On the measurement of two-photon single-mode coupling efficiency in parametric downconversion photon sources

S Castelletto¹, I P Degiovanni¹, A Migdall² and M Ware²

¹Istituto Elettrotecnico Nazionale G Ferraris, Strada delle Cacce 91-10135, Torino, Italy
²Optical Technology Division, National Institute of Standards and Technology, Gaithersburg, MD 20899-8441, USA

Email: castelle@ien.it and amigdall@nist.gov

Received 5 February 2004
Published 29 July 2004

Contents

• 1. Introduction
• 2. Definition of PDC field- and intensity-based collection efficiencies
  • 2.1. PDC single-mode field-based collection efficiency
  • 2.2. PDC single-mode intensity-based collection efficiency
• 3. Calculation of the field-based collection efficiency
• 4. Calculation of the single-mode intensity-based collection efficiency
• 5. Discussion of results
• 6. Conclusions
• Acknowledgments
• References

1. Introduction

The advent of photon-based quantum cryptography, communication, and computation schemes [1]-[10]; has increased the need for light sources that produce individual photons [11]. An ideal single-photon source would produce completely characterized single photons on demand. Because all of the currently available sources fall significantly short of this ideal (that is they do not produce photons 100% of the time and/or they do not produce only single photons), much effort has been focused on creating improved approximations of single-photon-on-demand sources (SPOD) [12]-[19]. Some of these schemes [17, 19] rely on optical parametric downconversion (PDC), because it produces photons two at a time, allowing one photon to herald the existence of the other. In a previous work, we proposed one such scheme where a multiplexed PDC array is used to make an improved SPOD source with increased probability of single-photon emission, while suppressing the probability of multi-photon generation [19]. Most PDC-based schemes (including ours) require that the PDC output be collected into a single spatial mode defined by an optical fibre. For these PDC schemes to reliably produce single photons, it is essential that the optical collection system efficiently gathers and detects the herald photon, and with minimal loss, sends its partner to the output of the system. In addition to SPOD applications, it is also important to understand collection efficiency for other applications such as PDC-based metrological applications, which are very sensitive to collection efficiency [20].

Various theoretical models have been developed to predict how the collection efficiency of PDC light in a `two-photon single-mode' can be improved [21]-[24], and in some cases these models have been used to improve coupling efficiency. In particular, it has been shown that the size of the pump beam focus affects the shape of the PDC output [21] and hence the coupling efficiency of the PDC light into a given spatial mode. In addition, a more detailed work [23] recently showed how increasing the crystal length and walk-off reduces the coupling efficiency for a pulsed broadband pump. However, in these works, the attempts to increase the single-mode fibre coupling efficiency were limited to a tightly focused pump beam and a thin crystal, which reduces the overall source brightness. This last restriction is forced on the calculations because the approximations used for the phase-mismatch function in terms of longitudinal wavevector mismatch are not valid in the long-crystal case. Moreover, these works deal only with either type I or II phase-matching conditions exclusively; they do not include the non-collinearity and/or walk-off of the emitted photons and the pump beam. The work done to date does however give some partial guidance for increasing coupling efficiency in certain experimental situations. For instance, the most practical results to date generally show that the collection efficiency for long crystals is optimized for properly matched large pump and collection waists. However the maximum efficiency is limited by walk-off in type II [23] phase-matching and by excessive crystal length in type I phase-matching [24]. Furthermore, the approximations done in the previous theory limited the validity of the results to a narrow range of crystal lengths and collection/pump waists.

A more general approach to the problem is needed to clarify open issues such as those mentioned above (i.e. overcoming the calculational limits and extending the theory), and to determine how we can best test the collection efficiency model. Here, we present one model to describe the coupling of the PDC source into single spatial modes. This method uses a field-based model to describe the coupling of the PDC field with single-mode fibre defined fields, as has been partially presented in other works along with some suggestive supporting data [23, 24]. We also consider an alternate intensity-based model, where an intensity projector operator represents the effect of a spatial filtering, as suggested in [21, 25]. We calculate the collection efficiencies using both methods for both type I and II PDC output into two single-modes defined by optical fibres, accounting for effects due to the crystal length, walk-off of extraordinary fields, non-collinear phase-matching, and pump transverse field distribution. We then analytically evaluate both efficiencies assuming negligible second-order terms in the transverse component of the wavevectors. The intensity-based approach exhibits counter-intuitive results, even in the thin crystal limit and may be more suitable for multi-mode collection. We propose an experimental test to distinguish between the two.

The work is organized as follows, in section 2 we define the field- and intensity-based collection efficiencies for a PDC field. In sections 3 and 4, we explicitly calculate the two efficiencies in terms of the parameters of the physical systems. In section 5, we compare the predictions of the two models.

2. Definition of PDC field- and intensity-based collection efficiencies

To determine the collection efficiency of the parametric downconversion output into two single spatial modes (defined in our setup by single-mode optical fibres and lenses as shown in figure 1), we use two very different approaches. In the first, we calculate the overlap of the PDC field with the field modes selected by the fibres and indirectly evaluate coincidences and singles. In the second, we calculate the PDC wave function over the spatial distribution of intensities as defined by single-mode fibres directly providing coincidences and associated singles counts. The main difference between the two approaches relates to the fibre-mode selection and associated detection process. The first assumes that the fibre plus the detector is a system able to distinguish single-mode fields, and the calculation is therefore performed in terms of fields rather than intensities. The second assumes that the fibres select the mode and the detectors measure the associated intensity.

2.1. PDC single-mode field-based collection efficiency

In the field-based approach, the first step is to calculate the two-photon PDC field [26] given by

where |0 is the vacuum state, and |ψ is the two-photon wavefunction, written as

with r_1,2 describing the positions of the two-photon at the instant t_1,2 and is the phase-matching function. The subscripts s, i and p indicate the signal, idler and pump, respectively. The fields

are the positive-frequency portions of the electric field operator evaluated at positions r_j, and times t_j. N_E is a normalization factor and is the photon annihilation operator. To determine the collection efficiency, we write A₁₂ as a coherent superposition of guided modes, _lm*(x,y), in the fibre as suggested in [23]

The field-based collection efficiency can then be written as

where is the fractional power [27] of the biphoton field coupled to the two single-mode fibres; i.e. the overlap between the PDC field with both collection modes. Likewise , and are the fractional powers of the biphoton field coupled to each single-mode fibre, defined by the overlap between the biphoton field and each collection mode:

This approach for obtaining the efficiency χ₁₂ is similar to classical photonics theory. However, the same result can obtained by a quantum mechanical approach with projectors given by

where

with j  =  1,2. The coincidences can then be calculated by projecting the wavefunction over two single-mode fibres:

and the singles are given by

2.2. PDC single-mode intensity-based collection efficiency

In the intensity-based approach, the projectors representing the spatial distribution of a single-mode fibre are given by

with j  =  1, 2. are the intensity profiles of the single-mode spatial distribution of the fields. This approach is similar to the approach of [25], which considers conditionally prepared photon states. The coincidences calculated by projecting the wavefunction over two single-mode fibres are then

and the singles are given by

The single-mode intensity-based collection efficiency is then defined by

3. Calculation of the field-based collection efficiency

The two-photon state at the output surface of a PDC crystal oriented with its face perpendicular to the z-axis is given by [26]

where Φ(k_s, k_i, ω_i, ω_s) is given by

where S is the cross sectional area of the crystal illuminated by the pump, N the normalization factor, and L the length of the crystal. The subscripts s, i and p indicate the signal, idler and pump. We denote the crystal axes in the lab frame by c_x, c_y, c_z, with the pump beam propagating along c_z. Because the signal and idler wavevectors do not generally point along the crystal axes, we identify the pump, signal, and idler k-vector directions as p_z, s_z, i_z, and their transverse components as p_x,y, s_x,y, i_x,y. Using the above notation, we evaluate Δk_x,y,z. Given that Δk_x,y,z  =  (k_p  -  k_s  -  k_i) c_x,y,z, we write Δk_x,y,z in terms of the pump, signal, and idler k-vectors. The longitudinal component of the pump k-vector is

where q_p is the transverse component of the pump k-vector and c is the speed of light. Analogous expressions give the signal and idler longitudinal components.

We now make some approximations to evaluate Δk_x,y,z. First, we assume that the pump, signal and idler have narrow transverse angular distributions, so we can adopt the paraxial approximation. We also rewrite the longitudinal k-vector components by expanding the index of refraction n_p,s,i(ω_p,s,i, ) around the central frequencies (Ω_s,i), and around the phase-matching angle _o. This last approximation holds only in cases where the considered field is an extraordinary wave (this of course depends on the type of phase-matching adopted, either type I or II). In all cases, we limit our calculation to the first perturbative order. We also assume small non-collinearity and small walk-off angles. When these two approximations hold, we can expand the sine and cosine terms, limited to first order. Finally, we write the overall Δk_x,y,z terms as

where θ_i,s are the emission angles of the idler and signal photons, K_i,s,p  =  n_i,s,p(Ω_i,s,p, ) Ω_i,s,p/c describe the directions of the central intensities of the wavevectors. The terms, and account for the effects on the refractive indexes of the pump and the signal due to the pump angular spread, which is responsible for a small deviation from the phase-matching angle _o. The other terms are defined as and ν  =  ω_s  -  Ω_s  =  Ω_i  -  ω_i. Note that for type I phase-matching. The first term in the expression for Δk_z accounts for the differential phase velocity between the signal and idler photons in the crystal, which is zero for type I degenerate; the second terms are responsible for the pump and signal walk-offs, respectively. (The signal and idler walkoff is generally zero for type I.) These terms account also for the pump transverse distribution as an angular spread around a principal direction, and the last two terms are associated with the non-collinear emission of the photons. We note that in the type I degenerate case, these last two terms cancel out.

The pump beam transverse field distribution is defined via the Fourier transform

We take this transverse distribution to be Gaussian with a waist of w_p at the crystal, and assume that the transverse crystal size is large relative to the pump beam, so we can take the cross section, S, to be infinite. Moreover, we assume that the pump beam has a narrow angular spectrum (transverse wavevector distribution) and that the signal and idler are observed only at points close to their central directions. The central frequencies are Ω_s,i. We also assume that the pump propagates with negligible diffraction inside the crystal, so that E_p(ρ) is independent of z. With these approximations, we calculate the valid ranges of w_p, L and w_o, the width of collection fibre mode as imaged at the crystal. In particular, L  K_pw_p²/2 to have negligible diffraction of the pump in the crystal. We can neglect second-order terms in the transverse wavevector component in (18) when w_o,p    (K_s,iθ_s,i) ^- 1.

Using these approximations we rewrite (15) as

To evaluate C₁₂ and S_1,2, we rewrite the state |ψ in terms of , using

and

Thus the Φ(k_s, k_i, ω_i, ω_s) in (16) becomes, in the new basis, its Fourier transform , as indicated formally in (2) and calculated at the output surface of the crystal (z_1,2  =  0). By performing the Fourier transforms and leaving the z integration for last, we obtain

where τ  =  t₁  -  t₂ and Π_DL(τ)  =  1 for 0 ≤ τ ≤ DL and 0 elsewhere. To guarantee that the biphoton state is properly normalized, the factor N₁ is determined from the condition , giving . To calculate the field-based collection efficiency, we assume the guided mode is a Gaussian field at the crystal output surface, defined by

The imaging optic is arranged to place the collection beam waist, w_o at the crystal. The intensity of the collection modes are normalized by setting ∫[*₁₀]²dx dy  =  1, which yields the coefficient in (24). The biphoton field is calculated using (1), (3) and (23), and the operator

to obtain

The single-mode field-based collection efficiency is then given by

with

and In the thin crystal limit this reduces to

as first calculated in [23, 24]. In this approach the fibres act to project the photons' state onto a specific propagation mode both in amplitude and phase, as indicated in (7) and (8). The spatial coherence of the single guided modes in the signal and idler arms should ultimately match the overall spatial coherence of the two-photon states.

4. Calculation of the single-mode intensity-based collection efficiency

To calculate the single-mode intensity-based collection efficiency, we assume the projectors , representing the fibre modes propagated back to the output surface of the crystal, are completely determined by the functions

(Note that in this approach, the intensity projection operator has a maximum of 1 because of its probabilistic nature.) We assume the conditions detailed following (19) are valid and also that w_s  =  w_i  =  w_o, and then calculate the single-mode intensity-based collection efficiency:

In the thin crystal limit, η₁₂ becomes

Note that there are several approximations other than L → 0 that lead to (32). For example, the assumptions following (19) preclude the case of an arbitrarily small pump waist at the crystal. The collection-mode waist is also restricted to modes that can be created by finite lenses that image fibres at a finite distance from the crystal.

In the intensity-based approach presented in this section, the collection mode can be thought of as spatially filtering the multi-mode input light. Thus, it is likely to be better suited for modelling multi-mode fibre collection. As the next section demonstrates, the predictions made by this model yield results different from the field-based model. The intensity-based model predicts that, for a fixed pump waist, the maximum collection efficiency is obtained when the fibre-defined collection mode (at the crystal) is large, i.e. all the pumped crystal volume is in a region of unit collection efficiency of the fibre/spatial filter system. With the field-based approach, the optics setup of figure (1), can be envisioned in an unfolded geometric arrangement [28] where one of the fibres acts as a single-mode source propagating back through a spatial filter (in this case the pumped crystal volume) to the other fibre. The maximum collection is achieved with a large pump waist, with respect to the collection beam waist at the crystal. If the pump waist is smaller than the fibre-defined collection beam waist, the collection efficiency is reduced. It is likely that the field-based approach gives the best answers for a single-mode fibre. However, moving to multi-mode fibres, the result should approach the intensity-based model, i.e. corresponding to a statistical optics physical description.

5. Discussion of results

The most immediately evident result here is that, in the thin crystal limit, η₁₂ and χ₁₂ provide totally different predictions. Figure 2 shows the field- and intensity-based collection efficiencies versus the collection-mode waists, for fixed pump waist. When the collection waist is much greater than the pump waist, η₁₂ asymptotically goes to 1, while χ₁₂ goes to 0; in the opposite condition (w_p    w_o), and χ₁₂ → 1. However, the thin-crystal approximation is far from valid unless the crystal is shorter than 0.1 mm for these collection beam diameters. In practical cases, crystal length ranges from 0.5 to 20 mm. This is one of the prime motivations for this work.

Figure 2. Field-based collection efficiency and intensity-based collection efficiency () in the thin crystal limit versus wo, for fixed values of wp.

To illustrate a practical lab setup, we consider a BBO crystal with type I and II phase-matching, pumped at 458 nm and look at downconversion at the degenerate wavelength (916 nm). For both type I and II, we describe the collinear and non-collinear cases with θ_s  =  θ_i 1.5°. To make valid predictions, we must ensure that the combination of L and w_p satisfy the assumptions made in deriving the expression. Figure 3 shows a graph of valid parameter combinations generally putting a lower limit on the size of the pump waist.

Figure 4 plots both η₁₂ and χ₁₂ versus w_o for fixed w_p  =  0.1 mm and crystal lengths of L  =  1 and 10 mm, with their limit values referred to the thin crystal approximation. Cases of collinear and non-collinear configurations for both type I (figure 4 (upper panel)) and type II (figure 4 (lower panel)) phase-matching are plotted. Again, note the different behaviours of η₁₂ and χ₁₂. When a crystal longer than 0.1 mm is used, both η₁₂ and χ₁₂ can be significantly reduced if the pump and collection waists are not properly matched. The way to match the waist for the optimum collection efficiency is different for η₁₂ and χ₁₂. In particular, η₁₂ can be optimized for a long crystal by increasing the collection waist at the crystal. The opposite holds for χ₁₂. Moreover, for η₁₂ in the case of type I phase-matching (figure 4 (upper panel)), we observe that the collinear configuration always guarantees a better coupling, whereas this is not the case for type II. This can be observed for η₁₂ evaluated at L  =  10 mm in figure 4 (lower panel): for w₀  <  0.5 mm the non-collinear configuration appears clearly more efficient than the collinear one. We can justify this behaviour, because in type II collinear phase-matching the walk-off angle of the signal is less than that in the non-collinear configuration.

Figure 4. Plots of η12 (solid lines) and χ12 (dashed lines) versus wo for fixed wp  =  100 μm and L  =  1 and 10 mm. We simulated the case of type I (upper panel) for collinear and non-collinear (with extra dots) conditions with γp  =  3.5°,  μm - 1 and type II (lower panel) for collinear and non-collinear conditions (extra dots),  μm - 1, and  μm - 1.

Figure 5 shows the effects of the crystal length on η₁₂ and χ₁₂ with w_p  =  w_o  =  0.2 mm and w_p  =  w_o  =  0.4 mm. We observe that, apart from their different values, η₁₂ and χ₁₂ present completely analogous dependences on the length of the crystal: all the chosen configurations (namely type I and II, collinear and non-collinear) converge to the thin crystal curve for very-short lengths. In fact, for L→∞, the factor in (28) approaches 1, thus the long crystal asymptotic values can be obtained directly from (31) and (27).

In general, the coupling efficiencies are higher in type I than II, and collinear geometry yields the best coupling efficiencies. It is noteworthy to observe that the dependence on the crystal length is actually rescalable, by scaling the pump and collection waists.

Figure 6 plots χ₁₂ and η₁₂ versus w_o and w_p for two different lengths of the crystal.

Because of other effects that can lower the efficiency in practice (such as crystal and optical losses, detector inefficiencies, and so on [29]), it is important to make an experimental test of the theory that is not sensitive to these extraneous types of losses. With this view, it is best to choose a configuration where the two models have opposite dependences on adjustable experimental parameters. For example, one could measure the collection efficiency for fixed crystal lengths, while varying either the pump waist or the collection waist, such as in central value range of figure 4.

6. Conclusions

We have presented an analytical model to quantify the collection efficiency in terms of adjustable experimental parameters with the goal of optimizing single-mode collection from two-photon sources. In addition, we have presented an alternative scheme that may have more validity for multi-mode collection arrangements. Our calculation was performed using generally accepted approximations such as relying mainly on a finite transverse distribution of the fields involved and neglecting second-order terms in the transverse wavevector's components. These calculations cover a wide range of experimental configurations and yield two formulas to quantify the collection efficiencies. We have pointed out experimental conditions that could be used to differentiate between the two models.

Acknowledgments

This work was supported in part by DARPA/QUIST, ARDA, ARO and partially by Elsag SpA/Qcrypt.

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