10
Generalized model for incoherent detection in confocal optical microscopy Rachid Hammoum,* Sidi Ould Saad Hamady, and Marc D. Fontana Laboratoire Matériaux Optiques, Photonique et Systèmes, EA 4423, Université Paul Verlaine et Supélec (Metz), 2 rue Edouard Belin, 57070 Metz, France *Corresponding author: [email protected] Received 9 December 2009; revised 21 February 2010; accepted 3 March 2010; posted 3 March 2010 (Doc. ID 120869); published 23 March 2010 We develop a generalized model in order to calculate the point spread functions in both the focal and the detection planes for the electric field strengths. In these calculations, based on the generalized Jones matrices, we introduce all of the interdependent parameters that could influence the spatial resolution of a confocal optical microscope. Our proposed model is more nearly complete, since we make no ap- proximations of the scattered electric fields. These results can be successfully applied to standard confocal optical techniques to get a better understanding for more quantitative interpretations of the probe. © 2010 Optical Society of America OCIS codes: 180.1790, 350.5730, 120.4820, 120.1880. 1. Introduction Confocal optical microscopy is widely used in various application areas, including chemical identification, optical and physical characterizations, and indus- trial process control. Generally, only light originating from a small volume (less than 1 μm 3 ) of a sample is probed with a confocal optical microscope. In this technique the sample is illuminated with a micro- metric laser spot (object plane), and the scattered light is detected through a variable pinhole aperture (detection plane) [13]. The detected signal in confo- cal optical microscopy is a complicated function of many parameters [4], including the numerical aper- ture of the lens objective, the laser wavelength, the polarization states, the absolute magnification of the confocal microscope, pinhole radius for incoher- ent detection, the refractive index mismatch between air and the sample, and the location of the latter com- pared with the focal plane in free space (z ¼ 0). Therefore, the spatial resolution is related to the quantitative accuracy with which a sample can be probed. For instance, it is necessary to use objectives with high numerical apertures (NAs) to obtain a better resolution in several optical systems [5,6]. The existing work on the detection part of the sys- tem has always performed theoretical calculations with some simplifying assumptions that considered only a part of the interdependent parameters for the confocal microscope. For instance, several studies have reported theoretical calculations for the electric field point spread functions (PSFs) [712] by using the well-established vectorial Debye theory with the introduction of the generalized Jones matrices. Nevertheless, the above treatments either have as- sumed that the dipole emitter (scattering source) must be many wavelengths from the first interface [711] or have made some inevitable approximations to overcome this difficulty, but remain valid only for the special cases studied [12]. The alternative ap- proach based on geometrical optics [13] lacks the influence of some of the important parameters. The angular spectrum representation [1416] of the scattered electromagnetic plane waves by a point source, embedded in a sample, is known to be an ef- fective method to approach the physical reality. This approach starts from deriving the exact solutions of Maxwells equations for each emitted plane wave. As a second step, the electric field at the pinhole plane of 0003-6935/10/160D96-10$15.00/0 © 2010 Optical Society of America D96 APPLIED OPTICS / Vol. 49, No. 16 / 1 June 2010

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Page 1: Generalized model for incoherent detection in confocal optical microscopy

Generalized model for incoherent detectionin confocal optical microscopy

Rachid Hammoum,* Sidi Ould Saad Hamady, and Marc D. FontanaLaboratoire Matériaux Optiques, Photonique et Systèmes, EA 4423, Université Paul

Verlaine et Supélec (Metz), 2 rue Edouard Belin, 57070 Metz, France

*Corresponding author: [email protected]

Received 9 December 2009; revised 21 February 2010; accepted 3 March 2010;posted 3 March 2010 (Doc. ID 120869); published 23 March 2010

We develop a generalized model in order to calculate the point spread functions in both the focal and thedetection planes for the electric field strengths. In these calculations, based on the generalized Jonesmatrices, we introduce all of the interdependent parameters that could influence the spatial resolutionof a confocal optical microscope. Our proposed model is more nearly complete, since we make no ap-proximations of the scattered electric fields. These results can be successfully applied to standardconfocal optical techniques to get a better understanding for more quantitative interpretations of theprobe. © 2010 Optical Society of America

OCIS codes: 180.1790, 350.5730, 120.4820, 120.1880.

1. Introduction

Confocal optical microscopy is widely used in variousapplication areas, including chemical identification,optical and physical characterizations, and indus-trial process control. Generally, only light originatingfrom a small volume (less than 1 μm3) of a sample isprobed with a confocal optical microscope. In thistechnique the sample is illuminated with a micro-metric laser spot (object plane), and the scatteredlight is detected through a variable pinhole aperture(detection plane) [1–3]. The detected signal in confo-cal optical microscopy is a complicated function ofmany parameters [4], including the numerical aper-ture of the lens objective, the laser wavelength, thepolarization states, the absolute magnification ofthe confocal microscope, pinhole radius for incoher-ent detection, the refractive index mismatch betweenair and the sample, and the location of the latter com-pared with the focal plane in free space (z ¼ 0).Therefore, the spatial resolution is related to thequantitative accuracy with which a sample can beprobed. For instance, it is necessary to use objectives

with high numerical apertures (NAs) to obtain abetter resolution in several optical systems [5,6].

The existing work on the detection part of the sys-tem has always performed theoretical calculationswith some simplifying assumptions that consideredonly a part of the interdependent parameters forthe confocal microscope. For instance, several studieshave reported theoretical calculations for the electricfield point spread functions (PSFs) [7–12] by usingthe well-established vectorial Debye theory withthe introduction of the generalized Jones matrices.Nevertheless, the above treatments either have as-sumed that the dipole emitter (scattering source)must be many wavelengths from the first interface[7–11] or have made some inevitable approximationsto overcome this difficulty, but remain valid only forthe special cases studied [12]. The alternative ap-proach based on geometrical optics [13] lacks theinfluence of some of the important parameters.

The angular spectrum representation [14–16] ofthe scattered electromagnetic plane waves by a pointsource, embedded in a sample, is known to be an ef-fective method to approach the physical reality. Thisapproach starts from deriving the exact solutions ofMaxwell’s equations for each emitted plane wave. Asa second step, the electric field at the pinhole plane of

0003-6935/10/160D96-10$15.00/0© 2010 Optical Society of America

D96 APPLIED OPTICS / Vol. 49, No. 16 / 1 June 2010

Page 2: Generalized model for incoherent detection in confocal optical microscopy

the detection system is derived in a consequent man-ner. Guo et al. [17,18] succeeded in deriving an elec-tromagnetic theory for a waveguide multilayeredoptical memory by using this kind of approach. Theirmethod was dedicated to deriving the exact solutionsof the object scattering patterns with the introduc-tion of the vector coherent transfer function [19],and the Lippman–Schwinger equation that usesthe dyadic Green’s functions. This latter includesthe whole interface effects, as the local refractiveindex changes. We note that this approach doesnot require that the dipole be several wavelengthsaway from the first interface.The approach recently introduced by Tang et al.

[20] is simpler, since, it takes advantage of the ele-gance of the generalized Jones matrices in order tomodel the detected electric field patterns. Tang etal. [20] considered an additional metal interface de-posited on the sample for surface plasmon-coupledemission microscopy, using the angular spectrum re-presentation of the scattered light as well.In our calculations, and according to this last pro-

cedure, for the present survey we take into accountonly a single diopter interface to focus our attentionon the other parameters. Our model, based on thislast procedure, could be easily extended to the caseof the light focused and collimated through a strati-fied medium [21] to analyze particular situations. Inmost existing studies, not all of the important param-eters in confocal optical microscopy are always takeninto account at the same time. Our work considersthis kind of omission, and puts forward the exact de-tected electric fields with no approximations for thescattering sources.This paper is organized in two main sections. We

start in Section 2 by the introduction of our theore-tical calculations of the illumination and the detec-tion parts as mentioned above. In Section 3, wediscuss some calculated patterns according to thepresented theory and mention briefly a comparisonof the detected PSF for the model of [9] and ours.Finally, conclusions from this study are drawn.

2. Theoretical Treatment

In a confocal optical microscope, with a backscatter-ing configuration, the emitted or radiated electricfield of the probe returns through the same lens objec-tive aperture with a finite sized detector (see Fig. 1).The most straightforward andmost used approach toanalyze this kind of optical microscopy is the general-ized vectorial Debye theory. Even though it is possibleto introduce electromagnetic calculations by usingseveral other models and techniques [12], those pro-cedures are known to bemore complex. Nevertheless,the Debye theory is known to be simple and moreusable, since it uses the generalized Jones matrices[12], and we have used this approach for this work.Besides, for theaimof simplicity,weassumeaplanaticmicroscope objective lenses to avoid inessential com-plications for the problem.

The conventional approach used, for instance, byTörök et al. [22] approximates the diffraction patternof the focused light by a superposition of plane waveswhose propagation direction is within the geometri-cal focal cone. The Debye approximation [23] predictsthat the electromagnetic field is cylindrically sym-metric around the optical axis (axis z) and takes intoaccount the polarization states of the fields, in con-trast to the scalar theory of light. The Debye ap-proach is valid if the Fresnel number Fn is muchlarger than unity. The Fresnel number depends par-ticularly on the lens objective numerical aperture,and in the vast majority of applications this assump-tion (Fn ≫ 1) is valid.

In this section we present a more complete conceptfor the vectorial ray-tracing applied to the opticalscattering process, and apply our study to the caseof a confocal pinhole detector. In our theoretical mod-el we perform a complete analysis of the scatteredlight including the effect of polarization, diopterinterface location, objective lens NA, wavelength,microscope magnification, and pinhole radius by as-suming a homogeneous refractive index for the sam-ple. We note that the objective lens is one of thecritical elements in a confocal optical microscope.It determines magnification, field of view, and reso-lution. For example, when the numerical aperture ofan objective becomes large, an arbitrary polarizedbeam becomes depolarized in the focal region. Suchan effect of depolarization can be easily described bythe proposed theory.

We introduce the general treatment for arbitrarypolarized states related to a particular orientationof a scattered Ramanmode, that is, to the vibrationaldirection of the dipole sources. This allows a com-plete view of the electric field patterns to be obtained,considering its different Cartesian components in theimage space of the confocal optical microscope, thedetected PSF. Finally, in order to validate our calcu-lations, comparison is made with the results reportedin [9].

A. Coherent Illumination

We consider a cylindrically symmetric optical systemwith an optical axis z (Fig. 1). This system focuses apoint source that is situated far from the object zone(focal region), at z ¼ −∞. It radiates arbitrarily polar-ized monochromatic and coherent electromagneticwaves in the form of a spherical wave towards thefocus. The origin O of the ðx; y; zÞ Cartesian coordi-nate system corresponds to the focus position in freespace along the positive direction of the optical axis ztowards the probed sample. The electric and mag-netic fields are hence well positioned at a particularpoint Q in the focal region. The aperture size of theobjective lens L1 and its focal distance f 1 are taken tobe large compared with the incident laser wave-length. We retain here that for a given wave vectors ¼ ðsx; sy; szÞ ¼ k=jkj is the unit vector along a typicalray written in the Cartesian coordinate system, andrQ ¼ ðx; y; zÞ is the position vector pointing from O to

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Q. Let Ei

eðrQ; tÞ ¼ Re½E0iðrÞ expð−iωtÞ�e (the subscript

i stands for medium i) to indicate the time-dependent electric field at point Q at time t, wheree ¼ E=jEj is a unit vector, ω is the phase velocity ofthe plan waves, and Re indicates the real part ofthe electric field E.First, by the introduction of the theoretical

approach for the illumination process by means ofgeneralized Jones matrices, we deduce the corre-sponding electric field strength expressions in med-ium 2. We reproduced the calculus reported by Töröket al. [12] in a more comprehensive form. The repro-duced results are written as follow:

Eðx;y;zÞ02 ¼ Aðϕ1Þ

2

264 Aþðτp cosϕ2 þ τsÞ þ ðτp cosϕ2 − τsÞðAþ cos 2θ þ iB sin 2θÞiBðτp cosϕ2 þ τsÞ þ ðτp cosϕ2 − τsÞð−iB cos 2θ þ Aþ sin 2θÞ

−2τp sinϕ2ðAþ cos θ þ iB sin θÞ

375; ð1Þ

where the angles θ and ϕi are the rotation anglearound the z axis compared with the x axis andthe angle between Si and the z axis in medium i(i ¼ 1 stands for air and i ¼ 2 stands for the sample),respectively; the function Aðϕ1Þ can be regarded asan apodization function on lens L1 [22]: Aðϕ1Þ ¼f 1l0cos1=2ϕ1, with f 1 the focal length of lens L1 in freespace and l0 is an amplitude function factor. Besides,in addition to [12], we assume that the laser beam ispartially defined by the term Dðϕ1Þ that accounts forthe amplitude transmittance at the pupil of the ob-jective lens L1, and one obtains

l0ðϕ1;NAÞ ¼�Fn

NA

�expð−2sin2ϕi=NA2Þ

¼ l01ðNAÞ ×Dðϕ1;NAÞ: ð2Þ

Here τs and τp are the Fresnel coefficients for trans-mission through the diopter interface air–sample fors and p polarization, respectively. Aþ and B are theelements defining the generalized Jones matrix for aBabinet–Soleil compensator [12] and are introducedhere for generality. After some straightforward inte-gration, the complete Cartesian electric field expres-sions for E2, inside medium 2, at an arbitrary point Qare given by a combination of the three functions I0,I1, and I2 [12,22]:

E2x ¼ −iK ½I0Aþ þ I2ðAþ cos 2θQ þ iB sin 2θQÞ�; ð3aÞ

E2y ¼ K ½I0B − I2ðB cos 2θQ þ iAþ sin 2θQÞ�; ð3bÞ

E2z ¼ −2KI1ðAþ cos θQ þ iB sin θQÞ; ð3cÞwhere the integrals I0, I1, and I2, are similar to thosereported in [12,22], except for the introduced factor

Dðϕ1Þ:

I0 ¼Z α

0Dðϕ1Þðcosϕ1Þ1=2ðsinϕ1Þ expðik0Ψðϕ1;ϕ2;−dÞÞ

× ðτs þ τp cosϕ2Þ × J0ðk1rQ sinϕQ sinϕ1Þ× expðik2rQ cosϕQ cosϕ2Þdϕ1;

I1 ¼Z α

0Dðϕ1Þðcosϕ1Þ1=2ðsinϕ1Þ expðik0Ψðϕ1;ϕ2;−dÞÞ

× τp sinϕ2 × J1ðk1rQ sinϕQ sinϕ1Þ× expðik2rQ cosϕQ cosϕ2Þdϕ1;

I2 ¼Z α

0Dðϕ1Þðcosϕ1Þ1=2ðsinϕ1Þ expðik0Ψðϕ1;ϕ2;−dÞÞ

× ðτs − τp cosϕ2Þ × J2ðk1rQ sinϕQ sinϕ1Þ× expðik2rQ cosϕQ cosϕ2Þdϕ1;

where the constant

K ¼ k21f l012k2

¼ πn21f l01λn2

;

and Ψ ¼ −dðn1 cosϕ1 − n2 cosϕ2Þ is the aberrationphase function introduced by the presence of the in-terface. The subscriptQ stands for an arbitrary pointlocated in the focal zone, and ni refers to the effectiverefractive index. The terms d and α are the nominaldisplacement (defocus) of the sample according to thefocal plane and the angle of the outer incident raytowards the sample, respectively.

B. Incoherent Detection of the Scattered Signal

The illumination profile, described by the last inte-grals [see Eqs. (3)], can occur at any point withinthe focal region. Usually, the analysis of the scatteredwaves is reduced to that originating from the surfaceof the material. Otherwise, the displacement due tothe point dipole scatterers is proportional to the ex-citing intensity distribution of the electric fieldsEðx;y;zÞ2 inside the isotropic sample. The electronic po-

larizabilities αij, generally expressed by a two-ranktensor, can be developed in normal coordinates qfor the chemical bonds of the sample through the in-duced dipole moment p ¼ α · Eðx;y;zÞ

2 (where we nor-malize αi¼j ¼ 1 and αi≠j ¼ 0). Thus, the scatteredRaman light is delivered in a small solid angle dΩaround the optical axis. Dipole waves are frequentlyused in physical optics to study focusing and scatter-

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ing of light. Hence, we present below the extendedtheory for our calculations and now consider aðxQ; yQ; zQÞ coordinate system centered at the loca-tion of the vibrating molecule.It is to bementioned that although Török [8] solved

the problem of the propagation of dipole wavesthrough dielectric interfaces, he assumed that thedistance of the dipoles from the first interface shouldbe of several wavelengths, and consequently the con-tribution of evanescent waves was ignored. However,research on near-field optics provides evidence thatevanescent waves often play a crucial role [14,15],and we do not use this assumption here. In our mod-el, the field produced by a dipole emitter is related tothe angular spectrum representation. After changingfrom Cartesian coordinates for the wave vector k tocylindrical coordinates k ¼ kρ þ kzez, the integrationby kρ using residue calculus is performed. The dipole,embedded in a medium of refractive index n2, oscil-lates harmonically as p

eðtÞ ¼ Re½p expð−iωtÞ�. Thecomplex magnitude EsðrÞ of the dipole-source fieldEs

eðr; tÞ ¼ Re½EsðrÞ expð−iωtÞ� in an infinite medium

can be represented as an angular spectrum, accord-ing to [14,15]

EsðrÞ ¼i

2πn22

Z þ∞

−∞

d2kρ1

k0ν2½k20n2

1p − ðp · kÞk�

× exp½ik · ðr −H · ezÞ� for z ≠ H; ð4Þ

where H stands for the depth location of the dipoleemitter and k0 ¼ ω=c is the modulus of the wave vec-tor in free space.

The angular spectrum representation is a superpo-sition of plane waves with wave vectors written aski ¼ kρ þ k0νi · signðz − dÞez, and here kρ indicatesthe component of k parallel to the xy plan. The inte-gration then runs over the entire kρ plane. In the lastequation, we note νi ¼ ðn2

i − η2Þ1=2 ¼ ni cosϕi, andη ¼ kp=k0 ¼ ni sinϕi. Both evanescent and travelingwaves are taken into consideration, and to obtain thescattered field propagated towards the air, it is moreconvenient to decompose the plane waves that arepropagating in the negative z direction, towardsthe collecting lens L1, into their p and s componentsso that we can use the Fresnel transmission coeffi-cients for both media and air. The unit vector forthe s and p polarizations after refraction are

es ¼1kρ

kρ∧ez; ep;t ¼ −1n1

�ν1�kρkρ

�þ η · ez

�: ð5Þ

The calculated strength components for the trans-mitted field EtðrÞ ¼ E1ðrÞ in terms of these unitvectors can then be written in the following way:

Fig. 1. Simplified schematic of the confocal microscope. (a) Schematic of the imaging geometry of the pinhole detector. (b) Labelingscheme for the scattered waves for a dipole embedded in a sample. (c) Labeling the s- and p unit vectors relative to the Cartesian coordinatesystem.

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Page 5: Generalized model for incoherent detection in confocal optical microscopy

E01 ¼24E01;ρE01;s

E01;z

35 ¼

24−ðν1=n1Þτpp · ep

τsp · es−ðη=n1Þτpp · ep

35

¼24 ðν1=n1n2Þτpðν2p · eρ þ ηp · ezÞ

τsp · esðη==n1n2Þτpðν2p · eρ þ ηpezÞ

35; ð6Þ

which denotes the electric field strength vector ofthe plane waves in medium 1, with epðρ; zÞ ¼− cosϕeρ − sinϕez, pρ ¼ p · eρ, and pz ¼ p · ez. As thewaves propagate towards collimation, the wavefrontaberration, due to the presence of the diopter inter-face, can be obtained as

iΦ ¼ idðk2s2z − k1s1zÞ ¼ idk0ðν2 − ν1Þ: ð7Þ

Note that the assumptions for an aplanatic opticalsystem stand for this subsection as well, as both theobjective lens L1 and the tube lens L0 are irrelevantfor our purpose and complicate the equations. Thisallows simplifying the problem of reflection and re-fraction as well. Hence, the electric field strengthafter L0 is

Eðx;y;zÞ00 ðγÞ ¼ Cðϕ1;ϕ0Þ · N · Rb · L0b · L1b · E

ðρ;s;zÞ01 ; ð8Þ

where the subindex b stands for the backward travel-ing waves, and the transmission coefficients τs;p areincluded in the expression for the field Eðρ;s;zÞ

01[Eq. (6)], ϕ1 is the angle of analysis, and Cðϕ1;ϕ0Þis the apodization factor for a system that satisfiesAbbe’s sine condition and is given by

Cðϕ1;ϕ0Þ ¼�cosϕ0

cosϕ1

�1=2

: ð9Þ

The matrices L1b and L0b describe the rotation ofpolarizations following the orientation of es:

L1b ¼24 cosϕ1 0 sinϕ1

0 1 0

− sinϕ1 0 cosϕ1

35;

L0b ¼24 cosϕ0 0 sinϕ0

0 1 0

− sinϕ0 0 cosϕ0

35: ð10Þ

In Eq. (8) Rb is the transformation matrix betweenthe ðρ;−s; zÞ coordinate system and the ðx; y; zÞ Car-tesian coordinate system, and N is the generalizedJones matrix of a polarizer whose axis makes an an-gle γ with the positive x direction. They are written asfollows:

Rb ¼24 cos θ sin θ 0

sin θ − cos θ 0

0 0 1

35;

N ¼24 cos2γ sin γ cos γ 0

sin γ cos γ sin2γ 0

0 0 1

35; ð11Þ

where θ is the azimuthal rotation angle of the wavevector. It should be noted that the NA of the detectorlens is, in practice, very low (<0:1). Hence, the opticalsetup is a true Fourier transforming arrangement,and the scalar field in the focal plane of the detectoris given by the truncated Fourier transform of thescalar field E0 after the lens L0. The detected electricfield is then given by the integral formula of Richardsand Wolf [23,24] transformed into spherical polar co-ordinates, which is written as follows:

Eðx;y;zÞ0 ¼ ik0

Zσ0

Z2π0

sinϕ0 · Eðx;y;zÞ00 exp½ik0ðrd sinϕ0

× sinϕd · cosðθ − θdÞ − zd cosϕ0Þ�× expðiΦÞdθdϕ0; ð12Þ

where σ is the maximum angle for the outer raytracing towards the detector and ðrd sinϕd ×cos θd; rd sinϕd sin θd; rd cosϕd ¼ zdÞ are the Carte-sian coordinates for a point Qdðxdyd; zdÞ in the imageplane of the low-aperture lens. Against the work re-ported by Tang et al. [20], we need here to express thedetected electric field components in Cartesian coor-dinate system for accommodation with the dipoleorientation vector p, which is introduced with its Car-tesian components px, py, and pz as well. When theintegration by θ is carried out analytically, we obtainthe detected electric field strengths as follows:

Ex0 ¼ ik0

2½pxKI

0 þ 2ipzKI1 cos θd

þ KI2ðpx cos2θd þ py sin 2θdÞ�; ð13aÞ

Ey0 ¼ ik0

2½pyKI

0 þ 2ipzKI1 sin θd

þ KI2ðpx sin 2θd − py cos 2θdÞ�; ð13bÞ

Ez0 ¼ k0½KII

1 · ðpx cos θd þ py sin θdÞ − ipzKII0 � ð13cÞ

with

KI0 ¼

Zσ0

�cosϕ0

cosϕ1

�1=2

ðτs þ τp cosϕ2 cosϕ0Þ sinϕ0

· J0ðk0ρ sinϕ0Þ exp½−iðk0zd cosϕ0 −ΦÞ�dϕ0;

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KI1 ¼

Zσ0

�cosϕ0

cosϕ1

�1=2

τp sinϕ2 cosϕ0 sinϕ0

· J1ðk0ρ sinϕ0Þ exp½−iðk0zd cosϕ0 −ΦÞ�dϕ0;

KI2 ¼

Zσ0

�cosϕ0

cosϕ1

�1=2

ðτs − τp cosϕ2 cosϕ0Þ sinϕ0

· J2ðk0ρ sinϕ0Þ exp½−iðk0zd cosϕ0 −ΦÞ�dϕ0;

KII1 ¼

Zσ0

�cosϕ0

cosϕ1

�1=2 τp

2cosϕ2 sin 2ϕ1 sin 2ϕ0

· J1ðk0ρ sinϕ0Þ exp½−iðk0zd cosϕ0 −ΦÞ�dϕ0;

KII0 ¼

Zσ0

�cosϕ0

cosϕ1

�1=2 τp

2sinϕ2 sin 2ϕ1 sin 2ϕ0

· J0ðk0ρ sinϕ0Þ exp½−iðk0zd cosϕ0 −ΦÞ�dϕ0:

Finally, in the presence of an analyzer the detectedelectric field polarizations are

ex0 ¼ cos2 γ Ex0 þ sin γ cos γEy

0; ð14aÞ

ey0 ¼ sin γ cos γEx0 þ sin2 γ Ey

0; ð14bÞ

ez0 ¼ Ez0: ð14cÞ

The quantities ρ and zd are the radial and axial dis-placements of the dipole, respectively, and the azi-muthal angle ϕ0 is related to the azimuthal angleϕ1 by the relationship [9]

k1 sinϕ1

k0 sinϕ0¼ sinϕ1

sinϕ0¼ M; rd ¼ −rQ ·M; ð15Þ

where M is the nominal differential (or transverse)magnification of the detector lens system for a defo-cus d ¼ 0 and k0 is the wave vector modulus in theimage space. The aperture size of the collector lensdetermines the degree of coherence of the imaging,and the defocus of the image is given by zd ¼−zQM2 for small diopter–interface displacement(d → 0) [25]. But, the extension should take this intoaccount, by the introduction of some geometrical op-tics calculus, to reach MðzQÞ [25]:

zd ¼ −zQM ·MðzQÞ; ð16Þ

with [25,26]

MðzQÞ ¼M

1þ ðzQ=tÞ ×M × ð1þMÞ ; ð17Þ

where t is the tube length of the microscope. Thus,this illustrates that the axial magnification is afunction of scatterer defocus. We note thatMðzQÞ ≠ Mð−zQÞ.

The intensity that is detected in the image space isknown to be expressed as follows:

I ∝ ðjEx0j2 þ jEy

0j2 þ jEz0j2Þ=PT ; ð18Þ

where the normalization factor PT is the total poweremitted by the dipole in the presence of an interface.Otherwise, and against the coherent detection meth-od, when the confocal pinhole (incoherent detection)is used, then the detected intensity is defined as

Id ¼ZDjE0j2SDdD ¼

ZR0

Z2π0

jE0j2f ðρdÞρddρddθp; ð19Þ

where SD is the sensitivity function andD is the areaof the detector, or a rotationally symmetrical detec-tor,R is its radius and f ðρdÞ is the sensitivity functionof the detector. Finally, the confocal PSF could be ob-tained as the product of the illumination PSF andthat of the detected one: PSFconf ¼ PSFill · PSFdet.

In the next section we present some numerical cal-culations based on the above model. The guiding ideais mainly to show how the spatial resolution isaffected by the evolution in some interdependentparameters.

3. Numerical Calculations and Discussion

We perform some numerical computations for thespatial resolution quantity relative to some fixedinterdependent parameters. The influence of the eva-nescent waves on the detected PSF is not discussedfurther here, since they are always filtered by themaximum aperture of the collecting lens. These nu-merical calculations were performed on a computerwith an Intel core 2 Duo processor and 3GbyteRAM. The programs were written in MATLAB soft-ware. For our system, the focusing occurs from air(n1 ¼ 1) into a transparent isotropic material withn2 ¼ 2:2 (close to those of the ferroelectric crystalLiNbO3, for instance). In our simulations, we as-sumed a linear x polarization for the illuminationPSF, and the detection PSFs are analyzed linearlyas well. Particularly, we show for illustration the di-mensions of the probed zone according to the as-sumed objective lenses. Nevertheless, when it isnot mentioned, we note that the used wavelengthis assumed to be the 514:5nm one, and the objectivelens is NA ¼ 0:9 with a 100× nominal magnification.The tube lens of the system is assumed to bet ¼ 250mm ¼ 2:5 × 105 μm, and all of the mentionedlengths are given in micrometers.

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A. Numerical Simulations

The probed zone and then the scattering volume ofthe sample is of first importance; we thus start todraw in Fig. 2 the illumination PSF as a functionof the interface defocus. Particularly, Fig. 2(a) showsthe distribution of the time-averaged electric energydensity excitation along the optical z axis (for r ¼ 0)as a function of the nominal probe depth d, assumingthe diopter interface system introduced above. Thisfigure is plotted on a linear gray scale and normal-ized so as to achieve maximum visibility. Figure 2(b)shows some profiles from Fig. 2(a) at some discreteprobe depths d. The center of gravity for the axialdistribution of the intensity is shifted comparedwith the initial nominal probe depth d. The full withat half-maximum (FWHM) of the main energy-density-distribution peak, profiled in Fig. 2(a), is re-ported in Fig. 2(c). This gives us an exact idea of theaxial and the lateral resolutions evolutions, for boththe 514.5 and the 633nm laser wavelengths, relativeto the defocus increase. Smaller wavelengths giverise to a better spatial resolution whose axial resolu-tion degrades faster than the lateral one. Forinstance, the degradation of the axial resolutionreaches a factor of four times the original value whenwe focus with d ≅ −80 μm [Fig. 2(c)]. Note that thelateral resolution shows a weaker dependence withthe probe depth location. For instance, for the514:5nm wavelength and starting from about

0:4 μm (d ¼ 0), the lateral resolution reaches a valueof about 0:73 μm at probe depth d ≅ −80 μm. Whenthe probe depth is increasing we observe the forma-tion of an additional lobe on the positive side of the zaxis, as well as the appearance of other smaller lobesthat remain far from the main one. Elsewhere, weobserve a much worse resolution for the 633nmwavelength, particularly when probing deeper inthe sample. The main lobe strength vanishes overthe defocus, and the intensity reduces to half ofits maximum value for d ≅ 25 μm [see Figs. 2(a)and 2(b)] and to a quarter of its maximum valuefor d ≅ 64 μm, for example.

We illustrate in Fig. 3 the detected signal originat-ing from electric dipole scatterers that are positionedthrough whole illuminating patterns of the illumina-tion PSF in the focal plan for d ¼ 0, with a densityrelated to a step of 0:01 μm for the lateral displace-ment r. These arranged dipoles are assumed to beoriented along a particular axis in the materialand induced by a preponderant incident x polariza-tion on the lens L1. The distribution of the detectedPSF due to electric dipoles polarized perpendicular(with αxx ¼ αyy ¼ 0 and αzz ¼ 1) and parallel (withαxx ¼ 1 and αyy ¼ αzz ¼ 0) to the interface are ob-tained. In both situations, the Ez component of thecalculated detection PSF is negligible relative tothose of the Ex and Ey components. They are of aboutthe same order of magnitude for the perpendicular

Fig. 2. Illumination PSF. (a) Calculated time-averaged electric energy density distribution. (b) Illumination PSF for four discretedisplacements d of the interface for two wavelengths. (c) Resolution versus probe depth illumination. Here we used an objective lens withNA ¼ 0:9 and a 100× magnification system. The black lines and the gray lines refer to the 633 and the 514:5nm laser wavelengths,respectively. The fluctuations present in (c) are due to the relatively small discrimination steps for our calculus.

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dipoles [Figs. 3(a)–3(c)] and for the x dominant com-ponent for the parallel dipoles [Figs. 3(d)–3(f)].Furthermore, Fig. 3 shows trivial symmetries inthe distribution of whole components. The detectedimages are larger in dipoles perpendicular to the in-terface than in dipoles parallel to the interface. Thespatial resolution shows dependence on the polariza-tion orientations of the dipoles, as well.Furthermore, the detected intensities are drawn

particularly to show the spatial resolution depen-dence on the NA of the objective lens and the polar-ization states as well (Fig. 5). Assuming that electricdipoles are positioned in the whole focal plane (z ¼ 0)for d ¼ 0, we show here the detected intensity as afunction of the pinhole radius for dipoles orientedperpendicularly to the interface. In this situation,we assume at least two interdependent parametersto change and leave the others fixed. As a first case[Fig. 5(a)] we compare the influence of parallel orcrossed polarizers, assuming an objective lens NA ¼0:5 with a nominal magnification of 50× [Fig. 5(a)],and then an objective lens NA ¼ 0:9 with nominalmagnifications 100× [Fig. 5(b)] is considered. Crossedpolarizers show much less sensitivity in term ofthe detected light intensities than parallel ones.Additionally, a smaller objective NA gives a largerdetected signal.

B. Discussion

It is of first importance to get the best possible knowl-edge about the detected polarization quantities inconfocal systems. The spatial resolution is the mainconcept that can limit the sensitivity of such appara-

tuses. In the light of these arguments, Fig. 2 showsthat the scattering volume is a function of the laserwavelength and the nominal defocus as well. Other-wise, the additional lobes shown are associated withthe increased spherical aberration when probingdeeper in a sample. This result is a confirmation forwhat has been reported until now in the literature.

Figure 3 shows how the z- and x-oriented dipolescan influence the detected intensities and their in-fluences on the pinhole radius optimization. Thesedependences are also due to the influence of dif-ferent contributions to the electric dipole moment

Fig. 3. Detection point spread function (PSF). (a)–(c) Detection PSF components, jExj2, jEyj2, and jEzj2, for electric dipoles orientedperpendicular to the interface. (d)–(f) Detection PSF, jExj2, jEyj2, and jEzj2, components for electric dipoles embedded in the xy plane,and oriented along the x axis. The dipoles are assumed to exist only in the whole focal plan for a focus in the air, d ¼ 0.

Fig. 4. Effective numerical aperture of a dry lens assuming afocal length f ¼ 250 μm. This is a function of the probing depth intothe sample. Comparison is made by assuming other values for therefractive index n2.

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intensities. For instance, the illumination PSF pat-terns contribute to those intensities as well. Suchan effect is very useful for particular exploitationin the interpretation of the origin of some particularfields. According to arbitrary assumptions, the con-figuration of the dipole emitters can be organizedin different patterns in order to reach the best pos-sible understanding for a particular case study.Elsewhere, the aberration phase function [see

Eq. (7)] represents simply the induced phase differ-ence for the real optical path length of the wave vec-tor trajectory relative to that optical path whenfocused towards z ¼ 0 in air. When the sample is in-troduced, the light collected by the objective lens L1is restricted by its effective numerical aperture,namely, NAeff . This means that not all the raysreaching the input face of the lens are confinedthrough it, so it is necessary to define the effectiveradius reff of the input face of the lens L1. Particu-larly, the derivation of NAeff , confined to the accuracyof Gaussian optics, is written as

NAeff ¼reffffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2eff þ f 2

q ⇒NA2eff

¼ ðf ·NAÞ2

ðf ·NAÞ2þ�d

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin22−NA2

qþðf −dÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1−NA2

p �2 ;

ð20Þ

where NA ¼ sinϕ1, NAeff ¼ sinϕeff , and ϕeff repre-sent a rim effective angle for the traveling wave vec-tors. The last equation indicates that NAeff dependsnot only on the source lens distance but also on theindex profile mismatch (see Fig. 4). Hence, the jumpin refractive index at the air–sample interface signif-icantly reduces the effective numerical aperture oflens L1, which increases the diffraction-limited spotsize. This argues for the fact that the evanescent

waves cannot be detected by microscope objectiveswhose outer incident rays arrive with an angle lessthan ϕac ≅ 87:2° (see [15]), even when the sample isshifted closer to the collecting lens. This value is notreached generally, particularly for our microscope ob-jectives, which is why we assumed NA ¼ 0:9 (whichgives ϕmax ≅ 64:2° < ϕac), for illustration. ϕac isdefined by the condition that permits the appearanceof evanescent waves propagating laterally afterrefraction.

Finally, we show in Fig. 5 that if the objective NA issmaller or/and polarizers are crossed, there is a lossin the spatial resolution in addition to as the loss inthe detection sensitivity for probed peculiarities.

4. Summary

We have described a generalized model for confocaloptical microscopy. This model takes into account dif-ferent parameters, such as the existence of the diop-ter interface, the polarization states, and the effect ofthe pinhole radius on the detection sensitivity. An in-coherent source region was considered, assuming anarbitrary polarized source dependent on the illumi-nating patterns. These different parameters haveto be taken into account so that an optimal spatialresolution can be achieved.

The present theory completes the existing theore-tical work reported in the literature. The simulatedresults reinforce the need for very important quanti-tative and qualitative studies for a better under-standing of the physical reality of confocal opticalmicroscopy. However, these calculated illustrationsgive us a richer concept of the spatial distributionand polarization orientations for particularapplications.

We sincerely thank Prof. Germano Montemezzani,from the Laboratoire MOPS—Univerity of Metz andSupélec—France, for his helpful, rich, and preciousdiscussions to improve this work.

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