Experimental analysis of image resolution of quantum imaging with undetected light through position correlations

Image resolution of quantum imaging with undetected photons is governed by the spatial correlations existing between the photons of a photon pair that has been generated in a nonlinear process. These correlations allow for obtaining an image of an object with light that never interacted with that object. Depending on the imaging configuration, either position or momentum correlations are exploited. We hereby experimentally analyze how the crystal length and pump waist affect the image resolution when using position correlations of photons that have been generated via spontaneous parametric down conversion in a nonlinear interferometer. Our results support existing theoretical models for the dependency of the resolution on the crystal length. In addition, we probe the resolution of our quantum imaging scheme for varying pump waists over one order of magnitude. This analysis reveals the intricate dependency of the resolution on the strength of the correlations within the biphoton states for parameter combinations in which the crystal lengths are much larger than the involved photon wavelengths. We extend the existing models in this parameter regime to properly take nontrivial effects of finite pump waists into account and demonstrate that they match the experimental results.

Figure 1

Near-field configuration setup. A pair of down-converted photons is generated at the ppKTP crystal in either the forward pass of the pump (path A) through the crystal or in the backward pass (path D). The pump beam is focused at the crystal with ${\mathrm{L}}_{\mathrm{p}}$. We change the ${\mathrm{L}}_{\mathrm{p}}$ focal length and its distance to the crystal accordingly to generate different pump waists at the center of the crystal. The undetected beam is reflected with DM1 towards the object while the detected and the pump beam are transmitted together. The DM2 reflects the detected beam towards the camera. The interferometer phase is varied by changing the position of the mirror M2 with a piezo translation stage. The lenses $\mathrm{L}1$ and $\mathrm{L}{1}_{\mathrm{u}}$ (with equal focal lengths, $f_1=125\mathrm{mm}$) image the generated modes in the crystal onto the object plane, and back onto the crystal after being reflected back by the mirror M2. This configuration allows us to exploit the position correlations of the photon pairs. Lens $\mathrm{L}{1}_{\mathrm{d}}$ also has focal length $f_1$. The lenses $\mathrm{L}{2}_{\mathrm{c}}$ ($f_2=75\mathrm{mm}$) and $\mathrm{L}{3}_{\mathrm{c}}$ ($f_3=200\mathrm{mm}$) image and magnify the object at the camera. Long pass and interference filters guarantee that only the $910\mathrm{nm}$ wavelength is detected. The insets show the different magnification systems present: magnifications of the undetected (detected) beam inside the interferometer $[M_{\mathrm{u},\mathrm{i}}$ ($M_{\mathrm{d},\mathrm{i}}$)], and the magnification of the detected beam before being detected on the camera ($M_{\mathrm{d},\mathrm{c}}$).

Figure 2

Spread of an edge analysis. (a) Cut from the visibility image taken to analyze the spread of an edge blocking the left side of the beam. (b) The edge profile (blue, error-function shape) is extracted from the image, and fit with an ESF (orange, error-function shape, barely distinguishable by eye from the edge profile). From the ESF, the LSF (green, Gaussian shape) and PSF (red, Gaussian shape) are calculated. LSF and PSF overlap. Example extracted from data taken with a 2-mm-long crystal and $214\mu \mathrm{m}$ pump waist.

Figure 3

Magnification-independent quality parameter. Ratio between the spread of the ESF derivative obtained from amplitude images (${\mathrm{\Delta }}_{{\mathrm{G}}_{\mathrm{ESF}}}$) of a sharp edge and the corresponding spread obtained from visibility images (${\mathrm{\Delta }}_{\mathrm{V}}$). The solid lines show the theoretical predictions for each crystal length $[2\mathrm{mm}$ in yellow (upper) curve; $5\mathrm{mm}$ in green (middle) curve; $10\mathrm{mm}$ in blue (lower) curve] when varying the pump waist. Experimental data are given in yellow circles ($2\mathrm{mm}$), green triangles ($5\mathrm{mm}$), and blue squares ($10\mathrm{mm}$). Since the experimental points fit the theory prediction, we conclude that the underlying physics is well described by our theory model and that the experimental uncertainties are kept within acceptable limits. Because this ratio does not depend on the system magnification, it is a useful quantity to describe the system performance purely induced by the underlying quantum-mechanical principles of the imaging scheme.

Figure 4

Magnification adjusted visibility spreads for images taken through position correlations existing between SPDC photon pairs for a sharp edge. Spread extracted from visibility images (resolution) of a sharp edge for three crystals with different lengths when varying the pump waist. Experimental data are given in yellow circles ($2\mathrm{mm}$), green triangles ($5\mathrm{mm}$), and blue squares ($10\mathrm{mm}$) and compared with the theory prediction $[2\mathrm{mm}$ in yellow (lower) solid curve; $5\mathrm{mm}$ in green (middle) solid curve; $10\mathrm{mm}$ in blue (upper) solid curve]. For comparison, we plot the limiting case of large pump waists as predicted in the literature [30] with dashed lines. For regimes where $w_{\mathrm{p}}^2\gg {\lambda }_{\mathrm{d}}^{2}L/\left({\lambda }_{\mathrm{d}}+{\lambda }_{\mathrm{u}}\right)$ our extended theory model (see Sec. 4) converges towards the simplified one, and resolution is mainly dependent on the crystal length. However, for smaller pump waists, the resolution worsens as the spatial correlations stored in the SPDC state worsen such that the biphoton state even becomes separable, i.e., the spatial correlations are lost for a specific parameter configuration (marked with dotted vertical lines for each crystal length).

Figure 5

Magnification adjusted spread ${\mathrm{\Delta }}_{{\mathrm{G}}_{\mathrm{ESF}}}$ from amplitude images. The solid blue upper line is the theoretical prediction for a $10\mathrm{mm}$ crystal while the solid green middle curve and the yellow lower curve show the predictions for a $5\mathrm{mm}$ crystal and a $2\mathrm{mm}$ crystal, respectively. In case the pump waist gets smaller, the information on the object gets gradually erased between detected and undetected beams. At the point where the SPDC state becomes separable [marked with a yellow dotted line (left vertical line) for a $2\mathrm{mm}$ crystal, in green (middle vertical line) for $5\mathrm{mm}$, and in blue (right vertical line) for $10\mathrm{mm}]$, the amplitude images only carry information on the detected beam which does not contain any spatial information about the object. The spread measured is then related to the detected beam size. For larger pump waists $[w_{\mathrm{p}}^2\gg {\lambda }_{\mathrm{u}}^{2}L/\left({\lambda }_{\mathrm{d}}+{\lambda }_{\mathrm{u}}\right)]$, the derivative of the amplitude ESF can be considered as a good approximation of the LSF which can be related to the PSF. Moreover, the image and visibility functions are approaching the same limiting function in the large pump waist regime. Therefore, for this regime, the ESF derivative value converges towards the resolution value given in Fig. 4 determined by the visibility images.

Figure 6

Near-field configuration setup. Sketch of the setup used for this work with detailed information on the lenses used and distance between them in order to exploit position correlations between the down-converted photons. Lenses are noted with $\mathrm{L}$, dichroic mirrors with $\mathrm{DM}$, $\mathrm{M}1$, and $\mathrm{M}2$ are the mirrors of the detected and undetected interferometer arms, respectively. Detected and undetected paths magnification value inside the interferometer are noted as ${\mathrm{M}}_{\mathrm{d},\mathrm{i}}$ and ${\mathrm{M}}_{\mathrm{u},\mathrm{i}}$, while ${\mathrm{M}}_{\mathrm{d},\mathrm{c}}$ refers to the magnification seen by the detected beam between the ppKTP crystal and the camera. Distances between the lenses are shown with arrows and specified in terms of the focal length of the corresponding lenses. Before the camera, we have an interference filter (IF) with $1.5\mathrm{nm}$ bandwidth (BW).

Figure 7

In-house made object used for the magnification measurements. The object is laser cut on a thin metal sheet. Therefore its transmission is either zero or one for the probing light. It represents the Jena skyline, and the tower windows are used as slits for the magnification measurements. The distance between the center of the windows (slit distance) is measured to be $133\pm 23\mu \mathrm{m}$. Where the uncertainty of the distance corresponds to the sum of the measuring device uncertainty and the manufacturing precision.

Figure 8

Object manufacturing precision measurement. The image of a window edge was analyzed by plotting the averaged intensity edge profile along the $x$ axis of the area marked in green. That profile is then fit with an error function and its spread evaluated at 1/e is taken as the manufacturing precision of one window edge. Notice that this value must be doubled to account for the two edges.

Figure 9

Object-image dimensions analysis. Image of the tower windows taken with our setup but illuminating the object with the detected wavelength to obtain a classical image. The windows are treated as slits and their intensity profile are fit by Gaussian functions. The distance between two slits (windows) is then taken as the distance between the maxima of the Gaussian fits. This example is taken with a 5-mm-long crystal and a pump waist of $214\mu \mathrm{m}$.