Second-order Nonlinear Medium Research Articles

Quantum speed limit of a single atom in a squeezed optical cavity mode

We theoretically study the quantum speed limit of a single atom trapped in a Fabry–Perot microresonator. The cavity mode will be squeezed when a driving laser is applied to the second-order nonlinear medium, and the effective Hamiltonian can be obtained under the Bogoliubov squeezing transformation. The analytical expression of the evolved atom state can be obtained by using the non-Hermitian Schrödinger equation for the initial excited state, and the quantum speed limit time coincides very well for both the analytical expression and the master equation method. From the perspective of quantum speed limit, it is more conducive to accelerate the evolution of the quantum state for the large detuning, strong driving, and coupling strength. For the case of the initial superposition state, the form of the initial state has more influence on the evolution speed. The quantum speed limit time is not only dependent on the system parameters but also determined by the initial state.

Speckle filtering through nonlinear wave mixing.

Light scattering by disordered media is a ubiquitous effect. After passing through them, the light acquires a random phase, masking or destroying associated information. Filtering this random phase is of paramount importance to many applications, such as sensing, imaging, and optical communication, to cite a few, and it is commonly achieved through computationally extensive post-processing using statistical correlation. In this work, we show that mixing noisy optical modes of various complexity in a second-order nonlinear medium can be used for efficient and straightforward filtering of a random wavefront under sum-frequency generation processes without utilizing correlation-based calculations.

Hybrid electro-optic modulator combining silicon photonic slot waveguides with high-k radio-frequency slotlines

Electro-optic (EO) modulators rely on the interaction of optical and electrical signals with second-order nonlinear media. For the optical signal, this interaction can be strongly enhanced using dielectric slot–waveguide structures that exploit a field discontinuity at the interface between a high-index waveguide core and the low-index EO cladding. In contrast to this, the electrical signal is usually applied through conductive regions in the direct vicinity of the optical waveguide. To avoid excessive optical loss, the conductivity of these regions is maintained at a moderate level, thus leading to inherent RC limitations of the modulation bandwidth. In this paper, we show that these limitations can be overcome by extending the slot–waveguide concept to the modulating radio-frequency (RF) signal. Our device combines an RF slotline that relies on B a T i O 3 as a high-k dielectric material with a conventional silicon photonic slot waveguide and a highly efficient organic EO cladding material. In a proof-of-concept experiment, we demonstrate a 1 mm long Mach–Zehnder modulator that offers a 3 dB bandwidth of 76 GHz and a 6 dB bandwidth of 110 GHz along with a small π voltage of 1.3 V ( U π L = 1.3 V m m ). We further demonstrate the viability of the device in a data-transmission experiment using four-state pulse-amplitude modulation (PAM4) at line rates up to 200 Gbit/s. Our first-generation devices leave vast room for further improvement and may open an attractive route towards highly efficient silicon photonic modulators that combine sub-1 mm device lengths with sub-1 V drive voltages and modulation bandwidths of more than 100 GHz.

Large and robust mechanical squeezing of optomechanical systems in a highly unresolved sideband regime via Duffing nonlinearity and intracavity squeezed light.

We propose a scheme to generate strong and robust mechanical squeezing in an optomechanical system in the highly unresolved sideband (HURSB) regime with the help of the Duffing nonlinearity and intracavity squeezed light. The system is formed by a standard optomechanical system with the Duffing nonlinearity (mechanical nonlinearity) and a second-order nonlinear medium (optical nonlinearity). In the resolved sideband regime, the second-order nonlinear medium may play a destructive role in the generation of mechanical squeezing. However, it can significantly increase the mechanical squeezing (larger than 3dB) in the HURSB regime when the parameters are chosen appropriately. Finally, we show the mechanical squeezing is robust against the thermal fluctuations of the mechanical resonator. The generation of large and robust mechanical squeezing in the HURSB regime is a combined effect of the mechanical and optical nonlinearities.

Photon blockade by enhancing coupling via a nonlinear medium.

A significantly low value of the single-photon coupling constant is a major challenge in the creation of a single-photon source via photon blockade. Here, we propose a photon blockade scheme composed of a weakly second-order nonlinear medium with an optical parametric amplification in a low-frequency cavity. Unlike the traditional weakly coupled system, the effective coupling strength in the proposed scheme can be significantly higher than the decay rate of the cavity mode. This can be achieved by adjusting the squeezing parameter even if the original coupling strength is weak. The thermal noise of the squeezed cavity mode can be suppressed by a squeezed vacuum field. Using a probability amplitude method, we obtain the optimal condition of photon blockade in the steady-state. By solving the master equation numerically in the steady-state, a strong photon antibunching effect that is consistent with the optimal conditions can be obtained in the cavity with low frequency. Besides, the photon blockade phenomenon and cross-correlation of two cavities can be significantly enhanced under a specific squeezing parameter. Our results may be useful for future studies on the characteristics of photon statistics.

Optical Frequency Combs in Quadratically Nonlinear Resonators.

Optical frequency combs are one of the most remarkable inventions in recent decades. Originally conceived as the spectral counterpart of the train of short pulses emitted by mode-locked lasers, frequency combs have also been subsequently generated in continuously pumped microresonators, through third-order parametric processes. Quite recently, direct generation of optical frequency combs has been demonstrated in continuous-wave laser-pumped optical resonators with a second-order nonlinear medium inside. Here, we present a concise introduction to such quadratic combs and the physical mechanism that underlies their formation. We mainly review our recent experimental and theoretical work on formation and dynamics of quadratic frequency combs. We experimentally demonstrated comb generation in two configurations: a cavity for second harmonic generation, where combs are generated both around the pump frequency and its second harmonic and a degenerate optical parametric oscillator, where combs are generated around the pump frequency and its subharmonic. The experiments have been supported by a thorough theoretical analysis, aimed at modelling the dynamics of quadratic combs, both in frequency and time domains, providing useful insights into the physics of this new class of optical frequency comb synthesizers. Quadratic combs establish a new class of efficient frequency comb synthesizers, with unique features, which could enable straightforward access to new spectral regions and stimulate novel applications.

Transient Second-Order Nonlinear Media: Breaking the Spatial Symmetry in the Time Domain via Hot-Electron Transfer

Second-order optical effects are essential to the active control of light and the generation of new spectral components. The inversion symmetry, however, prevents achieving a bulk χ^{(2)} response, limiting the portfolio of the second-order nonlinear materials. Here, we demonstrate subpicosecond conversion of a statically passive dielectric to a transient second-order nonlinear medium upon the ultrafast transfer of hot electrons. Induced by an optical switching signal, the amorphous dielectric with vanishing intrinsic χ^{(2)} develops dynamically tunable second-order nonlinear responses. By taking the second-harmonic generation as an example, we show that breaking the inversion symmetry through hot-electron dynamics can be leveraged to address the critical need for all-optical control of second-order nonlinearities in nanophotonics. Our approach can be generically adopted in a variety of material and device platforms, offering a new class of complex nonlinear media with promising potentials for all-optical information processing.

Effect of second-order nonlinearity on quantum coherent oscillations in a quantum dot embedded in a doubly resonant-semiconductor micro-cavity

The quantum Rabi oscillations in a coherently driven quantum dot embedded in a high-Q doubly resonant semiconductor micro-cavity in the presence of a strong second-order nonlinear medium is investigated theoretically. It is shown that the χ(2) nonlinearity effect the Rabi oscillations in a significant manner.

Simultaneous carrier extraction from wavelength-division-multiplexed phase-conjugated twin waves by using sum-frequency generation with second-order optical nonlinear medium for phase-sensitive optical amplifier repeaters

This paper proposes and numerically investigates simultaneous carrier extraction from wavelength-division multiplexed phase-conjugated twin waves (WDM-PCTWs) in optical phase-locked loop (OPLL) for phase-sensitive optical amplifier (PSA) repeaters on optical multi-relay transmission systems. First, it is mathematically derived that the proposed scheme contributes to a signal-to-noise ratio improvement of phase error signals between WDM-PCTWs and a pump wave for PSA. As a result, the quality of the pump wave is improved. Then, the influence of chromatic dispersion in a transmission fiber on the proposed scheme is numerically investigated. When a center optical frequency of WDM-PCTWs is equal to a zero-dispersion optical frequency of an 80 km transmission fiber, the extracted carrier power increases as the number of WDM channels increase. In contrast, an optical frequency difference between the center frequency of WDM-PCTWs and the zero-dispersion optical frequency of a transmission fiber severely impairs the extracted carrier power as the number of WDM channels increase.

Utilizing the plasmonic resonance to enhance three wave mixing effects in nano-scale cut-wire

In this paper, quantitative study of the enhanced three wave mixing effects from periodic array of nano-scale golden cut-wires embedded in nonlinear dielectric is performed. Geometrical parameters of the considered structure are chosen so that the plasmonic resonance occurs in the wavelength 1.5 μm. Plasmonic resonance of this structure gives rise to the localization factor significantly larger than that of the cut-wire structures with no plasmonic resonance. Considering the surrounding medium of the cut-wire a typical second-order nonlinear optical medium, enhancement of the nonlinearity of this nonlinear medium is illustrated. Since the mentioned structure can be considered a homogeneous nonlinear medium with the effective nonlinear susceptibility, nonlinear retrieval method is used to determine the effective nonlinear susceptibility for all four frequency combinations. Maximum enhancement of the effective susceptibility is obtained for the second harmonic generation of the applied wave with the frequency corresponding to the plasmonic resonance, with the enhancement up to two orders of magnitude. Efficiency of the second harmonic generation in this resonant structure surrounded with nonlinear dielectric is calculated to be 30 times more than that of the nonlinear dielectric with the same dimensions.

Investigation of anti-Stokes Raman processes at phonon-polariton resonance: from Raman oscillation, frequency upconversion to Raman amplification.

Raman oscillation, frequency upconversion, and Raman amplification can be achieved in a second-order nonlinear medium at the phonon-polariton resonance. By beating two optical fields, a second-order nonlinear polarization is generated inside the medium. Such a polarization induces a spatially uniform nonpropagating electric field at the beat frequency, which in turn mixes with the input optical field at the lower frequency to generate or amplify the anti-Stokes optical field. Raman oscillation can be efficiently reached for the copropagating configuration. In comparison, efficient frequency upconversion and large amplifications are achievable for the counterpropagating configuration. These Raman processes can be used to effectively remove transverse-optical phonons before decaying to lower-frequency phonons, achieve laser cooling, and significantly enhance coherent anti-Stokes Raman scattering. The counterpropagating configuration offers advantages for amplifying extremely weak signals.

Enhanced second order nonlinearity using plasmonic resonance of nano-scale metallic cut-wires

In this paper, we study quantitatively the enhanced second harmonic generation from periodic array of nano-scale golden cut-wires embedded in nonlinear dielectric. Geometrical parameters of this structure are varied so that the plasmonic resonance occurs in the wavelength of Nd:YAG laser (1064 nm), which its second-harmonic (with the wavelength 532 nm) is usually used in optical devices. Plasmonic resonance of these structures gives rise to the localization factor more than two times larger than that of the cut-wire structures with no plasmonic resonance. After then, surrounding medium of the cut-wire is considered a typical second-order nonlinear optical medium. In order to show the enhancement of the nonlinearity, unit cell of the cut-wires embedded in nonlinear medium is considered a homogeneous nonlinear medium with the effective nonlinear susceptibility. Nonlinear retrieval method, which is based on the transfer matrix approach, is used to determine the effective nonlinear susceptibility. The obtained results show the enhancement of the effective nonlinear susceptibility two orders of magnitude larger compared to the susceptibility of the embedding nonlinear dielectric. Efficiency of the second harmonic generation in these resonant structures surrounded with nonlinear dielectric is calculated to be 20 times more than that of the nonlinear dielectric with the same dimensions.

Restoration of blurred images due to phase distortion based on polarization-insensitive phase conjugation in second-order nonlinear medium

Broadband and polarization-insensitive phase conjugation, achieved based on difference-frequency generation in a second-order nonlinear composite consisting of stacked KTiOPO(4) (KTP) plates, was exploited to restore blurred images due to phase distortion as a novel scheme. Due to the quasi-phase matching in the stacked KTP crystals, our result reveals that the image restoration is insensitive to the polarization direction and wavelength of the input beam.

Phase conjugation based on single backward second-order nonlinear parametric process

We show that backward difference-frequency generation can be exploited to achieve phase conjugation in a second-order nonlinear medium. The backward configuration can be utilized to achieve broadband quasi-phase-matching, compared with the forward counterpart. Our calculation shows that a nonlinear reflectivity of close to 100% is achievable from a laser emitting an output power of ~1 mW. Such an efficient phase conjugator is made feasible by placing the nonlinear medium inside a pump laser cavity. In addition, a Fabry-Perot resonator at the input frequency is used to significantly improve the nonlinear reflectivity.

Terahertz intracavity generation from output coupler consisting of stacked GaP plates

We demonstrate intracavity terahertz generation from an output coupler of a compact dual-frequency solid state laser. The output coupler consisting of unbonded and stacked GaP plates is used as a second-order nonlinear medium where the difference-frequency generation takes place. When quasi-phase matching is achieved within the alternatively rotated GaP plates, terahertz output power is significantly enhanced compared with that for the corresponding external-cavity configuration.

Squeezing and photon antibunching in second harmonic generation: an analytical approach

With the help of second-order nonlinear interactions and hence Hamiltonian, we construct the equations of motion corresponding to pump and signal (second harmonic) fields in a second harmonic generation. The corresponding coupled differential equations involving the non-commuting field operators are not solvable in closed analytical forms. With the help of a perturbation method, we obtain analytical solutions of these field operators up to cubic orders in the interaction constant. In an appropriate limit (truncating the solution up to the second order in the interaction constant), these solutions lead to the existing solutions under the short-time approximation. The present analytical solutions are exploited to investigate the squeezing and the antibunching of photons of the input coherent light coupled to the said second-order nonlinear medium. We report the squeezing and antibunching effects of the pump field even for solutions up to the second order and hence the present results are consistent with earlier results. However, for the second harmonic mode, we report the squeezing involving the leading cubic term in the interaction constant.

Coherent phase-modulation transfer in counterpropagating parametric down-conversion

Distributed positive feedback established by a counterpropagating three-wave mixing process in a submicrometer structured second-order nonlinear medium leads to mirrorless parametric oscillation with unusual spectral properties. In this work, we demonstrate experimentally and theoretically that the phase modulation of the pump is coherently transferred to the co-propagating parametric wave, while the counterpropagating wave retains a narrow bandwidth and high coherence. The main mechanisms responsible for these properties are the maximized convective separation between the counterpropagating waves and the associated phase-locking between the pump and the co-propagating parametric wave. This suggests that mirrorless optical parametric oscillators can be pumped with incoherent light and still generate highly coherent backward-propagating radiation.

Sub-shot noise photon-number correlation in stimulated and spontaneous Raman processes

The light initially prepared in a coherent state runs the stimulated and spontaneous Raman processes in a second-order nonlinear medium. Using analytical expressions of field operators for the light modes, we determine the quantum statistical properties of the modes in the stimulated and spontaneous Raman processes. By a choice of phase, we obtain a boson-pair property correlating the pump and vibration modes and such a property correlating the pump and anti-Stokes modes. The Stokes and vibration modes exhibit the boson-pair property for all initial phase differences between the pump mode and the other ones. In the spontaneous Raman process only the Stokes and vibration modes show such a property.

40 GHz electro-optic modulation in hybrid silicon–organic slotted photonic crystal waveguides

In this Letter we demonstrate broadband electro-optic modulation with frequencies of up to 40 GHz in slotted photonic crystal waveguides based on silicon-on-insulator substrates covered and infiltrated with a nonlinear optical polymer. Two-dimensional photonic crystal waveguides in silicon enable integrated optical devices with an extremely small geometric footprint on the scale of micrometers. The slotted waveguide design optimizes the overlap of the optical and electric fields in the second-order nonlinear optical medium and, hence, the interaction of the optical and electric waves.

Efficient generation of high-frequency terahertz waves from highly lossy second-order nonlinear medium at polariton resonance under transverse-pumping geometry

Owing to strong coupling between transverse-optical phonons and high-frequency terahertz waves in zinc-blende semiconductors, second-order nonlinear coefficients can be dramatically enhanced within the forbidden band of the polariton resonance. However, linear absorption in this regime is also dramatically increased. We show that transverse-pumping geometry can be exploited for achieving an efficient terahertz generation at the polariton resonance. Our estimates illustrate that pump powers as low as 100 mW are sufficient for causing the significant depletion of the pump beams.