Page 166 - Photonics Essentials an introduction with experiments

Lasers

160 Photonic Devices

the emission of other photons. These stimulated photons contribute to
the electromagnetic field, creating gain. That is, they must have the
same wavelength and the same phase as the stimulating photon. If it
were otherwise, these photons would interfere destructively with the
electromagnetic field. In order for lasing to occur, the gain initiated by
a photon during one round trip circuit must be greater than the losses
incurred during the same circuit plus the fraction of the intensity that
is emitted through the reflecting mirrors. The lasing threshold is de-
fined as the point when the gain is equal to the loss.
We can treat any loss along the optical path by an effective absorp-
tion coefficient that we will denote by . The gain coefficient k be-
haves just like a loss with the opposite sign. The net laser gain can be
expressed as
R 1 R 2 e (k– )L (7.12)
R 1 and R 2 are the reflectivities of the mirrors at either end of the gain
region, and L is the cavity length. In the case of a semiconductor
laser, R 1 and R 2 are usually the same. During laser operation, the
only variable in this expression is the gain coefficient, which depends
ultimately on the forward current in the diode. Everything else re-
mains constant.
The laser threshold is reached when the net gain is unity. This also
defines the threshold gain:
Threshold gain = 1 = R 1 R 2 e th – )2L
(k
and

1 1
k th = + ln
2L R 1 R 2
In the case where R 1 and R 2 are the same
1
1
k th = + ln (7.13)
L R

Example 7.2
You can determine the necessary gain coefficient in order for laser ac-
tion to occur. A typical length for a semiconductor laser cavity is 400
m. Suppose that cavity losses are 30 cm –1 and the reflectivity of the
cleaved laser facet is 30%. The estimated gain coefficient needed to
reach threshold will be
1
1
k th = 30 + ln = 60 cm –1
0.04 0.3

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