Download Propagation characteristics of coherent optical waves in a

Tallinn University of Technology
Institute of Cybernetics
Propagation characteristics of coherent optical waves
in a stratified medium with Kerr nonlinearity
Koherentsete optiliste lainete levi karakteristikud
Kerri mittelineaarsusega kihilises keskkonnas
by
Liis Rebane
Thesis
Submitted to Tallinn University of Technology
for the Degree of
Bachelor of Science
Supervisors:
Dr. Pearu Peterson,
Institute of Cybernetics at TUT
Prof. Dr. Yasuo Tomita,
Dept. of Electronics Engineering,
University of Electro-Communications,
Tokyo, Japan
Tallinn 2004
Deklareerin, et käesolev väitekiri, mis on minu iseseisva töö tulemus, on
esitatud Tallinna Tehnikaülikooli bakalaureusekraadi taotlemiseks ja selle alusel
ei ole varem taotletud akadeemilist kraadi.
Bakalaureusekraadi taotleja L. Rebane:
Juhendaja Dr. P. Peterson:
Kokkuvõte
Käesolevas töös uuritakse koherentsete optiliste lainete levi Kerri mittelineaarsusega perioodilistes materjalides. Peatähelepanu all on mittelineaarsed efektid: bi- ja multistabiilsus ning stabiilne piirjuht, mille korral sisend-väljund intensiivsuste graafik on monotoonne
ja ülevalt tõkestatud. Analüüsi läbiviimiseks kasutatakse Maxwelli võrranditest tuletatud
paaris-võrrandite süsteemi, mille muutujateks on struktuuris edasi- ning tagasiiikuvate
lainete kompleks-amplituudid. Paarisvõrrandite süsteemi abil leitakse statsionaarsed lahendid perioodilises materjalis, mis koosneb kahest erinevate lineaarsete ja mittelineaarsete
murdumisnäitajatega kihide massiividest.
Töö kuuendas osas rakendatakse käsitletud teooriat CdSe-nano-osakeste lisandiga fotopolümeer struktuurile ning uuritakse optiliste lainete ülekande omadusi. Töös käsitletakse nelja erinevat murdumisnäitajate jaotust. Kõigi juhtude korral on sisend-väljund intensiivsuse graafik globaalselt multistabiilne. Selliseid struktuure kasutatakse kõige sagedamini optilisteks ümberlülitusteks madalamalt ülekandetasandilt kõrgemale. Näidatakse,
et käsitletud CdSe-lisandiga fotopolümeeris saavutatakse multistabiilne käitumine juba
madalatel sisendintensiivsustel. Töös jõutakse järeldusele, et stabiilse ülevalt tõkestatud
piirjuhu saavutamiseks tuleb muuta struktuuri murdumisnäitajate jaotust. See oleks antud materjali korral teostatav kasutades hologrammi kirjutamisel erineva intensiivsusega
valgust või muutes hologrammi kirjutamis-kiirte vahelist nurka.
Abstract
The propagation of coherent light through a nonlinear periodic optical structure with
periodic linear and nonlinear refractive indices is studied. Nonlinear coupled mode equations are derived from Maxwell equations to investigate input-output transmission regimes
of optical structures. Both stable and multi-stable behavior are analyzed including alloptical limiting — highly nonlinear stable transmission with monotonic and one-to-one
input-output power relationship that has upper threshold value. Periodic optical structures
can be applied in optical communication systems as uniform switches, logic elements, or
optical limiters.
In Chapter 6 this theory is applied to CdSe-nanoparticles dispersed photopolymer
holograms. We consider four different index distributions and find that all of them exhibit
globally multi-stable behavior. Such structures are usually applied as switching devices
between lower-transmissive and higher-transmissive states. We have shown that multistable behaviour at very low intensities can be achieved in given structure, especially for
longer gratings. Limiting behavior could be achieved by using different intensities for
hologram recording, or changing the angle between the writing beams.
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Acknowledgments
I am happy to take this opportunity to express my gratitude to my supervisor Prof. Yasuo
Tomita for introducing me to this subject and helping me during this research with his
ideas, explanations and comments.
Special thanks to my supervisor Dr. Pearu Peterson for his constant help and patience.
I look forward to working together in the future.
I also thank Mervi Sepp for valuable collaboration, Koji Furushima for his distribution
calculations, and Prof. Andrus Salupere and other staff of the Centre of Nonlinear Studies
for valuable discussions and help during the preparation of this thesis.
This research was supported by ETF grant 5767.
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Contents
1 Introduction
1.1 Optical communication systems . . . . . . . . .
1.1.1 Optical fibres . . . . . . . . . . . . . . .
1.1.2 Photonic crystals and their applications
systems . . . . . . . . . . . . . . . . . .
1.2 Statement of the problem . . . . . . . . . . . .
1.3 Outline . . . . . . . . . . . . . . . . . . . . . . .
2 General properties of wave dynamics
2.1 Introduction . . . . . . . . . . . . . . . . . .
2.2 Phase velocity, group velocity and dispersion
2.3 Nonlinearity . . . . . . . . . . . . . . . . . .
2.4 Solitons . . . . . . . . . . . . . . . . . . . .
3 Electromagnetic theory of light
3.1 Introduction . . . . . . . . . . . . .
3.2 Maxwell equations . . . . . . . . .
3.3 Electromagnetic waves in vacuum .
3.4 Electromagnetic waves in dielectrics
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4 Optical communication systems
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Optical fibres . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Linear theory of light propagation in fibres . . . . . . . .
4.2.2 Nonlinear theory of light propagation and optical solitons
4.3 All-optical signal processing . . . . . . . . . . . . . . . . . . . .
4.3.1 Photonic crystals . . . . . . . . . . . . . . . . . . . . . .
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in fibres
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5 Light propagation in periodic structures
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . .
5.2 Weak light assumption and linear periodic medium
5.2.1 Bragg reflection . . . . . . . . . . . . . . . .
5.2.2 Linear coupled mode equations . . . . . . .
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5.3
5.4
5.5
5.6
5.2.3 Dispersion relation and photonic band-gap . .
Intense light in a nonlinear periodic medium . . . . .
5.3.1 Kerr effect . . . . . . . . . . . . . . . . . . . .
5.3.2 Nonlinear coupled mode equations . . . . . .
5.3.3 Optical bistability . . . . . . . . . . . . . . . .
5.3.4 Bragg and gap solitons . . . . . . . . . . . . .
5.3.5 Nonlinear Schrödinger limit . . . . . . . . . .
Gratings with periodic nonlinear index . . . . . . . .
Stationary solutions . . . . . . . . . . . . . . . . . . .
5.5.1 Balanced nonlinearity management: nnl = 0 .
5.5.2 Unbalanced nonlinearity management: nnl 6= 0
5.5.3 Stationary transmission regimes . . . . . . . .
Numerical method . . . . . . . . . . . . . . . . . . .
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6 Transmission properties of nanoparticles (CdSe) dispersed photopolymer
hologram
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Holography with photopolymers . . . . . . . . . . . . . . . . . . . . . . . .
6.2.1 Grating formation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Mathematical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4 Data for numerical experiments . . . . . . . . . . . . . . . . . . . . . . . .
6.5 Results for nCdSe = 2.55 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5.1 Diffusion dominant case (R=10) . . . . . . . . . . . . . . . . . . . .
6.5.2 Photopolymerization dominant case (R=0.01) . . . . . . . . . . . .
6.6 Analyzes for nCdSe = 2.728 . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.6.1 Diffusion dominant case (R=10) . . . . . . . . . . . . . . . . . . . .
6.6.2 Photopolymerization dominant case (R=0.01) . . . . . . . . . . . .
6.7 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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7 Summary
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A Derivation of the expression for index modulation
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B Derivation of coupled mode equations
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C Derivation of stationary coupled mode system
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D Program listing
D.1 Analytical algorithm for balanced nonlinearity management (nnl = 0) . . .
D.2 General numerical algorithm based on backward finite difference scheme . .
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4
Chapter 1
Introduction
1.1
Optical communication systems
The objective of any communication system is the transfer of information from one point
to another. Most often, the information transfer is accomplished by modulating the information onto an electromagnetic carrier wave, which is then transmitted to the destination,
where the electromagnetic wave is received and the information is recovered. Such systems
are categorized by the location of carrier frequency in electromagnetic spectrum: radio
systems, microwave or millimeter systems, and optical communication systems. In optical
systems the carrier frequency is selected from the optical region, which usually includes
infrared and visible frequencies.
In any communication system, the amount of information transmitted is directly related
to the bandwidth of a modulated carrier (frequency extent, which shows how many pulses
per second could be sent into a fibre and be expected to emerge intact at the other end),
which is generally limited to a fixed portion to a carrier frequency itself. Thus, increasing
the carrier frequency also increases the available transmission bandwidth and therefore
the information capacity of the overall system. Using higher carrier frequencies generally
increases the capability of the system to achieve higher power densities, which leads to
improved performance, making the usage of optical frequencies in communication systems
to be a very appealing option [8].
1.1.1
Optical fibres
The intensive research for the use of light as a carrier of information began in 1960’s when
the laser technology as a source of coherent light became available. Initially the transmission distances were very short, but soon the enhanced techniques for manufacturing
high quality glass lead to the development of low-loss dielectric waveguides; and optical
communication systems became a serious alternative to electrical communications.
Optical fibres carry signals with much less energy loss than copper cables and with much
higher bandwidth, meaning that fibres can carry more channels of information over longer
distances and with fewer repeaters required. In addition to the high data transmission rate
5
and low transmission losses, the main advantages of the usage of optical fibers instead of
usual coaxial fibres are small size, low weight, flexibility, and high security.
In order to design fully integrated optical circuits, that would replace electro-optical
devices, many other optical components in addition to optical fibres are needed. These
optical components are used for swiching, pulse shaping, and amplifying of optical pulses.
The current information technology is based on our ability to control the flow of electrons
in a semiconductor. For all-optical signal processing we need to achieve a similar control
over photons [3, 8].
1.1.2
Photonic crystals and their applications in optical communication systems
Periodic optical structures or photonic crystals are dielectric arrays that selectively transmit or reflect light at various wavelengths. In the form of diffraction gratings they have
been used over hundred years to separate color components of a light beam. In optical
communication systems photonic crystals serve as optical equivalents to semiconductors in
electronics, both having energy band structure. Such photonic crystals could solve many
of the problems that currently limit the speed and capacity of optical-communication networks.
Nonlinear photonic crystals may bring to reality the vision of light controlling light
in micro-scale photonic circuits - the analog of present day electronic integrated circuits
where electrons control electrons. Given the impact that semiconductor materials have
had on every sector of society, photonic crystals could play even a greater role in future
optical-communications industry [14, 15].
1.2
Statement of the problem
In this the work the transmission properties of nonlinear periodic optical structures and
their applicability as all-optical signal-processing elements are studied. In particular the
main aims of this work are:
• To analyze the light transmission properties of a medium with periodic grating and
intensity dependent refractive index on the basis of Pelinovsky’s theory [14].
• To apply this theory to nano-particles dispersed photopolymer holograms that have
been proposed and intensively investigated by Prof. Yasuo Tomita’s Photonics Group
at University of Electro-Communications, Tokyo, Japan, and to examine their applicability in optical communication systems.
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1.3
Outline
This thesis consists of seven chapters and four appendices. In Chapter 2 some basic concepts of wave theory are explained. In Chapter 3, electromagnetic theory of light is introduced, including derivation of Maxwell’s wave equations for dielectrics. In Chapter 4
optical communication systems are described in more detail, including soliton propagation
in optical fibres and the concept of photonic crystals. In Chapter 5 light propagation in
periodic structures is analyzed including both linear and nonlinear theories for weak and
intense light, respectively. Maxwell’s wave equations are simplified further to a system for
complex amplitudes of forward and backward propagating electro-magnetic waves — coupled mode system. Chapter 6 contains the main results of this thesis. The theory, which
is described in Chapter 5, is applied to nanoparticles dispersed photopolymer holograms
and the transmission characteristics of these structures are calculated. Finally, Chapter 7
gives a brief summary and the further perspectives of the work.
7
Chapter 2
General properties of wave dynamics
2.1
Introduction
Light can be modelled as the propagation of electromagnetic waves in space. In this Chapter some terminology of wave dynamics, which is used in further analyzes, is introduced.
It is very complicated to define a wave in general. The most common and widelyused definition is: a wave is a disturbance propagating in a medium. Such definition is
not completely satisfactory since standing waves do not propagate. Another feature that
makes definition difficult is due to the interaction of wave with the medium through which
it is passing. A wave on the surface of water passes and leaves the medium unchanged. In
contrast, a chemical wave usually leaves the reacting species involved in a different chemical
state after it has passed by. For these reasons we can’t have a unique definition for a wave.
Wave phenomena is simply viewed as generic set of phenomena with many similarities [2]
or as a state moving into another state [11].
Media for wave propagation can be categorized as follows:
• Linear medium - the superposition principle holds - different waves at any particular
point in the medium can be added to form other solutions.
• Nonlinear medium - the solutions to the wave-equations do not form a vector space
and cannot be superposed to produce a new solutions.
• Bounded medium - finite in extent (otherwise unbounded).
• Uniform medium - its physical properties are unchanged at different points.
• Isotropic medium - its physical properties are the same in different directions.
While most of the waves need a medium for propagation, electromagnetic waves can also
exist in vacuum.
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2.2
Phase velocity, group velocity and dispersion
Phase velocity is the rate at which the phase of the wave propagates in space or the velocity
at which the phase of single frequency component of the wave will propagate. It is given
in terms of the wave’s frequency ω and wave vector k by:
v=
ω
.
k
(2.1)
The phase velocity generally describes only the propagation of monochromatic waves.
In general case waves occur as a mixture of a number of component frequencies, and the
motion of such a pulse is described by the group velocity, which is the rate that changes
in amplitude (known as the envelope of a wave) will propagate. Since the group velocity
is the velocity of maximum amplitude of the envelope, it can also be interpreted as the
velocity at which energy or information is conveyed along a wave. The group velocity is
defined in terms of the wave’s angular frequency ω and wave number k by:
vg ≡
dω
.
dk
(2.2)
If the phase-velocities of different frequency components are equal, then the component
frequencies and their superposition (or group) would travel with the same velocity and
the profile of the envelope wave remains constant. A medium in which phase velocity is
independent of frequency is called non-dispersive medium. If the phase-velocity is frequency
dependent, the envelope of the wave will become distorted as it propagates - this effect is
known as group velocity dispersion, which causes a short pulse of light to spread in time as
a result of different frequency components of the pulse travelling at different velocities. A
medium in which the phase velocity is frequency-dependent, is known as dispersive medium
and is described by a dispersion relation, which expresses the variation of ω as a function
of k.
Since ω = kv where v is the phase velocity, the group velocity can be given by:
vg =
d
dv
dv
dω
=
(kv) = v + k
=v−λ
dk
dk
dk
dλ
(2.3)
where k = 2π/λ is the wave number.
If dv/dλ is positive, so that vg < v, we have a normal dispersion; for negative dv/dλ,
and vg > v, we have anomalous dispersion.
The result of group velocity dispersion, whether negative or positive, is ultimately
temporal spreading of the pulse. This makes dispersion management extremely important
in optical communications systems based on optical fiber, since if dispersion is too high, a
group of pulses representing a bit-stream will spread in time and merge together, rendering
the bit-stream unintelligible [2].
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2.3
Nonlinearity
For linear systems we can use the superposition principle i.e. if y1 and y2 are solutions
of a linear equation, then so is a1 y1 + a2 y2 for any constants a1 and a2 . It means that
separation of variables and integral transform methods allow us to determine the solution.
For nonlinear systems superposition principle is not valid and the standard mathematical
techniques mentioned above, are not applicable [2].
The most interesting aspect of nonlinear phenomena is how the effect of a nonlinear
process is modified by that of a linear process (dispersion, diffusion). In general, linear
and nonlinear processes have opposing effects. In contrast to dispersion or diffusion, nonlinearity leads to the concentration of a disturbance. [9]
2.4
Solitons
Soliton is a stable wave structure that can exist due to a specific equilibrium of dispersive (linear) and focusing (nonlinear) effects. Solitons emerge unchanged from collisions
with each other, regaining their asymptotic shapes, magnitudes, and speeds. Solitons are
found in many non-linear physical phenomena, as they are solutions to many different nonlinear differential equations (the KdV equation, the Sine-Gordon equation, the nonlinear
Schrödinger equation etc.). Few examples of physical systems where solitons occur include
nonlinear optics, hydrodynamics, plasma physics, protein models, high energy physics, and
solid state physics [5].
Solitons in optics have been widely studied for their applications in distortionless signal
transmission in fibers and many other nonlinear applications in optical communication
systems. Optical solitons differ somewhat from the classical KdV soliton, which describe
the solitary wave of a wave; the optical soliton is the solitary wave of an envelope of a light
wave [9].
10
Chapter 3
Electromagnetic theory of light
3.1
Introduction
This Chapter introduces the electromagnetic theory of light, which is based on Maxwell
equations. The aim is to derive Maxwell wave equation that describes the propagation of
light waves in dielectrics, and will be later used to derive the equations for coupled mode
system — the main set of equations used in the analyzes of this thesis .
A consistent theoretical explanation of optical phenomena can be described jointly
by Maxwell electromagnetic theory and the quantum theory of light. While the quantum
theory describes the interactions between light and matter i.e. the absorption and emission
of light, Maxwell theory is most useful for treating the properties of propagation of light,
which is also the subject in this thesis.
According to electromagnetic theory, light is an electromagnetic radiation in the range
of wavelengths from infrared to ultraviolet. The three basic characteristics of light waves
are amplitude (intensity), frequency (color), and polarization (angle of vibration).
3.2
Maxwell equations
The propagation of electromagnetic waves is described by four Maxwell equations:
∇ · D = ρ,
∇ · B = 0,
(3.1)
(3.2)
∇×H = J+
(3.3)
∂D
∂t
∂B
,
∇×E = −
∂t
(3.4)
where ρ is the free electric charge density (in units of C/m3 ) without dipole charges bound
in a material, H is magnetic field strength (in units of A/m), J is current density, E is
electric field (in units of V /m), D is electric displacement field (in units of C/m2 ), and B
is magnetic flux density (in units of tesla, T ), also called as magnetic induction.
11
These four equations express how electric charges produce electric fields (Gauss law,
Eq.(3.1)), the experimental absence of magnetic charges (Eq.(3.2)), how currents produce
magnetic fields (Ampère law, Eq.(3.3)), and how changing magnetic field produces electric
field (Faraday law of induction, Eq.(3.4)), respectively.
3.3
Electromagnetic waves in vacuum
We consider electromagnetic waves in vacuum - a linear, isotropic, non-dispersive medium.
In vacuum, the D and H fields are related to E and B by:
D = ε0 E,
B = µ0 H,
(3.5)
(3.6)
where ε0 and µ0 are electrical permittivity and magnetic permeability, respectively, for
vacuum. Also there are no currents or electric charges present in vacuum (i.e. ρ = 0 and
J = 0). With these conditions we obtain the source-free Maxwell’s equations:
∇ · E = 0,
∇ · B = 0,
(3.7)
(3.8)
∇ × B = ε0 µ0
(3.9)
∂E
,
∂t
∂B
,
∇×E = −
∂t
(3.10)
This system has a trivial solution E = B = 0. Maxwell equations for vacuum can also be
written in a form of linear wave equations for E and B:
∇2 E =
1 ∂2E
,
c2 ∂t2
∇2 B =
1 ∂2B
c2 ∂t2
(3.11)
√
where c = 1/ µ0 ε0 . This is a set of linear differential equations which means that the
amplitude of two interacting waves is a simple superposition. Therefore the behavior of a
wave can be analyzed by breaking it up into components. The equation of this form has a
well known general solution (usually referred as d’Alembert solution):
E(r, t) = E+ (k · r − c|k|t) + E− (k · r + c|k|t)
(3.12)
where r is position vector, k is wave vector and E+ and E− are arbitrary vector functions
which can be determined, for example, from specifying initial conditions. This solution
represents two counter-propagating waves, which do not interact with themselves nor with
each-other. Both of these solution components can be represented as a superposition of
their Fourier’ components [2].
The argument (k · r + c|k|t) = p is called the phase of a wave. Evidently p is constant
in space-time if dp/dt = 0, i.e. kdr/dt = ±c|k|. An observer moving with velocity c in the
12
direction of a wave vector k will always notice the same phase, meaning that c represents
phase velocity of a light wave. For electromagnetic waves in a vacuum the velocity c is
√
given by: c = 1/ µ0 ε0 ≈ 3 × 108 m/s. This led Maxwell to a postulate (that was proved
later) that light is an electromagnetic wave [6].
Let us consider electromagnetic plane wave propagating in the direction of a wave vector
k. Then electric and magnetic fields are:
E(r, t) = E0 ei(kr−ωt) + c.c.
B(r, t) = B0 ei(kr−ωt) + c.c.
(3.13)
(3.14)
where we have denoted ω = c|k| and E0 and B0 are constant vectors.
Note that from
∇ · E = i(E0 · k)ei(kr−ωt) + c.c. = iE · k = 0
(3.15)
follows that E⊥k. Similarly from ∇ · B = 0 follows B⊥k. From Eq. (3.10) we have
∇ × E = i(k × E = −iωB.
(3.16)
So
1
(k × E)
(3.17)
ω
In conclusion, we have shown that the three vectors E, B and k form a pairwise perpendicular set of vectors (see Fig. 3.1)
B=
E
k
H
Figure 3.1: Electromagnetic field in vacuum
The k − E plane is called as plane of polarization. In special case when the plane of
polarization does not change, it is said that wave is linearly polarized. In general the plane
of polarization rotates as time t varies and then it is said that the wave is either circularly
or elliptically polarized [2].
13
3.4
Electromagnetic waves in dielectrics
Dielectrics are not passive carriers of electromagnetic waves and Maxwell equations need to
be modified in order to describe the wave dynamics in dielectrics. Although in dielectrics,
electrons are bound sufficiently tightly to the nuclei of the constituent atoms and can not
produce a current (J=0), the distribution of electrons around each nucleus is deformed
when the electromagnetic field is applied to the material. This leads to a modification of
the permittivity and magnetic permeability of the material and Maxwell equations become:
∇ · D = ρ,
∇ · B = 0,
(3.18)
(3.19)
∇×H = J+
(3.20)
∂D
,
∂t
∂B
,
∇×E = −
∂t
(3.21)
where
µ0 H = B − M, D = ε0 E + P,
(3.22)
P(E) = ε0 (χe E + χ(2) EE + χ(3) EEE + ...)
(3.23)
where P is electric dipole moment per unit volume or polarization and M is magnetic
dipole moment per unit volume or magnetization. In the following we assume that there
are no extra charges other than internal charges related to atomic structure, so ρ = 0.
For non-magnetic materials M = 0 and the response of the material depends on polarization P. In general, polarization depends on electric field. For small but finite fields we
can expand P into Taylor series
where χe is electric susceptibility of the material and χ(i) are i-th order nonlinear susceptibility tensors. For perfectly isotropic medium χ2 vanishes for symmetry reasons
P(−E) = −P(E). The third order nonlinearity (also known as Kerr nonlinearity) tensor χ(3) is nonzero and gives rise to nonlinear polarization effects. In the case of linear
polarization, that is, the vector of electric field stays in a polarization plane (say x − z
plane), we can write
P(E) ≈ ε0 (χe + χ(3) |E|2 )E = ε(z, |E|2)E − ε0 E,
(3.24)
where we have defined ε(z, |E|2 ) = ε0 (1 + χe + χ(3) |E|2 ).
So,
D = ε(z, |E|2 )E,
B = µH,
(3.25)
(3.26)
and
∇ · D = ε∇ · E + ∇ε · E = 0.
14
(3.27)
We consider the medium that is only slightly inhomogeneous along the z-axis, so that
∂ε/∂z ≪ ε and
∇ε
∇·E =−
· E ≈ 0.
(3.28)
ε
Now we can assume that E is perpendicular to the z-axis and take E = (E, 0, 0) where
E = E(z, t).
To obtain the wave equation for the electric field component E, we take the curl of
(3.21) and replace ∇ × B by using (3.19). This yields to relation:
 
1
2
2
∂ D
∂
2

0 ,
∇ × (∇ × E) = −µ0 2 = −µ0 2 ε(z, |E| )E ·
(3.29)
∂t
∂t
0
where ε(z, |E|2) is related to refractive index n of the material by ε(z, |E|2 ) = n2 (z, |E|2 )ε0 .
On the other hand,
 
1
2
∂
∇ × (∇ × E) = ∇(∇ · E) − ∇2 E = − 2 E ·  0  .
(3.30)
∂z
0
We assume that the intensity of the wave, |E|2 , is slowly varying compared to the variations
in the carrier of E, so that we arrive to nonlinear Maxwell wave equation in a form:
∂ 2 E n2 (z, |E|2 ) ∂ 2 E
−
=0
∂z 2
c2
∂t2
√
where E(z, t) is a scalar electric field and c = 1/ µ0 ε0 is the speed of light.
15
(3.31)
Chapter 4
Optical communication systems
4.1
Introduction
In this Chapter the structure of a general optical communication system is described and
the importance of the photonic crystal technology is explained. Complete all-optical communication is achievable when electro-optic components that are currently used in optical
communication systems, are replaced by all-optical signal-processing elements. For this
purpose, the development of new all-optical devices is necessary. Usage of photonic crystal
techniques is one of the most promising opportunities for these applications.
Advantages and problems of optical systems
The principal advantages in communicating at optical frequencies are: (i) wider available
transmission bandwidth due to higher carrier frequency, (ii) ability to concentrate the power
in extremely narrow beams, and (iii) significant reduction in component sizes because of
the extremely small optical wavelengths.
The main problems in development of optical networks are the technological difficulties
associated with reliable and cheap connections, and the development of an optical circuit
technology that would match the potential data-rates of the fibres. The speed of these
electronically controlled circuits is usually the limiting factor on the bit-rate of optical
systems. The difficulty of connection and high-cost of associated circuitry result in optical
fibres being used only in very high bit-rate communication in this moment [8].
4.2
Optical fibres
Most optical fibers consist of the core (cylindrical insulator), which is surrounded by the
cladding (a layer of another insulator with slightly lower refractive index). The material
is designed to be optically transparent and absorb as little light as possible. The main
advantages of optical fibres compared to widely used coaxial cables are [2, 8]:
16
• Potentially huge bandwidth. In practice it has not been realized yet and the bandwidth of individual fibre is almost the same as high quality coaxial cable. However,
it is possible to lay many hundreds of optical fibres in the same cable cross-section
as a single coaxial cable.
• Potentially high data transmission rate. Standard optical fibre transmitters can send
up to 109 bits of information per second by switching a laser beam on and off (data
transmission rates for coaxial cables are typically in the order of 106 bits per second).
• Small size, low weight, and high flexibility. Optical fibres have very small diameters
(about 120µm).
• Good electrical isolation. Optical fibres are almost completely immune to external
fields and do not suffer from cross-talk, radio interference, etc.
• High security. It is difficult to tap into an optical line.
• Low transmission losses (0.2 − 0.5 dB/km compared to about 7 dB/km for ordinary
coaxial cables).
4.2.1
Linear theory of light propagation in fibres
A plane wave, which is incident on the interface of two surfaces, will experience total
internal reflection when it is incident almost parallel to the interface. Otherwise, most of
the wave is transmitted into the second medium. Due to the wave nature of light, only
certain modes (standing oscillating electromagnetic waves, which are defined by the wave
geometry) would be guided by the fiber. A mono-mode fibre is a fibre that only has one
acceptable mode per frequency. A multi-mode fibre has a number of possible modes.
Light attenuation in a wave-guide has a number of sources. Absorption of light occurs in
the glass and this decreases with frequency. Scattering of light from internal imperfections
within the glass - Rayleigh scattering - increases with frequency. Waveguide imperfections
account for low-level loss that is approximately constant with wavelength. Bending the
waveguide changes the local angle of total internal reflection and loss increases through the
walls. A combination of all these effects results in a minimum absorption in the 0.8µm to
1.8µm wavelength region, that are used for transmission.
In addition to attenuation, optical waveguides also suffer from dispersion. The dispersion has two sources: firstly there is a dispersion due to the different modes propagating
in the fibre (modal dispersion) and material dispersion or chromatic dispersion, due to the
dielectric properties of glass. Dispersion effects cause the pulse spreading, which sets a
limit to the bandwidth of the fibre.
17
4.2.2
Nonlinear theory of light propagation and optical solitons
in fibres
The description of pulse transmission in optical fibres explained above was based on linear
theory. By making the use of nonlinear response of an optical fibre (this requires sufficiently
intense laser light), the effects from Kerr nonlinearity arise that tends to compress the
pulse while dispersion tends to broaden it. When these two competing effects balance, the
formation of stable optical pulses - optical solitons - is possible [2].
Optical solitons in fibers are different from the solitons of KdV equation, which describe
the solitary wave of a wave. In optical communications optical pulses are used and their
shape represents an envelope of a light wave. The soliton propagating in an optical fiber
is the solitary wave of an envelope of a light wave, not a soliton of a wave itself. These
solitons are referred as envelope solitons.
Soliton formation results from group velocity dispersion (its dependence on the wavelength of light) and the third order nonlinear effect (wavelength dependence on the intensity
of a wave). The model equation which describes the envelope soliton propagation is known
as the nonlinear or cubic Schrödinger equation.
The origin of nonlinear properties of a light wave comes from nonlinear Kerr effect,
which is caused by the deformation of the electron orbits in glass molecules due to the
electric field of light. As a result, the refractive index of a fibre becomes intensity dependent:
n(I) = nL + n(2) I
(4.1)
where nL is the linear component of refractive index. n(2) , which represents a change in
refractive index due to Kerr effect, has a very small value. The reason why such a small
change in refractive index becomes important, is that modulation frequency ∆ω, which is
determined by the inverse of a pulse width, is much smaller than the frequency of a wave
ω. Consequently, the group velocity dispersion, which is produced by ∆ω is also small.
As a result the relative change in the wave number due to the group dispersion becomes
comparable to the nonlinear change. In order for those effects to become significant, the
wave distortion due to the fiber loss should be less than these small effects. This requires
that the fiber loss rate per wavelength of light should be less than 10−10 , meaning that the
fiber loss rate of the fiber should be less than 1 dB per km.
The concept of optical solitons could be extremely useful in optical communications.
The width of an optical pulse used in most of the optical communication systems, which
are in practical use in this moment, is approximately 1 nanosecond. In this case the major
distortion of the pulse results from the fiber loss. We can overcome this problem by using
repeaters at every several tens of kilometers. When the pulse width is decreased in order
to increase the transmission rate to the level of 10 picoseconds, the separation between two
repeaters is no longer determined by the fiber loss, but by the group velocity dispersion of
the pulse. An optical soliton, which is produced by the balance between the nonlinear effect
and group velocity dispersion, produces no distortion caused by the dispersion. However,
soliton pulses are not immune to fiber losses and consequently the pulse width of a soliton
will expand. Due to that, a soliton transmission system also requires pulse to be repeated.
18
In contrast with the linear system, the reshaping can be achieved utilizing only optical
amplifiers (i.e. Raman effect) [9].
4.3
All-optical signal processing
Although optical fibers are widely used today, switching, pulse shaping, and amplifying
of optical pulses and optical data streams is most often accomplished by expensive optical/electronic devices, which convert optical pulses to electrical pulses, process the electrical pulse trains within electronic circuit and then convert the processed electrical pulses
back to light. In fiber-optic networks the use of all-optical processing instead of expensive
and slow electro-optic devices yields to huge savings in both power and cost. In contrast
with applications in information storage, where one could be limited, for example, to lowintensity regime, optical signal processing operation devices, which are uniformly stable for
all pertinent incident intensities, can be developed. All-optical devices include: optical amplifiers, optical limiters, optical switches, devices that can multiplex many independently
modulated wavelength channels onto a single physical medium, optical logic elements etc.
The development of all-optical signal-processing elements has been a widely studied
topic in recent years. There is considerable current debate as to whether optics could
completely replace the widely-used electronic technology [3, 14, 17].
4.3.1
Photonic crystals
Photonic crystals are highly periodic artificial structures that have attracted much attention in recent years. They serve as conducting media for electromagnetic waves and can
be designed to control and manipulate the propagation of light. In general they have a
similar role in optical communication systems as semiconductors in electronic signal processing, being even more flexible because we have far more control over the properties of
photonic crystals than we do over the electronic properties of semiconductors. Since the
periodic variation of photonic crystals can often be controlled and even engineered, limited
only by the available micro-fabrication technologies, such systems offer many promising
applications both for the purpose of studying fundamental effects in nonlinear dynamics
and optical device engineering [14, 16].
It is well known from the quantum theory of solids that the energy spectrum of an
electron in a solid consists of bands separated by gaps. The periodic arrangement of ions
on a lattice gives rise to the energy band structure in semiconductors. Energy bands
control the motion of charge carriers through the crystal. Similarly, in a photonic crystal,
the periodic arrangement of refractive index variation controls how photons are able to
move through the crystal. Replacing the ions on a lattice creates regions of low refractive
index within a high-refractive index material or vice-versa. Photons react to the refractive
index contrast in an analogous manner to the way electrons react when confronted with a
periodic potential of ions. In both cases a range of allowed energies and a band structure
characterized by an energy gap or photonic band gap appears. The basic physical reason for
19
the rise of gaps lies in the coherent multiple scattering and interference of waves inside the
crystal – Bragg resonance of the waves with the crystal structure. The basic mathematical
theory that describes how the band gaps arise in periodic dielectric and acoustic media
was essentially constructed by using the Floquet-Bloch theory [1].
Since photonic band-gaps occur in the spectrum of light waves due to periodicity of
the medium and described Bragg resonance between incident light and the grating, lowintensity light with frequencies inside the photonic bandgap can not propagate. If the
medium is nonlinear (a nonlinear photonic crystal) and sufficiently intense light is used,
the photonic band gaps are shifted and light still can be transmitted through the grating
due to the effects of Kerr nonlinearity. Nonlinear photonic crystals can be used to control
the propagation of light. The most useful applications include:
• Formation of a narrow band filter for selecting a particular wavelength.
• Pulse shaping and compression using Bragg soliton effects.
• Photonic crystal defects which can slow or even stop light pulses propagating in the
crystal.
• The use of temporal solitons — stable pulses which result from combination of Kerr
nonlinearity and chromatic dispersion.
By using these effects, the ultimate goal of nonlinear photonic crystal technique - the
all-optical processing of light pulses - will hopefully be achievable [17].
20
Chapter 5
Light propagation in periodic
structures
5.1
Introduction
In this chapter light propagation in periodic structures is compared with light propagation
in uniform media. The aim is to derive the coupled-mode system from Maxwell equations
that allows us to analyze the transmission properties of periodic optical structures (photonic crystals). Modelling of time-dependent responses of photonic crystals in three spatial
dimensions can be analytically difficult in the framework of Maxwell equations and is considerably simplified when using the coupled-mode approach. Coupled-mode equations are
typically derived for two counter-propagating waves in one spatial dimension and at the
lowest band gap of Bragg resonances.
One-dimensional periodic medium
In the following calculations it is assumed that a laser pulse is incident on a sample of
infinite extent whose surface coincides with the xy plane. If the wave fronts are infinite
planes parallel to the xy-plane, then the simple one-dimensional picture can be used for
describing the evolution of the pulse [12]. It is assumed that medium has uniform average
Figure 5.1: Linear periodic structure
21
index of refraction n0 and periodic index modulation ∆n with grating period Λ (see Fig.
5.1). In many cases (including calculations in this thesis) it is assumed that the modulation
amplitude of the index of refraction is small (∆n ≪ n0 ) that will allow several further
simplifications.
5.2
5.2.1
Weak light assumption and linear periodic medium
Bragg reflection
In the case of weak incident light the properties of prescribed periodic medium and the
medium with uniform index are very similar: the light, which is reflected off from various
interfaces is mutually out of phase and, as a consequence, the light propagates through
the structure essentially unimpeded as is the case for uniform medium. The crucial difference occurs when the lights wavelength satisfies the Bragg condition - the condition of
cumulative reflections - which is given by:
λB = 2n0 Λ,
(5.1)
where λB is Bragg wavelength, n0 is average linear refractive index of the material and Λ is
the grating wavelength. The Bragg law specifies that when the light wavelength equals to
Bragg wavelength (λ = λB ), exactly half of a wavelength of light in medium fits into each
period of the grating. Consequently, light-beams that are Fresnel reflected off interfaces of
the grating, are all in phase and this leads to a strong reflected wave. Light, which exactly
satisfies the Bragg condition, can not propagate inside the structure because it is entirely
reflected off. Light with a wavelength nearly satisfying the Bragg condition slows down
since it is reflected back and forth between the periodic layers instead of being transmitted
directly through the material. This slowing down strongly depends on the wavelength
of light - different colors propagate with different velocities, i.e. a very strong dispersion
occurs. The range of wavelengths, where the incident light is strongly reflected is called
photonic bandgap and its width is determined by the index contrast, as is shown later (Eqs.
(5.13) and (5.9)).
The properties of periodic structures are often given in terms of the (angular) frequency
ω rather than wavelength. The expression for Bragg frequency can be found from Eq.(5.1)
in a form:
ω0 =
πc
no Λ
(5.2)
where c is the speed of light in vacuum [16].
5.2.2
Linear coupled mode equations
Let us consider an one-dimensional optical structure and assume the incident light is coherent and linearly polarized. The direction of light propagation is chosen to be along the
22
z-axis. The scalar electric field is described by the linear Maxwell wave equation, which is
derived in Section 3.4:
∂ 2 E n2 (z) ∂ 2 E
− 2
= 0,
(5.3)
∂z 2
c
∂t2
where n2 (z) is a periodic function. We assume nearly monochromatic electric field that
will be represented as a slowly varying envelope of a highly oscillatory carrier wave.
The slow modulation in space and time of the plane wave envelope in an uniform
medium is induced by a weak periodicity component of the refractive index, so that:
n(z) = nln + ∆n(z).
(5.4)
Where nln is average index of refraction and ∆n is periodical variation. Since ∆n is a
periodic function, it can be expanded into the Fourier series. In the following analyzes we
consider only the first order of the expansion (derivation of Fourier’ expansion for refractive
index in nonlinear periodic medium is given in Appendix B) and ∆n can be expressed as:
∆n(z) ≈ 2n0k cos(2k0 z),
(5.5)
where k0 = 2π/λB denotes Bragg wave-number [17].
If we would neglect the periodicity of the refractive index, the solution to (5.3) would
be:
E(z, t) = E+ ei(k0 z−ω0 t) + E− ei(k0 z+ω0 t) + c.c,
(5.6)
where E+ and E− are the amplitudes of the field components propagating to the right and
left, respectively.
With the coupled mode approximation we can use the same form of a solution, but
include small variation in E+ and E− , because the small perturbation of ε could scatter
the wave going to the right causing it to go to the left and vice versa. So the solution to
(5.3) is searched in the following form [16]:
E(z, t) = E+ (z, t)ei(k0 z−ω0 t) + E− (z, t)ei(k0 z+ω0 t) + c.c.
(5.7)
Using these assumption, a linear coupled mode system can be derived (see Appendix B
and linearize)[17]:
∂E+
nln ∂E+
+i
+ κE− = 0,
∂z
c ∂t
nln ∂E−
∂E−
+i
+ κE+ = 0,
−i
∂z
c ∂t
i
where
κ≡
ω0 n20k
.
2nln c
23
(5.8)
(5.9)
5.2.3
Dispersion relation and photonic band-gap
To obtain the dispersion relation for the linear system (5.8), we search for the envelope
functions of the form:
E± (z, t) = A± ei(Qz−Ωt)
(5.10)
where Q ≡ k − ko is the wave number and Ω ≡ ω − ω0 is the frequency of the envelope.
Coupled mode equations then reduce to a set of algebraic linear equations:
A+
nln Ω/c − Q
κ
= 0.
(5.11)
A−
κ
nln Ω/c + Q
Solving this eigenvalue problem leads to dispersion relation:
p
nln Ω/c ≡ ∆ = ± κ2 + Q2 .
(5.12)
This relation is plotted in Figure 5.2 and clearly shows that there is a frequency region (a
band gap)in which plane wave solutions of the form Eq. (5.10) cannot exist.
Ω=ω−ω0
δω
Q=k−k0
Figure 5.2: Dispersion relation
From the dispersion relation we can also determine the width of the gap:
δω = 2κc/nln .
(5.13)
Combining it with Eq.(5.9), δω turns out to be proportional to the lowest Fourier’ component of the grating [16].
We can also define the normalized group velocity of the wave by:
v± =
p
dω
d∆
=
= ± 1 − κ2 /∆2
dt
dQ
24
(5.14)
At the edges of the photonic band gap (∆ → κ), the group velocity is v = 0; and far
from the gap (∆ → ∞) it equals to the group velocity of the uniform medium (normalized
group velocity is 1). For intermediate values of the detuning, the group velocity has a value
between these extreme values.
For each eigenvalue in Eq. (5.11), there is an eigenvector, which describes the mixing
of the forward and backward propagating mode by the grating. Eigenvectors describe the
eigen-states of the field inside the grating. Since the grating is periodic, the eigen-states
are Bloch functions. We define the eigenvectors in terms of group velocity v in upper and
lower branch. The Bloch functions are then given by:
p
p
p
p
1 + v/2, 1 − v/2 , v− =
1 − v/2, − 1 + v/2 .
v+ =
(5.15)
Note that the Bloch functions are normalized and that Bloch functions on the two different
branches and at the same Q are orthogonal. Far from the Bragg resonance (v = 1) and
the effect of the grating is small, and and the eigen-states of the system are decoupled
forward- and backward propagating waves. At the band edges (v = 0), the Bloch functions
are standing waves [17].
Since the group velocity is now frequency dependent, the grating is dispersive. For
typical gratings, the group velocity varies between 0 and v within a wavelength range of
less than a nanometer, leading to a huge dispersion near the edges of the photonic bandgap.
5.3
5.3.1
Intense light in a nonlinear periodic medium
Kerr effect
Intense laser sources allow us to generate optical fields that may change dielectric constants in a photonic crystal. It makes possible to tune the photonic crystal reflection and
transmission bands by simply varying the intensity of the incident light.
In this work we have considered third order nonlinear material. The influence of
third order nonlinearities to light propagation is called nonlinear Kerr effect, and is often described by nonlinear refractive index n2 . The expression for refractive index of
one-dimensional Kerr nonlinear photonic crystal is given by:
n(z) = n0 + n2 I + ∆n(z),
(5.16)
where n0 is the linear refractive index, n2 is the Kerr refractive index, I is the light intensity,
and ∆n(z) is the linear periodic refractive index modulation along the z-axis. This is quite
good approximation for a light propagating through the optical structure with a grating
written in the core (optical fibers and optical elements), or for a waveguide with a periodic
variation in its thickness [17].
Kerr nonlinearity causes the intensity dependent influences of the wave on itself, so
called self-interaction effect, wherein a pulse changes the medium within which it propagates [12].
25
5.3.2
Nonlinear coupled mode equations
Described structure now incorporates both photonic band dispersion and optical Kerr
nonlinearity. The envelope of slowly varying fields with appropriate carrier frequency and
amplitude are governed by nonlinear coupled mode equations, which can be obtained from
the nonlinear Maxwell wave equation Eq. (3.31):
∂ 2 E n2 (z, |E|2 ) ∂ 2 E
−
= 0,
∂z 2
c2
∂t2
(5.17)
where the refractive index n(z, |E|2 ) is given by:
n(z, |E|2 ) = n0 + nnl |E|2 + 2n0k cos 2k0 z.
(5.18)
The derivation is given in Appendix A (here we have temporarily neglected the periodic
variation in nonlinear refractive index, but it canpbe easily included). Note that nnl =
(n0 /(2η0 )n2 where n2 is given in SI units and η0 ≡ µ0 /ε0 ∼ 377Ω is the wave impendence
in vacuum.
Light, that is incident to the medium is assumed to lie within the linear forbidden band
and then the two counter-propagating waves are strongly coupled. The intensity-dependent
refraction of the optical material supports the resulting transmission of light through the
periodic structure. All the effects due to deep gratings are neglected and the nonlinear
coupled mode equations are simply the generalization of Eq. (5.8) (derivation is given in
Appendix B) and have a general form :
∂E+
n0 ∂E+
+i
+ κE− + Γ(|E+ |2 + 2|E− |2 )E+ = 0
∂z
c ∂t
n0 ∂E−
∂E−
+i
+ κE+ + Γ(|E− |2 + 2|E+ |2 )E− = 0,
−i
∂z
c ∂t
i
(5.19)
where Γ = 3ω02 χ(3) /(2k0 c2 ) is the nonlinear coefficient. Nonlinear terms in (5.19) describe
the influence of nonlinear changes in the grating refractive index to the wave propagation of interest. In Kerr-nonlinear material waves affect their own propagation (self-phase
modulation) and the wave propagating in the opposite direction (cross-phase modulation)
[16].
5.3.3
Optical bistability
The calculations of Winful et al. [23] show that the field inside the grating (the solution to
the nonlinear coupled mode equations) can be written in terms of Jacobi elliptic functions.
At the center of the band gap, the solution has a form:
I/S = nd[2 cosh (χ)κz|1/ cosh2 (χ)],
(5.20)
where I = |E+ |2 + |E− |2 is the total intensity, and S = |E+ |2 − |E− |2 is the energy flow
through the grating. nd(z|m) is one of the Jacobi elliptic functions and χ satisfies the
26
equation: sinh χ = 3ΓS/(4κ). For real arguments, nd is a real positive periodic function
that varies between 1 and 1/ tanh χ. If the field’s envelope function is periodic, then so
must be the electric field, up to a phase detuning. Therefore, if the period of the field
envelope equals the length of the grating, then the fields at the front and the back of the
grating are identical, resulting in the zero reflectivity vanishes at certain powers or lengths
of the medium.
Described behaviour shows that the structure becomes transparent at certain intensities
(S = I), while for other values of the intensity S < I. Detailed analyzes shows that the
transmissivity of the structure is no longer a single-valued function, but consists of a lowertransmission branch corresponding to linear regime and of a higher transmission branch.
Such effect is termed as bistability in the input-output power relationship and means that
the state of the system is not uniquely determined by the input.
Stability of these solutions must be determined by the analyses of full time-dependent
equations. This shows that the lower branch is always stable, but the upper branch is
often unstable. For cases in which the upper branch is stable, the system exhibits bistable
switching [17].
5.3.4
Bragg and gap solitons
Another class of solutions to coupled mode equations is a class of pulse-like solutions that
are referred as Bragg solitons. They result from the combination of linear dispersion of the
grating and nonlinear change in dielectric constant at a high light intensity. These pulses
do not change in shape in spite of the strong dispersion (that otherwise would quickly
broaden an arbitrary pulse at low intensities).
As shown previously, the uniform Kerr nonlinearity leads to optical bistability and to
switching pulsations between lower-transmissive and higher-transmissive stationary states.
The grating may become transparent in the gap, switching from total reflection to high
transmissivity. The switchings occur as regular and, sometimes, irregular oscillations,
arising at the left end of the grating, where the light is illuminated, and travelling to
the right end of the gratings. These pulsations of light in the optical grating display a
series of localized pulses travelling across the periodic structure with constant speed, which
are called Bragg solitons. The input pulse is transmitted through the periodic structure
without any change in its shape and amplitude.
In simple, one-dimensional geometries Bragg solitons are very similar to the regular
solitons in uniform optical fibers. In both of the structures the wave gains its stability
through a counter-balancing effect of the group velocity dispersion and the effect of the
nonlinearity. The difference is that for the solitons of uniform medium the group-velocity
dispersion is primarily due to the underlying dispersion of the uniform material, while for
a gap soliton, it is due to the photonic band structure. Optical dispersion for wavelengths
near to the photonic bandgap are nearly 6 orders of magnitude larger than for propagation
in a uniform material. Large dispersion with nonlinear changes in the refractive index
results in soliton formation in length scales of only in centimeters.
27
Gap soliton is an optical pulse that propagates at the wavelength within the photonic
bandgap for long distances without distortion at any velocity 0 − c.
Bragg soliton is formed when the pulse wavelengths are either inside or partially
outside the gap (even at wavelengths nearly outside the gap).
The first experimental observation of Bragg solitons in a fiber with Bragg grating was
performed in 1996 [4] under laser pulse irradiation at a frequency near (but not inside)
the photonic band gap. The nonlinear optical pulses can be used for all-optical switching,
soliton lasers, pulse compressors, optical buffers, and storing devices [16, 17].
5.3.5
Nonlinear Schrödinger limit
Nonlinear coupled mode equations can be reduced to nonlinear Schrödinger equation
(NLSE) by using the method of multiple scales [10] if the regime of weak nonlinearities
is assumed. The electric field of nonlinear coupled mode equations is assumed to be the
perturbation of linear coupled mode equations.
The solutions to nonlinear coupled mode equation are constructed as slow modulation
of Bloch waves on appropriately long spatial and temporal scales. With small dimensionless parameter µ, which is the measure of the amplitude of electric field E± , spatial and
temporal variables are scaled as:
tj = µj t, zj = µj z, j = 1, 2, ...
(5.21)
Finally we search for the solution of the nonlinear coupled mode equations (5.19) of the
form:
E = (µa(z1 , z2 ; ..., t1 , t2 , ...)v+ + µ2 b(z1 , z2 , ...; t1 , t2 , ...)v− )ei(Qz−V ∆+ t) ,
(5.22)
where E is a vector with components E± , a and b are the envelope functions and the
v+ mode dominates the v− mode. By finding the derivatives up to the order µ3 , we can
achieve the NLSE in a form:
ic ∂a ω2 ∂ 2 a Γ
+
+ (3 − v 2 )|a|2 a = 0
2
n0 ∂τ
2 ∂ξ
2
(5.23)
where ω2 is the quadratic dispersion, and τ = t and ξ = z − v ncln t are the moving coordinates.
Equation (5.23) is regarded as the NLSE, because it has a structure of the Schrödinger
equation with Γ2 (3 − v 2 )|a|2 as a self-induced potential. Since a is a complex function,
travelling wave solution is expected to have an oscillatory modulation. The fundamental
soliton solution of the NLSE is a sech-shaped wave, which acts as an envelope to the highly
oscillatory carrier wave.
The NLSE plays an extremely important role in the theory of evolution of slowly varying
wave trains in stable weakly nonlinear systems and it occurs in many other physical areas,
not only in nonlinear optics. Similarly to the KdV and Sine-Gordon equations its travelling
wave envelope-solutions are solitons [5, 17].
28
5.4
Gratings with periodic nonlinear index
In previous section we used an assumption that the analyzed structure has a periodic
linear and uniform nonlinear index of refraction. A number of entirely new applications
are possible if both linear and nonlinear components of the refractive index are modulated.
Periodically nonlinear optical materials generally exhibit the phenomena of bistability
and multi-stability, and do not lead to memoryless operation. They do not exhibit saturation of the transmitted intensity and may undergo chaotic behavior. Many operations in
signal-processing are required to be independent of the past state of the channel [14, 15].
In structures with periodical variation in nonlinear refractive index, stable limiting
behavior is possible in their input-output transmission characteristic: the transmitted intensity is bounded by the asymptotic limiting value and no switching to a state of higher
transmittance takes place. This stable highly-nonlinear effect is termed as all-optical limiting.
Structures with a periodic and nonlinear refractive index variation, which exhibit stable
all-optical limiting, represent an important example of optical signal-processing elements
- passive optical limiters, which are most commonly used as protective devices, but also in
optical logic circuits, optical signal processing, optical sensing, and optical fiber communications [3].
Generalization of coupled mode theory
Mathematical theory for gratings with a periodic and nonlinear index variation is very
similar to the coupled mode theory discussed previously in this chapter. The expression
for the refractive index now has an additional term for nonlinear periodic index and is
expressed as:
n(Z, |E|2 ) = nln + nnl |E|2 + 2n0k cos kZ + 2n2k |E|2 cos kZ
(5.24)
(derivation is given in Appendix A). The structure can still be analyzed with the coupled
mode system, which now has a form:
∂E+ ∂E+
i
+ n0k E− + nnl (|E+ |2 + 2|E− |2 )E+
(5.25)
+
∂Z
∂T
+n2k [(2|E+ |2 + |E− |2 )E− + E+2 E−∗ ] = 0,
∂E− ∂E−
−i
+ n0k E+ + nnl (2|E+ |2 + |E− |2 )E− +
−
∂Z
∂T
+n2k [(|E+ |2 + 2|E− |2 )E+ + E−2 E+∗ ] = 0.
p
where E± are the normalized complex amplitudes for the electric field defined as E±′ / 2η0 I0 /nln ,
where E±′ are the complex amplitudes of the electric field and I0 is reference intensity, and
Z = ω0 z/c and T = ω0 t/nln are the normalized spatial coordinate and time, respectively. Also, note that nnl and n2k are normalized such that they are non-dimensional
′
′
′
′
as nnl = nnl I0 2η0 /nln and n2k = n2k I0 2η0 /nln , where nnl and n2k are nonlinear index
coefficients used in previous chapters (Derived in Appendix B).
29
5.5
Stationary solutions
In this Section we study the stationary solutions of light transmission in finite optical
grating. The normalized length of the grating is L such that Z ∈ [0, L]. The boundary
conditions are given at both ends of the structure (Z = 0 and Z = L) by:
|E+ (0)|2 = Iin , |E− (0)|2 = Iref ,
|E+ (L)|2 = Iout , |E− (L)|2 = 0,
(5.26)
where Iin , Iref , and Iout are the normalized intensities of incident, reflected, and transmitted
waves, respectively. The backward propagating wave vanishes at the right end of the
grating.
The intensity flow through the structure is conserved and satisfies the equation:
|E+ (Z)|2 − |E− (Z)|2 = Iin − Iref = Iout
(5.27)
Stationary coupled mode system is obtained by assuming that ∂E± /∂T = 0 in (5.25).
It can be written in terms of the intensity of reflected wave Q = |E(Z)|2 and the phase
mismatch Ψ = Arg(E− (Z)) − Arg(E+ (Z)) in a form:
p
∂Q
= −2 Q(Iout + Q) sin Ψ[n0k + n2k (Iout + 2Q)],
(5.28)
∂Z
∂Ψ
cos Ψ
2
[n0k (Iout + 2Q) + n2k (Iout
+ 8Iout Q + 8Q2 )]
= −3nnl (Iout + 2Q) − p
∂Z
Q(Iout + Q)
(see derivation in Appendix C).
The stationary coupled mode system can also be written in Hamiltonian form:
dE±
∂H
= ±i ∗ ,
dZ
∂E±
(5.29)
where the real-valued Hamiltonian is given by:
1
H = [n0k +n2k (|E+ |2 +|E− |2 )](E+∗ E− +E+ E−∗ )+ nnl (|E+ |4 +4|E+ |2 |E− |2 +|E− |4 ). (5.30)
2
If we replace E± in Eq. (5.30) by Eq. (C.1), then the Hamiltonian can be written in terms
of Q and Ψ:
p
1
2
H = 2 Q(Iout + Q) cos Ψ[n0k + n2k (Iout + 2Q)] + 3nnl Q(Iout + Q) + nnl Iout
. (5.31)
2
2
The boundary conditions are satisfied for H = 1/2nnl Iout
when Q(z) and Ψ(z) are related
by the equation:
p
−3nnl Q(Iout + Q)
.
(5.32)
cos Ψ =
2[n0k + n2k (Iout + 2Q)]
Coupled mode system given by Eq. (5.28) can be written in a single equation either
for Q(z) or for Ψ(z) by using Eq. (5.32). Such a reduced equation depends on several
parameters: nnl , n0k , n2k and Iout and we consider different possibilities separately.
30
5.5.1
Balanced nonlinearity management: nnl = 0
From Eq. (5.32) follows that cos Ψ = 0 whenever nnl = 0, meaning that Ψ = ±π/2. At the
right end of the structure the boundary conditions Q(L) = 0 and Q(Z) ≥ 0 for Eq. (5.28)
must be satisfied. The sign of dQ/dZ at Z = L depends only on the signs of n0k + n2k Iout
and cos Ψ at Z = L. As the slope must be negative at the right boundary, the phase factor
needs to satisfy the following boundary conditions:
π
, if n0k + n2k Iout ≥ 0,
2
π
Ψ(z) = − , if n0k + n2k Iout < 0.
2
Ψ(z) =
(5.33)
We may assume without loss of generality that the first layer is focusing and the second
layer is defocusing - i.e. nnl1 > 0 and nnl2 < 0. In this assumption, according to Eq.(A.7)
n2k always has a positive value. Now we can interpret negative values of n0k as out of
phase matching, and positive values of n0k as in-phase matching between linear and Kerr
nonlinear refractive indices.
Slightly differently from analyzes of Pelinovsky [14] we consider two cases defined in
Eq. (5.33) separately.
The periodic structure with zero net average Kerr nonlinearity is now analytically
tractable.
Solution for n0k ≤ n2k Iout and Ψ(Z) =
π
2
The coupled mode system Eq. (5.28) is reduced to the ordinary differential equation for
Q:
p
dQ
(5.34)
= −2 Q(Iout + Q)[n0k + n2k (Iout + 2Q)].
dZ
Now we can separate the variables:
dQ
p
= −4n2k dZ,
Q(Iout + Q)(Q + a)
(5.35)
where we have denoted a = ( nn0k
+ Iout )/2.
2k
Direct integration of Eq. (5.35) gives:
1
p
a(Iout − a)
sin
−1
L
(Iout − 2a)(Q + a) + 2a(a − Iout ) = −4n2k (L − Z)
Iout |Q + a|
Q(Z)
(5.36)
Now we can acquire the analytical solution for reflected intensity Q(z):
Q(z) =
where θ =
p
Iout (n0k + n2k Iout ) sin2 θ
,
n2k Iout cos 2θ − n0k
2
n22k Iout
− n20k (L − Z).
31
(5.37)
Q(z) is defined if the denominator of Eq. (5.37) is nonzero. Output intensity Iout needs
to satisfy the condition Iout ≤ Ilim , where Ilim is the upper threshold for output-intensity
and satisfies the equation:
q
n0k
2
≤ 1.
(5.38)
−1 ≤ cos (2 n22k Ilim
− n20k L) =
n2k Ilim
It is seen from the Figure 5.3 below that the reflected intensity Q is monotonically decreasing between Z = 0 and Z = L as expected.
−4
3
x 10
Reflected intensity
2.5
2
1.5
1
0.5
0
0
5
10
Z
15
20
Figure 5.3: Z-dependence of reflected intensity through the structure (L=20), where nnl =
0, n0k = −0.02, n2k = 1 and Iout = 0.025
Solution for n0k > n2k Iout and Ψ(Z) = −π/2
The coupled mode system given by Eq. (5.28) is now reduced to:
p
dQ
= 2 Q(Iout + Q)[n0k + n2k (Iout + 2Q)],
dZ
(5.39)
and the solution has a generalized form:
Q(z) =
Iout (n0k + n2k Iout ) sinh2 φ
,
n0k − n2k Iout cosh 2φ
(5.40)
p
2
(L − Z). With the change of variables Φ = −iθ the
where φ = −iθ = n20k − n22k Iout
solution (5.40) reduces to the solution (5.37) for n0k ≤ n2k Iout .
32
Transmission characteristics
Transmission curves for balanced nonlinearity management can be calculated either analytically (using the Matlab program in Appendix D.1) or by general numerical algorithm
(described in Section 5.6 and D.2).
0.03
0.025
n =−0.02
0k
n =0
0k
n0k=0.02
Iout
0.02
0.015
0.01
0.005
0
0
0.01
0.02
0.03
0.04
0.05
Iin
Figure 5.4: Transmission curves for different linear gratings where nnl = 0, n2k = 1 and
L = 50
1
Iout/Ilim
0.8
L=20
L=50
L=200
0.6
0.4
0.2
0
0
0.5
1
Iin/Ilim
1.5
2
2.5
Figure 5.5: Transmission curves for different grating lengths. nnl = 0 n0k = −0.02, and
n2k = 1
In-phase gratings (lower curves in Figure 5.4) exhibit simple all-optical limiting, while
for out of phase gratings (curve for n0k = −0.02) the dependence Iout (Iin ) exhibits the
S-shaped profile, which is still one-to-one function and is bounded by its limiting value
Ilim .
33
Transmission properties also depend on the grating length L. As it can be seen from
the Figure 5.5, case L = 20 is suitable for optical limiting. When the structure is sufficiently long (e.g. L=200), the S-shaped transmission curve becomes a simple two-step map
between the lower-transmissive and higher-transmissive limiting state and is suitable for
logic operations.
Generally bistability never occurs in the limit nnl = 0.
5.5.2
Unbalanced nonlinearity management: nnl 6= 0
Combining (5.28) and (5.32), we obtain a single differential equation for reflected intensity
Q(Z):
2
dQ
= Q(Iout + Q)(4[n0k + n2k (Iout + 2Q)]2 − 9n2nl Q(Iout + Q)).
(5.41)
dZ
Explicit solutions to Eq. (5.41) are given in terms of elliptic functions [16] and can be
analyzed numerically.
The condition for the limiting behavior, i.e. Q(0) → ∞ for Iout → Ilim < ∞, can be
found from the connecting relation (5.32). True all-optical limiting regime exists when the
condition | cos Ψ| ≤ 1 is satisfied, i.e when
n2k ≥
3nnl
4
(5.42)
Otherwise, no limiting regime is possible and the system is either bistable or multi-stable.
Transmission characteristics
Curves, which fit to the domain given by Eq. (5.42), show limiting behaviour, while curves
which violate it, exhibit multi-stable behaviour (illustrated in Figure 5.6).
For sufficiently strong out-of-phase gratings which satisfy the condition (5.42), multistability takes place in low and high intensities, but transmission curve still has a limiting
value (n0k = −0.04). This behavior is illustrated on Figure 5.7 and is called locally multistable limiting. It occurs for n0k ≤ −0.03
Generally for nnl 6= 0 true all-optical limiting is supported within domain given by Eq.
(5.42) by the in-phase and weakly out of phase gratings [15].
5.5.3
Stationary transmission regimes
As a conclusion, the described system has three types of stationary transmission regimes:
• Stable limiting regime, which is uniform for all pertinent incident intensities. Transmission curve Iout (Iin ) is a one-to-one function. Occurs for gratings with balanced
nonlinearity management (nnl = 0) and unbalanced nonlinear gratings (nnl 6= 0) in
the domain (5.42) for in-phase or weak out-of-phase gratings [15, 13].
34
0.4
III
0.35
Transmitted intensity
0.3
0.25
II
0.2
Ib
0.15
0.1
Ia
0.05
0
0
0.2
0.4
0.6
Incident intensity
0.8
Figure 5.6: Transmission curves with parameters: L = 20, n2k = 1, and (Ia ) nnl = 1,
n0k = 0, (Ib ) nnl = 0, n0k = −0.15, (II) nnl = 1, n0k = −0.15, and (III) nnl = 1.4,
n0k = 0.
0.08
0.07
n0k=−0.04
Transmitted intensity
0.06
0.05
0.04
n0k=−0.02
0.03
0.02
n0k=0
0.01
0
0
n0k=0.02
0.02
0.04
0.06
Incident intensity
0.08
Figure 5.7: Unbalanced nonlinearity management with different linear gratings, where
nnl = 1, n2k = 1 and L = 50. n0k = −0.04 illustrates locally multi-stable limiting.
35
• Locally multi-stable limiting, which exhibits branching for small and intermediate
values of Iin , but is bounded by the limiting value of Iout for large values of Iin .
Occurs in strongly out of phase unbalanced nonlinear gratings where the constraint
(5.42) is still held.
• Multi-stable regime has several stationary transmission regimes for the same value
of incident intensity Iin . Occurs in strongly unbalanced nonlinear gratings when
constraint (5.42) is violated.
5.6
Numerical method
Backward finite difference scheme was used to find the time independent solutions of E± (Z)
of the coupled mode system, starting from the boundary conditions at the right end Z = L
and iterating back to the left end Z = 0 [15].
The complex valued stationary coupled mode system for E± (Z) can be decomposed
into the real and imaginary parts by substituting:
E+ = u + iω, E− = v + iy
(5.43)
The coupled mode system can now be written in terms of real functions u, ω, v and y in
a form:
∂u
+ n0k y + fu
∂z
∂y
+ n0k u + fy
−
∂z
∂ω
−
+ n0k v + fω
∂z
∂v
+ n0k ω + fv
∂z
= 0
(5.44)
= 0
(5.45)
= 0
(5.46)
= 0,
(5.47)
where the nonlinear functions are given by:
fu = nnl (u2 + ω 2 + 2v 2 + 2y 2)ω + n2k [(u2 + 3ω 2 + v 2 + y 2)y + 2uωv]
fy = nnl (2u2 + 2ω 2 + v 2 + y 2)v + n2k [(u2 + ω 2 + 3v 2 + y 2)u + 2ωvy]
fω = nnl (u2 + ω 2 + 2v 2 + 2y 2 )u + n2k [(3u2 + ω 2 + v 2 + y 2)v + 2uωy]
fv = nnl (2u2 + 2ω 2 + v 2 + y 2 )y + n2k [(u2 + ω 2 + v 2 + 3y 2)ω + 2uvy]
(5.48)
(5.49)
(5.50)
(5.51)
The functions u, ω, v, and y are calculated on the grid points of spatial coordinate z:
z = zn = n δz, n = 0, 1, ..., N, (N + 1)
(5.52)
where δz = L/(N + 1) is the space step size and the grid has N interior points and two
end points at z = 0 and z = L = (N + 1)δz.
36
The boundary conditions at the right end of the structure (z = L) are given by:
p
u(L) =
Iout ,
ω(L) = 0
(5.53)
v(L) = 0,
y(L) = 0
(5.54)
Now the coupled mode system can be solved by iteration method (program code is given
in Appendix D.2). [15]
37
Chapter 6
Transmission properties of
nanoparticles (CdSe) dispersed
photopolymer hologram
6.1
Introduction
In this Chapter we analyze the transmission properties of a photopolymer material that
is dispersed with semiconductor nanoparticles (CdSe - cadmium selenide), reported by
Tomita et. al. [20, 22, 18, 19], which is holographically illuminated by Nd:YAG laser
to form a stable structure with periodical variation in its index of refraction. The wave
dynamics in this structure is analogous to a photonic crystal and could be analyzed by the
computational model of Pelinovsky, described in Ref. [14] (see Chapter 5).
6.2
6.2.1
Holography with photopolymers
Grating formation
A conventional photopolymer (without nanoparticles) consists of mobile light-sensitive
monomers, immobile polymers, and light-insensitive immobile binders. During the holographic exposure monomers are inhomogeneously polymerized depending on the intensity
pattern of the incident light. In bright regions monomers polymerize more intensively,
leading to the density gradient of monomers between the bright and dark regions. As a
result monomers diffuse from the dark to the bright regions. Index grating (a hologram)
is formed by periodical changes in monomer concentration (transient grating) and polymer concentration (permanent grating). Photopolymers with radical photopolymerization
have high refractive index contrast (∆n ≈ 10−2 ), which is determined by the difference
of refractive indices between un-reacted and reacted components, but at the same time it
possesses undesired large photopolymerization shrinkage (∼ 10%).
38
If we added secondary mobile but photo-insensitive component (i.e., inorganic nanoparticles) to monomers, ∆n is considerably increased [18, 19, 20, 22]. Now, in addition to
the above-mentioned photopolymerization process, photo-insensitive mobile nanoparticles
would diffuse from the dark to the bright regions due to the chemical potential difference
of monomers and nanoparticles in space[19]. Due to a large difference in refractive indices
of polymers and nanoparticles (where we assume high contrast materials), a high-contrast
grating (a hologram) can be formed.
In Figure 6.2 blue squares represent mobile nanoparticles. Spatial mass-transfer of both
monomers and nanoparticles during holographic exposure is responsible for the observed
grating formation.
Figure 6.1: Formation of a hologram in conventional photopolymer structure
Figure 6.2: Formation of a hologram in nanoparticles-dispersed photopolymer structure
Since nanoparticles such as II-VI semiconductor (e.g., CdSe) quantum dots with large
third-order nonlinearities (optical Kerr effect) are employed, then the hologram would be
nonlinear - i.e., its background and spatially modulated indices of refraction have optical
Kerr nonlinearities[21]. The functional form of such linear and nonlinear refractive indices
will depend on the density distributions of polymers and nanoparticles after holographic
exposure [7]. In the following calculations it is assumed that the index modulation has
sinusoidal form (instead of its distorted form) for simplicity. Such a structure may be
described by the computational model of Pelinovsky [14].
6.2.2
Applications
Holographic recording using photopolymers is well known for its high refractive-index contrast (∆n ≈ 10−2 ) and high recording sensitivity. Holography with the usage of pho39
topolymers has found many important applications in photonics, including holographic
optical elements, holographic memories, optical interconnects, narrow-band optical filters,
waveguides, electrically switchable Bragg gratings, and photonic crystals [21].
Inclusion of inorganic nanoparticles as functional components for holographic applications would increase the index contrast further (∆n ≥ 10−1 ). This allows us to use smaller
writing-intensities, the suppression of polymerization shrinkage, the increased dimensional
stability, and the addition of new functionalities to photopolymers (e.g., nonlinearities)[21].
6.3
Mathematical model
The system of the photopolymer hologram is mathematically identical to an optical device
that has finite length and a number of alternating layers with different linear refractive
indices and different Kerr nonlinearities (the properties of such structures are described
in Section 5.3). Total length of the structure is given by l = NΛ, where Λ is a grating
period, and N is the number of layers. Using averaged approximation for the weak fields,
the refractive index is given by:
n(z, |E|2 ) = n0 (z) + nnl (z)|E|2 + O(|E|4)
(6.1)
where n0 (z) is a linear refractive index and nnl (z) is an intensity dependent index-modulation
caused by Kerr nonlinearity. n0 (z) and nnl (z) are considered to be constant within each
layer.
Figure 6.3: Mathematical model of the nano-particles dispersed photopolymer hologram
For two-layer model, the Fourier coefficients of (n(z, |E|2 )) can be evaluated from constant linear and nonlinear indices in each layer.
Coupled mode system (5.25) (derived in Appendix B) is used to model the transmission properties of CdSe-dispersed photopolymer holograms. Since this structure does not
have balanced nonlinearity management (nnl 6= 0), the explicit solutions to coupled mode
equations are given in terms of elliptic functions [16] and can be solved only numerically
40
(numerical model is described in Section 5.6 and the Matlab code is given in Appendix
D.2).
6.4
Data for numerical experiments
Second harmonic component of Nd:YAG laser (green light, λ =532nm) was used during
hologram formation. Resultant grating spacing after holographic exposure is Λ = λm / cos q,
where q is the half angle between two recording beams and λm = λ/nln is the recording
wavelength in the medium. nln is average index of refraction and equals about 1.9 in the
given material. By choosing the appropriate value for q, hologram can be written for both
fundamental wavelength of Nd:YAG laser (λ = 1064nm) and second harmonic component
(λ = 532nm). Corresponding grating spacing equals to either 0.28µm (for λ = 1064nm)
or 0.14µm (for λ = 532 nm). Realistic grating lengths would be in the range of 30µm to
few millimeters.
For the study of transmission properties, intensities of 1, 10 and 100 MW/cm2 are
considered. These values are experimentally obtainable by using high intensity light pulses
from Q-switched Nd:YAG laser with pulse-width of about few nanoseconds. Since such a
pulse width covers the whole structure, the pulse can be interpreted as quasi-continuous
incident wave in the model.
In CdSe-dispersed photopolymer holograms there are two types of index distributions:
the diffusion-dominant case (R = 10) and the photopolymerization dominant case (R =
0.01). Parameter R is defined as the ratio of the diffusion rate to photopolymerization
rate, i.e. large values R implies the dominance of diffusion of monomers and nanoparticles
compared to the photopolymerization. R depends on material parameters and recording
intensity (R decreases with the increase of recording intensity). Also we need to consider
two different cases for the values of refractive indices, because the index of refraction of
CdSe nanoparticles has been reported to be either 2.55 or 2.728 in the green. In following
numerical calculations we consider all four combinations of R and nCdSe . Distributions of
monomers, polymers, and nanoparticles were calculated by mutual diffusion model [7].
Normalized parameters
The nonlinear coupled mode equations, given by Eq. (5.25) are given in normalized space
and time variables (Z and T ). Linear uniform refractive index nln only appears in normalized time variable T = ω0 t/nln , so that nln influences only time dependent dynamics
of the system, but stationary solutions are nln -independent.
Spatial variable Z is also normalized as Z = zω0 /c (see Appendix B) and the normalized
grating length L is related to grating length l by L = lω0 /c = 2πl/λ = kl, where ω0 is the
light frequency, λ is the optical wavelength and k is the wave-number of the incident light.
To match with the realistic grating lengths defined above, we need to consider normalized
grating lengths within the interval 200 - 6000 (in this analyzes we have considered the cases
L = 200 and L = 500).
41
As shown in Section 5.4, nonlinear refractive indices and intensities (I) are normalized
with respect to the reference intensity I0 = 100MW/cm2 . Note that the following relations
are used: nnl = II0 (n2 )nl and n2k = II0 (n2 )2k , where (n2 )nl and (n2 )2k are average and
periodic components of nonlinear index n2 . In numerical experiments normalized intensities
of I = 0.01, I = 0.1 and I = 1, are used (corresponding to actual intensities of 1, 10 and
100 MW/cm2 , respectively).
6.5
6.5.1
Results for nCdSe = 2.55
Diffusion dominant case (R=10)
n0k = 0.10402
2
nonlinear indices in cm /W
I0 n2
(n2 )nl
8.0608 × 10−10
0.080608
(n2 )2k
2.5813 × 10−10
0.025813
1
Transmitted intensity
0.8
0.6
0.4
0.2
0
0
0.5
1
Incident intensity
1.5
Figure 6.4: Multi-stable transmission for L = 200. See Figs. 6.5 and 6.6 for the behavior
near the origin of the transmitted intensity.
42
−17
7
x 10
Transmitted intensity
6
5
4
3
2
1
0
0
0.5
1
Incident intensity
1.5
Figure 6.5: The first multi-stable branch for low output intensities (part of the Fig. 6.4).
−14
5
x 10
Transmitted intensity
4
3
2
1
0
0
0.02
0.04
0.06
Incident intensity
0.08
0.1
Figure 6.6: The second multi-stable branch for low output intensities (part of the Fig. 6.4).
We can see, that multistability occurs for input intensities as low as Iin = 0.01
43
2.5
I
f
I
b
Intensity
2
1.5
1
0.5
0
0
50
100
Distance (Z)
150
200
Figure 6.7: Intensities of forward- (If ) and backward (Ib ) propagating waves inside the
grating. L = 200, Iin = 1, Iout = 0.986. The values correspond to the positive slope of
multi-stable curve. The light transmission at high transmissivity exhibits high amplitude
oscillations inside the structure.
1.8
I
f
I
1.6
b
1.4
Intensity
1.2
1
0.8
0.6
0.4
0.2
0
0
50
100
Distance (Z)
150
200
Figure 6.8: Intensities of forward- (If ) and backward (Ib ) propagating waves inside the
grating. L = 200, Iin = 1, Iout = 0.448. The values correspond to the negative slope of
multi-stable curve and give unstable solutions.
44
1
Intensity
0.8
0.6
0.4
0.2
0
0
50
100
Distance (Z)
150
200
Figure 6.9: Intensities of forward- (If ) and backward (Ib ) propagating waves inside the
grating at lower intensity transmissive state. The graphs of If and Ib are overlapping in
the figure (Ib ≈ If ). L = 200,Iin = 0.1 and Iout = 2.8 · 10−18 .
1.6
1.4
If
Ib
1.2
Intensity
1
0.8
0.6
0.4
0.2
0
0
50
100
Distance (Z)
150
200
Figure 6.10: Intensity of the forward- and backward propagating waves inside the grating
for L = 200, Iin = 0.1, and Iout = 0.0155. Oscillations occur at higher intensity transmissive
state
45
Longer gratings
1
Transmitted intensity
0.8
0.6
0.4
0.2
0
0
0.5
1
Incident intensity
1.5
Figure 6.11: Multi-stable transmission in longer gratings — L = 500. Generally longer
gratings have higher density of multi-stable branches.
−22
Transmitted intensity
x 10
4
3
2
1
0
0
0.5
1
Incident intensity
1.5
Figure 6.12: Multi-stable transmission for L = 500 at low output intensities (part of Fig.
6.11). Multistability occurs at lower output intensities in longer gratings.
46
6.5.2
Photopolymerization dominant case (R=0.01)
n0k = 0.024955
nonlinear indices in cm2 /W
I0 n2
(n2 )nl
8.0958 × 10−10
0.080958
(n2 )2k
6.1929 × 10−11
0.0061929
1
Transmitted intensity
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
Incident intensity
0.8
1
Figure 6.13: Multi-stable transmission for L = 200. See Fig. 6.14 for the behavior near
the origin of the transmitted intensity.
47
−3
1
x 10
Transmitted intensity
0.8
0.6
0.4
0.2
0
0
0.05
0.1
0.15
Incident intensity
0.2
0.25
Figure 6.14: Multi-stable transmission for L = 200 at low output intensities (part of the
Fig. 6.13). Multistability doesn’t occur at Iin = 0.01 (compare with Fig. 6.6 for diffusion
dominant gratings where such behavior is observable).
1
Transmitted intensity
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
Incident intensity
0.8
1
Figure 6.15: Multi-stable transmission in longer grating (L = 500). Transmission curve
has higher density of multistability branches compared to Fig. 6.13 for L = 200. Such
effect is also observable in diffusion dominant gratings (see Figs. 6.4 and 6.11)
48
0.12
Intensity of the forward wave
0.1
0.08
0.06
0.04
0.02
0
0
50
100
Distance (Z)
150
200
Figure 6.16: Intensities of forward- (If ) and backward (Ib ) propagating waves inside the
grating for L = 200, Iin = 0.1, and Iout = 1.9 · 10−5 at low intensity transmissive state.
The graphs of If and Ib are overlapping in the figure (Ib ≈ If ) and the structure transmits
only a very small fraction of incident light.
Intensity of the forward wave
0.25
0.2
0.15
0.1
0.05
0
0
50
100
Distance (Z)
150
200
Figure 6.17: Intensities of forward- (If ) and backward (Ib ) propagating waves inside the
grating for L = 200, Iin = 0.1, and Iout = 3.1 · 10−4 at first multi-stable branch state. The
graphs of If and Ib are overlapping in the figure (Ib ≈ If ).
49
6.6
Analyzes for nCdSe = 2.728
The behaviour is qualitatively identical to the case nCdSe = 2.55. In this Section only
transmission figures are plotted, discussions are analogical to previous Section.
6.6.1
Diffusion dominant case (R=10)
n0k = 0.1233
nonlinear indices in cm2 /W
I0 n2
(n2 )nl
7.0358 × 10−10
0.070358
(n2 )2k
2.253 × 10−10
0.02253
1
Transmitted intensity
0.8
0.6
0.4
0.2
0
0
0.5
1
Incident intensity
1.5
Figure 6.18: Multi-stable transmission for L = 200. See Figs. 6.19 and 6.20 for the
behavior near the origin of the transmitted intensity
50
−8
1
x 10
Transmitted intensity
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
Incident intensity
2
Figure 6.19: Multi-stable transmission for L = 200 at first multistability branch (part of
the Fig 6.18).
−6
3
x 10
Transmitted intensity
2.5
2
1.5
1
0.5
0
0
0.02
0.04
0.06
Incident intensity
0.08
0.1
Figure 6.20: Multi-stable transmission for L = 200 at low output intensities (part of the
Fig 6.18). Multistability is not achievable for Iin = 0.01 for such index distribution.
51
6.6.2
Photopolymerization dominant case (R=0.01)
n0k = 0.029582
nonlinear indices in cm2 /W
I0 n2
(n2 )nl
7.0664 × 10−10
0.070664
(n2 )2k
5.4054 × 10−11
0.0054054
1
Transmitted intensity
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
Incident intensity
1
1.2
Figure 6.21: Multi-stable transmission for L = 200. See Figs. 6.22 and 6.23 for the
behavior near the origin of the transmitted intensity.
52
−4
Transmitted intensity
x 10
2
1
0
0
0.1
0.2
0.3
Incident intensity
0.4
Figure 6.22: Multi-stable transmission for photopolymerization dominant grating with
nCdSe = 2.55, L = 200 at low output intensities (part of the Fig. 6.21).
0.01
Transmitted intensity
0.008
0.006
0.004
0.002
0
0
0.02
0.04
0.06
Incident intensity
0.08
0.1
Figure 6.23: Multi-stable transmission for photopolymerization dominant grating with
nCdSe = 2.55, L = 200 at low output intensities (part of the Fig. 6.21).
6.7
Discussions
All the analyzed structures clearly exhibit globally multi-stable behaviour. This is in good
agreement with analytical model described in Ref. [14]. For nnl 6= 0 the limiting behavior
is possible if the condition
3|nnl |
n2k ≥
(6.2)
4
is satisfied.
53
Evaluating n2k and nnl from Sections 6.5 and 6.6 clearly shows that condition given
by Eq. (6.2) is not satisfied for considered CdSe-dispersed photopolymer hologram with
strongly unbalanced nonlinear grating. Stable limiting regime does not occur for these
material parameters and this multi-stable behaviour is also predictable by the analytical
model.
We have shown that given structure generates efficient multi-stabilities at very low
intensities and long grating lengths that are practically achievable. During wave transmission through the grating, intensities of forward- and backward propagating waves are
monotonic functions at lowest transmissive state of multi-stable transmission curve. but
show oscillatory behaviour at higher transmissive states.
Such behaviour could be usable for bistable or multi-stable switching, all-optical logic,
or memory operations. However, we have not examined their stabilities yet and some
of the transmission regimes might be unstable, leading to chaotic behaviors. Only full
stability analyzes would show whether observed light transmission does indeed survive
under real-life disturbances.
We found that longer grating lengths lead to increased number of multistability branches
in input-output power relationship. Also multistability occurs at smaller input intensities.
In this work we have analyzed only shorter gratings (normalized lengths of 200 and 500),
but in practice grating lengths up to few millimeters (L = 6000) is achievable. Such long
gratings should be analyzed in future. My experience has shown that these computation
require very high accuracy and longer computation time.
Stable all-optical limiting can’t be achieved in given material without changing the
material parameters and index distributions, namely, we should increase n2k compared to
nnl (condition given by Eq. (6.2) must be satisfied). Index distributions are generated
during holographic exposure and resultant grating is characterized by the parameter R,
as described in Section 6.4. Since R depends on hologram recording intensity and grating
spacing, it can be varied during hologram writing and it is thus possible to engineer the
material in order to achieve the behaviour of interest.
Finally it needs to be mentioned that actually the nonlinear uniform index of refraction
influences the Bragg-matching condition - i.e. the Bragg-matching frequency depends on
the average index of refraction of a periodic structure. In previously described mathematical model, which we used, the Bragg-mismatch term ”∆k” has been neglected in the
coupled-wave equations, as the nonlinear refractive index is assumed to be much smaller
than nonlinear refractive index. Since in considered structure the values for average linear
refractive indices are between 1.9 and 2, while the maximum value for (n2 )nl I0 is about
0.08, nonlinear index is about 4% of the linear index. It might slightly affect the result
because the value of nonlinear index is more than 1% of linear index.
54
Chapter 7
Summary
We have studied the coherent light transmission in Kerr nonlinear periodic structures and
analyzed light transmission properties in stationary regime — true all optical limiting
and multi-stable transmission. True all-optical limiting is best achieved when the Kerr
nonlinearity is compensated exactly across the alternating layers, i.e., when the net average
nonlinearity is zero. It can also be achieved for gratings with unbalanced nonlinearity
management. Generally there are three factors that affect the performance of these optical
structures: mismatch between linear and nonlinear refractive indices, the length of the
structure, and net average nonlinearity.
We have analyzed transmission properties of CdSe-nanoparticles dispersed photopolymer holograms and found that they exhibit global multistability in input-output power
relationship, including lower intensities. Such structures can be used as optical switches
or logic elements. Also we can model the material parameters during hologram recording
in order to meet the conditions that are necessary for all-optical limiting behaviour.
Further perspectives of this work include: (i) performing the stability analyzes in order
to estimate the applicability of the structure in real life disturbances; and (ii) improving
the numerical algorithm in order to study longer gratings (currently such calculations are
very time-consuming).
I have completed the task of numerically analyzing light transmission properties in
CdSe-dispersed photopolymer holograms and performed the theoretical analyzes of nonlinear periodic structures and their applications in optical communication systems, which
gives me sufficient background for further studies at this field.
55
Appendix A
Derivation of the expression for
index modulation
If we consider the structure described on Figure 6.3, where the index modulation has a
rectangular form, and make the assumption that the variations in the refractive index of
the grating are much smaller than the average index and the spacial variations of indices
n0 (z) and n2 (z) have the same spatial symmetry centered at Z = 0 (index modulation is
an even function of Z), the index modulation can be described by the Fourier’ series of a
square wave.
General expression for index modulation is given by:
n(Z, |E|2 ) = n0 (Z) + n2 (Z)|E|2
(A.1)
Fourier expansion for even square wave with the grating period Λ is given by:
∞
∞
4 X cos (2m − 1)2πZ/Λ
4 X cos (2m − 1)kZ
J(Z) =
=
π m=1
2m − 1
π j=1
2m − 1
(A.2)
is the grating wave number.
where k = 2π
Λ
Now the linear and nonlinear refractive indices can be written in a form:
(
4k+1
;
Λ, p ∈ Z
n01 , if Z ∈ 4k−1
n01 + n02 n01 − n02
2
2
n0 (Z) =
(A.3)
+
J(Z) =
2
2
n02 , elsewhere
(
4k+1
nnl1 , if Z ∈ 4k−1
;
λ, p ∈ Z
nnl1 + nnl2 nnl1 − nnl2
2
2
n2 (Z) =
+
J(Z) =
(A.4)
2
2
nnl2 , elsewhere
Inserting these values to general index expression (A.1) and considering only m = 1 values
of J(Z) gives:
n(Z, |E|2 ) = nln + nnl |E|2 + 2n0k cos kZ + 2n2k |E|2 cos kZ
56
(A.5)
where:
n01 + n02
,
2
n01 − n02
=
π
nln =
n0k
nnl1 + nnl2
,
2
nnl1 − nnl2
=
.
π
nnl =
n2k
57
(A.6)
(A.7)
Appendix B
Derivation of coupled mode equations
Let us consider the nonlinear Maxwell’s wave equation (derived in Section 3.4)
∂ 2 E n2 (z, |E|2 ) ∂ 2 E
−
= 0,
∂z 2
c2
∂t2
(B.1)
with refractive index (derived in Appendix A):
n(z, |E|2 ) = nln + 2n0k cos kz + nnl |E|2 + 2n2k |E|2 cos kz,
(B.2)
where k = 2π/Λ and Λ is the grating period. We assume that the electric field to consist
of two counter-propagating components with slowly varying amplitudes:
E(z, t) = E+ (z, t) exp[i(k0 z − ω0 t)] + E− (z, t) exp[i(k0 z + ω0 t)],
(B.3)
where ω0 = ck0 /nln is the frequency of incident light and k0 = 2πnln /λ is the corresponding
wave number.
Maxwell wave equation simplifies when the spatial variation of linear grating and the
variations due to nonlinearity are small compared to the average index of refraction nln , i.e.
|n0k |, |nnl ||E|2, |n2k ||E|2 ≪ nln . If we also assume that E± is slowly varying in space and
time: |∂E± /∂z|, |∂E± /∂z| ≪ |E± |, then the amplitudes E± (z, t) satisfy the coupled-mode
system near the resonance frequency (when the optical wavelength λ matches the period
of the structure: λ = 2nln Λ). First let us find the derivatives:
∂2E
∂E+
∂E−
= (2ik0
− ko2 E+ )ei(k0 z−ω0 t) + (−2ik0
− k02 E− )ei(k0 z+ω0 t) + c.c. (B.4)
2
∂z
∂z
∂z
∂E+
∂E−
∂2E
= (−2iω0
− ω02 E+ )ei(k0 z−ω0 t) + (−2iω0
− ω02 E− )ei(k0 z+ω0 t) + c.c.
2
∂t
∂t
∂t
Here we have neglected the second derivatives because of the slow variation approximation.
We also approximate the expression for n2 :
n2 ≈ n2ln + 2nln (n0k cos kz + nnl |E|2 + n2k |E|2 cos kz)
58
(B.5)
The Bragg condition is satisfied if k = 2k0 . We now insert Eq.(B.4) and Eq.(B.5) to
Eq.(B.1) and collect the terms with ei(k0 z−ω0 t) and ei(k0 z+ω0 t) . The coupled mode equations
can be given in a form:
∂E+ ∂E+
i
+ n0k E− + nnl (|E+ |2 + 2|E− |2 )E+
(B.6)
+
∂Z
∂T
+n2k [(2|E+ |2 + |E− |2 )E− + E+2 E−∗ ] = 0
∂E− ∂E−
−i
+ n0k E+ + nnl (2|E+ |2 + |E− |2 )E− +
−
∂Z
∂T
+n2k [(|E+ |2 + 2|E− |2 )E+ + E−2 E+∗ ] = 0
where Z = ω0 z/c and T = ωt/nln are the normalized spatial coordinate and time parameter, respectively. For n2k = 0 and nnl = 0, the system represents coupled mode equations
for linear Bragg gratings.
59
Appendix C
Derivation of stationary coupled
mode system
The coherent light transmission through the photonic grating is stationary when the amplitudes E± do not depend on time, i.e. ∂E± /∂T = 0 in coupled mode equations (B.6).
The coupled mode system can be written in terms of the intensity of the reflected wave
Q(Z) and the phase mismatch Ψ(Z) when using the change of variables:
p
Iout + Qei(Φ−Ψ) ,
(C.1)
E+ (z) =
p iΦ
E− (z) =
Qe .
Substituting (C.1) in the coupled mode system (B.6) gives:
p
∂Q
∂Φ ∂Ψ p
1
Iout + Q + n0k QeiΨ +
−
−
i √
∂Z
∂Z
2 I + Q ∂Z
p out
p
nnl (Iout + 3Q) Iout + Q + n2k Q[(2Iout + 3Q)eiΨ + (Iout + Q)e−iΨ )] = 0
p
p
1 ∂Q p ∂Φ
−i √
+ Q
+ n0k Iout + Qe−iΨ + nnl Q(2Iout + 3Q)
∂Z
2 Q ∂Z
p
+n2k Iout + Q[(Iout + 3Q)e−iΨ + QeiΨ ] = 0
(C.2)
(C.3)
Imaginary part of (C.2) is:
1
p
p
∂Q
+ n0k Q sin Ψ + n2k Q[(2Iout + 3Q) sin Ψ − (Iout + Q) sin Ψ] = 0 (C.4)
2 Iout + Q ∂Z
√
that yields to differential equation for reflected intensity Q:
p
∂Q
= −2 Q(Iout + Q) sin Ψ[n0k + n2k (Iout + 2Q)]
∂Z
60
(C.5)
The real part of Eqs. (C.2) and (C.3) give a system:
s
∂Φ ∂Ψ
Q
−
+
=
cos Ψ[n0k + n2k (3Iout + 4Q)] − nnl (Iout + 3Q),
(C.6)
∂Z ∂Z
Iout + Q
s
Iout + Q
∂Φ
= −
cos Ψ(n0k + n2k (Iout + 4Q)) − nnl (2Iout + 3Q). (C.7)
∂Z
Q
Which can be reduced to a single differential equation for Ψ in a form:
∂Ψ
cos Ψ
2
[n0k (Iout + 2Q) + n2k (Iout
+ 8Iout Q + 8Q2 )]. (C.8)
= −3nnl (Iout + 2Q) − p
∂Z
Q(Iout + Q)
Equations (C.5) and (C.8) form a system of stationary coupled mode system for Q and
Ψ:
p
∂Q
(C.9)
= −2 Q(Iout + Q) sin Ψ[n0k + n2k (Iout + 2Q)],
∂Z
∂Ψ
cos Ψ
2
= −3nnl (Iout + 2Q) − p
[n0k (Iout + 2Q) + n2k (Iout
+ 8Iout Q + 8Q2 )].
∂Z
Q(Iout + Q)
61
Appendix D
Program listing
D.1
Analytical algorithm for balanced nonlinearity management (nnl = 0)
clear
format short
n_2k=1;%modulation of nonlinear refractive index
n_0k=-0.02;%modulation of linear refractive index
Z=0;
L=50;%length of the structure
I_out=linspace(0.0001,0.0283);%vector for output intensities
%calculating transmission (input-output) relationship
of n_0k for k=1:100
if(abs(n_0k)<=n_2k*I_out)
I_in(k)=(I_out(k)*(n_0k+n_2k*I_out(k))*(sin(sqrt(n_2k^2*I_out(k)^2-n_0k^2)...
*(L-Z)))^2)/(n_2k*I_out(k)*cos(2*sqrt(n_2k^2*I_out(k)^2-n_0k^2)*...
(L-Z))-n_0k)+I_out(k);
else
I_in(k)=(I_out(k)*(n_0k+n_2k*I_out(k))*(sinh(sqrt(-n_2k^2*I_out(k)^2+n_0k^2)...
*(L-Z)))^2)/(-n_2k*I_out(k)*cosh(2*sqrt(-n_2k^2*I_out(k)^2+n_0k^2)*...
(L-Z))+n_0k)+I_out(k);
end
end
plot(I_in,I_out)
xlabel(’I_{in}’)
ylabel(’I_{out}’)
62
D.2
General numerical algorithm based on backward
finite difference scheme
clear
L =100;
n_0k= 0.024955; %modulation of linear refractive index
n_2k = 0.0061929; %modulation of nonlinear refractive index
n_nl = 0.080958; %average nonlinear index
N = 400; %number of mesh points for the grating
K =400; %number of points for output intensity
I_start =0; %the starting value for transmitted intensity (|A_{+}(0)|^2)
I_fin =1; %output intensity (|A_{+}(L)|^2)
dz = L/(N + 1); % the space step size
z = 0 : dz : L; % the vector for the grid points between z = 0 and z = L
dI = (I_fin-I_start)/K; %transmitted intensity step size
I_out = (I_start + dI) : dI : I_fin; %the vector for transmitted intensities
% boundary conditions at the right boundary (output intensity)
u(N+2,:) = sqrt(I_out);
w(N+2,:) = zeros(1,K);
v(N+2,:) = zeros(1,K);
y(N+2,:) = zeros(1,K);
for n = (N+1):-1:1
uu = u(n+1,:); ww = w(n+1,:); vv = v(n+1,:); yy = y(n+1,:);
% steady-state coupled mode equations:
u(n,:) = uu + dz*( n_0k*yy + n_nl*(uu.^2+ww.^2+2*vv.^2+2*yy.^2).*ww
n_2k*((uu.^2+3*ww.^2+vv.^2+yy.^2).*yy+2*uu.*ww.*vv));
w(n,:) = ww - dz*( n_0k*vv + n_nl*(uu.^2+ww.^2+2*vv.^2+2*yy.^2).*uu
n_2k*((3*uu.^2+ww.^2+vv.^2+yy.^2).*vv+2*uu.*ww.*yy));
v(n,:) = vv - dz*( n_0k*ww + n_nl*(2*uu.^2+2*ww.^2+vv.^2+yy.^2).*yy
n_2k*((uu.^2+ww.^2+vv.^2+3*yy.^2).*ww+2*uu.*vv.*yy));
y(n,:) = yy + dz*( n_0k*uu + n_nl*(2*uu.^2+2*ww.^2+vv.^2+yy.^2).*vv
n_2k*((uu.^2+ww.^2+3*vv.^2+yy.^2).*uu+2*ww.*vv.*yy));
end
I_in = u(1,:).^2 + w(1,:).^2;
x=linspace(0,1);
plot(I_in,I_out,’k’,x,x,’k-.’);
xlabel(’Incident intensity’)
ylabel(’Transmitted intensity’)
63
+
+
+
+
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