2 ⴱ I0 兲 2 jI1 j2
(7)
Equation (7) is valid for small displacements of the focus position from the geometrical focus, where the second-order expansion of the intensity distribution is a good approximation to describe the focal behavior. Therefore, the position of the maximum intensity is given by the coordinates 共0, uF 兲. Analogously to the development by Sheppard and Hegedus, expressions for the transverse (GT ) and axial (GA) gains corresponding to complex pupil functions can be obtained from the second-order expansion of the intensity with
(10)
These generalized superresolution factors def ined for complex pupil functions have the same meaning as those def ined by Sheppard and Hegedus9 for amplitude pupil functions. In fact, our expressions contain those proposed by Sheppard and Hegedus. To show the performance of our designing procedure, we are interested in applying our factors to f ind superresolving phase-only pupil f ilters. Since our goal is to obtain simple f ilters, so that they can be reproduced by a deformable mirror, we begin by choosing a slowly varying phase function such as f共r兲 苷 a sin共2pbr兲, where a and b are two parameters to be fitted. In our analysis, the parameters will be fitted to fulfill restrictive conditions such as small displacement of the focus position, a high Strehl ratio, and low sidelobe intensity Ir . Consequently, a and b are obtained from the following system of equations: juF 共a, b兲j # uF 0 ,
(11)
GT 共a, b兲 2 GT 0 苷 0 ,
(12)
S共a, b兲 2 S 苷 0 ,
(13)
0
where uF 0 is the maximum displacement of focus allowed, GT 0 is the desired transverse gain, and S 0 is the desired Strehl ratio. A Mathcad mathematical environment was used to solve this system of equations, since it provides user-friendly functions. The main advantage of this package is that it selects an appropriate method from a group of algorithms. The algorithms available for solving nonlinear systems are conjugate gradient, quasi-Newton, and the Levenberg– Marquardt algorithms taken from the public-domain MINPACK10 library. This method requires a guess value for each unknown at the beginning of the search process. For systems with more than one solution, the guess values determine the particular solution that is obtained. Table 1 shows some of the results obtained by this family of f ilters corresponding to particular pairs of parameters (a, b). For each solution, we give the values of uF , GT , S, and Ir . Furthermore, we show the values of the spot size, G, which also gives a measure of the superresolution performance and is def ined as the ratio of the position of the first zero of intensity for the superresolution pattern to that for the Airy pattern. We use this figure of merit to compare our results with those of others’ designs, because the f igure is broadly used in the literature to describe the performance of superresolving filters. The three f ilters
April 15, 2003 / Vol. 28, No. 8 / OPTICS LETTERS Table 1. Performance of Phase Filters with the Form f共共r, a, b兲兲 5 a sin共共2p br兲兲 Filter Parameter
1
2
3
a b uF GT G S Ir
4.622 0.391 0.94 1.20 0.86 0.62 0.056
12.066 0.339 0.76 1.35 0.78 0.38 0.050
19.112 0.316 20.78 1.38 0.75 0.29 0.041
Fig. 2. Normalized transverse PSF produced by the superresolving phase filter with parameters a and b given in the legend (solid curve) compared with the Airy PSF (dashed curve).
Table 2. Comparison of the Performance of Phase Filters Obtained with Three Methods Method
G
S
Ir
Airy DSE DSE GLUSA GLUSA GLUSA
1 0.8 0.71 0.87 0.75 0.65
1 0.42 0.37 0.63 0.38 0.28
0.016 0.127 0.297 0.029 0.066 0.125
shown in Table 1 produce superresolution in the transverse direction. The transverse point-spread function (PSF) for a 苷 8.364 and b 苷 0.352 is shown in Fig. 2. It can be seen that the method that we propose works properly, since we obtain superresolution with high Strehl ratios and low sidelobe intensity values. Table 1 also shows that the normalized spot size obtained is directly related to the desired Strehl ratio value. As a rule, an increase in resolution is accompa-
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nied by a decrease of the Strehl ratio and an increase of the sidelobe intensity. Table 2 compares the performance of phase f ilters obtained from different methods. One can see that our phase prof iles offer better results than the two- and three-zone annular designs (DSE) of Sales and Morris.4,5 For a comparable level of Strehl ratio, we achieve lower spot sizes and a significant reduction of the sidelobe intensities. In the case of the filters generated with the GLUSA algorithm, one can see that, for the same values of Strehl ratio, our f ilters provide lower values of the sidelobe intensities, at the cost of slightly greater spot sizes. To summarize, with a continuous and slowly varying phase prof ile such as a sine function, we have designed f ilters with a superresolution performance comparable with that obtained with the complex phase prof iles generated with the GLUSA algorithm, created by summation of a great number of cosine phase functions. Because of their simplicity, the proposed phase profiles are easy to produce and hence are applicable to any superresolution optical setup. In conclusion, we have developed a new method for searching pupil f ilters that allows us to achieve great control over the desired PSF characteristics. In the focal plane we can control the transverse gain and the Strehl ratio. Along the axial direction we are able to control the depth of focus by means of the axial gain and its displacement from the position of the geometrical focus, making use of parameter uF . As an example, we used our method to design a phase-only pupil filter that gives superresolution in the focal plane. Our procedure compares advantageously with others that are already known. Furthermore, one can also apply our procedure to obtain phase masks, producing a large focus to make the optical system insensitive to defocus aberration, or in any application where control of the spot size is required. This research was supported by Ministerio de Ciencia y Tecnología grant AYA2000-1565-C02. D. M. de Juana’s e-mail address is
[email protected]. References 1. G. Toraldo di Francia, Nuovo Cimento Suppl. 9, 426 (1952). 2. C. J. R. Sheppard, Optik 48, 329 (1977). 3. I. J. Cox, C. J. R. Sheppard, and T. Wilson, J. Opt. Soc. Am. 72, 1287 (1982). 4. T. R. M. Sales and G. M. Morris, Opt. Lett. 22, 582 (1997). 5. T. R. M. Sales and G. M. Morris, J. Opt. Soc. Am. 14, 1637 (1997). 6. H. Wang, Z. Chen, and F. Gan, Opt. Eng. 40, 991 (2001). 7. H. Wang, Z. Chen, and F. Gan, Appl. Opt. 40, 5658 (2001). 8. J. Zhai, Y. Yan, D. Huang, M. Wu, and G. Jin, Proc. SPIE 3429, 177 (1998). 9. C. J. R. Sheppard and Z. S. Hegedus, J. Opt. Soc. Am. 5, 643 (1988). 10. J. J. More, B. S. Garbow, and K. E. Hillstrom, User’s Guide to Minpack I, publ. ANL-80-74 (Argonne National Laboratory, Argonne, Ill., 1980).
