Optics Wave Behavior in Optics

Transcription

1 Optics Wave Behavior in Optics Lana Sheridan De Anza College June 19, 2017

2 Last time mirrors image terminology images formed by spherical mirrors images formed by refraction images formed by lenses (images formed by lens combinations)

3 Overview Huygen s Principle Interference of light: the Double-Slit experiment diffraction gratings diffraction patterns diffraction and interference resolution and Raleigh s criterion

4 Moving beyond Ray Optics Ray optics can tell us about the formation of images, but it cannot tell us about the behavior of light that is specific to waves. The ray approximation assumes that light will travel out from a source in straight lines, unless it encounters a change of medium. However light does also behave a wave and can change direction at apertures: diffraction and interference.

5 Huygens s Principle Applied to Reflection and Refraction Huygens Principle We now derive the laws of reflection and refraction, using Huygens s principle. For the law of reflection, refer to Figure The line AB represents a plane wave front This of the is aincident geometric light principle just as ray 1 tostrikes construct the surface. a wavefront At this instant, at a later the wave at A sends pointout a time Huygens fromwavelet a wavefront (appearing at at ana earlier later time one. as the light brown circular arc passing through D); the reflected light makes an angle g9 with the surface. At the A The new wave front is drawn tangent to the circular wavelets radiating from the point sources on the original wave front. B Th se 1 c t 2 Old wave front c t New wave front Old wave front New wave front 1 A B a Figure Huygens s construction for (a) a plane wave b Fig

6 front of the incident light just as ray 1 strikes the surface. At this instant, the wave at A sends out a Huygens wavelet (appearing at a later time as the light brown circular arc passing through D); the reflected light makes an angle g9 with the surface. At the Huygens Principle A The new wave front is drawn tangent to the circular wavelets radiating from the point sources on the original wave front. B This wave sent out b 1 from po c t 2 Old wave front c t New wave front Old wave front New wave front 1 u 1 u A B A Huygens Principle a Figure Huygens s construction for (a) a plane wave propagating to the right and (b) a spherical wave propagating to the right. Every point on a propagating wavefront can be thought of as the source of a new spherical secondary wavelet. The wavefront formed at a later time is the surface tangent (or envelope) of all these wavelets. b Figure 35 struction f reflection.

7 ras, and eyeglasses. Huygens Principle odel: Wave Under Reflection Huygens motivation for this principle was the idea of the existence of an ether: each part of the light wave would interact with particles composing the ether. cept of reflection of waves in a discussion of waves on s with waves on strings, when a light ray traveling in one oundary with another medium, part of the incident light Now we know there is no ether. ue e alid. When l d, the rays When l.. d, the opening This principle is still useful, since it gives us the correct idea of spread out after passing behaves as a point source what through happens the opening. variousemitting circumstances, spherical waves. including diffraction. l d l.. d Figure 35.4 A plane wave of wavelength l is incident on a barrier in which there is an opening b c of diameter d. See page in the textbook for a derivation of Snell s Law from Huygens Principle.

8 Huygens-Fresnel Principle Fresnel added the concept of interference to Huygens Principle, which explained the paths that rays of light seem to take. Kirchhoff then showed that this new Huygens-Frensel Principle was a consequence of the wave equation, so it can be expressed in a rigorous mathematical way. In fact, working from Kirchhoff s mathematical approach, this idea can be used to relate wavefronts at a light source to an image on a screen using Fourier Transforms.

9 Young s Double-Slit Experiment Thomas Young in 1801 did the first experiment that conclusively showed the interference of light from 2 sources. He filtered sunlight to make as source of red light, then shone the light through a series of narrow apertures (slits). The first screen blocks all but tiny source of light. This is called the collimating slit, and ensure the light reaching the other two slits will be coherent. 1

10 Young s Double-Slit Experiment The filtered coherent light then goes through two slits cut from the same mask. The light from these two sources interferes. The light strikes a screen where bright and dark areas can be seen.

11 Young s Double-Slit Experiment If light could be modeled as a particle, then one would expect to see two bright patches, one for each slit. This is not what Young observed. 1

12 Young s Double-Slit Experiment A pattern of light and dark fringes (stripes of light and darkness) appear on the screen. Zoomed in view:

13 Young s Experiment: Interference 37 Wave Optics e slits s on es Constructive interference occurs at point O when the waves combine. Constructive interference also occurs at point P. Destructive interference occurs at point R when the two waves combine because the lower wave falls one-half a wavelength behind the upper wave. S 1 S 2 Bright O fringe S 1 S 2 P O Bright fringe S 1 S 2 P R Dark fringe O Viewing screen a b In places where the path lengths differ by a whole number of wavelengths (mλ) there is constructive interference. the upper one by exactly one wavelength, they still arrive in phase at P and a second bright fringe appears at this location. At point R in Figure 37.3c, however, between points O and P, the lower wave has fallen half a wavelength behind the upper wave and a crest of the upper wave overlaps a trough of the lower wave, giving rise to destructive interference at point R. A dark fringe is therefore observed at this 1 Serway location. & Jewett. c

14 Young s Experiment: Finding the Maxima d 5 r 2 2 r 1 5 d sin u The value of d determines whether the two waves are in phase w Consider point light P. If that d is either is diffracted zero or through some each integer slit multiple and an angle of the θ. wavelen P r 1 S 1 y S 1 d Q S 2 u u u d O d r 2 S 2 d r 2 L Viewing screen When we as parallel to r difference b rays is r 2

15 Young s Experiment: Finding the Maxima If the distance to the screen is very much larger that the distance between the slits (it will be), then φ is very small. P S 1 r 1 y d Q u u r 2 O S 2 d L Viewing screen cos φ 1 and r 1 = r 2 δ. The path difference is δ!

16 37.5b), which is approximately true if L is much greater than d, then d is given by Young s Experiment: Finding the Maxima d 5 r 2 2 r 1 5 d sin u (37.1) The value of d determines whether the two waves are in phase when they arrive at point Effectively, P. If d is either the two zero rays or some areinteger parallel. multiple of the wavelength, the two waves P r 1 r 1 S 1 y S 1 r 2 d Q S 2 u u r 2 d u O d S 2 d r 2 r 1 d sin u L Viewing screen When we assume r 1 is parallel to r 2, the path difference between the two rays is r 2 r 1 d sin u. a Looking at the right triangle with hypotenuse d (the slit separation distance): δ = d sin θ. b

17 Young s Experiment: Finding the Angles of the Maxima Maxima (bright fringes) occur when d sin θ max = mλ where m Z Minima (dark fringes) occur when d sin θ min = ( m + 1 ) λ 2 where m Z These expressions give us the angles (measured outward from the axis that passes through the midpoint of the slits) where the bright and dark fringes occur.

18 Order Number m is the order number. The central bright fringe is the 0th order fringe, the neighboring ones are the 1st order fringes, etc. 1 Figure from Quantum Mechanics and the Multiverse by Thomas D. Le.

19 ence d (Greek letter delta). If we assume the rays labeled r 1 and r 2 ar 37.5b), which is approximately true if L is much greater than d, then d Young s Experiment: Finding the Position of the Maxima d 5 r 2 2 r 1 5 d sin u We canthe alsovalue predict of d the determines distance whether from the center two of waves the are screen, in phase y, when in terms point of the P. If distance d either from zero the or slits some tointeger the screen, multiple L. of the wavelength P r r 1 S 1 y S 1 d Q r 2 u u u d O S 2 d S 2 d r 2 r 1 L tan θ = y L Viewing screen When we assum parallel to r 2, th difference betwe rays is r 2 r 1

20 Young s Experiment: Finding the Position of the Maxima Maxima (bright fringes) occur at y max = L tan θ max Minima (dark fringes) occur at y min = L tan θ min

21 Young s Experiment: Finding the Position of the Maxima When θ is also small, sin θ tan θ, and we can use our earlier expressions for the fringe angles. Maxima (bright fringes) occur at y max = L mλ d (small θ) Minima (dark fringes) occur at y min = L ( m + 1 2) λ d (small θ)

22 Young s Experiment Question Quick Quiz Which of the following causes the fringes in a two-slit interference pattern to move farther apart? (A) decreasing the wavelength of the light (B) decreasing the screen distance L (C) decreasing the slit spacing d (D) immersing the entire apparatus in water 1 Serway & Jewett, page 1138.

23 Double-Slit Fringes of Two Wavelengths, Ex 37.2 A light source emits visible light of two wavelengths: λ = 430 nm and λ = 510 nm. The source is used in a double-slit interference experiment in which L = 1.50 m and d = mm. Find the separation distance between the third-order bright fringes for the two wavelengths.

24 Young s Experiment: Intensity Distribution Consider the electric field at a certain point P on the screen from each slit: E 1 = E 0 sin(ωt) E 2 = E 0 sin(ωt + φ)

25 Young s Experiment: Intensity Distribution Consider the electric field at a certain point P on the screen from each slit: E 1 = E 0 sin(ωt) E 2 = E 0 sin(ωt + φ) Using ( ) ( ) α β α + β sin α + sin β = 2 cos sin 2 2 We see that the net E-field at that point P is: E P = E 1 + E [ 2 ( )] ( φ = 2E 0 cos sin ωt + φ ) 2 2 Amplitude Time-fluctuation

26 Young s Experiment: Intensity Distribution E P,max = 2E 0 cos ( ) φ 2 Relating E-field to intensity, recall (Ch. 34): I E 2 P,max So, ) I = I max cos 2 ( φ 2 The phase difference is related to δ, the path difference by: φ 2π = δ λ So, using δ = d sin θ (and φ = 2πd sin θ/λ) ( ) πd sin θ I = I max cos 2 λ

27 Young s Experiment: Intensity Distribution ( ) πd sin θ I = I max cos 2 λ Q uick Quiz 37.2 Using Figure from six slits. N 2 N 3 N 4 I I max N 5 2l l 0 l 2l d sin u N 10 2l l Figure 37.6 Light intensity versus d sin u for a Figure 37.7 Mul

28 d E 2 are represented by phasors φ E 0 at Young s angular Experiment: speed v. The values Intensity Distribution ω t e corresponding phasors on the ( ) their projections at an arbitrary φ E = 2E 0 cos (a) e phasor for E 2 1 has a rotation ngle UBLE-SLIT vt f (it INTERFERENCE is phase-shifted 971 n on the vertcal axis varies with ns of Eqs and vary at any E 2 point P in Fig , we β 13b.The magnitude E ω 0 of the vector E φ oint E 1 P,and that wave has a cer- E 1 Fig b,we φ first E 0 note that the β E 0 e opposite ω equal-length t sides of ωt at an exterior angle (here f, as o opposite interior angles Phasors (here ( ) that represent φ ave (a) I = I (b) max cos waves can 2 be 2 Fig. added to (a) Phasors repres E 2 Phasors that PART waves 4 can be find the net w E 0 ω

29 Interference with Three Slits E 1 = E 0 sin(ωt), E 2 = E 0 sin(ωt + φ), E 3 = E 0 sin(ωt + 2φ)

30 ry large number of slits in a device called a diffraction grat Interference with Three Slits 37.2 Using Figure 37.7 as a model, sketch the interferen lits. I I max N 2 Primary maximum Secondary maximum N 3

31 Interference Patterns from Many Slits I I max N 2 Primary maximum Secondary maximum N 3 N 4 N 5 For any value intensity in m right of the ce indicated by t is due to diffra the individual discussed in C N 10 2l l 0 l 2l d sin u

32 Diffraction Grating A diffraction grating is a device that works in a similar way to Young s two slits, but produces a brighter set of fringes for the same source, and the bright fringes are narrower. It breaks the light from a source up into very, very many coherent sources. (Young s slit does the same, but only breaks the light into 2 sources.) It is used mainly for spectroscopy (determining the spectrum of a type of atom or molecule) and in monochromators (devices that select a particular frequency of light).

33 Interference Pattern from a Diffraction Grating A diffraction grating has so many slits that effectively N. With monochromatic light, the peaks are sharp and well-separated. m l d l d 0 l d 2l d Figure Intensity versus sin u For light that is composed of several frequencies, the peaks for each will be separated out. sin u for a diffraction grating. The We c spacing lengths wavelen The fir ship sin angle u angles u The monoch princip bright f should decreas

34 Diffraction Gratings There are two types of diffraction grating. Transmission gratings: Incoming plane wave of light Diffraction grating P P First-order maximum (m 1) Central or zeroth-order maximum (m 0) First-order maximum (m 1) Many slits allow light to pass through.

35 Diffraction Gratings Reflection gratings: Light reflects off of a series of mirrored surfaces. 1 lclose/a302/lecture14/lecture 14.html

36 Diffraction Grating We can find the maxima (bright fringes) of the pattern produced in action Patterns and Polarization a diffraction grating in exactly the same way we did for Young s slits. Incoming plane wave of light P First-order maximum (m 1) Diffraction grating P Central or zeroth-order maximum (m 0) First-order maximum (m 1) u d u d d sin u

37 Diffraction Grating Once again, light from different slits interferes constructively when the path differnce δ = mλ (m is an integer). δ = d sin θ Maxima (bright fringes) occur when d sin θ max = mλ where m Z

38 such as telescopes, cameras, and eyeglasses. Diffraction 35.4 Analysis Model: Wave Under Reflection We already know that light and other waves that travel through a small We introduced gap (< λ) the diverge, concept of and reflection that the of waves smaller in a the discussion gap, the of waves moreon strings in Section As with waves on strings, when a light ray traveling in one divergence. medium encounters a boundary with another medium, part of the incident light When l,, d, the rays continue in a straight-line path and the ray approximation remains valid. When l d, the rays spread out after passing through the opening. When l.. d, the opening behaves as a point source emitting spherical waves. d l,, d l d l.. d a b c The intensity of light in each direction is not the same however. Figure wavelen rier in of diam

39 Diffraction Patterns alternating with dark fringes. Incoming wave Slit min max min u max min max L min Viewing screen a b

40 Diffraction Spikes 1 NASA, ESA, and H. Richer (University of British Columbia); Svon Halenbach

41 Diffraction Spikes in Camera Apertures Iris diaphragms adjust the amount of light allowed into a camera body. They cause characteristic diffraction patterns on photos taken of bright lights. 1 Wikipedia user Cmglee

42 Diffraction Patterns: Arago Spot Directly in the center of the shadow produced by a round object lit with coherent light, a spot of light can be observed! This is called the Arago spot, Fresnel bright spot, or Poisson spot. 1 Photo taken at Exploratorium in SF, own work.

43 by much weaker maxima Understanding the Diffraction alternating Pattern with dark fromfringes. a Single Slit Incoming wave Slit min max min u max min max L min Viewing screen

44 Diffraction and Huygens Principle The new wave front is drawn tangent to When we have a slit or aperture the illuminated circular wavelets by coherent radiating light, from eachthe part of the aperture acts as a point sources of on spherical the original wavelets. front. A B c t Old wave front c t New wave front Old wave front N f A B a b These wavelets interfere to produce a diffraction pattern.

45 Understanding the Diffraction Pattern from a right Single of th Slit Each portion of the slit acts as difference a point source of light waves. Consider a series of point sources in different parts of the slit. between The r slit has width a. difference the pairs o 5 lation occu 4 because th the upper a a/2 a/2 u a sin u or, if we c below,

46 Understanding the Diffraction Pattern from a Single Slit We can find minima (dark fringes) in the pattern by breaking up our point sources into pairs that cancel each other out. Matching point sources in the top half of the slit with ones in the bottom half, the source separation distances will be d = a/2. This will be a fringe dark when: δ = a 2 sin θ = λ 2

47 Understanding the Diffraction Pattern from a Single Slit However, we could also break the slit up into 4 equal parts and match sources from the 1st and 2nd, and match from the 3rd and 4th. This will be dark when: δ = a 4 sin θ = λ 2 If we break the slit up into 6 equal parts and match sources from the 1st and 2nd, the 3rd and 4th, and the 5th and 6th. This will be dark when: δ = a 6 sin θ = λ 2

48 Understanding the Diffraction Pattern from a Single Slit In general we expect dark fringes when: sin θ min = m λ a where m = ±1, ±2, ±3,...

49 Question by much weaker maxima alternating with dark fringes. Incoming wave Slit min max min u max min max L min Viewing screen a Quick Quiz Suppose the slit width in the figure is made a Fraunhofer half as wide. diffraction Doespattern. the central A bright fringeis observed along the axis at u 5 with alternating dark and bright fringes on each side of the central bright fringe. Until (A) now, become we have wider, assumed slits are point sources of light. In this section, w abandon (B) that remain assumption the same, and or see how the finite width of slits is the basis for unde standing Fraunhofer diffraction. We can explain some important features of th phenomenon (C) become by examining narrower? waves coming from various portions of the slit as show in Figure According to Huygens s principle, each portion of the slit acts as 2 Serway & Jewett, page b

50 The Intensity Pattern from a Single Slit For a single slit: ( ) sin((πa/λ) sin θ) 2 I = I max (πa/λ) sin θ) This is a sinc-function squared. Just to check, we said the minima should be sin θ min = m λ a This corresponds to sin((πa/λ) sin θ min ) = 0 (πa/λ) sin θ min = mπ

51 996 CHAPTER 36 DIFFRACTION The Intensity Pattern from a Single Slit A θ E θ Here, with an even larger phase difference, they add to give a small amplitude and thus a small intensity. (d) The last phasor is out of phase with the first phasor by 2 π rad (full circle). E θ = 0 (c) Here, with a larger phase difference, the phasors add to give zero amplitude and thus a minimum in the pattern. I E θ (b) Here the phasors have a small phase difference and add to give a smaller amplitude and thus less intensity in the pattern. E θ (= E m ) Fig Phasor diagrams for N 18 phasors, corresponding to the division of a single slit into 18 zones. Resultant amplitudes E u are shown for (a) the central maximum at u 0, (b) a point on the screen lying at a small angle u to the central axis, (c) the first minimum,and (d) the first side maximum. Phasor for top ray E (a) Phasor for bottom ray The phasors from the 18 zones in the slit are in phase and add to give a maximum amplitude and thus the central maximum in the diffraction pattern.

52 Two Slits that have some width Suppose two slits each have width a and their centers are a distance d apart. What intensity pattern do they create?

53 Two Slits that have some width 38.2 Diffraction Patterns from Interference fringes I The diffraction pattern acts as an envelope (the blue dashed curve) that controls the intensity of the regularly spaced interference maxima. F e i d p a Diffraction minima 3p 2p p p 2p 3p p a sin u l

54 Resolution and the Rayleigh criterion We use optical systems like our eyes, a camera, or a telescope to view distant objects and interpret them. Imagine two distant stars. If they are very close together, or so far away that they make a very small angle to each other in the sky, they may look like only one star. When can we distinguish them as two separate points?

55 Many optical systems use circular apertures rather than slits. The diffraction pattern of a circular aperture as shown in the photographs of Figure 38.9 consists of Resolution and the Rayleigh criterion The angle subtended by the sources at the slit is large enough for the diffraction patterns to be distinguishable. The angle subtended by the sources is so small that their diffraction patterns overlap, and the images are not well resolved. S 1 S 2 u S 1 S 2 u a Slit Viewing screen Slit Viewing screen When the central maximum of one falls on the first dark fringe of the other, the two images begin to look like they came from one object. θ min sin θ min = λ a b

56 Resolution and the Rayleigh criterion 38.3 Resolution of Single-Slit and Circular Ap The sources are closer together such that the angular separation satisfies Rayleigh s criterion, and the patterns are just resolved. Figure 3 patterns curves) a (dashed separatio passes th In each c sum of th sources are part, and atterns are resolved. The sources so close toge that the patt are not reso a b c

57 Resolution and the Rayleigh criterion For circular points being resolved, the Rayleigh Criterion is: θ min = 1.22 λ D where D is the diameter of the aperture. The 1.22 factor comes from a full analysis of the 2-dimensional diffraction pattern for a circular aperture.

58 Summary Huygen s principle two-slit interference multiple slit interference diffraction gratings diffraction patterns diffraction and interference resolution and Raleigh s criterion Final Exam 6:15-8:15pm, Monday, June 26. Homework Serway & Jewett: Ch 37, onward from page OQs: 3, 9; CQs: 3, 5; Probs: 1, 3, 5, 13, 19, 21, 25, 51, 60 Ch 38, onward from page OQs: 3, 5; CQs: 5; Probs: 1, 7, 10, 15, 21, 25, 60