使用傅里叶变换测量夫劳恩霍夫衍射条纹宽度的改进
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2021-03-29
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Improvements of measuring the width of
Fraunhofer diffraction fringes using Fourier
transform
Yicheng Wu, Jialin Ma, Yi Yang and Ping Sun*
Department of Physics, Beijing Normal University, Beijing 100875, China
Email: pingsun@bnu.edu.cn
Abstract
We proposed an improved method for calculating the fringe width of Fraunhofer diffraction. The
diffraction patterns is equidistant when the diffraction angle is small, so the width of fringes can be
determined through finding out the spatial frequency of the fringes using Fourier transform. We applied
the method to measure the diameters of some filaments using a hand-made setup. This study will help
students to understand the wide application of Fourier transform in physical experiments.
Introduction
Diffraction is a common phenomenon in our daily lives and of much greater importance in the optical
course of university students. If wavefronts arriving at the aperture and observation screen can be
considered plane, we speak of Fraunhofer diffraction [1]. Nowadays, the Fraunhofer diffraction has
been widely applied to measure the diameter of filaments [2, 3]. Especially when the charge-coupled
device (CCD) is adopted [4, 5], the real-time and noninvasive measurement is realized.
In general, the traditional method to measure the diameter of filaments is to find the extrema of
intensity in every period of diffraction patterns and then get the average of the fringe widths. This
method demands a high-quality laser and CCD. In addition, the speckle noises of laser will influence
the locations of extrema. Therefore, the
filtering process is necessary to improve the signals of
diffraction pattern [6], which is quite difficult for undergraduate students. In this study, we use Fourier
transform [7], a basic means in mathematics, to find out the
spatial frequency of the diffraction pattern
and calculate the width of fringes, which is precise and convenient.
Basic theory
Figure 1 shows the diagram of light propagation of
single-slit Fraunhofer diffraction. All rays passing
through the slit are parallel. A convex lens is used to
converge parallel light rays to a focus P in the focal
plane of the lens. The focal length of convex lens is f.
Suppose that the light is diffracted by a slit of width a at
an angle θ, the intensity of point P can be written
2
0
sin( sin / )
sin /
a
II
a
πθλ
πθλ
⎡⎤
=
⎢⎥
⎣⎦
(1)
where
λ
is the incident wavelength, I
0
is the intensity at the central position x=0. The destructive
interference will occur when
sin , 1, 2, 3,
m
amm
θ
λ
=
=± ± ± " (2)
If the diffraction angle θ
m
is very small, there is
Figure 1. Diagram of light propagation of single-sli
t
Fraunhofer diffraction.
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