## Nano-lens diffraction around a single heated nano particle |

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

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]

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]

*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]

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]

*μ*m-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]

## 2. The thermal lens

*T*(

**r**) =

*T*

_{0}+ Δ

*TR/r*which decays with the inverse distance

*r*from the particle of radius

*R*to the unperturbed ambient temperature

*T*

_{0}. 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 Ω

*≈ 1MHz, depending on the thermal conductivity*

_{m}*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. 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]

*R*

_{th}= [2

*κ*/(

*C*Ω

*)]*

_{m}^{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]

*n*(

*T*), quanti-fied by its thermorefractive coefficient (d

*n*/d

*T*), a corresponding refractive index profile

*n*(

**r**) is established as indicated in Fig. 1(a): with Δ

*n*= Δ

*T*(d

*n*/d

*T*) being the heating induced refractive index contrast relative to the unperturbed refractive index

*n*

_{0}=

*n*(

*T*

_{0}) at infinite distance. The amount of energy absorbed by the particle is determined by its absorption cross-section

*σ*

_{abs}and the intensity of the heating laser

*I*

_{0,}

*is the heating beam peak intensity,*

_{h}*z*

*its Rayleigh-range and Δ*

_{R,h}*z*

*is the offset relative to the probing beam (see Fig. 1(b)). Together with the thermal conductivity*

_{f}*κ*and the radius

*R*of the particle this controls the induced temperature and thus the contrast Δ

*n*of the lens The lens described by Eq. (1) decays to half its maximum perturbation Δ

*n*at a distance of

*r*= 2

*R*, making it a nanoscopic object. Nonetheless, it has infinite extent. It will only be limited by the probing beam which will for most cases be a focused beam of diffraction-limited extent. As Δ

*n*will quantify the photothermal single particle signal, the knowledge of the heating beam intensity and the thermal properties of the embedding bulk material will then allow the determination of the particles’ absorption cross-section

*σ*

_{abs}. It is thus necessary to obtain expressions for the photothermal single particle signal from the induced refractive contrast Δ

*n*. This is the purpose of the next section.

## 3. The diffraction integral

*U*

*(*

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

*χ*for each component-wave as a result of the thermal lens (see Fig. 1(a)) [24].

*r*will be assumed. This corresponds to the case of a heated nano-particle positioned along the optical axis, see Fig. 1(a).

*χ*(

*ρ*) for a single ray passing the lens immersed in a sample slab of thickness 2

*L*at a distance

*ρ*from the optical axis can be approximated via a straight ray calculation already utilized for collimated beams in reference [24] 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]

*L*≫

*ρ*were used after the integration. Although the integration in the distance

*ρ*extends to infinity, the weighting by the Gaussian field amplitude

*U*

*(*

_{a}*ρ*) will ensure the validity of the inequality. The

*ρ*-dependent part of the phase factor exp(−

*i*

*χ*), i.e.

*ρ*

^{2iRΔnk0}, will later be used in the explicit integration, while the factor (4

*L*

^{2})

^{−iRΔnk0}, which is in magnitude equal to unity and has units of [m

^{2}] ensures unitless-ness of the expressions encountered during the derivations.

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

*z*

*from the center of the heated particle: where the beam-waist in the aperture plane is*

_{p}*ζ*

*(*

_{G}*z*

*) = tan*

_{p}^{−1}(

*z*

_{p}*/z*

*). The Rayleigh-range*

_{R}*z*

*is connected to the beam-waist*

_{R}*ω*

_{0}and the wave-number

*k*=

*k*

_{0}

*n*

_{0}by

*k*

_{0}= 2

*π*

*/*

*λ*represents the vacuum wave-vector for the probing wavelength

*λ*.

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

*I*∝ |

*U*|

^{2}within the image-plane relative to the constant much larger background of the unperturbed field: This relative signal will be independent of the probe power

*𝒫*

*as long as no additional heating is induced, meaning that Δ*

_{d}*n*= const. with

*𝒫*

*. In case of a single laser being diffracted by its own induced thermal lens, the relative signal will be proportional to*

_{d}*𝒫*

*.*

_{d}*I/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

*z*

*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*

_{p}*ρ*. The found expressions which are exemplarily visualized in the plots of Fig. 2 allow the study of (finite NA)

*z*

*-scans in photothermal microscopy setups similar to the thin sample slab studies in [4*

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

*θ*

_{max}= arcsin (NA/

*n*

_{0}), while the illuminating objective determines the beam waist(s).

## 4. Comparison to rigorous vectorial electromagnetic (EM) model

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

*a*

_{N}_{+1}and

*b*

_{N}_{+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]

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]

*𝒫*

*of the total electromagnetic field (*

_{d}**E**

*,*

^{t}**H**

*) detected in the far-field under a scattering-angle*

^{t}*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

*ACS Nano*, DOI: [CrossRef] [PubMed]

**S**

*under a certain angle*

^{t}*θ*with respect to the optical axis (Fig. 3(a)):

**E**

*and*

^{i}**H**

*with complex expansion coefficients*

^{i}*g*

*, and an outgoing scattered field*

_{n}**E**

*and*

^{s}**H**

*described by complex scatter coefficients*

^{s}*a*

*and*

_{n}*b*

*. Thereby, the total field consists of the incident field and the scattered field,*

_{n}**E**

*=*

^{t}**E**

*+*

^{i}**E**

*and*

^{s}**H**

*=*

^{t}**H**

*+*

^{i}**H**

*. The scattered far-field is commonly expressed via the scattering amplitudes*

^{s}*S*

_{1,2}through

*𝒫*

*= d*

_{d}*𝒫*

_{inc}+ d

*𝒫*

_{sca}+ d

*𝒫*

_{ext}. Now, the time-averaged projected Poynting vector may be computed via

*g*

*, the so-called beam shape coefficients (BSCs). These coefficients reduce to*

_{n}*g*

*= 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*

_{n}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]

*Q*= (1 +

*is*

*γ*)

^{−1}where the beam-confinement factor

*s*is defined through

*s*=

*ω*

_{0}/(2

*z*

*) and the defocussing parameter*

_{R}*γ*= 2

*z*

_{p}*/*

*ω*

_{0}describes the displacement of the particle relative to the beam-waist

*ω*

_{0}. As before,

*z*

*< 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*

_{p}*ϕ*carried out in the far-field can now be expressed as differential cross-sections in any forward/backward polar angle

*θ*: d

*𝒫*

_{ext}(

*θ*) = −d

*σ*

_{ext}(

*θ*)

*I*

_{0}, d

*𝒫*

_{sca}(

*θ*) = d

*σ*

_{sca}(

*θ*)

*I*

_{0}and d

*𝒫*

_{inc}(

*θ*) = d

*σ*

_{inc}(

*θ*)

*I*

_{0}, where the Gaussian beam focus intensity

*S*

_{12}(

*θ*) =

*S*

_{1}(

*θ*) +

*S*

_{2}(

*θ*) and

*M*(

*θ*) were introduced: To obtain the total cross-sections, one needs to compute for instance

*σ*

_{abs}=

*σ*

_{ext}–

*σ*

_{sca}with: For a Gaussian beam (on-axis) the integrated flux of the collected beam can be done analytically, i.e. ∫

**S**

*· d*

^{i}**A**with

*σ*

_{inc}

*given by: wherein Δ*

_{,n}*≡ Π*

_{n}*–*

_{n}*τ*

*and ∑*

_{n}*≡ Π*

_{n}*+*

_{n}*τ*

*. To ensure numerical stability for small angles, a direct recursive determination of Δ*

_{n}*≡ Π*

_{n}*–*

_{n}*τ*

*may be used [40*

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

*σ*

_{sca,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 (

*ω*

_{0}≫

*λ*). 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

*z*

*-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 [*

_{p}*θ*

_{min},

*θ*

_{max}] = [21°, 31°]. To test this prediction, a central beam-stop experiment has been carried out which is detailed in the following section.

*z*

*as observed experimentally in our recent study [17*

_{f}*ACS Nano*, DOI: [CrossRef] [PubMed]

*θ*

_{max}= 30°. The focal displacement Δ

*z*

*that is included in Eq. (2) has been varied from −3*

_{f}*z*

*to 3*

_{R,d}*z*

*. The effect in the photothermal signal can be seen in Fig. 5(a). Depending on Δ*

_{R,d}*z*

*, either the positive signal is enhanced while the negative one is decreased (for a positive Δ*

_{f}*z*

*) 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 Δ*

_{f}*z*

*shows a reasonable agreement (see Fig. 5(b)), when the scaling factor of order unity is included. The influence of Δ*

_{f}*z*

*on the maxima and minima position is depicted in Fig. 5(c). The axial distance between maximum and minimum is the shortest for Δ*

_{f}*z*

*= 0. In this region the predictions of both theories are nearly identical. Only for a large focal displacement Δ*

_{f}*z*

*> 1.5*

_{f}*z*

*, the deviation becomes noticable.*

_{R,d}## 5. Signal inversion of an axial single particle scan

*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

*ACS Nano*, DOI: [CrossRef] [PubMed]

*D*

*= 10 mm was thereby reduced to an annular ring of*

_{o}*D*

*= 9 mm inner diameter corresponding to an angle of*

_{i}*θ*

_{min}= arcsin (NA

*D*

*/ (*

_{i}*D*

_{o}*n*

_{0})) = 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

*ω*

_{0,}

*= 281 nm and*

_{d}*ω*

_{0}

*= 233 nm, respectively. The beams were offset in axial direction by Δ*

_{,h}*z*

*= 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*

_{f}*ACS Nano*, DOI: [CrossRef] [PubMed]

## 6. Conclusion

## Acknowledgments

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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 |

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 |

39. | G. Gouesbet, J. A. Lock, and G. Grehan, “Partial-wave representations of laser-beams for use in light-scattering calculations,” Appl. Opt. |

40. | G. H. Meeten, “Computation of s1–s2 in mie scattering-theory,” J. Phys. D: Appl. Phys. |

41. | J. A. Lock and E. A. Hovenac, “Diffraction of a gaussian-beam by a spherical obstacle,” Am. J. Phys. |

**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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