Download - 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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Photonics Communications Engineering, OPTI 500B, Lectures 17 and 18
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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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Photonics Communications Engineering, OPTI 500B, Lectures 17 and 18
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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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Photonics Communications Engineering, OPTI 500B, Lectures 17 and 18
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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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Photonics Communications Engineering, OPTI 500B, Lectures 17 and 18
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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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Photonics Communications Engineering, OPTI 500B, Lectures 17 and 18
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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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Photonics Communications Engineering, OPTI 500B, Lectures 17 and 18
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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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Photonics Communications Engineering, OPTI 500B, Lectures 17 and 18
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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.