Effective electric-field force for a photon in a synthetic frequency lattice created in a waveguide modulator

Chengzhi Qin, Luqi Yuan, Bing Wang, Shanhui Fan, and Peixiang Lu

Phys. Rev. A 97, 063838 – Published 18 June 2018

Abstract

Here we create an effective electric field force for photon in a synthetic frequency lattice to control the spectrum of light. The frequency lattice is created based on an optical waveguide modulator in which the dynamic index modulation can induce photonic transitions between adjacent lattice sites. We show that the wave vector mismatch during photonic transitions and periodic distribution of the modulation phase can be mapped into a linear-varying and a periodically driven gauge potential, which gives rise to a constant and a harmonic oscillating force, respectively. Under different combinations of the constant and oscillating forces, we can realize the effects of frequency Bloch oscillations, anharmonic Bloch oscillations, super-Bloch oscillations, and directional frequency shift. With an appropriate choice of the magnitude of the oscillating force, we also achieve dynamic localization in the frequency domain. The realization of effective force provides a mechanism to control the spectrum of light, which may find applications in frequency perfect imaging, efficient shifting, and precise transduction.

Received 9 March 2018

DOI:https://doi.org/10.1103/PhysRevA.97.063838

Physics Subject Headings (PhySH)

Dynamical localization Electric field effects Interference & diffraction of light Synthetic gauge fields

Frequency combs & self-phase locking

Mode coupling theory

Nonlinear DynamicsInterdisciplinary Physics

Authors & Affiliations

Chengzhi Qin¹, Luqi Yuan², Bing Wang^1,*, Shanhui Fan², and Peixiang Lu^1,3,†

¹School of Physics and Wuhan National Laboratory for Optoelectronics, Huazhong University of Science and Technology, Wuhan, 430074, China
²Department of Electrical Engineering, and Ginzton Laboratory, Stanford University, Stanford, California 94305, USA
³Laboratory for Optical Information Technology, Wuhan Institute of Technology, Wuhan, 430205, China

^*wangbing@hust.edu.cn
^†lupeixiang@hust.edu.cn

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Vol. 97, Iss. 6 — June 2018

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Images

Figure 1
(a) Schematic diagram of the waveguide modulator with periodic electrodes. The substrate is constituted with a $LiNb O_{3}$ slab waveguide with thickness $d = 0.5 μ m$ , background dielectric constant $ɛ_{d} = 4.58$ ( $n_{0} = 2.14$ ), and dielectric modulation amplitude $Δ ɛ = 5 \times 10^{- 3}$ . (b) Photonic intraband transitions in the $T E_{0}$ band (blue curved arrows). The red and blue curves represent $T E_{0}$ and $T E_{1}$ bands with the gray region denoting the light cone. Both frequencies and wave numbers are normalized with respect to $a = 1 μ m$ . We choose $ω_{0} / (2 π c / a) = 0.6451$ ( $λ_{0} = 1.55 μ m$ ) and $Ω / 2 π = 20 GHz$ . $Δ q = q - q_{m}$ is the wave vector mismatch with $q_{m}$ and $q$ being the modulation wave number and wave number difference between adjacent order modes. $\pm ϕ (z)$ is the $z$ -dependent, nonreciprocal phase shift accompanying photonic transitions.
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Figure 2
(a), (b) Spectral evolutions of frequency BOs for a frequency comb (a) and a single frequency (b) input. The width of Gaussian envelope is $W = 5 Ω$ . The modulation length is $L = 3 Z_{B} / 2$ with black dashed lines denoting $z = Z_{B}$ . The modulation phase with respect to the initial Bloch momentum is $ϕ - ϕ_{0} = π / 2$ . (c), (d) Spectral evolutions of frequency ABOs for a frequency comb (c) and a single frequency (d) input. The parameters are $W = 8 Ω$ , $C_{1} = C_{2}$ and $ϕ_{1} - ϕ_{0} = ϕ_{2} - ϕ_{0} = π$ . (e), (f) Spectral evolutions of frequency SBOs for a frequency comb (e) and a single frequency (f) input. We choose $W = 50 Ω$ , $Δ ϕ_{m} = π / 2$ , $ϕ - ϕ_{0} = ϕ_{m} - ϕ_{0} = 0$ . The wave vector of ac force is $Q_{m} = (Δ q - Δ Q) / m$ with $m = 1$ and $Δ Q = Δ q / 20$ , thus we have $Z_{SB} = 20 Z_{B}$ . The modulation length is $L = 3 Z_{SB} / 2$ , and the black dashed lines denote $z = Z_{SB}$ . The red solid lines denote the theoretical trajectories.
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Figure 3
(a) Output spectral evolution under a frequency comb input as $ϕ - ϕ_{0}$ varies from 0 to 2π as $L = 3 Z_{SB} / 2$ . We choose $W = 100 Ω$ , $Q_{m} = (Δ q - Δ Q) / m$ with $m = 1$ and $Δ Q = Δ q / 20$ such that $Z_{SB} = 20 Z_{B}$ . The blue, red, and black dashed lines denote $ϕ - ϕ_{0} = π / 2$ , $3 π / 2$ and $π$ , respectively. (b)–(d) Spectral evolutions for a frequency comb input with $ϕ - ϕ_{0} = 0$ , $3 π / 2$ , and $π / 2$ , respectively. The red solid lines denote the respective theoretical trajectories. (e), (f) The original and equivalent amplitude spectral evolutions under a single frequency input. The red solid lines denote the theoretical results.
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Figure 4
The maximum frequency shift $Δ ω$ versus $Δ ϕ_{m}$ for both frequency directional shift (red) and SBOs (blue), respectively. We choose $L = Z_{SB}$ with other parameters kept the same with those in Fig. 2. Insert (a), spectral evolution for frequency directional shift under a single frequency input as $Δ ϕ_{m} = 2.40483$ (red circle a). Insert (b), spectral evolution for frequency SBOs as $Δ ϕ_{m} = 3.83171$ (blue circle b). The blue circle c and red circle d denote $Δ ϕ_{m} = π / 2$ , which represent the cases in Fig. 2 and Fig. 3, respectively.
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Figure 5
(a), (b) Theoretical and numerical spectral evolutions of frequency BOs for a single frequency and a frequency comb input. Here we choose $Ω / 2 π = 4 THz$ , $Δ ɛ = 0.5$ with other parameters kept the same with those in Fig. 2. (c) Theoretical and numerical spectral evolutions of frequency ABOs under a single frequency and a frequency comb input. The red dashed lines and black circles represent the numerical and theoretical spectra at $z = 0$ , $Z_{B} / 4$ , $Z_{B} / 2$ , $3 Z_{B} / 4$ , and $Z_{B}$ , which are obtained by first-principle simulations using COMSOL and by solving the coupled-mode equation of Eq. (1).
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Figure 6
Theoretical and numerical spectral evolutions of frequency BOs under a single ultrashort pulse input. We choose $ω_{0} / 2 π = 193.4 THz$ ( $λ_{0} = 1.55 μ m$ ), $t_{0} = 50 ps$ , and $Δ t = 50 fs$ , and thus the Gaussian spectral width is $Δ ω / 2 π = 40 THz$ . The modulation phase with respect to the pulse arrival time is $ϕ = 0$ , $π$ , $- π / 2$ , $π / 2$ in panels (a)–(d), respectively. The red solid lines denote the numerical spectra distribution at $z = 0$ , $Z_{B} / 4$ , $Z_{B} / 2$ , $3 Z_{B} / 4$ , which are obtained by using COMSOL Multiphysics.
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