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Through the von Neumann interaction followed by post-selection, we canextract not only the eigenvalue of an observable of the measured system butalso the weak value. In this post-selected von Neumann measurement, the initialpointer state of the measuring device is assumed to be a fundamental Gaussianwave function. By considering the optical implementation of the post-selectedvon Neumann measurement, higher-order Gaussian modes can be used. In thispaper, we consider the Hermite--Gaussian (HG) and Laguerre--Gaussian (LG) modesas pointer states and calculate the average shift of the pointer states of thepost-selected von Neumann measurement by assuming the system observable$\hat{A}$ with $\hat{A}^{2}=\hat{I}$ and $\hat{A}^{2}=\hat{A}$ for an arbitraryinteraction strength, where $\hat{I}$ represents the identity operator. Ourresults show that the HG and LG pointer states for a given coupling directionhave advantages and disadvantages over the fundamental Gaussian mode inimproving the signal-to-noise ratio (SNR). We expect that our general treatmentof the weak values will be helpful for understanding the connection betweenweak- and strong-measurement regimes and may be used to propose newexperimental setups with higher-order Gaussian beams to investigate further theapplications of weak measurement in optical systems such as the optical vortex.

Post-selected von Neumann measurement with Hermite-Gaussian and Laguerre-Gaussian pointer states