Theoretical analysis of numerical aperture increasing lens microscopy

Transcription

1 JOURNAL OF APPLIED PHYSICS 97, Theoretical aalysis of umerical aperture icreasig les microscopy S. B. Ippolito, B. B. Goldberg, ad M. S. Ülü Departmets of Physics ad Electrical ad Computer Egieerig ad the Photoics Ceter, Bosto Uiversity, 8 Sait Mary s Street, Bosto, Massachusetts Received 3 August 2004; accepted 14 December 2004; published olie 11 February 2005 We preset a detailed theoretical aalysis ad experimetal results o a subsurface microscopy techique that sigificatly improves the light-gatherig, resolvig, ad magifyig power of a covetioal optical microscope. The umerical aperture icreasig les NAIL is a plao-covex les placed o the plaar surface of a object to ehace the amout of light coupled from subsurface structures withi the object. I particular, a NAIL allows for the collectio of otherwise iaccessible light at agles beyod the critical agle of the plaar surface of the object. Therefore, the limit o umerical aperture icreases from uity for covetioal subsurface microscopy to the refractive idex of the object for NAIL microscopy. Spherical aberratio associated with covetioal subsurface microscopy is also elimiated by the NAIL. Cosequetly, both the amout of light collected ad diffractio-limited spatial resolutio are improved beyod the limits of covetioal subsurface microscopy. A theoretical optical model for imagig structures below the plaar surface of a object, both with ad without a NAIL, is preseted. Experimetal results demostratig the predicted improvemets i resolutio of subsurface imagig are also preseted America Istitute of Physics. DOI: / I. INTRODUCTION May subsurface optical microscopy applicatios, such as Si itegrated circuit IC failure aalysis, spectroscopy of sigle quatum dots, ad optical data storage, require operatio ear the ultimate limits of light-gatherig ad resolvig power, beyod the capability of covetioal subsurface microscopy. I the absece of aberratios, the diffractio of light limits the spatial resolutio of a covetioal optical microscope to 0 /2 si a laterally ad 0 / 1 cos a logitudially, where 0 is the free space wavelegth of light, is the refractive idex of the object space, ad a is the agular semiaperture i the object space. 1 The umerical aperture NA, si a, is the most commo measure of light-gatherig power, because of its iverse relatioship with the diffractio limit of lateral spatial resolutio. Icreasig a improves the spatial resolutio ad icreases the amout of light collected. Reducig the wavelegth of light by either decreasig 0 or icreasig also improves the spatial resolutio. Covetioal optical microscopy may be divided ito two modes, accordig to the locatio of the focus i the object space: surface optical microscopy ad subsurface optical microscopy. I Sec. I A, surface optical microscopy techiques ad a brief review of experimetal results are preseted, providig a backgroud for the termiology established for the umerical aperture icreasig les NAIL subsurface optical microscopy techique preseted i Sec. I B. I Sec. II, a theoretical optical model for imagig subsurface structures i a object with a plaar surface, first without, the with a NAIL, is preseted i Secs. II A ad II B, respectively. I Sec. II C, the theoretical results are compared, ad fially, i Sec. III, agreemet betwee the theoretical ad experimetal results is cofirmed. A. Surface optical microscopy Covetioal optical microscopy of the surface of a object is illustrated schematically i Fig. 1 a. The refractive idex of air i the object space is uity, which limits the NA to uity ad ultimately limits the lateral spatial resolutio to 0 /2 ad the logitudial spatial resolutio to 0. May surface optical microscopy techiques have bee developed to improve spatial resolutio beyod the limits of covetioal optical microscopy, by placig additioal elemets i the optical ear-field of the surface of a object. Immersio techiques itroduce a medium with a high refractive idex ito the object space, above the surface of a object. This icreases the limit o the NA to, ad improves the ultimate limit o the lateral spatial resolutio to 0 /2 ad the logitudial spatial resolutio to 0 /. I liquid immersio microscopy, the regio betwee a liquid immersio objective les ad the surface of a object is filled with a immersio liquid, such as water or oil with a refractive idex up to 1.6, as illustrated i Fig. 1 b. Hece, the FIG. 1. Imagig the surface of a object with a a covetioal optical microscope, b a oil immersio microscope, ad c a solid immersio les microscope /2005/97 5 /053105/12/$ , America Istitute of Physics

2 Ippolito, Goldberg, ad Ülü J. Appl. Phys. 97, optical ear-field of the object is immersed i liquid. Despite the sigificat improvemets i the NA ad spatial resolutio, the drawbacks of liquid immersio microscopy iclude the cotamiatio of a object ad a small workig distace. The solid immersio les SIL techique places a plao-covex les o the surface of a object, as show i Fig. 1 c. 2 I this case, the optical ear-field of the object is immersed i a solid. The SIL ca have a refractive idex up to 4, depedig o the free space wavelegth of light ad the material selected. To date, Si, GaP, GaAs, sapphire, ad glass SILs have bee successfully fabricated ad experimetally demostrated. The SIL techique was first proposed ad demostrated by Masfield ad Kio to clearly resolve 100 m lies ad spaces i photoresist, with light at a 0 of 436 m, usig a glass SIL. 2 Sice the, the SIL techique has bee applied to a variety of applicatios to improve spatial resolutio. I optical data storage, the SIL techique icreased iformatio desity by a factor of Low temperature photolumiescece microscopy studies of low dimesioal semicoductor structures, usig a SIL, have revealed importat physical pheomea ad material properties The SIL techique has bee used i ultrafast pump-probe spectroscopy of a quatum well structure to study lateral carrier trasport. 15 I fluorescece microscopy, the SIL has ot oly yielded icreased spatial resolutio but also icreased the amout of light collected The SIL techique has also bee used i micro-rama spectroscopy of pattered Si, 19 ad has bee suggested as a way of icreasig the resolutio of photolithography, 20 as well as beig used to measure the laser power of highly diverget beams. 21 For most of the aforemetioed experimets, the diameter of the SIL was a few millimeters. However, a microfabricated SIL with a diameter of 15 µm has also bee demostrated Despite the sigificat improvemets i NA ad spatial resolutio, the major drawback of SIL microscopy is that direct physical cotact with a object is ecessary. I summary, both solid ad liquid immersio techiques have bee show to improve spatial resolutio by itroducig a medium with a high refractive idex ito the object space. B. Subsurface optical microscopy If a object cosists of structures buried withi a trasparet ad homogeeous medium with a optical quality surface, the its subsurface structures may be imaged with a optical microscope. The ecapsulatig medium, through which light must propagate, provides additioal optical costraits the rage of 0 where it is trasparet, its refractive idex, as well as the reflectio ad refractio at its surface. Oce the lowest possible value of 0 / for the ecapsulatig medium is selected, the spatial resolutio ca oly be improved by alterig the surface geometry of the ecapsulatig medium. Ideally, the surface geometry should maximize the agular semiaperture i the object space a, ad itroduce o sigificat aberratios, to permit operatio at the diffractio limit. A plaar surface geometry is the most commo type of surface geometry, as it is either characteristic of a object or FIG. 2. Imagig a subsurface structure i a object with a a covetioal optical microscope, b a cetral NAIL, ad c a aplaatic NAIL. producible by polishig or cleavig. Figure 2 a illustrates a covetioal optical microscope imagig subsurface structures i a object with a plaar surface. Reflectio ad refractio at the plaar surface of a object limits a covetioal optical microscope s light-gatherig power from subsurface structures withi the object. Reflectio at the plaar surface of a object reduces the amout of light that escapes from the object, allowig o light to escape above the critical agle c, which is, by defiitio, si 1 1/. Accordig to Sell s law, refractio at the plaar surface of a object reduces si a for a give objective les by a factor of ; however, the NA remais the same because of the icreased refractive idex i the object space. Refractio also imparts spherical aberratio that icreases mootoically with agle, thereby requirig correctio to operate at the diffractio limit. The absece of light from above the critical agle ultimately limits the lateral spatial resolutio to 0 /2 ad the logitudial spatial resolutio to 0 1+cos c. Thus, despite the advatage of havig a higher refractive idex of the ecapsulatig medium i relatio to the refractive idex of air, the subsurface spatial resolutio is always worse tha the surface spatial resolutio i covetioal optical microscopy of a object with plaar surface geometry. Circumvetig this plaar surface barrier would yield sigificat improvemet i the amout of light collected ad the spatial resolutio. Despite these limitatios, because of coveiece, a covetioal optical microscope is commoly used for subsurface imagig of objects with plaar surface geometry. As illustrated i Figs. 2 b ad 2 c, the additio of a NAIL, a plao-covex les, to the plaar surface of a object icreases the covetioal optical microscope s lightgatherig ad resolvig power from that object, without itroducig ay aberratios. 25 Whe first proposed ad demostrated, the NAIL was termed a trucated SIL; however, the terms SIL ad NAIL were later adopted to delieate betwee applicatio of a plao-covex les for surface ad subsurface microscopy, respectively. 2,26 I surface microscopy with a SIL, the refractive idex i the object space is altered by immersio. I subsurface microscopy, the refractive idex i the object space may be high, but caot be altered sice the optical ear-field of the object is ot accessible, thus a NAIL simply icreases the NA or light-gatherig power. Icreasig the NA, which is ultimately limited to, improves the spatial resolutio, which is ultimately limited to 0 /2 laterally

3 Ippolito, Goldberg, ad Ülü J. Appl. Phys. 97, FIG. 3. Parameters of the object ad NAIL. ad 0 / logitudially. Preferably, the NAIL is made of a material with a refractive idex matchig that of the ecapsulatig medium. The plaar surface of the NAIL closely matches that of the object, so that light propagates from the object ito the NAIL by evaescetly couplig across ay potetial fiite gap betwee them. After propagatig through the NAIL, the light the refracts at the spherical surface ito air ad is collected by a covetioal optical microscope. A geeral set of experimetal procedures for usig a NAIL i a covetioal optical microscope is described by Ippolito. 27 The key parameters of the object ad NAIL are show i Fig. 3. The subsurface structure withi the object to be imaged is located at a depth of X below the plaar surface of the object. The covex surface of the NAIL is spherical with a radius of curvature of R. The ceter thickess of the NAIL, the distace from the top of the spherical surface to the plaar surface, is D. There are two types of NAILs that create stigmatic or aberratio-free images. A cetral NAIL, as illustrated i Fig. 2 b, is desiged to have the subsurface structures i a object at the geometrical ceter of the spherical surface of the NAIL, so D must equal R X. A aplaatic NAIL, as illustrated i Fig. 2 c, is desiged to have the subsurface structures i a object at the aplaatic poits of the spherical surface of the NAIL, so D must equal R 1 +1/ X. 28 Similarly, there are two types of SILs that create stigmatic images. A cetral SIL, which is usually referred to as a hemispherical SIL, is desiged to have the surface of a object at the geometrical ceter of the spherical surface of the SIL, so D must equal R. A aplaatic SIL, which is usually referred to as a Weierstrass or hyper-hemispherical SIL, is desiged to have the subsurface structures i a object at the aplaatic poits of the spherical surface of the SIL, so D must equal R 1+1/. II. THEORETICAL OPTICAL MODELING FIG. 4. Imagig parameters for a object with a plaar surface. I this sectio, a theoretical optical model for imagig subsurface structures i a object with a plaar surface, both without ad with a NAIL, will be described ad the results will be compared. The priciples of electromagetic optics will be used to characterize the behavior of light at the surfaces of the object ad NAIL, as well as at the focus i the object space. For mathematical simplicity the priciples of geometrical optics will be used to characterize the behavior of light i the remaiig areas of the object ad NAIL. Two approximatios are made i the theoretical optical model. The first approximatio is that the object space cosists of a homogeeous medium, eve though a subsurface structure may be preset. The secod approximatio is that the reflectios from the surfaces of the object ad NAIL do ot sigificatly cotribute to the electromagetic fields at the focus i the object space. I order to preset a example i additio to the geeral model, umerical values will be calculated for ear-ifrared ispectio microscopy of a Si IC, both without ad with a Si NAIL, at a 0 of 1.3 µm, where the refractive idex of Si is 3.5. A. Imagig subsurface structures without a NAIL A theoretical optical model for imagig subsurface structures i a object with a plaar surface, without a NAIL, will be preseted begiig with the Gaussia optics approximatio of imagig. The plaar dielectric boudary betwee the object, with a refractive idex of, ad air, with a refractive idex of uity, passes through the origi. The lateral locatio of the origi is arbitrary, because of the stratified ature of the object alog the logitudial directio. As illustrated i Fig. 4, a object poit P A with cylidrical coordiates A, A,z A is imaged by refractio to a virtual image poit P B with cylidrical coordiates B, B,z B. Withi the realm of Gaussia optics, B is equal to A, B is equal to A, ad z B is equal to z A /. The lateral magificatio is equal to uity ad the logitudial magificatio is equal to 1/. The primary logitudial chromatic aberratio, the logitudial displacemet of the object poit to first order due to the chage i refractive idex d of the dispersive object, is equal to z A d/. The geometrical aberratios that become sigificat as icreases will ow be foud. Ray 1, show i Fig. 4, origiates from P A, passes through P B, ad is parallel to the optical axis, which is the z axis. Ray 1 is defied as the referece ray, ad the exit pupil is defied as the plaar surface of the object. Ray 2 origiates from P A, itersects the plaar boudary at P D, ad has a directio i the object space defied i spherical coordiates,. The distaces from P A to P D ad P B to P D are defied as M ad N, give i Eqs. 1 ad 2, respectively. The optical path legth OPL from P A to P D ad back to P B is give i Eq. 3, ad the resultig wave aberratio fuctio is give i Eq. 4. The wave aberratio fuctio is idepedet of A, A, ad, because of the plaar geometry, so oly spherical aberratio propor-

4 Ippolito, Goldberg, ad Ülü J. Appl. Phys. 97, FIG. 5. Wave aberratio fuctios, as fuctio of. tioal to z A is preset. The primary spherical aberratio 040 term i the power series expasio of the wave aberratio fuctio, as a fuctio of si, is give i Eq. 5. Figure 5 shows the wave aberratio fuctio, ad primary spherical aberratio at z A equals 130 µm, i Si at a 0 of 1.3 µm, o both logarithmic ad liear scales, as a fuctio of, upto the critical agle c of 0.29 rad. Figure 5 clearly demostrates that the primary spherical aberratio is ot a good estimate for the wave aberratio fuctio i this agular domai, 29 M = z A cos, N = zb 2 z A 2 + M 2, 1 2 OPL za, = M N = z A cos cos , 3 = OPL za, OPL za,0 = z A 1 1 cos 1 +1 cos 2 1 2, = z A si 4. 5 The trasmissio coefficiets t s ad t p ad the resultig trasmissivities for refractio at the plaar surface of the object, where the agle of icidece is, are give by the Fresel formulae. 28 Figure 6 shows the trasmissivity T for s ad p polarized light i Si at a 0 of 1.3 µm, as a fuctio of. For a uiform emitter, which has a costat itesity per uit solid agle, total iteral reflectio above the critical agle greatly reduces the amout of light collected. The collectio efficiecy CE is equal to the ratio of the power trasmitted ito air P t to the power icidet o the plaar boudary of the object P i. Equatio 6 itegrates the trasmissivities to yield the collectio efficiecy for a uiform emitter of upolarized light i the object space. Assumig a full /2 agular semiaperture, for Si at a 0 of 1.3 µm the collectio efficiecy is 0.03, which leaves sigificat room for improvemet by a NAIL, CE = P t = 1 a Is + I p si d. 6 P i 2 0 Cosideratio of a high agular semiaperture beam focused at the origi i the object space requires a model of diffractio defied by vector field theory, rather tha scalar field theory. 30 A agular spectrum of plae waves method, as described by Wolf, ca be used to fid the electromagetic fields at ay poit ear the focus, with cylidrical coordiates,,z. 31 Ichimura et al. formulated the electromagetic fields for a optical system, cosistig of a SIL ad a mageto-optical disk, based o a formulatio for a aplaatic system by Richards ad Wolf. 26,32 Parts of the Wolf ad Ichimura formulatios are used to costruct the followig formulatio, i which the illumiatio is uiform, the polarizatio cross terms are equal to zero, ad a additioal phase chage of k 0 is itroduced to accout for the geometrical aberratios. The resultig complex amplitudes E ad H are give by Eqs The resultig time-averaged Poytig vector S ad the itesity S are give i Eqs. 12 ad 13, respectively. I this model, the icidet beam is polarized; however, if upolarized light is cosidered the itesity will be the same, 0 i I 0 + I 2 cos 2 E = E ii 2 si 2, 7 2I 1 cos H = E 0 FIG. 6. Trasmissivity T, as a fuctio of. ii 2 si 2 i I 0 I 2 cos 2, 8 2I 1 si I 0 = 0 a ts + t p cos si J 0 k 0 si e ik 0 z cos +ik 0 d, 9 I 1 = 0 a tp si 2 J 1 k 0 si e ik 0 z cos +ik 0 d, 10

5 Ippolito, Goldberg, ad Ülü J. Appl. Phys. 97, TABLE I. Imagig characteristics without a NAIL at several values of z A. Spatial resolutio z A µm Strehl ratio Lateral µm Logitudial µm Corrected FIG. 7. Itesity alog the optical axis at equals zero, as a fuctio of z. I 2 = 0 a ts t p cos si J 2 k 0 si e ik 0 z cos +ik 0 d, S = 1 2 Re E H* = E cos Im I1 * I 0 I 2 2 si Im I * 1 I 0 I 2 I 0 2 I , 12 S = E 0 2 I 0 2 I Im I * 2 1 I 0 I Usig this vector field formulatio, the itesity is calculated i the viciity of P A, for several values of z A i Si at a 0 of 1.3 µm. We assume the illumiatio of P B is upolarized ad uiform, ad the agular semiaperture is c. The wave aberratio fuctio for this geometry was give i Eq. 4, ad the focus coordiate z is equal to z A z. Figure 7 shows the itesity alog the optical axis at equals zero, as a fuctio of z. Figure 8 shows the itesity i the lateral plae of peak itesity, as a fuctio of. Table I shows the imagig characteristics at several values of z A. The spatial resolutio is defied as the full-width-at-half maximum of the itesity, accordig to the Housto criterio. 33 The Strehl ratio is the ratio of the maximum itesities, with ad without aberratios. If the Strehl ratio is greater tha or equal to 0.8, the spatial resolutio is said to be diffractio limited. 28 For the most practical depths of greater tha 65 µm isiata 0 of 1.3 µm, the spherical aberratio must be corrected by the objective les or elsewhere, to operate at diffractio limited spatial resolutio. To achieve these values of spatial resolutio ad collectio efficiecy would require a objective les with a NA of uity, so i practice the values are sigificatly degraded. Rigorous aalytical modelig ad umerical simulatio of light, focused through a plaar dielectric boudary, were previously coducted by Török et al B. Imagig subsurface structures with a NAIL We ow describe a theoretical optical model for imagig subsurface structures i a object with a plaar surface usig a NAIL. A geometrical optics model of a sphere, with a refractive idex of, i air, with a refractive idex of uity, will be used to model to the object ad NAIL. As illustrated i Fig. 9, the dielectric boudary betwee the sphere ad air FIG. 8. Itesity i the lateral plae of peak itesity, as a fuctio of. FIG. 9. Imagig parameters for a sphere.

6 Ippolito, Goldberg, ad Ülü J. Appl. Phys. 97, has a radius of curvature of R ad is cetered at the origi P C. A object poit P A with cylidrical coordiates A, A, A is imaged by refractio to a virtual image poit P B with cylidrical coordiates B, B, B, accordig to Eqs , withi the realm of Gaussia optics. B = A R 1 +1A, 14 B = A R 1 +1 A, B = A, lateral magificatio = A R 1 +1, 17 logitudial magificatio = A. R Next, the geometrical aberratios that become sigificat, as icreases, are foud. Ray 1, show i Fig. 9, origiates from P A, passes through P C, ad is colliear with P B,i accordace with the projective ature of Eqs Ray 1 is defied as the referece ray, ad the exit pupil is defied as the surface of the sphere. Ray 2 origiates from P A, itersects the spherical boudary at P D, ad has a directio i the object space defied i spherical coordiates,. Ray 2 forms a agle of, with respect to Ray 1, ad a agle of, with respect to the lie from P C to P D. The distaces from P A to P C ad P B to P C are defied as K ad L, respectively. The distaces from P A to P D ad P B to P D are defied as M ad N, give i Eqs. 19 ad 20, respectively. The optical path legth from P A to P D ad back to P B is give i Eq. 21, ad the resultig wave aberratio fuctio is give i Eq. 22, M = K cos + R 2 K 2 si 2, N = R 2 + L 2 +2L M cos K, OPL K,L, = M N = K cos + R 2 K 2 si 2 R 2 + L 2 +2L K si 2 + cos R 2 K 2 si 2, 21 = OPL K,L, OPL K,L,0 = R 1 + K cos 1 + L + R 2 K 2 si 2 R 2 + L 2 +2L K si 2 + cos R 2 K 2 si So far, the wave aberratio fuctio has bee derived purely from geometry, ad the Gaussia optics relatioships betwee K ad L, Eqs , have ot bee applied. This was doe to show that there are two solutios for K ad L, where the wave aberratio fuctio is equal to zero for all, ad therefore the imagig is stigmatic. I the cetral solutio, both K ad L are equal to zero, which occurs whe both P A ad P B are at P C. For the cetral solutio, the rays are ormal to the spherical boudary, ad refractio does ot chage their directio. I the aplaatic solutio, K ad L are equal to R/ ad R, respectively, ad as previously metioed, P A,P B, ad P C are colliear. The poits o the spherical surface, defied by K equal to R/ i the aplaatic solutio, are termed the aplaatic poits of the spherical boudary. The Gaussia optics relatioships betwee K ad L, Eqs , yield the values for K,L,, ad give by Eqs , respectively, i terms of the object poit ad Ray 2 coordiates, K = A 2 + A 2, 23 L = A A 2 R A, 24 cos = Asi cos A + A cos, 25 A A si = si A A. 26 R The geometrical optics model of the sphere i air ca be exteded to the object ad NAIL by icludig three additioal effects. The first effect is the trasmissio for refractio at the spherical surface of the NAIL, which is govered by the Fresel formulas. The secod effect is the trasmissio across a air gap that exists i practice betwee the object ad NAIL. The object, air gap, ad NAIL are modeled as a stratified medium cosistig of; a gap layer with a thickess of h ad a refractive idex of uity for air, betwee two bulk layers with a refractive idex of for the object ad NAIL. The characteristic matrix method is the used to solve for the trasmissio coefficiets ad the resultig trasmissivities ad phase chages o trasmissio. 28 Figure 10 shows the trasmissivity ad phase chage o trasmissio t for s ad p polarized light across a air gap betwee a Si object ad NAIL at a 0 of 1.3 µm, as a fuctio of. Simulatios for the trasmissio across a air gap betwee a object ad a SIL, i the visible spectrum, were previously coducted by Milster et al.. 37,38 The trasmissio coefficiets for refractio at the spherical surface of the NAIL ad the trasmissio coefficiets across a air gap betwee the object ad NAIL ca be multiplied, to yield the total trasmissio coefficiets of t s ad t p, for poits alog the optical axis,

7 Ippolito, Goldberg, ad Ülü J. Appl. Phys. 97, FIG. 12. Itesity alog the optical axis at equals zero, as a fuctio of z. 1. Cetral NAIL FIG. 10. Trasmissivity T ad phase chage o trasmissio t, as a fuctio of, for a air gap height h of a 0 /20 ad b 0 /80. because the plaes of icidece for both refractios are the same. The third effect is the agular semiaperture defied by the regio of optical cotact betwee the object ad NAIL. If the plaar surface of the object exteds beyod the plaar surface of the NAIL, as show i Fig. 3, the agular semiaperture is give by Eq. 27, X si a = /2 R 2 X A The cetral NAIL is based o the cetral stigmatic solutio, where A ad B are both equal to zero, so D must equal R X. For a cetral NAIL, R should be selected to be less tha the workig distace of the objective les. The lateral ad logitudial magificatios are both equal to. The primary logitudial chromatic aberratio is equal to zero. Accordig to Eq. 26, ifa ad A are much smaller tha R, the is approximately equal to zero. Thus, the trasmissio coefficiet ad the resultig trasmissivity for refractio at the spherical surface of the cetral NAIL, where the agle of icidece is, are give by the Fresel formula for ormal icidece. 28 Assumig a full /2 agular semiaperture ad o air gap, for Si at a 0 of 1.3 µm the collectio efficiecy is At the cetral stigmatic solutio, the oly primary aberratios are astigmatism ad distortio. The variable will be the logitudial displacemet from the stigmatic solutio, so for the cetral NAIL, A is equal to. The first term c i the power series expasio of the wave aberratio fuctio alog the optical axis aroud the cetral stigmatic solutio, as a fuctio of, is give i Eq. 28, which is equivalet to the expressio give by Baba et al.. 10 Agai, i this agular domai the primary spherical aberratio 040, give i Eq. 29, is ot a good estimate for the wave aberratio fuctio. 29 These wave aberratio fuctios are show i Fig. 11 for a cetral NAIL with R equal to 1.61 mm ad FIG. 11. Wave aberratio fuctios, as fuctio of. FIG. 13. Itesity i the lateral plae of peak itesity, as a fuctio of.

8 Ippolito, Goldberg, ad Ülü J. Appl. Phys. 97, TABLE II. Imagig characteristics with a cetral NAIL for varyig logitudial displacemet from the stigmatic solutio. Spatial resolutio µm Strehl ratio Lateral µm Logitudial µm equal to 50 m. The wave aberratio fuctios, ad c, are approximately the same at this value of, c =2 1 2 R si4 2, = 1 2 8R si4. 29 The itesity, usig a cetral NAIL with R equal to 1.61 mm, is calculated i the viciity of P A for several values of, isiata 0 of 1.3 µm. We assume the illumiatio of P B is upolarized ad uiform, the agular semiaperture is a full /2, ad o air gap exists. The wave aberratio fuctio was give i Eq. 22, ad the focus coordiate z is equal to z. Figure 12 shows the itesity alog the optical axis at equals zero, as a fuctio of z. Figure 13 shows the itesity i the lateral plae of peak itesity, as a fuctio of. Table II shows the imagig characteristics at several values of. Thus, the absolute value of must be less tha 22 µm to operate at diffractio limited spatial resolutio. However, sigificat gais i spatial resolutio still exist, as the Strehl ratio is equal to 0.4 whe is equal to 32 µm. To achieve these values of spatial resolutio ad collectio efficiecy would require a objective les with a NA of uity, so i practice the values are sigificatly degraded. 2. Aplaatic NAIL FIG. 15. Wave aberratio fuctios, as fuctio of. The aplaatic NAIL is based o the aplaatic stigmatic solutio, where A ad B are equal to R/ ad R, respectively, so D must equal R 1+1/ X. For a aplaatic NAIL, R should be selected such that R +1 is less tha the workig distace of the objective les. The lateral ad logitudial magificatios are equal to 2 ad 3, respectively. The primary logitudial chromatic aberratio is equal to R +1 d/ 3. Accordig to Eqs. 25 ad 26, ifa is equal to R/ ad A is much smaller tha A, the ad si are approximately equal to ad si /, respectively. The trasmissio coefficiets ad the resultig trasmissivity for refractio at the spherical surface of the aplaatic NAIL, where the agle of icidece is, are give by the Fresel formulas. Figure 14 shows the trasmissivity for refractio at the spherical surface of the NAIL for s ad p polarized light i Si at a 0 of 1.3 µm, as a fuctio of. Assumig a full /2 agular semiaperture ad o air gap, for Si at a 0 of 1.3 µm the collectio efficiecy is At the aplaatic stigmatic solutio, the oly primary aberratios are curvature of field ad distortio. Agai, is the logitudial displacemet from the stigmatic solutio, so for the aplaatic NAIL, A is equal to R/+. The first term a i the power series expasio of the wave aberratio fuctio alog the optical axis aroud the aplaatic stigmatic solutio, as a fuctio of, is give i Eq. 30. As show i Fig. 15, the wave aberratio fuctios, ad a, for a aplaatic NAIL with R equal to 1.61 mm ad equal to 8 m, are approximately the same, a = cos si FIG. 14. Trasmissivity T, as a fuctio of. FIG. 16. Itesity alog the optical axis at equals zero, as a fuctio of z.

9 Ippolito, Goldberg, ad Ülü J. Appl. Phys. 97, FIG. 17. Itesity i the lateral plae of peak itesity, as a fuctio of. The itesity, usig a aplaatic NAIL with R equal to 1.61 mm, is calculated i the viciity of P A for several values of, isiata 0 of 1.3 µm. We assume the illumiatio of P B is upolarized ad uiform, the agular semiaperture is a full /2, ad o air gap exists. The wave aberratio fuctio was give i Eq. 22, ad the focus coordiate z is equal to R/ z. Figure 16 shows the itesity alog the optical axis at equals zero, as a fuctio of z. Figure 17 shows the itesity i the lateral plae of peak itesity, as a fuctio of. Table III shows the imagig characteristics at several values of. Thus, the absolute value of must be less tha 1 µm to operate at diffractio limited spatial resolutio. However, sigificat gais i spatial resolutio still exist, as the Strehl ratio is equal to 0.4 whe is equal to 3.2 µm. To achieve these values of spatial resolutio ad collectio efficiecy requires a objective les with a NA of 1/, which is usually available i practice. The itesity, usig a aplaatic NAIL with R equal to 1.61 mm, is calculated i the viciity of P A for several values of air gap height h, isiata 0 of 1.3 µm. We assume the illumiatio of P B is upolarized ad uiform, ad the agular semiaperture is a full /2. We assume the focus is at the aplaatic stigmatic solutio, so ad the wave aberratio fuctio are equal to zero, ad the focus coordiate z is equal to R/ z. Figure 18 shows the itesity alog the optical axis at equals zero, as a fuctio of z. Figure 19 shows the itesity i the lateral plae of peak itesity, as a fuctio of. Table IV shows the imagig characteristics at several FIG. 18. Itesity alog the optical axis at equals zero, as a fuctio of z. values of air gap height h, icludig the resultig chages i collectio efficiecy, as defied by Eq. 6, for Si at a 0 of 1.3 µm. Thus, the air gap height must be less tha 7.8 m to operate at diffractio limited spatial resolutio. However, sigificat gais i spatial resolutio still exist, as the Strehl ratio is equal to 0.4 whe the air gap height is equal to 41 m. C. Compariso of theoretical results The theoretical optical models for imagig subsurface structures i a object with a plaar surface, both with ad without a NAIL, will ow be compared. Numerical values, assumig there is o air gap, will also be compared for earifrared ispectio microscopy, two other Si IC failure aalysis applicatios, spectroscopy of sigle quatum dots, ad optical data storage. Agai, i the first example of earifrared ispectio microscopy of a Si IC at a 0 of 1.3 µm, the refractive idex of Si is 3.5. Numerical values are calculated for a 0.5 mm thick Si IC, both without ad with a Si NAIL havig R equal to 1.61 mm. 25 The secod example is thermal emissio microscopy of a1mmthick Si IC, usig a ISb detector, both without ad with a Si NAIL havig R equal to 1.61 mm. 39 I this example, most of the light is at free space wavelegths betwee 4 µm ad 5 µm, where the refractive idex of Si mootoically varies from to , respectively. The third example is picosecod imag- TABLE III. Imagig characteristics with a aplaatic NAIL for varyig logitudial displacemet from the stigmatic solutio. Spatial resolutio µm Strehl ratio Lateral µm Logitudial µm FIG. 19. Itesity i the lateral plae of peak itesity, as a fuctio of.

10 Ippolito, Goldberg, ad Ülü J. Appl. Phys. 97, TABLE IV. Imagig characteristics with a aplaatic NAIL for varyig air gap height h. Spatial resolutio h h m Strehl ratio Lateral µm Logitudial µm Collectio efficiecy / / / / / / / / FIG. 20. Itesity i the object plae at z equals zero, as a fuctio of. ig circuit aalysis PICA of a 0.1 mm thick Si IC, usig a IGaAs avalache photodiode or a supercoductig siglephoto detector, both without ad with a Si NAIL havig R equal to 0.4 mm I this example, most of the light is at free space wavelegths betwee 1.2 µm ad 1.53 µm, where the refractive idex of Si mootoically varies from to 3.478, respectively. The fourth example is photolumiescece microscopy of idividual IGaAs quatum dots o a 0.5 mm thick GaAs substrate, both without ad with a GaAs NAIL havig R equal to 1.61 mm. 43 I this example, a spectrum is acquired betwee the free space wavelegths of 0.94 µm ad 0.96 µm, where the refractive idex of GaAs mootoically varies from to 3.475, respectively. The fifth example is optical data storage o a Blu-ray Disc at a 0 of 405 m, where the refractive idex of the 0.1 mm thick cover layer o the disc is 1.5, both without ad with a glass NAIL havig R equal to 0.2 mm ad a matchig refractive idex equal to The calculated lateral ad logitudial spatial resolutio values for each example, both without ad with a aplaatic NAIL ad a cetral NAIL, are show i Table V ad compare well to the approximatios for the ultimate limit o spatial resolutio due to the diffractio of light. Note that to achieve the stated values of spatial resolutio for covetioal subsurface microscopy requires a objective les that corrects for the spherical aberratio itroduced by refractio at the plaar surface of the object. For compariso i Si at a 0 of 1.3 µm, assumig a full /2 agular semiaperture, Fig. 20 shows the itesity i the object plae at z equals to zero, as a fuctio of, ad Fig. 21 shows the itesity alog the optical axis at equals zero, as a fuctio of z. The itesity for covetioal subsurface microscopy plaar/corrected has to be placed o a separate scale because the lightgatherig power is so low. The lateral ad logitudial spatial resolutio improvemets resultig from the NAIL are approximately factors of ad 2 1+cos c, which are 3.5 ad 24, respectively, i Si at a 0 of 1.3 µm. The effective volume of the focus is reduced by a factor of 4 1 +cos c, which i Si at a 0 of 1.3 µm is 294. The calculated collectio efficiecy values for selected examples, as defied by Eq. 6, are show i Table VI. I the example of earifrared ispectio microscopy, the improvemet i collectio efficiecy is approximately a factor of 10. To achieve the stated values of spatial resolutio ad collectio efficiecy with a covetioal optical microscope, both without ad with a cetral NAIL, would require a objective les with a NA of uity; so i practice, the values will be sigificatly degraded. However, the stated values of spatial resolutio ad collectio efficiecy ca be achieved i practice with a aplaatic NAIL ad a objective les with a NA of 1/. A further advatage of usig a NAIL is the additioal lateral ad logitudial magificatio it provides, as show i Table VII for selected examples. If the object ad NAIL are affixed to each other, the the additioal magificatio greatly reduces the positioig oise due to vibratio ad drift, ad causes a sca relaxatio that allows less accurate, ad therefore less costly, scaig stages to be used for positioig of the object ad NAIL. Aother advatage of usig a NAIL is a reductio i logitudial chromatic aberratio, as show i Table VIII. For polychromatic applicatios TABLE V. Compariso of lateral ad logitudial spatial resolutio. Corrected covetioal Aplaatic NAIL Cetral NAIL Lateral µm Logitudial µm Lateral µm Logitudial µm Lateral µm Logitudial µm Approximatio 0 /2 0 1+cos c 0 /2 0 / 0 /2 0 / Near-ifrared ispectio Thermal emissio PICA Quatum dot Optical data storage

11 Ippolito, Goldberg, ad Ülü J. Appl. Phys. 97, TABLE VI. Compariso of collectio efficiecy. Covetioal Aplaatic NAIL Cetral NAIL Thermal emissio PICA Quatum dot FIG. 21. Itesity alog the optical axis at equals zero, as a fuctio of z. the cetral NAIL is ofte the best choice, because it completely elimiates chromatic aberratios. I PICA measuremets, a GaAs or GaP NAIL may be used to reduce the amout of chromatic aberratio ad absorptio caused by a Si NAIL. The theoretical optical modelig idicates that both the cetral ad aplaatic NAILs provide sigificat improvemet i resolvig ad light-gatherig power. Furthermore, improvemets i magificatio ad lower NA requiremets for the objective les provide sigificat cost savigs, whe usig a NAIL. A choice betwee the cetral or aplaatic NAIL should ivolve cosideratio of all the cotributig factors discussed i this theoretical compariso. III. CONCLUSIONS Both the cetral ad aplaatic NAIL have experimetally demostrated improvemets i light-gatherig ad resolvig power that compare well with the results of our theoretical optical modelig. The cetral NAIL has bee used to improve the desity of optical data storage at a 0 of 546 m. 29 The theoretical example of optical data storage o a Blu-ray Disc at a 0 of 405 m was preseted to show the improvemet i desity by greater tha a factor of 2, available with curret techology. The aplaatic NAIL has bee used i may applicatios, but was first demostrated i ear-ifrared ispectio microscopy of Si IC usig a Si NAIL. A lateral spatial resolutio of better tha 0.23 µm ad a logitudial spatial resolutio of better tha 1.7 µm were experimetally demostrated at a 0 of 1.05 µm. 25 These experimetal spatial resolutios compare well to the theoretical lateral spatial resolutio of 0.16 µm ad logitudial spatial resolutio of 0.77 µm predicted by modelig. These experimetal spatial resolutios represet a sigificat improvemet over the theoretical lateral spatial resolutio limit of 0.5 µm ad logitudial spatial resolutio limit of 6 µm for corrected covetioal subsurface microscopy. I a secod experimet, as described i the first theoretical example, a lateral spatial resolutio of about 0.3 µm was demostrated o a commercial ear-ifrared ispectio microscope at a 0 of 1.3 µm. 25,27 This experimetal lateral spatial resolutio is cosiderably larger tha the theoretical lateral spatial resolutio of 0.16 µm predicted by modelig, but represets a sigificat improvemet over the theoretical lateral spatial resolutio limit of 0.5 µm for corrected covetioal subsurface microscopy. The aplaatic NAIL was also experimetally demostrated i thermal emissio microscopy, as described i the secod theoretical example. A lateral spatial resolutio of 1.4 µm ad a logitudial spatial resolutio of 7.4 µm were experimetally demostrated at free space wavelegths up to 5 µm. 39 These experimetal spatial resolutios compare well to the theoretical lateral spatial resolutio of 0.90 µm ad logitudial spatial resolutio of 2.92 µm predicted by modelig. These experimetal spatial resolutios represet a sigificat improvemet over the theoretical lateral spatial resolutio limit of 2.74 µm ad logitudial spatial resolutio limit of 31.6 µm for corrected covetioal subsurface microscopy. The third theoretical example of PICA was give, because we believe a NAIL will soo experimetally yield the great improvemets i light-gatherig ad resolvig power as predicted by modelig. A aplaatic Si NAIL has also bee used to improve the spatial resolutio ad amout of light collected i scaig far-ifrared microscopy of cyclotro emissios from quatum Hall devices. 44 A aplaatic GaAs NAIL has allowed photolumiescece spectroscopy of a idividual IGaAs quatum dot by improvig the lateral spatial resolutio ad icreasig the light-gatherig power, as described i the fourth theoretical example. 43 These experimetal results are curretly beig prepared for publicatio. I all of these applicatio examples, the theoretical ad experimetal results are i good agreemet, ad we believe NAIL microscopy promises may future improvemets i other applicatios as well. TABLE VII. Compariso of lateral ad logitudial magificatio. Covetioal Aplaatic NAIL Cetral NAIL Lateral Logitudial Lateral Logitudial Lateral Logitudial I geeral 1 1/ 2 3 Near-ifrared ispectio Optical data storage

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