# High-Fidelity Harmonic Generation in Optical Micro-Resonators Using BFGS Algorithm

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## Abstract

**:**

## 1. Introduction

## 2. Nonlinear Wave Interaction in Optical Micro-Resonators

_{i}is the polarization density component at the ith resonance frequency, γ

_{i}is the polarization damping rate associated with the ith resonance frequency, ω

_{i}is the ith angular resonance frequency, e is the electron charge, d is the atomic diameter, σ is the medium conductivity, and ${\epsilon}_{\infty}$ is the background permittivity. The number of electrons oscillating at each resonance is indicated by ${Q}_{i}$ and they are related to the number of electrons per unit volume (electron density) $Q$ via the oscillation strength ${\xi}_{i}$ such that

_{i}is the dipole moment at the ith resonance.

## 3. Non-Linear Programming for Efficient Harmonic Generation

#### 3.1. BFGS Algorithm-Based Optimization

#### 3.2. Determining the Penalty Weights

## 4. Numerical Simulations

#### 4.1. Simulation1: Intense Harmonic Generation in the Ultraviolet (UV) Frequency Range

- $\mathrm{Spatial}\mathrm{and}\mathrm{temporal}\mathrm{intervals}\mathrm{of}\mathrm{the}\mathrm{simulation}:0\le x\le 10\mathsf{\mu}\mathrm{m},0\le t\le 10\mathrm{ps}$
- $\mathrm{Resonance}\mathrm{frequencies}\mathrm{of}\mathrm{the}\mathrm{interaction}\mathrm{medium}:{f}_{r}=\left\{4\times {10}^{14}\mathrm{Hz},6.3\times {10}^{14}\mathrm{Hz},8.8\times {10}^{14}\mathrm{Hz}\right\}$
- $\mathrm{Polarization}\mathrm{damping}\mathrm{rates}\mathrm{of}\mathrm{the}\mathrm{interaction}\mathrm{medium}:\gamma =\left\{2\times {10}^{9}\mathrm{Hz},3\times {10}^{10}\mathrm{Hz},1\times {10}^{11}\mathrm{Hz}\right\}$
- $\mathrm{Relative}\mathrm{permittivity}\mathrm{of}\mathrm{the}\mathrm{interaction}\mathrm{medium}:\left({\epsilon}_{r}\right)=10({\mu}_{r}=1)$
- $\mathrm{Location}\mathrm{of}\mathrm{the}\mathrm{optical}\mathrm{isolator}:x=0\mathsf{\mu}\mathrm{m},\mathrm{Bandpass}\mathrm{filter}\mathrm{location}:x=10\mathsf{\mu}\mathrm{m}$
- $\mathrm{Spatial}\mathrm{range}\mathrm{of}\mathrm{the}\mathrm{interaction}\mathrm{material}:0\mathsf{\mu}\mathrm{m}x10\mathsf{\mu}\mathrm{m}$$\mathrm{Density}\mathrm{of}\mathrm{electrons}:Q=3.5\times \frac{{10}^{28}}{{m}^{3}},\mathrm{Atom}\mathrm{diameter}:d=0.3\mathrm{nanometers}$
- Resonance probabilities: $\xi =\left\{0.3,0.4,0.3\right\},\mathrm{Cos}\mathrm{t}\mathrm{function}\mathrm{to}\mathrm{be}\mathrm{maximized}:C$

**Bandpass filtering**: Frequency selective cavity wall (right port) is fixed at $\mathrm{x}=10\mathsf{\mu}\mathrm{m}$

**Cost function:**$C\left({\nu}_{1},{\nu}_{2}\right)=\left|E\left(\nu =820\mathrm{THz}\right)\right|-{{\displaystyle \sum}}_{i=1}^{2}\{{\delta}_{i,1}{\left({\nu}_{i}-{\nu}_{\mathrm{max}}\right)}^{2}+{\delta}_{i,2}{\left({\nu}_{\mathrm{min}}-{\nu}_{i}\right)}^{2}\}$

**FDTD Equations:**Discretization of Equations (1)–(4) via finite difference time domain is as follows [27,28,29,30]

- $P=$${P}_{1}+{P}_{2}+{P}_{3}$
- x: Space coordinate, t: Time, k: Iteration, ${E}_{k}\left(x,t\right)={E}_{k}\left(i\u2206x,j\u2206t\right)\to {E}_{k}\left(i,j\right)$

**Optimization via BFGS algorithm:**Choose the identity matrix as the initial Hessian matrix

- ${\mathit{p}}_{k}=-$${\mathit{H}}_{k}$$\mathit{\nabla}{\mathit{C}}_{k}$, ${\mathit{\nu}}_{\mathit{k}+1}={\mathit{\nu}}_{\mathit{k}}+{\mathit{\alpha}}_{\mathit{k}}$${\mathit{p}}_{k}$, ${\mathit{s}}_{k}={\mathit{v}}_{k+1}-$${\mathit{v}}_{k},$${\mathit{\nu}}_{\mathit{k}}=\left[\begin{array}{c}{\nu}_{1,k}\\ {\nu}_{2,k}\end{array}\right]$
- ${\mathit{y}}_{k}=$$\mathit{\nabla}{\mathit{C}}_{k+1}-$$\mathit{\nabla}{\mathit{C}}_{k}$${\rho}_{k}=\frac{1}{{y}_{k}{}^{T}{s}_{k}}$

^{8}V/m.

^{2}Hz) as seen in Figure 6.

#### 4.2. Simulation 2: Intense Quasi-Monochromatic Yellow-Light Generation Around 515THz

- $\mathrm{Spatial}\mathrm{and}\mathrm{temporal}\mathrm{intervals}\mathrm{of}\mathrm{the}\mathrm{simulation}:0\le x\le 10\mathsf{\mu}\mathrm{m},0\le t\le 10\mathrm{ps}$
- $\mathrm{Resonance}\mathrm{frequencies}\mathrm{of}\mathrm{the}\mathrm{interaction}\mathrm{medium}:{f}_{r}=\left\{3\times {10}^{14}\mathrm{Hz},4.4\times {10}^{14}\mathrm{Hz},6.3\times {10}^{14}\mathrm{Hz}\right\}$
- $\mathrm{Polarization}\mathrm{damping}\mathrm{rates}\mathrm{of}\mathrm{the}\mathrm{interaction}\mathrm{medium}:\gamma =\left\{1\times {10}^{10}\mathrm{Hz},2.5\times {10}^{10}\mathrm{Hz},1\times {10}^{11}\mathrm{Hz}\right\}$
- Resonance probabilities: $\xi =\left\{\frac{1}{3},\frac{1}{3},\frac{1}{3}\right\}$, $\mathrm{Permittivity}\mathrm{of}\mathrm{the}\mathrm{interaction}\mathrm{medium}:\left({\epsilon}_{r}\right)=12({\mu}_{r}=1)$
- $\mathrm{Location}\mathrm{of}\mathrm{the}\mathrm{optical}\mathrm{isolator}\left(\mathrm{input}\mathrm{port}\right):x=0\mathsf{\mu}\mathrm{m},\mathrm{Filter}\left(\mathrm{output}\mathrm{port}\right)\mathrm{location}:x=10\mathsf{\mu}\mathrm{m}$
- $\mathrm{Spatial}\mathrm{range}\mathrm{of}\mathrm{the}\mathrm{interaction}\mathrm{material}:0\mathsf{\mu}\mathrm{m}x10\mathsf{\mu}\mathrm{m}$, $\mathrm{Density}\mathrm{of}\mathrm{electrons}:N=3.5\times \frac{{10}^{28}}{{m}^{3}}$
- $\mathrm{Atom}\mathrm{diameter}:d=0.3\mathrm{nanometers},\mathrm{Cos}\mathrm{t}\mathrm{function}\mathrm{to}\mathrm{be}\mathrm{maximized}:C$

**Initial conditions of the electric field and polarization density**: (Prime sign refers to the time derivative)

**Band-pass filtering**: Frequency selective cavity wall (right port) is fixed at $\mathrm{x}=10\mathsf{\mu}\mathrm{m}$

**Cost function:**$C\left({\nu}_{1},{\nu}_{2}\right)=\left|E\left(\nu =515\mathrm{THz}\right)\right|-{{\displaystyle \sum}}_{i=1}^{2}\{{\delta}_{i,1}{\left({\nu}_{i}-{\nu}_{max}\right)}^{2}+{\delta}_{i,2}{\left({\nu}_{min}-{\nu}_{i}\right)}^{2}\}$

**Optimization via BFGS algorithm:**Choose the identity matrix as the initial Hessian matrix

- ${p}_{k}=-$${\mathit{H}}_{k}$$\mathit{\nabla}{\mathit{C}}_{k}$, ${\nu}_{k+1}={\nu}_{k}+{\alpha}_{k}$${\mathit{p}}_{k}$, ${\mathit{s}}_{k}={\mathit{f}}_{p,k+1}-$${\mathit{f}}_{p,k},$${\mathit{\nu}}_{\mathit{k}}=\left[\begin{array}{c}{\nu}_{1,k}\\ {\nu}_{2,k}\end{array}\right]$, ${\mathit{y}}_{k}=$$\mathit{\nabla}{\mathit{C}}_{k+1}-$$\mathit{\nabla}{\mathit{C}}_{k}$, ${\rho}_{k}=\frac{1}{{y}_{k}{}^{T}{s}_{k}}$
- BFGS recursion:${\mathit{H}}_{k+1}=\left(\mathit{I}-{\rho}_{k}{\mathit{s}}_{k}{\mathit{y}}_{k}{}^{T}\right){\mathit{H}}_{k}\left(\mathit{I}-{\rho}_{k}{\mathit{y}}_{k}{\mathit{s}}_{k}{}^{T}\right)+{\rho}_{k}{\mathit{s}}_{k}{\mathit{s}}_{k}{}^{T}$$\mathit{I}:\mathrm{Identity}\mathrm{matrix}$

^{8}V/m.

^{2}Hz) as seen in Figure 10.

## 5. Testing the Model Accuracy via Comparison with Experimental Results

#### Example 5.1: Second Harmonic Generation by Nonlinear Wave Mixing

- ${\mathit{E}}_{1}\left(x=2.4\mathsf{\mu}\mathrm{m},t\right)={A}_{1}\times \mathrm{sin}\left(2\pi \left(1\times {10}^{14}\right)t+{\phi}_{1}\right)\mathrm{V}/\mathrm{m}$$\left({\phi}_{1}=0\right)$
- Spatial range and duration of the computation: $0\le x\le 10\mathsf{\mu}\mathrm{m},0\le t\le 30\mathrm{ps}$
- $\mathrm{Resonance}\mathrm{frequencies}\mathrm{of}\mathrm{the}\mathrm{interaction}\mathrm{medium}:{\mathit{f}}_{\mathit{r}}=\left\{7.8\times {10}^{14}\mathrm{Hz},9.5\times {10}^{14}\mathrm{Hz},1.4\times {10}^{15}\mathrm{Hz}\right\}$
- $\mathrm{Damping}\mathrm{coefficients}\mathrm{of}\mathrm{the}\mathrm{interaction}\mathrm{medium}:\mathit{\gamma}=\left\{4\times {10}^{12}\mathrm{Hz},3\times {10}^{12}\mathrm{Hz},1\times {10}^{12}\mathrm{Hz}\right\}$
- $\mathrm{Interaction}\mathrm{medium}\mathrm{background}\mathrm{permitttivity}\left({\epsilon}_{\infty}\right)=1+\chi =12({\mu}_{r}=1)$
- $\mathrm{Left}\mathrm{absorption}\mathrm{domain}\mathrm{ranges}\mathrm{from}x=0tox=2.35\mathsf{\mu}\mathrm{m}\left(\mathrm{absorbing}\mathrm{boundary}\right)$
- $\mathrm{Right}\mathrm{absorption}\mathrm{domain}\mathrm{ranges}\mathrm{from}x=7.65\mathsf{\mu}\mathrm{m}tox=10\mathsf{\mu}\mathrm{m}\left(\mathrm{absorbing}\mathrm{boundary}\right)$

- d = Nonlinearity coefficient, $\eta $ = Medium impedance, n = Index of refraction
- ${A}_{1}$ = Input wave amplitude, L = Length of the medium

- L = Length of the interaction medium = 3.33 $\mathsf{\mu}\mathrm{m}$ (ranging from x = 3.33 $\mathsf{\mu}\mathrm{m}$ to 6.66 $\mathsf{\mu}\mathrm{m}$)
- ${\omega}_{2}=\mathrm{Second}\mathrm{harmonic}\mathrm{angular}\mathrm{frequency}=2\pi \times 200\mathrm{THz},$n$=\sqrt{12}$
- ${A}_{1}$ = Input wave amplitude (Varied from ${10}^{8}\frac{\mathrm{V}}{\mathrm{m}}to2.5\times {10}^{9}\mathrm{V}/\mathrm{m}$, in increment of ${10}^{8}\mathrm{V}/\mathrm{m}$)

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Chung, I.; Song, J.H.; Jang, J.I.; Freeman, A.J.; Kanatzidis, M.G. Na
_{2}Ge_{2}Se_{5}: A highly nonlinear optical material. J. Solid State Chem.**2012**, 195, 161–165. [Google Scholar] [CrossRef] - Almeida, G.F.; Santos, S.N.; Siqueira, J.P.; Dipold, J.; Voss, T.; Mendonça, C.R. Third-Order Nonlinear Spectrum of GaN under Femtosecond-Pulse Excitation from the Visible to the Near Infrared. Photonics J.
**2019**, 6, 69. [Google Scholar] [CrossRef] [Green Version] - Rout, A.; Boltaev, G.S.; Ganeev, R.A.; Fu, Y.; Maurya, S.K.; Kim, V.V.; Rao, K.S.; Guo, C. Nonlinear Optical Studies of Gold Nanoparticle Films. Nanomaterials
**2019**, 9, 291. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Wu, R.; Collins, J.; Canham, L.T.; Kaplan, A. The Influence of Quantum Confinement on Third-Order Nonlinearities in Porous Silicon Thin Films. Appl. Sci.
**2018**, 8, 1810. [Google Scholar] [CrossRef] [Green Version] - Sakhno, O.; Yezhov, P.; Hryn, V.; Rudenko, V.; Smirnova, T. Optical and Nonlinear Properties of Photonic Polymer Nanocomposites and Holographic Gratings Modified with Noble Metal Nanoparticles. Polymers
**2020**, 12, 480. [Google Scholar] [CrossRef] [Green Version] - Reinke, C.M.; Jafarpour, A.; Momeni, B.; Soltani, M.; Khorasani, S.; Adibi, A.; Xu, Y.; Lee, R.K. Nonlinear finite-difference time-domain method for the simulation of anisotropic, /spl chi//sup (2)/, and /spl chi//sup (3)/ optical effects. J. Lightwave Technol.
**2006**, 24, 624–634. [Google Scholar] [CrossRef] - Varin, C.; Emms, R.; Bart, G.; Fennel, T.; Brabec, T. Explicit formulation of second and third order optical nonlinearity in the FDTD framework. Comput. Phys. Commun.
**2018**, 222, 70–83. [Google Scholar] [CrossRef] [Green Version] - Zygiridis, T.T.; Kantartzis, N.V. Finite-Difference Modeling of Nonlinear Phenomena in Time-Domain Electromagnetics: A Review. In Applications of Nonlinear Analysis. Springer Optimization and Its Applications; Rassias, T., Ed.; Springer: Cham, The Netherland, 2018; Volume 134. [Google Scholar]
- Sahakyan, A.T.; Starodub, A.N. A simplified formula for calculation of second-harmonic generation efficiency for type I synchronism. In Journal of Physics: Conference Series; IOP Science: Bristol, UK, 2019. [Google Scholar]
- Xu, L.; Rahmani, M.; Smirnova, D.; Zangeneh Kamali, K.; Zhang, G.; Neshev, D.; Miroshnichenko, A.E. Highly-Efficient Longitudinal Second-Harmonic Generation from Doubly-Resonant AlGaAs Nanoantennas. Photonics
**2018**, 5, 29. [Google Scholar] [CrossRef] [Green Version] - De Ceglia, D.; Carletti, L.; Vincenti, M.A.; De Angelis, C.; Scalora, M. Second-Harmonic Generation in Mie-Resonant GaAs Nanowires. Appl. Sci.
**2019**, 9, 3381. [Google Scholar] [CrossRef] [Green Version] - Rocco, D.; Vincenti, M.A.; De Angelis, C. Boosting Second Harmonic Radiation from AlGaAs Nanoantennas with Epsilon-Near-Zero Materials. Appl. Sci.
**2018**, 8, 2212. [Google Scholar] [CrossRef] [Green Version] - Nguyen, D.T.T.; Lai, N.D. Deterministic Insertion of KTP Nanoparticles into Polymeric Structures for Efficient Second-Harmonic Generation. Crystals
**2019**, 9, 365. [Google Scholar] [CrossRef] [Green Version] - Huang, Z.; Lu, H.; Xiong, H.; Li, Y.; Chen, H.; Qiu, W.; Guan, H.; Dong, J.; Zhu, W.; Yu, J.; et al. Fano Resonance on Nanostructured Lithium Niobate for Highly Efficient and Tunable Second Harmonic Generation. Nanomaterials
**2019**, 9, 69. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Cheng, T.; Xiao, Y.; Li, S.; Yan, X.; Zhang, X.; Suzuki, T.; Ohishi, Y. Highly efficient second-harmonic generation in a tellurite optical fiber. Opt. Lett.
**2019**, 44, 4686–4689. [Google Scholar] [CrossRef] [PubMed] - Alsunaidi, M.A.; Al-Hajiri, F.S. Efficient NL-FDTD Solution Schemes for the Phase-Sensitive Second Harmonic Generation Problem. J. Lightwave Technol.
**2009**, 27, 4964–4969. [Google Scholar] [CrossRef] - Reed, M.K.; Steiner-Shepard, M.K.; Armas, M.S.; Negus, D.K. Microjoule-energy ultrafast optical parametric amplifiers. J. Opt. Soc. Am. B
**1995**, 12, 2229–2236. [Google Scholar] [CrossRef] - Ciriolo, A.G.; Negro, M.; Devetta, M.; Cinquanta, E.; Faccialà, D.; Pusala, A.; De Silvestri, S.; Stagira, S.; Vozzi, C. Optical Parametric Amplification Techniques for the Generation of High-Energy Few-Optical-Cycles IR Pulses for Strong Field Applications. Appl. Sci.
**2017**, 7, 265. [Google Scholar] [CrossRef] [Green Version] - Migal, E.A.; Potemkin, F.V.; Gordienko, V.M. Highly efficient optical parametric amplifier tunable from near-to mid-IR for driving extreme nonlinear optics in solids. Opt. Lett.
**2017**, 42, 5218–5221. [Google Scholar] [CrossRef] - Wu, C.; Fan, J.; Chen, G.; Jia, S. Symmetry-breaking-induced dynamics in a nonlinear microresonator. Opt. Express
**2019**, 27, 28133–28142. [Google Scholar] [CrossRef] - Aşırım, Ö.E.; Kuzuoğlu, M. Numerical Study of Resonant Optical Parametric Amplification via Gain Factor Optimization in Dispersive Micro-resonators. Photonics
**2020**, 7, 5. [Google Scholar] [CrossRef] [Green Version] - Aşırım, Ö.E.; Kuzuoglu, M. Enhancement of Optical Parametric Amplification in Micro-resonators via Gain Medium Parameter Selection and Mean Cavity Wall Reflectivity Adjustment. J. Phys. B At. Mol. Opt. Phys.
**2020**, 53, 7–12. [Google Scholar] - Aşırım, Ö.E.; Kuzuoğlu, M. Super-Gain Optical Parametric Amplification in Dielectric Micro-Resonators via BFGS Algorithm-Based Non-Linear Programming. Appl. Sci.
**2020**, 10, 1770. [Google Scholar] [CrossRef] [Green Version] - Hasan, M.H.; Alsaleem, F.; Ramini, A. Voltage and deflection amplification via double resonance excitation in a cantilever microstructure. Sensors
**2019**, 19, 380. [Google Scholar] [CrossRef] [Green Version] - Ramini, A.; Ibrahim, A.I.; Younis, M.I. Mixed frequency excitation of an electrostatically actuated resonator. Microsyst. Technol.
**2016**, 22, 1967–1974. [Google Scholar] [CrossRef] - Jaber, N.; Ramini, A.; Younis, M.I. Multifrequency excitation of a clamped–clamped microbeam: Analytical and experimental investigation. Microsyst. Nanoeng.
**2016**, 2, 16002. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Shi, Y.; Shin, W.; Fan, S. Multi-frequency finite-difference frequencydomain algorithm for active nanophotonic device simulations. Optica
**2016**, 3, 1256–1259. [Google Scholar] [CrossRef] - Ji, H.; Zhang, B.; Wang, G.; Wang, W.; Shen, J. Photo-excited multifrequency terahertz switch based on a composite metamaterial structure. Opt. Commun.
**2018**, 412, 37–40. [Google Scholar] [CrossRef] - Asfaw, A.T.; Sigillito, A.J.; Tyryshkin, A.M.; Schenkel, T.; Lyon, S.A. Multifrequency spin manipulation using rapidly tunable superconducting coplanar waveguide microresonators. Appl. Phys. Lett.
**2017**, 111, 032601. [Google Scholar] [CrossRef] [Green Version] - Ogata, Y.; Vorobyev, A.; Guo, C. Optical Third Harmonic Generation Using Nickel Nanostructure-Covered Microcube Structures. Materials
**2018**, 11, 501. [Google Scholar] [CrossRef] [Green Version] - Chen, S.; Li, K.F.; Li, G.; Cheah, K.W.; Zhang, S. Gigantic electric-field-induced second harmonic generation from an organic conjugated polymer enhanced by a band-edge effect. Light Sci. Appl.
**2019**, 8, 17. [Google Scholar] [CrossRef] - Luo, R.; Jiang, H.; Rogers, S.; Liang, H.; He, Y.; Lin, Q. On-chip second-harmonic generation and broadband parametric down-conversion in a lithium niobate microresonator. Opt. Express
**2017**, 25, 24531–24539. [Google Scholar] [CrossRef] - Roland, I.; Gromovyi, M.; Zeng, Y.; El Kurdi, M.; Sauvage, S.; Brimont, C.; Guillet, T.; Gayral, B.; Semond, F.; Duboz, J.Y.; et al. Phase-matched second harmonic generation with on-chip GaN-on-Si microdisks. Sci. Rep.
**2016**, 6, 34191. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Lin, J.; Yao, N.; Hao, Z.; Zhang, J.; Mao, W.; Wang, M.; Chu, W.; Wu, R.; Fang, Z.; Qiao, L.; et al. Broadband Quasi-Phase-Matched Harmonic Generation in an On-Chip Monocrystalline Lithium Niobate Microdisk Resonator. Phys. Rev. Lett.
**2019**, 122, 173903. [Google Scholar] [CrossRef] [PubMed] - Ciriolo, A.G.; Vázquez, R.M.; Tosa, V.; Frezzotti, A.; Crippa, G.; Devetta, M.; Faccialá, D.; Frassetto, F.; Poletto, L.; Pusala, A.; et al. High-order harmonic generation in a microfluidic glass device. J. Phys. Photonics
**2020**, 2, 024005. [Google Scholar] [CrossRef] - Lin, J.; Xu, Y.; Fang, Z.; Wang, M.; Fang, W.; Cheng, Y. Efficient second harmonic generation in an on-chip high-Q crystalline microresonator fabricated by femtosecond laser. In Proceedings of the SPIE, Laser Resonators, Microresonators, and Beam Control XVIII, San Francisco, CA, USA, 22 April 2016; Volume 9727, p. 972710. [Google Scholar] [CrossRef]
- Soavi, G.; Wang, G.; Rostami, H.; Purdie, D.G.; De Fazio, D.; Ma, T.; Luo, B.; Wang, J.; Ott, A.K.; Yoon, D.; et al. Broadband, electrically tunable third-harmonic generation in graphene. Nat. Nanotechnol.
**2018**, 13, 583–588. Available online: www.nature.com/articles/s41565-018-0145-8 (accessed on 22 April 2020). [CrossRef] [PubMed] [Green Version] - Yoshikawa, N.; Tamaya, T.; Tanaka, K. High-harmonic generation in graphene enhanced by elliptically polarized light excitation. Science
**2017**, 365, 736–738. Available online: science.sciencemag.org/content/356/6339/736 (accessed on 22 April 2020). [CrossRef] - Mateen, F.; Boales, J.; Erramilli, S.; Mohanty, P. Micromechanical resonator with dielectric nonlinearity. Microsyst. Nanoeng. Nat.
**2018**, 4, 14. Available online: www.nature.com/articles/s41378-018-0013-6 (accessed on 22 April 2020). [CrossRef] - Boyd, R.W. Nonlinear Optics; Academic Press: New York, NY, USA, 2008; pp. 105–107. [Google Scholar]
- Mark, F. Optical Properties of Solids; Oxford University Press: New York, NY, USA, 2002; pp. 237–239. [Google Scholar]
- Bahaa, E.A. Saleh, Malvin Carl Teich, Fundamentals of Photonics; Wiley-Interscience: New York, NY, USA, 2007; pp. 885–917. [Google Scholar]
- Silfvast, W.T. Laser Fundamentals; Cambridge University Press: New York, NY, USA, 2004; pp. 24–35. [Google Scholar]
- Satsuma, J.; Yajima, N. Initial Value Problems of One-Dimensional Self-Modulation of Nonlinear Waves in Dispersive Media. Prog. Theor. Phys. Suppl.
**1974**, 55, 284–306. [Google Scholar] [CrossRef] - Taflove, A.; Hagness, S.C. Computational Electrodynamics: The Finite-Difference Time-Domain Method; Artech House: Boston, MA, USA, 2005; pp. 353–361. [Google Scholar]

**Figure 1.**Excitation of an optical microcavity using M different ultrashort pulses, whose frequencies are to be tuned for high-intensity targeted harmonic generation via intracavity energy maximization.

**Figure 3.**Nonlinear mixing of two ultrashort pulses in an optical microcavity, whose frequencies were tuned for high-intensity targeted harmonic generation at 820THz via intracavity energy maximization.

**Figure 4.**Time variation of the maximum electric field amplitude at the bandpass filter output (820 THz).

**Figure 5.**Spectrum of the total wave inside the cavity measured at x = 5.73 µm, t = 10 ps, at (

**a**) 6th, (

**b**) 8th, (

**c**) 10th, and (

**d**) 13th iteration of the optimization process for harmonic generation at 820 THz.

**Figure 6.**Intensity spectral density at the band-pass filter output for harmonic generation at 820 THz.

**Figure 7.**Nonlinear mixing of two ultrashort pulses in an optical microcavity, whose frequencies were tuned for high-intensity targeted harmonic generation at 515THz.

**Figure 8.**Time variation of the maximum amplitude at the band-pass filter output (harmonic generation at 515 THz).

**Figure 9.**Spectrum of the total wave inside the cavity, measured at x = 5.73 µm, t = 10 ps, at (

**a**) 6th, (

**b**) 14th, (

**c**) 24th, and (

**d**) 33rd iteration of the optimization process for harmonic generation at 515 THz.

**Figure 10.**Intensity spectral density at the band-pass filter output for generation of a wave at 515 THz.

**Figure 12.**Comparison of the theoretical and the computational second harmonic generation efficiencies for ${\mathrm{f}}_{\mathrm{input}}$ = 100 THz and d = $1.21\times {10}^{-21}$ C/V, versus the source wave amplitude.

$\left|{\mathit{E}}_{\mathit{\nu}=820\mathbf{THz}}\right|$ | ${\mathit{\nu}}_{1}$ | ${\mathit{\nu}}_{2}$ | ${\mathit{W}}_{\mathit{e},\mathit{p}}\left(\frac{\mathit{J}}{{\mathit{m}}^{3}}\right)$ | ${\mathit{P}}_{\mathit{p}\mathit{u}\mathit{m}\mathit{p}}\left(\frac{\mathit{C}}{{\mathit{m}}^{2}}\right)$ | k (Iteration #) |
---|---|---|---|---|---|

5.8 $\times {10}^{3}\mathrm{V}/\mathrm{m}$ | 250 THz | 225 THz | 2.9 $\times {10}^{7}$ | 0.09 | 1 |

2.9 $\times {10}^{3}\mathrm{V}/\mathrm{m}$ | 245 THz | 220 THz | 1 $\times {10}^{7}$ | 0.05 | 2 |

2.7 $\times {10}^{5}\mathrm{V}/\mathrm{m}$ | 249.4 THz | 226.1 THz | 8.66 $\times {10}^{7}$ | 0.14 | 3 |

2.0 $\times {10}^{4}\mathrm{V}/\mathrm{m}$ | 257.2 THz | 231.3 THz | 5.49 $\times {10}^{7}$ | 0.13 | 4 |

1.3 $\times {10}^{3}\mathrm{V}/\mathrm{m}$ | 264.9 THz | 226.6 THz | 1.31 $\times {10}^{7}$ | 0.06 | 5 |

7.6 $\times {10}^{3}\mathrm{V}/\mathrm{m}$ | 276.0 THz | 222.2 THz | 1.76 $\times {10}^{7}$ | 0.07 | 6 |

1.4 $\times {10}^{4}\mathrm{V}/\mathrm{m}$ | 286.7 THz | 231.8 THz | 1.73 $\times {10}^{7}$ | 0.07 | 7 |

1.1 $\times {10}^{6}\mathrm{V}/\mathrm{m}$ | 279.7 THz | 218.0 THz | 3.06 $\times {10}^{7}$ | 0.10 | 8 |

9.8 $\times {10}^{5}\mathrm{V}/\mathrm{m}$ | 272.1 THz | 236.9 THz | 2.49 $\times {10}^{7}$ | 0.10 | 9 |

5.1 $\times {10}^{6}\mathrm{V}/\mathrm{m}$ | 263.5 THz | 285.1 THz | 3.30 $\times {10}^{7}$ | 0.10 | 10 |

1.2 $\times {10}^{7}\mathrm{V}/\mathrm{m}$ | 273.4 THz | 310.7 THz | 8.21 $\times {10}^{7}$ | 0.16 | 11 |

3.2 $\times {10}^{6}\mathrm{V}/\mathrm{m}$ | 275.1 THz | 288.8 THz | 3.48 $\times {10}^{7}$ | 0.12 | 12 |

7.6 $\times {10}^{7}\mathrm{V}/\mathrm{m}$ | 273.2 THz | 284.7 THz | 5.75 $\times {10}^{7}$ | 0.17 | 13 |

$\left|{\mathit{E}}_{\mathit{\nu}=515\mathbf{THz}}\right|$ | ${\mathit{\nu}}_{1}$ | ${\mathit{\nu}}_{2}$ | ${\mathit{W}}_{\mathit{e},\mathit{p}}\left(\frac{\mathit{J}}{{\mathit{m}}^{3}}\right)$ | ${\mathit{P}}_{\mathit{p}\mathit{u}\mathit{m}\mathit{p}}\left(\frac{\mathit{C}}{{\mathit{m}}^{2}}\right)$ | k (iteration #) |
---|---|---|---|---|---|

1.9 $\times {10}^{3}\mathrm{V}/\mathrm{m}$ | 250 THz | 225 THz | 2.4 $\times {10}^{7}$ | 0.20 | 1 |

2.6 $\times {10}^{3}\mathrm{V}/\mathrm{m}$ | 245 THz | 220 THz | $2.5\times {10}^{7}$ | 0.18 | 2 |

$3\times {10}^{3}\mathrm{V}/\mathrm{m}$ | 248.6 THz | 226.1 THz | 2.2 $\times {10}^{7}$ | 0.19 | 4 |

$5\times {10}^{3}\mathrm{V}/\mathrm{m}$ | 243.8 THz | 258.5 THz | 2.7 $\times {10}^{7}$ | 0.18 | 6 |

$7\times {10}^{3}\mathrm{V}/\mathrm{m}$ | 234.3 THz | 288.7 THz | 2.3 $\times {10}^{7}$ | 0.18 | 9 |

1.3 $\times {10}^{4}\mathrm{V}/\mathrm{m}$ | 249.8 THz | 341.4 THz | 3.5 $\times {10}^{7}$ | 0.18 | 12 |

3.4 $\times {10}^{4}\mathrm{V}/\mathrm{m}$ | 259.9 THz | 331.9 THz | 5.2 $\times {10}^{7}$ | 0.21 | 15 |

2.2 $\times {10}^{5}\mathrm{V}/\mathrm{m}$ | 272.8 THz | 363.6 THz | 4.6 $\times {10}^{7}$ | 0.21 | 18 |

9.4 $\times {10}^{5}\mathrm{V}/\mathrm{m}$ | 277.7 THz | 411.1 THz | 3.9 $\times {10}^{7}$ | 0.18 | 21 |

4.3 $\times {10}^{6}\mathrm{V}/\mathrm{m}$ | 270.4 THz | 448.2 THz | 6.5 $\times {10}^{7}$ | 0.18 | 24 |

1.1 $\times {10}^{7}\mathrm{V}/\mathrm{m}$ | 261.5 THz | 475.3 THz | 8.8 $\times {10}^{7}$ | 0.20 | 27 |

2.7 $\times {10}^{7}\mathrm{V}/\mathrm{m}$ | 265.3 THz | 475.8 THz | 7.1 $\times {10}^{7}$ | 0.22 | 30 |

9.1 $\times {10}^{7}\mathrm{V}/\mathrm{m}$ | 266.2 THz | 476.5 THz | $7.7\times {10}^{7}$ | 0.22 | 33 |

**Table 3.**Comparison of the numerical and experimental results for different excitation amplitudes with the estimated nonlinearity coefficient of ${d}_{est}=1.21\times {10}^{-21}$ C/V.

Excitation Wave Amplitude (V/m) | Theoretical Efficiency | Numerical Efficiency | Error Percentage |
---|---|---|---|

$1\times {10}^{8}$ | $1.93\times {10}^{-5}$ | $1.92\times {10}^{-5}$ | 0.5 |

$3\times {10}^{8}$ | $1.74\times {10}^{-4}$ | $1.74\times {10}^{-4}$ | 0.4 |

$5\times {10}^{8}$ | $4.82\times {10}^{-4}$ | $4.88\times {10}^{-4}$ | 1.2 |

$1\times {10}^{9}$ | $1.93\times {10}^{-3}$ | $1.96\times {10}^{-3}$ | 1.55 |

$1.5\times {10}^{9}$ | $4.33\times {10}^{-3}$ | $4.39\times {10}^{-3}$ | 1.38 |

$2\times {10}^{9}$ | $7.68\times {10}^{-3}$ | $7.76\times {10}^{-3}$ | 1.04 |

$2.5\times {10}^{9}$ | $1.20\times {10}^{-2}$ | $1.21\times {10}^{-2}$ | 0.83 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Aşırım, Ö.E.; Yolalmaz, A.; Kuzuoğlu, M.
High-Fidelity Harmonic Generation in Optical Micro-Resonators Using BFGS Algorithm. *Micromachines* **2020**, *11*, 686.
https://doi.org/10.3390/mi11070686

**AMA Style**

Aşırım ÖE, Yolalmaz A, Kuzuoğlu M.
High-Fidelity Harmonic Generation in Optical Micro-Resonators Using BFGS Algorithm. *Micromachines*. 2020; 11(7):686.
https://doi.org/10.3390/mi11070686

**Chicago/Turabian Style**

Aşırım, Özüm Emre, Alim Yolalmaz, and Mustafa Kuzuoğlu.
2020. "High-Fidelity Harmonic Generation in Optical Micro-Resonators Using BFGS Algorithm" *Micromachines* 11, no. 7: 686.
https://doi.org/10.3390/mi11070686