## Abstract

We demonstrate quantum-noise correlations between the spatial frequencies of a parametrically amplified signal image and the generated conjugate (idler) image. Test images were amplified by an optical parametric amplifier that can be operated either as a low-pass or a band-pass amplifier for spatial frequencies. Direct difference detection of the signal and idler spatial frequencies at ±16 mm^{-1} resulted in noise that fell below the shot-noise level by ≃5 dB. Parametric-gain and phase-mismatch dependence of the observed quantum-noise reduction is in good agreement with the theory of a spatially-broadband optical parametric amplifier.

© Optical Society of America

## 1. Introduction

Research involving squeezed states of light has evolved from demonstrations of novel generation schemes to applications where performance can be enhanced beyond the shot-noise limit. While applications in the temporal domain^{1-8} have been reported over the years, there has been little attention devoted to the applications of squeezed light to phenomena in the spatial domain, such as optical imaging, diffraction, and holography. However, it has been proposed that spatially-broadband squeezed light can be used to image faint objects with sensitivity exceeding the shot-noise limit.^{9,10} In this scheme, the phase object to be imaged is placed in one arm of a Mach-Zehnder interferometer, whose normally-unused input port is illuminated with spatially-broadband squeezed light generated by a traveling-wave optical-parametric amplifier (OPA).^{11}

We have recently begun to experimentally investigate the noise properties of parametrically-amplified images.^{12} Since an OPA can generate correlated photons over a broad band of spatial frequencies, it is an ideal device for image amplification. Early work in this field concentrated on parametric up-conversion of infrared images to visible wavelengths.^{13-15} More recently, parametric amplification of images^{16-19} has been shown to have practical applications in time-gated image recovery.^{20} An example is the amplification of ballistic photons through a turbid medium for biomedical imaging.^{21-23}

In this paper, we expand on our recently-reported first observation^{24} of quantum-noise correlations in parametric image amplification. An object is imaged into the OPA, where the real image is amplified in a twin-beams configuration. Quantum noise is measured by means of direct difference detection of the twin beams.^{25} In particular, we show that there are strong quantum correlations between the appropriate spatial frequencies of a parametrically-amplified image and its generated conjugate (idler) image. Portions of the amplified signal and idler images that are at different spatial frequencies, however, are not correlated.

Previous experiments have produced squeezed light with uniform features in the spatial domain. Quantum-noise reduction of more than 6 dB has been achieved using twin beams generated by means of a traveling-wave OPA.^{25} Subtraction of signal and idler photocurrents in this twin-beams configuration shows quantum-noise reduction below the shot-noise limit as measured when an equivalent coherent-state photon flux is incident on each of the two detectors. In the preceding experiment, the correlated signal and idler photons were created at a spatial frequency of zero.

In the case of an image, the signal input is composed of a spectrum of spatial frequencies, which are amplified by the OPA within the limits of its spatial bandwidth. Both horizontal and vertical components of the spatial frequency must be included for a true two-dimensional image. For simplicity, however, we consider only one-dimensional images. Since the spatial frequencies are the transverse components of the wavevectors, for a plane-wave pump beam with no transverse components, the phase-matching condition in parametric down-conversion dictates that a signal photon at a spatial frequency of *q* be co-generated with a conjugate idler photon at -*q*. Therefore, to observe quantum correlations in a parametrically-amplified image, we need to sample both the signal and the idler photons at the same magnitude of the spatial frequency.

## 2. Theory

A simple theoretical description for quantum-noise reduction as a function of spatial frequency can be constructed from the standard equations of optical-parametric amplification. The equations governing a traveling-wave OPA are

where *â*_{s} and *â*_{i}
are the input and *b̂*_{s}
and *b̂*_{i}
are the output annihilation operators for the signal and idler fields, respectively. In our experiment, there is no idler input field so that 〈*â*_{i}
〉 = 〈${\widehat{a}}_{i}^{\u2020}$〉 = 0. After taking into account the effect of non-ideal detection efficiency *η*, the expressions for the parametric gain and the quantum-noise reduction in the case of a collinear twin-beams experiment^{25} can be written as

In the non-collinear case, *g* and *R* have the same forms as in the collinear case, but the coupling coefficients *μ* and *ν* are given by^{26}

where $h\equiv \frac{1}{2}\sqrt{{\kappa}^{2}-{\left(\Delta {k}_{\mathrm{eff}}\right)}^{2}},\Delta {k}_{\mathrm{eff}}\equiv \left({\overrightarrow{k}}_{p}-{\overrightarrow{k}}_{s}-{\overrightarrow{k}}_{i}\right)\bullet {\overrightarrow{i}}_{z}$ is the effective phase mismatch within the crystal along the nominal propagation direction *z*, *κ* is the parametric-gain coefficient which is proportional to the intensity of the pump beam, and *ℓ* is the length of the χ^{(2)}-nonlinear medium—a 5.21-mm long KTP crystal in our experiment. Optimum amplification occurs when Δ*k*
_{eff} = 0. At the spatial frequency of *q* = 0, this phase matching condition is fulfilled for Δ*k* = *k*_{p}
- *k*_{s}
- *k*_{i}
= 0. However, at higher spatial frequencies, phase matching occurs only when Δ*k* ≠ 0. Using the paraxial approximation,^{26} it can be shown that the effective phase mismatch for a spatial frequency *q* is given by

Therefore, by making Δ*k* become progressively more negative, it is possible to bring increasingly higher spatial frequencies into the phase match condition.

Since *μ* and *ν* depend on *q* through Δ*k*
_{eff}, we can evaluate the signal and idler outputs, and the quantum-noise reduction, as a function of the spatial frequency for any given signal input. The simplest case is for an input signal with a small spread centered at *q* = 0, as shown in Fig. 1(a). We define *ξ* = *q*/2*π* so that the spatial frequency *ξ* is in units of mm^{-1}. Here the phase matching condition is satisfied for Δ*k* = 0 and we have chosen the OPA gain *g* ≡ |*μ*(0)|^{2} = 4. As expected, the signal and idler outputs as well as the noise reduction are maximized for *ξ* = 0. From the noise-reduction curve, we estimate that the spatial bandwidth of our OPA is approximately 15mm^{-1} (HWHM). In this configuration, the OPA functions as a low-pass amplifier for spatial frequencies.

In the experiments, we are interested in observing the quantum correlations at a spatial frequency of 16mm^{-1}. From Fig. 1(a), it is evident that a signal input at +16 mm^{-1} (or -16 mm^{-1}) will be amplified very little when Δ*k* = 0. For this spatial frequency, the phase-matching condition is fulfilled for Δ*k* = -0.95rad/mm for the parameters of our KTP crystal. In practice, Δ*k* can be adjusted by rotating the azimuthal angle of the KTP crystal in the OPA so that the incidence angle of the signal changes while that of the pump remains fixed. Note that *q* remains unchanged in this process since the polar angle is kept fixed. As shown in Fig. 1(b), when Δ*k* = -0.95rad/mm, the signal at +16 mm^{-1} is amplified with an OPA gain of *g* = 4, and the conjugate idler is generated at -16mm^{-1}. Since the maximum gain and the noise reduction depend on the ability to achieve optimum phase matching for the appropriately chosen Δ*k*, we can obtain a gain of 4 for the same value of *κ* as in the low-pass configuration (Δ*k* = 0). Therefore, the noise reduction at *ξ* = ±16 mm^{-1} is not diminished from that at *ξ* = 0 in the low-pass case, although the spatial-frequency bandwidth is somewhat reduced.

For values of Δ*k* < 0, the OPA acts like a band-pass amplifier, allowing us to amplify higher spatial frequencies more effectively. This feature of optical parametric amplification makes it possible to optimize phase matching, and hence quantum-noise correlations, for a single incident spatial frequency of our choosing. To observe correlations at many different spatial frequencies simultaneously, an OPA with a spatial bandwidth covering all the desired frequencies would be required. Also, a band-pass OPA can be utilized for edge and contrast enhancement in classical parametric amplification of images.^{16,23}

## 3. Experiments

The layout of our parametric image amplification experiment is depicted in Fig. 2. A 5.21-mm long KTP crystal (the OPA) is pumped by a Q-switched, mode-locked, and frequency-doubled Nd:YAG laser. The IR (1064 nm) signal input and the green (532 nm) pump are each *p* polarized (parallel to the crystal *z*-axis) for type II phase-matching in the crystal. The object is placed in the signal-beam path in front of the OPA. A real image of this object is formed in the center of the KTP crystal by a ×1 telescope consisting of two 10-cm focal-length lenses. The spatial frequencies of this image are amplified by the pump beam, which is made coincident with the signal beam using a dichroic beamsplitter. The green pump is blocked after the crystal by using a filter which passes only the IR. CCD cameras are placed in the output image as well as the Fourier planes of a 20-cm focal-length lens that is placed after the filter. The generated idler is orthogonally polarized relative to the amplified signal because of type II phase matching. Therefore, a half-wave plate followed by a polarizing beamsplitter placed after the 20-cm lens allows us to observe either the signal or the idler output in the image as well as the Fourier planes by simply rotating the half-wave plate.

For parametric image amplification, we used a negative test pattern of three vertical lines with a uniform spacing of 62.5*μ*m (16 lines/mm). The horizontal Fourier transform of this object consists of three main peaks at *ξ* = 0, ±16 mm^{-1} with two smaller peaks in between at *ξ* = ±8mm^{-1}. As recorded in the output image plane, real images of the bare signal (i.e., with the pump turned off), the amplified signal, and the generated idler are shown in Fig. 3(a) for an OPA gain of ≃1.2. The transverse pattern of the bare signal as recorded in the output Fourier plane is shown in Fig. 3(b). Figure 3(c) shows the transverse pattern of the amplified signal in the output Fourier plane when the OPA was aligned in the low-pass configuration (Δ*k* = 0) and the pump power adjusted for a gain of ≃4. As shown, the central peak (*ξ* = 0) was strongly amplified with little amplification occurring at the side peaks (*ξ* = +16mm^{-1}). Transverse pattern in the output Fourier plane for band-pass amplification with Δ*k* = -0.95rad/mm is shown in Fig. 3(d). Here, the pump power was the same as in Fig. 3(c) and the OPA was aligned for maximum amplification of the two side peaks at ±16 mm^{-1}. These results compare favorably with the theoretical predictions presented in Fig. 1 above.

For the measurement of quantum correlations, our goal was to observe noise reduction at a single spatial frequency. First, the OPA was optimized for maximum gain by aligning the signal and idler patterns to be simultaneously coincident in both the real-image *and* the Fourier planes. By placing an iris in the Fourier plane that is in front of the OPA (halfway in between the two lenses of the ×1 telescope), we blocked all spatial-frequency components of the input signal pattern except the peak centered at *ξ* = +16 mm^{-1}. Thus the input signal had a well defined spatial frequency. The OPA was adjusted for maximum gain at ±16 mm^{-1}, corresponding to an azimuthal rotation of the KTP crystal of about 0.85° from the angle for phase matching at *ξ* = 0. In this way the input signal Fourier component was band-pass amplified, and a conjugate Fourier component at *ξ* = -16mm^{-1} of the idler beam was generated. Since the amplified signal Fourier component at +16 mm^{-1} exits the OPA at an angle of 2 × 17 = 34 mrad with respect to the idler Fourier component at -16 mm^{-1}, it was easy to separate the two with use of plane mirrors. The mirror that sent the beams to the CCD cameras was removed (cf. Fig. 2), and each beam was focused onto a separate photodetector located in the far-field (i.e., in the Fourier plane). Therefore, one detector saw the amplified signal Fourier component at +16 mm^{-1}, while the other detected the idler component at -16 mm^{-1}. Twin-beams type of noise measurements were made using direct difference detection, similar to those described in Ref. 25. Noise power of the difference photocurrent was measured at 27 MHz with a 3 MHz resolution bandwidth.

Figure 4(a) shows the quantum-noise reduction below the shot-noise level for the detected signal and idler Fourier components as a function of the signal gain. For OPA gains below ≃4, the data is in good agreement with the theory (solid curve), once a detection quantum efficiency of 0.76 is taken into account. We are in the process of investigating the reasons for the discrepancy between the theory and the experiment for OPA gains above ≃4. We have also measured the quantum-noise reduction as a function of the phase mismatch Δ*k* for various values of the OPA gain. Results from one set of data for an OPA gain of ≃3.9 are shown in Fig. 4(b). As pointed out previously, Δ*k* was varied by rotating the azimuthal angle of the KTP crystal in the OPA. The micrometer readings on the KTP rotation stage were calibrated and converted into units of Δ*k* for comparison with the theory. As shown, the experimental data are once again in good agreement with the theory (solid curve), once the detection quantum efficiency is taken into account.

## 4. Conclusion

In conclusion, we have demonstrated quantum-noise reduction of almost 5 dB in direct difference detection of the correlated spatial frequencies of parametrically-amplified signal and idler images. Test images were amplified by a spatially-broadband OPA that can be operated either as a low-pass or a band-pass amplifier. Parametric-gain and phase-mismatch dependence of the observed quantum-noise reduction is in good agreement with the theory of a spatially-broadband optical parametric amplifier. The observation of such a large quantum-noise reduction suggests that this amplifier can be configured to produce spatially-broadband squeezed light with a bandwidth that is on the order of 10 mm^{-1}, thus making it a potential squeezer for enhancing the imaging of faint objects.^{9,10} In order to image two-dimensional objects, it would be necessary to place a pair of two-dimensional detector arrays in the Fourier plane, match the pair of detectors at each spatial frequency to subtract the corresponding photocurrents, and process the resulting Fourier image to recover the real image. The spatial resolution of the final image will depend on the resolution of the detector arrays placed in the Fourier plane. Future experiments will explore this approach, using a pair of one-dimensional detector arrays to study the imaging of one-dimensional objects.

This work was supported in part by the U. S. Office of Naval Research. We are grateful to M. Vasilyev for useful discussions.

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