Generation of nearly nondiffracting Bessel beams with a Fabry Perot interferometer

Transcription

1 Horváth et al. Vol. 14, No. 11/November 1997/J. Opt. Soc. Am. A 3009 Generation of nearly nondiffracting Beel beam with a Fabry Perot interferometer Z. L. Horváth, M. Erdélyi, G. Szabó, and Z. Bor Department of Optic and Quantum Electronic, JATE Univerity H-670 Szeged, Dóm tér 9, Hungary F. K. Tittel and J. R. Cavallaro Rice Univerity, Electrical & Computer Engineering Department, MS Main, Houton, Texa Received October 9, 1996; revied manucript received April 8, 1997; accepted May 15, 1997 A new concept for generating zero-order Beel beam wa tudied theoretically. The patial intenity ditribution wa calculated numerically uing a wave optic model. Approximate analytical expreion were derived to decribe the radial intenity ditribution in plane perpendicular to the optical axi of an imaging len Optical Society of America [S (97) ] 1. INTRODUCTION An ideal zero-order Beel beam conit of a uperpoition of monochromatic plane wave with wave vector lying on a conical urface having the ame magnitude. Several experiment have been reported that achieve uch a uperpoition of plane wave. For example, thi type of angular pectrum can be obtained by applying an annular lit in the focal plane of a len 1 or by the ue of an axicon, a holographic proce, 3,4 a Fabry Perot interferometer, 5,6 or a pecial type of laer cavity. 7,8 A novel concept for the generation of nearly nondiffracting Beel beam a applied to microlithography wa decribed in Ref. 9. The experimental arrangement i hown in Fig. 1. A pointlike ource wa formed by focuingahe Ne laer beam ( 63.8 nm). Such a point ource illuminated a canning Fabry Perot interferometer, which produced a concentric ring ytem in front of a len. The image produced by the len wa magnified by two microcope objective, and the intenity ditribution wa oberved with a CCD camera. The len aperture wa adjuted o that it tranmitted only the firt Fabry Perot ring and blocked all other ring. The meaured intenity ditribution in plane perpendicular to the optical axi i given by the J 0 function. Thi reult wa expected becaue a century ago 10 1 it wa recognized that the diffraction pattern of a narrow annular aperture can be decribed by the zero-order Beel function. Becaue of the annular illumination of the len, the depth of focu increaed and the tranvere reolution could be improved by a factor of 1.6. Thi paper report on an analytical wave optic decription of experimental reult obtained in Ref. 9.. THEORY Figure depict a monochromatic pherical wave generated by a point ource that illuminate a Fabry Perot interferometer; the light paing through the interferometer i incident on a thin len with focal length f at wavelength. Becaue of multiple reflection in the interferometer, the electric field in front of the len i the ame a the field generated by a equence of point ource I 0, I 1,..., I m,.... The eparation between two adjacent ource i d and their intenity ratio i R, where d i the eparation of the etalon of mirror and R i the reflectivity of the mirror. The len tranform the incoming pherical wave front generated by the mth ource into a pherical wave front. The radiu (q m ) of the wave front immediately after the exit urface of the len i given by 1/q m 1/f 1/p m, (1) where p m i the radiu of the incoming pherical wave front at the entrance urface of the len. The len aperture truncate the outgoing pherical wave o that the wave front i a pherical calotte (a egment of a pherical urface). The electric field produced by the mth ource beyond the len at point P can be obtained by calculating the diffraction integral 13 over the pherical calotte SC (ee Fig. ): E m r, z i A m expik p m nd p m q m expik p m nd q m p m i A m expikq m q m SC SC expik ds expik ds, () where r and z are the cylindrical coordinate of point P, A m /p m i the amplitude, and kp m i the phae of the incoming wave generated by the mth ource, k i the wave number, and knd i the phae hift caued by the len (n i the refractive index, and D i the axial thickne). The integral between the bracket ha already been calculated in Ref. 13, and it lead to the well-known threedimenional Airy pattern, /97/ $ Optical Society of America

2 3010 J. Opt. Soc. Am. A/Vol. 14, No. 11/November 1997 Horváth et al. The total electric field behind the len i the um of the field of the individual ource, Er, z ika 0 a expikz 0 m0 R expi m p m q m Fig. 1. Schematic diagram of the experimental etup for generating zero-order Beel beam. A pointlike ource illuminate a Fabry Perot interferometer, which produce a concentric ring ytem in front of an imaging len. If the aperture i adjuted o that it tranmit the firt Fabry Perot ring only and block all other, a zero-order Beel beam i generated beyond the len. Fig.. Notation ued for the calculation. Becaue of multiple reflection, the electric field in front of the len i the ame a the field generated by a equence of point ource I 0, I 1,..., I m,.... Beyond the len the electric field i the uperpoition of the field produced by virtual point ource. i A m expikq m q m SC ika ma q m expik ds expiq m /a u m Cu m, v m isu m, v m, (3) where a i the radiu of the len aperture, C and S function can be calculated by the Lommel function, 13 and u m and v m are dimenionle variable given by u m ka/q m f z q m, v m ka/q m r. (4) Inerting Eq. (3) and Eq. (4) into Eq. (), we obtain E m r, z ika m a expik pm nd f z p m q m Cu m, v m isu m, v m. (5) Since A m R m A 0 and p m p 0 md, Eq. (5) may be written a E m r, z ikrm a A 0 expik p0 nd f z md p m q m Cu m, v m isu m, v m. (6) Cu m, v m isu m, v m, (7) where 0 k( p 0 nd f ) i an unimportant phae factor and kd 4d/ (8) i the phae hift introduced by the Fabry Perot interferometer. In the derivation of Eq. (7), we aumed that the phae change due to internal reflection in the etalon wa zero. Thi effect can be taken into account by ubtituting R exp(i) for the reflectivity R or increaing the etalon eparation with a ditance of /(), where i the phae change on internal reflection. Eq. (8) yield the phae difference between two adjacent virtual ource. If denote the larget integral value that i le than or equal to d/, then one can define the reduced phae hift r a r 4d d 0 K, (9) where d 0 / and K i a newly introduced parameter given by 4(d d 0 )/. Thu K varie from 0 to. A variation of d of / lead to a change of in the phae difference. For uch a mall variation of d, the poition of image point of the virtual ource remain practically unchanged. Therefore r and d can be regarded a independent variable. Thi fact i important from the experimental point of view becaue it i difficult to meaure the etalon eparation with an accuracy of. 3. RESULTS AND DISCUSSION If the len i illuminated by a point ource, the depth of focu of the image (DOF) i given by DOF NA 1 M (10) and i defined a the ditance between the principal intenity maximum and the firt intenity minimum on the optical axi, where NA a/f i the numerical aperture of the len and M i the magnification. The ditance between the image point of the virtual ource approximately equal dm. The relative image denity defined by N DOF dm 1 M d NAM (11) i an important quantity for determining the hape of the axial intenity ditribution. 9 In the previouly reported experiment, 9 four different cae were tudied. The focal length and the numerical aperture of the len ued in the experiment were 50 mm and , repectively. The meaured value of DOF

3 Horváth et al. Vol. 14, No. 11 /November 1997/J. Opt. Soc. Am. A 3011 wa 0 m. For comparion thee parameter are ued. In thi cae from Eq. (10), the magnification i M and the ditance of ource point I 0 from the len i given by p 0 f f/m mm. Figure 3 how the intenity ditribution for variou value of image denity N. The axial intenity ditribution were fitted to the meaured curve. The calculation were done by uing Eq. (7) with the following parameter: (a) d m (N 0.47), K 1.501; (b) d 3100 m (N 1.13), K 0.35; (c) d 1091 m (N 3.1), K 0.; (d) d m (N 8.0), K The reflectivity R wa aumed to be Thee value of the parameter agree with their meaured value within the accuracy of the meaurement. The inet how the comparion of the meaured (dot) and calculated (olid curve) intenity ditribution on the optical axi. In cae (a) the ditance between the image point on the optical axi i large compared with the DOF. Thu eparate harp peak are clearly evident. By increaing the image denity (i.e., decreaing d), we can make the ocillation on the optical axi diappear and the intenity curve become moother. The number adjacent to the peak how the value of the peak intenity. In agreement with the law of conervation of energy, the peak intenity increae with increaing N. For certain circumtance the intenity ditribution in a plane perpendicular to the optical axi can be decribed by a zero-order Beel function. The radiu of the interference ring i different for different cae and lightly increae with increaing z, a hown in Fig. 3. The detailed analyi how that the radiu of the interference ring depend trongly on the phae difference. The intenity ditribution [calculated from Eq. (7)] i plotted for variou value of r, auming N (d m). The value of coefficient K were 0, 0.15, 0.5, and 1.5 for cae depicted in Fig. 4(a), (b), (c), and (d), repectively. A in Fig. 3, the number adjacent to the peak diplay the value of the peak intenity. The inet in the topright corner how the intenity immediately in front of the len. For cae (d) it hould be noted that a different cale wa ued becaue the intenity immediately in front Fig. 3. The patial intenity ditribution a calculated from Eq. (7) for different value of image denity N and reduced phae difference r [ee Eq. (11) and (9)]. The inet how a comparion of the meaured and calculated axial intenity ditribution. The number adjacent to the peak diplay the intenity maxima in arbitrary unit.

4 301 J. Opt. Soc. Am. A/Vol. 14, No. 11/November 1997 Horváth et al. Fig. 4. The patial intenity ditribution, a calculated from Eq. (7), a a function of the reduced phae difference r, auming approximately contant image denity (N ). The number adjacent to the peak diplay the intenity maxima in arbitrary unit. The inet on the right ide how the illumination of the len. The inet on the left diplay a comparion of the radial intenity ditribution calculated from Eq. (7) and from approximate analytical expreion [Eq. (1) and (14)]. In cae (d) the intenity i decreaed coniderably, and therefore a different cale wa ued. of the len wa much le than that for cae (a) (c). The electric field in front of the len can be calculated a the um of the electric field produced by point ource I 0, I 1,..., I m,.... The inet in the top-left corner how the radial intenity ditribution in a plane perpendicular to z. The olid curve indicate the radial intenity ditribution calculated from Eq. (7), and the dot diplay the reult of the approximate analytical expreion derived below. In cae (a), where r 0(di a multiple of /), a mall, nearly homogeneou bright pot can be een on the center of the len, and the firt Fabry Perot ring i beyond the len aperture. The intenity ditribution i like a top-hat profile; thu, the diffraction pattern in a plane perpendicular to z i imilar to an Airy-type diffraction pattern given by Ref. 13, I A v I A0 J 1v v, (1) where I A0 i the intenity on the axi at a point z and v l A r. (13) f z In Eq. (13) l A i the radiu of the illuminated area. In Fig. 4(a) the circle how an Airy-type diffraction pattern calculated from Eq. (1) with l A 0.81 mm. Thi i the ditance at which the amplitude i one half of that on the axi immediately in front of the len. When r i increaed, the center of the pot become dented, and finally a ring i formed. When r i further increaed, the radiu of the ring increae and therefore the interference ring hrink in a plane perpendicular to the axi [ee Fig. 4(b) and 4(c)]. Then the radial intenity ditribution can approximately be decribed by I B r I B0 J 0 l B z r, (14) where I B0 i the intenity on the axi at point z and

5 Horváth et al. Vol. 14, No. 11/November 1997/J. Opt. Soc. Am. A 3013 l B f K 1 1 f tan. (15) Eq. (14) wa plotted a dot in Fig. 4(b) and 4(c). The radial intenity ditribution can be explained with a imple model. The Fabry Perot interferometer tranmit the light in direction m given by co m m/(d/), where m i an integer between 1 and d/. The integral value of correpond to the mallet angle. The light incident on the len in direction i collected by the len to a bright interference fringe in the focal plane. The radiu of the fringe i given by l B f tan. Uing tan 1/co 1 and the definition of K, one can obtain Eq. (15) for l B. Only one fringe i formed in the focal plane becaue the len aperture i adjuted o that it tranmit only the firt Fabry Perot ring and block all other. At a point z the light arrive from a bright narrow ring lying in the focal plane. The radial intenity ditribution of the diffraction pattern of a narrow ring can be decribed by Eq. (14). When r i further increaed, the ring move beyond the len aperture and the intenity in front of the len decreae coniderably. The illumination of the len i again homogeneou, and therefore the intenity ditribution reemble a three-dimenional Airy pattern [Fig. 4(d)]. The inet on the left ide of Fig. 4(d) how a comparion of the radial intenity ditribution calculated from Eq. (7) (olid curve) and from Eq. (1) (dot) with l A a in a plane given by z 8914 m, where the intenity reache it maximum on the axi. 4. CONCLUSION A new concept for generating nearly nondiffracting Beel beam ha been tudied theoretically. The patial intenity ditribution ha been calculated with a wave optic model for variou value of the image denity and phae difference. Approximate analytical expreion have been derived to decribe the radial intenity ditribution in plane perpendicular to the optical axi. ACKNOWLEDGMENTS Thi work wa upported in part by Texa Intrument, by the National Science Foundation under grant DMI and INT , and by the OTKA Foundation of the Hungarian Academy of Science (grant T0910, F00889, and W01539). The author may be reached a follow: Z. L. Horváth, Z.Horvath@phyx.u-zeged.hu. F. K. Tittel can be reached by tel: and fkt@rice.edu. REFERENCES 1. J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, Diffractionfree beam, Phy. Rev. Lett. 58, (1987).. R. Arimoto, C. Saloma, T. Tanaka, and S. Kawata, Imaging propertie of axicon in a canning optical ytem, Appl. Opt. 31, (199). 3. J. Turunen, A. Vaara, and A. T. Friberg, Holographic generation of diffraction-free beam, Appl. Opt. 7, (1988). 4. A. J. Cox and D. C. Dibble, Holographic reproduction of a diffraction-free beam, Appl. Opt. 30, (1991). 5. A. J. Cox and D. C. Dibble, Nondiffracting beam from a patially filtered Fabry Perot reonator, J. Opt. Soc. Am. A 9, 8 86 (199). 6. G. Indebetouw, Nondiffracting optical field: ome remark on their analyi and ynthei, J. Opt. Soc. Am. A 6, (1989). 7. J. K. Jabczynki, A diffraction-free reonator, Opt. Commun. 77, 9 94 (1990). 8. K. Uehara and H. Kikuchi, Generation of nearly diffraction-free laer beam, Appl. Phy. B 48, (1989). 9. M. Erdélyi, Z. L. Horváth, G. Szabó, Z. Bor, F. K. Tittel, J. R. Cavallaro, and M. C. Smayling, Generation of diffraction-free beam for application in optical microlithography, J. Vac. Sci. Technol. B 15, 87 9 (1997). 10. G. B. Airy, On the diffraction of an annular aperture, Philo. Mag. 18, January E. H. Linfoot and E. Wolf, Diffraction image in ytem with annular aperture, Proc. Phy. Soc. B 66, (1953). 1. C. A. Taylor and B. J. Thompon, Attempt to invetigate experimentally the intenity ditribution near the focu in error-free diffraction pattern of circular and annular aperture, J. Opt. Soc. Am. 48, (1958). 13. M. Born and E. Wolf, The three-dimenional light ditribution near focu, Principle of Optic, 6th [corrected] ed. (Pergamon, Oxford, 1989), Chap. 8.8.