34. Wave Optics

A diffraction grating has 15,000 rulings in its 1.9 cm width. Determine (a) its resolving power in first and second orders, and (b) the minimum wavelength resolution (∆λ) it can yield for λ = 410 nm.

When yellow sodium light, λ = 589nm, falls on a diffraction grating, its first-order peak on a screen 62.0 cm away falls 3.32 cm from the central peak. Another source produces a line 3.71 cm from the central peak. What is its wavelength? How many slits/cm are on the grating?

A He–Ne gas laser which produces monochromatic light of wavelength λ = 6.328 x 10^-7 m is used to calibrate a reflection grating in a spectroscope. The first-order diffraction line is found at an angle of 18.5° to the incident beam. How many lines per meter are imprinted on this reflecting diffraction grating?

How many slits per centimeter must a grating have if there is to be no second-order spectrum for any visible wavelength?

If X-ray diffraction peaks corresponding to the first three orders ( m = 1, 2, and 3) are measured, can both the X-ray wavelength λ and lattice spacing d be determined? Prove your answer.

(a) Derive an expression for the intensity in the interference pattern for three equally spaced slits. Express in terms of δ = 2πd sin θ / λ where d is the distance between adjacent slits and assume the slit width D ≈ λ.

(b) Show that there is only one secondary maximum between principal peaks.

Suppose the angles measured in Problem 42 were produced when the spectrometer (but not the source) was submerged in water. What then would be the wavelengths (in air)?

Red laser light from a He–Ne laser (λ = 632.8 nm) creates a second-order fringe at 53.2° after passing through a grating. What is the wavelength λ of light that creates a first-order fringe at 21.2°?

The wings of a certain beetle have a series of parallel lines across them. When normally incident 520-nm light is reflected from the wing, the wing appears bright when viewed at an angle of 56°. How far apart are the lines?

(a) How far away can a human eye distinguish two car headlights 2.0 m apart? Consider only diffraction effects and assume an eye pupil diameter of 6.0 mm and a wavelength of 560 nm. (b) What is the minimum angular separation an eye could resolve when viewing two stars, considering only diffraction effects? In reality, it is about of arc. Why is it not equal to your answer in (b)?

A beam of 125-eV electrons is scattered from a crystal, as in X-ray diffraction, and a first-order peak is observed at θ = 43°. What is the spacing between planes in the diffracting crystal? (See Section 35–11.)

What is the highest spectral order that can be seen if a grating with 6800 slits per cm is illuminated with 633-nm laser light? Assume normal incidence.

A laser beam of wavelength λ = 632.8 nm shines at normal incidence on the reflective side of a compact disc. (a) The tracks of tiny pits in which information is coded onto the CD are 1.60 μm apart. For what angles of reflection (measured from the normal) will the intensity of light be maximum? (b) On a DVD, the tracks are only 0.740 μm apart. Repeat the calculation of part (a) for the DVD.

(III) Derive an expression for the intensity in the interference pattern for three equally spaced slits. Express in terms of δ = 2πd sin θ / λ where d is the distance between adjacent slits and assume the slit width D ≈ λ . Show that there is only one secondary maximum between principal peaks.

Two rectangular pieces of plane glass are laid one upon the other on a table. A thin strip of paper is placed between them at one edge so that a very thin wedge of air is formed. The plates are illuminated at normal incidence by 546-nm light from a mercury-vapor lamp. Interference fringes are formed, with 15.0 fringes per centimeter. Find the angle of the wedge.