## Abstract

Multi-mode interference (MMI) devices fabricated in silicon oxynitride (SiON) with a refractive index contrast of 2.4% provide a highly compact and stable platform for multi-photon non-classical interference. MMI devices can introduce which-path information for photons propagating in the multi-mode section which can result in degradation of this non-classical interference. We theoretically derive the visibility of quantum interference of two photons injected in a MMI device and predict near unity visibility for compact SiON devices. We complement the theoretical results by experimentally demonstrating visibilities of up to 97.7% in 2×2 MMI devices without the requirement of narrow-band photons.

© 2013 OSA

## 1. Introduction

Quantum information technologies utilise uniquely quantum mechanical effects such as superposition and entanglement to gain advantages over classical systems in areas such as secure key distribution [1], metrology [2], computational efficiency [3] and simulation of quantum systems[4]. Single photons are an ideal choice for observing quantum mechanical phenomena [5] due to their low noise properties and ease of manipulation.

Non-classical interference of single photons [6] is at the heart of emerging quantum technologies and the field of quantum photonics utilizes this phenomenon to design and implement protocols for quantum information and computation schemes. Non-classical interference of single photons has been demonstrated on integrated platforms [7], enabling the miniaturisation and enhanced stability of linear-optical components. Single-mode (SM) waveguide devices have been widely used as building blocks of complex quantum circuits, the simplest being a 2×2 directional coupler which relies on evanescently coupled SM waveguides to act as a beam splitter[8]. Multimode interference (MMI) devices have less stringent fabrication requirements than directional couplers [9] due to better tolerances to variation in wavelength of guided photons and critical dimensions of the device design and provide an efficient and stable way of producing large N×M multi-port devices, offering a viable alternative to SM waveguide architectures. Recently, quantum interference was demonstrated with integrated MMI devices [10, 11], albeit with reduced visibility, and the coherent dynamics of two-photon path-entangled states were investigated in a two-mirror, tunable planar multi-mode waveguide configuration [12].

In this paper we present a general expression for the visibility of two-photon quantum interference in 2×2 MMI devices. From this we show theoretically that near unity visibilities can be attained with these devices without the need for narrow-band photons which can induce a decrease in the single photon rate when filtering, or require narrow-band sources. We complement our theoretical results with experimental measurements yielding visibilities of up to 97.7%.

## 2. Theory

A MMI device comprises three main connected regions: entry SM waveguides, a homogeneous slab waveguide which supports multiple modes and exit SM waveguides. For a 2×2 MMI device, the entry and exit SM waveguides are connected parallel to, and symmetrically about an axis bisecting the device. The dynamics describing the propagation of many modes, with different propagation constants, in the slab waveguide allows an input field to be reproduced in single and multiple images at periodic intervals in the direction of propagation; a phenomenon known as self imaging [13]. As such, the length of the multimode waveguide can be set to give a balanced 2×2 MMI coupler which reproduces the input field into a double-image coincident with the positions of exit SM waveguides.

While MMI devices have been well studied for classical operation [14], much less is known about their capability for quantum operation, where interference occurs between photons as well as self-interference. For any real MMI device, the self imaging conditions are not perfectly satisfied, leading to non-optimum performance. Small deviations in the accumulated phases of the guided modes, unbalancing of the output field or high losses of the devices pose limitations on the performance of MMI devices. This imposes a trade-off between balancing on the output waveguides and losses. The length of the MMI device corresponding to a double-image can be chosen to give a balanced but lossy output: *r _{i}* =

*t*≠ 1/2,

_{i}*r*+

_{i}*t*≠ 1, where

_{i}*r*and

_{i}*t*are the reflected and transmitted power through the device for input

_{i}*i*respectively. For photonics the main goals that drive MMI device design and fabrication optimisation are the balancing of light at the output ports, the minimisation of losses and the trade-off between the two [15].

Conversely, the central motivation in the field of quantum photonics is the realisation of repeated, consistently high quality non-classical interference of many photons, in many modes. Non-classical interference of indistinguishable photons incident on two entry ports of a beam splitter [6] has been investigated and serves as a basis for future quantum photonic technologies. A lossless 2×2 MMI device or beam splitter can be described simply by the unitary matrix:

*r*is the

*reflectivity*,

*t*= 1 −

*r*is the

*transmittivity*and

*r*

_{1}=

*r*

_{2}=

*r*,

*t*

_{1}=

*t*

_{2}=

*t*for the different input modes of the device (see Fig. 1).

For an ideal, balanced beam-splitter (for which *r* = 1/2), two of the four classical outcomes are indistinguishable: both photons transmitted and both reflected. The probability amplitudes for these alternatives will interfere destructively to output the two-photon “NOON” state
$(|20\u3009+|02\u3009)/\sqrt{2}$, so that no coincidental detection is observed at the output of the device. In contrast, if these two alternatives are in principle completely distinguishable, say by introducing a delay larger than the photon coherence length, then no interference occurs and the rate of classical coincidental detections can be calculated from the product of probabilities for each alternative. The visibility—defined as the normalised difference between the quantum rate of coincidental detections *Q* and the classical rate *C*, *V* = (*C* − *Q*)/*C*—depends on the degree of indistinguishability of the interfering photons and the parameters of the beam-splitting device and gives a quality measure for the operation of the device: an ideal device gives *V* = 1.

In the presence of losses, the transition matrix of the MMI device is transformed. The 2×2 matrix that relates input to output does not describe the full device since there exist lossy modes.

*e*is a relative phase between the reflected and transmitted output photons.

^{iφ}This 2 × 2 matrix is no longer unitary and can be considered as a sub-matrix of a larger unitary that includes the lossy modes. In this case the phase *φ* ≠ *π*/2 [11] and the maximum visibility has an upper limit lower than one. On the other hand, if the MMI device is designed to have low losses—losing the balancing—we will have *r _{i}* +

*t*≃ 1 while

_{i}*r*≠

_{i}*t*for reflected and transmitted powers through the device for input

_{i}*i*. Also in this case the visibility cannot reach a unit value.

A second effect to be considered in real MMI devices is intermodal dispersion. An expression for the visibility of non-classical interference between two Gaussian modes (i.e. Gaussian in their temporal shape) in a 2×2 port device is given by:

*ω*is the bandwidth,

*δτ*is the relative temporal delay and

*φ*is the relative phase between two modes. A 2×2 MMI device supports multiple guided modes with different group velocities in the multimode waveguide, making Eq. (3) a simplified picture for considering the visibility function, since Eq. (3) accounts only for two single modes interfering. A single photon at an input port excites allowed guided modes within the multimode waveguide: which is a superposition state of lateral guided modes where

*j*are the modes propagating in the multimode section for length

*z*,

*p*are the mode excitation coefficients, and

_{j}*β*are the mode propagation constants with each mode having a different group velocity.

_{j}Intermodal group dispersion between the photon modes will induce temporal distinguishability which will decohere the state. In order to maintain high quantum coherence it is necessary to minimise dispersion. One way of working around this problem is to use narrow-band filters to increase the coherence length of the photons (as described in [10]), although loss increases, or use narrow-band sources. An alternative route is to use an integrated platform with high refractive index contrast between the guiding structures and the substrate. This enables the fabrication of compact devices, shortening the required length for the multimode waveguide and providing stronger confinement of fewer lower-order modes that propagate the transmitted power, thus minimising the effects of intermodal dispersion.

The 2×2 MMI devices investigated here were fabricated with a SiON core and silicon oxide (SiO_{2}) cladding, which provides high refractive index contrast (the core-cladding refractive index contrast is
$({n}_{\mathit{core}}^{2}-{n}_{\mathit{cladding}}^{2})/2{n}_{\mathit{core}}^{2}=2.4\%$) allowing further device miniaturisation compared to some platforms, in turn promising high visibilities due to smaller intermodal dispersion.

The detailed scattering matrix of the MMI devices was obtained using the FIMMWAVE (http://www.photond.com/products/fimmwave.htm) simulation platform, giving full information about the reflectivities, relative phases and losses for the devices. Furthermore, the different modes that are excited in the multimode waveguide with their associated mode overlap (SM waveguide to multimode waveguide) were calculated and the different time delays due to mode dispersion extracted. Simulation shows that only a few low-order guided modes propagate most of the transmitted power.

This gives a complete characterisation of the process and gives insight into the effect of modal dispersion on the visibility of the non-classical interference. Since the input field does not excite all modes equally there is a population weight associated with each photon mode during propagation in the multimode region. From this information a general expression for the expected visibility can be formulated from Eq. (3) and Eq. (4):

*i*and

*j*are the mode numbers for each photon and

*p*and

_{i}*p*are the associated mode excitation coefficients of those modes whose product

_{j}*p*gives the weight of the corresponding visibility for that pair and

_{i}p_{j}*δτ*is now given by the intermodal dispersion. We now have a complete expression for multimode quantum interference in 2×2 MMI devices. We use this expression to investigate device concatenation which gives an important insight into the suitability of these devices for large quantum photonic networks. By fixing the width and length to make the concatenated devices identical, we calculate the dependence of the maximum visibility on intermodal dispersion (see Fig. 2). These results demonstrate that even for complex networks of up to 20 concatenated devices, high degree visibility of non-classical interference is maintained with 0.5 nm bandwidth photons, resulting in visibilities above 99%, critical for large scale implementations of quantum photonic circuits. Despite the degradation of the visibility when considering 3.1 nm bandwidth photons, the visibility remains high for a significant number of concatenated devices, showing promise for present day experiments.

_{i,j}## 3. Fabrication and experimental setup

The devices measured and presented in this work were fabricated in SiON technology. The waveguide layer was grown on thermally oxidised (8*μ*m) silicon (Si〈100〉) wafers with deposition performed in an Oxford Instruments parallel plate PECVD reactor utilising SiH_{4} and N_{2}O precursors. After annealing channel waveguides are obtained by standard lithography and reactive ion etching in CHF_{3}/O_{2} chemistry. The channel structures are covered by a PECVD silicon oxide (SiO_{2}) cladding layer. The core index was slightly less than the designed index of 1.4905 at 800 nm, confirmed by measurements of directional couplers fabricated on the same wafer as the MMI devices.

Photon pairs were produced in a Type-I spontaneous parametric downconversion (SPDC) source using a 2 mm thick *χ*^{(2)} non-linear crystal pumped by a 404 nm continuous wave (CW) 60 mW diode laser. Degenerate 808 nm photon pairs were produced non-deterministically with low probability at an opening angle of 3° from the crystal and a rate of 40 – 50 kHz of pairs collected in polarization-maintaining fibres (PMFs)—we observed visibilities of up to 97.9 ± 0.4% directly from the source using a balanced fibre beam-splitter. These photon pairs were fed into a PMF array. Each PMF in the array is coupled to entry waveguides on the chip for each MMI device. Once the photons have traversed the MMI device they are collected using a multi-mode fibres (MMF) array coupled to the exit waveguides on the rear facet. Photons were detected using avalanche photo-diode (APD) single-photon counting modules (SPCMs). Coincidence counting logic was managed by a field-programmable gate array (FPGA). A pair of 3.1 nm interference filters, centred on 808 nm, were used just after the non-linear crystal.

## 4. Experimental results

By injecting single photons into input ports 1 and 2 of the MMI devices separately and using the photon count ratios measured by the APDs at the output ports the maximum expected visibilities *V _{max}*(

*r*) were calculated.

_{exp}*V*(

_{max}*r*) depends on the measured reflectivity only since losses from the ports of the device algebraically cancel out:

_{exp}*V*= 2

_{max}*r*(1 −

*r*)/(2

*r*

^{2}− 2

*r*+ 1) where the subscript “exp” has been left out for convenience. These measurements were corroborated by injecting 810 nm CW diode laser light.

In our Hong-Ou-Mandel (HOM) type interference measurements [6], the relative free-space delay was varied to scan a region ±0.5 mm of the zero delay position for the arrival time of the twin photons and the visibility *V _{exp}* was calculated by fitting of the experimental data (for example see Fig. 3). Since

*V*depends only on the measured reflectivity and not on the relative phase,

_{max}*ϕ*, of the output modes in the device (which is assumed in this case to equal

_{exp}*π*/2), then

*V*/

_{exp}*V*further from 100% indicates that

_{max}*ϕ*is further from the ideal of

_{exp}*π*/2.

Each device was measured once and the measurements consisted of forty steps of length 25 *μ*m. The steps in the relative free-space delay of the twin photons were centred around the zero delay position. Each step was integrated for a time of 20 s. We measured a total of twenty 2×2 MMI devices which were grouped into four sets of five devices. The sets are differentiated by widths of: 20 *μ*m, 16 *μ*m, 10 *μ*m, and 8 *μ*m. In each set are five devices with incrementally different lengths of 15 *μ*m. The range of lengths covered the extent of the self-imaging double-image region. This allowed us to investigate performance of the devices in the vicinity of the optimum length and how this performance was affected by different device widths.

The results are summarised in Fig. 4. The coupler with the largest maximum expected visibility was the MMI (device) of width (W) 20 *μ*m and length (L) 515 *μ*m, with *V _{max}* = 99.98%. The measured visibility for this device was

*V*= 93.32%, an agreement of

_{exp}*V*/

_{exp}*V*= 93.34%. The MMI with

_{max}*W*= 8

*μm*and

*L*= 335

*μm*exhibited the best agreement with prediction:

*V*/

_{exp}*V*= 99.06% which shows an almost ideal behaviour of the MMI devices at the single photon level, considering the multimode dynamics of these devices. The highest net visibility was measured on MMI with

_{max}*W*= 8

*μm*and

*L*= 320

*μm*, with

*V*= 97.69%. An average agreement for all twenty MMI couplers is

_{exp}*V*/

_{exp}*V*= 96.44 ± 2.34%, showing close to optimum performance of the measured devices, with low intrinsic loss.

_{max}These results demonstrate that in spite of intermodal dispersion MMI devices can nonetheless maintain a very high degree of quantum coherence without the need for additional lossy narrow-band filtering apparata or narrow-band sources. Indeed the most important factors for maintaining a high degree of quantum coherence are the reflectivity and relative phase, which are determined by how well optimised the self imaging is in the device. This means that couplers designed for the first double-image region, with different widths and lengths, will not give significant changes in visibility because the difference in intermodal dispersion is small.

However, devices with different widths have modes with different propagation constant, corresponding to different mode evolutions in the MMI waveguide. Various realizations of MMI devices with fixed width and different lengths can be used to experimentally search the device with the best trade-off between losses and splitting ratio. Our results reflect this and although there is no clear dependence of the maximum visibility on the width, we can observe how the visibility is optimum for certain dimensions.

## 5. Conclusion

Single mode waveguide structures are at the heart of quantum information and computation experimental demonstrations. MMI devices provide an alternative route to integrated single mode waveguide photonic circuits and comparisons between the different technologies have previously been done [16] with classical photonic circuits. For certain photonic implementations, such as multi-port N×M devices, MMI devices have an inherit advantage to single mode waveguides, providing a less complex and more error-tolerant fabrication methods. As such, the performance of MMI devices in the single photon regime needs to be investigated and their potential as quantum multiport building blocks evaluated.

In this work we investigated the performance of 2×2 MMI devices in the quantum regime. From theoretical simulations it is obvious that intermodal dispersion does not have a great effect in degradation of the non-classical interference of single photons. Rather it is the reflectivity and the internal phase of the device that is crucial for demonstrating and maintaining high quality non-classical interference. Devices with larger widths exhibit an increase in the number of guided modes in the multi-mode waveguide, however this does not have a noticeable impact on device performance for the widths investigated in this work.

We have presented an expression for the visibility of two-photon multimode quantum interference which considers the reflectivity, relative phase, mode excitation coefficients and inter-modal dispersion effects inherent to 2×2 MMI devices, which can be used in device design for quantum photonics. SiON 2×2 MMI devices have a high refractive index contrast which is a desirable feature since it enables compact devices to be made and mode dispersion effects minimised. Simulations suggest that our devices transmit most of the power in a subset of low-order guided modes which exhibit small intermodal dispersion, allowing near unity visibilities of non-classical interference without the need for narrow-band photons.

Further to this, MMI device performance is known to be more robust against fabrication errors in refractive indices and waveguide dimensions, and experimental errors in the wavelength of guided photons. As explained in this Letter, non-perfect self imaging introduces additional losses that have to be considered when using the devices for schemes that require low overall efficiencies, and in the context of loss error correction [17]. Even in this case, depending on the technology used, MMI devices can compare favourably to the directional coupler configuration, thanks to the smaller footprint that reduces propagation losses. MMI devices can also be designed with restricted interference [14] to be more compact than SM devices. We experimentally show that 2×2 MMI devices are a viable and ready alternative for integrated quantum-photonic networking. Indeed our theoretical calculations show that device concatenation does not lead to significant photon decoherence, which means 2×2 MMI devices are scalable making them candidate building-block components for future quantum-photonic networking.

## Acknowledgments

The authors would like to thank Damien Bonneau and Anthony Laing for helpful discussions. This work was supported by EPSRC, ERC and QUANTIP. J.L.OB. acknowledges a Royal Society Wolfson Merit Award and a Royal Academy of Engineering Chair in Emerging Technologies.

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