THE THEORY OF PHOTOTHERMAL EFFECT IN FLUIDS

PHOTOTHERMAL INVESTIGATIONS OF SOLIDS AND FLUIDS

CHAPTER 3

THE THEORY OF PHOTOTHERMAL EFFECT IN FLUIDS R.

GUPTA

Department of Physics University of Arkansas Fayetteville, AR

I. Formation of the Thermal Image A. A Simple Model B. Thermal Diffusion in the Presence of Fluid Flow C. Solution of the Differential Equation for a Pulsed Source D . Solution of the Differential Equation for a CW Source II. Detection of the Thermal Image A. Photothermal Phase-Shift Spectroscopy B. Photothermal Deflection Spectroscopy C. Photothermal Lensing Spectroscopy References

82 82 84 86 90 93 94 101 112 126

In photothermal spectroscopy, absorption of a laser beam (pump beam) results in an increase in the temperature of the irradiated region. The increase in temperature is normally accompanied by a decrease in the refractive index of the medium. The changes in the refractive index can be monitored in several different ways. For example, if the absorbing medium is placed in one beam of an interferometer (or inside a F a b r y - P e r o t cavity), the change in the refractive index would cause a fringe shift that can conveniently be detected as an intensity change of the central fringe. The nonuniform refractive index produced by the absorption of the p u m p beam can also be detected by the deflection of a probe laser beam passing through this medium. The nonuniform refractive index may also produce a lensing effect, that is, a probe beam passing through the medium (as well as the p u m p b e a m itself) may change shape. This can be detected as a change in the intensity of the probe (or the pump) beam passing through a pinhole. T h e theory of photothermal spectroscopy of fluids naturally breaks down into two parts: formation of the thermal image and the optical detection of the thermal image. The former subject is discussed in Section I, while the latter is discussed in Section II. 81 Copyright © 1988 by Academic Press, Inc. All rights of reproduction in any form reserved ISBN 0-12-636345-5

R. Gupta

82

I. Formation of the Thermal Image In this section, an expression for the temperature distribution T(r, t) produced by the absorption of a laser beam (pump beam) and the subsequent evolution of this temperature distribution with time will be derived. A derivation based on a simple model appropriate for a short laser pulse ( G u p t a , 1987) will be given in Section I.A. This model has the advantage over the rigorous solution that it gives good physical insight into the problem. A rigorous solution to the problem (Rose, Vyas, and Gupta, 1986) will be given in Sections I.Β and I.C. A solution for the case of a continuous-wave (CW) laser beam (Vyas et al., 1988) will be derived in Section I.D.

A . A SIMPLE M O D E L

Let us assume that the pump-laser beam propagates through the medium in the z-direction, and the medium is flowing with velocity vx in the x-direction. The p u m p beam is assumed to be centered at the origin of the coordinate system. If we assume that the medium is weakly absorbing, the heat produced per unit time per unit volume by the absorption of laser energy is given by Q(x,

y, t)

=

al(x,

y, t) for

0 < t < t0

where a is the absorption coefficient of the medium, and I(x, y, t) is the intensity of the laser beam. Implicit in Eq. (1) is the assumption that all of the absorbed laser energy is converted to heat and only a negligible portion is radiated as fluorescence. It is further assumed that this conversion of optical energy into heat takes place on a time scale much shorter than the typical thermal diffusion and convection times in that medium. The total energy in each laser pulse is assumed to be E0. The spatial profile of the b e a m is assumed to be a Gaussian with l / e 2 - r a d i u s a. It is further assumed that the laser pulse turns on sharply at / = 0 and turns off sharply at t = t0. The assumption of a rectangular temporal profile is a good one if the rise and the fall times of the laser pulse are very short compared to the thermal diffusion and convection times.

83

THE THEORY OF PHOTOTHERMAL EFFECT IN FLUIDS

In this simple model, we assume the laser pulse to be so short that during the period that the laser pulse is on, no appreciable heat is transferred out of the laser-irradiated region either by thermal diffusion or by the flow of the medium. The temperature rise of the laser-irradiated region above the ambient is then given by = -ί-

T(x,y)

=

f'°Q(x,y,t)dt

2«£„

,

— p

,

2

2

)

,2

-2(x +y )/a

Here ρ is the density and C p is the specific heat at constant pressure of the medium. Equation (2) shows that a cylindrical volume of heated medium with a Gaussian cross-section is produced by the absorption of the laser pulse. After the laser pulse is over, this "line of heat" moves downstream with the flow of the medium, and as it does so, it broadens spatially due to thermal diffusion. By ignoring the effect of the thermal diffusion for the moment, the spatial and temporal behavior of the heated region may simply be written by replacing χ by (x - vx(t — t0)) as 2aE T(x,

y91)

2

—^e-2K*-M'-'o)) +>V<

=

(3)

πα p C p

N o w consider the effect of the thermal diffusion. At t = r 0 , the laser-irradi2 ated region has a Gaussian spatial profile with a l / e - r a d i u s a. At t > t0, the heated region broadens, and its new radius a'(t) is given by a'(t)=[a

2

1/2

(4)

+ vD(t-t0)} , 2

where D is the thermal diffusivity of the medium ( m / s ) and η is a constant. Equation (4) has been written down simply from dimensional arguments. Rigorous calculations show that η = 8 (see Section I.C). Therefore, to take the effect of thermal diffusion into account, the radius of the heated region a in Eq. (3) must be replaced with a'(t). The result is 2aE0 Τ(γ

ν

A

=

-

-2{[x-M'-'o)r+r}/["

2

+ 8£(>-'o)]

(5) Figure 1 shows the spatial profile of the heat pulse in a flowing gas at various times after the end of the laser pulse, as calculated from Eq. (5)

< 2

R. Gupta

84

*

1.5-v x= 0 . 5 m / s

LU D

h-

1.0"

<

CL

2

t=2ms

0.5--

LU

0

4a

8a

12a

16a

20a

Distance FIG. 1. Spatial profile of the heat pulse in a flowing gas at different times after the end of the excitation pulse. The medium was assumed to be N 2 at room temperature seeded with 1000 1 ppm of N 0 2 (absorption coefficient a = 0.39 m " ) and flowing with velocity vx = 0.5 m / s . The pump laser was assumed to have an energy E0 = 1 mJ in a pulse of / 0 = l-ju,s duration. 2 ( l / e ) - r a d i u s of the pump beam was 0.35 mm. (From Rose, Vyas, and Gupta, 1986.)

(Rose, Vyas, and Gupta, 1986). The distance has been plotted in units of 2 the l / e - r a d i u s of the pump beam. As expected, the heat pulse moves downstream with the flow, and as it does, it broadens due to thermal diffusion. Various parameters used in this calculation are given in the figure caption.

B . THERMAL D I F F U S I O N IN THE PRESENCE OF F L U I D F L O W

In this section, a general equation describing the temperature distribution T(r, t) in a fluid medium will be derived. The fluid is assumed to be moving with velocity ν in an arbitrary direction, and the heat is supplied by a source <2(r, / ) . Consider a small element of volume dV = dxdydz at x, y,z with temperature at its center T(x, y, z, t), as shown in Fig. 2. To begin with, assume that the medium is stationary and that there is no source of heat present. Let us consider the flow of heat in the x-direction. T h e temperature at the two boundaries of the volume element perpendicular to the x-direction is given by

(6)

THE THEORY OF PHOTOTHERMAL EFFECT I N FLUIDS

85

dy

dx

dz , / ~ \ J .-•'/ ΙΧ^-Τ(χ,ν,ζ,ϋ ·' ^ x v

FIG. 2. Transport of heat through a volume element dx dy dz due to thermal conduction and forced convection.

T h e rate of heat transport across the two boundaries is then given by dq + dt

=

-kdA

dT

Λ

dx

d = -kdydz—\T± dx\

I

(7)

dT dx — — dx2

where k is the thermal conductivity of the medium. The net heat gain of the volume element due to thermal conduction is then the difference between dq_/dt and dq+/dt: dq —

dt

d2T =

(8)

kdV—z.

dx2

N o w consider the heat gain of the volume element due to the flow of the m e d i u m in the jc-direction with velocity component vx. The rate at which the heat flows in and out of the volume element is given by dq\

-•

(

_ dT dx

. = C dyd »lT+m

P

p

Z

Y

(9)

where the plus sign applies to heat flow out of the volume element, and the minus sign applies to heat flow into the volume element. The net gain of

86

R. Gupta

heat by the volume element due to the flow is then dq\

dT

= -pCrdV—ox.

(10)

The total heat gain due to thermal conduction and the flow is then given by the sum of Eqs. (8) and (10): da

d2T

3T

Eq. (11) can easily be generalized to three dimensions: ^ = kdW2Tdt

pCOdV\·

v

(12)

VT.

If there is a source of heat present that deposits an amount of heat Q(r, t) per unit volume per second, then Eq. (12) modifies to — = kdW2Tdt

pC„dV\

p

· νΓ+

QdV.

(13)

N o w , since the heat gain of the volume element is related to the temperature rise by dq

dT

the final equation for the temperature distribution T(r, t) is given by

d^/;^ dt

= D v r ( r , 0 - ν - v r ( r , 0 + - ^ ß ( r , t).

T h e constant D = k/pCp

2

pCp

(15)

is called the thermal diffusivity of the medium.

C . S O L U T I O N OF THE DIFFERENTIAL EQUATION FOR A P U L S E D S O U R C E

A rigorous solution of the differential equation, Eq. (15), will now be attempted in order to determine the temperature distribution in the medium. For the conditions described earlier in Section I.A, that is, for the laser b e a m propagating through the medium in the z-direction and the flow velocity confined to the jc-direction, the differential equation for T(x, y, t)

THE THEORY OF PHOTOTHERMAL EFFECT IN FLUIDS

87

is dT(x,y,t) 8t

2

= Dv T{x,y,t)

-

vx

dT(x,y,t) dx (16)

T h e first, second, and third terms on the right in Eq. (16) represent, respectively, the effects of thermal diffusion, fluid flow, and the heating due to p u m p - b e a m absorption. The two-dimensional differential equation implies that any inhomogeneities in the medium along the p u m p beam are negligible. T h e heat source is assumed to be again given by Eq. (1), i.e., we again assume the medium to be weakly absorbing, with the p u m p beam having a Gaussian spatial and rectangular temporal profile, etc. In other words, all assumptions leading to Eq. (1) are still valid. The assumption of short laser pulse is now relaxed, however, and the laser pulse width is now arbitrary. Equation (16) is essentially the equation of conservation of energy. For a general solution to the problem at hand, one must also consider the conservation of mass and momentum. It is, however, not necessary to consider the latter two if the effect of the pressure pulse (photoacoustic effect) accompanying the photothermal signal (Rose, Salamo, and Gupta, 1984) is ignored. The pressure pulse occurs on a much shorter time scale than the thermal diffusion (Héritier, 1983), and the energy carried off by the pressure pulse is negligible (Tarn and Coufal, 1983). Therefore, the effect of the pressure pulse on the photothermal signal will be neglected and the conservation of mass and momentum will not be considered. T h e following boundary conditions are assumed to hold: T(x,y,t)\t=o

= 0

T(x,y,t)\x=±x=0

T'(x,y,t)\t=0 T(x,y,t)\y=±x

= 0, == 0.

(17)

H e r e V is the gradient of the temperature. These boundary conditions imply that we have taken the ambient temperature to be zero, and therefore, T(x, y, t) in Eq. (16) may be considered to be the temperature above the ambient. W e shall use the Green's function method (used earlier by Rose, Vyas, and G u p t a , 1986) for a solution to Eq. (16). The solution is given by (see for example, Arfken, 1985) "+•

/ ——

O

f /

+

O

O O RVΊ J — cr\

/ *0

/ 0· 0

Q(t,V,r)G(x/Ç;y/r,t/T)dtdridT,

(18)

88

R. Gupta

where the functional form of <2(£, η, τ ) is given by Eq. (1). The Green's function satisfies the differential equation 2

-Dv xyG

dG 3G 1 + vx— + — = —

8(x - ξ) S(y - η) S(t - τ ) , (19)

with the b o u n d a r y conditions G ( + o o / £ ; y/η; t/τ) = 0, ± ο ο / η ; ί / τ ) = 0,

G(x/t\ G(x/£;

y/ψ,Ο/τ)

(20)

=0.

T h e solution to Eq. (19) can b e found conveniently b y taking the Fourier transform of this equation (Arfken, 1985). The resulting equation in the ω χ , ωγ space is / \ 2 2 (ω + u y)DGF

- iuxvfi¥

dGF 1 ( + — = -^*'

ω Α + ω η ) δ

( ' - Ο - (21)

*

H e r e GF is the Fourier transform of the Green's function. An examination of E q . (21) shows the simplification afforded by this transformation: Differential operators have been reduced to algebraic factors. Equation (21) can be further simplified by taking its Laplace transform from /-space to its complementary s-space. The result is 2

2

{ω χ + o^ y)DGFL

- io>xvxGFL

+ * G FL =

)

where G F L is the Laplace transform of G F . Eq. (22) is simply an algebraic equation with the solution,

^FL

=

Γ—1—9 I

27TPCp[D(œ x

T\

Γ*

(^)

+ ω ; ) - ίωχνχ + s\

N o w GF can b e obtained by taking the inverse Laplace transform of Eq. (23). This can b e accomplished conveniently by recognizing that GFL is of the form

(

2

2

89

THE THEORY OF PHOTOTHERMAL EFFECT IN FLUIDS

where both C and k are complex functions, and that (Arfken, 1985) Lie")

-

^

(25)

By using Eqs. (24) and (25), we get GF = L-i(GFL) =

=

L-l[Ce^L(ek')]

(26)

CHT(t)ek('~T\

where H (t) is the unit step function: T

t V

\1

7

for

t > τ.

By inserting the values of C and k in Eq. (26), we obtain e^^e^H

(t)

T h e original Green's function is now obtained by taking the inverse Fourier transform of Eq. (28): HAt) G

=

/>O0 T

—-— 4

7

7

^

j

^

-

io>At 0

e+ v x ( t - T ) ) e - " x D ( t - T ) e - i < » x x

ei u > y

V- oe > y D ( t - T ) e - i u > y y

j

Both integrals in convolution theorem (Arfken, 1985):

conveniently by use of the

fCC

OO

F(Ux)GMe-'»''dUx -

(29)

-y

—™ Eq. (29) can b e evaluated

/

du

0

/ • OO OO

/

Τ

= ]

oo

(30)

f(X)g(x-X)d\, 0

0

where functions / and g are the inverse Fourier transforms of the functions F a n d G, respectively. We obtain H (t) G

=

T

-L1

-[x-a

+ vx(t-r))]2/4D(i-r)

-[y-v]2/4D(t-r) e

4 7 T PC pZ ) ( / - T )

where we have used the following two relationships, 1

Ζ00 *

/Q\

'

V

;

R. Gupta

90 and b)/2

Finally, the temperature distribution T(x, y, t) is obtained by substituting the Green's function, Eq. (31), in Eq. (18). Integrations over £ and η can be performed in a straightforward manner, with the result 2 -2{[x-vx(t-r)]2+y p }/{%D(t-T)

I m F

TV T

{

X

' y>

Λ t }

-

0

a2}

Λ

vt0pCpJ0

for t >

+

f°_ [W(t-r)

+ a2]

'd

r

(32)

t0.

Unfortunately, the integration over τ cannot be performed analytically and must be done numerically. This integration may be conveniently performed by using one of the commonly available subroutines in IMSL (International Mathematical and Statistical Library). O n e very useful case in which the temperature distribution T(x, y, t) can be written in a closed form is when the laser pulse is very short. In this case, lim Γ ° / ( τ ) < / τ = / ( 0 ) ί ο ,

(33)

and 2ccE T(x, V

, y

)

yt }

-Wx-ux')2+y2lAa*

=

vpCp(SDt

+ *Dt]m

e

+ a) 2

) }

This result is in agreement with that of a simple model, Eq. (5), and the results are shown in Fig. 1. Rose, Vyas, and G u p t a (1986) have investigated the range of validity of Eq. (34) by comparing the prediction of Eq. (34) with that of Eq. (32) for different laser pulse widths t0. They conclude that for pulse widths t0 < 10 /is, Eq. (34) gives almost exact agreement with Eq. (32), but caution must be used for longer pulse widths.

D.

S O L U T I O N OF THE DIFFERENTIAL EQUATION FOR A C W SOURCE

In many situations, it is more desirable to use a CW laser source for the p u m p beam. The photothermal signal can be measured conveniently if the C W laser is amplitude-modulated at some frequency ω, because then a phase-sensitive detection can be used, that is, a signal at frequency ω and having a definite phase with respect to the source can be detected. The source term Q(x, y, t) in this case may be written as

( V

3

4

THE THEORY OF PHOTOTHERMAL EFFECT IN FLUIDS

Q(x,y,t)

=

-—rle-^' ^" ]^ 1

1

+

(35)

cos ωή,

where P a v is the average power of the p u m p beam in the presence of the modulation. The p u m p laser is sinusoidally modulated with the degree of modulation being 100%. The spatial profile of the p u m p beam is again assumed to be Gaussian with 1/e2-width a. The temperature distribution is again given by the solution of Eq. (16), that is, it is given by Eq. (18) with the functional form of ζ)(ξ, η, τ ) given by Eq. (35). The Green's function is independent of the source term and it is still given by Eq. (31). As a matter of fact, the strength of the Green's function method lies precisely in the fact that a solution for T(x, y, t) can be found for any source term that can be expressed analytically (although the final result may not be in the closed form). In the present case, the result is

(36) -2{[x-ox(l-T))2+y2}/[a2 Xe

+ 8D(t-T))

l

d

The integration over τ must again be performed numerically. A few typical temperature distributions produced by a CW laser beam are shown in Fig. 3 (Vyas et al., 1988). The integration in Eq. (36) was performed by the method of 64-point Gaussian quadrature (Vyas et ai, 1988). In this calculation, the modulation frequency ω was set equal to zero. T h e temperature T(x, y) has been plotted as a function of the distance χ from the center of the p u m p beam. This distance has been expressed in units of the l / e 2 - r a d i u s a. Positive χ is measured in the direction of the gas flow. Because we have assumed our medium to have no boundaries, the temperature of the medium never attains a true steady state. However, for points close to the p u m p beam, a quasi-steady state is established, that is, the rate of temperature increase becomes negligible after a certain time (a few seconds for a stationary medium). The approach of this quasi-steady state becomes faster as the flow velocity of the medium increases. Monson, Vyas, and G u p t a (1988) have discussed the temporal evolution of the temperature in more detail. The curves shown in Fig. 3 have been computed for t = 5 sec, when a quasi-steady state has been attained. The b o t t o m curve is for a stationary medium (vx = 0), and the other curves are for vx = 1 c m / s , 10 c m / s , and 1 m / s , as labeled. All other relevant parameters used in this calculation are given in the figure caption. For vx = 0, the temperature distribution extends far outside the laser beam due to thermal diffusion. As the flow velocity increases, the temperature distribution becomes more and more asymmetric, and the peak value of the temperature rise becomes smaller because the heat is carried by the medium in the direction of the flow.

R. Gupta

92

Α

LU Od

an LU Û_

FIG. 3. Temperature distribution in a medium for zero-modulation frequency and for vx = 0, 1 c m / s e c , 10 c m / s e c , and 1 m / s e c , as noted on the diagram. In this computation, the 1 medium was assumed to be N 2 seeded with 1000 ppm of N 0 2 ( a = 0.39 m " ) , laser power was 1 W, and the pump-beam radius a was taken to be 0.5 mm. Abscissa is plotted in units of a (0.5 mm). (From Vyas et al, 1988.)

t = 0 . 1 0 sLU DC I<

ce

4-

t=0.13S

t = 0 . 1 7 S-

x/a

*

FIG. 4. Variation of the temperature distribution with time in a medium with flow velocity vx = 10 c m / s , when the pump beam is modulated at 10 Hz. The four curves represent five different times in the modulation cycle of the pump beam as noted on the diagram. Time t = 0.10 sec corresponds to the peak of the laser power, while t = 0.15 sec corresponds to the minimum of laser power. All parameters used in this computation are given in the caption to Fig. 3. (From Vyas et ai, 1988.)

THE THEORY OF PHOTOTHERMAL EFFECT IN FLUIDS

93

Figure 4 shows the effect of the modulation on the temperature distribution (Vyas et al, 1988). Four curves are shown in the figure for / = 10 Hz and vx = 10 c m / s . The four curves correspond to t = 0.10, 0.13, 0.15, and 0.17 sec. Time t = 0.10 corresponds to the peak of the laser power, whereas t = 0.15 corresponds to the minimum of laser power. As can be seen from these curves, the temperature distribution goes through drastic changes in shape as the laser power oscillates.

II. Detection of the Thermal Image T h e change in the refractive index of the medium produced by the absorption of the pump beam may be detected in a number of different ways. For example, a probe-laser beam passing through the sample will experience a change in the optical path length. The change in the optical p a t h length may conveniently be detected as a fringe shift in an interferometer (Stone, 1972; Stone, 1973; Davis, 1980; Davis and Petuchowski, 1981). In this method of detection, the signal is proportional to the change in refractive index n(r, / ) , and we shall refer to this technique as photothermal phase-shift spectroscopy (PTPS). PTPS will be discussed in Section H.A. Another convenient method for the detection of the thermal image relies on the measurement of the gradient of the refractive index, dn(r, t)/dr. The change in the refractive index of the medium due to the absorption of the p u m p b e a m follows the spatial profile of the p u m p beam (which is generally assumed to be a Gaussian). A probe-laser beam passing through the inhomogeneous refractive index suffers a deflection that can easily be measured by a position-sensitive detector (Boccara, Fournier, and Badoz, 1980; Boccara et al, 1980; Fournier et al, 1980; Jackson et al, 1981). This technique is called photothermal deflection spectroscopy (PTDS) and will be discussed in Section II.B. The nonuniform change in the refractive index may also produce a lensing effect in the medium. A probe beam passing through the medium may change shape (the sample may act as a lens) resulting in a change in the intensity of the probe beam passing through a pinhole ( G o r d o n et al, 1965; H u and Whinnery, 1973). In this technique, the signal is proportional to the second derivative of the refractive index, d2n(r, t)/dr2. This technique is called the photothermal lensing spectroscopy (PTLS) and is discussed in Section U.C. F o r each of the three methods mentioned above, we shall consider two cases, collinear and transverse. Figure 5 shows the typical p u m p - and probe-beam configuration. The p u m p beam propagates in the z-direction, and the flow velocity of the medium is in the x-direction. For the transverse case, the p r o b e beam propagates in the ^-direction, whereas for the collinear case, the probe beam propagates in the z-direction. Pump- and probe-

R. Gupta

94

Χ

PROBE

α

PUMP

(a) FIG. 5.

PUMP

(b)

Pump- and probe-beam configurations for (a) transverse and (b) collinear geometries.

b e a m axes may or may not intersect. In all cases, the probe-beam radius is assumed to be much smaller than the pump-beam radius.

A . PHOTOTHERMAL PHASE - SHIFT SPECTROSCOPY

In this section, expressions for PTPS signal size and shape for a sample placed in one arm of a Michelson interferometer will be derived. Similar expressions for other types of interferometers (e.g., F a b r y - P e r o t interferometer) can be derived analogously (Campillo et ai, 1982). A typical experimental arrangement is shown in Fig. 6(a). M4 and M5 are dielectric mirrors that totally transmit the probe beam and totally reflect the p u m p beam. Broken lines show transverse PTPS, whereas the solid lines show the collinear PTPS. A general expression for the PTPS signal in terms of the change in refractive index Δ « ( γ , t) is derived in Section I I . A . l , and explicit expressions for pulsed and CW PTPS signals are derived in Sections II.A.2 and I I . A . 3 , respectively (Monson, Vyas, and Gupta, 1988).

1. THERMALLY I N D U C E D PHASE-SHIFT A N D INTENSITY CHANGE IN AN INTERFEROMETER

Let us assume that the two interfering waves have amplitudes A and Β and phases φΑ and φΒ. An additional phase difference y(t) is introduced into one of the waves by the photothermal effect. The interfering waves may be written as a = A sin(kx — ωί + φΑ) and

b = Β nn(kx - ωί + φ + γ ( 0 ) · Β

T h e resultant wave is then c = a + b = R sin(kx

- ωί + δ ( / ) ) ,

( ) 3 7

THE THEORY OF PHOTOTHERMAL EFFECT IN FLUIDS M6

LASER

95

M1

^ —\ ν• ^ PUMP • BEAM

M4

M7 M5

S A M P L E CELL

<4-

LASER

M3

PROBE BEAM

~7

M2

(a) APERTURE PHOTODETECTOR

(b)

(Φ Α"Φ Β)

—

FIG. 6. (a) Schematic illustration of a PTPS experiment. The sample cell is placed in one arm of a Michelson interferometer. The pump beam passes through the cell either collinearly (solid line) or transversely (dotted Une), (b) Intensity variation observed at the photodetector as a function of the phase difference (φ^ - φΒ).

where R and δ(ί) are given by R cos 8(t)

= ^cosqp^ + Bcos(
+ γ(0)>

and R s i n S ( i ) = A sin
= R2 = A2 + B2 + 2ABcos[(
-
(38)

Here (φ^ — φΒ) is the constant phase difference between the two waves (in the absence of the photothermal signal). The "operating point" on the intensity curve is determined by (φ^ - φ#), as shown in Fig. 6(b). If (Φβ ~ Ψβ) = W 7 >7 t ne operating point is either Ρ or P', and the intensity is quite insensitive to small changes in γ ( / ) . This is the least desirable situation. O n the other hand, if (φ^ -
96

R. Gupta

terms of the temperature change Τ produced by the absorption of the p u m p beam: y(x,y,t)

=

A ^path

(39)

àn(x,y,t)ds,

where the integration has been carried out over the path of the probe beam, and we have assumed the probe beam to be infinitesimally thin. Since T(x, y, t) / \

= (n0-l)

Hn(x,y,t)

(40)

where TA is the ambient temperature and n0 is the refractive index at the ambient temperature, Eq. (39) becomes y(x9

y,t)

4π (n0 - I) = / A

r

(41)

T(x9 y, t) ds. J

path

Assuming that we have chosen our operating point to be such that (φ^ Ψ b ) = ( m + 1/2) π, we get for the signal: / 477 ( / l 0 - 1) , ÔV(x9 y, t) α 2ΑΒ sin — — ^ / \

A

\ Τ(χ,

./path

ΤΑ

(42)

y, t)ds\. /

O n e convenient way to calibrate the detector is to simply note the difference between the maximum and the minimum voltage at the detector, V, as (φ^ — φΒ) is taken through a change of m (in the absence of the thermal signal). F r o m Eq. (38), V is given by V=Vmax-

(43)

Vmina4AB.

Equation (42), along with Eq. (43), then gives the desired result, / 477" ( H 0 8V(x9

\

2.

1)

r

/

y9 t) = \Vsm\A

1A

T(x9 y9 t) ds

•'path

PULSED PHOTOTHERMAL PHASE-SHIFT SPECTROSCOPY

The collinear PTPS signal is given by 8VL(x,

y, t) = \Vsin

= ^sin

/ 477 ( « n - 1) r °/

V T

°^

T(x,y,t)dz

T(x,y,i)\,

97

THE THEORY OF PHOTOTHERMAL EFFECT I N FLUIDS

where / is the length of interaction between the p u m p a n d the probe beams. By substituting for T(x, y, t) from Eq. (32), we get

8VL(x,y,

t) =

±Vsm

4T7 (n0 - 1)1 λ

2aE0

TA

vt0pCρ

2 2 2 e-2[(x-vx(t-T)) +y ]/[a X

(46)

„Ι 2

2 +

+ SD(t-r)]

2

/0°

\ T

[a +SD(t-r)]

d

O n e very useful special case where one can write the result in closed form is when the laser pulse length i 0 is very short. In this case, by using Eq. (33), we get 8VL(x,

y,t)

= ±Fsin

4π (n0 - 1 ) / 2aE0

- 2 l (e x - v x

2

(a

2

) l +y W

+ SD,)'

+ 8Dt) (47)

T h e transverse PTPS signal is given by 8VT(x,î)

= ^Vsm

I 4ττV (n0- 1) °^ fT(x,y,t)dy^ T

(48)

where T(x, y, t) is given by Eq. (32). T h e ^-integration is to be carried out over the interaction length of the p u m p and the probe beams. However, since the ^-integrand is nearly zero outside this interaction length, the integration may b e performed from - oo to + 0 0 . The result is

8VT(x,

t) =

\Vnn

4π (nQ — 1) ~X

TAA

2aE0 v 2 7 T i 0p C p

(49)

2 2 2[(x-vx(t-T)) ]/[a

0

+

2

[a + SD(t - τ ) ]

SD(t-r)]

1/2

As in the collinear case, for a short laser pulse, this result may be written in a closed form using Eq. (33) as /

8VT(x,t)

= ^Fsin

477 ( « 0 - l )

2aE0 /ÏÏTpCp

-Äx-vxtY/[a

2

2

e

[a + Wt]

+

1/2

*DtA

(50)

Figure 7 shows the PTPS signals for the collinear case calculated by using Eq. (46) for a stationary medium (vx = 0) for the parameters given in the

R. Gupta

98

COLLINEAR PTPS

—

U)

•Ρ

x=a

Γ

L Ο

p - ^ x . = a / 2

CO

x=0 1

-1

1

1

1

1

1

3 5 TIME (ms)

1

1

7

1

1

9

FIG. 7. Typical pulsed collinear PTPS signals in a stationary medium for several p u m p - p r o b e separations (expressed in units of a); s i n y ( f ) has been plotted as a function of time, and one division corresponds to sin γ = 1. The top curve has been expanded by a factor of 5 for clarity. Parameters used in this computation are t0 = 1 J I T S, E0 = 6 mJ, a — 0.39 m " 1 , a = 0.5 mm, 1=1 cm, 7^ = 300 K, and λ = 490 nm. (From Monson, Vyas, and Gupta, 1988.)

figure caption (Monson, Vyas, and Gupta, 1988). Four curves are shown, for χ = 0, a/2, a, and 2a. The signal has the largest amplitude for χ = 0, since the change in the refractive index is the maximum at this position. T h e intensity of light at the detector suffers a transient change when the p u m p laser is fired and returns to its original value as the heat diffuses out of the region. As the distance between the p u m p and the probe beams is increased, the signal becomes smaller and broader. When the probe beam is outside the p u m p beam (x = 2a), the heat arrives at the probe-beam position via thermal diffusion, and therefore the peak signal occurs later in time and is weak and broad. PTPS signals for the transverse case in a stationary medium have the same general shape as those in Fig. 7, except that they are broader and, of course, are smaller in amplitude due to smaller interaction length (Monson, Vyas, and Gupta, 1988). Signal shapes for a flowing medium (collinear case) are shown in Fig. 8. T h e flow velocity of the medium is assumed to be vx = 2 m / s , and all other parameters used in this calculation are the same as those used for Fig. 7. A positive value of χ corresponds to the probe being downstream from the p u m p beam. Signal shape is essentially the spatial profile of the p u m p beam broadened by thermal diffusion. Transverse P T D S signals have essentially the same shape as the collinear ones, except for a smaller amplitude.

99

THE THEORY OF PHOTOTHERMAL EFFECT IN FLUIDS COLLINEAR PTPS

-125

375

875

1375

1875

2375

TIME(MS) FIG. 8. Pulsed collinear PTPS signals in a flowing medium (vx = 2 m / s ) . One division corresponds to s i n y ( i ) = 1. All parameters used in this computation are given in the caption to Fig. 7. (From Monson, Vyas, and Gupta, 1988.)

PHOTOTHERMAL PHASE-SHIFT SPECTROSCOPY

3. C W

Substitution of the equation for temperature distribution, Eq. (36), in Eq. (45) leads to the expression for collinear CW PTPS: Ä

Ν

W

8VL(x,

! T,

.

/

47 7

-

« ^ a v 2 ft

TA

— / r 2 R Y
Κ

y, t) = ±Ksin \ λ

(1

-2{[x-vx(t-T)]2+y2}/[a2 Xe

+

ωτ)

cos

+ SD(t-T)]

(51) j

d

Substitution of Eq. (36) into Eq. (48) and integration over y leads to the expression for transverse CW PTPS signal: , , ι τ, · δν (χ ί)Ν = \Vsm FIT

τ

9

ί

4 77

("ο - 1)

™ ρ— 2aP

/ — (1

π

~2[x-vx(t-T)]2/[a2 Xe

+ cos

ωτ)

+ SD(t-r)]

^

R. Gupta

100

COLLINEAR PTPS

-ρω χιίο <

CO CO

Έ CC

FIG. 9. CW collinear PTPS signals as a function of the pump-probe distance for four different flow velocities of the medium. R M S values of s i n y ( / ) have been plotted for a modulation frequency of 10 Hz and an interaction length of 1 cm. All other parameters are given in the caption of Fig. 3. One division corresponds to the rms value of sin γ = 0.5. The top curve has been expanded by a factor of 5. (From Monson, Vyas, and Gupta, 1988.)

TRANSVERSE PTPS f = 1 0 Hz X5

i

-6

1

1

-4

-2

1

0 χ/α

1

1

2

4

6

FIG. 10. CW transverse PTPS signals computed with the same parameters as in Fig. 9. One division corresponds to the rms value of sin γ = 0.05. The top curve has been expanded by a factor of 5 for clarity. (From Monson, Vyas, and Gupta, 1988.)

THE THEORY OF PHOTOTHERMAL EFFECT IN FLUIDS

101

Some of the predictions of CW collinear PTPS are shown in Fig. 9. R M S values of the PTPS signals have been plotted against the distance between the p u m p and the probe beams, x, for a modulation frequency of 10 Hz. 2 This distance has been expressed in units of the l / e - r a d i u s of the p u m p beam. The distance χ is taken to be positive downstream. Four curves are shown for υχ = 0, 1 c m / s e c , 10 c m / s e c , and 1 m / s e c . As the velocity increases, these curves become more and more asymmetric, with the signal extending far to the right side (downstream). Low- and zero-velocity curves show interesting undulations. These undulations are a direct consequence of the change in temperature distribution as the pump-beam intensity oscillates (see Fig. 4 and Monson, Vyas, and Gupta, 1988). Figure 10 shows curves analogous to those of Fig. 9 for transverse PTPS, that is, predictions of Eq. (52). These curves have no undulations, and of course, they are smaller in amplitude.

B . PHOTOTHERMAL DEFLECTION SPECTROSCOPY

In this section, expressions for the deflection of a probe-laser beam when passing through a medium with an inhomogeneous refractive index created by the absorption of the pump-laser beam will be derived. A general ray equation describing the propagation of light in an inhomogeneous medium will be derived in Section II.B.l, explicit expressions for the pulsed photothermal signal (Rose, Vyas, and Gupta, 1986) will be derived in Section II.B.2, and those for the CW photothermal signals (Vyas et al, 1988) will be derived in Section II.B.3.

1. T H E R A Y EQUATION

W e wish to derive an equation for the propagation of an optical ray between points A and Β if the (inhomogeneous) refractive index of the m e d i u m in this region is n(r). From Fermat's principle, the ray path s would be such that the optical path length between A and Β would be an extremum, that is, (53) A

Here, ds = (dx

2

+ dy

2

+

2

dz2 \)

1 2

/

_

[l

2

2 1/2

+x +y ] dz,

(54)

102

R. Gupta

where the dots represent differentiation with respect to z. Therefore Eq. (53) becomes B

8 J[ n(x,

2

+ x

y, z)(l

2

dz = 0.

+ y)

(55)

A

Eq. (55) will be solved using Lagrangian mechanics (Ghatak and Thyagarajan, 1978). According to Hamilton's principle, the trajectory of a particle between times t1 and t2 is given by (Goldstein, 1959) h

= 0,

8 ( Ldt

(56)

where L is the Lagrangian of the particle. Comparison of Eq. (56) with Eq. (55) shows that the Lagrangian in this case is 2

2 l/2

(57)

L = n(x,y,z)(l+x +y ) ,

and ζ plays the role played by / in Eq. (56). The Lagrangian satisfies the equations d

ι dL\

dz \ dx j

dL

= Tdx

(58a)

and dldL\

dL

dz\dyj

dy

(58b)

By substituting Eq. (57) in Eq. (58a), we get 1 (1 + x

2

d 2 l/1

+ y)

η

dz \ ( i + & + y2)

1/2

dx\

dn

dz j

dχ

By using Eq. (54), this can be written as d

I

ds\ Similarly, by using (58b), one gets

dx\

dn

ds j

dx

(59a)

THE THEORY OF PHOTOTHERMAL EFFECT IN FLUIDS

103

PROBE 'BEAM

MEDIUM FIG. 11. Diagram showing the relationship between the probe-beam path s, perpendicular displacement δ, and the deflection angle φ.

By taking the z-direction to be approximately 11), Eq. (59) may be written as

the ray direction (see Fig.

(60) where δ is the perpendicular displacement of the beam from its original direction as shown in Fig. 11, and V±n is the gradient of the refractive index perpendicular to the beam path. Equation (60) is the ray equation we h a d set out to derive. T h e refractive index «(r, t) is a function of both space and time. We may replace «(r, t) by « 0 , where n0 is the uniform (unperturbed) refractive index of the medium on the left-hand side of Eq. (60): (61) This approximation is justified because any variations of η along s will not deflect the beam. The refractive index n(r, / ) is related to the unperturbed refractive index n0 by (62) where TA stands for the ambient temperature. Using Eq. (62) in Eq. (61), we get (63) where the integration is carried out over the path of the probe beam. For our geometry (Fig. 5), and for small deflections, the deflection angle φ in

104

R. Gupta

the x-direction may be written as dT(x,

y, t)

(64)

ds.

Again, two cases will be considered: the probe beam propagating in the j - d i r e c t i o n shown in Fig. 5a (transverse PTDS), and the probe beam propagating in the z-direction shown in Fig. 5b (collinear PTDS). In either case, deflection of the probe beam in the x-direction, φ, is observed. The general case, where the p u m p and probe beams make an arbitrary angle θ with respect to each other, has been considered by Rose, Vyas, and G u p t a (1986) and will not be considered here. The probe beam is again considered to be infinitesimally thin. For transverse and collinear PTDS, Eq. (64) reduces, respectively, to (65) and (66)

2.

PULSED PHOTOTHERMAL DEFLECTION

In this section, explicit expressions for pulsed photothermal deflection will be derived by using Eq. (32) in Eqs. (65) and (66). By differentiating Eq. (32) with respect to χ inside the τ-integral, we obtain

-2{[x-vx(t-T)]2+y2}/{8D(t-T) Xe

+ a2}

for

fa

t > t0

(67)

Collinear P T D S is then given by 3n

=

—

8a£0

^

n 0 dTirt0pCpJ0 -2{[x-uxU-T)]2+y2}/[SD(,-T) Xe

„0

{x - vx(t (8D(i -

T

+ a2)

- τ)} ) + J T>

a2}2

(68)

THE THEORY OF PHOTOTHERMAL EFFECT IN FLUIDS

where / is the interaction length. Transverse PTDS is given by

n0 dT 7Tt0pCp

Jq

x - vx(t

r)

-

|/-OO (69) _e-2[x-vx(t-T)]'/[SD(t-r)

X

[SD(t-r)

+ a']^

2 :

+

a]

The integrand in the ^-integral vanishes outside an interaction length, 2 l/1 which is of the order of (a + %Dt) . It is therefore permissible to choose the Hmits as ± 0 0 on this integral. The integral can be solved analytically, with the result _ J _ 3n_

Ν φ

τ

[

Χ

'

η

8aE0 &

n0dT

0

p C

nQ p

i

- τ)]

[x - vx(i

2

0

[δΤ)(ί-τ) + * ]

3 /2

(70)

2

2

^ - 2 [ χ - ^ ( / - τ ) ] / [ 8 / ) ( / - τ ) +α ] j

χ

T

The T-integral in Eqs. (68) and (70) must be evaluated numerically, which can be done conveniently by using the IMSL subroutine D C A D R E . However, in a few special cases, dT/dx may be written in closed form. In the following, we shall consider these cases. a. PTDS Signal in the Absence of a Flow (vx = 0) In this case, Eq. (67) may be integrated to yield: dT(x,

y, t) _ " dx

aE0x 2 2

_e-Kx +y )/[a

, 2 e 2vtoPCpD(x +y ){

2

2

2

2

2(x +r )/[" + 80(f-i„)]

2

(71)

+ SDt] j _

Explicit expressions for φ τ and
1

Ix

aE0 2

a

+ W{t

2

2

^ !/ '

-

t0) (72)

2

-erf

2x a

2

+ SDt

1/2

R. Gupta

106 and I I dn\

αΕ0χ

2(x2+y2)/[a2

+ _e-2(x2+y2)/[a2

SD(t-t0)]

y) (73)

+ SDt)

b . PTDS Signals in Flow-Dominated

+

2

Conditions

If the flow velocity is very high and P T D S signal is observed downstream on such a short time scale that no appreciable thermal diffusion takes place, we may set D = 0 in Eq. (67). This equation can then be integrated analytically, with the result: dT(x,y9t)

2aE0e-2y2/a2

~2[x-vx(t-t0)]2/a2

7Tt0pCpa2vx

dx

(74)

_e-2[x-vxt)2/a2Y Explicit expressions for φ τ and (p L are then given by 1 / dn \

2aEn 2aE0

-2[x-vx(t-t0))2/a2

ν 2 τ τ / 0 ρ ς av

n0\dT)

ρ

(75)

X

_e-2[x-vxt]2/a2^

and / dn «o

d T

2aE0e-ly2/a2 vt0pCpa vx 2

-llx-uAt-h)]2/"2

(76)

_e-2[x-vxt]2/a2

Equations (75) and (76) are most useful in liquids where the diffusion rates are low.

107

THE THEORY OF PHOTOTHERMAL EFFECT IN FLUIDS

c. PTDS Signal in the Impulse

Approximation

If the laser pulse duration t0 is very short, Eq. (34) may be used to determine dT/dx. The expressions for φ τ and
τΚ

}

dn

SaEn

n0 dT^npCp

(x - vj) (SDt +

„

af 2

/2

and I 3n SaE0 n0 dT vpCp 2 2

-2[(x-vxt) +y ]/[a Xe

2

(x (sDt

vxt) +

2)

fl

^

+ SDt]^

Rose, Vyas, and G u p t a (1986) have discussed the range of validity of impulse approximation. Figure 12 shows some of the results predicted by Eq. (70). The photothermal deflection has been plotted as a function of time for different positions of the p r o b e beam. The center-to-center distance between the p u m p and the p r o b e beams is represented by x. A negative value of χ corresponds to the p r o b e b e a m being upstream from the p u m p beam, whereas a positive value of χ corresponds to the probe beam being downstream. We note that as the p r o b e b e a m is moved upstream, the signal quickly disappears because the heat is unable to diffuse against the gas flow. As the probe beam is moved downstream, the signal gets stronger at first and then acquires a shape that is essentially the derivative of the spatial profile of the p u m p beam (which is assumed to be a Gaussian here). This shape, of course, can be understood easily. As the heat pulse passes by, the probe beam at first experiences a positive gradient of the heat, followed by a negative gradient. As χ is increased further, the signal becomes broader and smaller due to thermal diffusion. Flow velocity of the medium has been measured in several different ways by using these signals. The simplest method is based on the measurement of the time-of-flight of the heat pulse between two probe-beam positions downstream from the p u m p beam (Sontag and Tarn, 1985; Rose and G u p t a , 1985; Sell, 1985). One may also determine the flow velocity by fitting the shape of the PTDS signal (Sell, 1985; Weimer and Dovichi, 1985). Yet another way is to use the amplitude of the signal that depends on the flow velocity as described below (Sell, 1984; Nie, Hane, and Gupta, 1986). Figure 13 shows results similar to that of Fig. 12 but in a stationary medium (vx = 0 in Eq. (70)). In this case, the signal is zero for χ = 0,

R. Gupta

108

νχ =2.0

m/s

_ / \

Η

Ο

Χ =-0.33mm

1

1

0.4

1

1

0.8

1

Ι-

1.2

TIME (ms) FIG. 12. Pulsed transverse PTDS signal shapes in a flowing medium for several different pump-to-probe-beam distances, χ = 0 corresponds to the case when the axes of the two beams intersect. Negative χ corresponds to the probe beam being upstream from the pump beam, and positive χ corresponds to it being downstream. The deflection is expressed in microradians. Flow velocity of the medium ( N 2 seeded with 1000-ppm N 0 2 ) was assumed to be 2.0 m / s . Laser pulse energy was assumed to be 1.65 mJ, beam radius a was 0.33 mm, and all the other parameters used in this computation are given in the caption for Fig. 1. (From Rose, Vyas, and Gupta, 1986.)

because dT/dx is zero at this point, and reverses sign as χ changes sign, as expected. F o r small values of χ (0 < χ < α), the signal consists of a sharp deflection of the probe beam shortly after the pump-laser firing, followed by a gradual return of the probe beam to its original position on the time scale of the diffusion time of the heat out of the probe region. The signal attains its maximum value for χ = α / 2 , where the gradient of T(x, y, t) is a maximum. For larger values of x, the peak of the signal occurs later in time, and the signal is broader and weaker, as expected. F o r collinear PTDS (Eq. (68)), the signal shapes are similar to those shown in Figs. 12 and 13. However, the amplitude of φL is larger than that of φ τ and it depends on the interaction length. Moreover, φ χ ( / ) in a stationary medium is broader than

THE THEORY OF PHOTOTHERMAL EFFECT IN FLUIDS

0

1

2 TIME (ms)

3

4

109

5

V

FIG. 13. Pulsed transverse PTDS signal in a stationary medium (ux = 0) for several probe-to-pump-beam distances as indicated above. Laser energy was assumed to be E0 = 1 mJ, and all the other parameters used in this calculation are the same as for Fig. 12. Two curves on the bottom have been expanded by the indicated factors for clarity. (From Rose, Vyas, and Gupta, 1986.)

collinear P T D S , the width of the p u m p beam in the j-direction (for an infinitely thin probe beam) does not contribute significantly to the width of ( p L ( / ) . However, in the case of transverse PTDS, the probe beam samples heat from different parts of the heat source in the ^-direction. Since it takes longer for the heat to diffuse out of the probe beam for y Φ 0 than for y = 0, the φ χ ( 0 signal is broader. For a flowing medium, the difference in the widths of φ τ ( 0 and ( p L ( / ) is very small. Figure 14 shows the peak value of the probe-beam deflection

R. Gupta

110

<

oc

e g Ü

ο

-1

LU

-2

LU Q

-3

<

-4

LU CL

V x= 0

—

v =o. 0.5m/s x

2.0m/s

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1.0

D I S T A N C E B E T W E E N PUMP A N D P R O B E B E A M S , X ( m m ) FIG. 14. Peak value of the probe-beam deflection φ ΓΏ plotted as a function of the p u m p - p r o b e distance χ for vx = 0, vx = 0.5 m / s , and vx = 2.0 m / s . The pump laser was assumed to have a pulse length of 1 / A S and a pulse energy equal to 1.4 mJ, and the radius of the pump beam was assumed to be 0.3 mm. The medium was assumed to be N 2 at atmospheric pressure with 1000-ppm N 0 2 (absorption coefficient a = 0.39 m " 1 ) . (From Nie, Hane, and Gupta, 1986.)

it is antisymmetric about χ = 0. For larger values of vx, we find that the symmetry of the curve about χ = 0 is broken, as expected, and a nonzero deflection signal is obtained at χ = 0. As a matter of fact, the signal at χ = 0 rises monotonically with vx until it reaches saturation (Nie et al., 1986). This fact can be used to measure very small flow velocities, that is, velocities so low that the transit time method is inappropriate due to thermal diffusion broadening of the signal. We also note in Fig. 14 that the curve splits into two branches for υx > 0 and χ > 0. This can be understood with the aid of Fig. 12. The negative and positive peaks of the signal in Fig. 12 correspond to the negative and positive branches in Fig. 14 for vx > 0 and JC > 0.

3. C W PHOTOTHERMAL DEFLECTION SPECTROSCOPY

In this section, explicit expressions for C W photothermal deflection will be derived by using Eq. (36) in Eqs. (65) and (66). The expressions for Q P L

111

THE THEORY OF PHOTOTHERMAL EFFECT IN FLUIDS

and
/

-

τ))

(79) -2{(x-vx(t-r))2+y2}/[a2 Xe

+ SD(t-T)]

fa

and 1 dn Φ

τ =

8 a P av

(1 + cos ωτ)(χ

~~n~ JrW^C~Jo 0

X

-e 2 { ( x - M f - T ) )

2

[ö2 +

- vx(t - τ ) )

8Z)(/-T)]

3 /2

(80)

} / V + 8Z)(/-T)]

Figures 15 and 16 show CW PTDS signal shapes for collinear and transverse cases evaluated using Eqs. (79) and (80), respectively (Vyas et al., 1988). The method of 64-point Gaussian quadrature was used to perform the integration. These figures show the equilibrium values of the PTDS signals. Even though the temperature distribution for cw excitation does not reach an equilibrium value, the PTDS signals do reach equilibrium (Vyas, et al, 1988). RMS values of the deflection have been plotted as a function

COLLINEAR PTDS

χ/α FIG. 15. CW collinear PTDS signals for modulation frequencies of 10 Hz (solid lines) and 100 Hz (broken lines) for four different flow velocities. R M S values of the deflection in μ radians have been plotted as a function of the pump-probe distance (in units of a). The top curves have been expanded by the indicated factors. The interaction length is assumed to be 1 cm, and all other parameters used in this calculation are given in the caption to Fig. 3. (From Vyas et al., 1988.)

112

R. Gupta TRANSVERSE PTDS

χ/α FIG. 16. CW PTDS signals, similar to those of Fig. 15, except for the transverse geometry. (From Vyas et ai, 1988.)

of the distance χ between the p u m p and the probe beams for four velocities υχ = 0, vx = 1 c m / s e c , vx = 10 c m / s e c , and vx = 1 m / s e c . Solid lines are for a modulation frequency of 10 Hz, whereas the dotted lines are for 100-Hz modulation frequency. Consider Fig. 15 first. As the flow velocity of the medium increases, the curves become more and more asymmetric, as expected. N o t e that for large velocities, the signal downstream becomes very small even though a significant temperature distribution (above ambient) exists (see Fig. 3). This is because the gradient of the temperature in this region is very small. In order to understand the dip in the signal at χ — a(vx = 1 m / s ) , one must examine the change in the temperature gradient as the pump-laser intensity oscillates (Vyas et ai, 1988). We also note that as the modulation frequency increases, the signal amplitude in general decreases, and for vx = 0, the signal peaks occur closer to χ = a/2 because the heat is able to diffuse only a shorter distance during the modulation cycle. The transverse PTDS signals shown in Fig. 16 are similar to those of collinear PTDS, except that their magnitudes are smaller.

C . PHOTOTHERMAL LENSING SPECTROSCOPY

In this section, expressions for the change in intensity of a probe laser passing through a thermal lens will be derived (Fang and Swofford, 1983;

THE THEORY OF PHOTOTHERMAL EFFECT IN FLUIDS

113

Harris and Dovichi, 1980; Dovichi, Nolan, and Weimer, 1984). The thermal lens, of course, is created by the nonuniform refractive index, which is caused by the absorption of the p u m p beam. An expression for the signal strength in terms of the focal length of the thermal lens is derived in Section I I . C . l . General expressions for the focal length of the thermal lens are derived in Section II.C.2. Explicit expressions for the pulsed and CW lensing signals are derived in Sections II.C.3 and II.C.4, respectively.

1.

DETECTION OF THE THERMAL LENS

Figure 17 shows a typical configuration for the detection of a thermal lens (Fang and Swofford, 1983). The thermal lens is placed a distance zx in front of the probe-beam waist. A screen with a pinhole is placed at a distance z 2 in front of thermal lens. Intensity of the probe beam passing through the pinhole of radius b is observed by a photodetector; w 0 is the 2 l / e - r a d i u s of the probe beam at its waist; wx and w2 are the radii at the position of the thermal lens and the screen, respectively. When the thermal lens is activated (by turning on the p u m p beam), w 2 changes, resulting in a change of intensity at the detector. Our aim is to derive an expression for the signal s(t) in terms of the focal length f(t) of the thermal lens, where s(t) is defined as s

W

=

^ , « = o)

·

<

81)

FIG. 17. Detection of the photothermal lensing effect by observation of the change in intensity of the probe beam passing through an aperture. (Adapted from Swofford and Morrell, 1978.)

R. Gupta

114

where Pdet(t) is the power at the detector at time /, with t = 0 meaning an instant before the laser is turned on. For the CW case, the time / at which observation is made is generally large compared to the thermal diffusion time. For the pulsed case, generally the observation is made at t > t0. The radial intensity distribution of the probe beam at the screen is given by IP

/(,·) =

-2r2/wj

(82)

where Ρ is the power of the probe beam. Then rb

1

irrdr

p(r)2

det

(83)

mb2

-

2P-

,2 '

The signal s(t) is then s(t)

=

- w 22( Q

j(0)

W

w 22( 0 w 2 2 (0) -

(84) wj(t)

where we have replaced w2(t) by w 2(0) in the denominator because the change in the radius of the beam is small. W e can find w2(t) by using the ABCD law (Yariv, 1976). According to the ABCD law, the complex beam parameter q2 at the position of the screen is given in terms of the parameter q0 at the beam waist by Ago + Β 42

=

(85)

Cq^D'

where A, B, C, and D are the elements of the transformation matrix representing translation by a distance z 1 ; focusing by a lens of focal length / , and a translation by distance z2, that is, /

1 1

7

0\ 1

0

1

(86)

THE THEORY OF PHOTOTHERMAL EFFECT IN FLUIDS

115

Explicitly, the matrix elements are A = l-

z2/f,

Β = ζλ + z 2 -

ζλζ2//,

1 c

- - - r

and D

= l-

(87)

zx/f.

The complex beam parameter q(z) is defined by 1

1

q(z)

R(z)

λ

(88)

2

vnw (z) 2

where R(z) is the radius of the phase front, w ( z ) is the l / e - r a d i u s of the intensity, η is refractive index, and λ is the wavelength of the probe radiation. At the waist of the beam, the radius R0 = oo and q0 is simply given by (89)

qo = i*o>

where z 0 = TTHW^/X is the confocal distance (distance in which the beam diameter increases by a factor of i/2 ). By substituting Eq. (89) into Eq. (85) and by using Eq. (88), we find that 1\

mnw2

z0(BC-AD)

= Im I

z2

QA

\qi)

+ Β

But AD - BC = 1 from Eq. (87). Therefore,

?(0 = w L +^). 2

2

0

(90)

By substituting for A and Β from Eq. (87) into Eq. (90), and by making the approximation z2^> zx (which can generally be arranged), we get

^ - Α^ποϊ'i( - M) l

(91)

Equation (91) can be further simplified if z 0 « : z2. Then

" - v i - m\ M

1

(92)

R. Gupta

116

Remembering that / ( / = 0) = oo, substitution of Eq. (92) into Eq. (84) yields the signal s(t): s{,}

Generally, f(t)

' m ' m

(93)

» zv In that case, Eq. (93) simplifies to

.«) - ^ .

<*)

which is the relationship we had set out to derive. A general equation for s(t) without the approximation is given by Vyas and Gupta, (1988). In many situations, the thermal lens may be astigmatic. In these cases, the b e a m radius w2 is different in two orthogonal directions. In these cases, Eq. (83) modifies to 7Tb2 ^det

= 2P

TTWxWy

,

(95)

where wx and wy are the beam radii in the x- and the ^-directions, respectively. The signal s(t) is then given by s

(')

=

—T^—7^—

·

(96)

A straightforward calculation similar to the one leading to Eq. (94) gives

=m 2.

+

m

(97)

FOCAL L E N G T H OF THE THERMAL LENS

In order to evaluate Eq. (94) or Eq. (97), we need to know the focal length of the thermal lens. An expression for the focal length can be derived easily with the aid of Fig. 18 (Fang and Swofford, 1983). We shall follow F a n g and Swofford's treatment for the cylindrically symmetric configuration, and will generalize it later to include the astigmatic thermal lens. A more sophisticated treatment of this subject is given by Vyas and G u p t a (1988). Consider a medium of length / with refractive index n(r) that increases radially. We shall make the thin-lens approximation, that is, we shall assume that / is very small compared with focal length / . Consider an

THE THEORY OF PHOTOTHERMAL EFFECT IN FLUIDS

117

ι

Ο

\ Β

A F

— -:^ r ^ ^

> 'kx)

r

— f

FIG. 18. Diagram showing the focal length of the thermal lens. (Adapted from Fang and Swofford, 1983.)

optical ray AB incident normally at the left interface of the medium. Inside the medium, the ray follows a curved trajectory BC of radius R. The ray arrives at the right interface making an angle φ χ with the normal to the interface. D u e to the refraction at the right interface, the ray actually travels along C E , making an angle φ 2 with the normal. The effective focal length of this medium then is / , as shown in the diagram. In the derivation below, small angle approximation will be made throughout. F r o m the geometry, Φι =

BC

I

R

R

Therefore, r rR / ' = - - = - — Φι /

(98)

but / 77 /

Φι

=-

Φ2

1

= -.

n

W

where the last step is given by SnelPs law. By using Eq. (98) in Eq. (99), we get r R / = - - T ; (100) η I

118

R. Gupta

R can easily b e found from the ray equation, Eq. (61),

j

d

di\

^ o ^ J = V F o r small deflection, d8/ds

= φν and άψλ 1 n0

ds But ds = Rdyx,

n ( r , 0 .

±

and therefore

i =l R

V

±

n0

«.

(101)

In m a n y cases, the refractive index is cylindrically symmetric, i.e., the refractive index depends only on the radial distance from the axis (for example, if the pump-laser beam has a Gaussian spatial profile and the medium is stationary). Moreover, we assume the probe beam to propagate collinearly at r = 0. In this case, it is convenient to expand the refractive index n(r) in a MacLaurin series, I

(dn\ "

{

r

' "

)

m

+

r

( ^ )

r

. „

+

i

r

d2n\

[ ^ )

r

- o

+

" ' -

<

1)

Because of the cylindrical symmetry, (dn/dr)r==0 = 0. Therefore the leading nonuniform term is a quadratic term that makes the medium act like a lens. By using Eq. (102) in Eq. (101), we get 1

1 dn(r)

j ^ ~jr =

0

=

r I

^ [j^j 0

d2n\

(103)

r=0

T h e focal length may now be found by using Eq. (103) in Eq. (100): 1

/ θ2η

A t ï U

r

(104)

where we have set η - n0 in Eq. (100). If (d2n/dr2)r=0 is not constant over the p a t h of the probe beam, Eq. (104) may be generalized as follows: 1 -

/

, f

= - J

(32n\ - dS

^ 2

V t h \ or

j

r=0

W

0

2

119

THE THEORY OF PHOTOTHERMAL EFFECT IN FLUIDS

By using Eq. (62), Eq. (105) may finally be written as 1

dn

1"

dT

2

{

r

λ

3Τ

2

(106)

ds.

dr

•^ath(

r= 0 2

This is the desired equation. We rind (d T/dr )r=0 from Eq. (32) or Eq. (36), as appropriate. Equation (106), when substituted in Eq. (94), yields the desired signal s(t). Now, we shall consider both transverse and collinear beam configurations, as we have done before in connection with PTPS and PTDS. For the beam configurations shown in Fig. 5, the probe beam propagates along the z-axis for the collinear case, whereas it propagates along the j^-axis for the transverse case. It is obvious that a cylindrical thermal lens is experienced by the probe beam in the case of transverse PTLS. Even in the case of collinear PTLS, if the medium is not stationary, (υχ Φ 0), a n d / o r the p r o b e b e a m is not placed at χ = 0 and y = 0, an astigmatic lens is experienced by the probe beam. Therefore, we must distinguish between focal lengths fx and fy. Thus, it is necessary to expand the refractive index n(x, y) in a Taylor series about a point x\ y' (a point on the axis of the p r o b e beam): dn \ n(x,

y) = n(x\

/ dn

y') + (x - x')\ J^Jx=x'

(χ -

x'Y l

+ (y ~ f

(y - y )

2

dn 2

x=x

2

X=X

/)

y = v'

2

I dn dy

7

2

I

χ = χ'

(107)

y=y'

2

dn dxdy

x

=x

] I y=y By following the steps leading from Eq. (102) to Eq. (106), we get 1

dn ç

Jx

~dfL

1

2

dT

(108)

path 2

dT

(109)

and (110) N o t e that in this case, the first derivatives of the refractive index (dn/dx) and (dn/dy) in Eq. (107) are not necessarily zero. These terms result in the

120

R. Gupta

deflection of the probe beam, as discussed in Section II.Β (see Vyas and G u p t a , 1988). For the collinear case, Fig. 5b, we may simplify Eqs. (108) and (109) as 1

i

dn

2

dT

Ä--«='M;:.'

(m)

and 1

7

2

dn

I3 T

= ~- ^~df'\Ty / | — U'-

("2)

=

y = 0

Similarly, for the transverse case, Fig. 5a, we have 1

2

3n

ä--«=/(

and

1

{dr T\
= 0.

(114)

Jζ

3. PULSED PHOTOTHERMAL LENSING SPECTROSCOPY

In this section, explicit expressions for pulsed PTLS signals will be derived (Vyas and Gupta, 1988). We start with the collinear case. Substitution of Eq. (32) for T(x, y, t) in Eqs. ( I l l ) and (112) leads to SaE0l

fx

I dn

^pCpt0\dTjJo 2

-2[x-ox(t-T)f/la Xe

1

[a2 +

1 -

&D(t-r)Y

+ &D(t-T)]

A{x - „x(t - τ ) ) 2

[a

2

+ SD(t - τ ) ] (115)

r

d

and

τ =— - h - Γ

-->— ^ ( h ) d

6

where we have dropped the primes on χ for convenience. Substitution into

121

THE THEORY OF PHOTOTHERMAL EFFECT I N FLUIDS

Eq. (97) gives = ? ιΛ

]

8 a £ 0 f e 1 / dn \ „„ *pCpt0

1 [2 +

\ dTJJo

a

_ 4 [ x - v (t - τ ) ] ζ x

_ T) ] \

[a + SD(t - τ ) ] /

2

8

Ζ

)

(ί

2

(117)

Similarly, using Eq. (32) in Eq. (113), and using Eqs. (113) and (114) in Eq. (97) gives the transverse PTLS: 8 α £ ' 0ζ 1

=

M

/ dn \ , ,

^pCp/oUrjio

0

^ X

[α2+8Ζ)(ί-τ)]3

/2

4[χ-„,(/-τ)]2|

(118)

[a +SD(t-r)]j

< 1

X e

1

0

2

2 l x - v x( ' - r ) ] 2/ [ a

:t

fa

+ SD(t~T))

These a r e t h e final equations that we wanted to derive. F o r a short laser pulse, Eqs. (117) a n d (118) can b e written in a closed form, by using Eq. (33), as 1

SaE0lzlldn\ *pCp χ^

\ar/[

f l

2

/ 8

+

] Z2 ^)

4(x-vxt)2\ ( ^

i

+

8/)/)/

(119)

- 2 ( χ - ί ; Λ0 2 / ( α 2 + 8 £ > / )

and

'τ(0 = -^-TTrl^L

2 * p C ^ \ a r / [

„ f

l2

„ . +

8

,3/2 i

Z ,) ] 3 / ^

1

(

2 0

+ 8 £ / ) /

(120)

- 2 ( x - i ; xr ) 2/ ( a 2 + 8 ^ 0

Figure 19 shows a few typical pulsed PTLS signals in a stationary medium for the transverse case. The fractional change in the power at the detector, sT(t), h a s been plotted against the time. A negative value of the signal corresponds to a diverging lens, whereas a positive value corresponds to a

R. Gupta

122 O

q

*

τ

-x/a=0

ο CO I

v =0 x

ο

cvî

I

Co γ

ο ο -

|x/a|=2

ο 1-4

+

ο

cvi +

-1

"T" o

- r 3

2 ί

- r 4

6

( m s )

FIG. 19. Transverse photothermal lensing signals for a pulsed laser and a stationary medium. The five curves are for five different pump-probe separations, as labeled. The curves have been computed for the following parameters: E0 = 6 mJ, / 0 = 1 /is, a = 0.5 mm, and - 1 a = 0.39 m , corresponding to 1000-ppm N 0 2 in N 2 . (From Vyas and Gupta, 1988).

converging lens. For |x| < 0.5a, the photothermal lens behaves like a diverging lens. For |JC| = 0.5a the signal vanishes at t = t0 (because the second derivative of temperature is zero at this place). However, for t > r 0 , some signal gets generated by the heat diffusing out from the inside of the p u m p beam. For |x| > 0.5a, the photothermal lens behaves like a converging lens. Figure 20 shows pulsed PTLS signals in a flowing medium for the transverse case. In this case, the thermal lens travels downstream with the flow of the medium. The signal at χ = 3 a shows the shape of the signal due to the entire lens. The thermal lens is a diverging lens in the center, whereas it is a converging lens in the wings. The wings are asymmetric due to the thermal diffusion. Collinear PTLS signal shapes are similar to those of Figs. 19 and 20; however, the signal amplitudes are larger (by about a factor of 30), and the zeros of signal occur at χ = ± 0.707a (see Vyas and Gupta, 1988).

123

THE THEORY OF PHOTOTHERMAL EFFECT IN FLUIDS

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

t (ms) FIG. 20. Transverse photothermal lensing signals for a pulsed laser and a medium flowing with a velocity vx = 2 m / s . All parameters used in this computation are the same as those for Fig. 19. (From Vyas and Gupta, 1988).

4.

PHOTOTHERMAL LENSING SPECTROSCOPY

CW

Explicit expressions for CW PTLS signals (Vyas and Gupta, 1988) are obtained in a manner completely analogous to that of the last section by using Eqs. (36) in Eqs. (111)-(114) and by using Eq. (97). The results are

{

ΐ)

~

TrpCp

[JTJJO

[

a

2

+

S

_D T

{ ]) t

(121) 4[x - vx(t [a2+

-

r)]2

8Ζ)(ί-τ)]

z e-2[x-vx(t-T)]'/[a

+ SD(t-r)]

fa

R. Gupta

124

FIG.21. RMS values of the CW PTLS signals for the transverse geometry plotted against the pump-probe distance for four different flow velocities. The modulation frequency was assumed to be 10 Hz, and all other parameters used in this computation are given in the caption to Fig. 3. (From Vyas and Gupta, 1988).

and

*τ(0 =

SaPavz1

[α

1

}/ïîrpCO Χ 1 -

[(1 + cos ω τ ) ]

I on dn \ π

4[x - vx(t

-

+ SD(t - τ ) ] τ)]2

2

2

e-2[x-vx(t-T)] /[a

3/2

(122) + SD(t-T)}

fa

[α2 + 82)(f - τ ) ]

Figure 21 shows the CW PTLS signals for the transverse case. The equilibrium values of the R M S signals are plotted against the p u m p - p r o b e distance for modulation frequency of 10 Hz. Whereas the temperature of the medium for CW excitation does not reach equilibrium, the PTLS signals

THE THEORY OF PHOTOTHERMAL EFFECT IN FLUIDS

125

CM

x/a FIG. 22. R M S values of the CW PTLS signals for the collinear case. The interaction length was assumed to be 1 cm, and all other parameters are the same as for Fig. 21. (From Vyas and Gupta, 1988).

d o reach equilibrium. A stationary medium, and flow velocities of 1 c m / s , 10 c m / s , and 1 m / s are considered. The signal is symmetric about χ = 0 for vx = 0, and this symmetry is lost as vx is increased, as expected. The apparent symmetry of the signal for D x = 1 m / s is fortuitous. For ux = 0, the signal consists of three peaks; the central peak corresponds to a diverging lens, whereas the side peaks correspond to a converging lens. For higher velocities, (vx > 10 c m / s ) , the peaks on the left correspond to converging lenses, whereas those on the right correspond to diverging lenses. W h e n R M S values are measured, this information is difficult to obtain except by measuring the phase of the signal. Figure 22 shows the corresponding signals for the collinear case. As expected, these signals are larger in magnitude and have a less pronounced structure. Vyas and G u p t a (1988) have also investigated the dependence of the P T L S signals on modulation frequency and they find that very significant changes in signal shapes can occur as the modulation frequency is varied.

126

R. Gupta ACKNOWLEDGMENTS

Contents of this chapter are based on work done in collaboration with Reeta Vyas, Allen Rose, and Brian Monson, which was supported in part by Air Force Wright Aeronautical Laboratories.

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