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University of California, Berkeley Physics H7C Spring 2013 (Yury Kolomensky)

PROBLEM SET 5 Interference, Diffraction Maximum score: 200 points Due: March 11, 2013 (note date !)

1. (20 points) (Hecht 9.4) ment, a thin film, Michelson’s interferometer, etc.) with Will we get an interference pattern in Young’s experi- a light at a mean wavelength of λ0 = 500 nm, having ment (Fig. 9.8) if we replace the source S with a single a linewidth of ∆λ = 2.5 × 10−3 nm. At approximately long-filament lightbulb ? What would occur if we re- what optical path length difference can you expect the placed the slits S1 and S2 with the same lightbulbs ? fringes to vanish ?

2. (30 points) (Hecht 9.3) 6. (20 points) (Hecht 9.47)

Return to Fig. 2.22 and prove that if two electromag- A glass camera lens with an index of refraction of ng = netic plane waves making an angle θ have the same in- 1.55 is to be coated with a creolite film (nc ≈ 1.30) to tensity I0, the resulting interference pattern on the yx- decrease the reflection of normally incident green light plane is a cosine-squared irradiance distribution given (λ0 = 500 nm). What thickness should be deposited on by the lens ?

2 2 π 7. (40 points) I(y) = 4I0 cos y sin θ  λ  Find conditions under which a charged relativistic par- Locate the zeros of irradiance. What is the value of ticle, moving with constant velocity v ∼ c in medium fringe separation ? What happens to the separation as of refraction index n, would radiate coherently. This θ increases ? Compare you analysis with that leading to effect is called Vavilov-Cherenkov radiation (or just Eq. (9.17). Cherenkov radiation, for uninitiated westerners), and its discovery and interpretation earned Pavel Cherenkov (a 3. (20 points) (Hecht 9.10) graduate student under Sergei Vavilov at the time of dis- White light falling on two long narrow slits emerges covery in 1934) and his theory colleagues Ilya Frank and is observed on a distant screen. If red light (λ0 = and Igor Tamm Nobel Prize in Physics in 1958. Con- 780 nm) in the first-order fringe overlaps violet in the ceptually, the effect is similar to sonic boom for sound second-order fringe, what is the latter’s wavelength ? waves. 4. (30 points) (Hecht 9.12) Hint: consider interference between light emitted by With regard to Young’s experiment, derive a general ex- the particle at different points on its trajectory. pression for the shift in the vertical position of the m-th 8. (20 points) (Hecht 10.9) maximum as a result of placing a thin parallel sheet of A collimated beam of microwaves impinges on a metal glass of index n and thickness d directly over one of the screen that contains a long horizontal slit that is 20 cm slits. Identify your assumptions. wide. A detector moving parallel to the screen in the 5. (30 points) (Hecht 9.17) far-field region detects the first minimum of irradiance It is our intention to produce interference fringes by il- at an angle 36.87◦ above the central axis. Determine the luminating some sort of arrangement (Young’s experi- wavelength of the radiation. 2

9. (40 points) (Hecht 10.16) refraction index n. The back side of the plate is dark- From symmetry considerations, create a rough sketch ened, except for a small round opening. The radius of of the Fraunhofer diffraction patterns of an equilateral the opening matches the first Fresnel zone r1 for some triangular aperture, and an aperture in the form of a plus point P behind the plate. In the middle of the opening, sign. the glass has a round cavity of depth h and diameter 0.5r1. Find the depth h such that the irradiance at point 10. (30 points) (Hecht 10.27) P is maximal and find that irradiance. The neoimpressionist painter Georges Seurat was a member of the pointillist school. His paintings consist λ of an enormous number of closely spaced small dots (≈ 1/10 inch) of pure pigment. The illusion of color mixing is produced only in the eye of an observer. How far from such a painting should one stand in order to achieve the desired blending of color ? n

11. (50 points) A plane wave of wavelength λ impinges on a glass h transmission grating shown below. The index of refrac- tion of glass is n and the step size is a. Find the min- imum depth h for which the irradiance at the central 0.5r1 Fraunhofer’s maximum is zero. Find the angle corre- sponding to the first diffraction maximum. r1 λ

n h P a a

12. (50 points) A monochromatic plane wave of wavelength λ and ir- radiance I0 impinges normally on a large glass plate,