## Abstract

We introduce a method for shaping the spectral response of nonlinear light sources by tailoring the quasi-phase matching. Our algorithm relies on engineering the poling to accurately trace a generated target signal field amplitude to determine the desired nonlinearity profile. The proposed poling algorithm results in a poling pattern that is more robust to manufacture, as all domain inversions are of equal width. The poling pattern is verified using a nonlinear beam propagation method simulation. This approach is applied to achieve Gaussian-shaped phase matching along a potassium titanyl phosphate (KTP) crystal in order to generate pure heralded single photons of spectral purity ~0.996—this is highly desirable for heralded single photon quantum optics.

© 2016 Optical Society of America

## 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

*L*is the length of the nonlinear region,

*ω*,

_{s}*are the signal and idler frequencies,*

_{i}*g*(

*x*) is the normalized nonlinearity profile, and the wavenumber mismatch is Δ

*k*≡ Δ

*k*(

*ω*) =

_{s}, ω_{i}*k*(

_{p}*ω*+

_{s}*ω*) −

_{i}*k*(

_{s}*ω*) −

_{s}*k*(

_{i}*ω*). One can gain insight into the strength of potential sidelobes in the phase matching function by considering instead the evolution of, say, the signal field amplitude with a very weak (essentially noise-seeded) fixed idler frequency in the presence of an undepleted pump

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

*x*) ≡ sin(

*x*)

*/x*and * is the convolution operator, and the box function (windowing function), Π

*(*

_{L}*x*), is defined in App. A. The box function windows the relative nonlinearity along the crystal, and is responsible for defining the start and end of the crystal, abruptly nulling the relatively nonlinearity at either end of the crystal. The unwindowed normalized effective nonlinearity profile refers to a nonlinearity profile of an infinitely long crystal (one with no beginning or end) that, therefore, has no abrupt nulling. From (3), we see that the spectral response at the output of the crystal is directly proportional to the Fourier Transform (FT) in Δ

*k*of the unwindowed nonlinearity profile in

*x*,

*g*(

*x*); this means that one can design a crystal with a desired spectral response by choosing an appropriate

*g*(

*x*). Equation (3) also indicates that there is an unavoidable sinc component to the spectrum of the crystal that is proportional to the length of the crystal.

### 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*).

### 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_{Δ}* _{k}*(

*g*(

*x*)) to be a Gaussian [15, 21] centered at 2

*π/*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

*σ*is the width of the Gaussian function. There is a trade off between output signal and idler power and sidelobe suppression; the smaller one makes the width (to increase the sidelobe filtering) the more the output signal and idler power drops. To adequately suppress the sidelobes of the sinc function,

*g*(0) ≈ 0 and

*g*(

*L*) ≈ 0, these conditions can be practically imposed by setting, as an example,

*σ*≈

*L/*4. For the rest of this paper,

*σ*≈

*L/*4; thus achieving a high heralded photon purity at the expense of generation efficiency—different widths result in different trade-offs. We now make use of (2) to calculate the field amplitude along the crystal for the unwindowed nonlinearity profile in (4) (shown in c) of Fig. 1)

*g*(

*x*) that is phase matched. This result tells us that if a Gaussian spectral response is desired, the field amplitude must have an error function profile along the crystal.

#### 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 = 2*π/*Δ*k*, there will be *N* = 2*L/*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*).

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 Δ*ū* = 2*w/π*, one inverts the second of the two poling sections (poling block b) shows this) and if one wishes to decrease *ū*(*x*) by Δ*ū*, 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 ±Δ*ū* at *x* + *w* and by ±2Δ*ū* at *x* + 2*w*. Provided that the gradient of *u*_{target} does not change between positive and negative over a length of 4*w*, 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 2*w*. 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* + 2*w*) and *ū*(*x* + 2*w*)

*ū*; the following list presents the three possible outcomes:

- if −Δ
*ū*≤*e*(*x*+ 2*w*) ≤ Δ*ū*, poling block a) is placed at*x*; - if Δ
*ū*<*e*(*x*+ 2*w*), poling block b) is placed at*x*; and - if
*e*(*x*+ 2*w*) < −Δ*ū*, poling block c) is placed at*x*.

#### 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 100*w* 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).

#### 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.

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]

*ω*

_{1}⟩

*(|*

_{y}*w*

_{2}⟩

*) is a single-photon states with frequencies*

_{z}*ω*

_{1}(

*ω*

_{2}) polarized in

*y*(

*z*),

*α*(

*ω*+

_{y}*ω*) describes the pump spectrum, and

_{z}*ϕ*(Δ

*k*) describes the phase matching spectrum of the crystal.

Around 2*π/*Λ (far from the harmonics of the true square wave poling pattern), the phase matching spectrum, *ϕ*(Δ*k*), is related to the nonlinearity profile along the crystal, *g*(*x*), by a Fourier transform [17]; *ϕ*(Δ*k*) is directly proportional to *A _{s}*,

*(*

_{i}*L*, Δ

*k*) from subsection 2.1

*ϕ*(Δ

*k*) can be attributed to the finite length of the crystal effectively acting as a window—

*χ*

^{(2)}vanishes outside the crystal, and abruptly changes when one transitions between outside and inside the crystal (or vice-versa).

The joint spectral amplitude (JSA) is found by calculating *α*(*ω*_{1} + *ω*_{2})*ϕ*(Δ*k*). 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*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].

_{i}For ordinary QPM, *ϕ*(Δ*k*) ∝ sinc((Δ*k* −*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

*ϕ*(Δ

*k*) with suppressed sinc sidelobes and hence a JSA indicating higher purity.

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.

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]).

## Appendix A: Background

The standard equations describing the amplitude of two modes, a signal and idler mode, in a difference frequency generation (DFG) process are

*k*≡ Δ

*k*(

*λ*) =

_{s}, λ_{i}*k*−

_{p}*k*−

_{s}*k*= 2

_{i}*π*(

*n*(

_{p}*λ*)

_{p}*/λ*−

_{p}*n*(

_{s}*λ*)

_{s}*/λ*−

_{s}*n*(

_{i}*λ*)

_{i}*/λ*) is the wavenumber mismatch,

_{i}*d*(

*x*) is the value of the second order nonlinearity along the crystal and can either be ±

*d*where

_{ij}*d*is the

_{ij}*χ*

^{(2)}tensor component in use, and the pump is assumed to not deplete [28].

The general case of down-conversion is described by (12) and (13); the general solution to these leads to exponential increase in the signal and idler field amplitudes. In practice, even if there is no external input signal or idler field, there is still signal and idler noise power,
${P}_{{N}_{s,i}}$, present in the crystal that is constant along the crystal. The field amplitude that varies in *z* represents the generated stimulated field amplitude and will be denoted *A _{s}*,

*(*

_{i}*x*). The total signal and idler field amplitude is a combination of the noise and stimulated field such that ${{A}^{\prime}}_{s,i}(x,\mathrm{\Delta}k)=\sqrt{{P}_{{N}_{s,i}}}+{A}_{s,i}(x,\mathrm{\Delta}k)$. Because we are operating under low pump power conditions, the noise is much greater than the generated field along the crystal such that $\sqrt{{P}_{{N}_{s,i}}}\gg {A}_{s,i}(x)$; thus, one can simplify (12) and (13) to

*x*-invariant constant,

*g*(

*x*)Π

*(*

_{L}*x*−

*L/*2) =

*d*(

*x*)

*/d*consists of a box function, Π

_{ij}*(*

_{L}*x*−

*L/*2), denoting the start and end of the crystal (Π

*(*

_{L}*x*−

*L/*2) is 1 from 0 →

*L*and 0 elsewhere), and

*g*(

*x*) represents an unwindowed normalized second order nonlinearity function that is not restricted by the length of the crystal and that one has much freedom in choosing. Any down-converted field generated under the low-pump power constraint is field that is generated spontaneously, and not from stimulated emission.

One can determine the amplitude of the signal and the spectrum of the signal (in Δ*k*) at any point along the crystal, *x*, by integrating (14) and (15) up to *x*

*s*↔

*i*.

## Funding

Australian Research Council (ARC)

## References and links

**1. **C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, “Mixed-state entanglement and quantum error correction,” Phys. Rev. A **54**, 3824–3851 (1996). [CrossRef] [PubMed]

**2. **D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A **64**, 052312 (2001). [CrossRef]

**3. **M. Nielsen and I. Chuang, Quantum computation and quantum information (2000).

**4. **L. Mandel and E. Wolf, *Optical coherence and quantum optics* (Cambridge university, 1995). [CrossRef]

**5. **M. Fox, *Quantum Optics: An Introduction: An Introduction*, vol. 6 (Oxford university, 2006).

**6. **C. K. Hong and L. Mandel, “Theory of parametric frequency down conversion of light,” Phys. Rev. A **31**, 2409–2418 (1985). [CrossRef]

**7. **S. Du, “Quantum-state purity of heralded single photons produced from frequency-anticorrelated biphotons,” Phys. Rev. A **92**, 043836 (2015). [CrossRef]

**8. **A. M. BraÅĎczyk, T. C. Ralph, W. Helwig, and C. Silberhorn, “Optimized generation of heralded Fock states using parametric down-conversion,” New J. Phys. **12**, 063001 (2010). [CrossRef]

**9. **C. I. Osorio, N. Sangouard, and R. T. Thew, “On the purity and indistinguishability of down-converted photons,” J. Phys. B **46**, 055501 (2013). [CrossRef]

**10. **G. Harder, V. Ansari, B. Brecht, T. Dirmeier, C. Marquardt, and C. Silberhorn, “An optimized photon pair source for quantum circuits,” Opt. Express **21**, 13975–13985 (2013). [CrossRef] [PubMed]

**11. **W. P. Grice, A. B. U’Ren, and I. A. Walmsley, “Eliminating frequency and space-time correlations in multiphoton states,” Phys. Rev. A **64**, 063815 (2001). [CrossRef]

**12. **F. KÃűnig and F. N. C. Wong, “Extended phase matching of second-harmonic generation in periodically poled KTiOPO_{4} with zero group-velocity mismatch,” Appl. Phys. Lett. **84**, 1644 (2004). [CrossRef]

**13. **A. B. U’Ren, C. Silberhorn, R. Erdmann, K. Banaszek, W. P. Grice, I. A. Walmsley, and M. G. Raymer, “Generation of pure-state single-photon wavepackets by conditional preparation based on spontaneous parametric downconversion,” arXiv preprint quant-ph/0611019 (2006).

**14. **L. Zhang, C. SÃűller, O. Cohen, B. J. Smith, and I. A. Walmsley, “Heralded generation of single photons in pure quantum states,” J. Mod. Opt. **59**, 1525–1537 (2012). [CrossRef]

**15. **R. B. Jin, R. Shimizu, K. Wakui, H. Benichi, and M. Sasaki, “Widely tunable single photon source with high purity at telecom wavelength,” Opt. Express **21**, 10659–10666 (2013). [CrossRef] [PubMed]

**16. **A. M. BraÅĎczyk, A. Fedrizzi, T. M. Stace, T. C. Ralph, and A. G. White, “Engineered optical nonlinearity for quantum light sources,” Opt. Express **19**, 55–65 (2011). [CrossRef]

**17. **P. B. Dixon, J. H. Shapiro, and F. N. C. Wong, “Spectral engineering by Gaussian phase-matching for quantum photonics,” Opt. Express **21**, 5879–5890 (2013). [CrossRef] [PubMed]

**18. **A. Dosseva, L. Cincio, and A. M. Brańczyk, “Shaping the joint spectrum of down-converted photons through optimized custom poling,” Phys. Rev. A **93**, 013801 (2016). [CrossRef]

**19. **L. G. Helt, M. Liscidini, and J. E. Sipe, “How does it scale? Comparing quantum and classical nonlinear optical processes in integrated devices,” J. Opt. Soc. Am B **29**, 2199–2212 (2012). [CrossRef]

**20. **M. Liscidini and J. E. Sipe, “Stimulated emission tomography,” Phys. Rev. Lett. **111**, 193602 (2013). [CrossRef] [PubMed]

**21. **K. N. Cassemiro, K. Laiho, and C. Silberhorn, “Accessing the purity of a single photon by the width of the Hong-Ou-Mandel interference,” New J. Phys. **12**, 113052 (2010). [CrossRef]

**22. **H. M. Masoudi and J. M. Arnold, “Modeling second-order nonlinear effects in optical waveguides using a parallel-processing beam propagation method,” IEEE J. Quant. Electron. **31**, 2107–2113 (1995). [CrossRef]

**23. **K. Kato and E. Takaoka, “Sellmeier and thermo-optic dispersion formulas for KTP,” Appl. Opt. **41**, 5040–5044 (2002). [CrossRef] [PubMed]

**24. **M. H. Rubin, D. N. Klyshko, Y. H. Shih, and A. V. Sergienko, “Theory of two-photon entanglement in type-ii optical parametric down-conversion,” Phys. Rev. A **50**, 5122–5133 (1994). [CrossRef] [PubMed]

**25. **G. Harder, V. Ansari, B. Brecht, T. Dirmeier, C. Marquardt, and C. Silberhorn, “An optimized photon pair source for quantum circuits,” Opt. Express **21**, 13975–13985 (2013). [CrossRef] [PubMed]

**26. **M. Bentivegna, N. Spagnolo, C. Vitelli, F. Flamini, N. Viggianiello, L. Latmiral, P. Mataloni, D. J. Brod, E. F. Galvão, A. Crespi, R. Ramponi, R. Osellame, and F. Sciarrino, “Experimental scattershot boson sampling,” Sci. Adv.1 (2015). [CrossRef] [PubMed]

**27. **J. Yamawaku, H. Takara, T. Ohara, K. Sato, A. Takada, T. Morioka, O. Tadanaga, H. Miyazawa, and M. Asobe, “Simultaneous 25 GHz-spaced DWDM wavelength conversion of 1.03 Tbit/s (103× 10 Gbit/s) signals in PPLN waveguide,” Electron. Lett. **39**, 1 (2003). [CrossRef]

**28. **R. W. Boyd, *Nonlinear Optics* (Academic Press, Burlington, MA, 2008), 3rd ed.