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An experimental test of conical diffraction theory
cm __
__ BB
ELSFZVIER
15 February 1997
OPTICS COMMUNICATIONS Optics Communications 135 (1997) 189- 192
An experimental test of conical diffraction theory Richard A. Watts, J. Roy Sambles, J. Blair Harris Thin Film Photonics Group, Universiry of Exeter, Department of Physics, Stocker Road, Exeter, Devon, .&X4 4QL, I/K Received 22 July 1996; accepted 2 October 1996
Abstract A test of a modelling theory for conical optical diffraction in which both real diffracted orders and evanescent fields are present is given. Data in the form of reflectivities from a metallic grating, as a function of angle of incidence for different azimuthal orientations and different radiation polarisations, is completely fitted by one simple model of the surface profile for a fixed complex metal permittivity. The form of the surface found from this fit to optical data is in extremely good agreement with the form found by mechanical profilometry.
1. Introduction The development of the differential formalism of Chandezon et al. [I] to rigorously mode1 the optical response of diffraction gratings has spawned a considerable body of theory for diffractive structures based upon this original work. These theories permit the modelling of: Gratings in the conical mount [2], (where the incident wavevector is not required to be parallel to the gratings Bragg vector); conical diffraction from corrugated multilayered isotropic media [3]; and conical diffraction from corrugated multilayered uniaxial media [4]. In parallel with this a similar body of work has emerged referring to doubly periodic (bi-grating) structures [S--8]. In this study we compare, in detail, predictions from the conical diffraction theory [2] based on Chandezon’s formalism with experimental dam recorded for different radiation polarisations and different azimuthal orientations of the grating. The system chosen to study in order to test the conical diffraction model is a corrugated gold-air interface with a period of 1078 nm. For incident radiation of 632.8 nm a grating of this period may be used to couple to a maximum of six surface plasmon resonances (SPRs) these are the f 1, + 2 and k 3 SPRs. (Where we adopt the convention of labelling the resonance with the number and sign of the diffraction coupling to the mode.) These resonances produce very distinct reflectivity features which are specifically sensitive to the profile of the grating surface, especially in the conical mount where polarisation conversion
is observed [9]. A number of workers have exploited this sensitivity and used the in-plane reflectivities (incident wavevector parallel to the Bragg vector) to characterise grating surfaces [lo- 121. This characterisation involves matching to the experimental dam a theoretical model reflectivity which includes a function quantifying the grating shape. Here we test the modelling theory by taking a number of experimental dam sets for both in-plane and conical geometries from the same grating surface, and show that one set of parameters representing the grating shape and the dielectric functions of the two media at the interface will produce theoretically modelled reflectivities which agree with the experimental reflectivities recorded at different azimuths for different polarisations. We also compare the functional shape of the surface obtained from fitting the optical data to the theoretical model with measurements taken using a profilometer and find excellent agreement between the two.
2. Experimental 2.1. Grating manufacture The grating used in this study was manufactured by spin coating a silica disk with Shipley S1400-17 photoresist. This blank was then dried in an oven at 60°C for 30 minutes, left to cool, then exposed to two expanded collimated interfering beams of 457.9 nm laser light, thus
0030~4018/97/$17.00 Copyright 0 1997 Elsevier Science B.V. All rights reserved. PII SOO30-4018(96)00637-2
190
RA. Watts et al./Optics
Communications 135 (1997) 189-192
recording the interference fringes as a solubility profile in the photoresist. Development removes regions of more soluble (exposed) resist, producing a nominally sinusoidal surface relief diffraction grating, which is then illuminated with a UV source to fully expose any remaining resist before being baked at 90°C for one hour. In order to produce a robust diffraction grating capable of withstanding aggressive experimental procedures the photoresist sample was then atom beam etched to transfer the groove profile through into the silica substrate. To facilitate coupling to SPRs on a gold air interface the silica grating was coated with an optically thick, N 350 nm, layer of gold by thermal evaporation. Gold was chosen over silver which gives a sharper SPR at 632.8 nm because it is inert in air and thus has optical properties which will not vary over the time scale of the experiment. 2.2. Experimental
geometry
We define the co-ordinate system shown in Fig. 1 to describe the experimental geometry. The x direction is parallel to the grating Bragg vector with y normal to the average plane of the grating surface. The polar angle 6 is the angle of incidence measured from the normal, y. and the azimuthal angle, cp, the rotation of the incident plane from the x-axis. For the purposes of our theoretical model, because the gold is optically opaque, this system may be represented as a single interface between semi-infinite air and gold. Also because the area of the substrate is large compared to the spot size of the incident beam (2 mm’), which is in turn large when compared to the grating period, the theoretical model represents the light as an infinite plane wave incident upon a corrugated surface of infinite extent. Two polarisers were used to set the polarisation of both the incident beam and the detected (specular) beam. These are aligned to pass either p (transverse magnetic), or s (transverse electric) polarisations, defined with respect to the plane of incidence, enabling the measurement of R,,, R,, and R,, reflectivities, where the subscripts refer R toPge setting of the incident and detector polarisers respectively. The source used was the 632.8 nm line from a He-Ne laser.
3. Results The following reflectivities were chosen as they exhibit a variety of different SPR features, and recorded as polar angle scans at a fixed azimuth: p=OO”:
R
cp=30°:
b”pp’ R ps* R spr R ss
~p=45=‘:
R,,,
rp= 90”:
R,,
R ps’ R spy R ss
z
Semi - infinite
Ail
Fig. 1. The co-ordinate system describing the orientation of the grating grooves to the ‘incident wavevector k,.
These reflectivities were then fitted by matching the theoretical model to them using an iterative least squares fitting routine. The model used the sum of cosine waves shown in Eq. (1) to represent the grating’s surface, A(x)=q
cos k,x+a,
cos 2k,x+a,
cos 3k,x.
(1)
Here k, (= 27r/h,) is the grating vector, ao, al and a2 the fundamental amplitude, first and second harmonics respectively. The parameters obtained from the fitting using this functional form of the surface were: a0 = 49.37 f 0.06 nm, a, = -8.30 It 0.05 nm, a2 = -2.28 f 0.03 nm and h, = 1078 nm. The air permittivity was set as E, = 1.0006 and the gold permittivity found to be E, = - 10.78 + 1.28i. This value for gold is in good agreement with previous experiments using SPRs to characterise gold at this wavelength [ 13,141. Fig. 2 shows the excellent agreement between theory and experiment for all ten sets of data. To confirm the fitted surface shape a profilometer was used to produce an alternative quantification of the grating’s surface. This was achieved by recording a profilometer trace across several grooves, this data being subsequently Fourier analysed yielding an equivalent surface function. Some 100 grooves were sampled so that the average profile could be determined. Such a step is essential in comparing the profiles determined by these two methods because the laser spot in the optical method, because of its size, samples many grating periods. This analysis gave the functional form of the surface as: u0 = 49.8 + 0.6 nm, a, = - 5.6 + 0.3 nm, a, = = - 1.0 + 0.1 nm and a3 = 0.6 + 0.1 nm. Fig. 3 compares these two surface profiles. An apparent narrowing of the grooves is recorded in the profilometer trace. This would be expected
RA. Watts et al./Optics Communications 135 (1997) 189-192
191
Fig. 2. The experimental reflectivities compared to theoretical model reflectivities created from the single set of fitted parameters.
-
Profile
derived
from fitting
-----
Profile
derived
from
reflectivities
profilometer
on this scale of measurement because of the finite width of the stylus which is used to contact the surface
to theory.
measurements.
4. Conclusion
1’ -20 -
-4x-
Horizontal
Displacement
(nm)
Fig. 3. A comparison of the grating groove profiles deduced from fitting the experimental reflectivities to theory, and profilometer
We have demonstrated that a single set of parameters describing the surface shape of a grating together with its optical permittivity produces convincing agreement between model reflectivities generated from this parameter set using conical diffraction theory, and experimental data taken at a number of azimuths and polarisations. Both the dielectric function of the gold and tire shape of the grating have been shown to be in excellent accord with other measurements, the latter by comparison with profilometer measurements. This is a convincing test of conical diffraction modelling theory, dealing with an experimental sys-
192
RA. Watts et al/Optics
Communications
tern in which both real diffracted and evanescent fields are strongly excited.
Acknowledgements
The authors would like to acknowledge the financial support of the National Physical Laboratory, the Defence Research Agency (Malvem) and the Engineering and Physical Sciences Research Council.
References [l] J. Chandezon, M.T. Dupuis, G. Comet and D. Maystre, J. Opt. Sot. Am. 72 (1982) 839. [2] S.J. Elston, G.P. Bryan-Brown and J.R. Sambles, Phys. Rev. B 44 (1991) 6393.
135 (19971189-192
[3] Lifeng Li, J. Opt. Sot. Am. A 11 (1994) 2829. [4] J.B. Harris, T.W. Preist, E.L. Wood and J.R. Sambles, J. Opt. Sot. Am. A 13 (1996) 803. [5] G.H. Derrick, R.C. McPhedran, D. Maystre and M. Neviere, Appl. Phys. 18 (1979) 39. [61 G.H. Derrick and R.C. McPhedran, J. Optics (Paris) 15 (1984) 69. 171 G. Granet, Pure Appl. Optics 4 (1995) 777. 181 J.B. Harris, T.W. Preist, J.R. Sambles, R.N. Thorpe and R.A. Watts, The optical response of bi-gratings J. Opt. Sot. Am. A, accepted for publication. [9] G.P. Bryan-Brown, J.R. Sambles and M.C. Hutley, J. Mod. Optics 37 (1990) 1227. [lo] E.H. Rosengart and I. Pockrand, Optics L&t. 1 (1977) 194. [I 11 G.P. Bryan-Brown, M.C. Jory, S.J. Elston and J.R. Sambles, J. Mod. Optics 40 (1993) 959. [12] E.L. Wood, J.R. Sambles, N.P. Cotter and S.C. Kitson, J. Mod. Optics 42 (1995) 1343. 1131 G.P. Bryan-Brown, S.J. Elston and J.R. Sambles, J. Mod. Optics 38 (1991) 1181. 1141 R.A. Innes and J.R. Sambles, J. Phys. F 17 (1987) 277.
__ BB
ELSFZVIER
15 February 1997
OPTICS COMMUNICATIONS Optics Communications 135 (1997) 189- 192
An experimental test of conical diffraction theory Richard A. Watts, J. Roy Sambles, J. Blair Harris Thin Film Photonics Group, Universiry of Exeter, Department of Physics, Stocker Road, Exeter, Devon, .&X4 4QL, I/K Received 22 July 1996; accepted 2 October 1996
Abstract A test of a modelling theory for conical optical diffraction in which both real diffracted orders and evanescent fields are present is given. Data in the form of reflectivities from a metallic grating, as a function of angle of incidence for different azimuthal orientations and different radiation polarisations, is completely fitted by one simple model of the surface profile for a fixed complex metal permittivity. The form of the surface found from this fit to optical data is in extremely good agreement with the form found by mechanical profilometry.
1. Introduction The development of the differential formalism of Chandezon et al. [I] to rigorously mode1 the optical response of diffraction gratings has spawned a considerable body of theory for diffractive structures based upon this original work. These theories permit the modelling of: Gratings in the conical mount [2], (where the incident wavevector is not required to be parallel to the gratings Bragg vector); conical diffraction from corrugated multilayered isotropic media [3]; and conical diffraction from corrugated multilayered uniaxial media [4]. In parallel with this a similar body of work has emerged referring to doubly periodic (bi-grating) structures [S--8]. In this study we compare, in detail, predictions from the conical diffraction theory [2] based on Chandezon’s formalism with experimental dam recorded for different radiation polarisations and different azimuthal orientations of the grating. The system chosen to study in order to test the conical diffraction model is a corrugated gold-air interface with a period of 1078 nm. For incident radiation of 632.8 nm a grating of this period may be used to couple to a maximum of six surface plasmon resonances (SPRs) these are the f 1, + 2 and k 3 SPRs. (Where we adopt the convention of labelling the resonance with the number and sign of the diffraction coupling to the mode.) These resonances produce very distinct reflectivity features which are specifically sensitive to the profile of the grating surface, especially in the conical mount where polarisation conversion
is observed [9]. A number of workers have exploited this sensitivity and used the in-plane reflectivities (incident wavevector parallel to the Bragg vector) to characterise grating surfaces [lo- 121. This characterisation involves matching to the experimental dam a theoretical model reflectivity which includes a function quantifying the grating shape. Here we test the modelling theory by taking a number of experimental dam sets for both in-plane and conical geometries from the same grating surface, and show that one set of parameters representing the grating shape and the dielectric functions of the two media at the interface will produce theoretically modelled reflectivities which agree with the experimental reflectivities recorded at different azimuths for different polarisations. We also compare the functional shape of the surface obtained from fitting the optical data to the theoretical model with measurements taken using a profilometer and find excellent agreement between the two.
2. Experimental 2.1. Grating manufacture The grating used in this study was manufactured by spin coating a silica disk with Shipley S1400-17 photoresist. This blank was then dried in an oven at 60°C for 30 minutes, left to cool, then exposed to two expanded collimated interfering beams of 457.9 nm laser light, thus
0030~4018/97/$17.00 Copyright 0 1997 Elsevier Science B.V. All rights reserved. PII SOO30-4018(96)00637-2
190
RA. Watts et al./Optics
Communications 135 (1997) 189-192
recording the interference fringes as a solubility profile in the photoresist. Development removes regions of more soluble (exposed) resist, producing a nominally sinusoidal surface relief diffraction grating, which is then illuminated with a UV source to fully expose any remaining resist before being baked at 90°C for one hour. In order to produce a robust diffraction grating capable of withstanding aggressive experimental procedures the photoresist sample was then atom beam etched to transfer the groove profile through into the silica substrate. To facilitate coupling to SPRs on a gold air interface the silica grating was coated with an optically thick, N 350 nm, layer of gold by thermal evaporation. Gold was chosen over silver which gives a sharper SPR at 632.8 nm because it is inert in air and thus has optical properties which will not vary over the time scale of the experiment. 2.2. Experimental
geometry
We define the co-ordinate system shown in Fig. 1 to describe the experimental geometry. The x direction is parallel to the grating Bragg vector with y normal to the average plane of the grating surface. The polar angle 6 is the angle of incidence measured from the normal, y. and the azimuthal angle, cp, the rotation of the incident plane from the x-axis. For the purposes of our theoretical model, because the gold is optically opaque, this system may be represented as a single interface between semi-infinite air and gold. Also because the area of the substrate is large compared to the spot size of the incident beam (2 mm’), which is in turn large when compared to the grating period, the theoretical model represents the light as an infinite plane wave incident upon a corrugated surface of infinite extent. Two polarisers were used to set the polarisation of both the incident beam and the detected (specular) beam. These are aligned to pass either p (transverse magnetic), or s (transverse electric) polarisations, defined with respect to the plane of incidence, enabling the measurement of R,,, R,, and R,, reflectivities, where the subscripts refer R toPge setting of the incident and detector polarisers respectively. The source used was the 632.8 nm line from a He-Ne laser.
3. Results The following reflectivities were chosen as they exhibit a variety of different SPR features, and recorded as polar angle scans at a fixed azimuth: p=OO”:
R
cp=30°:
b”pp’ R ps* R spr R ss
~p=45=‘:
R,,,
rp= 90”:
R,,
R ps’ R spy R ss
z
Semi - infinite
Ail
Fig. 1. The co-ordinate system describing the orientation of the grating grooves to the ‘incident wavevector k,.
These reflectivities were then fitted by matching the theoretical model to them using an iterative least squares fitting routine. The model used the sum of cosine waves shown in Eq. (1) to represent the grating’s surface, A(x)=q
cos k,x+a,
cos 2k,x+a,
cos 3k,x.
(1)
Here k, (= 27r/h,) is the grating vector, ao, al and a2 the fundamental amplitude, first and second harmonics respectively. The parameters obtained from the fitting using this functional form of the surface were: a0 = 49.37 f 0.06 nm, a, = -8.30 It 0.05 nm, a2 = -2.28 f 0.03 nm and h, = 1078 nm. The air permittivity was set as E, = 1.0006 and the gold permittivity found to be E, = - 10.78 + 1.28i. This value for gold is in good agreement with previous experiments using SPRs to characterise gold at this wavelength [ 13,141. Fig. 2 shows the excellent agreement between theory and experiment for all ten sets of data. To confirm the fitted surface shape a profilometer was used to produce an alternative quantification of the grating’s surface. This was achieved by recording a profilometer trace across several grooves, this data being subsequently Fourier analysed yielding an equivalent surface function. Some 100 grooves were sampled so that the average profile could be determined. Such a step is essential in comparing the profiles determined by these two methods because the laser spot in the optical method, because of its size, samples many grating periods. This analysis gave the functional form of the surface as: u0 = 49.8 + 0.6 nm, a, = - 5.6 + 0.3 nm, a, = = - 1.0 + 0.1 nm and a3 = 0.6 + 0.1 nm. Fig. 3 compares these two surface profiles. An apparent narrowing of the grooves is recorded in the profilometer trace. This would be expected
RA. Watts et al./Optics Communications 135 (1997) 189-192
191
Fig. 2. The experimental reflectivities compared to theoretical model reflectivities created from the single set of fitted parameters.
-
Profile
derived
from fitting
-----
Profile
derived
from
reflectivities
profilometer
on this scale of measurement because of the finite width of the stylus which is used to contact the surface
to theory.
measurements.
4. Conclusion
1’ -20 -
-4x-
Horizontal
Displacement
(nm)
Fig. 3. A comparison of the grating groove profiles deduced from fitting the experimental reflectivities to theory, and profilometer
We have demonstrated that a single set of parameters describing the surface shape of a grating together with its optical permittivity produces convincing agreement between model reflectivities generated from this parameter set using conical diffraction theory, and experimental data taken at a number of azimuths and polarisations. Both the dielectric function of the gold and tire shape of the grating have been shown to be in excellent accord with other measurements, the latter by comparison with profilometer measurements. This is a convincing test of conical diffraction modelling theory, dealing with an experimental sys-
192
RA. Watts et al/Optics
Communications
tern in which both real diffracted and evanescent fields are strongly excited.
Acknowledgements
The authors would like to acknowledge the financial support of the National Physical Laboratory, the Defence Research Agency (Malvem) and the Engineering and Physical Sciences Research Council.
References [l] J. Chandezon, M.T. Dupuis, G. Comet and D. Maystre, J. Opt. Sot. Am. 72 (1982) 839. [2] S.J. Elston, G.P. Bryan-Brown and J.R. Sambles, Phys. Rev. B 44 (1991) 6393.
135 (19971189-192
[3] Lifeng Li, J. Opt. Sot. Am. A 11 (1994) 2829. [4] J.B. Harris, T.W. Preist, E.L. Wood and J.R. Sambles, J. Opt. Sot. Am. A 13 (1996) 803. [5] G.H. Derrick, R.C. McPhedran, D. Maystre and M. Neviere, Appl. Phys. 18 (1979) 39. [61 G.H. Derrick and R.C. McPhedran, J. Optics (Paris) 15 (1984) 69. 171 G. Granet, Pure Appl. Optics 4 (1995) 777. 181 J.B. Harris, T.W. Preist, J.R. Sambles, R.N. Thorpe and R.A. Watts, The optical response of bi-gratings J. Opt. Sot. Am. A, accepted for publication. [9] G.P. Bryan-Brown, J.R. Sambles and M.C. Hutley, J. Mod. Optics 37 (1990) 1227. [lo] E.H. Rosengart and I. Pockrand, Optics L&t. 1 (1977) 194. [I 11 G.P. Bryan-Brown, M.C. Jory, S.J. Elston and J.R. Sambles, J. Mod. Optics 40 (1993) 959. [12] E.L. Wood, J.R. Sambles, N.P. Cotter and S.C. Kitson, J. Mod. Optics 42 (1995) 1343. 1131 G.P. Bryan-Brown, S.J. Elston and J.R. Sambles, J. Mod. Optics 38 (1991) 1181. 1141 R.A. Innes and J.R. Sambles, J. Phys. F 17 (1987) 277.