time-resolved, photothermal-deflection spectrometry with step optical excitation
TRANSCRIPT
Zhao et al. Vol. 21, No. 5 /May 2004/J. Opt. Soc. Am. B 1065
Time-resolved, photothermal-deflectionspectrometry with step optical excitation
Jianhua Zhao, Jun Shen, Cheng Hu, and Jianqin Zhou
National Research Council of Canada, Institute for Fuel Cell Innovation, 3250 East Mall, Vancouver, BritishColumbia V6T 1W5, Canada
Mauro Luciano Baesso
Departamento de Fısica, Universidade Estadual de Maringa, Avenida Colombo, 5790, 87020-900 Maringa,Parana Brazil
Received September 19, 2003; revised manuscript received December 15, 2003; accepted January 12, 2004
A theoretical model of a novel time-resolved, photothermal-deflection spectrometry with step optical excitation(a rectangular pulse of a finite duration) is presented. To make the theory practicable, i.e., able to be used todetermine the values of optical and thermal properties of the sample by fitting experimental data to the theory,approximations are introduced. The numerical simulations show that when the value of the thermal conduc-tivity of the deflecting medium (e.g., air) is small, the adiabatic approximation for the heat conduction at bothfront and rear surfaces of a sample is effective for both transparent and opaque samples, while the adiabaticand isothermal approximations for the heat conduction at the front and rear surfaces, respectively, are validonly for an opaque sample. With these approximations, the optical and thermal properties—such as opticalabsorption coefficient and thermal diffusivity—of solid materials (optically transparent or opaque) can be di-rectly measured by this novel method. © 2004 Optical Society of America
OCIS codes: 300.6430, 350.5340, 120.4290, 300.6500, 190.4870, 260.2160.
1. INTRODUCTIONSince 1970, photothermal techniques have been devel-oped for nondestructive material characterization, suchas measuring optical and thermophysical properties ofdifferent materials and depth profiling, that are impor-tant because of their high sensitivity and versatility.1–3
These techniques can be classified into two main catego-ries by the criterion whether the samples contact or donot contact the detection system.3 Photoacoustic andphotopyroelectric methods are examples of the contacttechniques. Remote (noncontact) techniques, such asphotothermal deflection (namely, optical beam deflection)and thermal lens methods, are suitable for studyingsamples in a severe environment.
In transverse photothermal-deflection (PD) (or mirageeffect) spectrometry, a probe beam senses the gradient ofthe optical refractive index in the ambient medium adja-cent to a sample surface, resulting in deflection of theprobe beam. The gradient of the optical refractive indexis induced by the temperature gradient from the samplesurface to the medium, and the temperature rise in thesample is the result of conversion of the absorbed electro-magnetic excitation radiation into heat. The optical andthermophysical properties of the sample, therefore, can bemeasured by monitoring the frequency dependence (forfrequency-modulated excitation4,5) or the time depen-dence (for impulse excitation6,7) of the PD signal. Thetheory, and applications of PD spectrometry can be foundin Refs. 1–7, and the reported thermophysical propertymeasurement with the PD technique were achievedmainly by use of frequency-modulated excitation.3 One
0740-3224/2004/051065-08$15.00 ©
may measure thermal diffusivity in a frequency-modulated experiment by using the linear relationshipbetween two measurable parameters whose slope can beused to retrieve the thermal diffusivity value. However,a careful management of the linear relations is necessarysince some of them are strictly valid only under ideal con-ditions that are very difficult to achieve in practice.8
With no requirement to find any linear relations, time-resolved, thermal-lens spectrometry by means of step op-tical excitation (i.e., the rectangular-pulse optical excita-tion) has enjoyed great success in the determination ofthe thermal diffusivities of transparent samples.9–11 Theresults are absolute measurements, and the thermal dif-fusivity values can be obtained directly by a convenientprocedure of fitting experimental data to a theoreticalmodel. In this paper, we propose theoretically a noveltime-resolved, photothermal-deflection spectrometry withstep optical excitation. Similar to the thermal-lens tech-nique, this PD technique can be used not only for measur-ing optical properties of materials, but can be employedfor determining thermophysical properties. Further-more, this technique can be employed to study both trans-parent and opaque samples, while thermal-lens spectrom-etry can be applied only to transparent samples.
In this work, we start with the calculation of the one-dimensional temperature rise due to the absorbed energyof step optical excitation in a thin, solid slab sample of fi-nite thickness with boundary conditions.12 Then wework out the temperature-rise distribution in the deflect-ing medium owing to the heat conduction from the frontsurface of the sample to the medium. Finally, we derive
2004 Optical Society of America
1066 J. Opt. Soc. Am. B/Vol. 21, No. 5 /May 2004 Zhao et al.
the time-resolved PD signal based on the temperaturegradient in the deflecting medium.
To calculate the temperature rise in the sample and itssurrounding medium after the absorption of the step op-tical excitation by the sample, scientists have usedLaplace transforms,2 such as in the theoretical work inRef. 13. Only in some special cases could the tempera-ture rise (in the time domain, not in s space) in the gasmedium adjacent to the sample’s front surface be ob-tained with an analytical expression. Green’s functionshave also been employed to solve the equation of heat con-duction and find the temperature rise in the sample; onemay find an example in the theoretical work of Ref. 14.Although the Green’s function method might be an ap-proach to calculate the temperature increases after thestep optical excitation in both the sample and the sur-rounding media, as a complete treatment (not just for spe-cial cases), to the best of our knowledge, there is littlework reported in which this approach was used.
In using Green’s functions to solve the heat conductionequations, we present here a complete theoretical treat-ment of photothermal-deflection spectrometry with stepoptical excitation, aiming to obtain a practicable theory,i.e., one that can be used to determine the values of opti-cal and thermal properties of the sample by fitting experi-mental data to the theory. However, there is an obstacleto the theory’s being practicable: To use this theory prac-ticably to measure optical and thermal properties, oneneeds to know the thermal property to be measured tosolve a transcendental equation. To remove this ob-stacle, we introduce approximations. Numerical simula-tions show that when the thermal conductivity of the de-flecting medium is small (e.g., air), the adiabaticapproximation for the heat conduction at both front andrear surfaces of the sample is effective for both transpar-ent and opaque samples, while the adiabatic and isother-mal approximations for heat conduction at the front andrear surfaces, respectively, are valid only for an opaquesample. Consequently, these approximations make thisnovel time-resolved, photothermal-deflection spectrom-etry practicable.
2. THEORETICAL MODELIn a PD experiment as shown in Fig. 1 an isotropic, ho-
Fig. 1. Schematic diagram of the photothermal-deflection spec-trometry of a thin-slab solid sample in a one-dimensional treat-ment.
mogenous, thin slab sample is placed in a transparentand homogeneous deflecting medium. The thickness ofthe sample is ls, and the optical absorption coefficient ofthe sample is b. The distance between the probe beamand the sample surface is a. A heating beam of spot sized is uniformly incident on the sample surface at x 5 0with intensity I0 and a pulse duration tp (step heating):
I~t ! 5 H I0 , ~t < tp , active heating interval!
0, ~t . tp , relaxation interval!.
(1)
For the convenience of later discussion, all variables withsubscript a are for the active heating interval (t < tp),while those with subscript r are for the relaxation inter-val (t . tp). If we assume the absorbed optical energy isconverted into heat through a nonradiative process com-pletely and instantaneously, the heat due to the opticalabsorption per unit volume per unit time at x in thesample is
Q~x, t ! 5 bI~t !exp~2bx !. (2)
The differential equations of heat conduction in thesample and the deflecting medium are given by
]2Ts~x, t !
]x22
1
as
]Ts~x, t !
]t5 2
Q~x, t !
ks, (3)
]Tg2~x, t !
]x22
1
ag
]Tg~x, t !
]t5 0. (4)
Here, we define Ti(x, t) as the temperature rise for boththe active interval and relaxation interval; the actualtemperature is the sum of the temperature rise Ti(x, t)and the ambient temperature T` . ki is the thermal con-ductivity, a i 5 ki /r ici is the thermal diffusivity, and r iand ci are the density and the specific heat, respectively.i [ s or g stands for the sample or the deflecting medium,respectively. The initial condition is
Ti~x, 0! 5 0. (5)
The boundary conditions are the continuities of the tem-perature and heat fluxes at the boundaries of x 5 0 andx 5 ls :
Ts~0, t ! 5 Tg~0, t !, (6a)
Ts~ls , t ! 5 Tg~ls , t !, (6b)
ks
]Ts~x, t !
]xU
x50
5 kg
]Tg~x, t !
]xU
x50
, (6c)
ks
]Ts~x, t !
]xU
x5ls
5 kg
]Tg~x, t !
]xU
x5ls
. (6d)
These conditions assume that the heat transfer fromthe sample to the deflecting medium adjacent to thesample surfaces is dominated by heat conduction. Theheat conduction from the sample surface to a location x5 2lg in Fig. 1 in the deflecting medium takes time tg' lg
2/4ag (Ref. 2, p. 106). We may call this thin layer ofmedium in contact with a sample surface with thicknesslg a thin-skin layer (Ref. 15, p. 20). When t ! tg the
Zhao et al. Vol. 21, No. 5 /May 2004/J. Opt. Soc. Am. B 1067
temperature increases Tg(2lg , t) and Tg(ls 1 lg , t) arezero, i.e., the actual temperature at x < 2lg and x > ls1 lg is the ambient temperature T` . By neglecting theheat capacities of this thin skin layer on the front samplesurface x 5 0 and following Refs. 9 and 16, as a first ap-proximation we have
kg
]Tg~x, t !
]xU
x50
' kg
Tg~0, t ! 2 Tg~2lg , t !
lg
5kg
lgTg~0, t !. (7)
Relation (7) is the rate of heat flow through the thin-skinlayer (Ref. 15, p. 20) when t ! tg , and at that timeTg(2lg , t) 5 0. By substituting the boundary conditionEq. (6a) of the temperature continuity at the boundary x5 0 into Eq. (7), the heat flux continuity of Eq. (6c) canthen be replaced by the boundary condition for the linearheat transfer through the thin-skin layer into a mediumat zero temperature rise16
ks
]Ts~x, t !
]xU
x50
2 h1Ts~0, t ! 5 0. (6c8)
Similarly, one can find the boundary condition at x 5 lsas
ks
]Ts~x, t !
]xU
x5ls
1 h2Ts~ls , t ! 5 0. (6d8)
Here, h1 5 h2 5 kg /lg are the heat transfer coefficientsfor the boundary condition of the third kind.12 One canfind a similar expression in Ref. 15, p. 20. When t! tg , Eqs. (6a), (6b), (6c8), and (6d8) are the boundaryconditions of the differential Eqs. (3) and (4) of heat con-duction in the sample and the deflecting medium.
The Green’s function for the heat conduction Eq. (3)with the initial condition of Eq. (5) and the boundary con-ditions of Eqs. (6c8) and (6d8) is given by
G~x, x8, t, t! 5 (m51
`
AmZ~x, x8!exp@2asdm2~t 2 t!#,
(8)
(not valid for both h1 5 0 and h2 5 0; Ref. 15, p. 360),where
dm is the root of the following transcendental equation:
tan dmls 5dmks~h1 1 h2!
ks2dm
2 2 h1h2
. (11)
The temperature rise in the sample due to the absorptionof the step optical excitation is then given by
Am 52~ks
2dm2 1 h2
2!
~ks2dm
2 1 h12!@ls~ks
2dm2 1 h2
2! 1 ksh2# 1 ks
Z~x, x8! 5 ~ksdm cos dmx 1 h1 sin dmx !~ksdm cos dmx8 1 h
Ts,a~x, t ! 5 E0
tE0
ls Q~x8, t!
rscsG~x, x8, t, t!dx8dt,
t < tp , (12a)
Ts,r~x, t ! 5 E0
tpE0
ls Q~x8, t!
rscsG~x, x8, t, t!dx8dt,
t . tp . (12b)
Ts,a(x, t) stands for the temperature rise in the activeheating interval and Ts,r(x, t) the temperature rise in therelaxation interval. Let um 5 dmls . By substitutingEqs. (2) and (8) into Eqs. (12), we have
Ts,a~x, t ! 5 (m51
`
BmS ksum
lscos
umx
ls1 h1 sin
umx
lsD
3 H 1 2 expF2S um
pD 2 t
tsG J , t < tp ,
(13)
Ts,r~x, t ! 5 (m51
`
BmS ksum
lscos
umx
ls1 h1 sin
umx
lsD
3 H expF S um
pD 2 tp
tsG 2 1J expF2S um
pD 2 t
tsG ,
t . tp , (14)
where
Bm 5bI0Amls
2
ksum2~um
2 1 b2ls2!
$ksum
3 @exp~2bls!~um sin um 2 bls cos um! 1 bls#
2 h1ls@exp~2bls!~um cos um 1 bls sin um!
2 um#%. (15)
In Eqs. (13) and (14), ts 5 ls2/p2as is the characteristic
diffusion time constant of the sample.17 The physicalmeaning of ts is that for a semi-infinite solid of thermaldiffusivity as heated by a Dirac delta heating impulse onits surface plane x 5 0 at t 5 0, the temperature rise atx 5 ls in the sample due to heat conduction is less than
10% of the surface magnitude when t 5 ts .From Eqs. (13) and (14), we have the temperature rise
on the sample surface x 5 0 as
Ts,a~0, t ! 5 (m51
` Bmksum
lsH 1 2 expF2S um
pD 2 t
tsG J ,
t < tp , (16)
ks2dm
2 1 h22!
, (9)
dmx8!. (10)
h1~
1 sin
1068 J. Opt. Soc. Am. B/Vol. 21, No. 5 /May 2004 Zhao et al.
Ts,r~0, t ! 5 (m 5 1
` Bmksum
lsH expF S um
pD 2 tp
tsG
2 1J expF2S um
pD 2 t
tsG , t . tp . (17)
Equations (16) and (17) will be used as the boundary con-ditions in calculating the temperature rise in the deflect-ing medium.
The deflecting medium can be regarded as semi-infinite, so the temperature rise with the above initial andboundary conditions Eqs. (5) and (6a) is given by
Tg~x, t ! 52
ApE
x/2Aagt
`
fS t 2x2
4agj2D exp~2j2!dj
(18)
(Ref. 15, p. 63), where f(t) 5 Ts,a(0, t) in the active heat-ing interval and f(t) 5 Ts,r(0, t) in the relaxation inter-val. By substituting Eqs. (16) and (17) into Eq. (18), wecan obtain the temperature rise in the deflecting mediumas
Tg,a~x, t ! 5 (m51
` Bmksum
2lsH 2 erfcS 2
x
2AagtD
2 expF2S um
pD 2 t
tsG
3F expS jAas
ag
umx
lsD erfcS 2
x
2Aagt
2 jum
pA t
tsD
1 expS 2jAas
ag
umx
lsD erfcS 2
x
2Aagt
1 jum
pA t
tsD G J , t < tp , (19)
Tg,r~x, t ! 5 Tg,a~x, t ! 2 Tg,a~x, t 2 tp!,
t . tp . (20)
Here erfc(z) is the complementary error function.From the geometric optics theory of photothermal-
deflection spectroscopy, the probe beam deflection angle inone dimension can be written as18
w~x, t ! 5 21
n
]n
]T
]T~x, t !
]xd, (21)
where n is the optical refractive index of the deflectingmedium. ]n/]T is the temperature coefficient of the re-fractive index of the medium, which can be regarded asconstant (usually negative) when the temperature rise issmall. With Eqs. (19) to (21), the deflection angle can beexpressed as
wa~x, t ! 51
n
]n
]Td (
m51
` jksBmum2
2ls2
Aas
agexpF2S um
pD 2 t
tsG
3 H expS jAas
ag
umx
lsD
3 erfcS 2x
2Aagt2 j
um
pA t
tsD
2 expS 2jAas
ag
umx
lsD
3 erfcS 2x
2Aagt1 j
um
pA t
tsD J ,
t < tp , (22)
wr~x, t ! 5 wa~x, t ! 2 wa~x, t 2 tp!, t . tp . (23)
3. DISCUSSIONTo use the above theory to measure the optical and ther-mal properties of the sample, one needs to solve transcen-dental Eq. (11) that is associated with the thermal con-ductivity ks of the sample. This in turn means that oneneeds to know the thermal conductivity to be measured toapply the theory for measuring the thermal properties ofthe sample, which renders the theory impracticable forexperimental measurements. It is, therefore, necessaryto find an approach to simplify Eq. (11) so it can be easilysolved without knowledge of the sample’s thermal proper-ties. Furthermore, with the real boundary conditionsEqs. (6c8) and (6d8), the mathematical expressions of thetemperature rise in the sample and the deflecting me-dium, as well as the probe-beam deflection angle (PD sig-nal), are complicated. It will be helpful for processingthe experimental data if the mathematical expressionscan be simplified. In the discussion below, we will try todevelop an approach and determine its validity and theconditions for the approach and simplification based onsome reasonable approximations.
A. Boundary ConditionsFor optically opaque samples, the temperature rise is pro-duced mainly at the front surface x 5 0 in Fig. 1. Withthe physical meaning of ts , one can infer that when t! ts , the heat has not reached the rear surface of thesample,17 i.e., the temperature rise at the rear surface x5 ls is then zero; that is, the actual temperature is theambient temperature, which is equivalent to isothermalcontact with the deflecting medium, i.e., h2 → `. At thefront surface of the sample x 5 0, if the thermal conduc-tivity value of the deflecting medium is low, the heat flowcrossing the boundary at x 5 0 will be insignificant. Sothe boundary condition at the front surface of the samplemay be approximated by adiabatic insulation h1 → 0when t is short and the thickness lg ' A4agt of the thin-skin layer in the deflecting medium adjacent to thesample front surface is thin enough that the thermal ca-
Zhao et al. Vol. 21, No. 5 /May 2004/J. Opt. Soc. Am. B 1069
Table 1. Optical and Thermal Properties of Glassy Carbon and Lime Glass
SampleDensity(kg/m3)
SpecificHeat
(J/kg K)
ThermalConductivity
(W/m K)
AbsorptionCoefficient
(m21)
ThermalDiffusivity
(m2/s)
ThermalEffusivity
(Ws1/2/m2 K)
GlassyCarbona
1.45 3 103 1.1 3 103 8 106 5.02 3 1026 3.57 3 103
LimeGlassb
— — 1.0 49c 4.6 3 1027 1.5 3 103
a See Ref. 19.b See Ref. 10; rc 5 2.2 3 106 J/m3 K.c At 647.1 nm.
pacity of the thin-layer medium is negligible and the ap-proximation of Eq. (7) is applicable.
For optically transparent samples, the optical absorp-tion occurs through the sample from x 5 0 to ls , and sodoes the temperature rise. The adiabatic approximationmight be applied to both front and rear surfaces of thetransparent sample if t were short. Air is the commondeflecting medium in PD experiments. In this medium,one may expect that the adiabatic–isothermal (AI) ap-proximation for opaque samples and the adiabatic–adiabatic (AA) approximation for transparent sampleswill be effective when t is short because of the low conduc-tivity and consequent small value of the heat transfer co-efficient h with air (Ref. 15, p. 20).
With the AI approximation, we can easily solve Eq. (11)without knowledge of sample properties:
dm 5 ~2m 2 1 !p/2ls , m 5 1, 2, 3,... . (24)
Similarly, we can determine the solution for Eq. (11) withthe AA approximation:
dm 5 mp/ls , m 5 1, 2, 3,... . (25)
The mathematical expressions of the temperature rise inthe sample and the medium, as well as the PD signal withthese two important approximations, are also simplified.The analytical expressions with the AA approximationcan be found in Appendix A.
B. SimulationsIn the following simulations, we will discuss the condi-tions and validity of the AI and AA approximations.Glassy carbon19 will be used as an opaque sample andlime glass (72% SiO2 , 18% Na2O, and 10% CaO inweight)10 as a transparent sample in the simulations.The optical and thermal properties of theses two samplescan be found in Table 1. In the simulations, the totaltime, i.e., the sum of the active heating interval (t < tp)and the relaxation interval (t . tp), does not exceed thecharacteristic diffusion time constant ts of the sample.The thickness lg of the thin-skin layer in the deflectionmedium is estimated by lg ' A4agts. The parameterswe used in the simulations with air and water as the de-flecting media are listed in Table 2. The thermal proper-ties of the media can be found in Ref. 20. In our simula-tions, we will discuss three possible boundary conditioncases: (1) the normal (real) boundary case (h1 5 h25 kg /lg), (2) the AA approximation (h1 5 h2 → 0), and(3) the AI approximation (h1 → 0 and h2 → `). In thesimulation, the criterion for ceasing the summation in
Eqs. (13), (14), (19), (20), (22), and (23) will be the ratiomth term:first term ,1023.
1. Temperature Rise in the SamplesThe temperature rise in the samples in the active heatinginterval (tp 5 1ts) for different times t is shown in Fig. 2.For glassy carbon in air, the temperature rises in thethree cases at the front surface of the sample are almostthe same [see Fig. 2(a)]. The relative difference betweencase (1) and case (2) and between case (1) and case (3) isless than 0.5%. However, at the rear surface of thesample, the AI approximation keeps the temperature risealways at zero, accounting for the difference between case(1) and case (3) when t . 0.5ts . For an opaque samplesuch as glassy carbon, the heat source produced after ab-sorption of the heating beam energy is located very closeto the front surface of the sample. When t ! ts , the heatflow does not reach the rear surface; as a result there isno temperature rise at the rear surface. Thus the AI ap-proximation is valid only when t ! ts .
For the AA approximation, the temperature rise in thesample is almost the same as in case (1) with small rela-tive difference ,1% because of the low value of the ther-mal conductivity kg of air and consequent low heat fluxfrom the sample to air, which is close to the adiabatic situ-ation. Figure 2(b) shows the temperature rise in theglassy carbon sample in water. There is no big differencebetween case (2) and case (3) at the front surface of thesample. However, there are differences of 4%, 9%, and13%, between case (1) and case (2) or case (3) when t5 0.1ts , 0.5ts , and 1.0ts , respectively, because of thehigh value of the thermal conductivity of water. Onemay notice that at the rear surface the difference betweencase (1) and case (2) is not as great as the differences atthe front surface. This smaller difference may be due tothe temperature rise at the rear surface and the conse-quent heat flux from the sample to the medium, which aremuch lower than that at the front surface.
Table 2. Parameters Used in the Simulations
Sample Deflection Mediumls
(mm)h
(W/m2 K)ts
(s)
Glassy Carbon Air 1.0 12.44 0.020Glassy Carbon Water 1.0 3540 0.020Lime Glass Air 3.0 1.255 1.98Lime Glass Water 3.0 357.4 1.98
1070 J. Opt. Soc. Am. B/Vol. 21, No. 5 /May 2004 Zhao et al.
As opposed to the glassy carbon sample, lime glass isoptically transparent, and the heat source is spread overthe sample in the direction of its thickness. The lowvalue of kg and the small difference in the temperaturerises at the front and rear surfaces are the reasons thetemperature rises calculated in case (1) and in the AA ap-proximation [case (2)] are almost the same (difference,0.5%) for lime glass in air, as shown in Fig. 2(c). Thereis a big difference at the rear surface between case (1) andcase (3) because the AI approximation keeps the tempera-ture rise at zero at the rear surface of the sample, whichis unrealistic for the transparent sample, even with shortt. For lime glass in water, Fig. 2(d), the temperature riseat the middle of the sample along the thickness directionis higher than that at its surfaces, and all the approxima-tions deviate from case (1) because of the high thermalconductivity of water and the consequent high heat fluxfrom the glass to water. For t < 0.1ts , the difference be-tween case (1) and case (2) is smaller than 8%.
2. Temperature Rise and Photothermal-Deflection Anglein the Deflecting MediaThe temperature rise in the deflecting medium is depen-dent on the temperature rise at the front surface of thesample, as predicted by Eq. (18). The photothermal-deflection signal is proportional to the derivative of thetemperature rise in the medium, as predicted by Eq. (21).One may expect that if there is good agreement betweencase (1) and an approximation [case (2) or case (3)] in thetemperature rise at the front surface, there will be goodagreement in the prediction of the temperature rise in themedium and the PD signal. Figures 3 and 4, in general,meet this expectation. Figure 3 shows the temperaturerise in the deflecting media with tp 5 0.5ts for the activeand relaxation intervals. The temperature rises at thefront surfaces of the samples in Figs. 3(a) and 3(c) reach amaximum when t 5 0.5ts and start to fall thereafter.The temperature rise in air falls faster with glassy carbonthan it does with lime glass because of the different heat
Fig. 2. Temperature rise in the samples during the active heat-ing interval with the heating beam pulse duration tp 5 1ts fordifferent time t. (a) Glassy carbon in air, (b) glassy carbon inwater, (c) lime glass in air, (d) lime glass in water. Solid curves,case (1); diamonds, case (2); squares, case (3).
source distributions. It is worth noting that for bothsamples, the temperature rises at x 5 A4agts in the de-flecting medium (air) deviate more from zero with in-crease of time t, which means that Eqs. (7), (6c8), and (6d8)are not applicable when t > ts .
Figure 4 shows the time-resolved PD signals with tp5 0.5ts at the different positions in the deflecting media.The prediction by AA approximation [case (2)] is in goodagreement with that by AI for both samples in air, asshown in Figs. 4(a) and 4(c). It is interesting to note thatwhen the heating beam shuts off, the PD signal does notdecrease instantaneously, and the maxima of the signalswith different positions in the deflecting media take placeat different times. This phenomenon is probably due to
Fig. 3. Temperature rise in the deflecting media with the heat-ing beam pulse duration tp 5 0.5ts for different time t; lg
' A4agts. (a) Glassy carbon in air, (b) glassy carbon in water,(c) lime glass in air, and (d) lime glass in water. Solid curves,case (1); diamonds, case (2); squares, case (3).
Fig. 4. Time-resolved, photothermal-deflection signals with theheating beam pulse duration tp 5 0.5ts for different positions inthe deflecting media; lg ' A4agts. All results are normalized to2d/n]n/]T. (a) Glassy carbon in air, (b) glassy carbon in water,(c) lime glass in air, (d) lime glass in water. Solid curves, case(1); diamonds, case (2); squares, case (3).
Zhao et al. Vol. 21, No. 5 /May 2004/J. Opt. Soc. Am. B 1071
the time delay for the probe beam to sense the effect ofthe termination of the heating beam, depending on thedistance a between the probe beam and the sample frontsurface in Fig. 1. When the distance a is small, e.g., x5 0.1A4agts, the PD signal starts to fall almost immedi-ately. For a larger distance a, the time delay is longer be-cause it takes more time for the heat to diffuse through alonger distance. For the lime glass sample, the time-resolved PD signal decays slower after the termination ofthe heating beam than the PD signal does with the glassycarbon sample. For the transparent lime glass, the heatsource is distributed throughout the sample, while for theopaque glassy carbon it is only within a thin layer close tothe front surface. Heat conduction from the inside of thelime glass sample to the deflecting media after termina-tion of the heating beam slows down the PD signal decay.Figure 4(c) shows that the prediction by the AI approxi-mation is not as good as that by the AA approximation forthe lime glass sample; this is because of the unrealisticisothermal approximation for the transparent sample.
Concerning the temperature rise in water, Figs. 3(b)and 3(d), neither approximation is very close to case (1)for both samples because of the high value of the thermalconductivity of water and the consequent high heat fluxfrom the samples to water, especially when t . 0.1ts .Correspondingly, for the PD signals in Figs. 4(b) and 4(d),relative differences of more than 8% can be found at x5 20.5A4agts and t 5 0.5ts for both approximationscompared with case (1). The difference increases withthe decrease of uxu in the medium.
4. CONCLUSIONSA novel time-resolved, photothermal-deflection spectrom-etry with step optical excitation has been proposed with aset of closed-form analytical solutions. For the purposeof the practical application and the simplification of thetheory, the adiabatic–adiabatic (AA) and adiabatic–isothermal (AI) approximations have been fully exploredto determine their validity. The numerical simulationsreveal that for deflecting media with low values of ther-mal conductivity, such as air, the AA approximation iswell suited for both opaque and transparent samples usedin the simulations if t is short, while the AI approximationis applicable only to the opaque sample. For media withhigh thermal conductivity, e.g., water, neither approxima-tion is very effective. We conclude that these approxima-tions can be effectively used for characterizing solid ma-terials, such as measuring their optical and thermalproperties, with this novel spectrometry.
APPENDIX AWith the AA approximation h1 → 0 and h2 → 0
dm 5 mp/ls , m 5 1, 2, 3,... . (A1)
With these boundary conditions, the Green’s function is(Ref. 15, p. 361)
G~x, x8, t, t!
51
lsF1 1 2 (
m51
`
expS 2m2t
tsD cos
mpx
lscos
mpx8
lsG .
(A2)
By using Eq. (12), the temperature rise in the sample isfound to be
Ta~x, t ! 5 qt 1 (m51
`
Em@1 2 exp~2m2t/ts!#cosmpx
ls,
(A3)
Tr~x, t ! 5 qtp 1 (m51
`
Em@exp~m2tp /ts! 2 1#
3 exp~2m2t/ts!cosmpx
ls. (A4)
Here
Em 52b2I0@1 2 ~21 !m exp~2bls!#
ksls~mp/ls!2@~mp/ls!
2 1 b2#, (A5)
q 5I0
rscsls@1 2 exp~2bls!#. (A6)
The temperature rise in the deflection medium is
Tg,a~x, t ! 5 qtF S 1 1x2
2agt D erfcS 2x
2AagtD
1x
ApagtexpS 2
x2
4agt D G1 (
m51
` Em
2 H 2 erfcS 2x
2AagtD
2 expS 2m2t
tsD
3 F expS jmpAas
ag
x
lsD erfcS 2
x
2Aagt
2 jmA t
tsD
1 expS 2jmpAas
ag
x
lsD erfcS 2
x
2Aagt
1 jmA t
tsD G J , t < tp (A7)
Tg,r~x, t ! 5 Tg,a~x, t ! 2 Tg,a~x, t 2 tp!, t . tp .(A8)
The deflection angles in the active and relaxation inter-vals are, respectively,
1072 J. Opt. Soc. Am. B/Vol. 21, No. 5 /May 2004 Zhao et al.
wa~x, t ! 51
n
]n
]TdqtF2
x
agterfcS 2
x
2AagtD
22
ApagtexpS 2
x2
4agt D G1 (
m51
`
Dm expS 2m2t
tsD
3F expS jmpAas
ag
x
lsD erfcS 2
x
2Aagt
2 jmA t
tsD
2 expS 2jmpAas
ag
x
lsD erfcS 2
x
2Aagt
1 jmA t
tsD G , t < tp , (A9)
wr~x, t ! 5 wa~x, t ! 2 wa~x, t 2 tp!, t . tp .(A10)
Here
Dm 5j
2
d
n
]n
]T
mp
lsEm , (A11)
and n is the optical refractive index of the deflecting me-dium.
Corresponding author J. Shen’s e-mail address [email protected].
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