This document discusses optical resonator design for diode-pumped solid-state lasers (DPSS). It begins by explaining how laser diode end-pumping allows efficient power transfer to the laser medium by shaping the pumping beam to match the fundamental TEM00 mode volume. The key equations for Gaussian beam propagation are presented, which relate the beam waist size and positions of cavity mirrors. Examples show how to calculate the required curvature of output couplers for empty and real cavities containing active materials. The coupling of the pumping beam to the TEM00 mode volume is also analyzed, relating beam focusing, material absorption coefficients, and ensuring most power is absorbed within the mode volume.

1. 1
Optical resonator design
for DPSS lasers
by: Mario Monico

2. 2
Laser resonator design – Technical note
Laser diode end-pumping is the most effective way to transfer power efficiently to a solid
state laser active medium. The main advantage with respect to side-pumping scheme
consists of the possibility to shape properly the pumping beam so as to match the pumped
volume inside the crystal with the fundamental mode TEM00. This is a quite
straightforward way to obtain a laser emitting a Gaussian beam, without need of inserting
any aperture in the optical cavity to select the lowest order oscillating mode. Typically,
the design of an end-pumped laser starts with the calculation of the TEM00 beam waist
cross section for the specific resonator configuration; then, a proper focusing lens or
group of lenses is selected in order to concentrate the pumping power coming from the
laser diode, directly or through a multimode fiber, within the volume occupied by TEM00
mode. Reversing the procedure, we could design the optical resonator configuration so as
to get a specific value of the TEM00 spot size in its waist, which again shall match the
section covered by the focused pumping beam.
A TEM00 mode is characterized by an intensity distribution whose spatial profile is given
by a Gaussian function. Propagation of Gaussian beams is ruled by the following
fundamental equations:
(1)
2
2
0
0 1)( 








w
z
wzw


(2)















22
0
1)(
z
w
zzR


where:
z is the distance from the beam waist
w(z) is the radius of the beam cross section at distance z
w0 = w(0) is the radius of the beam cross section in its waist
R(z) is the radius of curvature of the beam wavefront at distance z
λ is the laser wavelength
Equation (2) is particularly important, as it can be demonstrated that the radius of
curvature of the wavefront of a Gaussian beam at certain positions z1 and z2 is equal to
the radius of curvature that the two mirrors of the cavity, placed at the same positions z1
and z2, must have in order for the cavity to sustain such a Gaussian beam. This means that,
once fixed the beam waist size and the positions of the cavity mirrors with respect to
waist, equation (2) allows calculating consequently the curvature of the two mirrors.
The most common optical configuration for an end-pumped cavity is given by a plano-
concave resonator, where one of the two mirrors (usually the rear one, through which

3. 3
pumping power is transmitted) is flat. From equation (2) it is evident that the only finite
position corresponding to a flat beam wavefront is z = 0, that is in the beam waist, where
R becomes infinite. See also figure 1, where the distance zR, corresponding to the
maximum curvature of the beam wavefront, is equal to

 2
0w
.
Figure 1
Example 1
We want to estimate the curvature of the output coupler of an empty cavity in its
plano-concave configuration, whose length L is 160mm, for a wavelength of
1064nm, in order to get a Gaussian beam with a cross section diameter D in its
waist of 200µm.
The parameter w0 of the beam waist is equal to D/2, that is 100µm. Solving
equation (2) imposing z = L and λ = 1064nm, we obtain R(L) = 165.45mm, which
is exactly the radius of curvature of the required output coupler for the empty
cavity.
In the previous example, we have considered the simplified case of an empty cavity,
without any other optical component between the two end mirrors. In case of a real laser,
where as a minimum the active medium is placed inside the cavity, we need to consider
the length of the equivalent empty cavity before calculating the curvature of the output
coupler. The length of this empty cavity, equivalent to a real cavity having a physical
length L, is given by the following general equation:
(3) air
i i
i
eq L
n
L
L  
where:
Leq is the length of the equivalent empty cavity
Li is the physical length of the i-th component present in the real cavity

4. 4
ni is the index of refraction of the i-th component at the laser wavelength
Lair is the total length of the optical path in air inside the real cavity
Of course we have also:
(4) air
i
i LLL   or 
i
iair LLL
Example 2
Assuming the same data of example 1, let’s consider in addition a real cavity,
including a Nd:YAG rod, 10mm long with n = 1.82, and an AOM fused quartz
crystal, 30mm long with n = 1.457. We want to asses the new value of the
curvature of the output coupler in order to have again a beam waist diameter of
200μm.
We have L1 = 10mm with n1 = 1.82, L2 = 30mm with n2 = 1.457 and from
equation (4) Lair = L - L1 - L2 = 120mm. So from equation (3) we get Leq =
146.08mm. Solving again equation (2) imposing z = Leq and λ = 1064nm, we
obtain this time R(Leq) = 152.05mm, which is exactly the radius of curvature of
the required output coupler for the real cavity including a Nd:YAG crystal.
Example 3
Assuming the same data of example 2, let’s replace the Nd:YAG rod with a
Vanadate crystal, 7mm long with n = 2.1652. We want to calculate the new value
of the curvature of the output coupler.
This time we have L1 = 7mm with n1 = 2.1652, again L2 = 30mm with n2 = 1.457,
and from equation (4) Lair = L - L1 - L2 = 123mm. So from equation (3) we get Leq
= 146.82mm. Solving again equation (2) imposing z = Leq and λ = 1064nm, we
obtain this time R(Leq) = 152.76mm, which is exactly the radius of curvature of
the required output coupler for the real cavity including a Vanadate crystal.
Example 4
Assuming the same cavity configuration of example 3, we want to calculate the
beam waist diameter D if we use a concave output coupler with a radius of
curvature R of 2m.
Imposing R = 2m, Leq = 146.82mm, and solving equation (2) with respect to w0
we obtain:

5. 5
(5) )(22 0 eqeq LRLwD 


from which D = 0.84mm.
Once defined the optical configuration of the cavity, we need to focus the pumping beam
in such a way that most of its power is absorbed within the volume occupied by TEM00
mode. This depends on several parameters. First of all we consider the absorption
coefficient σ of the laser medium at the pumping beam wavelength. In general the
fraction X of input power absorbed by the crystal over a depth of propagation d is given
by the following expression:
(6) d
eX 
1
Example 5
Vanadate has an absorption coefficient of 31cm-1
at 808nm. We want to calculate
the depth of propagation corresponding to a total absorption of 99%.
We have σ = 31cm-1
and X = 0.99. Solving equation (6) with respect to d we
obtain:
(7)

)1ln( X
d


that is d = 1.48mm.
Example 6
Nd:YAG has an absorption coefficient of 9.5cm-1
at 808nm. We want to calculate
the total absorption after propagation in the crystal of 1.48mm of depth.
We have σ = 9.5cm-1
and d = 1.48mm. Through equation (6) we obtain X = 0.756.
This means that at the same propagation distance, corresponding to a total
absorption of 99% in Vanadate, Nd:YAG absorbs only 75.6%.
The choice of the material as a laser active medium, depending on its absorption
coefficient, affects the coupling geometry of the pumping beam with the TEM00 mode.
With reference to figure 2, as a first approximation we can represent the fundamental
mode with a cylinder, whose circular cross section size is given by the beam waist
diameter, that is 2w0; at the same time we can represent the focused pumping beam with a
cone whose axis is coincident with cylinder’s one and whose circular cross section size at

6. 6
the entrance of the crystal is also equal to 2w0. From this simplified geometrical scheme it
is evident that, in order to get a single mode oscillation from the laser resonator, it is
fundamental to absorb most of the pumping power within the distance d of propagation
through the active material, where the cone of the focused beam is included in the
cylindrical volume of the mode. From examples 5 and 6, we see that, focusing the
pumping beam in such a way that the value of d is greater than or equal to 1.48mm, in
case of Vanadate all the power will be absorbed within the volume of TEM00 mode,
whilst in case of Nd:YAG about one quarter of pumping power will be absorbed after the
distance of 1.48mm, with the unwanted consequence that part of this power will be
finally absorbed in a volume external to the cylinder occupied by the fundamental mode;
this means that, if the pumping power is strong enough, higher order modes could start
lasing. Once known the values of w0 and d, the numerical aperture NA of the focused
pumping beam can be approximated in most practical situations by the following
expression:
(8)
d
nw
NA 02

where n is the index of refraction of the laser crystal at the wavelength of pumping beam.
The above geometrical analysis does not consider the finite size of the focal spot inside
the crystal due to diffraction. Moreover, given that the pumping source is not collimated,
coming from a laser diode or an optical fiber, what we get in the focal point is actually
the image of the emitting area of the diode or fiber. The dimension of this image inside
the active medium is obviously a function of both the emitting area and the magnification
of imaging optics.
d
TEM00 mode
Pumping beam
Figure 2
Rear mirror
2w0

7. 7
Example 7
Let’s consider a Vanadate laser whose cavity is configured as follows:
Cavity physical length: L = 155mm
Vanadate crystal physical length: L1 = 7mm
Index of refraction of Vanadate at 1064nm: n1’ = 2.1652
Index of refraction of Vanadate at 808nm: n1” = 2.1858
Absorption coefficient of Vanadate at 808nm: σ = 31cm-1
AOM crystal length: L2 = 30mm
Index of refraction of fused quartz in AOM: n2 = 1.457
Rear mirror: flat (rear surface of Vanadate crystal)
Output coupler: concave, radius of curvature R = 170mm
The laser is pumped by a diode whose power is carried by an optical fiber with
the following characteristics:
Core diameter: a = 400μm
Numerical aperture: NA = 0.22
The imaging optics, focusing the pumping beam into Vanadate, is given by an
aspheric doublet, conjugating object and image with a magnification ratio M =
2.44 (for example P/N C230220P-B from Thorlabs).
We want to assess the depth d of propagation in Vanadate, within which the
focused pumping beam is contained in the volume of TEM00 mode, and the
relevant fraction X of absorbed power.
We start with the calculation of the beam waist diameter D for the TEM00 mode.
On the basis of the above data, from equations (3) and (4) we get:
Leq = 141,82mm
From equation (5), we obtain:
D = 2w0 = 292.65μm
The aspheric doublet shall be oriented so as to get a demagnified image of the
output face of the fiber; the numerical aperture of the focused beam is
consequently given by the NA of the fiber times the magnification M of the
doublet:
NA = 0.54

8. 8
Similarly, the size of image of the end face of the fiber can be calculated by
dividing the fiber core diameter a by the magnification M:
a / M = 163.64μm
which is lower than D, as required in order to have the pumping beam matched
with TEM00 mode.
From equation (8), we have:
(9)
NA
wn
d 012


so that we obtain as a final result:
d = 1.18mm
Replacing this value in equation (6) we eventually obtain:
X = 97.42%
which is lower than 99%, as expected given that the value of d we have just
estimated is shorter than 1.48mm, value corresponding to an absorption of 99% in
Vanadate as shown in example 5.
For a more detailed discussion of the subjects contained in this technical note, the
following reference books are recommended:
Orazio Svelto: Principles of Lasers, 3rd
edn. (Plenum, New York 1989)
Walter Koechner: Solid State Laser Engineering, 4th
edn. (Springer, Heidelberg 1996)