## Abstract

Interference phenomena play a fundamental role in physics, and interferometric techniques have helped advance science and technology significantly. In this Letter, we observe spatial fringes in the interference of two beams, which are controlled by a third beam through the phenomenon of induced coherence without induced emission. We show that the interference pattern depends on the alignment of this third beam in an analogous way, as fringes created in a traditional division of amplitude interferometer depend on the relative alignment of the two interfering beams. We demonstrate that the pattern is characterized by an equivalent wavelength, which corresponds to a combination of the wavelengths of the involved light beams. Our results open up the possibility of developing new techniques, such as wavefront sensing and holography at wavelengths for which no suitable single-photon cameras are available.

© 2017 Optical Society of America

The relationship between path information and interference is fundamental to quantum physics [1] and has been studied in various contexts [2–4]. A particularly remarkable manifestation of this relationship is the phenomenon of induced coherence without induced emission [5,6]. This phenomenon was used for fundamental tests of complementarity [7,8] and led to applications in imaging [9,10], metrology [11], spectrum shaping [12], and spectroscopy [13,14]. If two spatially separated nonlinear crystals produce photon pairs (signal and idler) by spontaneous parametric down-conversion (SPDC) [15,16], the emitted signal beams in general do not interfere, even if the two pump beams are mutually coherent. This is due to the fact that the idler beams carry information about which source a down-converted pair originated from. By aligning the two idler beams, this information can be suppressed, and lowest-order interference can be observed between the two signal beams. It has been shown that path distinguishability can be introduced through a time delay between the two idler beams [17], due to a difference in their transverse sizes [18,19], or by attenuating the idler beam from one source using a partially transmissive filter. In the last case, the resulting visibility was quantitatively connected to the transmission coefficient [5,6].

Here, we analyze a situation in which distinguishability is introduced by marginally misaligning the idler beams, either by tilting or by defocusing one with respect to the other. One could conjecture that this has the analogous effect of merely reduced visibility. However, our experiment shows that because the misalignment can be described by a transverse phase gradient, it results in the observation of spatial interference fringes. This interference pattern leads to a reduction of visibility when the intensity is obtained by integrating over a transverse section of the beam.

We demonstrate analogies and differences between these fringes created by *induced* coherence and those created in traditional interferometry by division of amplitude, i.e., by interfering two beams, which are derived from a single beam by splitting it on a semi-reflecting surface. We further show that the obtained fringe pattern is characterized neither by the signal nor by the idler wavelength alone, but by a combination of them. We attribute this effect to the momentum correlation between signal and idler photons.

Our setup is depicted in Fig. 1. Two nonlinear crystals (NL1 and NL2) are pumped coherently and produce photon pairs by nondegenerate SPDC. The two signal beams are superposed at a beam splitter and subsequently detected by a camera. Lens systems ensure that if the signal fields at the two crystals are decomposed into plane-wave components, one wave vector (spatial mode) of each signal field is detected at one point on the camera. In other words, the camera detects the transverse Fourier transform of the superposed signal field. The idler beam emerging from NL1 is directed through NL2 and aligned with the idler beam generated at NL2. A $4f$ lens system is inserted between the two crystals, which produces an image of the idler beam cross section from NL1 at NL2. In this way, each spatial mode of the idler beam after NL2 has almost an equal probability of being populated with a photon generated by NL1 or by NL2. If the path lengths are chosen accordingly, an idler photon after NL2 does not carry any information about which crystal it emerged from. Therefore, it is impossible to infer from which crystal its partner signal photon arrives at the beam splitter. As a consequence, interference between the two signal beams can be observed in each spatial mode, i.e., at each point on the camera. No heralding or coincidence detection is used. If the idler beam is blocked between NL1 and NL2, the signal beams do not interfere. The pair production rate in our experiment is low enough to render the possibility of stimulated emission negligible, as the probability of two idler photons being present in the setup at the same time is practically zero.

In the case of perfect alignment, the lens systems effectively cancel the effect of free-space propagation between the crystals. The observed interference pattern exhibits a uniform intensity modulation along the beam cross section. That is, by scanning the position-independent interferometric phase, all points on the camera simultaneously undergo the same transition from maximum to minimum intensity [see Fig. 2(a)]. In our experiment, we investigate situations in which the relative alignment of the two idler beams is modified. By maintaining perfect alignment of the two signal beams, a small tilt of the idler beam generated at NL1 relative to that generated at NL2 causes parallel interference fringes to appear [Fig. 2(b)]. This tilt is implemented by translating a lens of the imaging system perpendicular to the beam propagation axis. A circular interference pattern is created if, instead, the imaging system from NL1 to NL2 is defocused [Fig. 2(c)].

The observation of a spatially dependent interference pattern in the superposition of signal beams can be understood by considering the quantum state of light in our experiment. Each crystal NL$j$ produces photon pairs in a set of spatial modes labeled by the wave vectors ${\mathbf{k}}_{S}$ and ${\mathbf{k}}_{I}$,

We consider the beams to be paraxial and the detection plane to be perpendicular to the optical axis. The superposed signal beam is detected after a 3 nm bandpass filter centered at the signal wavelength ${\lambda}_{S}$. We assume a well-collimated pump beam and neglect any transverse phase mismatch. In this case, the transverse wave vectors ${\mathbf{q}}_{S}$ and ${\mathbf{q}}_{I}$ of a signal and an idler photon belonging to the same pair are related by the phase-matching condition ${\mathbf{q}}_{S}+{\mathbf{q}}_{I}={\mathbf{q}}_{P}\approx 0$, where ${\mathbf{q}}_{P}$ denotes a transverse wave vector of the pump field. The implied perfect transverse momentum correlation is reflected in the coefficients $C({\mathbf{k}}_{S},{\mathbf{k}}_{I})$. It follows from Eq. (2) that in this case, a spatially dependent phase introduced in the idler beam between the two crystals is observed in the interference pattern of the two signal beams. Note that such spatial fringes can occur only if some correlation between the momenta of the signal and idler photons exists. The relationship between momentum correlation and fringe visibility is the subject of a separate paper [21] and is analyzed theoretically in [20].

In common optical interferometers, in which two interfering beams are created by division of amplitude [22], circular fringes occur if the optical path length of one of the beams is extended with respect to the other. These fringes are often referred to as fringes of equal inclination, or Haidinger fringes [23]. Their transverse spacing at a given relative distance is determined by the wavelength of the interfering light. In our interferometer, the wavelengths of the signal (${\lambda}_{S}=810\text{\hspace{0.17em}}\mathrm{nm}$) and idler (${\lambda}_{I}=1550\text{\hspace{0.17em}}\mathrm{nm}$) beams differ significantly from each other. This gives rise to the question: which wavelength governs the spatial fringes in our experiment?

In order to create the fringes, the imaging system in the idler beam was defocused by translating a lens (Li1 in Fig. 1) about a distance $\delta $ along the beam propagation axis. The resulting phase shift can be approximated by the phase shift introduced by free-space propagation about a distance $d=({f}^{2}\delta )/({f}^{2}+{\delta}^{2})$ (see Supplement 1). Here, $f$ denotes the focal length of the lenses in the idler beam (Li1 and Li2 in Fig. 1). Under this approximation, the introduced phase shift can be expressed as ${\varphi}_{I}({\theta}_{I})\approx \pi d{\theta}_{I}^{2}/{\lambda}_{I}$, where ${\theta}_{I}$ is the angle that a wave vector of the idler beam subtends with the optical axis.

As our sources produce nondegenerate photon pairs, correlated signal and idler plane wave components leave the crystals at different angles from the optical axis (${\theta}_{S}$ and ${\theta}_{I}$, respectively) as a consequence of the phase-matching condition. For small angles, ${\theta}_{S}$ and ${\theta}_{I}$ are related to the wavelengths of the signal and idler beams as ${\theta}_{S}/{\theta}_{I}\approx {\lambda}_{S}/{\lambda}_{I}$.

A wave vector of the superposed signal beam is observed on the camera at a transverse distance $\rho \approx {f}_{c}{\theta}_{S}$ from the optical axis. Here, ${f}_{c}$ denotes the focal length of the lens in front of the camera (Lc in Fig. 1). As a consequence of Eq. (2) and the phase-matching condition, the phase introduced in the idler beam at ${\theta}_{I}$ modulates the intensity in the superposed signal beam at ${\theta}_{S}$. This modulation is observed in the camera at the transverse distance from the beam center $\rho \approx {f}_{c}{\theta}_{I}{\lambda}_{S}/{\lambda}_{I}$. Therefore, the observed interference pattern depends on both the signal and idler wavelengths.

It can be shown that the resulting fringes are circular and that their radii ${\rho}_{n}$ obey the following condition (cf. [20]),

where the intensity maxima correspond to the integer $n=0,1,2,3,\dots $ and the minima to $n=0.5,1.5,2.5,\dots $. Here, $\phi $ is a phase offset, which is constant across the beam cross section. Equation (3) closely resembles the condition for the maxima and minima of fringes of equal inclination in classical interferometry [22]. However, the fringes in our experiment are characterized by an “equivalent wavelength,” which governs the number of minima and maxima within a certain radial distance on the camera for a given value of $d$.Figure 3(a) shows examples of the resulting fringe patterns at the camera when phase shifts corresponding to different propagation distances $d$ are introduced. The images were analyzed using a computer algorithm, which evaluated the radii ${\rho}_{n}$ of bright and dark fringes. It follows from Eq. (3) that these radii are subject to the condition $a{\rho}_{n}^{2}+{\phi}^{\prime}=n$, where $a=d/(2{f}_{c}^{2}{\lambda}_{\mathrm{eq}})$. Figure 3(b) shows experimentally obtained pairs of $({\rho}_{n},n)$ for three different values of $d$. The coefficient $a$ was evaluated using second-order polynomial fits to the data. In Fig. 3(c), the obtained values of $a$ are plotted for different propagation distances $d$ in comparison to the theoretical prediction. The equivalent wavelength was determined from the dependence of $a$ on $d$. The result, ${\lambda}_{\mathrm{eq}}=420\pm 7\text{\hspace{0.17em}}\mathrm{nm}$, agrees well with the predicted value of 423 nm. This shows that the circular fringe pattern created through induced coherence without induced emission is governed by a combination of the signal and idler wavelengths and not by either of the two alone.

By measuring the equivalent wavelength, it is possible to determine the wavelengths of both the idler and pump beams from the spatial structure of the fringes in the signal beam, given the signal wavelength is known. No idler photon needs to be detected for this measurement. For our choice of pump wavelength and crystal parameters, the equivalent wavelength is smaller than any of the involved physical wavelengths (of the signal, idler, or pump). This fact could potentially be useful for applications in metrology.

The appearance of circular fringes governed by the equivalent wavelength is not limited to the quantum effect of induced coherence without induced emission. For example, if the setup were to be operated in the high-gain regime, idler photons from NL1 stimulate the emission of photon pairs in NL2, which also would cause coherence between the two signal beams. In this case, the two emitted signal beams would display many of the observed features, although with a different visibility (see also [24]). In particular, similar fringes could be observed if an appropriate phase shift is introduced in the idler beam between the two crystals. However, if the idler beam is blocked between the two crystals, no stimulated emission can occur in the second crystal, leading to a reduced emission rate of the signal photons there. This stands in contrast to our experiment, where the intensities of the individual signal beams do not change when the idler beam is either blocked or misaligned.

The formation of striped and circular interference fringes is familiar from traditional interferometry, where similar patterns are produced, e.g., by introducing a tilt or an additional propagation distance in one of the interfering beams. In our experiment, the interference pattern is created by manipulating only the undetected idler beam. Neither of the two interfering beams traverses the lens system, which is used to produce and control the fringes. Nevertheless, the obtained fringe pattern resembles that of a traditional interferometer, in which the same manipulation is performed in one of the interfering beams. However, in contrast to a traditional interferometer, the pattern is characterized by a combination of the wavelengths of both photons. This is a consequence of the phase-matching condition and thus a signature of the momentum correlation between signal and idler photons.

As long as the idler beam in our experiment is blocked or not aligned, the signal beams are mutually incoherent and do not interfere. Therefore, it is impossible to attribute a deterministic phase difference to the two signal beams. However, the alignment of the respective idler beams induces coherence between the signal beams. The spatial dependence of the induced phase difference can be controlled in the undetected idler beam in an analogous way, as a spatially dependent phase shift can be controlled in the interfering beams of a classical interferometer.

Of particular interest for future work could be the investigation of spatially varying phase changes at photon wavelengths, for which no suitable detector is available. This opens up possible applications in wavefront sensing or holography with undetected photons.

## Funding

Austrian Science Fund (FWF) (SFB40 (FOQUS), W1210-2 (CoQuS)); Austrian Academy of Sciences (OAW); Narodowe Centrum Nauki (NCN) (2015/16/S/ST2/00424, 2015/17/D/ST2/03471).

## Acknowledgment

The authors thank F. Steinlechner for the helpful discussions.

See Supplement 1 for supporting content.

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