Department of Physics and Astronomy The University of Sheffield

argumentwildlifeΠολεοδομικά Έργα

16 Νοε 2013 (πριν από 3 χρόνια και 10 μήνες)

74 εμφανίσεις

Department of Physics and Astronomy

The University of Sheffield

1

4
x
4
Transfer Matrix and Reflectivity Calculations


Study the effect of using a thick substrate
(incoherent back reflections)




2


The aims of this work are


To derive expression of
4
×
4
Transfer matrix at a normal
incidence of light for a model of circularly birefringent
materials.


To calculate the reflectivity spectra in the case of

circularly polarised light for these structures.


To calculate the reflectance magneto
-
circular dichroism
(RMCD) , the Kerr and Faraday rotations.


To study the effect of using a thick substrate



(incoherent back reflections)
.

3


In recognizing real experimental magneto
-
optical data.




Magneto photonic structures play a key role in controlling
the optical properties and in enhancing the magneto optical
effect
(
Lourtioz

et al.
,
2008
).

4


Magneto optical studies have importance in understanding
the electronic structure of magnetic media
(
Reim

and
Schoenes
,
1990
).

In forming novel structures that utilise the optical property
sensitivity of photonic crystal to small variations in the
refractive index of the material from which it is fabricated.

5

Electromagnetic wave propagation inside multilayer structures
obeys Maxwell's equations.

in source free
J
=
0
and =
0

It is composed of periodic layers which have varied refractive
index or dielectric constant in one
-
dimension (
1
D).





The layer thickness is a quarter
-
wavelength




(
Joannopoulos

et al.
,
2008
)

6

and Kerr rotation as

Sato (
1981
) defined


the reflectance magneto
-
circular dichroism (RMCD) as


http://www.enzim.hu/~szia
/cddemo/edemo
16
.htm









7

The T
-
matrix matrix links E and B fields in different layers of
the structure
(Whittaker and
Culshaw
,
1999
), (Hecht,
2002
)






For a number of layers (multilayer film), the T
-

matrix is computed
as the product of the matrix for every layer, which means,

(
Whittaker and Culshaw,
1999
)

Hecht (
2002
)

8

The constitutive relation at a normal incidence for lossless
media that display a circular birefringence in an applied
magnetic field is given in matrix form by



(
Orfanidis
,
2008
).

9

The superscripts indicate to two values of
q
.

The eigenvector components are
circularly polarised state:




Starting from Maxwell's equations, the magnitude of wave vectors
are calculated
at normal incidence

In addition,

the expression of
4
x
4
transfer matrix is derived
for
these media




M

where
M

is a
4
x
4
transfer matrix of a single layer, and includes
2
x
2
block
.

matrices , are given by



(
1
)


10

11

For multilayer structures such as quarter wave stack and by applying
the boundary conditions at an interface between couple of layers,
equation
(
1
)

can be written as

M

here the superscripts
1

and
N

refer to the initial and final layers, respectively. The
resultant matrix
M

is
4
×
4
matrix.

This matrix is used to calculate the reflectivity spectra for both
right and left circularly polarised lights using computational
codes, which are written by FORTRAN

program
.

12

The reflectivity spectra for both left, and right circularly polarised light at
normal incidence

13

was taken from

(Dong et. al.,
2010
)

The reflectivity spectrum,

14

was taken from

(Dong et. al.,
2010
)

The RMCD against the wavelength

15

The Kerr and Faraday Rotations against the wavelength

16

17


the structure was taken
from (Dong et. al.,
2010
)


Reflectivity Spectrum for cavity structure

18

19

The RMCD against the wavelength

20

The Kerr and Faraday Rotations against the wavelength

At
629
nm
, the maximum is
4.73

compared with
0.0192
for film,

in Kerr rotation


Simulated Spectra for

Simulated Spectra

Dong et al. (
2010
)

Simulated Spectra

(this work)

21

, here we set n
s
=
1.0

Question has been raised
about the effect of use a
thick substrate

22

Those studies considered the coherent and incoherent
multiple reflections and transmissions for
isotropic

structures
to deal with this situations

23

As Previous studies pointed out that the spectra with a fine
Fabry
-
Perot fringes result, when one layer has a thicker
thickness than others. The resulted spectra are not realistic .
e.g.
(Harbecke,
1986
) ;(
Whittaker and
Gehring

2010
)

24

The total R for fully polarisation are given by

Whittaker and
Gehring

(
2010
)

front

back

(
Whittaker and
Gehring
,
2010
)

25

26

The reflectivity spectra for left circularly polarised light at normal incidence

27

The RMCD against the wavelength

28

3
.
multiple
incoherent back

reflections

2
.
Single
incoherent back
reflections

1
.
without incoherent
back reflections


a thick substrate

The equations of total and are calculated individually as


for x
-
polarised state

29

In a similar way, for y
-
polarised state

where


30

and are the matrices of linear x and y polarisations, respectively
(
Pedrotti

and
Pedrott
,
1993
)



The Kerr rotation is found as following

31

32

The Kerr Rotation against the wavelength

At
629
nm
, the maximum is
4.73
without
incoherent back reflections compared with
1.368
with incoherent back reflections

33

The Faraday Rotation against the wavelength


A multilayer structure of photonic crystal was modelled for anisotropic
materials that display a circular birefringence


Maxwell's equations were used to derive expression of
4
x
4
T
-
matrix

for these media


In circularly birefringent media, the reflectivity spectra and magneto
-
optical effect (RMCD, Kerr and Faraday rotations) were calculated.


There was a significant contribution of incoherent back reflections
….
from substrate . A thick substrate should be studied in real system.

34

35