1. Wave Nature of Light
Questions and Problems
1. Gaussian beam A particular HeNe laser beam at 633 nm has a spot size of 0.8 mm. Assuming a
Gaussian beam, what is the divergence of the beam? What are its Rayleigh range and the beam width at 10
m?
2. Gaussian beam in a cavity with spherical mirrors
Consider an optical cavity formed b two aligned spherical mirrors facing each other as shown in Figure
1.1. Such an optical cavity is called a spherical mirror resonator, and is most commonly used in gas
lasers. Sometimes, one of the reflectors is a plane mirror. The two spherical mirrors and the space between
them form an optical resonator because only certain light waves with certain frequencies can exit in this
optical cavity. The radiation inside a spherical mirror cavity is a Gaussian beam. The actual or particular
Gaussian beam that fits into the cavity is that beam whose wavefronts at the mirrors match the curvature
of the mirrors. Consider the symmetric resonator shown in Figure 1.1 in which the mirrors have the same
radius of curvature R. When a wave starts at A, its wavefront is the same as the curvature of A. In the
middle of the cavity it has the minimum width and at B the wave again has the same curvature as B. Such
a wave in the cavity can replicate itself (and hence exist in the cavity) as it travels between the mirrors
provided that it has right beam characteristics, that is the right curvature at the mirrors. The radius of
curvature R of a Gaussian bean wavefront at a distance z along its axis is given by
R(z) = z[1 + (zo/z)2] ; zo = wo2/.is the Rayleigh range
Consider a confocal symmetric optical cavity in wso that the mirrors are separated by L = R.
a
Show that the waist of the beam 2wo is fully determined only by the radius of curvature R of the
mirrors.
b
Show that the cavity length L is 2zo, that is, it is the same as the Rayleigh range, which is the
reason the latter is called the confocal length.
c
If the cavity length L = 50 cm, find R, the waist of the beam, and the beam width 2w at the mirrors.
Figure 1.1 Two spherical mirrors reflect waves to and from each other. The optical cavity contains a
Gaussian beam. This particular optical cavity is symmetric and confocal; the two focal points coincide at
F.
Chapter 1
1.2
3. Refractive index, reflection and the Brewster angle
a
Consider light of free-space wavelength 1300 nm traveling in pure silica medium. Calculate the
phase velocity and group velocity of light in this medium. Is the group velocity ever greater than the phase
velocity?
b
What is the Brewster angle (the polarization angle p) and the critical angle (c) for total internal
reflection when the light wave traveling in this silica medium is incident on a silica/air interface. What
happens at the polarization angle?
c
What is the reflection coefficient and reflectance at normal incidence when the light beam
traveling in the silica medium is incident on a silica/air interface?
d
What is the reflection coefficient and reflectance at normal incidence when a light beam traveling
in air is incident on an air/silica interface? How do these compare with part (c) and what is your
conclusion?
4. Antireflection (AR) coating
a
Consider three dielectric media with flat and parallel boundaries with refractive indices n1, n2, and
n3. Show that for normal incidence the reflection coefficient between layers 1 and 2 is the same as that
between layers 2 and 3 if n2 = [n1n3]. What is the significance of this result?
b
Consider a Si photodiode that is designed for operation at 900 nm. Given a choice of two possible
antireflection coatings, SiO2 with a refractive index of 1.5 and TiO2 with a refractive index of 2.3, which
would you use and what would be the thickness of the antireflection coating? The refractive index of Si is
3.5. Explain your decision.
c
Consider a Ge photodiode that is designed for operation around 1200 nm. What is the best AR
refractive index and its thickness if the refractive index of Ge is about 4.0?
5. Reflection at glass-glass and air-glass interfaces
A ray of light that is traveling in a glass medium of refractive index n1 = 1.460 becomes incident on a less
dense glass medium of refractive index n2 = 1.430. Suppose that the free space wavelength of the light ray
is 850nm.
a
What should the minimum incidence angle for TIR be?
(b)
What is the phase change in the reflected wave when the angle of incidence i = 85° and
when i = 90°?
b
90°?
What is the penetration depth of the evanescent wave into medium 2 when i = 85° and when i =
c
What is the reflection coefficient and reflectance at normal incidence (i = 0°) when the light beam
traveling in the silica medium (n = 1.455) is incident on a silica/air interface?
d
What is the reflection coefficient and reflectance at normal incidence when a light beam traveling
in air is incident on an air/silica (n = 1.455) interface? How do these compare with part (c) and what is
your conclusion?
6. Reflection and transmission at a semiconductor-semiconductor interface
Chapter 1
1.3
A light wave with a free space wavelength of 890 nm (free space wavelength) that is propagating in
GaAs becomes incident on AlGaAs. The refractive index of GaAs is 3.60, that of AlGaAs is 3.30.
a
Consider normal incidence. What are the reflection and transmission coefficients and the
reflectance and transmittance? (From GaAs into AlGaAs)
b
What is the Brewster angle (the polarization angle p) and the critical angle (c) for total internal
reflection for the wave in a; the wave that is traveling in GaAs and incident on the GaAs/AlGaAs
interface.
c
What is the reflection coefficient and the phase change in the reflected wave when the angle of
incidence i = 79?
d
What is the penetration depth of the evanescent wave into medium 2 when i = 79 and when i =
89? What is your conclusion?