Block-Wise Recurrence for Non-Metal Mirror Design
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Solution Overview
Problem
Existing methods for creating omnidirectional non-metallic mirrors face challenges in efficiently calculating and producing structures with non-periodic heterostructures, leading to prohibitive calculation complexity and potential errors due to cumulative rounding errors, while prior solutions restrict the flexibility of layer thicknesses and structures.
Innovation Solution
A method using a block-wise recurrence technique to calculate the product of elementary matrices based on a substitutive sequence, allowing for the generation of a structure with a variable number of layers and physical parameters, enabling the determination of the transmission coefficient and optimization of the mirror's structure without prior constraints.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If traditional methods are used to calculate the product of elementary matrices for non-periodic heterostructures, then the structure can be determined, but the calculation complexity becomes prohibitive and cumulative rounding errors occur
Solution Approach 1:
The patent segments the calculation process by introducing a block-wise recurrence method that divides the product of elementary matrices into manageable blocks. Instead of calculating the entire product sequentially, the method recursively computes products of blocks, reducing the overall computational complexity from O(N) to O(log N) where N is the number of layers.
Solution Approach 2:
The patent applies nesting by implementing a recursive algorithm where the product of matrices is calculated by nesting smaller matrix products within larger ones. The block-wise recurrence relation An+1 = A1,n × A2,n × ... × Ak,n nests the calculation of smaller blocks (An) within the computation of larger structures, efficiently managing the complexity of non-periodic heterostructures.
2Reliability
If the number of layers and physical parameters are increased to improve mirror performance, then the reflection quality improves, but the calculation time and complexity increase significantly
Solution Approach 1:
The patent performs preliminary action by pre-defining the block structure and recurrence relations before actual calculation. The method establishes the block-wise recurrence framework An+1 = A1,n × A2,n × ... × Ak,n in advance, allowing for efficient computation when the number of layers is increased. This preliminary structuring enables the system to handle larger numbers of layers without proportionally increasing calculation time.
3Device complexity
If prior constraints are imposed on layer thicknesses and structures to simplify calculations, then the calculation complexity reduces, but the flexibility and adaptability of the mirror design is restricted
Solution Approach 1:
The patent applies dynamics by creating a flexible block-wise recurrence framework that can adapt to different layer configurations without requiring fixed constraints. The method dynamically adjusts to various numbers of layers (k) and different substitutive sequences, allowing the mirror design to be versatile while maintaining computational efficiency. The recurrence relation An+1 = A1,n × A2,n × ... × Ak,n naturally accommodates different structural configurations.
Applied Scientific Principles
This section explains which scientific principles are used to turn an abstract innovation direction into a practical engineering solution.
Function Achieved in This Case
This approach significantly reduces calculation time and complexity, allowing for the reliable design of omnidirectional non-metallic mirrors with greater flexibility in layer arrangements and physical parameters, avoiding the limitations of prior methods and achieving accurate reflection across a wide range of angles and wavelengths.
Implementation Method 1
The appropriate solution of Maxwell's Equations which governs the propagation takes the form in this stratified medium of a 2×2 matrix termed the 'transfer matrix' MN, which relates the properties of light in the entrance plane of the multilayer and in the exit plane
Implementation Method 2
A multilayer non-metallic mirror comprises two or more layers that may differ in their composition and/or their thickness... such a mirror can also consist of two or more of these layers repeated according to non-periodic substitutive sequences
Data Source
AI summary
A method for structuring an omnidirectional non-metal mirror for any predetermined wavelength or range of wavelengths. The mirror having at least two layers of different non-metal materials, with an elementary matrix associated to each layer, including physical parameters of the layer and parameters of the light passing through the layer.