Bloch oscillation and unidirectional translation of frequency in a dynamically modulated ring resonator

The ability to control the spectrum of light is of fundamental significance in many applications of light. We consider a dynamically modulated ring resonator that supports a set of resonant modes equally spaced in their resonant frequencies, and is modulated at a frequency that is slightly detuned from the modal frequency spacing. We find that such a system can be mapped into a tight-binding model of photons under a constant effective force, and, as a result, the system exhibits Bloch oscillation along the frequency axis. The sign of the detuning is mapped to the sign of the effective force and hence controls the direction of the Bloch oscillation. We also show that a periodic switching of the detuning can lead to unidirectional transport of light along the frequency axis. Our work points to a new capability for manipulating the frequencies of light using dynamic microcavity structures. © 2016 Optical Society


INTRODUCTION
Microring resonators have been extensively used for explorations of fundamental physics and in a wide range of applications including spectroscopy, sensing, and information processing [1][2][3][4][5].In particular, the resonant modes of a ring cavity form a frequency comb, and hence the ring can be used as the basis for compact and robust frequency comb sources [6][7][8].In addition, active modulation of microring cavities can couple these resonant modes together, leading to new possibilities for manipulating the spectrum of light [9][10][11].
In the absence of group velocity dispersion in the ring waveguide, a ring cavity supports an equally spaced frequency comb [6].In the on-resonance modulation case, in which the modulation frequency Ω is chosen to be equal to the frequency spacing Ω R between the resonant modes in the comb, the system exhibits photonic transition [12][13][14] and can be mapped into a tightbinding model with each mode corresponding to one lattice site in the model [15,16].Building upon this notion, in this paper we consider the off-resonance modulation case, in which the modulation frequency is slightly different from the frequency spacing.We show that, in such a case, the system can be mapped into a tight-binding model subject to a constant effective force for photons.Thus, the photon wave amplitudes in the system undergo Bloch oscillation along the frequency axis.Moreover, we find that a unidirectional transport of the light along the frequency axis can be achieved by periodically changing the modulation frequency Ω.
The Bloch oscillation of light has been previously explored in waveguide or waveguide array systems, with the oscillation occurring either in the spatial [17][18][19][20][21] or the spectral domain [22,23].Our work here points to a novel manifestation of the effect of Bloch oscillation [24,25] and establishes a connection between the physics of microresonators and the physics of coherent dynamics of quantum particles under an external field.Moreover, the ability to achieve unidirectional transport of light along the frequency axis naturally leads to the capability of photon frequency translation, which is of interest for both classical and quantum information processing [26][27][28].

THEORETICAL ANALYSIS
To illustrate the essential physics, we first consider a simple physical model of a ring resonator undergoing dynamic refractive index modulation (Fig. 1).Suppose the waveguide forming the ring has no group velocity dispersion; the frequency of the mth resonant mode is then located at ω m ω 0 mΩ R ; where Ω R is the free spectral range.Suppose we place an electrooptic modulator into the ring, with a modulation frequency Ω ≈ Ω R .A simple Hamiltonian for this system is then where a † m and a m are the creation and annihilation operators, respectively, for the mth resonant mode in the resonator.g is the strength of modulation.We assume that the modes are all nondegenerate, and there is no backscattering in the waveguide forming the ring.Defining c m ≡ a m e −iω m t and taking the rotating wave approximation, which is valid since Ω ≈ Ω R , we transform H in Eq. ( 2) to where Δ ≡ Ω − Ω R is the detuning.Equation ( 3) is equivalent to a time-independent Hamiltonian describing a photon in a one-dimensional lattice under a constant effective force: as can be seen with the gauge transformation [29] jΨi with v m ṽm being the amplitude of the field at the mth mode.jΨ Ψi satisfies the Schrödinger equation, i∂∕∂tjΨ Ψi H RWA H RWA jΨ Ψi.In Eq. ( 4), we note that the frequency of the mode at the mth site is proportional to m, signifying the presence of a constant force, as schematically represented in Fig. 1(c).The strength of the effective force is F jΔ∕Ω R j, which leads to a Bloch oscillation with a period T B 2π∕Δ [24,30].We now proceed with a more realistic model of a ring under dynamic modulation [16,31].We expand the electric field inside the resonator as [32] Et; r ⊥ ; z where z denotes the propagation direction along the waveguide that forms the ring resonator, r ⊥ gives the directions perpendicular to z, and E m r ⊥ is the modal profile.E m t; z is the modal amplitude for the mth resonant mode, which satisfies the following equation: as can be derived from Maxwell's equations under the slowly varying envelope approximation.Here β is the wavevector in the z direction and n g is the group index.The geometry of the ring imposes a periodic boundary condition E m t; z L E m t; z, where L is the circumference of the ring.
The electric field inside the resonator undergoes a dynamic phase modulation when it passes through an electro-optic phase modulator located at z 0 , as described by Et ; r ⊥ ; z 0 Et − ; r ⊥ ; z 0 e iα cosΩt ; (8 where α is the modulation amplitude and t t 0 .The modulator imposes an additional time-dependent phase as the field passes through the modulator.We then use the rotating wave approximation to describe this process in terms of E m t; z and obtain [33] where Δ Ω − Ω R and J q is the Bessel function of the qth order. To relate the physical model here to the simple model of Eq. ( 2), we keep only the coupling between nearest neighbor sidebands in Eq. ( 9), (i.e., keep only q 1 terms), and also assume that J 0 α ≈ 1 and J 1 α ≈ α∕2, which are valid in the weak modulation limit α → 0. Therefore, the variation of E m in Eq. ( 9) after it passes the modulator is The total change of E m over the propagation over one loop is where T R is the round-trip propagation time around the ring [34].By defining We therefore see that, in the appropriate limit, Eqs. ( 7) and ( 9) describe a dynamics that is the same as the dynamics of the Hamiltonian of Eq. ( 2).Therefore, a ring under modulation can indeed exhibit Bloch oscillation physics in the frequency domain as described in Eq. (2).

NUMERICAL DEMONSTRATIONS OF BLOCH OSCILLATION AND TRANSLATION OF FREQUENCY
As a numerical confirmation of the analytical argument above, we solve Eqs. ( 7) and ( 9) to obtain the solution of the propagation of the electric field inside a ring resonator modulated by an electrooptic modulator (with α 0.2), without making the approximation that led to Eq. ( 10) [16].The simulation involves solving for the dynamics for a finite set of modes around ω 0 as labeled by m.Let n 0 nω m denote the effective refractive index for all resonant modes, so we have Ω R 2π c∕n 0 L. We inject the excitation into the system at the position z s by the equation Here we assume that the excitation source is inside the ring.Throughout the paper, we choose where t is in the unit of n 0 L∕c.Here S m is a pulse with its frequency centered at ω m and its full width at half-maximum (FWHM) Δω ≪ Ω R .f m is the peak excitation amplitude for the mth mode.A given S m essentially excites only the mth mode.In the first set of simulations, for the excitation, we choose f 0 1 and f m≠0 0 in Eq. ( 13).We consider two modulation frequencies corresponding to detunings Δ 0.1 c∕n 0 L and Δ 0.5 c∕n 0 L. In the simulation we include 17 resonant modes labeled with m −8; …; 8.We plot in Fig. 2 the temporal evolution of the total energy for each resonant mode (I m t≡ R ring jE m t; zj 2 dz).With a small detuning of Δ 0.1 c∕n 0 L, we see the regular Bloch oscillation in the frequency dimension with the period T B 20π n 0 L∕c.When the detuning is large (Δ 0.5 c∕n 0 L), we no longer observe the spectral Bloch oscillation because the modulation cannot couple the nearby resonant modes efficiently.In Fig. 2(a), we observe that the oscillation pattern is symmetric around the m 0 axis, in spite of the fact that the choice of Δ ≠ 0 gives a directional effective force along the frequency axis, as seen in Eq. ( 4).This effect was analytically studied in [35].Physically, the excitation of a single site corresponds to a uniform excitation over the entire band in k f -space (where k f is the wavevector reciprocal to the frequency axis).The application of a force shifts the entire band and, hence, cannot create an asymmetric oscillation.
To observe an asymmetric Bloch oscillation pattern, we now consider an excitation profile by setting in Eq. ( 13) with ω c ω 0 .We again perform the simulation including 17 resonant modes.We consider the cases with the positive detuning Δ 0.055 c∕n 0 L and the negative detuning Δ −0.055 c∕n 0 L, respectively.The total energy for each mode I m t is shown in Fig. 3. Unlike the case of Fig. 2, here the oscillation pattern no longer has a mirror symmetry around the m 0 axis.The Gaussian excitation profile [Eq.( 14)] excites only a portion of the band around k f 0. The effective force can then directionally shift the field amplitude profile in the k f space as well as the frequency space.Changing the sign of the detuning Δ flips the oscillation pattern with respect to the m 0 axis.In Fig. 3, we see that, under a constant effective force corresponding to a constant detuning Δ, the total intensity oscillates around the m 0 point in the frequency axis.Next, we show that by varying the direction of the effective force periodically in time, as can be achieved by varying the detuning Δ periodically in time, we can achieve a unidirectional transport of light in the frequency space.In the simulation, we periodically switch Δ between 0.055 c∕n 0 L at every time interval of t 55 n 0 L∕c ∼ π∕jΔj [see Fig. 4(b)].In the simulation we include 29 resonant modes (m −14; …; 14), to accommodate the range of frequency shift in the unidirectional transport process.The excitation follows Eqs. ( 13) and ( 14) with ω c ω −12 .In Fig. 4(a), we plot the total energy for each mode I m t.The Bloch oscillation is interrupted every time a sudden change in the modulation frequencies occurs, resulting in a unidirectional shift in the frequency space as a function of time.
In experiments, a ring is typically excited through the coupling to external waveguides.Here we demonstrate unidirectional frequency transport in the presence of external coupling waveguides (Fig. 5).We assume that the coupling occurs at a position at z 1;2 on the ring circumference with the external waveguide 1 or 2, as described by where E 1;2 m is the amplitude of the field component at the frequency ω m in external waveguide 1 or 2. γ 1;2 t is the coupling strength between the ring resonator and waveguide 1 or 2. For optimal observation of unidirectional transport along the frequency axis, it is preferable that the cavity does not leak into the waveguide during the modulation process.We therefore choose a time-dependent waveguide-cavity coupling constant in the form γ 1 t 0.1H 40 − t and γ 2 t 0.1H t − 200 in the simulation, where H t is the Heaviside step function.Such an on-off behavior in coupling can be achieved with electrooptic tuning of waveguide coupler [32].With such a choice, the input source is injected into the ring from waveguide 1 at t < 40 n 0 L∕c, and, after modulation, the field in the ring leaks out into waveguide 2 at t > 200 n 0 L∕c and gets detected.We first perform the simulation with the input field into external waveguide 1 obeying Eqs. ( 13) and (14) with ω c ω −12 .We turn on the dynamic modulation at t 0 with the same periodic switch of the modulation frequency following Fig. 4(b), but turn it off at various times t f (t f 50; 100; 150; 200 n 0 L∕c).In Fig. 5(b), we plot the output spectrum from external waveguide 2 after the modulation is switched off.The top panel of Fig. 5(b) shows the profile of the input spectrum.Compared to it, the output spectra shift to higher frequencies.As t f gets larger, the magnitude of the shift becomes larger, demonstrating the unidirectional frequency shift in this system.In contrast, in Fig. 5(c), we choose the same input field except ω c ω 0 in Eq. ( 14), and we apply a constant modulation frequency with Δ 0.055 c∕n 0 L instead over the same periods from 0 to the various t f , as discussed above.For all values of t f , the output spectra are located somewhat near ω 0 , and do not exhibit the unidirectional shift as we observe in Fig. 5(b).

DISCUSSION AND CONCLUSION
We discuss a few experimental aspects for on-chip implementation of our theoretical predictions.Modulation-induced coupling between different resonant modes separated by a free spectral range was demonstrated in [31], which used a silicon ring resonator with a radius of 445 μm modulated at a frequency of Ω 26 GHz.Our theory can be implemented in a similar system.The maximum coupling strength between the modes, as observed in Ref. [31], translates to α ≈ 0.8 in Eq. ( 11) as applied to this system.Therefore, our choice of α 0.2 in the simulation is within what one can accomplish experimentally.To observe a period of Bloch oscillation, the photon lifetime τ p in the ring needs to be larger than the period of a single Bloch oscillation 2π∕Δ, which, with a choice of Δ ≈ 0.1Ω R ≈ 2 GHz, translates into a requirement of the quality factor of the ring of the order of 10 5 -10 6 .Such a quality factor may be achievable in silicon ring modulators [31].Alternatively, one can consider other material systems that have significant electro-optic effects, and where high-Q resonators have already been achieved [36].To demonstrate the Hamiltonian of Eq. ( 2), there needs to be sufficient bandwidth over which the group velocity dispersion is small.For a silicon waveguide, near-zero group velocity dispersion has been observed over a bandwidth of 670 nm [37], which is far larger than what we require here.
Our study in general points to interesting opportunities for the control of light spectra through the use of dynamic modulation inside the cavity.While the simulation above is for a classical wave, since the phase modulator as described by Eq. ( 8) is a linear and lossless device, the same physics should occur in the quantum regime at the few photon level, as well.The unidirectional shift here is, therefore, potentially useful for photon frequency translation [27].Importantly, here the magnitude of frequency translation can be far larger than the modulation frequency.Our work here also indicates other interesting new opportunities for manipulating the spectrum of light in microring resonators.For example, with a different choice of parameters, an effect of dynamic localization [29,35,38,39] in the frequency domain may be achieved.Our work here points to interesting new opportunities for manipulating the spectrum of light in microring resonators.
Funding.Air Force Office of Scientific Research (AFOSR) (FA9550-12-1-0488).The ring is modulated for a period from 0 to t f .The output spectra, after the modulation is turned off, are plotted in the remaining panels.In (b), the modulation frequency periodically switched between Δ 0.055 c∕n 0 L as described in Fig. 4(b).In (c), the modulation frequency is maintained constant at Δ 0.055 c∕n 0 L.

Fig. 1 .
Fig. 1.(a) Ring resonator modulated by an electro-optic modulator (EOM).(b) In the absence of group velocity dispersion, the ring in (a) supports a set of resonant modes (blue dashed lines) with their resonant frequencies equally spaced by Ω R .The modulation frequency Ω is chosen to be slightly different from Ω R .For the modes that are located at ω 0 , the modulation generates sidebands that are detuned from the other resonant modes.(c) The system as described in (b) can be mapped onto a tight-binding model of a photon under an effective force [Eq.(4)].

Fig. 2 .
Fig. 2. Evolution in time of the total energy for each resonant mode (I m t ≡ R ring jE m t; zj 2 dz): (a) Δ 0.1 c∕n 0 L, and (b) Δ 0.5 c∕n 0 L, when only the m 0 mode is excited.

Fig. 3 .
Fig. 3. Evolution in time of the total energy for each resonant mode (I m t ≡ R ring jE m t; zj 2 dz): (a) Δ 0.055 c∕n 0 L, and (b) Δ −0.055 c∕n 0 L, when several modes are initially excited.

Fig. 4 .
Fig. 4. (a) The evolution in time of the total energy for each resonant mode I m t, as the modulation frequency is periodically switched between Δ 0.055 c∕n 0 L, as described in (b).

Fig. 5 .
Fig. 5. (a) Ring resonator under dynamic modulation by the electrooptic modulator (EOM), and in addition coupled to two external waveguides.(b) and (c) Detected signals in external waveguide 2: the top panels are the input spectra injected through external waveguide 1.The ring is modulated for a period from 0 to t f .The output spectra, after the modulation is turned off, are plotted in the remaining panels.In (b), the modulation frequency periodically switched between Δ 0.055 c∕n 0 L as described in Fig.4(b).In (c), the modulation frequency is maintained constant at Δ 0.055 c∕n 0 L.