Exercises Laser Physics 2007
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)
Answer: a) 3x103 W/m2, 4x105 W/m2, b) 1.5x109 Hz
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 spontaneous 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.).
Answer: a) 3.2x1010 Hz, 2.8x108 Hz b) 2.1x10-17 m2
Chapter 4
4-1. The experimental setup below is used to determine the diameter of a TEM00 Gaussian
beam.
Laser
Photo
meter
Iris
diaphragm
Determine the power transmitted by the iris as a function of the beam diameter (2w) and
iris diameter (2b). The diameter of the iris can be continuously varied. Explain how the
setup can then be used to determine the beam diameter.
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.
1
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 the beam radius at
the waist 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 

1  L / f L 
w12
 (b) with z1 
Answer: (a) 
Rayleigh length before the lens;
1 

  1/ f
2w2  2
Lf
(c) 94 cm, 24 cm, 0.5 mm, (d) 1m, 0.4 mm, 20 m (e) d  f#(d) 1023
z1
W/m2 (c) 8.8 1012 V/m, 5.1 1011 V/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
2
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) MHz.
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.
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?
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 
  1/ f
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
3
M4
L1
L2
L1
M3
f
L2
M1
M2
2L1
5.6 We consider a Nd-YAG laser. The laser wavelength is = 1064 nm. The refractive
index in the YAG-rod is nYAG= 1.82. The cavity is designed as shown in the figure below:
L= 80 cm
R2=1 m
R1=5 m
It contains two spherical mirrors with radius of curvature 1 m and 5 m and reflection
coefficients 90% and 100% respectively.
1) For which cavity lengths is the cavity stable?
We choose a cavity length L=0.8 m.
2) What are the beam radii and radii of curvature at the mirrors?
3) Calculate the position of the beam waist and the beam radius at the beam waist.
4) What is the frequency difference between two consecutive transverse modes and two
consecutive longitudinal modes?
A 1 cm long cell containing a solution with refractive index n=1.5 is placed close to mirror
1. A 6 cm long Nd-YAG rod is placed so that its centre is 76 cm from mirror 1. (see figure
below).
76 cm
L= 80 cm
R1=1 m
R2=5 m
5) Write the ABCD-matrix for propagation inside the laser cavity (not including the
mirrors and for a single pass). Show that the effect of the cell and the rod can be taken
4
into account in a simple way by using a modified (effective) cavity length. What is the
value of this cavity length?
6) What are the "new" beam radii and radii of curvature at the mirrors? ("new" means for
the cavity including the cell and the rod, by opposition to "old" as in the first problem)
7) Where is the "new" focus?
8) Where should the rod be placed so that the "new" focus is at the centre of the rod?
Answer: L<1, 5<L<6; 780 m, R1, 381 m, R2; 76 cm from mirror 1, 379 m; 68 MHz, 187
a
c


b d
a
c
1
MHz. 
na
nc
 , Leff  n  b  n  d =77 cm; 746 m, R1, 389 m, R2; 73.5 cm
0

a
c
1


from mirror 1; 74.6 cm from mirror 1.
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

 (W p  W ) N 2  W p N t ; (b) N 2 
;(c)
;
W  Wp
dz W  W p
dt
 I ( z )  I ( z )  I (0)
 
 g 0 z ; I<<Is I(z)=I(0)exp(g0z);
(d) g0=Nt, Is=hWp/; (e) Ln
Is
 I (0) 
I>>Is I(z)=I(0)+Isg0z; (f) I=Isg02L; Iout= I
5
7. 2 A Nd-YAG laser is based upon a four-level scheme as shown in the figure. The pump
rate is Rp. The lifetime of level 2 is
purely radiative with a branching
ratio =0.5 for the transition from
3
level 2 to level 1 (This means that
the rate for spontaneous emision
2
from level 2 to 1 is 2, where 2 is
2
Rp
the lifetime of level 2). The
1
transition from level 3 to 2 is very
1
fast so that the population in level 3
0
is assumed to be negligible. Some
data are given below:

= 1064 nm: laser wavelength
= 234 s: Lifetime for level 2
= 657 ps: Lifetime for level 1
0= 4.5 cm-1: Emission bandwidth
e= 2.8x 10-19 cm2: Cross-section for stimulated emission
1) We first examine the condition for the validity of the four-level scheme approximation.
Write the rate equations for the populations in level 1 and 2, using , number of
photons in the cavity and B, stimulated transition rate per photon and per mode.
2) Assume that the laser is below threshold and in steady-state. How short must be 1 so
that N1/N2 < 1%. Is this condition fulfilled for the Nd-YAG transition (see data
above)?
3) We consider a cavity containing a Nd-YAG rod (assumed not to be pumped), with
length 6 cm, a mirror with 100% reflection efficiency and a mirror with 90% reflection
efficiency. The length of the cavity is 80 cm. The internal losses per single passage are
estimated to 2 %. What is the photon lifetime in the cavity? (1p)
4) Write the equation for the variation of the intensity in the cavity including a Nd-YAG
rod pumped to a fraction x (x<1) of the laser pump threshold value (such that the gain
dI
I
 .
is equal to x times of its value at threshold). Show that it can be written as
dt
x
What is the value of x? How does x vary when the pumping rate approaches
threshold? (2p)
5) Calculate the population inversion at threshold. (1p)
 dN 2
 dt  R p  BN 2  BN 1  N 2 /  2
 c
Answer: 
; 1 < 5 s, yes; 39 ns;  x 
 ;
x 1
 dN1  BN  BN  N /   N / 
2
1
2
2
1
1
 dt
4.3 1022 m-3.
6
Chapter 8
8.1. 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.
8.2 We consider the continuous-wave and transient operations of a Nd-YAG laser (see data
above). The cavity contains a Nd-YAG rod with length 6 cm, a mirror with 100%
reflection efficiency and a mirror with 90% reflection efficiency. The length of the cavity is
80 cm. The internal losses per single passage are estimated to 2 %.
7
1) The laser mode has a beam radius w0=380 m. The pump threshold is 200 W.
Calculate the pump power required to obtain an output power equal to 10 W (We
assume that the laser is extracted through mirror 2). What is the slope efficiency?
2) The laser is passively mode-locked with a fast saturable absorber. What is the repetition
rate of the pulse train originating from the laser?
3) The average output power is the same as in the continuous-wave operation (10 W).
What is the energy per pulse?
4) What is the shortest pulse duration that could be expected from the laser if all the
modes inside the emission bandwidth were phase-locked? (assume that the mode
distribution is Gaussian) ?
5) In reality, the pulse duration p is 10 ps and the intensity profile can be written as
I (t )  I o sec h 2 (t /  p ) , where sech is the secant hyperbolic function (sech(x)=1/(ex+e-x)). p
is related to the full width at half maximum of the intensity profile p by p=p/1.76.
What is the peak pulse power? (Hint: you can use the integral
Answer: 6.1 kW, 0.17%; 176 MHz; 57 nJ; 3.3 ps; 5 kW.
8



sec h( x) 2 dx  2 ).