Novel results on collinear mirage deflection

Novel results on collinear mirage deflection

Agustin Salazar †, Milen Gateshki, Gerardo Gutiérez-Juárez1 and Agustin Sánchez-Lavega Departamento de Física Aplicada I, Escuela Superior de Ingenieros, Universidad del País Vasco, Alameda

Urquijo s/n, 48013 Bilbao, Spain.

Whye T. Ang

Computational and Mathematical Sciences Group, University of Malaysia Sarawak, Kota Samarahan 94300, Malaysia

In this paper we report on the strong influence of the polarization state of the probe beam on the amplitude and phase of the collinear mirage deflection. It is shown that the photoelastic effect, i.e. the change of the refractive index with stresses, is the responsible of this influence. Theoretical and experimental studies on isotropic materials have been performed. The influence on the measurement of the thermal diffusivity of these materials is analyzed.

Keywords thermal diffusivity, mirage technique, photoelasticity, thermoelasticity

(Received on June 29, 2000, accepted on October 11, 2000)

In a conventional collinear mirage setup a modulated heating beam is sent perpendicular to the sample surface while a low-power probe beam is directed parallel to the exciting beam and is periodically deflected inside the sample because of the dependence of the refractive index on temperature. One of the applications of the collinear mirage technique is the measurement of the thermal diffusivity of semitransparent materials. The method is based on the linear relation between the phase of the collinear mirage deflection Ψ(Φc) and the distance between heating and probe beams. Its slope (m) is related to the thermal diffusivity of the sample (D) through the expression1

mf

D= −

π (1)

where f is the modulation frequency of the heating beam. In this way the thermal diffusivity of polymers, glasses and liquids has been measured.2 However, it has been shown recently that the polarization state of the probe beam greatly influences the amplitude and phase of the collinear mirage deflection due to the influence of the photoelastic effect.3 In particular, the slope m in Eq. (1) depends on the polarization state of the probe beam. This result raises the question of whether the collinear mirage technique provides a reliable method to measure the thermal diffusivity of semitransparent materials. __________________________________________ † To whom correspondence should be addressed. 1 On leave from Universidad de Guanajuato (México). In this paper we study theoretically and experimentally the influence of the photoelastic effect in the collinear mirage

deflection. We focus on the conditions under which the linear method to retrieve the thermal diffusivity can be applied. Thermal model The geometry of the collinear mirage deflection is shown in Fig. 1. Considering a pure thermal model, i.e. n = n(T), the collinear mirage deflection is written as

Φc n

dn

dTT dz k= ∇ ×

−∫

1

0 (2)

where n is the refractive index of the sample and is the sample thickness. Fig.1. Geometry of the collinear mirage deflection.

Probe beam

y

z Heating beam

x

r

Φ c 0 -

x x

y y

Heating beam

Probe beam

Heating beam

Probe beam

Φcsph Φc

cyl

2001 © The Japan Society for Analytical Chemistry

s95ANALYTICAL SCIENCES APRIL 2001, VOL.17 Special Issue

Fig. 2. Mirage deflection in a transparent sample. Left: Heating beam focused by a spherical lens. Right: Heating beam focused by a cylindrical lens. For simplicity we present the theory for weakly absorbing materials although the conclusions have general validity. Two experimental configurations are considered: (a) If the heating beam is focused by a spherical lens, cylindrical thermal waves are generated and T T r= ( ) ; while (b) if the heating beam is focused by a cylindrical lens, planar thermal waves are generated and T T x= ( ) (see Fig. 2). The corresponding expressions for the collinear deflections are respectively:

Φcsph

n

dn

dT

dT

drj= − (3a)

Φccyl

n

dn

dT

dT

dxj= − (3b)

It is worth noting that when dn/dT < 0 (e.g. liquids and polymers) the probe beam is deflected outward from the heating source, i.e. as in Fig. 2. However, when dn/dT > 0 (e.g. most glasses) it is directed inward to the heating source. Since Φc is proportional to the temperature gradient its phase decreases linearly with the separation between heating and probe beams and the slope gives the thermal diffusivity of the material according to Eq. (1). Influence of the photoelastic effect Thermal waves produce a nonuniform temperature field and therefore inhomogeneous thermal stresses (σ) are developed in the sample (thermoelastic effect) that produce a change of the refractive index (photoelastic effect).4,5 Consequently, there is a direct modification of the index of refraction due to the temperature change, and an indirect modification due to the stresses n n(T, )= σ . In this section we account for the additional mirage deflection produced by the photoelastic effect. The main consequence is that the collinear deflection changes with the polarization state of the probe beam. Three polarization states of the probe beam are considered: linearly polarized along the x axis (||), linearly polarized along the y axis (⊥), and unpolarized (*). When the heating beam is focused by a spherical lens the collinear mirage deflection for the three polarization states we are working with are given by3

Φcsph

nA

dT

drC

d

dr rTrdr j⊥ =

−+ ∫

1

2 (4a)

Φcsph

nB

dT

drC

d

dr rTrdr j|| =

−− ∫

1

2 (4b)

ΦΦ Φ

csph c

sphc

sph

n

A B dT

drj* =

⊥ +=

− +| |

2 2 (4c)

Similarly, when the heating beam is focused by a cylindrical lens the collinear mirage deflection writes:

Φccyl

nA

dT

dxj⊥ = − (5a)

Φccyl

nB

dT

dxj|| = − (5b)

ΦΦ Φ

ccyl c

cylc

cyl

n

A B dT

dxj* =

⊥ +=

− +| |

2 2 (5c)

where

An

T

n E= +

+

−

∂

∂

α π π

ν

3

2

11 12

1

( )

Bn

T

n E= +

−

∂

∂

α π

ν

3

2

2 12

1

Cn E

=−

−

3

2

12 11

1

α π π

ν

( )

being α the thermal expansion coefficient, E the Young modulus, ν the Poisson ratio and πij the stress-optical coefficients of the sample. (a) Cylindrical lens It can be seen from a comparison of Eqs. (5) and (3b) that although the amplitude of the collinear deflection is affected by the polarization state of the probe beam, its phase remains unchanged with respect to the pure thermal model. This is due to the fact that A and B are real parameters and consequently the phase of the mirage deflection is governed by the temperature gradient, as in the pure thermal model. Thus, when using a cylindrical lens to focus the heating beam the thermal diffusivity can be obtained using Eq. (1). This is shown in Fig. 3, where mirage measurements performed on a polycarbonate (PC) sample are presented. A conventional collinear mirage setup1 was used where an Ar+ laser focused by a cylindrical lens of 7 cm focal length was employed as heating beam. The polarization of the 10 mW He-Ne laser used as probe beam was selected by a linear polarizer. As expected the phase of the mirage deflection is only slightly affected by the polarization state of the probe beam.

s96 ANALYTICAL SCIENCES APRIL 2001, VOL.17 Special Issue

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-200

-150

-100

-50

0

0 0.05 0.1 0.15 0.2 0.25

Ψ(Φ

)

x (mm)

cPC

Fig. 3. Measurements of the phase of the collinear mirage deflection using a cylindrical lens to focus the heating beam (f = 11.1 Hz). The polarization state of the probe beam was changed by a linear polarizer: Dots = (*), squares = (||) and crosses = (⊥). We have also measured the thermal diffusivity of a set of semitransparent materials using Eq. (1) for the three polarizations of the probe beam we are working with. The results are summarized on the left of Table 1. The influence of the polarization is computed by the parameter η defined as:

η =⊥ −

×D D

D

||

*100 (6)

The fact that η ≠ 0, against the prediction of the model, can be explained because actually the shape of the heating beam is not an infinite straight line but an elongated ellipse of 4 mm major axis and 70 µm minor axis. This explanation is confirmed by the fact that η decreases when the heating laser is expanded before reaching the cylindrical lens, e.g. for the red filter η = 10% (D⊥ = 0.54 mm2/s, D|| = 0.49 mm2/s and D* = 0.50 mm2/s). Moreover note that for the three glasses η > 0 indicating that D⊥ > D||, while for the two polymers η < 0 indicating that D⊥ < D||. These results are related to the behavior of dn/dT, that is positive for most of glasses and negative for polymers. (b) Spherical lens From a comparison of Eqs. (4) and (3a) the following conclusions can be obtained. (a) The amplitude of the collinear mirage deflection is different for each polarization differs from the pure thermal model. (b) Moreover, Ψ(Φc) also depends on the polarization state of the probe beam due to the terms containing an integral of the temperature. (c) However, for an unpolarized probe beam the phase is the same as in the pure thermal model, i.e. proportional to dT/dr. Consequently, it can be used to measure the thermal diffusivity using Eq. (1). This situation is illustrated in Fig. 4 for a BK7 glass (D = 0.50 mm2/s, n = 1.517, dn/dT = 1.6x10-6 K-1, α = 7.1x10-6 K-1, E = 81x109 N/m2, ν = 0.208, π11 = 0.363x10-12 m2/N and π12 = 2.25x10-12 m2/N) and for polycarbonate (PC) (D = 0.148 mm2/s, n = 1.585, dn/dT = -1.07x10-4 K-1, α = 65.5x10-6 K-1, E = 2.48x109 N/m2, ν = 0.409, π11 = -4.6x10-12 m2/N and π12 = 34.6x10-12 m2/N). Dots represent measurements performed using the same mirage setup as before with a spherical lens of 5 cm

focal length to focus the heating beam. Unpolarized results are intermediate between perpendicular and parallel and are not shown for clarity. The 180o phase difference between PC and BK7 is due to the different sign of dn/dT, responsible of the opposite deflection of the probe beam.

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-400

-350

-300

-250

-200

-150

-100

-50

0

0 0.1 0.2 0.3

Ψ(Φ

)

r (mm)

c

PC

BK7

II

II

T

T

Fig. 4. Measurements of the phase of the collinear mirage deflection on a BK7 glass (f = 21.4 Hz) and on PC (f = 11.2 Hz) using a spherical lens to focus the heating beam. Continuous lines represent theoretical calculations taking into account the contribution of the photoelastic effect (Eqs. 4). The thermal diffusivity of the same set of semitransparent materials as in the previous section has been measured using Eq. (1) for the three polarizations of the probe beam we are working with. The results are listed on the right of Table 1. Note that now η is large due to the huge influence of the photoelastic effect. For water η = 0 because of stresses cannot be stored in a liquid and therefore there is no photoelastic effect. It is interesting to point out that for all measured samples D*cyl < D*sph. This can be explained as follows. The phase of the collinear deflection near the heating beam is a curve that becomes a straight line as separation increases. Application of Eq. (1) requires to reach the linear region but it is difficult to decide where this linear region starts. As a result it usually happens that a curved part of Ψ(Φc) is considered as linear, so the slope m is lower than expected and therefore D is overestimated. This problem is reduced when the cylindrical lens is used because the curved part of Ψ(Φc) is smaller. In Fig. 5 the unusual shape of the collinear mirage deflection that was found when we were trying to measure the thermal diffusivity of the red filter at low temperatures using an elliptically polarized probe beam is shown. The double peak structure is related to directional changes in the deflection of the probe beam. From r = -0.6 mm to r = -0.15 mm rightward, from r = -0.15 mm to r = 0 leftward, from r = 0 to r = 0.15 mm

s97ANALYTICAL SCIENCES APRIL 2001, VOL.17 Special Issue

Table 1.- Thermal diffusivity in mm2/s for several semitransparent materials. Errors are between 2 and 4%. _________________________________________________________________________________________________________ Sample Cylindrical lens Spherical lens _________________________________________________________________________________________________________ Red filter D⊥ = 0.57 D|| = 0.48 D* = 0.50 η = 18% D⊥ = 0.86 D|| = 0.40 D* = 0.53 η = 86% Orange filter D⊥ = 0.64 D|| = 0.53 D* = 0.58 η = 19% D⊥ = 1.05 D|| = 0.49 D* = 0.65 η = 90% IR filter D⊥ = 0.47 D|| = 0.43 D* = 0.46 η = 8% D⊥ = 0.63 D|| = 0.45 D* = 0.52 η = 35% PEEK D⊥ = 0.156 D|| = 0.168 D* = 0.162 η = -7% D⊥ = 0.171 D|| = 0.317 D* = 0.218 η = -67% PC D⊥ = 0.148 D|| = 0.158 D* = 0.153 η = -6% D⊥ = 0.158 D|| = 0.232 D* = 0.174 η = -42% Water D⊥ = 0.156 D|| = 0.156 D* = 0.159 η = 0% D⊥ = 0.172 D|| = 0.173 D* = 0.175 η < 1% _________________________________________________________________________________________________________ rightward and from r = 0.15 mm to r = 0.6 mm leftward. The deflection of the elliptically polarized probe beam we have used can be expressed as

ΦΦ Φ

celliptical

sph csph

csph

nG

dT

drH

d

dr rTrdr j

=⊥ +

=−

+ ∫

0 35

1 35

1

2

.

.

| |

(6)

where G = (0.35A+B)/1.35 and H = -0.65C/1.35. In Fig. 6 the numerical evaluation of Eq. (6) is presented. In it G is kept fixed while H is varied. As it can be seen, for H between -20 and -30 the double peak structure can be qualitatively explained.

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0

1

2

3

4

5

6

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

Φ

r (mm)

c(m

V)

Fig. 5. Amplitude of the collinear mirage deflection on the red filter at 77.1 K. A spherical lens to focus the heating beam and an elliptically polarized probe beam were used (f = 20 Hz). Conclusions It has been demonstrated that the photoelastic effect greatly influences the collinear mirage deflection when using polarized probe beams and is the responsible in producing peculiar results as the double peak structure. In particular, when using the collinear mirage technique to measure the thermal diffusivity of isotropic materials care must be taken on the polarization state of the probe beam. The use of an unpolarized probe beam guarantees that the phase method can be applied to measure the thermal diffusivity of isotropic materials (care must

be taken in using tilted optical elements in the path of the probe beam since they introduce a certain degree of polarization). On the other hand, the use of a cylindrical lens to focus the heating beam is vividly recommended while the use of a spherical lens leads to an overestimation of the thermal diffusivity. The study of the collinear mirage deflection on anisotropic materials is currently under progress.

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0

2

4

6

8

10

0 0.5 1 1.5 2 2.5 3

Φ

r/

(a.u

.)c

µ

H = -15

H = -20

H = -25

H = -30

Fig. 6. Numerical evaluation of the amplitude of Φc for an elliptically polarized probe beam, according to Eq. (6). Here µ is the thermal diffusion length and G = 10. Acknowledgements This work was supported by the University of the Basque Country through the research grant No. EB108/97. References 1 A. Salazar, A. Sánchez-Lavega, and J. Fernández, J. Appl. Phys., 1993, 74, 1539. 2 A. Salazar and A. Sánchez-Lavega, Rev. Sci. Instrum., 1994, 65, 2896. 3 A. Salazar, M. Gateshki and A. Sánchez-Lavega, Appl. Phys. Lett., 2000, 76, 2665. 4 B.A. Boley and J.H. Weiner, “Theory of Thermal Stresses”, 1960, John Wiley and Sons, New York. Chap. 9. 5 J.F. Nye, “Physical Properties of Crystals”, 1969, 5th ed., Oxford University Press, London, Chap. 13.

s98 ANALYTICAL SCIENCES APRIL 2001, VOL.17 Special Issue

Novel results on collinear mirage deflection

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