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Quantum Optics Journal of the European Optical Society Part B

Optical amplification and squeezed-light generation in fibre interferometers performing degenerate four-wave mixing

M.E. Marhic, Chung-Ho Hsia

Published Dec 1, 1991• 19 citations
Publisher

Abstract

Degenerate four-wave mixing (DFWM) in optical fibres can be used to obtain gain and to generate squeezed states, by utilizing a balanced Mach-Zehnder fibre interferometer (MZFI) with two non-linear arms, which behaves as a degenerate parametric amplifier. In a classical derivation for the optical gain; pump depletion and propagation loss are considered. Maximum achievable gain is calculated in the presence of loss. A numerical study compares the signal gain under different input pump-to-signal field ratios and its limit (undepleted pump), which is the assumption for the derivation of squeezing. Pump depletion and fibre loss hinder the validity of using large non-linear phase shifts to generate squeezed states. A semiclassical derivation provides the correct expression for the degree of squeezing in the absence of loss and pump depletion. Quantum-mechanical solutions provide the same result. The squeezing limitation due to loss is studied by a lumped-parameter model.

DOI
10.1088/0954-8998/3/6/004

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Degenerate four-wave mixing (DFWM) in optical fibres can be used to obtain gain and to generate squeezed states, by utilizing a balanced Mach-Zehnder fibre interferometer (MZFI) with two non-linear arms, which behaves as a degenerate parametric amplifier. In a classical derivation for the optical gain; pump depletion and propagation loss are considered. Maximum achievable gain is calculated in the presence of loss. A numerical study compares the signal gain under different input pump-to-signal field ratios and its limit (undepleted pump), which is the assumption for the derivation of squeezing. Pump depletion and fibre loss hinder the validity of using large non-linear phase shifts to generate squeezed states. A semiclassical derivation provides the correct expression for the degree of squeezing in the absence of loss and pump depletion. Quantum-mechanical solutions provide the same result. The squeezing limitation due to loss is studied by a lumped-parameter model.
Optical amplification in optical fibers by means of degenerate four-wave mixing (DFWM) is studied by classical equations. A balanced interferometer is used to mix and separate the signal and the pump. Exact solutions are obtained even with pump depletion. If the cross-phase modulation is small, the signal power gain varies as the square of the pump self-phase shift. This gain can be sizable for moderate fiber lengths and laser powers, before the onset of stimulated Raman scattering. We have experimentally verified these calculations with a Sagnac interferometer, comprising a 300 m long polarization-preserving fiber, emitting pulses with 100-200 W peak power and 100-200 nsec duration. We have measured a signal power gain of 22 (13.3 dB); this value correlated well with a pump self-phase shift of 3π/2 measured in a separate two-wavelength experiment. This system can be used for squeezing fluctuations of the signal quadrature which is not amplified. We have performed a classical derivation of squeezing, based upon a geometrical study of the transformation of the signal phasor. The results are in agreement with those of more elaborate quantum-mechanical derivations.
A nonlinear Mach-Zehnder fiber-optic interferometer as shown in Fig. 1 can be used for light squeezing1 and amplification.2 The mechanism can be viewed either as the Kerr effect,1 or degenerate four-wave mixing (DFWM).2 Using the classical DFWM approach, one finds that if fields of the same frequency with phasors E s and E p (′signal′ and ′pump′) are fed into the inputs, then one output phasor is given by where the nonlinear phase shift is given by δφnl= QL |E S E p | cosδ, Q is the fiber nonlinear coefficient, L is the fiber length, and δ is the phase difference between the inputs.
The suitability of degenerate four-wave mixing (DFWM) as a source of squeezed-state (two-photon coherent state) light is investigated, both theoretically and experimentally. In previous semiclassical theory, which neglected pump quantization and loss in the nonlinear medium, Yuen and Shapiro [Opt. Lett. 4, 334 (1979)] showed that such states would be generated by 50%-50% combination of the transmitted probe (TP) and the phase-conjugate (PC) output beams of DFWM. Here the effects of quantum fluctuations in the amplitudes and phases of the pump beams are calculated in a traveling-wave finite-interaction-length DFWM configuration. It is shown that in the limit of strong pump beams and weak nonlinear coupling, the classical-pumps assumption of Yuen and Shapiro is valid. With use of this assumption, loss in the nonlinear medium is incorporated into the model by coupling the interacting modes to a set of reservoir modes. It is shown that loss presents an absolute limit on the degree of squeezing that can be achieved with DFWM. Also, for comparison with experimental work, the photoelectron statistics of the TP, PC, and 50%-50% combination modes are calculated for lossy DFWM; significant deviations are found from the predictions for lossless operation. Quantum-noise measurements on the PC and TP beams generated in sodiumvapor DFWM are described. The mixer is pumped and probed by the nearly transform-limited pulses produced by a continuous-wave-oscillator-pulse-amplifier chain dye laser at 589 nm wave-length. The normalized second-factorial moment g2{g}_{2}g2​ of the PC and TP beams is determined from photoelectron-counting measurements. Inability to simultaneously achieve high DFWM reflectivity and acceptably low scattered-plus-fluorescent background, coupled with low photomultiplier quantum efficiency, restrict the experiment to a DFWM regime wherein theory predicts Poisson statistics. Experimental g2{g}_{2}g2​ values corroborate this prediction, indicating that systematic and excess noises have been eliminated. No attempt was made to measure the quantum statistics of the 50%-50% combination mode.
A nonlinear symmetric Mach–Zehnder interferometer is analyzed. It performs squeezing and phase-sensitive amplification through forward degenerate four-wave mixing with automatic phase matching. Since the balanced interferometer is frequency insensitive, the operation is broadband. The squeezing performance is computed numerically by means of the quantum theory without approximation. This result is compared with the squeezing characteristics derived from the linearized representation. It is found that squeezed states can be produced without excess noise. Noise reduction of an interferometric measurement and phase-preserving amplification with this squeezer are also analyzed.
This thesis is primarily a theoretical study of degenerate parametric amplification as a means of generating squeezed-state light. i) A wideband traveling-wave formalism is developed for analyzing quantum mechanically a degenerate parametric amplifier. The formalism is based on spatial differential equations — spatial Langevin equations — that propagate temporal Fourier components of the field through the nonlinear medium. In addition to the parametric nonlinearity, the Langevin equations include absorption and associated fluctuations, dispersion, and pump quantum fluctuations. The dominant effects of dispersion and pump quantum fluctuations on the squeezing produced by a degenerate parametric amplifier are analyzed. ii) The wideband formalism of i) is used to carry out a more detailed analysis of the effects of phase mismatching. With the assumption of a lossless medium and a classical pump, we find that parametric amplification is capable of generating squeezed-state light over a wide band if materials with large χ(2) nonlinearities can be found, and that the squeezing bandwidth can be enhanced by phase mismatching away from degeneracy. iii) We consider again the effect of pump quantum fluctuations on the squeezing produced by parametric amplification. We perform discrete-mode calculations for a parametric amplifier with a quantum pump, and discuss some of the limitations of calculations of this sort in quantum optics. We derive stochastic differential equations (SDEs) for one- and two-mode parametric amplifiers, and from them obtain an iterative solution showing that pump quantum fluctuations impose a limitation on the degree of squeezing obtainable from a parametric amplifier. iv) A possible application of squeezing is considered; in particular, we study the effects of squeezing the intracavity noise in a laser oscillator. We solve the classical noise problem of a realistic laser model by making a bold — and possibly unrealizable — assumption, that the in-phase and quadrature Langevin sources which are responsible for the "noisiness" of the laser can be squeezed. We show that the effect of squeezing the in-phase quadrature is to reduce the phase noise, including the linewidth, of the laser but, due to amplitude-phase coupling, not to eliminate them altogether.
The squeezing properties of non-degenerate parametric amplification and oscillation with pump depletion and non-homogeneous absorption losses are studied. A method based in the Wigner representation is used which allows one, within a linearization scheme, to derive directly the Langevin equations for the involved optical fields from their quantum Heisenberg equations. Non-homogeneous absorption losses for the three beams are included in order to analyse the sensitivity of the squeezing spectra to the asymmetry of the losses. Intracavity squeezing spectra are calculated within this method both below- and above-threshold regimes. Special attention is given to the squeezing properties of the phase sum and amplitude difference of the correlated beams and amplitude of a single beam above threshold, where the role of loss asymmetry and pump losses is analysed in detail.