Optics

Herring and related fish have a brilliant silvery appearance that camouflages them while they are swimming in a sunlit ocean. The silveriness is due to platelets attached to the surfaces of these fish. Each platelet is made up of several alternating layers of crystalline guanine 1n = 1.802 and of cytoplasm (n = 1.333, the same as water), with a guanine layer on the outside in contact with the surrounding water (Fig.). In one typical platelet, the guanine layers are 74 nm thick and the cytoplasm layers are 100 nm thick. (a) For light striking the platelet surface at normal incidence, for which vacuum wavelengths of visible light will all of the reflections R1 , R2 , R3 , R4 , and R5 , shown in Fig., be approximately in phase? If white light is shone on this platelet,

what color will be most strongly reflected (see Fig. 32.4)? The surface of a herring has very many platelets side by side with layers of different thickness, so that all visible wavelengths are reflected. (b) Explain why such a “stack” of layers is more reflective than a single layer of guanine with cytoplasm underneath. (A stack of five guanine layers separated by cytoplasm layers

reflects more than 80% of incident light at the wavelength for which it is “tuned.”) (c) The color that is most strongly reflected

from a platelet depends on the angle at which it is viewed. Explain why this should be so. (You can see these changes in color by examining a herring from different angles. Most of the platelets on these fish are oriented in the same way, so that they are vertical when the fish is swimming.)

After a laser beam passes through two thin parallel slits, the first completely dark fringes occur at ±19.0° with the original direction of the beam, as viewed on a screen far from the slits. (a) What is the ratio of the distance between the slits to the wavelength of the light illuminating the slits? (b) What is the smallest angle, relative to the original direction of the laser beam, at which the intensity of the light is 1/10 the maximum intensity on the screen? 

In your summer job at an optics company, you are asked to measure the wavelength l of the light that is produced by a laser. To do so, you pass the laser light through two narrow slits that are separated by a distance d. You observe the interference pattern on a screen that is 0.900 m from the slits and measure the separation ∆y between adjacent bright fringes in the portion of the pattern that is near the center of the screen. Using a microscope, you measure d. But both ∆y and d are small and difficult to measure accurately, so you repeat the measurements for several pairs of slits, each with a different value of d. Your results are shown in Fig. P35.52, where you have plotted ∆y versus 1>d. The line in the graph is the best-fit straight line for the data.
(a) Explain why the data points plotted this way fall close to a straight line. (b) Use Fig. P35.52 to calculate l.

Short-wave radio antennas A and B are connected to the same transmitter and emit coherent waves in phase and with the same frequency f . You must determine the value of f and the placement of the antennas that produce a maximum intensity through constructive interference at a receiving antenna that is located at point P, which is at the corner of your garage. First you place antenna A at a point 240.0 m due east of P. Next you place antenna B on the line that connects A and P, a distance x due east of P, where x < 240.0 m. Then you measure that a maximum in the total intensity from the two antennas occurs when x = 210.0 m, 216.0 m, and 222.0 m. You don’t investigate smaller or larger values of x. (Treat the antennas as point sources.) (a) What is the frequency f of the waves that are emitted by the antennas? (b) What is the greatest value of x, with x < 240.0 m, for which the interference at P is destructive?

In your research lab, a very thin, flat piece of glass with refractive index 1.40 and uniform thickness covers the opening of a chamber that holds a gas sample. The refractive indexes of the gases on either side of the glass are very close to unity. To determine the thickness of the glass, you shine coherent light of wavelength l0 in vacuum at normal incidence onto the surface of the glass. When l0 = 496 nm, constructive interference occurs for light that is reflected at the two surfaces of the glass. You find that the next shorter wavelength in vacuum for which there is constructive interference is 386 nm. (a) Use these measurements to calculate the thickness of the glass. (b) What is the longest wavelength in vacuum for which there is constructive interference for the reflected light?

The index of refraction of a glass rod is 1.48 at T = 20.0°C and varies linearly with temperature, with a coefficient of 2.50 * 10-5/C°. The coefficient of linear expansion of the glass is 5.00 * 10-6/C°. At 20.0°C the length of the rod is 3.00 cm. A Michelson interferometer has this glass rod in one arm, and the rod is being heated so that its temperature increases at a rate of 5.00 C°/min. The light source has wavelength l = 589 nm, and the rod initially is at T = 20.0°C. How many fringes cross the field of view each minute?

Figure P35.56 shows an interferometer known as Fresnel’s biprism. The magnitude of the prism angle A is extremely small. (a) If S0 is a very narrow source slit, show that the separation of the two virtual coherent sources S1 and S2 is given by d = 2aA(n – 1), where n is the index of refraction of the material of the prism. (b) Calculate the spacing of the fringes of green light with wavelength 500 nm on a screen 2.00 m from the biprism. Take a = 0.200 m, A = 3.50 mrad, and n = 1.50

The professor then adjusts the apparatus. The frequency that you hear does not change, but the loudness decreases. Now all of your fellow students can hear the tone. What did the professor do? (a) She turned off the oscillator. (b) She turned down the volume of the speakers. (c) She changed the phase relationship of the speakers. (d) She disconnected one speaker.

The professor returns the apparatus to the original setting. She then adjusts the speakers again. All of the students who had heard nothing originally now hear a loud tone, while you and the others who had originally heard the loud tone hear nothing. What did the professor do? (a) She turned off the oscillator. (b) She turned down the volume of the speakers. (c) She changed the phase relationship of the speakers. (d) She disconnected one speaker.

The professor again returns the apparatus to its original setting, so you again hear the original loud tone. She then slowly moves one speaker away from you until it reaches a point at which you can no longer hear the tone. If she has moved the speaker by 0.34 m (farther from you), what is the frequency of the tone? (a) 1000 Hz; (b) 2000 Hz; (c) 500 Hz; (d) 250 Hz.

The professor once again returns the apparatus to its original setting, but now she adjusts the oscillator to produce sound waves of half the original frequency. What happens? (a) The students who originally heard a loud tone again hear a loud tone, and the students who originally heard nothing still hear nothing.(b) The students who originally heard a loud tone now hear nothing, and the students who originally heard nothing now hear a loud tone. (c) Some of the students who originally heard a loud tone again hear a loud tone, but others in that group now hear nothing. (d) Among the students who originally heard nothing, some still hear nothing but others now hear a loud tone.

Why can we readily observe diffraction effects for sound waves and water waves, but not for light? Is this because light travels so much faster than these other waves? Explain.

What is the difference between Fresnel and Fraunhofer diffraction? Are they different physical processes? Explain.

You use a lens of diameter  and light of wavelength  and frequency  to form an image of two closely spaced and distant objects. Which of the following will increase the resolving power? (a) Use a lens with a smaller diameter; (b) use light of higher frequency; (c) use light of longer wavelength. In each case justify your answer.

Light of wavelength $\mathit{\lambda }$ and frequency v passes through a single slit of width a. The diffraction pattern is observed on a screen a distance x from the slit. Which of the following will decrease the width of the central maximum? (a) Decrease the slit width; (b) decrease the frequency v of the light; (c) decrease the wavelength $\mathit{\lambda }$ of the light; (d) decrease the distance   of the screen from the slit. In each case justify your answer.

Question: Monochromatic light from a distant source is incident on a slit 0.750 mm wide. On a screen 2.00 m away, the distance from the central maximum of the diffraction pattern to the first minimum is measured to be 1.35 mm . Calculate the wavelength of the light.

Parallel rays of green mercury light with a wavelength of 546 nm pass through a slit covering a lens with a focal length of 60.0 cm. In the focal plane of the lens, the distance from the central maximum to the first minimum is 8.65 mm. What is the width of the slit?

Question: Light of wavelength 585 nm falls on a slit 0.0666 mm wide. (a) On a very large and distant screen, how many totally dark fringes (indicating complete cancellation) will there be, including both sides of the central bright spot? Solve this problem without calculating all the angles! (Hint: What is the largest that sin u can be? What does this tell you is the largest that m can be?) (b) At what angle will the dark fringe that is most distant from the central bright fringe occur?

Light of wavelength 633 nm from a distant source is incident on a slit 0.750 mm wide, and the resulting diffraction pattern is observed on a screen 3.50 m away. What is the distance between the two dark fringes on either side of the central bright fringe?

In a diffraction experiment with waves of wavelength $\mathit{\lambda }$, there will be no intensity minima (that is, no dark fringes) if the slit width is small enough. What is the maximum slit width for which this occurs? Explain your answer.

Question: Diffraction occurs for all types of waves, including sound waves. High-frequency sound from a distant source with wavelength 9.00 cm  passes through a slit 12.0 cm wide. A microphone is placed  8.00 m directly in front of the center of the slit. The microphone is then moved in a direction perpendicular to the line from the center of the slit to point O. At what distances from O will the intensity detected by the microphone be zero?

On December 26, 2004, a violent earthquake of magnitude 9.1 occurred off the coast of Sumatra. This quake triggered a huge tsunami (similar to a tidal wave) that killed more than 150,000 people. Scientists observing the wave on the open ocean measured the time between crests to be 1.0 h and the speed of the wave to be 800 km>h. Computer models of the evolution of this enormous wave showed that it bent around the continents and spread to all the oceans of the earth. When the wave reached the gaps between continents, it diffracted between them as through a slit.

(a) What was the wavelength of this tsunami? 

(b) The distance between the southern tip of Africa and northern Antarctica is about 4500 km, while the distance between the southern end of Australia and Antarctica is about 3700 km. As an approximation, we can model this wave’s behavior by using Fraunhofer diffraction. Find the smallest angle away from the central maximum for which the waves would cancel after going through each of these continental gaps.

Tsunami! On December 26, 2004, a violent earthquake of magnitude 9.1 occurred off the coast of Sumatra. This quake triggered a huge tsunami (similar to a tidal wave) that killed more than 150,000 people. Scientists observing the wave on the open ocean measured the time between crests to be 1.0 h and the speed of the wave to be 800 km/h. Computer models of the evolution of this enormous wave showed that it bent around the continents and spread to all the oceans of the earth. When the wave reached the gaps between continents, it diffracted between them as through a slit. (a) What was the wavelength of this tsunami? (b) The distance between the southern tip of Africa and northern Antarctica is about 4500 km, while the distance between the southern end of Australia and Antarctica is about 3700 km. As an approximation, we can model this wave’s behavior by using Fraunhofer diffraction. Find the smallest angle away from the central maximum for which the waves would cancel after going through each of these continental gaps

An interference pattern is produced by eight equally spaced narrow slits. The caption for Fig. 36.14 claims that minima occur for f = $\mathbf{3}\mathbf{\pi }\mathbf{/}\mathbf{4}$, $\mathbf{\pi }\mathbf{/}\mathbf{4}$, $\mathbf{3}\mathbf{\pi }\mathbf{/}\mathbf{2}$ and $\mathbf{7}\mathbf{\pi }$ /4. Draw the phasor diagram for each of these four cases, and explain why each diagram proves that there is in fact a minimum. In each case, for which pairs of slits are there totally destructive interference?

A rainbow ordinarily shows a range of colors. But if the water droplets that form the rainbow are small enough, the rainbow will appear white. Explain why, using diffraction ideas. How small do you think the raindrops would have to be for this to occur?

Some loudspeaker horns for outdoor concerts (at which the entire audience is seated on the ground) are wider vertically than horizontally. Use diffraction ideas to explain why this is more efficient at spreading the sound uniformly over the audience than either a square speaker horn or a horn that is wider horizontally than vertically. Would this still be the case if the audience were seated at different elevations, as in an amphitheater? Why or why not?

Figure 31.12 (Section 31.2) shows a loudspeaker system. Low-frequency sounds are produced by the woofer, which is a speaker with a large diameter; the tweeter, a speaker with a smaller diameter, produces high-frequency sounds. Use diffraction ideas to explain why the tweeter is more effective for distributing high-frequency sounds uniformly over a room than is the woofer.

Information is stored on an audio compact disc, CD-ROM, or DVD disc in a series of pits on the disc. These pits are scanned by a laser beam. An important limitation on the amount of information that can be stored on such a disc is the width of the laser beam. Explain why this should be, and explain how using a shorter wavelength laser allows more information to be stored on a disc of the same size.

With which color of light can the Hubble Space Telescope see finer detail in a distant astronomical object: red, blue, or ultraviolet? Explain your answer.

Explain, using phasor diagrams, why each statement is true. (a) A minimum occurs whenever ϕ is an integral multiple of 2π/N, except when ϕ is an integral multiple of 2π (which gives a principal maximum). (b) There are (N-1) minima between each pair of principal maxima.

Could x-ray diffraction effects with crystals be observed by using visible light instead of x-rays? Why or why not?

Why is a diffraction grating better than a two-slit setup for measuring wavelengths of light?

One sometimes sees rows of evenly spaced radio antenna towers. A student remarked that these act like diffraction gratings. What did she mean? Why would one want them to act as a diffraction grating?

If a hologram is made using 600-nm light and then viewed with 500-nm light, how will the images look compare to those observed when viewed with 600-nm light? Explain.

A hologram is made using 600-nm light and then viewed by using white light from an incandescent bulb. What will be seen? Explain.

Ordinary photographic film reverses black and white, in the sense that the most brightly illuminated areas become blackest upon development (hence the term negative). Suppose a hologram negative is viewed directly, without making a positive transparency. How will the resulting images differ from those obtained with the positive? Explain.

Red light of wavelength 633 nm from a helium-neon laser passes through a slit 0.350 mm wide. The diffraction pattern is observed on a screen 3.00 m away. Define the width of a bright fringe as the distance between the minima on either side. 

(a) What is the width of the central bright fringe? 

(b) What is the width of the first bright fringe on either side of the central one?

Public Radio station KXPR-FM in Sacramento broadcasts at 88.9 MHz. The radio waves pass between two tall skyscrapers that are 15.0 m apart along their closest walls. 

(a) At what horizontal angles, relative to the original direction of the waves, will a distant antenna not receive any signal from this station? 

(b) If the maximum intensity is 3.50 W>m2 at the antenna, what is the intensity at _5.00_ from the center of the central maximum at the distant antenna?

Monochromatic light of wavelength 580 nm passes through a single slit and the diffraction pattern is observed on a screen. Both the source and screen are far enough from the slit for Fraunhofer diffraction to apply. 

(a) If the first diffraction minima are at _90.0_, so the central maximum completely fills the screen, what is the width of the slit? 

(b) For the width of the slit as calculated in part (a), what is the ratio of the intensity at u = 45.0_ to the intensity at u = 0?

Monochromatic light of wavelength λ = 620 nm from a distant source passes through a slit 0.450 mm wide. The diffraction pattern is observed on a screen 3.00 m from the slit. In terms of the intensity I0 at the peak of the central maximum, what is the intensity of the light at the screen the following distances from the center of the central maximum: 

(a) 1.00 mm; 

(b) 3.00 mm; 

(c) 5.00 mm?

A slit 0.240 mm wide is illuminated by parallel light rays of wavelength 540 nm. The diffraction pattern is observed on a screen that is 3.00 m from the slit. The intensity at the center of the central maximum 1u = 0_2 is 6.00 * 10-6 W>m2. 

(a) What is the distance on the screen from the center of the central maximum to the first minimum? 

(b) What is the intensity at a point on the screen midway between the center of the central maximum and the first minimum?

Monochromatic light of wavelength 592 nm from a distant source passes through a slit that is 0.0290 mm wide. In the resulting diffraction pattern, the intensity at the center of the central maximum  = 0° is 4.00 * 10-5 W/m^2 . What is the intensity at a point on the screen that corresponds to  = 1.20°?

A single-slit diffraction pattern is formed by monochromatic electromagnetic radiation from a distant source passing through a slit wide 0.105 mm. At the point in the pattern $\mathbf{3}\mathbf{.}\mathbf{25}\mathbf{°}$ from the center of the central maximum, the total phase difference between wavelets from the top and bottom of the slit is 56.0 rad. (a) What is the wavelength of the radiation? (b) What is the intensity at this point, if the intensity at the center of the central maximum is ${\mathit{l}}_{\mathbf{0}}$?

Number of Fringes in a Diffraction Maximum. In Fig.36.12 c the central diffraction maximum contains exactly seven interference fringes, and in this case d =a =4. (a) What must the ratio d /a be if the central maximum contains exactly five fringes? (b) In the case considered in part (a), how many fringes are contained within the first diffraction maximum on one side of the central maximum?

Diffraction and Interference Combined. Consider the interference pattern produced by two parallel slits of width $\mathbf{\alpha }$ and separation d, in which d =3 a. The slits are illuminated by normally incident light of wavelength $\mathit{\lambda }$. (a) First, we ignore diffraction effects due to the slit width. At what angles $\mathbf{\theta }$ from the central maximum will the next four maxima in the two slit interference pattern occurs? Your answer will be in terms of d and $\mathbf{\lambda }$. (b) Now we include the effects of diffraction. If the intensity at $\mathbf{\theta }\mathbf{=}\mathbf{0}\mathbf{°}$ is ${\mathbf{I}}_{\mathbf{0}}$, what is the intensity at each of the angles in part (a)? (c) Which double-slit interference maxima are missing in the pattern? (d) Compare your results to those illustrated in Fig. 36.12 c. In what ways are your results different?

An interference pattern is produced by light of wavelength from a distant source incident on two identical parallel slits separated by a distance (between centers) of 0.530 mm. (a) If the slits are very narrow, what would be the angular positions of the first-order and second-order, two-slit interference maxima? (b) Let the slits have width 0.320mm. In terms of the intensity ${\mathit{I}}_{\mathbf{0}}$ at the center of the central maximum, what is the intensity at each of the angular positions in part (a)?

Laser light of wavelength 500.0 nm illuminates two identical slits, producing an interference pattern on a screen 90.0 cm from the slits. The bright bands are 1.00 cm apart, and the third bright bands on either side of the central maximum are missing in the pattern. Find the width and the separation of the two slits.

When laser light of wavelength 632.8nm passes through a diffraction grating, the first bright spots occur at $\mathbf{\pm }\mathbf{17}\mathbf{.}\mathbf{8}\mathbf{°}$ from the central maximum. (a) What is the line density (in lines/cm) of this grating? (b) How many additional bright spots are there beyond the first bright spots, and at what angles do they occur?

Monochromatic light is at normal incidence on a plane transmission grating. The first-order maximum in the interference pattern is at an angle of $\mathbf{11}\mathbf{.}\mathbf{3}\mathbf{°}$ . What is the angular position of the fourth-order maximum?

If a diffraction grating produces its third-order bright band at an angle of $\mathbf{78}\mathbf{.}\mathbf{4}\mathbf{°}$ for light of wavelength 681nm, find (a) the number of slits per centimeter for the grating and (b) the angular location of the first-order and second-order bright bands. (c) Will there be a fourth-order bright band? Explain.