Diffraction and interference (slits)
Consider a single slit diffraction pattern for a slit of width d. Find the value of d if the angle between the first minimum and the central maximum is 4*10-3 radians and the wavelength of the light is 633 nm.
A beam of monochromatic light with a wavelength of 500 nm
is directed through an absorber having 5 equally narrow slits separated by 20
μm between adjacent slits. The
resulting diffraction pattern is observed on a screen that is perpendicular to
the direction of light and 5 m from the slits. The intensity of the central
maximum is 1.3 W/m2.
(a) What are the distances from the central maximum to the first and second principal maxima on the screen?
(b) What will be the intensity of the central maximum if there are only 4 equally narrow slits (of the same width as in part a), separated by 20 μm between adjacent slits?
(b) The central intensity is proportional to the square
of the number of sources.
If there are only 4 sources, then the intensity of the central maximum will be
1.3*16/25 W/m2 = 0.832 W/m2.
Two gratings A and B have slit separations dA and dB,
They are used with the same light source and the same observation screen. When grating A is replaced with grating B, it is observed that the first-order maximum of A is exactly replaced by the second-order maximum of B.
(a) Determine the ratio dB/dA.
(b) Find the next two principal maxima of grating A and the principal maxima of grating B that exactly replace them when the gratings are switched. Identify these maxima by their order numbers.
Light of wavelength 640 nm is shone on a double-slit apparatus and the interference pattern is observed on a screen. When one of the slits is covered with a very thin sheet of plastic of index of refraction n = 1.6, the center point on the screen becomes dark. What is the minimum thickness of the plastic?
Suppose light from a Helium-Neon laser (wavelength λ = 633 nm) is expanded and collimated into a beam with a diameter of 2 cm, and then split into two beams that intersect as shown in the drawing below. Let the polarization of the beam be perpendicular to the plane of the drawing.
(a) If the full angle between the two overlapping beams is 10o,
how many fringes appear in the overlapping region?
(b) When the angle between the overlapping beams is increased to 30o, does the fringe spacing in the overlap region increase or decrease? By how much?
k1x = kcosθ, k2x = kcosθ, k1y = ksinθ, k2y
In the regions where the beams overlap we have E = E1 + E2. Let E = Ek.
E(r,t) = A1cos(k1∙r-ωt) + A2cos(k2∙r-ωt).
E(r,t)2 = (A1cos(k1∙r-ωt))2 + (A2cos(k2∙r-ωt))2 + 2A1cos(k1∙r-ωt)A2cos(k2∙r-ωt).
cosAcosB = ½[cos(A+B) + cos(A-B)]
cos(k1∙r-ωt+k2∙r-ωt) = cos(2kcosθ x - 2ωt).
cos(k1∙r-ωt-k2∙r+ωt) = cos(2ksinθ y)
E(r,t)2 = (A1cos(k1∙r-ωt))2 + (A2cos(k2∙r-ωt))2 + A1A2[cos(2kcosθ x - 2ωt)+cos(2k1sinθ y)]
The average values of cos2(k1∙r-ωt) and cos2(k2∙r-ωt) are ½, and cos(2kcosθ x - 2ωt) is zero, when averaged over a large number of periods.
<E(r,t)2> = A12/2 +A22/2 + A1A2cos(2ksinθ y)]
<I> = <I1> + <I2> + 2(<I1><I2>)½cos(2ksinθ y)]
The intensity therefore varies with y as C1 + C2cos((2π/λ')y) = C1 + C2cos(2ksinθ y).
λ' = π/ksinθ.
How many λ' fit into d, the maximum width of the overlapping region?
Φ = 90o - 2θ, w = b/cosΦ = b/sin(2θ), d/2 = wsinθ, d = 2bsinθ/sin(2θ)
The maximum width of the overlap region is 4 cm*sinθ/sin(2θ) = 2 cm/cos(θ).
The maximum number of fringes is (2 cm/cos(θ))/(π/ksinθ) = 4*tan(θ)/6.33*10-5.
When θ = 5o, then λ' = 3.63*10-6 m, we observe 5529 fringes.
(b) When θ = 15o, then λ' = 1.22 10-6 m, we observe 16932 fringes
Two radio-frequency sources, A and B, a distance 4 m apart, are radiating in phase radiation of wave length λ = 1 m. You move a detector on the circular path of radius R >> 4 m around the two sources in a plane containing them. How many maxima do you detect?
A monochromatic wave of wavelength λ illuminates an opaque mask with two slits as shown in the
figure. The diffraction pattern is recorded on a screen a distance L from the
mask. You may assume that λ << d. D << L.
(a) What is the distance Λ between adjacent interference fringes observed on the screen?
(b) What is the width Δx of the central lobe of the interference pattern on the screen?
Circular apertures, resolution
A spy satellite travels at a distance of 50 km above Earth's surface. How large must the lens be so that it can resolve objects with a size of 2 mm and thus read a newspaper? Assume the light has a wavelength of 400 nm.
For an ideal telescope, the limiting resolution is the size of the Airy disk. However, turbulence in Earth's atmosphere produces pockets of denser air which refract light rays from distant stars, causing them to strike a telescope detector at slightly different locations. At sea level, this imposes an effective resolution limit of 1 arc-second (1/3600 degree) regardless of the size of the telescope. For what diameter of telescope does this represent the diffraction limit for blue light (λ = 400 nanometers)?
(a) An incident monochromatic x-ray beam with wavelength
λ = 1.9 Å is
reflected from the (111) plane in a 3D solid with a Bragg angle of 32o
for the n = 1 reflection. Compute the distance (in
Å) between adjacent (111) planes.
(b) Assuming that the solid has a fcc lattice, use the result from part (a) to compute the lattice constant a (in Å).
(111) planes, FCC lattice