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Multipole optimization of light focusing by silicon nanosphere structures

Nikita Ustimenko, Kseniia V. Baryshnikova, Roman Melnikov, Danil Kornovan, Vladimir Ulyantsev, Boris N. Chichkov, and Andrey B. Evlyukhin

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Abstract

We investigate the applicability of the coupled multipole model and its modification in the framework of the zero-order Born approximation for modeling of light focusing by finite-size nanostructures of silicon nanospheres, supporting electric and magnetic dipole and quadrupole resonances. The results based on the analytical approximations are verified by comparison with the numerical simulations performed by the T-matrix method. Using the evolutionary algorithm optimization, we apply the developed approach to design silicon nanosphere metalenses with predefined focusing properties. The obtained results demonstrate a strong optimization potential of the suggested calculation scheme for engineering ultrathin metalenses.

© 2021 Optical Society of America

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Data Availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (6)
Fig. 1.
Fig. 1. (a) Schematic view of the investigated structure, which is a ring consisting of Si nanospheres with a diameter $d = {200}\;{\rm nm}$. The ring is placed in the $z = 0$ plane. (b) Simulated multipole decomposition of the scattering efficiency for a single Si sphere of diameter $d = {200}\;{\rm nm}$ in air. The scattering efficiency and multipole decomposition were calculated using the Mie theory [33].

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Fig. 2.
Fig. 2. Normalized intensity calculated using the CMM for a single ring [see Fig. 1(a)] of radius $R = {2}\;{{\unicode{x00B5}{\rm m}}}$ and number of particles $N = 62$ at the wavelength (a) ${\lambda _{{\rm MD}}} = 770 \; {\rm nm}$ of the MD resonance and (c) ${\lambda _{{\rm MQ}}} = 574 \; {\rm nm}$ of the MQ resonance. The intensities (a) and (c) illustrate that the ring of 62 equally separated nanospheres focuses light at both resonant wavelengths. (b) and (d) Simulated normalized intensity along the $z$ axis from the T-matrix method [60] (blue solid lines) and CMM Eq. (15) (red dashed lines). The normalization factor is the intensity of the incident plane wave.

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Fig. 3.
Fig. 3. (a) Focal length calculated using the ZBA as a function of the ring radius and inter-particle (center-center) distance at wavelength of the single-particle MD resonance. The focal length is defined as a distance between the ring plane and the point on the $z$ axis corresponding to the global maximum of light intensity. (b) Intensity (normalized on the maximum value) of the fields generated by two rings ($N = 32$) along the $z$ axis calculated at the MD resonance in the ZBA [Eq. (B1) in Appendix B]. Red and blue arrows in (b) indicate the focal lengths of rings with the radius $R/{\lambda _{{\rm MD}}} = 6$ and $R/{\lambda _{{\rm MD}}} = 5$, respectively. All dimensional values in (a) and (b) are normalized by the MD resonance wavelength ${\lambda _{{\rm MD}}} = 770 \; {\rm nm}$.

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Fig. 4.
Fig. 4. Focal length error $\Delta f$ of the ZBA as a function of the ring radius and inter-particle (center-center) distance at wavelengths of (a) the MD (${\lambda _{{\rm MD}}} = 770 \; {\rm nm}$) and (b) the MQ (${\lambda _{{\rm MQ}}} = 574 \; {\rm nm}$) resonances. Focal intensity error $\Delta {{I}_f}$ of ZBA for (c) MD and (d) MQ resonances. The white dashed lines indicate the limiting distances: for all larger distances, the corresponding errors of ZBA are the order or less than 10%.

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Fig. 5.
Fig. 5. Optimized silicon nanosphere structures for light focusing (metalenses), and their normalized intensity profiles at wavelengths of the (a)–(c) MD resonance and (d)–(f) MQ resonance. (a) Scheme of particles’ distribution in the optimized design. (b), (c) Intensity profiles of the design, calculated in (b) ZBA and (c) by T-matrix method. (d)–(f) The same at wavelength of the MQ resonance (574 nm). The structures provide light focusing at the target position of 5 µm.

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Fig. 6.
Fig. 6. Normalized intensity profiles for the optimized structures in Si particles (a) with and (b) without absorption losses. The red dashed and blue lines correspond to the MD [Fig. 5(a)] and MQ [Fig. 5(d)] resonance structures, respectively. The intensity profiles were computed in the ZBA by Eq. (B1) in Appendix B.

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Tables (1)

Tables Icon

Table 1. Summary of the Actual Focal Length f and Focal Intensity I f / I 0 for Optimized Designs of Metalenses from Figs. 5(a) and 5(d)a

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Equations (28)

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p j = α p E l o c ( r j ) , m j = α m H l o c ( r j ) ,
Q ^ j = α Q 2 [ ∇ j ⊗ E l o c ( r j ) + ( ∇ j ⊗ E l o c ( r j ) ) T ] ,
M ^ j = α M 2 [ ∇ j ⊗ H l o c ( r j ) + ( ∇ j ⊗ H l o c ( r j ) ) T ] ,
E l o c ( r j ) = E i n c ( r j ) + E p ′ ( r j ) + E m ′ ( r j ) + E Q ′ ( r j ) + E M ′ ( r j ) ,
H l o c ( r j ) = H i n c ( r j ) + H p ′ ( r j ) + H m ′ ( r j ) + H Q ′ ( r j ) + H M ′ ( r j ) .
E p ′ ( r j ) = k 0 2 ε 0 ∑ l = 1 , l ≠ j N G ^ j l p p l , H p ′ ( r j ) = c k 0 i ∑ l = 1 , l ≠ j N g j l × p l , E m ′ ( r j ) = i k 0 c ε 0 ∑ l = 1 , l ≠ j N g j l × m l , H m ′ ( r j ) = k S 2 ∑ l = 1 , l ≠ j N G ^ j l p m l , E Q ′ ( r j ) = k 0 2 ε 0 ∑ l = 1 , l ≠ j N G ^ j l Q ( Q ^ l n l j ) , H Q ′ ( r j ) = c k 0 i ∑ l = 1 , l ≠ j N q j l × ( Q ^ l n l j ) , E M ′ ( r j ) = 3 i k 0 c ε 0 ∑ l = 1 , l ≠ j N q j l × ( M ^ l n l j ) , H M ′ ( r j ) = 3 k S 2 ∑ l = 1 , l ≠ j N G ^ j l Q ( M ^ l n l j ) ,
g j l × p j = ∇ j × G ^ j l p p j ,
q j l × ( Q ^ l n l j ) = ∇ j × G ^ j l Q ( Q ^ l n l j ) .
E ( r ) = E i n c ( r ) + k 0 2 ε 0 ∑ j = 1 N { G ^ p ( r , r j ) p j + i c k 0 [ g ( r , r j ) × m j ] + G ^ Q ( r , r j ) ( Q ^ j n j ) + 3 i c k 0 [ q ( r , r j ) × ( M ^ j n j ) ] } ,
H ( r ) = H i n c ( r ) + k 0 2 ∑ j = 1 N { c i k 0 [ g ( r , r j ) × p j ] + ε S G ^ p ( r , r j ) m j + c i k 0 [ q ( r , r j ) × ( Q ^ j n j ) ] + 3 ε S G ^ Q ( r , r j ) ( M ^ j n j ) } ,
α p = i 6 π ε 0 ε S k S 3 a 1 , α m = i 6 π k S 3 b 1 , α Q = i 120 π ε 0 ε S k S 5 a 2 , α M = i 40 π k S 5 b 2 .
Y = Y 0 + VY ^ ,
Y = [ p x 1 , … , p z N , m x 1 , … , m z N , Q x x 1 , … , Q z z N , M x x 1 , … , M z z N ] T ,
Y 0 = [ p 0 , x 1 , … , p 0 , z N , m 0 , x 1 , … , m 0 , z N , Q 0 , x x 1 , … , Q 0 , z z N , M 0 , x x 1 , … , M 0 , z z N ] T .
Y = ( I ^ − V ^ ) − 1 Y 0 ,
Y = Y 0 + V ^ Y 0 + V ^ 2 Y 0 + … .
Y = Y 0 .
Y n = Y 0 + V ^ Y n − 1 .
Δ V = | V ( C M M ) − V ( Z B A ) | V ( C M M ) × 100 % ,
∂ ∂ γ G α β p ( r , r 0 ) = e i k S l 4 π l { k S ( i − 2 k S l − 3 i k S 2 l 2 + 3 k S 3 l 3 ) δ α β n γ + k S ( − i + 4 k S l + 9 i k S 2 l 2 − 9 k S 3 l 3 ) n α n β n γ + ( − 1 − 3 i k S l + 3 k S 2 l 2 ) ( ∂ n α ∂ γ n β + n α ∂ n β ∂ γ ) } ,
∂ ∂ γ g α ( r , r 0 ) = e i k S l 4 π { ( − k S 2 l − 2 i k S l 2 + 2 l 3 ) n α n γ + ( i k S l − 1 l 2 ) ∂ n α ∂ γ } ,
∂ ∂ γ G α β Q ( r , r 0 ) = i k S e i k S l 24 π l { k S ( − i + 4 k S l + 12 i k S 2 l 2 − 24 k S 3 l 3 − 24 i k S 4 l 4 ) δ α β n γ + k S ( i − 7 k S l − 27 i k S 2 l 2 + 60 k S 3 l 3 + 60 i k S 4 l 4 ) n α n β n γ + ( 1 + 6 i k S l − 15 k S 2 l 2 − 15 i k S 3 l 3 ) ( ∂ n α ∂ γ n β + n α ∂ n β ∂ γ ) } ,
∂ ∂ γ q α ( r , r 0 ) = k S 2 e i k S l 24 π l { ( 1 + 3 i k S l − 3 k S 2 l 2 ) ∂ n α ∂ γ + k S ( i − 4 k S l − 9 i k S 2 l 2 + 9 k S 3 l 3 ) n α n γ } ,
∂ n α ∂ β = δ α β − n α n β l .
I ( z ) / I 0 = I ~ E ( z ) + I ~ H ( z ) ,
I ~ E ( z ) = 1 2 | e i k S z + N e i k S l 4 π l { α p ε 0 ε S [ A ( l ) + B ( l ) R 2 2 l 2 ] + α m C ( l ) z l + α Q 12 ε 0 ε S z l [ D ( l ) + F ( l ) R 2 l 2 ] + α M 4 G ( l ) R 2 / 2 − z 2 l 2 } | 2 ,
I ~ H ( z ) = 1 2 | e i k S z + N e i k S l 4 π l { α m [ A ( l ) + B ( l ) R 2 2 l 2 ] + α p ε 0 ε S C ( l ) z l + α M 4 z l [ D ( l ) + F ( l ) R 2 l 2 ] + α Q 12 ε 0 ε S G ( l ) R 2 / 2 − z 2 l 2 } | 2 ,
A ( l ) = k S 2 + i k S l − 1 l 2 , B ( l ) = − k S 2 − 3 i k S l + 3 i l 2 , C ( l ) = k S 2 + i k S l , D ( l ) = k S 4 + 3 i k S 3 l − 6 k S 2 l 2 − 6 i k S l 3 , F ( l ) = − k S 4 − 6 i k S 3 l + 15 k S 2 l 2 + 15 i k S l 3 , G ( l ) = − k S 4 − 3 i k S 3 l + 3 k S 2 l 2 .

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