# 2 nd & 3 th N.U.T.S. Workshops Gulu University Naples FEDERICO II University 5 – Diffraction.

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2 nd & 3 th N.U.T.S. Workshops Gulu University Naples FEDERICO II University 5 – Diffraction

5- Diffraction 2nd & 3th NUTS Workshop ( Jan 2010) 2 Huygens(–Fresnel) Principle 1 The Huygens principle is a method of analysis applied to problems of wave propagation. It recognizes that each point of an advancing wave front is in fact the centre of a new secondary disturbance and the source of a new train of spherical waves. The advancing wave as a whole may be regarded as the sum of all the secondary spherical waves arising from points in the medium already traversed. The Huygens view of wave propagation helps better understand a variety of wave phenomena, such as reflection, refraction, diffraction, interference, …

5- Diffraction 2nd & 3th NUTS Workshop ( Jan 2010) 3 Huygens Principle 2 12 At time t0 a plane wave front, propagating from left to right at a speed v, reach the position 1 All the points of position 1 became sources of spherical waves expanding at speed v After a time interval  t, the radius of the spherical waves will be v  t; the sum (envelope) of all the secondary spherical waves will give the new position of the plane wave front at 2 The same is in the case of spherical wave front

5- Diffraction 2nd & 3th NUTS Workshop ( Jan 2010) 4 Huygens Principle 3 -Any small area dΣ of wave front is source of secondary spherical waves with the same frequency f of the primary wave -Amplitude of secondary wave is proportional to that of primary wave and area dΣ; varies as cos(  - The wave front is superposition of all spherical waves reaching P; wave front = envelope of all wave fronts of secondary waves

5- Diffraction 2nd & 3th NUTS Workshop ( Jan 2010) 5 Schematic representation of a plane wave arriving on a single slit Huygens Principle 4

5- Diffraction 2nd & 3th NUTS Workshop ( Jan 2010) 6 Interference The term interference usually refers to the interaction of waves which are coherent with each other (because they come from the same splitted source) and have the same or nearly the same frequency. The interaction depends on total phase difference = sum of path difference and initial phase difference (if the waves are from 2 or more different sources) If the waves reaching a point P are in phase the result is constructive interference, if out of phase destructive interference occur. W1W1 W2W2 W 1 +W 2

5- Diffraction 2nd & 3th NUTS Workshop ( Jan 2010) 7 Destructive Interference  d1d1 d2d2 At the intersection of two directions the crest of blue wave and the through of red wave meet: it is destructive interference There we have (d 1 -d 2 )=n λ/2 (n integer odd)

5- Diffraction 2nd & 3th NUTS Workshop ( Jan 2010) 8 Constructive Interference  d1d1 d2d2 At the intersection of two directions the crests of blue wave and of red wave superpose: it is constructive interference at maxima (d 1 -d 2 )=m λ (m integer)

5- Diffraction 2nd & 3th NUTS Workshop ( Jan 2010) 9 Diffraction Light does not always travel in a straight line. It tends to bend around objects. This phenomenon is called diffraction. Any wave will diffract, including matter and acoustic waves. Shadow of a hand illuminated by a Helium-Neon laser. Note the bright and dark lines around the fingers, they are caused by diffraction.

5- Diffraction 2nd & 3th NUTS Workshop ( Jan 2010) 10 Diffraction and Huygens principle Diffraction Applet

5- Diffraction 2nd & 3th NUTS Workshop ( Jan 2010) 11 Slit Diffraction: the Idea λ r1r1 r2r2 a if (r 1 -r 2 ) = m λ  constructive interference (m integer) if (r 1 -r 2 ) = n λ/2  destructive interference (n integer odd) Problem: we have to account for all the points (infinite) of the slit !!!

5- Diffraction 2nd & 3th NUTS Workshop ( Jan 2010) 12 Slit Diffraction: the Experiment λ r a x x′ L θ

5- Diffraction 2nd & 3th NUTS Workshop ( Jan 2010) 13 Slit Diffraction: the Solution

5- Diffraction 2nd & 3th NUTS Workshop ( Jan 2010) 14 Slit Diffraction: Maxima (& Minima) Maxima: Minima: Slit Diffraction Applet General behaviors: - The smaller the slit width, the 'wider' the resulting diffraction pattern; - Diffraction angles are invariant under scaling: they depend only on the ratio wavelength/size of the slit; - Different λ ‘s peak at different angles. m positive or negative integer; there is a m max ? n positive or negative integer; except n = 0 where there is the main maximum

5- Diffraction 2nd & 3th NUTS Workshop ( Jan 2010) 15 Slit Diffraction: Different λ Different λ  different location and width of maxima with white light  no diffraction pattern but … blurred image

5- Diffraction 2nd & 3th NUTS Workshop ( Jan 2010) 16 Nice Diffraction Diffraction by spider netDiffraction by squared slit (using expanded red laser pointer)

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