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IEEE Journal of Quantum Electronics

Noise in interferometric optical systems: an optical Nyquist theorem

William H. Glenn

Published Jun 1, 1989• 74 citations
Publisher

Abstract

The index of refraction of an optical medium at a temperature above absolute zero undergoes statistical fluctuations, which in turn introduce fluctuations in the phase of an optical signal propagating in the medium. The magnitude and spectral density of these phase fluctuations are calculated, and it is shown that they can be larger than the phase uncertainty due to quantum noise. Barring the onset of nonlinearities, these fluctuations and not quantum noise set the limit to the measurement of optical phase.>

DOI
10.1109/3.29251

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The index of refraction of an optical medium at a temperature above absolute zero undergoes statistical fluctuations, which in turn introduce fluctuations in the phase of an optical signal propagating in the medium. The magnitude and spectral density of these phase fluctuations are calculated, and it is shown that they can be larger than the phase uncertainty due to quantum noise. Barring the onset of nonlinearities, these fluctuations and not quantum noise set the limit to the measurement of optical phase.>
A stabilized laser emits a light beam in a coherent state: a light field with properties very similar to a classical field, but with quantum zero-point fluctuations (vacuum noise) of the field superposed. The nonlinear refractive index of optical fibers can be used to alter the quantum statistics of the light to produce a beam with non-classical properties: a squeezed state. Upon detection, squeezed states exhibit a noise level which is phase-dependent and which can be well below the coherent state level ("shot-noise limit"). The nonlinear refractive index also provides a means for experimental demonstration of the related concept of "quantum non-demolition detection": The measurement of a quantum observable without adding uncertainty to that variable.
This paper examines thermal (light-scattering) fluctuations as a dominant noise source in nonlinear optical processes. Fundamental limits to conjugate wave fidelity and signal power requirement are obtained through the statistical thermodynamic treatment of light-scattering noise in four-wave mixing based on the fluctuation-dissipation theorem. Several types of nonlinear media are examined including artificial Kerr suspensions, isotropic Kerr media, and fluids near a critical point. Where measurements are available, excellent quantitative agreement is obtained between theory and experiment.
Optical solitons are pulses that propagate in optical fibers without temporal or spectral distortion, owing to balancing of the second-order dispersion of the refractive index by nonlinear self-phase modulation induced by the intensity-dependent refractive index. Classically, this stationary distortion-free propagation only requires the proper choice of pulse width and shape for a given pulse energy. However, quantum fluctuations do not undergo stationary propagation but rather evolve due to dispersion and self-phase modulation. As a soliton pulse propagates, zero-point or vacuum noise associated with the pulse evolves into squeezed noise. Vacuum fluctuations are responsible for the fundamental detector noise floor known as the shot noise limit. Reduction of the noise in the detection of squeezed soliton pulses has achieved a noise level 34% below this limit. In optical fibers, thermal fluctuation of the refractive index limit the noise reduction that can be achieved by adding phase fluctuations in excess of the vacuum noise. At low frequencies thermal noise originates from local structural dynamics of fused silica which produces a 1/f power spectrum and from guided acoustic wage Brillouin scattering which modulates the refractive index at frequencies characteristic of the fiber structure. The bandwidth of these fluctuations is ~2 GHz; and for soliton pulses with spectral bandwidth much >2 GHz, we have shown that this noise can be made insignificant. For very short pulses, however, noise from Raman scattering must be considered. Recent calculations incorporating this noise mechanism are compared with experiment.
An improvement in precision beyond the limit set by the vacuum-state or zero-point fluctuations of the electromagnetic field is reported for the measurement of phase modulation in an optical interferometer. The experiment makes use of squeezed light to reduce the level of fluctuations below the shot-noise limit. An increase in the signal-to-noise ratio of 3.0 dB relative to the shot-noise limit is demonstrated, with the improvement currently limited by losses in propagation and detection and not by the degree of available squeezing.
The influence of quantum fluctuations of the electromagnetic field on the propagation of a polarized light wave in a nonlinear dielectric is investigated. It is shown that in some cases the fluctuations couple to the optical nonlinearities of the medium and make its refractive index dependent on the polarization of the propagating wave. As a consequence, two light waves propagating in the same direction but with different polarizations will have different speeds, so there will be a Kerr-effect birefringence induced by quantum fluctuations. We consider the case of the electromagnetic field in a squeezed vacuum state, where there are regions where the mean squared electric field can be negative and the birefringence effect has the opposite sign to the Kerr effect in classical physics. We give some estimates for the magnitude of these effects, and discuss their possible observability.
Classical optical solitons propagate in optical fibers without temporal or spectral distortion as a result of a balance between phase shifts due to group velocity dispersion and those due to nonlinear self-phase modulation. This balance is achieved by the proper choice of pulse shape and duration for a given pulse energy. The quantum fluctuations associated with a coherent soliton do not undergo stationary propagation, however, and a number of effects are predicted, including the evolution of vacuum noise associated with the input coherent pulse into squeezed fluctuations and the temporal spreading of the soliton pulse because of dispersion acting on quantum fluctuations of the pulse frequency.