POSITIVE AND NEGATIVE LATERAL GOOS
HÄNCHEN SHIFT IN TRANSMISSION AND
REFLECTION
By
SHAKIL AHMAD
MUHMMAD HANIF
MUHAMMAD SALMAN
DEPARTMENT OF PHYSICS GOVT DEGREE
COLLEGE LALQILLA AFFILIATED WITH
UNIVERSITY OF MALAKAND, DIR (L), PAKISTAN
2022
POSITIVE AND NEGATIVE LATERAL GOOS
HÄNCHEN SHIFT IN TRANSMISSION AND
REFLECTION
By
SHAKIL AHMAD
MUHMMAD HANIF
MUHAMMAD SALMAN
Supervised By
Mr. ATTA ULLAH
Thesis submitted to the Department of Physics University of Malakand Dir
Lower in partial fulfillment of the requirements for the Degree of BS in Physics
DEPARTMENT OF PHYSICS GOVT DEGREE
COLLEGE LALQILLA AFFIEATED WITH
UNIVERSITY OF MALAKAND, DIR (L), PAKISTAN
CERTIFICATE OF APPROVAL
This is to certify that the research work presented in this thesis entitled “Positive
and Negative Lateral Goos Hänchen Shift In Transmission And Reflection”
was conducted by Mr. Shakil Ahmad, Mr. Muhammad Hanif and Mr. Muhammad
Salman under the supervision of Mr. Atta Ullah. This thesis is submitted to the
department of Physics Govt. Degree College Lal Qilla, Dir (Lower) affiliated with
university of Malakand in partial fulfilment of the requirements for the degree of
BS (4 years) in the field of Physics.
Internal Examiner
Signature.......................................
Chairman Department of Physics
Signature........................................
External Examiner
Signature…………………………
Acknowledgement
Foremost, we would like to pay our deepest gratitude to Almighty ALLAH”,
Lord of the galaxies and the earth, creator of this mysterious universe, who
is merciful and worthy of all praises, who is the internal source of
knowledge that has bestowed us with potential to contribute a drop of a few
words in the sea of knowledge gave senses to observe.
Next, we offer our earnest gratefulness to the Holy Prophet Muhammad
(peace be upon Him) who is a source of guidance and knowledge for
humanity.
It is an honor for us to express our deep sense of gratitude to our learned
Supervisor Mr.Atta Ullah Department of Physics, Government Degree
College Lalqilla for his invaluable guidance, constant persuasion and
efficient supervision at each and every stage of this research work. We are
grateful to Mr.Arif Ullah for his support, remarkable suggestions, constant
encouragement, supervision, and cooperation in writing this thesis.
Thanks to, Mr.Abdullah for his thorough guidance during this period of
time. We extend our thanks to all of our respectable Teachers, Colleagues
and Friends. We appreciate greatly the staff of the laboratory, of government
Degree College Lalqilla for their cooperation in the theoretical work. Last
but not the least; we are thankful to our Parents for their prayers and
constant support, without which we would have not completed this work.
Shakil Ahmad
Muhammad Hanif
Muhammad Salman
Table of Contents
ABSTRACT ........................................................................................................ i
CHAPTER 1 INTRODUCTION ..................................................................... 1
CHAPTER 2 LITERATURE REVIEW ......................................................... 3
2.1 Rabi Oscillation .............................................................................................. 3
2.1.1 Vacuum Field Rabi Oscillation ................................................................... 5
2.2 Sodium............................................................................................................ 7
2.3 Permitivity ...................................................................................................... 9
2.3.1 Linear Permittivity ...................................................................................... 9
2.4 Negative And Positive GH-Shifts ................................................................ 11
CHAPTER 3 METHOD AND CALCULATIONS ....................................... 14
CHAPTER 4 RESULTS AND DISCUSSION ............................................... 19
CHAPTER 5 CONCLUSIONS ....................................................................... 27
REFERENCES ................................................................................................. 27
List of Figures
2.1 Rabi oscillation depicted for various detunings ............................................ 9
2.2 Rabi oscillation ............................................................................................... 9
2.4 Calculated effictive permittivity and permeability ......................................... 9
2.5 Shifts in transmission and reflection ............................................................ 19
3.1 Schematic diagram of Sodium atomic system ............................................... 9
4.1(a,b) Absorption and dispersion vs Δ𝑝 ⁄𝛾 and Ω𝑚 ⁄𝛾 ..................................... 9
4.1(c,d) Absorption and dispersion vs Δ1⁄𝛾 and 𝜑 ............................................ 20
4.2(a,b) Transmission and reflection vs Δ𝑝 ⁄𝛾 and Ω𝑚 ⁄𝛾 ................................. 22
4.2(c,d) Transmission and reflection vs Δ1⁄𝛾 and 𝜑.......................................... 22
4.3(a,b) Shifts in transmission and reflection vs Δ𝑝 ⁄𝛾 and Ω𝑚 ⁄𝛾 ...................... 9
4.3(c,d) Shifts in transmission and reflection vs ∆1⁄𝛾 and 𝜑 ............................. 9
Abstract
The lateral positive and negative lateral Goos-Hänchen (GH) Shift in transmission and
reflection beams are investigated in this work using sodium atomic medium. The
transmission, absorption and reflection in sodium medium will be the functions of ∆ 𝑝
probe detuning, control fields Rabi frequencies and collective phase. In certain cases
the sum of absolute values of absorption, transmission and reflection is |T|+|R|+|A| < 1.
This show that |T|+|R|+|A| is less than unit value and have not 100% efficiency. The
remained fraction is due to the shifted beams in reflection and transmission. The
positive and negative shifts are measured in the reflection and transmission beams. The
GH-shift in transmission will be positive and reflection will be negative in the
anomalous dispersion regions, while GH-shift in transmission will be negative and
reflection will be positive in the normal dispersion regions. The results show significant
application in optical sensors and optical waveguide.
i
CHAPTER 1
Introduction
A tiny lateral shift of light beam from incident or emerging point of the medium during
reflection and transmission is known as tiny lateral GH-shifts. The Goos–Hänchen
(GH) shift, being a nonspecular effect, is a lateral displacement of a beam of light from
its predictable geometrical path at the interface of two media having different refractive
indices. The interface between different dielectric materials having small absorption or
transmission allows a small but recognizable shift of the reflected beam [1].This lateral
displacement or shift was first time experimentally observed by Goos and Hänchen in
1947 [2]. They further refined their results by introducing polarization effects of the
light beam in 1949 [3]. Since then a lot of interest has emerged to study the GH shift in
different systems. People have noticed a positive GH shift in total internal reflection
[4], when the two media are right handed [5], when the reflection and refraction of light
at a plane interface [6], and in the multilayered and periodic structures [7]. However, a
relatively more interesting observation is the negative Goos Hänchen shift which has
been observed in absorbing media [8,9] and negative refractive media [10,11]. The
coherent control of the negative and positive GH shifts in the reflected light was
investigated for the first time by Wang et al. in 2008 [12]. Further Ziauddin et al,
considered a two-level atomic medium was in a cavity and control of the negative and
positive GH shift in the reflected light was investigated by modifying the susceptibility
of the atomic medium with an external control field [13]. Since then several proposals
have been investigated using different atomic media inside a cavity [14–18].
1
The Goos Hänchen shift shows certain applications in, optical heterodyne sensors
which can measure various quantities such as beam angle, refractive index,
displacement, temperature, and film thickness [19].Also, phenomenon of GH shift can
be used for the characterization of the permeability and permittivity of the materials
[20]. Because of the potentials applications in integrated optics [21], optical waveguide
switch [22], and optical sensors [23], the GH shifts including other three non specular
effects such as angular deflection, focal shift, and waist-width modification have been
extensively investigated in partial reflection [24-26], attenuated total reflection [27,28],
and frustrated total internal reflection (FTIR) [29-36]. Since then a lot of interest has
emerged to study the GH shift in different systems. People have noticed a positive GH
shift in total internal reflection [37-39], when the two media are right handed [40], when
the reflection and refraction of light at a plane interface [41], and in the multilayered
and periodic structures [42]. However, a relatively more interesting observation is the
negative Goos Hänchen shift which has been observed in absorbing media [43,44] and
negative refractive media [45,46].Artmann explained this effect by considering a beam
with finite width of which the plane-wave components have different transverse wave
vectors. Therefore, the reflected beam is a superposition of all the components, with
each component undergoing a different phase change. As a result, a longitudinal shift
occurs in the plane of incidence [47, 48]. In my best knowledge there is no work present
in the research articles in which the positive and negative tiny lateral GH-Shift in
transmission and reflection beams have been modified in atomic medium by control
fields intensities , their collective phases and detunings . In this work a four level system
is used to modify the positive and negative GH-Shifts simultaneously in transmission
and reflection beams. We show that how to control field intensity, collective phases and
detuning significantly influence the positive and negative GH-Shifts in transmission
and reflection.
2
CHAPTER 2
Literature Review
2.1 Rabi oscillations
Rabi oscillations are a population difference process that occurs at the Rabi frequency
for time scales shorter than the dephasing time. The incident radiations are producing
oscillations between atomic energy states. The induced oscillations was first time
originally studied by Rabi [49, 50]. When the population of a resonant is equal to 𝛿1 =
0, the population alternates between absolute concentration, total inversion, and state.
As the detuning progresses, the oscillation amplitude decreases, while the flopping
frequency rises. Electric and magnetic fields combine in microwave radiation. The
87Rb ground state is split into two sub-levels by the hyperfine interaction 𝐹1 = 1 and 𝐹2
= 2. The angular frequency difference between the ground state levels is zero in the
absence of an external magnetic field.
𝜔1|𝐹1 =1i↔|𝐹2 =2i = 2π · 6, 834, 682, 610 𝐻𝑧
(2.1)
This transition is thought to be very stable, and it could be used as an atomic clock
reference frequency standard.
Ω ≅ 2𝜇̃011 𝐴1 /  =
3
𝐵0

(𝜇𝐵𝑔1𝑠 − 𝜇𝑁𝑔𝐼 )
(2.2)
The intensity of the field can be given in terms of the amplitude I = c𝐵02/2π. Therefore
the Rabi frequency is proportional to √𝐼 and according to the generalised Rabi
frequency, the amplitude of the oscillations decreases as the detuning increases.
Fig.2.1 Rabi oscillations depicted for various detunings. The amplitude of the oscillations decreases as
detuning increases, while the flopping frequency rises.
4
The Rabi oscillations was first time observed in an active wave guide [51]. They can
also be described as a sinusoidal population difference on pulse area dependence (the
time-integrated Rabi frequency).
Fig.2.2 Rabi oscillation
There are many types of Rabi oscillations, one of them are,
2.1.1 Vacuum-field Rabi oscillation
Rabi oscillations of the atoms are induced by the external field, which is well known.
The external field are well known, Rabi oscillations of the atoms are induced by the
external field [52, 53]. For instance, consider the interaction of a two-level system with
states 1 and 2
separated by a plane field of frequency ω. As a result, the probability
𝑝1 of transitioning from excited to ground state is given by
𝑝1 = 𝑠𝑖𝑛2
Ω𝑡
2
2
, Ω = ℏd.𝜀
(2.3)
So the frequency Ω oscillates atomic population between the excited and ground states.
The electric field is proportional to the frequency of oscillation. The most important
and observable effects are the external field induced oscillations in Raman scattering,
5
resonance fluorescence [54]. Extensive research has been done on external fieldinduced oscillations, which are known to cause significant observable effects in
resonance fluorescence, Raman scattering, and other areas.
Fig.2.3 The full curves are for the single-atom case, whereas the dashed curves are for a
cooperative system of two atoms.
The transition probabilities in Eq (2.3) were calculated using a classical treatment of
the external field. Now the question is what happens if the external field is extremely
weak. For example, the field could have a small number of photons. In an empty cavity,
this would undoubtedly be the case. The field e must be treated quantum mechanically
in this case. The interaction of a two-level system with a single mode of the quantized
electromagnetic field of frequency is studied by Jayne and Cummings using a simplemodel Hamiltonian (c). Permanent Rabi oscillations are enabled by a physical
mechanism that dispels exaction polarton condensates in semiconductor micro cavities
subjected to external magnetic fields. The scattering of stimulated exactions from the
6
incoherent reservoir is the basis for this method. Due to the equal time symmetry of the
coupled exaction photon system realised in specific authority of pumping to the
excitons state and depletion of the reservoir, permanent non decomposing oscillations
may occur. At non-zero exciton photons with unequal amplitudes of exciton and photon
components, strong permanent Rabi oscillations occur. At non-zero exciton detuning,
robust proper, stable Rabi oscillations occur with unequal amplitudes of exciton and
photon components. The characteristics of the emitted radiation are determined by the
presence of vacuum-field Rabi oscillations in a variety of atomic-correlation functions
[89].
2.2
Sodium
Sodium is a chemical element with the atomic number 11 and the symbol Na (from
Latin natrium). It's a soft, silvery-white metal with a high reactivity. Because it has a
single electron in its outer shell, which it readily donates, sodium is an alkali metal,
belonging to group 1 of the periodic table. This results in a positively charged ion, the
Na+ cation. It only has one stable isotope, 23Na. All animals and some plants require
sodium to survive. Atomic number 11, atomic weight 22.99, melting point 97.7°C,
boiling point 883°C, specific gravity 0.971, valence 1, melting point 97.7°C, boiling
point 883°C, specific gravity 0.971, valence 1, melting point 97.7°C, boiling point
883°C, specific gravity 0.971, The major cation in the extracellular fluid (ECF) is
sodium, which contributes significantly to the ECF osmotic pressure and compartment
volume.The free metal isn't found in nature, so it has to be made from compounds.
Sodium is the sixth most abundant element in the Earth's crust, and it can be found in
feldspars, sodalite, and rock salt, among other minerals (NaCl). Many sodium salts are
highly water soluble: sodium ions have been leached from the Earth's minerals by the
action of water over aeons, and sodium and chlorine are the most common dissolved
7
elements in the oceans by weight. Titanium, zirconium, and a variety of other chemicals
are made by combining sodium with other metals. It's used to make tetraethyl lead, and
some power plants even use it to cool nuclear reactors in liquid form. Mice and roaches
are poisoned with sodium fluoride, which is also used to make ceramics. Chile salt
pepper, which is the compound sodium nitrate, is also used as a fertiliser. Sodium is
also necessary for maintaining normal fluid balance and other psychological functions
in the body. It's also used to make vapour lamps. Finally, we can conclude that sodium
is a versatile element after looking at its uses. Potassium and sodium compounds are
very similar. They can be substituted for each other because they are so similar. Both
are used in the industry, but sodium is more commonly used due to its lower cost.
Humphry Davy was the first to isolate sodium by electrolysis of sodium hydroxide in
1807. Sodium hydroxide (lye), for example, is used in soap production, and sodium
chloride (edible salt) is a de-icing agent and a nutrient for animals, including humans.
When water is lost from the ECF compartment, the sodium concentration rises, causing
hypernatremia. In a condition known as ECF hypovolemia, isotonic loss of water and
sodium from the ECF compartment reduces the compartment's size. All animals and
some plants require sodium to survive. The major cation in the extracellular fluid (ECF)
is sodium, which contributes significantly to the ECF osmotic pressure and
compartment volume. When water is lost from the ECF compartment, the sodium
concentration rises, causing hypernatremia. In a condition known as ECF hypovolemia,
isotonic loss of water and sodium from the ECF compartment reduces the
compartment's size. Living human cells use the sodium-potassium pump to pump three
sodium ions out of the cell in exchange for two potassium ions pumped in. When
comparing ion concentrations inside and outside the cell membrane, potassium is about
40:1, and sodium is about 1:10. When an electrical charge crosses the cell membrane
of a nerve cell, it allows the nerve impulse an action potential to be transmitted. When
the charge is dissipated, sodium plays an important role [55-57].
8
2.3 Permittivity
The ability of a medium to polarise in response to an electric field determines
permittivity. The absolute permittivity, also known as permittivity and denoted by the
Greek letter (epsilon), is a measure of a dielectric's electric polarizability in
electromagnetism. In response to an applied electric field, a material with a high
permittivity polarises more than a material with a low permittivity, storing more energy
in the electric field. The permittivity of a capacitor is important in electrostatics because
it determines its capacitance. The electric displacement field D resulting from an
applied electric field E is the simplest case.
𝐷 = 𝑒 𝐸. 𝐷 = 𝜀 𝐸. 𝐷 = 𝜀 𝐸
(2.4)
Furthermore, permittivity is a state-dependent thermodynamic function. The lateral
shift of the transmitted beam as it passes through media with varying permittivity.
Periodic change is important in optical and materials engineering, as is well known.
2.3.1 Linear permittivity
The linear permittivity of a homogeneous material is typically expressed as a relative
permittivity 𝜀𝑟 (also called dielectric constant, although this term is deprecated and
often only refers to the static, zero-frequency relative permittivity) (also called
dielectric constant, although this term is deprecated and often only refers to the static,
zero-frequency relative permittivity) (also called dielectric constant, although this term
is deprecated and sometimes only refers to the static. The relative permittivity of an
anisotropic content can be a tensor, resulting in birefringence. After that, multiply the
relative permittivity by 𝜀0 to get the real permittivity.
𝜀0 𝜀 = 𝜀𝑟 .𝜀0 = (1+𝜒) 𝜀0
(2.5)
Where 𝜒 is the material's electric susceptibility. The susceptibility is defined as the
proportionality constant (which can be a tensor) relating an electric field E to the
induced dielectric polarisation density P such that P= 𝜀0 𝜒E, where 𝜀0 is the free space
9
electric permittivity. A medium's susceptibility is proportional to its relative
permittivity 𝜀𝑟 by the formula 𝜒=𝜀𝑟 − 1 [58, 59]. The description of effective material
parameters permittivity and permeability for composite layers containing only one-two
parallel arrays of complex shaped inclusions is critical for the design of novel
metamaterials, as the realizable layers frequently have only one or two layers of
particles around the sample thickness. The averaged induced polarizations are
described by effective parameters. For the normal plane-wave occurrence, the electric
and magnetic dipole moments induced in the structure, as well as the resulting reflection
and transmission coefficients, are determined using the local field approach, and
effective parameters are introduced into the averaged fields and polarizations [60].
Fig.2.4 Calculated effective permittivity and permeability.
10
The
permittivity
for
a
binary
gas
mixture,
𝜀𝐸
is
𝜀 𝐸 = 𝜖𝑚 - (𝜒1 𝜀1 +𝜒2 𝜖2 )
as
follows:
(2.6)
Where 𝜀𝑚 denotes the mixture's permittivity and 𝜒1 and 𝜀1 denote the mole fraction and
permittivity of the variable I respectively.
2.4
Negative and positive Goos Hänchen shifts in transmission and
reflection
Over 5000 massive negative and positive images Changes in the Goos-Hanchen (GH)
Shifts When a beam is fully reflected from a dielectric-decorated substrate only grating,
operating wavelength times are measured [61]. A lateral shift exists between the
reflected and incident beams when a beam experiences total internal reflection. Named
after its discoverers Goos and Hänchen. New materials have become active, such as
weakly absorbing dielectrics [62], metal dielectric composites, inclined uniaxial
crystals, and negative index metamaterials [63-65], and GH changes have resurfaced.
Wan et al. [66] were successful. A large positive GH shift occurs when surface Bloch
waves are excited. These engineering applications pique people's interest in this
phenomenon beyond scientific curiosity. Because the sensitivity of these sensors is
proportional to the amount of GH shift, achieving a large (positive or negative) GH
shift is of practical interest [67,68]. The negative GH shift [69] is used to propose
designs of metamaterial waveguides that support a "frozen mode," a waveguide mode
with no net energy propagation. When left-handed media are used, positive GooseHänchen Shifts are more common than negative Shifts. The negative Goose-Hänchen
shifts occurs when a wave (here a Gaussian beam) represents a boundary between an
isotropic right arm half-spot and an isotropic left arm half-spot [70]. The GH-Shifts in
transmission and reflection can change from positive to negative by increasing the
wavelength. These negative and positive GH-Shifts can also be enhanced by
transmission vibrations when the frequency in the DP is far away [71].
11
d p, s
𝐺𝐻 =  0
2 d
𝐺𝐻 = −
d Im[rp , s ]
dRe [rp , s ]
0 1
{
R
[
r
]

Im[
r
]
}
e
p
,
s
p
,
s
2 r 2
d
d
p,s
(2.7)
(2.8)
r p , s and  p, s they correspond to the Fresnel reflection coefficient's modulus and phase.
Low-loss photonic crystal (PC) mirrors exhibit positive and negative Goos-Hänchen
shifts due to the strong angular and wavelength dependencies of their reflected phase
(GHS). The presence of large positive and negative GHS in PC mirrors is revealed in
this letter using theoretical, numerical, and experimental methods. For angle-shifting
blue (red) resonances, a simple algebraic relationship shows that positive effective
thickness produces positive (negative) GHS, while the opposite is true for negative
effective thickness interfaces [72].
12
Fig.2.5 Shifts in transmission and reflection
Significant negative GH shifts were obtained in the reflected probe light beam (solid
line) and positive GH shifts in the transmitted light at certain incident angles, as shown
in Fig. [73-78].
13
CHAPTER 3
Method & Calculations
Our proposed four level sodium atomic system is shown in Fig.(3.1) .In this
configuration the ground state |1⟩, state |2⟩ and state |3⟩ resided below the upper
excited state |4⟩.The lower energy level |1⟩ is coupled to the upper energy level |4⟩
by a probe field 𝐸𝑝 having Rabi frequency Ω𝑝 .State |2⟩ is connected with levels |3⟩
and |4⟩ by a magnetic field 𝐸𝑚 having Rabi frequency Ω𝑚 and a control field 𝐸2
having Rabi frequency Ω2 respectively. State |3⟩ is coupled to excited level |4⟩ with
control field 𝐸1 having Rabi frequency Ω1 .The corresponding decay rates
are 𝛾41,𝛾32,𝛾42 and 𝛾43 respectively.
Fig.3.1 Schematic diagram of four-level Sodium atomic system
14
The optical response of the proposed atomic system subjected to the probe field and
the three control fields. The pure Hamiltonian for the system can be written is
𝐻0 = ℏ𝜔1 |1⟩⟨1| +ℏ𝜔2 |2⟩⟨2| +ℏ𝜔3 |3⟩⟨3| +ℏ𝜔4 |4⟩⟨4|
(1)
To discuss the system dynamics the interaction picture Hamiltonian of sodium atomic
medium is proceeded in the dipole and rotating wave approximations:
ℏ
𝐻𝑖 = − 2 [Ω1 𝑒 −𝑖Δ1 𝑡 |3⟩ ⟨4| + Ω2 𝑒 −𝑖Δ2 𝑡 |2⟩ ⟨4| + Ω𝑚 𝑒 −𝑖Δ𝑚𝑡 |2⟩ ⟨3| +
Ω𝑝 𝑒 −𝑖Δ𝑝 𝑡 |1⟩ ⟨4| ] + 𝐻. 𝐶
(2)
The density matrix equation is used for the time evaluation of the proposed atomic
system is written as
𝑑
𝑖
1
𝜌 = − [𝐻𝑖𝑛𝑡 , 𝜌] − ∑ 𝛾𝑖𝑗 (−2𝑏𝜌𝑏 † + 𝑏 † 𝑏𝜌 + 𝜌𝑏 † 𝑏)
𝑑𝑡
ℏ
2
(3)
Where the 𝜌 are the density-matrix-operator elements. 𝑏 † and b are the ladder up and
down operators respectively and 𝛾𝑖𝑗 (𝑖, 𝑗 = 1,2,3,4) represents decay rates. The
explicitly time independent coupling-rate equations are the following:
.
𝑖 ∼
𝑖 ∼
𝑖
∼
∼
∼
∼
𝜌14 = 𝐴1 𝜌14 − Ω1 𝜌13 − Ω2 𝜌12 + Ω𝑝 (𝜌44 − 𝜌11 )
2
2
2
(4)
.
∼
𝜌13
𝑖 ∼
𝑖
𝑖
∼
∼
∼
= 𝐴2 𝜌13 − Ω1∗ 𝜌14 − Ω𝑚 𝜌12 + Ω𝑝 𝜌43
2
2
2
(5)
.
∼
𝜌12
𝑖 ∼
𝑖
𝑖
∼
∼
∼
= 𝐴3 𝜌12 − Ω∗2 𝜌14 − Ω∗𝑚 𝜌13 + Ω𝑝 𝜌42
2
2
2
(6)
Where
1
𝐴1 = 𝑖Δ𝑝 − 𝛾41
2
(7)
1
𝐴2 = 𝑖(Δ𝑝 − Δ1 ) − 2 (𝛾43 + 𝛾41 )
1
𝐴3 = 𝑖(Δ𝑝 − Δ2 ) − 2 (𝛾42 + 𝛾32 + 𝛾41 )
15
(8)
(9)
As initially, the atoms are populated in the ground state |1⟩, this implies that its
∼ (0)
density element 𝜌11 = 1. Therefore, initially the population of atoms in excited states
∼ (0)
∼ (0)
∼ (0)
are assumed to be zero. Therefore, 𝜌44 = 𝜌43 = 𝜌42 = 0. Employing the first-order
perturbation approximations, we can solve density matrix equations with the help of
the following equation
𝑋(𝑡) = 𝑌 −1 𝑀
(10)
Where 𝑋(𝑡) and 𝑀 are column matrices, and 𝑌 is a 3 × 3 matrix.
∼
The calculated probe coherence term 𝜌14 is given by
∼
𝜌14
𝑖(4𝐴2 𝐴3 + Ω2𝑚 )Ω𝑝
=
(11)
2(𝐴3 Ω12 + 𝐴2 Ω22 + 𝐴1 (4𝐴2 𝐴3 + Ω2𝑚 ) + 𝑖Ω1 Ω2 Ω𝑚 cos(𝜙1 − 𝜙2 + 𝜙𝑚 ))
The complex susceptibility for the proposed four-level sodium atomic system is
determined as:
𝜒=
2𝑁℘214 ∼ (1)
𝜌
𝜖0 ℏΩ𝑝 14
(12)
Where 𝑁 represents atomic density.
The reflection and transmission coefficients are the following:
𝑅=
(𝛽02 −𝛽12 )𝛽1 𝛽2 sin 2𝛼1 cos(𝛼2 )+𝑣3 sin(𝛼2 )
𝛽1 𝛽2 𝑣1 cos(𝛼2 )+𝑣2 sin(𝛼2 )
2𝑖𝛽0 𝛽2 𝛽12
1 𝛽2 𝑣1 cos(𝛼2 )+𝑣2 sin(𝛼2 )
𝑇=𝛽
16
(13)
(14)
Such that
𝑣1 = 2𝑖𝛽0 𝛽1 cos 2𝛼1 + (𝛽02 + 𝛽12 ) sin 2𝛼1
𝑣2 = 𝛽12 (𝛽02 + 𝛽22 ) cos 2 𝛼1 − (𝛽14 + 𝛽02 𝛽22 ) sin2 𝛼1 − 𝑖𝛽0 𝛽1 (𝛽12 + 𝛽22 ) sin 2𝛼1
𝑣3 = 𝛽12 (𝛽02 − 𝛽22 ) cos2 𝛼1 + (𝛽14 − 𝛽02 𝛽22 ) sin2 𝛼1
Where
𝜖2 = 1 + 𝜒, 𝛽0,1 = √𝜖0,1 − sin2 𝜃 , 𝛽2 = √𝜖2 − sin2 𝜃,
𝛼1 =
2𝜋
𝜆𝑝
𝑑1 √𝜖1 − sin2 𝜃, 𝛼2 =
2𝜋
𝜆𝑝
𝑑2 𝛽2.
The GH shifts in reflected and transmitted probe field as written as:
𝜆
𝜕
𝜕
𝑆𝑅 = − 2𝜋|𝑅|2 [𝑅𝑒(𝑅) 𝜕𝜃 (𝐼𝑚(𝑅)) − 𝐼𝑚(𝑅) 𝜕𝜃 𝑅𝑒(𝑅)]
𝜆
𝜕
𝜕
𝑆𝑇 = − 2𝜋|𝑇|2 [𝑅𝑒(𝑇) 𝜕𝜃 (𝐼𝑚(𝑇)) − 𝐼𝑚(𝑇) 𝜕𝜃 𝑅𝑒(𝑇)]
17
(15)
(16)
CHAPTER 4
Results & Discussion
The absorption, dispersion, reflection, transmission and corresponding GH shift are
controlled and modified while using sodium atomic medium tapped in a cavity. The
decay rates 𝛾 = 1𝐺𝐻𝑧 and other parameters of frequency are scaled to this decay rate
𝛾. Furthermore other units are taken in atomic units. The permittivity of free space is
𝜖0 , permittivity of cavity medium is 𝜖1 and permittivity of sodium medium is 𝜖2 = 1 +
𝜒 while 𝑑1 = 1𝑐𝑚 , 𝑑2 = 1.5𝑐𝑚, collective phase of control field Rabi frequency 𝜑 =
𝜋
𝜑1 − 𝜑2 + 𝜑3 the other parameters are𝛾41,42,43,32 = 2𝛾, 𝜑 = 2 , Δ1,2 = 𝑜𝛾 ,𝜃 =
𝜋
4
𝑎𝑛𝑑
Δ𝑝 = 0.5𝛾. Rabi frequency Ω1 = |Ω1 |𝑒 𝑖𝜑1 , Ω1∗ = |Ω1 |𝑒 −𝑖𝜑1 ,
Ω2 = |Ω2 |𝑒 𝑖𝜑2 , Ω∗2 = |Ω2 |𝑒 −𝑖𝜑2 , And Ω𝑚 = |Ω𝑚 |𝑒 𝑖𝜑𝑚 , Ω∗𝑚 = |Ω𝑚 |𝑒 −𝑖𝜑𝑚 .
In Fig.4.1 (a,b) the plots are traced for absorption and dispersion spectrum versus probe
detuning ∆𝑝 ⁄𝛾 and control filed Rabi frequency Ω𝑚 ⁄𝛾 . The Rabi frequency |Ω1 | is
varied from 40𝛾 to 30𝛾 and then to 20𝛾 stepwise. The absorption is minimized at
around resonance point Δ𝑝 = 0𝛾 in the range −25 ≤ Δ𝑝 ≤ 25𝛾 and at the control field
Rabi frequency range 0𝛾 ≤ Ω𝑚 ≤ 20𝛾 . As the control field Rabi frequency |Ω1 |
decreased from 40𝛾 to 30𝛾 and then to 20𝛾, the absorption becomes zero and the
transparency width decreased around the resonance point.
18
Fig.4.1 (a,b) Absorption and dispersion vs Δ𝑝 ⁄𝛾 and control field Rabi frequency Ω𝑚 ⁄𝛾 such that Ω2 =
2𝛾, |Ω1 | = 40𝛾, 30𝛾, 20𝛾. (c,d) Absorption and dispersion vs Δ1 ⁄𝛾 and collective phase 𝜑 such that
Δ2 = 0𝛾, 2𝛾, −2𝛾.
The doublet absorption peaks occur at the range of probe detuning ±20𝛾 ≤ Δ𝑝 ≤ ±40𝛾
and with control field Rabi frequency at the range of 0𝛾 ≤ Ω𝑚 ≤ 20𝛾 when |Ω1 | =
40𝛾. These peaks are shifted towards the resonance point ∆𝑝 = 0𝛾 as |Ω1 | decrease to
30𝛾 and then to 20𝛾, further the doublet absorption peaks become lower with
decreasing the intensity of control field Rabi frequency |Ω1 | as shown in Fig.4.1 (a).
The slope of dispersion is normal in the transparency region where the absorption
minimum and anomalous in the absorption peaks regions. The transparency region is
19
at resonance point Δ𝑝 = 0𝛾 and around the resonance point at the range −25 ≤ Δ𝑝 ≤
25𝛾. As the intensity of control field Rabi frequency |Ω1 | decreases the transparency
width decreases and hence increases the sharpness of normal dispersion. The normal
dispersion region is associated to positive group index, positive group velocity and
delay time in the medium. The anomalous dispersion region is associated to negative
group index, negative group velocity and advance time in the medium. The normal and
anomalous behaviors of dispersion is fluctuated with the intensity of control field Rabi
frequency Ω𝑚 ⁄𝛾 as shown in the Fig.4.1 (b).
Fig.4.1 (c,d) shows the absorption and dispersion spectrums with Δ1/𝛾 and collective
phase 𝜑 of the control fields. In this condition the detuning Δ2 is varied from Δ2 = 0𝛾
to ±2𝛾. The absorption is minimized at resonance point Δ1= 0𝛾 of control field. In this
case the control field detuning Δ1 variation is in the range of -20𝛾 ≤ Δ1 ≤ 20𝛾 and
collective phase variation of 0 ≤ 𝜑 ≤ 2𝜋. As the detuning Δ2 is varied from 0𝛾 to ±2𝛾
the absorption decreases to 50% and the transparency width is decreased around the
resonance point of control field detuning Δ1 = 0𝛾. The absorption maxima is shifted
towards the resonance point Δ1 = 0 𝛾 with detuning Δ2 . Further the absorption is
fluctuated with the collective phase 𝜑 of the control fields as shown in Fig.4.1 (c). The
dispersion is anomalous in the absorption region and normal in the in the transparency
region of control field detuning Δ1= 0𝛾. The positive and negative values of control
field detuning Δ2 increase the steepness of anomalous dispersion and hence enhance
superluminality at low absorption. The normal and anomalous behaviors of dispersion
is fluctuated with the collective phase 𝜑 of control fields as shown in Fig.4.1 (d).
20
Fig.4.2 (a,b) Transmission and reflection vs ∆𝑝 ⁄𝛾 and control field Rabi frequency Ω𝑚 ⁄𝛾 such
that Ω2 = 2𝛾, |Ω1 | = 40𝛾, 30𝛾, 20𝛾, (c,d) Transmission and reflection vs Δ1 ⁄𝛾 and collective phase 𝜑
such that Δ2 = 0𝛾, 2𝛾, − 2𝛾 the other parameters are 𝛾41,42,43,32 = 2𝛾, 𝜑 = 0,2𝜋.
Fig.4.2 (a,b) shows the absolute values of transmission and reflection coefficients
versus probe field detuning ∆𝑝 ⁄𝛾 and control field Rabi frequency Ω𝑚 ⁄𝛾. In this
condition the variation in the Rabi frequency |Ω1 | is from 40𝛾 to 30𝛾 and then 20𝛾.
The sum of the absolute values of reflection and transmission approaches to unit value
at the region of transparency such that |𝑇| + |𝑅|~1, if |𝐴|~0. In other words |𝑇| +
|𝑅| + |𝐴| = 1. At the region of ∆𝑝 = 0𝛾 the absorption coefficient |𝐴|~0 see Fig.4.1
(a), the transmission coefficient |𝑇|~60% reflection coefficient |𝑅|~40% see in
21
Fig.4.2 (a,b). At the other point of probe detuning and control field Rabi frequency the
same normalization condition |𝑇| + |𝑅| + |𝐴| = 1 is satisfied as shown in Fig.4.2
(a,b).
In Fig.4.2 (c,d) The plot are traced for absolute values of transmission and reflection
coefficients vs detuning field Δ1 / 𝛾 and control field collective phase 𝜑 . In this
condition Δ2 is varied from Δ2 = 0𝛾 to ±2𝛾. The absorption coefficient is minimized to
|A| ∼ 50% at resonance point Δ1=0𝛾see Fig.4.1 (c) and the transmission coefficient |T|
∼ 3% see Fig .4.2(c) reflection coefficient |R| ∼ 36% see Fig.4.2 (d). The remaining ∼
11% are the GH-shifted reflection and transmission beams. The sum of absolute values
of absorption, transmission and reflection is approached to unit value at the region of
transparency such as |T| + |R| + |A| = 1. As the detuning Δ2 is varied from 0𝛾 to ±2𝛾 the
absorption decreases to 50% and the transparency width is decreased around the
resonance point of control field detuning Δ1 = 0𝛾 and the widths of transmission,
reflection is also decreases. As the absorption, reflection decreases and minimized to
small value at the resonance point transmission is increased at this point see in
Fig.4.1(c) and Fig.4.2 (c,d). At the other point of detuning Δ1 /𝛾 and control field
collective phase 𝜑 the same normalization condition |T| + |R| + |A| = 1 is satisfied as
shown in Fig.4.1 (c) and Fig.4.2 (c,d).
In Fig.4.3 (a,b) the plots are traced for GH shift in transmission 𝑆𝑇 ⁄𝜆 and reflection
𝑆𝑅 ⁄𝜆 beams. The GH shift in transmission and reflection beams are the function of
probe field detuning Δ𝑝 ⁄𝛾 and control field Rabi frequency Ω𝑚 ⁄𝛾 . Positive and
negative shifts are investigated in the transmission and reflection beams. The GH shift
in transmission is negative and reflection is positive in the normal dispersion regions,
where absorption is minimum. The GH shift in transmission is positive and reflection
is negative in the anomalous dispersion regions, where absorption is maximum. The
negative GH shift in transmission and positive GH shift in reflection are shifted towards
22
the resonance point ∆𝑝 = 0𝛾 with the intensity of control field Rabi frequency Ω𝑚 ⁄𝛾
and fluctuated with intensity of control field |Ω1 |⁄𝛾 as shown in Fig.4.3 (a,b).
The GH-shifts in transmission and reflection beams are the function of detuning Δ1/γ
and control field collective phase 𝜑. The GH-shift in transmission is positive value at
Δ2 = 0γ, 2γ in the range of -20γ ≤ Δ1≤-3γ. The GH-shift in transmission decreases from
positive to negative value at Δ2 =2γ in the range of -20γ ≤ Δ1 ≤3γ. The GH-shift in
transmission suddenly reduce to negative value at Δ2 = 0γ, ±2γ in Δ1= 3γ. The negative
value of GH-shift in transmission is continually remained in the range of 3γ ≤ Δ1≤ 5γ
and 𝜑 ≤ 2 radian. The GH-shift continually increases from negative to positive value as
3γ ≤ Δ1≤ 20γ at all detuning of Δ2 =0γ, ±2γ and gradually fluctuated with the phase 𝜑.
In the ranges of Δ1 =-5γ and 2≤ 𝜑 ≤ 5 there is negative hole in the GH-shift in
transmission beam. Further the GH-shift in transmission oscillates between +2λ
and−4𝜆. The GH-shift in reflection is negative value in the range of 20γ ≤ Δ1 ≤ 20γ and
Δ1= 0γ, ±2γ. The reflection shift is minimum near the resonance point control field and
have value of 0.6λ. GH-shift in reflection fluctuated with the phase 𝜑 of control field
as shown in Fig.4.3 (c,d).
23
Fig.4.3 (a,b) Shifts in Transmission and reflection vs ∆𝑝 ⁄𝛾 and control field Rabi frequency Ω𝑚 ⁄𝛾 such
that Ω2 = 2𝛾, |Ω1 | = 40𝛾, 30𝛾, 20𝛾, (c,d) Shifts in Transmission and reflection vs Δ1 ⁄𝛾 and collective
phase 𝜑 such that Δ2 = 0𝛾, 2𝛾, − 2𝛾 the other parameters are 𝛾41,42,43,32 = 2𝛾, 𝜑 = 0,2𝜋.
24
CHAPTER 5
Conclusions
The positive and negative lateral Goos-Hänchen (GH) Shifts in transmission and
reflection beams are investigated in this research work using sodium atomic medium.
It is noted that the absorption, reflection and transmission in sodium medium are the
functions of probe detuningΔ𝑝 , control fields Rabi frequencies |Ω1,2,𝑚 |, collective phase
𝜑 of control fields. It is reported that the sum of absolute values of absorption,
transmission and reflection is approached to unit value at the region of transparency
such as |T|+|R|+|A| = 1. In certain cases the sum of absolute values of absorption,
transmission and reflection is |T|+|R|+|A| < 1. This show that |T|+|R|+|A| is less than
unit and have not 100% efficiency. The remained fraction is due to the shifted beams
in reflection and transmission. The lateral GH-shifts in corresponding reflection and
transmission beams are strongly influenced by these parameters. The positive and
negative shifts are measured in the reflection and transmission beams. The GH-shift in
transmission is positive and reflection is negative in the anomalous dispersion regions,
where absorption is minimum. The GH-shift in transmission is negative and reflection
is positive in the normal dispersion regions, where absorption is maximum. The results
show potential application in optical sensing, measurement of beam angle, refractive
index, irregularities and roughness of the surface of a medium, quantum mechanics,
plasma physics, acoustics, micro-optics and Nano-optics.
25
References
[1]
H.Iqbal, M.Idrees, M.Javed, B.A Bacha, S.Khan, & S.A Ullah. “Goos–Hänchen
Shift from Cold and Hot Atomic Media Using Kerr Nonlinearity”. J. Russian
Laser Research, 38(5), 426-436, (2017).
[2]
F.Goos & H. Hänchen. “A new and fundamental experiment on total reflection”.
J. Ann. Phy. (Leipzig), 1(7-8), 333-346. (1947).
[3]
F.Goos & H.Hänchen. “Neumessung des strahlversetzungseffektes bei
totalreflexion”. J. Annalen der Phy, 440(3), 251-252. (1949).
[4]
H.Rémi. "Total Reflection: A New Evaluation of the Goos–Hänchen Shift," J.
Opt. Soc. Am. 54, 1190-1197, (1964).
[5]
D.Q-Kui & C.Gang. "Goos–Hänchen shifts at the interfaces between left- and
right-handed media," J. Opt. Lett. 29, 872-874, (2004).
[6]
K.Helmut, V. Lotsch. "Reflection and Refraction of a Beam of Light at a Plane
Interface," J. Opt. Soc. Am. 58, 551-561, (1968).
[7]
T.Tamir & H.L.Bertoni. "Lateral Displacement of Optical Beams at
Multilayered and Periodic Structures," J. Opt. Soc. Am. 61, 1397-1413, (1971).
[8]
S.Chu, & S.Wong. “Linear pulse propagation in an absorbing medium”. J. Phy.
Rev.Lett. 48(11), 738, (1982).
[9]
J.L.Birman, D.N.Pattanayak & A.Puri, “Prediction of a resonance-enhanced
laser-beam displacement at total internal reflection in semiconductors”. J. Phy.
Rev.Lett. 50(21), 1664, (1983).
[10]
P.R.Berman “Goos-Hänchen shift in negatively refractive media”. J. Phy.Rev.
E, 66(6), 067603, (2002).
26
[11]
A.Lakhtakia, “On planewave remittances and Goos–Hänchen shifts of planar
slabs with negative real permittivity andpermeability”. Electromagnetics, 23(1),
71-75, (2003).
[12]
L.G.Wang, M.Ikram & M.S.Zubairy, “Control of the Goos-Hänchen shift of a
light beam via a coherent driving field”.J. Phy.Rev. A, 77(2), 023811, (2008).
[13]
C.Ziauddin, & R.K.Lee, “Negative and positive Goos-Hänchen shifts of
partially coherent light fields”. J. Phy.Rev. A, 91(1), 013803, (2015).
[14]
S.Qamar, & M.S.Zubairy, “Coherent control of the Goos-Hänchen
shift”.J. Phy.Rev. A, 81(2), 02382, (2010).
[15]
S.Qamar, “Gain-assisted control of the Goos-Hänchen shift”.J.Phy.Rev.
A, 84(5), 053844, (2011).
[16]
S.Qamar, “Control of the Goos-Hänchen shift using a duplicated two-level
atomic medium”.J. Phy.Rev. A, 85(5), 055804, (2012).
[17]
Q.S.Ziauddin, & M.Abbas, “Amplitude control of the Goos-Hänchen shift via
Kerr nonlinearity”. J.Laser Phy.Lett. 11, 015201, (2014).
[18]
S.Qamar, “Effect of width of incident Gaussian beam on the longitudinal shifts
and distortion in the reflected beam”.J.Opt.Communications. 319, 1-7, (2014).
[19]
T.Hashimoto & T.Yoshino, "Optical heterodyne sensor using the Goos–
Hänchen shift," J. Opt.Lett. 14, 913-915, (1989).
[20]
H.Xiaolong, H.Yidong, W.Zhang, D.K. Qing & P.Jiangde, "Opposite Goos–
Hänchen shifts for transverse-electric and transverse-magnetic beams at the
interface associated with single-negative materials," J.Opt.Lett. 30, 899-901,
(2005).
[21]
H.K.Lotsch, “Beam displacement at total reflection: The Goos-Hanchen effect
I”. J.Optik. 32(2), 116-137, (1970).
27
[22]
T.Sakata, H.Togo, & F.Shimokawa, “Reflection-type 2× 2 optical waveguide
switch using the Goos–Hänchen shift effect”. J.App.Phy.Lett. 76(20), 28412843, (2000).
[23]
C.W.Chen, W.C.Lin, L.S.Liao, Z.H.Lin, H.P.Chiang, P.T.Leung & W.S.Tse,
“Optical
temperature
sensing based
on the Goos-Hänchen
effect”.
J.App.Opt. 46(22), 5347-5351, (2007).
[24]
C.W,Hsue, & T.Tamir, “Lateral displacement and distortion of beams incident
upon a transmitting-layer configuration”. J.OSA. A, 2(6), 978-988, (1985).
[25]
R.P.Riesz, & R.Simon, “Reflection of a Gaussian beam from a dielectric
slab”. J.OSA. A, 2(11), 1809-1817, (1985).
[26]
T.Tamir, “Nonspecular phenomena in beam fields reflected by multilayered
media”. J.OSA. A, 3(4), 558-565. (1986).
[27]
C.F.Li, “Negative lateral shift of a light beam transmitted through a dielectric
slab and interaction of boundary effects”. J.Phy.Rev.letters, 91(13), 133903,
(2003).
[28]
D.Müller, D.Tharanga, A.A.Stahlhofen, & G.Nimtz, “Nonspecular shifts of
microwaves in partial reflection”. J. EPL.73(4), 526, (2006).
[29]
X.Chen, C.F.Li, R.R.Wei & Y.Zhang, “Goos-Hänchen shifts in frustrated total
internal reflection studied with wave-packet propagation”. J.Phy.Rev. A, 80(1),
015803, (2009).
[30]
X.Yin, L.Hesselink, Z.Liu, N. Fang & X.Zhang “Large positive and negative
lateral optical beam displacements due to surface plasmon resonance”.
J. App.Phy.Lett. 85(3), 372-374. (2004).
[31]
F.Pillon, H.Gilles, S.Girard, M.Laroche, R.Kaiser, & A.Gazibegovic, “GoosHaenchen and Imbert-Fedorov shifts for leaky guided modes”. J. OSA B, 22(6),
1290-1299, (2005).
28
[32]
J.J.Cowan, & B,Aničin, “Longitudinal and transverse displacements of a
bounded microwave beam at total internal reflection”. J.OSA. 67(10), 1307131, (1977).
[33]
J.L.Agudin, & A.M.Platzeck, “Trajectories of the reflected and transmitted
beams in the oblique tunnel effect”. J. Journal of Optics, 9(3), 187, (1978).
[34]
A.K.Ghatak, M.R.Shenoy, I,C.Goyal, & K.Thyagarajan, “Beam propagation
under frustrated total reflection”. J.Opt.Communications. 56(5), 313-317,
(1986).
[35]
A.Haibel, G.Nimtz, & A.A.Stahlhofen, “Frustrated total reflection: The doubleprism revisited”. J. Phy.Rev. E, 63(4), 047601, (2001).
[36]
J.Broe, & O.Keller, “Quantum-well enhancement of the Goos–Hänchen shift
for p-polarized beams in a two-prism configuration”. J.OSA. A, 19(6), 12121222, (2002).
[37]
T.Hashimoto, & T.Yoshino, “Optical heterodyne sensor using the Goos–
Hänchen shift”. J.Opt.lett, 14(17), 913-915, (1989).
[38]
T.Hashimoto, & T.Yoshino, “Optical heterodyne sensor using the Goos–
Hänchen shift”.J.Opt.lett, 14(17), 913-915, (1989).
[39]
J.L.Birman, D.N.Pattanayak, & A.Puri, “Prediction of a resonance-enhanced
laser-beam displacement at total internal reflection in semiconductors”.
J.Phy.Rev. Lett, 50(21), 1664, (1983).
[40]
H.R.Rémi, "Total Reflection: A New Evaluation of the Goos–Hänchen Shift,"
J.Opt.Soc. Am. 54, 1190-1197, (1964).
[41]
D.K.Qing & G.Chen, "Goos–Hänchen shifts at the interfaces between left- and
right-handed media," J.Opt.Lett. 29, 872-874, (2004).
[42]
K.V.L.Helmut, "Reflection and Refraction of a Beam of Light at a Plane
Interface," J.Opt.Soc.Am. 58, 551-561, (1968).
29
[43]
T.Tamir & H.L.Bertoni, "Lateral Displacement of Optical Beams at
Multilayered and Periodic Structures". J.Opt.Soc.Am. 61, 1397-1413, (1971).
[44]
S.Chu, & S.Wong, “Linear pulse propagation in an absorbing medium”.
J.Phy.Rev. lett, 48(11), 738, (1982).
[45]
J.L.Birman, D.N.Pattanayak, & A.Puri, “Prediction of a resonance-enhanced
laser-beam displacement at total internal reflection in semiconductors”.
J.Phy.Rev. Lett, 50(21), 1664, (1983).
[46]
P.R.Berman, “Goos-Hänchen shift in negatively refractive media”. J. Phy.Rev.
E, 66(6), 067603, (2002).
[47]
A.Lakhtakia, “On planewave remittances and Goos–Hänchen shifts of planar
slabs
with
negative
real
permittivity
and
permeability”. Electromagnetics, 23(1), 71-75, (2003).
[48]
K.Artmann, “Berechnung der Seitenversetzung des totalreflektierten Strahles”.
J. Annalen der Physik, 437(1-2), 87-102, (1948).
[49]
I. I. Rabi, "Space Quantization in a Gyrating Magnetic Field".J.Phy. Rev. 51
652, (1937).
[50]
H.Choi, V.M.Gkortsas, L.Dieh, D.Bour, S.Corzine, J.Zhu, G.Ho¨flfler,
F.Capasso, and T. B. Norris," Ultrafast Rabi, flopping and coherent pulse
propagation in a quantum cascade laser".J. Nat. Photon. 4, 706 (2010).
[51]
L. Allen and J. H. Eberly, “Optical Resonance and Two Level Atoms”. Wiley,
New York 58, (1975).
[52]
S. Swain, in “Advances in Atomic and Molecular Physics”, D. R. Bates and B.
Bederson, eds. (Academic, New York), Vol. 16, p. 159; P. Knight and
PMiloni, Phy. Rep. 6621, (1980).
30
[53]
S. Swain, in “Advances in Atomic and Molecular Physics”, D. R. Bates
and B. Bederson, eds. Academic, New York, Vol. 16, p. 159; P. Knight
and P.Miloni, Phy.Rep. 6621, (1980).
[54]
P.S.Linsay, "Period Doubling and Chaotic Behavior in a Driven Anharmonic
Oscillator". Phys.Rev.Lett. 47, 1349, (1981).
[55]
C.Zhang, Q.Cheng, J.Yang, J.Zhao, and T.J.Cui, "Broadband metamaterial for
optical transparency and microwave absorption" J.App.Phy.Lett. 110, 143511,
(2017).
[56]
Meija, Juris; "Atomic weights of the elements” Pure and Applied
Chemistry. 88 (3): 26591, (2016).
[57]
‘Magnetic susceptibility of the elements and inorganic compounds”, in
D.R.Lide, ed. CRC Handbook of Chemistry and Physics (86th ed), (2005).
[58]
O.Schmidt, R.Wynands, Z.Hussein, and D. Meschede," Steep dispersion and
group velocity below c /3000 in coherent population trapping". Phy.Rev.A,
vol. 53, P. R27, (1996).
[59]
D.J.Goorskey, H.Wang, W.H.Burkett & Min Xiao. J. modern optics, vol. 49,
PP. 305–317, (2002).
[60]
M.M.Hawkeye and M. J. Brett, "Narrow bandpass optical filters fabricated
with one-dimensionally periodic inhomogeneous thin films" J.App.Phy. 100,
044322, (2006).
[61]
S.Mukamel, "Roadmap on quantum light spectroscopy" J. Phy. B: At., Mol.
Opt.Phys. 53, 072002, (2020).
[62]
R.Feynman, “Theres plenty of room at the bottom”. Caltech Eng. Sci, 23, 2236, (1960).
31
[63]
R.Yang, W.Zhu, and J.Li," Giant positive and negative Goos-Hänchen shift on
dielectric gratings caused by guided mode resonance" J.The optical society
(OSA), Vol. 22, No. 2, (2014).
[64]
L.Wang, H.Chen, S.Zhu, "Large negative Goos-H€ anchen shift from a
weakly absorbing dielectric slab".J.Opt.Lett. 30 (21) 2936-2938 (2005).
[65]
B.Zhao, L.Gao, "Temperature-dependent Goos-H€ anchen shift on the
interface of metal/dielectric composites".J.Optic Express 17 (24), PP. 2143321441, (2009).
[66]
X.Wu, "Goos-Hanchen shifts in tilted uniaxial crystals".J.Optic.Commun. 416
2018 181-184, (2018).
[67]
L.Ju, B.Geng, J.Horng, C.Girit, M.Martin, Z.Hao, "Graphene plasmonics for
tunable terahertz metamaterials", Nat. Nanotechnol. 6 (10), (2011).
[68]
Y.Wan, Z.Zheng, W.Kong, X.Zhao, Y.Liu, Y.Bian, "Nearly three orders of
magnitude enhancement of Goos-Hanchen shift by exciting Bloch surface
wave".J.Optic. Express 20 (8), 8998-9003, (2012).
[69]
K.Artmann, “Annalen der physik 6,” Band 2, 87, (1948).
[70]
T.Tamir, “Nonspecular phenomena in beam fifields reflflected by multilayered
media,” J.Opt.Soc.Am. A 3, 558–565. (1986).
[71]
K.Y.Bliokh and A.Aiello, “Goos-h¨achen and imbert-fedorov beam shifts: an
overview,” J.Opt.15, 014001, (2013).
[72]
T.M.Grazegorczyk, X.Chen, J.pacheco, J.Chen, I.Wu and J. Kong, J.
Electromagnetics research, PIER, Vol. 51, pp. 83-118, (2005).
[73]
X.Chen, L.Gang and C.F.Li," Transmission gap, Bragg-like reflection, and
Goos-Hänchen shifts near the Dirac point inside a negative-zero-positive index
metamaterial slab" J.Phy.Rev.A, Vol. 80, P 043839, (2009).
[74]
Y.P.Wong, Y.Miao, J.Skarda and O.Solgaard," Large negative and positive
optical Goos–Hänchen shift in photonic crystals" J.Opt.Lett.Vol. 43, No. 12, P
2804 (2018).
32
[75]
K.Y.Bliokh and A.Aiello, “Goos-hachen and imbert-fedorov beam shifts: an
overview,” J.Opt.15, 014001, (2013).
[76]
J.A.Arnaud, "Degenerate Optical Cavities". vol. 8, P.1 applied optics, (1969).
[77]
L.Weisheng and F.Xiaohui, "The Phenomena and Applications of Light
Dispersion ". J. AISC.116, pp. 273–280, (2009).
[78]
J.Rhim, S.Hong, H.Park and P. K. W. Ng," The Phenomena and Applications
of Light Dispersion". Agric. Food Chem. 54, 5814-5822, (2006).
33