- Email: [email protected]
Light scattering in random planar structures supporting guiding modes
G Model
ARTICLE IN PRESS
IJLEO 56702 1–4
Optik xxx (2015) xxx–xxx
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Light scattering in random planar structures supporting guiding modes
1
2
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Q1
Rabi Rabady a,∗ , Ivan Avrutsky b a
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b
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Jordan University of Science and Technology, Irbid, Jordan Wayne State University, Detroit, MI, United States
6
a r t i c l e
7 19
i n f o
a b s t r a c t
8
Article history: Received 15 February 2015 Accepted 30 October 2015 Available online xxx
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Keywords: Waveguide gratings Optical resonance Scattering Guided modes
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1. Introduction
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We study light scattering by surface roughness assisted by excitation of guided modes in dielectric layers. The same roughness might result in a very different scattering efficiency depending on whether and how the guided modes participate in scattering. Enhanced scattering to modes with a different modal number and to modes propagating backward is predicted and observed. © 2015 Published by Elsevier GmbH.
Study of light scattering in planar structures supporting guided modes confined in dielectric layers is important from practical point of view due to their application in optoelectronic devices used in communication and optical sensor systems. The scattering problem is also related to fundamental issues such as weak localization in two- and three-dimensional systems, local field enhancement resulting in giant nonlinear response, etc. Moreover, light scattering problem became attractive for the lasing in random media. It was rather surprising that random scattering and random trajectories of light waves not necessarily would result in de-phasing of the scattered waves [1]. Light scattering in waveguides-gratings structures has been studied by many researchers [2–8]. Various approaches like the Fourier–Bloch mode method [8] and the guided mode expansion methods [9] were developed in order to understand such effect. Moreover, the backscattered light that is induced by the waveguide surface roughness was also modeled for planar and channel waveguides [10,11]. In this paper we present an experimental evidence of enhanced scattering in two-dimensional systems. Namely, we show that a light wave that is confined to the guiding layer would more likely scatter to another guided wave rather than scattered out of the waveguide. We also show a case of interference-induced suppression of scattering out of the waveguide. It means that even
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Q2
∗ Corresponding author. Tel.: +962 796622279. E-mail address: [email protected] (R. Rabady).
in a random medium trajectory of scattered waves stays mainly in the plane of the waveguide. Intuitively, it would be easier to get closed trajectories in two dimensions compared to the threedimensional case. Thus, the random lasers that are realized in planar waveguides will more likely show lower threshold and, due to light confinement in the waveguide, exhibit higher brightness. This justifies the importance of studying the scattering in planar waveguides.. 2. Theory Enhanced scattering itself is usually explained by stronger light intensity at the surface associated with excitation of guided modes. However, it is only a half of explanation. Really, the Fourier components of random surface roughness in general would produce relatively uniform scattering in wide range of angles. The field enhancement cased by excitation of guided modes can explain the stronger scattering but it cannot be responsible for the arc-shaped scattering. The arcs indicate that there is a strong scattering within the plane of the guiding structure with subsequent out-coupling of the scattered waves. In other words, if a given Fourier component of the surface roughness happens to provide a resonant coupling between the modes of a planar structure, the scattering is strong. Another Fourier component with close period and orientation, but beyond the resonant condition, may provide only a negligibly small contribution to scattering because there is no suitable guided mode final state for the scattering wave. There could be also the Fourier components that provide scattering to waves propagating in free space. This scattering must be much weaker
http://dx.doi.org/10.1016/j.ijleo.2015.10.226 0030-4026/© 2015 Published by Elsevier GmbH.
Please cite this article in press as: R. Rabady, I. Avrutsky, Light scattering in random planar structures supporting guiding modes, Optik - Int. J. Light Electron Opt. (2015), http://dx.doi.org/10.1016/j.ijleo.2015.10.226
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compared to the scattering to guided modes in order to achieve the arc-shaped scattering picture. In other words, intensity of scattering on surface grating strongly depends on the final state of the scattered wave, namely, whether the scattered wave is a guided mode or wave in the threedimensional space. There is a clear analogy for this phenomenon in quantum mechanics. The gold Fermi rule for electron transitions in a periodical field is probably the first known formulation of the relationship between the probability of the process and the density of states in the final state. According to this rule, the probability of electron transition is proportional to the density of states in the final state. Consequently, in a random field characterized by wide spectrum of field oscillations, only certain spectral components of the perturbation field affect the quantum system. These components provide transition to a localized energy level with delta-like density of states. Similar rules are applied to many different processes involving different kinds of waves. For example, intensity of spontaneous emission is proportional to photon density of states in the final state and hence becomes controlled in microcavities and photonic bandgap structures. Scattering of phonons in superlattices is also controlled by a similar mechanism. In the case of light scattering of guided modes, the surface roughness plays the role of the perturbation that serves as a coupling between the incident and the scattered wave. One can treat a rough surface as a superposition of the Fourier components of the surface profile, that is, as a set of diffraction grating with all possible periods and orientations. The gratings with very similar periods and orientation can nevertheless have very different contribution to the scattering of the guided mode depending on whether or not the final state is another guided mode. The enhanced scattering is easily observed and, undoubtedly, has been noticed by researchers working with dielectric waveguides. The arcs around the reflected beam appear in both prism and grating coupler schemes. They often used as a clear indicator of a waveguide excitation. What we emphasize in this paper is that the shape of the scattering pattern provides a proof of guided mode to guided mode scattering being much stronger than scattering out of the guiding film. The non-resonant and resonant scattering cases are illustrated by wave-vector diagrams in Fig. 1. The direction of scattered wave is determined by the projection of incident wave-vector on the plane of the waveguide and the roughness wavevector. When no guided mode involved (Fig. 1a), it simply gives
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k s = k i + k r
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(1)
We refer to this case as direct scattering on the surface roughness. It is certainly non-resonant process and there is no reason
to expect that small variations of the direction of magnitude of k r could significantly affect the intensity of scattering. In the presence of a coupling grating, a waveguide mode with · k0 n∗ is excited in the first order of diffraction at wavevector k w = w the coupling grating with wavevector K when the phase matching condition is satisfied. The excitation of guided modes condition is as follow: · k0 sin() = w · n∗ k0 , K + p
= 2/ is the grating’s wavevector, is the grating’s where |K| period, k0 = 2/ is the vacuum wavevector of light, is the wavelength, is the resonant angle measured from the normal to the is the unit vector along the projection of the incident sample, p is the unit wavevector wavevector on the plane of the sample, w along the guided mode propagation direction, and n* is the modal index. The wave scattered by the surface roughness component with wave-vector k r becomes observable due to the negative first order of diffraction (Fig. 1b), which eventually results in the same relation between the wave-vectors of the incident and the scattered waves as in the case of the direct scattering as in Eq. (1). Although this process involves excitation of the waveguide mode, we refer to it as non-resonant since the final state of scattering by k r is not a waveguide mode. Finally, there could be a situation, when k w + k r = k w which is the wavevector of another waveguide mode, propagating at different direction. The scattering direction (Fig. 1c) is still determined by Eq. (1), but the intensity is expected to be much stronger in accor | = k n∗ means dance with the above consideration. Note, that |k w 0 that the ends of vectors kw are placed along the arc with radius k0 n*, which eventually is transferred into another arc in the directions of scattered waves k s = k w − k g shown as a dashed line in Fig. 1c. Note also, that non-resonant scattering shown in Fig. 2a and the resonant one illustrated by Fig. 1c have only one significant difference: the surface roughness component k r in the non-resonant case couples three-dimensional waves in the free space (k i and k s ), while in the resonant case it couples two-dimensional guided modes ). The arc-shaped scattering confirms that the resonant (k w and k w mechanism is much stronger despite the fact that it involves diffraction on the coupling grating, diffraction on the surface roughness, and diffraction on the coupling grating while the non-resonant scattering directly couples the incident and the scattered waves. This is rather important observation: the same kind of surface roughness may cause either negligible scattering or very strong scattering depending on whether and how the guided modes are involved into the scattering process. The above qualitative explanation can be also supported by a simple numerical consideration. In the non-resonant state, the electric field strength of scattered wave is proportional to k0 r , where r is the amplitude of a Fourier component of surface roughness with wave-vector k r . Efficiency of the non-resonant scattering NR becomes proportional to the second order of surface roughness amplitude: NR ∝ (k0 r )
Fig. 1. (a) Wavevectors representation that corresponds waveguide mode excitation in planar waveguide with imperfections; (b) wavevectors representation for the mode coupling by the diffractive gratings; (c) wavevectors representation for the evolution of the arc-shape scattered light that associates the excitation of guided mode.
(2)
2
(3)
In general, light scattering by surface roughness treated as Fraunhoffer diffraction is one of optical approaches for the evaluation of surface roughness. Exact formula would contain few other factors taking in order to account for the polarization of the waves, angles of incidence and scattering, indexes of refraction and thickness of the layers composing the structure. Those details are beyond the scope of this paper, so we keep only the leading term relating the efficiency of scattering to the roughness amplitude. In the case of the resonant scattering, when both incident and scattered waves are the guided modes, the Fourier component of the surface
Please cite this article in press as: R. Rabady, I. Avrutsky, Light scattering in random planar structures supporting guiding modes, Optik - Int. J. Light Electron Opt. (2015), http://dx.doi.org/10.1016/j.ijleo.2015.10.226
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the same plane. Thus, the scattering process becomes distributed in time and space, and it is accumulated during propagation of the guided mode along the rough surface. Similar enhanced scattering is expected to take place in a variety of different two-dimensional physical systems such as surface acoustic waves, electrons and holes in quantum wells, etc.
3. Experiments and results
Fig. 2. (a) The scattering arc shape pattern of sample #1 when the TE1 mode was excited at incident angle 1.25◦ , grating period = 394 nm, the two effective refractive index of the electric polarized n∗TE0 = 1.70, n∗TE1 = 1.628. (b) Resonant and nonresonant scattering intensity that resulted from slicing the image in (a) horizontally at pixel number 80. Image resolution is 220 × 150 pixels.
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roughness provides efficient Bragg coupling between them. The coupling coefficient is proportional to k0 r /h* where h* is the effective thickness of the waveguide, typically of the order of or, maybe, several times greater for weakly guiding structures; we again omit unimportant factors. The coupling efficiency depends on the grating length L as = tanh2 (L), which results in a leading factor of resonant scattering R ∝
2 r k0
h∗
L
(4)
Comparison of (3) and (4) gives R ∝ NR
L 2 h∗
(5)
In the case of rough surface the parameter L must be of the order of correlation length of surface roughness. Formula (5) shows enhancement of the resonant scattering. Exact formulas similar for (3)–(5) can be derived using approaches developed in integrated optics, and it will be published elsewhere. This could eventually result in a new method for quantitative analysis of surface roughness in structures supporting guided modes. The very origin of the scattering enhancement is in the dimensionality of the scattering. In the case of the direct scattering an incident wave in three-dimensional space hits the surface with roughness. As long as the Poynting vector of the incident wave has non-zero component normal to the surface, the scattering becomes a momentarily event. In the two-dimensional case, the incident and the scattered guided modes as well as the roughness itself areall in
In the experiment, we used TiO2 /SiO2 waveguides prepared by RF magnetron sputtering from a dual source. Relative content of silica and titania is controlled by relative RF power supplied to the sources. We found that the mix of titania and silica with relatively low content of titania has very uniform amorphous structure, and the waveguides made of this material reveal low optical losses [12]. After post-deposition annealing the losses become less than 1 dB/cm. At higher concentrations of titania the layer tends to be polycrystalline so that light scattering becomes very strong. Guided track in such waveguides may be as short as a few hundred micrometers, which corresponds to several dozens to few hundred dB/cm of scattering loss. For study of random planar structures we choose highly scattering waveguides. To couple light to the waveguides we fabricate a diffraction grating on each sample using deep-UV holographic scheme. The resistive grating was dry-etched in CF4 plasma, and residual photoresist was removed in oxygen plasma. To observe scattering, we install the sample on a rotation stage and adjust incident angle to provide phase matching in excitation of guided modes by a He–Ne laser beam. The scattering is then observed as very specific arcs around the reflected and transmitted beam. When incident angle is about one degree off the resonance condition given in Eq. (2), light is scattered by the nonuniformities of the sample; such scattering is rather isotropic and weak. Once the guided mode is excited, the scattering becomes much stronger and shows the arcs. Although the scattering mechanism is a multistage which includes the excitation of guided mode, scattering inside a waveguide, and re-emission of the scattered waves, the efficiency of scattering may be much stronger compared to direct scattering at the surface roughness. The above consideration is very general and it allows for predicting of the scattering peculiarities in planar structures supporting guided modes. For example, if the waveguide excitation according to (1) happens to be at relatively small angle , one might observe both forward propagating and backward propagating waves in the scattered field. The diagram in Fig. 2(a) illustrate how it happens; only the mode propagating from left to right in the figure is resonantly excited by the diffractive gratings for an electric polarized light. The backward propagating mode appears due to strong scattering of the resonantly excited mode on the waveguide imperfections. Then the +1st order of diffraction of the grating brings light to the arc that is shown in the scattered field. Depending on whether the grating’s wavevector is shorter or longer than the mode wavevector, the arcs may cross or separate. Fig. 2(b) the relative intensity of the resonant and nonresonant scattering versus horizontal pixel location, which is taken by reading the 220 × 150 pixels image in Fig. 2(a) using Matlab software and then plotting a horizontal slice that is taken from raw number 80 of the resulted matrix. Another interesting case is shown in Fig. 3(a); in this sample the fundamental mode is excited exactly at normal incidence angle for an electric polarized light. Thus, the forward and backward propagating modes were excited simultaneously in the ±1st orders of diffraction. The second mode is clearly shown by the pair of external arcs. The remarkable feature is the suppression of scattering for the fundamental mode at small angles. In the crossing area the arcs virtually vanish. More accurately, the out-coupling
Please cite this article in press as: R. Rabady, I. Avrutsky, Light scattering in random planar structures supporting guiding modes, Optik - Int. J. Light Electron Opt. (2015), http://dx.doi.org/10.1016/j.ijleo.2015.10.226
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ARTICLE IN PRESS R. Rabady, I. Avrutsky / Optik xxx (2015) xxx–xxx
efficiency in second order distributed feedback lasers. Fig. 3(b) and (c), the relative intensity of the resonant and nonresonant scattering versus horizontal pixel location, which is taken by reading the 430 × 160 pixels image in Fig. 3(a) using Matlab software and then plotting a horizontal slice take from raw number 130 and 260 of the resulted matrix, respectively. In Fig. 2(b) the intensity of the resonant scattering is clipped because of the imaging limitation, however, by simple extrapolation the peaks were restored, and it was found that the resonant scattering can be stronger than the nonresonant scattering by a factor of 21; whereas, for the second case, shown in Fig. 3 (b) and (c), where the resonant scattering arcs partially overlap destructively, such factor does not exceed 3.
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4. Conclusion In conclusion, we have demonstrated experimentally that in a random planar structure that supports guided modes, once excited, the guided mode is likely to be scattered to another guided mode rather than scattered out of the structure. This observation is in line with the wave scattering theories, which predicts that the efficiency of scattering is proportional to the density of states in the final state. The variety of scattering patterns around the reflected and transmitted beam is well explained by the vector diagrams. Enhanced scattering to a waveguide mode with a different modal number as well as the back scattering were predicted and observed. Possible practical application of the discussed phenomenon includes quantitative evaluation of surface roughness. For example, measuring of weak Fraunhoffer diffraction could be difficult for very smooth surface, while scattering involving modes of dielectric layers is strongly enhanced. The enhanced scattering might play essential role in many experiments dealing with planar structures supporting guided modes in dielectric layers. The reported results expect an improved performance of the random planar lasers compared to the three-dimensional random lasers.
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References
Fig. 3. (a) The arc shape scattering pattern for sample #2 when the TE0 mode was excited at incident angle 5.36◦ , grating period = 394 nm, the two effective refractive index of the electric polarized light n∗TE0 = 1.70, n∗TE1 = 1.633. (b) Resonant and nonresonant scattering intensity that resulted from slicing the image in (a) horizontally at pixel number 130, (c) resonant and nonresonant scattering intensity that resulted from slicing the image in (a) horizontally at pixel number 260. Image resolution is 430 × 160 pixels.
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of scattered light is vanishing due to perfect phase relationship between the forward and backward propagating modes and hence between the scattered waves produce by these modes. Similar phenomenon is known to be responsible for low first order emission
[1] H. Cao, Lasing in random media, Top. Rev. Waves Random Media 13 (2003) R1–R39. [2] T. Barwicz, H. Haus, Three-dimensional analysis of scattering losses due to sidewall roughness in microphotonic waveguides, J. Lightwave Technol. 23 (2005) 2719–2732. [3] D. Marcuse, Mode conversion caused by surface imperfections of a dielectric slab waveguide, Bell Syst. Tech. J. 48 (1969) 3187–3215. [4] C. Ciminelli, F. Dell’Olio, V. Passaro, M. Armenise, Fully three dimensional accurate modeling of scattering loss in optical waveguides, Opt. Quant. Electron. 41 (2009) 285–298. [5] D.G. Hall, Scattering of optical guided waves by waveguide surface roughness: a three-dimensional treatment, Opt. Lett. 6 (1981) 601–603. [6] S. Mazoyer, J.P. Hugonin, P. Lalanne, Disorder-induced multiple scattering in photonic-crystal waveguides, Phys. Rev. Lett. 103 (2009) 063903. [7] V. Savona, Electromagnetic modes of a disordered photonic crystal, Phys. Rev. B 83 (2011) 085301. [8] G. Lecamp, J.P. Hugonin, P. Lalanne, Theoretical and computational concepts for periodic optical waveguides, Opt. Express 15 (2007) 11042–11060. [9] D. Gerace, L.C. Andreani, Disorder-induced losses in photonic crystal waveguides with line defects, Opt. Lett. 29 (2004) 1897–1899. [10] F. Ladouceur, L. Poladian, Surface roughness and backscattering, Opt. Lett. 21 (1996) 1833–1835. [11] D. Melati, A. Melloni, F. Morichetti, Real photonic waveguides: guiding light through imperfections, Adv. Opt. Photonics 6 (2) (2014) 156–224. [12] R. Rabady, I. Avrutsky, Titania, silica and tantalum pentoxide waveguides and optical resonant filters prepared by RF magnetron sputtering and annealing, Appl. Opt. 44 (January (3)) (2005) 378–383.
Please cite this article in press as: R. Rabady, I. Avrutsky, Light scattering in random planar structures supporting guiding modes, Optik - Int. J. Light Electron Opt. (2015), http://dx.doi.org/10.1016/j.ijleo.2015.10.226
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ARTICLE IN PRESS
IJLEO 56702 1–4
Optik xxx (2015) xxx–xxx
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Light scattering in random planar structures supporting guiding modes
1
2
3
Q1
Rabi Rabady a,∗ , Ivan Avrutsky b a
4
b
5
Jordan University of Science and Technology, Irbid, Jordan Wayne State University, Detroit, MI, United States
6
a r t i c l e
7 19
i n f o
a b s t r a c t
8
Article history: Received 15 February 2015 Accepted 30 October 2015 Available online xxx
9 10 11 12 13
18
Keywords: Waveguide gratings Optical resonance Scattering Guided modes
20
1. Introduction
14 15 16 17
We study light scattering by surface roughness assisted by excitation of guided modes in dielectric layers. The same roughness might result in a very different scattering efficiency depending on whether and how the guided modes participate in scattering. Enhanced scattering to modes with a different modal number and to modes propagating backward is predicted and observed. © 2015 Published by Elsevier GmbH.
Study of light scattering in planar structures supporting guided modes confined in dielectric layers is important from practical point of view due to their application in optoelectronic devices used in communication and optical sensor systems. The scattering problem is also related to fundamental issues such as weak localization in two- and three-dimensional systems, local field enhancement resulting in giant nonlinear response, etc. Moreover, light scattering problem became attractive for the lasing in random media. It was rather surprising that random scattering and random trajectories of light waves not necessarily would result in de-phasing of the scattered waves [1]. Light scattering in waveguides-gratings structures has been studied by many researchers [2–8]. Various approaches like the Fourier–Bloch mode method [8] and the guided mode expansion methods [9] were developed in order to understand such effect. Moreover, the backscattered light that is induced by the waveguide surface roughness was also modeled for planar and channel waveguides [10,11]. In this paper we present an experimental evidence of enhanced scattering in two-dimensional systems. Namely, we show that a light wave that is confined to the guiding layer would more likely scatter to another guided wave rather than scattered out of the waveguide. We also show a case of interference-induced suppression of scattering out of the waveguide. It means that even
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Q2
∗ Corresponding author. Tel.: +962 796622279. E-mail address: [email protected] (R. Rabady).
in a random medium trajectory of scattered waves stays mainly in the plane of the waveguide. Intuitively, it would be easier to get closed trajectories in two dimensions compared to the threedimensional case. Thus, the random lasers that are realized in planar waveguides will more likely show lower threshold and, due to light confinement in the waveguide, exhibit higher brightness. This justifies the importance of studying the scattering in planar waveguides.. 2. Theory Enhanced scattering itself is usually explained by stronger light intensity at the surface associated with excitation of guided modes. However, it is only a half of explanation. Really, the Fourier components of random surface roughness in general would produce relatively uniform scattering in wide range of angles. The field enhancement cased by excitation of guided modes can explain the stronger scattering but it cannot be responsible for the arc-shaped scattering. The arcs indicate that there is a strong scattering within the plane of the guiding structure with subsequent out-coupling of the scattered waves. In other words, if a given Fourier component of the surface roughness happens to provide a resonant coupling between the modes of a planar structure, the scattering is strong. Another Fourier component with close period and orientation, but beyond the resonant condition, may provide only a negligibly small contribution to scattering because there is no suitable guided mode final state for the scattering wave. There could be also the Fourier components that provide scattering to waves propagating in free space. This scattering must be much weaker
http://dx.doi.org/10.1016/j.ijleo.2015.10.226 0030-4026/© 2015 Published by Elsevier GmbH.
Please cite this article in press as: R. Rabady, I. Avrutsky, Light scattering in random planar structures supporting guiding modes, Optik - Int. J. Light Electron Opt. (2015), http://dx.doi.org/10.1016/j.ijleo.2015.10.226
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compared to the scattering to guided modes in order to achieve the arc-shaped scattering picture. In other words, intensity of scattering on surface grating strongly depends on the final state of the scattered wave, namely, whether the scattered wave is a guided mode or wave in the threedimensional space. There is a clear analogy for this phenomenon in quantum mechanics. The gold Fermi rule for electron transitions in a periodical field is probably the first known formulation of the relationship between the probability of the process and the density of states in the final state. According to this rule, the probability of electron transition is proportional to the density of states in the final state. Consequently, in a random field characterized by wide spectrum of field oscillations, only certain spectral components of the perturbation field affect the quantum system. These components provide transition to a localized energy level with delta-like density of states. Similar rules are applied to many different processes involving different kinds of waves. For example, intensity of spontaneous emission is proportional to photon density of states in the final state and hence becomes controlled in microcavities and photonic bandgap structures. Scattering of phonons in superlattices is also controlled by a similar mechanism. In the case of light scattering of guided modes, the surface roughness plays the role of the perturbation that serves as a coupling between the incident and the scattered wave. One can treat a rough surface as a superposition of the Fourier components of the surface profile, that is, as a set of diffraction grating with all possible periods and orientations. The gratings with very similar periods and orientation can nevertheless have very different contribution to the scattering of the guided mode depending on whether or not the final state is another guided mode. The enhanced scattering is easily observed and, undoubtedly, has been noticed by researchers working with dielectric waveguides. The arcs around the reflected beam appear in both prism and grating coupler schemes. They often used as a clear indicator of a waveguide excitation. What we emphasize in this paper is that the shape of the scattering pattern provides a proof of guided mode to guided mode scattering being much stronger than scattering out of the guiding film. The non-resonant and resonant scattering cases are illustrated by wave-vector diagrams in Fig. 1. The direction of scattered wave is determined by the projection of incident wave-vector on the plane of the waveguide and the roughness wavevector. When no guided mode involved (Fig. 1a), it simply gives
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k s = k i + k r
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(1)
We refer to this case as direct scattering on the surface roughness. It is certainly non-resonant process and there is no reason
to expect that small variations of the direction of magnitude of k r could significantly affect the intensity of scattering. In the presence of a coupling grating, a waveguide mode with · k0 n∗ is excited in the first order of diffraction at wavevector k w = w the coupling grating with wavevector K when the phase matching condition is satisfied. The excitation of guided modes condition is as follow: · k0 sin() = w · n∗ k0 , K + p
= 2/ is the grating’s wavevector, is the grating’s where |K| period, k0 = 2/ is the vacuum wavevector of light, is the wavelength, is the resonant angle measured from the normal to the is the unit vector along the projection of the incident sample, p is the unit wavevector wavevector on the plane of the sample, w along the guided mode propagation direction, and n* is the modal index. The wave scattered by the surface roughness component with wave-vector k r becomes observable due to the negative first order of diffraction (Fig. 1b), which eventually results in the same relation between the wave-vectors of the incident and the scattered waves as in the case of the direct scattering as in Eq. (1). Although this process involves excitation of the waveguide mode, we refer to it as non-resonant since the final state of scattering by k r is not a waveguide mode. Finally, there could be a situation, when k w + k r = k w which is the wavevector of another waveguide mode, propagating at different direction. The scattering direction (Fig. 1c) is still determined by Eq. (1), but the intensity is expected to be much stronger in accor | = k n∗ means dance with the above consideration. Note, that |k w 0 that the ends of vectors kw are placed along the arc with radius k0 n*, which eventually is transferred into another arc in the directions of scattered waves k s = k w − k g shown as a dashed line in Fig. 1c. Note also, that non-resonant scattering shown in Fig. 2a and the resonant one illustrated by Fig. 1c have only one significant difference: the surface roughness component k r in the non-resonant case couples three-dimensional waves in the free space (k i and k s ), while in the resonant case it couples two-dimensional guided modes ). The arc-shaped scattering confirms that the resonant (k w and k w mechanism is much stronger despite the fact that it involves diffraction on the coupling grating, diffraction on the surface roughness, and diffraction on the coupling grating while the non-resonant scattering directly couples the incident and the scattered waves. This is rather important observation: the same kind of surface roughness may cause either negligible scattering or very strong scattering depending on whether and how the guided modes are involved into the scattering process. The above qualitative explanation can be also supported by a simple numerical consideration. In the non-resonant state, the electric field strength of scattered wave is proportional to k0 r , where r is the amplitude of a Fourier component of surface roughness with wave-vector k r . Efficiency of the non-resonant scattering NR becomes proportional to the second order of surface roughness amplitude: NR ∝ (k0 r )
Fig. 1. (a) Wavevectors representation that corresponds waveguide mode excitation in planar waveguide with imperfections; (b) wavevectors representation for the mode coupling by the diffractive gratings; (c) wavevectors representation for the evolution of the arc-shape scattered light that associates the excitation of guided mode.
(2)
2
(3)
In general, light scattering by surface roughness treated as Fraunhoffer diffraction is one of optical approaches for the evaluation of surface roughness. Exact formula would contain few other factors taking in order to account for the polarization of the waves, angles of incidence and scattering, indexes of refraction and thickness of the layers composing the structure. Those details are beyond the scope of this paper, so we keep only the leading term relating the efficiency of scattering to the roughness amplitude. In the case of the resonant scattering, when both incident and scattered waves are the guided modes, the Fourier component of the surface
Please cite this article in press as: R. Rabady, I. Avrutsky, Light scattering in random planar structures supporting guiding modes, Optik - Int. J. Light Electron Opt. (2015), http://dx.doi.org/10.1016/j.ijleo.2015.10.226
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ARTICLE IN PRESS R. Rabady, I. Avrutsky / Optik xxx (2015) xxx–xxx
3
the same plane. Thus, the scattering process becomes distributed in time and space, and it is accumulated during propagation of the guided mode along the rough surface. Similar enhanced scattering is expected to take place in a variety of different two-dimensional physical systems such as surface acoustic waves, electrons and holes in quantum wells, etc.
3. Experiments and results
Fig. 2. (a) The scattering arc shape pattern of sample #1 when the TE1 mode was excited at incident angle 1.25◦ , grating period = 394 nm, the two effective refractive index of the electric polarized n∗TE0 = 1.70, n∗TE1 = 1.628. (b) Resonant and nonresonant scattering intensity that resulted from slicing the image in (a) horizontally at pixel number 80. Image resolution is 220 × 150 pixels.
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roughness provides efficient Bragg coupling between them. The coupling coefficient is proportional to k0 r /h* where h* is the effective thickness of the waveguide, typically of the order of or, maybe, several times greater for weakly guiding structures; we again omit unimportant factors. The coupling efficiency depends on the grating length L as = tanh2 (L), which results in a leading factor of resonant scattering R ∝
2 r k0
h∗
L
(4)
Comparison of (3) and (4) gives R ∝ NR
L 2 h∗
(5)
In the case of rough surface the parameter L must be of the order of correlation length of surface roughness. Formula (5) shows enhancement of the resonant scattering. Exact formulas similar for (3)–(5) can be derived using approaches developed in integrated optics, and it will be published elsewhere. This could eventually result in a new method for quantitative analysis of surface roughness in structures supporting guided modes. The very origin of the scattering enhancement is in the dimensionality of the scattering. In the case of the direct scattering an incident wave in three-dimensional space hits the surface with roughness. As long as the Poynting vector of the incident wave has non-zero component normal to the surface, the scattering becomes a momentarily event. In the two-dimensional case, the incident and the scattered guided modes as well as the roughness itself areall in
In the experiment, we used TiO2 /SiO2 waveguides prepared by RF magnetron sputtering from a dual source. Relative content of silica and titania is controlled by relative RF power supplied to the sources. We found that the mix of titania and silica with relatively low content of titania has very uniform amorphous structure, and the waveguides made of this material reveal low optical losses [12]. After post-deposition annealing the losses become less than 1 dB/cm. At higher concentrations of titania the layer tends to be polycrystalline so that light scattering becomes very strong. Guided track in such waveguides may be as short as a few hundred micrometers, which corresponds to several dozens to few hundred dB/cm of scattering loss. For study of random planar structures we choose highly scattering waveguides. To couple light to the waveguides we fabricate a diffraction grating on each sample using deep-UV holographic scheme. The resistive grating was dry-etched in CF4 plasma, and residual photoresist was removed in oxygen plasma. To observe scattering, we install the sample on a rotation stage and adjust incident angle to provide phase matching in excitation of guided modes by a He–Ne laser beam. The scattering is then observed as very specific arcs around the reflected and transmitted beam. When incident angle is about one degree off the resonance condition given in Eq. (2), light is scattered by the nonuniformities of the sample; such scattering is rather isotropic and weak. Once the guided mode is excited, the scattering becomes much stronger and shows the arcs. Although the scattering mechanism is a multistage which includes the excitation of guided mode, scattering inside a waveguide, and re-emission of the scattered waves, the efficiency of scattering may be much stronger compared to direct scattering at the surface roughness. The above consideration is very general and it allows for predicting of the scattering peculiarities in planar structures supporting guided modes. For example, if the waveguide excitation according to (1) happens to be at relatively small angle , one might observe both forward propagating and backward propagating waves in the scattered field. The diagram in Fig. 2(a) illustrate how it happens; only the mode propagating from left to right in the figure is resonantly excited by the diffractive gratings for an electric polarized light. The backward propagating mode appears due to strong scattering of the resonantly excited mode on the waveguide imperfections. Then the +1st order of diffraction of the grating brings light to the arc that is shown in the scattered field. Depending on whether the grating’s wavevector is shorter or longer than the mode wavevector, the arcs may cross or separate. Fig. 2(b) the relative intensity of the resonant and nonresonant scattering versus horizontal pixel location, which is taken by reading the 220 × 150 pixels image in Fig. 2(a) using Matlab software and then plotting a horizontal slice that is taken from raw number 80 of the resulted matrix. Another interesting case is shown in Fig. 3(a); in this sample the fundamental mode is excited exactly at normal incidence angle for an electric polarized light. Thus, the forward and backward propagating modes were excited simultaneously in the ±1st orders of diffraction. The second mode is clearly shown by the pair of external arcs. The remarkable feature is the suppression of scattering for the fundamental mode at small angles. In the crossing area the arcs virtually vanish. More accurately, the out-coupling
Please cite this article in press as: R. Rabady, I. Avrutsky, Light scattering in random planar structures supporting guiding modes, Optik - Int. J. Light Electron Opt. (2015), http://dx.doi.org/10.1016/j.ijleo.2015.10.226
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ARTICLE IN PRESS R. Rabady, I. Avrutsky / Optik xxx (2015) xxx–xxx
efficiency in second order distributed feedback lasers. Fig. 3(b) and (c), the relative intensity of the resonant and nonresonant scattering versus horizontal pixel location, which is taken by reading the 430 × 160 pixels image in Fig. 3(a) using Matlab software and then plotting a horizontal slice take from raw number 130 and 260 of the resulted matrix, respectively. In Fig. 2(b) the intensity of the resonant scattering is clipped because of the imaging limitation, however, by simple extrapolation the peaks were restored, and it was found that the resonant scattering can be stronger than the nonresonant scattering by a factor of 21; whereas, for the second case, shown in Fig. 3 (b) and (c), where the resonant scattering arcs partially overlap destructively, such factor does not exceed 3.
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4. Conclusion In conclusion, we have demonstrated experimentally that in a random planar structure that supports guided modes, once excited, the guided mode is likely to be scattered to another guided mode rather than scattered out of the structure. This observation is in line with the wave scattering theories, which predicts that the efficiency of scattering is proportional to the density of states in the final state. The variety of scattering patterns around the reflected and transmitted beam is well explained by the vector diagrams. Enhanced scattering to a waveguide mode with a different modal number as well as the back scattering were predicted and observed. Possible practical application of the discussed phenomenon includes quantitative evaluation of surface roughness. For example, measuring of weak Fraunhoffer diffraction could be difficult for very smooth surface, while scattering involving modes of dielectric layers is strongly enhanced. The enhanced scattering might play essential role in many experiments dealing with planar structures supporting guided modes in dielectric layers. The reported results expect an improved performance of the random planar lasers compared to the three-dimensional random lasers.
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References
Fig. 3. (a) The arc shape scattering pattern for sample #2 when the TE0 mode was excited at incident angle 5.36◦ , grating period = 394 nm, the two effective refractive index of the electric polarized light n∗TE0 = 1.70, n∗TE1 = 1.633. (b) Resonant and nonresonant scattering intensity that resulted from slicing the image in (a) horizontally at pixel number 130, (c) resonant and nonresonant scattering intensity that resulted from slicing the image in (a) horizontally at pixel number 260. Image resolution is 430 × 160 pixels.
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of scattered light is vanishing due to perfect phase relationship between the forward and backward propagating modes and hence between the scattered waves produce by these modes. Similar phenomenon is known to be responsible for low first order emission
[1] H. Cao, Lasing in random media, Top. Rev. Waves Random Media 13 (2003) R1–R39. [2] T. Barwicz, H. Haus, Three-dimensional analysis of scattering losses due to sidewall roughness in microphotonic waveguides, J. Lightwave Technol. 23 (2005) 2719–2732. [3] D. Marcuse, Mode conversion caused by surface imperfections of a dielectric slab waveguide, Bell Syst. Tech. J. 48 (1969) 3187–3215. [4] C. Ciminelli, F. Dell’Olio, V. Passaro, M. Armenise, Fully three dimensional accurate modeling of scattering loss in optical waveguides, Opt. Quant. Electron. 41 (2009) 285–298. [5] D.G. Hall, Scattering of optical guided waves by waveguide surface roughness: a three-dimensional treatment, Opt. Lett. 6 (1981) 601–603. [6] S. Mazoyer, J.P. Hugonin, P. Lalanne, Disorder-induced multiple scattering in photonic-crystal waveguides, Phys. Rev. Lett. 103 (2009) 063903. [7] V. Savona, Electromagnetic modes of a disordered photonic crystal, Phys. Rev. B 83 (2011) 085301. [8] G. Lecamp, J.P. Hugonin, P. Lalanne, Theoretical and computational concepts for periodic optical waveguides, Opt. Express 15 (2007) 11042–11060. [9] D. Gerace, L.C. Andreani, Disorder-induced losses in photonic crystal waveguides with line defects, Opt. Lett. 29 (2004) 1897–1899. [10] F. Ladouceur, L. Poladian, Surface roughness and backscattering, Opt. Lett. 21 (1996) 1833–1835. [11] D. Melati, A. Melloni, F. Morichetti, Real photonic waveguides: guiding light through imperfections, Adv. Opt. Photonics 6 (2) (2014) 156–224. [12] R. Rabady, I. Avrutsky, Titania, silica and tantalum pentoxide waveguides and optical resonant filters prepared by RF magnetron sputtering and annealing, Appl. Opt. 44 (January (3)) (2005) 378–383.
Please cite this article in press as: R. Rabady, I. Avrutsky, Light scattering in random planar structures supporting guiding modes, Optik - Int. J. Light Electron Opt. (2015), http://dx.doi.org/10.1016/j.ijleo.2015.10.226
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