IN
DEGREE PROJECT MATHEMATICS, SECOND CYCLE, 30 CREDITS
STOCKHOLM SWEDEN 2020 ,
Periodic and Non-Periodic Filter Structures in Lasers
LEO ENGE
KTH ROYAL INSTITUTE OF TECHNOLOGY
SCHOOL OF ENGINEERING SCIENCES
Periodic and Non-Periodic Filter Structures in Lasers
LEO ENGE
Degree Projects in Mathematics (30 ECTS credits) Degree Programme in Mathematics (120 credits) KTH Royal Institute of Technology year 2020 Supervisor at II-VI Incorporated: Jan-Olov Wesström Supervisor at KTH: Anders Szepessy
Examiner at KTH: Anders Szepessy
TRITA-SCI-GRU 2020:379 MAT-E 2020:086
Royal Institute of Technology School of Engineering Sciences KTH SCI
SE-100 44 Stockholm, Sweden
URL: www.kth.se/sci
Abstract
Communication using fiber optics is an integral part of modern societies and one of the most important parts of this is the grating filter of a laser. In this report we introduce both the periodic and the non-periodic grating filter and discuss how there can be resonance in these structures. We then provide an exact method for calculating the spectrum of these grating filters and study three different methods to calculate this approximately. The first one is the Fourier approximation which is very simple. For the studied filters the fundamental form of the results for this method is correct, even though the details are not. The second method consists of calculating the spectrum exactly for some values and then use interpolation by splines. This method gives satisfactory results for the types of gratings analysed. Finally a method of perturbation is provided for the periodic grating filter as well as an outline for how this can be extended to the non-periodic grating filter. For the studied filters the results of this method are very promising. The method of perturbations may also give a deeper understanding of how a filter works and we therefore conclude that it would be of interest to study the method of perturbations further, while all the studied methods can be useful for computation of the spectrum depending on the required precision.
Keywords
Laser, Grating Filters, Perturbation Theory, WKB Approximation, Bi-cubic
Splines.
Sammanfattning
Fiberoptisk kommunikation utgör en viktig del i moderna samhällen och en av de grudläggande delarna av detta är Bragg-filter i lasrar. I den här rapporten introducerar vi både det periodiska och det icke-periodiska Bragg-filtret och diskuterar hur resonans kan uppstå i dessa. Vi presenterar sedan en exakt metod för att beräkna spektrumet av dessa filter samt studerar tre approximativa metoder för att beräkna spektrumet. Den första metoden är Fourier-approximationen som är väldigt enkel. För de studerade filtrena blir de grundläggande formerna korrekta med Fourier-approximationen, medan detaljerna är fel. Den andra metoden består av att räkna ut spektrumet exakt för några punkter och sedan interpolera med hjälp av splines. Den här metoden ger mycket bra resultat för de studerade filtrena. Till sist presenteras en metod baserad på störningsteori för det periodiska filtret, samt en översikt över hur det här kan utökas till det icke-periodiska filtret. Denna metod ger mycket lovande resulat och den kan även ge djupare insikt i hur ett filter fungerar. Vi sluter oss därför till att det vore intressant att vidare studera metoder med störningar, men även att alla studerade metoder kan vara användabara för beräkningen av spektra beroende på vilken precision som krävs.
Nyckelord
Laser, optiska filter, störningsteori, WKB-approximation, bi-kubiska splines.
Acknowledgements
I would like to thank my two supervisors, Anders Szepessy at KTH and Jan-Olov Wesström at II-VI Inc., for their help and support throughout the whole project.
I am also grateful to the whole of II-VI Inc. for suggesting and then hosting this
project.
Contents
1 Introduction 1
2 Foundations in Electromagnetism and Optics 2
2.1 Light Propagation . . . . 2
2.1.1 Dispersion . . . . 4
2.2 Matrix Representation of Scattering and Transmission . . . . 4
2.3 Fabry-Perot Interferometer . . . . 6
3 Grating Reflector 11 3.1 Periodic Grating . . . . 11
3.2 Non-Periodic Grating . . . . 16
3.3 Tuning of a Grating . . . . 21
4 An Overview of Laser 23 4.1 Gain . . . . 23
4.2 The Laser Cavity . . . . 25
4.3 Tuning and Operation of a Laser . . . . 26
5 The Fourier Approximation 28 5.1 Periodic grating . . . . 29
5.2 Implementation . . . . 31
5.2.1 Periodic Grating . . . . 31
5.2.2 Non-Periodic Grating . . . . 33
6 Interpolation Methods 36 6.1 Cubic Splines . . . . 37
6.2 Bicubic Splines . . . . 38
6.3 Implementation for Grating Reflection . . . . 41
CONTENTS
6.3.1 Interpolation Over the Frequency . . . . 41
6.3.2 Interpolation Over Frequency and Current . . . . 46
7 Perturbation Methods and Coupled-Mode Theory 51 7.1 The Periodic Grating as a Perturbation Problem . . . . 51
7.1.1 Implementation . . . . 58
7.2 Non-Periodic Grating and WKB Analysis . . . . 59
7.2.1 Effective Medium and WKB Approximation . . . . 63
8 Conclusions 64
References 66
Chapter 1 Introduction
Optical communication is widely used in the world. In the development of this technology the lasers involved are becoming more and more complex. The methods to model these lasers must therefore evolve as well.
One fundamental component of many modern lasers is the so called grating filter or grating reflector, which is a structure of multiple reflecting surfaces interfering.
The structure can be either periodic, or a structure which is slightly perturbed from a periodic one.
There are well-known methods for modelling the behaviour of a periodic grating reflector in a computationally efficient way. In [1] an exact method is given and the approximative Coupled-mode theory is presented in [2], [3] and [4]. For the non- periodic case a suggested method, namely the WKB method is presented in [5] and [6], which is a development of the Coupled-mode theory, or a method of lattice filters as in [7].
The objective of this report is to introduce both the periodic and non-periodic grating
and discuss shortly the filtering properties of these gratings. We then aim to study
three different approximative approaches to model the behaviour of both periodic and
non-periodic Bragg gratings. The first one is the Fourier approximation, the second is
a method based on interpolation methods as found in [8] and the last one is a method
using perturbation theory, which is closely related to the Coupled-mode theory.
Chapter 2
Foundations in Electromagnetism and Optics
In this chapter we introduce some simple concepts in optics that are necessary to be able to understand the rest of the report. This includes the basic wave propagation, refractive indices, scattering and transmission matrices and the Fabry- Perot interferometer. For a more complete background reference, see for example [9].
2.1 Light Propagation
In vacuum, light propagates with the speed c ≈ 3 · 10 8 m s −1 , whereas in other medium it may propagate with a different speed, v. The ratio between the two is called the refractive index, n, of the material
n = c
v . (2.1)
The frequency, f , of light is the same in all mediums, but then since the wavelength, λ, of the light is equal to v/f , it must depend on the medium. If we let λ v be the wavelength in vacuum and λ be the wavelength in the current medium, they are related by
λ = λ v
n . (2.2)
CHAPTER 2. FOUNDATIONS IN ELECTROMAGNETISM AND OPTICS
n 1 n 2
E in E transmitted = t · E in
E reflected = r · E in
Figure 2.1.1: The reflection coefficient r and transmission coefficient t at an interface between two different materials.
Mathematical representation of light Light travelling along the x-axis is a transverse wave and can be written as a vector field E
E(x, t) = ˆ y E(x, t) = ˆ y E 0 e i(ωt − ˜ βx) (2.3)
where E 0 is the amplitude, ω = 2πf is the angular frequency and β = k + ˜ i g − α
2 = 2πn
λ v + i g − α
2 . (2.4)
Here, α is a loss in the amplitude of the light over space and g represents an amplification, or so called gain, which is explained more in Chapter 4, and i is the imaginary unit. Unless stated otherwise, we assume that g = 0. Looking only at the phase part of ˜ β, we can write
k = 2πn λ v = 2π
λ (2.5)
which is just the conventional real wave number.
Reflection and transmission As light E in propagates through an interface between two materials of different refractive indices n 1 and n 2 some of the light will be reflected and some will be transmitted, see Figure 2.1.1. The reflection coefficient r and transmission coefficient t shows how much of the light is reflected and how much is transmitted as follows
E transmitted = t · E in , E reflected = r · E in (2.6)
and the reflection coefficient and transmission coefficient are given by
r = n 2 − n 1
n 2 + n 1 , t = √
1 − r 2 . (2.7)
CHAPTER 2. FOUNDATIONS IN ELECTROMAGNETISM AND OPTICS
2.1.1 Dispersion
The frequency of the light depends on the wavelength, hence ω = ω(k). This relation between ω and k is called the dispersion relation. The speed of light mentioned earlier is the so called phase velocity, v phase , the speed with which the phase of the wave propagates through space. This is for example the speed any given peak of the light wave travels with. The phase velocity is given by
v phase = ω
k = f λ (2.8)
just as claimed earlier. If ω(k) is linear in k we see that the phase velocity is the same for all frequencies or equivalently that the relation between wavelength and frequency is always the same, i.e. f λ is constant. However, for many materials, this is not the case. There are materials where the refractive index depends on the frequency of the light and since v phase = c/n(f ) the phase velocity will also depend on the frequency of the light. This means that if the light is superposition of multiple frequencies traveling at different speeds the wave will not only move over time, but the shape of the wave will change over time.
2.2 Matrix Representation of Scattering and Transmission
If we have an optical structure with multiple interfaces where a i is the optical input and b i the optical output at interface i, as in figure 2.2.1, then we define the scattering matrix to be the matrix S such that
b 1
.. . b n
=
S 11 . . . S 1n .. . . . . .. . S n1 . . . S nn
a 1
.. . a n
. (2.9)
For a structure with only two interfaces, let A i be the right going wave and B i the left-
going wave at interface i = 1, 2, as in Figure 2.2.2. Then the transmission matrix is the
CHAPTER 2. FOUNDATIONS IN ELECTROMAGNETISM AND OPTICS
a 1
b 1
b 2 a 2
Figure 2.2.1: An optical structure with inputs a i and outputs b i .
matrix T such that
A 1 B 1
=
T 11 T 12 T 21 T 22
A 2 B 2
(2.10)
We can easily transform between the two types of matrices, S and T . For example we
A 1
B 1
A 2
B 2
Figure 2.2.2: An optical structure with right-going wave A i and left-going wave B i at interface i.
can find S 11 in terms of T as follows
S 11 = b 1 a 1
a 2 =0
= B 1 A 1
B 2 =0
= 1
A 1 T 21 A 2 = 1 A 1
T 21 A 1
T 11 = T 21 T 11 .
Doing this calculation for all components we find that
S 11 S 12 S 21 S 22
= 1 T 11
T 21 |T | 1 −T 12
, (2.11)
T 11 T 12 T 21 T 22
= 1 S 21
1 −S 22
S 11 −|S|
, (2.12)
where |T | denotes the determinant of, |T | = T 11 T 22 − T 12 T 21 . One of the advantages of
using the transmission matrix instead of the scattering matrix is that it works well for
CHAPTER 2. FOUNDATIONS IN ELECTROMAGNETISM AND OPTICS
multiple consecutive sections. For a situation like the one in Figure 2.2.3 we get
A 1 B 1
= T 1
A 2 B 2
= T 1 T 2
A 3 B 3
= T 1 T 2 . . . T n
A n+1 B n+1
(2.13)
A 1 A 2 A 3 A 4
B 1 B 2 B 3 B 4
T 1 T 2 T 3
Figure 2.2.3: Stacking multiple transmission matrices after each other.
2.3 Fabry-Perot Interferometer
One of the most fundamental optical structures is the Fabry-Perot interferometer. It consists of two interfaces where light is both transmitted and reflected and between them there is material of length L and refractive index n through which the light can propagate. See Figure 2.3.1.
a 1
b 1
b 2
a 2
t 1 t 2
r 1 r 2
−r 1 −r 2
L n
Figure 2.3.1: Fabry-Perot interferometer
We now look to find the scattering matrix of this structure, which is given by
b 1 b 2
=
S 11 S 12 S 21 S 22
a 1 a 2
. (2.14)
We see that
S 11 = b 1 a 1
a 2 =0
(2.15)
CHAPTER 2. FOUNDATIONS IN ELECTROMAGNETISM AND OPTICS
which is the reflection coefficient for the whole structure. So to find S 11 we derive an expression for how b 1 depends on a 1 .
First there is one term, −r 1 a 1 which is due to reflection in the first surface. However, some light will be transmitted through the first surface, then propagate to the second surface, reflect, propagate back, transmit through the first surface and rejoin with the other light after having made one round-trip in the interferometer. This will contribute to b 1 with the term
t 1 e −i ˜ βL r 2 e −i ˜ βL t 1 a 1 = t 2 1 r 2 e −2i ˜ βL a 1 . (2.16) Here we have used that light that propagates length L through a medium of refractive index n will change according to
E(x + L)
E(x) = e −i ˜ β(x+L)
e −i ˜ βx = e −i ˜ βL . (2.17) In the same way, some light will instead make two round-trips in the interferometer before joining with the rest of b 1 . The term for this light will be
t 2 1 r 2 2 r 1 e −4i ˜ βL a 1 . (2.18)
In general, the term for light making k round-trips in the interferometer will be
t 2 1 r k 2 r k 1 −1 e −2ki ˜ βL = t 2 1 r 1
r 1 r 2 e −2i ˜ βL
k
. (2.19)
Summing over k we get
b 1
a 2 =0 = −r 1 a 1 + a 1
t 2 1 r 1
X ∞ k=1
r 1 r 2 e −2i ˜ βL
k
=
= −r 1 a 1 + a 1 t 2 1
r 1 −1 + X ∞ k=0
r 1 r 2 e −2i ˜ βL
k
! .
(2.20)
The sum is over a geometric series so we can rewrite this as
b 1
a 2 =0 = −r 1 a 1 + t 2 1 a 1 r 1
−1 + 1
1 − r 1 r 2 e −2i ˜ βL
=
= −r 1 a 1 + t 2 1 a 1 r 1
−1 + r 1 r 2 e −2i ˜ βL + 1 1 − r 1 r 2 e −2i ˜ βL
!
= −r 1 + t 2 1 r 2 e −2i ˜ βL 1 − r 1 r 2 e −2i ˜ βL
! a 1
(2.21)
CHAPTER 2. FOUNDATIONS IN ELECTROMAGNETISM AND OPTICS
0.5 1 1.5 2
L / 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
|S 11 |
r = 0.9
r = 0.5
r = 0.3
Figure 2.3.2: Magnitude of S 11 as a function of the wavelength of the incident light plotted for some different values of the r 1 = r 2 = r.
and we get
S 11 = −r 1 + t 2 1 r 2 e −2 ˜ βL
1 − r 1 r 2 e −2i ˜ βL . (2.22)
In general S 11 will be a complex number and the amplitude of the reflected light will change with the factor |S 11 | and the phase of the light will shift corresponding to the phase of S 11 . In figure 2.3.2 and 2.3.3 we see how the magnitude and phase of S 11 depends on the wavelength of the incident light. By the same kind of calculations, the other components in the scattering matrix become
S 21 = b 2
a 1
a 2 =0
= t 1 t 2 e −i ˜ βL
1 − r 1 r 2 e −2j ˜ βL , (2.23)
S 22 = b 2 a 2
a 1 =0
= −r 2 + t 2 2 r 1 e −2 ˜ βL
1 − r 1 r 2 e −2i ˜ βL , (2.24) S 12 = b 1
a 2
a 1 =0
= S 21 (2.25)
CHAPTER 2. FOUNDATIONS IN ELECTROMAGNETISM AND OPTICS
0.5 1 1.5 2
L / 0
/2 3 /2 2
arg(S 11 )
r = 0.3 r = 0.9
Figure 2.3.3: Phase of S 11 as a function of the wavelength of the incident light plotted
for some different values of the r 1 = r 2 = r.
CHAPTER 2. FOUNDATIONS IN ELECTROMAGNETISM AND OPTICS
We can use (2.12) to find the transmission matrix of a Fabry-Perot interferometer
T 11 = 1 t 1 t 2
h
e i ˜ βL − r 1 r 2 e −i ˜ βL i
,
T 21 = − 1 t 1 t 2
h
r 1 e i ˜ βL − r 2 e −i ˜ βL i
,
T 12 = − 1 t 1 t 2
h
r 1 e −i ˜ βL − r 2 e i ˜ βL i
,
T 22 = 1 t 1 t 2
h
e −i ˜ βL − r 1 r 2 e i ˜ βL i
.
(2.26)
Chapter 3
Grating Reflector
We now turn to the central part of this report, the grating reflector. In this chapter we will explain what a grating is, its most important properties and what cause these properties. Note that there are many different names for the grating reflector: for example we may use Bragg grating, Bragg reflector, Fiber Bragg grating, Bragg filter, grating filter, or just reflector, grating or filter.
3.1 Periodic Grating
n 1 n 2 n 1 n 2 n 1 n 2 n 1 n 2
L 1 L 2
Figure 3.1.1: Grating structure.
A grating is a optical structure with alternating refractive indices, n 1 and n 2 , as in Figure 3.1.1. In the interfaces between the two indices there will be both reflection and transmission, according to r = ±(n 2 − n 1 )/(n 1 + n 2 ) and t = √
1 − r 2 . Each of the parts confined by reflecting interfaces will act as a small Fabry-Perot interferometer, where there will be interference between the reflected light depending on the wavelength of the light.
For a periodic grating structure each part of index n 1 will be of the same length, L 1 , and
each part of of index n 2 will be of the same length L 2 . For wavelengths with a strong
CHAPTER 3. GRATING REFLECTOR
resonance in the structure, there will also be a strong reflection through constructive interference. Other wavelengths will propagate through the structure more freely, less effected by the grating. If we let ¯ n be the average of the two refractive indices and Λ = L 1 + L 2 the length of a full period, then there will be a resonating mode in each period for the Bragg wavelength λ 0 satisfying the Bragg condition
λ 0 = 2¯ nΛ. (3.1)
This will lead to a resonating mode in the whole grating as well. Apart from this mode there will be other smaller modes for multiples of the Bragg wavelength.
We now want to find an expression for the reflection, r G , for the whole grating structure. We will do this by finding the transmission matrix, T G , of the grating.
T G =
T G,11 T G,12 T G,21 T G,22
(3.2)
and then as before let
r G = T G,21
T G,11 . (3.3)
We begin by looking at the transmission matrix T for one period of the structure. One period of the grating is just a Fabry-Perot interferometer followed by a transmission through a section of refractive index n 2 and length L.
Using (2.26) the transmission matrix T F B of the Fabry-Perot interferometer can be written as
T FB,11 = 1 t 2
h
e i ˜ β 1 L 1 − r 2 e −i ˜ β 1 L 1 i
,
T FB,21 = − r t 2
h
e i ˜ β 1 L 1 − e −i ˜ β 1 L 1 i ,
T FB,12 = − r t 2
h
e −i ˜ β 1 L 1 − e i ˜ β 1 L 1 i ,
T FB,22 = 1 t 2
h
e −i ˜ β 1 L 1 − r 2 e i ˜ β 1 L 1 i
.
(3.4)
where as before
β ˜ i = k i − i α
2 = 2πn i λ − i α
2 . (3.5)
CHAPTER 3. GRATING REFLECTOR
The transmission matrix T 2 through the part of length L 2 and refractive index n 2 is
T 2 =
e i ˜ β 2 L 2 0 0 e −i ˜ β 2 L 2
. (3.6)
Note that this transmission includes no reflections, since both reflections of a period are included in the transmission matrix for the Fabry-Perot interferometer. Now, multiplying these two matrices together we get the transmission matrix for one period, T = T F B T 2 . The components are
T 11 = 1 t 2
h
e i( ˜ β 1 L 1 + ˜ β 2 L 2 ) − r 2 e −i( ˜ β 1 L 1 − ˜ β 2 L 2 ) i
= 1 t 2
e iϕ+ − r 2 e −iϕ− ,
T 21 = r t 2
h
e i( ˜ β 1 L 1 + ˜ β 2 L 2 ) − e −i( ˜ β 1 L 1 − ˜ β 2 L 2 ) i
= 1 t 2
e iϕ+ − e −iϕ− ,
T 12 = r t 2
h
e −i( ˜ β 1 L 1 + ˜ β 2 L 2 ) − e i( ˜ β 1 L 1 + ˜ β 2 L 2 ) i
= 1 t 2
e −iϕ+ − e iϕ− ,
T 22 = 1 t 2
h
e −i( ˜ β 1 L 1 + ˜ β 2 L 2 ) − r 2 e i( ˜ β 1 L 1 − ˜ β 2 L 2 ) i
= 1 t 2
e −iϕ+ − r 2 e iϕ − ,
(3.7)
where we have set
ϕ + = ˜ β 1 L 1 + ˜ β 2 L 2 ϕ − = ˜ β 1 L 1 − ˜β 2 L 2
(3.8)
If we have a grating consisting of m periods, we can find the transmission matrix, T G for the whole grating through the multiplication
T G = T m (3.9)
From this we can then determine r G in (3.3). This can be a computationally heavy calculation. Since the structure is periodic, it is reasonable to believe that there is a way to calculate the reflection in a more computationally efficient way, using that the same period repeats itself over and over.
The matrix T is a self-adjoint or Hermitian matrix. Every finite self-adjoint matrix
is diagonalizable, this suggests looking at the eigenvalues of the matrix. If we have
the two eigenvalues λ 1 and λ 2 we can write ξ = ln λ 1 , where then e ξ = λ 1 will be an
eigenvalue. Here we must have ξ = a + inπ for some real number a and some integer
CHAPTER 3. GRATING REFLECTOR
n since the eigenvalues of a self-adjoint matrix are real. For any matrix we also know that
λ 1 λ 2 = |T | (3.10)
and calculating the determinant of T we get
|T | = T 11 T 22 − T 21 T 12 =
= 1 t 4
e iϕ+ − r 2 e −iϕ−
e −iϕ+ − r 2 e iϕ −
− r 2 t 4
e iϕ+ − e −iϕ−
e −iϕ+ − e iϕ −
=
= 1 t 4
¶ 1 − r 2
e i[(ϕ+)+(ϕ −)] + e −i[(ϕ+)+(ϕ−)] + r 4 ©
− 1 t 4
¶ 2r 2 − r 2
e i[(ϕ+)+(ϕ−)] + e −i[(ϕ+)+(ϕ−)] ©
=
= 1 + r 4 − 2r 2
t 4 = (1 − r 2 ) 2 t 4 = 1.
Therefore λ 1 λ 2 = 1. Then, since λ 1 = e ξ , it follows that λ 2 = e −ξ . We now look for an expression for the eigenvalues, by solving |T − e ±ξ I | = 0,
0 =
T 11 − e ±ξ T 12 T 21 T 22 − e ±ξ
= T 11 T 22 − e ±ξ (T 11 + T 22 ) + e ±2ξ − T 12 T 21 =
=
e ±ξ − 1
2 (T 11 + T 22 )
2
− 1
4 (T 11 + T 22 ) 2 + T 11 T 22 − T 12 T 21 which implies
e ±ξ = 1
2 (T 11 + T 22 ) ±
… 1
4 (T 11 + T 22 ) 2 − 1 (3.11) where we have used that T 11 T 22 − T 12 T 21 = 1. Let us now write the eigenvectors corresponding the eigenvalues as
v ± =
A ± B ±
. (3.12)
Then for one period of the grating
T 11 T 12 T 21 T 22
A ± B ±
= e ±ξ
A ± B ±
(3.13)
CHAPTER 3. GRATING REFLECTOR
and for the whole grating
T G,11 T G,12 T G,21 T G,22
A ± B ±
=
T 11 T 12 T 21 T 22
m
A ± B ±
= e ±mξ
A ± B ±
. (3.14)
The equations for A ± in (3.13) and (3.14) can be rearranged as following B ±
A ± = e ±ξ − T 11
T 12 = e ±mξ − T G,11
T G,12 . (3.15)
By taking B A +
+ − B A − − , we get e ξ − T 11
T 12 − e −ξ − T 11
T 12 = e mξ − T G,11
T G,12 − e −mξ − T G,11
T G,12 ,
sinh ξ
T 12 = sinh mξ
T G,12 . (3.16)
Instead taking B A +
+ + B A −
− , we get
cosh ξ − T 11
T 12 = cosh mξ − T G,11
T G,12 . (3.17)
Equations (3.16) and (3.17) can be rearranged into
T G,12 = sinh mξ
sinh ξ T 12 (3.18)
T G,11 = cosh mξ − T G,12
T 12 (cosh ξ − T 11 ) = cosh mξ − sinh mξ
sinh ξ (cosh ξ − T 11 ) =
= sinh mξ
sinh ξ T 11 − sinh mξ cosh ξ − sinh ξ cosh mξ
sinh ξ =
= sinh mξ
sinh ξ T 11 − sinh (m − 1)ξ sinh ξ .
(3.19)
The same kind of expressions can be derived for T G,21 and T G,22 as well.,
T G,21 = sinh mξ
sinh ξ T 21 , (3.20)
T G,22 = sinh mξ
sinh ξ T 22 − sinh (m − 1)ξ
sinh ξ . (3.21)
CHAPTER 3. GRATING REFLECTOR
1.934 1.936 1.938 1.94 1.942 1.944 1.946
f 10 14
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
|r G |
Figure 3.1.2: Magnitude of the reflection coefficient r G in a periodic grating structure.
Using these expressions the reflection coefficient of the grating becomes
r G = T G,21
T G,11 = T 21 T 11
1
1 − sinh (m T 11 sinh mξ −1)ξ . (3.22) In general the reflection coefficient is a complex number and in Figure 3.1.2 and 3.1.3 we can see the magnitude and phase of r G against the frequency of the reflected light. Throughout the report, when we use a periodic grating we will use the grating represented in Figure 3.1.2 and 3.1.3.
3.2 Non-Periodic Grating
If we consider a grating where the lengths of parts with the same refractive index are
not equal the grating is no longer periodic and much of the theory above fails. For each
part i, we can still calculate the transmission matrix T i . The transmission matrix for
CHAPTER 3. GRATING REFLECTOR
1.934 1.936 1.938 1.94 1.942 1.944 1.946
f 10 14
0 /2 3 /2 2
arg(r G )
Figure 3.1.3: Argument of the reflection coefficient r G for a periodic grating structure.
The phase of the reflected light will be shifted compared to the incident light by the
corresponding to the argument of r G
CHAPTER 3. GRATING REFLECTOR
the whole grating is then
T = T 1 T 2 . . . T m . (3.23)
This can be compared with the simpler counterpart for a periodic grating in (3.9).
Using equation (3.23), the reflection r G , for a non-period grating can be calculated
r G = T 21
T 11 , (3.24)
which provides an exact value for the reflection, in the sense that it is free from approximations. Calculating T will involve determining and then multiplying m matrices, so this calculation can be computationally heavy for a large m.
In the case of a periodic grating, we had an average refractive index ¯ n and a period length Λ and a Bragg wavelength λ 0 related to these. For a non-periodic grating, both ¯ n and Λ can vary over the length of the grating and so the Bragg wavelength will also vary.
We therefore get a local Bragg wavelength, λ 0 (x), where x is the point in the grating.
This means that as light propagates through the grating it only interact noticeably with the grating in regions where the local Bragg wavelength is approximately equal to the wavelength of the light. In all other regions, the light will propagate almost freely.
So far, we have considered gratings where the refractive index varies in a discrete way.
But we could have a grating where the index varies continuously, for example
n(x) = n 0
1 + a cos
2π Λ x
(3.25)
where a is some constant. This is still a periodic grating with period Λ. We can now introduce one type of non-periodic grating, the linearly chirped grating, defined by
n(x) = n 0
1 + a cos
2π
Λ 0 x(1 + cx)
(3.26) where c is called the chirp parameter. We see that the period length of this grating becomes shorter the further into the grating we get, so the local Bragg wavelength gets shorter in a corresponding way. There are many other types of non-periodic gratings.
In [10] and [11], some of these are defined and discussed. For the rest of the report we will as a non-periodic grating use a grating which has the reflection spectrum that can be seen in Figure 3.2.1 and 3.2.2.
We see that the there are multiple peaks in the reflection spectrum of this grating and
CHAPTER 3. GRATING REFLECTOR
1.9 1.92 1.94 1.96 1.98 2 2.02
f [Hz] 10 14
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
|r|
Figure 3.2.1: Magnitude of the reflection coefficient r G for a non-periodic grating
structure.
CHAPTER 3. GRATING REFLECTOR
1.9 1.92 1.94 1.96 1.98 2 2.02
f [Hz] 10 14
0 /2 3 /2 2
arg(r)
Figure 3.2.2: Phase of the reflection coefficient r G for a non-periodic grating structure.
CHAPTER 3. GRATING REFLECTOR
1.9 1.91 1.92 1.93 1.94 1.95 1.96 1.97 1.98 1.99 2
f [Hz] 10 14
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
|r|
I = 0 mA I = 4 mA I = 8 mA
Figure 3.3.1: Magnitude of the reflection coefficient r G for a non-periodic grating structure and three different detuning currents I.
that the peaks are much narrower than in the periodic case. This can be a desired property of a grating, and a reason why non-periodic gratings are used instead of periodic ones. Compared to the periodic case there are also many more lines in the plot of the argument of the reflection and their behaviour is seemingly more erratic than for the periodic grating.
3.3 Tuning of a Grating
The refractive indices of the grating can be changed by applying a current I to the
grating. As a current is applied, the refractive indices decreases and therefore the
optical path length in each part of the grating gets shorter. Therefore it is reasonable
to believe that the wavelength of a given mode in the grating gets shorter, or in other
words that the frequency increases. In Figure 3.3.1 we see how this can look, when
the grating is tuned with some currents. The exact relation for how the refractive
indices change as a functions of the applied current is left out. We will only care about
CHAPTER 3. GRATING REFLECTOR
calculating the reflection spectrum of a grating for a given set of properties, such as the
refractive indices and the lengths of each part, as correctly as possible. Details on how
the properties came about are beyond the scope of this report.
Chapter 4
An Overview of Laser
There are many different types of lasers, intended for completely different purposes.
This means that the desired properties of lasers can differ and therefore also the way in which they work. The lasers we will consider are supposed to emit coherent light (i.e.
light which is completely in phase) and of a single frequency. This single frequency should be tunable, while operating the laser.
We will describe a greatly simplified explanation on how this is achieved to provide some necessary context for the rest of the report. This exposition follows the one in [1]
and for a more complete discussion on the laser, this is a great reference.
4.1 Gain
As mentioned earlier, light propagating along the x-axis can be written as a transverse wave on the form
E(x, t) = ˆ y E(x, t) = ˆ y E 0 e i(ωt − ˜ βx) (4.1) where E 0 is the amplitude, ω = 2πf is the angular frequency and
β = k + ˜ i g − α
2 = 2πn
λ v + i g − α
2 . (4.2)
We have explained what the wave number k is, but left the parameters g and α more or
less unmentioned. As light travels through any material, some of the energy in the light
is transferred to the medium and so there will be losses. The parameter α represents
CHAPTER 4. AN OVERVIEW OF LASER
these losses. Looking only at the time-independent wave, we can write
E(x) = E 0 e −i ˜ βx = E 0 e g −α 2 x e −i 2πn λv x . (4.3)
So the amplitude of the wave decreases as it propagates, by the relation e −αx/2 . At the same time there is an increase in the amplitude, depending on the parameter g. This is called the gain. For most wave guides, the gain is zero. But it is possible to construct media where the gain is not zero. A component where the gain is non-zero is called a gain section. As light travels through a gain section the amplitude of the light increases depending on the relation
g − α
2 (4.4)
The gain g does in general depend on the frequency of the light, so certain frequencies of the light will be amplified more than others, as in Figure 4.1.1 for example.
1.85 1.9 1.95 2 2.05 2.1
f [Hz] 10 14
-1.5 -1 -0.5 0 0.5 1 1.5
g [m -1 ]
10 4
Figure 4.1.1: Example of the gain g as a function of the frequency f .
CHAPTER 4. AN OVERVIEW OF LASER
Figure 4.2.1: The laser cavity.
4.2 The Laser Cavity
The fundamental part of a laser is the laser cavity. This is where the light is generated and amplified. The laser cavity is a gain section enclosed by two reflecting surfaces, as in Figure 4.2.1. The reflecting surfaces can be simple surfaces with a reflectivity r which is constant for all frequencies, but one or both of the surfaces can also be a grating as explored in chapter 3, or something even more complex. The laser cavity is therefore similar to the Fabry-Perot interferometer described in chapter 2, but with generalised surface reflections r.
In each surface some light will be reflected and some transmitted. The output of the laser is the light transmitted through one of the two surfaces. Light which is reflected will continue to propagate in the cavity. If we follow light starting at some point of the cavity which is reflected in both surfaces, and propagates along the length of the cavity in both directions, then this light will after having completed one such round-trip have changed according to the round-trip reflection
X(f ) = r 1 e −i ˜ βL r 2 e −i ˜ βL (4.5)
The condition for a frequency f las to lase, is that X(f las ) = 1. This means that the light is unchanged by a round-trip in the laser cavity, which can be split into two separate conditions, one for the argument of the round-trip reflection, and one for the magnitude. The condition for the argument is
arg(X(f las )) ≡ 0 ( mod 2π) (4.6)
CHAPTER 4. AN OVERVIEW OF LASER
which is called the phase condition, and the condition for the magnitude is
1 = |X(f las ) | = |r 1 (f las ) ||r 2 (f las ) |e g(flas)−α 2 2L (4.7)
which is called the amplitude condition, and can be rewritten as
g(f las ) = α + 1 L ln
1
r 1 (f las )r 2 (f las )
(4.8)
Apart from short periods of time, it is not possible for the round-trip reflection to satisfy the phase condition and have |X(f)| > 1, because the gain g will decrease in such a way that |X(f)| = 1, see [1] for a more detailed explanation.
The lasing conditions implies that the intensity of the light of frequency f las , is the only light that is stable in the laser cavity. Light of any other frequency will be attenuated either by constructive interference if it does not satisfy the phase condition, or by attenuation in the magnitude if it does not satisfy the amplitude condition.
Ideally there is only one frequency satisfying the laser condition and then the intensity of this frequency will be substantially larger than for any other frequency. As some of the light is then transmitted through the output side of the laser, the output light will be almost of a single frequency.
4.3 Tuning and Operation of a Laser
The lasing frequency can be changed while operating the laser, by tuning of the laser.
This is done by modifying the properties of the components in the laser. Specifically modifying the refractive indices of a component will change the transmitting or reflecting spectrum that component. The refractive index can be changed in multiple ways, but we will only consider tuning by electric current I. As current is applied to a component, not only is the refractive index changed, but the losses α in the component will also increase.
Consider the laser in Figure 4.3.1. The gain section is the laser cavity as in Figure
4.2.1, but the reflecting surfaces of the laser cavity has been expanded into the separate
components they consist of. Both reflectors consist of some kind of grating reflector
as explained in Chapter 3. The right side also includes a so called phase section. Light
travels through this component while the phase of the light is shifted.
CHAPTER 4. AN OVERVIEW OF LASER
Figure 4.3.1: The laser as a system of multiple components.
Current can be applied to any of the four components in Figure 4.3.1. The current
applied to the gain section, will not affect what frequency is lasing to any large extent,
it mainly affects the power of the emitted light. Current applied to any of the other
three sections will change the lasing frequency and by choosing a combination of the
three, the desired lasing frequency can be achieved.
Chapter 5
The Fourier Approximation
The first approximative technique for calculating the grating reflection that we explore is the so called Fourier approximation. For a periodic grating this involves a sum that looks much like a Fourier sum, hence the name.
For many gratings, the reflection coefficient for each interface in the grating is extremely small, from the order 10 −3 down to 10 −5 , and the reflectivity of the grating as a whole is due to the fact that there are thousands of these interfaces whose reflectivity is added up. The Fourier approximation assumes that this interface reflectivity is so small that all but the linear terms of it are negligible.
r n ≈ 0 for all n ≥ 2. (5.1)
Applying the approximation to the interface transmitivity t, we see that t 2 = 1 −r 2 ≈ 1.
The physical interpretation of the approximation (5.1) is that the light is transmitted freely through all surfaces, but is reflected only once. Any multiple reflections are disregarded. It is important however to note that all light is not reflected in the same surface. We get situation as in Figure 5.0.1.
Figure 5.0.1: The Fourier approximation where light is reflected only once. For all
surfaces, t i = 1.
CHAPTER 5. THE FOURIER APPROXIMATION
If we let r i , n i and L i be the interface reflectivity, refractive index and length respectively of the the i th section of the grating, we get the following expression for the grating reflection r G
r G = r 1 + r 2 e −i 2πn12L1 λ + r 3 e −i 2πn12L1 λ e −i 2πn22L2 λ + · · · +r m e −2 iπn12L1 λ · · · e −2 2πnm2Lm λ =
= X m k=1
r k Y k l=1
e −i 2πnl2Ll λ .
(5.2)
This expression can be used directly to calculate an approximation for the grating reflection. For a periodic grating however, this expression can be simplified.
5.1 Periodic grating
We have a periodic grating as in Figure 5.1.1. As well as being a periodic grating, we also assume that
L 1 = λ 0 4n 1
, L 2 = λ 0 4n 2
(5.3) for some wavelength λ 0 , which we call the center wavelength.
n 1 n 2 n 1 n 2 n 1 n 2 n 1 n 2
L 1 L 2 L 1 L 2
Figure 5.1.1: A periodic grating structure
We can now add the contributions of each interface reflection to the total grating reflection r G
r G = −r + re −2iβ 1 L 1 − re −2i(β 1 L 1 +β 2 L 2 ) + re −2iβ 1 L 1 e −2i(β 1 L 1 +β 2 L 2 )
−re −4i(β 1 L 1 +β 2 L 2 ) + re −2iβ 1 L 1 e −4i(β 1 L 1 +β 2 L 2 ) − · · · =
= r −1 + e −2iβ 1 L 1 m X −1
k 0
e −2ki(β 1 L 1 +β 2 L 2 )
(5.4)
CHAPTER 5. THE FOURIER APPROXIMATION
Now, let
Λ = L 1 + L 2 , δ = β 1 L 1 + β 2 L 2 − π
Λ (5.5)
where we call δ the detuning parameter. Using (5.3), it is obvious that β 1 L 1 = β 2 L 2 , and it follows that δΛ + π = 2β 1 L 1 = 2β 2 L 2 , which we can insert into the expression (5.4) for r G . We get
r G = r −1 + e −iδΛ e iπ m X −1
k=0
e −2kiδΛ e −2kiπ = −r 1 + e −iδΛ m X −1
k=0
e −2kiδΛ =
= −r 1 + e −iδΛ 1 − e −2imδΛ
1 − e −2iδΛ = −r
e i δΛ 2 + e −i δΛ 2
e −i δΛ 2 e imδΛ − e −imδΛ
e −imδΛ (e iδΛ − e −iδΛ ) e −iδΛ =
= −2r cos
δΛ 2
e −i ( m − 1 2 ) δΛ sin (mδΛ) sin (δΛ) .
(5.6) If we look closer at the detuning parameter δ, we see that
δΛ = β 1 L 1 + β 2 L 2 − π = 2π
λ (n 1 L 1 + n 2 L 2 ) − π = 2π λ
λ 0
4 + λ 0
4
− π =
= 2π f c
c 4f 0 + c
4f 0
− π = π f
f 0 − π = π f − f 0
f 0
(5.7)
where we have used the relation λ = c/f , between wavelength and frequency. So δΛ is a measure on how close the frequency f is to some center frequency f 0 . This is in fact the frequency of the large peak in the reflection spectrum of a periodic grating, and we are interested in frequencies that are quite close to this center frequency. Therefore δΛ will be very small, and so sin (δΛ) ≈ δΛ. However, the number of periods, m, can be very large and therefore sin (mδΛ) ̸≈ mδΛ.
Since Λ = L 1 + L 2 is the length of one period and m is the number of periods in the grating, we have mΛ = L g , the length of the whole grating. Using these observations, the expression for r G can be written as
r G = −2r e −iδmΛ ( m − 1 2 ) /m m
m · sin (δmΛ)
δΛ =
= −2mr e −iδL g ( m− 1 2 ) /m sin (δL g ) δL g
.
(5.8)
CHAPTER 5. THE FOURIER APPROXIMATION
5.2 Implementation
5.2.1 Periodic Grating
In Figures 5.2.1 and 5.2.2 a comparison of the reflection in a periodic grating calculated using the Fourier approximation and the exact calculation is plotted for two different interface reflectivities, r interf ace = 3 ·10 −4 and r interf ace = 3 ·10 −5 . We can see that the two methods are much more similar, both in terms of magnitude and phase, for the smaller interface reflectivity. For this reflectivity, the graph of the Fourier approximation is indistinguishable from that of the exact calculation. For the larger reflectivity the difference is substantial, and the largest magnitude of the Fourier reflection is larger than 1, which impossible since this would mean that the reflection amplifies the light.
1.937 1.938 1.939 1.94 1.941 1.942 1.943
Frequency (Hz) 10
140 0.2 0.4 0.6 0.8 1 1.2
|r|
Exact calculation Fourier approximation
r
interface= 3 10
-4(a) The magnitude of the reflection.
1.937 1.938 1.939 1.94 1.941 1.942 1.943
Frequency (Hz) 10
140 1 2 3 4 5 6 7
arg(r) (mod 2 )
Exact calculation Fourier approximation
