Optical Communications Part 03 Light Propagation in Optical Fiber (2)

1. Taiz University, YEMEN
4 October 2018

2. Taiz University, YEMEN
 Fiber types according to dimensions and mechanism
of propagation:
4 October 2018 2
Types of Fibers
Step Index Fibers
Single-mode
Multimode
Graded Index Fiber

3. Taiz University, YEMEN
4 October 2018 3
The refractive index profile and ray transmission in step index fibers:
(a) multimode step index fiber; (b) single-mode step index fiber

4. Taiz University, YEMEN
• The optical fiber with a core of constant refractive
index n1 and a cladding of a slightly lower refractive
index n2 is known as step index fiber.
• This is because the refractive index profile for this type
of fiber makes a step change at the core-cladding
interface as indicated in previous Figure which
illustrates the two major types of step index fiber.
• The refractive index profile may be defined as:
𝑛 𝑟 =
𝑛1 𝑟 < 𝑎 (𝑐𝑜𝑟𝑒)
𝑛2 𝑟 ≥ 𝑎 (𝑐𝑙𝑎𝑑𝑑𝑖𝑛𝑔)
4 October 2018 4

5. Taiz University, YEMEN
• Figure shows a single mode or monomode step index fiber
which allows the propagation of only one transverse
electromagnetic mode and hence the core diameter must
be of the order of 8-12 µm.
• The advantage of the propagation of a single mode within
an optical fiber is that the signal dispersion caused by the
delay differences between different modes in a multimode
fiber may be avoided. Thus achieving a large bandwidth.
4 October 2018 5

6. Taiz University, YEMEN
• In describing SMF, a parameter known as mode-field
diameter (MFD) or spot size is used. In a SMF light
travels mostly within the core and partially within the
cladding. MFD is a function of the wavelength.
• Spot size given as:
• It can be shown that for small values of V (normalize
frequency), the light beam extends into the cladding.
Therefore susceptible to bend losses. To reduce this, V
is made normally between 2 - 2.4
4 October 2018 6
6
2
3
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2
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1
65
.
0 



 V
V
a
wo

7. Taiz University, YEMEN
• Mode-field diameter (MFD)
• Distribution of the beam’s intensity in a single mode fiber.
Core diameter 8.3 μm and MFD is typically 9.3 μm.
4 October 2018 7

8. Taiz University, YEMEN
• The beams travel at distinct propagating angles ranging from zero to
critical value.
• These different beams are called modes.
• The smaller the propagating angle, the lower the mode.
• The mode traveling precisely along the axis is the zero-order mode
or the fundamental
• Hence for the transmission of a single mode the fiber must be
designed to allow propagation of only one mode, whilst all other
modes are attenuated by leakage or absorption.
• This may be achieved through a suitable choice of normalized
frequency, V for the fiber. For single mode operation, only the
fundamental TE01 mode can exist. The cut-off normalized frequency
for the TE01 mode occurs at V = 2.405. Thus single mode
propagation is possible over the range:
4 October 2018 8

9. Taiz University, YEMEN
• This parameter can also be expressed in terms of numerical aperture and
relative refractive index difference Δ as:
where a = core radius
4 October 2018 9
405
.
2
0 
 V
)
(
2
NA
a
V



2
1
1 )
2
(
2

 n
a
V


2
2
2
1
2
n
n
a
V 




10. Taiz University, YEMEN
• Figure shows a multimode step index fiber with a core diameter of
around 50 µm or greater, which is large enough to allow the
propagation of many modes within the fiber core
• Multimode step index fibers allow the propagation of a finite
number of guided modes along the channel.
• The number of guided modes is dependent upon the physical
parameters (i.e. relative refractive index difference, core radius) of
the fiber and the wavelength of the transmitted light.
4 October 2018 10

11. Taiz University, YEMEN
• It can be shown that the total number of guided modes (or mode volume)
Ms, for a step index fiber is related to the V value for the fiber by the
approximate expression:
𝑀𝑠 ≅
𝑉2
2
• which allows an estimate of the number of guided modes propagating in a
particular multimode step index fiber.
Lower bandwidth applications multimode fibers have several advantages over
single mode fibers:
1. The use of spatially incoherent optical sources (e.g. most light emitting
diodes) which cannot be efficiently coupled to single mode fibers.
2. Larger numerical apertures, as well as core diameters, facilitating easier
coupling to optical sources
3. Lower tolerance requirements on fiber connectors.
4 October 2018 11

12. Taiz University, YEMEN
• GRIN fibers do not have a constant refractive index in the core but a
decreasing core index n(r) with a radial distance from a maximum value of
n1 at the axis to a constant value n2 beyond the core radius, a in the
cladding.
• This index variation may be represented as:
4 October 2018 12
   
 
 













)
(
;
2
1
)
(
;
/
2
1
2
2
1
1
2
1
1
cladding
a
r
n
n
core
a
r
a
r
n
r
n


13. Taiz University, YEMEN
• Δ is the relative refractive index difference and α
is the profile parameter which gives the refractive
index profile of the fiber core.
• The equation above is a convenient method of
expressing the refractive index profile of the fiber
core as a variation of α allows representation of
• Step index profile when α = ∞, a parabolic
profile when α = 2 and a triangular profile when α
= 1.
• This range of refractive index profiles is illustrated
in next figure
4 October 2018 13

14. Taiz University, YEMEN
• Possible fiber refractive index profile for
different values of α
4 October 2018 14

15. Taiz University, YEMEN
• The refractive index profile and ray
transmission in a multimode GRIN fiber
4 October 2018 15

16. Taiz University, YEMEN
• The graded index profiles which at present produce the best
results for multimode optical propagation have a near
parabolic refractive index profile core with α= 2.
• A multimode graded index fiber with a parabolic index profile
core is illustrated in previous figure. It may be observed that
the meridional rays shown appear to follow curved paths
through the fiber core.
• Using the concepts of geometric optics, the gradual decrease
in refractive index from the center of the core creates many
refractions of the rays as they are effectively incident on a
large number of high to low index interfaces.
4 October 2018 16

17. Taiz University, YEMEN
• This mechanism is illustrated in below Figure where a ray is shown to be
gradually curved, with an ever-increasing angle of incidence, until the
conditions for total internal reflection are met, and the ray travels back
towards the core axis again being continuously refracted.
4 October 2018 17
An expanded ray diagram showing refraction at the various high to
low index interface within a graded index fiber given an overall curved ray path

18. Taiz University, YEMEN
• Parameter defined for the step index fiber may also be applied to
graded index fibers and give a comparison between them.
• However, for graded index fibers the situation is more complicated
since the numerical aperture is a function of the radial distance
from the fiber axis.
• Graded index fiber therefore accept less light than corresponding
step index fibers with same relative refractive index difference.
• To support single mode transmission in a graded index fiber, the
normalized frequency is:
4 October 2018 18
2
1
2
1
405
.
2 








g
V

19. Taiz University, YEMEN
For the parabolic profile, the numerical aperture is given
by;
𝑁𝐴 = 𝑛1(2𝛥)
1
2 1 − (
𝑟
𝑎
)2
This shows that the NA is a function of the radial distance
from the fiber axis (r/a).
For n1 = 1.48 and n2 = 1.46, the axial NA is 0.24 and this is
equal to the NA of a step index fiber with same n1 and
n2.
The NA drops to zero at the edge of the core
4 October 2018 19

20. Taiz University, YEMEN
• Therefore, it is possible to determine fiber
parameters which will give single mode operation.
• For multimode graded index fibers, the total number
of guided modes, Mg is also related to the V value for
the fiber by the approximate expression:
4 October 2018 20
















2
2
2
V
M g



21. Taiz University, YEMEN
A multimode step index fiber with a core diameter of 80 μm and a
relative index difference of 1.5% is operating at a wavelength of
0.85μm. If the core refractive index is 1.48, estimate: (a) the normalize
frequency of fiber; (b) the number of guided modes.
Answer: (a) 75.8, (b) 2873
A graded index fiber has a core with a parabolic refractive index profile
which has a diameter of 50μm. The fiber has a numerical aperture of
0.2. Estimate the total number of guided modes propagating in the
fiber when it is operating at wavelength of 1 μm
Answer: V=3.14, Mg=247 guided modes
4 October 2018 21
Example (1)
Example (2)

22. Taiz University, YEMEN
A graded index fiber with a parabolic refractive index profile core has a
refractive index at the core axis of 1.5 and a relative index difference
of 1%. Estimate the maximum possible core diameter which allows
single-mode operation at a wavelength of 1.3 μm
Answer: V=2.4(2)0.5, a=3.3 μm, the diameter= 6.6 μm
4 October 2018 22
Example (3)

23. Taiz University, YEMEN
• The number of modes Ms , transmitted by a step-index fiber,
depends on NA, core diameter D, and the wavelength of light, λ
• Maximum allowable core diameter D for single-mode transmission
is
• To ensure single-mode transmission
 Making the core thin enough
 Making the refractive index difference between core and cladding
small enough
 Using a longer wavelength
4 October 2018 23
 
2
2
2
1
2
2
2
5
.
0
.
2
5
.
0
2
n
n
D
NA
a
V
Ms 



















2
2
2
1
405
.
2
n
n
D





24. Taiz University, YEMEN
• The cut-off wavelength is wavelength above which a
particular fiber becomes single mode. If Vc is the cut-
off normalized frequency and V is the normalized
frequency corresponding to wavelength λ, then the
cut-off wavelength λc, is given by:
𝜆𝑐
𝜆
=
𝑉
𝑉
𝑐
• For a step index fiber the cut-off wavelength is given
by:
𝜆𝑐 =
𝑉𝜆
2.405
=
2𝜋𝑎. 𝑁𝐴
2.405
4 October 2018 24

25. Taiz University, YEMEN
Determine the cutoff wavelength for a step index fiber to exhibit single
mode operation when the core refractive index and radius are 1.46
and 4.5 μm, respectively, with the relative index difference being
0.25%.
Answer: λc= 1214 nm
Hence the fiber is single-moded to a wavelength of 1214 nm
What is the maximum allowable core diameter for a step-index single-
mode fiber operating at 1.3 μm, with core index of 1.5 and cladding
index of 1.0003? If the operating wavelength is 1.5 μm, will the fiber
designed above still be operating as single-mode?
Answer: D=0.89 μm
If the operating wavelength is 1.5 μm, the above fiber is still single-mode
4 October 2018 25
Example (1)
Example (2)

26. Taiz University, YEMEN 26
4 October 2018