1. Introduction

Sources of pure state photons are a necessary requirement in the fields of optical quantum communication [1], optical quantum measurement [2] and optical quantum computation [3]. One of the most common methods for generating single photons is via spontaneous parametric down-conversion (SPDC) in χ⁽²⁾ nonlinear optical crystals [4–6]. SPDC sources convert higher energy photons into lower energy photon pairs, typically labelled the signal and idler photons. A pair of SPDC photons is useful, as a pair can be used for heralding single photons by detecting a photon from the pair; thus, indicating the existence of its partner. Unfortunately, due to the relatively long duration of detector response times compared to photon pair correlation times, the detection of a photon in the pair can leave the other photon of the pair in a partially mixed state [7]—this is undesirable for many applications and can be avoided by removing the spectral correlations between the signal and idler photons (increasing their spectral purity) [8].

The traditional technique for achieving high spectral purity SPDC photon pairs is to pass them through narrow-band filters; this has the effect of filtering out photon pairs that are spectrally correlated [9, 10]. The downside to narrowband filtering is that it requires additional optical components, hindering the ability to integrate photon sources with other integrated devices, and more importantly, lowering the heralding efficiency. An integrated method for improving purity without compromising heralding efficiency is to use the group velocity matching properties of the material to generate SPDC photon pairs that are less spectrally correlated [11]. In the group velocity matching regime, one seeks to have either the signal or idler photon travelling at the same group velocity (or faster) than the pump pulse, and the other photon of the pair to be trailing the pump pulse. In order to obtain signal and idler photons travelling at different group velocities in the crystal, a type-II down-conversion scheme is often used with one polarization experiencing a higher refractive index than the other. It has been shown that one can generate C-band SPDC photon pairs, one of which lags a 780 nm pump, and the other that leads it in potassium titanyl phosphate (KTP) [12]. A further requirement of this scheme is that the length of the crystal and pump pulse duration be adequately designed [13, 14].

When the group velocity matching scheme is combined with birefringent phase matching or quasi-phase matching (QPM) in χ⁽²⁾ crystals, one can obtain a high flux of high purity photons; however, when using a uniformly poled crystal, the abrupt transition in relative nonlinearity at the input and output interfaces of the crystal leads to a sinc-shaped spectral response of the QPM crystal. The sinc-shaped spectral response consists of spectral lobes corresponding to the wavelength bands in which photons are likely to be generated; there is a main lobe centred at the nominal phase matching wavelength (where the probability of photon generation is highest), and sidelobes centred at wavelengths to either side (where the probability of photon generation is less than the main lobe, but significant). These sidelobes lead to an increase in the spectral correlations between the signal and idler photons, limiting the heralded photon purity. To increase the signal and idler spectral purity, the sidelobes should be suppressed; this can be done by using an external Gaussian filter [15]. One may opt for an integrated alternative and suppress the sidelobes by apodizing the relative nonlinearity along the crystal, softening the relative nonlinearity transition between the outside and inside of the crystal [16].

Recently, tailored QPM poling profiles have been used to control the strength of the nonlinearity along a crystal [16–18]. It has been shown that a gradual Gaussian nonlinearity profile suffices in moderating the sinc sidelobes leading to a decrease in spectral correlations. Methods for achieving a Gaussian nonlinearity profile include using higher order QPM spectral components [16] and varying the duty-cycle of the inverted QPM domains [17]—both approaches have their shortcomings. Because there is a significant difference between the magnitude of the fundamental and first harmonic of a square wave function, the higher order QPM scheme will result in large discrete steps when attempting to approximate larger relative nonlinearities. The drawback of duty-cycle modulation is that the shortest achievable inverted domain width limits the duty-cycle modulation; consequently, relatively small nonlinearities become challenging to produce. Another approach to achieve the Gaussian nonlinearity profile was achieved by using a fixed domain width and an optimization algorithm that attempts to find where the domains should be inverted along the crystal length [18]. The idea of using a fixed domain width is attractive—especially for ease of fabrication—however, the previously proposed algorithm to determine the poling function is stochastic, relies on heuristically determined cost function parameters, and does not necessarily converge to the optimal solution.

In this paper, a new method is presented to determine the poling pattern to achieve a desired nonlinearity profile. The proposed algorithm monitors the generated field amplitude along the crystal, and while iterating along the crystal, selects the appropriate next poling block to direct the generated field amplitude in the direction required to track a predefined target field amplitude. The proposed method of designing arbitrary nonlinearity profiles along the crystal is not only intuitive, it is also easily implementable, predictable and avoids the manufacturing difficulties inherent to schemes using variable domain widths; for example, Dixon et al. [17] used poled sections as small as 4.7 µm and as large as 41.3 µm, which can be challenging to achieve as the small domains may collapse and the large domains may grow too quickly and over-shoot their target size—these fabrication difficulties also lead to a limit in to the upper and lower bounds on the minimum and maximum achievable duty-cycles. Using the proposed method, we design a device in KTP with a Gaussian nonlinearity profile and confirm its accuracy using a nonlinear beam propagation method (BPM) simulation. Finally, we show that the designed device can produce heralded signal photons of spectral purity ~0.996.

2. Analysis

The background theory is outlined before the poling design algorithm is presented. The poling design algorithm is applied to an easily visualizable poling design, and then to a more realizable design in KTP.

2.1. Background

Although the end goal is improvement of a quantum nonlinear optical process, it has recently been shown that appropriate classical nonlinear optical processes can provide remarkable insight into their quantum counterparts [19, 20]. The phase matching function for the type-II SPDC process of interest is

In the special case where we are interested in the spectrum at the end of the crystal (x = L), one finds that

2.1.1. General solution

The goal is to determine how g(x) should ideally vary to obtain the desired spectral response at the output of the crystal, and then to determine a realistic poling function that approximates g(x). Unfortunately, in reality, the value of g(x) at some position along the crystal cannot be chosen to be any value from −1 to 1; rather, only a value of exactly −1 or 1 can be implemented. We shall denote the physically implementable representation of g(x), p(x) ∈ {−1, 1}, where p(x) is an unwindowed poling function. Because p(x) can only take on the aforementioned values, it cannot exactly replicate any desired g(x).

As a compromise, one may seek to approximate g(x) by determining an adequate p(x). The high-level description of the scheme that we employ to obtain p(x) for a Gaussian spectral response is presented in Fig. 1. One starts with the desired spectral response, a), and then takes its inverse Fourier transform (IFT) to determine the ideal nonlinearity profile, g(x), required to achieve this response. At this point, one could directly proceed to d) and determine a poling function, p(x), that approximates b) using, for example, the aforementioned higher order phase matching [16] or duty-cycle modulation [17], or perhaps by even using chirping or a combination thereof; however, we wish to determine a p(x) that adequately approximates b) consisting only of fixed width domain inversions—it is challenging to do this analytically or directly apply an algorithm to b) to obtain a fixed-width poling function. We show that the poling function can be determined from the field amplitude; thus, (2) is used to calculate the field amplitude shown in c). From the field amplitude in c), an algorithm (presented in subsection 2.2) will be applied to yield d), the fixed width poling function that approximates g(x).

 

Fig. 1 Illustration of the strategy to determine an appropriate poling function that well approximates a desired spectral response, in this case, a Gaussian spectral response. a) The FT of g(x) in Δk. b) The ideal, unwindowed nonlinearity profile. c) The field amplitude that the ideal nonlinearity profile of b) would generate. d) The unknown poling function to approximate c) and, consequently, b) and a). The arrows between the panes indicate the function required to obtain the result shown in one pane, from that of the previous pane. To obtain b) from a), one needs to apply an IFT. To obtain c) from b), one needs to integrate. To obtain d) from c), one needs to apply our algorithm outlined in subsection 2.2.

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2.1.2. Gaussian modulation

In order to achieve a gradual nonlinearity profile along the crystal, the sidelobes introduced by the sinc function in (3) need to be suppressed; we therefore choose ℱ_Δk(g (x)) to be a Gaussian [15, 21] centered at 2π/Λ.

We can now apply the strategy outlined in Fig. 1. Given the desired spectral response, we find g(x), shown in Fig. 1b), by taking the IFT of the target spectral response

2.2. Algorithm

We shall now present our algorithm that bridges the gap between panes c) and d) of Fig. 1; thus, systematically determining an appropriate fixed width poling function given a known, desired field amplitude. We start by defining the normalized field amplitude, u_target(x), along the crystal for the case of Gaussian modulation introduced in subsubsection 2.1.2

We now describe the necessary dimensions of the crystal: each inverted domain is to be of equal width, so the crystal is discretized to obtain two domains per QPM period. For a crystal of length L, with a QPM period Λ = 2π/Δk, there will be N = 2L/Λ domains of width w = Λ/2. The crystal can have three possible poling configurations: a), b) and c) which are defined in Fig. 2. The boxes represent domains of width w, the arrows indicate the sign of the crystal domain second-order nonlinearity (d_ij), the dark curves show the field amplitude that the corresponding domain block will induce, and the grey horizontal lines show the average generated normalized field amplitude over a single domain, ū(x). We seek to piece these three poling blocks together in such a fashion to achieve ū(x) ≈ u_target(x).

 

Fig. 2 An arbitrary poling configuration is shown and the three poling blocks are labelled: a), b) and c). The oscillatory waveform is the field amplitude and the horizontal lines through the oscillatory waveform indicates the average value of the oscillatory waveform over the domain width, w. Poling block a) does not increase the average field amplitude, poling block b) increases the average level of the field amplitude, and c) is the same as b) except that it decreases the average field amplitude.

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The poling configurations have different effects on ū(x): they can increase or decrease ū(x) by some quantized amount, or they can leave ū(x) unchanged. Figure 2 pictorially outlines the various possibilities. If one wishes for the average level of the generated field amplitude to remain unchanged between for two domains, one simply does not perform QPM (poling block a) shows this); whereas, if one wishes to increase ū(x) by Δū = 2w/π, one inverts the second of the two poling sections (poling block b) shows this) and if one wishes to decrease ū(x) by Δū, one inverts the first of the two poling sections (poling block c) shows this). If the poling block b) or c) is used at x, ū(x) will change by ±Δū at x + w and by ±2Δū at x + 2w. Provided that the gradient of u_target does not change between positive and negative over a length of 4w, poling blocks b) and c) will never be adjacent to one another—all inverted domains will be of fixed width. The maximum slope that u_target can have and be adequately tracked by ū(x) is

The poling pattern, p(x), is generated using a simple algorithm that traverses the length of the crystal once, with a step-size of 2w. The algorithm determines which poling block from Fig. 2 should be inserted to track u_target. The poling block to be inserted at x is determined from the error between u_target(x + 2w) and ū(x + 2w)

2.3. Application to Gaussian spectral engineering

In this section, the poling algorithm is applied to achieve a Gaussian nonlinearity QPM profile. The algorithm is applied to a short crystal to visually show how the algorithm works.

For illustrative purposes, we have conducted our design for a crystal 100w long. The results are shown in Fig. 3 where u_target and the field amplitude generated from the poling, u_poling, is shown in a); and the calculated poling profile is shown in b). One can see that the field amplitude generated from the poling tracks the target field amplitude indicating that the desired g(x) has been well approximated. The small oscillations that may be observed in u_poling are to be expected as they are naturally present in QPM (caused by the higher order phase matching spectral components intrinsic to square-wave QPM profiles).

 

Fig. 3 a): The target field amplitude, u_target, and the true, generated field amplitude that the designed poling would generate, u_poling. The width used for the Gaussian profile is σ = L/4; one can see that the efficiency of the source drops by 40% compared to ordinary QPM at this given σ. b): The calculated normalized poling function consisting of inverted domains of a fixed width.

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2.4. Full design

In the previous section, the field amplitude was not modelled; it was inferred from the poling function. This section employs the poling algorithm to a KTP crystal and then verifies that the poling produces an error function shaped field amplitude by performing a nonlinear BPM simulation of the design. Unlike coupled amplitude theory based analyses, which are to an extent approximate in that they neglect radiation modes, the in-house developed nonlinear BPM based on an explicit finite difference approximation [22] models the lateral field amplitudes of the pump, signal and idler fields independently.

We design the poling for a type-II KTP crystal 12 mm long, with a straight waveguide 2 µm wide (assuming an index contrast of 0.01), and a target nonlinearity function given by (4) using σ = L/4 = 3 mm. We excite the eigenmode of the waveguide structure with a 775 nm pump and investigate its behavior, along with the 1550 nm field amplitude’s behavior. A waveguide of width 2 µm was used because only the fundamental mode of 1550 nm and 775 nm fields are supported at such a width, simplifying the nonlinear BPM simulation results; however, there should be no problem using a width for which the 775 nm field is multimode.

The results of the nonlinear BPM for the aforementioned structure are shown in Fig. 4; they were generated using a strong pump and a very weak signal and idler field and using Sellmeier equations for KTP [23]. One can see from Fig. 4 that the generated 1550 nm field amplitude increases in an error function manner (as expected from (5))—initially, there is little conversion, towards the center of the device there is significant conversion and at the end of the device there is, again, little conversion. The error function profile of the field amplitude implies that g(x) is Gaussian in shape.

 

Fig. 4 a): The field amplitude of the y-polarized pump. A nonlinear BPM simulation of a waveguide in KTP 2 µm wide centered at y = 0. All simulations are normalized to 1. b): The y-polarized signal field amplitude. c): The z-polarized idler field amplitude. d): A 2-D slice of a) at y = 0 showing the depletion in magnitude of the pump field amplitude. e): A 2-D slice of b) at y = 0 showing that the signal field amplitude increases as an error function. f): A 2-D slice of c) at y = 0 showing that the idler field amplitude increases as an error function.

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The fuzziness that can can be seen in Figs. 4(e) and 4(f) and that increases with x is caused by reflections from the simulation walls at ±40 µm. When the pump field generates the signal and idler fields, there is a slight mode mismatch between it and the signal and idler fields causing some coupling to radiation modes which then reflect off the simulator’s boundaries and interfere with the primary guided mode.

3. Quantum analysis

In this section, we present a quantum analysis of the generated poling profile showing that tailoring the poling using the previously described algorithm can be used to obtain high purity heralded photons in KTP.

The photon state generated from an SPDC is [24]

Around 2π/Λ (far from the harmonics of the true square wave poling pattern), the phase matching spectrum, ϕ(Δk), is related to the nonlinearity profile along the crystal, g(x), by a Fourier transform [17]; ϕ(Δk) is directly proportional to A_s,_i(L, Δk) from subsection 2.1

The joint spectral amplitude (JSA) is found by calculating α(ω₁ + ω₂)ϕ(Δk). It is common to visually assess the purity of a photon pair through the JSA of the pair [13]. The JSA visually indicates the purity in ω_s and ω_i of the photon state by its shape; the greater the purity of the photon pair, the more the JSA resembles a 2-D Gaussian in shape [14,25].

For ordinary QPM, ϕ(Δk) ∝ sinc((Δk −k_QPM) L/2) resulting in a JSA with spectral correlations between the generated photons. To increase the purity above that of ordinary QPM, the spectral correlation between the signal and idler should be removed by removing the correlation visible in the ordinary QPM JSA—it is apparent from the JSA that the correlation is the result of the sinc function’s sidelobes [13].

As can be seen from (10), engineering the phase matching spectrum allows one to choose a function that softens the nonlinearity transition at the ends of the crystal, consequently suppressing the sinc function’s sidelobes. When we use a Gaussian nonlinearity profile, we see that the phase matching function becomes less sinc-like and more Gaussian-like

Three JSAs were simulated to show the affect of our proposed poling algorithm [13]. All simulations are based on the Sellmeier equations from [23], model the effective nonlinearity of the QPM and utilise a singular value decomposition (SVD) to calculate the purity. Figure 5a) shows the JSA of an unmodulated QPM; the sidelobes are clearly visible. The purity is calculated from the JSA as ~0.833. Dixon et al.’s [17] duty-cycle modulated scheme is shown in Fig. 5b), where it can be seen that the sidelobes are suppressed, increasing the purity to 0.971. These results are in accordance with Dixon et al.’s calculations, offering validation of our simulation approach.

 

Fig. 5 The simulated JSAs for a KTP crystal of length 12 mm pumped with pulses centered about a wavelength of 775 nm. a): The simulated JSA for ordinary QPM. The purity [3] was found to be ~0.833 assuming a pump pulse with a FWHM of 1.0 ps. b) The simulated JSA of Dixon et al’s design using a 1.4 nm = 0.63 ps pulse. We simulated the purity of the design to be ~0.971. c): The simulated JSA using the poling algorithm and a pump pulse with a FWHM of 0.80 ps. The purity was found to be ~0.996.

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Figure 5c) shows the proposed poling scheme; it can be seen that the sidelobes are highly suppressed, leading to a purity of ~0.996. This improved purity can be attributed to our proposed poling algorithm enabling small and large relative non-linearities to be achieved, allowing tight Gaussian relative non-linearity profiles to be implemented that well-suppress the sinc sidelobes. The proposed algorithm also overcomes the limitation of Dixon’s scheme where minimum and maximum duty-cycles of 20% and 80% restrict the range of relative non-linearities that can be achieved.

4. Conclusion

We have proposed a deterministic, simple and intuitive method for engineering the QPM nonlinearity profile in a way that avoids small domains. The suggested scheme was used to design poling with a Gaussian nonlinearity profile over a short length for illustrative purposes, and then the scheme was applied to a realistic design in KTP at the group velocity matching conditions verified using a nonlinear BPM simulation. The BPM showed an error function field amplitude profile indicating that the Gaussian nonlinearity had been attained as expected. Lastly, JSA simulations were performed and the JSAs based on our poling method and ordinary QPM were compared to show the expected increase in heralded photon purity of ~0.996. It is expected that there be applications for the proposed poling scheme for filtering in quantum optics (such as scatter-shot boson sampling [26]) and classical optics fields (such as in high speed telecommunication systems [27]).