optical characteristic of waveguide in fused silica written by femtosecond laser.

1. Dielectric
waveguide
Light guide by low index cladding
waveguide characterisation
- Uma Shankar

2. Dielectric wave guide: glass processing
Photonics crystal fibre also called micro structured fibre are of two types,
One is photonic band gap fibre confine light using band gap rather than index guiding
and another is index – guiding photonic crystal fibre
in which the periodic structure is not employed for its band gap but rather to form an effective low index cladding around the core.
Figure: three example of photonic crystal fibre (a) Bragg fibre or photonic band gap fibre, 1-d periodic cladding of
concentric layers.
(b) 2-d periodic structure, triangular lattice of air holes and (c) holey fibre that confines light in a solid by index
guiding

3. • Dielectric waveguide or Optical fibre based on modes will be of three type.
• 1. Single mode optical fibre
• 2. multi-mode optical fibre
• 3. multi-mode graded index fibre
•

4. Here, we will characterise only single mode optical fibre by index guiding, in single mode
wave guide refractive index of core will be greater than refractive index of cladding.
CCore > cladding

5. • Modes in waveguides
• By Maxwell equation for hollow metallic waveguides, only transverse electric mode (TE) and transverse
magnetic (TM) mode are found. But for optical fiber core cladding boundary condition led to coupling
between the electric and magnetic field components.
• This will lead to hybrid modes, which make optical waveguide analysis more complex than metallic
wave guide. the hybrid mode are designed as HE or EH modes, depending on weather the transverse
electric field ( E field ) or transverse magnetic field ( H field ) is larger for that mode.
• Modes in fibre
• Fibre usually constructed such that the difference in core and cladding indices is exceedingly small i.e.
• n1-n2 <<1
• where n1 is core refractive index and n2 is clad refractive index
• mode LP (linearly polarized)

6. weakly guiding modes in fibres
i.e. n1-n2<<1
ELP = Elm(r, φ) . Exp j(ωt - ßlmZ)
Field pattern travelling wave
Eand B mutually perpendicular to each other and travelling along – Z
Where l is number of variations along azimuthal direction like number of
maxima along r starting from core center.
and m number of variations along the radial direction.
Fundamental mode is LP01 mode, l=0 and m=0,
propagate through a single mode fiber and intensity distribution of this LP01
mode is like gaussian distribution.

7. Numerical aperture
NA = 𝑛2
𝑐𝑜𝑟𝑒 − 𝑛𝑐𝑙𝑎𝑑𝑑𝑖𝑛𝑔
2
; for straight fiber
Maximum acceptance angle αmax is that which just gives total internal reflection at the core cladding interface
i.e., when α= αmax then θ=θc Rays with α > αmax become refracted and penetrate the cladding and eventually lost.
2αmax =total acceptance angle
Sin αmax =
𝒏𝟐
𝒄𝒐𝒓𝒆−𝒏𝒄𝒍𝒂𝒅𝒅𝒊𝒏𝒈
𝟐
𝒏𝒂𝒊𝒓

8. Cut off wavelength
The Cut off wavelength for single mode fiber is source wavelength at which the fiber allow only
a single mode LP01 to propagate.
if λ<λc ; then fiber allow higher order of modes propagate.
if λ>𝜆c ; then only mode LP01 will propagate
Normalized frequency V
V-number, V=
2𝜋𝑎.𝑁𝐴
𝜆
=
2𝜋𝑎. 𝑛2
𝑐𝑜𝑟𝑒−𝑛𝑐𝑙𝑎𝑑𝑑𝑖𝑛𝑔
2
𝜆
For V-number
If V < 2.405 only 1 mode exist, fundamental mode LP01
Or, V< 2.405 and λ>λc single mode fiber
And V> 2.405 multiple mode allowed to propagate LP01 , LP21 , LP11, LP02
Number of modes, M = V2/2

9. Cut off wavelength
For rectangular waveguide
λ c =
2
𝑚2
𝑎2 +
𝑛2
𝑏2
and it will be for circular waveguide 𝜆𝑐 =
2
𝑚2+𝑛2
𝑎2
Calculation of NA and V-number of fiber
If cut off wavelength is 1260 nm
For source wavelength 650 nm (λ<λc fibre allow higher order of modes propagate)
Core index 1.45
Cladding index 1.44
NA (numerical aperture) 0.17
V-number 8.2
Cut off wavelength is 1260 nm

10. For source wavelength 650 nm( give multiple mode only because
Sr no. Radius of core (in µm) V-number
,
V=
2𝜋𝑎.𝑁𝐴
𝜆
V=1.642·a
Number of mode
M=V2/2(approx.)
1 5 8.21 33
2 10 16.42 134
3 15 24.63 303
4 20 32.84 539
5 25 41.05 842
6 30 49.26 1213

11. For source wavelength 1260 nm (gives multiple mode above 5µm radius)
Sr no. Radius of core (in µm) V-number
,
V=
2𝜋𝑎.𝑁𝐴
𝜆
V=0.847·a
Number of mode
M=V2/2(approx.)
1 5 4.235 8
2 10 8.47 35
3 15 12.7 80
4 20 16.94 143
5 25 21.7 235
6 30 25.41 322

12. If the core radius is lesser than 5 micrometre, we will get single mode optical fiber
Like for 2 micrometre V= 1.694 which is lesser than 2.405 , so there will be only one
fundamental mode LP01( single mode propagation )
For a single mode fibre, we can calculate mode field diameter by
MFD= 2.a.(0.65 +1.619/V1.5 +2.879/V6)
Here, given core radius 2 micrometre
V- number is 1.694.
Therefore, MFD = 2 × 2 ×10-6 × { (0.65 + 1.619/(1.694)1.5+2.879/(1.694)6}
= 6 × 10-6
=6 micrometre
A low V- number makes fiber sensitive to micro bend losses and to absorption losses to
cladding, however a high V number may increase scattering losses in core or core cladding
interfaces.

13. If cut off wavelength is 1260 nm
Then for source wavelength 1260 nm
For 1260 nm , the refractive index of SiO2 is ncore= 1.4474
And the refractive index of cladding is ncladding= 1.4446
Then nuclear aperture NA= 𝑛2
𝑐𝑜𝑟𝑒 − 𝑛𝑐𝑙𝑎𝑑𝑑𝑖𝑛𝑔
2
= 1.4474 2 − (1.4446)2=0.08997
V-number for 1 micron radius V=
2𝜋𝑎.𝑁𝐴
𝜆
=
2×3.14×1×10−6×0.08997
1260×10−9 =0.448

14. For source wavelength 1260 nm, refractive index ncore=1.4476, ncladding=1.4446, NA= 0.08997
Sr no. Core radius (in µm) V-number
V=
2𝜋𝑎.𝑁𝐴
𝜆
V=0.448·a
No of modes
M=V2/2(approx.)
Type of mode
1 1 0.448 (0.1) No mode -
2 2 0.896 (0.8) No mode -
3 3 1.344 (0.9) no mode -
4 4 1.792 (1.6) one mode One fundamental mode(single mode
fiber)
5 5 2.24 (2.5) two modes Single mode fiber
6 6 2.688 (3.61) three modes Multiple mode fiber
7 7 3.136 (4.91) four modes Multiple mode fiber
8 8 3.584 (6.42) six modes Multiple mode fiber
9 9 4.032 (8.12) eight modes Multiple mode fiber
10 10 4.48 10 modes Multiple mode fiber