Fiber optics ray theory

Describes Fiber Optics using Optical Ray Theory.
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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 n
core
surrounded by
cladding of index n
cladding
where n
cladding
< n
core
.
SOLO Optical Fiber – Ray Theory
Cladding
Core
axis
q
0q
iq
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 θ
0
is refracted at an
angle θ, and then reflected back at the
boundary between core and cladding,
if the angle of incidence θ
i is greater
than the critical angle θ
c
.
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
axis
q
0q
iq
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 -<-=== qqqq
r
core
cladding
i
n
n
qq sinsin=
If there is no tunneling from core to cladding. 1sin:sin £=>
c
core
cladding
i
n
n
qq
Since we have

90=+
iqq

qq sinsin
0
1
coreair nn =
Therefore total internal reflection will occur if:
2
22
0
1sin
÷
÷
ø
ö
ç
ç
è
æ
-=-<
core
cladding
corecladdingcore
n
n
nnnq

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
Coreaxis
Cladding
z
q
fq
f1
r1
z1
.constnn
corecladding =<
Meridional ray in a grated-index core
Core
axis
Cladding
Skew ray in a grated-index core of fiber
()rnn
core
=
Core axis
Cladding
zq
fq
r
r1
f1
• step-index core fiber
where the index of
refraction in core is
constant and changes
by a step in the cladding
such that
corecladdingnn <
• graded-index core fiber
where the index of
refraction in core changes
as function of radius r
such that ()rnn
core=

6. 6
For a graded-index core fiber n
core
= n ( r ) let develop the ray equation:
SOLO Optical Fiber – Ray Theory
() () ()rrn
rd
d
rn
sd
rd
rn
sd
d
1
ray
=Ñ=
÷
÷
ø
ö
ç
ç
è
æ

zzrrr 11
ray
+=

where:
rayr

- ray vector
ray
rdsd

=
Assuming a cylindrical core fiber we will use cylindrical coordinates
zzddrrrdrd 111
ray
++= ff

Graded-index Fiber
sz
sd
zd
sd
d
rr
sd
rd
sd
rd
1:111
ray
=++= f
f

ï
ï
î
ï
ï
í
ì
=
-=
=
01
11
11

zd
rdd
drd
ff
ff
011111 =-== z
sd
d
r
sd
d
sd
d
sd
d
r
sd
d f
ff
f
ï
ï
î
ï
ï
í
ì
=
+-=
+=
zz
yx
yxr
11
1cos1sin1
1sin1cos1
fff
ff
to describe the ray vector:
( ) ( ) ( ) ( )
2222
2/1
zddrrdrdrdsd
rayray
++=×= f
ray propagation direction
See S. Hermelin, “Foundation of Geometrical Optics”

7. 7
SOLO Optical Fiber – Ray Theory
Skew ray in core of fiber
z
q
f
q
f1
r1
z1
r
Q
P
zrrr
zzz
1cos1cossin1sinsin1
ray
qqqqq
ff
++=
r
f
q
Core
Q'axis
Core
axis
Cladding
z
q
fq
r
r1
f1
ray1r
()rnn
core=
() () ()rrn
rd
d
rn
sd
rd
rn
sd
d
1
ray
=Ñ=
ú
û
ù
ê
ë
é

Graded-index Fiber (continue – 1(
z
sd
zd
sd
d
rr
sd
rd
sd
rd
111
ray
++= f
f

()
() ()
() ()
() ()

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
+
ú
û
ù
ê
ë
é
+
+
ú
û
ù
ê
ë
é
+
+
ú
û
ù
ê
ë
é
=
ú
û
ù
ê
ë
é
ff
f
f

() () () () () 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
ú
û
ù
ê
ë
é
+
÷
÷
ø
ö
ç
ç
è
æ
-
ú
û
ù
ê
ë
é
++
ú
û
ù
ê
ë
é
=
f
f
f
f
f
() () () () () ()
()
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
=
ú
û
ù
ê
ë
é
+
þ
ý
ü
î
í
ì
+
ú
û
ù
ê
ë
é
+
ï
þ
ï
ý
ü
ï
î
ï
í
ì
÷
÷
ø
ö
ç
ç
è
æ
-
ú
û
ù
ê
ë
é
=
ú
û
ù
ê
ë
é
f
fff

011111 =-== z
sd
d
r
sd
d
sd
d
sd
d
r
sd
d f
ff
f

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
=
ú
û
ù
ê
ë
é
+
þ
ý
ü
î
í
ì
+
ú
û
ù
ê
ë
é
+
ï
þ
ï
ý
ü
ï
î
ï
í
ì
÷
÷
ø
ö
ç
ç
è
æ
-
ú
û
ù
ê
ë
é
=
ú
û
ù
ê
ë
é
f
fff

From this equation we obtain the following three
equations:
() ()
()
rd
rnd
sd
d
rnr
sd
rd
rn
sd
d
=
÷
÷
ø
ö
ç
ç
è
æ
-
ú
û
ù
ê
ë
é
2
f
()
()
02 =+
ú
û
ù
ê
ë
é
sd
d
sd
rd
r
rn
sd
d
rn
sd
d ff
() 0=
ú
û
ù
ê
ë
é
sd
zd
rn
sd
d
() () 02
2
=+
ú
û
ù
ê
ë
é
sd
d
sd
rd
rrn
sd
d
rn
sd
d
r
ff
2
r´
() 0
2
=
ú
û
ù
ê
ë
é
sd
d
rnr
sd
d f
() const
sd
zd
rn ==b () .
2
constl
sd
d
rnr ==r
f
Integration
Integration
where:
l,b - dimensionless constants (ray invariants( to be defined
r - 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 qb 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
f
f1
r1 z1
Q
P
zrs
zzz 1cos1cossin1sinsin1 qfqqqq
ff ++=
Core
Q'
axis
Core
axis
Cladding
zq
s
sd
rd
1:
ray
=
f
f1
f
q
r
r1
f1inner
caustic
outer
caustic
s1
z1
z
q
()rnn
core
=
sz
sd
zd
sd
d
rr
sd
rd
sd
rd
1111
ray
=++= f
f

()rnsd
zdb
=
()rnr
l
sd
d
2
rf
=
() () sd
rd
z
rnrnr
l
r
sd
rd
s
ray
1111

=++=
b
f
r
( )
sd
rd
zrs
zz
ray
1cos1cos1sinsin1

=++= qfqqq
ff
Let write also as a function of two geometric parameterss1 f
qq,
z
f
q - skew angle
z
q - angle between and
s1 z1
()rnr
l
z
r
qq
f
=cossin ()
f
qq
r
cossin
z
rn
r
l=
(3’) (2’)

10. 10
() ()
zrn
sd
zd
rn qb cos==
SOLO Optical Fiber – Ray Theory
Graded-index Fiber (continue – 4(
We found
f
q
r
r1
f1
inner
caustic
intesects
ray path
outer
caustic
intersects
ray path
0=
f
q
0=
f
q
The skew rays take a helical path, as seen from the cross-section figure.
()
f
qq
r
cossin
z
rn
r
l=
() () () ()
22222
cos
sin
cos
b
r
q
r
q
r
q
f
-
=
-
==
rn
l
r
rnrn
l
rrn
l
r
z
z
() ()0==
ocic
rr
ff
qq
A particular family of skew ray will not come closer to the fiber axis than the
inner caustic cylindrical surface of radius r
ic
and further from the axis than the
outer caustic cylindrical surface of radius r
oc
. 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
===
-
f
q
b
r
rn
l
r
or () () 0:
2
2
222
=--=
r
lrnrg
r
b () ()0==
ocic rgrg

11. 11
SOLO Optical Fiber – Ray Theory
Graded-index Fiber (continue – 5(
We obtained:
()rnsd
zdb
=
()rnr
l
sd
d
2
rf
=
()zd
d
rnzd
d
sd
zd
sd
d b
==
()
()
()
()
()
()
()rn
rd
rnd
rnr
l
rnr
zd
rd
rn
rn
zd
d
rn
´=
÷
÷
ø
ö
ç
ç
è
æ
-
ú
û
ù
ê
ë
é
2
2
rbb
()
2
2
3
2
2
2
2
2
2
1
rd
rnd
r
l
zd
rd
=-
r
b
Define:
zd
rd
r=:'
rd
rd
r
zd
rd
rd
d
zd
rd
zd
rd
zd
d
zd
rd '
'
2
2
=
÷
÷
ø
ö
ç
ç
è
æ
=
÷
÷
ø
ö
ç
ç
è
æ
=
()
2
2
3
2
22
2
1
'
rd
rnd
r
l
rd
dr
r =-
r
b
Integration
()constrn
r
l
zd
rd
+=+
÷
÷
ø
ö
ç
ç
è
æ
2
2
2
2
2
2
2
1
2
1
2
1 r
b
() const
sd
zd
rn ==b(3’) () .
2
constl
sd
d
rnr ==r
f
(2’)
() ()
()
rd
rnd
sd
d
rnr
sd
rd
rn
sd
d
=
÷
÷
ø
ö
ç
ç
è
æ
-
ú
û
ù
ê
ë
é
2
f
(1)
()
()
2
2
2
222
2
2
2 b
r
bb +×+--=
÷
÷
ø
ö
ç
ç
è
æ
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 b
r
bb +×+--=
÷
÷
ø
ö
ç
ç
è
æ
const
r
lrn
zd
rd
rg

f
q
r
r1
f1
inner
caustic
intesects
ray path
outer
caustic
intersects
ray path
0=
f
q
0=
f
q
To determine the constant we use the fact that at
the caustic we have
therefore
() () 0&0
2
2
222
=--==
r
lrnrg
zd
rd r
b
02
2
=+× bconst
Finally we obtain the ray path equation:
() ()
2
2
222
2
2
:
r
lrnrg
zd
rd r
bb --==
÷
÷
ø
ö
ç
ç
è
æ
Since a ray path exists only in the regions where0
2
2
³
÷
÷
ø
ö
ç
ç
è
æ
zd
rd
b ()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 r
bb --==
÷
÷
ø
ö
ç
ç
è
æ
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
ic
<r < r
oc
and g (r)<0 for r ≥ ρ
r
r
ocr
icr
2
2
2
r
l
r
cladding
core
0¹l
()rg
skew ray
b<
cladding
n()
ociccore rrrrn ££>b
()
ociccorecladding rrrrnn ££<<b
r
r
oc
r
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 r
bb --==
÷
÷
ø
ö
ç
ç
è
æ
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
rr
icr
2
2
2
r
l
r
cladding
core
0¹l
()rg
skew ray
222
ln
cladding
+>b

15. 15
SOLO Optical Fiber – Ray Theory
Graded-index Fiber (continue – 9(
Analysis of: () ()
2
2
222
2
2
:
r
lrnrg
zd
rd r
bb --==
÷
÷
ø
ö
ç
ç
è
æ
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<r
rad
and g (r)>0 for r ≥ r
rad
222
ln
cladding
+<b
rr
ocr
icr
2
2
2
r
l
r
cladding
core
0¹l
()rg
skew ray
radr
b>
cladding
n
22
ln
cladding
+<< bb
( ) 0
2
2
222
=--=
rad
claddingrade
r
lnrg
r
b
22
b
r
-
=
cladding
rad
n
l
r
The energy leaks from the core to
the cladding region.

16. 16
For a step-index core
fiber n
core
= constant.
SOLO Optical Fiber – Ray Theory
Core axis
Cladding
Skew ray in core of fiber
z
q
f
q
s1
f1
r1
z1
r
Q
P
zrrs
zzz 1cos1cossin1sinsin1 qqqqq
ff ++=
r
f
q
Core
P
Q
Q'axis
P
Q'
r
f
qrsin2'=PQ
fq
fq
icr
f
qrcos=
ic
r
fq
inner
caustic
.constnn
corecladding
=<
Step-index Fiber
() ()
zrn
sd
zd
rn qb cos==
()
f
qq
r
cossin
z
rn
r
l=
() ()
2
2
222
2
2
:
r
lrnrg
zd
rd r
bb --==
÷
÷
ø
ö
ç
ç
è
æ
()
î
í
ì
³=
<=
=
r
r
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 r
bb --==
÷
÷
ø
ö
ç
ç
è
æ
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
b<
cladding
nb>
core
n
corecladding nn <<b
rr
22
b
r
-
=
core
ic
n
l
r
2
2
2
r
l
r
claddingcore
0¹l
()rg
skew ray
22
b-
core
n
22
b-
cladding
n
corenn= claddingnn=
0
222
>--= lng
core
b
0
222
<--= lng
clad dingb
r
r
0=l
claddingcore
()rg
meridional ray
0
22
<-= b
clad din gng
0
22
>-= b
core
ng
co re
nn=
cla dd in g
nn=
()
î
í
ì
³=
<=
=
r
r
rconstn
rconstn
rn
cladding
core
2
1
()0=
ic
rg f
qqr
qb
qr
b
r
f
cos
cossin
cos22
zcore
zcore
nl
n
core
ic
n
l
r
=
=
=
-
=
P Q'
r
f
qrsin2'=PQ
f
q
fq
i c
r
f
qrcos=
i c
r
fq
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 r
bb --==
÷
÷
ø
ö
ç
ç
è
æ
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
ln
cladding
+>b
()
î
í
ì
³=
<=
=
r
r
rconstn
rconstn
rn
cladding
core
2
1
rr
22
b
r
-
=
core
ic
n
l
r
2
2
2
r
l
r
claddingcore
0¹l
()rg
skew ray
22
b-
coren
22
b-
cladding
n
corenn= clad dingnn=
0
222
>--= lng
coreb
0
222
>--= lng
cla d din gb

19. 19
SOLO Optical Fiber – Ray Theory
Step-index Fiber (continue – 9(
Analysis of: () ()
2
2
222
2
2
:
r
lrnrg
zd
rd r
bb --==
÷
÷
ø
ö
ç
ç
è
æ
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<r
rad
and g (r)>0 for r ≥ r
rad
222
ln
cladding
+<b b>
cladding
n
22
ln
cladding
+<< bb
( ) 0
2
2
222
=--=
rad
claddingrade
r
lnrg
r
b
22
b
r
-
=
cladding
rad
n
l
r
The energy leaks from the core to
the cladding region.
()
î
í
ì
³=
<=
=
r
r
rconstn
rconstn
rn
cladding
core
2
1
rr
22
b
r
-
=
core
ic
n
l
r
2
2
2
r
l
r
claddingcore
0¹l
()rg
skew ray
22
b-
coren
22
b-
cladding
n
core
nn= cladding
nn=
22
b
r
-
=
cladding
rad
n
l
r
0
222
>-- ln
coreb
0
222
<-- ln
claddingb

20. 20
For a step-index core
fiber n
core
= constant.
SOLO Optical Fiber – Ray Theory
P Q'
r
fqrsin2'=PQ
f
q
f
q
ic
r
f
qrcos=
i c
r
fq
inner
caustic
Step-index Fiber
() ()
2
2
222
2
2
:
r
lrnrg
zd
rd r
bb --==
÷
÷
ø
ö
ç
ç
è
æ
()
î
í
ì
³=
<=
=
r
r
rconstn
rconstn
rn
cladding
core
2
1
rr
22
b
r
-
=
co re
ic
n
l
r
2
2
2
r
l
r
claddingcore
0¹l
()rg
skew ray
22
b-
core
n
22
b-
cl ad d ing
n
co renn= cla d di ng
nn=
0
222
>--= lng
c oreb
0
222
<--= lng
cla d di ng
b
r
r
0=l
claddingcore
()rg
meridional ray
0
22
<-= b
c l add i n gng
0
22
>-= b
coreng
co renn=
cl a ddi n gnn=
corecladding nn <<b
rr
22
b
r
-
=
co re
i c
n
l
r
2
2
2
r
l
r
claddingcore
0¹l
()rg
skew ray
22
b-
co re
n
22
b-
cl a dd ingn
co renn= cl ad di ngnn=
0
222
>--= lng
co reb
0
222
>--= lng
c la dd i ngb
rr
22
b
r
-
=
co re
i c
n
l
r
2
2
2
r
l
r
claddingcore
0¹l
()rg
skew ray
22
b-
co ren
22
b-
cl a dd ingn
co renn= cl ad di ngnn=
22
b
r
-
=
cl a ddi n g
rad
n
l
r
0
222
>-- ln
coreb
0
222
<-- ln
cla dd i ngb
1. Bounded rays
2. Refracted rays
222
ln
cladding
+>b
3. Tunneling rays
22
ln
cladding
+<< bb

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