Chapter 1. Introduction. 1.1 The scanning optical microscope

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1 Chapte 1 This thesis descibes the fomulation of a mathematical model descibing the signal geneation pocess in both the eflectance and magneto-optic, Type 1 and Type 2, scanning optical micoscopes. The model has been implemented in compute code and allows the analysis and investigation of a vaiety of imaging techniques commonly used in the field of scanning optical micoscopy. Two sepaate signal geneation techniques, the 'diect calculation' and 'tansfe function' appoaches ae pesented and thei applications studied. 1.1 The scanning optical micoscope The conventional micoscope is a vesatile instument commonly used in many scientific fields fo poducing high magnification images of objects. In the conventional micoscope an aea of the object is illuminated by a spot of light and is simultaneously imaged by the eye, o viewed diectly on a sceen. An altenative micoscope aangement, the scanning optical micoscope (SOM) was fist illustated by Robets and Young in the 1950s. Thei 'flying spot' micoscope employed a cathode ay tube to geneate a scanning spot of light which was then tansmitted though the optics of the conventional micoscope in evese to poduce a scanning spot on the suface of the object. The light tansmitted though the object was then detected using a photocell and the electical signal was displayed on a TV sceen [2]. In 1969 the fist scanning lase micoscope (SLM) was demonstated by Davidovits and Egge [3]. In thei system a 5mW HeNe lase light souce was scanned acoss the suface of the object in a aste like fashion using a -y-z scanning objective. The intoduction of the SLM lead to the implementation of a vaiety of futhe imaging techniques. A pime eample is the confocal scanning micoscope 15

2 which is often used in biological sphees and offes impotant depth discimination popeties and impoved esolution ove the conventional micoscope. The scanning lase micoscope offes many advantages ove the conventional micoscope. These include : Impoved esolution using the confocal scanning micoscope, 1.4 that of the conventional micoscope. No chomatic abeations since the lase is monochomatic. High numeical apetues can be used at all magnifications. Electonic pocessing and optical esolution impovement techniques may be implemented with geate ease than is possible with the conventional micoscope. Image bightness and contast is contolled electonically athe than by the optics. Numeous imaging modes possible. Figue 1.1 illustates the optical layouts of the common foms of scanning optical micoscope. Figue 1.1 : Scanning optical micoscope aangements : a) the conventional micoscope, b) the Type 1a scanning micoscope, c) the Type 1b scanning micoscope, d) the Type 2, o confocal, scanning micoscope. 16

3 Figue 1.1 (a) illustates the optical aangement of the conventional micoscope. In this configuation the object is illuminated by a patch of light fom an etended souce though a condense lens. The object is then imaged though an objective lens and viewed using a eyepiece. The esolution of the conventional micoscope is detemined pimaily by the quality of the objective lens. Figue 1.1 (b) illustates the optical aangement of a scanning micoscope based aound the aangement of the conventional micoscope; the Type 1a scanning micoscope. Hee, a point detecto is scanned though the image plane, such that it detects a small egion of the image at any instance in time. The image is constucted point by point and displayed on a colou monito. Figue 1.1 (c) illustates the optical aangement of the Type 1b scanning micoscope. Hee, a point souce is focused onto the object though an objective lens. The detecto measues the total amount of light that is tansmitted though, o eflected fom, the object. The image is constucted by scanning the point souce acoss the object, thus constucting an image point by point. Altenatively, the object is scanned with espect to the stationay focused spot. The Type 1b configuation is commonly efeed to simply as the Type 1 scanning micoscope. Figue 1.1 (d) illustates the optical aangement of the Type 2, o confocal, scanning micoscope. Hee, a point souce is again focused onto the object though an objective lens. Afte inteaction with the object the light passes to a point detecto. An image is constucted by scanning the point souce and point detecto in synchonism acoss the object. The tem confocal is used to indicate that both the objective and collecto lenses ae focused on the same point on the object [4]. Many authos have illustated the imaging similaities between the conventional and the Type 1 scanning micoscopes [4,5,6]. It can be shown that the imaging chaacteistics of the two configuations ae simila poviding the popeties of the objective lenses ae identical and that the collecto lens of the Type 1 scanning micoscope and the condense lens of the conventional micoscope have the same pupil chaacteistics. This elationship is tue poviding the effective souce in the 17

4 scanning micoscope is infinitesimal, i.e. the incident illumination is coheent, as with a lase. The confocal configuation, howeve, offes vey diffeent imaging chaacteistics compaed with the conventional micoscope, and has impoved esolution chaacteistics ove the Type 1 configuation. In the confocal system both the objective and collecto lenses detemine the imaging chaacteistics of the optical system. The confocal configuation has impotant esolution enhancement and depth discimination popeties. Figue 1.2 : The depth discimination popeties of the confocal scanning micoscope. The depth discimination effect is used to geat advantage in biological applications whee 3-D images of tanspaent o semi-tanspaent objects ae geneated. Its oigins ae illustated in Fig. 1.2 whee it can be seen that only light oiginating fom within the focal plane of the lens is bought to focus in the plane of the pinhole and popagates though the pinhole apetue to each the photo-detecto. Light fom above the focal plane is bought to focus afte the plane of the pinhole, hence, vey little light popagates though the pinhole apetue towads the detecto. Light fom below the focal plane is bought to focus befoe the plane of the pinhole, and again vey little light popagates though the pinhole apetue towads the detecto. Hence, the signal fom the detecto in the confocal configuation is etemely sensitive to the degee of focus. 18

5 Due to the vesatility of the scanning optical micoscope it has an eceptionally wide ange of applications and may be used to investigate the popeties of objects using a vaiety of imaging and contast techniques. A paticula use of the scanning micoscope is in the field of magneto-optics [8,9,10]. It has been shown that by the modification of the optics of the odinay scanning micoscope and by using a coheent polaised souce, such as a lase, the scanning micoscope may be used to image magneto-optic (MO) contast. Such an instument has been shown to have paticula advantages in the investigation of the popeties of a ange of thin films and devices such as magneto-optic stoage mateials, magneto-esistive sensos and magnetic multi-layes [11,12,13,14,15]. The study of MO techniques foms a majo pat of the wok of this thesis. 1.2 The magneto-optic effects The magneto-optic effects ae chaacteised by a change in the state of linealy polaised light upon eflection fom, o tansmission though, a magnetic suface. The fist inteaction between light and magnetic mateials was ecoded by Michael Faaday in Faaday obseved that linealy polaised light was otated upon tansmission though lead glass in a magnetic field [16], as illustated in Fig. 1.3 (a). Late in 1897, John Ke discoveed that the plane of polaisation of linealy polaised light was otated upon eflected fom the poles of an magnet, the magnitude of the otation being popotional to the net magnetisation [17]. The fom of the magneto-optic inteaction depends upon the diection of polaisation with espect to the plane of incidence and the diection of magnetisation. When the incident light is linealy polaised in a diection paallel, o pependicula, to the plane of incidence, then the pola Ke and longitudinal Ke effects poduce a small, magnetisation sensitive, otation of polaisation. The pola Ke signal is maimised when the polaised light is nomal to the magnetic suface, as illustated in Fig. 1.3 (d). The longitudinal Ke signal is maimised when the polaised light is incident at an angle of 60 to the nomal of the magnetic suface, as illustated in Fig. 1.3 (c). A futhe magneto-optic effect, the tansvese Ke effect, is chaacteised by the 19

6 attenuation of the magnitude of linealy polaised light upon eflection fom a magnetic suface, and is illustated in Fig. 1.3 (d) [18,19,20,21]. Figue 1.3 : The magneto-optic effects : a) the Faaday effect, b) the pola Ke effect, c) the longitudinal Ke effect and d) the tansvese Ke effect. The magneto-optics effects can be descibed in tems of quantum mechanical inteactions between light and the effective magnetic field due to the spin-obit inteaction of atomic atoms [22]. Howeve, such a desciption is beyond the scope of this thesis. Instead, a phenomenological appoach is used fo which it is useful to fist gasp the popeties of the polaised optical field The polaised optical field A monochomatic, unifom plane wave popagating in fee space, o an isotopic medium, can be epessed in the fom (, y, z, t) = ( $ y) ( j( kz t) + $ y ep ) o ( ) oy ( y) ψ ψ ψ ω ( ψ ep jφ $ jψ ep jφ y$ ) ep j( kz ωt ) ( ) = + (1.1) whee ψ and ψ y ae othogonal field components and it is assumed the wave is popagating in the z diection in a Catesian co-odinate system, whee $ and $y ae 20

7 unit vectos. Equation (1.1) descibes the popagating electic field E, also efeed to as the optical field. The polaisation state of the popagating optical field is detemined by the elative amplitudes of ψ o and ψ oy, and thei elative phase diffeence, ( φ φ ). Geneally eq. (1.1) epesents an electic field vecto, the tip of y which descibes an ellipse in the (, y) plane, otating though one complete cycle as the wave popagates one wavelength in the z diection. Linea polaisation If the phase diffeence between ψ and ψ y is zeo o ± 2nπ, whee n is an intege, the wave is said to be linealy polaised since the optical field oscillates along a line which is at an angle θ with the - ais, whee θ is given by θ ψ = actan ± ψ o oy (1.2) as illustated in Fig Figue 1.4 : Linealy polaised optical field. Cicula polaisation If the magnitude of ψ and ψ y is equal, i.e. ψo = ψ oy = ψ o, and the phase diffeence, ( φ φ ), is π 2 ± 2n π, such that y (, y, z, t) ( $ jy$ ) ep( j( kz t) ) ψ = ψ ± ω (1.3) o 21

8 then the optical field is of constant scala amplitude at any paticula value of z, and the tip of the vecto descibes a cicle in the (, y) plane. The plus / minus sign indicates the diection of otation. Plus ( + ) indicates anticlockwise otation, looking back at the souce, and is efeed to as left ciculaly polaised (LCP), and negative ( ) indicates clockwise otation and is efeed to as ight ciculaly polaised (RCP). Figue 1.5 illustates the LCP optical field. Figue 1.5 : Left ciculaly polaised (LCP) optical field. It is evident fom eq. (1.3) that linealy polaised light can be fomed by combinations of RCP and LCP components. Adding o subtacting equal amplitude RCP and LCP components esults in linea, hoizontally and vetically polaised light espectively. Adding o subtacting equal amplitude RCP and LCP components with an abitay phase diffeence, ± φ, yields linea polaisation but oientated at an angle ± φ / 2 to the -ais. Elliptical polaisation As aleady discussed, eq. (1.1) epesents the geneal case of elliptically polaised light. Reaanging eq. (1.1) gives ψ ψ y oy 2 2 ψ + ψ o ψ y ψ 2 2 φ φ ψ cos = sin (1.4) oy ψ o 22

9 whee φ is the phase diffeence between the othogonal components, i.e. φ = ( φ φy ). Equation (1.4) epesents an ellipse whose majo ais lies at an angle θ with the - ais, whee θ is given by 2ψ oψ oy tan( 2θ ) = 2 2 cosφ ψ + ψ o oy. (1.5) Again, an elliptically polaised optical field can be fomed using RCP and LCP combinations of diffeing magnitude and phase Phenomenological desciption of the magneto-optic effects The pola and longitudinal Ke effects ae the most commonly used in imaging applications. A simple phenomenological desciption of thei oigins is discussed below. In magnetic mateials the RCP and LCP components of a polaised field display diffeent efactive indices, n ±, and thei Fesnel amplitude eflection coefficients, + and - espectively, will be diffeent [23], such that ± = n n ± ± = { j } ep φ (1.6) ± ± and = ep{ j( φ φ ) } (1.7) whee the incident medium is fee space. If the incident light is linealy polaised, i.e. the RCP and LCP components ae of equal magnitude, then two limiting cases ae defined. Fistly, when + = and φ φ which esults in magnetic cicula + biefingence (MCB), whee the eflected field is also linealy polaised but otated by + an amount θ ( φ φ ) k = + / 2. The second case is when and φ = φ + which esults in magnetic cicula dichoism (MCD), whee the eflected field is elliptically polaised, with the majo ais aligned with the incident polaisation. Howeve, in geneal the eflected field is detemined by both MCB and MCD effects. 23

10 Thus, the eflected field is elliptically polaised with its majo ais aligned at an angle θ k to the incident polaisation. It is often convenient to epesent the eflected light by Catesian eflection coefficients, and y. Hence, the eflected field is compised of a component which is polaised in the same diection as the incident polaisation, the magnitude and phase of which is detemined by, and a magneto-optically induced component, the magnitude and phase of which is detemined by [24] y. It is simple to switch between the two epesentations of the eflection coefficients, using the elationships = + j, y = (1.8) If the incident illumination is linealy polaised along the - ais, then upon eflection fom the magnetic sample the light will be elliptically polaised, as illustated in Fig Figue 1.6 : Elliptically polaised light, geneated upon eflection fom an MO suface. The magneto-optically induced Ke signal illustated in Fig. 1.6 can be descibed using the elations tan k tan cos cos 2 2 y ( 2θ ) = ( 2α ) ( φ) = 2 ( φ) tan sin k sin sin y ( α) =, ( 2ε ) = ( 2α ) ( φ) y (1.9) whee θ k is the Ke otation angle, φ is the phase diffeence between the othogonal eflected components and ε k is the Ke ellipticity, the tangent of which is defined as 24

11 ellipticity (atio of majo to mino ais). The Ke signal is maimised, and ellipticity is emoved, when the phase diffeence between the othogonal components is zeo, such that the plane of polaisation is meely otated upon inteaction with the sample. Since the Ke otation is chaacteistically small, aound 0.2 to 0.4 fo ae eathtansition metal (RE-TM) media [25], it is common to epess the magneto-optically induced signal using small angles, i.e. ( ) ( ) θ = α cos φ ε = α sin φ θ k k k ε k = α = 0 φ 0. (1.10) φ = 0 The Ke otation, θ k, is popotional to the net magnetisation of the magnetic sample. The small otation of the plane of polaisation intoduced by the magneto-optic effect can be measued using a vaiety of detection techniques [19,20,21,26,27,28,29,30,31]. The two most common MO detection stategies, the single detecto MO configuation and the diffeential detecto MO configuation, will be descibed in detail in Chapte Optical stoage systems and the esolution limit In optical stoage systems a high infomation stoage density is achieved using an optical aangement simila to that of the scanning optical micoscope [7,32,33,34]. The esolution of the scanning micoscope is detemined pimaily by the diffaction limit of the optical system and the fom of the incident illumination. The pactical esolution limit is usually defined as the minimum distance sepaating two point objects at which they can just be esolved. Two citeia have been poposed to measue the so called two point esolution of an optical imaging system : the Rayleigh citeion and the Spaow citeion. The Rayleigh citeion states that two point souces ae esolvable when the maimum of the illuminance poduced by the fist point souce falls on the fist minima of the illuminance poduced by the second point souce. This is often epessed as the sepaation whee the intensity at the midpoint 25

12 between the two point souces is times the maimum intensity, and can be epessed as esolution 0. 6 λ Rayleigh (1.11) NA whee λ is the wavelength of illumination and NA is the numeical apetue of the lens. As such the Rayleigh citeion is a abitay limit. The spaow citeion is concened with the ate of change of the slope of the image at the midpoint [4,23,35,36,37]. 1.4 The spatial fequency esponse of the scanning optical micoscope The spatial fequency esponse of an optical system is analogous to the tempoal fequency esponse commonly used to chaacteise electical systems. Spatial fequency is epessed in invese metes (m -1 ), o often in lines/mm. Fo eample, a uled gating with a line peiod p g has a spatial fequency f g given simply by f g = 1. (1.12) p g A plane wave incident on such a gating will be split up into a stong zeoth-ode plane wave and two weak fist-ode plane waves whose diections α N ae given by Nλ sinα N sin α o = ( N = 1, 0, 1 ) (1.13) p whee α o is the angle of incidence and N is the ode of the wave. Highe diffaction odes ae pesent fo low spatial fequencies of the gating. Howeve, the contibution due to the highe diffaction odes is small compaed to the zeoth and fist odes and so ae geneally ignoed [32,38]. The amplitude and phase vaiations of the diffacted odes depends pimaily on the geomety of the gating and its position. Geneally, as the peiod of the gating is educed the amplitude of the diffacted ode emains unchanged. Howeve, the phase of the diffacted odes changes, and is given by φ N g Nuo = 2 π (1.14) p g 26

13 whee φ N is the phase of the diffacted ode N and u o is the displacement of the gating. In the scanning micoscope the detecto is consideed to be a unit cicle, infomation egading the gating position is detemined by the aea of ovelap of the zeoth ode and the fist diffacted odes which falls within the detecto aea, as illustated in Fig Figue 1.7 : Aea of ovelap of the zeo and fist diffacted odes epesents the shift of the fist diffacted ode and is given by λ pg λf g = = NA NA whee NA is the numeical apetue of the objective lens. (1.15) Hence, it can be seen that at low fequencies, whee the angle of diffaction of the fist diffacted odes is small, a lage popotion of ovelap will occu between the zeoth and fist diffacted odes and the esulting signal fom the detecto will be high. Howeve, at highe spatial fequencies, whee the angle of diffaction of the fist diffacted odes is lage, the ovelap will be significantly educed and so the esulting signal fom the detecto will be educed also. The point at which the fequency will not be measued is given by the displacement of the fist diffacted odes equal to the adius of the objective apetue. At this point the diffacted odes no longe ovelap and this coesponds to the maimum obsevable spatial fequency and is given by the invese of eq. (1.11), i.e. NA f = 2 ma. (1.15) λ 27

14 1.5 Summay and Objectives This chapte has pesented a bief eview of the scanning optical micoscope. The diffeent optical configuations and thei popeties have been biefly discussed. The inteaction between light and a magnetic suface, the magneto-optic effect, has been pesented, illustating how a scanning micoscope can be developed which is sensitive to magneto-optic contast. The fequency esponse of the scanning micoscope, with efeence to the diffaction gating, has been intoduced. The emainde of this thesis concentates on the development of a mathematical model that descibes, in detail, the signal geneation pocess in the scanning micoscope, which employs a coheent, monochomatic, souce, such as a lase. In chapte 2 scala diffaction theoy and Fouie imaging will be pesented. An epession is developed that descibes the fa-field diffaction patten and this is seen to be simila to that epession epesenting the Fouie tansfom. Diffaction due to finite apetues and the thin lens will also be intoduced. Chapte 3 concentates on the development of a mathematical famewok descibing the signal geneated in the odinay eflectance (non magneto-optic) scanning micoscope. This is then be applied to imaging in the coheent, incoheent and Type 1 optical systems. A tansfe function epesentation is developed that descibes the signal geneated by the Type 1 eflectance scanning micoscope in a fom whee the popeties of the optical channel ae distinct fom the popeties of the sample. The patially coheent tansfe function and the medium function will be intoduced. A simila analysis will be applied fo epesenting the signal geneated in the Type 2, o confocal, eflectance scanning micoscope. In chapte 4 the single detecto and diffeential detecto magneto-optic scanning micoscopes ae illustated. Epessions epesenting the signal geneated in the magneto-optic scanning micoscopes ae developed, fo both the Type 1 and confocal configuations, using an etension of the analysis developed in chapte 3. 28

15 In chapte 5 a computational method, the diect calculation appoach, fo modelling the signal geneated in the Type 1 eflectance, single detecto MO and diffeential detecto MO scanning micoscopes is pesented. The diect calculation appoach is shown to be paticulaly beneficial fo geneating theoetical two-dimensional images of objects. In chapte 6 the computational implementation of the tansfe function appoach is pesented fo geneating the theoetical one-dimensional esponse fom all the scanning micoscope configuations discussed. In paticula the effects of pinhole size in the confocal configuations and analyse and wave-plate misalignment in the magneto-optic configuations on the eadout signal will be pesented. Chapte 7 illustates how the computational models may be used to investigate the signal geneation pocess in optical stoage systems. In paticula the signal geneation pocess in the CD-ROM, phase-change and magneto-optic stoage systems, as well as futue geneation optical stoage systems, such as digital vesatile disk (DVD) is analysed. Chapte 8 illustates how the effects of common foms of abeations obseved in optical stoage systems may be incopoated into the computational models. The effects of defocus, spheical abeation and astigmatism (as a esult of substate biefingence) ae discussed. In chapte 9 epeimental esults ae pesented to veify impotant theoetical esults obtained in the pevious chaptes. 29