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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 7 — Mar. 26, 2012
  • pp: 8055–8070
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Nano-lens diffraction around a single heated nano particle

Markus Selmke, Marco Braun, and Frank Cichos  »View Author Affiliations


Optics Express, Vol. 20, Issue 7, pp. 8055-8070 (2012)
http://dx.doi.org/10.1364/OE.20.008055


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Abstract

The action of a nanoscopic spherically symmetric refractive index profile on a focused Gaussian beam may easily be envisaged as the action of a phase-modifying element, i.e. a lens: Rays traversing the inhomogeneous refractive index field n(r) collect an additional phase along their trajectory which advances or retards their phase with respect to the unperturbed ray. This lens-like action has long been understood as being the mechanism behind the signal of thin sample photothermal absorption measurements [Appl. Opt. 34, 41–50 (1995)], [Jpn. J. Appl. Phys. 45, 7141–7151 (2006)], where a cylindrical symmetry and a different lengthscale is present. In photothermal single (nano-)particle microscopy, however, a complicated, though prediction-wise limited, electrodynamic scattering treatment was established [Phys. Rev. B 73, 045424 (2006)] during the emergence of this new technique. Our recent study [ACS Nano, DOI: 10.1021/nn300181h] extended this approach into a full ab-initio model and showed for the first time that the mechanism behind the signal, despite its nanoscopic origin, is also the lens-like action of the induced refractive index profile only hidden in the complicated guise of the theoretical generalized Mie-like framework. The diffraction model proposed here yields succinct analytical expressions for the axial photothermal signal shape and magnitude and its angular distribution, all showing the clear lens-signature. It is further demonstrated, that the Gouy-phase of a Gaussian beam does not contribute to the relative photothermal signal in forward direction, a fact which is not easily evident from the more rigorous EM treatment. The presented model may thus be used to estimate the signal shape and magnitude in photothermal single particle microscopy.

© 2012 OSA

1. Introduction

Photothermal lens spectroscopy (PLS) has become a valuable tool in the study of solids and liquids [1

1. X. S. Xie, S. J. Lu, W. Min, S. S. Chong, and G. R. Holtom, “Label-free imaging of heme proteins with two-photon excited photothermal lens microscopy,” Appl. Phys. Lett. 96, 113701 (2010). [CrossRef]

3

3. A. V. Brusnichkin, D. A. Nedosekin, M. A. Proskurnin, and V. P. Zharov, “Photothermal lens detection of gold nanoparticles: Theory and experiments,” Appl. Spectrosc. 61, 1191–1201 (2007). [CrossRef] [PubMed]

]. Recent publications include the study of non-linear effects [4

4. G. Battaglin, P. Calvelli, E. Cattaruzza, F. Gonella, R. Polloni, G. Mattei, and P. Mazzoldi, “Z-scan study on the nonlinear refractive index of copper nanocluster composite silica glass,” Appl. Phys. Lett. 78, 3953–3955 (2001). [CrossRef]

] and nanoparticles in solution [3

3. A. V. Brusnichkin, D. A. Nedosekin, M. A. Proskurnin, and V. P. Zharov, “Photothermal lens detection of gold nanoparticles: Theory and experiments,” Appl. Spectrosc. 61, 1191–1201 (2007). [CrossRef] [PubMed]

, 5

5. D. Rings, R. Schachoff, M. Selmke, F. Cichos, and K. Kroy, “Hot Brownian Motion,” Phys. Rev. Lett. 105, 090604 (2010). [CrossRef] [PubMed]

]. Many authors have focused on the theoretical description of the thin-sample slab geometry which is common in such macroscopic lensing experiments, often providing numerical equations which may be used to obtain absorption coefficients [6

6. F. Jurgensen and W. Schroer, “Studies on the diffraction image of a thermal lens,” Appl. Opt. 34, 41–50 (1995). [CrossRef] [PubMed]

, 7

7. J. Moreau and V. Loriette, “Confocal dual-beam thermal-lens microscope: Model and experimental results,” Jpn. J. Appl. Phys. 45, 7141–7151 (2006). [CrossRef]

]. In all these models the thermal lens induced originates from the absorbed power of a heating laser which constitutes a spatially extended cylindrically symmetric heat source in the heat equation. The solution obtained is then used to study the effect on the propagation of a probing laser beam. In thermal lens spectroscopy (TL) the probing beam is coaxial (possibly offset) with the heating beam [7

7. J. Moreau and V. Loriette, “Confocal dual-beam thermal-lens microscope: Model and experimental results,” Jpn. J. Appl. Phys. 45, 7141–7151 (2006). [CrossRef]

, 8

8. J. Moreau and V. Loriette, “Confocal thermal-lens microscope,” Opt. Lett. 29, 1488–1490 (2004). [CrossRef] [PubMed]

] while in beam deflection spectroscopy [9

9. M. Harada, T. Kitamori, and T. Sawada, “Phase signal of optical beam deflection from single microparticles -theory and experiment,” J. Appl. Phys. 73, 2264–2271 (1993). [CrossRef]

, 10

10. J. Q. Wu, T. Kitamori, and T. Sawada, “Theory of optical beam deflection for single microparticles,” J. of Appl. Phys. 69, 7015–7020 (1991). [CrossRef]

] (BDS) the probing and the heating beams are aligned perpendicularly to each other. Both methods have in principle the same sensitivity [11

11. W. B. Jackson, N. M. Amer, A. C. Boccara, and D. Fournier, “Photothermal deflection spectroscopy and detection,” Appl. Opt. 20, 1333–1344 (1981). [CrossRef] [PubMed]

].

In contrast to these macroscopic techniques, a new microscopic approach has been developed in the 1990’s by Harada and Kitamori [12

12. M. Harada, K. Iwamotok, T. Kitamori, and T. Sawada, “Photothermal microscopy with excitation and probe beams coaxial under the microscope and its application to microparticle ananlysis,” Anal. Chem. 65, 2938–2940 (1993). [CrossRef]

, 13

13. K. Uchiyama, A. Hibara, H. Kimura, T. Sawada, and T. Kitamori, “Thermal lens microscope,” Jpn. J. Appl. Phys. 39, 5316–5322 (2000). [CrossRef]

] termed photothermal lens microscopy. The developement of single particle photothermal microscopy [14

14. A. Gaiduk, M. Yorulmaz, P. V. Ruijgrok, and M. Orrit, “Room-temperature detection of a single molecule’s absorption by photothermal contrast,” Science 330, 353–356 (2010). [CrossRef] [PubMed]

16

16. D. Boyer, P. Tamarat, A. Maali, B. Lounis, and M. Orrit, “Photothermal imaging of nanometer-sized metal particles among scatterers,” Science 297, 1160–1163 (2002). [CrossRef] [PubMed]

] followed and detects a very different kind of thermal lens with a modified standard confocal fluorescence microscope: Instead of an axially symmetric refractive index profile a spherically symmetric profile n(r), created by the point-like heat-source which is the absorbing nano-particle, is probed. Also, instead of a profile that decays on the length-scale of the heating beam focus, in PT single particle microscopy a lens is probed which decays to half its value on the length-scale of the nanoparticle. On the other hand, the profile extends infinitely as 1/r and thus misses a characteristic length-scale. This is the reason, why the description presented in this paper and within the recent electrodynamic (EM) study [17

17. M. Selmke, M. Braun, and F. Cichos, “Photothermal Single Particle Microscopy: Detection of a Nano-Lens,” ACS Nano, DOI: [CrossRef] [PubMed]

] provide no evidence for a role-play of the Gouy-phase which is otherwise important for the probing of small scatterers as shown by Hwang and Moerner [18

18. J. Hwang and W. E. Moerner, “Interferometry of a single nanoparticle using the gouy phase of a focused laser beam,” Opt. Commun. 280, 487–491 (2007). [CrossRef]

]. While models for spherical absorbers have been put forward [19

19. G. C. K. Chen, M. Andika, and S. Vasudevan, “Excitation temporal pulse shape and probe beam size effect on pulsed photothermal lens of single particle,” J. Opt. Soc. Am. B 27, 796–805 (2010). [CrossRef]

, 20

20. M. Harada, M. Shibata, T. Kitamori, and T. Sawada, “Application of coaxial beam photothermal microscopy to the analysis of a single biological cell in water,” Anal. Chim. Acta. 299, 343–347 (1995). [CrossRef]

], these were numerical in nature and targeted for the μm-size regime. The theoretical description of the nanoscopic photothermal lens has been first given by Berciaud et al. [21

21. S. Berciaud, D. Lasne, G. Blab, L. Cognet, and B. Lounis, “Photothermal heterodyne imaging of individual metallic nanoparticles: Theory versus experiment,” Phys. Rev. B 73, 045424 (2006). [CrossRef]

] in a scattering treatment considering an extinction mechanism. Our recent ab-initio theoretical description of the electrodynamic problem has shown, however, that instead a simple lensing mechanism is responsible for the photothermal signal of nanoscopic absorbers showing a clear angular thermal diffraction signature and a double-lobe lens signature in axial scans. We have also demonstrated the importance of both the interference- (extinction) and scatter-term in forward detection appearing in the general situation which complicated an intuitive understanding within such a scattering framework.

It is the aim of this paper to show that both worlds, the vast literature on macroscopic thermal lens spectroscopy and the recent emerging tool of single particle thermal lens microscopy are very similar. The diffraction picture which has been a successful tool in the first domain will be shown to yield analytical and easily tractable expressions in the nano-scopic domain. They will allow for a quantitative assessment of absorption cross-sections of single nano-object based on standard photothermal measurements and will be able to explain the main phenomena of photothermal microscopy qualitatively as well as quantitatively, while providing an intuitive picture of the working mechanism. The quality of the simple model is checked against the more elaborate electromagnetic model within the extended scattering description. Axial scans and angular patterns of single heated nano-particles will be described and compared. The angular diffraction pattern will be shown to explain the signal inversion observed for the first time for a single nano-particle upon the introduction of an inverse aperture in the detection path.

2. The thermal lens

The lens to be considered in single particle generated nano-lens experiments such as photothermal microscopy originates from the absorption of electromagnetic power provided by a focused laser beam by a single nano-particle. The absorbing particle can be treated as a point-like heat-source, yielding the steady-state temperature profile T(r) = T0 + ΔTR/r which decays with the inverse distance r from the particle of radius R to the unperturbed ambient temperature T0. In the case of modulated heating, as utilized in the lock-in approach common to photothermal single particle microscopy, the assumption remains valid as long as the modulation frequencies used remain below Ωm ≈ 1MHz, depending on the thermal conductivity κ and the heat capacity C per unit volume of the medium [17

17. M. Selmke, M. Braun, and F. Cichos, “Photothermal Single Particle Microscopy: Detection of a Nano-Lens,” ACS Nano, DOI: [CrossRef] [PubMed]

, 21

21. S. Berciaud, D. Lasne, G. Blab, L. Cognet, and B. Lounis, “Photothermal heterodyne imaging of individual metallic nanoparticles: Theory versus experiment,” Phys. Rev. B 73, 045424 (2006). [CrossRef]

]. Only in case of higher modulation frequencies where the thermal length scale Rth = [2κ/(CΩm)]1/2 becomes comparable to the diffraction limit of the focusing optics the detailed form of the time-dependent temperature profile needs to be considered. As the signal of photothermal single particle microscopy has been shown to decrease considerably above this limiting frequency [21

21. S. Berciaud, D. Lasne, G. Blab, L. Cognet, and B. Lounis, “Photothermal heterodyne imaging of individual metallic nanoparticles: Theory versus experiment,” Phys. Rev. B 73, 045424 (2006). [CrossRef]

], we postpone a detailed discussion of this issue to a different publication.

Fig. 1 a) Geometry for the computation of the phase advance iχ (ρ), Eq. (4). b) Geometry for the diffraction integral, Eq. (3). The shading in the image-plane corresponds to the difference in intensities between the diffraction of a cold and a hot nano-particle, Eq. 8, and shows the configuration for a negative detected rel. PT signal Φ, Eq. (6).

3. The diffraction integral

The Frauenhofer diffraction integral for circular apertures in the Fresnel-grade approximation connects the complex field-amplitude of the probe beam Ua (ρ) in the aperture plane at z = 0 to the complex field amplitude U (r,z) in the image plane at a distance z (see Fig. 1(b)) [22

22. Z. L. Horvath and Z. Bor, “Focusing of truncated gaussian beams,” Opt. Commun. 222, 51–68 (2003). [CrossRef]

,23

23. S. Teng, T. Zhou, and C. Cheng, “Fresnel diffraction of truncated gaussian beam,” Optik 118, 435–439 (2007). [CrossRef]

]. A further factor is included which represents the collected phase Δχ for each component-wave as a result of the thermal lens (see Fig. 1(a)) [24

24. A. A. Vigasin, “Diffraction of light by absorbing inclusions in solids,” Kvant. Elektron. (Moscow) [Sov. J. Quantum Electron.] 4, 662–666 (1977).

].
U(r,z)=kizexp(ikr22zikz)RUa(ρ)exp(ikρ22z)J0(kρrz)exp(iΔχ(ρ))ρdρ
(3)

To simplify matters, we will consider an axially symmetric scenario only, i.e. the refractive index profile will be symmetric with respect to the optical axis and the steady-state or low-frequency limit (as defined above) of the temperature profile 1/r will be assumed. This corresponds to the case of a heated nano-particle positioned along the optical axis, see Fig. 1(a).

The phase advance Δχ (ρ) for a single ray passing the lens immersed in a sample slab of thickness 2L at a distance ρ from the optical axis can be approximated via a straight ray calculation already utilized for collimated beams in reference [24

24. A. A. Vigasin, “Diffraction of light by absorbing inclusions in solids,” Kvant. Elektron. (Moscow) [Sov. J. Quantum Electron.] 4, 662–666 (1977).

] and similarly in reference [25

25. G. Baffou, P. Bon, J. Savatier, J. Polleux, M. Zhu, M. Merlin, H. Rigneault, and S. Monneret, “Thermal Imaging of Nanostructures by Quantitative Optical Phase Analysis,” ACS Nano , DOI: [CrossRef] [PubMed]

]:
Δχ(ρ)=k0LL[n0+RΔnz2+ρ2]dz2k0[Ln0RΔnln(ρ2L)],
(4)
where sinh1(z)=log(z+1+z2) and Lρ were used after the integration. Although the integration in the distance ρ extends to infinity, the weighting by the Gaussian field amplitude Ua (ρ) will ensure the validity of the inequality. The ρ-dependent part of the phase factor exp(−iχ), i.e. ρ2iRΔnk0, will later be used in the explicit integration, while the factor (4L2)iRΔnk0, which is in magnitude equal to unity and has units of [m2] ensures unitless-ness of the expressions encountered during the derivations.

The field amplitude in the aperture plane will be taken to be the complex field of the probing Gaussian beam [26

26. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (John Wiley and Sons, Inc., 1991). [CrossRef]

] having a focus displaced by zp from the center of the heated particle:
Ua(ρ)=U0ω0ω(zp)exp(ρ2ω2(zp))exp(ikzpikρ22RC(zp)+iζG(zp)),
(5)
where the beam-waist in the aperture plane is ω(zp)=ω0[1+zp2/zR2]1/2, the curvature is given by RC(zp)=zp[1+zR2/zp2] and the Gouy-phase ζG (zp) = tan−1 (zp/zR). The Rayleigh-range zR is connected to the beam-waist ω0 and the wave-number k = k0n0 by zR=kω02/2, where k0 = 2π/λ represents the vacuum wave-vector for the probing wavelength λ.

To put photothermal single particle microscopy on a quantitative footing, we have introduced the relative PT signal Φ [17

17. M. Selmke, M. Braun, and F. Cichos, “Photothermal Single Particle Microscopy: Detection of a Nano-Lens,” ACS Nano, DOI: [CrossRef] [PubMed]

]. It is the change in intensity I ∝ |U|2 within the image-plane relative to the constant much larger background of the unperturbed field:
Φ(r,z)=[|U(r,z)|Δn(ΔT(zp))2|U(r,z)|Δn=02]/|U(r=0,z)|Δn=02
(6)
This relative signal will be independent of the probe power 𝒫d as long as no additional heating is induced, meaning that Δn = const. with 𝒫d. In case of a single laser being diffracted by its own induced thermal lens, the relative signal will be proportional to 𝒫d.

Fig. 2 a) zp-Scan of the rel. photothermal signal Φ for on-axis detection, i.e. θ = 0°, Eq. (7). The chosen laser-offset of Δzf = 0 results in a symmetric shape. b) Angular pattern of the photothermal signal for zp = {0, −zR, −2zR, −3zR}, i.e. Φ(θ,zp), Eq. (8). The plot has been normalized to 1.0 on the optical axis. The following parameters have been used for the calculations: R = 10 nm, ΔT0 = 100K, n0 = 1.46, dn/dT = −3.6 × 10−3, λ = 635 nm, ω0 = 281 nm, λh = 532 nm, ω0,h = 233 nm, defining a beam divergence angle θdiv = arctan (ω0/zR) ≈ 26° determining the approximate angular extent of the feature.

Equations (7), (8) and (9) present the main results of this paper and give the background normalized intensity change Φ = ΔI/I detected either on the optical axis or detected under a finite angle θ with respect to the optical axis upon the introduction of the lens n(r), Eq. (1), on the optical axis. Hereby, the particle/lens was assumed to be displaced by a distance zp with respect to the probing beam-waist position. In this approach the Gouy-phase terms, while present in the fields in the aperture plane, cancel each other in the relative photothermal signal since they do not depend on the integration variable ρ. The found expressions which are exemplarily visualized in the plots of Fig. 2 allow the study of (finite NA) zp-scans in photothermal microscopy setups similar to the thin sample slab studies in [4

4. G. Battaglin, P. Calvelli, E. Cattaruzza, F. Gonella, R. Polloni, G. Mattei, and P. Mazzoldi, “Z-scan study on the nonlinear refractive index of copper nanocluster composite silica glass,” Appl. Phys. Lett. 78, 3953–3955 (2001). [CrossRef]

, 29

29. R. Escalona, “Comparative study between interferometric and z-scan techniques for thermal lensing characterization,” Opt. Commun. 281, 1323–1330 (2008). [CrossRef]

33

33. R. Polloni, B. F. Scremin, P. Calvelli, E. Cattaruzza, G. Battaglin, and G. Mattei, “Metal nanoparticles-silica composites: Z-scan determination of non-linear refractive index,” J. Non-Cryst. Solids 322, 300–305 (2003). [CrossRef]

]. To this end, the collection angle depends on the collecting microscope objectives’ numerical aperture through θmax = arcsin (NA/n0), while the illuminating objective determines the beam waist(s).

4. Comparison to rigorous vectorial electromagnetic (EM) model

To compare the found results Eq. (7) and (8) to a more rigorous solution of the problem, a full vectorial electromagnetic treatment will be used in the following. Therefore, the problem at hand may be expressed as the scattering of a shaped incident probe field interacting with a multilayered scatterer. The scatterer described by Eq. (1) may be viewed as an unbounded gradient refractive index lens (GRIN). The scattering of such an object has been studied in the literature and a publicly available C-code is attainable through reference [34

34. O. Pena and U. Pal, “Scattering of electromagnetic radiation by a multilayered sphere,” Comput. Phys. Commun. 180, 2348–2354 (2009). [CrossRef]

] providing the scattering coefficients of the GRIN aN+1 and bN+1 when the scatterer is discretized into N concentric spherical shells as shown in Fig. 3(b). The appropriate scattering framework is the generalized Lorenz-Mie theory (GLMT), which is applicable to any spherically symmetric scatterer [35

35. F. Onofri, G. Grehan, and G. Gouesbet, “Electromagnetic scattering from a multilayered sphere located in an arbitrary beam,” Appl. Opt. 34, 7113–7124 (1995). [CrossRef] [PubMed]

] positioned in an arbitrarily shaped beam [36

36. G. Gouesbet, G. Grehan, and B. Maheu, “Scattering of a gaussian-beam by a mie scatter center using a bromwich formalism,” J. Optics-Nouvelle Revue D Optique 16, 83–93 (1985).

38

38. G. Gouesbet, J. Lock, and G. Grehan, “Generalized lorenz-mie theories and description of electromagnetic arbitrary shaped beams: Localized approximations and localized beam models, a review,” J. Quant. Spectrosc. Radiat. Transfer 112, 1–27 (2011). [CrossRef]

]. We have therefore adopted and modified it to yield an analytical expression for the power 𝒫d of the total electromagnetic field (Et, Ht) detected in the far-field under a scattering-angle θ. This treatment has been verified by exhaustive comparison to single gold nanoparticle scattering and photothermal microscopy and is described in more detail in our reference [17

17. M. Selmke, M. Braun, and F. Cichos, “Photothermal Single Particle Microscopy: Detection of a Nano-Lens,” ACS Nano, DOI: [CrossRef] [PubMed]

]. The quantity of interest is the projected total fields’ Poynting-vector St under a certain angle θ with respect to the optical axis (Fig. 3(a)):
d𝒫d(θ,ϕ)=StdA=StdA.
(11)

Fig. 3 a) Schematic of the Poynting-vector integration within the GLMT framework. b) Discretization of the refractive index profile n(r) into concentric spherical shells for the computation of the multilayer Mie scatter coefficients [34].

Within the GLMT formalism the total field is mathematically decomposed into series expressions of the incidence field Ei and Hi with complex expansion coefficients gn, and an outgoing scattered field Es and Hs described by complex scatter coefficients an and bn. Thereby, the total field consists of the incident field and the scattered field, Et = Ei + Es and Ht = Hi + Hs. The scattered far-field is commonly expressed via the scattering amplitudes S1,2 through Eθs=iE0krexp(ikr)cos(ϕ)S2(θ), Hθs=H0E0Eϕs, Eϕs=iE0krexp(ikr)sin(ϕ)S1(θ), Hϕs=H0E0Esθ. These amplitudes are:
S12=n=12n+1n(n+1)gn[anΠn(cosθ)τn(cosθ)+bnτn(cosθ)Πn(cosθ)],
(12)
wherein the angular functions are Πnm(cosθ)=Pnm(cosθ)/sinθ, τnm(cosθ)=dPnm(cosθ)/dθ and Pmm are the associated Legendre polynomials. This artificially decomposes the detected power into three parts which are not individually detectable in the scattering situation: d𝒫d = d𝒫inc + d𝒫sca + d𝒫ext. Now, the time-averaged projected Poynting vector may be computed via St=12Re(EθtHϕt*EϕtHθt*), yielding three terms:
2St=Re(EθiHϕi*EϕiHθi*)(incidencefield flux)+Re(EθsHϕs*EϕsHθs*)(scattered field flux)+Re(EθiHϕs*+EθsHϕi*EϕiHθs*EϕsHθi*)(interferenceflux).
(13)
The shape of the incidence field is determined by complex-valued expansion coefficients gn, the so-called beam shape coefficients (BSCs). These coefficients reduce to gn = 1 for plane-wave illumination, i.e. regular Mie Theory. For a particle illuminated on-axis by a weakly focused Gaussian beam the modified local approximation (MLA) has been developed [39

39. G. Gouesbet, J. A. Lock, and G. Grehan, “Partial-wave representations of laser-beams for use in light-scattering calculations,” Appl. Opt. 34, 2133–2143 (1995). [CrossRef] [PubMed]

]:
gn(zp)=Qexp(Qs2(n1)(n+2))exp(iγs1/2).
(14)
Herein, Q = (1 + isγ)−1 where the beam-confinement factor s is defined through s = ω0/(2zR) and the defocussing parameter γ = 2zp/ω0 describes the displacement of the particle relative to the beam-waist ω0. As before, zp < 0 corresponds to the situation where the focus is between the particle and the collecting objective. The result of Eq. (11), with the integration over the azimuthal angle ϕ carried out in the far-field can now be expressed as differential cross-sections in any forward/backward polar angle θ : d𝒫ext (θ) = −dσext (θ)I0, d𝒫sca (θ) = dσsca (θ)I0 and d𝒫inc (θ) = dσinc (θ)I0, where the Gaussian beam focus intensity I0=2P0/πω02 was used:
dσsca(θ)=πk2(|S1(θ)|2+|S2(θ)|2),
(15)
dσext(θ)=πk2[Re(M)Re(S12)+Im(M)Im(S12)].
(16)
The auxilliary functions S12 (θ) = S1 (θ) + S2 (θ) and M (θ) were introduced:
M(θ)=n=12n+1n(n+1)gn[Πn(cosθ)+τn(cosθ)].
(17)
To obtain the total cross-sections, one needs to compute for instance σsca=0πdσsca(θ)sin(θ)dθ. The absorption is then computed from σabs = σextσsca with:
σsca=2πk2n=1(2n+1)|gn|2(|an|2+|bn|2),σext=2πk2n=1(2n+1)|gn|2Re(an+bn).
(18)
For a Gaussian beam (on-axis) the integrated flux of the collected beam can be done analytically, i.e. ∫ Si · dA with 2Si=Re(EθiHϕi*EϕiHθi*)/2. The result may be written as a Cauchysum σinc=π2k2n=1σinc,n with summands σinc,n given by:
σinc,n=m=1nNmgmNnm+1gnm+1*0θmax[mnm+1(1)nΔmΔnm+1]sinθdθ,
(19)
wherein Δn ≡ Πnτn and ∑n ≡ Πn + τn. To ensure numerical stability for small angles, a direct recursive determination of Δn ≡ Πnτn may be used [40

40. G. H. Meeten, “Computation of s1–s2 in mie scattering-theory,” J. Phys. D: Appl. Phys. 17, L89–L91 (1984). [CrossRef]

]. The above expressions may now be used to obtain the angular spectrum of the difference signal, i.e. the PT signal analogously defined to Eq. (6).
Φ(θ)=d𝒫dhotd𝒫dcoldd𝒫inccold
(20)
=[dσsca(θ)dσext(θ)]anL+1,bnL+1n(r),nAu[dσsca(θ)dσext(θ)]an,bnn0,nAudσincd(θ)
(21)

To obtain the relative PT signal collected with a finite detection aperture, the integrated differential cross-sections dσsca,ext have to be used instead of the differential ones. This corresponds then to Eq. (8) in the diffraction model and to Eq. (9) when integrated accordingly. The predictions of both models are displayed in Fig. 4. The best agreement is found in the low-focusing regime (ω0λ). In the strong focusing case, the GLMT may be expected to deviate since the beam shape coefficients used here rely on the low focusing expansion of the paraxial field, while within the diffraction treatment the straight ray phase advance approximation, Eq. (4), becomes less accurate. Both frameworks agree for typical experimental parameters within a factor of the order of unity. A direct scaling-free comparison of the on-axis diffraction formula Eq. (7) and the Gaussian GLMT prediction with Eq. (21) is shown in Fig. 4(b), where the solid thin black line corresponds to Eq. (7) and the red dashed line corresponds to Eq. (21). The qualitative agreement is illustrated by further zp-scan examples with finite numerical collection apertures (see Fig. 4(c), NA = 0.3 and NA = 0.75), i.e. Eq. (9), scaled by 1.5 to match the GLMT predictions. The angular pattern in Fig. 4(a) of the photothermal signal has two consequences: The relative change in intensity as compared to the unperturbed beam, i.e. the relative photothermal signal, is maximal, if the detection is on-axis or in a small angular detection domain around the forward direction (see Fig. 4(b) and 4(c)). Further, the relative as well as the absolute PT signal will decrease if the angular collection domain extends across the zero-crossing of the angular pattern. Also, both frameworks predict a signal inversion for the collection of an annular region of detection angles (Fig. 4(d)), for example for [θmin, θmax] = [21°, 31°]. To test this prediction, a central beam-stop experiment has been carried out which is detailed in the following section.

Fig. 4 Comparison of the diffraction (black) and Gaussian GLMT model (red). Parameters used for calculations are detailed in the caption of Fig. 2. The diffraction model results have been scaled by a constant factor of 1.5 except for (d) where the factor is 0.8. a) PT signal angular distribution with positive (blue) and negative (red) signal. b) On-axis z-scan (NA = 0, θ = 0°) of PT signal. The black dotted curve is the unscaled prediction of the diffraction model. c) On-axis zp-scan for a finite NA detection. d) On-axis zp-scan with central beam stop (inverse aperture). The grey curve corresponds to no central beam-stop (NA = 0.75 from (c)).

Notably, the diffraction framework Eq. (9) also predicts the dependence of the spatial photothermal signal shape on the axial displacement of the heating and the detection beam focus Δzf as observed experimentally in our recent study [17

17. M. Selmke, M. Braun, and F. Cichos, “Photothermal Single Particle Microscopy: Detection of a Nano-Lens,” ACS Nano, DOI: [CrossRef] [PubMed]

]. Once again, the calculations have been carried out in both frameworks. In Fig. 5, the GLMT calculations are shown by the solid lines, while the diffration framework corresponds to the dashed-solid lines. The numerical aperture of the detection lens for these calculations is NA = 0.75, resulting in a maximum collection angle of θmax = 30°. The focal displacement Δzf that is included in Eq. (2) has been varied from −3zR,d to 3zR,d. The effect in the photothermal signal can be seen in Fig. 5(a). Depending on Δzf, either the positive signal is enhanced while the negative one is decreased (for a positive Δzf) or vice versa. However, the position of the zero-intersection, when the detection beam is focused to the center of the thermal lens, is not influenced, which is covered by both frameworks. By scaling the curves obtained by diffraction theory again by a constant factor of 1.6, the overall shape matches very well with the curves predicted by GLMT. Also, the dependence of the maximum and the minimum signal on the focal displacement Δzf shows a reasonable agreement (see Fig. 5(b)), when the scaling factor of order unity is included. The influence of Δzf on the maxima and minima position is depicted in Fig. 5(c). The axial distance between maximum and minimum is the shortest for Δzf = 0. In this region the predictions of both theories are nearly identical. Only for a large focal displacement Δzf > 1.5zR,d, the deviation becomes noticable.

Fig. 5 On-axis scans of the rel. PT signal Φ (zp) for a finite detection numerical aperture NA for Δzf = −3zR,d, −2zR,d,..., 3zR,d (red to blue). a) Comparison of the diffraction (dashed-solid) and Gaussian GLMT model (solid). The diffraction model results have been scaled by a constant factor of 1.6 (thick dashed-solid). The unscaled prediction of the diffraction model is thin dashed-solid. Insets: PT signal angular distribution Φ (θ)/Φ (0) for GLMT and diffraction model for decreasing zp < 0. b) Peak amplitudes of the PT signal versus axial beam displacement Δzf. c) Axial position of the peaks.

In summary, the comparison of the simple and intuitive diffraction-picture, as put forward in the preceding section, and the rigorous vectorial treatment of a Gaussian beam scattered by the GRIN lens shows a near-perfect qualitative agreement, while a quantitative agreement within a factor of order unity is found.

5. Signal inversion of an axial single particle scan

Having demonstrated the equivalence of the results obtainable within the generalized Lorenz-Mie framework and in the simple diffraction model, the following part of the paper will show a direct consequence of the above findings. Whereas the collection of the probe-beam in an angular domain around the forward-direction will show a dispersion-like lens-signature as displayed in Fig. 2, the collection of an annular region can invert the detected signal. Indeed, the measurement of the photothermal signal of a single gold nanoparticle of R = 30 nm radius shows this effect. To this end, a central beam-stop was introduced into the detection beam path of a photothermal microscope as described in detail in [17

17. M. Selmke, M. Braun, and F. Cichos, “Photothermal Single Particle Microscopy: Detection of a Nano-Lens,” ACS Nano, DOI: [CrossRef] [PubMed]

]. The collimated beam of diameter Do = 10 mm was thereby reduced to an annular ring of Di = 9 mm inner diameter corresponding to an angle of θmin = arcsin (NA Di/ (Don0)) = 27° as given by the Abbe sine condition and the used numerical aperture of the detection objective. The illumination objective was a high-NA oil immersion objective and the resulting probe and heating beam waists were ω0,d = 281 nm and ω0,h = 233 nm, respectively. The beams were offset in axial direction by Δzf = 350 nm to achieve a symmetric signal configuration with the aberrated beams. The observed z- and xz-scans (Fig. 6(a)) of the signal measured correspond very well to the calculated scans (Fig. 6(b)) and clearly show the inversion of the signal. In addition to a pure change in sign of the signal, the zero-crossing shifts. Although these details are only reconcilable with the predictions of the exact GLMT description (see supplement of [17

17. M. Selmke, M. Braun, and F. Cichos, “Photothermal Single Particle Microscopy: Detection of a Nano-Lens,” ACS Nano, DOI: [CrossRef] [PubMed]

]), which includes the details of the aberrated beams, the basic principle and the prediction of the effect is understood based on the angular diffraction pattern described in section 3.

Fig. 6 Normalized photothermal single particle signal zp-scan (left) and xz-scan (right) of a gold-nanoparticle (R = 30 nm) measured (a) and computed (b) with the exact beam shape co-efficients in the GLMT framework for a full NA = 0.75 detection (black curve on the left and upper pictures on the right) and with a inverse aperture up to an detection angle of θmax = 31° (dashed blue on the left and lower pictures on the right). The parameters used in the calculation are the same as those in our reference [17].

6. Conclusion

While a close relation of Gaussian beam diffraction and scattering has been shown by J. Lock et al. [41

41. J. A. Lock and E. A. Hovenac, “Diffraction of a gaussian-beam by a spherical obstacle,” Am. J. Phys. 61, 698–707 (1993). [CrossRef]

] for the case of spherical dielectric particles, the more complicated scattering approach was thus far the only theoretical approach for the emerging technique of single particle photothermal microscopy. Although this ansatz may be used, the signature of a simple lensing mechanism (e.g. in zp-scans) suggests a more intuitive model. Employing the scalar diffraction formalism common in thin sample slab absorption spectroscopy we have demonstrated that the signal obtained in photothermal single (nano-)particle microscopy can indeed be understood as the signal of a phase-modifying element, i.e. a lens, despite the microscopic origin of the latter. In contrast to the former, the spherical symmetry of the heated single-particle lens allows a direct analytical evaluation of the relative photothermal signal. The analytical model presented here could be verified in shape and absolute value when compared to experiments and a full electromagnetic vectorial treatment as published by the authors earlier [17

17. M. Selmke, M. Braun, and F. Cichos, “Photothermal Single Particle Microscopy: Detection of a Nano-Lens,” ACS Nano, DOI: [CrossRef] [PubMed]

]. Further, it could be shown, that the Gouy-phase of a Gaussian detection beam does not contribute to the relative photothermal signal in forward direction. The understanding of the mechanism and angular distribution of the photothermal signal was then shown to explain the observed signal inversion upon the collection of an angular domain corresponding to the outer angles only. This introduces a simple and intuitive model for single particle absorption measurements and allows the quantitative assessment of nano-particle absorption cross sections.

Acknowledgments

Financial support by the DFG research unit 877 and the graduate school BuildMoNa as well as funding by the European Union and the Free State of Saxony is acknowledged.

References and links

1.

X. S. Xie, S. J. Lu, W. Min, S. S. Chong, and G. R. Holtom, “Label-free imaging of heme proteins with two-photon excited photothermal lens microscopy,” Appl. Phys. Lett. 96, 113701 (2010). [CrossRef]

2.

S. Sinha, A. Ray, and K. Dasgupta, “Solvent dependent nonlinear refraction in organic dye solution,” J. Appl. Phys. 87, 3222–3226 (2000). [CrossRef]

3.

A. V. Brusnichkin, D. A. Nedosekin, M. A. Proskurnin, and V. P. Zharov, “Photothermal lens detection of gold nanoparticles: Theory and experiments,” Appl. Spectrosc. 61, 1191–1201 (2007). [CrossRef] [PubMed]

4.

G. Battaglin, P. Calvelli, E. Cattaruzza, F. Gonella, R. Polloni, G. Mattei, and P. Mazzoldi, “Z-scan study on the nonlinear refractive index of copper nanocluster composite silica glass,” Appl. Phys. Lett. 78, 3953–3955 (2001). [CrossRef]

5.

D. Rings, R. Schachoff, M. Selmke, F. Cichos, and K. Kroy, “Hot Brownian Motion,” Phys. Rev. Lett. 105, 090604 (2010). [CrossRef] [PubMed]

6.

F. Jurgensen and W. Schroer, “Studies on the diffraction image of a thermal lens,” Appl. Opt. 34, 41–50 (1995). [CrossRef] [PubMed]

7.

J. Moreau and V. Loriette, “Confocal dual-beam thermal-lens microscope: Model and experimental results,” Jpn. J. Appl. Phys. 45, 7141–7151 (2006). [CrossRef]

8.

J. Moreau and V. Loriette, “Confocal thermal-lens microscope,” Opt. Lett. 29, 1488–1490 (2004). [CrossRef] [PubMed]

9.

M. Harada, T. Kitamori, and T. Sawada, “Phase signal of optical beam deflection from single microparticles -theory and experiment,” J. Appl. Phys. 73, 2264–2271 (1993). [CrossRef]

10.

J. Q. Wu, T. Kitamori, and T. Sawada, “Theory of optical beam deflection for single microparticles,” J. of Appl. Phys. 69, 7015–7020 (1991). [CrossRef]

11.

W. B. Jackson, N. M. Amer, A. C. Boccara, and D. Fournier, “Photothermal deflection spectroscopy and detection,” Appl. Opt. 20, 1333–1344 (1981). [CrossRef] [PubMed]

12.

M. Harada, K. Iwamotok, T. Kitamori, and T. Sawada, “Photothermal microscopy with excitation and probe beams coaxial under the microscope and its application to microparticle ananlysis,” Anal. Chem. 65, 2938–2940 (1993). [CrossRef]

13.

K. Uchiyama, A. Hibara, H. Kimura, T. Sawada, and T. Kitamori, “Thermal lens microscope,” Jpn. J. Appl. Phys. 39, 5316–5322 (2000). [CrossRef]

14.

A. Gaiduk, M. Yorulmaz, P. V. Ruijgrok, and M. Orrit, “Room-temperature detection of a single molecule’s absorption by photothermal contrast,” Science 330, 353–356 (2010). [CrossRef] [PubMed]

15.

S. Berciaud, L. Cognet, G. Blab, and B. Lounis, “Photothermal heterodyne imaging of individual nonfluorescent nanoclusters and nanocrystals,” Phys. Rev. Lett. 93, 257402 (2004). [CrossRef]

16.

D. Boyer, P. Tamarat, A. Maali, B. Lounis, and M. Orrit, “Photothermal imaging of nanometer-sized metal particles among scatterers,” Science 297, 1160–1163 (2002). [CrossRef] [PubMed]

17.

M. Selmke, M. Braun, and F. Cichos, “Photothermal Single Particle Microscopy: Detection of a Nano-Lens,” ACS Nano, DOI: [CrossRef] [PubMed]

18.

J. Hwang and W. E. Moerner, “Interferometry of a single nanoparticle using the gouy phase of a focused laser beam,” Opt. Commun. 280, 487–491 (2007). [CrossRef]

19.

G. C. K. Chen, M. Andika, and S. Vasudevan, “Excitation temporal pulse shape and probe beam size effect on pulsed photothermal lens of single particle,” J. Opt. Soc. Am. B 27, 796–805 (2010). [CrossRef]

20.

M. Harada, M. Shibata, T. Kitamori, and T. Sawada, “Application of coaxial beam photothermal microscopy to the analysis of a single biological cell in water,” Anal. Chim. Acta. 299, 343–347 (1995). [CrossRef]

21.

S. Berciaud, D. Lasne, G. Blab, L. Cognet, and B. Lounis, “Photothermal heterodyne imaging of individual metallic nanoparticles: Theory versus experiment,” Phys. Rev. B 73, 045424 (2006). [CrossRef]

22.

Z. L. Horvath and Z. Bor, “Focusing of truncated gaussian beams,” Opt. Commun. 222, 51–68 (2003). [CrossRef]

23.

S. Teng, T. Zhou, and C. Cheng, “Fresnel diffraction of truncated gaussian beam,” Optik 118, 435–439 (2007). [CrossRef]

24.

A. A. Vigasin, “Diffraction of light by absorbing inclusions in solids,” Kvant. Elektron. (Moscow) [Sov. J. Quantum Electron.] 4, 662–666 (1977).

25.

G. Baffou, P. Bon, J. Savatier, J. Polleux, M. Zhu, M. Merlin, H. Rigneault, and S. Monneret, “Thermal Imaging of Nanostructures by Quantitative Optical Phase Analysis,” ACS Nano , DOI: [CrossRef] [PubMed]

26.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (John Wiley and Sons, Inc., 1991). [CrossRef]

27.

C. Hu and J. R. Whinnery, “New thermooptical measurement method and a comparison with other methods,” Appl. Opt. 12, 72–79 (1973). [CrossRef] [PubMed]

28.

M. Selmke, M. Braun, and F. Cichos, “Photonic Rutherford Scattering,” in preparation (2012).

29.

R. Escalona, “Comparative study between interferometric and z-scan techniques for thermal lensing characterization,” Opt. Commun. 281, 1323–1330 (2008). [CrossRef]

30.

S. M. Mian, S. B. McGee, and N. Melikechi, “Experimental and theoretical investigation of thermal lensing effects in mode-locked femtosecond z-scan experiments,” Opt. Commun. 207, 339–345 (2002). [CrossRef]

31.

A. Gnoli, L. Razzari, and M. Righini, “Z-scan measurements using high repetition rate lasers: how to manage thermal effects,” Opt. Express 13, 7976–7981 (2005). [CrossRef] [PubMed]

32.

A. Gnoli, A. M. Paoletti, G. Pennesi, G. Rossi, and M. Righini, “High-accuracy z-scan measurements of the optical nonlinearity of bis-phthalocyanines,” J. Porphyrins Phthalocyanines 11, 481–486 (2007). [CrossRef]

33.

R. Polloni, B. F. Scremin, P. Calvelli, E. Cattaruzza, G. Battaglin, and G. Mattei, “Metal nanoparticles-silica composites: Z-scan determination of non-linear refractive index,” J. Non-Cryst. Solids 322, 300–305 (2003). [CrossRef]

34.

O. Pena and U. Pal, “Scattering of electromagnetic radiation by a multilayered sphere,” Comput. Phys. Commun. 180, 2348–2354 (2009). [CrossRef]

35.

F. Onofri, G. Grehan, and G. Gouesbet, “Electromagnetic scattering from a multilayered sphere located in an arbitrary beam,” Appl. Opt. 34, 7113–7124 (1995). [CrossRef] [PubMed]

36.

G. Gouesbet, G. Grehan, and B. Maheu, “Scattering of a gaussian-beam by a mie scatter center using a bromwich formalism,” J. Optics-Nouvelle Revue D Optique 16, 83–93 (1985).

37.

G. Gouesbet, B. Maheu, and G. Grehan, “Light-scattering from a sphere arbitrarily located in a gaussian-beam, using a bromwich formulation,” J. Opt. Soc. Am. A 5, 1427–1443 (1988). [CrossRef]

38.

G. Gouesbet, J. Lock, and G. Grehan, “Generalized lorenz-mie theories and description of electromagnetic arbitrary shaped beams: Localized approximations and localized beam models, a review,” J. Quant. Spectrosc. Radiat. Transfer 112, 1–27 (2011). [CrossRef]

39.

G. Gouesbet, J. A. Lock, and G. Grehan, “Partial-wave representations of laser-beams for use in light-scattering calculations,” Appl. Opt. 34, 2133–2143 (1995). [CrossRef] [PubMed]

40.

G. H. Meeten, “Computation of s1–s2 in mie scattering-theory,” J. Phys. D: Appl. Phys. 17, L89–L91 (1984). [CrossRef]

41.

J. A. Lock and E. A. Hovenac, “Diffraction of a gaussian-beam by a spherical obstacle,” Am. J. Phys. 61, 698–707 (1993). [CrossRef]

OCIS Codes
(050.5080) Diffraction and gratings : Phase shift
(110.6820) Imaging systems : Thermal imaging
(180.5810) Microscopy : Scanning microscopy
(190.4870) Nonlinear optics : Photothermal effects
(260.1960) Physical optics : Diffraction theory
(350.4990) Other areas of optics : Particles
(050.1965) Diffraction and gratings : Diffractive lenses
(350.4238) Other areas of optics : Nanophotonics and photonic crystals

ToC Category:
Physical Optics

History
Original Manuscript: January 18, 2012
Revised Manuscript: February 29, 2012
Manuscript Accepted: February 29, 2012
Published: March 22, 2012

Virtual Issues
Vol. 7, Iss. 5 Virtual Journal for Biomedical Optics

Citation
Markus Selmke, Marco Braun, and Frank Cichos, "Nano-lens diffraction around a single heated nano particle," Opt. Express 20, 8055-8070 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-7-8055


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References

  1. X. S. Xie, S. J. Lu, W. Min, S. S. Chong, and G. R. Holtom, “Label-free imaging of heme proteins with two-photon excited photothermal lens microscopy,” Appl. Phys. Lett.96, 113701 (2010). [CrossRef]
  2. S. Sinha, A. Ray, and K. Dasgupta, “Solvent dependent nonlinear refraction in organic dye solution,” J. Appl. Phys.87, 3222–3226 (2000). [CrossRef]
  3. A. V. Brusnichkin, D. A. Nedosekin, M. A. Proskurnin, and V. P. Zharov, “Photothermal lens detection of gold nanoparticles: Theory and experiments,” Appl. Spectrosc.61, 1191–1201 (2007). [CrossRef] [PubMed]
  4. G. Battaglin, P. Calvelli, E. Cattaruzza, F. Gonella, R. Polloni, G. Mattei, and P. Mazzoldi, “Z-scan study on the nonlinear refractive index of copper nanocluster composite silica glass,” Appl. Phys. Lett.78, 3953–3955 (2001). [CrossRef]
  5. D. Rings, R. Schachoff, M. Selmke, F. Cichos, and K. Kroy, “Hot Brownian Motion,” Phys. Rev. Lett.105, 090604 (2010). [CrossRef] [PubMed]
  6. F. Jurgensen and W. Schroer, “Studies on the diffraction image of a thermal lens,” Appl. Opt.34, 41–50 (1995). [CrossRef] [PubMed]
  7. J. Moreau and V. Loriette, “Confocal dual-beam thermal-lens microscope: Model and experimental results,” Jpn. J. Appl. Phys.45, 7141–7151 (2006). [CrossRef]
  8. J. Moreau and V. Loriette, “Confocal thermal-lens microscope,” Opt. Lett.29, 1488–1490 (2004). [CrossRef] [PubMed]
  9. M. Harada, T. Kitamori, and T. Sawada, “Phase signal of optical beam deflection from single microparticles -theory and experiment,” J. Appl. Phys.73, 2264–2271 (1993). [CrossRef]
  10. J. Q. Wu, T. Kitamori, and T. Sawada, “Theory of optical beam deflection for single microparticles,” J. of Appl. Phys.69, 7015–7020 (1991). [CrossRef]
  11. W. B. Jackson, N. M. Amer, A. C. Boccara, and D. Fournier, “Photothermal deflection spectroscopy and detection,” Appl. Opt.20, 1333–1344 (1981). [CrossRef] [PubMed]
  12. M. Harada, K. Iwamotok, T. Kitamori, and T. Sawada, “Photothermal microscopy with excitation and probe beams coaxial under the microscope and its application to microparticle ananlysis,” Anal. Chem.65, 2938–2940 (1993). [CrossRef]
  13. K. Uchiyama, A. Hibara, H. Kimura, T. Sawada, and T. Kitamori, “Thermal lens microscope,” Jpn. J. Appl. Phys.39, 5316–5322 (2000). [CrossRef]
  14. A. Gaiduk, M. Yorulmaz, P. V. Ruijgrok, and M. Orrit, “Room-temperature detection of a single molecule’s absorption by photothermal contrast,” Science330, 353–356 (2010). [CrossRef] [PubMed]
  15. S. Berciaud, L. Cognet, G. Blab, and B. Lounis, “Photothermal heterodyne imaging of individual nonfluorescent nanoclusters and nanocrystals,” Phys. Rev. Lett.93, 257402 (2004). [CrossRef]
  16. D. Boyer, P. Tamarat, A. Maali, B. Lounis, and M. Orrit, “Photothermal imaging of nanometer-sized metal particles among scatterers,” Science297, 1160–1163 (2002). [CrossRef] [PubMed]
  17. M. Selmke, M. Braun, and F. Cichos, “Photothermal Single Particle Microscopy: Detection of a Nano-Lens,” ACS Nano, DOI: [CrossRef] [PubMed]
  18. J. Hwang and W. E. Moerner, “Interferometry of a single nanoparticle using the gouy phase of a focused laser beam,” Opt. Commun.280, 487–491 (2007). [CrossRef]
  19. G. C. K. Chen, M. Andika, and S. Vasudevan, “Excitation temporal pulse shape and probe beam size effect on pulsed photothermal lens of single particle,” J. Opt. Soc. Am. B27, 796–805 (2010). [CrossRef]
  20. M. Harada, M. Shibata, T. Kitamori, and T. Sawada, “Application of coaxial beam photothermal microscopy to the analysis of a single biological cell in water,” Anal. Chim. Acta.299, 343–347 (1995). [CrossRef]
  21. S. Berciaud, D. Lasne, G. Blab, L. Cognet, and B. Lounis, “Photothermal heterodyne imaging of individual metallic nanoparticles: Theory versus experiment,” Phys. Rev. B73, 045424 (2006). [CrossRef]
  22. Z. L. Horvath and Z. Bor, “Focusing of truncated gaussian beams,” Opt. Commun.222, 51–68 (2003). [CrossRef]
  23. S. Teng, T. Zhou, and C. Cheng, “Fresnel diffraction of truncated gaussian beam,” Optik118, 435–439 (2007). [CrossRef]
  24. A. A. Vigasin, “Diffraction of light by absorbing inclusions in solids,” Kvant. Elektron. (Moscow) [Sov. J. Quantum Electron.]4, 662–666 (1977).
  25. G. Baffou, P. Bon, J. Savatier, J. Polleux, M. Zhu, M. Merlin, H. Rigneault, and S. Monneret, “Thermal Imaging of Nanostructures by Quantitative Optical Phase Analysis,” ACS Nano, DOI: [CrossRef] [PubMed]
  26. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (John Wiley and Sons, Inc., 1991). [CrossRef]
  27. C. Hu and J. R. Whinnery, “New thermooptical measurement method and a comparison with other methods,” Appl. Opt.12, 72–79 (1973). [CrossRef] [PubMed]
  28. M. Selmke, M. Braun, and F. Cichos, “Photonic Rutherford Scattering,” in preparation (2012).
  29. R. Escalona, “Comparative study between interferometric and z-scan techniques for thermal lensing characterization,” Opt. Commun.281, 1323–1330 (2008). [CrossRef]
  30. S. M. Mian, S. B. McGee, and N. Melikechi, “Experimental and theoretical investigation of thermal lensing effects in mode-locked femtosecond z-scan experiments,” Opt. Commun.207, 339–345 (2002). [CrossRef]
  31. A. Gnoli, L. Razzari, and M. Righini, “Z-scan measurements using high repetition rate lasers: how to manage thermal effects,” Opt. Express13, 7976–7981 (2005). [CrossRef] [PubMed]
  32. A. Gnoli, A. M. Paoletti, G. Pennesi, G. Rossi, and M. Righini, “High-accuracy z-scan measurements of the optical nonlinearity of bis-phthalocyanines,” J. Porphyrins Phthalocyanines11, 481–486 (2007). [CrossRef]
  33. R. Polloni, B. F. Scremin, P. Calvelli, E. Cattaruzza, G. Battaglin, and G. Mattei, “Metal nanoparticles-silica composites: Z-scan determination of non-linear refractive index,” J. Non-Cryst. Solids322, 300–305 (2003). [CrossRef]
  34. O. Pena and U. Pal, “Scattering of electromagnetic radiation by a multilayered sphere,” Comput. Phys. Commun.180, 2348–2354 (2009). [CrossRef]
  35. F. Onofri, G. Grehan, and G. Gouesbet, “Electromagnetic scattering from a multilayered sphere located in an arbitrary beam,” Appl. Opt.34, 7113–7124 (1995). [CrossRef] [PubMed]
  36. G. Gouesbet, G. Grehan, and B. Maheu, “Scattering of a gaussian-beam by a mie scatter center using a bromwich formalism,” J. Optics-Nouvelle Revue D Optique16, 83–93 (1985).
  37. G. Gouesbet, B. Maheu, and G. Grehan, “Light-scattering from a sphere arbitrarily located in a gaussian-beam, using a bromwich formulation,” J. Opt. Soc. Am. A5, 1427–1443 (1988). [CrossRef]
  38. G. Gouesbet, J. Lock, and G. Grehan, “Generalized lorenz-mie theories and description of electromagnetic arbitrary shaped beams: Localized approximations and localized beam models, a review,” J. Quant. Spectrosc. Radiat. Transfer112, 1–27 (2011). [CrossRef]
  39. G. Gouesbet, J. A. Lock, and G. Grehan, “Partial-wave representations of laser-beams for use in light-scattering calculations,” Appl. Opt.34, 2133–2143 (1995). [CrossRef] [PubMed]
  40. G. H. Meeten, “Computation of s1–s2 in mie scattering-theory,” J. Phys. D: Appl. Phys.17, L89–L91 (1984). [CrossRef]
  41. J. A. Lock and E. A. Hovenac, “Diffraction of a gaussian-beam by a spherical obstacle,” Am. J. Phys.61, 698–707 (1993). [CrossRef]

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