Biophotonics_Concepts_to_Applications

resistance, the ratio of the output electrical power at the frequencyωto the electric
power at zero modulation is

Ratioelec¼10 log

PelecðxÞ

Pelecð 0 Þ



¼10 log

i^2 ðxÞ

i^2 ð 0 Þ



ð 4 : 5 Þ

where i(ω) is the electrical current in the detection circuitry. The electrical 3-dB
point occurs at that frequency point where the detected electrical power
Pelec(ω)=Pelec(0)/2. This happens when

i^2 ðxÞ

i^2 ð 0 Þ

¼

i^2 out

i^2 in

¼

1

2

ð 4 : 6 Þ

or at the point where i(ω)/i(0) = iout/iin=1/√ 2 =0.707, as is illustrated in Fig.4.10.
Sometimes, the modulation bandwidth of an LED is given in terms of the 3-dB
bandwidth of the modulated optical power P(ω); that is, it is specified at the fre-
quency where P(ω)=P 0 /2 where P 0 is the optical power at zero modulation. In this
case, the 3-dB bandwidth is determined from the ratio of the optical power at
frequencyωto the unmodulated value of the optical power. Since the detected
current is directly proportional to the optical power P, this ratio is

Ratiooptical¼10log

PðxÞ

P 0



¼10log

iðxÞ

i(0)



¼10log

iout

iin



ð 4 : 7 Þ

Theoptical 3-dB pointoccurs at that frequency where the ratio of the currents is
equal to 1/2. As shown in Fig.4.10, this corresponds to an electrical power
attenuation of 6 dB.

Example 4.4Consider the particular LED described in Example4.3, which

has a 5-ns injected carrier lifetime.

(a) What is the 3-dB optical bandwidth of this device?

Electrical 3-dB point

Optical 3-dB point

Electrical bandwidth

Optical bandwidth

Current ratio i

out

/iin

Modulation frequency

1.000

0.707

0.500

Fig. 4.10 Frequency Small-signal region
response of an optical source
showing the electrical and
optical 3-dB-bandwidth
points

104 4 Fundamentals of Light Sources