Complex Refractive Index Determination for Uniaxial ...

Complex Refractive Index Determination for Uniaxial Anisotropy with the Use of Kramers±Kronig Analysis KIYOSHI YAMAMOTO* and HATSUO ISHIDA≤ Department of Macromolecular Science, Case Western Reserve University, Cleveland, Ohio 44106-7202

Kramers±Kronig analysis of re¯ection spectra from a single interface with perpendicular (s) and parallel ( p) polarization has been theoretically studied. The re¯ection spectr a have been sim ulated from the complex refractive indices based on disper sion theory by using Fresnel equations for an anisotropic material whose optical axis is normal to the surface. The errors in anisotropic complex refractive indices obtained from simu lated re¯ection spectra by Kramers±Kronig analysis have been examined for such techniques as external and total internal re¯ection spectroscopies. Index Headings: Uniaxial anisotropy; Kramers±Kronig analysis; Phase correction.

INTRODUCTION Anisotropic properties of materials have been studied by the combination of measurements with perpendicular (s) and parallel ( p) polarization because p polarization has components of the electric ®eld both parallel and normal to the surface, while s polarization has only a component parallel to the surface. For such a purpose, the Fresnel equations for an anisotropic material have been given by Drude1 and Schopper.2 Schopper’s formulas were checked for their validity by Tomar and Srivastava3 and have been used to modify ellipsometry for application to anisotropic materials.3±6 The Fresnel equations for an anisotropic material have been also derived in more convenient forms for the complex calculation directly from the Maxwell equations and boundary conditions.7 With the use of these equations and re¯ection measurements at various angles, a new method was developed to obtain complex refractive indices parallel and normal to the surface for uniaxial absorbing crystals. These equations have also been used for ellipsometric spectroscopy of oriented molecular layers.8 Using Drude equations, AbeleÁs et al.9 evaluated the reliability of the complex refractive index calculated from re¯ectivity alone, and Ishino and Ishida10 simulated the spectra for uniaxially oriented Langmuir±Blodgett monolayers. On the other hand, Kramers±Kronig analysis of re¯ection spectra has been widely used to determine the complex refractive index of materials in the infrared region,11±32 where a well-collimated beam is dif®cult to obtain, and an angle of incidence measured with high accuracy is dif®cult to determine. Kramers±Kronig analysis with s polarization needs only a spectrum observed at an angle of incidence and does not need an iterative calculation or a least-squares re®nement for a single interface system. This advantage is pronounced, especially for the Received 26 April 1996; accepted 18 February 1997. * Present address: Research Center, Asahi Glass Co., Ltd., Hazawa 1150, Kanagawa-ku, Yokohama 221, Japan. ≤ Author to whom correspondence should be sent.

Volume 51, Number 9, 1997

measurement near normal incidence where the incident angle dependence of re¯ectivity is less sensitive than that at larger angles of incidence. As the observed spectrum is limited to a ®nite frequency region, the extrapolation error to the complex refractive index was estimated for Kramers±Kronig analysis of a re¯ection spectrum.30 However, Kramers±Kronig analysis of an external re¯ection spectrum should not be applied to thin ®lm because of the re¯ection from the back side of sample. Consequently, Kramers±Kronig analysis of attenuated total re¯ection (ATR) spectra has been developed in which the knowledge about the refractive index of material at a frequency with no dispersion is necessary for correcting the phase shift of re¯ection due to the singularity on the imaginary axis in the complex plane.13,22±25,28±29,31±32 With the combination of Fresnel equations for anisotropy with Kramers±Kronig analysis with s and p polarization, a new method was developed to obtain complex refractive indices parallel and normal to the surface of an anisotropic material from ATR spectra by Bardwell and Dignam28 and Dignam and Mamiche-afara.29 The advantage of this method is that measurements at only a single angle of incidence are necessary, and a least-squares re®nement calculation is not needed (unlike the techniques mentioned previously). However, these authors did not deal with the external re¯ection–a condition that is better for this technique–or with the question of how this analysis affects the obtained complex refractive index. The purpose of this paper is to estimate the errors in the calculated complex refractive indices theoretically. After the re¯ection spectra are simulated from the complex refractive indices on the basis of the dispersion theory using Fresnel equations for an anisotropic material, anisotropic complex refractive indices are obtained by Kramers±Kronig analyses with s and p polarization. From the comparison of the calculated spectra with the original ones, the potential of this analysis is evaluated. While experimental errors such as the signal-to-noise (S/N) ratio of the spectrum and the accuracy in determining the angle of incidence and the beam divergence may produce error greater than theoretically intrinsic error, an estimate of theoretically intrinsic error is important to obtain complex refractive indices by the analysis described in this paper. EXPERIMENTAL Calculation Methods. In this paper, the s and p components of the electric ®eld, for which the electric vectors are perpendicular and parallel to the plane of incidence, respectively, are considered. The directions of the electric vector for incident and re¯ected light, E i and E r, are de®ned in Fig. 1a. In this de®nition, it is obvious that the sign of E rp is different from that of E ip . This situation can

0003-7028 / 97 / 5109-1287$2.00 / 0 q 1997 Society for Applied Spectroscopy

APPLIED SPECTROSCOPY

1287

FIG. 1 (a) De®nition of electric vectors for s and p polarizations: E is, E rs for s polarization and E ip , Erp for p polarization where i and r denote incident and re¯ected lights. Directions for both E is and E rs are normal to the plane of incidence. (b) Optical con®guration and de®nition of anisotropic complex refractive indices: n0, refractive index of incident media; u 0 , angle of incidence; u 1 , angle of refraction and nÃx , nÃy , and nÃz, complex refractive indices along the x, y, and z axes, respectively.

be solved by introducing phase shift, p, between E ip and E rp as discussed in our previous papers.31,32 The optical con®guration, in which only a single interface is considered, is shown in Fig. 1b. The anisotropic complex refractive indices, nÃx, nÃy, and nÃz, are also de®ned in Fig. 1b. Among the complex refractive indices, nÃz is the only component normal to the surface. It is noted here that the angle of refraction, uÃ 1, is a function of the complex refractive index so that the angle of refraction for s polarization, uÃ 1s, is different from that for p polarization, uÃ 1p, in an anisotropic material. Anisotropy. In order to obtain nÃy, the Fresnel equation

FIG. 3 Complex refractive index for an isotropic model based on dispersion theory: (a) a real part, n, and (b) an imaginary part, k.

and Snell’s law, taking into account the anisotropy for s polarization, must be considered.7,8,29 rÃs 5

E rs n0cos u 0 2 nÃycos uÃ 1s , 5 E is n0cos u 0 1 nÃycos uÃ 1s

(1)

and n0sin u 0 5 nÃysin uÃ 1s , (2) where the incident medium is assumed to be transparent, so that n0 is not a complex number. Since these equations are identical with that for an isotropic material, the procedure of Kramers±Kronig analysis with s polarization31 can be directly applied. By rotating the sample 908 around the z axis, one can obtain nÃx in the same manner as for nÃy. If the sample has only uniaxial anisotropy along the z axis, nÃx is equal to nÃy. The procedure to obtain nÃz is more complicated. The Fresnel equation accounting for anisotropy is given for p polarization as follows.7,8,29 rÃp 5

E rp nÃxcos u 0 2 n0cos uÃ 19p 5 E ip nÃxcos u 0 1 n0cos uÃ 19p

(3)

and n0sin u 0 5 nÃzsin uÃ 19p , (4) where uÃ 1p is introduced for convenience but has no physical meaning.8 It is evident that Eqs. 3 and 4 become TABLE I. Dispersion parameters for the complex refractive indixes of isotropic and anisotropic models. Anisotropic

FIG. 2 Calculated re¯ectivities for s and p polarization, Rs and Rp : broken line for s polarization and solid line for p polarization. (a) External re¯ection: n0 5 1.0, and n1 5 1.5 and (b) internal re¯ection: n0 5 2.4, n1 5 1.5.

1288


n 0(cm-1) g(cm-1) Î} Î}-Î0 n}

Isotropic

x and y axes

z axis

1500 40 2.25 0.015 1.5

1500 40 2.25 0.015 1.5

1600 20 2.25 0.015 1.5

rÃp can be obtained by Kramers±Kronig analysis with p polarization (Eq. 8) because rÃp is expressed as Eq. 9 with the use of energy re¯ectivity, Rp, and the phase shift for re¯ection, dp.

dKK p (n 9) 5

n9 p

and

E

`

0

lnRp (n ) 2 lnRp (n 9) dn n 2 2 n 92

(8)

rÃp 5 ÏRp e i d p

(9)

where dkk p is the phase shift calculated from Rp by Kramers±Kronig analysis. Taking into account the correction term due to singularities, the actual phase shift, dact p , is given as follows:31,32 KK corr dact p 5 dp 1 dp ,

(10)

where dcorr is the correction term. As a result, rÃp is obp tained by replacing dp in Eq. 9 with Eq. 10. Since the anisotropy is considered only in the infrared (IR) region here, it is assumed that nz 5 nx 5 n1 at a frequency high enough to neglect the dispersion in the mid-IR region. Therefore, the phase correction terms for this case, dp 5 0 or p, are obtained in the same manner as for isotropic material.32 For total internal re¯ection, u 0 . u c, Eq. 11 is given instead of Eq. 6. FIG. 4 Simulated re¯ectivities for the isotropic model: (a, b) for external re¯ection (n0 5 1.0) and (c, d) for internal re¯ection (n0 5 2.4); (a, c) with s polarization and (b, d) with p polarization. Angles of incidence are indicated in the ®gures.

equations for an isotropic material when nÃx 5 nÃz. The critical angle, u c, is expressed as follows from Eq. 4, and nÃz is de®ned as nÃz 5 nz 1 ikz, where i 5 Ï21, and nz and kz represent the refractive index and the extinction coef®cient along the z axis, respectively.

u c 5 sin 21

12

nz . n0

(5)

As shown in previous papers, there is no imaginary part in rÃp at u 0 , u c when no dispersion is considered, but rÃp becomes a complex quantity at u 0 . u c even with no dispersion. Therefore, two cases, u 0 , u c and u 0 . u c, must be considered. For the ®rst case (u 0 , u c), which includes external re¯ection, Eq. 6 is given from Eq. 4. 31,32

cos uÃ 19p 5

!

12

n02 2 sin u 0 . nz2

(6)

Substituting Eq. 6 into Eq. 3 and solving for nÃz leads to the following equation: nÃz 5

!

n0sin u 0

.

(7)

n 2cos 2u (1 2 rÃp ) 2 12 x 2 0 n0 (1 1 rÃp ) 2 In Eq. 7, n0 and u 0 are the conditions of measurement, and nÃx can be directly obtained by Kramers±Kronig analysis with s polarization as shown before. Furthermore,

!

n2 cos uÃ 1p9 5 i 02 sin 2u 0 2 1 . nz

(11)

Solving Eqs. 3 and 11 for nÃz leads to Eq. 12, which is identical with Eq. 7. nÃz 5

n0sin u 0

!

n cos 2u 0 (1 2 rÃp ) 2 12 n02 (1 1 rÃp ) 2 2 x

.

(12)

Therefore, nÃz can be obtained by the same procedure as for the case, u 0 , u c. The phase correction term for this case is obtained as follows. In order to obtain the phase correction term due to the singularities, no dispersion, kx 5 kz 5 0, is considered.31,32 Thus, the next equation is given from Eqs. 3 and 11. rÃp 5

5[

nx2cos 2u 0 2 n02

2 2n0 nxcos u 0i

[

1

2

n02 2 sin u 0 2 1 nz2

!

4 nx2cos 2u 0 1 n02

n02 2 sin u 0 2 1 nz2

1

]

n02 2 sin u 0 2 1 nz2

2]6

.

(13)

It is obvious in Eq. 13 that there is an imaginary part, even when no dispersion is considered. The phase correction term, dcorr p , is obtained for this case from Eq. 13 as follows:


1289

FIG. 5 Results of Kramers±Kronig analysis for the isotropic model (a±d) for external re¯ection (u 0 5 308, n0 5 1.0) and (e±h) for total interal re¯ection (u 0 5 458, n0 5 2.4): (a, e) re¯ectivities; (b, f) calculated actual phase shift, dact; errors in (c, g) the real part, n, and (d, h) the imaginary part, k, of complex refractive index. The broken line is for s polarization and the solid line is for p polarization in (a), (b), (e), and (f). The broken line is for the x or y axis and the solid line is for the z axis in (c) and (d). The solid line overlaps on the broken line in (g) and (h). The actual phase shift, dact s , is shifted by 1808 for direct comparison with that for p polarization.

dcorr 5 tan 21 p

and

F

22n0 nxcos u 0

nx2cos 2u 0 1 n02

5 2 tan 21

1290

!

F

n0

!

n02 2 sin u 0 2 1 nz2

1

2

n02 2 sin u 0 2 1 nz2

G

n02 2 sin u 0 2 1 nz2 2p. nxcos u 0

G

Equation 14 is identical with the phase correction term shown by Dignam and Mamiche-afara.29 In practice, it may be assumed that nz 5 nx 5 n1 at frequencies high enough to ensure that dispersion does not occur so that the effects of dispersion are neglected, as mentioned previously. Therefore, Eq. 14 becomes Eq. 15, which is the same as that for isotropic material with p polarization.32


dcorr 5 2 tan21 p (14)

F

!

n0

G

n02 2 sin u 0 2 1 n12 2p. n1cos u 0

(15)

As shown here, the refractive index, n1, must be known at a frequency with no dispersion for the total internal re¯ection technique. At this point, rÃp is known through

FIG. 7 Simulated re¯ectivities for the anisotropic model: (a, b) for external re¯ection (n0 5 1.0) and (c, d) for internal re¯ection (n0 5 2.4); (a, c) with s polarization and (b, d) with p polarization. Angles of incidence are indicated in the ®gures. FIG. 6 Complex refractive indexes for an anisotropic model based on dispersion theory: (a) the real part, n, and (b) the imaginary part, k. The solid line is for both nÃx and nÃy , and the dotted line is for nÃz.

Eqs. 8, 9, 10, and 15, so that we can calculate nÃz by putting rÃp into Eq. 17. It is noted that nÃz is explicitly solved in this method, although nÃ1 for an isotropic material cannot be explicitly expressed by rÃp .32 Complex Refractive Index Based on Dispersion Theory. The complex refractive indices along the x, y, and z axes, nÃx , nÃ y and nÃz, for an anisotropic model material are derived from dispersion theory. The calculation procedures have been shown in a previous paper.31 With the use of complex refractive indices based on dispersion theory, energy re¯ectivities for s and p polarization are calculated in Eqs. 1, 2, and 16 and Eqs. 3, 4, and 17. Rs 5 zrÃsz2

(16)

Rp 5 zrÃpz2 .

(17)

and Methods. The programs used in this paper have been written in Fortran 77, and all calculations have been done on Micro VAX with the VMS operating system. The spectrum range used during integration for Kramers± Kronig analysis is the same as that shown in later ®gures. The procedures of integration and extrapolation for the Kramers±Kronig analysis have been reported elsewhere.30 RESULTS AND DISCUSSION In order to check the validity of this method and to estimate the error in the obtained complex refractive index, an isotropic model is used ®rst and an anisotropic model is examined next. It is noted that the use of an isotropic model has meaning only to validate this technique since nÃz must be equal to nÃy for the isotropic model. In this paper, it is assumed that n, one of the dispersion parameters, is always equals to 1.5, because these models simulate organic materials. The refractive indices for in-

cident media considered here are n0 5 1.0 for external re¯ection and n0 5 2.4 for internal re¯ection. For nonabsorbing substrates, the angular dependence of energy re¯ectivity for s and p polarization, Rs and Rp, calculated from Eqs. 1, 2, and 16 and Eqs. 3, 4, and 17, is shown in Fig. 2. For external re¯ection in Fig. 2a, Rs increases monotonically while Rp decreases at u 0 , u B 5 tan21(n1/ n0), Brewster’s angle, and rapidly increases at u 0 . u B. For internal re¯ection in Fig. 2b, Rs and Rp have the same tendency as that for external re¯ection, but both equal 1.0 at u c; that is, total internal re¯ection is observed at u 0 . u c 5 sin21 (n0 /n1 ), the critical angle. When dispersion occurs, the situation becomes complex. For an isotropic model that has a moderate dispersion, the noise-free complex refractive index, nÃi, shown in Fig. 3 is calculated from dispersion theory using dispersion parameters listed in Table I. Since nÃx 5 nÃy 5 nÃz 5 nÃi in an isotropic material, the re¯ection spectra for s and p polarization are calculated in Fig. 4 from Eqs. 1, 2, and 16 and Eqs. 3, 4, and 17, respectively. For external re¯ection with s polarization, Fig. 4a shows that the dispersive shape appears on the re¯ectivities that monotonically increase with angle of incidence. For external re¯ection with p polarization in Fig. 4b, the re¯ectivities decrease, contrary to that for s polarization, as shown for the nonabsorbing substrate. It is interesting to note that the shape of the dispersion reverses at u 0 5 608 (.u B 5 56.318 for this case) because of the phase shift due to singularity in the case of u 0 . u B.32 For internal re¯ection with s polarization in Fig. 4c, the re¯ectivities show the same tendency as that for external re¯ection, but ATR occurs at u 0 . u c 5 41.818. The dispersive shape reverses to that for external re¯ection because of the singularity in the case of n0 . n1 for internal re¯ection. For internal re¯ection with p polarization in Fig. 4d, ATR also occurs at u 0 . u c, but the absorption peaks are greater than that for s polarization because of the difference in Fresnel transmissivity and are not due to the difference in the penetration depth.33 It is also shown that the shape of dispersion is reversed at u 0 , u B (556.318)


1291

FIG. 8 Results of Kramers±Kronig analysis for the anisotropic model; (a±d) for external re¯ection (u 0 5 308, n0 5 1.0) and (e±h) for total internal re¯ection (u 0 5 458, n0 5 2.4); (a, e) re¯ectivities; (b, f) calculated actual phase shift, dact; errors in (c, g) the real part, n, and (d, h) the imaginary part, k, of complex refractive index. The broken line is for s polarization and the solid line for p polarization in (a), (b), (e), and (f). The broken line is for the x or y axis and the solid line for the z axis in (c), (d), (g), and (h). The actual phase shift, dact s , is shifted by 1808 for direct comparison with that for p polarization.

because of the singularity in the case of n0 . n1, and reverses again at u 0 . u B.32 In Fig. 4, the spectra at u 0 5 308 for external re¯ection and u 0 5 458 for internal re¯ection are used to obtain nÃy and nÃz because they have appropriate re¯ectivity. Internal re¯ection at u 0 , u c is not considered here, because it is not practical. The calculation result is shown in Fig. 5. For external re¯ection, simulated re¯ectivities, Rs and Rp , and actual phase shift, dact, calculated by Kramers± Kronig analysis are shown in Figs. 5a and 5b. The complex refractive indices, nÃy and nÃz (nÃkk 5 nkk 1 ikkk), calculated from these spectra for the isotropic model, where nÃy must be equal to nÃz, are almost identical with the original ones, nÃorig 5 norig 1 ikorig, shown in Fig. 3. It is shown in Figs. 5c and 5d that the errors, (nkk 2 norig) and (kkk 2 korig), are small enough to allow one to obtain nÃz. Although the error is slightly larger in nÃz than in nÃx , both are obtained within about 1% accuracy. Figures 5e±5h,

1292


which include the same information as in Figs. 5a±5d, show that nÃz can also be obtained accurately from internal re¯ection. In Figs. 5g±5h, the error in nÃz completely overlaps on that in nÃx. The situation becomes more complex for an anisotropic model. The noise-free complex refractive indices, nÃx and nÃz, shown in Fig. 6, are calculated from dispersion theory using dispersion parameters listed in Table I. For simple interpretation, it is assumed that optical conditions are the same as for the isotropic model, nÃx 5 nÃy ± nÃz (uniaxial anisotropy) and that nÃx and nÃy are equal to nÃi, so that the results from the previous section can be compared. Although there does not exist a material that has such complex refractive indices, it is convenient to evaluate the error due to calculations by using a ®ctitious material. For s polarization, calculated re¯ectivities are identical with that for the isotropic model shown in Figs. 4a and 4c because only nÃy affects the re¯ectivity. The

re¯ectivities, calculated for external re¯ection with p polarization in Fig. 7a, show that the effect of nÃz at 1600 cm21 on re¯ectivity increases with angle of incidence from 0 at u 0 5 08, comparable to that of nÃx (at 1500 cm21) at u 0 5 608. There is the same tendency in Fig. 7b for internal re¯ection with p polarization, but the effect of nÃz on re¯ectivity is more pronounced than in external re¯ection because the refractive angle, u 1, is greater for internal re¯ection even when u 0 is identical. Therefore, ATR may be a better technique to obtain nÃz. Calculation results are shown in Fig. 8. In Fig. 8b, it is interesting to note that the phase shift is increased by dispersion along x axis while decreased by that along the z axis. Figures 8c and 8d show that the errors, (nkk 2 norig) and (kkk 2 korig), are small enough to allow one to obtain nÃz. It is also noted that the error in nÃx hardly affects the error in nÃz in Fig. 8c. Figures 8e±8h include the same information as Figs. 8a±8d for external re¯ection, demonstrating that nÃz can be obtained accurately from ATR spectra. CONCLUSION The re¯ection spectra from a single interface have been simulated by using Fresnel equations for an anisotropic material. The errors in anisotropic complex refractive indices obtained by Kramers±Kronig analysis with s and p polarization have been theoretically studied. Both external and total internal re¯ection can be used to obtain the anisotropic complex refractive indices. The theoretical intrinsic errors can be ,1% for the technique described here. It is important to note that this analysis does not need the iterative calculation or a least-squares re®nement necessary for the other techniques to extract the complex refractive index from re¯ection spectra at several angles of incidence. Each technique was theoretically shown to have some intrinsic advantages. The external re¯ection technique does not require the knowledge of n1 at some frequency, but it cannot be applied to thin-®lm samples because the re¯ection from the back surface cannot be avoided. The total internal re¯ection technique may have better S/N ratio than the external re¯ection spectrum and can be applied to thin ®lms. However, to calculate the phasecorrection term, the analysis for the total internal re¯ection technique requires n1, which may be determined by ellipsometry in the visible or near-infrared region. For

external re¯ection and ATR, u 0 5 45±608 is appropriate to extract nÃz from a re¯ection spectrum. In this paper, the method has been validated only for an organic model, but it should be possible to extend the considerations toward the application for an inorganic model on the analogy of the results in this and previous papers.31,32 ACKNOWLEDGMENT The authors gratefully acknowledge the partial ®nancial support of Asahi Glass Co., Ltd. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33.

P. Drude, Ann. Phys. Chem. 32, 584 (1887). H. Schopper, Z. Phys. 132, 146 (1952). M. S. Tomar and V. K. Srivastava, Thin Solid Films 15, 207 (1973). M. S. Tomar and V. K. Srivastava, J. Appl. Phys. 45, 1849 (1974). M. S. Tomar, J. Phys. Chem. 82, 2726 (1978). A. R. Elsharkawi and K. C. Kao, J. Opt. Soc. Am. 65, 1269 (1975). L. P. Mosteller Jr. and F. Wooten, J. Opt. Soc. Am. 58, 511 (1968). M. J. Dignam, M. Moskovits, and R. W. Stobie, Trans Faraday Soc. 67, 3306 (1971). F. AbeleÁs, H. A. Washburn, and H. H. Soonpaa, J. Opt. Soc. Am. 63, 104 (1973). Y. Ishino and H. Ishida, Langmuir 4, 1341 (1988). T. S. Robinson, Proc. Phys. Soc. 910, (1952). T. S. Robinson and W. C. Price, Proc. Phys. Soc. LXVI, 969 (1953). J. S. Plaskett and P. N. Schartz, J. Chem. Phys. 38, 612 (1963). D. M. Roessler, Brit. J. Appl. Phys. 16, 1119 (1965). D. M. Roessler, Brit. J. Appl. Phys. 16, 1359 (1965). D. W. Berreman, Appl. Opt. 6, 1519 (1967). G. M. Hale, W. E. Holland, and M. R. Querry, Appl. Opt. 12, 48 (1973). M. R. Querry and E. Holland, Appl. Opt. 13, 595 (1974). A. H. Kachare, W. G. Spitzer, F. K. Euler, and A. Kahan, J. Appl. Phys. 45, 2938 (1974). E. G. Makarova and U. N. Morozov, Opt. Spectrosc. 40, 138 (1976). R. E. Prange, H. D. Drew, and J. B. Restorff, J. Phys. C: Solid State Phys. 10, 5083 (1977). R. Young, J. Opt. Soc. Am. 67, 520 (1977). J. A. Bardwell and M. J. Dignam, J. Chem. Phys. 83, 5468 (1985). J. A. Bardwell and M. J. Dignam, Anal. Chim. Acta 172, 101 (1985). J. A. Bardwell and M. J. Dignam, Anal. Chim. Acta 181, 253 (1986). B. Harbecke, Appl. Phys. A40, 151 (1986). K. Jezierski, J. Phys. C: Solid State Phys. 19, 2103 (1986). J. A. Bardwell and M. J. Dignam, J. Colloid Interface Sci. 116, 1 (1987). M. J. Dignam and S. Mamiche-Afara, Spectrochim. Acta 44A, 1435 (1988). K. Yamamoto and A. Masui, Proc. Euroanalysis VII B2, L-8 (1990). K. Yamamoto, A. Masui, and H. Ishida, Appl. Opt. 33, 6285 (1994). K. Yamamoto and H. Ishida, Spectrochim. Acta 50A, 2079 (1994). A. Masui, K. Yamamoto, and K. Ohta, Bunseki Kagaku 41, T49 (1991).


1293