Faraday and Pockels Effect

7/27/2019 Faraday and Pockels Effect

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F a r a d a y a n d P o c k e l s E e c t

Benjamin Seeber Thierry Fredrich

9th march 09

The Faraday and the Pockels effect has been observed since physicist were ableto control electric and magnetic fields. But until the modern Quantum Mechanics

were established these effect were not understood.

The electric and the magnetic fields influences the optical properties of matter. Pock-

els observed this for the electric field and Faraday analogous for the magnetic field.

On this labor-day we did two experiments to become familiar with these effects.


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C o n t e n t s

1 P a r t A P o c k e l s - E e c t 4

1.1 Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3 Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 P a r t B F a r a d a y - E e c t 1 1

2.1 Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 S u m m a r i z i n g 1 5

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1 P a r t A P o c k e l s - E e c t

The electrooptic effect or the so called Pockels Effect contributes to the fact that dielectric con-

stant is only an approximation. More exactly is an tensor and defined by:

=D

E(1)

I.e. the displacement field D is not (!!!) a linear function of the electric field. So as usual we

describe it with its series expansion:

D = aE+ bE2 + cE3 + (2)

were a, b and c are some constants. We assume for our puspose that the higher terms have

negligible contribution. So becomes:

=dD

dE= a + 2bE+ 3cE2 +

(3)

If we know remember that the refraction index n is defined by n =rr were r and r are the

electric and magnetic constants of the material, it is obvious that well get some minor changes

in n by some changes in E. Thats what we call electrooptic effect.

Especially for the Pockels effect it is the linear term in (3) which is in involved.

1 . 1 P h y s i c s

1 . 1 . 1 B i - r e f r a c t i o n a n d t h e I n d e x e l l i p s o i d

The propagation of light in matter could either happen with the same velocity in all direction or

with different speed for each direction. In the first case we call the medium isotropic and in thesecond anisotropic. In general the isotropic are only the ones with cubic latice and anisotropic

all the others.

In the anisotropic case the propagation depends on the polarization and the direction of the k. Todescribe velocity in dependence of the direction and the polarization it is handy to introduce the

index ellipsoid as Fresnel did. Take a right handed coordinate system with Axis X1, X2, X3 and

at least one parallel to a latice axis so the index ellipsoid is:

X21

n21+

X22

n22+

X23

n23= c2(X21 v

21 +X

22 v

22 +X

23 v

23) = 1 (4)

vi velocity in direction i

ni refraction index for direction i

For instance: are we interested in the two perpendicular oscillating linear polarized components

of an nonpolarized beam of light that goes in direction OP than we just have to put this vectorin the ellipsoid and check out the perpendicular plain through the origin which is intersecting in

an ellipse. The two major-axis of this ellipse are pointing in the direction parallel to the linear

polarized parts of the wave and their length is the magnitude of the corresponding refraction

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Figure 1: Visualizing of the described example [Her77]

indexes n and n.

It is possible to derive the formula (4) exactly from the Maxwell- Equations (see [Wik09a]. For

an isotropic crystal the dielectric displacement is given by

D = 0rE (5)

For the anisotropic case equation (5) has to be replaced by an tensor- equation because of the

direction dependence:

Di = 0i jEj (6)

Equation (6) and the Maxwell- Equations leads to the fact that there are two waves with different

velocities in the same direction. These waves are perpendicular to each other and linear polarized

as describe for the index ellipsoid.

1 . 1 . 2 P i e z o e l e c t r i c - a n d r e v e r s e p i e z o e l e c t r i c e e c t

Piezoelectric is the ability of some materials (notably crystals and certain ceramics, including

bone) to generate an electric potential in response to applied mechanical stress. This may take

the form of a separation of electric charge across the crystal lattice. If the material is not short-

circuited, the applied charge induces a voltage across the material.

The piezoelectric effect is reversible in that materials exhibiting the direct piezoelectric effect

(the production of electricity when stress is applied) also exhibit the reverse piezoelectric effect

(the production of stress and/or strain when an electric field is applied). For example, the KD+P

crystal used in our experiment.

1 . 1 . 3 E l e c t r o o p t i c e e c t

Up to know we didnt consider the constrains which does or doesnt apply to the latice. We have

two cases to distinguish:

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Figure 2: Schematic diagram of the Pockelscell [Her77]

An mathematically computation (see [Her77]) with this formula, the fact that the crystall is

quadratic in shape and U = Ud

shows that

z63 =d

2 l n3

1 U2

(13)

: wavelength of the light

d width of the crystall

l length of the crystall

n1 refraction index of the centered direction

U2

current which is needed to get a phase shift of

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1 . 2 T a s k s

According to our instruction we do the following:

adjust the setup and the Pockelscell

measuring U/2 in two diffrent ways

computation ofz63

1 . 3 P r o c e s s i n g

The setup was completely installed on a fixed plate as seen on the picture.

Figure 3: schematic diagram of the experiment [Her77]

We put on our glasses and adjusted the laser beam to focus the detector. After getting familiar

with the aperture we tried some settings on the Oscilloscope to get a focused screen. And then

started our measurements according to the manual.

1 . 3 . 1 M e a s u r e m e n t s

At first we switched to the sawtooth generator an adjusted the following picture on the Oscillo-

scope. To get more accuracy the peak to peak method is used.The results are:

scale units scale value absolute value

amplitude of sawtooth As (4.70.1) SCAL 1 SCAL=1V (4.70.1) Volthalf period of sawtooth T1/2 (2.80.1) SCAL 1 SCAL=5ms (140.5) msamplitude of sine Asin (3.10.1) SCAL 1 SCAL=500mV (1.550.05) Volthalf period of sin 1/2 (3.10.1) SCAL 1 SCAL=2ms (6.20.2) ms

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Figure 4: Screenshot of the oscilloscope

1 . 3 . 2 C o m p u t a t i o n s

We compute now the U/2 on two different ways:

O n e w a y We know that the sawtooth takes 14 ms from 500V to 0V and the period of the sine

is also known. So that we can compute:

U/2 =500V

T1/2 1/2 =

500 V

14 ms6.2 ms = 221.43 V (14)

If we assume the sawtooth to be without error we get to as error estimation:

SU/2 =

(U/2T1/2

ST1/2 )2 + (

U/21/2

S1/2 )2

(500V1/2

T21/2ST1/2 )

2 + ( 500V1/2S1/2 )

2

(500V 6.2ms 0.5ms

(14ms)2)2 + ( 500V 0.2ms

14ms)2 = 10.66V

With formula (13) we can compute the electrooptic coefficient. Here we are assuming that all

the error of the given constants are small compared to our measurement ofU/2.

z63 = d

U/2 L n31

and Sz63 =SU/2

U/2z63 (15)

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with constants: L = 10.6 102mn1 = 1.5= 6328 1010md= 3.5 103m

So we get:

z63 = (27.961.35) 1012m

V

T h e o t h e r w a y Taking the mean of our 20 measurements which you can see in the appendix

we get U/2=250.97. Knowing the error for a single measurement we get via error propagation

the error for U/2 to be

5/

20V 0.5V.Again we compute the electrooptic coefficient with formula (15). Here the result is:

z63 = (24.670.05) 1012m

V

1 . 4 C o n c l u s i o n s

We have two results:

z63 = (27.961.35) 1012m

V

and

z63 = (24.670.05) 1012m

V

Because of the so different errors it make no sense to add this results up to one. If you do youll

get nearly exact the second result because it has a so little error. Maybe we under- or in case of

the second overestimated the errors of the oscilloscope.

An other source of errors could be that we neglect changes of temperature which the laserbeam

will cause in the crystall. Also the optical path is not 100% symmetric so we get some optical

errors such as aberrations and astigmatism.

Nevertheless matches our first result literature value of Kleen/Mller1 in a 2 error range.

126.4 1012 Voltm

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2 P a r t B F a r a d a y - E e c t

Already 1846 M. Faraday observed that a magnetic field influences the propagation of light in an

isotropic medium. This is a kind of induced bi-refraction of circular polarization what is named

the Faraday-effect. If you put a lucent isotropic material in a strong magnetic field and let a beamof light travel along the magnetic flux lines than its oscillating plane will be turned. The rotation

depends on the direction of the applied field. Therefor the angel must be proportional to the first

potency (or at least a odd potency) of the field. Its is also naturally that the angel depends on the

length of the material. If we take this two considerations into account we get following law:

= V l H (16)

The constant V is called the Verdetconstant and depends not on the field and on the thickness l.

It only depends on the wavelength of the used light.

The aim of this experiment is to verify this law.

2 . 1 P h y s i c s

You can describe the magnetrotation by the circular bi-refraction. The linear polarized beam can

be expanded in two contranary circular polarized waves. Each of those has a different refraction

index and therefore a different propagation speed in the glass. The general case is that the one

which goes along the direction of the current in the loop has the higher velocity.

A linear wave in the propagation in z direction in some matter can be describe by their two

components in the x-y plane:

x

y=

Fcos(z) cos(w(t zv

))

Fsin(z) cos(w(tz

v )) (17)

In which w is the angular velocity, v the speed of propagation and the angel which the

polarization is turned per length unit.

(18)

Now we substitute to get the equation for circular polarized light with:

w

v+=

w

v (19)

w

v =

w

v + (20)

Adding (19) and (20) gives us and expression for in dependence of the propagation speed

which we can assign.

2= wv w

v+(21)

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According to our former formula we have know a expression for turning the polarization by

on the distance l:

= l = w l2

(1

v 1

v+) =

w l2 c (nn+) (22)

By the approximation that


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Figure 5: Look through the telescope (from our mobile phone)

current[A]-6 -4 -2 0 2 4 6

]

angle

ofrotation[

-15

-10

-5

0

5

10

15

faraday.csv / ndf2 1.13 / 19p0 0.054670.4471

p1 0.01807-2.602

/ ndf2 1.13 / 19

p0 0.054670.4471

p1 0.01807-2.602

faraday.csv

Figure 6: Fit of the measured values

2 . 3 . 2 C o m p u t a t i o n s

C o m p u t e a n d c o m p a r e t h e m a g n e t i c e l d s

To compute the Verdet constat it is necessaryto know the magnetic field as function of the position. It is not!!! sufficient to use the approxi-

mation of an infinite solenoid.

It is a simple exercise to calculate the field with the Biot-Savart law. It is:

dH = 1/4Idl

r2sin() (24)

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The integration can be seen in [Her77].

H(z) =N I

2 L(x2

x1)

(Lz) lgx2 +

(Lz)2 +x22

x1 +(Lz)2 +x2

1

+z lgx2 +

z2 +x22

x1 +z2 +x21

(25)

Notice that this formula is linear in I!

length of the solenoid: 175mm

total number of coil 3600

diameter in 2x1 20mm

diameter out 2x2 150mm

Looking for the field in the center and applying 1A we plug these values in the formula and get

H(L/2) = 8212A/m

In discrepancy to the approximated formula for an infinite solenoid:

H(L/2) = const. =N I

L= 20571A/m

You see the difference so we have to use the exact form.

R o t a t i o n o f t h e p l a n e o f p o l a r i z a t i o n To get rotation angle we have to integrate again:

d= V H(z) dz = VL+l

2

Ll2

H(z) dz

Again we just give the result for the dimension of your solenoid. As you can imagine the exact

derivation is given in [Her77]. It is= VI2556 (26)

If you solve the upper equation to V you get a short formula to calculate V.

V =

I2556 (27)

From our data follows that we get a rotation of 26.4 for 10 Ampere.

V =26.45

10A 2556 = 1.035 103 Degree

Ampere(28)

The manufacture reference for the Schwertflint is given in an other unit so we have to change:

1A 1 Oe m79.59 =

1 Oe cm 10079.59 1.2564Oe cm

1Degree = 60 Min

V = 1.035 103 60Min1.2564 Oe cm

= 0.0494Min

Oe cm(29)

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2 . 4 C o n c l u s i o n s

In a first part we calculated the magnetic field in the approximation of an infinite solenoid and a

finite solenoid. Because of the dimensions of our experiment we saw that the approximated for-

mula differs too much from the exact one. Hence you used the exact formula for the calculation

of the Verdet constant in the second part.

Because we did just one series of measurement we skip the error calculation because it is not a

representative sample.

3 S u m m a r i z i n g

In the first part we computed the matrixelement z63 to

z63 = (27.961.35) 1012m

V.

which matches the result of Kleen/Mller in two standard deviation.

In the second part we determine the Verdet constant of a Schwertflint to

V = 0.0494Min

Oe cm.

The manufacturer reference for the Verdet constant of a Schwertflint is 0 .05 MinOe cm

according to

[Her77]. Because we did no error estimation for this result we can only say the same order of

magnitude. The manufacture took not in account that the constant depends on the wavelength.

Maybe they used an other one and thats the reason of the difference!

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L i s t o f F i g u r e s

1 Visualizing of the described example [Her77] . . . . . . . . . . . . . . . . . . 5

2 Schematic diagram of the Pockelscell [Her77] . . . . . . . . . . . . . . . . . . 7

3 schematic diagram of the experiment [Her77] . . . . . . . . . . . . . . . . . . 84 Screenshot of the oscilloscope . . . . . . . . . . . . . . . . . . . . . . . . . . 9

5 Look through the telescope (from our mobile phone) . . . . . . . . . . . . . . 13

6 Fit of the measured values . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

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Faraday and Pockels Effect