Frequency diffraction management through arbitrary engineering of photonic band structures

Chengzhi Qin; Bing Wang; Peixiang Lu

doi:10.1364/OE.26.025721

1. Introduction

Diffraction is a ubiquitous phenomenon in optics which refers to the transverse shift and wave packet broadening for a space-limited light beam. In general, it is governed by the uncertainty principle between the transverse coordinate and wavenumber, leading to the different phase accumulation rate for each wavenumber component [1-4]. Basically, diffraction can be managed by engineering the dispersion relation (or band structure) ε(k), with ε and k being the respective photon energy and wavenumber. From the band structure, one can obtain the group velocity v_g(k) = ∇_kε(k) and diffraction coefficient $D (k) = \nabla_{k}^{2} ε (k)$ [5-8], which determine the transverse shift and wave packet broadening, respectively. However, either for continuous diffraction in homogenous medium or discrete diffraction in periodic lattice structure, the band structures are usually fixed once the medium or lattice structure is chosen, making the diffraction properties unchangeable. Particularly in photonic lattice structure, such as coupled waveguide arrays, photon hopping is usually dominated by the nearest-neighbor coupling and the diffraction is governed by the tight-binding approximation. As a result, the band structure manifests a fixed cosine shape with the diffraction exhibiting the conventional cone-like pattern [5, 9-12]. So it remains an open question as to whether we can break the tight-binding limitation and realize nonlocal diffraction phenomena.

To overcome the tight-binding limitation, recent efforts have been dedicated to transferring the concept of diffraction from real space to synthetic ones, such as in the frequency [13-20], temporal [21-26] and momentum spaces [27-30]. For example, by introducing multiple pumped light fields in a four-wave mixing process, a synthetic spectral lattice with long-range couplings can be created for the signal light [19]. Therefore, new diffraction phenomena have been achieved such as the asymmetric discrete diffraction for a single frequency incidence and the generalized discrete Talbot effect for periodic spectrum input, all of which have no counterparts in the conventional tight-binding waveguide arrays. Based on this work, it is straightforward to further improve the capability of band structure engineering by including more higher-order long-range couplings. While for the four-wave mixing process adopted in [19], the increase of more long-range coupling orders requires the precise control of the phase and intensity of more independent pumped fields, which will dramatically increase the system complexity. Recently, it is demonstrated that the method of time modulation can also induce the discrete frequency diffraction [15, 17]. Since the coupling in frequency lattice is mediated by external modulation, this scheme can inherently break the tight-binding limitation and enables arbitrary engineering of lattice band structures.

In this work, we propose to manage the frequency diffraction with arbitrary band structure engineering using time modulation method. Without relying on the independent control of each order long-range couplings, we apply the periodic modulation signals in the optical phase modulators to simultaneously excite all orders of long-range couplings. The band structure can thus be arbitrarily engineered by controlling the modulation waveform. Particularly, by using sawtooth, triangular and circular modulation waveforms, we create linear, bilinear and circular band structures and realize the diffraction-free directional, bidirectional and omnidirectional frequency shifts, respectively. Moreover, we revisit the frequency discrete Talbot effect under long-range couplings and generalize the allowed incident period to an arbitrary integer by using sawtooth and triangular wave modulations. Additionally, as the frequency transition of each order also carries a wave vector mismatch, an effective electric field will emerge, through which we can realize the effect of frequency Bloch oscillations. By engineering the band structure, an input frequency comb can oscillate along arbitrary trajectory and a single frequency manifests self-imaging with arbitrary breathing pattern. Finally, due to the asymmetric wave vector mismatches in the forward and backward directions, the discrete frequency diffraction under arbitrary band structure engineering manifests inherent nonreciprocal properties.

2. Results and Discussions

2.1 Band structure engineering in a synthetic frequency lattice

A synthetic frequency lattice can be constructed from a LiNbO₃ electro-optical phase modulator (PM). As shown schematically in Fig. 1(a), each PM is driven by a periodic radiofrequency signal synthesized by an arbitrary waveform generator (AWG). Generally, a periodic signal satisfies f(t) = f(t + T), where T = 2π/Ω is the time period with Ω being the fundamental modulation frequency. We consider the periodic signal is applied on the PM with sufficient duration time, the spectrum of which contains a series of discrete Fourier harmonics. Each harmonic component contributes to a sinusoidal travelling-wave modulation in the PM, giving rise to the total refractive index distribution

n (z, t) = n_{0} + \sum_{m} Δ n_{m} \cos (m Ω t - m Q z + ϕ_{m}),

where n₀ is the background refractive index, Δn_m, mΩ, mQ and ϕ_m (m = 1, 2,…) are the amplitude, frequency, wavenumber, and initial phase of the mth-order index modulation wave. As shown in Fig. 1(b), the fundamental modulation wave of m = 1 creates a synthetic frequency lattice with lattice constant Ω in the TE₀ band of the slab waveguide. All orders of modulation waves (m = 1, 2,…) can induce long-range couplings in this frequency lattice. The total field distribution is thus E(z, t) = Σ_na_n(z)exp[i(ω_nt − β_nz)], where ω_n = ω₀ + nΩ and β_n = β₀ + nq (n = 0, ± 1, ± 2,…) are the frequency and propagation constant of nth-order optical mode, q denotes the wavenumber difference between adjacent order modes. a_n(z) is the mode amplitude, which is governed by the coupled-mode equation (see appendix for detailed derivation)

i \frac{\partial a_{n} (z)}{\partial z} = \sum_{m} C_{m} [e^{i (m Δ q z + ϕ_{m})} a_{n - m} (z) + e^{- i (m Δ q z + ϕ_{m})} a_{n + m} (z)],

where C_m = Δn_mk₀/2 and ϕ_m are the strength and phase of the mth-order long-range coupling. Δq = q – Q is the wavenumber mismatch for the nearest-neighbour coupling m = 1. As will be discussed in Part 2.3, this mismatched wavenumber plays the role of an effective gauge electric field, which can induce the effect of frequency Bloch oscillations [14, 17].

Fig. 1 (a) Schematic of two cascaded phase modulators (PMs) driven by periodic RF signals with sinusoidal, sawtooth, triangular and semicircular waveforms generated by the arbitrary waveform generator (AWG). The phase shifter (PS) is utilized to tune the phase difference of modulation in the two PMs. (b) Schematic of different order photonic transitions. Ω and q denote the frequency and wavenumber difference between nearest-neighbor order modes. C_m, ± ϕ_m are the amplitude and phase shift of mth-order long-range coupling (m = 1, 2, 3, …). The modulation wavenumber of mth-order long-range coupling is mQ, corresponding to a mismatched wavenumber of mΔq = m(q − Q). The wavenumber mismatch corresponds to an effective electric field E_eff = − Δq applied in the frequency lattice.

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Firstly in sections 2.1 and 2.2, we consider the phase-matching condition with Δq = 0, such that the electric field vanishes. The eigen Bloch mode in the frequency lattice is an infinite-width frequency comb a_n(z) = a₀exp(ink_ωΩ)exp(ik_zz), where k_ω, k_z are Bloch wave vector in the frequency dimension and collective propagation constant in the z direction. Substituting the Bloch mode into Eq. (2), we can obtain the band structure

k_{z} (k_{ω}) = - 2 \sum_{m = 1}^{M} C_{m} \cos (m k_{ω} Ω - ϕ_{m}),

where we truncate the order of long-range couplings to the maximum m = M. For a finite-width frequency comb centered at a Bloch wave vector k_ω, the frequency-domain group velocity is

v_{g, ω} (k_{ω}) = - \frac{\partial k_{z} (k_{ω})}{\partial k_{ω}} = - 2 Ω \sum_{m = 1}^{M} m C_{m} \sin (m k_{ω} Ω - ϕ_{m}),

from which we can obtain the central shift for the frequency comb. Due to the finite width for the spectrum envelope, the frequency comb will also experience bandwidth broadening, which can be described by the frequency-dimension diffraction coefficient

D (k_{ω}) = \frac{\partial^{2} k_{z} (k_{ω})}{\partial k_{ω}^{2}} = 2 Ω^{2} \sum_{m = 1}^{M} m^{2} C_{m} \cos (m k_{ω} Ω - ϕ_{m}),

So the frequency discrete diffraction can be managed by engineering the band structure through controlling the amplitudes and phases of all long-range couplings. In the following, we choose several typical modulation waveforms and realize special diffraction phenomena, including the directional, bidirectional and omnidirectional frequency diffraction, all of which can’t find counterparts in the tight-binding lattice systems.

In practical applications, the precise conversion from one frequency to another is very important in the multiple-wavelength source generation and wavelength-division-multiplexing communications [31-34]. Conventional approach to realize frequency conversion is to input a single frequency into a PM modulated by a sinusoidal RF signal. Note that a single frequency contains all Bloch mode components covering the entire Brillouin zone. As shown in Fig. 2(a), each Bloch mode component possesses different group velocity and diffraction coefficient, leading to the cone-like diffraction pattern for the single frequency. So it is impossible to realize a non-diffraction conversion from one frequency to another using the conventional sinusoidal wave modulation. To achieve diffraction-free frequency conversion, linear band structures are required to cancel the diffraction. Here we can adopt a periodic sawtooth wave modulation which corresponds to the mth-order long-range coupling strength and phase C_m = C₁(−1)^m⁺¹/m and ϕ_m = − π/2. According to Eqs. (3)-(5), the band structure, group velocity and diffraction coefficient are given by

k_{z} (k_{ω}) = C_{1} Ω k_{ω}, v_{g, ω} (k_{ω}) = - C_{1} Ω, D (k_{ω}) = 0,

As shown in Fig. 2(b), the band structure is linear in the Brillouin zone, so there is only one group velocity for all Bloch mode components. Note that D(k_ω) = 0, the input single frequency will exhibit diffraction-free evolution, with the total accumulated frequency shift given by

Δ ω = v_{g, ω} (k_{ω}) L = - \frac{m_{φ 1} Ω}{2},

where m_φ1 = 2C₁L is the first-order phase modulation depth with L being the total length of two PMs. Moreover, we can also achieve bidirectional diffraction-free frequency shift by choosing the periodic triangular modulation waveforms with C_m = C₁sin²(mπ/2)/m² and ϕ_m = 0. The corresponding band structure, group velocity and diffraction coefficient are

k_{z} (k_{ω}) = \frac{C_{1} π^{2}}{2} (| \frac{Ω k_{ω}}{π} | - \frac{1}{2}), v_{g, ω} (k_{ω}) = \mp \frac{π C_{1} Ω}{2}, D (k_{ω}) = 0,

Where “−” and “+” represent the Bloch mode component with 0 < k_ω < π/Ω and − π/Ω < k_ω < 0, respectively. As plotted in Fig. 2(c), the band structure is bilinear in the entire Brillouin zone, leading to the existence of only two group velocities for all Bloch mode components. So a single frequency will experience spectral self-splitting and manifests both blue and red shifts in the PMs, with each shifting amount given by

Δ ω = v_{g, ω} (k_{ω}) L = \mp \frac{m_{φ 1} Ω π}{4},

It is very interesting to compare the diffraction-free evolutions here to those in the tight-binding waveguide arrays [1, 2, 5]. The tight-binding waveguide array is described by a cosine shape band structure shown in Fig. 2(a). The diffraction-free evolution is achieved only for the Bloch wave packet centered at Bloch momentum ± π/2 with D = 0. While for the linear and bilinear band structures, the diffraction-free evolution can be achieved for the Bloch wave packet with arbitrary Bloch momentum and even for a single frequency containing all Bloch modes in the entire Brillouin zone. So the excitation conditions for the diffraction-free evolution can be dramatically relaxed with proper band structure engineering.

Fig. 2 (a)-(d) Cosine, linear, bilinear and circular band structures synthesized using sinusoidal, sawtooth, triangular and semicircular modulation waveforms. The blue and red curves represent the band structures in the two PMs under out-of-phase modulations for each order wave. The black arrows denote the group velocities in the band structures. (e-h) Frequency diffraction patterns for a single frequency input under the above four modulation waveforms: (e) conventional discrete diffraction pattern, (f) diffraction-free directional frequency shift, (g) diffraction-free frequency shift, (h) omnidirectional frequency diffraction. (i-l) Frequency perfect imaging for a single frequency input in the spectral superlens under out-of-phase modulations in the two PMs. The modulation waveforms in (i) (j) (k) and (l) are sinusoidal, sawtooth, triangular and semicircular, respectively.

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In stark contrast to the above directional and bidirectional frequency shifts where the diffraction or bandwidth expansion is completely suppressed, we can also achieve the other extreme situation in which a single frequency experiences omnidirectional shifting and maximum diffraction. This can be achieved by a direct analogue with the omnidirectional diffraction of a point light source in the free space. Here we use a periodic semicircular modulation waveform with C₀ = C₁I₀/I₁, C_m = C₁I_m/I₁ (m ≥ 1), ϕ_m = 0, in which $I_{0} = \int_{0}^{π} \sin^{2} (θ) d θ / 4, I_{m} = \int_{0}^{π / 2} \sin^{2} (θ) \cos [m π \cos (θ)] d θ$ . The corresponding band structure is

k_{z} (k_{ω}) = \frac{C_{1}}{I_{1}} (\frac{π}{4} - \sqrt{1 - \frac{Ω^{2} k_{ω}^{2}}{π^{2}}}),

As plotted in Fig. 2(d), the band structure exhibits the same semicircular shape with the modulation waveform. Under such modulation, the ω-z plane can be regarded as an effective free space in which the input single frequency will experience omnidirectional diffraction, which is analogous to the situation of point source diffraction in real space.

Apart from realizing the above frequency diffraction phenomena, we can also achieve frequency refraction by using different modulations in the two cascaded PMs. Particularly, we consider the two PMs are modulated with the same waveforms but with different initial phases. An input frequency comb will experience a refraction at the boundary of two PMs, with the effective relative refractive index given by

n_{12} (k_{ω}) = \frac{k_{z, 1} (k_{ω})}{k_{z, 2} (k_{ω})} = \frac{\sum_{m = 1}^{M} C_{m} \cos (m k_{ω} Ω - ϕ_{m})}{\sum_{m = 1}^{M} C_{m} \cos (m k_{ω} Ω - ϕ_{m} + Δ ϕ_{m})},

where Δϕ_m is the phase difference of the mth-order harmonic modulation waves. Here we consider a special case where the modulation waveforms are opposite in the two PMs with f₂(t) = − f₁(t), which corresponds to the out-of-phase modulation Δϕ_m = π in the two PMs. The relative refractive index is thus n₁₂(k_ω) = −1, which manifests the analogous characteristics with the spatial superlens [15, 16, 35-37]. A single frequency will experience discrete diffraction in PM1 and then exhibit a time-reversal process in PM2, ultimately converging back onto a single frequency. Alternatively, the imaging process with discrete diffraction have been also achieved in the finite-number waveguide arrays [38, 39] with a different physical mechanism. In [38, 39], the imaging is realized via mode switching between distinct supermodes with opposite propagation constants through the lumped π phase shift in each other waveguides at the center of propagation distance. So the time-reversal is realized with the lumped phase shift while our scheme is based on the out-of-phase modulation in the whole propagation process. Moreover, since our scheme can be generalized to the long-range coupling situations, indicating that it is a more universal imaging approach.

To verify all the above theoretical analysis, we can perform numerical simulations for the frequency diffraction under different modulation waveforms, by solving the coupled-mode equation of Eq. (1). In the simulations, the PM is modeled by a LiNbO₃ slab waveguide with thickness of d = 0.5 μm and background and first-order modulation amplitude of ε_d = 4.58 (n₀ = 2.14) [40] and Δε₁ = 2 × 10⁻³. The modulation frequency is Ω/2π = 10 GHz and the input single frequency is ω₀/2π = 193.2 THz, corresponding to the telecommunication wavelength of λ₀ = 1.55 μm. For comparision, we firstly review the conventional discrete frequency diffraction under sinusoidal signal modulation. As shown in Fig. 2(e), the cone-like pattern is the direct counterpart of discrete diffraction in spatial waveguide arrays under a single site excitation. While for the sawtooth wave modulation, as shown Fig. 2(f), the single frequency exhibits a directional shift along a straight line with the diffraction being completely suppressed. By choosing m_φ1 = 8π, the maximum spectral shift is |Δω|/Ω = 4π, which is in precise agreement with the theoretical prediction in Eq. (7). Figure 2(g) shows the spectral evolution under the triangular wave modulation, where the single frequency manifests the self-splitting effect and exhibits non-diffraction bidirectional shifts along two straight lines. For m_φ1 = 8π, the maximum red or blue spectral shift is |Δω|/Ω = 2π², which is also consistent with the theoretical prediction in Eq. (9). In Fig. 2(h), we utilize the periodic semicircular wave modulation. The input single frequency manifests the omnidirectional shift and form the circular shape wave front, which is analogous to the point source diffraction in the free space.

The frequency refraction phenomena using spectral superlens can also be verified with numerical simulations. To construct spectral superlens, as indicated in Eq. (11), the two PMs are driven under the out-of-phase modulations with opposite modulation waveforms. Figures 2(i)-2(l) show the refraction processes with sinusoidal, sawtooth, triangular and semicircular waveforms, respectively. In all scenarios, the diffraction patterns in PM2 manifest the mirror symmetries with respect to those in PM1, verifying the negative refraction and perfect focusing for the single frequency. Specifically, under sawtooth and triangular wave modulations shown in Figs. 2(j) and 2(k), the single frequency exhibits a “zigzag” diffraction trajectories that can circumvent the central frequency. So they could be regarded as the frequency analogues of “cloaking” [41, 42] in which the initial frequency component can’t be detected in the central propagating regions. Finally, for the semicircular wave modulation shown in Fig. 2(l), the single frequency firstly exhibits a diverging pattern in PM1 and then a time-reversal converging process in PM2, ultimately focusing onto the single frequency at the end of PM2, verifying that it is the spectral analogue of a spatial superlens. Although here we only choose four typical periodic modulation waveforms and consider only the single frequency input, the spectral superlens can be constructed using any kinds of periodic modulation waveforms and are applicable to arbitrarily input spectra.

2.2 Frequency discrete Talbot effect with band structure engineering

As another important diffraction phenomenon under periodic input field, the Talbot effect has also attracted considerable attention both in the continuous and discrete optical systems [19, 25, 43-47]. Specifically, the discrete Talbot effect has been studied in tight-binding waveguide arrays where the allowed period is limited to a specific set of N ∈ {1, 2, 3, 4, 6} [43, 44]. In [19], the introduction of second-order long-range coupling can generalize the allowed period to N = 5. Here we can take a step further to generalize the allowed period to an arbitrary integer with proper engineering of band structures. Moreover, in practical applications, the condition of periodic spectrum in the Talbot effect is the same with the wavelength-division-multiplexing (WDM) optical communications, where the information is encoded in the equally distanced frequency channels [31-34]. So the frequency discrete Talbot effect could be applied to manage the frequency channels to control the WDM optical communications.

Generally, the amplitude of a periodic spectrum satisfies a_n(0) = a_n₊_N(0), where N is the input period in the frequency dimension. Due to the periodic boundary condition, the spectrum contains a discrete set of Bloch mode components

k_{ω, l} = \frac{2 π l}{N Ω}, (l = 1, 2, ..., N)

where these N Bloch mode components are equally distributed in the entire Brillouin zone, each of which corresponds to an effective mode wavelength

λ_{l} = \frac{2 π}{| k_{z, l} (k_{ω, l}) |} = \frac{π}{| \sum_{m = 1}^{M} C_{m} \cos (\frac{2 m l π}{N} - ϕ_{m}) |},

The Talbot distance is the lowest common multiple (LCM) of all effective mode wavelengths

Z_{T} = lcm (λ_{1}, λ_{2}, ..., λ_{l}, ..., λ_{N}),

A sufficient but not necessary condition for discrete Talbot effect is that each mode wavelength λ_l is itself a rational number, such that the LCM always exists. This can be readily accomplished by choosing linear or bilinear band structures. Firstly, for the linear band structure under sawtooth wave modulation shown in Eq. (6), the effective mode wavelength is

λ_{l} = \frac{2 π}{| k_{z, l} (k_{ω, l}) |} = \frac{N}{C_{1} l},

from which we have

Z_{T} = \frac{N}{C_{1}} lcm (\frac{1}{1}, \frac{1}{2}, ..., \frac{1}{l}, ..., \frac{1}{N}) = \frac{N}{C_{1}},

The Talbot distance corresponds to the effective phase modulation depth of m_φ1,_T = 2C₁Z_T = 2N. So the allowed period can be chosen as an arbitrary integer, and the Talbot distance can take the value of an arbitrary even number. Analogously, for the bilinear band structure with triangular modulation waveform shown in Eq. (8), the effective mode wavelength is

λ_{l} = \frac{2 π}{| k_{z, l} (k_{ω, l}) |} = \frac{2 N}{π C_{1} l},

Which corresponds to the Talbot distance

Z_{T} = \frac{2 N}{π C_{1}},

So the effective phase modulation depth is m_φ1,_T = 2C₁Z_T = 4N/π. For the semicircular wave modulation with the band structure given by Eq. (10), the effective mode wavelength is generally an irrational number, so the discrete Talbot effect will not occur.

The above theoretical analysis can also be verified by using numerical simulations. Though the Talbot effect requires the periodic spectrum input, it has no requirement of the specific amplitude distribution for the periodic spectrum. Here we choose the simplest binary format of “1” and “0” for each discrete frequency. It is worthy note that the binary “1” and “0” could be regarded as the digital information encoded in the discrete frequency dimension, and the Talbot imaging process can be regarded as the reconstruction process of the digital information [47]. Figures 3(a)-3(d) show the spectral evolutions under sawtooth wave modulations. In Fig. 3(a), the input period is N = 4 and the intensity pattern is {10001, …}. From the Talbot diffraction carpet, we can obtain the Talbot distance m_φ1,_T = 8, which is consistent with the theoretical prediction in Eq. (16). In Fig. 3(b), we keep the input period N = 4 unchanged and input another intensity pattern {11001,…}. The Talbot distance is unchanged, indicating that it is independent of the specific input pattern. Figure 3(c) shows the Talbot diffraction carpet under the input period of N = 5 and intensity pattern {100001,…}. The corresponding Talbot distance is m_φ1,_T = 10, which also agrees well with the theoretical result. In Fig. 3(d), we choose out-of-phase modulation in the PMs and investigate the discrete Talbot effect in the spectral superlens. The diffraction carpet in PM2 is mirror symmetric with respect to that in PM1, exhibiting the time-reversal imaging process of discrete Talbot effect in spectral superlens. For comparison, we also simulate the situations of triangular wave modulations in Figs. 3(e)-3(h). Figures 3(e) and 3(f) denote the Talbot diffraction carpets under the input period N = 8 and intensity patterns of {100000001…} and {101000001…}. The effective Talbot distance is m_φ1,_T/π = 4N/π² ≈3.24, which agrees well with the numerical results. In Fig. 3(g), by choosing the input period of N = 9 and intensity pattern {1000000001…}, the Talbot distance becomes m_φ1,_T/π = 4N/π² ≈3.64. Figure 3(h) shows the diffraction carpet in the spectral superlens, which also manifests the time-reversal imaging process for the periodic input pattern.

Fig. 3 Spectral evolutions of frequency discrete Talbot effect under the sawtooth and triangular wave modulations. The red dashed lines represent the positions of the Talbot distances. (a)-(d) The input periods are N = 4 in (a) (b) and N = 5 in (c) (d) with the PMs being modulated with the sawtooth waveforms. The input patterns are given by (a) {10001,…}, (b) {11001,…}, (c) {100001,…}, (d) {100001,…}. The two PMs in (d) are subject to out-of-phase modulations with opposite sawtooth waveforms. (e)-(f) The input periods are N = 8 in (e) (f) and N = 9 in (g) (h) under the triangular modulation waveforms, with the input patterns given by (a) {100000001,…}, (b) {101000001,…}, (c) {1000000001,…}, (d) {1000000001,…}. The two PMs in (h) are under out-of-phase modulations by the opposite triangular waveform.

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2.3 Frequency Bloch oscillations with band structure engineering

In previous sections, we consider frequency diffraction phenomena in the absence of effective electric field with Δq = 0. In this section, we turn to the situation where there exists a wave vector mismatch during each order of photonic transitions with Δq ≠ 0. As is shown in the following, this wave vector mismatch can act as an effective electric field and induce frequency Bloch oscillations. To date, the frequency Bloch oscillations have been realized either in the dynamically modulated resonator [14], waveguide [17] or in the optical fiber under the cross phase modulation between signal and periodic distributed pumped light waves [48]. While both cases are limited by the tight-binding approximation where the Bloch wave packet oscillates along a fixed cosine trajectory. Here with band structure engineering, the Bloch oscillations can be generalized to manifest arbitrary oscillation trajectories and self-imaging effects with arbitrary breathing patterns. In the presence of Δq ≠ 0, the band structure reads

k_{z} [k_{ω} (z)] = - 2 \sum_{m = 1}^{M} C_{m} \cos [m k_{ω} (z) Ω - ϕ_{m}],

where the z-dependent Bloch wave vector is given by

k_{ω} (z) = k_{ω} (0) - \frac{Δ q z}{Ω},

Where ϕ₀ = k_ω(0)Ω is the initial Bloch momentum. In waveguide system, the z coordinate plays the role of time t via z = ct. By choosing the light speed c = 1, the time-dependent Bloch momentum is ϕ_ω(t) = k_ω(t)Ω = ϕ₀ − Δqt, corresponding to an effective electric field [17]

E_{e f f} = \frac{\partial ϕ_{ω} (t)}{\partial t} = - Δ q,

The presence of effective electric field can induce the continuous shift of the Bloch momentum, giving rise to the effect of frequency Bloch oscillations [14, 17]. Considering a frequency comb input with initial Bloch momentum ϕ₀, the frequency-dimension group velocity is periodically varying with respect to the z coordinate

v_{g, ω} (z) = - \frac{\partial k_{z} [k_{ω} (z)]}{\partial k_{ω} (z)} = - 2 Ω \sum_{m = 1}^{M} m C_{m} \sin (m ϕ_{0} - m Δ q z - ϕ_{m}),

The periodic frequency shift is thus

Δ ω (z) = \int_{0}^{z} v_{g, ω} (z^{'}) d z^{'}

, which reads

Δ ω (z) = \frac{2 Ω}{Δ q} \sum_{m = 1}^{M} C_{m} [\cos (ϕ_{m} - m ϕ_{0}) - \cos (ϕ_{m} - m ϕ_{0} + m Δ q z)],

The frequency evolution is thus the superposition of Bloch oscillations contributed from all orders of long-range couplings. The oscillation period is Z_B = 2π/|Δq|, which is inversely proportional to the magnitude of the effective electric field. By engineering the band structure through using different modulation waveforms, an input frequency comb packet will evolve along any predesigned trajectories. It should be mentioned that since there is only one band in the band structure, the dynamics evolving interband transitions such as Landau-Zener tunneling will not occur [49, 50], making the frequency Bloch oscillations undamped in the system.

Figures 4(a)-4(d) show the dynamics of frequency Bloch oscillations for the frequency comb input under the sinusoidal, sawtooth, triangular and semicircular modulation waveforms, respectively. The numerical (blue spectra) and theoretical (red lines) results can agree well with each other. As indicated in Figs. 2(a)-2(d), the band structure shares the same shape with the modulation waveform, and the frequency comb evolution traces out the trajectory with the same shape of the band structure. So the modulation waveform, band structure and the spectral evolution trajectory are closely related with one another and share the same shape. An input frequency comb can thus be designed to evolve along any predetermined trajectory, giving rise to the effect of spectral arbitrary routing. In Figs. 4(e)-4(h), we keep the modulation waveforms unchanged and input a single frequency instead of a frequency comb into the PMs. In all sceneries, the single frequency experiences a periodically breathing pattern and exhibits perfect self-focusing at integer multiples of the oscillation period of Z_B. Since a single frequency input represents the excitation of all Bloch mode components, the breathing pattern is thus the interference result of the frequency diffraction of all frequency combs. The spectral self-focusing effect is due to the periodic oscillatory nature for the Bloch oscillations, which can be generalized to the spectral imaging for arbitrarily input spectra [17].

Fig. 4 Spectral evolutions of frequency Bloch oscillations with different modulation waveforms. (a)-(d) Spectral evolutions for a frequency comb input under (a) sinusoidal, (b) sawtooth, (c) triangular, (d) semicircular modulation waveforms, respectively. The input frequency comb has a Gaussian envelope with width of W = 5 Ω and initial central Bloch momentum ϕ₀ = 0. The red solid lines denote the theoretical trajectories obtained by Eq. (23). (e)-(h) Frequency Bloch oscillations under a single frequency input under the four modulation waveforms. The black dashed lines denote the positions for a Bloch oscillation period of Z_B.

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Finally, we consider the nonreciprocal properties for the frequency discrete diffraction in the presence of long-rang couplings. The nonreciprocal properties stem from the presence of distinct magnitude of the electric field as the light propagates in opposite directions. Generally in the forward and backward directions, as shown in Fig. 5(a), the wave vector mismatches for the nearest-neighbour coupling are q⁺ = q − Q and Δq⁻ = q + Q, corresponding to the wave vector mismatches of mΔq⁺ and mΔq⁻ for the mth-order long-range couplings. The periods of frequency Bloch oscillations are thus Z_B⁺ = 2π/|q − Q| and Z_B⁻ = 2π/|q + Q|, leading to the asymmetric evolutions for the frequency Bloch oscillations in forward and backward directions. To verify the analysis, we choose the triangular wave modulation as an example and input a single frequency forward and backward into the PMs, as shown in In Fig. 5(b). The modulation wavenumber is Q = q/3, such that Z_B⁺ = 2Z_B⁻. With the total lengths of two PMs chosen as L = Z_B⁻, the single frequency can accomplish one oscillation period in the backward direction while only half period in the forward. The asymmetric oscillation patterns indicate the inherent nonreciprocal properties for the frequency discrete diffraction, as endowed in dynamically modulation systems [15, 51-54].

Fig. 5 (a) Schematic of photonic transitions as the light propagates in the forward and backward directions. Ω and Q denote the modulation frequency and wavenumber with q being the wave number difference between adjacent order modes. The mismatched wavenumbers are thus Δq⁺ = q − Q and Δq⁻ = q + Q in the forward and backward directions. (b) (c) Nonreciprocal spectral evolutions of frequency Bloch oscillations under a single frequency input in backward and forward directions, as denoted by the green arrows. We choose Δε₁ = 4 × 10⁻³ with other parameters kept the same with the above figures.

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3. Conclusions and discussions

In summary, we investigate the discrete frequency diffraction in PMs where the band structure of frequency lattice can be arbitrarily engineered with long-range couplings. By choosing the sawtooth, triangular and semicircular modulation waveforms, we synthesize the linear, bilinear and circular shapes of band structures and realize diffraction-free directional, bidirectional and omnidirectional frequency shifts and spectral superlens. We also revisit the frequency discrete Talbot effect in the presence of long-range couplings and find that the allowed input periods can be generalized to arbitrary integers. By controlling the wavenumber mismatches in the frequency transitions, we can also introduce an effective gauge electric field for photons and realize the effects of frequency Bloch oscillations, which manifest the effects of arbitrary spectral routing and self-imaging. Finally, we show that the discrete frequency diffraction manifests inherent nonreciprocal properties due to the asymmetric wave number mismatches in forward and backward directions. The study may find applications in the precise spectral manipulations for both optical communications and signal processing.

Now we discuss the feasibility for experimental realizations. Though PMs are conventional devices widely used in optical communications and signal processing, the experimental difficulties lie in the stringent bandwidth requirements. The periodic modulation waveforms contain a series of higher-order Fourier harmonics, which can far surpass the bandwidth of PMs and other RF components. To fulfill the bandwidth requirement, we can reduce the modulation frequency from ~10 GHz to ~100 MHz, such that the total frequency shifting amount will be scaling-down reduced. However, the scaling down of frequency lattice requires the dramatically higher spectral resolution for the optical spectrum analyzer. So the experiments could be doable by using the optical spectrum analyzer with very high spectral resolution.

Appendix Coupled-mode equation in the presence of long-range couplings

In the appendix, we provide the detailed derivation of coupled-mode equation of Eq. (2) in the presence of long-range couplings and the matrix algorithm to solve the equation. Consider an optical phase modulator driven by a time-periodic RF signal, the dielectric distribution is

ε (z, t) = ε_{d} + \sum_{m} Δ ε_{m} \cos [m (Ω t - Q z) + ϕ_{m}],

where ε_d is the background dielectric constant. Δε_m, mΩ, mQ and ϕ_m are amplitude, frequency, wave vector, and initial phase of mth-order modulation wave. The electric field distribution is

E (x, z, t) = \sum_{n} a_{n} (z) e^{i (ω_{n} t - β_{n} z)},

where ω_n = ω₀ + nΩ and β_n = β₀ + nq (n = 0, ± 1, ± 2,…) are the frequency and propagation constant of nth-order TE₀ mode. The electric field distribution satisfies

\nabla^{2} E (x, z, t) - \frac{1}{c^{2}} \frac{\partial^{2}}{\partial t^{2}} [ε (z, t) E (x, z, t)] = 0,

Plugging Eqs. (24) and (25) into (26), we can obtain the master equation

\frac{\partial^{2}}{\partial z^{2}} E (x, z, t) - \frac{ε_{d}}{c^{2}} \frac{\partial^{2}}{\partial t^{2}} E (x, z, t) = \frac{1}{c^{2}} \frac{\partial^{2}}{\partial t^{2}} (\sum_{m} Δ ε_{m} \cos [m (Ω t - Q z) + ϕ_{m}] E (x, z, t)),

By applying the slowly varing amplitude approximation, the left side of Eq. (27) is

- \sum_{n} 2 i β_{n} \frac{\partial a_{n} (z)}{\partial z} e^{i (ω_{n} t - β_{n} z)},

By denoting ω_n_±_m → ω_n_±_m, β_n_±_m → β_n_±_m, where q is the wave number difference between adjacent order modes. The right side of Eq. (27) thus reads

\begin{array}{l} \frac{\partial^{2}}{\partial t^{2}} {\frac{1}{2} \sum_{m} Δ ε_{m} [e^{i (m Ω t - m Q z + ϕ_{m})} + e^{- i (m Ω t - m Q z + ϕ_{m})}] \sum_{n} a_{n} (z) e^{i (ω_{n} t - β_{n} z)}} \\ = \frac{1}{2} \frac{\partial^{2}}{\partial t^{2}} \sum_{n} a_{n} (z) \sum_{m} Δ ε_{m} [e^{i (ω_{n} + m Ω) t} e^{- i (β_{n} + m q) z} e^{i (m q - m Q z + ϕ_{m})} + e^{i (ω_{n} - m Ω) t} e^{- i (β_{n} - m q) z} e^{- i (m q - m Q z + ϕ_{m})}] \\ = - \frac{1}{2} \sum_{n} a_{n} (z) \sum_{m} Δ ε_{m} [ω_{n + m}^{2} e^{i ω_{n + m} t} e^{- i β_{n + m} z} e^{i (m Δ q z + ϕ_{m})} + ω_{n - m}^{2} e^{i ω_{n - m} t} e^{- i β_{n - m} z} e^{- i (m Δ q z + ϕ_{m})}], \end{array}

where Δq = q – Q is wave number mismatch during photonic transition between adjacent order modes. By substituting (n ± m) by n, we have

\sum_{n} a_{n} (z) \sum_{m} ω_{n \pm m}^{2} e^{i ω_{n \pm m} t} e^{- i β_{n \pm m} z} = \sum_{n} a_{n \mp m} (z) \sum_{m} ω_{n}^{2} e^{i ω_{n} t} e^{- i β_{n} z},

So the coupled-mode equation for the nth-order mode reads

i \frac{\partial a_{n} (z)}{\partial z} = \sum_{m} C_{n, m} [e^{i (m Δ q z + ϕ_{m})} a_{n - m} (z) + e^{- i (m Δ q z + ϕ_{m})} a_{n + m} (z)],

where the coupling strength between the nth and (n ± m)th-order mode is given by

C_{n, m} = \frac{Δ ε_{m} ω_{n}^{2}}{4 β_{n} c^{2}} = \frac{Δ n_{m} k_{0}}{2},

Since Ω << ω₀, we can denote C_n.m = C_0,_m = C_m = Δε_mω₀²/(4β₀c²). To solve Eq. (31), we can truncate the Fourier harmonics to a maximum order M and matrix dimension to N. The mode amplitudes can thus be denoted as |φ(z)〉 = [a₁(z), a₂(z),…, a_N(z)]^T. Equation (31) can be rewritten as the time-dependent Schrodinger-like equation

i \frac{\partial | φ (z) 〉}{\partial z} = H (z) | φ (z) 〉,

with the z-dependent matrix H(z) given by

H (z) = (\begin{matrix} 0 & C_{1} e^{- i (Δ q z + ϕ_{1})} & \dots & C_{M} e^{- i (M Δ q z + ϕ_{M})} & 0 & \dots & 0 \\ C_{1} e^{i (Δ q z + ϕ_{1})} & 0 & C_{1} e^{- i (Δ q z + ϕ_{1})} & \dots & C_{M} e^{- i (M Δ q z + ϕ_{M})} & 0 & 0 \\ ⋮ & C_{1} e^{i (Δ q z + ϕ_{1})} & 0 & C_{1} e^{- i (Δ q z + ϕ_{1})} & \dots & C_{M} e^{- i (M Δ q z + ϕ_{M})} & 0 \\ C_{M} e^{i (M Δ q z + ϕ_{M})} & ⋮ & C_{1} e^{i (Δ q z + ϕ_{1})} & 0 & C_{1} e^{- i (Δ q z + ϕ_{1})} & \dots & C_{M} e^{- i (M Δ q z + ϕ_{M})} \\ 0 & C_{M} e^{i (M Δ q z + ϕ_{M})} & ⋮ & C_{1} e^{i (Δ q z + ϕ_{1})} & ⋱ & \dots & ⋮ \\ ⋮ & 0 & C_{M} e^{i (M Δ q z + ϕ_{M})} & ⋮ & ⋮ & 0 & C_{1} e^{- i (Δ q z + ϕ_{1})} \\ 0 & 0 & 0 & C_{M} e^{i (M Δ q z + ϕ_{M})} & \dots & C_{1} e^{i (Δ q z + ϕ_{1})} & 0 \end{matrix}),

Generally, the solution of Eq. (33) is given by |φ(z)〉 = T(z)|φ(0)〉, where |φ(0)〉 = [a₁(0), a₂(0),…, a_N(0)]^T is the input state, T(z) is the amplitude transfer matrix obtained by numerically solving Eq. (33). In particular, for the phase-matching condition Δq = 0, the matrix can reduce to

H = (\begin{matrix} 0 & C_{1} e^{- i ϕ_{1}} & \dots & C_{M} e^{- i ϕ_{M}} & 0 & \dots & 0 \\ C_{1} e^{i ϕ_{1}} & 0 & C_{1} e^{- i ϕ_{1}} & \dots & C_{M} e^{- i ϕ_{M}} & 0 & 0 \\ ⋮ & C_{1} e^{i ϕ_{1}} & 0 & C_{1} e^{- i ϕ_{1}} & \dots & C_{M} e^{- i ϕ_{M}} & 0 \\ C_{M} e^{i ϕ_{M}} & ⋮ & C_{1} e^{i ϕ_{1}} & 0 & C_{1} e^{- i ϕ_{1}} & \dots & C_{M} e^{- i ϕ_{M}} \\ 0 & C_{M} e^{i ϕ_{M}} & ⋮ & C_{1} e^{i ϕ_{1}} & ⋱ & \dots & ⋮ \\ ⋮ & 0 & C_{M} e^{i ϕ_{M}} & ⋮ & ⋮ & 0 & C_{1} e^{- i ϕ_{1}} \\ 0 & 0 & 0 & C_{M} e^{i ϕ_{M}} & \dots & C_{1} e^{i ϕ_{1}} & 0 \end{matrix}),

The eigen values of H are denoted by λ₁, λ₂,…, λ_N₋₁, λ_N and the eigenvector matrix is P. The solution of Eq. (33) is |φ(z)〉 = T(z)|φ(0)〉, with the transfer matrix T(z) given by

T (z) = P [\begin{matrix} e^{- i λ_{1} z} & 0 & \dots & 0 \\ 0 & e^{- i λ_{2} z} & \dots & 0 \\ ⋮ & ⋮ & ⋱ & 0 \\ 0 & 0 & 0 & e^{- i λ_{N} z} \end{matrix}] P^{- 1},

Funding

Program 973 (2014CB921301); National Natural Science Foundation of China (11674117); Graduates’ Innovation Fund, Huazhong University of Science and Technology (5003012009).

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Frequency diffraction management through arbitrary engineering of photonic band structures

Abstract

1. Introduction

2. Results and Discussions

2.1 Band structure engineering in a synthetic frequency lattice

2.2 Frequency discrete Talbot effect with band structure engineering

2.3 Frequency Bloch oscillations with band structure engineering

3. Conclusions and discussions

Appendix Coupled-mode equation in the presence of long-range couplings

Funding

References and links

Cited By

Figures (5)

Equations (36)

Optics Express