International audienceWe propose a comprehensive treatment of theta microscopy based on dipole emission, which better describes fluorescence emission than the isotropic emission model, as fluorescence emission is often polarized. Formulas describing the point spread function for polarized confocal fluorescence theta microscopy are given. Examples are given and some advantages of polarized theta fluorescence microscopy are presented

Hiromitsu Furukawa

Koji Tenjimbayashi

Olivier Haeberlé

Comptes Rendus Physique

C. R. Physique 3 (2002) 1445–1450Électromagnétisme, optique/Electromagnetism, opticsPHYSIQUE/PHYSICSPolarized confocal theta microscopyOlivier Haeberlé a∗, Hiromitsu Furukawa b, Koji Tenjimbayashi ba Groupe LabEl – Laboratoire MIPS, Université de Haute-Alsace IUT Mulhouse, 61, rue A. Camus,68093 Mulhouse cedex, Franceb OptMec, Photonics Research Institute, AIST, Namiki 1-2, Tsukuba, Ibaraki305-8564, JapanReceived 15 July 2002; accepted 6 September 2002Note presented by Ionel Solomon.Abstract We propose a comprehensive treatment of theta microscopy based on dipole emission,which better describes fluorescence emission than the isotropic emission model, asfluorescence emission is often polarized. Formulas describing the point spread function forpolarized confocal fluorescence theta microscopy are given. Examples are given and someadvantages of polarized theta fluorescence microscopy are presented. To cite this article:O. Haeberlé et al., C. R. Physique 3 (2002) 1445–1450. 2002 Académie des sciences/Éditions scientifiques et médicales Elsevier SASfluorescence microscopy / reflection microscopy / theta microscopyMicroscopie confocale theta polariséeRésumé Nous présentons un modèle pour la microscopie de fluorescence theta basé sur lerayonnement dipolaire, qui décrit mieux le phénomène de fluorescence que le modèleisotropique, car l’émission de fluorescence est souvent polarisée. Les formules décrivantla tache de diffraction pour la microscopie de fluorescence polarisée en montage thetasont données. Des exemples sont donnés, et certains avantages à utiliser la polarisationen microscopie de fluorescence theta sont présentés. Pour citer cet article : O. Haeberlé etal., C. R. Physique 3 (2002) 1445–1450. 2002 Académie des sciences/Éditions scientifiques et médicales Elsevier SASmicroscopie de fluorescence / microscopie en réflexion / microscopie thetaVersion française abrégéeEn microscopie classique et confocale [1,2], un compromis entre résolution et distance de travail doit êtreaccepté, les objectifs à grande distance de travail ayant une faible ouverture numérique, et donc une faiblerésolution. Le microscope theta a été proposé pour pallier à cet inconvénient. Il consiste à utiliser deuxobjectifs de microscope pour l’excitation et la détection, dont les axes optiques sont perpendiculaires [3].La Fig. 1 en décrit le principe. L’objectif 1 est utilisé pour illuminer le specimen, et la détection se fait autravers de l’objectif 2.Nous proposons un traitement détaillé du microscope theta, basé sur la théorie vectorielle de la diffractionpour les objectifs à grande ouverture numérique, et sur le modèle d’émission dipolaire.∗ Correspondence and reprints.E-mail address:o.haeberle@uha.fr (O. Haeberlé). 2002 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservésS1631-0705(02)01406-8 /FLA 1445O. Haeberlé et al. / C. R. Physique 3 (2002) 1445–1450La configuration du microscope theta oblige à utiliser des objectifs sans lamelle, la RIO d’excitationest donc calculée à l’aide du modèle de Richards et Wolf [1]. Les équations (1), (2) donnent le champélectromagnétique au point focal. La RIO du microscope theta est alors souvent calculée à l’aide del’équation (3), en considérant une émission de fluorescence isotrope et non-polarisée [3]. L’émission defluorescence étant polarisée [4,5], il est plus rigoureux de considérer le champ émis par un dipole [6,7].Le moment et le champ dipolaires sont alors donnés par les équations (4) et (5). Le champ est capté parl’objectif 2, et refocalisé sur le détecteur. Avec un objectif de refocalisation de faible ouverture numérique,le détecteur quadratique enregistre alors le carré de la transformée de Fourier du champ électromagnétique :équation (7), les équations (6a) et (6b) décrivant le champ, calculé à l’aide des intégrales (8). La formule(9) lie les angles entre l’objectif de détection 2 et la lentille de refocalisation [6,7].Nous avons calculés les RIO pour un microscope theta utilisant des objectifs d’ouverture numériqueN.A. = 0,8 à immersion à eau [8], avec une excitation à λ = 400 nm et une détection λ = 450 nm(Cascade Blue from Molecular Probes). La Fig. 2 montre les résultats obtenus. La Fig. 2(a) montre laRIO calculée avec le modèle isotrope. Les Figs. 2(b) et 2(c) montrent la RIO calculée avec le modèledipolaire, pour une excitation x-polarisée et une détection x-polarisée et pour une excitation z-polarisée etune détection x-polarisée, respectivement. Le modèle isotrope a tendance à surestimer la résolution en x ,et sous-estimer la résolution en z et en y . La Fig. 3 montre les RIO pour une détection y-polarisée. Nonseulement les formes des RIO sont très différentes, mais surtout l’intensité est environ 400x plus faible. LesRIOs montrent la forme caractéristique en feuille de trèfle aussi prévue en microscopie confocale polarisée[6,7,9,11] (Figs. 3(a) et 3(c)). Notons cependant que les RIO sont maintenant asymétriques, comme lemontrent les Figs. 3(b), 3(d) et 3(e), qui présentent des coupes x−y et x−z à divers plans en profondeur etlatéralement.1. IntroductionIn classical and even confocal microscopy [1,2], a trade-off between resolution and working distancemust always be made. Long working distance objectives have a low numerical aperture, and therefore a lowresolution, especially along the optical axis. High numerical aperture objectives permit a high resolution,but at the price of a usually very short working distance.Theta microscopy has been proposed as an effective solution to combine high resolution with longworking distances. Theta microscopy therefore provides the unique feature of allowing the study of largebiological sample while keeping a good resolution.In previous work, the Point Spread Function of theta microscopy was calculated using an isotropicemission model, with unpolarized excitation. However, fluorescence is known to be polarized. As aconsequence, dipole emission describes better the fluorescence process.We propose a comprehensive treatment of polarized fluorescence microscopy based on the electromag-netic theory of high numerical aperture objective, taking into account dipole emission by the fluorescentdye.2. Polarized fluorescence theta microscopyThe idea of theta microscopy is to diminish the size of the Point Spread Function by using two microscopeobjectives with crossed optical axes [3]. Doing so, the lower resolution along the optical axis of the detectionobjective is compensated by the fact that this axis corresponds to a lateral axis for the illumination objective.Fig. 1 describes the setup considered. Objective 1 is used for fluorescence excitation, and objective 2 is forconfocal detection.This optical configuration does not allow us to use glass slides, so, the Richard and Wolf model ofdiffraction describes well the illumination PSF formation [1]. Considering a plane wave, linearly polarizedalong the x-axis, being focused by objective 1, the electric field at point P with coordinates x = x ′, y = −z′,1446To cite this article: O. Haeberlé et al., C. R. Physique 3 (2002) 1445–1450Figure 1. Sketch of a confocal theta microscope. Objective 1 is used for sample excitation. Objective 2, with opticalaxis at 90◦ with respect to objective 1, is used for detecting the fluorescence signal.Figure 1. Schéma de principe d’un microscope confocal theta. L’objectif1 sert à l’excitation du specimen.L’objectif 2, dont l’axe optique est à90◦ de l’objectif 1, sert à la detection du signal de fluorescence.z = y ′ (see Fig. 1: (x, y, z) coordinates are for detection arm and (x ′, y ′, z′) coordinates are for illuminationarm) in the focal region is written as:Ex = −i(I0 + I2 cos 2φ), Ey = 2I1 cosφ, Ez = −i(I2 sin 2φ) (1)with:I0 =∫ α0a(θ) sin θ(1 + cosθ)J0(k√x2 + z2 sin θ)exp(−iky cos θ)dθ,I1 =∫ α0a(θ) sin2 θJ1(k√x2 + z2 sin θ) exp(−iky cosθ)dθ, (2)I2 =∫ α0a(θ) sin θ(1 − cosθ)J2(k√x2 + z2 sin θ)exp(−iky cos θ)dθ,the Bessel function of first kind of order n. The apodization function in Eqs. (2) is given by a(θ) = cos1/2 θ ,φ the angle between the direction of observation and the x-axis (see Fig. 1), α is the objective lens apertureangle, Jn is the Bessel function of first kind or order n, and k = k0 · n with n being the immersion mediumrefraction index.The classical treatment of confocal or theta confocal microscopy consists in considering that the excitedmolecule acts as an isotropic radiator. The final Point Spread Function is simply calculated as the productof the illumination PSF by the detection PSF. For a theta setup, it is written as [3]:PSFtheta(x, y, z) = PSF(x, y, z) × PSF(−z, y, x) (3)as the difference in wavelength between illumination and detection is often neglected for simplicity. In thatapproach, the illumination and detection PSF are computed using the Richards and Wolf model. However,this model is for a plane wave focused into the medium, thus valid to compute the illumination PSF, butcalculating the detection PSF as an illumination PSF constitutes another approximation.Furthermore, fluorescence is known to be polarized [4,5]. A more rigorous approach therefore consistsin considering a point scatterer whose far field emission is that of an electric dipole, which moment p is1447O. Haeberlé et al. / C. R. Physique 3 (2002) 1445–1450proportional to the illumination field [6,7]. Hence, the Cartesian components of the dipole moment aregiven by:px = −i(I0 + I2 cos 2φ), py = 2I1 cosφ, pz = −i(I2 sin 2φ), (4)the electric field in the radiation zone being given by:Edp = −r × (r × p). (5)The dipole electric vector is collimated by the detection objective, then passed through an analyzer andrefocused onto the detector. Considering a low aperture detector lens, a quadratic detector therefore recordsthe intensity I = |Ed |2, with Ed being the Fourier transform of the scalar field [6]:Edx = px(Id0 + Id2 cos 2φp) + pyId2 sin 2φp − 2iId1 pz cosφp, (6a)Edy = pxId2 sin 2φp + py(Id0 − Id2 cos 2φp) − 2iId1 pz sin φp (6b)and then one obtains the intensity PSF after the polarizer as [6]:PSF(x, y, z) = ∣∣cos2 γEdx + sinγ cosγEdy∣∣2 + ∣∣cosγ sinγEdx + sin2 γEdy∣∣2 (7)with:Id0 =∫ αd0√cosθ1 sin 2θdJ0(kρ sin θd)(1 + cosθ1) exp(−ikzs cosθ1)dθd,Id1 =∫ αd0√cosθ1 sin 2θdJ1(kρ sin θd) sin θ1 exp(−ik1zs cosθ1)dθd, (8)Id2 =∫ αd0√cosθ1 sin 2θdJ2(kρ sin θd)(1 − cosθ1) exp(−ik1zs cosθ1)dθd,γ being the angle between the analyzer axis and the x-axis. The angular aperture of the detector lens αdand the coordinate angle θd are linked to the objective lens angular aperture α1 and the coordinate θ1 by therelation [6,7]:n1 sinα1nd sinαd= n1 sin θ1nd sin θd= β, (9)where β is the nominal magnification of the objective lens, and n1 and nd are the indices of refraction ofthe immersion medium and the image space, respectively [6,7].3. ExamplesFig. 2 shows computed point spread functions. An Olympus Universal Plan Fluorite 40x water-im-mersion objective with N.A. = 0.8 is considered. This objective has a working distance of 3.3 mm andwould be well suited for a theta setup, after a slight modification of the enclosure [8]. The excitationPSF is computed at λ = 400 nm and we consider emission at λ = 450 nm (corresponding to excitation andemission of Cascade Blue from Molecular Probes). The specimen is scanned through the focus. For the sakeof simplicity, a point-like incoherent detector is considered [6,7]. Fig. 2(a) shows the PSF computed usingthe classical isotropic emission model, for unpolarized excitation and detection, using Eq. (3). Fig. 2(b)shows the PSF using the dipole model for x-polarized excitation and x-polarized detection. It is shown thatthe classical model tends to slightly overestimate the x-lateral resolution, while simultaneously slightlyunderstimates the y-lateral and z-longitudinal resolution. The dipole emission models predicts a moreisotropic PSF, with better separation of the lateral lobes (more noticeable differences are expected if larger1448Pour citer cet article : O. Haeberlé et al., C. R. Physique 3 (2002) 1445–1450Figure 2. Point Spread Functions for a theta microscope using water immersion objectives with N.A. = 0.8, excitationat λ = 400 nm and detection at λ = 450 nm (Cascade Blue – Molecular Probes). (a) conventional unpolarizedisotropic radiation model. (b), (c) dipole emission model. (b) x-polarized excitation, x-polarized detection.(c) z-polarized excitation, x-polarized detection.Figure 2. Reponse Impulsionelle Optique pour un microscope theta utilisant des objectifs à immersion à eau(N.A. = 0,8). Excitation àλ = 400 nm et détection àλ = 450 nm (Cascade Blue – Molecular Probes). (a) modèleclassique d’émission isotrope non polarisée.(b), (c) modèle dipolaire.(b) excitationx-polarisée, détectionx-polarisée.(c) excitationz-polarisée, détectionx-polarisée.N.A. objectives may be used [6,7,9]). The x-resolution is 220 nm, the y- and z-resolution being of 270 nm(versus 210 nm and 290 nm for the isotropic emission model).An inherent problem of polarized detection in confocal microscopy is that the detection of moleculesexcited by light with z-axis polarization is limited by the fact that the excitation (and therefore the availablesignal) is much weaker than for polarization along the x-axis or the y-axis [9,10]. Typically the signal is100 times lower [9]. A possible approach is to use phase filters to enhance the z polarized component ofthe excitation and detection [10].Theta microscopy offers another possibility: as the illumination and detection axes are crossed, the opticalaxis for the detection arm corresponds to a lateral axis of the illumination arm, for which the componentis expected to be of much higher intensity. As shown on Fig. 1, choosing a x ′- or y ′-polarized illuminationwave corresponds to enhancing detection for the x- or z-component, respectively. The equations giving theillumination field, therefore the dipole moment, are readily obtained from Eqs. (1), (2) by a 90◦ rotationaround the (z′ = −y)-axis, thus permuting the roles of the variable x and −z. Fig. 2(c) shows the PSFobtained for z-polarized excitation and x-polarized detection. Note that the maxima of the PSFs are the samein x-polarized or z-polarized excitation. Polarized theta microscopy may therefore constitute an interestingalternative to z-polarized confocal microscopy for large specimen [10].Fig. 3 shows PSFs computed for crossed polars, with y-polarized detection. Not only are the shapesdifferent, but also the relative intensities of the PSFs, the detection for crossed-polarization being typically400 times weaker. Fig. 3(a) shows the computed PSF whith x-polarized excitation and y-polarized detection(in the z = 0 µm and y = 0 µm planes, respectively). It exhibits the characteristic four-leave pattern alsoobserved in confocal polarized microscopy [6,7,9,11], but now in both planes. The pattern is however nowasymetric along the z-axis, as can be seen on Fig. 3(b), showing x−z cuts along the y = −0.18 µm andy = +0.18 µm, respectively. Fig. 3(c) shows the PSF computed for z-polarized excitation and y-polarizeddetection (in the z = 0 µm and y = 0 µm planes, respectively). The PSF has now a very different shape,as can be seen on Figs. 3(d) and 3(c), which show cuts along various y- and z-planes. Note that a centralsymmetry is conserved.1449O. Haeberlé et al. / C. R. Physique 3 (2002) 1445–1450Figure 3. Same as Fig. 2, but for y-polarized detection. (a) x-polarized excitation, y-polarized detection. (b) same as(a)-bottom, but in the y = −0.18 µm and y = +0.18 µm plane respectively. Note the assymetry. (c) z-polarizedexcitation, y-polarized detection. (d), (e) same as (c) but in the z = ±0.15 µm and y = ±0.15 µm plane. Note thecomplex shape of the PSF in that case.Figure 3. Idem à Fig.2, mais pour une détectiony-polarisée.(a) excitationx-polarisée, détectiony-polarisée.(b) identique à(a)-bas dans les plansy = −0,18 µm ety = +0,18 µm. Notez l’assymétrie.(c) excitationz-polarisée,détectiony-polarisée.(d), (e) identique à(c) mais dans les plansz = ±0,15 µm et y = ±0,15 µm. Notez la formecomplexe de la RIO dans ce cas.To favor the y-polarized component, one could either symetryze the detection and illumination arms,or use a second illumination objective, placed perpendicularly to the theta setup (along the x-axis), in aconfiguration similar to Multiple Objective Microscopy [12] or Multiple Imaging Axis Microscopy [13].4. ConclusionWe have given a rigorous electromagnetic treatment of polarized confocal theta microscopy, using thedipole emission model to describe the fluorescence process. Formulas to compute Point Spread Functionsfor this type of microscopy are given as well as typical examples of PSF. The dipole emission model predictsa slightly more isotropic resolution than the isotropic emission one. The main advantage is the possibilityof using strongly z-polarized excitation. Taking into account polarization properties of the fluorescenceemission may permit to obtain additional information about the distribution and orientation of the dipolesin a large specimen with a good spatial resolution.Acknowledgements.This work was made possible with the financial support of the New Energy and IndustrialTechnology Development Organization (NEDO) of Japan under an Industrial Technology Research Grant Program.References[1] B. Richards, E. Wolf, Proc. Roy. Soc. London Ser. A 253 (1959) 349.[2] M. Minsky, Scanning 10 (1988) 128.[3] E.H.K. Stelzer, S. Lindek, Opt. Commun. 111 (1994) 536.[4] F. Perrin, Ann. Phys. (Paris) 12 (1929) 169.[5] P. Soleillet, Ann. Phys. (Paris) 12 (1929) 23.[6] P. Török, P.D. Higdon, T. Wilson, Opt. Commun. 148 (1998) 12.[7] P. Török, P.D. Higdon, T. Wilson, J. Mod. Opt. 45 (1998) 1681.[8] Olympus France, private communication.[9] C.J.R Sheppard, P. Török, Bioimaging 5 (1997) 205.[10] N. Huse, A. Schönle, S.W. Hell, J. Biomed. Opt. 6 (2001) 480.[11] T. Wilson, R. Juskaitis, P. Higdon, Opt. Commun. 141 (1997) 298.[12] O. Haeberlé, C. Xu, A. Dieterlen, S. Jacquey, C. R. Acad. Sci. Paris, Série IV 2 (2001) 1509.[13] E.H.K. Stelzer, J. Swoger, J. Huisken, Multiple imaging axis microscopie (MIAM), in: Technical Digest of Focuson Microscopy 2002, Kaohsiung, 2002, p. 58.1450

Polarized confocal theta microscopy

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Polarized confocal theta microscopy

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