Nonparaxial singular beams inside the focal region of a high numerical-aperture lens

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1 Nonparaxial singular beas inside the focal region of a high nuerical-aperture lens Vladlen G. Shvedov Departent of hysics Taurida National V. Vernandsky University Siferopol Criea Ukraine Received: Abstract. Key optical technologies including lithography data storage optical tweezers icroscopy and ultrafast laser aterials processing rely on strongly focused light beas. Such beas are often used to exploit a vectorial nature of light and so detailed knowledge of the field structure inside a tight focus becoes increasingly iportant. So far theoretical studies of intra-focal optical field coponents have been ainly concentrated on spatially hoogeneous states of light polarisation. In this work we present a new developent in the calculations of local polarisation structure of tightly focused singular beas including radially and aziuthally polarised hollow beas. Keywords: singular bea vector optical field tight focus ACS: 4.5.Ja 4.60.Jf 78.0.Bh UDC: A rapidly increasing nuber of various practical applications in the last few years have attracted significant interest to studies related to three-diensional structure of the field inside a focus of high nuerical-aperture optical systes [ 5]. xperiental intrafocal apping of the squared electric field coponents is usually indirectly achieved with near field probes [6] point scatterers [7] fluorescent olecules and beads [8 9] and the knife edge ethod [0]. Recently a new approach has been developed [] for visualising nanoscale structure of the electric field inside the focal volue of tightly focused singular beas which is based on peranent iprinting of the field in transparent edia. Fro the theoretical point of view the first coprehensive studies of tightly focused linearly and circularly polarised Laguerre-Gaussian beas have been perfored in Refs. [ 3]. In Refs. [ 4] the authors have attepted to analyse the focal spot of both radially and aziuthally polarised beas. The proble has been qualitatively solved in Ref. [4] for the case of non-apodised cylindrical vector beas hence providing basic understanding of the focal structure of real beas. The general ethod allowing one to solve such probles is based on Debye vector integrals. A coplexity of the proble for soe types of vector beas e.g. radially or aziuthally polarised beas is related to finding an integrand which would satisfy the conditions for the paraxial waves ( ( 4 i ) 0 ) and siultaneously describe the vector field coponents. In the z above expression xex ye y and u / u with ex e y being the Cartesian unit vectors and the wavelength of light. In this work we wish to show that the coplete vector function for radially or aziuthally polarised beas can be obtained by first finding solutions for linearly and circularly polarised optical vortices in which case the choice of the integrand does not pose any difficulties. We start Ukr. J. hys. Opt. 0 V 3 09

2 Vladlen G. Shvedov with finding the solutions of the Debye proble for linearly polarised optical vortices following the foralis suggested in Refs. [ ]. The coplex aplitude of a scalar Laguerre-Gaussian bea at its waist ay be written as [] p 0 0 ( r z 0) A ( r / w ) L ( r / w )exp( r / w )exp( i) () where А 0 is the characteristic aplitude at the peak of intensity r the radial distance fro the bea axis the aziuthal angle in the entrance plane (see Fig. ) L ( x ) the Laguerre polynoial with the radial index p and the aziuthal index and w 0 the bea waist. The radial coordinate can also be presented as r f sin( ) where is the angle between the optical axis and the light ray (see Fig. ). Then the ratio r/w 0 becoes r a sin sin. () w w NA sin Here NA a / f sin stands for the nuerical aperture of the focusing lens (fro now on the refractive index of the ediu in which the bea is focused is assued to be equal to unity) is the axiu angle defining the bea convergence ( ) and the ratio a / w of the entrance aperture radius a and the bea waist w represents the truncation paraeter denoting the function of the bea inside the physical aperture. It defines the fraction of the bea which goes through the lens. The lens is overfilled if and underfilled if [ ]. ax p Fig.. Representation of focal geoetry for singular beas. (x y) and (x p y p ) are the Cartesian coordinates in a plane of physical aperture and a focal plane respectively r p denotes the radial distance of point fro z axis a the physical aperture a the radius of non-transparent disk which increases efficiency of the physical aperture the nuerical aperture (NA) of an objective the angular diension of circular bea stop of radius a w the paraeter defining size of an incident bea and f the focal length. In our case which deals with an optical vortex incident on a lens the radial index p = 0. Upon taking q. () into account q. () ay be written as [4] where ( r z 0) A ( )exp( i) (3) 0 sin sin ( ) A0 exp A sin sin 0 Ukr. J. hys. Opt. 0 V 3

3 Nonparaxial singular describes the aplitude function of a colliated optical vortex. According to the focusing geoetry shown in Fig. the field distribution in the focal region is given by 0 (4) ( r z) ( ia/ ) A ( ) ( ) A ( )exp( i) expikr sin cos( ) exp ikz cos sin dd where А is a coefficient that depends on the optical paraeters of the syste [6] the light wavelength the ter ( ) defines the polarisation distribution [4] at the entrance pupil and A gives the apodising function which is equal to cos for aplanatic lenses [5]. The polarisation distribution of the incident bea in the entrance pupil plane ay be presented as [4 ] ( ) b (cos cos sin sin cos ) b (cos sin cos ) b sin cos b sin sin where b and b denote the strengths of the x- and y-linearly polarised incident beas respectively. Then q. (4) can be reduced to a single integral by using the identity exp( i)exp( ikr sin cos( )) d i J( kr sin )exp( i ) (6) 0 where J ( kr sin ) is the Bessel function of the first kind and the th order. Taking into consideration the above arguents one can write the electric field coponents in the focal region of the x-polarised vortex bea as [] x y exp( ( ) ) exp( ( ) ) i exp( i ) I i exp( i( ) ) I i exp( i( ) ) I i i i I i i i I z i exp( i( ) ) I i exp( i( ) ) I. For coparison when the incident vortex is y-polarised the coponents of its electric field in the focal region are given by Here x exp( ( ) ) exp( ( ) ) y i i i I i i i I i exp( i ) I i exp( i( ) ) I i exp( i( ) ) I z i i exp( i( ) ) I i i exp( i( ) ) I. I ( r z) ( ia/ ) A ( ) cos ( cos ) J ( kr sin )exp ikz cossin d I ( r z) ( ia/ ) A ( ) cos J ( kr sin )exp ikz cos sin d I( r z) ( ia/ ) A ( ) cos ( cos ) J( kr sin )exp ikz cossind. The analysis of the above expressions shows that all of the three electric field coponents are present in the focal region of the optical vortex and the light intensity is generally nonzero at the optical axis: (5) (7) (8) Ukr. J. hys. Opt. 0 V 3

4 Vladlen G. Shvedov x ( ) y ( ) z ( ) I r z r z r z. The solutions found above also allow one to exaine focusing of elliptically polarised optical vortices with the coplex field aplitude in the entrance pupil plane given by: i ( ) A ( )exp( i) e (9) where x and y stand for the x and y vector field coponents and is the phase shift between the. (Notice that the bold type indicates a vector variable). A circularly polarised vortex is produced when x y and /. The following sign convention for the circularly polarised beas is adopted hereafter: the circular polarisation is referred to as right-handed if / and left-handed if / i.e. i σ. Here σ ( xi y )/ denotes the unit vector of circular polarisation x y and x y the unit Cartesian vectors. Using the sign convention introduced we can add the field coponents in qs. (7) and (8) taking into account the phase shift given by q. (9). Then the Cartesian coponents of the righthanded field in the vicinity of the focus take the following fors: x exp( ) exp( ( ) ) y exp( ) exp( ( ) ) i i I i i I i i i I i i i I z 4 exp( ( ) ) i i I. Siilarly for the left-handed light field we obtain x i exp( i ) I i exp( i( ) ) I y exp( ) exp( ( ) ) i i i I i i i I z 4 exp( ( ) ) i i I. Using the sae notations we get i expi I i expi I x y i i exp( i ) I i exp( i( ) ) I 4i expi I. z for the both vortices. These relations allow one to deduce the salient features of the electric field in the focal region. For instance the fundaental Gaussian bea at the focal point (z = 0 r = 0) always retains the initial state of polarisation []. At the sae tie the optical vortices with the topological charges = have in general nonzero electric fields at this point due to contribution fro the longitudinal coponents of the field. When the topological charge is greater than three (i.e. 3 ) the focal intensity is always zero at the centre of the bea [ 6]. Of particular interest is the case of circularly polarised vortices with. According to qs. (0) the vortex preserves zero intensity along the whole optical axis when the sign of the topological charge coincides with that of the handedness ( ). When the charge and the polarisation have the opposite signs ( ) an intensity axiu appears at the focus owing to the longitudinal coponent of the electric field z (see Fig. a). A detailed analysis of the properties of circularly polarised and ore coplex singular odes is easier to perfor using the polar basic vectors of the circular cylindrical coordinates (see q. (9)): x y (0) Ukr. J. hys. Opt. 0 V 3

5 Nonparaxial singular e x cos y sin r e y cos x sin where e r and e are the unit vectors along the radial and aziuthal directions. The field coponents in q. (9) transfor to the new coordinate syste as follows: r exp( ( ) ) exp( ( ) ) i i I i i I i i exp( i( ) ) I i exp( i( ) ) I z 4 i exp( i( ) ) I. After siplification they can be rewritten as r i I I expi ii I I expi 4i expi I. z In order to obtain solutions for the radial and aziuthal odes at the focus we use the following reasoning. Radial and aziuthal beas with coinciding waists and the sae divergences acquired while propagating along the optical axis z of a focusing syste can be presented as TM r / w0 G e r / 0 where G A0 exp r / w0 Ukr. J. hys. Opt. 0 V 3 3 () () T r w G e (3) describes the fundaental Gaussian bea given by q. (). The circularly polarised optical vortices with a zero orbital angular oentu ( ; ) ay be presented as a superposition of the two odes given by qs. (3) [7 8]: TM T 0 i r / w Gσ exp( i) (4) where σ ( er i e )exp( i ) is the unit vector of circular polarisation in the cylindrical coordinates. Hence by knowing the focal field distribution of the circularly polarised vortices () it is possible to reconstruct with the aid of q. (4) the field of the corresponding radial and aziuthal odes. Indeed fro the syste of equations we find where TM i T TM i T TM. (5) T i In this case the fields given by qs. () ay be expressed as r r i I I i I I i I I i I I z 4 0 z 4 0 I I (6)

6 Vladlen G. Shvedov I ( r z) ( ia/ ) A ( ) cos ( cos ) J ( kr sin )exp ikz cossin d I ( r z) ( ia/ ) A ( ) cos ( cos ) J ( kr sin )exp ikz cos sin d I ( r z) ( ia/ ) A ( ) cos J ( kr sin )exp ikz cossin d 0 0 I ( r z) ( ia/ ) A ( ) cos ( cos ) J ( kr sin )exp ikz cossin d I ( r z) ( ia/ ) A ( ) cos ( cos ) J ( kr sin )exp ikz cossin d. By applying the well-known identity J( x) J( x) to the above integrals for the circularly polarised single-charged vortices one can reduce qs. (6) to the following: where r ii 4 ii z I0 r 4 r z ii r 4 ii 4 I I ( ia/ ) A ( ) cos J ( kr sin )exp ikz cossin d r I ( ia/ ) A ( ) cos J ( kr sin )exp ikz cossin d I ( ia/ ) A ( ) cos J ( kr sin )exp ikz cossin d. 0 0 Substituting qs. (7) into qs. (5) we find the electric field coponents of the focused radial and aziuthal odes: 4 ii 0 8 I 0 8 ii 0. r( TM ) r ( TM ) z( TM ) 0 r( T) ( T) z( T) In the vector representation the focal field coponents of the radial and aziuthal odes becoe as follows: TM 4 iir er I0ez. (9) T 8iI e One can easily see fro these expressions that the aziuthal ode preserves both its polarisation and zero intensity along the whole optical axis even in the regie of tight focusing whereas the intensity and the polarisation structure of the radial ode undergo significant transforations (see Fig. ). This difference in the behaviours of the two odes inside the tight focus can be exploited when solving various practical probles. For instance conservation of the zero intensity in the aziuthal ode facilitates significantly optical iplosion of aterials. This novel phenoenon has recently been deonstrated in the experients on copressing fused silica with double-charge fetosecond vortex laser pulses [9 0]. On the other hand the longitudinal coponent of the electric field is also crucial for soe iportant applications [5]. Radially polarised beas focused with high nuerical-aperture optics are now routinely used in icroscopy [] second-haronic generation [ 3] Raan spectroscopy [4] and particle trapping or anipulation [5]. (7) (8) 4 Ukr. J. hys. Opt. 0 V 3

7 Nonparaxial singular Fig.. Siulated intensity distributions in the focal plane for an optical vortex ; ((a) and (d)) and aziuthally ((b) and (e)) or radially ((c) and (f)) polarised beas. Solid lines in (d) (f) show total intensity distribution dotted lines correspond to intensity distributions in the transferred polarisation state and dashed lines give intensity distributions for the longitudinal polarisation state inside the beas. The paraeters of the optical syste and the beas are as follows: NA = 0.9 = 780 n a =. and a =. Acknowledgeent I thank Dr. Cyril Hnatovsky and rof. Alexander Volyar for useful discussions. References. Wang H Shi L Lukyanchuk B Sheppard C and Chong C 008. Creation of a needle of longitudinally polarized light in vacuu using binary optics. Nature hotonics. : Hnatovsky C Shvedov V Krolikowski W and Rode A 0. Revealing local field structure of focused ultrashort pulses. hys. Rev. Lett. 06: Wang Z Guo W Li L Luk'yanchuk B Khan A Liu Z Chen Z and Hong Z 0. Optical virtual iaging at 50 n lateral resolution with a white-light nanoscope. Nature Coun. : Helseth L 006. Sallest focal hole Opt. Co. 57: Zhan Q 009. Cylindrical vector beas: fro atheatical concepts to applications. Adv. Opt. hotonics. : Rhodes S Nugent K and Roberts A 00. recision easureent of the electroagnetic fields in the focal region of a high-nuerical-aperture lens using a tapered fiber probe. J. Opt. Soc. Aer. A. 9: Wilson T Juškaitis R and Higdon 997. The iaging of dielectric point scatterers in conventional and confocal polarisation icroscopes. Opt. Coun. 4: Novotny L Beversluis M Youngworth K and Brown T 00. Longitudinal field odes probed by single olecules. hys. Rev. Lett. 86: Bokor N Iketaki Y Watanabe T and Fujii M 005. Investigation of polarization effects for high-nuerical-aperture first-order Laguerre-Gaussian beas by D scanning with a single fluorescent icrobead. Opt. xpress. 3: Ukr. J. hys. Opt. 0 V 3 5

8 Vladlen G. Shvedov 0. Dorn R Quabis S and Leuchs G 003. Sharper focus for a radially polarized light bea. hys. Rev. Lett. 9: Torok and Munro 004. The use of Gauss-Laguerre vector beas in STD icroscopy. Opt. xpress. : Singh R Senthilkuaran and Singh K 008. ffect of priary spherical aberration on highnuerical aperture of a Laguerre-Gaussian bea. J. Opt. Soc. Aer. A. 5: Volyar A Shvedov V and Fadeyeva T 00. The structure of a nonparaxial Gaussian bea near the focus: II. Optical vortices. Opt. Spectrosc. 90 (): Youngworth K and Brown T 000. Focusing of high nuerical aperture cylindrical vector beas. Opt. xpress. 7: Richards B and Wolf 959. lectroagnetic diffraction in the optical systes.. Structure of the iage field in an aplanatic syste. roc. Roy. Soc. London Ser. A. 53: Izdebskaya Y Shvedov V and Volyar A 005. Focusing of wedge-generated higher-order optical vortices. Opt. Lett. 30: Fadeyeva T Shvedov V Izdebskaya Y Volyar A Brasselet Neshev D Desyatnikov A Krolikowski W and Kivshar Y 00. Spatially engineered polarization states and optical vortices in uniaxial crystals. Opt. xpress. 30: Fadeyeva T Shvedov V Shostka N Alexeyev C and Volyar A 00. Natural shaping of the cylindrically polarized beas. Opt. Lett. 35: Shvedov V Hnatovsky C Krolikowski W and Rode A 00. fficient bea converter for the generation of high-power fetosecond vortices. Opt. Lett. 35: Hnatovsky C Shvedov V Krolikowski W and Rode A 00. Materials processing with a tightly focused fetosecond laser vortex pulse. Opt. Lett. 35: Watanabe K Horiguchi N and Kano H 007. Optiized easureent probe of the localized surface plason icroscope by using radially polarized illuination. Appl. Opt. 46: Biss D and Brown T 003. olarization-vortex-driven second-haronic generation. Opt. Lett. 8: Yew and Sheppard C 007. Second haronic generation polarization icroscopy with tightly focused linearly and radially polarized beas. Opt. Coun. 75: Hayazawa N Saito Y and Kawata S 004. Detection and characterization of longitudinal field for tip-enhanced Raan spectroscopy. Appl. hys. Lett. 85: Zhan Q 004. Trapping etallic Rayleigh particles with radial polarization. Opt. xpress. : V.G. Shvedov 0. Nonparaxial singular beas inside the focal region of a high nuericalaperture lens Ukr.J.hys.Opt. : Анотація Основні оптичні технології зокрема такі як літографія запис даних використання оптичних пінцетів мікроскопія та надшвидка обробка лазерних матеріалів базуються на сильному фокусуванні світлових променів. Використання цих променів здебільшого базується на векторній природі світла тому детальна інформація про структуру поля всередині фокуса є особливо важливою. Однак теоретичне дослідження внутрішньо фокусних оптичних компонент поля досі переважно концентрувалося на просторово однорідних станах поляризації світла. У даній роботі представлено нові результати розрахунку структури локальної поляризації сильно сфокусованого сингулярного променя включаючи випадки радіально та азимутально поляризованого пустотілого променя. 6 Ukr. J. hys. Opt. 0 V 3

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