Generally Fibonacci series and Lucas series are the same, they converge to golden ratio. After I read Fibonacci series, I thought, is there or are there any series which converges to golden ratio. Because of that I ex...Generally Fibonacci series and Lucas series are the same, they converge to golden ratio. After I read Fibonacci series, I thought, is there or are there any series which converges to golden ratio. Because of that I explored the inter relations of Fibonacci series when I was intent on Fibonacci series in my difference parallelogram. In which, I found there is no degeneration on Fibonacci series. In my thought, Pascal triangle seemed like a lower triangular matrix, so I tried to find the inverse for that. In inverse form, there is no change against original form of Pascal elements matrix. One day I played with ring magnets, which forms hexagonal shapes. Number of rings which forms Hexagonal shape gives Hex series. In this paper, I give the general formula for generating various types of Fibonacci series and its non-degeneration, how Pascal elements maintain its identities and which shapes formed by hex numbers by difference and matrices.展开更多
Zone plates play an important role in X-ray and optics fileds.In this work,we proposed a generalized Fibonacci zone plates(GFiZP)is proposed,which enables the manipulation of focusing properties for X-ray and light.Co...Zone plates play an important role in X-ray and optics fileds.In this work,we proposed a generalized Fibonacci zone plates(GFiZP)is proposed,which enables the manipulation of focusing properties for X-ray and light.Compared to conventional Fibonacci zone plates(FiZPs)and Cantor-type fractal zone plates(FraZPs),this device offers additional degrees of freedom for engineering focusing performance.In comparison with FraZPs,GFiZPs are constructed with inflation rule instead of extraction rule,thus avoiding sparse structures at high recursion orders,the sparse structures would lead to issues in fabrication.Notably,numerical simulations demonstrate that,similar to FraZPs,GFiZPs retain two critical functionalities:multi-focal capability and self-similar intensity distribution.Furthermore,we conducted a detailed investigation into the axial irradiance profiles of GFiZPs with varying structural parameters.Given these favorable properties,this novel zone plate design holds great potential for expanded applications in areas such as optical manufacturing and optical manipulation of microparticles.展开更多
文摘Generally Fibonacci series and Lucas series are the same, they converge to golden ratio. After I read Fibonacci series, I thought, is there or are there any series which converges to golden ratio. Because of that I explored the inter relations of Fibonacci series when I was intent on Fibonacci series in my difference parallelogram. In which, I found there is no degeneration on Fibonacci series. In my thought, Pascal triangle seemed like a lower triangular matrix, so I tried to find the inverse for that. In inverse form, there is no change against original form of Pascal elements matrix. One day I played with ring magnets, which forms hexagonal shapes. Number of rings which forms Hexagonal shape gives Hex series. In this paper, I give the general formula for generating various types of Fibonacci series and its non-degeneration, how Pascal elements maintain its identities and which shapes formed by hex numbers by difference and matrices.
文摘Zone plates play an important role in X-ray and optics fileds.In this work,we proposed a generalized Fibonacci zone plates(GFiZP)is proposed,which enables the manipulation of focusing properties for X-ray and light.Compared to conventional Fibonacci zone plates(FiZPs)and Cantor-type fractal zone plates(FraZPs),this device offers additional degrees of freedom for engineering focusing performance.In comparison with FraZPs,GFiZPs are constructed with inflation rule instead of extraction rule,thus avoiding sparse structures at high recursion orders,the sparse structures would lead to issues in fabrication.Notably,numerical simulations demonstrate that,similar to FraZPs,GFiZPs retain two critical functionalities:multi-focal capability and self-similar intensity distribution.Furthermore,we conducted a detailed investigation into the axial irradiance profiles of GFiZPs with varying structural parameters.Given these favorable properties,this novel zone plate design holds great potential for expanded applications in areas such as optical manufacturing and optical manipulation of microparticles.
