PHYS 410 Optics Homework Set #1 (Ch. 2)
(Due date: March 4, 2024)
1. [5 pts] Show that 𝜓(𝑥, 𝑦, 𝑧, 𝑡) = 𝑓 [𝑘(𝛼𝑥 + 𝛽𝑦 + 𝛾𝑧) − 𝜔𝑡] is a plane wave solution of the
three-dimensional (3-D) wave equation, where 𝑓 is an arbitrary twice-differentiable function.
!(#$%&)
2. [5 pts] Show that 𝜓(𝑟, 𝑡) = # is the spherical wave solution (centered at the origin
and moving away from it with a speed 𝑣) of the 3-D wave equation. Here 𝑓(𝑟 − 𝑣𝑡) is an
arbitrary twice-differentiable function.
(
3. Consider the disturbance profile 𝜓(𝑦, 0) = )* ! +,.
(a) [3 pts] Write the expression for the corresponding progressive wave moving with a speed
of 5 m/s in the positive 𝑦-direction.
(b) [2 pts] Verify that the answer above is indeed a solution to the wave equation.
4. Consider two plane waves, 𝜓, (𝑧, 𝑡) = 𝐴 cos(𝑘𝑧 − 𝜔𝑡) and 𝜓) (𝑧, 𝑡) = 𝐴 cos(𝑘𝑧 + 𝜔𝑡).
(a) [3 pts] When the two waves overlap in some region of space, what is the resulting wave?
(b) [2 pts] In what direction is the resulting wave propagating?
5. [5 pts] Hecht 2.36
6. [5 pts] Hecht 2.41
7. [5 pts] Hecht 2.49
8. [5 pts] Hecht 2.51
9. [5 pts] Hecht 2.59
.
10. The speed of light (phase velocity) in a medium is given as 𝑣- = /, where c is the speed
of light in vacuum and n is the refractive index of the medium. The refractive index of a
medium varies with frequency or wavelength, i.e., 𝑛(𝜔) or 𝑛(𝜆), and this phenomenon is
.
0
called dispersion. The phase velocity is therefore 𝑣- = / = 1 . The group velocity is given as,
𝑑𝜔
𝑑
𝑑 𝑘𝑐
𝑐 𝑐𝑘 𝑑𝑛
𝑘 𝑑𝑛
𝑣2 =
=
?𝑘𝑣- @ =
A C= − )
= 𝑣- A1 −
C.
𝑑𝑘 𝑑𝑘
𝑑𝑘 𝑛
𝑛 𝑛 𝑑𝑘
𝑛 𝑑𝑘
Derive the following.
3%
(a) [5 pts] 𝑣2 = 𝑣- − 𝜆 34" .
,
4# 3/
"
. 34#
(b) [10 pts] 𝑣2 = A% −
$,
C , where 𝜆5 is the vacuum wavelength.