Frustrated Total Internal Reflection from Thin-Layer Structures with a Metal Film

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1 ISSN 3-4X, Optics and Spectroscopy, 29, Vol. 16, No. 5, pp Pleiades Publishing, Ltd., 29. Original ussian Text N.D. Goldina, 29, published in Optika i Spektroskopiya, 29, Vol. 16, No. 5, pp PHYSICAL OPTICS Frustrated Total Internal eflection from Thin-Layer Structures with a Metal Film N. D. Goldina Institute of Laser Physics, Siberian Branch, ussian Academy of Sciences, Novosibirsk, 639 ussia ngold@laser.nsc.ru eceived September 25, 28; in final form, December 29, 28 Abstract Formulas of the model of a conducting surface for calculating the reflection coefficient and phase shift of light reflected from a thin metal film at the interface between two dielectrics under the conditions of frustrated total internal reflection are analyzed. The effect of additional dielectric layers is discussed. PACS numbers: Bs DOI: /S34X9521X INTODUCTION In optics, there are a number of problems related to multilayer structures containing thin metal films. ecently, these structures have found applications in biosensors with the use of frustrated total internal reflection (FTI) [1 5]. The FTI occurs if a thin metal film is located at the interface between two dielectric media. The refractive index of the initial medium n 1 is higher than the refractive index of the final medium n 2. The high sensitivity of the angular or spectral dependences of the amplitude (or the phase) of a reflected wave linearly polarized in the plane of incidence to small variations in the refractive index of biochemical media allows one to achieve a low ( ) threshold for detecting these variations [1]. In this study, we analyze the angular characteristics of the reflection coefficient of a thin-layer structure with one metal film in the case of FTI. The effect of additional dielectric layers is considered. The analysis is performed in terms of the Fresnel theory using the model of a complex conducting surface for a thin metal film [6]. BASIC FOMULAS OF THE CONDUCTING- SUFACE MODEL (CSM) An inhomogeneous wave refracted at an angle exceeding the TI angle, θ > θ cr, where θ cr = arcsinn 2 /n 1, penetrates a medium to be analyzed to a depth on the order of the wavelength. The complex reflection coefficient demonstrates a strong dependence on the refractive index of the second medium near θ cr in the case of the choice of the appropriate metal film parameters, namely, its optical constants n and k and thickness d. Under the FTI conditions for a thin metal film at the interface between two nonabsorbing dielectrics, the absorption of the energy of the incident wave is related only to the boundary conditions. In [6], the use of two parameters ξ' and ξ'' to characterize thin (d λ) metal films lead to a good agreement between calculated and experimental data for the normal incidence. The parameters ξ' = 2nkγ and ξ'' = (n 2 k 2 )γ, where γ = 2πd/λ, are the active and reactive components of the complex surface conductivity ξ. The reflection and transmission coefficients of the metal film, which is considered to be a conducting surface at the interface between two dielectric media, can be found from approximate boundary conditions by the ordinary method of equating the tangent components of the strengths of the electric and magnetic fields. In this study, we consider the model of complex conducting surface for an oblique incidence of light on the absorbing film. We restrict our consideration to the case of plane harmonic waves and omit the exponential dependence of the amplitudes on time. For the amplitudes of the reflected light, for two orthogonal polarizations, we can write n r 1 cosθ 1 n 2 cosθ 2 ξ s = n 1 cosθ 1 + n 2 cosθ 2 + ξ, r p = ( η 1 η 2 ξ)/ ( η 1 + η 2 + ξ). (1) (2) Here, we used the notation η 1 = n 1 /cosθ 1 and η 2 = n 2 /cosθ 2. If n 1 > n 2, then, at an angle θ 1 > θ cr (where θ cr is the angle of FTI), the angle θ 2 becomes complex. From (1), (2), the intensity of the reflected light = rr* can be written as 748

2 FUSTATED TOTAL INTENAL EFLECTION 749 [ n 1 cosθ 1 e( n 2 cosθ 2 ) ξ' ] 2 + [ Im( n 2 cosθ 2 ) + ξ'' ] 2 s = , [ n 1 cosθ 1 + e( n 2 cosθ 2 ) + ξ' ] 2 + [ Im( n 2 cosθ 2 ) + ξ'' ] 2 (3) ( η 1 e η 2 ξ' ) 2 + ( Im η 2 + ξ'' ) 2 p = (4) ( η 1 + e η 2 + ξ' ) 2 + ( Im η 2 + ξ'' ) 2 For the phases of the reflected light ψ = arctan(imr/er), we obtain the following expressions from (1) and (2): 2n ψ 1 cosθ 1 [ Im( n 2 cosθ) + ξ'' ] s = arctan ( n 1 cosθ 1 ) 2 [ e( n 2 cosθ 2 ) + ξ' ] 2 [ Im( n 2 cosθ 2 ) + ξ'' ] 2, 2η ψ 1 ( Im η 2 + ξ'' ) p = arctan η 1 ( e η 2 + ξ' ) 2 ( Imη 2 + ξ'' ) 2. (5) (6) EXPEIMENTAL VALIDATION The applicability of the model of a complex conducting surface for oblique incidence can be proved by experimental data. We measured the refractive index of a nickel film on a quartz substrate for two orthogonal components of linearly polarized light (λ =.63 µm) in the range of angles from to 85. A beam of a He Ne laser with a diameter of 2 mm irradiated a sample (with the film diameter of 16 mm) placed on a special table. To the same table at a distance of 25 cm in front of the sample, we attached a photodiode, which was rotated simultaneously with the sample by a double angle, so that the reflected beam does not shift with respect the detector aperture. This allowed us to rapidly perform measurements, thus minimizing the error related to the variations in the power of the radiation source. In addition, this setup provides the possibility to fix the pseudo-brewster angle, at which p reaches a minimum. The angles were measured by a limb with an error not exceeding 1. The relative error in the measurement of the reflection coefficient was.5% for the angles of incidence smaller than <3 and, about 2%, for angles in the range 3 7. At larger angles of incidence (above 8 ), the measurement accuracy decreases due to a rapidly increasing area of the illuminated sample surface and to alignment errors. The measured s and p for the nickel film are shown by dots in Fig. 1. This figure also presents the angular dependences of s and p calculated by the homogeneous-layer model (HLM) with parameters n, k, and d and by the CSM by formulas (3), (4), which are simplified for the air metal quartz structure because Imη 2 =. The ξ' and ξ'' parameters were measured at the normal incidence of light by the method described in [6] and were found to be ξ' = 1.44 and ξ'' =.24. The thickness of the Ni film d = 1 nm was determined using a Fizeau interferometer illuminated by radiation of a He Ne laser. The n and k parameters were calculated from the formulas for ξ' and ξ'' and proved to be n = 2.88 and k = 2.5. In Fig. 1, the results of calculation by traditional formulas in terms of HLM are shown by solid lines, and the results obtained using approximate formulas (2) and (3) of the CSM are shown by the dashed lines. For the entire range of angles, the coincidence of the curves can be considered to be very good. The experimental points fall well on the calculated curves, except for high values of s for angles exceeding >7 because of a large experimental error. As a whole, we can conclude that the CSM can be used to calculate the angular characteristics of absorbing films. DEPENDENCES p (θ) AND ψ p (θ) AT n 1 > n 2 The presence of a metal film at the interface between two dielectrics with n 1 > n 2 leads to the FTI. Near θ cr, the reflection coefficient p and the phase ψ p undergo sharp changes, which are very sensitive to the variations in the refractive index of the external medium. Biosensors usually contain a silver film of a certain thickness (~5 nm) deposited onto a prism made of a Fig. 1. Angular dependences of p and s for the nickel film. The solid and dashed curves correspond to calculations by the HLM and CSM, respectively. Experimental data are shown by points. s p OPTICS AND SPECTOSCOPY Vol. 16 No. 5 29

3 75 GOLDINA p.8 (a).8 (b) Fig. 2. Dependences p (θ) calculated by (solid curves) the HLM and (dashed curves) the CSM for (a) silver and (b) aluminum. material with a high refractive index, namely, quartz (n 1 = 1.46), TF-5 (n 1 = 1.755), and SF-11 (n 1 = 1.785) [1 4]. If the external medium is an aqueous solution (n 2 ~ 1.33), the angle θ cr = arcsin( n 2 /n 1 ) is close to 65.6, 49.2, and 48.1, respectively. The film thickness is chosen so that p near θ cr reaches a deep minimum ( p ~ ), the position of which depends strongly on the angle of incidence and the wavelength. Based on the experimental validation of the CSM calculations for the metal film at the air quartz interface, let us perform calculations for the case of FTI using the CSM and compare them with the HLM calculations. We will mainly consider the p (θ) and ψ p (θ) dependences in order to determine their sensitivity to variations in the refractive index of the external medium. In this study, we analyze the behavior of p near θ cr for thin films of two metals, Al and Ag. The parameters of the Ag film are taken from our experimental work [6], ξ' = 1.54 and ξ'' =.6 for λ = 63 nm. The optical constants recalculated for a homogeneous layer with d = 1 nm are n = 1.3 and k = The parameters of the Al film are taken from the work by Abeles [7], n = 1.3, k = 7.11, d = 12 nm, and λ = 65 nm. These parameters correspond to ξ' = 2.14 and ξ'' = 5.67 in the CSM calculations. Figure 2 shows the dependences p (θ) for Ag and Al films at the quartz air interface. It is clearly seen that the calculations for these films by the two models, HLM (solid curves) and CSM (dashed lines), coincide well with each other. Since formula (4) has a relatively transparent form, the CSM allows us to estimate the parameters at which p changes considerably with varying angles. Let us demonstrate this based on the example of the aluminum film (Fig. 2b). On a large scale, the angular range in the vicinity of θ cr is reproduced in Fig. 3 and 4. The dependence p (θ) is shown by the upper solid line. The maximum p = 97.7% is reached at θ cr = arcsin(1./1.46) = 43.23, when Imη 2 is maximum and eη 2 =. The minimum p = 2% is observed at θ = 44.6, when the condition (Imη 2 + ξ'') = is fulfilled. The thin solid line in Fig. 4 shows a portion of the angular dependence of the sum (Imη 2 + ξ''). It is seen that this line intersects the zero line at θ = θ min. The depth of the minimum depends on the value of ξ', and pmin tends to zero when the matching condition ξ' = η 1 is fulfilled. In our case, this condition is well satisfied since η 1 = for angles of 4 45 and ξ' = 2.14 for Al. An important role in the formation of a narrow minimum is played by the negative and large value of ξ''; i.e., the ratio k/n must be high. The value of ξ'' in Fig. 2a is small (.6) and the curves show no narrow dip because the minimum of p is well spaced from the maximum, and Imη 2 decreases not too sharply. For the Al film, ξ'' = 5.67 and the condition Imη 2 + ξ'' = is fulfilled on the steep part of the Imη 2 (θ) curve, which results in the narrow dip in the dependence p (θ). It is of interest to study the change in the phase of the reflected light. Figure 3 shows the dependences ψ p (θ) for the HLM (solid lines) and the CSM (dashed lines); the curves are similar in shape, but the curves corresponding to the CSM are shifted to smaller angles from the curves calculated by the HLM. The curves in Fig. 3 are calculated for the Al film with quartz and air as dielectric media. As follows from formula (6), in the vicinity of angles exceeding θ cr, an important role is played by the term (Imη 2 + ξ''). For this angular range, i.e., at eη 2 =, formula (6) can be rewritten in the simplified form ψ p arctan( 2η 1 /Imη ( 2 + ξ'' )), (7) assuming that the matching condition ξ' = η 1 is fulfilled. For the case considered, ξ' η 1 2. The dashed-and-dotted line in Fig. 3 corresponds to the ψ p calculated by formula (7). It coincides well with the dashed line, except for the region of the extrema, which are smoothed in the case of curves calculated by the CSM. As follows from the simplified model (for- OPTICS AND SPECTOSCOPY Vol. 16 No. 5 29

4 FUSTATED TOTAL INTENAL EFLECTION 751 p ψ p ψ t Fig. 4. Calculated dependences of (solid curve) p and (dashed curve) s on the thickness of a dielectric layer deposited on the Al film. EFFECT OF DIELECTIC LAYES Fig. 3. Dependences p (θ) and ψ p (θ) for Al calculated (solid curves) by the HLM, (dashed curves) by the CSM, and (dashed-and -dotted curves) by simplified formulas (7) and (8). The thin solid line shows the sharp angular dependence of the sum (Imη 2 + ξ''). mula (7)), the term (Imη 2 + ξ'') changes the sign at θ = θ min, and the phase jumps by π. In the case of the CSM, the jump is smaller. The steepest portion of the phase dependence belongs to the angular range between the extrema of p. Formula (4) can also be presented in the simplified form ( Im η 2 + ξ'' ) 2 1 p = = cos 2 ψ ( 2η 1 ) 2 + ( Im η 2 + ξ'' ) 2 1 tan 2 p. + ψ p (8) The dependence p (θ) calculated by formula (8) is shown by the dashed-and-dotted line in Fig. 3 and coincides well with the curve calculated by the CSM, i.e., by formula (4) (the dashed line). It is interesting to compare the steepness of the curves p (θ) and ψ p (θ) in this simple model. From (8), we have ψ p = arccos p, ψ p / θ = ( 1 p ) 1 p / θ. It follows from this that the steepness of ψ p is much stronger than the steepness of p in the region of the p maximum, since the coefficient K = (1 p ) 1/2 increases with increasing p. In the region of the p minimum, the steepness of ψ p and p is almost the same since K tends to unity. Dielectric layers can be placed on either side of a metal layer. Variations in the refractive index or thickness of the dielectric layer results in a great variety of the angular dependences p (θ). Figure 4 presents the calculated dependences of s and p on the thickness (in units of quarter-wave thickness t) of a dielectric layer with the refractive index n c = 2.3 located between s, p Fig. 5. Calculated angular dependences of p for the aluminum film (thick solid curve) without and (thin solid curve) with a dielectric layer with n c = 2.3 and t = 2.3. The dashed curve shows the calculated dependence s (θ) for the film with the dielectric layer n c = 1.5, t = 2.3. The dashed-anddotted line shows the curve p (θ) shifted due to a change in the refractive index of the external medium n 2. OPTICS AND SPECTOSCOPY Vol. 16 No. 5 29

5 752 GOLDINA the medium with a refractive index of 1.33 and the Al film. The calculation is fulfilled by the HLM for the critical angle for two media 1.785/1.33. From Fig. 4, it is possible to determine the optimal thickness of the dielectric layer at which the steepness of the curve p (θ) is maximum. It is noteworthy that, with the help of the dielectric layer, it is also possible to obtain curves s (θ) with a sharp minimum. To demonstrate these possibilities, Fig. 5 presents the calculated angular dependences of s and p for several variants: (a) without a dielectric layer; (b) with a dielectric layer with n c = 2.3, t = 2.3; and (c) with a dielectric layer with n c = 1.5, t = 2.3. The steepness of the curve p (θ) (thin solid line) for case (b) is stronger than for case (a) (thick solid line). In case (c), a narrow minimum is obtained for s (θ), which is shown by the dashed line. SHIFT OF CUVES WITH CHANGING EFACTIVE INDEX n 2 It remains to consider how a change in n 2 affects the extrema of p. The dashed-and-dotted line in Fig. 5 shows the curve p (θ), which is shifted to the right due to a change in n 2 by.1. For calculations, we used n 1 = and n 2 = 1.33 and The value of the shift (~.5 ) depends only on n 1 /n 2. To detect the shifts of curves by replacing the medium analyzed, it is possible to use not only the sharp minima, but also the sharp maxima, as in Fig. 2a. CONCLUSIONS The formulas obtained by the CSM are experimentally validated for the angular dependence of the refractive coefficient of a nickel film 1 nm thick at the air quartz interface. The experimental points lie well on the dependences of s (θ) and p (θ) calculated by the CSM. In addition, they also coincide with the dependences calculated by the HLM. The angular dependence of very thin (d λ) metal films under conditions of FTI at the interface between two dielectric media with n 1 > n 2 is analyzed in terms of the HLM and CSM. Based on the example of two metals, i.e., silver and aluminum, it is shown that the CSM well describes the p (θ) and ψ p (θ) dependences. This allows one to hypothetically evaluate the behavior of p and ψ p in the vicinity of the critical angle under the condition that the parameters of thin metal films are measured at the normal incidence. This is necessary for their optimal application as biosensors. It is demonstrated that, with the help of an additional dielectric layer between the metal and the external medium, the steepness of the angular dependences p (θ) and ψ p (θ) can be increased. In addition, it is possible to obtain the dependence s (θ) with a sharp minimum. EFEENCES 1. V. E. Kochergin, A. A. Beloglazov, M. V. Valeœko, and P. I. Nikitin, Kvantovaya Élektron. (Moscow) 25 (5), 457 (1998). 2. P. Lecaruyer, V. Canva, and J. olland, Appl. Opt. 46 (12), 2361 (27). 3. T. Wakamatsu and K. Saito, J. Opt. Soc. Am. B 24 (9), 237 (27). 4. W. Yuan, H. P. Ho, Y. K. Suen, S. K. Kong, and C. Lin, Appl. Opt. 46 (33), 868 (27). 5. Surface Polaritons: Electromagnetic Waves at Surfaces and Interfaces, Ed. by V. M. Agranovich and D. L. Mills (North-Holland, Amsterdam, 1982; Nauka, Moscow, 1985). 6. N. D. Goldina, M. I. Zakharov, and Yu. V. Troitskiœ, Avtometriya, No. 3, 17 (1975). 7. F. Abeles, Phys. Thin Films 6, 151 (1969). Translated by M. Basieva OPTICS AND SPECTOSCOPY Vol. 16 No. 5 29