Nonambiguous determination of optical constants forabsorbing materials
Franco Gori and Giorgio Guattari
A recently proposed method for determining the optical constants of absorbing materials starting fromreflectance measurements is analytically treated. The conditions for a nonambiguous determination of theoptical constants are established. The values of the parameters satisfying these conditions are explicitlyevaluated.
1. IntroductionThe knowledge of the optical constants for absorb-
ing materials is of fundamental importance in a greatnumber of applications. A variety of methods havebeen envisaged for the determination of the complexrefractive index.'-' 4 Among them, those based onmeasurements of reflectivity are largely used for theirexperimental simplicity.6-9
Very often the material under investigation isstrongly absorbing. This happens, for example, in thetechnology of solar cells. For these cases, Gupta re-cently proposed a method in which the optical con-stant determination is based on the measurement ofthe reflectivity in the presence and absence of a trans-parent overcoat film.' 5 In Ref. 15, a graphicalrepresentation was used for determining the opticalconstants starting from the measured data. Shortlyafter, Spink and Thomas treated the Gupta methodfrom an analytical point of view and found an explicitexpression for the Fresnel reflection coefficient of theuncoated material.' 6 According to Ref. 16, the solu-tion is affected by an ambiguity that can be removedthrough a third measurement. Almost at the sametime, Siqueiros et al. proposed a method for determin-ing the optical constants of absorbing materials
Both authors are with University of Rome, Rome, Italy: F. Gori isin the Physics Department, 2 Piazzale Aldo Moro, I-00185, and G.Guattari is with the Electronics Department, 18 Via Eudossiana, I-00184.
through two reflectivity measurements. 7 These aremade when the sample is coated with a transparentdielectric in correspondence with two different opticalthicknesses. In particular, it is shown in Ref. 17 thatthe use of a X/4 optical thickness leads to a nonambi-guous determination of the optical constants. Thisresult demonstrates that the ambiguity cited by Spinkand Thomas does not always occur. It must be ob-served, however, that the coating optical thickness canbe exactly X/4 only at one precise wavelength, while itwould be useful to determine the optical constants as afunction of X using a single coating thickness. In Ref.17, the optical constants for wavelengths correspond-ing to optical thicknesses different from X/4 are evalu-ated through a numerical procedure, but the problemremains of establishing in which conditions nonambi-guous solutions can be obtained.
In this paper we study analytically the Gupta meth-od following a different approach with respect to Ref.16. The explicit expressions for the optical constantsof the material are written as a function of the reflec-tivity of the coated and uncoated samples. The resultsshow that a range for the coating optical thicknessexists inside which a nonambiguous determination ofthe optical constants is possible. We refer to such arange as the useful interval. For optical thicknessesoutside the useful interval, two different solutions forthe refractive index are compatible with the measuredreflectivities. As a consequence, the refractive indexdetermination is mathematically ambiguous, althoughsometimes the ambiguity can be removed throughphysical considerations or some a priori knowledge.We calculated explicitly the optical thicknesses limit-ing the useful interval as a function of the coatingrefractive index. They turn out to depend also on theunknown optical constants of the material so that,strictly speaking, they cannot be determined beforethe measurements are made. We find, however, that
such a dependence is rather weak so that an approxi-mate a priori estimate of the unknown quantities issufficient for determining the useful interval with ade-quate confidence. In addition, the useful interval iswide enough to keep the optical thickness of the mate-rial inside it also in correspondence with significantvariations of the wavelength or other parameters.
II. Statement of the ProblemLet us consider an absorbing material whose com-
plex refractive index will be denoted by n, = n - ik.When a monochromatic plane wave of wavelength Xfalls onto a plane boundary between the material andthe air (whose refractive index is assumed equal toone), the normal incidence reflectivity is given by'8
_ 1-n 2
1+nl '
where the asterisk denotes complex conjugate and2
1 + n2 + (1 - n2) COS(,y)
(1 - n2) sin(y)
n211 + n2 + (1 - n2) cos(1)}(7)
Furthermore, let us introduce the following nota-tions:
(8)n = N exp(i), N = n2 + k, = artan(--);
z =X-iY,
X 2
(1)
where the absorbing material has been supposed sothick that the light reflected from the material-sub-strate interface does not reach the incident side.
When a homogeneous plane-parallel nonabsorbingfilm of known real refractive index n2 and thickness dis deposited on the reflecting surface of the absorbingmaterial, the reflectivity of the coated material is givenby18
_ r2 + r2 exp(-iy) 2- 1 + r2r12 exp(-iy) ,
where1 - n2
r =+ 2
(2)
(1 - n') sin(zy)Y=2
n2[1 + n 2 + (1 - n2) cos(y)]'
z = Z exp(-iM), Z = X2 + Y2, 0 = arctan( ). (10)
By making use of Eqs. (8) and (10), Eqs. (1) and (6)become
respectively.(3) Starting from Eqs. (11) and (12), we can solve our
inverse problem. From Eq. (11) we obtain
(9)
(4)2N cos(M) = h1(1 + N2),
where1 -R1(5) h, 1 = R1 + R1
If the indices n, and n2 are known and the same istrue for d and X, Eqs. (1) and (2) can be used tocalculate the reflectivities of the uncoated and coatedmaterial. On the other hand, we are interested in theinverse problem, i.e., in finding n, starting from themeasured values of R1 and R2 (where, of course, n2, d,and X are assumed to be known).
Gupta's method15 should require that a unique solu-tion exists for the problem, but this cannot always betaken for granted. The consistency of different solu-tions with one and the same set of experimental dataprevents the refractive index from being determinedunivocally. Our aim is, in fact, to find out in whichconditions such an ambiguity does not occur.
II1. Solution of the Inverse ProblemBy making use of Eqs. (3) and (4), Eq. (2) can be
written in the following different manner
I 1-n 1 z 2
1 + n1z' (6)
Furthermore, by developing the cosinusoidal termsin Eq. (12) we obtain
As n and k are positive quantities, angle 0 must beincluded inside the interval (-7r/2,0). As a conse-quence, the following equation can be deduced fromEq. (13)
2N sin(o) = -4N2-hl(1 + N2)2. (17)
By making use of Eqs. (9), (10), (13), and (17), Eq.(15) becomes
N2(h 22 - h1X) + (h2 - hX)
= -h2Y N2(4-2hl)-hlN 4 -hl. (18)
This equation has to be solved with respect to N2.
To this purpose, let us consider also the followingequation:
N2(hZ' - hX) + (h, - hX)
= hY N2 (4 -2h2)-hlN4
-hl. (19)
The only difference between Eqs. (18) and (19) is thesign of their right-hand sides. By squaring both sidesof Eq. (18) or (19), we obtain the following equation ofthe second degree for N2:
aN4 + 2bN2 + c = 0, (20)
where
a = (h2Z'-hX)' + h h2Y2,
b = (h2 - hX)(h2Z' - hX) - (2 - h2)hY 2, (21)
c = (h2-hjX) + hhY2 .
Let us denote the two solutions of Eq. (20) by super-scripts I and II, i.e.,
(N2)I =1-b + }a~,a (22)
(N2)II = -{ - b a}ja
Owing to Eqs. (8), (13) and (17), the real and imagi-nary parts n and k of the complex refractive index njcan be deduced from the solutions for N2. We obtain
(n)II = h 1 + (N2)"11, (23)2
(k)"J = 2i)'-h1 + (24)2
respectively. In Eqs. (23) and (24), superscript I or IImust be assumed simultaneously.
The two solutions obtained in this manner should beinserted back into Eqs. (11) and (12) to verify that theyfurnish the right measured values for R1 and R2. Infact, some solutions of Eq. (20) could originate fromEq. (19) rather than from Eq. (18). More precisely, RIis reproduced exactly in any case, because Eq. (19)originates from Eq. (17) by changing 0 into -0 and thisdoes not affect the value of R1 [see Eq. (11)]. On theother hand, R2 changes when 4 is transformed into -0[see Eq. (12)]. In Sec. IV we discuss this problem indetail.
IV. Discussion of the SolutionsLet us preliminarily observe that Eq. (18) simplifies
when Y = 0, because its right-hand side vanishes. Ascan be seen from the third of Eqs. (9), this happenswhen -y = 0 (or y equals an integer multiple of 27r) andwhen y = 7r (or y equals an odd integer multiple of 7r).In the first case, however, as X = 1, Z = X, and h1 = h2,the left-hand side of Eq. (18) also vanishes and theequation becomes meaningless. On the other hand,when y = r (or y equals an odd integer multiple of 7r)Eq. (18) gives rise to a first degree equation for N2 andit is easily seen from Eqs. (9), (10), and (18) that thefollowing unique solution exists for N2:
h, -h 2~N2 = n2 h h2n2h2 - hn2
(25)
The above choice for -y corresponds to that performedin Ref. 17.
Let us now pass to the general case of y $ 0 and gr, in which case Eq. (18) is really a second degreeequation for N2. As N is a real quantity, the solutionsof Eq. (20) must be real (distinct or coincident).
When b2 = ac in Eqs. (22), the solutions for N2 arereal and coincident. Obviously, in this case, the quan-tities n and k are univocally determined. It is immedi-ately verified that -y = 7r (or equals an odd integermultiple of 7r) satisfies the condition b2 = ac and asecond y-value satisfying the same condition could becalculated. However, for a fixed coating optical thick-ness, both these -y-values correspond to one precisewavelength and cannot be used for obtaining nonambi-guous solutions in the whole wavelength range. Inaddition, the second y-value is actually useless from anexperimental point of view, because it depends on theunknown optical constants of the material.
When the solutions for N2 are real and distinct, twocases may take place. In the first case, one solutionsatisfies Eq. (18), while the other satisfies Eq. (19). Inthis case, the unknown refractive index n1 is univocallydetermined because only one pair n,k reproduces theright values for R1 and R2. On the other hand, in thesecond case, both solutions satisfy Eq. (18), while Eq.(19) has no solution. In this case, two different pairs,let us say (n)I,(k)' and (n)II,(k)II, reproduce the rightvalues for R and R2 and the determination of theunknown refractive index n is ambiguous.
To go into details, let us examine the practical exam-ple considered in Ref. 15. For that case, n = 1.61, k =0.42, and n2 = 1.95. By making use of Eqs. (1) and (2),together with Eqs. (3) and (4), reflectivities R1 and R2can be calculated for every y. These values can thenbe used for calculating parameters h and h2 from Eqs.(14) and (16). These two parameters, together withthe others obtainable from Eqs. (9) and (10), complete-ly specify coefficients a, b, and c given by Eqs. (21), andsolutions (2)III of Eq. (20) as given by Eqs. (22) can befound. Finally, by making use of Eqs. (23) and (24),the real and imaginary parts (n)I II and (k)III of thecomplex refractive index can be deduced from (N2)' I".In Fig. 1, the two pairs (n)I,(k)I and (n)II,(k)II are givenas a function of y. Figure 2 plots the true reflectivityR2 and the reflectivities, let us say (R2)I and (R2)II, ascalculated starting from (n)l,(k)I and (n)I,(k)II, respec-tively.
Although the exact behavior of the quantities shownin Figs. 1 and 2 depends on n, k, and n2, some impor-tant characteristics appear to be always the same.From Fig. 1 it is seen that, for y included inside theintervals (0,2.083) and (7r,27r), the pair (n)I,(k)I equalsthe right values of n and k, while the pair (n)II,(k)II doesnot; the opposite is true for y included inside theinterval (2.083,7r). From Fig. 2 it is seen that, for yincluded inside the interval (0,2.6), both pairs (n)I,(k)Iand (n)II,(k)II reproduce the right reflectivity, i.e., R2 =
(2.6,5.9) is the useful interval defined before, insiden which the unknown optical constants of the material
can be univocally determined. It must be noted thatthe value of y = 4.43 used in Ref. 15 is definitely insidethe useful interval and this explains the attainment of
.. . nonambiguous results.Let us now consider, in the general case, the explicit
*- k1I -. evaluation of the extreme points of the useful intervaldenoted by Ay and Y2. We observe that these points
. coincide with the y-values corresponding to which onek' \ / "I.,of the two solutions of Eq. (20) satisfies both Eqs. (18)
k' \ / -. . and (19). Obviously, in this case, the right-hand sidesv of Eqs. (18) and (19) are required to vanish. This may
1 2 3 4 5 6 well happen, also when Y 5d 0, in correspondence withcurves represent the pairs of values (n)',(k)l (solid the particular N 2-value that makes the radicand on the,(k) (dotted lines) of real part n and imaginary part k right-hand side of Eq. (18) vanish. By taking inton complex refractive index as evaluated starting from account Eq. (24), it is immediately seen that the pre-reflectivities. Both pairs of solutions are drawn as a ceding condition is obtained when k = 0. Therefore,e phase shift y undergone by the reflected radiation for y = 1Yi (or y = -Y2) the same reflectivity is producedling to a round trip through the coating material. by the absorbing material whose optical constants are
(n)II and (k)II [or (n)I and (k)'] and by a suitable nonab-sorbing material whose real refractive index is denotedby nr, By setting the radicand on the right-hand side
R'D of Eq. (18) equal to zero, nr can be calculated from
/ 2nr = i - R, (1 + n),1+1?, r
(26)
where Eq. (14) has been used and the positive nature ofnr has been taken into account.
By making use of Eq. (1), the values of nr can beexplicitly determined as a function of the optical con-stants n and k of the corresponding absorbing material
S7 n
1 2 3 4 5 6Fig. 2. Three curves represent true reflectivity R2 (solid line) andreflectivites (R)' (dotted line) and (R)II (dashed line) as calculatedstarting from (n)l,(k)I and (n)l",(k)l, respectively. All the reflectivi-ties are drawn as a function of phase shift y undergone by thereflected radiation corresponding to a round trip through the coatingmaterial. Depending on the range ofy that is considered, two curves
or all three are coincident.
1 +n'+k± 1 +n'+k'-4n'
2n(27)
Finally, ey and 7Y2 can be explicitly calculated startingfrom the condition of equal reflectivity R2 correspond-ing to the pairs n,k, and nr,0, that is,
r2 + r12 exp(-iy) 2 r2 + rr exp(-iy) 2
1 + r2r12 exp(-iy) 1 + r2rr exp(-i-y)(28)
where
(R2)I = (R2)II, and the optical constant determinationis ambiguous. The only exception to this is when y =2.083, for which the solutions of Eq. (20) are real andcoincident. For -y included inside the interval (2.6,ir),only the pair (n)II,(k)II reproduces the right reflectiv-ity, i.e., R2 = (R2)II ` (R2)I, and the optical constantscan be univocally determined. As already noted, thevalue of y = 7r leads to the second case for which onlythe solution given by Eq. (25) exists for Eq. (20). For yincluded inside the interval (7r,5.9), only the pair(n)',(k)I reproduces the right reflectivity, i.e., R2 =
(R2)I id (R2)II, and the optical constants can be univo-cally determined. Finally, for y included inside theinterval (5.9,2-r), the optical constant determination isagain ambiguous.
The above described behavior shows that, for thepractical example we examined, the y-interval
(29)n2 + nr
Ir n2 + nr
Equation (28) must be solved with respect to y.This requires some lengthy and tedious algebra andthe final resulting expressions for -yj and 72 are givenby
1 2 3 4 5 6Fig.5. Two curves represent Y2 (solid line) and y (dotted line) as afunction of real refractive index n2 of the coating material for fixed n
= 1.61 and k = 0.42.
ambiguity or nonambiguity of the refractive index de-termination can be always tested a posteriori by in-serting the pairs (n)I,(k)I, and (n)II,(k)II into Eq. (2)and by comparing the resulting reflectivities (R2)' and(R2)", with the effective measured reflectivity R2.
Finally, let us observe that, in many cases, the math-ematical ambiguity of the solution can be removed on aphysical basis. For example, with reference to thecases in Figs. 1 and 2, when nonmetallic materials areconsidered, in correspondence with 0 < y < 1.21 and4.15 < y < 27r, one solution can be rejected because thereal part of the refractive index is less than one.
1 2 3 4 5 6k
Fig. 4. Two curves represent Y2 (solid line) and y1 (dotted line) as afunction of imaginary part k of the unknown complex refractive
index for fixed n = 1.61 and 2 = 1.95.
V. Some Practical Remarks
The Gupta method can be advantageously used todetermine unambiguously the optical constants of anabsorbing material in a given range of wavelength,provided y remains within the useful interval (1,y2).
It must be observed that, as shown by Eqs. (30) and(31), 7l and Y2 depend on n and k, i.e., on the unknownquantities to be calculated. As a consequence, yl and72 cannot be known exactly before the measurement ismade. However, their dependence on n and k is suchthat an approximate a priori estimate is sufficient inpractice to determine the useful interval (71,7Y2) and,then, to evaluate the optical thickness of the coatingmaterial to be used for the measurements. This isshown in Figs. 3 and 4, where -yl and 72 are drawn asfunctions of n (Fig. 3) and k (Fig. 4) for the practicalcase examined in Sec. IV.
In Fig. 5, an example of dependence of yl and 72 onn2 is also given. Curves of this kind can be useful tochoose the refractive index of the coating material. Ascan be seen, the dependence is quite weak so that thechoice of the value of 2 is not critical. Moreover, the
VI. ConclusionsIn this paper, the analytical relationship between
the optical constants of an absorbing material and thereflectivities measured through the Gupta method hasbeen established. It has been demonstrated that themathematical ambiguity of the solution depends onthe optical thickness of the coating material. In par-ticular, it has been shown that, when the optical thick-ness is included inside a certain interval (that we calledthe useful interval), only one solution of the unknownrefractive index is compatible with the experimentaldata and the unknown optical constants of the materi-al can be univocally determined. The limiting valuesof the useful interval have been explicitly determined.Their dependence on the unknown optical constants ofthe material has been shown to be such that an approx-imate a priori knowledge of them is sufficient for apractical determination of the useful interval. By ex-ploiting these results, the optical constants of an ab-sorbing material can be determined without ambiguityover a large range of wavelengths.
The mathematical ambiguity of the solution wespoke about is not to be confused with the uncertaintyarising from the measurement errors. Such an uncer-tainty was not considered in this paper.
The present work was developed under MPI 40%grant Materiali, dispositivi e tecnologie per la fotonica.
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