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Degenerate long range surface modes, supported on thin nickel films
I December1996 OPTICS COM MUN ICATIONS ELSEVIER
Optics Communications132 (1996) 212-216
Degenerate long range surface modes, supported on thin nickel films E.L. Wood, J.R. Sambles, F.A. Pudonin i, V. Yakovlev 2 Thin Film Photonics Group, Departmentof Physics, Universityof Exeter, Exeter, EX4 4QL UK
Received25 January 1996;revisedversionreceived24 April 1996;accepted2 May 1996
Abstract
In this work islandised nickel films have been placed in a symmetric environment to allow optical excitation of long-range surface modes via attenuated total reflection. Comparison of modelling theory to the experimental angle-dependent refleetivity data shows that it is not possible to determine the optical permittivity of the thin nickel layer without knowledge of its thickness. The reflectivity data has a set of degenerate fits; the deduced optical permittivities for this degeneracy describing a circle when plotted as a Cole-Cole plot. This circle is naively predicted from an analytical three.media theory which ignores the coupling prism. Moreover, further investigations show that the degenerate circle is a general result of fitting refiectivity curves in the vicinity of a long range mode.
1, Introduction Long-range optical modes supported by very thin layers have been classified in a number of ways. These classifications reflect the nature of the thin layer which supports their excitation. If for example the exciting medium is metallic in character, the resonance is generally described as a long-range sufface-plasmon polariton (LRSPP) [1], whilst a dielectric with low levels of absorption may support excitation of a guided mode (GM). Similarly, if the real part of the optical permittivity is equal to zero whilst the imaginary part is large, the long-range sufface-exciton polariton (LRSEP) is excited [2]. I Lebedev Physical Institute, Russian Academyof Sciences. Leninskiiprosp, 53, Moscow117924,Russia 2 Institute of Spectroscopy, Russian Academyof Sciences, Troitzk, Moscowreg. 142092, 142092 Russia.
Such modes have momenta greater than that of polaritons in the surrounding dielectric and may be coupled in real systems using a prism to enhance the incident momentum. The mode manifests itself as a dip in the transverse magnetic (p) polarized reflectivity as the incident angle, hence momentum is scanned. In this paper we study the long range modes excited in the attenuated total reflection geometry using a thin nickel film (d ffi 2 nm) at a wavelength of 632.8 nm. It becomes clear that the angle-dependent reflectivity data does not allow a unique characterization of the system; indeed, there are an infinite number of degenerate fits to the data which give rise to a circular trace if the deduced optical permittivity is plotted in a Cole-Cole geometry [3]. The circular trace may be very closely reproduced by using simple analytical long-range mode theory [4]. This work then shows not only that, in principle, a vast range of
0030-4018/96/$12.00 Copyright© 1996ElsevierScienceB,V. All rights reserved, PI! S0030-4018(96)00334-3
213
E.L. Wood et a l . / Optics Communicatiora ~132 (1996) 212-216
1.2,
optical permittivities may produce long-range modes in thin films, but it also clearly illustrates that the optical permittivity for such a thin film may not be determined unless the thickness is known.
Theory 1.0
•
Data
0.8
2. Experimental A long-range mode on a thin film is a resonance which requires for its excitation a close to symmetric environment. In a symmetric system containing three media, where the second has a thickness much smaller than the wavelength of light, it is possible to excite surface modes on both interfaces simultaneously using light of the appropriate momentum. Since both modes have identical energy and momentum, they couple together strongly to form two new coupled surface modes which have symmetric and antisymmetric wavefunctions respectively. The difference in electric field profile determines the propagation length of the resonance and the two coupled modes are classified either as long- or short-range modes. They require different momenta for their excitation and hence may be studied separately. Long-range surface modes provide a means of supporting, on a highly absorbing film, a resonance which has a propagation length significantly larger than the equivalent resonance in a single interface system. The prediction, observation and properties of these modes are well documented [1,2,4-9]. Work by Yang et al. [9] showed that LRSEP modes may be excited in islandised silver films using visible radiation. This islandised film has optical permittivities appropriate for the excitation of the mode. In this work we investigate long-range modes supported by islandised nickel and to this end, a very thin nickel film (nominal thickness 2 nm) was prepared by sputter deposition on to a silica substrate. The nickel film was placed in a symmetric environment by immersing the free side of the film in silica indexmatching fluid which was made from suitable proportions of methyl benzoate and cycloheptane. Light of wavelength 632.8 nm was coupled into the system using a 60 ° prism of refractive index 1,8. A schematic of the experimental arrangement is shown in the inset to Fig. 1. Rpp data were obtained for a variety of coupling gaps (D), the data at optimum coupling being denoted by the crosses in Fig. !.
O r,*
0.6 Matching Fluid
11.4 [
Glass
1
0.2
0'052
3
54 Internal Angle
56
Fig. I. p-polarized reflectivity data and a typical theoretical fit to the data obtained from a thin layer of nickel in the symmetrical environment denoted by the inset to the figure.
Theoretical fits to the data were made using multilayer optical modelling theory [10,11] and a least squares minimization routine. The modelling theory is a scattering matrix formulation, which is capable of dealing with any number of arbitrarily oriented biaxial layers. In general, fitting with this routine will lead to possibly degenerate solutions, to avoid this some of the fitting parameters were determined independently, for example the refractive indices of the prism and substrate glasses were well known, while the matching fluid, which was a volatile mixture, was given a refractive index which could vary within small bounds to allow for the fact that its refractive index might have changed by some small amount between experiments. However, the thickness of the coupling gap D, the thickness of the nickel film d and the optical permittivity of the nickel film were allowed to fit as free variables. If d is not closely constrained in the fitting procedure, the algorithm allows the parameters to wander until d encounters its upper or lower bounds. For a given fixed value of d, solutions are found which yield similar values of D, ranging from 1.37 to 1.42 p,m, virtually identical matching fluid permittivities, ranging from 2.1363 to 2.1372, and similar sum of squares. However the fits give large variations in the values of the metal permittivity as d is varied. A typical fit to the data is denoted by the solid line in
E.L. Wood et aL / Optics Communications 132 (1996)212-216
214 2oj
imaginary parts respectively of the mode momenta are given by the following expressions [4]:
15
k , = k 0 8 [ / ~ I + 2 [ '~o ]
(8) + 82 - 8,8r) ~- 8~82 ]
~i 10
× 5
ki =
( - 7 + 82) ~
ko8[/282( vrd 128 i ( 8 r 2 + 8 2 -- 8 1 P r ) ~o I
0
-.
-5
_
0
Er
I
5
10
Fig. 2. The crosses show the variation of the real and imaginary optical permittivities obtained from the degenerate fits to the data in Fig. 1, whilst the solid line is produced from the three-media analytical theory.
J,
(8~ + 82) 5
(2)
where 8 t represents the real optical permittivity of matched media 1 and 3, Ao is the wavelength of the incident light, and 8 r + i 8 i is the complex permittivity of the second medium which in this case is islandised nickel. Rearranging (1) and (2) to eliminate d gives:
(8~ + 82- 8,8,) ~- 8~8i~ = 2c8,8,(8~ + 87 - 8, 8r), Fig. 1. If the variation of values obtained for the real permittivity 8 r is plotted against those obtained for the imaginary optical permittivity, et, then the resulting curve, shown by the symbols in Fig. 2, is circular. This shows that the reflectivity associated with the excitation of a long-range mode may not be used under these circumstances to determine the optical parameters of the system. Moreover. even the nature of the resonance supported remains unknown. Unless the thickness d of the thin film is known, it is not possible to say from the reflectivity traces whether the long-range mode is a LRSPP, LRSEP or simply a GM resonance. It seems at first rather surprising that the degenerate fitting describes a circle when plotted in a ColeCole form. To explain this effect it is necessary to consider the analytical theory for long range modes in a semi-infinite three media system. This is at first sight unreasonable, since the prism so strongly perturbs the mode. However bearing in mind that the prism perturbs equally all resonances with the same initial k ( = k r + i k i ) , then the initial, unperturbed complex mode momenta are the determining factors. As we see below there are an infinity of values for 8 r and ~i which give the same kr, k i provided d is chosen appropriately. The equations for the real and
(l)
(s)
where C = ( k r - ko s [/2 ) / ki. This expression further reduces to:
e,~+8~-e,~,-e,8,(C+~+C~)=O.
(4)
This represents two part-circles in the positive 8~ half-space with radii ½81[1 + (C + ~/1 + C 2 )2]1/2 and centres ~8 ' I, ½ 8 , ( c + f / + c 2 ) respectively. Although both roots of Eq. (4) are solutions, the negative root represents values of k r which are < k ~ l / 2 and so are not evanescent and which therefore are not long-range coupled surface modes. Hence hereafter we consider only the positive roots. The parameter C which is the ratio of the shift in real momentum from kie[/2 to the imaginary mode momentum determines the radius and centre of the part circle which always cuts the 8r axis at 0 and e I. Using a set of permittivities determined from the degenerate fitting, a value of C ffi 3.809 is calculated for the nickel system. The circle predicted from Eq. (4) using the three-media analytical theory with this value of C is denoted by the solid line in Fig. 2. This gives very good agreement to the optical permittivities determined from fitting the experimental data. Note that, if instead of using the mode momenta calculated from the analytical theory, the real and
E.L. Wood et al./ Optics Communications 132 (1996) 212-216
imaginary momenta are found directly from a measurement of the width and excitation angle of the long-range mode, then the value of C = 1.101 obtained in this way predicts a circle for which there is extremely poor agreement with the degenerate fit curve. This is because the coupling prism dramatically perturbs the reflectivity trace, both shifting and broadening the apparent resonance. This perturbed mode momentum does not have the same degeneracy as the unperturbed momentum. What we have done is to fit in effect to kr and k i of the unperturbed system, the modelling taking into account the prism perturbation. In the analytic theory any one of the three unknown parameters may be eliminated between Eqs. (1) and (2) (in this case we have chosen to eliminate d) hence there will always be a degeneracy between the three parameters, d, er and ~i which give identical values of kr and k i. Therefore, though we obtain a precise value for the complex wavevector of the mode, it is impossible to use reflectivity data corresponding to the excitation of a long-range mode to determine which sort of long-range mode is being excited (eg LRSPP, LRSEP or GM), or the independent optical permittivity. Some other technique must be used to establish one of these parameters independently. In order to test for the effect of the coupling prism on the values of k, and kt, a number of theoretical reflectivity traces were generated using again the multilayer optics modelling theory. Six reflectivities were generated for which the only difference was the coupling gap D, these are shown in Fig. 3a, D being varied between 1.2 ~m and 2.2 p,m. The variation in D produces very different traces, yet when submitted to the fitting procedure, once more degenerate solutions result. When these are plotted in a Cole-Cole form, the solutions coalesce on to approximately the same circle shown by the symbols in Fig. 3b. The circle predicted from the symmetric, three-media analytical approach is shown by the solid line, and once again the agreement is good. A similar result is obtained by varying d, generating reflectivity curves and obtaining degenerate fits. Once again there is good agreement between successive degenerate fits, and the predictions from the analytical three-media theory. Clearly there are limits to the dimensions of degenerate circles which it is possible to find. To examine this, theoretical reflectivity curves are gen-
"[
215
(a)
1.0
r
0.8
> o m
o Q li=
0.6
0.4
0.2
o
I
|
0"53.0
|
5&5
I
i
54.0
I
I
t
54.5
55.0
Angle 20
(b)
°,
:,=,,°
0
'
" -5
" ~ 0 6r
"IV ' 5
'
' 10
Fig. 3. (a) Six reflectivities generated from multilayer opiical modelling theory with values of D which vary from 1.2 I~m to 2.2 itm, where the wavelength of the incident light is 632.8 rim. the thickness of the nickel is 5.5 nm and its optical permittivity is 0.01 + 18.0i, (b) Symbols denote the results of degenerate fitting to the six datasets shown in (a). The solid line represents the three-media analytical theory.
erated with optical permittivity values which produce degenerate circles of different size. In each case the coupling gap D and the thickness d for the modelled reflectivity are kept constant, and the parameters allowed to vary in the fit are simply the optical permittivities of the thin film, and D. The degenerate fits obtained are depicted by the crosses shown in
216
E.L. Wo(~ etaL/ Optics Communications 132 f1996) 212-216 40-
30
t' i 20'
.
~-~"
"
.
•
1o
o
-20
-10
0
t0
20
Er
Fig. 4. The symbols show the optical permittivities obtained from fitting to theory, seven theorttically generated datasets whose input parameters art chosen to give a variety of degenerate circle sizes. The solid lines are the circles predicted from the thrte-media theory.
dent characterization of t,e film requires the use of an effective medium theory). The form of the fit degeneracy is interesting in that it describes a circle when ploted as a Cole-Cole plot. This circle is predicted from an analytical three-media theo W modelling the unperturbed mode momentum, ignoring the influence of the coupling prism. This is because the fit to the data will be the same for a range of film thicknesses and permittivities provided they give the same complex mode momenta, since to first order the coupling prism perturbs the mode momenta by the same amount regardless of the parameters which cause it. Further investigations show that the degenerate circle is a general result of fitting reflectivity curves in the vicinity of a long range mode. They indicate that a large range of optical permittivities will provide, for a thin film, coupled modes which have long propagation lengths.
Acknowledgements Fig. 4 whilst in each case the solid lines represent the circle predicted by the analytical theory. The fits obtained all had sums of squares which were within acceptable limits, but it is worth remarking that the smaller the radius of the degenerate fit circle in epsilon space, the broader and shallower the mode, and the more impressive the fit. However for the largest value of C ( - 7 ) , the fitting degeneracy begins to disappear and for this reason the fitted ~r and ~ values no longer describe a circle.
3. Conclusions In this work islandised nickel films have been placed in a symmetric environment to allow coupling to long-range surface modes. Comparison of theory to the experimental data shows that it is not possible to optically characterize the optical permittivity of this system without knowledge of the thickness of the islandised nickel layer. (In sltch a layer however the determination of the optical permittivity and thickness becomes complicated since any indepen-
The authors are grateful to EPSRC and DRA Malvern for their support of this work.
References [I] D. Sarid, Phys. Rev. Lett. 47 (1981) 1927. [2] Fuzi Yang, J.R. Sambles and G.W. Bradberry, Phys. Rev. Lctt, 64 (1990) 559.
[3] K.S. Cole and R.H. Cole, J. Chem. Phys. 9 (1941) 341. [4] Fuzi Yang, LR. Sambles and O.W. Bradberry, Phys. Rev, B 44 (1991) 5855. [5] S. Hayashi, T. Yamada and H. Kanamori, Optics Comm. 36 (1981) 195. [6] Y. Kuwamura, M. Fukuri and O. Tada, J. Phys. Soc. Jpn. 52 (1983) 2350. [7] J.C. Quail, J.G. Rabo and H. Simon, Optics Lett, 8 (1983) 377. [8] A.E, Craig, G.A. Olson and D. Sarid, Optics Lett, 8 (I 983) 383. [9] Fuzi Yang, G.W. Bradberry and J.R. Sambles, Phys. Rev. Leu. 66 (1991) 2030. [10] R.M.A. Azzam and N.M. Bashara, Ellipsometry and Polafised Light (North Holland, Amsterdam, 1979). [I 1] D.Y.K. Ko and J.R. Sambles. J. Opt. Soc. Am A 5 (1988) 1863.
Optics Communications132 (1996) 212-216
Degenerate long range surface modes, supported on thin nickel films E.L. Wood, J.R. Sambles, F.A. Pudonin i, V. Yakovlev 2 Thin Film Photonics Group, Departmentof Physics, Universityof Exeter, Exeter, EX4 4QL UK
Received25 January 1996;revisedversionreceived24 April 1996;accepted2 May 1996
Abstract
In this work islandised nickel films have been placed in a symmetric environment to allow optical excitation of long-range surface modes via attenuated total reflection. Comparison of modelling theory to the experimental angle-dependent refleetivity data shows that it is not possible to determine the optical permittivity of the thin nickel layer without knowledge of its thickness. The reflectivity data has a set of degenerate fits; the deduced optical permittivities for this degeneracy describing a circle when plotted as a Cole-Cole plot. This circle is naively predicted from an analytical three.media theory which ignores the coupling prism. Moreover, further investigations show that the degenerate circle is a general result of fitting refiectivity curves in the vicinity of a long range mode.
1, Introduction Long-range optical modes supported by very thin layers have been classified in a number of ways. These classifications reflect the nature of the thin layer which supports their excitation. If for example the exciting medium is metallic in character, the resonance is generally described as a long-range sufface-plasmon polariton (LRSPP) [1], whilst a dielectric with low levels of absorption may support excitation of a guided mode (GM). Similarly, if the real part of the optical permittivity is equal to zero whilst the imaginary part is large, the long-range sufface-exciton polariton (LRSEP) is excited [2]. I Lebedev Physical Institute, Russian Academyof Sciences. Leninskiiprosp, 53, Moscow117924,Russia 2 Institute of Spectroscopy, Russian Academyof Sciences, Troitzk, Moscowreg. 142092, 142092 Russia.
Such modes have momenta greater than that of polaritons in the surrounding dielectric and may be coupled in real systems using a prism to enhance the incident momentum. The mode manifests itself as a dip in the transverse magnetic (p) polarized reflectivity as the incident angle, hence momentum is scanned. In this paper we study the long range modes excited in the attenuated total reflection geometry using a thin nickel film (d ffi 2 nm) at a wavelength of 632.8 nm. It becomes clear that the angle-dependent reflectivity data does not allow a unique characterization of the system; indeed, there are an infinite number of degenerate fits to the data which give rise to a circular trace if the deduced optical permittivity is plotted in a Cole-Cole geometry [3]. The circular trace may be very closely reproduced by using simple analytical long-range mode theory [4]. This work then shows not only that, in principle, a vast range of
0030-4018/96/$12.00 Copyright© 1996ElsevierScienceB,V. All rights reserved, PI! S0030-4018(96)00334-3
213
E.L. Wood et a l . / Optics Communicatiora ~132 (1996) 212-216
1.2,
optical permittivities may produce long-range modes in thin films, but it also clearly illustrates that the optical permittivity for such a thin film may not be determined unless the thickness is known.
Theory 1.0
•
Data
0.8
2. Experimental A long-range mode on a thin film is a resonance which requires for its excitation a close to symmetric environment. In a symmetric system containing three media, where the second has a thickness much smaller than the wavelength of light, it is possible to excite surface modes on both interfaces simultaneously using light of the appropriate momentum. Since both modes have identical energy and momentum, they couple together strongly to form two new coupled surface modes which have symmetric and antisymmetric wavefunctions respectively. The difference in electric field profile determines the propagation length of the resonance and the two coupled modes are classified either as long- or short-range modes. They require different momenta for their excitation and hence may be studied separately. Long-range surface modes provide a means of supporting, on a highly absorbing film, a resonance which has a propagation length significantly larger than the equivalent resonance in a single interface system. The prediction, observation and properties of these modes are well documented [1,2,4-9]. Work by Yang et al. [9] showed that LRSEP modes may be excited in islandised silver films using visible radiation. This islandised film has optical permittivities appropriate for the excitation of the mode. In this work we investigate long-range modes supported by islandised nickel and to this end, a very thin nickel film (nominal thickness 2 nm) was prepared by sputter deposition on to a silica substrate. The nickel film was placed in a symmetric environment by immersing the free side of the film in silica indexmatching fluid which was made from suitable proportions of methyl benzoate and cycloheptane. Light of wavelength 632.8 nm was coupled into the system using a 60 ° prism of refractive index 1,8. A schematic of the experimental arrangement is shown in the inset to Fig. 1. Rpp data were obtained for a variety of coupling gaps (D), the data at optimum coupling being denoted by the crosses in Fig. !.
O r,*
0.6 Matching Fluid
11.4 [
Glass
1
0.2
0'052
3
54 Internal Angle
56
Fig. I. p-polarized reflectivity data and a typical theoretical fit to the data obtained from a thin layer of nickel in the symmetrical environment denoted by the inset to the figure.
Theoretical fits to the data were made using multilayer optical modelling theory [10,11] and a least squares minimization routine. The modelling theory is a scattering matrix formulation, which is capable of dealing with any number of arbitrarily oriented biaxial layers. In general, fitting with this routine will lead to possibly degenerate solutions, to avoid this some of the fitting parameters were determined independently, for example the refractive indices of the prism and substrate glasses were well known, while the matching fluid, which was a volatile mixture, was given a refractive index which could vary within small bounds to allow for the fact that its refractive index might have changed by some small amount between experiments. However, the thickness of the coupling gap D, the thickness of the nickel film d and the optical permittivity of the nickel film were allowed to fit as free variables. If d is not closely constrained in the fitting procedure, the algorithm allows the parameters to wander until d encounters its upper or lower bounds. For a given fixed value of d, solutions are found which yield similar values of D, ranging from 1.37 to 1.42 p,m, virtually identical matching fluid permittivities, ranging from 2.1363 to 2.1372, and similar sum of squares. However the fits give large variations in the values of the metal permittivity as d is varied. A typical fit to the data is denoted by the solid line in
E.L. Wood et aL / Optics Communications 132 (1996)212-216
214 2oj
imaginary parts respectively of the mode momenta are given by the following expressions [4]:
15
k , = k 0 8 [ / ~ I + 2 [ '~o ]
(8) + 82 - 8,8r) ~- 8~82 ]
~i 10
× 5
ki =
( - 7 + 82) ~
ko8[/282( vrd 128 i ( 8 r 2 + 8 2 -- 8 1 P r ) ~o I
0
-.
-5
_
0
Er
I
5
10
Fig. 2. The crosses show the variation of the real and imaginary optical permittivities obtained from the degenerate fits to the data in Fig. 1, whilst the solid line is produced from the three-media analytical theory.
J,
(8~ + 82) 5
(2)
where 8 t represents the real optical permittivity of matched media 1 and 3, Ao is the wavelength of the incident light, and 8 r + i 8 i is the complex permittivity of the second medium which in this case is islandised nickel. Rearranging (1) and (2) to eliminate d gives:
(8~ + 82- 8,8,) ~- 8~8i~ = 2c8,8,(8~ + 87 - 8, 8r), Fig. 1. If the variation of values obtained for the real permittivity 8 r is plotted against those obtained for the imaginary optical permittivity, et, then the resulting curve, shown by the symbols in Fig. 2, is circular. This shows that the reflectivity associated with the excitation of a long-range mode may not be used under these circumstances to determine the optical parameters of the system. Moreover. even the nature of the resonance supported remains unknown. Unless the thickness d of the thin film is known, it is not possible to say from the reflectivity traces whether the long-range mode is a LRSPP, LRSEP or simply a GM resonance. It seems at first rather surprising that the degenerate fitting describes a circle when plotted in a ColeCole form. To explain this effect it is necessary to consider the analytical theory for long range modes in a semi-infinite three media system. This is at first sight unreasonable, since the prism so strongly perturbs the mode. However bearing in mind that the prism perturbs equally all resonances with the same initial k ( = k r + i k i ) , then the initial, unperturbed complex mode momenta are the determining factors. As we see below there are an infinity of values for 8 r and ~i which give the same kr, k i provided d is chosen appropriately. The equations for the real and
(l)
(s)
where C = ( k r - ko s [/2 ) / ki. This expression further reduces to:
e,~+8~-e,~,-e,8,(C+~+C~)=O.
(4)
This represents two part-circles in the positive 8~ half-space with radii ½81[1 + (C + ~/1 + C 2 )2]1/2 and centres ~8 ' I, ½ 8 , ( c + f / + c 2 ) respectively. Although both roots of Eq. (4) are solutions, the negative root represents values of k r which are < k ~ l / 2 and so are not evanescent and which therefore are not long-range coupled surface modes. Hence hereafter we consider only the positive roots. The parameter C which is the ratio of the shift in real momentum from kie[/2 to the imaginary mode momentum determines the radius and centre of the part circle which always cuts the 8r axis at 0 and e I. Using a set of permittivities determined from the degenerate fitting, a value of C ffi 3.809 is calculated for the nickel system. The circle predicted from Eq. (4) using the three-media analytical theory with this value of C is denoted by the solid line in Fig. 2. This gives very good agreement to the optical permittivities determined from fitting the experimental data. Note that, if instead of using the mode momenta calculated from the analytical theory, the real and
E.L. Wood et al./ Optics Communications 132 (1996) 212-216
imaginary momenta are found directly from a measurement of the width and excitation angle of the long-range mode, then the value of C = 1.101 obtained in this way predicts a circle for which there is extremely poor agreement with the degenerate fit curve. This is because the coupling prism dramatically perturbs the reflectivity trace, both shifting and broadening the apparent resonance. This perturbed mode momentum does not have the same degeneracy as the unperturbed momentum. What we have done is to fit in effect to kr and k i of the unperturbed system, the modelling taking into account the prism perturbation. In the analytic theory any one of the three unknown parameters may be eliminated between Eqs. (1) and (2) (in this case we have chosen to eliminate d) hence there will always be a degeneracy between the three parameters, d, er and ~i which give identical values of kr and k i. Therefore, though we obtain a precise value for the complex wavevector of the mode, it is impossible to use reflectivity data corresponding to the excitation of a long-range mode to determine which sort of long-range mode is being excited (eg LRSPP, LRSEP or GM), or the independent optical permittivity. Some other technique must be used to establish one of these parameters independently. In order to test for the effect of the coupling prism on the values of k, and kt, a number of theoretical reflectivity traces were generated using again the multilayer optics modelling theory. Six reflectivities were generated for which the only difference was the coupling gap D, these are shown in Fig. 3a, D being varied between 1.2 ~m and 2.2 p,m. The variation in D produces very different traces, yet when submitted to the fitting procedure, once more degenerate solutions result. When these are plotted in a Cole-Cole form, the solutions coalesce on to approximately the same circle shown by the symbols in Fig. 3b. The circle predicted from the symmetric, three-media analytical approach is shown by the solid line, and once again the agreement is good. A similar result is obtained by varying d, generating reflectivity curves and obtaining degenerate fits. Once again there is good agreement between successive degenerate fits, and the predictions from the analytical three-media theory. Clearly there are limits to the dimensions of degenerate circles which it is possible to find. To examine this, theoretical reflectivity curves are gen-
"[
215
(a)
1.0
r
0.8
> o m
o Q li=
0.6
0.4
0.2
o
I
|
0"53.0
|
5&5
I
i
54.0
I
I
t
54.5
55.0
Angle 20
(b)
°,
:,=,,°
0
'
" -5
" ~ 0 6r
"IV ' 5
'
' 10
Fig. 3. (a) Six reflectivities generated from multilayer opiical modelling theory with values of D which vary from 1.2 I~m to 2.2 itm, where the wavelength of the incident light is 632.8 rim. the thickness of the nickel is 5.5 nm and its optical permittivity is 0.01 + 18.0i, (b) Symbols denote the results of degenerate fitting to the six datasets shown in (a). The solid line represents the three-media analytical theory.
erated with optical permittivity values which produce degenerate circles of different size. In each case the coupling gap D and the thickness d for the modelled reflectivity are kept constant, and the parameters allowed to vary in the fit are simply the optical permittivities of the thin film, and D. The degenerate fits obtained are depicted by the crosses shown in
216
E.L. Wo(~ etaL/ Optics Communications 132 f1996) 212-216 40-
30
t' i 20'
.
~-~"
"
.
•
1o
o
-20
-10
0
t0
20
Er
Fig. 4. The symbols show the optical permittivities obtained from fitting to theory, seven theorttically generated datasets whose input parameters art chosen to give a variety of degenerate circle sizes. The solid lines are the circles predicted from the thrte-media theory.
dent characterization of t,e film requires the use of an effective medium theory). The form of the fit degeneracy is interesting in that it describes a circle when ploted as a Cole-Cole plot. This circle is predicted from an analytical three-media theo W modelling the unperturbed mode momentum, ignoring the influence of the coupling prism. This is because the fit to the data will be the same for a range of film thicknesses and permittivities provided they give the same complex mode momenta, since to first order the coupling prism perturbs the mode momenta by the same amount regardless of the parameters which cause it. Further investigations show that the degenerate circle is a general result of fitting reflectivity curves in the vicinity of a long range mode. They indicate that a large range of optical permittivities will provide, for a thin film, coupled modes which have long propagation lengths.
Acknowledgements Fig. 4 whilst in each case the solid lines represent the circle predicted by the analytical theory. The fits obtained all had sums of squares which were within acceptable limits, but it is worth remarking that the smaller the radius of the degenerate fit circle in epsilon space, the broader and shallower the mode, and the more impressive the fit. However for the largest value of C ( - 7 ) , the fitting degeneracy begins to disappear and for this reason the fitted ~r and ~ values no longer describe a circle.
3. Conclusions In this work islandised nickel films have been placed in a symmetric environment to allow coupling to long-range surface modes. Comparison of theory to the experimental data shows that it is not possible to optically characterize the optical permittivity of this system without knowledge of the thickness of the islandised nickel layer. (In sltch a layer however the determination of the optical permittivity and thickness becomes complicated since any indepen-
The authors are grateful to EPSRC and DRA Malvern for their support of this work.
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