Exercises Laser Physics 2003
Chapter 2
2-1. A He-Ne laser has an output power of 10 mW at 633 nm. The beam radius is assumed
to be approximately constant, equal to 1.0 mm. The outcoupling mirror has a
transmission of 1.5 %, the other has a reflexion coefficient of 100%.
a) Calculate the intensity outside and inside the resonator. (Assume that there are two
travelling waves into the resonator, going into different directions).
b) Calculate the Doppler width for the transition. (The temperature in the discharge is
400 K; the transition is realised by the neon atoms)
c) Compare the laser intensity outside the resonator with the intensity on the surface of
the sun at the same wavelength and over the same bandwidth. (The sun is considered
as a black body with a temperature of 6000 K. Assume that the laser bandwidth is
one tenth of the Doppler one. The relation between spectral intensity and spectral
energy density for a black body is I=c/4n.)
d) Calculate the ratio B / A for the laser outside the resonator. (Use the formula
A/B=8h/3. The relation between intensity and spectral energy density for a plane
wave is I=c/n).
e) What is the temperature of a black body with the same energy density per frequency
unit at =633 nm as the laser?
Answer: a) 3x103 W/m2, 4x105 W/m2, b) 1.5x109 Hz c) 17 W/m2 d) 1 e) 33 000 K.
2-2. The transition energy between the ground state of helium (1s2) and the first excited
state (1s2p) is 21.3 eV.
a) Calculate the Doppler width for the transition at room temperature. Compare with
the natural linewidth, given by spontneous emission. The lifetime of the 1s2p-state is
0.57 ns).
b) Calculate the absorption cross section at the top of the line profile. (Since the 1s2pstate consists of three degenerate levels, the cross section must be multiplied by
three. Use formula 2.4.13b and 2.4.45 to express the cross section as a function of
the lifetime.).
c) Assume that the light used for the excitation has a width of 0.01 nm. What is the
ratio between the total number of photons and the number of photons useful for the
excitation? (Assume here that this number is proportional to the ratio of Doppler
width by the light bandwidth)
d) Assume that the light source gives 105 photons per pulse, that the density of helium
atoms is 1013 cm-3 over l=1 mm length along the propagation direction of the light.
Give an estimation of the number of excited atoms produced at each laser pulse.
(Start from NHe* = F V where is the atomic density, V the volume and  the
pulse duration. One gets NHe* = Nuseful photons l).
Answer: a) 3.2x1010 Hz, 2.8x108 Hz b) 2.1x10-17 m2, c) 0.036 d) 800 at/puls
1
Chapter 4
4-1. How can the experimental setup below be used to determine the diameter of a TEM00
Gaussian beam?
Laser
Photo
meter
Iris
diaphragm
(Determine the power emitted by the beam within a diameter equal to 2b)
Answer: The power emitted by the iris with a diameter equal that of the beam is equal to
0.865 the total emitted power.
4-2. En CO2 laser (10.6 m) is used to drill a hole into a coppar plate with a width of 1.0
mm. The beam waist is located in the middle of the plate.
a) What is the smallest beam diameter which can be obtained at the surface and what is
the corresponding diameter at the beam waist? (Write w(z) as a function of w0 and the
width of the plate. Calculate the derivative and make it equal to zero).
b) Sometimes it is better to make a hole with little variation in diameter. What is the
diameter at the surface and at the beam waist, if the difference between the two must
not be larger than 0.5%.
Answer: (a) 116 m, 82 m, (b) 260 m, 261 m
4-3 A Titanium Sapphire laser beam (0.8 m), assumed to be parallel with a beam
diameter of 2 mm, is focused by a thin lens with a focal length f=1 m.
a) Calculate the ABCD matrix for propagation through the lens to the beam waist.
b) Apply the ABCD-law for Gaussian beams: It is convenient to write the law for 1/q
and to work with the Rayleigh lengths. Separate the real and imaginary parts of the
equation and solve the coupled equations to obtain an expression for the distance from
the lens to the beam waist and the diameter at the beam waist.
c) Calculate this distance, the Rayleigh length and the diameter at the beam waist.
d) Same question if the diameter is 5 cm.
e) Find a way to answer question c) without using matrices: Express the beam diameter
and the radius of curvature just after the lens as function of L and w0. Approximate
these expressions by assuming that L is much larger than the Rayleigh length. Justify
the approximation afterwards. Show that the diameter can be simply expressed as a
function of f#=f/D, where D is the beam diameter before the lens.
f) What is the peak intensity at focus in case d) if the pulse duration is 30 fs and the
energy per pulse 500 mJ ?

2 I 
g) What is the corresponding electric field  E 
 ? Compare it to that binding
c

0




e
.
the 1s electron in a H atom  E 
2 
4

a
0 0 

2
1  L / f L 
w12
f
 (b) L 
Answer: (a) 
with
Rayleigh length before the
z

1
1

1  f 2 / z12
  1/ f
Lf
lens; 2w2  2
(c) 94 cm, 24 cm, 0.5 mm, (d) 1m, 0.4 mm, 20 m (e) d  f#(d)
z1
1023 W/m2 (c) 8.8 1012 V/m, 5.1 1011 V/m
4-4 En laser beam with a beam waist of 500 m and a wavelength of 514 nm is used to
pump a dye laser (600 nm). It is focused in a dye cell with the help of a lins with f = 1
m. Where should the cell and the lins be located relatively to the pump laser beam
waist, so that the dimensions of the dye laser and that of the pump laser be the same at
focus (= 300 m) ("mode matching")?
w1
w2
L1
L2
(Write the matrix for propagation from the first beam waist to the second. Then apply
the ABCD-law for Gaussian beams. It is convenient to write the law for 1/q and to work
with the Rayleigh lengths z1, z2. Separate the real and imaginary parts. Solve these
coupled equations to get L1 and L2. There are two solutions for L1, L2. The physically
possible solution is such that L1, L2 > f).
Answer: 1.67 m, 1.24 m.
Chapter 5
5-1. Besides 5.5.8, 5.5.9, another important formula is that giving the distance to the waist:
Lg 2 (1  g1 )
. The exercise presents a concrete application of these formulas.
zm 
g1  g 2  2 g1 g 2
A resonator consists of two concave spherical mirrors with radius R= 4 m, R= 1.5 m
and a distance between the two mirrors of 1 m. The laser wavelength is 514 nm (Ar ion
laser). Is the cavity stable? Calculate the position of the beam waist and the diameter of
the beam at the position of the mirrors and at the beam waist.
Answer: Yes,0.14 m, 0.70 mm, 1.08 mm, 0.72 mm
5-2. We consider a laser cavity with a distance between the mirrors equal to 1 m. The laser
wavelength is =633 nm. The mirrors have a radius of curvature equal to 1 m.
a) What type of cavity is it? Is it stable?
b) Calculate the position of the beam waist and the beam radius there.
c) Calculate the mode frequencies.
Answer: confocal, at the threshold, at the middle, 317m, 150 (n+(l+m+1)/2) Hz.
5-3. We want to build an oscillator for a He-Ne laser (633 nm) with a variable-diameter
output beam. We have at our disposal a plane mirror with 100% reflection och a
concave mirror with a radius of curvature R=0.50 m.
a) Write the stability condition for the cavity.
3
b) Give an expression for the beam diameters at the mirrors as a function of the cavity
length L= R-  ( is supposed to be small. Use a Taylor expansion as a function of
/R).
c) We want to vary the beam diameter outside the cavity between 1.0 och 2.0 mm. How
should the length of the cavity be adjusted?
d) How does the laser divergence vary just outside the cavity and at a long distance
from the cavity when the beam diameter varies as in question c)
 R    
 R   R 
Answer: a) L<R b) w1  
   ; w2  
   c) between 5.07 mm
   R
    
and 8.1 cm d) The laser divergence is the same just outside the cavity and at a long
distance. It varies between 1 and 2 mrad.
1/ 2
1/ 4
1/ 2
1/ 4
5-4. A resonator for a He-Ne laser (633 nm) consists of a tube with plane windows which
contains the gas mixture and two concave mirrors with radius of curvature 5.0 m,
placed at a distance of 1.0 m. One mirror has a transmission coefficient of 0%, the other
3%. The fractional internal loss per pass is 2%.
a) Calculate the photon lifetime in the resonator and the resonator's linewidth.
b) Calculate the frequency difference between two consecutive longitudinal modes and
between two consecutive transversal modes.
Answer: a) 95 ns, 1.7 MHz b) 150 MHz, 31 MHz
5-5. A ring cavity (see figure below) includes four plane mirrors M1, M2, M3, M4 and a thin
lens with focal length f. The total length of the ring is L. The laser wavelength is 0.6
m.
a) Determine the cavity ray-matrix (hint: Begin at the lins).
b) When is the cavity stable?
c) To which symmetric two mirror- cavity is the ring cavity equivalent to? (i.e. with a
length equal to L and with the same stability condition).
d) With f=20 cm, L1=5 cm, L2=10 cm, calculate the position and diameter of the beam
waist.
e) Calculate the beam diameter on the mirrors, as well as on the lins.
f) The laser beam is extracted through mirror M3. What is its divergence at the mirror
and at a long distance from the cavity?
g) Calculate the frequency difference between two consecutive longitudinal modes.
1  L / f L 
 (b) L<4f (c)A two mirror cavity with radius of curvature for
Answer: (a) 

1
/
f
1


the mirrors: R=2f (d) Between M1 and M2, 390 m ( e) 403 m (M1, M2) ,489 m
(M3, M4),553 m (lens) (f) 0.59 mrad, 0.98 mrad (g) 750 MHz
M4
L2
L1
L1
f
L2
M1
2L1
M3
M2
4
Chapter 7
7.1. A laser amplifier works according to a 4-level scheme. The population of level 3, resp.
1 is assumed to decay rapidly to level 2, resp. 0 so that the populations in levels 1 and 3
can be neglected. Atoms are excited from 0 to 3 with a pumping rate per atom Wp. The
transition responsible for the amplification is between 1 and 2. Spontaneous emission
and other nonradiative processes are neglected. The total population density is Nt.
3
2
Wp
W
1
0
a) Write the equation describing the variation of the population's inversion.
b) Give the stationary solution as a function of the pump rate Wp, the stimulated
emission rate W and the total population's density Nt.
c) Write the equation describing the variation of the intensity through the amplifier.
d) Show that it can be written as dI/dz = g0I /(1+I/Is). Give the expression of g0
(unsaturated gain) and Is (saturation intensity).
e) Solve the equation in c) by separating the variables I and z. Consider in particular the
two cases I<< Is and I>> Is. The length of the medium is L.
f) Assume that the amplifier is placed in a cavity with two mirrors, one with a
reflection coefficient of 100 % and the other with R=95%. Assume also that I>> Is.
What is the increase in intensity through the amplifier at each round trip? What is the
output intensity if the losses are only due to the outcoupling mirror?
Answer: a)
Wp Nt
dN 2
dI IW p N t

dz ;
;(c)
 (W p  W ) N 2  W p N t ; (b) N 2 
dz W  W p
W  Wp
dt
 I ( z )  I ( z )  I (0)
 
(d) g0=Nt, Is=hWp/; (e) Ln
 g 0 z ; I<<Is I(z)=I(0)exp(g0z);
Is
 I (0) 
I>>Is I(z)=I(0)+Isg0z; (f) I=Isg02L; Iout= I
7.2 A dye laser works according to the 4-level scheme shown below.
4
2
Wp
W2
A2
1
3
A2 and W2 are the stimulated, resp. spontaneous emission rates. The transitions from 4 to 2
and 1 to 3 are so fast that the populations in levels 1 and 4 can be neglected. The pump
process is described with a pump rate per molecule W p . The density of molecules is Nt.
5
The number of molecules per volume unit participating to the process is supposed to be
much less than Nt .
a) Write the rate equations for the population density in level 2.
We assume that a stationary state is reached.
b) Give an expression for the population density N 2 for a given intensity (I) in the
medium. Use the stimulated emission cross-section , the laser frequency , W p , Nt,
and the lifetime  of level 2.
We want to calculate the amplification through a thin medium with thickness d.
c) Calculate the variation in intensity through the amplifier and express I(d) as a function
of I(0).
d) The amplification is expressed by a  E ( d ) / E (0) , where E is the electric field and
approximated by a  1   med   1 . Spontaneous emission is neglected. Give an
expression for .
One mirror of the cavity has a reflection coefficient for the electric field equal to
r2  1   with   1 . The other losses can be neglected. We assume that only a plane
wave oscillates in the cavity.
e) Give an expression of  as a function of  .
f) Give an expression for the intensity I in the cavity, as well as the intensity I' which
leaves the cavity.
g) With the following data: Nt= 1019 cm-3, =4x10-16 cm2, = 3 ns, Wp= 108 s-1,
= 0.02, d= 200 m,  = 0.6 m, calculate I and I’, and show that it was a good
approximation to neglect spontaneous emission.
Wp Nt
dN 2
Answer: a)
;
 (W2  A2 ) N 2  W p N t ; (b) N 2 
I 1
dt

h 
I (0)W p N t d
W p N t dh
W p N t dh
(c) I (d )  I (0) 
;(d)  
;(e) =2;(f) I 
, I’=2I;
I 1
2I
(
0
)


h 
I
1

(g) I= 3.3 1012 W/m2; I’=1.3 1011 W/m2;
h

Chapter 8
8.1. When an intense 800 nm laser pulse is focused in a gas medium, odd harmonics are
generated (even harmonics are not produced because of symmetry reasons). They have
approximately the same intensity. Assume that the 21st to the 61st can be filtered out.
a) Which energy domain cover these harmonics (use eV).
b) Assume that these harmonics are in phase. What is the time structure corresponding
to these harmonics (qualitatively)? What is the time difference between two pulses?
What is the duration of one pulse?
c) If one pulse can be extracted, what is its bandwidth?
Hint: Do an analogy between harmonics and mode-locked modes in a laser.
Answer: a) 62 eV b) 1.3 fs, 67 attoseconds (10–18 s) d) 1.5x1016 Hz.
6
8.2. a) By numerically solving the rate equations for the population inversion and the
photon flux, we obtain figure 1 below where x  I / chN c ,   t /  c , y  N / N c ,
N is the population inversion after Q-switching, Nc is the critical population
inversion, c is the resonator's lifetime. The initial population inversion at t=0 (when
Q-switching occurs) is assumed to be N ( 0)  2 N c . Using the figure, estimate the
pulse duration and the pulse intensity at the maximum.
b) The intensity can sometimes look like in figure 2. What happens? What is the
pulse duration of each pulse? What is the time difference between two pulses?
c) A Fabry-Perot etalon is placed in the cavity close to normal incidence, in order to
change the pulse duration obtained in question b) Which pulse duration is thus obtained?
Data:
= 1064 nm: Laser wavelength
Nt= 1,38x 1020 cm-3: Number of Nd ions /cm3
= 230 s: Lifetime for the highest level
0= 4,5 cm-1: Emission bandwidth
e= 2,8x 10-19 cm2: Cross-section for stimulated emission
nYAG= 1,82: refractive index in the YAG-rod
Components
En Nd-YAG rod, with length 8 cm and diameter 5 mm.
Two mirrors with reflection coefficients 100% resp. 97%, separated by a distance 0.5 m.
A Fabry-Perot etalon made of glass (refractive index 1.5) with a thickness of 100 m and a
finesse of 150.
,,
Answer: (a) t=550 ns; 1.1 10 12 W/m2; (b) Mode locking, 3.3 ps, 3.8 ns; (c) 66 ps.
7