Subtomographic imaging of a polarisation sensitive phase pattern localised in phase space

Abstract

A transparent polarisation-sensitive phase pattern changes the phase of transmitted light without absorption, whereas this change of phase depends on the polarisation of incident light. A position-localised polarisation-dependent phase pattern is imprinted onto the phase-space of atoms by using atomic state dependent velocity-selective hole burning. A phase-space localised pattern is a higher dimensional generalisation of patterns localised in the position-space. Such a pattern cannot be imaged with a lens. The imprinted pattern is localised in a unique three-dimensional subspace of the six-dimensional phase-space of atoms. The phase-space localised pattern transforms the polarisation of light transmitting through it. This pattern is tomographically imaged at room temperature by measuring the intensity of the transmitted imaging laser beam of variable frequency with a camera after its polarisation analysis. Two sub-tomographs of the imprinted phase-space localised pattern are constructed. This paper presents a concept and experiment of imprinting and imaging of a polarisation-sensitive phase pattern localised in the phase-space.

Figure 1

Three different light absorbing patterns, $\mathbf{C}$, $\mathbf{A}$ and $\mathbf{T}$, which were initially localised in 2D position-space separately, are forming a single 3D localised pattern in the phase-space. This pattern is completely delocalised in the other 3D subspaces.

Figure 2

(a) A position-space localised transparent polarisation-sensitive phase pattern with phase shift $\varphi (x,z)$, where the width (W) and height (H) of the pattern are both 8.2 mm. (b) Transverse intensity profile of the retro-reflected laser beam measured by a camera at a distance about 45 cm from the pattern without polarisation analyser.

Figure 3

(a) A schematic diagram of the subtomographic imaging experiment in phase-space. (b) A part of the experiment to measure a frequency difference of lasers by time domain interference on a fast response photodetector. The vertical x-axis is perpendicular to the plane of page.

Figure 4

(a) Transmittance of the atomic medium when both laser beams are linearly polarised. (b) Photodetector $D_1$ output voltage, when the imprinting laser beam is ${\widehat{\sigma }}^{+}$ polarised. (c) When the imprinting laser beam is ${\widehat{\sigma }}^{-}$ polarised. The dotted line represents the same plot without the imprinting laser beam. The imaging laser beam is linearly polarised in all plots.

Figure 5

Two sub-tomographs of a 3D phase-space localised pattern. Each plot is an experimentally constructed subtomographic image, $p(x,y,\delta \nu )=-ln(I_r(x,y,\delta \nu )/I_i(x,y,\delta \nu ))$. (a) For $\delta \nu =-11.4$ MHz. (b) An inverted image for $\delta \nu =+8.6$ MHz. Where the image levels of images (a) and (b) are inverted w.r.t. each other.