Calorimetric measurements of weakly absorbing materials:theory

P. J. Severin

The time-dependent heat equation is solved in a cylindrical geometry of finite length with heat loss by radia-tion and conduction. Exact expressions are derived for the time constants of radial and longitudinal modes.The steady-state solution is obtained in longitudinal modes and used as the initial state for the decay. Asimple expression is presented for the ratio of the amplitude of the first-order longitudinal order mode Aland the corresponding time constant rl fit with the separate expressions for A1 and Ti. These parametersare experimentally readily accessible and directly yield the absorption coefficient a of rod-shaped compoundglass from which optical fibers are to be drawn.

1. Introduction

For optical communication networks, transmissionlines with glass fibers as a wave-supporting medium arecurrently the subject of intensive investigations.Though most research is concentrated on quartz glassfibers prepared with CVD techniques, compound glassresearch could be more rewarding in the long run.These aspects have been described at length else-where.",2

The compound glass used for this purpose shouldsatisfy several requirements which are orders of mag-nitude more severe than have ever before been formu-lated for glass. The first parameter to be reduced to avery small value is the absorption coefficient a. Withcompound glass this quantity can be determined in thestarting material from which a fiber is to be drawn.This paper deals with the calorimeter used to performsuch measurements.

The glass sample to be measured has the shape of acylindrical rod of radius R 1 and length 2L and is coax-ially positioned in a cylindrical cavity of radius R2 andthe same length. Laser light, of power Po and a wave-length for which the absorption coefficient a is to bemeasured, is transmitted along the z axis of the rod, andthe temperature T(t) of the surface of the rod is mea-sured at the center with a thermocouple. The enclosingcavity can be evacuated or filled with a gas of specificheat at constant pressure c2, density P2, heat conduc-tivityX2, and diffusivity k2 = X2/c2p2.

The author is with Philips Research Laboratories, Eindhoven, TheNetherlands.

Received 6 November 1978.0003-6935/79/101546-09$00.50/0.© 1979 Optical Society of America.

The difficulties to be overcome in this work turn outto be threefold being related to the experimental con-ditions, the external temperature stabilization, and thetheory. In this paper only the latter problem is ad-dressed. The experimental aspects will be the subjectof a companion paper,3 which will deal with the exper-imental test of the dependence of the relaxation timeand the steady-state temperature T on the gas thermalproperties and on the exposed length 2L, as derived inthis paper. It will also be shown that the glass thermalparameters can be derived without any other instru-ments.

It was probably Daglish and North 4 who first men-tioned the calorimetric method as a tool for measuringthe absorption coefficient in compound glass. Theyused a small rectangular sample, silvered to reduce ra-diation losses and supported by the thermocouple wires.Witte5 suggested a method in which a cylindrical rod,coated with a metal film to ensure uniform temperatureand low losses, is heated in vacuum by a beam of muchsmaller radius. The dimensions were calculated suchthat the only relevant losses were through the thermo-couple wires. In another proposal 6 the dimensions werediscussed such that the thermocouple losses were only1% of the sample radiation losses. Experimental workwas presented 7 using a cylindrical sample, 3 cm long and1 cm across, with clamped ends in a copper cell invacuum.

A full analysis of the temperature distributionT(r,z,t) in a cylindrical rod without a heat source invacuum was given by White and Midwinter.8 Assum-ing a linear source on the axis the radial temperaturedistribution is then averaged, and the surface temper-ature is measured with a thermocouple with wireswound around the sample over some length. Both theinitial slope and the full heating curve are used to

1546 APPLIED OPTICS / Vol. 18, No. 10 / 15 May 1979

measure a. Pinnow and Rich9 described an arrange-ment in which the sample is positioned inside a 1.06-gimNd:YAG laser cavity. Cooling is assumed to take placeby radiation and free convection with a typical tem-perature rise of a few degrees. Zaganiaris' 0 used amodulated heat source to derive a from the shape of theresulting surface temperature variation in vacuum.

Interest in the theory of the calorimeteric measure-ment is also stimulated by the fact that the same ar-rangement is useful for widely differing applications,each of which allows for specific approximations.

The most related application is in fiber calorime-try1 l-15 where the central rod is a thin capillary in whichthe fiber can be inserted. Thus the thermal propertiesof the rod are practically unaffected by the thermalproperties of the fiber. Because the infinite lengthapproximation is allowed, the temperature change canbe measured directly all over the length with a platinumresistance wire. In this way the absorption loss in thefiber can be distinguished from the scattering loss,which is included in the total loss, measured with aspectrophotometer. Such an instrument has been built,and the results will be presented at length elsewhere.16

The theoretical effects of axial heating are discussed inthis paper.

An important application of very weakly absorbingmaterials is their use as high-power laser windows. Tosatisfy the greatly increasing demands on laser power,these should be almost as transparent as fiber glass.The absorption coefficients may be measured in a cal-orimeter, generally using shorter samples.17-' 9

Perhaps the most interesting application is in thespectrophone. Many papers have recently been pub-lished on optoacoustic effects in solids,2 0'21 liquids,2 2 23

and gases.2 4 Various arrangements have been de-scribed, although a quantitative corroboration of theoryor the explanation of observed effects appears to bedifficult.22 Basically the gas is heated either directlythrough irradiation with light of a wavelength close toan absorption line, or indirectly by heating a solid, andthe ensuing pressure rise is detected with a micro-phone.

In theoretical work either the radial25 26 or the axial27

dependence of the temperature in the solid and the gasspace is neglected. In this paper the 3-D temperaturedistribution in the gas space is shown to be of para-mount importance. The boundary condition is onlyapplied through one heat transfer coefficient so thatradiation and conduction cannot be separated,252 8

whereas they have widely different time constants, andonly conduction generates a diffusion length. Thediscrimination between surface and bulk absorptioneffects presents theoretical and experimental prob-lems.29 30 It is the author's contention that any theoryon ac effects should be preceded by a 3-D analysis of thedc heat flow as presented in this paper. It should beexperimentally verified that the model chosen yields themeasured dependence on geometry and thermal prop-erties of gas and solid. It is then possible to calculatethe ac heat flow which is superimposed on the heat flowdue to the time-averaged heat input. The most ad-

I, L7

A// ///////.,© . _ . . ..

m /zi//,/X,/X //S,iH

. OfOf to f O @t / H;R2

4iL4-_

Fig. 1. The geometry of the cylindrical rod and cavity, with ther-mally floating ends (a) and thermally clamped ends (b).

vanced application of the spectrophone is in the de-tection of minute concentrations of gases. For themeasurement of small absorption coefficients down to10-6 cm-1 in solids, the dc calorimetric method appearsto be perfectly adequate. The extension of the presenttheory to include the effects of a modulated laser powerinput will be discussed in a subsequent paper.

11. Formulation of the Problem

The sample to be measured has the shape of a cylin-drical rod of length 2LI and radius RI and is positionedcoaxially in a similarly shaped cavity of length 2L2 andradius R2. The origin of the cylindrical coordinatesystem is in the center. The ends of the rod are eitherthermally floating, as shown in Fig. 1(a), or thermallyclamped in the end walls of the cavity at the cavitytemperature Tm = 0, as shown in Fig. 1(b). For thetime being the discussion will be limited to the lattercondition where LI = L2 = L.

The cylindrical rod is heated by absorption of laserlight with incident power Po and beam radius R0 < RI.With the sample absorption coefficient a small, so thataL << 1, the dissipated power per unit length is Poa.This heat is carried away from the surface of the rod byradiation with a heat transfer coefficient fr and byconduction through a gas of thermal conductivity X2,specific heat c2, density P2, and diffusivity k2 = X2/c2p2.With the thermal properties of the sample c1, pl, Xi, andk 1, the heating gives rise to a temperature distributionTl(r,z,t) in the rod (0 < r S R,), which, assuming uni-form heating and hence Ro = R,, is the solution of

aT 1 (92 T12 T1 + ITl\ Poa

at .z 2 Or2 r r c lpvrRl

and T 2(r,z,t) in the gas-filled space (R, S r S R2), whichis the solution of

,9T2 (O2 T2 T2 19T 21-=k 2I-+-+ rOt9 ( 2 9r2 r Or (2)

Three boundary conditions are to be satisfied in theradial direction. Assuming that the metal enclosure hasa very large thermal conductivity and that the metal-gastransition does not involve any boundary layer, the firstboundary condition reads at the gas space outer wall

T 2(R2 ,z,t) = 0, (3)

15 May 1979 / Vol. 18, No. 10 / APPLIED OPTICS 1547

rI I

�,Z// III/X/l//z/Z�1,�,

/ / 11L1111' - 11-, -- ____ ___ 1� ,-

Table 1. Thermal Properties of Some Glasses and Gases used in theCalorimeter at 260 C

cp ( 3 W se ) (102 ) k O3CM2

Soda-lime silicate 2000 1.15 5.7Germano-silicate 2300 1.0 4.3Pyrex 1700 1.04 6.1Silica 1600 1.4 8.7Hydrogen 1.06 0.186 1760Helium 0.93 0.15 1600Neon 0.93 0.046 500Air 1.2 0.026 214Argon 0.94 0.0176 187

The second condition states that the heat loss from thesample is divided into a radiation current and a con-duction current through the gas

-pX1 IaTj) = \A2 (dT2) + rTi(Ri). (4)

The third boundary condition reads

T1(Rj,z,t) = T2 (Rl,z,t). (5)

It could be argued that the transition from the sampleto the gas involves not only a discontinuity in the de-rivative but also in the temperature itself due to a sur-face layer with a nonzero heat transfer coefficient. Sucha thin layer may be important in optoacoustical effectsof frequency if the thickness is comparable to (2k2/w)1/2. This will not be considered here.

In the following sections Eqs. (1) and (2) are solvedwith two simplifications. First, it is assumed that theheat supply is switched off at t = 0 and that the sampleand the gas decay from the steady-state condition to thefinal temperature T = 0. Second, this steady-statecondition is calculated. In the fifth section these twosolutions are combined. The results obtained in thenext sections are illustrated by examples to check thelimits of the validity of the approximations made. Tothis end Table I gives the relevant thermal data for anumber of glasses and gases.

The heat transfer coefficient for radiation fir can beexpressed in terms of the blackbody coefficient fro4o-T3 for small excursions of the temperature as

Or = el2or0, (6)

where, for a coaxial geometry, the emission coefficiente12 is a function of the emission coefficients e and e2and the radii R and R2 as

ele2

e2 + (R1/R2) ei(1 - e2)

Numerically the following data apply to the experi-mental situation of a glass rod in a worked copper en-closure el = 0.9, e 2 = 0.67, R = 0.33 cm, R 2 = 1.5 cm.Hence, e2 = 0.82 and ro = 6.12 X 10-4 W/cm 2 K at T= 300 K, so that fir = 5.02 X 10-4 W/cm 2 K. Oftenir/X = h is specified; hRl is called the Biot number for

radial heat conduction, here equal to 1.59 X 10-2.It is important to note here that Eqs. (6) and (7) are

valid only for coaxial geometry of infinite length; the end

walls of the cavity draw an extra radiation current fromthe rod theoretically from any point and in practicemainly from points closer than R2 near the ends. Theextra conduction current drawn from the rod near theend is fully accounted for in the theory presented in thispaper, but the radiation current makes an extremelycomplicated contribution in a finite length coaxialsystem with closed ends. For a constant temperaturerod radiating toward the inside of a finite length cylin-der the expression for the configuration factors isknown;31 for the radiation from the rod to the closed endit can be calculated. 3 2 However in the problem dis-cussed in this paper the temperature is not uniform, andhence the configuration factors and the radiation cur-rent should be specified as a function of z. To go intothis complicated theory is beyond the scope of thispaper.

111. Time-Dependent Solution Without Heat Source

A. Solution and the Boundary ConditionsThe solutions to Eq. (1) without heat supply and to

Eq. (2) can be written as the product of a time-depen-dent and two space coordinate-dependent factors interms of the zero-order Bessel function Jo and Neu-mann function No:

Tl(r,z,t) = E [AmnJo(.tmx)m,n

+ BmnNO(imX)l cos'YniZ exp(-t/i-mn 2 ),

T2(r,z,t) = [CmnJo(vmr)m,n

+ DmnNo(vmr)] cosYn2z exp(-t/mn 2 )

(8)

(9)

The summations extend over the series of roots to bedetermined from the boundary conditions. Because theboundary conditions should be independent of t and ofz for T and T!, the t- and z-dependent factors in Eqs.(8) and (9) are equal, and the subscripts 1 and 2 can bedropped in -Yn and mn so that

1-= k (Tm 2 + 2)= k2('V2 + ).Tmn

(10)

Because r = 0 is included in the range of validity ofEq. (8), the coefficient Bmn = 0. With the boundarycondition Eq. (3), Dmn can be expressed in Cmn, andfrom Eqs. (4) and (5) the relation between gm and vmthen follows:

Ji(mRl) f2XiA.l O )=/r + RF 2 R )' (1

where the finite radial dimensions are taken into ac-count by the aspect function

1Fi(v ,R1,R2 ) =vmRi

x J0 (,mRj)No(vmR 2) - J((vmR2)N0(V. 12)J1j(vmRi)No(vmR2) - Jo(imR 2)N1(vmR1)

From Eqs. (10) and (11) the coefficients Arm and mcan be found; the subscript m refers to the order of theroots satisfying these equations. The function

1548 APPLIED OPTICS / Vol. 18, No. 10 / 15 May 1979

e12 =

I _

0.

I I ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Io 1 2 3 4 50 0.2 - 0.4 0.6 0.8 2 1

Fig. 2. The function 1/Fi(v 2 ,Rl,R 2 ) = 1/F2 (-v 2 ,R 1 ,R 2 ) against v2,

for v2 < 0 with R1 = 0.3 cm and R2 = 1.5 cm. The expanded scalerefers to the lower curve.

bulk absorption. It has been found indeed that end-surface heating spoils the bulk absorption measure-ment, when the fiber-glass rod is supported in the cal-orimeter with floating ends, as shown in Fig. 1(a). Al-though, in principle this can be taken into account in atheory, it may be subject to wide variations because itdepends on surface preparation. That is why thermalclamping is used, as shown in Fig. 1(b), over a lengthlong enough with respect to the diameter that the axialboundary condition reads with reasonable accuracy

(14)Tl(r,t,+L) = T2(r,t,+L) = 0,

and hence that

,yn= (2n - 1)[7r/(2L)]. (14a)

Fj'(v,R,,R 2) is plotted in Fig. 2 for R1 = 0.3 cm and R2= 1.5 cm as a function of -v2, which is positive, as willbe made clear below. If v1R, and v1R2 are small enoughthe Bessel functions in Eq. (12) may be expanded, and,as shown in Fig. 2, a linear approximation in v2 holds:

1 1

Fi(v',R,,R2) ln(R2/Rl)

X J_ I IR2-_R, - 2R 2 1 + ln )L} (12a)1 4 [1.(R2/Ri) \ 1J

formally written asFj1 = a - bvP.

Similarly in general gRI < 1, and the expansion of Eq.(11) will be allowed to determine the lowest root Rfrom

(AtRi)2(1 +-8 = A1 [r + R1 F(RR 2 )1. (1a)

It will be shown in the next section that with m > 1 thevalue of Am increases so much that Tmn becomes veryshort and that the contributions of higher order radialmodes can thus be neglected.

It is interesting to note that for R2 - the aspectfunction Fj-' = vR /2, which effectively means in Eq.(lla) that g2 depends on radiation only.

As can be seen from Table I, the heat diffuses muchmore rapidly in the gas than in the glass, k2 >> k1.Therefore it is fairly safe to conclude from Eq. (10) thatto a first approximation

2 2pi = 1, _y2(13)

and, therefore, provided y 2 is known, F,(-_y2) can bedetermined and thus A and rll. For the axial boundaryconditions the temperature of the glass rod and the gasat z = LL is to be considered.

It is well known that a polished surface of glass mayshow a substantial surface-absorption effect. This hasgiven rise to confusion as to what is actually measuredwith a spectrophone. With very low absorption coef-ficients and short length this effect may dominate the

B. Some Applications

1. Infinite Length in VacuumThe simplest assumption is that the rod has infinite

length and that the enclosure is evacuated. This ap-proximation has been discussed by several authors.Then Yn = 0, and the subscript n in Eq. (8) can bedropped. Equation (9) makes no sense, and Eq. (4)with X2 = 0 is retained as the only boundary condition.With h = frXi, the roots A-mR1 follow from

(gmRl)Jl(mRl) = hRlJo(AmRl). (15)

In a Pyrex rod with R, = 0.332 cm, fr = 5.06 X 10-4W/cm2 K, Xi = 1.05 X 10-2 W/cm K, and hence hR, =1.6 X 10-2, it follows that only AR 1 shows dependenceon hR 1, while 1U2R1, ,A3R,, etc. are constants: 1u2R, -3.83; ,u3 R, = 7.02, etc. From expansion of Eq. (15) itfollows that up to hR = 0.1 the equation

, = 2(h/R1 )

is valid. Hence A2 = 0.29 cm- 2 , and

1/71 = k = 2[r/(clplRl)],

(1Sa)

(15b)

which, for a Pyrex rod, equals about 565 sec. The sec-ond order relaxation time is orders of magnitudesmaller: 2 = 7 sec. Also it is evident that there ishardly any radial variation in temperature over the rodeven for the lowest mode: g 1r = 0.18r/R 1, let alone forthe higher modes.

2. Infinite Length, in GasWhen cooling takes place by radiation and by con-

duction in a gas, it follows from Eq. (10) that

2 = (k2 /kl). (lOa)

Using this relation in Eq. (lla) and disregarding thehigher order term in ,1R 1 the value of v2 can be foundgraphically from

cip, Rl2 Mlri clpl l v2 Sr-l = 1 * (llb)C2P2 2 A2 Fl(vl)

As shown in Eq. (12a), for small v2 the function F71 can

15 May 1979 / Vol. 18, No. 10 / APPLIED OPTICS 1549

be approximated as FL` = a - bv2, and hence using Eqs.(lOa) and (la) it is found that the infinite length re-laxation time rl is given by

1 k a + rRI/X 2

Tr 1 + klbc/k 2

where c = (2X2)/(XR 2). To bring this expression intoagreement with Eq. (10) the right-hand side should bereferred to as k A', .

With the example used above the approximationreads FL1 = 0.662-0.143v2, and it is found that T = 80sec in helium and 283 sec in air. The correspondingvalues of 1uL, are 2.05 cm- 2 and 0.58 cm- 2 .

3. Finite Length, in VacuumIn vacuum for a finite length the same value of t is

found, whereas Yn follows from Eq. (14a). With lengthL = 5 cm, hence y2 = 0.1 cm- 2 , it is found for the ex-ample used above that T1 = 420 sec. The second andthird-order longitudinal modes decay with T2 = 138 secand r3 = 59 sec.

4. Finite Length, in GasProceeding in the same way as for the infinite length

in gas, A2 can be written with Eq. (10) as

Al pi I 2-l)I (lob)

and hence, with Eq. (la), one can again find .v2 graph-ically. Using the same linear approximation to FL' forsmall v, from Eqs. (lOb) and (la) the expression for

2,ui can be found to beR kAfl + a + by2 1-

= C

1 + bc-

IV. Steady-State Solution with Heat Source

A. Extreme CasesIn the preceding sections Eqs. (1) and (2) have been

solved without a heat source to determine the temper-ature relaxation time. The relative contributions of thevarious modes could not be determined because theinitial condition for decay at t = 0, the steady-state so-lution with heat source, was unknown. That solutionwill be calculated in this section.

It is much easier to obtain a solution for the case ofvacuum than for a gas, because in the former case nosolution is needed outside the rod. In the next sub-section this will be discussed, and conduction will beincluded in Sec. IV.C. The general solutions needed arepresented in this subsection.

The sample is heated by a laser beam of radius Rl,and any power density profile over the laser beam willbe disregarded. An objection could be that the resultwould be different if the beam does not heat the rod overthe whole cross section 7rR2, but only through a cylinderof radius Ro. In Sec. IV.D the steady-state temperatureis derived assuming linear axial heating of the rod. Thedifference between the results using these two extremeapproximations will be shown to be negligible.

Two extreme cases can most easily be solved: theinfinitely long cylinder and the extremely short cylinderdescribed by the simplified versions of Eqs. (1) and(2):

O2T 1 T 1 Pa+ -- = ,

Or2 r r x 7rRI

2T + I Pa0Z A - 2= 0.

Oz2 Xi rRi

(17)

(18)

X JIR' [R1, r + a + by2

I X (lic)8

C

The second factor embodies the second-order approx-imation in ,lR 1, arising from the expansion of the Besselfunctions in Eq. (la) ). It turns out to reduce gu2 byabout 3% in He and by less than 1% in air. Neglectingthis second order term in ,uRl, the finite length relax-ation time may be written with Eq. (10) as

I [1+bc y2 a +R1l/X2c)= ki c 72 + a CrX (16b)

Ti 1 + bckI/k2 1 + bckl/k 2 (1b

It can easily be verified that the term bkl/k 2 may beignored but that bc cannot; it equals 0.356 in He and0.064 in air. In the example used above, Eq. (lib) reads

- = k(,y 2 1.37 + n) for helium and -rj = kQ(-y2 1.06+ 2_) for air, which yields 75 sec and 238 sec for L = 5and 57 sec and 132 sec for L = 2 in helium and air, e-spectively.

It is important to realize that for finite length in gasthe condition at the ends enters into the resultant i-1directly by the term y 2 and indirectly by the depen-dence of 2 on F(v2) Fl(--y2) manifested in the pa-rameter b.

For radiation only, the solution to the first equationreads, taking into account the boundary condition Eq.(4) with hR, 10-2,

T P =2a [1 + hRi (1 r2 o (19)

27iRiI1r I 2 2RI 2irRifir

For cooling by radiation and conduction the situationis slightly more complicated. If the steady-state tem-perature is maintained by the equality of input powerPoa/7rRl and outward heat flow by radiation and con-duction, the solution to Eq. (17) reads in the rod

Poa r2

Ti=- -+A,7rRIX 4

and in the gas with T2(R2) = 0,

T2 = B ln(r/R2).

(20)

(21)

Using the boundary conditions Eqs. (4) and (5) it isfound that

Te= PI (Ri r 2 + [(Rll)/2] I

rR 1 4 RiJ Or + IX2 /[R1 ln(R 2 /R1)1 ,]

Poar (R1/2) ln(R 2/r)T2 = -_ _ _ _ _ _ _22_ _

7rR ( 2 /R1 ) + fr ln(R2/Ri)The solution to Eq. (18) reads, taking into account the

boundary conditions Eq. (14) at z = L,

1550 APPLIED OPTICS / Vol. 18, No. 10 / 15 May 1979

Poa L2 Z2

T = ° I _LZ27rRi Xit

(23)

This solution is valid for cooling by radiation and byradiation and conduction combined, because no radialboundary conditions are involved.

B. Finite Length, in Vacuum

If the amount of radiation flowing out of the cylinderby radiation is not dominant as in Eq. (17) or negligibleas in Eq. (18), the intermediate situation can be dis-cussed only by solving the full steady-state version ofEq. (1):

O2 T + O2Ti + Ti P = 0.(2- ~ ~ ~ ~~=.(24)Oz

2Or

2 r r 7rRi i

The solution should be the sum of the particular so-lution of the inhomogeneous equation and a generalsolution of the homogeneous equation. For the formerEq. (19) will be used, which yields results more easilythan Eq. (23) would. The general solution reads

T = Ei A. [exp(+mz) + exp(-1tmz)]Jo(umr). (25)

The particular solution Eq. (19) has been determinedso that it satisfies the radiation boundary condition Eq.(4), which reads for the general solution as Eq. (15).Therefore, the roots Ag are as found in Sec. III.B.2.The boundary condition T,(r, +L) = 0 should be satis-fied by the sum of the particular and the general solu-tions, hence

Z Amnexp(+MumL) + exp(-MuL)]Jo(nmr)

-Poa h 1[ 2 ( r 2

- 1 -Il- - l27rRil~r [ 2 2 l/

(26)

In order to find Am the right-hand side should be ex-panded in terms of Jo(,mRi). Multiplying both sidesby rJo(gmr) and integrating over 0 S r S R1, it is foundthat

A,,[exp(+,anL) + exp(- tmL)] 2 (h2 + A )j2-a- h2 ( + 1,4,)Jn(gRi)

-Poa + hRi\ R1JltR)2-Rr\ 2 A.m

Poca hR 1 R.+ --2 R2 I R r3Jo(Amr)dr.27rRi/3r 2 R 20~

(27)

Using hR, << 1 and ignoring the contribution of the in-tegral term, this equation simplifies to

Poa R1 2J( Jl(An.Ri)

27rRlr g. R1(h2 + g (g)J.(j R 1 )

T Pa 1 2 piRi Ji(AiRi)27rRilr L Rl(h 2 + ,t2) J2(itRi)

exp(+,ulz) + exp(-Aiz) Jo(Air)exp(+lL) + exp(-,u1 L) I

(29)

Using Eq. (15a) and approximating the Bessel func-tions, it is found that at r = R 1

P°a ( coslg, z .27fRir cosh LI

(30)

This equation is only valid in vacuum and equals Eqs.(19) and (23) in the approximations AiL >> 1 andAL <<1. The values of AR, and ,lL can be found in Sec.III.B.

C. Finite Length in Gas, Uniform Heating

The steady-state temperature distribution of a finitelength cylinder in gas, including radiation and con-duction, presents the most general of the situationsdiscussed in this section. The solutions T,(r,z) of Eq.(24) in the rod and T2(r,z) of the same equation withoutthe heat source are to be matched at r = RI, both intemperature and in heat flow. Therefore, the z-de-pendence should be the same, and because T 2 (r,L+L) =O only a goniometric function can be used. The r-dependence then automatically involves modifiedBessel functions:

T2 (r,z) = E [cjo(yr) + D,,Ko(Ynr)] cosynz. (31)

The solution in the rod should be the sum of a particularsolution and of a general solution of the homogeneousequation. For the former, Eq. (23) can now mostreadily be used which satisfies the boundary conditionat z = +L so that the total solution reads with 'Yn = (2n- 1)7r/2L.

PociL2 I z2 \

Ti(r,z) = Y AnIO(-Ynr) coslynZ + - (32)27rR2Xi . 2

In order to match the two solutions at r = R, for any zthe particular solution should be expanded into aFourier series

z2

1 - - = Z Bn CoS-YnZ(L2 n

Because cos'ynz is periodic over the interval -2L S z< 2L, the parabola should be extended beyond therange of physical definition, and the continuation at thepoints z = LL should be chosen with the same sym-metry as cos'ynz in order to have vanishing uneventerms. These requirements are satisfied if the parabolicexpression is replaced by a function g(z) defined as

(z) = 1-- for -1 < - < 1L2 L

= (L- - 2) -1for +1<-<+2

= (-+ 21 - 1 for -2 <-< -1 .

(34)

(28)exp(+MmL) + exp(-AmL)

Retaining only the first term Al of the general solutionand adding this to the particular solution Eq. (19), it isfound that-

Then the coefficients of the desired expansion are foundas

32 (-)n-11r3 (2n - 1)3

(35)

15 May 1979 / Vol. 18, No. 10 / APPLIED OPTICS 1551

(33)

x

Using the boundary conditions, Eqs. (3), (4), and (5)the coefficients An can be determined in terms of B,.Proceeding as in Sec. III.A, upon elimination of C, it isfound that

A, = -B, 1 y1 { /rfl + (36)Il(,Rl) [2/(RIF2)] + Or I&(nY)

where F2(CY2,R,,R 2) = F(-,y2,R 1,R2 ) is the aspectfunction defined in Eq. (12), and hence that

Tl(rz) PoaL*27rRiX

J F, -Io(Ynr) 1

XJ'Yn + 0O('YnR,) Il(,y.Rl)i[2/(RF2)] + 1r Il('YnR,)

(-J)n-1 32

(2n-1)3 -3 cosyz

The temperature experimentally measured at r =Rreads

32 PaL2 (-)n-:T1R 3z 27rR i n (2n -1) 3

X{1 1.i-Yn _l('Yn~) 1 |cos-Ynz- (38)

1 + XlYn Ii(nRi)J[X2/(R1 F2 )1 + 1r Io(ynRl)

In order to gain some insight into the nature ofsolution it is worth making various approximation.L is so large that not only yRl << 1, but also the hiorder terms, as far as they contribute to the sum, reismall enough, the value of F2 (y) does not change mand the linear approximation holds over the relerange in 'Yn, hence F`L (y2) = a + by2 . Then retaionly the first-order terms in the Bessel functionsusing the notation in a, b, and c, introduced in SecB.2, Eq. (38) reads

Tj(R ,z) = Poa 8 (-1)nTi(R R 2)= R x ir2n -cosy,z

1 2 OS rRiy(1 +bc)+ a+ c

'thiss. IfghernainLuch,

'2 a + !3rR,/X21 + bc

With X2 = 0, Eq. (39b) reduces to Eq. (30) valid in vac-uum. In gas the linear approximation holds for F- 1 butonly for small arguments. In calculating the sum thisapproximation has been used for larger arguments too.This approach has not been used in deriving Eq. (30).Therefore Eq. (30) is a better approximation in vacuumthan Eq. (39b) in gas is.

D. Finite Length, in Gas, Linear Axial Heating

In a more approximate treatment the heat source canbe introduced as a boundary condition. The sample isassumed to be heated by a laser beam of radius R0,which will be made vanishingly small later. In theheated region 0 < r Ro the power density equalsPoa/7rR2, and the radial flow along the cylinder shouldsatisfy the boundary condition at r = Ro for the heatcurrent per unit length through the surface r = R0, av-eraged over a cross section

x2T

, 2 9TiT Poaz2 Ro Or rRo

(40)

where Tl(r,z) is the steady-state solution in the glassRo < r S R, without heat source. This solutionreads -

Tl(r,z) = E [AnIo(�ynr) + BnKo(-Ynr)] cos'Ynz,n

and in the gas space similarly

T2 (r,z) = YI [CnIo('Ynr) + DnKo(-Ynr)] CoS-YnZ

(41)

(42)

vant The z-dependence has been taken a goniometric func-ning tion because in the gas space for any R, < r < R2 theand temperature should vanish at z = L, and Eq. (14a)

. III. applies. The boundary conditions Eqs. (4) and (5)should be valid at any z, and therefore Tl(r,z) is chosensimilarly. The radial dependence then automaticallyinvolves modified Bessel functions.

With the boundary conditions Eqs. (3), (4), and (5)the following relation between A, and Bn is obtained:

(39)

Because the influence of higher order terms decr(roughly as (2n -1)-3 the error due to the expansiiBessel functions with not small enough argumersmall. Using the Fourier expansion

4L22

cosh/uz 4 (-)n- 72

coshjtL 7r n 2n - 1 4L2

(2n,-1)2 +-_ 2

X cos(2n -1)--2 L

this expression can be written asPoa 1 ( cosh,4'z~

Ti(Rl,z) - 2 1 '(a + X -

whee2where

Bn Io('YnRi) + GnIl('ynR)

A,, GnKl(YR 1 ) - Ko(-YnRi)

where the function Gn is defined as

(43a)

=XlJn R (RF2 I

and F2 is a function defined with Eq. (36). The secondrelation between An and Bn is provided by the boundarycondition Eq. (40), which yields

(39a) A,1y,, [1 Ij(ynRo) -YnIo(YnRo)j

+ -BnYn LR-Kl(YnRo) + YnKO(YnRO)I

(39b) Poa 4 (-l)n7rRX -7r 2n - 1

(43b)

where the right-hand side is an expansion of the con-stant term in cos'ynz. Because, notwithstanding the

1552 APPLIED OPTICS / Vol. 18, No. 10 / 15 May 1979

increasing values of yn with n, the radius Ro can bechosen so small that YnRo << 1 in the relevant range ofthe sum, the Bessel functions in Eq. (43b) can be ex-panded to yield

Bn-Poa 4 (-1)n (3B,, = 2 A_ -2 __ _ * (43c)2-,rX, r 2n - 1

Using Eqs. (43a) and (43c) in Eq. (41) the desired tem-perature dependence in the glass rod at r = R1 reads

Poa 4 (-1)n- G,Tl(Rl,z) = 27rXj r 2n-1 'YnR1

G,,1(yR 1) + Io(7YR 1) (44a)

where Gn is defined with Eq. (43a) and is related to Anby Eq. (Ila) as Gn = 2-yn/RI, 2(y2). Applying again-YnR1 << 1, this equation can be expanded to

Ti(R 1,z) = 2 , - )' s A2(y2 (44b)zrRlXl nr 2n -1 yn, n '(~

which reduces to the expression valid for the fullyheated rod, Eq. (39). The validity of this expression islimited for the same reason as discussed in the case ofEq. (39).

V. Time-Dependent and Steady-State SolutionsCombined

In the two preceding sections the basic Eqs. (1) and(2) have been solved with the aid of simplifying as-sumptions. In Sec. III it was assumed that the heatsource is switched off at t = 0 from the steady-statedistribution, and the time constants -rmn are obtained.In Sec. IV it was assumed that the stationary state andthe equilibrium temperature distribution have beenreached. In the present section the two solutions willbe combined by calculating T(r,z,t) and the slope dT/dtat t = 0.

From the most general solutions at r = R,, Eq. (8) att = 0 and Eq. (39) follow the coefficients An of cos-ynzin Eq. (8). It should be noted that in Eq. (8) thesecoefficients appear as the sum over the roots labeled m,which do not occur in Eq. (39). However, by expandingEq. (11) it is implied that only the first root AjR, is re-tained. It is only close to t = 0 that the effects of thesummation can be seen, and these are far more due tohigher order longitudinal modes in Yn than radialmodes in Am.

It has been made clear in the preceding section thatthe first term in the series Eq. (39) is the most accurate,because there the right value 7Y is used in F2 , whereasthe use of ,yj in F2 in the higher order terms becomesincreasingly less accurate, particularly for large X2 andshort length. On the other hand, it is also clear that thefirst-order mode in yj dominates the decay processeverywhere, apart from at the beginning. The higherorder modes disappear more rapidly and are of smalleramplitude. Thus, experimentally the first mode decaytime rl and the amplitude T,, are most reliable and

readily accessible, the latter being obtained by extrap-olating the exponential temperature decay to t = 0 tofind the amplitude T,, = Aiyo(AoRi).

Equating the first-order terms in Eq. (8) at t = 0 andin Eq. (39) and disregarding the summation over theradial roots, as done throughout this paper, it followsthat at z = 0 and r =R

Poa 6 1 1

27rRXir Jo(/iRj) y'(1 + bc) + ca + [(/rRl)/X2](

which, using Eq. (16b), may be expressed asT u Poa 4 1

T1 7rRl2Clpl 7r 1 + bckl/k2(46)

This result does not depend on gas parameters becausethe term involving k /k2 may be left out of account.

In principle the initial slope of the temperature riseor decay curve is also an easily accessible parameter.Using Eq. (8) and ignoring the summation over radialroots it follows that

dt - - cosy,,zJo(AjiRi). (47)

The expressions for Tn can in principle be derived fromEqs. (10) and (11), but, as explained earlier, higher orderterms are incorrectly taken into account by assuminga linear approximation to FL'.

Using the values derived for An in Eq. (39) and for Tn

in Eqs. (10) and (la) it is found that

OTil -Poa 8 (-L)n-1dt -z i =0 n 2n-1 cOs YnZ,1

=- 27rRe1 7r ,,2n 1(47a)

and because at z = 0 the sum is known to be equal tor/4, the final expression reads

0T/t Poa

dtX-0R 7rR l~(47b)

Although this expression has been derived usingsometimes debatable approximations, it is valid withoutany restrictions. This can easily be seen by consideringEq. (1) at the moment of switching off the heat source.It is also an attractive expression because it does notinvolve any properties of the surrounding medium.However, it relies on a measurement at the most delicatepart of the decay curve, as experience shows. There-fore, the use of Eq. (46) is preferred for practical use.This will be discussed in the companion paper.3

The constructive advice of J. Geurst at some criticalpoints in this analysis is gratefully acknowledged.

References1. C. M. G. Jochem, T. P. M. Meeuwsen, F. Meyer, P. J. W. Severin,

and G. A. C. M. Spierings, in Proceedings 4th European Con-ference Optical Communications, (Instituto Internazionale delleComunicazioni, Via Pertinace, Villa Piaggio, Genova, Italy, 1978),p. 2.

2. H. M. van Ass, P. Geittner, R. G. Gossink, D. Kuppers, and P. J.Severin, Philips Tech. Rev. 36, 182 (1976).

3. P. J. Severin and H. van Esveld, to be published.

15 May 1979 / Vol. 18, No. 10 / APPLIED OPTICS 1553

4. H. N. Daglish and J. C. North, at 9th International Congress onGlass, (Institut du Verre, 75-Paris 16, France, 1971), Vol I, p.769.

5. H. H. Witte and R. Khan, Optik 36, 202 (1972).6. H. H. Witte, J. Opt. Soc. Am. 63, 332 (1973).7. H. H. Witte, Appl. Phys. 4, 109 (1974).8. K. I. White and J. W. Midwinter, Opto-electronics 5, 323

(1973).9. D. A. Pinnow and T. C. Rich, in Digest of Topical Meeting on

Integrated Optics I (Optical Society of America, Washington,D.C., 1972), paper TuA 4; Appl. Opt. 12, 984 (1973).

10. A. Zaganiaris, Verres Refract. 28, 113 (1974); Appl. Phys. Lett.25, 345 (1974); Phys. Chem. Glasses 17,83 (1976); Analusis 4,249(1976).

11. R. L. Cohen, Appl. Opt. 13,2518 (1974); R. L. Cohen, K. W. West,P. D. Lazy, and J. Simpson, Appl. Opt. 13, 2522 (1974).

12. F. A. Pinnow and T. C. Rich, Appl. Opt. 14, 1258 (1975).13. K. J. White, Opt. Quantum Electron. 8, 73 (1976).14. J. A. Lewis, Appl. Opt. 15, 1304 (1976).15. F. T. Stone, W. B. Gardner, and C. R. Lovelace, Opt. Lett. 2, 48

(1978).16. P. J. Severin and H. van Haren, to be published.17. S. S. Mitra and B. Bendow, Eds., Proceedings of the International

Conference on Optical Properties of Highly Transparent Solids(Plenum, New York, 1975).

18. M. Hass, J. W. Davisson, P. H. Klein, and L. L. Boyer, J. Appl.Phys. 45,3959 (1974); M. Hass, J. W. Davisson, H. B. Rosenstock,and J. Babiskin, Appl. Opt. 14, 1128 (1975).

19. D. C. Johnson, Appl. Opt. 12, 2192 (1973).20. A. Rosencwaig, Opt. Commun. 7, 305 (1973); Science 181, 657

(1973}; Anal. Chem. 47, 548 (1975); Phys. Today 28, No. 9, 23(1975).

21. A. Rosencwaig and A. Gersho, J. Appl. Phys. 47, 64 (1976).22. G. C. Wetzel and F. A. McDonald, Appl. Phys. Lett. 30, 252

(1977).23. J. F. McClelland and R. N. Kniseley, Appl. Opt. 15, 2658

(1976).24. L. Rosengren, Appl. Opt. 14, 1960 (1975).25. H. S. Bennett and R. A. Forman, Appl. Opt. 14,3031 (1975); 15,

1313 (1976).26. H. S. Bennett, J. Res. Nat. Bur. Stand. Sect. A: 79, 641 (1975).27. L. C. Aamondt, J. C. Murphy, and J. G. Parker, J. Appl. Phys. 48,

927 (1977).28. E. G. Bernal, Appl. Opt. 14, 314 (1975).29. J. G. Parker, Appl. Opt. 12, 2974 (1973).30. H. S. Bennett and R. A. Forman, J. Appl. Phys. 48, 1432

(1977).31. R. Siegel and J. R. Howell, Thermal Radiation Heat Transfer

(McGraw-Hill, New York, 1972).32. E. M. Sparrow and R. D. Cess, Radiation Heat Transfer

(Brooks/Cole, Belmont, Calif., 1970).

Books continued from page 1545

Self-Study Manual on Optical Radiation Measurements, Part1: Concepts; Chapters 4,5, and 6. Edited by F. E. NICODEMUSand H. J. KOSTKOWSKI. U.S. Government Printing Office,Washington, D.C., 1977/78. Chaps. 4 and 5, NBS Tech. Note 910-2$3.00; Chapt. 6, NBS Tech. Note 910-3. $3.00.

This review concerns a nonbook, which should be a book. TheSelf-Study Manual on Optical Radiation Measurements at presentconsists of six chapters bound in three separate soft-cover parts ofNBS Technical Note Series 910. The Manual, which is being pub-lished serially, was conceived by H. J. Kostkowski and is edited byF. E. Nicodemus. The material available to-date was written mostlyby these men, with contributions from A. T. Hattenburg and J. B.Shumaker. When complete, the Manual will consist of three parts:1. Concepts; 2. Instrumentation; and 3. Applications. The materialpublished so far, all of which belongs to the Concepts Section, dealswith the basics of radiation distributions with respect to position,direction, spectrum, and polarization. Also included is a chapter onthe central topic of the Manual, the measurement equation, whichrelates instrument output to both radiation and instrument param-eters. Other chapters are planned for the Concepts Section.

To appreciate the Manual, a reader should understand its purpose,its many facets and limitations, as well as its contents. The title ac-curately implies that this is a tutorial presentation. It seeks to coverwide ranges of wavelength, fields of use, and degrees of reader so-phistication. The stated spectral coverage is approximately 0.2-20mn, that is, the near UV, the visible, and part of the IR regions. The

Manual is written expressly for those who must do accurate radiationmeasurements within this range in diverse fields, including astronomy,mechanical heat transfer illumination engineering, photometry,photochemistry and biology, optical pyrometry, remote sensing, andmilitary applications. It is also clearly of use to other areas of re-search, development, and testing, plasma diagnostics being one fur-ther example. Users of the Manual can vary widely in the levels ofboth their backgrounds and needs. Even the most demanding

readers, those with little experience who must learn to perform ac-curate radiation measurements, should find the Manual accessibleand useful. The level and clarity of presentation, the numerous di-agrams, the tables of radiometry and photometry units, appendicesfull of details, and the extensive bibliography combine to serve a wideaudience.

Two criticisms of the Self-Study Manual can be cited. The first,which is largely a matter of style, concerns the order of presentation.It appears that the material was not published in the best sequence.Information on position and direction of radiation fields constitutesboth Chaps. 2 and 4, while Chap. 3 is on spectral parameters. Theexcellent Chap. 6 on polarization follows the chapter on the mea-surement equation, which requires some backtracking. However,awareness of this situation is sufficient to avoid frustration. Thesecond concern with the Manual deals with its scope and contents.The presentation is limited to geometrical (ray) optics and excludesphysical (wave) and quantum (particle) optics. Hence important andcurrent topics such as coherence and the interaction of radiation withmatter are slighted. But clearly, producing a Manual within theconfines of geometrical optics is a long-term task (4 years so far), andalso the field of coherence vis d vis radiometry is still developing. Aspecialist who examines this Manual can find omissions in his favoritearea. For example, a spectoscopist might wish to find formulas forinterconversion of spectra between energy, wavelength, and quantabases in the chapter on spectral distributions. However, suchomissions do not seem to be numerous. The Manual errs on the sideof detail (completeness), if anything.

The Concepts Section of the Self-Study Manual certainly shouldbe completed and made into a book. (Then it can be reviewed again!)Instrumentation and applications might be covered by companionvolumes. While we wait for such books, it is good to have availablemuch of the fundamental material.

DAVID J. NAGEL

continued on page 1599

1554 APPLIED OPTICS / Vol. 18, No. 10 / 15 May 1979