## Abstract

Randomly poled nonlinear crystals are shown to be able to emit intense ultra-broadband photon-pair fields with properties comparable to those coming from chirped periodically-poled crystals. Their intensities scale linearly with the number of domains. Also photon pairs extending over intervals with durations comparable to one optical cycle can be generated in these crystals.

© 2010 Optical Society of America

## 1. Introduction

Spontaneous parametric down-conversion (SPDC) with its production of photon pairs belongs to the most fascinating nonlinear optical effects. Two photons comprising a photon pair can be entangled in their degrees of freedom as it was first observed by Hong, Ou, and Mandel [1] for temporal correlations. Nonlinear bulk crystals served nearly exclusively as sources of these photon pairs for many years. However, many highly nonlinear crystals could not be used due to impossibility to achieve natural phase-matching conditions. Here, the concept of additional periodic modulation of nonlinear susceptibility as introduced by Armstrong [2] has become fruitful and resulted in the invention of poling of nonlinear crystals [3]. Highly nonlinear materials in which quasi-phase-matching conditions [4] are met can be efficiently used since then. They have allowed the construction of bright and versatile sources of photon pairs.

Moreover, periodical poling has also allowed to tailor the properties of emitted photon pairs using nonlinear domains with variable lengths (chirped periodical poling). Domains of different lengths in an ordered structure allow an efficient nonlinear interaction in an ultra-wide spectral region extending typically over several hundreds of nm [5–8]. It has been shown that such photon pairs can posses quantum temporal correlations at the timescale of fs and so can be extraordinarily useful, e.g., in metrology (quantum optical coherence tomography [9]) or quantum-information processing [10].

Alternatively, domains of different lengths can be ordered randomly. The nonlinear interaction can be efficient even in this case [stochastic quasi-phase-matching] as studies of the process of second-harmonic generation indicate [4, 11–14]. Here, we show that these structures despite their randomness are able to generate spectrally ultra-wide photon pairs at generation rates comparable to chirped periodically-poled crystals (CPPC). This shows that contributions from the coherent summation of photon-pair fields coming from ordered domains of different lengths in CPPCs are considerably lower than generally accepted. When properly phase compensated photon pairs with correlation times at the fs timescale can be emitted from these structures. Moreover, these structures have usually smaller requirements with respect to fields’ polarization properties and orientation of the nonlinear medium [15]. Also fabrication tolerances are less strict in this case [4]. Practically, technology of poling produces structures with relatively large declinations from an ideal geometry which results in reduction of photon-pair generation rates. There is the hope that the fabricated randomly poled structures can even outperform their chirped periodically-poled counterparts. The extension to 2D or 3D geometries offers additional possibilities including naturally poled materials [15].

## 2. Spontaneous parametric down-conversion in poled nonlinear crystals

Quantum state |*ψ*〉 describing a photon pair generated in the process of SPDC in a poled crystal can be written as follows [1]:

*ω*and its idler twin at frequency

_{s}*ω*. It can be obtained in the form:

_{i}*ξ*, frequency ${\omega}_{p}^{0}$ and wave-vector

_{p}*k*. Creation operator ${\widehat{a}}_{a}^{\u2020}({\omega}_{a})$ in Eq. (1) generates a photon with wave-vector

_{p}*k*and frequency

_{a}*ω*(

_{a}*a*=

*s, i*) into the vacuum state |vac〉. Coupling constant

*g*is given as $g({\omega}_{s},{\omega}_{i})={\chi}^{(2)}(0)\sqrt{{\omega}_{s}{\omega}_{i}}/[ic\pi \sqrt{{n}_{s}({\omega}_{s}){n}_{i}({\omega}_{i})}]$ where

*χ*

^{(2)}denotes second-order susceptibility,

*n*(

_{a}*ω*) means index of refraction, and

_{a}*c*is speed of light in vacuum. Symbol Δ

*k*stands for nonlinear phase mismatch (Δ

*k*=

*k*–

_{p}*k*–

_{s}*k*). The stochastic phase-matching function

_{i}*F*introduced in Eq. (2) takes the following form:

*N*gives the number of domains and

_{L}*n*-th domain extends from

*z*=

*z*

_{n}_{–1}to

*z*=

*z*. Positions

_{n}*z*of domain boundaries are random and can be described as

_{n}*z*=

_{n}*z*

_{n}_{–1}+

*l*

_{0}+

*δl*(

_{n}*n*= 1,...,

*N*,

_{L}*z*

_{0}= 0) using stochastic declinations

*δl*. The basic domain length

_{n}*l*

_{0}is determined such that quasi-phase-matching is reached, i.e.

*l*

_{0}=

*π*/Δ

*k*

_{0}, $\Delta {k}_{0}\equiv \Delta k\left({\omega}_{s}^{0},{\omega}_{i}^{0}\right)$ and ${\omega}_{a}^{0}$ stands for the central frequency of field

*a*. Independent random declinations

*δl*are assumed to obey the joint Gaussian probability distribution

_{n}*P*:

**B**is diagonal and its nonzero elements are equal to 1/

*σ*

^{2}. Stochastic vector

*δ*

**L**is composed of declinations

*δl*; symbol

_{n}*stands for transposition.*

^{T}Integration of the expression in Eq. (3) leaves us with the following simple formula (valid for *N _{L}* ≫ 1):

*χ*

^{(2)}with doubled amplitudes and domains with negative susceptibility

*χ*

^{(2)}play only the role of a ’linear’ filler. This elucidates why two neighbouring domains form basic elementary units for the interpretation of properties of photon-pairs [7].

The emitted photon pairs can be characterized by mean spectral density *n*(*ω _{s}*,

*ω*) of the number of photon pairs. The density

_{i}*n*is defined along the formula $n({\omega}_{s},{\omega}_{i})={\u3008\u3008\psi |{\widehat{a}}_{s}^{\u2020}({\omega}_{s}){\widehat{a}}_{s}({\omega}_{s}){\widehat{a}}_{i}^{\u2020}({\omega}_{i}){\widehat{a}}_{i}({\omega}_{i})|\psi \u3009\u3009}_{\text{av}}$ where the symbol 〈〉

_{av}means stochastic averaging over an ensemble of random realizations of the crystal. In practice mean values can correspond to averaging over states of photon pairs emitted at different positions in the transverse plane of a naturally-poled material [15]. Or they can just serve as an indicator of expected values for individual realizations. The spectral density

*n*in quantum state |

*ψ*〉 reads:

In the considered random structures, the averaged squared modulus of phase-matching function *F*, defined as ∫*dδ***L***P*(*δ***L**)|*F*(Δ*k*)|^{2}, is obtained as follows:

*δk*(

*ω*,

_{s}*ω*) = Δ

_{i}*k*(

*ω*,

_{s}*ω*) – Δ

_{i}*k*

_{0}. Symbol c.c. replaces the complex-conjugated term and

*H*(

*δk*) = exp[

*iδkl*

_{0}] exp[–(

*σ*Δ

*k*)

^{2}/4]. A detailed analysis of the formula in Eq. (7) shows that 〈|

*F*(Δ

*k*)|

^{2}〉

_{av}is peaked around

*δk*= 0. The larger the deviation

*σ*the broader the peak.

We consider CPPC for comparison. In this case, the ordered positions of boundaries are given as
${z}_{n}=n{l}_{0}+{\zeta}^{\prime}{(n-{N}_{L}/2)}^{2}{l}_{0}^{2}$, *ζ*′ = *ζ*/Δ*k*_{0}, and *ζ* stands for a chirping parameter. The phase-matching function *F*^{chirp} can then be derived in the form [6]:

*ζ*the broader the phase-matching function

*F*

^{chirp}(Δ

*k*).

## 3. Photon-pair generation rates and intensity spectra

In our investigation, we use spectrally degenerate collinear down-conversion from a poled LiNbO_{3} crystal pumped at the wavelength
${\lambda}_{p}^{0}=775\text{nm}$. The signal and idler photons occur at the fiber-optics communication wavelength *λ _{s}* =

*λ*= 1.55

_{i}*μ*m. The crystal optical axis is perpendicular to the fields’ propagation direction and is parallel to the vertical direction. All fields are vertically polarized and so the largest nonlinear element ${\chi}_{33}^{(2)}$ is exploited. The basic domain length

*l*

_{0}equals to 9.515

*μ*m. A structure composed of

*N*= 2000 layers is roughly 19 mm long and typically delivers 2 × 10

_{L}^{7}photon pairs per second per 100 mW of pumping for ordered positions of boundaries.

The most striking feature of random structures is that the photon-pair generation rate *N* increases linearly with the number *N _{L}* of domains (see Fig. 1). On the other hand, standard deviation

*σ*plays the central role in the determination of spectral widths Δ

*S*and Δ

_{s}*S*of the signal and idler fields. The larger the deviation

_{i}*σ*the broader the signal- and idler-field spectra

*S*and

_{s}*S*(see Fig. 2a and Fig. 2b for

_{i}*σ*–

*ζ*transformation). This can be understood from the behavior of spatial spectrum of the

*χ*

^{(2)}(

*z*) modulation participating in phase matching conditions: the larger the deviation

*σ*the broader the spatial spectrum of

*χ*

^{(2)}(

*z*). However, the spectral broadening is at the expense of photon-pair generation rates

*N*(see Fig. 2c).

Similar behavior as observed in random structures has been found in CPPCs [6, 7] considering the chirping parameter *ζ* instead of the deviation *σ*. Our investigations have revealed that this similarity is both qualitative and quantitative (see Fig. 2). For any value of the chirping parameter *ζ* there exists a value of the standard deviation *σ* such that the spectral widths Δ*S _{s}* and Δ

*S*equal. Moreover, also the photon-pair generation rates

_{i}*N*are comparable. This behavior is illustrated in Fig. 2 for structures with

*N*= 2000 domains. The signal-field spectra

_{L}*S*in CPPCs are extraordinarily wide (larger that 1

_{s}*μ*m) for sufficiently large values of parameter

*ζ*(see Fig. 2a). The signal-field spectra

*S*coming from random structures can have the same widths assuming sufficiently strong randomness: the transformation curve between the standard deviation

_{s}*σ*and chirping parameter

*ζ*conditioned by equal spectral widths Δ

*S*is plotted in Fig. 2b. The photon-pair generation rates

_{s}*N*for random and chirped crystals are compared in Fig. 2c. Photon-pair generation rates

*N*in random crystals reach at least 80 % of those of CPPCs.

Comparison of the signal-field spectra *S _{s}* for one typical realization of the random structure and a CPPC (see Fig. 3) reveals that spectra of random structures cover a larger range of frequencies and are composed of many local peaks. Nevertheless, the averaged spectrum of the random crystal has the same FWHM as the considered CPPC (see Fig. 3b).

## 4. Temporal correlations

These ultra-wide spectra allow to generate photon pairs with extremely short temporal correlations that can be measured in a Hong-Ou-Mandel interferometer [1]. The coincidence-count rate *R _{n}* in this interferometer as a function of relative time delay

*τ*between two photons forms a typical dip (see Fig. 4) that can be described by the formula:

*R*in Fig. 4a. They mainly differ in amplitudes of oscillations occurring at the shoulders.

_{n}The behavior observed in the interferometer shows the potential to generate photon pairs with wave-packets extending over the period of only several fs provided that a good phase compensation is reached. The temporal profile of a wave-packet is given by the fourth-order stationary correlation function *I*_{Φ}(*τ*) defined as

*T*denotes detection duration. Function

*I*

_{Φ}gives the probability of detecting an idler photon in the instant that precedes the instant of signal-photon detection by

*τ*. In experiment, function

*I*

_{Φ}(

*τ*) is obtained from the measurement of intensity

*I*

^{sum}of the sum-frequency field generated by two photons mutually delayed by

*τ*[

*I*

_{Φ}(

*τ*) ∝

*I*

^{sum}(

*τ*)]. Intensities

*I*

^{sum}created by photon pairs coming from the considered ideally phase-compensated structures are plotted in Fig. 4b and demonstrate the principal ability to detect both photons in a window ≈ 5 fs wide. Restricting ourselves to quadratic phase compensation [6, 16, 17], this window is roughly two times wider. Quadratic compensation is more powerful in CPPCs, however, the difference is not large. This emphasizes the potential of the studied photon pairs as a tool of diagnostics of ultra-fast processes in physics, biology or chemistry.

## 5. Conclusion

Quantitative similarity in properties of photon pairs generated from randomly poled and chirped periodically-poled crystals has been found. Namely ultra-wide signal and idler fields can be emitted from randomly poled crystals. Two photons in a pair can in principle occur together in a temporal window that characterizes one optical cycle. Importantly, photon-pair generation rates are comparable and depend linearly on the number of domains. Contrary to chirped periodically-poled crystals the randomly poled crystals are quite tolerant in fabrication. This gives a great promise for the use of randomly poled crystals as standard bright ultra-wide spectral sources of entangled photon pairs.

## Acknowledgments

Support by projects IAA100100713 of GA AVČR, COST OC 09026, 1M06002 of MŠMT and PrF-2010-009 of Palacky University is acknowledged. The authors thank O. Haderka for discussions.

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