Indirect transitions of a signal interacting with a moving refractive index front

Transcription

1 Invited Paper Indirect transitions o a signal interacting with a moving reractive index ront Michel Castellanos Muñoz, Alexander Yu. Petrov, Liam O Faolain, Juntao Li 3, Thomas F. Krauss 4, and Manred Eich * Institute o Optical and Electronic Materials, Hamburg University o Technology, Hamburg 073, Germany SUPA, School o Physics and Astronomy, University o St. Andrews, St. Andrews, Fie KY6 9SS, United Kingdom 3 State Key Laboratory o Optoelectronic Materials & Technology, Sun Yat-Sen University, Guangzhou 5075, China 4 Department o Physics, University o York, Heslington, York, YO0 5DD, United Kingdom ABSTRACT The dynamic manipulation o light can be achieved by the interaction o a signal pulse propagating through or relected rom a reractive index ront. Both the requency and the wave vector o the signal are changed in this case, which is generally reerred to as an indirect transition. We have developed a theory to describe such transitions in integrated photonic crystal waveguides. Through indirect transitions, the ollowing eects can be envisaged: large requency shits and light stopping and order o magnitude pulse compression and broadening without center requency shit. All eects can be potentially realized with a reractive index modulation as small as For the experimental realization, we have used slow light photonic crystal waveguides in silicon. The reractive index ront was obtained by ree carriers generation with a switching pulse co-propagating with the signal in the same slow light waveguide. The group velocities o the signal and the ront could be varied arbitrarily by choosing the right requencies o the signal and switching pulses. The indirect transition was unambiguously demonstrated by considering two situations: a) the ront overtaking the signal and b) the signal overtaking the ront. In both cases, a blue shit o the signal requency was observed. This blue shit can only be explained by the occurrence o the expected indirect transition and not by a direct transition without wave vector variation. Keywords: indirect transition, dynamic requency shit, photonic crystal waveguides, on-chip, silicon photonics. INTRODUCTION The process o an optical signal undergoing a transition between two modes o a photonic structure is reerred to as a photonic transition [ 5]. Photonic transitions can be direct, i the optical signal experiences a shit in requency but not in wave vector, or indirect, i both requency and wave vector o the signal are changed. It was previously shown that light conined to a photonic structure can be shited in requency i a reractive index change is applied to the entire structure while the light is still conined within it. Such a process has been reerred to as dynamic or adiabatic control o light in the literature [6 0]. The magnitude o the resulting optical requency shit is proportional to the induced index change n and was irst shown or light travelling in a millimeter-long semiconductor slab [] and later or light conined in microphotonic resonators [6 8] and photonic crystal waveguides [9, 0]. These cases can be classiied as direct photonic transitions, since the reractive index change causes a transition in requency ω but leaves the wave vector k o the light signal unaltered. The required ast change o reractive index is achieved in these cases by generating ree carriers in silicon, which leads to a reractive index change via the carrier plasma dispersion eect []. Dierent methods or generating the ree carriers have been proposed and implemented, or instance by electrical injection with a p-i-n junction [3], by the linear absorption o an optical switching pulse which is incident on the device rom the top [, 7, 9, 0], and by nonlinear two-photon absorption o an optical switching pulse which co-propagates with the signal through the photonic structure, or the irst time realized by us [8]. *m.eich@tuhh.de; phone ; /oem Active Photonic Materials VI, edited by Ganapathi S. Subramania, Stavroula Foteinopoulou, Proc. o SPIE Vol. 96, 96W 04 SPIE CCC code: X/4/$8 doi: 0.7/ Proc. o SPIE Vol W-

2 Indirect photonic transitions, on the other hand, imply both a change o requency and wave vector o the optical signal, and are not only o undamental physical interest, but may constitute, or instance, the operating principle or the realization o integrated non-reciprocal optical isolators without using magneto-optical or non-linear eects [4]. Further, they oer exciting opportunities or ultraast delay applications [5, 4]. In this context, indirect photonic transitions between modes that belong to dierent photonic bands, called indirect interband transitions, have attracted attention, since a cascade o transitions, and thus the appearance o multiple requency components, can be avoided [5, 4]. Indirect photonic transitions can be achieved in dierent ways. For example, it was shown that an indirect photonic transition between two modes (ω 0,k 0 ) and (ω,k ) separated by a requency Ω= ω - ω 0 and a wave vector q= k - k 0 can be achieved by modulating the reractive index o the structure both temporally with a requency Ω, and spatially with a spatial requency q [ 4, 5]. This process can be explained as a three wave mixing process in a nonlinear system, where one o the waves is an electric signal with requency Ω and spatial requency q that modulates the reractive index [5]. However, while the modulation can be designed to match the mode separation (Ω,q) or a particular optical requency ω 0 in principle, the matching no longer exists or neighbouring requencies ω 0 + ω i the bands involved in the transition are not completely parallel throughout the desired requency bandwidth [ 4]. There exists, thus, an intrinsic limitation on the bandwidth over which a transition is possible, imposed by the dierence in the slopes o the bands, thus their group velocity mismatch. There are other practical limits: the requency shit Ω is limited by the achievable modulation requencies to a maximum o tens o GHz, and the typical wave vector dierence q requires a spatial periodicity o the modulation that must be encoded in the geometry o the device, since it cannot be matched by the spatial periodicity o a travelling GHz electric signal [4]. Thus, tuning o the indirect photonic transition is only possible by varying the geometry o the device. Quasi-indirect photonic transitions in a photonic crystal waveguide have now also been demonstrated in a two-step process, where the requency shit is realized irst, ollowed by a shit o the wave vector [5]. This approach is still applicable or tunable delay but does not have all the properties o the indirect transition. It cannot, or example, transer light to a state o zero group velocity or produce large requency shits. From microwave engineering, it is known that an indirect photonic transitions can be induced in a single step by the interaction o electromagnetic radiation with a moving perturbation [6]. This perturbation can be any kind o travelling discontinuity o the electromagnetic properties o the medium where the radiation propagates. Such a perturbation can be, or instance, a moving ree-carrier plasma ront in a semiconductor [7, 8], which leads to a moving reractive index ront via the carrier-plasma dispersion eect []. Indirect photonic transitions induced upon interaction o an optical signal with a reractive index ront moving along a photonic crystal waveguide have been employed to realize a tunable optical delay [4]. Furthermore, it was theoretically shown that indirect photonic transitions induced upon interaction with a mechanical shock ront moving along a Bragg stack can lead to the generation o inverse Doppler shited requencies [9]. Here we present the theory o the indirect transition through the pulse interaction with a moving reractive index ront. This transition can lead to a palette o signal pulse modiications which will be discussed in this paper. Finally we present the experimental results o our recent work on indirect transition with moving ront in slow light waveguides [0, ].. THEORY OF INDIRECT TRANSITION WITH MOVING FRONT We consider the signal wave beore and ater its interaction with the ront. The signal ater the interaction is assumed to have a new requency and wavevector. The relationship between the change in requency and wave vector o the signal with respect to the moving ront can be understood by irst studying the process in the rest rame o the ront. In this rame, the ront can be chosen to lie at the position z =0 and the requency is conserved. I ω, k, ω, k denote the requencies and wave vectors o the signal on each side o the ront, the phase dierence on each side is: ' ' ' ' ' ' ( ct', z ) = ( ω ω ) t' ( k k ) 0 ' ' ω ω ' ', k = k z c, () where c is the light velocity in vacuum and t is the time coordinate in the rest rame o the ront. This is a our vector product which is invariant under Lorentz transormation. Thus, the time and space dependent phase o the signal is continuous at the position o the moving ront, independently o the reerence rame o the observer. I we now denote Proc. o SPIE Vol W-

3 the requencies and wave vectors in the laboratory rame by ω, k, ω, k, the velocity o the reractive index ront in the laboratory rame by v and its position by z, we ind: ( ω ω ) t ( k k ) ω z ω = k k t Eq. indicates that, as observed in the laboratory rame, the ratio o the changes o requency and wave vector induced by the interaction with the moving ront is identical to the velocity at which the ront propagates. In order to determine the magnitude o the indirect photonic transition induced by the ront we use a graphical representation known rom microwave research [, 3] and shown in Fig. b. The lower solid and the upper dashed curves in Fig. b schematically show the dispersion bands o a silicon photonic crystal waveguide or silicon reractive indices n si and n si + n FC, respectively. At the input o the structure, the signal pulse travels in a waveguide with silicon reractive index o n si, and is represented by a point (ω, k ) lying on the corresponding dispersion curve. Next, we draw all points (ω, k) which ulill the phase continuity condition rom Eq. by plotting a straight phase continuity line with a slope equal to the travelling velocity o the ront v. Ater being overtaken by the reractive index ront, the signal pulse travels in a waveguide with silicon reractive index o n si + n FC. Its inal requency and wave vector (ω, k ) are determined graphically rom the crossing point o the grey phase continuity line and the upper dashed dispersion curve. tv = t z = v = 0 () n si + n FC (reractive index ront) n si t t >t switch / ront signal angular requency ω n si n si + n FC dω/dk = v ω,k modiied signal ω,k (a) wave vector k π/a (b) Figure. (a) Schematic representation o the experiment perormed in [0]. A switching pulse with high peak power generates ree carriers in the silicon by two-photon absorption and consequently induces a change o reractive index whose ront propagates with the velocity o the switching pulse. Here, the switching pulse is aster than the signal. (b) Schematic representation o the indirect photonic transition induced. Using this graphical representation, one easily recognizes undamental dierences rom the concepts discussed beore, where the reractive index is modulated with ixed temporal and spatial modulation requencies [ 4]. In our case, the velocity o the reractive index ront does not unambiguously determine one, but rather a whole set o possible inal states (ω, k). Out o this set, such transitions will take place or which the inal state corresponds to a photonic mode o the system. As long as this second condition is satisied, there are no intrinsic limitations on the bandwidth over which a perect mode requency and wave vector matching can be achieved. Furthermore, the velocity o the reractive index ront, which is identical to the velocity o the switching pulse, can be tuned via the operating point o the slow-light waveguide. Thus, the indirect photonic transitions are tunable without the need or any geometrical modiications. Proc. o SPIE Vol W-3

4 3. POSSIBLE TRANSITIONS As will be show later, our experimental results conirm the theoretical prediction that the magnitude o the induced transitions depends on the propagation velocity o the ront and on the photonic bands o the structure, as represented in Fig. b. Thereore, by exploiting the lexibility in dispersion design provided by slow-light waveguides [4, 5], our results open up new versatile possibilities or light control, including non-reciprocal optical isolation, tunable optical requency shits, optical bandwidth compression and broadening, and tunable optical delays. For instance, the authors have proposed a theoretical concept or broadening or compressing the bandwidth o an optical signal upon relection rom a moving ront o a photonic crystal [6]. Some o the other possible eects are presented in Fig.. Eects such as light stopping, signal reversal, large requency shits can all be predicted. The realisation o these eects depends on the velocity o the ront and the dispersion relation o the medium or waveguide where the ront and signal are propagating. Figure. Schematic presentation o dierent eects achieved with indirect transition close to the band edge. Initial group velocities o the signal and o the ront are counter directed. The velocity o the ront deines the inal requency and the direction o propagation o the modiied signal. The dispersion o the initial n medium and inal n medium is shown. Depending on the group velocity, the point o zero group velocity () can be reached, thereby eectively stopping the light. Point () corresponds to a reversed signal propagating behind the ront in the inal medium. Transition (3) describes the relection o the ront with a large requency shit. 4. FREQUENCY SHIFT IN SLOW LIGHT WAVEGUIDES We have, or the irst time, experimentally demonstrated indirect photonic transitions that are driven by a reractive index ront travelling along a slow-light photonic crystal waveguide [0]. Fig. 3a shows a schematic representation o the experiment we perormed. A high power switching pulse propagating through a silicon photonic crystal waveguide generates ree carriers by two-photon absorption, and induces a corresponding change o the reractive index n FC due to the ree-carrier plasma dispersion eect []. Accordingly, a reractive index ront moves with the group velocity o the switching pulse and interacts with a slower signal co-propagating in the waveguide. Here, the dierence between the group velocities o the ront and the signal is chosen to be large compared to the pulse duration and the propagation length, such that the reractive index ront completely overtakes the signal. The group velocity o both pulses is chosen by the requency o light. Due to dispersion o the photonic crystal waveguide group indices rom 0 to 30 are accessible. In Fig. 3, the schematic transition and the experimental results are presented or the case when the signal is aster than the switching pulse. The signal overtakes the ront and comes rom the area with the reractive index shited by the ree carriers into the area without reractive index change. The schematic presentation o the indirect transition in Fig. 3(a) predicts a blue shit o the signal requency, which cannot be obtained by the direct transition. It should be mentioned that we transer signal rom the medium with ree carriers to a medium without ree carriers. Thus, the direct transition in this case would produce a strong red shit, according to the relation ω~- n. Fig. 3(b) presents the wavelength shit as the unction o the initial delay between the signal and the switching pulse. The dynamic transition takes place only at the delays when the signal has time to overtake the switching pulse within the dimensions o the slow light waveguide. A clear blue shit is observed. Proc. o SPIE Vol W-4

5 Figure 3. (a) Schematic presentation o the indirect transition when signal is aster than ront. A blue shit is expected rom the band diagram. (b) Blue shit measured as the delay variation between signal and switching pulse. This measurement supports the indirect transition prediction. 5. CONCLUSION We have presented the theory o indirect photonic transitions obtained by the pulse interaction with the reractive index ront. Dierent eects such as requency shit, light stopping and time reversals are predicted. Experimental demonstration o the requency shit in the slow light waveguide is presented, where the ront is realized as a ree carrier injection caused by a high power switching pulse. The eect o indirect transition is conirmed by the measurement. 6. ACKNOWLEDGEMENTS This research was supported by the German Research Foundation DFG (EI 39/-). The authors would like acknowledge the support rom CST, Darmstadt, Germany with their Microwave Studio sotware. REFERENCES [] J. N. Winn, S. Fan, J. D. Joannopoulos, and E. P. Ippen, "Interband transitions in photonic crystals," Phys. Rev. B 59, (999). [] Z. Yu and S. Fan, "Complete optical isolation created by indirect interband photonic transitions," Nature Photon. 3, 9 94 (009). [3] Z. Yu and S. Fan, "Integrated Nonmagnetic Optical Isolators Based on Photonic Transitions," IEEE J. Sel. Top. Quantum Electron. 6, (00). [4] H. Lira, Z. Yu, S. Fan, and M. Lipson, "Electrically Driven Nonreciprocity Induced by Interband Photonic Transition on a Silicon Chip," Phys. Rev. Lett. 09, 3390 (0). [5] D. M. Beggs, I. H. Rey, T. Kamprath, N. Rotenberg, and L. Kuipers, et al., "Ultraast Tunable Optical Delay Line Based on Indirect Photonic Transitions," Phys. Rev. Lett 08, 390 (0). [6] M. Notomi and S. Mitsugi, "Wavelength conversion via dynamic reractive index tuning o a cavity," Phys. Rev. A 73, 5803 (May, 006). [7] S. F. Preble, Q. Xu, and M. Lipson, "Changing the colour o light in a silicon resonator," Nature Photon., (May, 007). [8] M. Castellanos Muñoz, A. Y. Petrov, and M. Eich, "All-optical on-chip dynamic requency conversion," Appl. Phys. Lett. 0, 49 4 (0). [9] J. Upham, Y. Tanaka, T. Asano, and S. Noda, "On-the-Fly Wavelength Conversion o Photons by Dynamic Control o Photonic Waveguides," Appl. Phys. Express 3, 600 (00). [0] T. Kamprath, D. M. Beggs, T. P. White, A. Melloni, and T. F. Krauss, et al., "Ultraast adiabatic manipulation o slow light in a photonic crystal," Phys. Rev. A 8, (April, 00). [] I. Geltner, Y. Avitzour, and S. Suckewer, "Picosecond pulse requency upshiting by rapid ree-carrier creation in ZnSe," Appl. Phys. Lett 8, 6 8 (00). Proc. o SPIE Vol W-5

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