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1
Fiber Optics
Ray Theory
SOLO HERMELIN
Updated: 17.06.06
http://www.solohermelin.com
2
SOLO Optical Fibre – Ray Theory
http://www.datacottage.com/nch/fibre.htm
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
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
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
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
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
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
( ) ( ) 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
( ) ( ) 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
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
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
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
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
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
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
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
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
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
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
21
22
SOLO
References
C.C. Davis, “Laser and Electro-Optics”, Cambridge University Press, 1996,
OPTICS
S. Hermelin, “Foundation of Geometrical Optics”
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

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Fiber optics ray theory

  • 1. 1 Fiber Optics Ray Theory SOLO HERMELIN Updated: 17.06.06 http://www.solohermelin.com
  • 2. 2 SOLO Optical Fibre – Ray Theory http://www.datacottage.com/nch/fibre.htm
  • 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
  • 21. 21
  • 22. 22 SOLO References C.C. Davis, “Laser and Electro-Optics”, Cambridge University Press, 1996, OPTICS S. Hermelin, “Foundation of Geometrical Optics”
  • 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

Editor's Notes

  1. S. Hermelin, “Foundation of Geometrical Optics”