Expand this Topic clickable element to expand a topic
Skip to content
Optica Publishing Group
Journal Home About Issues in Progress Current Issue All Issues Feature Issues

Numerical far field simulations with the fast Fourier transformation and Fourier space interpolation

Raoul-Amadeus Lorbeer, Heiko Meyer, and Tammo Ripken

Open Access Open Access

Abstract

As more complicated microscope systems are engineered, the amount of effects taken into account rises steadily. In this context we experienced the need for a simulation approach, that will deliver the intensity distribution in space and time for scanning laser microscopes. To achieve this goal, the frequency space representation of microscope objectives was used and adapted to determine their solution of the electromagnetic wave equation. We describe the steps necessary to efficiently implement an approach to simulate multidimensional solutions of the wave equation. This includes the connection between the back focal plane and the Fourier space representation as well as a proper interpolation method for the latter. The error-potential of our least erroneous interpolation, the power of hann (POH) interpolation, is compared to other common interpolation methods. Finally we demonstrate the current potential of the approach by simulating an “expanding” optical vortex focus.

© 2015 Optical Society of America

Full Article  |  PDF Article
More Like This
Nonuniform fast Fourier transform method for numerical diffraction simulation on tilted planes

Yu Xiao, Xiahui Tang, Yingxiong Qin, Hao Peng, Wei Wang, and Lijing Zhong
J. Opt. Soc. Am. A 33(10) 2027-2033 (2016)

Band-Limited Angular Spectrum Method for Numerical Simulation of Free-Space Propagation in Far and Near Fields

Kyoji Matsushima and Tomoyoshi Shimobaba
Opt. Express 17(22) 19662-19673 (2009)

Fourier-based interpolation bias prediction in digital image correlation

Yong Su, Qingchuan Zhang, Zeren Gao, Xiaohai Xu, and Xiaoping Wu
Opt. Express 23(15) 19242-19260 (2015)

View More...

Supplementary Material (1)

Media 1: MOV (3269 KB)     

Cited By

Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)
Figure 1
Figure 1 Impulse response for multiple widths of the POH interpolation. a) Several POH kernels at the central grid position. b) Impulse response after Fourier transformation with identical color code. The resulting normed amplitude show the behavior of a POH H H − 2 2 kernel for H = 4, 6 and 12. The first drop in frequency domain stays consistent on the Nyquist frequency 0.5 1 grid   points. For bigger H the kernel broadens and drops at steeper rates for extreme frequencies above Nyquist.

Download Full Size | PDF

Fig. 2
Fig. 2 Impulse response for multiple positions pi of the POH interpolation. a) POH 6 2 kernels at three different positions. The dots indicate the values used for the interpolation. b) Impulse response after Fourier transformation with identical color code. The resulting amplitude and phase distributions differ from position to position. For the amplitude distribution a constant (0 dB) would represent the ideal frequency response. The phase response should result in a linear phase with a slope proportional to the displacement Δ spanning from −90 to 90 deg for Δ = ±0.5. Differences from the ideal expected behavior have to be compensated by normalization. This approach is limited by the variations e.g. in the extreme frequencies of the normed amplitude.

Download Full Size | PDF

Fig. 3
Fig. 3 Amplitude and phase error at all frequencies after a one dimensional FFT with normalization to constant errors. a) Maximum deviation from the ideal amplitude of 1 in any of the 500 tested interpolation positions after normalization to average amplitude after the FFT. b) Maximum deviation from the ideal phase −i(kr) in any of the 500 tested positions after subtraction of the average phase error after the FFT.

Download Full Size | PDF

Fig. 4
Fig. 4 Top: POH 6 2 interpolation method compared to oversampled version. Bottom: POH 6 2 interpolation method compared to equidistant sampling of an analytical | E ˜ → ( k → , ω ) |. a) & d): Results of the POH 6 2 interpolation method. b) & e) The absolute error calculated as the difference between the standard simulation and the reference simulation, both with their maximum intensity normed to one. c) & f) Relative error calculated by dividing the absolute errors (b & e) by their reference simulations.

Download Full Size | PDF

Fig. 5
Fig. 5 Simulation of a spectral vortex. Side by side view of the x,z-plane and the x,y-plane (indicated by the red dashed lines) with the time evolving from top left to bottom right. The diameter of the vortex reduces over time. After a complete collapse it starts expanding again. Scale bar represents 10 μm. (see Media 1

Download Full Size | PDF

Equations (15)

Equations on this page are rendered with MathJax. Learn more.

Δ E → ( r → , t ) = n 2 c 2 ∂ t 2 E → ( r → , t )
E → ( r → , t ) = ∫ − ∞ ∞ d k → d ω E ˜ → ( k → , ω ) e i ( k → r → − ω t )
k → ( ρ → ) = ( cos ϕ sin Θ sin ϕ sin Θ cos Θ )     | k → | = ( ρ x f ρ y f 1 − ( ρ f ) 2 ) ω ⋅ n ( ω ) c ,
E ˜ → ( k → ( ρ → ) , ω ) = S ( ρ → ) E → BFP ( ρ → , ω )
V i = F I ( s = x des − x i Δ x )
F I ( s ) = { NN ( s ) = { 1 − 1 2 ≦ s < 1 2 0 otherwise CS ( s ) = { 0 | s | ≧ 2 1 6 ( 2 − | s | ) 3 1 ≦ | s | < 2 2 3 − 1 2 | s | 2 ( 2 − | s | ) | s | < 1 SWS H , ξ ( s ) = { sin ( π s ξ ) π s ∑ i = 0 n a i cos ( 2 π i s H ) | s | < H 2 0 otherwise POH H L ( s ) = { cos 2 L ( π s H ) | s | < H 2 0 otherwise
δ S BFP = | ∂ ρ x ρ → × ∂ ρ y ρ → | = 1 f 2
δ S KS = | ∂ ρ x k → × ∂ ρ y k → | = | k → | f 2 cos ( θ )
P ∝ δ S BFP | E → BFP | 2 = δ S KS | E → KS | 2
| E → KS | = | E → BFP | S ( ρ → ) = | E → BFP | δ S BFP δ S KS = | E → BFP | cos ( θ ) | k → |
P ∝ δ S BFP ( | E → BFP | num δ S BFP δ S ) 2 = δ S KS ( | E → KS | num δ S KS δ S ) 2
| E → KS | num = | E → BFP | num S num ( ρ → ) = | E → BFP | num δ S K S δ S BFP = | E → BFP | num | k → | cos ( θ )
E ( k , r ) = ∫ − ∞ ∞ d k x δ ( k x − k ) e − i ( k x r ) = e − i ( k r )
| E ˜ → ( k → , ω ) | = cos θ | k → | ⋅ sin 2 ( π k − k min k max − k min ) ⋅ cos 2 ( π 2 sin θ NA ) ,
NN < CS < POH 6 2 < SSW 11 < SSW 31 < POH 12 5
Select as filters
Select Topics Cancel
Top
       
© Copyright 2024 | Optica Publishing Group. All rights reserved, including rights for text and data mining and training of artificial technologies or similar technologies.
Privacy | Terms of Use