Lecture 17 Optical Fiber Modes

8/13/2019 Lecture 17 Optical Fiber Modes

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Photonics Communications Engineering, OPTI 500B, Lectures 17 and 18

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Theory of Optical Modes in Step Index Fibers


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inside the core

inside the cladding

core

clad

nn

n

=

r z

r z

E E r E E z

H H r H H z

= + +

= + +

We find the modes by looking for solutions of:

( ) ( )

( )

( )( )

2 22 2

0 02 2

22

02 2

22

0

2

2

0

r

r r

r

z z

E E

E nk E E nk E r r r

EEE nk E

r r

E nk E z

+ = +

+ + +

+ + =

The equations have a simple physical interpretation.

(from Pollock)


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Since the equations for Er and E are coupled, we firstsolve for Ez. Hz is a solution of the same Helmholtzequation and its solutions have the same form. We find allother field components from Ez and Hz using Maxwellsequations.

We look for solutions of the form:

( ) ( ) ( )zE R r Z z=

(from Pollock)


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In the core we find:

( )

( )( ) ( ) ( )

( )22 2

0where , and 0,1,2...

j z j z

j j

core

Z z ae be

ce de

R r eJ r fN r

n k

= +

= +

= +

= =

We can simplify these noting that:

Often we have only forward going waves (b=0) The N(r) solution goes to minus infinity at r = 0 so it

is unphysical (f=0)

We need both the ejand e-jterms to describe the dependence of the eigenmodes, but we can limit the

discussion to the ejsolution with the understanding


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that a mode with e-jdependence can be found from

the ejmode by rotating the fiber.

Then we can write:

( )

( )

. .

. .

j j z

z

j j z

z

E AJ r e e c c

H BJ r e e c c

= +

= +

In the cladding region we find:

( )

( )

. .

. .

j j z

z

j j z

z

E CK r e e c c

H DK r e e c c

= +

= +

where ( )22 2

0cladn k =

From Pollock and Lipson


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From Izuka

Characteristic Equation for an Optical Fiber

We insist on continuity of the tangential field components

Ez, E, Hz, and Hand find:

( )( )

( )( )

( )( )

( )( )

22 2

2 2 2

2 2 2 2

0 0

1 1

core clad

a

J a K a k n J a k n K a

J a K a J a K a

+

= + +


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This characteristic equation can be used with:

( ) ( )2 22 2 2

0, where core cladV a a V k a n n = + =

to find values for , , , and neff.

Meridional Modes (=0):

For modes that correspond to bouncing meridional rays,

there is no dependence. Modes are of two types TE0and TM0with =1,2, .

( )( )

( )( )

( )( )

( )( )

2 2 2 2

0 0

If we set this term = 0, If we set this term = 0,0 and this is a TE mode 0 and this is a TM modez r z r

core clad

E E H H

J a K a k n J a k n K a

J a K a J a K a

= = = =

+ +

0=

Skew Modes (0):

These modes have radial structure. The modes have both

Ez0 and Hz0 and thus are called hybrid modes. The

hybrid modes are of two types labeled EHand HE,depending on the whether Ezor Hzis dominant,respectively.


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Field Distributions in Optical Fibers

Lets examine the mode profiles in the plane z=0:

TE Modes: TM Modes:

( )

( )1 0

1 0

0

0

r

r

E

E J r

H J r

H

( )

( )

1 0

1 0

0

0

r

r

E J r

E

H

H J r

There is no azimuthal variation for either type of mode.

Example, TM01Mode:

Figure 11.21. All figures (unless noted) and the table in this lecture are fromElements of Photonics, Volume II.

J1(01r) has a zero at the origin and one maximum in thecore.


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EHModes:

( )( )

( )

( )

1

1

1

1

cos

sin

sin

cos

r

r

E J r

E J r

H J r

H J r

+

+

+

+

HEModes:

( )

( )

( )

( )

1

1

1

1

cos

sin

sin

cos

r

r

E J r

E J r

H J r

H J r


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Example - the HE21mode:

( )

( )( )

( )

1 21

1 21

1 21

1 21

cos2

sin2

sin2

cos2

r

r

E J r

E J r

H J r

H J r

E is purely radial for = 0, /2, , and 3/2.E is purely azimuthal for = /4, 3/4, 5/4, and 7/4.H looks like E rotated counter clockwise by /4.J1(K21r) has a zero at the origin and one maximum in thecore.

Field is purely

radial here Field is purely

azimuthal here

Fields are

zero here

Fields have

a maximum

here

Field is purely

radial here Field is purely

azimuthal here

Fields are

zero here

Fields have

a maximum

here

Figure 11.21.


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Can we construct a set of linearly polarized modes?

Yes. This is good because polarized light from a laser

would excite these superpositions of true fiber modes.

HE11is already linearly polarized.

Figure 11.21 in Elements of Photonics, Volume II.

Other LP modes can be constructed from sums of the EHand HE modes that have the same propagation constant.


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+

=


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Construction and Labeling Rules:

LP0= HE1

LP1

= HE2

+ TE0

or HE2

+ TM0

LPm= HEm+1,+ EHm-1,


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Fiber Mode Degeneracy and Number of Modes

Degeneracy of the Hybrid Modes

From Electromagnetic Theory for Microwaves and Optoelectronics, Keqian Zhang and Dejie Li

The TE0and TM0modes are not degenerate.

The hybrid EHand HEmodes are two-fold degenerate.

Degeneracy of the LP Modes

The LP0modes are the HE1modes, so they are two-folddegenerate.


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The LP1modes are formed by summing HE2+ TE0or

HE2+ TM0, so they are four-fold degenerate.

The LPmmodes with m > 1 are formed by summing

HEm+1,+ EHm-1,, so they are four-fold degenerate.

Two of the 4 LP21modes that can be formed from HE31and EH11modes.

From Electromagnetic Theory for Microwaves and Optoelectronics, Keqian Zhang and Dejie Li


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From Introduction to Fiber Optics, Ghatak and Thyagarajan

Number of Modes

For large V, the number of LP or hybrid of modes is 4V

2/2.

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Lecture 17 Optical Fiber Modes

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