Chapter 38: Diffraction Patterns and Polarization

38.1 Introduction to Diffraction Patterns Light of wavelength width of a slit ( >> a) Light of wavelength comparable to or larger than the width of a slit ( >> a) spreads out in all forward directions upon passing through the slit diffraction This phenomena is called diffraction This indicates that light spreads beyond the narrow path defined by the slit into regions that would be in shadow if light traveled in straight lines

Diffraction Pattern, Narrow Slit single slit diffraction pattern A single slit placed between a distant light source and a screen produces a diffraction pattern central maximum It will have a broad, intense central band, called the central maximum side maxima secondary maxima The central band will be flanked by a series of narrower, less intense secondary bands, called side maxima or secondary maxima minima The central band will also be flanked by a series of dark bands, called minima

Diffraction Pattern, Object Edge diffraction pattern edge of an opaque object This shows a diffraction pattern associated with light from a single source passing by the edge of an opaque object is vertical The diffraction pattern is vertical with the central maximum at the bottom

Diffraction Pattern, Illuminated Penny diffraction pattern This shows a diffraction pattern created by the illumination of a penny, positioned midway between screen and light source. The bright spot at the center. It is a constructive interference The center of the shadow is dark. The part of the viewing screen is completely shielded by the penny

38.2 Diffraction Pattern from Narrow Slits Fraunhofer diffraction pattern A Fraunhofer diffraction pattern occurs when the rays leave the diffracting object in parallel directions Screen very far from the slit Could be accomplished by a converging lens (  = 0) A bright fringe is seen along the axis (  = 0) brightdark Alternating bright and dark fringes are seen on each side

Active Figure 38.4 (SLIDESHOW MODE ONLY)

Diffraction vs. Diffraction Pattern Diffraction Diffraction refers to the general behavior of waves spreading out as they pass through a slit diffraction pattern A diffraction pattern is actually a misnomer (a mistake in naming a thing) that is deeply well-established in the language of physics interference The pattern seen on the screen is actually another interference pattern The interference incident light regions of the slit The interference is between parts of the incident light illuminating different regions of the slit

Single-Slit Diffraction The finite width the basis Fraunhofer diffraction The finite width of slits is the basis for understanding Fraunhofer diffraction According to Huygens’s principle: Each portion of the slit acts as a source of light waves Each portion of the slit acts as a source of light waves light from one portioninterfere light from another portion Therefore, light from one portion of the slit can interfere with light from another portion

Single-Slit Diffraction, 2 resultant light intensity depends direction  The resultant light intensity on a viewing screen depends on the direction  The diffraction pattern interference pattern The diffraction pattern is actually an interference pattern The different sources of light are different portions of the single slit The different sources of light are different portions of the single slit

Single-Slit Diffraction, 3 All the waves originate at the slitare in phase All the waves that originate at the slit are in phase travels farther path difference Wave 1 travels farther than wave 3 by an amount equal to the path difference (a/2) sin 

Single-Slit Diffraction, Final path difference (± /2)cancel each otherdestructive interference If this path difference is exactly half of a wavelength, (± /2) the two waves cancel each other and destructive interference results, so: (a/2) sin  = ± /2  sin  = ± /a sin  = ± 2 /a If the slit is divided in 4 equal parts: sin  = ± 2 /a sin  = ± 3 /a If the slit is divided in 6 equal parts: sin  = ± 3 /a destructive interference a In general, destructive interference occurs for a single slit of width a when(38.1)

Single-Slit Diffraction, Intensity intensity distribution The general features of the intensity distribution are shown A broad central bright fringe is flanked by much weaker bright fringes alternating with dark fringes A broad central bright fringe is flanked by much weaker bright fringes alternating with dark fringes Each bright fringe peak lies approximately halfway between the dark fringes Each bright fringe peak lies approximately halfway between the dark fringes The central bright maximum is twice as wide as the secondary maxima The central bright maximum is twice as wide as the secondary maxima

Example 38.1 Where Are the Dark Fringes? Light with = 580 nm reaches a slit with a = 0.3 mm. The viewing screen is at 2.0 m from the slit. Find: A) The position of the first dark fringes B) The width of the central bright fringe

Intensity of Single-Slit Diffraction Patterns Phasors light intensity distribution single-slit diffraction pattern Phasors can be used to determine the light intensity distribution for a single-slit diffraction pattern Slit width (a) Slit width (a) can be thought of as being divided into zones The zones have a width of  y The zones have a width of  y coherent radiation Each zone acts as a source of coherent radiation

Intensity of Single-Slit Diffraction Patterns, 2 Each zone incremental electric field  E Each zone contributes an incremental electric field  E at some point on the screen total electric field The total electric field can be found by summing the contributions from each zone out of phase  The incremental fields between adjacent zones are out of phase with one another by an amount (38.2)

Phasor for E R,  = 0  = 0 and all phasors are aligned E o = N  E The field at the center is E o = N  E N is the number of zones N is the number of zones E R is a maximum E R is a maximum

Phasor for E R, Small  differs in phase  Each phasor differs in phase from an adjacent one by an amount of  E R is the vector sum E R is the vector sum of the incremental magnitudes E R < E o The total phase difference is: The total phase difference is: (38.3) (38.3)

Phasor for E R,  = 2   increases closed path As  increases, the set of phasors eventually forms a closed path E R =0 The vector sum is zero, so E R =0 This corresponds to a minimum on the screen This corresponds to a minimum on the screen  = N  = 2π At this point,  = N  = 2π Therefore, from (38.3): sin  dark = / a a = N  y Remember, a = N  y is the width of the slit This is the first minimum This is the first minimum

Phasor for E R,  > 2π larger values of  spiral chain tightens At larger values of , the spiral chain of the phasors tightens This represents the second maximum This represents the second maximum  = 360 o o = 540 o = 3π rad A second minimum would occur at 4π rad A second minimum would occur at 4π rad

Resultant E, General limiting condition  y becoming infinitesimal (dy) N  ∞ Consider the limiting condition of  y becoming infinitesimal (dy) and N  ∞ The phasor chains become the red curve The phasor chains become the red curve From Figure: E o = R  (arc length) E R = 2Rsin(  /2) (chord length) Therefore:

Intensity light intensity square of E R (38.4) The light intensity at a point on the screen is proportional to the square of E R : (38.4) I max intensity  = 0 I max is the intensity at  = 0 This is the central maximum This is the central maximum Substitution of (38.3) into (38.4) gives:(38.5) Minima are at Minima are at

Intensity, final Most of the light intensity is concentrated in the central maximum Most of the light intensity is concentrated in the central maximum The graph shows a plot of light intensity I vs.  /2 The graph shows a plot of light intensity I vs.  /2

Example 38.2 Relative Intensities of the Maxima Find the ratio of intensities of the secondary maxima to the intensity of the central maxima for the single-slit Fraunhofer diffraction pattern. From the figure we see that the secondary maxima occur at  /2 = 3  /2, 5  /2, 7  /2, … Using (38.4)

Examples to Read!!! Examples to Read!!! NONE NONE Homework to be solved in Class!!! Homework to be solved in Class!!! Problems: 2, 9 Problems: 2, 9 Material for the Final Exam