Optics

Question: If a diffraction grating produces a third-order bright spot for red light (of wavelength 700 nm ) at $\mathbf{65}\mathbf{.}\mathbf{0}\mathbf{°}$ from the central maximum, at what angle will the second-order bright spot be for violet light (of wavelength 400 nm )?

Question: Visible light passes through a diffraction grating that has 900 slits / cm, and the interference pattern is observed on a screen that is 2.50 m from the grating. (a) Is the angular position of the first-order spectrum small enough for $\mathit{s}\mathit{i}\mathit{n}\mathbf{ }\mathit{\theta }\mathbf{\approx }\mathit{\theta }$ to be a good approximation? (b) In the first-order spectrum, the maxima for two different wavelengths are separated on the screen by 3.00 mm. What is the difference in these wavelengths?

The wavelength range of the visible spectrum is approximately 380 - 750 nm. White light falls at normal incidence on a diffraction grating that has 350 slits/mm. Find the angular width of the visible spectrum in (a) the first order and (b) the third order. (Note: An advantage of working in higher orders is the greater angular spread and better resolution. A disadvantage is the overlapping of different orders, as shown in Example 36.4.)

(a) What is the wavelength of light that is deviated in the first order through an angle of $\mathbf{13}\mathbf{.}\mathbf{5}\mathbf{°}$ by a transmission grating having 5000 slits / cm? (b) What is the second-order deviation of this wavelength? Assume normal incidence.

Monochromatic light with wavelength 620 nm passes through a circular aperture with diameter 7.4  $\mathit{\mu }$m. The resulting diffraction pattern is observed on a screen that is 4.5 m from the aperture. What is the diameter of the Airy disk on the screen?

Monochromatic light with wavelength 490 nm passes through a circular aperture, and a diffraction pattern is observed on a screen that is 1.20 m from the aperture. If the distance on the screen between the first and second dark rings is 1.65 mm, what is the diameter of the aperture?

If you can read the bottom row of your doctor’s eye chart, your eye has a resolving power of 1 arcminute, equal to $\frac{\mathbf{1}}{\mathbf{60}}$ degree. If this resolving power is diffraction-limited, to what effective diameter of your eye’s optical system does this correspond? Use Rayleigh’s criterion and assume $\mathbf{\lambda }\mathbf{=}\mathbf{550}\mathbf{ }\mathbf{nm}$.

The VLBA (Very Long Baseline Array) uses a number of individual radio telescopes to make one unit having an equivalent diameter of about 8000 km. When this radio telescope is focusing radio waves of wavelength 2.0 cm, what would have to be the diameter of the mirror of a visible-light telescope focusing the light of wavelength 550 nm so that the visible-light telescope has the same resolution as the radio telescope?

If an optical telescope focusing light of wavelength 550 nm has a perfectly ground mirror, what would the minimum mirror diameter have to be so that the telescope could resolve a Jupiter-size planet around our nearest star, Alpha Centauri, which is about 4.3 lightyears from earth? (Consult Appendix F.)

The Hubble Space Telescope has an aperture of 2.4 m and focuses visible light (380–750 nm). The Arecibo radio telescope in Puerto Rico is 305 m (1000 ft) in diameter (it is built in a mountain valley) and focuses radio waves of wavelength 75 cm. (a) Under optimal viewing conditions, what is the smallest crater that each of these telescopes could resolve on our moon? (b) If the Hubble Space Telescope were to be converted to surveillance use, what is the highest orbit above the surface of the earth it could have and still be able to resolve the license plate (not the letters, just the plate) of a car on the ground? Assume optimal viewing conditions, so that the resolution is diffraction limited.

A wildlife photographer uses a moderate telephoto lens of focal length 135 mm and maximum aperture f/4.00 to photograph a bear that is 11.5 m away. Assume the wavelength is 550 nm. (a) What is the width of the smallest feature on the bear that this lens can resolve if it is opened to its maximum aperture? (b) If, to gain depth of field, the photographer stops the lens down to f/22.0, what would be the width of the smallest resolvable feature on the bear?

You are asked to design a space telescope for earth orbit. When Jupiter is $\mathbf{5}\mathbf{.}\mathbf{93}\mathbf{\times }{\mathbf{10}}^{\mathbf{8}}\mathbf{km}$away (its closest approach to the earth), the telescope is to resolve, by Rayleigh’s criterion, features on Jupiter that are 250 km apart. What minimum-diameter mirror is required? Assume a wavelength of 500 nm.

Coherent monochromatic light of wavelength $\mathit{\lambda }$passes through a narrow slit of width a, and a diffraction pattern is observed on a screen that is a distance x from the slit. On the screen, the width w of the central diffraction maximum is twice the distance x. What is the ratio a/$\mathit{\lambda }$of the width of the slit to the wavelength of the light?

Although we have discussed single-slit diffraction only for a slit, a similar result holds when light bends around a straight, thin object, such as a strand of hair. In that case, a is the width of the strand. From actual laboratory measurements on a human hair, it was found that when a beam of light of wavelength 632.8 nm was shone on a single strand of hair, and the diffracted light was viewed on a screen 1.25 m away, the first dark fringes on either side of the central bright spot were 5.22 cm apart. How thick was this strand of hair?

A loudspeaker with a diaphragm that vibrates at 960 Hz is traveling at 80.0 m/s directly toward a pair of holes in a very large wall. The speed of sound in the region is 344 m/s. Far from the wall, you observe that the sound coming through the openings first cancels at ±11.4° with respect to the direction in which the speaker is moving. (a) How far apart are the two openings? (b) At what angles would the sound first cancel if the source stopped moving?

Laser light of wavelength 632.8 nm falls normally on a slit that is 0.0250 mm wide. The transmitted light is viewed on a distant screen where the intensity at the center of the central bright fringe is 8.50 W/${\mathbf{m}}^{\mathbf{2}}$. (a) Find the maximum number of totally dark fringes on the screen, assuming the screen is large enough to show them all. (b) At what angle does the dark fringe that is most distant from the center occur? (c) What is the maximum intensity of the bright fringe that occurs immediately before the dark fringe in part (b)? Approximate the angle at which this fringe occurs by assuming it is midway between the angles to the dark fringes on either side of it.

Your boss asks you to design a diffraction grating that will disperse the first-order visible spectrum through an angular range of 27.0°. (See Example 36.4 in Section 36.5.) (a) What must be the number of slits per centimeter for this grating? (b) At what angles will the first-order visible spectrum begin and end?

A thin slit illuminated by light of frequency f produces its first dark band at ±38.2° in air. When the entire apparatus (slit, screen, and space in between) is immersed in an unknown transparent liquid, the slit’s first dark bands occur instead at ±21.6°. Find the refractive index of the liquid.

An underwater camera has a lens with focal length in air of 35.0 mm and a maximum aperture of f/2.80. The film it uses has an emulsion that is sensitive to light of frequency $\mathbf{6}\mathbf{.}\mathbf{00}\mathbf{\times }{\mathbf{10}}^{\mathbf{14}}\mathbf{Hz}$. If the photographer takes a picture of an object 2.75 m in front of the camera with the lens wide open, what is the width of the smallest resolvable detail on the subject if the object is (a) a fish underwater with the camera in the water and (b) a person on the beach with the camera out of the water?

The intensity of light in the Fraunhofer diffraction pattern of a single slit is given by Eq. (36.5). Let $\mathit{\gamma }\mathbf{=}\mathit{\beta }\mathbf{/}\mathbf{2}$. (a) Show that the equation for the values of $\mathit{\gamma }$ at which I is a maximum is tan $\mathit{\gamma }\mathbf{=}\mathit{\gamma }$ . (b) Determine the two smallest positive values of $\mathit{\gamma }$ that are solutions of this equation. (Hint: You can use a trial-and-error procedure. Guess a value of $\mathit{\gamma }$ and adjust your guess to bring tan $\mathit{\gamma }$ closer to $\mathit{\gamma }$. A graphical solution of the equation is very helpful in locating the solutions approximately, to get good initial guesses.) (c) What are the positive values of $\mathit{\gamma }$ for the first, second, and third minima on one side of the central maximum? Are the $\mathit{\gamma }$ values in part (b) precisely halfway between the $\mathit{\gamma }$ values for adjacent minima? (d) If a = 12 $\mathit{\lambda }$, what are the angles $\mathit{\theta }$ (in degrees) that locate the first minimum, the first maximum beyond the central maximum, and the second minimum?

A slit 0.360 mm wide is illuminated by parallel rays of light that have a wavelength of 540 nm. The diffraction pattern is observed on a screen that is 1.20 m from the slit. The intensity at the center of the central maximum $\left(\theta -0°\right)$ is ${\mathit{l}}_{\mathbf{0}}$ . (a) What is the distance on the screen from the center of the central maximum to the first minimum? (b) What is the distance on the screen from the center of the central maximum to the point where the intensity has fallen to ${\mathit{l}}_{\mathbf{0}}$/2?

In a large vacuum chamber, monochromatic laser light passes through a narrow slit in a thin aluminum plate and forms a diffraction pattern on a screen that is 0.620 m from the slit. When the aluminum plate has a temperature of 20.0°C, the width of the central maximum in the diffraction pattern is 2.75 mm. What is the change in the width of the central maximum when the temperature of the plate is raised to 520.0°C? Does the width of the central diffraction maximum increase or decrease when the temperature is increased?

In a laboratory, light from a particular spectrum line of helium passes through a diffraction grating and the second-order maximum is at 18.9° from the center of the central bright fringe. The same grating is then used for light from a distant galaxy that is moving away from the earth with a speed of $\mathbf{2}\mathbf{.}\mathbf{65}\mathbf{\times }{\mathbf{10}}^{\mathbf{7}}\mathit{m}\mathbf{/}\mathit{s}$. For the light from the galaxy, what is the angular location of the second-order maximum for the same spectral line as was observed in the lab?

What is the longest wavelength that can be observed in the third order for a transmission grating having ? Assume normal incidence.

Two satellites at an altitude of 1200 km are separated by 28 km. If they broadcast 3.6-cm microwaves, what minimum receiving-dish diameter is needed to resolve (by Rayleigh’s criterion) the two transmissions?

 .. It has been proposed to use an array of infrared telescopes spread over thousands of kilometers of space to observeplanets orbiting other stars. Consider such an array that has an effective diameter of  and observes infrared radiation at a wavelength of . If it is used to observe a planet orbiting the star Virginis, which is   from our solar system, what is the size of the smallest details that the array might resolve on the planet? How does this compare to the diameter of the planet, which is assumed to be similar to that of Jupiter ? (Although the planet of  Virginis is thought to be at least  more massive than Jupiter, its radius is probably not too different from that of Jupiter. Such large planets are thought to be composed primarily of gases, not rocky material, and hence can be greatly compressed by the mutual gravitational attraction of different parts of the planet.)

A diffraction grating has . What is the highest order that contains the entire visible spectrum? (The wavelength range of the visible spectrum is approximately.)

.. Quasars, an abbreviation for quasi-stellar radio sources, are distant objects that look like stars through a telescope but that

emit far more electromagnetic radiation than an entire normal galaxy of stars. An example is the bright object below and to the

left of center in Fig. P36.60; the other elongated objects in this image are normal galaxies. The leading model for the structure

of a quasar is a galaxy with a supermassive black hole at its center. In this model, the radiation is emitted by interstellar gas and dust within the galaxy as this material falls toward the black hole. The radiation is thought to emanate from a region just a few light-years in diameter. (The diffuse glow surrounding the bright quasar shown in Fig. P36.60 is thought to be this quasar’s host galaxy.) To investigate this model of quasars and to study other exotic astronomical objects, the Russian Space Agency plans to place a radio telescope in an orbit that extends to from the earth. When the signals from this telescope are combined with signals from the ground-based telescopes of the VLBA, the resolution will be that of a single radio telescope in diameter. What is the size of the smallest detail that this arrangement could resolve in quasar , which is light-years from earth, using radio waves at a frequency of ? (Hint: Use Rayleigh’s criterion.) Give your answer in light-years and in kilometres. Figure P36.60

.. A glass sheet is covered by a very thin opaque coating. In the middle of this sheet there is a thin scratch  thick. The sheet is totally immersed beneath the surface of a liquid. Parallel rays of monochromatic coherent light with wavelengthin air strike the sheet perpendicular to its surface and pass through the scratch. A screen is placed in the liquid a distance of  away from the sheet and parallel to it. You observe that the first dark fringes on either side of the central bright fringe on the screen are apart. What is the refractive index of the liquid?

.. BIO Resolution of the Eye. The maximum resolution of the eye depends on the diameter of the opening of the pupil (a diffraction effect) and the size of the retinal cells. The size of the retinal cells (about  in diameter) limits the size of an

object at the near point (25 cm) of the eye to a height of about . (To get a reasonable estimate without having to go through complicated calculations, we shall ignore the effect of the fluid in the eye.) (a) Given that the diameter of the human pupil is about , does the Rayleigh criterion allow us to resolve a tall object at  from the eye with light of wavelength  (b) According to the Rayleigh criterion, what is the shortest object we could resolve at the near point with light of wavelength ? (c) What angle would the object in part (b) subtend at the eye? Express your answer in minutes , and compare it with the experimental value of about 1 min. (d) Which effect is more important in limiting the resolution of our eyes: diffraction or the size of the retinal cells?

.. DATA While researching the use of laser pointers, you conduct a diffraction experiment with two thin parallel slits.

Your result is the pattern of closely spaced bright and dark fringes shown in Fig. P36.63. (Only the central portion of the pattern is shown.) You measure that the bright spots are equally spaced at centre to centre (except for the missing spots) on a screen that is  from the slits. The light source was a helium–neon laser producing a wavelength of . (a) How far apart are the two slits? (b) How wide is each one?

.. DATA Your physics study partner tells you that the width of the central bright band in a single-slit diffraction pattern is inversely proportional to the width of the slit. This means that the width of the central maximum increases when the width of the slit decreases. The claim seems counterintuitive to you, so you make measurements to test it. You shine monochromatic laser light with wavelength l onto a very narrow slit of width a and measure the width w of the central maximum in the diffraction pattern that is produced on a screen 1.50 m from the slit. (By “width,” you mean the distance on the screen between the two minima on either side of the central maximum.) Your measurements are given in the table.

a 1Mm2 0.78 0.91 1.04 1.82 3.12 5.20 7.80 10.40 15.60

w 1m2 2.68 2.09 1.73 0.89 0.51 0.30 0.20 0.15 0.10

(a) If w is inversely proportional to a, then the product law is constant, independent of a. For the data in the table, graph aw versus a. Explain why aw is not constant for smaller values of a. (b) Use your graph in part (a) to calculate the wavelength l of the laser

light. (c) What is the angular position of the first minimum in the diffraction pattern for (i) a = 0.78 mm and (ii) a = 15.60 mm?

.. DATA At the metal fabrication company where you work, you are asked to measure the diameter D of a very small circular hole in a thin, vertical metal plate. To do so, you pass coherent monochromatic light with wavelength   through the hole and observe the diffraction pattern on a screen that is a distance x from the hole. You measure the radius r of the first dark ring in the diffraction pattern (see Fig. 36.26). You make the measurements for four values of x. Your results are given in

the table.

(a) Use each set of measurements to calculate D. Because the measurements contain some error, calculate the average of the four values of D and take that to be your reported result. (b) For , what are the radii of the second and third dark rings in the diffraction pattern?