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Optics Express

Two-color unidirectional reflections by modulating the spatial susceptibility in a homogeneous atomic medium

Xiaoshan Pei, Hanxiao Zhang, Mengmei Pan, Yue Geng, Tianming Li, Hong Yang

Published Apr 5, 2023• 8 citations
Publisher

Abstract

Non-reciprocal reflections of optical signals are unusual yet fascinating to achieve the imminent applications of non-reciprocal photonic devices and circuits. The complete non-reciprocal reflection (unidirectional reflection) was recently found to be achievable in a homogeneous medium, if the real and imaginary parts of the probe susceptibility satisfy the spatial Kramers-Kronig (KK) relation. We propose a coherent four-level tripod model for realizing dynamically tunable two-color non-reciprocal reflections by applying two control fields with linearly modulated intensities. We found that, the unidirectional reflection can be obtained if the non-reciprocal frequency regions are located in the electromagnetically induced transparency (EIT) windows. This mechanism is to break the spatial symmetry by the spatial modulation of susceptibility to induce unidirectional reflections, the real and imaginary parts of the probe susceptibility are no longer required to satisfy the spatial KK relation.

DOI
10.1364/oe.488247

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Non-reciprocal reflections of optical signals are unusual yet fascinating to achieve the imminent applications of non-reciprocal photonic devices and circuits. The complete non-reciprocal reflection (unidirectional reflection) was recently found to be achievable in a homogeneous medium, if the real and imaginary parts of the probe susceptibility satisfy the spatial Kramers-Kronig (KK) relation. We propose a coherent four-level tripod model for realizing dynamically tunable two-color non-reciprocal reflections by applying two control fields with linearly modulated intensities. We found that, the unidirectional reflection can be obtained if the non-reciprocal frequency regions are located in the electromagnetically induced transparency (EIT) windows. This mechanism is to break the spatial symmetry by the spatial modulation of susceptibility to induce unidirectional reflections, the real and imaginary parts of the probe susceptibility are no longer required to satisfy the spatial KK relation.
It is known that the Kramers-Kronig (KK) relation between real and imaginary parts of the optical susceptibility in the frequency domain can also be realized in the space domain, as first proposed in [Nat. Photonics9(7), 436 (2015)10.1038/nphoton.2015.106]. We here study a mechanism to implement spatial KK relations in a cold atomic sample and use it to control unidirectional reflectionless for probe light incident from either the left or right side of the sample at will. In our model, the complex frequency dependent atomic susceptibility is mapped into a spatially dependent one, employing a far-detuned driving field of intensity linearly varied in space. The reflection of an incident light from one side of the sample can then be set to vanish over a specific frequency band directly by changing the driving field parameters, such as its intensity and frequency. Also, by incorporating the Bragg scattering into the spatial KK relation, the reflectivity from the opposite side of the sample, though typically small for realistic atomic densities, can be made to increase to improve the reflectivity contrast. The present scheme bears potentials for all-optical network applications that require controllable unidirectional light propagation.
Control of unidirectional light propagation is of paramount importantance to optical signal processing and optical communication. Especially, the amplified optical signal can isolate noise well that may provide more applications. In this work, we propose a dynamically modulated regime to realize unidirectional reflection amplification in a short and dense uniform atomic medium, and all atoms are driven into four-level double-Λ type by two coupling fields with linearly varied intensities along x direction and two weak probe fields. Based on four-wave mixing resonance and the broken spatial symmetry, the complete nonreciprocal reflection (unidirectional reflection) can be amplified with reflectivity more than 2.0, even to 6.0. In addition, the width, height, and position of the unidirectional reflection bands can be tunable. Thus, our regime is feasible and may inspire further applications in all-optical networks that require controllable unidirectional light amplification.
We propose a scheme for realizing spatial Kramers-Kronig (KK) relation of probe susceptibility in a cold atomic sample driven by a linearly modulated control field. The sample is found to exhibit broken, transitional, and unbroken regimes, where the non-Hermitian KK relation is well satisfied, foreshortened, and fully destroyed in order. The unbroken and transitional regimes are of special interest because they allow unidirectional reflection at one and both sample ends, respectively. Bragg scattering can also be incorporated into spatial KK relation to largely enhance the nonzero reflectivity, leading thus to a high forward-backward reflectivity contrast.
Kramers-Kronig (KK) relation between the dispersion and absorption responses of a signal field can be mapped from the frequency domain into the space domain via the dipole-dipole interactions between a homogeneous sample of target atoms and a control atom. This is achieved by establishing an effective two-level configuration for the three-level target atoms in the single-photon far-detuned driving regime while maintaining a high Rydberg excitation for the three-level control atom in the single-photon resonant driving regime. We find in particular that it is viable to realize a dynamically tunable spatial KK relation supporting asymmetric and even unidirectional reflection for appropriate signal frequencies in a controlled range. Taking a periodic lattice of target atoms instead, multiple Bragg scattering can be further incorporated into spatial KK relation to largely enhance the nonzero reflectivity yet without breaking the asymmetric or unidirectional reflection.
Kramers-Kronig (KK) relation between the dispersion and absorption responses of a signal field can be mapped from the frequency domain into the space domain via the dipole-dipole interactions between a sample of homogeneous target atoms and a ring of equally spaced control atoms. This is achieved by establishing an effective two-level configuration for the target atoms in the far-detuned driving regime of both lower and upper transitions while maintaining the maximal Rydberg excitations for the control atoms via an antiblockade driving scheme of the lower and upper transitions. We find, in particular, that it is viable to realize a tunable spatial KK relation supporting asymmetric and even unidirectional reflection for appropriate signal frequencies in a desired range depending on the frequency of a coupling field. Taking a periodic lattice of target atoms instead, the multiple Bragg scattering can be incorporated into the spatial KK relation in order to enhance the nonzero reflectivity yet without breaking the asymmetric or unidirectional reflection.
In order to achieve the tunable unidirectional reflection amplification in a uniform atomic medium that is of vital importance to design high-quality nonreciprocal photonic devices, we propose a coherent closed three-level Δ-type atomic system by applying a microwave field, and a strong coupling field of linear variation along the x direction to control a probe field. In our scheme, the linearly increased coupling field destroys the spatial symmetry of probe susceptibility and effectively suppresses the reflection of one side; the microwave field constructs closed loop transitions to amplify the probe field and causes phase changes. The numerical simulation indicates that the unidirectional reflection amplification is sensitive to the relative phase ϕ and the coupling detuning Δ c . Our results will open a new route toward harnessing optical non-reciprocity, which can provide more convenience and possibilities in the experimental realization.