Optics Lab – Department of Physics – Baghdad University 2012 – Exp. No. 10 1

Polarization by Reflection at a Dielectric and verifying Fresnel’s

Equations

Principle Plane-polarized light is reflected at a glass surface. Both the rotation of the plane of polarization and the intensity of the reflected light are to be determined and compared with Fresnel's formulae for reflection. Tasks

i. The reflection coefficients for light polarized perpendicular and parallel to the plane of incidence are to be determined as a function of the angle of incidence and plotted graphically.

ii. The refractive index of the flint glass prism is to be found. iii. The reflection coefficients are to be calculated using Fresnel's formulae and

compared with the measured curves. iv. The reflection factor for the flint glass prism is to be calculated. v. The rotation of the polarization plane for plane polarized light when

reflected is to be determined as a function of the angle of incidence and presented graphically. It is then to be compared with values calculated using Fresnel's formulas.

What you can learn about ▪ Electromagnetic theory of light ▪ Reflection coefficient, Reflection factor ▪ Brewster's law ▪ Law of refraction ▪ Polarization, Polarization level

Theory: When unpolarized light is incident on a dielectric medium such as a glass, the reflected rays are partially plane – polarized, i.e., it is a mixture of a plane polarized and unpolarized (ordinary) light. Let a plane light wave strike a boundary between two media having refractive indexes n1 and n2. The angles , , and are the angles of incidence, reflection, and refraction, respectively. In this case:

Optics Lab – Department of Physics – Baghdad University 2012 – Exp. No. 10 2

| | | | (the law of reflection) and

(the law of refraction) The fraction of light reflected or refracted depends on:

i. The refractive indices of the media ii. The angles of incidence

iii. The state of polarization of the incident light At a certain angle of incidence , the reflected light is totally plane – polarized, (Figure 1). This angle is called the “Brewster’s Angle” or “Polarizing Angle”, and it is related to the refractive index of the medium by:

This relation is called “Brewster’s law”. Also, for this angle the reflected electric vector E is normal to the plane of incidence, i.e., the plane containing the incident ray and the normal at the point of incidence. Let the plane of vibration makes an angle

with the normal to the plane of incidence (figure 2). This angle is called the orientation angle whether it refers to light vibration in the incident, reflected or refracted light. The vector E in the incident, reflected or refracted light can be resolved into two amplitude components:

𝜑𝐵 𝜑𝐵

𝜑 𝑛

𝑛

90

Reflected ray

(polarized,𝐸⊥ )

Normal

Incident ray

(Unpolarized)

Refracted ray

(polarized)

Figure 1: Unpolarized light incident at the polarizing angle

𝐸𝑝

𝐸𝑠

𝐸

𝜑

𝜑 80

Figure 2: Plane - polarized light externally reflected from a glass surface at 80 degrees angle of

incidence.

𝑅𝑝

𝑅𝑠

𝑅

𝜓

Normal

Optics Lab – Department of Physics – Baghdad University 2012 – Exp. No. 10 3

i. Components parallel to the plane of incidence Ep, Rp, . These amplitude

components are called the p – components. ii. Components normal to the plane of incidence Es, Rs,

. They are called the s – components.

For unpolarized light, Ep=Es. According to the electromagnetic theory, the amplitudes, phase, and state of polarization of the reflected and refracted waves for the p – vibrations and s – vibrations are related to the corresponding characteristics of the incident wave by Fresnel’s equations:

for the reflected amplitudes.

for the refracted amplitudes.

It follows from equations (2 - 5) that for any value of the angles the signs of and

and the signs of and coincide. This means that the

phases also coincide; that is, in all cases the refracted wave retains the phase of the incident wave. For the components of the reflected wave (Rp and Rs), the phase relations depend on , n1, and n2. For example, if = 0, then when n2 > n1 the phase of the reflected wave will be shifted by π.

From 2 and 3, one gets:

This is the tangent of the angle , i.e.,

In experiments, rather than measuring the amplitude, scientists usually measure the intensity of a light wave, that is, the energy flux carried by it, which is proportional to the square of the amplitude. The ratios of the average energy

Optics Lab – Department of Physics – Baghdad University 2012 – Exp. No. 10 4

fluxes over a period of time in the reflected and refracted waves to the average energy flux in the incident wave are called the reflection coefficient r and the transmission coefficient t. We obtain from equations (2 - 5) the Fresnel equations that define the reflection and transmission coefficients for the s- and p-components of the incident wave:

(

)

In the absence of light absorption, rs + ts = 1 and rp + tp = 1, in accordance with the law of the conservation of energy. If natural light is incident on the boundary, that is, if all directions of oscillations of the electric vector are equally probable, then, one half of the wave’s energy is accounted for by p – oscillations and the other half by s – oscillations. In this case, the total reflection coefficient is:

( )

[

]

If 90 and , then rp = 0; that is, light that is polarized such that its electric vector lies in the plane of incidence under these conditions will not be reflected at all from the boundary. The reflected light, when natural light is incident at this angle, will be completely polarized.

For normal incidence of light on the boundary between two media ( = 0), the Fresnel equations for the amplitudes of reflected and refracted waves can be reduced to the form:

(

) (

)

(

)

(

)

Optics Lab – Department of Physics – Baghdad University 2012 – Exp. No. 10 5

In such a case, the difference between the components s and p disappears, since the concept of plane of incidence loses meaning. In this case, we obtain, in particular,

(

)

It follows from the above two equations that the reflection of light at the boundary is greater the larger the absolute value of the difference n2 – n1; the coefficients r and t do not depend on which side of the boundary the incident light wave arrives from.

A condition for applicability of the Fresnel equations is that the refractive index of the medium be independent of the amplitude of the electric field strength of the light wave. This condition, which is trivial in classical (linear) optics, is not satisfied for high – power radiant fluxes, such as those radiated by lasers. In these cases, the Fresnel equations do not provide a satisfactory description of the observed phenomena, and consequently the methods and concepts of nonlinear optics must be used.

Procedure

Brewster’s angle and the index of refraction n.

1. Arrange all the optical components on the optical bench as in figure 3.

Laser

Polarizer

Prism or

glass plate

360 0 - graduated

rotatable stage

Intensity meter

Figure 3 – Experimental setup for reflectance measurements as a function of angle of incidence.

Analyzer

Optics Lab – Department of Physics – Baghdad University 2012 – Exp. No. 10 6

2. Make the transmission axis of the polarizer in the horizontal orientation,

i.e., the transmitted laser beam will have horizontal polarization with the

vibration of the electric field vector component E parallel to the plane of

incidence ( ). Remove the analyzer from the optical bench.

3. Align the incident laser beam spot with the 0 mark on the graduated

rotating stage.

4. Align the photocell with the incident polarized laser spot and read the

initial intensity (I0) as registered by the attached multimeter.

5. Put the prism on the rotating stage which must be level and make the laser

beam incident normally on the prism. This is done whenever the reflected

and incident laser spots coincide with each other. If not, simply rotate the

prism (not the stage!) clockwise or anticlockwise direction around the

vertical axis until the reflected laser spot precisely becomes aligned with

the incident beam.

6. Slowly turn the graduated rotatable stage clockwise in 10 – degree steps.

At each orientation, rotate the Photocell arm slightly to get into the beam.

Read the intensity (I) as registered by the attached multimeter. Record the

measured data in a table 1 below.

7. Repeat step 3 by rotating the stage of the prism in 10 – degree angle

increments until reaching the value 80 0 (or even 90 0, if possible).

8. Using the same procedure as before, take measurements with the

transmission axis of the polarizer in the vertical orientation, i.e., the

I0= mV at =0 degree

degrees

Ip mV

rp=Ip/I0. Is

mV rs=Is/I0

10

20

30 40

.

.

90

Optics Lab – Department of Physics – Baghdad University 2012 – Exp. No. 10 7

transmitted laser beam will have vertical polarization: the electric field

component vibrating perpendicularly to the plane of incidence ( ⊥).

Analysis

1. Take the ratio Ip/Io, graph it versus angle . Determine the Brewster’s

angle.

2. Use Brewster's angle to calculate the index of refraction of glass prism

using equation 1. Use n1 = 1.000308

3. Calculate the parallel and perpendicular reflectances using Fresnel

equations.

Verification of Fresnel’s equations

1. Record the value of the analyzer scale at which the laser beam has

totally been extinct. Notice that there are always two analyzer positions

to give the extinction. Use the one closest to the zero reading of the

analyzer.

2. Rotate the analyzer exactly 450 from its initial reading Remove the

prism and return the analyzer arm to 180 0. Insert the polarizer between

the laser and the prism (on the optical bench) and rotate it until the

light is extinguished. The transmission axis of the polarizer is now 45 0 to

the horizontal and vertical directions. Hence, Ep=Es. Don’t change this

polarizer setting during the remaining part of the experiment.

3. Turn the analyzer arm to position 170 0 ( 8 . Place the prism on

the stage. Rotate this stage until the laser beam falls on the analyzer.

4. Rotate the transmission axis of the analyzer until the light is

extinguished. The difference between the current analyzer reading

and the zeroth reading recorded previously at the Brewster’s angle

gives the angle . Be sure that is closest to the zeroth reading of the

analyzer scale.

5. Repeat steps 6 and 7 for rotating stage angles of 160, 150, 140, 130,

120, 110, and 100. Tabulate your readings as in table 2 below.

Optics Lab – Department of Physics – Baghdad University 2012 – Exp. No. 10 8

Analysis

1. Use Snell’s law and the evaluated refractive index n to calculate the

angle of refraction for each angle of incidence

2. Calculate the value of using equation 7.

3. Plot a graph of vs. Compare it with that of Jenkins and White.

Questions

1. Would Brewster's angle be more or less for light in air reflecting off water?

2. How would data look like for an arrangement with vertical square

polarizer?

3. How do polarized sunglasses reduce glare? Which direction is the axis of

polarization in a pair of polarized sunglasses? How could you check this?

170 160

.

. 100

85 80 . .

50