The conventional finite-difference time-domain (FDTD) algorithm, based on 2nd-order finite difference (FD) approximations to the derivatives in Maxwell’s equations, is a simple and flexible methodology that can be used to solve a wide class of problems, but its accuracy is low unless a very fine grid is used. For grid spacing h=Δx=Δy=Δz the error is (epsilon) ~ (h/λ)4 where λ is the wavelength. Putting h → h/2, reduces the error by a factor of 16 but the computation cost rises 16-fold (in three dimensions) because the time step must scale with h to maintain numerical stability. In principle, higher-order FD approximations would improve the accuracy, but they not only complicate the algorithm, but can also render it numerically unstable. We introduced an 8th-order accurate FDTD algorithm with respect to basis function solutions of Maxwell's equations by superposing 2nd-order FDs. This methodology, originally applied to monochromatic propagation, is extended to broadband computations. We validate our methodology on a problem with a known solution.
Nanophotonic devices, such as CMOS image sensor (CIS) pixels, are formed by stacking multiple layers of semiconductor materials. The complex refractive indices of these materials vary with the wavelength of light. Currently, industrial development of photonic devices includes a design step where light propagation is simulated using numerical methods, such as finite-difference time-domain (FDTD). Such simulations require that the refractive indices of the constituent materials be known accurately.
Most commonly employed methods for computing the real and imaginary parts of the dispersive refractive indices are based either on the evaluation of the Kramers-Kronig (K-K) integral, or on the use of theoretical models of permittivity. These methods rely on the experimentally measured reflectivity or transmissivity spectra of thin films of a material to determine its refractive indices.
In the first part of this paper, we describe the computation of the dispersive refractive indices of certain materials using an optimization routine based on a genetic algorithm and the coherent reflectivity and transmissivity spectra of thin-films. This approach finds the global optimum unlike earlier methods based on local optimization techniques. In the second part of the paper, we evaluated the K-K integral and used the Lorentz model of permittivity to compute the real part of the refractive index of Rhodamine B from its imaginary part. The imaginary part was determined from the transmission spectrum of a thin film of Rhodamine B. Recently, we used a similar strategy to compute the dispersive refractive index of an on-chip color filter commonly used for CIS pixel.
The finite difference time domain (FDTD) method computes the time evolution of a system at discrete time steps, and periodically visualizing the results lets us view its time evolution, yielding valuable physical insights. FDTD and FDTD-like methods can be used to solve a wide variety of problems, including—but not limited to—the wave equation, Maxwell’s equations, and the Schrödinger equation. In addition to introducing useful new methodologies, this book provides readers with analytical background and simulation examples that will help them develop their own methodologies to solve yet-to-be-posed problems. The book is written for students, engineers, and researchers grappling with problems that cannot be solved analytically. It could also be used as a textbook for a mathematical physics or engineering class. Accompanying online materials provide supplemental Mathcad and pseudocode programs.
Traditionally, liquid crystal display (LCD) systems employ color filters that are fabricated using organic dye and pigment based colorants. As a result, conventional color filters can lower the system performance by removing substantial amount of incident light through absorption. Also, the transmission bandwidth can be unacceptably large. Furthermore, there is a need to combine functions of multiple optical elements on one, facilitating miniaturization and compactness. Metal-insulator- metal (MIM) nanoresonators that can combine the functions of color filtering and polarizing can provide a useful solution to some of these issues. An MIM nanoresonator structure is proposed for use as color filters. However, the proposed structure uses high refractive index, inorganic materials in the insulator layer. Also, the bandwidth of transmission is not narrow enough to generate saturated color. Here, we simulated some MIM nanoresonator structures that might be realized using relatively low refractive index, polymeric materials and can function as polarizing, color filters in transmission mode. These structures might also yield narrower bandwidths of transmission. The simulations are carried out using a monochromatic version of RC-FDTD. This algorithm uses the 1st order Drude model to evaluate the convolution operation needed to make FDTD stable for metals for which the real part of permittivity is negative. Unlike the conventional RC-FDTD [3], the Drude parameters are computed at each wavelength of the incident light using the corresponding handbook value of permittivity. Hence, this version of RC-FDTD allows us to use the handbook permittivity values at all wavelengths of operation.
The simple source model used in the conventional finite difference time domain (FDTD) algorithm gives rise to large
errors. Conventional second-order FDTD has large errors (order h**2/ 12), h = grid spacing), and the errors due to the
source model further increase this error. Nonstandard (NS) FDTD, based on a superposition of second-order finite
differences, has been demonstrated to give much higher accuracy than conventional FDTD for the sourceless wave
equation and Maxwell’s equations (h**6 / 24192). Since the Green’s function for the wave equation in free space is
known, we can compute the field due to a point source. This analytical solution is inserted into the NS finite difference
(FD) model and the parameters of the source model are adjusted so that the FDTD solution matches the analytical one.
To derive the scattered field source model, we use the NS-FD model of the total field and of the incident field to deduce
the correct source model. We find that sources that generate a scattered field must be modeled differently from ones
radiate into free space. We demonstrate the high accuracy of our source models by comparing with analytical solutions.
This approach yields a significant improvement inaccuracy, especially for the scattered field, where we verified the
results against Mie theory. The computation time and memory requirements are about the same as for conventional
FDTD. We apply these developments to solve propagation problems in subwavelength structures.
We dispersed silver nanospheres of diameter 5nm in a homogeneous binder. Films were spin-coated on glass substrates.
The transmission spectra of such films are measured as particle concentration is varied. The transmission spectra show
deeper and wider minima in the shorter wavelength side as the concentration of nanoparticles increase. This might be
explained by the formation of aggregates of nanoparticles and the coherent interaction among the constituent elements of
the aggregates. The coherent interaction can include coupling among the localized surface plasmon resonance (LSPR)
modes of individual particles. To explain the dependence of transmission spectra on the concentration of particles we
computed the scattering properties of particle aggregates.
The scattering properties of a single spherical particle can be computed analytically using Mie theory. No analytical
computation method is available for aggregates of nanoparticles. Numerical methods, like finite-difference time-domain
(FDTD) method can be used.
We computed the scattering properties of aggregates of silver nanospheres using a monochromatic version of
recursive convolution finite-difference time-domain (RC-FDTD) method. In contrast with the conventional broadband
RC-FDTD [3], the monochromatic version allows one to use the handbook values of permittivity of the material of the
particles at every simulation wavelength. The algorithm employs the 1st order Drude model to make it stable for metals
with negative real part of permittivity. The particle-aggregates are generated using a random number generator that
distributes nanospheres uniformly throughout a larger sphere made of the homogeneous binder medium.
A special class of finite-difference time-domain (FDTD) algorithm, called recursive convolution (RC) FDTD, is
developed to simulate metals in visible domain. Conventionally, RC-FDTD is implemented for pulsed light sources. This
requires that the analytical model of permittivity should fit the handbook permittivities (derived from experimental
measurements) closely over broadband of wavelengths. This is not an easy task and the choice of a particular model
depends on the metal being simulated.
We developed a monochromatic version of RC-FDTD. This algorithm uses the 1st order Drude model to evaluate
the convolution operation needed to make FDTD stable for metals for which the real part of permittivity is negative.
Unlike the conventional RC-FDTD, the Drude parameters are computed at each wavelength of the incident light using
the corresponding handbook value of permittivity. Hence, this version of RC-FDTD allows us to use the handbook
permittivity values at all wavelengths of operation.
Here, we study the dependence of localized surface plasmon resonance (LSPR) properties of arrays of nano-particles
on the shape, size and number of particles in the array and the interparticle distance. We compute the extinction spectra
of linear nano-particle arrays using the RC-FDTD method. For such arrays, the peak of the extinction spectra shifts
toward the longer wavelengths as the interparticle distance decreases.
The arrays of metal nano-particles can support localized surface plasmon resonance (LSPR) modes, making them
suitable for coloring applications. The LSPR peaks of such arrays can be tuned by changing the structural parameters,
such as, shape, size, of each particle and the interparticle distance. In this paper we study the dependence of LSPR
properties of arrays on the structural parameters.
The extinction spectra of a spherical particle can be computed analytically using Mie theory. No analytical computation
method is available for particle arrays. Numerical methods, e.g., discrete dipole approximation, finite-difference
time-domain (FDTD) method are used. Here, we compute the extinction spectra of linear arrays of nano-particles using a
monochromatic version of recursive convolution (RC) FDTD method. We developed this method to be able to use the
handbook values of permittivity of the material of the particles at each wavelength.
The simulations indicate that the position and size of the peaks of the extinction spectra are determined by the
interparticle distance between any two particles and the number of particles in the array. In case of linear arrays of
infinite, silver nano-cylinders, the LSPR peak can be shifted toward the longer wavelengths (red-shift) by reducing the
interparticle distance. The red-shift increases as the interparticle distance becomes smaller. The peaks of extinction
spectra become larger as the number of particles in the array increases.
The finite-difference time-domain (FDTD) method as realized in the Yee algorithm is very useful for solving the analytically intractable diffraction problems that arise in the design of diffractive optical elements (DOE). The error, ε, of the Yee algorithm is rather large. We introduce a new high accuracy version of the Yee algorithm based on nonstandard finite differences. We have verified its performance by comparing with known analytical solutions. We have used this algorithm to study the near field (0 - 10 wavelengths) diffraction patterns of three binary phase gratings. Grating (1) has a constant period, while grating (2) has a variable one. Grating (3) is generated by interlacing grating (1) with (2). The period of grating (1) is on the order of a wavelength. Some of these computed fields display interesting and possibly useful Fourier spectra.
This paper is in tune with our efforts to develop a systematic method for multicomponent lens design. Our aim is to find a suitable starting point in the final configuration space, so that popular local search methods like damped least squares (DLS) may directly lead to a useful solution. For 'ab initio' design problems, a thin lens layout specifying the powers of the individual components and the intercomponent separations are worked out analytically. Requirements of central aberration targets for the individual components in order to satisfy the prespecified primary aberration targets for the overall system are then determined by nonlinear optimization. The next step involves structural design of the individual components by optimization techniques. This general method may be adapted for the design of triplets and their derivatives. However, for the thin lens design of a Cooke triplet composed of three airspaced singlets, the two steps of optimization mentioned above may be combined into a single optimization procedure. The optimum configuration for each of the single set, catering to the required Gaussian specification and primary aberration targets for the Cooke triplet, are determined by an application of genetic algorithm (GA). Our implementation of this algorithm is based on simulations of some complex tools of natural evolution, like selection, crossover and mutation. Our version of GA may or may not converge to a unique optimum, depending on some of the algorithm specific parameter values. With our algorithm, practically useful solutions are always available, although convergence to a global optimum can not be guaranteed. This is perfectly in keeping with our need to allow 'floating' of aberration targets in the subproblem level. Some numerical results dealing with our preliminary investigations on this problem are presented.
'Ab initio' design of a Cooke triplet usually necessitates a heuristic preselection of optical glasses for the three lens elements. Though some rules of thumb are available for specific cases in general, this is a tricky problem, and often one has to take recourse to a trial and error approach. We propose to tackle the complex problem of design of a Cooke triplet lens with prespecified aberration targets by reducing it to relatively simpler problems of design of individual components with required amounts of primary spherical aberration, central coma and longitudinal chromatic aberration. This reduction is implemented by a global optimization technique. Our method of global optimization is developed along the lines of the well-known method of simulated annealing. Some new features like constrained random walk have been incorporated for facilitating the solution of our problem. A followup procedure also based on global optimization seeks singlet lenses for the individual components. In case of nonavailability of suitable singlet for a component, one seeks to satisfy requirements of that particular component by using a suitable doublet lens. Indeed, the approach provides a systematic method for the development of triplet derivatives as and when required.
Method of generalized simulated annealing with constrained random walk is used for global optimization of the structural design of cemented/broken contact doublet lenses in accordance with prespecified paraxial characteristics and primary aberration targets.
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