Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Computational fluid dynamics wikipedia , lookup
Mathematical optimization wikipedia , lookup
Computational complexity theory wikipedia , lookup
Knapsack problem wikipedia , lookup
Inverse problem wikipedia , lookup
Perturbation theory wikipedia , lookup
Multiple-criteria decision analysis wikipedia , lookup
CHAPTER 37 INTERFERENCE AND DIFFRACTION . -- Section 37-2: - - Section 37-3: Multiple-Slit Interference and Diffraction Gratings Double-Slit Interference Problem 3. A double-slit experiment has slit spacing 0.12 mm. (a) What should be the slit-tc-screen distance L if the bright fringes are to be 5.0 mm apart when the slits are illuminated with 633-nm laser light? (b) What will be the fringe spacing with 480-nm light? Solution The particular geometry of this type of double-slit experiment is described in the paragraphs preceding Equations 37-2a and b. (a) The spacing of bright fringes on the screen is Ay = XL/d, so L = (0.12 mm)(5 __-mm)/(633 nm] = 94.8 cm. (b) For two different wavelengths, the ratio of the spacings is a y f / A y = Af/A; therefore Ay' = (5 mm)(480/633) = 3.79 mm. Problem 7. Light shines on a pair of slits whose spacing is three times the wavelength. Find the locations of the first- and second-order bright fringes on a screen 50 cm from the slits. Hint: Do Equations 37-2 apply7 Solution Since d = 3X, the angles are not small, and Equations 37-2 do not apply. The interference maxima occur a t angles given by Equation 37-la, 6' = sin-](mX/d) = sill-'(m/3), so only two orders are present, for values of m = 1 and 2 (6' < 90'). If we assume that the slit/screen geometry is as shown in Fig. 37-6, then y = L tan 6' = L tan(sin-' (77213)) = ~ m / (Consider a right triangle with hypotenuse of 3 and opposite side m, or use tan 6' = sin 6'1 d m . )For m = 1 and 2, and L = 50 cm, this gives yl = (50 cm)(l/&) = 17.7 cm, and yz = (50 c m ) ( 2 / 4 ) = 44.7 cm. Jw. -- - Problem 9. For a double-slit experiment with slit spacing 0.25 mm and wavelength 600 nm, at what angular position is the path difference equal to one-fourth of the wavelength? Solution If we set the path difference equal to a quarter wavelength, we obtain d sin 6' = X/4, or 6' w sin 6' = 600 nm/4(0.25 mm) = 6 x 1 0 ~rad~ 0.0344". - Problem 13. In a 5-slit system, how many minima lie betwee* the zeroth-order and first-order maxima? Solution In an N-slit system with slit separation d (illuminated by normally incident plane waves), the main maxima occur for angles sin 0 = mX/d, and minima for sin 8 = mJX/Nd (excluding m' equal to zero or multiples of N ). Between two adjacent maxima, say m' = m N and (m 1)N, there are N - 1 minima. (The number of integers between mN and (m + l ) N is (m l ) N - m N - 1 = N - 1, because the limits are not included.) For N = 5, the number of minima is 4. + + Problem 17. Green light at 520 nm is diffracted by a grating with 3000 lines per cm. Through what angle is the light diffracted in (a) first and (b) fifth order? Solution For light normally incident on a diffraction grating, maxima occur a t angles 6' = sin-'(mX/d), where d is the grating spacing (equal to the reciprocal of the number of lines per meter), and m is the order. (a) In first order, 6'' = sin-'(520 nm x 3000/cm) = 8.97", and (b) in fifth order, O5 = sinq1(5 sin 8.97') = 51.3'. Problem 23. Estimate the number of lines per cm in the grating used to produce Fig. 37-15. Solution The number of lines per cm (lid in cm-') is easily estimated from the angular position of the central 550-nm line in a particular order, as shown in the figure; that is, l/d = sin 6'lmA. For example, in fifth order, this line is at 6' = 61' (average of right and left values), so l/d = sin61°/5(550 nm) = 3.18x103/cm or about 3200 lines/cm. I -qoO I, -40. I -30. 0 8 1 .I 30. dob FIGURE 37-15 Problem 23 Solution. 90.