Describes Fiber Optics using Optical Ray Theory.
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Optical fiber communication Part 1 Optical Fiber FundamentalsMadhumita Tamhane
Optical fiber systems grew from combination of semiconductor technology, which provided necessary light sources and photodetectors and optical waveguide technology. It has significant inherent advantages over conventional copper systems- low transmission loss, wide BW, light weight and size, immunity to interferences, signal security to name a few. One principle characteristic of optical fiber is its attenuation as a function of wavelength. Hence it is operated in two major low attenuation wavelength windows 800-900nm and 1100-1600nm . Light travels inside optical fiber waveguide on principle of total internal reflection. Fiber is available as single mode and multiple mode, step index and graded index depending on applications and expenditures. Principle of fiber can be understood by ray theory or mode theory. ...
Optical fiber communication Part 1 Optical Fiber FundamentalsMadhumita Tamhane
Optical fiber systems grew from combination of semiconductor technology, which provided necessary light sources and photodetectors and optical waveguide technology. It has significant inherent advantages over conventional copper systems- low transmission loss, wide BW, light weight and size, immunity to interferences, signal security to name a few. One principle characteristic of optical fiber is its attenuation as a function of wavelength. Hence it is operated in two major low attenuation wavelength windows 800-900nm and 1100-1600nm . Light travels inside optical fiber waveguide on principle of total internal reflection. Fiber is available as single mode and multiple mode, step index and graded index depending on applications and expenditures. Principle of fiber can be understood by ray theory or mode theory. ...
This narrated power point presentation attempts to explain the various dispersion mechanisms that are observed in optical fibers. Some fundamental terms and concepts are also discussed. The material will be useful for KTU final year B Tech students who prepare for the subject EC 405, Optical Communications.
The attached narrated power point presentation attempts to explain the methods of computation of total power loss and system rise time in a fiber optic link. The material will be useful for KTU final year B Tech students who prepare for the subject EC 405, Optical Communications.
The following ppt gives overview about Optical Communication and the underlying principle with the general overview of all the contents for optical communication
The attached narrated power point presentation mentions the various figures of merit and the types of noise associated with photodetectors. The material will be useful for KTU final year B Tech students who prepare for the subject EC 405, Optical Communications.
A Klystron is a vacuum tube that can be used either as a generator or as an amplifier or as an oscillator, at microwave frequencies.The Klystron is a linear beam device; that is, the electron flow is in a straight line focused by an axial magnetic field.
This narrated power point presentation attempts to examine the losses due to non-linear effects in optical fibers. The material will be useful for KTU final year B Tech students who prepare for the subject EC 405, Optical Communications.
In radio and electronics, an antenna is an electrical device which converts electric power into radio waves, and vice versa. It is usually used with a radio transmitter or radio receiver. In transmission, a radio transmitter supplies an electric current oscillating at radio frequency to the antenna's terminals, and the antenna radiates the energy from the current as electromagnetic waves (radio waves). In reception, an antenna intercepts some of the power of an electromagnetic wave in order to produce a tiny voltage at its terminals, that is applied to a receiver to be amplified.
This narrated power point presentation attempts to explain the various dispersion mechanisms that are observed in optical fibers. Some fundamental terms and concepts are also discussed. The material will be useful for KTU final year B Tech students who prepare for the subject EC 405, Optical Communications.
The attached narrated power point presentation attempts to explain the methods of computation of total power loss and system rise time in a fiber optic link. The material will be useful for KTU final year B Tech students who prepare for the subject EC 405, Optical Communications.
The following ppt gives overview about Optical Communication and the underlying principle with the general overview of all the contents for optical communication
The attached narrated power point presentation mentions the various figures of merit and the types of noise associated with photodetectors. The material will be useful for KTU final year B Tech students who prepare for the subject EC 405, Optical Communications.
A Klystron is a vacuum tube that can be used either as a generator or as an amplifier or as an oscillator, at microwave frequencies.The Klystron is a linear beam device; that is, the electron flow is in a straight line focused by an axial magnetic field.
This narrated power point presentation attempts to examine the losses due to non-linear effects in optical fibers. The material will be useful for KTU final year B Tech students who prepare for the subject EC 405, Optical Communications.
In radio and electronics, an antenna is an electrical device which converts electric power into radio waves, and vice versa. It is usually used with a radio transmitter or radio receiver. In transmission, a radio transmitter supplies an electric current oscillating at radio frequency to the antenna's terminals, and the antenna radiates the energy from the current as electromagnetic waves (radio waves). In reception, an antenna intercepts some of the power of an electromagnetic wave in order to produce a tiny voltage at its terminals, that is applied to a receiver to be amplified.
Estimate the hidden States, Parameters, Signals of a Linear Dynamic Stochastic System from Noisy Measurements. It requires knowledge of probability theory. Presentation at graduate level in math., engineering
For comments please contact me at solo.hermelin@gmail.com.
For more presentations on different subjects visit my website at http://solohermelin.com. Since a few Figure were not downloaded I recommend to see the presentation on my website at RADAR Folder, Tracking subfolder.
This presentation is intended for undergraduate students in physics and engineering.
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For more presentations on different subjects please visit my homepage at http://www.solohermelin.com.
This presentation is in the Physics folder.
Introduction to elasticity part 1 of 2 is a presentation at undergraduate in science (physics, mathematics, engineering) level. For comments or improvement suggestions please contact me at solo.hermelin@gmail.com.
For more presentations on different subjects please visit my website at http://www.solohermelin.com.
This presentation is in the Elasticity folder.
Describes Radar Tracking Loops in Range, Doppler and Angles.
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Optical Aberration is the phenomenon of Image Distortion due to Optics Imperfection.
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For more presentations visit my website at http://www.solohermelin.com.
This presentation is in the Optics Folder. Since some of the Figures were not downloaded I recommend to see the presentation on my website.
The Cramér-Rao Lower Bound on the Variance of the Estimator of continuous SISO, MIMO and Digital Systems.
Probability and Estimation are prerequisite.
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For more presentations on different subjects visit my website at http://solohermelin.com.
Maxwell equations and propagation in anisotropic mediaSolo Hermelin
Describes the propagation of electromagnetic waves in anisotropic electrical media.
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For more presentations on different subjects visit my website at http://www.solohermelin.com.
The presentation is not properly downloaded. Please go to my website and open it in the Optics Folder.
Equation of motion of a variable mass system2Solo Hermelin
This is the second of three presentations (self content) for derivation of equations of motions of a variable mass system containing moving solids (rotors, pistons,..) and elastic parts. It uses the Reynolds' Transport Theorem. It is recommended to see the first presentation before this one. Each presentation uses a different method of derivation.
The presentation is at undergraduate (physics, engineering) level.
Please sent comments for improvements to solo.hermelin@gmail.com. Thanks!
For more presentations on different subjects please visit my website at http://www.solohermelin.com
Presents the Tracking methods of moving targets by sensors (radar, electro optics,..).
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For more presentations visit my website at http://www.solohermelin.com.
I recommend to view this presentation on my website at RADAR folder, Tracking Systems subfolder.
Fighter Aircraft Performance, Part II of two, describes the parameters that affect aircraft performance.
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For more presentations on different subjects visit my website at http://www.solohermelin.com.
Fighter Aircraft Performance, Part I of two, describes the parameters that affect aircraft performance.
For comments please contact me at solo.hermelin@gmail.com.
For more presentations on different subjects visit my website at http://www.solohermelin.com.
Lens history and physics.
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For more presentations visit my website at http://www.solohermelin.com.
This presentation is in the Optics folder.
Aircraft Susceptibility and Vulnerability.
This is from the last presentations from my side. Medical Problems prevent me to continue with new presentations.Please do not contact me.
Describes concepts and development of flying cars and other flying vehicles. Reference are given including to YouTube movies. At the end my view of Main Requirements and the related Design Requirements for a SkyCar are given. The main conclusion is that technologically we are ready to develop and product such a SkyCar in a few years.
For comments please contact me at solo.hermelin@gmail.com.
For more presentations on different subjects visit my website at http://www.solohermelin.com.
This presentation gives example of "Calculus of Variations" problems that can be solved analytical. "Calculus of Variations" presentation is prerequisite to this one.
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For more presentations on different subjects visit my website at http://www.solohermelin.com.
Describes the mathematics of the Calculus of Variations.
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Air Combat History describes the main air combats and fighter aircraft, from the beginning of aviation. The additional Youtube links are an important part of the presentation. A list of Air-to-Air Missile from different countries. is also given
For comments please contact me at solo.hermelin@gmail.com.
For more presentations visit my website at http://www.solohermelin.com.
This presentation explores a brief idea about the structural and functional attributes of nucleotides, the structure and function of genetic materials along with the impact of UV rays and pH upon them.
The ability to recreate computational results with minimal effort and actionable metrics provides a solid foundation for scientific research and software development. When people can replicate an analysis at the touch of a button using open-source software, open data, and methods to assess and compare proposals, it significantly eases verification of results, engagement with a diverse range of contributors, and progress. However, we have yet to fully achieve this; there are still many sociotechnical frictions.
Inspired by David Donoho's vision, this talk aims to revisit the three crucial pillars of frictionless reproducibility (data sharing, code sharing, and competitive challenges) with the perspective of deep software variability.
Our observation is that multiple layers — hardware, operating systems, third-party libraries, software versions, input data, compile-time options, and parameters — are subject to variability that exacerbates frictions but is also essential for achieving robust, generalizable results and fostering innovation. I will first review the literature, providing evidence of how the complex variability interactions across these layers affect qualitative and quantitative software properties, thereby complicating the reproduction and replication of scientific studies in various fields.
I will then present some software engineering and AI techniques that can support the strategic exploration of variability spaces. These include the use of abstractions and models (e.g., feature models), sampling strategies (e.g., uniform, random), cost-effective measurements (e.g., incremental build of software configurations), and dimensionality reduction methods (e.g., transfer learning, feature selection, software debloating).
I will finally argue that deep variability is both the problem and solution of frictionless reproducibility, calling the software science community to develop new methods and tools to manage variability and foster reproducibility in software systems.
Exposé invité Journées Nationales du GDR GPL 2024
Observation of Io’s Resurfacing via Plume Deposition Using Ground-based Adapt...Sérgio Sacani
Since volcanic activity was first discovered on Io from Voyager images in 1979, changes
on Io’s surface have been monitored from both spacecraft and ground-based telescopes.
Here, we present the highest spatial resolution images of Io ever obtained from a groundbased telescope. These images, acquired by the SHARK-VIS instrument on the Large
Binocular Telescope, show evidence of a major resurfacing event on Io’s trailing hemisphere. When compared to the most recent spacecraft images, the SHARK-VIS images
show that a plume deposit from a powerful eruption at Pillan Patera has covered part
of the long-lived Pele plume deposit. Although this type of resurfacing event may be common on Io, few have been detected due to the rarity of spacecraft visits and the previously low spatial resolution available from Earth-based telescopes. The SHARK-VIS instrument ushers in a new era of high resolution imaging of Io’s surface using adaptive
optics at visible wavelengths.
Richard's aventures in two entangled wonderlandsRichard Gill
Since the loophole-free Bell experiments of 2020 and the Nobel prizes in physics of 2022, critics of Bell's work have retreated to the fortress of super-determinism. Now, super-determinism is a derogatory word - it just means "determinism". Palmer, Hance and Hossenfelder argue that quantum mechanics and determinism are not incompatible, using a sophisticated mathematical construction based on a subtle thinning of allowed states and measurements in quantum mechanics, such that what is left appears to make Bell's argument fail, without altering the empirical predictions of quantum mechanics. I think however that it is a smoke screen, and the slogan "lost in math" comes to my mind. I will discuss some other recent disproofs of Bell's theorem using the language of causality based on causal graphs. Causal thinking is also central to law and justice. I will mention surprising connections to my work on serial killer nurse cases, in particular the Dutch case of Lucia de Berk and the current UK case of Lucy Letby.
ESR spectroscopy in liquid food and beverages.pptxPRIYANKA PATEL
With increasing population, people need to rely on packaged food stuffs. Packaging of food materials requires the preservation of food. There are various methods for the treatment of food to preserve them and irradiation treatment of food is one of them. It is the most common and the most harmless method for the food preservation as it does not alter the necessary micronutrients of food materials. Although irradiated food doesn’t cause any harm to the human health but still the quality assessment of food is required to provide consumers with necessary information about the food. ESR spectroscopy is the most sophisticated way to investigate the quality of the food and the free radicals induced during the processing of the food. ESR spin trapping technique is useful for the detection of highly unstable radicals in the food. The antioxidant capability of liquid food and beverages in mainly performed by spin trapping technique.
Nucleophilic Addition of carbonyl compounds.pptxSSR02
Nucleophilic addition is the most important reaction of carbonyls. Not just aldehydes and ketones, but also carboxylic acid derivatives in general.
Carbonyls undergo addition reactions with a large range of nucleophiles.
Comparing the relative basicity of the nucleophile and the product is extremely helpful in determining how reversible the addition reaction is. Reactions with Grignards and hydrides are irreversible. Reactions with weak bases like halides and carboxylates generally don’t happen.
Electronic effects (inductive effects, electron donation) have a large impact on reactivity.
Large groups adjacent to the carbonyl will slow the rate of reaction.
Neutral nucleophiles can also add to carbonyls, although their additions are generally slower and more reversible. Acid catalysis is sometimes employed to increase the rate of addition.
The use of Nauplii and metanauplii artemia in aquaculture (brine shrimp).pptxMAGOTI ERNEST
Although Artemia has been known to man for centuries, its use as a food for the culture of larval organisms apparently began only in the 1930s, when several investigators found that it made an excellent food for newly hatched fish larvae (Litvinenko et al., 2023). As aquaculture developed in the 1960s and ‘70s, the use of Artemia also became more widespread, due both to its convenience and to its nutritional value for larval organisms (Arenas-Pardo et al., 2024). The fact that Artemia dormant cysts can be stored for long periods in cans, and then used as an off-the-shelf food requiring only 24 h of incubation makes them the most convenient, least labor-intensive, live food available for aquaculture (Sorgeloos & Roubach, 2021). The nutritional value of Artemia, especially for marine organisms, is not constant, but varies both geographically and temporally. During the last decade, however, both the causes of Artemia nutritional variability and methods to improve poorquality Artemia have been identified (Loufi et al., 2024).
Brine shrimp (Artemia spp.) are used in marine aquaculture worldwide. Annually, more than 2,000 metric tons of dry cysts are used for cultivation of fish, crustacean, and shellfish larva. Brine shrimp are important to aquaculture because newly hatched brine shrimp nauplii (larvae) provide a food source for many fish fry (Mozanzadeh et al., 2021). Culture and harvesting of brine shrimp eggs represents another aspect of the aquaculture industry. Nauplii and metanauplii of Artemia, commonly known as brine shrimp, play a crucial role in aquaculture due to their nutritional value and suitability as live feed for many aquatic species, particularly in larval stages (Sorgeloos & Roubach, 2021).
Seminar of U.V. Spectroscopy by SAMIR PANDASAMIR PANDA
Spectroscopy is a branch of science dealing the study of interaction of electromagnetic radiation with matter.
Ultraviolet-visible spectroscopy refers to absorption spectroscopy or reflect spectroscopy in the UV-VIS spectral region.
Ultraviolet-visible spectroscopy is an analytical method that can measure the amount of light received by the analyte.
Comparing Evolved Extractive Text Summary Scores of Bidirectional Encoder Rep...University of Maribor
Slides from:
11th International Conference on Electrical, Electronics and Computer Engineering (IcETRAN), Niš, 3-6 June 2024
Track: Artificial Intelligence
https://www.etran.rs/2024/en/home-english/
Remote Sensing and Computational, Evolutionary, Supercomputing, and Intellige...University of Maribor
Slides from talk:
Aleš Zamuda: Remote Sensing and Computational, Evolutionary, Supercomputing, and Intelligent Systems.
11th International Conference on Electrical, Electronics and Computer Engineering (IcETRAN), Niš, 3-6 June 2024
Inter-Society Networking Panel GRSS/MTT-S/CIS Panel Session: Promoting Connection and Cooperation
https://www.etran.rs/2024/en/home-english/
ANAMOLOUS SECONDARY GROWTH IN DICOT ROOTS.pptxRASHMI M G
Abnormal or anomalous secondary growth in plants. It defines secondary growth as an increase in plant girth due to vascular cambium or cork cambium. Anomalous secondary growth does not follow the normal pattern of a single vascular cambium producing xylem internally and phloem externally.
3. 3
A step-index cylindrical fiber has a central core of index ncore surrounded by
cladding of index ncladding where ncladding < ncore.
SOLO Optical Fiber – Ray Theory
Cladding
Core
axisθ
0θ
i
θ
Core axis
Cladding
Skew ray in core of fiber
Meridional ray in core
with two reflexions
When a ray of light enters such a
fiber at an angle θ0 is refracted at an
angle θ, and then reflected back at the
boundary between core and cladding,
if the angle of incidence θi is greater
than the critical angle θc.
Two distinct rays can travel inside
the fiber in this way:
• meridional rays remain in a plan that contains fiber axis
• skew rays travel in a non-planar zig-zag path and never cross the fiber axis
4. 4
For the meridional ray
SOLO Optical Fiber – Ray Theory
Cladding
Core
axisθ
0θ
iθ
Meridional ray in core
with two reflexions
Snell’s Law at the fiber enter
If the ray is refracted from the core
to the cladding than according to
Snell’s Law:
222
0
sin1cossinsin claddingcoreicoreicorecore
nnnnn −<−=== θθθθ
r
core
cladding
i
n
n
θθ sinsin =
If there is no tunneling from core to cladding.1sin:sin ≤=> c
core
cladding
i
n
n
θθ
Since we have
90=+ i
θθ
θθ sinsin 0
1
coreair nn =
Therefore total internal reflection will occur if:
2
22
0
1sin
−=−<
core
cladding
corecladdingcore
n
n
nnnθ
5. 5
We consider only two
types of optical fibers:
SOLO Optical Fiber – Ray Theory
Skew ray in step-index
core fiber
Meridional ray in step-index
core fiber
Core axis
Cladding
Core axis
Cladding
zθ
φθ
φ1
r1z1.constnn corecladding =<
Meridional ray in a grated-index core
Core
axis
Cladding
Skew ray in a grated-index core of fiber
( )rnncore =
Core axis
Cladding
zθ
φθ
r
r1
φ1
• step-index core fiber
where the index of
refraction in core is
constant and changes
by a step in the cladding
such that
corecladding nn <
• graded-index core fiber
where the index of
refraction in core changes
as function of radius r
such that ( )rnncore =
6. 6
For a graded-index core fiber ncore = n ( r ) let develop the ray equation:
SOLO Optical Fiber – Ray Theory
( ) ( ) ( ) rrn
rd
d
rn
sd
rd
rn
sd
d
1
ray
=∇=
zzrrr 11ray +=
where:
rayr
-ray vector
rayrdsd
=
Assuming a cylindrical core fiber we will use cylindrical coordinates
zzddrrrdrd 111ray ++= φφ
Graded-index Fiber
sz
sd
zd
sd
d
rr
sd
rd
sd
rd
1:111
ray
=++= φ
φ
=
−=
=
01
11
11
zd
rdd
drd
φφ
φφ
011111 =−== z
sd
d
r
sd
d
sd
d
sd
d
r
sd
d φ
φφ
φ
=
+−=
+=
zz
yx
yxr
11
1cos1sin1
1sin1cos1
φφφ
φφ
to describe the ray vector:
( ) ( ) ( ) ( )22222/1
zddrrdrdrdsd rayray ++=⋅= φ
ray propagation direction
See S. Hermelin, “Foundation of Geometrical Optics”
7. 7
SOLO Optical Fiber – Ray Theory
Skew ray in core of fiber
z
θ
φθ
φ1
r1
z1
ρ
Q
P
zrrr zzz
1cos1cossin1sinsin1 ray
θθθθθ φφ ++=
ρ
φθ
Core
Q' axis
Core
axis
Cladding
zθ
φθ
r
r1
φ1
ray1r
( )rnncore =
( ) ( ) ( ) rrn
rd
d
rn
sd
rd
rn
sd
d
1
ray
=∇=
Graded-index Fiber (continue – 1(
z
sd
zd
sd
d
rr
sd
rd
sd
rd
111
ray
++= φ
φ
( )
( ) ( )
( ) ( )
( ) ( )
0
ray
1
1
1
1
1
1
sd
zd
sd
zd
rnz
sd
zd
rn
sd
d
sd
d
sd
d
rrn
sd
d
rrn
sd
d
sd
rd
sd
rd
rnr
sd
rd
rn
sd
d
sd
rd
rn
sd
d
+
+
+
+
+
=
φφ
φ
φ
( ) ( ) ( ) ( ) ( ) z
sd
zd
rn
sd
d
r
sd
d
rnr
sd
d
rnr
sd
d
sd
d
sd
rd
rnr
sd
rd
rn
sd
d
11111
2
+
−
++
=
φ
φ
φ
φ
φ
( ) ( ) ( ) ( ) ( ) ( ) ( ) r
rd
rnd
z
sd
zd
rn
sd
d
sd
d
sd
rd
rn
sd
d
rn
sd
d
r
sd
d
rnr
sd
rd
rn
sd
d
sd
rd
rn
sd
d
11121
2
ray
=
+
+
+
−
=
φ
φφφ
011111 =−== z
sd
d
r
sd
d
sd
d
sd
d
r
sd
d φ
φφ
φ
8. 8
SOLO Optical Fiber – Ray Theory
Graded-index Fiber (continue – 2(
( ) ( ) ( ) ( ) ( ) ( ) ( ) r
rd
rnd
z
sd
zd
rn
sd
d
sd
d
sd
rd
rn
sd
d
rn
sd
d
r
sd
d
rnr
sd
rd
rn
sd
d
sd
rd
rn
sd
d
11121
2
ray
=
+
+
+
−
=
φ
φφφ
From this equation we obtain the following three
equations:
( ) ( ) ( )
rd
rnd
sd
d
rnr
sd
rd
rn
sd
d
=
−
2
φ
( ) ( ) 02 =+
sd
d
sd
rd
r
rn
sd
d
rn
sd
d φφ
( ) 0=
sd
zd
rn
sd
d
( ) ( ) 022
=+
sd
d
sd
rd
rrn
sd
d
rn
sd
d
r
φφ
2
r×
( ) 02
=
sd
d
rnr
sd
d φ
( ) const
sd
zd
rn == β ( ) .2
constl
sd
d
rnr == ρ
φ
Integration
Integration
where:
l,β -dimensionless constants (ray invariants( to be defined
ρ -radius of the boundary between core and cladding
By integrating the last two equation we obtain:
(1)
(2)
(3)
(3’) (2’)
9. 9
( ) ( ) zrn
sd
zd
rn θβ cos==
SOLO Optical Fiber – Ray Theory
Graded-index Fiber (continue – 3(
We found that the ray
propagation vector is
Skew ray in core of fiber
φ
φ1
r1 z1
Q
P
zrs zzz 1cos1cossin1sinsin1 θφθθθθ φφ ++=
Core
Q'
axis
Core
axis
Cladding
zθ
s
sd
rd
1:ray
=
φ
φ1
φθ
r
r1
φ1inner
caustic
outer
caustic
s1
z1
zθ
( )rnncore =
sz
sd
zd
sd
d
rr
sd
rd
sd
rd
1111
ray
=++= φ
φ
( )rnsd
zd β
=
( )rnr
l
sd
d
2
ρφ
=
( ) ( ) sd
rd
z
rnrnr
l
r
sd
rd
s
ray
1111
=++=
β
φ
ρ
( ) sd
rd
zrs zz
ray
1cos1cos1sinsin1
=++= θφθθθ φφ
Let write also as a function of two geometric parameterss1 φθθ ,z
φθ -skew angle
zθ -angle between ands1 z1
( )rnr
l
z
ρ
θθ φ =cossin ( ) φθθ
ρ
cossin zrn
r
l =
(3’) (2’)
10. 10
( ) ( ) zrn
sd
zd
rn θβ cos==
SOLO Optical Fiber – Ray Theory
Graded-index Fiber (continue – 4(
We found
φθ
r
r1
φ1inner
caustic
intesects
ray path
outer
caustic
intersects
ray path
0=φθ
0=φθ
The skew rays take a helical path, as seen from the cross-section figure.
( ) φθθ
ρ
cossin zrn
r
l =
( ) ( ) ( ) ( ) 22222
cossin
cos
β
ρ
θ
ρ
θ
ρ
θφ
−
=
−
==
rn
l
rrnrn
l
rrn
l
r
z
z
( ) ( ) 0== ocic
rr φφ θθ
A particular family of skew ray will not come closer to the fiber axis than the
inner caustic cylindrical surface of radius ric and further from the axis than the
outer caustic cylindrical surface of radius roc. From the figure we can see that
at the intersection of ray path with the caustic surface
Therefore the caustic radiuses can be found by solving:
( )
( ) 10cos
22
===
−
φ
θ
β
ρ
rn
l
r
or ( ) ( ) 0: 2
2
222
=−−=
r
lrnrg
ρ
β ( ) ( ) 0== ocic
rgrg
11. 11
SOLO Optical Fiber – Ray Theory
Graded-index Fiber (continue – 5(
We obtained:
( )rnsd
zd β
= ( )rnr
l
sd
d
2
ρφ
=
( ) zd
d
rnzd
d
sd
zd
sd
d β
==
( )
( )
( )
( )
( )
( ) ( )rn
rd
rnd
rnr
l
rnr
zd
rd
rn
rn
zd
d
rn
×=
−
2
2
ρββ
( )
2
2
3
2
2
2
2
2
2
1
rd
rnd
r
l
zd
rd
=−
ρ
β
Define:
zd
rd
r =:'
rd
rd
r
zd
rd
rd
d
zd
rd
zd
rd
zd
d
zd
rd '
'2
2
=
=
=
( )
2
2
3
2
22
2
1
'
rd
rnd
r
l
rd
dr
r =−
ρ
β
Integration
( ) constrn
r
l
zd
rd
+=+
2
2
2
2
2
2
2
1
2
1
2
1 ρ
β
( ) const
sd
zd
rn == β(3’) ( ) .2
constl
sd
d
rnr == ρ
φ
(2’)
( ) ( ) ( )
rd
rnd
sd
d
rnr
sd
rd
rn
sd
d
=
−
2
φ
(1)
( )
( )
2
2
2
222
2
2
2 β
ρ
ββ +⋅+−−=
const
r
lrn
zd
rd
rg
12. 12
SOLO Optical Fiber – Ray Theory
Graded-index Fiber (continue – 6(
We obtained:
( )
( )
2
2
2
222
2
2
2 β
ρ
ββ +⋅+−−=
const
r
lrn
zd
rd
rg
φθ
r
r1
φ1inner
caustic
intesects
ray path
outer
caustic
intersects
ray path
0=φθ
0=φθ
To determine the constant we use the fact that at
the caustic we have
therefore
( ) ( ) 0&0 2
2
222
=−−==
r
lrnrg
zd
rd ρ
β
02 2
=+⋅ βconst
Finally we obtain the ray path equation:
( ) ( ) 2
2
222
2
2
:
r
lrnrg
zd
rd ρ
ββ −−==
Since a ray path exists only in the regions where0
2
2
≥
zd
rd
β ( ) 0>rg
13. 13
SOLO Optical Fiber – Ray Theory
Graded-index Fiber (continue – 7(
Analysis of: ( ) ( ) 2
2
222
2
2
:
r
lrnrg
zd
rd ρ
ββ −−==
A ray path exists only in the regions where ( ) 0>rg
1.Bounded rays
The rays are bounded in the core region iff:
g (r)>0 for ric<r < roc and g (r)<0 for r ≥ ρ
rρ
ocr
icr
2
2
2
r
l
ρ
cladding
core
0≠l
( )rg
skew ray
β<cladding
n( ) ociccore
rrrrn ≤≤> β
( ) ociccorecladding
rrrrnn ≤≤<< β
rρ
ocr
0=l
cladding
core
( )rg
meridional ray
14. 14
SOLO Optical Fiber – Ray Theory
Graded-index Fiber (continue – 8(
Analysis of: ( ) ( ) 2
2
222
2
2
:
r
lrnrg
zd
rd ρ
ββ −−==
A ray path exists only in the regions where ( ) 0>rg
2.Refracted rays
The rays are refracted from the core in the
cladding region iff:
g (r)>0 for r ≥ ρ
rρ
icr
2
2
2
r
l
ρ
cladding
core
0≠l
( )rg
skew ray
222
lncladding
+> β
15. 15
SOLO Optical Fiber – Ray Theory
Graded-index Fiber (continue – 9(
Analysis of: ( ) ( ) 2
2
222
2
2
:
r
lrnrg
zd
rd ρ
ββ −−==
A ray path exists only in the regions where ( ) 0>rg
3.Tunneling rays
The rays escape in the cladding region iff:
g (r)<0 for ρ <r<rrad and g (r)>0 for r ≥ rrad
222
lncladding
+< β
rρ
ocr
ic
r
2
2
2
r
l
ρ
cladding
core
0≠l
( )rg
skew ray
radr
β>cladding
n
22
lncladding
+<< ββ
( ) 02
2
222
=−−=
rad
claddingrade
r
lnrg
ρ
β 22
β
ρ
−
=
cladding
rad
n
l
r
The energy leaks from the core to
the cladding region.
16. 16
For a step-index core
fiber ncore = constant.
SOLO Optical Fiber – Ray Theory
Core axis
Cladding
Skew ray in core of fiber
z
θ
φθ
s1
φ1
r1
z1
ρ
Q
P
zrrs zzz 1cos1cossin1sinsin1 θθθθθ φφ ++=
ρ
φθ
Core
P
Q
Q' axis
P Q'
ρ
φθρ sin2' =PQ
φθ
φθ
icr
φ
θρ cos=ic
r
φθ
inner
caustic
.constnn corecladding =<
Step-index Fiber
( ) ( ) zrn
sd
zd
rn θβ cos==
( ) φθθ
ρ
cossin zrn
r
l =
( ) ( ) 2
2
222
2
2
:
r
lrnrg
zd
rd ρ
ββ −−==
( )
≥=
<=
=
ρ
ρ
rconstn
rconstn
rn
cladding
core
2
1
17. 17
SOLO Optical Fiber – Ray Theory
Step-index Fiber (continue – 7(
Analysis of: ( ) ( ) 2
2
222
2
2
:
r
lrnrg
zd
rd ρ
ββ −−==
A ray path exists only in the regions where ( ) 0>rg
1.Bounded rays
The rays are bounded in the core region iff:
g (r)>0 for r = ρ- ε and g (r)<0 for r = ρ+ε
β<cladding
nβ>core
n
corecladding
nn << β
rρ
22
β
ρ
−
=
core
ic
n
l
r
2
2
2
r
l
ρ
claddingcore
0≠l
( )rg
skew ray
22
β−core
n
22
β−cladding
n
corenn = claddingnn =
0222
>−−= lng core
β
0222
<−−= lng cladding β
rρ
0=l
claddingcore
( )rg
meridional ray
022
<−= βcladdingng
022
>−= βcore
ng
corenn = claddingnn =
( )
≥=
<=
=
ρ
ρ
rconstn
rconstn
rn
cladding
core
2
1
( ) 0=ic
rg φ
θθρ
θβ
θρ
β
ρ φ
cos
cossin
cos22
zcore
zcore
nl
n
core
ic
n
l
r
=
=
=
−
=
P Q'
ρ
φθρsin2' =PQ
φθ
φθ
icr
φθρ cos=icr
φθ
inner
caustic
18. 18
SOLO Optical Fiber – Ray Theory
Step-index Fiber (continue – 8(
Analysis of: ( ) ( ) 2
2
222
2
2
:
r
lrnrg
zd
rd ρ
ββ −−==
A ray path exists only in the regions where ( ) 0>rg
2.Refracted rays
The rays are refracted from the core in the
cladding region iff:
g (r)>0 for r ≥ ρ
22
lncladding
+> β
( )
≥=
<=
=
ρ
ρ
rconstn
rconstn
rn
cladding
core
2
1
rρ
22
β
ρ
−
=
core
ic
n
l
r
2
2
2
r
l
ρ
claddingcore
0≠l
( )rg
skew ray
22
β−coren
22
β−cladding
n
corenn = claddingnn =
0222
>−−= lng core β
0222
>−−= lng cladding β
19. 19
SOLO Optical Fiber – Ray Theory
Step-index Fiber (continue – 9(
Analysis of: ( ) ( ) 2
2
222
2
2
:
r
lrnrg
zd
rd ρ
ββ −−==
A ray path exists only in the regions where ( ) 0>rg
3.Tunneling rays
The rays escape in the cladding region iff:
g (r)<0 for ρ <r<rrad and g (r)>0 for r ≥ rrad
222
lncladding
+< β β>cladding
n
22
lncladding
+<< ββ
( ) 02
2
222
=−−=
rad
claddingrade
r
lnrg
ρ
β 22
β
ρ
−
=
cladding
rad
n
l
r
The energy leaks from the core to
the cladding region.
( )
≥=
<=
=
ρ
ρ
rconstn
rconstn
rn
cladding
core
2
1
rρ
22
β
ρ
−
=
core
ic
n
l
r
2
2
2
r
l
ρ
claddingcore
0≠l
( )rg
skew ray
22
β−core
n
22
β−claddingn
core
nn = cladding
nn =
22
β
ρ
−
=
cladding
rad
n
l
r
0222
>−− lncore β
0222
<−− lncladding β
20. 20
For a step-index core
fiber ncore = constant.
SOLO Optical Fiber – Ray Theory
P Q'
ρ
φθρ sin2' =PQ
φθ
φθ
ic
r
φθρ cos=ic
r
φθ
inner
caustic
Step-index Fiber
( ) ( ) 2
2
222
2
2
:
r
lrnrg
zd
rd ρ
ββ −−==
( )
≥=
<=
=
ρ
ρ
rconstn
rconstn
rn
cladding
core
2
1
rρ
22
β
ρ
−
=
core
ic
n
l
r
2
2
2
r
l
ρ
claddingcore
0≠l
( )rg
skew ray
22
β−core
n
22
β−cladding
n
corenn = claddingnn =
0222
>−−= lng core β
0222
<−−= lng cladding β
rρ
0=l
claddingcore
( )rg
meridional ray
022
<−= βcladdingng
022
>−= βcoreng
corenn = claddingnn =
corecladding
nn << β
rρ
22
β
ρ
−
=
core
ic
n
l
r
2
2
2
r
l
ρ
claddingcore
0≠l
( )rg
skew ray
22
β−core
n
22
β−claddingn
corenn = claddingnn =
0222
>−−= lng core β
0222
>−−= lng cladding β
rρ
22
β
ρ
−
=
core
ic
n
l
r
2
2
2
r
l
ρ
claddingcore
0≠l
( )rg
skew ray
22
β−coren
22
β−claddingn
core
nn = claddingnn =
22
β
ρ
−
=
cladding
rad
n
l
r
0222
>−− lncore
β
0222
<−− lncladding β
1.Bounded rays
2.Refracted rays
222
lncladding
+> β
3.Tunneling rays
22
lncladding
+<< ββ
23. January 9, 2015 23
SOLO
Technion
Israeli Institute of Technology
1964 – 1968 BSc EE
1968 – 1971 MSc EE
Israeli Air Force
1970 – 1974
RAFAEL
Israeli Armament Development Authority
1974 – 2013
Stanford University
1983 – 1986 PhD AA