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Free carrier front induced indirect photonic transitions: A new paradigm for frequency manipulation on chip Mahmoud Gaafar, Alexander Yu. Petrov, and Manfred Eich ACS Photonics, Just Accepted Manuscript • DOI: 10.1021/acsphotonics.7b00750 • Publication Date (Web): 02 Oct 2017 Downloaded from http://pubs.acs.org on October 5, 2017
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Free carrier front induced indirect photonic transitions: A new paradigm for frequency manipulation on chip Mahmoud A. Gaafar,1,2,*Alexander Yu. Petrov,1,3 and Manfred Eich1,4 1
Institute of Optical and Electronic Materials, Hamburg University of Technology, Hamburg 21073, Germany
2
Department of Physics, Faculty of Science, Menoufia University, Menoufia, Egypt
3
ITMO University, 49 Kronverkskii Ave., 197101, St. Petersburg, Russia
4
Institute of Materials Research, Helmholtz-Zentrum Geesthacht, Max-Planck-Strasse 1, Geesthacht, D-21502, Germany
*
[email protected] Abstract Nonlinear degenerate four wave mixing and cross phase modulation are established approaches for all optical frequency manipulation in a silicon chip. These approaches require exact group velocity and/or phase velocity matching of pump, signal and idler. On the other hand several experimental demonstrations were presented recently where frequency of light was changed by a free carrier front propagating in a silicon waveguide. This Doppler-like effect is less known, but has important advantages for frequency manipulation on chip. It requires no phase velocity matching and is not dependent on the shape and duration of the pump pulse. It also allows packet switching and can operate in a pump power independent regime. Here, we shortly review the work on front induced indirect transitions in silicon slow light waveguides. We consider three possible interaction regimes: transmission through the front, reflection from the front and moving with the front called surfing. We derive analytical equations for the front with a linearly rising edge, which provide a unified description of the frequency shift in all three regimes. Finally, we compare the front induced dynamic frequency conversion to the frequency shifting based on nonlinear effects like cross-phase modulation and four wave mixing. Keywords: Indirect photonic transitions, Silicon photonics, Frequency manipulation on chip, Nonlinear optics, and Slow light waveguides.
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Frequency manipulation of optical signals plays an important role in wavelength-division multiplexing (WDM) based optical communication,1,2 thus its on-chip implementation is of great interest. All optical switching based on nonlinear effects such as cross-phase modulation (XPM), 3–5
and four-wave mixing (FWM)6–9 have been successfully demonstrated and thoroughly
investigated in silicon-based nanophotonic devices. On the other hand, several contributions have demonstrated frequency conversion through dynamic modulation of light confined in the photonic structures. Dynamic frequency conversion in cavities was first proposed by Notomi et. al.10 and has been experimentally demonstrated by generating free carriers (FCs) in silicon via a pump pulse at visible wavelength which is incident on the structure from the top.11–13 This case corresponds to a refractive index modulation in time only which is otherwise spatially uniform. Frequency shift in such a configuration was also realized in photonic crystal (PhC) waveguides14– 16
where the refractive index was changed while the signal was propagating in the slow light
waveguide. However, the magnitude of the resulting frequency shift is proportional to, thus limited to the maximum induced index change ∆ in such configuration and the scheme is not
fully integrated.
Furthermore, the dynamic frequency conversion of a light signal by a FC front copropagates on-chip with the signal has been experimentally investigated in PhC cavities17 and in a slow-light PhC waveguides.18–20 The front is generated by the two photon absorption (TPA) of the pump pulse. In case of PhC cavities which confine the optical field in a small volume still just a time dependent modulation via the change of the cavity’s refractive index is observed. However, in case of PhC waveguides, the signal transiting the front experiences a modulation of the refractive index in time and space and thus undergoes frequency and wave vector shifts which depend on the velocity of the front, the so called indirect photonic transition. Front induced Indirect Transitions (FITs) represent an original effect distinct from other nonlinear interactions and dynamic transitions and requires separate consideration. FITs allow realisation of frequency shifts much larger than that can be achieved by direct transitions for the same index change
∆.18,20 The dynamic manipulation of light can be produced by the interaction of a signal wave not only propagating through but also reflected from an index front. Recently, we have
experimentally realised a case of an “intraband” indirect transition where the signal wave was forward reflected from the co-propagating index front 20. This effect is an optical analogue of an event horizon, which, in a different approach, was demonstrated by employing the Kerr effect.21,22 ACS Paragon Plus Environment
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The frequency shifting by FWM and XPM are usually considered for signal and pump propagating with the same group velocity. Thus, a group velocity mismatch limits the efficiency of FWM conversion8 or the frequency shift in XPM configuration.4 Also the group velocity mismatch is usually difficult to take into account in the standard theoretical approaches of nonlinear interactions.23 On the contrary, the group velocity mismatch is the central characteristics of indirect photonic transitions caused by the interaction of a signal with a moving FC front. Theoretical approaches were developed to describe this interaction including group velocity changes of the signal.18–20,24 In this paper, we shortly review the work on FITs in silicon slow light waveguides. We discuss in more detail the theory of frequency conversion through moving FIT and present simple analytical equations to calculate the accumulated frequency shift. Finally, we compare indirect photonic transitions to optical switching based on nonlinear effects like XPM and FWM. Several advantages of indirect transitions are presented. First, the pump frequency can be chosen far away from the signal window. Second, the pump pulse does not contribute to spectral broadening of the switched signal. Third, the front can overtake a packet of signal pulses thus enabling for packet switching. Fourth, the reflection regime20 can be used to obtain a power independent frequency shift. Background We consider here a moving refractive index front. Such an index front, for example, can be realised by FC injection in silicon. A high power pump pulse propagating through a silicon
waveguide with a refractive index n1 generates FCs in the silicon by TPA, and induces a change
of the refractive index ∆ () < 0 and therefore a moving refractive index front due to FC
plasma dispersion effect:25
2 (t) = 1 + ∆FC (t)
(1)
The duration of the rising edge of the front is defined by the pump pulse duration. We assume that the life time of FC in the waveguide (approx. 100-1000 ps26,27) is much longer than the signal propagation time in the waveguide. Thus we do not discuss the carrier recombination induced falling edge of the FC front. We start with a sharp front assuming infinite steepness. In case of a signal wave propagating and interacting with the index front, the change of its frequency and wave vector ACS Paragon Plus Environment
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upon interaction can be identified by applying phase continuity at the front. If 1 , 1 , 2 , 2
denote the frequencies and wave vectors of the signal on each side of the front, and ∆ , ∆ are the spatial differences between two points of arbitrary phase (green and pink colors) located on each side of the front, as shown in Fig. 1(a). Then: ∆1 1 2 = = ∆2 2 1
(2)
where 1 and 2 are the wavelengths of the signal before and after interaction with the front,
respectively. The phase velocity of the signal wave defines the shift of a point of constant phase in direction of propagation, as illustrated in Fig. 1(b), therefore ∆ and ∆ can be written as: ∆ = ∆ − ∆ = − !∆,
∆ = ∆ − ∆ = ( − )∆
(3)
where #" is the group velocity of the front, &$ℎ1 and &$ℎ2 are the phase velocities of the signal wave on each side of the front. From Eqs. (2) and (3), we find: #" − (1⁄1 )
#"
− (2⁄2 )
#" =
=
2 , 1
2 − 1 ∆ = 2 − 1 ∆
(4)
Equation (4) indicates that the ratio of the signal frequency change to the wave vector change induced by the interaction with the moving front is identical to the velocity at which the front propagates. This relation can be derived from the Doppler equation,21 using the phase conservation under Lorentz transformation,18,28 or phase evolution integrals.24 Therefore, the shifts of frequency and wave vector are determined, on the one hand, by the propagation velocity of the front, and, on the other hand, by the dispersion of the system,18,28 where the dispersion of
the system plays a key role as the 1, 1 and , should be points available in the dispersion
relation of the waveguide modes without and with free carriers present. Here we have considered
a simplified situation of a sharp front. But this model can be used to describe a smooth transition as a staircase of many small sharp transitions leading to the same final result. Thus, the frequency
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and wave vector changes induced by a moving front are independent of the rise time of the front, provided that the complete signal interacts with the front within the limited waveguide length. Fig. 2(a) shows a schematic representation of the indirect transition in the band diagram.
The solid curve represents the dispersion band of a waveguide mode with refractive index 1 , while the dashed curve indicates the switched state with refractive index 1 + ∆)* . In this
schematic example, the initial group velocities of the front and of the signal are counter directed. The grey line represents the phase continuity line with a slope equal to the group velocity of the front/pump chosen at a frequency-wavenumber combination marked by the black dot. The red and blue dots indicate the initial and final states of the signal wave, respectively. According to Equation (4) the final state of the signal is determined graphically from the phase continuity line connecting the starting point of the unswitched state with the upper dashed dispersion curve of the switched state. Depending on the velocity of the front and the shape of the modal dispersion relation of the waveguide within which front and signal are propagating, possible effects –as presented in Fig. 2(b)-can be realized such as light stopping, signal reversal, and signal reflection with large frequency shift.18,24
Fig. 1. Schematic representation of the signal wave before ( , ) and after ( , ) the transition from one dispersion relation to another dispersion relation. The bold solid curve shows how the pump pulse transform the
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waveguide medium from a refractive index to a refractive index = + ∆ . We assume that the time function of the pump pulse power is very steep, thus the refractive index front is also steep at any position in the waveguide. Green and pink dots represent two points of arbitrary phase.
The relative group velocity difference between the front and the signal waves in combination with the dispersion relation play an important role for the induced frequency shift and constitute a powerful mechanism for the control of light.18,20 Fig. 3(Left) shows the interactions between the index front and a co-propagating signal pulse inside a PhC waveguide, while Fig. 3(Right) represents a band diagram schematic of the corresponding induced indirect transitions. In case of large group velocity mismatch, either the front completely overtakes the signal, as shown in Fig. 3(a), or vice versa, the signal overtakes the front as shown in Fig. 3(b).18 In both cases, the induced frequency shift of the signal is achieved upon transmission through the front, leading to an interband indirect transition. In case of the front completely overtaking the
signal, the final state of the signal lies on the band corresponding to 1 + ∆)* . In case of the
signal completely overtaking the front, the initial state of the signal lies on the switched band, while the final state lies on the original band.18 On the other hand, a frequency shift of the signal can be also achieved upon reflection from a co-propagating front, however under certain conditions.20 For a special waveguide dispersion relation and a specific pump and signal locations on the band diagram, the signal – which is initially propagating ahead of the front with a slightly smaller group velocity- cannot find states on the band of the switched PhC behind the front after interaction, and therefore the new state of the signal will remain in the initial band, which means that an intraband transition takes place. This can happen provided that the phase continuity line can reach the shorter wavelength position in the same band without cutting into the shifted band.20 This intraband transition manifests itself as a forward reflection from the front, as schematically shown in Fig. 3(c). This reflection is accompanied by a larger frequency shift than interband transition.
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Fig. 2. (a) Schematic representation of the indirect photonic transition. The solid curve represents the dispersion band of a waveguide mode with refractive index , while the dashed curve indicates the switched state with refractive index + ∆ . Initial group velocities of the signal (red dot) and of the front (black dot) are counter directed. The grey line represents the phase continuity line with a slope equal to the group velocity of the pump pulse (slope at the black dot). The blue dot indicates the final state of the signal, (b) Different effects achieved with indirect transitions. The velocity of the pump (front) defines the final frequency and wave vector as well as the direction of propagation of the modified signal. A point of zero group velocity can be reached, thereby effectively stopping the light, or a signal can be reversed and propagating behind the front in the switched medium. Furthermore, reflection at the front associated with a large frequency shift can be also achieved.
The signal and pump pulses can also move with the same group velocity and overlap within the silicon waveguide. In this case, the frequency shift induced by the XPM can be derived from the difference in the phase shift accumulated by different portions of the signal pulse.29 In silicon additional effects of FCs can be taken into account.3–5,30,31 Dekker et. al. reported a large wavelength shift (10 nm red due to Kerr effect and 15 nm blue shifts due to FC effect) of a pulsed signal wave caused by XPM using a 300 fs pump and signal pulses in a silicon-on-insulator waveguide.4 It is not possible to use the phase continuity line to determine the phase shift as the dispersion band is parallel to phase continuity line and no intersection point can be defined (Fig. 3d). The signal is surfing with the pump and continuously shifted along the waveguide and the final frequency shift is limited by the length of the structure. At the same time real systems will always have a finite dispersion and group velocity mismatch thus the signal pulse will at some point exit the interaction zone of the pump. In this case, the maximal frequency shift will be defined by the intersection of the phase continuity line with the dispersion curves. The band diagram approach also fails for incomplete dynamic transitions when the waveguide length is insufficient to allow the signal to shift to its final position on the switched state band diagram. Thus, a theory involving the length of the waveguide and front properties should be developed. The exact equations for change of the frequency and wave number were presented by Kondo et al.24 using the phase accumulated by the signal in the waveguide.19 The equations can be integrated over propagation distance but they are difficult to interpret and cannot give direct insight into the signal evolution. At the same time, intermediate bands can be drawn, which correspond to partial refractive index shifts inside the front. The signal wave penetrates into the front and thus move from one intermediate band to the other along the phase continuity line.18,24 The speed of motion along the phase continuity line is connected to the front properties. We present here a simplified derivation of this speed for a front with a linear slope.
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Fig. 3. Left: schematic of the interaction between front and signal. A pump pulse with high power generates FCs in the silicon by TPA and consequently induces a change of refractive index which propagates with the velocity of the pump. (a) and (b) either the pump pulse will overtake the signal or the signal pulse will overtake the pump, respectively. These are indirect interband photonic transitions. (c) Reflection of the signal pulse by the pump pulse, due to intraband transition. This is indirect intraband photonic transition. (d) Signal surfing; when both the pump and signal pulses are moving with the same velocity at the input of the structure. Right: schematic of the corresponding induced indirect transitions. Again the solid and dashed curves represent the original and switched dispersion bands, respectively. The grey line represents the phase continuity line. In case of surfing, we neglected the dispersion for simplicity. Substantial dispersion will inevitably limit the effective distance surfing can take place.
When a small perturbation of the refractive index is applied, the frequency of an eigenmode in the band structure will change according to:32 2
3(/01)2 4/01 - ∆.(/01) 2001 ∆+, = − 2 2 01) 2001 3(/01)2 4/01 - .(/
(5)
For homogeneous spatial distributions for ∆.(/1) and .(/1) one gets from equation (5) the relation: ∆+, ∆ =−
(6)
Special attention should be paid to carrier diffusion in silicon when considering the dynamic
switching configuration in a PhC waveguide. The spatial distribution of ∆ is restricted to the ACS Paragon Plus Environment
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center region of the waveguide up to the surrounding first row of holes, which partly inhibits carrier diffusion out of the center region of the waveguide.33 Assuming a refractive index change ∆ which is constant over the center region and zero elsewhere, one gets from equation (4): ∆+, ∆ =− ∙6
(7)
where 6 is the fraction of mode energy - .|3| in the perturbed regions. From this equation we can estimate the expected frequency shift of the optical mode according to the maximum refractive index change ∆. The portion of the mode energy inside the center of PhC line defect
waveguide changes slightly with frequency and wave number.33 Thus the relative frequency shift can vary along the dispersion curve. Still for a small part of the dispersion curve where the mode field profile does not change substantially a constant relative frequency shift can be assumed. As an example we consider a dispersion engineered waveguide with shifted rows of holes.34 In case
of a small ∆ -in the order of 5⋅10-3 which can be generated by a FC density of 5⋅1017/cm³- there
is no considerable change of the slope of the shifted band diagram (not presented here). Therefore, the fraction of light in the perturbed region is approximately constant. Frequency change is directly proportional to the time spent inside the front
A moving FC front reduces the refractive index of silicon by a quantity ∆, which in turn
changes, e.g. blue shifts the dispersion curve. Figure 4(a) shows a schematic of the spatial change of the waveguide’s refractive index induced by the pump pulse at a given point of time. In this representation we again assume that the refractive index changes linearly in the front. The spatial and temporal dependence of the refractive index can be then written as: (, ) = ∆ ∙ ( − ):8 + 9
(8)
where 8# is the length over which the index front changes from its initial to its final value. When a
signal wave with a different group velocity contacts the moving front, it will first penetrate into the zone of reduced refractive index and thus experiences an indirect transition to the new frequency and wave vector. In order to know how the signal wave trajectory inside the front will behave and what is the expected frequency change of the signal wave along this trajectory, a useful approximation of the trajectory can be derived for the case where the band of the
unswitched waveguide upon switching is just vertically shifted in frequency ; ?@
∙ ( 4 − 4) =
∆> ?@
∙ ( − )4
(9)
where &" = 4⁄4 is the local group velocity of the signal wave. The corresponding band diagram shift in frequency can be expressed as: 4
