Intensity of parametric fluorescence pumped by ultrashort pulses
We investigate intensity of the parametric down conversion of ultrashort, ultraviolet pulse both in a low- and high-conversion regime. In the first regime, we develop a simple analytical expressions for the photon flux of the fluorescence. Numerical simulations using three-dimensional stochastic Wigner method provide results in the latter regime. We find that the perturbative approximation for the photon flux as a function of the wavelength of the signal and the observation angle, quantitatively describes even the non-perturbative regime. For short pump pulses the intensity of the fluorescence is highest at frequencies where the group velocities of the pump and the down-conversion are equal. For longer pump pulses this requirement is gradually relaxed. Additionally, for small pump beams, the intensity strongly depends on the spatial divergence between interacting fields.
pacs:42.65.Lm, 42.65.Yj, 42.65.Re
Quantum parametric fluorescence is one of the most frequently used sources of nonclassical radiation. In the low gain regime, the parametric process is used as a robust source of single photons and photon pairs. The parametric fluorescence–based sources allowed for a number of fundamental experiments. Examples include Bell inequality breaking Kwiat et al. (1995), quantum cryptography Gisin et al. (2002) and quantum teleportation Bouwmeester et al. (1997). Nowadays the effort focuses on engineering the correlation properties Grice et al. (2001); Giovannetti et al. (2002); U’Ren et al. (2003); Tsang and Psaltis (2005) of such sources. On the other hand, in some experiments intense pump pulses are used, which results in high parametric gain. This regime is characterized by an appreciable amount of multiple pair creation events and most conveniently described by recalling the notion of squeezing Scully and Zubairy (1997). Single or two-mode squeezing described in textbooks can be used to describe realistic multimode parametric amplifier Wasilewski et al. (2006) and it turns out that a laboratory device produces a superposition of squeezed states in modes defined both by the nonlinear medium and the pump pulse. High-gain parametric amplifiers are nowadays also frequently used for amplification of ultrashort pulses Wilhelm et al. (1997). In this application the parametric fluorescence is an important and troublesome source of noise.
In this paper we calculate the average angular and spectral intensity of a spontaneous parametric fluorescence. Since parametric fluorescence has quantum origin, its intensity cannot be calculated using classical nonlinear optics. Below, we adopt both perturbative and stochastic approach for this task. With a few approximations we provide a simple formula for the intensity of the fluorescence in single-pair generation regime. Numerical simulations show, that it qualitatively reproduces main features of the donwnconversion spectrum in the high gain regime as well.
Our results could be used as guidelines for constructing brighter photon-pair sources. Other application would be optimization of high-gain parametric amplifiers for a minimal contribution of parametric fluorescence Tavella et al. (2006).
Let us mention that a process analogous to parametric down-conversion is observed in physics of cold atoms. When a pair Bose-Einstein condensates collide, a halo of spontaneously scattered of atoms is observed Vogels et al. (2002); Perrin et al. (2007). Stochastic methods used for description of this process proved extremely successful Norrie et al. (2005); Deuar and Drummond (2007) and we adopted them to obtain results presented below.
The paper is organized as follows. In Section II we recall some basics of the theory of the parametric down-conversion. In Section III, we calculate the photon flux in the perturbative regime. In Section IV, we compare these results with numerical predictions in the high-gain regime. We conclude in Section V.
Ii Theory of the parametric down-conversion
A process of parametric down-conversion takes place in a crystal with non-zero second order nonlinear susceptibility tensor when it is irradiated with a pump beam. During pass through the nonlinear medium each of the pump pulses gives rise to photon pairs which form the parametric fluorescence. Below, we formulate a theoretical model describing the nonlinear interaction which allows for calculation of the fluorescence intensity.
We assume, that the medium interfaces are parallel to – plane, while fields propagate along the -coordinate, as depicted in Fig. 1. We expand all the fields involved in the interaction using an approach from nonlinear optics. At each an expansion in a basis of plane waves with frequency and wave-vector , is given, while parametrizes the development of the evolution. To shorten the notation, let’s denote the triple parametrizing each plane-wave component of the fields by . For concreteness, we consider Type-I interaction in BBO (-Barium Borate) in which pump propagates as an -ray while the fluorescence as an -ray. Results can however be easily generalized for any nonlinear crystal by substituting its particular dispersion relation in the formulas below.
Let’s consider a regime in which the pump consists of intense pulses described by a classical envelope in the frequency domain, spanned by . The evolution of the pump pulse along is determined by the dispersion relation of the crystal. A nonlinear interaction with the fluorescence field can be neglected, since the intensity of the pumping field is large and its depletion due to parametric down conversion is minimal. This condition is readily satisfied in many experiments. Note that the intensity of the pump is also assumed large enough to neglect any quantum fluctuations and describe it by a classical field.
For a pump propagating as an -ray in a crystal cut at an angle to the -axis the dispersion relation of the crystal reads:
where and are ordinary and extraordinary refractive indices of the crystal. For each plane wave component of pump field with given the above relation fixes the -component of the wavevector at a value . During propagation along each of the plane-wave components will acquire phase equal to . Thus, the equation of evolution of the pump pulse can be written as Band and Trippenbach (1996):
On the contrary, the parametric fluorescence has to be described by a quantum field. We expand this field in a basis of plane waves parametrized by , which corresponds to the following expression for the field operator in the position space:
Here form a complete set of bosonic annihilation operators at each while is the index of refraction of the medium. The evolution of the parametric fluorescence is also largely determined by the crystal dispersion, which determines for each plane wave component with given at a value ,
Additionally, the signal field interacts with the pump pulse via the second order nonlinearity. Classical equations, describing propagation of the fluorescence field through the crystal are well known Band et al. (1994). Quantization of the signal field corresponds to replacement of c-number amplitudes and their complex conjugates of plane-wave modes by annihilation and creation operators and and leads to the Heisenberg equations for their evolution:
Here, is the maximal amplitude of the pump pulse in time-position space , while is a nonlinear length, a constant which determines a characteristic length over which nonlinear interaction produces a few photon pairs, defined as:
For a crystal of a length and given pump pulses at the crystal entrance we can find Green functions and of the Eq. (5) and express the fluorescence field at the crystal exit face in terms of that on the input face :
The above equation gives a complete quantum description of the parametric amplifier. In our case it is highly redundant, since we are interested in the average intensity of the parametric fluorescence as a function of the frequency and direction, both contained in . In the following section, we use Eq. (7) only as a backbone for developing more specialized results.
Iii Perturbative approximation
In this section we discuss the properties of the average intensity of the parametric fluorescence calculated in the first order approximation with respect to the strength of the nonlinearity . This approximation physically corresponds to a situation when the fluorescence field consists of a single or a few pairs of photons emitted into distinct frequency or angular regions. It is expected that as soon as the pump becomes so intense that the photons are emitted in bunches, perturbative expansion fails. However, as we show, some results inferred from the perturbative approach, can be directly applied even in the high-gain regime.
Basic expressions derived in this section are well known and have been introduced in works on spontaneous parametric down-conversion in the context of photon sources Grice et al. (2001). However, the intensity of such sources have, surprisingly, received very little theoretical attention.
In order to calculate which describes the intensity of the parametric fluorescence, we first obtain an approximate Green functions and . This is done by transforming into the interaction picture, which is accomplished by substituting:
into Eq.(5). The resulting equation of evolution of contains only nonlinear term and can be easily solved perturbatively with respect to Wasilewski et al. (2006). The first order solution is next compared with a general form given by Eq. (7). The function in the lowest order describes simple dispersive evolution of the fluorescence filed passing through the crystal:
while the describes the process of pair generation:
where is a wavevector mismatch between plane-wave components of the fluorescence characterized by spatio-temporal frequencies and and coupled plane wave component of the pump characterized by spatio-temporal frequency :
The above result can be used to calculate the intensity of spontaneous parametric fluorescence emitted at a certain direction in a certain frequency, corresponding to a vector :
The above integral can be given a meaningful physical interpretation. Observation of a signal photon testifies generation of the idler photon. In our case the first is detected using narrow-band filters and is nearly plane-wave and monochromatic, characterized by vector . Thus, all the frequency and direction distribution present in the pump is transferred to the idler photon, as depicted in Fig. 2. Each plane-wave component of the idler characterized by can support the production of signal characterized by if the pair in question is phase-matched with a pump component characterized by , i.e. the phase mismatch is small.
The integral given in Eq. (12) cannot be evaluated analytically in case of pulsed pumping, when is nonzero over a finite range of . However, the integrand can be approximated with a Gaussian function. First, let us assume a Gaussian form of the pump amplitude at the crystal entrance face , corresponding to a pulse of duration propagating in a beam of width , centered around frequency :
Concentration of the pump energy around given together with requirement of phase matching reduces the range of spatio-temporal frequencies and over which a significant amount of parametric fluorescence is generated. To analyze this effect, let us consider perfectly phase-matched parametric down-conversion of the central plane-wave component of the pump with . This part of the fluorescence consists of a plane wave component-pairs characterized by and which satisfy the following set of equations:
The first equation corresponds to a requirement that both components of the down-conversion couple to the central component of the pump, while the second equation is the perfect phase matching condition. This is in fact a set of four equations for six components of and which defines a two-dimensional surface. Assuming as before, from the first equation we get . Then the solution of the second equation can be cast into the following form:
This is an axially-symmetric surface in three-dimensional space of , describing plane-wave components of the parametric fluorescence which can be produced from the central component of the pump in a perfectly phase-matched process. In Fig. 3 we plot a section through this surface in the – plane, for parametric down conversion in a BBO crystal pumped at nm and four different cut angles .
As the spatio-temporal frequency of the fluorescence deviates from the perfect phase matching surface its intensity diminishes. In the region of in which the fluorescence intensity is large, the sinc function can be approximated by a Gaussian Dragan (2004). Such an approximation fails only for very thick crystals, in which intricacies of the phase matching may play important role, but the fluorescence emitted in such cases contains very complicated correlations, which render it useless in most of the known applications.
The next step towards Gaussian approximation of the integrand in Eq. (12) is a linearization of the phase mismatch given by Eq. (11) around points on perfect phase-matching surface. Since it is axially symmetric, we choose points in and half-plane. For each fixed observation frequency we expand phase mismatch around and up to linear terms. This gives
where the expansion coefficients equal:
The above coefficients have simple physical meaning. is a difference of inverse group velocities between the pump pulse and the fluorescence component characterized by , while give the amount of spatial displacement per unit length between pump pulses and the fluorescence component characterized by or in or direction. Note, that all the above coefficients depend on the choice of the observation frequency .
Finally, in Eq. (12) we approximate the sinc function by a Gaussian
and obtain the following expression for the fluorescence intensity
This integral can be evaluated analytically. Let us substitute which describes the perfect phase-matching line. Then we obtain an estimate of the intensity of the fluorescence emitted at a frequency at an angle to the axis in plane:
Let us consider the dependence of the intensity of the fluorescence along the perfect phase-matching line on the frequency at which we observe it. Note again, that observation of the fluorescence signal at the frequency testifies generation of the conjugate (idler) photons at the frequency . It turns out, that the propagation properties of the idler photons influence the intensity of the signal. In particular three parameters, which vary with come into play. The first is the ratio of the temporal walk-off between pump pulse and conjugate photons over the crystal length to the pump pulse duration . The other two measure the ratio of the transverse displacement between the pump pulse and conjugate photons and to the pump pulse width .
This can be understood by reconsidering picture drawn in Fig. 2. As we argued before, observation of a nearly plane-wave signal transfers the frequency and wavevector distribution of the pump to the idler photon, which takes a form of a short pulse and propagates according to the crystal dispersion. Usually, it quickly diverges from the pump pulse — the expression in the square brackets in Eq. (20) is large. However, for some frequencies of the signal the idler propagates with the pump for a longer distance which increases the nonlinear interaction and the photon flux .
We also plot the intensity of the fluorescence along the perfect phase-matching line given in Eq. (20) as a function of the wavelength of observation in Fig. 6 for a few typical pump pulse durations and widths.
It is worthwhile to consider the features displayed by this plots. For short pump pulses, when temporal walk-off between pump and conjugate photons is more pronounced, the fluorescence intensity reaches local maximum values at a wavelength where this walk-off vanishes . For longer pulses the temporal walk-off loses its importance, and the intensity of the fluorescence increases with wavelength due to reduced spatial displacement between the pump and conjugate photons, i.e. .
Iv Three dimensional stochastic model
In the previous section we calculated the intensity of the parametric fluorescence in the single pair generation limit. Now we switch to intense-pumping regime, when photons are generated in bunches and perturbative approximation breaks down. Estimation of the fluorescence intensity is of a great importance in novel practical applications of parametric amplifiers. For example, when using this kind of nonlinear interaction to amplify broadband chirped pulses, the parametric fluorescence is a source of large and difficult to avoid noise Tavella et al. (2006).
In this section we describe a numerical method of solving the dynamics of a coupled pump-signal system given by Eqs.(2) and (5) and estimating the intensity of the parametric fluorescence. We use so-called stochastic Wigner method, which can in principle provide exact solutions for linear quantum systems, such as described by Eq. (5). If the Wigner function of the field at the entrance is Gaussian, in particular representing vacuum state, the Wigner function of the final state of the parametric fluorescence will be Gaussian too. It is easily deduced from a general form of the input-output relations given by Eqs.(7).
A crucial observation is that if the Heisenberg equation is linear, the evolution of the Wigner function is the same as a classical Liouville equation for the density probability of finding the system in a state with particular electric field. Thus the Wigner function can be interpreted as a probability density of finding the system with a fluorescence field described by a spectral amplitude .
Therefore, to find the Wigner function of the fluorescence at the crystal exit face we repeat the following procedure. First we draw an input spectral amplitude with probability given be the vacuum Wigner function
where we assumed a finite number of discrete modes indexed by . Next we solve classical equations of propagation of the signal field:
This way we obtain a possible output spectral amplitude . By repeating this procedure many times, we obtain a set of probable output field amplitudes , which approximates Wigner function . This is schematically illustrated in Fig. 7.
This can be used to calculate the mean photon flux at a given direction and frequency. It equals a mean averaged with the Wigner function minus vacuum average. Using the set of output amplitudes this is calculated as:
We performed the procedure outlined above for interaction in 2mm long BBO crystal. The pump intensity was adjusted so that the total number of photons in the fluorescence field is near . This corresponds to the typical conditions found in noncollinear-OPA (NOPA) used for amplification of ultrashort pulses Wilhelm et al. (1997).
For each of the various pump pulse durations, widths and crystal cut angles we performed ten independent simulations of classical fluorescence evolution. The average photon flux was computed according to the formula (23). Additionally, we assumed that is almost independent of the angle around axis and we averaged the results over this angle. This last step made ten independent simulations sufficient to calculate smooth intensities.
Figure 8 shows the results of the simulations. The area of high intensity winds around the perfect phase-matching curve pictured in Fig.3, as we argued before. Moreover, qualitatively, the intensity along this curve changes as predicted by the perturbation theory and plotted in Fig. 6. This can be considered an unexpected success of the perturbation theory and qualitatively explained as follows. The process of parametric amplification with intense pumping leads to intense fluorescence fields, which exhibit mainly classical properties. In particular 1/2 in Eq. (23) can be neglected, which makes the process virtually identical to a classical amplification of a weak, white noise. In the course of interaction the components of the fluorescence become pairwise correlated. To some extend, the noise-like fluorescence can be separated into pairs of peaks which have approximately opposite and and frequencies summing up to the frequency of the pump, . Most of such pairs quickly diverge away from the pump pulse, but some propagate with it enjoying exponential growth. This is very similar to the mechanism behind the intensity of the fluorescence in a single pair generation regime, hence the intensity profiles are very similar.
It is worthwhile to compare the numerical results from Fig. 8 with Figures 4 and 5. We find that for a short pump pulses the temporal walk-off between the pump and the fluorescence must be very small to achieve intense generation. For longer pulses, this requirement is relaxed and our calculations reveal broadband parametric generation.
In this article we adopted perturbative and Wigner stochastic approaches to the task of calculating an intensity of the parametric fluorescence. In a single pair generation limit, some analytical expressions for the down-converted field can be derived. By linearizing the phase-matching function we found simple formulas for the intensity of the fluorescence in this regime. Photon flux of the signal depends on the temporal and spatial walk-off between the idler and the pump. In case of type I interaction in BBO this leads to peaks in the spectrum of the downconversion signal at the frequencies where group velocity of the idler matches up with the pump.
In an intense pumping case we performed a number of numerical simulations and found that the spectrum of the fluorescence exhibits qualitatively the same features as predicted by the perturbative approximation. The contrast between the wavelengths at which the fluorescence is intense and those at which is weak is more apparent due to exponential character of amplification in non-perturbative regime.
The above results could be used as guidelines for constructing more bright photon-pair sources. They could be also used for optimization of parametric amplifiers, in which one seeks minimal contribution from the parametric fluorescence.
We acknowledge insightful discussions with Konrad Banaszek, Marek Trippenbach and Czesław Radzewicz, as well as financial support from Polish Government scientific grant (2007-2009).
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