Fibre Parameters using Variational Formalism

1. By..
Prof Dr. Pratap Kumar Bandyopadhyay
“ SIMPLE APPROACH OF FINDING OUT
FIBER PARAMETERS AND
CHARACTERISTICS USING MARCUSE SPOT
SIZES USING VARIATIONAL FORMALISM”

2. 2
CONTENTS
AIM & INTRODUCTION
ANALYSIS
RESULTS AND SISCUSSIONS
CONCLUSION
ACKNOWLEDGMENT
REFERENCES

3. 3
AIM & INTRODUCTION
Aim: To find out fiber parameters and characteristics….
When?
We have only known spot size for particular normalized
frequency of a single mode fiber
How?
By applying Variational Method along with Marcuse
formulation

4. 4








1
;
1
);
(
)
(
)
( 2
2
2
2
2
1
2
1
2
R
n
R
R
f
n
n
n
R
n






1
;
1
1
;
)
(
R
R
R
R
f
q
Fig: Refractive index profile
n1
n2
a b
r
n(r)
R=r/a, Normalized radius
q profile exponent
ANALYSIS

5. 5












0
2
0
2
1
2
1
0
2
1
2
2
2
RdR
RdR
RdR
V
dR
R
V
U
q




Normalized propagation constant, U is
calculated using the following expression:
Our chosen wave function:
)
exp(
)
( 2
2
V
w
R
A
R 


wV : Fiber spot size

6. 6
Variational expressions:
2
2 2
1
ln
2
1
V
V
w
V
U 



for step profile:
for parabolic profile:   


2
1
2
2
2
2


 
e
V
UV
2
1
V
w


2
2
1
_ V
w
e
co
f



To calculate the Power
in the core:

7. 7
Marcuse Formulae:-
6
2
3
879
.
2
619
.
1
65
.
0
V
V
a
wM



for step profile:
for graded profile: 6
2
3
)
2
(
2
V
C
V
B
V
A
a
w
q
M




 






































418
.
0
19
.
4
077
.
0
298
.
0
2
1
6
5
76
.
3
1
478
.
1
1
2
4
1
(
5
2
q
q
q
e
C
e
e
B
q
A

8. 8
V UN UM UV
1.5 1.316888 1.276084 1.345708
1.9 1.492849 1.460036 1.511194
2.2 1.591060 1.567540 1.605277
2.4 1.645312 1.628303 1.658595
2.8 1.734204 1.730295 1.749068
TABLE: 1
COMPARISON OF DIFFERENT U VALUES
FOR DIFFERENT V VALUES FOR STEP INDEX FIBER
RESULTS AND DISCUSSIONS

9. 9
V UN UM UV
2.1 1.9077395 1.945949 2.050669
2.5 2.1431626 2.170102 2.220538
3.0 2.3941375 2.411598 2.414642
3.5 2.6124428 2.624616 2.625601
TABLE: 2
COMPARISON OF DIFFERENT U
VALUES FOR DIFFERENT V VALUES
FOR PARABOLIC INDEX FIBER

10. 10
Fig.2: Variation of Power in core (f_co) with Normalized
Frequency (V) for Parabolic index Fiber

11. 11
CONCLUSION
Variational formulation coupled to Marcuse
spot size is shown to predict propagation
characteristics of single mode fiber in simple
and effective way when only spot size is known
theoretically or experimentally.

12. 12
[1] A. Ghatak and K. Thyagarajan, “Introduction to Fiber Optics”,
Cambridge University Press, 1998.
[2] A. W. Snyder and J. D. Love, “Optical Waveguide Theory”, Chapman
and Hall Ltd., New York (1983).
[3]S.K. Khijwania, V.M. Nair and S.N. Sarkar, Propagation Characteristics
of Single–Mode Graded-Index Elliptical Core Linear and Nonlinear Fiber
Using Super-Gaussian Approximation, Appl. Opt., 48, G156-G162, (2009).
[4] D. Marcuse, “Gaussian approximation of the fundamental modes of
graded-index fibers”, J. Opt. Soc. Am., Vol. 68, No. 1, 103-109, (1978).
[5]A. Sharma, S. I. Hosain and A. K. Ghatak “The fundamental mode of
graded-index fibers: simple and accurate variational methods,” Optical
and Quantum Electronics,. vol. 14, 7-15, (1982).
REFERENCES

13. 13