Light propagation in optical fibers and power loss

mechanisms

Optical communications 2005. Henrik Fredriksson ISY/ek

Outline• The optical band• Electro magnetic wave theory • Reflection and reflection• Propagation modes • The structure of optical fibers• Dispersion• Fresnel losses and angle of acceptance• Loss mechanisms• Loss vs. frequency

Optical band

• An optical view or ray view of elements can be used only when the dielectric constant varies slowly over the wavelength of light.

Maxwell• Maxwell equations

• For a neutral material • For a non-magnetic material• An electrical wave propagating in the z

direction can be expressed as

• Where kz is wave number k=ω/v

( )

( )( )( )403

2

10

=⋅∇=⋅∇

=∂∂

+×∇

=∂∂

+×∇

BD

JtDH

tBE

ρ

( )( )( )765

EJHBED

σµε

===

0=ρ)1( 0 == rµµµ

( ) ( ) ( ) ( )( ) ( )8Recosˆˆ, 000 wtzkizyx zeEtzkyExEtzE −=−+= ω

James Clerk Maxwell 13 June 1831 –5 Nov 1879

Plane wave

• For vacuum is εr=1, µr=1, and σ=0

• For a non magnetic isolator• nr is called the refraction index

( ) ( ) ( ) ( ) ( )( ) ( ) ( )109,

)9(2,7,6,5,122

0

2

22

EiEEkeEEtE

tEE

trki ωσµεµω

σµεµω −−=−⇒=

∂∂

+∂∂

=∇⇒×∇−⋅

000 ωε

µσµεµεω rrrik +=

cvv

k

k==

=

=⇒

00

00 1µε

ωµεω

rr ncck ωεω ==

Plane wave cont. and surface boundary condition

• The B field can be solved in a similar way

• For a plane wave traveling along the z-axis

• At the surface between two materials

( ) ( )11,1 instancefor yxxy BkEBkEωω

−==⇒

)(02

22 , trkieBB

tB

tBB ωσµεµ −⋅=

∂∂

+∂∂

=∇

( )( )

( ) ( )( )( ) ( )( ) ( )130ˆ

ˆ43

=⋅−=⋅−

⇒

−+

−+

yyByByyDyD sσ

( )( )

( ) ( )( )( ) ( )( ) ( )12ˆ

0ˆ21

sJyHyHyyEyEy

=−×=−×

⇒

−+

−+

nr2

nr1 x̂ ŷ

ẑ

Refraction and reflection

nr2

nr1

α φ

x̂ ŷ

ẑ

E

β

( ) ( )( )( ) ( ) ( )( )( )

( ) ( )( )( ) ( )14::

cossin2

cossincossin1

2

11

tyxkizr

tyxkityxkizr

BeEnCeAeEn

ωββ

ωφφωαα

−−

−+−−

=+=

( ) ( ) ( )( )( ) ( )( )( ) ( )( )( )( )150at 12,14 sinsinsin 211 txkitxkitxki BeCeAey ωβωφωα −−− =+⇒=

Transverse electric (TE) wave

Since (15) should be true for arbitrary x and t, we require all the phase terms to be equal

( ) ( ) ( ) ( )16 sinsinsin 211 βφα kkk ==

( )17 φα =( ) ( ) ( )( ) ( )18sin

sin sinsin1

2221

r

r

nn

kkkk ==⇔=

βαβα

Which givesLaw of reflection

Law of refraction (Snell’s Law)

Refraction and reflection cont.( ) ( ) ( )19 16,15 BCA =+⇒( ) ( ) ( )

( ) ( ) ( ) ( )( )

( ) ( ) ( )( )20

cos:

cos:

,14,11 ,6

cossin

0

22

cossinsin

0

11

0

21

111

−=

+−=

⇒=

−

ββ

ααα

βµε

αµε

εµω

yikxikxr

yikyikxikxr

BeeHn

CeAeeHn

k

( ) ( ) ( )( ) ( )

( )( ) ( )21 cos

cos

coscos20,12

1

2

0

2

0

1

BCA

BCA

αβ

εε

βµεα

µε

=−⇔

=+−⇒

( ) ( ) ( ) ( )22

cossincossin12

cossincossin12

21,19,18BC

BA

−=

+=

⇒

αββααββα

Fresnel’s first formula

Reflection and refraction TM mode

• In the same way as for TE mode we can derive Snell’s law and Fresnel’s formula for transverse magnetic (TM ) waves.

• Snell’s law stays the same but Fresnel’sformula for TM mode is:

)23(

coscos

sinsin2

coscos

sinsin2

BC

BA

−=

+=

αβ

βα

αβ

βα

Reflected amplitude vs. angle • Using Fresnel’s formulas

we can plot the relative reflected amplitude vs. angle.

• For the angle α=αtot relative reflected amplitude is 1, We have total internal reflection.

• Beyond α=αtott the reflected wave will be phase shifted due to a complex angle β.

Relative reflected amplitude for TE and TM modes for a interface where nr1>nr2

Critical angle• The angle of total internal reflection can be

derived from Snell’s law.( )( )

=⇒

=

= −1

21

01

2

sin90

sinsin

r

rcr

r

nn

nn

αβ

βα

Optical modes

• When we are dealing with optical elements where the refraction index does change significantly over the dimensions of a wavelength, we must use s full wave description to understand optical phenomenon. The assumption of a plane wave is not valid.

Optical modes in planar waveguides

≥≤≤−+

−≤=

2/2/2/)sin()cos(

2/

dxDedxdxkCxkB

dxAeE

xxx

x

yγ

γ

20

22

0

consantn propagatio

2

kn

ck

r−=

=

==

βγ

β

ωλπ

*) Jasprit Singh: Optoelectronics An intorduction to materials and devices 1996

It can be shown* that the electric field for a TE polarized wave is given by

where

Applying boundary conditions at d/2 and –d/2 gives a condition for allowed modes

22tan

2ddkdk xx γ=

Optical modes in planar waveguides cont.

Example of different modes in a plane waveguide

The number of allowed modes depend on the thickness d and wavelength λ.Note the modes has different propagation constants β and therefore different modes travel at different speeds

Structure of optical fiber

• Optical fiber are based on total internal reflection. Having a core (nr1) surrounded by the cladding (nr2 < nr1) and a surrounding protective cover, Coating or buffer.

Types of fibers

Types of material used in optical fibers

• Silica fibers (SiO2), glass, and plastics• Plastic is cheap but has high attenuation. • The refraction index of Silica fibers can be fine

tuned by adding metal oxides such as TiO2, Al2O3 or GeO2

• Plastic are only used for multimode step index fibers

• Silica fibers are used for single mode step index fibers and multimode fibers (almost always gradient index fibers)

Dispersion

• Dispersion cause light pulses to stretch out in time.

• Dispersion is caused by the fact that different parts of a light pulse travel at different speeds through a fiber

Effects that cause dispersion• Mode dispersion. Different propagation modes

propagates at different speed (only present in multimode fibers.

• Chromatic dispersion. The refraction index differ slightly for different wavelengths (nr(λ)) and is always present.

• The way to minimize dispersion is to use a gradient fiber or single mode fiber and to use a light source with narrow spectral width

Fresnel losses

• Recall the reflected amplitude vs angle.

Relative reflected amplitude for TE and TM modes for a interface where nr1>nr2

•Even for α=0 is the reflected amplitude not zeroThis will cause a slight fraction of the in-sent light to be reflected in the fiber ends which causing losses.

Angle of acceptance

• At the fiber end. The light can be sent in at the most at angle γ that cause the internal reflection to be at angle αc.

• The angle of acceptance is given by

• na is called the numerical aperture of the fiber.

nannnnn =−=

−= 22

21

2

1

21 1)sin(γ

Loss mechanisms• As seen above, light sent in at an angle larger then the

angle of acceptance will not benefit from total internal reflection and a fraction of the light will leak to the core at each reflection.

• Other loss mechanisms include scattering, absorption, surface imperfections, and bending.

Loss vs. frequency

Summary• We can derive ray optics from Maxwell’s equations • We need to solve wave equations to understand the modes in single

mode fibers.• Total internal reflection is used to contain light in a optic fiber. • To propagate under total internal reflection the light has to be sent

into the fiber-end at specific angles• Dispersion cause light pulses to stretch out in time. Dispersion can

be minimized by using single mode fibers or gradient index fibers optimized for the used wavelength. Fibers with a small wavelength dependent relative refraction index for the used wavelength bandcan be used to minimize chromatic dispersion.

• Attenuation in the fiber is minimized by minimizing fiber imperfections such as surface imperfections, local changes in refraction index (scattering), minimize absorbing impurities andmechanical stress on the fiber