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Light scattering from a multimode waveguide of planar metallic walls
Optics Communications 227 (2003) 227–235 www.elsevier.com/locate/optcom
Light scattering from a multimode waveguide of planar metallic walls Aldo S. Ramırez-Duverger a
a,b,*
, Ra ul Garcıa-Llamas
b
Programa de Doctorado en Ciencias (Fısica), Universidad de Sonora, Apdo. Postal 5-88, Hermosillo, Sonora, Mexico b Departamento de Investigaci on en Fısica, Universidad de Sonora, Apdo. Postal 5-88, Hermosillo, Sonora, Mexico Received 6 June 2003; received in revised form 29 August 2003; accepted 22 September 2003
Abstract The experimental results of the reflection at a MgF2 waveguide between Ag metallic walls and the light scattering from the same system for p and s linearly polarized incident light are reported. The thickness of the dielectric film was chosen in such a way that the system supports six transverse magnetic and six transverse electric-guided modes at a given wavelength. We found dips in the reflection as a function of the incident angle as well as a function of the wavelength that are due to excitation of modes in the MgF2 film. From the spectral reflection curves the dispersion relation of the modes is obtained. The scattered light shows peaks at angles corresponding to the measured excitation of the waveguide modes. These peaks are because of single-order scattering and occur at any angle of the incident light. Enhancement peaks in the scattering response are found in resonant conditions, and the efficiency of injecting light into the guide is reduced. Ó 2003 Published by Elsevier B.V. PACS: 42.82 Et; 42.68.Mj; 82.45.Mp Keywords: Waveguide; Scattering; Thin Films
1. Introduction Waveguides are of great importance in passive optical components such as arrayed waveguide gratings, couplers and splitters. Planar waveguides
*
Corresponding author. Tel.: +526622-59-21-56; fax: +526622-12-66-49. E-mail addresses: [email protected] (A.S. Ramırez-Duverger), [email protected] (R. GarcıaLlamas). 0030-4018/$ - see front matter Ó 2003 Published by Elsevier B.V. doi:10.1016/j.optcom.2003.09.057
have been used in the development of optical communications almost since it is inception. Until recently, this potential was unrealized, however, several prevalent planar optical waveguide technologies and their application to solving optical network problems have emerged since last decade. Photonic networks based on wavelength division multiplexing systems have been developed in response to the growth of the internet devices. New functions are required for light networks, and planar light wave circuits are considered to meet this need.
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Thin films typically form waveguide and as a result of that, films are generally uniform, but can have roughness that will contribute to scattering losses. The attenuation is directly proportional to the root mean square roughness, which meets our expectation that the rougher the waveguides are, the more they will scatter. Recently, a novel mechanism to inject light into a dielectric guide of metallic wall was studied numerically [1]. The energy of the incident radiation can be almost completely injected to the planar waveguide of metallic walls. This theoretical calculation of the light injection was presented as a minimum in the angular (spectral) reflection or like a maximum in the modulus of the Poynting vector into the guide. The experimental reflection at the air/Ag/Al2 O3 /Ag system that displayed only a minimum was presented as evidence of the phenomenon. This minimum was associated to the excitation of the so-named oscillatory modes. Other authors have studied electromagnetic modes in metal–insulator–metal structure [2] using the air/Ag/CaF2 /Ag system, similar to that presented in [1]. The reflectance of such structure was measured at several angles of incidence in the 1–5 eV spectral range and a minimum was detected. In this case the excitated mode was named Fabry– Perot. It is possible to show that the oscillatory mode is the same as the Fabry–Perot one. Under resonance condition, that is when the reflection reaches a minimum, light can be partially guided minimizing the leaky rate. At the same time, the flow of energy along the guide is attenuated due to the losses in the metallic walls of finite conductivity. It happens that the TM wave, which has an electric field component perpendicular to the planar walls, decays faster than the TE waves [3]. In fact, such decaying properties have been used to design polarization sensitive devices as waveguide polarizers [4,5] and in-line fiber polarizers [6]. Although the distance of propagation of modes in this system is about 5 lm, in the visible range, its potential application in nanostructure technology, as a simple way to inject light to an optical circuit, motivates the present study. In this paper we show experimental results of reflection (spectral and angular) at an air/Ag/
MgF2 /Ag system and light scattering from the mentioned system supporting electromagnetic modes. This novel and easy mechanism to inject light, associated to the reflection minima into a waveguide and its roughness effects, is investigated. The spectral reflection, for both polarizations s and p, at the air/Ag/MgF2 /Ag system shows dips, which are associated to the excitation of some guided modes. The dip positions are shifted toward lower wavelength as the incidence angle is increased. From the spectral reflection, the dispersion relation of the electromagnetic modes of the optical cavity was obtained. The angular reflection shows two minima, and basically the same mentioned above explanation can be done. The depth of a minimum strongly depends on the thickness of the metallic film; this means that, the reflection reaches a zero depth for a given metal thickness and for some polarization of the incident light. On the other hand, the scattering is due to the natural roughness of each interface of the system, whereas the resonant character is due to the excitation of the waveguide modes and its interactions with the interface roughness. When the reflection reaches a minimum at some angular (spectral) position, the diffuse light is enhanced, and this is named resonant scattering.
2. The waveguide A multimode planar waveguide was grown as follows: First, a very thick film of Ag (250.0 nm) was evaporated thermically onto a glass substrate; second, a 1273.0 nm MgF2 film and a 36.0 nm Ag metallic film were grown. Using an oscillating quartz controller and an optical monitor, the thickness of each film was monitored. The vacuum pressure was 106 mb during the evaporation. The evaporation rates were in the range of 0.5 nm/s. The experimental angular reflection and scattering measurements were obtained by using an experimental setup reported elsewhere [7], while the dispersion relation was obtained from the spectral reflection curves between 450 and 900 nm wavelength range.
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3. Results
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higher than those for Ag, so we assume that most of the roughness comes from the Ag/MgF2 interface.
The system supports six TM waveguide modes and the same number for TE at k ¼ 632:8 nm. The magnesium fluoride film is surrounded by a thicker (substrate) and thinner silver metallic films. The thicker silver metallic film acts like an opaque mirror and the other film like a semitransparent one. This system resembles an opaque etalon or Fabry–Perot interferometer, so the explanation of the experimental results shown in this paper are only based on optical physic. It has nothing to do with geometrical optics at all. The roughness is attributed to the material as well as the grown method used. At this point, the properties of the Ag and MgF2 interface roughness as a function of thickness were previously studied by Mayani et al. [8]. They showed that the roughness spectrum of the MgF2 films is not a perfect Gaussian function. Other authors [9] showed that the light scattering depends on the power spectrum, and using a Gaussian roughness spectrum approximates it. The rms roughness values reported for MgF2 are
3.1. Spectra reflection Fig. 1 shows the reflection as a function of wavelength for an incident angle equal to 45.0°. The experimental results are represented by circles for p polarization, and by triangles for s polarization of the incident light, the theoretical results by solid and dashed curves for p and s polarization, respectively. The experimental minima deðpÞ ðpÞ ðpÞ tected at k4 ¼ 859:4 nm, k5 ¼ 688:1 nm, k6 ¼ ðpÞ 575:6 nm and k7 ¼ 498:2 nm correspond to the excitation of the fourth, fifth, sixth and seventh TM waveguide modes for p polarization of the incident light. The lower (higher) modes cannot be seen because they lay out of the maximum (minimum) wavelength used in this experiment. The ðsÞ ðsÞ minima detected at k4 ¼ 848:6 nm, k5 ¼ 680:7 ðsÞ ðsÞ nm, k6 ¼ 566:5 nm and k7 ¼ 490:9 nm correspond to the excitation of the fourth, fifth, sixth and seventh TE mode for s polarization.
θi= 45.
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Fig. 1. Reflection versus wavelength for an incident angle hi ¼ 45:0° for p and s polarization of the incident light. The system is the planar air/Ag(36.0 nm)/MgF2 (1273.0 nm)/Ag(250.0 nm substrate). The circles and the triangles represent the experimental results for p and s polarization of the incident light, respectively. The solid and dashed curves correspond to the theoretical data for p and s polarization, respectively. The minima displayed correspond to excitation of fourth, fifth, sixth and seventh guided modes for both p and s polarization.
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The theoretical reflection given by Eq. (4) in [7] was calculated by using a double layer model, air/ metal/dielectric/metal, with planar, parallel and smooth interfaces. We neglected the last term. The dielectric material was considered non-dispersive and its refractive and absorption indexes were fixed to nMgF2 ¼ 1:46 and kMgF2 ¼ 104 , respectively, and its thickness d3 ¼ 1273:0 nm. The optical constants of silver were taken from Palik [10]. The numerical calculations were carried out to fit the fifth mode for each polarization. The actual refractive index of MgF2 changes when the wavelength is varied. That means, it is weakly dispersive. Therefore, there is a slightly difference between positions of the minima k6 and k7 displayed by theory and experiment. When the incident angle is increased, the positions of the minima are shifted toward lower wavelength. 3.2. Dispersion relation The positions of the minima displayed in the spectra reflection curve can be used to obtain the ‘‘optical’’ dispersion relation. Several spectral curves were obtained. The incident angle was varied from 10° up to 65° with an increment of 5°. Fig. 2(a) shows the TM dispersion relation of the electromagnetic modes of the metal–dielectric– metal system. Circles represent the experiment and the solid curve represents the theory. The straight solid and dashed lines represent the light line in the dielectric medium and in the vacuum, respectively. The circles were obtained from the spectral positions of the minima, for example, the excitation ðpÞ wavelength of the fifth TM mode is k5 ¼ 688:1 nm, see Fig. 1, the value of wavelength is changed to ðpÞ energy with the relation energy (eV) ¼ 1240.0/k5 , ðpÞ where k5 is in nanometer. Then, the wave number of the mode is obtained by using the coupling ðpÞ condition q ðnm1 Þ ¼ ½2p=k5 ðnmÞ sin hi , where hi is the incident angle, whose value is 45.0°. And so on for each minimum. The theoretical dispersion relation is obtained by using Eq. (1) in [1] (see also [11] ). In this model semi-infinite metallic media surrounds the dielectric film, with planar, parallel and smooth inter-
faces. The numerical curve was obtained by using the same parameters as in the section above. The ‘‘optical’’ dispersion relation and the theoretical one are in excellent agreement, without being modified by the roughness. This direct illumination method to inject light into the guide is restricted by the excitation of modes at the left of the light line in vacuum, oscillatory mode and in this paper we are only interested in them. Other methods, prism coupling, grating coupling or lateral injection, must be used to excite modes at the right of this line and those are not considered in this paper. Fig. 2(b) shows the dispersion relation of the TE electromagnetic modes. For this case all modes are waveguide-type and for TM there is one surface-plasmon-type, which appears on the right of the light line at the dielectric medium and it is observed only in the theoretical curve of Fig. 2(a). Similar behavior is observed experimentally in agreement with the theory. 3.3. Angular reflection The reflection as a function of the incidence angle is shown in Fig. 3 for a wavelength k ¼ 632:8 nm. The experimental results are represented by circles for p polarization, and by triangles for s polarization of the incident light. The theoretical results are represented by solid and dashed curves for p and s polarizations, respectively. The theoretical data were calculated by using Eq. (4) of [7] for a rms height of the roughness h3 ¼ 1:0 nm and a correlation length r3 ¼ 20:0 nm. The dielectric 2 constant of silver is eAg ¼ ð0:134 þ i3:985Þ for the wavelength used and its thickness is dAg ¼ 36:0 nm. The remainder parameters used to fit the experimental data were given in Section 3.1. ðpÞ ðpÞ The minima at h6 ¼ 17:4° and h5 ¼ 61:04° correspond to the excitation of the sixth and the fifth TM waveguide modes supported by the sysðpÞ tem, while, the minima at h6 ¼ 16:0° and ðpÞ h5 ¼ 56:9° are related to the excitation of the sixth and the fifth TE waveguide modes. The minima are not deeper for s polarization because the samples are optimized to reach very deep minima for p polarization. Therefore, the fifth minimum for s polarization is shallower. If we
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(b) Fig. 2. (a)The dispersion relation of the TM electromagnetic modes of a Ag/MgF2 (1273.0 nm)/Ag system is shown. Circles represent the experiment and the solid curve represents the theory. The straight solid and dashed lines represent the light line in the dielectric medium and in the air, respectively. The modes on left of the light line in the dielectric medium are the surface-planon type-modes. (b) Same as in (a) but for TE modes. In this case all modes are waveguided-type.
compare theory with experiment, the angular positions of the minima are approximately the same. The width of the experimental curve is greater than those of theoretical one, probably because the dielectric film is not quite uniform. The resonant condition occurs when the reflection at the system reaches a minimum at some specific angular position. At this point, the incident energy divides into several parts: a small
fraction is reflected, then part of the energy is absorbed by the system and is transformed into heat, then another part is conducted along the dielectric layer and finally the remaining part is scattered from the rough interfaces of the system. In this condition the scattering is named as resonant scattering. Important comments about the behavior of the theoretical specular reflection follow. By fixing the
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Fig. 3. Reflection as a function of the incidence angle at air/Ag(36.0 nm)/MgF2 (1273.0 nm)/Ag(250.0 nm substrate) system for a wavelength of 632.8 nm. The circles and the triangles represent the experimental results for p and s polarization of the incident light, respectively. The solid curve and the dashed one correspond to the theoretical data for p and s polarization. The minima (circles) at ðpÞ ðpÞ h5 ¼ 61:04° and h6 ¼ 17:42° correspond to excitation of the fifth and sixth TM-guided modes, respectively. The dips (triangles) ðsÞ ðsÞ at h5 ¼ 56:94° and h6 ¼ 15:95° are related to excitation of the fifth and sixth TE-guided mode, respectively.
dielectric thickness, an increment of nMgF2 makes the positions of the minima change toward greater angles and the separation between them increases. On the other hand, the angular positions of the excited guided modes are shifted toward larger angles when the value of dMgF2 film is increased. The kMgF2 does not cause any significant variation in the form of the curve neither in the position nor in the depth of the minima. If the thickness of the thinner Ag film is increased the width of the minima diminishes, whereas its deepness decreases. As a matter of fact there is not a unique solution. 3.4. Scattered light for p polarization A comment on the experimental measurements of scattered light is necessary: The scatterometer cannot measure the incident and reflected beam directions. Around these positions the experimental curve looks flat because the signal is very intense and it exceeds the maximum sensibility allowed for the device. Based on Eq. (27) of [9], the scattered theoretical data were calculated by using a second-order perturbation approach and a Gaussian roughness
spectrum, so it only takes account of single scattering effects. In this model, the roughness is at the inner dielectric–metal interface with a rms height h3 ¼ 1:0 nm and a correlation length r3 ¼ 20:0 nm. The remainder parameters were given in the preceding section. Fig. 4(a) shows the light scattering as function of the scattered angle, for a fixed incident angle ðpÞ of h6 ¼ 17:42°, for p polarization of the incident light with wavelength k ¼ 632:8 nm (the red visible line of a He–Ne laser at 10 mW). This angle corresponds to the direct excitation of the sixth TM mode supported by the guide. The small circles represent the experimental data and the solid curve represents the theoretical one. The intensity of the scattered light increases toward positive angles, that is, at the direction of the reflected beam. The peaks that are supposed to be noticeable at the positions of the incident and reflected beams could not be seen in the experimental curve because they are located at the directions of the incident and the reflected light, and they are well described by the theoretical data. The peaks at ðpÞ hs ¼ h5 ¼ 61:04° correspond to the excitation, via roughness, of the fifth TM modes traveling in
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Fig. 4. (a) Light scattering versus scattering angle for an incident angle hi ¼ h6 ¼ 17:42°, for p polarization of the incident light and for a wavelength of 632.8 nm. Both experimental (circles) and theoretical (solid curve) data are displayed. This angle corresponds to the ðpÞ direct excitation of the sixth TM-guided mode. We observed peaks at hs ¼ h5 ¼ 61:04° due to single scattering effect and they are related to the indirect excitation of the fifth TM-guided mode via the roughness of the system. (b) Same as (a) but for hi ¼ 0°. The angle ðpÞ ðpÞ does not correspond to the excitation of any mode. Anyway, we observed four weak peaks at h5 ¼ 61:04° and at h6 ¼ 17:42° due to single scattering effect. They are related to the indirect excitation of the sixth and fifth TM-guided mode.
opposite directions, and they are displayed on both experimental and theoretical curves. From the experimental point of view, the ‘‘peaks’’ are fringes with ring shapes that are out of the plane of incidence. These rings are not modeled on the theoretical solution because the roughness surface profile is assumed as one-dimensional, whereas the ‘‘natural’’ roughness at the
interfaces of the sample is two-dimensional. Further calculation must be performed to take account of this effect. Single scattering, as stated for the second-order perturbation in the surface profile theoretical solution, produces the peaks corresponding to the excitation of the guided modes; this excitation produces the enhancement of the scattering as a
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consequence of the resonant condition. Then, a large part of the input energy is observed outside the system as scattered energy into the reflection medium. This energy is proportional to the square of the rms height and wavelength ratio. No evidence of second-order scattering effect was detected. To obtain this kind of effect, the rms height must be increased artificially. Although light scattering occurs in all of incidence angles, the existence of guided modes and its excitation amplify the scattering at the resonant angles, being the reason of the signalÕs enhancement. The roughness parameters were inferred by fitting the theoretical curve to the experimental data. The parameters obtained this way are consistent with those reported in the literature. We did not expect to obtain very accurate data because the simplicity of the model, which considers only one interface with 1D-roughness. Fig. 4(b) plots the same as marks Fig. 4(a) but for normal incidence. This angle does not correspond to the direct excitation of any guide modes and only weak light scattering is expected. In this figure, the experiment (circles) and fit (solid curve) are shown, and it is possible to observe four low ðpÞ intensity peaks, at hs ¼ h5 ¼ 61:04° and hs ¼ ðpÞ h6 ¼ 17:42°. Experimentally only in this situ-
3.5. Scattered light for s polarization In this section we analyzed the same sample by changing only the polarization of the incident θi=-56.94
-2
10
Scattered light reflection (s pol)
ation the four peaks can be seen. On the other hand the theoretical data always displayed them. For normal incidence, the incident and reflected beams are detected in the same direction, and the spurious signal is detected in a minor angular arc, about 6.0° around 0°. In this situation, the interaction light-guided mode is weak, and the heights of the peaks are very low. Therefore, the excited waveguide modes are detected just above the background signal. For this reason, the excited modes are not detected with good definition. If we compare the two measured curves in resonance to the curve off resonance, we see that the scattered light increases when the light arrives at any angle of excitation of the waveguide modes; the peaks are very sharp and defined. Out of resonance, the intensity of the scattered light and the heights of the peaks are reduced. We observe that the angular positions of the peaks in the two Figs. 4(a) and (b) are alike, which demonstrates that the angular positions of the excited waveguide modes do not change when the incident angle is varied.
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Fig. 5. Same as Fig. 4(a) but for hi ¼ h5 ¼ 56:94° and for s polarization. This angle corresponds to the direct coupling of the fifth ðsÞ TE-guided mode and incident light. As in p polarization case, we observed peaks at hs ¼ h6 ¼ 15:95° due to single scattering effect.
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light. All TE electromagnetic modes are waveguide-type. Scattered light versus the scattered angle is shown in Fig. 5 for s polarization of the incident ðsÞ light and for a fixed incident angle of h5 ¼ 56:94°, there is a resonant condition of excitation of the fifth TE waveguide mode. Circles represent the experimental curve and the solid line represents the fit. The peaks that correspond to the positions of excitation of sixth TE waveguide mode are ðsÞ shown at hs ¼ h6 ¼ 15:95°, for both experimental and theoretical curves. The remainder modes cannot be excited because they are on the right of the light line at vacuum. The enhancement of scattering found in resonant condition diminished the efficiency for injecting light into the guide. Scattering light reflection in p polarization is one order of magnitude higher than the same response in s polarization because the thickness of the metallic film in the system is optimized for p polarization. This effect can be used to create reflection polarizer to some specific angle; for example, if the system is lit by non-polarized light to an incident angle of 61.04°, it is possible to obtain s-polarized reflection light.
the thickness of the metallic wall, we can separate an incident light from a reflected and guided wave with any desired ratio of reflected light to guide it. The angular distribution of the scattered light shows peaks caused by single-order scattering effects. The diffused light from the system shows ðpÞ four peaks at angles given by h5 ¼ 61:04°, ðpÞ ðsÞ ðsÞ h6 ¼ 17:42° (or h5 ¼ 56:94° and h6 ¼ 15:95°) due to the coupling between the fifth and sixth TM (or TE) waveguide modes and p (or s) polarized incident light via roughness of the interfaces. For s and p polarizations, the angular positions of the peaks are independent on the incident angle and due to single-order scattering. All features in the scattering response are enhanced by resonant conditions, and the efficiency for injecting light into the guide is reduced.
4. Conclusions
References
The experimental results of the reflection at a planar guide air/Ag/MgF2 /Ag structure and the light scattering from the mentioned system for p and s polarized light are reported We found dips in the reflection as a function of the incident angle as well as the wavelength. This is because of excitation of waveguide modes in the MgF2 film. The dispersion relation of such structure was measured in 450–900 nm wavelength range and theory matches experiment quite well. The experimental ‘‘optical’’ dispersion curves of modes are calculated through the spectral positions of the minima displayed in the reflection curve. The deepness of these minima is strongly dependent on the thickness of the metal layer, and this parameter allows controlling the amount of energy injected to the guide. By selecting appropriately
[1] R. Garcıa-Llamas, J.A. Gaspar-Armenta, F. Ramos-Mendieta, R. L opez, R.F. Haglund, R. Ruiz, Design, manufacturing, and testing of planar optical waveguide devices, in: R.A Norwood (Ed.), Proceedings of SPIE, vol. 4439, 2001, pp. 88–94. [2] F. Villa, T. Lopez-Rios, L.E. Regalado, Phys. Rev. B 63 (2001) 165103. [3] J.N. Polky, G.L. Mitchell, J. Opt. Soc. Am. 64 (1974) 274. [4] Z.H. Wang, S.R. Sheshadri, J. Opt. Soc. Am. A 6 (1989) 142. [5] C. Ma, S. Liu, J. Opt. Soc. Am. A 7 (1990) 1577. [6] R.B. Dyott, J. Bello, V.A. Handerek, Opt. Lett. 12 (1987) 287. [7] A. Ramırez-Duverger, J. Gaspar-Armenta, R. GarcıaLlamas, J. Opt. Soc. Am. A 20 (2003) 1582. [8] N. Mayani, F. Varnier, G. Rasigni, J. Opt. Soc. Am. A 7 (1990) 191. [9] R. Garcıa-Llamas, L.E. Regalado, C. Amra, J. Soc. Am. A 16 (1999) 2713. [10] E.D. Palik, Handbook of Optical Constants of Solids I, Academic Press, New York, 1991. [11] J. Kane, H. Osterberg, J. Opt. Soc Am. 54 (1964) 347.
Acknowledgements This work was partially supported by the Consejo Nacional de Ciencia y Tecnologıa under Grant 35223-E.
Light scattering from a multimode waveguide of planar metallic walls Aldo S. Ramırez-Duverger a
a,b,*
, Ra ul Garcıa-Llamas
b
Programa de Doctorado en Ciencias (Fısica), Universidad de Sonora, Apdo. Postal 5-88, Hermosillo, Sonora, Mexico b Departamento de Investigaci on en Fısica, Universidad de Sonora, Apdo. Postal 5-88, Hermosillo, Sonora, Mexico Received 6 June 2003; received in revised form 29 August 2003; accepted 22 September 2003
Abstract The experimental results of the reflection at a MgF2 waveguide between Ag metallic walls and the light scattering from the same system for p and s linearly polarized incident light are reported. The thickness of the dielectric film was chosen in such a way that the system supports six transverse magnetic and six transverse electric-guided modes at a given wavelength. We found dips in the reflection as a function of the incident angle as well as a function of the wavelength that are due to excitation of modes in the MgF2 film. From the spectral reflection curves the dispersion relation of the modes is obtained. The scattered light shows peaks at angles corresponding to the measured excitation of the waveguide modes. These peaks are because of single-order scattering and occur at any angle of the incident light. Enhancement peaks in the scattering response are found in resonant conditions, and the efficiency of injecting light into the guide is reduced. Ó 2003 Published by Elsevier B.V. PACS: 42.82 Et; 42.68.Mj; 82.45.Mp Keywords: Waveguide; Scattering; Thin Films
1. Introduction Waveguides are of great importance in passive optical components such as arrayed waveguide gratings, couplers and splitters. Planar waveguides
*
Corresponding author. Tel.: +526622-59-21-56; fax: +526622-12-66-49. E-mail addresses: [email protected] (A.S. Ramırez-Duverger), [email protected] (R. GarcıaLlamas). 0030-4018/$ - see front matter Ó 2003 Published by Elsevier B.V. doi:10.1016/j.optcom.2003.09.057
have been used in the development of optical communications almost since it is inception. Until recently, this potential was unrealized, however, several prevalent planar optical waveguide technologies and their application to solving optical network problems have emerged since last decade. Photonic networks based on wavelength division multiplexing systems have been developed in response to the growth of the internet devices. New functions are required for light networks, and planar light wave circuits are considered to meet this need.
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Thin films typically form waveguide and as a result of that, films are generally uniform, but can have roughness that will contribute to scattering losses. The attenuation is directly proportional to the root mean square roughness, which meets our expectation that the rougher the waveguides are, the more they will scatter. Recently, a novel mechanism to inject light into a dielectric guide of metallic wall was studied numerically [1]. The energy of the incident radiation can be almost completely injected to the planar waveguide of metallic walls. This theoretical calculation of the light injection was presented as a minimum in the angular (spectral) reflection or like a maximum in the modulus of the Poynting vector into the guide. The experimental reflection at the air/Ag/Al2 O3 /Ag system that displayed only a minimum was presented as evidence of the phenomenon. This minimum was associated to the excitation of the so-named oscillatory modes. Other authors have studied electromagnetic modes in metal–insulator–metal structure [2] using the air/Ag/CaF2 /Ag system, similar to that presented in [1]. The reflectance of such structure was measured at several angles of incidence in the 1–5 eV spectral range and a minimum was detected. In this case the excitated mode was named Fabry– Perot. It is possible to show that the oscillatory mode is the same as the Fabry–Perot one. Under resonance condition, that is when the reflection reaches a minimum, light can be partially guided minimizing the leaky rate. At the same time, the flow of energy along the guide is attenuated due to the losses in the metallic walls of finite conductivity. It happens that the TM wave, which has an electric field component perpendicular to the planar walls, decays faster than the TE waves [3]. In fact, such decaying properties have been used to design polarization sensitive devices as waveguide polarizers [4,5] and in-line fiber polarizers [6]. Although the distance of propagation of modes in this system is about 5 lm, in the visible range, its potential application in nanostructure technology, as a simple way to inject light to an optical circuit, motivates the present study. In this paper we show experimental results of reflection (spectral and angular) at an air/Ag/
MgF2 /Ag system and light scattering from the mentioned system supporting electromagnetic modes. This novel and easy mechanism to inject light, associated to the reflection minima into a waveguide and its roughness effects, is investigated. The spectral reflection, for both polarizations s and p, at the air/Ag/MgF2 /Ag system shows dips, which are associated to the excitation of some guided modes. The dip positions are shifted toward lower wavelength as the incidence angle is increased. From the spectral reflection, the dispersion relation of the electromagnetic modes of the optical cavity was obtained. The angular reflection shows two minima, and basically the same mentioned above explanation can be done. The depth of a minimum strongly depends on the thickness of the metallic film; this means that, the reflection reaches a zero depth for a given metal thickness and for some polarization of the incident light. On the other hand, the scattering is due to the natural roughness of each interface of the system, whereas the resonant character is due to the excitation of the waveguide modes and its interactions with the interface roughness. When the reflection reaches a minimum at some angular (spectral) position, the diffuse light is enhanced, and this is named resonant scattering.
2. The waveguide A multimode planar waveguide was grown as follows: First, a very thick film of Ag (250.0 nm) was evaporated thermically onto a glass substrate; second, a 1273.0 nm MgF2 film and a 36.0 nm Ag metallic film were grown. Using an oscillating quartz controller and an optical monitor, the thickness of each film was monitored. The vacuum pressure was 106 mb during the evaporation. The evaporation rates were in the range of 0.5 nm/s. The experimental angular reflection and scattering measurements were obtained by using an experimental setup reported elsewhere [7], while the dispersion relation was obtained from the spectral reflection curves between 450 and 900 nm wavelength range.
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3. Results
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higher than those for Ag, so we assume that most of the roughness comes from the Ag/MgF2 interface.
The system supports six TM waveguide modes and the same number for TE at k ¼ 632:8 nm. The magnesium fluoride film is surrounded by a thicker (substrate) and thinner silver metallic films. The thicker silver metallic film acts like an opaque mirror and the other film like a semitransparent one. This system resembles an opaque etalon or Fabry–Perot interferometer, so the explanation of the experimental results shown in this paper are only based on optical physic. It has nothing to do with geometrical optics at all. The roughness is attributed to the material as well as the grown method used. At this point, the properties of the Ag and MgF2 interface roughness as a function of thickness were previously studied by Mayani et al. [8]. They showed that the roughness spectrum of the MgF2 films is not a perfect Gaussian function. Other authors [9] showed that the light scattering depends on the power spectrum, and using a Gaussian roughness spectrum approximates it. The rms roughness values reported for MgF2 are
3.1. Spectra reflection Fig. 1 shows the reflection as a function of wavelength for an incident angle equal to 45.0°. The experimental results are represented by circles for p polarization, and by triangles for s polarization of the incident light, the theoretical results by solid and dashed curves for p and s polarization, respectively. The experimental minima deðpÞ ðpÞ ðpÞ tected at k4 ¼ 859:4 nm, k5 ¼ 688:1 nm, k6 ¼ ðpÞ 575:6 nm and k7 ¼ 498:2 nm correspond to the excitation of the fourth, fifth, sixth and seventh TM waveguide modes for p polarization of the incident light. The lower (higher) modes cannot be seen because they lay out of the maximum (minimum) wavelength used in this experiment. The ðsÞ ðsÞ minima detected at k4 ¼ 848:6 nm, k5 ¼ 680:7 ðsÞ ðsÞ nm, k6 ¼ 566:5 nm and k7 ¼ 490:9 nm correspond to the excitation of the fourth, fifth, sixth and seventh TE mode for s polarization.
θi= 45.
1.0 0.9 0.8
Reflection
0.7 0.6 0.5
4
5
0.4
6
7
0.3 0.2 0.1 0.0 450
500
550
600
650
700
750
800
850
900
Fig. 1. Reflection versus wavelength for an incident angle hi ¼ 45:0° for p and s polarization of the incident light. The system is the planar air/Ag(36.0 nm)/MgF2 (1273.0 nm)/Ag(250.0 nm substrate). The circles and the triangles represent the experimental results for p and s polarization of the incident light, respectively. The solid and dashed curves correspond to the theoretical data for p and s polarization, respectively. The minima displayed correspond to excitation of fourth, fifth, sixth and seventh guided modes for both p and s polarization.
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The theoretical reflection given by Eq. (4) in [7] was calculated by using a double layer model, air/ metal/dielectric/metal, with planar, parallel and smooth interfaces. We neglected the last term. The dielectric material was considered non-dispersive and its refractive and absorption indexes were fixed to nMgF2 ¼ 1:46 and kMgF2 ¼ 104 , respectively, and its thickness d3 ¼ 1273:0 nm. The optical constants of silver were taken from Palik [10]. The numerical calculations were carried out to fit the fifth mode for each polarization. The actual refractive index of MgF2 changes when the wavelength is varied. That means, it is weakly dispersive. Therefore, there is a slightly difference between positions of the minima k6 and k7 displayed by theory and experiment. When the incident angle is increased, the positions of the minima are shifted toward lower wavelength. 3.2. Dispersion relation The positions of the minima displayed in the spectra reflection curve can be used to obtain the ‘‘optical’’ dispersion relation. Several spectral curves were obtained. The incident angle was varied from 10° up to 65° with an increment of 5°. Fig. 2(a) shows the TM dispersion relation of the electromagnetic modes of the metal–dielectric– metal system. Circles represent the experiment and the solid curve represents the theory. The straight solid and dashed lines represent the light line in the dielectric medium and in the vacuum, respectively. The circles were obtained from the spectral positions of the minima, for example, the excitation ðpÞ wavelength of the fifth TM mode is k5 ¼ 688:1 nm, see Fig. 1, the value of wavelength is changed to ðpÞ energy with the relation energy (eV) ¼ 1240.0/k5 , ðpÞ where k5 is in nanometer. Then, the wave number of the mode is obtained by using the coupling ðpÞ condition q ðnm1 Þ ¼ ½2p=k5 ðnmÞ sin hi , where hi is the incident angle, whose value is 45.0°. And so on for each minimum. The theoretical dispersion relation is obtained by using Eq. (1) in [1] (see also [11] ). In this model semi-infinite metallic media surrounds the dielectric film, with planar, parallel and smooth inter-
faces. The numerical curve was obtained by using the same parameters as in the section above. The ‘‘optical’’ dispersion relation and the theoretical one are in excellent agreement, without being modified by the roughness. This direct illumination method to inject light into the guide is restricted by the excitation of modes at the left of the light line in vacuum, oscillatory mode and in this paper we are only interested in them. Other methods, prism coupling, grating coupling or lateral injection, must be used to excite modes at the right of this line and those are not considered in this paper. Fig. 2(b) shows the dispersion relation of the TE electromagnetic modes. For this case all modes are waveguide-type and for TM there is one surface-plasmon-type, which appears on the right of the light line at the dielectric medium and it is observed only in the theoretical curve of Fig. 2(a). Similar behavior is observed experimentally in agreement with the theory. 3.3. Angular reflection The reflection as a function of the incidence angle is shown in Fig. 3 for a wavelength k ¼ 632:8 nm. The experimental results are represented by circles for p polarization, and by triangles for s polarization of the incident light. The theoretical results are represented by solid and dashed curves for p and s polarizations, respectively. The theoretical data were calculated by using Eq. (4) of [7] for a rms height of the roughness h3 ¼ 1:0 nm and a correlation length r3 ¼ 20:0 nm. The dielectric 2 constant of silver is eAg ¼ ð0:134 þ i3:985Þ for the wavelength used and its thickness is dAg ¼ 36:0 nm. The remainder parameters used to fit the experimental data were given in Section 3.1. ðpÞ ðpÞ The minima at h6 ¼ 17:4° and h5 ¼ 61:04° correspond to the excitation of the sixth and the fifth TM waveguide modes supported by the sysðpÞ tem, while, the minima at h6 ¼ 16:0° and ðpÞ h5 ¼ 56:9° are related to the excitation of the sixth and the fifth TE waveguide modes. The minima are not deeper for s polarization because the samples are optimized to reach very deep minima for p polarization. Therefore, the fifth minimum for s polarization is shallower. If we
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th
8
3.00
231
th
7 6th th 5 th 4
2.75 2.50 2.25
Energy (eV)
2.00 1.75 1.50 1.25 1.00 0.75 0.50 0.25 0.00 0.000
0.002
0.004
0.006
0.008
0.010
0.012
0.014
0.016
-1
q(nm )
(a)
TE modes
th
8
3.00
th
7
2.75 2.50
th
6 th 5 th 4
2.25
Energy (eV)
2.00 1.75 1.50 1.25 1.00 0.75 0.50 0.25 0.00 0.000
0.002
0.004
0.006
0.008
0.010
0.012
0.014
0.016
(b) Fig. 2. (a)The dispersion relation of the TM electromagnetic modes of a Ag/MgF2 (1273.0 nm)/Ag system is shown. Circles represent the experiment and the solid curve represents the theory. The straight solid and dashed lines represent the light line in the dielectric medium and in the air, respectively. The modes on left of the light line in the dielectric medium are the surface-planon type-modes. (b) Same as in (a) but for TE modes. In this case all modes are waveguided-type.
compare theory with experiment, the angular positions of the minima are approximately the same. The width of the experimental curve is greater than those of theoretical one, probably because the dielectric film is not quite uniform. The resonant condition occurs when the reflection at the system reaches a minimum at some specific angular position. At this point, the incident energy divides into several parts: a small
fraction is reflected, then part of the energy is absorbed by the system and is transformed into heat, then another part is conducted along the dielectric layer and finally the remaining part is scattered from the rough interfaces of the system. In this condition the scattering is named as resonant scattering. Important comments about the behavior of the theoretical specular reflection follow. By fixing the
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0.9 0.8
Reflection
0.7 0.6
0
56.94
0.5 0
61.04
0.4 0
15.95
0.3
0
17.42
0.2 0.1 0.0 5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
Fig. 3. Reflection as a function of the incidence angle at air/Ag(36.0 nm)/MgF2 (1273.0 nm)/Ag(250.0 nm substrate) system for a wavelength of 632.8 nm. The circles and the triangles represent the experimental results for p and s polarization of the incident light, respectively. The solid curve and the dashed one correspond to the theoretical data for p and s polarization. The minima (circles) at ðpÞ ðpÞ h5 ¼ 61:04° and h6 ¼ 17:42° correspond to excitation of the fifth and sixth TM-guided modes, respectively. The dips (triangles) ðsÞ ðsÞ at h5 ¼ 56:94° and h6 ¼ 15:95° are related to excitation of the fifth and sixth TE-guided mode, respectively.
dielectric thickness, an increment of nMgF2 makes the positions of the minima change toward greater angles and the separation between them increases. On the other hand, the angular positions of the excited guided modes are shifted toward larger angles when the value of dMgF2 film is increased. The kMgF2 does not cause any significant variation in the form of the curve neither in the position nor in the depth of the minima. If the thickness of the thinner Ag film is increased the width of the minima diminishes, whereas its deepness decreases. As a matter of fact there is not a unique solution. 3.4. Scattered light for p polarization A comment on the experimental measurements of scattered light is necessary: The scatterometer cannot measure the incident and reflected beam directions. Around these positions the experimental curve looks flat because the signal is very intense and it exceeds the maximum sensibility allowed for the device. Based on Eq. (27) of [9], the scattered theoretical data were calculated by using a second-order perturbation approach and a Gaussian roughness
spectrum, so it only takes account of single scattering effects. In this model, the roughness is at the inner dielectric–metal interface with a rms height h3 ¼ 1:0 nm and a correlation length r3 ¼ 20:0 nm. The remainder parameters were given in the preceding section. Fig. 4(a) shows the light scattering as function of the scattered angle, for a fixed incident angle ðpÞ of h6 ¼ 17:42°, for p polarization of the incident light with wavelength k ¼ 632:8 nm (the red visible line of a He–Ne laser at 10 mW). This angle corresponds to the direct excitation of the sixth TM mode supported by the guide. The small circles represent the experimental data and the solid curve represents the theoretical one. The intensity of the scattered light increases toward positive angles, that is, at the direction of the reflected beam. The peaks that are supposed to be noticeable at the positions of the incident and reflected beams could not be seen in the experimental curve because they are located at the directions of the incident and the reflected light, and they are well described by the theoretical data. The peaks at ðpÞ hs ¼ h5 ¼ 61:04° correspond to the excitation, via roughness, of the fifth TM modes traveling in
A.S. Ramırez-Duverger, R. Garcıa-Llamas / Optics Communications 227 (2003) 227–235
θi=-17.42
-2
Scattered light reflection (p pol)
10
-3
10
233
0
-θi
+θi
-61.04
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61.04
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-6
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(a)
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θi=0
-2
Scattered light reflection (p pol)
10
-3
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17.42
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-5
10
-6
10
-7
10
-90 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 90
(b) ðpÞ
Fig. 4. (a) Light scattering versus scattering angle for an incident angle hi ¼ h6 ¼ 17:42°, for p polarization of the incident light and for a wavelength of 632.8 nm. Both experimental (circles) and theoretical (solid curve) data are displayed. This angle corresponds to the ðpÞ direct excitation of the sixth TM-guided mode. We observed peaks at hs ¼ h5 ¼ 61:04° due to single scattering effect and they are related to the indirect excitation of the fifth TM-guided mode via the roughness of the system. (b) Same as (a) but for hi ¼ 0°. The angle ðpÞ ðpÞ does not correspond to the excitation of any mode. Anyway, we observed four weak peaks at h5 ¼ 61:04° and at h6 ¼ 17:42° due to single scattering effect. They are related to the indirect excitation of the sixth and fifth TM-guided mode.
opposite directions, and they are displayed on both experimental and theoretical curves. From the experimental point of view, the ‘‘peaks’’ are fringes with ring shapes that are out of the plane of incidence. These rings are not modeled on the theoretical solution because the roughness surface profile is assumed as one-dimensional, whereas the ‘‘natural’’ roughness at the
interfaces of the sample is two-dimensional. Further calculation must be performed to take account of this effect. Single scattering, as stated for the second-order perturbation in the surface profile theoretical solution, produces the peaks corresponding to the excitation of the guided modes; this excitation produces the enhancement of the scattering as a
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consequence of the resonant condition. Then, a large part of the input energy is observed outside the system as scattered energy into the reflection medium. This energy is proportional to the square of the rms height and wavelength ratio. No evidence of second-order scattering effect was detected. To obtain this kind of effect, the rms height must be increased artificially. Although light scattering occurs in all of incidence angles, the existence of guided modes and its excitation amplify the scattering at the resonant angles, being the reason of the signalÕs enhancement. The roughness parameters were inferred by fitting the theoretical curve to the experimental data. The parameters obtained this way are consistent with those reported in the literature. We did not expect to obtain very accurate data because the simplicity of the model, which considers only one interface with 1D-roughness. Fig. 4(b) plots the same as marks Fig. 4(a) but for normal incidence. This angle does not correspond to the direct excitation of any guide modes and only weak light scattering is expected. In this figure, the experiment (circles) and fit (solid curve) are shown, and it is possible to observe four low ðpÞ intensity peaks, at hs ¼ h5 ¼ 61:04° and hs ¼ ðpÞ h6 ¼ 17:42°. Experimentally only in this situ-
3.5. Scattered light for s polarization In this section we analyzed the same sample by changing only the polarization of the incident θi=-56.94
-2
10
Scattered light reflection (s pol)
ation the four peaks can be seen. On the other hand the theoretical data always displayed them. For normal incidence, the incident and reflected beams are detected in the same direction, and the spurious signal is detected in a minor angular arc, about 6.0° around 0°. In this situation, the interaction light-guided mode is weak, and the heights of the peaks are very low. Therefore, the excited waveguide modes are detected just above the background signal. For this reason, the excited modes are not detected with good definition. If we compare the two measured curves in resonance to the curve off resonance, we see that the scattered light increases when the light arrives at any angle of excitation of the waveguide modes; the peaks are very sharp and defined. Out of resonance, the intensity of the scattered light and the heights of the peaks are reduced. We observe that the angular positions of the peaks in the two Figs. 4(a) and (b) are alike, which demonstrates that the angular positions of the excited waveguide modes do not change when the incident angle is varied.
-θi
+θi
-3
10
0
0
15.95
-15.95 -4
10
-5
10
-6
10
-7
10
-80
-60
-40
-20
0
20
40
60
80
egrees) ðsÞ
Fig. 5. Same as Fig. 4(a) but for hi ¼ h5 ¼ 56:94° and for s polarization. This angle corresponds to the direct coupling of the fifth ðsÞ TE-guided mode and incident light. As in p polarization case, we observed peaks at hs ¼ h6 ¼ 15:95° due to single scattering effect.
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light. All TE electromagnetic modes are waveguide-type. Scattered light versus the scattered angle is shown in Fig. 5 for s polarization of the incident ðsÞ light and for a fixed incident angle of h5 ¼ 56:94°, there is a resonant condition of excitation of the fifth TE waveguide mode. Circles represent the experimental curve and the solid line represents the fit. The peaks that correspond to the positions of excitation of sixth TE waveguide mode are ðsÞ shown at hs ¼ h6 ¼ 15:95°, for both experimental and theoretical curves. The remainder modes cannot be excited because they are on the right of the light line at vacuum. The enhancement of scattering found in resonant condition diminished the efficiency for injecting light into the guide. Scattering light reflection in p polarization is one order of magnitude higher than the same response in s polarization because the thickness of the metallic film in the system is optimized for p polarization. This effect can be used to create reflection polarizer to some specific angle; for example, if the system is lit by non-polarized light to an incident angle of 61.04°, it is possible to obtain s-polarized reflection light.
the thickness of the metallic wall, we can separate an incident light from a reflected and guided wave with any desired ratio of reflected light to guide it. The angular distribution of the scattered light shows peaks caused by single-order scattering effects. The diffused light from the system shows ðpÞ four peaks at angles given by h5 ¼ 61:04°, ðpÞ ðsÞ ðsÞ h6 ¼ 17:42° (or h5 ¼ 56:94° and h6 ¼ 15:95°) due to the coupling between the fifth and sixth TM (or TE) waveguide modes and p (or s) polarized incident light via roughness of the interfaces. For s and p polarizations, the angular positions of the peaks are independent on the incident angle and due to single-order scattering. All features in the scattering response are enhanced by resonant conditions, and the efficiency for injecting light into the guide is reduced.
4. Conclusions
References
The experimental results of the reflection at a planar guide air/Ag/MgF2 /Ag structure and the light scattering from the mentioned system for p and s polarized light are reported We found dips in the reflection as a function of the incident angle as well as the wavelength. This is because of excitation of waveguide modes in the MgF2 film. The dispersion relation of such structure was measured in 450–900 nm wavelength range and theory matches experiment quite well. The experimental ‘‘optical’’ dispersion curves of modes are calculated through the spectral positions of the minima displayed in the reflection curve. The deepness of these minima is strongly dependent on the thickness of the metal layer, and this parameter allows controlling the amount of energy injected to the guide. By selecting appropriately
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Acknowledgements This work was partially supported by the Consejo Nacional de Ciencia y Tecnologıa under Grant 35223-E.