To see how a sharp edge or an aperture affect light. To analyze single-slit diffraction and calculate the intensity of the light

Transcription

1 Diffraction

2 Goals for lecture To see how a sharp edge or an aperture affect light To analyze single-slit diffraction and calculate the intensity of the light To investigate the effect on light of many closely spaced slits To learn how scientists use diffraction gratings To see what x-ray diffraction tells us about crystals To learn how diffraction places limits on the resolution of a telescope Introduction How can we use coherent light to visually see the difference in pit density on CDs, DVDs and Blu-Ray disks? Why does light from a point source form light and dark fringes? We will continue our exploration of the wave nature of light with diffraction.and we will see how to form threedimensional images using a hologram. Diffraction According to geometric optics, a light source shining on an object in front of a screen should cast a sharp shadow. Surprisingly, this does not occur because of diffraction.

3 Diffraction and Huygen s Principle Huygens s principle can be used to analyze diffraction. Fresnel diffraction: Source, screen, and obstacle are close together. Fraunhofer diffraction: Source, screen, and obstacle are far apart Diffraction from a single slit In Figure below, the prediction of geometric optics in (a) does not occur. Instead, a diffraction pattern is produced, as in (b). The narrower the slit, the broader the diffraction pattern. Fresnel and Fraunhofer diffraction by a single slit Figure below shows Fresnel (near-field) and Frauenhofer (far-field) diffraction for a single slit.

4 Locating the dark fringes Follow the single-slit diffraction discussion in the text. Figure below shows the geometry for Fraunhofer diffraction An example of single-slit diffraction Figure 36.6 (bottom left) is a photograph of a Fraunhofer pattern of a single horizontal slit. Example 36.1: You pass 633-nm light through a narrow slit and observe the diffraction pattern on a screen 6.0 m away. The distance at the screen between the center and the first minima on either side is 32 mm long. How wide is the slit? y m xm (6000 mm)( mm) a 0.24 mm a 32 mm

5 Intensity in the single-slit pattern Follow the text discussion of the intensity in the single-slit pattern using the phasor diagrams in Figure below.

6 Quantitative Intensity in the single-slit pattern Follow the text discussion of the intensity in the single-slit pattern using the phasor diagrams in Figure below.the angle b is the phase angle of the ray from the top of the slit, while the phase angle from the bottom of the slit is 0. The vectors lie along a circle whose center is at C, so E p is a chord of the circle. The arc length E 0 is subtended by this same angle β, so the radius of the circle is E 0 /β. From the diagram since E p E sin / sin E0 2 / 2 2 asin We have I a sin sin / I 0 a sin / 2 (sinc function)

7 Intensity maxima in a single-slit pattern Figure at the right shows the intensity versus angle in a single-slit diffraction pattern. The minima occur when βis a multiple of 2π, i.e. at asin m ( m 1, 2, 3,...) Width of the single-slit pattern The single-slit diffraction pattern depends on the ratio of the slit width a to the wavelength Example : (a) The intensity at the center of a single-slit diffraction pattern is I 0. What is the intensity at a point in the pattern where there is a 66-radian phase difference between wavelets from the two edges of the slit? (b) If this point is 7 degrees from the central maximum, how many wavelengths across is the slit? a) b) a 2 2 sin sin / sin 33 rad I I I I a sin / 33 rad a sin 33 rad 33 rad a 86 sin 7

8 Two slits of finite width When we discussed two-slit interference, we ignored the width of each slit. When we demonstrated it, however, we saw clearly the effect of the slit widths. The overall pattern of two finite-width slits is the product of the two patterns, i.e. 2 2 I I0sinc cos sinc x 2 2 sin x x

9 Several slits In Figure below, a lens is used to give a Fraunhofer pattern on a nearby screen. It s function is to allow the pattern to be seen nearby, without having the screen really distant. The phasor diagrams show the electric vectors from each slit at different screen locations. Interference pattern of several slits The figure below shows the interference pattern for 2, 8, and 16 equally spaced narrow slits. By making the slits really close together, the maxima become more separated. If the light falling on the slits contains more than one wavelength (color), there will be more than one pattern, separated more or less according to wavelength, although all colors have a maximum at m = 0.This means that the different orders make rainbows separating wavelengths into a spectrum, with the separation being greater for greater order m.

10 The diffraction grating A diffraction grating is an array of a large number of slits having the same width and equal spacing. The intensity maxima occur at dsin m Example 36.4: The wavelengths of the visible spectrum are approximately 380 nm (violet) to 750 nm (red). (a) Find the angular limits of the first-order visible spectrum produced by a plane grating with 600 slits per millimeter when white light falls normally on the grating. (b) Do the first order and second order spectra overlap? What about the 2 nd and 3 rd orders? (a)distance between slits is Violet light for 1 st order occurs at Red light for 1 st order occurs at (b) recalculate for m = 2 and m = 3 1 mm m d 600 slits 7 6 d arcsin / arcsin / d arcsin / arcsin / The 2 nd -order spectrum extends from while the 3 rd order is from

11 Grating spectrographs A diffraction grating can be used to disperse light into a spectrum. The greater the number of slits, the better the resolution. Figure (a) below shows our sun in visible light, and in (b) dispersed into a spectrum by a diffraction grating. Diagram of a grating spectrograph Figure below shows a diagram of a diffraction-grating spectrograph for use in astronomy.

12 X-ray diffraction When x rays pass through a crystal, the crystal behaves like a diffraction grating, causing x-ray diffraction. Figure below illustrates this phenomenon. A simple model of x-ray diffraction Follow the text analysis using Figure below. The Bragg condition for constructive interference 2d sin = m.

13 Circular apertures An aperture of any shape forms a diffraction pattern Figures below illustrate diffraction by a circular aperture. The airy disk is the central bright spot. Figures below illustrate diffraction by a circular aperture. The airy disk is the central bright spot. Diffraction limits the resolution of optical equipment, such as telescopes. The larger the aperture, the better the resolution. Figure (right) illustrates this effect.

14 Bigger telescope, better resolution Because of diffraction, large-diameter telescopes, such as the VLA radio telescope below, give sharper images than small ones What is holography? By using a beam splitter and mirrors, coherent laser light illuminates an object from different perspectives. Interference effects provide the depth that makes a three-dimensional image from two-dimensional views. Figure below illustrates this process.