Chapter 3. Mathematical analysis of the ordinary reflectance scanning microscope. 3.1 The ordinary reflectance scanning microscope

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1 Mathematical analsis f the rdinar reflectance scanning micrscpe Chapter 3 Mathematical analsis f the rdinar reflectance scanning micrscpe In this chapter the mathematical framewrk fr mdelling the reflectance scanning micrscpe is described in detail. Man authrs have presented mathematical analsis describing imaging in ptical sstems, the frerunner f which was Hpkins wh described the imaging prcess in partiall cherent ptical sstems [44,45]. Hpkins wrk was further adapted and applied t the case f imaging in rdinar (simple reflectance and transmissin) scanning laser micrscpes b Sheppard and Wilsn and their c-wrkers in the 970s and 980s [4,5,6,46]. 3. The rdinar reflectance scanning micrscpe The analsis begins with an intrductin f the simple scanning micrscpe illustrated in Fig. 3., a cnfiguratin that has been etensivel studied b thers [4,5,6,44,45,46]. Hwever, it is instructive t begin with a review f such a simple cnfiguratin since this intrduces man f the basic cncepts used t treat the mre cmplicated imaging sstems f interest in this thesis. Optical cnfiguratin Illustrated in Fig. 3. is the ptical laut f the simple rdinar reflectance scanning micrscpe. The standard ptical head f the reflectance scanning micrscpe cmprises an bjective lens (Obj), cllectr lens (Cl), and large area pht-detectr (Det). Althugh Fig. 3. illustrates the micrscpe perating in transmissin, the same analsis can be applied t a sstem perating in reflectin, in which case the bjective 44

2 Mathematical analsis f the rdinar reflectance scanning micrscpe and cllectr lenses are ne and the same, and a beamsplitter is placed prir t the bjective lens t separate the incident and reflected beams. Figure 3. : The ptical laut f the rdinar reflectance scanning micrscpe. Scanning is perfrmed in scanning micrscpes using tw main techniques: beam scanning and sample scanning. In beam scanning sstems the fcused spt is scanned acrss a statinar sample, whereas in sample scanning sstems the sample is scanned under a statinar fcused spt. In the current analsis it is assumed that the sample scanning technique is used t image the sample, thus eliminating the ptical cmpleities evident in beam scanning sstems [8]. The bjective lens, with assciated aperture pupil functin p(, ), fcuses a cllimated plane wave nt the surface f the sample. The fcused field interacts with the sample and then prpagates t the cllectr lens b the diffractin prcess. The frm f the resultant field after interactin with the sample depends upn the reflectin / transmissin prperties f the sample abut the scan psitin. At the plane f the cllectr lens the field is mdified b the aperture pupil functin f the cllectr lens, p(, ), is cllimated and prpagates, withut further mdificatin, t the large area pht-detectr where the signal is generated b the integral, ver the area f the pht-detectr, f the square magnitude f the incident amplitude field distributin. 45

3 Mathematical analsis f the rdinar reflectance scanning micrscpe Image calculatin The aim f the mdelling prcess is t calculate the electric field distributin, ψ (, ), in the plane f the pht-detectr fr a knwn incident field distributin, ψ (, ), n the bjective aperture. The field distributin, ψ (, ), in the plane f the sample (the fcal pint f the bjective lens) is given b the tw-dimensinal Furier transfrm f the field, ψ (, ), immediatel after the plane f the bjective aperture, which fllwing the analsis described in sec..4 gives ψ ( ), = Ψ, = h (, ) (3.) λ f λ f λ f where h (, ) is the amplitude pint spread functin f the bjective lens, and all ther smbls have their usual meaning. An imprtant nte is that in the current analsis it is assumed that the bjective lens is in perfect fcus and is free f aberratins; detractins frm such ideal cnditins that result in signal degradatin which will be treated in detail in later chapters. The field immediatel after interactin with the sample, whether perating in reflectin r transmissin, is given b ( ) ( ) (, ) (, ) ψ (, ) = ψ, r, s s = h r s s (3.) where r(, ) represents the cmple amplitude reflectance, r transmittance, f the sample abut the scan psitin, ( s, s ). After reflectin frm the surface f the sample the field prpagates twards the cllectr lens b the diffractin prcess. The field, ψ (, ), in the plane f the cllectr lens is given b the tw-dimensinal Furier transfrm f the field, ψ (, ), immediatel after interactin with the sample. Hence, the field at the cllectr lens [4,7,0] is given b jk ψ (, ) = ψ (, ) ep ( ) λ f + dd f (3.3) jk = h (, ) r ( s, s ) ep ( + ) dd λ f 46

4 Mathematical analsis f the rdinar reflectance scanning micrscpe in which phase factrs have been ignred, f is the fcal length f the cllectr lens, which is assumed t be equal t the fcal length f the bjective lens, and all ther smbls have their usual meaning. The effect f the cllectr lens is t cllimate the diverging reflected wave b intrducing a phase factr int the ptical field. Hwever, t simplif the current analsis the phase intrduced b the lens has been ignred. The field, ψ (, ), immediatel after the cllectr lens is given b the multiplicatin f the incident field distributin, ψ (, ), and the cllectr aperture pupil functin, p (, ), i.e. ( ) ψ ( ) p ( ) ψ, =,, (3.4) this field then prpagates, withut further mdificatin, twards the large area phtdetectr. The signal frm the pht-detectr, as a functin f scan psitin, is given b the integral, ver the area f the pht-detectr, f the square magnitude f the incident field distributin, i.e. (, ) = ψ (, ) (, ) I R d d s s d d d d d =,d = (3.5) where R(, ) represents the respnsivit f the pht-detectr. Assuming the d d respnsivit f the pht-detectr is unifrm (and set t unit) then the signal frm the pht-detectr, as a functin f scan psitin, is given b which b epanding gives ( s, s) = (, ) (, ) I p dd (3.6) = ψ I( s, s ) ψ (, ) ψ *(, ) p(, ) dd. (3.7) Substituting fr ψ (, ) frm eq. (3.3) gives I(, ) = p (, ) s s jk h (, ) r(, ) ep ( + ) d d f λ s s jk h *(, ) r *(, ) ep λf d d d d ( + ) s s (3.8) 47

5 Mathematical analsis f the rdinar reflectance scanning micrscpe where and are dumm variables intrduced t satisf the multiplicatin f ψ and ψ *. Rearranging eq. (3.8) gives I(, ) = h (, ) r(, ) h *(, ) r *(, ) s s s s s s jk p(, ) ep ( ( ) + ( )) dd (3.9) d d d d where cnstant scaling factrs have been ignred. Equatin (3.9) can be further simplified t give ( ) I, = h (, ) r(, ) s s s s h *( ', ' ) r *( ', ' ) s s (3.0) g ( ', ' ) d d d ' d ' where g (, ) is the pint spread functin assciated with the square magnitude f the cllectr aperture pupil functin [4,7,0], given b jk g(, ) = p(, ) ep ( + ) d d (3.) where f is the fcal length f the cllectr lens and all ther smbls have their usual meaning. Equatin (3.0) is a cnvlutin tpe prcess invlving the bjective amplitude pint spread functin, h, the bject reflectance, r, and the pint spread functin assciated with the square mdulus f the cllectr pupil functin, g. It is useful at this pint t cnsider tw etreme cases f cllectr aperture size that gvern the peratin f the reflectance scanning micrscpe. These tw cases are referred t as cherent and incherent imaging [4,7,0]. 48

6 Mathematical analsis f the rdinar reflectance scanning micrscpe 3.. The cherent ptical channel Cnsider the situatin where the cllectr aperture is ver small, such that it ma be described b a delta functin [4,7,0]. In this case, the amplitude pint spread functin, g, becmes g (, ) = (3.) which b substituting int eq. (3.0) and rearranging gives I( s, s ) = h (, ) r( s, s ) dd h *( ', ' ) r *( ' s, ' s ) d ' d ' - Equatin (3.3) can be epressed in the simplified frm. (3.3) I(, ) = h (, ) r(, ) s s s s s s (3.4) where represents cnvlutin. The resulting signal frm the reflectance scanning micrscpe is nw linear in field amplitude and is generall termed cherent imaging. The negative signs in eq. (3.4) indicate that strictl eq. (3.3) is in the frm f a crrelatin integral. Equatin (3.4) illustrates that the respnse f the cherent scanning micrscpe is generated b calculating the square magnitude f the cnvlutin f the bjective amplitude pint spread functin and the reflectance characteristics f the sample. Step respnse Figure 3. illustrates the nrmalised step respnse f the cherent scanning micrscpe fr a clear, aberratin free, circular bjective aperture under unifrm illuminatin. The respnse has been generated using the transfer functin mdel described in chapter 6. An imprtant characteristic f cherent imaging is evident in Fig. 3.. The respnse f the cherent imaging sstem ehibits undesirable ringing and a lag in the respnse t a straight edge. It has been the wrk f man researchers t appl apdizatin 49

7 Mathematical analsis f the rdinar reflectance scanning micrscpe techniques t remve the undesired ringing frm the cherent respnse whilst maintaining the fastest rise time [4,7]. 0.8 Nrmalised respnse Nrmalised distance frm step - λ/na Figure 3. : The nrmalised step respnse f the cherent scanning micrscpe, fr a clear, aberratin free, circular bjective under unifrm illuminatin. 3.. The incherent ptical channel Cnsider the situatin where the cllectr aperture is infinitel large [4,7,0], such that the aperture pupil functin, p (, ), is cnstant, and the assciated amplitude pint spread functin is effectivel a delta functin, Substituting int eq. (3.0) gives g (, ) = δ (, ). (3.5) I(, ) = h (, ) r (, ) s s s s which can be epressed in the simplified frm h *(, ) r *(, ) d d s s ( ) ( ) I(, ) = h, r, s s s s s s (3.6) (3.7) where represents cnvlutin. It can be seen frm eq. (3.7) that the signal frm the reflectance scanning micrscpe is nw linear in the square mdulus f the electric field amplitude, the irradiance, and s is generall termed incherent imaging. Equatin (3.7) illustrates that the respnse f the incherent scanning micrscpe is 50

8 Mathematical analsis f the rdinar reflectance scanning micrscpe generated b the cnvlutin (strictl crrelatin) f the square magnitude f the bjective pint spread functin and the square magnitude f the reflectance prperties f the sample. Step respnse Figure 3.3 illustrates the nrmalised step respnse f the incherent scanning micrscpe fr a clear, aberratin free, circular bjective aperture under unifrm illuminatin. The respnse has been generated using the transfer functin mdel described in chapter 6. It can be seen that the respnse is ver different t that f the cherent sstem. The respnse f the incherent imaging sstem ehibits nne f the undesirable ringing characteristics f the cherent imaging sstem, and, unlike the cherent sstem, the respnse t a straight edge is smmetrical abut the centre f the edge. 0.8 Nrmalised respnse Nrmalised distance frm step - λ/na Figure 3.3 : The nrmalised step respnse f the incherent scanning micrscpe, fr a clear, aberratin free, circular bjective under unifrm illuminatin. The simple incherent imaging mdel, ften referred t as the cnvlutinal mdel, is ften used b researchers t calculate the theretical readut signal in ptical disc sstems, due t the fact that eperimental bservatins agree ver clsel with predictins f the incherent mdel [8,48,49,50,5]. 5

9 Mathematical analsis f the rdinar reflectance scanning micrscpe The incherent transfer functin It is cmmn t characterise the perfrmance f electrical sstems in terms f their impulse respnse, step respnse r frequenc respnse. In electrical sstems the variable f interest is tempral frequenc, and it is with this that transfer functin representatins f electrical circuits can be develped. In rder t evaluate the frequenc respnse f the ptical imaging sstem, a transfer functin representatin has t be develped. In ptical sstems the variable f interest is spatial frequenc, and hence this can be used t characterise their transfer functin behaviur. The ptical impulse can be represented b a pint bject, which can be visualised as a delta functin, i.e. (, ) (, ) r which b substituting int eq. (3.7) gives = δ (3.8) ( ) I(, ) h, s s = s s (3.9) which is the impulse respnse f the incherent imaging sstem. If the bjective is a clear, aberratin free, circular aperture under unifrm illuminatin, then the impulse respnse f the incherent sstem is given b the well knwn Air disc pattern f eq. (.3) which is illustrated in Fig. (.) [4,6,7,0,46]. The ptical transfer functin, OTF, is the spatial frequenc representatin f the frequenc respnse f the ptical imaging sstem [40,44]. The incherent OTF is given b the tw-dimensinal Furier transfrm f the incherent impulse respnse. Therefre, the incherent OTF is given b { } ( ν, ν ) = (, ) ep π ( ν + ν ) O h j d d (3.0) where ν and ν are spatial frequenc cmpnents in the and directins respectivel. Hwever, the amplitude pint spread functin is simpl the twdimensinal Furier transfrm f the aperture pupil functin f the bjective lens. Therefre, substituting fr h (, ) frm eq. (3.) int eq. (3.0) gives 5

10 Mathematical analsis f the rdinar reflectance scanning micrscpe O ( ν ν ) P, =, P *, λ f λ f λ f λ f λ f { π ( ν ν )} ep j + d d (3.) where p and P are Furier transfrm pairs and * represents cmple cnjugatin. Using Furier transfrm therems eq. (3.) can be represented as a functin f the bjective aperture pupil functin and its cmple cnjugate [4,4,43], i.e. r in shrt frm, ( ν ν ) = ( + ν λ + ν λ ) O, p f, f p * (, ) dd (3.) ( ν, ν ) = ( ν λ, ν λ ) * ( ν λ, ν λ ) O p f f p f f (3.3) where represents tw-dimensinal cnvlutin and all ther smbls have the usual meaning. It can seen that the incherent ptical transfer functin is given b the twdimensinal cnvlutin f the aperture pupil functin and its reversed cmple cnjugate. Fr a clear, aberratin free, circular bjective aperture under unifrm illuminatin, and a specific pair f spatial frequencies ν and ν, the incherent OTF is equivalent t the area f verlap f tw circles, ne centred abut the rigin and the ther displaced b ν λf in the - directin, and ν λf in the - directin, as illustrated in Fig Area f verlap. Figure 3.4 : Generatin f the incherent ptical transfer functin b the calculatin f the area f verlap f the displaced circular bjective pupil functin and its cmple cnjugate centred abut the rigin. 53

11 Mathematical analsis f the rdinar reflectance scanning micrscpe In this eample where the bjective is devid f an aberratins, the OTF is purel real. Hwever, in aberrated sstems the OTF is in fact cmple, having bth magnitude and phase. The magnitude f the OTF is referred t as the mdulatin transfer functin, MTF, and the phase f the OTF is referred t as the phase transfer functin, PTF [3,43,5]. Figure 3.5 illustrates the calculated incherent OTF fr a clear, aberratin free, circular bjective aperture under unifrm illuminatin, where the frequenc aes are given in nrmalised spatial frequenc cmpnents ν N ν λ = (3.4) NA where λ is the wavelength f illuminatin, and NA is the numerical aperture f the bjective lens [4,0]. The cut-ff in the OTF (i.e. O( ν, ν ) = 0) ccurs at the characteristic spatial frequenc ν = ν + ν = c NA (3.5) λ which crrespnds t a displacement f p f ne diameter in an directin, at which pint the circles in Fig. 3.4 n lnger verlap. The incherent OTF illustrated in Fig. 3.5 has the well knwn frm given b O, = cs π ( ν ν ) ν ν ν ( ν ν ) λf ν ; ν = + d (3.6) where d is the diameter f the circular bjective aperture, and all ther smbls have their usual meaning. Figure 3.6 illustrates the effects that the frm f the incident illuminatin can have n the spatial frequenc characteristics f the incherent imaging sstem. Plts are illustrated alng the ν ais f the incherent OTF, fr a clear, aberratin free, circular bjective aperture, under bth unifrm illuminatin and Gaussian illuminatin ( w = a e /, a - aperture radius). 54

12 Mathematical analsis f the rdinar reflectance scanning micrscpe Nrmalised OTF ν Ν νn Figure 3.5 : The incherent ptical transfer functin, fr a clear, aberratin free, circular bjective aperture under unifrm illuminatin. 0.8 Nrmalised OTF Nrmalised Spatial Frequenc - ν N Figure 3.6 : Plt alng the ν ais f the incherent OTF fr unifrm illuminatin (slid line) and Gaussian illuminatin ( w e = a/) (dashed line), fr a clear, aberratin free, circular bjective aperture. It is clear that the respnse and the shape f the incherent OTF depends significantl n the frm f the incident illuminatin and the shape f the bjective aperture. In the case f unifrm illuminatin and a circular bjective aperture (slid line) the transfer functin rlls ff at an apprimatel linear rate (actuall as in eq. (3.6)) with 55

13 Mathematical analsis f the rdinar reflectance scanning micrscpe increase in spatial frequenc. Hwever, in the case f Gaussian illuminatin ( w e = a/) and a circular aperture the shape f the OTF has changed dramaticall, with a bst f the lw spatial frequenc respnse and the attenuatin f the high spatial frequenc respnse Cherence rati The cherence rati [3], γ, is defined as the rati f the numerical aperture f the cllectr lens t the numerical aperture f the bjective lens, i.e. γ = NA NA C O (3.7) where the subscripts O and C refer t the bjective and cllectr lenses respectivel. Fr the cherent ptical sstem where the cllectr aperture pupil functin is infinitesimall small, the cherence rati is γ = 0, and fr the incherent imaging sstem where the cllectr aperture pupil functin is infinitel large, the cherence rati is γ =. It can be shwn that the spatial frequenc cut-ff f the OTF is a functin f the cherence rati f the imaging sstem, and can be epressed as ν ν c c NA = ( + γ ) 0 γ λ. (3.8) NA = γ λ Figure 3.7 illustrates the effect that the size f the cllectr aperture has n the spatial frequenc respnse and the reslving pwer f the rdinar reflectance sstem. It can be seen that the cherent imaging sstem cuts ff at half the spatial frequenc f the incherent sstem and that it has a flat frequenc respnse within the lw spatial frequenc regin. An interesting result is that fr a cherence factr f γ the spatial frequenc cut-ff f the imaging sstem remains cnstant. A cherence factr f γ = is the case where the numerical apertures f the bjective and cllectr lenses are the same. The case where 0 < γ < crrespnds t the case 56

14 Mathematical analsis f the rdinar reflectance scanning micrscpe f the mre general partiall cherent imaging sstem which will be treated in detail in sec γ= γ = 0 Nrmalised OTF Nrmalised spatial frequenc - ν N Figure 3.7 : Plt alng the ν ais f the OTF fr a cherence factr f γ=0 (bld dashed line) (cherent sstem) and a cherence factr f γ= (slid line). 3. The Tpe reflectance scanning micrscpe It has been shwn that when the reflectance scanning micrscpe empls an infinitesimall small cllectr aperture r an infinitel large cllectr aperture, then the characteristics f the imaging sstem are ver different. Hwever, it is f mre interest t cnsider the case when the cllectr aperture is neither infinite in etent nr infinitesimall small in etent, in which case the imaging is neither incherent r cherent, but partiall cherent. The ptical sstem illustrated in Fig. 3. is ften referred t as the Tpe rdinar, reflectance scanning micrscpe, and it has been discussed in Chap. that it has imaging characteristics similar t that f the cnventinal micrscpe. Fllwing the previus analsis, it can be seen that the epressin representing the signal frm the Tpe reflectance scanning micrscpe is given b eq. (3.0), where the cllectr aperture pupil functin, p (, ) is f finite areal size. It is als useful t develp a transfer functin descriptin fr this mre general, partiall cherent, case. 57

15 Mathematical analsis f the rdinar reflectance scanning micrscpe Transfer functin representatin T derive a transfer functin representatin fr the signal generated in the Tpe reflectance scanning micrscpe, eq. (3.0), then the reflectance prperties f the sample must be epressed in terms f its spatial frequenc spectrum, Γ( ν, ν ), i.e. and its cmple cnjugate (, ) = (, ) ep( [ + ] ) r Γ ν ν π j ν ν dν dν (3.9) (, ) = *(, ) ep( [ + ] ) * r Γ ν ν π j ν ν dν dν (3.30) where ν and ν are spatial frequencies in the and directins respectivel. Substituting fr r(, ) and r *(, ) int eq. (3.0) and rearranging gives I( s, s ) = h(, ) h *(, ) g (, ) { π ( ν ν ν ν )} ep j + d d d d { j( )} ( ν, ν ) Γ *( ν, ν ) ep π ( ν ν ) ( ν ν ) Γ + + s s (3.3) dν dν dν dν where the smbls have their usual meaning. Replacing the term in the square brackets f eq.(3.3) with the functin C( ν, ν ; ν, ν ), the signal frm the Tpe reflectance sstem can be epressed in the frm ( s, s ) = ( ν, ν ; ν, ν ) ( ν, ν ; ν, ν ) I C M { [( ) ( ) ]} ep π j ν ν + ν ν dν dν dν dν s s (3.3) where the functin C( ν, ν ; ν, ν ) is termed the partiall cherent transfer functin, PCTF, which is a functin f the prperties f the ptical sstem itself. It describes the intensit f the cmpnents in the image with spatial frequencies equal t ( ν ν ) in the directin, and ( ν ν ) in the directin. The term 58

16 Mathematical analsis f the rdinar reflectance scanning micrscpe M ν ν ν ν (, ;, ) is called the medium functin and describes the spatial frequenc prperties f the sample, in this case given b ( ν, ν ; ν, ν ) = ( ν, ν ) *( ν, ν ) M Γ Γ. (3.33) The Tpe PCTF Cmparing eq. (3.3) and eq. (3.3), it can be seen that the PCTF fr the Tpe sstem is given b ( ν, ν ; ν, ν ) = (, ) *(, ) (, ) ep{ π j( ν + ν )} ep{ π j( ν + ν )} C h h g (3.34) d d d d which b substituting fr g (, )frm eq. (3.) and rearranging gives ( ν, ν ; ν, ν ) = (, ) *(, ) (, ) C h h p { π j( ν ν )} ep π j( ν ν ) π j ep ( + ) λ f { } ep + +. (3.35) d d d d d d Cmbining the terms in the epnentials and separating the variables, allws eq. (3.35) t be recast in the frm ( ν, ν ; ν, ν ) = (, ) C p h (, ) ep π j ν + λ f h *(, ) ep π j ν ν λ f + d d d d λ f (3.36) Using the shift and cnvlutin therems [4,43] nw allws the Tpe PCTF t be epressed in terms f the bjective and cllectr aperture pupil functins, i.e. 59 ν d d λ f.

17 Mathematical analsis f the rdinar reflectance scanning micrscpe ( ν, ν ; ν, ν ) = ( ν λ, ν λ ) * ( ν λ, ν λ ) C p f f p f f (, ) p d d where scaling terms have been ignred. (3.37) If it assumed that the bjective and cllectr apertures are smmetrical abut the and aes, then the time reversal f p and p * in eq. (3.37) can be ignred, such that the Tpe PCTF ma be written in the mre usual frm ( ν, ν ; ν, ν ) = ( + ν λ, + ν λ ) *( + ν λ, + ν λ ) C p f f p f f (, ) p d d where the smbls have their usual meaning. (3.38) It can be seen frm eq. (3.38) that the generatin f the Tpe PCTF is effectivel a crrelatin tpe prcess invlving the bjective aperture pupil functin, its cmple cnjugate and the square magnitude f the cllectr aperture pupil functin [4,7,0,34]. A cmputatinal prcedure fr generating the Tpe PCTF will be described in detail in sec. 6.. The respnse f the Tpe reflectance scanning micrscpe can nw be calculated in tw was, either directl via eq. (3.0) r using the transfer functin representatin f eq. (3.3). These methds have been implemented in cmputer cde and are termed the direct calculatin apprach and the transfer functin apprach. In the direct calculatin apprach the electric field is calculated as it prpagates thrugh the ptical sstem as the sample is scanned beneath the fcused spt. It ma be used fr generating the respnse t simple ne-dimensinal and tw-dimensinal reflectance tpe bjects, and is described in detail in sec. 5.. In the transfer functin apprach the respnse is generated using the transfer functin representatin f the signal and is described in sec. 6. fr generating the respnse t simple ne-dimensinal reflectance tpe bjects. 60

18 Mathematical analsis f the rdinar reflectance scanning micrscpe 3.3 The Tpe, r cnfcal, reflectance scanning micrscpe It has been described in chapter that an alternative scanning micrscpe arrangement can be cnfigured b placing a pinhle arrangement int the detectin arm f the instrument. The Tpe, r cnfcal scanning micrscpe, as it is cmmnl referred, has fund widespread use in bilgical spheres, where its depth discriminatin prperties are primaril useful [4,53]. The intrductin f a cnfcal pinhle int the detectin arm f the instrument leads t a mdificatin f the imaging characteristics f the scanning micrscpe. T understand the imaging prcess in the cnfcal micrscpe, and t cmpare its imaging characteristics with thse f the Tpe cnfiguratin, it is useful t develp a similar mathematical descriptin f imaging in such an ptical cnfiguratin. The analsis fllws directl frm that presented fr the Tpe cnfiguratin, as described in sec. 3., and invlves the calculatin f the electric field as it prpagates thrugh the ptical sstem. Optical cnfiguratins The cnfcal reflectance scanning micrscpe can be implemented using the tw cnfiguratins which are illustrated in Figs. 3.8 and 3.9. Figure 3.8 : The ptical laut f the rdinar Tpe, r cnfcal, reflectance scanning micrscpe, empling a pinhle pht-detectr. 6

19 Mathematical analsis f the rdinar reflectance scanning micrscpe Figure 3.8 illustrates the cnfcal reflectance scanning micrscpe where the pinhle aperture is intrduced b replacing the large area pht-detectr f the Tpe sstem with an auiliar lens and pinhle pht-detectr arrangement. The auiliar lens is used t fcus the prpagating field that is reflected frm the sample, nt the plane f the pinhle pht-detectr. The pinhle pht-detectr can be visualised as either a pht-detectr f infinitesimall small etent, r a large area pht-detectr with a pinhle aperture placed immediatel in frnt f it. Figure 3.9 : The ptical laut f the rdinar Tpe, r cnfcal, reflectance scanning micrscpe, empling a pinhle aperture and auiliar lens arrangement. Figure 3.9 illustrates an alternative cnfcal detectin arrangement where a pinhle aperture is placed in the cmbined fcal pint f tw auiliar lenses. The prpagating field that is reflected frm the sample is brught t fcus in the plane f the pinhle aperture b the first auiliar lens, the field then prpagates thrugh the pinhle and diverges twards t secnd auiliar lens. Here the field is cllimated and cntinues t prpagate, withut further mdificatin, t the large area phtdetectr, which is assumed t be f unifrm respnsivit. Due t the cmpleities invlved in fabricating a pinhle pht-detectr, the cnfcal arrangement illustrated in Fig. 3.9 is the mre realistic sstem t implement practicall. It is instructive t analse bth the cnfcal imaging sstems illustrated in Fig. 3.8 and Fig. 3.9 t determine if bth sstems ehibit identical imaging characteristics. 6

20 Mathematical analsis f the rdinar reflectance scanning micrscpe Image calculatin In rder t analse the signal generatin prcess in the cnfcal reflectance scanning micrscpe, the field distributin incident n the pht-detectr has t be calculated fr a knwn incident field distributin n the bjective aperture, as in the Tpe cnfiguratin. Cnsider first the cnfcal cnfiguratin f Fig The field, ψ (, ), in the plane f the auiliar lens is as calculated previusl in the Tpe sstem, as given b eq. (3.4). Immediatel after the auiliar lens the field distributin, ψ (, ), is mdified b the aperture pupil functin f the auiliar lens, p3(, ), t give ( ) ψ ( ) p( ) p3( ) ψ, =,,,. (3.39) The field distributin, ψ ( d, d ), incident n the pht-detectr is given b the twdimensinal Furier transfrm f the field distributin ψ (, ), immediatel after the auiliar lens, i.e. ψ λf ( ) ( d, d ) = ψ (, ) p (, ) p3(, ) A jk ep ( d + d ) d d A (3.40) where f A is the fcal length f the auiliar lens, { d, d} is the plane f the phtdetectr and all ther smbls have their usual meaning. The signal frm the phtdetectr is again given b the integral, ver the area f the pht-detectr, f the square magnitude f the incident distributin, i.e. (, ) = ψ (, ) (, ) I R d d s s d d d d d d (3.4) r (, ) = (, ) *(, ) (, ) I ψ ψ R d d (3.4) s s d d d d d d d d where R(, ) represents the respnsivit f the pht-detectr and * represents d d cmple cnjugatin. Substituting fr ψ ( d, d ) frm eq. (3.40) allws eq. (3.4) t be recast in the frm 63

21 Mathematical analsis f the rdinar reflectance scanning micrscpe ( s, s) = ψ (, ) ψ *(, ) I (, ) *(, ) (, ) *(, ) p p p p 3 3 jk R( d, d ) ep ( ( ) + ( )) d d A d d d d (3.43) d d d d which simplifies t give ( s, s ) = ψ (, ) ψ * (, ) c(, ) c * (, ) I p p (, ) G d d d d (3.44) where G(, ) is the Furier transfrm f the pinhle pht-detectr respnsivit, i.e. (, ) = ( d, d ) ep ( + ) G R jk A d d d d d d (3.45) and p ( c, ) is the cmbined aperture pupil functin f the cllectr and auiliar lenses, i.e. (, ) (, ) (, ) p = p p. (3.46) c 3 Substituting fr ψ (, ) frm eq. (3.3) allws eq. (3.44) t be epressed in terms f the reflectance prperties f the sample, i.e. ( s, s) = (, ) *(, ) ( s, s) *( s, s) I h h r r (, ) *(, ) (, ) pc pc G. (3.47) jk jk ep ( + ) ep ( + ) ddd d d d d d This can be rewritten as ( s, s) = (, ) *(, ) ( s, s) *( s, s) I h h r r where g(, ;, ) is given b (, ;, ) g d d d d (3.48) 64

22 Mathematical analsis f the rdinar reflectance scanning micrscpe (, ;, ) = c(, ) c * (, ) (, ) g p p G jk ep + ( ) ep ( + ) f jk d d d d (3.49) where all smbls have their usual meaning and cnstant scaling factrs have been ignred. It can be seen that the signal generatin prcess fr the cnfcal reflectance sstem is again a cnvlutin tpe peratin. Hwever, it is mre cmplicated than the Tpe imaging prcess in s much as it nw incrprates prperties f the auiliar lens and pinhle pht-detectr. Presenting nw the analsis fr the alternative cnfcal reflectance sstem illustrated in Fig 3.9. The field, ψ (, ), immediatel after auiliar lens is given b ( ) ψ ( ) p ( ) p ( ) ψ, =,,, (3.50) 3 where p3(, ) represents the aperture pupil functin f auiliar lens, and p(, ) is the cllectr aperture pupil functin. The field distributin, ψ (, ) 3 3, in the plane f the pinhle is given b the tw-dimensinal Furier transfrm f the field distributin, ψ (, ), immediatel after auiliar lens, i.e. ψ λf ( ) ( 3, 3) = ψ (, ) p (, ) p3(, ) A jk ep ( + ) d d A 3 3 (3.5) where f A is the fcal length f auiliar lens, and all the ther smbls have their usual meaning. Immediatel after the pinhle aperture the field is mdified b the pupil functin f the pinhle, p p ( 3, 3 ), t give ( 3 3) p p( 3 3) ψ ( 3 3) ψ, =,,. (3.5) The field, ψ ( 4, 4 ), in the plane f auiliar lens is given b the tw-dimensinal Furier transfrm f the field,ψ ( 3, 3 ), immediatel after the pinhle, i.e. λf jk A d d ψ (, ) = ψ (, ) ep ( + ) 4 4 A (3.53) 65

23 Mathematical analsis f the rdinar reflectance scanning micrscpe where f A is the fcal length f auiliar lens, and all the ther smbls have their usual meaning. Immediatel after auiliar lens the field is mdified b the aperture pupil functin f auiliar lens, p4( 4, 4 ), t give ( ) p ( ) ψ ( ) ψ, =,,. (3.54) The field then prpagates, withut further mdificatin, twards the pht-detectr. The signal frm the pht-detectr is calculated as previusl, and is given b s s d d (, ) = ψ ( 4, 4) I d d (3.55) where it is assumed the respnsivit is unifrm (equal t unit) ver the area f the pht-detectr. Substituting fr ψ ( 4, 4 ) frm eq. (3.54) int eq. (3.55) gives ( s, s ) = ψ ( 4, 4) ψ *( 4, 4) 4( 4, 4) I p d4d (3.56) 4 which b substituting fr ψ ( 4, 4 ) using eq. (3.53) and rearranging gives ( s, s) = ψ ( 3, 3) ψ *( 3, 3 ) I ( ) jk p4( 4, 4) ep ( ) + ( ) d d A (3.57) d d d d If it assumed that auiliar lens cllects all the field that prpagates thrugh the cnfcal pinhle, and that it des nt bstruct the field in an wa, then p (, ) can be assumed t be unifrm (equal t unit) fr all { 4, 4} and eq. (3.57) simplifies t give * ( s, s) = ψ (, ) ψ (, ) I d d (3.58) which substituting fr ψ (, ) 3 3 frm eq. (3.5) gives ( s, s) = ψ ( 3, 3) ψ *( 3, 3) p( 3, 3) I p d3d3. (3.59) Substituting fr ψ (, ) frm eq. (3.5) and rearranging gives

24 Mathematical analsis f the rdinar reflectance scanning micrscpe ( s, s) = ψ (, ) ψ *(, ) I (, ) *(, ) (, ) *(, ) p p p p 3 3 ( ) jk pp( 3, 3) ep ( ) + ( ) d d A (3.60) d d d d which can be further simplified t give ( s, s) = ψ (, ) ψ *(, ) c(, ) c *(, ) I p p (, ) G d d d d (3.6) where G(, ) is the Furier transfrm f the square magnitude f the pinhle aperture pupil functin, i.e. (, ) = p ( 3, 3 ) ep ( + ) G p jk A d d (3.6) and p ( c, ) is the cmbined aperture pupil functin f the cllectr and auiliar lenses as given b eq. (3.46). Substituting fr ψ (, ) frm eq. (3.3) allws eq. (3.6) t be epressed in terms f the reflectance prperties f the sample, i.e. ( s, s) = (, ) *(, ) ( s, s) *( s, s) I h h r r (, ) *(, ) (, ) p p G jk ep + c c jk ( ) ep ( + ) d d d d (3.63) d d d d which can be rewritten as ( s, s) = (, ) *(, ) ( s, s) *( s, s) I h h r r (, ;, ) g d d d d (3.64) where g(, ;, ) is given b eq. (3.49) with G(, ) given b eq. (3.6), all ther smbls have their usual meaning and cnstant scaling factrs have been ignred. 67

25 Mathematical analsis f the rdinar reflectance scanning micrscpe Cmparing eq. (3.48) and eq. (3.64) it can be seen that the tw cnfcal cnfiguratins illustrated in Figs 3.8 and 3.9 have eactl the same imaging characteristics, prviding auiliar lens in Fig. 3.9 cllects all the field that prpagates thrugh the pinhle. It is instructive at this pint t cnsider tw etreme cases f pinhle size fr the cnfiguratin f Fig If the pinhle aperture is infinitel large, such that pp( 3, 3) = fr all {, }, then 3 3 G(, ) (, ) = δ (3.65) and eq. (3.64) reduces t the same frm as that fr the Tpe reflectance sstem, eq. (3.0), as wuld be epected. If the pinhle is infinitesimall small, such that pp( 3, 3) = δ ( 3, 3), then and ( ) G, = (3.66) (, ;, ) (, ) *(, ) g = h h (3.67) c c which allws the signal t be epressed in the frm (, ) = (, ) (, ) (, ) I h h r d d s s c s s (3.68) and the cnfcal imaging sstem ehibits cherent imaging prperties, and is termed the ideal cnfcal sstem. Nte, the frm f the cllectr lens pint spread functin nw effects the imaging prcess, unlike the Tpe cnfiguratin. Transfer functin representatin As in the Tpe reflectance scanning micrscpe analsis, it is pssible t develp a transfer functin representatin f the signal frm the cnfcal reflectance scanning micrscpe. Recalling eq. (3.9) and eq. (3.30), the reflectance prperties f the sample can be epressed in terms f its spatial frequenc spectrum, Γ(, ) ν ν, i.e. 68

26 Mathematical analsis f the rdinar reflectance scanning micrscpe and its cmple cnjugate (, ) = (, ) ep( [ + ] ) r Γ ν ν π j ν ν dν dν (3.9) (, ) *(, ) ep( [ ] ) * r = Γ ν ν π j ν + ν dν dν (3.30) which substituting int eq. (3.64) gives I( s, s ) = h (, ) h *(, ) g (, ;, ) { π ( ν ν ν ν )} ep j + d d d d { j( )} ( ν, ν ) Γ *( ν, ν ) ep π ( ν ν ) ( ν ν ) Γ + s s (3.69) dν dν dν dν where the smbls have their usual meaning. Replacing the term in square brackets f eq. (3.69) with C( ν, ν ; ν, ν ), the signal frm the cnfcal reflectance sstem can be epressed in the characteristic frm given previusl in eq. (3.3), i.e. ( s, s ) = ( ν, ν ; ν, ν ) ( ν, ν ; ν, ν ) I C M { [( ) ( ) ]} ep π j ν ν + ν ν dν dν dν dν s s (3.3) where fr the case f the cnfcal sstem the term C( ν, ν ; ν, ν ) represents the cnfcal PCTF, and the medium functin M( ν, ν ; ν, ν ) remains unchanged. The cnfcal PCTF Cmparing eq. (3.3) and eq. (3.69) it can be seen that the cnfcal PCTF is given b ( ν ν ν ν ) ep{ π ( ν ν )} ep π ( ν ν ) C, ;, = h (, ) h *(, ) g (, ;, ) { } j + j + d d d d (3.70) 69

27 Mathematical analsis f the rdinar reflectance scanning micrscpe that b substituting fr g (, ;, ) using eq. (3.49) and rearranging gives ( ν ν ν ν ) C, ;, = h (, ) h *(, ) (, ) *(, ) (, ) p p G c jk ep + ( ) ep ( + ) { π j ( ν ν ) } c ep + jk d d d d { ( )} ep π j ν + ν d d d d. Gruping the terms in the epnentials and rearranging gives (3.7) ( ν, ν ; ν, ν ) = c (, ) c *(, ) (, ) C p p G h (, ) ep πj ν λf + d d λf ν h *(, ) ep πj ν ν λ + λ d d f f d d d d (3.7) which b using the shift and cnvlutin therems [4,43] allws the cnfcal PCTF t be epressed in the frm ( ν, ν ; ν, ν ) = ( + ν λ, + ν λ ) *( + ν λ, + ν λ ) C p f f p f f (, ) *(, ) (, ) p p G ddd d c c (3.73) where G(, ) is given b eq. (3.6) and all ther smbls have their usual meaning and it is assumed the apertures are smmetrical abut the and aes. It can be seen that the epressins representing the signal frm bth the Tpe and cnfcal reflectance scanning micrscpes are f identical frm. Hwever, the PCTFs f the tw imaging cnfiguratins are ver different. Cmparing eq. (3.38) and eq. (3.73) it can be seen that the cnfcal PCTF is mre cmplicated cmpared t the Tpe PCTF, in s much that the cnfcal PCTF is a functin f the Furier 70

28 Mathematical analsis f the rdinar reflectance scanning micrscpe transfrm f the square mdulus f the pinhle aperture pupil functin, and is a cnvlutin prcess invlving fur variables. Again cnsider tw etreme cases f cnfcal pinhle aperture size. Fr the case where the cnfcal pinhle is infinitesimall small, such that p p ( 3, 3 ) = δ ( 3, 3 ), then G(, ) = and eq. (3.73) reduces t the frm, ( ν, ν ; ν, ν ) = ( + νλ, + ν λ ) c(, ) C p f f p dd (3.74) and the cnfcal PCTF is effectivel given b the square magnitude f the incherent OTF [4]. Fr the case where the cnfcal pinhle is infinitel large, such that pp( 3, 3) = fr all {, }, then in such a case G (, ) is a delta functin and eq. (3.73) 3 3 reduces t the frm, ( ν, ν; ν, ν ) = ( + νλ, + νλ ) *( + ν λ, + ν λ ) C p f f p f f (, ) *(, ) p p dd c which is the same frm as fr the Tpe case, as wuld be epected. c (3.75) Chapters 5 and 6 described cmputatinal prcedures fr evaluating the imaging equatins develped thrughut this current chapter. 7