1. Introduction

The functionality to image samples in the mid-infrared (mid-IR) offers a new perspective for problems of tremendous biological and industrial relevance. By exploiting the highly specific vibrational and rotational ‘fingerprints’ of molecules as contrast mechanisms, one can obtain insights into the (bio-)chemical and molecular structure inaccessible in traditional microscopy [1–3]. The principle limitation, however, remains one of detection, with mid-IR imaging technology being prohibitively expensive while suffering from limited spectral response, spatial resolution or sensitivity. This absence of suitable detection options has lead to Raman-based imaging techniques [4,5] that require raster scanning, which together with the comparatively low signal yield of about $10^6$ photons/s per W excitation power, often renders the technique too slow for many applications, or, in the case of CARS-imaging, require sophisticated laser systems. Other approaches employ nonlinear wavelength conversion to the visible regime, where one can enjoy the comparable maturity of charge-coupled device (CCD) and complementary metal-oxide-semiconductor (CMOS) technology driven by the life sciences. Frequency up-conversion imaging shifts the detection frequency from the IR light to the desired visible while retaining the spatial and spectral information. This up-conversion imaging technique has been realised in the near- and mid-IR [6–9].

An alternative approach and that which we consider here exploits quantum imaging with undetected photons (QIUP) [10]. Though its initial realisation was based on induced coherence without induced emission [11], it can equivalently be described as an implementation of nonlinear interferometry [12]: two nonlinear crystals are pumped sequentially and coherently with laser light, generating photon pairs through spontaneous parametric down conversation (SPDC). When the two processes are aligned such that any information distinguishing whether the biphoton was born on the first or second crystal is erased, the two processes interfere. The strength (visibility) and phase of this interference signature can serve as a contrast mechanism for imaging - analogously so for related applications in optical sensing tasks, such as optical coherence tomography [13–15], spectroscopy [16–20] and polarimetry [21]. In contrast to ghost imaging schemes [22–24], in QIUP the detection of only one of the photons of the pair suffices to yield the information imprinted on the other. The strong spatial entanglement shared between the signal and idler pair, allows inference of the idler via the measurement of the signal [25,26], facilitating multi-mode (widefield) imaging. Crucially, this allows for the decoupling of the sensing and detection wavelengths when using a non-degenerate SPDC process [10]. An appealing way to use this advantage is to realise sensing at wavelengths for which advanced multi-pixel detection technologies are lacking. In this way, QIUP can shift the detection into the visible or near-IR wavelength ranges, where the far superior CMOS- and CCD-based sensor technologies operate.

This potential of QUIP for varied imaging applications, spanning wavelengths and applications [27,28] motivates its relevance for the life sciences. Recently, QIUP was tailored to image at the microscopic length scale, so-called quantum microscopy with undetected photons (QMUP) [29,30]. All prior realisations of QIUP and QMUP have exclusively exploited the momentum anti-correlations, imaging in the far-field of the crystal [10,25–27,29,30]. Here, we shift focus (both literately and figuratively) to instead examine imaging with the complementary position correlations that exist in the image plane [31,32]. While the image plane has previously been studied in the context of both spatial EPR demonstrations and ghost imaging [23,33–35], here we realise its use for the first time in QIUP and demonstrate its advantage for microscopy in the mid-IR. In addition to demonstrating the viability of position correlations for imaging, we show that for microscopy, this approach dramatically simplifies the required optical overhead, obtaining sufficiently high resolutions with relative ease.

2. Theory

All existing QIUP and QMUP experimental implementations have utilised the momentum anti-correlations that arise due to transverse momentum conservation in the pair production process. However, the same spatial entanglement that gives rise to the aforementioned anti-correlations in momentum space, equally gives rise to strong correlations in the conjugate position space. Physically, these correlations stem from the tight position localisation created when the signal and idler pairs are born from the annihilation of a pump photon. Accordingly, the field of view (FoV) of the imaging system is then specified by the waist of the pump beam that illuminates the crystal, essentially defining an aperture within which the SPDC process can occur. For a Gaussian pump beam, the FoV accordingly is a Gaussian distribution with a full width at half maximum (FWHM) given by:

Here, $\theta _i$ is half the idler emission angle at FWHM, which relates to the crystal length through $\theta _i=\lambda _i\sqrt {\frac {2.78}{\pi L}\frac {n_s n_i}{n_s \lambda _i+n_i\lambda _s}}$. One can equivalently and perhaps, more intuitively, consider the resolution to be a limitation arising from the thickness of the down-conversion crystal itself; the longer the crystal the more the ambiguity that arises regarding the birthplace of the signal and idler pair. The ratio of the FoV and resolution allows us to now approximate the number of spatial modes per direction as

Optimisation of the imaging capabilities thus requires maximising the pump waist and minimising the crystal length, while also balancing the required illumination per mode. When inserting our experimentally determined values of $w_p=431 \pm 6 \mu m, \theta _i=0.0491 \pm 0.0003 \rm {rad}, \lambda _i=3.74 \pm 0.02 \mu m$ into eqns. 1,2 and 3, we obtain $FOV_{IP}=127 \pm 2 \mu m, res_{IP}=7.9 \pm 0.1 \mu m, m_{IP}=16.1 \pm 0.3$. By comparison, the corresponding far-field implementation [29] theoretically predicts almost 50 % more spatial modes per axis for the same pump light and crystal specifications. This is a consequence of some of the entanglement directly accessible in the far-field migrating into the imaginary part of the amplitude in the near-field, rendering it inaccessible with intensity measurements alone [37–39]. Despite this reduction in available spatial modes, here we show that image plane imaging offers a practical advantage: imaging at microscopic length scales with large reduction in optical complexity. Requiring less magnification, fewer optical elements and consequently shorter interferometric arms, the demonstrated resolution in the image plane is thrice superior to that of achieved in the far field [29]. Alongside this practical consideration of optical complexity, the image plane does not suffer from the same degradation in correlations that arises for the Fourier plane when the illuminating light is not sufficiently monochromatic [40]. While this phenomenon is immediately apparent for non-collinear emission, it is also relevant for collinear schemes when the transverse momentum is appreciable, increasingly blurring the resolution off-axis. This could prove a concrete advantage of the image plane over the Fourier plane, where the photon flux on the camera would not need to be traded off against resolution.

3. Experiment

The experimental setup is detailed in Fig. 1 and exploits a Michelson-type configuration to realise the nonlinear interferometer. A 660 nm CW pump laser illuminates the ppKTP crystal, with a 431 µm waist maximally covering the crystal aperture. The ppKTP crystal is quasi-phase matched for a collinear type-0 process and specifically engineered [41] to produce simultaneously highly non-degenerate and broadband photon pairs. The broadband SPDC emission is due to the group velocity matching of the signal and idler, with spectral widths of 780-830 nm and 3.4-4.3 $\mu$m respectively at room temperature [41]. After the crystal, an off-axis parabolic mirror (OPM) is placed at its focal distance for achromatic collimation (focusing) of the emerging (returning) signal and idler. Using a dichroic mirror (DM), the idler is then split from the signal and pump, with the pump subsequently back-reflected using a cold mirror to preserve the desired imaging condition. The idler and signal fields are then each focused to align the image plane of the crystal with their respective end mirrors. The sample to be imaged is placed on the low-e slide that serves as an end mirror for the idler arm. The reflected idler and signal fields then back-propagate and are focused into the crystal, interfering with signal and idler fields generated upon the second pass of the pump through the crystal. The idler and pump light emerging from the second pass of the crystal are then discarded and the signal light is imaged onto a CMOS camera. Prior to detection, the signal field is filtered using a band-pass filter (3.5 nm FWHM) and a telescopic arrangement is used to position the CMOS camera in the image plane of the crystal.

 

Fig. 1. Experimental Setup: A continuous wave pump laser at 660 nm stimulates a highly non-degenerate, collinear SPDC process in a Michelson-type interferometer. The signal (800 nm) and idler (3.7 $\mu$m) fields generated on the first pass are split via a dichroic mirror, allowing the idler to probe the sample, before being recombined and traveling collinear with the coherent pump field back into the crystal. The pump is independently reflected back via a separate cold mirror in the signal arm. The signal field emerging after the second pass of the crystal is spectrally filtered via a band-pass filter (BPF) and then imaged onto a CMOS camera, revealing the spatial information obtained by the idler when probing the sample.

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It is crucial to carefully align the setup to ensure optimal spatial overlap between the biphoton fields generated in the two passes through the crystal. This ensures indistinguishably that will be manifested in the visibility of the interferometric image captured. This requirement is achieved by simultaneously matching the interferometric arms within the coherence length of the detected signal light and carefully tuning the optical components to fulfill the imaging conditions. Any deviation from those requirements will result in departure from optimal visibility and potentially, resolution, of the imaging system.

4. Results

Fig. 2 presents the characterisation of our imaging system at a magnification of $M=4$. The resolution, obtained via an edge knife response, is $9\pm 1\mu m$ (FWHM) and the obtained FoV is $161 \pm 1\mu m$ (FWHM). These values compare favourably to the theoretical values for the resolution and FoV of $7.8 \mu m$ and $127 \mu m$ respectively, albeit at a lower than anticipated magnification, as the corresponding number of spatial modes – $18\pm 2$ per axis – is in agreement with the theoretical value.

 

Fig. 2. Characterization of the imaging arrangements. (A) FoV. The measured data (pink points) was fitted with a Gaussian (purple line) yielding a FoV of $161\mu m$ (FWHM). (B) Edge response in the top row, the measured data (pink points) was fitted with an error function (purple line), differentiation of the error function gives a Gaussian shaped (marked in blue) point spread function (PSF). The resolution, determined by the FWHM of the PSF is $9\mu m$.

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The experimental results are summarised in Table 1. We attribute the systematic deviation of the larger obtained FoV and resolution to a reduced magnification realised in practice for the 4f system, when compared to the anticipated magnification of exactly 4. As predicted by the theory, QMUP via position correlations allows one to access high resolutions with a simplified optical system, notably when compared to realisations in the Fourier plane [29]. This inherent characteristic of Fourier plane imaging is due to the divergence of the biphoton field at the crystal exit, resulting in a large illumination spot (FoV) in the far field.

Table 1. QMUP in the image plane - experiment vs. theory

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Due to the interferometric nature of the imaging technique, a distortion of the biphoton wavefront can result in ambiguities; for a single image it can be ambiguous whether a dark region references absorption, or rather, destructive interference. This is particularly relevant for biological or industrial applications, where complex morphologies underpin the imaging motivation. By scanning axially, well within the coherence length, one can nevertheless obtain a pixel-wise visibility of the sample, allowing pixel-wise reconstruction of both the transmissivity and the phase [29]. This has a secondary advantage of increasing the ‘effective’ FoV of the image; in contrast to the intensity distribution of the illumination spot itself, the visibility distribution is considerably flatter. The effective number of spatial modes in the visibility images is here approximately double, resulting in roughly $30\times 30 = 900$ spatial modes for the 2D widefield imaging arrangement.

Using this scanning technique, a thin unstained slice of a mouse heart tissue was mounted on a low-e slide and imaged. Complementary to the above characterised imaging system (M=4), a lower magnification (M=2) configuration was also used to acquire a larger scale absorption image (Fig. 3(B)). To enable comparison, a bright field image acquired with a standard visible microscope is presented in Fig. 3(A). The left ventricle of the mouse heart and surrounding structure is visible in the images, revealing various morphological features. In Fig. 3(C) two smaller regions in the sample were characterised using the larger magnification arrangement (M=4), with this increased resolution revealing additional features that are not visible in the lower magnification images. The additional morphological information revealed in the mid-IR image is inaccessible in the bright field image, which can be attributed to the considerably reduced scattering at mid-IR wavelengths.

 

Fig. 3. Histology sample of a mouse heart imaged with (A) bright field microscopy with visible light for illustration which part of the sample we investigated with our method. (B) Mid-IR microscopy of the same sample with undetected photons for absorption imaging with a 2-fold magnification. (C) Higher resolution absorption images, taken with the a 4-fold magnification arrangement. Images are formed by stitching roughly 7 widefield absorption images (translating the sample transversely), each absorption image was reconstructed by averaging 6 images at 1 s integration time for 6 axial positions within the coherence length of the biphoton (longitudinal scan). The visibility is between 20-25%.

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5. Discussion

One apparent difference of QMUP in the image plane is the reduced homogeneity of the illumination distribution. In the image plane, we illuminate with the photon birth zone itself, revealing inhomogeneities arising from small defects in the crystal, dust or imperfect pump modes. By contrast, the Fourier plane provides a very homogeneous illumination distribution, its uniformity being a consequence of the smoothness of the phase-matching curve itself. This disadvantage is reminiscent of classical microscopy, where schemes usually avoid illumination in the image plane of the light source to circumvent the analogous noise contributions [42]. Here, where the correlations between the planes are indispensable, it cannot be easily avoided, but can be eliminated with care towards the crystal and pump. The higher noise in the image plane is one potential contributor to the reduction of visibility, as the background noise in the image plane is inverted on the second pass (and thus does not cancel out). A second source of potential visibility reduction stems from the high magnification, which results in a lower depth of focus and thus a higher sensitivity to alignment errors. Both of these sources of reduced visibility are technical in nature and not fundamental limitations.

In contrast to our prior work in the Fourier plane microscopy, we suffer no degradation from the theoretical performance of our system despite imaging at considerably smaller length scales. While imperfect matching of the image plane at the sample and camera will likely degrade the resolution of the imaging system, the short depth of focus aids in matching these conditions very precisely. The 9 $\mu$m resolution presented here was realised in a simple 4f setup with a standard aspheric lens (f = +25 mm, Thorlabs) serving as the magnifying lens. This basic 4f arrangement affords superior compactness and stability, and with targeted optical engineering, should attain diffraction-limited resolutions at the state-of-the-art of mid-IR imaging. For example, employing a technically feasible 4-fold larger crystal aperture and a magnification of 10, would theoretically result in a wavelength-scale resolution of around 3 $\mu$m with almost 10000 2D resolved elements within a 300 $\mu$m FoV. Furthermore, for a Fourier plane implementation with a similar number of modes and spectral bandwidth, degradation in the off-axis resolution and thus the effective mode number would be apparent. The image plane does not suffer this same constraint on the illumination per spatial mode and thus the corresponding signal-to-noise [40].

The clear success of this approach for imaging and microscopy in the mid-IR motivates the question of its limitations - can we envisage this new approach for widefield imaging stretching beyond the near and mid-IR, towards the far IR and even terahertz wavelengths [20], where detection technologies are even more limited? Fig. 4 presents a theoretical analysis of this question framed in terms of available spatial modes. Utilising a realistic biphoton wavefunction [43] and a fixed pump wavelength, we numerically calculated the spatial entanglement (via the Schmidt decomposition) as a function of increasing idler wavelength. The longitudinal phase-mismatch was assumed to be zero and for simplicity the refractive index was set to 1.5 for all three participating wavelengths.

 

Fig. 4. Numerical simulation of the number of spatial modes available for widefield imaging (as characterised by the Schmidt number) as a function of increasing idler wavelength for four different ‘effective’ crystal apertures, V. For instance, the four decreasing values of V (= 1.0, 0.71, 0.45 and 0.3 mm$^{-1/2})$ correspond to increasing values of crystal length, L (= 1, 2, 4 and 11 mm) for a pump waist of $\omega _p$ = 1 mm. The pump wavelength of the pump light is fixed at 660 nm and for simplicity the refractive indices are assumed to be 1.5 across the examined range. The pump wavelength and degeneracy point (1.32 $\mu$m) are highlighted by vertical dashed lines. The dashing of curves lines indicates a parameter regime where these results, derived under the paraxial approximation, may no longer faithfully describe the system. The imaging capacity of the system presented here ($\omega _p$ = 430 $\mu$m and L = 2 mm) is indicated by a star.

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The results presented in Fig. 4 show a decrease in the widefield imaging capacity QIUP with increasing (undetected) wavelength. The increasing non-degeneracy between the signal and idler energy that enables ‘silicon imaging’ at silicon incompatible wavelengths, must be traded off against a decrease in the available spatial entanglement and thus widefield imaging capacity. These results are also consistent with heuristic approach of eqn. 3, where the mode number scales inversely with $\sqrt {\lambda _i+\lambda _s}$, showing that at large non-degeneracy, the imaging power is limited by the size of the longest wavelength, regardless of whether the light is undetected or detected. While widefield QIUP applications remain promising into the far IR, they become increasingly unfavourable as we approach the terahertz regime. Furthermore, this analysis does not consider additional adversarial factors including material absorption, parasitic seeding and the necessity for long crystals that complicate the imaging at very long wavelengths. This, however, does not preclude single-pixel based scanning approaches to imaging tasks at these wavelengths.

The QIUP-approach presented here also opens up the possibility of an imaging regime previously inaccessible at mid and far-infrared wavelengths: facilitating shot-noise limited imaging with high quantum efficiencies at exceptionally low illumination powers. This is the consequence of the intrinsic (theoretical unity) efficiency of the nonlinear interferometer in the low-gain regime, where any (IR) idler photon has its (silicon-compatible) partner. Accordingly, any imaging information carried by that idler photon can be transferred perfectly to the signal photon. Therefore, in the absence of additional loss and mode mismatch, the noise performance of the mid-IR imaging is determined by the properties of the silicon camera, where shot noise–limited images are accessible with only a few 1000s of photons per pixel per second or less. Here, the images were obtained at mid-IR illumination levels of only 15 pW – with the light within our current detection bandwidth amounting to only 2-3 pW. Such low illuminations could find application in specific tasks, such as the longitudinal imaging of light sensitive cells, where the sensing illumination itself can otherwise interfere with the cellular and molecular mechanisms one seeks to understand.

6. Conclusion

In conclusion, we have presented the first experimental realisation of QIUP via position correlations. We have shown that imaging with position correlations presents valuable advantages over its predecessors for the task of microscopy, allowing easy access to resolutions below 10 $\mu m$ at a sensing wavelength of 3.7 $\mu m$ with no observable deviation from theoretical predictions. This improved resolution permitted mid-IR imaging of an unstained tissue from a mouse heart with several orders of magnitude less light than any comparative methods. The presented results further extend the growing toolbox of QMUP and take us another step closer toward real-world applications.