Скачать или смотреть Diffraction - Positions of dark and bright fringes

The diffraction angle θ for a point B on the diffraction pattern is the angle that lies between the incident beam and the line joining the center of the slit and B. The diffraction angle at the center of a dark fringe (zero light intensity) is given by: sin (θn) = (n λ)/a.
λ is the wavelength of the emitted radiation in m. a is the width of the slit in m.
n is the order of the dark fringe: n = ±1, ±2, ±3 , …
For small angles sin θ ≅ θ(rad), then: θn = (n λ)/a.
For example:
The center of the first dark fringe on the positive side of the central bright fringe (C.B.F) corresponds to n = 1, so θ1 = ( λ)/a
The center of the first dark fringe on the negative side of the central bright fringe corresponds to n = 1 , so θ-1 = ( λ)/( a)
The center of the second dark fringe on the positive side of the central bright fringe corresponds to n = 2 , so θ2 = (2 λ)/a .
The diffraction angle at the center of the central bright fringe is θ(C.B.F) = 0
The angular width α of the central fringe is the angle through which the central fringe is seen from the center of the slit: α = 2θ1, then for small angles: α = (2 λ)/a
The linear width L of the central bright fringe is : L = α D = (2 λ D)/a
As the wavelength of the incident radiation increases or/and the dimensions of the aperture decreases, the width L increases and the diffraction pattern becomes broader.
Positions of the dark fringes: Take the center of the central bright fringe as the origin O of the x-axis. The position of the center of a dark fringe relative to O is given by : Xn = D tan θn . For small angles, Xn = D θn. But θn = (n λ)/a , then Xn = (n λ D)/( a )
For example:
The position of the center of the first dark fringe on the positive side corresponds to n = 1 , so X1 = ( λ D)/a
The position of the center of the first dark fringe on the negative side corresponds to n = -1 , so X-1 = (-λ D)/a
The position of the center of the second dark fringe on the positive side corresponds to n = 2 , so X2 = ( 2 λ D)/a
Remark: L = (2 λ D)/a , then Xn = n L/2
If the angles of diffraction are small, the central bright fringe would be twice as broad as other bright fringes; therefore, the width of a bright fringe is L/2.
The center of a bright fringe is approximately halfway between the centers of its bording dark fringes. Therefore, the distance between the center of a dark fringe and the center of the next bright fringe is approximately L/4.
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