Modeling of Random Quasi-Phase-Matching in Birefringent Disordered Media

Jolanda S. Müller, Andrea Morandi, Rachel Grange, and Romolo Savo

Phys. Rev. Applied 15, 064070 – Published 29 June 2021

Abstract

We provide a vectorial model to simulate second-harmonic generation (SHG) in birefringent, transparent media with an arbitrary configuration of nonlinear ( $χ^{(2)}$ ) crystalline domains. We apply this model on disordered assemblies of ${Li Nb O}_{3}$ and ${Ba Ti O}_{3}$ to identify the influence of the birefringence on the random quasi-phase-matching process. We show that in monodispersed assemblies, the birefringence relaxes the domain size dependence of the SHG efficiency. In polydispersed assemblies with sufficiently large domains, we find that the birefringence introduces a SHG efficiency enhancement of up to 54% compared to isotropic reference crystals. This enhancement is domain size independent in non-phase-matchable materials, while it increases linearly with the domain size if the domains can be phase matched. These two different scaling behaviors are used in Kurtz and Perry’s (KP) powder technique to identify the phase matchability of a material. We show on the example of ${Li Nb O}_{3}$ and ammonium dihydrogen phosphate that the KP technique cannot be applied to domains smaller than the coherence length, because then the SHG scaling with the domain size becomes material specific.

Received 20 November 2020
Revised 17 March 2021
Accepted 30 April 2021

DOI:https://doi.org/10.1103/PhysRevApplied.15.064070

Physics Subject Headings (PhySH)

Birefringence Nanophotonics Nonlinear optics

Nanoparticles Optical materials & elements Polycrystalline materials

Electromagnetic wave theory Optical second-harmonic generation

Condensed Matter, Materials & Applied PhysicsGeneral Physics

Authors & Affiliations

Jolanda S. Müller, Andrea Morandi, Rachel Grange , and Romolo Savo ^*

Optical Nanomaterial Group, Institute for Quantum Electronics, Department of Physics, ETH Zurich, Zurich, Switzerland

^*savor@phys.ethz.ch

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Issue

Vol. 15, Iss. 6 — June 2021

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Images

Figure 1
(a) Sketch of a three-dimensional disordered assembly of second-order nonlinear domains, consisting of multiple parallel one-dimensional sticks. (b) Each stick contains domains of varying size $X_{n}$ and orientation $(φ, ϑ, γ)_{n}$ , represented by the variation in color. (c) Depiction of the angles between the crystal frame ( $x$ , $y$ , $z$ ) and the laboratory frame ( $x^{'}$ , $y^{'}$ , $z^{'}$ ) with the transformation matrix $R$ . $β$ is the polarization angle of the input beam in case of a linearly polarized beam ( $ϕ_{x^{'}} = ϕ_{y^{'}}$ ). The ordinary and extraordinary axes are indicated with $o$ and $e$ . (d) Schematics of the process to propagate, generate and interfere the beams. The input electric fields (left) are transformed into the crystal frame and propagated to the end of the domain. They are then summed with the generated SHG from the current domain, and transformed back into the laboratory frame. (e) Phasor ( $a_{n}$ ) representation of the interference between the second-harmonic waves generated by the domains within the stick. Each stick corresponds to a single random walk in the SHG complex plane, in which the step length is the amplitude of the second-harmonic field.
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Figure 2
Scaling of the SHG intensities with the number of domains $N$ in birefringent ${Li Nb O}_{3}$ and in its isotropic analog at a pump wavelength of 930 nm. SHG intensities are averaged over 1000 sticks, leading to a linear growth. Its slope represents the SHG efficiency for a certain domain size, and the shaded area shows the possible efficiencies in monodispersed assemblies. The insets explicitly show the SHG intensity in monodispersed assemblies only as a function of the domain size $X$ for $N = 100$ . In polydispersed assemblies ( $σ = 30 %$ ) with large domains ( $\bar{X} > 3 L_{c}^{iso}$ in the isotropic and $\bar{X} > 8 L_{c}^{iso}$ in the birefringent material) the efficiency has a stable value corresponding to the dotted line. (a) The isotropic analog has zero-efficiency minima when $\bar{X}$ is an even multiple of $L_{c}^{iso}$ , and efficiency maxima when $\bar{X}$ is an odd multiple of $L_{c}^{iso}$ . (b) The birefringent assembly has its minimum at $\bar{X} = 9.25 L_{c}^{iso}$ and its maximum at $\bar{X} = 3.47 L_{c}^{iso}$ .
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Figure 3
(a) Scaling of the SHG intensity with the average domain size for disordered assemblies of isotropic ${Li Nb O}_{3}$ , birefringent ${Li Nb O}_{3}$ , and birefringent ${Li Nb O}_{3}$ in the single domain (SD) approximation (single-domain average $\times N$ ). The assemblies have a domain size polydispersity $σ = 30$ %, a fixed number of domains per stick $N = 100$ , an averaging over 1000 sticks, and are simulated at a pump wavelength of 930 nm. At this wavelength, phase matching in ${Li Nb O}_{3}$ is not possible. (b) Probability distributions of the SHG intensities generated by the individual domains within the assembly ( $\bar{X}$ = 3 $L_{c}^{iso}$ and $σ = 30$ %) for isotropic ${Li Nb O}_{3}$ , birefringent ${Li Nb O}_{3}$ , and birefringent ${Li Nb O}_{3}$ in the SD approximation ( $N = 1$ , linearly polarized input beam). Probabilities are calculated as the relative frequency of appearance of a SHG intensity value over $10^{7}$ domains. The SHG binning is 0.002. The SHG intensities are multiplied by 100 to be comparable with the intensities of the 100-domain assemblies in (a).
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Figure 4
Scaling of the SHG intensity with the average domain size $\bar{X}$ in disordered assemblies of ${Li Nb O}_{3}$ and ADP and their isotropic analogs at a pump wavelength of $λ = 1200$ nm. At this wavelength, phase matching is possible in both materials. The total length of the sticks is kept constant and the domains have a polydispersity $σ = 30$ %. (a) The total length of the sticks is fixed at 1000 $L_{c}^{iso}$ . When changing $X$ , the number of domains per stick varies between 1000 domains (at $\bar{X} = 1 L_{c}^{iso}$ ) to 10 domains (at $\bar{X} = 100 L_{c}^{iso}$ ). The phase-matchable materials show a different scaling in comparison to their not phase-matchable isotropic analogs. Non-phase-matchable ${Li Nb O}_{3}$ can also be simulated by using a shorter pump wavelength, which is shown in Sec. VI within the Supplemental Material [23]. (b) Higher-resolution simulation of the small-domain-size regime, performed at a reduced stick length of 50 $L_{c}^{iso}$ for computational efficiency. The shift in the $y$ scale compared to (a) is due to the different stick length. In this small-domain regime the scaling of the SHG intensity is not determined solely by the phase matchability, but becomes material specific.
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