---
title: "Phase-matched third-harmonic generation via doubly resonant optical surface modes in 1D photonic crystals."
authors: ["Valery Konopsky", "Elena Alieva", "Sergey Alyatkin", "Alexey Melnikov", "Sergey Chekalin", "Vladimir Agranovich"]
affiliation: "Institute of Spectroscopy, Fizicheskaya, 5, Troitsk, Moscow, 142190, Russia."
journal: "Light: Science and Applications"
year: 2016
volume: ""
issue: ""
article_number: ""
pages: ""
doi: ""
type: journal-article
site_group: ""
url_abstract: ""
url_pdf: "https://valery.konopsky.com/kvnlocal/lsa2016168a.pdf"
language: en
source_tex: "Z:\\ValeryData\\Valery_New\\my_articles\\LSA2016\\LSA\\THG_final.tex"
source_pdf: "Z:\\ValeryData\\Valery_New\\my_articles\\LSA2016\\LSA\\published\\lsa2016168a.pdf"
---
## Abstract

%\textsf {\normalsize Efficient nonlinear conversion requires that interacting optical waves maintain a consistent phase relationships when travelling in a medium despite its dispersion. Birefringent phase-matching, often used to compensate the dispersion, is inapplicable to optically isotropic nonlinear materials. Here we present a one-dimensional photonic crystal structure that allows the propagation of optical surface waves, both at the fundamental and third-harmonic frequencies, as an efficient medium for phase-matched third-harmonic generation. A unique advantage of this structure is the similarity of the effective refractive indices for the surface waves to refractive index of air, at both frequencies. This allows phase-matching between the first and third harmonics, and a visible collinear beam of the third harmonic is produced at the prism-coupled output. Moreover, these optical surface waves propagate over long distances even if a lossy nonlinear nanofilm is deposited onto the photonic crystal surface. We provide experimental results for third-harmonic generation at wavelength 410\,nm for a bare dielectric $\mathsf{Ta_2O_5}$/$\mathsf{SiO_2}$ multilayer structure and for the same structure coated with a 15\,nm $\mathsf{GaAs}$ film. } \noindent\textbf{Keywords:} nonlinear optics; optical surface waves; photonic crystals

**INTRODUCTION**\
Nonlinear photonic devices, particularly those achievable by planar processing technology, show great promise for future all-optical signal processing. Large nonlinear susceptibilities, in configurations that perform efficient nonlinear conversion, are crucial for such photonic technologies. The nonlinear optical susceptibility of a material can be defined by expanding its polarization as a power series, in terms of the applied optical electric field $\mathbf{E}$:
``` math
\begin{equation}
 \mathbf{P}(t) = \chi^{(1)} \mathbf{E}(t) + \chi^{(2)} \mathbf{E}^2(t) + \chi^{(3)} \mathbf{E}^3(t) + \ldots\ , 
 %\vspace*{-0.1cm}
\end{equation}
```
where $\chi^{(1)}$ is the linear susceptibility, the second-order susceptibility $\chi^{(2)}$ describes processes such as second-harmonic generation, and the third-order susceptibility $\chi^{(3)}$ describes, e.g., third-harmonic generation (THG) and the intensity-dependent addition to the refractive index $n_2I$ (where $n_2=12\pi^2\chi^{(3)}/n_0^2$) [1]. The coefficient $\chi^{(2)}$ is nonzero only in noncentrosymmetric materials, whereas $\chi^{(3)}$ is nonzero in all optical materials, regardless of their symmetry.

However, though $\chi^{(3)}$ is non-vanishing in all media, its small value makes the implementation of third-order nonlinearity in integrated planar photonic devices a formidable challenge. One way to increase $\chi^{(3)}$ in planar systems is based on hybrid (organic-semiconductor) structures [2]. It was shown that a substantial enhancement of $\chi^{(3)}$ could be achieved both in strongly coupled hybrid structures [3] and in hybrid structures with weak coupling [4]. These systems may also allow an optical modulation of the nonlinearity via pumping of the organic material [5].

Besides enhanced nonlinear susceptibilities, an intelligent design of planar configurations that provide phase-matching of interacting optical waves is crucial for experimental development of these hybrid structures. Third-order nonlinearity manifests itself primarily in THG. In this case, however, phase-matching is difficult or impossible to realise, because of the large interval between wavelengths.

Here we demonstrate THG in a planar multilayer structure, where the effective refractive index (RI) for each surface wave (the first and third harmonics) is very close to that of air ($\mathrm{n_{air}\simeq 1}$) and the phases of the interacting waves match. This one-dimensional (1D) photonic crystal (PC) structure can accommodate further semiconductor and organic nanolayers deposited on its external surface. This makes such a system ideal for investigating different planar hybrid schemes with enhanced $\chi^{(3)}$.

**MATERIALS AND METHODS**\
**Photonic crystal structures**\
Our 1D PC had the following structure: substrate/$H(LH)^{14}L'$/air, where $L$ represents a $SiO_2$ layer (thickness $d_1=321.7$ nm), $H$ is a $Ta_2O_5$ layer ($d_2=146.7$ nm), and $L'$ is a $SiO_2$ layer ($d_3=211.7$ nm). This structure was also used with an additional GaAs coating (thickness 15 nm) applied to the 1D PC surface.\
**Film deposition**\
The $Ta_2O_5/SiO_2$ multilayer was deposited by magnetron sputtering. An additional GaAs nanofilm was applied by metalorganic chemical vapor deposition. The prism and substrate were made from fused silica. The RI of the prism, $SiO_2$, $Ta_2O_5$ layers and air at $\lambda=1230$ nm, are $n_0=n_1=n_3=1.46$, $n_2=2.05$ and $n_{air}=1.0003$, respectively. The RI at other wavelengths were derived using dispersion data given by Palik [6].\
**Laser sources**\
Two different laser sources, for pico- and femtosecond pulses, were used in the experiments. *Picosecond* infrared radiation at wavelength 1230 nm was generated by an optical parametric oscillator (MIRA-OPO, APE GmbH) that was pumped synchronously by a picosecond mode-locked Ti:Sapphire laser (MIRA 900D, Coherent) at a repetition rate of 76.4 MHz. To ensure the stability of infrared radiation power during the experiment, the OPO cavity length was stabilised actively. *Femtosecond* pulses at 1230 nm were generated by a parametric amplifier (Topas, Light Conversion Ltd.) pumped by a Ti:sapphire regenerative amplifier (Spitfire Pro, Spectra Physics) at a repetition rate of 1 kHz. Signal (1230 nm) and idler (2330 nm) beams were separated by a specialised wavelength separator (Light Conversion Ltd.). Then, the signal beam was guided through an achromatic half-wave plate and a variable continuous neutral-density filter (both Thorlabs) to prepare p-polarized 1 $\mu$J, 100 fs pulses of wavelength 1230 nm, incident on the prism.\
**Detectors**\
The generated *picosecond* signals near 410 nm were detected using a back-illuminated EMCCD camera (Andor iXon3 897), thermoelectrically cooled to $-100^0$C. A standard camera lens of 50 mm focal length formed the image on a 512$\times$<!-- -->512 format matrix with pixel size 16 $\mu$m. *Femtosecond* excitation produced a directional emission of blue light, visible with the naked eye. The spectrum for this emission was registered using a grating CCD spectrometer with a fibre-optic input (ASP-100MF, Avesta Ltd.).

**RESULTS AND DISCUSSION**\
The experiment is outlined in Fig. 1. A laser beam of wavelength $\mathrm{\lambda_1=1230}$ nm excites photonic crystal surface waves (PC SWs) on the external surface of a $\mathrm{Ta_2O_5}$/$\mathrm{SiO_2}$ multilayer structure. The confinement of the PC SW field near the interface results from the photonic band gap in the multilayer structure on one side of the external surface and from total internal reflection (TIR) on the other side. TIR requires that the effective RI for a PC SW exceeds RI of the external medium. The effective RI (or, equivalently, the wave vector) of the incoming light can be increased in several ways, e.g. via grating coupling or end-fire coupling. In this work, we used a Kretschmann-like scheme to excite PC SWs through a prism, in the angle range displaying TIR.

![Outline of experiment.](media/LSA2016/setup_1.eps)

*Outline of experiment.*

The 1D PC structure is designed so that both the fundamental and third-harmonic wavelengths are within the photonic band gaps, where the PC SWs are located. Fig. 2 shows the calculated dispersion of the structure together with measurements of the PC SW excitation. Clearly, both dispersion curves approach the light line ($\mathrm{\rho=n_{air}}$). In contrast to most optical materials, the RI of air is almost wavelength-independent, so that an angular range exists near the TIR threshold, where $\mathrm{\rho_\mathrm{SW_1}=\rho_\mathrm{SW_3}}$, indicating phase matching between the first and third harmonics. In this structure, this phase matching occurs at $\rho\simeq 1.04$, as shown in Fig. 2 by the white pentagrams.

![Calculated and measured dispersions of the 1D PC structure. The dispersion is presented as the logarithm of the optical-field enhancement factor (i.e. log10[Ie/I0]) in the external medium near the structure. A general view of the dispersion (b) is shown in the coordinates 1/λ(ρ), while a more detailed view of the first- (c) and third-harmonic (a) band gaps is shown in the coordinates λ(ρ). The photonic band gaps are clearly apparent as the dark blue regions with an enhancement much less than 1. The optical surface mode is shown as the red curves with an enhancement about 100 inside the band gaps. The measured points for the PC SWs excitation at different wavelengths and angles are shown for the first (c, cyan squares) and third harmonics (a, magenta diamonds). An angular parameter ρ = n0sin (θ0), at which the excitation of the surface mode occurs, is equal to the effective RI of the mode.](media/LSA2016/disp1total3.eps)

*Calculated and measured dispersions of the 1D PC structure. The dispersion is presented as the logarithm of the optical-field enhancement factor (i.e. log10[Ie/I0]) in the external medium near the structure. A general view of the dispersion (b) is shown in the coordinates 1/λ(ρ), while a more detailed view of the first- (c) and third-harmonic (a) band gaps is shown in the coordinates λ(ρ). The photonic band gaps are clearly apparent as the dark blue regions with an enhancement much less than 1. The optical surface mode is shown as the red curves with an enhancement about 100 inside the band gaps. The measured points for the PC SWs excitation at different wavelengths and angles are shown for the first (c, cyan squares) and third harmonics (a, magenta diamonds). An angular parameter ρ = n0sin (θ0), at which the excitation of the surface mode occurs, is equal to the effective RI of the mode.*

The third-harmonic intensity was dramatically enhanced when the angle (Fig. 3a) and wavelength (Fig. 3b) of the picosecond-laser excitation beam approached this phase-matching point. To illustrate this situation, the dispersion of the third harmonics and the frequency-shifted ($\lambda_1/3$) dispersion of the first harmonics are superimposed in Fig. 3c. The measurements ($\lambda, \rho$) for the PC SW excitation, for both infrared and visible dispersion curves (the same as in Fig. 2 a,c) are also displayed together (cyan squares and magenta diamonds for the first and third harmonics, respectively).

![THG intensity near the phase-matching point. a, Angular and, b, wavelength dependences of THG. c, Measurements and contour plot of the PC SW. The angular and wavelength ranges from a and b plots shown in c as red and cyan solid lines, respectively.](media/LSA2016/angle_lambda_plus3c.eps)

*THG intensity near the phase-matching point. a, Angular and, b, wavelength dependences of THG. c, Measurements and contour plot of the PC SW. The angular and wavelength ranges from a and b plots shown in c as red and cyan solid lines, respectively.*

Based on the results in Fig. 3, we conclude that the observed third-harmonic enhancement in our 1D PC results from phase matching. To verify that the observed signal is generated by a nonlinear process involving $\chi^{(3)}$, we also measured the dependence of THG on the picosecond pump pulse energy, shown in the log-log plot in Fig. 4. The slope of the linear fit equals $3.1\pm0.3$, confirming third-order nonlinearity.

![Pump power dependence of THG. Filled circles: experimental points; solid line: linear fit in logarithmic coordinates, with slope 3.1 ± 0.3.](media/LSA2016/THG1L.eps)

*Pump power dependence of THG. Filled circles: experimental points; solid line: linear fit in logarithmic coordinates, with slope 3.1 ± 0.3.*

The above measurements were performed using picosecond-laser excitation, while the third-harmonic radiation was detected with a CCD camera. Femtosecond laser excitation makes blue-light emission visible with the naked eye. The blue light beam thus produced is highly directional, making its divergence difficult to measure. This means that THG is produced by a long-range propagated surface wave. This directional blue beam appears on the exit side of the coupled prism, at a slightly different angle from that of the first harmonic, as shown schematically in Fig. 1. This difference in angles is due to the fact that $n_3>n_1$, and therefore $\theta_3<\theta_1$ at the phase-matching point, where $\rho_3=\rho_1$. Both spots (the fundamental and third-harmonic beams) are clearly visible on an IR visualizer and their angular separation in air can be measured. The measured angle ($\Delta\theta=1.37^0$) is very close to the expected value, deduced from the quartz prism dispersion.

Besides field enhancement and phase matching, the presented 1D PC structure has an additional unique property: it supports long-range propagation of the PC SWs, even if the surface is coated with a nanofilm made of strongly absorbing material. This ultra-long-range propagation was predicted and experimentally confirmed for a metal-coated 1D PC, when the effective refractive index of the optical-surface mode is very close to the RI of the external media (e.g. air) [7]. The effect was confirmed not only for nanofilms made from “plasmonic metals" (e.g. gold), but also for nanofilms made from very lossy materials (e.g. palladium) [8].

The usefulness of this feature for developing nonlinear devices arises from the fact that many nonlinear materials with a large $\chi^{(3)}$ are also very lossy, which limits nonlinear interactions. For example, some semiconductors, such as Si and GaAs, have a very large $\chi^{(3)}$ (an order of magnitude greater than $\mathrm{LiNbO_3}$), but their RI also has a large imaginary part, especially for the third harmonic, which is usually produced in the spectral region of intrinsic absorption of a semiconductor. To demonstrate this feature, we deposited a 15-nm-thick GaAs nanofilm on the external side of our multilayer structure.

| **Material** | $\mathbf{n_0}$ | $\mathbf{\chi^{(3)}}$(esu) | $\mathbf{n_2} \mathrm{(m^2/W)}$ |
|:--:|:--:|:--:|:--:|
| $\mathrm{SiO_2}$ | 1.46 | $1.8\!\times\!10^{-14}$ | $3.3\!\times\!10^{-20}$ |
| $\mathrm{Ta_2O_5}$ | 2.05 | $6.1\!\times\!10^{-13}$ | $5.7\!\times\!10^{-19}$ |
| GaAs | 3.43 | $1.0\!\times\!10^{-10}$ | $3.3\!\times\!10^{-17}$ |

**Third-order optical nonlinear coefficients of materials in our 1D PC.**

Table 1 lists the nonlinear coefficients of the materials constituting our 1D PC [9, 10, 11]. Clearly, without the GaAs nanofilm, a major contributor to THG is $\mathrm{Ta_2O_5}$, for which $\chi^{(3)}$ is an order of magnitude greater than for $\mathrm{SiO_2}$. GaAs has a cubic structure and is therefore an optically isotropic material. In contrast to birefringent nonlinear crystals, frequency conversion in isotropic materials is limited by the difficulty of achieving phase matching, so more complicated schemes of quasi-phase-matching are used in a GaAs [12].

Besides, the band gap of GaAs is 1.42 eV, so that strong intrinsic absorption occurs for wavelengths shorter than 870 nm. In our structure, the deposition of the 15 nm GaAs nanofilm additionally increases the THG intensity and shifts the wavelength of the third harmonic slightly by +1.6 nm (see Fig. 5). A small additional spectral feature near 430 nm is THG at the photonic band-gap edge, which is excited by femtosecond pulses of wide bandwidth ($1230 \pm 25$ nm). An image of the third harmonic beam is shown in the inset of Fig. 5.

![THG spectrum, with and without the GaAs nanofilm, under femtosecond excitation. The acquisition time is 100 ms. The blue spot in the inset is an image of the third harmonic beam.](media/LSA2016/GaAs_bare_spot.eps)

*THG spectrum, with and without the GaAs nanofilm, under femtosecond excitation. The acquisition time is 100 ms. The blue spot in the inset is an image of the third harmonic beam.*

These spectra were recorded using femtosecond excitation at 1230 nm with an average beam power of 1 mW and a repetition rate of 1 kHz. The fundamental wave was weakly focused by a lens of focal distance 300 mm. The third harmonic radiation produced at 411.6 nm had an intensity of approximately 5 nW. Therefore, the estimated conversion efficiency is $5\!\times\!10^{-6}$ for the current configuration. THG efficiency can be further increased by optimising the 1D PC structure to accommodate the full bandwidth of the femtosecond pulses, improving the composition of the GaAs nanofilms, and creating resonant hybrid (semiconductor-organic) structures on the surface.

**CONCLUSION**\
We demonstrated THG in a single step via optical surface modes on a 1D PC when both the fundamental and third harmonic frequencies resonate with the corresponding surface modes. Three key properties of the 1D PC structure increase the otherwise poor efficiency of THG:\
(i) localisation and enhancement of the optical field near the surface via PC SW excitation;\
(ii) phase-matching between the first and the third harmonic SWs, both of which have an effective RI close to that of virtually dispersionless air;\
(iii) long-range propagation of surface waves in a nonlinear nanofilm deposited onto the 1D PC, even if considerable optical losses are present at the third-harmonic frequency for this material.

It is important that the proposed 1D design be simple to produce and compatible with planar processing technology. In recent years, several high-efficient THG schemes have been proposed, based on, e.g. 3D PC [13], 2D photonic waveguides [14], or nano-size silicon-based plasmonic waveguides [15]. However, these approaches require nanofabrication facilities, whereas our design is achievable using standard multilayer-coating equipment.

**ACKNOWLEDGEMENTS**\
The present study was supported by the Russian Foundation for Basic Research, research project No. 14-29-07132. Authors thank A.A. Padalitsa for the preparation of 15-nm GaAs nanofilms.

10 url urlprefix

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Konopsky, V. N. & Alieva, E. V. Long-range propagation of plasmon polaritons in a thin metal film on a one-dimensional photonic crystal surface. *Phys. Rev. Lett.* **97**, 253 904 (2006).

Konopsky, V. N. & Alieva, E. V. Long-range plasmons in lossy metal films on photonic crystal surfaces. *Opt. Lett.* **34**, 479–481 (2009).

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Tai, C.-Y. *et al.* Determination of nonlinear refractive index in a Ta2O5 rib waveguide using self-phase modulation. *Opt. Express* **12**, 5110–5116 (2004).

Kuo, P. S., Bravo-Abad, J. & Solomon, G. S. Second-harmonic generation using quasi-phasematching in a gaas whispering-gallery-mode microcavity. *Nature Communications* **5** (2014).

Markowicz, P. P. *et al.* Dramatic enhancement of third-harmonic generation in three-dimensional photonic crystals. *Phys. Rev. Lett.* **92**, 083903 (2004).

Corcoran, B. *et al.* Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides. *Nature Photon.* **3**, 206–210 (2009).

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## References

1. Boyd, R. W. newblock *Nonlinear Optics (Academic Press, San Diego, 1992).

2. Agranovich, V., Gartstein, Y. N. & Litinskaya, M. Hybrid resonant organic--inorganic nanostructures for optoelectronic applications. newblock *Chemical Reviews **111, 5179--5214 (2011).

3. Agranovich, V., Basko, D., La Rocca, G. & Bassani, F. Excitons and optical nonlinearities in hybrid organic-inorganic nanostructures. newblock *Journal of Physics: Condensed Matter **10, 9369 (1998).

4. Agranovich, V. & La Rocca, G. Organic--inorganic heterostructures for nonlinear optics. newblock *Journal of Luminescence (2015).

5. Agranovich, V., Basko, D. & La Rocca, G. Efficient optical pumping of organic-inorganic heterostructures for nonlinear optics. newblock *Phys. Rev. B **86, 165204 (2012).

6. Palik, E. D. newblock *Handbook of Optical Constants of Solids (Academic, London, 1985).

7. Konopsky, V. N. & Alieva, E. V. Long-range propagation of plasmon polaritons in a thin metal film on a one-dimensional photonic crystal surface. newblock *Phys. Rev. Lett. **97, 253,904 (2006).

8. Konopsky, V. N. & Alieva, E. V. Long-range plasmons in lossy metal films on photonic crystal surfaces. newblock *Opt. Lett. **34, 479--481 (2009).

9. Boyd, R. & Fischer, G. Nonlinear optical materials. newblock In Buschow, K. J. *et al. (eds.) *Encyclopedia of Materials: Science and Technology (Second Edition), 6237--6244 (Elsevier, Oxford, 2001).

10. Hashimoto, T. & Yoko, T. Third-order nonlinear optical properties of sol-gel-derived V2O5, Nb2O5, and Ta2O5 thin films. newblock *Appl. Opt. **34, 2941--2948 (1995).

11. Tai, C.-Y. *et al. Determination of nonlinear refractive index in a Ta2O5 rib waveguide using self-phase modulation. newblock *Opt. Express **12, 5110--5116 (2004).

12. Kuo, P. S., Bravo-Abad, J. & Solomon, G. S. Second-harmonic generation using quasi-phasematching in a gaas whispering-gallery-mode microcavity. newblock *Nature Communications **5 (2014).

13. Markowicz, P. P. *et al. Dramatic enhancement of third-harmonic generation in three-dimensional photonic crystals. newblock *Phys. Rev. Lett. **92, 083903 (2004).

14. Corcoran, B. *et al. Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides. newblock *Nature Photon. **3, 206--210 (2009).

15. Sederberg, S. & Elezzabi, A. Coherent visible-light-generation enhancement in silicon-based nanoplasmonic waveguides via third-harmonic conversion. newblock *Phys. Rev. Lett. **114, 227401 (2015). thebibliography.

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