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In this lecture let us continue our discussion
on laser diodes. What we had seen in the previous
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lecture that the requirement for lasing is
that we need to lift sufficient number of
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atoms from lower laser level to the upper
laser level, and the upper laser level is
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metastable level.
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So, once we have the population inversion
then the system can provide amplification
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by stimulated emission of radiation. However,
this amplification should be sufficient enough
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to overcome the loss of the system, and once
the gain coefficient is such that the gain
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of the amplifier is such that it is able to
overcome the losses then oscillations will
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start or the lasing action will start and
we will have laser.
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Since we have an amplifier and to convert
this amplifier into an oscillator, we need
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to provide feedback and in order to provide
the feedback we enclose the active medium
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between a pair of mirrors, in this way we
form a resonator cavity and this cavity has
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the losses. So, lasing start when one round
trip gain is greater than 1. So, let us now
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look at the resonator cavity.
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We have the active medium and let us say the
length of this active medium is d, the gain
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provided by this medium is gamma, and the
loss coefficient is alpha which constitutes
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scattering and absorption losses.
How to make it a cavity? We enclose it between
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2 mirrors of reflectivities R1 and R2 mirror
1 of reflectivity R1, and mirror 2 of reflectivity
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R2, in a beam of light it starts from here
and it goes through this active medium it
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makes a pass here, then it gets reflected
from mirror 2 makes another pass from 2 to
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1 and then get reflected from mirror 1. So,
from here to here there is one round trip,
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and we should have the net gain of this cavity
greater than 1 in one round trip. If in one
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round trip we have the gain greater than 1
then in successive round trips the gain will
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multiply, and it will give you a net gain
which is always greater than 1 and we will
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have lasing.
So, let us find out what is the net gain.
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So, we start from here with intensity I0 and
by the time it reaches here just before hitting
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this mirror 2, the light intensity gets amplified
by gain coefficient gamma and it is attenuated
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with loss coefficient alpha. The intensity
just before hitting this mirror is I0 times
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e to the power gamma d times e to the power
minus alpha d. When it gets reflected from
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this mirror, then because of the finite reflectivity
of this mirror 2 which is R2, the intensity
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becomes I0 e to the power gamma d times e
to the power minus alpha d times R2.
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Now, this intensity starts from this mirror
and then goes towards mirror 1 then just before
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hitting this mirror 1 the intensity will become
I0 e to the power gamma times 2d times e to
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the power minus alpha times 2d times R2, and
after getting reflected from mirror 1 it will
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be multiplied by this reflection coefficient
R1. So, after one round trip the net intensity
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the total intensity would of the light beam
would be this. So, we start with I0 and after
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one round trip the intensity becomes this.
So, what is the gain? So, one round trip gain
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is of course, this divided by I0. So, the
one round trip net gain would be e to the
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power gamma times 2d times e to the power
minus alpha times 2d times R2 times R1.
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So, for lasing to happen this round trip gain
should be greater than 1 which means this
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should be greater than 1 or I can say that
the gain coefficient gamma should be greater
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than a particular value gamma th which I call
threshold gain. What is threshold gain? The
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threshold gain comes from the equality sign
from here. So, from here I can get what is
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the threshold gain. Sometimes even if you
do not put mirrors here mirror 1 and mirror
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2 of reflectivities R1 and R2, then the cleave
surfaces of this medium can also act as mirrors
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of reflectivity R is equal to n minus 1 divided
by n plus 1 whole square, where n is the refractive
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index of this medium and outside medium is
a R. So, this reflection coefficient is nothing,
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but the final reflection coefficient.
So, this reflectivity can also provide feedback
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to the cavity and if the round trip gain is
greater than 1 then it will start lasing.
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So, what is the requirement for lasing? Requirement
for lasing is gamma should be greater than
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gamma th, what is the requirement for amplification?
Requirement for amplification is there should
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be population inversion, but this is not sufficient.
Our population inversion should be so much
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that the gain provided gamma should be greater
than a threshold value, which is able to overcome
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the losses.
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So, this threshold gain is that gain that
exactly compensate for losses, which means
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that e to the power gamma th if this equality
sign is a gamma is equal to gamma th, then
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e to the power gamma th times 2d times e to
the power minus alpha 2d times R2R1 should
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be greater than should be equal to 1 or e
to the power gamma th 2d should be equal to
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e to the power alpha 2d divided by R2R1 or
I can write it in this fashion that gamma
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th is equal to alpha minus 1 over 2d log R1
R2.
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So, what I see here this gain coefficient
gamma th is nothing, but the loss coefficient
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is scattering at absorption loss coefficient
plus the finite reflectivity of the mirror,
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the loss coefficient due to finite reflectivity
of the mirror. Please see that this minus
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sign does not mean that I am subtracting the
loss due to finite reflectivities of the mirror
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from scattering and absorption losses no because
R1 and R2 r smaller than 1. So, this log R1R2
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is minus. So, overall effect is this plus
this. Correspondingly I should have a current
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density to acquire this much of gain and that
current density is known as threshold current
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density, and correspondingly I should have
threshold injection current.
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So, what I should do in my laser diode? I
should provide injection current and when
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I provide injection current if the injection
current is low then the population inversion
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is not sufficient the gain coefficient is
not sufficient to overcome the losses, and
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lasing does not start. When I slowly increase
this current then as soon as the gain provided
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overcomes the losses, as soon as the gain
provided is greater than gamma th, the lasing
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starts.
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So, in a typical laser diode the pumping is
by injection current. So, you provide forward
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bias current to this and when this current
is greater than the threshold current or the
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game provided is greater than the threshold
gain, then the lasing start. You have stimulated
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amplification and then because of feedback
oscillation. So, laser starts working
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Typically for gallium arsenide laser diode
d is about half a millimeter, and if you do
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not put even the mirrors at the end of the
cavity, the cavity is formed just by cleave
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surfaces then R1 R2 have typical value 30
percent, and scattering absorption loss coefficient
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is 5 millimeter inverse, then if you calculate
the threshold gain then it comes out to be
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about 74 centimeter inverse. So, if I look
at now the output power works the injection
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current characteristic of the laser diode,
then what I would see initially when the current
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is low then I will have very small power coming
out.
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As I increase the current the power would
increase, but is still it would not be very
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high and if I examine this then this radiation
is not coherent radiation, which means this
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is not a laser this light is coming out due
to the process of a spontaneous emission.
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So, in this region the laser diode works just
like an LED. As soon as the current crosses
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the threshold value the power rises sharply
and this is laser, this is laser action this
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is the power which is coming out of the process
of stimulated emission.
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So, this region below threshold this is the
region in which the laser diode operates as
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an LED, and in this region it operates as
an LD or laser diode. Typically ith is 25
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to 250 milliamps and the output power that
you get is 1 to 10 milliwatts. And important
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characteristic of a laser diode is how much
power would you produce per unit injection
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current. There is what is the slope of this
line and this is known as slope efficiency
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denoted by dP over di, another important characteristic
is differential external quantum efficiency
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what it is? It is simply how many photons
you would produce per unit injected electrons.
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So, if you increase current around some value
by an amount di and correspondingly the output
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power increases by an amount dP, then the
number of photons that are produced in this
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is dP over h nu, and the number of electrons
injected per unit time are di over e. So,
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this differential external quantum efficiency
is the number of photons produced per unit
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time divided by the number of electrons injected
per unit time, and it has to be smaller than
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1 because we know that not all the electrons
injected will result in radiative recombination.
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So, not all the electrons injected will result
into the photons. So, this gives you a finite
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is smaller than 1 differential external quantum
efficiency. So, this I can write as by just
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rearranging the terms as e over h nu dP over
di. So, I can write this as in terms of dP
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over di or slope efficiency. Now what happens
if I change the temperature of a laser diode?
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So, what I do I take a laser diode, and then
see how the P-i characteristics look like
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at 0 degree temperature, then I increase the
temperature to 10 degrees, 30 degrees, 50
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degrees and so on and see the output characteristics
the P-i characteristics, and what do I notice
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is that when I increase the temperature the
threshold current increases.
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And if I fit an empirical relation between
the temperature and the threshold current
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then this relationship fits very well this
empirical relationship fits very well to the
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experimentally observed data and it is ith
at temperature T is equal to i0 e to the power
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T over T naught, where i naught and t naught
are the characteristics of a particular material
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or laser diode. T0 is known as characteristic
temperature and it denotes the temperature
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dependency of ith on T of course, higher the
value of T naught lower would be the temperature
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dependency.
Typically in gallium aluminum arsenide laser
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diode if I change the temperature by 1 degree,
then ith increases by 0.6 to 1 percent, and
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in indium gallium arsenide phosphide laser
diode one degree change in temperature results
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in the change in ith by 1.2 to 2 percent.
Now let us look at output spectrum of a diode
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laser. So, I take a diode laser and observe
its spectrum. So, what I do? I take a diode
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laser and couple its output to a spectrum
analyzer, and then I slowly increase the current
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of this diode laser. So, initially when the
current is small my optical spectrum analyzer
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shows this kind of graph.
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So, what I see? I see a very broad spectrum
and the power level is not very high, when
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I increase the injection current then I again
have the broad spectrum which slowly narrows
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down and the peak power increases, I further
increase the injection current they spectrum
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again slightly narrows down and then the peak
power again increases. But in all these values
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of current I still below the threshold, and
this is typical output characteristic of an
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LED. So, below threshold current I will get
out of a spectrum which is similar to that
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of an LED.
When I increase the current beyond threshold
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then I see this kind of spectrum. So, this
output in spectrum now transforms to these
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kind of sharp peaks equi-spaced sharp peaks,
and the power level is much much higher than
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the power level here. This is typical output
spectrum of a diode laser, what would these
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peaks correspond to? These peaks correspond
to multi longitudinal modes of the cavity
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it is nothing just Fabry-perot cavity actually.
You have a resonator you have optical energy
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confined between 2 mirrors then you basically
make a Fabry-perot cavity.
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And you get an interference pattern something
like this you will have certain frequencies
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you have certain frequencies which have 2
pipe a shift in a round trip and they go like
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this they are nothing, but just like if you
take the analogy of a stretched string, then
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they are just like the modes of a stretched
string what is the frequency of a longitudinal
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mode.
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So, this can be very easily shown that the
frequency of a mode can be given by c by 2
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and d times q where q as an integer, and n
is the refractive index of the medium and
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it is at frequency nu, d is the length of
the cavity.
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What is the spacing can I find the spacing
between the 2 modes? Yes because I know what
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is the frequency of a mode then I can find
out the spacing easily. So, if this particular
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line for example, occurs that the value of
q is equal to q naught, then this nu would
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be c by 2 n nu d times q naught or if I just
put all the frequency dependent terms on one
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side, then this nu times n nu would be equal
to c over to d times q naught. The next one
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the immediate next one would occur it q naught
plus 1 and let us say the frequency of this
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is nu plus delta nu that is line spacing is
delta nu.
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Then this new plus delta nu would be c by
2 n nu plus delta nu times b times q naught
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plus 1. Again I put all the frequency dependent
terms on this side, and I get nu plus delta
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nu times n of nu plus delta nu is equal to
c over 2d times q naught plus 1. Let me subtract
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this from this and. So, I will get nu plus
delta nu times n nu plus delta nu minus nu,
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nu is equal to c over 2 d, and now the task
is to find out delta nu from this side.
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So, for that what I can do? I know this delta
nu is much much smaller than nu then I can
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make a Taylor series expansion of n nu plus
delta nu around n nu. So, it would be n nu
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plus delta nu d n over d nu, and I retain
only first order term because the delta nu
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is much much smaller than nu.
Then this which I have in the previous slide
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nu plus delta nu times nu plus delta nu minus
nu and nu is equal to c over 2d. So, if I
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put this and nu plus delta nu here, and simplify
then open the bracket then what do I get?
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I get this. Now I can see here that this n
nu times nu would get cancelled from this
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term, and this delta nu is very small. So,
this term I can approximate to 0 I can neglect
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this term. So, this gives me delta nu times
n plus nu plus n plus nu times dn over d nu
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is equal to c over 2d or if I take n outside.
So, delta nu times n would be 1 plus nu by
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n dn over d nu is equal to c over 2d or delta
nu is equal to c over 2 and d times 1 plus
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nu over and d and over d nu inverse.
So, this is how I can get the line spacing;
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of course, if I neglect the dependence of
refractive index of the medium on frequency
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then it is simply c by 2 and d, but if this
dependence is strong enough then I will have
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to take this factor also into account. Typically
n is 3.6, d is 250 micron and if nu over and
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dn over d nu is 0.38 around lambda is equal
to 0.85 then delta nu would be 121 gigahertz.
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If I put all these values here, I can also
convert it into delta lambda because when
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I look at the spectrum in a spectrum analyzer
I see power as a function of wavelength. So,
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it is more appealing to look into delta lambda
as compared to delta nu.
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So, this delta lambda would be around 0.3
nanometers. How can I get a single longitudinal
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mode laser? Of course, now I have a laser,
but it has got several lines, ideally I should
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have a laser which has only one wavelength
and line width as narrow as possible. So,
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how can I have single longitudinal mode laser.
Well if I look at this frequency spacing line
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spacing then it is given by this. So, I can
have a single longitudinal mode laser by 2
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ways, one way is to increase the line separation.
If I increase delta nu in such a way that
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only one mode crosses the loss line in the
gain profile, then only one mode would oscillate.
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And how can I increase this line separation?
I can increase this line separation by reducing
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the value of d, but the problem is that if
I reduce the value of d, I also reduce the
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volume of the gain medium. So, I will have
to increase threshold current.
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If I look at this example here this green
curve shows the gain profile and the red line
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shows the loss line. So, all the boards which
are above this loss line will be oscillating.
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Now if I increase the line spacing such that,
only one mode falls above this line then only
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this one will survive. Typical example is
that if gain bandwidth above loss line is
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delta nu g that is this, which is 500 gigahertz
and nu over and dn over d nu is 0 0.38, then
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to a single longitudinal mode line spacing
is delta nu here delta nu here. So, this 2
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delta nu should be greater than delta nu g
then only one mode would be above this line.
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Accordingly your d should be less than c over
n delta nu g times this, and if you plug in
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all these numbers. So, this will give you
the value of d which should be smaller than
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120 micrometer. So, this reduces the volume
of the gain medium significantly another way
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of doing.
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This is that you use mirrors which are highly
wavelength selective. You use highly wavelength
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selective mirrors and we know from our previous
knowledge of Bragg gratings, that if I have
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a Bragg grating then this Bragg grating reflect
a particular wavelength and wavelength selectivity
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is very good ok.
I can have line width which is about 0.1 nanometer
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or 0.2 nanometers. So, I can put these kind
of gratings Bragg gratings either at the end
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of the gain medium, outside the gain medium
or I can integrate it over the entire gain
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medium. This kind of structure is known as
distributed Bragg reflector DBR laser, and
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this kind of a structure is known as distributed
feedback laser or DFB laser. What kind of
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periodicity you will require here? If you
have lambda be around 1550 nanometer and an
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effective is typically 3.6 which is close
to the refractive index of the medium, then
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the periodicity would be around 215 nanometers
and typical line width you will get here is
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of the order of 0.1 or 0.2 nanometers.
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Another characteristic of a laser diode is
its radiation pattern. So, how of course,
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we know the laser is highly directional, how
directional the laser is that we get out of
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laser diode, how directional the laser diode
is. So, if I look at this the light coming
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out of this then because of the structure
of this, this is the multilayer structure
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and light is coming from a very thin region
here very small region. So, this is the lazing
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spot and since it is a small then it diffract.
So, in this direction the dimensions are smaller.
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So, diffraction would be larger in this direction
the dimensions are larger. So, diffraction
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would be less. So, accordingly I will have
the radiation pattern which has theta parallel
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about 5 to 10 degrees, and theta perpendicular
which is 30 to 50 degrees which is of course,
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much much better than that we get in the case
of LED. The intensity pattern is Gaussian,
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so, if I look at the mode field distribution.
So, it is given by psi xy is equal to psi
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0 exponential minus x square over wL square,
minus y square over wT square. So, in both
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the directions x and y direction I have a
Gaussian, but the width is different typically
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wL is 0.5 to 1 micrometer and wT is 1 to 2
micrometer.
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The last thing is that if I want to use this
laser diode in communication, I need to modulate
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it. How do I modulate it? So, for that I look
at P-i characteristics and if my signal current
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goes like this, then when this signal current
crosses this ith my laser diode is switched
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on and when it goes below this then the laser
diode is switched off, and accordingly I will
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get the modulated power like this. So, the
current pulses will get converted into optical
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pulses, but here you see to switch this on
I will have to increase the current to a much
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higher level.
So, my amplitude of the signal has to be very
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high. So, in order to avoid that what I can
do? I can bias this laser diode around threshold
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and then write the signal over this, then
it would be much more efficient. So, for efficient
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modulation of LD. I should always modulate
it near the threshold.
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So, with this I finish the discussion on laser
diode. In the next lecture I will start studying
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the photo detectors the receiver and of the
communication system.
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Thank you.