The Wave Nature of Light - Sections 24.4 - 24.6

Announcements Review Diffraction by a Single Slit Final QuestionsReading AssignmentRead section 24.10Homework Assignment 10Homework for Chapter 24 (due at the beginning of class on Friday, November 5)Q: 4, 5, 10, 17, 22, 30P: 4, 20, 30, 58PP: 37.2, 38.4The Wave Nature of Light

Announcements Review Diffraction by a Single Slit Final QuestionsYoung’s double-slit experiment: interference patternThe slits S 1 and S 2 are separated by a distance d, and the source is monochromaticAssume that the viewing screen is located a perpendicular distance L ≫ d from the barrier containingthese slitsTo reach any arbitrary point P in the upper half of the screen, a wave from the lower slit must travelfarther than a wave from the upper slit by a distance (called the path distance ∆D)∆D = D 1 − D 2 = d sinθThe value of ∆D determines whether the two waves are in phase when they arrive at point PIf ∆D is zero or some integer multiple of the wavelength, the two waves are in phase at point P andconstructive interference occursTherefore, the condition for bright fringes (or constructive interference) at point P isd sinθ bright = mλ m = 0,±1,±2,...When ∆D is an odd multiple of λ/2, the two waves arriving at point P are 180 ◦ out of phase and giverise to destructive interferenceTherefore, the condition for dark fringes (or destructive interference) at point P is(d sinθ dark = m + 1 )λ m = 0,±1,±2,...2The Wave Nature of Light

Announcements Review Diffraction by a Single Slit Final QuestionsQuestionA viewing screen is separated from a double slit by 4.80 m. The distance between the two slits is 0.0300 mm.Monochromatic light is directed toward the double slit and forms and interference pattern on the screen. The firstdark fringe is 4.50 cm from the center line on the screen. Determine the wavelength of the light.The Wave Nature of Light

Announcements Review Diffraction by a Single Slit Final QuestionsQuestionA viewing screen is separated from a double slit by 4.80 m. The distance between the two slits is 0.0300 mm.Monochromatic light is directed toward the double slit and forms and interference pattern on the screen. The firstdark fringe is 4.50 cm from the center line on the screen. Determine the wavelength of the light.AnswerLet y = 4.50 cm be the distance of the first dark fringe from the center lineThen, tanθ dark = y/L, where L = 4.80 m is the distance to the viewing screenSolving for θ dark , we find thatθ dark = tan −1 ( yL) ( )= tan −1 4.50 × 10−2 m= 0.537 ◦4.80 mFurthermore, dark fringes are caused by destructive interference so(d sinθ dark = m + 1 )λ2Solving for λ (and substituting m = 0), we find thatλ = 2d sinθ dark = 2(0.0300 × 10 −3 m)sin(0.537 ◦ ) = 5.62 × 10 −7 m = 562 nmThe Wave Nature of Light

Announcements Review Diffraction by a Single Slit Final QuestionsThe wave theory of light: a historical perspectiveNewton’s view of light – that it consisted of a stream of particles – was the prevailing view in mostscientific circles of the early 19th centuryDuring Newton’s lifetime, however, there was another theory, advanced by the Dutch physicist ChristianHuygens in 1678, that light was a wave (Huygens’ principle)In 1801 – 123 years after Hugyens first proposed that light was a wave – Thomas Young experimentallydemonstrated the wave nature of light (the double slit experiment) by demonstrating interferenceHowever, even with this remarkable demonstration, the scientific community was slow to adopt thewave-theory explanationIn 1819, the French Academy of Sciences organized a competition for an essay on diffraction – AugustinFresnel, a French physicist and supporter of the wave-theory of light, won (Huygens-Fresnel principle)The Academy, however, was unconvinced, with the preeminent French mathematician Siméon Poissonpointing out this “strange result”:If Fresnel’s theories were correct, then light waves should diffractinto the shadow region of a sphere as they pass its edge, producinga bright spot at the center of the shadowHowever, the French physicist and mathematician Dominique Arago soon experimentally verified thisstartling prediction (a Poisson bright spot)The Wave Nature of Light

Announcements Review Diffraction by a Single Slit Final QuestionsThe Wave Nature of Light

Announcements Review Diffraction by a Single Slit Final QuestionsSingle-slit diffractionPreviously we saw what happens when monochromatic light from a distant source passes through twonarrow slits (an interference pattern is formed)Now we will examine what happens when monochromatic light from a distant source passes through onenarrow slitRemarkably, the diffraction pattern formed on a viewing screen consists of a very bright central maximumand a number of narrower and less bright secondary maxima to both sidesThis pattern would be total unexpected in geometrical opticsKeep in mind that the diffraction of light is not limited to situations of light passing through a narrowopening; it also occurs when light passes an edgeThe Wave Nature of Light

Announcements Review Diffraction by a Single Slit Final QuestionsSingle-slit diffraction: locating the minimaWe will now examine how coherent, monochromatic light is diffracted by a single narrow slit of width aWhen the diffracted light reaches the viewing screen, waves from different points within the slit undergointerference and produce a diffraction pattern of bright and dark fringesThe single-slit experiment is more mathematically challenging than the double-slit experiment; as a result,we will find equations for the dark fringes onlyTo find the dark fringes, we will pair up all of the rays coming through the slit in a clever way, and thendetermine what conditions cause each pair to cancel each otherUsing this procedure, we find that the condition for minima (dark fringes) isasinθ = mλ m = 1,2,3,...In other words, in a single-slit diffraction experiment, dark fringes are produces when the path lengthdifferences (asinθ) between the top and bottom rays are positive integer multiples of λThis result may seem counterintuitive, because the waves of those two particles rays will be exactly inphase with each otherHowever, the result holds, because each of these rays will be part of a pair of waves that are exactly out ofphase with each otherThe Wave Nature of Light

Announcements Review Diffraction by a Single Slit Final QuestionsThe Wave Nature of Light

Announcements Review Diffraction by a Single Slit Final QuestionsQuestionLight of wavelength 633 nm is incident on a narrow slit. The angle between the first diffraction minimum on oneside of the central maximum and the first minimum on the other side is 1.20 ◦ . What is the width of the slit?The Wave Nature of Light

Announcements Review Diffraction by a Single Slit Final QuestionsQuestionLight of wavelength 633 nm is incident on a narrow slit. The angle between the first diffraction minimum on oneside of the central maximum and the first minimum on the other side is 1.20 ◦ . What is the width of the slit?AnswerThe width of the slit is given bywhere θ is measured from the center linea = mλsinθIn this problem, m = 1 and θ = 0.60 ◦ (why?)Therefore, solving for a, we find thatm = 1,2,3,...a = (1)(6.33 × 10−7 m)sin(0.60 ◦ )= 6.04 × 10 −5 m = 60.4 µmThe Wave Nature of Light

Announcements Review Diffraction by a Single Slit Final QuestionsReading AssignmentRead section 24.10Homework Assignment 10Homework for Chapter 24 (due at the beginning of class on Friday, November 5)Q: 4, 5, 10, 17, 22, 30P: 4, 20, 30, 58PP: 37.2, 38.4The Wave Nature of Light