Rationalized diffraction calculations for high accuracy and high speed with few bits

Soma Fujimori; Tomoyoshi Ito; Tomoyoshi Shimobaba

doi:10.1364/JOSAA.510884

Journal of the Optical Society of America A
Vol. 41,
Issue 2,
pp. 303-310
(2024)
•https://doi.org/10.1364/JOSAA.510884

Rationalized diffraction calculations for high accuracy and high speed with few bits

Soma Fujimori, Tomoyoshi Ito, and Tomoyoshi Shimobaba

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Soma Fujimori, Tomoyoshi Ito, and Tomoyoshi Shimobaba, "Rationalized diffraction calculations for high accuracy and high speed with few bits," J. Opt. Soc. Am. A 41, 303-310 (2024)

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- Diffraction and Gratings
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The topics in this list come from the Optics and Photonics Topics applied to this article.

About this Article
History
- Original Manuscript: November 8, 2023
- Revised Manuscript: December 20, 2023
- Manuscript Accepted: December 23, 2023
- Published: January 23, 2024

Abstract

Diffraction calculations in few-bit formats, such as single-precision floating-point and fixed-point numbers, are important because they yield faster calculations and lower memory usage. However, these methods suffer from low accuracy owing to the loss of trailing digits. Fresnel diffraction is widely known to prevent the loss of trailing digits. However, it can only be used when the paraxial approximation is valid. In this study, a few-bit diffraction calculation method that achieves high accuracy without using any approximation is proposed. The proposed method is derived only by rationalizing the numerator of conventional formulas. Even for scenarios requiring double-precision floating-point numbers using conventional methods, the proposed method exhibits higher accuracy and faster computation time using single-precision floating-point numbers.

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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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Equations (34)

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	RSC	ASM
Naïve-single (naïve-double)	$\exp (i 2 π \frac{z}{λ} (\sqrt{1 + X_{r}^{2}} - 1))$	$\exp (i 2 π \frac{z}{λ} (\sqrt{1 - X_{f}^{2}} - 1))$
Fresnel	$\exp (i 2 π \frac{z}{λ} (\frac{1}{2} X_{r}^{2}))$	$\exp (- i 2 π \frac{z}{λ} (\frac{1}{2} X_{f}^{2}))$
Polynomial-4	$\exp (i 2 π \frac{z}{λ} (\frac{1}{2} X_{r}^{2} - \frac{1}{8} X_{r}^{4}))$	$\exp (- i 2 π \frac{z}{λ} (\frac{1}{2} X_{f}^{2} + \frac{1}{8} X_{f}^{4}))$
Rationalized	$\exp (i 2 π \frac{z}{λ} \frac{X_{r}^{2}}{\sqrt{1 + X_{r}^{2}} + 1})$	$\exp (- i 2 π \frac{z}{λ} \frac{X_{f}^{2}}{\sqrt{1 - X_{f}^{2}} + 1})$

	Plane Wave	Spherical Wave
Sampling number $N \times N$	$1024 \times 1024$	$4096 \times 4096$
Sampling pitch $p$ [µm]	10.0	0.5
Wavelength $λ [n m]$	500	500
Propagation distance $z$ $[m]$	4.096	$2.048 \times 10^{- 3}$

	RSC	ASM
Naïve-single (naïve-double)	$\frac{z}{r^{2}} (\frac{1}{2 π r} + \frac{1}{i λ}) \exp (i 2 π \frac{z}{λ} (\sqrt{1 + X_{r}^{2}}))$	$\exp (i 2 π \frac{z}{λ} (\sqrt{1 - X_{f}^{2}}))$
Fresnel	$\frac{1}{i λ z} \exp (i 2 π \frac{z}{λ} (\frac{1}{2} X_{r}^{2}))$	$\exp (- i 2 π \frac{z}{λ} (\frac{1}{2} X_{f}^{2}))$
Polynomial-4	$\frac{z}{r^{2}} (\frac{1}{2 π r} + \frac{1}{i λ}) \exp (i 2 π \frac{z}{λ} (\frac{1}{2} X_{r}^{2} - \frac{1}{8} X_{r}^{4}))$	$\exp (- i 2 π \frac{z}{λ} (\frac{1}{2} X_{f}^{2} + \frac{1}{8} X_{f}^{4}))$
Rationalized	$\frac{z}{r^{2}} (\frac{1}{2 π r} + \frac{1}{i λ}) \exp (i 2 π \frac{z}{λ} \frac{X_{r}^{2}}{\sqrt{1 + X_{r}^{2}} + 1})$	$\exp (- i 2 π \frac{z}{λ} \frac{X_{f}^{2}}{\sqrt{1 - X_{f}^{2}} + 1})$

CPU	Intel Core i9-11900K 3.50 GHz
GPU	NVIDIA GeForce RTX 3060
CPU memory	DDR4-3200 64 GB
Operating system	Windows10 (Windows Subsystem for Linux, Ubuntu 20.04LTS)
C $+ +$	GNU Compiler Collection 9.3
CUDA	NVIDIA CUDA compiler driver 11.8

	RSC	ASM
Naïve-single (naïve-double)	$\exp (i 2 π \frac{z}{λ} (\sqrt{1 + X_{r}^{2}} - 1))$	$\exp (i 2 π \frac{z}{λ} (\sqrt{1 - X_{f}^{2}} - 1))$
Fresnel	$\exp (i 2 π \frac{z}{λ} (\frac{1}{2} X_{r}^{2}))$	$\exp (- i 2 π \frac{z}{λ} (\frac{1}{2} X_{f}^{2}))$
Polynomial-4	$\exp (i 2 π \frac{z}{λ} (\frac{1}{2} X_{r}^{2} - \frac{1}{8} X_{r}^{4}))$	$\exp (- i 2 π \frac{z}{λ} (\frac{1}{2} X_{f}^{2} + \frac{1}{8} X_{f}^{4}))$
Rationalized	$\exp (i 2 π \frac{z}{λ} \frac{X_{r}^{2}}{\sqrt{1 + X_{r}^{2}} + 1})$	$\exp (- i 2 π \frac{z}{λ} \frac{X_{f}^{2}}{\sqrt{1 - X_{f}^{2}} + 1})$

Abstract

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Figures (10)

Tables (4)

Equations (34)

Journal of the Optical Society of America A