Abstract

In our publication [Opt. Express, 20(7), 8055–8070 (2012)] a convergence issue resulted in a discrepancy between the relative photothermal signal of two models: the paraxial scalar diffraction model and the accurate vectorial generalized multilayer Lorenz-Mie scattering theory which served as a reference. The resolution yields the expected agreement.

© 2013 OSA

Discrepancy by a factor of order unity between the two models in Fig. (4) and the discussion following Eq. (21)

Unfortunately, the discretizations of the thermal lens scatterer n(r) used in the GLMT calculations of our article [1] were not carried out to large enough sizes, i.e. the outmost layer was not chosen sufficiently large as compared to the probing beam waist. As a result, in Fig. 4 of Ref. [1], and in the discussions based on these plots, an unexpected discrepancy by a factor of the order 𝒪 (1) was found between the results of the diffraction and the GLMT model. The problem resides in a convergence issue of the signal in the GLMT model [3]: the thermal lens must be discretized up to a size of about rL > 5ω 0, see Fig. 1. If this is done, the expected equivalence between both approaches is realized for weak and even moderate focusing, see the new Fig. 2 correcting former Fig. (4) of the article. This also corrects the discrepancy of Fig. (5) of Ref. [1], i.e. giving the same picture but without the scaling by a factor 1.6. Now, the solution to the paraxial approximation of the Helmholtz equation, i.e. Fresnel diffraction, and the exact solution for a focused beam which approximates the Gaussian beam (the Davis beam) match perfectly as they should. Expectedly, minor discrepancies remain for large numerical apertures or inverse apertures, see the new Fig. 2d).

 

Fig. 1 TL around a R = 10nm AuNP in PDSM (n 0 = 1.46) with Δn = −3.60×10−2, wavelength λ = 635nm, beam-waist ω 0 = 281nm. The graph shows the rel. transmitted power contributions of scattering (blue), extinction (red), and their sum (black), for a numerical detection aperture NAd = 0.75 at zp = −zR/2, plotted against the thermal lens cut-off radius normalized to the probing beam-waist rL 0. The computed total detectable signal ΔPd saturates for a clipping size of the lens for rL ≈ 5ω 0 [3].

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Fig. 2 (correcting Fig. 4 of [1]) Comparison of the diffraction (black) and Gaussian GLMT model (red). Parameters used for calculations are detailed in the caption of Fig. 2 of the original article [1]. b) On-axis z-scan NAd = 0 of the rel. PT signal Φzp. The superimposed grey dashed curve is the approximation Eq. (1) of this errata. c) Scan for NAd = 0.75 (solid and dashed) and NAd = 0.3 (dashed solid and double-dashed solid). d) Scan with central beam stop (inverse aperture), i.e. NAd = [0.5, 0.75]. The semi-transparent curves corresponds to no central beam-stop, NAd = 0.75 from c)

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Simplification of Eq. (7)

Eq. (7) of Ref. [1] is correct but can be simplified further to first order in the perturbation Δn/n 0, see also Ref. [2, 3] on photonic Rutherford scattering:

Φ(zp)=2k0RΔnarctan(zp/zR)
This matches the accurate calculations well as seen in Fig. 2b) (gray dashed vs. black line).

Typo in Eq. (19)

Eq. (19) in Ref. [1] has a typo. The factor (−1)n should read (−1)n +1, i.e. the correct equation reads [3]:

σinc,n=m=1nNmgmNnm+1gnm+1*θminθmax[ΣmΣnm+1(1)n+1ΔmΔnm+1]sin(θ)dθ,
where one may set θ min = 0. The calculations in Ref. [1] were all done using this correct equation. In view of practicability, we here also provide a more compact form [3] for Gaussian (zero-order Davis) beams based on the Gaussian intensity profile:
σinc=π2ω02[eϑr(θmin)eϑr(θmax)],ϑr(θ)=2tan2(θ)θdiv2,θdiv=2kω0

References and links

1. M. Selmke, M. Braun, and F. Cichos, “Nano-lens diffraction around a single heated nano particle,” Opt. Express 20(7), 8055–8070 (2012). [CrossRef]   [PubMed]  

2. M. Selmke and F. Cichos, “Photothermal single particle Rutherford scattering microscopy,” Phys. Rev. Lett. 110, 103901 (2013). [CrossRef]   [PubMed]  

3. M. Selmke, “Photothermal single particle detection in theory & experiments,” Dissertation, Universität Leipzig, Institute for experimental physics I, (2013).

References

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  1. M. Selmke, M. Braun, and F. Cichos, “Nano-lens diffraction around a single heated nano particle,” Opt. Express20(7), 8055–8070 (2012).
    [CrossRef] [PubMed]
  2. M. Selmke and F. Cichos, “Photothermal single particle Rutherford scattering microscopy,” Phys. Rev. Lett.110, 103901 (2013).
    [CrossRef] [PubMed]
  3. M. Selmke, “Photothermal single particle detection in theory & experiments,” Dissertation, Universität Leipzig, Institute for experimental physics I, (2013).

2013

M. Selmke and F. Cichos, “Photothermal single particle Rutherford scattering microscopy,” Phys. Rev. Lett.110, 103901 (2013).
[CrossRef] [PubMed]

2012

Braun, M.

Cichos, F.

M. Selmke and F. Cichos, “Photothermal single particle Rutherford scattering microscopy,” Phys. Rev. Lett.110, 103901 (2013).
[CrossRef] [PubMed]

M. Selmke, M. Braun, and F. Cichos, “Nano-lens diffraction around a single heated nano particle,” Opt. Express20(7), 8055–8070 (2012).
[CrossRef] [PubMed]

Selmke, M.

M. Selmke and F. Cichos, “Photothermal single particle Rutherford scattering microscopy,” Phys. Rev. Lett.110, 103901 (2013).
[CrossRef] [PubMed]

M. Selmke, M. Braun, and F. Cichos, “Nano-lens diffraction around a single heated nano particle,” Opt. Express20(7), 8055–8070 (2012).
[CrossRef] [PubMed]

M. Selmke, “Photothermal single particle detection in theory & experiments,” Dissertation, Universität Leipzig, Institute for experimental physics I, (2013).

Opt. Express

Phys. Rev. Lett.

M. Selmke and F. Cichos, “Photothermal single particle Rutherford scattering microscopy,” Phys. Rev. Lett.110, 103901 (2013).
[CrossRef] [PubMed]

Other

M. Selmke, “Photothermal single particle detection in theory & experiments,” Dissertation, Universität Leipzig, Institute for experimental physics I, (2013).

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Figures (2)

Fig. 1
Fig. 1

TL around a R = 10nm AuNP in PDSM (n 0 = 1.46) with Δn = −3.60×10−2, wavelength λ = 635nm, beam-waist ω 0 = 281nm. The graph shows the rel. transmitted power contributions of scattering (blue), extinction (red), and their sum (black), for a numerical detection aperture NA d = 0.75 at zp = −zR /2, plotted against the thermal lens cut-off radius normalized to the probing beam-waist rL 0. The computed total detectable signal ΔPd saturates for a clipping size of the lens for rL ≈ 5ω 0 [3].

Fig. 2
Fig. 2

(correcting Fig. 4 of [1]) Comparison of the diffraction (black) and Gaussian GLMT model (red). Parameters used for calculations are detailed in the caption of Fig. 2 of the original article [1]. b) On-axis z-scan NA d = 0 of the rel. PT signal Φzp . The superimposed grey dashed curve is the approximation Eq. (1) of this errata. c) Scan for NA d = 0.75 (solid and dashed) and NA d = 0.3 (dashed solid and double-dashed solid). d) Scan with central beam stop (inverse aperture), i.e. NA d = [0.5, 0.75]. The semi-transparent curves corresponds to no central beam-stop, NA d = 0.75 from c)

Equations (3)

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Φ ( z p ) = 2 k 0 R Δ n arctan ( z p / z R )
σ inc , n = m = 1 n N m g m N n m + 1 g n m + 1 * θ min θ max [ Σ m Σ n m + 1 ( 1 ) n + 1 Δ m Δ n m + 1 ] sin ( θ ) d θ ,
σ inc = π 2 ω 0 2 [ e ϑ r ( θ min ) e ϑ r ( θ max ) ] , ϑ r ( θ ) = 2 tan 2 ( θ ) θ div 2 , θ div = 2 k ω 0

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