## Abstract

Generating polarization-entangled photon pairs on chip is generally complicated by the birefringence of waveguides. In this work, we propose a technique that uses waveguide birefringence and lends itself to simple device designs. The technique relies on two orthogonal spontaneous four-wave mixing processes. We employ the full quantum optics theory and dispersion analysis, and show that the technique can produce highly entangled states, with concurrence as high as 0.976 and covering the entire C-band.

© 2017 Optical Society of America

## 1. Introduction

Photonic polarization-entangled states have applications in quantum information science and quantum sensing [1–6]. Migrating these demonstrations to an integrated format is required for practical realization and deployment. However, the polarization entanglement is difficult to prepare from an integrated waveguide due to waveguide birefringence that leads to distinguishability. Though such sources can be implemented in semiconductor waveguides by utilizing the *χ*^{(2)} nonlinearity, satisfying phase matching is a challenge that can lead to sophisticated fabrication requirements, for example the Bragg reflection waveguides [7]. On the other hand, the *χ*^{(3)}-based sources are readily phase-matched even with simple waveguide designs and material composition. Nonlinear silicon waveguides, fabricated in the standard ~200-nm-thick silicon-on-insulator (SOI) wafer, employ such an approach [8–10]. However, these sources are still affected by severe birefringence; to avoid this problem only one guided mode is used, usually the transverse electric (TE) mode. Producing polarization entanglement then requires an additional stage to rotate the polarization state of the photon pair, which can be accomplished with either an on-chip device [8] or a off-chip setup [11].

The additional polarization rotation step is not necessary if the birefringence is properly engineered. In particular, the direct generation of a polarization-entangled state is possible when exploiting two spontaneous four-wave mixing (SFWM) processes that occur independently and simultaneously: one process is associated with the TE mode and the other with the TM mode. This approach, herein called the orthogonally pumped SFWM technique, was once deployed in a 220-nm-thick, 500-nm-wide silicon waveguide to generate the state [9]. In this study, the birefringence effect was overlooked, and the two-photon interference visibility was not high (<70% raw visibility). Recently, we experimentally demonstrated the generation and characterization of a direct polarization-entangled state and a Bell state in a deep etched AlGaAs waveguide via the orthogonally pumped SFWM technique [12]. We pointed out that the technique requires a non-vanishing birefringence that simultaneously suppresses the spectral distinguishability and state factorizability. However, the deep etched structure does not allow an independent control over the dispersion property of the TE and TM modes where the birefringence rises.

In this work, we fully explore the benefit of birefringence in a rectangular waveguide geometry that would allow us to tune the TE mode dispersion (via the waveguide w idth) and the TM mode dispersion (via the waveguide thickness) essentially independently. We then employ an extended version of the backward Heisenberg framework to calculate the polarization entanglement via biphoton wavefunctions. From the spectral profile of the entanglement measure, we identify the most suitable waveguide dimension and later discuss means to fabricate the device.

## 2. Basic idea

Inside an integrated waveguide, photons in the transverse electric (TE) mode have a horizontal, linear polarization state (oriented perpendicular to the growth direction), denoted by *H*. Those in the transverse magnetic (TM) mode are linearly polarized in the vertical direction and represented by *V*. Due to dispersion, FWM interactions of same-mode photons occur more favorably as a result of their ready phase matching. For instance, a pump field in the TE mode will result in a correlated photon pair where both the signal and idler photons are in the TE mode. The pair can be represented by a quantum state |*HH*〉 for its polarization degree of freedom. In a similar manner, the TM mode pump field produces a photon pair in a state |*VV*〉.

We can exploit this fact to create photonic polarization-entangled states as follows. The two processes occur independently if the pump photons exist in both the TE and TM modes. The power of the pump field in a typical implementation is low such that the photon pair generation rate (or probability), *µ*, is ≪1 to avoid generating multipair states [13]. Therefore, at the waveguide exit, either no pair is generated, or one photon pair is created in *either |HH*〉 *or* |*VV*〉. Consequently, the resultant quantum state of the photon pair is entangled in the polarization degree of freedom, and it can be represented by

*HH*〉 and |

*VV*〉 with probabilities |

*a*|

^{2}and |

*b*|

^{2}, which are affected by pump powers of the modes, and a relative phase

*θ*, which is determined by the birefringence. Hence, with this orthogonally pumped SFWM scheme, we can prepare polarization-entangled states by utilizing both the TE and TM modes of integrated waveguides. The following section will detail how dispersion and other SFWM processes could affect the polarization entanglement.

## 3. Quantum theory of orthogonally pumped SFWM

We employ a quantum optics treatment to characterize the state that would be generated from an integrated waveguide via the orthogonally pumped SFWM scheme with a targeted state expressed in Eq. (1). In particular, the biphoton wavefunctions, which fully describe the photon pair state, of all the possible SFWM processes are derived. We focus on degenerate SFWM where the two pump photons have identical central frequencies, but the theory can be extended to non-degenerate cases as well. Due to the symmetry of AlGaAs, the third-order susceptibility only allows a total of eight SFWM processes that we identify with a four-letter notation *stuv* where *s, t, u, v* ∈ {*H, V* } and the letters respectively specify the polarization state (or waveguide mode) of the signal, idler, and the two pump photons. The resultant polarization quantum states of the photon pair are listed in the left column of Table 1, and their contributing SFWM processes are in the right column. For instance, the state |*HH*〉 is a result from two independent SFWM processes, HHHH and HHVV. The processes HHHH and VVVV are called scalar processes since the photons are in the same mode; other processes are then considered vectorial.

In deriving the biphoton wavefunction of each SFWM process, we follow the backward Heisenberg approach [14, 15] but explicitly extend the input pump field to consist of two independent coherent states, one for the TE mode and the other for the TM mode. In particular, the input pump field can be expressed as

*f*denotes the central frequency of the pump photons and the subscript

*w*∈ {

*H, V*} represents the polarization mode of the pump field. The pump spectral profile is described by the spectral density

*ϕ*(

_{wf}*k*), and the number of the pump photon is proportional to |

*α*|

_{wf}^{2}. We focus on degenerate SFWM such that the subscript

*f*can be dropped. In a typical experiment, the generated polarization-entangled state is collected and passes through a series of filters to remove the pump photons. Hence, the resultant state prior to detection becomes

*is called herein the specific biphoton wavefunction of the*

_{stuv}*stuv*SFWM process at the exit of the waveguide with a length of

*L*. The factor

*K*collects all other constants, and

*µ*can be used to find the occurrence probability. The function

_{stuv}*F*(

*ω*

_{1},

*ω*

_{2}) describes the filtering in the signal and idler channels. The first phase factor accounts for the signal and idler photons propagating from the waveguide center to the exit whereas the phase factor inside the integral takes care of the pump photon propagation from the waveguide entrance to the center. The phase mismatch is ∆

*k*=

_{stuv}*k*

_{s}_{1}+

*k*

_{t}_{2}−

*k*

_{u}_{3}−

*k*

_{v}_{4}. The delta function captures the conservation of energy. In the actual computation, the integral over the

*k*space is transformed to the integral over the

*ω*space with appropriate conversion factors that preserve normalization in both domains [15].

The birefringence could lead to distinguishability between the photon pairs that are created by the HHHH and the VVVV processes, and therefore poor entanglement. We can quantify the degree of entanglement by the concurrence *C* [16], which has a monotonic relation with entanglement. Given a density matrix *ρ*, the concurrence is found from *C*(*ρ*) = max{0, *r*_{1} −*r*_{2} − *r*_{3} − *r*_{4}} where *r _{x}* is an eigenvalue, in decreasing order, of the matrix $\sqrt{\sqrt{\rho}\tilde{\rho}\sqrt{\rho}}$ and the spin-flipped state is defined as $\tilde{\rho}=({\sigma}_{y}\otimes {\sigma}_{y})\rho *({\sigma}_{y}\otimes {\sigma}_{y})$. The density matrix

*ρ*

_{0}= |

*ψ*

_{det}〉〈

*ψ*

_{det}| of our state generated via the orthogonally pumped SFWM is

*m, n, o, p*∈ {

*H, V*}. To simplify the notation, the numeric subscripts for the

*ρ*elements are defined such that 1, 2, 3, and 4 mean

*HH*,

*VH*,

*HV*, and

*VV*, respectively. The pump field ratio defined as

*r*=

*α*can be chosen to vary the values of the diagonal elements

_{v}/α_{u}*ρ*

_{11}and

*ρ*

_{44}. The magnitudes of

*ρ*

_{14}and

*ρ*

_{41}, imply the degree of entanglement in one way and reduce when the phase difference between the Φ

*and Φ*

_{HH}*functions, termed here*

_{VV}*spectral phase beating*(SPB), varies rapidly within the integral domain. In a physical sense, this is the spectral distinguishability [17]. If a pulsed excitation source is used, the time domain explanation, i.e. the temporal walk-off, applies where the birefringence causes a group velocity mismatch (GVM) that subsequently leads to the wavepackets of the |

*HH*〉 and |

*VV*〉 states becoming more separate in time.

Hence, the birefringence is generally regarded as undesirable, and one is limited to a short waveguide, or to a spectrally narrow mode, or has to use complicated waveguide design to erase distinguishability. If the birefringence can be eliminated by making the dispersion of the TE and TM modes identical, it is certain that the spectral distinguishability is nullified. However, eliminating birefringence would generally mean eliminating birefringence, which would lead to all the possible SFWM processes be phase matched. We would consequently produce a fully factorizable state $|{\psi}_{\text{gen}}\u3009=|HH\u3009+{e}^{i{\theta}_{s}}|VH\u3009+{e}^{i{\theta}_{i}}|HV\u3009+{e}^{i({\theta}_{s}+{\theta}_{i})}|VV\u3009={(|H\u3009+{e}^{i{\theta}_{s}}|V\u3009)}_{s}\otimes {(|H\u3009+{e}^{i{\theta}_{i}}|V\u3009)}_{i}$ and the entanglement is completely lost. Therefore, one realizes that the birefringence is indeed *required* to prepare the polarization-entangled state via the orthogonally pumped SFWM technique.

## 4. Waveguide dispersion and phase matching functions

The dispersion properties of the TE and TM modes are of paramount importance in producing the polarization-entangled state according to our technique. The typical deep etched structure of AlGaAs waveguides, shown in Fig. 1(a), is limited in that the dispersion of the TM mode is not engineered due to the low index contrast in the vertical direction [18]. Only the TE mode is such structure experiences a high index contrast due to the interface between AlGaAs materials and air. In this paper, we study rectangular waveguides, which allow independent control of the TE and TM modes such that they can be tuned from equivalence to a stark contrast. Fig. 1(b) shows a cross section of such structure. The waveguide cross section has a thickness *t* and a width *w* and is surrounded by air on all sides, to achieve a high index contrast in two dimensions.

To avoid deleterious effects from two-photon absorption, the aluminum concentration of the AlGaAs core material should be 0.20 [19], and we apply this in our following analysis. The waveguide thickness varies from 300 nm to 1000 nm with an increment of 100 nm, and for each thickness the width starts from the same value and is increased up to 2000 nm in steps of 100 nm. The fundamental modes of these waveguides are calculated using Lumerical Mode. The effective index *n*_{eff}, group index *n _{g}*, and the second-order dispersion coefficient

*k*

^{(2)}determined at a wavelength of 1550 nm are displayed in Fig. 2, for both the TE and TM polarizations.

The TM mode is more dependent on the thickness of the waveguide while the TE mode is more affected by the waveguide width. As the waveguide dimension approaches 300 nm × 300 nm, both the fundamental modes approach a cut-off frequency where the optical field is barely guided in the waveguide. Additionally, the TE dispersion coefficient *k*^{(2)} behaves very similarly in all the waveguide thicknesses, and crosses from the normal to the anomalous regime at a waveguide width of 700 nm, which is consistent with those of a typical deeply etched structure [20–22]. The displayed *n _{g}* and

*k*

^{(2)}curves offer a key guide to understanding their effects on temporal walk-off (or equivalently SPB) and FWM bandwidths that would eventually feature in the biphoton wavefunction.

We consider in detail a waveguide thickness of 700 nm (corresponding to the red curves in Fig. 2) since it later turns out to have the optimum performance for generating polarization-entangled state. Starting from a 700 nm × 700 nm square waveguide, the 1548.5-nm pump wavelength is in the anomalous dispersion regime for both the TE and TM modes. As the waveguide widens, the zero dispersion wavelength (ZDW, the wavelength at which *k*^{(2)} changes sign) of the TE mode moves to a longer wavelength; the pump wavelength correspondingly transitions from being in the anomalous dispersion to being in the normal dispersion regime at the width of ∼900 nm. The ZDW of the TM mode also changes but much less sensitively to the waveguide width, and as a result, the 1548.5-nm TM mode remains anomalous.

If we expand the propagation constant *k* up to the second-order term around the angular frequency of the pump *ω*_{3} and properly considering birefringence, the phase mismatch of the *stuv* SFWM process becomes

*ω*=

*ω*

_{1}−

*ω*

_{3}is the signal-pump angular frequency detuning. We assume here the pump degeneracy

*ω*

_{3}=

*ω*

_{4}and the conservation of energy such that

*ω*

_{2}= 2

*ω*

_{3}−

*ω*

_{1}. The first term in fact disappears for all the SFWM processes except for the HHVV and VVHH processes. Therefore, these two processes are the hardest to phase-match and can be safely neglected with an exception of the square waveguide geometry. The second term highlights the GVM. It is significant in the HVHV, HVVH, VHHV, and VHVH processes but disappears in the scalar processes. These scalar processes, HHHH and VVVV, are solely influenced by the second-order dispersion coefficient

*k*

^{(2)}.

In Fig. 3, the magnitudes of the sinc(∆*kL/*2) phase matching function of all the eight SFWM processes are shown for four representative waveguides at a fixed thickness of 700 nm and waveguide widths of 700, 800, 1100, and 2000 nm. The phase matching functions are calculated using the full dispersion (i.e. without Taylor expansion), the pump wavelength of 1548.5 nm, and the waveguide length *L* of 5 mm were used. The square waveguide exhibits identical phase matching functions for all the processes since the TE and TM modes behave identically. As the waveguide geometry deviates from a square, the HHVV and VVHH quickly become extremely phase-mismatched over a broad range such that their contributions are negligible. As the waveguide width reaches 1100 nm, the spectral profiles of the scalar processes (HHHH and VVVV) become almost identical; yet, note that at the pump wavelength, the TE mode is in the normal dispersion regime with *k*^{(2)} = 0.34 ps^{2}m^{−1} whereas the TM mode is anomalous with *k*^{(2)} = −0.30 ps^{2}m^{−1}. As the width grows wider, the spectral profiles of the scalar processes drift apart from one another and one can easily measure them using a classical continuous-wave four-wave mixing experiment.

## 5. Producing polarization-entangled states

Using the biphoton wavefunction expression in Eq. (4) and simulated dispersion, we can predict the upper bound of the concurrence. In the following analysis, the pump wavelength is again set to *λp* = 1548.5 nm and the waveguide length is assumed *L* = 5 mm. The post-generation filtering function has a passband of ∆*ν* = 0.1 THz, emulating real DWDM filters, for both the signal and idler channels. The signal central frequency is varied in the blue-shifted side of the spectrum with respect to the pump wavelength. The idler central frequency is determined from the conservation of energy. We also neglect the case where the signal-pump frequency detuning *ω*_{1} − *ω*_{3} is less than 1.5 Trad (2 nm in wavelength) because in the actual implementation the signal and idler channels should be reasonably far from the pump frequency for the pump photon removal. The field ratio *r* is sought such that |*ρ*_{11}| = |*ρ*_{44}|.

The achievable maximum concurrence (AMC) from each waveguide dimension corresponds to the highest concurrence as the signal wavelength changes, and it is shown on the left of Fig. 4. It generally decreases as the waveguide narrows due to the escalating birefringence. For each thickness, the AMC is zero as expected at the square waveguide geometry since the generated state is fully factorizable. It quickly rises to a peak value when the waveguide is slightly off-square where the SPB and factorizability are optimally balanced. The AMC then slightly decreases as the waveguide widens as a result of increasing birefringence.

Even though the slightly off-square dimension provides the highest AMC for each thickness, it does not guarantee broadband state generation. The spectral profiles for the concurrence are plotted in Fig. 4 (right) for the last three waveguide representatives (widths of 800, 1100, and 2000 nm). The concurrence of the square waveguide (700 nm × 700 nm) is very low, in the order of 10^{−5}, and therefore omitted from the plot. The plot can be divided into three regions with low, medium and large frequency detunings. In the low detuning region, the concurrence rises from zero with several deep ripples as the detuning increases. This is a result of the side lobes in the phase matching functions of the contaminating HVHV, HVVH, VHHV, and VHVH processes. The medium detuning region starts when the difference in the phase matching magnitude of the scalar processes and that of the contaminating processes reaches a certain threshold. When the threshold is reached, the contributions from the contaminating processes are negligible (even though their phase matching values do not vanish) compared to the scalar processes in terms of the probability of generating their respective photon pair states. The middle region terminates with the first notch, for instance at ∆*ω* = 23.5 Trad for the 700 nm × 2000 nm waveguide (green curve). The notch corresponds to the point of vanishing phase matching function of one of the scalar processes as shown in Fig. 3, i.e. the HHHH process for the 700 nm × 2000 nm waveguide. The notch in the blue curve, at ∆*ω* = 28.5 Trad, corresponds to the VVVV process for the 700 nm × 800 nm waveguide. For large detunings beyond this point, the entangled state is created from the side lobes of the scalar-process phase matching functions. The resultant concurrence of this region also depends on the difference in the phase matching magnitudes of the scalar processes and of the contaminating processes. However, as the phase matching magnitude of the side lobes is significantly less than that of the main lobe, the state generation in this region is less efficient.

Therefore, the middle detuning region suitably represents the operational spectrum of the state generation with high concurrence and brightness. The region is bounded at the lower limit by how fast the phase matching functions of the contaminating processes diminish, and this demands a certain level of birefringence according to Eq. (6). It is bounded at the upper limit by the scalar SFWM bandwidths, hence requiring the pump photons of both the TE and TM modes to be located as close the ZDW as possible. However, the two modes must not become identical; otherwise the generated state becomes factorizable and the entanglement is lost. We quantify the state generation bandwidth in the following way. The bandwidth is found where the concurrence drops by 0.01 (reflecting a measurement error from quantum state tomography experiment [12]) from its maximum value. The generation bandwidths of three chosen waveguide thicknesses (500, 700, and 900 nm) are displayed in Fig. 5. The calculation shows that the 700-nm-thick waveguide provides the broadest bandwidth that peaks around the waveguide width of 1100 nm. The 700 nm × 1100 nm waveguide has its concurrence profile shown in Fig. 4 (right), and it has a very broad generation bandwidth of ∆*ω*_{BW} = 20 Trad (∆*ν*_{BW} = 3.18 THz) and the AMC of 0.976. The reason is that the pump photon in the TM mode is located near the ZDW at the thickness of 700 nm with the *k*^{(2)} of −0.30 ps^{2}m^{−1}, yielding a broad main lobe of the VVVV phase matching function. In order to avoid the spectral distinguishability while maintaining a broad main lobe of the HHHH phase matching function, the pump photon in the TE mode should also operate near the ZDW but with the opposite *k*^{(2)} of 0.34 ps^{2}m^{−1}, as satisfied in the 700 nm × 1100 nm waveguide.

The pump power ratio between the TE and the TM modes is related to the field ratio by |*r*|^{2}, and it is plotted in Fig. 5 for the three waveguide representatives discussed in Fig. 4. Peaks and notches in the power ratio correspond to when the phase matching functions of the scalar processes reach zero. It is larger than 0 dB if the VVVV process is less efficient than the HHHH process, for instance in the 700 nm × 800 nm waveguide. It approaches infinity when the VVVV phase matching function becomes zero. The power ratio is less than 0 dB if the HHHH process is less efficient as such in the 700 nm × 2000 nm waveguide, and therefore it approaches zero when the HHHH phase matching function vanishes. The very large (≫ 1) or very small (≪1) power ratios indicate a situation where the entanglement generation is impractical. For example, if the polarizer is used to prepare a linearly polarized pump light before being coupled to the waveguide, the extinction ratio of the polarizer might not be sufficient to guarantee the power ratio of 100:1 or the polarizer mount lacks a sufficient position resolution. Hence, it is important to restrict the generation to a practical value of the field ratio.

## 6. Waveguide fabrication possibilities

Rectangular waveguides can be fabricated from a multi-layer AlGaAs wafer, which is typically used to make deep etched AlGaAs waveguides, using a combination of electron beam lithography and dry etching. An additional wet etching step is required to remove the upper and lower cladding layers, while leaving to waveguide core intact. Hence, it is necessary to choose the core and cladding materials that react very differently to with a specific etchant. One possibility is to use Al_{0.2}Ga_{0.8}As as the core, Al_{0.7}Ga_{0.3}As as the cladding, and a dilute hydrofluoric acid (HF) as an etchant [23]. The HF solution attacks Al_{0.7}Ga_{0.3}As with a much faster rate compared to removing Al_{0.2}Ga_{0.8}As. This technique has been used to fabricate a range of devices such as microdisks [24, 25], photonics crystals [26], and suspended beams [27, 28]. The resultant waveguides are surrounded by air, which then provides a high index contrast and tight mode confinement. This approach then requires mechanical supports to properly lift the waveguide and they should induce as minimum loss as possible. Additionally, the multi-layer AlGaAs platform offers potential of both the monolithic and vertical integration. Another possibility is the AlGaAs-on-insulator approach [29] that results in a rectangular waveguide resting on top of the insulator layer. Waveguides fabricated in this fashion have been shown to have very low propagation loss and provide efficient SFWM interactions. However, this approach is limited to the monolithic integration.

## 7. Conclusion

The generation of the polarization-entangled photon pair states via the orthogonally pumped SFWM technique is numerically studied in rectangular AlGaAs waveguides. The entanglement of the polarization-entangled state is predicted using the biphoton wavefunction of all the eight allowed SFWM processes as calculated by the backward Heisenberg approach with the full dispersion of the TE and TM modes. The features of the concurrence profile have been identified in relation to the phase mismatch ∆*k _{stuv}* and phase matching function of each SFWM process. The 700 nm × 1100 nm waveguide is the waveguide dimension that could simultaneously deliver a high concurrence at a maximum of 0.976 and over a broad bandwidth of ∆

*ν*

_{BW}= 3.18 THz. The possible means of realizing rectangular AlGaAs waveguides is suggested by employing lithography, dry etching, and wet etching.

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