Physics 2 - Lecture 6: Diffraction - Pham Tan Thi

Nội dung text: Physics 2 - Lecture 6: Diffraction - Pham Tan Thi

1. Diffraction Pham Tan Thi, Ph.D. Department of Biomedical Engineering Faculty of Applied Science Ho Chi Minh University of Technology
2. Effects of Materials on Light Materials can be classified based on how it responds to light incident on them: 1. Opaque materials - absorb light; do not let light to pass through 2. Transparent materials - allow light to easily pass through them 3. Translucent materials - allow light to pass through but distort the light during the passage
3. Diffraction of Light When a narrow opaque (aperture) is placed between a source of light and a screen, light bends around the corners of the aperture. This encroachment of light is called “diffraction”. For diffraction, the size of the aperture is small (comparable to the wavelength). As a result of diffraction, the edges of the shadow (or illuminated region) are not sharp, but the intensity is distributed in a certain way depending on the nature of the aperture.
4. Diffraction and Hyugen’s Principle Hyugen’s principle can be used to analyze the diffraction Diffraction pattern of a razor blade
5. Diffraction of Light No diffraction; No spreading after passing through slit Weak diffraction; Weak spreading after passing through slit Diffraction
6. Types of Diffraction Diffraction phenomena can be classified either as Fresnel diffraction or Fraunhofer diffraction The observable difference: Fresnel diffraction The viewing screen and the aperture are located close together, the image of the aperture is clearly recognizable despite slight fringing around its periphery. As the separation between the screen and the aperture increases, the image of the aperture becomes increasingly more structured; fringes become more prominent. Fraunhofer diffraction The viewing screen and the aperture separated by a large distance, the projected pattern bears little or no resemblance to the aperture. As the separation increases, the size of the pattern changes but not its shape.
7. Fraunhofer’s Diffraction In the case of Fraunhofer’s diffraction, the source of light or screen are effectively at infinite distance from the diffracting aperture (obstacle). This is achieved by placing the source and screen in the focal planes of two lenses (require lenses). The incident wavefront is plane.
8. Difference between Fraunhofer and Fresnel Diffraction Fraunhofer Diffraction Fresnel Diffraction intensity pattern intensity pattern The maxima and minima are well defined The maxima and minima are not well defined
9. Fraunhofer’s Diffraction at a Single Slit In terms of wave theory, a plane wavefront is incident on the slit AB. According to the Huygens’ principle, each point in AB sends out secondary wavelets in all directions. The rays proceeding in the same direction as the incident rays focused at O; while those diffracted through an angle θ are focused at P. Let us find the resultant intensity at P. Let AK be perpendicular to BP. As the optical paths from the plane AK to P are equal, the path difference between wavelets from A to B in the direction θ is BK = AB sinθ = e sinθ The corresponding phase difference = (2π/λ)e sinθ Let the width AB of the slit be divided into n equal parts. The amplitude of vibration at P due to the waves from each part will be the same (= a)
10. Fraunhofer’s Diffraction at a Single Slit Asin↵ Let na = A then R = ↵ The resultant intensity at P is sin↵ 2 I = R2 = A2 ↵ ✓ ◆
11. Fraunhofer’s Diffraction at a Single Slit Direction of Maxima: To find the direction of maximum intensity, let us differentiate the intensity with respect to α and equate it to zero dI/dα = 0 2sin↵ ↵cos↵ sin↵ A2 =0 ↵ ↵2 ✓ ◆ ↵cos↵ sin↵ =0 ↵2 ↵cos↵ sin↵ =0 ↵ = tan↵ This equation is solved graphically by plotting the curves: y = α y = tan α
12. Fraunhofer’sContinued Diffraction at a Single Slit TheTheprincipal principal maximamaxima occursoccur atat α =α( =π (eπsin e sinθ)/θλ)/=λ 0or, θor = θ0=, i.e.,0 the principal maxima occur at the i.e. the principal maxima occurs at same direction of light. the same direction of light The diTheffractiondiffraction pattern consistspattern consistsof a brightof principala bright maximumprincipal maximumin the directionin the of incidentdirection light,of incidenthas alternativelylight, having minimaalternately and minimaweak subsidiaryand weak maximasubsidiary of rapidly decreasing intensity on maxima of rapidly decreasing either side of it. intensity on either side of it. TheTheminima minimalie atlie αat= α± =π ,±±π2, π±2, π , .
13. Fraunhofer’s Diffraction of a Double Slit Let a parallel beam of monochromatic light of wavelength λ be incident normally on two parallel slits AB and CD, each of width e and separate by opaque space d. The distance between the corresponding points of the two slits is (e+d). Let diffracted light be focused by a convex lens L on a screen XY placed in the focal plane of the lens.
14. Fraunhofer’s Diffraction of a Double Slit The resultant amplitude at a point P on the screen will be the result of interference between two waves of sample amplitude (Asinα)/α, and having a phase difference δ Let us draw S1K perpendicular to S2K. The path difference between the wavelets from S1 and S2 in the direction θ is S2K = (e + d)sinθ Hence the phase difference is δ = (2π/λ) x path difference = (2π/λ) x (e + d) sinθ The resultant amplitude R at P can be determined by the vector amplitude diagram, which gives OB2 = OA2 + AB2 + 2(OA)(AB)cos(BAC)
15. Fraunhofer’s Diffraction of a Double Slit A2sin2↵ I = R2 =4A2 cos2 ↵2 The intensity in the resultant pattern depends of two factors: (i) (sin2α)/α2, which gives the diffraction pattern due to each individual slit (ii) cos2β, which gives interference pattern due to diffracted light waves from the two slits
16. Continued . (b) The interference term, cos2 β, gives a set of equidistant dark and bright fringes,CalculationFraunhofer’sas in ofYoung’s Opticaldouble DiPathffraction -Dislitfferenceinterference of a between Doubleexperiment TwoSlit Waves. The bright(ii) Thefringes interference(maxima) termare, cosobtained2β, which ingivesthe adirections set of equidistantgiven by dark and bright fringes, as in Young’s double slit interference 2 experiment.cos β = 1 The bright fringesβ =(maxima)± nπ are obtained in the directions given by 2 cos(βπ =/λ 1,) (e+d) sin θ = β± =n ±nππ (π/λ)(e+d)sinθ = ±nπ (e+d) sin θ = ± nλ, Where n = 0,1,2, (e+d)sinθ = ±nλ (n = 0,1,2,3 ) The various maxima corresponding to n = 0, 1,2, are zero-order, first- The various maxima corresponding to n = 0,1,2, are zero-order, order, secondfirst order,-order second order, maxima maxima
17. The double slit intensity distribution corresponding to e = 0.0088 cm, λ = 6.328 x 10-5 cm, and d = 0.035 cm and d = 0.07 cm 2 sin 2 sin ↵ 2 I 4I0 cos 2 e = 0.0088 cm I =4Io 2 cos ↵2 d =0.035 cm = 6.328 10-5 cm e = 0.0088 cm d =0.07 cm = 6.328 10-5 cm The double-slit intensity distribution corresponding to e = 0.0088 cm, = 6.328 x 10–5 cm, and d = 0.035 cm and d = 0.07cm respectively.
18. Fraunhofer’s Diffraction at an N Slit (Grating) The* diTheffractiondiffraction gratinggrating is an immenselyis an immensely useful tooluseful for separationtool for ofthe the separationspectral linesof the associatedspectral lines withassociated atomic transitions.with atomic transitions. Sometimes* Sometimes this thisdiffractiondiffraction gratinggrating is alsois also calledcalled “super‘super prismprism’” . The* conditionThe condition for maximumfor maximum intensityintensity isis thethe samesame asas thatthat forfor thethe double doubleslit or slitmultiple or multipleslits, but slits,with buta large withnumber a largeof numberslits the ofintensity slits themaximum is intensityvery sharp maximumand narrow, is very providingsharp andthe narrow,high resolution providingfor thespectroscopic high resolution for spectroscopic applications. applications. The tracks of a compact disc act as a diff*ractionThe tracks grating,of a producingcompact disc a act as a diffraction separationgrating, ofproducing the colorsa separationof white light.of the colours of white light.
19. Fraunhofer’s Diffraction at an N Slit (Grating) According to Huygens’ principle, all the points in each slit send out secondary wavelets in all directions. According to Fraunhofer’s diffraction at a single slit, the wavelets from all points in a slit diffracted in a direction θ are equivalent to a single wave of amplitude (Asinα)/α, where α = (π/λ)e sinθ If N is the total number of slits in the grating, the diffracted rays from all the slits are equivalent to N parallel rays, one each from middle points S1, S2, S3, of the slits. After calculating the path difference and corresponding phase difference, the resultant amplitude in the direction θ is sin↵ sinN R = A ↵ sin ⇡ Where = (e + d)sin✓
20. Continued . Fraunhofer’s Diffraction at an N Slit (Grating) It is clear that m = 0 gives a principal maximum, m = 1,2,3, .(N-1) give It is clear that m = 0 gives a principal maximum; m = 1,2,3, (N-1) give minimaminimaand, andm m= =N Ngives givesagain againa aprincipal principalmaximum maximum ThusThus therethere areare (N-1)(N- 1minima) minima betweenbetween two consecutivetwo consecutive principalprincipal maxima. maxima.
21. Diffraction - The Central Maximum There are several bright spots and dark areas in between. The spot in the middle if the brightest and thus the central maximum. We call these spots being fringes. We also have additional bright spots, yet the intensity is a bit less. These additional bright spots are denoted as orders. So the first bright spot on either side of the central maximum is called the first order bright fringe. The intensity of the orders as we move farther from the bright central maximum.
22. Diffraction - Bright Fringes: Path Difference These two waves are in phase. When they hit the screen, they both hit at the same relative position at the bottom of a crest. How much farther did the red wavelength have to travel? Exactly — One Wavelength This extra difference is the path difference. The path difference and the order of a fringe help to form a pattern.
23. Diffraction - Dark Fringes We see a definite decrease in the intensity between the bright fringes. In the pattern, we visible notice the dark region. These are areas where destructive interference occurs. We call these areas being dark fringes or minima.
24. Diffraction - Dark Fringes On either side of the bright central maximum, we see areas that are dark or minimum intensity. The blue wave has to travel farther than the red wave to reach the screen. They are said to destructively build or that they are out of phase. How much farther did the blue wave have to travel so that they both hit the screen out of phase?
25. Diffraction - Bright Fringes: Path Difference The dark fringes you see on either side of the central maximum are multiple wavelengths from the bright central. The multiple is the order. The path different is equal to the order plus half, times the wavelength P.D. = (m+1/2)λ (Destructive)
26. Problem In Fraunhofer’s diffraction due to a narrow slit, a screen is placed 2 m away from the lens to obtain the pattern. If the slit width is 0.2 mm and the first minima lie 5 mm on either side of the central maximum, find the wave length of light?