Download Exercice N°1 : Laser à 4 niveaux

Institut d'Optique, 2nd year and Master 1
François BALEMBOIS
Laser examination 17/12/2009
Laser Examination
3 hours
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Pocket calculator authorized
Problem: Power balance of a continuous wave laser
We consider a Nd:YVO4 laser pumped in continuous wave either at 808 nm or at 914 nm
(fig.1). The laser effect occurs at 1064 nm. With a pumping at 808 nm, the Nd:YVO4 is a four level
system. The transitions 0-3, 0-2, and 1-2 are purely radiative whereas the transitions 3-2 and 1-0
are non radiative (heat generation).
Without pumping, the level 0 is the only one which is populated.
The purpose of this problem is to calculate the balance (or budget) of all the powers emitted
by the laser, considered here as a black box.
Figure 1: simplified scheme of the energy levels in Nd:YVO 4. On the left : levels and wavelengths. On the
right: the different rates and population densities.
We use the following quantities and values :
S : beam surface (pump and signal) in the gain medium,
w : beam radius (S=w2), w=100µm,
d : length of the gain medium, d=1cm,
pass: cavity passive losses, pass=0,01,
T : output mirror transmission, T=0,05,
 : laser wavelength, =1064 nm,
p : pump wavelength, p=808 nm or 914 nm,
A : Einstein coefficient for spontaneous emission, 1/A=100µs,
 : probability per second to decay from level 3 to level 2 and from level 1 to level 0 (this rate is
very high compared to the other rates),
 : cross section at 1064 nm, =7,6 10-19 cm2,
p : pump cross section (at 808 nm or at 914 nm),
IP: pump intensity in photons/s/m2 (at 808 nm or at 914 nm),
IS: saturation intensity at 1064nm,
h : Planck constant (h=6,62 10-34 Js),
n0 : population inversion density (without laser action). It is supposed to be uniform in the
medium,
nth : population inversion density at the laser threshold. It is supposed to be uniform in the
medium,
nt : total population density,
Pabs : absorbed pump power. Pabs =1W in both cases (808 nm and 914 nm),
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Institut d'Optique, 2nd year and Master 1
François BALEMBOIS
Laser examination 17/12/2009
We suppose that the pump power is uniformly absorbed in the gain medium such that all the
population densities are independent of the position in the medium. The pumping is longitudinal
such that the pump beam and the cavity beam have exactly the same shape and the same size
(the overlap between the two beams is perfect).
1) What is the maximum output power achievable from this laser with a pumping at 808 nm and
with a pumping at 914 nm?
2) Give all the ways for the power to go out of the « laser box ».
3) In the tutorial 1, we have calculated the expression of the population inversion density in the
case of a four level system (in the steady state regime and under a continuous wave pumping) :
with
.
and
Give the expression of the population inversion density in the case of a 914 nm pumping (in the
steady state regime and under a continuous wave pumping). Note that the three level-system
considered here differs from the three-level system studied in the course.
In the following part of this problem, we suppose that p IP <<A (at 808 nm and at 914 nm).
4) We suppose that, without laser action, every absorbed pump photon gives a spontaneous
photon at 1064 nm. Show that the relation between n0 and Pabs is
n0 
1 P
PAbs
ASd hc
.
5) Under laser action, give the value of the population inversion density versus nth.
 The cavity is a linear cavity in which
6) We suppose that the small losses approximation is correct.
we neglect the interference effects between the two counter propagating waves.
6.1) Give the expression of the output intensity (the demonstration for this expression is not
requested).
6.2)
expression.

Th
PAbs
is the absorbed pump power necessary to reach the laser threshold. Give its
Pout 
6.3) Show that the output power can be written as :
T
P
Th
PAbsPAbs


T pass 
Th
6.4) Evaluate P Abs numerically (in watts) for a pumping at 808 nm and at 914 nm.
a pumping at 808 nm and at 914 nm.
6.5) Evaluate Pout numerically (in watts) for
7) Give 
the expression of the power
action.
Pspont
corresponding to spontaneous emission under laser
8) Give the expression of Ploss : the laser power lost on the passive losses.
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Institut d'Optique, 2nd year and Master 1
François BALEMBOIS
Laser examination 17/12/2009
9) Give the expression of Pheat : the power lost in heat in the gain medium.
10) Evaluate all the outputs of the laser “box” numerically and present the different values in a
table (under a pumping at 808 nm and at 914 nm). Comment and compare the values.
11) Additional question : we are looking for the highest population density possible n0. We now
consider that we have a high pump power intensity such that p IP >>A (at 808 nm and at 914 nm).
11.1) Give the expression of n0 for a pumping at 808 nm and for a pumping at 914 nm.
11.2) How to explain the difference between the two cases?
Exercise N°1: Realization of a constant population inversion density
We consider a four level system that is pumped longitudinally. The signal and the pump beam have
the same profile and the same size and propagate in the same direction. It is a solid state medium.
The pumping intensity on the pumped face of the gain medium is denoted as IP0.
z is the propagation axis of the beams. The origin of the axis is on this pumped face of the medium.
nt is the total population density in the medium.
All the other quantities are the same as in the problem.
 , the probability per second to decay from level 3 to level 2 and from level 1 to level 0, is very high
compared to the other rates.
1) Preliminary questions
1.1) Demonstrate that in a four level system, the population inversion density n is
approximately equal to the population density of the upper level n2.
dIp
 An0 (z)
1.2) Show that dz
.
2) In the case p I p <<A,
2.1) Give
 the expression of I p (z).
2.2) Give the expression of n0(z), the population inversion density without laser action.
2.3) The previous questions show that the amount of pump intensity deposited per unit lenght
(dIP
/dz) varies. What is the consequence on the evolution on the z axis of the heating of the
 perpendicular to the propagation axis ?
medium in each slice
3) In order to realize both a uniform population inversion density (without laser action) and a
uniform repartition of the heat in the medium, we elaborate the medium with a variable
concentration nt (z).
We are looking for the expression of nt(z) that leads to n0(z)=K (constant).
3.1) Give nt(z) in the case p I p <<A as a function of IP0
3.2) We want an absorption of 90% over a medium length of d. Knowing the concentration
of the medium at the beginning, nt(0), give the concentration of the medium at the end nt(d), in the
case p I p <<A.

3.3) Give nt(z) without the hypothesis p I p <<A.
3.4) When IP0 is very high, show that there is no use to have a concentration gradient on n t.
To which case does it physically correspond ?


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Institut d'Optique, 2nd year and Master 1
François BALEMBOIS
Laser examination 17/12/2009
Exercise N°2: Single frequency oscillator with a very short cavity
We consider a laser oscillator with a linear cavity formed by two mirrors (M1 and M2) with
reflection coefficients R1 and R2, respectively. The optical cavity length is denoted as L.
The spectral shape of the gain G0() is a bell shape centered on the central frequency 0. Its full
width at half maximum is . The spectral broadening is inhomogeneous.
In this exercise, we would like to find trends and orders of magnitude.
1) Using the gain condition for laser action, explain graphically how it is possible to have the
oscillation on only 4 consecutive longitudinal modes.
2) If the reflection coefficient of the mirror M2 is decreasing, how is the number of oscillating
modes modified?
3) If the cavity length is decreasing, how is the number of oscillating modes modified?
4) =100GHz. Give an order of magnitude of the optical length of the cavity L such that only
one mode can oscillate.
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