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Fourier transforms method for measuring thermal lens induced in diluted liquid samples

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Fourier transforms method for measuring thermal lensinduced in diluted liquid samples
L. Rodriguez
*
, R. Escalona
Laboratorio de O´ ptica e Interferometrı´a, Departamento de Fı´sica, Universidad Simo´n Bolı´var, A.P. 89000, Caracas 1080-A, Venezuela
Received 25 October 2006; received in revised form 23 April 2007; accepted 30 April 2007
Abstract
Drawing on interferometry and Fourier analysis, this paper describes the use of a two-beam thermal lens technique for measuringthermo-optical properties in optical materials. The procedure consists of yield interference patterns deformed by a localized photother-mal eﬀect. The photothermal phase shift is locally induced by the pump beam focused on a tested sample located on an on-axis probebeam, which is the ﬁrst arm of a Mach–Zehnder interferometer. The plane where the eﬀect is localized is imaged onto a CCD camera.Then two interferograms are recorded: one without eﬀect and the other one with the induced photothermal phase. Fourier analysis per-formed on these interferograms allow us to plot the thermal lens map and, therefore, to estimate thermo-optic constant of MalachiteGreen in water solution. The method is applied to measure low linear absorptions of a diluted sample of Rhodamine B in water solutionat 633 nm, showing that the proposed technique allows to measure photothermal phase shift as low as 3.1 mrad at 8 mW of input powerin diluted materials.
2007 Elsevier B.V. All rights reserved.
PACS:
42.25.Hz; 42.30.
d; 78.20.Nv
Keywords:
Thermo-optical and photothermal eﬀects; Interference; Fourier analysis
1. Introduction
Thermal lens spectrometry has drawn attention due toapplications in measurements of thermo-optical propertiesin optical materials [1–3]. Several methods have beenproposed for detecting the thermal lens signal with highsensitivity [4–11]. Among them, the thermal lens
Z
-scan(TLZ-scan) technique is one of the most versatile [9].TLZ-scan consists in detecting changes of the wave frontsof a probe beam in the far ﬁeld, when the linear mediumis scanned through the focal plane of the pump beam.The sign and magnitude of temperature change refractiveindex
ð
d
n
=
d
T
Þ
can be deduced by plotting the transmit-tance of a small aperture as a function of the sample posi-tion. Despite its sensitivity and experimental simplicityTLZ-scan has limitations it is time-consuming, labor-inten-sive, and requires large amounts of thermal lens signal foreach sample position. This paper proposes an alternative toTLZ-scan, namely a simple interference method based onFourier transform thermal lens (FTTL) to measure
ð
d
n
=
d
T
Þ
and low linear absorption. FTTL consists of apump–probe experiment, in which the wave front of aprobe beam changes due to the photothermal phase shiftinduced by the incident pump beam. A Mach–Zehnderinterferometer is used to produce the interferograms. Itsﬁrst arm is the probe beam and the second is the referencebeam. Deformed interference patterns are recorded on aCCD camera and treated by classical Fourier transformanalysis, which produces the photothermal phase shift intwo dimensions (2D). In this way, the thermal lens phaseis deduced by recording and processing only one deformedinterferogram.
0030-4018/$ - see front matter
2007 Elsevier B.V. All rights reserved.doi:10.1016/j.optcom.2007.04.039
*
Corresponding author. Tel.: +58 212 906 3522; fax: +58 212 906 3601.
E-mail addresses:
rluis@usb.ve, lrodigu@yahoo.com (L. Rodriguez).
www.elsevier.com/locate/optcom
Optics Communications 277 (2007) 57–62
The proposed method is a variant of holographic inter-ference technique [12], where image-processing procedureswere used in order to estimate nonlinear refractive index inoptical materials. The measurements are validated by usingdiluted samples of Malachite Green (MG) and RhodamineB (RhB) in water solutions, where
ð
d
n
=
d
T
Þ
and linearabsorption are calculated, respectively. The advantages of this method are simple experimental alignment, the absenceof mechanical movements during the measurement andgood sensitivity in phase measurement (up to
p
/1000), allof which allow to estimate quickly low absorptioncoeﬃcients.
2. Theoretical considerations
Let us consider a two-beam interferometer, where eachbeam is described by the complex electric ﬁeld
E
i
ð
r
Þ ¼
A
i
ð
r
Þ
exp
½
j
u
i
;
i
¼
1
;
2. Where
u
i
and
A
i
(
r
) are,respectively, phases and complex amplitudes of the ﬁelds,the radial distance from on-axis is represented by vector
r.
We also suppose that beams
E
1
(
r
) and
E
2
(
r
) are the probeand reference beams, respectively. The probe beamimpinges onto optical media and its power is so weak thatno photothermal eﬀect is induced. Lacking the pumpbeam, the complex electric ﬁeld of the probe beam at theexit face of the sample is only aﬀected by the absorbance
A
ð
E
1
ð
r
Þ ¼
exp
ð
A
=
2
Þ
E
1
ð
r
ÞÞ
, therefore the intensity distri-bution without photothermal phase shift, linear interfer-ence, is described as follows:
i
0
ð
r
Þ ¼ j
E
1
ð
r
Þ þ
E
2
ð
r
Þj
2
¼
a
ð
r
Þ þ
b
ð
r
Þ½
exp
ð
j
D
u
Þ þ
exp
ð
j
D
u
Þ
;
ð
1
Þ
a
ð
r
Þ ¼
exp
ð
A
Þ
A
1
ð
r
Þ
2
þ
A
2
ð
r
Þ
2
;
ð
1a
Þ
b
ð
r
Þ ¼
exp
ð
A
=
2
Þ
A
1
ð
r
Þ
A
2
ð
r
Þ
;
ð
1b
Þ
where functions
a
(
r
) and
b
(
r
) determine the varying back-ground intensity and the local contrast of the interferencepattern, respectively. The linear phase of interference pat-terns is given by the phase diﬀerence
u
1
u
2
¼
2
p
x
=
s
,where 1/
s
is the carrier frequency of fringes relative to
x
-axis.When the pump beam, stronger than both
E
1
(
r
) and
E
2
(
r
), is focused on the sample, it produces changes in itsrefractive index (thermal lens) inducing deformations onthe wave front of probe beam at the exit surface of the sam-ple. These alterations in the wave front can be described asfollows:
E
U
1
ð
r
Þ ¼
exp
ð
A
=
2
Þ
exp
ð
j
U
ð
r
ÞÞ
E
1
ð
r
Þ
;
ð
2
Þ
where
U
(
r
) is the additional photothermal phase shift in-duced by the thermal lens. This equation describes the ac-tion of the absorbance and the thermal lens of the mediumon the probe beam.A simpliﬁed two-dimensional thermal lens model [7]assumes that a Gaussian beam induces photothermal phaseshift when it goes through the absorbing sample in whichthe heat ﬂow is radial and convection eﬀects are negligible.For a purely thermo-optical eﬀect, the photothermal phaseshift at the steady state can be estimated by the followingexpression:
U
ð
r
Þ
AP
2
jk
d
n
d
T
X
4
n
¼
1
a
n
exp
ð
b
n
r
2
=
w
2
e
Þ
;
ð
3
Þ
where
a
n
¼ f
1
:
44
;
2
:
047
;
1
:
596
;
5
:
083
g
,
b
n
¼ f
1
:
027
;
0
:
0382
;
0
:
214
;
0
:
0015
g
,
w
e
is the waist,
P
is the incidentpower of the pump beam and
k
is the wavelength of theprobe beam. The parameters
A
,
j
and
ð
d
n
=
d
T
Þ
are theabsorbance, the thermal conductivity and the photother-mal coeﬃcient of the sample, respectively. We deﬁne themagnitude of the photothermal phase shift as the inducedphase shift between the center of the pump beam and the
e
2
radius of the power proﬁle. By doing
r
=
w
e
in Eq.(3) the photothermal phase scales linearly with the meanpower:
U
ð
P
Þ ¼ ½
0
:
66
A
ð
d
n
=
d
T
Þ
=
ð
jk
Þ
P
;
ð
4
Þ
where 87% of the beam energy is included in this range.This function provides a pretty accurate description of the photothermal phase shift at exit face of the sample.We can use it to calculate photothermal coeﬃcients andlow absorption from the tested samples with fewparameters.The transversal intensity distribution of photothermalinterference pattern, given by ﬁelds
E
U
1
ð
r
Þ
and
E
2
ð
r
Þ
, isdescribed as follows:
i
U
ð
r
Þ ¼ j
E
U
1
ð
r
Þ þ
E
2
ð
r
Þj
2
¼
a
ð
r
Þ þ
b
ð
r
Þ
exp
ð
j
U
ð
r
ÞÞf
exp
ð
j
ð
D
u
ÞÞþ
exp
ð
j
ð
D
u
ÞÞg
:
ð
5
Þ
The function
U
(
r
) indicates a local phase variation in theinterference patterns at pixel
r
. To extract this photother-mal phase shift, the 2D Fourier Transform (FT) of theintensity distributions (1) and (5) are conducted, obtainingthe following equations:
I
0
ð
u
Þ ¼
A
ð
u
Þ þ
B
ð
u
Þ½ð
u
D
u
Þ þ
B
ð
u
Þ½ð
u
þ
D
u
Þ
;
ð
6
Þ
I
U
ð
u
Þ ¼
A
ð
u
Þ þ ð
FT
½
exp
ð
j
U
ð
r
ÞÞ
B
ð
u
ÞÞ½ð
u
D
u
Þþ ð
FT
½
exp
ð
j
U
ð
r
ÞÞ
B
ð
u
ÞÞ½ð
u
þ
D
u
Þ
;
ð
7
Þ
where
A
(
u
) and
B
(
u
) are FT of functions
a
(
r
) and
b
(
r
),respectively. The symbol
is the convolution product intwo dimensions;
u
is spatial frequency variable, and
D
u
isthe frequency displacement depending on the linear phase.The frequency spectrum of ﬁrst term represents the lowspatial frequency and it is centralized at the adjacent do-main of the srcin, while the others terms are two pseu-do-Dirac’s distributions placed in ±
D
u
symmetrical to thesrcin. When processing the Fourier analysis [13], one of these pseudo-Dirac’s spots centered at ±
D
u
is extractedfrom the frequency spectrum:
G
0
ð
u
Þ ¼
B
ð
u
Þ½ð
u
Þ
;
ð
8
Þ
G
U
ð
u
Þ ¼ ð
FT
½
exp
ð
j
U
ð
r
ÞÞ
B
ð
u
ÞÞ½ð
u
Þ
;
ð
9
Þ
58
L. Rodriguez, R. Escalona / Optics Communications 277 (2007) 57–62
these selected spots carry information about linear phaseand photothermal phase shift. The deconvolution of thesefunctions using inverse Fourier Transform in two dimen-sions renders the following complex functions:
g
0
ð
r
Þ ¼
b
ð
r
Þ
2 exp
ð
j
D
u
Þ
;
ð
10
Þ
g
U
ð
r
Þ ¼
b
ð
r
Þ
2 exp
ð
j
D
u
Þ
exp
ð
j
U
ð
r
ÞÞ
:
ð
11
Þ
The photothermal phase shift is ﬁnally calculated from theargument of resulting complex function:
g
U
ð
r
Þ
g
0
ð
r
Þ ¼
exp
ð
j
U
ð
r
ÞÞ
;
ð
12
Þ
where amplitudes and linear phase have been eliminated,remaining only the photothermal phase shift. As a resultof the previous analysis, the FTTL technique requires onlytwo experimental images to be recorded in order to deter-mine the photothermal phase shift.
3. Experimental details and results
The experimental setup is shown in Fig. 1. Both pumpand probe beams are taken from a CW Helium–Neonlaser, 40 mW of power at 633 nm of wavelength. The beamis split by means of the beam splitter BS
1
. The reﬂectedbeam has approximately 4% of the total power and it isused to produce the interference patterns by means of aclassical Mach–Zehnder interferometer, which is composedof beam splitters BS
2
and BS
3
, and mirrors M
2
and M
3
.The strong transmitted beam
h
3
i
, pump beam, is focusedby positive lens L
1
onto the sample cell S. Its power canbe adjusted at diﬀerent power values by means of neutraldensity ﬁlters (F
1
). This beam is stopped (SB) after passageby the sample to avoid overlapping with other beams onthe camera.The sample is located on-axis of the probe beam
h
1
i
andits exit face is imaged onto the CCD camera (15
·
15
l
m
2
/pixel size cell and 255 gray level), by means of the positivelens L
3
. Under these experimental conditions, the beam
h
1
i
carries the photothermal phase created by the strong beam
h
3
i
, while beam
h
2
i
is used as reference carrying just the lin-ear phase diﬀerence in the interference pattern. In order tovalidate the technique we use samples of MG and RhB inwater solution (10
4
M), contained in 1-mm path-lengthglass cells. MG is selected as calibration sample due to itsabsorbance at 633 nm (
A
= 0.35), while at the same wave-length the solution of RhB exhibits very low absorbanceand we used it as tested sample.The insertion of calibrated neutral density ﬁlters F
2
andF
3
into the beam path avoids saturation of the camera,improving the resolution and contrast between the interfer-ence fringes. During the experiment, ﬁlters F
2
and F
3
areﬁxed and two experimental images are acquired. The ﬁrstrecord is the linear interferogram in absence of the photo-thermal phase (
U
(
r
) = 0, TL-oﬀ), shown in Fig. 2a. Thesecond image is the photothermal interferogram acquired
Fig. 1. Experimental setup, CCD camera allows to record both linear andphotothermal interferograms. The thermal lens is induced by pump beamfocused into the sample.Fig. 2. Typical images for MG sample acquired with experimental setup:(a) rectilinear fringes are observed in absence of thermal lens (TL-oﬀ) and(b) local deformation of fringes due to induced thermal lens (TL-on) whenthe pump beam impinges in the sample.
L. Rodriguez, R. Escalona / Optics Communications 277 (2007) 57–62
59
by passing the beam
h
3
i
through the sample (TL-on),shown in Fig. 2b.Fig. 3a shows intensity proﬁles traced at position
y
= 125 from images given in Fig. 2a and b, respectively.In this ﬁgure we can observe that the linear phase is shiftedto the left of the ﬁgure in the region where the pump beamimpinges onto the sample, while outside of this region thephase does not change. Due to the relatively weak eﬀect,this local displacement in the fringes produces changes inthe phase alone, not in its carrier frequency (1/
s
) (seeFig. 3b). Hence, we can determine the magnitude of photo-thermal phase shift given by Eq. (4) using FTTL method.In order to extract this photothermal information fromimages given in Fig. 2, ﬁrst it is necessary to eliminate back-ground and high frequency noise. This is done using stan-dard Fourier signal processing, like FFT ﬁlters. For each of these processed interferograms, the Fast Fourier Trans-form in two dimensions (2D-FFT) is calculated obtainingboth linear and photothermal pseudo-Dirac’s distributions.As a typical example, Fig. 4a and b shows the linear andphotothermal power spectrum obtained for the interfero-grams given in Fig. 2a and b. In these ﬁgures we can seethe three pseudo-Dirac’s delta distributions described byexpressions (6) and (7). Similar power spectrums areobtained for RhB sample. A selective 2D band pass ﬁlter-ing is indicated by the squares, indicating the pseudo-Dir-ac’s delta centered at
D
u
that have been selected toobtain the function (8) and (9). This selected ﬁlter is easilyimplemented by using software to crop the interest regionfrom processed image. By applying the inverse 2D-FFTto these selected pseudo-Dirac’s spots we obtain the com-plex distributions
g
0
ð
r
Þ
and
g
U
ð
r
Þ
, and ﬁnally the photo-thermal phase is extracted by calculating the argument of resulting complex function (12). Fig. 5a and b shows the
maps of calculated photothermal phase and the processedspatial power distribution of the pump beam, respectively.The incident spatial power distribution of the pump beamon the sample is previously measured by taking an imagewith the CCD camera located at the same position of thesample.Working with distributions Phase(
x
,
y
) and Power(
x
,
y
)shown in Fig. 5, we plot the parametric graphic pixel bypixel of phase versus radial incident pump power forMG, obtaining 650 experimental data points (seeFig. 6a). From this graphic we obtain the slope of 5.7
·
10
2
rad/mW by linear regression, which depends of thermo-optic parameters by means of
m
= (
0.66
A
(d
n
/d
T
)/(
jk
)). Using this slope and the following values for
A
¼
0
:
35
;
k
¼
633 nm
;
j
¼
5
:
92 mW cm
1
K
1
, the magni-tude and sign of
ð
d
n
=
d
T
Þ
is estimated at
9
:
23
10
5
K
1
, which is the same result that is obtained withdistilled water [7]. Because water is the solvent used to pre-pare the samples and they are diluted samples
ð½
C
¼
10
4
M
Þ
, we assume that the thermo-optical coeﬃ-cient is the same for both samples. It is necessary ﬁrst todetermine the value of (d
n
/d
T
) as calibration of our exper-iment. After this calibration is performed we replace GMsample with RhB sample in order to test the sensitivityof the FTTL method. We calculate the absorbance of thissample at the worked wavelength. In Fig. 6b we showexperimental data obtained and the calculated slope is3
:
9
10
4
rad
=
mW. We use this slope value,
ð
d
n
=
d
T
Þ
pre-viously calculated and the same parameters given above for
k
and
j
, to estimate the absorbance of RhB in 2.4
·
10
3
at
k
= 633 nm. Taking into account this experimental datashown in Fig. 6b, the induced photothermal phase changeis 3
:
1 mrad at 8 mW of incident power, which indicates theinferior limit of the phase shift that can be measured byFTTL method. This limit is approximately
p
=
1000 radusing only one photothermal interferogram from dilutedsample. This limit in the detected phase shift is due tothe low resolution of the CCD camera (320
·
240 pixels)used in our experiment. Sensitivity would be increased if a camera CCD with greater resolution than the oneemployed in our work is used. It is important to emphasizethat in our experiment we do not use neither current ampli-ﬁers nor large optical path length to increase the ratio sig-nal–noise as usually one becomes in the measurements byusing
Z
-scan technique [9,10]. However, although
Z
-scansignal is very sensitive and relatively easy to obtain exper-imentally, its analysis and interpretation requires a model
609012015018021024027004812160481216
TL-off TL-on
x (pixel)
L i n e a r I n t e n s i t y ( a . u . ) x 1 0 T L I n t e n s i t y ( a . u . ) x 1 0
0.020.040.060.080.1051015202530
TL-off TL-on
u (1/pixel)
L i n e a r P o w e r S p e c t r u m x 1 0
246810121416
T L P o w e r S p e c t r u m x 1 0
TL-off TL-on
3 3
TL-off TL-on
3 3
ab
Fig. 3. (a) Intensity proﬁles interferograms traced from experimentalimages of Fig. 2a and b at position
y
= 125. We can observe thephotothermal shift phase on the fringes located near to the area where thethermal lens is induced (TL-on). (b) Power spectrums indicated byexpressions (6) and (7) are located at 1/
s
frequency.60
L. Rodriguez, R. Escalona / Optics Communications 277 (2007) 57–62
of propagation of distorted wave front, so the physicalbehavior of TL is not directly related to the
Z
-scan signalbut through the proposed model [4,7,9]. Interferometryprovides several accurate techniques for refractive indexdistribution measurements [8,14]. Unlike
Z
-scan tech-niques, the interferometric techniques do not need a modelto describe the general behavior of the TL, so the diﬀeren-tial refractive index spatial distribution can be measureddirectly, and hence valuable and original comparisonsbetween nonlinear absorbing processes, thermal diﬀusiontheory and other measurements obtained by
Z
-scan tech-niques can be made. Note that only the photothermalphase shift induced by the pump beam is taken intoaccount as shown in expression (12). The instrumentalparameters, for example imperfections in the cell walls,do not aﬀect the results. Another important characteristicthat we would like to point out is that the results obtaineddo not depend on the laser intensity distribution
A
1
(
r
) and
A
2
(
r
), the experimental geometry, the waist of probe beamor sample positions, like TLZ-scan. This experimentalmethod is also suitable for measurement of two photonabsorption coeﬃcient. In this case the photothermal phaseshift
U
(
r
) is induced by using high energy pulsed laser asexcitation source, experiments are currently being con-ducted to demonstrate this important application of theproposed method.
0.02 0.04 0.06 0.080.020.040.06
uv
0.02 0.04 0.06 0.080.020.040.06
uv
05101520
A m p l i t u d e ( a . u . ) x 1 0
3
v
A m p l i t u d e ( a . ) i t
a b
Fig. 4. Linear and photothermal 2D power spectrums of the experimental interferograms show in Fig. 2. Three pseudo-Dirac’s distributions indicated byexpression (6) are observed, which contain information about linear and photothermal phases. The 2D band pass ﬁltering is schematized by the square inthis last ﬁgure.Fig. 5. (a) Photothermal phase 2D-map extracted by Fourier analysisfrom MG sample. (b) Incident power distribution.Fig. 6. Phase(
x
;
y
) as function of Power(
x
;
y
) plots, for MG and RhBsamples. From slopes calculated by linear regressions is possible toestimate the refractive index change with temperature and low linearabsorption.
L. Rodriguez, R. Escalona / Optics Communications 277 (2007) 57–62
61

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