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Applied Optics

Weak value amplification of an optical Faraday differential refraction effect

Allen Parks, Scott E. Spence

Published May 29, 2012• 8 citations
Publisher

Abstract

In the presence of a longitudinal magnetic field B, a beam of linearly polarized light incident from a Faraday medium of Verdet constant V refracts at its interface with a medium of negligible Verdet constant and emerges as two opposite circularly polarized beams that are separated by a small divergence angle δ that is proportional to the product BV. Judicious postselection of the polarization state of the emergent light can be used to amplify the measured value of δ by several orders of magnitude. This technique makes it possible to optically measure either very small V values when B is known or small magnetic fields when V is known.

DOI
10.1364/ao.51.003364

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In the presence of a longitudinal magnetic field B, a beam of linearly polarized light incident from a Faraday medium of Verdet constant V refracts at its interface with a medium of negligible Verdet constant and emerges as two opposite circularly polarized beams that are separated by a small divergence angle δ that is proportional to the product BV. Judicious postselection of the polarization state of the emergent light can be used to amplify the measured value of δ by several orders of magnitude. This technique makes it possible to optically measure either very small V values when B is known or small magnetic fields when V is known.
Three calculations of optical Faraday rotation are presented in which a linearly polarized field is incident on a medium of harmonic oscillators in the presence of a longitudinal magnetic field. The rotation of the plane of polarization of the field is evaluated using classical oscillators and the Lorentz force equation, quantum oscillators and the Heisenberg equations of motion, and quantum oscillators and a Schrödinger equation approach. It is shown that a simple argument, based on the assumption that a circularly polarized field drives either Δm=1 or Δm=−1 transitions on absorption (m is the magnetic quantum number), leads to an incorrect result for the Verdet constant.
Generally a measurement yields (eigen)values that can be distinguished and recorded by a suitable measuring device. Aharanov, Albert, and Vaidman (AAV) considered the case of weak measurements in quantum mechanics, where the eigenvaluespectrum is not properly resolved. AAV showed that in this case it is helpful to choose a suitable postselection state, which should be almost orthogonal to the initial state, and which is able to give rise to arbitrarily large expectation values. Here, we discuss how weak value amplification can be adapted to measure the optical activity of chiral liquids. The hallmark of optical activity is that the liquid has different refractive indices for right and left circularly polarized light, which causes the rotation of the plane of polarization of a linearly polarized light beam traversing the solution. This small difference (<< 10-7) in refractive indices may, not only be detected in transmission (polarimetry), but can also be detected in refraction. A linearly polarized light beam incident at an interface between a chiral and an achiral medium will split into its two circular polarization components, as the two components refract with different angles of refraction. A position sensitive detector may be used to register the difference in beam positions. For chemical and pharmaceutical applications it is of interest to measure small optical activities (with correspondingly small refractive index differences) in minute liquid samples. In order to increase the sensitivity of the measurement it becomes necessary to increase the separation between the two split circular beam components. We show that conventional weak value amplification, as reported in the literature before, does not provide sufficient information to determine the handedness of a chiral liquid. Instead we describe how weak value amplification combined with a heterodyne detection scheme can measure both the concentration and the handedness of the solution. The Faraday Effect in a glass prism is used as a model system for optical activity. The linear polarized beam is split into its circular polarization components at the interface.. To amplify these small angular displacements (typically a few nanoradians) of the two circular beam components we have implemented a weak measurement with a suitable post-selection scheme. It is shown, that the signal is proportional to the circular birefringence and furthermore sensitive to the sign of the optical activity.
Nonlinear in laser power, near-resonant light-atom interaction leads to a number of spectacular modifications of a well-known effect -polarization plane rotation in a magnetic field applied along the direction of light propagation.In particular, the rotation can be enhanced by up to nine orders of magnitude compared to the linear case.Faraday effect [ 1 ] is the rotation of the plane of linear light polarization by a medium in a magnetic field applied in the direction of light propagation.In 1898, Macaluso and Corbino eJ, studying absorption spectra of the alkali atoms in the presence of magnetic fields, discovered that the Faraday effect in the vicinity of resonance absorption lines also had a distinct resonant character.The essence of the Macaluso-Corbino effect can be illustrated by considering a simple case of a F= 1 ~ F'=O transition (Fig. 1 ).Linearly polarized light incident on the sample can be resolved into two counter-rotating circular components ( cr~.In the absence of a magnetic field, the M=±l sublevels are degenerate and the optical resonance
We study the amplification of weak magnetic field with Weak-value technique based on Faraday magneto-optic effect. By using a different scheme to perform the Sagnac interferometer with the probe in momentum space, we have demonstrated the new weak measure protocol to detect the small weak magnetic field by amplifying the phase shift of Faraday magneto-optic effect. At the given the maximum incident intensity of the initial spectrum, the detection limit of the intensity of the spectrometer and the accuracy of detecting weak magnetic field, we can theoretically give the appropriate pre-selection, post-selection and others optical structure before experiment. Our numerical results show our scheme with Weak-value technique is effective and feasible to detect weak magnetic field with magnetic field intensity lower than 10−1010^{-10}10−10 T.
The extraordinary concept of weak value amplification and a number of other seemingly counter-intuitive wave phenomena are manifestations of fine interference effects. In this report, we show that weak value amplification naturally arises in the cases of nearly destructive interference between the two orthogonal states of the observable which mimics the corresponding near destructive interference between the eigenstates of the measuring observable as a consequence of nearly mutually orthogonal pre and post selections of the system states. In order to elucidate this, we first experimentally demonstrate a weak measurement concept that uses near destructive interference of two paths of an interferometer having slightly rotated linear polarization states of light. By generating nearly destructive interference between the two paths with small amplitude and phase offsets of the waves, we observe both real and imaginary (respectively) weak value amplification of the small polarization rotation leading to large rotation of the polarization vector orientation and dramatic changes in the circular polarization descriptor fourth Stokes vector element of light, respectively. We go on to demonstrate that the giant enhancement of magneto-optical Kerr rotation, by which we can claim that we have found a non-modulation based and cost-effective magneto-optical Kerr effect (MOKE) measurement. At the end, we show successful weak value amplification of spin Hall shift in magneto-optical sample in both longitudinal and polar MOKE. Our ongoing work is to find out the relation between magneto-optical Kerr rotation and spin Hall shift of light in magnetic material.
We propose a modified difference weak measurement scheme that permits precise measurements of the magneto-optical Faraday effect. By making normalized difference processing for a set of post-selected light intensity, a linear-response regime with a significant weak-value amplification effect is established. In the proof-of-principle experiment, we measure the magnetic intensity using the polarization system and achieve precision at the order of ∼10−7 T. Our scheme can be applied to measure other magneto-optical effects, providing a method for future ultra-sensitive sensing and metrology in magnetic physics.