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1	
EECS	285A	HW1	SOLUTIONS	
	
Agrawal	Fiber-Optic	Communication	Systems	4th	Edition	
Chapter	3	Questions
	
	
	
	
	
	
	
	
	
	
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	 2	
Extra	Problem	to	explain	solution	of	Problem	3.9:	
	
	
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	 3	
	
	 	
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	 4	
Q-1.	Consider	a	semiconductor	LED	operating	at	800nm	with	3.5	refractive	index	
value,	radiative	recombination	time	of	10ns	and	total	non-radiative	recombination	
time	of	50ns.	50mA	current	is	injected	into	the	diode.		
Calculate:		
a.	Internal	quantum	efficiency,	and	external	quantum	efficiency
	
b.	Total	optical	output	power
	
c.	Steady	state	carrier	density	at	50mA	current	injection	rate	(assume	volume	of	the	
active	region	is	10-4	cm3).
	
d.	Modulation	bandwidth	if	AC	current	is	applied.	
	
a) 	
𝜂"#$ =
𝑅''
𝑅'' + 𝑅#'
=
1
10
1
10 +
1
50
=
5
6 = 𝟖𝟑. 𝟑%	
𝜂12$ = 𝑛45(𝑛 + 1)48 =
1
3.5 ×
1
4.58 = 𝟏. 𝟒%	
b) 		
𝑃"#$ = 𝜂"#$ℏ𝜔
𝐼
𝑞 =
5
6
1.24
0.8 	𝑒𝑉
50𝑚𝐴
𝑞 = 64.58	𝑚𝑊	
𝑃KL$ = 𝜂12$𝑃"# =
1.4
100 	64.58	𝑚𝑊 = 𝟎.𝟗	𝒎𝑾	
c) 		
𝜂QQ =
𝐼
𝑞 𝜏	
𝜏45 =
1
𝜏''
+
1
𝜏#'
=
1
10 +
1
50 =
6
50	𝑛𝑠
45	
𝜂QQ =
𝐼
𝑞 𝜏 =
50 × 104T
1.6 × 1045U
𝑐𝑎𝑟𝑟𝑖𝑒𝑟𝑠
𝑠 	
50
6 × 10
4U𝑠 = 2.6 × 10U	𝑐𝑎𝑟𝑟𝑖𝑒𝑟𝑠	
𝜂 =
2.6 × 10U
104Z
𝑐𝑎𝑟𝑟𝑖𝑒𝑟𝑠
𝑐𝑚T = 2.6 × 10
5T 𝑐𝑎𝑟𝑟𝑖𝑒𝑟
𝑐𝑚T 	
d) Optical	bandwidth	
𝑓T\] = √3(2𝜋𝜏`)45 =
√3
2𝜋 506 𝑛𝑠
= 33	𝑀𝐻𝑧	
	
	
	
	 	
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	 5	
Q-2.	Consider	a	Mach-Zhender	modulator	with	two	Y	branches	with	equal	splitting	
ratios.	Derive	the	transmission	function	of	the	modulator	when	it	is	operating	in	
push-push	and	push	pull	modes.	
General	transfer	function	of	MZM	when	perfect	50/50	splitting	is	assumed	will	be	as	
follow		
𝐸KL$(𝑡)
𝐸"#(𝑡)
=
1
2	
fexpf𝑗𝜙5(𝑡)l + expf𝑗𝜙8(𝑡)ll	
where	𝜙"(𝑡) = 𝑢"(𝑡)
n
op
,	assuming	same	𝑉n 	for	both	branches.	
Push-push	mode	operation:	𝑢5(𝑡) = 𝑢8(𝑡) → 𝜙5(𝑡) = 𝜙8(𝑡) = 𝜙(𝑡),	therefore	
𝐸KL$(𝑡)
𝐸"#(𝑡)
= 𝐞𝐣𝛟(𝐭) → 𝑝𝑢𝑟𝑒	𝑃𝑀	
𝑃KL$(𝑡)
𝑃"#(𝑡)
= 𝟏	
Push-pull	mode	operation:	𝑢5(𝑡) = −𝑢8(𝑡) =
L($)
8
→ 𝜙5(𝑡) = −𝜙8(𝑡) =
x($)
8
,	
therefore	
𝐸KL$(𝑡)
𝐸"#(𝑡)
=
1
2yexp y
𝑗𝜙(𝑡)
2 z + exp y−
𝑗𝜙(𝑡)
2 zz = cosy
𝜙(𝑡)
2 z = 𝐜𝐨𝐬 y
𝒖(𝒕)
𝟐𝑽𝝅
𝝅z	
𝑃KL$(𝑡)
𝑃"#(𝑡)
= cos8 y
𝑢(𝑡)
2𝑉n
𝜋z =
𝟏
𝟐 +
𝟏
𝟐𝐜𝐨𝐬 y
𝒖(𝒕)
𝑽𝝅
𝝅z =
1
2 +
1
2 cos
(𝜙(𝑡))	
	
	
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