Analytical relations between the phase of the photothermal signal and the thermalwavelengthAngelamaria Figari Citation: Journal of Applied Physics 71, 3138 (1992); doi: 10.1063/1.350980 View online: http://dx.doi.org/10.1063/1.350980 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/71/7?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Optical signal storage and switching between two wavelengths Appl. Phys. Lett. 91, 071106 (2007); 10.1063/1.2770965 The influence of the spectral emissivity on the signal phase at photothermal radiometry Rev. Sci. Instrum. 67, 3658 (1996); 10.1063/1.1147131 Thermal diffusivity measurements using linear relations from photothermal wave experiments Rev. Sci. Instrum. 65, 2896 (1994); 10.1063/1.1144635 Relation between photoluminescence wavelength and lattice mismatch in metalorganic vaporphaseepitaxy InGaAs/InP J. Appl. Phys. 73, 4599 (1993); 10.1063/1.352751 Injected currentrelated distortion of photothermal signals from a photovoltaic cell Appl. Phys. Lett. 49, 1351 (1986); 10.1063/1.97375

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Analytical relations between the phase of the photothermal signal and the thermal wavelength

Angelamaria Figari CISE Tecnologie Innovative, Via Reggio Emilia 39, 20090 Segrate (Milano), Italy

(Received 24 June 199 1; accepted for publication 9 December 199 1)

Analytical expressions are derived for the photothermal signal, as the probe beam explores the thermal wave generated by the pump beam in the sample, grazing along its surface at a distance x from the heating spot. The relation between the phase of the signal and the thermal wavelength &, is pointed out, providing an analytical foundation to a refined photothermal technique for measuring the thermal diffusivity a. In particular, the analysis shows that both ;Ith and (r can be measured using arbitrary phase shifts relative to the heating spot center, which improves the measurement accuracy and the resolution in the application of this technique to imaging.

I. INTRODUCTION

The mirage effectId is the basis for an absolute and noncontact measurement of the thermal diffusivity a of materials. 7V8 This refined app lication of the photothermal technique is briefly summarized here. Consider a radiation beam (pump beam) for which the intensity is modulated with frequency o = 2rf using a chopper, and then focused to a region of the sample surface (Fig. 1). The sample is assumed to be photothermally saturated.’ A “thermal wave” is generated in the sample [as described by the so- lution of the periodical diffusion equation, exhibiting the behavior of a strongly damped wave (see, e.g., Ref. lo)]. The thermal wavelength is defined as7*i1

&h = 27r( 2cY/w) 1’2. (1)

The probe beam runs parallel to the sample and as close as possible to its surface. Furthermore, the distance x from the beam to the center of the heated spot can be varied. Once the position x is fixed, the probe beam is periodically deflected, due to the thermal gradient induced in the air close to the sample, with the components of the deflection vector in directions both perpendicular and parallel to the surface. It is this latter component which is measured by a position sensor and a two-phase lock-in amplifier. Accord- ing to Refs. 7 and 8, &, is determined by measuring the separation x0 between the points corresponding to a phase shift of AS-/~ with respect to the spot center x = 0 (Fig. 1) . Taking into account the pump beam diameter via the constant d, one can write

x0 z/&/2 + d. (2)

By combining Eqs. ( 1) and (2) one obtains the simple relation

x0- @if - 1’2 + d. (3)

According to Eq. (3), the thermal diffusivity a can be obtained from the slope ,,/& of the x0 vs l/ $f plot. This technique has already given interesting results7 and seems to be promising even for photothermal microscopy applications. l2

In Refs. 7 and 8 the experimental procedure was jus- tified by heuristic considerations and supported by numer- ical experimentation. In this paper:

( 1) Analytical expressions are derived for the photo- thermal signal, as the probe beam explores the thermal wave generated by the pump beam in the sample grazing along its surface at a distance x from the heating spot. These results provide insight into the link between the measurement of the phase $M of the signal and the deter- mination of the thermal wavelength jlth. They give an an- alytical foundation to the experimental procedure for mea- suring the thermal wavelength & and a! via a scanning in the x variable.

(2) In this theoretical frame, the wavelength ;Ith (and a) can be determined by measuring the separation x0(+*) between the points corresponding to an arbitrary phase shift $* of the signal with respect to the heating spot center x = 0.

II. THEORY The thermal diffusion length in the medium13 (or re-

duced wavelength”) is defined as

/..l = a&2%-. (4)

The expression describing the parallel component M of the probe-beam deflection is’

M(~,~) =, ge - iute - iCo2/2$ s

m XdA 0

e-44 Xsin(Lx)e- (X2d2’4)F(a) 7,

s (5)

where B is a constant factor, the integration variable X forms a Fourier variable pair with x (k/25- is the spatial frequency), 13 is the radius of the probe beam, d is the radius of the pump beam, a is the sample thickness, h is the probe-beam distance from the surface, and

(1 +Rb) + (1 -RRb)ee2”P F(a)sCl +.Rg)(l+R,) - (1 -R,)(l -RRb)e-2X#’

(6) with

3138 J. Appl. Phys. 71 (7), 1 April 1992 0021-8979/92/073138-05$04.00 @ 1992 American Institute of Physics 3138

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PUMP BEAM

SENSOR

FIG. 1. Experimental setup.

Rb=k&dk&, (7)

R, = k&&4 (8)

A,= (X2 - 2i/pf) “2. (I = s,g,b) (9)

Indices s&, and b refer to solid, gas, and backing, respec- tively, and k is the thermal conductivity.

Expression (5) can also be obtained by combining the equations of Refs. 6 and 14 and integrating across the pro- file of the probe beam. Furthermore, a recent Raper by LBrinczt2 discusses the temperature distribution T(x,h) in a metal. In our notation, Eq. (10) of Ref. 12 can be written as

T^(x,h) = $

(10)

where P is the incident power and h is the depth in the metal. Now, taking a derivative of the temperature field ( 10) with respect to x, which gives the parallel component of the mirage effect, one obtains an expression equivalent to Eq. (5), provided that (a) the Bessel function is approxi- mated by Jc(Xx) z cos(Xx); (b) the argument of the exponent, - A&, is replaced by - ,&Jr for the photother- ma1 signal in air; (c) the effect of a backing is taken into account, so that F#l; (d) the effect of a finite probe beam is introduced via the exponential factor exp( - id/2pi). Starting from expression (5) and tak- ing the limits d-0, h-+0, and a-+ CO the deflection M is proportional to the integral

ii(x) = s

m k(X2 + P’) - 1’2 siri(kx (11) 0

where

f12= - 2i/p f . (12)

Now the integral ( 11) can be solved’5”6 to obtain

kx) = PK,(xP). (13)

In other words,

&(x) =--,-rw4 z K1 ($emiT14), (14)

where Kt is a first-order Bessel function whose argument has its phase equal to - 7r/4, namely, a Kelvin function of order unity.

The first order of the Kelvin function depends on the structure of the generalized thermal wave number X, [Eq. (9>]: the function A(X2 + fi2) - 1’2 is the kernel of the Fou- rier sine transform Eq. ( 11) and one can demonstrateI that the transform of the function X(X2 + B2) ‘- “2 gener- ates the Kelvin function K,, + , (xfl).

The expression

ZdZX/ps (15)

represents a corrected ratio of distance x to the thermal diffusion length.

Since z is real, Kt (ze - i”‘4) can be represented by the relation’“‘9

ker,(z) - ikeii(z) = eir’2K,(ze-iV’4) (16)

with the functions ker, (z) and keit (z) being real: they are defined through series expansion (reported in the appen- dix) and calculated, e.g., in Ref. 18.

Relation ( 14) yields the useful formula

k(z) =iY,+&z)e’h [kert(z) - ikeii(z)], (17)

where &(z) is an amplitude, e’b a constant phase factor. With the help of the decomposition”

the

l--l(z) = Nl(z) cos 4,(z), (18)

keil (z) = IV1 (z) sin 4, (z),

integral M(z) can be written as

M(z) = pM(Z)ei~oeih(Z), (19)

wherepw(z) is an amplitude and +i (z) a well-known func- tion (tabulated, e.g., in Ref. 18, p. 433).

Hence it turns out that the phase 4M of M depends only on ~1,. Furthermore, Eq. ( 19) gives an analytical de- scription of the experimental procedure for measuring the thermal wavelength jlth via a scanning in the x variable. Actually, one sees that the phase $M is a function of the ratio x/p, = (27r/,&)x. The physical meaning of 4, (z) is a phase shift of the signal with respect to the phase 4, (0) at z = 0. Therefore the phase offset of the first-order Kelvin function at z = 0, 4,(O) = - 2.356194 radians” can be suitably eliminated by a clockwise rotation of the reference frame by an angle 4, (0). This rotation corresponds to fix- ing the Cartesian axes of the lock-in so that the phase of the signal vanishes at z = 0, i.e., the signal is a real quantity in the origin of axes. The components of Kt (ze - ilr/4) must be recalculated relative to the new reference frame: the transformed components of the Kelvin function will be indicated with an asterisk. The results are reported in Figs. 2-7.

3139 J. Appt. Phys., Vol. 71, No. 7, 1 April 1992 Angelamaria Figari 3139 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

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T 0

8 b a -20

8

23 -40 * a.

-100

-120

-140

-160

T .$ -0.4 2

Y ‘;j

-0.8 *- ‘8,

-1.2

-1.6

-2.0

-2.4

-2.8

FIG. 2. Phase 4*(z) of the photothermal signal.

III. RESULTS

Figures 2 and 3 show the phase 4*(z) and modulus p*(z), respectively; Figs. 4 and 5 show the real part ker: z and the imaginary part kei: z. Finally, the meaning- ful ratio of real to imaginary component of the signal M(x) is reported (Figs. 6 and 7):

Re(M) keryz ----=-=cotan[$*(z)]. Im(M) keiyz (20)

From the definitions of z [Eq. ( 15>], ,uL, [Eq. (4)], x0 ( = 2x) and taking into account the pump-beam diameter via the constant d, one can write

x0(4*) = A(4*) &h/2) + d (21)

* ig a 8 0 r 7

6

FIG. 3. Modulus p*(z) of the first-order Kelvin function calculated rel- FIG. 5. Enlarged part of Fig. 4: z = 24.5. kery z = 0 at z= 2.666; ative to the reference frame in which K,(O) is real. moreover, kery z = keiy z at z-3.804.

5

4

3

2.5

2

1.5

1

0.5

0.0

-1

ker,* z

\

L&e=77

kei;z

1 I I I

FIG. 4. Real part (kery z) and imaginary part (key? z) of the first-order Kelvin function, calculated relative to the reference frame in which key: = 0. They are proportional to the real and imaginary components of the photothermal signal, through the same proportionality factor.

with

A(+*) = (vmr)z.

From the definition of &, [Eq. (l)] one obtains

(22)

3140 J. Appl. Phys., Vol. 71, No. 7, 1 April 1992 Angelamaria Figari 3140

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ker,*z

kei ,* z

-lO-

-15-

-2o-

/

FIG. 6. Ratio of real to imaginary component of the photothermal signal (Fig. 6 is obtained from Figs. 4 and 5).

x0(4*) = A(t$*) ,/ii&f - “* + d. (23)

Let us examine some particular cases: (1) In Figs. 4-7, the points z at which the plots of the

functions go to zero correspond to a purely imaginary sig- nal, i.e., values I#J*(z) = - rr/2. This is the case of the “classic” experiment,7*8 and Eq. ( 19) gives the value z u 2.666 (Ref. 18 and Figs. 2, 4-7). The definition of the factor A [Eq. (22)] gives A u 1.2; this term was equal to 1 in Eqs. (2) and (3), derived in a heuristic way.7

(2) For 4*(z) = - 7r/4, Eq. ( 19) gives the value z z 1.496 (Ref. 18 and Fig. 2) and A ~0.673.

(3) For 4*(z) = - (3/4)7r, Eq. (19) gives the value z-3.804 and A- 1.713. In this case ker:(z) = keiy(z), which corresponds to the crossing in Fig. 5.

(4) If x0 equals the thermal wavelength, i.e., x0 = &,, then 4*(z) = - 2.8 radians and z-4.443.

FIG. 7. Enlarged part of Fig. 6: z = Z-3.2. The function is zero at z Therefore, with h-0 and low chopper frequency, one =2.666. is allowed to neglect the influence of the gas. Conversely, in

The general recipe for obtaining the factor A ( t$* ) is the following: if 4*(z) is fixed, Eq. ( 19) gives the correspond- ing value z (Ref. 18 and Fig. 2) and Eq. (22) gives A.

The developed theory shows that one can measure the thermal wavelength /2th and the thermal diffusivity a using arbitrary signal phase shifts relative to the heating spot center. This procedure offers three advantages for applica- tions to imaging with respect to the standard technique:

(1) & and a can be obtained as mean values calcu- lated for different measured values jlrh ($* ) and a (r#~*) cor- responding to arbitrary phase shifts +*. The measurement accuracy should be greatly improved.

(2) Taking into account Eqs. (2 1) and (23)) &, and a can be obtained without scanning the whole distance x0 corresponding to phase shifts $* = *r/2 with respect to the spot center x = 0 [Eqs. (2) and (3)]; any smaller phase shift +* and consequently any smaller distance x0( $*) can be chosen to evaluate ilrh and a. Clearly, measurements made closer to the heating spot should improve the reso- lution in applications to imaging.

( 3 ) As a consequence of point ( 2 ), a higher scan speed can be obtained.

A final remark concerns an approximation of the tech- nique. The temperature exhibits various profiles depending on the value of the thermal diffusivity; this temperature profile will heat the air in the immediate vicinity of the sample. One would expect that the magnitude of the signal in a particular measurement will be a function of both the sample diffusivity and the air diffusivity. However, we ob- serve that the signal depends on the diffusivity of air through two terms: ( 1) F(a) and (2) exp( - AJr) [see Eq. (511.

( 1) Omitting the dependence of the signal on backing, which is not relevant to this problem, F(a) can be written

1 F(a) = 1 + k&/k&, * (24)

Even if the values for thermal diffusivities of the sample and air are comparable, but kg < k, the term F(a) can be neglected.

(2) Consider now the term exp( - k&z). Measuring the temperature gradient in the air extremely close to the surface, with h-0, gives

exp( -@)-I (25)

allowing one to neglect this term also. One can consider the question from another point of

view: generally the measurement of the value of As in air can be “disturbed” by the thermal diffusion from the sam- ple into the air at the interface. For a low chopper fre- quency, the thermal wavelength & of air is high, and h is a small fraction of the thermal diffusion length in air. In other words, the temperature variation cannot be appreci- ated on a scale h much smaller than the thermal diffusion length in air.

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the high-frequency limit, necessary for thermal-wave microscopes,‘* the thermal diffusion coefficient of the gas cannot be omitted.

IV. CONCLUSION W) = 0,

Meaningful analytical expressions describing the pho- tothermal signal within the thermal wavelength have been derived for photothermally saturated samples. In particu- lar, these relations permit one to measure the thermal wavelength and the thermal diffusivity choosing arbitrary signal phase shifts relative to the heating spot center. This procedure offers definite advantages with respect to the standard technique in terms of measurement accuracy and resolution. The analysis developed in this paper can be useful in view of a future evolution of the photothermal technique in imaging applications such as microscopy and tomography.

beri(z) = ,z, ~~)~~)i, cos (3 + 2X)77

4 ,

m (z/2)%+’ (3 +2&r beit = c xc0 x!(l+x)!sin 4 *

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APPENDIX

kerl(z) = - lnii-‘j’ berl(z) +$nbei,(z) -Gi ( )

(3 + 2X)n x cos

4 ’

1 VQl -4qberi(z) -y;

+z p. //!(I +A)! IW) +4(1 +A)1

X sin (3 -I- 2&n

4 ’

with

y(Euler’s constant) = 0.577215...,

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Interscience, New York, 1980). “J Opsal, A. Rsosencwaig, and D. L. Willenborg, Appl. Opt. 22, 3169

(;983). “J S Gradshteyn and J. M. Ryzhik, Tables of Integrals, Series and

&acts (Academic, New York, 1980). 16A. Erdtlyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Tables of

Integral Trangbrms, Vol. 1 (McGraw-Hill New York, 1954). “A. Erdtlyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher

Transcendentai Functions, Vol. 2 (McGraw-Hill, New York, 1953). ‘*M Abramovitc and I. Stegun, Handbook of Mathematical Functions,

8th ed. (Dover, New York, 1972). I9 G N Watson, A Treatise on the Theory of Bessel Functions (Cambridge

University, Cambridge, 1966).

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