4.7 Holography

If a hologram is recorded using monochromatic light at one wavelength but its image is viewed at another wavelength, say $10 \%$ shorter, what will you see? What if it is viewed using light of exactly half the original wavelength?

White light falls on two narrow slits separated by 0.40 mm. The interference pattern is observed on a screen 3.0 m away. (a) What is the separation between the first maxima for red light $\left(\lambda = 700 \text{nm}\right)$ and violet light $\left(\lambda = 400 \text{nm}\right) ?$ (b) At what point nearest the central maximum will a maximum for yellow light $\left(\lambda = 600 \text{nm}\right)$ coincide with a maximum for violet light? Identify the order for each maximum.

Quasars, or quasi-stellar radio sources, are astronomical objects discovered in 1960. They are distant but strong emitters of radio waves with angular size so small, they were originally unresolved, the same as stars. The quasar 3C405 is actually two discrete radio sources that subtend an angle of 82 arcsec. If this object is studied using radio emissions at a frequency of 410 MHz, what is the min…

A microwave of an unknown wavelength is incident on a single slit of width 6 cm. The angular width of the central peak is found to be $25 °$. Find the wavelength.

A light ray of wavelength 461.9 nm emerges from a 2-mm circular aperture of a krypton ion laser. Due to diffraction, the beam expands as it moves out. How large is the central bright spot at (a) 1 m, (b) 1 km, (c) 1000 km, and (d) at the surface of the moon at a distance of 400,000 km from Earth.

How far apart must two objects be on the moon to be resolvable by the 8.1-m-diameter Gemini North telescope at Mauna Kea, Hawaii, if only the diffraction effects of the telescope aperture limit the resolution? Assume 550 nm for the wavelength of light and 400,000 km for the distance to the moon.

Monochromatic light of wavelength 530 nm passes through a horizontal single slit of width $1.5 \mu \text{m}$ in an opaque plate. A screen of dimensions $2.0 \text{m} \times 2.0 \text{m}$ is 1.2 m away from the slit. (a) Which way is the diffraction pattern spread out on the screen? (b) What are the angles of the minima with respect to the center? (c) What are the angles of the maxima? (d) How wide is the central bright fringe on the screen? (e) H…

A source of light having two wavelengths 550 nm and 600 nm of equal intensity is incident on a slit of width $1.8 \mu \text{m}$. Find the separation of the $m = 1$ bright spots of the two wavelengths on a screen 30.0 cm away.

A single slit of width $3.0 \mu \text{m}$ is illuminated by a sodium yellow light of wavelength 589 nm. Find the intensity at a $15 °$ angle to the axis in terms of the intensity of the central maximum.

A diffraction grating produces a second maximum that is 89.7 cm from the central maximum on a screen 2.0 m away. If the grating has 6000 lines per centimeter, what is the wavelength of the light that produces the diffraction pattern?

A diffraction grating with 2000 lines per centimeter is used to measure the wavelengths emitted by a hydrogen gas discharge tube. (a) At what angles will you find the maxima of the two first-order blue lines of wavelengths 410 and 434 nm? (b) The maxima of two other first-order lines are found at $\theta_{1} = 0.097 \text{rad}$ and $\theta_{2} = 0.132 \text{rad}$. What are the wavelengths of these lines?

How many complete orders of the visible spectrum $\left(400 \text{nm} < \lambda < 700 \text{nm}\right)$ can be produced with a diffraction grating that contains 5000 lines per centimeter?

On a bright clear day, you are at the top of a mountain and looking at a city 12 km away. There are two tall towers 20.0 m apart in the city. Can your eye resolve the two towers if the diameter of the pupil is 4.0 mm? If not, what should be the minimum magnification power of the telescope needed to resolve the two towers? In your calculations use 550 nm for the wavelength of the light.

Calculate the wavelength of light that produces its first minimum at an angle of $36.9 °$ when falling on a single slit of width $1.00 \mu \text{m}$.

As an example of diffraction by apertures of everyday dimensions, consider a doorway of width 1.0 m. (a) What is the angular position of the first minimum in the diffraction pattern of 600-nm light? (b) Repeat this calculation for a musical note of frequency 440 Hz (A above middle C). Take the speed of sound to be 343 m/s.

How far would you place a screen from the slit of the previous problem so that the second minimum is a distance of 2.5 mm from the center of the diffraction pattern?

Suppose that the central peak of a single-slit diffraction pattern is so wide that the first minima can be assumed to occur at angular positions of $\pm 90 °$. For this case, what is the ratio of the slit width to the wavelength of the light?

Determine the intensities of three interference peaks other than the central peak in the central maximum of the diffraction, if possible, when a light of wavelength 500 nm is incident normally on a double slit of width 1000 nm and separation 1500 nm. Use the intensity of the central spot to be $\left(1 \text{mW}/\text{cm}\right)^{2}$.

Structures on a bird feather act like a reflection grating having 8000 lines per centimeter. What is the angle of the first-order maximum for 600-nm light?

(a) What visible wavelength has its fourth-order maximum at an angle of $25.0 °$ when projected on a 25,000-line per centimeter diffraction grating? (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

An amateur astronomer wants to build a telescope with a diffraction limit that will allow him to see if there are people on the moons of Jupiter. (a) What diameter mirror is needed to be able to see 1.00-m detail on a Jovian moon at a distance of $7.50 \times 10^{8} \text{km}$ from Earth? The wavelength of light averages 600 nm. (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent…

(a) Assume that the maxima are halfway between the minima of a single-slit diffraction pattern. The use the diameter and circumference of the phasor diagram, as described in Intensity in Single-Slit Diffraction, to determine the intensities of the third and fourth maxima in terms of the intensity of the central maximum. (b) Do the same calculation, using the reference.

What is the maximum number of lines per centimeter a diffraction grating can have and produce a complete first-order spectrum for visible light?

A He-Ne laser beam is reflected from the surface of a CD onto a wall. The brightest spot is the reflected beam at an angle equal to the angle of incidence. However, fringes are also observed. If the wall is 1.50 m from the CD, and the first fringe is 0.600 m from the central maximum, what is the spacing of grooves on the CD?