Tight Focusing of Optical Beams; A Review - Part 1web.iitd.ac.in/~psenthil/Conferences/InvertJSTP1-2009.pdf · Tight Focusing of Optical Beams; A Review - Part I 197 Tight Focusing

Tight Focusing of Optical Beams; A Review - Part I

197

Tight Focusing of Optical Beams; A Review - Part 1

RAKESH KUMAR SINGH, P. SENTHILKUMARAN and KEHAR SINGH*Indian Institute of Technology Delhi, New Delhi - 110 016, India

*E-mail : [email protected]

Abstract

Complex amplitude and polarization distribution of an optical beam plays a dominant role in shaping thefocused structure of the beam. It is therefore possible to engineer the focal spot using the pupil functionmanipulation. Helical phase structure arising due to phase singularity in the wave front plays an important rolein shaping the focal spot. Tight focusing of an optical beam produces intensity distribution in the focal volumedifferent from the well-known results based on scalar theory, and polarization distribution shows space variantcharacteristics. In the present paper, roles of the amplitude-, phase-, and polarization distribution on thetightly focused structure of the optical beams are reviewed. Impact of the helical phase structure in the pupilfunction engineering and subsequently on the focused structure is discussed with special reference to theauthors' investigations at IIT Delhi. Certain applications in which tight focusing is desired are briefly discussed.Some miscellaneous investigations are also mentioned.

Keywords : Debye-Wolf integral, polarization distribution, singular optics, optical vortices.

1. Introduction

Beam propagation has been a subject ofinvestigations for a long period and studies havebeen carried out for paraxial (Kogelnik 1965,Kogelnik and Li 1966, Siegman 1986, Seshadri2006, 2008a, Zhou et al 2007a [1-6]), and non-paraxial regimes (Lax et al 1975, Agrawal andPattanayak 1979, Agrawal and Lax 1983, Takenakaet al 1985, Wünsche 1992 [7-11]), (Laabs1998,Sheppard and Saghafi 1999, Martinez-Herrero etal 2001, 2008, Chen et al 2002 [12-16]), (Ciattoniet al 2002, Deng et al 2008, Seshadri 2008b[17-19]). Interest in the propagation dynamics ofvectorial beams has been generated in recent years(Tervo and Turunen 2001a,b , Borghi et al 2002,Duan and Lu 2004a,b [20-24]), [Chaumet 2006,Hernandez-Aranda et al 2006, Kang and Lu 2006,Sepke and Umstadter 2006 a-c [25-30]), (Zhou2006a, 2008a, Zhou et al 2006[31-33], Zhou etal2007a[6], Zhou et al 2007b,c [34-35]), (Zhouand Chu 2007, Wu et al 2008, Zhou and Zheng

Invertis Journal of Science & Technology Vol. 1, No. 4, 2008 ; pp. 197-230

2008 [36-38]). Exact analytical solution to theMaxwell equation has been derived by Sepke andUmstadter (2006a-c [28-30], 2006d[39]) fordifferent situations. Importance of the aperturedbeam and its focusing has been realized for the sakeof its practical application and theory, and studieshave been carried out both for paraxial (Stamnes1981, Stamnes and Spjelkavik 1981, Belland andCrenn 1982, Tanaka et al 1985a,b [40-44]),(Stamnes 1986, Born and Wolf 1989, Stamnes andEide 1998, Eide and Stamnes 1998a,b [45-49]),(Almeida et al 1999, Mahajan 2001, Kopeika 2003,Shamir 2003, Goodman 2007 [50-54]) and non-paraxial regimes (Duan and Lu 2003, Lu and Duan2003, Duan and Lu 2004c,d [55-58], Zhou 2008[38]). Propagation of vectorial apertured beamshas drawn attention of several research groups(Ling and Lee 1984 [59], Lu and Duan 2003 [56],Duan and Lu 2004b [24], Gao and Lu 2006, 2007[60-61]), (Liu et al 2006, Zheng et al 2007 [62-63]). Focusing characteristics of the mocrolenseswith long focal depth and under multiple wavelength

Rakesh Kumar Singh et al

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illumination has been investigated by (Wang et al2007 [64]). Vector diffraction theory of refractionof light by a spherical surface has been investigatedby (Guha and Gillen 2007 [65]).

As is well-known, diffraction pattern of auniformly illuminated circular aperture shows brighthigh intensity ring surrounded by low intensity ringsin the far field. This intensity distribution is referredto as the 'Airy pattern'. A focusing system convertsthe plane wave front into a converging sphericalwave front with common center located at the focalpoint. Diffraction pattern at the focal plane of anaberration-free optical system is the Airy pattern,and Lommel functions are used to describe the fielddistribution in the focal volume. Engineering of theAiry pattern using pupil function manipulation hasdrawn attention of several researchers (Linfoot andWolf 1953, Welford 1960, Singh and Dhillon 1969,Singh and Gupta 1970, 1971 [66-70]), (Sheppardand Wilson 1979 [71], Born and Wolf 1989 [46],Horikawa 1994 [72], Luo and Zhou 2004 [73]).Positional shift of the high intensity regions fromthe geometrical focal point for low Fresnel numberoptical systems has also become an area of interestof some research groups (Li and Wolf 1981, Li1982, Carter 1982, Li and Platzer 1983 [74-77],Stamnes 1986 [45]), (Carter and Aburdene 1987,Poon 1988, Mahajan 1994 [78-80], Mahajan2001[51]). Positional shift of the high intensity regionfrom the focal point is referred to as 'focal shift'.

For various reasons, focusing of optical beamsby high numerical aperture (NA) systems hasattracted considerable interest in optics(Ingnatowsky 1919, 1920, Hopkins 1943, 1945,Richards 1955[81-85]), (Richards and Wolf 1956,1959, Burtin 1956, Focke 1957, Wolf 1959 [86-90]), (Barakat 1961, Barakat and Levi 1963,Carswell 1965, Innes and Bloom 1966, Luneburg1966 [91-95]), (Tremblay and Boivin 1966, Boivinand Wolf 1965, Boivin et al 1967, 1978, Carter1973a,b [96-101]), (Hardy and Treves 1973,Harrach 1973, Yoshida and Asakura1974a,b, vanden Berg and Zieren 1980 [102-106]), (Sheppardand Wilson 1981,1982, Sheppard 1978, 1988,1997[107-111]),(Sheppard 2001, 2004a,b, 2007

[112-115]), (Sheppard and Torok 1997a-c, 1998,2003, Sheppard and Yew 2008 [116-121]),(Sheppard et al 1977, 1994, 2008, Barakat 1987[122-125]), (Sheppard and Matthews 1987, Kant1991, 1993a,b,1995 [126-130]), (Kant 1996,2000, 2004, 2005, Mansuripur 1986 [131-135]),(Mansuripur 1991, Sheppard and Gu 1991,1993,Visser et al 1991, Visser and Wiersma 1991 [136-140]) (Visser and Wiersma 1992-1994, Flagelloand Milster 1992,Flagello and Rosenbluth 1992[141-145]), (Flagello and Rosenbluth 1993, Flagello1993, Hsu and Barakat 1994, Visser and Wiersma1994, Muller and Brakenhoff 1995 [146-150]),(Flagello et al 1996, Dhayalan 1996, Mittleman etal 1996, Dhayalan et al 1997, Karman et al 1998[151-155]), (Torok and Varga 1997, Torok andWilson 1997, Jiang and Stamnes 1998, Gu 1996a,Gu 1996b [156-160]), (Gu 2000, Müller et al 1997,Bahlmann and Hell 2000, Cronin et al 2000,Dhayalan and Stamnes 2000 [161-165]),(Sheppard and Aguilar 2000, Torok and Gu 2000,Varga and Torok 2000a,b, Drechsler et al2001[166-170]), (Lieb and Meixner 2001, Rhodes2001, Shimura and Milster 2001, Stamnes 2001,Rhodes et al 2002 [171-175]), (Helseth 2001-2004,2006, Varga 2002, Arnison and Sheppard2002 [176-180]), (Blanca and Hell 2002, Chon etal 2002, Hess and Webb 2002, Zhan and Leger2002, Dorn et al 2003a [181-185]), (Ganic et al2003, Ruckstuhl and Seeger 2003, Alvarez-Cabanillas et al 2004, Davidson and Bokor 2004,Diehl and Visser 2004 [186-190]), (Haeberle 2003,Sherif and Torok 2004, 2005, Torok and Munro2004, Torok 1998 [191-195]), (Torok 2004, vande Nes et al 2004a-b, Flagello and Progler 2005a,b[196-200]), (Munro and Torok 2004, 2005,2007,Bomzon and Gu 2006, 2007 [201-205]), (Guo etal 2006a,b, Bokor and Davidson 2007a,b,Botcherby et al 2007 [206-210]), (Bouchelier etal 2007, Hao and Leger 2007, Iketaki et al 2007,Juskaitis 2007, Khoptyar et al 2008 [211-215]),(Lindlein et al 2007, Liu and Lu 2007, Moore et al2007, Yew and Sheppard 2007a, Zhao C-.L. et al2007 [216-220]), (Alali and Massoumian 2008,Ando et al 2008, Bowlan et al 2008, Foreman et al2008, Gao 2008 [221-225]), (Jabbour and Kuebler2006-2008, Jabbour et al 2008, Kauert et al 2008


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[226-230]), (Kim et al 2008, Lerman and Uriel2007, Lerman and Levy 2008, Leutenegger andLasser 2008, Li, J.-s. et al 2008 [231-235]), (Olivierand Beaurepaire 2008, Romero and Hernandez2008, Rydberg 2008, Schonbrun and Crozier2008, Sheppard and Martinez-Corral 2008 [236-240]), (Sheppard and Yew 2008 [121], Sherif et al2008, Stadler et al 2008, Torok et al 1995a-c [241-245]), (Torok et al 1996a, b,1997,2006, 2008[246-250]), (Vamivakas et al 2007, 2008,Wang etal 2008, Zapata-Rodriguez 2008, Zhan 2006 [251-255]), (Zhang et al 2008a-d, Zhao Y et al 2008,Zhao Z-g et al 2008 [256-261]).

Literature on the high NA focusing bydiffractive lenses is also available (Sergienko et al2001, Menon et al 2006. Oron et al 2000a [262-264]) have demonstrated that a diffractive lensbehaves differently than an aplanatic lens.Investigations on the high NA focusing of beamshave been carried out by making the modificationin the scalar diffraction theory or by using thevectorial characteristics of the beam. (Ingnatowsky1919, 1920 [81-82]) started work on high NAfocusing and (Hopkins 1943, 1945[83-84], Burtin1956 and Focke 1957 [88-89]) also madecontributions in this direction. However thedetailed framework to analyze the focused structureof the beam using the vectorial characteristics wasinitiated by (Wolf 1959 [90] and Richards and Wolf1959 [87]). The scalar Huygens-Fresnel principlehas also been reformulated to take into accountthe vector nature of light Marathay and (McCalmont2001 [265]). The Debye approach for high NAfocusing systems (Richards and Wolf 1959 [87])begins to give error in the case of stronglyconvergent beam and study on the subject has beencarried out by (Duan and Lu 2004c [57]) based onRayleigh-Sommerfeld diffraction integral.

As is well-known, a plane wave is purelytransversal. However it possible to generate thelongitudinal polarization components by providingspherical curvature in the wavefront using high NAsystems (Richards and Wolf 1959 [87]). Restrictionon the spatial expansion of the beam is responsiblefor the mechanical strength even in a plane wave

and this logic has been used to explain the origin ofmechanical force in Beth's experiment (Allen et al2003[266]). Optical beam focused by high NAsystems possesses all three polarizationcomponents namely x-, y-, and z in contrast totransverse field in the low NA systems.Contribution of the longitudinal polarizationcomponent increases with an increase in the angleof convergence. As a result, high NA focusing ofthe optical beams has attracted much attentionduring the last decade. Polarization distribution ofthe beam becomes important in shaping the pointspread function (PSF) in high NA focusing(Youngworth and Brown 2000, Youngworth 2002[267-268], Zhan and Leger 2002 [184], Biss andBrown 2001, 2003 [269-270]), (Sun and Liu 2003,Sheppard and Choudhury 2004, Biss 2005,Beversluis et al 2006, Chen and Zhan 2006 [271-275]), (Grosjean and Courjon 2007, Iglesian andVohnsen 2007, [276-277], Zhan 2006 [255]) andmanipulation of the PSF by polarization distributionis referred to as 'polarization engineering' (Hao etal 2008 [278]). Initial studies on the subject werelimited to the beam with homogeneous polarizationdistributions such as linear, circular and ellipticalpolarizations. However, spatially inhomogeneouspolarization has drawn attention of researchers inrecent years due to its increasing number ofapplications (Scully and Zubairy 1991, Niziev andNesterov 1999, Salamin and Keitel 2002, Dorn etal 2003b[279-282], Davidson and Bokor 2004[189]), (Quabis et al 2000, 2001, 2005, Ferrari etal 2007, Gupta et al 2007 [283-287]), (Salamin2007a,b [288-289], Zhao Z-g et al 2008 [261]).One of the reasons for increasing interest in theinhomogeneous polarization is due to availability oftechniques for the generation of such beams.Radially and azimuthally polarized beams areexamples of inhomogeneous polarizationdistribution, and these are also referred to ascylindrical vector (CV) beams (Hall 1996 [290]).Wigner function of high convergent and non-paraxialwave has also been investigated (Sheppard andLarkin 2001a,b [291-292]).

With ever increasing number of applications,


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the three-dimensional structure of the beam in thefocal region of high NA systems has attracted agreat deal of attention in recent years. Prominentexamples are applications in data storage (Day andGu 1998, Guo et al 2008, Yuan et al 2007[293-295]), microscopy (Kawata et al 1994 [296],Ruckstuhl and Seeger 2003 [187], Torok and Kao2003, Boruah and Neil 2007, Tang et al 2007 [297-299]), (Yew and Sheppard 2007b, Mondal andDiaspro 2008 [300-301], Olivier and Beaurepaire2008 [236]), and optical trapping (Ganic et al2004a,b, Zhan 2004a, Marenda et al 2007, Padgett2007 [302-306], Zhao C-L et al 2007 [220]) etc.Due to the dominating role of polarization in thehigh NA systems, the scalar diffraction theory doesnot give accurate results (Richard and Wolf 1959[87]). Tight focusing of a singular beam has alsoemerged as a promising alternative in particletrapping (Gahagan and Swartzlander 1996, 1999[307-308], Allen et al 2003[266], Jeffries et al 2007,Nieminen et al 2008 [309-310], Zhao et al 2008[260]) and in stimulated emission depletion (STED)microscopy because of the beam's inherent darkcore in the center (Torok and Munro 2004 [194],Willig et al 2006a,b, Iketaki et al 2006 [311-313],Iketaki et al 2007 [213]), (Bokor et al 2005, 2007,2008 [314-316]).

Deviation of the spherical wavefront from itsideal shape, and its impact on the diffraction patternhas been investigated extensively. This deviation isreferred to as aberration and its effect on thediffraction pattern in the focal region has beeninvestigated by several groups. In a number ofpapers, researchers have studied the effects ofaberrations in terms of optical transfer function(OTF) and/or PSF (Kapany and Burke 1962, Barakatand Houston 1966, Hunt et al 1976, Gupta et al1977, 1978 [317-321]), (Gupta and Singh 1978a,b, Yoshida 1982, Biswas and Villeneuve 1985, 1986[322-326]), (Born and Wolf 1989 [46], Williamsand Becklund 1989, Cojocaru et al 1990 [327-328],Mahajan 2001 [51], Sanyal and Ghosh 2002 [329],Goodman 2007 [54]). Deviation in the shape ofthe ideal wavefront is also possible due to deepfocusing of the beam into media of differentrefractive indices (Ling and Lee 1984 [59], Torok

et al 1995a-c, 1996a-b [243-247], Wiersma et al1997[330]). Subject has attracted considerableinterest in the recent years due to increasingnumber of applications in imaging (Sheppard andGu 1991[137], Sheppard et al 1994, Booth andWilson 2000, Ganic et al 2000 [331-333], Gu 2000[161]), (Fwo et al 2005 [334], Botcherby et al2007 [210], Escobar et al 2008 [335], Mondal andDiaspro 2008[301], Olivier and Beaurepaire 2008[236]), data storage (Day and Gu 1998 [293],Wang H-f et al 2007[336]), and optical trapping(Ke and Gu 1998 , Rohrbach and Stelzer 2002 [337-338], Ganic et al 2004a [302],Theofanidou et al2004, Reihani et al 2006[339-340]). Investigationson the compensation of the effect of sphericalaberration have also been subject of interest (Ganicet al 2000 [333], Theofanidou et al 2004 [339],Escobar et al 2006 [341]). Investigation on tightfocusing of astigmatic Gaussian beam has beencarried out by (Gregor and Enderlein 2005[342]).

A singular beam has an azimuthal angulardependence of the complex form exp (im φ), whereφ is the azimuthal coordinate in the transverse planeand m is an integer (Allen et al 2003[266]). Presenceof the azimuthal phase factor creates helical phasestructure with a point of undefined phase in theheart of the wavefront. The point of undefinedphase is referred to as 'singular point' (opticalvortex), and the accumulated phase around thispoint is 2mπ, where m is called the 'topologicalcharge'. When an optical vortex is hosted within aGaussian beam, the resulting beam exhibits anannular intensity profile with a dark core, whilstmaintaining a helical phase structure. Free-spacepropagation of singular beams in the paraxial andnon-paraxial regimes has been investigated byseveral authors (Kogelnik and Li 1966, Siegman1986[2-3], Allen et al 1992, Seshadri 2002[343-344], Allen et al 2003[266]),(Duan et al 2005,Zhangrong and Daomu 2007, Kang et al 2007, Meiand Zhao 2007, Yan and Yao 2008[345-349]),(Zhou 2006b, 2008b[350-351], Zhou and Zheng2008 [38]. Zhou 2006a [31]) has investigated theanalytical vectorial structure of the Laguerre-Gaussian beam in the far field. Effect of sphericalaberration arising due to refractive index mismatch


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on the tightly focused structure of linearly polarizedsingular beam has also been investigated (Zhang etal 2008b[257]). Interest in the structure of theapertured singular beams has increased in recentyears due to their various applications. Some ofthese applications are in astronomy, microscopy,optical trapping, and lithography etc (Swartzlander2001[352], Allen et al 2003[266], Torok and Munro2004[194], Levenson et al 2004, Iketaki et al 2005[353-354], Iketaki et al 2006 [313], Iketaki et al2007[213]), (Andrews 2008 , Davila Romero andAndrews 2008, Juzeliunas and Ohberg 2008,Padgett and Leach 2008, Spalding et al 2008,Swartzlander et al 2008 [355-360]).

Size and shape of the dark core in the diffractionpattern of the singular beam plays an important rolein many applications; for example the size of thefocused annular ring influences the transfer ofmechanical strength to the trapped particles(Courtial and Padgett 2000[361]), and to givesmallest fluorescent spot in the case of stimulatedemission and depletion microscopy (Torok andMunro 2004[194], Bokor et al 2007[315], 2008[316]). Structural change in the focused structureof the singular beam has also been investigated dueto refractive index mismatch (Zhang, Z-m et al2008a [256]). However role of the aberrations onthe focused structure of the singular beam has notbeen discussed in detail or studies have been limitedto low NA systems. It has been observed thateven with optics considered being well-corrected,the intensity distribution of a vortex beam getsdistorted in the presence of small amount ofazimuthally-dependent aberrations, in comparisonto that of the non-vortex beam (Boruah and Neil2006 [362]). Structural modifications in the focusedstructure of the singular beam due to primaryaberrations in low NA system have been carriedout in recent years (Roichman et al 2006, Fatemiand Bashkansky 2007, Singh et al 2007a-d, 2008a,b [363-370]). However these studies are not usefulfor high NA systems due to dominating role of thevectorial characteristics of the beam.

It is impossible to avoid presence of opticalaberrations completely in the focusing systems.

Hence investigations have been carried out tomeasure aberrations in the focusing systems andcompensation of the effect of aberrations bydifferent means (Tanaka and Yoshikawa 1992, Strandand Werlich 1994, vander Avoort et al 2005[371-373], Roichman et al 2006 [363], Wang et al2006a-c [374-376]). (Tanaka and Yoshikawa1992[371]) proposed a method for novel

calculation of aberrations in objective lenses inthe context of optical disk systems. (Mahajan2001[51]) has carried out extensive study on thecompensation of effect of aberrations using theZernike polynomials. Zenike representations havebeen used for variable numerical aperture systemsand in the presence of aberrations (Braat et al 2003,2005, Janssen et al 2008 [377-379]). It is well-known that possibility of aberrations can not beruled out even for well-corrected high NA systems.Investigations in this direction have been initiatedin the past by several groups (Sheppard 1988,1997[110-111], Visser and Wiersma 1991 [140],1992 [141], Kant 1993a [128]), (Kant 1995, 1996,2000 [130-132], Braat et at 2003 [377], Biss andBrown 2004 [380], Lan and Tien 2007 [381]). Focalshift in the high NA systems and vectorial beamhas also been investigated by several researchgroups (Sheppard and Torok 2003 [120], Li Y-J2005a, 2005b, 2008, Wang, X-e et al 2006[382-385]). However no systematic study on theeffect of aberrations seems to have been carriedout on the beams with phase singularity for highNA systems except by (Braat et al 2003, 2005[377-378]) who used extended Nijboer- Zernikerepresentation for evaluation of intensity in the focalregion of an aberrated high NA system. In view ofthe importance of the high NA focusing of vortexbeams, and their structure in the focal region, wehave undertaken a comprehensive study on thesubject (Singh et al 2008c-e [386-388]).

The present review is aimed at discussing brieflythe tight focusing of optical beams and theirapplications. The article is arranged as follows.Section 1 is introductory in nature and besidesoutlining the historical perspective, provides a briefreview of the tight focusing of optical beams. In


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section 2, basic theory and properties of opticalbeams in the high NA systems are outlined. Section3 discusses the numerical methods used to evaluatethe diffraction integrals. Section 4 providesinformation on the singular beams and theirstructure in the focal region of high NA systems.Section 5 describes applications of tightly focusedbeams and few of these applications are discussedbriefly. Section 6 mentions some miscellaneousinvestigations. Finally in section 7, we presentsummary and conclusions.

2. Theoretical background

When a positive lens projects real image of apoint source, it produces converging wavefrontwith a common center of conversion. However,image of a point never becomes as precise as thepoint source itself and spreading takes place aroundthe image point. Spreading around the image pointis referred to as 'point spread function' (PSF).Spreading results due to finite size of the focusingsystem, and referred to as diffraction limitedsituation. We have considered the optical geometryshown in Fig.1. The position of a point in the exitpupil-, and observation planes, is denoted by polarcoordinates (ρ, φ, f) and (rP,φP,z) respectively. ρ isthe normalized position coordinate written asρ = r/ r0 where r is the distance of a point fromthe center on the exit pupil plane of radius r0. Thediffraction image centered at the Gaussian imagepoint P/ is aberration-free, if converging waveemanating from the exit pupil has center of curvatureat point P/. In the presence of aberration, center ofcurvature of wave shifts from point P/ because ofthe deviation of the actual wave front from the idealwave front at the exit pupil (Born and Wolf 1989[46], Mahajan 2001 [51]).

Paraxial or Gaussian optics is only approximatelycorrect in describing the diffraction pattern forsmall angles. In this situation, approximations likesinθ θ≈ and cosθ ≈1are correct and next term intheir expansion can be ignored. If the next term isconsidered in the expansion, then we have third-

order approximation, in which case sin /!θ θ θ≈ − 3 3

and cos /θ θ= −1 22 are the lowest order terms. The

aberrations that affect image quality are generallymost important in third-order approximation(Mahajan 2001 [51], Trappe et al 2003 [389]).These aberrations have been classified by von Seideland are referred to as 'Seidel aberrations' (Pedrotti1993[390]). For coherent systems the phaseerrors due to distorted wavefront can also beexpanded in a series of Zernike polynomials (Bornand Wolf 1989 [46], Mahajan 2001 [51]). Advantageof the Zernike polynomials lies with their usefulnessin balancing the aberrations to reduce the effect ofaberrations on image quality. The Seidel (or primary)aberrations are divided into five categories, namelyspherical aberration, coma, astigmatism, curvatureof the field, and distortion.

As is well-known, light is transverse in natureand in the case of optical systems with smallnumerical aperture, only small twist of relativepolarizations takes place. For these, it is reasonableapproximation to use scalar theory of diffractionor ignore vectorial characteristics of the beam inthe evaluation of diffraction pattern (Wolf 1959 [90],Richards and Wolf 1959[87], Boivin and Wolf 1965[97], McCutchen 1965 [391]). However vectorialcharacteristics of the beam in high NA focusingbecomes dominating and the diffraction evaluationcan be categorized into two parts: one based onvectorial properties and the other on scalar. Thescalar theory of diffraction for converging beambegins to breakdown for NA of about 1/ 2 andfocusing is characterized as high NA focusing(Sheppard and Matthews 1987 [126]). Howevermodifications in the scalar theory have been madefor high NA systems (Miks et al 2007 [392]).

2.1 Debye-Wolf diffraction integral

With dominating role of the vectorialcharacteristics of the beam in high NA focusing, itis desirable to use diffraction integral which canincorporate the vectorial characteristics of theincident beam in the diffraction integral. Debye-Wolf integral is suitable for this purpose and it givesthe same results for low NA systems as obtainedusing the scalar diffraction theory. Investigationson the Debye integral and its validity have also beena subject of much interest (Wolf and Li 1981,


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Pedersen and Stamnes 1983, 1992 [393-395],Stamnes 1986 [45], Gu 2000 [161], Sheppard 2000[396], Zapata-Rodriguez 2007 [397]). The fielddistribution at a point in the image space is given(Wolf 1959 [90], Richards and Wolf 1959 [87]) as:

E x y zik a s s

se ds dsP P P

x y

z

i k s s s rx y

x y( , , )( , ) [ ( , ) . ]=− ∫∫ −

2π Ω

Φ (1)

H x y zik b s s

se ds dsP P P

x y

z

i k s s s rx y

x y( , , )( , ) [ ( , ) . ]=− −∫∫2π

Φ

Ω (2)

b s a= × (3)

Here E and H are respectively the electric andmagnetic vectors, a and b and are strength factorsat the exit pupil. The strength factor includescontribution from the polarization, phase, and

amplitude distribution at the exit pupil. k( )= 2πλ

is

the wave number, λ is wavelength, Φ is aberrationfunction of the system, s is unit vector along atypical ray in the image space (with its positivedirection in the direction of propagation of thebeam) and Ω is the solid angle formed by all thegeometrical rays which pass through the exit pupilof the system.

2.2 Focusing by an aberrated high NA system

Following (Richards and Wolf 1959 [87]), thefield distribution in the focal volume of a high NAoptical system is given by the diffraction integral(also known now as the Debye-Wolf integral) as:

E Pik a s s

sik s s s r P ds dsx y

zx y x y( )

( , )exp { ( , ) . ( )}=− −⎡⎣ ⎤⎦∫∫2π Ω

Φ (4)

where a is a strength factor, r(P) is the radiusvector connecting the point P with the Gaussianfocus which is also the origin of the coordinate

system (Fig. 1), s s s sx y z= ( , , ) is the direction vectorof a typical ray in the image space,

Φ

is the waveaberration function which denotes the deviation ofthe actual wavefront from the ideal one, and

k( )= 2πλ

. The integral is taken over the entire

surface of the wavefront leaving the exit pupil. Inthe presence of the aberrations, the radial distanceof any point on the wavefront depends on the angularcoordinates and written (Kant 1995 [130]) as:

r f W( , )θ φ = + Δ

r P r P i j kP P P P P( ) ( )(sin cos sin sin cos )= + +θ φ θ φ θ (5)

Φ = + +A A As a cρ ρ φ ρ φ4 2 2 3cos cos

where ρθα

= sinsin

is zonal radius, is maximum angle

of convergence (Fig. 1b), f is the focal length ofthe optical system, Φ is phase aberration, As,Ad, Aa and Acare the spherical aberration, defocusing, astigmatic,and comatic aberration coefficients respectively inthe units of wavelength, and ( , , )rP P Pθ φ are theposition coordinates of a point on the observationplane.

Using the concept of the two orthogonaltangent vectors in the polar and azimuthaldirections, the unit normal to the aberratedwavefront is given as:

sr

rr

r

s

x

y

= − ∂∂

+ ∂∂

⎛⎝⎜

⎞⎠⎟

=

1 1 1

1

σθ φ

θθ φ

θ φφ

σθ

sin cos cos cossin

sin

sin ssin cos sinsin

cos

cos sin

φθ

θ φθ φ

φ

σθ θ

− ∂∂

− ∂∂

⎛⎝⎜

⎞⎠⎟

= + ∂

1 1

1 1

rr

rr

srz

rr∂

⎛⎝⎜

⎞⎠⎟θ

(6)

where σis the normalization factor given as :

σθ θ φ

= + ∂∂

⎛⎝⎜

⎞⎠⎟

+ ∂∂

⎛⎝⎜

⎞⎠⎟

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

⎡

⎣⎢⎢

⎤

⎦⎥⎥

11 12

2

2

21 2

rr r

sin

/

Using the binomial expansion and ignoring thehigher order derivative of the position vector dueto small value of the aberrations, we write

11

σ≅ − +Θ Θo( )


204

where Θ = ∂∂

⎛⎝⎜

⎞⎠⎟

+ ∂∂

⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢⎢

⎤

⎦⎥⎥

12

12

2

2

2

rr rθ θ φsin

Using the approach of (Richards and Wolf 1959[87]) for the linearly polarized beam and of (Visserand Wiersma 1991 [140]) for the arbitrarypolarization, one can derive the contribution of the

Fig. 1. (a) Schematic representation of the wavefront deviation and (b) geometrical configuration of the focusing system


205

polarization factor in the evaluation of strengthfactor a.

a s k s s

A s s s B s s s s s

A sP x y

y x z x y x y z

( , ) ( . ) /( )

( ) ( )

(/θ φ = +

+ + − +

−1 2 2 2

2 2

xx y x y z x y z

x y x y

s s s s B s s s

A s B s s s

+ + +

− + − +

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

) ( )

[ (( ) ( )]( )

2 2

2 2

⎥⎥⎥⎥⎥ (7)

or a A PP = 2 ( ) ( , )θ θ φ

where A s k sz21 2 1 2( ) ( . ) ( )/ /θ = = , and

Ps s

A s s s B s s s s s

A s s s s sx y

y x z x y x y z

x y x y( , )( )

( ) ( )

(θ φ =+

+ + − +

− +12 2

2 2

zz x y z

x y x y

B s s s

A s B s s s

) ( )

[ ( ) ( )]( )

+ +

− + − +

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

2 2

2 2 (8)

Here A( , )θ φ , B( , )θ φ are respectively thestrengths of the x-, and y- polarized input beams,A2(θ ) corresponds to the apodization factor equalto cos /1 2θ for an aplanatic lens (Richards and Wolf1959 [87], Stamnes 1986 [45], Gu 2000 [161]).In the aberration-free case, the position coordinatesat the wavefront are independent of the polar andradial coordinates, and in this situation the unitnormal to the wavefront is transformed to the caseof Richards and Wolf. The polarization distributionin the aberration-free case is transformed intopolarization matrix of (Helseth 2004 [177]). Foraberration-free case and x polarization ( )B = 0 , thepolarization distribution at the exit pupil istransformed into results of the (Richards and Wolf1959 [87] and Boivin and Wolf 1965 [97]).Expression for the unit normal to the aberratedwavefront can be written as the sum of two termsrepresenting the vector along the unit normal tothe ideal wavefront and other representing thedeviation such as:

s n F= − +( ) ( , )11

Θσ

θ φ

where n i j k= + +sin cos sin sin cosθ φ θ φ θ

and

Fr

rr

r

Fr

r

x

y

( , ) ( cos cossin

sin )

( , ) ( co

θ φθ

θ φθ φ

φ

θ φθ

= − ∂∂

+ ∂∂

= − ∂∂

1 1

1ss sin

sincos )

( , ) ( sin )

θ φθ φ

φ

θ φ θθ

− ∂∂

= ∂∂

⎧

⎨

⎪⎪⎪

⎩

⎪⎪⎪

1

1r

r

Fr

rz

(9)

Expressing the ds dsx y as a function of d dθ φ

and using Eqs. (5)-(8), Eq. (4) can be written, as:

E u vikf

A im A P

ik n

( , ) ( )exp( ) ( ) ( , )exp

[ (

=−

+ −

∫∫2

1

1 20

2

0πθ φ θ θ φ

σ

πα

Φ Θ FF r P iu

iv

). ( )] expcossin

exp

sinsin

c

⎧⎨⎩

⎫⎬⎭

× −⎛⎝⎜

⎞⎠⎟

−⎛⎝⎜

⎞⎠⎟

θα

θα

2

oos( )φ φ φ θ−⎡⎣⎢

⎤⎦⎥

P J d d

(10)

where the optical coordinates ( , )v u are defined as:

v k r

u k rP P

P P

=

=

sin sin

cos sin ,

θ α

θ α2

and Js s s sx y x y= ∂

∂∂∂

− ∂∂

∂∂

⎛

⎝⎜

⎞

⎠⎟θ φ φ θ

2.2 Focusing by aberration-free high NA systems

In the aberration-free case ( )Φ = 0 , the unitnormal to the wavefront is given as:

s n i j k= = + +sin cos ˘ sin sin ˘ cos ˘θ φ θ φ θ

where (˘,˘,˘)i j k are the unit vectors in the Cartesiancoordinate systems. The field distribution in thefocal region for an aberration-free case is given(Richards and Wolf 1959 [87], Helseth 2004 [177])as:


206

E u v if A A P

im iv

( , ) ( / ) ( ) ( ) ( , )exp

( ) exp[sin

si

= −

−

∫∫λ θ θ θ φ

φα

πα

10

2

02

nn cos( )]

exp(sin

cos )sin

θ φ φ

αθ θ θ φ

−

−

P

iu

d d2

(11)

where f is focal length of the optical system and λis the wavelength of light in the medium withrefractive index nin the focal region. P( , )θ φ represents the

polarization distribution of the input beam, E0( , )θ φrepresents the complex amplitude of the inputbeam, A1( )θ represents amplitude distribution of the

beam, A 2( )θ and is the apodization factor. Thepolarization distribution of the input beam can bewritten (Helseth 2004 [177]) as :

P

A B

A( , )

[cos cos sin ]

[cos sin cos sin cos ]

[cos coθ φ

θ φ φθ φ φ φ φθ=

+ +−

2 2

ss sin sin cos ]

[cos sin cos ]

sin cos sin sin

φ φ φ φ

θ φ φθ φ θ φ

− ++

− −

⎡

⎣

B

A B

2 2

⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

(12)

where A( , )θ φ and B( , )θ φ are the strengths of thex-, and y- polarized incident beams respectively. Inthe case of uniform polarization the strength ofthe x-, and y- polarized incident beams areindependent of the angular coordinates, whereasin the non-uniform polarization case the strengthfactors depend on the coordinates. Azimuthal andradial polarization distributions are example of non-uniform or inhomogeneous polarizationdistribution. The strength factor A ( , ) cosθ φ φ= ,B( , ) sinθ φ φ= for radial case and B( , ) sinθ φ φ= ,B( , ) sinθ φ φ= in the in the azimuthal polarizationcase respectively (Helseth 2004 [177]). Fig. 2shows how the linear polarization state istransformed into radially and azimuthally polarizedbeam using the liquid crystal devices. A polarizationconverter is commercially available from ARCoptix(Switzerland) and can be used to convert the linearlypolarized beam into radial or azimuthal polarization.

Intensity distributions of a Gaussian beam atthe focal plane for various polarization distributionsare shown in Fig. 3. For the linear polarizationdistribution, the focal spot shows elliptical structure

for an aplanatic system with α =75 (Fig. 3a), and

Fig. 2. Polarization distribution of the linear, radial andazimuthal states (web page of ARCoptix Switzerland)


207

circular symmetry is achieved for incident beamswith radial and azimuthal polarization as shown inFigs. 3b & 3c. In case of a low numerical aperturesystem, circular symmetry in the intensitydistribution is also obtained even for a linearlypolarized beam (Fig. 3d). Fig. 3d shows the intensitydistribution of the linearly polarized (x

polarized) Gaussian beam at the focal plane of an

aplanatic lens withα =30 . These results show thatthe circular symmetry vanishes in the tight focusingof the optical beam, and symmetry can be achievedfor radial and azimuthal polarizations even in caseof tight focusing. Removal of the circular symmetryfor the linearly polarized beam in a high NA focusing

Fig. 3. Intensity distribution of the Gaussian beam focused by an air aplanatic lens with α =75 for

(a) linear (b) radial and (c) azimuthal polarizations ; for α =30 (d) for linear polarization


208

results due to depolarization effect. The square ofthe polarization components for the x polarizedGaussian beam focused by an aplanatic lens with

are α =75 shown in Fig. 4. The electric field at thefocal point is purely x-polarized (Fig. 4a) and thecontribution of different polarization componentsvaries along different directions. |Ey|

2 and |Ez|2

possess respectively four fold (Fig. 4b) and twofold (Fig. 4c) symmetry about the coordinate axes.In order to obtain a circular focal spot, thepolarization distribution of the input beam must berotationally symmetric, as is the case e.g. forradially or azimuthally polarized fields (Quabiset al 2005 [285]).

3. Numerical methods for evaluation of theintegral

Analytical solution of the diffraction integrals isdifficult in the presence of aberrations. Hencenumerical methods are preferred for theirevaluation, and a fair amount of literature is availableon the subject (Hopkins 1957, Barakat 1964, 1980,Hopkins and Yzuel 1970, Matsui et al 1976[398-402]), (Yzuel and Arlegui 1980, Gravelsaeterand Stamnes 1982, Stamnes and Spjelkavik 1983,Hazra 1988, Mendlovic et al 1997 [403-407]),

(Stamnes and Heier 1998, Cooper et al 2002,Cooper and Sheppard 2003, Engelberg and Ruschin2004, Leutenegger et al 2006 [408-412]), (Xin etal 2007, Kaddour et al 2008, Shimobaba et al 2008[413-415]). To avoid the numerical difficultiesseveral assumptions are made like paraxialapproximation and number of samples. Even thoughthe paraxial approximation is not a limitation in theintegral in vectorial diffraction, direct numericalevaluation of the integral is time consuming. Herewe discuss some of the numerical methods usefulin the evaluation of the diffraction integrals on thebasis of speed and accuracies.

3.1 Hopkins' Method

(Hopkins 1957 [398]) presented a numericalmethod for the frequency response of opticalsystems based on dividing the region of integration intosmall rectangles. Let us consider the diffraction integralin the form (Hopkins 1957 [398] as given below:

I if x y dx dyS

= { }∫∫ exp ( , ) (13)

Let the region of the rectangle be defined by

x x

y yp x

p y

= ±

= ±

ε

ε

,

Fig. 4. Distribution and contour lines of the squares of x-,y-, and z- polarization components at thefocal plane of an aplanatic lens with α=75º for an x- polarized Gaussian beam (m=0) with γ=1.

For As=0.0;(a) |Ex|2, (b) |Ey|

2(c) |Ez|2


209

The mean value of the integrand of (13) over thisrectangle will be given by

14ε ε ε

ε

x yp q x p q p

x

x

if x y if x y x x dxp x

p x

exp ( , ) exp ( , )( )

exp

{ } −{ } ×−

+

∫

iif x y y y dyy p q qy

y

p y

p y

( , )( )−{ }−

+

∫ε

ε

i.e exp ( , )sin ( , )

( , )

sin ( , )if x y

f x y

f x y

f x yp q

x x p q

x x p q

y y p q{ } { }{ }

{ε

ε

ε }}{ }ε y y p qf x y( , ) (14)

The value of the integral (14) is given by

I if x yf x y

f x yx y p qq

p

x x p q

x x p q

= { } { }{ }∑∑( ) exp ( , )sin ( , )

( , )

si4ε ε

ε

ε

nn ( , )

( , )

i f x y

f x yy p q

y y p q

{ }{ }ε

where 4ε εx y is the area of elementary rectangle. In

the limiting case of ε x → 0 and ε y → 0 , the righthand side of Eq. (14) converges to the correctvalue of the integral and it simply becomes thedefinition of integration. However increasing thenumber of rectangles results into increasing the timeof evaluation. Limitation and drawback of themethod is associated with the fact that boundaryof the integration domain should be in rectangularform and inaccuracy arises due to mismatchbetween boundary of the sampled rectangles andboundary of the integration. One can accuratelyhandle rectangular aperture by using Cartesiancoordinates, and circular and elliptical apertures byusing polar coordinates in this apporach, but notaperture of general shape, unless the number ofsubdivisions is greatly increased (Stamnes et al1983[405]).

3.2 Linear Phase and Amplitude Approximation

(Gravelsaeter and Stamnes 1982 [404])proposed a method based on linear phase andamplitude approximation as used by Hopkins forspeedy evaluation even in the high NA systems.Let us consider the diffraction integral in the formas given below:

I g x y if x y dx dyx

x

y

y

L

U

L

U

= ∫∫ ( , )exp[ ( , )]

The technique is implemented by dividing theintegration domain in M x N rectangular subdomains, so that

I Im nm

M

n

N

===

∑∑ ,11

where I g x y i f x y dx dym n

x

x

y

y

mL

mU

nL

nU

, ( , )exp[ ( , )]= ∫∫

Let us introduce the quantities (with n =1, 2…, N;m=1, 2,…..M)

Δ Δx x x y y y

x x x ym m

UmL

n nU

nL

mA

mU

mL

nA

= − = −

= + =

1 2 1 2

1 2 1 2

/ ( ), / ( )

/ ( ), / ( yy ynU

nL+ )

Performing first the x integration and using thelinearization in the phase function, we can write

f x y f x y a y b yx x

xx x xm m m

mA

mmL

mU( , ) ( , ) ( ) ( ) ;≈ = + − < <

Δ

where

a f x y f x y f x y

b f x y f xm m

AmU

mL

m mU

= + +

= −

2 3 1 6

1 2

/ ( , ) / ([ ( , ) ( , )],

/ [ ( , ) ( mmL y, )]

The coefficients am and bm are found by requiringsize such that in each subdomain phase errorΔ= −f x y f x ym( , ) ( , ) at the mid point be equal to halfof the negative phase error at either end point.This approach requires fewer subdomians incomparison to the (Hopkins approach 1957 [398])which results in less time for the computation. Anassumption is made that the amplitude g(x,y) variesso slowly over each subinterval that it may bereplaced by its value at the mid point and theexpanded function is substituted in the integral.Similarly the integration with respect to the ycoordinates can be carried out first by expandingboth the phase and amplitude functions such as:


210

I G y iF y dym n m m

y

y

nL

nU

, ( )exp[ ( )]= ∫

and

G y G y c dy yy

F y F y e fy y

m m n m n m nnA

n

m m n m n m nn

( ) ( )

( ) ( )

, , ,

, , ,

≈ = + −

≈ = + −

ΔAA

nyΔ

The coefficients , , and areobtained by the same as described for coefficientsam and bm. In the first integration (involving thevariable x) the amplitude is set equal to a constantin each subdomain, and the phase is linearalized; inthe second integration both the phase andamplitude are linearalized. In these methods, theboundary of integration is approximately zig-zagdue to division of the area of integration intorectangular subdomains and numerical integrationusing the polar coordinates provides less error incomparison to Cartesian coordinates. The numericalevaluation of the diffraction integral is still a timeconsuming process and it depends on the numberof sampling points in the quadrature method.Another most important numerical method for thediffraction evaluation is the fast Fourier transform(FFT) which is very useful in terms of the time ofcomputation. (Leutenegger et al 2006 [412]) havepresented a fast field calculation based on FFT forhigh NA systems. Instead of direct integration,vectorial Wolf-Debye integral is evaluated with theFFT for the evaluation of complex field in the focalregion of a high NA system.

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Tight Focusing of Optical Beams; A Review - Part 1web.iitd.ac.in/~psenthil/Conferences/InvertJSTP1-2009.pdf · Tight Focusing of Optical Beams; A Review - Part I 197 Tight Focusing