Lecture 13: Fraunhofer diffraction by two slits
Lecture aims to explain:
1. Calculation of the diffraction pattern for light diffracted by two slits
2. Properties of diffraction pattern for light diffracted by two slits
3. “Missing orders” in diffraction pattern produced by two slits Calculation of the diffraction pattern for light diffracted by two slits , The Double Slit integral Electric field (compare with Lecture 11): b / 2 a+b / 2 E = C ∫ F( z )dz + C ∫ F( z )dz −b / 2 a−b / 2 where: F( z ) = sin[ωt − k( R − z sinθ )] 2π phase difference: ( R − z sinθ ) ≡ k( R − z sinθ ) λ
Using result for a single slit (see Lecture 12): sin β E = bC [sin(ωt − kR ) + sin(ωt − kR + 2α )] β Simplifying further: α ≡ ( ka / 2 )sinθ sin β E = 2bC cosα sin(ωt − kR +α ) β ≡ ( kb / 2 )sinθ β Analytical expression for diffraction by two slits
The pattern formed due to diffraction by two slits is given by: 2 sin β 2 I(θ ) = I(0) cos α β
α ≡ (πa / λ)sinθ b- slit width, a- slit separation β ≡ (πb / λ)sinθ
Term interference will be used for those cases in which a modification of the amplitude is determined by the superposition of a finite number of beams (or wavelets).
Term diffraction is used for those cases in which the amplitude is determined by an integration over the infinitesimal elements of the wavefront. Properties of the diffraction pattern for light diffracted by two slits 1.0 Diffraction Zoom-in at
pattern small angles produced by Intensity of
two slits: diffracted light zoom-in for 0.5 small angles
0.0 -0.04 -0.02 0.00 0.02 0.04 sin (θ)
2 sin β 2 Zeros: β = ±π ,±2π ,±3π ... I(θ ) = I(0) cos α β α = ±π / 2,±3π / 2,±5π / 2... Diffraction pattern produced by two slits: full view 2 sin β 2 I(θ ) = I(0) cos α Single slit β
Intensity maxima: α = ±π ,±2π ,±3π... asinθ = mλ
a- slit separation, m-integer Double slit
-0.15 -0.10 -0.05 0.00 0.05 0.10 0.1 sin (θ) “Missing orders” in diffraction by two slits “Missing orders” 0.2 0.6
0.4 a=5b
0.0 0.2 0.05 0.10 0.15 0.2 sin (θ) Intensity of Diffracted Light 0.0 2 0.00 0.02 0.04 0.06 sin β 2 sin (θ) I(θ ) = I(0) cos α β A missing order occurs when the “diffraction minimum” overlaps with the “interference maximum” Example 13.1
A parallel beam of monochromatic light falls normally on a screen containing two parallel slits each of width 0.1 mm and 0.8 mm distance between centres. Which Fraunhofer diffraction orders are missing? Example 13.2
A screen with two slits with adjustable width is illuminated at normal incidence with monochromatic light. The centres of the slits are separated by 0.5 mm. A convex lens with a focal length of 1 m is positioned behind the screen. Another screen is placed in the focal plane of the lens and is used for imaging the diffraction pattern produced by the slits.
(i) One of the slits is blocked. The other is opened to 50 micron. Find the full width of the central intensity maximum in the diffraction pattern obtained in the focal plane of the lens if the slit is illuminated with light having wavelength λ=500 nm.
(ii) Calculate how the width of the central intensity maximum will change if the second slit, also opened to 50 micron, is unblocked.
(iii) Which interference orders are clearly observed if the width of both slits is increased to 100 micron? Illustrate your answer with a graph depicting the intensity of the diffracted light in the focal plane of the lens. Example 13.3
In the intensity distribution of light diffracted by two slits calculate the full width at half maximum (FWHM) of the interference fringe. SUMMARY The Fraunhofer diffraction pattern formed by two slits is given by: α ≡ (πa / λ)sinθ 2 sin β 2 β ≡ (πb / λ)sinθ I(θ ) = I(0) cos α β b- slit width, a- slit separation Sharp maxima occur due to Intensity maxima occur: constructive interference of α = ±π ,±2π ,±3π... light emerging from the two slits. Their intensity is modulated by asinθ = mλ the envelope due to diffraction by each individual slit. 0.6
0.4 -0.15 -0.10 -0.05 0.00 0.05 A missing order occurs when the sin (θ) 0.2 “diffraction minimum” overlaps with the “interference maximum” Intensity of Diffracted Light0.0 0.00 0.02 0.04 0.06 sin (θ)