Download CALCULATION OF THE FOCAL LENGTH OF A THERMAL LENS

Volume
21, number
CALCULATION
J. WESFREID
June 1977
OPTICS COMMUNICATIONS
3
OF THE FOCAL LENGTH OF A THERMAL LENS INSIDE A LASER CAVITY
*, A. BURGOS **, H. MANCINI and E. QUEL
Grupo Laser, CITEFA,
Zufriategui y Varela, Villa Martelli, Pcia. de Buenos Aires, Argentina
Received
1976
9 December
A formula has been found for the value of the induced thermal lens focus in a cell containing liquids with low absorption coefficients,
inside a laser cavity. Refraction
laws, and in extent, transport
equations for progressive and regressive
waves with gaussian profile, have been obtained in the geometrical
optics approximation.
The stationary
case is considered,
with energy transfer by conduction.
The relationship
found is compared with results from other works and discussed.
1. Introduction
The transient defocusing of a beam produced by
liquids placed in the cavity of a He-Ne laser was
first described by Gordon et al. [ 11. The radiation
field induces a localized increment of temperature by
absorption in the liquid producing a gradient of the
refractive index (thermal lens effect). They used an
approximate model in order to calculate the focal
length F of the thermal lens, assuming low absorbent
liquids placed in a “thin” intracavity cell,
rrknOw;
F = blP(dnldT)
’
(1)
where k = thermal conductivity, no = refractive index
of the liquid, w. = spot size of the beam, dn/dT =
change of the refractive index with temperature, b =
absorption coefficient, 1 = cell length and P = total
power of the beam.
The effect and the model mentioned above were
widely used by several authors for the determination
of e.g. small absorption coefficients, (-lop4
cm-l).
This was due to the feasibility to measure the change
in some cavity parameters produced by the formation of the inner thermal lens: the value of the beam
radius on the mirrors [2-41, and the shift of the beat
frequency between modes [5-71. This model was also
used with cells outside the laser cavity [8-131.
The object of this work is to check the validity of
the relationship (1) because there is a great disparity
* CIT, UTN.
** CONICET
- CITEFA.
of criteria about the meaning of its parameters. Besides, the model was developed for an effect observed
in an intracavity cell which was not considered in its
first formulation.
2. Geometrical
optics approximation
Let us consider a medium with variable dielectric
constant due to the change in the temperature by absorption of radiation. Consequently, the refractive
index will be variable too. But e = n2 for optical
frequencies (Jo = l), so
E = ni + 2(dn/dT)
AT,
(2)
AT bein the increment of temperature. The term in
(dn/dT_! 8 has been neglected because (n - no) is of
the order of 10e5 for weakly absorbent liquids.
From Maxwell’s equations for a medium free of
charges and with conductivity u (responsible for
the ohmic losses produced by heating) results
-+a
0
[E-VT]
+V2E=--eE
2n~
2
!!?
o dT
+2n0
,2
dnAT
dT
a_T
at
CE
at2
’
1
aE-+at
dn
a2T
4
at2
a2E
_
c4
at2
dT
(3)
In most liquids several transitory effects occur when
the thermal lens effect is produced. The typical time .
413
Volume
21, number
3
OPTICS COMMUNICATIONS
for these is of the order of one second or less, after
which no variations are observed of the parameters
which depend on the focus of the thermal lens. In
this way we arrive at a stationary situation where
aT/at and a2T/at2 in (3) will be zero. On the other
hand, the first term at the left-hand side of (3)
called the depolarization, is of second order for weakly absorbent liquids and can be neglected. We have
now the equation for a damped wave
: a2E
2n0 dn
02E,n -_+__~T~E+ko~
,2
dT
c2 at2
at2
,2
at 7
(4)
E = 6 [Aexp {i(wt - kz - ks)}
(5)
where s(r, z) is the increment of the eikonal, while
A(r, z) and B(r, z) are the real amplitudes of both
waves.
Paraxial beams occur when the thermal lenses have
a large focal length, which is precisely our case. Moreover, as the cell is thin, the slow change of the amplitudes with z is justified. Hence V12is more important
than 7;. Then, the terms in d2/az2 and (d/az)2 can
also be neglected.
By substituting (5) in (4) with a2 m c2k2/n2, we
have
G(A, B, s, AT) t L(A, B, s, u)/ik t M(A, B, s)/k2 = 0,
(6)
where
-%I, B, s, a)
-ik(z+s)
and
2
where the last term is small due to the low losses of
the system.
A solution of (4) is a linearly polarized quasi-plane
wave progapating along the optical axis of the cavity,
which slow amplitude changes. If we consider an intracavity cell with travelling waves in both ways through
it, this solution will be of the form
+ B exp {i(wt + kz t ks)} ]
M(A, B, s) = &! + i ?$ _ L!Ie-ik(z+,
ar2
+
From now on the two coefficients of L(A, B, s, u) at
the righthand side will be called L l(A, s, u) and
L2(B, s, o).
The openings of the cavity and the radii of the cell
and of the beam are much larger than the wavelength.
Thus, we can use the geometrical optics approximation,
i.e. k + m (h -+ 0). Then, we can neglect the L and M
terms in (6) if they are not excessively large:
G(A, B, s, AT) = 0. This assumption is valid if in the
media convergent thick lenses are not formed (I <F),
and, in consequence, there will be no self-focusing
with A and B -+ m. On the other hand, we study
regions far away from the caustics which are in thin
lenses outside these.
Putting the real and imaginary parts of G equal to
zero, we have
2(aqaz)
+ Beik(Z+d),
t (as/ar)2
= 2(dn/dT)AT/no.
t e(ae/ar)
= (dn/dT)(aT/ar)/no.
(8)
From the geomtrical optics approximation and (6)
as a natural extension of the geometrical optics concept,
for 1AMI < IL 1,we get the relationship for the amplitudes (transport equation): L(A, B, s, u) = 0. By
separating the real and imaginary parts of L we obtain
(Lr - L2) cos [k(z + s)] = 0,
(L1 + L2)sin
414
(7)
We can study the increment of temperature starting
from the heat conduction equation, obtaining an expression for aT/ar. The laws of geometrical optics
[ 141 show for paraxial propagation that sin 8 = 0 =
as/%. Then it is convenient to take the derivative of
(7), introducing the variable 0 = as/& with 0 = 0(r, z).
(aela2)
x (Ae-ik(Z+S)
June 1977
[k(z +s)] = 0.
June 1977
OPTICS COMMUNICATIONS
Volume 21, number 3
This is valid for all z and r if L1 = L2 = 0.
Then multiplying L, by A and L2 by B we get the
following system of differential equations:
Then the shape of the modes will be
2
A24
- w1
B2 =B;
2
_??-
2r2
exp -bz - w2t4
,
exp
,
(14)
w2(4
In order to
absorption we
No convection
cause they are
power. Then
kV2T+
obtain the value of aT/ar produced by
will use the heat conduction equation.
or radiation will be considered beof second order for small levels of
(Q, = 0,
(11)
where (Q) is the average density of heat given by the
laser beam. From the relationship between the average
density of absorbed energy and the field amplitudes
results
<Q> = (cnob/8n)(A2
+ B2).
(12)
For amplitudes of cylinder symmetry, as are the
transverse modes with I= 0 of order p, TEM,&, and
having in mind that the changes with z are slow, we
have
(k/r)(a/ar)(raT/ar)
+ cnob(A2
t B2)/8n = 0.
(13)
This equation together with (8), (9) and (10) form
a system of four differential equations. They describe
the angle 8 of the beam trajectory as a function of
the electric field considering the temperature distribution. In other words, they justify the thermic origin
of the called thermal lens effect.
where w1 is the radius of the beam waist and b = 47ro/
020 [15].
By substituting (14) in (9) and (15) in (10) we
obtain the unique result
tl de
7+dr+
4r2
4rB
dw
w2(z) dz
w2(z)
2
dw_O
’
w(z) dz -
06)
the solution of which is
e(r, Z) = L- @
w(z) dz
.
(17)
From the trajectory
aT
-=
ar
equation
*or
d2W
w(z)(dn/dT)
dz2
(8) we have
(18)
that shows the influence of the temperature gradient
on the change of the beam radius in the medium and
in consequence the change of the trajectory. Now, if
we substitute eq. (18) for aT/&, and the amplitudes
(14) and (15) for A and B, respectively, in eq. (13),
the following equation is obtained
bc(dn/dT)wf
-d2w =-_
16?rkw(z)
dz2
X
2r2
exp - __
w2(4
( 1
.
1
(19)
3. Solution for TEMOw modes
4. Calculation of the focal length
We can obtain solutions for A and B if the laser
modes when the
oscillates in the transverse TEM,
focal length of the thermal lens is f arge enough.
The damped wave equation (4) will have solutions
that will change exponentially with z. This can also
be shown by using Akhmanov et al.‘s suggestion [ 121
of replacing in the system of four differential equations the unknown functions u(r, z) and u(r, z) that
satisfy, respectively:
A2 = (l/r)(&/ar),
B2 = (1 /r)(au/&).
The thermal lens effect occurs when a cell containing a low absorbent liquid is placed inside the
cavity of a laser. The output power of this is measured outside the output mirror. From now on we will
consider that the cell is evenly located inside the cavity,
near to this mirror, in order to avoid, in the calculations, to consider travelling waves in the laser active
medium. See fig. 1.
We have a progressive wave at the output of the
laser with intensity
415
Volume
21, number
w3
;(qexp
I=I,
(
3
OPTICS COMMUNICATIONS
-- &)
1
(20)
,
where w3 is the radius of the waist of the field distribution outside the cavity (obviously, this corresponds to
the spot size on the output mirror) and besides, there
is not exponential damping of the wave along z.
The power on the detector will be
r
Pout(r) =
June 1977
put power, assuming that the spot size is the same on
both faces of the mirror,
0;
c; = 16P,,t/cw;T,
If the cell is placed in the cavity at Brewster angle,
we have tan i^= n o, and if the effect of the thin glass
walls of the cell is not considered (ng e no): n&4’ =
C2 and niB2 = D2. Hence,
Ai = 16Pout/cn~w~T,
$cnout s12rdr
= 16P,,tR/cw;T.
Bi = 16P out R/cniwTT.
0
(23)
With these values eq. (19) results in:
= &I~w~cnout[l
(21)
-exp(-51.
with nout = 1. The input radius of the detector, R, is
muc larger than w3; then, from (21)
PouP= cI;w2,/16
(22)
showing the relationship between 1; and Pout.
Inside the cavity, in the space between the output
mirror and the cell, there are progressive and regressive
waves (see fig. 1) with amplitude, respectively
w2
C=CoG)exp
(
r
w(z)
-1
’
w2
D=Do G)exp
where w2 is the radius of the waist of the field distribution in the space considered.
For the output mirror we have C2T = 12, D2 = RC2,
with T and R the transmission and reflection coefficient, respectively. Thus we have the following relationship between the internal amplitudes and the out-
n
/
Fig. 1. Scheme
416
P,,t(dnldQ
dw
_=_
dz
+Rebz]exp(-5)).
R(ebz - 1) t (1 - eebz)
7rkniTwl
DETECTOR
r-l
dl
.
of the geometrical
1
(25)
C
1
I
0
[eehi
(24)
The change of w(z) with respect to w1 has a
negligible effect on the field amplitude because we
work with weakly defocusing lenses. However, it is
important for the behavior of the phase B = as/h and
the formula (17). For this reason we may consider at
the right-hand side of eq. (24) that w(z) = wl. In this
case eq. (24) can easily be solved ; 12- 131. The
boundary condition at z = 0 is from (17) (dw/dz)z=o =
eow(0)/ro with 19~the angle formed by the incident
ray and the optical axis at the entrance of the cell at
the distance r = ro. Then
MIRROR
CELL
i),,
d2w =--_._
__
b~~$~~
dz2
2
I
arrangement
around
the cell.
There is a value of r that corresponds to a maximum
value of 8, but we must take into account that the
largest amount of power will be concentrated in a zone
with r < WI. We are particularly interested to know
the deflection of the rays in this region. Hence, we must
use the solution with r + 0.
The angle 19(z) for weakly absorbent media and thin
cells, bz Q 1, will be
@)=e
bP
(dn/dT)rz
war
_-out
O w(z)ru
rrkn;Tw 1w(z)
[l + R].
(26)
In order to calculate the position of the principal
image focus we shall use eq. (26) for incident rays
with B. = 0. As refraction occurs at the output of the
cell, for small defocusing we have n,e(Z) = e,,,, and
the position of the image focus measured from the
principal plane of the lens will be
(27)
P’ = ro/80ti,
with F = -F’ in symmetric
(26) in (27) we have
F = rrknow~T/bl(dn/dT)Pout(
June 1977
OPTICS COMMUNICATIONS
Volume 21, number 3
media. By substituting
1 + R).
(28)
If dn/dT < 0, as occurs for most liquids, the value of
the focus is negative and we have thermal defocusing.
The astigmatism produced by a thermal lens
placed at Brewster’s angle inside the cavity is not
considered here [ 161.
5. Conclusions
We have developed the above model in order to explain the self-refraction induced by laser radiation
on a inner cell containing weakly absorbent liquids.
We have explicitly used an intracavity cell with progressive and regressive waves. On the other hand, in
ref. [l], only progressive waves are considered and
in refs. [ 121 and [ 171, the situation is treated with
cells outside the cavity.
The results obtained do not depend on the integration of the heat equation and are independent of the
other equations of the system, not like in ref. [ 11.
Hence, we avoid the objections made to the original
model [l] due to the chosen boundary conditions
[ 151 or to the failure of the model when T + 00 for
t+m [6, 181.
Our result is equivalent to the one obtained by
[l] if
Pinside =Pout(l
+ R)/T
and w. = w1 in (28). In general R M 1, and Pinside =
2Pou,/T, a value already suggested in [3], but not considered in refs. [4,19].
In ref. [ 151 it is suggested to use (1) without the
value of no, because it is observed correctly that the
refraction on the wall of the cell was not considered
in the deduction of the formula. This form was used
in [5,8,20]. We had this in mind before, in the deduction of the model, but the value of no should be kept
in the formula of the focus, with intracavity cell,
because the measurement of the power is made outside
the laser cavity with nout = 1 # no as can be seen in
(21).
This work allows us to clarify the meaning of w1
in (28). It is the radius of the waist of the field distribution in the cell, see (14) and (15). In [l] w. is used,
and it is not clear whether this is the radius of the beam
on the cell or of the of waist in the cavity, because it is
alternatively called characteristic radius of the beam
and spot size. In [2] the formula (1) is used considering w as the radius of the spot on the farthest mirror
from the cell; in [3] w was considered as the spot size
on the output mirror. As change of the radius of the
beam is small in weakly absorbent media, it is possible to approximate w1 by the radius of the spot on
the output mirror. But this is only valid if the cell is
close to the output mirror [7].
Thus, in order to calculate absorption coefficients,
the formula to be used must be
nknowfT
b = FZ(dn/dt)Pout( 1 + R)
(29)
or
nknowf
b = Fl(dn/dT)Pi,
.
In any attempt to correlate different values of measured absorption coefficients one must have in mind
the necessary corrections to the value of w considered
here, the power used and the omission of no.
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417
Volume 21, number 3
OPTICS COMMUNICATIONS
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June 1977
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