Prediksi Efisien dan Analisis Perangkap Optik pada Skala Nano melalui Metode Robek dan Interkoneksi Elemen Hingga

Hasil dan Diskusi

Tiga contoh disajikan untuk menunjukkan efektivitas metode yang diusulkan. Karena logam mulia biasanya digunakan untuk merangsang plasmon permukaan, kami memilih bahan emas dan perak yang representatif untuk analisis. Contoh pertama menghitung gaya optik nanopartikel perak untuk memverifikasi keakuratan metode yang diusulkan. Contoh kedua dan ketiga mensimulasikan dan mendiskusikan perangkap optik nanopartikel emas. Untuk semua contoh, domain tak hingga dipotong dengan ABC, dan jarak antara ABC dan objek diatur menjadi satu panjang gelombang, yang cukup untuk mencapai akurasi yang dapat diterima. Semua perhitungan dilakukan pada workstation Dell yang dilengkapi dengan prosesor Intel Xeon 3,6 GHz.

Nanokapsul Perak

Objek nanokapsul perak pertama kali dipertimbangkan untuk menguji akurasi dan efisiensi metode FETI-DP yang diusulkan dalam memprediksi gaya optik. Gambar 3 a dan b menyajikan konfigurasi dan dimensinya. Parameter konstitutif perak adalah semua nilai terukur yang diambil dari [32]. Untuk mengimplementasikan skema FETI-DP, seluruh domain analisis terlebih dahulu dibagi menjadi 24 subdomain. Jaring yang lebih padat diperlukan di dekat permukaan logam untuk memodelkan efek peningkatan medan lokal plasmonik. Elemen tetrahedral diadopsi untuk diskritisasi, yang menghasilkan total 6,9 × 10 ⁵ tidak diketahui, termasuk 4.1 × 10 ⁴ ganda tidak diketahui dan 313 sudut tidak diketahui. Lampu insiden menerangi sepanjang arah +z , sedangkan arah polarisasi medan listrik adalah x .

Konfigurasi struktur nanokapsul perak. a tampilan 3D. b Tampilan depan dan dimensi, di mana R = 30 nm dan h = 60 nm

Pertama, kita ubah panjang gelombang cahaya datang λ dari 200 nm hingga 400 nm untuk mensimulasikan gaya optik yang diberikan pada nanokapsul. Karena FETI-DP bekerja dalam domain frekuensi, gaya optik dihitung pada 15 titik frekuensi sampling. Gambar 4 menunjukkan kurva terhitung gaya optik yang diberikan pada nanokapsul perak. Untuk menunjukkan keakuratan FETI-DP, hasil gaya optik FETI-DP dibandingkan dengan perangkat lunak komersial Lumerical FDTD Solutions [33], dan kesepakatan yang baik dapat diamati.

Hasil gaya optik yang diberikan pada nanokapsul perak, bervariasi dengan panjang gelombang λ cahaya insiden, termasuk hasil FETI-DP dan solusi FDTD perangkat lunak komersial

Then, the performance of FETI-DP is tested for different numbers of subdomains. We increase the number of subdomains from 4 to 24 by keeping the discretization density. We assign each processor to deal with one subdomain. Table 1 reports the time used for the construction of global interface Eq. (6) and the total solution time. It can be seen that the FETI-DP can fully exploit parallel computing resources and significantly improve the solution efficiency. Besides, the accuracy of the FETI-DP with the number of subdomains increasing is also examined and reported in Table 1. Here, the accuracy is defined by the 2-norm relative error of the optical force as δ _OF  = ‖OF _i  − OF _ref ‖/‖OF _ref ‖, where OF _i is the optical force using i subdomains and OF _ref denotes the reference optical force using two subdomains. It can be seen that the accuracy keeps almost constant with the number of subdomains increasing.

Gold Nanosphere Dimer

The second example analyzes the optical trapping of a gold nanosphere by using a gold nanosphere dimer. The plasmonic effects at the dimer gap can effectively enhance the optical force for trapping nanoparticle. Figure 5 a and b gives the configuration and dimensions of this system. The constitutive parameters of gold are all measured values taken from [32]. The surrounding medium is water with a relative refractive index of n  = 1.33. The incident light is a plane wave with the power of 10 mW/μm ² , the electric field polarization direction is +x , and the incident direction is −z . The optical force exerted on the object nanosphere is calculated by the FETI-DP method. For the FETI-DP implementation, the whole computational domain is divided into 32 subdomains and discretized by tetrahedral meshes, which results in 3.5 × 10 ⁶ unknowns, including 1.6 × 10 ⁵ dual unknowns and 1738 corner unknowns.

Configuration of an optical trapping system of a gold nansphere dimer in water. a 3D view. b Front view and dimensions, where R = 25 nm, r = 5 nm, and g = 2 nm

First, we test the parallel performance of the proposed FETI-DP by using various numbers of processors. Table 2 reports the solution time for Eq. (6) as well as the total solution time. Besides, the speedups for the parallel computation are also provided in Table 2. Here, the speedup is defined by

where \( {T}_{N_p} \) denotes the total wall-clock time using N _p processors. It can be seen that the FETI-DP significantly improves the solution efficiency and exhibits good parallel speedup. For this large number of unknowns, the total memory usage of all the processors is only 57.2 GB.

Then, the effectiveness of the low-rank sparsification approach is examined. With the low-rank sparsification, the subdomain matrix can be factorized by data-sparse algorithm and stored as data-sparse matrices. The construction time and memory usage are only 18 s and 0.5 GB, while they are 67 s and 1.7 GB by conventional matrix algorithm. It can be seen that we get 72% time saving and 70% memory compression. Related to the memory usage, the subsequent MVPs can also get 70% time-saving.

Next, the FETI-DP is tested for the optical force calculation with the wavelength λ varying from 277 nm to 818 nm. In practice, the analyses of optical force under incident light of different wavelengths are often necessary for searching the plasmonic resonance wavelength, where drastic field enhancement occurs and the strongest optical force can be obtained. Two cases are considered with the nanosphere located at (0, 0, 20 nm) and (0, 0, − 20 nm). Figure 6 a and b plots the calculated optical forces exerted on the nanosphere for different λ . It can be seen that the maximum optical force occurs at λ  = 472 nm, which is the plasmonic resonance wavelength. The optical force at this resonance wavelength enhanced by nearly 40 times as against that at non-resonance wavelength. Moreover, the optical force always points to the dimer gap, as shown in Fig. 6, where the electric field intensity is strongest. It is also the direction of gradient force to trap the object. Figure 7 a and b shows the calculated electric field enhancement distributions at the non-resonance wavelength of λ = 300 nm and the resonance wavelength of λ = 472 nm, respectively. It can be seen that the electric field intensity has been increased by almost 250 times due to the plasmonic resonance effect.

Calculated results of optical forces exerted on the nanosphere in the system of gold nanosphere dimer, varying with the wavelength λ of incident light. a The object nanosphere is located at (0, 0, 20 nm). b The object nanosphere is located at (0, 0, − 20 nm)

The electric field enhancement distributions on the xoz plane for the system of gold nanosphere dimer. a λ = 300 nm (non-resonance wavelength). b λ  = 472 nm (resonance wavelength)

Besides, the optical force and optical potential are calculated with the nanosphere moving from (0, 0, − 30 nm) to (0, 0, − 17 nm) along the z -sumbu. Since the most typical and interesting behavior of trapping forces and potentials are those acting along z -direction, we here consider the axial trapping potential by integration along the z -sumbu. Because the motion of the nanoparticle is restricted to one dimension, the definition of an optical potential is unambiguous from (16), even though the total optical force from (15) is non-conservative. As shown in Fig. 8 a, b, with the nanosphere moving to the dimer gap, the optical force and optical potential depth obviously increase. At the position of (0, 0, − 17 nm), an optical potential depth of 4.6 k _B T is produced, which is sufficient to overcome the Brownian motion in water to achieve stable optical trapping.

The optical forces and optical potentials exerted on the nanosphere in the system of gold nanosphere dimer, when the nanosphere moves from (0, 0, − 30 nm) to (0, 0, − 17 nm). a The optical forces. b The optical potentials

Finally, we test the effects of the dielectric substrate for this example. The optical forces are calculated with and without a substrate, respectively. For both two cases, the nanosphere is located at (0, 0, − 20 nm) and the incident wavelength is chosen as the resonance wavelength. For the case without substrate, the calculated result of the optical force is |F ₀ | = 0.769 pN. For the case with a substrate, the gold nanosphere dimer is put on a dielectric substrate with a thickness of 60 nm and a relative permittivity of ε _r = 2.25. The calculated result of the optical force is |F ₁ | = 0.761 pN. The relative error between these two results of optical forces is about 1.0 × 10 ⁻² , which is defined as |F ₁  − F ₀ |/|F ₀ |.

Gold Truncated Cone Dimer

The third example deals with the optical trapping of a gold nanosphere by using a gold truncated cone dimer. Figure 9 gives the configuration and dimensions of this system. The constitutive parameters of gold are taken from [32]. The dielectric substrate has a relative permittivity of ε _r  = 2.25. The surrounding medium is water with a relative refractive index of n  = 1.33. The incident light is plane wave with the power of 10 mW/μm ² , the electric field polarization direction is +x , and the incident direction is −z . The whole computational domain is divided into 32 subdomains and discretized by tetrahedral meshes, which leads to 3.1 × 10 ⁶ unknowns, including 1.3 × 10 ⁵ dual unknowns and 1227 corner unknowns.

Configuration of an optical trapping system of a gold truncated cone dimer based on a dielectric substrate in water. a 3D view. b Front view and dimensions, where UR = 20 nm, LR = 30 nm, h = 35 nm, and g = 2 nm

First, we analyze the optical forces by changing λ from 277 nm to 818 nm. Figure 10 plots the calculated optical forces exerted on the nanosphere for different λ . The nanosphere is located at (0, 0, 35 nm). It can be seen that the maximum optical force occurs at λ  = 464 nm, which is the plasmonic resonance wavelength, and the optical force here is enhanced by nearly 30 times at non-resonance wavelength. Moreover, the total optical force always points to −z , as shown in Fig. 10, which is the direction of the gradient force. This confirms that the gradient force is greater than the scattering force, which is one of the conditions that the nanosphere can be stably trapped. Figure 11 a and b presents the calculated electric field distributions at the non-resonance wavelength of λ =300 nm and the resonance wavelength of λ = 464 nm, respectively. It can be seen that electric field intensity has been increased by almost 500 times due to the localized surface plasmon resonance.

Calculated results of optical forces exerted on the nanosphere in the system of gold truncated cone dimer, varying with λ . The nanosphere is located at (0, 0, 35 nm)

The electric field enhancement distributions on the xoz plane for the system of gold truncated cone dimer. a λ =300 nm (non-resonance wavelength). b λ =464 nm (resonance wavelength)

Then, the location of the nanosphere is changed to 0, 5, and 35 nm to observe the optical force. Figure 12 gives the calculated optical forces exerted on the nanosphere, where obvious y -component of optical force can be observed, while greater z -component of optical force exists. The total optical force still points to the position with the strongest electric field to trap the nanosphere.

Calculated results of optical forces exerted on the nanosphere in the system of gold truncated cone dimer varying λ . The nanosphere is located at (0, 5 nm, 35 nm)

Furthermore, we analyze the optical potential with the nanosphere moving from (0, 0, 50 nm) to (0, 0, 20 nm) along the z -sumbu. Here, we consider the axial trapping potential along z -direction, which restricts the motion of the nanoparticle to one dimension and leads to an unambiguous definition of optical potential. Both the optical force and potential are calculated. As can be observed from Fig. 13 a, b, with the nanosphere moving to the dimer gap, the optical force and the optical potential depth obviously increase. At (0, 0, 20 nm), an optical potential depth of 3.8 k _B T is obtained, which is sufficient to overcome the Brownian motion in water to achieve stable optical trapping.

The optical forces and optical potentials exerted on the nanosphere in the system of gold truncated cone dimer, when the nanosphere moves from (0, 0, 50 nm) to (0, 0, 20 nm). a The optical forces. b The optical potentials

Finally, we test the computational costs of the FETI-DP by changing the number of unknowns from 1.0 million to 3.2 million based on different mesh size. In practice, the tests under different mesh density are usually necessary to meet different accuracy requirements. Such a large-scale complex problem brings great challenges to conventional numerical methods. However, the FETI-DP can easily handle this problem. Thirty-two processors are employed for the FETI-DP simulation, while each processor deals with a subdomain. Table 3 reports the computational costs of the FETI-DP. It can be seen that the FETI-DP exhibits high simulation efficiency and low memory requirement.