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Langmuir 2007, 23, 9330-9334
Quasi-isotropic Analysis of Anisotropic Thin Films on Optical Waveguides Robert Horvath* and Jeremy J. Ramsden Nanotechnology Centre, Department of Materials, Cranfield UniVersity, Bedfordshire MK45 0AL, United Kingdom ReceiVed May 15, 2007. In Final Form: June 23, 2007 Thin films assembled on a substrate are often anisotropic. Nevertheless, because of experimental limitations, sufficient parameters to characterize the anisotropy, even in the simplest (and perhaps most common) case of uniaxial thin films, which are birefringent, are not usually available. This paper examines the consequences of treating them as isotropic thin films, with particular reference to their characterization via perturbation of the propagation constants (effective refractive indices) of optical waveguides. It is shown that the refractive index and geometrical thickness of a thin film thus calculated are often unrealistic (especially when the thin film is positively birefringent), but the mass per unit area may be quite precise, depending on the sign and magnitude of the birefringence.
I. Introduction Optical waveguides offer a highly sensitive way to determine the optogeometrical properties of thin films.1 Compared with other optical techniques (scanning angle reflectometry and ellipsometry) or optoelectronic techniques (surface plasmon resonance), waveguides offer substantially higher sensitivity.2,3 Furthermore, two orthogonal polarizations are available for every mode (compared with only one for surface plasmon resonance), hence, doubling the number of parameters available. Quasiuniaxial thin films are very common. The anisotropy may arise through the presence of two or more components with individually distinct refractive indices, structured in layers or columns (form anisotropy), or through the nonrandom orientation of individual molecules self-assembled into a thin film. In such cases, at least three parameters are required to fully characterize the filmsthe geometrical thickness and the ordinary and extraordinary refractive indices, whereas for the more familiar, but not necessarily more common, isotropic thin films just two parameters, geometrical thickness and refractive index, are required. Thin films assembled from complicated precursors, e.g., proteins, possibly forming multilayers, are very likely to show a nonuniform refractive index profile perpendicular to the plane of the film. This nonuniformity can in addition be characterized by Mann’s F factor.4 Although therefore three or four parameters are required to characterize the most common kind of films, at the same time, the so-called monomode optical waveguides, just thick enough to allow the propagation of two zeroth order modes, are the most sensitive to perturbations in the adlayer.5 Furthermore, interferometric methods of measuring the waveguide parameters can, in practice, yield at most two modes (when the two polarizations are measured separately, rather than allowed to interfere with one another). In some special cases, the thickness of an anisotropic layer may be determined independently (e.g., lipid bilayers6 or * To whom correspondence should be addressed. E-mail: r.horvath@ cranfield.ac.uk. (1) Swalen, J. D.; Tacke, M.; Santo, R.; Rieckhoff, K. E.; Fischer, J. HelV. Chim. Acta 1978, 61, 960. (2) Lukosz, W. Sens. Actuators B 1995, 29, 37. (3) Mann, E. K. Langmuir 2001, 17, 5872. (4) Mann, E. K.; Heinrich, L.; Voegel, J. C.; Schaaf, P. J. Chem. Phys. 1996, 105, 6082. (5) Tiefenthaler, K.; Lukosz, W. J. Opt. Soc. Am. 1989, 6, 209.
polyelectrolyte multilayers7), hence allowing the measured propagation constants of two modes to be used to determine the ordinary and extraordinary refractive indices, but in many other systems, e.g., in thin films self-assembled in situ from particles in suspension during the waveguide propagation constant measurement, this is not practicable. In such cases, thicker waveguides allowing the propagation of both zeroth- and firstorder modes, allowing four parameters can be determined, should be used. However, this might not be practicable for various reasons, e.g., technically it might be more difficult to make thicker waveguides, in any case the sensitivity is somewhat less,5 and using Mach-Zehnder8 or dual polarization interferometry9 more modes are anyway inaccessible. The purpose of this paper is to push the analysis of the two parameters obtainable from monomode waveguides further than hitherto and investigate what information about an anisotropic thin film can be inferred when only two modes can be measured. The errors arising when the two parameters are used to determine particle coverage or total added mass are also revealed, when for example the thin film is assembled from particles in suspension and the uniform thin-film approximation is applied. The structure of this paper is as follows. First the mode equations for a waveguide coated by a uniaxial thin film are presented, and the perturbation of the propagation constants (expressed throughout the paper as effective refractive indices), due to the presence of a thin film, is analyzed. New quasiparameters of the adlayer are derived assuming the thin film to be isotropic. The analysis of these quasiparameters may be used to deduce the presence of anisotropy under certain conditions. We also analyze the errors in the particle coverage or deposited mass that arise when attempting to calculate it for an anisotropic thin film assuming isotropy.
II. Mode Equations for a Waveguide Coated by a Uniaxial Thin Film In the following, we suppose that a thin uniaxial adlayer with ordinary and extraordinary refractive indices no, ne, and thickness (6) Ramsden, J. J. Phil. Mag. B 1999, 79, 381. (7) Ramsden, J. J.; Lvov, Y. M.; Decher, G. Thin Solid Films 1995, 254, 246. (8) Ramsden, J. J. Front. Drug Des. DiscoVery 2006, 2, 211. (9) Freeman, N. In Proteins at Solid-Liquid Interfaces; Dejardin, Ph., Ed.; Springer-Verlag: Heidelberg, 2006; p 75. (10) Picart, C.; Gergely, C.; Arntz, Y.; Voegel, J. C.; Schaaf, P.; Cuisinier, F. J. G.; Senger, B. Biosens. Bioelectron. 2004, 20, 553.
10.1021/la701405n CCC: $37.00 © 2007 American Chemical Society Published on Web 08/07/2007
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By implicit partial differentiation of eq 1 we have5,11
[
nF2 - N2 no2 - nC2 (N/nC)2 + (N/n˜ 2) - 1 ∂N/∂dA ) Ndeff n 2 - n 2 (N/n )2 + (N/n 2) - 1 F C C F
]
F
(5)
where deff is the so-called effective thickness of the mode, i.e.
deff ) dF +
∑
∆zF,J
(6)
J)S,C
Figure 1. Waveguide structure (not drawn to scale) used in the analysis. The layers are (top to bottom): C, cover; A, adlayer; F, waveguiding film; and S, substrate. The refractive indices are denoted by n and the geometrical thicknesses by d.
where the parameters
dA is deposited on the surface of a waveguide (see Figure 1.). We assume that the extraordinary axis is perpendicular to the waveguiding film, i.e., along the z axis. We suppose that the adlayer is thin, i.e., dA , λ, where λ is the vacuum wavelength of the light (in typical cases, we suggest that dA should not exceed 30-50 nm for λ ) 600-650 nm, but note that the limit of applicability of the thin adlayer model strongly depends on the refractive index of the adlayer; for a thorough analysis of the thin adlayer limit the reader should consult ref 10). Then, the 4-layer mode equation that determines the modes’ effective refractive indices, N, can be linearized5,11
are the modes’ penetration depths into the substrate (J ) S) and into the cover medium (J ) S). In the case of homogeneous and isotropic adlayers with refractive index nA, obviously no ) ne ) n˜ ) nA. If only two parameters (typically effective refractive indices) are determined experimentally, it is evidently impossible to calculate the full set of opto-geometrical parameters characterizing an anisotropic adlayer, for which no, ne, and dA would be required if the adlayer is uniaxial. Very often, however, using the most sensitive so-called “monomode” waveguides that only support the zeroth-order modes (m ) 0), or waveguide interferometry, only two experimental parameters are available from the waveguide experiment and there is no other independently obtained information about the adlayer. In many published reports, any possible anisotropy is simply neglected, and values for the adlayer thickness and refractive index are calculated by assuming that the thin adlayer is isotropic.10,12-14 In the following section, we investigate the relation between the parameters determined using that assumption and the true opto-geometrical parameters of a uniaxial adlayer, and determine the errors in the calculated opto-geometrical parameters of the adlayer arising from neglect of the actual anisotropy.
πm =
2π n 2 - N2(dF + ∆dF) λ x F
[( )
nF arctan nC
2F
xN2 - nC2 xnF2 - N2
where
∆dF )
no2 - nC2 nF2 - nC2
] [( ) nF - arctan nS
[
( ) () () ()
[] 1-
n˜ ) no 2
2
1-
]
xN2 - nS2 xnF2 - N2
]
F N2 -1 n˜ dA N 2 -1 nF
N 2 + nC N 2 + nC
and
2F
(1)
(2)
nC2
2πxN - nJ [(N/nF)2 + (N/nJ2) - 1]F
(7)
2
In the “pseudo-isotropic” analysis of anisotropic adlayers, the experimentally measured effective refractive index shifts due to the presence of the adlayer are equated to the effective index shifts originating from an isotropic adlayer with thickness and refractive index d˜ A and n˜ A respectively, i.e.
(3)
2
ne
and m ) 0, 1, 2, ... is the mode order, and F ) 0 for the transverse electric (TE) modes and F ) 1 for the transverse magnetic (TM) modes. Notice that the TE mode equation depends on the adlayer thickness and ordinary refractive index only, but the TM mode equation depends on both the ordinary and extraordinary refractive indices of the film.
III. Shifts in Effective Refractive Index Due to the Presence of a Thin Uniaxial Adlayer For thin adlayers (dA , λ), the effective refractive index shifts due to the presence of the adlayer can be approximated by
∆N ) (∂N/∂dA)dA
λ 2
IV. Treating the Uniaxial Adlayer as an Isotropic Film
nC2 no2
∆zF,J )
(4)
(11) Horvath, R.; Fricsovszky, G.; Papp, E. Biosens. Bioelectron. 2003, 18, 415.
∆N(no,ne,dA) ) ∆N(n˜ A,d˜ A)
(8)
Since the effective refractive index shifts can be expressed for both the TE and TM modes, we have two independent equations. Using eqs 4 and 5 for the TE mode, we get
(no2 - nC2)dA ) (n˜ A2 - nC2)d˜ A
(9a)
and for the TM mode
(no2 - nC2)[(NTM/nC)2 + (NTM/n˜ 2) - 1]dA ) (n˜ A2 - nC2)[(NTM/nC)2 + (NTM/n˜ A2) - 1]d˜ A (9b) This system of eqs 9a and 9b can be solved for n˜ A2 and d˜ A, yielding the following analytical formulas: (12) Voros, J.; Ramsden, J. J.; Csucs, G.; Szendro, I.; De Paul, S. M.; Textor, M.; Spencer, N. D. Biomaterials 2002, 23, 3699. (13) Calonder, C.; Van Tassel, P. R. Langmuir 2001, 17, 4392. (14) Horvath, R.; Pedersen, H. C.; Cuisinier, F. J. G. Appl. Phys. Lett. 2006, 88, 111102.
[]
9332 Langmuir, Vol. 23, No. 18, 2007
n˜ A2 )
ne2(no2 - nC2) (ne2 - nC2)
and
d˜ A )
1-
) no2
no2 - nC2 n˜ A2 - nC2
1-
HorVath and Ramsden
nC2 no2
nC2
) n˜ 2
(10)
ne2
dA
(11)
Therefore, the adlayer refractive index calculated from the pseudoisotropic analysis depends on the ordinary and extraordinary refractive indices of the adlayer but not on the adlayer thickness. The calculated adlayer thickness, however, depends on the full set of opto-geometrical parameters of the adlayer. It is important to emphasize that eqs 10 and 11 are independent of the effective refractive index N of the mode. This means that the behavior of n˜ A2 and d˜ A can be studied in general, independently of the waveguide design used in the experiments.
V. Application to Particle Counting When the adlayer is assembled by the addition of discrete particles dissolved or suspended in the bulk phase to the surface of the waveguide, it is of great practical importance to be able to determine the number of particles (or their mass) deposited per unit area. Mann has previously shown that provided the particle diameter rA is much less than the wavelength of the light (i.e. dA , λ), the adlayer can be treated as if it were uniform.15 The number of particles (or mass) per unit area can then be calculated from the measured adlayer thickness and refractive index applicable to homogeneous and isotropic adlayers.5,12 Using eqs 10 and 11 the mass per unit area takes the form
M ˜)
n˜ A2 - nC2 no2 - nC2 d˜ A ) dA d/dc d/dc
(12)
where d/dc is the dielectric constant increment of the solution. This parameter is usually independently determined by measuring the refractive indices of bulk solutions containing different concentrations of the particles.16 We note that the above mass depends only on the ordinary index of the adlayer but is independent from the extraordinary refractive index. The same result can be deduced from analyzing the adlayer using optical invariants.3 In the analytical calculations, it is easier to deal with the squares of the refractive indices (i.e., the dielectric constants). However, since practically it is usually the refractive indices of the bulk solutions that are measured, the refractive index increment dn/dc is usually used.16 In this case, the mass per unit area takes the form
n˜ A - nC no2 - nC2 d d˜ ) M ˜ RI ) dn/dc A (dn/dc)(n˜ A + nC) A
(13)
This change results in a significant difference: in contrast to eq 12, the mass per unit area calculated using dn/dc depends on the full set of opto-geometrical parameters of the uniaxial adlayer. For isotropic adlayers one can use either definition. As a rule, the relationship between solution refractive index and bulk particle (15) Mann, E. K.; Heinrich, L.; Schaaf, P. Langmuir 1997, 13, 4906. (16) Ball, V.; Ramsden, J. J. Biopolymers 1998, 46, 489. (17) Lukosz, W. Biosens. Bioelectron. 1997, 12, 175.
Figure 2. Solution refractive index and solution dielectric constant as a function of particle concentration, with n(c ) 0) ) 1.331.
concentration is linear, and the relationship between the solution dielectric constant and the bulk particle concentration is linear to a very good approximation, as shown in Figure 2, in which a value of dn/dc ) 0.183 cm3/g, typical for many serum proteins in moderately concentrated sodium chloride solutions,16 was used to calculate the solution refractive index as a function of concentration according to n ) n(c ) 0) + c(dn/dc). The dielectric constant of the solution was then calculated according to ) n2. The deviation from linearity over this concentration range is barely perceptible; the maximum concentration is typical of that found in an adsorbed protein layer (∼1.38 g/cm3). In the next section, we investigate the errors in the optogeometrical parameters arising from the pseudoisotropic treatment of anisotropic adlayers.
VI. Errors in the Calculated Adlayer Refractive Index, Thickness, and Mass Due to the Pseudoisotropic Treatment of Uniaxial Adlayers Adlayer Refractive Index. First of all, we investigate the errors introduced in the refractive index of the adlayer due to neglecting the anisotropy. The volume-averaged refractive index of the uniaxial adlayer is
nA )
x
2no2 + ne2 3
(14)
Therefore, the error in adlayer refractive index due to the pseudoisotropic treatment is δnA ) nA - n˜ A, and the relative error is
n˜ A δnA )1nA nA
(15)
Looking at eq 10 (see also ref 17), it is trivial to see that in the case of a negative uniaxial film (no > ne) eq 16 holds
n˜ A > no > nA > ne
(16)
For positive uniaxial adlayers (no < ne), we find
n˜ A < no < nA < ne
(17)
This means that δnA is negative for no > ne and positive for no < ne. The adlayer refractive index calculated from the pseudoisotropic treatment over- or under-estimates the averaged refractive index of the adlayer (nA). Figure 3a shows the relative error δnA/nA in adlayer refractive index when the extraordinary refractive index of the adlayer is fixed at 1.36, 1.40, or 1.45 and the birefringence (no - ne) is tuned from -0.01 to +0.01. In this range the relative errors can be as high as 15% for the lowest refractive index film (typical of a highly hydrated protein). In general the relative error increases for decreasing extraordinary refractive index, e.g., for less dense films.
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Langmuir, Vol. 23, No. 18, 2007 9333
Figure 3. Relative errors in adlayer refractive index (a) and adlayer refractive index (b) using the homogeneous and isotropic analysis for anisotropic films with fixed extraordinary refractive indices, using nC ) 1.331 in the calculations. Figure 5. Relative error in adlayer thickness (a) and calculated adlayer thickness (b) using the homogeneous and isotropic treatment for a 1 nm thick anisotropic film with anisotropy tuned from -0.01 to +0.01 and nC ) 1.331.
thickness. The origin of this error is easily seen from eq 11: the adlayer thickness goes to infinity at n˜ A ) nC. The error and relative error in adlayer thickness can be written as
(
Figure 4. Adlayer birefringence required for the apparent refractive index calculated according to the pseudo-isotropic approximation to be equal to the cover medium refractive index, as a function of the extraordinary refractive index of the adlayer, using nC ) 1.331.
It is important to note that in the case of no < ne the relative errors can be so large that the adlayer refractive index calculated using the pseudoisotropic approximation (n˜ A) may be equal to or even less than that of the cover medium (nC). This is further emphasized in Figure 3b, where n˜ A is plotted as a function of the birefringence for the same three values for the extraordinary refractive index. Since the ordinary refractive index of the adlayer is in almost all practical cases greater than the cover medium index nC, n˜ A cannot be treated in any sense as the “averaged” refractive index of the adlayer. Neglecting the anisotropy leads to an unphysical result under this set of circumstances. Looking at eq 10, it is easy to see that the condition n˜ A ) nC can only be fulfilled if no < ne. Figure 4 shows the values of the birefringence as a function of the extraordinary refractive index required to fulfill this condition. We emphasize that it can occur at typical birefringences likely to be encountered, i.e., not in extreme or unusual situations. Adlayer Thickness. The above-mentioned condition n˜ A ) nC introduces even more pronounced errors in the calculated adlayer
δdA ) dA - d˜ A ) 1 -
no2 - nC2 n˜ A2 - nC2
)
dA
(18)
and
no2 - nC2 δdA )1- 2 dA n˜ - n 2 A
(19)
C
respectively. The relative error is plotted as a function of the birefringence in Figure 5a. Three well-separated zones can be observed. For no - ne > 0, the relative error is positive but less than 100% (this is easily deduced from eq 19 using eq 16). In the zone corresponding to no - ne < 0 and n˜ A > nC, the relative error is negative. In this case the calculated thickness overestimates the real thickness. At n˜ A ) nC, the relative error has a singularity (see eqs 18 and 19) and upon decreasing the anisotropy further the relative error changes its sign and became positive again, with relative errors larger than 100%. Close to the singularity, the calculated thickness can deviate very strongly from the real thickness. This is further emphasized in Figure 5b, where the adlayer thickness calculated from the homogeneous analysis is plotted for an adlayer with 1 nm thickness and anisotropy
9334 Langmuir, Vol. 23, No. 18, 2007
HorVath and Ramsden
MRI )
(nA - nC)dA dn/dc
(23)
Therefore, the pseudo-isotropic analysis implies an error of
no2 - nC2 n˜ A + nC dA dn/dc
nA - nC ˜ RI ) δMRI ) MRI - M
(24)
In this case the relative error is
no2 - nC2 δMRI )1MRI (nA - nC)(n˜ A + nC) Figure 6. Relative errors in the adlayer masses per unit area using the pseudo-isotropic analysis for extraordinary refractive indices of 1.36, 1.4, and 1.45 while the anisotropy is tuned from -0.01 to +0.01, with nC ) 1.331. Two types of mass definitions were used, solid lines: mass per unit area calculated using adlayer refractive index; dashed lines: mass per unit area calculated using the adlayer dielectric constant.
between -0.01 and +0.01. We note that the condition n˜ A ) nC is fulfilled at higher anisotropy values for denser films (cf. Figures 4 and 5). Mass per Unit Area. The analytical formulas derived above reveal that the n˜ A - nC term that causes the thickness to diverge at n˜ A ) nC cancels out in both definitions of adlayer mass per unit area (see eqs 12 and 13). This suggests that the errors introduced in refractive index and thickness are in some way able to compensate each other (as inferred by Lukosz using numerical simulations18). At the same time, eqs 12 and 13 differ in some important ways, which are analyzed in detail below. Using the mass definition based on dielectric constants (eq 12), the real total mass per unit area takes the form
2no2 + ne2 - 3nC2 M ) dA 3d/dc
(20)
The error in mass per surface area is
ne2 - no2 ˜) δM ) M - M d 3d/dc A
(21)
and the relative error takes the form
ne2 - no2 δM ) M 2no2 + ne2 - 3nC2
(22)
This relative error is plotted in Figure 6 for extraordinary refractive indices 1.36, 1.4, and 1.45 while tuning the birefringence from -0.01 to +0.01. It is important to note that this relative error with a given birefringence strongly depends on the mean refractive index of the film. For a film with a low refractive index of 1.36 (e.g., a highly hydrated protein film deposited from aqueous solution), the relative error can be larger than 6%, if the birefringence is 0.06. Using the other mass definition based on refractive indices (eq 13), the real total mass per unit area takes the form (18) Clerc, D.; Lukosz, W. Biosens. Bioelectron. 1997, 12, 185.
(25)
which is smaller than in the other case, see Figure 6.
VII. Conclusions Interpreting the results from the perturbation of the propagation constants (effective refractive indices) of optical waveguides due to the presence of a thin film of unknown structure on the waveguide surface as if the film were isotropic leads to errors in the mean refractive index n and geometrical thickness d of the film, and the adlayer mass per unit area calculated from n and d. Considering uniaxial films, when the birefringence is negative no > ne (which can arise from a structure of layers of one substance interleaved with a substance of a different refractive index),19 the mean refractive index is always overestimated and the geometrical thickness is underestimated, typically by 10% or more. The mass per unit area on the other hand benefits from some fortuitous cancellation of errors and may be in error by less than 1%. When on the other hand the birefringence is positive no < ne (which can arise from a structure of columns of one substance embedded in a substance of different refractive index), the mean refractive index is underestimated, and the geometrical thickness can be over- or under-estimated, by up to several 100%, depending on the refractive index of the cover medium and the actual (ordinary and extraordinary) refractive indices of the adlayer. Furthermore, the mass per unit area is subject to greater errors than in the case of negatively birefringent films. Hence, caution is needed when interpreting the results from investigations using optical waveguides in order to determine the opto-geometrical parameters of thin films. Ideally of course sufficient parameters should be determined, as is of course perfectly possible with the technique,3 but if for any practical reason that is impossible, or impossible with adequate sensitivity, the approach adopted in this paper may be applied. By proceeding with caution along the lines outlined in this paper acceptable results may nevertheless be obtained, at least as far as the mass per unit area of the film is concerned, even in the absence of independently determined missing parameters. Acknowledgment. R.H. gratefully acknowledges the award of a Marie Curie Fellowship by the European Commission. LA701405N (19) Born, M; Wolf, E. Principles of Optics, 7th ed.; Cambridge Univ. Press: New York, 1999. (20) Horvath, R.; Kerekgyarto, T.; Csucs, G.; Gaspar, S.; Illyes, P.; Ronto, G.; Papp, E. Biosens. Bioelectron. 2001, 16, 17.
