Zone plate for arbitrarily high focal depth

Zone plate for arbitrarily high focal depth
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  Zone plate for arbitrarily high focal depth Jorge Ojeda-Castaneda and L. R. Berriel-Valdos We show that a zone plate with a prespecified number of foci, which are separated axially by Rayleigh's limit ofresolution, can create an arbitrarily high focal depth when used as an apodizer over an otherwise clear aperture. We discuss the resolution and light gathering power of this method. 1. Introduction Various efforts are going on involving the use of apodizers for designing modern optical instruments with novel imaging characteristics.1- 7 Imaging systems used in microelectronics, medicaldiagnosis, and machine vision require high focal depth.In a previous publication we indicated that the conceptof superresolution can be used to achieve an arbitrarilyhigh focal depth. 6 Furthermore, in a recent Letter, 7 we reported that an optical system with a double focuscan be used to increase focal depth, provided that the two axial amplitudes from each focus are axially sepa- rated by a distance equal to or less than Rayleigh'slimit of resolution.Our aim here is to discuss a method for achieving anarbitrarily high focal depth by the use of a zone platethat generates a prespecified number of identical mul-tiple foci. The amplitude distribution along the opti- cal axis of any two adjacent foci of the zone plate are at Rayleigh's limit of resolution. Our approach can berephrased heuristically with the help of Fig. 1 as fol-lows: We propose using a zone plate that creates aprespecified number of multiple foci that have identi-cal amplitude responses along the optical axis. Ifthese axial amplitude responses are separated by Ray-leigh's limit of resolution, their superimposition gener-ates within a prespecified interval a practically contin-uous axial distribution.In Sec. II we describe the design of the multiple focizone plate. In Sec. III, we discuss the light gatheringpower and the resolution of our method. The authors are with National Institute of Astrophysics, Optics, &Electronics, Apdo.Postal 216, Puebla 72000 Pue, Mexico. Received 23 March 1989.0003-6935/90/070994-04$02.00/0.C 1990 Optical Society of America. II. Zone Plate Design Let us consider the complex amplitude p(r;W 20 ) atthe image of a point source as produced by a 2-Doptical system with radial symmetry and sufferingfrom focus errors: p(r;W 20 ) = 27r J p)JO 2rrp) exp[i2r(p/U) 2 W 2 0 ]pdp. (1) In Eq. (1) r is the radial coordinate in the image plane;W 2 0 is the defocus coefficient measured in wavelengthunits; p(p) is the complex amplitude at the exit pupil;Jo denotes the Bessel function of zero order and of thefirst kind; and p is the radial spatial frequency at theexit pupil whose maximum value is Q.It is convenient to rewrite Eq. (1) for r = 0 as follows: p(r = ,W 20 ) = 7rU2 exp(irW 20 )q(W 20 ), (2) where q(W 20 ) = J5(p) X expti2rW 20 [(p/Q) 2 - 0.5]ld[(p/) 2 ] = J 4(t) rect(r) exp(i2rW 20 t)dD. (3) In Eq. (3) we define = (p/Q) 2 - 0.5, q(t) rect() = p(p). (4) For a clear circular aperture the axial amplitude response q(W 20 ) is q(W 20 ) = sinc(W 20 ), (5) where sinc(O) = sin(7r0)/(7rO). We propose using a zoneplate over the exit pupil to replicate several times theaxial amplitude response in Eq. (5). The separationbetween any two adjacent axial amplitude responses isRayleigh's limit of resolution, in this case along theoptical axis.Consequently, we aim to obtain with a zone plate thefollowing axial amplitude response: 994 APPLIED OPTICS / Vol. 29, No. 7 / 1 March 1990  = /Xf N\ EXIT PUPIL IMAGE PLANE W 20 Fig. 1. Schematic diagram that shows he superposition of multiple axial amplitude responses. I (n D 0.8 00 n 0.6 0 cr 0.4 a:-J -0.20.0 0.1 Fig. 2. Streb 0 1.0 ~ 0.9 , 0.8 U Z 0.7 - t 0.6 U 0.5 - z < 0.4- W 0.3- s 0.2- 0.1 - 2 -0.0 -0.1 - -0.2 - -0.3 - 0.00.1 0.2 0.3 0.4 0.5 DIMENSIONLESS VARIABLE Fig. 3. Amplitude transmittance of the equivalent 1-D pupil func-tion. 1.0 0.8 w   z U z 0I- Q. 1.0 2.0 3.0 4.0 5.0 6.0 DEFOCUS COEFFICIENT W 20 ratio vs defocus of the proposed apodizer for anincreased number of terms. Mq(W 20 ) = E 6(W20 -m)*sinc(W 2 0), (6)m=-M where the asterisk denotes convolution. The sum in Eq. (6) reduces to Eq. (5), q(W 20 ) = sinc(W 20 ) for M = 0, while if M - , according to the sampling theorem, 8 the sum approaches a uniform background. In Fig. 2,we display the Strehl ratio, S( W 20 ) = ( W 20 ) l 24 q(0)l 2, when M =0,1,2,3,4,5, and 10. As can be appreciatedfrom Fig. 2, the focal depth can be extended arbitrarilyin this fashion. We next determine the apodizer'samplitude profile.From Eqs. (2), (3), and (6) we find that ) = 2M + 1Y~' 1 + E cos(2rmt)] rect(r), (7) and from Eqs. (3) and (7) we obtain p(p) = (2M + 1)-' r M- X 1 + -l)m cos(2IrmIP2/2) circ(p/I2). (8) m=l In Figs. 3 and 4 we display the curves of the functions inEqs. (7) and (8), respectively. Note that when M = 10, the curve in Fig. 3 tends to be a sinc function. One 0.2 0.4 0.6 08 1.0 RADIAL SPATIAL FREQUENCY: /Q Fig. 4. True amplitude transmittance of the 2-D circular symmet- ric apodizer. expects then that if M-- , then (r) -6 ). In otherwords, our method is another way of implementing athin annular aperture by properly weighing the ampli-tude of the pupil aperture trying to keep lateral resolu-tion, as shown in Sec. III. In Figs. 5(a), (b), and (c), we show as a grey level picture the irradiance point spread functions for thein-focus plane W 20 = 0, and for the out-of-focus planes W 2 0 = 1.0 and W 20 = 2.0, obtained when using theproposed zone plate, for an increasing number of foci forM= 0,M= 1,M= 2,M= 3,M= 4,M= 5, andM= 10. From Fig. 5, it is clear that when using our proposedzone plate, as the number of foci increase, so does the depth of focus. However, as we show next, the incre- ment in focal depth is achieved by sacrificing lightgathering power. Ill. Light Throughput and Resolution Using the change of variable in Eq. (4), the normal-ized light throughput can be written as any of the twoexpressions: 1 March 1990 / Vol. 29, No. 7 / APPLIED OPTICS 995 M=2t M =It ~~Yl 1v-\ AV V", Z ' M= 10 'M=5 I 'M=4 11 M=3 ,M= 2 , 'r" = I M=O  a) W 2 0=O b)W20= IC ) W 20 = 2 M=O M=I M=2M=3 M=4 M=5 M=10 Fig. 5. Grey level picture of the point spread functions at various focal planes (columns) and various foci (lines) of the zone plate. T = (2r2)-2r I Lb p)I pdp =f qt)2¢ (9) By substituting Eq.(7) in Eq. (9) we obtain T = (2M + 1)-2 1+ 4 1 cos(2rmt)dt +4>3 >3 1/2 +4 Y Y 1/2 os(2rmt) cos(2rnt)d] = (2M + 1)-1. T I .0 08 I 50.6 0 0.4 I 0 2 00 M=O M=IM=2 2 4 6 Fig. 6. Light throughput vs number of foci. 8 M a)W 2 0=0.0 b)W 2 o=i.o C)W20 = 2.0 7 - M=3 M=4 (10) This formula is plotted in Fig. 6. As M increases, thereduction of light throughput, shown in Fig. 6, is equalto the loss of light throughput in the method reported in Ref. 6. Note, however, that, as pointed out in Ref. 6, the resolution for our present method is higher thanthat obtained with a clear aperture of equal lightthroughput. To support this statement, we show inFig. 7 a gray level picture of the optical transfer func-tions (OTFs) for a variable defocus, associated with M=5 M=10 Fig. 7. Groy 1vol picturo of tho OTFa aogociated with tho point spread functions in Fig. 5. 996 APPLIED OPTICS / Vol. 29, No. 7 / 1 March 1990 I -------- --------  use of the proposed zone plate, for various foci: M = 0, 1, 2, 3, 4, 5, and 10. From the above results and discussion, we claim thatour method increases focal depth without substantial-ly reducing lateral resolution for a given light through-put. IV. Conclusions We describe a zone plate that produces arbitrarilyhigh focal depth when used as an apodizer over anotherwise clear optical exit pupil. The proposed zoneplate creates a prespecified number of foci, all havingidentical axial amplitude responses. Any two adja-cent axial amplitude responses are separated axially byRayleigh's limit of resolution. Computer calculatedcurves (Strehl ratio vs defocus, point spread functions,and OTFs) show that as the number of foci increases,so does the depth of focus. This property is achievedby sacrificing light throughput but without substan-tially decreasing lateral resolution.We are indebted to Alberto Gomez for numericallyevaluating the grey level pictures associated with thepoint spread functions and OTFs for various focuserrors and foci. The financial support of COSNET 704.87 is gratefully acknowledged. References 1. P. Jaquinot and B. Roizen-Dossier, "Apodization," in Progress in Optics, Vol. 3, E. Wolf, Ed. (North-Holland, Amsterdam, 1964),p. 29.2. W. B. Wetherell, "The Calculation of Image Quality," in Applied Optics and Optical Engineering, Vol. 3, R. R. Shannon and J. C. Wyant, Eds. (Academic, New York, 1980).3. G. Indebetouw and H. X. Bai, "Imaging with Fresnel Zone PupilMasks: Extended Depth of Field," Appl. Opt. 23, 4299-4302(1984).4. J. Ojeda-Castaneda, L. R. Berriel-Valdos, and E. Montes, "Line- Spread Function Relatively Insensitive to Defocus," Opt. Lett. 8, 458-460 (1983).5. J. Ojeda-Castaneda, L. R. Berriel-Valdos, and E. Montes, "Spa-tial Filter for Increasing the Depth of Focus," Opt. Lett. 10, 520-522 (1985).6. J. Ojeda-Castaneda and L. R. Berriel-Valdos, "Arbitrarily High Focal Depth with Finite Apertures," Opt. Lett. 13, 183-185 (1988).7. J. Ojeda-Castaneda and A. Diaz, "High Focal Depth by Quasibi focus," Appl. Opt. 27, 4163-4165 (1988). 8. R. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1978), p. 189. 1 March 1990 / Vol. 29, No. 7 / APPLIED OPTICS 997
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