Lecture 22 Interference, Diffraction

Lecture 22 Interference, DiffractionLooking forward at what happens when two waves combine, or interfere, in space. how to understand the interference pattern formed by the interference of twocoherent light waves. What is the pattern of interference from two slits, when the screen at which weproject it is very far away: double slit (Young’s) experiment. what is a single-slit diffraction pattern. what happens when coherent light shines on an array of narrow, closely spaced slits. how x-ray diffraction reveals the arrangement of atoms in a crystal. how diffraction sets limits on the smallest details that can be seen with an opticalsystem. 2016 Pearson Education Inc.

Introduction Why do soap bubbles showvibrant color patterns, eventhough soapy water iscolorless? What causes the multicoloredreflections from DVDs? We will now look at optical effects, such as interference, thatdepend on the wave nature of light. 2016 Pearson Education Inc.

Principle of superposition The term interference refers to any situation in which two ormore waves overlap in space. When this occurs, the total wave at any point at any instant oftime is governed by the principle of superposition:When two or more waves overlap, the resultantdisplacement at any point and at any instant is found byadding the instantaneous displacements that would beproduced at the point by the individual waves if eachwere present alone.For EM waves: displacements- E and B fields 2016 Pearson Education Inc.

Wave fronts from a single source Interference effects aremost easily seen when wecombine sinusoidal waveswith a single frequency andwavelength. Shown is a “snapshot” of asingle source S1 ofsinusoidal waves and someof the wave fronts producedby this source. 2016 Pearson Education Inc.

Constructive and destructive interference Shown are two identicalsources of monochromaticwaves, S1 and S2. The two sources arepermanently in phase; theyvibrate in unison. Constructive interferenceoccurs at point a(equidistant from the twosources). 2016 Pearson Education Inc.

Conditions for constructive interference The distance from S2to point b is exactlytwo wavelengthsgreater than thedistance from S1 to b. The two waves arrivein phase, and theyreinforce each other. This is calledconstructiveinterference. 2016 Pearson Education Inc.

Conditions for destructive interference The distance from S1to point c is a halfintegral number ofwavelengths greaterthan the distance fromS2 to c. The two waves cancelor partly cancel eachother. This is calleddestructiveinterference. 2016 Pearson Education Inc.

Constructive and destructive interference Shown are two identical sources ofmonochromatic waves, S1 and S2,which are in phase and at a distancemλ d (m 1)λ The red curves show all positionswhere constructive interferenceoccurs; these curves are calledantinodal curves. Not shown are the nodal curves,which are the curves that show wheredestructive interference occurs. The destructive interference is notperfect for ! 0 and small distances

Two-source interference of light Shown below is one of the earliest quantitative experimentsto reveal the interference of light from two sources, firstperformed by Thomas Young. The interference of waves from slits S1 and S2 produces apattern on the screen. 2016 Pearson Education Inc.

Two-source interference of light (b) shows the actual geometry of Young’s experiment. If the distance R to the screen is much greater than thedistance d between the slits, we can use the approximategeometry shown in (c). 2016 Pearson Education Inc.

Interference from two slits Constructive interference (reinforcement) occurs at pointswhere the path difference is an integral number ofwavelengths, mλ. Here λ is the wavelength in So the bright regions on the screen occur at angles θ for which Similarly, destructive interference (cancellation) occurs,forming dark regions on the screen, at points for which thepath difference is a half-integral number of wavelengths. 2016 Pearson Education Inc.

Interference in thin films 2016 Pearson Education Inc.

Phase shifts during reflection 2016 Pearson Education Inc.

Interference in thin films For light of normal incidence on a thin film with wavelength λ in the film, inwhich neither or both of the reflected waves have a half-cycle phase shift: When only one of the reflected waves has a half-cycle phase shift : Δ" &'2*( 2 , Dividing by&'(we get: Here λ is the wavelength in the film! λ λ0/n , where n is the index ofrefraction of the film.

Nonreflective coatings 2016 Pearson Education Inc.

Diffraction from a single slit 2016 Pearson Education Inc.

Diffraction from a single slit 2016 Pearson Education Inc.

Fresnel diffraction by a single slit 2016 Pearson Education Inc.

Frensel/Poisson/Arago spotThe weird phenomenon that convinced French academy that lightis a wave.D – diameter of diskL– distance to screenD 2mmL 1mD 1mmL 1m

Fraunhofer diffraction by a single slit 2016 Pearson Education Inc.

Locating the dark fringes Shown is the Fraunhofer diffraction patternfrom a single horizontal slit. It is characterized by a central bright fringecentered at θ 0, surrounded by a series ofdark fringes. The central bright fringe is twice as wideas the other bright fringes. Opposite to the 2-slit interference: 2016 Pearson Education Inc.

Intensity maxima in a single-slit pattern Shown is the intensity versusangle in a single-slitdiffraction pattern. Most of the wave power goesinto the central intensity peak(between the m 1 andm 1 intensity minima).(For illustration only, won’t appear onthe Final) 2016 Pearson Education Inc.

Several slits Shown is an array ofeight narrow slits, withdistance d betweenadjacent slits. Constructive interferenceoccurs for rays at angle θto the normal that arriveat point P with a pathdifference betweenadjacent slits equal to aninteger number ofwavelengths. 2016 Pearson Education Inc.

The diffraction grating An array of a large number of parallelslits is called a diffraction grating. In the figure,is a cross sectionof a transmission grating. The slits are perpendicular to theplane of the page. The diagram shows only six slits; anactual grating may contain severalthousand. 2016 Pearson Education Inc.

Diagram of a grating spectrograph 2016 Pearson Education Inc.

X-ray diffraction When x rays pass through a crystal, the crystal behaves like adiffraction grating, causing x-ray diffraction. 2016 Pearson Education Inc.

A simple model of x-ray diffraction To better understand x-raydiffraction, we consider atwo-dimensional scatteringsituation. The path length from sourceto observer is the same forall the scatterers in a singlerow if θa θr θ. 2016 Pearson Education Inc.

Circular apertures The diffraction pattern formed by a circular aperture consistsof a central bright spot surrounded by a series of bright anddark rings. 2016 Pearson Education Inc.

Diffraction by a circular aperture The central bright spot in thediffraction pattern of acircular aperture is called theAiry disk. We can describe the radius ofthe Airy disk by the angularradius θ1 of the first darkring: 2016 Pearson Education Inc.

Diffraction and image formation Diffraction limits the resolution ofoptical equipment, such as telescopes. The larger the aperture, the better theresolution. A widely used criterion for resolutionof two point objects, is calledRayleigh’s criterion:- Two objects are just barely resolved(that is, distinguishable) if the center ofone diffraction pattern coincides withthe first minimum of the other. 2016 Pearson Education Inc.

Lecture 22 Interference, Diffraction Looking forward at what happens when two waves combine, or interfere, in space. how to understand the interference pattern formed by the interference of two