Fiber optics ray theory

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Fiber OpticsRay Theory

SOLO HERMELIN

Updated: 17.06.06http://www.solohermelin.com

http://www.solohermelin.com/

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SOLO Optical Fibre – Ray Theory

http://www.datacottage.com/nch/fibre.htm

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A step-index cylindrical fiber has a central core of index ncore surrounded bycladding of index ncladding where ncladding < ncore.

SOLO Optical Fiber – Ray Theory

Cladding

Coreaxisθ

0θ

iθ

Core axisCladding

Skew ray in core of fiber

Meridional ray in corewith two reflexions

When a ray of light enters such afiber at an angle θ0 is refracted at anangle 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

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For the meridional ray


Cladding

Coreaxisθ

0θ

iθ

Meridional ray in corewith 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

nθθ sinsin =

If there is no tunneling from core to cladding. 1sin:sin ≤=> c

core

cladding

i n

nθθ

Since we have90=+ iθθ

θθ sinsin 0

1

coreair nn =

Therefore total internal reflection will occur if:2

22

0 1sin

−=−<

core

cladding

corecladdingcore n

nnnnθ

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We consider only two types of optical fibers:


Skew ray in step-indexcore fiber

Meridional ray in step-indexcore fiber

Core axisCladding

Core axisCladding

zθ

φθφ1

r1z1.constnn corecladding =<

Meridional ray in a grated-index core

Core

axisCladding

Skew ray in a grated-index core of fiber

( )rnncore =

Core axisCladding

zθφθ

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 =

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For a graded-index core fiber ncore = n ( r ) let develop the ray equation:


( ) ( ) ( ) rrnrd

drn

sd

rdrn

sd

d1ray =∇=

zzrrr 11ray +=

where:rayr

- ray vector

rayrdsd=

Assuming a cylindrical core fiber we will use cylindrical coordinates

zzddrrrdrd 111ray ++= φφ

Graded-index Fiber

szsd

zd

sd

drr

sd

rd

sd

rd1:111ray =++= φφ

=

−=

=

01

11

11

zd

rdd

drd

φφ

φφ

011111 =−== zsd

dr

sd

d

sd

d

sd

dr

sd

d φφφφ

=

+−=

+=

zz

yx

yxr

11

1cos1sin1

1sin1cos1

φφφ

φφ

to describe the ray vector:

( ) ( ) ( ) ( ) 22222/1zddrrdrdrdsd rayray ++=⋅= φ

ray propagation direction

See S. Hermelin, “Foundation of Geometrical Optics”

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zθ

φθ

φ1

r1

z1

ρ

Q

P

zrrr zzz 1cos1cossin1sinsin1 ray θθθθθ φφ ++=

ρφθ

CoreQ' axis

Core

axisCladding

zθφθ

r

r1

φ1

ray1r

( )rnncore =

( ) ( ) ( ) rrnrd

drn

sd

rdrn

sd

d1ray =∇=

Graded-index Fiber (continue – 1(

zsd

zd

sd

drr

sd

rd

sd

rd111ray ++= φφ

( )

( ) ( )

( ) ( )

( ) ( )0

ray

11

11

11

sd

zd

sd

zdrnz

sd

zdrn

sd

d

sd

d

sd

drrn

sd

drrn

sd

d

sd

rd

sd

rdrnr

sd

rdrn

sd

d

sd

rdrn

sd

d

+

+

+

+

+

=

φφφφ

( ) ( ) ( ) ( ) ( ) zsd

zdrn

sd

dr

sd

drnr

sd

drnr

sd

d

sd

d

sd

rdrnr

sd

rdrn

sd

d11111

2

+

−

++

= φφφφφ

( ) ( ) ( ) ( ) ( ) ( ) ( )r

rd

rndz

sd

zdrn

sd

d

sd

d

sd

rdrn

sd

drn

sd

dr

sd

drnr

sd

rdrn

sd

d

sd

rdrn

sd

d11121

2

ray =

+

+

+

−

=

φφφφ

011111 =−== zsd

dr

sd

d

sd

d

sd

dr

sd

d φφφφ

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SOLO Optical Fiber – Ray TheoryGraded-index Fiber (continue – 2(

( ) ( ) ( ) ( ) ( ) ( ) ( )r

rd

rndz

sd

zdrn

sd

d

sd

d

sd

rdrn

sd

drn

sd

dr

sd

drnr

sd

rdrn

sd

d

sd

rdrn

sd

d11121

2

ray =

+

+

+

−

=

φφφφ

From this equation we obtain the following three equations:

( ) ( ) ( )rd

rnd

sd

drnr

sd

rdrn

sd

d =

−

2

φ

( ) ( )02 =+

sd

d

sd

rd

r

rn

sd

drn

sd

d φφ

( ) 0=

sd

zdrn

sd

d

( ) ( ) 022 =+

sd

d

sd

rdrrn

sd

drn

sd

dr

φφ2r×

( ) 02 =

sd

drnr

sd

d φ

( ) constsd

zdrn == β ( ) .2 constl

sd

drnr == ρφ

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’)

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( ) ( ) zrnsd

zdrn θβ cos==


We found that the ray propagation vector is


φ

φ1

r1 z1

Q

P

zrs zzz 1cos1cossin1sinsin1 θφθθθθ φφ ++=Core

Q' axis

Core

axis

Cladding

zθ

ssd

rd1:ray =

φ

φ1

φθ

r

r1

φ1innercaustic

outercaustic

s1

z1

zθ

( )rnncore =

szsd

zd

sd

drr

sd

rd

sd

rd1111ray =++= φφ

( )rnsd

zd β= ( )rnrl

sd

d2

ρφ =

( ) ( ) sd

rdz

rnrnr

lr

sd

rds ray1111

=++= βφρ

( )sd

rdzrs zz

ray1cos1cos1sinsin1

=++= θφθθθ φφ

Let write also as a function of two geometric parameterss1 φθθ ,z

φθ - skew angle

zθ - angle between ands1 z1

( )rnrl

z

ρθθ φ =cossin ( ) φθθρ

cossin zrnr

l =

(3’) (2’)

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( ) ( ) zrnsd

zdrn θβ cos==


We found

φθ

r

r1

φ1innercaustic

intesectsray path

outercaustic

intersectsray path

0=φθ

0=φθ

The skew rays take a helical path, as seen from the cross-section figure.

( ) φθθρ

cossin zrnr

l =

( ) ( ) ( ) ( ) 22222 cossincos

β

ρ

θ

ρθ

ρθ φ−

=−

==rn

l

rrnrn

l

rrn

l

rz

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:( )

( ) 10cos22

===−

φθβ

ρ

rn

l

r

or ( ) ( ) 0:2

2222 =−−=r

lrnrgρβ ( ) ( ) 0== ocic rgrg

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We obtained:

( )rnsd

zd β= ( )rnrl

sd

d2

ρφ =

( ) zd

d

rnzd

d

sd

zd

sd

d β==

( ) ( ) ( ) ( ) ( )( ) ( )rnrd

rnd

rnr

lrnr

zd

rd

rnrn

zd

d

rn×=

−

2

2

ρββ

( )2

2

3

22

2

22

2

1

rd

rnd

rl

zd

rd =− ρβ

Define:zd

rdr =:'

rd

rdr

zd

rd

rd

d

zd

rd

zd

rd

zd

d

zd

rd ''

2

2

=

=

=

( )2

2

3

222

2

1'

rd

rnd

rl

rd

drr =− ρβ Integration ( ) constrn

rl

zd

rd +=+

2

2

22

2

2

2

1

2

1

2

1 ρβ

( ) constsd

zdrn == β(3’) ( ) .2 constl

sd

drnr == ρφ

(2’)

( ) ( ) ( )rd

rnd

sd

drnr

sd

rdrn

sd

d =

−

2

φ(1)

( )( )

2

2

2222

2

2 2 βρββ +⋅+−−=

const

rlrn

zd

rd

rg

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We obtained: ( )( )

2

2

2222

2

2 2 βρββ +⋅+−−=

const

rlrn

zd

rd

rg

φθ

r

r1

φ1innercaustic

intesectsray path

outercaustic

intersectsray path

0=φθ

0=φθ

To determine the constant we use the fact that at

the caustic we havetherefore

( ) ( ) 0&02

2222 =−−==r

lrnrgzd

rd ρβ

02 2 =+⋅ βconst

Finally we obtain the ray path equation:

( ) ( )2

2222

2

2 :r

lrnrgzd

rd ρββ −−==

Since a ray path exists only in the regions where0

2

2 ≥

zd

rdβ ( ) 0>rg

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Analysis of: ( ) ( )2

2222

2

2 :r

lrnrgzd

rd ρββ −−==

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

ocricr

2

22

rl

ρ

cladding

core

0≠l

( )rg

skew ray

β<claddingn( ) ociccore rrrrn ≤≤> β

( ) ociccorecladding rrrrnn ≤≤<< β

rρ

ocr

0=l

cladding

core( )rg

meridional ray

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2222

2

2 :r

lrnrgzd

rd ρββ −−==


2. Refracted rays

The rays are refracted from the core in the cladding region iff:

g (r)>0 for r ≥ ρ

rρicr

2

22

rl

ρ

cladding

core

0≠l

( )rg

skew ray

222 lncladding +> β

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2222

2

2 :r

lrnrgzd

rd ρββ −−==


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

icr

2

22

rl

ρ

cladding

core

0≠l

( )rg

skew ray

radr

β>claddingn

22 lncladding +<< ββ

( ) 02

2222 =−−=

rad

claddingrader

lnrgρβ

22 β

ρ

−=

cladding

rad

n

lr

The energy leaks from the core tothe cladding region.

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For a step-index core fiber ncore = constant.


Core axisCladding


zθ

φθ

s1

φ1

r1

z1

ρ

Q

P

zrrs zzz 1cos1cossin1sinsin1 θθθθθ φφ ++=

ρφθ

Core

PQ

Q' axis

P Q'ρ

φθρ sin2' =PQ

φθ

φθ

icr

φθρ cos=icr

φθ

innercaustic

.constnn corecladding =<

Step-index Fiber

( ) ( ) zrnsd

zdrn θβ cos==

( ) φθθρ

cossin zrnr

l =

( ) ( )2

2222

2

2 :r

lrnrgzd

rd ρββ −−==

( )

≥=<=

=ρρ

rconstn

rconstnrn

cladding

core

2

1

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SOLO Optical Fiber – Ray TheoryStep-index Fiber (continue – 7(


2222

2

2 :r

lrnrgzd

rd ρββ −−==


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<claddingnβ>coren

corecladding nn << β

rρ22 β

ρ

−=

core

ic

n

lr

2

22

rl

ρ

claddingcore

0≠l

( )rg

skew ray

22 β−coren

22 β−claddingn

corenn = claddingnn =

0222 >−−= lng core β

0222 <−−= lng cladding β

rρ

0=l

claddingcore( )rg

meridional ray

022 <−= βcladdingng

022 >−= βcoreng


( )

≥=<=

=ρρ

rconstn

rconstnrn

cladding

core

2

1

( ) 0=icrg φ

θθρ

θβθρ

β

ρ φ

coscossin

cos22

zcore

zcore

nl

n

core

ic

n

lr

=

==

−=

P Q'ρ

φθρ sin2' =PQ

φθ

φθ

icr

φθρ cos=icr

φθ

innercaustic

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2222

2

2 :r

lrnrgzd

rd ρββ −−==


2. Refracted rays

The rays are refracted from the core in the cladding region iff:

g (r)>0 for r ≥ ρ

22 lncladding +> β

( )

≥=<=

=ρρ

rconstn

rconstnrn

cladding

core

2

1

rρ22 β

ρ

−=

core

ic

n

lr

2

22

rl

ρ

claddingcore

0≠l

( )rg

skew ray

22 β−coren

22 β−claddingn



0222 >−−= lng cladding β

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2222

2

2 :r

lrnrgzd

rd ρββ −−==


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>claddingn


( ) 02

2222 =−−=

rad

claddingrader

lnrgρβ

22 β

ρ

−=

cladding

rad

n

lr

The energy leaks from the core tothe cladding region.

( )

≥=<=

=ρρ

rconstn

rconstnrn

cladding

core

2

1

rρ22 βρ

−=

core

ic

n

lr

2

22

rl

ρ

claddingcore

0≠l

( )rg

skew ray

22 β−coren

22 β−claddingn


22 β

ρ

−=

cladding

rad

n

lr

0222 >−− lncore β

0222 <−− lncladding β

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For a step-index core fiber ncore = constant.


P Q'ρ

φθρ sin2' =PQ

φθ

φθ

icr

φθρ cos=icr

φθ

innercaustic

Step-index Fiber

( ) ( )2

2222

2

2 :r

lrnrgzd

rd ρββ −−==

( )

≥=<=

=ρρ

rconstn

rconstnrn

cladding

core

2

1

rρ22 β

ρ

−=

core

ic

n

lr

2

22

rl

ρ

claddingcore

0≠l

( )rg

skew ray

22 β−coren

22 β−claddingn



0222 <−−= lng cladding β

rρ

0=l

claddingcore( )rg

meridional ray

022 <−= βcladdingng

022 >−= βcoreng


corecladding nn << β

rρ22 β

ρ

−=

core

ic

n

lr

2

22

rl

ρ

claddingcore

0≠l

( )rg

skew ray

22 β−coren

22 β−claddingn



0222 >−−= lng cladding β

rρ22 β

ρ

−=

core

ic

n

lr

2

22

rl

ρ

claddingcore

0≠l

( )rg

skew ray

22 β−coren

22 β−claddingn


22 β

ρ

−=

cladding

rad

n

lr

0222 >−− lncore β

0222 <−− lncladding β

1. Bounded rays

2. Refracted rays

222 lncladding +> β

3. Tunneling rays


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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

TechnionIsraeli Institute of Technology

1964 – 1968 BSc EE1968 – 1971 MSc EE

Israeli Air Force1970 – 1974

RAFAELIsraeli Armament Development Authority

1974 – 2013

Stanford University1983 – 1986 PhD AA

http://upload.wikimedia.org/wikipedia/en/a/ae/RAFAEL_logo.png

Science

Fiber optics ray theory