Homework for G.P III, Set #2

1. Hecht’s book（4th edition-electronic version）, problem 3.46

2. Hecht’s book, problem4.5.

3. Hecht’s book, problem 4.25.

4. This problem is related to the formation of rainbow, see the figure:

The incoming light is a plane wave (say the light coming from the sun, it acts like a plane wave

for the droplet of water), it shines on a spherical shaped water droplet. The reflected light from the

inner surface of the droplet would generally have a different direction of propagation. The angle

difference between the directions of the output and incoming light is called deviation angle .

Please show that 1) For a certain incident angle , there is a minimum value of the deviation

angle (first find the geometric relations between , and other angles, then find the from

minimizing , you need Snell’s law) 2) The refractive index of water for red and violet light are:

n(700nm)=1.331, n(400nm)=1.343, what are the minimum deviation angles for these two light? 3)

For the broad illumination like the sun, the reflected light form the droplet would be limited inside

a cone, find the angles of this light-cone for the red and violet light.

5. Hecht’s book, problem 4.40. Add one question, what is the angle between the electric field of

the incident and reflected light?

6. A point source (the light from it is a spherical wave) is placed in air (n0), and it is at distance

of r0 from a flat surface that separate air and a media with refractive index of n1 :

(1) Prove for the light that forms very small angles with the normal direction of the

surface, the refracted light after the surface will also be a spherical wave, but its

origin (the spherical center) is shifted.

(2) Find the origin of the refracted light. (If you work out this correct, you just proved

image formation by flat surface under paraxial condition)

7. (No points if no derivation)Prove that for the internal reflection (ni > nt), at the incident angle

ii larger or equal to the critical angle, the reflection coefficient r, for the S and P components

are complex numbers in forms of ApExp (-iδp), AsExp (-iδs), with amplitude A=1, and the

phase δ is given by equation 10.27 in Zhao’s book, pg 254.

8. A circular polarized light is defined in Zhao’s book, pg242，and Hecht’s 8.1.2. Basically it

can decompose into two orthogonal components, the two orthogonal components can be the

Vertical and Horizontal ones, or the oscillation along the y, x direction as in Zhao’s book,

where , x , y , z are unit vectors along the positive direction of the Cartesian

coordinate system, z is the direction of light propagation. (Another way to put it, is that the circular polarization is a superposition of two perpendicular harmonic oscillation with equal

amplitude but π/2 phase difference) When looking into (facing) the direction of the

propagation of light, if the phase difference between the Vertical and Horizontal components

is -π/2, i.e. if the Vertical is leading in phase by π/2, it is a right-circular polarized light; if the

Vertical is lagging in phase by π/2, (phase difference is π/2) it is a left-circular polarized light.

Now the question is: if the incident light is a right-circular polarized light, at normal incidence

angle (ii=0) at an interface between (ni, nt), what is the polarization state of the reflected light?

Does it matter whether it is internal or external reflection? ( As to the respective coordinate

systems for the incident and reflected light, you can define the S component corresponds to

the Vertical, the P component corresponds to the Horizontal, the coordinate system is

specified in the class as well as in Zhao’s book, Figure 10-1) Give out your reasoning (80% of

the points), not just the answer.

Other recommended practice: Hecht’s 4.3, 4.20,4.22, 4.33,4.44, 4.50, 4.56, 4.65

Zhao’s book: Question 5,6, Pg 22; problems 3,5,6,12, 14, 16, Pg 23-25; Question7 (Pg 263);

Problem 5, 9 (pg 264-265)