---
title: "Enhanced third harmonic generation for s- and p- polarized optical surface modes of 1D photonic crystal structure."
authors: ["Valery Konopsky", "Alexey Melnikov", "Elena Alieva", "Sergey Chekalin"]
affiliation: "Institute of Spectroscopy, Fizicheskaya, 5, Troitsk, Moscow, 108840, Russia."
journal: "Journal of the Optical Society of America B"
year: 2019
volume: ""
issue: ""
article_number: ""
pages: ""
doi: "10.1364/JOSAB.36.002871"
type: journal-article
site_group: ""
url_abstract: ""
url_pdf: "https://valery.konopsky.com/kvnlocal/Konopsky_JOSA_B_2019.pdf"
language: en
source_tex: "Z:\\ValeryData\\Valery_New\\my_articles\\JOSA_B_2019_THG_510_1530s\\JOSAb\\sent\\zip\\THG2JOSAb.tex"
source_pdf: "Z:\\ValeryData\\Valery_New\\my_articles\\JOSA_B_2019_THG_510_1530s\\JOSAb\\published\\Konopsky_JOSA_B_2019.pdf"
---
## Abstract
Generation of the third harmonic radiation, visible to the naked eye as a collimated green beam, is obtained via femtosecond excitation of the optical surface modes (SMs) propagating along a one-dimensional (1D) photonic crystal (PC) for \textsl{s-} and \textsl{p-} polarization separately. For both polarizations, the PC\,SMs exist at the fundamental and third harmonic frequencies, that allows efficient nonlinear conversion at the phase-matching points. The pattern of the third harmonic surface wave scattering is detected and modeled, and it is shown that this pattern reveals the mode structure of the PC. Applications of the studied 1D PC structure for the experimental testing of 2D nonlinear materials are discussed.
# Introduction
Nonlinear conversion of light in planar multilayer structures is one of the most important processes that can be employed for the optical signal processing in integrated photonic devices. However, the low values of nonlinear susceptibilities of common optical materials make phase matching necessary for efficient nonlinear frequency conversion [1]. To achieve highly efficient third harmonic generation (THG) is especially challenging: first, third-order susceptibility $\chi^{(3)}$ is extremely small for most optical materials, and second, phase-matching is difficult to realize because of the large difference between refractive indices (RIs) at wavelengths of the first harmonic (FH) and the third harmonic (TH). The third-order susceptibility can potentially be increased by using new 2D materials such as graphene [2, 3], hybrid (organic-semiconductor) structures [4, 5], layers of nanotubes, etc. (for a recent review of 2D nonlinear materials see [6]). The problem of phase-matching can be solved in one-dimensional (1D) PC structures, where FH and TH waves excite photonic crystal surface modes (PC SMs), as we demonstrated in [7]. It is important that such an approach is compatible with planar processing technology, and 2D nonlinear materials (such as graphene) can be deposited on top of this multilayer planar structure. In recent years, several highly efficient THG schemes have been proposed, based on, e.g. 3D PC [8], 2D photonic waveguides [9], or nanosized silicon-based plasmonic waveguides [10]. However, these methods require nanofabrication facilities, whereas 1D PC design is achievable using standard multilayer-coating equipment.
To test the third-order susceptibility of 2D nonlinear materials, deposited on top of 1D PCs, it is very useful to have 1D PCs that support propagation and phase-matching for both *s-* and *p-*polarized PC SMs. A comparison of THG efficiency for the two polarizations will provide important information on the inherently anisotropic nonlinearity of 2D materials. In this paper we present such a 1D PC structure, which supports PC SMs at FH and TH wavelengths, and, moreover, phase-matching is possible close to the angle of the total internal reflection (TIR) for *s-* and *p-*polarization separately.
# Experimental setup
image
Layout of the experiment. In the color inset a photograph of the reflected beams of the first and third harmonics is presented (s-polarization).
Experimental layout is shown in Fig. 1. *S-* (*p-*) polarized PC SMs on the external surface of a $\mathrm{TiO_2}$/$\mathrm{SiO_2}$ multilayer structure are excited by a femtosecond laser beam at the wavelength of $\lambda_{1(s)}=1548$ nm ($\lambda_{1(p)}=1524$ nm). The confinement of the PC SM field near the interface is achieved due to the presence of the photonic band gap in the multilayer structure on one side of the external surface and 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 incident 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 SMs through a prism, in the range of angles, for which TIR is realized. The prism was fixed on a rotating base, making it possible to precisely set the angle of incidence required for the excitation of PC SMs.
Femtosecond pulses with *s-* (*p-*) polarization at the fundamental wavelength of 1548 nm (1524 nm) were produced by a parametric amplifier (TOPAS, Light Conversion Ltd.) pumped by a Ti:sapphire regenerative amplifier (Spitfire Pro, Spectra Physics). The duration of the pulses was about 70 fs and the repetition rate 1 kHz. The idler beam was filtered out with a specialized wavelength separator (Light Conversion Ltd.) and the residual signal beam at the fundamental wavelength was weakly focused into the prism by a fused silica lens with a focal length of 50 cm (the focal point was located 10 cm farther than the prism). In order to rotate the polarization of femtosecond pulses we used an achromatic half-wave plate. The pump power required for the measurements was set by attenuating the beam with a variable metallic neutral density filter. The spectrum of the third harmonic radiation was measured with a grating CCD spectrometer (ASP-100MF, Avesta Ltd.).
# 1D photonic crystal structure
The 1D PC structure is designed so that both the fundamental and third-harmonic wavelengths for both polarizations fall inside photonic band gaps, where the PC SMs are located. The following 1D PC structure, supporting *s-* and *p-*polarized PC SMs at FH and TH wavelengths was developed: prism/$(HL)^{7}H'$/air, where $H$ is a $TiO_2$ layer (thickness $d_2=83.05$ nm), $L$ is a $SiO_2$ layer (thickness $d_1=596.95$ nm), and $H'$ is a final $TiO_2$ layer (thickness $d_3=344.13$ nm). The prism was made from fused silica. The $TiO_2/SiO_2$ 15-layer structure, with $TiO_2$ as the first and the last layer, was deposited directly onto the prism base by the SYRUSpro 710 optical vacuum coater (Buhler Leybold Optics, Alzenau, Germany) via electron-beam evaporation and plasma ion-assisted deposition. The RIs of the prism, $SiO_2$, $TiO_2$ layers and air at $\lambda=1548$ nm, are $n_0=1.444$, $n_1=1.477$, $n_2=n_3=2.248$ and $n_\mathrm{air}=1.0003$, respectively. The RIs at other wavelengths were derived using the dispersion data given by Palik [11].
image
Calculated dispersions of the 1D PC structure for (a) p-polarization and (b) s-polarization. Phase-matching points are indicated by white pentagrams and are connected by white dotted lines.
The simulated dispersions of the 1D PC structure are presented in Fig. 2, both for *p*-polarization (a) and *s*-polarization (b). For *p*-polarization phase-matching point occurs at the effective RI $\rho=1.128$ for $\lambda_{1(p)}=1524$ nm and $\lambda_{3(p)}=508$ nm, whereas for *s*-polarization this takes place at $\rho=1.0199$ for $\lambda_{1(s)}=1548$ nm and $\lambda_{3(s)}=516$ nm.
# Results
image
TH spectra for (a) p- and (b) s-polarization.
A directional green beam of the third harmonic is readily seen by the naked eye when the pump beam impinges on the surface of the prism at a specific phase-matching angle, and PC SMs are excited simultaneously at wavelengths of the first and third harmonics. A photograph of the generated TH beam and the reflected FH beam is shown in the color inset to Fig. 1. The green beam of the TH appears at the exit side of the coupled prism, at an angle slightly different from that of the first harmonic, as shown schematically in Fig. 1. This difference in angles is due to the fact that $n_0(\lambda_3)=1.461>n_0(\lambda_1)=1.444$, and therefore $\theta(\lambda_3)<\theta(\lambda_1)$ at the phase-matching point, where $\rho(\lambda_3)=\rho(\lambda_1)$. Both spots (the fundamental and the third-harmonic beams) are clearly visible on an IR visualizer and their angular separation in air can be measured. The experimentally obtained angle ($\Delta\theta=1.0^0$) is very close to the expected value, that was deduced from the quartz prism dispersion.
The correspondent spectra of the TH for both polarizations are shown in Fig. 3. To verify that the observed signal is generated by the nonlinear process involving $\chi^{(3)}$, we measured the dependence of the TH power on the femtosecond pump power, shown in the log-log plot in Fig. 4. The slope of the linear fit is $3.1\pm0.5$, confirming third-order nonlinearity.
image
Pump power dependence of THG (s-polarization). Unfilled circles: experimental points; solid line: linear fit in logarithmic coordinate, with a slope 3.1.
## FH to TH conversion efficiency for *p*-polarization
The efficiency of THG in our system is estimated to be $2\cdot10^{-26}~[\mathrm{cm}^2/\mathrm{W}]^2$ for *p*-polarization. As an example, for the average pump power of 40 mW and the pump beam cross-sectional area of $0.96\cdot10^{-2}$ cm$^2$ (which corresponds to the pulse peak intensity of $6\cdot10^{10}~\mathrm{W}/\mathrm{cm}^2$), the conversion efficiency is $7.5\!\times\!10^{-5}$, and we obtain 3 $\mu$W of *p*-polaraized third harmonic radiation generated at 508 nm.
## FH to TH conversion efficiency for *s*-polarization
For *s*-polarization, the efficiency of THG is estimated to be $4.8\cdot10^{-27}~[\mathrm{cm}^2/\mathrm{W}]^2$. As an example, for the average pump power of 35 mW and the pump beam cross-sectional area of $0.96\cdot10^{-2}$ cm$^2$ (which corresponds to the pulse peak intensity of $5.2\cdot10^{10}~\mathrm{W}/\mathrm{cm}^2$), the conversion efficiency is $1.3\!\times\!10^{-5}$, and we obtain 0.45 $\mu$W of *s*-polaraized third harmonic radiation generated at 516 nm.
## Scattering pattern of TH on rough surface and THG from scattered FH
In addition to the two bright spots of TH and the reflected FH, several diffuse arcs can be seen on the screen opposite to the exit side of the prism, as schematically shown in Fig. 1. To see these arcs more clearly, we took photographs of the screen with a long exposure time. Images of the scattered TH pattern for *s*-polarization are shown in Fig. 5(a,b). Note that the exposure time for these photographs was 30 seconds, i.e. much longer than the one for the color inset photograph in Fig. 1. As a result, the spots of the TH beam and the reflected FH beam are overexposed, however the TH scattering arcs become visible on the photographs.
image
Scattering pattern of TH at 516 nm.(a) Ordinary scattering and (b) increased scattering after breathing onto the surface. (c) The simulated angular spectrum of the intensity of the optical field at the external surface of the 1D PC. (d) TH scattering lines when the TH is the sum of three unscattered FH wavevectors and when it is the sum of two unscattered and one scattered wavevector of the FH.
Most of these arcs result from the scattering of TH waves on the rough surface. The propagating TH surface wave interacts with surface irregularities and is scattered into modes with high density of states. The most probable is the scattering into the same surface mode, but with a different direction of propagation. This is the arc that contains the bright spot of the TH. The second arc, which is located inside the optical bandgap, is the result of scattering into the surface mode with *p*-polarization. The other arcs are the result of scattering into the modes that constitute edges of the optical bandgap. Simulated intensities of the optical fields at the outer surface of the 1D PC for various propagation constants $\rho$ are shown in Fig. 5(c).
In Fig. 5(b) an additional arc with a different curvature is distinguishable. This photograph was taken during breathing onto the surface and, therefore, strong scattering on small drops of condensed water occurred. This arc is the result of THG by one scattered PC SW of FH and two unscattered PC SWs of FH. An illustration of the origin of this arc is shown in Fig. 5(d). A more accurate comparison of the scattering pattern with theoretical modeling is made in the Appendix.
# Discussion
Optical field enhancement at the surface that results from excitation of the optical surface modes is widely used to study surface nonlinear optical effects [16]. As a rule, surface plasmons existing at metal-dielectric interfaces are utilized for this purpose [17]. Recent progress in the development of 2D non-linear materials increases interest in this topic [6]. PC SMs are a good alternative to surface plasmons due to their long-range propagation, absence of strong damping and the possibility to design the PC structure for any desired wavelength.
The maxima of the FH and TH fields are located near the external surface of the PC, and, therefore, deposition of an additional layer of a 2D nonlinear material will allow an immediate experimental evaluation of the third-order nonlinearity of the 2D material under study. There are many theoretical papers that predict high third-order nonlinearity of 2D materials, such as graphene [18]. On the other hand, there are theoretical works the authors of which state that these expectations are overly optimistic (see, for example, the work [19] entitled “Graphene — A rather ordinary nonlinear optical material” and Comment [20] with Response [21]). Thus, a convenient and reliable technique for the experimental verification of the nonlinearity of such materials for in- and out-of-plane polarizations of light would be very useful. In particular, THG from TE-polarized graphene modes [22] can be tested in such PC structure.
# Conclusions
We have developed a multilayer structure that allows one to routinely obtain enhanced phase-matched THG for both *p*- and *s*-polarized optical waves. Such a structure can be used for experimental verification of the nonlinearity of 2D materials and of its anisotropy.
The scattering pattern of TH, which reveals the mode structure of the PC, was detected and modeled. An additional scattering curve of TH, which is a nonlinear sum of one scattered FH and two unscattered FH waves, was also detected and simulated.
# Funding Information
This work was performed as a part of the State assignment for the Institute of Spectroscopy of the Russian Academy of Sciences.
## Appendix
For the theoretical simulation of the scattering pattern of the third-harmonic wave shown in Fig. 5 (a,b), we took into account the refraction of the output beam at the output edge of the prism for inclined incidence. An illustration of the model of this refraction is shown in Fig. 6. All the parameters necessary for the calculation of (x, y) coordinates of the scattered and then refracted beam are indicated in Fig. 6.
image
Illustration of the model used for the simulation of refraction of the reradiated wave, which is formed after surface scattering of the third harmonic outside the plane of incidence of the first harmonic.
image
(a) Simulated angular pattern of the scattered and refracted waves. (b) Superimposed simulated and experimental angular patterns.
The coordinates of the scattering arcs are:
``` math
\begin{eqnarray}
x_N(\alpha)=R\tan({\delta_1})\cos({\gamma})\\
y_N(\alpha)=R\tan({\delta_1})\sin({\gamma})
\end{eqnarray}
```
where
``` math
\begin{eqnarray}
\delta_1&=&\arcsin\left(n_0\sin({\delta_0})\right)\\
\delta_0&=&\arctan\left({\frac {2\,{\mathrm{ |AS|}}\,\sin(\beta)}{\sin({\gamma})a}}\right)\\
\gamma&=&\arctan\left(\frac{\sin(\beta)}{\cos(\beta)-{\frac {a}{2\,{\mathrm{|AS|}}}}}\right)\\
\beta&=&\arctan\left(\frac{\sqrt {2}}{2}\,\tan(\alpha)\right)\\
\mathrm{|AS|}&=&\frac {a\sqrt {2}\sin(\theta_N)}{2\,\sin(\theta_N+\varphi)}\\
\varphi&=&\arctan\left (\left (\cos(\alpha)\right )^{-1}\right)\\
\theta_N&=&\arcsin(\rho_N/n_0)
\end{eqnarray}
```
and $\rho_N= [0.8206,\; 0.8875,\; 0.9445,\; 0.9898,\; 1.0199,\; 1.0992,\; \\ 1.1662,\; 1.1892,\; 1.2153,\; 1.2402,\; 1.2621,\; 1.2798,\; 1.2915]$ is the set of density of states maxima, containing PC SMs $\rho_\mathrm{SM}(s)=1.0199$ and $\rho_\mathrm{SM}(p)=1.0992$.
The coordinates ($x_{[sc+2]}(\alpha_1)$, $y_{[sc+2]}(\alpha_1)$) of the arc, which is the result of scattering of the first harmonic (shown in Fig. 5(b)), can be obtained replacing the argument $\alpha$ by the argument $\alpha_1$:
``` math
\begin{eqnarray}
\alpha=\arctan\left({\frac {\sin({\alpha_1})}{2+\cos({\alpha_1})}}\right)\\
\theta_{{[sc+2]}}=\arcsin\left(\frac{\sqrt {5+4\,\cos({\alpha_1})}}{3}\,\sin\left( \arcsin(\rho_\mathrm{{SM(s)}}/n_0) \right)\right)\, .
\end{eqnarray}
```
The results of simulations for the set of $\rho_N$ data are shown in Fig. 7(a) for $-20^0<\alpha<20^0$ and $-60^0<\alpha_1<60^0$. In Fig. 7(b) the result of modeling is superimposed on the photograph. One can see that the model reproduces the experimentally observed pattern relatively well.
10
R. W. Boyd, *Nonlinear Optics* (Academic Press, San Diego, 1992).
G. Soavi, G. Wang, H. Rostami, D. G. Purdie, D. De Fazio, T. Ma, B. Luo, J. Wang, A. K. Ott, D. Yoon *et al.*, “Broadband, electrically tunable third-harmonic generation in graphene,” **13**, 583 (2018).
S.-Y. Hong, J. I. Dadap, N. Petrone, P.-C. Yeh, J. Hone, and R. M. Osgood Jr, “Optical third-harmonic generation in graphene,” **3**, 021014 (2013).
V. Agranovich, Y. N. Gartstein, and M. Litinskaya, “Hybrid resonant organic–inorganic nanostructures for optoelectronic applications,” **111**, 5179–5214 (2011).
V. Agranovich and G. La Rocca, “Organic–inorganic heterostructures for nonlinear optics,” (2015).
S. Yamashita, “Nonlinear optics in carbon nanotube, graphene, and related 2d materials,” **4**, 034301 (2019).
V. N. Konopsky, E. V. Alieva, S. Y. Alyatkin, A. A. Melnikov, S. V. Chekalin, and V. M. Agranovich, “Phase-matched third-harmonic generation via doubly resonant optical surface modes in 1d photonic crystals,” **5**, e16168 (2016).
P. P. Markowicz, H. Tiryaki, H. Pudavar, P. N. Prasad, N. N. Lepeshkin, and R. W. Boyd, “Dramatic enhancement of third-harmonic generation in three-dimensional photonic crystals,” **92**, 083903 (2004).
B. Corcoran, C. Monat, C. Grillet, D. Moss, B. Eggleton, T. White, L. O’Faolain, and T. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” **3**, 206–210 (2009).
S. Sederberg and A. Elezzabi, “Coherent visible-light-generation enhancement in silicon-based nanoplasmonic waveguides via third-harmonic conversion,” **114**, 227401 (2015).
E. D. Palik, *Handbook of Optical Constants of Solids* (Academic, London, 1985).
V. N. Konopsky and E. V. Alieva, “Long-range propagation of plasmon polaritons in a thin metal film on a one-dimensional photonic crystal surface,” **97**, 253904 (2006).
V. N. Konopsky and E. V. Alieva, “Photonic crystal surface waves for optical biosensors,” **79**, 4729–4735 (2007).
D. L. Mills, “Attenuation of surface polaritons by surface roughness,” **12**, 4036–4046 (1975).
H. Raether, “Surface plasmons,” **111**, 1 (1988).
Y. Shen, “Surface nonlinear optics,” **28**, A56–A66 (2011).
E. Van der Ham, Q. Vrehen, E. Eliel, V. Yakovlev, E. Alieva, L. Kuzik, J. Petrov, V. Sychugov, and A. Van Der Meer, “Giant enhancement of sum-frequency yield by surface-plasmon excitation,” **16**, 1146–1152 (1999).
X. Yao and A. Belyanin, “Giant optical nonlinearity of graphene in a strong magnetic field,” **108**, 255503 (2012).
J. Khurgin, “Graphene - A rather ordinary nonlinear optical material,” **104**, 161116 (2014).
S. A. Mikhailov, “Comment on “Graphene - A rather ordinary nonlinear optical material”,” **111**, 106101 (2017).
J. B. Khurgin, “Response to “Comment on ‘Graphene - A rather ordinary nonlinear optical material’”,” **111**, 106102 (2017).
I. Degli-Eredi, J. Sipe, and N. Vermeulen, “TE-polarized graphene modes sustained by photonic crystal structures,” **40**, 2076–2079 (2015).
## References
1. %Y. Zhang, S. Qiao, L. Sun, Q. W. Shi, W. Huang, %L. Li, and Z. Yang, % Photoinduced active terahertz metamaterials with nanostructured %vanadium dioxide film deposited by sol-gel method, Opt. Express **22, %11070--11078 (2014). %thebibliography.
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