Simultaneous Measurement of Multiple Parameters of a Subwavelength Structure Based on the Weak Value Formalism

Abstract

A mathematical extension of the weak value formalism to the simultaneous measurement of multiple parameters is presented in the context of an optical focused vector beam scatterometry experiment. In this example, preselection and postselection are achieved via spatially-varying polarization control, which can be tailored to optimize the sensitivity to parameter variations. Initial experiments for the two-parameter case demonstrate that this method can be used to measure physical parameters with resolutions at least 1000 times smaller than the wavelength of illumination.The concepts of weak value and weak measurement were introduced by Aharonov, Albert and Vaidman in 1988 [1][2][3] as an alternative to the standard measurement formalism of quantum mechanics. For a quantity associated with an operator B, a standard measurement is related to the expected value Φ|B|Φ / Φ|Φ , where Φ is the state vector for the quantum state being measured. Since the state is normalized, the inner product in the denominator is typically taken as unity. Clearly, for Hermitian operators, this expected value is real and limited to the range of values spanned by the eigenvalues of B. On the other hand, weak measurements are based on weak values defined as Φ post |B|Φ pre / Φ post |Φ pre , where Φ pre and Φ post are preselected and postselected states. It is easy to see that there is no bound to a weak value since the denominator can be made arbitrarily small by appropriate preselection and postselection. In fact, weak values need not even be real-valued. Weak measurements have been employed, for example, to measure very small angular deviations with great precision [4][5][6][7].

Geometric Phase in Optics: From Wavefront Manipulation to Waveguiding

“…These beams have important applications, such as deep sub‐wavelength microscopy based upon the weak measurement. [ 166 ] When tightly focused, these structured beams show a complex 3D dynamics of their polarization, with the optical field describing a Möbius strip while evolving in time. [ 167,168 ] Furthermore, ‐plates have been used for several applications where vector beams are required, such as mode division multiplexing, [ 169,170 ] STED microscopy, [ 171 ] nonlinear propagation of structured beams, [ 172 ] nanostructuring of surfaces via femtosecond ablation, [ 114,173,174 ] radial polarizers for ultrashort pulses [ 175 ] high‐harmonic generation with angular momentum, [ 176 ] and nodal areas in coherent beams.…”

Maximum likelihood estimation in the context of an optical measurement

“…The focused beam scatterometry experiment in Ref. [16] operates on the same principle but with a spatially-varying polarization distribution, resulting in an output intensity of the form I ∝ | n a n (x)[p n −p n (x)]| 2 , where the functions a n (x) characterize the sample under test and the functionsp n (x) (which determine the required input polarization) can be tailored to optimize the sensitivity to each parameter. As an example of this type of measurement for the one-parameter case, consider the intensity distribution…”

“…Examples of such applications include the measurement of small optical beam shifts [23,24] and the focused beam scatterometry experiment discussed in Ref. [16]. Fig.…”

“…This is a slightly simplified example of the distribution considered in Ref. [16], with the contributions from each parameter adding incoherently (i.e., in intensity) rather than coherently (i.e., in electric field). Despite this difference, similar statistical behavior is observed in either case.…”

Enhanced weak-value amplification via photon recycling

“…Introduction-Weak-value [1] amplification has been successfully implemented in a variety of optical platforms to sensitively measure a number of system parameters [2]. This method has measured the optical spin Hall shift of 1 Å [3], 4-pm displacement or 400-frad angular tilt measurements [4], frequency measurements of 130 kHz [5], velocity measurements of 400 fm/s [6], temperature shifts of 0.2 mK precision [7], glucose concentration of 9 × 10 −5 g/L [8], magnetic field sensitivities of 7 fT [9], simultaneous multiparameter measurement [10], fine-tuned beam displacements [11], among many other experiments. The method is inspired by a quantum effect where the shift of a quantum meter is amplified by the weak value of an operator, but with the sacrifice of the count rate by the probability of postselection on a given result of a subsequent measurement.…”