8/10/2019 EE 164 Problem Set 5 Solution

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EE 164 Problem Set 5 Spring 2005

Problem 1: For a GRIN fiber, let 1 1.5n = , 0.01 = , 2= , and 50a m= .

a) Plot ( )n r within the core to scale.

( ) ( ) ( ) ( )2

1 1 2 / 1.5 1 2 / 50 0.01a

n r n r a r = = b) Repeat on the same graph, changing to 10.

( ) ( ) ( )10

1.5 1 2 / 50 0.01n r r=

c) Repeat on the same graph with 0.001 = and 2=

( ) ( ) ( )2

1.5 1 2 / 50 0.001n r r=

/r a ( )a ( )b ( )c 0 1.5 1.5 1.5

0.1 1.4998 1.5 1.4999

0.2 1.4994 1.5 1.49990.3 1.4986 1.5 1.49990.4 1.4976 1.5 1.4998

0.5 1.4962 1.5 1.4996

0.6 1.4946 1.4999 1.4995

0.7 1.4926 1.4996 1.49930.8 1.4904 1.4984 1.4990

0.9 1.4878 1.4948 1.4988

1.0 1.4849 1.4849 1.4985

0 0.2 0.4 0.6 0.8 1.0

1.485

1.490

1.495

1.500

( )a( )b

( )c

/r a

( )r

Problem 2: For a single-mode and a multimode GRIN fiber, 1 1.48n = , and 2 1.46n = ,

wavelength = 0.82 m , linewidth = 20 nm.

1 21.48, 1.46, 0.82 , 20 , 0.0135n n m nm = = = = =

M= material dispersion (from figure in text)

gM = waveguide dispersion (assumed negligible).

a) Compute the 3-dB bandwidth-length product. Neglect waveguide dispersion.

( ) ( ) ( ) ( )/ 110 20 2200gDISL M M ps nm x km nm ps km = + = =

( ) ( )( )1 10 1sin 2 sin 1.5 2 0.01 12.247n = = =

( ) ( ) ( )2 2

/ 2.2456TOTAL DIS MOD

L L L ns km = + =

8/10/2019 EE 164 Problem Set 5 Solution

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( ) ( )

3

1 10.227

2 2 2.2db

DIS

f L GHz kmL ns km

= = =

(single mode)

( ) ( )3

1 10.2227

2 2 2.2456db

TOT

f L GHz kmL

= = =

(multimode)

b) Repeat if linewidth = 1 nm.

c) Repeat of wavelength = 1.5 m , linewidth = 50 nm.

d) Repeat of wavelength = 1.5 m , linewidth = 1 nm.

( )m

( )nm

M

ps

nmkm

( )

( )

/DIS

L

ns km

( )

( )mod

/L

ns km

( )

( )

/TOT

L

ns km

Single

Mode

( )3dbf L

GHz km

Multimod

e

( )3dbf L

GHz km

( )a 0.82 20 110 2.2 0.496 2.25 0.227 0.223

( )b 0.82 1 110 0.11 0.45 0.46 4.55 1.08

( )c 1.5 50 -15 0.75 0.45 0.87 0.667 0.574( )d 1.5 1 -15 0.015 0.45 0.45 33.3 1.11

Note the advantage of the single mode fiber except where an LED is used at the shortwavelength.

Problem 3: A fiber has an NA = 0.2588. A light source coupled to it emits 75% of its light

into a60 full-cone angle, 50% into a30 cone, and 25% in to a15 cone. What is

the coupling efficiency when this source and fiber are connected?

0.2588 sinNA = =

15 ,=

fiber acceptance angelThe source emits 50% of its light into this 30 cone.Efficiency 50% 3dB= =

Problem 4: Compute the bandwidth in Hertz of the S-band, the C-band, and the L-band that

is compute the range from the lowest to the highest frequency in each band.

S-Band: 1460-1530 nm

C-Band: 1530-1565 nmL-Band: 1565-1625 nm

f c =

( )nm ( )f Hz 1460 142.0547 10 1530 141.9608 10 1565 141.9169 10 1625 141.846 10

S-Band: 14 14 122.0547 10 1.9608 10 9.39 10 9.39Hz = = THzC-Band: 4.39 THz

L-Band: 7.09 THz