Phase Conjugate Wave Research Articles

Nonlinearity-tolerant transmission using modified phase-conjugate twin wave for M-PSK signals

Abstract Using modified phase-conjugate twin wave (M-PCTW) scheme, in which additional information bits are modulated one of the conjugated wave, is helpful to increase the spectral efficiency (SE) of the conventional PCTW scheme. In this paper, we show that the phase modulated formats, but not the conventional square QAM formats are more suitable to M-PCTW scheme. Results of 1600 km nonlinear transmission show that for polarization multiplexed (PM) 8-PSK format, performance improvement from M-PCTW is about 3.3 dB, while it is about 2.2 dB for PM 8-QAM formats. For PM 16-PSK (QAM) modulation, performance improvement from M-PCTW is about 3.4 dB and 1.6 dB, respectively.

Four-wave mixing model for centrosymmetric constellation shaping in twin-wave-based coherent OFDM system including intersymbol interference

The four-wave-mixing (FWM) noises are modeled when walk-off and intersymbol interference (ISI) are taken into account for any centrosymmetric constellation shaped signal in orthogonal frequency-division multiplexed (OFDM) with phase-conjugated twin waves (PCTW) scheme. Bit error rate (BER) and the equivalent Q-factor for any constellation shaped signal are also modeled when the FWM and amplified spontaneous emission (ASE) noises are taken into account as additive white Gaussian noise. By using the newly derived semi-analytic models, example calculations are carried out for 16-ary signal in PCTW OFDM system with four cases classified according to some proposed constellation shaping schemes adopted or not. A hybrid constellation shaping scheme consisted of Huffman-coded probabilistic shaping and radius-optimized geometric shaping is proposed. The performance gain in terms of the equivalent Q-factor is calculated to be about 0.563 dB for 16-ary signal in PCTW OFDM system with the proposed hybrid constellation shaping scheme over the square conventional 16QAM signal. The performance degradation for such system due to the effects of ISI on the FWM noise is also evaluated by using the semi-analytic calculation models.

Quantum Langevin theory for two coupled phase-conjugated electromagnetic waves

While loss-gain-induced Langevin noises have been intensively studied in quantum optics, the effect of a complex-valued nonlinear coupling coefficient on the noises of two coupled phase-conjugated optical fields has never been questioned before. Here, we provide a general macroscopic phenomenological formula of quantum Langevin equations for two coupled phase-conjugated fields with linear loss (gain) and complex nonlinear coupling coefficient. The macroscopic phenomenological formula is obtained from the coupling matrix to preserve the field commutation relations and correlations, which does not require knowing the microscopic details of light-matter interaction and internal atomic structures. To validate this phenomenological formula, we take spontaneous four-wave mixing in a double-$\Lambda$ four-level atomic system as an example to numerically confirm that our macroscopic phenomenological result is consistent with that obtained from the microscopic Heisenberg-Langevin theory. Finally, we apply the quantum Langevin equations to study the effects of linear gain and loss, complex phase mismatching, as well as complex nonlinear coupling coefficient in entangled photon pair (biphoton) generation, particularly to their temporal quantum correlations.

Phase Conjugation of Sound Beams in Piezoelectric Semiconductors in an Alternating Magnetic Field

The propagation of a linearly polarized transverse sound wave in a semiconductor layer with a certain crystallographic structure with a sufficiently high concentration of conduction electrons is considered. The semiconductor layer is located in a spatially uniform magnetic field directed perpendicular to the plane of incidence of the sound wave and modulated with the sound frequency. The relationship between the plasma subsystem and acoustic oscillations in the layer is achieved due to the internal piezoelectric field. It is shown that propagation of a sound beam under these conditions is accompanied by a counterpropagating sound wave with a conjugated wave front. The conversion coefficient of direct and phase-conjugated waves is found.

Modeling four-wave mixing noises for any signal constellation in twin-wave-based coherent OFDM system

Semi-analytic models of four-wave-mixing (FWM) noise for any possible signal constellation, called as general FWM models, are derived in this paper for coherent orthogonal frequency-division multiplexed (OFDM) system with three kinds of phase-conjugated twin waves (PCTW) schemes. The PCTW scheme, used to reduce the impact of FWM noise by co-propagating a signal with its phase-conjugated copy along optical fiber link, can be implemented in polarization, time or subcarrier domains. Based on the derived general FWM models, example calculations are carried out to evaluate the performance of OFDM system with polarization-domain PCTW scheme for 16-ary signal with several kinds of signal constellations when the FWM and the amplified spontaneous emission (ASE) noises are included. Three kinds of 16-ary signal constellations are defined here. The first kind, written as (N1, N2), has N1 signal points on the inner circle and N2 points on the outer circle. The second kind, written as (1, N1, N2), has N1 signal points on the inner circle, N2 points on the outer circle and a signal point in the center. The third kind, written as (N1, N2, N3), has N1 signal points on the innermost circle, N2 signal points on the media circle and N3 points on the outermost circle. Among all 16-ary signal constellations discussed here, the performance of (5, 11) is best and that of the 16-ary phase-shift keying (PSK) is worst. The performance of (5, 11) is close to that of (1, 5, 10) and 16-ary quadrature amplitude modulation (QAM).

Phase-Conjugate Wave in Disperse Orange 3-Doped Polymer

Phase-Conjugate Wave in Disperse Orange 3-Doped Polymer

Degenerate Four-Wave Mixing in Phycoerythrin Dye-Doped Nanoparticles

We have generated a phase-conjugate (PC) wave from nanoparticles with a new microscopic system proposed. The microscope includes a confocal system with a degenerate four-wave mixing (DFWM) system, which plays a major role in generating the phase-conjugate wave to compensate phase distortion in the optical path toward targets. The proposed optical system detects feeble PC wave and imagines 3D particles while improving the inplane contrast resolution of the microscopic image.

Autofocusing of in-line holography based on compressive sensing

Holographic reconstruction is affected by the phase-conjugate wave arising from the symmetry of the complex field. Compressive sensing (CS) has been used in in-line digital holography (DH) to eliminate noise, especially the interference from twin images. Herein, CS with total variation regularization combining autofocusing is presented. It provides an autofocusing function from a single-exposure hologram and obtains reconstructed objects without twin image noise. A series of images at a fixed interval within a reconstruction distance are processed using a two-step iterative shrinkage/thresholding algorithm in CS. It can calculate the focus distance in a larger range around the focal plane using twin-image-free reconstruction, so it can achieve a higher focusing accuracy than traditional focusing methods, including the Laplace operator, absolute gradient operator, and Tamura coefficient. The proposed method is a simple combination of algorithms and a powerful extension that can effectively improve simulated and experimental image quality and handle difficult datasets, which existing algorithms cannot.

Numerical simulation of the interaction of wave phase conjugation with bubble clouds

The interaction of the Wave Phase Conjugation (WPC) process with bubble cloud dynamics is investigated via numerical modeling in application to bubble size measurements. The WPC simulator by Modarreszadeh and Timofeev (Computers and Fluids 197, 104353) is used as a starting point. In the simulator, a modified version of the high-order Nodal Discontinuous Galerkin method is used for modeling of acoustic wave propagation in the active WPC conjugator and the surrounding linear or non-linear fluid in 2D/axisymmetric domains. The effects of bubble dynamics are incorporated with the help of induced volume fractions. The modified Keller–Miksis model is employed to simulate vibrations of bubbles subjected to acoustic waves. After thorough verification and validation of the numerical scheme and the physical model, the interaction of various bubble clouds, including poly-disperse ones, with the WPC process is examined with the help of some existing and newly suggested descriptive indicators, e.g., the amplification factor and the absolute and relative signal intensities. The results show that signals scattered from bubble-plane clouds, in which the bubbles are non-physically synced, are strong enough to affect the modulation process if the stimulation frequency is close to the natural frequency of bubbles. For more realistic bubble clouds, the modulation process is almost intact. However, conjugate waves do carry some signatures of bubble-cloud dynamics, e.g., signals with frequencies higher than the natural frequency of bubbles are significantly diminished. The study demonstrates that WPC-based techniques have a promising potential to be used for measuring bubble dimensions.

Effect of shift in refractive index of dispersive elements in waveband-shift-free optical-phase conjugator based on DFG

In this study, the effect of the shift in the refractive index between two dispersive elements on the output phase-conjugated (PC) wave power of waveband-shift-free optical-phase conjugators based on difference-frequency generation (DFG-OPCs) is analyzed. First, a numerical model of the DFG-OPC is built by considering the shift ∆n in the refractive index. The derived formula shows that the output PC wave power of DFG-OPCs depends on the shift in the refractive index. Subsequently, the dependence of the output PC wave power on the shift is confirmed analytically. Calculations indicate that the output PC wave power varies according to the shift. For a relatively fractional shift ∆n = 1.0e-6, the output PC power is degraded by 3 dB. Finally, acceptable values of the shift and temperature difference are calculated as functions of the length of the dispersive element. It is illustrated in this study that the acceptable value of shift in the refractive index increases as the length of the dispersive element is reduced and that the output PC wave power can be stabilized against changes in temperature.

Performance analysis of WDM-SDM system with employing Phase-Conjugated twin waves technique

In this paper, the phase conjugated twin waves (PCTWs) technique is proposed to reduce the nonlinear phase noise in wavelength division multiplexing (WDM) - spatially division multiplexing (SDM) system. In the proposed system, the m-ary quadrature amplitude modulation (mQAM) signals are generated and multiplexed to produce mQAM WDM signal. Similarly, the second WDM signal is created by multiplexing the phase conjugated copies of the mQAM signals and then they are spatially multiplexed by two fiber links. At receiver, they are firstly demultiplexed and then superimposed coherently to revoke the nonlinear phase noise and improve the signal to noise ratio (SNR). Moreover, the analytical model that best describes the phase noise alleviation is developed. Analytical results show that the proposed scheme significantly reduces the variance of the phase noise. Furthermore, to show the effectiveness of proposed scheme, 16-channel 4QAM WDM-SDM system with and without PCTWs method is numerically investigated. Numerical results reveal that the PCTWs scheme enhances the SNR of the received signal by more than 2.7 dB and extends the achievable transmission distance by 39.4% at 10 −5 bit error rate (BER).

Self-focusing in the ocean using out-of-band phase conjugation

Phase conjugation methods have been used in underwater acoustics because of their remarkable spatial self-focusing performance in inhomogeneous, dynamically changing environments. However, phase conjugation devices have stringent constraints such as acoustic transmitters and receivers working simultaneously in the same frequency band, which complicates the implementation of these devices and prevents their wide deployment in the ocean. Here, we introduce the out-of-band phase conjugation (OBPC) for ocean acoustics through which phase conjugation waves are generated at integer multiples of the incident wave frequency. The different bands translate into remarkably simple implementations while maintaining excellent spatial focusing in challenging shallow ocean environments. Furthermore, the resolution of OBPC is significantly higher than that of conventional phase conjugation. An interesting phenomenon typical to out-of-band phase conjugation is the generation of phantom paths along which some of the energy in the phase conjugated wave is focused.

Interaction of Phase-Conjugate Acoustic Waves with Discrete Modes of Elastic Vibrations in a Magnet

Experimental investigation is given to overthreshold parametric phase conjugation of ultrasound waves with the frequency of 22 MHz under resonant excitation of discrete acoustic modes with the frequencies of 0.214 and 1.248 MHz in the antiferromagnetic single crystal of α-Fe2O3. It is shown that nonlinear interaction with discrete modes is accompanied by amplitude and phase modulation of traveling phase-conjugate ultrasound waves. In processes of Raman scattering by discrete modes the modulation depth can amount to 100%.

Nonlinear Distortion Cancellation in Spatial Division Multiplexing System Based on RZ-Coded PCTWs Technique

spatial division multiplexing (SDM) system has been considered as promising systems due to its ability to transport a higher bit rate for longer transmission distance. However, nonlinear phase noise (NPN (degrades SDM system performance. In this paper, we propose return-to-zero (RZ)-coded phase-conjugated twin waves (PCTWs) to improve the effectiveness of nonlinear distortion cancellation in the SDM system. In this approach, the PCTWs are modulated by m-array quadrature amplitude modulation (mQAM) then RZ encoded. After that, RZ-mQAM PCTWs are co-propagated over two fibres links. The received signals are superimposed to suppress the NPN. An analytical model that characterizes the performance of spatial-multiplexed (SM) RZ-coded PCTWs scheme is developed. Moreover, we numerically investigate the system performance with 4QAM format at 20Gsymbol/s rate. The results display that the performance is substantially better for the proposed scheme. Transmission reaches of both 4QAM PCTWs and RZ-4QAM PCTWs schemes are extended by 77.8% and 100%, respectively, in contrast with the 4QAM scheme.

Efficiency enhancement of phase-conjugated twin waves technique by shaping envelopes of subcarriers in all-optical OFDM systems

All-optical orthogonal frequency division multiplexing (AO-OFDM) transmission, which employs high-order modulation formats, opens up a new field of advanced optical transmission. However, a major limitation is the impact of Kerr nonlinearity on AO-OFDM systems that induces nonlinear phase noise (NPN). In this work, we proposed a new method for suppressing NPN in AO-OFDM systems based on shaping envelopes of phase-conjugated twin waves (PCTWs). In the proposed method, each subcarrier and its phase-conjugated replica are firstly modulated with m-array quadrature amplitude modulation (mQAM) format and then shaped by return-to-zero (RZ) coding format. Sequentially, the subcarriers are multiplexed to form twin AO-OFDM signals, which are sent into two similar fiber links. At receiver, both waveforms are separately detected and twin subcarriers are coherently added for suppressing the NPN. An analytical model that characterizes NPN in proposed system is also developed. In addition, the performance of proposed system with 15 subcarriers that modulated by a RZ-4QAM format is numerically demonstrated. The results show that the NPN is considerably suppressed and system performance is highly improved. A signal-to-noise ratio (SNR) is enhanced by 5.7 dB and transmission length is lengthened by 150% as compared to conventional 4QAM-AO-OFDM system.

Waveband-shift-free optical phase conjugator based on difference-frequency generation.

In this paper, a waveband-shift-free optical phase conjugator based on difference-frequency generation (DFG-OPC) that enables broadband operation is proposed and theoretically investigated. First, the principle of phase-conjugated (PC) wave generation with the DFG-OPC is mathematically described. Using a Sagnac loop interferometer with a χ(2) optical nonlinear material and two dispersive elements (DEs), a PC wave with the same wavelength as a signal can be generated. Subsequently, the required DE length difference for the PC wave generation is theoretically calculated. The calculation results indicate the minimal DE length difference is 20.0 μm, and this is because the DFG-OPC enables broadband operation. Second, the wavelength characteristics of the DFG-OPC are investigated through numerical simulation. The operation bandwidth of the DFG-OPC depends on the DE length difference, and an operating bandwidth of the DFG-OPC of 54.5 nm can be achieved when the DE length difference is less than 0.01 m. Finally, the influence of the splitting ratio of an optical 3-dB coupler in the DFG-OPC is numerically studied. The results indicate that tolerance of the optical coupler splitting ratio is equal to or less than ±6% for the DFG-OPC.

High-order numerical simulation of axisymmetric wave phase conjugation

In the present paper, a general physico-mathematical model of magneto-acoustic Wave Phase Conjugation (WPC) is proposed, which includes wave propagation both in solid zones (the conjugator itself and other solid regions, if present) and in fluid zones surrounding the conjugator. Acoustic waves in the solid zones are assumed to behave linearly while in the fluid zones the waves may be either linear or non-linear. The axisymmetric governing equations of the model are discretized on an unstructured triangular mesh with a modified version of the Nodal Discontinuous Galerkin (NDG) method, which is a compact, high-order technique based on non-collocated solution and flux bases. The third-order Strong Stability-Preserving (SSP) Runge-Kutta (RK) scheme is used for integration in time. To avoid any false reflections from the outer boundaries of the computational domain the Nearly Perfectly Matched Layer (NPML) technique is adopted. A comprehensive set of test problems addressing all facets of the proposed numerical model is used for verification via comparison with analytical solutions. After the accuracy, performance, and reliability of the proposed model are established, two WPC problems are numerically simulated for the linear and weakly non-linear regimes. The main WPC properties, such as retro-focusing and parametric resonance, are observed in the numerical experiments with good accuracy.

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.

Numerical Investigation of Phase-Conjugate Wave Generation as a Pulse Cleaner in Femtosecond Petawatt Laser Systems

We report on comprehensive research regarding the potential of phase-conjugate wave (PCW) generation via backward degenerate four-wave mixing process as a pulse cleaner in femtosecond high-power laser systems. We analyzed the performance of this technique by investigating the contrast enhancement capability, gain bandwidth, and energy conversion efficiency. All of these parameters are important in pulse cleaning. The numerical calculations verified that a considerably large efficiency bandwidth product and an excellent contrast enhancement capability can be realized by finding a balance between each physical effect. The simulation results based on this study provide guidance for the application of this technique in laser systems with different bandwidths. The method, which is based on PCW generation, provides new insights for femtosecond petawatt laser systems as pulse cleaners.

Modeling inter-subcarrier four-wave mixing noises in QAM coherent OFDM system using phase-conjugated twin waves with diversity implement domains

For coherent orthogonal frequency-division multiplexed (CO-OFDM) system, four-wave-mixing (FWM) noise is a major limiting factor. A scheme of phase-conjugated twin waves (PCTW) which co-propagates a signal with its phase-conjugated copy along optical fiber link, can be used to reduce the impact of FWM noise. Including walk-off between subcarriers, semi-analytic models of FWM noise are derived here for quadrature amplitude modulation (QAM) CO-OFDM system with three PCTW schemes implemented in time, subcarrier and polarization domains, respectively. We show that many phase-conjugated twin FWM components can be obtained in such system and they can be added destructively at the receiver, so the total FWM noise can be reduced, but some FWM components have not their phase-conjugated twins for the subcarrier-domain PCTW scheme. Some phase-conjugated twin FWM components are generated from different FWM collision processes for the time-domain PCTW scheme, as will reduce the effect of the PCTW scheme The calculative results obtained by using our semi-analytical model show that the polarization-domain PCTW scheme excels the other two schemes in improving the nonlinear tolerance of such system and the three PCTW schemes are all better than a partial carrier filling (PCF) technique.