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Paper Review: The third-order nonlinear optical susceptibility of gold – Boyd et. al.


When I started this blog, I decided that I’d regularly post reviews of interesting papers and articles on a regular basis. Then several things (projects, sub-projects, courses, and qualifying exams to name a few) happened, and this issue had to be put on hold. Now that I have a little time, I’ll attempt to give this a shot.

The first paper I’ll review is on nonlinear optics.

I decided to be thorough and wanted to get a permission from the journal to use the figures and tables on my website, for which, I was asked for a sum of $149 dollars. Hence, the review will be devoid of any figures or tables. This presents a challenge, now, because I’ll have to explain the physics without resorting to graphs, which will be interesting.

So, without further ado, here it is.

The third-order nonlinear optical susceptibility of gold

Authors: Robert W. Boyd, Zhimin Shi, Israel De Leon


This is a critique of the paper “The third-order nonlinear optical susceptibility of gold”, by Boyd et. al., published in Optics Communications in 2014. This paper goes over several experimental papers dealing with the nonlinear optics of bulk gold films, points out how largely the measured χ(3) vary with the frequency of the laser used and the pulse duration, and provides a feasible physical explanation for the variance, which has been overlooked in the previous papers. I chose this paper because it is very closely related to part of my Ph.D. work, which involves the characterization of nonlinear optical properties of new plasmonic materials.

Summary of the Paper

The third order nonlinearities of gold can arise mainly from three mechanisms:

  1. The contribution of free electrons

Free electrons do not have a strong contribution to the nonlinear response of bulk gold in the electric dipole approximation. Since there is no restoring force, there can be no nonlinearities in the restoring force. Also, the ponderomotive nonlinearity of free electrons also plays a negligible role in the case of gold because of strong interband transition. When electrons are confined in a small region, e.g. in a nanosphere, they display a nonlinear response because of quantum size effects. But this effect diminishes as the sample gets larger in size.

  1. Interband transitions

This is the dominant effect in bulk gold and arises from the transitions from the 5d valence band to the 6sp band. This provides for the lowest order contribution to the saturation of the absorption associated with this transition.


where A is an angular factor, T1 and T2 are, respectively, the energy lifetime and the dephasing time for the two-level system describing the interband transition, J(ω) is the joint density of states, and P is a constant associated with the momentum operator between the two states. This is the predominant effect that is used to explain the nonlinear coefficients of bulk gold.

2. The hot-electron contribution

This is the contribution to non-linearity that arises from the excitation of the 5d electrons to the 6sp conduction band through laser excitation. This causes the electrons in the conduction band to heat up. This heating causes the population of energies above the Fermi level to increase and that below the Fermi level to decrease, resulting in a change in the dielectric function of gold, in a largely frequency dependent manner. This is also known as the Fermi Smearing contribution. Typical values of this effect range around 10-16 m2/V2.  This has a slower response time because it takes about 500 fs for the electrons to heat up and several picoseconds to relax, after which the effect disappears. This is also highly frequency-dependent (for wavelengths ranging from 300 to 800nm).

Boyd argues that the third effect, namely the effect of hot-carriers excited through laser absorption, is a dominant effect in the cases where the nonlinearity observed was larger than typical values of the χ(3) (10-19 m2/V2).

Ranging from the first reported study of the third order nonlinear responses of gold [1], the authors go through several experiments taking note of the laser power and the pulse duration used in the experiments, and the observed χ(3). For papers which did not have the χ(3) calculated, the authors assumed the real part of the refractive index to be zero and used the formula starting from basic equations of permittivity and refractive index.

It was seen that for experiments where long pulses (tens of picoseconds or higher) of lasers were used, the values of χ(3) obtained were in the order of 10-16 m2/V2. This was seen in the papers of Smith et. al., Xenogiannopolou et. al., and Wang et. al. [2,3,6]. Whereas, when short-duration (lower than 1 ps) pulses were used, the values of χ(3) obtained were in the order of 10-19 m2/V2, several orders of magnitudes smaller. These were seen in the papers of Bloembergen et. al., the van Driel group, and Renger et. al. [1,4,5,7]. 

For the papers reporting very large nonlinear indices, the effect of hot-electrons has not been used to explain the phenomenon. Boyd and his co-authors propose that hot-carrier induced refractive index change may be causing the large value of χ(3). The hot carrier induced χ(3) is also highly frequency dependent, which is also observed in the experiments conducted by the van Driel group, where with a change in the laser wavelength, the χ(3) changed by a factor of 100.


This paper covered the origin of the nonlinear optical properties of bulk gold quite comprehensively and provided an explanation for the large discrepancies of the nonlinearities that are seen across publications by different groups. By looking at the results obtained by a large number of groups, they have also noticed a trend in the dependence of the nonlinearity of a gold film with the energy and the pulse duration of the laser used to characterize it. Their explanation for the phenomenon is supported by several experiments cited in the paper.

All that being said, the paper has some drawbacks which I should point out.

Some of the papers cited in the article do not seem to add any useful information to support the authors’ point. For example, the authors mention that the paper by Wang et. al. has an error in the calculation of the nonlinearities. But this does not serve to push forward their own point regarding hot-electron-assisted nonlinearity. The paper by Smith et. al. involved the z-scan measurement of the nonlinear absorption of gold composite media. The χ(3)  computed in the paper was (-1+5i) x 10-16 m2/V2. Using the same experimental results, Boyd et. al. calculated the real and imaginary parts of χ(3)  = (-9.5+2.3i) x 10-15 m2/V2. There is a large difference between the χ(3) computed by Boyd’s group and Smith’s group although they are based on the same experimental data, but the authors do not make a comment on that.

The authors also do not elaborate on why the sign of the nonlinearity varies between papers.

In conclusion, I believe that a good way to really set the hypothesis regarding hot-electron assisted nonlinearity in gold would be for one group performing an array of experiments on several gold films by varying respectively the pump frequency and the pump width. If the χ(3)  is indeed high for longer pulse durations as well as has a large frequency-dependence, it must have a strong contributing factor from hot-electrons.

Overall, I found the organization of details in this paper to be quite easy to follow, and the background provided was sufficient to give even someone unfamiliar with the topic a clear picture of what was being discussed.


[1] W.K. Burns, N. Bloembergen, Phys. Rev. B 4 (1971) 3437.

[2] D.D. Smith, Y.K. Yoon, R.W. Boyd, J.K. Campbell, L.A. Baker, R.M. Crooks, M. George, J. Appl. Phys. 86 (1999) 6200.

[3] P. Wang, Y. Lu, L. Tang, J. Zhang, H. Ming, J. Xie, F. Ho, H. Chang, H. Lin, Di. Tsai, Opt. Commun. 229 (2004) 425.

[4] T.K. Lee, A.D. Bristow, J. Hübner, H.M. van Driel, J. Opt. Soc. Am. B 23 (2006) 2142.

[5] N. Rotenberg, A.D. Bristow, M. Pfeiffer, M. Betz, H.M. van Driel, Phys. Rev. B 75 (2007) 155426.

[6] E. Xenogiannopoulou, P. Aloukos, S. Couris, E. Kaminska, A. Piotrowsk, E. Dynowska, Opt. Commun. 275 (2007) 217.

[7] J. Renger, R. Quidant, N. van Hulst, L. Novotny, Phys. Rev. Lett. 104 (2010) 046803.


Disclaimer 1: I am responsible for the explanations, interpretations, and opinions presented in the review; these do not reflect the views of my group or my mentors.

Disclaimer 2: I have tried to explain everything to the best of my abilities. Feel free to leave a comment if you think something is unclear or incorrect. I will try to address it as soon as possible.